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+{"text":"\\section{Introduction and Conclusions}\n\n\\subsection{Introduction}\n\nIn the last years several proposals for the non-perturbative glueball propagators of $QCD$-like confining asymptotically-free gauge theories have been advanced, based\non the $AdS$ String\/ large-$N$ Gauge Theory correspondence \\cite{Mal} and more recently on a Topological Field Theory ($TFT$) underlying the large-$N$ limit of the pure Yang-Mills ($YM$) theory \\cite{boch:quasi_pbs} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\cite{Top}. \\par\nAbove all these proposal aim to elucidate, at least in the large-$N$ limit, the most fundamental feature of the infrared of large-$N$ $QCD$-like confining asymptotically-free gauge theories, i.e. the existence of a mass gap in the pure glue sector, as opposed to the massless spectrum of gluons in perturbation theory. \\par\nHowever, these proposals predict a variety of spectra for large-$N$ $QCD$ different among themselves, asymptotically quadratic for large masses \\cite{Mal}  \\cite{Witten} \\cite{Brower1} \\cite{PS} \\cite{Brower2} or exactly linear \\cite{Softwall} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\footnote{Exact linearity in the $TFT$ refers to the joint large-$N$ spectrum of scalar and pseudoscalar glueballs. The $TFT$ in its present formulation does not contain information about higher spin glueballs.}\nin the square of the glueball masses and in general do not agree about the qualitative and quantitative details of the low-energy spectrum but for the existence of the mass gap.  \\par\nIn view of the importance of the problem that these proposals aim to answer and in order to discriminate between the various proposals it is worth investigating whether there is any constraint that we know by the fundamental principles of any confining asymptotically-free gauge theory that any supposed answer for the non-perturbative glueball propagators has to satisfy. \\par\nIn fact, we do know with certainty the implications of the asymptotic freedom for the large-momentum asymptotic behavior of any gauge invariant correlation function. \\par\nIn this paper we do not discuss at all the theoretical justification of the various proposal that we examine, leaving it to the original papers.\nWe limit ourselves to check whether or not the constraint that follows by the asymptotic freedom and by the renormalization group in the ultraviolet ($UV$) is satisfied by any given proposal. Indeed, the importance of this constraint has been pointed out since the early days of large-$N$ $QCD$ \\cite{migdal:multicolor}, see also \\cite{polyakov:gauge}. \\par\nIn fact the purpose of this paper is threefold. \\par\n\n\\subsection{Implications of the renormalization group and of the asymptotic freedom}\n\nFirstly, in sect.(2) we point out that perturbation theory in conjunction with the renormalization group ($RG$) severely constraints the asymptotic behavior of glueball propagators in pure $SU(N)$ Yang-Mills,\nin $QCD$ and in $\\mathcal{N}$ $=1$ $SUSY$ $QCD$ with massless quarks, or in any confining asymptotically-free gauge theory massless to every order of perturbation theory. \\par\nIndeed, we show in this paper, on the basis of $RG$ estimates, that the most fundamental object involved in the problem of the mass gap \\footnote{The lightest glueball is believed to be a scalar\nin pure $YM$ and in the 't Hooft large-$N$ limit of $QCD$.}, the scalar ($S$) glueball propagator in any (confining) asymptotically-free gauge theory with no perturbative physical mass scale, up to unphysical contact terms, i.e. distributions supported at coinciding points,\nhas the following universal, i.e. renormalization-scheme independent, large-momentum asymptotic behavior:\n\\begin{align}\\label{eqn:corr_scalare_inizio}\n&\\int\\langle \\frac{\\beta(g)}{gN}tr\\bigl(\\sum_{\\alpha\\beta}{F_{\\alpha\\beta}^2}(x)\\bigr)\\frac{\\beta(g)}{gN} tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}^2(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n= & C_S p^4\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align} \nAnalogously for the pseudoscalar ($PS$) propagator:\n\\begin{align}\n&\\int\\langle \\frac{g^2}{N}  tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}\\tilde{F}_{\\alpha\\beta}(x)\\bigr) \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}\\tilde{F}_{\\alpha\\beta}(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n=& C_{PS} p^4 \\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align}\nand for a certain linear combination as well, the anti-seldual ($ASD$) propagator:\n\\begin{align}\\label{eqn:corr_asd}\n& \\frac{1}{2}\\int\\langle \\frac{g^2}{N}tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n= &  C_{ADS} p^4 \\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align}\nwhere $F_{\\alpha\\beta}^-=F_{\\alpha\\beta}-\\tilde F_{\\alpha\\beta}$ and $\\tilde F_{\\alpha\\beta}=\\frac{1}{2} \\epsilon_{\\alpha \\beta \\gamma \\delta} F_{\\gamma \\delta}$. \\par\nThe explicit dependence on the particular $\\Lambda_{\\overline{MS}}$ scale in Eq.(\\ref{eqn:corr_scalare_inizio})-Eq.(\\ref{eqn:corr_asd}) is illusory. A change of scheme affects only the $O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)$ terms.\nThe coincidence of the asymptotic behavior, up to the overall normalization constants that are computed in sect.(3), $C_S, C_{PS},C_{ASD}$, is due to the coincidence of the naive dimension in energy, $4$, and of the one-loop  anomalous dimension, $\\gamma(g)=-2 \\beta_0 g^2+\\cdots$, of these operators deprived of the factors of $\\frac{\\beta(g)}{g}$ or of $g^2$. Euclidean signature is always understood in this paper unless otherwise specified. \\par\n\n\\subsection{Perturbative check of the $RG$ estimates}\nSecondly, in sect.(3) we check the correctness of our $RG$ estimate on the basis of an explicit very remarkable three-loop computation \\footnote{The earlier two-loop computation was performed in \\cite{Kataev:1981gr}.}\nperformed by Chetyrkin et al.\\cite{chetyrkin:scalar}\n\\cite{chetyrkin:pseudoscalar}\nin pure $SU(N)$ $YM$ and in $SU(3)$ $QCD$ with $n_f$ massless Dirac fermions in the fundamental representation. \nFor example, we show that in pure $SU(N)$ $YM$ Chetyrkin et al. result \\cite{chetyrkin:scalar}\n\\cite{chetyrkin:pseudoscalar} can be rewritten by elementary methods as:\n\\begin{align}\\label{eqn:prologo:ris_sommato}\n& \\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F_{\\alpha\\beta}^{-2}}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F^{-2}_{\\alpha\\beta}}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x  \\nonumber\\\\\n&=(1-\\frac{1}{N^2})\\frac{p^4}{2\\pi^2\\beta_0} \\bigl(2g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambda_{\\overline{MS}}^2})-2g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2}) \\nonumber\\\\\n& +\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(\\frac{p^2}{\\Lambda_{\\overline{MS}}^2})\n-\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2})\\bigr)+O(g^6)\n\\end{align} \nwhere $a$ and $\\tilde{a}$ are two scheme-dependent constants that are defined in sect.(3.5) and $g_{\\overline{MS}}$ is the 't Hooft coupling constant in the $\\overline{MS}$ scheme.\nIn Eq.(\\ref{eqn:prologo:ris_sommato}) the terms that depend on $g(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2})$ correspond in the coordinate representation to distributions supported at coincident points (contact terms), and therefore they have no physical meaning.\nRemarkably, the correlator without the contact terms does not in fact depend on the arbitrary scale $\\mu$ (within $O(g^6)$ accuracy) as it should be.\nThe running coupling constant $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ occurs in Eq.(\\ref{eqn:prologo:ris_sommato}) with two-loop accuracy and it is given by:\n\\begin{align}\\label{eqn:prologo:g_pert}\n&g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})=g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\bigl(1-\\beta_0g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&-\\beta_1g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2} +\\beta_0^2g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2}) \\log^2\\frac{p^2}{\\mu^2}\\bigr) + \\cdots\n\\end{align}\nTherefore, the perturbative computation furnishes an expansion of the correlator in powers of $g_{\\overline{MS}}^2(\\mu)$ and of logarithms. This expansion has been rearranged by elementary methods in terms of the two-loop running coupling $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ in Eq.(\\ref{eqn:prologo:ris_sommato}). \\par\nAt this point our basic strategy to check the $RG$ estimates of sect.(2) consists in substituting in Eq(\\ref{eqn:prologo:ris_sommato}) instead of Eq.(\\ref{eqn:prologo:g_pert}) the $RG$-improved expression for $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ given by:\n\\begin{equation}\\label{eqn:prologo:rg_g}\ng_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})=\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggl[1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggr]\n+O\\biggl(\\frac{\\log^2\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log^3\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggr)\n\\end{equation}\nThe $\\overline{MS}$ scheme is indeed defined \\cite{chetyrkin:schema} in such a way to cancel the term of order of $\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}$ that would occur in Eq.(\\ref{eqn:prologo:rg_g}) in other schemes.\nBy subtracting the unphysical contact terms and by substituting the $RG$-improved two-loop asymptotic expression for $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ it follows the actual large-momentum scheme-independent asymptotic behavior of Eq.(\\ref{eqn:prologo:ris_sommato}):\n\\begin{align}\\label{eqn:prologo_comportamento_pert}\n& \\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x  \\nonumber\\\\\n= & (1-\\frac{1}{N^2})\\frac{p^4}{\\pi^2\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align} \nas opposed to the perturbative behavior that would follow by Eq.(\\ref{eqn:prologo:g_pert}).\nThe asymptotic result in the other cases is checked similarly. \\par\n\n\\subsection{ $AdS$\/Large-$N$ Gauge Theory correspondence and disagreement with the $RG$ estimates}\n\nThirdly, in this subsection and in the next one, we inquire whether the large-$N$ non-perturbative scalar or pseudoscalar propagators actually computed in the literature agree or disagree with the $RG$ estimate. \\par\nWe find, to the best of our knowledge, that all the scalar propagators presently computed in the literature in the framework of the $AdS$ String\/ large-$N$ Gauge Theory correspondence disagree with the universal asymptotic behavior. \\par\nWe should mention that the comparison of the asymptotics of the scalar glueball propagators in the $AdS$ approach with $YM$ or with $QCD$ at the lowest non-trivial order of perturbation theory has been already performed in \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd}, but with somehow different conclusions.\nThe reasons is that in \\cite{forkel}  \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd} the comparison has been performed only with the one-loop result for the scalar glueball propagator, i.e. only with the first term in Eq.(\\ref{eqn:corr_pert_scalar_3l}), that is conformal in the $UV$. No higher order of perturbation theory and no $RG$ improvement has been taken into account in the comparison, as instead we do in this paper. \\par\nHere we enumerate the models based on the $AdS$\/Gauge Theory correspondence for which we could find explicit computations of the scalar glueball propagator in the literature.  \\par\nIn the Hard Wall model (Polchinski-Strassler background \\cite{PS} in the so called bottom-up approach):\n\\[\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x \\sim  p^4 \\biggl[2\\frac{K_1(\\frac{p}{ \\mu})}{I_1(\\frac{p}{\\mu})}-\\log\\frac{p^2}{\\mu^2}\\biggr]\n\\]\nwhere $K_1,I_1$ are the modified Bessel functions \\cite{forkel}. The asymptotic behavior \\cite{forkel} is conformal in the $UV$:\n\\begin{align*}\n&p^4 \\biggl[2\\frac{K_1(\\frac{p}{ \\mu})}{I_1(\\frac{p}{ \\mu})}-\\log\\frac{p^2}{\\mu^2}\\biggr]\n \\sim - p^4\\biggl[\\log\\frac{p^2}{\\mu^2}+ O(e^{- 2 \\frac{p}{\\mu}}) \\biggr]\n\\end{align*}\nwith $p=\\sqrt{p^2}$. Indeed, as recalled in appendix A, in the coordinate representation:\n\\[\n- \\int p^4 \\log\\frac{p^2}{\\mu^2} e^{i p \\cdot x}\\frac{d^4p}{(2\\pi)^4} \\sim \\frac{1}{x^8}\n\\]\nand, as observed in \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd}, it matches the one-loop large-$N$ $QCD$ result for the perturbative glueball propagator displayed in the first term of Eq.(\\ref{eqn:corr_pert_scalar_3l}). Nevertheless, it disagrees by a factor of $(\\log p)^2$ with the correct asymptotic behavior in Eq.(\\ref{eqn:corr_scalare_inizio}). \\par\nThe Soft Wall model (bottom-up approach) \\cite{Softwall} implies the same leading conformal asymptotic behavior \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd} in the $UV$ for the scalar glueball propagator:\n\\[\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x  \n \\sim - p^4\\biggl[\\log\\frac{p^2}{\\mu^2}+O(\\frac{\\mu^2}{p^2})\\biggr]\n\\] \nthat therefore disagrees in the $UV$ by the same factor of $(\\log p)^2$. \\par\nA more interesting example of the $AdS$ string \/ large-$N$ Gauge Theory correspondence from the point of view of first principles applies to the cascading $\\mathcal{N}=1$ $SUSY$ $YM$ theory (top-down approach) \\cite{KS1} \\cite{KS2}, because in this case the correct asymptotically-free $\\beta$ function of the cascading theory is exactly reproduced in the supergravity approximation in the Klebanov-Strassler background \\cite{KS1} \\cite{KS2}.\nNevertheless, the asymptotic behavior of the scalar correlator is \\cite{krasnitz:cascading2} \\cite{krasnitz:cascading}:\n\\begin{align*}\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x\\sim p^4 \\log^3 \\frac{p^2}{\\mu^2}\n\\end{align*}\nthat disagrees by a factor of $(\\log p)^4$ with the correct asymptotic behavior in Eq.(\\ref{eqn:corr_scalare_inizio}). \\par\n\n\\subsection{Topological Field Theory and agreement with the $RG$ estimates}\n\nFinally, in sect.(4) we prove that in the large-$N$ limit of pure $SU(N)$ $YM$ the $ASD$ glueball propagator computed in \\cite{boch:glueball_prop}\\cite{boch:crit_points}\\cite{MB0} \\footnote{We use here a manifestly covariant notation as opposed to the one of the $TFT$ employed in \\cite{boch:glueball_prop}\\cite{boch:crit_points}\\cite{MB0}.}:\n\\begin{equation}\\label{eqn:formula_prologo}\n\\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{-2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{-2}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x =\n \\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}\n\\end{equation}\nagrees with the universal $RG$ estimate in Eq.(\\ref{eqn:prologo_comportamento_pert}).  \\par\nSince the proposal for the $TFT$ underlying large-$N$ $YM$ is recent and not widely known we add here a few explanations, but for the purposes of this paper the reader can consider Eq(\\ref{eqn:formula_prologo}) just as a phenomenological model\nfactorizing the $ASD$ glueball propagator on a spectrum linear in the masses squared with certain residues. \\par \nYet, to say it in a nutshell, the rationale behind Eq(\\ref{eqn:formula_prologo}) is as follows. In \\cite{boch:quasi_pbs} \\cite{boch:crit_points} \\cite{Top} it is shown that there is a $TFT$ trivial \\cite{boch:crit_points}  \\cite{Top} at $N=\\infty$ underlying the large-$N$ limit of $YM$. At $N=\\infty$ the $TFT$ is localized on critical points \\cite{MB0} \\cite{boch:crit_points}. However, at the first non-trivial $\\frac{1}{N}$ order the $ASD$ propagator of the $TFT$ arises computing non-trivial fluctuations around the critical points of the $TFT$ \\cite{MB0} \\cite{boch:glueball_prop}. \\par\nIn Eq.(\\ref{eqn:formula_prologo}) $F^-_{\\alpha\\beta}$ is the anti-selfdual part of the curvature $F_{\\alpha\\beta}=\\partial_\\alpha A_\\beta-\\partial_\\beta A_\\alpha +i\\frac{g}{\\sqrt{N}}[A_\\alpha,A_\\beta]$ with the canonical normalization defined in Eq.(\\ref{eqn:F_canonical}), $\\Lambda_{\\overline{W}}$ is the renormalization group invariant scale in the scheme in which it coincides with the mass gap and $g_k=g(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}=k)$ is the 't Hooft running coupling constant at the scale of the pole (in Minkowski space-time) in the scheme defined in \\cite{boch:quasi_pbs}, that is recalled in sect.(4). In fact, the analysis of the $UV$ behavior of Eq.(\\ref{eqn:formula_prologo}) has already been performed at the order of the leading logarithm occurring in Eq.(\\ref{eqn:corr_asd}) in \\cite{boch:glueball_prop}.\nHere we go one step further comparing Eq.(\\ref{eqn:formula_prologo}) with Eq.(\\ref{eqn:corr_asd}) at the order of the next-to-leading logarithm.\nOur basic strategy to obtain the large momentum asymptotics of Eq.(\\ref{eqn:formula_prologo}) is as follows. We write the RHS of Eq.(\\ref{eqn:formula_prologo}) as a sum of physical terms and contact terms according to \\cite{boch:glueball_prop}: \n\\begin{align}\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}=\n\\frac{1}{\\pi^2}p^4 \\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2} +\n \\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}g_k^4\\Lambda^2_{\\overline{W}}(k\\Lambda^2_{\\overline{W}}-p^2)\n\\end{align}\nThe first sum contains the physical terms that in Minkowski space-time carry the pole singularities, while the second sum contains the contact terms, that we ignore in the following.\nWe now consider only the physical terms and to find the leading $UV$ behavior we use the Euler-McLaurin formula according to the technique first introduced by Migdal \\cite{migdal:meromorphization} \\footnote{We understand that Migdal technique has been known to him for decades.} and employed in \\cite{boch:glueball_prop} :\n\\begin{equation}\n\\sum_{k=k_1}^{\\infty}G_k(p)=\n\\int_{k_1}^{\\infty}G_k(p)dk - \\sum_{j=1}^{\\infty}\\frac{B_j}{j!}\\left[\\partial_k^{j-1}G_k(p)\\right]_{k=k_1}\n\\end{equation}\nwhere $B_j$ are the Bernoulli numbers.\nIn our case the terms proportional to the Bernoulli numbers involve negative powers of $p$ and they are therefore suppressed with respect to the first term which behaves as the inverse of a logarithm, so that we ignore them as well. \nWe have:\n\\begin{align}\\label{eqn:prologo_hom_intermedio}\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}\\sim \n& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2}dk \\nonumber\\\\\n\\sim& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{1}{\\beta_0^2\\log^2\\frac{k}{c}}\n\\biggl(1-\\frac{2\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\frac{dk}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}\n\\end{align} \nwhere we have used the $RG$-improved asymptotic behavior for large $k$ of the running coupling constant $g_k$ at the scale of the $k$-th pole, i.e. on shell (in Minkowski space-time):\n\\begin{equation}\ng^2_k\\sim \\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\end{equation}\nThe constant $c$ is related to the scheme that occurs in the non-perturbative calculation \\cite{boch:quasi_pbs}\\cite{MB0}\\cite{boch:crit_points}\\cite{boch:glueball_prop}. The actual value of $c$ is not relevant in this paper since we study only the universal asymptotic behavior.\nIn sect.(4) we compute the universal leading and next-to-leading behavior of the integral in Eq.(\\ref{eqn:prologo_hom_intermedio}) and the result is:\n\\begin{align}\n& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{1}{\\beta_0^2\\log^2\\frac{k}{c}}\n\\biggl(1-\\frac{2\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\frac{dk}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}\\nonumber\\\\\n&= \\frac{1}{\\pi^2\\beta_0}p^4 \\biggl[\\frac{1}{\\beta_0\\log\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda_{\\overline{W}}^2}\\bigr)}-\\frac{\\beta_1}{\\beta_0^3}\\frac{\\log\\log\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda^2_{\\overline{W}}}\\bigr)}{\\log^2\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggr]\n+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\nonumber\\\\\n&=\\frac{1}{\\pi^2\\beta_0}p^4\\biggl[\\frac{1}{\\beta_0\\log\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}{\\log^2\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggr)\n\\biggr]+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\n\\end{align} \n\n\\subsection{Conclusions}\n\nThe preceding result, for the $ASD$ glueball propagator computed in the $TFT$ underlying large-$N$ pure $YM$, agrees perfectly in the large-$N$ limit with the universal part of the renormalization group improved expression of the perturbative result Eq.(\\ref{eqn:prologo_comportamento_pert}). \\par\nThe agreement is due to the conspiracy between the residues of the poles, that are proportional to the fourth power of the coupling constant renormalized on shell times the fourth power of the glueball mass at the pole, and the exact linearity of the joint scalar and pseudoscalar spectrum of the square of the mass of the glueballs in the $ASD$ correlator of the $TFT$. \\par\nTo the best of our knowledge this is the only non-perturbative result for the scalar or pseudoscalar glueball propagator proposed in the literature that agrees with large-$N$ $YM$ perturbation theory and the renormalization group. \\par\nWhile this agreement is not by itself a guarantee of correctness of Eq.(\\ref{eqn:formula_prologo})  it deserves further investigations, both at theoretical level\nand of further checks. \\par\nBesides, our analysis shows that the $AdS$\/Large-$N$ Gauge Theory correspondence in any of its present strong coupling incarnations, the bottom-up or the top-down approach, for which scalar glueball propagators are available in the literature,\ndoes not capture, not even approximatively, the fundamental ultraviolet feature of $YM$ or of $QCD$ or of any large-$N$  confining asymptotically-free gauge theory in the pure glue sector. \\par\nWhile this conclusion is certainly known to some experts (see for just one example \\cite{KS2}), we think that it is not widely recognized that constructing theories that are conformal in the ultraviolet, as the Hard or the Soft Wall models, or even with the correct beta\nfunction but in the strong coupling phase, as the Klebanov-Strassler supergravity background, is not at all a good approximation for the correct result in the ultraviolet. In this paper, for the first time with leading and next-to-leading logarithmic accuracy, we have computed quantitatively the measure of the disagreement. \\par\nFinally, given the disagreement between the propagators of the $TFT$ and the propagators of the $AdS$\/Large-$N$ Gauge Theory correspondence\nin the infrared for the first few lower-mass glueballs, a careful critical analysis of the two approaches at level of numerical lattice data is needed, and also at theoretical level of further constraints arising by the $OPE$ and by the \nlow-energy theorems of Shifman-Vainshtein-Zakharov ($SVZ$).\n\n\\section{Renormalization group estimates on the universal behavior of correlators}\n\n\\subsection{Definitions}\n\nThe  $SU(N)$ pure $YM$ theory is defined by the partition function:\n\\begin{equation}\\label{eqn:Z1}\nZ=\\int \\mathcal{D}A\\, e^{-\\frac{1}{2g_{YM}^2}\\int tr F^2(x) d^4x}\n\\end{equation}\nwhere we use the simplified notation $tr F^2(x)= \\sum_{\\alpha\\beta} tr \\bigl(F_{\\alpha\\beta}^2\\bigr)$.\nIntroducing the 't Hooft coupling constant $g$ \\cite{'t hooft:large_n}:\n\\begin{equation}\ng^2=g^2_{YM}N\n\\end{equation}\nthe partition function reads:\n\\begin{equation}\nZ=\\int \\mathcal{D}A\\, e^{-\\frac{N}{2g^2}\\int tr F^2(x) d^4x}\n\\end{equation} \nAccording to 't Hooft \\cite{'t hooft:large_n} the large-$N$ limit is defined with $g$ fixed when $N\\rightarrow \\infty$. \\par\nFor the structure of large-$N$ glueball propagators see \\cite{migdal:multicolor}\nand for reviews of the large-$N$ limit see \\cite{polyakov:gauge} and \\cite{makeenko:large_n}.\nThe normalization of the action in Eq.(\\ref{eqn:Z1}) corresponds to choosing the gauge field $A_{\\alpha}$ in the fundamental representation of the Lie algebra, with generators normalized as:\n\\begin{equation}\ntr\\, (t^a t^b)=\\frac{1}{2}\\delta^{ab}\n\\end{equation}\nIn Eq.(\\ref{eqn:Z1}) $F_{\\alpha\\beta}$ is defined by:\n\\begin{equation}\\label{eqn:F_wilsonian}\nF_{\\alpha\\beta}(x)=\\partial_\\alpha A_\\beta-\\partial_\\beta A_\\alpha + i[A_\\alpha,A_\\beta]\n\\end{equation}\nWe refer to the normalization of the action in Eq.(\\ref{eqn:Z1}) as the Wilsonian normalization.\nPerturbation theory is formulated with the canonical normalization, obtained rescaling the field $A_\\alpha$ in Eq.(\\ref{eqn:Z1}) by the coupling constant $g_{YM}=\\frac{g}{\\sqrt{N}}$:\n\\begin{align}\nA_\\alpha \\rightarrow g_{YM}A^{can}_\\alpha\n\\end{align}\nin such a way that in the action the kinetic term becomes independent on $g$:\n\\begin{equation}\n\\frac{1}{2}\\int tr (F^2(A^{can})) (x) d^4 x\n\\end{equation}\nwhere:\n\\begin{align}\\label{eqn:F_canonical}\nF_{\\alpha\\beta}= \\partial_\\beta A^{can}_\\alpha - \\partial_\\alpha A^{can}_\\beta +ig_{YM}[A^{can}_\\alpha , A^{can}_\\beta]\n\\end{align}\nFrom now on we will simply write $F_{\\alpha \\beta}$ for the curvature as a function of the canonical field, without displaying the superscript $can$.\n\n\n\\subsection{A short summary of perturbation theory and of the renormalization group}\n\nWe recall the relation between bare and renormalized two-point connected correlators of a multiplicatively renormalizable gauge-invariant scalar operator $\\mathcal{O}$ of naive dimension in energy $D$:\n\\begin{equation}\nG^{(2)}(p,\\mu,g(\\mu))=Z_{\\mathcal{O}}^{2}(\\frac{\\Lambda}{\\mu}, g(\\Lambda)) G_0^{(2)}(p, \\Lambda, g(\\Lambda))\n\\end{equation} \nwhere $G_0^{(2)}$ is the bare connected correlator in momentum space, computed in some regularization scheme with cutoff $\\Lambda$, and $\\mu$ is the renormalization scale:\n\\begin{equation}\nG_0^{(2)}(p, \\Lambda, g(\\Lambda))= \\int \\braket{\\mathcal{O}(x)\\mathcal{O}(0)}_{conn} e^{i p \\cdot x} d^4x  \\equiv\\braket{\\mathcal{O}(p)\\mathcal{O}(-p)}_{conn}\n\\end{equation} \nSince $YM$ or $QCD$ with massless quarks or $\\mathcal{N}$ $=1$ $SUSY$ $YM$ with massless quarks is massless to every order of perturbation theory and since $\\mathcal{O}$ has naive dimension $D$ we can write:\n\\begin{equation}\nG^{(2)}(p,\\mu, g(\\mu))=p^{2D-4}G_{DL}^{(2)}(\\frac{p}{\\mu}, g(\\mu))\n\\end{equation}  \nwhere $G_{DL}^{(2)}$ is a dimensionless function.\nThe Callan-Symanzik equation for the dimensionless two-point renormalized correlator \nexpresses the independence of the bare two-point correlator with respect to the subtraction point $\\mu$:\n\\begin{equation}\n\\frac{\\mathrm{d} G_0^{(2)}}{\\mathrm{d}\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}=0\n\\end{equation}\\begin{equation}\\label{eqn:RG_eq}\n\\left(\\frac{\\partial}{\\partial\\log\\mu}+\\beta(g)\\frac{\\partial}{\\partial g(\\mu)}+2\\gamma_{\\mathcal{O}}(g)\\right)G_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=0\n\\end{equation}\nwhere we have defined the beta function with respect to the renormalized coupling $g(\\mu)$: \n\\begin{equation}\n\\beta(g)=\\frac{\\partial g}{\\partial\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}\n\\end{equation}\nand the anomalous dimension:\n\\begin{equation}\n\\gamma_{\\mathcal{O}}(g)= - \\frac{\\partial\\log Z_{\\mathcal{O}}}{\\partial\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}\n\\end{equation}\nEq.(\\ref{eqn:RG_eq}) can be rewritten taking into account the dependence of $G_{DL}^{(2)}$ on the momentum $p=\\sqrt {p^2}$:\n\\begin{equation}\\label{eqn:RG_eq_p}\n\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g}-2\\gamma_{\\mathcal{O}}(g)\\right)G_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=0\n\\end{equation}\nThe general solution of Eq.(\\ref{eqn:RG_eq_p}) is:\n\\begin{equation}\\label{eqn:solu_rg}\nG_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=\\mathcal{G}(g(\\frac{p}{\\mu},g(\\mu)))\\, e^{2\\int_{g(\\mu)}^{g(p)}\\frac{\\gamma_{\\mathcal{O}}(g)}{\\beta(g)}dg} \\equiv\n\\mathcal{G}_{\\mathcal{O}}(g(p)) \\, Z^2_\\mathcal{O}(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}\nThe running coupling $g(\\frac{p}{\\mu},g(\\mu))$, that we briefly denote by $g(p)$, solves:\n\\begin{equation}\\label{eqn:eq_rg_flow}\n\\frac{\\partial g(p)}{\\partial \\log p}=\\beta(g(p))\n\\end{equation}\nwith the initial condition $g(1,g(\\mu))=g(\\mu)$. \\par \nThe multiplicative renormalized factor $Z_\\mathcal{O}(\\frac{p}{\\mu},g(\\mu))$ satisfies:\n\\begin{equation}\n\\gamma_{\\mathcal{O}}(g)= - \\frac{\\partial\\log Z_{\\mathcal{O}}}{\\partial\\log \\mu}\n\\end{equation}\nand from now on it is thought as a finite dimensionless function of $g(\\mu)$ and $g(p)$ only $Z_\\mathcal{O}(g(p),g(\\mu))$:\n\\begin{equation}\nZ_{\\mathcal{O}}=e^{\\int_{g(\\mu)}^{g(p)}\\frac{\\gamma_{\\mathcal{O}}(g)}{\\beta(g)}dg}\n\\end{equation}\nEq.(\\ref{eqn:solu_rg}) expresses the solution of the $RG$ equation as a product of a $RG$ invariant ($RGI$) function $\\mathcal{G}_{\\mathcal{O}}$ of $g(p)$ only and of a multiplicative factor $Z_{\\mathcal{O}}^2$ that is determined by the anomalous dimension $\\gamma_{\\mathcal{O}}(g)$ and by the beta function $\\beta(g)$. $\\mathcal{G}_{\\mathcal{O}}$ and $Z_{\\mathcal{O}}$ can be computed order by order in renormalized perturbation theory. \\par\nFrom Eq.(\\ref{eqn:eq_rg_flow}), that represents the coupling constant flow as a function of the momentum, we obtain the well-known behavior of the $RG$-improved 't Hooft running coupling constant with one- and two-loop accuracy, starting from the one- and two-loop perturbative beta function:\n\\begin{equation}\n\\beta(g)=-\\beta_0 g^3 - \\beta_1 g^5 + \\cdots\n\\end{equation}\nwhere for pure $YM$:\n\\begin{align}\n&\\beta_0=\\frac{11}{3}\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n&\\beta_1=\\frac{34}{3}\\frac{1}{(4\\pi)^4}\n\\end{align}\nWith two-loop accuracy we get:\n\\begin{align}\\label{eqn:g_2loop}\n&\\frac{d g}{d\\log p}=-\\beta_0 g^3 - \\beta_1 g^5 \\nonumber\\\\\n\\Rightarrow & \\int_{g(\\mu)}^{g(p)}\\frac{1}{\\beta_0 g^3}(1-\\frac{\\beta_1}{\\beta_0}g^2)dg=-\\log\\frac{p}{\\mu} \\nonumber\\\\\n\\Rightarrow & \\frac{1}{\\beta_0}(\\frac{1}{2g(\\mu)^2}-\\frac{1}{2g(p)^2})\n-\\frac{\\beta_1}{\\beta_0^2}\\log\\frac{g(p)}{g(\\mu)}=-\\log\\frac{p}{\\mu}\\nonumber\\\\\n\\Rightarrow & g^2(p)=\\frac{g^2(\\mu)}{1+2\\beta_0 g^2(\\mu)\\log\\frac{p}{\\mu}-2\\frac{\\beta_1}{\\beta_0}g^2(\\mu)\\log\\frac{g(p)}{g(\\mu)}}\\nonumber\\\\\n\\sim & \\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1+\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\frac{g(p)}{g(\\mu)}}{\\log\\frac{p}{\\mu}}\\right) \\sim\n\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1-\\frac{\\beta_1}{2\\beta_0^2}\\frac{\\log\\frac{g^2(\\mu)}{g^2(p)}}{\\log\\frac{p}{\\mu}}\\right)\\nonumber\\\\\n= & \\frac{1}{\\beta_0\\log\\frac{p^2}{\\mu^2}}\n\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\mu^2}}{\\log\\frac{p^2}{\\mu^2}}\\right)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\mu^2}}\\biggr)\n\\end{align}\nThis is the well known actual $UV$ asymptotic behavior of the running coupling constant.\nThe function $\\mathcal{G}_{\\mathcal{O}}$ in Eq.(\\ref{eqn:solu_rg}) is not known from general principles but can be computed in perturbation theory as a function of $g(\\mu)$ and then expressed in terms of $g(p)$, since  $\\mathcal{G}_{\\mathcal{O}}$ is $RGI$. \nSimilarly, we can evaluate $Z_{\\mathcal{O}}$ using again the one-loop or two-loop perturbative expressions for $\\beta(g)$ and $\\gamma_{\\mathcal{O}}(g)$:\n\\begin{equation}\n\\gamma_\\mathcal{O}(g)=-\\gamma_{0(\\mathcal{O})} g^2 -\\gamma_{1(\\mathcal{O})}g^4 + \\cdots\n\\end{equation}\nWith one-loop accuracy:\n\\begin{equation}\nZ^2_{\\mathcal{O}}\\sim\\left(\\frac{g^2(p)}{g^2(\\mu)}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\\sim\n\\left(\\log(\\frac{p}{\\mu})\\right)^{-\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\n\\end{equation}\nand with two-loop accuracy we have:\n\\begin{align}\\label{eqn:z_pert}\nZ^2_{\\mathcal{O}} \\sim &\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\ne^{\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{\\beta_0^2}\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]}\\nonumber\\\\\n\\sim &\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}} \n\\left(1-\\frac{\\gamma_{0(\\mathcal{O})}\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^3\\log\\frac{p}{\\mu}}\\right)\\nonumber\\\\\n\\times &\\left\\{1+\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{\\beta_0^2}\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]\\right\\}\\nonumber\\\\\n\\sim &\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{2\\beta_0}}\n\\Biggl(1-\\frac{\\gamma_{0(\\mathcal{O})}\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^3\\log\\frac{p}{\\mu}}+\n\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{2\\beta_0^3\\log\\frac{p}{\\mu}}\\nonumber\\\\\n&-\\frac{(\\gamma_{0(\\mathcal{O})}\\gamma_{1(\\mathcal{O})}\\beta_0\\beta_1-(\\gamma_{0(\\mathcal{O})}\\beta_1)^2)\\log\\log\\frac{p}{\\mu}}{4\\beta_0^6(\\log\\frac{p}{\\mu})^2}\n\\Biggr) \n\\end{align}\nIn evaluating the last two expressions we have used the two-loop $RG$-improved expression for $g(p)$ given by Eq.(\\ref{eqn:g_2loop}).\nFrom the two-loop $RG$-improved expression in Eq.(\\ref{eqn:z_pert}) it follows that the leading and next-to-leading logarithms for $Z_{\\mathcal{O}}$ are determined only by $\\beta_0$, $\\beta_1$  and by $\\gamma_{0(\\mathcal{O})}$, that are in fact universal, i.e. scheme independent. Indeed, the two-loop coefficient of the anomalous dimension $\\gamma_{1(\\mathcal{O})}$ does not occur in the first $\\log\\log\\frac{p}{\\mu}$ term, but only in terms that have a subleading behavior as powers of  logarithms.\nKeeping only up to the next-to-leading term in $Z^2_{\\mathcal{O}}$, we obtain for the universal logarithmic behavior of the dimensionless two-point correlator:\n\\begin{equation}\\label{eqn:pert_th_next_to_lead}\nG_{DL}^{(2)}(\\frac{p}{\\mu})\\sim\\left[\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)\n\\Biggl(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\Biggr)\\right]^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\n\\mathcal{G}_{\\mathcal O}(g(p))\n\\end{equation}\nThus our aim, in order to get asymptotic estimates, is to determine the one-loop coefficient of the anomalous dimension $\\gamma_{0 (\\mathcal{O})}$ and the $RGI$ function $\\mathcal{G}_{\\mathcal O}$ for our operators $\\mathcal{O}$.\n \n\\subsection{Anomalous dimension of $tr F^2$ and of $trF\\tilde{F}$}\n\nThe operator $\\frac{\\beta(g)}{g}tr F^2$\nis proportional to the conformal anomaly, that is the functional derivative with respect to a conformal rescaling of the metric of the renormalized effective action that must be $RGI$.\nTherefore $\\frac{\\beta(g)}{g}tr F^2$ is $RGI$ as well. \nHence its anomalous dimension vanishes and, using the notation of the previous section, the form of its correlator, ignoring possible contact terms that will be taken into account in sect.(3), is:\n\\begin{equation}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\np^4\\mathcal{G}_{\\frac{\\beta(g)}{g}F^2}(g(p)) \n\\end{equation} \nOn the other hand $tr F^2$  is not $RGI$ and therefore its correlator is:\n\\begin{equation}\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=\np^4\\mathcal{G}_{F^2}(g(p))Z_{F^2}^2(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}  \nSince the relation between $G^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))$ and $G^{(2)}_{F^2}(p,\\mu,g(\\mu))$ is:\n\\begin{equation}\\label{eqn:relationGg}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\\left(\\frac{\\beta(g)}{g}\\right)^2 G^{(2)}_{F^2}(p,\\mu,g(\\mu)) \n\\end{equation}\nit follows that $\\left(\\frac{\\beta(g)}{g}\\right)^2$ and $Z^2(\\frac{p}{\\mu}, g(\\mu))$ must combine in such a way to obtain a function of $g(p)$ only:\n\\begin{equation}\n\\left(\\frac{\\beta(g(\\mu))}{g(\\mu)}\\right)^2\\, Z_{F^2}^2(\\frac{p}{\\mu}, g(\\mu))\\, \\mathcal{G}_{F^2}(g(p)) =\n\\mathcal{G}_{\\frac{\\beta(g)}{g}F^2}(g(p)) \n\\end{equation}\nTo find the anomalous dimension $\\gamma_{ F^2}$ of $tr F^2$ we exploit once again the property of $\\frac{\\beta(g)}{g}tr F^2$ being $RGI$.\nIts two-point correlator must indeed satisfy the equation:\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4\\right)G^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=0\n\\end{equation}\nwhere the last term occurs because we are considering the complete correlator and not its dimensionless part.\nUsing Eq.(\\ref{eqn:relationGg}) we find the anomalous dimension of $tr F^2$:\n\\begin{align}\n&\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g}-4\\right)\n\\biggl[\\left(\\frac{\\beta(g)}{g}\\right)^2 G^{(2)}_{F^2}(p,\\mu,g(\\mu))\\biggr]=0 \\nonumber\\\\\n\\Rightarrow &\\biggl[\\left(\\frac{\\beta(g)}{g}\\right)^2\\frac{\\partial}{\\partial\\log p}\n-\\left(\\frac{\\beta(g)}{g}\\right)^2\\beta(g)\\frac{\\partial}{\\partial g}\\nonumber\\\\\n&-2\\beta(g)(\\frac{\\beta(g)}{g})\\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right) - 4 \\left(\\frac{\\beta(g)}{g}\\right)^2 \\biggr]\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=0 \\nonumber\\\\\n\\Rightarrow & \\biggl[\\frac{\\partial}{\\partial\\log p}\n-\\beta(g)\\frac{\\partial}{\\partial g}\n-2g\\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right) - 4\\biggr]\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=0\n\\end{align}\nFrom this equation it follows:\n\\begin{equation}\n\\gamma_{F^2}(g)=g \\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right)\n\\end{equation}\nWith two-loop accuracy this expression reads:\n\\begin{equation}\\label{eqn:dim_anomala_scalar}\n\\gamma_{tr F^2}(g)=-2\\beta_0 g^2-4\\beta_1 g^4 +\\cdots\n\\end{equation} \nKeeping only the first term, we can derive the expression for $Z^2(\\frac{p}{\\mu},g(\\mu))$ with one-loop accuracy:\n\\begin{equation}\nZ^2(\\frac{p}{\\mu},g(\\mu))\\sim\\frac{g^4(p)}{g^4(\\mu)}\n\\end{equation}\nFinally, the correlator of $\\frac{\\beta(g)}{g}trF^2$, with one-loop accuracy, is:\n\\begin{equation}\\label{eqn:corr_l2_rg}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\np^4\\beta_0^2 g^4(\\mu)\\,\\frac{g^4(p)}{g^4(\\mu)} \\, \\mathcal{G}_{F^2}(g(p))=\np^4\\beta_0^2 g^4(p)\\, \\mathcal{G}_{F^2}(g(p))\n\\end{equation}\nWe can repeat similar calculations for the operator $tr F\\tilde{F}$ in order to compute its anomalous dimension, using the property of $g^2 tr F\\tilde{F}$ being $RGI$.\nIndeed $g^2 tr F\\tilde{F}$ is the density of the second Chern class or topological charge.\nThe Callan-Symanzik equation is:\n\\begin{align}\n&\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4\\right)G^{(2)}_{g^2 F\\tilde{F}}(p,\\mu,g(\\mu))\\nonumber\\\\\n&=\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4 \\right)\\biggl[g^4 G^{(2)}_{F\\tilde{F}}(p,\\mu,g(\\mu))\\biggr]=0\n\\end{align}\nfrom which we obtain the anomalous dimension of $trF\\tilde{F}$:\n\\begin{equation}\\label{eqn:dim_anomala_pseudoscalar}\n\\gamma_{F\\tilde{F}}(g)=2\\frac{\\beta(g)}{g}=-2\\beta_0 g^2 -2\\beta_1 g^4 +\\cdots\n\\end{equation}\nWe notice that while the one-loop anomalous dimensions of $trF^2$ and of $tr F\\tilde{F}$ coincide, the two-loop anomalous dimensions are different.\nThis means that the operator $tr {F^-}^2$ has a well defined anomalous dimension only at one loop, in agreement with the fact that it belongs to the large-$N$ one-loop integrable sector of Ferretti-Heise-Zarembo \\cite{ferretti:new_struct}.\nTherefore, only the universal part of its correlator, that is determined by the one-loop anomalous dimension and by the two-loop $\\beta$ function, can be meaningfully compared with the non-perturbative computation in Eq.(\\ref{eqn:intro_formula_L2}).\n\n\n\n\\subsection{Universal behavior of correlators}\\thispagestyle{empty}\n\n\nKnowing the naive dimension $D$ and the anomalous dimension of a (scalar) operator $\\mathcal{O}_D$, we can write the asymptotic form for $p>>\\mu$ of its correlator obtained by the $RG$ theory.\nIndeed, as we recalled in sect.(2.2), assuming multiplicative renormalizability, the $RG$-improved form of the Fourier transform of the correlator is given by:\n\\begin{equation}\\label{eqn:pert_general_behavior}\nG^{(2)}(p^2)=\\int\\braket{\\mathcal{O}_D(x)\\mathcal{O}_D(0)}_{\\mathit{conn}} e^{i p \\cdot x} d^4x=\np^{2D-4}\\,\\mathcal{G}_{\\mathcal{O}_D}(g(p)) \\, Z^2_{\\mathcal{O}_D}(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}\nwhere the power of $p$ is implied by dimensional analysis, $\\mathcal{G}_{\\mathcal{O}_D}$ is a dimensionless function that depends only on the running coupling $g(p)$, and $Z^2_{\\mathcal{O}_D}$ is the contribution from the anomalous dimension.\nBut in fact in general the correlator of $\\mathcal{O}_D$ is not even multiplicatively renormalizable because of the presence of contact terms. These terms would affect the $UV$ asymptotic behavior, but they are non-physical and therefore they must be subtracted. In fact, they spoil the positivity of the correlator in Euclidean space in the momentum representation, that is required by the Kallen-Lehmann representation (see the comment below Eq.(\\ref{eqn:rg_improved_scalar_2l})). \\par\nIn the coordinate representation of the correlator, for $x \\neq 0$, contact terms do not occur. Therefore, a strategy to avoid that contact terms interfere with the $RG$ improvement is to pass to the coordinate scheme \\cite{chetyrkin:TF}, where the correlator is multiplicatively renormalizable, to compute its $RG$-improved expression, to go back to the momentum representation, and eventually to subtract the contact terms. \\par\nIn the coordinate representation for $x \\neq 0$ the solution of the Callan-Symanzik equation reads:\n\\begin{equation}\\label{eqn:pert_general_behavior_x}\nG_{\\mathcal{O}_D}^{(2)}(x)=\\braket{\\mathcal{O}_D(x)\\mathcal{O}_D(0)}_{\\mathit{conn}}=\n{\\bigl(\\frac{1}{x^2}\\bigr)}^{D}\\,\\mathcal{G}_{\\mathcal{O}_D}(g(x))\\, Z^2_{\\mathcal{O}_D}(x \\mu ,g(\\mu))\n\\end{equation}\nwith $x=\\sqrt {x^2}$, where we have denoted by $g(x)$ the running coupling in the coordinate scheme \\cite{chetyrkin:TF} and by an abuse of notation we have used the same names $\\mathcal{G}$ and $Z$ for the $RGI$ function\nand renormalization factor in the coordinate and momentum representation. \\par\nThe function $\\mathcal{G}(g(p))$ can be guessed at the lowest non-trivial order,\nsince the correlator must be conformal at the lowest non-trivial order in perturbation theory, that implies $\\mathcal{G}(g(x)) \\sim const$. Hence:\n\\begin{equation}\n\\mathcal{G}(\\frac{p}{\\mu}) \\sim const \\,\\log\\frac{p}{\\mu}\n\\end{equation}\nIndeed, in appendix A we show that $ \\int p^{2D-4} \\log\\frac{p}{\\mu} e^{i p \\cdot x} d^4p = const (\\frac{1}{x^2})^{D} $ that is conformal in the coordinate representation. The explicit dependence on $\\mu$,\nthat contradicts $RG$ invariance in the momentum representation, is due to the fact that the correlator in the momentum representation, as opposed to the coordinate representation, is not really multiplicatively renormalizable because (scale dependent) contact terms arise. This can be understood observing that in the coordinate representation for $x \\neq 0$  the lowest-order correlator is independent on the scale $\\mu$\nbut it is not an integrable function, in such a way that its Fourier transform needs a regularization, that introduces the arbitrary scale $\\mu$. \\par\nNaively, we can already derive the leading $UV$ asymptotic behavior:\n\\begin{align}\\label{eqn:naive_rg}\n&G^{(2)}_{\\mathcal{O}_D}(p^2)\\sim p^{2D-4}\\log\\frac{p^2}{\\mu^2}\n\\biggl(\\frac{g^2(p)}{g^2(\\mu)}\\biggr)^{\\frac{\\gamma_{0 (\\mathcal{O}_D)}}{\\beta_0}}\\sim \np^{2D-4}\n(g^2(p))^{\\frac{\\gamma_{0 (\\mathcal{O}_D)}}{\\beta_0}-1}\n\\end{align}\nwhere we have used the fact that $g^2(p)\\sim\\frac{1}{\\log(\\frac{p}{\\mu})}$. It easy to check that for $D=4$ and $\\gamma_{0 (\\mathcal{O}_D)}=2 \\beta_0$ this estimate coincides with Eq.(\\ref{eqn:corr_scalare_inizio})-Eq(\\ref{eqn:corr_asd}). \\par However, this estimate assumes multiplicative renormalizability in the momentum representation and it does not take into account the occurrence of contact terms in the momentum representation of the correlators. \\par Nevertheless, in the next section we confirm by direct computation that after subtracting the contact terms the actual behavior of the scalar and of the pseudoscalar correlator agrees with the estimate in Eq.(\\ref{eqn:naive_rg}). \n\n\\section{Perturbative check of the universal behavior of correlators}\n\nIn this section we obtain the explicit form of the three-loop correlators of $tr F^2$ and of $tr F\\tilde{F}$ starting from their imaginary parts that have been computed in \\cite{chetyrkin:scalar} \\cite{chetyrkin:pseudoscalar} in the $\\overline{MS}$ scheme.\nThe $\\overline{MS}$ scheme can be defined as the scheme in which the two-loop $RG$-improved running coupling does not contain\n$\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_s^2}}$ contributions \\cite{chetyrkin:schema}.\nMore precisely, we consider the equation for the running coupling constant that follows from the two-loop $\\beta$ function:\n\\begin{equation}\n\\log\\frac{p}{\\Lambda_s}=\\int_{g(\\Lambda_s)}^{g(p)}\\frac{dg}{\\beta(g)}=\\frac{1}{2\\beta_0 g^2(p)}+\\frac{\\beta_1}{\\beta_0^2}\\log\\bigl(g(p)\\bigr)+ C +\\cdots\n\\end{equation}\nwhere $C$ is an arbitrary integration constant and $\\Lambda_s$ is the $RGI$ scale in a generic scheme $s$.\nThe value of $C$ in the $\\overline{MS}$ scheme is chosen in such a way to cancel the $\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_s^2}}$ term in the solution:\n\\begin{align}\n&g_s^2(p)=\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_s^2}}\\biggl[1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log(\\beta_0\\log\\frac{p^2}{\\Lambda_s^2})}{\\log\\frac{p^2}{\\Lambda_s^2}}+\\frac{C}{\\log\\frac{p^2}{\\Lambda_s^2}}\\biggr] \\nonumber\\\\\n\\Rightarrow &C=\\frac{\\beta_1}{\\beta_0^2}\\log(\\beta_0)\n\\end{align}\nThe result reported in \\cite{chetyrkin:scalar}, for $tr F^2$ in the $SU(3)$ $YM$ theory, is:\n\\begin{align}\\label{eqn:im_scalare}\n\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}&=\n\\frac{8}{4\\pi}p^4\n\\biggl\\{1+\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi}\\biggl[\\frac{73}{4}-\\frac{11}{2}\\log\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&+(\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi})^2\\biggl[\\frac{37631}{96}\n-\\frac{363}{8}\\zeta(2)-\\frac{495}{8}\\zeta(3)\\nonumber\\\\\n&-\\frac{2817}{16}\\log\\frac{p^2}{\\mu^2}+\\frac{363}{16}\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\biggr\\}\n\\end{align}\nwhere $\\alpha_s=\\frac{g^2_{YM}}{4\\pi}$ and  $\\alpha_{\\overline{MS}}$ is $\\alpha_s$ in the $\\overline{MS}$ scheme.\nFirstly, we want to find from Eq.(\\ref{eqn:im_scalare}) the result for the $SU(N)$  $YM$ theory and we want to express the result in terms of the 't Hooft coupling in the $\\overline{MS}$ scheme $g_{\\overline{MS}}$.\nIn fact, this operation is quite easy since it is known, and it can be checked in \\cite{chetyrkin:pseudoscalar}, that at this order of perturbation theory the rank of the gauge group enters the result only through the Casimir factor $C_A=N$.\nTherefore, to obtain the general result it is sufficient to divide by 3 and to multiply by $N$ the coefficient of $\\alpha_{\\overline{MS}}$ and to divide by 9 and to multiply by $N^2$ the coefficient of $\\alpha_{\\overline{MS}}^2$ .\nThe factors of $N$ and of $N^2$ are then absorbed in the definition of the 't Hooft coupling constant.\nWe obtain:\n\\begin{align}\\label{eqn:ris_chet}\n&\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}\\nonumber\\\\\n&=\\frac{N^2-1}{4\\pi}p^4\n\\biggl\\{1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\frac{73}{3(4\\pi)^2}-2\\frac{11}{3(4\\pi)^2}\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl[\\frac{37631}{54(4\\pi)^4}\n-\\frac{242}{3(4\\pi)^4}\\zeta(2)-\\frac{110}{(4\\pi)^4}\\zeta(3)\\nonumber\\\\\n&-\\frac{313}{(4\\pi)^4}\\log\\frac{p^2}{\\mu^2}+\\frac{121}{3(4\\pi)^4}\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\biggr\\}\n\\end{align}\nFrom Eq.(\\ref{eqn:ris_chet}) we derive the complete expression of the correlator,\nassuming the correlator in the form:\n\\begin{align}\\label{eqn:ipotesi}\n\\braket{tr F^2(p)tr F^2(-p)}_{conn}&=\n-\\frac{N^2-1}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+f_3\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nWe extract the imaginary part of Eq.(\\ref{eqn:ipotesi}) that arises from the imaginary part of the logarithm in Minkowski signature, $\\log(-\\frac{p^2}{\\mu^2})=\\log\\frac{p^2}{\\mu^2}-i\\pi$. We obtain:\n\\begin{align}\\label{eqn:ris_ipotesi}\n&\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}\\nonumber\\\\\n&=\\frac{(N^2-1)}{4\\pi} p^4\\biggl[1+f_0g^2(\\mu)+(f_1-f_3\\pi^2)g^4(\\mu)\\nonumber\\\\\n&-2\\beta_0g^2(\\mu)\\log\\frac{p^2}{\\mu^2}+2f_2g^4(\\mu)\\log\\frac{p^2}{\\mu^2}+3f_3g^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nFinally, comparing Eq.(\\ref{eqn:ris_chet}) with Eq.(\\ref{eqn:ris_ipotesi}) we determine the values of the coefficients $f_i$:\n\\begin{align}\nf_0&=\\frac{73}{3(4\\pi)^2}\\nonumber\\\\\nf_1-f_3\\pi^2&=(\\frac{37631}{54}-\\frac{242}{3}\\zeta(2)-110\\zeta(3))\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n-2\\beta_0&=-2\\frac{11}{3(4\\pi)^2}\\nonumber\\\\\n2f_2&=-\\frac{313}{(4\\pi)^4}\\Rightarrow f_2=-\\frac{313}{2(4\\pi)^4}\\nonumber\\\\\n3f_3&=\\frac{121}{3(4\\pi)^4}\\Rightarrow f_3=\\frac{121}{9(4\\pi)^4}\\Rightarrow f_3=\\beta_0^2\\nonumber\\\\\n\\nonumber\\\\\n\\Rightarrow f_1&=(\\frac{37631}{54}-110\\zeta(3))\\frac{1}{(4\\pi)^4}\n\\end{align}\nTherefore, the correlator is:\n\\begin{align}\\label{eqn:corr_pert_scalar_3l}\n\\braket{tr F^2(p)tr F^2(-p)}_{conn}=\n&-\\frac{(N^2-1)}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nSimilarly, the imaginary part of the correlator of $tr F\\tilde{F}$, already written in \\cite{chetyrkin:pseudoscalar} for the gauge group $SU(N)$, is:\n\\begin{align}\n\\Im\\braket{tr F\\tilde{F}(p)tr F\\tilde{F}(-p)}_{conn}=\n&\\frac{N^2-1}{4\\pi}p^4\n\\biggl\\{1+\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi}\\biggl[N\\biggl(\\frac{97}{12}-\\frac{11}{6}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\nonumber\\\\\n&+(\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi})^2\\biggl[N^2\\biggl(\\frac{51959}{864}-\\frac{121}{24}\\zeta(2)\n-\\frac{55}{8}\\zeta(3)\\nonumber\\\\\n&-\\frac{1135}{48}\\log\\frac{p^2}{\\mu^2}+\\frac{121}{48}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\biggr\\}\n\\end{align} \nWe obtain:\n\\begin{align}\\label{eqn:corr_pert_pseudoscalar_3l}\n\\braket{tr F\\tilde{F}(p)tr F\\tilde{F}(-p)}_{conn}=\n&-\\frac{(N^2-1)}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\tilde{f}_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(\\tilde{f}_1+\\tilde{f}_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nwhere:\n\\begin{align}\n\\tilde{f}_0&=\\frac{97}{3(4\\pi)^2}\\nonumber\\\\\n\\tilde{f}_1&=(\\frac{51959}{54}-110\\zeta(3))\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n-2\\beta_0&=-2\\frac{11}{3(4\\pi)^2}\\nonumber\\\\\n2\\tilde{f}_2&=-\\frac{1135}{3(4\\pi)^4}\\Rightarrow \\tilde{f}_2=-\\frac{1135}{6(4\\pi)^4}\\nonumber\\\\\n\\end{align} \n\n\\subsection{Correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(N)$ $YM$ (two loops)}\n\nWe now determine the $UV$ asymptotic behavior for the correlators by employing their $RG$-improved expression.\nFirstly, we recall that in every generic scheme labelled by $a$ the relation between the coupling constant at two different scales is, with  one-loop accuracy:\n\\begin{align}\\label{eqn:scale_relation}\n&\\frac{1}{g_{a}^2(\\mu)}=\\frac{1}{g_{a}^2(p)}-\\beta_0\\log\\frac{p^2}{\\mu^2} \n\\end{align}\nThis relation is necessary to express the correlators in their $RG$-improved form.\nAs a starting simplified example we consider the two-loop expression of the correlator of $\\beta_0 \\frac{g^2}{N}trF^2$ (for the moment we skip overall positive numerical factors in the normalization of the correlator):\n\\begin{align}\\label{eqn:2l_perturbative_corr}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim -\\beta_0^2 p^4 g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nThis expression is renormalization group invariant with one-loop accuracy, since the factor $(\\frac{\\beta(g)}{g})^2$  is $\\beta_0^2 g^4$ if we employ the one-loop $\\beta$ function. \nThe finite term $f_0 g_{\\overline{MS}}^2(\\mu)$ can be absorbed in a change of scheme.\nIndeed, defining:\n\\begin{equation}\ng^2_{a}(\\mu)=g_{\\overline{MS}}^2(\\mu)(1+ag_{\\overline{MS}}^2(\\mu))\n\\end{equation}\nit follows:\n\\begin{align}\\label{eqn:cambio_schema}\ng^4_{a}(\\mu)&=g_{\\overline{MS}}^4(1+2ag_{\\overline{MS}}^2(\\mu)+a^2g_{\\overline{MS}}^4(\\mu))+ O(g^{10})\\nonumber\\\\\ng_{\\overline{MS}}^2(\\mu)&=g^2_{a}(\\mu)(1-ag_{\\overline{MS}}^2(\\mu)+a^2g_{\\overline{MS}}^4(\\mu))+O(g^8)\\nonumber\\\\\n&=g^2_{a}(\\mu)(1-a g_{a}^2(\\mu)+2a^2g^4_{a}(\\mu))+O(g^8)\\nonumber\\\\\ng_{\\overline{MS}}^4(\\mu)&=g^4_{a}(\\mu)(1-2a g^2_{a}(\\mu)+5a^2g^4_{a}(\\mu))+O(g^{10})\n\\end{align}\nWe obtain for the correlator:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n& \\sim -\\beta_0^2 p^4\\log\\frac{p^2}{\\mu^2}g^4_{a}(\\mu)\n\\biggl[1+(f_0-2a)g^2_{a}(\\mu)-\\beta_0g^2_{a}(\\mu)\\log\\frac{p^2}{\\mu^2}+O(g^4\\log\\frac{p^2}{\\mu^2})\\biggr]\n\\end{align}\nTo cancel the finite term it is sufficient to put:\n\\begin{equation}\na=\\frac{f_0}{2}\n\\end{equation}\nHence we obtain:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0^2 p^4\\log\\frac{p^2}{\\mu^2}g^4_{a}(\\mu)\\biggl[1-\\beta_0g^2_{a}(\\mu)\\log\\frac{p^2}{\\mu^2}+O(g^4\\log\\frac{p^2}{\\mu^2})\\biggr]\n\\end{align} \nAt this order the term in square brackets is precisely the renormalization factor necessary to renormalize two powers of $g_{a}(\\mu)$.\nWe obtain:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0^2 p^4g^2_{a}(\\mu)g^2_{a}(p)\\log\\frac{p^2}{\\mu^2}\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\n\\end{align} \nFrom  Eq.(\\ref{eqn:scale_relation}) we express the logarithm in terms of the coupling constant:\n\\begin{equation}\n\\beta_0\\log\\frac{p^2}{\\mu^2}=\\frac{1}{g^2_{a}(p)}-\\frac{1}{g^2_{a}(\\mu)}\n\\end{equation}\nThe correlator becomes:\n\\begin{align}\\label{eqn:rg_improved_scalar_2l}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0 p^4g^2_{a}(\\mu)g^2_{a}(p)\\bigl(\\frac{1}{g^2_{a}(p)}-\\frac{1}{g^2_{a}(\\mu)}\\bigr)\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\\nonumber\\\\\n&=\\beta_0 p^4\\bigl(g_{a}^2(p)-g_{a}^2(\\mu)\\bigr)\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\n\\end{align}\nThe second term in the last line is in fact a contact term that has no physical meaning, therefore it may depend on the arbitrary scale $\\mu$ since it must be subtracted anyway.\nThe physical term is positive, despite the correlator that we started with was negative. \nThis is an important feature, since a negative physical term would have been in contrast with the Kallen-Lehmann representation, that requires a positive spectral function.\n\\newline\n\n\\subsection{Correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now consider the three-loop result Eq.(\\ref{eqn:corr_pert_scalar_3l}), this time including also the correct normalization factors:\n\\begin{align}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)g_{\\overline{MS}}^4(\\mu)\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nThis correlator is not supposed to be $RGI$, because the factor of $(\\frac{\\beta(g)}{g})^2=g^4\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2\\bigr)^2$ is missing.\nWe can eliminate the finite terms in the correlator by a redefinition of the scheme:\n\\begin{align}\\label{eqn:cambio_schema2}\ng^2_{ab}(\\mu)&=  g_{\\overline{MS}}^2(\\mu) \\bigl(1+ag_{\\overline{MS}}^2(\\mu)+bg_{\\overline{MS}}^4(\\mu)\\bigr)\\nonumber\\\\\n\\Rightarrow g_{\\overline{MS}}^4(\\mu)&=g^4_{ab}(\\mu)(1-2a g^2_{ab}(\\mu)+(2b+5a^2)g^4_{ab}(\\mu))+O(g^{10})\n\\end{align}\nSubstituting we obtain:\n\\begin{align}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\\bigl(1-2a g^2_{ab}(\\mu)+(2b+5a^2)g^4_{ab}(\\mu)\\bigr)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+f_0 g^2_{ab}(\\mu)(1-ag^2_{ab}(\\mu))\n-\\beta_0 g^2_{ab}(\\mu)(1-ag^2_{ab}(\\mu))\\log\\frac{p^2}{\\mu^2}+f_1 g_{ab}^4(\\mu)\\nonumber\\\\\n&+ f_2 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+(f_0-2a) g^2_{ab}(\\mu)\n-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}+(f_1+5a^2+2b-af_0) g_{ab}^4(\\mu)\\nonumber\\\\\n&+(f_2+3\\beta_0 a) g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nWe eliminate the two finite terms choosing:\n\\begin{align}\n&a=\\frac{f_0}{2}\\nonumber\\\\\n&f_1+5(\\frac{f_0}{2})^2+2b-\\frac{f_0^2}{2}=0 \\nonumber\\\\\n\\Rightarrow & b=\\frac{3}{8}f_0^2-\\frac{f_1}{2}\n\\end{align}  \nWith this choice of $a$ the coefficient of the $g^4\\log\\frac{p^2}{\\mu^2}$ term becomes:\n\\begin{equation}\nf_2+3\\beta_0 a = f_2+\\frac{3}{2}f_0\\beta_0=-\\frac{68}{3(4\\pi)^4}=-2\\beta_1\n\\end{equation}\nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:corr_scalare_intermedio}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}=\n-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nWe notice that the expression in square brackets is the two-loop $Z$ factor determined by the anomalous dimension of $tr F^2$ according to Eq.(\\ref{eqn:dim_anomala_scalar}).\nThe coefficient $2\\beta_1$ should become $\\beta_1$ if we multiply the correlator in Eq.(\\ref{eqn:corr_scalare_intermedio}) by the factor of $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)^2$, in order to make the correlator $RGI$:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0^2 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)^2\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0^2 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)}\\quad\\nonumber\\\\\n\\times\\quad &\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nwhere we have multiplied and divided by $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)$ in order to exploit the two-loop relation:\n\\begin{equation}\n\\beta_0(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(p))\\log\\frac{p^2}{\\mu^2} =\\frac{1}{g_{ab}^2(p)}-\\frac{1}{g_{ab}^2(\\mu)}\n\\end{equation}\nWe now evaluate separately:\n\\begin{align}\n&\\frac{(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu))}{(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(p))}\\nonumber\\\\\n&=\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu)\\bigr)\n\\bigl(1-\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu)+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\frac{\\beta_1^2}{\\beta_0^2}g_{ab}^4(\\mu)\\bigr)+O(g^6\\log\\frac{p^2}{\\mu^2})\\nonumber\\\\\n&= 1+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +O(g^6\\log\\frac{p^2}{\\mu^2})\n\\end{align}\nPutting all together  we get:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\quad \\nonumber\\\\\n\\times\\quad & \\bigl(1+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\bigr)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\bigr)\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2} \\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\bigr)\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nThe factor in square brackets in the last line is now precisely the renormalization factor for two powers of $g_{ab}$. Hence the correlator reads:\n\\begin{align}\\label{corr_scalare_rgi_3_loop_improved}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2} g^2_{ab}(\\mu)g^2_{ab}(p)\n\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(p)}\\bigr)\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2} \\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(g_{ab}^2(\\mu)-g_{ab}^2(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2}\n\\biggl[g_{ab}^2(p)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)-g_{ab}^2(\\mu)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\\biggr]\n\\end{align}\nThe second term in the last line is a contact term, but the first term depends on $g_{ab}(\\mu)$, therefore it is not $RGI$.\nHence Eq.(\\ref{corr_scalare_rgi_3_loop_improved}) is not exactly $RGI$ even after subtracting the contact terms.\nThe scale dependence in the physical term is due to the fact that the correlator is not exact but it is computed to a finite order of perturbation theory.\nWe notice that the scale dependence occurs at order of $g^4$ only and in any case it does not affect the structure of the universal $UV$ behavior but only the overall coefficient in the $RG$ estimate.\nYet it is interesting to determine the precise overall coefficient of the asymptotic behavior. This is done for the correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(3)$ $QCD$ in sect.(3.7) by assuming its $RG$-invariance, instead of checking it to a finite order of perturbation theory as we just did.\n\n\\subsection{Correlator of $\\frac{g^2}{N} trF^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now present the result for the correlator of $\\frac{g^2}{N} trF^2$. \nWe recall that in this case we do not expect to get a $RGI$ function to all orders in perturbation theory.\nWe start from Eq.(\\ref{eqn:corr_scalare_intermedio}) and we write it as:\n\\begin{align}\\label{eqn:rg_not_improved_scalar_3l}\n&\\braket{\\frac{g^2}{N}tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{ab}(\\mu)\n\\frac{1}{\\beta_0} \\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)}\n\\frac{1}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}\n\\biggl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\biggr)\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{ab}(\\mu) \n\\frac{1}{\\beta_0}\n\\frac{1}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}\n\\biggl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\biggr)\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0} \n\\bigl(1-\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)+\\frac{\\beta_1^2}{\\beta_0^2}g^4_{ab}(\\mu)\\bigr)\n\\bigl(g^2_{ab}(\\mu)-g^2_{ab}(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{ab}(p)-g^2_{ab}(\\mu)+\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)-\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)g^2_{ab}(\\mu)\\bigr)\n\\end{align}\nSurprisingly we notice that the term that depends on the product $g_{ab}(\\mu)g_{ab}(p)$, that is not $RGI$, is of the same order of $g^4$ as the non-$RGI$ terms in the correlator in Eq.(\\ref{corr_scalare_rgi_3_loop_improved}), that must be $RGI$.\n\n\\subsection{Correlator of $\\frac{g^2}{N} trF\\tilde{F}$ in $SU(N)$ $YM$ (three loops)}\n\nWe repeat the same steps to find the $RG$-improved expression for the correlator of $\\frac{g^2}{N}tr F\\tilde{F}$, that is $RGI$.\nThe three-loop correlator reads:\n\\begin{align}\n\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}=\n&-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2} g_{\\overline{MS}}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad &\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\tilde{f}_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(\\tilde{f}_1+\\tilde{f}_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nNow we perform a generic change of scheme as in Eq.(\\ref{eqn:cambio_schema2}):\n\\begin{equation}\ng_{\\tilde{ab}}^2=g_{\\overline{MS}}^2(\\mu)\\bigl(1+\\tilde{a}g_{\\overline{MS}}^2(\\mu)+\\tilde{b}g_{\\overline{MS}}^4(\\mu)\\bigr)\n\\end{equation}\nThe correlator becomes:\n\\begin{align}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{\\tilde{ab}}(\\mu)\n\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1+(\\tilde{f}_0-2\\tilde{a}) g^2_{\\tilde{ab}}(\\mu)\n-\\beta_0g^2_{\\tilde{ab}}(\\mu)\\log\\frac{p^2}{\\mu^2}+(\\tilde{f}_1+5\\tilde{a}^2+2\\tilde{b}-\\tilde{a}\\tilde{f}_0) g_{\\tilde{ab}}^4(\\mu)\\nonumber\\\\\n&+(\\tilde{f}_2+3\\beta_0 \\tilde{a}) g_{\\tilde{ab}}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{\\tilde{ab}}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nAgain we impose the conditions to eliminate the finite terms:\n\\begin{align}\n&\\tilde{a}=\\frac{\\tilde{f}_0}{2}\\nonumber\\\\\n&\\tilde{f}_1+5(\\frac{\\tilde{f}_0}{2})^2+2\\tilde{b}-\\frac{\\tilde{f}_0^2}{2}=0 \\nonumber\\\\\n\\Rightarrow & \\tilde{b}=\\frac{3}{8}\\tilde{f}_0^2-\\frac{\\tilde{f}_1}{2}\n\\end{align}  \nWith this choice of $\\tilde{a}$ the coefficient of the $g^4\\log\\frac{p^2}{\\mu^2}$ term becomes:\n\\begin{equation}\n\\tilde{f}_2+3\\beta_0 \\tilde{a} = \\tilde{f}_2+\\frac{3}{2}\\tilde{f}_0\\beta_0=-\\frac{34}{3(4\\pi)^4}=-\\beta_1\n\\end{equation}\nSubstituting in the correlator we get:\n\\begin{align}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{\\tilde{ab}}(\\mu)\n\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times \\quad&\\biggl[1-\\beta_0g^2_{\\tilde{ab}}(\\mu)\\log\\frac{p^2}{\\mu^2}-\\beta_1 g_{\\tilde{ab}}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{\\tilde{ab}}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nWe notice that the expression in square brackets is the two-loop $Z$ factor implied by the anomalous dimension of $trF\\tilde{F}$ computed in Eq.(\\ref{eqn:dim_anomala_pseudoscalar}). It renormalizes two powers of $g(\\mu)$.\nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:rg_improved_pseudo_3l}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^2_{\\tilde{ab}}(\\mu) g^2_{\\tilde{ab}}(p)\n\\log\\frac{p^2}{\\mu^2}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^2_{\\tilde{ab}}(\\mu) g^2_{\\tilde{ab}}(p)\\frac{1}{\\beta_0}\\biggl(\\frac{1}{g_{\\tilde{ab}}^2(p)}-\\frac{1}{g^2_{\\tilde{ab}}(\\mu)}\\biggr)\n\\frac{1}{1+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(\\mu)-g^2_{\\tilde{ab}}(p)\\bigr)\\frac{1}{1+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(\\mu)-g^2_{\\tilde{ab}}(p)\\bigr)\\bigl(1-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)+\\frac{\\beta_1^2}{\\beta_0^2}g^4_{\\tilde{ab}}(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(p)+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)g^2_{\\tilde{ab}}(\\mu)-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)-g^2_{\\tilde{ab}}(\\mu)\\bigr) \n\\end{align}\nThe second term in the last line is scale dependent to the order of $g^4$ as the term that occurs in the correlator of $\\frac{g^2}{N}tr F^2$ in Eq.(\\ref{eqn:rg_not_improved_scalar_3l}).\n\n\n\\subsection{Correlator of $\\frac{g^2}{N} tr {F^-}^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now sum the two results for the correlators of $trF^2$ and of $tr F\\tilde{F}$ to obtain the correlator of $tr {F^-}^2$.\nIndeed, we recall that:\n\\begin{equation}\n\\frac{1}{2}\\braket{tr{F^-}^2(p)tr{F^-}^2(-p)}_{conn}=\n2\\braket{trF^2(p)trF^2(-p)}_{conn}+2\\braket{trF\\tilde{F}(p)trF\\tilde{F}(-p)}_{conn}\n\\end{equation}\nSumming the two results in Eq.(\\ref{eqn:rg_improved_pseudo_3l}) and in Eq.(\\ref{eqn:rg_not_improved_scalar_3l}) we obtain:\n\\begin{align}\\label{eqn:corr_asd_pert}\n&\\frac{1}{2}\\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}=\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2 \\pi^2}\\frac{1}{\\beta_0} \\bigl(g^2_{ab}(p) + g^2_{\\tilde{ab}}(p) -g^2_{ab}(\\mu) -g^2_{\\tilde{ab}}(\\mu) +\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)\\nonumber\\\\\n&-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)g^2_{ab}(\\mu)-\\frac{\\beta_1}{\\beta_0}g^2_{\\tilde{ab}}(p)g^2_{\\tilde{ab}}(\\mu)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{ab}(p) + g^2_{\\tilde{ab}}(p) -g^2_{ab}(\\mu) -g^2_{\\tilde{ab}}(\\mu) +\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)\\nonumber\\\\\n&-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)+O(g^6)\\bigr)\n\\end{align}\nwhere surprisingly the mixed terms  $g^2(p)g^2(\\mu)$ cancel to the order of $g^6$.\nThere is no perturbative explanation for such cancellation, but conjecturally the cancellation occurs because of the $RG$ invariance of the non-perturbative formula Eq.(\\ref{eqn:formula_prologo}) in the $TFT$ for the $L=2$ ground state \\cite{boch:crit_points} \\cite{boch:glueball_prop} of the large-$N$ one-loop integrable sector of Ferretti-Heise-Zarembo (see sect.(4)).\nWe can express the last result in terms of the coupling constant in the $\\overline{MS}$ scheme:\n\\begin{align}\\label{eqn:corr_asd_ms}\n&\\frac{1}{2}\\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2\\pi^2}\\frac{1}{\\beta_0} \\bigl(2g_{\\overline{MS}}^2(p)-2g_{\\overline{MS}}^2(\\mu) +\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(p)\\nonumber\\\\\n&+\\bigl(\\frac{\\beta_1}{\\beta_0}-a-\\tilde{a}\\bigr) g_{\\overline{MS}}^4(\\mu)\\bigr)+O(g^6)\n\\end{align}\nthat coincides with Eq.(\\ref{eqn:prologo:ris_sommato}).\n\n\\subsection{Scalar correlators in $SU(3)$ $QCD$ with $n_l$ massless Dirac fermions}\n\n\nIn this section we derive the $RG$-improved expression for the correlators of $trF^2$ and of $\\frac{\\beta(g_{YM})}{g_{YM}}trF^2$ in $QCD$. \\par\nThe three-loop perturbative result for the imaginary part of the correlator of $tr F^2$ in $QCD$ with $n_l$ massless Dirac femions is \\cite{chetyrkin:scalar}:\n\\begin{align}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\Biggl\\{1+\\frac{\\alpha_s(\\mu)}{\\pi}\\biggl[\\biggl(\\frac{73}{4}-\\frac{11}{2}\\log\\frac{p^2}{\\mu^2}\\biggr)\n-n_l\\biggl(\\frac{7}{6}-\\frac{1}{3}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\nonumber\\\\\n&+\\Bigl(\\frac{\\alpha_s(\\mu)}{\\pi}\\Bigr)^2\\biggr[\\frac{37631}{96}-\\frac{363}{8}\\zeta(2)\n-\\frac{495}{8}\\zeta(3)-\\frac{2817}{16}\\log\\frac{p^2}{\\mu^2} +\\frac{363}{16}\\log^2\\frac{p^2}{\\mu^2}\\nonumber\\\\\n&+n_l\\biggl(-\\frac{7189}{144}+\\frac{11}{2}\\zeta(2)+\\frac{5}{4}\\zeta(3)+\\frac{263}{12}\\log\\frac{p^2}{\\mu^2}-\\frac{11}{4}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l^2\\biggl(\\frac{127}{108}-\\frac{1}{6}\\zeta(2)-\\frac{7}{12}\\log\\frac{p^2}{\\mu^2} +\\frac{1}{12}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\Biggr\\}\n\\end{align}\nWe write the correlator in terms of the coupling $g_{YM}$ in the $\\overline{MS}$ scheme instead of $\\alpha_s$:\n\\begin{align}\\label{eqn:im_pert}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\Biggl\\{1+g_{YM}^2(\\mu)\\biggl[\\biggl(73-22\\log\\frac{p^2}{\\mu^2}\\biggr)\n-n_l\\biggl(\\frac{14}{3}-\\frac{4}{3}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\biggr[\\biggl(\\frac{37361}{6}-726\\zeta(2)-990\\zeta(3)\n-2817\\log\\frac{p^2}{\\mu^2} +363\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l\\biggl(-\\frac{7189}{9}+88\\zeta(2)+20\\zeta(3)+\\frac{1052}{3}\\log\\frac{p^2}{\\mu^2}-44\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l^2\\biggl(\\frac{508}{27}-\\frac{8}{3}\\zeta(2)-\\frac{28}{3}\\log\\frac{p^2}{\\mu^2} +\\frac{4}{3}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\frac{1}{(4\\pi)^4}\\Biggr\\}\n\\end{align}\nIf we suppose the correlator to be of the form:\n\\begin{align}\n&\\braket{trF^2(p)trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{YM}^2(\\mu)\\Bigl(h_0+h_1\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +h_4 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr]\n\\end{align}\nits imaginary part is:\n\\begin{align}\\label{eqn:im_ipotesi}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\biggl[1+h_0 g_{YM}^2(\\mu) + 2h_1 g_{YM}^2(\\mu)\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&+(h_2-\\pi^2 h_4)g_{YM}^4(\\mu) +2h_3 g_{YM}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +3h_4 g_{YM}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nComparing Eq.(\\ref{eqn:im_ipotesi}) and Eq.(\\ref{eqn:im_pert}) we get:\n\\begin{align}\nh_0&=\\biggl(73-\\frac{14}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n2h_1&=\\biggl(-22+\\frac{4}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n\\Rightarrow h_1&=\\biggl(-11+\\frac{2}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}=-\\tilde{\\beta}_0\\nonumber\\\\\nh_2-\\pi^2 h_4 &=\\biggl[\\frac{37361}{6}-726\\zeta(2)\n-990\\zeta(3)\n+n_l\\Bigl(-\\frac{7189}{9}+88\\zeta(2)+20\\zeta(3)\\Bigr)\\nonumber\\\\\n&+n_l^2\\Bigl(\\frac{508}{27}-\\frac{8}{3}\\zeta(2)\\Bigr)\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n2h_3 &=\\biggl[-2817+\\frac{1052}{3}n_l-\\frac{28}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n\\Rightarrow h_3 &=\n\\biggl[-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n3h_4&=\\biggl[363- 44n_l+\\frac{4}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n\\Rightarrow h_4&=\\biggl[121- \\frac{44}{3}n_l+\\frac{4}{9}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}=\\tilde{\\beta}_0^2\n\\end{align}\nNow we repeat the same steps as in the $n_l=0$ case. \nWe change renormalization scheme in order to cancel the finite parts:\n\\begin{equation}\ng_{uv}^2(\\mu)=g_{YM}^2(\\mu)\\bigl(1+u g_{YM}^2(\\mu)+vg_{YM}^4(\\mu)\\bigr)\n\\end{equation} \nWe use the perturbative expression for the renormalized coupling constant with two-loop accuracy:\n\\begin{align*}\n&g_{YM}^2(p)=g_{YM}^2(\\mu)\\Bigl(1-\\tilde{\\beta}_0 g_{YM}^2(\\mu)\\log\\frac{p^2}{\\mu^2} -\\tilde{\\beta}_1 g_{YM}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\nonumber\\\\ &+\\tilde{\\beta}_0^2g^4_{YM}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\Bigr)\n\\end{align*} \nwhere the tilde refers to the $QCD$ coefficients of the $\\beta$ function: \n\\begin{align}\n\\tilde{\\beta}_0&=\\Bigl(11-\\frac{2}{3}n_l\\Bigr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n\\tilde{\\beta}_1 &=\\Bigl(102-\\frac{38}{3}n_l\\Bigr)\\frac{1}{(4\\pi)^4}\n\\end{align}\n\\par\nWe consider now the correlator of $g_{YM}^2 tr F^2$:\n\\begin{align}\n&\\braket{g_{YM}^2 trF^2(p)g_{YM}^2trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2g_{YM}^4(\\mu)}{\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{YM}^2(\\mu)\\Bigl(h_0-\\tilde{\\beta}_0\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +\\tilde{\\beta}_0^2 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr]\\nonumber\\\\\n&=-\\frac{2g^4_{uv}(\\mu)}{\\pi^2}p^4\n\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+(h_0-2u) g^2_{uv}(\\mu)-\n\\tilde{\\beta}_0 g^2_{uv}(\\mu)\\log\\frac{p^2}{\\mu^2}+(h_2+5u^2+2v-uh_0) g_{uv}^4(\\mu)\\nonumber\\\\\n&+(h_3+3\\tilde{\\beta}_0 u) g_{uv}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\tilde{\\beta}_0^2 g_{uv}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nChoosing $u=\\frac{h_0}{2}$ to cancel the finite term of order of $g^2$ in the square brackets we get for the coefficient of the term of order of $g^4 \\log\\frac{p^2}{\\mu^2}$:\n\\begin{align}\n&h_3+3\\tilde{\\beta}_0 u\\nonumber\\\\\n&=h_3+\\frac{3}{2}\\tilde{\\beta}_0 h_0\\nonumber\\\\\n&=\\biggl(-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2\\biggr)\\frac{1}{(4\\pi)^4}\n+\\frac{3}{2}\\biggl(73-\\frac{14}{3}n_l\\biggr)\\biggl(11-\\frac{2}{3}n_l\\biggr)\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n&=\\biggl(-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2+\\frac{2409}{2}-73n_l-77n_l+\\frac{14}{3}n_l^2\\biggr)\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n&=-204+\\frac{76}{3}n_l\\nonumber\\\\\n&=-2\\tilde{\\beta}_1\n\\end{align}\nas predicted by Eq.(\\ref{eqn:dim_anomala_scalar}) and by the computational experience gained in the pure $YM$ case. \nTo cancel the finite term of order of $g^4$ we put:\n\\begin{equation}\nh_2+\\frac{5}{2}h_0^2+2v-\\frac{h_0^2}{2}=0\n\\end{equation} \nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:corr_scalare_qcd}\n&\\braket{g_{YM}^2trF^2(p)g_{YM}^2trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2g^4_{uv}(\\mu)}{\\pi^2}p^4\n\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\tilde{\\beta}_0 g^2_{uv}(\\mu)\\log\\frac{p^2}{\\mu^2}-2\\tilde{\\beta}_1 g_{uv}^4(\\mu)\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&+\\tilde{\\beta}_0^2 g_{uv}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nNow we follow the same steps as in the $n_l=0$ case. The only differences are the coefficients of the $\\beta$ function and the parameters $u,v$ that define the new renormalization scheme.\nThe result is:\n\\begin{align}\n&\\braket{g_{YM}^2trF^2(p)g_{YM}^2trF^2(-p)}\\nonumber\\\\\n&=\\frac{2}{\\tilde{\\beta}_0\\pi^2}p^4\\Bigl(g^2_{uv}(p)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{uv}(p)g^2_{uv}(\\mu)-g^2_{uv}(\\mu)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{uv}(\\mu)\\Bigr)\n\\end{align}\n\\newline\nHence:\n\\begin{align}\n\\biggl(\\frac{\\beta(g_{uv})}{g_{uv}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\n=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{uv}^2(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{uv}(\\mu)g_{uv}^2(p)-g_{uv}^2(\\mu)\n-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{uv}(\\mu)\\biggr)\n\\end{align}\nWe recall that $u=\\frac{h_0}{2}$, therefore:\n\\begin{equation}\ng_{uv}^2(\\mu)=g_{YM}^2(\\mu)\\bigl(1+\\frac{h_0}{2}g_{YM}^2(\\mu)+vg_{YM}^4(\\mu)\\bigr)\n\\end{equation}\nHence we get:\n\\begin{align}\\label{eqn:ris_vecchio}\n&\\biggl(\\frac{\\beta(g_{uv})}{g_{uv}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\\nonumber\\\\\n&=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{YM}^2(p)+\\frac{h_0}{2}g_{YM}^4(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{YM}(\\mu)g_{YM}^2(p)+\\nonumber\\\\\n&-g_{YM}^2(\\mu)\n-\\frac{h_0}{2}g_{YM}^4(\\mu)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{YM}(\\mu)+O(g^6)\\biggr)\n\\end{align}\nNow we  multiply the $RHS$ of Eq.(\\ref{eqn:ris_vecchio}) by \n$\\frac{\\bigl(\\frac{\\beta(g_{YM})}{g_{YM}}\\bigr)^2}{\\bigl(\\frac{\\beta(g_{uv})}{g_{uv}}\\bigr)^2}$.\nIndeed, this is necessary to take into account the change of scheme performed to compute Eq.(\\ref{eqn:ris_vecchio}).\nThe additional factor is:\n\\begin{align}\n&\\frac{\\bigl(\\frac{\\beta(g_{YM})}{g_{YM}}\\bigr)^2}{\\bigl(\\frac{\\beta(g_{uv})}{g_{uv}}\\bigr)^2}=\n\\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{YM}^2(\\mu)\\bigr)^2}{(1+\\frac{\\beta_1}{\\beta_0}g_{uv}^2(\\mu)\\bigr)^2}\\nonumber\\\\\n&=\\bigl(1+2\\frac{\\beta_1}{\\beta_0}g_{YM}^2(\\mu)+\\frac{\\beta_1^2}{\\beta_0^2}g_{YM}^4(\\mu)\\bigr)\n\\bigl(1-2\\frac{\\beta_1}{\\beta_0}g_{uv}^2(\\mu)+3\\frac{\\beta_1^2}{\\beta_0^2}g_{YM}^4(\\mu)\\bigr)\\nonumber\\\\\n&=1-h_0\\frac{\\beta_1}{\\beta_0}g^4_{YM}(\\mu) +O(g^6)\n\\end{align}\nTherefore, the correlator in Eq.(\\ref{eqn:ris_vecchio}) becomes:\n\\begin{align}\n&\\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\\nonumber\\\\\n&=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{YM}^2(p)+\\frac{h_0}{2}g_{YM}^4(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{YM}(\\mu)g_{YM}^2(p)+\\nonumber\\\\\n&-g_{YM}^2(\\mu)\n-\\frac{h_0}{2}g_{YM}^4(\\mu)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{YM}(\\mu)-h_0\\frac{\\beta_1}{\\beta_0}g^4_{YM}(\\mu)+O(g^6)\\biggr)\n\\end{align}\nthat has some dependence on the scale $\\mu$ even after subtracting the contact terms. In the next section we get rid of this dependence \nby using a different method, that assumes the $RG$ invariance of the correlator instead of checking it. \\par\nIn any case the universal $UV$ asymptotic behavior is in agreement with the $RG$ estimate, i.e.:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{YM})}{g_{YM}}tr F^2(p)\\frac{\\beta(g_{YM})}{g_{YM}}tr F^2(-p)}_{conn}\n\\sim \\frac{p^4}{\\tilde{\\beta}_0 \\log\\frac{p^2}{\\Lambdams^2}}\\Biggl(1-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambdams^2}}{\\log\\frac{p^2}{\\Lambdams^2}}\\Biggr)\n\\end{align}\n\n\\subsection{$RG$-invariant scalar correlator in $SU(3)$ $QCD$ with $n_l$ massless Dirac fermions}\n\nFirstly, we check the correctness of the finite parts of the scalar correlator in $QCD$, reconstructed in sect.(3.6) from its imaginary part, thanks to another result reported in \\cite{chet:tensore}:\n\\begin{align}\\label{eqn:der_chet}\np^2\\frac{d}{d p^2} \\Pi(p)\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}=\n\\frac{1}{\\pi^2}\\biggl[-2+\\frac{\\alpha_s}{\\pi}\\biggl(-\\frac{73}{2}+\\frac{7}{3}n_l\\biggr)+\\nonumber\\\\\n+\\frac{\\alpha_s^2}{\\pi^2}\\biggl(-\\frac{37631}{48}+\\frac{495}{4}\\zeta(3)+n_l\\Bigl(\\frac{7189}{72}-\\frac{5}{2}\\zeta(3)\\Bigr)-\\frac{127}{54}n_l^2\\biggr)\\biggr]\n\\end{align}\nwith:\n\\begin{equation}\np^4 \\Pi(p)=\\braket{trF^2(p)trF^2(-p)}\n\\end{equation}\nwhere we have changed the overall normalization factor of the correlator with respect to \\cite{chet:tensore} to be coherent with the one used in this paper.\nWe perform the derivative at $p^2=\\mu^2$ of the correlator obtained in sect.(3.6):\n\\begin{align}\\label{eqn:correlator}\n&p^2\\frac{d}{d p^2} \\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{d}{d\\log p^2}\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\biggl[ \\frac{2}{\\pi^2}\\log\\frac{p^2}{\\mu^2}\n\\biggl(1+g_{YM}^2(\\mu)\\Bigl(h_0+h_1\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +h_4 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr)\\biggr] \\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl(1+g_{YM}^2(\\mu) h_0+g_{YM}^4(\\mu) h_2\\biggr) \n\\end{align}\nWe recall that:\n\\begin{align}\\label{eqn:der_mia}\nh_0&=\\biggl(73-\\frac{14}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\nh_2&=\\frac{37361}{6}-990\\zeta(3) + n_l\\Bigl(-\\frac{7189}{9}+20\\zeta(3)\\Bigr)+\\frac{508}{27}n_l^2\n\\end{align}\nIt is easy to verify that Eq.(\\ref{eqn:der_chet}) and Eq.(\\ref{eqn:der_mia}) are in agreement.\nIndeed:\n\\begin{align}\n&\\frac{1}{\\pi^2}\\biggl[-2+\\frac{\\alpha_s}{\\pi}\\biggl(-\\frac{73}{2}+\\frac{7}{3}n_l\\biggr)\\nonumber\\\\\n&+\\frac{\\alpha_s^2}{\\pi^2}\\biggl(-\\frac{37631}{48}+\\frac{495}{4}\\zeta(3)+n_l\\Bigl(\\frac{7189}{72}-\\frac{5}{2}\\zeta(3)\\Bigr)-\\frac{127}{54}n_l^2\\biggr)\\biggr]\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl[1+\\frac{g_{YM}^2}{4\\pi^2}\\biggl(+\\frac{73}{4}-\\frac{7}{6}n_l\\biggr)\\nonumber\\\\\n&+\\frac{g_{YM}^4}{(4\\pi^2)^2}\\biggl(+\\frac{37631}{96}-\\frac{495}{8}\\zeta(3)+n_l\\Bigl(-\\frac{7189}{144}+\\frac{5}{4}\\zeta(3)\\Bigr)+\\frac{127}{108}n_l^2\\biggr)\\biggr]\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl[1+g_{YM}^2h_0+g_{YM}^4 h_2\\biggr]\n\\end{align}\nFrom Eq.(\\ref{eqn:correlator}) it follows the derivative of the correlator of $\\frac{\\beta(g_{YM})}{g_{YM}} trF^2$ with two-loop accuracy:\n\\begin{align}\\label{eqn:der_corr_rgi}\n&p^2\\frac{d}{d p^2} \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}\\biggr)^2\\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 g^4_{YM}(\\mu)\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2(\\mu)\\bigr)^2\\biggl[1+g_{YM}^2(\\mu)h_0+g_{YM}^4(\\mu) h_2\\biggr]\n\\end{align}\nSecondly, we write $g_{YM}(p)$ instead of $g_{YM}(\\mu)$  in Eq.(\\ref{eqn:der_corr_rgi}) since $\\log\\frac{p^2}{\\mu^2}=0\\Rightarrow p^2=\\mu^2$ in order to get \nthe large-momentum correlator in a manifestly $RGI$ form.\nExploiting the definition of the $\\beta$ function we can express $d\\log p^2$ in terms of $dg(p)$:\n\\begin{align}\n&\\frac{dg}{d\\log p}=\\beta(g)\n\\Rightarrow  d\\log(p^2)=2\\frac{dg}{\\beta(g)} = \\frac{d(g^2)}{g\\beta(g)}\n\\end{align} \nWe integrate Eq.(\\ref{eqn:der_corr_rgi}) to obtain:\n\\begin{align}\\label{eqn:ris_nuovo}\n&\\frac{d}{d \\log p^2} \\biggl(\\frac{\\beta(g)}{g}\\biggr)^2\\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 g^4_{YM}(p)\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2(p)\\bigr)^2\\biggl[1+g_{YM}^2(p)h_0+g_{YM}^4(p) h_2\\biggr] \\nonumber\\\\\n\\Rightarrow & \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}(p)\\biggr)^2 \\Pi(\\frac{p}{\\mu}) - \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}(\\mu)\\biggr)^2\\Pi(1) \\nonumber\\\\\n&=\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 \\int_{g_{YM}^2(\\mu)}^{g^2_{YM}(p)}g_{YM}^4\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2\\bigr)^2\\biggl[1+g_{YM}^2h_0+g_{YM}^4 h_2\\biggr]\\frac{d(g_{YM}^2)}{\\tilde{\\beta_0}g_{YM}^4(1+\\frac{\\tilde{\\beta_1}}{\\tilde{\\beta_0}}g_{YM}^2)}\\nonumber\\\\\n&=\\frac{2}{\\pi^2}\\tilde{\\beta}_0\\biggl[g_{YM}^2(p)-g_{YM}^2(\\mu)+\\biggl(\\frac{\\tilde{\\beta}_1}{2\\tilde{\\beta}_0}+\\frac{h_0}{2}\\biggr)\\biggl(g_{YM}^4(p)-g_{YM}^4(\\mu)\\biggr)+ O(g^6)\\biggr]\n\\end{align}\nEq.(\\ref{eqn:ris_nuovo}) gives the manifestly $RGI$ form of the correlator after subtracting the $\\mu$-dependent contact terms.\n\n\n\\subsection{Correlators in the coordinate representation}\n\nIn this section we find the $RG$-improved expression for the perturbative correlators in the coordinate representation.\nThis procedure has the main advantage that in the coordinate representation the contact terms do not occur, since they are eliminated by the Fourier transform.\nIndeed, the Fourier transform of $p^4$ is:\n\\begin{equation}\n\\int p^4  e^{ip\\cdot x} \\frac{d^4 p}{(2\\pi)^4}=\n\\Delta^2 \\delta (x) \n\\end{equation}\nthat is supported only at $x=0$. This implies that at points different from zero the contact terms do not occur. \nThe $RG$ improvement and the Fourier transform must commute up to perhaps finite scheme-dependent terms. Therefore, in this way we get another check of the asymptotic behavior.\nIn appendix A we compute the Fourier transforms necessary to pass from the momentum to the coordinate representation.\nIn particular we use the following results:\n\\begin{align}\n\\int {(p^2)}^2\\log\\frac{p^2}{\\mu^2} e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n-\\frac{2^6\\cdot 3}{\\pi^2 x^8}\\nonumber\\\\\n\\int {(p^2)}^2\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^2 e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n\\frac{2^7\\cdot 3}{\\pi^2 x^8}\\bigl(-\\frac{10}{3}+2\\gamma_E -\\log\\frac{4}{x^2\\mu^2}\\bigr)\\nonumber\\\\\n\\int {(p^2)}^2\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^3 e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n\\frac{2^6\\cdot 3}{\\pi^2 x^8}\\bigl(-\\frac{51}{2}+40\\gamma_E-12\\gamma_E^2\\nonumber\\\\\n&-(20-12\\gamma_E)\\log\\frac{4}{x^2\\mu^2} -3\\log^2\\frac{4}{x^2\\mu^2}\\bigr)\\nonumber\\\\\n\\end{align}\nUsing these formulae to compute the Fourier transform of the two-loop perturbative result in Eq.(\\ref{eqn:2l_perturbative_corr}) we get, disregarding the finite parts in Eq.(\\ref{eqn:2l_perturbative_corr}):\n\\begin{align}\\label{eqn:int_corr_2l}\n&-\\int g_{\\overline{MS}}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{3\\cdot 2^6}{\\pi^2 x^8} g_{\\overline{MS}}^4(\\mu)\n+\\beta_0 g_{\\overline{MS}}^6(\\mu)\\frac{3\\cdot 2^6}{\\pi^2 x^8}\\bigl[-2\\log\\frac{4}{x^2\\mu^2} +4\\gamma_E - \\frac{20}{3}\\bigr]\\nonumber\\\\\n&= \\frac{3\\cdot 2^6}{\\pi^2 x^8}g_{\\overline{MS}}^4(\\mu)\\biggl[1+\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr) g_{\\overline{MS}}^2(\\mu)-2\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nFirstly, the Fourier transform has produced a new finite part. \nSecondly, the coefficient of the logarithm in the square brackets is multiplied by two after the Fourier transform. This implies that the factor in the square brackets renormalizes four powers of $g_{\\overline{MS}}(\\mu)$, as opposed to the momentum representation, where only two powers of the coupling constant were renormalized. \nThis is as expected, since in the coordinate representation the correlator is multiplicatively renormalizable as implied by Eq.(\\ref{eqn:pert_general_behavior_x}). \\par\nTo eliminate the finite term arising from the Fourier transform we change scheme defining:\n\\begin{equation}\ng^2_s(\\mu)= g^2(\\mu)\\bigl[1+\\frac{1}{2}\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr)g^2(\\mu)\\bigr]\n\\end{equation} \nThe integral in Eq.(\\ref{eqn:int_corr_2l}) reads:\n\\begin{align}\n&-\\int g_{\\overline{MS}}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}=\n\\frac{3\\cdot 2^6}{\\pi^2 x^8} g^4(x)\\nonumber\\\\\n\\end{align} \nwhere $g(x)$ is the one-loop running coupling in the coordinate scheme \\cite{chetyrkin:TF}:\n\\begin{align}\ng^2(x)=g^2(\\mu)\\biggl[1-\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nTherefore, the renormalization group improved one-loop asymptotic expression for the correlator is:\n\\begin{equation}\\label{eqn:rg_improved_2l_{a}}\n\\braket{\\frac{g^2}{N} trF^2(x) \\frac{g^2}{N} tr F^2(0)}_{conn}\\sim\n \\bigl(1-\\frac{1}{N^2}\\bigr) \\frac{3\\cdot 2^6}{\\pi^2 x^8} \\frac{1}{\\log^2\\frac{4}{x^2\\mu^2}}\n\\end{equation}\nThe Fourier transform provides automatically the change in sign necessary to obtain a positive expression.\nThis is due to the fact that in the coordinate representation contact terms do not occur.\nWe now go one step further performing the Fourier transform of the three-loop propagators in Eq.(\\ref{eqn:corr_pert_scalar_3l}) and in Eq.(\\ref{eqn:corr_pert_pseudoscalar_3l}). \nWe start with the scalar correlator up to the overall normalization:\n \\begin{align}\\label{eqn:int_corr_3l}\n&-\\int g_{ab}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&\\times\\biggl[1-\\beta_0 (\\mu)g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} -2\\beta_1 g^4_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{ab}^4(\\mu)\\biggl[1+\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr)g^2_{ab}(\\mu)\n-2\\beta_0 g^2_{ab}(\\mu)\\log\\frac{4}{x^2\\mu^2} \\nonumber\\\\\n&+\\bigl(8\\beta_1\\gamma_E-\\frac{40}{3}\\beta_1+\\frac{51}{2}\\beta_0^2 -40\\beta_0^2\\gamma_E+12\\beta_0^2\\gamma_E^2\\bigr)g_{ab}^4(\\mu)\n-4\\beta_1 g_{ab}^4(\\mu)\\log\\frac{4}{x^2\\mu^2} \\nonumber\\\\\n&-\\beta_0^2\\bigl(12\\gamma_E-20\\bigr)g^4_{ab}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n+3\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nThe following scheme redefinition:\n\\begin{equation}\ng^2_{st}(\\mu)= g^2_{ab}(\\mu)\\bigl(1+(2\\beta_0\\gamma_E-\\frac{10}{3}\\beta_0)g^2_{ab}+t g_{ab}^4(\\mu)\\bigr)\n\\end{equation}\ncancels the finite term of order of $g^2$ in the square brackets and some terms of order of $g^4\\log\\frac{4}{x^2\\mu^2}$, leaving only the term proportional to $-4\\beta_1$.\nMoreover, the finite term of order of $g^4$ in the square brackets is cancelled by a suitable choice of $t$, as in the previous section.\nEq.(\\ref{eqn:int_corr_3l}) now reads:\n\\begin{align}\\label{eqn:corr_scalare_3l_revisited}\n&-\\int g_{ab}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&\\times\\biggl[1-\\beta_0 (\\mu)g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} -2\\beta_1 g^4_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(\\mu)\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}\\quad \\nonumber\\\\\n\\times\\quad &\\biggl[1-2\\beta_0 g^2_{st}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-4\\beta_1 g_{st}^4(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_0^2 g^4_{st}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(\\mu)\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}\\quad \\nonumber\\\\\n\\times\\quad & \\biggl[1-2\\beta_0 g^2_{st}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-2\\beta_1 g_{st}^4(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_0^2 g^4_{st}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)\n-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)\\Bigr)\n\\end{align}\nThe scale dependent term in Eq.(\\ref{eqn:corr_scalare_3l_revisited}) occurs now at the order of $g^6$, while in the  momentum representation occurred at the order of $g^4$.\nNow we multiply both sides of Eq.(\\ref{eqn:corr_scalare_3l_revisited}) by $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{st}^2(\\mu)\\bigr)^2$, i.e. by the factor necessary to make the correlator $RGI$.\nReinserting the overall normalization, we obtain:\n\\begin{align}\n&\\int \\braket{\\frac{\\beta(g)}{Ng} trF^2(p)\\frac{\\beta(g)}{Ng}trF^2(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\ \n&= \\frac{1}{4 \\pi^2} \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+\\frac{\\beta_1}{\\beta_0}g_{st}^2(\\mu)\\Bigr)^2\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)\n-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)\\Bigr)\\nonumber\\\\\n&= \\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x) +O(g^4)\\Bigr)\n\\end{align}\nAs a result the possible scale dependence is of order of $g^8$. \\par\nPerforming the same steps for the pseudoscalar correlator in Eq.(\\ref{eqn:corr_pert_pseudoscalar_3l}) we get:\n\\begin{align}\\label{eqn:corr_pseudoscalare_3l_revisited}\n&\\int \\braket{\\frac{g^2}{N}trF\\tilde{F}(p) \\frac{g^2}{N}trF\\tilde{F}(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr) \\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{\\tilde{st}}^4(\\mu)\\biggl[1-2\\beta_0 g^2_{\\tilde{st}}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-2\\beta_1 g^4_{\\tilde{st}}(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_1^2g^4_{\\tilde{st}}\\log^2\\frac{4}{x^2\\mu^2}\\biggr] \\nonumber\\\\\n&= \\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{\\tilde{st}}^4(x) \n\\end{align}\nThe correlator in Eq.(\\ref{eqn:corr_pseudoscalare_3l_revisited}) is $RGI$ in the coordinate representation with three-loop accuracy, while in the momentum representation scale-dependent terms of the order of $g^4$ occurred in Eq.(\\ref{eqn:rg_improved_pseudo_3l}). \nAs in the momentum representation we find the correlator of $tr{F^-}^2$ summing the double of the scalar Eq.(\\ref{eqn:corr_scalare_3l_revisited}) and pseudoscalar Eq.(\\ref{eqn:corr_pseudoscalare_3l_revisited}) correlators.\nWe obtain:\n\\begin{align}\\label{eqn:corr_asd_{a}}\n&\\frac{1}{2}\\int \\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{1}{4 \\pi^2} \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^7\\cdot 3}{\\pi^2 x^8}\\bigl(g^4_{st}(x)+g^4_{\\tilde{st}}(x)+2\\frac{\\beta_1}{\\beta_0}g^6_{st}(x)-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)g^4_{st}(x)\\bigr)\\nonumber\\\\\n\\end{align}\nThe scale dependence enters the term of order of $g^6$ as in the momentum representation in Eq.(\\ref{eqn:corr_asd_pert}). \\par\nWe check the correctness of the separation of the contact terms performed in the momentum representation.\nWe verify to the order of the leading logarithm that the Fourier transform of the $RG$-improved expression in the momentum representation in Eq.(\\ref{eqn:corr_asd_ms}) is equal to Eq.(\\ref{eqn:corr_asd_{a}}) in the coordinate representation.\nWithin the leading logarithmic accuracy it is sufficient to put $g^2(p)$:\n\\begin{equation}\ng^2(p)=\\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}}\n\\end{equation}\nTherefore, the Fourier transform of the correlator in Eq.(\\ref{eqn:corr_asd_ms}) can be computed reducing it to a series of positive powers of logarithms:\n\\begin{align}\\label{eqn:serie_p_da_trasformare}\n& \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2\\beta_0}\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&= \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\sum_{l=0}^{\\infty}(-1)^l\\int p^4 g^2(\\mu)\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\bigr)^l e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\n\\end{align}\nWe extract the leading logarithms of this Fourier transform.\nBy leading we mean terms that have the highest power of logarithm with the power of $g$ fixed.\nWe use  Eq.(\\ref{eqn:ft_leading}) that furnishes the leading logarithm of the Fourier transform:\n\\begin{equation}\n\\int p^4\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^l e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} =\n-\\frac{l\\Gamma(4) 2^{5}}{\\pi^2}\n\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}+\\cdots\n\\end{equation}\nInserting it in Eq.(\\ref{eqn:serie_p_da_trasformare}) we obtain for the leading logarithms: \n\\begin{align}\\label{eqn:x_series_da_tf}\n& \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&= \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\sum_{l=0}^{\\infty}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}\n\\end{align}  \nWe compare it with the $ASD$ correlator in the coordinate representation Eq.(\\ref{eqn:corr_asd_{a}}):\n\\begin{align}\\label{eqn:rg_improved_2l_{a}_series}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4) g^4(x)\\sim\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4) \\biggl(\\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}}\\biggr)^2\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n=0}^{\\infty}\\sum_{l=0}^{\\infty}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{x^2\\mu^2}{4}\\bigr)^{n+l}\n\\end{align} \nWe want to prove that the two series in Eq.(\\ref{eqn:x_series_da_tf}) and in Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) are equal. \nThe proof is by induction.\nWe prove it for the first non trivial term, i.e. for $l=1$ in Eq.(\\ref{eqn:x_series_da_tf}):\n\\begin{equation}\n\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)\ng^{4}(\\mu)\n\\end{equation} \nthat is equal to the term obtained from Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) putting $n=l=0$.\nAssuming that the equality is valid up to the order of $\\biggl(\\log\\frac{x^2\\mu^2}{4} \\biggr)^{m-1}$, we show that it holds at the order of $\\biggl(\\log\\frac{x^2\\mu^2}{4} \\biggr)^{m}$.\nIndeed, the $m$-power of the logarithm occurs in Eq.(\\ref{eqn:x_series_da_tf}) for $l=m+1$:\n\\begin{align}\\label{eqn:dim_induzione_intermedio}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4} \\nonumber\\\\\n&\\sim \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\sum_{l=0}^{m}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}+\\nonumber\\\\\n&+(-1)^m\\frac{(m+1)\\cdot 2^5}{\\pi^2 x^8}\\beta_0^{m} \\Gamma(4)\ng^{2m+4}(\\mu)\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{m} \\bigg]\n\\end{align} \nThe $m$-th power of the logarithm in Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) occurs for the $m+1$ couples $(n,l)$ such that $l+n=m$:\n\\begin{align}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(x)\\nonumber\\\\\n&\\sim \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{n+l}+\\nonumber\\\\\n&+\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l=m}}(-1)^{m}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{m} \\bigg]\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{n+l}+\\nonumber\\\\\n&+(-1)^m\\frac{(m+1)\\cdot 2^5}{\\pi^2 x^8}\\beta_0^{m} \\Gamma(4)\ng^{2m+4}(\\mu)\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{m} \\bigg]\n\\end{align}\nFor the inductive hypothesis the first term in the last expression is equal to the first one in Eq.(\\ref{eqn:dim_induzione_intermedio}), i.e.:\n\\begin{align}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{x^2\\mu^2}{4}\\bigr)^{n+l}  \\bigg]\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\sum_{l=0}^{m}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1} \\bigg]\n\\end{align}\nThe remaining terms, i.e. the terms of order of $\\log^m\\frac{4}{x^2\\mu^2}$, are equal and therefore\nthe proof by induction is complete.\n\n\n\n\n\\section{ $ASD$ correlator in the Topological Field Theory}\\thispagestyle{empty} \n\nWe briefly summarize the results for the glueball propagators in the $TFT$ underlying large-$N$ $YM$ \\cite{boch:quasi_pbs} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\cite{Top}. \nFor the $ASD$ glueball propagator \\cite{boch:glueball_prop}\\cite{boch:crit_points} \\footnote{We use here a manifestly covariant notation\nas opposed to the one in the $TFT$ \\cite{boch:glueball_prop}\\cite{boch:crit_points}.}:\n\\begin{equation}\\label{eqn:intro_formula_L2}\n\\frac{1}{2}\\braket{\\frac{g^2}{N}tr\\bigl(F^{-2}(p)\\bigr) \\frac{g^2}{N}tr \\bigl(F^{-2}(-p)\\bigr)}_{conn} =\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{k^2 g_k^4\\Lambda_{\\overline{W}}^6 }{p^2+k\\Lambda_{\\overline{W}}^2} + ...\n\\end{equation}\nBesides, in the $TFT$ the two-point correlators of certain scalar operators $\\mathcal{O}_{2L}$ of naive dimension $D=2L$ that are homogeneous polynomials of degree $L$\nin the $ASD$ curvature $F^-$\\cite{boch:glueball_prop}\\cite{boch:crit_points} can be computed asymptotically for large $L$:\n\\begin{align}\\label{eqn:formula}\n&\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\n=const\\sum_{k=1}^{\\infty}\\frac{k^{2L-2} Z_k^{-L}\\Lambda_{\\overline{W}}^2 \\Lambda_{\\overline{W}}^{4L-4}}{p^2+k\\Lambda_{\\overline{W}}^2} \n\\end{align}\nThe operators $\\mathcal{O}_{2L}$ occur as the ground state in the integrable sector of large-$N$ $YM$ of Ferreti-Heise-Zarembo \\cite{ferretti:new_struct} asymptotically for large $L$.\nFerreti-Heise-Zarembo have computed their one-loop anomalous dimension for large $L$ \\cite{ferretti:new_struct}:\n\\begin{align}\n\\gamma_{0 (\\mathcal{O}_{2L})}= \\frac{1}{(4\\pi)^2}\\frac{5}{3} L+O(\\frac{1}{L})\n\\end{align}\nThe ground state for $L=2$ is the $ASD$ operator that occurs in Eq.(\\ref{eqn:intro_formula_L2}) for which $\\gamma_{0 (\\mathcal{O}_{4})}=2 \\beta_0$ exactly. \\par\nIn Eq.(\\ref{eqn:intro_formula_L2}) and in Eq.(\\ref{eqn:formula}) $\\Lambda_{\\overline{W}}$ is the $RG$ invariant scale in the scheme in which it coincides with the mass gap. The functions $g^2(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})$ and $Z(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})$ are the solutions of the differential equations:\n\\begin{align} \\label{eqn:eq_def_gk}\n\\frac{\\partial g}{\\partial \\log p}\n&=\\frac{-\\beta_0 g^3+\\frac{1}{(4\\pi)^2}g^3\\frac{\\partial \\log Z}{\\partial \\log p}}{1-\\frac{4}{(4\\pi)^2}g^2} \\nonumber \\\\\n\\frac{\\partial\\log Z}{\\partial\\log p}\n&=2\\gamma_0 g^2 +\\cdots \\nonumber \\\\\n\\gamma_{0}&=\\frac{1}{(4\\pi)^2}\\frac{5}{3}\n\\end{align}\nwhere $p$ is equal to the square root of $p^2$.\nThe definitions of $g_k$ and $Z_k$ are:\n\\begin{align}\n&g_k=g(k)\\\\\n&Z_k=Z(k)\n\\end{align}\nIn \\cite{boch:quasi_pbs} it is shown that Eq.(\\ref{eqn:eq_def_gk}) reproduces the correct universal one-loop and two-loop coefficients of the perturbative $\\beta$ function of pure $YM$. Indeed, substituting in Eq.(\\ref{eqn:eq_def_gk}) we get:\n\\begin{align}\\label{eqn:matching_beta_pert}\n\\frac{\\partial g}{\\partial \\log p}\n&=\\frac{-\\beta_0 g^3+\\frac{2\\gamma_0 }{(4\\pi)^2}g^5}{1-\\frac{4}{(4\\pi)^2}g^2}+\\cdots \\nonumber \\\\\n&=\\bigl(-\\beta_0 g^3+\\frac{2\\gamma_0}{(4\\pi)^2}g^5\\bigr)\\bigl(1+\\frac{4}{(4\\pi)^2}g^2\\bigr)+\\cdots \\nonumber\\\\\n&=-\\beta_0 g^3+\\frac{2\\gamma_0}{(4\\pi)^2}  g^5-\\frac{4\\beta_0}{(4\\pi)^2}g^5+\\cdots \\nonumber\\\\\n&=-\\beta_0 g^3+\\frac{1}{(4\\pi)^4}\\frac{10}{3}g^5 - \\frac{44}{3}\\frac{1}{(4\\pi)^4}g^5+\\cdots\\nonumber\\\\\n&=-\\beta_0 g^3-\\beta_1 g^5+\\cdots\n\\end{align}\nwhere:\n\\begin{align}\n&\\beta_0=\\frac{1}{(4\\pi)^2}\\frac{11}{3}\\\\\n&\\beta_1=\\frac{1}{(4\\pi)^4}\\frac{34}{3}\n\\end{align}\nThese are the correct one- and two-loop coefficients that arise in perturbation theory of pure $YM$ for the 't Hooft coupling.\nTherefore, the renormalization-group improved universal asymptotic behavior of $g_k$ is:\n\\begin{equation}\\label{eqn:gk_as_behav}\ng^2_k\\sim\\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n+O\\biggl(\\frac{1}{\\log^2\\frac{k}{c}}\\biggr)\n\\end{equation}\nand the renormalization group improved universal asymptotic behavior of $Z_k^{-1}$ is:\n\\begin{equation}\\label{eqn:zk_as_behav}\nZ_k^{-1}\\sim (g^2_k)^{\\frac{\\gamma_0}{\\beta_0}}\\sim\n\\Biggl(\\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{k}{c}} \\biggr)  \\Biggr)^{\\frac{\\gamma_0}{\\beta_0}}\n\\end{equation}\nIn this section we find the asymptotics of the $ASD$ propagator in Eqs.(\\ref{eqn:intro_formula_L2}) and of the large-$L$ propagator in Eq.(\\ref{eqn:formula}) at the order of the leading and of the next-to-leading logarithms following the technique employed in \\cite{boch:glueball_prop} at the order of the leading logarithm. \\par\nTo find the asymptotics of the glueball propagator for large $L$ in Eq.(\\ref{eqn:formula}) we follow the strategy explained in sect.(1.4) for the\n$ASD$ correlator. Firstly, we highlight the physical terms contained in Eq.(\\ref{eqn:formula}) neglecting the non-physical contact terms. Secondly, we extract the asymptotic behavior writing the sum in Eq.(\\ref{eqn:formula}) as an integral \\cite{boch:glueball_prop}. Finally, we use the leading and next-to-leading expression for $Z_k^{-1}$ in Eq.(\\ref{eqn:zk_as_behav}) to compare Eq.(\\ref{eqn:formula}) with $RG$-improved perturbation theory.\n\nWe write Eq.(\\ref{eqn:formula}) as \\cite{boch:glueball_prop}\\cite{boch:crit_points}:\n\\begin{align}\n&\\sum_{k=1}^{\\infty}\n\\frac{k^{2(L-1)}Z_k^{-L}\\Lambda_{\\overline{W}}^2 \\Lambda_{\\overline{W}}^{4(L-1)}}{p^2+k\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n&=\\sum_{k=1}^{\\infty}\n\\frac{((k\\Lambda_{\\overline{W}}^2+p^2)(k\\Lambda_{\\overline{W}}^2-p^2)+p^4)^{L-1} Z_k^{-L}\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n&=p^{4L-4}\\sum_{k=1}^{\\infty}\\frac{Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2} \\nonumber\\\\\n&+\\sum_{k=1}^{\\infty}\\sum_{m=1}^{L-1}\\binom{L-1}{m}p^{4(L-1-m)}(k\\Lambda_{\\overline{W}}^2+p^2)^{m-1}(k\\Lambda_{\\overline{W}}^2-p^2)^m  Z_k^{-L}\\Lambda_{\\overline{W}}^2\\nonumber\\\\ \n&\\sim p^{4L-4}\\sum_{k=1}^{\\infty}\\frac{ Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2}+\\dots\n\\end{align}\nwhere the dots stand for contact terms.\n\nAs in sect.(1.4) we use the Euler-McLaurin formula to approximate the sum to an integral \\cite{boch:glueball_prop}\\cite{boch:crit_points}:\n\\begin{equation}\n\\sum_{k=k_1}^{\\infty}G_k(p)=\n\\int_{k_1}^{\\infty}G_k(p)dk - \\sum_{j=1}^{\\infty}\\frac{B_j}{j!} \\left[\\partial_k^{j-1}G_k(p)\\right]_{k=k_1}\n\\end{equation}\nIn our case the terms proportional to the Bernoulli numbers involve negative powers of $p$ and they are therefore subleading with respect to the first term, hence we ignore them. \n\nWe obtain: \n\\begin{equation}\\label{eqn:int_fond}\n\\sum_{k=1}^{\\infty}\\frac{Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2}\\sim\n\\int_{1}^\\infty\\frac{Z_k^{-L}}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}dk\n\\end{equation}\nIn order to compare Eq.(\\ref{eqn:int_fond}) to the $RG$-improved perturbation theory, we substitute for $Z_k^{-1}$ its leading and next-to-leading logarithmic behavior given by Eq.(\\ref{eqn:zk_as_behav}).\nWe define:\n\\begin{equation}\n\\gamma'=\\frac{\\gamma_0}{\\beta_0}L\n\\end{equation}\nand:\n\\begin{equation}\n\\nu=\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\n\\end{equation}\nThe integral that determines the leading asymptotic behavior is:\n\\begin{equation}\\label{eqn:int_fond2}\nI_c^{1}(\\nu)=\\int_{1}^\\infty\\biggl(\\frac{1}{\\beta_0\\log(\\frac{k}{c})}\\biggr)^{\\gamma'}\\frac{dk}{k+\\nu}\n\\end{equation}\nThe next-to-leading logarithmic behavior is determined by:\n\\begin{equation}\\label{eqn:int_fond3}\nI^{2}_c(\\nu)=\\int_{1}^\\infty\\left(\\frac{1}{\\beta_0\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\n\\end{equation}\n$\\gamma'=2$ for the $ASD$ correlator and $\\gamma'=\\frac{\\gamma_0}{\\beta_0}L$ for the large-$L$ correlator.\nWe show in the following that the leading and next-to-leading behavior of $I^{2}_c(\\nu)$ is: \n\\begin{align}\n&I^{2}_c(\\nu)\\sim\\frac{1}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambdawb^2}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}{\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1} \n\\end{align}\nTherefore, the asymptotic behavior of the correlator of the $TFT$ for large-$L$ is:\n\\begin{equation}\n\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\\sim p^{4L-4}\\frac{1}{\\gamma_0 L-\\beta_0}\\bigl(g^2(p)\\bigr)^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{equation}\nIt agrees with the naive $RG$ estimate Eq.(\\ref{eqn:naive_rg}).\n\n\\subsection{Asymptotic series to the order of the leading logarithm}\n\nWe now perform an explicit expansion in series of $I_c^{1}(\\nu)$.\nFirstly, we change variables from $k$ to $k+\\nu$:\n\\begin{equation}\nI_c^{1}(\\nu)=\\int_{1+\\nu}^\\infty\\biggl(\\frac{1}{\\beta_0\\log(\\frac{k-\\nu}{c})}\\biggr)^{\\gamma'}\\frac{dk}{k}\n\\end{equation}\nWe have that: \n\\begin{equation}\n[\\log(\\frac{k'-\\nu}{c})]^{-\\gamma'}=\n[\\log(\\frac{k'}{c})]^{-\\gamma'}\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\n\\end{equation}\nIt is easy to see that if $c<1$:\n\\begin{equation}\n\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}<1\n\\end{equation}\nWe define:\n\\begin{equation}\n\\epsilon=\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\n\\end{equation}\nand we exploit the binomial formula \\cite{BIN}:\n\\begin{align}\\label{eqn:espansione_bin}\n(1+\\epsilon)^{-\\gamma'}&=\\sum_{r=0}^{\\infty}\\binom{\\gamma'+r-1}{r}(-1)^r\\epsilon^r\n\\end{align}\nto obtain a series expansion.\nWe proceed order by order in $\\epsilon$.\nAt the order of $\\epsilon^1$ the only contribution is:\n\\begin{equation}\n-\\gamma'\\epsilon\n\\end{equation}\n$\\epsilon$ can be further expanded in powers of $\\eta=\\frac{\\nu}{k'}$, since in the integration domain $\\eta<1$:\n\\begin{align}\\label{eqn:espansione_log}\n\\log(1-\\eta)&=\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}}{m}\\eta^m\n\\end{align}\nUp to the order of $\\eta^1$ this expansion reads: \n\\begin{equation}\n-\\gamma'\\epsilon\\sim\\gamma'\\frac{\\nu}{k'\\log(\\frac{k'}{c})} \n\\end{equation} \nSubstituting in $I_c^{1}(\\nu)$ we get:\n\\begin{align}\\label{eqn: int1}\n&\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}dk'\\nonumber\\\\\n&\\sim \\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'} \n\\biggl[1+\\gamma'\\frac{\\nu}{k'\\log(\\frac{k'}{c})}\\biggr]dk'\\nonumber\\\\\n&=\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}dk'+\n\\gamma' \\nu\\int_{1+\\nu}^\\infty\\frac{1}{k'^2}\\beta_0^{-\\gamma'}[\\log(\\frac{k'}{c})]^{-\\gamma'-1}dk' \n\\end{align}\nFrom the first integral it follows the leading asymptotic behavior \\cite{boch:glueball_prop}:\n\\begin{equation}\\label{eqn:sol_int_leading}\n\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}dk'=\n\\frac{1}{\\gamma'-1} \\beta_0^{-\\gamma'}\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'+1}\n\\end{equation}\nSince for large $\\nu$:\n\\begin{equation}\n\\biggl(\\log\\left(\\frac{1+\\nu}{c}\\right)\\biggr)^{-1}\\sim\n\\bigl(\\log\\nu\\bigr)^{-1} \n\\end{equation}\nit follows the leading asymptotic behavior of Eq.(\\ref{eqn:formula}) \\cite{boch:glueball_prop}:\n\\begin{align}\n\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\\sim\\frac{p^{4L-4}}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambdawb^2}}\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{align} \nPerforming the same steps for the $ASD$ correlator, i.e. for $L=2$ and $\\gamma'=2$, we get:\n\\begin{equation}\n\\frac{1}{\\pi^2}\\int_1^{\\infty}\\frac{\\bigl(\\beta_0\\log\\frac{k}{c}\\bigr)^{-2}}{k+\\nu}\\sim\n\\frac{1}{\\pi^2\\beta_0}\\Biggl(\\beta_0\\log\\biggl(\\frac{1+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}{c}\\biggr)\\Biggr)^{-1}\\sim\n\\frac{1}{\\pi^2\\beta_0}\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\n\\end{equation}\nthat agrees with the leading logarithm of the asymptotic behavior in Eq.(\\ref{eqn:corr_asd}). \\par\nWe now compute the second term in the last line of Eq.(\\ref{eqn: int1}), that is the first subleading term. \nWe write it as:\n\\begin{equation}\\label{eqn:int_ord1}\n\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^\\infty\\frac{1}{k^2}[\\log(k)]^{-\\gamma'-1}dk \n\\end{equation}\nand we integrate by parts:\n\\begin{align}\n&\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{1}{k^2}[\\log(k)]^{-\\gamma'-1}dk \\nonumber\\\\\n&=\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\biggl[-\\frac{[\\log(k)]^{-\\gamma'-1}}{k}\\bigg|_{\\frac{1+\\nu}{c}}^{\\infty}\n-(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-2}\\biggr]\\nonumber\\\\\n&=\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\biggl[\\frac{c}{1+\\nu}[\\log(\\frac{1+\\nu}{c})]^{-\\gamma'-1}\n-(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-2}\\biggr]\n\\end{align}\nWe notice that the second term in the last line has the same structure as the original integral but with a more negative power of the logarithm.\nThis implies that it is a less relevant term.\nFurthermore, since performing integration by parts repeatedly we always obtain integrals with the same structure, we can derive a possibly asymptotic series expansion for Eq.(\\ref{eqn:int_ord1}): \n\\begin{align}\n&\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-1}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\frac{\\nu}{1+\\nu}\\sum_{s=0}^{\\infty}(-1)^s\\left(\\prod_{t=0}^{s}(\\gamma'+t)\\right)[\\log(\\frac{1+\\nu}{c})]^{-\\gamma'-1-s}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\frac{p^2}{p^2+\\Lambda_{\\overline{W}}^2}\\sum_{s=0}^{\\infty}(-1)^s\\left(\\prod_{t=0}^{s}(\\gamma'+t)\\right)\n\\biggl[\\log\\biggl(\\frac{1+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}{c}\\biggr)\\biggr]^{-\\gamma'-1-s}\n\\end{align}\n\nNow that we have understood the technique, we derive a complete expression taking into account all the terms coming from the expansion of the logarithm in Eq.(\\ref{eqn:espansione_log}),\nsimply substituting it in $I_c^{1}(\\nu)$: \n\\begin{align}\n&\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1-\\gamma'\\frac{\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m k^m}}{\\log(\\frac{k'}{c})}\\biggr]\\nonumber\\\\\n&=\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\\nonumber\\\\\n&-\\gamma'\\beta_0^{-\\gamma'}\n\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m}\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'^{m+1}}\\left[\\log(\\frac{k'}{c})\\right]^{-\\gamma'-1}\n\\end{align}\nFocusing on the second term:\n\\begin{align}\\label{eqn:ord1_eps}\n&\\gamma'\\beta_0^{-\\gamma'}\n\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m}\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'^{m+1}}\\left[\\log(\\frac{k'}{c})\\right]^{-\\gamma'-1}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\gamma'\\sum_{m=1}^{\\infty}\\frac{(-1)\\nu^m}{m c^m}\\left[-\\frac{\\left[\\log(k)\\right]^{-\\gamma'-1}}{m k^{m}}\n\\bigg|_{\\frac{1+\\nu}{c}}^{\\infty}-\n(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}dk\\frac{\\left[\\log(k)\\right]^{-\\gamma'-2}}{mk^{m+1}}\\right]\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\frac{(-1)\\nu^m}{m c^m}\\left[\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{m^{s+1}}\n\\frac{c^m\\prod_{t=0}^{s}(\\gamma'+t)}{(1+\\nu)^m}\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-1-s}\\right]\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\sum_{s=0}^{\\infty}(-1)^{s+1}\\left(\\frac{\\nu}{1+\\nu}\\right)^m\\frac{\\prod_{t=0}^{s}(\\gamma'+t)}{m^{s+2}}\n\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-s-1}\t\n\\end{align}\nTherefore, at the first order in $\\epsilon$ we get:\n\\begin{align}\\label{eqn:int_lead_temp}\n&\\int_{1+\\nu}^\\infty dk'\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\\nonumber\\\\\n&\\sim \\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'} \\nonumber\\\\\n&+\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\sum_{s=0}^{\\infty}(-1)^{s}\\left(\\frac{\\nu}{1+\\nu}\\right)^m\\frac{\\prod_{t=0}^{s}(\\gamma'+t)}{m^{s+2}}\n\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-s-1}\t \n\\end{align}\nWe find the subleading behavior keeping only the terms with $s=0$ in Eq.(\\ref{eqn:int_lead_temp}).\nWe obtain in the large $\\nu$ limit: \n \\begin{align}\\label{eqn:correzione_serie_as}\n&\\int_{1+\\nu}^\\infty dk'\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\\nonumber\\\\\n& \\sim \\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n+\\gamma'\\beta_0^{-\\gamma'}\\left[\\log \\nu \\right]^{-\\gamma'-1}\\sum_{m=0}^{\\infty}\\frac{1}{m^2}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma'-1}\\beta_0^{-\\gamma'}\\left[\\log\\nu\\right]^{-\\gamma'+1}\n+\\gamma'\\beta_0^{-\\gamma'}\\zeta(2)[\\log \\nu]^{-\\gamma'-1}\n\\end{align} \nIt is interesting to notice that the transcendental function $\\zeta(2)=\\frac{\\pi^2}{6}$ occurs, as it often does in Feynman-graph computations.\n\n\n\\subsection{Asymptotic series to the order of the next-to-leading logarithm}\n\nWe now perform a series expansion of $I_c^{2}(\\nu)$:\n\\begin{align}\nI_c^{2}(\\nu)&=\\int_1^{\\infty}\\beta_0^{-\\gamma'}\\left(\\frac{1}{\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left(\\frac{1}{\\log(\\frac{k-\\nu}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k-\\nu}{c})}{\\log(\\frac{k-\\nu}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k}\\nonumber\\\\\n&\\sim \\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'}\n\\left(1-\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k-\\nu}{c})}{\\log(\\frac{k-\\nu}{c})}\\right)\\frac{dk}{k}\\nonumber\\\\\n&\\sim  \\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'}\\frac{dk}{k}+\\nonumber\\\\\n&-\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\n\\end{align}\nThe first integral has been evaluated in the previous section and the second term is the new contribution. We evaluate it at the leading order by changing variables and integrating by parts:\n\\begin{align}\\label{eqn:int_next-to-leading}\n&\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\\nonumber\\\\\n&\\sim  \\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k}{c})\n\\frac{dk}{k}\\nonumber\\\\\n&=\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1+\\nu}{c}}^{\\infty}t^{-\\gamma'-1}\\log(t)dt\\nonumber\\\\\n&=\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\left[\\frac{1}{\\gamma'}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\log\\log(\\frac{1+\\nu}{c})+\n\\frac{1}{\\gamma'^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\right]\n\\end{align}\nThe second term in brackets is subleading with respect to the first one. \nPutting together  Eq.(\\ref{eqn:int_next-to-leading}) and Eq.(\\ref{eqn:sol_int_leading}) we get for $I_c^{2}(\\nu)$ :\n\\begin{align}\\label{eqn:esp_ntl}\n&\\beta_0^{-\\gamma'}\\int_1^{\\infty}\\left(\\frac{1}{\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma'-1}\\beta_0^{-\\gamma'}\\left(\\log\\frac{1+\\nu}{c}\\right)^{-\\gamma'+1}-\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\log\\log(\\frac{1+\\nu}{c})\\nonumber\\\\\n&=\\frac{\\beta_0^{-\\gamma'}}{\\gamma'-1}\\biggl(\\log\\frac{1+\\nu}{c}\\biggr)^{-\\gamma'+1}\\left[1-\\frac{\\beta_1(\\gamma'-1)}{\\beta_0^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-1}\\log\\log(\\frac{1+\\nu}{c})\\right]\\nonumber\\\\\n&\\sim \\frac{1}{\\beta_0(\\gamma'-1)}\\left(\\beta_0\\log\\frac{1+\\nu}{c}\\right)^{-\\gamma'+1}\\left[1-\\frac{\\beta_1}{\\beta_0^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-1}\\log\\log(\\frac{1+\\nu}{c})\\right]^{\\gamma'-1}\\nonumber\\\\\n&\\sim \\frac{1}{\\beta_0(\\gamma'-1)}(g^2(p))^{\\gamma'-1}+ O\\biggl(\\Bigl(\\frac{1}{\\log\\frac{p^2}{\\Lambda_{\\overline{W}}}}\\Bigr)^{\\gamma'}\\biggr)\n\\end{align}\nThis result agrees with the $RGI$ perturbative estimate in Eq.(\\ref{eqn:naive_rg}).\nRepeating the same steps for the $ASD$ correlator we get: \n\\begin{align}\n&\\frac{1}{\\pi^2}\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-2}dk'\n-\\frac{\\beta_1}{\\pi^2\\beta_0^4}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-3}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\\nonumber\\\\\n& \\sim \\frac{1}{\\pi^2\\beta_0} g^2(p)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_{\\overline{W}}}}\\biggr)\n\\end{align}\nAgain this result agrees with the universal behavior of the $RG$-improved perturbation theory in Eq.(\\ref{eqn:rg_improved_scalar_2l}).\n\n\\newpage\n\\thispagestyle{empty}\n\n\\subsection{Link with the Lerch transcendent and the polylogarithmic function}\n\nWe may obtain the asymptotic behavior by a different method as an independent check, relating the relevant integrals\nto special functions and employing the known asymptotic behavior of the special functions. \\par\n\nWe briefly recall the definition of the Lerch Zeta function \\cite{handbook, wikilerch}:\n\\begin{equation}\nL(\\lambda,s,a)=\\sum_{n=0}^{\\infty}\\frac{e^{2\\pi i\\lambda n}}{(n+a)^s}\n\\end{equation}\nSetting $z=e^{2\\pi i \\lambda}$, we obtain the Lerch transcendent \\cite{handbook, wikilerch}:\n\\begin{equation}\n\\Phi(z,s,a)=\\sum_{n=0}^{\\infty}\\frac{z^n}{(n+a)^s}\n\\end{equation}\nThe Lerch transcendent admits the integral representation:\n\\begin{equation}\\label{eqn:lerch_int_repr}\n\\Phi(z,s,a)=\\frac{1}{\\Gamma(s)}\\int_0^{\\infty}\\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt\n\\end{equation}\nwhich is valid for $\\Re (a)>0 \\,\\wedge \\, \\Re (s)>0 \\,\\wedge \\,|z|<1$ or $\\Re (a)>0 \\,\\wedge \\, \\Re (s)>1 \\,\\wedge \\,|z|=1$.\nThe Lerch transcendent can be analytically continued to the region \\cite{analytic}:\n\\begin{equation}\n\\mathcal{M}=\\{(z,s,a)\\in (\\mathbb{C}\\setminus \\{0\\})\\times\\mathbb{C}\\times (\\mathbb{C}\\setminus\\mathbb{Z})\\}\n\\end{equation} \nMoreover, we exploit the following recursive formula:\n\\begin{equation}\\label{eqn:lerch_recursive}\n\\Phi(z,s,a)=z^l \\Phi(z,s,a+l)+\\sum_{k=0}^{l-1}\\frac{z^k}{(a+k)^s}\n\\end{equation}\nFinally, we use the relationship between the Lerch transcendent and the polylogarithmic function \\cite{handbook, wikipoly}:\n\\begin{equation}\n\\mathrm{Li}_s(z)=z\\Phi(z,s,1)\n\\end{equation}\nwhere the polylogarithmic function is defined by:\n\\begin{equation}\n\\mathrm{Li}_s(z)=\\sum_{k=1}^{\\infty}\\frac{z^k}{k^s}\n\\end{equation}\n\n\\subsection{Asymptotic behavior and polylogarithmic function}\n\nWe start performing the change of variables $t=\\log\\frac{k}{c}$ in the integral in Eq.(\\ref{eqn:int_fond2}):\n\\begin{equation}\nI^1_c(\\nu)=\\int_{1}^\\infty\\frac{[\\beta_0\\log(\\frac{k}{c})]^{-\\gamma'}}{k+\\nu}dk=\nc\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^{\\infty}\\frac{t^{-\\gamma'}}{c+\\nu e^{-t}}dt\n\\end{equation}\nSetting $c=e^{-\\epsilon}$ in the limit $\\epsilon\\rightarrow 0$ we get the upper bound:\n\\begin{equation}\nI^1_c(\\nu)=\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^{\\infty}\\frac{t^{-\\gamma'}}{1+\\frac{\\nu}{c} e^{-t}}dt\\leq\n\\beta_0^{-\\gamma'}\\int_{\\epsilon}^{\\infty}\\frac{t^{-\\gamma'}}{1+\\frac{\\nu}{e^{-\\epsilon}} e^{-t}}dt = I^1_{1-\\epsilon}(\\nu)\n\\end{equation}\nbut the upper bound is in fact asymptotic since varying $c$  is equivalent to a change of scheme.\nTherefore, we take the limit $\\epsilon\\rightarrow 0$ in order to express $I^1_1$ in terms of the integral representation of the Lerch transcendent in Eq.(\\ref{eqn:lerch_int_repr}). We get:\n\\begin{equation}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\Phi(-\\nu,-\\gamma'+1,0)\n\\end{equation}\nWe now exploit the relation in Eq.(\\ref{eqn:lerch_recursive}) with\n$n=1$, $a=0$, $z=-\\nu$ and $s=-\\gamma'+1$: \n\\begin{equation}\n\\Phi(-\\nu,-\\gamma'+1,0)=z \\Phi(-\\nu,-\\gamma'+1,1)\n\\end{equation}\nFinally, we find the relation with the polylogarithmic function:\n\\begin{equation}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\mathrm{Li}_{-\\gamma'+1}(-\\nu)\n\\end{equation}\nNow we use the following asymptotic expansion of $\\mathrm{Li}_s$ \\cite{wikipoly}:\n\\begin{equation}\n\\mathrm{Li}_s(z)=\\sum_{j=0}^{\\infty}(-1)^j(1-2^{1-2j})(2\\pi)^{2j}\\frac{B_{2j}}{(2j)!}\\frac{[\\log(-z)]^{s-2j}}{\\Gamma(s+1-2k)}\n\\end{equation}\nto find an asymptotic expansion for $I^1_1(\\nu)$:\n\\begin{equation}\\label{eqn:esp_polylog}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\sum_{j=0}^{\\infty}(-1)^j(1-2^{1-2j})(2\\pi)^{2j}\\frac{B_{2j}}{(2j)!}\n\\frac{[\\log\\nu]^{-\\gamma'+1-2j}}{\\Gamma(-\\gamma'+2-2j)}\n\\end{equation}\nWe get the leading behavior of $I_1^1(\\nu)$ from the $j=0$ term in Eq.(\\ref{eqn:esp_polylog}):\n\\begin{equation}\\label{eqn:leading_polylog}\nI_1^1(\\nu)\\sim -\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\frac{[\\log \\nu]^{-\\gamma'+1}}{\\Gamma(-\\gamma'+2)}=\n\\frac{[\\beta_0\\log \\nu]^{-\\gamma'+1}}{\\beta_0(\\gamma'-1)}\n\\end{equation}\nKeeping also the $j=1$ term we obtain: \n\\begin{equation}\\label{eqn:subleading_polylog}\nI_1^1(\\nu)\\sim \\frac{[\\beta_0\\log \\nu]^{-\\gamma'+1}}{\\beta_0(\\gamma'-1)}+\n\\gamma'\\beta_0^{-\\gamma'}\\frac{\\pi^2}{6}[\\log \\nu]^{-\\gamma'-1} \n\\end{equation}\nin perfect agreement with Eq.(\\ref{eqn:correzione_serie_as}) since $\\zeta(2)=\\frac{\\pi^2}{6}$.\nReinserting the momentum $p$ in the definition of $\\nu$ the asymptotic result is:\n\\begin{equation}\nI^1_c\\left(\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\\right)\\sim \\frac{[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L+1}}{\\gamma_0 L-\\beta_0}+\n\\gamma_0 L\\frac{\\pi^2}{6}[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L-1}  \n\\end{equation}\nUsing the same technique we find the next-to-leading logarithmic behavior of $I^2_c$. \nIndeed, also in this case we obtain an upper bound putting $c=e^{-\\epsilon}$ and taking the limit $\\epsilon\\rightarrow 0$:\n\\begin{align}\nI^{2}_c(\\nu)&=c\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^\\infty\\left(\\frac{1}{t}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\right)^{\\gamma'}\\frac{dt}{c+\\nu e^{-t}}\\\\\n\\leq & \\beta_0^{-\\gamma'}\\int_{\\epsilon}^\\infty\\left(\\frac{1}{t}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\right)^{\\gamma'}\\frac{dt}{1+\\frac{\\nu}{e^{-\\epsilon}} e^{-t}}=I^{2}_{1-\\epsilon}(\\nu)\n\\end{align}\nbut the upper bound is in fact asymptotic since varying $c$  is equivalent to a change of scheme.\nWe now expand $I^{2}_{1-\\epsilon}(\\nu)$:\n\\begin{equation}\nI^{2}_{1-\\epsilon}(\\nu)\\sim \\beta_0^{-\\gamma'}\\int_{\\epsilon}^{\\infty}\\frac{1}{t^{\\gamma'}}\\left(1-\\frac{\\beta_1\\gamma'}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\frac{dt}{1+\\frac{\\nu}{e^{-\\epsilon}}e^{-t} }\n\\end{equation}\nThe first term is equal to $I^1_{1-\\epsilon}(\\nu)$, while the second one is the new contribution.\nThis new term can be linked again to the polylogarithmic function using the relation:\n\\begin{equation}\nt^{-\\gamma'-1}\\log t=-\\frac{\\partial}{\\partial \\alpha} t^{-\\alpha}\\biggl|_{\\alpha=\\gamma'+1}\n\\end{equation} \nWe find :\n\\begin{equation}\nI_{1-\\epsilon}^{2}(\\nu,-\\gamma')\\sim I_{1-\\epsilon}^1(\\nu,-\\gamma')+\\frac{\\beta_1\\gamma'}{\\beta_0^2}\\frac{\\partial}{\\partial\\alpha}I^1_{1-\\epsilon}(\\nu,-\\alpha)\\biggl|_{\\alpha=\\gamma'+1}\n\\end{equation}\nWe take the limit $\\epsilon\\rightarrow 0$ and we perform the derivative in the asymptotic expression of $I_1^1(\\nu,-\\alpha)$ in Eq.(\\ref{eqn:esp_polylog}). Keeping only the leading contribution we obtain:\n\\begin{align}\n\\frac{\\partial}{\\partial\\alpha}I_1^1(\\nu,-\\alpha)\\biggl|_{\\alpha=\\gamma'+1}&=\n\\beta_0^{-\\gamma'}\\frac{\\Gamma(-\\gamma')}{\\Gamma(-\\gamma'+1)}(\\log \\nu)^{-\\gamma'}\\log\\log \\nu=\\\\\n&=-\\frac{\\beta_0^{-\\gamma'}}{\\gamma'}(\\log \\nu)^{-\\gamma'}\\log\\log \\nu\n\\end{align}\nThus the asymptotic behavior to the next-to-leading logarithmic order is:\n\\begin{align}\nI^{2}_c(\\nu)&\\sim  \\frac{[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L+1}}\n{\\gamma_0 L-\\beta_0}\n-\\frac{\\beta_1}{\\beta_0^2}(\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{W}}^2})^{-\\frac{\\gamma_0}{\\beta_0}L}\\log\\log\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}{\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{align}\nthat agrees perfectly with the $RG$ estimate Eq.(\\ref{eqn:esp_ntl}).\n\n\\thispagestyle{empty}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe structure of nuclei involving $\\alpha$ clusters continues to be a subject of very active studies (see \\cite{Freer:2017gip} \nfor a recent review, \\cite{brink2008history} for a historical perspective, \\cite{Arriola:2014lxa} for a discussion of clustering mass formulas and form factors as \nmanifestations of the geometric structure,\nand \\cite{brink1965alpha,freer2007clustered,ikeda2010clusters,Okolowicz:2012kv,beck2012clusters,Beck:2014fja} for additional \ninformation), exploring the ideas dating back to Gamow's original clusterization proposal~\\cite{gamow1931constitution} \nwith modern theoretical~\\cite{Funaki:2006gt,Chernykh:2007zz,KanadaEn'yo:2006ze} and\ncomputational~\\cite{PhysRevLett.109.052501,Barrett:2013nh,Epelbaum:2012qn,Pieper:2002ne,Wiringa:2013ala,Lonardoni:2017egu} methods, as well as with \nanticipated new experimental prospects~\\cite{Yamaguchi:2012sz,Zarubin,Fritsch:2017rxc,Guo:2017tco}. \n\nA few years ago a possible approach of investigating $\\alpha$ clustering in light nuclei via studies of ultra-relativistic nuclear \ncollisions was proposed in Ref.~\\cite{Broniowski:2013dia} and explored in further \ndetail for the ${}^{12}$C nucleus in~Ref.~\\cite{Bozek:2014cva}. Quite remarkably, the experimental application of the method could reveal information on \nthe {\\em ground state} of a light clustered nucleus, i.e. on the lowest possible energy state, via the highest-energy nuclear collisions, such as those carried out at \nultra-relativistic accelerators: the CERN Super Proton Synchrotron (SPS), BNL Relativistic Heavy-Ion Collider (RHIC), or the CERN Large Hadron Collider (LHC).\nIn the first part of this paper we extend the results of Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva} obtained for  ${}^{12}$C  to other light nuclei, \nnamely ${}^{7}$Be, ${}^{9}$Be, and ${}^{16}$O, which are believed to have a prominent cluster structure in their ground states, see Fig.~\\ref{fig:structure}. \n\n\n\\begin{figure}[tb]\n\\centering\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.40 \\textwidth]{structures.pdf}\n\\vspace{-4mm}\n\\caption{Schematic view of the cluster structure of light nuclei. The dark blobs indicate $\\alpha$ clusters (in the case of ${}^7$Be, also the ${}^3$He cluster). The \nadditional dot in ${}^9$Be indicates the extra neutron.\n\\label{fig:structure}}\n\\end{figure}\n\nWe recall the basic concepts of Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva}: Spatial correlations in the ground state of a light nucleus, such as the \npresence of clusters, lead to an intrinsic deformation. When colliding with a heavy nucleus (${}^{208}$Pb,  ${}^{197}$Au) at a very high energy, \nwhere due to the Lorentz contraction the collision time is much shorter than any characteristic nuclear time scale, \na reduction of the wave function occurs and a correlated spatial distribution of participant nucleons is formed.\nThis, via individual nucleon-nucleon collisions between the colliding nuclei in the applied Glauber \npicture~\\cite{Glauber:1959aa,Czyz:1968zop,Bialas:1976ed}, leads to an initial distribution of entropy in the \ntransverse plane, whose {\\em eccentricity} reflects the deformation of the ground-state due to correlations. \nIn short, the deformed intrinsic shape of the light nucleus, when hitting a ``wall'' of a heavy target, yields a \ndeformed fireball in the transverse plane.\n\nAs an example, \nif the intrinsic state of the ${}^{12}$C nucleus is a triangle made of three $\\alpha$ particles, then the shape of the initial \nfireball in the transverse plane reflects this triangular geometry. Next, the {\\em shape-flow transmutation} mechanism (cf. Fig.~\\ref{fig:concept}), \na key geometric concept in the phenomenology of ultra-relativistic heavy-ion collisions~\\cite{Ollitrault:1992bk}, generates a large collective \ntriangular flow through the dynamics in the later stages of the evolution, modeled via hydrodynamics (for recent reviews see~\\cite{Heinz:2013th,Gale:2013da,Jeon:2016uym}), \nor transport~\\cite{Lin:2004en}.  As a result, one observes the azimuthal asymmetry of the transverse momentum distributions\nof produced hadrons. Similarly, the dumbbell \nintrinsic shape of the ground states of the ${}^{7,9}$Be nuclei, which occurs when  these nuclei are clustered, leads to a large elliptic flow. \n\nWe remark that the methodology applied in Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva} and in the present work, \nwas used successfully to describe harmonic flow in d+Au collisions~\\cite{Bozek:2011if} (small dumbbells)  \nand in $^3$He+Au collisions~\\cite{Nagle:2013lja,Bozek:2014cya} (small triangles), \nand the predictions later experimentally confirmed in~\\cite{Adare:2013piz,Adare:2015ctn}.\n\n\n\\begin{figure}[tb]\n\\centering\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.45 \\textwidth]{concept.pdf}\n\\vspace{-1mm}\n\\caption{Cartoon of ultra-relativistic $^{7,9}$Be+$^{208}$Pb collisions.\nThe clustered beryllium creates a fireball whose initial transverse shape reflects the deformed intrinsic shape of the projectile (left panel). \nSubsequent collective evolution leads to faster expansion along the direction perpendicular to the symmetry axis of the beryllium, and slower expansion \nalong this axis, as indicated by the arrows (right panel). The effect generates specific signatures in the harmonic flow patterns in spectra of the produced hadrons in the final state.\n\\label{fig:concept}}\n\\end{figure}\n\nAs the positions of the nucleons in the colliding nuclei fluctuate, being distributed according to their wave functions, the initial eccentricity, \nand in consequence the harmonic flow, always receives an additional contribution from \nthese random fluctuations~\\cite{Miller:2003kd,Alver:2006wh,Voloshin:2006gz,Broniowski:2007ft,Hama:2007dq,Luzum:2011mm,Bhalerao:2014xra} (the shape fluctuations are indicated \nwith a warped surface of the fireball in Fig.~\\ref{fig:concept}). For that reason the applied measures of the harmonic flow should be able to \ndiscriminate between these two components.\n\nTo a good  approximation, the measured harmonic flow coefficients $v_n$ in the spectra of produced hadrons are linear in the corresponding initial eccentricities \n$\\epsilon_n$ (see, e.g., ~\\cite{Gardim:2011xv,Niemi:2012aj,Bzdak:2013rya}).  This allows for a construction of flow measures given \nin Sec.~\\ref{sec:signatures}, which are independent the of details of the dynamics of the later stages of the collision, and thus carry \ninformation pertaining to the initial eccentricities. We describe such measures in Sect.~\\ref{sec:signatures}.\nWe note that another measure, involving the ratio of the triangular and elliptic flow coefficients, \nhas been recently proposed in Ref.~\\cite{Zhang:2017xda} \nfor the case of ${}^{12}$C, and tested within the AMPT~\\cite{Lin:2004en} transport model.\n\nTo have realistic nuclear distributions with clusters, yet simple enough to be implemented in a Monte Carlo simulation,\nwe apply a procedure explained in Sec.~\\ref{sec:making}, where positions of nucleons are determined within clusters of a given size, whereas \nthe clusters themselves are arranged in an appropriate shape (for instance, triangular for ${}^{12}$C.  \nThe parameters, determining the separation distance between the clusters and their sizes,  \nare fixed in such a way that the resulting one-body nucleon densities compare well to the state-of-the-art Variational Monte Carlo \n(VMC)~\\cite{Wiringa:2013ala,Lonardoni:2017egu} simulations. \nThe simulations for clustered nuclei are compared to the base-line case, where no clustering is present.\n\nOur basic findings, presented in Sec.~\\ref{sec:signatures}, are that clusterization in light nuclei leads to sizable effects in the harmonic flow \npattern in collisions with heavy nuclei. The effect is most manifest for the highest-multiplicity collisions, where additional fluctuations \nfrom the random distribution of nucleons are reduced. For the dumbbell shaped ${}^{7,9}$Be, the measures of \nthe elliptic flow are affected, whereas for the triangular ${}^{12}$C and tetrahedral ${}^{16}$O there are significant imprints of clusterization in \nthe triangular flow. These effects, when observed experimentally, could be promptly \nused to assess the degree of clusterization in light nuclei.\n\nIn the second part of this paper we examine a novel possibility of observing the intrinsic deformation  resulting from clusterization \n of light nuclei with spin, such as ${}^{7,9}$Be, when these are collided with ultra-relativistic {\\em protons}. \nThis interesting but exploratory proposal would require a magnetically \n{\\em polarized} ${}^{7,9}$Be nuclei, which in the ground state have $J^P=3\/2^-$. \n\nIn this case the geometric mechanism is as follows:\nWhen the dumbbell shaped nucleus in $m=1\/2$ ground state is polarized along the proton beam direction, there is a much higher chance \nfor the proton to collide with more nucleons (as it can pass through both clusters) than in the case where it is polarized perpendicular to the beam axis \n(where it would pass through a single cluster only). Thus more participants are formed in the former case. \nThe effect is opposite for the $m=3\/2$ state, as explained in Sect.~\\ref{sec:pA}.\n\nOne could thus investigate the distribution of participant nucleons, $N_w$, for various magnetic numbers $m$ and geometric orientations.\nWe find from our simulations a factor of two effects for $N_w = 4 $ and an order of magnitude effect for $N_w \\ge 6$, when \ncomparing the cases of \\mbox{$m=3\/2$} and \\mbox{$m=1\/2$}, or changing of the direction of the beam relative \nto the polarization axis. We discuss the mechanism and the relevant issues in Sec.~\\ref{sec:pA}.\n\n\n\n\\section{Nucleon distributions in clustered light nuclei}\n\\label{sec:making}\n\n\nTo model the collision process in the applied Glauber framework~\\cite{Glauber:1959aa,Czyz:1968zop,Bialas:1976ed}, we \nfirst need the distributions of centers of nucleons in the considered nuclei. We have adopted a simple and practical procedure \nwhere these distributions are generated randomly in clusters placed at preassigned positions in such a way that \nthe one-body density reproduces the shapes obtained from state-of-the-art \nVariational Monte Carlo (VMC)~\\cite{Buendia:2004yt,Wiringa:2013ala,Lonardoni:2017egu} studies.\n\nExplicitly, our steps are as follows: We set the positions of clusters according to the geometry of Fig.~\\ref{fig:structure}, separating their centers\nfrom each other with the distance $l$. The distribution of the nucleons in each cluster is randomly generated according to the Gaussian function\n\\begin{eqnarray}\nf_i(\\vec{r})=A \\exp \\left (- \\frac{3}{2} \\, \\frac{(\\vec{r}-\\vec{c_i})^2}{r_c^2} \\right ),   \n\\end{eqnarray}\nwhere $\\vec{r}$ is the 3D coordinate of the nucleon, $\\vec{c_i}$ is the position of the center of the cluster $i$, and $r_c$ is the rms radius \nof the cluster, which equals $r_\\alpha$ or  $r_{{}^3{\\rm He}}$ depending on the cluster type. \nWe generate the positions of the nucleons in sequence, alternating the number of the cluster: 1, 2,\\dots, 1, 2,\\dots, until all the nucleons are \nplaced.\n\n\n\n\\begin{table}[tb]\n\\caption{\\label{tab:param} Parameters used in the GLISSANDO simulations to obtain the nuclear distribution: $l$ is the distance between the centers \nof clusters, arranged according to the geometry shown in Fig.~\\ref{fig:structure}, $r_\\alpha$ is the size of the $\\alpha$ cluster, $r_{{}^3{\\rm He}}$ \nis the size of the  ${}^3{\\rm He}$ cluster in \\ensuremath{{}^{7}{\\rm Be}}, and $r_n$ determines the distribution of the extra neutron in \\ensuremath{{}^{9}{\\rm Be}}.}\n\\vspace{3mm}\n\\begin{tabular}{|c|cccc|}\n\\hline\n Nucleus & $l$ [fm] & $r_\\alpha$ [fm] &  $r_{{}^3{\\rm He}}$ [fm]  & $r_n$ [fm]\n\\\\\n\\hline \n\\ensuremath{{}^{7}{\\rm Be}}   & 3.2 & 1.2 & 1.4 & - \\\\\n\\ensuremath{{}^{9}{\\rm Be}}   & 3.6 & 1.1 & -  & 1.9  \\\\\n\\ensuremath{{}^{12}{\\rm C}}     & 2.8 & 1.1 & - & - \\\\\n\\ensuremath{{}^{16}{\\rm O}}    & 3.2 & 1.1 & - & - \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[angle=0,width=0.48\\textwidth]{density_points.pdf}\n\\caption{(Color online) Nuclear density profiles of the considered light nuclei. The points correspond to our Monte Carlo generation of the nuclear distributions \nin GLISSANDO, with parameters listed of Table~\\ref{tab:param} adjusted in such a way that the results from  \nVariational Monte Carlo (VMC)~\\cite{Wiringa:2013ala,Lonardoni:2017egu}(dashed lines)  are properly reproduced. \nWe use the normalization $4\\pi \\int_0^\\infty r^2dr \\, \\rho(r)=1$. \\label{fig:density}}\n\\end{figure}\n\n\nFor ${}^{9}$Be, we add the extra neutron on top of the two $\\alpha$ clusters according to a distribution with a hole in the middle, \n\\begin{eqnarray}\nf_n(\\vec{r})=A' r^2 \\,\\exp \\left (- \\frac{3}{2} \\frac{r^2}{r_n^2} \\right ).\n\\end{eqnarray}\n\nThe short-distance\nnucleon-nucleon repulsion is incorporated by precluding the centers of each pair of nucleons to be closer than the expulsion distance of \n0.9~fm, which is a customary prescription~\\cite{Broniowski:2010jd} in preparing nuclei for the Glauber model in ultra-relativistic nuclear collisions. \nAt the end of the procedure the distributions are shifted such that their center of mass is \nplaced at the origin of the coordinate frame.\nAs a result, we get the Monte Carlo distributions with the built-in cluster correlations.\n\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_7Be.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_7Be.pdf}\n\\caption{(Color online) Scaled standard deviations of rank-$n$ flow coefficients (panel (a)) and ratios of the four-particle to two-particle \ncumulants (panel (b)), plotted as functions of the total number of the wounded nucleons. Clustered nuclei (thick lines) are compared with \nthe case where the nucleons are distributed uniformly with the same one-body radial distributions (thin lines). \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions.\nThe vertical lines indicate the multiplicity percentiles (centralities) corresponding to the indicated values of $N_w$. \n}\n\\label{fig:7Be}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_9Be.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_9Be.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{9}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:9Be}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_12C.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_12C.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{12}{\\rm C}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:12C}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_16O.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_16O.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{16}{\\rm O}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:16O}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_7Be_rap.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_7Be_rap.pdf}\n\\caption{(Color online) Scaled standard deviations of rank-$n$ flow coefficients (panel (a)) and ratios of the four-particle to two-particle \ncumulants (panel (b)) simulated for the backward, central, and forward rapidity regions, plotted as functions of the total number of the \nwounded nucleons. \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions, clustered nuclei case. Thick lines correspond to the clustered case, and thin lines to the \nuniform distributions.}\n\\label{fig:7Berap}\n\\end{figure*}\n\n\nTo fix the parameters listed in Table~\\ref{tab:param}, we use specific reference radial distribution obtained from \nVMC simulations, which use the Argonne~v18 two-nucleon and Urbana~X three-nucleon potentials, as provided in \n{\\small \\url{http:\/\/www.phy.anl.gov\/theory\/research\/density}}~\\cite{Wiringa:2013ala,Lonardoni:2017egu}. Our distribution parameters \nare then optimized such that the one particle densities $\\rho(r)$ from VMC  \nare properly reproduced.  Thus the radial density of the centers on nucleons serves as a constraint for building our clustered \ndistributions. Figure~\\ref{fig:density} shows the quality of our fit to the one-body densities, which is satisfactory in the \ncontext of modeling ultra-relativistic nuclear collisions.  We note from  Fig.~\\ref{fig:density} that\nthe distributions (except for $^{7}$Be nucleus) develop a dip in the center.\nThe parameters used in our simulations are collected in Table~\\ref{tab:param}.\n\nAs we are interested in specific effects of clusterization, as a ``null result'' we use the {\\em uniform} distributions, i.e., \nwith no  clusters.  We prepare such distributions with exactly the same radial density as the clustered ones. \nThis is achieved easily with a trick, where we randomly re-generate the spherical angles of the nucleons from \nthe clustered distributions, while leaving the radial coordinates intact.\n\n\n\\section{Harmonic flow in relativistic light-heavy collisions}\n\\label{sec:signatures}\n\nAs already mentioned in the Introduction, we use the so-called Glauber approach to model the early stage of the collision. The Glauber model~\\cite{Glauber:1959aa}\nformulated almost sixty years ago to model the elastic scattering amplitude in high-energy collisions, was later \nextended to inelastic collisions~\\cite{Czyz:1968zop}, and subsequently led to the widely used wounded-nucleon model~\\cite{Bialas:1976ed}.\nThe model assumes that \nthe trajectories of nucleons are straight lines and the individual nucleons at impact parameter $b$ interact with a probability $P(b)$, where \n$\\int d^2b \\, P(b) =\\sigma_{\\rm inel}$ is the total inelastic nucleon-nucleon cross section. We use a Gaussian form of $P(b)$, \nwhich for the studied heavy-ion observables is of sufficient accuracy~\\cite{Rybczynski:2011wv}.\n\nGenerally, in the Glauber framework, at the initial stage of the collision the interacting nucleons deposit \nentropy (or energy) in the transverse plane. Such deposition occurs from wounded nucleons, but also from binary collisions.\nSuch an admixture of binary collisions is necessary to obtain proper multiplicity distributions~\\cite{Kharzeev:2000ph,Back:2001xy}.\nIn this model the transverse distribution of entropy takes the form \n\\begin{eqnarray}\n\\rho(x,y)&=&\\frac{1-\\alpha}{2}\\rho_W(x,y) + \\alpha \\rho_{\\rm bin}(x,y), \\label{eq:rho0}\n\\end{eqnarray}\nwhere $\\rho_W(x,y)$ is the distribution of the wounded nucleons, \n$\\rho_{\\rm bin}(x,y)$ is the distribution of the binary collisions, and  $\\alpha$ is the parameter controlling the relative weight of the wounded to binary sources. \nIn our simulations we use $\\alpha=0.12$ (the value fitting the multiplicity distributions at the SPS collision energies). \nThe sources forming the distributions are smeared with a Gaussian of a width of 0.4~fm.\n\nIn the following we show the numerical results of our GLISSANDO~\\cite{Broniowski:2007nz,Rybczynski:2013yba} \nsimulations of collisions of the described above nuclei composed \nof $\\alpha$-clusters with ${}^{208}$Pb nucleus at $\\sqrt{s_{NN}}=17$~GeV, where the corresponding inelastic nucleon-nucleon \ncross section is $\\sigma_{\\rm inel}=32$~mb. Such collision energies are available at SPS and the considered reactions are possible to study in the on-going \nNA61\/SHINE experiment with ${}^{208}$Pb or proton beams. A variety of targets and secondary beams are available in this experiment~\\cite{Abgrall:2014xwa}.\nTherefore the present study may be thought of as a case study for possible NA61\/SHINE investigations.\n\nTo analyze the effects of clusterization in the considered light nuclei on the harmonic flow coefficients\nin the reactions with  ${}^{208}$Pb nuclei, one needs to use appropriate flow measures. The eccentricity coefficients, $\\epsilon_n$, are designed\nas measures of the harmonic deformation in the initial state. They are defined for each collision event as\n\\begin{eqnarray} \n\\epsilon_n e^{i n \\Phi_n} = - \\frac{\\int \\rho(x,y) e^{i n\\phi}  (x^2+y^2)^{n\/2} dx dy}{\\int \\rho(x,y) (x^2+y^2)^{n\/2}  dx dy}, \\label{eq:eps}\n\\end{eqnarray}\nfor $n=2,3,\\dots$, with $\\phi=\\arctan(y\/x)$ and $\\Phi_n$ denoting the angle of the principal axes in the transverse plane $(x,y)$.\n\nThe subsequent collective evolution with hydrodynamic~\\cite{Heinz:2013th,Gale:2013da,Jeon:2016uym} \nor transport~\\cite{Lin:2004en} has a shape-flow transmutation feature: The deformation of shape in the initial stage leads to harmonic \nflow of the hadrons produced in the late stage. The effect is manifest in an approximate proportionality of the flow \ncoefficients $v_n$ to the eccentricities  $\\epsilon_n$, which holds for $n=2$ and $3$ (for higher rank non-linear coupling effects may be present):\n\\begin{equation}\nv_n = \\kappa_n \\epsilon_n \\ , \n\\label{eq:linear}\n\\end{equation} \nThe cumulant coefficients follow an analogous relation: \n\\begin{eqnarray}\nv_n\\{m\\}=\\kappa_n \\epsilon_n\\{m\\}.\n\\end{eqnarray}\nThe proportionality coefficients $\\kappa_n$ depend on various features of the colliding system (centrality, collision energy), \nbut are to a good approximation independent of the eccentricity itself, hence the above relations are linear.\nTo get rid of the influence of the (generally) unknown $\\kappa_n$ coefficients on the results, \none may consider the ratios of cumulants of different order $m$ for a given rank-$n$ flow coefficient $v_n$, e.g., \n\\begin{eqnarray}\n\\frac{v_n\\{m\\}}{v_n\\{2\\}}=\n\\frac{\\epsilon_n\\{m\\}}{\\epsilon_n\\{2\\}} \\ . \\label{eq:ratios} \n\\end{eqnarray}\nTherefore the ratios of the flow cumulants can be directly compared to the corresponding ratios of the eccentricity cumulants.\nIn our work we also use the scaled event-by event standard deviation, ${\\sigma(\\epsilon_n)}\/{\\langle \\epsilon_n \\rangle}$, where\n\\begin{eqnarray}\n \\frac{\\sigma(\\epsilon_n)}{\\langle \\epsilon_n \\rangle} \\simeq \\frac{\\sigma(v_n)}{\\langle v_n \\rangle}. \\label{eq:ev}\n\\end{eqnarray}\n\nIn order to find the specific effects of clusterization, we always compare the obtained results to those corresponding to the  ``uniform'' case, where \nthe nucleons are distributed without clusterization (see Sect.~\\ref{sec:making}).\n\nIn Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be} we show the event-by-event scaled standard deviations of the elliptic ($n=2$), \ntriangular ($n=3$), and quadrangular ($n=4$) flow coefficients, as well as the ratios of the four-particle to \ntwo-particle cumulants, plotted as functions of the total number of wounded nucleons. Since clusters in ${}^{7,9}$Be nuclei form a dumbbell \nshape, the influence of clusterization is, as expected, visible in the $n=2$ (elliptic) coefficients. \nThe behavior seen in panels (a) is easy to explain qualitatively: At large numbers of wounded nucleons the beryllium is oriented in such a way that it \nhits the wall of ${}^{208}$Pb side-wise, as drawn in Fig.~\\ref{fig:concept}. Then the eccentricity of the created fireball, which is an imprint of the \nintrinsic shape of beryllium, is largest. Hence the scaled variance decreases (note division with $\\langle \\epsilon_n \\rangle$ in Eq.~(\\ref{eq:ev})) with $N_w$.\nThe feature is clearly seen from Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be}. Of course, this is not the case for the uniform distributions, where at large \n$N_w$ the scaled standard deviations for all $n$ acquire similar values. A detailed quantitative \nunderstanding of the dependence on $N_w$ requires simulations, as one needs to assess the influence of the random fluctuations on eccentricities, or account for effects \nwhen the beryllium hits the edge of ${}^{208}$Pb. The size of the effect in panels (a) starts to be significant for the 10\\% of the highest-multiplicity events.\n\nThe results for the $v_n\\{4\\}\/v_n\\{2\\}$ (panels (b) of Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be}) are complementary.\nWe note that for high multiplicity collisions\nthe ratio is significantly larger for the clustered case compared to the uniform distributions. This is because the \ntwo-particle cumulants are more sensitive to the random fluctuations than the four-particle cumulants.\n\nFor the case of ${}^{12}$C+${}^{208}$Pb and ${}^{16}$O+${}^{208}$Pb collisions, the significant influence of clusters \nas compared to ``uniform'' case is visible for the rank-3 (triangular) coefficients, see Figs.~\\ref{fig:12C} and \\ref{fig:16O}. \nThis is mainly caused by the triangular and tetrahedral arrangements of clusters in ${}^{12}$C and ${}^{16}$O, respectively.\nThe qualitative understanding is as for the beryllium case, with the replacement of $n=2$ with $n=3$. The case of ${}^{12}$C \nhas also been thoroughly discussed in Ref.~\\cite{Bozek:2014cva}.\n\nAll previously shown simulations were carried out at the mid-rapidity, $y\\sim 0$, region. To study the dependence on rapidity, \nwe apply a model with rapidity-dependent emission functions of the entropy sources. Such an approach is necessary,\nsince in most fixed-target experiments the detectors measure particles produced in rapidity regions which are away from the mid-rapidity\ndomain. \nTaking this into account, we apply the model described in Refs.~\\cite{Bialas:2004su,Bozek:2010bi}. There, the initial density of the fireball \nin the space-time rapidity $\\eta_\\parallel=\\frac{1}{2} \\log (t+z)(t-z)$ and the transverse coordinates ($x, y$) is described by the function:\n\\begin{eqnarray}\n\\rho(\\eta_\\parallel,x,y)&=&(1-\\alpha)[\\rho_A(x,y) f_+(\\eta_\\parallel)\n+ \\rho_B(x,y) f_-(\\eta_\\parallel)] \\nonumber \\\\\n&+& \\alpha\n\\rho_{\\rm bin}(x,y) \\left [ f_+(\\eta_\\parallel) + f_-(\\eta_\\parallel) \\right ].\n\\label{eq:em}\n\\end{eqnarray}\nwhich straightforwardly generalizes Eq.~(\\ref{eq:rho0}), assuming factorized profiles from a given source.\nHere $\\rho_{A,B}(x,y)$ denotes the transverse density of the wounded sources from the nuclei $A$ and $B$, \nwhich move in the forward and backward directions, respectively. The entropy emission functions  \n$f_{\\pm}(\\eta_\\parallel)$ are given explicitly in~\\cite{Bozek:2010bi}. They are peaked in the forward or backward \ndirections, respectively, reflecting the fact that a wounded nucleon emits preferentially in its own forward hemisphere.\n\nIn the Fig.~\\ref{fig:7Berap} we plot, as functions of $N_w$, the scaled standard deviations of the rank-2 and 3 flow coefficients and ratios \nof the four-particle to two-particle cumulants calculated in backward ($\\eta_\\parallel = -2.5 $), central ($\\eta_\\parallel = 0$), and forward ($\\eta_\\parallel = 2.5 $) \nrapidity regions (at the SPS collision energy of $\\sqrt{s_{NN}}=17$~GeV the rapidity of the beam is $\\sim 2.9$). \nWe focus on results here for \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions, as for the other light clustered nuclei the results\nare qualitatively similar. The centrality dependence \nof the scaled standard deviation of second-rank flow coefficients (panel (a)) is similar for all considered regions of phase-space, however its magnitude \ngrows when we move from the backward (${}^{208}$Pb) to the forward (beryllium) hemisphere. The effect has to do with a \na much larger number of wounded nucleons in the backward compared to the forward hemisphere in the applied model \nof the rapidity dependence. This makes the random fluctuations smaller in the backward compared to the forward hemisphere, giving the effect seen in \nFig.~\\ref{fig:7Berap}.\n\nWe note that the previously discussed difference of the behavior of eccentricities between the clustered and uniform cases holds also \nfor other regions in rapidity, which makes the effect possible to study also in the fixed-target experiments with detectors covering the \nforward rapidity region.\n\n\\section{Proton-polarized light nucleus scattering}\n\\label{sec:pA}\n\nIn this section we present a more exploratory study, as the investigation needs the magnetic field \nto polarize the beryllium nuclei along a chosen direction. Polarized nuclear targets or beams have not yet been\nused in ultra-relativistic collisions. Nevertheless, our novel effect, also geometric in its origin, is worth \npresenting as a possibility for future experiments.\n\nSince the ground states of ${}^{7,9}$Be nuclei have $J^P=3\/2^-$, they can be polarized. Then, due to their\ncluster nature, the intrinsic symmetry axis correlated to the polarization axis in a specific way described in detail below. \nOne can thus control (to a certain degree)  the orientation of the intrinsic dumbbell shape. This, in turn, can be probed in \nultra-relativistic collisions with protons, as more particles are produced when the proton goes along the dumbbell, compared to the case when \nit collides perpendicular to the symmetry axis. \n\nWe wish to consider the beryllium nuclei polarized in magnetic field, therefore the first task is to obtain \nstates of good quantum numbers in our model approach, where we prepare intrinsic states \nwith the method described in Sect.~\\ref{sec:making}. \nWe use the Peierls-Yoccoz projection (see, e.g., ~\\cite{ring}), which is a \nstandard tool in nuclear physics of (heavy) deformed nuclei. The basic formula to pass from an intrinsic wave function $\\Psi^{\\rm intr}_k(\\Omega)$,\nwhere $\\Omega$ is the spherical angle of the symmetry axis and $k$ is the intrinsic spin projection, to the state of good quantum numbers $|j,m\\rangle$ has the form\n\\begin{eqnarray}\n|j,m\\rangle = \\sum_k \\int d\\Omega D^{j}_{m,k}(\\Omega) |\\Psi_k^{\\rm intr}(\\Omega) \\rangle, \\label{eq:PY}\n\\end{eqnarray}\nwhere $ D^{j}_{m,k}(\\Omega)$ is the Wigner $D$ function.\n\n\nThe ${}^7$Be nucleus has the following cluster decomposition and angular momentum decomposition between the spin of the clusters and the orbital angular momentum of the clusters:\n\\begin{eqnarray}\n{}^7{\\rm Be} &=& {}^4{\\rm He} + {}^3{\\rm He}, \\label{eq:7Be} \\\\  \n\\frac{3}{2}^-  &=& 0^+ + \\tfrac{1}{2}^+ + 1^-, \\nonumber\n\\end{eqnarray}\nwhere $0^+$ is the $J^P$ of the $\\alpha$ particle,  $ \\tfrac{1}{2}^+$ of ${}^3{\\rm He}$, and $ 1^-$ is the orbital angular \nmomentum. Similarly, for ${}^9$Be \n\\begin{eqnarray}\n{}^9{\\rm Be} &=& {}^4{\\rm He} + {}^4{\\rm He} + n, \\label{eq:9Be} \\\\  \n\\frac{3}{2}^-  &=& 0^+ + 0^+ +  \\tfrac{1}{2}^+ + 1^-, \\nonumber\n\\end{eqnarray}\nwhere the neutron is assumed to be in an $S$ state, and the $J^P$ of the angular motion of the two $\\alpha$ clusters is $1^-$. The Clebsch-Gordan decomposition is \n\\begin{eqnarray}\n|\\tfrac{3}{2},m &=& \\tfrac{3}{2}\\rangle = |\\tfrac{1}{2},\\tfrac{1}{2}\\rangle \\otimes |1,1\\rangle, \\label{eq:CG} \\\\\n|\\tfrac{3}{2},m &=& \\tfrac{1}{2}\\rangle = \n\\sqrt{\\tfrac{2}{3}}|\\tfrac{1}{2},\\tfrac{1}{2}\\rangle \\otimes |1,0\\rangle + \\sqrt{\\tfrac{1}{3}} |\\tfrac{1}{2},-\\tfrac{1}{2}\\rangle \\otimes |1,1\\rangle. \\nonumber \n\\end{eqnarray}\nIn the intrinsic frame, where the clusters are at rest, the angular momentum comes from the spin of $ {}^3{\\rm He}$ or $n$ in the cases of  ${}^7$Be \nor ${}^9$Be, respectively, hence the available values of $k$ are $\\pm \\tfrac{1}{2}$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{wig2.pdf} \n\\caption{The distributions of the intrinsic symmetry axis of ${}^{7,9}$Be in the polar angle $\\theta$, Eq.~(\\ref{eq:tilt}), \nfollowing from the Peierls-Yoccoz projection method. \\label{fig:wig}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0,width=0.23 \\textwidth]{pl32_t.pdf} \\includegraphics[angle=0,width=0.23 \\textwidth]{pl12_t.pdf} \\\\\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.23 \\textwidth]{pl32_t_p.pdf} \\includegraphics[angle=0,width=0.23 \\textwidth]{pl12_t_p.pdf} \n\\vspace{-5mm}\n\\caption{Schematic representation of collisions of protons with polarized $^{7,9}$Be. The sphere presents the $\\alpha$ or $^3$He clusters \nand the clouds indicate the quantum washing out of the symmetry axis of the intrinsic states, in accordance to Eq.~(\\ref{eq:PY}). The tube represents \nthe proton beam with the area given by the total inelastic proton-proton cross section. Arrows show the direction of the magnetic field which corresponds \nto the quantization axis of spin. Details in the text. \\label{fig:geom}}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{7z.pdf} ~~ \\includegraphics[angle=0,width=0.4 \\textwidth]{7y.pdf}\n\\caption{Results of Monte Carlo simulations of p+$^7$Be collisions. We note that for $N_w \\ge 3 $  the probability of wounding \n$N_w$ nucleons is higher for $m=1\/2$ than for $m=3\/2$ in the case when $\\vec{B}$ is parallel to z axis (panel (a)). For situation when $\\vec{B}$ \nis perpendicular to z, we observed more wounded nucleons for $m=3\/2$ than for $m=1\/2$ (panel (b)).   \\label{fig:distr1}}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{9z.pdf} ~~ \\includegraphics[angle=0,width=0.4 \\textwidth]{9y.pdf} \n\\caption{The same as in Fig.~\\ref{fig:distr1} but for p+$^9$Be collisions. \\label{fig:distr2}}\n\\end{figure*}\n\n\nAccording to Eq.~(\\ref{eq:PY}), we have for both nuclei \n\\begin{eqnarray}\n|\\tfrac{3}{2},m\\rangle = \\sum_{k=\\pm \\tfrac{1}{2}} \\int d\\Omega D^{3\/2}_{m,k}(\\Omega) |\\Psi_k^{\\rm intr}(\\Omega) \\rangle. \\label{eq:PY32}\n\\end{eqnarray}\nUnder the assumptions $\\langle \\Psi_{\\rm intr}(\\Omega') | \\Psi_{\\rm intr}(\\Omega) \\rangle \\simeq \\delta(\\Omega-\\Omega')$, which becomes \nexact in the limit of many nucleons, but still holds to a sufficiently good accuracy for 7 or 9 nucleons, we find \n\\begin{eqnarray}\n\\left | \\langle \\theta, \\phi |\\tfrac{3}{2},m\\rangle \\right |^2 =  [D^{3\/2}_{m,1\/2}(\\theta,\\phi)]^2+[D^{3\/2}_{m,-1\/2}(\\theta,\\phi)]^2. \n\\end{eqnarray}\nExplicitly, \n\\begin{eqnarray}\n&& \\left | \\langle \\theta, \\phi |\\tfrac{3}{2},\\tfrac{3}{2} \\rangle \\right |^2 = |Y_{11}(\\theta,\\phi)|^2= \\frac{3}{8\\pi} \\sin^2 \\theta, \\label{eq:tilt} \\\\\n&& \\left | \\langle \\theta, \\phi |\\tfrac{3}{2},\\tfrac{1}{2} \\rangle \\right |^2 = \\tfrac{2}{3} |Y_{10}(\\theta,\\phi)|^2 + \\tfrac{1}{3} |Y_{11}(\\theta,\\phi)|^2 \\nonumber \\\\\n&& \\hspace{2cm} =  \\frac{1}{8\\pi} \\left ( 1+3\\cos^2 \\theta \\right ), \\nonumber\n\\end{eqnarray}\nin accordance to Eq.~(\\ref{eq:CG}). The distributions (\\ref{eq:tilt}), which depend on the polar angle $\\theta$ and \nnot on the azimuthal angle $\\phi$, are shown in Fig.~\\ref{fig:wig}. \n\n\nThe prescription for the Monte Carlo simulations that follows from the above derivation is that the symmetry axes of ${}^{7,9}$Be should be \nrandomly tilted in each collision event according to the distributions (\\ref{eq:tilt}). We note that the $m=1\/2$ state is approximately aligned \nalong the spin projection axis (the distribution peaks at $\\theta=0$ or $\\theta= \\pi$), \nwhereas the $m=3\/2$ state is distributed near the equatorial plane (with the maximum at $\\theta=\\pi\/2$). \n\nSuppose that the targets of ${}^{7,9}$Be are 100\\% polarized along the direction of the magnetic field $B$ and consider collisions with a proton beam \nparallel or perpendicular to $B$. Then the geometry of the collision is influenced by the distributions of the intrinsic symmetry axis, \nas pictorially displayed in Fig.~\\ref{fig:geom}.\n \nThe figure shows schematically the collisions of protons with a polarized $^{7,9}$Be target, with the spheres representing the $\\alpha$ or $^3$He \nclusters and the clouds indicating the quantum distribution of the symmetry axis of the intrinsic states, in accordance to Eq.~(\\ref{eq:PY}). In the two \nleft panels of the Fig.~\\ref{fig:geom}, corresponding to $m=3\/2$ states the clusters are distributed near the equatorial plane, whereas in the two right \npanels, corresponding to $m=1\/2$ states the distribution of the clusters is approximately align along the quantization axis given by the magnetic field \ndirection B. The tubes represent the proton beam, drawn in such a way that the area of the tube is given by the total inelastic proton-proton cross section.\nWe can distinguish several geometric cases in the top panels of Fig.~\\ref{fig:geom} the proton beam is parallel to the direction of $\\vec{B}$, we notice \nthat for the $m=1\/2$ case the chance of hitting two clusters, thus wounding more nucleons, is higher than for $m=3\/2$ case. The effect is opposite \nwhen the proton beam is perpendicular to $\\vec{B}$ as can be seen from the two bottom panels. \n\nThe above discussed simple geometric mechanism finds its realization in numerical Monte Carlo simulations. Distribution of the number of \nwounded nucleons (in a logarithmic scale) is shown in Fig.~\\ref{fig:distr1} and in Fig.~\\ref{fig:distr2}. We note from panels (a) that in the \ncase of $\\vec{B}$ parallel to z (beam direction) indeed the probability of wounding more nucleons $N_w\\ge 3$ is larger for $m=1\/2$ than \nfor $m=3\/2$. The effect for $N_w=5$ reaches about a factor of 5, and increases for higher $N_w$. Note however, that at higher $N_w$ the collisions \nbecome very rare, thus statistical errors would preclude measurements.\nIn the case when $\\vec{B}$ is perpendicular to z (panels (b)) the effect is opposite \nwith higher probability of wounding more nucleons for $m=3\/2$ than for $m=1\/2$.\n\n\n\\section{Summary and conclusions}\n\\label{sec:summa}\n\nWe have shown that clusterization in light nuclei leads to characteristic signatures which could be studied in ultra-relativistic nuclear collisions. \nThe presence of clusters leads to specific intrinsic geometric deformation, which in collisions with a heavy nucleus \ngenerates hallmark harmonic flow patterns, especially \nfor the  collisions of highest multiplicity of the produced particles. \nAs the phenomenology of flow and the corresponding data analysis methods are standard, we believe that the proposal \nis experimentally feasible, requiring collisions with appropriate beams and then using the well developed and tested data analysis \ntechniques. We note that in the NA61\/SHINE experiment the beryllium beams and targets, studied in this paper, \nhave already been used~\\cite{Abgrall:2014xwa}. \n\nWe have also explored an opportunity following from the fact that the ground states of  ${}^{7,9}$Be have a non-zero spin, which allows for their \npolarization in an external magnetic field. Then, clusterization leads to significant effects in the spectra of participant (or spectator) nucleons in ultra-relativistic \ncollisions with the protons. We have found a factor of two effects for $N_w = 4 $ and an order of magnitude effect for $N_w \\ge 6$, when changing the orientation of the direction of the beam \nrelative to the polarization axis, or when comparing the spin states \\mbox{$m=3\/2$} and \\mbox{$m=1\/2$}. As the polarized nuclei have not, \nup to now, been used in ultra-relativistic nuclear collisions, our proposal is to be considered in future experimental proposals.\n\nFinally, we note that the effects of $\\alpha$ clusterization for heavier nuclei are small in the sense that the resulting intrinsic eccentricities are much \nsmaller than in the light systems considered in this paper. Therefore the investigations with the ${}^{7,9}$Be, ${}^{12}$C, and ${}^{16}$O nuclei \nwould be most promising. \n\n\\begin{acknowledgments}\nThe numerical simulations were carried out in laboratories created under the project\n``Development of research base of specialized laboratories of public universities in Swietokrzyskie region'',\nPOIG 02.2.00-26-023\/08, 19 May 2009.\nMR was supported by the Polish National Science Centre (NCN) grant 2016\/23\/B\/ST2\/00692 and WB by NCN grant 2015\/19\/B\/ST2\/00937.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSupersymmetry (SUSY) is one of the most promising extensions of the\nStandard Model (SM) \\cite{MSSM} since, among other things, it solves\nthe hierarchy problem, provides a natural candidate for dark matter\n\\textit{etc}. It also introduces many new sources of CP violation\nthat may be needed to explain baryon asymmetry of the universe.\nThese phases, if large $\\mathcal{O}(1)$, can cause problems with\nsatisfying experimental bounds on lepton, neutron and mercury\nEDMs~\\cite{susycp}. This can be overcome by pushing sfermion spectra\nabove a TeV scale or arranging internal cancelations\n\\cite{Ibrahim:2007fb}.\n\nMost unambiguous way to detect the presence of CP-violating phases\nwould be to study CP-odd observables measurable at future\naccelerators --- the LHC and the ILC. Such observables in the\nchargino sector are, for example, triple products of momenta of\ninitial electrons, charginos and their decay products\n\\cite{Kittel:2004kd}. However they require polarized initial\nelectron\/positron beams or measurement of chargino polarization.\n\nIn this talk we present another possibility of detecting\nCP-violating phases in the chargino sector. As it was recently\npointed out \\cite{Osland:2007xw,my}, in non-diagonal chargino pair\nproduction\n\\begin{equation}\ne^+e^-\\to\\tilde{\\chi}_1^\\pm\\tilde{\\chi}_2^\\mp \\label{produkcja}\n\\end{equation}\na CP-odd observable can be constructed beyond tree-level from\nproduction cross section without polarized $e^+e^-$ beams or\nmeasurement of chargino polarization. We show here the results of\nthe full one-loop calculation of this effect. In the\nreaction~(\\ref{produkcja}) the CP violation can be induced by the\ncomplex higgsino mass parameter $\\mu$ or complex trilinear coupling\nin top squark sector $A_t$. Since these asymmetries can reach a few\npercent, they can be detected in simple event-counting experiments\nat future colliders.\n\n\\section{CP-odd asymmetry at one loop}\n\nIn $e^+e^-$ collisions charginos are produced at tree-level via the\n$s$-channel $\\gamma,Z$ exchange and $t$-channel $\\tilde{\\nu}_e$\nexchange. As it was shown in \\cite{Choi:1998ei} no CP violation\neffects can be observed at the tree-level for the production\nprocesses of diagonal $\\tilde{\\chi}_i^+ \\tilde{\\chi}_i^-$ and\nnon-diagonal $\\tilde{\\chi}_i^+ \\tilde{\\chi}_j^-$ chargino pairs\nwithout the measurement of polarization of final chargino. However\nthe situation is different for non-diagonal production if we go\nbeyond tree-level approximation.\n\nRadiative corrections to the chargino pair production include the\nfollowing generic one-loop Feynman diagrams: the virtual vertex\ncorrections, the self-energy corrections to the $\\tilde{\\nu}$, $Z$\nand $\\gamma$ propagators, and the box diagrams contributions. We\nalso have to include corrections on external chargino legs.\n\nOne-loop corrected matrix element squared is given by\n\\begin{eqnarray}\n|\\mathcal{M}_{\\mathrm{loop}}|^2 = |\\mathcal{M}_{\\mathrm{tree}}|^2 +\n2 \\mathrm{Re}(\\mathcal{M}_{\\mathrm{tree}}^*\n\\mathcal{M}_{\\mathrm{loop}} )\\, .\n\\end{eqnarray}\nAccordingly, the one-loop CP asymmetry for the non-diagonal chargino\npair is defined as\n\\begin{eqnarray}\n&& A_{12}=\\frac{\\sigma^{12}_{\\rm loop}-\n  \\sigma^{21}_{\\rm loop}}{\\sigma^{12}_{\\rm tree}+\n  \\sigma^{21}_{\\rm tree}}\\, ,\n  \\label{CPasym}\n\\end{eqnarray}\nwhere $\\sigma^{12}$, $\\sigma^{21}$ denote cross sections for\nproduction of $\\tilde{\\chi}_1^+ \\tilde{\\chi}_2^-$ and\n$\\tilde{\\chi}_2^+ \\tilde{\\chi}_1^-$, respectively. Since the\nasymmetry vanishes at tree-level it has to be finite at one loop,\nhence no renormalization is needed.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.9]{Rolbiecki_fig1a.eps}\\hskip 1cm\n\\includegraphics[scale=0.35]{Rolbiecki_fig1b.eps}\n\\end{center}\n\\caption{Box diagram with selectron exchange and its contribution to\nthe asymmetry $A_{12}$ vs.\\ center of mass energy. The selectron\nmass is 403~GeV.\\label{fig:thrs_scan}}\n\\end{figure}\n\nThe CP asymmetry Eq.~(\\ref{CPasym}) arises due to the interference\nbetween complex couplings, which in our case are due to complex\nmixing matrices of charginos or stops, and non-trivial imaginary\npart from Feynman diagrams --- the absorptive part. Such\ncontributions appear when some of the intermediate state particles\nin loop diagrams go on-shell. This is illustrated in\nFig.~\\ref{fig:thrs_scan} where the contribution to $A_{12}$ from\ndouble selectron exchange appears at the threshold for selectron\npair production at $\\sqrt{s}=806$~GeV.\n\n\n\\section{Numerical results}\n\nFor the numerical results in this section we use two parameter sets\n(A) and (B) with gaugino\/higgsino mass parameters defined as follows\nat the low scale:\n\\begin{eqnarray}\n&\\mbox{A:}&\\quad |M_1| = 100\\mbox{ GeV},\\quad M_2 = 200\\mbox{ GeV},\n\\quad|\\mu| = 400\\mbox{ GeV},\\nonumber \\\\\n&\\mbox{B:}&\\quad |M_1| = 250\\mbox{ GeV}, \\quad M_2 = 200\\mbox{ GeV},\n\\quad |\\mu| = 300\\mbox{ GeV},\\nonumber\n\\end{eqnarray}\nand with $\\tan\\beta=10$. This gives the following chargino masses:\n\\begin{eqnarray*}\n&\\mbox{A:}&\\quad m_{\\tilde{\\chi}^-_1} = 186.7 \\mbox{ GeV},\\quad\nm_{\\tilde{\\chi}^-_2} = 421.8 \\mbox{ GeV},\\\\\n&\\mbox{B:}&\\quad m_{\\tilde{\\chi}^-_1} = 175.6 \\mbox{ GeV},\\quad\nm_{\\tilde{\\chi}^-_2} = 334.5 \\mbox{ GeV}.\n\\end{eqnarray*}\nFor the sfermion mass parameters in scenario (A) we assume\n\\begin{eqnarray*}\n&&m_{\\tilde{q}}\\equiv M_{\\tilde{Q}_{1,2}}=M_{\\tilde{U}_{1,2}}=M_{\\tilde{D}_{1,2}}=450\\mbox{ GeV},\\nonumber\\\\\n&&M_{\\tilde{Q}}\\equiv M_{\\tilde{Q}_{3}}=M_{\\tilde{U}_{3}}=M_{\\tilde{D}_{3}}=300\\mbox{ GeV},\\\\\n&&m_{\\tilde{l}}\\equiv\nM_{\\tilde{L}_{1,2,3}}=M_{\\tilde{E}_{1,2,3}}=150\\mbox{ GeV},\n\\end{eqnarray*}\nand for the sfermion trilinear coupling:\n$|A_{t}|=-A_{b}=-A_{\\tau}=A=400\\mbox{ GeV}$. Scenario (B) is for\ncomparison with Ref.~\\cite{Osland:2007xw} for which we take\n$$M_{S}= M_{\\tilde{Q}}= M_{\\tilde{U}} =M_{\\tilde{D}} =M_{\\tilde{L}}=\nM_{\\tilde{E}}=10\\mbox{ TeV}.$$\n\nIn our numerical analysis we consider the dependence of the\nasymmetry (\\ref{CPasym}) on the phase of the higgsino mass parameter\n$\\mu = |\\mu| e^{i \\Phi_\\mu}$ and soft trilinear top squark coupling\n$A_t = |A_t| e^{i \\Phi_t}$. In Fig.~\\ref{fig2} we show the CP\nasymmetry in scenario (A) as a function of the phase of $\\mu$ and\n$A_t$, left and middle panel, respectively.  Contributions due to\nbox corrections, vertex corrections and self energy corrections have\nbeen plotted in addition to the full result. In this scenario the\nasymmetry can reach $\\sim 1\\%$ for the $\\mu$ parameter and $\\sim\n6\\%$ for $A_t$, respectively. We note that for the asymmetry due to\nthe non-zero phase of the higgsino mass parameter there are\nsignificant cancelations among various contributions. In addition,\nwe also show in the right panel of Fig.~\\ref{fig2} the dependence of\nthe asymmetry due to $A_t$ as a function of $\\tan\\beta$.\n\nFor the asymmetry generated by the $\\mu$ parameter all possible\none-loop diagrams containing absorptive part contribute. The\nsituation is different for the phase of  the trilinear coupling\n$A_t$ --- when chargino mixing matrices remain real. In this case\nonly vertex and self-energy diagrams containing stop lines\ncontribute to the asymmetry~\\cite{my}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.365]{Rolbiecki_fig2a.eps}\n\\includegraphics[scale=0.35]{Rolbiecki_fig2b.eps}\n\\includegraphics[scale=0.365]{Rolbiecki_fig2c.eps}\n\\end{center}\\vspace{-0.1cm}\n\\caption{Asymmetry $A_{12}$ in scenario (A) as a function of the\nphase of $\\mu$ parameter (left), the phase of $A_t$ (middle), and as\na function of $\\tan\\beta$ with $\\Phi_t=\\pi\/3$ (right). Different\nlines denote full asymmetry (full line) and contributions from box\n(dashed), vertex (dotted) and self energy (dash-dotted) diagrams.\n\\label{fig2}}\n\\end{figure}\n\n\nWe present also the results for the heavy sfermion scenario~(B).\nThis is to compare with~\\cite{Osland:2007xw} where only box diagrams\nwith $\\gamma$, $W$, $Z$ exchanges have been calculated neglecting\nall sfermion contributions. As can be seen in the left panel of\nFig.~\\ref{fig3} these gauge-box diagrams constitute the main part of\nthe asymmetry $A_{12}$, however this is due to partial cancelation\nof vertex and self-energy contributions. For lower values of the\nuniversal scalar mass $M_S$ the discrepancy between full and\napproximate result of~\\cite{Osland:2007xw} increases significantly.\nThis is illustrated in the middle and right panel of Fig.~\\ref{fig3}\nwhere we show two paths of approaching of the full result to the\ngauge-box approximation as the function of $M_S$. As can be seen\nthese paths depend strongly on the center of mass energy.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.335]{Rolbiecki_fig3a.eps}\n\\includegraphics[scale=0.32]{Rolbiecki_fig3b.eps}\n\\includegraphics[scale=0.32]{Rolbiecki_fig3c.eps}\n\\end{center}\n\\caption{Left: Asymmetry $A_{12}$ in scenario (B) as a function of\nthe phase $\\Phi_\\mu$. Different lines denote full asymmetry (full\nline) and contributions from box (dashed), vertex (dotted) and self\nenergy (dash-dotted) diagrams. Middle and Right: Asymmetry $A_{12}$\nas a function of the universal scalar mass $M_S$ with\n$\\Phi_\\mu=\\pi\/2$ at different cms. The full lines denote full result\nand dashed lines show only the box contributions after neglecting\ndiagrams with slepton exchange.\\label{fig3}}\n\\end{figure}\n\n\n\n\n\\section{Summary}\nIt has been shown that CP-odd asymmetry can be generated in\nnon-diagonal chargino pair production with unpolarized\nelectron\/positron beams. The asymmetry is pure one-loop effect and\nis generated by interference between complex couplings and\nabsorptive parts of one loop integrals. The effect is significant\nfor the phases of the higgsino mass parameter $\\mu$ and the\ntrilinear coupling in stop sector $A_t$. At future linear collider\nit may give information about CP violation in chargino and stop\nsectors.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe flow around objects in a bulk medium is a classical problem in fluid dynamics which occurs in numerous natural phenomena and has many practical applications. The majority of research concerns high Reynolds number flows, where vortex shedding plays an important role, and mixing is greatly enhanced~\\cite{Choi2008}. Until recently, and even though it was pointed out more than 60 years ago, much less attention has been given to mixing due to the drift volume, a mechanism described by potential flow known as Darwin's drift~\\cite{Darwin1953}. One of the recent studies has shown that for many marine animals Darwin's drift is the dominant contribution to mixing by swimming~\\cite{Katija2009}. Darwin's original proposition states that the drift volume induced by an object, which was started infinitely far away from an imaginary plane and moves infinitely far to the other side of the plane is equal to the added mass volume of that object~\\cite{Darwin1953}. This prediction is based on potential flow, and has been confirmed two decades ago by Eames \\emph{et al.}~\\cite{Eames1994}, who added specific information concerning the method of evaluating the integrals to calculate the drift volume and of taking into account partial drift for finite systems. An open question is how the drift volume is influenced in the case of a less smooth object, or at higher Reynolds numbers, when flow separation starts to play a role. This problem is related to the drift volume of a vortex ring, which was considered analytically for a continuously expanding vortex~\\cite{Turner1964} and experimentally for a steady vortex ring~\\cite{Dabiri2006}.\n\nWe approach this problem experimentally by impulsively starting a disc from the interface of two immiscible liquids, oil and water, similarly and with the same setup as we did before when studying the dynamics of an air-water interface when a disc would impact it~\\cite{Bergmann2006,Bergmann2009,Gekle2009a,Gekle2010,Gekle2010d}. Besides being a practical solution to studying drift volumes, this method has relevance for, e.g., the mechanism of oil dispersion in water~\\cite{Murphy2015}. In our experiment, when the disc moves down, it drags along the oil, which then obtains a particular funnel-shaped profile (see, e.g., Fig.~\\ref{fig:sequence_thick_layer}). Both gravity and surface tension are deforming the shapes, and we determine how these profiles depend on the velocity of the disc. We observe that there exists a universal profile which becomes more prominent for higher velocities, for which this universal shape will extend deeper into the fluid. Surprisingly, however, despite the observed universal behavior, these profiles do not agree with potential flow simulations we performed using a boundary integral technique. We attribute this difference to the formation of a vortex ring, which distorts the potential flow. We specifically show that the drift volume is \\emph{larger} than that predicted by potential flow.\n\n\\section{Experimental setup}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=70mm]{fig01}\n    \\caption{\\label{fig:setup} Schematic view of the experiment, with disc radius $R_D$ and disc velocity $V_D$. We start with the bottom of the disc at rest at the interface between a deep layer of oil (45 mm) on top of a deep layer of water (25 cm), after which we pull down the disc at constant speed $V_D$. We define $z=0$ at the undisturbed oil-water interface.}\n\\end{figure}\nThe experimental setup (Fig.~\\ref{fig:setup}) consisted of a water reservoir with a cross section of 15 cm by 15 cm and a height of 50 cm. A linear motor that was mounted below the tank pulls a disc with a radius $R_D=20~\\mathrm{mm}$ through the water surface at a constant speed $V_D$, by means of a thin rod (radius 3 mm) connecting the linear motor with the disc. The disc was accelerated with a maximum acceleration of $42~\\mathrm{m\/s^2}$ until the desired velocity ($V_D$) was reached. The events were recorded with a Photron SA2 high-speed color camera at frame rates ranging from $1$ to $8~\\text{kHz}$. The main control parameter of the experiment is an effective Froude number $\\Froude^*$, which is similar to the regular Froude number defined as the disc speed $V_D$, made dimensionless using the disc radius $R_D$ and the gravitational acceleration $g$. Only we replace $g$ by the effective gravitational acceleration $g^*$ of the oil phase inside the water phase, as one would use to determine the wave speed of gravitational waves on a density interface $g^*=g(\\rho_w-\\rho_o)\/(\\rho_w+\\rho_o)$ \\cite{Kundu2004_waves}, yielding\n\\begin{equation}\n    \\Froude^*=\\frac{V_D^2}{gR_D}\\left(\\frac{\\rho_w+\\rho_o}{\\rho_w-\\rho_o}\\right),\n\\end{equation}\nwhere $\\rho_w$ and $\\rho_o$ are the densities of water and oil respectively. In our experiments we used sunflower oil, which has a density $\\rho_o=900~\\mathrm{kg\/m^3}$ and a viscosity $\\nu\\sim50\\cdot10^{-6}\\mathrm{m^2\/s}$. Next to demineralized water we used a solution of table salt in water to increase the density of the water phase. We dissolved $1.0~\\mathrm{kg}$ of table salt in $5000~\\mathrm{ml}$ water, resulting in $\\rho_{sw}=1140~\\mathrm{kg\/m^3}$. The thickness of the oil layer in these experiments was $45~\\mathrm{mm}$, which was thick enough to be considered as infinite. We verified this by performing the same experiment with increased oil layer thicknesses of $90~\\mathrm{mm}$ and $135~\\mathrm{mm}$, which did not influence our results.\n\n\n\\section{Results}\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=86mm]{fig02}\n    \\caption{\\label{fig:sequence_thick_layer}\n    Snapshots from experiments with two different values of the effective Froude number $\\Froude^*=2.7$ ($a$i-$c$i), and $\\Froude^*=43$ ($a$ii-$c$ii). Corresponding pictures have been taken at the same dimensionless times $\\tau=2,~\\tau=3,~\\tau=4$. In the top experiment gravity has a clear influence on the shape of the entrained oil column. The bottom experiment is in the inertial regime, where gravity has negligible influence. A vortex ring is formed above the disc, which grows with the same dimensionless rate, independent of the Froude number. Black dashed line is the result from boundary integral simulations.\n    }\n\\end{figure}\nFigure \\ref{fig:sequence_thick_layer} shows two experiments in which we pulled down the disc from the oil-water interface at different velocities $V_D = 0.25$ and $1.0~\\mathrm{m\/s}$. Initially the disc was at rest and its lower surface was aligned with the oil-water interface. Then, the disc was set into motion and in a short period of time obtained a constant speed $V_D$ \\footnote{With an acceleration of $42~\\mathrm{m\/s^2}$, it takes $0.024~\\mathrm{s}$ to reach $V_D=1.0~\\mathrm{m\/s}$ ($\\Froude^*=43$). The duration of the experiment in that case is $0.08~\\mathrm{s}$. The acceleration does not significantly influence the experiment, as can be appreciated in Fig.~\\ref{fig:all_profiles}}. A vortex ring appeared just above the disc, along with a smooth profile in the center which connects to the thick oil layer at the top. We first focus on this smooth profile, which has similarities with the shapes seen in \\cite{Lian1989}, although that study  concentrated on the formation of the vortex ring.\n\nIn order to compare the experiments for different disc speeds, we define a dimensionless time $\\tau=z(t)\/R_D$, where z(t) is the depth the disc has reached at time $t$ after starting from $z=0$ at $t=0$, i.e., at equal dimensionless times the disc has reached the same vertical position below the undisturbed oil-water interface, measured in units of the disc radius $R_D$. By varying the acceleration we verified that at equal target velocities $V_D$ and dimensionless times $\\tau$ the observed column shapes are independent of the precise value of the acceleration used in experiments~\\footnote{We have used accelerations ranging from $21$ to $42~\\mathrm{m\/s^2}$ and found no discernible differences in the shape of the entrained oil column, up to the point in time that gravity becomes significant, where small differences would be introduced due to the somewhat longer time that is needed to reach a depth $z(t)$ at small accelerations.}. If the disc would be moving at a constant velocity $V_D$ all the time, i.e., if the acceleration phase would be infinitely small, then $\\tau$ and $t$ would be related as $\\tau=tV_D\/R_D$. In the remainder of the article we will for simplicity ignore the existence of the acceleration phase and take $z(t) = V_D t$.\n\nComparing Fig.~\\ref{fig:sequence_thick_layer}(\\emph{a}i) and (\\emph{a}ii), we see that at $\\tau=2$ the shape of the entrained oil is very similar for $\\Froude^*=2.7$ and $\\Froude^*=43$, although the amount of vorticity in the vortex ring appears to be\nmuch larger for the higher speed. At $\\tau=3$ the effect of buoyancy becomes visible for the lower Froude number, where a difference in the shape of the entrained oil between Fig.~\\ref{fig:sequence_thick_layer}(\\emph{b}i) and (\\emph{b}ii) is appreciable. In the last frame, Fig.~\\ref{fig:sequence_thick_layer}(\\emph{c}i) and (\\emph{c}ii) at $\\tau=4$, the oil in the case of $\\Froude^*=2.7$  has clearly moved back up due to buoyancy, leaving only a relatively straight cylinder of oil behind. For $\\Froude^*=43$, the shape is still unaffected by gravity at this point in time.\n\nIn order to see what the effect of the density difference is on the shape that we obtain at high Froude numbers, we performed experiments with demineralized water ($\\rho=998~\\mathrm{kg\/m^3}$) at $V_D=1~\\mathrm{m\/s}$ ($\\Froude^*=99$) and with salt water ($\\rho=1140~\\mathrm{kg\/m^3}$) at $V_D=1.5~\\mathrm{m\/s}$ ($\\Froude^*=97$). Although there is a factor two in the density difference between the oil and the water phases (demineralized vs. salt water), the difference in the profiles is negligible, which leads us to conclude that the shape of the entrained oil column does not strongly depend on the relative density difference between the fluids. The use of salt water does however have an experimental advantage: The oil-water interface became less contaminated with oil and water droplets after the experiment is finished, which reduced the time that we had to wait between two experiments until the surface was smooth enough to clearly observe formation of the profile of the entrained oil. For experimental convenience, we used salt water in all experiments, except noted otherwise.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=85mm]{fig03}\n    \\caption{\\label{fig:all_profiles}\n    Profiles of the entrained oil, for effective Froude numbers ranging from $2.7$ to $97$. We observe universal profiles for $\\Froude^*\\rightarrow\\infty$, each color in the image corresponds to one Froude number. (\\emph{a}) at $\\tau=2$, all shapes collapse. (\\emph{b}) at $\\tau=3$, a difference becomes visible for $\\Froude^*=2.7$ and $\\Froude^*=5.3$ (\\emph{c}) $\\tau=4$, for increasing $\\Froude^*$, the shapes of the entrained oil converge to a single universal profile independent of $\\Froude^*$. ($d$-$f$) shows the same profiles (only retaining shapes fulfilling the universality criterium $\\tau^2 < Fr^*$) together with the boundary integral simulation results (dotted lines). The position and size of the disc is indicated by the black rectangle. The profile of the oil-water interface is not shown in the vicinity of the disc because the profile was not visible due to the vortex ring, see Fig. \\ref{fig:sequence_thick_layer}.\n    }\n\\end{figure}\n\nIn Fig.~\\ref{fig:all_profiles} we compare the profiles of the entrained oil at equal dimensionless times $\\tau$ and for a wide range of Froude numbers. Every experiment in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}) consists of several repetitions of the experiment, sometimes using two different disc accelerations ($21$ and $42~\\mathrm{m\/s^2}$), indicating the excellent reproducibility of the experiment and the irrelevance of the initial startup motion of the disc and the actual acceleration that is used in this startup phase.\n\nThe appearance of differences in the shapes shown in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}) are a result of gravity that is pushing the oil phase upwards. This will only happen if the time is long enough for gravity to become more important than the inertia that is pulling the oil phase down. We can predict the moment that differences appear by comparing the inertial time scale $t_{in}\\equiv R_D\/V_D$ to the gravitational time scale $t_g \\equiv\\sqrt{R_D\/g^*}$. Gravity is expected to play a role only if $t\\gtrsim t_g$, which, after dividing both sides by the inertial time scale can be written as\n\\begin{equation}\n    \\tau^2\\gtrsim\\Froude^*,\n    \\label{eq:OilTimeScales}\n\\end{equation}\nwhere we have used $t\/t_{in}=\\tau$. If we now again look at Fig.~\\ref{fig:all_profiles}(\\emph{a-c}), we expect according to Eq.~(\\ref{eq:OilTimeScales}) to see a difference for $\\Froude^*\\lesssim4$ at $\\tau=2$, for $\\Froude^*\\lesssim9$ at $\\tau=3$, and for $\\Froude^*\\lesssim16$ at $\\tau=4$. These predictions agree well with the moment that we observe differences in the experimental profiles in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}).\n\nWe now proceed to compare our experimental findings to potential flow solutions. For this, we use the boundary integral method as described in~\\cite{Oguz1993,Bergmann2009,Gekle2010d,Gekle2011a,Pozrikidis2011}. We performed boundary integral simulations of an impulsively started disc in an infinite bath of fluid, moving at a constant speed $V_D$. The disc had the same size and thickness as in the experiment. We injected tracers at the position corresponding to the initial oil-water interface, \\textit{i.e.}, aligned with the bottom of the disc. The tracers were then advected with the flow field around the disc. Because we used tracers in an infinite bath of a single liquid, the motion of the tracers corresponds to the case of $Fr^*\\rightarrow\\infty$. To validate the numerical code, we verified our simulations by calculating the displaced volume in the case of a sphere. We found good agreement with the analytical results of Eames~\\textit{et al.}~\\cite{Eames1994} for $\\rho_{max}\/|x_0|\\rightarrow\\infty$, where $\\rho_{max}$ is the radius of the reference plane that is taken into account in the calculation of the drift volume and $x_0$ the initial axial distance of the sphere from this plane (see Eames~\\textit{et al.}~\\cite{Eames1994} for details). We note that in our experiments the initial position of the disc is at the reference plane (the oil-water interface), such that $x_0\\rightarrow0$, and, consequently, $\\rho_{max}\/|x_0|\\rightarrow\\infty$.\n\nFig.~\\ref{fig:all_profiles}\\emph{d-f} shows that the shapes are approximated by our simulation results, but also that there exists a significant difference close to the undisturbed surface. A closer inspection reveals that the discrepancy increases as the disc moves further down: while at $\\tau=2$ ($d$) there is a reasonable agreement between the simulation and experiment, at $\\tau=4$ ($f$) the difference is much larger. The same discrepancy is also illustrated in Fig.~\\ref{fig:sequence_thick_layer}($a$ii-$c$ii).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig04}\n    \\caption{\\label{fig:radiusTime}\n    Radius of the entrained column oil as a function of dimensionless time $\\tau$ at different depths $z$ for the experiment (closed symbols) and the simulation (solid lines). Blue circles: $z\/R_D=-0.5$, green diamonds: $z\/R_D=-1.0$, red triangles: $z\/R_D=-1.5$, black squares: $z\/R_D=-2.0$.}\n\\end{figure}\n\nWe further investigate the development of this difference between the experiments and simulation in Fig.~\\ref{fig:radiusTime}, where we plot the radius of the oil\/water interface at different depths $z$ below the location of the undisturbed interface at $z=0$. First, we observe that the larger the distance to the undisturbed surface is, the smaller the discrepancy becomes. The difference in radius even appears to switch sign at the deepest position plotted here (black squares), although this difference is close to the experimental uncertainty. Second, the figure shows an apparent qualitative difference between the experiments and simulations at $z\/R_D=-0.5$, for which the simulation shows an initial decrease of the radius, which then seemingly asymptotes to a constant value. The experiment on the other hand shows a significant increase in radius after the initial decrease~\\footnote{Actually, also in the simulations we observe such an increase, only much smaller ($<2\\%$ of the minimum radius) than in the experiment}.\n\nA qualitative explanation for both these observations originates from the formation and growth of the vortex which introduces an additional downward velocity (and velocity gradient) inside the entrained volume of oil, since oil will continuously flow into the vortex. There where the shape of the entrained oil volume $r(z,t)$ is column-like (i.e., $\\partial r\/\\partial z$ is small), this will lead to simple stretching and thus a decreasing radius of the column~\\cite{Eggers2008}. However, where there exists a large gradient in the radius, the overall downward translation may introduce an increase in radius in the lab frame. This can be seen as follows. Because the volume $\\propto r^2v_z$ of the entrained oil is conserved, the interface $r(z,t)$ obeys the PDE\n\\begin{equation}\n\t\\frac{\\partial r^2}{\\partial t} + \\frac{\\partial}{\\partial z}(r^2 v_z)=0,\n\\end{equation}\nwhere $v_z(z,t)$ ($\\geq 0$) is the downward velocity in the vertical direction inside the column, which for simplicity is assumed to be independent of the radial coordinate. This can immediately be rewritten as\n\\begin{equation}\n\t\\frac{\\partial r}{\\partial t} + v_z\\frac{\\partial r}{\\partial z} + \\frac{1}{2}r\\frac{\\partial v_z}{\\partial z}=0.\n\\end{equation}\nNote that for such a stretching flow the second term in the above expression is always negative, since $\\partial r\/\\partial z < 0$ for the measured profiles, whereas the third (or stretching) term is always positive because $\\partial v_z\/\\partial z \\geq 0$. In case of a purely columnar shape, the second term is zero, and therefore $\\partial r\/\\partial t$ is negative. In case of a funnel-shaped profile with a strong radial gradient ($\\partial r\/\\partial z \\ll 0$) and sufficient downward velocity, the magnitude of the second term may well\nbe larger than the stretching term, which will result in a positive $\\partial r\/\\partial t$.\n\nThe remaining question is why we still observe universal shapes even though there is a clear influence from the vortex. The reason is that for early times ($\\tau\\lesssim8$) the\ndimensionless size and strength of a vortex behind a impulsively started disc is independent of the disc speed, as was shown in~\\cite{Yang2012}. That the same holds for our experiments can clearly be seen in Fig.~\\ref{fig:sequence_thick_layer}, where the vortex for $Fr^*=2.7$ has the same size as that for $Fr^*=43$, when they are compared at the same dimensionless time.\n\nWe now proceed to providing an estimate of the magnitude of the downward velocity $v_z$ due to the presence of the vortex. Given the circulation $\\Gamma$ of a vortex line in a closed loop $C$, the velocity $\\bf v$ at position $\\bf r$ can be obtained from the Biot-Savart law\n\\begin{equation}\n\t{\\bf v}({\\bf r}) = \\frac{\\Gamma}{4\\pi}\\oint_C\\frac{d{\\bf l}\\times({\\bf l} - {\\bf r})}{|{\\bf l} - {\\bf r}|^3},\n\t\\label{eq:BiotSavart}\n\\end{equation}\nwhere $\\bf l$ is the core of the vortex line. For a ring-shaped vortex with a radius equal to the disc radius $R_D$ we can write for points $\\bf r$ on the axis of symmetry that, taking into account that the velocity is purely vertical, $|{\\bf l} - {\\bf r}|=\\sqrt{R_D^2+(z')^2}$ and ${\\bf \\hat{e}_z} \\cdot (d{\\bf l}\\times({\\bf l} - {\\bf r})) = R_D dl$, with $\\bf \\hat{e}_z$ the unit vector in the $z$-direction, $dl=R_Dd\\theta$, and $z'=z-z_c$, where $z_c$ is the vertical position of the core of the vortex. The $z$-component of the integral of Eq.~(\\ref{eq:BiotSavart}) can now be evaluated straightforwardly as\n\\begin{equation}\n\tv_z(z') = \\frac{\\Gamma}{4\\pi}\\int_0^{2\\pi}\\frac{R_D^2}{(R_D^2 + z'^2)^{3\/2}}d\\theta\n\\end{equation}\nand gives the vertical component of the velocity $v_z$ as\n\\begin{equation}\n\tv_z = \\frac{R_D^2\\Gamma}{2(R_D^2+z'^2)^{3\/2}},\n\\end{equation}\nor, in dimensionless form,\n\\begin{equation}\n\t\\tilde v_z = \\frac{\\tilde\\Gamma}{2(1+\\tilde z'^2)^{3\/2}},\n\\end{equation}\nwhere $\\tilde v_z = v_z\/V_D$, $\\tilde z = z\/R_D$, and the dimensionless, time-dependent circulation $\\tilde\\Gamma=\\Gamma\/(R_DV_D)$ is independent of the disc speed~\\cite{Yang2012}. In Fig.~\\ref{fig:vzVortex} we compare the vertical velocity on the axis of symmetry resulting from our potential flow calculation to the one induced by the vortex ring at different instances of time. We have used the empirical relation from Yang~\\emph{et al.}~\\cite{Yang2012} for $\\tilde\\Gamma(\\tau)$~\\footnote{$\\tilde\\Gamma\\approx-4.67(1-0.75e^{-0.416\\tau} - (1-0.75)e^{-24.927\\tau})$} and approximated the velocity of the vortex core to be linear (i.e., $\\tilde z'=\\tilde z_D + 0.25\\tau$). The latter approximately matches the position of the vortex core in the experiments, where we observe that the core is about one disc radius above the disc at $\\tau=4$ (see Fig.~\\ref{fig:sequence_thick_layer}). Clearly, the velocities and velocity gradients are greatly enhanced by the starting vortex, and an influence on the shape of the entrained oil column is therefore expected. Most importantly note that the velocity profiles of Fig.~\\ref{fig:vzVortex} are independent of the disc velocity, resulting in a mechanism through which the universality of the shapes shown in Fig.~\\ref{fig:all_profiles} are preserved.\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig05}\n    \\caption{\\label{fig:vzVortex}\n    Comparison of the estimated non-dimensional velocity $\\tilde v_z = v_z\/V_D$ on the axis of symmetry as a function of the non-dimensional distance $\\tilde z -\\tilde z_D = (z-z_D)\/R_D$ to the center of the disc for the potential flow solution (solid line) and the time-dependent contribution from the vortex ring (dotted lines) at four different instances in dimensionless time $\\tau = tR_D\/V_D$.}\n\\end{figure}\n\nTo further quantify the influence of the vortex on the drift volume, we calculate the entrained volume of oil in both the simulations and the experiment. Because a part of the interface is masked by the vortex, we do not calculate the complete displaced volume~\\cite{Eames1994}, but only compare the part that is accessible in both the experiment and the simulation. In the simulations we find, at $\\tau=4$, between the depths $z\/R_D=-0.2$ and $z\/R_D=-1.0$ (with $z=0$ at the unperturbed oil\/water interface), an entrained volume of 4.24 ml, while in the experiment 7.25 ml of oil is entrained. Extending the range to $z\/R_D=-2.0$ gives 5.57 ml and 8.66 ml for the simulation and experiment respectively. Clearly, the experimental volume of entrained oil is significantly larger than the volume predicted by the potential flow simulations.\n\n\n\\section{Conclusions and outlook}\nWe have performed experiments where we started a disc at an oil-water interface and pulled it down at a constant speed. We have shown that at high speeds, gravity and surface tension can be neglected, and the  entrained oil obtains a universal funnel shape, independent of the Froude number. However, a vortex ring is formed at the disc edge, which influences the shape of the entrained oil resulting in a qualitative and quantitative difference compared to the potential flow solution. Nonetheless, the universality of the funnel shape is conserved. The shape also appears insensitive to the relative density difference, at least for the density differences studied.\n\nThe effect might be investigated further for flows at lower Reynolds numbers, exploring the limit where the vortex disappears. In our current setup this regime is inaccessible due to the influence of gravity at low disc speeds. A consequence of our finding is that the displaced volume as predicted by \\citep{Darwin1953} is underestimated in cases where flow separation is of importance. This introduces a non-trivial shape-effect on the entrainment in the wake of, for example, ocean life \\cite{Katija2009}. The preserved universality in such cases, however, may help in simplifying and generalizing analysis in these situations.\n\n\\begin{acknowledgments}\nWe acknowledge financial support from FOM and the NWO-Spinoza program.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA number of fundamental problems in algebraic topology can be\ndescribed as measuring the extent to which a given functor \\ \n$T:{\\mathcal C}\\to{\\mathcal D}$ \\ between model categories induces an equivalence of\nhomotopy categories: more specifically, which objects (or maps) from\n${\\mathcal D}$ are in the image of $T$, and in how many different ways. For example\\vspace{2 mm}:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item How does one distinguish between different topological spaces\nwith the same homology groups, or with chain-homotopy\nequivalent chain complexes?  How can one realize a given map of\nchain complexes up to homotopy\\vspace{2 mm}?\n\\item When do two simply-connected topological spaces have the same\nrational homotopy type\\vspace{2 mm}?  \n\\item When is a given topological space a suspension, up to homotopy?\nDually, how many distinct loop space structures, if any, can a given topological\nspace carry\\vspace{2 mm}?\n\\item Is a given $\\Pi$-algebra\\ (that is, a graded group with an action\nof the primary homotopy operations) realizable as the homotopy\ngroups of a topological space, and if so, in how many ways\\vspace{2 mm}? \n\\end{enumerate}\n\nOur goal is to describe a unified approach to such problems that\nworks for functors between \\emph{spherical} model categories, for which\nseveral familiar concepts and constructions are available. These\ninclude a set ${\\mathcal A}$ of \\emph{models} (to play the role of spheres, in\nparticular determining the corresponding homotopy groups \\ $\\pinC{\\ast}$), \\ \nPostnikov systems, and $k$-invariants. If a functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\\nrespects this additional structure, we obtain a natural long exact\nsequence of the form: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eles}\n\\dotsc\\to \\Gamma_{n}X~\\xra{s}~\\pinC{n} X~\\xra{h}~\n\\pinD{n} TX~\\xra{\\partial}~\\Gamma_{n-1}X\\dotsc~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which generalizes the EHP sequence, J.H.C. Whitehead's ``certain exact\nsequence'', and the spiral exact sequence of Dwyer, Kan, and\nStover. See \\ \\eqref{efive} \\ below.\n\nUnder these hypotheses, given an object $G$ in ${\\mathcal D}$, we want to find\nan object $X$ in ${\\mathcal C}$ with \\ $TX\\simeq G$. \\ The key step is to choose \\ \n$\\pinC{\\ast} X$ \\ which fits into \\ \\eqref{eles}. \\ We describe an inductive \nprocedure for doing this, using the Postnikov systems in both\ncategories, together with an obstruction theory for lifting $G$ to\n${\\mathcal C}$, along the following lines: \n\n\\begin{tha}\nGiven \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ and \\ $G\\in{\\mathcal D}$ \\ as above, for each \\ $X\\in{\\mathcal C}$ \\ \nwith \\ $TX\\simeq G$, \\ there is a tower of fibrations in ${\\mathcal C}$:\n$$\n\\dotsb \\xra{p\\q{n+1}} \\Xpn{n+1} \\xra{p\\q{n}} \\Xpn{n} \\xra{p\\q{n-1}} \\dotsb\n\\xra{p\\q{0}} \\Xpn{0}~,\n$$\ncalled the \\emph{modified Postikov tower for} $X$ (Def.\\ \\ref{dmps}), \nwith $G$ mapping compatibly to \\ $T\\Xpn{n}$ \\ for each $n$, and \\\n$X\\simeq\\operatorname{holim}_{n} \\Xpn{n}$.\n   \nConversely, given such a tower up to level $n$, the obstruction to\nextending it to level \\ $n+1$ \\ lies in \\\n$\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}$, \\ and the choices for \\ $\\Xpn{n+1}$ \\ \nare classified by:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{$\\bullet$~}\n\\item a class in \\ $\\HL{n+2}{G}{\\Gamma_{n+1}{\\Xpn{n}}}$;  \n\\item a class in \\ $\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$, \\ where \\ \n$K_{n+1}:=\\operatorname{Coker}\\,\\pi_{n+2}\\rho\\q{n}$, \\ for \\ \n$\\rho\\q{n}:P_{n+2}G\\to P_{n+2}T\\Xpn{n}$.\n\\end{enumerate}\n\\end{tha}\n\nSee Theorem \\ref{tfour}.\n\n\\subsection{Related work}\n\\label{srw}\\stepcounter{thm}\n\nThe comparison problems discussed above are familiar ones in \nalgebraic topology:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item The question of the realizability of a graded algebra as a\n  cohomology ring was first raised explicitly by Steenrod in\n  \\cite{SteCA}, but it goes back to Hopf (in \\cite{HopfT}) in the\n  rational case. The ``Steenrod problem'' of realizing a given \\\n  $\\pi_{1}$-action in homology has been studied, for example, \n  in \\cite{ThP,JRSmiT1}. \n\\item The comparison between integral and rational homotopy type \n  was implicit in the notion of a Serre class (cf.\\ \\cite{SerG,ACuH}),\n  although an explicit formulation was only possible after the \n  construction of the rationalization functors of Quillen and Sullivan\n  in \\cite{QuiR,SulG}.\n\\item Possible loop space structures on a given $H$-space were\n  analyzed extensively, starting with the work of Sugawara and\n  Stasheff (cf.\\ \\cite{SugG,StaH}). The dual question on\n  identifying suspensions has also been studied (see, e.g., \\cite{BHilS}).\n\\item The question of the realizability of homotopy groups goes back\n  to J.H.C.~Whitehead, in \\cite{JWhR} (see also \\cite{JWhSH}), and has\n  reappeared in recent years in the context of $\\Pi$-algebra s (cf.\\\n  \\cite{DKStE,DKStB}). The relationship between homology and homotopy\n  groups, which is relevant to the realization problem for both, was\n  studied in  \\cite{JWhSB,JWhC} (in which the ``certain exact sequence'' \n  was introduced)\\vspace{2 mm}.\n\\end{enumerate}\n\nIn \\cite{BauCF}, H.-J.\\ Baues gave what appears to be the first \ngeneral theory covering a wide spectrum of such realization problems.\nThis was an outgrowth of his earlier work on classifying\nhomotopy types of finite dimensional CW complexes in\n\\cite{BauCHF,BauHH} (which in turn builds on \\cite{JWhSF}). \n\nHis initial setting consists of a homological cofibration \ncategory ${\\mathcal C}$ (corresponding to, and extending, the notion of a\nresolution model category) under a theory  of coactions $\\mathbb T$\n(corresponding to the category \\ $\\Pi_{\\A}$ \\ of \\S \\ref{ssmod}). \nBaues then constructs a generalized ``certain exact sequence'' \nsimilar to \\ \\eqref{eles}, \\ and\nprovides an inductive obstruction theory for realizing a chain complex\n(or a chain map) by a $\\mathbb T$-complex (corresponding to a CW complex, or\nmore generally a cofibrant object in ${\\mathcal C}$) \\ -- \\ see \n\\cite[VI, (2.2-2.3)]{BauCF}). \n\nThese results apply inter alia to the problem of realizing \na chain complex by a topological space (the motivating example \nfor Baues's approach), as well as to the realization of a $\\Pi$-algebra\\ \n(cf.\\ \\cite[D, (7.9)]{BauCF}).  However, here we consider \nfunctors between two different model categories that \nare not covered by \\cite{BauCF}. In particular, our original\nmotivating example \\ -- \\ the realization of a \\emph{simplicial} $\\Pi$-algebra\\\n(by a simplicial space) \\ -- \\ shows that in the relative context a\nmore refined obstruction theory may be necessary:  \ncompare Theorem (2.3) of \\cite[VI]{BauCF} with Theorem \\ref{tfour} below.\n\n\\begin{remark}\\stepcounter{subsection}\nAnother set of closely related questions  \\ -- \\ which do not quite fit\ninto the framework described here, though they can also be stated as\nrealization problems \\ -- \\  arise in categories of structured ring\nspectra; see for example \\cite{RobO} and \\cite[Cor.\\ 5.9]{GHopkM}\\vspace{2 mm} . \n\\end{remark}\n\n\\subsection{Notation and conventions}\n\\label{snac}\\stepcounter{thm}\n\n$\\TT_{\\ast}$ \\ denotes the category of pointed connected topological\nspaces; \\ $\\Set_{\\ast}$ \\ that of pointed sets, and \\ ${\\EuScript Gp}$ \\ that of\ngroups. \\ For any category ${\\mathcal C}$, \\ $\\operatorname{gr}{\\mathcal C}$ \\ denotes the category of \nnon-negatively graded objects over ${\\mathcal C}$, and \\ $s{\\mathcal C}$ \\ the category of\nsimplicial objects over ${\\mathcal C}$. \\ $s{\\EuScript Set}$ \\ is denoted by ${\\mathcal S}$, \\\n$s\\Set_{\\ast}$ \\ by \\ $\\Ss_{\\ast}$, \\ and \\ $s{\\EuScript Gp}$ \\ by \\ ${\\mathcal G}$. \\ The constant\nsimplicial object an an object \\ $X\\in{\\mathcal C}$ \\ is written \\ $\\co{X}\\in s{\\mathcal C}$. \n\nIf ${\\mathcal C}$ has all coproducts, then given \\ $A\\in{\\mathcal S}$ \\ and \\ $X\\in{\\mathcal C}$, \\\nwe define \\ $X\\hat{\\otimes} A\\in s{\\mathcal C}$ \\ by \\ \n$(X\\hat{\\otimes} A)_{n}:=\\coprod_{a\\in A_{n}} X$, \\ \nwith face and degeneracy maps induced from those of $A$. \nFor \\ $Y\\in s{\\mathcal C}$, \\ define \\ $Y\\otimes A\\in s{\\mathcal C}$ \\ by \\ \n$(Y\\otimes A)_{n}:= \\coprod_{a\\in A_{n}} Y_{n}$ \\ (the diagonal of the\nbisimplicial object \\ $Y\\hat{\\otimes} A$) \\ -- \\ so that for \\ $X\\in{\\mathcal C}$ \\\nwe have \\ $X\\hat{\\otimes} A=\\co{X}\\otimes A$.\n\nThe category of chain complexes of $R$-modules is denoted by \\\n${\\EuScript Chain}_{R}$ \\ (or simply \\ ${\\EuScript Chain}$, \\ for $R=\\mathbb Z$).\n\n\\subsection{Organization:}\n\\label{sorg}\\stepcounter{thm}\n\nIn Section \\ref{csmc} we define \\emph{spherical} model categories,\nhaving the additional structure mentioned above. \nMost examples of such categories are in particular \\emph{resolution} \nmodel categories, which are described in Section \\ref{crmc}; we\nexplain how to produce the needed structure for them in Section \\ref{csp}.\nWe define \\emph{spherical functors} between such categories, and\nconstruct the comparison exact sequence for them, in\nSection \\ref{csf}. This is applied in Section \\ref{ccps} to study the\neffect of a spherical functor on Postnikov systems. Finally, in\nSection \\ref{cfib} we construct an obstruction theory as above for the\nfiber of a spherical functor. In Section \\ref{cat} we indicate how the\ntheory works for the above examples.\n\n\\subsection{Acknowledgements}\n\\label{sack}\\stepcounter{thm}\n\nI would like to thank Paul Goerss for many hours of discussion on various\nissues connected with this paper, and especially for his essential\nhelp with Sections \\ref{ccps}-\\ref{cfib}, the technical core of this\nnote. I would also like to thank Hans Baues for explaining the\nrelevance of his work in \\cite{BauCF} to me.\n\n\\setcounter{thm}{0}\\section{Spherical model categories}\n\\label{csmc}\n\nBefore defining the additional structure we shall need, we briefly\nrecapitulate the relevant homotopical algebra:\n\n\\subsection{Model categories}\n\\label{smc}\\stepcounter{thm}\n\nRecall that a \\emph{model category} is a bicomplete category ${\\mathcal C}$\nequipped with three classes of maps: weak equivalences,\nfibrations, and cofibrations, related by appropriate lifting\nproperties. By inverting the weak equivalences we obtain the\nassociated homotopy category \\ $\\operatorname{ho}{\\mathcal C}$, \\ with morphism set \\ \n$[X,Y]=[X,Y]_{{\\mathcal C}}$. \\ We shall concentrate on \\emph{pointed} model\ncategories (with null object $\\ast$). \\ See \\cite{QuiH} or \\cite{PHirM}.   \n\n\\subsection{The set of models}\n\\label{ssmod}\\stepcounter{thm}\n\nThe additional initial data that we shall require for our model\ncategory consists of a set ${\\mathcal A}$ of cofibrant homotopy cogroup objects\nin ${\\mathcal C}$, called \\emph{models} (playing the role of the spheres in $\\TT_{\\ast}$).\nGiven such a set ${\\mathcal A}$, let \\ $\\Pi_{\\A}$ \\ denote the smallest subcategory of ${\\mathcal C}$\ncontaining ${\\mathcal A}$ and closed under weak equivalences, arbitrary\ncoproducts, and suspensions. Note that every object in \\ $\\Pi_{\\A}$ \\ is a\nhomotopy cogroup object, too.  \n\n\\begin{example}\\label{esmod}\\stepcounter{subsection}\nLet \\ ${\\mathcal C}={\\mathcal G}$ \\ be the category of simplicial groups, \\\n$S^{k}=\\Delta[k]\/\\partial\\Delta[k]$ \\ the standard simplicial $k$-sphere\nin\\ $\\Ss_{\\ast}$, \\ $G:\\Ss_{\\ast}\\to{\\mathcal G}$ \\ the Kan's loop functor (cf.\\ \\cite[\\S 26.3]{MayS}), \nand \\ $F:\\Ss_{\\ast}\\to{\\mathcal G}$ \\ the free group functor. For each \\ $n\\geq 1$,  \\  \n$\\gS{n}:= GS^{n}\\in{\\mathcal G}\\cong FS^{n-1}$ \\ will be called the\n\\textit{$n$-dimensional ${\\mathcal G}$-sphere}, \\ with \\\n$\\Sigma^{k}\\gS{n}\\simeq\\gS{n+k}$. \\  These, and their coproducts, are\ncofibrant strict cogroup objects for ${\\mathcal G}$. \\ Here \\\n${\\mathcal A}:=\\{\\gS{1}=\\co{\\mathbb Z}\\}$; \\ in fact, throughout this paper ${\\mathcal A}$ will\nbe either a singleton, or countable. \n\\end{example}\n\n\\begin{remark}\\label{ssg}\\stepcounter{subsection}\nThe adjoint pairs of functors:\n$$\n\\TT_{\\ast}\\ \\ \\substack{S\\\\ \\rightleftharpoons\\\\ \\|-\\|}\\ \\ \\Ss_{\\ast}\\ \\ \n\\substack{G\\\\ \\rightleftharpoons\\\\ \\bar{W}}\\ \\ {\\mathcal G}\n$$\n\\noindent induce equivalences of the corresponding homotopy categories \\ -- \\ \nwhere \\ $\\bar{W}:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ is the \nEilenberg-Mac~Lane classifying space functor, \\ $S:\\TT_{\\ast}\\to\\Ss_{\\ast}$ \\ is the \nsingular set functor, and \\ $\\|-\\|:\\Ss_{\\ast}\\to\\TT_{\\ast}$ \\ is the geometric realization \nfunctor (cf.\\ \\cite[\\S 14,23]{MayS}). Thus to study the usual homotopy category\nof (pointed connected) topological spaces, we can work in ${\\mathcal G}$ (or \\ $\\Ss_{\\ast}$), \\ \nrather than \\ $\\TT_{\\ast}$. \n\\end{remark}\n\n\\begin{defn}\\label{dpis}\\stepcounter{subsection}\nIf ${\\mathcal A}$ is a set of models for ${\\mathcal C}$, then given \\ $X\\in{\\mathcal C}$, \\  for\neach \\ $A\\in{\\mathcal A}$ \\ let \\ $\\pia{k}(X):=[\\Sigma^{k}A,X']_{{\\mathcal C}}$, \\\nwhere \\ $X'\\to X$ \\ is a (functorial) fibrant replacement. We write \\ \n$\\pinC{k} X$ \\ for \\ $(\\pia{k}X)_{A\\in{\\mathcal A}}$, \\ and \\ \n$\\pinC{\\ast} X:=(\\pinC{k}X)_{k=0}^{\\infty}$.\n\\end{defn}\n\n\\subsection{Theories and algebras}\n\\label{staa}\\stepcounter{thm}\n\nRecall that a \\emph{theory} is a small category  $\\Theta$ with \nfinite products (so in particular, an FP-sketch \\ -- \\ cf.\\ \n\\cite[\\S 5.6]{BorcH2}), and a $\\Theta$-\\emph{algebra} (or\n\\emph{model}) is a product-preserving functor \\ $\\Theta\\to{\\EuScript Set}$. \\ \nThink of $\\Theta$ as encoding the operations and relations for \na ``variety of universal algebras'', the category \\ $\\Alg{\\Theta}$ \\ \nof $\\Theta$-algebras (which is \\emph{sketched} by $\\Theta$).\n\nFor example, the obvious category $\\mathfrak{G}$, which sketches groups, is\nequivalent to the opposite of the homotopy category of (finite) wedges\nof circles. An $\\mathfrak{G}$-\\emph{theory} $\\Theta$\n(cf.\\ \\cite[\\S 2]{BPescF}) is one equipped with a map of theories \\ \n$\\coprod_{S}\\,\\mathfrak{G}\\to\\Theta$ \\ (coproduct taken in the category of\ntheories, over some index set $S$) which is bijective on objects. This\nimplies that each $\\Theta$-algebra has the underlying \nstructure of an $S$-graded group, so that \\ $\\Alg{\\Theta}$ \\  \ncan be thought of as a ``variety of (graded) groups with operators'' \n(cf.\\ \\cite[I, (2.5)]{BauCF}).\n\n\\begin{remark}\\label{rpis}\\stepcounter{subsection}\nWe will assume that all the functors \\ $\\pinC{n}$ \\ ($n\\geq 0$) \\ take\nvalue in a category \\ $\\Alg{\\PiC}$ \\ sketched by a $\\mathfrak{G}$-theory $\\Theta$, \nand thus equipped with a faithful forgetful\nfunctor \\ $U_{{\\mathcal C}}:\\Alg{\\PiC}\\to{\\EuScript Gp}^{{\\mathcal A}}$ \\ into the category of\n${\\mathcal A}$-graded groups. \nThe objects of \\ $\\Alg{\\PiC}$ \\ are called \\ \\emph{$\\PiC$-algebra s}.\n\nFor topological spaces, with \\ ${\\mathcal A}=\\{\\bS{1}\\}$, \\ the $\\PiC$-algebra s are simply\ngroups. If we use rational spheres as the models, then \\ $\\Alg{\\PiC}$ \\ \nis the category of $\\mathbb Q$-vector spaces. A more interesting example\nappears in \\S \\ref{dpa} below. \n\\end{remark}\n\n\\subsection{Constructions based on models}\n\\label{scbm}\\stepcounter{thm}\n\nThere are a number of familiar constructions for topological spaces which we\nrequire for our purposes.  We can \\emph{define} them once we are given\na set of models ${\\mathcal A}$ as above, although they do not always exist (see\n\\S \\ref{snsrmc} below).  \n\n\\begin{defn}\\label{dfpt}\\stepcounter{subsection}\nA \\emph{Postnikov tower} (with respect to ${\\mathcal A}$) is a functor that\nassigns to each \\ $Y\\in{\\mathcal C}$ \\ a tower of fibrations: \n$$\n\\dotsc \\to P^{{\\mathcal A}}_{n}Y\\xra{p\\q{n}}P^{{\\mathcal A}}_{n-1}Y\\xra{p\\q{n-1}}\\dots \n\\to P^{{\\mathcal A}}_{0}Y~,\n$$\nas well as a weak equivalence \\ $r:Y\\to P^{{\\mathcal A}}_{\\infty}Y:=\\lim_{n}P^{{\\mathcal A}}_{n}Y$ \\ \nand fibrations \\ $P^{{\\mathcal A}}_{\\infty}Y\\xra{r\\q{n}}P^{{\\mathcal A}}_{n}Y$ \\ such that \\\n$r\\q{n-1}=p\\q{n}\\circ r\\q{n}$ \\ for all $n$. \\ Finally, \\ \n$(r\\q{n}\\circ r)_{\\#}:\\pinC{k}Y\\to\\pinC{k}(P^{{\\mathcal A}}_{n}Y)$ \\ \nis an isomorphism for \\ $k\\leq n$, \\ and \\ $\\pinC{k}(P^{{\\mathcal A}}_{n}Y)$ \\\nis zero for \\ $k>n$. \\ \n\nWhen ${\\mathcal A}$ is clear from the context, we denote \\ $P_{n}^{{\\mathcal A}}$ \\\nsimply by \\ $P_{n}$.\n\\end{defn}\n\n\\begin{example}\\label{epost}\\stepcounter{subsection}\nFor a free chain complex \\ $C_{\\ast}\\in{\\EuScript Chain}_{R}$ \\ of modules over a\nring $R$, we may take \\ $C'_{\\ast}:=P_{n}C_{\\ast}$ \\ where \\\n$C'_{i}=C_{i}$ \\ for \\ $i\\leq n+1$, \\ $C'_{n+2}=Z_{n+1}C_{\\ast})$, \\ and \\ \n$C'_{i}=0$ \\ for \\ $i\\geq n+3$. \\ The map \\ $r\\q{n}:C_{\\ast}\\to C'_{\\ast}$ \\ \nis defined by \\ $r\\q{n}_{n+2}:=\\partial_{n+2}:C_{n+2}\\to Z_{n+1}C_{\\ast}$. \n\\end{example}\n\n\\begin{defn}\\label{drem}\\stepcounter{subsection}\nGiven an $\\PiC$-algebra\\ \\ $\\Lambda$, \\ a \\emph{classifying object} \\ $B_{\\C}\\Lambda$ \\\n(or simply \\ $B\\Lambda$) \\ for $\\Lambda$ is any \\ $B\\in s{\\mathcal C}$ \\ such that \\\n$B\\simeq P_{0}K$ \\ and \\ $\\pinC{0}B\\cong\\Lambda$.\n\\end{defn}\n\nThe name is used by analogy with the classifying space of a group,\nwhich classifies $G$-bundles. One can interpret \\ $B_{\\C}\\Lambda$ \\ similarly,\nthough perhaps less naturally (see, e.g., \\cite[\\S 4.6]{BJTurR}).\n\n\\begin{defn}\\label{dmod}\\stepcounter{subsection}\nA  \\emph{module over a $\\PiC$-algebra} $\\Lambda$ is an abelian group object \nin \\ $\\Alg{\\PiC}\/\\Lambda$ (cf.\\ \\cite[\\S 2]{QuiC}), and the\ncategory of such is denoted by \\ $\\RM{\\Lambda}$. \n\\end{defn}\n\n\\begin{remark}\\label{rmodule}\\stepcounter{subsection}\nSince any $\\PiC$-algebra\\ is in particular a (graded) group, if \\\n$p:Y\\to\\Lambda$ \\ is a module, then \\ $Y=K\\times\\Lambda$  \\ \n(as sets!) for \\ $K:=\\operatorname{Ker}\\,(p)$, \\ with an appropriate $\\PiC$-algebra\\ structure \n(cf.\\ \\cite[\\S 3]{BlaG}). \\ We may call $K$ itself a\n$\\Lambda$-\\emph{module} (which corresponds to the traditional\ndescription of an $R$-module, for a ring $R$). \n\\end{remark}\n\n\\begin{example}\\label{emodule}\\stepcounter{subsection}\nFor any object \\ $X\\in{\\mathcal C}$ \\ as above, the \\ ${\\mathcal A}\\times\\mathbb N$-graded group \\ \n$\\pinC{\\ast} X$ \\ has an action of the ${\\mathcal A}$-primary homotopy\noperations, corepresented by the maps in \\ $\\operatorname{ho}\\Pi_{\\A}$ \\ (see \n\\S \\ref{dpa} below). In particular, one of these operations,\ncorresponding to the action of the fundamental group on the higher\nhomotopy groups, makes each \\ $\\pinC{n}X$ \\ ($n\\geq 1$) \\ into a\nmodule over $\\pinC{0}X$ \\ (see Fact \\ref{ffour} below).\n\\end{example}\n\n\\begin{defn}\\label{dem}\\stepcounter{subsection}\nGiven an abelian $\\PiC$-algebra\\ $M$ and an integer \\ \n$n\\geq 1$, \\ an \\emph{$n$-dimensional $M$-Eilenberg-Mac~Lane object} \\ \n$\\EC{M}{n}$ \\ (or simply \\ $\\EK{}{}{M}{n}$) \\ is any \\ $E\\in s{\\mathcal C}$ \\ such that \\ \n$\\pinC{n}E\\cong M$ \\ and \\ $\\pinC{k}E=0$ \\ for \\ $k\\neq n$.\n\\end{defn}\n\n\\begin{defn}\\label{deaem}\\stepcounter{subsection}\nGiven a $\\PiC$-algebra\\ $\\Lambda$, \\ a module $M$ over $\\Lambda$, and an integer \\ \n$n\\geq 1$, \\ an \\emph{$n$-dimensional extended $M$-Eilenberg-Mac~Lane object} \\ \n$\\ECL{M}{n}$ \\ (or simply \\ $\\EL{M}{n}$) \\ is any homotopy abelian group object \\ \n$E\\in s{\\mathcal C}\/\\Lambda$, \\ equipped with a section $s$ for \\ \n$p\\q{0}:E\\to P_{0}E\\simeqB\\Lambda$, \\ such that \\ \n$\\pinC{n}E\\cong M$ \\ as modules over $\\Lambda$; \\ and \\ $\\pinC{k}E=0$ \\ \nfor \\ $k\\neq 0,n$.\n\\end{defn}\n\n\\begin{defn}\\label{dki}\\stepcounter{subsection}\nGiven a Postnikov tower functor as in \\S \\ref{dfpt}, an $n$-th \n\\emph{$k$-invariant square} (with respect to ${\\mathcal A}$)  \nis a functor that assigns to each \\ $Y\\in{\\mathcal C}$ \\ a homotopy pull-back square:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ezero}\n\\xymatrix@R=25pt{\\ar @{} [dr] |<<<{\\framebox{\\scriptsize{PB}}}\nP^{{\\mathcal A}}_{n+1}Y \\ar[r]^{p\\q{n+1}} \\ar[d] &\n  P^{{\\mathcal A}}_{n}Y \\ar[d]^{k_{n}}\\\\ B\\Lambda \\ar[r] & \\EL{M}{n+2}\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent for \\ $\\Lambda:=\\pinC{0}Y$ \\ and \\ $M:=\\pinC{n+1}Y$, \\ where\n\\ $p\\q{n+1}:P_{n+1}Y\\to P_{n}Y$ \\ is the given fibration of the\nPostnikov tower. \n\nThe map \\ $k_{n}:P_{n}Y\\to\\EL{M}{n+2}$ \\ is the $n$-th\n(functorial) $k$-\\emph{invariant} for $Y$. \n\\end{defn}\n\n\\begin{example}\\label{ekinv}\\stepcounter{subsection}\nIf \\ $C_{\\ast}$ \\ is a chain complex of $R$-modules, and \\ \n$P_{n}C_{\\ast}=C'_{\\ast}$ \\ as in \\S \\ref{epost}, we may take \\\n$\\EK{}{}{H_{n+1}C_{\\ast}}{n+2}=E_{\\ast}$, \\ where \\ $E_{i}=0$ \\ for \\\n$i<n+2$, \\ $E_{n+2}=Z_{n+1}C_{\\ast}$, \\ and \\  \n$E_{n+3}=B_{n+1}C_{\\ast}$. \\ Then \\ $k_{n}:C'_{\\ast}\\to E_{\\ast}$ \\ is\ndefined by \\ $\\operatorname{Id}:C'_{n+1}\\to E_{n+1}$.\n\nOf course, if $R$ is a principle ideal domain (or a hereditary ring),\nsuch as $\\mathbb Z$, then the $k$-invariants for \\ $C_{\\ast}$ \\ are trivial,\nsince in that case any two free (or projective) chain complexes with\nthe same homology are homotopy equivalent, by \\cite[Prop.\\ 3.5]{DolH}. \nBut this need not hold for an arbitrary ring $R$.\n\\end{example}\n\n\\subsection{Spherical models}\n\\label{dsmod}\\stepcounter{thm}\n\nA set of objects \\ ${\\mathcal A}:=\\{A\\}_{A\\in{\\mathcal A}}$ \\ in a model category\n${\\mathcal C}$ is called a collection of \\emph{spherical models} if the\nfollowing axioms hold\\vspace{2 mm} : \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{Ax~\\arabic{enumi}.~}\n\\item Each \\ $\\Sigma^{n}A$ \\ ($A\\in{\\mathcal A}$, \\ $n\\in\\mathbb N$) \\ is a\n  cofibrant homotopy cogroup object in ${\\mathcal C}$\\vspace{2 mm}.\n\\item For any \\ $X\\in{\\mathcal C}$ \\ and \\ $n\\geq 1$, \\ $\\pinC{n}X$ \\ has a\n  natural structure of a module over \\ $\\pinC{0}X$\\vspace{2 mm}.\n\\item A map \\ $f:X\\to Y$ is a weak equivalence if and only if \\ $\\pia{n}f$ \\ is a \nweak equivalence for each \\ $A\\in{\\mathcal A}$ \\ and \\ $n\\in\\mathbb N$\\vspace{2 mm}.\n\\item ${\\mathcal C}$ has Postnikov towers with respect to ${\\mathcal A}$\\vspace{2 mm}.\n\\item For every $\\PiC$-algebra\\ $\\Lambda$ \\ and module $M$ over $\\Lambda$, the\n  classifying object \\ $B\\Lambda$ \\ and extended $M$-Eilenberg-Mac~Lane\n  object \\ $\\EL{M}{n}$ \\ exist (and are unique up to homotopy) for\n  each \\ $n\\geq 1$\\vspace{2 mm}.\n\\item ${\\mathcal C}$ has $k$-invariant squares with respect to ${\\mathcal A}$ for\n  each \\ $n\\geq 0$\\vspace{2 mm}. \n\\end{enumerate}\n\nIf each model \\ $\\Sigma^{k}A$ \\ ($A\\in{\\mathcal A}$, \\ $k\\in\\mathbb N$) \\ is a \ncofibrant \\emph{strict} cogroup object \\ -- \\ which implies that \nevery object in \\ $\\Pi_{\\A}$ \\ is such, up to weak equivalence \\ -- \\ we\ncall ${\\mathcal A}$ a collection of \\emph{strict} spherical models.\n\nA pointed simplicial model category \\ ${\\mathcal C}$ equipped with\na collection \\ ${\\mathcal A}:=\\{A\\}_{A\\in{\\mathcal A}}$ \\ of spherical models\nis called a \\emph{spherical model category}, \\ and we denote it\nby \\ $\\lra{{\\mathcal C};{\\mathcal A}}$. \\ Such a category is \\emph{stratified} in the\nsense of Spali\\'{n}ski (cf.\\ \\cite{SpalSM}). \n\n\\begin{example}\\label{esmc}\\stepcounter{subsection}\nThe category \\ $\\Ss_{\\ast}$ \\ of pointed simplicial sets, as well as the\ncategory \\ $\\TT_{\\ast}$ \\ of pointed connected topological spaces, have\nspherical model category structures with \\ ${\\mathcal A}=\\{\\bS{1}\\}$. \\ \n(Functorial $k$-invariants in these categories are provided by the\nconstruction of \\cite[\\S 5]{BDGoeR}; \\ in both cases \\\n$\\Alg{\\PiC}\\approx{\\EuScript Gp}$).  \\ Similarly for the category \\ ${\\EuScript Chain}_{R}$ \\ of\nchain complexes over $R$, with the constructions indicated in \\S \\ref{epost} and\n\\S \\ref{ekinv}.\n\\end{example}\n\nIn the examples we have in mind, our model categories enjoy\nadditional useful properties, which we can summarize in the following:\n\n\\begin{defn}\\label{dssmc}\\stepcounter{subsection}\nA spherical model category \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ as above is called\n\\emph{strict} if the following axioms hold\\vspace{2 mm} :  \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{Ax~\\arabic{enumi}.~}\n\\item ${\\mathcal C}$ \\ is a pointed right-proper cofibrantly generated simplicial \nmodel category (cf.\\ \\cite[11.1, 13.1]{PHirM}), in which every object is fibrant.\n\\item ${\\mathcal C}$ is equipped with a faithful forgetful functor \\\n$\\hat{U}:{\\mathcal C}\\to{\\mathcal D}$, \\ with left adjoint $\\hat{F}$ \\ -- \\ where \\ ${\\mathcal D}$\n  is one of the ``categories of groups'' \\ ${\\mathcal D}={\\EuScript Gp}$, \\ $gr{\\EuScript Gp}$, \\\n  ${\\mathcal G}$, \\ $\\RM{R}$, \\ or \\ $s\\RM{R}$, \\ for some ring $R$.\n\\item The adjoint pair \\ $(\\hat{U},\\hat{F})$ \\ \\emph{create} the \nmodel category structure on ${\\mathcal C}$ in the sense of \\cite[\\S 4.13]{BlaN} \\ -- \\  \nso in particular $\\hat{U}$ creates all limits in ${\\mathcal C}$.\n\\item ${\\mathcal A}$ is a collection of strict spherical models, each of which lies\nin the image of the composite \\ $\\hat{F}\\circ F':{\\mathcal S}\\to {\\mathcal C}$, \\ \nwhere \\ $F':{\\mathcal S}\\to{\\mathcal D}$ \\ is adjoint to the forgetful functor \\\n$U':{\\mathcal D}\\to{\\mathcal S}$, \\ with the group structure on \\ $\\operatorname{Hom}_{{\\mathcal C}}(A,X)$ \\\ninduced from that of \\ $\\hat{U}(X)$.\n\\end{enumerate}\n\\end{defn}\n\n\\setcounter{thm}{0}\\section{Resolution model categories}\n\\label{crmc}\n\nMany examples of spherical model categories fit into the\nframework originally conceived by Dwyer, Kan and Stover in\n\\cite{DKStB} under the name of ``$E^{2}$ model categories,'' and later \ngeneralized by Bousfield (see \\cite{BousCR,JardBE}. A slightly\ndifferent generalization is given by Baues in \n\\cite[Ch.~D, \\S 2]{BauCF} under the name of \\emph{spiral} model categories.\n\nFirst, some preliminary concepts:  \n \n\\begin{defn}\\label{dmat}\\stepcounter{subsection}\nThe $n$-th \\emph{matching object} for a simplicial object $X$ over\n${\\mathcal C}$ is defined by\n$$\nM_{n}X=\\{(x_{0},\\dotsc,x_{n})\\in(X_{n-1})^{n+1}~|\\ \nd_{i}x_{j}=d_{j-1}x_{i} \\text{\\ \\ for all\\ }0\\leq i<j\\leq n\\}\n$$\n(see \\cite[X,\\S 4.5]{BKaH}). Note that each face map \\\n$d_{k}:X_{n}\\to X_{n-1}$ \\ factors through the obvious map \\ \n$\\delta_{n}:X_{n}\\to M_{n}X$.\n\\end{defn}\n\n\\begin{defn}\\label{dlat}\\stepcounter{subsection}\nThe $n$-th \\emph{latching object} of a simplicial object $X$ over ${\\mathcal C}$\nis defined \\ $L_{n}X:=\\coprod_{0\\leq i\\leq n-1} X_{n-1}\/\\sim$, \\ where for any \\ \n$x\\in X_{n-k-1}$ \\ and \\ $0\\leq i\\leq j \\leq n-1$ \\ we set \\ \n$s_{j_{1}}s_{j_{2}}\\dotsc s_{j_{k}}x$ \\ in the $i$-th copy of \\ $X_{n-1}$ \\ \nequivalent to \\ $s_{i_{1}}s_{i_{2}}\\dotsc s_{i_{k}}x$ \\ in the $j$-th copy \nof \\ $X_{n-1}$ \\ whenever the simplicial identity \\ \n$s_{i}s_{j_{1}}s_{j_{2}}\\dotsc s_{j_{k}}=s_{j}s_{i_{1}}s_{i_{2}}\\dotsc\ns_{i_{k}}$ \\ holds. The map \\ $\\sigma_{n}:L_{n}X\\to X_{n}$ \\ is defined\n\\ $\\sigma_{n}(x)_{i}=s_{i}x$, \\ where \\ $(x)_{i}\\in (X_{n-1})_{i}$.\n\\end{defn}\n\nThere are two canonical ways to extend a given model category structure on \n$\\hat{\\C}$ to \\ ${\\mathcal C}:=s\\hat{\\C}$:\n\n\\subsection{The Reedy model structure}\n\\label{sreed}\\stepcounter{thm}\n\nThis is defined by letting a simplicial map \\ $f:X\\to Y$ \\ in \\\n${\\mathcal C}:=s\\hat{\\C}$ \\ be:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\roman{enumi})}\n\\item a weak equivalence if \\ $f_{n}:X_{n}\\to Y_{n}$ \\ is a weak equivalence \nin $\\hat{\\C}$ \\ for each \\ $n\\geq 0$;\n\\item a (trivial) cofibration if \\ \n$f_{n}\\amalg \\sigma_{n}:X_{n}\\amalg_{L_{n}X}L_{n}Y\\to Y_{n}$ \\ is a \n(trivial) cofibration in $\\hat{\\C}$ \\ for each \\ $n\\geq 0$;\n\\item a (trivial) fibration if \\ \n$f_{n}\\times \\delta_{n}:X_{n}\\to Y_{n}\\times_{M_{n}Y}M_{n}X$ \\ is a \n(trivial) fibration in $\\hat{\\C}$ \\ for each \\ $n\\geq 0$.\n\\end{enumerate}\n\nSee \\cite[15.3]{PHirM}.\n\n\\subsection{The resolution model category}\n\\label{srmc}\\stepcounter{thm}\n\nLet $\\hat{\\C}$ be a pointed cofibrantly generated right proper model category\n(in our cases, every object will be fibrant, though this is not needed\nin general  \\ -- \\ cf.\\ \\cite{JardBE}). Given a \nset $\\hat{\\A}$ of models for $\\hat{\\C}$ \\ (\\S \\ref{ssmod}), we let \\ \n${\\mathcal A}:=\\{\\co{\\Sigma^{k}\\hat{A}}\\}_{k\\in\\mathbb N,\\hat{A}\\in\\hat{\\A}}$ \\ \n(the constant simplicial objects on \\ $\\Sigma^{k}\\hat{A}\\in\\hat{\\C}$) \\ be\nthe set of models for ${\\mathcal C}$. \\ Note that \\ \n$\\Sigma^{n}\\co{\\Sigma^{k}\\hat{A}}:=\\co{\\Sigma^{k}\\hat{A}}\\hat{\\otimes} S^{n}$ \\ \n(\\S \\ref{ssmod}), \\ so we shall generally reserve the notation \\\n$\\Sigma^{k}$ \\ for (internal) suspension in $\\hat{\\C}$, and \\\n$-\\otimes S^{n}$ \\ for the (simplicial) suspension in \\ ${\\mathcal C}=s\\hat{\\C}$. \n\n\\begin{remark}\\label{rmods}\\stepcounter{subsection}\nIf we do not assume that each \\ $\\hat{A}\\in\\hat{\\A}$ \\ is a homotopy\ncogroup object in $\\hat{\\C}$, we take \\\n${\\mathcal A}:=\\{(\\Sigma^{k}\\hat{A})\\otimes S^{1}\\}_{A\\in{\\mathcal A}}$ \\  as our\ncollection of models for ${\\mathcal C}$.  \n\\end{remark}\n\n\\begin{defn}\\label{dffm}\\stepcounter{subsection}\nA map \\ $f:V\\to Y$ \\ in  \\ ${\\mathcal C}=s\\hat{\\C}$ \\ is called \\emph{homotopically\n$\\hat{\\A}$-free} if for each \\ $n\\geq 0$, \\ there is \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item a cofibrant object \\ $W_{n}$ \\ in \\ $\\Pi_{\\hat{\\A}}\\subset\\hat{\\C}$, \\ and\n\\item a map \\ $\\varphi_{n}:W_{n}\\to Y_{n}$ \\ in ${\\mathcal C}$ inducing a trivial \ncofibration \\ $(V_{n}\\amalg_{L_{n}V}L_{n}Y)\\amalg W_{n}\\to Y_{n}$.\n\\end{enumerate}\n\\end{defn}\n\nWe define the \\emph{resolution model category structure} on \\ $s\\hat{\\C}$ \\ \ndetermined by $\\hat{\\A}$, by letting a simplicial map \\ $f:X\\to Y$ \\ be:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\roman{enumi})}\n\\item a \\emph{weak equivalence} if \\ $\\pia{n}f$ \\ is a weak equivalence \nof simplicial groups for each \\ $A\\in{\\mathcal A}$ \\ and \\ $n\\geq 0$.\n\\item a \\emph{cofibration} if it is a retract of a homotopically $\\hat{\\A}$-free map;\n\\item a \\emph{fibration} if it is a Reedy fibration (\\S \\ref{sreed}(iii)) \nand \\ $\\pia{n}f$ \\ is a fibration of simplicial groups for each \\ \n$A\\in{\\mathcal A}$ \\ and \\ $n\\geq 0$\n\\end{enumerate}\n\n\\begin{defn}\\label{dnc}\\stepcounter{subsection}\nGiven a fibrant \\ $X\\in s\\hat{\\C}$, \\ define its $n$-\\emph{cycles} object \\ \n$Z_{n}X$ \\ to be \\ $\\{ x\\in X_{n}\\,|\\ d_{i}x=0 \\ \\text{for}\\\ni=0,\\dotsc,n\\}$ \\ (the fiber of \\ $\\delta_{n}:X_{n}\\to M_{n}X$ \\ of \n\\S \\ref{dmat}). \\ Similarly, the $n$-\\emph{chains} object for $X$ is \\ \n$C_{n}X=\\{ x\\in X_{n}\\,|\\ d_{i}x=0 \\ \\text{for}\\ i=1,\\dotsc,n\\}$.\n\\end{defn}\n\nIf $X$ is fibrant, the map \\ \n$\\mathbf{d}_{0}=\\mathbf{d}_{0}^{n}:=d_{0}\\rest{C_{n}X}:C_{n}X\\to Z_{n-1}X$ \\ fits into a \nfibration sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwo}\n\\dotsb \\Omega Z_{n}X\\to Z_{n+1}X\\xra{j^{X}_{n+1}}C_{n+1}X \\xra{\\mathbf{d}_{0}^{n+1}} Z_{n}X\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent (see \\cite[Prop.\\ 5.7]{DKStB}). \n\n\\begin{defn}\\label{dpa}\\stepcounter{subsection}\nA \\emph{$\\PiA$-algebra} \\ is a product-preserving functor from \\\n$(\\operatorname{ho}\\Pi_{\\A})^{\\operatorname{op}}$ \\ to sets. The category of $\\PiA$-algebra s is denoted by \\ $\\Alg{\\PiA}$.\n\nEquivalently, we can think of an $\\PiA$-algebra\\ $\\Lambda$ as an \\ \n$\\mathbb N\\times{\\mathcal A}$-graded group equipped with an action of the ${\\mathcal A}$-primary\nhomotopy operations (corepresented by the maps in \\ $\\operatorname{ho}\\Pi_{\\A}$). \n\nThus we can think of the functor \\ $\\pinC{\\ast}$ \\ as taking value in \\ $\\Alg{\\PiA}$.\nThis explains the additional $\\PiC$-algebra\\ structure on the ${\\mathcal A}$-graded groups \\ \n$\\pinC{n}X$, \\ mentioned in \\S \\ref{rpis}: when \\ ${\\mathcal C}=s\\hat{\\C}$, \\ we\nhave \\ $\\Alg{\\PiC}:=\\Alg{\\Pi_{\\hA}}$.\n\\end{defn}\n\n\\begin{example}\\label{epa}\\stepcounter{subsection}\nWhen \\ ${\\mathcal C}={\\mathcal G}$, \\ and \\ ${\\mathcal A}=\\{\\gS{1}\\}$ \\ -- \\ so \\ $\\Pi_{\\A}$ \\ is the\ncategory of wedges of ${\\mathcal G}$-spheres (\\S \\ref{esmod}) \\ -- \\  then (up \nto indexing) \\ $\\Alg{\\PiA}$ \\ is the usual category of $\\Pi$-algebra s (see\n\\cite[\\S 2]{StoV}): \\ graded groups equipped with an action of the\nprimary homotopy operations (Whitehead products and compositions). \n\\end{example}\n\n\\subsection{Examples of resolution model categories}\n\\label{ermc}\\stepcounter{thm}\n\nIn this paper we shall be interested mainly in the following instances\nof resolution model categories:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item Let \\ $\\hat{\\C}={\\EuScript Gp}$ \\ with the \\emph{trivial} model category structure: \\ \ni.e., only isomorphisms are weak equivalences, and every map is both a fibration\nand a cofibration. Let \\ $\\hat{\\A}=\\{\\mathbb Z\\}$ \\ consist of the free cyclic group \n(whose coproducts are the cogroup objects in \\ ${\\EuScript Gp}$). \\ \nThe resulting resolution model category\nstructure on \\ ${\\mathcal G}:= s\\hat{\\C}$ \\ is the usual one (cf.\\ \\cite[II,\\S 3]{QuiH}. \\ \nHere \\ $\\Alg{\\PiC}\\approx {\\EuScript Gp}$ \\ -- \\ there is no extra structure on the\nindividual homotopy groups of a simplicial group.\n\nNote that if we tried to do the same for \\ $\\hat{\\C}={\\EuScript Set}$, \\ there are no \nnontrivial cogroup objects, while in \\ ${\\mathcal S}$ \\ not all objects are fibrant.\nNote also that the category \\ $\\TT_{\\ast}$ \\ of pointed topological spaces, \nwhich is one of the main examples we have in mind, has a spherical\nmodel category structure which is not strict (\\S \\ref{dssmc}). This\nexplains the significance of Remark \\ref{ssg} in our context\\vspace{2 mm}.\n\n\\item The previous example extends to any category $\\hat{\\C}$ of (possibly graded) \nuniversal algebras with an underlying group structure \\ -- \\ such as\nrings, $R$-modules,  associative algebras, Lie algebras, and so on \\ -- \\ \nso that ${\\mathcal C}$ is corepresented by a $\\mathfrak{G}$-theory $\\Theta$, in the\nlanguage of \\cite[\\S 4]{BPescF}. Here ${\\mathcal A}$ consists\nof free monogenic algebras (one for each isomorphism class), and\nthus once more \\ $\\Alg{\\PiC}\\approx{\\mathcal C}$. \n\\item We can iterate the process by taking ${\\mathcal G}$ for $\\hat{\\C}$, and letting \\ \n$\\hat{\\A}:=\\{\\gS{n}\\}_{n=1}^{\\infty}$ \\ (\\S \\ref{esmod}). We thus obtain\na resolution model category structure on \\ $s{\\mathcal G}$ \\ (or equivalently, on\nthe category of simplicial spaces). \n\nIn this case the homotopy groups \\ $\\pi^{s{\\mathcal G}}_{k,n}X$, \\ denoted\nbriefly by \\ $\\pi^{\\natural}_{n}X$, \\ are the ``bigraded groups'' of\n\\cite{DKStB}, and Proposition 5.8 there shows that, for a fibrant\nsimplicial space \\ $X\\in s{\\mathcal G}$, \\ we have \\ \n$\\pia{n}{X}\\cong\\pi_{0}\\,\\operatorname{map}(A\\otimes S^{n},X)$\\vspace{2 mm}.\n\\item If ${\\mathcal C}$ is a resolution model category and $I$ is some small\ncategory, the category \\ ${\\mathcal C}^{I}$ \\ of $I$-diagrams in ${\\mathcal C}$ also has\na resolution model category structure, in which the models consists of\nall free $I$-diagrams \\ $F[A,i]$ \\ for $i\\in\\operatorname{Obj} I$ \\ and \\ \n$A\\in{\\mathcal A}$, \\ where \\ $F[A,i](j):=\\coprod_{\\operatorname{Hom}_{I}(i,j)}~A$. \\ \nSee \\cite[\\S 1]{BJTurR}).\n\\end{enumerate}\n\n\\begin{remark}\\label{rbgg}\\stepcounter{subsection}\nIn all these examples, if \\ $Y\\in {\\mathcal C}=s\\hat{\\C}$, \\ is fibrant, then for\neach \\ $n\\geq 0$ \\ we have an exact sequence: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ethree}\n\\pi_{\\ast}^{\\hat{\\C}}C_{n+1}X~\\xra{(\\mathbf{d}_{0}^{n+1})_{\\#}}~\n\\pi_{\\ast}^{\\hat{\\C}}Z_{n}X~\\xra{\\hat{\\vartheta}_{n}}~ \n\\pinC{n}{Y}\\to 0.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{remark}\n\n\\setcounter{thm}{0}\\section{Constructions in resolution model categories}\n\\label{csp}\n\nNot all spherical model categories are resolution model categories \n(see \\S \\ref{esmc}), but all known examples appear to be Quillen\nequivalent to such. Conversely, the examples of resolution model categories \\ \n$\\lra{{\\mathcal C}=s\\hat{\\C};{\\mathcal A}}$ \\ we are interested in are spherical (though this\ndoes not hold in general \\ -- \\ see \\S \\ref{snsrmc} below). We briefly\nindicate why this is so. \n\n\\subsection{Postnikov sections}\n\\label{sps}\\stepcounter{thm}\n\nGiven \\ $Y\\in s\\hat{\\C}$, for each \\ $n\\geq 0$ \\ define \\ $Y\\q{n}\\in s\\hat{\\C}$ \\ \nby setting \\ $Y\\q{n}_{k}:= Y_{k}$ \\ for \\ $k\\leq n+1$ \\ and \\ \n$Y\\q{n}_{k}:= M_{k}(Y\\q{n})$ \\ (\\S \\ref{dmat}) for \\ $k\\geq n+2$. \\ \nNote that for any \\ $X\\in s{\\mathcal C}$, \\ $M_{k}X$ \\ depends only on $X$\nthrough dimension \\ $(k-1)$, \\ so this definition is valid inductively.\nDenote the obvious maps by \\ $r\\q{n}:Y\\to Y\\q{n}$ \\ and \\ \n$p\\q{n}:Y\\q{n+1}\\to Y\\q{n}$ \\ (see \\cite[\\S 1.2]{DKanO}).\n\nNow for any \\ $X\\in s\\hat{\\C}$, \\ choose a functorial fibrant replacement\n$Y$, and set \\ $P_{n}X := Y\\q{n}$, \\ with \\ \n$\\varphi\\q{n}:X\\to P_{n}X$ \\ defined to be the composite of \\ $r\\q{n}$ \\ \nwith the trivial cofibration \\ $i:X\\to Y$, \\ and \\ \n$p\\q{n}:P_{n+1}X\\to P_{n}X$ \\ defined as above. \n\n\\begin{remark}\\label{rcsk}\\stepcounter{subsection}\nThe functor \\ $-\\q{n}:{\\mathcal C}\\to{\\mathcal C}$ \\ is right adjoint to the \\\n$(n+1)$-skeleton functor \\ $\\sk{n+1}$, \\ so \\ $P_{n}X$ \\ depends only\non \\ $\\sk{n+1}X$, \\ even if $X$ is not fibrant. If $X$ is fibrant, we can find \\ \n$Y\\simeq P_{n}X$ \\ with \\ $\\sk{n+1}Y=\\sk{n+1}X$.\n\\end{remark}\n\n\\begin{fact}\\label{fone}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), the tower:\n$$\nX\\to\\dotsc \\to P_{n+1}X\\xra{p\\q{n}} P_{n}X\\to\\dotsc \\to P_{0}X\n$$\nis a functorial Postnikov tower for \\ ${\\mathcal C}=s\\hat{\\C}$ \\ with respect to ${\\mathcal A}$ \\ \n(\\S \\ref{dfpt}).  \n\\end{fact}\n\n\\begin{proof}\nFrom \\S \\ref{sreed} and \\S \\ref{sps} it follows that if  \\ $Y\\in\ns\\hat{\\C}$ \\ is fibrant, then \\ so is each \\ $Y\\q{n}$, \\ and  \nfor each $n$, \\ $Y\\q{n+1}\\to Y\\q{n}$ \\ is a fibration, \\ \n$Z_{k}Y\\q{n}=0$ \\ and \\ $C_{k}Y\\q{n}\\xra{\\mathbf{d}_{0}}Z_{k-1}Y$ \\ is an\nisomorphism for \\ $k\\geq n+2$. \\ The claim then follows from \\eqref{ethree}.\n\\end{proof}\n\n\\begin{fact}\\label{ftwo}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ there is a\nclassifying object \\ $B\\Lambda$ \\ for any $\\PiA$-algebra\\ \\ $\\Lambda$, \\ and it is\nunique up to homotopy. \n\\end{fact}\n\n\\begin{proof}\nIn the algebraic cases of \\S \\ref{ermc}(a-b), \\ we may take \\ $B\\Lambda$ \\\nto be (a cofibrant model for) the constant simplicial object on\n$\\Lambda$. For simplicial spaces, \\ $B\\Lambda$ \\ may be constructed as for\ntopological spaces, using generators and relations (see \\cite[\\S 8.9]{BDGoeR}). \nThe extension to the diagram case of \\S \\ref{ermc}(d) is objectwise.\n\\end{proof}\n\n\\begin{fact}\\label{fthree}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $n\\geq 1$ \\ \nthere is an $n$-dimensional $M$-Eilenberg-Mac~Lane object \\ \n$\\EK{}{}{M}{n}$ \\ for any abelian $\\PiA$-algebra\\ $M$, \\ and there is an\n$n$-dimensional extended $M$-Eilenberg-Mac~Lane object \\ \n$\\EL{M}{n}$ \\ for any $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over\n$\\Lambda$. \\ Each of these is unique up to homotopy. \n\\end{fact}\n\n\\begin{proof}\nIn the algebraic cases of \\S \\ref{ermc}(a-b), \\ we may take \\\n$\\EK{}{}{M}{n}$ \\ to be the iterated Eilenberg-Mac Lane construction\n$\\bar{W}$ on \\ $BM$, \\ while \\ $\\EL{M}{n}$ \\ is a semi-direct product \\ \n$\\EK{}{}{M}{n}\\ltimesB\\Lambda$ (see \\cite[Prop.\\ 2.2]{BDGoeR}).\nFor simplicial spaces, use the explicit construction of \\cite[\\S 8.9]{BDGoeR} \\\nThe extension to the diagram case is again objectwise.\n\\end{proof}\n\n\\begin{fact}\\label{ffour}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $n\\geq 1$ \\ \nand \\ $X\\in{\\mathcal C}$, \\ $\\pinC{n}X$ \\ has a natural structure of a \nmodule over \\ $\\pinC{0}X$.\n\\end{fact}\n\n\\begin{proof}\nNote that by \\cite[II,1,(6)]{QuiH} we have \\ \n$\\operatorname{map}(A\\otimes S^{n},X)\\cong\\operatorname{map}_{{\\mathcal S}}(S^{n},\\operatorname{map}(A,X))$ \\ \n(unpointed maps), \\ so \\ $\\pinC{n}X\\to\\pinC{0}X$ \\ associates to each \\ \n$f:A\\otimes S^{n}\\to X$ \\ its component in \\ $\\operatorname{map}(A,X)$. \\ This defines \nan \\emph{abelian} algebra over \\ $\\pinC{0}X$ \\ by \\cite[Prop.\\ 6.26]{BPescF}).\n\\end{proof}\n\n\\begin{fact}\\label{ffive}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $X\\in s{\\mathcal C}$, \\ \n$\\Lambda:=\\pinC{0}X$ \\ and \\ $n\\geq 1$, \\ the commutative square\nobtained by applying the functor \\ $P_{n+2}$ \\ to the pushout diagram:\n$$\n\\xymatrix@C=40pt{\\ar @{} [dr] |>>>>>{\\framebox{\\scriptsize{PO}}}\nP_{n+1}X  \\ar[r]^-{p\\q{n+1}} \n\\ar[d] & P_{n}X \\ar[d]^{k_{n}}\\\\ B\\Lambda \\ar[r] & Y\n}\n$$\nis an $n$-th $k$-invariant square (Def.\\ \\ref{dki}) \\ -- \\ that is, \\ \n$P_{n+2}Y\\simeq\\EL{\\pia{n+1}X}{n+2}$.\n\\end{fact}\n\n\\begin{proof}\nSee \\cite[\\S 5]{BDGoeR}.\n\\end{proof}\n\nWe may summarize these facts in the following:\n\n\\begin{thm}\\label{tone}\\stepcounter{subsection}\nThe following resolution model categories (cf.\\ \\S \\ref{ermc})\nare strict spherical model categories:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item The category \\ ${\\mathcal C}=s\\hy{\\Theta}{\\Set_{\\ast}}$ \\ of simplicial\n  $\\Theta$-algebras for any $\\mathfrak{G}$-theory $\\Theta$,\n  with $\\hat{\\A}$ consisting of monogenic free $\\Theta$-algebras;\n\\item In particular, the category \\ ${\\mathcal C}={\\mathcal G}$ \\ of simplicial groups, with \\\n  $\\hat{\\A}=\\{\\mathbb Z\\}$; \n\\item The category \\ $s{\\mathcal G}$ \\ of bisimplicial groups (``simplicial\n  spaces''), with \\ $\\hat{\\A}=\\{\\gS{1}\\otimes S^{k}\\}_{k=0}^{\\infty}$.\n\\item The category \\ ${\\mathcal C}^{I}$ \\ of $I$-diagrams in a strict spherical\n  model category ${\\mathcal C}$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{thm}\\label{ttwo}\\stepcounter{subsection}\nThe following are spherical model categories (which are not strict):\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item The category \\ $\\Ss_{\\ast}$ \\ of pointed simplicial sets, with \\ ${\\mathcal A}=\\{S^{1}\\}$; \n\\item The category \\ $\\TT_{\\ast}$ \\ of pointed topological spaces, with \\ \n      ${\\mathcal A}=\\{\\bS{1}\\}$; \n\\item The category \\ $s\\TT_{\\ast}$ \\ of simplicial pointed topological spaces, with \\ \n      $\\hat{\\A}=\\{\\bS{1}\\otimes S^{k}\\}_{k=1}^{\\infty}$.\n\\end{enumerate}\n\\end{thm}\n\n\\subsection{Non-spherical model categories}\n\\label{snsrmc}\\stepcounter{thm}\n\nConsider the trivial model category structure on \\ $\\hat{\\C}={\\EuScript Gp}$, \\ with \\ \n$\\hat{\\A}:=\\{A=\\mathbb Z\/p\\}$ \\ (for $p$ an odd prime). \\ This defines a\nresolution model category structure on ${\\mathcal G}$ \\ -- \\ or equivalently, on \\ \n$\\TT_{\\ast}$ \\ (see Remark \\ref{rmods}). \\ Note that \\ $-\\otimes S^{n}$ \\ \ncorresponds to suspension of simplicial sets, not\nsimplicial abelian group, so the model \\ $A\\otimes S^{n}\\in{\\mathcal G}$ \\\ncorresponds to the $n$-dimensional mod $p$ Moore space \\ \n$\\bS{n-1}\\cup_{p}\\be{n}$. \n\nThus \\ $\\pia{k}X:=[A\\otimes S^{k},X]$ \\ is by definition the $k$-th\n\\emph{mod $p$ homotopy group} of $X$ \\ -- \\  denoted by \\ $\\pi_{k}(X;\\mathbb Z\/p)$ \\ \nin \\cite[Def.\\ 1.2]{NeiP} \\ -- \\  which fits into a short exact sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efour}\n0\\to\\pi_{k}X\\otimes\\mathbb Z\/p \\to\n        \\pi_{k}(X;\\mathbb Z\/p)\\to\\operatorname{Tor}^{\\mathbb Z}_{1}(\\pi_{k-1}X,\\mathbb Z\/p)\\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent for \\ $k\\geq 2$ \\ (see \\cite[Prop.\\ 1.4]{NeiP}). \\ In particular, \nfor \\ $Y:=A\\otimes S^{n}$ \\ ($n\\geq 4$) \\ we have\n$$\n\\pi_{i}(Y;\\mathbb Z\/p)=\\begin{cases}\\mathbb Z\/p & \\text{for \\ }i=n-1,n,\\\\\n             0     & \\text{for \\ } 2\\leq i<n-1 \\text{\\ or \\ } i=n+1,\n\\end{cases}\n$$\n\\noindent with the two non-trivial groups connected by a Bockstein \n(cf.\\ \\cite[\\S 1]{NeiP})\\vspace{2 mm}.\n\nHowever, the resolution model category structure on ${\\mathcal G}$ determined by ${\\mathcal A}$ is \nnot spherical: \\ if it were, in particular there would be Postnikov functors \\\n$P_{k}=P_{k}^{{\\mathcal A}}$ \\ for all \\ $k\\geq 1$ \\ (Def.\\ \\ref{dfpt}). \\ \nFrom \\eqref{efour} we see that, disregarding torsion prime to $p$,  \nbecause of the Bockstein we must have \\ \n$P_{n-1}Y\\simeq E(\\mathbb Z,n-1)$ \\ and \\ $P_{n}Y\\simeq E(\\mathbb Z\/p,n-1)$ \\ \n(for \\ $Y=S^{n-1}\\cup_{p} e^{n}$). \\ But then there is no\nnon-trivial map \\ $P_{n}Y\\to P_{n-1}Y$. \n\n\\subsection{Cohomology in spherical model categories}\n\\label{scpa}\\stepcounter{thm}\n\nNote that the  $k$-invariants of a simplicial object\nactually take value in cohomology groups, as expected:\n\n\\begin{prop}\\label{ptwo}\\stepcounter{subsection}\nFor each $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over $\\Lambda$, the functors \\\n$D^{n}:{\\mathcal C}\/B\\Lambda\\to{\\Ab{\\EuScript Gp}}$ \\ \\textup{($n>0$)}, \\ defined \\ \n$D^{n}(X):=[X,\\EL{M}{n}]_{B\\Lambda}$, \\ \nare \\emph{cohomology functors} on ${\\mathcal C}$ \\ -- \\ that is, they \nare homotopy invariant, take arbitrary coproducts to products, vanish \non the spherical models \\ $\\Sigma^{n}A$, \\ except in degree $n$, \nand have Mayer-Vietoris sequences for homotopy pushouts.\n\\end{prop}\n\nWe therefore denote \\ $[X,\\EL{M}{n}]_{B\\Lambda}$ \\ by \\ $\\HL{n}{X}{M}$.\n\n\\begin{proof}\nSee \\cite[Thm.\\ 7.14]{BPescF}.\n\\end{proof}\n\n\nFact \\ref{fthree} then follows from Brown Representability, since \\\n$\\EL{M}{n}$ \\ represents the $n$-th Andr\\'{e}-Quillen cohomology\ngroup in ${\\mathcal C}$; see  \\cite[\\S 6.7]{BDGoeR} and \\cite[\\S 4]{BlaG}.  \n\n\\setcounter{thm}{0}\\section{Spherical functors}\n\\label{csf}\n\nOur objective is to study functors between model categories, and\ninvestigate the extent to which they induce an equivalence of homotopy\ncategories. Our methods work only for functors between spherical model\ncategories which take models to models, in the following sense:\n\n\\begin{defn}\\label{dsfunc}\\stepcounter{subsection}\nLet \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ and \\ $\\lra{{\\mathcal D};{\\mathcal B}}$ \\ be two spherical model\ncategories. A functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is called \\emph{spherical} if \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item $T$ defines a bijection \\ ${\\mathcal A}\\to{\\mathcal B}$;\n\\item $T\\rest{\\Pi_{\\A}}$ \\ preserves coproducts and suspensions;\n\\item $T$ induces an equivalence of categories \\ $\\Alg{\\PiC}\\approx\\Alg{\\PiD}$ \\ \n(in fact, it suffices that \\ $\\Alg{\\PiD}$ \\ be a full subcategory of \\ $\\Alg{\\PiC}$).\n\\end{enumerate}\n\\end{defn}\n\n\\subsection{Examples of spherical functors}\n\\label{ssfunc}\\stepcounter{thm}\n\nIn the cases we shall be considering (those mentioned in the\nintroduction), ${\\mathcal C}$ and ${\\mathcal D}$ will be strict spherical resolution model\ncategories, with \\ ${\\mathcal C}=s\\hat{\\C}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}$, \\ and $T$ will \nbe prolonged from a functor \\ $\\hat{T}:\\hat{\\C}\\to\\hat{\\D}$\\vspace{2 mm} . \n\nThe four examples:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{{\\Ab{\\EuScript Gp}};\\{\\mathbb Z\\}}$, \\ let \\ $\\hat{T}={\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\\nbe the abelianization functor.\n\nHere \\ ${\\mathcal C}=s\\hat{\\C}={\\mathcal G}$, \\ so \\ $\\operatorname{ho}{\\mathcal C}$ \\ is equivalent to the homotopy\ncategory of pointed connected topological spaces (\\S \\ref{ssg}), \nwhile \\ ${\\mathcal D}=s\\hat{\\D}$, \\ the category of simplicial abelian groups,\nis equivalent to the category of chain complexes under the\nDold-Kan correspondence (see \\cite[\\S 1]{DolH}). Thus \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\\nrepresents the singular chain complex functor \\ $C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$.\n\nNote that \\ $\\Alg{\\PiC}={\\EuScript Gp}$, \\ while \\ $\\Alg{\\PiD}={\\Ab{\\EuScript Gp}}$, \\ in this case, so \nstrictly speaking $T$ does not induce an equivalence of categories. \\\nBut since \\ ${\\Ab{\\EuScript Gp}}$ \\ is a full subcategory of \\ ${\\EuScript Gp}$, \\ we can in\nfact think of \\ $\\pi^{\\natural}$ \\ as taking values in groups\\vspace{2 mm} .\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{{\\EuScript Hopf};\\{H\\}}$, \\ where \\ ${\\EuScript Hopf}$ \\ is the\ncategory of complete Hopf algebras over $\\mathbb Q$, \\ $H$ is the\nmonogenic free object in this category, let \\ $\\hat{Q}:{\\EuScript Gp}\\to{\\EuScript Hopf}$ \\ \nbe the functor which associates to a group $G$ the completion of the\ngroup ring \\ $\\mathbb Q[G]$ \\ by powers of the augmentation ideal. \n\nAgain, \\ ${\\mathcal C}=s\\hat{\\C}$ \\ is a model category for connected \ntopological spaces, while \\ ${\\mathcal D}=s\\hat{\\D}$ \\ is a model category for the\nrational simply-connected spaces (see \\cite{QuiR}); \\ $Q$ (when\nrestricted to connected simplicial groups) represents the\nrationalization functor. \\ Once more, \\ $\\Alg{\\PiC}={\\EuScript Gp}$, \\ while \\\n$\\Alg{\\PiD}$ \\ is the subcategory of vector spaces over $\\mathbb Q$\\vspace{2 mm}. \n\\item For \\ $\\lra{\\hat{\\C},\\hat{\\A}}=\\lra{\\Set_{\\ast};\\{S^{0}\\}}$ \\ \n(so that \\ $\\lra{{\\mathcal C},{\\mathcal A}}=\\lra{{\\mathcal S};\\{S^{1}\\}}$, \\ by Remark\n\\ref{rmods}), and \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$, \\ let \\\n$\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ be the free group functor.\n\nAgain, we think of both \\ ${\\mathcal C}=s\\hat{\\C}={\\mathcal G}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}=\\Ss_{\\ast}$ \\ as\nmodel categories for pointed topological spaces, (under the respective\nequivalences of \\S \\ref{ssg}) \\ -- \\ so $F$ here represents \nthe suspension functor \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$ \\ (rather than \\\n$\\Omega\\Sigma$, \\ as one might think at first glance)\\vspace{2 mm}.\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\mathcal G};\\{\\gS{k}\\}_{k=0}^{\\infty}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{\\Alg{\\Pi};\\{\\pi_{\\ast}\\gS{k}\\}_{k=0}^{\\infty}}$, \\ let \\ \n$\\widehat{\\pi_{\\ast}}:{\\mathcal G}\\to\\Alg{\\Pi}$ \\ be the graded homotopy group functor \\ $X\\mapsto\\pi_{\\ast} X$. \\ \nHere \\ ${\\mathcal C}=s{\\mathcal G}$ \\ is a model category for simplicial spaces\\vspace{2 mm}.\n\\end{enumerate}\n\n\\begin{thm}\\label{tthree}\\stepcounter{subsection}\nLet \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ and \\ $\\lra{{\\mathcal D};{\\mathcal B}}$ \\ be spherical model\ncategories,  and let \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ be a spherical functor. Then for\neach \\ $X\\in{\\mathcal C}$ \\ and \\ $A\\in{\\mathcal A}$ \\ there is a natural long exact\nsequence of $\\PiC$-algebra s: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efive}\n\\dotsc\\to \\Gamma^{X}_{\\alpha,n}X~\\xra{s_{n}^{X}}~\\pia{n}\nX~\\xra{h_{n}^{X}}~\\pinC{T_{\\ast}(\\alpha),n}\nTX~\\xra{\\partial_{n}^{X}}~\\Gamma^{T}_{\\alpha,n-1}X\\dotsc~. \n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{thm}\n\nWe call \\eqref{efive} the \\emph{comparison exact sequence} for\n$T$. Compare \\cite[V, (5.4)]{BauCF}.\n\n\\begin{proof}\nIf \\ $\\tilde{X}\\to X$ \\ is a functorial fibrant replacement, the functor \n$T$ induces a natural transformation \\ \n$\\tau:\\operatorname{map}_{{\\mathcal C}}(A,\\tilde{X})\\to \\operatorname{map}_{{\\mathcal D}}(TA,\\widehat{T\\tilde{X}})$, \\ which\nwe may functorially change to a fibration of simplicial sets, \nwith fiber \\ $F(X)$. \\ Setting \\ $\\Gamma^{T}_{\\alpha,n}:=\\pi_{n}F(X)$, \\ \nthe corresponding long exact sequence in homotopy is\n\\eqref{efive}. \n\nNote that the map \\ $h^{X}_{n}=h^{X}$ \\ is also natural in the\nvariable \\ $A$, \\ so the graded map \\ $h^{X}_{\\ast}:\\pinC{n}X\\to\\pinD{n}TX$ \\ \nis a morphism of $\\PiC$-algebra s (i.e., \\ $\\Pi_{\\hA}$-algebras).\n\\end{proof}\n\n\\subsection{Applications of Theorem \\protect{\\ref{tthree}}}\n\\label{satt}\\stepcounter{thm}\n\nThe Theorem is not very useful in this generality. However, in all \nthe examples of \\S \\ref{ssfunc}, we obtain\ninteresting (though mostly known) exact sequences\\vspace{2 mm} :\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For \\ $\\hat{T}={\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\ the abelianization functor, where \\ \n$T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$ \\ represents the singular chain complex functor \\ \n$C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$ \\ (cf.\\ \\S \\ref{ssfunc}(a)), the sequence \n\\eqref{efive} is the ``certain exact sequence'' of J.H.C. Whitehead:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyfive}\n\\dotsc\\to \\Gamma_{n} X \\to\\pi_{n} X~\\xra{h_{n}}~ H_{n}(X;\\mathbb Z)\\to\n\\Gamma_{n-1} X \\dotsc\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\n(See \\cite{JWhC}).  In particular, the third term in this sequence, \\\n$\\Gamma^{{\\mathcal A}}_{n}(X)$, \\ is simply the $n$-th homotopy group of the\ncommutator subgroup of \\ $GX$\\vspace{2 mm}. \n\\item For \\ $Q:{\\mathcal G}\\to s{\\EuScript Hopf}$ \\ of \\S \\ref{ssfunc}(b), representing \nthe rationalization functor, \\ we obtain a long exact sequence\nrelating the integral and rational homotopy groups of a\nsimply-connected space $X$. The third term in \\eqref{efive} may be\ndescribed in terms of the torsion subgroup of \\ $\\pi_{\\ast} X$ \\ together\nwith \\ $\\pi_{\\ast} X\\otimes\\mathbb Q\/\\mathbb Z$\\vspace{2 mm}. \n\n\\item The free group functor \\ $\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ of \\S\n\\ref{ssfunc}(c) represents the suspension \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ \nand indeed for \\ $K\\in\\Ss_{\\ast}$ \\ the map \\ $h^{K}$, \\ which is the composite:\n\n\\begin{equation*}\n\\begin{split}\n\\pi_{n}K~=~\\pi_{0}\\operatorname{map}_{\\Ss_{\\ast}}(S^{n},K)~&~\n\\longrightarrow~\\pi_{0}\\,\\operatorname{map}_{{\\mathcal G}}(FS^{n},FK)\\\\\n\\xra{\\cong}~& \\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(\\Sigma S^{n},\\Sigma K)~=~\\pi_{n+1}\\Sigma K~,\n\\end{split}\n\\end{equation*}\n\\noindent is the suspension homomorphism, so \\eqref{efive} is a\ngeneralized EHP sequence (cf.\\ \\cite{BauR,GanG,NomE})\\vspace{2 mm}. \n\\item For \\ $\\pi_{\\ast}:s{\\mathcal G}\\to s\\Alg{\\Pi}$ \\ as in \\S \\ref{ssfunc}(d), it\nturns out that for any simplicial space \\ $X\\in s{\\mathcal G}$, \\ the induced map \\\n$h^{X}_{n}$ \\ is the ``Hurewicz homomorphism'' \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi_{n}\\pi_{\\ast} X$ \\ of \\cite[7.1]{DKStB}, \\ \nwhile \\ $\\Gamma^{T}_{n}X$ \\ is just \\ $\\Omega\\pi^{\\natural}_{n-1}X$ \\ -- \\ that is, \\ \n$\\Gamma^{T}_{i,n}X=\\pi^{\\natural}_{i+1,n-1}X$ \\ for each $i$. Thus \\ \n\\eqref{efive} \\ is the \\emph{spiral long exact sequence}:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efifteen}\n\\dotsc \\pi_{n+1}\\pi_{\\ast} X \\xra{\\partial^{\\star}_{n+1}} \\Omega\\pi^{\\natural}_{n-1}X\n\\xra{s_{n}} \\pi^{\\natural}_{n}X \\xra{h_{n}} \\pi_{n}\\pi_{\\ast} X\\to\n\\dotsb \\pi^{\\natural}_{0}X\\xra{h_{0}}\\pi_{0}\\pi_{\\ast} X \\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent  of \\cite[8.1]{DKStB}. Of course, \\ $\\pi^{\\natural}_{-1}{X}=0$, \\ \nso \\ $h_{0}$ \\ is an isomorphism\\vspace{2 mm} .\n\\end{enumerate}\n\nNote that for \\ $T:{\\mathcal C}\\to{\\mathcal D}$ as above, the homotopy groups \\ $\\pinD{n}TX$ \\ for\nany \\ $X\\in{\\mathcal C}=s\\hat{\\C}$ \\ may be computed using the Moore chains \\\n$C_{\\ast}TX$ \\ as in \\S \\ref{dnc}; each  \\ $\\pinD{n}TX$ \\ is a \\ $\\Pi_{\\D}$-algebra,\nabelian for \\ $n\\geq 1$.  \n\n\\subsection{Explicit construction of the spiral exact sequence}\n\\label{sses}\\stepcounter{thm}\n\nIt may be helpful to inspect in detail the construction of last long\nexact sequence, since it is perhaps the least familar of the four.\nSpecificializing to \\ $\\hat{\\C}={\\mathcal G}$ \\ and \\ $T=\\pi_{\\ast}$, \\ we have:\n\n\\begin{lemma}\\label{lone}\\stepcounter{subsection}\nFor fibrant \\ $X\\in{\\mathcal C}$, \\ the inclusion \\ $\\iota:C_{n}X\\hookrightarrow X_{n}$ \\ \ninduces an isomorphism \\ $\\iota_{\\star}:\\pi_{\\ast} C_{n}X\\cong C_{n}(\\pi_{\\ast} X)$ \\ \nfor each \\ $n\\geq 0$. \n\\end{lemma}\n\n\\begin{proof}\nSee \\cite[Prop.\\ 2.11]{BlaCW}.\n\\end{proof}\n\nTogether with \\eqref{ethree}, this yields a commuting diagram:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyseven}\n\\xymatrix{\n\\pi_{\\ast} C_{n+1}X \\rto^{(\\mathbf{d}_{0})_{\\#}} \\dto_{\\iota_{\\star}}^{\\cong} & \n\\pi_{\\ast} Z_{n}X \\ar@{->>}[r]^{\\hat{\\vartheta}_{n}} \\dto^{\\hat{\\iota}_{\\star}} & \n\\pi^{\\natural}_{n}X \\ar@{.>}[d]^{h_{n}} \\\\\nC_{n+1}(\\pi_{\\ast} X) \\rto^{d_{0}^{\\pi_{\\ast} X}} & \nZ_{n}(\\pi_{\\ast} X) \\ar@{->>}[r]^{\\vartheta_{n}} & \\pi^{\\natural}_{n}\\pi_{\\ast} X\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which defines the dotted morphism of $\\Pi$-algebra s \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi_{n}(\\pi_{\\ast} X)$. \\ Note that for \\ $n=0$ \\ the map \\ \n$\\hat{\\iota}_{\\star}$ \\ is an isomorphism, so $h$ is, too.\n\nIf  \\ $X\\in s{\\mathcal G}$ \\ is fibrant, applying \\ $\\pi_{\\ast}$ \\ to the fibration sequence \n\\eqref{etwo} yields a long exact sequence, with connecting homomorphism \\ \n$\\partial_{n}:\\Omega\\pi_{\\ast} Z_{n} X=\\pi_{\\ast}\\Omega Z_{n} X \\to\\pi_{\\ast} Z_{n+1} X $; \\ \n\\eqref{ethree} then implies that\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efourteen}\n\\Omega\\pi^{\\natural}_{n}X=\\Omega\\operatorname{Coker}\\,(\\mathbf{d}_{0}^{n+1})_{\\#}\\cong\\operatorname{Im}\\,\\partial_{n}\n\\cong \\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\subseteq\\pi_{\\ast} Z_{n+1}X,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent and the map \\ $s_{n+1}:\\Omega\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n+1}X$ \\ in\n\\eqref{efour} is then obtained by composing the inclusion \\ \n$\\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\hookrightarrow\\pi_{\\ast} Z_{n+1}X$ \\ with the quotient map \\ \n$\\hat{\\vartheta}_{n+1}:\\pi_{\\ast} Z_{n+1}X\\to\\pi^{\\natural}_{n+1}X$ \\ of \\eqref{ethree}.\n\nSimilarly, \\ $h_{n}:\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n}\\pi_{\\ast} X$ \\ is induced by the \ninclusion \\ \n$(j^{X}_{n})_{\\#}:\\pi_{\\ast} Z_{n}X\\to Z_{n}\\pi_{\\ast} X\\subseteq C_{n}\\pi_{\\ast} X$, \\ and \\ \n$\\partial^{\\star}_{n+2}:\\pi^{\\natural}_{n+2}\\pi_{\\ast} X\\to\\Omega\\pi^{\\natural}_{n}X$ \\ is induced by \\ \nthe composite \\ \n$$\nZ_{n+2}\\pi_{\\ast} X\\subseteq C_{n+2}\\pi_{\\ast} X\\cong \\pi_{\\ast} C_{n+2}X\n\\xra{(\\mathbf{d}_{0}^{n+2})_{\\#}}Z_{n+1}\\pi_{\\ast} X,\n$$\n\\noindent which actually lands in \\ \n$\\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\cong \\Omega\\pi^{\\natural}_{n}X$ \\ by the exactness of the \nlong exact sequence for the fibration.\n\nMoreover, for each \\ $n\\geq 0$, \\ \\eqref{etwentyseven} may be extended\n(after rotating by $90^{\\circ}$) \\ to a commutative diagram with exact\nrows and columns: \n$$\n\\xymatrix{\n& 0 \\dto & 0\\dto & 0\\dto & & \\\\ \n0 \\rto & \\operatorname{Ker}\\, s_{n} \\ar@{|->}[r] \\dto & B_{n+1}X \\dto\n\\ar@{->>}[r]^>>>>>{(j_{n})_{\\ast}} & B_{n+1}\\pi_{\\ast} X_{n+2}\\rto\\dto & 0\\dto & \\\\\n0 \\rto & \\Omega\\pi^{\\natural}_{n-1}X \\ar@{|->}[r] \\dto & \n\\pi_{\\ast} Z_{n}X \\dto^{\\hat{\\vartheta}_{n}} \\rto^{(j_{n})_{\\ast}} & \nZ_{n}\\pi_{\\ast} X \\ar@{->>}[r] \\dto^{\\vartheta_{n}} & \\operatorname{Coker}\\, h_{n} \\rto \\dto^{=} & 0 \\\\\n0 \\rto & \\operatorname{Ker}\\, h_{n} \\ar@{|->}[r] \\dto & \\pi^{\\natural}_{n}X \\dto \\rto^{h_{n}} & \n\\pi_{n}\\pi_{\\ast} X \\ar@{->>}[r] \\dto & \\operatorname{Coker}\\, h_{n}\\rto\\dto & 0 \\\\\n & 0 & 0 & 0 & 0 &\n}\n$$\n\\noindent in which \\ \n$B_{n+1} X:=\\operatorname{Im}\\,(\\mathbf{d}_{0}^{X_{n+2}})_{\\#}\\subseteq\\pi_{\\ast} Z_{n} X$ \\ and \\  \n$B_{n+1}\\pi_{\\ast} X_{n+2}:=\\operatorname{Im}\\,\\mathbf{d}_{0}^{\\pi_{\\ast} X_{n+2}}$ \\ are the respective\nboundary objects.\n\nThe maps \\ $\\partial^{\\star}_{n+1}$, \\ $s_{n}$, \\ and \\ $h_{n}$, \\ \nas defined above, form the spiral long exact sequence.\n\n\\subsection{Inverse spherical functors}\n\\label{sisf}\\stepcounter{thm}\n\nWe may sometimes be interested in functors between spherical model\ncategories which are not quite spherical. Thus, if \\ \n$T:\\lra{{\\mathcal C};{\\mathcal A}}\\to\\lra{{\\mathcal D};{\\mathcal B}}$ \\ is a spherical functor as in \\S\n\\ref{dsfunc}, a functor \\ $V:{\\mathcal D}\\to{\\mathcal C}$ \\ equipped with a natural\ntransformation \\ $\\vartheta:\\operatorname{Id}_{{\\mathcal C}}\\to VT$ \\ is called an\n\\emph{inverse spherical functor} to $T$. \n\n\\begin{example}\\label{eisf}\\stepcounter{subsection}\nFor the free group functor \\ $F:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ of \\S \\ref{ssfunc}(c),\nthe forgetful functor \\ $\\hat{U}:{\\EuScript Gp}\\to\\Set_{\\ast}$ \\ (right adjoint to $F$)\nwith the adjunction counit \\ $\\eta:\\operatorname{Id}\\to UF$ \\ as the natural\ntransformation $\\vartheta$, yields the inverse spherical functor \\ \n$U:{\\mathcal G}\\to\\Ss_{\\ast}$. \\ Here we do not think of ${\\mathcal G}$ as a model for $\\TT_{\\ast}$ \\ -- \\ \nrather, $U$ represents the forgetful functor from loop spaces\n(topological groups) to spaces.\n\nSimilarly, the adjoint to the abelianization functor \\ ${\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\ \nis the inclusion \\ $\\hat{I}:{\\Ab{\\EuScript Gp}}\\to{\\EuScript Gp}$, \\ and the corresponding functor \\ \n$I:s{\\Ab{\\EuScript Gp}}\\to{\\mathcal G}$ \\ represents the factorization of the Dold-Thom infinite\nsymmetric product functor \\ $SP^{\\infty}:\\TT_{\\ast}\\to\\TT_{\\ast}$ \\ \nthrough \\ ${\\EuScript Chain}$.\n\\end{example}\n\n\\begin{prop}\\label{pfour}\\stepcounter{subsection}\nIf \\ $V:{\\mathcal D}\\to{\\mathcal C}$ \\ is an inverse spherical functor to $T$,\nthen for each \\ $Y\\in{\\mathcal D}$ \\ and \\ $B\\in{\\mathcal B}$ \\ there is a natural \nlong exact sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{esix}\n\\dotsc\\to \\Delta^{V}_{B,n}Y\\to\\pinD{B,n} Y\\xra{V_{\\#}} \n\\pinC{V_{\\ast}(B),n} VY\\to \\Delta^{V}_{B,n-1}Y\\dotsc\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{prop}\n\n\\begin{proof}\nIf $V$ is an inverse spherical functor, because \\ $T\\rest{{\\mathcal A}}$ \\ is a\nbijection onto ${\\mathcal B}$, there is an \\ $A\\in{\\mathcal A}$ \\ such that \\ \n$B=TA$. \\ As before, $V$ induces a natural transformation \\ \n$\\nu:\\operatorname{map}_{{\\mathcal D}}(B,\\tilde{Y})\\to \\operatorname{map}_{{\\mathcal D}}(VB,\\widehat{V\\tilde{Y}})$ \\ and the\nnatural transformation \\ $\\vartheta:A\\to VTA$ \\ yields \\ \n$\\vartheta^{\\#}:\\operatorname{map}_{{\\mathcal D}}(VTA,\\widehat{V\\tilde{Y}})\\to\n\\operatorname{map}_{{\\mathcal D}}(A,\\widehat{V\\tilde{Y}})$ \\ so we get a composite map \\ \n$\\operatorname{map}_{{\\mathcal D}}(B,\\tilde{Y})\\to\\operatorname{map}_{{\\mathcal D}}(A,\\widehat{V\\tilde{Y}})$, \\ with homotopy\nfiber \\ $E(Y)$. \\ If we let \\ $\\Delta^{V}_{\\beta,n}Y:=\\pi_{n} E(Y)$, \\ \nthe fibration long exact sequence is \\eqref{esix}. \n\\end{proof}\n\n\\begin{remark}\\label{rlesisf}\\stepcounter{subsection}\nNote that in contradistinction to Theorem \\ref{tthree}, \\ \n$V_{\\#}$ \\ of \\eqref{esix} need not respect any operations, since we\nonly have a bijection \\ $T\\rest{{\\mathcal A}}:{\\mathcal A}\\to{\\mathcal B}$, \\ not a functor. \n\nFor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ as in \\S \\ref{eisf}, we may assume \\ $X\\in{\\mathcal G}$ \\\nis of the form \\ $X\\simeq GK$ \\ for \\ $K\\in\\Ss_{\\ast}$, \\ and then \\ \n$V_{\\#}$ \\ is the identity:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentythree}\n\\begin{split}\n\\pi_{n}K~=~\\pi^{\\natural}_{n}X&=~\\pi_{0}\\,\\operatorname{map}_{{\\mathcal G}}(FS^{n-1},GK)~\\to~\n\\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(UFS^{n-1},UGK)\\\\\n& \\xra{\\eta^{\\#}}~\\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(S^{n-1},UGK)~=~\\pi_{n}K~,\n\\end{split}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent so \\eqref{esix} is not interesting in this case. \n\\end{remark}\n\n\\setcounter{thm}{0}\\section{Comparing Postnikov systems}\n\\label{ccps}\n\nThe basic problem under consideration in this paper may be formulated as\nfollows\\vspace{2 mm} : \n\n\\subsection*{Question}\nGiven a spherical functor \\ $T:\\lra{{\\mathcal C};{\\mathcal A}}\\to\\lra{{\\mathcal D};{\\mathcal B}}$ \\\nand an object \\ $G\\in{\\mathcal D}$, \\ what are the different objects \\ $X\\in{\\mathcal C}$ \\ \n(up to homotopy) such that \\ $TX\\simeq G$\\vspace{2 mm} ?\n\nAs shown in the previous section, such a pair \\ \n$\\lra{X,G}$ \\ must be connected by a comparison exact sequence. Thus,\nin order to reconstruct $X$ from $G$, we first try to determine \\ \n$\\pinC{\\ast}X$, \\ and its relation to \\ $\\pinD{\\ast}G$. \n\nIn order to proceed further, we must make an additional assumption on\n$T$, contained in the following:\n\n\\begin{defn}\\label{dspecial}\\stepcounter{subsection}\nA spherical (or inverse spherical) functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is called\n\\emph{special} if:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item ${\\mathcal C}=s\\hat{\\C}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}$ \\ are spherical resolution model\n  categories, and $T$ is prolonged from a functor \\ $\\hat{T}:\\hat{\\C}\\to\\hat{\\D}$. \\ \n\\item For any $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over $\\Lambda$, \\ $T$ induces a \nhomomorphism of (graded) groups \\ $\\phi_{T}:\\Lambda\\to\\pinD{0}TB_{\\C}\\Lambda$. \\ \n\\item This \\ $\\phi_{T}$ \\ induces a functor \\ \n$\\hat{T}:\\RM{\\Lambda}\\to\\RM{\\phi_{T}\\Lambda}$ \\ which is an\nisomorphism on $\\Lambda$-modules (see Remark \\ref{rmodule}).\n\\item For each \\ $n\\geq 1$ \\ and $n$-dimensional extended\n$M$-Eilenberg-Mac~Lane object \\ $E=\\ECL{M}{n}$, \\ there is a natural\nisomorphism \\ $\\pinD{n}TE\\cong M$ \\ which respects $\\hat{T}$ in the\nobvious sense.\n\\item The natural map\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyfour}\n[X,\\ECL{M}{n}]_{B_{\\C}\\Lambda}\\to [TX,E\\sp{\\hat{T}\\Lambda}\\sb{{\\mathcal D}}(M,n)]_{B_{{\\mathcal D}}\\hat{T}L}~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent defined by composition with the projection\n$$\n\\rho:T\\ECL{M}{n}\\to P_{n}T\\ECL{M}{n}~=~E\\sp{\\hat{T}\\Lambda}\\sb{{\\mathcal D}}(M,n)~, \n$$ \nis an isomorphism.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{example}\\label{especial}\\stepcounter{subsection}\nAll the functors we have considered hitherto, except for the rationalization\nfunctor \\ $Q:{\\mathcal G}\\to s{\\EuScript Hopf}$ \\ of \\S \\ref{ssfunc}(b), are special\\vspace{2 mm}:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For the singular chain functor \\ $T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$, induced by \nabelianization, this follows from the Hurewicz Theorem (recall that \\\n$\\pinC{0}X$ \\ is the fundamental group, in our indexing for \\ $X\\in{\\mathcal G}$).\n\\item For the suspension \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ induced by \nthe free group functor \\ $F:\\Set_{\\ast}\\to{\\EuScript Gp}$, \\ this follows (in the\nsimply connected case) from the Freudenthal Suspension Theorem.\n\\item For the homotopy groups functor \\ $\\pi_{\\ast}:s{\\mathcal G}\\to s\\Alg{\\Pi}$, \\ (i)-(iii)\nfollow by inspecting the spiral long exact sequence \\eqref{efifteen},\nwhile (iv) is \\cite[Prop.\\ 8.7]{BDGoeR}.\n\\item For the inverse spherical functor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ of \\S \\ref{eisf},\ninduced by the forgetful functor \\ $\\hat{U}:{\\EuScript Gp}\\to\\Set_{\\ast}$, \\ this is\nimmediate from \\eqref{etwentythree}\\vspace{2 mm}.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{ltwo}\\stepcounter{subsection}\nAny special spherical functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ as above \\emph{respects\nPostnikov systems} \\ -- \\ that is, for any \\ $X\\in{\\mathcal C}$ \\ \nand \\ $n\\geq 0$ \\ we have:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eeight}\nP^{{\\mathcal D}}_{n}T P^{{\\mathcal C}}_{n} X\\cong P^{{\\mathcal D}}_{n}T X \\ - \\ \n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\nso that \\ $\\pinC{k}T X\\cong \\pinD{k}TP_{n}X$ \\ and \\ \n$\\Gamma_{k}X\\cong \\Gamma_{k}P_{n}X$ \\ for \\ $k\\leq n$. \n\\end{lemma}\n\n\\begin{proof}\nThis follows from the constructions in \\S \\ref{sps} and the proof of\nTheorem \\ref{tthree}.\n\\end{proof}\n\n\\subsection{Postnikov systems and spherical functors}\n\\label{spssf}\\stepcounter{thm}\n\nFrom now on, assume \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor.\nUltimately, for each object \\ $G\\in{\\mathcal D}$, \\ we would like find any and all \\\n$X\\in{\\mathcal C}$ \\ such that \\ $TX\\simeq G$. \\ First, however, we try to\ndiscover what can be said about \\ $TX$ \\ and its Postnikov systems\nfor a given \\ $X\\in{\\mathcal C}$. \\ Using the comparison exact sequence for $T$\nand Lemma \\ref{ltwo}, we see that:  \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{enine}\n\\pinD{k}TP_{n}X\\cong\\begin{cases}\n        \\pinD{k}TX & \\text{for \\ } k\\leq n,\\\\\n        \\operatorname{Coker}\\,\\{h^{X}_{n+1}:\\pinC{n+1}X\\to\\pinD{n+1}T X\\} & \\text{for \\ }\n          k=n+1,\\\\\n        \\Gamma_{k-1}P_{n}X & \\text{for \\ }k\\geq n+2~.\n\\end{cases}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\n\\begin{fact}\\label{fsix}\\stepcounter{subsection}\nIf \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor, applying \\\n$\\pinC{n+2}$ \\ to the $n$-th $k$-invariant \\\n$k_{n}:P_{n}X\\to \\ECL{\\pinC{n+1}X}{n+2}$ \\  \nyields the homomorphism \\ $s^{X}_{n+1}:\\Gamma_{n+1}X\\to\\pinC{n+1}X$.\n\\end{fact}\n\n\\begin{proof}\nSince $T$ is special, \\ $\\pinD{n+2}T\\ECL{\\pi^{\\natural}_{n+1}X}{n+2}\\cong\\pinC{n+1}X$, \\ \nand \\ $\\pinD{n+2}TP_{n}X\\cong\\Gamma_{n+1}X$ \\ from \n\\eqref{enine}, so this follows from the naturality of the comparison\nexact sequence, applied to the maps in \\eqref{ezero}.\n\\end{proof}\n\n\\begin{lemma}\\label{lthree}\\stepcounter{subsection}\nIf \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor, for any \\ $X\\in{\\mathcal C}$, \n$$\n\\xymatrix{\nP_{n+1}TP_{n}X \\rto \\dto & P_{n+1}TP_{n-1}X \\dto \\\\ \nP_{n+1}TB_{\\C}\\Lambda \\rto^-{\\protect{Tk_{n}}} & P_{n+1}T\\ECL{\\pinC{n}X}{n+2}\n}\n$$\n\\noindent is a homotopy pullback square in \\ ${\\mathcal D}\/TB_{\\C}\\Lambda$, \\ where \\ \n$\\Lambda:=\\pinC{0}X$. \n\\end{lemma}\n\n\\begin{proof}\nSet \\ $E:=T\\ECL{J}{n+1}$, \\ $M^{n-1}:=TP_{n-1}X$, \\ and \\ $M^{n}:=TP_{n}X$. \\ \nThe naturality of the comparison exact sequence, applied to the maps\nin \\eqref{ezero}, \\ combined with Fact \\ref{fsix}, imply that the\nvertical maps in the following commutative diagram are isomorphisms:\n$$\n\\xymatrix{\n\\pinD{n+2}E \\rto \\dto^{\\cong} & \\pinD{n+1}M^{n} \\rto \\dto^{\\cong} &\n\\pinD{n+1}M^{n-1} \\rto^>>>>{Tk_{n-1}} \\dto^{\\cong} & \n\\pinD{n+1}E \\rto \\dto^{\\cong} &\n\\pinD{n}M^{n} \\rto \\dto^{\\cong} & \\pinD{n}M^{n-1} \\dto^{\\cong} \\\\\n0\\rto & \\operatorname{Coker}\\, h^{X}_{n+1} \\ar@{|->}[r] & \\Gamma_{n}X \\rto^{s^{X}_{n}} &\n\\pinC{n}X \\rto^{h^{X}_{n}} & \\pinD{n}T X \\ar@{->>}[r] & \\operatorname{Coker}\\, h_{n}^{T}\n}\n$$\n\\noindent and since the bottom row is part of the comparison long\nexact sequence, and the rest of the top sequence to the right \nis exact for by \\eqref{eeight}, the $k$-invariant square \\eqref{ezero} \ninduces a long exact sequence after applying \\ $\\pi^{\\natural}$ \\ (except in the\nbottom dimensions). \\ The obvious map from \\ $M^{n}$ \\ to the fiber of \\\n$Tk_{n-1}$ \\ is thus a weak equivalence in \\ ${\\mathcal D}\/TB_{\\C}\\Lambda$ \\ through\ndimension \\ $n+1$.\n\\end{proof}\n\n\\begin{cor}\\label{czero}\\stepcounter{subsection}\nFor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ as above, \\ for any \\ $X\\in{\\mathcal C}$ \\ and \\ $n\\geq 1$ \\ \nthe natural map \\ $r\\q{n}:X\\to P_{n}X$ \\ of \\S \\ref{sps} induces an \\ \nisomorphism \\ $\\Gamma_{k}X\\cong\\Gamma_{k}P_{n}X$ \\ for \\ $k\\leq n+1$.\n\\end{cor}\n\n\\begin{proof}\nFor each \\ $A\\in{\\mathcal A}$, \\ take fibers vertically and horizontally of\nthe commutative square:\n$$\n\\xymatrix{\n\\operatorname{map}_{B_{\\C}\\Lambda}(A,P_{n}X) \\rto^-{\\protect{h^{P_{n}X}}} \\dto^{(k_{n})_{\\ast}} & \n\\operatorname{map}_{B_{\\D}\\Lambda}(TA,TP_{n}X) \\dto^{(Tk_{n})_{\\ast}} \\\\ \n\\operatorname{map}_{B_{\\C}\\Lambda}(A,\\ECL{\\pinC{n+1}X}{n+2}) \\rto^-{\\protect{h^{E}}} & \n\\operatorname{map}_{B_{\\D}\\Lambda}(TA,T\\ECL{\\pinC{n+1}X}{n+2})~,\\\\\n}\n$$\nand use Lemma \\ref{lthree} and \\S \\ref{dspecial}(iv).\n\\end{proof}\n\n\\begin{remark}\\label{rhem}\\stepcounter{subsection}\nFor \\ ${\\mathcal C}=s{\\mathcal G}$ \\ this follows from the fact that \\\n$\\Gamma_{n}X\\cong\\Omega\\pi^{\\natural}_{n-1}X$, \\ while for the algebraic cases\nof \\S \\ref{ermc}(i-ii), this follows from the fact that \\\n$H_{n+1}(K(\\pi,n);\\mathbb Z)=0$ \\ for \\ $n\\geq 1$.\n\\end{remark}\n\n\\subsection{The extension}\n\\label{sext}\\stepcounter{thm}\n\nThe map \\ $r\\q{n}:X\\to P_{n}X$ \\ induces a map of comparison exact\nsequences: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentysix}\n\\xymatrix{\n\\pinC{n+2}X \\dto \\rto^{h^{X}_{n+2}} & \n\\pinC{n+2}TX \\dto^{\\pinD{n+2}Tr\\q{n}} \\rto^{\\partial^{\\star}_{n+2}} & \n\\Gamma_{n+1}X \\dto^{=} \\rto^{s_{n+1}} & \\pinC{n+1}X \\dto \\rto^{h^{X}_{n+1}} & \n\\pinD{n+1}TX \\dto^{\\pinD{n+1}Tr\\q{n}} \\rto^{\\partial^{\\star}_{n+1}} & \n\\Gamma_{n}X \\dto^{=} \\\\\n0 \\rto & \\pinD{n+2}M^{n} \\rto^{\\cong} & \\Gamma_{n+1}P_{n}X \\rto & 0 \\rto &\n\\pinD{n+1}M^{n} \\rto & \\Gamma_{n}P_{n}X\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent so that \\ $\\pinC{n+1}X$ \\ fits into a short exact sequence \nof $\\PiA$-algebra s:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eten}\n0\\to \\operatorname{Coker}\\,\\pinD{n+2}Tr\\q{n} \\to \\pinC{n+1}X \\to\n\\operatorname{Ker}\\,\\pinD{n+1}Tr\\q{n} \\to 0,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent where \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyone}\n\\operatorname{Coker}\\,\\pinD{n+2}Tr\\q{n}\\cong \\operatorname{Ker}\\, h^{X}_{n+1}\\hspace{2 mm} \\text{and}\\hspace{2 mm} \n\\operatorname{Ker}\\,\\pinD{n+1}Tr\\q{n} \\cong \\operatorname{Im}\\, h^{X}_{n+1}.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nSince \\ $h^{X}_{n+1}$ \\ is a map of modules over \\ $\\Lambda:=\\pinC{0}X$, \\ \nby Theorem \\ref{tone}, \\eqref{eten} is actually a short exact sequence\nof modules over $\\Lambda$, and we can classify the possible values of \\ \n$J\\in\\RM{\\Lambda}$ \\ (the candidates for \\ $\\pinC{n+1}X$) \\ using the following:\n\\begin{prop}\\label{peleven}\\stepcounter{subsection}\nGiven \\ $Tr\\q{n}:T X\\to T P_{n}X$, \\ a choice for the \nisomorphism class of \\ $\\pinC{n+1}X$ \\ uniquely determines an element of \\ \n$$\n\\operatorname{Ext}_{\\RM{\\Lambda}}(\\operatorname{Ker}\\,(Tr\\q{n})_{n+1}, \\operatorname{Coker}\\,(Tr\\q{n})_{n+2}).\n$$\n\\end{prop}\n\n\\begin{proof}\nSince \\ $\\RM{\\Lambda}$ \\ is an abelian category, with a set  \\ \n$\\{A_{\\operatorname{ab}}\\otimes S^{n}\\amalgB_{\\D}\\Lambda\\}_{A\\in{\\mathcal A},n\\in\\mathbb N}$ \\ \nof projective generators, the argument of \\cite[III]{MacH} carries\nover to our setting. \n\\end{proof}\n\n\\begin{remark}\\label{rinfo}\\stepcounter{subsection}\nObserve that given \\ $P_{n}X$, \\ we know the comparison exact sequence \n\\eqref{efive} for \\ $X$ \\ only from \\ \n$s_{n}:\\Gamma_{n-1}X\\to\\pinC{n}X$ \\ down. However, if \\ \n$\\pinD{i}Tr\\q{n}:\\pinD{i}TX\\to\\pinD{i}M^{n}$ \\ (for \\ $i\\geq 0$) \\ and the \nextension \\eqref{eten} are also known,  all we need in order to\ndetermine \\eqref{efive} for $X$ from \\ \n$\\partial_{n+3}^{\\star}:\\pinD{n+3}TX\\to\\Gamma_{n+1}X$ \\ down is the \nhomomorphism\n$$\n\\pinD{n+3}Tr\\q{n+1}:\\pinD{n+3}TX\\to\\pinD{n+3}TP_{n+1}X~,\n$$\nwhich is just \\ $\\partial_{n+3}^{\\star}$, \\ as one can see \nfrom \\eqref{etwentysix}.\n\\end{remark}\n\n\\begin{prop}\\label{ptwelve}\\stepcounter{subsection}\nFor any \\ $\\Lambda\\in{\\mathcal D}$, \\ $J',J''\\in\\RM{\\Lambda}$, \\ and \\ \n$n\\geq 2$, \\ there is a natural isomorphism \\ \n$\\operatorname{Ext}_{\\RM{\\Lambda}}(J'',J')\\cong\\HL{n+1}{\\EDL{J''}{n}}{J'}$.\n\\end{prop}\n\nIn particular, this implies that \\ $\\HL{n+1}{\\EDL{-}{n}}{-}$ \\ is \n\\emph{stable} \\ -- \\ i.e., independent of $n$.\n\n\\begin{proof}\nBy Proposition \\ref{ptwo}\\emph{ff.} there is a natural isomorphism \\ \n$$\n\\HL{n+1}{\\EDL{J''}{n}}{J'}\n\\cong[\\EDL{J''}{n},\\EDL{J'}{n+1}]_{s{\\mathcal D}\/B_{\\D}\\Lambda},\n$$\n\\noindent and given a map \\ $\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$, \\ we can \nform the fibration sequence over \\ $B_{\\D}\\Lambda$ \\ (that is, pullback square\nas in \\eqref{ezero}):\n$$\n\\Omega\\EDL{J''}{n}\\xra{\\Omega\\psi}\\Omega\\EDL{J'}{n+1}\\simeq\\EDL{J'}{n}\n\\to F\\to\\EDL{J''}{n}\\xra{\\psi}\\EDL{J'}{n+1}.\n$$\n\\noindent From the corresponding long exact sequence in homotopy \nfor this sequence in ${\\mathcal D}$, we obtain a short exact sequence of \nmodules over $\\Lambda$:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eeleven}\n0\\to J'\\to J\\to J''\\to 0.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nOn the other hand, given a short exact sequence \\eqref{eeleven} in \\ \n$\\RM{\\Lambda}$, \\ we can construct a map \\ \n$\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$ \\ over \\ $B_{\\D}\\Lambda$ \\ as follows:\n\nAssume \\ $E:=\\EDL{J''}{n}$ \\ is constructed  starting with \\ \n$\\sk{n-1}\\EDL{J''}:=\\sk{n-1}B_{\\D}\\Lambda$, \\ and \\ \n$E_{n}\\simeq W\\amalg L_{n}B_{\\D}\\Lambda$ \\ (cf.\\ \\S \\ref{dlat}), \\ where $W$ is free, \nequipped with a surjection \\ $\\phi:W\\to J''$. \\ Because \\ $J\\to\\hspace{-5 mm}\\to J''$ \\ is \na surjection, and $W$ is free, we can lift $\\phi$ to \\ \n$\\phi':W\\to J$, \\ defining a map \\ $\\tilde{\\phi}':Z_{n}\\EDL{J''}{n}\\to J$. \\ \nSince \\  $\\pinD{n}\\EDL{J''}{n}=J''$, \\ the restriction of \\ \n$\\tilde{\\phi}'$ \\ to \\ $B_{n}\\EDL{J''}{n}=\\operatorname{Ker}\\,\\{Z_{n}\\EDL{J''}{n}\\to J''\\}$ \\ \nfactors through \\ $\\psi:B_{n}\\EDL{J''}{n}\\to J'=\\operatorname{Ker}\\,\\{J\\to J''\\}$. \\\nPrecomposing with \\ $\\mathbf{d}_{0}:C_{n+1}\\EDL{J''}{n}\\to B_{n}\\EDL{J''}{n}$ \\ defines \\ \n$\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$, \\ which classifies \\eqref{eeleven} \nas before.\n\\end{proof}\n\n\\begin{cor}\\label{cone}\\stepcounter{subsection}\nFor $\\Lambda$, $J'$, and $J''$ as above, there is a natural isomorphism:\n$$\n\\operatorname{Ext}_{\\RM{\\Lambda}}(J'',J')\\cong\\HL{n+1}{\\ECL{J''}{n}}{J'}.\n$$\n\\end{cor}\n\n\\begin{proof}\nThis follows from \\eqref{etwentyfour}-\\eqref{enine} and the \nnaturality of \\ $P^{{\\mathcal D}}_{n+1}$.\n\\end{proof}\n\n\\begin{defn}\\label{dmps}\\stepcounter{subsection}\nGiven \\ $X\\in{\\mathcal C}$, \\ its \\emph{$n$-th modified Postnikov section}, \ndenoted by \\ $\\bPa{n}X$, \\ is defined as follows\\vspace{2 mm} :\n\nLet \\ ${\\EuScript K}:=\\{f:A\\otimes S^{n+1}\\to X~|\\ \nA\\in{\\mathcal A},~[f]\\in\\operatorname{Ker}\\, h_{n+1}^{T}\\subset\\pinC{n+1}X\\}$, \\ and let\n$C$ be the cofiber of the obvious map \\ \n$\\Phi:\\bigvee_{f\\in{\\EuScript K}}~A\\otimes S^{n+1}\\to X$ \\  \n(so that \\ $\\pinC{n+1}C\\cong\\operatorname{Coker}\\,\\Phi$), \\ with \\ $\\bPa{n}X:= P_{n+1}C$. \\ \nThere are then natural maps \\ $\\bpa{n+1}:P_{n+1}X\\to\\bPa{n}X$ \\ \n(induced by \\ $X\\to C$), \\ as well\nas \\ $\\bpc{n}:\\bPa{n}X\\to P_{n}X$ \\ (which is just \\ \n$p\\q{n}_{C}:P_{n+1}C\\to P_{n}C\\cong P_{n}X$), \\ with \\ \n$\\bpc{n}\\circ\\bpa{n}=p\\q{n}_{X}:P_{n+1}X\\to P_{n}X$. \\ \nNote that \\ $\\pinC{n+1}\\bPa{n}X\\cong\\operatorname{Im}\\, h^{X}_{n+1}$, \\ and \\ \n$P_{n}\\bPa{n}X\\cong P_{n}X$.\n\\end{defn}\n\nThe map \\ $\\brp{n}:=\\bpa{n}\\circ r\\q{n}:X\\to\\bPa{n}X$ \\ induces a map of \ncomparison exact sequences:\n$$\n\\xymatrix{\n\\pinC{n+2}X \\rto^{h^{X}_{n+2}} \\dto & \n\\pinD{n+2}T X \\rto^{\\partial^{\\star}_{n+2}} \\dto^{\\pinD{n+2}T\\brp{n}} &\n\\Gamma_{n+1}X \\rto^{s_{n+1}} \\dto^{=} &\n\\pinC{n+1}X \\rto^{h^{X}_{n+1}} \\dto &\n\\pinD{n+1}T X \\rto^{\\partial_{n+1}} \\dto_{=}^{\\pinD{n+1}T\\brp{n}}\n& \\Gamma_{n}X \\dto^{=} \\\\\n0 \\rto & \\pinD{n+2}T\\bPa{n}X \\rto^{\\cong} & \\Gamma_{n+1}\\bPa{n}X \\rto^{0} & \n\\pinC{n+1}\\bPa{n}X \\ar@{|->}[r] & \\pinD{n+1}T\\bPa{n}X \\rto & \\Gamma_{n}\\bPa{n}X\n}\n$$\n\\noindent so that:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwenty}\n\\pinD{k}T\\bPa{n}X\\cong\\begin{cases}\n        \\pinD{k}TX& \\text{for \\ } k\\leq n+1,\\\\\n        \\Gamma_{n+1}X & \\text{for \\ }k=n+2,\\\\\n        \\Gamma_{k-1}\\bPa{n}X & \\text{for \\ }k\\geq n+3~.\n\\end{cases}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nThus \\ $\\brp{n}$ \\ induces a weak equivalence \\ $P_{n+1}TX\\simeq\nP_{n+1}T\\bPa{n}X$, \\ which, together with the existence of the\nappropriate maps \\ $P_{n+1}X\\xra{\\bpa{n}}\\bPa{n}X\\xra{\\bpc{n}}P_{n}X$, \\ \ndetermines \\ $\\bPa{n}X$ \\ up to homotopy. In fact we have:\n\n\\begin{prop}\\label{pfive}\\stepcounter{subsection}\n$\\bPa{n}X$ \\ is determined uniquely (up to weak equivalence) by \\ \n$P_{n}X$ \\ and the map \\ $\\rho:=P_{n+1}Tr\\q{n}:P_{n+1}TX\\to P_{n+1}TP_{n}X$.\n\\end{prop}\n\n\\begin{proof}\nNote that \\ $I_{n+1}:=\\operatorname{Ker}\\, \\pinD{n+1}\\rho$ \\ is isomorphic to \\ \n$\\operatorname{Im}\\, h^{X}_{n+1}$ \\ and \\ $C_{n+1}:=\\operatorname{Im}\\, \\pinD{n+1}\\rho$ \\ \nis isomorphic to \\ $\\operatorname{Coker}\\, h^{X}_{n+1}$ \\ by \\eqref{etwentyone}. \\ \n\nWe construct \\ $Y\\simeq\\bPa{n}X$ \\ as follows, starting with \\ \n$\\sk{n+1}Y:=\\sk{n+1}P_{n}X$; \\ by Remark \\ref{rcsk}, we may \nassume \\ $\\sk{n+1}TX=\\sk{n+1}TP_{n}X$, \\ so that \\ \n$P_{n}TX\\cong P_{n}TP_{n}X$. \\ By Fact \\ref{ffive}), the lower right \nhand square in Figure \\ref{fig6} commutes in ${\\mathcal D}$, thus inducing the\nrest of the diagram, in which the rows and columns are fibration\nsequences over \\ $B_{\\D}\\Lambda$. \n\n\\begin{center}\n\\setcounter{figure}{\\value{thm}}\\stepcounter{subsection}\n\\begin{figure}[hbtp]\n\\begin{picture}(300,110)(25,-5)\n\\put(15,100){$F$}\n\\multiput(35,105)(3,0){34}{\\circle*{.5}}\n\\put(137,105){\\vector(1,0){2}}\n\\put(85,110){$\\hat{\\rho}$}\n\\put(145,100){$P_{n+1}TP_{n}X$}\n\\multiput(205,105)(3,0){13}{\\circle*{.5}}\n\\put(245,105){\\vector(1,0){2}}\n\\put(217,110){$\\tkp{n}$}\n\\put(255,100){$\\EDL{I_{n+1}}{n+2}$}\n\\put(20,92){\\vector(0,-1){30}}\n\\put(8,75){$\\simeq$}\n\\put(25,75){$\\lambda$}\n\\put(170,92){\\vector(0,-1){30}}\n\\put(175,75){$p\\q{n}_{TP_{n}X}$}\n\\put(290,92){\\vector(0,-1){30}}\n\\put(295,75){$i_{\\ast}$}\n\\put(0,50){$P_{n+1}TX$}\n\\put(45,55){\\vector(1,0){55}}\n\\put(65,62){$p\\q{n}_{TX}$}\n\\put(105,50){$P_{n}TX\\cong P_{n}TP_{n}X$}\n\\put(205,55){\\vector(1,0){45}}\n\\put(220,61){$k_{n}^{TX}$}\n\\put(255,50){$\\EDL{\\pinD{n+1}TX}{n+2}$}\n\\put(20,42){\\vector(0,-1){30}}\n\\put(170,42){\\vector(0,-1){30}}\n\\put(175,28){$k_{n}^{TP_{n}X}$}\n\\put(290,42){\\vector(0,-1){30}}\n\\put(295,28){$q_{\\ast}$}\n\\put(10,0){$B_{\\D}\\Lambda$}\n\\put(55,5){\\vector(1,0){65}}\n\\put(125,0){$\\EDL{C_{n+1}}{n+2}$}\n\\put(215,5){\\vector(1,0){35}}\n\\put(230,10){$=$}\n\\put(255,0){$\\EDL{C_{n+1}}{n+2}$}\n\\end{picture}\n\\caption[fig6]{}\n\\label{fig6}\n\\end{figure}\n\\setcounter{thm}{\\value{figure}}\n\\end{center}\n\nIn particular, the induced map \\ \n$\\tkp{n}:P_{n+1}TP_{n}X\\to\\EL{I_{n+1}}{n+2}$ \\ provides a canonical lifting of:\n$$\nk_{n}^{TX}\\circ p\\q{n}_{TP_{n}X}:P_{n+1}TP_{n}X\\to\\EDL{\\pinD{n+1}TX}{n+2}\n$$\n\\noindent to \\ $\\EDL{I_{n+1}}{n+2}$. \\ Composing it with the natural map \\ \n$r\\q{n+1}:TP_{n}X\\to P_{n+1}TP_{n}X$ \\ defines an element in: \n$$\n[TP_{n}X,\\EDL{I_{n+1}}{n+2}]\\cong\\HL{n+2}{P_{n}X}{I_{n+1}}~,\n$$\n\\noindent which we call the \\emph{$n$-th modified $k$-invariant} for $X$\\vspace{2 mm}.\n\nIf \\ $\\hk{n}:P_{n}X\\to\\ECL{I_{n+1}}{n+2}$ \\ is the map corresponding to\n$\\tkp{n}$ \\ under \\eqref{etwentyfour}), then its homotopy fiber $Y$ is \n(weakly equivalent to) \\ $\\bPa{n}X$, \\ as one can verify by calculating \\ \n$\\pinC{\\ast}Y$. \\ Note that Lemma \\ref{lthree} implies that \\ \n$F\\simeq P_{n+1}TP_{n}X$, \\ so that $\\lambda$ is the homotopy inverse\nof the weak equivalence \\ $P_{n+1}\\rho:TX\\to TP_{n}X$, \\ which\ncompletes the construction. \n\\end{proof}\n\n\\begin{remark}\\label{rmin}\\stepcounter{subsection}\nNote that there is a certain indeterminacy in our description of \\ \n$\\tkp{n}$, \\ and thus of \\ $\\hk{n}$, \\ since we must make the lower right\ncorner of Figure \\ref{fig6} into a strict commuting diagram of fibrations,\nrather than one which commutes only up to homotopy. However,\n\\end{remark}\n\n\\begin{fact}\\label{fseven}\\stepcounter{subsection}\nThe indeterminacy for \\ $\\tkp{n}$ \\ as an induced map is contained in \nthe indeterminacy for \\ $\\tkp{n}$ \\ as a $k$-invariant\nfor \\ $P_{n+1}TX=P_{n+1}TY$.\n\\end{fact}\n\n\\begin{proof}\nLet \\ $M:=TP_{n}X$. \\ Making the lower right corner of Figure\n\\ref{fig6} commute on the nose (assuming \\ $q_{\\ast}$ \\ is already a\nfibration) requires the choice of a homotopy\n$$\nH:P_{n}TX\\to\\Omega\\EDL{C_{n+1}}{n+2}=\\EDL{C_{n+1}}{n+1}~,\n$$\n\\noindent so the indeterminacy for \\ $\\tkp{n}$ \\ as defined above is \\ \n$\\psi_{\\ast}p^{\\ast}[P_{n}TX,\\EDL{C_{n+1}}{n+1}]$, \\ where \\ \n$\\psi:\\EDL{C_{n+1}}{n+1}\\to\\EDL{I_{n+1}}{n+2}$ \\ classifies the extension \\ \n$$\n0\\to I_{n+1}\\to\\pinD{n+1}TX\\to C_{n+1}\\to 0\n$$\n\\noindent (Proposition \\ref{ptwelve}), and \\ \n$p=p\\q{n}_{M}:P_{n+1}M\\to P_{n}M=P_{n}TX$. \n\nOn the other hand, the $k$-invariant \\ \n$\\tkp{n}^{M}:P_{n+1}M\\to\\EDL{I_{n+1}}{n+2}$ \\ \nfor \\ $P_{n+1}TP_{n}X$ \\ (which is \\ $P_{n+1}TX$) \\ is \ndetermined only up to the actions of the group \\ $\\operatorname{haut}_{\\Lambda}(P_{n+1}M)$ \\\nof homotopy self-equivalences of \\ $P_{n+1}M$ \\ over \\\n$B_{\\D}\\Lambda$, \\ and of \\ $\\Aut_{\\Lambda}(I_{n+1})$, \\ the group of automorphisms of\nmodules over $\\Lambda$ of \\ $I_{n+1}$, \\ in \\\n$[P_{n+1}M,\\EDL{I_{n+1}}{n+2}]$. \nThus given a map \\ $f:P_{n}M\\to\\EDL{C_{n+1}}{n+1}$, \\ we obtain a \nself-map \\ $g:P_{n+1}M\\to P_{n+1}M$ \\ such that \\ \n$P_{n}g=\\operatorname{Id}_{P_{n}M}$ \\ and \\ $\\pinD{n+1}g=\\operatorname{Id}$, \\ by letting \\ \n$g=\\operatorname{Id}+i_{\\ast}p^{\\ast}(f)$, \\ for \\ $i:\\EDL{C_{n+1}}{n+1}\\to P_{n+1}M$ \\ \nthe inclusion of the fiber. It is readily verified that \n$g$ induces the identity on \\ $\\pinD{\\ast}P_{n+1}M$, \\ so \\ \n$[g]\\in \\operatorname{haut}_{\\Lambda}(P_{n+1}M)$, \\ and that \\ \n$\\tkp{n}+\\psi_{\\ast}p^{\\ast}(f)$ \\ is obtained from \\ $\\tkp{n}$ \\ under\nthe action of \\ $[g]$ \\ on \\ $\\HL{n+2}{P_{n+1}M}{I_{n+1}}$.\n\\end{proof}\n\n\\begin{notation}\\label{nbpa}\\stepcounter{subsection}\nGiven \\ $W\\simeq P_{n}X$ \\ and \\ $\\rho:P_{n+1}TX\\to P_{n+1}TW$, \\ \nProposition \\ref{pfive} allows us to write \\ $\\bPa{n}(W,\\rho)$, \\ or\nsimply \\ $\\bPa{n}W$ \\ for \\ $\\bPa{n}X\\in{\\mathcal C}$, \\ which\nthey determine up to homotopy. This comes equipped with a weak equivalence \\ \n$\\rho:P_{n+1}TX\\to P_{n+1}T\\bPa{n}W$ \\ lifting $\\rho$.\n\\end{notation}\n\n\\begin{cor}\\label{ctwo}\\stepcounter{subsection}\nThe weak equivalence \\ $\\rho:P_{n+1}TX\\to P_{n+1}T\\bPa{n}W$ \\ is\nwell-defined up to homotopy.\n\\end{cor}\n\n\\begin{proof}\nThe map $\\rho$ is inverse to $\\lambda$ in Figure \\ref{fig6}, which is induced\nby the upper right hand square, which is determined by \\ $\\tkp{n}$ \\ \nand thus up to a self-equivalence \\ $g:P_{n+1}TW\\to P_{n+1}TW$, \\ \naccording to Fact \\ref{fseven}. \\ But such a $g$ induces a canonical\nself-equivalence \\ $g':F'\\to F$, \\ where \\ \n$F':=\\operatorname{Fib}\\,(\\tkp{n}\\circ g)$, \\ and the resulting \\ \n$\\lambda':F'\\simeq P_{n+1}TX$ \\ satisfies \\ \n$\\lambda\\circ g'\\simeq\\lambda'$.\n\\end{proof}\n\n\\begin{defn}\\label{dae}\\stepcounter{subsection}\nFor $W\\simeq P_{n}X$ \\ and \\ $\\rho:P_{n+1}TX\\to P_{n+1}TW$ \\ as above, \nan extension\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ethirteen}\n0\\to \\operatorname{Coker}\\, \\pinD{n+2}\\rho\\hookrightarrow J\\to\\hspace{-5 mm}\\to\\operatorname{Ker}\\,\\pinD{n+1}\\rho \\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent is called \\emph{allowable} if its classifying cohomology class \\ \n$$\n[\\psi]\\in\\HL{n+3}{\\EDL{\\operatorname{Coker}\\, \\pi_{n+2}\\rho}{n+2}}{\\operatorname{Ker}\\, \\pi_{n+1}\\rho }\n$$\n(cf.\\ Proposition \\ref{ptwelve}) satisfies \\ $[\\psi]\\circ\\hk{n}=0$.\n\\end{defn}\n\n\\begin{prop}\\label{pthirteen}\\stepcounter{subsection}\nFor any \\ $X\\in{\\mathcal C}$, \\ the extension \\eqref{eten} is\nallowable.\n\\end{prop}\n\n\\begin{proof}\nWriting \\ $V\\simeq P_{n+1}X$ \\ and \\ $Y\\simeq\\bPa{n}X$, \\ by\nnaturality we have a commutative square:\n$$\n\\xymatrix{\nP_{n}V \\rto^-{\\protect{k_{n}}} \\dto^{=} & \\ECL{\\pinC{n+1}V}{n+2}\n\\dto^{q_{\\ast}} \\\\ \nP_{n}Y \\rto^-{\\protect{k_{n}}} & \\ECL{\\operatorname{Ker}\\,\\pi_{n+1}Tr\\q{n}}{n+2}. \\\\\n}\n$$\n\nLemma \\ref{lthree} and \\eqref{etwentyfour} then yield the following\ncommuting diagram in ${\\mathcal D}$ in which the rows and columns are all\nfibration sequences over \\ $B_{\\D}\\Lambda$: \n\n\\begin{center}\n\\begin{picture}(380,170)(0,-50)\n\\put(0,100){$\\EDL{\\operatorname{Ker}\\, h_{n}}{n+1}$}\n\\put(100,105){\\vector(1,0){45}}\n\\put(150,100){$B_{\\D}\\Lambda$}\n\\put(180,105){\\vector(1,0){40}}\n\\put(225,100){$\\EDL{\\operatorname{Ker}\\, h_{n+1}}{n+2}$}\n\\put(50,95){\\vector(0,-1){33}}\n\\put(160,95){\\vector(0,-1){30}}\n\\put(270,95){\\vector(0,-1){30}}\n\\put(30,50){$TP_{n+1}X$}\n\\put(79,55){\\vector(1,0){60}}\n\\put(95,62){$Tr\\q{n}$}\n\\put(143,50){$TP_{n}X$}\n\\put(182,55){\\vector(1,0){40}}\n\\put(195,61){$k$}\n\\put(225,50){$\\EDL{\\pinC{n+1}X}{n+2}$}\n\\put(50,45){\\vector(0,-1){33}}\n\\put(55,28){$r\\q{n+2}$}\n\\put(160,45){\\vector(0,-1){30}}\n\\put(165,28){$=$}\n\\put(270,45){\\vector(0,-1){30}}\n\\put(255,28){$q_{\\ast}$}\n\\put(40,0){$TY$}\n\\put(70,5){\\vector(1,0){70}}\n\\put(143,0){$TP_{n}X$}\n\\put(177,5){\\vector(1,0){40}}\n\\put(200,11){$\\tkp{}$}\n\\put(225,0){$\\EDL{\\operatorname{Im}\\, h_{n+1}}{n+2}$}\n\\put(270,-7){\\vector(0,-1){23}}\n\\put(257,-18){$\\psi$}\n\\put(225,-43){$\\EDL{\\operatorname{Ker}\\, h_{n+1}}{n+3}$}\n\\end{picture}\n\\end{center}\n\n\\noindent The map $k$ is induced by \\ $k_{n}$, \\ and \\ $\\tkp{}$ \\ is\ninduced by \\ $\\hk{n}$. \\ The claim then follows from the\nuniversal property for fibrations. \n\\end{proof}\n\n\\setcounter{thm}{0}\\section{The fiber of a special spherical functor}\n\\label{cfib}\n\nLet  \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ be a special spherical functor.\nWe would like to use the results of Section \\ref{ccps}  in order to\ndetermine whether a given \\ $G\\in{\\mathcal D}$ \\ is (up to homotopy) of the form \\ \n$TX$ \\ for some \\ $X\\in{\\mathcal C}$ \\ -- \\  and if so, how we can distinguish \nbetween such \\emph{realizations}, or liftings. \n\n\\subsection{Lifting objects of ${\\mathcal D}$}\n\\label{srsp}\\stepcounter{thm}\n\nLet us assume for simplicity that \\ $\\Lambda:=\\pinD{0}G$ \\ is a $\\PiC$-algebra,\nand that the map \\ $\\phi_{T}:\\Lambda\\to\\pinD{0}TB_{\\C}\\Lambda$ \\ of \\S\n\\ref{dspecial}(i) is an isomorphism. [In the general case, we are\nfaced with an additional, purely algebraic, problem of determining the\nfiber of the functor \\ $T_{\\ast}:\\Alg{\\PiC}\\to\\Alg{\\PiD}$ \\ \n(compare \\cite{BPescF}); \\ we bypassed this question in \\S \\ref{dsfunc}(iv).  \n   \nWe want a map \\ $\\varphi:TX\\to G$ \\ inducing isomorphisms \\\n$\\pinD{i}TX\\to\\pinD{i}G$ \\ for \\ $i\\geq 0$. \\ \nOur approach is inductive: we are trying to define a tower in ${\\mathcal C}$:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eseven}\n\\dotsb \\xra{p\\q{n+1}} \\Xpn{n+1} \\xra{p\\q{n}} \\Xpn{n} \\xra{p\\q{n-1}} \\dotsb\n\\xra{p\\q{0}} \\Xpn{0}\\simeqB_{\\C}\\Lambda\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which are to serve as the modified Postnikov tower of the\n(putative) \\ $X\\in{\\mathcal C}$ -- \\ so that in the end we will have \\ \n$X:=\\operatorname{holim}_{n}\\Xpn{n}$.\n\nAt the $n$-th stage \\ ($n\\geq 0$), we have constructed \\ $\\Xpn{n}$ \\ as our \ncandidate for \\ $\\bPa{n}X$ \\ -- \\ so in particular if we let \\ \n$\\Xn{n}:=P_{n}\\Xpn{n}$, \\ (our candidate for the ordinary $n$-th Postnikov \nsection of \\ $X$), \\ then \\ $T\\Xn{n}$ \\ satisfies \\eqref{enine}, \\ \n$T\\Xpn{n}$ \\ satisfies \\eqref{etwenty}, \\ and of course \\ \n$\\Xpn{n}=P_{n+1}\\Xpn{n}$.\n\nAssume also, as part of our inductive hypothesis, a given weak equivalence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eseventeen}\n\\hr{n}:P_{n+1}G\\xra{\\simeq}P_{n+1}T\\Xpn{n}.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nWe start the induction with \\ $\\Xn{0}:=B_{\\C}\\Lambda$. \\ The natural\nmap \\ $r\\q{1}:G\\to P_{1} TB_{\\C}\\Lambda=B_{\\D}\\Lambda$ \\ allows us to define \\\n$\\Xpn{0}$, \\ together with \\ $\\hr{0}:P_{1}G\\xra{\\simeq}P_{1}T\\Xpn{0}$, \\ \nas in Definition \\ref{dmps} (see \\S \\ref{rmin}).\n\n\\subsection{Lifting $\\rho\\q{n}$}\n\\label{slrpn}\\stepcounter{thm}\n\nThe first stage in the inductive step occurs in ${\\mathcal D}$: we must lift \\ \n$\\hr{n}$ \\ to \\ $\\rho\\q{n}:P_{n+2}G\\to P_{n+2}T\\Xpn{n}$. \\ \nNote that by Remark \\ref{rinfo} and Fact \\ref{fsix}, we already know the \ncomparison exact sequence \\eqref{efive} for the putative $X$ from \\ \n$h_{n+1}$ \\ down; the lifting \\ $\\rho:=\\rho\\q{n}$ \\ will determine \\ \n$\\partial_{n+2}:\\pinD{n+2}G\\to\\Gamma_{n+1}\\Xpn{n}$ \\ in addition, \nsince this is just \\ $\\pi_{n+2}\\rho$, \\ so that \\ \n$C_{n+2}:=\\operatorname{Im}\\, \\pinD{n+2}\\rho$ \\ is our candidate for \\ \n$\\operatorname{Coker}\\, h_{n+2}^{X}$, \\ while \\ $K_{n+1}:=\\operatorname{Coker}\\, \\pinD{n+2}\\rho$ \\ is our \ncandidate for \\ $\\operatorname{Ker}\\, h_{n+1}^{X}$.\n\nFrom \\eqref{etwenty} we see that the obstruction is the class:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{enineteen}\n\\chi_{n}:=k_{n+1}^{T\\Xpn{n}}\\circ\\rho\\q{n}\\in\n\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent and the different liftings are classified by \\ \n$\\HL{n+2}{G}{\\Gamma_{n+1}\\Xpn{n}}$.\n\n\\subsection{Constructing \\ $\\Xn{n+1}$}\n\\label{scpx}\\stepcounter{thm}\n\nThe next step is to choose a cohomology class \\ \n$\\hk{n}$ \\ in \\ $\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$. \\ This fits into a\ncommutative diagram with rows and fibers all fibration sequences over \\ $B_{\\C}\\Lambda$:\n$$\n\\xymatrix{\nB_{\\C}\\Lambda \\rto \\dto & \\ECL{I_{n+1}}{n+1} \\rto^{=} \\dto^{i} &\n\\ECL{I_{n+1}}{n+1} \\dto^{\\psi} \\\\\n\\Xn{n+1} \\rto^{\\bpa{n}} \\dto & \\Xpn{n} \\rto^>>>>>>>{\\hk{n}}\\dto & \n\\ECL{K_{n+1}}{n+2} \\dto^{j_{\\ast}} \\\\\n\\Xn{n+1}\\rto & \\Xn{n} \\ar@{.>}[r]^>>>>>>>>>>{k_{n}} & \\ECL{J }{n+2}\n}\n$$\nfor the bottom fibration sequence \\ $\\Xn{n+1}\\to\\Xn{n}\\to\\ECL{J }{n+2}$ \\ \nas indicated (though we shall not need this).\n\nNote that $J$, our candidate for \\ $\\pinC{n+1}X$, \\ fits into the short \nexact sequence of modules over $\\Lambda$:\n$$\n0\\to K_{n+1}\\hookrightarrow J\\to\\hspace{-5 mm}\\to I_{n+1}\\to 0,\n$$\n\\noindent as in \\eqref{eten}, and is classified by \\ \n$\\psi:=\\hk{n}\\circ i\\in\\HL{n+2}{\\ECL{I_{n+1}}{n+1}}{K_{n+1}}$, \\ \nas in Corollary \\ref{cone}. Moreover, this extension is\nobviously allowable in the sense of \\S \\ref{dae}.\n\n\\subsection{Lifting $\\rho$}\n\\label{slr}\\stepcounter{thm}\n\nTo complete the induction on \\eqref{eseventeen}, we must lift \\ \n$\\rho:G\\to P_{n+2}T\\Xpn{n}$. \\ This will be done in two steps\\vspace{2 mm} :\n\nFirst, note that we obtain a commuting diagram:\n\n\\begin{center}\n$$\n\\begin{picture}(350,180)(0,0)\n\\put(15,170){$P_{n+2}G$}\n\\put(60,175){\\vector(1,0){205}}\n\\put(140,180){$\\rho$}\n\\put(270,170){$P_{n+2}T\\Xpn{n}$}\n\\multiput(60,168)(2,-1){25}{\\circle*{.5}}\n\\put(110,143){\\vector(2,-1){2}}\n\\put(78,143){$\\rho$}\n\\put(115,135){$P_{n+2}T\\Xn{n+1}$}\n\\put(205,142){\\vector(2,1){55}}\n\\put(230,143){$\\tilde{i}_{\\ast}$}\n\\put(35,162){\\vector(0,-1){62}}\n\\put(3,130){$p\\q{n+1}_{G}$}\n\\put(150,128){\\vector(0,-1){30}}\n\\put(155,112){$p\\q{n+1}_{T\\Xn{n+1}}$}\n\\put(290,162){\\vector(0,-1){62}}\n\\put(295,130){$p\\q{n+1}_{T\\Xpn{n}}$}\n\\put(15,85){$P_{n+1}G$}\n\\put(60,90){\\vector(1,0){50}}\n\\put(80,95){$f$}\n\\put(80,82){$\\simeq$}\n\\put(115,85){$P_{n+1}T\\Xn{n+1}$}\n\\put(208,90){\\vector(1,0){55}}\n\\put(230,95){$g$}\n\\put(230,82){$\\simeq$}\n\\put(270,85){$P_{n+1}T\\Xpn{n}$}\n\\put(35,77){\\vector(0,-1){62}}\n\\put(15,40){$k_{n+1}^{G}$}\n\\put(150,77){\\vector(0,-1){28}}\n\\put(155,60){$k_{n+1}^{T\\Xn{n+1}}$}\n\\put(290,77){\\vector(0,-1){62}}\n\\put(295,40){$k_{n+1}^{T\\Xpn{n}}$}\n\\put(-30,0){$\\EDL{\\pi^{\\natural}_{n+2}G}{n+3}$}\n\\put(75,5){\\vector(1,0){190}}\n\\put(145,11){$(\\pi^{\\natural}_{n+2}\\rho)_{\\ast}$}\n\\multiput(70,10)(2,1){23}{\\circle*{.5}}\n\\put(116,33){\\vector(2,1){2}}\n\\put(90,30){$q_{\\ast}$}\n\\put(123,35){$\\EDL{C_{n+2}}{n+3}$}\n\\put(213,37){\\vector(2,-1){53}}\n\\put(235,30){$i_{\\ast}$}\n\\put(270,0){$\\EDL{\\pinD{n+1}TX}{n+3}$}\n\\end{picture}\n$$\n\\end{center}\n\n\\noindent in which the columns are fibration sequences over \\\n$B_{\\C}\\Lambda$, \\ since by definition \\ \n$$\n\\pinD{n+2}\\rho:\\pinD{n+2}G \\to\\pinD{n+1}T\\Xpn{n}=\\pinD{n+1}TX\n$$\nfactors through \\ $C_{n+2}:=\\operatorname{Im}\\, \\pinD{n+2}\\rho$, \\ so that the\nbottom triangle  commutes. \n\nSince the natural $K$-invariant \\ \\ $k_{n+1}^{G}$ \\ is given,\nthe other two $k$-invariants in the diagram above are determined by\ninverting the given homotopy equivalences \\ \n$f:P_{n+1}G\\to P_{n+1}T\\Xn{n+1}$ \\ and \\ \n$g:P_{n+1}G\\to P_{n+1}T\\Xpn{n}$ \\ (assuming all objects in ${\\mathcal D}$ are\nfibrant and cofibrant), \\ and letting \\ \n$k_{n+1}^{T\\Xn{n+1}}:=q_{\\ast}\\circ k_{n+1}^{G}\\circ f^{-1}$ \\ and \\ \n$k_{n+1}^{T\\Xpn{n}}:=i_{\\ast}\\circ k_{n+1}^{G}\\circ g^{-1}$, \\ \nusing Fact \\ref{ffive}.\n\nTherefore, the map \\ $\\rho:G\\to P_{n+2}T\\Xpn{n}$ \\ \nlifts to \\ $\\rho:P_{n+2}G\\to P_{n+2}T\\Xn{n+1}$ \\ (which is induced by \\ \n$q_{\\ast}$). \\ In fact, the lifting $\\rho$ is unique up to homotopy.\nMoreover, from the proof of Proposition \\ref{pfive} we see that this suffices \nto define \\ $\\Xpn{n+1}$, \\ as well as determining a lifting of $\\rho$ to \na weak equivalence \\ $\\hr{n+1}:P_{n+2}G\\to P_{n+2}T\\Xpn{n+1}$\\vspace{2 mm} . \n\nWe may summarize our results in:\n\n\\begin{thm}\\label{tfour}\\stepcounter{subsection}\nGiven \\ $G\\in{\\mathcal D}$, \\ there is an object \\ $X\\in{\\mathcal C}$ \\ \nsuch that \\ $TX\\simeq G$ \\ if and only if there is a tower as in \n\\eqref{eseven}, serving as the modified Postnikov tower for $X$. \\ If we \nhave constructed \\ $\\Xpn{n}$ \\ satisfying \\eqref{eseventeen} for \\ $n$, \\\na necessary and sufficient condition for the existence of an \\ $\\Xpn{n+1}$ \\ \nsatisfying \\eqref{eseventeen} for \\ $n+1$ \\ is the vanishing of \\ \n$\\chi_{n}\\in\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}$. \\ The choices are classified by: \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{$\\bullet$~}\n\\item $\\HL{n+2}{G}{\\Gamma_{n+1}{\\Xpn{n}}}$ \\ (distinguishing the\n  liftings of \\ $\\hr{n}$ \\ to \\ $P_{n+2}T\\Xpn{n}$); \\ and \n\\item $\\hk{n}\\in\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$, \\ where \\ \n$K_{n+1}:=\\operatorname{Coker}\\, \\pi_{n+2}\\rho\\q{n}$, \\ \nup to self-homotopy equivalences of \\ $\\Xpn{n}$ \\ over \\ $B_{\\C}\\Lambda$ \\ and \\ \n$\\Aut_{\\Lambda}(K_{n+1})$. \\ In particular, this distinguishes the class of \\ \n$\\pinC{n+1}X$ \\ in \\ \n$\\operatorname{Ext}_{\\RM{\\Lambda}}(\\operatorname{Ker}\\,(Tr\\q{n})_{n+1}, \\operatorname{Coker}\\,(Tr\\q{n})_{n+2})$.\n\\end{enumerate}\n\\end{thm}\n\nNote that \\ $\\Gamma_{n+1}\\Xpn{n}=\\Gamma_{n+1}\\Xpn{n+1}=\\Gamma_{n+1}X$, \\ \nby Corollary \\ref{czero}.\n\n\\subsection{Moduli spaces}\n\\label{sreal}\\stepcounter{thm}\n\nIt is possible to refine the statement of our fundamental problem of\nlifting \\ $G\\in{\\mathcal D}$ \\ to ${\\mathcal C}$ in terms of \\emph{moduli} spaces\\vspace{2 mm} : \n\nGiven a model category ${\\mathcal C}$, let $\\mathfrak{W}$ be a homotopically small \nsubcategory of ${\\mathcal C}$,  such that all maps in $\\mathfrak{W}$ are weak equivalences,\nand if \\ $f:X\\to Y$ \\ is a weak equivalence in ${\\mathcal C}$ with either $X$ or\n$Y$ in $\\mathfrak{W}$, then \\ $f\\in\\mathfrak{W}$. \\  \nRecall from \\cite[\\S 2.1]{DKanCD} that the nerve \\ $B\\mathfrak{W}$ \\ of such a\ncategory is called a \\emph{classification complex}. Its components are\nin one-to-one correspondence with the weak homotopy types (in ${\\mathcal C}$) \nof the objects of $\\mathfrak{W}$, and the component containing \\ $X\\in{\\mathcal C}$ \\ is\nweakly equivalent to the classifying space \\ $B\\operatorname{haut} X$ \\ of the\nmonoid of self-weak equivalences of $X$. \n\n\\begin{defn}\\label{drealsp}\\stepcounter{subsection}\nGiven a spherical functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ and \\ $G\\in{\\mathcal D}$, \\ we denote \nby \\ ${\\EuScript M}(G)$ \\ the category of objects in ${\\mathcal D}$ weakly equivalent to \n$G$ (with weak equivalences as morphisms), and by \\ ${\\EuScript TM}(G)$ \\ \nthe category of objects \\ $X\\in{\\mathcal C}$ \\ such that \\ \n$TX\\in{\\EuScript M}(G)$ \\ (again, with weak equivalences in ${\\mathcal C}$ as \nmorphisms). The ``pointed'' version is denoted\nby \\ $\\R{}(G)$ \\ -- \\ the category of pairs \\ $(X,\\rho)$, \\ \nwhere \\ $X\\in{\\mathcal C}$ \\ and \\ $\\rho:G\\to TX$ \\ is a specified weak equivalence.\n\\end{defn}\n\nIn all our examples the obvious functors \\ $\\R{}(G)\\xra{F}{\\EuScript TM}(G)\\xra{T} {\\EuScript M}(G)$ \\  \npreserve fibrant and cofibrant objects, and thus induce a homotopy \npullback diagram:\n$$\n\\xymatrix{\nB\\R{}(G) \\rto^{BF} \\dto &\n\tB{\\EuScript TM}(G) \\dto^{BT} \\\\\n\\{\\operatorname{Id}_{G}\\} \\rto & B{\\EuScript M}(G)\\\\\n}\n$$\n\\noindent and there are weak equivalences \\ \n$B{\\EuScript TM}(G) \\simeq \\coprod_{X\\in \\pi_{0}{\\EuScript TM}(G)} B\\operatorname{haut} X$, \\ where \\ \n$B{\\EuScript M}(G) \\simeq B\\operatorname{Aut}(G)$ \\ for \\ $\\operatorname{Aut}(G)$ \\ the monoid \nof self weak equivalences of $G$.\n\n\\subsection{Towers of moduli spaces}\n\\label{streal}\\stepcounter{thm}\n\nAlthough \\ $B{\\EuScript TM}(G)$ \\ is the more natural object of interest in \nour context, it is more convenient to study \\ $B\\R{}(G)$ \\ by means of a \ntower of fibrations, corresponding to the Postnikov system of \\ $X\\in\\R{}(G)$:\n\nLet \\ $\\R{n}(G)$ \\ denote the category whose objects are pairs \\ \n$(\\Xpn{n},\\rho')$, \\ where \\ $\\Xpn{n}\\in{\\mathcal C}$ \\ has \\ \n$P_{n+1}\\Xpn{n}\\simeq\\Xpn{n}$ \\ and \\ \n$\\rho':P_{n+1}G\\to P_{n+1}T\\Xpn{n}$ \\ is a weak equivalence.\nThe maps of \\ $\\R{n}(G)$ \\ are weak equivalences compatible \nwith the maps \\ $p\\q{n}$.\t\n\nAs in \\cite[Thm.\\ 9.4]{BDGoeR}, one can show that \\ \n$B\\R{}(G)\\simeq \\operatorname{holim}_{n} B\\R{n}(G)$, \\ so we may try to obtain \ninformation about the moduli space \\ ${\\EuScript TM}(G)$ \\ by studying the \nsuccessive stages in the tower:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{esixteen}\n\\dotsc B\\R{n+1}(G) \\xra{BF_{n}}B\\R{n}(G) \\xra{BF_{n-1}}\\dotsc\n\\to B\\R{1}(G).\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nHowever, from the discussion above we see that we need several \nintermediate steps in the study of \\ $B\\R{n+1}(G)\\to B\\R{n}(G)$, \\ \ncorresponding to the additional choices made in obtaining \\ \n$\\bPa{n+1}X$ \\ and \\ \n$p\\q{n+1}:P_{n+2}G\\xra{\\simeq}P_{n+2}T\\bPa{n+1}X$ \\ from \\ \n$\\bPa{n}X$ \\ and \\ $p\\q{n}:P_{n+1}G\\xra{\\simeq}P_{n+1}T\\bPa{n}X$. \\ \nAs a result one obtains a refinement of the tower \\ \\eqref{esixteen}, \\ \nwhere the successive fibers $F$ are either empty, or else generalized\nEilenerg-Mac Lane spaces, whose homotopy groups may be described in\nterms of appropriate Quillen cohomology groups. We leave the details\nto the reader; compare \\cite[Thm.\\ 9.6]{BDGoeR}.\n\n\\setcounter{thm}{0}\\section{Applying the theory}\n\\label{cat}\n\nThe approach to the lifting problem for a spherical functor \\ \n$T:{\\mathcal C}\\to{\\mathcal D}$ \\ described in the previous section is somewhat\nunwieldy. However, in specific applications it may simplify in various\nways.  We illustrate this by a number of examples:\n\n\\subsection{Singular chains}\n\\label{ssc}\\stepcounter{thm}\n\nConsider the singular chain functor \\ $C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$, \\ which\nin the form \\ $T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$ \\ is induced by abelianization \n(see \\S \\ref{ssfunc}(a)). Thus, given a chain complex \\ $G_{\\ast}$, \\\nwe would like to find all topological spaces $X$ (if any) with \\ \n$C_{\\ast}X\\simeq G_{\\ast}$. \\ Over $\\mathbb Z$, this is equivalent to the\nquestion of realizing a given sequence of homology groups. \n\nOur approach uses  Whitehead's exact sequence \\\n\\eqref{etwentyfive} \\ to relate the (trivial) Postnikov system for the\nchain complex \\ $G_{\\ast}$ \\ to the modified Postnikov system for the\nspace $X$,  in which we attach at each stage not a single new homotopy\ngroup, but a pair of groups in adjacent dimensions, corresponding to\nthe image and kernel respectively of the Hurewicz homomorphism.\n\nIt should be observed that the functor $T$ involves only \n``algebraic'' categories \\ ${\\mathcal C}=s\\hat{\\C}$, \\ where $\\hat{\\C}$ \\ -- \\ in our\ncase, \\ ${\\EuScript Gp}$ \\ or \\ ${\\Ab{\\EuScript Gp}}$ \\ -- \\ has a trivial model category\nstructure, as in \\S \\ref{ermc}(a-b). \\ The analysis \nin Section \\ref{cfib} then simplifies considerably, in as much as the \ncategories of $\\PiC$-algebra s and $\\PiD$-algebra s are simply \\ ${\\EuScript Gp}$ \\ and \\ ${\\Ab{\\EuScript Gp}}$, \\\nrespectively. \n\nAs noted in the Introduction, Baues's \\cite[VI, (2.3)]{BauCF} is\nactually a generalization the obstruction theory described here for\nthis case. His earlier approach in \\cite{BauHH} (as well as that of \nBenkhalifa in \\cite{BenkT} is parallel to this, though not framed in the\nsame cohomological language. See \\cite{MandE} for another viewpoint. \n\n\\subsection{Rationalization}\n\\label{srat}\\stepcounter{thm}\n\nOn the other hand, the rationalization functor \\\n$(-)_{\\mathbb Q}:{\\mathcal T}\\to{\\mathcal T}_{\\mathbb Q}$, \\ induced by the completed group ring\nfunctor \\ $\\hat{Q}:{\\EuScript Gp}\\to{\\EuScript Hopf}$ \\ (cf.\\ \\S \\ref{ssfunc}(b)), is spherical\nbut not special (Def.\\ \\ref{dspecial}), and so the theory described\nhere does not apply as is. In fact, one can see why if one\nconsiders the comparison exact sequence for $\\hat{Q}$ \\ (\\S \\ref{satt}(b)):\ngiven a (simply-connected) rational space \\ $G\\in{\\mathcal T}_{\\mathbb Q}$, \\ for\neach $\\mathbb Q$-vector space \\ $\\pi_{n}G$, \\ we need an abelian group \\  \n$A=\\pi_{n}X$ \\ such that \\ $A\\otimes\\mathbb Q\\cong\\pi_{n}G$, \\ and then lift\nthe rational $k$-invariants for $X$ to integral ones. \\ Thus, much of\nthe indeterminacy for $X$ is algebraic.\n\n\\subsection{Suspension}\n\\label{ssusp}\\stepcounter{thm}\n\nThe suspension functor \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ induced by the free\ngroup functor \\ $\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ as in \\S \\ref{ssfunc}(c), is\nsimilar to singular chains, with the generalized EHP sequence\nreplacing the ``certain long exact sequence'', and the modified\nPostniov systems involve the kernel and image of the suspension\nhomomorphism \\ $E:\\pi_{n}X\\to\\pi_{n+1}\\Sigma X$.\n\n\\subsection{Homotopy groups}\n\\label{shg}\\stepcounter{thm}\n\nThe motivating example for the treatment in this paper \\ -- \\ and the\nonly one which requires the full force of Section \\ref{cfib} \\ -- \\ \nis the functor \\ $\\pi_{\\ast}:\\TT_{\\ast}\\to\\Alg{\\Pi}$, \\ prolonged to simplicial spaces\n(as in as in \\S \\ref{ssfunc}(d)). \\  However, even this case\nsimplifies greatly if we want to realize a single $\\Pi$-algebra\\ $\\Lambda$ \\ -- \\\nthat is, we take \\ $G\\in s\\Alg{\\Pi}$ \\ to be the constant simplicial $\\Pi$-algebra \\ $B\\Lambda$.\n\nIndeed, given a simplicial space $X$ with \\ $\\pi_{\\ast} X\\simeqB\\Lambda$ \\ (which\nimplies that \\ $\\pi_{\\ast} \\|X\\|\\cong G$), from the spiral exact\nsequence \\eqref{efifteen} we find that \\\n$\\pi^{\\natural}_{n}X\\cong\\Omega^{n}\\Lambda$ \\ for all \\ $n\\geq 0$, \\ so that \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n}\\pi_{\\ast} X$ \\ is trivial for \\ $n>0$. \\ We do\nnot need the modified Postnikov system in this case: the\nobstructions to realizing $\\Lambda$ (or $G$) are just the classes \\ \n$\\chi_{n}\\in H^{n+3}(\\Lambda;\\Omega^{n+1}\\Lambda)$, \\ and the \ndifference obstructions distinguishing between the different\nrealizations are \\ $\\delta_{n}\\in H^{n+2}(\\Lambda;\\Omega^{n+1}\\Lambda)$ \\ \n($n\\geq 1$). \\ See \\cite{BDGoeR} and \\cite[\\S 5]{BJTurR} for two\ndescriptions of this case.\n\n\\begin{remark}\\label{risf}\\stepcounter{subsection}\nOur obstruction theory is irrelevant, of course, for the inverse\nspherical functor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ (see \\S \\ref{eisf}) \\ -- \\ that is, \nin determining loop structures on a given topological space.\nNevertheless, from \\eqref{etwentythree} we can easily recover the\nwell-known fact that \\ $X\\simeq\\Omega Y$ \\ is a loop space if and only if its\n$k$-invariants are suspensions of those of $Y$ (cf.\\ \\cite{AHKaneN}).\n\\end{remark}\n\n\\subsection{Lifting morphisms}\n\\label{slm}\\stepcounter{thm}\n\nIn all of the above examples, one can ask the analogous question\nregarding the lifting of \\emph{maps}, or more complicated diagrams,\nfrom ${\\mathcal D}$ to ${\\mathcal C}$. This can be addresses via Theorem \\ref{tfour} by\ntransfering the spherical structure from ${\\mathcal C}$ and ${\\mathcal D}$ to the diagram\ncategories \\ ${\\mathcal C}^{I}$ \\ and \\ ${\\mathcal D}^{I}$ \\ (cf.\\ \\S \\ref{ermc}(d)).\nSee \\cite[\\S 8]{BJTurR} for a detailed example. \n\nNote that the $k$-invariants for a map of chain complexes are not\ntrivial (cf.\\ \\cite[(3.8)]{DolH}), so the theory for realizing chain maps in \\ \n$\\TT_{\\ast}$ \\ is correspondingly more complicated.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nCoherent systems, defined as existing in a superposition of different states, form the backbone of the second quantum revolution brought about by the advent of quantum information science and technology \\cite{nielsen_chuang_2010,doi:10.1098\/rsta.2003.1227}. Formalized in terms of a quantum resource theory \\cite{aberg2006quantifying,PhysRevLett.113.140401,PhysRevLett.116.120404,RevModPhys.89.041003,PhysRevLett.119.230401,https:\/\/doi.org\/10.1002\/qute.202100040} the equivalence between coherence and entanglement (the fuel behind applications such as quantum dense coding \\cite{PhysRevLett.69.2881}, unhackable cryptography \\cite{PhysRevLett.67.661} and teleportation \\cite{PhysRevLett.70.1895}) was recognized early on \\cite{PhysRevLett.115.020403,PhysRevLett.117.020402,PhysRevLett.128.160402}. Recently, the role of coherence and its depletion during the execution of quantum algorithms has received increasing attention \\cite{PhysRevA.93.012111,arxiv.2205.13610,arxiv.2203.10632,arxiv.2112.10867,Pan_2022,PhysRevA.95.032307,e21030260}. Coherence plays a part in other physical contexts as well, such as in quantum metrology \\cite{PhysRevA.94.052324,Giorda_2017}, thermodynamics \\cite{Lostaglio2015,PhysRevLett.115.210403,PhysRevX.5.021001,Narasimhachar2015,Korzekwa_2016} and even possibly in biological processes \\cite{Lloyd_2011,doi:10.1080\/00405000.2013.829687}. Because of its usefulness as a resource, it is of particular interest to study the conditions under which coherence can be extracted or generated from other systems \\cite{PhysRevLett.113.150402,PhysRevA.101.042325,Swelling,NK:DM}, as well as devise methods for its protection \\cite{PhysRevA.99.022107,PhysRevA.104.052405,Miller_2022} against the decohering effects of the environment \\cite{PhysRevA.51.992,buchleitner2002coherent,schlosshauer}.\n\nIn this report we will examine the ability of a massive quantum field to generate or destroy coherence in a two-level Unruh-DeWitt (UDW) detector under an instantaneous interaction \\cite{Simidzija,https:\/\/doi.org\/10.48550\/arxiv.2002.01994,PhysRevD.101.036014,PhysRevD.105.065011,PhysRevD.104.125017, PhysRevD.105.085011,https:\/\/doi.org\/10.48550\/arxiv.2204.02983} (for a study of the coherence present in the field under a different context see \\cite{Huang2018,Du2017,Wu_2021,PhysRevA.105.052403}). To accomplish this we will determine the cohering and decohering power of the quantum evolution channel induced by the action of the field on the detector \\cite{PhysRevA.92.032331,BuXiong,10.5555\/3179439.3179441,BU20171670,PhysRevA.105.L060401}. Compared to other approaches that study coherence in a relativistic setting in a perturbative manner \\cite{https:\/\/doi.org\/10.48550\/arxiv.2111.01358,Feng_2022,PhysRevA.93.062105,Swelling,NK:DM,Huang2022,coh:axions}, an instantaneous interaction permits an exact solution of the final state of the detector for arbitrary coupling strengths. This provides the opportunity for a better understanding of the effects that different parameters such as the size of the detector, the energy of the field or its temperature have in the creation and destruction of coherence, free from any need for use of approximations. An example is given in Section \\ref{SecIV}, where it is observed that for specific values of the detector's radius, the amount of coherence generated by a coherent field vanishes, an effect which in a perturbative treatment would have otherwise remained unnoticed.\n\nThe reasons for considering a scalar field with mass will become apparent in Section \\ref{SecV}, where the decohering power of a thermal field with inverse temperature $\\beta$ is presented. The ability of the field to preserve part of the coherence present in a maximally coherent state of the detector is enhanced for increasing values of its mass. This observation is in line with similar perturbative results about the coherent behaviour of an atom immersed in a massive field \\cite{Huang2022} and the advantages of mass in entanglement harvesting  \\cite{Harvest:mass,JHEPentmass,https:\/\/doi.org\/10.48550\/arxiv.2206.06381} and sensing \\cite{Quantaccel,QFItherm}. Since decoherence is currently a major hurdle in practical uses of quantum computation such results may be of interest and could perhaps be leveraged with the use of massive electromagnetic fields in Proca metamaterials \\cite{Procameta}. \n\nIn \\cite{coh:axions} the authors considered the possibility of using the coherence of the detector as a means of probing the mass of axion dark matter. We show how, under a suitable choice of parameters, it is similarly possible to infer the mass of a scalar field by either measuring the cohering power of a coherent or the decohering power of a thermal state of the field. In this case changes in coherence are easier to detect since they are orders of magnitude larger than what is possible in a weak coupling treatment.\n\nWe begin by giving a short introduction in Section \\ref{SecII} to the resource theory of quantum coherence and the UDW detector model in Section \\ref{SecII}. As is common practice, throughout the manuscript we employ a natural system of units in which $\\hbar=c=k_B=1$.\n\\section{Cohering and decohering power of quantum channels}\\label{SecII}\nCoherence, i.e. the degree of superposition of a quantum system \\cite{aberg2006quantifying,RevModPhys.89.041003,PhysRevLett.119.230401}, is dependent on the choice of basis of the underlying Hilbert space in which we decide to express the state $\\rho$ of the system.  For a state of the form\n\\begin{equation}\n    \\rho=\\sum_{i,j}\\rho_{ij}\\ketbra{i}{j},\n\\end{equation}\nwhere $\\{\\ket{i}\\}_{i=0}^{d-1}$ is a finite set of basis spanning the $d$-dimensional Hilbert space $\\mathbb{C}^d$, we say that $\\rho$ represents a \\emph{coherent state} with respect to this basis, if there exists at least one pair of indices $i\\neq j$ such that $\\rho_{ij}\\neq 0$. A system which is \\emph{incoherent} is represented by a diagonal matrix and satisfies\n\\begin{equation}\n    \\Delta(\\rho) = \\rho\n\\end{equation}\nwhere \n\\begin{equation}\n    \\Delta(\\rho)=\\sum_{i}\\rho_{ii}\\ketbra{i}\n\\end{equation}\ndenotes the \\emph{dephasing operation} in the chosen basis.\n\nThe set of quantum operations acting on a state is similarly divided into those that can and those that cannot create coherence. The so called \\emph{maximally incoherent operations} (MIO) are defined as those completely positive and trace preserving operations $\\Phi$ that map the set of incoherent states $\\mathcal{I}$ onto a subset of itself\n\\begin{equation}\\label{MIO}\n    \\Phi(\\mathcal{I})\\subseteq\\mathcal{I}.\n\\end{equation}\nThe ability of a quantum channel $\\Phi$ to generate coherence out of incoherent states can be determined by calculating its \\emph{cohering power}. \\cite{PhysRevA.92.032331,BuXiong,10.5555\/3179439.3179441,BU20171670,PhysRevA.105.L060401}. In order to define the latter it is necessary first to introduce the notion of a \\emph{coherence measure}. This is a non-negative real valued function $C$ on the set of density matrices with the following properties:\n\\begin{enumerate}\n    \\item[i)] $C(\\rho)\\geq0$ with equality if and only if $\\rho\\in\\mathcal{I}.$\n    \\item[ii)] $C(\\Phi(\\rho))\\leq C(\\rho)$ for every $\\Phi$ $\\in$ (MIO).\n    \\item[iii)] $C(\\sum_ip_i\\rho_i)\\leq\\sum_ip_iC(\\rho_i)$.\n\\end{enumerate}\nThe first property requires the measure to be \\emph{faithful} so that it can distinguish between coherent and incoherent states. The second property reflects the restrictions of the theory. Since by definition (MIO)'s cannot generate coherent out of incoherent states it makes sense to require the measure to be \\emph{monotonic}, the amount of coherence in a state after the action of a (MIO) operation should therefore always be less than before. This property is what gives the theory the structure of a \\emph{quantum resource} \\cite{RevModPhys.91.025001}. The final property, which imposes \\emph{convexity} on the measure, states that it is not possible to increase the average amount of coherence in a quantum ensemble $\\{p_i,\\rho_i\\}$, where $p_i$ is the probability of obtaining state $\\rho_i$, by simply mixing its elements.\n\nArmed with a valid measure of coherence we are now able to define the cohering power of the channel as the maximum amount of coherence created when $\\Phi$ acts on the set of incoherent states\n\\begin{equation}\\label{continous coh power}\n    \\mathcal{C}(\\Phi) = \\max_{\\rho\\in\\mathcal{I}}C(\\Phi(\\rho)).\n\\end{equation}\nBecause of convexity the maximum on the right hand side is actually reached by acting $\\Phi$ on one of the basis states. This simplifies considerably the calculation since the required optimization is now performed over a discrete instead of a continuous set. In this case\n\\begin{equation}\\label{cohering power}\n    \\mathcal{C}(\\Phi) = \\max_{\\ket{i}}C(\\Phi(\\ketbra{i})).\n\\end{equation}\n\nAnother property of interest for a quantum channel is the amount of coherence that it destroys when it is applied on a \\emph{maximally coherent state}, i.e. a uniform superposition, of the form\n\\begin{equation}\\label{maximally coherent}\n    \\psi_d(\\boldsymbol\\theta)=\\frac{1}{\\sqrt{d}}\\sum_{j=0}^{d-1}e^{i\\theta_j}\\ket{j}.\n\\end{equation}\nSimilar to Eq. (\\ref{cohering power}) we define the \\emph{decohering power} of the channel as the maximum possible difference in the amount of coherence before and after its action on the maximally coherent state\n\\begin{equation}\n    \\mathcal{D}(\\Phi) = \\max_{\\boldsymbol\\theta}[C(\\psi_d(\\boldsymbol\\theta))-C(\\Phi(\\psi_d(\\boldsymbol\\theta)))].\n\\end{equation}\n\nIn what follows we will employ the oft-used $\\ell_1$-norm of coherence as our measure. This is defined as the sum of the absolute values of the non-diagonal elements of the density matrix\n\\begin{equation}\\label{l1-norm}\n    C_{\\ell_1}(\\rho) = \\sum_{i\\neq j}\\abs{\\rho_{ij}}.\n\\end{equation}\nFor the set of maximally coherent states \n\\begin{equation}\n    C_{\\ell_1}(\\psi_d(\\boldsymbol\\theta))=d-1\n\\end{equation} \nso in this case\n\\begin{equation}\\label{decohering power}\n    \\mathcal{D}_{\\ell_1}(\\Phi) = d-1 -\\min_{\\boldsymbol\\theta}C_{\\ell_1}(\\Phi(\\psi_d(\\boldsymbol\\theta))).\n\\end{equation}\n\n\\section{The Unruh-DeWitt detector model}\\label{Sec III}\nThe UDW detector model is frequently employed as a means of studying the interaction between a two-level system (the detector) and a quantum field \\cite{Unruh,DeWitt,birrell}. The interaction induces transitions between the detector's excited $\\ket{e}$ and ground $\\ket{g}$ states, which depend on the initial state of the field $\\sigma_\\phi$ as well as on the trajectory of the detector and the structure of the underlying spacetime. Coupling the \\emph{monopole} operator of the detector\n\\begin{equation}\\label{monopole}\n    \\hat{\\mu}(t)=e^{i\\Omega t}\\ketbra{e}{g}+ e^{-i\\Omega t}\\ketbra{g}{e},\n\\end{equation}\nto the field operator $\\hat\\varphi(t,\\mathbf{x})$ evaluated at the detector's position $\\mathbf{x}$ at time $t$ defines the UDW interaction Hamiltonian\n\\begin{equation}\\label{UDW:interaction}\n    \\hat{H}_{\\text{int}}(t)=\\chi(t)\\hat{\\mu}(t)\\otimes\\hat{\\varphi}(t,\\mathbf{x}).\n\\end{equation}\nwhere $\\Omega$ is the energy gap between the detector's levels, and the real valued \\emph{switching function} function $\\chi(t)$ describes the strength of the interaction at each instant in time. For a scalar field with mass $m$ the field operator in flat Miknowski spacetime is equal to\n\\begin{equation}\n    \\hat\\varphi(t,\\mathbf{x})=\\int\\frac{d^3\\mathbf{k}}{\\sqrt{(2\\pi)^32\\omega(\\mathbf{k})}}\\left(\\hat{a}_{\\mathbf{k}}e^{i(\\mathbf{k}\\cdot\\mathbf{x}-\\omega(\\mathbf{k})t)}+\\text{H.c.}\\right)\n\\end{equation}\nwhere $\\hat a_{\\mathbf{k}}$ and $\\hat a_{\\mathbf{k}}^\\dag$ denote the annihilation and creation operators respectively, of a field mode with momentum $\\mathbf{k}$ and energy $\\omega(\\mathbf{k})=\\sqrt{|\\mathbf{k}|^2+m^2}$, that satisfy the canonical commutation relations\n\\begin{equation}\\label{commut:rel}\n    [\\hat{a}_{\\mathbf{k}},\\hat{a}_{\\mathbf{k}'}]=[\\hat{a}_{\\mathbf{k}}^\\dag,\\hat{a}_{\\mathbf{k}'}^\\dag]=0,\\quad\n    [\\hat{a}_{\\mathbf{k}},\\hat{a}_{\\mathbf{k}'}^\\dag]=\\delta(\\mathbf{k}-\\mathbf{k}').\n\\end{equation}\n\nThe UDW Hamiltonian describes a point-like interaction in which the field interacts with the detector at a single point in space each time. It is possible to take into account the finite size of the detector by averaging over a region in a neighborhood of the detector's position. For a detector at rest at position $\\mathbf{x}$ \\footnote{Equation (\\ref{smeared_interaction}) can be easily extended in the case of a moving detector by making use of a Fermi-Walker coordinate system \\cite{MTW}.}, Eq. (\\ref{UDW:interaction}) is replaced by\n\\begin{equation}\\label{smeared_interaction}\n    \\hat{H}_{\\text{int}}(t)=\\chi(t)\\hat{\\mu}(t)\\otimes\\int f(\\mathbf{x}-\\mathbf{x} ')\\hat{\\varphi}(t,\\mathbf{x} ')d^3\\mathbf{x} '.\n\\end{equation}\nThe real valued  \\emph{smearing function} function $f(\\mathbf{x})$ with dimensions $(length)^{-3}$ reflects the shape and size of the detector \\cite{Schlicht_2004,Louko_2006,Wavepacket:det,CHLI2} with a mean effective radius $R$ equal to\n\\begin{equation}\\label{radius}\n    R=\\int\\abs{\\mathbf{x}}f(\\mathbf{x})d^3\\mathbf{x}.\n\\end{equation}\nBy taking the pointlike limit $f(\\mathbf{x})=\\delta(\\mathbf{x})$, (i.e., $R\\to 0$), Eq. (\\ref{UDW:interaction}) is immediately recovered. Setting\n\\begin{equation}\\label{Fourier}\n    F(\\mathbf{k})=\\int f(\\mathbf{x})e^{i\\mathbf{k}\\cdot\\mathbf{x}}d^3\\mathbf{x}, \n\\end{equation}\nfor the Fourier transform of the smearing function,  we can rewrite Eq. (\\ref{smeared_interaction}) as\n\\begin{equation}\\label{interaction}\n    \\hat{H}_{\\text{int}}(t)=\\chi(t)\\hat{\\mu}(t)\\otimes\\hat{\\varphi}_f(t,\\mathbf{x})\n\\end{equation}\nwith a `smeared' field operator of the form\n\\begin{equation}\\label{mod_smeared}\n\\hat{\\varphi}_f(t,\\mathbf{x})=\\int\\frac{d^3\\mathbf{k}}{\\sqrt{(2\\pi)^32\\omega(\\mathbf{k})}}\\left(F(\\mathbf{k})\\hat a_{\\mathbf{k}}e^{i(\\mathbf{k}\\cdot\\mathbf{x}-\\omega(\\mathbf{k})t)}+\\text{H.c.}\\right).\n\\end{equation}\n\n\\subsection{Evolution under an instantaneous interaction}\nIn order to obtain the final state of the detector after the interaction with the field has been switched off, we must first evolve the combined system of detector and field with the unitary operator $\\hat{U}$ generated by the time integral of the interaction Hamiltonian\n\\begin{equation}\\label{evolution}\n    \\hat{U}=\\mathcal{T}\\text{exp}\\left(-i\\int\\limits_{-\\infty}^{+\\infty}\\hat{H}_{\\text{int}}(t)dt\\right),\n\\end{equation}\nwhere $\\mathcal{T}$ denotes the time ordering operator. Tracing out the field degrees of freedom, induces a quantum evolution channel on the initial state $\\rho$ of the detector defined by\n\\begin{equation}\\label{Phi definition}\n\\Phi(\\rho)=\\tr_{\\varphi}[\\hat{U}(\\rho\\otimes\\sigma_\\varphi)\\hat{U}^\\dagger].\n\\end{equation}\nUnder a delta switching function centered around $t_0$,\n\\begin{equation}\n    \\chi(t)=\\lambda\\delta(t-t_0),\n\\end{equation}\nwith $\\lambda$ a coupling constant with the same dimensions as length, it is possible to drop the time ordering in \\eqref{evolution} \\cite{Simidzija,https:\/\/doi.org\/10.48550\/arxiv.2002.01994,PhysRevD.101.036014,PhysRevD.105.065011,PhysRevD.104.125017, PhysRevD.105.085011,https:\/\/doi.org\/10.48550\/arxiv.2204.02983}. In this case\n\\begin{equation}\n    \\hat{U}=\\exp[-i\\lambda\\hat\\mu_0\\otimes\\hat\\varphi_{f_0}],\n\\end{equation}\nwhere $\\hat\\mu_0=\\hat\\mu(t_0)$ and $\\hat\\varphi_{f_0}=\\hat\\varphi_{f}(t_0,\\mathbf{x})$. With a little bit of algebra it is easy to show that since $\\hat\\mu_0^2=I$\n\\begin{equation}\\label{exact unitary}\n    \\hat{U}=\\frac{I-\\hat\\mu_0}{2}\\otimes\\exp(i\\lambda\\hat\\varphi_{f_0})+\\frac{I+\\hat\\mu_0}{2}\\otimes\\exp(-i\\lambda\\hat\\varphi_{f_0}).\n\\end{equation}\n\nInserting Eq. (\\ref{exact unitary}) into Eq. (\\ref{Phi definition}), we find that the action of the channel on the detector is equal to a convex combination of a bit flip channel \\cite{nielsen_chuang_2010}\n\\begin{equation}\n    B(\\rho)=\\frac{\\rho+\\hat\\mu_0\\rho\\hat\\mu_0}{2}\n\\end{equation}\nand a unitary rotation,\n\\begin{equation}\n    V = \\sqrt{\\frac{\\abs{z}+\\Re z}{2\\abs{z}}}\\hat{I}+\\sqrt{\\frac{\\abs{z}-\\Re z}{2\\abs{z}}}\\hat\\mu_0\n\\end{equation}\n\\begin{equation}\\label{final state}\n    \\Phi(\\rho)=(1-\\abs{z})B(\\rho)+\\abs{z}V\\rho V^\\dagger\n\\end{equation}\nwhere\n\\begin{equation}\\label{z}\n    z=\\tr_\\varphi[e^{i2\\lambda\\hat\\varphi_{f_0}}\\sigma_\\varphi].\n\\end{equation}\n\\subsection{Cohering and decohering power of scalar fields}\nAccording to Eqs (\\ref{cohering power}) and (\\ref{l1-norm}), the $\\ell_1$-cohering power of the channel induced by the UDW interaction of the detector with the massive field, is equal to the maximum amount of coherence obtained by acting $\\Phi$ on either the ground or excited state. In both cases this amount is the same and equal to\n\\begin{equation}\\label{Phi cohering power}\n    \\mathcal{C}_{\\ell_1}(\\Phi) = \\abs{\\langle\\sin(2\\lambda\\hat\\varphi_{f_0})\\rangle},\n\\end{equation}\nwhere $\\langle\\hat X\\rangle=\\tr_\\varphi (\\hat X\\sigma_\\varphi)$ denotes the expectation value of field operator $\\hat{X}$. This is actually equal to the maximum amount of coherence that can be obtained by acting $\\Phi$ on the whole set of states (for more details consult Appendix  \\ref{first appendix}). \n\nTo obtain the $\\ell_1$-decohering power requires a little more effort. Replacing the maximally coherent state\n\\begin{equation}\n    \\psi_2(\\theta) = \\frac{1}{\\sqrt{2}}(\\ket{g}+e^{i\\theta}\\ket{e})\n\\end{equation}\nin Eq. (\\ref{final state}) we see that the coherence of the final state of the detector is equal to\n\\begin{multline}\\label{remaining coh}\n    C_{\\ell_1}(\\Phi(\\psi_2(\\theta))) =\\\\ \\sqrt{\\cos^2\\left(\\frac{\\theta-\\Omega t_0}{2}\\right)+(\\Re z)^2\\sin^2\\left(\\frac{\\theta-\\Omega t_0}{2}\\right)}.\n\\end{multline}\nNote that for a maximally coherent state with $\\theta=\\Omega t_0$ the amount of coherence before and after the interaction has taken place is frozen \\cite{PhysRevLett.114.210401}. For this choice of phase the state is a fixed point of the evolution channel. This observation holds in general and is independent of details such as the mass of the field, its initial state or the size of the detector.\n\nIt is straightforward to show that the minimum in Eq. (\\ref{remaining coh}) is obtained by setting $\\theta=\\pi+\\Omega t_0$. With the help of Eq. (\\ref{decohering power}) we therefore find that the $\\ell_1$-decohering power of the field induced channel is equal to\n\\begin{equation}\\label{Phi decohering power}\n    \\mathcal{D}_{\\ell_1}(\\Phi)=1-\\abs{\\langle\\cos(2\\lambda\\hat\\varphi_{f_0})\\rangle}.\n\\end{equation}\n\nWe now proceed to study the cohering and decohering power of a field in a coherent and a thermal state respectively.\n\n\\section{Cohering power of coherent scalar fields}\\label{SecIV}\n\\begin{figure*}\\begin{minipage}{\\textwidth}\n\\subfloat[${\\lambda}=0.2\\lambda_C$]{\\includegraphics[width=0.45\\textwidth]{graphs\/c02.pdf}}\\hspace{0.5cm}\n\\subfloat[${\\lambda}=\\lambda_C$]{\\includegraphics[width=0.45\\textwidth]{graphs\/c1.pdf}}\\\\\n\\subfloat[${\\lambda}=5\\lambda_C$]{\\includegraphics[width=0.45\\textwidth]{graphs\/c5.pdf}}\\hspace{0.5cm}\n\\subfloat[${\\lambda}=\\lambda_C$]{\\includegraphics[scale=0.45]{graphs\/osc.png}}\n\\end{minipage}\n\\caption{(a)-(c): $\\ell_1$-cohering power of a quantum evolution channel acting on a static Unruh-DeWitt detector as a result of its interaction with a massive scalar field in a coherent state under an instantaneous interaction, as a function of the energy of the field in field mass units, $E\/m$, and the effective radius of the detector in Compton wavelength units $R\/\\lambda_C$ for three different values of the coupling constant between detector and field. (d): Oscillatory behavior of $\\ell_1$-cohering power for $\\lambda=\\lambda_C$.}\n\\label{fig:1}\n\\end{figure*}\n\nA coherent state $\\ket{a}$ of the field is equivalent to a complex valued \\emph{coherent amplitude distribution $a(\\mathbf{k})$} such that the action of the annihilation operator $\\hat{a}_\\mathbf{k}$ on the state is equal to \\cite{PhysRev.130.2529,Simidzija,PhysRevD.96.025020}\n\\begin{equation}\n    \\hat{a}_{\\mathbf{k}}\\ket{a}=a(\\mathbf{k})\\ket{a}.\n\\end{equation} \nLet us now decompose the field into parts\n\\begin{equation}\\label{decomposed field}\n    \\hat\\varphi_{f_0} = \\hat{\\mathrm{a}} + \\hat{\\mathrm{a}}^\\dagger\n\\end{equation}\neach containing only annihilation or creation operators\n\\begin{subequations}\n\\begin{equation}\\label{phi-a}\n    \\hat{\\mathrm{a}} = \\int\\frac{d^3\\mathbf{k}}{\\sqrt{(2\\pi)^32\\omega(\\mathbf{k})}}F(\\mathbf{k})\\hat a_{\\mathbf{k}}e^{i(\\mathbf{k}\\cdot\\mathbf{x}-\\omega(\\mathbf{k})t_0)},\n\\end{equation}\n\\begin{equation}\n    \\hat{\\mathrm{a}}^\\dagger = \\int\\frac{d^3\\mathbf{k}}{\\sqrt{(2\\pi)^32\\omega(\\mathbf{k})}}F^*(\\mathbf{k})\\hat a^\\dagger_{\\mathbf{k}}e^{-i(\\mathbf{k}\\cdot\\mathbf{x}-\\omega(\\mathbf{k})t_0)}.\n\\end{equation}\n\\end{subequations}\nBy employing the Baker-Campbell-Hausdorff formula\n\\begin{equation}\n    e^{\\hat{X}+\\hat{Y}}=e^{\\hat{X}}e^{\\hat{Y}}e^{-\\frac{1}{2}[\\hat{X},\\hat{Y}]},\n\\end{equation}\nwhich holds true when both $[\\hat{X},[\\hat{X},\\hat{Y}]]=0$ and $[\\hat{Y},[\\hat{X},\\hat{Y}]]=0$, it can be shown that\n\\begin{align}\\label{zBCH}\n    \\langle e^{i2\\lambda\\hat\\varphi_{f_0}}\\rangle_a&=e^{-2\\lambda^2[\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]}\\langle e^{i2\\lambda \\hat{\\mathrm{a}}^\\dagger}e^{i2\\lambda\\hat{\\mathrm{a}}}\\rangle_a.\\nonumber\\\\\n    &=e^{-2\\lambda^2[\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]}e^{i4\\lambda\\Re \\langle\\hat{\\mathrm{a}}\\rangle_a}\n\\end{align}\nwhere\n\\begin{equation}\n    [\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]=\\frac{1}{(2\\pi)^3}\\int\\frac{\\abs{F(\\mathbf{k})}^2}{2\\omega(\\mathbf{k})}d^3\\mathbf{k},\n\\end{equation}\nand the subscript in the expectation value of the field operator is included in order to indicate its dependence on the coherent amplitude distribution.\n\nFrom Eq. (\\ref{Phi cohering power}) it follows that the $\\ell_1$-cohering power of a coherent scalar field is equal to\n\\begin{equation}\\label{coher:exp}\n    \\mathcal{C}_{\\ell_1}(\\Phi)= e^{-2\\lambda^2[\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]}\\abs{\\sin({4\\lambda\\Re \\langle\\hat{\\mathrm{a}}\\rangle_a})}.\n\\end{equation}\nAssuming a static detector with a Gaussian smearing function and a mean effective radius equal to $R$\n\\begin{equation}\\label{smear:gauss}\nf(\\mathbf{x})=\\frac{\\exp[-\\frac{4\\abs{{\\mathbf{x}}}^2}{\\pi R^2}]}{(\\pi R\/2)^3},\n\\end{equation}\nwith a corresponding Fourier transform of the form\n\\begin{equation}\n    F(\\mathbf{k})= \\text{exp}\\left[-\\frac{\\pi \\abs{\\mathbf{k}}^2R^2}{16}\\right],\n\\end{equation}\nthe commutator between $\\hat{\\mathrm{a}}$ and $\\hat{\\mathrm{a}}^\\dagger$ will now dependson the radius and mass of the field and will be equal to\n\\begin{align}\\label{mass-com}\n    [\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]&=\\frac{1}{(4\\pi^2)}\\int_{0}^\\infty\\frac{k^2e^{-\\frac{\\pi k^2R^2}{8}}}{\\sqrt{k^2+m^2}}dk\\nonumber\\\\\n    &=\\frac{\\sqrt{\\pi}}{4\\lambda_C^2}U\\left(\\frac32,2,\\frac{\\pi^3 R^2}{2\\lambda_C^2}\\right),\n\\end{align}\nwhere\n\\begin{equation}\n    U(a,b,z)=\\frac{2}{\\Gamma(a)}\\int_0^\\infty e^{-zt^2}t^{2a-1}(1+t^2)^{b-a-1}dt\n\\end{equation}\ndenotes \\emph{ Tricomi's confluent hypergeometric function} \\cite{gradshteyn2014table} and\n\\begin{equation}\n    \\lambda_C=\\frac{2\\pi}{m}\n\\end{equation}\nis the Compton wavelength of a particle with mass $m$.\n\nIn a similar fashion by defining a Gaussian coherent amplitude distribution \n\\begin{equation}\na(\\mathbf{k})=\\sqrt{\\frac{|\\mathbf{k}|}{\\omega(\\mathbf{k})}}\\frac{\\exp(-\\frac{2|\\mathbf{k}|^2}{\\pi E^2})}{(\\pi E\/2)^{3\/2}},\n\\end{equation}\nwith mean energy $E$ equal to the expectation value of the field Hamiltonian\n\\begin{equation}\n    \\hat{H}_\\phi=\\int \\omega(\\mathbf{k})\\hat a^\\dagger_{\\mathbf{k}}\\hat a_{\\mathbf{k}}d^3\\mathbf{k},\n\\end{equation} \nthe mean value of the real part of Eq. (\\ref{phi-a}) for a detector at rest at the origin of the coordinate system at a time $t_0=0$, depends on the the mean effective radius of the detector, the mean field energy and the mass of the field\n\\begin{align}\\label{mass-re}\n    \\Re \\langle\\hat{\\mathrm{a}}\\rangle_a &=\\sqrt{\\frac{8}{\\pi^4E^3}}\\int_0^\\infty{\\frac{k^{5\/2}}{\\sqrt{k^2+m^2}}}\\exp[-\\frac{k^2}{2\\sigma^2}]dk,\\\\\\nonumber\n    &=m\\sqrt{\\frac{2m^3}{\\pi^4 E^3}}\\Gamma\\left(\\frac74\\right)U\\left(\\frac74,\\frac94,\\frac{m^2}{2\\sigma^2}\\right),\n\\end{align}\nwhere for ease of notation we define\n\\begin{equation}\n\\frac{1}{\\sigma^2}=\\frac{4}{\\pi E^2}+\\frac{\\pi R^2}{8}.\n\\end{equation}\n\nIn Fig. \\ref{fig:1} (a)-(c) we present the $\\ell_1$-cohering power of the field as a function of the detector's radius, and the mean energy of the field for various values of coupling constant. We observe that the ability of the field to generate coherence in an initially incoherent detector is reduced for increasing values of the coupling strength, and tends to zero in the asymptotic limit of a large detector radius and field energy. More importantly there are regions of the parameter space where the cohering power is identically equal to zero even in the non-asymptotic limit due to its oscillatory behavior.  Because of damping, these regions are hard to spot in the figure but are easily visible once the exponential factor in Eq. (\\ref{coher:exp}) is removed such as in Fig. \\ref{fig:1} (d) for example.\n\\begin{figure}\n{\\includegraphics[width=\\linewidth]{graphs\/mass.png}}\n\\caption{Dependence of $\\ell_1$-cohering power of the field as a function of mass (measured with respect to the detector's energy gap $\\Omega$) for a field in a coherent state with mean energy $E=\\Omega$ and a detector with mean radius equal to $R=1\/\\Omega$.}\n\\label{mass:comp}\n\\end{figure}\n\nFrom Eqs. (\\ref{mass-com}) and (\\ref{mass-re}) it is also evident that for a fixed coupling strength and a specific value of the detector's effective radius and energy of the field, a massive field's cohering power is damped less and is phase-shifted towards smaller values compared to the massless case. In Fig. \\ref{mass:comp} we demonstrate the dependence of the cohering power on the field mass for different values of the coupling constant for a field with mean energy $E=\\Omega$ and a detector with mean effective radius equal to $R=1\/\\Omega$. For couplings below $\\lambda\\Omega=2$ the cohering power of the field is in one-to-one correspondence with its mass.\n\\section{Decohering power of thermal fields}\\label{SecV}\nFor a thermal field at an inverse temperature $\\beta$\n\\begin{equation}\n    \\sigma_\\phi=\\frac{e^{-\\beta\\hat{H}_\\phi}}{Z}\n\\end{equation}\nwith partition function $Z=\\tr_\\varphi e^{-\\beta\\hat{H}_\\phi}$, let $\\langle\\hat X\\rangle_\\beta$ denote the dependence of the expectation value of field operator $\\hat{X}$ on the temperature. Employing the same decomposition as in Eq. (\\ref{decomposed field}) it can be shown that in this case\n\\begin{equation}\\label{th-z}\n    \\langle e^{i2\\lambda\\hat\\varphi_{f_0}}\\rangle_\\beta = e^{-2\\lambda^2\\langle\\hat\\varphi^2_{f_0}\\rangle_\\beta}.\n\\end{equation}\nTo see this we must first rewrite the left hand side following the same steps that led to Eq. (\\ref{zBCH})\n\\begin{equation}\\label{decomposed z}\n    \\langle e^{i2\\lambda\\hat\\varphi_{f_0}}\\rangle_\\beta=e^{-2\\lambda^2[\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]}\\langle e^{i2\\lambda \\hat{\\mathrm{a}}^\\dagger}e^{i2\\lambda\\hat{\\mathrm{a}}}\\rangle_\\beta.\n\\end{equation}\nTo compute the expectation value on the right hand side we now Taylor expand $e^{i\\lambda\\hat{\\mathrm{a}}}$ and $e^{i\\lambda\\hat{\\mathrm{a}}^\\dagger}$ to obtain\n\\begin{equation}\n    \\langle e^{i2\\lambda \\hat{\\mathrm{a}}^\\dagger}e^{i2\\lambda\\hat{\\mathrm{a}}}\\rangle_\\beta=\\\\\\sum_{m,m'=0}^\\infty\\frac{(i2\\lambda)^{m+m'}}{(m!)(m'!)}\\langle(\\hat{\\mathrm{a}}^\\dagger)^m(\\hat{\\mathrm{a}})^{m'}\\rangle_\\beta.\n\\end{equation} \nNoting that because the field is diagonal in the energy basis we only need consider terms where $m=m'$ since all the rest will equal zero. We will now show that\n\\begin{equation}\\label{identity}\n    \\langle(\\hat{\\mathrm{a}}^\\dagger)^m(\\hat{\\mathrm{a}})^{m}\\rangle_\\beta = m!(\\langle\\hat{\\mathrm{a}}^\\dagger\\hat{\\mathrm{a}}\\rangle_\\beta)^m.\n\\end{equation}\nWith the help of the following identity\n\\begin{equation}\n    e^{\\hat{X}}\\hat{Y}e^{-\\hat{X}} = \\hat{Y} +\\frac{1}{2!}[\\hat{X},\\hat{Y}]+\\frac{1}{3!}[\\hat{X},[\\hat{X},\\hat{Y}]]+\\cdots\n\\end{equation}\nand the commutation relation between $\\hat{a}_{\\mathbf{k}}$ and the field Hamiltonian\n\\begin{equation}\n    [\\hat{a}_{\\mathbf{k}},\\hat{H}_\\varphi]=\\omega(\\mathbf{k})\\hat{a}_{\\mathbf{k}},\n\\end{equation}\nwe find that\n\\begin{equation}\n    \\hat{a}_\\mathbf{k} e^{-\\beta\\hat{H}_\\varphi}=e^{-\\beta \\omega(\\mathbf{k})}e^{-\\beta\\hat{H}_\\varphi}\\hat{a}_\\mathbf{k}.\n\\end{equation}\nUsing this and Eq. (\\ref{commut:rel}) it is straightforward to show that\n\\begin{equation}\n    \\langle \\hat{a}^\\dagger_\\mathbf{k} \\hat{a}_{\\mathbf{k}'}\\rangle_\\beta = \\frac{e^{-\\beta \\omega(\\mathbf{k}')}}{1-e^{-\\beta \\omega(\\mathbf{k}')}}\\delta(\\mathbf{k}-\\mathbf{k}'),\n\\end{equation}\nwhich implies by induction\n\\begin{equation}\\label{recursion}\n    \\langle\\prod_{i=1}^m\\hat{{a}}_{\\mathbf{k}_i}^\\dagger\\prod_{j=1}^m\\hat{{a}}_{\\mathbf{k}'_j}\\rangle_\\beta =\\sum_{i=1}^m\\langle \\hat{a}^\\dagger_{\\mathbf{k}_{i}} \\hat{a}_{\\mathbf{k}'_m}\\rangle_\\beta\\langle\\prod_{i\\neq i'}\\hat{{a}}_{\\mathbf{k}_i'}^\\dagger\\prod_{j=1}^{m-1}\\hat{{a}}_{\\mathbf{k}'_j}\\rangle_\\beta.\n\\end{equation}\nIt follows that\n\\begin{equation}\n    \\langle(\\hat{\\mathrm{a}}^\\dagger)^m(\\hat{\\mathrm{a}})^{m'}\\rangle_\\beta =m\\langle\\hat{\\mathrm{a}}^\\dagger\\hat{\\mathrm{a}}\\rangle_\\beta\\langle(\\hat{\\mathrm{a}}^\\dagger)^{m-1}(\\hat{\\mathrm{a}})^{m-1}\\rangle_\\beta.\n\\end{equation}\nfrom which Eq. (\\ref{identity}) can be obtained recursively. Finally\n\\begin{equation}\n    \\langle e^{i2\\lambda \\hat{\\mathrm{a}}^\\dagger}e^{i2\\lambda\\hat{\\mathrm{a}}}\\rangle_\\beta =  e^{-4\\lambda^2\\langle\\hat{\\mathrm{a}}^\\dagger\\hat{\\mathrm{a}}\\rangle_\\beta}=e^{2\\lambda^2[\\hat{\\mathrm{a}},\\hat{\\mathrm{a}}^\\dagger]}e^{-2\\lambda^2\\langle\\hat{\\varphi}^2_{f_0}\\rangle_\\beta}\n\\end{equation}\nwhich completes the proof.\n\nLooking back at Eq. (\\ref{th-z}) and noting that $\\mathcal{C}_{\\ell_1}(\\Phi)=\\abs{\\Im z}$ we observe, perhaps not surprisingly, that a thermal field is incapable of generating coherence through an instantaneous interaction. On the other hand its decohering power is equal to\n\\begin{equation}\n    \\mathcal{D}_{\\ell_1}(\\Phi)=1-e^{-\\lambda^2I(\\beta)}\n\\end{equation}\nwhere\n\\begin{equation}\n    I(\\beta) = \\frac{1}{(2\\pi)^3}\\int\\frac{\\abs{F(\\mathbf{k})}^2}{\\omega(\\mathbf{k})}\\coth\\left( \\frac{\\beta\\omega(\\mathbf{k})}{2}\\right)d^3\\mathbf{k}.\n\\end{equation}\n\nSince\n\\begin{equation}\n    \\frac{\\partial{I(\\beta)}}{\\partial R}<0, \\quad \\frac{\\partial{I(\\beta)}}{\\partial m}<0, \\quad\n    \\frac{\\partial{I(\\beta)}}{\\partial \\beta}<0,\n\\end{equation}\nthe decohering power decreases for increasing values of the detector's radius and the mass of the field, while it increases with temperature. This is evident in Fig. \\ref{Fig:Thermal2} (a) and (b) where we present the $\\ell_1$-decohering power of a thermal field as a function of the detector's mean radius and the field's temperature for a detector with the same Gaussian smearing function as in Eq. (\\ref{smear:gauss}).\n\\begin{figure}\\begin{minipage}{\\columnwidth}\n\\subfloat[]{\\includegraphics[width=\\textwidth]{graphs\/therm3dR.pdf}}\\\\\n\\subfloat[]{\\includegraphics[width=\\textwidth]{graphs\/therm3dT.pdf}}\n\\end{minipage}\n\\caption{$\\ell_1$-decohering power of a massive scalar field with respect to a) the radius of the detector for a field with inverse temperature $\\beta=2\/\\Omega$ and a coupling constant $\\lambda=\\Omega$ and b) with respect to temperature for a detector with radius equal to $R=1\/\\Omega$ and a coupling constant $\\lambda=2\\Omega$.}\\label{Fig:Thermal2}\n\\end{figure}\n\\section{Discussion}\nEmploying an instantaneous interaction between a two-level UDW detector and a massive scalar field we investigate the ability of the field to generate or destroy coherence in the detector. This non-perturbative approach allows for an exact examination of the effects that different parameters (such as the strength of the coupling constant, the size of the detector, the energy of the field and its temperature for example) have on the cohering and decohering power of the induced quantum evolution channel.\n\nIn the case of coherence generation by a coherent field state it was shown that the success of the process depends on the size of the detector. More specifically, apart from the point-like $R<<R_0$, and macroscopic limit $R>>R_0$, where $R_0$ is some characteristic wavelength (e.g., the transition wavelength $\\frac{2\\pi}{\\Omega}$ in the massless and Compton wavelength $\\frac{2\\pi}{m}$ in the massive case) there exist non-trivial values of the detector's radius for which it is impossible to generate any amount of coherence between its energy levels (Fig. \\ref{fig:1} (d)). This phenomenon which manifests itself even in the case of a moderately weak coupling, as in Fig. \\ref{fig:1} (a), demonstrates how the size of the system, which we wish to bring into a superposition of states, needs to be taken into consideration when designing experiments. It is expected that such effects will also be present in the generation of other quantum resources from the field, as in entanglement harvesting  \\cite{VALENTINI1991321,Reznik2003,PhysRevA.71.042104,PhysRevD.92.064042}, for example. \n\nBy calculating the decohering power of a thermal field, we investigated its effect on the set of maximally coherent states of the detector. In Fig. \\ref{Fig:Thermal2}, it was demonstrated that for fixed values of the detector's radius, the field's temperature, and the coupling constant between the two, a massive field performs better than a massless one. Under a perturbative approach, similar results have also been reported in the limiting case of a point-like detector interacting weakly with the field in \\cite{Huang2022}. \n\nFor a suitable choice of detector radius, field energy and coupling strength it is possible to infer the mass of the field by measuring the amount of coherence present in the detector. This is evident from Fig. \\ref{mass:comp}, where for a detector with $R=1\/\\Omega$, a field with energy $E=\\Omega$ and coupling constants below $\\lambda=2\/\\Omega$ the cohering power of the field is in one-to-one correspondence with its mass. The same conclusion can be made by studying the decohering power of a thermal field. Similar approaches have previously been employed for distinguishing the kinematic state of a detector interacting with a massive field by measuring its transition probability \\cite{Quantaccel}, for determining the distance of closest approach for two accelerating detectors by studying the amount of entanglement that they can harvest from the field \\cite{Salton_2015} and for probing the mass of an axion dark matter field by measuring the coherence stored in a detector \\cite{coh:axions}.\n\nThe UDW Hamiltonian in Eq. \\eqref{monopole} contains all of the essential features of the interaction of matter with an electromagnetic field \\cite{Wavepacket:det,CHLI2}, where the analogue of a massive field in this case is a Proca field \\cite{jackson1999classical}. Since massive electromagnetic theory is equivalent to Maxwell theory in Proca metamaterials \\cite{Procameta}, the above results could find application for the construction of novel technologies such as new types of quantum memories communication channels and sensors. For this reason a more complete investigation of cohering and decohering effects, for detectors interacting continuously with massive fields, by making use of other non-perturbative methods \\cite{bruschi2013,MartinezNonPert} would certainly be of interest.\n\\acknowledgments{}\nN. K. K. acknowledges the support and hospitality of the Kenneth S. Masters foundation during preparation of this manuscript. D. M. 's research has been co-financed by Greece and the  European Union (European Social Fund-ESF) through the Operational Programme ``Human Resources Development, Education and Lifelong Learning\" in the context of the project ``Reinforcement of Postdoctoral Researchers - 2nd Cycle\" (MIS-5033021), implemented by the State Scholarships Foundation (IKY).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{intro}Introduction} \n\nMassive stars are essential constituents of stellar populations and galaxies in the near and far Universe. They are among the most important sources of ionizing photons, energy, and some chemical species, which are ejected into the interstellar medium through powerful stellar winds and during their extraordinary deaths as supernovae (SN) and long gamma-ray bursts (GRB). For these reasons, massive stars are often depicted as cosmic engines, because they are directly or indirectly related to most of the major areas of astrophysical research.\n\nDespite their importance, our current understanding of massive stars is still limited. This inconvenient shortcoming can be explained by many reasons on which we elaborate below. First, the physics of star formation mean that massive stars are rare \\citep{salpeter55}. Moreover, their lifetime is short, of a few to tens of millions of years \\citep[e. g.,][]{ekstrom12,langer12}. These factors make it challenging to construct evolutionary sequences and relate different classes of massive stars. This is in sharp contrast to what can be done for low-mass stars.\n\nSecond, one can also argue that the evolution of massive stars is extremely sensitive to the effects of some physical processes, such as mass loss and rotation \\citep{maeder_araa00,heger00}, that have relatively less impact on the evolution of low-mass stars. However, the current implementation of rotation in one-dimensional codes relies on parametrized formulas, and the choice of the diffusion coefficients has a key impact on the evolution \\citep{meynet13a}. Likewise, mass-loss recipes arising from first principles are only available for main sequence (MS) objects \\citep{vink00,vink01} and a restricted range of Wolf-Rayet (WR) star parameters \\citep{grafener08}. Third, binarity seems to affect the evolution of massive stars, given that a large portion of them are in binary systems that will interact during the evolution \\citep{sana12}.\n\nFourth, our understanding of different classes of stars is often built by comparing evolutionary models and observations. However, mass loss may affect the spectra, magnitudes, and colors of massive stars, thus making the comparison between evolutionary models and observations a challenge. In addition to luminosity, effective temperature, and surface gravity, the observables of massive stars can be strongly influenced by a radiatively driven stellar wind that is characteristic of these stars. The effects of mass loss on the observables depend on the initial mass and metallicity, since they are in general more noticeable in MS stars with large initial masses, during the post-MS phase, and at high metallicities. When the wind density is significant,  the mass-loss rate, wind clumping, wind terminal velocity, and velocity law have a strong impact on the spectral morphology. This makes the analysis of a fraction of massive stars a difficult task, and obtaining their fundamental parameters, such as luminosity and effective temperature, is subject to the uncertainties that comes from our limited understanding of mass loss and clumping.  Furthermore, the definition of effective temperature of massive stars with dense winds is problematic and, while referring to an optical depth surface, it does not relate to a hydrostatic surface. This is caused by the atmosphere becoming extended, with the extension being larger the stronger the wind is.  Stellar evolution models are able to predict the stellar parameters only up to the stellar hydrostatic surface, which is not directly reached  by the observations of massive stars when a dense stellar wind is present. Since current evolutionary models do not  thoroughly simulate the physical mechanisms happening at the atmosphere and wind, model predictions of the evolution of massive stars are difficult to be directly compared to observed quantities, such as a spectrum or a photometric measurement.\n\nThe main driver of this paper is to improve the comparison between models and observations of massive stars. To properly understand the Physics that govern massive stars, it is urgently necessary to combine stellar evolutionary calculations to radiative transfer models of the stellar atmosphere. Essentially, the atmospheric models allow the physical quantities predicted by the stellar evolution model to be directly compared to observed features. We build on \nearlier studies in this direction that were made by the Geneva group, which  coupled an earlier version of the Geneva stellar evolution code with the ISAWIND atmospheric code  \\citep{dekoter93,dekoter97}, creating the Costar models \\citep{schaerer96a}. These models focused mainly on the spectroscopic evolution during the Main Sequence (MS; \\citealt{schaerer96b,schaerer97a}), and on the effects of mass loss on the evolution and envelope structure of Wolf-Rayet (WR) stars \\citep{schaerer96wr}. \n\nFrom the Costar models up to now, significant improvements both in the stellar evolution and atmospheric models have been accomplished. From the atmospheric modeling perspective, the main advances have been the inclusion of full line blanketing and line overlap, updated and extended atomic data, and wind clumping.  Solving the radiative transfer across the atmosphere of massive stars is a complex and demanding task, and adequate atmospheric codes became available only in the past decade. Complex radiative transfer models such as CMFGEN \\citep{hm98}, PoWR \\citep{hamann06}, and FASTWIND \\citep{puls06}, that take into account the necessary physics to study the radiation transport across the atmosphere and wind, have been separately employed to analyze observations of O stars (e.g., \\citealt{hillier03,martins05,bouret03,bouret05,puls06,marcolino09,najarro11,repolust04,mokiem05,mokiem07b,tramper11}), LBVs (e.g., \\citealt{hillier01,ghd06,ghd09,ghd11,ghm12,najarro09}), and WRs (e.g., \\citealt{hm99,dessart00,grafener05,sander12}). From the stellar evolution perspective, the main advances have been to include the effects of rotation and magnetic fields, and improve opacities and mass-loss recipes.\n\nAs of yet, however, no code has been capable of studying the evolution of the spectra of massive stars throughout their entire evolution, since the modern atmospheric\/wind models have never been coupled to stellar evolutionary models. Here we bridge this gap by performing, for the first time, coupled calculations of stellar evolution with the Geneva code and atmospheric and wind modeling with the CMFGEN code. This approach opens up the possibility to investigate stellar evolution based not only on interior properties, but also from a spectroscopic point of view. This allow us to relate interior properties of the star with its appearance to the observer. Our ultimate goal is to provide improved comparison between models and observations of massive stars. \n\nIn this first paper of a series, we analyze the spectroscopic and photometric evolution of a non-rotating 60~\\msun\\ star at solar metallicity, from the zero-age main sequence (ZAMS) until the pre-SN stage. We choose a non-rotating model to properly disentangle the effects that mass loss and rotation have on the evolution of the spectra of massive stars. The reasons for using a 60~\\msun\\ star are twofold. First, this initial mass is representative of the qualitative evolution of the most massive stellar models, in the range 50--120~\\msun. Second, stars with this initial mass do not evolve through a red supergiant (RSG) phase, which allow us to use a single atmospheric code (CMFGEN) to analyze the whole evolution. \n\nThe same modeling approach described here has been employed in previous papers from our group that analyzed the properties of massive stars just before the SN explosion \\citep{gme13,gmg13,gge13}. In \\citet{gme13}, we found that rotating stars with initial mass (\\mini) in the range 20--25 ~\\msun\\ end their lives as luminous blue variable (LBV) stars. The fate of single massive stars with $\\mini=9-120~\\msun$ was investigated in \\citet{gmg13}, where we showed that massive stars, depending on their initial mass and rotation, can explode as red supergiants (RSG), yellow hypergiants (YHG), LBVs, and Wolf-Rayet (WR) stars of the WN and WO subtype. We applied these models to investigate the nature of the candidate progenitor of the SN Ib iPTF13bvn, concluding that a single WR star with initial mass $\\sim31-35~\\msun$ could explain the properties of the progenitor \\citep{gge13}. These analyses showed that it is crucial to produce an output spectrum out of evolutionary calculations to properly interpret the observations.\n\nThis paper is organized as follows. In Sect.~\\ref{model} we describe our modeling approach, while we discuss our definitions of evolutionary and spectroscopic phases in Sect. \\ref{evolspec}. We analyze the evolution of a non-rotating 60~\\msun\\ star in Sect.~\\ref{broadevol}. In Sect. \\ref{lifetimes} we investigate the lifetimes of different evolutionary  and spectroscopic phases, while in Sect. \\ref{origspec} we discuss how evolutionary phases are linked to different spectroscopic phases. Sect. \\ref{specevol} analyzes the evolution of the spectra, magnitudes, colors, and ionizing flux across the star's lifetime. Caveats of our analysis are discussed in Sect. \\ref{caveat}, and our concluding remarks are presented in Sect. \\ref{conc}.\n\nIn a series of forthcoming papers, we will present the results for a larger initial mass range and investigate the effects of rotation, metallicity, and magnetic fields on the spectroscopic evolution.\n\n\\section{\\label{model}Physics of the models}\n\nWe compute coupled models using the Geneva stellar evolution and the CMFGEN atmosphere\/wind radiative transfer codes. Evolutionary models from the ZAMS to the pre-SN stage comprises tens of thousands of calculations of stellar structures. Given that a typical CMFGEN model takes about half day of CPU time to finish, it is impracticable to compute full atmosphere\/wind modeling at each timestep of the evolution and still produce a grid of evolution models. Given the huge computing effort involved, our strategy is to perform post-processing atmospheric\/wind radiative transfer on already existing evolutionary calculations. We apply this procedure to 53 stages that are carefully selected to sample the full evolution (see Table \\ref{model log}). The advantages are that we can benefit from the Physics included in two of the most up-to-date and advanced codes, produce a grid of evolution models with output spectra, and analyze the grid in a manageable amount of time. The main disadvantage is that our models do not include feedback effects of the wind on the evolution \\citep{schaerer96wr} and envelope structure \\citep{grafener13}. Below we describe the Geneva stellar evolution code (Sect. \\ref{evolcode}), the CMFGEN atmospheric\/wind code (Sect. \\ref{cmfgen}), how the two codes are combined (Sect.~\\ref{combine}), and the criteria used for spectroscopic classification (Sect. \\ref{classifyspec}).\n\n\\subsection{\\label{evolcode} Stellar evolution}\n\nThe evolutionary model of a 60~\\msun\\ star discussed here has been computed by \\citet{ekstrom12}  with the Geneva stellar evolution code, as a part of a large grid of models. We refer the interested reader to the aforementioned paper for further details. The main characteristics of the code are summarized below.\n\nThe model assumes solar metallicity (Z=0.014) and initial abundances from \\citet{asplund09}. For the H-core and He-core burning phases, an overshoot parameter of $d_\\mathrm{over}= 0.10 H_P$ is assumed\\footnote{This value of $d_\\mathrm{over}$ is chosen to reproduce the MS width between 1.35 and 9~\\msun, and is smaller than in other evolutionary codes (see comparison in \\citealt{martins13}).}, where $H_P$ is the local pressure scale height. Mass loss is a key ingredient in the models, affecting not only the evolution throughout the Hertzprung-Russel (HR) diagram but also the emerging spectrum. Therefore, most conclusions achieved in this paper depend on the mass-loss recipe used in the computations. To select the most suitable mass-loss recipe, criteria based on chemical abundances and the effective temperature estimated by the Geneva code (\\teffg; see Sect. \\ref{temp}) are used. The relevant criteria and respective mass-loss recipes relevant for the 60~\\msun\\ model discussed here are: \\\\\n-- surface H abundance (\\xsurf)  $> 0.3$ and $\\log(\\teffg\/\\K)>3.9)$: \\citet{vink01};\\\\\n-- $\\xsurf < 0.3$ and $3.900 < \\log(\\teffg\/\\K) \\leq 4.000$): \\citet{vink01};\\\\\n-- $\\xsurf < 0.3$ and $4.000 < \\log(\\teffg\/\\K) \\leq 4.477$): \\citet{nl00};\\\\\n-- $\\xsurf < 0.3$ and $4.477 < \\log(\\teffg\/\\K) \\leq 4.845$): \\citet{grafener08} or \\citet{vink01}, whichever gives the highest value of $\\mdot$;\\\\\n-- $\\xsurf < 0.3$ and $4.845 \\geq \\log(\\teffg\/\\K)$: \\citet{nl00};\\\\\n-- $\\log(\\teffg\/\\K) \\leq 3.9)$: \\citet{dejager88}.\n\nThe 60~\\msun\\ star evolution model used here has been computed up to the Si burning phase. Here we discuss the results up to the end of C burning, since no appreciable changes in the surface properties are seen beyond this phase up to core collapse.\n\n\\subsection{\\label{cmfgen} Atmospheric and wind modeling}\n\nTo compute the output spectra\\footnote{The spectra computed here (in vacuum wavelengths) and evolutionary models are public available through the webpage  \\url{http:\/\/obswww.unige.ch\/Recherche\/evol\/-Database-}} we use the atmospheric radiative transfer code CMFGEN \\citep{hm98}. CMFGEN is a  spherically-symmetric, fully line blanketed code that computes line and continuum formation in non-local thermodynamical equilibrium. Since the evolutionary model discussed here has no rotation, the use of spherical symmetry seems justified\\footnote{Note, however, that wind inhomogeneities could break the spherical symmetry if they have a large scale length.}. CMFGEN computes a self-consistent radiative transfer including the stellar hydrostatic surface and the wind. CMFGEN is suitable for stars with $\\teffg > 7500~\\K$,  a condition that is satisfied during almost all the lifetime of a non-rotating 60~\\msun\\ star. For the $10^4$ years when $\\teffg < 7500~\\K$, no spectra is computed, and the synthetic photometry (Sect. \\ref{specevol}) is linearly interpolated in age between stages 27 and 28 (see Sect. \\ref{broadevol}). \n\nA CMFGEN model needs as input the luminosity (\\lstar), effective temperature at a reference optical depth, mass (\\mstar), and surface chemical abundances. For consistency, we adopt in CMFGEN the same $\\mdot$ recipe as that used by the Geneva evolution code.The momentum equation of the wind is not solved and a velocity structure $v(r)$ needs to be  adopted. For the wind part, we assume a standard $\\beta$-type law, with $\\beta=1$ if $\\teffg > 21000~\\K$ and $\\beta=2.5$ otherwise. A hydrostatic solution is computed iteratively for the subsonic portion and is applied up to 0.75 of the sonic speed, where the hydrostatic and wind solutions are merged. This scheme has been computed for all models except those with $\\teffg < 9000~\\K$. These models are cool LBVs that have dense winds and an optical depth larger than 100 at the sonic point. In these cases, the computation of the hydrostatic structure failed and we employed an approximate solution following a scale-height approach. The wind terminal velocity ($\\vinf$) is computed using the parametrization from \\citet{kudritzki00} for OB stars and LBVs, and from \\citet{nl00} for WR stars of the WN and WC type. For WO stars, an iterative scheme is adopted. We initially compute a spectrum with the value of $\\vinf$ as given by the \\citet{nl00} recipe, which is typically at most $\\sim2800~\\kms$. If a WO-type spectrum arises, we recompute a spectrum with $\\vinf=5000$~\\kms\\, which is more representative of the observed Galactic WO stars \\citep{drew04,sander12}.\n\nOptically-thin wind clumping is included via a volume filling factor ($f$) approach, which assumes dense clumps and a void interclump medium. The wind is also assumed to be unclumped close to the stellar surface and to acquire full clumpiness at large radii. The variation of $f$ as a function of distance from the center of the star ($r$) is given by \n\\begin{equation}\n\\label{clump}\nf(r)=f_\\infty+(1-f_\\infty)\\exp[-v(r)\/v_c]\\,\\,, \n\\end{equation} \nwhere $f_\\infty$ is the filling factor and $v_c$  is the velocity at which clumps start to form. For O stars we assume $f_\\infty=0.2$ and $v_c=30~\\kms$. The assumed $f$ is what is typically needed to bring in agreement the observed wind momentum of O stars with the \\citealt{vink01} \\citep{repolust04,mokiem07b}. For subsequent evolutionary phases we assume $f_\\infty=0.1$, which is characteristic of WR stars. We use $v_c=20~\\kms$  for B stars and LBVs, and $v_c=200~\\kms$ for WR stars. We discuss how wind clumping affects our results in Sect. \\ref{mdotclump}.\n\nX-rays are included only for OB-type stars and, for simplicity, we assume fixed x-ray temperatures and filling factors for all OB star models. A two-component plasma is assumed (see \\citealt{hm98,pauldrach94,najarro11}), one with temperature $T=2\\times10^6\\K$ and filling factor of 6.0$\\times10^{-2}$, and the other with $T=6\\times10^6\\K$ and filling factor 8.0$\\times10^{-3}$. For both components, we assume a velocity of 500~\\kms\\ for the shocks to become important.\n\n\\begin{figure}\n\\center\n\\resizebox{0.99\\hsize}{!}{\\includegraphics{cmfgen_evol_merge_temp_den_struct.pdf}}\\\\\n\\caption{\\label{tstruct} Temperature (panel $a$) and density ($b$) as a function of radius for a non-rotating star with initial mass of 60~\\msun\\ at metallicity $Z=0.014$ for a O I model at stage 3 (as defined in Sect. \\ref{broadevol}). The solution obtained by the Geneva code is shown in black, while the final CMFGEN solution is displayed in solid red. A one-iteration CMFGEN solution used for initial guessing of the T structure is shown by the dashed blue line. A zoom-in on the connecting region of the Geneva and CMFGEN solutions (at $\\tauross=10$) is shown in the inset. The temperature and density structures of a model at stage 48 (WC) is displayed in panels $c$ and $d$, respectively.}\n\\end{figure}\n\n\\subsection{\\label{combine} Combining the Geneva code and CMFGEN solutions}\n\nThe stellar structure calculations with the Geneva code produce as output \\teffg,  \\lstar, \\mstar, and surface abundances, among others. They are used as input for the CMFGEN model and  we use the temperature and density structures of the stellar envelope to merge the CMFGEN and stellar structure solutions. They are merged at a Rosseland optical depth (\\tauross) of 10, to ensure that the envelope solution computed by the Geneva code is used in the inner part of the atmosphere ($\\tauross \\geq 10$), and that the CMFGEN solution is employed in the outer atmosphere\/wind ($\\tauross < 10$).\n\nIn practice, the merging is achieved by running a CMFGEN model for one iteration, assuming for this initial iteration only that \\teffg\\ corresponds to the effective temperature at \\tauross=10. The output T structure from the one-iteration CMFGEN model is then compared to the T structure of the Geneva code. An inconsistent and non-continuous solution will generally be obtained, which means that the radius of the CMFGEN model needs to be adjusted in order that the two T solutions match at $\\tauross=10$. The procedure is illustrated in Fig. \\ref{tstruct} for an O star model at the MS and a WR model of the WC subtype at the He-core burning phase. As one can see, our approach guarantees that the temperature and density structure are continuous from the center of the star through the envelope up to the stellar wind, at several hundred \\rstar.\n\nThe procedure above allows us to compute the values of the effective temperature out of evolutionary models in a novel way. We use the CMFGEN results to determine $\\tauross (r)$ and, with that, we find the photospheric radius as $\\reff=r(\\tauross=2\/3)$. The effective temperature is calculated as\n\\begin{equation}\n\\label{teffeq}\n\\teffc=\\left({\\frac{\\lstar}{4 \\pi \\sigma \\reff^2}}\\right)^{1\/4},\n\\end{equation}\nwhere $\\sigma$ is the Steffan-Boltzmann constant.\n\n \\subsubsection{\\label{temp} Comparison with previous determinations of the effective temperature by the Geneva code}\n\nPrevious papers using the Geneva code present evolutionary tracks using $\\teffg$  \\citep[e.g.,][for the more recent ones]{ekstrom12,georgy12a}. In this subsection, we investigate how it compares with our revised $\\teffc$.  In particular, the optical depth ($\\tau$) scale computed by CMFGEN may differ from that obtained using the simplified approach from the Geneva code. Let us recall that \\teffg\\ is one of the boundary conditions needed to solve the usual set of stellar structure equations and is determined using an iterative, numerical scheme. A plane parallel, gray-atmosphere is assumed by the Geneva code. We refer the reader to Chapter 24 of \\citet{maeder09} and Chapters 11 and 12 of \\citet{kip13} for thorough reviews.\n\nWhen the optical depth of the wind is appreciable, for instance when the star is a WR, the $\\tau=2\/3$ surface moves outwards in comparison with the optically-thin case \\citep{deloore82}. To correct for this effect, the Geneva code uses a simple scheme that takes into account both electron-scattering and line opacities. The correction is based on the relationship from \\citet{langer89}\\footnote{Note that the original results from \\citet{langer89} assume electron-scattering opacity only.} to compute the radius where $\\tau=2\/3$ as:\n\\begin{equation}\nR_\\mathrm{phot,GVA}=R_{\\star,GVA}+3 \\bar{\\kappa} \\mdot\/(8 \\pi \\vinf),\n\\end{equation}\nwhere $R_{\\star,GVA}$ is the radius associated with \\teffg\\ via the Steffan-Boltzmann relation. This equation is valid for $\\beta=2$, and $\\vinf=2000~\\kms$ is assumed in the Geneva code. Here, $\\bar{\\kappa}$ is a flux-weighted mean opacity, which is computed using the modified radiation-driven wind theory \\citep{kudritzki89} as $\\bar{\\kappa}=\\sigma_{\\mathrm e} (1+M)$, where $\\sigma_{\\mathrm e}$ is the electron-scattering opacity and $M$ is the force-multiplier parameter \\citep{cak}. The value of $M$ depends on the line-force parameters $k$, $\\alpha$, and $\\delta$, which are assumed constant in the Geneva code with values $k=0.124$, $\\alpha=0.64$, and $\\delta=0.07$\\footnote{These values correspond to those of an O star with $\\teff=50000~\\K$ \\citep{pauldrach86}.}. The effective temperature corrected for the wind optical depth is then estimated as \n\\begin{equation}\n\\label{testeq}\n\\teffgcorr=\\left({\\frac{\\lstar}{4 \\pi \\sigma R_\\mathrm{phot,GVA}^2}}\\right)^{1\/4}.\n\\end{equation}\nIn practice, \\teffgcorr\\ is computed by the Geneva code only at phases when the H surface abundance is less than 0.3 by mass, corresponding roughly to the WR phase \\citep{ekstrom12}. We refer the reader also to Sect. 2.7 of \\citet{schaller92} for further details.\n\nFigure \\ref{teffcomp} presents the 60~\\msun\\ evolutionary track in the HR diagram computed with the three effective temperature definitions described above. Important differences can be readily noted. First, in the regime where the winds are optically thin in the continuum, the values of $\\teffc$ are similar to those of $\\teffg$. This occurs from the beginning of the evolution up to stage 9. Second, throughout the rest of the evolution, $\\teffc$ is lower than $\\teffg$, indicating that the optical depth of the wind is non-negligible. Third, the wind optical depth becomes important even when $\\xsurf>0.3$. This means that \\teffgcorr\\ is not applied at all phases where a dense wind is present, such as when the star is an LBV. Fourth, in the regime of optically-thick winds, \\teffgcorr\\  is systematically lower than \\teffc. The difference amounts from thousands to several tens of thousands of K, depending on the parameter space. The reasons for the difference are the simple assumptions behind the computation of \\teffgcorr. Among them, we highlight that the opacity is computed using constant values of $k$, $\\alpha$, and $\\delta$, which are more appropriate to O stars \\citep{pauldrach86} and may not apply to WR stars \\citep{schmutz97,grafener05}. In addition, a high value of $\\beta=2$ is assumed when calculating \\teffgcorr, while the computation of $\\teffc$ uses $\\beta=1$. We note that detailed spectroscopic analyses  \\citep[e.g.,][]{hm99,sander12} and hydrodynamical models of WR winds \\citep{grafener05} indicate that $\\beta=1$ is preferred. A higher value of $\\beta$ produces a shallow density structure at the base of the wind, shifting the photosphere outwards compared to the $\\beta=1$ case. \n\n\\begin{figure}\n\\center\n\\resizebox{0.980\\hsize}{!}{\\includegraphics{teff_comp.pdf}}\\\\\n\\caption{\\label{teffcomp} Evolutionary tracks of a non-rotating 60~\\msun\\ star at $Z=0.014$. The track with effective temperatures computed using our revised procedure (\\teffc) is shown in red, while previous estimates using the Geneva code are shown in dashed black (\\teffg; not corrected for the wind optical depth) and blue (\\teffgcorr; corrected for the wind optical depth). The filled circles correspond to timesteps for which a CMFGEN model was computed.}\n\\end{figure}\n\n{\\it In summary, \\teffgcorr\\ does not seem to provide a good estimate of the effects of a dense wind on the effective temperature.} We also reinforce that $\\teffg$ does not  correspond to T at a fixed $\\tau$, since the wind contribution to the $\\tau$ scale is variable during the evolution. Therefore, \\teffg\\ should not be compared to $T$ derived from atmospheric modeling at high optical depth (e.g., \\tauross=10). Instead, whenever available, the values of $\\teffc$ should be preferred over \\teffgcorr\\ or \\teffg\\ when comparing evolutionary models to observations of massive stars. Since CMFGEN models could not be computed if $\\teffg<7500~\\K$, we assume throughout the rest of this paper\n\\[\n\\teff = \n\\left\\{\n\\begin{array}{ll}\n      \\teffg & \\mathrm{if}~ \\teffg \\leq 7500 \\K \\\\\n      \\teffc & \\mathrm{if }~\\teffg > 7500 \\K \\\\\n\\end{array} \n\\right.\n\\]\n\n\\subsection{\\label{classifyspec} Spectroscopic classification}\n\nTo classify the synthetic spectra, we use well-established quantitative criteria for O and WR  stars. For O stars, we use the calibrations from \\citet{mathys88}, which relates the \\ion{He}{i} $\\lambda4473$\/ \\ion{He}{ii} $\\lambda4543$ line ratio to spectral types\\footnote{We quote vacuum wavelengths in this paper.}. The luminosity classes I, III, and V are related to the ratio of \\ion{Si}{iv} $\\lambda4090$ to \\ion{He}{i} $\\lambda4714$ for spectral types later than O7, and to the strength of \\ion{He}{ii} $\\lambda4687$ for spectral types earlier than O7 \\citep{contialschuler71,contileep74,mathys88}. We supplement the quantitative spectral types with the morphological nomenclature devised by \\citet{walborn71,walborn73}  and \\citet{walborn02}, and recently updated by \\citet{sota11}. For WR stars, we employ the criteria from \\citet{crowther98} for WO and WC stars, and those from \\citet{ssm96} and \\citet{crowther95a} for WN stars. \n\nFor early B stars, we employ the morphological classification from \\citet{walborn71} and \\citet{walborn90}. These are based on the relative strength between \\ion{Si}{iv} $\\lambda4117$ and \\ion{He}{i} $4122$,  \\ion{Si}{iii} $\\lambda4554$ to \\ion{He}{i} $\\lambda4389$, and  \\ion{Si}{iii} $\\lambda4554$ to \\ion{Si}{iv} $\\lambda4090$. As is common in the literature, we associate the Ia$^+$ luminosity class to B-type hypergiants (BHG; \\citealt{vg82b}). BHGs are differentiated from BSGs based on the strength of the H Balmer line emission.  BHG have these lines in emission, usually with a P-Cygni profile, while normal BSGs have Balmer lines in absorption, with the possible exception of H$\\alpha$ \\citep{lennon92}.\n\nHowever, we argue here that the BHG classification do not include all stars with $\\teff < 25000~\\K$ and dense winds. For increasing wind densities, stars in the $\\teff$ range of what would be called a `B star'  develop strong (P Cygni) emission line spectrum, which are only seen in LBVs. It would be misleading to classify these stars as BHGs, since LBVs have spectral and wind properties markedly distinct from BHGs \\citep{clark12a}. Despite having similar $\\teff$ as BHGs, LBVs typically have lower $\\vinf$ and higher $\\mdot$ than BHGs \\citep{clark12a}. However, there seems to be a smooth transition (and even some overlap) between BHG and LBV properties, which makes it challenging to draw a firm line dividing the two classes. It seems clear that there is a progression in wind density from B supergiants to B hypergiants to LBVs \\citep{clark12a}. Here we arbitrarily choose the strength of H$\\alpha$ as a proxy for the wind density, and classify models that have the intensity of the H$\\alpha$ peak above 4 times the continuum as LBVs. \n\nTherefore, models that have spectrum similar to observed bona-fide LBVs, such as AG Car, P Cygni, and HR Car, have their spectral type listed as LBVs. While we recognize that formally there is no ``LBV'' spectral classification, we opted to use this classification since there is no objective spectral classification criteria of stars with winds denser than those of BHGs and that have $8000 \\K \\lesssim \\teff \\lesssim 25000 \\K$. The spectra of these stars have  been commonly referred to in the literature as ``P Cygni-type'', ``iron'', and ``slash'' stars (see, e.g., \\citealt{wf2000,clark12a}). In addition, we subdivide the LBVs in `hot LBV' and `cool LBV' according to the spectral morphology, following in general lines the classification scheme from \\citet{massey07} and \\citet{clark12a}. Hot LBVs have strong or moderate \\ion{He}{i} lines showing P Cygni profiles. Cool LBVs have weak or no  \\ion{He}{i} absorption lines and strong emission from low-ionization species such as  \\ion{Fe}{ii},  \\ion{N}{i}, and  \\ion{Ti}{ii}.\n\n\n\\section{\\label{evolspec} Evolutionary and spectroscopic phases}\n\nUnfortunately, often the terms ``evolutionary phase\" and ``spectroscopic phase\" have been indiscriminately used in the literature, causing confusion. Here we would like to ask the reader to appreciate the difference between the two terms, which sometimes have had different meanings for stellar evolution theorists, stellar atmosphere theorists, and observers working on massive stars. In particular, we note that spectroscopic phases such as RSG, WR, or LBV, should {\\it not} be used as a synonym of evolutionary phase.\n\n{\\it Evolutionary phases} are defined here in terms of nuclear burning stages and the stellar structure, which are properties inherent to the stellar interior. As such, these are not directly accessible by observations and, to determine the evolutionary phases of stars, one necessarily has to rely on stellar evolution models. The evolutionary phase names are self-explanatory and are H-core burning (or Main Sequence), H-shell burning, He-core plus H-shell burning, He-core burning, He-shell burning, C-core plus He-shell burning, C-core plus He-shell burning plus H-shell burning, and so on. From stellar evolution theory, evolutionary phases can be unambiguously determined and chronologically ordered.\n\n\\begin{figure*}\n\\center\n\\resizebox{0.980\\hsize}{!}{\\includegraphics{fig1_montage.pdf}}\\\\\n\\caption{\\label{hrd1} {\\it (a):} HR diagram showing the evolutionary track of a non-rotating star with initial mass of 60~\\msun\\ at  metallicity $Z=0.014$,  using our revised values of \\teff. The color code corresponds to the evolutionary phases of a massive star, with H-core burning in blue, He-core burning  in orange, C-core burning in green, and H and\/or He-shell burning in gray.  {\\it (b):} Similar to $a$, but color coded according to the spectroscopic phases. Lifetimes of each phase are indicated in parenthesis. {\\it (c):} Evolution of $\\teff$ as a function of age. The color code is the same as in {\\it a}. {\\it (d):} Surface abundances of H (black), He (red), C (magenta), N (green), and O (blue) as a function of age. {\\it (e):} Mass-loss rate (left axis) and mass (right axis) as a function of age. Color code is the same as in $a$. {\\it (f):} Central abundances as a function of age, with the same color-coding as in $d$. Spectral types at selected timesteps are indicated. }\n\\end{figure*}\n\n{\\it Spectroscopic phases} are related to the duration when a certain spectral type appears during the evolution of a star, i.e.,  they are defined here in terms of the spectral appearance of stars.  Spectroscopic phases are in general determined by the surface properties, i.\\, e., the physical conditions in the atmosphere and wind. Atmospheric properties are comprised by quantities such as $\\teff$, surface gravity, and surface abundances, and wind properties by $\\mdot$, $\\vinf$, velocity law, and  wind clumping. Spectroscopic phases are labelled here using the spectral type and luminosity class when necessary, corresponding to:\n\\begin{itemize}\n\\item O-type spectroscopic phase, B-type, etc (following the usual MK spectral type determination), which can be subdivided according to the luminosity class (I, Ia, Ia$^+$,Ib, Iab, II, III, IV, V). More generally, they can be grouped in O supergiants (all O stars with luminosity class I, Ia, Ib, Iab), or B-type supergiants (BSG), or yellow supergiants (all AFG supergiants), or red supergiants (all KM supergiants); or B-type hypergiants (BHG; B stars with luminosity class Ia$^+$), or yellow hypergiants (YHG, luminosity class Ia$^+$); \n\\item Wolf-Rayet (WR) phase, which is subdivided in WN (with N lines), WC (with C lines), or WO (with strong O and C lines). WN and WC stars are further subdivided in WNE (early WN), corresponding to spectral types WN1 to WN5; WNL (late WN, spectral types WN6 to WN 11); WCE (early WC, subtypes WC1 to WC5), and WCL (late WC, subtypes WC6 to WC9), following \\citet{crowther07};\n\\item Luminous Blue variable (LBV) phase, when the spectrum has similar characteristics to those of well-known LBVs such as AG Car, P Cyg, Eta Car, and S Dor. Depending on the spectral lines that are present, they are subdivided here in hot or cool LBVs, as described in Sect. \\ref{classifyspec}.\n\\end{itemize}\n\nTherefore, spectroscopic phases reflect the surface conditions of the star, and may be linked to different evolutionary phases depending on the initial mass, metallicity, rotation, and magnetic fields. To relate evolutionary and spectroscopic phases, a calibration is needed by computing output spectra from stellar evolution models, which is one of the goals of this paper.\n\n\\section{\\label{broadevol} The evolution of a non-rotating 60~\\msun\\ star and its various spectral types }\n\n\\begin{figure*}\n\\center\n\\resizebox{0.95\\hsize}{!}{\\includegraphics{60msun_spectra_evol_v3}}\\\\\n\\resizebox{0.95\\hsize}{!}{\\includegraphics{60msun_spectra_evol_v3_optical}}\\\\\n\\caption{\\label{spec1} Evolution of the ultraviolet (a) and optical spectra (b) of a non-rotating 60~\\msun\\ star. The evolution proceeds from top to bottom, with labels indicating the evolutionary phase, spectral type, scale factor when appropriate, age, and model stage according to Table \\ref{model log}. Note that certain spectra have been scaled for the sake of displaying the full range of UV and optical emission lines.}\n\\end{figure*}\n\nIn this Section we investigate how a star with \\mini=60~\\msun\\ evolves from a spectroscopic point of view and how the surface properties are linked with the evolution of the stellar interior. There is a long record in the literature about the interior evolution of non-rotating massive stars with $\\mini\\sim60~\\msun$, and the interior evolution described here is qualitatively similar to that described in, e.g., \\citet{chiosi86}. A number of recent papers discuss this subject, including the latest advances in modeling of the stellar interior, and we refer the reader to \\citet{maeder11}, \\citet{brott11}, \\citet{ekstrom12}, \\citet{georgy12a}, \\citet{langer12}, and \\citet{chieffi13}.\n\nFigure \\ref{hrd1}a shows the evolutionary track of a 60~\\msun\\ star in the Hertzsprung-Russell (HR) diagram. Spectra were computed at 53 stages, which are indicated in Fig. \\ref{hrd1}a together with the corresponding spectral type. Table \\ref{model log} presents the fundamental parameters of the models at each stage. Figure \\ref{hrd1}b illustrates the effects of the stellar wind on the determination of \\teff. Because of the wind optical depth, $\\teff <\\teffg$, and the difference between the two depends on $\\mdot$, \\vinf, \\lstar, and \\rstar. These values of \\teff\\ can be directly compared to those determined from observations of massive stars, as long as the observed value of \\teff\\ refers to the surface where \\tauross=2\/3.  \n\nFigure \\ref{hrd1} also shows the evolution of the surface temperature as a function of age (panel $c$), the surface abundances  in mass fraction\\footnote{Abundances are quoted in mass fraction in this paper.} ($d$), mass-loss rate and mass ($e$), and central abundances in mass fraction ($f$). Selected stages and corresponding spectral types are indicated, focusing on stages of either relative long duration or related to significant stages of the interior evolution, such as the different nuclear burning phases. \n \nFigure \\ref{spec1} displays sections of the ultraviolet and optical spectrum at selected stages, illustrating the broad evolution. Regarding the evolution of the optical spectrum, we find that the H and He absorption lines that dominate the O-type spectra (black, red, and blue tracing in Fig. \\ref{spec1}b) progressively turn into emission as the star evolves and the stellar wind becomes denser (green, orange, and cyan). The H emission lines disappear at the WNE phase because of the H exhaustion at the surface (purple), leaving \\ion{He}{ii} and \\ion{N}{iv} lines as the dominant features in the spectrum. When the star evolves to the WC phase (green), N lines vanish and C lines appear as a result of the He burning products (C and O) being exposed at the surface. At the end of its life, the star shows a WO spectral type (dark gray), with strong \\ion{O}{vi} lines dominating the spectrum as a result of the extremely high surface temperatures due to the surface contraction.\n\n\nWe find the following evolutionary sequence for a non-rotating 60~\\msun\\ star, where we quote the main spectroscopic phases only and their respective evolutionary phase in parenthesis:\\\\\n\\\\\n{\\it O3 I (ZAMS) $\\rightarrow$ O4 I (mid H-core burning) $\\rightarrow$ hot LBV (end H-core burning) $\\rightarrow$ cool LBV (start He-core burning) $\\rightarrow$ WNL $\\rightarrow$ WNE $\\rightarrow$ WC (mid He-core burning) $\\rightarrow$ WO (end He-core burning until core collapse). }\n \\\\\n \nIn the next subsections, we analyze in detail the different evolutionary stages and the spectroscopic phases associated to them.\n\n\\subsection{\\label{msdetail} H-core burning: the Main Sequence evolution (stages 1--15)}\n\nThe MS is characterized by H-core burning, which in the case of the non-rotating 60~\\msun\\ star takes 3.53 Myr to finish \\citep{ekstrom12}. As we elaborate below, we find that when this model is in the MS evolutionary phase, the following spectroscopic phases appear: O I, BSG, BHG, and LBV (Fig. \\ref{msfig}g,h).\n\n\\subsubsection{\\label{oi} An O3 I during 40\\% of the MS phase (stages 1--3)}\nOur models indicate that the star appears at the Zero Age Main Sequence (ZAMS) as an O3 If$^*$c star ($\\teff=48390~\\K$, $\\lstar=504582~\\lsun$, stage 1, Fig. \\ref{spec1}), and not as a luminosity class V star. This is because the luminosity criterium for early O stars is based on the equivalent width of the \\ion{He}{ii} $\\lambda$4687 line \\citep{contialschuler71,walborn71}, which in our model is filled by wind emission even at the ZAMS for a 60~\\msun\\ star. Therefore, the star has a supergiant appearance even at the ZAMS. It is also worth noting that the star has strong \\ion{N}{iv} \\lam4058 and \\ion{C}{iii} \\lam\\lam\\lam4658,4651,4653 emissions, which justifies the O3 If$^*$c classification. The supergiant appearance is caused by the high value of \\lstar\\ since, at this regime, $\\mdot \\propto \\lstar^2$ \\citep[][see also e.\\,g. \\citealt{dejager88,vink00}]{abbott82}. The strong wind affects the spectrum, with lines turning progressively from absorption into emission. Here we show that this effect starts to occur for a 60~\\msun\\ star when using the \\citet{vink01} mass-loss rate prescription and $f_\\infty=0.2$. This result is also in agreement with the observational findings from \\citet{crowther10} that very massive stars (above 100~\\msun) have an O I or even WNh spectral type at the ZAMS at the LMC metallicity (see also \\citealt{dekoter97,martins08}). In our models, \\ion{He}{ii} $\\lambda$4687 would be partially filled by wind emission even assuming an unclumped wind, and one would infer a luminosity class III at the ZAMS (see discussion in Sect. \\ref{mdotclump}.)\n\n\\begin{figure*}\n\\center\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page1}} \n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page2}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page3}}\\\\\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page4}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page5}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{ms_evol_age-page6}}\\\\\n\\resizebox{0.90\\hsize}{!}{\\includegraphics{60msun_spectra_evol_v2}}\n\\caption{\\label{msfig} {\\it (a)} through {\\it (e)}: Similar to Fig. \\ref{hrd1}, but focusing on the MS evolution. {\\it (f)}: Evolution of the Eddington parameter at the surface, computed for electron-scattering opacity. {\\it (g)} and {\\it (h)}: Similar to Fig. \\ref{spec1}, but focusing on the MS evolution.}\n\\end{figure*}\n\nDuring the MS (stages 1 through 15 in Fig. \\ref{msfig}a), the convective core contracts, the stellar envelope expands, and \\rstar\\ increases. There is also an increase in $\\lstar$ (Fig. \\ref{msfig}a), which is caused by the increase in mean molecular weight as H burns into He. As a combination of these two effects, \\teff\\ decreases (Fig. \\ref{msfig}b) and the spectral type shifts to later types (Fig. \\ref{msfig}g). Because of the relatively high \\lstar\\ and \\mdot\\, the \\ion{He}{ii} $\\lambda$4687 line remains contaminated by wind emission, meaning that the star appears as a supergiant with luminosity class I. During the MS, the \\mdot\\ behavior is regulated by its dependence on $\\lstar$ and $\\teff$ (see \\citealt{vink00}). Up to stage 6, $\\mdot$ increases because of the increase in $\\lstar$ and the shallow dependence of $\\mdot$ on $\\teff$ in the range 35000--50000~\\K. \nBetween stages 6 and 9 and until the star reaches the bistability limit (Fig. \\ref{msfig}e), $\\mdot$ decreases because $\\lstar$ is roughly constant and $\\teff$ decreases. In addition, since $\\vinf$ is thought to depend on $\\vesc$ \\citep{cak}, $\\vinf$ decreases as a result of the decrease in $\\vesc$ as \\rstar\\ increases. Mass loss and the increase in \\lstar\\ during the MS causes the Eddington parameter to increase \\citep[Fig. \\ref{msfig}f;][]{vink99,vink11}, which also makes $\\vinf$ smaller.\n\n\n\n\\begin{figure*}\n\\center\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page1}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page2}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page3}}\\\\\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page4}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page5}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hshell_evol_age-page6}}\\\\\n\\resizebox{0.99\\hsize}{!}{\\includegraphics{60Msun_spectra_evol-page3}}\n\\caption{\\label{hshellfig} Similar to Fig. \\ref{msfig}, but focusing on the evolution during H shell burning. }\n\\end{figure*}\n\nWe obtain that the increase in $\\lstar$ and decrease in \\teff\\ is modest for the first 1.44 Myr (stages 1 through 3), and the small changes are not enough to change its spectral type. The star remains with an O3 If$^*$c spectral type for about $40\\%$ of the MS lifetime (36\\% of the total lifetime). However, important changes occur in the stellar interior, with an increase in He abundance, at the expense of the decrease of the H content, as H burns (Fig. \\ref{msfig}d).  \n\n\\subsubsection{From O4 I to B0.2 I during the next 50\\% of the MS (stages 4--7)}\nUp to the middle of the MS (age=1.8 Myr, stage 4),  the spectroscopic appearance of a 60~\\msun\\ star does not change significantly. We determine an O4 If$^*$c spectral type, which lasts  for 0.66 Myr (Fig. \\ref{msfig}g).  It is interesting to notice that while the surface abundances and temperature have changed little (Fig. \\ref{msfig}b,c) , the interior properties have been greatly modified. The H abundance at the center (X$_c$) has decreased from 0.72 to 0.45, while the He abundance at the center (Y$_c$) has increased from 0.26 to 0.53 (Fig. \\ref{msfig}d).\n \nAs the star evolves on the MS, it becomes even more luminous and cooler, showing an O5 Ifc (for 0.47 Myr) and O6 Iafc (for 0.32 Myr) spectral types (Fig. \\ref{msfig}g). The total duration of the phase when the star shows an early O I spectral type (O3 I to O6 I) is thus 2.89 Myr (stages 1 through 6), when the star loses 7.6~\\msun\\ due to its stellar wind (Fig. \\ref{msfig}e).\n\nEventually, the increased $\\lstar$ and $\\mdot$ produces an O7.5 Iafc spectral type (stage 7), with \\ion{He}{ii} $\\lambda$4687 still significantly filled by wind emission (Fig. \\ref{msfig}g). Note that the $fc$ morphological classification lasts for a significant amount of time during the O I phase, since the surface C abundance still remains close to the solar values. The star continues to increase in $\\lstar$ (Fig. \\ref{msfig}a), with $\\mdot$ slightly decreasing as $\\teff$ decreases (Fig. \\ref{msfig}e), shifting to later spectral types (Fig. \\ref{msfig}g). For instance, it shows an O9~Iab spectral type when its age is t=3.18 Myr (stage 8). Therefore, the non-rotating 60~\\msun\\ star is a late O supergiant star (luminosity class I) for 0.36 Myr, with the star losing 0.9~\\msun\\ during this period (stages 6--8; Fig. \\ref{msfig}e). At the end of the late O I spectroscopic phase (stage 8), $\\lstar$ and \\mdot\\ have increased by 50\\% and 10\\% relatively to the initial values, respectively (Fig. \\ref{msfig}a,e), while $\\vinf$ has decreased to 45\\% of the ZAMS value. The surface composition remains essentially unchanged compared to the initial state (Fig. \\ref{msfig}c), while at the center, H continues to decrease (X$_c$=0.12) and He to increase (Y$_c$=0.86) (Fig. \\ref{msfig}d). Later, the decrease in \\teff\\ produces a B0.2 Ia spectral type (stage 9). We estimate that the B supergiant phase is short, lasting for about 0.039 Myr and during which the star loses 0.09~\\msun. \n\n\\subsubsection{The encounter of the bistability limit: producing BHG and LBV spectra in the last 10\\% of the MS phase (stages 10--15)}\n\\label{bistability}\n\nDue to the core contraction and corresponding envelope expansion, at some point the surface of the star is cool enough to encounter the bistability limit of line-driven winds.  Note that reaching the bistability limit during the MS is dependent on the choice of the overshooting parameter, mass-loss recipe, and initial rotation, since all these factors affect the width of the MS band (see e\\,g. the reviews by \\citealt{chiosi86} and \\citealt{maeder_araa00}).\n\nThis is a crucial point in the evolution of a massive star because \\mdot\\ increases significantly according to the \\citet{vink00} prescription. The first bistability jump occurs when $\\teff\\sim21000-25000~\\K$ \\citep{pauldrach90,vink99,vink00,vink02,ghd09,ghd11}, when Fe recombines from Fe$^{3+}$ to Fe$^{2+}$ in the inner wind, resulting in increased $\\mdot$. According to the \\citet{vink99,vink00} parametrization incorporated in our stellar evolution models, $\\mdot$ increases by a factor of $\\sim10$ as the star crosses from the hot (stage 9) to the cool side (stage 10) of the bistability. The factor of $\\sim10$ arises because the Geneva models follow the \\citet{vink00} recommendation to change the ratio $\\vinf\/\\vesc$ from 2.6 (hot side) to 1.3 (cool side) of the bistability limit \\citep{vink99,lamers95}. For the non-rotating 60~\\msun\\ star, this occurs at t=3.27 Myr, when the star becomes a BHG with a B0.2--0.5 Ia$^+$ spectral type (Fig. \\ref{msfig}g). We estimate that the star remains as a BHG for 0.079 Myr and loses 1.3~\\msun\\ during this period.\n\nFor the non-rotating 60~\\msun\\ model, we find that when the star crosses the bistability limit, the wind optical depth increases, the photosphere becomes extended and is formed in an expanding layer. Most of the spectral lines are affected by the presence of the wind, with some of them developing a classical P-Cygni profile. Because of the high \\mdot\\ and low $\\vinf$, coupled with the relatively low $\\teff$, the spectrum markedly resembles that of a hot LBV (stages 11 and 12). This suggests that some stars with spectra similar to LBVs may still be in the end stages of H-core burning, MS objects. When the star first shows an LBV spectrum, the H content at the center is X=0.06 (Fig. \\ref{msfig}d). \n\nFurther on, the star crosses the second bistability jump,  when Fe$^{2+}$ recombines to Fe$^+$ in the inner wind and another increase by factor of $~\\sim10$ occurs (\\citealt{vink01}; stage 12, Fig. \\ref{msfig}e). At this point, the star has reached its coolest point in the MS at t=3.41 Myr (stage 13), showing a cool LBV spectral type (Fig. \\ref{msfig}g) and having $\\mdot=1.4\\times10^{-4}~\\msunyr$ and $\\vinf=303~\\kms$.\n\nUp to this point, the surface abundances are essentially unchanged compared to the initial values on the ZAMS (Fig. \\ref{msfig}c), with $X_\\mathrm{sur}=0.72$, $Y_\\mathrm{sur}=0.27$, $C_\\mathrm{sur}=2.3\\times10^{-3}$, $N_\\mathrm{sur}=6.6\\times10^{-4}$, and $O_\\mathrm{sur}=5.7\\times10^{-3}$ (Fig. \\ref{msfig}c). However, because of the increase in $\\mdot$ that occurs when the star crosses to the cool side of the bistability limit, \\mstar\\ rapidly decreases and the surface abundances begin to change from this point in the evolution onwards.  First, N is rapidly enriched at the surface while C is depleted, so the star will appear as an LBV with solar N and C abundance only for a very short amount of time. For instance, at stage 13, when the star has crossed the second bistability jump, severe modifications have already occurred, with  $C_\\mathrm{sur}=3.3\\times10^{-5}$, $N_\\mathrm{sur}=4.0\\times10^{-3}$, and $O_\\mathrm{sur}=5.0\\times10^{-3}$. The H and He surface abundances take a longer timescale to change, and we find H and He surface abundances similar to the initial values, with $X_\\mathrm{sur}=0.72$ and  $Y_\\mathrm{sur}=0.27$ (Fig. \\ref{msfig}c). \n\nWhen the star returns to the blue side of the HR diagram (stage 14) after reaching its coolest point on the MS (stage 13), the surface shows the products of the CNO cycle, with an enrichment of He and N and depleted with H, C and O (Fig. \\ref{msfig}c).  Because of the increase in $\\mdot$ (Fig. \\ref{msfig}e), the Eddington parameter  at the hydrostatic radius becomes noticeably higher than during the preceding stages of the MS, reaching $\\Gamma=0.40-0.52$ (Fig. \\ref{msfig}f). The star shows a hot LBV spectrum for the remaining of its MS lifetime, up to t=5.53 Myr (Fig. \\ref{msfig}g), but with a progressively increase in He and decrease of H contents at the surface (Fig. \\ref{msfig}c). At the end of the MS (between stages 14 and 15), the star still has a hot LBV spectrum (Fig. \\ref{msfig}g) and the surface abundances shows the products of the CNO cycle, with $\\xsurf=0.49$, $\\ysurf=0.49$, $\\csurf=5.6\\times10^{-5}$, $\\nsurf=8.2\\times10^{-3}$, and $\\osurf=1.3\\times10^{-4}$ (Fig. \\ref{msfig}c). The total duration of the LBV phase during the MS in our model is 0.15 Myr (see discussion in Sect. \\ref{lifetimes}). During the LBV phase that occurs during the MS, the star loses 14.0~\\msun\\ of material though steady stellar winds (Fig. \\ref{msfig}e).\n\nThus, the appearance of an LBV spectrum during the end of the MS is a consequence of the increase in $\\mdot$ that occurs when the star crosses the bistability limit. We reinforce that the LBV phase is thus a {\\it prediction} of our models and not an assumption put by hand in the modeling.\n\n\\subsection{H-shell burning (stages 15--27)}\n\nWhen H burning at the center halts (stage 15), the core contracts and the envelope expands (stages 15 through 27). Since this occurs on a Kelvin-Helmholtz timescale (0.007 Myr for the non-rotating 60~\\msun\\ star), $\\lstar$ remains roughly constant (Fig. \\ref{hshellfig}a). As the core contracts, H begins to burn in a shell around the core. As a result of the envelope expansion, the star becomes cooler at the surface (Fig. \\ref{hshellfig}b). The star crosses the second bistability limit, which causes further increase in $\\mdot$ and decrease in $\\vinf$ (Fig. \\ref{hshellfig}e). As a consequence, the star remains as an LBV during H-shell burning, with the spectrum shifting from hot LBV to cool LBV as the envelope expands (Fig. \\ref{hshellfig}g). The total mass lost during the H-shell burning phase is 4.0~\\msun\\ (Fig. \\ref{hshellfig}e).\n\nDuring the H-shell burning, because of the extreme \\mdot, the surface abundances vary from $\\xsurf=0.49$ and $\\ysurf=0.49$ at the beginning to $\\xsurf=0.37$ and $\\ysurf=0.62$ at the end of this phase (Fig. \\ref{hshellfig}c). No noticeable changes in the central abundances occur, since H is burnt in a shell (Fig. \\ref{hshellfig}d).\n\nAt the end of the He-core contraction (between stages 27 and 28), immediately before the He-core burning begins, the star shows a cool LBV spectrum (Fig. \\ref{spec1}). The CNO abundances have already reached the equilibrium value of the CNO cycle and remain the same as those of the end of the MS ($\\csurf=5.6\\times10^{-5}$, $\\nsurf=8.2\\times10^{-3}$, and $\\osurf=1.3\\times10^{-4}$; Fig. \\ref{hrd1}d).\n\n\\subsection{\\label{hecore}He-core burning (stages 28--51)}\n\nAs the He-core contracts, the central T increases. When the central T is high enough, He-core burning begins. At this point (stage 28), a sizable amount of H is still present on top of the core (Fig. \\ref{hecorefig}c), and part of that burns in a shell around the He core. Thus, the star initiates He-core burning with a layer of H that burns in a shell, and below we discuss its consequences in the spectrum.\n\n\\subsubsection{From cool LBV to WN: losing the H envelope (stages 28--45)}\n\nThe star enters this phase showing an extremely cool LBV spectrum (Fig. \\ref{hespec}). As soon as He-core burning starts, the surface temperature increases (Fig. \\ref{hecorefig}b) and the star rapidly evolves to the blue (Fig. \\ref{hecorefig}a). This is caused by the decrease of the opacity as the chemical composition changes, and by the reduction of the inflation of the envelope as the mass of the H-shell burning layer decreases. The star shows a cool LBV spectrum for a brief period of time (0.010 Myr, stages 28 through 33), but substantial mass loss occurs in this period, with the star losing 5.94~\\msun\\ (Fig. \\ref{hecorefig}e). The star proceeds its evolution presenting quickly-evolving WNL spectral types that, for the non-rotating 60~\\msun\\ star, occurs with small amounts of H still present on the surface ($\\xsurf=0.13$; Fig. \\ref{hecorefig}c). This warrants a WN(h) classification \\citep{ssm96}. The spectral type evolves from WN11(h) (stage 34) to WN7(h) (stage 36), as shown in Fig. \\ref{hespec}. This is a brief phase that lasts for 0.005 Myr, during which 0.31~\\msun\\ of material is lost (Fig. \\ref{hecorefig}e).\n\nThe evolution to the blue causes a rapid display of spectral types and, on a short timescale ($\\sim0.030$ Myr, stages 33 to 43), the star presents spectra of a cool LBV, hot LBV, WNL, and WNE with small amounts of H (Fig. \\ref{hespec}). The surface abundances present little evolution (Fig. \\ref{hecorefig}c), and the main changes in spectral type arise from the progressive increase in the surface temperature.\n\n\\begin{figure*}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page1}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page2}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page3}}\\\\\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page4}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page5}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics{hecore_advstage_evol_age-page6}}\\\\\n\\caption{\\label{hecorefig} Similar to Fig. \\ref{hrd1}, but focusing on the evolution during He-core burning and advanced stages.\n }\n\\end{figure*}\n\n\\begin{figure*}\n\\center\n\\resizebox{0.89\\hsize}{!}{\\includegraphics{60Msun_spectra_evol_heburn}}\\\\\n\\caption{\\label{hespec} Evolution of the optical spectra of a non-rotating 60~\\msun\\ star during the He-core burning and advanced stages. The evolution proceeds from top to bottom, with labels indicating the spectral type, age, and model ID according to Table \\ref{model log}. In panel $(c)$, the fluxes are shown in logarithm scale for the purpose of displaying the full range of optical emission lines.}\n\\end{figure*}\n\nThe size of the convective core increases during He burning, with the envelope contracting and the surface temperature increasing (Fig. \\ref{hecorefig}b). This causes the star to eventually display early WN spectral types (Fig. \\ref{hespec}). Interestingly, the model shows a rather constant WN5(h) spectral type throughout this phase\\footnote{Except for a short amount of time when a WN2(h) spectrum appears, which is caused by a momentary diminishing of $\\mdot$.} (Fig. \\ref{hespec}). The first 40\\% of the WNE phase (0.043 Myr) develops with H still present on the surface ($\\xsurf=0.13$; Fig. \\ref{hecorefig}c), with the star losing 1.94~\\msun\\ (Fig. \\ref{hecorefig}e). The strong mass loss reduces the mass of the H-burning shell, reducing \\lstar. The remainder 60\\% (0.060 Myr) occurs without H at the surface (Fig. \\ref{hecorefig}c) and 3.81~\\msun\\ is lost (Fig. \\ref{hecorefig}e). We note that there is a decrease in $\\lstar$ during the WNE phase (Fig. \\ref{hecorefig}a) because of the significant decrease in $\\mstar$ due to strong mass loss (Fig. \\ref{hecorefig}e). \n\n\\subsubsection{A long-lived WC and short WO phases (stages 46--51)}\n\nBecause of mass loss, eventually the layers rich in He, C, and O appear at the surface, while the N abundance dramatically drops at the surface (stage 46, Fig. \\ref{hecorefig}c). These changes in the surface abundances mark the end of the WN phase. At this point, the star is in a short transition phase that lasts for 0.006 Myr, and a WO4 spectral type appears (Fig. \\ref{hespec}). This is caused by a momentary reduction of $\\mdot$. Once \\mdot\\ increases again, It quickly evolves to an early WC (WC4) spectral type that lasts for a significant amount of time (0.244 Myr; stages 47 through 49). During this period, the surface temperature steadily increases  (Fig. \\ref{hecorefig}b), while $\\lstar$ decreases (Fig. \\ref{hecorefig}a) since the star loses 7.02~\\msun\\ due to the action of strong stellar winds (Fig. \\ref{hecorefig}e). The main change in the surface abundances is the increase in O (Fig. \\ref{hecorefig}c).\n\nClose to end of He-core burning (stage 50), several changes in the fundamental properties of the star occurs. There is an increase in $\\teff$ (Fig. \\ref{hecorefig}b) and O abundance at the surface (Fig. \\ref{hecorefig}c), and decrease of $\\mdot$ (Fig. \\ref{hecorefig}e). These changes cause the star to show a WO4 and subsequently a WO1 spectral type (Fig. \\ref{hecorefig}b), which is a brief phase that lasts 0.030 Myr, when the star loses 0.57~\\msun\\ (Fig. \\ref{hecorefig}e).\n\nThe sequence WO -- WC -- WO seen at the end of the evolution indicates that the difference between WC and WO stars results mainly from ionization effects, which are regulated by $\\teff$ and $\\mdot$, rather than by a large increase of the O abundance at the surface as the star evolves from WC to WO. This result is in line with suggestions from observational studies of WO stars \\citep{crowther99b,crowther00,tramper13}.\n\n\\subsection{He-shell burning,  C-core burning, and advanced stages}\n\nAt the end of He-core burning (stage 51), the core and envelope contract, with He being ignited in a shell. This is an extremely short evolutionary phase (0.006 Myr; stage 51), where the contraction causes an increase in the surface temperature (Fig. \\ref{hecorefig}b). In combination with the high O surface abundance, this causes the star shows a WO1 spectral type (Fig. \\ref{hespec}). When the central temperature is high enough, C-core burning begins. It lasts for 0.002 Myr, the star loses 0.02~\\msun, and the spectral type continuing to reflect early WO subtypes (WO1; Fig. \\ref{hespec}). Beyond C-core burning (stage 53), our models indicate only modest changes of 0.01 dex in $\\log \\teff$ and $\\log \\lstar$, showing that the star will remain as a WO1 until core collapse \\citep{gmg13}.\n\n\\begin{sidewaystable*}\n\\begin{minipage}{\\textwidth}\n\\caption{Fundamental properties of the non-rotating 60 \\msun\\ star at Z=0.014 at selected phases of its evolution. From left to right, the columns correspond to the stage ID, evolutionary phase,  spectral type, age, mass (\\mstar), bolometric luminosity (\\lstar), effective temperature estimated by the Geneva code (\\teffg), effective temperature computed by CMFGEN at $\\tauross=2\/3$ (\\teff), radius of the hydrostatic surface (\\rstar), mass-loss rate (\\mdot), wind terminal velocity (\\vinf), steepness of the velocity law ($\\beta$),  and H, He, C, N, and O abundances at the surface and center, in mass fraction. The code for the evolutionary phases are: H-c=H-core burning, H-sh=H-shell burning, He-c=He-core burning, and C-c=C-core burning. }\n\n\\label{model log}\n\\scriptsize\n\\centering\n\\vspace{0.1cm}\n\\begin{tabular}{l c c c c c c c c c c c c c c c c c c c c c c}\n\\hline\\hline\nID &  Evol. & Spec. & Age & \\mstar & \\lstar & \\teffg & \\teff & \\rstar & \\mdot & \\vinf & $\\beta$ & X$_{sur}$ & Y$_{sur}$ & C$_{sur}$ & N$_{sur}$ & O$_{sur}$ & X$_{c}$ & Y$_{c}$ & C$_{c}$ & N$_{c}$ & O$_{c}$ \\\\ \n& Phase & Type  & (yr) & (\\msun) & (\\lsun) & (\\K) & (\\K) &(\\rsun) & (\\msunyr) & (\\kms) &  & & & & & & & & & &    \\\\\n\\hline \n      1 & H-c & O3If$^*$c &        29532 & 59.95 &  504821 &  48853 &  48390 &    9.95 &   1.73E-06 &    3543 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.72 &  0.27 &   3.55E-05 &   4.02E-03 &   4.90E-03 \\\\\n      2 & H-c & O3If$^*$c &       551447 & 58.97 &  528380 &  47031 &  47300 &   10.98 &   2.02E-06 &    3314 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.65 &  0.34 &   6.67E-05 &   7.71E-03 &   6.41E-04 \\\\\n      3 & H-c & O3If$^*$c &      1059410 & 57.86 &  555083 &  45796 &  45800 &   11.87 &   2.33E-06 &    3122 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.57 &  0.41 &   7.12E-05 &   8.09E-03 &   1.97E-04 \\\\\n      4 & H-c & O4If$^*$c &      1753470 & 56.06 &  597948 &  43261 &  43260 &   13.80 &   2.88E-06 &    2795 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.45 &  0.53 &   7.46E-05 &   8.15E-03 &   1.27E-04 \\\\\n      5 & H-c & O5Ifc &      2360274 & 54.16 &  645563 &  39375 &  39380 &   17.31 &   3.36E-06 &    2396 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.33 &  0.65 &   7.76E-05 &   8.15E-03 &   1.20E-04 \\\\\n      6 & H-c & O6Iafc &      2711788 & 52.97 &  680682 &  35749 &  35750 &   21.57 &   3.35E-06 &    2084 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.25 &  0.74 &   8.04E-05 &   8.15E-03 &   1.17E-04 \\\\\n      7 & H-c & O7.5Iafc &      3019444 & 52.02 &  719178 &  31088 &  31090 &   29.31 &   2.67E-06 &    1737 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.17 &  0.82 &   8.40E-05 &   8.15E-03 &   1.12E-04 \\\\\n      8 & H-c & O9Iab &      3178984 & 51.65 &  743097 &  27864 &  27810 &   37.09 &   1.90E-06 &    1520 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.12 &  0.86 &   8.67E-05 &   8.15E-03 &   1.09E-04 \\\\\n      9 & H-c & B0.2 Ia &      3262672 & 51.51 &  757148 &  25884 &  25310 &   43.39 &   1.40E-06 &    1394 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.10 &  0.89 &   8.88E-05 &   8.15E-03 &   1.06E-04 \\\\\n     10 & H-c & B0.5 Ia$^+$ &      3300425 & 50.91 &  759715 &  24443 &  24070 &   48.74 &   1.87E-05 &    1301 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.08 &  0.90 &   8.97E-05 &   8.15E-03 &   1.05E-04 \\\\\n     11 & H-c & hot LBV &      3371211 & 49.65 &  766547 &  21072 &  20700 &   65.87 &   1.69E-05 &    1093 &   1.0 &  0.72 &  0.27 &   2.28E-03 &   6.59E-04 &   5.72E-03 &  0.06 &  0.92 &   9.18E-05 &   8.15E-03 &   1.03E-04 \\\\\n     12 & H-c & hot LBV &      3418874 & 48.88 &  772591 &  18573 &  17150 &   85.12 &   1.53E-05 &     500 &   1.0 &  0.72 &  0.27 &   2.27E-03 &   6.60E-04 &   5.72E-03 &  0.05 &  0.94 &   9.39E-05 &   8.15E-03 &   1.00E-04 \\\\\n     13 & H-c & cool LBV &      3449657 & 44.32 &  755429 &  12105 &   8564 &  198.14 &   1.43E-04 &     303 &   1.0 &  0.72 &  0.27 &   3.28E-05 &   3.96E-03 &   4.98E-03 &  0.04 &  0.95 &   9.57E-05 &   8.14E-03 &   9.87E-05 \\\\\n     14 & H-c & hot LBV &      3499862 & 37.61 &  741291 &  19271 &  16850 &   77.45 &   2.02E-05 &     440 &   2.5 &  0.54 &  0.45 &   5.75E-05 &   8.15E-03 &   1.54E-04 &  0.02 &  0.97 &   9.99E-05 &   8.14E-03 &   9.54E-05 \\\\\n     15 & H-c & hot LBV &      3559650 & 36.26 &  814797 &  25785 &  20050 &   45.35 &   3.59E-05 &    1018 &   1.0 &  0.49 &  0.49 &   5.62E-05 &   8.17E-03 &   1.34E-04 &  0.00 &  0.99 &   1.48E-04 &   8.08E-03 &   8.79E-05 \\\\\n     16 & H-sh & hot LBV &      3561340 & 36.20 &  818839 &  20457 &  16770 &   72.24 &   2.86E-05 &     424 &   2.5 &  0.49 &  0.49 &   5.61E-05 &   8.17E-03 &   1.34E-04 &  0.00 &  0.99 &   1.49E-04 &   8.07E-03 &   9.84E-05 \\\\\n     17 & H-sh & cool LBV &      3561847 & 36.03 &  827076 &  16164 &   8918 &  116.29 &   3.04E-04 &     331 &   2.5 &  0.49 &  0.50 &   5.60E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.50E-04 &   8.07E-03 &   1.12E-04 \\\\\n     18 & H-sh & cool LBV &      3562100 & 35.96 &  834422 &  12785 &   8099 &  186.68 &   2.43E-04 &     260 &   2.5 &  0.48 &  0.50 &   5.59E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.51E-04 &   8.06E-03 &   1.22E-04 \\\\\n     19 & H-sh & cool LBV &      3562142 & 35.95 &  835687 &  12130 &   8348 &  207.57 &   2.31E-04 &     246 &   2.5 &  0.48 &  0.50 &   5.59E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.24E-04 \\\\\n     20 & H-sh & cool LBV &      3562185 & 35.94 &  836924 &  11444 &   8231 &  233.36 &   2.18E-04 &     232 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.25E-04 \\\\\n     21 & H-sh & cool LBV &      3562201 & 35.94 &  837454 &  11141 &   8176 &  246.28 &   2.11E-04 &     226 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.26E-04 \\\\\n     22 & H-sh & cool LBV &      3562235 & 35.93 &  838434 &  10594 &   8069 &  272.52 &   2.00E-04 &     214 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.27E-04 \\\\\n     23 & H-sh & cool LBV &      3562286 & 35.92 &  839972 &   9664 &   7750 &  327.83 &   1.82E-04 &     139 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.30E-04 \\\\\n     24 & H-sh & cool LBV &      3562303 & 35.92 &  840531 &   9317 &   7696 &  352.77 &   1.77E-04 &     134 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.30E-04 \\\\\n     25 & H-sh & cool LBV &      3562320 & 35.91 &  840990 &   9038 &   7673 &  375.03 &   1.69E-04 &     130 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.52E-04 &   8.06E-03 &   1.31E-04 \\\\\n     26 & H-sh & cool LBV &      3562354 & 35.91 &  841934 &   8373 &   7593 &  437.23 &   1.87E-04 &     120 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.53E-04 &   8.06E-03 &   1.32E-04 \\\\\n     27 & H-sh & cool LBV &      3562379 & 35.90 &  842525 &   8011 &   7610 &  477.81 &   1.77E-04 &     115 &   2.5 &  0.48 &  0.50 &   5.58E-05 &   8.17E-03 &   1.32E-04 &  0.00 &  0.99 &   1.53E-04 &   8.06E-03 &   1.32E-04 \\\\\n     28 & He-c+H-sh & cool LBV &      3572156 & 28.65 & 1027971 &  10827 &   7958 &  288.92 &   5.12E-04 &     118 &   2.5 &  0.14 &  0.85 &   9.28E-05 &   8.14E-03 &   1.09E-04 &  0.00 &  0.97 &   9.37E-03 &   2.18E-06 &   1.59E-04 \\\\\n     29 & He-c+H-sh & cool LBV &      3574142 & 27.60 & 1005669 &  11606 &   8077 &  248.72 &   5.52E-04 &     124 &   2.5 &  0.14 &  0.85 &   9.78E-05 &   8.13E-03 &   1.08E-04 &  0.00 &  0.97 &   1.63E-02 &   1.04E-06 &   1.94E-04 \\\\\n     30 & He-c+H-sh & cool LBV &      3574434 & 27.43 & 1001146 &  12182 &   8184 &  225.23 &   5.81E-04 &     129 &   2.5 &  0.14 &  0.85 &   9.81E-05 &   8.13E-03 &   1.08E-04 &  0.00 &  0.96 &   1.73E-02 &   1.07E-05 &   2.01E-04 \\\\\n     31 & He-c+H-sh & cool LBV &      3574979 & 27.10 &  989874 &  13648 &   8607 &  178.44 &   6.50E-04 &      20 &   2.5 &  0.14 &  0.85 &   9.68E-05 &   8.13E-03 &   1.08E-04 &  0.00 &  0.96 &   1.93E-02 &   2.62E-06 &   2.15E-04 \\\\\n     32 & He-c+H-sh & cool LBV &      3575461 & 26.76 &  976661 &  15961 &   8479 &  129.60 &   7.59E-04 &      20 &   2.5 &  0.14 &  0.85 &   9.54E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.96 &   2.09E-02 &   5.91E-06 &   2.28E-04 \\\\\n     33 & He-c+H-sh & cool LBV &      3575803 & 26.48 &  964242 &  19166 &   8964 &   89.30 &   9.10E-04 &     203 &   2.5 &  0.13 &  0.85 &   9.39E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.96 &   2.21E-02 &   2.00E-06 &   2.39E-04 \\\\\n     34 & He-c+H-sh & WN11(h) &      3576316 & 26.24 &  954291 &  23678 &  19030 &   58.21 &   9.82E-05 &     351 &   1.0 &  0.13 &  0.85 &   9.48E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.96 &   2.38E-02 &   2.00E-05 &   2.55E-04 \\\\\n     35 & He-c, H-sh & WN8(h) &      3576966 & 26.18 &  941859 &  30783 &  24210 &   34.21 &   7.69E-05 &     461 &   1.0 &  0.13 &  0.85 &   9.41E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.96 &   2.60E-02 &   2.85E-06 &   2.77E-04 \\\\\n     36 & He-c, H-sh & WN7(h) &      3578756 & 26.10 &  927045 &  38708 &  36090 &   21.47 &   3.27E-05 &    1113 &   1.0 &  0.13 &  0.85 &   9.32E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.95 &   3.23E-02 &   4.55E-07 &   3.50E-04 \\\\\n     37 & He-c, H-sh & WN5(h) &      3582601 & 26.00 &  916946 &  45243 &  44100 &   15.63 &   2.33E-05 &    1310 &   1.0 &  0.13 &  0.85 &   9.12E-05 &   8.14E-03 &   1.08E-04 &  0.00 &  0.94 &   4.56E-02 &   6.15E-08 &   5.62E-04 \\\\\n     38 & He-c, H-sh & WN2(h) &      3590081 & 25.85 &  897212 &  56419 &  56170 &    9.94 &   1.68E-05 &    1658 &   1.0 &  0.13 &  0.85 &   8.76E-05 &   8.15E-03 &   1.08E-04 &  0.00 &  0.91 &   7.18E-02 &   1.72E-06 &   1.21E-03 \\\\\n     39 & He-c, H-sh & WN5(h) &      3595309 & 25.77 &  883519 &  63351 &  60350 &    7.82 &   1.17E-05 &    1884 &   1.0 &  0.13 &  0.85 &   8.60E-05 &   8.15E-03 &   1.08E-04 &  0.00 &  0.89 &   9.01E-02 &   7.07E-10 &   1.85E-03 \\\\\n     40 & He-c, H-sh & WN5(h) &      3600871 & 25.72 &  869168 &  70075 &  42180 &    6.34 &   8.45E-05 &    2000 &   1.0 &  0.13 &  0.85 &   8.40E-05 &   8.15E-03 &   1.08E-04 &  0.00 &  0.87 &   1.09E-01 &   3.70E-07 &   2.69E-03 \\\\\n     41 & He-c, H-sh & WN5(h) &      3603786 & 25.48 &  835922 &  81012 &  44170 &    4.65 &   7.93E-05 &    2000 &   1.0 &  0.13 &  0.85 &   6.81E-05 &   8.18E-03 &   1.08E-04 &  0.00 &  0.86 &   1.18E-01 &   6.16E-07 &   3.20E-03 \\\\\n     42 & He-c, H-sh & WN5(h) &      3607017 & 25.23 &  791919 &  92368 &  46260 &    3.48 &   7.26E-05 &    2000 &   1.0 &  0.13 &  0.85 &   6.04E-05 &   8.19E-03 &   1.08E-04 &  0.00 &  0.85 &   1.28E-01 &   2.90E-06 &   3.79E-03 \\\\\n     43 & He-c, H-sh & WN5(h) &      3613846 & 24.78 &  717440 & 106349 &  49580 &    2.50 &   6.24E-05 &    2000 &   1.0 &  0.13 &  0.86 &   6.56E-05 &   8.18E-03 &   1.10E-04 &  0.00 &  0.83 &   1.47E-01 &   1.93E-07 &   5.20E-03 \\\\\n     44 & He-c, H-sh & WN5 &      3625652 & 24.02 &  682455 & 122874 &  48170 &    1.83 &   7.31E-05 &    2000 &   1.0 &  0.03 &  0.95 &   7.49E-05 &   8.18E-03 &   9.42E-05 &  0.00 &  0.79 &   1.84E-01 &   2.00E-11 &   8.36E-03 \\\\\n     45 & He-c & WN5 &      3663742 & 21.40 &  584484 & 125165 &  51330 &    1.63 &   6.12E-05 &    2000 &   1.0 &  0.00 &  0.99 &   1.39E-04 &   8.09E-03 &   8.79E-05 &  0.00 &  0.66 &   2.92E-01 &   1.21E-13 &   2.45E-02 \\\\\n     46 & He-c & WO4 &      3686605 & 20.20 &  545448 & 126228 &  82330 &    1.55 &   2.99E-05 &    2475 &   1.0 &  0.00 &  0.87 &   1.10E-01 &   2.41E-03 &   4.21E-03 &  0.00 &  0.59 &   3.50E-01 &   6.83E-15 &   3.93E-02 \\\\\n     47 & He-c & WC4 &      3715449 & 19.02 &  507199 & 133655 &  70130 &    1.33 &   4.16E-05 &    2394 &   1.0 &  0.00 &  0.74 &   2.31E-01 &   1.32E-10 &   1.40E-02 &  0.00 &  0.50 &   4.13E-01 &   1.88E-16 &   6.47E-02 \\\\\n     48 & He-c & WC4 &      3771686 & 16.75 &  434538 & 136007 &  66620 &    1.19 &   3.76E-05 &    2185 &   1.0 &  0.00 &  0.62 &   3.30E-01 &   6.61E-14 &   3.34E-02 &  0.00 &  0.35 &   4.94E-01 &   2.07E-19 &   1.39E-01 \\\\\n     49 & He-c & WC4 &      3873640 & 13.90 &  350243 & 140542 &  86090 &    1.00 &   1.92E-05 &    1928 &   1.0 &  0.00 &  0.38 &   4.82E-01 &   1.75E-17 &   1.21E-01 &  0.00 &  0.13 &   4.67E-01 &   7.87E-22 &   3.88E-01 \\\\\n     50 & He-c & WO4 &      3954404 & 12.68 &  332128 & 151199 & 105900 &    0.84 &   1.20E-05 &    1918 &   1.0 &  0.00 &  0.29 &   5.10E-01 &   3.81E-19 &   1.84E-01 &  0.00 &  0.01 &   2.57E-01 &   2.78E-29 &   7.13E-01 \\\\\n     51 & He-c & WO1 &      3963096 & 12.58 &  355836 & 165881 & 146200 &    0.72 &   1.22E-05 &    5000 &   1.0 &  0.00 &  0.28 &   5.11E-01 &   2.53E-19 &   1.91E-01 &  0.00 &  0.00 &   2.34E-01 &   1.23E-29 &   7.41E-01 \\\\\n     52 & He-sh & WO1 &      3965312 & 12.55 &  390822 & 182276 & 153000 &    0.63 &   1.30E-05 &    5000 &   1.0 &  0.00 &  0.28 &   5.12E-01 &   2.27E-19 &   1.93E-01 &  0.00 &  0.00 &   2.34E-01 &   3.26E-29 &   7.41E-01 \\\\\n     53 & end C-c & WO1 &      3969319 & 12.50 &  483177 & 224855 & 164600 &    0.46 &   1.52E-05 &    5000 &   1.0 &  0.00 &  0.27 &   5.12E-01 &   1.77E-19 &   1.97E-01 &  0.00 &  0.00 &   1.76E-05 &   3.26E-29 &   6.01E-01 \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{sidewaystable*}\n\n\n\\section{\\label{lifetimes} Lifetimes of different evolutionary phases: spectra vs. surface chemical abundance and $\\teff$ criteria}\n\nUp to now, in the absence of a spectrum, the duration of different spectroscopic phases has been estimated based on chemical abundance and $\\teff$ criteria \\citep{smithmaeder91,meynet03}. Let us analyze the lifetime of the spectroscopic phases that we found based on the computation of the spectrum, and how those compares to the previous estimates. Here we compare with the results from \\citet{georgy12a}, which used the same 60~\\msun\\ evolutionary model employed here but employed chemical abundance and $\\teff$ criteria. Our results for the lifetimes are shown in Table \\ref{duration}, while Table \\ref{compareduration} compares the duration of the spectroscopic phases of our models with previous studies.\n\nWe obtained that the O-type spectroscopic phase lasts for $3.22\\times10^6$ yr, which is similar to the value estimated based on the $\\teff$ and abundance criteria ($3.00\\times10^6$ yr). This small difference arises because \\citet{georgy12a} assumes that O stars have $\\log (\\teff\/K) > 4.5$, while we find that the latest O supergiant models have $\\log (\\teff\/K)  \\simeq 4.45$ (in agreement with \\citealt{martins05}). This makes the O-type phase slightly longer.\n\n\\begin{table*}\n\\center\n\\caption{Spectral types, duration, and mass lost at different evolutionary stages of a non-rotating 60~\\msun\\ star. }\n\\footnotesize\n\\label{duration}\n\\begin{tabular}{llcccr}\n\\hline\n\\hline         \nEvolutionary stage \t\t& Spectral type & Duration & Duration    & Mass lost & Mass lost\\\\\n                           \t\t          &                         &  (Myr)        & (\\% of total)& (\\msun)     & (\\% of total loss)\\\\                         \n\\hline\nMS (core H burning)  & early O I               &   2.894 &    72.9 &    7.62 &    16.1  \\\\\n\"                    & late O I                &   0.329 &     8.3 &    0.81 &     1.7  \\\\\n\"                    & BSG                     &   0.039 &     1.0 &    0.06 &     0.1  \\\\\n\"\t   \t\t\t     & BHG                     &   0.079 &     2.0 &    1.34 &     2.8  \\\\\n\"        \t\t\t & LBV                     &   0.217 &     5.5 &   13.85 &    29.3  \\\\\nH shell burning      & LBV                     &   0.009 &     0.2 &    4.60 &     9.7  \\\\\ncore He burning      & LBV                     &   0.009 &     0.2 &    5.37 &    11.4  \\\\\n\"\t\t\t\t     &  WNL with H             &   0.005 &     0.1 &    0.31 &     0.7  \\\\\n\"\t\t\t       \t &  WNE with H             &   0.045 &     1.1 &    2.02 &     4.3  \\\\\n\"\t\t\t\t     &  WNE no H               &   0.055 &     1.4 &    3.62 &     7.7  \\\\\n\"\t\t\t    \t &  WO          \t       &   0.006 &     0.2 &    0.23 &     0.5  \\\\\n\"\t\t\t     \t &  WCE \t\t           &   0.257 &     6.5 &    7.38 &    15.6  \\\\\n\" \t\t\t         &  WO                     &   0.030 &     0.7 &    0.57 &     0.9 \\\\\nHe shell burning\t &  WO          \t\t&         0.006  &0.1                  &    0.07 & 0.1       \\\\\ncore C burning  and beyond        &  WO     &        0.002   & <0.1\t\t     &   0.02   & <0.1  \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table*}\n\nAccording to our new models, the LBV phase of the non-rotating 60~\\msun\\ star lasts for $2.35\\times10^5$~yr. This is $~\\sim7\\%$ of the lifetime of the O-type phase and  $~\\sim59\\%$ of the WR lifetime for that particular initial mass and rotation. The lifetime of the LBV phase in our models is $\\sim5 $ times longer than the  $\\sim4\\times10^4$ yr that is usually assumed based on the number counting of known LBVs compared to WR stars in the LMC \\citep{bohannan97}. However, we caution that an analysis of models comprising the full range of initial masses and rotation is needed to draw firm conclusions about the LBV lifetime. For instance, we foresee that our most massive models ($120~\\msun$) will have much shorter LBV phases, and so will our rotating models. Thus, all we can safely say at the moment is that the duration of the LBV phase depends on the initial mass and rotation. Also, with the criteria employed here to classify a star as  an LBV (Sect. \\ref{classifyspec}), the lifetime of the LBV phase also depends on the mass-loss recipe and clumping factor used. As discussed in Sect. \\ref{bistability}, the presence of a bistability jump in \\mdot\\ is what causes a BHG\/LBV spectrum to appear. If the jump in \\mdot\\  is weaker than theoretically predicted, as suggested by \\citep{crowther06}, this would favor the presence of BSGs, BHGs, and A-type supergiants and hypergiants, decreasing the duration of the  LBV phase.\n\nWe found that the duration of the WR phase is also slightly modified when a spectroscopic criterium is used.  We find a slightly shorter lifetime of $3.95\\times10^5$ yr compared to \\citet{georgy12a}, because part of the track that was classified as WNL by \\citet{georgy12a} at the beginning of He-core burning are actually LBVs in our models. The difference in the total lifetime of the WR phase is small because the evolution of the surface properties is fast at the beginning of He-core burning (Sect. \\ref{hecore}). The duration of the WN phase is also similar to what has been obtained before using an abundance criterium.\n\nHowever, the duration of the WNE and WNL phases are strongly modified using the spectroscopic criterium. The WNL lifetime decreases by a factor of 15.25, while the WNE duration increases by a factor of 2. These huge changes occur because the H abundance at the surface, which has been employed in previous stellar evolution studies as a diagnostic \\citep[e.\\,g.][]{georgy12a}, is not a good tracer of the WNE or WNL status. The WNL or WNE appearance is regulated by the ionization structure of the wind, which is determined by $\\teff$ and \\lstar, and ultimately sets whether a WNE or WNL spectrum arises \\citep[e.g.,][]{hillier87a}. If the decrease of the WNL and increase of the WNE lifetime is confirmed for other initial masses and rotation, this could help to reconcile the theoretical and observed WNL\/WNE ratio, which was found to be too high by \\citet{georgy12a}.\n\nOur models indicate that the WC lifetime is slightly reduced when spectroscopic criteria are used. The main reason is that a certain surface C abundance is required in order to have a WC spectrum, and that occurs at a higher C abundance than what is assumed by the chemical abundance criterium of \\citet{georgy12a}. We also note that no WN\/WC transition stars are seen in our models, in agreement with previous observational and theoretical studies \\citep{crowther95b,meynet03}.\n\nWe found that a short WO phase exists at the end of the evolution of a 60~\\msun\\ star. This was inexistent when an abundance criterium was used,  but our atmospheric calculations show that because of the extremely high \\teff\\ and O abundance, a WO spectral type arises at the very end of the evolution of the most massive stars until core collapse \\citep{gmg13}.\n\n\\begin{table}\n\\caption{Lifetime of the spectroscopic phases of a non-rotating 60~\\msun\\ star. Column 2 refers to the previous values found using chemical abundance and $\\teff$ criteria (from \\citealt{georgy12a}), while  column 3 shows the revised durations obtained in this work through a classification of the synthetic spectra from the evolutionary models. The timescales of the LBV ($\\tau_\\mathrm{LBV,60,0}$), WR ($\\tau_\\mathrm{WR,60,0}$), and O-type phases ($\\tau_\\mathrm{O,60,0}$) for a non-rotating 60~\\msun\\ star are indicated at the bottom of the table.}\n\\footnotesize\n\\begin{center}\n\\label{compareduration}\n\\begin{tabular}{lcr}\n\\hline\n\\hline         \nSpectral type & Previous duration & Revised duration \\\\\n                         &  (chem. abund., yr )        &  (spec. class., yr)   \\\\                         \n\\hline\nO                   &       $3.00\\times10^6$\t          & $3.22\\times10^6$\t                  \\\\\nBHG             &      $-$                                       & $0.79\\times 10^5$                  \\\\\nLBV     \t   &       $-$                                      & $2.35\\times10^5$\t\t\t\\\\\nWR                &       $3.97\\times10^5$           & $3.95\\times10^5$\t                  \\\\\nWNL              &       $6.13\\times10^4$           & $0.49\\times10^4$\t                  \\\\\nWNE              &       $5.15\\times10^4$           & $10.31\\times10^4$\t         \\\\\nWC                &       $2.85\\times10^5$           & $2.57\\times10^5$\t                  \\\\\nWO                &       $-$                                   & $3.77\\times10^4$\t                  \\\\\n\\hline\n$\\tau_\\mathrm{LBV,60,0}\/\\tau_\\mathrm{O,60,0}=0.073$ &&\\\\\n$\\tau_\\mathrm{WR,60,0}\/\\tau_\\mathrm{O,60,0}=0.120$ &&\\\\\n$\\tau_\\mathrm{LBV,60,0}\/\\tau_\\mathrm{WR,60,0}=0.594$ &&\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{\\label{origspec} The origin of the different spectroscopic classes of massive stars}\n\nIn this paper we analyze the evolution of the interior and spectra of a star with only one initial mass (60~\\msun) and rotation speed (0~\\kms). Nevertheless, we are already able to make inferences about what causes different spectroscopic classes to appear during the evolution of a massive star. In particular, we argue that a given spectroscopic phase can be linked to different evolutionary phases depending on initial mass, rotation, and metallicity. We stress that the conclusions presented here are valid at solar metallicity and are dependent on the mass-loss rates across the evolution, since the spectral morphology of massive stars is significantly affected by \\mdot. Moreover, the conclusions do not cover all possible single star evolutionary scenarios, and will be augmented as the analysis of models with different initial masses, rotation, and metallicity becomes available. \n\nThe O-star phase seems to be exclusively linked to the H-core burning evolutionary phase, i.e., all O stars are in the MS. This is a direct result of the stellar structure and evolution equations, which determines the value of \\teff\\ and \\lstar\\ for a given \\mstar. The fact that the star appears as an O-type star during the MS  is also dependent on \\mdot.\n\nThe BHG phase can be linked either to the end portion of the MS or to post-MS evolutionary phases, depending on the initial mass and rotation. For a 60~\\msun\\ star without rotation,  the BHG spectroscopic phase is intimately linked to the crossing from the hot side to the cool side of the first bistability limit of line-driven winds around $\\teff=21000-25000~\\K$. When this occurs, \\mdot\\ increases by a factor of $\\sim10$ according to the \\citet{vink01} recipe, and a BHG spectrum appears. For lower initial masses (e.g., 20--25~\\msun\\ with rotation), we found that the BHG spectroscopic phase occurs only after the RSG phase, during He-core burning \\citep{gme13}. This result was obtained using the same technique of combined stellar evolution and atmospheric modeling to obtain spectral types described in this paper. Because BHGs are pre- or post-RSGs, this illustrates that a given spectroscopic phase can be linked to different evolutionary phases, and using spectroscopic phases as a synonym for evolutionary phases will lead to confusion.\n\nLikewise, the LBV spectroscopic phase also seem to be linked to different evolutionary stages, depending on the initial mass and rotation. In the case of a 60~\\msun\\ star without rotation, our models predict that LBVs are linked to the end of the MS (i.e., H-core burning), H-shell burning, and beginning of He-core burning, before the star evolves and show a WR spectra. We find that the LBV phase is also linked with the star having a high \\mdot\\ due to the crossing of the bistability limit. However, if the initial mass is lower, LBVs can appear only at the advanced stages of He-core, C-core burning and core collapse ($\\sim20-25$~\\msun; \\citealt{gme13}; see also, e.g., \\citealt{kv06}).\n\nConcerning the WN phase with H, the non-rotating 60~\\msun\\ model indicates that this phase appears at beginning of He-core burning, when there is still a shell of H burning on top of the He core. Since at this stage $X_\\mathrm{sur}=0.13$, a spectral type of WN(h) is warranted \\footnote{We thank P. Crowther for this remark.}. This is in contrast with the hydrogen-rich WNh spectroscopic phase, which seems to appear during the MS if the initial mass and\/or \\mdot\\ are high enough (likely above 100~\\msun; \\citealt{dekoter97,martins08,sc08,crowther10}).\n\n\\begin{figure*}\n\\center\n\\resizebox{0.79\\hsize}{!}{\\includegraphics{magnitudes_evol.pdf}}\n\\caption{\\label{magevol} Evolution of absolute magnitudes (a), $B-V$ and $V-K$ colors (b), and number of \\ion{H}{}, \\ion{He}{i}, and \\ion{He}{ii} ionizing photons emitted per second (c) throughout the lifetime of a non-rotating 60~\\msun\\ star. At the top we indicate the duration of evolutionary and spectroscopic phases as reference. To make the post-MS evolution clearer, the ordinate scale was changed at 3.5 Myr.}\n\\end{figure*}\n\n\\section{\\label{specevol} Evolution of the absolute magnitudes, colors, bolometric correction, and ionizing fluxes }\n\nOur combined stellar evolution and atmospheric modeling allows us to investigate the temporal evolution of the observables of massive stars, such as the spectrum in different wavelength ranges, absolute magnitudes in various filters, colors, bolometric corrections, ionizing fluxes, and \\teff. This complements the output from classical stellar evolution models \\citep[e.g.,][]{ekstrom12}, which provide the evolution of \\teffg\\ and \\lstar. In the future, the temporal evolution of the observables will be used as input to produce theoretical isochrones and, in more general terms, as input to stellar population synthesis models.\n\nOur models provide as output the high-resolution spectrum from the extreme UV to radio wavelengths. Synthetic photometry was performed using the Chorizos code \\citep{chorizos04}, adopting its built-in passband and zero point definitions that were obtained from \\citet{cohen03,ma06,holberg06,ma07}.\n\nFollowing \\citet{gmg13}, the absolute magnitudes ($M_P$)  in the modified Vega magnitude system,  for a given filter $P$, are\n\\begin{equation}\nM_{P} =  -2.5\\log_{10}\\left(\\frac{\\int P(\\lambda)F_{\\lambda}(\\lambda)\\lambda\\,d\\lambda}\n                         {\\int P(\\lambda)F_{\\lambda,\\mathrm{Vega}}(\\lambda)\\lambda\\,d\\lambda}\\right)\n                       + {\\rm ZP}_{P},\n\\label{absmageq}\n\\end{equation}\nwhere  $\\lambda$ is the wavelength, $P(\\lambda)$ is the sensitivity curve of the system, $F_{\\lambda}$ is the model flux at 10 pc, $F_{\\lambda,\\mathrm{Vega}}$ is the flux of Vega scaled to a distance of 10pc, and ${\\rm ZP}_{P}$ is the zero point.\n\nWe use the usual relationship to compute the bolometric magnitudes (\\mbol), assuming that the solar \\mbol\\ is 4.74 mag, \n\\begin{equation}\nM_{bol} =   -2.5\\log_{10} (\\lstar\/\\lsun) + 4.74.\n\\label{mboleq}\n\\end{equation}\nTo obtain bolometric corrections in a given filter $P$ (BC$_P$), we use\n\\begin{equation}\n\\mathrm{BC}_{P} = \\mbol - M_{P}.\n\\label{bceq}\n\\end{equation}\nValues of BC$_P$ for each of the 53 stages discussed here are shown in Tables~\\ref{absmagbc1} and \\ref{absmagbc2}.\n\nMassive stars are the main sources of input of ionizing photons into the interstellar medium. Because stars evolve, the number of ionizing photons emitted by a massive star is expected to vary as the surface conditions change during the evolution. Here we quantify this effect by computing the number of photons capable of ionizing \\ion{H}{} ($Q_0$), \\ion{He}{i} ($Q_1$), and \\ion{He}{ii} ($Q_2$) as follows,\n\n\\begin{equation}\nQ_i=4\\pi R_{\\star} \\int^{\\lambda_i}_0 \\frac{ \\pi \\lambda F_\\lambda }{hc} \\mathrm{d}\\lambda \\,,\n\\end{equation}\nwhere $F_\\lambda$ is the stellar flux at \\rstar\\ and $\\lambda_\\mathrm{0}$, $\\lambda_\\mathrm{1}$, and $\\lambda_\\mathrm{2}$ are the wavelength corresponding to the ionization edges of  \\ion{H}{} (912~{\\AA}), \\ion{He}{i}  (574~{\\AA}), and \\ion{He}{ii}  (228~{\\AA}), respectively.\n\nOne of the advantages of our models is the flexibility to produce synthetic photometry in different filter systems, avoiding errors that stem from converting between different magnitude systems. Here we quote results in the Johnson-Cousins $UBVRI$ and 2MASS $J$, $H$, and $K_S$ (Table \\ref{absmagbc1}), and the  {\\it Hubble Space Telescope (HST)}\/Wide Field Planetary Camera 2 (WFPC2) $F170W$, $F300W$, $F450W$, $F606W$, $F814W$ filters (Table \\ref{absmagbc2}).\n\nThe evolution of the non-rotating 60~\\msun\\ star analyzed in this paper covers a large range of surface temperatures (6700 to 225000~\\K; Fig. \\ref{hrd1}a), luminosities (0.3 to $1.0\\times10^6~\\lsun$; Fig. \\ref{hrd1}a), mass-loss rates ($10^{-5}$ to $10^{-3}$~\\msunyr; Fig. \\ref{hrd1}e), and surface abundances (Fig. \\ref{hrd1}d). As such, we would expect significant variations of the spectral energy distribution, absolute magnitudes, and colors throughout the evolution.\n\n\nFigure \\ref{magevol}a shows the absolute magnitudes in the WFPC2\/$F170W$ and $UBVRIJHK_S$ filters throughout the evolution of a non-rotating 60~\\msun\\ star. Indeed, one can readily see that absolute magnitudes present strong variations of up to 6 mag during the star's lifetime. The behavior of the absolute magnitudes and bolometric corrections as a function of time is then regulated by how much flux from the star falls within the passband of a given filter. This depends on the stellar (\\teff\\ and \\lstar) and wind properties (\\mdot\\ and \\vinf). Let us analyze the absolute magnitudes and how they broadly vary during the evolution. For this purpose, we refer to the stages  overplotted in Fig. \\ref{magevol} (as defined in Fig.~\\ref{hrd1}).\n\nDuring the majority of the MS evolution (stages 1 through 13), there is a brightening in all UV, optical and near-IR filters. This is ultimately caused by the core contraction due to H being burnt into He, which causes an expansion of the envelope and increase of the global mean molecular weight and opacity. As a result, \\lstar\\ increases and \\teff\\ decreases, which shifts the spectral energy distribution (SED) towards longer wavelengths and increase the flux in the $UBVRIJHK$ filters. During the OIII and OI spectroscopic phases, the star is much brighter in $UBV$ than $JHK$. When it becomes an LBV (stage 13), the color indexes become closer to 0. At the final portion of the MS (stages 13 through 15), the model becomes fainter in all UV, optical and near-IR filters. This is caused by the increase in $\\teff$ as the model returns to the blue. \n \nWhen the H-shell burning stage begins (stage 15), the star becomes brighter again, reaching its maximum brightness at stage 28.  This is caused by the contraction of the core, which is not burning H anymore. As a result of the rapid core contraction, the envelope expands with constant \\lstar\\ and decreasing $\\teff$, shifting the SED to longer wavelengths and increasing the flux in the optical and near-IR filters.\n\nWhen He-core burning begins (stage 28), there is another rapid fainting of the model up to when the star becomes a WNEh (stage 37). The ultimate reason is the huge mass loss, which removes the outer layers of the star and decreases the size of the H-shell burning layer. This decreases the inflation of the stellar envelope, increasing $\\teff$ and making the star fainter in the optical and near-IR.\n\n During He-core burning (stages 28 to 51), there is a progressive decrease in the optical an near-IR fluxes as the star becomes hotter and less luminous. This is ultimately caused by mass loss, which diminishes $\\mstar$ and, as a consequence, $\\lstar$ as well. Mass loss also removes the outer layers of the star, making $\\teff$ higher. These factors contribute to shifting the SED to the blue and decreasing the brightness in the optical and near-IR. The decrease in the $U$ filter is much less pronounced than in the other filters. The star becomes even fainter at the end of He-core burning (stages 49 through 50), which is caused by the overall contraction of the star.\n\nFrom the end of He-core burning (stage 50) until core collapse (stage 53), there is a significant decrease in the optical and near-IR fluxes. This is caused by the contraction of the star as He-core burning ends. There are two competing effects happening. There is an increase in $\\lstar$, which has the tendency to make the star brighter in the optical and near-IR filters, but there is an increase in \\teff\\ as well, which has the tendency to make the star fainter in these filters. We found that the $\\teff$ effect dominates and the star becomes much fainter at core collapse than it was at the end stages of He-core burning, This phenomenon is relevant for investigating the detectability of massive stars just before the SN explosion, and explains the overall faintness of SN Ic progenitors \\citep[][see also \\citealt{yoon12}]{gmg13}.\n\nThe variation of the $B-V$ and $V-K$ colors as a function of age is displayed in Fig. \\ref{magevol}b. We see that $B-V$ and $V-K$ are roughly constant during the O-type phase, and rapidily change when the star reach the LBV phase. From the beginning of He-core burning onwards, these broadband colors are affected by the presence of emission lines, which dominate the flux in these filters. Therefore, erratic variations are seen depending on the stellar parameters when the star is a WR.\n\nFigure \\ref{magevol}c shows the evolution of the number of  \\ion{H}{} ($Q_0$), \\ion{He}{i} ($Q_1$), and \\ion{He}{ii} ($Q_2$) ionizing photons emitted per unit time. We found that $Q_0$ is roughly constant until the end of the MS ($Q_0=10^{49.5}$~photon\/s), when the star becomes an LBV with low $\\teff$. Our models indicate that $Q_0$ increases again at the beginning of He-core burning, remaining roughly constant until the pre-SN stage.$Q_1$ follows a similar qualitative trend, while $Q_2$ rapidly decreases during the MS and only becomes significant at the transition between the WNE and WCE phase, and before core-collapse, when the star is a WO. However, we stress that the behavior of $Q_2$ and the wind ionization structure are thought to be severely affected by the presence of x-rays \\citep{hillier93,pauldrach94,feldmeier97}. Here we include the effects of x-rays only for O-type stars  (see Sect. \\ref{cmfgen}), so the values of $Q_2$ should be taken as lower limits during the WR stages.\n\n\\begin{sidewaystable*}\n\\begin{minipage}{\\textwidth}\n\\scriptsize\n\\caption{Evolution of the absolute magnitudes and bolometric corrections of a non-rotating 60~\\msun\\ star in the Johnson-Cousins $UBVRI$ and 2MASS $JHK_S$ filters.}\n\\label{absmagbc1}\n\\centering\n\\vspace{0.1cm}\n\\begin{tabular}{l c c c c c c c c c c c c c c c c c c}\n\\hline\\hline\nStage & Evol. phase &  Sp. Type & $M_U$& $M_B$ & $M_V$ & $M_R$ & $M_I$ & $M_J$ & $M_H$ & $M_K$ & BC$_U$& BC$_B$ & BC$_V$ & BC$_R$ & BC$_I$ &  $BC_J$ & $BC_H$ & $BC_K$  \\\\ \n& & & (mag) & (mag)& (mag)& (mag)& (mag)& (mag)& (mag)& (mag)& (mag)& (mag) & (mag)& (mag)& (mag)& (mag)& (mag) & (mag)\\\\\n\\\\  \\hline\n \\hline\n     1 & H-c & O3If$^*$c &  -6.73 &  -5.53 &  -5.24 &  -5.11 &  -4.93 &  -5.29 &  -4.55 &  -4.41 &  -2.79 &  -3.99 &  -4.28 &  -4.41 &  -4.59 &  -4.23 &  -4.97 &  -5.11 \\\\\n      2 & H-c & O3If$^*$c &  -6.83 &  -5.64 &  -5.35 &  -5.23 &  -5.05 &  -5.41 &  -4.68 &  -4.53 &  -2.73 &  -3.92 &  -4.21 &  -4.34 &  -4.52 &  -4.16 &  -4.89 &  -5.03 \\\\\n      3 & H-c & O3If$^*$c &  -6.98 &  -5.79 &  -5.51 &  -5.38 &  -5.20 &  -5.56 &  -4.84 &  -4.70 &  -2.64 &  -3.83 &  -4.11 &  -4.24 &  -4.42 &  -4.06 &  -4.78 &  -4.92 \\\\\n      4 & H-c & O4If$^*$c &  -7.22 &  -6.05 &  -5.77 &  -5.65 &  -5.47 &  -5.82 &  -5.11 &  -4.97 &  -2.48 &  -3.66 &  -3.93 &  -4.05 &  -4.23 &  -3.88 &  -4.59 &  -4.73 \\\\\n      5 & H-c & O5Ifc &  -7.55 &  -6.40 &  -6.13 &  -6.02 &  -5.85 &  -6.18 &  -5.51 &  -5.37 &  -2.23 &  -3.38 &  -3.65 &  -3.77 &  -3.93 &  -3.60 &  -4.28 &  -4.41 \\\\\n      6 & H-c & O6Iafc &  -7.87 &  -6.73 &  -6.48 &  -6.37 &  -6.22 &  -6.53 &  -5.88 &  -5.75 &  -1.97 &  -3.11 &  -3.36 &  -3.47 &  -3.62 &  -3.31 &  -3.96 &  -4.09 \\\\\n      7 & H-c & O7.5Iafc &  -8.28 &  -7.17 &  -6.94 &  -6.84 &  -6.71 &  -6.99 &  -6.39 &  -6.27 &  -1.62 &  -2.73 &  -2.96 &  -3.06 &  -3.20 &  -2.92 &  -3.51 &  -3.64 \\\\\n      8 & H-c & O9Iab &  -8.58 &  -7.49 &  -7.28 &  -7.19 &  -7.06 &  -7.32 &  -6.76 &  -6.63 &  -1.35 &  -2.45 &  -2.66 &  -2.75 &  -2.88 &  -2.62 &  -3.18 &  -3.31 \\\\  \n      9 & H-c & B0.2 Ia &  -8.81 &  -7.73 &  -7.54 &  -7.44 &  -7.31 &  -7.57 &  -7.02 &  -6.90 &  -1.14 &  -2.22 &  -2.42 &  -2.51 &  -2.64 &  -2.38 &  -2.94 &  -3.06 \\\\\n     10 & H-c & B0.5 Ia$^+$ &  -8.91 &  -7.86 &  -7.67 &  -7.64 &  -7.50 &  -7.72 &  -7.32 &  -7.27 &  -1.06 &  -2.10 &  -2.29 &  -2.32 &  -2.46 &  -2.24 &  -2.64 &  -2.69 \\\\\n     11 & H-c & hot LBV &  -9.22 &  -8.19 &  -8.03 &  -8.02 &  -7.91 &  -8.08 &  -7.78 &  -7.77 &  -0.75 &  -1.78 &  -1.94 &  -1.95 &  -2.06 &  -1.89 &  -2.19 &  -2.20 \\\\\n     12 & H-c & hot LBV &  -9.48 &  -8.47 &  -8.35 &  -8.37 &  -8.31 &  -8.39 &  -8.29 &  -8.35 &  -0.50 &  -1.51 &  -1.63 &  -1.61 &  -1.67 &  -1.59 &  -1.69 &  -1.63 \\\\\n     13 & H-c & cool LBV & -10.31 &  -9.53 &  -9.64 &  -9.83 &  -9.96 &  -9.67 & -10.19 & -10.32 &   0.36 &  -0.43 &  -0.31 &  -0.13 &   0.01 &  -0.29 &   0.23 &   0.36 \\\\\n     14 & H-c & hot LBV &  -9.42 &  -8.39 &  -8.28 &  -8.33 &  -8.28 &  -8.32 &  -8.30 &  -8.39 &  -0.51 &  -1.55 &  -1.66 &  -1.61 &  -1.66 &  -1.61 &  -1.63 &  -1.55 \\\\\n     15 & H-c & hot LBV &  -9.14 &  -8.08 &  -8.00 &  -8.10 &  -8.04 &  -8.05 &  -8.11 &  -8.18 &  -0.90 &  -1.96 &  -2.04 &  -1.93 &  -2.00 &  -1.99 &  -1.93 &  -1.86 \\\\\n     16 & H-sh & hot LBV &  -9.49 &  -8.44 &  -8.35 &  -8.44 &  -8.39 &  -8.40 &  -8.47 &  -8.58 &  -0.56 &  -1.60 &  -1.69 &  -1.61 &  -1.65 &  -1.64 &  -1.57 &  -1.46 \\\\\n     17 & H-sh & cool LBV & -10.46 &  -9.62 &  -9.67 &  -9.86 &  -9.98 &  -9.71 & -10.19 & -10.37 &   0.40 &  -0.43 &  -0.39 &  -0.19 &  -0.08 &  -0.35 &   0.14 &   0.31 \\\\\n     18 & H-sh & cool LBV & -10.43 &  -9.71 &  -9.85 & -10.02 & -10.17 &  -9.87 & -10.40 & -10.53 &   0.37 &  -0.35 &  -0.21 &  -0.04 &   0.11 &  -0.19 &   0.34 &   0.47 \\\\\n     19 & H-sh & cool LBV & -10.42 &  -9.70 &  -9.83 &  -9.99 & -10.13 &  -9.85 & -10.35 & -10.47 &   0.36 &  -0.36 &  -0.24 &  -0.07 &   0.06 &  -0.22 &   0.28 &   0.41 \\\\\n     20 & H-sh & cool LBV & -10.42 &  -9.71 &  -9.83 & -10.00 & -10.13 &  -9.86 & -10.35 & -10.48 &   0.35 &  -0.35 &  -0.23 &  -0.07 &   0.07 &  -0.21 &   0.29 &   0.41 \\\\\n     21 & H-sh & cool LBV & -10.42 &  -9.72 &  -9.84 & -10.00 & -10.14 &  -9.86 & -10.35 & -10.48 &   0.35 &  -0.35 &  -0.23 &  -0.07 &   0.07 &  -0.20 &   0.29 &   0.41 \\\\\n     22 & H-sh & cool LBV & -10.42 &  -9.74 &  -9.86 & -10.01 & -10.15 &  -9.88 & -10.36 & -10.48 &   0.35 &  -0.33 &  -0.21 &  -0.06 &   0.08 &  -0.19 &   0.29 &   0.41 \\\\\n     23 & H-sh & cool LBV & -10.40 &  -9.80 &  -9.92 & -10.06 & -10.19 &  -9.94 & -10.38 & -10.50 &   0.33 &  -0.27 &  -0.15 &  -0.01 &   0.12 &  -0.13 &   0.31 &   0.43 \\\\\n     24 & H-sh & cool LBV & -10.38 &  -9.83 &  -9.95 & -10.08 & -10.21 &  -9.97 & -10.39 & -10.50 &   0.31 &  -0.24 &  -0.12 &   0.01 &   0.14 &  -0.10 &   0.32 &   0.43 \\\\\n     25 & H-sh & cool LBV & -10.36 &  -9.86 &  -9.97 & -10.09 & -10.22 &  -9.99 & -10.39 & -10.49 &   0.29 &  -0.22 &  -0.10 &   0.02 &   0.15 &  -0.08 &   0.31 &   0.42 \\\\\n     26 & H-sh & cool LBV & -10.24 &  -9.92 & -10.06 & -10.17 & -10.29 & -10.08 & -10.42 & -10.51 &   0.17 &  -0.15 &  -0.01 &   0.10 &   0.22 &   0.01 &   0.35 &   0.44 \\\\\n     27 & H-sh & cool LBV & -10.16 &  -9.97 & -10.12 & -10.22 & -10.33 & -10.14 & -10.45 & -10.52 &   0.08 &  -0.10 &   0.05 &   0.15 &   0.26 &   0.06 &   0.38 &   0.44 \\\\\n     28 & He-c+H-sh & cool LBV & -10.78 & -10.06 & -10.14 & -10.25 & -10.38 & -10.16 & -10.52 & -10.65 &   0.49 &  -0.23 &  -0.15 &  -0.04 &   0.09 &  -0.13 &   0.23 &   0.36 \\\\\n     29 & He-c+H-sh & cool LBV & -10.77 & -10.02 & -10.10 & -10.21 & -10.35 & -10.12 & -10.50 & -10.63 &   0.50 &  -0.25 &  -0.17 &  -0.05 &   0.08 &  -0.15 &   0.23 &   0.36 \\\\\n     30 & He-c+H-sh & cool LBV & -10.76 & -10.01 & -10.09 & -10.20 & -10.33 & -10.10 & -10.48 & -10.62 &   0.50 &  -0.26 &  -0.18 &  -0.06 &   0.07 &  -0.16 &   0.22 &   0.35 \\\\\n     31 & He-c+H-sh & cool LBV & -10.76 &  -9.94 &  -9.99 & -10.11 & -10.24 & -10.02 & -10.42 & -10.57 &   0.51 &  -0.31 &  -0.25 &  -0.14 &  -0.01 &  -0.23 &   0.17 &   0.32 \\\\\n     32 & He-c+H-sh & cool LBV & -10.89 & -10.07 & -10.11 & -10.24 & -10.35 & -10.13 & -10.52 & -10.68 &   0.65 &  -0.17 &  -0.13 &   0.00 &   0.12 &  -0.10 &   0.29 &   0.45 \\\\\n     33 & He-c+H-sh & cool LBV & -10.69 &  -9.92 &  -9.96 & -10.06 & -10.18 &  -9.98 & -10.34 & -10.49 &   0.47 &  -0.30 &  -0.26 &  -0.16 &  -0.04 &  -0.24 &   0.12 &   0.27 \\\\\n     34 & He-c+H-sh & WN11(h) &  -9.33 &  -8.23 &  -8.07 &  -8.08 &  -7.93 &  -8.12 &  -7.85 &  -7.86 &  -0.88 &  -1.98 &  -2.14 &  -2.13 &  -2.28 &  -2.09 &  -2.36 &  -2.35 \\\\\n     35 & He-c, H-sh & WN8(h) &  -8.70 &  -7.62 &  -7.39 &  -7.40 &  -7.30 &  -7.47 &  -7.20 &  -7.22 &  -1.49 &  -2.58 &  -2.80 &  -2.80 &  -2.90 &  -2.73 &  -2.99 &  -2.98 \\\\\n     36 & He-c, H-sh & WN7(h) &  -8.06 &  -6.97 &  -6.71 &  -6.69 &  -6.58 &  -6.80 &  -6.48 &  -6.50 &  -2.12 &  -3.21 &  -3.47 &  -3.49 &  -3.60 &  -3.38 &  -3.70 &  -3.68 \\\\\n     37 & He-c, H-sh & WN5(h) &  -7.57 &  -6.49 &  -6.18 &  -6.16 &  -6.05 &  -6.30 &  -6.00 &  -6.03 &  -2.60 &  -3.68 &  -3.98 &  -4.00 &  -4.12 &  -3.87 &  -4.17 &  -4.14 \\\\\n     38 & He-c, H-sh & WN2(h) &  -6.79 &  -5.59 &  -5.29 &  -5.19 &  -5.04 &  -5.37 &  -4.79 &  -4.73 &  -3.35 &  -4.55 &  -4.86 &  -4.95 &  -5.10 &  -4.78 &  -5.36 &  -5.41 \\\\\n     39 & He-c, H-sh & WN5(h) &  -6.72 &  -5.59 &  -5.26 &  -5.26 &  -5.13 &  -5.41 &  -5.11 &  -5.15 &  -3.41 &  -4.54 &  -4.86 &  -4.86 &  -4.99 &  -4.72 &  -5.01 &  -4.98 \\\\\n     40 & He-c, H-sh & WN5(h) &  -7.46 &  -6.58 &  -6.26 &  -6.38 &  -6.32 &  -6.44 &  -6.47 &  -6.55 &  -2.64 &  -3.53 &  -3.84 &  -3.73 &  -3.79 &  -3.66 &  -3.64 &  -3.56 \\\\\n     41 & He-c, H-sh & WN5(h) &  -7.31 &  -6.42 &  -6.12 &  -6.25 &  -6.19 &  -6.31 &  -6.36 &  -6.45 &  -2.75 &  -3.64 &  -3.95 &  -3.82 &  -3.88 &  -3.75 &  -3.70 &  -3.61 \\\\\n     42 & He-c, H-sh & WN5(h) &  -7.16 &  -6.26 &  -5.96 &  -6.10 &  -6.04 &  -6.17 &  -6.23 &  -6.33 &  -2.85 &  -3.75 &  -4.05 &  -3.91 &  -3.97 &  -3.84 &  -3.77 &  -3.68 \\\\\n     43 & He-c, H-sh & WN5(h) &  -6.93 &  -6.03 &  -5.73 &  -5.88 &  -5.81 &  -5.96 &  -6.02 &  -6.11 &  -2.97 &  -3.87 &  -4.16 &  -4.02 &  -4.09 &  -3.94 &  -3.88 &  -3.79 \\\\\n     44 & He-c, H-sh & WN5 &  -6.96 &  -6.08 &  -5.81 &  -5.93 &  -5.88 &  -6.02 &  -6.08 &  -6.18 &  -2.88 &  -3.77 &  -4.04 &  -3.91 &  -3.97 &  -3.82 &  -3.76 &  -3.66 \\\\\n     45 & He-c & WN5 &  -6.72 &  -5.83 &  -5.59 &  -5.71 &  -5.63 &  -5.80 &  -5.83 &  -5.93 &  -2.96 &  -3.84 &  -4.09 &  -3.97 &  -4.05 &  -3.88 &  -3.85 &  -3.75 \\\\\n     46 & He-c & WO4 &  -5.71 &  -4.86 &  -4.76 &  -4.91 &  -4.73 &  -4.98 &  -4.94 &  -5.03 &  -3.89 &  -4.74 &  -4.84 &  -4.69 &  -4.87 &  -4.62 &  -4.66 &  -4.57 \\\\\n     47 & He-c & WC4 &  -6.07 &  -5.39 &  -5.82 &  -6.14 &  -5.23 &  -6.01 &  -5.38 &  -5.46 &  -3.45 &  -4.13 &  -3.70 &  -3.39 &  -4.30 &  -3.51 &  -4.15 &  -4.06 \\\\\n     48 & He-c & WC4 &  -6.34 &  -5.61 &  -5.39 &  -5.60 &  -5.18 &  -5.74 &  -5.18 &  -5.25 &  -3.01 &  -3.75 &  -3.97 &  -3.76 &  -4.17 &  -3.62 &  -4.18 &  -4.11 \\\\\n     49 & He-c & WC4 &  -5.91 &  -4.88 &  -5.01 &  -5.23 &  -4.55 &  -5.25 &  -4.51 &  -4.60 &  -3.21 &  -4.24 &  -4.11 &  -3.89 &  -4.57 &  -3.87 &  -4.61 &  -4.53 \\\\\n     50 & He-c & WO4 &  -5.41 &  -4.13 &  -4.78 &  -5.04 &  -3.95 &  -4.90 &  -4.05 &  -4.11 &  -3.65 &  -4.94 &  -4.28 &  -4.02 &  -5.11 &  -4.16 &  -5.02 &  -4.96 \\\\\n     51 & He-c & WO1 &  -4.58 &  -3.14 &  -2.72 &  -2.73 &  -2.81 &  -2.88 &  -2.87 &  -2.96 &  -4.55 &  -6.00 &  -6.41 &  -6.40 &  -6.33 &  -6.26 &  -6.27 &  -6.18 \\\\\n     52 & He-sh & WO1 &  -4.75 &  -3.26 &  -2.86 &  -2.83 &  -2.97 &  -2.98 &  -2.98 &  -3.09 &  -4.49 &  -5.98 &  -6.38 &  -6.41 &  -6.27 &  -6.26 &  -6.26 &  -6.15 \\\\\n     53 & end C-c & WO1 &  -3.79 &  -2.69 &  -2.58 &  -2.61 &  -2.71 &  -2.66 &  -2.85 &  -3.07 &  -5.68 &  -6.78 &  -6.89 &  -6.86 &  -6.76 &  -6.81 &  -6.62 &  -6.40 \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{sidewaystable*}\n\n\\begin{sidewaystable*}\n\\begin{minipage}{\\textwidth}\n\\scriptsize\n\\caption{Evolution of the absolute magnitudes and bolometric corrections of a non-rotating 60~\\msun\\ star in the {\\it HST}\/WFPC2 F170, F336W, F450W, F555W, F606W, and F814W filters.}\n\\label{absmagbc2}\n\\centering\n\\vspace{0.1cm}\n\\begin{tabular}{l c c c c c c c c c c c c c c}\n\\hline\\hline\nStage & Evol. phase &  Sp. Type &$M_{170}$ & $M_{336}$  & $M_{450}$ & $M_{555}$ & $M_{606}$ & $M_{814}$ & BC$_{170}$ & BC$_{336}$  & BC$_{450}$ & BC$_{555}$ & BC$_{606}$ & BC$_{814}$ \\\\ \n & & & (mag) & (mag)& (mag)& (mag)& (mag)& (mag) & (mag)& (mag)& (mag)& (mag)& (mag) & (mag)\\\\\n \\hline\n \\hline\n      1 & H-c & O3If$^*$c &  -8.93 &  -7.27 &  -5.52 &  -5.29 &  -5.21 &  -4.95&  -0.59 &  -2.26 &  -4.00 &  -4.23 &  -4.31 &  -4.58 \\\\\n      2 & H-c & O3If$^*$c &  -9.01 &  -7.37 &  -5.63 &  -5.41 &  -5.33 &  -5.06&  -0.55 &  -2.20 &  -3.93 &  -4.16 &  -4.24 &  -4.50 \\\\\n      3 & H-c & O3If$^*$c &  -9.13 &  -7.51 &  -5.78 &  -5.56 &  -5.49 &  -5.22&  -0.49 &  -2.11 &  -3.84 &  -4.06 &  -4.14 &  -4.40 \\\\\n      4 & H-c & O4If$^*$c &  -9.32 &  -7.75 &  -6.04 &  -5.82 &  -5.75 &  -5.49&  -0.38 &  -1.95 &  -3.66 &  -3.88 &  -3.95 &  -4.21 \\\\\n      5 & H-c & O5Ifc &  -9.57 &  -8.08 &  -6.40 &  -6.18 &  -6.12 &  -5.87&  -0.22 &  -1.70 &  -3.39 &  -3.60 &  -3.67 &  -3.92 \\\\\n      6 & H-c & O6Iafc &  -9.79 &  -8.39 &  -6.73 &  -6.53 &  -6.46 &  -6.23&  -0.05 &  -1.45 &  -3.11 &  -3.31 &  -3.38 &  -3.61 \\\\\n      7 & H-c & O7.5Iafc & -10.07 &  -8.80 &  -7.17 &  -6.99 &  -6.93 &  -6.72&   0.16 &  -1.11 &  -2.73 &  -2.92 &  -2.97 &  -3.18 \\\\  \n      9 & H-c & B0.2 Ia & -10.38 &  -9.31 &  -7.74 &  -7.57 &  -7.53 &  -7.33&   0.42 &  -0.65 &  -2.22 &  -2.38 &  -2.43 &  -2.63 \\\\\n     10 & H-c & B0.5 Ia$^+$ & -10.26 &  -9.39 &  -7.87 &  -7.72 &  -7.71 &  -7.52&   0.30 &  -0.57 &  -2.09 &  -2.24 &  -2.25 &  -2.44 \\\\\n     11 & H-c & hot LBV & -10.27 &  -9.69 &  -8.20 &  -8.08 &  -8.08 &  -7.92&   0.30 &  -0.28 &  -1.77 &  -1.89 &  -1.89 &  -2.05 \\\\\n     12 & H-c & hot LBV & -10.26 &  -9.94 &  -8.49 &  -8.39 &  -8.41 &  -8.33&   0.28 &  -0.04 &  -1.49 &  -1.59 &  -1.57 &  -1.65 \\\\\n     13 & H-c & cool LBV &  -8.75 & -10.61 &  -9.59 &  -9.67 &  -9.79 &  -9.96&  -1.21 &   0.66 &  -0.36 &  -0.29 &  -0.17 &   0.01 \\\\\n     14 & H-c & hot LBV & -10.25 &  -9.88 &  -8.42 &  -8.32 &  -8.36 &  -8.29&   0.32 &  -0.06 &  -1.52 &  -1.61 &  -1.58 &  -1.64 \\\\\n     15 & H-c & hot LBV & -10.24 &  -9.61 &  -8.12 &  -8.05 &  -8.11 &  -8.05&   0.20 &  -0.43 &  -1.92 &  -1.99 &  -1.93 &  -1.98 \\\\\n     16 & H-sh & hot LBV & -10.37 &  -9.95 &  -8.47 &  -8.40 &  -8.45 &  -8.41&   0.33 &  -0.09 &  -1.57 &  -1.64 &  -1.59 &  -1.64 \\\\\n     17 & H-sh & cool LBV &  -9.21 & -10.79 &  -9.68 &  -9.71 &  -9.84 &  -9.97&  -0.84 &   0.74 &  -0.38 &  -0.35 &  -0.22 &  -0.08 \\\\\n     18 & H-sh & cool LBV &  -8.76 & -10.68 &  -9.78 &  -9.87 &  -9.98 & -10.17&  -1.30 &   0.61 &  -0.29 &  -0.19 &  -0.08 &   0.11 \\\\\n     19 & H-sh & cool LBV &  -8.65 & -10.67 &  -9.76 &  -9.85 &  -9.96 & -10.13&  -1.41 &   0.61 &  -0.30 &  -0.22 &  -0.11 &   0.06 \\\\\n     20 & H-sh & cool LBV &  -8.61 & -10.65 &  -9.77 &  -9.86 &  -9.96 & -10.14&  -1.45 &   0.59 &  -0.29 &  -0.21 &  -0.10 &   0.07 \\\\\n     21 & H-sh & cool LBV &  -8.59 & -10.65 &  -9.78 &  -9.86 &  -9.97 & -10.14&  -1.47 &   0.58 &  -0.29 &  -0.20 &  -0.10 &   0.07 \\\\\n     22 & H-sh & cool LBV &  -8.55 & -10.63 &  -9.80 &  -9.88 &  -9.98 & -10.15&  -1.52 &   0.56 &  -0.27 &  -0.19 &  -0.09 &   0.08 \\\\\n     23 & H-sh & cool LBV &  -8.44 & -10.54 &  -9.86 &  -9.94 & -10.03 & -10.19&  -1.63 &   0.47 &  -0.21 &  -0.13 &  -0.04 &   0.12 \\\\\n     24 & H-sh & cool LBV &  -8.38 & -10.50 &  -9.89 &  -9.97 & -10.05 & -10.21&  -1.69 &   0.43 &  -0.18 &  -0.10 &  -0.02 &   0.14 \\\\\n     25 & H-sh & cool LBV &  -8.32 & -10.44 &  -9.91 &  -9.99 & -10.07 & -10.22&  -1.75 &   0.37 &  -0.16 &  -0.08 &   0.00 &   0.15 \\\\\n     26 & H-sh & cool LBV &  -7.92 & -10.18 &  -9.99 & -10.08 & -10.16 & -10.30&  -2.15 &   0.10 &  -0.08 &   0.01 &   0.08 &   0.23 \\\\\n     27 & H-sh & cool LBV &  -7.77 &  -9.98 & -10.04 & -10.14 & -10.21 & -10.34&  -2.30 &  -0.10 &  -0.03 &   0.06 &   0.13 &   0.27 \\\\\n     28 & He-c+H-sh & cool LBV &  -8.66 & -10.98 & -10.10 & -10.16 & -10.23 & -10.38&  -1.63 &   0.69 &  -0.19 &  -0.13 &  -0.06 &   0.09 \\\\\n     29 & He-c+H-sh & cool LBV &  -8.70 & -10.98 & -10.06 & -10.12 & -10.20 & -10.35&  -1.57 &   0.72 &  -0.20 &  -0.15 &  -0.07 &   0.09 \\\\\n     30 & He-c+H-sh & cool LBV &  -8.71 & -10.98 & -10.05 & -10.10 & -10.18 & -10.34&  -1.55 &   0.71 &  -0.21 &  -0.16 &  -0.08 &   0.08 \\\\\n     31 & He-c+H-sh & cool LBV &  -9.11 & -11.04 &  -9.98 & -10.02 & -10.09 & -10.24&  -1.14 &   0.80 &  -0.27 &  -0.23 &  -0.16 &  -0.00 \\\\\n     32 & He-c+H-sh & cool LBV &  -8.33 & -11.15 & -10.11 & -10.13 & -10.22 & -10.35&  -1.90 &   0.92 &  -0.13 &  -0.10 &  -0.01 &   0.12 \\\\\n     33 & He-c+H-sh & cool LBV &  -9.00 & -10.99 &  -9.96 &  -9.98 & -10.06 & -10.19&  -1.22 &   0.77 &  -0.26 &  -0.24 &  -0.16 &  -0.03 \\\\\n     34 & He-c+H-sh & WN11(h) & -10.40 &  -9.81 &  -8.25 &  -8.12 &  -8.13 &  -7.96&   0.19 &  -0.40 &  -1.96 &  -2.09 &  -2.08 &  -2.25 \\\\\n     35 & He-c, H-sh & WN8(h) & -10.14 &  -9.22 &  -7.63 &  -7.47 &  -7.44 &  -7.32&  -0.06 &  -0.98 &  -2.57 &  -2.73 &  -2.75 &  -2.88 \\\\\n     36 & He-c, H-sh & WN7(h) &  -9.90 &  -8.58 &  -6.98 &  -6.80 &  -6.75 &  -6.61&  -0.28 &  -1.60 &  -3.20 &  -3.38 &  -3.43 &  -3.57 \\\\\n     37 & He-c, H-sh & WN5(h) &  -9.58 &  -8.12 &  -6.50 &  -6.30 &  -6.24 &  -6.11&  -0.59 &  -2.04 &  -3.66 &  -3.87 &  -3.93 &  -4.06 \\\\\n     38 & He-c, H-sh & WN2(h) &  -8.99 &  -7.34 &  -5.59 &  -5.37 &  -5.29 &  -5.07&  -1.16 &  -2.80 &  -4.55 &  -4.78 &  -4.86 &  -5.07 \\\\\n     39 & He-c, H-sh & WN5(h) &  -8.92 &  -7.30 &  -5.62 &  -5.41 &  -5.34 &  -5.22&  -1.21 &  -2.83 &  -4.50 &  -4.72 &  -4.79 &  -4.91 \\\\\n     40 & He-c, H-sh & WN5(h) &  -9.42 &  -8.04 &  -6.61 &  -6.44 &  -6.41 &  -6.41&  -0.69 &  -2.07 &  -3.49 &  -3.66 &  -3.70 &  -3.70 \\\\\n     41 & He-c, H-sh & WN5(h) &  -9.29 &  -7.90 &  -6.47 &  -6.31 &  -6.27 &  -6.29&  -0.78 &  -2.16 &  -3.59 &  -3.75 &  -3.79 &  -3.77 \\\\\n     42 & He-c, H-sh & WN5(h) &  -9.14 &  -7.76 &  -6.32 &  -6.17 &  -6.13 &  -6.16&  -0.87 &  -2.25 &  -3.68 &  -3.84 &  -3.88 &  -3.85 \\\\\n     43 & He-c, H-sh & WN5(h) &  -8.92 &  -7.55 &  -6.10 &  -5.96 &  -5.91 &  -5.94&  -0.98 &  -2.35 &  -3.80 &  -3.94 &  -3.99 &  -3.96 \\\\\n     44 & He-c, H-sh & WN5 &  -8.96 &  -7.58 &  -6.15 &  -6.02 &  -5.96 &  -6.00&  -0.88 &  -2.26 &  -3.69 &  -3.82 &  -3.88 &  -3.85 \\\\\n     45 & He-c & WN5 &  -8.75 &  -7.35 &  -5.91 &  -5.80 &  -5.73 &  -5.75&  -0.93 &  -2.33 &  -3.77 &  -3.88 &  -3.95 &  -3.93 \\\\\n     46 & He-c & WO4 &  -9.18 &  -6.29 &  -4.98 &  -4.98 &  -4.85 &  -4.74&  -0.43 &  -3.32 &  -4.62 &  -4.62 &  -4.75 &  -4.86 \\\\\n     47 & He-c & WC4 &  -8.51 &  -6.63 &  -5.53 &  -6.01 &  -5.84 &  -5.22&  -1.02 &  -2.89 &  -4.00 &  -3.51 &  -3.69 &  -4.31 \\\\\n     48 & He-c & WC4 &  -8.53 &  -6.96 &  -5.73 &  -5.74 &  -5.43 &  -5.17&  -0.83 &  -2.40 &  -3.63 &  -3.62 &  -3.93 &  -4.18 \\\\\n     49 & He-c & WC4 &  -7.88 &  -6.56 &  -5.00 &  -5.25 &  -5.00 &  -4.53&  -1.24 &  -2.56 &  -4.12 &  -3.87 &  -4.12 &  -4.59 \\\\\n     50 & He-c & WO4 &  -7.24 &  -6.00 &  -4.24 &  -4.90 &  -4.74 &  -3.94&  -1.82 &  -3.07 &  -4.82 &  -4.16 &  -4.33 &  -5.12 \\\\\n     51 & He-c & WO1 &  -6.27 &  -4.43 &  -3.23 &  -2.88 &  -2.76 &  -2.80&  -2.86 &  -4.70 &  -5.91 &  -6.26 &  -6.38 &  -6.34 \\\\\n     52 & He-sh & WO1 &  -6.32 &  -4.61 &  -3.33 &  -2.98 &  -2.87 &  -2.95&  -2.92 &  -4.63 &  -5.91 &  -6.26 &  -6.37 &  -6.29 \\\\\n     53 & end C-c & WO1 &  -5.84 &  -4.22 &  -2.74 &  -2.66 &  -2.62 &  -2.71&  -3.63 &  -5.25 &  -6.73 &  -6.81 &  -6.85 &  -6.76 \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{sidewaystable*}\n\n\\section{Caveats}\n\\label{caveat}\nOne should have in mind that the results discussed in the previous sections are dependent on a number of physical ingredients of the models, concerning both the interior and atmosphere\/wind of massive stars. Changes in any physical ingredient that has an impact on the position of the star on the HR diagram will affect the output spectrum, and possibly some of the conclusions reached here. For instance, the overshooting parameter is well known to affect the position of the star on the HR diagram \\citep{maeder75}. \n\nLikewise, the evolution of a massive star is strongly dependent on mass loss, which affects not only the tracks in the HR diagram but also the spectroscopic appearance. The effects of mass loss on the interior evolution are well documented in the literature (see \\citealt{chiosi86} and references therein), and exploring its effects on the spectroscopic appearance is computationally expensive and beyond the scope of this paper. We note though that a reduction in $\\mdot$ during the MS would favor the appearance of O stars with luminosity class V and III and, instead of a long LBV phase,  BSGs\/BHGs and A-type supergiants\/hypergiants would be favored as well.\n\nAs noted before, wind clumping can also have a significant effect on the spectra of massive stars \\citep{hillier91,schmutz97}. In the next subsection, we discuss how the choice of the clumping parameters ($f_\\infty$ and $v_c$) affect our results.\n\n\\subsection{Effects of clumping}\n\\label{mdotclump}\n\nWe focus on the optical spectra of O stars and four spectral lines (H$\\alpha$, \\ion{He}{ii} \\lam4687, \\ion{He}{i} $\\lambda4473$, and  \\ion{He}{ii} $\\lambda4543$) that are key for spectral type and $\\mdot$ determinations. The effects of clumping on these and other spectral lines are well known and have been extensively discussed in the literature \\citep{crowther02,bouret03,bouret05,bouret13,hillier03,fullerton06,martins07,martins09,martins12,oskinova07,sundqvist10,sundqvist11,surlan12,zsargo08}.\n\nFor O stars, our models use the mass-loss rate recipe from \\citet{vink01}. Assuming that this prescription is not affected by clumping, \\citet{repolust04} and \\citet{mokiem07b} find that the observed H$\\alpha$ strengths are consistent with the \\citet{vink01} mass-loss rates if $f_\\infty=0.2$ and clumps are formed immediately above the photosphere. For this reason, we adopt $f=0.2$ and $v_c=30~\\kms$ for our O star models. Note, however, that clumping may affect the $\\mdot$ prescription depending on the clumping scale length \\citep{muijres11}.\n\nTo investigate the effects of clumping at the O-type stage, we computed models for stages 1 through 10 with the \\citet{vink01} mass-loss rates, $v_c=30~\\kms$,  but using different values of $f_\\infty$ (0.1, 0.2, and 1.0). We show results for stages 1 (ZAMS) and 7 ($X_c=0.17$) in Fig. \\ref{lineclump}, but models computed for the intermediate stages show similar behavior. Since $f$ is radially dependent, spectral lines are affected differently by changes in $f$ according to their formation region (e.g., \\citealt{puls06}). Therefore, lines formed close to the photosphere, at low velocities, are less affected by clumping than lines formed in the wind. The strength of the H$\\alpha$ emission, which is a recombination process, increases as $f_\\infty$ decreases, as illustrated in Fig. \\ref{lineclump}a,b. One can see that the H$\\alpha$ line predicted by our evolutionary model with a fixed $\\mdot$ is significantly affected by clumping.\n\nIn early O stars, the main criterium for determining the luminosity class is the strength of \\ion{He}{ii} \\lam4687 \\citep{contialschuler71,walborn71}. Our models show that the \\ion{He}{ii} \\lam4687 profile is affected by clumping (Fig. \\ref{lineclump}c,d), with stronger  \\ion{He}{ii} \\lam4687 emission seen as $f_\\infty$ decreases. As a consequence, models with $f_\\infty=0.1$ and $0.2$ both yield a luminosity class I at the ZAMS, while the unclumped model yield a luminosity class III.  At later stages in the MS (stage 7), the choice of $f_\\infty$ does not change the luminosity class (both clumped and unclumped models yield a class I), although the shape of the \\ion{He}{ii} \\lam4687 line varies depending on $f_\\infty$ .\n\nThis result suggests that once a certain amount of clumping is present, the luminosity class determination is not severely affected by the choice of $f_\\infty$. Likewise, while the choice of $v_c$ can affect the amount of emission in H$\\alpha$ and \\ion{He}{ii} \\lam4687, the luminosity class determination should be weakly affected by the precise choice of $v_c$ within reasonable values ($\\sim20-300~\\kms$), since significant wind contamination of \\ion{He}{ii} \\lam4687 is already present in unclumped models.\n\nLet us look at how the diagnostic lines for spectral types of O stars (\\ion{He}{i} $\\lambda4473$ and  \\ion{He}{ii} $\\lambda4543$) are affected by clumping. These are lines formed mostly close to the photosphere, at low velocities, where clumps have not yet started to form.  Although a weak filling of the core of the line may occur, clumping does not affect these lines to an amount that would cause changes in the spectral type (Fig. \\ref{lineclump}e,fg,h). For instance, both clumped and unclumped models  yield a spectral type O3 at the ZAMS and O7.5 at stage 7. \n\n\\begin{figure}\n\\center\n\\resizebox{0.980\\hsize}{!}{\\includegraphics{clump_line_montage}}\\\\\n\\caption{\\label{lineclump} Normalized line profiles of H$\\alpha$ (panels $a$ and $b$), \\ion{He}{ii} \\lam4687 ($c,d$) \\ion{He}{i} $\\lambda4473$ ($e,f$) and  \\ion{He}{ii} $\\lambda4543$ ($g,h$) for the evolutionary stages 1 (ZAMS; left panels) and 7 ($X_c=0.17$; right panels) using the \\citet{vink01} mass-loss rates, but $f_\\infty=0.1$ (dashed blue) , 0.2 (solid red), and 1.0 (solid black)). All models assume  $v_c=30~\\kms$.\n}\n\\end{figure}\n\nFor stars with dense winds, such as LBV and WRs, most or all lines are formed in the clumped stellar wind. As such, changes in $f$ significantly affect the output spectrum \\citep{hillier91}. For a fixed $\\mdot$, the strength of the emission lines in general increase with decreasing $f$, as long as no significant changes in the ionization structure occur. Thus, for small changes in $f_\\infty$,  the spectrum, magnitudes, and colors are affected, but the spectroscopic classification (BHG, LBV, or WR) remains weakly affected. Large reductions in $f_\\infty$ may cause a BHG to appear as an LBV or a WR to appear as an LBV. Therefore, the timescales for the different spectroscopic phases derived in this paper (Table \\ref{duration}) should be taken with caution, as they depend on the choice of the clumping parameter. \n\nFinally, we stress that the real physical conditions in stellar winds are certainly more complicated than our simple exponential clumping law models. Wind clumping in O stars likely originates from line-driven instabilities \\citep{owocki88}. Hydrodynamical simulations of line-driven instability predict non-monotonic, complex $f(r)$ \\citep{runacres02, sundqvist12}, which were shown to have a significant effect in line profiles of O stars \\citep{puls06, najarro11}. However, little is known as to the dependence of the clumping (structure) on stellar properties (but see \\citealt{cantiello09}). These uncertainties warrant the simple clumping prescription adopted in our models.\n\n\\section{\\label{conc} Concluding remarks}\n\nIn this paper we investigated the evolution of the interior, spectra, and photometry of a non-rotating 60~\\msun\\ star. For the first time, coupled stellar evolution and atmospheric modelings covering the full evolution of a massive star were developed. We employed the Geneva stellar evolution code and the CMFGEN atmospheric\/wind code to produce observables out of stellar evolutionary calculations. Our main conclusions are summarized below.\n\n\\begin{enumerate}\n\n\\item We find that a non-rotating 60~\\msun\\ star has the following evolutionary sequence of spectral types: O3 I (ZAMS) $\\rightarrow$ O4 I (mid H-core burning) $\\rightarrow$ hot LBV (end H-core burning) $\\rightarrow$ cool LBV (start He-core burning) $\\rightarrow$ WNL $\\rightarrow$ WNE $\\rightarrow$ WC (mid He-core burning) $\\rightarrow$ WO (end He-core burning until core collapse). \n\n\\item During the MS evolutionary phase, when H-core burning occurs, the star has the following spectral types (durations in parenthesis): early O I (2.89 Myr), late O I (0.36 Myr), BSG (0.04 Myr),  BHG (0.08 Myr), and LBV (0.23 Myr). Thus, our models indicate that some LBVs can be H-core burning objects. During the MS, the star loses $8.5~\\msun$ when it has an O-type spectrum, $1.3~\\msun$ as a BHG, and $14.0~\\msun$ as an LBV. The huge mass loss as an LBV is a consequence of the star crossing the bistability limit of line-drive winds, which occurs around 21000 K \\citep{vink01}.\n\n\\item The post-MS evolution of a non-rotating 60~\\msun\\ star is comprised by a short H-shell burning, long He-core burning, and short He-shell burning, C-core burning, and multiple shell-burning stages. \n\n\\item During the H-shell burning (0.007 Myr), the star is an LBV and loses 4~\\msun. During the He-core burning, the surface temperature and chemical composition change dramatically, causing the star to show a variety of spectral types. It begins the He-core burning phase as an LBV and, in 0.10~Myr, 5.94~\\msun\\ is lost. It goes rapidly through spectroscopic phases of WNL with small amounts of H, WNE with small amounts of H, and WNE without H (WN 5). Most of the He-core burning (0.244 Myr) is spent as an early WC (WC 4), when 7.02~\\msun\\ of material is lost.\n\n\\item At the end of He-core burning, the surface temperature increases as the star contracts. Combined with the increase in O abundance at the surface, this implies a WO spectral type, with strong \\ion{O}{vi} $\\lambda3811$ emission. The star remains as  a WO, with extremely high $\\teff$, until core-collapse.\n\n\\item With our approach of computing spectra out of evolutionary models, we investigated the duration of the different spectroscopic phases in a direct way. Compared to the results from \\citet{georgy12a}, which employed the same evolutionary model but chemical abundance and $\\teff$ criteria, we find a similar duration for the O and WR phases. However, we find that the star has a spectral type similar to LBVs for 0.235 Myr, which is relatively longer than commonly assumed for LBVs (a few 0.010 Myr; \\citealt{bohannan97}). We stress that this result is valid only for a 60~\\msun\\ star without rotation and we anticipate that rotating models will have a much shorter duration for the LBV phase. Also, the long duration is extremely dependent on our spectroscopic definition of LBV and the \\citealt{vink01} mass-loss recipe assumed.\n\n\\item Although we find similar duration as previous studies for the WR phase, the lifetimes of the different subtypes of WR is significantly different. We find that the duration of the WNL phase is about 20 times shorter than when using chemical abundance criteria. The duration of the WNE phase is increased by a factor of 2. The duration of the WC phase is also reduced by 16\\%. Finally, we find that the endstage is characterized by a short WO phase that lasts for 0.038 Myr.\n\n\\item We present the photometric evolution of a non-rotating 60~\\msun\\ star from the ZAMS to the pre-SN stage. We find that the star becomes progressively brighter in all optical and near-IR filters during the MS, and reaches the peak of its brightness as an LBV during H-shell burning. It becomes rapidly faint in these filters as the He-core burning phase begins and $\\teff$ increases. At the end of He-core burning and subsequent advanced phases, the star becomes even fainter in the optical and IR as $\\teff$ surpasses 100000~\\K. This has serious consequences for detecting progenitors of SN Ic in pre-explosion images, predicting that these progenitors  should be undetectable with the current magnitude limits \\citep{gmg13}.\n\n\\item We computed the evolution of  number of  \\ion{H}{} ($Q_0$), \\ion{He}{i} ($Q_1$), and \\ion{He}{ii} ($Q_2$) ionizing photons emitted per unit time. We found that $Q_0$ is roughly constant until the end of the MS ($Q_0=10^{49.5}$~photon\/s), when the star becomes an LBV with low $\\teff$. Our models indicate that $Q_0$ increases again at the beginning of He-core burning, remaining roughly constant until the pre-SN stage. $Q_1$ follows a similar qualitative trend, while $Q_2$ rapidly decreases during the MS and only becomes significant at the transition between the WNE and WCE phase, and before core-collapse, when the star is a WO.\n\n\\end{enumerate}\n\n\\begin{acknowledgements}\nWe thank the referee, Alex de Koter, for a careful review and a constructive report that improved the quality of the original manuscript. We are also thankful to John Hillier and Paco Najarro for discussions, sharing models, and making CMFGEN available, and Paul Crowther for comments, advices, and discussions. We thank Jes\\'us Ma\\'{i}z-Appel\\'aniz for making CHORIZOS available. JHG is supported by an Ambizione Fellowship of the Swiss National Science Foundation. CG acknowledges support from the European Research Council under the EU Seventh Framework Program (FP\/2007-2013) \/ ERC Grant Agreement n. 306901.\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNRGR \\cite{NRGR}, an Effective Field Theory (EFT) approach to gravity, has emerged\nas a powerful tool to systematically describe the dynamics of finite size objects in General Relativity (GR). It has been utilized to calculate higher order spin corrections in the PN expansion \\cite{eih,comment}, dissipative effects for non-spinning \\cite{dis1} as well spinning objects \\cite{dis2},  \nradiation reaction effects in the extremal limit \\cite{chad} and corrections to thermodynamic\nquantities in caged black holes \\cite{cbh,kol}. In this paper we will extend the formalism for spin in NRGR originally developed in \\cite{nrgr3}. In particular we will demonstrate how to calculate the equations of motion (EOM) using the Routhian formalism discussed in \\cite{nrgr5}.  \nThe leading order (LO) spin--spin and spin--orbit potentials were shown in \\cite{nrgr3}  to reproduce known results \\cite{spin,will,will2,kidd} within the Newton--Wigner (NW) and covariant SSCs. In \\cite{eih},  the previously uncalculated 3PN spin--spin potential was obtained using the NW SSC at the level of the action. In \\cite{nrgr5}, it was argued that within this approach,  the Hamiltonian method is accurate up to 4PN in the ${\\bf S}_1{\\bf S}_2$ sector, when curvature effects start to play a role, and the canonical structure in the reduced phase space, $({\\bf x},{\\cal P},{\\bf S})$, is modified. However, in order to calculate the ${\\cal O}({\\bf S}_1 {\\bf S}_2)$ contributions to the EOM at 3PN, the spin--spin potential in \\cite{eih} is not sufficient. This fact was made clear by an independent calculation in \\cite{Schafer3pn} where the complete potential was computed using a more traditional methodology. Within our approach, we must also include a correction stemming from a subleading effect due to spin--orbit interactions \\cite{comment}. Once this extra piece is included, the results in \\cite{eih, comment} and \\cite{Schafer3pn} agree. As we will see, the Routhian formalism provides yet another independent cross check of the new results to 3PN. \n\nWhile working within the NW SSC at the level of the action is relatively simple for LO effects, and ${\\cal O}({\\bf S}_1{\\bf S}_2)$ corrections up to 4PN, calculating subleading ${\\cal O}({\\bf S}_q)$,  or ${\\cal O}({\\bf S}^2_q)$ effects can be cumbersome within this methodology. Recall that  the NW SSC leads to a canonical structure in the reduced phase space only in a flat spacetime background \\cite{nrgr5}. For ${\\cal O}({\\bf S}_q)$ effects this structure is lost already at 2.5PN. It is thus desirable to have a technique where the SSC is not imposed until the end of the calculation thus avoiding complicated algebraic structures. Here we elaborate upon such an  approach, presented in  \\cite{nrgr5},  and compute the spin--spin potential to 3PN in the covariant SSC.  \n\nWithin the NRGR formalism spin--spin, or spin--orbit, refer to the type of diagrams contributing to the potential \\cite{nrgr3,eih}. Since we will postpone the SSC to the later stages of the computation, our results for the potentials and EOM will be written in terms of the spin tensor in the local frame, e.g. $S^{ab}$. Therefore, a spin--orbit term proportional to $S^{j0}$ in the EOM can contribute at ${\\cal O}({\\bf S}_1{\\bf S}_2)$ once the SSC is enforced, the spin tensor is reduced to a three vector and the velocity in the local frame is transformed to the global PN frame. Furthermore,  we will show that the spin--spin potential to 3PN also depends on $S^{j0}$, and therefore it will contribute at higher orders in the ${\\bf S}^2_1{\\bf S}_2$ and ${\\bf S}_1{\\bf S}_2^2$ sectors.\\\\\n\nHere we will present some of the details of the calculation of the 3PN potential, as well as including the EOM for the spin of the constituents.  We will first calculate the LO EOM due to the spin--orbit potential and show that we reproduce the well known results before moving on to the spin--spin 3PN computation. For completeness, finally we show the equivalence with our previous results using the NW SSC at the level of the action of \\cite{eih,comment}, by constructing an effective potential which agrees with the results in \\cite{eih,comment}, and from which the EOM follow via the canonical methods, thereby providing a formal proof of the claims in \\cite{eih,comment, nrgr5}.\\\\\n \n \n\\section{Review of Spin in GR}\nThe extension of NRGR to include spin effects was\nachieved in \\cite{nrgr3} by adding world--line degrees of freedom $\\Lambda_a^J(\\lambda)$, which is the boost that transforms the locally flat frame (labelled by small Roman letters),  to the co-rotating frame labelled by capital Roman letters. The generalized angular velocity is given by\n$\\Omega^{\\mu\\nu}=e^{\\mu J}\\frac{De^\\nu_J}{d\\lambda}$, where $e^\\mu_I= e^\\mu_a \\Lambda^a_I$ and\n$e^\\mu_a$ are the co--rotating and locally flat basis respectively (verbeins) and  $e^a_{\\mu}e^b_{\\nu} g^{\\mu\\nu} = \\eta^{ab}$.\nThe spin $S_{\\mu\\nu}$ is introduced as the conjugate momentum to $\\Omega_{\\mu \\nu}$.  The form of\nthe world--line action is then fixed by reparametrization\ninvariance \\cite{eih},\n\\begin{equation} \\label{action}\nS=-\\sum_i \\left(\\int p^\\mu_i u^i_{\\mu}d\\lambda_i + \\int\n\\frac{1}{2}S_i^{IJ}\\Omega^i_{IJ} d\\lambda_i\\right),\n\\end{equation}\nwhere the sum extends over the consituents , and $S^{IJ} \\equiv S^{\\mu\\nu}e^I_\\mu e^J_\\nu$. Here we have not included higher dimensional operators which account for finite size effects.  Corrections due to finite size effects are reported in \\cite{nos} . The Mathisson--Papapetrou (MP) equations \\cite{papa} follow from (\\ref{action}) \\cite{nrgr3,eih}. The spin--gravity coupling in (\\ref{action}) can be rewritten by introducing the Ricci rotation coefficients, $\\omega_\\mu^{ab} = e^b_\\nu D_\\mu e^{a\\nu}$, as \\cite{nrgr3,nrgr5}\n\\begin{equation} \\label{act2}\nS_{spin-gravity} =  -\\frac{1}{2} \\int S_{Lab}\\omega^{ab}_\\mu u^\\mu\nd\\lambda,\n\\end{equation}\nwith $S_L^{ab} \\equiv S^{\\mu\\nu}e_{\\mu}^a e_{\\nu}^b$, the spin in the locally flat frame (we drop the \n$L$ from now on). By further expanding (\\ref{act2}) in the weak gravity limit one obtains the Feynman rules \\cite{nrgr3,eih}. Let us emphasize that the SSC is imposed in the local  frame.\n\n\\section{The Routhian approach for spinning bodies in NRGR}\n\nA Routhian formalism\\footnote{A similar Routhian was originally proposed in \\cite{yee} with $S^{ab}u_b=0$ as SSC, which is equivalent to ours at 3PN. See appendix A.} was introduced within the covariant SSC in \\cite{nrgr5}.  In what follows we will adopt this framework and compute the 3PN corrections to the potential.  \nThe virtue of the Routhian formalism is that it allows us to consistently impose, and preserve upon evolution, the SSC in a canonical framework, and properly account for $S^2$ corrections to the potential. The price to pay is that we will work with a spin tensor, $S^{ab}$, rather than a three vector. However, we will show later on that an effective potential in terms of (${\\bf x},{\\bf v},{\\bf S}$) exists, which turns out to be equivalent to our previous results in \\cite{eih,comment}.\\\\\n\nSince the spin is a conjugate momentum, we would like to treat the spin within a Hamiltonian formalism. Whereas, for the worldline position we would like to work within the Lagrangian formalism.  That is, we would like to Legendre transform the Lagrangian with respect to the wordline spin degrees of freedom only. This is done within what is called the ``Routhian\" formalism \\cite{LL}. We will work in the covariant SSC,\n\\begin{equation}\n\\label{p.s}\np_a S^{ab} = p_\\mu S^{\\mu \\nu}=0,\n\\end{equation}\nwith $p_\\mu$ the coordinate momentum of the particle. To dynamically maintain this conditions we need to impose\n\\begin{equation}\n\\frac{D}{D\\lambda}(p_\\mu S^{\\mu \\nu})=0,\n\\end{equation}\nand utilizing the MP equations (which follow from (\\ref{action}))\n\\begin{equation}\n\\frac{D S^{\\mu\\nu}}{D\\lambda} \n=p^\\mu u^\\nu-p^\\nu u^\\mu , ~~~~~~ \\frac{D p^\\gamma}{D\\lambda}= -\\frac{1}{2} R^\\gamma_{\\rho\\alpha\\beta}S^{\\alpha\\beta}u^\\rho,\n\\end{equation}\nyields the momentum\n\\begin{equation}\np^{\\alpha}= \\frac{1}{\\sqrt{u^2}}\\left(m\nu^{\\alpha}+\\frac{1}{2m}R_{\\beta\\nu\\rho\\sigma}S^{\\alpha\\beta}S^{\\rho\\sigma}u^{\\nu}\\right)\\label{up}.\n\\end{equation}\nNotice that $p \\cdot u = m$  once the SSC is\nimposed. \n\nWe introduce now the following Routhian  \\cite{nrgr5}\n\\begin{equation} \\label{actR}\n{\\cal R} =-\\sum_i \\left( m_i \\sqrt{u^2_i} + \n\\frac{1}{2}S_{i}^{ab}\\omega_{ab\\mu} u^\\mu_i\n+\\frac{1}{2m_i}R_{d\ne a b}(x_i)S^{c d}_{i} S^{a b}_{i} \\frac{u^e_i u_{ic}}{\\sqrt{u^2}}\n+\\ldots\\right),\n\\end{equation}\nwhere the ellipses represent curvature  terms necessary to account for the mismatch between $p$ and $u$ in (\\ref{p.s}). These terms  contribute \n beyond the  3PN order we work in this paper\\footnote{ In other words, to our level of accuracy, we can consider $S^{ab}u_b = 0$.}. In addition there are finite size corrections to (\\ref{up}) which are not shown in (\\ref{actR})  but can be consistently included (for details  see appendix A) when going to higher orders in the  PN expansion. \n \n The EOM follow from\n\\begin{equation}\n\\frac{\\delta }{\\delta x^\\mu}\\int {\\cal R} d\\lambda=0, \\;\\;\\; \\frac{d\nS^{ab}}{d\\lambda} = \\{S^{ab},{\\cal R}\\}\\label{eom1},\n\\end{equation}\nwhere the algebra for the phase space variables\n$(x^\\mu,p^\\nu,S^{ab})$ is given by\n\\begin{eqnarray}\n\\{x^\\mu ,{\\cal P}_\\alpha\\}&=& \\delta^\\mu_\\alpha,\\;\\;\\; \\{x^\\mu,p_\\alpha\\}=\\delta^\\mu_\\alpha,  \\;\\;\\; \\{{\\cal P}^\\alpha,{\\cal P}^\\beta\\}= 0, \\\\\n \\{x^\\mu,x^\\nu\\} &=& 0, \\;\\;\\; \\{p^\\alpha,p^\\beta\\} = \\frac{1}{2}{R^{\\alpha\\beta}}_{ab}S^{ab}, \\label{pp}\\\\\n\\{x^\\mu ,S^{ab}\\}&=& 0, \\;\\;\\; \\{p_\\alpha,S^{ab}\\}= \\omega^{c[a}_\\alpha S^{b]c} , \\;\\;\\; \\{{\\cal P}^\\alpha,S^{ab}\\}= 0 \\label{ps} \\\\\n\\{S^{ab},S^{cd}\\} &=& \\eta^{ac} S^{bd}\n+\\eta^{bd}S^{ac}-\\eta^{ad} S^{bc}-\\eta^{bc}\nS^{ad} \\label{als}\n\\end{eqnarray}\nwith $p^\\mu$ related to the canonical momentum by $ {\\cal\nP}^\\mu = p^\\mu + \\frac{1}{2}\\omega^\\mu_{ab} S^{ab}$. It can be shown that the EOM are equivalent to the MP equations and that the extra term in (\\ref{actR}) guarantees the preservation  of the covariant SSC.\nNotice that with our choice of metric convention $(+,-,-,-)$,  the spin vector algebra differs from the canonical SO(3) algebra by a minus sign. We have compensated for this convention choice by the overall minus sign in the Routhian of (\\ref{actR}) which allows us to treat ${\\cal R}$ as the usual Lagrangian and keep the spinless feynman rules untouched. Therefore, the relationship between the potential and the Routhian stays as before for the Lagrangian, namely \\begin{equation} V = - {\\cal R}, \\end{equation} \nand therefore the spin EOM in terms of the potential take the form,\n\\begin{equation}\n\\frac{dS^{ab}}{d\\lambda} = \\{V, S^{ab}\\}\\label{eomV}.\n\\end{equation}\n\nIn practice the EOM for spin can be derived from\n\\begin{equation}\n\\label{trads}\n\\frac{d {\\bf S}}{d t} = \\frac{\\partial V}{\\partial {\\bf S}} \\times {\\bf S}\n\\end{equation}\nas one would expect, plus corrections from the $S^{0i}$ components. We will study an example in detail later on.\n  \nAccording to the program developed in \\cite{NRGR}, to calculate the potential we first need to\ngenerate a set of Feynman rules. The potential will then follow by including the appropriate\nset of Feynman diagrams.  Once we have the potential in terms of the spin, position and velocities of the binary constituents, we can calculate the EOM using (\\ref{eomV}). \n\n\\subsection{The effective action}\n\nLet us elaborate upon the manipulations leading to the potential. \nThe EFT approach is built to separate physics at different scales.\nGiven that the radiation and potential modes have a ratio of  wavelengths of order $v$, \nthis allows us to cleanly separate the physics of radiation from that of potentials in\na systematic fashion. This is not to say that potential modes have no effect on radiation.\nIndeed, the tail effect arises from the coupling of a radiation graviton to a potential graviton, and\nthe EFT reproduces known results \\cite{unpublished}. The same can be said for the so--called ``memory effect\". However, if we are interested in pure potentials we may completely ignore\nthe radiation mode in the effective action\\footnote{The LO radiation effects were computed for spinless and spinning bodies in \\cite{NRGR} and \\cite{unp2} respectively, and shown to agree with known results.}. As  discussed in \\cite{NRGR}, for spinless objects the effective NRGR action follows from the path integral ($q=1,2$)\n\\begin{equation}\n\\label{PI} \\mbox{exp}\\left[i S_{NRGR}[x^i_q]\\right] = \\int {\\cal D}H_{\\mu\\nu} \\mbox{exp}\\left[i S[H_{\\mu\\nu},x^i_q] + i S_{GF}\\right],\n\\end{equation}\nwhich accounts for the vacuum to vacuum amplitude in the presence of sources, in our case the binary.\nIn the expression above $S=S_{EH}+S_{pp}$, that is the Einstein--Hilbert action plus the wordline sources, and $S_{GF}$ is a suitable gauge fixing term\\footnote{The gauge chosen in \\cite{NRGR} corresponds to an harmonic condition up to ${\\cal O}(G^2)$ corrections (see Eqs. (62)--(65) in \\cite{NRGR}).}  \\cite{NRGR}. By expanding the Einstein-Hilbert action in the weak gravity limit we can immediately read\noff Feynman rules \\cite{NRGR}. Once we compute $S_{NRGR}[x^i_q]$ the EOM follow from a minimal action principle \\cite{coleman}, since we have yet to perform the path integral over the sources, namely the wordlines. Notice that the kinetic term is a pure phase which factors out of the path integral.  Therefore, by summing Feynman diagrams effectively we are calculating the potential energy \\cite{coleman}.\\\\ \n\nWithin this framework the inclusion of spin is straightforward. The Routhian ${\\cal R}$ will replace the worldline $S_{pp}$ action (recall ${\\cal R}=-V$) and the path integral in (\\ref{PI}) will produce the effective potential, $-V_{NRGR}$, from which the EOM follow via (\\ref{eomV}).  Since we are not imposing the SSC until the EOM is obtained, we are always dealing with a canonical structure, although we pay the price of having a spin tensor $S^{ab}$, rather than a three vector. The latter follows once the SSC imposed at the level of the EOM. For the spin dynamics we directly compute the potential and no kinetic piece is necessary. In a sense the spin dynamics has a more direct contact with the usual interpretation of the path integral as providing the energy of the `vacuum' in the presence of the sources. \n\nFinally note that the extra terms in the action proportional to the SSC effectively act as Lagrange\nmultipliers, and enforces the conservation of the SSC upon evolution. Given that we are dealing with second class constraints in the SSC, the Lagrange multipliers are  fixed by  preservation of the constraints \\cite{teit}. \nMoreover, since the extra term is proportional to the SSC itself, we are\nfree to use the equations of motion (i.e. perform an implicit coordinate shift)  \n to replace $dp\/d\\lambda$ using (5) in the Routhian as we do in (\\ref{actR}). \n\n\n\\section{An Example: The spin--orbit effects at leading order}\n\nAs a warm up let us start by computing the LO effects in the EOM due to the spin--orbit coupling \\footnote{Recall that the LO spin--spin potential does not include the troublesome term proportional to $S^{a0}$ \\cite{nrgr3}.}. In this section we will quote the contribution from each diagram to the potential. \nThe relevant Feynman rules have been relegated to appendix B.\nEach diagrams contribution is of the form $ -iV$.\n\nThe  LO spin--orbit potential is found by the instantaneous one graviton exchange\ndiagram, with the LO spin vertex on one world-line and a mass vertex on the other, as discussed in \\cite{nrgr3},\n\\begin{equation}\n\\label{so15pn}\nV^{so}_{1.5PN}= \\frac{G_Nm_2}{r^2}n^j\\left(S^{j0}_1+S^{jk}_1(v_1^k-2v^k_2)\\right)\n+ 1 \\leftrightarrow 2,\n\\end{equation}\nwith $n^j=\\frac{(x_1-x_2)^j}{\\mid x_1-x_2\\mid}$, hence ${\\cal R}^{so}_{1.5pn} = -V^{so}_{1.5pn}$.\n\nApplying the algebra (\\ref{als}) we find \n\\begin{equation}\n\\frac{dS_{lk}}{dt}= \\frac{Gm_2}{r^2}\\left[ (n_iv_{1j}-2n_iv_{2j})   (\\delta_{li}S_{kj}+\\delta_{kj}S_{li}-\\delta_{lj}S_{ki}-\\delta_{ki}S_{lj} )+(n_{l}S_{k0}-n_{k}S_{l0} \n)\\right]. \n\\end{equation}\nIn terms of the spin `vector', $S^l=\\frac{1}{2}\\epsilon^{ljk}S^{jk}$, we have in the covariant SSC,\n\\begin{equation}\n\\frac{d{\\bf S} _1}{dt} = 2\\frac{m_2\nG_N}{r^2}({\\bf n} \\times{\\bf v} )\\times{\\bf S} _1 +\n\\frac{m_2 G_N}{r^2}({\\bf S} _1\\times{\\bf n} )\\times {\\bf v}_1 \\label{ds1dt}\n\\end{equation} \nwith ${\\bf v} = {\\bf v}_1-{\\bf v}_2$. The expression in (\\ref{ds1dt}) agrees with the known spin precession  \\cite{will,kidd} \n\\begin{equation}\n\\frac{d{\\tilde {\\bf S}} _1}{dt} = \\left(2+\\frac{3m_2}{2m_1}\\right)\\frac{\\mu\nG_N}{r^2}({\\bf n} \\times{\\bf v} )\\times{\\tilde {\\bf S}} _1,  \\label{ds1dtn}\n\\end{equation} \nwith $\\mu$ the reduced mass, after the transformation \\cite{nrgr3,nrgr5}\n\\begin{equation}\n{\\tilde {\\bf S}} _1 = (1-\\frac{1}{2}{\\bf \\tilde v} _1^2){\\bf S} _1 + \\frac{1}{2}{\\bf \\tilde v} _1({\\bf \\tilde v} _1\\cdot{\\bf S} _1)\\label{pnshift}.\n\\end{equation}\n\nIn the expression of (\\ref{pnshift}) ${\\bf \\tilde v}_1$ is the velocity in the local frame, which agrees with the coordinate veolocity in the PN (global) frame  at LO.\nLet us add a few comments about this distinction. First of all, we are dealing with the local spin, therefore to the order we are working at (recall $p^a \\sim m u^a +\\ldots$), our SSC reads $S^{ab}u_b=0$, with $u^a=e^a_\\mu u^\\mu$. If  we choose $\\lambda=t$, we have $u^\\mu \\equiv (1,{\\bf v})$ and $u^a$ what we denote as $({\\tilde v}^0,{\\bf \\tilde v})$. For the spin--spin dynamics, the relevant (spin-dependent) part of this relationship is (for instance for particle one)\n\\begin{eqnarray}\n{\\tilde v}_1^{a=0} &=& 1 + \\ldots, \\\\\n{\\tilde v}_1^{a=j} &=& v_1^j  +  \\frac{G_N}{r^2} S_2^{jk}n^k + \\ldots, \n\\label{localv}\n\\end{eqnarray} \ntherefore  \n\\begin{equation}\n\\label{onshell}\nS_1^{i0}= S_1^{ij}v_1^j + S_1^{ij}e^j_0({\\bf x}_1) +\\ldots =({\\bf v}_1\\times{\\bf S}_1)^i  + \\frac{G_N}{r^2}\\left[({\\bf n} \\times {\\bf S}_2)\\times {\\bf S}_1\\right]^i +\\ldots ,\n\\end{equation} \nwhere we used $e^j_0 ({\\bf x}_1)= \\frac{G_N}{r^2} ({\\bf n}\\times {\\bf S}_2)^j$, which follows from the one point function, $\\langle H^j_0\\rangle\/2$, or simply inspection of the Kerr metric in harmonic coordinates.\nThis will add an extra piece in the spin--spin EOM from the LO spin--orbit term of (\\ref{so15pn}) (see (\\ref{eomspin})), since there is modification in the algebra given by\n\\begin{equation}\n\\{S_1^i,S_1^{j0}\\} = \\epsilon^{ijk}S_1^{0k}= S_1^iv_1^j-S_1^jv_1^i+\\frac{G_N}{r^2}(({\\bf n}\\times{\\bf S}_2)^jS_1^i-({\\bf n}\\times{\\bf S}_2)^iS_1^j), \n\\label{modal}\n\\end{equation}\nwhich will lead to a 3PN contribution in the potential\\footnote {The extra term in (\\ref{onshell}) also provides an extra piece in the potential within NW SSC \\cite{comment}.}. The expression in (\\ref{modal}) is the main reason why we need to keep track of spin--orbit terms in the potential which will wind up contributing at ${\\cal O}({\\bf S}_1{\\bf S}_2)$ in the EOM.\\\\\n\n\\subsection{The Equivalence of Methodologies}\n\n\\subsubsection{The PN frame versus the local frame in the covariant SSC}\n\nNaive comparison of the EOM in (\\ref{ds1dt}) with the spin EOM in the covariant SSC, for instance in \\cite{owen}, shows that they are indeed different expressions. To understand the discrepancy,  we have to transform from the locally flat frame where (\\ref{ds1dt}) is defined, to the commonly used PN frame by rotating the spin tensor using the vierbein and the metric at 1PN order \\cite{mtw}, \n\\begin{equation}\nS_1^{ij} = {\\bar S}_1^{ij} + {\\bar S}_1^{ik} \\frac{h^j_k}{2} - {\\bar S}_1^{jk}\\frac{h^i_k}{2} + \\ldots = {\\bar S}_1^{ij}+ 2 \\frac{G_Nm_2}{r}{\\bar S}_1^{ij}+\\ldots, \\label{spn}\n\\end{equation}\nwith ${\\bar S}^{ij}$ the spin tensor in the PN frame within the covariant SSC. One can now trace the disagreement back to the definition of the spin vector. In our calculations we introduced $S^{jk} = \\epsilon^{jkl}{\\bf S}^l$ in the local frame, however, more generally  we may define the spin four vector as\n\\begin{equation}\n{\\bar S}^{\\mu\\nu} = \\frac{1}{m\\sqrt{g}} \\epsilon^{\\mu\\nu\\alpha\\beta} p_\\beta {\\bar S}_\\beta . \\label{sp}\n\\end{equation}\n\nUsing now (\\ref{sp}) (for instance for $S_1$) we have in the PN frame \n\\begin{equation}\n\\label{sp2}\n{\\bar S}_1^{ij} = \\epsilon^{ijk} \\left[ \\left(1+\\frac{{\\bf v}_1^2}{2}-\\frac{G_Nm_2}{r}\\right) {\\bar S}_1^k - v^k_1 ({\\bf \\bar S}_1\\cdot {\\bf v}_1)\\right]+\\ldots\n\\end{equation}\n\nLeaving\n\\begin{equation}\n{\\bf S}_1 = \\left(1+2\\frac{G_Nm_2}{r}\\right) \\left[ \\left(1+\\frac{{\\bf v}_1^2}{2}-\\frac{G_Nm_2}{r}\\right) {\\bf \\bar S}_1 - {\\bf v}_1 ({\\bf \\bar S}_1\\cdot {\\bf v}_1)\\right]+\\ldots,\n\\end{equation}\nwhich we can expand at 1PN order,\n\\begin{equation}\n{\\bf S}_1 = \\left(1 +\\frac{{\\bf v}_1^2}{2}+\\frac{G_N m_2}{r} \\right) {\\bf \\bar S}_1 -  {\\bf  v}_1 ({\\bf \\bar S}_1\\cdot {\\bf v}_1)+\\ldots\n\\end{equation}\n\nThe EOM in terms of ${\\bf \\bar S}$ reads (at 1.5PN),\n\\begin{equation}\n\\frac{d{\\bf \\bar S} _1}{dt} =\\frac{m_2\nG_N}{r^2}\\left[  {\\bf \\bar S}_1 ({\\bf n} \\cdot {\\bf v}) - 2 {\\bf n}({\\bf \\bar S} _1\\cdot{\\bf v}) +  ({\\bf \\bar S}_1 \\cdot {\\bf n}) ({\\bf v}_1-2{\\bf v}_2) \\right], \n\\label{pns}\n\\end{equation} \nwhich agrees with the results in \\cite{owen, buon}.\\\\\n\n\\subsubsection{Imposing the SSC before or after calculating the EOM}\n\nIt is also instructive to see how applying the SSC prior to finding the EOM leads to the same results as in the Routhian approach, where the SSC is enforced after the EOM is obtained. In  \\cite{nrgr3} it was shown how the spin EOM in the covariant SSC follow from (\\ref{so15pn}) once the SSC is imposed and the non canonical algebra taken into account.  \nLet us see how this works for the EOM arising from the  the spin--orbit interaction. Recall the commutators after imposing the covariant SSC are ($Db$ stands for Dirac bracket) \\cite{regge}\n\\begin{eqnarray}\n{[x^i_q,x^j_q]}_{Db}&=&\\frac{S_q^{ij}}{m_q^2} \\label{dirac1}\\\\\n{[x^k_q,S_q^{ij}]}_{Db}&=&\\frac{1}{m_q}(S_q^{ik}v^j_q-S_q^{jk}v^i_q)\\label{dirac2},\\\\\n {[S_q^{ab},S_q^{cd}]}_{Db} &=& (\\eta^{ac}-\\frac{u_q^au_q^c}{u_q^2}) S_q^{bd} +(\\eta^{bd}-\\frac{u_q^bu_q^d}{u_q^2}) S_q^{ac}\n\\nn \\\\& & -(\\eta^{ad}-\\frac{u_q^au_q^d}{u_q^2}) S_q^{bc}-(\\eta^{bc}-\\frac{u_q^bu_q^c}{u_q^2})\nS^{ad}_q\\label{dirac3}\n\\end{eqnarray}\nwith $q=1,2$.\n(\\ref{dirac1})  contributes a non-canoncial  piece to the acceleration\n\\begin{equation}\n\\label{newt}\n\\delta {\\bf a}_1= \\frac{d}{dt}\\left(\\left[{\\bf x}_1,-\\frac{G_Nm_2}{r}\\right]_{Db}\\right)= \nG_N\\frac{m_2}{m_1}\\frac{d}{dt} \\left(\\frac{{\\bf n} \\times\n{\\bf S} _1}{r^2}\\right)+\\ldots\n\\end{equation}\n  In this equation we have left off the other contributions in the RHS.\n\nWithin the Routhian approach we are advocating here it is simple to show how (\\ref{newt}) arises. First of all we re--write (\\ref{so15pn}) as\n\\begin{equation}\n\\label{so215pn}\nV^{so}_{1.5pn} = \\frac{G_Nm_2}{r^2}n^j\\left(S_1^{jl}v_1^l+ S^{jk}_1(v_1^k-2v^k_2)\\right)\n+ \\frac{G_Nm_2}{r^2}n^j(S^{j0}_1- S_1^{jl}v_1^l)+ 1 \\leftrightarrow 2.\n\\end{equation}\nThe second piece would  have vanished had we imposed the covariant SSC and  is  thus responsible \nfor the new contribution on the RHS of (\\ref{newt}).  It is clear that the only term which does not cancel out once the covariant SSC is imposed, is the one coming from $\\frac{d}{dt}\\frac{\\partial V^{so}_{1.5pn}}{\\partial v_1^l}$. That contributes precisely the extra term in (\\ref{newt}). The resulting EOM reads (at 1.5PN)\n\\begin{eqnarray}\n {\\bf a}_1^{so}   &=& \\frac{G_N}{r^3} \\left\\{ \\frac{m_2}{m_1} \\left[ -3{\\bf v} \\times {\\bf S}_1 + 6 {\\bf n} ({\\bf v} \\times {\\bf S}_1)\\cdot {\\bf n} + 3 {\\bf n}\\cdot{\\bf v} ({\\bf n} \\times {\\bf S}_1) \\right] \\right. \\nn \\\\\n& & \\left. -4 {\\bf v} \\times {\\bf S}_2 + 6 {\\bf n} ({\\bf v} \\times {\\bf S}_2)\\cdot {\\bf n} + 6 {\\bf n}\\cdot{\\bf v} ({\\bf n} \\times {\\bf S}_2)\\right\\} \\label{covaso}.\n\\end{eqnarray}\n\nTo establish the full equivalence we need to show that  the spin equations also match.  If we impose the covariant SSC in (\\ref{so215pn}) and \nuse Hamilton's equations, including the correction due to (\\ref{dirac2}), we can show that the new piece due to the algebra is (for particle one)\n\\begin{equation}\n\\delta \\dot {\\bf S}_1 = \\frac{m_2 G_N}{ r^2}({\\bf S} _1\\times{\\bf n} )\\times {\\bf v}_1,\n\\end{equation}\nand the equivalence is thus proven. For more on the consistency of the Routhian approach see the Appendix.\\\\ \n\nAnother important point in the connection between the NW and covariant SSCs is that in addition to (\\ref{pnshift}) it also entails a coordinate transformation given by \\cite{kidd,regge}\n\\begin{equation}\n{\\bf x}_q \\to {\\bf x}_q - \\frac{1}{2m_q}({\\bf v}_q\\times {\\bf \\tilde S}_q) +\\ldots , \\\\\n\\label{pnshift2}\n\\end{equation}\nfor $q=1,2$. Equivalently we have\n\\begin{equation}\n{\\bf r} \\to {\\bf r} - \\frac{1}{2M}({\\bf v}\\times { \\tilde \\xi}),\n\\label{pnshiftv}\n\\end{equation}\nwith ${\\bf \\xi} = \\frac{m_2}{m_1}{\\bf S}_1+ \\frac{m_1}{m_2}{\\bf S}_2$.\nThis transformation, implemented in the LO EOM, allows us to transform the acceleration from the covariant ( \\ref{covaso} ) to the NW SSC. \n\n\\section{The 3PN spin--spin Potential}\n\nLet us now consider the 3PN potential. The result in \\cite{eih} was presented with the NW SSC imposed at the level of the action. Here we will derive in more detail the full expression for the potential in terms of the spin tensor, before imposing the covariant SSC. The resulting potential reproduces that of \\cite{eih} once the NW SSC is enforced. However, we will retain the expression in terms of $S^{ab}$ and obtain the EOM via (\\ref{eomV}). Only then will we impose the covariant SSC. As we will show later on, the EOM obtained with either procedure are equivalent in the ${\\bf S}_1{\\bf S}_2$ sector up to 4PN order as originally argued in \\cite{eih,nrgr5}.\n\nFollowing the usual rules we draw all possible Feynman diagrams which scale as $v^6$. Each one of this diagrams will contribute to the effective potential by the rule $-i \\int V dt  = \\mbox{diagram}$ \\cite{NRGR}, where only connected diagrams contribute. For simplicity in what follows we will suppress the factors of $\\int dt$.\\\\ \n\nTo calculate the one graviton exchange contribution we should in principle draw all diagrams with propagator which connects to vertices which have subleading scalings. These diagrams are collected in Figs.~\\ref{ss3}(a,b,c). This would be formally the correct way to do the calculation in the spirit of effective\nfield theory, in that each diagram would scale homogeneously. However, in practice it is sometimes\nsimpler to calculate the full covariant one graviton diagrams and then break it into its individual\npieces which scale homogeneously. This allows us to calculate multiple diagrams simultaneously.\nThe instantaneous one--graviton exchange can be combined into a single calculation stemming from the linear spin--gravity coupling. If we denote this contribution to the effective potential by $V_{1g}$, and include also the LO piece,  the combination of diagrams then reads  \n\\begin{equation}\n-iV_{1g}=\\left(  \\frac{i}{2 m_p^2}\\right)^2    S^{\\mu \\beta}_1 u^\\alpha_1   S^{\\lambda \\rho}_2 u^\\sigma_2  \\langle H_{\\alpha\\mu,\\beta}(x_1)H_{\\sigma\\lambda,\\rho}(x_2) \\rangle\n\\end{equation}\nwhere $\\langle ~ \\rangle$ represents the Wick contraction.\n\nLet us start by considering the LO contribution from the instantaneous propagator which comes from the  spatial components $c=i, d=j$, since temporal derivatives are down by $v$  and $S_{0i}$ is down by a factor of v. The result reads\n\\begin{equation}\n-iV^{LO}_{1g}=\\left(  \\frac{i}{2 m_p^2}\\right)^2    S^{ik}_1   S^{lj}_2   \\langle H_{0i,k}(x_1)H_{0l,j}(x_2) \\rangle\n\\end{equation}\nwith \n\\begin{equation}\n\\partial^{x_1}_i \\partial^{x_2}_j \\frac{1}{|{\\bf x}_1-{\\bf x}_2|} \\equiv \\partial_{ij} \\frac{1}{r} = \\frac{1}{r^3}\\left(\\delta_{ij} - 3 n_i n_j \\right).  \n\\end{equation}\n\nFrom here we read off the LO spin--spin potential (see Fig. \\ref{sslo})\n\\begin{equation}\nV^{s1s2}_{2PN}=-\\frac{G_N}{r^3}\\left({\\bf S}_1\\cdot {\\bf S}_2-3\\frac{{\\bf r}\n\\cdot {\\bf S}_1 {\\bf r} \\cdot {\\bf S}_2}{r^2}\\right). \\label{Ess} \\end{equation}\n\n\\begin{figure}[h!]\n   \\centering\n    \\includegraphics[width=4cm]{ss.eps}\n\\caption[1]{ Leading order spin--spin interaction.}\\label{sslo}\n\\end{figure}\n\nNow let us consider the subleading contributions. There are multiple terms at 3PN.\nA factor of $v$ arises from either a spatio-temporal component  $S_{0i}$, a temporal derivative, or an\nexplicit factor of $v$.  Expanding out and keeping only the terms which contribute at 3PN gives\n\\begin{eqnarray}\n& &\\left(\\frac{i}{2M_{pl}}\\right)^2 \\left[  S^{0i}_1 S^{0j}_2\\langle H_{00,i}H_{00,j}\\rangle+  S^{ij}_1 S^{nm}_2 v^k_1v^l_2\\langle H_{ki,j}H_{ln,m}\\rangle + \\left( S^{ij}_1 S^{0m}_2 v^k_1\\langle H_{ki,j}H_{00,m}\n\\rangle  \\right. \\right. \\nn \\\\\n & & + \\left. \\left. S^{i0}_1 S^{lk}_2\\langle H_{0i,0}H_{0l,k}\\rangle\n+S^{0i}_1 S^{mn}_2v^k_1\\langle H_{k0,i}H_{0m,n}\\rangle + 1 \\leftrightarrow 2 \\right) \\right] \\end{eqnarray}\n\n\n\\begin{figure}[h!]\n   \\centering\n    \\includegraphics[width=12cm]{ss3.eps}\n\\caption[1]{ Diagrams contributing to 3PN order which do not\ninvolve non--linear interactions. The blob represents a spin\ninsertion and the cross corresponds to a propagator\ncorrection.}\\label{ss3}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=10cm]{ss2.eps}\n\\caption[1]{Non--linear contributions to the 3PN spin--spin potential.}\\label{ss2}\n\\end{figure}\n\nThe evaluation of these integrals is straightforward. Here we evaluate  one particular integal which needs more delicate consideration.\nNamely the contribution where one $v$  comes  from a temporal derivative while the other\ncomes from $S^{0i}$. \nThis contribution is given by\n\\begin{eqnarray}\n S^{i0}_1S^{lk}_2 \\langle H_{0i,0}(x_1)H_{0l,k}(x_2)\\rangle\n&=& -\\frac{i}{2}S^{i0}_1S^{ik}_2  \\int \\frac{[d^4p]}{p^2}(\\partial^1_0e^{-ip_0(t_1(\\lambda_1)-t_2(\\lambda_2))})(\\partial^2_k e^{-i{\\bf p} \\cdot ({\\bf x}_1(\\lambda_1)-{\\bf x}_2(\\lambda_2))} )\\nn \\\\\n&=& -  \\frac{i}{2}S^{i0}_1S^{ik}_2  \\int \\frac{[d^4p]}{{\\bf p}^2}(\\partial^2_0e^{-ip_0(t_1(\\lambda_1)-t_2(\\lambda_2))})(\\partial^2_k e^{-i{\\bf p} \\cdot ({\\bf x}_1(\\lambda_1)-{\\bf x}_2(\\lambda_2))} )\\nn \\\\\n\\end{eqnarray}\nNow recall  that $\\frac{d S^i}{dt} \\sim v^2 \\frac{v}{r} S^i$ so  we can neglect the time variation of spin at the order we are working at \\cite{nrgr3}. It is therefore convenient to trade $\\partial_{t_1}$ for $\\partial_{t_2}$ picking up a minus sign, and integrating by parts with no net effect. Had we kept $\\partial_{t_1}$ we would have to deal with $\\frac{d}{dt} S^{0i}$ which we can not neglect. To see this notice that,  imposing  the SSC  would introduce an acceleration dependent piece into the Lagrangian, which can be eliminated using the LO EOM.  However doing so entails a change of coordinates \\cite{sch} which is not preferable\\footnote{Incidentally, had we insisted on keeping $\\partial_{t_1}$ and included this acceleration piece, it turns out that imposing the NW SSC at the level of the action would reproduce the exact same form for the 3PN Hamiltonian of \\cite{Schafer3pn}. That is actually the case due to a cancelation of this ${\\cal O}({G_N^2})$ acceleration piece with the extra term stemming from subleading corrections in the spin--orbit potential due to (\\ref{modal}). A similar result can be found in \\cite{levi}. Within the Routhian approach the trade for $\\partial_{t_2}$ is preferable.} . If we consider now the instantaneous interaction once again we get\n\\begin{eqnarray}\n S^{i0}_1S^{lk}_2 \\langle H_{0i,0}(x_1)H_{0l,k}(x_2)\\rangle\n&=&  \\delta(t_1-t_2)   \\frac{i}{2}S^{i0}_1S^{ik}_2 \\partial^2_k \\partial^2_0  \\int \\frac{[d^3p]}{{\\bf p}^2}(e^{-i{\\bf p} \\cdot ({\\bf x}_1(\\lambda_1)- {\\bf x}_2(\\lambda_2))} )\\nn \\\\\n&=& \\delta(t_1-t_2)   \\frac{i}{8\\pi r^3}S^{i0}_1S^{ik}_2(3n^k {\\bf n}\\cdot {\\bf v}_2-v_2^k).\n\\end{eqnarray}\n\nThe net result from the instantaneous one graviton exchange is then given by\n\\begin{eqnarray}\n-iV^{inst}_{1g}&=&\\frac{iG}{r^3}(\\delta^{ij}-3n^i n^j)\\left[ S^{i0}_1S^{j0}_2+{\\bf v}_1 \\cdot {\\bf v}_2 \nS^{in}_1S^{jn}_2+v_1^mv_2^k S^{ik}_1S^{jm}_2\\right.  \\\\\n& -& \\left. v_1^k v_2^m S^{ik}_1S^{jm}_2+\nS^{i0}_1S^{jk}_2(v_2^k-v_1^k)+S^{ik}_1S^{j0}_2(v_1^k-v_2^k)\\right. \\nonumber \\\\\n& & + \\left. (3n^l{\\bf v}_2 \\cdot{\\bf n} -v_2^l) S_1^{0k}S_2^{kl} + (3n^l{\\bf v}_1 \\cdot{\\bf n} -v_1^l) S_2^{0k}S_1^{kl}  \\right]\n\\end{eqnarray}\n\nThe first corrections to instantaneity comes from the diagram shown in  Fig.~\\ref{ss3}c.\nThis correction comes from expanding the graviton propapagtor to second order in the\nenergy which is suppressed by a factor of $v$ relative to the spatial momentum, i.e\n\\begin{equation}\n\\frac{1}{p_0^2-{\\bf p}^2} \\approx -\\frac{1}{{\\bf p}^2}+\\frac{p_0^2}{{\\bf p}^4}+...\n\\end{equation}\n The result from this diagram is very similar to the spinless case and we have\n\\begin{eqnarray}\n& & \\mbox{Fig~\\ref{ss3}c} = -i\\frac{G_N}{2r^3} S_1^{ki}S_2^{kj}\\left[{ \\bf v} _1\\cdot { \\bf v} _2(\\delta^{ij} - 3 n^i n^j) -3 { \\bf v} _1\\cdot{\\bf n}  { \\bf v} _2\\cdot{\\bf n}  (\\delta^{ij}-5n^in^j)\\right. \\nonumber \\\\ & & \\left. -3{\\bf v} _2\\cdot{ \\bf n} (v_1^jn^i +v_1^in^j) - 3 {\\bf v} _1\\cdot{\\bf n} (v_2^jn^i+v_2^in^j)+v_1^iv_2^j+v_2^iv_1^j\\right] \n\\end{eqnarray}\n\nLet us now consider  the terms involving non--linear graviton interactions, as shown in  Figs. \\ref{ss2}a and \\ref{ss2}b. Let us start with the so called ``seagull\" topology\n\\begin{equation}\n\\mbox{Fig~ \\ref{ss2}b} = \\frac{i m_1}{16m_p^2} S_1^{lm}S_2^{ij}\\langle T( H_{0l,m}(x_1) H_{00}(x_1)( H^{\\lambda}_j H_{0\\lambda,i}+ H_{0i,k}H_{kj}))\\rangle.\n\\end{equation}\n\nNote that for this diagram there are a few Wick contractions, that is, the two graviton\nvertex can contract in two ways with the mass and spin vertex on the opposing world line.\nHowever many of these contractions vanish since index structures vanish, e.g.\n\\begin{equation}\n\\langle H_{00}H_{0i}\\rangle =  0.\n\\end{equation}\nThe result is given by\n\\begin{eqnarray}\n\\mbox{Fig~\\ref{ss2}b}=  i  \\frac{m_1G_N^2}{r^4} S_1^{jk}S_2^{ij}(5n^kn^i-2\\delta^{ki})\n\\end{eqnarray}\n\nFinally we have the diagram with the three graviton interaction. Again there are mutiple ways of doing the Wick contractions.  As previously mentioned, the best way to handle these contraction is using\na symbolic manipulation program \\cite{workshop} where symmetrization is simply handled. The integrals are all variations of the same result used in the one graviton exchange diagram. The result for the diagram is\n\\begin{eqnarray}\n& & \\mbox{Fig ~\\ref{ss2}a}=  i m_1 \\frac{G_N^2}{r^4} S_1^{jk}S_2^{ij}(4n^kn^i-\\delta^{ki}) \\nonumber \\\\\n\\end{eqnarray}\nNote that in calculating this diagram we encounter multiple power divergent integrals. These divergences can be absorbed into pure counter-terms, since they just renormalize the mass and possibly other quadrupole moments.  It is simple to see how this occurs in a\ndiagramatic language. The divergences occur when one of the propagators ending on the\nline which has a mass insertion is cancelled by a power of ${\\bf k}^2$ arises from the momentum\ndepedence of the three graviton vertex. One of the lines in the diagrams then contracts to a point.\nThe resulting diagram looks like an interaction between a self energy (mass correction) and the\nspin on the opposite line. The result of this renormalization is that we may simply drop these divergent integrals. As was explained in \\cite{NRGR} no {\\it physical } logarithmic divergences occur until 5PN order for the case of spinless particles. As it was shown in \\cite{nrgr3} that is also the case for spinning bodies and logarithmic divergences due to finite size effects do no show up until ${\\cal O} (v^{10})$. This generalizes the so called ``effacement'' of internal structure \\cite{damour} to the case of spinning bodies \\cite{nrgr3}.\nThe logarithmic divergences are renormalized by absorption into finite size parameters which present a non trivial renormalization group flow. These are tidally induced effects which in turn do not contribute to the metric solution as it is expected from Birkhoff's theorem. However, there are other types of finite size effects, the so called self--induced effects, which do appear at lower orders as explained in \\cite{nrgr3}. This kind of effects are encoded in operators whose coefficients are fixed, like the mass, and can be generated by power law divergences \\cite{nrgr3}. For instance in the case of a rotating black hole, finite size corrections appear due to the quadrupole moment of the Kerr spacetime. The coefficient is set by the Kerr metric and it is proportional to $S^2$ \\cite{mtw}. The LO corrections (at 2PN) were computed in \\cite{nrgr3}, and subleading effects are reported in \\cite{nos}.\\\\ \n\nGathering all the pieces together, plus mirror images, we have the complete spin--spin potential to 3PN, prior to  imposing the covariant SSC, \n\\begin{eqnarray}\n\\label{ss312}\nV^{spin} &=& -\\frac{G_N}{r^3}\\left[(\\delta^{ij}-3n^i n^j)\\left( S^{i0}_1S^{j0}_2+\\frac{1}{2}{\\bf v}_1 \\cdot {\\bf v}_2 \nS^{ik}_1S^{jk}_2+v_1^mv_2^k S^{ik}_1S^{jm}_2- v_1^k v_2^m S^{ik}_1S^{jm}_2 \\right. \\right. \\nn \\\\ \n & & +  \\left.\nS^{i0}_1S^{jk}_2(v_2^k-v_1^k)+S^{ik}_1S^{j0}_2(v_1^k-v_2^k)\\right)+ \\frac{1}{2}S_1^{ki}S_2^{kj}\\left( 3 { \\bf v} _1\\cdot{\\bf n}  { \\bf v} _2\\cdot{\\bf n}  (\\delta^{ij}-5n^in^j) \\right. \\nn\\\\   & & + \\left. \n   3 {\\bf v} _1\\cdot{\\bf n} (v_2^jn^i+v_2^in^j)+3{\\bf v} _2\\cdot{ \\bf n} (v_1^jn^i +v_1^in^j) -v_1^iv_2^j-v_2^iv_1^j\\right) \\nn \\\\ \n & & + \\left. (3n^l{\\bf v}_2 \\cdot{\\bf n} -v_2^l) S_1^{0k}S_2^{kl} + (3n^l{\\bf v}_1 \\cdot{\\bf n} -v_1^l) S_2^{0k}S_1^{kl}  \\right] \\nn \\\\ & & +\\left(\\frac{G_N}{r^3}- \\frac{3M G_N^2}{r^4}\\right) S_1^{jk}S_2^{ji}(\\delta^{ki}-3 n^k n^i)\\nn \\\\\n & & + \\frac{G_Nm_2}{r^2}n^j\\left(S^{j0}_1+S^{jk}_1(v_1^k-2v^k_2)\\right) -\\frac{G_Nm_1}{r^2}n^j\\left(S^{j0}_2+S^{jk}_2(v_2^k-2v^k_1)\\right),\n\\end{eqnarray}\nwhere we included the LO spin--orbit term which will be relevant latter on due to (\\ref{onshell}) and (\\ref{modal}). The spin potential in (\\ref{ss312}) is the main result of the paper from which the EOM to 3PN order can obtained via (\\ref{eomV}).\n\n\\subsection{The spin Hamiltonian in the NW SSC to 3PN}\n\nNotice that the spin--spin part of the expression in (\\ref{ss312}) agrees with the result reported in \\cite{eih} if we impose the NW SSC\\footnote{It also provides the spin--orbit potential from which the precession equation follows \\cite{nrgr3}.}. However, as we mentioned earlier, to obtain all the contribution in the ${\\bf S}_1{\\bf S}_2$ sector we need to include subleading corrections in the spin--orbit potential  coming from (\\ref{onshell}). The extra term takes the form \\cite{comment}\n\\begin{equation}\n\\frac{G_N}{2r^2}\\left( m_2 n^i S_1^{ij}e^j_0({\\bf x}_1) - m_1 n^iS_2^{ij}e^j_0({\\bf x}_2)\\right)= \\frac{G^2_NM}{2r^4} \\left( ({\\bf S}_1\\times{\\bf n})\\cdot ({\\bf n}\\times{\\bf S}_2)\\right).\n\\end{equation}\n\nFor completeness, we present  the ${\\bf S}_1{\\bf S}_2$ potential in the NW SSC to 3PN order\n\\begin{eqnarray}\n& & V_{NW}^{s1s2} = -\\frac{G_N}{2r^3}\\left[ {\\bf S}_1 \\cdot {\\bf S}_2\\left({3\\over2}{\\bf v}_1\\cdot {\\bf v}_2-3{\\bf v}_1\\cdot {\\bf n} {\\bf v}_2\\cdot {\\bf n}\n-\\left({\\bf v}_1^2+{\\bf v}_2^2\\right)\\right)\n-{\\bf S}_1\\cdot {\\bf v}_1{\\bf S}_2\\cdot {\\bf v}_2 \\nn \\right. \\\\ &-&\\frac{3}{2}{\\bf S}_1\\cdot {\\bf v}_2 {\\bf S}_2\\cdot {\\bf v}_1+\n{\\bf S}_1\\cdot {\\bf v}_2 {\\bf S}_2\\cdot {\\bf v}_2\n+ {\\bf S}_2\\cdot {\\bf v}_1 {\\bf S}_1\\cdot {\\bf v}_1+\n3{\\bf S}_1\\cdot{\\bf n}{\\bf S}_2\\cdot {\\bf n}\n\\left({\\bf v}_1\\cdot{\\bf v}_2+5{\\bf v}_1\\cdot{\\bf n} {\\bf v}_2\\cdot{\\bf n}\\right) \\nn \\\\\n&-& 3{\\bf S}_1\\cdot{\\bf v}_1{\\bf S}_2\\cdot {\\bf n}{\\bf v}_2\\cdot {\\bf n}-\n3 {\\bf S}_2\\cdot{\\bf v}_2{\\bf S}_1\\cdot{\\bf n}{\\bf v}_1\\cdot{\\bf n} + 3({\\bf v}_2\\times{\\bf S}_1)\\cdot{\\bf n}({\\bf v}_2\\times {\\bf S}_2)\\cdot {\\bf n}\n\\nonumber\\\\\n&+& \\left.\n3( {\\bf v}_1\\times {\\bf S}_1)\\cdot {\\bf n}( {\\bf v}_1\\times {\\bf S}_2)\\cdot{\\bf n}-\n\\frac{3}{2}({\\bf v}_1\\times{\\bf S}_1)\\cdot{\\bf n}({\\bf v}_2\\times{\\bf S}_2)\\cdot{\\bf n} - 6({\\bf v}_1\\times{\\bf S}_2)\\cdot{\\bf n}({\\bf v}_2\\times{\\bf S}_1)\\cdot{\\bf n}\n\\right]  \\nonumber\\\\\n&+&  \n\\frac{G^2_N(m_1+m_2)}{2r^4}\\left(5{\\bf S}_1\\cdot{\\bf S}_2-17{\\bf S}_1\\cdot{\\bf n}{\\bf S}_2\\cdot{\\bf n}\\right) - \\frac{G_N}{r^3} \\left({\\bf S}_1\\cdot {\\bf S}_2 - 3{\\bf S}_1\\cdot{\\bf n}{\\bf S}_2\\cdot{\\bf n}\\right). \n\\label{nw3pn}\n\\end{eqnarray}\n\nAs it was argued in \\cite{nrgr5} the EOM in the ${\\bf S}_1{\\bf S}_2$ sector follow from (\\ref{nw3pn}) by means of the `traditional' Hamiltonian approach up to 4PN order \\cite{comment}. The spin dependent part of the Hamiltonian can be readily obtained from (\\ref{nw3pn}), (\\ref{Ess}) and (\\ref{so15pn}), and takes the form (ignoring 2.5PN spin-orbit and 3PN spin$^2$ terms) to 3PN\n\\begin{eqnarray}\n H_{NW}^{spin} &=& \\frac{G_N}{2m_1m_2r^3}\\left[ \\frac{3}{2} ({\\bf \\cal P}_1\\times {\\bf S}_1)\\cdot {\\bf n}({\\bf \\cal P}_2\\times {\\bf S}_2)\\cdot {\\bf n} +6 ({\\bf \\cal P}_2\\times {\\bf S}_1)\\cdot {\\bf n}({\\bf \\cal P}_1\\times {\\bf S}_2)\\cdot {\\bf n} \\right.  \\\\ & & -\n15  ({\\bf \\cal P}_1\\cdot {\\bf n})({\\bf \\cal P}_2\\cdot {\\bf n})({\\bf S}_1\\cdot {\\bf n})({\\bf S}_2\\cdot {\\bf n})+\\frac{3}{2}  ({\\bf \\cal P}_2\\cdot {\\bf S}_1)({\\bf \\cal P}_1\\cdot {\\bf S}_2) - \\frac{3}{2}  ({\\bf \\cal P}_2\\cdot {\\bf \\cal P}_1)({\\bf S}_1\\cdot {\\bf S}_2)\\nn \\\\ \n& & - 3({\\bf \\cal P}_1\\cdot {\\bf \\cal P}_2)({\\bf S}_1\\cdot {\\bf n})({\\bf S}_2\\cdot {\\bf n})+  3({\\bf \\cal P}_1\\cdot {\\bf S}_1)({\\bf n}\\cdot {\\bf \\cal P}_2)({\\bf S}_2\\cdot {\\bf n})+3({\\bf \\cal P}_2\\cdot {\\bf S}_2)({\\bf n}\\cdot {\\bf \\cal P}_1)({\\bf S}_1\\cdot {\\bf n})\\nn\\\\\n& & \\left.  + 3({\\bf \\cal P}_2\\cdot {\\bf n})({\\bf n}\\cdot {\\bf \\cal P}_1)({\\bf S}_1\\cdot {\\bf S}_2)+({\\bf \\cal P}_2\\cdot {\\bf S}_2) ({\\bf \\cal P}_1\\cdot {\\bf S}_1)\\right]\\nn \\\\ & & +\\frac{G_N}{2m_1^2r^3}\\left[{\\bf \\cal P}_1^2({\\bf S}_1\\cdot {\\bf S}_2) - 3 ({\\bf \\cal P}_1\\times {\\bf S}_1)\\cdot {\\bf n}({\\bf \\cal P}_1\\times {\\bf S}_2)\\cdot {\\bf n} - ({\\bf \\cal P}_1\\cdot {\\bf S}_2) ({\\bf \\cal P}_1\\cdot {\\bf S}_1)\\right] \\nn\\\\\n& & +\\frac{G_N}{2m_2^2r^3}\\left[{\\bf \\cal P}_2^2({\\bf S}_1\\cdot {\\bf S}_2) - 3 ({\\bf \\cal P}_2\\times {\\bf S}_1)\\cdot {\\bf n}({\\bf \\cal P}_2\\times {\\bf S}_2)\\cdot {\\bf n} - ({\\bf \\cal P}_2\\cdot {\\bf S}_2) ({\\bf \\cal P}_2\\cdot {\\bf S}_1)\\right] \\nn\\\\\n& & +  \n\\frac{G^2_N(m_1+m_2)}{2r^4}\\left(11{\\bf S}_1\\cdot{\\bf S}_2-23({\\bf S}_1\\cdot{\\bf n})({\\bf S}_2\\cdot{\\bf n})\\right) - \\frac{G_N}{r^3} \\left({\\bf S}_1\\cdot {\\bf S}_2 - 3({\\bf S}_1\\cdot{\\bf n})({\\bf S}_2\\cdot{\\bf n})\\right) \\nn \\\\ & & + \\frac{G_N}{r^2} \\left[ \\frac{3m_2}{2m_1} ({\\bf n}\\times {\\bf \\cal P}_1)\\cdot {\\bf S}_1 - 2({\\bf n}\\times {\\bf \\cal P}_2)\\cdot{\\bf S}_1 + 2({\\bf n}\\times {\\bf \\cal P}_1)\\cdot {\\bf S}_2  - \\frac{3m_1}{2m_2} ({\\bf n}\\times {\\bf \\cal P}_2)\\cdot {\\bf S}_2 \\right]\\nn,\n\\label{nwh3pn}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n{\\bf \\cal P}_1 = m_1 {\\bf v}_1 + 2 \\frac{G_N m_1}{r^2}{\\bf n}\\times {\\bf S}_2 + \\frac{3 G_N m_2}{2r^2}{\\bf n}\\times {\\bf S}_1,  \\:\\: 1 \\to 2.\n\\end{equation}\n\nTo obtain the EOM however we will proceed differently, and again we will not impose the SSC up until after we have solved for the EOM resulting from the potential in (\\ref{ss312}) using the Routhian approach. As we shall see the extra piece due to spin--orbit effects will come from (\\ref{modal}). We will explicitly show however that the results are equivalent. Let us remark that a Hamiltonian similar to that in (\\ref{nwh3pn}) was recently found in \\cite{Schafer3pn}, and shown to be equivalent in \\cite{comment} once the spin-orbit effect is included.\n\n\n\\section{The spin equation of motion to 3PN order}\n\nThe 3PN  contribution  to the EOM for spin follows from the potential  in a similar fashion to  the LO spin--orbit example. Let us proceed systematically for particle one. For the spin--spin part of the potential to 3PN in (\\ref{ss312}) we have two pieces, one depending on ${\\bf S}_1$,  \n\\begin{eqnarray}\nV_{{\\bf S}_1} &=& -\\frac{G_N}{r^3} \\left[ {\\bf S}_1\\cdot {\\bf S}_2 \\left( \\frac{1}{2} {\\bf v}_1\\cdot {\\bf v}_2 - {\\bf v}_2^2-\\frac{3}{2}{\\bf n}\\cdot {\\bf v}_1{\\bf n}\\cdot {\\bf v}_2\\right) + \\frac{3}{2} {\\bf n}\\cdot {\\bf S}_2{\\bf n}\\cdot {\\bf S}_1 \\left( {\\bf v}_1\\cdot {\\bf v}_2 +5 {\\bf n}\\cdot {\\bf v}_2{\\bf n}\\cdot {\\bf v}_1\\right) \\right. \\nonumber  \\\\ & &\n+{\\bf S}_1\\cdot {\\bf v}_2 {\\bf S}_2\\cdot {\\bf v}_2 - \\frac{1}{2} {\\bf S}_1\\cdot {\\bf v}_1 {\\bf S}_2\\cdot {\\bf v}_2 - \\frac{1}{2} {\\bf S}_2\\cdot {\\bf v}_1 {\\bf S}_1\\cdot {\\bf v}_2 -3 {\\bf n}\\cdot({\\bf v}_2\\times {\\bf S}_1){\\bf n}\\cdot({\\bf v}_1\\times {\\bf S}_2) \\nonumber \\\\ & & +  3 {\\bf n}\\cdot({\\bf v}_2\\times {\\bf S}_1){\\bf n}\\cdot({\\bf v}_2\\times {\\bf S}_2)\n+ \\frac{3}{2} {\\bf n}\\cdot{\\bf v}_1 {\\bf n}\\cdot{\\bf S}_2{\\bf S}_1\\cdot{\\bf v}_2-\\frac{3}{2} {\\bf n}\\cdot{\\bf v}_1 {\\bf n}\\cdot{\\bf S}_1{\\bf S}_2\\cdot{\\bf v}_2  \\nonumber \\\\ & & - \\left. \\frac{3}{2} {\\bf n}\\cdot{\\bf v}_2 {\\bf n}\\cdot{\\bf S}_1{\\bf S}_2\\cdot{\\bf v}_1 - \\frac{3}{2} {\\bf n}\\cdot{\\bf v}_2 {\\bf n}\\cdot{\\bf S}_2{\\bf S}_1\\cdot{\\bf v}_1 \\right] \\nonumber \\\\ & & + \\left(-\\frac{G_N}{r^3}+ \\frac{3MG_N^2}{r^4}\\right) \\left({\\bf S}_1\\cdot{\\bf S}_2 - 3{\\bf S}_1\\cdot{\\bf n}{\\bf S}_2\\cdot{\\bf n}\\right),\n\\end{eqnarray}\nand another one \n\\begin{eqnarray}\n\\label{w0}\nV_{S^{0i}_1} &=& - A^iS^{i0}_1 \\\\\n{\\bf A}_1 &=&  \\frac{G_N}{r^3}\\left\\{(3{\\bf v}_2-{\\bf v}_1)\\times{\\bf S}_2 - 3{\\bf n}\\cdot(2{\\bf v}_2-{\\bf v}_1)\\times{\\bf S}_2){\\bf n} -3 {\\bf n}\\cdot{\\bf v}_2({\\bf n}\\times{\\bf S}_2)\\right\\} = \\nonumber \\\\ &=& {\\bf \\tilde a}^{so}_{1(2)} +\\frac{G_N}{r^3}\\left(2 {\\bf v}_1\\times {\\bf S}_2-\n3  {\\bf n}\\cdot({\\bf v}_1\\times {\\bf S}_2) {\\bf n} -3 {\\bf n}\\cdot{\\bf v}_1({\\bf n}\\times{\\bf S}_2)\\right)\n\\end{eqnarray}\nwith ${\\bf \\tilde a}^{so}_{1(2)}$ the ${\\bf S}_2$ part of the acceleration in the local frame. \tThe latter is given by\n\\begin{equation}\n{\\bf \\tilde a}^{so}_1 = {\\bf a}_1^{so} + \\frac{d}{dt} \\left( \\frac{G_N}{r^2} {\\bf n} \\times {\\bf S}_2\\right)+\\ldots,  \n\\end{equation}\n where ${\\bf a}_1^{so}$ is the acceleration in the PN frame given in (\\ref{covaso}). Then (with $\\chi = {\\bf S}_2 + \\frac{m_2}{m_1}{\\bf S}_1$)\n\\begin{equation}\n\\label{covasoloc}\n{\\bf \\tilde a}^{so}_1 =  \\frac{G_N}{r^3} \\left[ -3{\\bf v} \\times \\chi + 6 {\\bf n} ({\\bf v} \\times \\chi)\\cdot {\\bf n} + 3 {\\bf n}\\cdot{\\bf v} ({\\bf n} \\times \\chi) \\right]. \n\\end{equation}\n\nNotice we also have \\begin{equation} {\\bf A}_1 = {\\bf \\tilde a}^{so}_{1(2)} + {\\bf v}_1\\times \\omega^{ss}_0,\\label{w0A}\\end{equation} with $\\omega^{ss}_0$ the LO spin--spin frequency. This expression will be useful later on to prove the equivalence with our previous results in \\cite{eih,comment}.\\\\\n \nUsing (\\ref{eomV}) the ${\\cal O}({\\bf S}_1{\\bf S}_2)$ part of the spin EOM ends up being  \n\\begin{equation}\n\\label{eomspin}\n\\frac{d{\\bf S}_1}{dt} = (\\omega^{ss}_0+\\omega^{ss}_1) \\times {\\bf S}_1 + ({\\bf v}_1\\times {\\bf S}_1)\\times{ \\bf A}_1 + \\frac{m_2G_N^2}{r^4} {\\bf n} \\times [ ({\\bf n}\\times{\\bf S}_2)\\times{\\bf S}_1],\n\\end{equation}\nwhere the  last term follows from the correction in the spin--orbit part of the potential in (\\ref{ss312}) due to (\\ref{modal}), and \n\\begin{eqnarray}\n\\omega^{ss}_0 &=& -\\frac{G_N}{r^3} \\left({\\bf S}_2 - 3{\\bf n}{\\bf S}_2\\cdot{\\bf n}\\right) \\\\\n\\omega^{ss}_1 &=& -\\frac{G_N}{r^3} \\left[ {\\bf S}_2\\left( \\frac{1}{2} {\\bf v}_1\\cdot {\\bf v}_2 - {\\bf v}_2^2-\\frac{3}{2}{\\bf n}\\cdot {\\bf v}_1{\\bf n}\\cdot {\\bf v}_2\\right) + \\frac{3}{2} {\\bf n} ({\\bf n}\\cdot {\\bf S}_2)\\left( {\\bf v}_1\\cdot {\\bf v}_2 +5 {\\bf n}\\cdot {\\bf v}_2{\\bf n}\\cdot {\\bf v}_1\\right) \\right. \\nonumber  \\\\ & &\n+ {\\bf v}_2 ({\\bf S}_2\\cdot {\\bf v}_2) - \\frac{1}{2}  {\\bf v}_1 ({\\bf S}_2\\cdot {\\bf v}_2) - \\frac{1}{2}  {\\bf v}_2 ({\\bf S}_2\\cdot {\\bf v}_1) -3 ({\\bf n}\\times {\\bf v}_2) {\\bf n}\\cdot({\\bf v}_1\\times {\\bf S}_2) \\nonumber \\\\ & & +  3 ({\\bf n}\\times {\\bf v}_2){\\bf n}\\cdot({\\bf v}_2\\times {\\bf S}_2)\n+ \\frac{3}{2}  {\\bf v}_2({\\bf n}\\cdot{\\bf v}_1) ({\\bf n}\\cdot{\\bf S}_2)-\\frac{3}{2} {\\bf n}({\\bf n}\\cdot{\\bf v}_1) ({\\bf S}_2\\cdot{\\bf v}_2)   \\nonumber \\\\ & & - \\left. \\frac{3}{2} {\\bf n} ({\\bf n}\\cdot{\\bf v}_2) ({\\bf S}_2\\cdot{\\bf v}_1) - \\frac{3}{2} {\\bf v}_1({\\bf n}\\cdot{\\bf v}_2) ({\\bf n}\\cdot{\\bf S}_2) \\right] + \\frac{3MG_N^2}{r^4} \\left({\\bf S}_2 - 3{\\bf n}{\\bf S}_2\\cdot{\\bf n}\\right)\n\\label{w1ss}\n\\end{eqnarray}\n\nIn what follows we will show how to reproduce the precession equation and the equivalence with the result of \\cite{eih,comment}.\n\n\\section{The Precession equation to 3PN and the equivalence with our previous results using the NW SSC}\n\nThe precession equation, \n\\begin{equation}\n\\label{eomnws}\n\\frac{d{\\bf S}_q}{dt} = \\omega^{nw}_q \\times {\\bf S}_q,\n\\end{equation} can be obtained from (\\ref{nwh3pn}) with $\\omega_q = \\frac{\\partial H^{spin}_{NW}}{\\partial {\\bf S}_q}$ ($q=1,2$), for instance for particle 1 (ignoring linear in spin, and also spin$^2$, terms),\n\\begin{eqnarray}\n\\omega^{nw}_1 &=& \n\\label{omegasnw} \n\\frac{G_N}{2r^3}\\left[ \\frac{3}{2} {\\bf n}\\times {\\bf  v}_1({\\bf v}_2\\times {\\bf S}_2)\\cdot {\\bf n} +6 {\\bf n}\\times {\\bf v}_2({\\bf v}_1\\times {\\bf S}_2)\\cdot {\\bf n} \\right. \\nn \\\\ & & -15 {\\bf n} ({\\bf v}_1\\cdot {\\bf n})({\\bf v}_2\\cdot {\\bf n})({\\bf S}_2\\cdot {\\bf n})+\\frac{3}{2}  {\\bf v}_2({\\bf v}_1\\cdot {\\bf S}_2) - \\frac{3}{2}  ({\\bf v}_2\\cdot {\\bf v}_1){\\bf S}_2 \\nn \\\\ \n& & - 3 {\\bf n}({\\bf v}_1\\cdot {\\bf v}_2)({\\bf S}_2\\cdot {\\bf n})+  3{\\bf v}_1({\\bf n}\\cdot {\\bf v}_2)({\\bf S}_2\\cdot {\\bf n})+3 {\\bf n}({\\bf v}_2\\cdot {\\bf S}_2)({\\bf n}\\cdot {\\bf v}_1)\\nn\\\\\n& & \\left.  + 3{\\bf S}_2({\\bf v}_2\\cdot {\\bf n})({\\bf n}\\cdot {\\bf v}_1)+{\\bf v}_1({\\bf v}_2\\cdot {\\bf S}_2) \\right]\\nn \\\\ & & +\\frac{G_N}{2r^3}\\left[{\\bf v}_1^2 {\\bf S}_2 - 3 {\\bf n}\\times {\\bf  v}_1 ({\\bf v}_1\\times {\\bf S}_2)\\cdot {\\bf n} - {\\bf v}_1({\\bf v}_1\\cdot {\\bf S}_2) \\right] \\nn\\\\\n& & +\\frac{G_N}{2r^3}\\left[{\\bf v}_2^2{\\bf S}_2 - 3 {\\bf n}\\times {\\bf v}_2({\\bf v}_2\\times {\\bf S}_2)\\cdot {\\bf n} - {\\bf v}_2 ({\\bf v}_2\\cdot {\\bf S}_2) \\right] \\nn\\\\\n& & +  \n\\frac{G^2_N(m_1+m_2)}{2r^4}\\left(5{\\bf S}_2-17{\\bf n}({\\bf S}_2\\cdot{\\bf n})\\right) - \\frac{G_N}{r^3} \\left({\\bf S}_2 - 3{\\bf n}({\\bf S}_2\\cdot{\\bf n})\\right). \n\\end{eqnarray}\n\nNotice that equivalently we have $\\omega_q = \\frac{\\partial V^{s1s2}_{NW}}{\\partial {\\bf S}_q}$ ($q=1,2$).\\\\ \n\nIn what follows we will show  (\\ref{eomnws}) is equivalent to (\\ref{eomspin}) up to ${\\cal O}({\\bf S}_1^2)$ effects. To transform the EOM in covariant SSC to NW SSC, as a first step we need to implement the shifts in (\\ref{pnshift}) and (\\ref{pnshift2}). Recall (\\ref{pnshift}) already transforms the LO spin--orbit part of the EOM into a precession equation (see (\\ref{ds1dt}) and (\\ref{ds1dtn})). Also the coordinate transformation in (\\ref{pnshift2}) shifts the form of the frequency in the precession equation from the spin--orbit part. The EOM in terms for ${\\bf \\tilde S}_1$ reads\n\\begin{equation}\n\\label{prewss}\n\\frac{d{\\bf \\tilde S}_1}{dt} = {\\tilde \\omega}^{ss}_1\\times {\\bf \\tilde S}_1\n\\end{equation}\nwith\n\\begin{equation}\n\\label{eqw1}\n{\\tilde \\omega}^{ss}_1= \\delta {\\omega}^{so}_1+\\omega^{ss}_0+\\delta \\omega^{ss}_0+ \\omega^{ss}_1 + \\frac{1}{2}{\\bf v}_1 \\times {\\bf \\tilde A}_1 + \\frac{1}{2} \\frac{m_2G_N^2}{r^4} \\left[({\\bf \\tilde S}_2\\times {\\bf n})\\times {\\bf n}\\right],\n\\end{equation} \n\\begin{eqnarray} \\delta {\\bf \\tilde \\omega}^{so}_1 &=&  \\frac{G_N}{2r^3}\\left\\{{\\bf n}\\times \\left(\\frac{9}{2}{\\bf v}_1-6{\\bf v}_2\\right)\\left[\\frac{m_2}{m_1}({\\bf n}\\times {\\bf v}_1)\\cdot {\\bf \\tilde S}_1 -({\\bf n}\\times {\\bf v}_2)\\cdot {\\bf \\tilde S}_2\\right] \\right. \\nonumber \\\\ & & \\left. + \\left( {\\bf v}_2\\times {\\bf \\tilde S}_2 -\\frac{m_2}{m_1}{\\bf v}_1\\times {\\bf \\tilde S}_1\\right)\\times \\left(\\frac{3}{2} {\\bf v}_1-2{\\bf v}_2\\right) \\right\\} \\nonumber\n\\\\ & & + \\frac{G_N^2m_2}{r^4} {\\bf n}\\times \\left(\\frac{3m_2}{4m_1}({\\bf n}\\times {\\bf \\tilde S}_1) + \\frac{m_1}{m_2}({\\bf n}\\times {\\bf \\tilde S}_2) \\right),\n\\label{dw1}\n\\end{eqnarray}\nand\n\\begin{equation}\n\\delta \\omega^{ss}_0 = -\\frac{G_N}{2r^3}\\left[ {\\bf v}^2_2 {\\bf \\tilde S}_2 - {\\bf v}_2 ({\\bf \\tilde S}_2\\cdot {\\bf v}_2)-\n3{\\bf n} ({\\bf n}\\cdot {\\bf \\tilde S}_2) {\\bf v}_2^2 + 3 {\\bf n} ({\\bf n}\\cdot {\\bf v}_2) ({\\bf v}_2\\cdot {\\bf \\tilde S}_2)\\right] \\label{deltaw0}.\n\\end{equation}\n\n\nFrom the expression in (\\ref{dw1}) we will only consider the ${\\bf S}_1{\\bf S}_2$ contributions.\nWhat we need now is to find an additional contributions (curvature effects) which would take the form of (\\ref{eqw1}) into the expression in (\\ref{omegasnw}). First of all notice that ${\\bf A}_1$ ends up effectively like in the NW SSC, due to the $\\frac{1}{2}$ in (\\ref{eqw1}). However, there is a piece which differs from the full NW form and comes from the $S^{j0}_2S_1^{i0}$ term in the potential. For the expression in $\\omega^{ss}_1$ the difference is just the factor of $\\frac{1}{2}$ for $S^{j0}_2$ in the NW SSC. Henceforth, we can split the terms in (\\ref{eqw1}) as\n\\begin{eqnarray}\n{\\bf \\tilde A}_1 &=& {\\bf \\tilde A}^{nw}_1 - \\frac{G_N}{2r^3} \\left(3{\\bf n} ({\\bf n}\\times {\\bf v}_2)\\cdot {\\bf \\tilde S}_2- {\\bf v}_2\\times{\\bf \\tilde S}_2\\right) \\\\\n\\omega^{ss}_1 &=& {\\hat \\omega}^{nw}_1 - \\frac{G_N}{2r^3} \\left[ ({\\bf v}_2\\times{\\bf \\tilde S}_2)\\times (2{\\bf v}_1-{\\bf v}_2) \n-3{\\bf n}\\times({\\bf v}_1-{\\bf v}_2) ({\\bf n}\\times {\\bf v}_2)\\cdot {\\bf \\tilde S}_2 \\right. \\nonumber \\\\ & & \\left. + 3{\\bf n}\\times({\\bf v}_2\\times{\\bf \\tilde S}_2)({\\bf n} \\cdot {\\bf v}_1)\\right]. \n\\end{eqnarray}\n\nNotice that $\\omega^{nw}_1 = \\frac{1}{2} {\\bf v}_1\\times {\\bf \\tilde A}^{nw}_1 + {\\hat \\omega}^{nw}_1$ and the EOM becomes\n\\begin{eqnarray}\n\\frac{d{\\bf \\tilde S}_1}{dt} &=& \\left. \\frac{d {\\bf S}_1}{dt}\\right|_{nw} - \\frac{G_N}{2r^3} \\left[-3({\\bf n}\\times {\\bf v}_2) ({\\bf n}\\times {\\bf v}_2)\\cdot {\\bf \\tilde S}_2+({\\bf v}_2\\times{\\bf \\tilde S}_2)\\times \\left({\\bf v}_1+{\\bf v}_2\\right)+ {\\bf v}^2_2 {\\bf \\tilde S}_2  \\right. \\nonumber \\\\ & & \\left. + 3{\\bf n}\\times({\\bf v}_2\\times{\\bf \\tilde S}_2)({\\bf n} \\cdot {\\bf v}_1) -  {\\bf v}_2 ({\\bf \\tilde S}_2\\cdot {\\bf v}_2) -\n3{\\bf n} ({\\bf n}\\cdot {\\bf \\tilde S}_2) {\\bf v}_2^2 + 3 {\\bf n} ({\\bf n}\\cdot {\\bf v}_2) ({\\bf v}_2\\cdot {\\bf \\tilde S}_2)\\right]\\times {\\bf \\tilde S}_1 \\nonumber \\\\ & & + \\frac{G_N^2m_1}{2r^4} \\left[{\\bf n}\\times \\left({\\bf n}\\times {\\bf \\tilde S}_2 \\right)\\right]\\times {\\bf \\tilde S}_1 \n\\end{eqnarray}\n\nThe extra shift we need to add to (\\ref{pnshift}) and transform away the undesired pieces ends up being \n\\begin{eqnarray}\n{\\bf S}_1^{nw}&=& {\\bf \\tilde S}_1 + \\frac{G_N}{2r^2} \\left( {\\bf \\tilde S}_2 ({\\bf v}_2\\cdot{\\bf n})- ({\\bf \\tilde S}_2\\cdot {\\bf n}){\\bf v}_2\\right) \\times {\\bf \\tilde S}_1 +\\dots = \\\\ &=& (1-\\frac{1}{2}{\\bf \\tilde v} _1^2){\\bf S} _1 + \\frac{1}{2}{\\bf \\tilde v} _1({\\bf \\tilde v}\\cdot {\\bf S}_1)  + \\frac{G_N}{2r^2}\\left[ ({\\bf v}_2 \\times {\\bf S}_2)\\times {\\bf n}\\right]\\times{\\bf S}_1+\\ldots \\label{nlonw},\n\\end{eqnarray} \nand the equivalence is thus formally proven. Notice that ${\\bf S}^{nw}={\\bf \\tilde S} + {\\cal O}(G_N)$. Recall that ${\\bf \\tilde S}$ reproduces the spin dynamics in NW gauge at LO, in particular the LO precession equation in (\\ref{ds1dtn}). However, at next to LO, to transform to the NW gauge we needed to take into account  curvature effects that modify the shift in (\\ref{pnshift}). Some of these effects were already included once the velocity in the local frame is transformed to the coordinate velocity in the PN frame (see (\\ref{pnshift}) and (\\ref{localv})). The extra term, necessary to account for the discrepancy between a flat and curved background, appears in (\\ref{nlonw}). Notice that in the limit $G_N \\to 0$ these contributions vanish. To avoid confusion, to 3PN one can skip the intermediate step in (\\ref{pnshift}) which defines ${\\bf \\tilde S}$, and use (\\ref{nlonw}) to relate ${\\bf S}$ with ${\\bf S}^{nw}$, the local spin in the covariant and NW  SSC respectively. The equivalence of results thus follows.\\\\\n\nTo show that the position dynamics is also recovered, once the spin EOM is reproduced, we can simply construct an effective potential as\n\\begin{equation}\nV_{eff} =  {\\bf \\tilde \\omega}^{ss}_1\\left({\\bf S}^{nw}_2\\right) \\cdot {\\bf S}^{nw}_1\n\\end{equation}\nfrom which the spin corrections to the position dynamics can be derived via the `traditional'  Hamiltonian approach. The above expression obviously reproduces the results of \\cite{eih,comment,Schafer3pn}. Nevertheless, the more traditional spin dynamics in covariant SSC is shown in (\\ref{eomspin}) with the spin defined in the local frame. To transform to the PN frame one can proceed as we did in (\\ref{spn})-(\\ref{pns}) for the LO case. \n\n\\subsection{Adding ${\\bf S}^2$ terms}\n\nAs we mentioned earlier, the spin EOM which follows from (\\ref{nw3pn}) fails to  reproduce all of the ${\\cal O}({\\bf S}_q^2)$ terms. These terms can be computed by working within the Routhian formalism by\n  adding  the corrections due to the Riemann dependent term in (\\ref{actR}). We may use this term as written or equivalently we may perform a field redefinition such that \n  \\begin{equation}\n\\frac{1}{2m_q}R_{d\ne a b} S^{c d} S^{a b} \\frac{u^e u_{c}}{\\sqrt{u^2}} \\to \\frac{1}{m_q}\\frac{Dp_d}{d\\lambda}\\frac{S^{dc}u_c}{\\sqrt{u^2}}\\label{dsu}.\n\\end{equation}\n\nThe procedure for calculating the potential follows the exact same steps as before and it can be shown that \\cite{nos} the potential due to this term in the covariant SSC takes the form\n\\begin{equation}\nV_{3PN}^{s^2} = \\ldots -\\left({\\bf \\tilde a}^{so}_{1(1)}\\right)^l S^{lc}_1v_{1c} + 1 \\to 2,\n\\end{equation}\nfrom which we get the following contribution to  the spin EOM \n\\begin{equation}\n\\frac{d{\\bf  S}_1}{dt}=\n\\ldots + ~ ({\\bf \\tilde a}^{so}_{1(1)}\\times {\\bf S}_1)\\times {\\bf v}_1 + \\ldots\n\\end{equation}\nwith ${\\bf \\tilde a}^{so}_{1(1)}$ the ${\\bf  S}_1$ dependent part of the spin--orbit acceleration in the local frame (see (\\ref{covasoloc})). \nThis is however not yet complete since we are still missing $S^2$ corrections stemming from finite size effects, as well as non--linear corrections ($\\sim G_N^2$) as in Fig.~\\ref{ss2}a, with two LO spin insertions on the same worldline\\footnote{Notice that the would be 3PN contribution from a seagull diagram similar to Fig.~\\ref{ss2}b, where a non--linear contribution from the mass worldline couples to two spin insertions in the companion worldline, vanishes.}. Finite size effects on the other hand are encoded in higher dimensional operators \\cite{NRGR,nrgr3}. For the case of self--induced spin effects the new term in the Routhian takes the form ($q=1,2$)\n\\begin{equation}\n\\frac{C^{(q)}_{ES^2}}{2m_q m_p}\\frac{E_{ab}}{\\sqrt{u_q^2}}\n{{\\cal S}_q}^a_c {\\cal S}_q^{cb},\\label{s2} \n\\end{equation} \nin the worldline \\cite{nrgr3,eih}, where ${\\cal S}^{ab}$ is defined as\\cite{yee}\n\\begin{equation}\n{\\cal S}^{ab} = S^{ab} + \\frac{u_c}{u^2}S^{c[a}u^{b]},\n\\end{equation}\n which guarantees the SSC is preserved in time (see appendix). In the expression of (\\ref{s2}) the Wilson coefficients, $C^{(q)}_{ES^2}$, are constants which are determined solely by the nature of the object \\cite{NRGR,nrgr3}, and $E_{ab}$ is the electric component of the Weyl tensor in the local frame. In the case of a rotating black hole we have $C_{ES^2}=1$, and this term represents the\nnon-vanishing quadrupole moment of the Kerr solution. The LO self--induced finite size contribution to the potential thus takes the form \\cite{eih,nrgr3}\n\\begin{equation}\nV_{2PN}^{s^2} = -C^{(1)}_{ES^2}\\frac{m_2}{2m_1r^3} \\left({\\bf S}_1\\cdot{\\bf S}_1 - 3{\\bf S}_1\\cdot{\\bf n}{\\bf S}_1\\cdot{\\bf n}\\right) + 1\\to 2 .\n\\end{equation}\n\nHigher order corrections will follow from (\\ref{s2}) in a similar manner. We report the full ${\\cal O}({\\bf S}_q^2)$ contribution in a companion publication \\cite{nos}.\n \n\\section{Conclusions}\n\nIn this paper we have presented the details of the calculation of the ${\\cal O}({\\bf S}_1{\\bf S}_2)$ effects to 3PN order. We computed the potential, and showed how to calculate using a Routhian approach, imposing the SSC only at the last stage of the calculation. The EOM follow from (\\ref{ss312}) via (\\ref{eomV}). We proved the equivalence of this methodology, with a covariant SSC, to that originally espoused in \\cite{eih,comment}, where we calculated within the NW imposing the SSC at the level of the action. \nIn this paper we have not included effects which go as $S^2$ such as finite size effects. The first non--zero finite size effects for spinless particles start out at 5PN \\cite{damour,NRGR}, whereas spin induced finite size effects show up at LO, e.g. 2PN \\cite{nrgr3}. Tidally induced finite size effects (logarithmic effects) first appear at 5PN for the case of spinning bodies \\cite{nrgr3}.\nIn a subsequent paper we will present the next to LO $S^2$ effects using the same formalism discussed here. \n\n\\vskip 0.5cm\n\nThis work was supported in part by the Department of Energy under Grants DOE-ER-40682-143 and DEAC02-6CH03000. RAP also acknowledges support from the Foundational Questions Institute (fqxi.org) under grant RPFI-06-18, and funds from the University of California.\n\n\n\\newpage\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Cross Section for Heavy Lepton Pair Production}\n\\chaptermark{Heavy Lepton Pair Production Cross Section}\n\\label{appa}\n\n\\section{$2\\rightarrow2$ body scattering in centre-of-mass frame}\n\nThe differential cross section for $2\\rightarrow2$ scattering in the centre-of-mass frame is given by\n\\begin{equation}\n\\frac{d\\sigma}{d\\Omega^*} = \\frac{1}{64\\pi^2 s} \\frac{p_\\text{f}^*}{p_\\text{i}^*} \n\\langle\\,|M_\\text{fi}|^2\\,\\rangle.\n\\label{diffxs}\n\\end{equation}\nIn the above expression, the asterisk indicates the centre-of-mass \nframe, subscripts i and f refer to initial- and final-state particles, and $s$ is the square of the total centre-of-mass energy.\n\nTherefore the major work in calculating the cross section is to determine the \nspin averaged square of the transition matrix element $\\langle\\,|M_\\text{fi}|^2\\,\\rangle$, evaluated in the centre-of-mass frame.\n\n\\section{Definitions and conventions}\n\\label{defs}\n\nThe 4-momenta of the particles involved are defined in this section.  Subscripts 1, 2, 3 and 4 refer to the quark, anti-quark, lepton and anti-lepton\nrespectively, and the angle $\\theta^*$ is between the out-going lepton and the \nincoming quark in the centre-of-mass frame.  Hence the $(E,p)$ 4-momenta are \nas follows (note that the incoming quarks are treated as massless):\n\\begin{equation}\n\\begin{split}\np_1 &= (E,0,0,E)\\,;\\\\[2mm]\np_2 &= (E,0,0,-E)\\,;\\\\[2mm]\np_3 &= (E_3,p_3 \\sin \\theta^*,0,p_3 \\cos \\theta^*)\\,;\\\\[2mm]\np_4 &= (E_4,-p_3 \\sin \\theta^*,0,-p_3 \\cos \\theta^*)\\,.\n\\end{split}\n\\end{equation}\n\nLater we will require the Lorentz dot products of several pairs of \nthese 4-vectors.  The useful ones are listed below:\n\\begin{equation}\n\\begin{split}\np_1.p_2 &= 2E^2\\,;\\\\[2mm]\np_1.p_3 &= EE_3-Ep_3\\cos\\theta^*\\,; \\\\[2mm]\np_2.p_4 &= EE_4-Ep_3\\cos\\theta^*\\,; \\\\[2mm]\np_1.p_4 &= EE_4+Ep_3\\cos\\theta^*\\,; \\\\[2mm]\np_2.p_3 &= EE_3+Ep_3\\cos\\theta^*.\n\\end{split}\n\\end{equation}\n\n\\section{Matrix element calculation}\n\nUsing the Feynman rules, the matrix element for Z boson exchange is\n\\begin{multline}\n-iM_\\text{fi}^\\text{Z} = \\left[\\bar{v}(p_2).-ig_\\text{Z}\\gamma^{\\mu}\\frac{1}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)\nu(p_1)\\right]\\frac{-ig_{\\mu\\nu}}{s-m_\\text{Z}^2+im_\\text{Z}\\Gamma_\\text{Z}}\\\\[3mm]\n\\times\\left[\\bar{u}(p_3).-ig_\\text{Z}\\gamma^{\\nu} \\frac{1}{2}\\left(c_\\text{V}^\\text{f}-c_\\text{A}^\\text{f}\\gamma^5\\right)v(p_4)\\right], \n\\end{multline}\nwhich gives\n\\begin{multline}\nM_\\text{fi}^\\text{Z} = -\\frac{g_\\text{Z}^2}{s-m_\\text{Z}^2+im_\\text{Z}\\Gamma_\\text{Z}}g_{\\mu\\nu}\n\\left[\\bar{v}(p_2)\\gamma^{\\mu}\\frac{1}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)u(p_1)\\right]\\\\[3mm]\n\\times\\left[\\bar{u}(p_3)\\gamma^{\\nu} \\frac{1}{2}\\left(c_\\text{V}^\\text{f}-c_\\text{A}^\\text{f}\\gamma^5\\right)v(p_4)\\right].\n\\end{multline}\nNote that the matrix element for $\\gamma$-exchange is easily \nobtained from this by setting $m_\\text{Z}$ and $c_\\text{A}$ to zero, $c_\\text{V}$ to 2, and \n$g_\\text{Z}^2$ to $q^\\text{i}q^\\text{f}e^2$.\n\nTaking into account Z- and $\\gamma$-exchange the full matrix element may be \nwritten as $M_\\text{fi}=M_\\text{fi}^\\text{Z}+M_\\text{fi}^{\\gamma}$.  Hence\n\\begin{equation}\n\\begin{split}\n|M_\\text{fi}|^2&=|M_\\text{fi}^\\text{Z}|^2+|M_\\text{fi}^{\\gamma}|^2+(M_\\text{fi}^\\text{Z})^*M_\\text{fi}^{\\gamma}\n+M_\\text{fi}^\\text{Z}(M_\\text{fi}^{\\gamma})^*\\\\[2mm]\n&=|M_\\text{fi}^\\text{Z}|^2+|M_\\text{fi}^{\\gamma}|^2+2 \\Re\\{(M_\\text{fi}^\\text{Z})^*M_\\text{fi}^{\\gamma}\\}\\,.\n\\end{split}\n\\end{equation}\nWe start by calculating the first term in this expression (from which the \nsecond term is easily obtained using the substitutions mentioned above).  \nFinally the third term which contains the Z-$\\gamma$ interference information \nis calculated.\n\nTo obtain the spin-averaged square of the transition matrix element which \nappears in equation~(\\ref{diffxs}), we must average over the spins of the \nincoming quarks and sum over the spins of out-going leptons such that\n\\begin{equation}\n\\langle\\,|M_\\text{fi}|^2\\,\\rangle = \\frac{1}{4} \\sum_\\text{spins} |M_\\text{fi}|^2.\n\\end{equation}\n\n\\subsection{Z-exchange}\n\nFirstly we want to calculate $\\sum_\\text{spins} |M_\\text{fi}^\\text{Z}|^2$.  This is given by\n\\begin{equation}\n\\begin{split}\n\\sum_\\text{spins} |M_\\text{fi}^\\text{Z}|^2 = \\sum_{r,r',s,s'}&\n\\frac{g_\\text{Z}^4}{(s-m_\\text{Z}^2)^2+m_\\text{Z}^2\\Gamma_\\text{Z}^2} \\\\[3mm]\n&\\times\\bar{v}_{r'}(p_4)\\left(c_\\text{V}^\\text{f}+c_\\text{A}^\\text{f}\\gamma^5\\right)\\frac{\\gamma^{\\mu}}{2}u_{s'}(p_3)\n\\left[\\bar{u}_s(p_1)\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac{\\gamma_{\\mu}}{2}v_r(p_2) \\right.\\\\[3mm]\n&\\times\\left.\\bar{v}_r(p_2)\\frac{\\gamma_{\\nu}}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)u_s(p_1)\\right]\n\\bar{u}_{s'}(p_3)\\frac{\\gamma^{\\nu}}{2}\\left(c_\\text{V}^\\text{f}-c_\\text{A}^\\text{f}\\gamma^5\\right)v_{r'}(p_4).\n\\end{split}\n\\label{zex1}\n\\end{equation}\nTo obtain the above expression the following have been used:\n\\begin{equation}\n\\begin{split}\n(\\bar{u}\\gamma^{\\nu}v)^*&=(u^{\\dagger}\\gamma^0\\gamma^{\\nu}v)^\\dagger\n=v^{\\dagger}(\\gamma^{\\nu})^{\\dagger}(\\gamma^0)^{\\dagger}u\n=v^{\\dagger}(\\gamma^0\\gamma^{\\nu}\\gamma^0)\\gamma^0u\n=\\bar{v}\\gamma^{\\nu}u\\,;\\\\[2mm]\n(\\bar{u}\\gamma^{\\nu}\\gamma^5v)^*&=(u^{\\dagger}\\gamma^0\\gamma^{\\nu}\n\\gamma^5v)^\\dagger\n=v^{\\dagger}(\\gamma^5)^{\\dagger}(\\gamma^{\\nu})^{\\dagger}(\\gamma^0)^{\\dagger}u\n=v^{\\dagger}\\gamma^5(\\gamma^0\\gamma^{\\nu}\\gamma^0)\\gamma^0u\\\\[2mm]\n&=v^{\\dagger}\\gamma^5\\gamma^0\\gamma^{\\nu}u\n=-v^{\\dagger}\\gamma^0\\gamma^5\\gamma^{\\nu}u\n=-\\bar{v}\\gamma^5\\gamma^{\\nu}u\\,.\n\\end{split}\n\\end{equation}\n\nWe now make use of familiar trace identities ($\\bar{u}_{\\alpha}\n\\Gamma_{\\alpha\\beta}u_{\\beta}= \\mbox{Tr}\\,(u\\bar{u}\\Gamma)$ etc. with \n$\\alpha,\\beta$ Dirac matrix indices) as well as the following spinor identities:\n\\begin{equation}\n\\begin{split}\n\\sum_\\text{spins}u(p)\\bar{u}(p)=\\gamma.p+m\\,; \\\\[2mm]\n\\sum_\\text{spins}v(p)\\bar{v}(p)=\\gamma.p-m\\,.\n\\end{split}\n\\end{equation}\nUsing these identities (and temporarily ignoring the propagators) eq.~(\\ref{zex1}) becomes\n\\begin{equation}\n\\begin{split}\n&\\sum_{r,r',s,s'}\n\\bar{v}_{r'}(p_4)\\left(c_\\text{V}^\\text{f}+c_\\text{A}^\\text{f}\\gamma^5\\right)\\frac{\\gamma^{\\mu}}{2}u_{s'}(p_3)\n\\left[\\bar{u}_s(p_1)\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac {\\gamma_{\\mu}}{2}v_r(p_2)\\right.\\\\[3mm]\n&\\quad\\:\\times\\left.\\bar{v}_r(p_2)\\frac{\\gamma_{\\nu}}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)u_s(p_1)\\right]\n\\bar{u}_{s'}(p_3)\\frac{\\gamma^{\\nu}}{2}\\left(c_\\text{V}^\\text{f}-c_\\text{A}^\\text{f}\\gamma^5\\right)v_{r'}(p_4)\\\\[3mm]\n&=\\mbox{Tr}\\biggl[\\gamma.p_1\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac{\\gamma_{\\mu}}{2}\\gamma.p_2\n\\frac{\\gamma_{\\nu}}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)\\biggr]\\\\[3mm]\n&\\quad\\:\\times\\mbox{Tr}\\left[(\\gamma.p_4-m_4)\\left(c_\\text{V}^\\text{f}+c_\\text{A}^\\text{f}\\gamma^5\\right)\\frac{\\gamma^{\\mu}}{2}(\\gamma.p_3\n+m_3)\\frac{\\gamma^{\\nu}}{2}\\left(c_\\text{V}^\\text{f}-c_\\text{A}^\\text{f}\\gamma^5\\right)\\right].\n\\end{split}\n\\end{equation}\n\\enlargethispage{-\\baselineskip}\n\nWe now evaluate the traces using the properties that the trace of \nthe product of an odd number of gamma matrices is zero, \n$\\mbox{Tr}\\,(\\gamma^{\\mu}\\gamma^{\\nu})=4g^{\\mu\\nu}$, $\\mbox{Tr}\\,(\\gamma^{\\mu}\\gamma^{\\nu}\n\\gamma^{\\lambda}\\gamma^{\\sigma})=4(g^{\\mu\\nu}g^{\\lambda\\sigma}-g^\n{\\mu\\lambda}g^{\\nu\\sigma}+g^{\\mu\\sigma}g^{\\nu\\lambda})$, \n$\\mbox{Tr}\\,(\\gamma^5\\gamma^{\\mu}\\gamma^{\\nu})=0$, and\n$\\mbox{Tr}\\,(\\gamma^5\\gamma^{\\mu}\\gamma^{\\nu}\\gamma^{\\lambda}\\gamma^{\\sigma})\n=4i\\epsilon^{\\mu\\nu\\lambda\\sigma}$.  Therefore the second trace becomes\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{4}\\left[\\left(c_\\text{V}^\\text{f}\\right)^2 \\mbox{Tr}\\,(\\gamma.p_4\\gamma^{\\mu}\\gamma.p_3\\gamma^{\\nu}\n-m_3m_4\\gamma^{\\mu}\\gamma^{\\nu})\\right.\\\\[3mm]\n&\\quad\\left.-\\left(c_\\text{A}^\\text{f}\\right)^2 \\mbox{Tr}\\,(\\gamma.p_4\\gamma^5\\gamma^{\\mu}\n\\gamma.p_3\\gamma^5 -m_3m_4\\gamma^5\\gamma^{\\mu}\\gamma^{\\nu}\\gamma^5)\n+2c_\\text{V}^\\text{f}c_\\text{A}^\\text{f} \\mbox{Tr}\\,(\\gamma.p_4\\gamma^5\\gamma^{\\mu}\\gamma.p_3\\gamma^{\\nu})\\right]\\\\[3mm]\n&=\\left[\\left(c_\\text{V}^\\text{f}\\right)^2+\\left(c_\\text{A}^\\text{f}\\right)^2\\right](p_4^{\\mu}p_3^{\\nu}+p_4^{\\nu}p_3^{\\mu}-\np_3.p_4 g^{\\mu\\nu})-\\left[\\left(c_\\text{V}^\\text{f}\\right)^2-\\left(c_\\text{A}^\\text{f}\\right)^2\\right]m_3m_4g^{\\mu\\nu}\\\\[3mm]\n&\\quad\\,-2c_\\text{V}^\\text{f}c_\\text{A}^\\text{f}i\\epsilon^{\\alpha\\mu\\beta\\nu}(p_4)_{\\alpha}(p_3)_{\\beta}\\,,\n\\end{split}\n\\end{equation}\nwhere the properties of $\\gamma^5$ (it squares to 1 and anti-commutes with the other $\\gamma$-matrices) have also been used.  The first trace is similar, but simpler as we are treating the incoming quarks\nas massless.  It becomes\n\\begin{equation}\n\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]\\left[(p_1)_{\\mu}(p_2)_{\\nu}+(p_1)_{\\nu}(p_2)_{\\mu}-\np_1.p_2 g_{\\mu\\nu}\\right]\n- 2c_\\text{V}^\\text{i}c_\\text{A}^\\text{i}i\\epsilon_{\\gamma\\mu\\delta\\nu}p_1^{\\gamma}p_2^{\\delta}\\,.\n\\end{equation}\n\nFinally we must multiply these two terms together, noting which \nterms do not contribute because of the anti-symmetry\/symmetry of the indices.\nWe obtain\n\\begin{equation}\n\\begin{split}\n&\\left[\\left(c_\\text{V}^\\text{f}\\right)^2+\\left(c_\\text{A}^\\text{f}\\right)^2\\right]\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]2\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)(p_2.p_4)\\,\\right]\\\\[3mm]\n&\\;+\\left[\\left(c_\\text{V}^\\text{f}\\right)^2-\\left(c_\\text{A}^\\text{f}\\right)^2\\right]\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]2m_3m_4p_1.p_2\n-4c_\\text{V}^\\text{f}c_\\text{A}^\\text{f}c_\\text{V}^\\text{i}c_\\text{A}^\\text{i}\\epsilon_{\\gamma\\mu\\delta\\nu}\\epsilon_{\\alpha\\mu\\beta\n\\nu}p_1^{\\gamma}p_2^{\\delta}(p_4)_{\\alpha}(p_3)_{\\beta}\\,.\n\\end{split}\n\\end{equation}\nTo complete the calculation we use the identity \n$\\epsilon_{\\mu\\nu\\alpha\\beta}\\epsilon_{\\mu\\nu\\gamma\\delta}=\n2(g_{\\alpha}^{\\delta}g_{\\beta}^{\\gamma}\n-g_{\\alpha}^{\\gamma}g_{\\beta}^{\\delta})$, which means that\n\\begin{equation}\n\\begin{split}\n\\epsilon_{\\gamma\\mu\\delta\\nu}\\epsilon_{\\alpha\\mu\\beta\\nu}\np_1^{\\gamma}p_2^{\\delta}(p_4)_{\\alpha}(p_3)_{\\beta}\n&=\\epsilon_{\\mu\\nu\\alpha\\beta}\\epsilon_{\\mu\\nu\\gamma\\delta}\np_1^{\\gamma}p_2^{\\delta}(p_4)_{\\alpha}(p_3)_{\\beta}\\\\[2mm]\n&=2\\left(g_{\\alpha}^{\\delta}g_{\\beta}^{\\gamma}-g_{\\alpha}^{\\gamma}\ng_{\\beta}^{\\delta}\\right)p_1^{\\gamma}p_2^{\\delta}(p_4)_{\\alpha}(p_3)_{\\beta}\\\\[2mm]\n&=2\\left[\\,(p_1.p_3)(p_2.p_4)-(p_1.p_4)(p_2.p_3)\\,\\right].\n\\end{split}\n\\end{equation}\n\\enlargethispage{\\baselineskip}Putting all of the above together gives us that\n\\begin{equation}\n\\begin{split}\n\\langle\\,|M_\\text{fi}^\\text{Z}|^2\\,\\rangle=&\\frac{1}{4}\\frac{g_\\text{Z}^4}{(s-m_\\text{Z}^2)^2+m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\\\[3mm]\n&\\times\\Bigl(\\left[\\left(c_\\text{V}^\\text{f}\\right)^2+\\left(c_\\text{A}^\\text{f}\\right)^2\\right]\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]2\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)\n(p_2.p_4)\\,\\right]\\\\[3mm]\n&\\quad\\quad+\\left[\\left(c_\\text{V}^\\text{f}\\right)^2-\\left(c_\\text{A}^\\text{f}\\right)^2\\right]\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]2m_3m_4p_1.p_2\\\\[3mm]\n&\\quad\\quad+8c_\\text{V}^\\text{f}c_\\text{A}^\\text{f}c_\\text{V}^\\text{i}c_\\text{A}^\\text{i}\\left[\\,(p_1.p_4)(p_2.p_3)-(p_1.p_3)(p_2.p_4)\\,\\right]\\Bigr).\n\\end{split}\n\\end{equation}\n\n\\subsection{$\\gamma$-exchange}\n\nUsing the replacements outlined above, it is immediately obvious that\n\\begin{equation}\n\\langle\\,|M_\\text{fi}^{\\gamma}|^2\\,\\rangle=\\frac{1}{4}\\frac{e^4(q^\\text{f})^2(q^\\text{i})^2}{s^2}32\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)(p_2.p_4)+m_3m_4p_1.p_2)\\,\\right].\n\\end{equation}\n\n\\subsection{Interference term}\n\nThe final part of the squared matrix element to be calculated is \n$\\Re\\{(M_\\text{fi}^\\text{Z})^*M_\\text{fi}^{\\gamma}\\}$.  Neglecting the propagators for now, the \nspin sum is\n\\begin{equation}\n\\begin{split}\n&\\sum_{r,r',s,s'}\n\\bar{v_{r'}}(p_4)\\left(c_\\text{V}^\\text{f}+c_\\text{A}^\\text{f}\\gamma^5\\right)\\frac{\\gamma^{\\mu}}{2}u_{s'}(p_3)\n\\left[\\bar{u_s}(p_1)\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac{\\gamma_{\\mu}}{2}v_r(p_2)\\right.\\\\[2mm]\n&\\quad\\:\\times\\Bigl.\\bar{v}_r(p_2)\\gamma_{\\nu}u(p_1)\\Bigr]\\bar{u}_{s'}(p_3)\\gamma^{\\nu}v_{r'}(p_4) \\\\[3mm]\n&=\\mbox{Tr}\\biggl[\\gamma.p_1\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac{\\gamma_{\\mu}}{2}\\gamma.p_2\n\\gamma_{\\nu}\\biggr]\\\\[3mm]\n&\\quad\\:\\times\\mbox{Tr}\\left[(\\gamma.p_4-m_4)\\left(c_\\text{V}^\\text{f}+c_\\text{A}^\\text{f}\\gamma^5\\right)\\frac{\\gamma^{\\mu}}{2}(\\gamma.p_3+m_3)\\gamma^{\\nu}\\right].\n\\end{split}\n\\end{equation}\nNow the second trace is\n\\begin{multline}\n\\frac{1}{2}\\left[c_\\text{V}^\\text{f} \\mbox{Tr}\\left(\\gamma.p_4\\gamma^{\\mu}\\gamma.p_2\\gamma^{\\nu}\n-m_3m_4\\gamma^{\\mu}\\gamma^{\\nu}\\right)\n+c_\\text{A}^\\text{f} \\mbox{Tr}\\left(\\gamma.p_4\\gamma^5\\gamma^{\\mu}\\gamma.p_3\\gamma^{\\nu}\\right)\\right]\\\\[2mm]\n=2c_\\text{V}^\\text{f}\\left(p_4^{\\mu}p_3^{\\nu}+p_4^{\\nu}p_3^{\\mu}-p_3.p_4 g^{\\mu\\nu}\n-m_3m_4g^{\\mu\\nu}\\right)-2c_\\text{A}^\\text{f}i\\epsilon^{\\alpha\\mu\n\\beta\\nu}(p_4)_{\\alpha}(p_3)_{\\beta}\\,,\n\\end{multline}\nand the first\n\\begin{equation}\n2c_\\text{V}^\\text{i}\\left[\\,(p_1)_{\\mu}(p_2)_{\\nu}+(p_1)_{\\nu}(p_2)_{\\mu}-p_1.p_2 g_{\\mu\\nu}\\right]\n- 2c_\\text{A}^\\text{i}i\\epsilon_{\\gamma\\mu\\delta\\nu}p_1^{\\gamma}p_2^{\\delta}\\,.\n\\end{equation}\nMultiplying these together and using the same identity as before for\nthe $\\epsilon$-tensor we obtain\n\\begin{equation}\n\\begin{split}\n\\langle(M_\\text{fi}^\\text{Z})^*M_\\text{fi}^{\\gamma}\\rangle=&\\frac{1}{4}\\frac{g_\\text{Z}^2e^2q^\\text{f}q^\\text{i}}\n{s(s-m_\\text{Z}^2-im_\\text{Z}\\Gamma_\\text{Z})}\\\\[3mm]\n&\\times8\\Bigl(c_\\text{V}^\\text{f}c_\\text{V}^\\text{i}\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)(p_2.p_4)+m_3m_4p_1.p_2\\right]\\Bigr.\\\\[3mm]\n&\\quad\\quad\\;\\Bigl.+c_\\text{A}^\\text{f}c_\\text{A}^\\text{i}\\left[\\,(p_1.p_4)(p_2.p_3)-(p_1.p_3)(p_2.p_4)\\right]\\Bigr).\n\\end{split}\n\\end{equation}\n\n\\subsection{Complete matrix element squared}\n\nMaking the change of variables \n\\begin{equation}\nd_\\text{V} = \\frac {c_\\text{V} g_\\text{Z}} {2e},\n\\end{equation}\nand remembering that the real part of the interference term \ncontributes twice, we obtain\n\\begin{multline}\n\\langle\\,|M_\\text{fi}|^2\\,\\rangle=\\frac{8e^4}{s^2}\\Bigl(C_1\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)(p_2.p_4)\\,\\right]+C_2m_3m_4p_1.p_2\\Bigr.\\\\[2mm]\n\\Bigl.+C_3\\left[\\,(p_1.p_4)(p_2.p_3)-(p_1.p_3)(p_2.p_4)\\,\\right]\\Bigr),\n\\end{multline}\nwhere\n\\begin{gather}\nC_1 = \\frac {\\left[\\left(d_\\text{V}^\\text{f}\\right)^2 + \\left(d_\\text{A}^\\text{f}\\right)^2 \n\\right] \\left[ \\left(d_\\text{V}^\\text{i}\\right)^2 + \\left(d_\\text{A}^\\text{i}\\right)^2 \\right]s^2 } \n{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \\left(q^\\text{f}\\right)^2\n\\left(q^\\text{i}\\right)^2 + \\frac {2 q^\\text{f} q^\\text{i} d_\\text{V}^\\text{f} d_\\text{V}^\\text{i}s\n\\left(s-m_\\text{Z}^2\\right)}{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,,\n\\\\[3mm]\nC_2 = \\frac {\\left[\\left(d_\\text{V}^\\text{f}\\right)^2 - \\left(d_\\text{A}^\\text{f}\\right)^2 \n\\right] \\left[ \\left(d_\\text{V}^\\text{i}\\right)^2 + \\left(d_\\text{A}^\\text{i}\\right)^2 \\right]s^2 } \n{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \\left(q^\\text{f}\\right)^2\n\\left(q^\\text{i}\\right)^2 + \\frac {2 q^\\text{f} q^\\text{i} d_\\text{V}^\\text{f} d_\\text{V}^\\text{i}s \n\\left(s-m_\\text{Z}^2\\right)}{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,,\n\\\\[3mm]\nC_3 = 2 \\left( \\frac {2 d_\\text{V}^\\text{f} d_\\text{A}^\\text{f} d_\\text{V}^\\text{i} d_\\text{A}^\\text{i}s^2} \n{\\left(s-m_\\text{Z}^2 \\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \n\\frac {q^\\text{f} q^\\text{i} d_\\text{A}^\\text{f} d_\\text{A}^\\text{i} s \\left(s-m_\\text{Z}^2\\right)} \n{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} \\right).\n\\end{gather}\n\n\\section{Differential cross section}\n\nSubstituting the squared matrix element into equation~(\\ref{diffxs}) we have\n\\begin{multline}\n\\frac{d\\sigma}{d\\Omega^*} = \\frac{e^4}{8\\pi^2}\\frac{1}{s^3} \\frac{p_3}{E} \n\\Bigl(C_1\\left[\\,(p_1.p_4)(p_3.p_2)+(p_1.p_3)(p_2.p_4)\\,\\right]\n+C_2m_3m_4p_1.p_2\\Bigr.\\\\[2mm]\n\\Bigl.+C_3\\left[\\,(p_1.p_4)(p_2.p_3)-(p_1.p_3)(p_2.p_4)\\,\\right]\\Bigr).\n\\end{multline}\nFrom \\secref{defs} we see that\n\\begin{equation}\n\\begin{split}\n(p_1.p_4)(p_2.p_3)&=E^2[E_3E_4+(E_3+E_4)p_3\\cos\\theta^*+p_3^2\\cos^2\\theta^*]\\,,\\\\[2mm]\n(p_1.p_3)(p_2.p_4)&=E^2[E_3E_4-(E_3+E_4)p_3\\cos\\theta^*+p_3^2\\cos^2\\theta^*]\\,,\n\\end{split}\n\\end{equation}\nand so, noting that $s=4E^2$ and $E_3+E_4=2E$,\n\\begin{equation}\n\\frac{d\\sigma}{d\\Omega^*} = \\frac{e^4}{16\\pi^2}\\frac{1}{s^2}\\frac{p_3}{E} \n\\left[C_1(E_3E_4+p_3^2\\cos^2\\theta^*)+C_2m_3m_4+C_3(2Ep_3\\cos\\theta^*)\\,\\right].\n\\end{equation}\n\nThe final result for the parton-level cross section (indicated by the `hat') \nincludes a factor of 1\/3 since the quarks must both be of the same colour if they are to annihilate.  So we have\n\\begin{equation}\n\\frac {d\\hat{\\sigma}}{d\\Omega^*} (\\text{q}\\bar{\\text{q}}\\rightarrow \\text{L}^-\\text{L}^+) \n= \\frac{e^4}{48\\pi^2} \\frac {1} {\\hat{s}^2} \\frac {p_3} {E}\n\\left[ C_1 \\left( E_3E_4 + p_3^2\\cos^2\\theta^*\\right) + C_2 m_3 m_4 +\n2 C_3 E p_3 \\cos\\theta^* \\right].\n\\end{equation}\nAlthough the notation is different, this expression is found to agree with the equivalent result in \\cite{Azuelos:1994qu}.\n\n\\section{Full cross section}\n\nTo integrate this we use $d\\Omega^*=2{\\pi}d(\\cos\\theta^*)$, and so obtain\n\\begin{equation}\n\\hat{\\sigma} (\\text{q}\\bar{\\text{q}}\\rightarrow \\text{L}^-\\text{L}^+)= \n\\frac {e^4} {12\\pi} \\frac {1} {\\hat{s}^2} \\frac {p_3} {E} \n\\left[ C_1 \\left(E_3E_4 + \\frac {p_3^2} {3}\\right) + C_2 m_3 m_4 \\right].\n\\end{equation}\n\n\\section{Slepton Drell-Yan cross section}\n\nThe slepton calculation is in many ways similar to the lepton one above, \nalthough it is made slightly simpler by the scalar coupling of the sleptons\nto Z and $\\gamma$.  Left-handed and right-handed scalar states can mix but we\njust consider the cross section for production of one or the other (L below).\n\nFor Z the vertex factor is then $i\\frac{g_\\text{Z}}{2}g_\\text{L}(p_3-p_4)^{\\mu}$, and hence\n\\begin{equation}\n-iM_\\text{fi}^\\text{Z} = \\left[\\bar{v}(p_2).-ig_\\text{Z}\\gamma^{\\mu}\\frac{1}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)\nu(p_1)\\right] \\frac{-ig_{\\mu\\nu}}{s-m_\\text{Z}^2+im_\\text{Z}\\Gamma_\\text{Z}}\n\\biggl[ig_\\text{Z}\\frac{g_\\text{L}}{2}(p_3-p_4)^{\\nu}\\biggr],\n\\end{equation}\nwhich gives\n\\begin{equation}\nM_\\text{fi}^\\text{Z} = \\frac{g_\\text{Z}^2}{s-m_\\text{Z}^2+im_\\text{Z}\\Gamma_\\text{Z}}g_{\\mu\\nu}\n\\left[\\bar{v}(p_2)\\gamma^{\\mu}\\frac{1}{2}\\left(c_\\text{V}^\\text{i}-c_\\text{A}^\\text{i}\\gamma^5\\right)u(p_1)\\right]\n\\biggl[\\frac{g_\\text{L}}{2}(p_3-p_4)^{\\nu}\\biggr].\n\\end{equation}\nNote that the first square bracket is exactly as previously, so \ncontributes the same factor when we evaluate $\\sum_\\text{spins} |M_\\text{fi}^\\text{Z}|^2$.  \nThe second square bracket contributes\n\\begin{equation}\n\\frac{g_\\text{L}^2}{4}(p_3-p_4)^{\\mu}(p_3-p_4)^{\\nu}.\n\\end{equation}\nMultiplying these together we obtain\n\\begin{equation}\n\\begin{split}\n\\langle\\,|M_\\text{fi}^\\text{Z}|^2\\,\\rangle&=\\frac{1}{4}\\frac{g_\\text{Z}^4}{(s-m_\\text{Z}^2)^2+m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\\\[3mm]\n&\\times\\Bigl(\\left[\\left(c_\\text{V}^\\text{i}\\right)^2+\\left(c_\\text{A}^\\text{i}\\right)^2\\right]\\frac{g_\\text{L}^2}{4}\n\\left[\\,2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\\,\\right]\\Bigr).\n\\end{split}\n\\end{equation}\n\\enlargethispage{-2\\baselineskip}\n\nThis time to obtain the $\\gamma$-exchange version it is necessary to make the\nadditional substitution of $g_\\text{L}=2$.  Therefore\n\\begin{equation}\n\\langle\\,|M_\\text{fi}^{\\gamma}|^2\\,\\rangle=\\frac{1}{4}\\frac{e^4(q^\\text{f})^2(q^\\text{i})^2}{s^2} 4\\left[\\,2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\\,\\right].\n\\end{equation}\n\nThe final term to be calculated is once again the interference term.  \nNeglecting the propagators, the spin sum is\n\\begin{multline}\n\\sum_{r,s}\n\\frac{g_\\text{L}}{2}(p_3-p_4)^{\\mu}\n\\left[\\bar{u_s}(p_1)\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac {\\gamma_{\\mu}}{2}.v_r(p_2)\n\\bar{v_r}(p_2)\\gamma_{\\nu}.u(p_1)\\right]\n(p_3-p_4)^{\\nu} \\\\[3mm]\n=\\mbox{Tr}\\left[\\gamma.p_1\\left(c_\\text{V}^\\text{i}+c_\\text{A}^\\text{i}\\gamma^5\\right)\\frac{\\gamma_{\\mu}}{2}\\gamma.p_2\n\\gamma_{\\nu}\\right]\\frac{g_\\text{L}}{2}(p_3-p_4)^{\\mu}(p_3-p_4)^{\\nu}.\n\\end{multline}\n\nMultiplying these terms (once again noting that the trace is the same as \nappeared in the lepton cross section) we obtain\n\\begin{multline}\n\\langle(M_\\text{fi}^\\text{Z})^*M_\\text{fi}^{\\gamma}\\rangle=\\frac{1}{4}\\frac{g_\\text{Z}^2e^2q^\\text{f}q^\\text{i}}\n{s(s-m_\\text{Z}^2-im_\\text{Z}\\Gamma_\\text{Z})}\\\\[3mm]\n\\times c_\\text{V}^\\text{i}g_\\text{L}\\left[\\,2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\\,\\right].\n\\end{multline}\n\nMaking the additional change of variables $h_\\text{L}=g_\\text{L}g_\\text{Z}\/2e$, we arrive at the total matrix element squared:\n\\begin{equation}\n\\langle\\,|M_\\text{fi}|^2\\,\\rangle=\\frac{e^4}{s^2}\nD\\left[\\,2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\\,\\right],\n\\end{equation}\nwhere\n\\begin{equation}\nD = \\frac {h_\\text{L}^2 \\left[\\left(d_\\text{V}^\\text{i}\\right)^2 +\\left(d_\\text{A}^\\text{i}\\right)^2 \\right] \ns^2} {\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \n\\left(q^\\text{f}\\right)^2 \\left(q^\\text{i}\\right)^2 +\n\\frac {2 q^\\text{f} q^\\text{i} h_\\text{L} d_\\text{V}^\\text{i}s\\left(s-m_\\text{Z}^2\\right)}\n{\\left(s-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,.\n\\end{equation}\n\nTherefore the differential cross section is\n\\begin{equation}\n\\frac{d\\sigma}{d\\Omega^*} = \\frac{e^4}{64\\pi^2}\\frac{1}{s^3} \\frac{p_3}{E} \nD\\left[\\,2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\\,\\right].\n\\end{equation}\nThe 4-momenta in \\secref{defs} give us\n\\begin{equation}\n\\begin{split}\np_1.(p_3-p_4)p_2.(p_3-p_4)=&E^2\\left[\\,(E_3-E_4)^2-4p_3^2\\cos^2\\theta^*\\right],\\\\[2mm]\np_1.p_2(p_3-p_4).(p_3-p_4)=&2E^2\\left[\\,(E_3-E_4)^2-4p_3^2\\right].\n\\end{split}\n\\end{equation}\nTherefore\n\\begin{equation}\n2p_1.(p_3-p_4)p_2.(p_3-p_4)-p_1.p_2(p_3-p_4).(p_3-p_4)\n=8E^2p_3^2\\sin^2\\theta^*\\,,\n\\end{equation}\nand the differential cross~section is\n\\begin{equation}\n\\frac{d\\sigma}{d\\Omega^*} = \\frac{e^4}{32\\pi^2}D\\frac{1}{s^2}\\frac{p_3}{E} \np_3^2\\sin^2\\theta^*.\n\\end{equation}\nThe final result for either left- or right-handed sleptons, including the \n1\/3 colour factor, is therefore\n\\begin{equation}\n\\frac {d\\hat{\\sigma}}{d\\Omega^*}(\\text{q}\\bar{\\text{q}}\\rightarrow\\tilde{\\text{l}}_\\text{L\/R}\n\\tilde{\\text{l}}^*_\\text{L\/R})=\\frac {e^4} {96\\pi^2} D \\frac {1}{\\hat{s}^2} \\frac {p_3} \n{E} p_3^2 \\sin^2\\theta^*.\n\\end{equation}\n\n\\chapter{Black Hole Master Equation}\n\\chaptermark{Black Hole Master Equation}\n\\label{appb}\n\nIn this appendix, Kanti's derivation of a master equation is reproduced from \\cite{Harris:2003eg}.  The Newman-Penrose formalism is used to obtain this equation describing the motion of a particle with spin $s$ in the background of a higher-dimensional,  non-rotating, neutral black\nhole projected onto a 3-brane. The corresponding 4-dimensional metric tensor is\ngiven in eq.~(\\ref{non-rot}). \n\nWe first need to choose a tetrad basis of null\nvectors $(\\ell^\\mu, n^\\mu, m^\\mu, \\bar m^\\mu)$, where $\\ell$ and $n$ are real\nvectors and $m$ and $\\bar m$ are a pair of complex conjugate vectors. They\nsatisfy the relations ${\\bf l} \\cdot {\\bf n}=1$ and ${\\bf m}\\cdot {\\bf \\bar m}=-1$,\nwith all other products being zero. Such a tetrad basis is given by\n\\begin{eqnarray}\n\\ell^\\mu=\\Bigl(\\frac{1}{h},\\,1, \\,0,\\,0\\Bigr)\\,, &\\quad &\nn^\\mu=\\Bigl(\\frac{1}{2}, \\,-\\frac{h}{2}, \\,0, \\,0\\Bigr)\\,, \\nonumber \\\\[3mm]\nm^\\mu=\\Bigl(0, \\,0, \\,1, \\,\\frac{i}{\\sin\\theta}\\Bigr)\\,\n\\frac{1}{\\sqrt{2} r}\\,, &\\quad&\n\\bar m^\\mu=\\Bigl(0, \\,0, \\,1, \\,\\frac{-i}{\\sin\\theta}\\Bigr)\\,\n\\frac{1}{\\sqrt{2} r}\\,. \n\\end{eqnarray}\n\nThe $\\lambda_{abc}$ coefficients, which are used to construct the spin coefficients, are defined by\n$\\lambda_{abc}=(e_b)_{i,j}\\Bigl[(e_a)^i (e_c)^j-(e_a)^j (e_c)^i \\Bigr]$,\nwhere $e_a$ stands for each one of the null vectors and $(i,j)$ denote the\ncomponents of each vector. Their non-vanishing components are found to be\n\\begin{equation}\n\\lambda_{122}=-\\frac{h'}{2}\\,, \\quad \\lambda_{134}=\\frac{1}{r}\\,,\n\\quad \\lambda_{234}=-\\frac{h}{2r}\\,, \\quad\n\\lambda_{334}=\\frac{\\cos\\theta}{\\sqrt{2} r \\sin\\theta}\\,.\n\\end{equation}\nThe above components must be supplemented by those that follow from the\nsymmetry $\\lambda_{abc}=-\\lambda_{cba}$ and the complex conjugates obtained\nby replacing an index 3 by 4 (or vice versa) or by interchanging 3 and 4.\n\nWe may now compute the spin coefficients defined by\n$\\gamma_{abc}=(\\lambda_{abc}+\\lambda_{cab} -\\lambda_{bca})\/2$. Particular\ncomponents, or combinations, of the spin coefficients can be directly used\nin the field equations \\cite{Newman:1962qr,chandra}. They are found to have the\nfollowing values:\n\\begin{eqnarray}\n\\kappa=\\sigma=\\lambda=\\nu=\\tau=\\pi=\\epsilon=0\\,; \\nonumber\n\\end{eqnarray}\n\\begin{equation}\n\\rho=-\\frac{1}{r}\\,; \\quad \\mu=-\\frac{h}{2r}\\,; \\quad \\gamma=\\frac{h'}{4}\\,; \n\\quad\\alpha=-\\beta=-\\frac{\\cot\\theta}{2\\sqrt{2} r}\\,.\n\\end{equation}\n\nIn what follows, we will also employ the Newman-Penrose operators\n\\begin{equation}\n\\hat D=\\frac{1}{h}\\,\\frac{\\partial \\,}{\\partial t} +\n\\frac{\\partial \\,}{\\partial r}\\,, \\qquad\n\\hat \\Delta = \\frac{1}{2}\\,\\frac{\\partial \\,}{\\partial t} -\n\\frac{h}{2}\\,\\frac{\\partial \\,}{\\partial r}\\,, \\qquad\n\\hat \\delta = \\frac{1}{\\sqrt{2} r}\\,\\Bigl(\\frac{\\partial \\,}{\\partial \\theta} +\n\\frac{i}{\\sin\\theta}\\,\\frac{\\partial \\,}{\\partial \\varphi}\\Bigr)\\,,\n\\end{equation}\nand make use of the field factorization\n\\begin{equation}\n\\Psi_s(t,r,\\theta,\\varphi)= e^{-i\\om t}\\,e^{i m \\varphi}\\,R_{s}(r)\n\\,{}_sS^{m}_{\\ell}(\\theta)\\,,\n\\label{facto}\n\\end{equation}\nwhere ${}_sY^m_{\\ell}(\\theta,\\varphi)=e^{i m \\varphi}\\, {}_sS^{m}_{\\ell}(\\theta)$ are the spin-weighted\nspherical harmonics \\cite{Goldberg:1967uu}. We will now consider each type of field\nseparately.\n\n\\section{Gauge bosons ($s=1$)}\n\nIn the Newman-Penrose formalism, there are only three `degrees of freedom'\nfor a gauge field (namely $\\Phi_0=F_{13}$, $\\Phi_1=(F_{12}+ F_{43})\/2$ and\n$\\Phi_2=F_{42}$) in terms of which the different components of the Yang-Mills \nequation for a massless gauge field are written as\n\\begin{align}\n(\\hat D-2 \\rho)\\,\\Phi_1 - (\\hat \\delta^*-2 \\alpha) \\,\\Phi_0 =& 0\\,, \n\\label{B1}\\\\%[3mm]\n\\hat \\delta\\,\\Phi_1 - (\\hat \\Delta + \\mu-2 \\gamma) \\,\\Phi_0 =& 0\\,, \n\\label{B2}\\\\%[3mm]\n(\\hat D-\\rho)\\,\\Phi_2 - \\hat \\delta^*\\,\\Phi_1 =& 0\\,, \\label{B3}\\\\%[3mm]\n(\\hat \\delta+2 \\beta)\\,\\Phi_2 - (\\hat \\Delta +2 \\mu) \\,\\Phi_1 =& 0\\,, \n\\label{B4}\n\\end{align}\nwhere $\\hat \\delta^*$ stands for the complex conjugate of $\\hat \\delta$.\nRearranging eqs. (\\ref{B1}) and (\\ref{B2}), we can see that $\\Phi_1$ decouples\nleaving behind an equation involving only $\\Phi_0$. Using the explicit forms\nof the operators and spin coefficients, as well as the factorized ansatz\nof eq.~(\\ref{facto}), this can be separated into an angular equation,\n\\begin{equation}\n\\frac{1}{\\sin\\theta}\\,\\frac{d \\,}{d \\theta}\\,\n\\biggl(\\sin\\theta\\,\\frac{d\\, {}_1S^m_{\\ell}}{d \\theta}\\biggr)\n+ \\biggl[ -\\frac{2 m \\cot\\theta}{\\sin\\theta} -\\frac{m^2}{\\sin^2\\theta}\n+ 1 - \\cot^2\\theta + {}_1\\lambda_{\\ell} \\biggl] {}_1S^m_{\\ell}(\\theta)=0\\,,\n\\end{equation}\nwith eigenvalue ${}_s\\lambda_{\\ell}=\\ell\\,(\\ell +1) - s\\,(s+1)$, and a radial\nequation,\n\\begin{equation}\n\\frac{1}{\\Delta}\\,\\frac{d \\,}{d r}\\biggl(\n\\Delta^2\\,\\frac{d R_1}{d r}\\biggr) + \\biggl[\n\\frac{\\omega^2 r^2}{h} + 2 i \\omega r -\\frac{i \\omega r^2 h'}{h}\n+ (\\Delta'' -2) -{}_1\\lambda_{\\ell}\\biggr]\\,R_1(r)=0\\,,\n\\end{equation}\nwhere $\\Delta=h r^2$. \n\n\\section{Fermion fields ($s=1\/2$)}\n\nFor a massless two-component spinor field, the Dirac equation can be\nwritten as\n\\begin{align}\n(\\hat \\delta^*-\\alpha)\\,\\chi_0 =& (\\hat D-\\rho)\\,\\chi_1\\,, \\\\%[3mm]\n(\\hat \\Delta + \\mu -\\gamma)\\,\\chi_0 =& (\\hat \\delta + \\beta)\\,\\chi_1\\,.\n\\end{align}\nPerforming a similar rearrangement as in the case of bosons, we find that\n$\\chi_1$ is decoupled and that the equation for $\\chi_0$ reduces to the\nfollowing set of angular,\n\\begin{equation}\n\\frac{1}{\\sin\\theta}\\,\\frac{d \\,}{d \\theta}\\,\n\\biggl(\\sin\\theta\\,\\frac{d\\, {}_{1\/2}S^m_{\\ell}}{d \\theta}\\biggr)\n+ \\biggl[ -\\frac{m \\cot\\theta}{\\sin\\theta} -\\frac{m^2}{\\sin^2\\theta}\n+ \\frac{1}{2} - \\frac{1}{4}\\,\\cot^2\\theta + {}_{1\/2}\\lambda_{\\ell} \\biggl] \n{}_{1\/2}S^m_{\\ell}(\\theta)=0\\,,\n\\end{equation}\nand radial,\n\\begin{equation}\n\\frac{1}{\\sqrt{\\Delta}}\\,\\frac{d \\,}{d r}\\biggl(\n\\Delta^{3\/2}\\,\\frac{d R_{1\/2}}{d r}\\biggr) + \\biggl[\n\\frac{\\omega^2 r^2}{h} + i \\omega r -\\frac{i \\omega r^2 h'}{2h}\n+ \\frac{1}{2}\\,(\\Delta'' -2) -{}_{1\/2}\\lambda_{\\ell}\\biggr]\\,R_{1\/2}(r)=0\\,,\n\\end{equation}\nequations, with the same definitions for $\\Delta$ and ${}_s\\lambda_{\\ell}$\nas before.  \n\n\\section{Scalar fields ($s=0$)}\n\nFor completeness the equation of motion for a scalar field\npropagating in the same background is included here. This equation can be determined quite\neasily by evaluating the double covariant derivative $g^{\\mu\\nu} D_\\mu D_\\nu$\nacting on the scalar field. It finally leads to this pair of equations:\n\\begin{equation}\n\\frac{1}{\\sin\\theta}\\,\\frac{d \\,}{d \\theta}\\,\\biggl(\\sin\\theta\\,\n\\frac{d\\, {}_0S^m_{\\ell}}{d \\theta}\\,\\biggr) + \\biggl[-\\frac{m^2}{\\sin^2\\theta}\n+ {}_0\\lambda_{\\ell} \\biggr]\\,{}_0S^m_{\\ell}=0\\,;\n\\end{equation}\n\\begin{equation}\n\\frac{d \\,}{dr}\\,\\biggl(\\Delta\\,\\frac{d R_0}{dr}\\biggr) +\n\\Bigl(\\frac{\\om^2 r^2}{h} - {}_0\\lambda_{\\ell}\\Bigr) R_0(r) =0\\,.\n\\label{scalar}\n\\end{equation}\nHere, ${}_0Y^m_{\\ell}(\\theta, \\varphi)=e^{i m \\varphi}\\,{}_0S^{m}_{\\ell}(\\theta)$ are the usual\nspherical harmonics $Y^m_\\ell(\\theta, \\varphi)$ and\n${}_0\\lambda_{\\ell}=\\ell (\\ell+1)$.\nThe above equations were used in \\cite{Kanti:2002nr} for the analytic determination\nof grey-body factors for the brane emission of scalar particles by higher-dimensional black holes.\n\n\\section{Master equation for a field with arbitrary spin}\n\nCombining the equations derived above for bosons, fermions and scalar fields, we may now rewrite them in the form of a master equation, valid for field types. The radial equation then takes\nthe form\n\\begin{equation}\n\\Delta^{-s}\\,\\frac{d \\,}{dr}\\,\\biggl(\\Delta^{s+1}\\,\\frac{d R_s}{dr}\\,\\biggr) +\n\\biggl(\\frac{\\om^2 r^2}{h} + 2i s\\,\\om\\,r -\\frac{i s \\om\\,r^2 h'}{h}\n+s\\,(\\Delta''-2) - {}_s\\lambda_{\\ell} \\biggr)\\,R_s(r)=0\\,,\n\\label{master1}\n\\end{equation}\nwhile the angular equation reads\n\\begin{equation}\n\\frac{1}{\\sin\\theta}\\,\\frac{d \\,}{d \\theta}\\,\\biggl(\\sin\\theta\\,\n\\frac{d S^m_{s,\\ell}}{d \\theta}\\,\\biggr) + \\biggl[-\\frac{2 m s \\cot\\theta}\n{\\sin\\theta} - \\frac{m^2}{\\sin^2\\theta} + s - s^2 \\cot^2\\theta \n+ {}_s\\lambda_{\\ell} \\biggr]\\,S^m_{s,\\ell}=0\\,.\n\\label{master2}\n\\end{equation}\n\\enlargethispage{-0.5\\baselineskip}\n\nThe latter equation is identical to the one derived by Teukolsky \\cite{Teukolsky:1973ha}\nin the case of a non-rotating, spherically-symmetric black hole. However the radial one\ndiffers by the extra factor $s\\,(\\Delta''-2)$ because for the metric\ntensor here this combination is not zero (unlike for the\n4-dimensional Schwarzschild and Kerr metrics). The $\\Delta''$-term can be removed\nif we make the redefinition $R_s=\\Delta^{-s}\\, P_s$.  Then, we obtain\n\\begin{equation}\n\\Delta^{s}\\,\\frac{d \\,}{dr}\\,\\biggl(\\Delta^{1-s}\\,\\frac{d P_s}{dr}\\,\\biggr) +\n\\biggl(\\frac{\\om^2 r^2}{h} + 2i s\\,\\om\\,r -\\frac{i s \\om\\,r^2 h'}{h}\n- {}_s\\Lambda_\\ell \\biggr)\\,P_s(r)=0\\,,\n\\label{master3}\n\\end{equation}\nwhere now ${}_s\\Lambda_\\ell={}_s\\lambda_{\\ell} +2 s =\\ell\\,(\\ell+1)-s\\,(s-1)$. The above\nform of the radial equation was used in \\cite{Kanti:2002ge} to determine analytically the grey-body factors and\nemission rates for fermions and gauge bosons on the brane.\n\\chapter{Black Hole Monte Carlo Method}\n\\label{appc}\n\n\\section{Monte Carlo calculation of the cross section}\n\nThe main features of the Monte Carlo (MC) method can be described by the following\nmathematical relationship:\n\\begin{equation}\nI=\\int_{x_1}^{x_2} f(x)\\, dx = (x_2-x_1) \\left<f(x)\\right> \\approx (x_2-x_1) \\frac{1}{N} \\sum_{i=1}^{N} f(x_i),\n\\label{mcdef}\n\\end{equation}\nwhere the $f(x_i)$ are values of $f(x)$ at the $N$ randomly chosen values of $x \\in[x_1,x_2]$.\n\nIn this case the integral we require is for the black hole production cross section $\\sigma$:\n\\begin{equation}\n\\sigma = \\int_{E_\\text{min}}^{E_\\text{max}} \\hat{\\sigma}(E) p(E)\\, dE, \n\\end{equation}\nwhere $E$ is the centre-of-mass energy of the partons involved and $\\hat{\\sigma}$ is the parton-level cross section.  Using the substitution $y(E)=dY\/dE$ we obtain\n\\begin{equation}\n\\sigma = \\int_{Y_1=Y(E_\\text{min})}^{Y_2=Y(E_\\text{max})} \\frac{\\hat{\\sigma}(E) p(E)}{y(E)}\\, dY.\n\\end{equation}\n\nThe MC procedure will be most efficient if the integrand is as flat as\npossible as a function of $E$.  The parton-level geometrical cross section for black hole production means that $\\hat{\\sigma} (E) \\sim E^{\\beta}$ where $\\beta = 2\/(n+1)$.  In addition we expect that $p(E) \\sim E^{-8}$ because of the behaviour of the parton distribution functions (PDFs).  Hence a sensible choice for $y(E)$ is\n\\begin{equation}\ny(E)=\\alpha E^{\\alpha -1} \\Rightarrow Y(E)=E^{\\alpha},\n\\end{equation}\nwhere $\\alpha = \\beta -7$.  Returning to eq.~(\\ref{mcdef}) we have\n\\begin{equation}\n\\sigma = (Y(E_\\text{max})-Y(E_\\text{min})) \\left<\\frac{\\hat{\\sigma} (E) p(E)}{y(E)} \\right> = \\frac{E_\\text{max}^{\\alpha}-E_\\text{min}^{\\alpha}}{\\alpha} \\left< \\frac{\\hat{\\sigma} (E) p(E)}{E^{\\alpha-1}}\\right>,\n\\end{equation}\nwhere $Y(E)$ is chosen at random from a uniform distribution between the maximum and minimum values.\n\\enlargethispage{-2\\baselineskip}\n\n$p(E)$ is calculated from the PDFs and is more commonly written as a function of $\\tau=\\hat{s}\/s$ (hence $\\tau \\propto E^2$).  Therefore $p(E)$ can be re-written as\n\\begin{equation}\np(E)=\\frac{2\\tau}{E} p(\\tau)=\\frac{2\\tau}{E} \\int_{\\tau}^{1} \\frac{dx}{x} f_1(x) f_2\\left(\\frac{\\tau}{x}\\right).\n\\end{equation}\nThis integral can also be evaluated by a MC procedure, and once again\na change of variable is useful.  We may conveniently write $p(\\tau)$ as\n\\begin{equation}\np(\\tau)=\\frac{1}{\\tau}\\int_{\\tau}^{1}\\frac{dx}{x} x f_1(x) \\frac{\\tau}{x} f_2\\left(\\frac{\\tau}{x}\\right)=\n\\frac{1}{\\tau}\\int_{\\tau}^{1} \\frac{dx}{x} h(x,\\tau)\\,,\n\\end{equation}\nand the substitution $z(x)=dZ\/dx$ gives us that\n\\begin{equation}\np(\\tau)=\\frac{1}{\\tau}\\int_{Z(\\tau)}^{Z(1)} \\frac{h(x,\\tau)}{x z(x)} \\, dZ\\,.\n\\end{equation}\nThe assumption that $h(x,\\tau)$ is fairly smooth (since $f(x)$ is usually of the approximate form $1\/x$) means that the MC procedure should be efficient if \n\\begin{equation}\nz(x)=\\frac{1}{x} \\Rightarrow Z(x)=\\ln x\\,.\n\\end{equation}\nSo, using eq.~(\\ref{mcdef}), we conclude that\n\\begin{equation}\np(\\tau)=\\frac{(Z(1)-Z(\\tau))}{\\tau}\\left<h(x,\\tau)\\right>= -\\frac{\\ln\\tau}{\\tau} \\left<h(x,\\tau)\\right>,\n\\end{equation}\nwhere $Z(x)$ is chosen at random from a uniform distribution between the maximum and minimum values.\n\nCombining these two MC procedures, a good approximation to the total cross section $\\sigma$ is given by\n\\begin{equation}\n\\frac{E_\\text{max}^{\\alpha}-E_\\text{min}^{\\alpha}}{\\alpha}\\left<\\frac{\\hat{\\sigma}}{E^{\\alpha-1}}\\frac{2}{E}(-\\ln\\tau) h(x,\\tau)\\right>,\n\\end{equation}\nwhere firstly $Y(E)$ is chosen at random ($\\rightarrow E \\mbox{ and hence } \\tau$) and then, given this value of $\\tau$, $Z(x)$ is chosen at random ($\\rightarrow x$).\n\n\\section{Monte Carlo generation of unweighted events}\n\nAs well as estimating the total cross section, the MC program must generate events drawn from this distribution.  This can be done using essentially the same procedure---$E$ and then $x$ are chosen and the cross section value is computed.  The calculated weight is then accepted or rejected against the maximum weight found by an initial search.  A weight $w_i$ is accepted with probability $w_i\/w_\\text{max}$.\n\\chapter{Introduction}\n\\chaptermark{Introduction}\n\\label{introch}\n\\section{The Standard Model}\n\\label{thesm}\n\nThe Standard Model (SM) of particle physics has been remarkably successful.  Many of the parameters are measured to great accuracy, and the Standard Model has made many predictions which have been verified experimentally.\nHowever there are several reasons why it is widely believed that, whilst working well in the energy regimes which have been investigated to date, the Standard Model does not give the full picture.  These issues, which are both technical and aesthetic, will be discussed further in the following sections.  It is the belief that there is physics `Beyond the Standard Model' (BSM) which motivates the continuation of large scale particle physics experiments, most notably the Large Hadron Collider (LHC) at CERN which will collide protons with a centre-of-mass energy of 14~TeV\\@.  Of course, the other great aim of the LHC is to discover the famous Higgs boson, the only missing particle of the Standard Model.  Although the Higgs mechanism is as yet unproved, much of the work in this thesis makes the implicit and sometimes explicit assumption that a Standard Model Higgs boson exists.\n\n\\subsection{Particle content}\n\nThe particle content of the Standard Model is intimately related to the symmetries of the Lagrangian which describes the physics.  The forces between the fermions are mediated by gauge boson fields which allow the Lagrangian to be invariant under gauge transformations---it is believed that all fundamental interactions are described by some form of gauge theory.  These symmetries imply the existence of conserved currents and charges in the theory.\n\nFirstly the situation before the breaking of electroweak symmetry is reviewed.  At this stage all particles are massless as the Higgs mechanism has not yet been invoked to give masses to the particles.\n\nThere are three main symmetries of the Standard Model Lagrangian.  Firstly there is\nthe U(1) hypercharge symmetry;  this is an Abelian symmetry and the corresponding gauge boson field is usually written as $B^{\\mu}$.  The conserved charge, the hypercharge, is most commonly called $Y$.  \n\nThere is also an SU(2) weak isospin symmetry which is sometimes referred to as SU(2)$_\\text{L}$ since the gauge field couples only to left-handed fermions.  This is a non-Abelian symmetry since the Lie group generators are matrices.  There are three accompanying gauge fields: $W_1^{\\mu}$, $W_2^{\\mu}$ and $W_3^{\\mu}$ and the conserved charge is $I_3$, the third component of the weak isospin $I$.\n\nFinally there is the SU(3) non-Abelian symmetry of Quantum Chromodynamics (QCD) which has a conserved colour charge.  The gauge bosons in this case are gluons and are described by the fields $A_a^{\\mu}$ where the colour index $a$ runs from 1 to 8.  The gluons themselves carry colour charge---red, green and blue.\n\nThe theory outlined above is clearly incomplete since we know that most observed particles are not massless.  It is the introduction of a Higgs field (a complex scalar field with a non-zero vacuum expectation value) to the theory which both gives most of the particles mass and causes the electroweak symmetry breaking.  The $W_3^{\\mu}$ from the SU(2) of isospin mixes with the $B^{\\mu}$ from the U(1) of hypercharge to produce a massless photon field and a massive\nZ boson.  At the same time the $W_1$ and $W_2$ states which can be identified with the W$^{\\pm}$ bosons acquire mass, as do all the fermions with the exception of the neutrinos.\\footnote{In the minimal Standard Model neutrinos are massless although recent evidence suggests that this is not the case.}  After this mixing there is a residual U(1)$_Q$ electromagnetic gauge symmetry; the coupling of fermions to the photon is found to be proportional to the electromagnetic charge $Q$, which is related to the hypercharge and weak isospin through $Q=Y+I_3$.\\footnote{It is not uncommon for $T$ to be used for the isospin rather than $I$, or for hypercharge to be defined such that this relationship becomes $Q=Y\/2+I_3$.}  The coupling to the Z boson is more complicated but is proportional to $I_3-Q\\sin^2\\theta_\\text{W}$ where $\\theta_\\text{W}$ is the Weinberg angle and $\\sin^2\\theta_\\text{W} \\sim 0.23$.  Since $I_3$ is different for the left and right-handed fermions, the coupling to the Z boson is also dependent on the fermion helicity.\n\nIn Table~\\ref{ewtable} the fermionic particle content of the Standard Model is summarized and the various quantum numbers of the different particles are shown.  From this table, in which L and R refer to left- and right-handed states respectively, it is clear why the isospin symmetry is often referred to as SU(2)$_\\text{L}$.\n\n\\begin{table}\n\\def1.0{1.25}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& \\multicolumn{2}{c|}{SU(2)$_\\text{L}$} & U(1)$_Y$ & U(1)$_Q$ \\\\\n\\cline{2-5}\n& $I$ & $I_3$ & $Y$ & $Q$\\\\\n\\hline\n$\\text{u}_\\text{R}$ & $0$ & $\\phantom{-}0$ & $\\phantom{-}\\frac{2}{3}$ & $\\phantom{-}\\frac{2}{3}$\\\\\n$\\text{d}_\\text{R}$ & $0$ & $\\phantom{-}0$ & $-\\frac{1}{3}$ & $-\\frac{1}{3}$\\\\\n\\hline\n$\\biggl(\\!\\begin{array}{c}\n\\text{u}\\\\\n\\text{d}\\\\\n\\end{array}\\!\\biggr)_{\\!\\text{L}}$ & $\\frac{1}{2}$ &\n$\\begin{array}{c}\n\\phantom{-}\\frac{1}{2}\\\\\n-\\frac{1}{2}\\\\\n\\end{array}$ &\n$\\phantom{-}\\frac{1}{6}$ &\n$\\begin{array}{r}\n\\phantom{-}\\frac{2}{3}\\\\\n-\\frac{1}{3}\\\\\n\\end{array}$\\\\\n\\hline\n$\\text{e}_\\text{R}$ & $0$ & $\\phantom{-}0$ & $-1$ & $-1$\\\\\n\\hline\n$\\biggl(\\!\\begin{array}{c}\n\\nu_\\text{e}\\\\\n\\text{e}\\\\\n\\end{array}\\!\\biggr)_{\\!\\text{L}}$ & $\\frac{1}{2}$ &\n$\\begin{array}{r}\n+\\frac{1}{2}\\\\\n-\\frac{1}{2}\\\\\n\\end{array}$ &\n$-\\frac{1}{2}$ &\n$\\begin{array}{r}\n0\\\\\n-1\\\\\n\\end{array}$\\\\\n\\hline\n\\end{tabular}\n\\capbox{Fermionic particle content of the Standard Model}{Fermionic particle content of the Standard Model showing the relevant quantum numbers.\\label{ewtable}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table} \n\nApart from the fermions (with spin $s=1\/2$) the SM particle content consists of the gauge bosons ($s=1$) and the Higgs boson ($s=0$).  The Higgs field introduced in the Higgs mechanism has four degrees of freedom, but three of these are `eaten' by the isospin gauge bosons to give mass to the W and Z bosons.  Therefore after the symmetry breaking there is a massive neutral scalar with one degree of freedom---this is the particle known as the Higgs boson.  Although the Higgs boson has not yet been discovered and the significance of hints during the final running of the Large Electron-Positron (LEP) collider has been reduced by more careful analyses \\cite{Barate:2003sz}, a low Higgs mass ($\\sim$~115 GeV) is still favoured.  This is because of tight constraints coming from precision measurements of electroweak radiative corrections.\n\nIn addition to the Standard Model gauge symmetries discussed above, there are two further `accidental symmetries' in that all the terms in the Lagrangian are found to conserve both lepton number $L$ and baryon number $B$.  Lepton number is defined to be $+1$ for leptons, $-1$ for anti-leptons and zero for all other particles; baryon number is defined to be $+1\/3$ for quarks, $-1\/3$ for anti-quarks and zero for all other particles (this means that baryons have $B=1$).  These symmetries were not imposed on the model, but appear naturally given the requirement that the terms in the Lagrangian should be renormalizable, Lorentz invariant and gauge invariant.  In fact $B$ and $L$ are not exactly conserved because they are violated by non-perturbative electroweak effects \\cite{'tHooft:1976fv,'tHooft:1976up}, but they can normally be considered as conserved quantum numbers.  Unfortunately for most theories of physics beyond the Standard Model the most general Lagrangian which can be written down includes terms which violate these symmetries.  Potentially this causes various problems, most notably that it can allow protons to decay with a much shorter lifetime than the present experimental limit.\n\n\\section{Motivations for new physics}\n\nIt has already been mentioned that there are various reasons for believing there is physics beyond the Standard Model.  The work in this thesis will address some, but by no means all, of these issues which are detailed below.\n\n\\subsection{Gravity}\n\nA full model for particle physics would be expected to describe all the fundamental forces between particles---electroweak, strong and gravitational.  However the Standard Model fails to describe any details of the gravitational force, and hence cannot be the full fundamental model.  At energy scales of order the Planck mass, $M_\\text{P}$, a theory of quantum gravitation will be required to describe the interactions between particles.  This shows that the Standard Model will need to be replaced by an alternative theory at very high energy scales, and it is reasonable to expect that a more complete model than the Standard Model might be required even at energy scales only moderately higher than those which have already been investigated in detail.\n\n\\subsection{Hierarchy problem}\n\nOne of the most common arguments for physics beyond the Standard Model is what is known as the hierarchy problem.  This can be expressed in two different ways which, whilst clearly related, are in some ways distinct.  They are referred to here as the `aesthetic hierarchy problem' and the `technical hierarchy problem'.\n\n\\subsubsection{Aesthetic hierarchy problem}\n\nAs has been discussed above, the Standard Model has nothing to say about gravitational interactions.  It also seems aesthetically unappealing that the mass scale with which particle physicists are most familiar (the electroweak mass scale, $\\sim 100$~GeV) is so many orders of magnitude below the Planck scale ($\\sim 10^{19}$~GeV).  If there is a fundamental `theory of everything' then it would seem more natural if all the energy scales were of the same order of magnitude.  Another way of expressing this is just to say that it seems unnatural for gravity to be so much weaker than the other gauge forces.  Why are there seemingly two different fundamental scales in nature?\n\n\\subsubsection{Technical hierarchy problem}\n\nMore technically, the hierarchy problem is the result of considering one-loop corrections to the Higgs mass.  It is relatively easy to calculate the correction to the Higgs mass from diagrams like that in Figure~\\ref{hmcf}, but since the diagram is quadratically divergent, a cut-off ($\\Lambda$) must be introduced in the loop momentum integral to regulate the divergence.  The correction to the `bare' Higgs mass is\n\\begin{equation}\n\\delta M^2_\\text{Hf}=\\frac{|\\lambda_\\text{f}|^2}{16\\pi^2}\\left[-2\\Lambda^2+6m_\\text{f}^2 \\ln(\\Lambda\/m_\\text{f})+\\cdots\\right].\n\\end{equation}  \nwhere $\\lambda_\\text{f}$ is the Higgs-fermion coupling and $m_\\text{f}$ the fermion mass.\n\\begin{figure}\n\\begin{center}\n\\begin{picture}(440,80)\n\\SetOffset(120,-10)\n\\DashArrowLine(40,50)(80,50){5}\n\\ArrowArcn(100,50)(20,0,180)\n\\ArrowArcn(100,50)(20,180,360)\n\\DashArrowLine(120,50)(160,50){5}\n\\Text(30,50)[c]{H$^0$}\n\\Text(170,50)[c]{H$^0$}\n\\Text(100,17)[c]{$\\bar{\\text{f}}$}\n\\Text(100,83)[c]{f}\n\\end{picture}\n\\capbox{Fermion one-loop correction to the Higgs mass}{Feynman diagram for the fermion one-loop correction to the Higgs boson mass.\\label{hmcf}} \n\\end{center}\n\\end{figure}\nIn itself this mass correction isn't necessarily a problem since the bare Higgs mass can have any value provided the physical Higgs mass is around the electroweak scale.   The Standard Model is only considered to be a low-energy effective theory and so a natural value for $\\Lambda$ would be the Planck scale.  However this means that the `bare Higgs mass' would have a natural value of order the Planck scale and therefore, if there is a Higgs boson with mass of order 100~GeV, there would need to be cancellation at the level of 1 part in $10^{16}$.  This is considered to require enormous `fine-tuning' to the parameters of the bare Lagrangian.\n\nAssuming we believe in the existence of the Standard Model Higgs boson and find the idea of fine-tuning unnatural this leave only one possibility: there must be new physics and\/or new particle content at energy scales $\\Lambda \\sim$~1~TeV\\@.  In fact even if there is no SM Higgs boson, new TeV-scale physics is still required in order to unitarize the W-W scattering cross section.\n\n\\subsection{Unification of couplings}\n\\label{couplings}\n\nIf the strong and electroweak forces are to unify into a single gauge theory at a high energy scale then the gauge couplings must also unify.  The particle content of a model determines how the couplings `run' with energy and although in the Standard Model they almost unify at a scale of about $10^{15}$~GeV, this unification isn't quite exact (as illustrated by Figure~\\ref{unifsm}).  This can be seen as a motivation for additional particle content or new physics which will modify the running of the couplings to make the unification more exact.\n\\begin{figure}[t]\n\\begin{center}\n\\epsfig{file=unif_sm.eps, angle=0, width=.55\\textwidth}\n\\capbox{Gauge coupling running with SM particles}{Running of the gauge couplings assuming only Standard Model particle content.  Values of the inverse couplings at energy $\\mu$ are determined using the one-loop renormalization group equations.  $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$ are the hypercharge, weak isospin and strong couplings respectively, but the conventional normalization means that $\\alpha_1\\equiv(5\/3)\\alpha_Y$. The width of the bands on the plot is due to experimental uncertainties. (Reproduced from \\cite{Dienes:1997du}).\\label{unifsm}}\n\\end{center}\n\\end{figure}\n\\section{Supersymmetry and intermediate scale models}\n\\label{ism}\n\nOne type of new physics which is relevant to the work presented in this thesis is supersymmetry (SUSY). This is one of the most popular choices for physics beyond the Standard Model and resolves several of the issues raised in the previous section.  The basic idea is that an additional symmetry is introduced to the theory such that the Lagrangian is invariant under transformations which convert fermions into bosons and vice versa.  This immediately doubles the number of particles and more importantly helps to solve the technical hierarchy problem.  This is because there will be cancellations between Feynman diagrams like that in Figure~\\ref{hmcf} and one-loop diagrams with S (the fermion's supersymmetric scalar partner, known as a sfermion) as shown in Figure~\\ref{hmcs}.  The Higgs mass correction in this case is\n\\begin{equation}\n\\delta M^2_\\text{HS}=\\frac{\\lambda_\\text{S}}{16\\pi^2}\\left[\\Lambda^2-2m_\\text{S}^2 \\ln(\\Lambda\/m_\\text{S})+\\cdots\\right],\n\\end{equation}  \nwith $\\lambda_\\text{S}$ the Higgs-scalar coupling and $m_\\text{S}$ the scalar mass.\n\\begin{figure}\n\\begin{center}\n\\begin{picture}(440,70)\n\\SetOffset(120,-50)\n\\DashArrowLine(40,50)(100,50){5}\n\\DashCArc(100,70)(20,0,180){5}\n\\DashCArc(100,70)(20,180,360){5}\n\\DashArrowLine(100,50)(160,50){5}\n\\Text(30,50)[c]{H$^0$}\n\\Text(170,50)[c]{H$^0$}\n\\Text(100,103)[c]{S}\n\\end{picture}\\\\[3mm]\n\\capbox{Scalar one-loop correction to the Higgs mass}{Feynman diagram for the scalar one-loop correction to the Higgs boson mass.\\label{hmcs}} \n\\end{center}\n\\end{figure}\nHence provided there are two scalar particles (with coupling $\\lambda_\\text{S}=|\\lambda_\\text{f}|^2$) for every fermion, there will be cancellation between the fermion and sfermion loop diagrams.  In fact the cancellation is only between the terms quadratic in $\\Lambda$ but the remaining Higgs mass corrections are only logarithmic in $\\Lambda$ and hence do not cause the same fine-tuning problems.  Although we know that, if it exists, supersymmetry must be broken (since no superpartners of Standard Model particles have yet been observed) this supersymmetry breaking must occur in such a way that the leading-order cancellation remains.  The requirement for the physical Higgs mass to be at the electroweak scale is now that the superparticles have masses of order 1~TeV or less.  If this is not the case then fine-tuning problems start to resurface and one of the main motivations for SUSY has been lost.  It is also difficult to construct SUSY models where the SM particle masses differ from those of their superpartners by more than about an order of magnitude.\n\nVarious different ways to break supersymmetry have been studied.  It has long been known that a viable way to break supersymmetry is in a hidden sector---the SUSY breaking in a particular model is then transmitted to the `outside world' by some means (usually gravitational or gauge interactions).  The supersymmetry breaking means that what is initially a very beautiful theory (and hence very attractive to particle theorists) becomes much less beautiful.  Many new parameters ($\\sim$103) are needed to describe a particular supersymmetric model, and only by making simplifying assumptions (e.g.\\ regarding various high-scale masses and couplings) can much headway be made in considering the phenomenology of these models.  This makes supersymmetry much less attractive, particularly to SUSY sceptics who argue that a region of parameter space can always be found to ensure that supersymmetry cannot be ruled out by either experimental or cosmological constraints.\n\nHowever there are several other arguments in favour of supersymmetry.  It can provide a dark matter candidate in the form of a lightest supersymmetric particle (LSP).  This is the result of the R-parity symmetry which is often imposed to prevent proton decay but which also has the effect of making the LSP stable since it is forbidden from decaying into only SM particles.  One of the other motivations for supersymmetry is that it improves the unification of gauge couplings discussed in \\secref{couplings} (although the present value of $\\alpha_\\text{S}(M_\\text{Z})$ seems to be slightly too low to allow perfect unification---see \\cite{deBoer:2003xm} for a recent discussion).  Figure~\\ref{unifsusy} shows the running of the gauge couplings with energy in the supersymmetric case.\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=unif_susy.eps, angle=0, width=.55\\textwidth}\n\\capbox{Gauge coupling running with MSSM particles}{Running of the gauge couplings assuming Minimal Supersymmetric Standard Model particle content.  For more details of the quantities plotted, see Figure~\\ref{unifsm}. (Reproduced from \\cite{Dienes:1997du}).\\label{unifsusy}}\n\\end{center}\n\\end{figure}\nHowever the couplings do not unify at $\\sim 10^{15}$~TeV in all supersymmetric scenarios.  One scenario, motivated by string theory, is that of an intermediate scale \\cite{Benakli:1998pw,Burgess:1998px,Aldazabal:1999tw,Aldazabal:2000sk}.  In this type of model, gauge coupling unification is assumed to occur at an `intermediate' energy scale ($\\sim 10^{11}$~GeV) which is the geometric mean of the electroweak and Planck energy scales.  This choice is motivated by the scale of supersymmetry breaking in hidden-sector gauge-mediated scenarios and also by the usual solution of the strong Charge Parity (CP) problem.  \n\\enlargethispage{\\baselineskip}\n\nThe strong CP problem is that there is an unnaturally small bound ($<10^{-9}$) on the $\\Theta$ parameter of the QCD Lagrangian.  The parameter is constrained to be this small because the $\\Theta$-term in the Lagrangian is CP-violating and would otherwise lead to a neutron electric dipole moment (on which there are strong experimental limits).  The Peccei-Quinn mechanism \\cite{Peccei:1977hh} solves this problem by introducing an additional global chiral symmetry, U(1)$_\\text{PQ}$, which is then spontaneously broken generating a Goldstone boson known as the axion field.  The axion acquires a small mass due to non-perturbative effects and and has a lifetime which exceeds the age of the Universe by many orders of magnitude.  The requirement that axion emission does not over-cool stars puts a lower bound bound on the axion decay constant $f_\\text{a}$, and an upper bound comes from dark matter constraints.  Therefore $f_\\text{a}$ is constrained to be in the approximate energy range $10^9$--$10^{12}$~GeV, as is naturally the case for scalars in intermediate scale string scenarios \\cite{Burgess:1998px}.  \n\\enlargethispage{\\baselineskip}\n\nThe intermediate scale is also consistent with some neutrino mass mechanisms, and is favoured by certain cosmological inflation models.  High-energy cosmic ray observations and certain non-thermal dark matter candidates are also supporting arguments for an intermediate energy scale.  For more details, see \\cite{Benakli:1998pw,Burgess:1998px} and the references contained.\n\nVarious intermediate scale models are motivated from different string models\nin references \\cite{Benakli:1998pw,Burgess:1998px,Aldazabal:1999tw,Aldazabal:2000sk}.  Some phenomenological \nconstraints on intermediate scale models are provided by \\cite{Abel:2000bj}, and \nconstraints from the muon anomalous magnetic moment have also been discussed \\cite{Baek:2001nz,Cerdeno:2001aj}.\nOne of the ways of achieving this intermediate scale unification is to include new particles (with $\\sim$~TeV masses) as part of extra supermultiplets added to the Minimal Supersymmetric Standard Model (MSSM).  Extra heavy leptons added in such a scenario are the subject of the work described in Chapter~\\ref{leptons}.\n\n\n\\section{Extra dimension models}\n\\label{edm}\n\nExtra dimension models are another class of model which have recently become quite fashionable and also provide a way of solving the hierarchy problem.\\footnote{Many would argue, with good reason, that extra dimensions really just \\emph{reformulate} the problem rather than solve it.}  The essence of these models is that the apparent weakness of gravity at macroscopic length scales is due to the presence of extra dimensions.  \n\nAll these models involve one or more $(3+1)$-dimensional slices of space-time (known as branes or 3-branes) which are embedded in the full extra-dimensional space-time (the bulk).  In most models the world as we know it and all the Standard Model particles are localized on the brane whilst gravitons can propagate in the bulk.  There are, though, some models where SM particles are not localized on the brane---one will be introduced below.  Another feature of extra dimension models is that the new dimensions are compactified or `curled up'---necessary to explain why they have never been observed in the same way as the infinite dimensions we are familiar with. \n\nDifferent models involve extra dimensions of different sizes but it is initially surprising to learn that they can be relatively large ($\\sim$1~mm) without being ruled out by experimental measurements of gravity.  Extra dimensions of this size imply that the Newtonian $1\/r^2$ force law will fail at these length scales.  However the force law has, even now, only been verified down to distances of $\\sim$~100 $\\mu$m (see \\secref{ssgrav}) and when these extra dimension models were first proposed in 1998, the limit was only about 1~mm.  \n\nAs well as providing a solution to the hierarchy problem, extra dimension models can be motivated by string theory.  When combined with supersymmetry, string theory is a candidate for a theory of quantum gravity but it requires the world to be 10- or 11-dimensional.  String theory can also naturally account for some fields (e.g.\\ SM particle fields) being fixed on $(3+1)$-dimensional surfaces or branes whilst others, like gravitons, propagate in the extra dimensions.  In both type-I and type-II string theories, matter and the Standard Model interactions are represented by open strings which start and end on a brane, whereas as gravity is described by a closed string which can propagate in all the spatial dimensions.  It is possible that supersymmetry plays an important r\\^{o}le in stabilizing the extra-dimensional geometry.  Unfortunately, in common with the majority of proposals for BSM physics, one prediction of most of these theories is a very large value for the vacuum energy density or cosmological constant; this is in contradiction with the very small value calculated by cosmologists.\n\nIn fact extra dimension models are not a new idea.  As early as the 1920s a model with one extra compactified dimension was suggested by Kaluza and Klein in an attempt to unify electromagnetism and gravity.  In such a scheme the photon is actually a component of the higher-dimensional graviton.  Although their theory fell out of fashion due to its failure to explain the weak and strong forces or the relative strength of gravity, their ideas seem to have come back into fashion.\n\nThe specific geometry of the extra dimensions varies in different models.  The work in this thesis considers phenomenology in large extra dimension models, but two other extra dimension scenarios are briefly discussed in this introductory chapter.\n\n\\subsection{Introducing different models and their phenomenology}\n\nBefore introducing the different extra dimension models we consider the wave equation for a scalar field in 5 dimensions; this is useful for demonstrating some of the main differences in phenomenology.  We consider the fifth dimension to be compactified on a circle of radius $R$ such that there is a periodic identification $x_5=x_5+2\\pi R$.  The scalar field equation for a particle of mass $m$ propagating in the bulk is\n\\begin{equation}\n\\left(\\frac{\\partial ^2}{\\partial t^2}-\\nabla^2_5+ m^2\\right)\\,\\Phi (x,y,z,x_5,t)=0\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\nabla^2_5=\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial y^2}+\\frac{\\partial ^2}{\\partial z^2}+\\frac{\\partial ^2}{\\partial x_5^2}\\,.\n\\end{equation}\nThe solution will be of the form\n\\begin{equation}\n\\Phi (x,y,z,x_5,t)=\\sum_{n \\in N} \\Phi_n(x,y,z,t)\\exp\\!\\left(\\frac{inx_5}{R}\\right),\n\\end{equation}\nwhich gives the following equation of motion:\n\\begin{equation}\n\\label{kk}\n\\sum_{n \\in \\mathbb{N}} \\left(\\frac{\\partial ^2}{\\partial t^2}-\\nabla^2_4+ m^2+\\frac{n^2}{R^2}\\right)\\Phi_n(x,y,z,t)\\,.\n\\end{equation}\nViewed from a four-dimensional perspective there is clearly a tower of states of different masses, with mass splitting $1\/R^2$;  this is the so-called Kaluza-Klein (KK) tower of states.  It is clear though that the Kaluza-Klein tower is simply a result of viewing a 5-dimensional model in 4-dimensions.  Taking a 5-dimensional view there is only one state of mass $m$ which in the equivalent spin-2 case will be the massless graviton.  \n\nAt this point it is worth pointing out another feature of most of these models: there is no requirement for momentum conservation in the extra-dimensions.  Energy-momentum conservation in the four-dimensional world is a consequence of the translational invariance of the four space-time dimensions.  However the 3-brane breaks translational invariance in the extra dimension and hence momentum is these directions need not be conserved in interactions between brane and bulk states.  In effect the brane will be considered as being infinitely heavy, or as being `nailed' in place.\\footnote{There are models in which this approximation is not valid, e.g.\\ \\cite{Murayama:2001av}.}\n\n\\subsubsection{`Large' extra dimensions}\n\\label{led}\n\nThe earliest extra dimensions models (at least in recent years) were those of Arkani-Hamed, Dimopoulos and Dvali (known as `large' extra dimension,\\footnote{The word `large' may seem to be something of a misnomer, but the extra dimensions are large relative to the fundamental energy scale in the models.} or ADD models \\cite{Arkani-Hamed:1998rs,Arkani-Hamed:1998nn,Antoniadis:1998ig}).  These are perhaps the simplest of the extra dimension models and are closest to the ideas of Kaluza and Klein.  \n\nGravity is the only force which propagates in the full volume of the extra dimensions (the bulk).  Hence the gravitational force in the four-dimensional world (on the brane) is `diluted' and gravity appears weak compared to the other forces which do not propagate in the extra dimensions.  In these models there is only one fundamental energy scale, the electroweak scale, and so there is no longer a fine tuning problem---$\\Lambda$ is now of order a TeV\\@.\n\\enlargethispage{-\\baselineskip}\n\nFor large extra dimensions, equation~(\\ref{kk}) shows that the mass splitting will be small, and all graviton modes (up to the mass scale involved in the process) will contribute to the process being considered.  Although each mode has a negligible contribution, the number of gravitons involved can be very large which means care must be taken that the total contribution isn't inconsistent with precision SM measurements.\n\nThere are two major classes of experimental signature: direct production of the KK excitations of gravitons (giving jets and missing energy) and virtual exchange of the KK excitations (modifying SM cross~sections and asymmetries).  The missing energy signatures are clearly different to those in other models with invisible particles (like SUSY) since from a four-dimensional point of view the graviton doesn't have a fixed mass---instead it appears to have a continuous mass distribution.\n\nThe work in the later chapters of this thesis is almost entirely in the framework of ADD models; as a result there is much further discussion of the details of these models in \\secref{add}.\n\n\\subsubsection{$\\mbox{TeV}^{-1}$ extra dimensions}\n\\label{tev-1}\n\nThe models with extra dimensions of size $\\mbox{TeV}^{-1}$ \\cite{Antoniadis:1990ew} are somewhat different to other models like those of ADD\\@.  These models do not themselves attempt to solve the hierarchy problem; the smaller size of the extra dimensions means that the volume of the bulk is not be large enough to explain the weakness of gravity (although there are models which include a mixture of large and $\\mbox{TeV}^{-1}$ extra dimensions).  In fact the extra dimensions are small enough to allow the gauge and Higgs bosons to propagate in the bulk without running into conflict with the constraints from the precision electroweak measurements made at LEP (for example the Z-width).\n\nAlthough they don't solve the hierarchy problem, there are various advantages of models like this which have the gauge and Higgs bosons propagating in the bulk.  Gauge bosons in the bulk provide a mechanism for early unification of the coupling constants because the logarithmic running is modified by the extra dimensions \\cite{Dienes:1998vg}.  If the Higgs field propagates in the bulk, the vacuum expectation value of the Higgs zero mode is able to generate spontaneous symmetry breaking.  Models with two branes with a $\\mbox{TeV}^{-1}$ separation can also suppress proton decay if the quarks are localized on one brane and the leptons on the other \\cite{Arkani-Hamed:1999za} (alternatively there may be a `thick' brane with quarks and leptons localized at different points but with the Higgs and gauge bosons propagating inside it \\cite{Arkani-Hamed:1999dc}).  Such models can also be useful to motivate the three generations of fermions and the hierarchy of fermion masses.\n\nIn these models, $R\\sim\\mbox{TeV}^{-1}$ and so, from equation~(\\ref{kk}), the mass splittings $M_\\text{c}$ are TeV-scale.  Precision electroweak data requires that $M_\\text{c} >$2--5~TeV, but the LHC should be able to probe $M_\\text{c}$ up to around 14~TeV by searching for leptonic signatures of KK excitations of the photon and the Z boson. \n\nA subset of these models which are slightly different are those of universal extra dimensions where all fields are allowed to propagate in the bulk \\cite{Appelquist:2000nn}.  These can be constructed to evade current electroweak limits and might provide a dark matter candidate \\cite{Cheng:2002ej}.  One particular feature of universal extra dimension models is that KK excitations are pair-produced; this could lead to some distinctive phenomenology at collider experiments.\n\n\\subsubsection{`Warped' extra dimensions}\n\nThe `warped' extra dimension models of Randall and Sundrum (RS) \\cite{Randall:1999ee,Randall:1999vf} solve the hierarchy problem in a different way to the ADD models.  Gravity is not `diluted' by the large volume of the bulk but by the strong curvature of the extra dimensions. RS models involve a second brane; gravity is strong on that brane but a `warp factor' in the metric ensures that gravity on our brane is weak.  These models may provide a way to explain why the cosmological constant seems to be so small.\n\nEquation~(\\ref{kk}) cannot strictly be applied in RS models (since the geometry is non-factorizable) but the extra dimensions are again small which corresponds to a large mass splitting.  This gives resonant graviton production where usually only the first graviton resonance is considered since the LHC is unlikely to be of high enough energy to study higher resonances.  Channels like $\\text{G}\\rightarrow\\text{e}^+\\text{e}^-$ and $\\text{G}\\rightarrow\\gamma\\gamma$ can be studied to find the mass of the first graviton resonance and to confirm its spin-2 nature \\cite{Allanach:2000nr}.\n\n\\section{Details of ADD models}\n\\label{add}\n\n\\subsection{Gauss' law in extra dimensions}\n\\label{gauss}\n\nWhilst it is relatively trivial to use Gauss' Law to derive the power dependence of the gravitational force in extra dimensions, slightly more care is required to determine constants of proportionality in the force law and to determine how these relate to the Planck scale.  The difficulties arise both because there are a number of different conventions in use and because it is not trivial to visualize geometries involving more than three spatial dimensions.  In all that follows in this thesis, $n$ toroidally compactified extra dimensions, all of radius $R$, are assumed.\\footnote{Much of the extra dimension phenomenology discussed in this and later chapters could be modified by compactification on more complicated manifolds.}\n\\enlargethispage{-\\baselineskip}\n\nIn extra dimensions, Gauss' Law can be written as\n\\begin{equation}\n\\int_s \\mathbf{F} \\cdot \\mathbf{dS} = \\Omega_{n+2} G_{(4+n)} M,\n\\end{equation}\nwhere $F$ is the force per unit mass, $G_{(4+n)}$ is the $(4+n)$-dimensional Newton constant,\\footnote{This is by no means unique as a way of defining the extra-dimensional Newton constant.} $M$ is the mass enclosed by the surface $S$ and $\\Omega_p$ (the area of a unit $p$-sphere) is given by\n\\begin{equation}\n\\Omega_p=\\frac{2\\pi^{\\frac{p+1}{2}}}{\\Gamma(\\frac{p+1}{2})}\\,.\n\\label{omegapdef}\n\\end{equation}\nAt short distances ($\\ll R$) the extra compactified dimensions are effectively flat and so we obtain\n\\begin{equation}\nF_{n+1}(r)=\\frac{G_{(4+n)} M}{r^{n+2}}\\,,\n\\end{equation}\nwhereas at long distances the usual 4-dimensional force law is obtained.\n\nThe above suggests that $G_{(4+n)} \\sim G_{(4)} R^n$ in order to get a smooth\nmatching between the two regimes.  However it is worth taking a little more care in deriving the correct relationship between these constants.  We follow the approach of ADD in one of their early papers \\cite{Arkani-Hamed:1998nn} and use the analogy of a line of point masses.  First consider an infinite two-dimensional line of masses, each of mass $m$ and separated by a distance $L$.  Provided we are at a large enough distance $r$ from the line, it will appear like a continuous line of mass and the problem has become two dimensional.\n\nUsing Gauss's law in three (spatial) dimensions and considering the flux through a cylinder of length $d$\nwe obtain\n\\begin{equation}\n(2 \\pi r d) F = 4\\pi G_{(4)} m \\left(\\frac{d}{L}\\right) \\longrightarrow F(r)=\\frac{2}{L} G_{(4)} \\frac{m}{r}\\,.\n\\end{equation}\nSo there is an effective three-dimensional Newton constant (applicable for two spatial dimensions) related to $G_{(4)}$ by\n\\begin{equation}\nG_{(3)}=\\frac{2}{L} G_{(4)}\\,,\n\\end{equation}\nwhich is more usefully written as\n\\begin{equation}\nG_{(3)}=\\frac{1}{L}\\frac{\\Omega_2}{\\Omega_1} G_{(4)}\\,.\n\\end{equation}\n\n\\enlargethispage{-\\baselineskip}\nThe situation with compactified extra dimensions is exactly analogous.  The periodic identification means that the equivalent uncompactified theory has lines of mirror masses $m$ with separation $L=2\\pi R$.  So for $r \\gg L$ there is an effective 4-dimensional Newton constant related to $G_{(4+n)}$ by\n\\begin{equation}\nG_{(4)}=\\frac{1}{L^n}\\frac{\\Omega_{n+2}}{\\Omega_2} G_{(4+n)}=\\frac{1}{L^n}\\frac{\\Omega_{n+2}}{4 \\pi} G_{(4+n)}\\,.\n\\label{g4ton}\n\\end{equation}\n\n\\subsection{Planck mass in extra dimensions}\n\nThe Planck mass is the scale at which gravity becomes strong, but there are varying definitions even in four dimensions.  Initially we follow the conventions of ADD who use a reduced Planck scale, $\\hat{M}_{(4+n)}^{n+2}$ given by\n\\begin{equation}\n\\hat{M}_{(4+n)}^{n+2}=\\frac{1}{G_{(4+n)}\\Omega_{n+2}}\\,,\n\\end{equation}\nand hence, using eq.~(\\ref{g4ton}),\n\\begin{equation}\n\\label{mplconv}\n\\hat{M}_{(4)}^2=(2 \\pi R)^n \\hat{M}_{(4+n)}^{n+2}=R^n M_{\\text{P}(4+n)}^{n+2}\\,,\n\\end{equation}\nwhere $M_{\\text{P}(4+n)}$ is the Planck mass in $4+n$ dimensions.  This is introduced because it is found to be this scale which is most directly constrained by experimental bounds and which is required to be $\\sim 1$~TeV in order to solve the hierarchy problem.  However in many ways the reduced Planck mass is more natural since the interactions of the canonically normalized field in the Lagrangian are suppressed by $\\sqrt{\\hat{M}_{(4+n)}}$\\,.  \n\nWe can always view interactions in two different ways.  From a 4-dimensional view-point the graviton coupling is very small ($\\sim 1\/\\sqrt{\\hat{M}_{(4)}}$) but many Kaluza-Klein modes contribute (for a process with energy scale $E$, there will be contributions from $\\sim (ER)^n$ modes).  However from a $(4+n)$-dimensional view-point the graviton coupling is much larger ($\\sim 1\/\\sqrt{\\hat{M}_{(4+n)}}$) but there is no tower of graviton states.\n\nAs mentioned above there are a number of other ways of defining the Planck scale, and at the same time a number of different definitions of the extra-dimensional Newton constant.  To compare most transparently with other conventions (which in part differ because of alternative definitions of the 4-dimensional Planck mass) equation~(\\ref{mplconv}) is more conveniently written as\n\\begin{equation}\n\\label{4pG}\nM_{\\text{P}(4+n)}^{n+2}=\\frac{1}{4 \\pi G_{(4)}}\\frac{1}{R^n}\\,.\n\\end{equation}\nFor ease of reference in later parts of this thesis, a summary of many of the popular conventions is shown in Table~\\ref{consum} (in all cases $n$ toroidally compactified extra dimensions, all of radius $R$, are assumed).\n\n\\begin{table}\n\\def1.0{1.5}\n\\begin{center}\n\\begin{tabular}{|l|c|l|}\n\\hline\n&Definition of $M_{\\text{P}(4+n)}^{n+2}$ & Comments\\\\\n\\hline\na & $\\frac{1}{4 \\pi G_{(4)}}\\frac{1}{R^n}$ & Used by ADD and also GT \\cite{Giddings:2001bu}\\\\\nb & $\\frac{1}{8 \\pi G_{(4)}}\\frac{1}{R^n}$ & Used by GRW \\cite{Giudice:2001ce}; often known as $M_*$\\\\\nc & $\\frac{1}{G_{(4)}}\\frac{1}{R^n}$ & Used by EOT-WASH \\cite{Adelberger:2002ic} \\\\\nd & $\\frac{1}{G_{(4)}}\\frac{1}{(2 \\pi R)^n}$ & Used by Dimopoulos and Landsberg \\cite{Dimopoulos:2001hw}\\\\\ne & $\\frac{1}{8 \\pi G_{(4)}}\\frac{1}{(2\\pi R)^n}$ & Used by Han, Kribs and McElrath \\cite{Han:2002yy}\\\\ \n\\hline\n\\end{tabular}\n\\capbox{Different conventions for the fundamental Planck scale}{Different conventions for the fundamental Planck scale $M_{\\text{P}(4+n)}$.  The last column indicates which conventions are used by the authors of some of the papers referenced in this thesis.\\label{consum}}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\subsection{Constraints on extra dimension models}\n\\label{edcon}\n\nThe differences between the conventions (particularly the first three) are usually relatively small since the conversion factors are close to unity for most allowed values of $n$.  When quoting astrophysical and cosmological limits where there are large uncertainties it is not crucial to be careful about which convention is being used.  However for collider limits the convention should be clearly stated.  For ease of comparison all the limits below are in the $M_*$ convention (convention `b' in Table~\\ref{consum}) which is also used by the Particle Data Group.\\footnote{These different conventions have unfortunately led to much confusion in the literature, and explain some apparent inconsistencies between the limits mentioned here and those in some other recent reviews, for example \\cite{Hewett:2002hv}.}\n\n\\subsubsection{Limits from short-scale gravity experiments}\n\\label{ssgrav}\n\nShort-scale gravity experiments provide limits on the size (i.e.\\ the compactification radius) of the extra dimensions and relations like those in Table~\\ref{consum} can be used to convert these to limits on the fundamental Planck scale (in a particular compactification scheme).  Seeking to find deviations from Newtonian gravity at short distances is difficult because at these distances gravity is not the dominant force, even between non-magnetic, electrically neutral interacting bodies.  This means it can be necessary to fully understand the Casimir force (taking into account finite conductivity, surface roughness and non-zero temperature corrections) and van der Waals forces which become important at these length scales.  There are also experimental considerations in seeking to isolate the experiments from mechanical, acoustic and thermal vibrations as well as electromagnetic fields.\n\nThe best constraints at present come from modern versions of the Cavendish torsion experiment.  These experiments now show no deviation from Newtonian gravity down to about 150~$\\mu$m \\cite{Adelberger:2002ic} which corresponds to $M_{*(4+2)} > 1.8$~TeV\\@.\n\n\n\\subsubsection{Limits from collider experiments and future sensitivity}\n\\label{collim}\n\nThe signature of real graviton production in the process $\\text{e}^{+}\\text{e}^{-}\\rightarrow \\text{G} \\gamma$ is a photon and missing energy.  Searches in this channel at the LEP collider were able to set limits on the fundamental Planck scale, usually given in the $M_*$ convention.  The best published limits range from $M_{*(4+2)} > 1.1$~TeV through $M_{*(4+4)} > 0.68$~TeV to $M_{*(4+6)} > 0.51$~TeV \\cite{Abreu:2000vk}, but some reviews quote values from the later higher energy running at LEP which gives $M_{*(4+2)} > 1.38$~TeV and $M_{*(4+6)} > 0.58$~TeV (DELPHI collaboration, quoted in \\cite{Landsberg:2001ma}) or even $M_{*(4+2)} > 1.45$~TeV and $M_{*(4+6)} > 0.61$~TeV (L3 collaboration, in \\cite{Hewett:2002hv}).\n\nThe Tevatron Run I limits only become competitive with the LEP II limits for the higher values of $n$.  The best limits from the Tevatron so far come from the CDF collaboration; they find, at 95\\% confidence level, that $M_{*(4+2)} > 1.00$~TeV but that $M_{*(4+6)} > 0.71$~TeV \\cite{Acosta:2003tz}.  These limits are obtained from a study of the `one or two high-energy jets + missing tranverse energy' signature (from processes like $\\text{gg}\\rightarrow \\text{Gg}$, $\\text{qg}\\rightarrow \\text{Gq}$ and $\n\\text{q}\\bar{\\text{q}}\\rightarrow \\text{Gg}$).  This is found to give a better sensitivity than the `photon + missing $E_\\text{T}$' signature (from $\\text{q}\\bar{\\text{q}}\\rightarrow \\text{G} \\gamma$)\\footnote{The `Z+ missing $E_\\text{T}$' signature is less significant because the Z can only be observed via its leptonic decay.} since rates are much lower than in the jet case.  However this signature could be used as confirmation in the event of a discovery in the jet channel.  It is expected that the Tevatron Run II will be able to improve slightly upon these limits.\n\nIt should be possible to reach much larger values of $M_*$ by studying the single-jet signature at the LHC (the dominant sub-process is $\\text{qg}\\rightarrow \\text{Gq}$).  With 100~$\\mbox{fb}^{-1}$ of integrated luminosity, the ATLAS collaboration expects to be able to probe $M_*$ in the following ranges: $M_{*(4+2)}=$4.0--9.1~TeV, $M_{*(4+3)}=$4.5--7.0~TeV and $M_{*(4+4)}=$5.0--6.0~TeV\\cite{Vacavant:2001sd}.  The lower limits reflect the fact that the theory is only an effective one and breaks down at energies $\\sim M_*$.  If the fundamental Planck scale is so low that a significant number of events would have $\\sqrt{\\hat{s}}>M_*$ then it is not possible to accurately extract information.  This also means that nothing can be said for the $n>4$ cases---in effect the LHC would have too high a centre-of-mass energy (this isn't an issue in the pessimistic case in which we are just setting exclusion limits on $M_*$).  The upper limits are due to the signal no longer being observable over the SM background.  In the ranges given, both $M_*$ and $n$ could be extracted at the LHC, but this would require running at two different centre-of-mass energies.\n\nSome limits can also be placed on ADD models by considering virtual graviton emission.  However the sum of KK states involved is found to be divergent; this means a cut-off has to be introduced which makes the process sensitive to ultra-violet effects.  Hence although it is possible to place limits on this cut-off scale (in various different conventions) it is not possible to convert them to direct limits on $M_*$.\n\n\\subsubsection{Limits from cosmic rays}\n\\label{crlim}\n\nModels with large extra dimensions would allow the production of miniature black holes at TeV-scale energies---a topic which will be properly introduced in Chapter~\\ref{bhintro}.  Ultra high-energy cosmic rays interact with centre-of-mass energies up to $\\sim$~400 TeV and so could produce black holes; their decays could then be observed by experiments like AGASA\\@.\\footnote{This provides reassurance to those who are unconvinced by the theoretical arguments and worry that black holes produced at the LHC might destroy the world.}  Although there are some uncertainties involved, ref.~\\cite{Anchordoqui:2001cg} finds limits on the fundamental scale which, for the larger values of $n$, are more restrictive than present collider limits (e.g.\\ $M_{*(4+4)}>$1.3--1.5~TeV and $M_{*(4+7)}>$1.6--1.8~TeV).\n\n\\subsubsection{Astrophysical and cosmological constraints}\n\\label{astcon}\n\n\\enlargethispage{\\baselineskip}\nSome of the strongest constraints on these extra-dimensional models are astrophysical and cosmological.  These put direct lower bounds on $M_{\\text{P}(4+n)}$ which cannot absolutely rule out extra dimension models but make them unattractive for two reasons:  firstly the whole motivation for the theory is lost if the fundamental Planck scale is constrained to be significantly larger than the electroweak scale; secondly there would be no obvious phenomenology at the next generation of colliders.\n\nThere are however various ways of managing values of $M_{\\text{P}(4+n)}$ up to of order 10~TeV\\@.  This is because although $\\Lambda\\sim M_{\\text{P}(4+n)}$, the exact form of this relationship depends on the theory being proposed.  For example if the extra dimension model is implemented within a type-I string theory, it can be shown that the scale of new physics ($\\Lambda$) is given by the string scale ($M_\\text{s}$) but that this can naturally be an order of magnitude smaller than $M_{\\text{P}(4+n)}$.  Hence the hierarchy problem is naturally resolved, and new physics is expected at energy scales accessible at the LHC\\@.\n\nThe first limits we consider here are those from the supernova remnant SN1987A\\@.  The rate at which the remnant can lose energy into the extra dimensions by graviton emission must not violate the observed cooling rate which is assumed to be mainly due to neutrinos (the flux of which has been measured by Kamiokande and IMB).  The predominant mechanism for graviton emission is known as `gravi-strahlung' ($\\text{N}+\\text{N}\\rightarrow \\text{N}+\\text{N}+\\text{G}$) and is enhanced over other processes by strong interaction effects.  The graviton emission rate is found to be proportional to $1\/(M_{\\text{P}(4+n)})^{n+2}$ so this can be used to set a limit on the fundamental Planck scale.  Using the $M_*$ convention once again this gives $M_{*(4+2)} >$25--45~TeV and $M_{*(4+3)} >$2--4~TeV \\cite{Cullen:1999hc,Hanhart:2000er} depending on the assumptions made and exactly what value is used for the supernova temperature. \n\nThere are also limits on the fundamental Planck scale from the distortion of the cosmic diffuse background radiation spectrum.  The gravitons produced both by gravi-strahlung and by neutrino annihilation can decay into SM particles and for the low-mass KK modes $\\text{G}\\rightarrow \\gamma\\gamma$ is kinematically favoured.\nObservations of the cosmic diffuse gamma radiation by EGRET and COMPTEL show that gravi-strahlung must account for less than about 1\\% of the total energy emitted by all the supernovae that have exploded during the history of the universe.  This is calculated to set limits of $M_{*(4+2)} > 70$~TeV and $M_{*(4+3)} > 5$~TeV \\cite{Hannestad:2001jv} (more recent calculations of the energy emitted by KK states trapped in neutron star haloes suggest $M_{*(4+2)} > 380$~TeV and $M_{*(4+3)} > 24$~TeV \\cite{Hannestad:2001xi}).\n\nOther authors have considered the contributions to the gamma radiation background from the gravitons produced by neutrino annihilation and obtained the limits $M_{*(4+2)} > 90$~TeV and $M_{*(4+3)} > 4$~TeV \\cite{Hall:1999mk}.  Tighter limits can be obtained by assuming that the universe doesn't enter the radiation epoch until after it has been reheated by the decay of a massive scalar field (or by some other means of entropy production).\n\nOther limits are derived by ensuring that the production of gravitons will not cause the universe to become matter dominated too early---this would lead to a value for the age of the universe which was too low.  The limits are $M_{*(4+2)} > 70$~TeV and $M_{*(4+3)} > 6$~TeV \\cite{Fairbairn:2001ct} with the assumption that the universe can be considered as `normal' up to temperatures of 100~MeV\\@.\n\nFinally there have been Hubble Space Telescope observations that the surface temperatures of some older neutron stars are higher than conventionally expected.  The explanation for this could be that they are heated by the decay products of gravitons trapped in the neutron star haloes during the supernova collapse.  However the requirement that this heating is not excessive puts limits on the fundamental scale: $M_{*(4+2)} > 1400$~TeV and $M_{*(4+3)} > 50$~TeV \\cite{Hannestad:2001xi}.  The equivalent limit for $n=4$ is of order a few TeV\\@.  These are clearly the most constraining of the bounds on the fundamental scale although they are subject to various cosmological uncertainties.  Even so, $n=2$ almost certainly seems to be ruled out and $n=3$ is disfavoured.\n\n\n\n\n\n\\chapter{Exotic Heavy Leptons at the LHC}\n\\chaptermark{Heavy Leptons}\n\\label{leptons}\n\\section{Introduction}\n\nWe have already seen in Chapter~\\ref{introch} that supersymmetry is a popular theory for new physics beyond the Standard Model.  It is seen as theoretically attractive, solves the hierarchy problem and improves the unification of the three gauge couplings at high energy scales.\n\n`Intermediate scale' models which can be motivated on various astrophysical and cosmological grounds (see \\secref{ism}) are constructed so that the gauge couplings unify at $\\sim 10^{11}$~GeV, an intermediate energy scale.  This intermediate scale unification can be achieved by including new leptons as part of extra supermultiplets added to the MSSM\\@.  Although \nthis choice is not unique, the extra leptons in \\cite{Abel:2000bj} are two left-handed SU(2) doublets and three right-handed singlets (all with vector-like copies).\n\\enlargethispage{-\\baselineskip}\n\nBy assumption there are no new Yukawa couplings for the model in \n\\cite{Abel:2000bj}.  This means that the lightest heavy\nlepton is stable and the others will decay into it.  This is possible because\nthe tree-level mass degeneracy of the charged and neutral leptons will be \ndestroyed by electroweak symmetry breaking.\n  \nAs in the MSSM, the renormalization group equations can be used to determine \nthe spectrum of supersymmetric particles which will have phenomenological \nimplications for the LHC\\@.  A study has also been performed \\cite{Allanach:2001qe} investigating how to distinguish intermediate scale models from other supersymmetric string scenarios at a future linear collider.  However a characteristic feature of this model is the existence of the new leptons and so this work concentrates on the phenomenology due to the extra leptons themselves.  More specifically it is considered whether it will be possible to detect the charged heavy leptons at the LHC, and for what range of masses.    \n\nPrevious limits on the masses of new quasi-stable charged leptons have \nbeen limited by the energy available for particle production in lepton \ncolliders.  The gauge unification arguments in intermediate scale models mean \nthat the extra leptons are expected to have masses in the TeV range.  At the \nLHC enough energy should be available to produce and detect these exotic heavy\ncharged leptons.  Since the charged leptons are expected to be quasi-stable they will decay outside the detector; therefore the experimental procedure is different to that in \\cite{Alexa:2001fz,CiezaMontalvo:2001cy} where limits are obtained by considering the decay products of the leptons.\n\nThis chapter starts (\\secref{litrev}) by reviewing the existing cosmological and experimental limits on heavy particles to ensure that leptons such as those incorporated in this intermediate scale model are not ruled out.  In section~\\ref{HERWIG} the theoretical input to the {\\small HERWIG} Monte Carlo event generator to take account of new heavy leptons is described.  Then, in \\secref{impl}, the use of a `time-of-flight' technique to detect the leptons at the LHC is considered: heavy leptons will arrive at the detector significantly later than relativistic particles, and the prospects for using this time delay as a method of detection are studied, for a range of masses.  There is also a study of how to distinguish such leptons from scalar leptons on the basis of their different angular distributions.  The results and conclusions are presented in \\secref{results} and \\secref{conchl} respectively.\n\n\\section{Cosmological and experimental constraints}\n\\label{litrev}\n\nThe mass degeneracy of the charged and neutral leptons will be broken by \nradiative corrections.  The electromagnetic self-energy correction for the \ncharged lepton ensures that its mass will be greater than that of the neutral \nlepton\\cite{Sher:1995tc}.  This means that the charged lepton can decay to its \nneutral partner producing either a real or virtual W, depending on the mass \ndifference.  The neutral heavy lepton can be assumed to be totally stable \n(i.e.\\ it has a lifetime orders of magnitude longer than that of the \nuniverse) because of the assumption that there are no new Yukawa couplings.\n\nThere are a number of cosmological and experimental limits on leptons \n(particularly charged leptons) as a function of both mass and lifetime:\n\n\\begin{itemize}\n\n\\item\nThe relic abundance of the leptons (calculated from the self-annihilation \ncross section) must not `over-close' the universe, i.e.\\ provide more \nthan the critical energy density ($\\sim 10^{-5}$ GeV cm$^{-3}$) which is \npresumed to be accounted for by dark matter\\cite{Wolfram:1979gp,Griest:1990wd}.\n\n\\item\nA stable, charged lepton must have a low enough relic abundance for it not to \nhave been detected in searches for exotic heavy isotopes in ordinary matter \n(see e.g.\\ \\cite{Smith:1982qu}).\n\n\\item\nThe massive lepton and any decays it may have must not significantly affect\nnucleosynthesis or the synthesized elemental abundances \\cite{Ellis:1992nb,Sarkar:1996dd}.\n\n\\item\nIf the lepton decays before the recombination era (at $\\sim 10^{12}$~s) it \nmust not distort the cosmic microwave background radiation \\cite{Ellis:1992nb}.\n\n\\item\nIf the lepton has a longer lifetime it must not contradict the limits from\ngamma-ray and neutrino background observations \\cite{Kribs:1997ac,Gondolo:1993rn}.\n\n\\item\nThe mass and lifetime of the new leptons must not be such that they would \nhave been detected in a previous collider experiment \\cite{Abbiendi:2003yd,Achard:2001qw}.\n\n\\end{itemize}\n\nMost of these limits are summarized in \\cite{Sher:1992yr} and no constraints on charged leptons are found for lifetimes less than $\\sim 1$~s, even for \nmasses up to the TeV scale.  The mass splitting between the neutral and \ncharged leptons is expected to be at least $\\sim 10$~MeV \\cite{Sher:1995tc} which means that the lifetime of the charged lepton is expected to be shorter than 1~s.\n\nThis means the only relevant constraints on the mass of the heavy leptons being studied are the experimental limits.  These are to some extent model-dependent, but the LEP collaborations put a lower limit (95\\% confidence) on the mass of a long-lived charged lepton at around 102 GeV \\cite{Abbiendi:2003yd,Achard:2001qw} (the Tevatron experiment has not improved upon the LEP limits).\n\n\\section{Theoretical input to {\\small HERWIG}} \n\\label{HERWIG}\n\n\\enlargethispage{-\\baselineskip}\nThe general-purpose Monte Carlo event generator {\\small HERWIG 6.3} includes \nsubroutines for both neutral and charged Drell-Yan processes in a hadron \ncollider \\cite{Marchesini:1992ch,Corcella:2000bw,Corcella:2001pi}.  As described in \\cite{Corcella:2000bw} the \ninitial-state parton showers in Drell-Yan processes are matched to the exact \n$\\mathcal{O}(\\alpha_\\text{S})$ matrix element result \\cite{Corcella:1999gs}.  However the \nneutral Drell-Yan processes all use the approximation that the two fermions \nproduced can be treated as massless.  It is reasonable that this is a valid \napproximation for the SM quarks and leptons but this will not necessarily be \nthe case for the proposed new heavy leptons discussed above which may have \nmasses of order 1~TeV\\@.\n\nFor this reason, full Born-level expressions for the Drell-Yan cross sections \nwere derived, taking into account the masses of the produced leptons.   The final results (for both neutral- and charged-current cases) were expressed in terms of vector and axial couplings to keep them as general as possible, although in all that follows Standard Model couplings will be assumed.  The details of the derivation are in Appendix~\\ref{appa}\n\n\\begin{figure}\n\\begin{center}\n\\begin{picture} (90,60)\n\\ArrowLine(30,35)(0,60)\n\\Text(-3,60)[r]{$\\bar{\\text{q}}$}\n\\ArrowLine(0,10)(30,35)\n\\Text(-3,10)[r]{q}\n\\Photon(30,35)(90,35){2}{8}\n\\Text(60,46)[c]{Z$^0$\/$\\gamma$}\n\\ArrowLine(120,60)(90,35)\n\\Text(123,60)[l]{L$^+$}\n\\ArrowLine(90,35)(120,10)\n\\Text(123,10)[l]{L$^-$}\n\\end{picture}\n\\capbox{Feynman diagram for Drell-Yan production of heavy leptons}{Feynman diagram for Drell-Yan production of heavy leptons.\\label{hlfeyn}}\n\\end{center}\n\\end{figure}\n\nThe derived expressions for both differential and total cross sections for the\nneutral-current Drell-Yan processes \n($\\text{q}\\bar{\\text{q}}\\rightarrow \\text{Z}^0\/\\gamma\\rightarrow \\text{L}^-\\text{L}^+$, see Figure~\\ref{hlfeyn}) are shown below.  The \nnotation is influenced by that already used in subroutines within \n{\\small HERWIG 6.3}.\n\\enlargethispage{-\\baselineskip}\n\nThe differential cross section is given by\n\\begin{equation}\n\\frac {d\\hat{\\sigma}}{d\\Omega} (\\text{q}\\bar{\\text{q}}\\rightarrow \\text{L}^-\\text{L}^+) \n= \\frac{e^4}{48\\pi^2} \\frac {1} {\\hat{s}^2} \\frac {p_3} {E}\n\\left[ C_1 \\left( E_3E_4 + p_3^2\\cos^2\\theta^*\\right) + C_2 m_3 m_4 +\n2 C_3 E p_3 \\cos\\theta^* \\right],\n\\label{dxs}\n\\end{equation}\nwhere\n\\begin{gather}\nC_1 = \\frac {\\left[\\left(d_\\text{V}^\\text{f}\\right)^2 + \\left(d_\\text{A}^\\text{f}\\right)^2 \n\\right] \\left[ \\left(d_\\text{V}^\\text{i}\\right)^2 + \\left(d_\\text{A}^\\text{i}\\right)^2 \\right]\\hat{s}^2 } \n{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \\left(q^\\text{f}\\right)^2\n\\left(q^\\text{i}\\right)^2 + \\frac {2 q^\\text{f} q^\\text{i} d_\\text{V}^\\text{f} d_\\text{V}^\\text{i}\\hat{s}\n\\left(\\hat{s}-m_\\text{Z}^2\\right)}{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,,\n\\\\[3mm]\nC_2 = \\frac {\\left[\\left(d_\\text{V}^\\text{f}\\right)^2 - \\left(d_\\text{A}^\\text{f}\\right)^2 \n\\right] \\left[ \\left(d_\\text{V}^\\text{i}\\right)^2 + \\left(d_\\text{A}^\\text{i}\\right)^2 \\right]\\hat{s}^2 } \n{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \\left(q^\\text{f}\\right)^2\n\\left(q^\\text{i}\\right)^2 + \\frac {2 q^\\text{f} q^\\text{i} d_\\text{V}^\\text{f} d_\\text{V}^\\text{i}\\hat{s} \n\\left(\\hat{s}-m_\\text{Z}^2\\right)}{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,,\n\\\\[3mm]\nC_3 = 2 \\left( \\frac {2 d_\\text{V}^\\text{f} d_\\text{A}^\\text{f} d_\\text{V}^\\text{i} d_\\text{A}^\\text{i}\\hat{s}^2} \n{\\left(\\hat{s}-m_\\text{Z}^2 \\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \n\\frac {q^\\text{f} q^\\text{i} d_\\text{A}^\\text{f} d_\\text{A}^\\text{i} \\hat{s} \\left(\\hat{s}-m_\\text{Z}^2\\right)} \n{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} \\right).\n\\end{gather}\nIn these expressions $q^\\text{i}$ and $q^\\text{f}$ are the charges (in units of\nthe electron charge) of the initial- and final-state particles respectively.  \nThe angle $\\theta^*$ is the angle between the out-going lepton (L$^-$) \ndirection and the incoming quark direction in the centre-of-mass frame.  \n$E_3$ and $p_3$ refer to the centre-of-mass energy and momentum magnitude\nof the produced lepton.  Similarly the subscript 4 refers to the produced\nanti-lepton. $E$ is the energy of both the colliding quarks in the \ncentre-of-mass frame (i.e.\\ $\\hat{s}=4E^2$).  \n\nThe couplings $d_\\text{V}$ and $d_\\text{A}$ are related to the normal vector and axial \ncoupling constants to the Z$^0$ ($c_\\text{V}$ and $c_\\text{A}$) by relations like\n\\begin{equation}\nd_\\text{V} = \\frac {c_\\text{V} g_\\text{Z}} {2e}\\,.\n\\end{equation}\nEquation~(\\ref{dxs}) is a general expression which includes the Z$^0\/\\gamma$ interference terms; the massless case is retrieved by\nsetting $p_3=E_3=E_4=E$ (as well as $m_3=m_4=0$).  The normal charged-current Drell-Yan case ($\\text{q}\\bar{\\text{q}}^\\prime \\rightarrow \\text{W}^\\pm\\rightarrow \n\\text{L}^\\pm \\text{L}^0$) will not be relevant in this work but can be obtained by setting $c_\\text{V}=c_\\text{A}=1$ and replacing $g_\\text{Z}$,\n$m_\\text{Z}$, $\\Gamma_\\text{Z}$ and $q^\\text{i\/f}$ by $g_\\text{W}\/\\sqrt{2}$, $m_\\text{W}$, $\\Gamma_\\text{W}$ and 0 respectively.\n\nIntegration of eq.~(\\ref{dxs}) gives the following expression for the parton-level cross section:\n\\begin{equation}\n\\hat{\\sigma} (\\text{q}\\bar{\\text{q}}\\rightarrow \\text{L}^-\\text{L}^+)= \n\\frac {e^4} {12\\pi} \\frac {1} {\\hat{s}^2} \\frac {p_3} {E} \n\\left[ C_1 \\left(E_3E_4 + \\frac {p_3^2} {3}\\right) + C_2 m_3 m_4 \\right].\n\\label{xs}\n\\end{equation}\n\nIn order to distinguish these heavy leptons from heavy supersymmetric \npartners of SM particles it would be necessary to consider the angular \ndistribution of the produced particles.  For the neutral-current production of\nleft\/right-handed scalars the differential cross section is also derived in Appendix~\\ref{appa} and is found to be\n\\begin{equation}\n\\frac {d\\hat{\\sigma}}{d\\Omega}(\\text{q}\\bar{\\text{q}}\\rightarrow\\tilde{\\text{l}}_\\text{L\/R}\n\\tilde{\\text{l}}^*_\\text{L\/R})=\\frac {e^4} {96\\pi^2} D \\frac {1}{\\hat{s}^2} \\frac {p_3} \n{E} p_3^2 \\sin^2\\theta^*,\n\\label{sdxs}\n\\end{equation}\nwhere\n\\begin{equation}\nD = \\frac {h_\\text{L\/R}^2 \\left[\\left(d_\\text{V}^\\text{i}\\right)^2 +\\left(d_\\text{A}^\\text{i}\\right)^2 \\right] \n\\hat{s}^2} {\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2} + \n\\left(q^\\text{f}\\right)^2 \\left(q^\\text{i}\\right)^2 +\n\\frac {2 q^\\text{f} q^\\text{i} h_\\text{L\/R} d_\\text{V}^\\text{i} \\hat{s}\\left(\\hat{s}-m_\\text{Z}^2\\right)}\n{\\left(\\hat{s}-m_\\text{Z}^2\\right)^2 + m_\\text{Z}^2\\Gamma_\\text{Z}^2}\\,.\n \\end{equation}\nThe couplings $d_\\text{V}$ and $d_\\text{A}$ are defined as above and $h_\\text{L\/R}= g_\\text{L\/R}g_\\text{Z}\/2e$\nwhere $g_\\text{L\/R}$ is the coupling to left\/right-handed sleptons at the gauge \nboson-slepton-slepton vertex.\n\nThe $\\sin^2\\theta^*$ angular distribution contrasts with the asymmetric \ndistribution for the heavy leptons seen in eq.~(\\ref{dxs}); this will \nbe studied in \\secref{angdist}.\n\n\\section{Implementation}\n\\label{impl}\n\n\\subsection{Modifications to {\\small HERWIG 6.3}}\n\nThe above formulae were incorporated into a new subroutine added to \n{\\small HERWIG 6.3} together with new particle entries (for the heavy \nleptons) and a new process code (\\texttt{IPROC}) for the new process.\n\nIncreasing the mass of the new leptons allowed the variation of cross section\nwith mass to be studied---the mass was varied over a range from of order the \ntop quark mass to a maximum of order 1~TeV\\@.  The lifetimes of the charged \nleptons produced were set to 1~s so that the number of decays occurring inside\nthe detector will be negligible.\n\nThe angular distribution of the produced leptons was studied over the same \nmass range to investigate the possibility of distinguishing heavy leptons \nfrom MSSM sleptons.\n\n\\subsection{Time-of-flight technique}\n\\label{tof}\n\nThe main aim was to consider the possible detection of charged heavy leptons \nat the LHC and for what mass ranges this might be practical.  This work \nrefers specifically to the technical specifications of the ATLAS detector \n\\cite{ATLASTDR1,ATLASTDR2,ATLASmTDR}; however it is believed that similar results will\nbe obtained for the {\\small CMS} experiment \\cite{CMS}.\n\nThe method used was a time-of-flight technique as discussed in \nref.~\\cite{Hinchliffe:1998ys}.  This method utilizes the fact that, when compared to \nrelativistic particles, there is a considerable time delay for heavy particles\nto reach the muon system.  Heavy charged leptons like those being considered \nin this work will be detected in both the central tracker and the muon \nchambers, and from the measured momentum and time delay it is possible to \nreconstruct the mass.\n\nImperfections in the time and momentum resolutions will broaden the mass peak.  Uncertainty over which bunch crossing a \nparticular detected particle comes from may also provide a background signal \n(from muons produced in Drell-Yan processes or in heavy quark decays).\nPlots from \\cite{ATLASmTDR} show some differential cross sections for the processes most likely to produce muons which might be mis-identified as heavy leptons.  These are reproduced in Figures~\\ref{muonpT} and \\ref{muonbg}.\n\n\\begin{figure}\n\\unitlength1cm\n\\begin{minipage}[t]{3.05in}\n\\begin{center}\n\\includegraphics[width=\\linewidth]{muonpT.eps}\n\\capbox{Transverse momentum dependence of the inclusive muon cross section}{Transverse momentum dependence of the inclusive muon cross section \nintegrated over $|\\eta|< 2.7$.  The horizontal scale is the transverse \nmomentum at production. (Reproduced from \\cite{ATLASmTDR}).\\label{muonpT}}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\begin{center}\n\\includegraphics[width=\\linewidth]{muonbg.eps}\n\\capbox{Rapidity-dependence of the inclusive muon cross sections}{Rapidity-dependence of the inclusive muon cross sections, integrated\nover \\mbox{3 $<p_\\text{T}<$ 50~GeV}. (Reproduced from \\cite{ATLASmTDR}).\n\\label{muonbg}}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nThe studies presented here do not attempt to model the background, which it is\nnot thought will be significant (there is no `physics' background, only \n`detector' background as discussed above), and use a simple cut \\cite{Hinchliffe:1998ys} on the range of time delays allowed.\n\nThe calculation of a reconstructed mass $m$ from the time delay $\\Delta t$ and \nmomentum is straightforward.  For a lepton hitting the radial part of the \nmuon spectrometer, the time delay with respect to a relativistic \n($\\beta=v\/c=1$) particle is given by\n\\begin{equation}\n\\Delta t = \\frac {r} {p_\\text{T}} \\left(E-p\\right), \n\\end{equation}\nwhere $r$ is the radius of the outer layer of the muon system, \n$p_\\text{T}$ the transverse momentum, and $E$ and $p$ the total energy and momentum \nrespectively.  Substituting for $E$ in the energy-momentum invariant \n$E^2-p^2=m^2$ gives the result\n\\begin{equation}\nm^2 = p^2 \\left( \\frac {1}{\\beta^2} - 1 \\right)\n= \\frac {p_\\text{T} \\Delta t} {r} \\left(2p + \\frac {p_\\text{T} \\Delta t} {r} \\right).\n\\label{masssq}\n\\end{equation}\nIt is also necessary to check for occasions when the lepton hits the\nend-cap of the detector by considering the magnitude of $p_\\text{T}\/p_z$\\@.\nThe muon system is modelled approximately as a cylinder of radius 10~m and with a half-length of 20~m \\cite{ATLASmTDR}.  A pseudo-rapidity cut requiring that \n$|\\eta|<2.7$ was applied to take account of the region close to the beam where\nparticles cannot be detected.\n\nFrom equation~(\\ref{masssq}), the time delay and measured momentum for \nany particle detected in the muon system can be used to calculate its mass.  \nAs in \\cite{Hinchliffe:1998ys} the time delay was smeared with a Gaussian (although a width \nof 0.7~ns  \\cite{ATLASmTDR} rather than 1~ns was used) and a cut on \nthe smeared time delay was applied such that \n$10 \\mbox{ ns} < \\Delta t < 50\\mbox{ ns}$.  Increasing the lower limit \non $\\Delta t$ would  \nreduce the efficiency but improve the mass resolution by removing many of the \nhigh-$\\beta$ leptons for which the time resolution is poor.\nThe upper limit eliminates very slow particles which lose most of their energy\nin the calorimeter.  An alternative upper limit of 25~ns, reflecting practical\nconcerns, is discussed in \\secref{masspeaks}.  \n\nThe most important contribution to the momentum uncertainty $\\Delta p$ is due\nto the measurement error on the sagitta (the deviation from a straight line of\na charged particle in a magnetic field).  $\\Delta p_\\text{sag}$ was taken into \naccount by using \n\\begin{equation}\n\\frac {\\Delta p_\\text{sag}}{p^2} = 1.1\\times10^{-4}, \n\\label{sagerror}\n\\end{equation}\nwhere $p$ is the total momentum in GeV\\@.  The constant in \neq.~(\\ref{sagerror}) was obtained from \\cite{ATLASTDR1} as an average over the\ndifferent parts of the ATLAS detector.  This expression should be valid near \nthe discovery limit for heavy leptons when multiple scattering is \ninsignificant---the maximum time delay cut is found to remove most of the low-$\\beta$ leptons for which multiple scattering would have more effect.  However\nto allow investigations over a full mass range a multiple scattering term \n($\\Delta p_\\text{ms}$) was also incorporated with  \n\\begin{equation}\n\\Delta p_\\text{ms} = 2\\times10^{-2}\\sqrt{p^2 + m^2}.\n\\label{mserror}\n\\end{equation}\nThe constant depends on the material distribution in the \nspectrometer---the above value is given in \\cite{ATLMUON} for the ATLAS muon \ndetector.\n\\enlargethispage{-\\baselineskip}\n\nThe two errors in eqs.~(\\ref{sagerror}) and (\\ref{mserror}) were taken to be \nindependent and hence combined in quadrature.  A third possible contribution \nto momentum resolution---due to fluctuation in the energy loss in the \ncalorimeter---is neglected as it is always dominated by one of the other \nterms.\n\nUsing the momentum and time resolution as described above it was possible to \nsatisfactorily reproduce the mass resolution as a function of $\\beta$ for \n101~GeV particles as given in \\cite{ATLASTDR1}.\n\nThe efficiency of the muon detection system is approximated to be 85\\%,\nindependent of the particle momentum \\cite{ATLASTDR1}.\n\n\\section{Results}\n\\label{results}\n\n\\subsection{Backgrounds}\n\\label{bg}\n\nAny `detector' background is greatly dependent on the \ndetails of the muon detection system.  Such a background would come from muons \nproduced either in the decays of heavy quarks or directly via the Drell-Yan process (both these processes \nhave very high cross sections compared to that for heavy lepton production).  \nIf timing inadequacies mean that some of these muons are mistakenly thought\nto have come from an earlier interaction, i.e.\\ they appear to have a \nlarge time delay, they could obscure the exotic lepton mass peak. For a \nbackground signal to look like a heavy lepton neutral-current signal, however,\ntwo opposite charge muons would have to be mis-identified at the same time \nwhich makes it very unlikely that background could be significant. \n\\enlargethispage{-2\\baselineskip}\n\nTo eliminate any possible background from heavy quark decays, a cut requiring \nthe transverse momentum to be greater than 50~GeV was made to produce the \nplots in \\secref{masspeaks}.  Figures~\\ref{bp} and \\ref{p250} show that a\n$p_\\text{T}$ cut at 50~GeV should remove most of the muons from bottom \nquark decay without significantly reducing the exotic heavy lepton signal.  It\ncan be seen that as the mass of the exotic heavy lepton increases, the peak in \nthe $p_\\text{T}$ spectrum becomes less well defined and shifts to higher values.  Top\nquark decays can give rise to muons at high $p_\\text{T}$ but the small top \ncross~section, combined with the improbability of mis-identifying two muons,\nshould suffice to eliminate this source of background. \n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.5}{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]{extra.ps}}}\n\\capbox{$p_\\text{T}$ spectrum for muons produced in b decays}{$p_\\text{T}$ spectrum for muons produced in b decays.\\label{bp}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\unitlength1cm\n\\begin{center}\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{pplot7.ps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{pplot11.ps}}}\n\\end{minipage}\n\\capbox{$p_\\text{T}$ spectra for 175 and 250~GeV heavy leptons from Drell-Yan \nprocesses}{$p_\\text{T}$ spectra for 175 and 250~GeV heavy leptons from Drell-Yan \nprocesses.\n\\label{p250}}\n\\end{center}\n\\end{figure}\n\n\\subsection{Heavy lepton mass peaks}\n\\label{masspeaks}\n\nThe results presented in this section show the reconstructed mass peaks for \nan integrated luminosity of 100~fb$^{-1}$ after the cuts outlined above have \nbeen applied. \n\nThe possibility of detection at the LHC is found to be entirely cross~section \nlimited---the decrease of cross~section with increasing lepton mass is shown\nin Figure~\\ref{cross}.  The fraction of leptons passing the cuts actually \nslightly increases with mass due to the longer time delays.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.48}{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]\n{cross.ps}}}\n\\capbox{Cross~section for Drell-Yan heavy lepton production}{Cross~section for Drell-Yan heavy lepton production as a function of\nthe lepton mass.\\label{cross}}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{1000} for the neutral-current Drell-Yan process makes it\nclear that for masses up to about 1~TeV it should be possible to detect new\ncharged heavy leptons at the LHC (particularly if a larger integrated \nluminosity can be obtained).  A discovery criterion of 10 leptons (from 5 \nevents) in the mass peak is found to give a mass limit of 950~GeV for the \ndetection of such leptons using an integrated luminosity of 100~fb$^{-1}$.\n\\enlargethispage{\\baselineskip}\n\nTo reduce possible backgrounds the plots are based on events from \nwhich both leptons produced passed the cuts imposed.\n\n\\begin{figure}[t]\n\\begin{center}\n\\unitlength1cm\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{2plot175ms.ps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{2plot500ms.ps}}}\n\\end{minipage}\n\n\\unitlength1cm\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{2plot750ms.ps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.6}{\\rotatebox{-90}{\\includegraphics{2plot900ms.ps}}}\n\\end{minipage}\n\\capbox{Reconstructed mass peaks for exotic leptons}{Reconstructed mass peaks for exotic leptons with masses of\n175~GeV, 500~GeV, 750~GeV and 900~GeV\\@.\n\\label{1000}}\n\\end{center}\n\\end{figure}\n\nThe long drift times in the muon chambers used by ATLAS allow the recording of events with time delays at arrival much greater than the 25~ns beam crossing interval of the LHC \\cite{ATLASmTDR}; following \\cite{Hinchliffe:1998ys}\na maximum time delay of 50~ns is assumed here. However, to \ncorrectly associate the delayed track in the muon system with the event \nrecorded in the other detector systems would require a specialized trigger\nbased, for example, on the presence of high-$p_\\text{T}$ tracks in the inner\ndetector (these would identify the true event time). Without such a trigger, the\nmaximum allowed time delay would be 25~ns, and the discovery reach\nwould be reduced to a lepton mass of 800~GeV\\@. The statistical\nsample available for the angular distribution analysis in \n\\secref{angdist} would also be reduced by a factor of about 2, depending on the \nlepton mass.\n\n\\subsection{Distinguishing leptons from sleptons}\n\\label{angdist}\n\nAs mentioned in \\secref{HERWIG} it should be possible to distinguish \nthese new heavy leptons from heavy scalar leptons by studying the angular \ndistribution with which they are produced.  The two different angular \ndistributions in the centre-of-mass frame are shown in eqs.~(\\ref{dxs}) and\n(\\ref{sdxs}).  The application of the cuts described in \\secref{tof} as \nwell as the effects of the mass and momentum resolution mean that the \nobserved angular distributions are somewhat different to these.\n\\enlargethispage{-\\baselineskip}\n\nThe forward-backward asymmetry for the exotic heavy leptons was not initially considered\nbecause of the difficulty in a p-p collider of distinguishing the quark and \nanti-quark in the centre-of-mass frame.  Hence only $|\\cos\\theta^*|$ was used\nwhen the angular distributions were compared.  Figure~\\ref{a400} shows the angular distribution for both exotic leptons and sleptons of mass 400~GeV\\@.  In an experimental situation, the exact quark direction is not known and so the angle used in this plot is not exactly $\\theta^*$---instead it is the angle between the out-going lepton and the incoming proton.  Examining the event records before and after the initial-state parton shower (in which the quark can acquire some transverse momentum) shows that the difference between the two angular distributions is small.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.48}{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]\n{aplot400sm.ps}}}\n\\capbox{Angular distributions of detected pairs of 400 GeV particles}{Angular distributions of detected pairs of 400 GeV particles.\n$\\theta^*$ is the angle between the out-going particle and the incoming quark in the centre-of-mass frame.\\label{a400}}\n\\end{center}\n\\end{figure}\n\nA chi-squared test was applied to find the probability that a scalar model \ndescribes the heavy lepton data.  The model was the angular distribution of a large Monte \nCarlo sample of scalars, and the total number of detected scalar pairs was \nrescaled to the expected number of detected pairs of heavy leptons.  An \n`average' data-set was obtained from a rescaled Monte Carlo sample of \nleptons, generated according to their angular distribution.  Both these \nrescaled distributions had negligible theoretical errors compared to the \n(Poisson distributed) statistical errors on the model for 100~fb$^{-1}$ of\nintegrated luminosity.\n\nFigure~\\ref{prob} shows that it will be possible to rule out scalar leptons \nat a confidence level of better than 90\\% up to a mass of 580~GeV\\@.  The \nangular distribution does not change significantly over the mass range of this\nplot---the decreasing probability of ruling out the scalar hypothesis as the\nmass increases is mainly because of the decreasing statistics.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.48}{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]{prob.ps}}}\n\\capbox{Probability of ruling out the scalar hypothesis}{Probability at which it will be possible to rule out the scalar\nhypothesis by studying the angular distribution of the detected leptons.\n\\label{prob}}\n\\end{center}\n\\end{figure}\n\n\\enlargethispage{-\\baselineskip}\nAn alternative approach to ruling out the scalar hypothesis is to try to measure the forward-backward asymmetry of the lepton pair.  To do this it is necessary to reliably discriminate between the quark and anti-quark directions; this is usually based (e.g.\\ in \\cite{Dittmar:1997my}) on the assumption that the quark is a valence quark (with relatively high momentum) whereas the anti-quark is a sea quark (with lower momentum).  A method based on the boost of the lepton pair was investigated for 500~GeV leptons and although the discriminator was found to be reliable for large boosts, the original asymmetry ($\\sim 0.3$) is not large enough to improve on the distinction discussed above.  Cuts which remove events where the lepton pair has only a small boost (and hence the quark direction is more uncertain) provided only a modest improvement because the increase in measured asymmetry is accompanied by a significant decrease in statistics.\n\n\\section{Conclusions}\n\\label{conchl}\n\nThis work was motivated by some of the new models of physics beyond the \nStandard Model in which the gauge couplings unify at an intermediate energy\nscale ($\\sim 10^{11}$~GeV)\\@.  These intermediate scale models can include\nadditional leptons added to the Minimal Supersymmetric Standard Model and \nwhich are expected to have TeV scale masses.\n\\enlargethispage{-\\baselineskip}\n\nIt is concluded that, assuming Standard Model couplings and a long enough \nlifetime, it should be possible to detect charged heavy leptons up to masses of 950~GeV with 100~fb$^{-1}$ of \nintegrated luminosity.  Background from mis-identified muons should be \nnegligible and can be reduced by the application of a \n$p_\\text{T}^\\text{min}$ cut at about 50~GeV\\@.  It will be difficult to use \nthe angular distribution of the produced particles to distinguish them from \nscalar leptons for masses above 580~GeV because the falling cross section \nlimits the available statistics.   \n\nThe absence of a specialized trigger in the inner detector could reduce the \ndiscovery limit to 800~GeV and would reduce the statistics available for the \nangular distribution analysis.\n\nNo present cosmological or experimental limits were found to rule out \nadditional exotic leptons at the masses considered in this work. \n\n\\chapter{Introduction to Black Holes}\n\\chaptermark{Black Hole Introduction}\n\\label{bhintro}\t\t\n\\let\\om=\\omega\t\n\\let\\si=\\sigma\n\\let\\Ga=\\Gamma\t\n\n\\section{Black holes in four dimensions}\n\nThe work in this thesis relates to extra-dimensional black holes, but it is useful to first summarize some of the properties and features of black holes in four dimensions, as well as the history of the theory describing them.  Black holes (or `frozen stars' as they were originally called) can only be correctly described by Einstein's 1915 theory of general relativity (GR).  Einstein himself was never completely convinced by the idea of black holes but, in 1916, Karl Schwarzschild applied GR to a static non-spinning massive object and obtained his well-known metric showing that an object of mass $M$ has a singularity at the Schwarzschild radius $r_\\text{S}=2GM\/c^2$.  Therefore an object with radius smaller than $r_\\text{S}$ is a black hole with an event horizon at this radius.  The Schwarzschild radius can be calculated for any object but, since most objects do not have the density required to be black holes, it is usually much smaller than the object's physical size---for example, the Schwarzschild radius of the earth is easily calculated to be approximately 1~cm.\n\\enlargethispage{-\\baselineskip}\n\nIt is entirely coincidental that the exact horizon radius can be derived using an `escape velocity' argument which would be expected to be only qualitative since it is entirely Newtonian and fails to take into account the relativistic effects.  However both Wheeler (1783) and Laplace (1796) independently used such an argument to postulate the existence of black holes as follows.  An object of mass $m$ propelled vertically upwards (with velocity $v$) from the surface of a mass $M$ with radius $r$ can only escape from the gravitational pull if\n\\begin{equation}\n\\frac{1}{2} m v^2 > \\frac{GMm}{r}\\,.\n\\end{equation}\nIf $r$ decreases, $v$ must be larger if the object is to escape.  The radius at which escape is only just possible even if travelling at the speed of light is found by setting $v=c$.  Hence we obtain\n\\begin{equation}\nr=\\frac{2GM}{c^2}\\,,\n\\end{equation}\nexactly as Schwarzschild derived using the later relativistic theory.\n\nThe above argument---that not even light can escape from a black hole---would seem to suggest that black holes can only grow.  However Hawking showed that the combination of general relativity and quantum mechanics predicts that black holes can evaporate by emitting Hawking radiation \\cite{Hawking:1975sw}.  The mechanism for this is that a particle anti-particle pair is created at the event horizon; one has positive energy and escapes the black hole's gravitational attractions but the other negative energy one falls back into the black hole leading to a net decrease in its mass. \n\nHawking found that the radiation spectrum is almost like that of a black body, and can be described by a characteristic Hawking temperature given by\n\\begin{equation}\nT_\\text{H}=\\frac{\\hbar c}{4\\pi k r_\\text{S}}\\,,\n\\end{equation}\nwhere $k$ is the Boltzmann constant.  Throughout the rest of this work natural units are used with the following definitions:\n\\begin{equation}\n\\hbar=c=k=1\\,.\n\\end{equation}\nThis makes it possible to use energy units (usually GeV) to express distances ($1\\mbox{ GeV}^{-1}=1.97\\times 10^{-16}$~m), time intervals ($1\\mbox{ GeV}^{-1}=6.58\\times 10^{-25}$~s) and temperatures ($1\\mbox{ GeV}=1.16\\times 10^{13}$~K).\n\nIt is important to remember that, despite the wide acceptance of the theory, Hawking radiation has not been experimentally verified.\\footnote{Compare this with the comments about the Higgs boson in \\secref{thesm}.}  In a four dimensional world we would only be able to observe Hawking radiation by studying astrophysical black holes.  In fact although there are many astrophysical black hole candidates, it is difficult to uniquely identify them as black holes (for a review of some of the difficulties see \\cite{Menou:1997qd} and the references therein).  The two main classes of black hole candidate are stellar black holes (with masses of order a few solar masses) in binary star systems, and super-massive black holes ($\\sim 10^{12}$ solar masses) at galactic centres which may power quasars and other active galactic nuclei.  \n\\enlargethispage{-\\baselineskip}\n\nThe masses of black hole candidates in binary systems can be estimated by studying the Doppler shift of emission lines from the secondary star and, since neutron stars have a maximum mass above which they will collapse to a black hole, this provides indirect evidence for the existence of astrophysical black holes.  Additionally, there seems to be confirmation that the primary stars are black holes from the emission spectra of Doppler-shifted X-rays.  These come from metals (e.g.\\ iron) which are stripped off the outer layer of the secondary star and are then ionized due to the high temperatures of the accretion disc around the black hole candidate.\n\nObservation of intense emission from super-massive black hole candidates is consistent with matter accretion by relatively small objects and hence also supports the black hole hypothesis.  The most significant difference between black holes and other massive compact objects is the presence of an event horizon.  Accretion models can provide evidence for both stellar and super-massive black holes by predicting how much energy falling on the central object from the accretion disc is `lost' (as for a black hole event horizon) and how much is re-radiated and observed (as in the case of an ordinary compact object).\n\nObserving astrophysical Hawking radiation would absolutely verify the existence of black holes, but unfortunately the huge masses of these objects make this impossible.  The large masses mean that even the highest black hole temperatures would be $\\sim$~nK---lower than the lowest temperatures achieved in the laboratory ($\\sim \\mu$K) and much lower than the temperature of the cosmic microwave background radiation (2.7~K).  This means that the observation of astronomical Hawking radiation is impossible: it would take $\\sim 10^{14}$~years for a single photon from the closest black hole candidate to hit the earth and these photons would be swamped by the X-ray emission from matter being accreted into the disc surrounding the black hole.  This is one of the reasons why the idea of observing Hawking radiation from miniature black holes produced in particle colliders is so exciting.\n\n\nIn spite of the difficulties in studying black holes there is a huge body of work in the mathematical literature considering their properties including the emitted radiation, the singularities, and the effect of charge and angular momentum.  A full review of this literature is not attempted here, but instead a few points particularly relevant for work on extra-dimensional black holes will be highlighted.\n\n\\subsection{Grey-body factors}\n\nBlack holes are often spoken about as if the Hawking radiation is exactly like that of a black body.  However this is not the case---Hawking's original work showed that the energy spectrum is in fact `grey-body'.  The spectrum is modified from that of a perfect black body by an energy-dependent grey-body factor; these factors are also spin-dependent and the main effect in four dimensions is to suppress the low-energy part of the spectrum (see Figure 1 of \\cite{MacGibbon:1990zk}).\n\nPerhaps counter-intuitively the grey-body factors for particle emission from black holes are equal to the absorption cross sections for the same particles incident on the black hole.  However it is this property which ensures that the grey-body factors do not destroy the thermal nature of the black hole; equilibrium with a heat bath at the same temperature is still possible.  Of course this equilibrium is unstable---black holes have a negative heat capacity so if the the temperature of the black hole is slightly below (above) that of the heat bath, there will be a net absorption (emission) of energy so the black hole mass will increase (decrease) with a corresponding further decrease (increase) in Hawking temperature.\n\nA substantial part of the work in this thesis (see Chapter~\\ref{greybody}) describes the numerical calculation of extra-dimensional grey-body factors.\n\n\\subsection{Super-radiance}\n\\enlargethispage{-2\\baselineskip}\n\nAs mentioned above, the grey-body factor for a particular particle can be calculated from the absorption cross section of that particle incident on the black hole.  However in the case of rotating black holes it was found that for incident bosons with low energies it is possible for the absorption cross section to be negative \\cite{Zeldovich:1971,Misner:1972kx,Press:1972}.  This is equivalent to the reflection coefficient being greater than unity which means that energy incident on the black hole can be amplified.  This amplification was found to be greatest when the black hole was maximally rotating, and for a particle of spin $s$ was found to occur in the mode with angular momentum quantum numbers $\\ell=m=\\max(s,1)$.  Perhaps most significant is the increase in magnitude of the super-radiant effect as $s$ increases; peak amplifications quoted in \\cite{Press:1974} are 0.3\\% for scalars, 4.4\\% for gauge bosons but 138\\% for gravitons.  In theory, the phenomenon of super-radiance makes it possible to construct a `black hole bomb' \\cite{Press:1972} by surrounding a rotating black hole with a spherical reflector; the trapped electromagnetic energy would grow exponentially and eventually the mirror would explode.\n\n\\subsection{Back reaction}\n\nThe calculation of the Hawking radiation emitted by a black hole relies on the assumption that the back reaction of the black hole metric can be ignored.  Essentially this is a requirement that the black hole mass $M_\\text{BH}$ (and equivalently the Hawking temperature $T_\\text{H}$) can be considered constant during the emission of each particle.  For this to be a good approximation requires that the energy of the emitted particle satisfies $\\omega \\ll M_\\text{BH}$.\\footnote{The spectrum for the emission of higher energy particles (with $\\omega \\sim M_\\text{BH}$) must clearly be modified from Hawking's grey-body spectrum if for no other reason than that $\\omega > M_\\text{BH}\/2$ is kinematically forbidden.}  This will be true for the majority of particle emissions provided that $T_\\text{H} \\ll M_\\text{BH}$ (equivalent to the requirement that $M_\\text{BH} \\gg M_{P}$ or that $S_\\text{BH} \\gg 1$ \\cite{Preskill:1991tb}, where $M_{P}$ and $S_\\text{BH}$ are the Planck mass and black hole entropy respectively).  Regardless of the initial black hole mass, the radiation will no longer be described by Hawking's formula in the later stages of the decay.\n\n\\subsection{Black hole charges}\n\nThe `no-hair' theorem states that the only meaningful characteristics of black holes are the charges related to continuous local symmetries.  This means that we expect the electric charge of a black hole to be a meaningful quantity, but the same is not true for baryon and lepton number, for example.  Black hole production and decay cannot violate Abelian gauge symmetries so electric charge must be conserved.  Coloured black holes have to be treated slightly differently to electrically charged holes because of the non-Abelian nature of the gauge symmetry (of course colour charges are irrelevant for very massive 4D black holes, but they will be more significant for the extra-dimensional holes to be introduced in \\secref{bhined}).  Although coloured black hole solutions can be found \\cite{Kanti:1997gs} the charge can never be observed by an observer at infinity, and so is not considered to characterize the black hole.\n\nThe Hawking emission process occurs just outside the event horizon where the metric is described only by the charge, mass and angular momentum of the black hole.  Hence the emission from, and decay of, the black hole cannot be dependent on any other properties of the black hole (or the matter from which it was originally made up).  This is equivalent to saying that these properties (e.g.\\ baryon number) are meaningless and will be violated in the decay.\n\n\\subsection{Information paradox}\n\\label{inpar}\n\\enlargethispage{-\\baselineskip}\n\nThere are many unanswered questions regarding the black hole information loss paradox, and these are intimately related to how the black hole decays, particularly in its final stages.  The essence of the problem is that since a black hole evaporates to a mixture of thermal radiation (according to Hawking) any information which has fallen into the black hole appears to be lost.  Since mass, angular momentum and charge are the only observable characteristics of a black hole, any other information carried by matter which becomes trapped would appear to be lost.\n\nMany of the suggested solutions to this problem relate to what happens once the black hole mass becomes comparable with the Planck mass.  Hawking's semi-classical calculations are no longer valid but without a quantum theory of gravity it is unclear what will happen to the remnant.  Some argue that information is not lost but might be stored in a stable remnant of the black hole.  A more popular idea is that information actually comes out of the black hole as non-thermal correlations within the Hawking radiation, whilst other theories accept and incorporate the loss of quantum information (for reviews of all these issues and possible resolutions, see \\cite{Preskill:1992tc,Page:1993up,Stephens:1994an,Strominger:1994tn,Banks:1995ph}).\n\n\\section{Black holes in extra dimensions}\n\\label{bhined}\n\nThe extra-dimensional models of Arkani-Hamed, Dimopoulos and Dvali (ADD) \\cite{Arkani-Hamed:1998rs,Arkani-Hamed:1998nn,Antoniadis:1998ig} and\nRandall and Sundrum (RS) \\cite{Randall:1999ee,Randall:1999vf} which were introduced in Chapter~\\ref{introch} were motivated by the desire to explain the hierarchy problem---that is, the sixteen orders of magnitude difference between the electroweak energy scale and the Planck scale.\n\n\nIn the standard version of those works the Standard Model fields are\nlocalized on a 3-brane, which plays the role of our 4-dimensional world,\nwhile gravity can propagate both on the brane and in the bulk---the\nspace-time transverse to the brane. \nIn theories with large extra dimensions the traditional reduced Planck scale, \n$\\hat{M}_{(4)}\\sim 10^{18}$ GeV, is only an effective energy scale derived\nfrom the fundamental higher-dimensional one, $M_{\\text{P}(4+n)}$, through the\nrelation\n\\begin{equation}\n\\hat{M}_{(4)}^2 \\sim  M_{\\text{P}(4+n)}^{n+2}\\,R^n.\n\\label{MPl}\n\\end{equation}\nThe above relation involves the volume of the extra dimensions, $V \\sim R^n$,\nunder the assumption that $R$ is the common size of all $n$ extra compact\ndimensions. Therefore, if the volume of the internal space is large (i.e.\\ if \n$R \\gg \\ell_\\text{P}$, where $\\ell_\\text{P} =10^{-33}$ cm is the Planck length) \nthen $M_{\\text{P}(4+n)}$ can be substantially lower than $\\hat{M}_{(4)}$.  Only if $M_{\\text{P}(4+n)}$ can be as low as $\\sim$~TeV does such a model provide a solution to the hierarchy problem (and permit study at the next generation of colliders).\n\nIn the regime $r\\ll R$, the extra dimensions `open up' and gravity becomes\nstrong.  Hence, Newton's law for the gravitational interactions in this\nregime is modified, with the gravitational potential assuming a $1\/r^{n+1}$\ndependence on the radial separation between two massive particles.   As a result there are limits on the fundamental scale from short-scale gravity experiments, but the most stringent constraints (at least for $n=2$ and 3) were found in \\secref{edcon} to be cosmological, specifically from observations of neutron stars.  Although there are many uncertainties these tend to exclude even the $n=3$ case, while allowing models with $M_{\\text{P}(4+n)} \\sim 1$~TeV for $n \\geq 4$.\n\nIf extra dimensions with $R \\gg \\ell_\\text{P}$  exist, then black holes with\na horizon radius $r_\\text{h}$ smaller than the size of the extra dimensions $R$\nare higher-dimensional objects centred on the brane and extending\nalong the extra dimensions. It is these black holes which are the subject of the remaining chapters of this thesis.  It has been shown that extra-dimensional black holes have modified properties, e.g.\\ they are larger and colder than four-dimensional black holes of the same mass \\cite{Argyres:1998qn}.\nOne striking consequence of the theories with large extra dimensions is that\nthe lowering of the fundamental gravity scale allows for the production of\nsuch miniature black holes during scattering processes\nwith centre-of-mass energy $\\sqrt{\\hat{s}} \\gg M_{\\text{P}(4+n)}$ \n\\cite{Banks:1999gd,Giddings:2001bu,Dimopoulos:2001hw,Voloshin:2001vs,Voloshin:2001fe,Giddings:2001ih,Dimopoulos:2001qe,Giudice:2001ce}. \n\nThere has already been a significant amount of work on production of black holes at future particle colliders \\cite{Hossenfelder:2001dn,Kim:2001pg,Cheung:2001ue,Bleicher:2001kh,Casadio:2001wh,Park:2001xc,Landsberg:2001sj,Ahn:2002mj,Rizzo:2002kb,Solodukhin:2002ui,Cardoso:2002ay,Cheung:2002aq,Konoplya:2002zu,Uehara:2002gv,Guedens:2002km,Kotwal:2002wg,Frolov:2002as,Chamblin:2002ad,Han:2002yy,Frolov:2002gf,Anchordoqui:2002cp,Frolov:2002xf,Cardoso:2002jr,Cardoso:2002pa,Frolov:2003en,Chamblin:2003wg,Mocioiu:2003gi,Konoplya:2003ii,Casadio:2003vk,Vasilenko:2003ak,Cavaglia:2003qk,Konoplya:2003dd} and that is also the focus of the work in this thesis.  However, as mentioned in \\secref{crlim}, a low fundamental energy scale would also make it possible for miniature black holes to be created in the earth's atmosphere \\cite{Goyal:2000ma,Feng:2001ib,Anchordoqui:2001ei,Emparan:2001kf,Anchordoqui:2001cg,Uehara:2001yk,Alvarez-Muniz:2002ga,Ringwald:2001vk,Kowalski:2002gb,Kazanas:2001ep,Jain:2002kf,Ringwald:2002if,Ahn:2003qn,Nicolaidis:2003hi,Anchordoqui:2003jr,Mironov:2003jw}.  For the highest energy cosmic neutrinos ($>10^4$~TeV) the black hole production cross section from neutrino-nucleon interactions would be larger than the Standard Model interaction rate.  There are several different techniques involved in detecting cosmic ray particle showers (for a review see \\cite{Anchordoqui:2002hs}) but showers from black hole production and decay are expected to be characterized by particular particle content and multiplicity, and a favoured quasi-horizontal direction.  Before the LHC becomes operational cosmic ray experiments should be able to improve their present limits \\cite{Anchordoqui:2001cg} on the Planck scale or, more optimistically, observe black hole production.  Recently it has been suggested \\cite{Mironov:2003jw} that black hole production could be one explanation for the hadron-rich `Centauro' cosmic ray events.  The theoretical results derived in Chapter~\\ref{greybody} should also be useful for the study of black hole decay in a cosmic ray context.\n\n\\subsection{Black hole production}\n\\label{bhprod}\n\nThere has been much discussion in the literature (e.g.\\ \\cite{Giddings:2001bu,Voloshin:2001vs,Giddings:2001ih,Voloshin:2001fe,Eardley:2002re}) about what the cross section for black hole production is, but the consensus opinion that the geometrical $\\sigma\\sim\\pi r_\\text{S}^2$ is valid seems to have been confirmed by the numerical work in \\cite{Yoshino:2002tx}.  More formally we can write\n\\begin{equation}\n\\label{fndef}\n\\sigma=F_n\\pi r_\\text{S}^2\\,.\n\\end{equation}\nIn order to obtain the coefficient $F_n$ it is necessary to consider the black hole production process in detail.  Various analytic techniques are available but for collisions with non-zero impact parameter in more than four dimensions, the collision must be modelled numerically \\cite{Yoshino:2002tx}.  This allows the determination of both $F_n$ and $b_\\text{max}$, the maximum impact parameter for which black hole formation will occur.  It is found that $F_n$ is of order 1 (for example $F_0=0.647$ and $F_7=1.883$) and that $b_\\text{max}\/r_\\text{S}$ ranges from 0.804 ($n=0$) to 1.37 ($n=7$).\n\nAlthough a full numerical relativistic approach was necessary to obtain these values, a simple model of the collision can give good agreement.  In four dimensions Thorne's hoop conjecture \\cite{Thorne:1972ji} predicts the creation of a black hole in the case where any two partons from the colliding particles pass within the horizon radius corresponding to their centre-of-mass energy.  In the extra-dimensional case there is probably an equivalent `volume conjecture' \\cite{Ida:2002hg,Yoshino:2002br}, but a similar prediction is expected.\n\nHere we follow the approach in \\cite{Ida:2002ez} and assume that a black hole will form if $b < 2r_\\text{h}(M_\\text{BH},J)$, where $r_\\text{h}$ is the horizon radius for a black hole with mass $M_{BH}$ and angular momentum $J$.  This differs from that in \\cite{Anchordoqui:2001cg} which assumes $b < r_\\text{h}(M_\\text{BH},J)$, but provides less good agreement with the numerical results.\\footnote{Reference \\cite{Anchordoqui:2001cg} was published before the numerical results in \\cite{Yoshino:2002tx}, whereas \\cite{Ida:2002ez} was published afterwards.}   The angular momentum $J$ can be usefully related to a dimensionless rotation parameter $a_*$ by\n\\begin{equation}\n\\label{astardef}\na_*=\\frac{(n+2)J}{2 r_\\text{h} M_\\text{BH}}\\,.\n\\end{equation}\nThere will be more discussion of this in \\secref{rotbh}, but $r_\\text{h}$ is found to be related to $r_\\text{S}$ by\n\\begin{equation}\nr_\\text{h}=\\frac{r_\\text{S}}{(1+a_*^2)^{\\frac{1}{n+1}}}\\,.\n\\end{equation}\nIt is therefore possible to determine $b_\\text{max}$ by setting $b=2r_\\text{h}$ and assuming that $J=bM_\\text{BH}\/2$.  This gives\n\\begin{equation}\nb_\\text{max}=2\\left[1+\\left(\\frac{n+2}{2}\\right)^2\\right]^{-\\frac{1}{n+1}}r_\\text{S}\\,,\n\\end{equation}\nwhich is in excellent agreement with \\cite{Yoshino:2002tx} for $n\\ge 1$.  It is then possible to estimate $F_n$ (assuming $\\sigma=\\pi b_\\text{max}^2$) and again the agreement is very good.\n\nAbove we have assumed that the mass of the black hole will be equal to the centre-of-mass energy of the colliding partons.  It is possible to put a lower bound (dependent on the ratio $b\/b_\\text{max}$) on the fraction of the energy which is trapped by the black hole.  Lower bounds were derived for head-on ($b=0$) collisions in \\cite{penrose,D'Eath:1992hb,D'Eath:1992hd,D'Eath:1992qu,D'Eath:1993gr} (for four dimensions) and in \\cite{Eardley:2002re} (for the extra-dimensional case).  However the value when the impact factor is close to its maximum is most important, since this is expected to give the largest contribution to the cross section.  The 4D case was studied in \\cite{Eardley:2002re} and then the extra-dimensional case in \\cite{Yoshino:2002tx} showing that the bound on the fraction of trapped energy can be as low as $\\sim 0.1$ for larger values of $n$.  A significant amount of angular momentum could also be lost at this stage.  More work is required in this area since if this bound is saturated the black hole decay discussed in this work would be significantly affected \\cite{Anchordoqui:2003ug} since only if the black hole mass is larger than a few times the fundamental Planck mass can these objects be treated semi-classically.  \n\nAt this point it is worth stressing the difference between black holes as a signature for extra dimensions and the graviton emission signatures briefly mentioned in \\secref{led} and \\secref{collim}.  The graviton emission signatures are necessarily in the `cis-Planckian' or `sub-Planckian' regime ($\\sqrt{\\hat{s}} \\ll M_{\\text{P}(4+n)}$) where the low-energy effective theory is valid and graviton emission can be reliably calculated with a perturbative expansion.  This means that if $M_{\\text{P}(4+n)}$ is too \\emph{low}, the ability of the LHC to extract information from these signatures will be severely limited.  In contrast, black holes would be produced in the `trans-Planckian' or `super-Planckian' regime ($\\sqrt{\\hat{s}} \\gg M_{\\text{P}(4+n)}$) where a semi-classical approach is valid.  If $M_{\\text{P}(4+n)}$ is too \\emph{high} it will not be possible to investigate this regime at the LHC, although it is hard to make a good estimate of what the minimum value of $\\sqrt{\\hat{s}}\/M_{\\text{P}(4+n)}$ should be if we are to be confident of examining trans-Planckian physics.  Black holes are not the only feature of such a regime---for colliding partons with larger impact parameter (i.e.\\ large compared to the horizon radius corresponding to their centre-of-mass energy) there will instead be gravitational elastic scattering with small momentum transfer.  This can be calculated using the eikonal approximation and would result in jet-jet production close to the beam and with high centre-of-mass energy \\cite{Giudice:2001ce}.  There have been some preliminary experimental studies of this by the ATLAS collaboration which suggest that it might be possible to extract information on $n$ and $M_{\\text{P}(4+n)}$ from these signatures \\cite{azuelos}.\n\nAnother possibility in the trans-Planckian regime is the production of `black' $p$-branes ($p$-dimensional, spatially extended solutions of gravitational theories in extra dimensions).  The production and decay of $p$-branes would exhibit some similar features to black holes (which can be considered as 0-branes) but the details of the decay are more model-dependent.  Although the cross section for the production of $p$-branes can dominate that for black holes this is only found to be the case in compactifications with a mixture of small and large extra dimensions \\cite{Ahn:2002mj,Ahn:2002zn,Cheung:2002aq,Cheung:2002uq}.  Qualitatively the explanation for this is that symmetric compactifications (with all extra dimensions of a similar size) favour the production of 0-branes (spherically-symmetric black holes) whereas non-spherically-symmetric $p$-branes are more likely to be produced in models with an asymmetric compactification, e.g.\\ \\cite{Lykken:1999ms}.  There will be no further discussion of $p$-brane production in this work, although it is an area in need of more study since string theory would seem to favour the existence of more than one compactification scale.\n\nThe trans-Planckian and cis-Planckian energy regimes in which it might be possible to extract parameters in a relatively model-independent way are separated by the Planckian regime in which quantum gravity effects become important.  A theory of quantum gravity is required to predict cross sections and experimental signals in this regime.  One possibility is string theory; this would introduce another energy scale, the string scale $M_\\text{s}$, which could naturally be slightly smaller than $M_{\\text{P}(4+n)}$ (see \\secref{astcon}).  Therefore, in the Planckian regime it might be possible for the colliding partons to be excited into string modes or even very excited `string balls'.\n\n\n\\subsection{Black hole decay}\n\nOnce produced, these miniature black holes are expected to decay almost\ninstantaneously (typical lifetimes are $\\sim 10^{-26}$~s).\\footnote{This is only true in the ADD model; in the RS model the black holes can be stable on collider time scales\\cite{Casadio:2001wh}.} According to\nrefs. \\cite{Giddings:2001bu,Giddings:2001ih}, the produced black holes will go through a number\nof phases before completely evaporating.\n\n\\begin{itemize}\n\n\\item\n{\\it Balding phase}\\,: The black hole emits mainly gravitational radiation\nand sheds the `hair' inherited from the original particles, and the asymmetry\ndue to the violent production process.\\footnote{This phase can also be considered as a production phase since it is predominantly the emission of the `junk energy' which is not trapped by the event horizon as the black hole forms.}\n\n\\item\n{\\it Spin-down phase}\\,: The typically non-zero impact parameter\nof the colliding partons leads to black holes with some angular momentum\nabout an axis perpendicular to the plane. During this phase, the black\nhole loses its angular momentum through the emission of Hawking radiation\n\\cite{Hawking:1975sw} and, possibly, through super-radiance.\n\n\\item\n{\\it Schwarzschild phase}\\,: A spherically-symmetric black hole loses\nenergy due to the emission of Hawking radiation. This results in the gradual\ndecrease of its mass and the increase of its temperature.\n\n\\item\n{\\it Planck phase}\\,: The mass and\/or the Hawking temperature approach\nthe Planck scale.  A theory of quantum gravity is necessary to study this\nphase in detail but it is suggested that the black hole will decay to a few quanta with Planck-scale energies \\cite{Giddings:2001bu}.\n\n\\end{itemize}\n\nAs in the 4-dimensional case \\cite{Page:1976df}, it is reasonable to expect\nthat the Schwarzschild\nphase in the life of a small higher-dimensional black hole will be the\nlongest one, and will account for the greatest proportion of the mass loss \nthrough the emission of Hawking radiation.  \n\n\\subsection{Experimental signatures}\n\\label{expsig}\n\nThe phases of black hole decay described in the previous section combine to produce distinctive experimental signatures which make it unlikely that black hole events would be mistaken for many other processes.  Some of these features are outlined below:\n\n\\begin{itemize}\n\\item{There is a very large total cross section, particularly at high centre-of-mass energies (because the parton-level cross section grows with energy);}\n\\item{The ratio of hadronic to leptonic activity is roughly 5:1 and as a result the amount of energy visible in the detector is large (the relative emission into gravitons is also relatively small);}\n\\item{Most events have a relatively high multiplicity with many hard jets and hard prompt leptons;}\n\\item{As the total event transverse energy increases the average multiplicity increases and the average energy of each primary emitted parton decreases---a manifestation of the infra-red ultra-violet connection of gravity;}\n\\item{Events have a high sphericity since most black holes are produced almost at rest;}\n\\item{Hard perturbative scattering processes are suppressed by the non-perturbative black hole production at high energies.}\n\\end{itemize}\n\nIt has been suggested \\cite{Anchordoqui:2002cp} that the high temperature of the black hole ($\\gg \\Lambda_\\text{QCD}$, the scale above which physics becomes perturbative) might cause a quark-gluon `chromosphere' to form around the decaying black hole; the Hawking radiation would then thermalize leading to a suppression of the number of hard hadronic jets.  However this would require $\\sim 10$ quarks to be emitted as primary partons \\cite{Anchordoqui:2002cp,Anchordoqui:2003ug} which is unlikely to be a problem for many black hole events at the LHC because of the lower entropy of higher-dimensional black holes (see the multiplicity discussion in \\secref{totfandp}).  For future higher energy particle colliders, the black hole signatures are more likely to be modified although there would still be hard hadronic jets from W, Z and Higgs bosons (as well as $\\tau$ leptons) which would penetrate the chromosphere before decaying \\cite{Landsberg:2002sa}.\n\nExperimental cuts proposed to isolate black hole events with negligible SM background include requiring the total energy deposited in the calorimeter to be $\\ge 1$~TeV and there to be $\\ge 4$ jets with energies above 100~GeV (including a high-energy lepton or photon which is useful for triggering) \\cite{Dimopoulos:2001hw}. The theoretical work in Chapter~\\ref{greybody} and the event generator described in Chapter~\\ref{generator} will eventually make it possible for more quantitative predictions to be made about typical black hole events, and for experimental cuts to be tailored accordingly.\n\n\\subsection{Proton decay}\n\\label{protonbh}\n\nProton decay is always a potential problem for models which describe physics beyond the Standard Model.  The present limits on proton lifetime require that, at a 90\\% confidence level, the partial lifetime for the decay $\\text{p}\\rightarrow \\text{e}^+ \\pi^0$ is greater than $1.6 \\times 10^{33}$~years \\cite{Shiozawa:1998si}.  In general introducing new physics at a scale of $\\Lambda$ can allow the decay of the proton (mass $m_\\text{p}$) with a width $\\Gamma \\sim (m_\\text{p}^5\/\\Lambda^4)$ unless the operators in the Lagrangian which would allow this are explicitly forbidden or at least heavily suppressed.\n\nTo stop present experimental limits on proton decay being violated in models of new physics it is usually necessary to artificially impose symmetries which forbid the problematic higher dimensions operators (as in the Standard Model this does not completely forbid proton decay since $B$ and $L$ symmetries can be broken by radiative effects).\n\nIn extra dimension models there is new physics at the fundamental Planck scale (so $\\Lambda\\sim M_{\\text{P}(4+n)} \\sim 1$~TeV) and the possible production of black holes in extra dimension models means that gravity itself could be responsible for proton decay.  This could occur through the production of virtual black holes in processes like $\\text{q} + \\text{q} \\rightarrow \\mbox{Black Hole} \\rightarrow \\bar{\\text{q}} + \\ell + \\ldots$, where the ellipsis represents possible additional particles (e.g.\\ gravitons, gluons, photons or neutrinos).  The consequence of the `no hair' theorem is that the only requirement for such processes is the conservation of charge, energy and angular momentum.\n\nThe usual method of suppressing proton decay in models of new physics is by imposing symmetries on the high-energy theory---for example, R-parity in supersymmetry.  However in the extra-dimensional case the high-energy theory is an unknown theory of quantum gravity so it is not clear whether it is possible to impose such a symmetry to suppress but not totally forbid proton decay.  The safest approach is to explicitly forbid proton decay by imposing a symmetry on the low-energy effective theory but again this is not trivial.  A global symmetry for baryon number does not seem to be viable since the black hole decay is not guaranteed to conserve quantum numbers associated with global symmetries.   This leaves discrete gauge symmetries involving combinations of $B$ and $L$ as apparently the only possibilities, since it is argued in \\cite{Krauss:1989zc} that quantum numbers associated with discrete gauge symmetries as well as continuous gauge symmetries will be conserved in the decay of black holes.  Care must be taken to ensure that any such symmetries introduced are anomaly free, and that they still allow the generation of baryon asymmetry and neutrino masses.\n\\enlargethispage{-\\baselineskip}\n\nOf course the above argument has made the usual assumption that quantum numbers associated with global symmetries can be violated, i.e.\\ information is lost in the black hole decay process.  Some would argue \\cite{Kobakhidze:2001yk} that the solution of the Hawking information loss paradox discussed in \\secref{inpar} might involve the conservation of global charges like baryon number in which case many of the above problems would be avoided.\n\nSome of the $\\mbox{TeV}^{-1}$ extra dimension models introduced in \\secref{tev-1} provide another way of avoiding proton decay issues.  The models involve physically separating the quarks and leptons \\cite{Arkani-Hamed:1999za,Arkani-Hamed:1999dc} which exponentially suppresses their wave-function overlap and hence also the possibility of proton decay.   To obtain normal SM physics the Higgs and gauge bosons must propagate in the bulk (or the `thick' brane) between the lepton and quark fields.  This requires the mass of the lowest KK modes for the bosons to be high enough to avoid the experimental constraints from the precision electroweak measurements, and translates to a lower limit on $M_{\\text{P}(4+n)}$ (at least of order 20~TeV \\cite{Han:2002yy}).  As a result such models are disfavoured phenomenologically because the high value of $M_{\\text{P}(4+n)}$ rules out black hole production at the next generation of colliders. \n\n\\section{Basic formulae and assumptions}\n\\label{basic}\n\nAlthough extra dimension models have only recently become popular among particle physicists, Myers and Perry \\cite{Myers:1986un} had previously worked on black holes in higher-dimensional space-times.  They considered the form of the gravitational background around an uncharged $(4+n)$-dimensional black hole.  In the non-rotating (Schwarzschild) case the line-element is found to be given by\n\\begin{equation}\nds^2=- h(r)\\,dt^2 + h(r)^{-1}\\,dr^2 + r^2\\,d \\Omega^2_{2+n}\\,,\n\\label{metric-D}\n\\end{equation}\nwhere \n\\begin{equation}\nh(r) = 1-\\biggl(\\frac{r_\\text{S}}{r}\\biggr)^{n+1},\n\\label{h-fun}\n\\end{equation}\nand\n\\begin{eqnarray}\nd\\Omega_{2+n}^2=d\\theta^2_{n+1} + \\sin^2\\theta_{n+1} \\,\\biggl(d\\theta_n^2 +\n\\sin^2\\theta_n\\,\\Bigl(\\,... + \\sin^2\\theta_2\\,(d\\theta_1^2 + \\sin^2 \\theta_1\n\\,d\\varphi^2)\\,...\\,\\Bigr)\\biggr).\n\\end{eqnarray}\nIn the above, $0 <\\varphi < 2 \\pi$ and $0< \\theta_i < \\pi$, for \n$i=1, ..., n+1$.  As shown in \\cite{Myers:1986un}, the extension of the usual 4-dimensional Schwarzschild calculation gives the following horizon radius: \n\\begin{equation}\nr_\\text{h} = r_\\text{S} =\\frac{1}{\\sqrt{\\pi}M_{P(4+n)}}\\left(\\frac{M_\\text{BH}}{M_{P(4+n)}}\\right)^\n{\\frac{1}{n+1}}\\left(\\frac{8\\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)\n^{\\frac{1}{n+1}},\n\\end{equation}\nwhere $M_{P(4+n)}$ is the fundamental $(4+n)$-dimensional Planck scale in convention `d' of Table~\\ref{consum}. \n\nThe black holes being considered in this work are assumed to have horizon radii satisfying the relation $\\ell_\\text{P} \\ll r_\\text{S} \\ll R$.\nThe former inequality guarantees that quantum corrections are not important\nin the calculations, while the latter is necessary for the black holes to\nbe considered as higher-dimensional objects (i.e.\\ the curvature of the extra dimensions can be ignored on the scale of the black hole).  The tension of the brane on which the black hole is centred is assumed to be much smaller than the black hole mass which means it can be neglected in this analysis, and a zero bulk cosmological constant is also assumed.  This is essentially equivalent to the assumption of the ADD scenario although much of what follows can also be applied to the RS models provided the bulk cosmological constant is small; this corresponds to a small warp factor and so the space-time can be considered as almost spherically symmetric.  If this is not the case, singularity problems emerge and the black hole solutions lose their spherical symmetry (the horizon becomes flattened on the brane in a pancake shape \\cite{Giddings:2000mu}).\n\nA black hole of a particular horizon radius is characterized by a\nHawking temperature related by\n\\begin{equation}\nT_\\text{H}=\\frac{(n+1)}{4\\pi\\,r_\\text{S}}\\,,\n\\end{equation}\nand the black hole entropy is given by\n\\begin{equation}\nS_\\text{BH}=\\frac{4\\pi r_\\text{S} M_\\text{BH}}{n+2}=\\left(\\frac{n+1}{n+2}\\right)\\frac{M_\\text{BH}}{T_\\text{H}}\\,.\n\\label{bhent}\n\\end{equation}\n\nThe emitted Hawking radiation is {\\it almost} like that of a black body at this temperature.  In fact the flux spectrum, i.e.\\ the number of particles emitted per unit time, is given by \\cite{Hawking:1975sw}\n\\begin{equation}\n\\label{3flux}\n \\frac{dN^{(s)}(\\om)}{dt} = \\sum_{\\ell} \\sigma^{(s)}_{\\ell}(\\om)\\,\n\\frac{1}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1} \n\\,\\frac{d^{n+3}p}{(2\\pi)^{n+3}}\\,,\n\\end{equation}\nwhile the power spectrum, i.e.\\ the energy emitted per unit time, is\n\\begin{equation}\n\\frac{dE^{(s)}(\\om)}{dt} = \\sum_{\\ell} \\sigma^{(s)}_{\\ell}(\\om)\\,\n\\frac{\\om}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1}\\,\\frac{d^{n+3}p}{(2\\pi)^{n+3}}\\,.\n\\label{3power}\n\\end{equation}\nIn the above, $s$ is the spin of the degree of freedom being considered\nand $\\ell$ is the angular momentum quantum number. The spin statistics factor\nin the denominator is $-1$ for bosons and $+1$ for fermions. For \nmassless particles $|p|=\\om$ and the phase-space integral reduces to\nan integral over $\\omega$. The term in\nfront, $\\sigma^{(s)}_{\\ell} (\\om) $, is the grey-body factor which encodes valuable information about the structure of the surrounding space-time including its dimensionality. Chapter~\\ref{greybody} will give more detail on how the grey-body factors are calculated.\n\nAs the decay progresses, the black hole mass decreases and the Hawking temperature rises.  The work performed here will allow some comments to be made on whether the usual `quasi-stationary' approach to the decay is valid; if such an approach is correct then the black hole has time to come into equilibrium at each new temperature before the next particle is emitted. \n\nAn important point to stress is that eqs.~(\\ref{3flux}) and (\\ref{3power}) \nrefer to individual degrees of freedom and not to elementary\nparticles, like electrons or quarks, which contain more than one polarization.\nCombining the necessary degrees of freedom and their corresponding flux\nor power spectra, the relative numbers of different elementary particles \nproduced, and the energy they carry, can be easily computed.  Since the higher-dimensional black holes which might be produced at the LHC have relatively high Hawking temperatures ($\\sim$100~GeV) in most cases it is possible for all elementary particles to be produced.  The numbers of degrees of freedom $d_s$ are summarized in Table~\\ref{pprobs}.  These take account of the fact that for each massive gauge bosons one of the degrees of freedom comes from the Higgs mechanism.\nThe numbers of degrees of freedom will be important in calculating total flux and power emission in \\secref{totfandp}, and for the construction of the black hole event generator (see Chapter~\\ref{generator}).\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nParticle type & $d_0$ & $d_{1\/2}$ & $d_1$  \\\\\n\\hline\nQuarks & 0 & 72 & 0\\\\\nGluons & 0 & 0 & 16\\\\\nCharged leptons & 0 & 12 & 0\\\\\nNeutrinos\\footnotemark & 0 & 6 & 0\\\\\nPhoton & 0 & 0 & 2\\\\\nZ$^0$ & 1 & 0 & 2\\\\\nW$^+$ and W$^-$ & 2 & 0 & 4\\\\\nHiggs boson & 1 & 0 & 0\\\\\n\\hline\nTotal & 4 & 90 & 24\\\\\n\\hline\n\\end{tabular}\n\\capbox{Degrees of freedom for different particle types}{Degrees of freedom for different particle types emitted from a black hole.\\label{pprobs}} \n\\end{center}\n\\end{table}\n\n\\footnotetext{It is possible that $d_{1\/2}=12$ should be used for the neutrinos since right-handed neutrinos, if they exist, will also be emitted by the black hole.}\\stepcounter{footnote}\n\\enlargethispage{-\\baselineskip}\n\nIn reference \\cite{Emparan:2000rs} it was argued that the majority of energy during\nthe emission of Hawking radiation from a higher-dimensional black hole is\nemitted into modes on the brane (i.e.\\ Standard Model fermions and gauge bosons,\nzero-mode gravitons and scalar fields). This argument was based on their\nresult that a single brane particle carries as much energy as the\nwhole Kaluza-Klein tower of massive excitations propagating in the bulk.  Essentially the reasoning is that the KK tower is simply the result of the projection onto four dimensions; from the extra-dimensional view-point there is only one graviton mode.  Since there are many brane modes for the Standard Model particles the majority of energy is expected to be emitted on the brane.\n\nChapter~\\ref{greybody} of this thesis is concerned with the numerical calculation of the extra-dimensional grey-body factors.  These will be a key part of the state-of-the-art black hole event generator described in Chapter \\ref{generator}.  The event generator simulates both the production and decay of small black holes at hadronic colliders and, by using the new results for the grey-body factors provides estimates for the spectra and relative numbers of the different types of elementary particles emitted.\n\n\\chapter{Grey-body Factors in $(4+n)$ Dimensions}\n\\chaptermark{Grey-body Factors}\n\\label{greybody}\t\t\n\\let\\om=\\omega\t\n\\let\\si=\\sigma\n\\let\\Ga=\\Gamma\t\n\n\\section{Introduction}\n\\label{gbfintro}\n\nThe idea of grey-body factors, both for four-dimensional and extra-dimensional black holes, was introduced in Chapter~\\ref{bhintro}.  Extra-dimensional black holes are of interest because, if scenarios like that of ADD \\cite{Arkani-Hamed:1998rs,Arkani-Hamed:1998nn,Antoniadis:1998ig} are realized in nature, they could possibly be produced at high-energy colliders.  To understand the particle emission probabilities and spectra for these black holes it is necessary to calculate the extra-dimensional grey-body factors.\n\nFor ease of reference, equations~(\\ref{3flux}) and (\\ref{3power}) describing the flux and power spectra are reproduced below:\n\\begin{equation}\n\\label{4flux}\n \\frac{dN^{(s)}(\\om)}{dt} = \\sum_{\\ell} \\sigma^{(s)}_{\\ell}(\\om)\\,\n\\frac{1}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1} \n\\,\\frac{d^{n+3}p}{(2\\pi)^{n+3}}\\,;\n\\end{equation}\nand\n\\begin{equation}\n\\frac{dE^{(s)}(\\om)}{dt} = \\sum_{\\ell} \\sigma^{(s)}_{\\ell}(\\om)\\,\n\\frac{\\om}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1}\\,\\frac{d^{n+3}p}{(2\\pi)^{n+3}}\\,.\n\\label{4power}\n\\end{equation}\n$\\sigma^{(s)}_{\\ell} (\\om) $ is the grey-body factor\\footnote{The quantity $\\sigma^{(s)}_{\\ell} (\\om) $ is alternatively called the absorption cross section. It is also common in the literature to refer to the absorption probability $|{\\cal A}^{(s)}_\\ell |^2$, related to $\\sigma^{(s)}_{\\ell} (\\om) $ through eq.~(\\ref{greydef}), as the grey-body factor.  The notation $\\Gamma_\\ell^{(s)}$ is sometimes used for $|{\\cal A}^{(s)}_\\ell|^2$.} which can be determined by solving the \nequation of motion of a particular degree of freedom in the gravitational \nbackground and computing the corresponding absorption coefficient\n${\\cal A}^{(s)}_\\ell$.  The absorption coefficient is related to the grey-body factor by \\cite{Gubser:1997yh} \n\\begin{equation}\n\\si^{(s)}_\\ell(\\om) = \\frac{2^{n}\\pi^{(n+1)\/2}\\,\\Ga[(n+1)\/2]}{ n!\\, \\om^{n+2}}\\,\n\\frac{(2\\ell+n+1)\\,(\\ell+n)!}{\\ell !}\\, |{\\cal A}^{(s)}_\\ell|^2.\n\\label{greydef}\n\\end{equation}\n \nAs already mentioned, the grey-body factors modify the spectra of emitted particles from that of a perfect thermal black body \\emph{even} in four dimensions \\cite{Hawking:1975sw}.  For a 4-dimensional Schwarzschild black hole, geometric arguments show that, in the high-energy ($\\omega\\gg T_\\text{H}$) regime, \n$\\Sigma_{\\ell}\\,|{\\cal A}_\\ell^{(s)}|^2 \\propto(\\omega r_\\text{h})^2$.  Therefore\nthe grey-body factor at high energies is independent of $\\omega$ and the\nspectrum is exactly like that of a black body for every particle species\n\\cite{misner,Sanchez:1978si,Sanchez:1978vz,Page:1976df}. The low-energy behaviour, on the other hand, is\nstrongly spin-dependent. A common feature for fields with spin $s=0$,\n1\/2 and 1 is that the grey-body factors reduce\nthe low-energy emission rate significantly below the geometrical optics value \n\\cite{Page:1976df,MacGibbon:1990zk}.  The result is that both the power and flux spectra peak at higher energies than those for a black body at the same temperature.  As $s$ increases, the low-energy suppression increases and so the emission probability decreases.  These particle emission probabilities have been available in the literature for over twenty years \\cite{Page:1976df,Sanchez:1978si,Sanchez:1978vz}.\n\nThese characteristics of the four-dimensional grey-body factors demonstrate why it is important to calculate the equivalent extra-dimensional factors.  Studying their dependence on the dimensionality of space-time is important if detected Hawking radiation emitted by small black holes is to be used to try to determine the value of $n$.  Both the energy spectra and the relative emissivities of scalars, fermions and gauge bosons will be potentially useful in this task.  The main motivation for this work was to obtain grey-body factors which could be used in the Monte Carlo event generator described in Chapter~\\ref{generator}, thereby allowing the possible determination of $n$ to be studied in detail.\n\nThe procedure for calculating grey-body factors needs to be generalized to include emission from small higher-dimensional black holes with $r_\\text{h} < R$;  there is an added complication in this case because black holes can emit radiation either on the brane or in the bulk.  The emission of particle modes on the brane is the most phenomenologically interesting effect since it involves Standard Model particles.  However gravitons will certainly be emitted into the bulk (since it is the propagation of gravity in the bulk which accounts for its apparent weakness and hence solves the hierarchy problem) and it is possible that a $(4+n)$-dimensional black hole will also emit bulk scalar modes.  These are potentially important because in some extra dimension models there are additional scalars propagating in the bulk; for example, supersymmetric models inspired by string theory involve graviton supermultiplets which can include bulk scalars (see e.g.\\ \\cite{Scherk:1979aj,Scherk:1979rh}).\\footnote{Scalars propagating in the bulk would affect many of the constraints described in \\secref{edcon}, although in model-dependent ways since the couplings to SM particles would not be universal (unlike those of the graviton).}  Numerical results for bulk and brane scalar modes are necessary to allow comparison of the total bulk and brane emissivities.  This would allow the usual heuristic argument for brane dominance (because there are many more SM brane degrees of freedom than there are\ndegrees of freedom in the bulk---see \\secref{basic}) to be confirmed or disproved.\n\nThere have already been some analytic studies for extra-dimensional black holes \\cite{Kanti:2002nr,Kanti:2002ge,Ida:2002ez} which have suggested that the grey-body factors can have a strong dependence on the number of extra dimensions.  However it is known that results from the power series expansions in these papers are only accurate in a very limited low-energy regime (especially in the case of scalar fields) and even the full analytic results are not necessarily reliable above the intermediate-energy regime.\n\nThe work described in this chapter focuses on the emission of energy from uncharged $(4+n)$-dimensional black holes; initially only Schwarzschild-like black holes are considered although there are also some results for rotating holes.  Section~\\ref{branegbf} gives some details of a master equation describing the motion of scalars, fermions and gauge bosons in the 4-dimensional induced background---the basic steps of Kanti's calculation \\cite{Harris:2003eg} are given in Appendix~\\ref{appb}.  This is particularly useful in that it helps to resolve ambiguities present in similar equations which have previously appeared in the literature.  The brane grey-body factors and emission rates are also defined in \\secref{branegbf}, and analytic and numerical methods for computing these quantities are discussed.  Exact numerical results for emission on the brane (for scalars, fermions and gauge bosons) are presented in \\secref{numres}.  This allows a comparison with  the earlier analytic studies of the Schwarzschild phase as well as a discussion of the relative emissivities and their dependence on $n$.  A calculation of the total power and flux emitted (and hence the lifetime and number of emitted particles expected for extra-dimensional black holes) is also possible in this section.  The emission of bulk scalar modes is studied in \\secref{embulk}, and the results for the relative bulk-to-brane emissivities are also presented in this section.  The analysis provides, for the first time, exact results for the relative bulk and brane emissivities in all energy regimes and for various values of $n$.  Finally section~\\ref{rotbh} includes a discussion of the emission of scalars from rotating black holes; this allows some comments to be made on super-radiance for extra-dimensional black holes.  The conclusions of the work in this chapter are summarized in \\secref{gbfconc}. \n\n\\section{Grey-body factors for emission on the brane}\n\\label{branegbf}\n\nThis section and \\secref{numres} focus on the emission of brane-localized modes, leaving the study of bulk emission and of relative bulk-to-brane emissivity until \\secref{embulk}.\n\nThe brane-localized modes propagate in a 4-dimensional black hole background\nwhich is the projection of the higher-dimensional one, given in eq.~(\\ref{metric-D}), onto the brane. The induced metric tensor follows by fixing\nthe values of the extra angular co-ordinates ($\\theta_i=\\pi\/2$ for $i \\geq 2$)\nand is found to have the form\n\\begin{equation}\nds^2=-h(r)\\,dt^2+h(r)^{-1}dr^2+r^2\\,(d\\theta^2 + \\sin^2\\theta\\,d \\varphi^2)\\,.\n\\label{non-rot}\n\\end{equation}\n\nThe grey-body factors are determined from the amplitudes of in-going and\nout-going waves at infinity so the essential requirement is to solve the\nequation of motion for a particle propagating in the above background. \nFor this purpose, a generalized {\\it master equation} was derived by Kanti (see Appendix~\\ref{appb}) for a particle with arbitrary spin $s$; this equation is similar to the 4-dimensional one derived by Teukolsky \\cite{Teukolsky:1973ha}.\nFor $s=$1\/2 and 1, the equation of motion was\nderived using the Newman-Penrose method \\cite{Newman:1962qr,chandra}, while for $s=0$\nthe corresponding equation follows by the evaluation of the double covariant\nderivative $g^{\\mu\\nu} D_\\mu D_\\nu$ acting on the scalar field. The derived\nmaster equation is separable in each case and, by using the factorization\n\\begin{equation}\n\\label{sepsoln}\n\\Psi_s=e^{-i\\omega t}\\,e^{im\\varphi}\\,R_s(r)\\,{}_sS^m_{\\ell}(\\theta)\\,,\n\\end{equation}\nwe obtain the radial equation\n\\begin{equation}\n\\label{radial}\n\\Delta^{-s} \\frac{d}{dr}\\left(\\Delta^{s+1}\\,\\frac{d R_s}{dr}\\right)+\n\\left(\\frac{\\omega^2 r^2}{h}+2i\\omega s r-\\frac{is\\omega r^2 h'}{h}+\ns(\\Delta''-2)-{}_s\\lambda_{\\ell} \\right)R_s (r)=0\\,,\n\\end{equation}\nwhere $\\Delta=hr^2$.  The corresponding angular equation has the form\n\\begin{equation}\n\\frac{1}{\\sin\\theta}\\,\\frac{d}{d\\theta}\\left(\\sin\\theta\\,\\,\n\\frac{d\\,{}_sS_{\\ell}^m}{d\\theta}\\right)+\n\\left(-\\frac{2ms\\cot\\theta}{\\sin\\theta}-\\frac{m^2}{\\sin^2\\theta}+\ns-s^2\\cot^2\\theta+{}_s\\lambda_{\\ell}\\right){}_sS^m_{\\ell}(\\theta)=0\\,,\n\\label{angular}\n\\end{equation}\nwhere $e^{im\\varphi}\\,{}_sS^m_{\\ell}(\\theta)={}_sY^m_{\\ell}(\\Omega_2)$ are known as the spin-weighted spherical harmonics and ${}_s\\lambda_{\\ell}$ is a separation constant which is found to have the value ${}_s\\lambda_{\\ell}=\\ell(\\ell+1)-s(s+1)$ \\cite{Goldberg:1967uu}.\n\nFor $s=0$, equation~(\\ref{radial}) reduces as expected to eq. (41) of \nref. \\cite{Kanti:2002nr} which was used for the analytical study of the emission of \nbrane-localized scalar modes from a spherically-symmetric higher-dimensional\nblack hole.  Under the redefinition $R_s=\\Delta^{-s} P_s$, eq.~(\\ref{radial})\nassumes a form similar to eq. (11) of ref. \\cite{Kanti:2002ge} which was\nused for the study of brane-localized fermion and gauge boson emission.\nThe two equations differ due to an extra term in the expression of the latter\none, which although vanishing for $s=$1\/2 and 1 (leading to the\ncorrect results for fermion and gauge boson fields)\ngives a non-vanishing contribution\nfor all other values of $s$. Therefore, the generalized equation derived by\nCvetic and Larsen \\cite{Cvetic:1998ap} cannot be considered as a master equation\nvalid for all types of fields.\\footnote{A similar equation was derived in \nref. \\cite{Ida:2002ez} but due to a typographical error the multiplicative factor $s$ in front of the $\\Delta''$-term was missing, leading to an apparently different equation for general $s$.} Hence the derivation of a consistent master equation was imperative, before addressing the question of the grey-body factors in the brane background.  This task was performed by Kanti and led to eqs.~(\\ref{radial}) and (\\ref{angular}) above.  \n\nFor the derivation of grey-body factors associated with the emission of\nfields from the projected black hole, we need to know the asymptotic solutions\nof eq.~(\\ref{radial}) both as $r\\rightarrow r_\\text{h}$ and as $r\\rightarrow \\infty$. \nIn the former case, the solution is of the form\n\\begin{equation}\n\\label{near}\nR_s^{(\\text{h})}=A_\\text{in}^{(\\text{h})}\\,\\Delta^{-s}\\,e^{-i\\omega r^{*}}+\nA_{out }^{(\\text{h})}\\,e^{i\\omega r^{*}},\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{rstarr}\n\\frac{dr^*}{dr}=\\frac{1}{h(r)}\\,.\n\\end{equation}\nWe impose the boundary condition that there is no out-going solution near the\nhorizon of the black hole, and therefore set $A_\\text{out}^{(\\text{h})}=0$.\nThe solution at infinity is of the form\n\\begin{equation}\n\\label{far}\nR_s^{(\\infty)}=A_\\text{in}^{(\\infty)}\\,\\frac{e^{-i\\omega r}}{r}+\nA_\\text{out}^{(\\infty)}\\,\\frac{e^{i\\omega r}}{r^{2s+1}}\\,,\n\\end{equation}\nand comprises both in-going and out-going modes. \n\\enlargethispage{-2\\baselineskip}\n\nThe grey-body factor $\\sigma^{(s)}_{\\ell}(\\om)$ for the emission of brane-localized\nmodes is related to the energy absorption coefficient ${\\cal A}_\\ell$\nthrough the simplified relation\n\\begin{equation}\n\\hat \\si^{(s)}_\\ell(\\om) =\\frac{\\pi}{\\om^2}\\,(2 \\ell +1)\\,\n|\\hat {\\cal A}^{(s)}_\\ell|^2,\n\\label{brane-loc}\n\\end{equation}\nwhere from now on a `hat' will denote quantities associated with\nthe emission of brane-localized modes. The above relation follows from\neq.~(\\ref{greydef}) by setting $n=0$ since the emission of brane-localized\nmodes is a 4-dimensional process.  The absorption coefficient itself is defined as\n\\begin{equation}\n|\\hat {\\cal A}^{(s)}_\\ell|^2=1-\\frac{{\\cal F}_\\text{out}^{(\\infty)}}\n{{\\cal F}_\\text{in}^{(\\infty)}}=\n\\frac{{\\cal F}_\\text{in}^{(\\text{h})}}{{\\cal F}_\\text{in}^{(\\infty)}}\\,,\n\\label{absorption}\n\\end{equation}\nin terms of the out-going and in-going energy fluxes evaluated either at infinity or at the horizon.\nThe two definitions are related by simple energy conservation and lead to\nthe same results.  The choice of which is used for the determination of the absorption coefficient may depend on numerical issues.  \n\nThe flux and power spectra (equations~(\\ref{4flux}) and (\\ref{4power}) respectively) of the Hawking radiation emitted on the brane can be computed, for massless particles, from the\n4-dimensional expressions: \n\\begin{align}\n\\frac{d \\hat N^{(s)}(\\om)}{dt}&= \\sum_{\\ell} (2\\ell +1) |\\hat {\\cal A}^{(s)}_\\ell|^2\\,\n\\frac{1}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1}\\,\\frac{d\\om}{2\\pi}\\,;\n\\label{fdecay-brane}\\\\\n\\frac{d \\hat E^{(s)}(\\om)}{dt}&= \\sum_{\\ell} (2\\ell +1) |\\hat {\\cal A}^{(s)}_\\ell|^2\\,\n\\frac{\\om}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1}\\,\\frac{d\\om}{2\\pi}\\,.\n\\label{pdecay-brane}\n\\end{align}\nNote, however, that the absorption coefficient $\\hat {\\cal A}_\\ell^{(s)}$ still depends on the number of extra dimensions since the projected metric tensor of eq.~(\\ref{non-rot}) carries a signature of the dimensionality of the bulk space-time through the expression of the metric function $h(r)$. \n\n\\subsection{Analytic calculation}\n\nThe grey-body factors have been determined analytically for the 4-dimensional\ncase in refs. \\cite{Page:1976df,Sanchez:1978si,Sanchez:1978vz} both for a rotating and non-rotating black\nhole. In the $(4+n)$-dimensional case, refs. \\cite{Kanti:2002nr,Kanti:2002ge} have provided\nanalytic expressions for grey-body factors for the emission of scalars,\nfermions and gauge bosons from a higher-dimensional Schwarzschild-like\nblack hole.  Both bulk and brane emission were considered for an arbitrary number of extra dimensions $n$. Reference \\cite{Ida:2002ez} presented\nanalytic results for brane-localized emission from a Kerr-like\nblack hole, but only in the particular case of $n=1$. All the above results were derived\nin the low-energy approximation where the procedure used is as follows:\n\n\\begin{itemize}\n\\item{Find an analytic solution in the near-horizon regime and expand it as\nin-going and out-going waves so that the $A^{(\\text{h})}$ coefficients can be extracted;}\n\\item{Apply the boundary condition on the horizon so that the wave is purely\nout-going;}\n\\item{Find an analytic solution in the far-field regime and again expand it as\nin-going and out-going waves;}\n\\item{Match the two solutions in the intermediate regime;}\n\\item{Extract $|{\\cal A}_\\ell|^2$ and expand it in powers of $\\omega r_\\text{h}$.}\n\\end{itemize}\n\nThe solution obtained by following the above approximate method is a power\nseries in $\\omega r_\\text{h}$, which is only valid for low energies and expected to\nsignificantly deviate from the exact result as the energy of the emitted\nparticle increases (this was pointed out in \\cite{Page:1976df,MacGibbon:1990zk} for the\n4-dimensional case). In \\cite{Kanti:2002ge}, the full analytic result for the absorption\ncoefficient (before the final expansion was made) was used for the evaluation\nof the emission rates for all particle species.\\footnote{This full analytic approach was pursued in \\cite{Kanti:2002ge} as a result of some of the numerical results presented in this thesis.} The range of validity of these\nresults, although improved compared to the power series, was still limited since the assumption of small\n$\\omega r_\\text{h}$ was {\\it still} made during the matching of the two solutions\nin the intermediate regime. As the equations of motion are too complicated to solve analytically for all values of $\\om r_\\text{h}$, it becomes clear that only an exact numerical analysis can give the full exact results for grey-body factors and emission rates. \n\n\n\\subsection{Numerical calculation}\n\\label{numcal}\n\nThere are various numerical issues which arise while trying to do the full\ncalculation of the absorption coefficient and the complexity of these problems\ndepends strongly on the spin $s$ of the emitted particle.  The usual numerical \nprocedure starts by applying the relevant horizon boundary condition (a \nvanishing out-going wave) to $R_s (r)$.  Then, the solution is integrated out \nto `infinity' and the asymptotic coefficient $A_\\text{in}^{(\\infty)}$ is extracted \nin terms of which ${\\cal A}_\\ell^{(s)}$ can be calculated.\n\nBy looking at the asymptotic solution at infinity in eq.~(\\ref{far}) we can\nsee that, for the scalar case, the in-going and out-going waves are of\ncomparable magnitude.  Therefore it is relatively easy to extract the\ncoefficients at infinity and thus determine the grey-body factor.  However for \nfields with non-vanishing spin, i.e.\\ fermions, vector bosons and\ngravitons, this is not an easy task. First of all, different components carry\na different\npart of the emitted field: the upper component $\\Psi_{+s}$ consists mainly of\nthe in-going wave with the out-going one being greatly suppressed, while for\nthe lower component $\\Psi_{-s}$ the situation is reversed (for a field with\nspin $s \\neq 0$, only the upper and lower components are radiative). \nDistinguishing between the two parts of the solution (in-going\nand out-going) is not an easy task no matter which component is used, and it\nbecomes more difficult as the magnitude of the spin increases.  In addition,\nthe choice of either positive or negative $s$ to extract the grey-body factor (i.e.\\ using either the upper or the lower component) affects\nthe numerical issues.\n\nIf $s$ is negative, then the horizon boundary condition is easy to apply \nbecause components of the in-going solution on the horizon will be \nexponentially suppressed, as can be seen from eq.~(\\ref{near}).  However eq.~(\\ref{far}) shows that negative $s$ also means the out-going solution at infinity is enhanced by $r^{-2s}$ \nwith respect to the in-going one; this makes accurate \ndetermination of $A_\\text{in}^{(\\infty)}$ very difficult.  Hence the negative $s$ \napproach is not used in this work.\n\nOn the other hand if $s$ is positive, it is easy to accurately extract \n$A_\\text{in}^{(\\infty)}$ because the out-going solution at infinity is suppressed by $r^{-2s}$.  However, close to the horizon the out-going solution is\nexponentially smaller than the in-going one. \nTherefore the solution for $R_s(r)$ can be easily contaminated by \ncomponents of the out-going solution. This problem becomes worse for larger \n$s$ and first becomes significant for $s=1$.\n\\enlargethispage{-\\baselineskip}\n\nFor $n=0$, various methods (see e.g.\\ \\cite{Press:1974}) have been used to \nsolve the numerical problems which arise in the gauge boson case.  The \napproach of Bardeen in \\cite{Press:1974} is also applicable for $n=1$; some details of this method are briefly outlined below.\n\nThe method starts from a mathematically equivalent form of the radial equation in Kerr in-going co-ordinates:\\footnote{This was derived, for general $n$, in \\cite{Kantiprivate} and for $n=0$ reduces to the result found in \\cite{Press:1974}.}\n\\begin{equation}\n\\Delta^{-s}\\frac{d}{dr} \\left(\\Delta^{s+1}\\frac{dR_s}{dr}\\right)-2 i r^2\\omega \\frac{dR_s}{dr}-\\left[2(2s+1)i\\omega r-s\\Delta''+{}_s\\Lambda_{\\ell}\\right]R_s=0\\,,\n\\end{equation}\nwhere ${}_s\\Lambda_{\\ell}=\\ell(\\ell+1)-s(s-1)$.  The asymptotic solutions are now different from those of eqs.~(\\ref{near}) and (\\ref{far}); they are given by\n\\begin{equation}\nR_s^{(\\text{h})}=B_\\text{in}^{(\\text{h})}+B_\\text{out}^{(\\text{h})}\\Delta^{-s} e^{2i\\omega r^{*}},\n\\end{equation}\nand\n\\begin{equation}\nR_s^{(\\infty)}=B_\\text{in}^{(\\infty)}\\frac{1}{r^{2s+1}}+B_\\text{out}^{(\\infty)} \\frac{e^{2i\\omega r}}{r}\\,. \n\\end{equation}\n\nWe can follow the method in Appendix A of \\cite{Press:1974} and apply the co-ordinate transformation\n\\begin{equation}\nx=\\frac{r}{r_\\text{h}}-1\\,.\n\\end{equation}\nThis transformation is easily applicable for either $n=0$ or $n=1$ since in both cases $\\Delta''=2$. In the case $s=-1$ (which is mathematically equivalent to the $s=1$ discussed earlier) applying this transformation for $n=0$ gives\n\\begin{equation}\n\\label{xradial}\nx(x+1)\\frac{d^2R_{-1}}{dx^2}-2i\\omega r_\\text{h}\\left(x^2+2x+1\\right)\\frac{dR_{-1}}{dx}-\\left[-2i\\omega r_\\text{h} (x+1)+\\lambda\\right]R_{-1}=0\\,,\n\\end{equation}\nwhere $\\lambda=\\ell(\\ell+1)$.  The asymptotic solutions can now be written in terms of $x$, with more care being taken over sub-leading terms in the in-going solution at the horizon:\n\\begin{equation}\nR_{-1}^{(\\text{h})}=B_\\text{in}^{(\\text{h})} (1+a_1x+a_2x^2)+B_\\text{out}^{(\\text{h})} r_\\text{h}^2 e^{2i\\omega r^*}x(1+b_1x+\\ldots)\\,.\n\\end{equation}\n\nIt is easy to obtain the coefficients $a_1$ and $a_2$ by substituting the in-going solution into the radial equation and comparing terms in different powers of $x$.  This leads to\n\\begin{equation}\na_1=1+i\\frac{\\lambda}{2\\omega r_\\text{h}}\\,,\n\\label{a1n0}\n\\end{equation}\nand\n\\begin{equation}\na_2=\\frac{1}{2(1-2i\\omega r_\\text{h})}\\left[(\\lambda+2i\\omega r_\\text{h})a_1-2i\\omega r_\\text{h}\\right].\n\\label{a2n0}\n\\end{equation}\nInstead of solving ${\\mathcal L}R=0$, the equation ${\\mathcal L}Y=f$ is solved, where $Y=R-(1+a_1x)$. This means that\n\\begin{equation}\nf=2(1-2i\\omega r_\\text{h})a_2x\\,.  \n\\label{fn0}\n\\end{equation}\nClose to the horizon the leading-order terms in $Y$ and $Y'$ are $a_2x^2$ and $2a_2x$---these are used to apply the horizon boundary condition.  Bardeen found that the equation in this form can now be stably integrated from $x=0$ to $x=\\infty$.  The $B_\\text{in}$ and $B_\\text{out}$ coefficients extracted can be used to obtain the grey-body factors using similar formulae to those discussed later for $A_\\text{in}$ and $A_\\text{out}$.\n\nFor $n=1$, equations~(\\ref{xradial}) and (\\ref{a1n0})--(\\ref{fn0}) are slightly modified and become\n\\begin{gather}\nx(x+2)\\frac{d^2R_{-1}}{dx^2}-2i\\omega r_\\text{h}\\left(x^2+2x+1\\right)\\frac{dR_{-1}}{dx}-\\left[-2i\\omega r_\\text{h} (x+1)+\\lambda\\right]R_{-1}=0\\,;\\\\\na_1=1+i\\frac{\\lambda}{2\\omega r_\\text{h}}\\,;\n\\label{a1n1}\\\\\na_2=\\frac{1}{4(1-i\\omega r_\\text{h})}\\left[(\\lambda+2i\\omega r_\\text{h})a_1-2i\\omega r_\\text{h}\\right];\n\\label{a2n1}\\\\\nf=4(1-i\\omega r_\\text{h})a_2x\\,.  \n\\label{fn1}\n\\end{gather}\n\nUnfortunately no analogous method has been found for $n \\geq 2$ because there is then no simple equivalent of eq.~(\\ref{xradial}).  Instead, an \nalternative transformation of the radial equation~(\\ref{radial}) was used.\nWriting $y=r\/r_\\text{h}$ and $R_1=y\\,F(y)\\,e^{-i\\omega r^*}$, the wave equation becomes\n\\begin{equation}\n(hy^2)\\,\\frac{d^2F}{dy^2}+2y\\,(h-i\\omega r_\\text{h} y)\\,\\frac{dF}{dy}-\\ell(\\ell+1)F=0\\,.\n\\end{equation}\nSince on the horizon $y=1$ and $h=0$, the boundary conditions become $F(1)=1$ \nand\n\\begin{equation}\n\\left. \\frac{dF}{dy} \\right|_{y=1}=\\frac{i \\ell(\\ell+1)}{2 \\omega r_\\text{h}}\\,.\n\\end{equation}\nUsing both this method and the Bardeen method for the $n=1$ case provided a useful cross-check on the spin-1 numerical results.\n\nFor fermions no special transformation was necessary and therefore the radial\nequation~(\\ref{radial}) was used.  However the application of the \nboundary condition at the horizon is made slightly easier by the \ntransformation $P_s=\\Delta^s R_s$ so that the asymptotic solution at\nthe horizon becomes\n\\begin{equation}\nP_s^{(\\text{h})}=A_\\text{in}^{(\\text{h})}\\,e^{-i\\omega r^{*}}+\nA_{out }^{(\\text{h})}\\,\\Delta^s \\,e^{i\\omega r^{*}}.\n\\label{nh-new}\n\\end{equation}\nSince we require $A_\\text{out}^{(\\text{h})}=0$, a suitable boundary condition to apply as $r \\rightarrow r_\\text{h}$ is that\n\\begin{equation}\nP_s=1\\,,\n\\end{equation}\nwhile using eq.~(\\ref{rstarr}) we also obtain\n\\begin{equation}\n\\frac{dP_s}{dr}=-i\\omega\\, \\frac{dr^*}{dr}=-\\frac{i\\omega}{h(r)}\\,.\n\\end{equation}\nThe above boundary conditions ensure that $|A_\\text{in}^{(\\text{h})}|^2=1$. \nThe asymptotic form for $P_s$ at infinity now looks like\n\\begin{equation}\nP_s^{(\\infty)}=A_\\text{in}^{(\\infty)}\\,\\frac{e^{-i\\omega r}}{r^{1-2s}}+\nA_\\text{out}^{(\\infty)}\\, \\frac{e^{i\\omega r}}{r}\\, \n\\label{ff-new}\n\\end{equation}\nsince $\\Delta \\rightarrow r^2$ as $r \\rightarrow \\infty$. \n\nThe numerical work described in this chapter was almost entirely performed using the \\texttt{NDSolve} package in \\texttt{Mathematica}.  However in some cases Fortran programs calling {\\small NAG} routines (specifically \\texttt{D02EJF} and its associated routines) were used as a check on the numerical robustness of the results obtained in \\texttt{Mathematica}.  The {\\small NAG} routines implement the Backward Differentiation Formulae (BDF) whilst in \\texttt{Mathematica} either the BDF method or the implicit Adams method is used.\n\nThere are various considerations which must be taken into account in order to\nobtain results to the desired accuracy (at least three significant figures).\nFirstly, although the horizon boundary condition cannot be applied exactly at\n$r_\\text{h}$ (due to singularities in the boundary condition and the differential \nequation) the error introduced by applying the condition at $r=r_\\text{n}$ (where $\\Delta r_\\text{h}=r_\\text{n}-r_\\text{h} \\ll r_\\text{h}$) must be small.  This can be investigated by studying changes in the\ngrey-body factors for order-of-magnitude changes in $\\Delta r_\\text{h}$.  Similarly it \nmust be verified that the value $r_{\\infty}$ used as an approximation for `infinity' \ndoes not introduce errors which will affect the accuracy of the result.  This is illustrated in Figure~\\ref{farstab} in which $\\hat{\\sigma}^{(0)}_\\text{abs}=\\sum_{\\ell}\\hat{\\sigma}^{(0)}_{\\ell}$ is plotted in units of $\\pi r_\\text{h}^2$.  The oscillations in the numerical result die away as the asymptotic form given in eq.~(\\ref{far}) is obtained.  A value of $r_{\\infty}=10000$ was used in the work that follows to ensure the grey-body factor was accurate to at least three significant figures.\n\n\\begin{figure}\n\\unitlength1cm\n\\psfrag{x}[][][1.4]{$r_{\\infty}$}\n\\psfrag{y}[][][1.4]{$\\hat{\\sigma}^{(0)}_\\text{abs}\/\\pi r_\\text{h}^2$}\n\\begin{center}\n\\begin{minipage}[t]{3.35in}\n\\scalebox{0.65}{\\rotatebox{0}{\\includegraphics{nfars0n0.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{2.75in}\n\\scalebox{0.65}{\\rotatebox{0}{\\includegraphics{nfars0n0mag.eps}}}\n\\end{minipage}\n\\capbox{Stability of grey-body factor for changes in `infinity'}{Stability of the $n=0$, $s=0$, $\\omega r_\\text{h}=0.5$ grey-body factor as $r_{\\infty}$ is changed.  The right plot is a magnified version of part of the left plot to illustrate the accuracy obtained.\\label{farstab}}\n\\end{center}\n\\end{figure}\n\nCare must also be taken that the numerical integration procedure is sufficiently accurate that significant integration errors are avoided even at large values of $r$.  Plots like those in Figure~\\ref{farstab} are also useful for investigating this because they can be used to confirm that there is no significant gradient on a straight line drawn through the centre of the oscillations.\n\nFinally, for each energy being considered, enough angular momentum \nmodes must be included in the summation so that only modes which do not\ncontribute significantly are neglected.  For any value of $\\ell$ the absorption coefficient will approach unity at high enough energies and the $(2\\ell+1)$ factor in eq.~(\\ref{brane-loc}) means that higher angular momentum modes will start to dominate the grey-body factor.  It was found that for the highest values of $\\omega r_\\text{h}$ considered in this work it was necessary to include contributions from in excess of ten angular momentum modes.\n\n\\section{Numerical results for brane emission}\n\\label{numres}\n\nIn this section, results are presented for grey-body factors and emission rates for brane-localized scalar, fermion and gauge boson\nfields, as obtained by numerically solving the corresponding equations of \nmotions.  The relationship between the absorption coefficient ${\\cal A}_\\ell^{(s)}$\nand the $A$ coefficients in equations~(\\ref{nh-new}) and (\\ref{ff-new}) is different in each case, so each will be considered separately.\n\n\\subsection{Spin 0 fields}\n\\label{numscalar}\n\nThe numerical integration of eq.~(\\ref{radial}) for $s=0$ yields the solution\nfor the radial function $R_0(r)$ which smoothly interpolates between the\nasymptotic solutions of eqs.~(\\ref{nh-new}) and (\\ref{ff-new}) in the near-horizon\nand far-field regimes respectively. The absorption coefficient is easily\ndefined in terms of the in-going and out-going energy fluxes at infinity,\nor equivalently by the corresponding wave amplitudes ($A_\\text{in}^{(\\infty)}$ and $A_\\text{out}^{(\\infty)}$) in the same asymptotic regime.\nTherefore eq.~(\\ref{absorption}) may be written in the form \\cite{Kanti:2002nr}\n\\begin{equation}\n|\\hat {\\cal A}^{(0)}_\\ell|^2=1-|\\hat {\\cal R}^{(0)}_\\ell|^2= \n1-\\Biggl|\\frac{A_\\text{out}^{(\\infty)}}{A_\\text{in}^{(\\infty)}}\\Biggr|^2,\n\\label{scalars}\n\\end{equation}\nwhere $\\hat {\\cal R}_\\ell$ is the corresponding reflection coefficient. \n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$\\hat{\\sigma}^{(0)}_\\text{abs}(\\om)\/\\pi r_\\text{h}^2$}\n\\scalebox{0.5}\n{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]{ngrey0.ps}}}\n\\capbox{Grey-body factors for scalars on the brane}{Grey-body factors for scalar emission on the brane from a $(4+n)$D\nblack hole.\\label{grey0}}\n\\end{center}\n\\end{figure}\n\nThe plot presented in Figure~\\ref{grey0} shows, for several values of $n$,\nthe grey-body factor for the emission of scalar fields on the brane (for completeness, the plots include values of $n$ ruled out\non astrophysical grounds, i.e.\\ $n=1$ and 2).\\footnote{Throughout the numerical analyses the horizon radius $r_\\text{h}$ is an\narbitrary input parameter which remains fixed.} The grey-body\nfactor is obtained by using eq.~(\\ref{brane-loc}) and summing over the angular momentum number $\\ell$.  \n\nFor $n=0$ and\n$\\omega r_\\text{h} \\rightarrow 0$, the grey-body factor assumes a non-zero value \nwhich is equal to $4\\pi r_\\text{h}^2$---that is, the grey-body factor for scalar \nfields with a very low energy is given exactly by the area of the black hole \nhorizon. As the energy increases, the factor soon starts oscillating\naround the geometrical optics\nlimit $\\hat{\\sigma}_\\text{g} = 27 \\pi r_\\text{h}^2\/4$ which corresponds to the spectrum of a\nblack body with an absorbing area of radius $r_\\text{c}=3 \\sqrt{3}\\,r_\\text{h}\/2$\n\\cite{Sanchez:1978si,Sanchez:1978vz,misner}.  The $n=0$ result agrees exactly with Page's result presented in Figure 1 of \\cite{MacGibbon:1990zk} (this is also found to be the case for the fermion and gauge boson grey-body factors shown later).  \n\\enlargethispage{-\\baselineskip}\n\nIn the extra-dimensional case, the grey-body factor has an asymptotic low-energy value which is the same for all values of $n$,\nand at high energies it again starts oscillating around a limiting value. This is always lower than the 4-dimensional geometrical optics limit because the effective radius $r_\\text{c}$ depends on the dimensionality of the bulk space-time through the metric tensor of the projected space-time in which the particle moves. For arbitrary $n$, it adopts the value \\cite{Emparan:2000rs}\n\\begin{equation}\nr_\\text{c}=\\biggl(\\frac{n+3}{2}\\biggr)^{1\/n+1}\\,\\sqrt{\\frac{n+3}{n+1}}\\,\\,r_\\text{h}\\,.\n\\label{effective}\n\\end{equation}\nThe above quantity is a strictly decreasing function of $n$ which causes the\nasymptotic grey-body factor, $\\hat{\\sigma}_\\text{g}=\\pi r_\\text{c}^2$\\,, to become more and more\nsuppressed as the number of extra dimensions projected onto the\nbrane increases.  The values of $\\hat{\\sigma}_\\text{g}$ are tabulated in Table~\\ref{brgolim} for different values of $n$.\\footnote{For the larger values of $n$ it was found that these asymptotic values are not approached until relatively high energies, typically $\\omega r_\\text{h} \\sim n$.}\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n$n$ & 0 & 1  & 2  & 3 & 4 & 5 & 6 & 7 \\\\\n\\hline\n$\\hat{\\sigma}_\\text{g}\/\\pi r_\\text{h}^2$ & 6.75 & 4 & 3.07 & 2.60 & 2.31 & 2.12 & 1.98 & 1.87\\\\\n\\hline \n\\end{tabular}\n\\capbox{High-energy limits of grey-body factors for brane emission}{High-energy limits of grey-body factors for brane emission, given in units of $\\pi r_\\text{h}^2$.\\label{brgolim}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\nThe power series expression of the grey-body factor determined in \\cite{Kanti:2002nr} matches the exact solution only in a very limited low-energy\nregime. In the limit $\\omega r_\\text{h} \\rightarrow 0$, the asymptotic\nvalue $4\\pi r_\\text{h}^2$ is recovered as expected; however, as the energy increases\nthe exact solution rapidly deviates from the behaviour dictated by the\ndominant term in the $\\omega r_\\text{h}$ expansion. This was confirmed in \\cite{Kanti:2002ge} where full analytic results for the grey-body\nfactors were plotted, although still in the low-energy approximation.  The behaviour depicted in Figure 3 of ref. \\cite{Kanti:2002ge} is much closer to the exact one, shown here in Figure~\\ref{grey0}, and\nsuccessfully reproduces some of the qualitative features including the \nsuppression of the grey-body factor as the dimensionality of the bulk space-time increases.  Nevertheless, as previously stated, even this result rapidly breaks down as the energy increases; it is particularly unreliable for the $n=0$ case and fails to reproduce the high-energy oscillations for any value of $n$.  This means that the exact numerical solution obtained in this work is the only reliable source of information concerning the form of the extra-dimensional grey-body factor throughout the energy regime.\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$r_\\text{h} d^2 \\hat{E}^{(0)}\/dtd\\om$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=23cm, height=15cm]{nscalars-dec.eps}}}\n\\capbox{Power emission for scalars on the brane}{Energy emission rates for scalar fields on the brane from\na $(4+n)$D black hole.\\label{sc-dec}}\n\\end{center}\n\\end{figure}\n\nThe numerical solution for the grey-body factor allows the computation of the energy emission rate for scalar fields on the brane.  Using eq.~(\\ref{pdecay-brane}) it is found that the suppression of the grey-body factor with $n$ does not lead to the suppression of the emission rate itself. The behaviour of the differential energy emission rate is given in Figure~\\ref{sc-dec}. As $n$ increases, the increase in the temperature of the black hole, and therefore in its emitting power, overcomes the decrease in the grey-body factor and leads to a substantial enhancement of the energy emission rate.  Figure~\\ref{sc-dec} clearly shows that the enhancement of the peak of the emission curve can be up to several orders of magnitude compared to the 4-dimensional case.  In addition the peak in the spectrum is shifted to higher values of the energy parameter $\\omega r_\\text{h}$ since for fixed $r_\\text{h}$ an increase in $n$ corresponds to an increase in the temperature of the radiating body.\n\nIn order to quantify the enhancement of the emission rate as the number of extra dimensions projected onto the brane increases, the total flux and power emissivities were computed, for a range of values of $n$, by integrating eqs.~(\\ref{4flux}) and (\\ref{4power}) with respect to $\\om$.  The results obtained are displayed in Table~\\ref{sc-tab}.  The relevant emissivities for different values of $n$ have been normalized in terms of those for $n=0$. From the entries of this table, the order-of-magnitude enhancement of both the flux and power radiated by the black hole as $n$ increases is again clear.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|}\n\\hline\n$n$& Flux  & Power\\\\\n\\hline\n0 & 1 & 1\\\\\n1 & 4.75 & 8.94\\\\\n2 & 13.0 & 36.0\\\\\n3 & 27.4 & 99.8\\\\\n4 & 49.3 & 222\\\\\n5 & 79.9 & 429\\\\\n6 & 121 & 749\\\\\n7 & 172 & 1220\\\\\n\\hline\n\\end{tabular}\n\\capbox{Emissivities for scalar fields on the brane}{Flux and power emissivities for scalar fields on the brane.\\label{sc-tab}}\n\\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Spin 1\/2 fields}\n\nUnlike the case of scalar fields, the study of the emission of fields with\nnon-vanishing spin involves, in principle, fields with more than\none component. Equation~(\\ref{radial}) depends on the helicity number $s$ which can be either $+$1\/2 or $-$1\/2 and this leads to radial equations for the two different components of the field. As mentioned in \\secref{numcal}, the upper ($+$1\/2) and lower ($-$1\/2) components carry mainly the in-going and out-going parts of the field respectively. Although knowledge of both components is necessary to construct the complete solution for the emitted field, the determination of either is sufficient to compute the absorption coefficient\n$\\hat {\\cal A}_j^{(s)}$ ($j$ is the total angular momentum number). For\nexample, if the in-going wave is known in the case of the emission of\nfields with spin $s=$1\/2, eq.~(\\ref{absorption})\nmay be directly written as \\cite{Kanti:2002ge,Cvetic:1998ap}\n\\begin{equation}\n|\\hat {\\cal A}^{(1\/2)}_j|^2=\\Biggl|\\frac{A_\\text{in}^{(\\text{h})}}\n{A_\\text{in}^{(\\infty)}}\\Biggr|^2.\n\\label{fermions}\n\\end{equation}\nThe above follows by defining the incoming flux of a fermionic field as the\nradial component of the conserved current, \n$J^\\mu=\\sqrt{2}\\,\\sigma^\\mu_{AB}\\,\\Psi^A\\,\\bar \\Psi^B$, integrated over a\ntwo-dimensional sphere and evaluated at both the horizon and infinity.\n\nThe grey-body factor is again related to the absorption probability through eq.~(\\ref{brane-loc}) with $\\ell$ now being replaced by the total angular momentum $j$.  Numerically solving the radial equation~(\\ref{radial}) and computing $\\hat {\\cal A}^{(1\/2)}_j$ gives the behaviour of the grey-body factor, in terms of the energy parameter $\\omega r_\\text{h}$ and the number of extra dimensions $n$.  This is shown in Figure~\\ref{grey05} for four different values of $n$. \n\nAs in the scalar case, at low energies the grey-body factor assumes a n-zero asymptotic value; this depends on the dimensionality of\nspace-time and increases with $n$. The enhancement of \n$\\hat{\\sigma}^{(1\/2)}_\\text{abs}(\\omega)$ with $n$ in the low-energy regime continues up to intermediate values of $\\omega r_\\text{h}$ where the situation is reversed: as $n$ becomes larger, the high-energy grey-body factor becomes more and more suppressed as was found in the scalar case.  The full analytic results\nderived in \\cite{Kanti:2002ge} provide a reasonable description of the low-energy behaviour except in the $n=0$ case; however, as expected, they fail to\ngive accurate information for the high-energy regime.  As for scalar fields, the high-energy grey-body factors for fermions are shown to oscillate around asymptotic values determined by the effective radius of eq.~(\\ref{effective}).\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$\\hat{\\sigma}^{(1\/2)}_\\text{abs}(\\om)\/\\pi r_\\text{h}^2$}\n\\scalebox{0.5}{\\rotatebox{-90}{\\includegraphics[width=\\textwidth]{ngrey05.ps}}}\n\\capbox{Grey-body factors for fermions on the brane}{Grey-body factors for fermion emission on the brane from a $(4+n)$D\nblack hole.\\label{grey05}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$r_\\text{h} d^2 \\hat{E}^{(1\/2)}\/dtd\\om$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=23cm, height=15cm]\n{nfermions-dec.eps}}}\n\\capbox{Power emission for fermions on the brane}{Energy emission rates for fermions on the brane from a $(4+n)$D black hole.\\label{fer-dec}}\n\\end{center}\n\\end{figure}\nThe energy emission rate for fermion fields on the brane is shown in Figure~\\ref{fer-dec} for various values of $n$.\nAs $n$ increases, the power emission is found to be significantly enhanced at both low and high energies. The emission curves show the same features as for the emission of scalar fields, i.e.\\ the peak increases in height by several orders of magnitude and shifts towards higher energies (due to the increasing black hole temperature).  Quantitative results regarding the enhancement with $n$ of both the flux and power fermion emission spectra were obtained by integrating the data in Figure~\\ref{fer-dec}; they are shown in Table~\\ref{fer-tab}. Once again, the enhancement is substantial, and in fact is even more significant than for scalar emission.\n\\begin{table}[b]\n\\begin{center}\n\\begin{tabular}{|l|c|c|}\n\\hline\n$n$& Flux  & Power\\\\\n\\hline\n0 & 1 & 1\\\\\n1 & 9.05 & 14.2\\\\\n2 & 27.6 & 59.5\\\\\n3 & 58.2 & 162\\\\\n4 & 103 & 352\\\\\n5 & 163 & 664\\\\\n6 & 240 & 1140\\\\\n7 & 335 & 1830\\\\\n\\hline\n\\end{tabular}\n\\capbox{Emissivities for fermions on the brane}{Flux and power emissivities for fermions on the brane.\\label{fer-tab}} \n\\end{center}\n\\end{table}\n\n\\subsection{Spin 1 fields}\n\nIn the case of the emission of gauge boson fields, the incoming flux can be\ncomputed from the energy-momentum tensor $T^{\\mu\\nu}=2 \\sigma^{\\mu}_{AA'} \\sigma^{\\nu}_{BB'} \\Psi^{AB}\\,\\bar\\Psi^{A'B'}$; the time-radial component is\nintegrated over a two-dimensional sphere and then evaluated at the horizon \nand at infinity. By making use of the solution for the in-going wave, the \nfollowing expression for the absorption probability of eq.~(\\ref{absorption}),\nis obtained \\cite{Kanti:2002ge,Cvetic:1998ap}:\n\\begin{equation}\n|\\hat {\\cal A}^{(1)}_j|^2=\\frac{1}{r_\\text{h}^2}\\,\n\\Biggl|\\frac{A_\\text{in}^{(\\text{h})}}{A_\\text{in}^{(\\infty)}}\\Biggr|^2.\n\\label{bosons}\n\\end{equation}\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$\\hat{\\sigma}^{(1)}_\\text{abs}(\\om)\/\\pi r_\\text{h}^2$}\n\\scalebox{0.5}{\\rotatebox{-90}{\\includegraphics{ngrey1.ps}}}\n\\capbox{Grey-body factors for gauge bosons on the brane}{Grey-body factors for gauge boson emission on the brane from a $(4+n)$D black hole.\\label{grey1}}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center} \n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$r_\\text{h} d^2 \\hat{E}^{(1)}\/dtd\\om$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=23cm, height=15cm]\n{ngauge-dec.eps}}}\n\\capbox{Power emission for gauge fields on the brane}{Energy emission rates for gauge fields on the brane from a $(4+n)$D\nblack hole.\\label{gauge-dec}}\n\\end{center}\n\\end{figure}\n\nThe exact results for the grey-body factors and energy emission rates for \ngauge boson fields are given in Figures~\\ref{grey1} and \\ref{gauge-dec}\nrespectively. A distinct feature of the grey-body factor for gauge fields,\npreviously observed in the 4-dimensional case, is that it vanishes when \n$\\omega r_\\text{h} \\rightarrow 0$. The same behaviour is observed for all values of $n$.  This leads to the suppression of the energy emission rate, in the low-energy regime, compared to those for scalar and fermion fields.  Up to intermediate energies the grey-body factors exhibit enhancement with increasing $n$ as in the case of fermion fields.  An asymptotic behaviour, similar to the previous cases, is observed in the high-energy regime with each grey-body factor assuming, after oscillation, the geometrical optics value (which decreases with increasing $n$). This result establishes the existence of a universal behaviour of all types of particles emitted by black holes at high energies. This behaviour is independent of the particle spin but dependent on the number of extra dimensions projected onto the brane. \n\\enlargethispage{-\\baselineskip}\n\nAgain the total flux and power emissivities for gauge fields on the brane can be obtained for different values of $n$.  The exact results obtained by numerically integrating with respect to energy are given in Table~\\ref{gauge-tab}.\nAs anticipated, the pattern of enhancement with $n$ is also observed \nfor the emission of gauge bosons. It is worth noting that \nthe relative enhancement observed in this case is the largest among all particle types---this is predominantly due to the large suppression of gauge boson emission in four dimensions.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|}\n\\hline\n$n$& Flux  & Power\\\\\n\\hline\n0 & 1 & 1\\\\\n1 & 19.2 & 27.1\\\\\n2 & 80.6 & 144\\\\\n3 & 204 & 441\\\\\n4 & 403 & 1020\\\\\n5 & 689 & 2000\\\\\n6 & 1070 & 3530\\\\\n7 & 1560 & 5740\\\\\n\\hline\n\\end{tabular}\n\\capbox{Emissivities for gauge fields on the brane}{Flux and power emissivities for gauge fields on the brane.\\label{gauge-tab}} \n\\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Relative emissivities for different species}\n\nIt is interesting to investigate how the relative numbers of scalars,\nfermions and gauge bosons emitted on the brane change\nas the number of extra dimensions projected onto the brane varies. In other\nwords, this means finding out what type of particles the black hole prefers\nto emit, for different values of $n$, and what fraction of the total energy\neach particular type of particle carries away during emission. \n\n\\begin{figure}\n\\unitlength1cm\n\\begin{center}\n\\psfrag{x}[][][1.25]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.25]{$r_\\text{h} d^2 \\hat{E}^{(0)}\/dtd\\om$}\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.75}{\\rotatebox{0} \n{\\includegraphics{nrelativen0.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.75}{\\rotatebox{0}\n{\\includegraphics{nrelativen6.eps}}}\n\\end{minipage}\n\\capbox{Power emission on the brane for $n=0$ and $n=6$}{Energy emission rates for the emission of scalars, fermions and gauge bosons on the brane with {\\bf(a)} $n=0$ and {\\bf(b)} $n=6$.\\label{relan6}}\n\\end{center}\n\\end{figure}\n\nComparing the energy emission rates for different types of particles and for \nfixed $n$, gives the qualitative behaviour.   Figures~\\ref{relan6}(a) and \\ref{relan6}(b) show (with a linear scale) the a power spectra for $n=0$ and $n=6$ respectively; these two figures demonstrate very clearly the effects already discussed, namely the orders-of-magnitude enhancement of the emission rates and the displacement of the peak to higher energies, as $n$ increases. Figure~\\ref{relan6}(a) reveals that, in the absence of any extra dimensions, most of the energy of the black hole emitted on the brane is in the form of scalar particles;\nthe next most important are the fermion fields, and less significant are the \ngauge bosons. As $n$ increases, the emission rates for all species are enhanced but at different rates.  Figure~\\ref{relan6}(b) clearly shows that, for a \nlarge number of extra dimensions, the most effective `channel' during\nthe emission of brane-localized modes is that of gauge bosons;\nthe scalar and fermion fields follow second and third respectively.\nThe changes in the flux spectra are similar as $n$ increases. \n\n\\begin{table}[b]\n\\begin{minipage}[t]{3.05in}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline {\\rule[-3mm]{0mm}{8mm} }\n$\\!\\!n$ & $s=0$ & $s=\\frac{1}{2}$ & $s=1$ \\\\\n\\hline\n0&1&0.37&0.11\\\\\n1&1&0.70&0.45\\\\\n2&1&0.77&0.69\\\\\n3&1&0.78&0.83\\\\\n4&1&0.76&0.91\\\\\n5&1&0.74&0.96\\\\\n6&1&0.73&0.99\\\\\n7&1&0.71&1.01\\\\\n\\hline \nBlack body&1&0.75&1\\\\\n\\hline\n\\end{tabular}\n\\capbox{Flux emission ratios}{Flux emission ratios.\\label{fratios}}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline {\\rule[-3mm]{0mm}{8mm} }\n$\\!\\!\\!n$ & $s=0$ & $s=\\frac{1}{2}$ & $s=1$ \\\\\n\\hline\n0&1&0.55&0.23\\\\\n1&1&0.87&0.69\\\\\n2&1&0.91&0.91\\\\\n3&1&0.89&1.00\\\\\n4&1&0.87&1.04\\\\\n5&1&0.85&1.06\\\\\n6&1&0.84&1.06\\\\\n7&1&0.82&1.07\\\\\n\\hline \nBlack body&1&0.87&1\\\\\n\\hline\n\\end{tabular}\n\\capbox{Power emission ratios}{Power emission ratios.\\label{pratios}}\n\\end{center}\n\\end{minipage}\n\\end{table}\n\nThe above behaviour can be more helpfully quantified by computing the relative emissivities for scalars, fermions and gauge bosons emitted on the brane by integrating the flux and power emission spectra. The relative emissivities obtained in this way are shown in Tables~\\ref{fratios} and \\ref{pratios} (they are normalized to the scalar values).  The ratios for $n\\geq 1$ are available for\nthe first time in the literature as a result of this numerical work, while\nthe $n=0$ results would appear to be the most accurate ones available.  From \\cite{MacGibbon:1990zk}, flux and power ratios are found to be 1\\,:\\,0.36\\,:\\,0.11 and 1\\,:\\,0.56\\,:\\,0.23 respectively, in good agreement with those ratios shown in Tables~\\ref{fratios} and \\ref{pratios}.  In \\cite{Giddings:2001bu} the power ratio for $n=0$ is given as 40\\,:\\,19\\,:\\,7.9 i.e.\\ 1\\,:\\,0.48\\,:\\,0.20 which is in less good agreement.  More careful examination shows that the relative power emitted by the $s=1\/2$ and $s=1$ degrees of freedom agrees exactly with Table~\\ref{pratios} and so the disagreement seems to come from the scalar value.  This is unsurprising since the scalar value quoted in \\cite{Giddings:2001bu} was only estimated from a plot in \\cite{Sanchez:1978si}.\n\\enlargethispage{-\\baselineskip}\n\nThe entries in these tables reflect the qualitative behaviour\nillustrated above for some extreme values of the number of extra dimensions.\nFor $n=0$, the scalar fields are the type of particle which are\nmost commonly produced and the ones which carry away most of the energy of\nthe black hole emitted on the brane; the fermion and gauge fields carry\napproximately 1\/2 and 1\/4, respectively, of the energy emitted in scalar fields, and their fluxes are only 1\/3 and 1\/10 of the scalar \nflux. For intermediate values of $n$, the fermion and\ngauge boson emissivities are considerably enhanced compared to\nthe scalar one and have become of approximately the same magnitude---e.g.\\\nfor $n=2$ the amount of energy spent by the black hole in the emission\nof fermions and gauge bosons is exactly the same, although the net number\nof gauge bosons is still subdominant. For large values of $n$, the situation\nis reversed: the gauge bosons dominate both flux and power spectra, with\nthe emission of fermions being the least effective channel both in\nterms of number of particles produced and energy emitted.  The reader is reminded that the above results refer to the emission of individual scalar,\nfermionic or bosonic degrees of freedom and not to the elementary particles.  However it is easy to combine the ratios in Table~\\ref{pratios} with the relevant numbers of degrees of freedom (see Table~\\ref{pprobs}) to obtain relative emissivities for different particle types. \n\nThese numerical results confirm that the relative emissivities of different particles produced by small, higher-dimensional black holes depend on the number of extra dimensions projected onto the brane.   Therefore, if Hawking radiation from such objects is detected, this could provide a way of determining the number of extra dimensions existing in nature.\n\n\\subsection{Total flux and power emitted}\n\\label{totfandp}\n\nWhilst it is often only necessary to know the relative emissivities of the fields of different spins, sometimes (for example, when estimating the black hole lifetime) the absolute values are required.  For the case of brane emission from a non-rotating black hole, the flux emitted in a field of spin $s$ can be written as\n\\begin{equation}\nF^{(s)}=\\int_0^{\\infty} \\frac{\\Gamma^{(s)} (\\omega r_\\text{h})}{\\exp\\left(\\omega\/T_\\text{H}\\right)\\mp1}\\frac{d\\omega}{2\\pi}=\\int_0^{\\infty} \\frac{\\Gamma^{(s)} (\\omega r_\\text{h})}{\\exp\\left(\\frac{4\\pi\\omega r_\\text{h}}{n+1}\\right)\\mp1}\\frac{d\\omega}{2\\pi}\\,.\n\\label{totflux}\n\\end{equation}\nThis expression is obtained by integrating equation~(\\ref{fdecay-brane}) using the definition\n\\begin{equation}\n\\Gamma^{(s)}(\\omega r_\\text{h})=\\sum_{\\ell} (2\\ell +1) |\\hat {\\cal A}^{(s)}_\\ell|^2.\n\\end{equation}\nSince the integrand in eq.~(\\ref{totflux}) is entirely a function of $\\omega r_\\text{h}$, the substitution $x=\\omega r_\\text{h}$ can be used to express $F^{(s)}$ as\n\\begin{equation}\nF^{(s)}=\\frac{1}{r_\\text{h}}\\int_0^{\\infty}\\frac{\\Gamma^{(s)}(x)}{\\exp\\left(\\frac{4\\pi x}{n+1}\\right)\\mp1}\\frac{dx}{2\\pi}\\,,\n\\end{equation}\nwhere the only dependence on the radius (or equivalently the mass) of the black hole is contained in the factor outside the integral.\n\nA similar procedure, starting from equation~(\\ref{4power}), shows the power emitted in a field of spin $s$ to be\n\\begin{equation}\nP^{(s)}=\\frac{1}{r_\\text{h}^2}\\int_0^{\\infty}\\frac{x\\Gamma^{(s)}(x)}{\\exp\\left(\\frac{4\\pi x}{n+1}\\right)\\mp1}\\frac{dx}{2\\pi}\\,.\n\\end{equation}\n\n\\enlargethispage{-\\baselineskip}\nThese results of these integrations are given in Tables~\\ref{fvalues} and \\ref{pvalues}.  The values in these tables can be combined with the total numbers of degrees of freedom for each spin.  These are given in the last row\nof Table~\\ref{pprobs} and are 4, 90 and 24 for $s=0$, 1\/2 and 1 respectively.\nTherefore the total flux $F$ and power $P$ emitted by black holes of different numbers of dimensions can be calculated---the results are shown in Table~\\ref{totfp}.\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n$n$& $r_\\text{h} F^{(0)}$  & $r_\\text{h} F^{(1\/2)}$ & $r_\\text{h} F^{(1)}$ \\\\\n\\hline\n0 & 0.00133 & 0.000486 & 0.000148\\\\\n1 & 0.00631 & 0.00439 & 0.00283\\\\ \n2 & 0.0173 & 0.0134 & 0.0119\\\\\n3 & 0.0364 & 0.0283 & 0.0301\\\\ \n4 & 0.0655 & 0.0499 & 0.0596\\\\\n5 & 0.106 & 0.0789 & 0.102\\\\\n6 & 0.160 & 0.116 & 0.159\\\\\n7 & 0.229 & 0.163 & 0.231\\\\\n\\hline\n\\end{tabular}\n\\capbox{Flux for different spin fields}{Total flux emitted in fields of different spins.\\label{fvalues}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n$n$& $r_\\text{h}^2 P^{(0)}$  & $r_\\text{h}^2 P^{(1\/2)}$ & $r_\\text{h}^2 P^{(1)}$ \\\\\n\\hline\n0 & 0.000298 & 0.000164 & 6.72$\\times 10^{-5}$\\\\\n1 & 0.00266 & 0.00232 & 0.00182\\\\\n2 & 0.0107 & 0.00973 & 0.00971\\\\\n3 & 0.0297 & 0.0265 & 0.0296\\\\\n4 & 0.0661 & 0.0575 & 0.0686\\\\\n5 & 0.128 & 0.109 & 0.135\\\\\n6 & 0.223 & 0.187 & 0.237\\\\\n7 & 0.362 & 0.299 & 0.386\\\\\n\\hline\n\\end{tabular}\n\\capbox{Power for different spin fields}{Total power emitted in fields of different spins.\\label{pvalues}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|l|c|c|}\n\\hline\n$n$& $r_\\text{h} F$  & $r_\\text{h}^2 P$\\\\\n\\hline\n0 & 0.0526 & 0.0175\\\\\n1 & 0.489 & 0.263\\\\\n2 & 1.56 & 1.15\\\\\n3 & 3.41 & 3.21\\\\\n4 & 6.18 & 7.09\\\\\n5 & 9.98 & 13.5\\\\\n6 & 14.9 & 23.4\\\\\n7 & 21.1 & 37.6\\\\\n\\hline\n\\end{tabular}\n\\capbox{Total flux and power emitted}{Total flux and power emitted from black holes of different numbers of dimensions.\\label{totfp}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\nIt is possible to use these values for the total flux emitted from extra-dimensional black holes to comment on the usual assumption that black hole decays can\nbe considered as quasi-stationary.  The simplest way of trying to verify this is to compare the typical time between emissions (given by $F^{-1}$) with $r_\\text{h}$ (the time for light to cross the black hole radius in the natural units being used).  To be sure that the quasi-stationary approach is valid, we would require $F^{-1}\\gg r_\\text{h}$ or equivalently $r_\\text{h} F \\ll 1$.  From Table~\\ref{totfp}, this can be seen to be true only for the cases $n=0$ and $n=1$.\n\n\\enlargethispage{-\\baselineskip}\nAnother requirement for the validity of a semi-classical description of black hole production and decay is that the lifetime $\\tau \\gg 1\/M_\\text{BH}$ so that the black hole is a well-defined resonance.  This assumption can also be tested here by using the values of $P$ in Table~\\ref{totfp}.  The relationship\n\\begin{equation}\nP=-\\frac{dM_\\text{BH}}{dt}=\\frac{p}{r_\\text{h}^2}\\,,\n\\end{equation}\ncan be integrated as follows:\n\\begin{equation}\np\\int_0^\\tau dt = - \\int_{M_\\text{BH}}^0 r_\\text{h}^2 dM_\\text{BH} = \\frac{1}{\\pi M_{\\text{P}(4+n)}^2}\\left(\\frac{8 \\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)^{\\frac{2}{n+1}} \\int_0^{M_\\text{BH}} \\left(\\frac{M_\\text{BH}}{M_{\\text{P}(4+n)}}\\right)^{\\frac{2}{n+1}} dM_\\text{BH}\\,. \n\\end{equation}\nIn the above, the final Planck phase and any complications caused by kinematically forbidden emissions have been ignored.  Proceeding with the integration gives\n\\begin{equation}\n\\tau M_\\text{BH} = \\frac{1}{\\pi p}\\left(\\frac{8 \\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)^{\\frac{2}{n+1}} \\frac{n+1}{n+3} \\left(\\frac{M_\\text{BH}}{M_{\\text{P}(4+n)}}\\right)^{\\frac{2(n+2)}{n+1}}.\n\\end{equation}\nValues of $\\tau M_\\text{BH}$ for different values of $n$ are shown in Table~\\ref{taum} both in the case $M_\\text{BH}=5M_{\\text{P}(4+n)}$ and in the case $M_\\text{BH}=10M_{\\text{P}(4+n)}$. It should be noted that switching to convention `a' for the definition of $M_{\\text{P}(4+n)}$ the high-$n$ values in the table would be significantly altered (for example, the $n=7$ values would be multiplied by a factor of 13.2).  The `long'-lifetime requirement is clearly different to the quasi-stationary issue discussed above since it depends on the initial mass of the black hole.  However given the limits on $M_{\\text{P}(4+n)}$ and the energy available at the LHC, it would again seem that this requirement may not be satisfied for the higher values of $n$.\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& \\multicolumn{2}{c|}{$\\tau M_\\text{BH}$} \\\\\n\\cline{2-3}\n$n$& $M_\\text{BH}=5M_{\\text{P}(4+n)}$  & $M_\\text{BH}=10M_{\\text{P}(4+n)}$\\\\\n\\hline\n0 & 47500 & 761000\\\\\n1 & 202 & 1610\\\\\n2 & 23.3 & 148\\\\\n3 & 6.60 & 37.4\\\\\n4 & 2.77 & 14.6\\\\\n5 & 1.43 & 7.23\\\\\n6 & 0.846 & 4.12\\\\\n7 & 0.544 & 2.59\\\\\n\\hline\n\\end{tabular}\n\\capbox{Values of $\\tau M_\\text{BH}$}{Values of $\\tau M_\\text{BH}$ for different values of $n$.\\label{taum}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\enlargethispage{\\baselineskip}\nThe results presented in Table~\\ref{taum} differ significantly from equivalent results which can be obtained from an expression given in \\cite{Dimopoulos:2001en}.  Partly this is due to inclusion of the grey-body factors here, but this does not account for order-of-magnitude differences.  It would appear that the lifetime is incorrect in \\cite{Dimopoulos:2001en} because of the failure to take all the degrees of freedom of the emitted particles into account.\n\nThe numerical results presented in Table~\\ref{totfp} can also be used to calculate the average particle multiplicity, again completely ignoring the complications of the Planck phase and kinematic constraints on the decay.  Defining $r_\\text{h} F=f$, the average particle multiplicity can be expressed as\n\\begin{equation}\n\\langle N\\rangle\\,=\\,\\frac{f}{p}\\int_0^{M_\\text{BH}} r_\\text{h} \\, dM_\\text{BH} =  \\frac{1}{\\sqrt{\\pi} M_{\\text{P}(4+n)}}\\left(\\frac{8 \\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)^{\\frac{1}{n+1}}\\!\\!\\int_0^{M_\\text{BH}} \\left(\\frac{M_\\text{BH}}{M_{\\text{P}(4+n)}}\\right)^{\\frac{1}{n+1}} dM_\\text{BH}\\,.\n\\end{equation}\nPerforming the integration gives\n\\begin{equation}\n\\langle N\\rangle\\,=\\,\\frac{f}{p}\\frac{1}{\\sqrt{\\pi}}\\left(\\frac{8 \\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)^{\\frac{1}{n+1}} \\frac{n+1}{n+2} \\left(\\frac{M_\\text{BH}}{M_{\\text{P}(4+n)}}\\right)^{\\frac{n+2}{n+1}}=\\frac{f}{p}\\frac{n+1}{4\\pi}S_\\text{BH}\\,,\n\\label{avn}\n\\end{equation}\nwhere the final identification has been made by referring back to equation~(\\ref{bhent}).  Table~\\ref{avntab} shows the average multiplicities obtained for different values of $n$ along with the (initial) entropy $S_\\text{BH}$.  The average multiplicity values given improve on those in \\cite{Dimopoulos:2001hw} (where $\\langle N\\rangle\\sim M_\\text{BH}\/2\\,T_\\text{H}$ was assumed based on the constant temperature approximation) and \\cite{Cavaglia:2003hg} (where a more careful approach was taken but still without the full numerical grey-body factors).\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& \\multicolumn{2}{c|}{$\\langle N\\rangle$} & \\multicolumn{2}{c|}{$S_\\text{BH}$}\\\\\n\\cline{2-5}\n$n$& $M_\\text{BH}=5M_{\\text{P}(4+n)}$  & $M_\\text{BH}=10M_{\\text{P}(4+n)}$ & $M_\\text{BH}=5M_{\\text{P}(4+n)}$ & $M_\\text{BH}=10M_{\\text{P}(4+n)}$\\\\\n\\hline\n0 & 75.3 & 301  & 314 & 1260 \\\\\n1 & 12.8 & 36.1 & 43.1 & 122\\\\\n2 & 6.82 & 17.2 & 21.0 & 52.9\\\\\n3 & 4.78 & 11.4 & 14.2 & 33.7\\\\\n4 & 3.80 & 8.75 & 11.0 & 25.2\\\\\n5 & 3.22 & 7.23 & 9.13 & 20.5\\\\\n6 & 2.80 & 6.20 & 7.92 & 17.5\\\\\n7 & 2.52 & 5.50 & 7.06 & 15.4\\\\\n\\hline\n\\end{tabular}\n\\capbox{Values of $\\langle N\\rangle$ and $S_\\text{BH}$}{Values of $\\langle N\\rangle$ and $S_\\text{BH}$ (calculated using eqs.~(\\ref{avn}) and (\\ref{bhent}) respectively) for different values of $n$.\\label{avntab}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\nThe average multiplicities in the first column clearly do not satisfy $\\langle N\\rangle\\gg 1$ for $n>3$ which means that a significant amount of the decay will be affected by kinematic and Planck phase considerations.  However using the alternative definition of $M_{\\text{P}(4+n)}$ these results would again be modified (for $n=7$ they would be multiplied by a factor of 3.64).  There is, of course, no fundamental difference between these conventions---setting $M_{\\text{P}(4+n)}=1$~TeV in convention `a' simply corresponds to a lower value of the Planck mass in convention `d' and hence for a fixed black hole mass the Hawking temperature is lower and the average multiplicity higher.  \n\nThe tabulated entropies allow comparison with the requirement that $1\/\\sqrt{S_\\text{BH}}$ should be much less than unity if statistical fluctuations of the number of micro-canonical degrees of freedom are to be small (as is necessary for a semi-classical description of the black hole decay to be valid).\n\n\n\n\\section{Emission in the bulk}\n\\label{embulk}\n\nAn important question regarding the emission of particles by\nhigher-dimensional black holes is how much of this energy is radiated\nonto the brane and how much is lost in the bulk. In the former case,\nthe emitted particles are zero-mode gravitons and scalar fields as \nwell as Standard Model fermions and gauge bosons, while in the latter\ncase all emitted energy is in the form of massive Kaluza-Klein gravitons\nand, as discussed in \\secref{gbfintro}, possibly also scalar fields. In \\cite{Emparan:2000rs}, it was shown that the whole\ntower of KK excitations of a given particle carries approximately the\nsame amount of energy as a massless particle emitted on the\nbrane. Combining this result with the observation that many more types of particles\nlive on the brane than in the bulk, it was concluded that most of the\nenergy of the black hole goes into brane modes. The results obtained\nin \\cite{Emparan:2000rs} were only approximate since the dependence of the grey-body factor on the energy of the emitted particle was ignored and the\ngeometric expression for the area of the horizon (valid in the high-energy regime) was used instead.  \n\nIn order to provide an accurate answer to the question of how much energy\nis emitted into the bulk compared to on the brane, it is necessary to take into account the dependence of the grey-body factor on both energy and number of extra dimensions.   There is a similar discussion in \\cite{Cavaglia:2003hg} but it is incomplete because full numerical results for the grey-body factor in all energy regimes were not available.  In this section the emission of scalar modes in the bulk is thoroughly investigated.  Exact numerical results are produced for the grey-body factors and energy emission rates in terms of the energy and number of extra dimensions.  This allows the questions mentioned above to be addressed by calculating the total bulk-to-brane relative emissivities for different values of $n$.\n\n\\subsection{Grey-body factors and emission rates}\n\nThe analysis in this section is relevant for gravitons and possibly scalar fields, and requires knowledge of the solutions of the\ncorresponding equations of motion in the bulk.  Only the case of scalar fields is considered here since the bulk equation of motion is known---the emission of bulk scalar modes was previously studied analytically, in the low-energy regime, in \\cite{Kanti:2002nr}.\\footnote{After this work was completed, some plots of grey-body factors for bulk scalars were discovered in Figure 1 of \\cite{Frolov:2002as}.  It is unclear how these were calculated and only the $\\ell=0$ contributions for $n=$1--3 are shown; however, they provided a useful cross-check on some of the results presented here.}\n\nA scalar field propagating in the background of a higher-dimensional,\nnon-rotating (Schwarzschild-like) black hole, with line-element given by eq.~(\\ref{metric-D}), satisfies the following equation of motion \\cite{Kanti:2002nr}\n\\begin{equation}\n\\frac{h(r)}{r^{n+2}}\\,\\frac{d \\,}{dr}\\,\n\\biggl[\\,h(r)\\,r^{n+2}\\,\\frac{d R}{dr}\\,\\biggr] +\n\\biggl[\\,\\om^2 - \\frac{h(r)}{r^2}\\,\\ell\\,(\\ell+n+1)\\,\\biggr] R =0 \\, .\n\\label{scalareqn}\n\\end{equation}\nThis radial equation was obtained by using a separable solution similar to equation~(\\ref{sepsoln}) but with ${}_sS^m_{\\ell}(\\theta)$ replaced by $S^m_{\\ell}(\\theta_1,\\theta_2,\\ldots,\\theta_n)$. The functions $e^{im\\varphi}\\,S^m_{\\ell}(\\theta_1,\\theta_2,\\ldots,\\theta_n)$ can be written as $\\tilde{Y}^m_{\\ell}(\\Omega_{2+n})$\\,; these are generalizations of the usual spherical harmonic functions to the case of $(3+n)$ spatial dimensions \\cite{muller} and account for the $\\ell(\\ell+n+1)$ term in eq.~(\\ref{scalareqn}).  \n\nAs for the emission of particles on the brane, the determination of the grey-body factor for bulk emission requires the above equation to be solved over the whole radial domain. The exact solution for the radial function must interpolate between the near-horizon and far-field asymptotic solutions, given by\n\\begin{equation}\n\\label{near-bulk}\nR^{(\\text{h})}=A_\\text{in}^{(\\text{h})}\\,e^{-i\\omega r^{*}}+ A_{out }^{(\\text{h})}\\,e^{i\\omega r^{*}},\n\\end{equation}\nand\n\\begin{equation}\n\\label{far-bulk}\nR^{(\\infty)}=A_\\text{in}^{(\\infty)}\\,\\frac{e^{-i\\omega r}}{\\sqrt{r^{n+2}}}+\nA_\\text{out}^{(\\infty)}\\,\\frac{e^{i\\omega r}}{\\sqrt{r^{n+2}}}\\,,\n\\end{equation}\nrespectively.  Again the boundary condition that no out-going solution should exist near the horizon of the black hole is imposed which means that $A_\\text{out}^{(\\text{h})}=0$. The solution at infinity comprises, as usual, both in-going and out-going modes. \n\nThe absorption probability $|\\tilde {\\cal A}_\\ell|^2$ may\nthen be calculated either by using eq.~(\\ref{scalars}) or from the ratio $|A_\\text{in}^{(\\text{h})}\/A_\\text{in}^{(\\infty)}|^2$ (this latter approach is found to be more numerically stable for low energies and\/or larger values of $n$).  A tilde will now denote bulk quantities, as opposed to brane quantities which carry a hat.  The grey-body factor $\\tilde \\si_\\ell(\\om)$ may be determined by using eq.~(\\ref{greydef}). The dimensionality of the grey-body factor changes as the number of extra dimensions varies; therefore, in order to be able to compare its values for different $n$, it is normalized to the horizon area of a $(4+n)$-dimensional black hole.  The discussion of the normalization procedure and the subsequent comments on the geometrical optics limit are due to Kanti \\cite{Kantiprivate,Harris:2003eg}.\n\nFirst equation~(\\ref{greydef}) is re-written in the form\n\\begin{equation}\n\\tilde \\si_\\ell(\\om) = \\frac{2^{n}}{\\pi}\\,\n\\Ga\\Bigl[\\frac{n+3}{2}\\Bigr]^2\\,\n\\frac{\\tilde A_\\text{h}}{(\\om r_\\text{h})^{n+2}}\\, \\tilde N_\\ell\\,\n|\\tilde {\\cal A}_\\ell|^2,\n\\label{greyb}\n\\end{equation}\nwhere $\\tilde N_\\ell$ is the multiplicity of states corresponding to the\nsame partial wave $\\ell$.  For a $(4+n)$-dimensional space-time it is given by\n\\begin{equation}\n\\tilde N_\\ell= \\frac{(2\\ell+n+1)\\,(\\ell+n)!}{\\ell! \\,(n+1)!}\\,,\n\\label{bulk-mult}\n\\end{equation}\nand the horizon area in the bulk is defined as\n\\begin{equation}\n\\begin{split}\n\\tilde A_\\text{h} =&\\, \nr_\\text{h}^{n+2}\\,\\int_0^{2 \\pi} \\,d \\varphi \\,\\prod_{k=1}^{n+1}\\,\n\\int_0^\\pi\\,\\sin^k\\theta_{n+1}\\,\n\\,d(\\sin\\theta_{n+1}) \\\\[3mm]\n=&\\, r_\\text{h}^{n+2}\\,(2\\pi)\\,\\prod_{k=1}^{n+1}\\,\\sqrt{\\pi}\\,\\,\\frac{\\Ga[(k+1)\/2]}\n{\\Ga[(k+2)\/2]}\\\\[3mm]\n=& \n\\,r_\\text{h}^{n+2}\\,(2\\pi)\\,\\pi^{(n+1)\/2}\\,\\Ga\\Bigl[\\frac{n+3}{2}\\Bigr]^{-1}.\n\\end{split}\n\\end{equation}\n\nEquation~(\\ref{greyb}) allows the computation of the grey-body factor's low-energy limit once the corresponding expression for the absorption\ncoefficient is determined. Analytic results for $\\tilde {\\cal A}_\\ell$\nwere derived in \\cite{Kanti:2002nr} by solving eq.~(\\ref{scalareqn}) in the two\nasymptotic regimes (near-horizon and far-field) and matching them in an intermediate zone. It was found that the low-energy expression of the $\\ell=0$ absorption coefficient has the form\n\\begin{equation}\n|\\tilde {\\cal A}_0|^2 = \\biggl(\\frac{\\omega r_\\text{h}}{2}\\biggl)^{n+2}\n\\,\\frac{4 \\pi}{\\Gamma[(n+3)\/2]^2} + \\ldots \\,,\n\\end{equation}\nwhere the ellipsis denotes higher-order terms in the $\\omega r_\\text{h}$ power series. These terms, as well as the corresponding expressions for higher partial waves, vanish quickly in the limit $\\omega r_\\text{h} \\rightarrow 0$, leaving the above term as the dominant one. Substituting\ninto eq.~(\\ref{greyb}) it is clear that, in the low-energy regime, the grey-body factor is given by the area $\\tilde A_\\text{h}$ of the black hole horizon. This behaviour is similar to the 4-dimensional case except that the area of the horizon now changes with $n$. \n\nIn the high-energy regime, it is anticipated that an equivalent of the geometrical optics limit discussed in \\secref{numscalar} will be recovered. In four dimensions, the low- and high-energy asymptotic limits are $4\\pi r_\\text{h}^2$ and $\\pi r_\\text{c}^2$ respectively.  This has led to the na\\\"{\\i}ve generalization that, in an arbitrary number of dimensions, the high-energy expression for the grey-body factor will be approximately $\\Omega_{n+2}\\,r_\\text{c}^{n+2}\/4$, where $\\Omega_{n+2}$ is defined as in eq.~(\\ref{omegapdef}). It will shortly be shown that this is in\nfact an over-estimate of the high-energy limit. \n\nAs in four dimensions it is assumed, for high energy particles, that the grey-body factor\nbecomes equal to the area of an absorptive body of radius $r_\\text{c}$, projected on a plane parallel to the orbit of the moving particle\n\\cite{misner}. According to ref. \\cite{Emparan:2000rs}, the value of the effective radius $r_\\text{c}$ remains the same for both bulk and brane particles and\nis given by eq.~(\\ref{effective}). The area of the absorptive body depends\nstrongly on the dimensionality of space-time and its calculation requires\none of the azimuthal angles to be set to $\\pi\/2$. A careful calculation\nreveals that the `projected' area is given by\n\\begin{equation}\n\\tilde A_\\text{p} = \\frac{2 \\pi}{(n+2)}\\,\\frac{\\pi^{n\/2}}{\\Gamma[(n+2)\/2]}\\,\nr_\\text{c}^{n+2} = \\frac{1}{n+2}\\,\\Omega_{n+1}\\,r_\\text{c}^{n+2}\\,.\n\\end{equation}\n\nThe above relation reduces to the usual 4-dimensional result\n($\\tilde A_\\text{p} = \\pi r_\\text{c}^2$) for $n=0$ but, compared to the \\,na\\\"{\\i}ve $\\Omega_{n+2}\\,r_\\text{c}^{n+2}\/4$, leads to values reduced by 50\\% for higher values of $n$. Assuming that the grey-body factor at high\nenergies becomes equal to the absorptive area $\\tilde A_\\text{p}$, it can be explicitly written as\n\\begin{equation}\n\\begin{split}\n\\tilde \\si_\\text{g} = & \\,\\frac{1}{n+2}\\,\\frac{\\Omega_{n+1}}{\\Omega_{n+2}}\\,\n\\biggl(\\frac{r_\\text{c}}{r_\\text{h}}\\biggr)^{n+2}\\,\\tilde A_\\text{h} \\\\[3mm]\n = &\\,\\frac{1}{\\sqrt{\\pi}\\,(n+2)}\\,\\frac{\\Gamma[(n+3)\/2]}{\\Gamma[(n+2)\/2]}\\,\n\\biggl(\\frac{n+3}{2}\\biggr)^{(n+2)\/(n+1)}\\,\n\\biggl(\\frac{n+3}{n+1}\\biggr)^{(n+2)\/2}\\,\\tilde A_\\text{h}\\,.\n\\label{high}\n\\end{split}\n\\end{equation}\nIn the above the same normalization is used as in the low-energy regime---that is, the normalization is in terms of $\\tilde{A}_\\text{h}$, the area of the $(4+n)$-dimensional horizon.  The values predicted by eq.~(\\ref{high}) are tabulated in Table~\\ref{golim}, along with the more na\\\"{\\i}ve $\\Omega_{n+2}\\,r_\\text{c}^{n+2}\/4$ prediction.\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|l|c|c|}\n\\hline\n$n$ & $\\phantom{\\Bigl.}\\tilde \\si_\\text{g}\/\\tilde{A}_\\text{h}\\phantom{\\Bigr.}$ & $\\Omega_{n+2}\\,r_\\text{c}^{n+2}\/4\\tilde{A}_\\text{h}$\\\\\n\\hline\n0 & 1.69 & 1.69\\\\\n1 & 1.70 & 2\\\\\n2 & 1.77 & 2.36\\\\\n3 & 1.85 & 2.72\\\\\n4 & 1.93 & 3.09\\\\\n5 & 2.01 & 3.45\\\\\n6 & 2.08 & 3.81\\\\\n7 & 2.16 & 4.17\\\\\n\\hline\n\\end{tabular}\n\\capbox{High-energy limits of grey-body factors for bulk emission}{High-energy limits of grey-body factors for bulk emission, given in units of the $(4+n)$-dimensional area $\\tilde{A}_\\text{h}$.\\label{golim}} \n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\nTurning now to the numerical analysis, the grey-body factors can be found by using eq.~(\\ref{greyb}) and the exact numerical results for the absorption\ncoefficients.  Their behaviour is shown in Figure~\\ref{grey0-bulk}. As\nit was anticipated after the above discussion, the normalized grey-body \nfactors, in the low-energy regime, tend to unity for all values of $n$\nas each one adopts the value of the black hole horizon area\nto which it has been normalized. As for the emission of \nscalar fields on the brane, the grey-body factors are suppressed with increasing $n$ in the low-energy regime, start oscillating at intermediate energies and then tend to their asymptotic high-energy limits. A simple numerical analysis shows that the na\\\"{\\i}ve expression\n$\\Omega_{n+2}\\,r_\\text{c}^{n+2}\/4$ fails to describe the high-energy asymptotic\nlimits for all values of $n$ larger than zero.  In contrast, eq.~(\\ref{high}) gives asymptotic values which are verified by the numerical results.  As in the brane emission case, it is found that relatively large values of $\\omega r_\\text{h}$ are required before the asymptotic values are approached for the higher values of $n$ (this can be seen in Figure~\\ref{grey0-bulk} in which the $n=4$ and 6 curves clearly do not attain the asymptotic values given in Table~\\ref{golim}).\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$\\tilde{\\sigma}^{(0)}_\\text{abs}(\\om)\/\\tilde{A}_\\text{h}$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=22cm, height=15.4cm]\n{ngrey-bulk.eps}}}\n\\capbox{Grey-body factors for scalars in the bulk}{Grey-body factors for scalar emission in the bulk from a $(4+n)$D\nblack hole.\\label{grey0-bulk}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.75]{$r_\\text{h} d^2\\tilde{E}^{(0)}\/dtd\\om$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=23cm, height=15cm]\n{nbulkn.eps}}}\n\\capbox{Power emission for scalars in the bulk}{Energy emission rates for scalar fields in the bulk from a $(4+n)$D\nblack hole.\\label{rate-bulk}}\n\\end{center}\n\\end{figure}\n\nIn general, the suppression of the grey-body factor for bulk emission at low\nenergies is milder than the one for brane emission. However this does not\nlead to higher emission rates for bulk modes compared to those for brane\nmodes: the integration over the phase-space in eq.~(\\ref{4power})\ninvolves powers of $\\om r_\\text{h}$ which cause an increasingly\nsuppressive effect in the low-energy regime as $n$ increases. Nevertheless,\nthe increase in the temperature of the black hole (still given by $T_\\text{H}=(n+1)\/4 \\pi r_\\text{h}$) eventually overcomes the decrease in the grey-body factor and causes the enhancement of the emission rate with $n$ at\nhigh energies (as well as shifting the spectrum peak to higher energies).  The behaviour of the differential energy emission rates as a function of the energy parameter $\\om r_\\text{h}$ is shown in Figure~\\ref{rate-bulk} for some indicative values of $n$.  Comparing these numerical results with the analytic results of ref. \\cite{Kanti:2002nr} shows that the earlier results were reasonably successful in describing the low-energy behaviour of both the grey-body factors and the energy emission rates for scalar fields in the bulk.\n\n\\subsection{Bulk-to-brane relative emissivities}\n\n\\enlargethispage{-3\\baselineskip}\nThe aim of this section is to perform an analysis which provides an answer\nto the question of the relative bulk-to-brane emissivity.  This requires the differential energy emission rates in the bulk and on the brane to be evaluated; the two quantities are then compared for different numbers of extra dimensions. \n\nEquation~(\\ref{4power}) for energy emission in the bulk may alternatively be\nwritten, in terms of the absorption coefficient, as\n\\begin{equation}\n\\frac{d \\tilde E(\\om)}{dt} = \n\\sum_{\\ell} \\tilde N_\\ell\\, |\\tilde {\\cal A}_\\ell|^2\\,\n\\frac{\\om}{\\exp\\left(\\om\/T_\\text{H}\\right) - 1}\\,\\,\\frac{d \\om}{2\\pi}\\,.\n\\label{alter-bulk}\n\\end{equation}\nThe above can be compared with the corresponding expression for the emission\nof brane-localized modes given, as in eq.~(\\ref{pdecay-brane}), by\n\\begin{eqnarray}\n\\frac{d \\hat E (\\om)}{dt} =  \\sum_{\\ell} \\hat N_\\ell\\, |\\hat {\\cal  A}_\\ell|^2\\,\n\\frac{\\om}{\\exp\\left(\\om\/T_\\text{H}\\right) - 1}\\,\\,\\frac{d \\om}{2\\pi}\\,,\n\\label{emission-br}\n\\end{eqnarray}\nwhere $\\hat N_\\ell =2\\ell +1$.\nSince the temperature is the same for both bulk and brane modes, the bulk-to-brane ratio of the two energy emission rates will be simply given by the expression \n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[t][b][1.75]{$\\omega r_\\text{h}$}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=23cm, height=15cm]\n{nratio-ex.eps}}}\n\\capbox{Bulk-to-brane relative emission rates for scalars}{Bulk-to-brane energy emission rates for scalar fields from a\n$(4+n)$D black hole.\\label{bb-ratio}}\n\\end{center}\n\\end{figure}\n\\begin{equation}\n\\frac{d \\tilde E\/dt}{d \\hat E\/dt} = \\frac{\\sum_{\\ell} \\tilde N_\\ell\\,\n|\\tilde {\\cal A}_\\ell|^2}{\\sum_{\\ell} \\hat N_\\ell\\,\n|\\hat {\\cal A}_\\ell|^2}\\,,\n\\label{ratio}\n\\end{equation}\n\\enlargethispage{-\\baselineskip}and will depend on the scaling of the multiplicities of states and the\nabsorption coefficients with $n$.\\footnote{The absorption coefficients are related to the grey-body factors through eq.~(\\ref{greydef}). This includes a multiplicative coefficient which depends on both $\\omega r_\\text{h}$ and $n$, and so the coefficients might have a completely different behaviour from the grey-body factors themselves.}  \n\nAs is clear from eq.~(\\ref{bulk-mult}), the\nmultiplicity of bulk modes $\\tilde N_\\ell$ increases quickly for increasing\n$n$, while $\\hat N_\\ell$ remains the same. However, it turns out that the\nenhancement with $n$ of the absorption probability $|\\hat {\\cal A}_\\ell|^2$ for\nbrane emission is considerably greater than\nthe one for bulk emission. This leads to the dominance of the emission of\nbrane-localized modes over bulk modes, particularly for intermediate values of $n$. The behaviour of this ratio is shown in Figure~\\ref{bb-ratio}.  This figure shows that, in the low-energy regime, the ratio for large $n$ is suppressed by many orders of magnitude, compared to the value of unity for $n=0$. In the high-energy regime on the other hand, the suppression becomes smaller and the ratio seems to approach unity.  A more careful examination reveals that bulk modes in fact dominate over brane modes in a limited high-energy regime which becomes broader as $n$ increases.\\footnote{For the very high energies for which this is true, it is not clear that the expressions for higher-dimensional Hawking radiation will remain valid.}\n\nA definite conclusion regarding the relative amount of energy emitted in the two `channels'---bulk and brane---can only be drawn if\nthe corresponding total energy emissivities are computed. By integrating\nthe areas under the bulk and brane energy emission rate curves, the relative energy emission rates are determined.  The results obtained, for values of $n$ from 1 to 7, are given in Table~\\ref{bb-ratios}.\n\n\\begin{table}\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n$n$ & 0 & 1  & 2  & 3 & 4 & 5 & 6 & 7 \\\\\n\\hline\nBulk\/Brane & 1 & 0.40 & 0.24 & 0.22 & 0.24 & 0.33 & 0.52 & 0.93\\\\\n\\hline \n\\end{tabular}\n\\capbox{Relative bulk-to-brane emission rates for scalars}{Relative bulk-to-brane energy emission rates for scalar fields.\n\\label{bb-ratios}}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\nFrom the entries of the Table~\\ref{bb-ratios}, it becomes clear that the emission of\nbrane-localized scalar modes is dominant, in terms of the energy emitted, for all the values of $n$ considered. As $n$ increases, the\nratio of bulk-to-brane emission gradually becomes smaller, and is\nparticularly suppressed for intermediate values, i.e.\\ $n=2$--5; in these cases, the total energy emitted in a bulk mode varies between approximately 1\/3 and 1\/4 of that emitted in\na brane mode. As $n$ increases further, the high-energy dominance of the bulk\nmodes mentioned above gives a boost to the value of the bulk-to-brane\nratio---nevertheless, the energy ratio never exceeds unity.\n\nThe above analysis provides exact results for the energy emission rates of brane and bulk scalar modes and gives considerable support to\nearlier, more heuristic, arguments \\cite{Emparan:2000rs}, according to which a\n$(4+n)$-dimensional black hole emits mainly brane modes. A complete confirmation would mean performing a similar analysis for the emission of gravitons, but there are still theoretical and numerical issues which prevent this. \n\n\\section{Rotating black holes}\n\\label{rotbh}\n\nWe have already seen that a black hole created in a hadron collider is expected to have some angular momentum $J$ about an axis perpendicular to the plane of parton collision.  \nUsing the same na\\\"{\\i}ve argument as in \\secref{bhprod}, we assume $J=bM_\\text{BH}\/2$ (where $b$ is the impact parameter of the colliding partons) and that $b<2r_\\text{h}$ if a black hole is to form.  The maximum possible value of the rotation parameter defined in eq.~(\\ref{astardef}) is then found to be\n\\begin{equation}\n\\label{astarmax}\na_*^\\text{max}=\\frac{n+2}{2}\\,.\n\\end{equation}\nThe equivalent of eqs.~(\\ref{fdecay-brane}) and (\\ref{pdecay-brane}) are now\n\\begin{equation}\n\\frac{d \\hat N^{(s)}(\\om)}{dt} = \\sum_{\\ell,m} |\\hat {\\cal A}^{(s)}_{\\ell,m}|^2\\,\n\\frac{1}{\\exp\\left[(\\om-m\\Omega)\/T_\\text{H}\\right] \\mp 1}\\,\\frac{d\\om}{2\\pi}\\,,\n\\label{rflux}\n\\end{equation}\nand\n\\begin{equation}\n\\frac{d \\hat N^{(s)}(\\om)}{dt} = \\sum_{\\ell,m} |\\hat {\\cal A}^{(s)}_{\\ell,m}|^2\\,\n\\frac{\\om}{\\exp\\left[(\\om-m\\Omega)\/T_\\text{H}\\right] \\mp 1}\\,\\frac{d\\om}{2\\pi}\\,,\n\\label{rpower}\n\\end{equation}\nwhere the Hawking temperature is now given by\n\\begin{equation}\nT_\\text{H}=\\frac{(n+1)+(n-1)a_*^2}{4\\pi(1+a_*^2)r_\\text{h}}\\,,\n\\end{equation}\nand $\\Omega$ is defined by\n\\begin{equation}\n\\Omega=\\frac{a_*}{(1+a_*^2)r_\\text{h}}\\,.\n\\end{equation}\n\nFor the rotating case, the metric takes on a more complicated form than previously.  In general $(4+n)$-dimensional objects are described by $(n+3)\/2$ angular momentum parameters; however, it is assumed that here there is only one non-zero parameter (about an axis in the brane).  This is reasonable because the partons which collide to produce the black hole are themselves on the brane.  Hence the metric reduces to\n\\begin{equation}\n\\begin{split}\nds^2=\\left(1-\\frac{\\mu}{\\Sigma r^{n-1}}\\right)dt^2&+\\frac{2 a\\mu\\sin^2\\theta}{\\Sigma r^{n-1}}dtd\\varphi-\\frac{\\Sigma}{\\Delta}dr^2 \\\\[3mm]\n&-\\Sigma d\\theta^2-\\left(r^2+a^2+\\frac{a^2\\mu\\sin^2\\theta}{\\Sigma r^{n-1}}\\right)\\sin^2\\theta d\\varphi^2, \n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Delta=r^2+a^2-\\frac{\\mu}{r^{n-1}} \\quad\\mbox{ and } \\quad\\Sigma=r^2+a^2\\cos^2\\theta,\n\\end{equation}\nwith $a=a_*r_\\text{h}$ and $\\mu=r_\\text{S}^{n+1}$ ($r_\\text{S}$ is the Schwarzschild radius of a non-rotating black hole of the same mass).\n\nAgain the field equation is separable using a solution of the same form as in the non-rotating case, but now the radial equation is\n\\begin{equation}\n\\Delta^{-s} \\frac{d}{dr}\\left(\\Delta^{s+1}\\frac{R_s}{dr}\\right)+\\left(\\frac{K^2-isK\\Delta'}{\\Delta}+4is\\omega r+s\\Delta''-{}_s\\Lambda^m_{\\ell} \\right)R_s=0\\,,\n\\end{equation}\nwhere\n\\begin{equation}\nK=(r^2+a^2)\\omega-am \\quad\\mbox{ and } \\quad{}_s\\Lambda^m_{\\ell}={}_sE^m_{\\ell}+2s+a^2\\omega^2-2am\\omega\\,.\n\\end{equation}\nThe angular equation is now\n\\begin{multline}\n\\label{spinang}\n\\frac{1}{\\sin\\theta} \\frac{d}{d\\theta}\\left(\\sin\\theta\\frac{d \\:{}_sT^m_{\\ell}(\\theta)}{d\\theta}\\right)+ \\\\[3mm]\n\\left(-\\frac{2ms\\cot\\theta}{\\sin\\theta}-\\frac{m^2}{\\sin^2\\theta}+a^2\\omega^2\\cos^2\\theta-2a\\omega s\\cos\\theta+s-s^2\\cot^2\\theta+{}_sE^m_{\\ell}\\right){}_sT^m_{\\ell}(\\theta)=0\\,,\n\\end{multline}\nwhere $e^{im\\varphi}{}_sT^m_{\\ell}(\\theta)$ are known as spin-weighted spheroidal harmonics.\n\nThe black hole horizon is given by solving $\\Delta(r)=0$.  Unlike the 4D Kerr black hole for which where are inner and outer solutions for $r_\\text{h}$, for $n\\geq1$ there is only one solution of this equation (as can easily be seen graphically---Figure~\\ref{deltacomp}).  In the $n=0$ case the maximum possible value of $a_*$ is 1 otherwise there are no solutions of $\\Delta=0$ (this is forbidden as it would mean there is no horizon and hence there is a naked ring singularity at $r=0$).  For $n>1$ there is no fundamental upper bound on $a_*$ but only the bound given in eq.~(\\ref{astarmax}) which was argued geometrically.  For general $n$, the horizon radius $r_\\text{h}$ is found to be\n\\begin{equation}\nr_\\text{h}=\\frac{r_\\text{S}}{(1+a_*^2)^{\\frac{1}{n+1}}}\\,.\n\\end{equation}\n\\begin{figure}\n\\unitlength1cm\n\\begin{center}\n\\begin{minipage}[t]{2.0in}\n\\scalebox{0.55}{\\rotatebox{0}{\\includegraphics{nd0.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{2.0in}\n\\scalebox{0.55}{\\rotatebox{0}{\\includegraphics{nd1.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{2.0in}\n\\scalebox{0.55}{\\rotatebox{0}{\\includegraphics{nd4.eps}}}\n\\end{minipage}\n\\capbox{Number of solutions of $\\Delta(r)=0$ for different $n$}{Number of solutions of $\\Delta(r)=0$ for $n=0,1 \\mbox{ and } 4$.  On all three plots, the two functions shown are $f(r)=r^2+a^2$ and $g(r)=\\mu r^{1-n}$ for particular values of $a$ and $\\mu$.\\label{deltacomp}}\n\\end{center}\n\\end{figure}\n\nThe solution of the radial equation to obtain the grey-body factor is carried out in exactly the same same way as for the non-rotating black hole.  There is a slight difference in the asymptotic solution close to the horizon in that the $\\omega$ in eq.~(\\ref{nh-new}) is replaced by $k$ where\n\\begin{equation}\nk=w-\\frac{ma}{r_\\text{h}^2+a^2}\\,,\n\\end{equation}\nand that the equivalent of eq.~(\\ref{rstarr}) is now\n\\begin{equation}\n\\frac{dr^*}{dr}=\\frac{r^2+a^2}{\\Delta(r)}\\,.\n\\end{equation}\n\n\\subsection{Calculation of the eigenvalue}\n\nThe only other complication is to determine ${}_sE^m_{\\ell}$ from the angular equation.  The $\\theta$ wave functions ${}_sT^m_{\\ell}$ are no longer like those of the spin-weighted spherical harmonics and so ${}_sE^m_{\\ell} \\neq {}_s\\lambda_{\\ell}$\\,.  The eigenvalues of the spin-weighted spheroidal harmonics are functions of $a\\omega$ and can be obtained by a continuation method \\cite{Wasserstrom:1972}.  This is a generalization of perturbation theory which can be applied for arbitrarily large changes in the initial Hamiltonian for which the eigenvalues are known.  Making the $a\\omega$-dependence of both $T$ and $E$ clear, eq.~(\\ref{spinang}) can be written as\n\\begin{equation}\n\\label{opang}\n({\\mathcal H}_0+{\\mathcal H}_1)\\, {}_sT^m_{\\ell}(\\theta,a\\omega) =-{}_sE^m_{\\ell} (a\\omega)\\, {}_sT^m_{\\ell}(\\theta,a\\omega)\\,, \n\\end{equation}\nwhere \n\\begin{equation}\n{\\mathcal H}_0=\\frac{1}{\\sin\\theta}\\frac{d}{d\\theta}\\left(\\sin\\theta\\frac{d}{d\\theta}\\right)+\\left(-\\frac{2ms\\cot\\theta}{\\sin\\theta}-\\frac{m^2}{\\sin^2\\theta}+s-s^2\\cot^2\\theta\\right),\n\\end{equation}\nand\n\\begin{equation}\n{\\mathcal H}_1=(a^2\\omega^2\\cos^2\\theta-2a\\omega s\\cos\\theta)\\,.\n\\end{equation}\nFrom these equations it is evident that \n\\begin{equation}\n{}_sT^m_{\\ell}(\\theta,0)= {}_sS^m_{\\ell}(\\theta)\\,, \n\\label{init1}\n\\end{equation}\nand that\n\\begin{equation}\n{}_sE^m_{\\ell}(0)={}_s\\lambda_{\\ell}=\\ell(\\ell+1)-s(s+1)\\,.\n\\label{init2}\n\\end{equation}\n\nThe continuation method was outlined in \\cite{Press:1973} although this paper unfortunately seems to contain a number of errors.  First consider the case in which $a\\omega$ is small and normal perturbation theory can be used.  This means we have\n\\begin{equation}\n\\label{pte}\n{}_sE^m_{\\ell}(a\\omega)={}_s\\lambda_{\\ell}-\\langle s\\ell m|{\\mathcal H}_1|s\\ell m\\rangle+\\ldots\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\label{pts}\n{}_sT^m_{\\ell}(a\\omega)={}_sS^m_{\\ell}-\\sum_{\\ell'\\neq l} \\frac{\\langle s\\ell'm|{\\mathcal H}_1|s\\ell m\\rangle}{\\ell(\\ell+1)-\\ell'(\\ell'+1)} {}_sS^m_{\\ell'}+\\ldots\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{prod}\n\\langle s\\ell'm|{\\mathcal H}_1|s\\ell m\\rangle \\equiv \\int d\\Omega\\,({}_sS_{\\ell'}^m)^* {}_sS_{\\ell}^m \\,{\\mathcal H}_1\\,. \n\\end{equation} \nCare must be taken about the signs of the terms in eqs.~(\\ref{pte}) and (\\ref{pts}) due to the sign of the eigenvalue ${}_sE^m_{\\ell}$ in eq.~(\\ref{opang}).  This seems to account for some differing results in \\cite{Press:1973}.\n\nThe product in eq.~(\\ref{prod}) can be split into terms for which the following standard formulae are useful:\n\\begin{equation}\n\\langle s\\ell'm|\\cos^2\\theta|s\\ell m\\rangle=\\frac{1}{3}\\delta_{\\ell\\ell'}+\\frac{2}{3}\\left(\\frac{2\\ell +1}{2\\ell'+1}\\right)^{\\frac{1}{2}}\\langle\\ell2m0|\\ell'm\\rangle\\langle\\ell2(\\!-\\!s)0|\\ell'(\\!-\\!s)\\rangle,\n\\end{equation}\nand\n\\begin{equation}\n\\langle s\\ell'm|\\cos\\theta|s\\ell m\\rangle=\\left(\\frac{2\\ell +1}{2\\ell'+1}\\right)^{\\frac{1}{2}}\\langle\\ell1m0|\\ell'm\\rangle\\langle\\ell1(\\!-\\!s)0|\\ell'(\\!-\\!s)\\rangle,\n\\end{equation}\nwhere $\\langle\\ell_1\\ell_2m_1m_2|LM\\rangle$ are Clebsch-Gordan coefficients.  Using the above we obtain\n\\begin{equation}\n{}_sE^m_{\\ell}= \\left\\{ \\begin{array}{ll}\n\t\t{}_s\\lambda_{\\ell}-2a\\omega \\frac{s^2m}{\\ell(\\ell+1)}+{\\mathcal O}[(a\\omega)^2] & \\mbox{if $s \\neq 0$}\\,;\\\\\n\\\\\n\t\t{}_s\\lambda_{\\ell}+a^2\\omega^2\\left[\\frac{2m^2+1-2\\ell(\\ell+1)}{(2\\ell-1)(2\\ell-3)}\\right]+{\\mathcal O}[(a\\omega)^4] & \\mbox{if $s=0$}\\,.\\end{array} \\right. \n\\end{equation}\nAgain this differs (in the $s=0$ case) from the result given in \\cite{Press:1973}.  However it agrees with \\cite{Seidel:1989ue} where several discrepancies in the literature are helpfully clarified.\n\nTo employ the continuation method we write the ${}_sT^m_{\\ell}(\\theta,a\\omega)$ functions in the basis of the $\\theta$-parts of the spin-weighted spherical harmonics:\n\\begin{equation}\n{}_sT^m_{\\ell}(\\theta,a\\omega)=\\sum_{\\ell'}{}_sB_{\\ell\\ell'}^m(a\\omega)S_{s\\ell'}^m(\\theta)\\,.\n\\end{equation}\nBy differentiating eq.~(\\ref{opang}) and applying the same techniques as in perturbation theory it is possible to obtain the results\n\\begin{equation}\n\\label{conte}\n\\frac{d \\, {}_sE_{\\ell}^m}{d(a\\omega)}=-\\sum_{\\alpha,\\beta}{}_sB_{\\ell\\alpha}^m \\,{}_sB_{\\ell\\beta}^m\\langle\\alpha|\\beta\\rangle,\n\\end{equation}\nand\n\\begin{equation}\n\\label{contb}\n\\frac{d \\, {}_sB_{\\ell\\ell'}}{d(a\\omega)}=-\\sum_{\\alpha,\\beta,\\gamma \\neq l}\\frac{{}_sB_{\\gamma\\alpha} \\, {}_sB_{\\ell\\beta}}{{}_sE^m_{\\ell}-{}_sE_{\\gamma}^m}\\langle\\alpha|\\beta\\rangle{}_sB_{\\gamma \\ell'},\n\\end{equation}\nwhere $\\langle\\alpha|\\beta\\rangle \\equiv \\langle s\\alpha m|d{\\mathcal H}_1\/d(a\\omega)|s\\beta m\\rangle$.  The initial conditions are obtained from eqs. (\\ref{init1}) and (\\ref{init2}) which imply that ${}_sB_{\\ell\\ell'}^m(0)=\\delta_{\\ell\\ell'}$.  By integrating eqs.~(\\ref{conte}) and (\\ref{contb}) it is possible to obtain the eigenvalues of the spin-weighted spheroidal functions for any $s,\\ell$ and $m$ and for arbitrarily large values of $a\\omega$.  This integration was performed by using the \\texttt{NDSolve} package in \\texttt{Mathematica} to implement a Runge-Kutta method.  The computational effort involved increases significantly for larger values of $\\ell$ as it is necessary to include a larger number of terms in the summations, increasing the number of differential equations to be solved.\n\nAlthough only the $s=0$ eigenvalues were required here, as a check the continuation method was also used to calculate the $s=1$ eigenvalues as a function of $a\\omega$.  This allowed comparison with the polynomial approximations given in \\cite{Press:1974}.  Excellent agreement was found---see Figure~\\ref{eigen} for an example.\n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.5]{$a\\omega$}\n\\psfrag{y}[][][1.5]{${}_sE^m_{\\ell}(a\\omega)$}\n\\scalebox{0.7}{\\rotatebox{0}{\n\\includegraphics[width=\\textwidth]{neigen.eps}}}\n\\capbox{Comparison of eigenvalue with previous result}{Comparison of the numerical result obtained in this work for the $s=1,\\ell=3,m=-2$ eigenvalue (solid line) with the polynomial approximation in \\cite{Press:1974} (dashed line).  The lines are exactly on top of each other (the maximum difference between them is $\\sim 10^{-5}$).\\label{eigen}}\n\\end{center}\n\\end{figure}\n\n\\subsection{Calculation of the grey-body factor}\n\nIn order to calculate the grey-body factors in the rotating case, it is necessary to know the equivalent of eqs.~(\\ref{scalars}), (\\ref{fermions}) and (\\ref{bosons}).  Unfortunately, for fermions and bosons these expressions are not available in the literature.  However for the scalar case, since eq.~(\\ref{scalars}) only involves coefficients at infinity, exactly the same formula holds even when the black hole is rotating.  This means that for scalars it has been possible to numerically calculate the grey-body factors for rotating black holes.\n\nSince the denominator in eqs.~(\\ref{rflux}) and (\\ref{rpower}) has $m$-dependence, it is less useful to plot an equivalent of the $\\hat{\\sigma}$ used in the non-rotating case,\\footnote{The absorption cross section $\\hat{\\sigma}_\\text{abs}$ summed over all angular momentum modes \\emph{can} still be calculated; the high-energy asymptotic value appears to increase with $a_*$, apparently in contradiction with \\cite{Page:1976df} which states that roughly the same value is expected as in the non-rotating case.} so only the power spectra are shown.  They are presented as a function of $\\omega r_\\text{h}$ for different values of $a_*$ and hence cannot be used to directly compare the spectra from two black holes of the same mass (but different $J$) since this is itself a function of $M_\\text{BH}$. \n\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.2]{$\\omega r_\\text{h}$}\n\\psfrag{y}[][][1.2]{$r_\\text{h} d^2 \\hat{E}^{(0)}\/dtd\\om$}\n\\scalebox{0.8}{\\rotatebox{0}{\\includegraphics[width=\\textwidth]{nefiopfig2.eps}}}\n\\capbox{Power spectra for scalar emission from rotating black holes ($n=1$)}{Power spectra for scalar emission from rotating black holes ($n=1$).\\label{s0a}}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{s0a} can be directly compared to Figure 2 in \\cite{Ida:2002ez} and shows markedly different behaviour for values of $\\omega r_\\text{h}$ larger than $\\sim 0.2$.  Almost all of this effect is due to the low-energy expansion used by those authors to obtain the grey-body factors.  Their approximation that ${}_sE^m_{\\ell}={}_s\\lambda_{\\ell}=\\ell(\\ell+1)-s(s+1)$ is much less significant except at the very highest values of $\\omega r_\\text{h}$.\n\nThese numerical results for the grey-body factors also allowed the statement in \\cite{Giddings:2001bu} that emission is dominated by modes with $\\ell=m$ to be tested.  It was found that a plot like Figure~\\ref{s0a} looks the same at the $\\sim$~90\\% level if only the the $\\ell=m$ modes are included in the sum of eq.~(\\ref{rpower})\n \n\\subsection{Super-radiance}\n\nThe numerical results obtained allow the possibility of super-radiance (i.e.\\ negative grey-body factors) to be investigated in an extra-dimensional context.  Previously an analytic approach \\cite{Frolov:2002xf} was used to confirm super-radiance for bulk scalars incident on 5D black holes, but there do not seem to be any other results in the literature for higher-dimensional black holes.  To study this phenomenon it is useful to consider angular momentum modes individually and, for comparison with the 4D scalar results presented in \\cite{Press:1972}, the quantity on the horizontal axis of the plots is\n\\begin{equation}\n\\frac{\\omega}{m \\Omega}=\\frac{(1+a_*^2)(\\omega r_\\text{h})}{m a_*}\\,,\n\\end{equation}\nrather than the $\\omega r_\\text{h}$ used in previous plots.  The vertical axis is the absorption probability (in fact $-|\\hat {\\cal A}_{\\ell,m}|^2$) expressed as a percentage so that it gives the percentage energy amplification of an incident wave.\n\\begin{figure}\n\\begin{center}\n\\psfrag{x}[][][1.5]{$\\omega\/m\\Omega$}\n\\psfrag{a}[][][1.5][18]{\\textsf{l=m=1}}\n\\psfrag{b}[][][1.5][30]{\\textsf{l=m=2}}\n\\psfrag{c}[][][1.5][37]{\\textsf{l=m=3}}\n\\psfrag{d}[][][1.5][40]{\\textsf{l=m=4}}\n\\psfrag{e}[][][1.5][45]{\\textsf{l=m=5}}\n\\psfrag{f}[][][1.5]{\\textsf{l=2, m=1}}\n\\scalebox{0.5}{\\rotatebox{0}{\\includegraphics[width=\\textwidth]{super1.eps}}}\n\\capbox{Super-radiant scattering by 4D black holes with $a_*=1$}{Super-radiant scattering of scalars by maximally rotating ($a_*=1$) 4-dimensional black holes.\\label{super1}}\n\\end{center}\n\\end{figure}\nFigure~\\ref{super1} shows excellent agreement with the results in Figure 1 of \\cite{Press:1972} which are for $n=0$ and $a_*=1$ (equivalent to $a=M_\\text{BH}$ in the 4D case).  Figure~\\ref{super2} is for $n=2$ and 6 but still with the same value of the rotation parameter.  It is clear that the peak amplification is more significant than previously and becomes comparable to that for gauge bosons in the 4D case.  It is still found that the $\\ell=m=1$ mode provides the greatest amplification.  However Figure~\\ref{super3} shows that for larger values of $a_*$ the maximum amplification can occur in modes with larger values of $\\ell$.  This trend continues as $n$ and $a_*$ increase further: for $n=6$, equation~(\\ref{astarmax}) shows that the geometric maximal rotation is $a_*^\\text{max}=4.0$ and for this value of the rotation parameter the maximum energy amplification is found to be just under 9\\% (and occurs in the $\\ell=m=7$ mode).\n\n\\begin{figure}\n\\unitlength1cm\n\\begin{center}\n\\psfrag{x}[][][0.8]{$\\omega\/m\\Omega$}\n\\psfrag{a}[][][0.8][18]{\\textsf{l=m=1}}\n\\psfrag{b}[][][0.8][28]{\\textsf{l=m=2}}\n\\psfrag{c}[][][0.8][35]{\\textsf{l=m=3}}\n\\psfrag{d}[][][0.8][40]{\\textsf{l=m=4}}\n\\psfrag{e}[][][0.8][43]{\\textsf{l=m=5}}\n\\psfrag{f}[][][0.8]{\\textsf{l=2, m=1}}\n\\psfrag{g}[][][0.8][20]{\\textsf{l=m=1}}\n\\psfrag{h}[][][0.8][30]{\\textsf{l=m=2}}\n\\psfrag{i}[][][0.8][37]{\\textsf{l=m=3}}\n\\psfrag{j}[][][0.8][42]{\\textsf{l=m=4}}\n\\psfrag{k}[][][0.8][45]{\\textsf{l=m=5}}\n\\psfrag{l}[][][0.8]{\\textsf{l=2, m=1}}\n\\begin{minipage}[t]{3.05in}\n{\\rotatebox{0}{\\includegraphics{super2.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n{\\rotatebox{0}{\\includegraphics{super3.eps}}}\n\\end{minipage}\n\\capbox{Super-radiant scattering by $(4+n)$D black holes with $a_*=1$}{Super-radiant scattering of scalars by $(4+n)$-dimensional black holes with $a_*=1$: {\\bf(a)} $n=2$ and {\\bf(b)} $n=6$.\\label{super2}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\unitlength1cm\n\\begin{center}\n\\psfrag{x}[][][1]{$\\omega\/m\\Omega$}\n\\psfrag{a}[r][][1]{\\textsf{l=m=1}}\n\\psfrag{b}[r][][1]{\\textsf{l=m=2}}\n\\psfrag{c}[][][1][60]{\\textsf{l=m=3}}\n\\psfrag{d}[][][1][60]{\\textsf{l=m=4}}\n\\psfrag{e}[l][][1]{\\textsf{l=m=5}}\n\\psfrag{f}[r][][1]{\\textsf{l=m=1}}\n\\psfrag{g}[r][][1]{\\textsf{l=m=2}}\n\\psfrag{h}[][][1]{\\textsf{l=m=3}}\n\\psfrag{i}[][][1][64]{\\textsf{l=m=4}}\n\\psfrag{j}[l][][1]{\\textsf{l=m=5}}\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.8}\n{\\rotatebox{0}{\\includegraphics{super4.eps}}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{3.05in}\n\\scalebox{0.8}\n{\\rotatebox{0}{\\includegraphics{super5.eps}}}\n\\end{minipage}\n\\capbox{Super-radiant scattering by $(4+n)$D black holes with $a_*>1$}{Super-radiant scattering of scalars by $(4+n)$-dimensional black holes: {\\bf(a)} $n=2$, $a_*=1.5$ and {\\bf(b)} $n=6$, $a_*=2$.\\label{super3}}\n\\end{center}\n\\end{figure}\n\nAs in the 4-dimensional case, the region in which the grey-body factors are negative exactly corresponds to the region in which the thermal factor in the denominator of equations like (\\ref{rflux}) and (\\ref{rpower}) is negative; hence the quantum emission rate always remains positive.  For all angular momentum modes the grey-body factor is positive for $\\omega > m \\Omega$ and the change from negative to positive at $\\omega = m \\Omega$ occurs such that the flux and energy spectra are smooth positive functions.  Therefore although super-radiance can in principle be used to extract energy from rotating black holes, its significance when considering the Hawking radiation spectra is limited.\n\n\\section{Conclusions}\n\\label{gbfconc}\n\nThe revival of the idea of extra space-like dimensions\nin nature has led to the formulation of theories which allow gravity to become strong at a significantly lower energy scale. This has opened the way for the proposal of the creation of miniature higher-dimensional black holes during collisions of energetic particles in the earth's atmosphere or at ground-based particle colliders.  The Hawking radiation emitted by such black holes is described by the relevant grey-body factors; a wide variety of these factors have been calculated numerically in the work described in this chapter and, at the same time, some open questions from previous analyses in the literature have been addressed. \n\nThe exact numerical results obtained in \\secref{numres} for the grey-body factors and emission rates of scalars, fermions and gauge bosons confirm that previous analytic results \\cite{Kanti:2002nr,Kanti:2002ge} were only valid in limited low-energy regimes.  The grey-body factors depend on both the dimensionality of space-time and the spin.  They adopt the same high-energy asymptotic values (which decrease as $n$ increases) for all particle species, confirming the geometrical optics limit of \\cite{Emparan:2000rs}.  In the low-energy limit the grey-body factors are found to be enhanced as $n$ increases for $s=1$ and 1\/2 and suppressed for $s=0$. \n\nUsing these grey-body factors it was possible to accurately calculate the flux and power emission spectra for the black holes.  As the number of extra dimensions projected on the brane increases there is a substantial (orders of magnitude) enhancement in the energy emission rate, mainly as a result of the increase in black hole temperature with $n$.  However the effect of including the grey-body factors is that the increase in the rate depends on the spin of the particle studied.  The computed relative emissivities show that scalar fields, which are the dominant degree of freedom emitted by $n=0$ black holes, are just outnumbered by gauge bosons for large values of $n$, with the fermions becoming the least effective emission channel.  Therefore the emission spectra and the relative particle emissivities may possibly both lead to the determination of the number of extra dimensions.\n\nThe emission spectra for modes of different spins were combined with the total number of degrees of freedom for all the particle types which could be emitted from an extra-dimensional black hole.  This made it possible to calculate the total flux and power emission and hence also the expected time for, and average multiplicity of, the black hole decay.  These calculations suggest that some of the assumptions usually made when modelling black hole decay by sequential emission of Hawking radiation may not be valid for the larger values of $n$.\n\\enlargethispage{-\\baselineskip}\n\nIn section \\ref{embulk}, the details of the emission of bulk scalar modes were investigated with the aim being to provide an accurate estimate for the amount of energy lost into the bulk.  Comparing the total bulk and brane emission rates for scalar modes, integrated over the whole energy regime, the conclusion is that more of the black hole energy is emitted in brane modes than bulk modes, for all values of $n$ considered. The total emissivity in the bulk is less than 1\/4 of that on the brane for $n=2$--4, while it becomes substantial for the extreme value of $n=7$ without, however, exceeding the brane value.  The accurate results presented in this chapter provide firm support to the heuristic arguments made in ref. \\cite{Emparan:2000rs}.\n\nNo results are presented in this work concerning the emission of gravitons either on the brane or in the bulk. The derivation of a consistent equation, which can describe the motion of gravitons in the induced black hole background on the brane, is still under investigation.  Nevertheless it is expected that, as for the emission of fields with spin $s=0$, 1\/2 and 1 the graviton emission becomes enhanced as the dimensionality of space-time increases, but remains subdominant compared to the other species at least for small values of $n$. \n\nThe work on rotating black holes in \\secref{rotbh} suggests that, at least for the emission of scalars, the power spectra can be significantly modified compared to those in the Schwarzschild-like case.  The spectra are dominated by the angular momentum modes with $\\ell=m$ and as the rotation parameter $a_*$ increases, the peak in the spectrum shifts to much larger values of $\\omega r_\\text{h}$.  The phenomenon of super-radiance was also observed for higher-dimensional rotating black holes.  The peak energy amplification was found to be significantly larger than in the four-dimensional case and occurred in higher angular momentum modes for larger values of the rotation parameter.\n\\enlargethispage{\\baselineskip}\n\nA final comment is appropriate here concerning the validity of the results obtained in this work. As pointed out in the text, the horizon of the black hole is an input parameter of the analysis.  The results are applicable for all values of $r_\\text{h}$ which are smaller than the size of the extra dimensions $R$, no matter how small or large $R$ is (as long as it remains considerably larger than $\\ell_\\text{P}$ to avoid quantum corrections).  The analysis therefore remains valid for all theories postulating the existence of flat extra dimensions and a fundamental scale of gravity even a few orders of magnitude lower than $M_\\text{P}$. If the emission of Hawking radiation from these black holes is successfully detected, either at next-generation colliders or in the more distant future, the distinctive features discussed here may help in the determination of the number of extra dimensions existing in nature.  Chapter~\\ref{generator} describes a black hole event generator which incorporates these grey-body effects and would be a useful tool in such an analysis.\n\\chapter{CHARYBDIS: A Black Hole Event Generator}\n\\chaptermark{Event Generation}\n\\label{generator}\n\\section{Introduction}\n\nThe work on black holes in this thesis was motivated by extra dimension models which allow the fundamental Planck scale to be of order a TeV\\@.  These models are attractive because they solve the hierarchy problem, but they are also exciting because they mean that gravity is strong at these energy scales and so black holes can be investigated at the LHC and other next-generation machines.  If these models are realized in nature, we would expect particle accelerators at TeV-scale energies to be able to produce miniature black holes.  These would then decay rapidly by Hawking evaporation, giving rise to characteristic high-multiplicity final states.\n\nThere has already been much discussion of this issue in the literature (for example, in many of the works cited in Chapters~\\ref{bhintro} and \\ref{greybody}) but \nlittle work has been done in trying to realistically simulate black holes at\nthe LHC\\@.  This chapter describes the implementation of a simple model of black hole production and decay which can be interfaced to existing Monte Carlo programs using the Les Houches accord \\cite{Boos:2001cv}.  The major new theoretical input to the generator is the inclusion of the results from Chapter~\\ref{greybody}---the grey-body factors for black holes in extra dimensions \\cite{Harris:2003eg}.  The recoil and change of temperature of the black hole during decay is also taken into account, and various models are provided for the termination of the decay process.\n\nSection~\\ref{proddec} reviews important aspects of the theory of extra-dimensional black holes.  Then, in \\secref{bhgen}, there is a description of the {\\small CHARYBDIS}\\footnote{In Greek mythology, Charybdis was a nymph daughter of Poseidon and Gaia who was turned into a monster by Zeus.  She lived in a cave at one side of the Strait of Messina (opposite the monster Scylla) and sucked water in and out three times each day.} event generator itself including a discussion of the theoretical assumptions involved, details of the Fortran code and instructions on how to use the program.  P.~Richardson was involved in the development of some of this code, specifically in ensuring baryon number and charge conservation, defining the colour flow required by general-purpose event generators, and making {\\small CHARYBDIS} compatible with the Les Houches accord.   An event display and some sample production cross sections, particle spectra and particle emissivities produced using the event generator are presented in \\secref{output}.  Details of some preliminary experimental studies are given in \\secref{expstud} and there is a concluding discussion in \\secref{discuss}.\n\n\\section{Black hole production and decay}\n\\label{proddec}\n\nMost of the details of the production and decay of black holes in extra dimension models have already been discussed in Chapter~\\ref{bhintro}.  Here some of the key features of the theory are re-iterated and some comments made on those which are particularly relevant for the task of event generation.\n\\enlargethispage{\\baselineskip}\n\nA fundamental Planck scale as low as $\\sim$ TeV means that it is possible for \ntiny black holes to be produced at the LHC when two partons pass within the \nhorizon radius set by their centre-of-mass energy, $\\sqrt{\\hat{s}}$.  Using the results of \\cite{Myers:1986un}, the horizon radius for a non-spinning black hole is found to be\n\\begin{equation}\nr_\\text{S}=\\frac{1}{\\sqrt{\\pi}M_{\\text{P}(4+n)}}\\left(\\frac{M_\\text{BH}}{M_{\\text{P}(4+n)}}\\right)^{\\frac{1}{n+1}}\\left(\\frac{8\\Gamma\\left(\\frac{n+3}{2}\\right)}{n+2}\\right)^{\\frac{1}{n+1}},\n\\end{equation}\nwhere $M_\\text{BH}$ is the mass of the black hole and, as throughout the rest of this chapter, $M_{\\text{P}(4+n)}$ is the fundamental $(4+n)$-dimensional Planck scale in convention `d' of Table~\\ref{consum}.  Numerical calculations, discussed in \\secref{bhprod}, show that $M_\\text{BH} \\le \\sqrt{\\hat{s}}$.  Provided there is enough energy to be in the trans-Planckian regime, the parton-level cross section is $F_n \\pi r_\\text{S}^2$ where $F_n \\sim 1$.\n\nBlack holes can be produced with any gauge and spin quantum numbers so to determine the p-p or $\\bar{\\text{p}}$-p production cross section it is necessary to sum over all possible quark and gluon pairings.  Although the parton-level cross sections grow with black hole mass, the parton distribution functions (PDFs) fall rapidly at high energies and so the cross section also falls off quickly.\n\nThese miniature black holes are expected to decay instantaneously on LHC detector time scales (typical lifetimes are $\\sim 10^{-26}$ s).  The decay is made up of four phases (discussed in more detail in Chapter~\\ref{bhintro}): the balding phase, the spin-down phase, the Schwarzschild phase (accounting for the greatest proportion of the mass loss in four dimensions \\cite{Page:1976df}) and the Planck phase.\n\nA black hole of a particular mass is characterized by a Hawking temperature and as the decay progresses the black hole mass falls and the temperature rises.  For an uncharged, non-rotating black hole the decay spectrum is described, as in eq.~(\\ref{fdecay-brane}), by\n\\begin{equation}\n\\label{spec}\n\\frac{d \\hat N^{(s)}(\\om)}{dt} = \\sum_{\\ell} (2l+1) |\\hat {\\cal A}^{(s)}_\\ell|^2\\,\n\\frac{1}{\\exp\\left(\\om\/T_\\text{H}\\right) \\mp 1}\\,\\frac{d\\om}{2\\pi}\\,,\n\\end{equation}\nwhere the Hawking temperature is given by\n\\begin{equation}\nT_\\text{H}=\\frac{n+1}{4\\pi r_\\text{S}}\\,,\n\\end{equation}\nand $|\\hat {\\cal A}^{(s)}_\\ell|^2$ is the absorption probability from which the grey-body factor can be obtained (see eq.~(\\ref{greydef}) in Chapter~\\ref{greybody}). Equation~(\\ref{spec}) can be used to determine the decay spectrum for\na particular particle, by considering the number of degrees of freedom (Table~\\ref{pprobs} shows the relevant numbers for all the particle types which the black hole can emit).  It is relevant for predicting both the spectra and the relative emission probabilities of the different particle types.\n\nThe grey-body factors modify the spectra of emitted particles from that of a\nperfect thermal black body\\cite{Hawking:1975sw}; their spin dependence means that they are also necessary to determine the relative emissivities of different particle types from a black hole.  Until the work in Chapter~\\ref{greybody}, the full numerical grey-body factors were only available in the literature for the 4D case \\cite{Page:1976df,Sanchez:1978si}.  \n\nIt is usually assumed that a quasi-stationary approach to the black hole decay is valid---that is, after each particle emission the black hole has time to come into equilibrium at the new temperature before the next particle is emitted.\\footnote{Note that the results of Chapter~\\ref{greybody} have cast doubt on this assumption for cases with $n \\ge 3$.}\n\nIt was argued in \\cite{Emparan:2000rs} that the majority of energy in Hawking \nradiation is emitted into modes on the brane (i.e.\\ as Standard Model \nparticles) but that a small amount is also emitted into modes in the bulk \n(i.e.\\ as gravitons).  The work described in Chapter~\\ref{greybody} provides support for this statement based on the relative emissivities of scalars in the bulk and on the brane.\n\n\\section{Event generator} \n\\label{bhgen}\n\n\\subsection{Features and assumptions of the event generator}\n\\label{egassump}\n\nThere are a number of features of the {\\small CHARYBDIS} generator which, within the uncertainties of much of the theory, allow reliable simulation of black hole events.  Most notable is that, unlike other generators (e.g.\\ \\cite{Dimopoulos:2001en}), the grey-body effects are fully included.  The dependence on both spin and the dimensionality of space-time means that grey-body factors must be taken into account in any attempt to determine the number of extra dimensions by studying the energy spectra and relative emission probabilities of particles from a black hole.  When studying black hole decay, other experimental variables may also be sensitive to these grey-body effects.\n\nThe generator also has an option to allow the black hole temperature to vary as the decay progresses and is designed for simulations with either p-p or $\\bar{\\text{p}}$-p.\n\n{\\small CHARYBDIS} only attempts to model in detail the Hawking evaporation phase which is expected to account for the majority of the mass loss.  To provide a further simplification only non-spinning black holes are modelled.  This is perhaps a less good approximation but comparison with the 4D situation suggests that most of the angular momentum will be lost in a relatively short spin-down phase \\cite{Page:1976ki}.  The balding phase is difficult to model and is neglected---this is equivalent to the assumption that $M_\\text{BH}=\\sqrt{\\hat{s}}$ in spite of the evidence that this will not necessarily be the case.  A related assumption is that the cross section calculation assumes $F_n$ of equation~(\\ref{fndef}) is equal to unity (i.e.\\ the parton-level cross section is assumed to be $\\pi r_\\text{S}^2$).\n\n\\enlargethispage{\\baselineskip}\nIt is hoped that the way in which the black hole decay is terminated will provide a reasonable approximation to the Planck phase of the decay.  The generator also assumes that energy lost from the black hole in graviton emission can be neglected, based on the work in \\cite{Emparan:2000rs} and in Chapter~\\ref{greybody}.\n\nAs discussed in \\secref{protonbh}, it is possible that black hole decay does not conserve baryon number.  However the treatment of processes which do not conserve baryon number is complicated, in both the QCD evolution and hadronization, and has only been studied for a few specific processes \\cite{Gibbs:1995cw,Gibbs:1995bt,Dreiner:1999qz,Sjostrand:2002ip}.  At the same time, the violation of baryon number is extremely difficult to detect experimentally and so the effect of including baryon number violation is not expected to be experimentally observable.  Therefore {\\small CHARYBDIS} conserves baryon number in black hole production and decay; lepton number, however, is not conserved.\n\n\\subsection{General description}\n\\label{gendes}\n\nThe black hole event generator developed attempts to model the theory as \noutlined in \\secref{proddec}, within the assumptions of \\secref{egassump}.  There are several related parameters and switches which can be set in the first part of the Les Houches subroutine \\texttt{UPINIT} \\cite{Boos:2001cv}.  No other part of the charybdis1000.F code should be modified.\n\nFirstly the properties of the beam particles must be specified.  \\texttt{IDBMUP(1)} and \\texttt{IDBMUP(2)} are their Particle Data Group (PDG) codes (only protons and anti-protons are allowed) and the corresponding energies are \\texttt{EBMUP(1)} and \\texttt{EBMUP(2)}.\n\nAs already discussed, the geometric parton-level cross section ($\\sigma=\\pi r_\\text{S}^2$) is used but the parameters \\texttt{MINMSS} and \\texttt{MAXMSS} allow the mass range for the black holes produced to be specified.  This means that it is possible to adjust the lower mass limit at which this cross section for the trans-Planckian production of black holes is thought to become valid.  The details of the Monte Carlo (MC) cross~section calculation are given in Appendix~\\ref{appc}.\n\nThree other parameters which must be set before using the event generator are\n\\texttt{TOTDIM}, \\texttt{MPLNCK} and \\texttt{MSSDEF}.  The total number of dimensions in the model being used is given by \\texttt{TOTDIM} (this must be set between \\texttt{6} and \\texttt{11}).  There are a number of different definitions of the Planck mass (set using \\texttt{MPLNCK}) but the parameter \\texttt{MSSDEF} can be set to three different values to allow easy interchange between the three conventions outlined in Appendix A of \\cite{Giddings:2001bu}.  The conversions between these conventions are summarized in Table~\\ref{massdefs} (in fact options \\texttt{1},\\texttt{2} and \\texttt{3} correspond to \\texttt{MPLNCK} being set in the conventions a, d and b of Table~\\ref{consum} respectively).\n\n\\begin{table}\n\\def1.0{1.2}\n\\begin{center}\n\\begin{tabular}{|c|l|c|}\n\\hline\n\\texttt{MSSDEF} & Conversion & Convention\\\\\n\\hline\n1 & $\\mbox{\\texttt{MPLNCK}}=(2^{n-2}\\pi^{n-1})^{\\frac{1}{n+2}}M_{\\text{P}(4+n)}$ & a\\\\\n2 & $\\mbox{\\texttt{MPLNCK}}=M_{\\text{P}(4+n)}$ & d\\\\\n3 & $\\mbox{\\texttt{MPLNCK}}=(2^{n-3}\\pi^{n-1})^{\\frac{1}{n+2}}M_{\\text{P}(4+n)} $ & b\\\\\n\\hline\n\\end{tabular}\n\\capbox{Definitions of the Planck mass}{\\label{massdefs}Definitions of the Planck mass.}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\nIt has been suggested that since black hole formation is a non-perturbative \nprocess, the momentum scale for evaluating the PDFs should be the inverse \nSchwarzschild radius rather than the black hole mass.  The switch \n\\texttt{GTSCA} should be set to \\texttt{.TRUE.}\\ for the first of these options\nand \\texttt{.FALSE.}\\ for the second.\\footnote{As confirmed in \\cite{Rizzoprivate}, the cross sections quoted in reference \\cite{Giddings:2001bu} were actually calculated with the latter PDF scale.}  The PDFs to be used are set using the Les Houches parameters \\texttt{PDFGUP} and \\texttt{PDFSUF}\\@.\n\nAs discussed in \\secref{proddec}, the Hawking temperature of the black \nhole will increase as the decay progresses so that later emissions will \ntypically be of higher energy.  However to allow comparison with other work \nwhich has ignored this effect, the \\texttt{TIMVAR} switch can be \nused to set the time-variation of the Hawking temperature as on \n(\\texttt{.TRUE.}) or off (\\texttt{.FALSE.}).  The indication in \\secref{numres} that a quasi-stationary approach to the black hole decay might not be valid for $n \\ge 3$ means that the \\texttt{.FALSE.}\\ option might in fact give a better description of the decay for higher values of $n$.\n\nThe emission probabilities of different types of particles are set according to the new theoretical results of Chapter~\\ref{greybody} (see particularly Table~\\ref{fratios}).  Heavy particle production is allowed and can be controlled by setting the value of the \\texttt{MSSDEC} parameter to \\texttt{2} for top quark, W and Z production, or \\texttt{3} to include the Higgs boson as well (\\texttt{MSSDEC=1} gives only light particles).  The production spectra for heavy particles may be unreliable if the initial Hawking temperature is below the rest mass of the particle in question.\n\nIf \\texttt{GRYBDY} is set as \\texttt{.TRUE.}\\ the particle types and energies are chosen according to the grey-body modified emission probabilities and spectra.  If instead the \\texttt{.FALSE.}\\ option is selected, the black-body emission probabilities and spectra are used.  The choice of energy is made in the rest frame of the black hole prior to the emission, which is isotropic in this frame.  As overall charge must be conserved, when a charged particle is to be emitted the particle or anti-particle is chosen such that the magnitude of the black hole charge decreases.  This reproduces some of the features of the charge-dependent emission spectra in \\cite{Page:1977um} whilst at the same time making it easier for the event generator to ensure that charge is conserved for the full decay.\n\nAlthough the Planck phase at the end of decay cannot be well modelled as it is\nnot well understood, the Monte Carlo event generator must have some way of\nterminating the decay.  There are two different possibilities for this, each with a range of options for the terminal multiplicity.  \n\nIf \\texttt{KINCUT=.TRUE.}\\ termination occurs when the chosen energy for an emitted particle is ruled out by the kinematics of a two-body decay.  At this point an isotropic \\texttt{NBODY} decay is performed on the black hole remnant where \\texttt{NBODY} can be set between \\texttt{2} and \\texttt{5}.  The \\texttt{NBODY} particles are chosen according to the same probabilities used for the first part of the decay. The selection is then accepted if charge and baryon number are conserved, otherwise a new set of particles is picked for the \\texttt{NBODY} decay. If this does not succeed in conserving charge and baryon number after \\texttt{NHTRY} attempts the whole decay is rejected and a new one generated.  If the whole decay process fails for \\texttt{MHTRY} attempts then the initial black hole state is rejected and a new one generated. \n\nIn the alternative termination of the decay (\\texttt{KINCUT=.FALSE.}) particles are emitted according to their energy  spectra until $M_\\text{BH}$ falls below \\texttt{MPLNCK}; then an \\texttt{NBODY} decay as described above is performed.  Any chosen energies which are kinematically forbidden are simply discarded. \n\nIn order to perform the parton evolution and hadronization the general-purpose event generators require a colour flow to be defined.  This colour flow is defined in the large number of colours ($N_\\text{c}$) limit in which a quark can be considered as a colour line, an anti-quark as an anti-colour line and a gluon both a colour and anti-colour line.  A simple algorithm is used to connect all the lines into a consistent colour flow.  This algorithm starts with a colour line (from either a quark or a gluon) and then randomly connects this line with one of the unconnected anti-colour lines (from either a gluon or an anti-quark).  If the selected partner is a gluon the procedure is repeated to find the partner for its colour line; if it is an anti-quark one of the other unconnected quark colour lines is selected.  If the starting particle was a gluon the colour line of the last parton is connected to the anti-colour line of the gluon.  Whilst there is no deep physical motivation for this algorithm it at least ensures that all the particles are colour-connected and the showering generator can proceed to evolve and hadronize the event.\n\nAfter the black hole decay, parton-level information is written into the Les Houches common block \\texttt{HEPEUP} to enable a general-purpose event generator to fragment all emitted coloured particles into hadron jets, and perform the decays of any unstable particles.\n\n\\subsection{Control switches, constants and options}\n\nThose parameters discussed in \\secref{gendes} which are designed to be set by the user are summarized in Table~\\ref{parameters}.\n\n\\begin{table}[t]\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|l|c|c|}\n\\hline\nName & Description & Values & Default\\\\\n\\hline\n\\texttt{IDBMUP(2)} & PDG codes of beam particles & $\\pm \\mbox{\\texttt{2212}}$ & \\texttt{2212}\\\\\n\\texttt{EBMUP(2)} & Energies of beam particles (GeV) & & \\texttt{7000.0}\\\\\n\\texttt{PDFGUP(2)} & {\\small PDFLIB} codes for PDF author group & & $-\\mbox{\\texttt{1}}$\\\\\n\\texttt{PDFSUP(2)} & {\\small PDFLIB} codes for PDF set & & $-\\mbox{\\texttt{1}}$\\\\  \n\\hline\n\\texttt{MINMSS} & Minimum mass of black holes (GeV) & $<\\mbox{\\texttt{MAXMSS}}$ & \\texttt{5000.0}\\\\\n\\texttt{MAXMSS} & Maximum mass of black holes (GeV) & $\\leq\\mbox{c.m. energy}$ & c.m. energy\\\\\n\\texttt{MPLNCK} & Planck mass (GeV) & $\\leq\\mbox{\\texttt{MINMSS}}$ & \\texttt{1000.0}\\\\\n\\texttt{MSSDEF} & Convention for \\texttt{MPLNCK} (see Table~\\ref{massdefs})& \\texttt{1}--\\texttt{3} & \\texttt{2}\\\\\n\\texttt{TOTDIM} & Total number of dimensions ($4+n$) & \\texttt{6}--\\texttt{11} & \\texttt{6}\\\\\n\\texttt{GTSCA} & Use $r_\\text{S}^{-1}$as the PDF momentum scale & \\texttt{LOGICAL} & \\texttt{.FALSE.}\\\\\n& rather than the black hole mass & &\\\\\n\\texttt{TIMVAR} & Allow $T_\\text{H}$ to change with time & \\texttt{LOGICAL}& \\texttt{.TRUE.}\\\\\n\\texttt{MSSDEC} & Choice of decay products & \\texttt{1}--\\texttt{3} & \\texttt{3}\\\\\n\\texttt{GRYBDY} & Include grey-body effects & \\texttt{LOGICAL} & \\texttt{.TRUE.}\\\\\n\\texttt{KINCUT} & Use a kinematic cut-off on the decay & \\texttt{LOGICAL} & \\texttt{.FALSE.}\\\\\n\\texttt{NBODY} & Number of particles in remnant decay & \\texttt{2}--\\texttt{5} & \\texttt{2}\\\\\n\\hline\n\\end{tabular}\n\\capbox{List of {\\small CHARYBDIS} parameters}{\\label{parameters}List of parameters with brief descriptions, allowed values and default settings.}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\n\\subsection{Using charybdis1000.F}\n\\enlargethispage{-\\baselineskip}\n\nThe generator itself only performs the production and parton-level decay of the black hole. It is interfaced, via the Les Houches accord, to either {\\small HERWIG} \\cite{Marchesini:1992ch,Corcella:2000bw} or {\\small PYTHIA} \\cite{Sjostrand:2000wi} to perform the parton shower evolution, hadronization and particle decays.  This means that it is also necessary to have a Les Houches accord compliant version of either {\\small HERWIG} or {\\small PYTHIA} with both the dummy Les Houches routines (\\texttt{UPINIT} and \\texttt{UPEVNT}) and the dummy {\\small PDFLIB} subroutines (\\texttt{PDFSET} and \\texttt{STRUCTM}) deleted.  For {\\small HERWIG}, the first Les Houches compliant version is {\\small HERWIG6.500} \\cite{Corcella:2002jc}; for {\\small PYTHIA}, version 6.220 \\cite{Sjostrand:2001yu} or above is required.\\footnote{Versions of {\\small PYTHIA} above 6.200 support the Les Houches accord but cannot handle more than 7 out-going particles, which is necessary in black hole decays.}\n\nThe black hole code is available\\footnote{\\url{ http:\/\/www.ippp.dur.ac.uk\/montecarlo\/leshouches\/generators\/charybdis\/}} as a gzipped tar file which includes\n\n\\begin{itemize}\n\\item{charybdis1000.F (code for the black hole generator),} \n\\item{dummy.F (dummy routines needed if not using PDFLIB),} \n\\item{mainpythia.f (example main program for {\\small PYTHIA}),} \n\\item{mainherwig.f (example main program for {\\small HERWIG}),} \n\\item{charybdis1000.inc (include file for the black hole generator).} \n\\end{itemize}\n\nThe Makefile must specify which general-purpose event generator is to be used (i.e.\\ \\texttt{GENERATOR=HERWIG} or \\texttt{GENERATOR=PYTHIA}) and also whether {\\small PDFLIB} is to be used (\\texttt{PDFLIB=PDFLIB} if required, otherwise \\texttt{PDFLIB=~}).  The name of the {\\small HERWIG} or {\\small PYTHIA} source and the location of the {\\small PDFLIB} library must also be included. \n\\enlargethispage{\\baselineskip}\n\nIf the code is extracted to be run separately then the following should be taken into account:\n\n\\begin{itemize}\n\\item{charybdis1000.F will produce the {\\small HERWIG} version by default when compiled; the flag \\texttt{-DPYTHIA} should be added if the {\\small PYTHIA} version is required.} \n\\item{dummy.F will by default produce the version for use without {\\small PDFLIB}; the flag \\texttt{-DPDFLIB} should be added if {\\small PDFLIB} is being used.} \n\\end{itemize}\n\n\\subsection{List of subroutines}\n\nTable~\\ref{subs} contains a list of all the subroutines of the generator along with their functions.  Those labelled by {\\small HW\/PY} are \n{\\small HERWIG\/PYTHIA} dependent and are pre-processed according to the \\texttt{GENERATOR} flag in the Makefile.  Many of the utility routines are identical to routines which appear in the {\\small HERWIG} program.\n\n\\begin{table}[p]\n\\def1.0{1.1}\n\\begin{center}\n\\begin{tabular}{|c|l|}\n\\hline\nName & Description\\\\\n\\hline\n& Les Houches routines\\\\\n\\hline\n\\texttt{UPINIT} & Initialization routine\\\\\n\\texttt{UPEVNT} & Event routine\\\\\n\\hline\n& Particle decays\\\\\n\\hline\n\\texttt{CHDFIV} & Generates a five-body decay\\\\\n\\texttt{CHDFOR} & Generates a four-body decay\\\\\n\\texttt{CHDTHR} & Generates a three-body decay\\\\\n\\texttt{CHDTWO} & Generates a two-body decay\\\\\n\\hline\n& Hard sub-process and related routines\\\\\n\\hline\n\\texttt{CHEVNT} & Main routine for black hole hard sub-process\\\\\n\\texttt{CHFCHG} & Returns charge of a SM particle\\\\\n\\texttt{CHFMAS} & Returns mass of a SM particle ({\\small HW\/PY})\\\\\n\\texttt{CHHBH1} & Chooses next particle type if \\texttt{MSSDEC=1}\\\\\n\\texttt{CHHBH2} & Chooses next particle type if \\texttt{MSSDEC=2}\\\\\n\\texttt{CHHBH3} & Chooses next particle type if \\texttt{MSSDEC=3}\\\\\n\\texttt{CHPDF} & Calculates the PDFs ({\\small HW\/PY})\\\\\n\\hline\n& Random number generators\\\\\n\\hline\n\\texttt{CHRAZM} & Randomly rotates a 2-vector\\\\\n\\texttt{CHRGEN} & Random number generator ({\\small HW\/PY})\\\\\n\\texttt{CHRLOG} & Random logical\\\\\n\\texttt{CHRUNI} & Random number: uniform\\\\\n\\hline\n& Miscellaneous utilities\\\\\n\\hline\n\\texttt{CHUBHS} & Chooses particle energy from spectrum\\\\\n\\texttt{CHULB4} & Boost: rest frame to lab, no masses assumed\\\\\n\\texttt{CHULOB} & Lorentz transformation: rest frame $\\rightarrow$ lab\\\\\n\\texttt{CHUMAS} & Puts mass in 5th component of vector\\\\\n\\texttt{CHUPCM} & Centre-of-mass momentum\\\\\n\\texttt{CHUROB} & Rotation by inverse of matrix {\\bf R}\\\\\n\\texttt{CHUROT} & Rotation by matrix {\\bf R}\\\\\n\\texttt{CHUSQR} & Square root with sign retention\\\\\n\\texttt{CHUTAB} & Interpolates in a table\\\\\n\\hline\n& Vector manipulation\\\\\n\\hline\n\\texttt{CHVDIF} & Vector difference\\\\\n\\texttt{CHVEQU} & Vector equality\\\\\n\\texttt{CHVSUM} & Vector sum\\\\\n\\hline\n\\end{tabular}\n\\capbox{List of {\\small CHARYBDIS} subroutines}{\\label{subs}List of {\\small CHARYBDIS} subroutines with brief descriptions.}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\section{Event generator output}\n\\label{output}\n\n\\subsection{Sample cross sections}\n\nIn this section some indicative black hole production cross sections are shown for various different settings of the {\\small CHARYBDIS} parameters.  These are chosen so as to allow direct comparison with cross sections quoted in \\cite{Giddings:2001bu} and \\cite{Dimopoulos:2001hw}.  The results in Table~\\ref{crosstab} are found to be in good agreement with the previously published values.\n\n\\begin{table}\n\n\\begin{center}\n\\def1.0{1.1}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\texttt{MPLNCK} & \\texttt{MINMSS} & \\texttt{TOTDIM} & Cross section (fb)\\\\\n\\hline\n1000.0 & 5000.0 & 8 & $0.155\\times10^6$\\\\\n1000.0 & 7000.0 & 8 & $6.0\\times10^3$\\\\\n1000.0 & 10000.0 & 8 & 6.9\\\\\n1000.0 & 5000.0 & 10 & $0.233\\times10^6$\\\\\n1000.0 & 7000.0 & 10 & $8.7\\times10^3$\\\\\n1000.0 & 10000.0 & 10 & 9.7\\\\\n\\hline\n2000.0 & 2000.0 & 11 & $0.511\\times10^6$\\\\\n6000.0 & 6000.0 & 7 & 0.12\\\\\n\\hline\n\\end{tabular}\n\\capbox{Black hole production cross sections}{\\label{crosstab} Black hole production cross sections for various different parameters.  Those in the upper part of the table can be directly compared with values quoted in \\cite{Giddings:2001bu} (they are calculated using \\texttt{MSSDEF}=1, \\texttt{PDFGUP(I)}=4 and \\texttt{PDFSUP(I)}=48) whereas those in the lower part can be compared with \\cite{Dimopoulos:2001hw} (they are calculated using \\texttt{MSSDEF}=2, \\texttt{PDFGUP(I)}=3 and \\texttt{PDFSUP(I)}=34).  In all cases \\texttt{MAXMSS}=14000.0.}\n\\end{center}\n\\def1.0{1.0}\n\\end{table}\n\n\\subsection{Sample event display}\n\n\\begin{figure}[p]\n\\begin{center}\n\\epsfig{file=bhevent1.ps, angle=0, width=0.95\\textwidth}\n\\capbox{Sample event display}{Sample event display (details in text).\\label{display}}\n\\end{center}\n\\end{figure} \n\nFigure~\\ref{display} is an example black hole event display produced from the {\\small CHARYBDIS} output using the {\\small ATLANTIS} program\\footnote{\\url{http:\/\/atlantis.web.cern.ch\/atlantis\/}} which is being developed for the ATLAS experiment.  The black hole event used was generated with an earlier version of {\\small CHARYBDIS} which did not fully take into account the grey-body effects; however it can still be considered as a `typical' black hole event which displays many of the expected features.  The event shown is for a black hole mass of $\\sim$ 8~TeV in a scenario with \\texttt{TOTDIM=10} and \\texttt{MPLNCK=1000.0} (\\texttt{MSSDEF=1}).  The event includes 9 quarks, a gluon, an electron, a positron and three neutrinos.\n\n\\subsection{Sample particle spectra and emissivities}\n\nFigures~\\ref{higgs}--\\ref{photon} show the results, at parton level, of neglecting the time-variation of the black hole temperature (\\texttt{TIMVAR=.FALSE.}, dashed line) or the grey-body factors (\\texttt{GRYBDY=.FALSE.}, dot-dashed line) for initial black hole masses in the range from \\texttt{MINMSS=5000.0} to \\texttt{MAXMSS=5500.0} (with the default values for the other parameters).  The solid line is for simulations with the default parameter settings (but with the same reduced range of initial black hole masses used in the other two cases).\n\nThe effect of time-variation is to harden the spectra of all particle species.\nHowever, the effect of the grey-body factors depends on the spin, in this case slightly softening the spectra of scalars and fermions but hardening the spectrum of gauge bosons.\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=nhiggs2.eps, angle=0, width=.78\\textwidth}\n\\capbox{Parton-level energy spectra of Higgs bosons, $m_\\text{H} = 115$~GeV}{Parton-level energy spectra of Higgs bosons, $m_\\text{H} = 115$~GeV. Solid: predicted energy spectrum of Higgs bosons from decay of black holes with initial masses 5.0--5.5 TeV.  Dashed: neglecting time-variation of temperature.  Dot-dashed: neglecting grey-body factors.\\label{higgs}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=nelectron2.eps, angle=0, width=.78\\textwidth}\n\\capbox{Parton-level energy spectra of electrons and positrons}{Parton-level energy spectra of electrons and positrons.  As Figure~\\ref{higgs} but for electron and positron spectra.\\label{electron}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=nphoton2.eps, angle=0, width=.78\\textwidth}\n\\capbox{Parton-level energy spectra of photons}{Parton-level energy spectra of photons.  As Figure~\\ref{higgs} but for photon spectra.\\label{photon}}\n\\end{center}\n\\end{figure}\n\nRelative numbers of primary emitted particles were also calculated and are shown as percentages in Table~\\ref{probtab}.  The same parameters were used as for the spectra, except that in both cases the time-variation of the temperature was included (\\texttt{TIMVAR=.TRUE.}).  For comparison, the theoretical values used by the event generator (and easily calculable using Tables~\\ref{pprobs} and \\ref{fratios}) are also shown.  The differences are due to any kinematic constraints on the massive particles and also the imposed conservation of charge and baryon number; however since the average particle multiplicity per event is relatively high in these examples, the decay of the black hole is not significantly constrained.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n& \\multicolumn{4}{c|}{Particle emissivity (\\%)} \\\\\n\\cline{2-5}\n& \\multicolumn{2}{c|}{\\texttt{GRYBDY=.TRUE.}} & \\multicolumn{2}{c|}{\\texttt{GRYBDY=.FALSE.}}\\\\\n\\cline{2-5}\nParticle type & Generator & Theory & Generator & Theory\\\\\n\\hline\nQuarks & 63.9 & 61.8 & 58.2 & 56.5\\\\\nGluons & 11.7 & 12.2 & 16.9 & 16.8\\\\\nCharged leptons  & 9.4 & 10.3 & 8.4 & 9.4\\\\\nNeutrinos & 5.1 & 5.2 & 4.6 & 4.7\\\\\nPhoton & 1.5 & 1.5 & 2.1 & 2.1\\\\\nZ$^0$ & 2.6 & 2.6 & 3.1 & 3.1\\\\\nW$^+$ and W$^-$ & 4.7 & 5.3 & 5.7 & 6.3\\\\\nHiggs boson & 1.1 & 1.1 & 1.0 & 1.1\\\\\n\\hline\n\\end{tabular}\n\\capbox{Relative emissivities of primary partons}{Comparison of relative emissivities of primary partons with their theoretical values.\\label{probtab}} \n\\end{center}\n\\end{table}\n\n\\section{Experimental studies}\n\\label{expstud}\n\nResults of a full experimental study of black hole production and decay at the LHC will be presented in \\cite{ali}.  However some of the experimental issues will be highlighted here with reference to the theoretical work of Chapter~\\ref{greybody}.\n\nIf extra dimensions do exist, then a priority will be to try to determine how many there are.  The black hole generator described in this chapter was constructed with the aim of being able to more realistically examine ways of doing this experimentally.  The grey-body factors of the previous chapter were incorporated for the same reason.\n\nOne method for determining $n$ is described in \\cite{Dimopoulos:2001hw}: Black hole events are isolated using the experimental cuts mentioned in \\secref{expsig}, events are put into 500~GeV bins based on the estimated black hole mass and then, for each mass bin, photon or electron energy spectra are produced.  If Hawking temperatures can be determined by fitting the spectra, then the correlation between $T_\\text{H}$ and black hole mass can be used to try to determine the number of extra dimensions (and potentially also the value of $M_{\\text{P}(4+n)}$).  \n\n\\enlargethispage{-\\baselineskip}\nIt was argued in \\cite{Dimopoulos:2001hw} that this method is quite successful but a number of unjustified approximations were made.  These include a failure to take account of the temperature increase as the decay progresses, the grey-body effects described in Chapter~\\ref{greybody}, and the secondary photons and electrons.  Isolation cuts can be used to reduce the problem of secondary particles, but using the {\\small CHARYBDIS} generator (with {\\small GETJET} providing a very simple detector simulation) it was found that the time-dependence of the Hawking temperature significantly reduces the viability of such a method.  The effect of the black hole not always being at rest is also found to have a significant effect on the energy spectra (although this is itself a matter of some uncertainty since the way in which a black hole recoils against high-energy emissions may be complicated \\cite{Page:1980tc,Frolov:2002as,Frolov:2002gf}).\n\nAttempts were made to perform a more sophisticated version of this method by using a more realistic spectrum, obtained by numerically integrating the Hawking spectrum, for the fitting procedure.  This approach was significantly more successful for $n=2$ but less promising for larger values of $n$.  This is because, as the emitted particle multiplicity decreases, it is difficult to know what cut-off to use when calculating the time-integrated spectrum, particularly without making assumptions about the value of $M_{\\text{P}(4+n)}$.  This highlights a problem which was found repeatedly when dealing with the larger values of $n$: the expected particle multiplicity at the LHC is not particularly high (see calculations in \\secref{totfandp}) and so it is difficult to extract information in a way which is not affected by the kinematic constraints on the decay, and the way in which the Planck phase is modelled.\n\nExperimentalists from the ATLAS collaboration have subsequently spent a significant amount of time trying to find event shape variables and other distributions which can be used to extract the number of extra dimensions from black hole events; however results are as yet inconclusive.  It has hoped that by using the different event generator options for modelling the Planck phase it will be possibly to confirm that any proposed method is relatively insensitive to this final phase of the decay.  However most of methods investigated have been limited by the problem discussed above---that is, the low multiplicity at the LHC for large numbers of extra dimensions (and with $M_{\\text{P}(4+n)}$ assumed to be $\\sim$~TeV).  A higher energy collider in the more distant future would alleviate many of these problems.  The rate of fractional change of $n$ decreases with $n$ which also makes it progressively more difficult to distinguish $n$ from $n+1$ extra dimensions.  It is unfortunate that these problems are most significant for large values of $n$ since these have the best theoretical motivation and are the least constrained astrophysically.\n\n\\section{Discussion}\n\\label{discuss}\n\nThe {\\small CHARYBDIS} program described in this chapter would appear to be the most sophisticated black hole event generator available.  Figures~\\ref{higgs}--\\ref{photon} and Table~\\ref{probtab} confirm that the inclusion of the grey-body factors can have a significant effect on the particle energy spectra and emissivities.\n\nHowever it will have become clear to the reader that there are many assumptions and approximations made in the way the production and decay of black holes is modelled.  Various improvements could be made, but most of them would be impossible without further theoretical work.\n\nA major advance would be to correctly model stages of the black hole decay other than the Schwarzschild phase.  The balding phase could be taken into account if the amount of energy trapped by the black hole horizon was more accurately known.  It was recently pointed out \\cite{Anchordoqui:2003ug} that this might have a significant effect on the black hole discovery potential of the LHC\\@.  Further numerical calculations to estimate the typical amount of trapped energy could also provide information about the expected distribution of initial angular momenta.  This information, together with more theoretical work on the grey-body factors for rotating black holes, would allow the spin-down phase of the decay to be simulated in the event generator.  If attempting to model the spin-down phase, the assumption of isotropic particle emission from the black hole would no longer be a good one.   Grey-body factors are calculated for individual values of $\\ell$ and $m$ and so it might be possible for the angular distribution of emitted particles to be modelled correctly (although to do this fully would also require the spin-weighted spheroidal harmonics of equation~(\\ref{spinang}) to be calculated).  \n\nThe assumption that the emission of energy into the bulk as gravitons (and possibly scalars) can be neglected is not ideal.  The emission of gravitons from the black hole could be taken into account but a full implementation of this would again require more theoretical work on the grey-body factors.\n\nOther possible improvements would be to more carefully account for the modifications in the energy spectra of massive gauge bosons and other heavy particles, or to allow the black holes to emit supersymmetric particles (since extra dimension models motivated from string theory are expected to be supersymmetric).\n\nOne relatively simple modification to {\\small CHARYBDIS} would be to try to simulate black holes in a separated fermions model \\cite{Arkani-Hamed:1999za,Arkani-Hamed:1999dc} which would avoid some of the proton decay issues (see discussion in \\secref{protonbh}).  A model like this would be implemented by altering the number of degrees of freedom available for decays as described in \\cite{Han:2002yy}.  However, since the values of $M_{\\text{P}(4+n)}$ required in such models make black hole production experimentally inaccessible at the next generation of colliders, there is little motivation for such a modification.\n\nIt would be desirable to have a better understanding of both the validity of the quasi-stationary approach to the decay of higher-dimensional black holes, and of the Planck phase of the decay.  In the latter case, it seems unlikely that this will be possible without a full theory of quantum gravity and in the meantime the best approach would seem to be to try to extract information which is independent of the way in which the Planck phase is modelled.\n\n\n\n\n\\chapter{Conclusions}\n\nThe individual chapters of this thesis contain their own conclusions, so here there is only a brief overview, and a few comments about the future of the field.  Two different possibilities for new physics beyond the Standard Model have been examined: supersymmetry and extra dimensions.  Although the phenomenology of both models is diverse the particular aspects addressed were the detection of heavy leptons in intermediate scale supersymmetric models, and the production and decay of black holes in extra dimension scenarios.  In both cases it was shown that the next generation of particle colliders will be able to investigate these models.\n\nThe study of the detection of charged heavy leptons at the LHC using a time-of-flight technique showed that, in the particular model considered, the discovery mass range extends up to 950~GeV\\@.  In addition it was found that it would be difficult to use their differing angular distributions to distinguish the heavy leptons from scalar leptons if the lepton mass was greater than 580~GeV\\@.\n\nThe majority of the thesis addressed various aspects of the production and decay of miniature black holes.  The grey-body factors which were calculated numerically in Chapter~\\ref{greybody} are useful in that they allow a more realistic Monte Carlo event generator to be constructed, as described in Chapter~\\ref{generator}.  Whilst the experimental signatures produced by black hole decays are expected to make events unmissable, it is important to model grey-body effects correctly if information (like the number of extra dimensions) is to be accurately extracted.  It was found that the grey-body factors modify the particle energy spectra and emissivities and so might be important in such a task.  The work on grey-body factors also allowed some of the usual assumptions and approximations made in black hole decay to be examined more carefully.  The assumption that the relative emission of particles into the bulk is small was shown to be correct, but some doubt was cast on the usual approximation that a higher-dimensional black hole decay can be treated in a quasi-stationary manner.  \n\nOne conclusion from this work was that if the number of extra dimensions is more than three or four, the centre-of-mass energy available at the LHC is barely sufficient for the semi-classical approximations used to be legitimate.  Therefore it might need a higher energy collider to investigate black hole production in detail.  The LHC would still be expected to provide the first indications of extra dimensions through graviton production, Planckian effects and the first signs of black hole production.\n\nThe question of how likely it is that any particular model for new physics is realized in nature is inevitably impossible to quantify.  Every physicist has their own prejudices and in general supersymmetric models seem to be the favoured option.  Extra dimension models are `wacky' enough to attract the attention of the popular science media and are quickly dismissed by some physicists.  However, as outlined in Chapter~\\ref{introch}, they can be well motivated theoretically.\n\nAlmost all high-energy physicists are convinced that the LHC will discover new physics of some kind.  Leaving aside all the theoretical arguments, it would be unprecedented in the history of the field if the order-of-magnitude increase in available energy did not reveal something new.  It may well be that nature surprises us and reveals physics that nobody has so far suggested.  Even if extra dimensions do exist the compactification scheme may make the physics much more complicated than the simple cases discussed here.  \n\nWith half a decade still to wait before CERN's Large Hadron Collider has produced a substantial amount of data, there is good reason for theorists to continue investigating these models, and refining and extending their phenomenological studies.\n\n\n\n\\chapter*{Preface}\n\nThis constitutes my research for a PhD degree in theoretical high-energy physics.  Chapter~\\ref{introch} contains a review of the relevant theoretical background.  It sets the context in which the work described in the rest of the thesis was carried out, and introduces the theoretical models on which the work is based.  Chapter~\\ref{leptons} describes work undertaken during the first year of my PhD on the prospects for detecting exotic heavy leptons at the Large Hadron Collider (LHC).  This was published \\cite{Allanach:2001sd} as a study performed by members of the Cambridge Supersymmetry Working Group, but the contents of Chapter~\\ref{leptons} are entirely my own work.  \n\nThe remainder of the \nthesis concentrates on the study of miniature black holes at the LHC---a topic which is introduced in Chapter~\\ref{bhintro}.  Chapter~\\ref{greybody} describes new theoretical work on black hole grey-body factors (building on previous work by Kanti and March-Russell); much of this work was published in \\cite{Harris:2003eg} and contributions from my collaborator, P.~Kanti, are clearly identified in Chapter~\\ref{greybody}.  The final chapter gives details of the {\\small CHARYBDIS} event generator written to simulate black hole production and decay.  This was published in \\cite{Harris:2003db} and, as detailed in Chapter~\\ref{generator}, involved some significant contributions from P.~Richardson (these were mainly involved in interfacing {\\small CHARYBDIS} to existing Monte Carlo programs).  The chapter also contains a brief discussion of some preliminary experimental studies using the event generator.  Some of this work will be published in \\cite{ali}, together with more detailed analyses by members of the ATLAS collaboration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nRecent experimental advances \\cite{Pendry2004} have put to living the\nspeculation about materials with negative refraction index, initiated\nin 1968 by Victor Veselago on purely theoretical grounds\n\\cite{Veselago1968}.  Such materials, starting to be at reach\nnowadays primarily for the microwave region, manifest exciting and\nunconventional phenomena ranging from the Inverse Doppler effect\n\\cite{Seddon2003} to the possibility of diffraction-free imaging\n\\cite{Grbic2004}. \n\nWere such ``left-handed'' materials, as Veselago called them, available\nfor any frequency domain, they would undoubtedly revolutionise optics.\nHowever, a negative refraction index requires simultaneously a\nnegative permittivity and a negative permeability, implying that the\nmaterial has dielectric and magnetic resonances in the same frequency\ndomain --- a property which turns out to be extremely rare and which is\npartly responsible why such materials have remained undiscovered for\nalmost three decades.\n\nUnlike to the optical domain, where the electric and magnetic\nresonances are generally separated in frequency by several orders of\nmagnitude, negatively refracting materials have recently been\nengineered for narrow bands in the microwave domain. One of the main\ningredients of these ``metamaterials'' are regular arrays of so-called\nsplit-ring resonators \\cite{Pendry2004}.\n\nIn this paper, we will focus on the dielectric part of the response of\narrays of similar resonators in square lattices. One goal is to\nprovide tools for the location of the resonances in the spectrum, a\ntask which cannot simply be performed by looking at the building\nblocks of such an array \\cite{Laurent2003}. At the same time, the\ntheoretical approaches presented in this paper allow for a deeper\nunderstanding of the resonance spectrum --- certainly an advantage in\nthe quest of materials with a tailored electromagnetic\nresponse. Throughout the paper, the analytical results are compared to\nnumerically calculated spectra, obtained either via Exact Numerical\nRenormalisation (ENR) \\cite{Vinograd,Sarych,Tort} or from the spectral\nmethod \\cite{Straley,Bergman,Milton} formulated with Green's functions\n\\cite{Clerc1996,Luck1998,Laurent2000}.\n\nAn archetype of the circuits to be dealt with in this paper is\ndepicted in Fig.~\\ref{fig:simple}. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.5\\cwidth]{simpleLC.eps}\n\\end{center}\n\\caption{\\label{fig:simple} One of the simplest non-trivial clusters,\nconsisting of two self-induction coils embedded in a capacitor\nnetwork.} \n\\end{figure}\nThis ``cluster'' consists of two coils in series (each of inductance\n$L$ and with no ohmic resistance, $R=0$) embedded in the simplest\nimaginable network made of three capacitors (each of capacity $C$). As\nwill be seen in the following, this circuit contains --- despite its\nsimplicity --- many of the essential features of the more complicated\narrays to be studied in the following sections.\n\nWith no external voltage applied to the plates, the state of the\nsystem can be obtained by straightforward application of Kirchhoff's\nrules: the loop rule yields a differential equation for the current\nthrough any of the coils $I_{k}$ in terms of the currents flowing\nthrough the vertical and the horizontal capacitor belonging to the\nsame loop; the latter currents are subsequently eliminated by\nKirchhoff's junction rule, resulting in a homogeneous system of two\nordinary differential equations for the currents through the coils.\nApplying a time-dependent external voltage $V_{\\rm ext}(t)$ to the\nplates introduces an inhomogeneity in the equations, which finally\nread\n\\begin{equation}\n\\label{simple:ode}\n\\frac{1}{\\wo^2} \\frac{\\dd^2}{\\dd t^2} \n\\begin{pmatrix} I_{1} \\\\ I_{2} \\end{pmatrix}\n\\,-\\,\n\\begin{pmatrix} -2 & \\;1 \\\\ \\;1 & -2 \\end{pmatrix}\n\\begin{pmatrix} I_{1} \\\\ I_{2} \\end{pmatrix}\n\\;=\\;\nI_{\\rm ext}(t) \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}\n\\text{,}\n\\end{equation} \nwith $\\wo=1\/\\sqrt{LC}$ and  $I_{\\rm ext}(t)$ the total net current\nflowing through the sample. \n\nEq.~(\\ref{simple:ode}) can be solved by standard means yielding the\nhomogeneous eigenmodes\n\\begin{equation}\n\\label{simple:Ih}\n\\V{I}_{\\rm h}(t)\n\\;=\\; \nc_{1} \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\e^{\\ic\\wo t}\n\\,+\\,\nc_{2} \\begin{pmatrix} \\;1 \\\\ -1 \\end{pmatrix} \\e^{\\ic \\sqrt{3}\\wo t}\n\\;\\mbox{,}\n\\end{equation}\nwhere $c_{1}$ and $c_{2}$ are arbitrary complex amplitudes. The\napplication of a sinusoidal external voltage of frequency $\\Omega$,\nimplying an external current $I_{\\rm ext}(t)=\\hat{I}_{\\rm\next}\\exp(\\ic\\Omega t)$, allows for {\\em one} additional solution\n\\begin{equation}\n\\label{simple:Iinh}\n\\V{I}_{\\rm inh}(t)\n\\;=\\;\nI_{\\rm ext}(t) \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\,\n\\begin{cases}\n\\frac{\\wo^{2}}{\\wo^{2}-\\Omega^{2}} \n& \\text{for $\\Omega\\neq\\wo$, } \\\\\n-\\frac{\\ic}{2} \\wo t \n& \\text{for $\\Omega=\\wo$.} \n\\end{cases}\n\\end{equation}\n\nFour observations can be made at this point: \n(i) in networks consisting of two species of components only, the\nstate of the system is fully described by a set of differential\nequations for the currents flowing through the minority components (in\nour case the coils). The contribution of the majority components\n(here, the capacitors) to the Kirchhoff rules is purely algebraic, and\nmay be eliminated from the system of equations.\n(ii) The external voltage can only excite the first resonance, of\nfrequency $\\wo$. The second resonance, $\\sqrt{3}\\wo$, is antisymmetric\n($I_{1}=-I_{2}$) and thus orthogonal to the currents which are always\ninduced symmetrically in the present configuration of the voltage\nplates.\n(iii) The applied voltage equals the sum of the two voltage drops in\nthe coils, {\\sl i.e.} $V_{\\rm ext}=L \\frac{\\dd}{\\dd\nt}(I_{1}+I_{2})$. Off resonance, this yields a total impedance of\n$Z={V_{\\rm ext}}\/{I_{\\rm ext}}=2\\ic L\\Omega\n\\wo^2\/(\\wo^{2}-\\Omega^{2})$ between the plates. This result may be\ngeneralised: as a resonance is approached, the {\\em internal} currents\nof the sample diverge for finite applied voltage. This may be\nassociated with an infinite {\\em conductivity} of the sample's\n(internal) components. At the same time, however, a divergent\nimpedance and thus zero conductivity is measured between the plates.\n(iv) In reality, the coils have a small but finite ohmic resistance\n$R$. The eigenfrequencies are thus shifted slightly in the complex\nplane towards small positive imaginary parts which damp out the\nsample's free modes, eq.~(\\ref{simple:Ih}). In addition to its\nimaginary part, the impedance $Z$ acquires a real part consisting of a\nnarrow lorentzian peak centred at the resonance frequency.\n\n\nThe paper is organised as follows: in Sec.~\\ref{sec:kirchhoff}, the\nresonance spectra of regular arrays with not too complicated unit\ncells are calculated directly from the solution of Kirchhoff's\nrules. In Sec.~\\ref{sec:transfer}, more complicated arrays are tackled\nwith an approach based on transfer matrices. Regular one-dimensional\n(1D) arrays in an infinite two-dimensional (2D) lattice are examined\nin Sec.~\\ref{sec:inflattice}, and the results compared to the formerly\ndiscussed 2D clusters. Finally, the Appendix~\\ref{app:dipole} is\ndevoted to a simple physical approximation, based on a dipole scenario\nin a 2D environment, which turns out to be helpful for the\ninterpretation of the spectra of linear clusters.\n\n\n\\section{\\label{sec:kirchhoff}Direct solution of Kirchhoff's rules}\n\nIn this section, we will tackle simple regular 1D and 2D binary arrays\nby a direct solution of Kirchhoff's rules. The results are then to be\ncompared to those obtained by Exact Numerical Renormalisation (ENR)\n\\cite{Vinograd,Sarych,Tort}.  The strategy of the latter algorithm\nconsists of eliminating the network sites one by one, while\nrenormalising the impedance between all couples of former neighbours\nof the eliminated site such that the global impedance remains\ninvariant, until the electrodes are connected by just one bond, which\nthen carries the whole network's impedance.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.7\\cwidth]{ladderL.eps}\n\\end{center}\n\\caption{\\label{fig:ladderL} A ladder-shaped circuit containing $N$\ncoils in series on one leg, and the same number of bonds with no\nimpedance on the other. All $N+1$ steps of the ladder are capacitors.}\n\\end{figure}\nTo begin with, we are going to summarise the solution of Kirchhoff's\nrules for the 1D array shown in Fig.~\\ref{fig:ladderL}. This\nladder-shaped circuit consists of a horizontal line of $N$ coils, each\nof inductance $L$, which is connected by $N+1$ capacitors, each of\ncapacity $C$, to another horizontal line with no resistance at\nall. The loop rule states that the voltage drop around a closed loop\nis zero; applying it to the $k.$th mesh and deriving with respect to\ntime yields\n${\\dd^2 I_{k}}\/{\\dd t^2}=\\wo^{2}(i_{k}-i_{k-1})$,\nwith $\\wo=1\/\\sqrt{LC}$. The currents through the capacitors, $i_{k}$,\ncan be eliminated using the junction rule $i_{k}+I_{k}=I_{k+1}$, where\n$I_{k}$ is assumed zero if $k$ is out of range. In total, we get\n\\begin{equation}\n\\label{eqL1}\n\\frac{\\dd^2}{\\dd t^2}\\V{I}\\;=\\;\\wo^2 \\M{D}\\V{I}\n\\text{,}\n\\end{equation}\nwhere $\\trans{\\V{I}=(I_{1},\\ldots,I_{N})}$ is the coil current vector,\nand $\\M{D}$ the 1D lattice Laplacian, {\\sl i.e.} a tridiagonal matrix\nwith entries $-2$ for all diagonal elements and $1$ for all non-zero\noff-diagonal elements.\n\nThe usual ansatz $\\V{I}\\sim\\exp{(\\ic\\w t)}$ converts the differential\nequation~(\\ref{eqL1}) into an eigenvalue problem for the tridiagonal\nmatrix $\\M{D}+\\wt^2\\M{1}$, where $\\wt=\\w\/\\wo$ stands for the frequency\nin units of $\\wo$. Its determinant can be calculated explicitly: for a\n$N$-dimensional tridiagonal matrix $\\M{M}_{N}(x)$ with diagonal\nelements $2x$, and first sub- and super-diagonal elements $1$,\nexpansion of the determinant by minors yields\n\\begin{equation}\n\\label{chebyshev}\nD_{N}(x)\\;=\\;2x D_{N-1}(x)\\,-\\,D_{N-2}(x)\n\\;\\text{,}\n\\end{equation}\nwhere $D_{N}(x)=\\det{\\M{M}_{N}(x)}$. We recognise in (\\ref{chebyshev})\nthe recurrence relation generating the Chebyshev polynomials. The\nrequired initial conditions, $D_{0}(x)=1$ and $D_{1}(x)=2x$, ties us\ndown to the Chebyshev polynomials of the second kind:\n\\begin{equation}\n\\label{chebyU}\nU_{N}(x=\\cos\\theta)\\;=\\;\\frac{\\sin\\left[(N+1)\\theta\\right]}{\\sin\\theta}\n\\end{equation}\nThe roots of $U_{N}(\\cos\\theta)$, which occur at\n$\\theta_{m}=\\frac{m\\pi}{N+1}$, determine the eigenfrequencies. In the\npresent example, where $2x=2\\cos\\theta_{m}=-2+\\wt^2$, after\nrenumbering the eigenstates according to their frequency, we have\n\\begin{equation}\n\\label{dispersL1}\n\\wt_{m}=\\pm 2\\sin\\frac{m\\pi}{2(N+1)}\n\\quad \\text{(for $m=1\\ldots N$).}\n\\end{equation}\nFor infinite ladder length, the dispersion relation $\\wt(k)$ {\\sl\nversus} wave number $k=\\frac{m\\pi}{N+1}$ is shown in the first graph\nof Fig.~\\ref{fig:dos}.\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=0.8\\twidth]{dos.eps}\n\\end{center}\n\\caption{\\label{fig:dos} Dispersion relations $\\wt(k)$ (left column)\nand corresponding densities of states $\\rho(\\wt)$ (right column). The\ntwo graphs in the first line correspond to the ladder of\nFig.~\\ref{fig:ladderL} (continuous lines --- eqs.~(\\ref{dispersL1}) and\n(\\ref{dosL1})) and to its 2D counterpart shown in\nFig.~\\ref{fig:unitcells}(a) (dashed lines). The two graphs in the\nsecond line where obtained for the arrays of\nFig.~\\ref{fig:unitcells}(b) (continuous lines ---\neqs.~(\\ref{dispersL2}) and (\\ref{dosL2})), and\nFig.~\\ref{fig:unitcells}(c) (dotted lines --- eq.~(\\ref{dispersLC1})).}\n\\end{figure*}\nThe associated density of states\n\\begin{equation}\n\\label{dosL1}\n\\rho(\\wt)\\,=\\,\\lim_{N\\to\\infty}\\frac{1}{2 N}\\sum\\limits_{m} \\delta(\\wt-\\wt_{m})\n\\,=\\, \\frac{1}{\\pi\\sqrt{4-\\wt^2}}\n\\;\\text{,}\n\\end{equation}\nshown in the second graph, is finite for $\\wt=0$, since the sample's\nlowest resonance frequencies go to zero for $N\\to\\infty$. The fact\nthat $\\rho(\\wt)$ displays Van Hove singularities for $\\wt=\\pm 2$\nshould not be misinterpreted in the sense that the sample mainly\nresonates in these frequencies when exposed to a multi-frequency\ninput: on the contrary, as will be pointed out in\nSec.~\\ref{sec:transfer}, for reasons of symmetry there is very little\noverlap between the voltage applied to the electrodes, generating an\noverall current with symmetry\n$\\trans{\\V{I}_{\\rm ext}\\sim (1,1,\\ldots,1)}$, and the highest excited\nstates, with $m$ close to $N$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.95\\cwidth]{array.eps}\n\\end{center}\n\\caption{\\label{fig:array} Staircase-array obtained by replicating the\nunit cell of Fig.~\\ref{fig:unitcells}(d) $N=3$ times in\n$x$-direction. The resulting structure is replicated in $y$-direction\nuntil the electrodes are fully covered. Periodic boundary conditions\nare assumed in $y$-direction.}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.9\\cwidth]{unitcells.eps}\n\\end{center}\n\\caption{\\label{fig:unitcells} Unit cells for several 2D arrays (left\ncolumn) and corresponding 1D building blocks (right column).}\n\\end{figure}\nIn the following, we are going to study regular 2D arrays generated\nfrom a unit cell which tiles the surface between the electrodes. At\nfirst, the unit cell is replicated $N$ times in $x$-direction, {\\sl\ni.e.} from the left towards the right electrode. The resulting\nhorizontal ladder is then stacked in $y$-direction until the width of\nthe electrodes is fully covered, assuming periodic boundary\nconditions. In other words, the 2D arrays to be studied are cylinders\nof length $N$ (times the length of the unit cell). At either end, they\nare connected to an electrode ring via at least one rank of pure\ncapacitor bonds, which avoids short circuiting and keeps the influence\nof the electrodes on the dielectric spectra as low as possible.\n\nThe building block which generates an array resembling the ladder in\nFig.~\\ref{fig:ladderL} is shown in Fig.~\\ref{fig:unitcells}(a). \nSimilar arrays can be generated from the other unit cells in\nFig.~\\ref{fig:unitcells}: the array shown in Fig.~\\ref{fig:array} was\nobtained from the unit cell represented in Fig.~\\ref{fig:unitcells}(d)\nwith $N=3$.\n\nAs illustrated in the right column of Fig.~\\ref{fig:unitcells}, these\n2D arrays can be mapped onto 1D ladders: translational invariance\nalong the electrodes in steps of an integer times the height of the\nunit cell allows to ``bend down'' the unbound components in the top\nrow and to attach them to the corresponding site at the\nbottom. Therefore, the only difference between the original ladder of\nFig.~\\ref{fig:ladderL} and the ladder in Fig.~\\ref{fig:unitcells}(a)\nresumes to doubling the vertical capacitors, and thus dividing the\nfrequency range by a factor of $\\sqrt{2}$ in eqs.~(\\ref{dispersL1})\nand (\\ref{dosL1}). (The corresponding dispersion relation and density\nof states are plotted with dashed lines in the first row of\nFig.~\\ref{fig:dos}).\n\nThe array in Fig.~\\ref{fig:unitcells}(b) can be tackled analogously.\nThe additional capacitors only contribute to the diagonal elements of\nthe tridiagonal matrix, and instead of eq.~(\\ref{eqL1}) one obtains\n$\\dd^{2}\\V{I}\/\\dd t^{2}=\\frac{\\wo^2}{2} [\\M{D}-2\\cdot\\M{1}]\\V{I}$. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.9\\cwidth]{renorm-5L.eps}\n\\end{center}\n\\caption{\\label{fig:renorm5L} \nImpedance between the electrodes for an array generated from the unit\ncell in Fig.~\\ref{fig:unitcells}(b). The sample consists of 6 lines,\nof $N=5$ inductors in series each, substituting the central piece of\nevery second row in a 12 by 12 capacitor lattice. The real part of the\nENR impedance, calculated for $L=C=1$ and $R_{\\rm coil}=10^{-4}$, is\nplotted with continuous lines. The dotted lines indicate the resonance\nfrequencies obtained analytically from eq.~(\\ref{dispersL2}) for $N=5$\nand $m=1\\ldots5$.}\n\\end{figure}\nThe resulting resonance frequencies,\n\\begin{equation}\n\\label{dispersL2}\n\\wt_{m}=\\pm \\sqrt{1+2\\sin^{2}\\frac{m\\pi}{2(N+1)}}\n\\quad \\text{(for $m=1\\ldots N$),}\n\\end{equation}\nare plotted, for $N=5$, with dotted lines in Fig.~\\ref{fig:renorm5L}.\nA corresponding ENR calculation for the total conductivity of the\narray between electrodes (continuous lines in Fig.~\\ref{fig:renorm5L})\ndetects the resonance frequencies with $m$ even, associated with\nsymmetric eigenmodes. The antisymmetric eigenmodes are orthogonal to\nthe current vector induced by the plates, and remain hence invisible\nin this calculation.\n\nThe dispersion relation obtained from eq.~(\\ref{dispersL2}) for\n$N\\to\\infty$, albeit with $k=\\frac{m\\pi}{N+1}$ finite, along with the\ncorresponding density of states,\n\\begin{equation}\n\\label{dosL2}\n\\rho(\\wt)\\,=\\, \n\\frac{\\abs{\\wt}}{\\pi\\,\\sqrt{1-\\left(2-\\wt^2\\right)^{2}}}\n\\;\\text{,}\n\\end{equation}\nis illustrated by the continuous lines in the graphs in the second row\nof Fig.~\\ref{fig:dos}.  This time, even for infinite chain length,\nthere are no low-lying resonances, and a gap spreads between $-1$ and\n$+1$.\n\nThe last array to be treated with this method can be obtained from a\npure capacitor lattice by replacing every second horizontal capacitor\non every second line by a coil. In this array, which corresponds to\nthe unit cell in Fig.~\\ref{fig:unitcells}(c), meshes with one coil and\nthree capacitors alternate horizontally with pure capacitor\nmeshes. For the latter, Kirchhoff's rules reduce to purely algebraic\nrelations; these can be removed from the system of differential\nequations, and one ends up with\n\\begin{equation}\n\\label{eqLC1}\n\\frac{\\dd^2}{\\dd t^2}\\V{I}\\;=\\;\n\\frac{\\wo^2}{12}\n\\begin{pmatrix} \n   -23 &      1 &      0 & \\cdots & 0      \\\\\n     1 &    -22 &      1 & \\ddots & \\vdots \\\\\n     0 & \\ddots & \\ddots & \\ddots & 0      \\\\\n\\vdots & \\ddots &      1 &    -22 & 1      \\\\\n     0 & \\cdots &      0 &      1 & -22\n\\end{pmatrix}\n\\V{I}\n\\text{\\;,}\n\\end{equation}\nwhere $\\V{I}$ is the current vector through the $N$ coils of the\ncluster. The matrix in eq.~(\\ref{eqLC1}) differs slightly from the\nhabitual form, since not all diagonal elements are the same.  The\nasymmetry in the upper left corner translates the fact that the\ncorresponding ladder (right column of Fig.~\\ref{fig:unitcells}(c))\nstarts with a loop of 3 capacitors and 1 coil, but ends with a\n4-capacitor loop.\n\nThe solution of eq.~(\\ref{eqLC1}) is analogous to the generic case,\neq.~(\\ref{eqL1}), albeit leading to a slightly more complicated\nrecursion relation.  A cumbersome but straightforward calculation\ngives the dispersion relation\n\\begin{equation}\n\\label{dispersLC1}\n\\wt_{m}=\\pm \\sqrt{\\frac{5}{3}+\\frac{1}{3}\\sin^{2}\\frac{m\\pi}{2 N+1}}\n\\quad \\text{(for $m=1\\ldots N$),}\n\\end{equation}\nwith an associated density of states showing narrow bands in the range\nof $\\abs{\\wt}=\\sqrt{5\/3}\\simeq1.29$ to $\\sqrt{2}\\simeq 1.41$. As can\nbe seen from the dotted curves in the last two graphs of\nFig.~\\ref{fig:dos}, the dispersion flattens substantially, and the\nbands in the density of states are squeezed with respect to the former\ncase, without pure capacitor meshes (continuous lines in the same\ngraphs, {\\sl cf.} eqs.~(\\ref{dispersL2}) and (\\ref{dosL2})).\n\nThis behaviour translates the tendency of the resonances to localise\non individual loops as more and more pure capacitor loops are inserted\nin the ladder. In the limit of infinite distance between coils, one is\nleft with $N$ decoupled LC circuits of the type\n\\begin{equation}\n\\label{LCinf}\n\\includegraphics[height=12ex]{LCinfty.eps}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Cinf}\n\\includegraphics[height=15ex]{Cinfty.eps}\n\\text{.}\n\\end{equation}\nThe solution of (\\ref{Cinf}) is\n$C_{\\rm inf}=\\frac{1}{2}C_{\\rm V}(1+\\sqrt{1+2C\/C_{\\rm V}})$\nwhich in our case, where $C_{\\rm V}=2C$, reduces to\n$C_{\\rm inf}=C(1+\\sqrt{2})$.  \nAll $N$ LC circuits of type (\\ref{LCinf}) resonate thus at the same\n$\\wt=\\pm\\sqrt{2\\sqrt{2}-1}\\simeq\\pm 1.35$, and the density of\nstates of the system reduces to delta functions at these frequencies\nwhich lie in the centre of the narrow dotted bands shown in the fourth\ngraph of Fig.~\\ref{fig:dos}.\n\nThe main advantage of the direct solution of Kirchhoff's rules,\npresented for several arrays in this section, are its analytical\nresults and the physical insight it provides into the structure of the\nresonance spectra. On the other hand, for increasingly complex arrays,\nthe method requires --- if feasible at all --- more and more cumbersome\ncalculations for the solution of the recursion relations it relies\non. We will therefore present an algorithm which circumvents this\nproblem in the next section.\n\n\\section{\\label{sec:transfer}Transfer matrix method}\n\nThe analysis of the resonance spectra of the arrays discussed so far\n--- which due to their translational invariance along the electrodes\nreduce to effective 1D problems --- can be rephrased very efficiently\nusing transfer matrices.\n\nTo illustrate the concept of a transfer matrix, consider the\nquadrupole\n\\begin{equation}\n\\label{quadrupole}\n\\includegraphics[width=18ex]{quadrupole.eps}\n\\;\\text{.}\n\\end{equation}\nIts incoming and outgoing currents and voltages are connected to each\nother via\n\\begin{equation}\n\\label{transfer}\n\\V{J}_{2}\\;=\\;\\M{T}\\V{J}_{1}\n\\;\\text{,}\n\\end{equation}\nwhere $\\trans{\\V{J}_{k}=(U_{k},I_{k},U_{k'},I_{k'})}$ is the vector of\nstate at each end $k=1,2$, {\\sl i.e.} a combination of currents $I$\nand voltages $U=V\\sigma_{0}$ (the latter, for convenience, multiplied\nby the conductivity $\\sigma_{0}$ of the lattice's majority component);\n$\\M{T}$ represents the dimensionless transfer matrix.\n\nFor quadrupoles exclusively made of passive elements, there are two\nconservation laws. The first states that the potentials are determined\nup to an arbitrary additive constant $U_{0}$. Increasing all incoming\npotentials by $U_{0}$ shifts the outgoing ones by the same amount:\nhence $\\trans{(1,0,1,0)}$ is a right eigenvector of the generally\nnon-symmetric matrix $\\M{T}$, with corresponding eigenvalue\n$\\mu=1$. Secondly, the continuity equation states that the total\nincoming and outgoing current have to be the same: hence, $(0,1,0,1)$\nis a left eigenvector of $\\M{T}$, again associated with $\\mu=1$.\n\nThese conservation laws may be easily verified since all passive\nquadrupoles can be assembled by combining two prototypes,\n\\begin{subequations}\n\\label{transferAB}\n\\begin{equation}\n\\label{transferABgraph}\n\\begin{tabular}{ccc}\n\\includegraphics[width=18ex]{quadrua.eps} & \\; & \n\\includegraphics[width=18ex]{quadrub.eps} \\\\\n\\text{(A)} && \\text{(B)}\n\\end{tabular}\n\\end{equation}\nwith corresponding transfer matrices\n\\begin{equation}\n\\label{transferABmatrix}\n\\M{T}_{\\rm A}=\n\\begin{pmatrix} \n1 & -\\frac{1}{\\eta} & 0 & 0                \\\\\n0 &               1 & 0 & 0                \\\\\n0 &               0 & 1 & -\\frac{1}{\\eta'} \\\\\n0 &               0 & 0 & 1                 \n\\end{pmatrix}\n\\;\\text{and}\\;\n\\M{T}_{\\rm B}=\n\\begin{pmatrix} \n1     &  0 &     0 & 0 \\\\\n-\\eta &  1 & \\eta  & 0 \\\\\n0     &  0 &     1 & 0 \\\\\n\\eta  &  0 & -\\eta & 1                 \n\\end{pmatrix}\n\\text{,}\n\\end{equation}\n\\end{subequations}\nwhere $\\eta=\\sigma\/\\sigma_{0}$ (and $\\eta'=\\sigma'\/\\sigma_{0}$) denote\nthe conductivities of the components (in units of\n$\\sigma_{0}$). Furthermore, since the transfer matrix $\\M{T}$ of any\npassive quadrupole can be written as a product of matrices of type\n$\\M{T}_{\\rm A}$ and $\\M{T}_{\\rm B}$, one always has $\\det\\M{T}=1$ ---\na relation that will be of some importance in the following.\n\nNaively one could define a resonance as ``if what goes in, comes\nout''. This amounts to resolving\n\\begin{equation}\n\\label{naivetransfer}\n\\begin{pmatrix}\nU_{2} \\\\ I_{2} \\\\ U_{2'} \\\\ I_{2'} \n\\end{pmatrix}\n=\n\\begin{pmatrix}\n0 \\\\ I_{1} \\\\ w \\\\ I_{1'} \n\\end{pmatrix}\n=\n\\M{T}\n\\left[\n\\begin{pmatrix}\n0 \\\\ I_{1} \\\\ w \\\\ I_{1'} \n\\end{pmatrix}\n+\\Delta U\n\\begin{pmatrix}\n1 \\\\ 0 \\\\ 1 \\\\ 0\n\\end{pmatrix}\n\\right]\n\\;\\text{,}\n\\end{equation}\nwhere (arbitrarily) leg $2$ has been grounded, $U_{2}=0$; $\\Delta U$\ndenotes the voltage drop between the two sides of the quadrupole, and\n$w$ stands for the potential difference between two legs on the same\nside of the quadrupole. When placed between electrodes, a resonance\nwould be seen as an impedance pole, implying that at finite $\\Delta U$\nthe net current through the sample is zero,\n$I_{1}+I_{1'}=I_{2}+I_{2'}=0$.  Apart from the technical difficulty\nthat (\\ref{naivetransfer}) relies on the solution of a singular system\nof equations, this method nicely yields the lowest eigenfrequency of\nthe quadrupole.\n\nBy contrast, the same strategy applied to a chain of $N$ identical\nquadrupoles in line (with transfer matrix $\\M{T}^{N}$), fails to\ndetect any additional eigenfrequency. To understand this failure, we\nrecall that eq.~(\\ref{naivetransfer}) implicitly attaches each of the\ntwo outgoing legs of the chain to the corresponding incoming leg, thus\nenforcing periodic boundary conditions under which the system acquires\nan additional translational symmetry along the chain. The lowest\neigenmode of the system possesses the full translational symmetry and\nshows the same current distribution for each of the $N$ quadrupoles,\nimplying an eigenvector collinear to the current induced by the\nelectrodes, $\\trans{\\V{I}_{\\rm ext}\\sim (1,1,\\ldots,1)}$. Since all\nother eigenvectors are orthogonal to the groundstate, they cannot be\nseen by simply looking ``if what goes in comes out''. (This fact is\nrigorous for periodic boundary conditions; for closed boundary\nconditions, as assumed in Sec.~\\ref{sec:kirchhoff}, the same mechanism\ngradually suppresses the response of the higher eigenmodes who have\nvery little overlap with the groundstate --- see\nFig.~\\ref{fig:renorm5L}.)\n\nWe will now present a new resonance condition which remedies these\nshortcomings; namely that, at resonance, all 4 eigenvalues $\\mu$ of\nthe transfer matrix of the entire chain should be unity. At this\npoint, one may think of it as a necessary condition which prevents the\nnorm of the state vector $\\V{J}$ from diverging or going to zero after\nmany transfers through the same quadrupole chain.  The proof that the\nnew resonance condition is equivalent to the usual definition --- {\\sl\ni.e.}  a divergent impedance between the electrodes (and thus zero net\ncurrent through the sample) --- is deferred to the end of this\nsection.\n\nIn order to simplify the following calculations, we introduce the\nhermitian and unitary matrix\n\\begin{equation}\n\\label{matrixP}\n\\M{P}=\\frac{1}{\\sqrt{2}}\n\\begin{pmatrix}\n1  &\\;  0 &\\;  1 &\\;  0 \\\\\n0  &\\;  1 &\\;  0 &\\;  1 \\\\\n1  &\\;  0 &   -1 &\\;  0 \\\\\n0  &\\;  1 &\\;  0 & -1                 \n\\end{pmatrix}\n\\end{equation}\nwhich allows for a transformation of the state vector to a symmetric\nbasis,\n$\\V{\\tilde{J}}_{k}=\\M{P}\\V{J}_{k}\n=\\trans{(U_{k}^{+},I_{k}^{+},U_{k}^{-},I_{k}^{-})}$\nwith components $U_{k}^{\\pm}=\\frac{1}{\\sqrt{2}}(U_{k}\\pm U_{k'})$ and\n$I_{k}^{\\pm}=\\frac{1}{\\sqrt{2}}(I_{k}\\pm I_{k'})$. The associated\ntransfer matrix of a single quadrupole in this basis is\n$\\M{S}=\\M{P}\\M{T}\\M{P}$, and thus of the general form\n\\begin{math}\n\\M{S}=\\left(\n\\begin{smallmatrix}\n\\M{A} & \\M{B} \\\\ \\M{C} & \\M{D}\n\\end{smallmatrix}\n\\right)\n\\end{math}\nwith the $2\\times 2$ submatrices\n\\begin{subequations}\n\\begin{align}\n\\label{matrixSsub}\n\\M{A}&=\\begin{pmatrix} 1 & a \\\\ 0 & 1 \\end{pmatrix}&\n\\M{B}&=\\begin{pmatrix} b_1 & b_2 \\\\ 0 & 0 \\end{pmatrix}\n\\\\\n\\M{C}&=\\begin{pmatrix} 0 & c_1 \\\\ 0 & c_2 \\end{pmatrix}&\n\\M{D}&=\\begin{pmatrix} d_1 & d_2 \\\\ d_3 & d_4 \\end{pmatrix}\n\\text{.}\n\\end{align}\n\\end{subequations}\nDue to the particular structure of the submatrices --- implying\n$\\M{A}^{n}=\n\\left(\\begin{smallmatrix} 1 & n a \\\\ 0 & 1\\end{smallmatrix}\\right)$,\n$\\M{A}\\M{B}=\\M{B}$, $\\M{C}\\M{A}=\\M{C}$, and $\\M{C}\\M{B}=\\M{0}$ --- the\ntransfer matrix of a chain of $N$ identical quadrupoles can be\ncalculated explicitly; as may be proved by induction, it reads\n\\begin{equation}\n\\label{matrixSN}\n\\M{S}^{N}=\n\\begin{pmatrix}\n\\M{A}^{N}+\\M{B}\\sum\\limits_{k=0}^{N-1}\\M{G}_{k}\\M{C} &\\;& \\M{B}\\M{G}_{N} \\\\\n\\M{G}_{N}\\M{C} && \\M{D}^{N}\n\\end{pmatrix}\n\\text{,}\n\\end{equation}\nwith the geometric series \n\\begin{equation}\n\\label{matrixG}\n\\M{G}_{k}\\,=\\,\\sum\\limits_{p=0}^{k-1} \\M{D}^{p}\n\\,=\\,\\left(\\M{1}-\\M{D}\\right)^{-1}\\left(\\M{1}-\\M{D}^{k}\\right)\n\\quad\\text{(using $\\M{G}_{0}=\\M{0}$).}\n\\end{equation}\n(Obviously, the second equality from the left holds only as long as\n$(\\M{1}-\\M{D})$ is invertible.)\n\n\n$\\M{S}^{N}$ has the same sparsity pattern, {\\sl i.e.} the same\ndistribution of zeros, as $\\M{S}$. In particular, its upper\nleft $2\\times 2$ submatrix reads\n$\\left(\\begin{smallmatrix} 1 & \\tilde{a} \\\\ 0 & 1\\end{smallmatrix}\\right)$,\nthus preserving the right eigenvector $\\trans{(1,0,0,0)}$, standing\nfor the invariance to a global voltage shift, and the left eigenvector\n$(0,1,0,0)$, responsible for current conservation, both with\neigenvalue $\\mu=1$.\n\nThe remaining two eigenvalues are $\\mu$ and $1\/\\mu$ (since\n$\\det\\M{S}^{N}=\\det\\M{S}=1$), and have their origin in the lower right\nsubmatrix, $\\M{D}^{N}$. Using\n$\\M{D}^{2}=\\M{D}\\,\\trace{\\M{D}} - \\M{1}\\det{\\M{D}}$\n(valid for any $2\\times 2$ matrix), together with $\\det{\\M{D}}=1$, we\nhave\n\\begin{subequations}\n\\begin{equation}\n\\label{matrixDNa}\n\\M{D}^{N}\\;=\\;\\M{D}^{N-2}\\M{D}^{2}\\;=\\;2\\xi\\,\\M{D}^{N-1}-\\M{D}^{N-2}\n\\;\\text{,}\n\\end{equation}\nwith $\\xi=\\frac{1}{2}\\trace{\\M{D}}$. The recurrence relation\n(\\ref{matrixDNa}) may be thought of as a matrix version of the one\ndefining the Chebyshev polynomials, eq.~(\\ref{chebyshev}). It may be\neasily verified that\n\\begin{equation}\n\\label{matrixDNb}\n\\M{D}^{N}\\;=\\;U_{N-1}(\\xi)\\M{D}\\,-\\,U_{N-2}(\\xi)\\M{1}\n\\;\\text{,}\n\\end{equation}\n\\end{subequations}\n(with $U_{n}(\\xi)$ the Chebyshev polynomials of the second kind,\neq.~(\\ref{chebyU})) fulfils eq.~(\\ref{matrixDNa}) and meets the initial\nconditions $\\M{D}^{0}=\\M{1}$ and $\\M{D}^{1}=\\M{D}$. Quite similarly,\nfor the trace of $\\M{D}^{N}$, (\\ref{matrixDNa}) gives\n\\begin{subequations}\n\\begin{equation}\n\\label{traceDNa}\n\\trace{\\M{D}^{N}}\\;=\\;2\\xi\\,\\trace{\\M{D}^{N-1}}-\\trace{\\M{D}^{N-2}}\n\\end{equation}\nwhich, along with the initial conditions\n$\\trace{\\M{D}^{0}}=\\trace{\\M{1}}=2$ and $\\trace{\\M{D}^{1}}=2\\xi$,\nyields\n\\begin{equation}\n\\label{traceDNb}\n\\trace{\\M{D}^{N}}\\;=\\;2 T_{N}(\\xi=\\cos\\theta)\\\n\\;\\equiv\\;2\\cos\\left[N\\theta\\right]\n\\;\\text{,}\n\\end{equation}\n\\end{subequations}\nwhere $T_{N}$ are the Chebyshev polynomials of the first kind.\n\n\nSince the trace of a matrix is invariant under basis transformation,\nthe resonance condition, $\\mu=1$, may be reformulated as\n$\\trace{\\M{D}^{N}}=\\mu+1\/\\mu=2$, insertion of which in\neq.~(\\ref{traceDNb}) requires $\\theta_{m}=\\frac{2\\pi m}{N}$ with\n$m=0\\ldots N-1$, and thus finally:\n\\begin{equation}\n\\label{traceresonance}\n\\frac{1}{2}\\trace{\\M{D}}=\\xi=\\cos\\theta_{m}=\\cos\\frac{2\\pi m}{N}\n\\end{equation}\nThe resonance condition (\\ref{traceresonance}) is the main result of\nthis section; it states that, in order to compute all resonances of a\nchain of $N$ quadrupoles under periodic boundary conditions, one\nsimply has to calculate the transfer matrix $\\M{S}$ of a single\nquadrupole --- which, of course, depends on the components constituting\nthe quadrupole and their setup --- and then resolve the algebraic\nequation (\\ref{traceresonance}) for the lower right $2\\times 2$\nsubmatrix $\\M{D}$.\n\n\\subsection{Applications}\n\nTo see this recipe at work, let us recalculate the resonances of an\narray obtained by replication of the unit cell of\nFig.~\\ref{fig:unitcells}(c). Using the capacitors' conductance as a\nreference, $\\sigma_{0}=\\sigma_{C}=\\ic\\w C$, the quadrupole's transfer\nmatrix $\\M{T}$ may be written (from right to left) as a product of the\ntransfer matrices of a vertical capacitor, $\\M{T}_{\\rm B}(2)$ (with\ntwice the capacity $C$, due to the periodic boundary conditions along\nthe electrodes), followed by an inductor and a capacitor in parallel,\n$\\M{T}_{\\rm A}(\\frac{\\sigma_{L}}{\\sigma_{C}},1)$, with\n$\\sigma_{L}\/\\sigma_{C}=-1\/(LC\\w^2)=-1\/\\wt^2$, another vertical\ncapacitor, and finally two capacitors in parallel, $\\M{T}_{\\rm\nA}(1,1)$:\n\\begin{eqnarray}\n\\label{matrixSunitcellc}\n\\M{S} &=&\n\\M{P}\\, \n\\M{T}_{\\rm A}(1,1)\\,\n\\M{T}_{\\rm B}(2)\\,\n\\M{T}_{\\rm A}(\\sfrac{\\sigma_{L}}{\\sigma_{C}},1)\\,\n\\M{T}_{\\rm B}(2)\\,\n\\M{P}\n\\nonumber\\\\\n&=&\n\\begin{pmatrix}\n1 \\;&-\\frac{3}{2}+\\frac{1}{2}\\wt^2 \\;& -2+2\\wt^2  \\;& \\frac{1}{2}+\\frac{1}{2}\\wt^2\\\\\n0 \\;& 1                            \\;& 0          \\;& 0                           \\\\\n0 \\;& \\frac{5}{2}+\\frac{5}{2}\\wt^2 \\;& 19-10\\wt^2 \\;&-\\frac{7}{2}+\\frac{5}{2}\\wt^2\\\\\n0 \\;& -2-2\\wt^2                    \\;& -16+8\\wt^2 \\;& 3-2\\wt^2\n\\end{pmatrix}\n\\end{eqnarray}\nA straightforward calculation shows that, in this case, the resonance\ncondition (\\ref{traceresonance}) reduces to\n\\begin{equation}\n\\label{dispersLC1period}\n\\wt_{m}=\\pm \\sqrt{\\frac{11}{6}-\\frac{1}{6}\\cos\\frac{2\\pi m}{N}}\n=\\pm \\sqrt{\\frac{5}{3}+\\frac{1}{3}\\sin^{2}\\frac{m\\pi}{N}}\n\\;\\text{,}\n\\end{equation}\nwith $m=0\\ldots N-1$. The only difference between the eigenfrequencies\n(\\ref{dispersLC1period}), calculated for periodic boundary conditions\nin both, $x$- and $y$-direction, and the former result,\neq.~(\\ref{dispersLC1}), obtained for a closed chain with periodic\nboundary conditions only in $y$-direction, resides in the argument of\nthe sine function whose symmetry about $\\frac{\\pi}{2}$ organises the\neigenfrequencies (\\ref{dispersLC1period}) in degenerate pairs (except\nfor $\\wt_{0}$ and, for even $N$, $\\wt_{N\/2}$).\n\nThe second example to be analysed with the present method is the array\nof staircases shown in Fig.~\\ref{fig:array}, and obtained from the\nunit cell in Fig.~\\ref{fig:unitcells}(d). For its transfer matrix,\n\\begin{equation}\n\\label{matrixSunitcelld}\n\\M{S} \\,=\\,\n\\M{P}\\, \n\\M{T}_{\\rm A}(\\sfrac{\\sigma_{L}}{\\sigma_{C}},1)\\,\n\\M{T}_{\\rm B}(1+\\sfrac{\\sigma_{L}}{\\sigma_{C}})\\,\n\\M{T}_{\\rm A}(1,\\sfrac{\\sigma_{L}}{\\sigma_{C}})\\,\n\\M{T}_{\\rm B}(1+\\sfrac{\\sigma_{L}}{\\sigma_{C}})\\,\n\\M{P}\n\\text{,}\n\\end{equation}\nthe resonance condition (\\ref{traceresonance}) is a fourth order\nequation in $\\wt^2$,\n\\begin{eqnarray*}\n\\wt^8-8\\wt^6+\\left(18-4\\beta_{m}^2\\right)\\wt^4-8\\wt^2+1&=& \n\\\\\n\\left[\\wt^4-\\left(4+2\\beta_{m}\\right)\\wt^2+1\\right]\n\\,\n\\left[\\wt^4-\\left(4-2\\beta_{m}\\right)\\wt^2+1\\right]&=&0\n\\,\\text{,}\n\\end{eqnarray*}\nwith $\\beta_{m}=\\pm\\cos\\frac{\\pi m}{N}$; its solutions are\n\\begin{equation}\n\\label{dispersstairs}\n\\wt_{m}^{2}\\,=\\,\\beta_{m}+2\\,\\pm\\,\\sqrt{\\left(\\beta_{m}+2\\right)^2-1}\n\\;\\text{,}\n\\end{equation}\n(where, again, $m\\in[0,N\/2]$ suffices, since the rest of the\nresonance frequencies is obtained by symmetry).\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.8\\cwidth]{renorm-3x2st.eps}\n\\end{center}\n\\caption{\\label{fig:renormstairs} \nImpedance between the electrodes for the setup shown in\nFig.~\\ref{fig:array}, with an array generated from the unit cell in\nFig.~\\ref{fig:unitcells}(d), replicated 3 times in $x$-direction and\ntwice in $y$-direction, embedded in a 9 by 4 capacitor lattice. The\nreal part of the ENR impedance, calculated for $L=C=1$ and $R_{\\rm\ncoil}=10^{-4}$, is plotted with continuous lines. The dotted lines\nindicate the resonance frequencies obtained analytically from\neq.~(\\ref{dispersstairs}) for $N=3$ and \n$m=0$ ($\\wt_{0}=1$ (doubly degenerate) \nand $\\wt_{1\\pm}=\\sqrt{3\\pm 2\\sqrt{2}}$),\nand $m=1$ ($\\wt_{2\\pm}=\\sqrt{{5}\/{2}\\pm{\\sqrt{21}}\/{2}}$ and\n$\\wt_{3\\pm}=\\sqrt{{3}\/{2}\\pm{\\sqrt{5}}\/{2}}$ }\n\\end{figure}\nFig.~\\ref{fig:renormstairs} displays the impedance of the staircase\narray of Fig.~\\ref{fig:array}, for which $N=3$. A comparison of the\nresonance frequencies (\\ref{dispersstairs}) (dotted lines) with an ENR\ncalculation for the same array (continuous lines) shows that\neq.~(\\ref{dispersstairs}) not only predicts the right number of\nresonances --- this result is non-trivial since, without degeneracy,\neq.~(\\ref{dispersstairs}) would produce 8 resonances ( 4 for $m=0$ and\nanother 4 for $m=1$) instead of 7 --- but also matches the position of\nmost of the resonances. The deviations, observed in particular for\n$\\wt_{2+}$ and $\\wt_{3+}$, are due to the assumption of periodic\nboundary conditions along the chain, an approximation used in the\nderivation of eq.~(\\ref{traceresonance}).\n\n\n\\subsection{Link to the usual resonance condition}\n\nUp to now, the resonance condition (\\ref{traceresonance}) had to be\nthought of as a necessary one, since $\\mu=1$ for all four eigenvalues\nof $\\M{S}^{N}$ was merely required for the norm of the eigenvectors\n$\\V{J}$ to remain finite after a great number of transfers through the\nsame chain of $N$ quadrupoles. The aim of this paragraph is to show\nthe equivalence of the new resonance condition (\\ref{traceresonance})\nwith the more familiar one, namely a vanishing total current through\nany cross-section of the chain,\n$I_{k}^{+}=\\frac{1}{\\sqrt{2}}(I_{k}+I_{k'})=0$, despite finite applied\nvoltage.\n\nAt resonance, $\\M{D}^{N}$ may be evaluated explicitly: from\neq.~(\\ref{traceresonance}), $\\theta_{m}=\\frac{2\\pi m}{N}$, the\nrecurrence relation (\\ref{matrixDNb}), and the Chebyshev polynomials\n(\\ref{chebyU}) evaluated at resonance,\n\\begin{align}\n\\label{chebyUres}\nU_{N-1}(\\cos\\theta_{m})&\n=\\frac{\\sin\\left[N\\theta_{m}\\right]}{\\sin\\theta_{m}}=\n\\begin{cases}\nN & \\text{if $m=0$,} \\\\\n0 & \\text{else}\n\\end{cases}\n\\nonumber\n\\\\\nU_{N-2}(\\cos\\theta_{m})&\n=\\frac{\\sin\\left[(N-1)\\theta_{m}\\right]}{\\sin\\theta_{m}}=\n\\begin{cases}\nN-1 & \\text{if $m=0$,} \\\\\n-1  & \\text{else}\n\\end{cases}\n\\nonumber\n\\end{align}\none obtains\n\\begin{equation}\n\\label{matrixDNres}\n\\M{D}^{N}=\n\\begin{cases} \nN\\M{D}\\,-\\,\\left(N-1\\right)\\M{1} & \\text{if $m=0$,}\\\\\n\\M{1} & \\text{else ({\\sl i.e.} $m=1\\ldots N-1$).}\n\\end{cases}\n\\end{equation}\n\nFor the excited eigenmodes, $m \\mod N\\neq 0$, $\\M{D}^{N}=\\M{1}$\nimplies $\\M{G}_{m}=\\M{0}$ (eq.~(\\ref{matrixG})), and hence, from\neq.~(\\ref{matrixSN}),\n\\begin{equation}\n\\label{matrixSNexcres}\n\\M{S}^{N}=\n\\begin{pmatrix}\n\\left(\\begin{smallmatrix} 1 & \\tilde{a} \\\\ 0 & 1 \\end{smallmatrix}\\right)\n &\\;& \\M{0} \\\\\n\\M{0} &\\;& \\M{1}\n\\end{pmatrix}\n\\text{.}\n\\end{equation}\nGenerally $\\tilde{a}$ is finite; it is thus obvious that {\\em any}\nvector $\\V{J}$ is a right eigenvector of $\\M{S}^{N}$ with eigenvalue\n$\\mu=1$, if (and only if) its second component, $I^{+}$, describing\nthe net current through the sample, vanishes.\n\nFor the groundstate, $m=0$, the line of reasoning is slightly more\nsubtle: the naive procedure of solving the matrix equation\n(\\ref{naivetransfer}) at resonance, $I_{1}+I_{1'}=0$, amounts in the\npresent language to solving\n\\begin{equation}\n\\M{S}^{N}\n\\left(\\begin{smallmatrix} U \\\\0 \\\\ x \\\\y \\end{smallmatrix}\\right)\n\\,=\\,\n\\left(\\begin{smallmatrix} U \\\\0 \\\\ x \\\\y \\end{smallmatrix}\\right)\n\\;\\text{.}\n\\end{equation}\nThe right eigenvector $\\trans{(U,0,0,0)}$ may be subtracted from the\nsystem of equations, and one is left with an eigenvalue problem for\nthe last two components of $\\V{J}$,\n$\\M{D}^{N}\\left(\\begin{smallmatrix} x \\\\y \\end{smallmatrix}\\right)\n=\\left(\\begin{smallmatrix} x \\\\y \\end{smallmatrix}\\right)$, which ---\nusing eq.~(\\ref{matrixDNres}) --- reduces to finding the right\neigenvector of the single quadrupole's $\\M{D}$, associated with\neigenvalue $\\mu=1$:\n\\begin{equation}\n\\M{D}\n\\left(\\begin{smallmatrix} x \\\\y \\end{smallmatrix}\\right)\n\\,=\\,\n\\left(\\begin{smallmatrix} x \\\\y \\end{smallmatrix}\\right)\n\\end{equation}\n(Note that, for the generally asymmetric matrix $\\M{D}$, the right\neigenspace contains only one eigenvector if --- as in the present case\nof a resonance --- the eigenvalues are degenerate.)\n\nIn all cases --- for the groundstate and for the excited modes --- the\ncondition that $\\M{S}^{N}$ shall only have eigenvalues $\\mu=1$ is\nequivalent to the usual definition of a divergent impedance between\nthe electrodes, implying $I_{k}+I_{k'}=0$ at finite applied voltage.\n\n\\section{\\label{sec:inflattice}Clusters in an infinite lattice}\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[height=0.45\\cwidth]{LClineLF.eps}\n\\end{center}\n\\caption{\\label{fig:LClineLF}Low-frequency renormalisation scheme for\nthe transformation of an infinite line of alternating coils and\ncapacitors in a 2D capacitor lattice with bonds of capacity $C$ (left\ngraph) to an infinite ladder of capacitors with only one horizontal\ncapacitor replaced by a coil (right graph). (In the second and third\ngraph, a bond with two capacitors in series {\\em always} represents\ntwo times $2C$ in series.)}\n\\end{figure*}\n\nThe systems studied in the preceding sections could be reduced to\neffective 1D problems, because (i) the arrays covered the whole width\nof the electrodes and (ii) periodic boundary conditions were assumed\nin $y$-direction. In this section, by contrast, we will examine simple\nregular 1D arrays in an infinite square lattice. The observed changes\nturn out to be considerable, and sometimes not only shift the\nfrequencies, but qualitatively change the spectrum.\n\nThe first example we want to inspect consists in a single line of\nalternating inductors and capacitors embedded in an otherwise pure\ncapacitor lattice, as shown in the leftmost graph of\nFig.~\\ref{fig:LClineLF}. Alternatively, this line is generated by\nreplicating the unit cell of Fig.~\\ref{fig:unitcells}(c) only in\n$x$-direction.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.95\\cwidth]{LCworm.eps}\n\\end{center}\n\\caption{\\label{fig:LCworm}Eigenfrequencies $\\w$ in units of $\\wo$ for\na line of $N$ alternating coils and capacitors embedded in an infinite\ncapacitor lattice. The circles represent numerically calculated\neigenfrequencies, obtained with the spectral method. The dashed lines\nare the analytical large $N$ asymptotes, $\\wt_{\\rm LF}\\simeq\n1\/\\sqrt{2\\sqrt{2}-1}\\simeq 0.7395$ and $\\wt_{\\rm HF}\\simeq\n1\/\\sqrt{\\sqrt{3}-1}\\simeq 1.1688$ discussed in the text.}\n\\end{figure}\nIts eigenfrequencies are obtained very accurately with the so-called\nspectral method, initially proposed by Straley~\\cite{Straley},\nBergman~\\cite{Bergman} and Milton~\\cite{Milton}, and later adapted to\n2D lattices by Clerc, Giraud, Luck and\nco-workers~\\cite{Clerc1996,Luck1998}, employing a Green's function\nformalism developped by McCrea and Whipple~\\cite{McCrea} in the\ncontext of random walks; (see also Spitzer~\\cite{Spitzer}).\nThis approach connects the impedance resonances of a (not necessarily\nordered) cluster to the eigenvalues of a non-symmetric square matrix,\nwhich --- in general --- have to be evaluated numerically. The latter\nversion of the spectral method assumes the cluster to be located in\nthe middle of an infinite lattice --- just as in the examples we wish\nto study. To obtain similar results from an ENR calculation one would\nhave to incorporate the cluster in a much larger capacitor network\n(compared to the cluster's size) --- a situation which is\ncomputationally very demanding.\n\nThe eigenfrequencies of the line of alternating coils and capacitors,\nobtained via the spectral method, are plotted with dots in\nFig.~\\ref{fig:LCworm} as a function of the number of coils, $N$. In\nthe limit of large $N$, the lowest eigenmode shows the full\ntranslational invariance of the system, and thus has the same current\ndistribution in every vertical stripe. This implies that the left and\nright boundary of the stripe can be thought of as attached\ntogether. In order to perform this operation, illustrated in\nFig.~\\ref{fig:LClineLF}, one has to (i) chose a stripe which is\nsymmetric about one of the coils, (ii) replace all horizontal\ncapacitors of capacity $C$ in all pure capacitor columns by two\ncapacitors in series, with capacity $2C$ each, (iii) cut between the\ndoubled capacitors, and (iv) tie the corresponding vacant ends of the\nstripe together. One ends up with an infinite ladder with horizontal\nsteps of $2C$ (two times $2C$ in series, the whole parallel to a\ncapacitor $C$), except for one step which contains a coil instead of\nthe single capacitor; the vertical legs of the ladder consist of\ncapacitors $C$ ({\\sl cf.} rightmost graph in Fig.~\\ref{fig:LClineLF}).\nThe capacities may then be summed up in a procedure analogous to\neqs.~(\\ref{LCinf}) and (\\ref{Cinf}), yielding a total capacity of\n$C(2\\sqrt{2}-1)$, and thus the low-frequency asymptote $\\wt_{\\rm\nLF}\\simeq 1\/\\sqrt{2\\sqrt{2}-1}\\simeq 0.7395$ (lower dashed line in\nFig.~\\ref{fig:LCworm}).\n\nThe position of the high-frequency asymptote $\\wt_{\\rm HF}\\simeq\n1\/\\sqrt{\\sqrt{3}-1}\\simeq 1.1688$ (upper dashed line in\nFig.~\\ref{fig:LCworm}), on the opposite side of the spectrum, can be\ncalculated almost analogously: in the limit of an infinite $LC$ chain\nin its highest eigenmode, the currents through two neighbouring coils\nare at any moment antiparallel, but of the same magnitude. The\ncapacitors separating the coils horizontally are thus located at\ncurrent nodes and can be omitted. The corresponding ladder looks the\nsame as the one shown in the right of Fig.~\\ref{fig:LClineLF}, except\nfor the fact that all capacitors of $2C$ (on the bent lines) have to\nbe removed.\n\nComparison to the case with periodic boundaries in $y$ direction,\nstudied in Sec.~\\ref{sec:kirchhoff} (eq.~(\\ref{dispersLC1}) in\nparticular), shows that the shift of a resonance depends on its\nlocation in the spectrum: at the low-frequency edge, formerly located\nat $\\wt\\simeq\\sqrt{5\/3}\\simeq 1.291$, now at $\\wt\\simeq 0.7395$, the\nrenormalisation is stronger than at the high-frequency boundary\n(formerly $\\wt\\simeq\\sqrt{2}\\simeq 1.414$, now $\\wt\\simeq\n1.1688$). The reason for this behaviour lies in the different\nrenormalisation schemes which --- as pointed out above --- couple the\ninductors to an equivalent capacity which is less enhanced for high\nfrequencies than for low ones.\n\n\nThe second cluster we are going to study in this section is a straight\nline of $N$ coils in series embedded in a pure capacitor environment.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.95\\cwidth]{Lworm.eps}\n\\end{center}\n\\caption{\\label{fig:Lworm}Eigenfrequencies $\\w$ in units of $\\wo$ for\na continuous line of $N$ coils embedded in an infinite capacitor\nlattice. The circles represent numerically calculated\neigenfrequencies, obtained with the spectral method. The dashed lines\nare the large $N$ asymptotes, $\\wt_{\\rm LF}\\simeq\\sqrt{1.16\/N}$ and\n$\\wt_{\\rm HF}\\simeq\\sqrt{1+\\sqrt{2}}\\simeq 1.5538$ (see text).}\n\\end{figure}\nThe resonance frequencies of this ``lattice worm'', obtained via the\nspectral method, are plotted with dots in Fig.~\\ref{fig:Lworm}. For\nlarge $N$ and at high frequencies, the cluster can again be reduced to\nalmost independent stripes.\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[height=0.45\\cwidth]{LlineHF.eps}\n\\end{center}\n\\caption{\\label{fig:LlineHF}High-frequency renormalisation scheme for\nan infinite line of coils in a 2D capacitor lattice (left graph),\nending up in an infinite capacitor ladder with only one coil (right\ngraph).}\n\\end{figure*}\nThe corresponding renormalisation scheme is illustrated in\nFig.~\\ref{fig:LlineHF}, and ends up in an infinite ladder with legs of\ncapacity $C\/2$ and steps of capacity $C$ (except for the central step\nwhich is a coil of inductance $L$). Summing up each semi-infinite\ncapacitor ladder yields\n$C_{\\rm inf}=\\frac{1}{2}C(1+\\sqrt{2})$\n({\\sl cf.} eqs~(\\ref{LCinf}) and (\\ref{Cinf})), and thus the\nhigh-frequency asymptote \n$\\wt_{\\rm HF}\\simeq \\sqrt{1+\\sqrt{2}}\\simeq 1.5538$\n(upper dashed line in Fig.~\\ref{fig:Lworm}). Comparison to the\nperiodic case, eq.~(\\ref{dispersL2}), where $\\wt\\to\\sqrt{3}$ for $m=N$\nand $N\\to\\infty$, shows that the renormalisation shifts are rather\nmodest at high frequencies.\n\nAt low frequencies, by contrast, the infinite environment has drastic\neffects: as can be seen from Fig.~\\ref{fig:Lworm}, the lowest\nresonance tends to zero frequency as the length of the worm\nincreases. The corresponding density of states therefore shows no gap\nat low frequencies --- as opposed to the density of states\n(\\ref{dosL2}) of the periodic system, illustrated in the last graph of\nFig.~\\ref{fig:dos}.\n\nThe reason for this behaviour can be qualitatively understood in a\nsimple picture representing the whole worm of $N$ coils as a dipole of\ntotal inductance $N L$. This dipole is thought to be coupled to a\nsingle capacitor bearing the entire capacity of the lattice. In this\nmodel, the lowest resonance would thus be found at\n$\\w\\simeq\\sqrt{1\/(N L C_{\\rm lat})}$.\nAs pointed out in Appendix~\\ref{app:dipole}, the dipole model\nevaluates the entire lattice capacity to roughly $C_{\\rm lat}\\simeq\nC$, a value which allows to reproduce the exact resonance frequency\n$\\wt=1$ for a single coil ($N=1$) in a 2D capacitor lattice. For\nincreasing $N$, $C_{\\rm lat}$ is found slightly reduced since more and\nmore capacitors are replaced by coils. In the limit of large $N$, the\nnumerically calculated groundstate fits very accurately to\n$\\wt\\simeq\\sqrt{1.16\/N}$ (lower dashed line in Fig.~\\ref{fig:Lworm}).\n\nIn the same manner, the dipole model predicts a resonance frequency of\n$\\wt_{m}\\propto\\sqrt{m\/N}$ for the $m$.th mode, with a proportionality\nconstant of the order of unity, since the current nodes split the\n``worm'' into $m$ almost independent segments of $N\/m$ coils\neach. Comparison to the numerically evaluated eigenfrequencies shows\nthat this scenario works well for $m\\lesssim 5$, but turns out to be\ntoo simplistic for the higher excited modes.\n\nOn more general grounds, and observing that second-order terms do not\nimprove the reproduction of the data, one may try an ansatz of the\nform\n\\begin{equation}\n\\label{dispersLfit}\n\\wt_{m}\\;\\simeq\\;\\sqrt{\\frac{\\alpha_{m}}{N}}\\;+\\;\\sqrt{\\frac{\\beta_{m}}{N}}^{3}\n\\;\\text{.}\n\\end{equation}\nA linear dependence of the parameters, $\\alpha_{m}=-0.40+1.57 m$ and\n$\\beta_{m}=-0.56+0.98 m$, deduced by a fit for $N=200\\ldots 1000$ and\n$m=1\\ldots 30$, reproduces the numerically calculated resonance\nfrequencies very accurately for a wide range of $N$ and $m$, as long\nas $N\/m$ is large ($\\gtrsim 5$).\n\nWe also note that the fitted $\\alpha$, with a slope of $1.57$,\nconcords with an analytical result obtained by Clerc {\\sl et al.}\n\\cite{Clerc1996}, stating that $\\wt_{m}\\sim\\sqrt{\\frac{\\pi}{2} m\/N}$ for\nthe bulk states of a long linear cluster, {\\sl i.e.} if both $m$ and\n$N$ are large.\n\n\\section{\\label{sec:conclusions}Conclusions and outlook}\n\nIn this article, we have studied the dielectric resonance spectra of\nordered passive arrays, typically --- although not necessarily ---\nconstituted of inductive and capacitive elements. Similar arrays,\nbased on split-ring resonators, have recently been used to assemble\nmetamaterials for the microwave regime, exhibiting negative refraction\nand other exotic properties~\\cite{Pendry2004,Seddon2003,Grbic2004}.\n\nIn the first part, we have calculated the resonance frequencies of\nseveral arrays by solving a system of differential equations deduced\ndirectly from Kirchhoff's rules --- a technique which allows for a\nstraightforward interpretation of the spectra and thus provides a good\nhandle for the influence of each of the array's parameters. On the\nother hand, each new circuit requires a separate individual analysis,\nrelying on the solution of increasingly complicated recurrence\nrelations as the unit cell of the array gets richer in structure.\n\nAn alternative approach, presented in Sec.~\\ref{sec:transfer}, deduces\nthe resonance frequencies from the array's transfer matrix, {\\sl i.e.}\na matrix connecting the state vectors (a combination of currents and\nvoltages at each vertex) at both ends of the cluster to each\nother. Within this formalism, the array is shown to be in resonance if\nall eigenvalues of its transfer matrix are unity for a given frequency\n--- a condition which is demonstrated to be equivalent to the more\nfamiliar definition of a divergent impedance for a cluster between two\nelectrodes, namely vanishing net current through the sample at finite\napplied voltage. Even large arrays, with complex unit cells, can be\neasily analysed with this algorithm since, in a handier reformulation,\nthe new resonance condition, eq.~(\\ref{traceresonance}), does not\nrequire the computation of the transfer matrix of the whole array, but\nonly of a single unit cell. The latter is most conveniently evaluated\nby multiplication of the transfer matrices of a few standard\nsituations --- two in our case, {\\sl cf.}  eq.~(\\ref{transferABgraph}),\nwhere the 2D arrays reduce to two-legged ladders due to the assumption\nof periodic boundary conditions along the electrodes --- a task which\ncan be performed using symbolic computation software. In its final\nform, the resonance frequencies are typically given as the roots of a\npolynomial whose degree depends on the complexity of the unit cell.\n\nIf, however, the cluster happens to be embedded in a much larger\nlattice, periodic boundary conditions are not a pertinent\napproximation, and the resonance frequencies are generically not\navailable in closed form anymore. In this case, major changes in the\nspectrum have to be expected, of which some can be understood in terms\nof renormalisation, while others cannot. Among the latter, one may\nrecall the example of a longer growing linear array: embedded in a\nperiodic medium, the spectrum remains gapped for any length of the\ncluster; if, by contrast, the same cluster is isolated in a\nhomogeneous network, the gap closes with increasing length and the\nresonance spectrum carries the signature of one or more rather\nindependent two-dimensional dipoles.\n\nThe tools discussed in this article may be useful for the design of\ncircuits with a custom electric response. Such devices could be used\nfor labelling --- just as bar codes --- which could be read by an\nautomated system.\n\nFor the future it would be undoubtedly desirable, especially in the\nperspective of the already mentioned development of metamaterials with\nnegative refraction, to take into account the magnetic part of the\noptical response, with the aim to create devices having both, tunable\ndielectric {\\em and} magnetic resonances.\n\n\n\n\n\\begin{acknowledgement}\nWe thank J.-M. Luck for very interesting discussions and remarks.\n\\end{acknowledgement}\n\n\n\\begin{appendix}\n\\section{\\label{app:dipole} Dipole approximation for linear clusters}\n\nThe scenario presented in the following is based on the idea that the\ncurrent flowing through a linear cluster isolated in a capacitor\nlattice flows back through the lattice with a current distribution\nwhich, in the lowest eigenmode, resembles the field lines of a dipole\nin 2D electrostatics (see Fig.~\\ref{fig:dipolefield}).\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.95\\cwidth]{dipolefield.eps}\n\\end{center}\n\\caption{\\label{fig:dipolefield} Current distribution (dashed lines)\nof a linear cluster of length $N=1$ in the dipole approximation. (The\nlower half-plane has been omitted for clarity.)}\n\\end{figure}\n\nThis idea is supported by the spectral method, in the framework of\nwhich the electric potential $V_{\\V{r}}$ induced by a single\nhorizontal coil, located at the origin, is known to be given by\n\\cite{Clerc1996}\n\\begin{equation}\n\\label{Clerc2.12}\n\\lambda V_{\\V{r}}\\;=\\; \n-W\\,\n\\left[ \nG_{\\V{r}+\\frac{1}{2}\\V{\\hat{x}}}- G_{\\V{r}-\\frac{1}{2}\\V{\\hat{x}}} \n\\right]\n\\;\\text{,}\n\\end{equation}\nwhere $\\lambda$ is a frequency-dependent proportionality constant, $W$\nthe voltage measured between the ends of the coil, and $G_{\\V{r}}$ the\n2D lattice Green's function. Far from the coil, for $\\abs{\\V{r}}\\gg\n1$, the lattice Green's function may be replaced by its continuous\ncounterpart, $G_{\\V{r}}\\simeq -\\frac{1}{2\\pi}\\ln\\abs{\\V{r}}$, and\neq.~(\\ref{Clerc2.12}) reduces to a 2D dipole potential\n\\begin{equation}\n\\label{dipolepotential}\n\\lambda V_{\\V{r}}\n\\;\\simeq\\;-W \\frac{\\partial}{\\partial x} G_{\\V{r}}\n\\;\\simeq\\;\\frac{W}{2\\pi}\\;\\frac{x}{x^{2}+y^{2}}\n\\;\\equiv\\;\\frac{W}{2\\pi}\\;\\frac{\\cos\\varphi}{\\abs{\\V{r}}}\n\\;\\text{.}\n\\end{equation}\n\nIn this approximation, the current distribution follows the field\nlines\n\\begin{equation}\n\\label{dipolefield}\nr(\\varphi)\\;=\\;\\frac{l}{\\pi}{\\sin{\\varphi}}\n\\end{equation}\nwhere $l$ is the length of the field line (in units of the lattice\nspacing). In order to calculate the resonance frequency of a cluster\nembedded in an infinite capacitor lattice, one has to assign an\nequivalent capacity, $C_{\\rm lat}$, to the entire lattice. Of course,\n$C_{\\rm lat}$ depends on the current distribution: in general,\ncapacitors almost perpendicular to the current lines contribute only\nlittle to $C_{\\rm lat}$, while capacitors following the current lines\ncontribute as if they were in series. At great distance from the\ncluster, the lattice behaves as if it were assembled by independent\nlines of capacitors, hence all in parallel, contributing each as the\ninverse of its length:\n\\begin{equation}\n\\label{Clat}\nC_{\\rm lat}\\;\\simeq\\;\\sum\\limits_{k}\\frac{C}{l_{k}}\n\\end{equation}\n\nThe sum in eq.~(\\ref{Clat}) runs over all possible current paths, and\n--- in order to avoid divergences --- the counting has to be done very\ncarefully. In our example of a single coil in an infinite lattice,\nthis may be achieved by supposing that the current can only leave the\n$x$-axis at spots where a vertical capacitor is present. Following the\nillustration of Fig.~\\ref{fig:dipolefield}, this amounts to retaining\nonly current lines passing through $(x=k+\\frac{1}{2},y=1)$ with\n$k=0,1,2,\\ldots$. In coordinates of the field lines\n(\\ref{dipolefield}), these points are described by\n\\begin{subequations}\n\\label{pinning}\n\\begin{eqnarray}\n\\label{pinningl}\nl_{k}&=&\\pi\\left[1+\\left(k+\\sfrac{1}{2}\\right)^2\\right] \\\\\n\\label{pinningphi}\n\\sin\\varphi_{k}&=&\\frac{1}{\\sqrt{1+\\left(k+\\frac{1}{2}\\right)^2}}\n\\;\\text{.}\n\\end{eqnarray}\n\\end{subequations}\nThe first equation can be used to select only the desired current\nlines, as shown in Fig.~\\ref{fig:dipolefield}. Pinning the field lines\nin this way has a twofold virtue, namely on the short side to set a\nlower boundary for the path length $l_{k}$ in (\\ref{Clat}), {\\em and}\nto avoid the logarithmic divergence on the long side,\n$l_{k}\\to\\infty$. After substitution in (\\ref{Clat}), and taking into\naccount contributions from the lower half-plane, we obtain\n\\begin{eqnarray}\n\\label{Clatdip}\nC_{\\rm lat}\n&\\simeq& \n\\frac{2 C}{\\pi} \\sum\\limits_{k=0}^{\\infty} \\frac{1}{1+(k+\\frac{1}{2})^{2}}\n\\nonumber \\\\\n&\\simeq& \n\\frac{2 C}{\\pi} \\int\\limits_{0}^{\\infty} \\frac{{\\rm d}k}{1+k^{2}}\n\\;=\\; C\n\\;\\text{.}\n\\end{eqnarray}\n(The sum in the first line may be thought of as a midpoint trapeze\napproximation for the integral in the second line).\n\nAccording to the dipole approximation, an isolated coil in a 2D\ncapacitor network thus resonates at\n$\\w=1\/\\sqrt{L C_{\\rm lat}}\\simeq 1\/\\sqrt{L C}\\equiv\\wo$ ---\nwhich is the exact result known from the spectral method.\n\n\\end{appendix}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{}\n\n\\bibliographystyle{elsarticle-num} \n\n\n\\section*{Appendix A.}\n\\label{app:theorem}\n\n\n\nIn this appendix we prove the following theorem from\nSection~6.2:\n\n\\noindent\n{\\bf Theorem} {\\it Let $u,v,w$ be discrete variables such that $v, w$ do\nnot co-occur with $u$ (i.e., $u\\neq0\\;\\Rightarrow \\;v=w=0$ in a given\ndataset $\\dataset$). Let $N_{v0},N_{w0}$ be the number of data points for\nwhich $v=0, w=0$ respectively, and let $I_{uv},I_{uw}$ be the\nrespective empirical mutual information values based on the sample\n$\\dataset$. Then\n\\[\n\tN_{v0} \\;>\\; N_{w0}\\;\\;\\Rightarrow\\;\\;I_{uv} \\;\\leq\\;I_{uw}\n\\]\nwith equality only if $u$ is identically 0.} \\hfill\\BlackBox\n\n\\noindent\n{\\bf Proof}. We use the notation:\n\\[\nP_v(i) \\;=\\;\\frac{N_v^i}{N},\\;\\;\\;i \\neq 0;\\;\\;\\;\nP_{v0}\\;\\equiv\\;P_v(0)\\; = \\;1 - \\sum_{i\\neq 0}P_v(i).\n\\]\nThese values represent the (empirical) probabilities of $v$\ntaking value $i\\neq 0$ and 0 respectively.  Entropies will be denoted\nby $H$. We aim to show that $\\fracpartial{I_{uv}}{P_{v0}} < 0$....\\\\\n\n\\section{Benchmark}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[Training Process in 150 epochs.]{\n\\includegraphics[width=0.45\\columnwidth]{figures\/mnist_top1_acc.pdf}\n}~~~~\n\\subfloat[Compression Ratio.]{\n\\includegraphics[width=0.45\\columnwidth]{figures\/mnist_cr.pdf}\n}\n\\caption{Experiments on UCF11. CR is short for Compression Ratio.}\n\\label{fig:ucf11}\n\\end{figure}\n\nUntil now, TDNs are mostly applied in computer vision field. Thus, aiming to validate performance of TedNet, we consider to conduct experiments on two datasets:\n\\begin{itemize}\n    \\item The UCF11 Dataset contains  1,600 video clips of a resolution $320 \\times 240$ and is divided into 11 action categories. Each category consists of 25 groups of videos, within more than 4 clips in one group. \n    \\item The Cifar10\/100 consists of 50,000 train images and 10,000 test images with size as $32 \\times 32 \\times 3$. CIFAR10 has 10 object classes and CIFAR100 has 100 categories.\n\\end{itemize}\n\n\n\n For the video classification task on UCF11, we adopt the same setting as described in literature \\cite{DBLP:conf\/aaai\/PanXWYWBX19}, where  we extract feature of dimension 2048 from each frame of a video by Inception-V3~\\cite{DBLP:conf\/cvpr\/SzegedyVISW16}. Then throw these features as step inputs into TD-LSTMs. Results are shown in Figure~\\ref{fig:ucf11}. Almost every tensor decomposition model can achieve better accuracy except Tucker-2.\n\n\n\n\n For the image classification task on Cifar10\/100, we employ ResNet-32 as the backbone network.  We show the results of  corresponding TD-ResNet-32 implementations with various tensor decomposition in Table~\\ref{tbl:cifar}.\n\n Note that the results shown in Table~\\ref{tbl:cifar} and Figure~\\ref{fig:ucf11} are obtained without fine tuning parameters, and are just used for verifying the correctness of these algorithms.  Thus the classification results does not mean the performance of these algorithms with the best parameter settings.\n\n\n\n\n\n\\begin{table}[t]\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{l|c|c|c|c|c|c|c} \n\\hline\n\\multicolumn{2}{l|}{} & \\multicolumn{3}{c|}{Cifar10} & \\multicolumn{3}{c}{Cifar100}  \\\\ \n\\hline\n\\hline\nModel         & Rank   & Params & CR & Accuracy          & Params & CR & Accuracy            \\\\ \n\\hline\nResNet-32     & -      & 0.46M &  1$\\times$  & 0.9228         & 0.47M &  1$\\times$  & 0.6804           \\\\ \n\\hline\nBTT-ResNet-32 & 4      & 0.08M  & 6$\\times$   & 0.8955         & 0.08M  &  6$\\times$  & 0.5661           \\\\ \n\\hline\nCP-ResNet-32  & 10     & 0.03M  &  18$\\times$  & 0.8802         & 0.03M  &  18$\\times$  & 0.4445           \\\\ \n\\hline\nTK2-ResNet-32 & 10     & 0.05M  &  9$\\times$  & 0.8915         & 0.06M  &  9$\\times$  & 0.5398           \\\\ \n\\hline\nTR-ResNet-32  & 10     & 0.09M  &  5$\\times$  & 0.9076         & 0.09M  & 5$\\times$   & 0.653            \\\\ \n\\hline\nTT-ResNet-32  & 10     & 0.09M  &  5$\\times$  & 0.9020         & 0.10M  &  5$\\times$  & 0.6386           \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Experiments on Cifar10\/100. Params denotes the number of parameters.  CR means compression rate.  The number of block-terms is set to 5 in BTT-ResNet-32.}\n\\label{tbl:cifar}\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we present a package named TedNet that is specially designed for TDNs.  TedNet is completely open-source and distributed under the MIT license. Compared with other related python packages, TedNet contains the most kinds of tensor decomposition.\n\n\\section*{Acknowledgements}\n\nThis paper was partially supported by the National Key Research and Development Program of China (No. 2018AAA0100204), and a key  program of fundamental research from Shenzhen Science and Technology Innovation Commission (No. JCYJ20200109113403826).\n\n\n\\section{TedNet Details}\n\nTedNet is designed with the goal of building TDNs by calling corresponding APIs, which can extremely simplify the process of constructing TDNs. As shown in Figure~\\ref{fig:frame}, TedNet adopts Pytorch as the training framework because of its auto differential function and convenience to build DNN models.  In addition, TedNet also uses NumPy~\\cite{DBLP:journals\/cse\/WaltCV11} to assist in tensor operations. The fundamental module of TedNet is \\textbf{\\_TNBase}, which is an abstracted class and inherits from \\textbf{torch.nn.Module}. Thus, TedNet models can be amicably combined with other Pytorch models. As an abstracted class,  \\textbf{\\_TNBase} requires sub-classes to implement 4 functions~\\footnote{https:\/\/github.com\/tnbar\/tednet\/blob\/main\/tednet\/tnn\/tn\\_module.py}.  On the right side of Figure~\\ref{fig:frame}, we show two main deep architectures of TedNet, namely TD ResNet and TD LSTM, which are  probably the most frequently used backbone in convolutional neural networks and recurrent neural networks, respectively.\n\n\n\n\n\nUsually, DNNs are constructed with CNNs and Linears. The weight of a CNN is a 4-mode tensor $\\ca{C} \\in \\mathbb{R}^{K\\times K \\times C_{in} \\times C_{out}}$, where $K$ means the convolutional window, $C_{in}$ denotes the input channel and $C_{out}$ represents the counterpart output channel. And a Linear is a matrix $\\mathbf{W} \\in \\mathbb{R}^{I \\times O}$, where $I$ and $O$ are length of input and output feature respectively. Similar to DNNs, TDNs consist of TD-CNNs and TD-Linears(For simplification, TD- denotes the corresponding tensor decomposition model), whose weights $\\ca{C}$ and $\\textbf{W}$ are factorized with tensor decomposition.  Following this pattern, \nthere are 5 frequently-used tensor decomposition (i.e. CP, Tucker-2, Block-Term Tucker, Tensor Train and Tensor Ring) in TedNet, which satisfies most of common situations. Notably,  TedNet is an open-source package which supports Tensor Ring Decomposition. Besides, based on TD-CNNs and TD-Linears, TedNet has built some tensor decomposition based Deep Neural Networks, e.g. TD-ResNets, TD-RNNs.\n\n\n\\section{Installation and Illustrative Examples}\n\nThere are two ways to install TedNet. For the sake that the source code of TedNet is submitted to GitHub, it is feasible to install from the downloaded code by command \\textbf{python setup.py install}. Compared with aforementioned fussy way, another one, the recommended way is to install TedNet trough PyPI~\\footnote{https:\/\/pypi.org\/project\/tednet} by command \\textbf{pip install tednet}. After installation, all tensor decomposition models of TedNet can be used.\n\n A simple MNIST~\\cite{lecunGradientbasedLearningApplied1998a} classifier based on tensor ring is shown in Listing~\\ref{ll:tr}. The tensor ring module can be used by importing \\textbf{tednet.tnn.tensor\\_ring}. We utilize two fundamental tensor ring layers (i.e., TRConv2D, TRLinear) to build the sample classifier. In addition, it is very convenient to build a whole tensor ring network with only one line of code, e.g., TR-LeNet5~\\footnote{https:\/\/tednet.readthedocs.io\/en\/latest\/quick\\_start.html}. The usage of other decomposition is the same and more details can be found in the Document~\\footnote{https:\/\/tednet.readthedocs.io}.\n\n\n\n\n\n\n\\section{Introduction}\n\nTensor Decomposition Networks (TDNs) are constructed by decomposing deep neural layers with tensor formats. For the reason that the original tensor of a layer can be recovered from tensor decomposition cores, TDNs are often regarded as a compression method for the corresponding networks. Compared with traditional networks like Convolution Neural Networks (CNNs) and Recurrent Neural Networks (RNNs), TDNs can be much smaller and occupy a little memory. For example, TT-LSTM~\\cite{DBLP:conf\/icml\/YangKT17}, BTT-LSTM~\\cite{DBLP:conf\/cvpr\/YeWLCZCX18,DBLP:journals\/nn\/YeLCYZX20}, TR-LSTM~\\cite{DBLP:conf\/aaai\/PanXWYWBX19,li2021heuristic} are able to reduce  17,554, 17,414 and 34,192 times parameters with a higher accuracy than the original models. With light-weight architectures and good performance,  TDNs are promising to be used in kinds of source-restricted scenes including mobile equipment and microcomputers.  Due to these advantages, TDNs can often achieve comparably high accuracy with huge parameter reduction in a number of tasks, such as action recognition~\\cite{DBLP:journals\/pieee\/PanagakisKCONAZ21,sun2020human}. TDNs have also been implemented in FPGA for fast inference with ultra memory reduction~\\cite{zhang2021fpga} and multi-task learning to improve the representing ability~\\cite{wang2020concatenated}. Under this background, we design TedNet package for providing convenience for researchers to explore on TDNs.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figures\/frame.pdf}\n\\caption{The framework of TedNet.  TedNet is based on Pytorch and adopts NumPy to process numerical calculations. Tensor decomposition (TD) can be applied to convolutional layers or linear layers.  We implemented 5 variants of tensor decomposition methods, namely CP, Tucker, Tensor Ring, Tensor Train, and Block-term Tucker. Tensor decomposition can be fulfilled in convolution neural networks. An illustration of two tensorial classical neural blocks(i.e., ResNet and  LSTM) that are built on the Tensor Decomposition Layer is shown in the right of the figure.}\n\\label{fig:frame}\n\\end{figure}\n\nThere are several related packages, such as T3F~\\cite{DBLP:journals\/jmlr\/NovikovIKFO20}, Tensorly~\\cite{DBLP:journals\/jmlr\/KossaifiPAP19}, TensorD~\\cite{DBLP:journals\/ijon\/HaoLYX18}, TensorNetwork~\\cite{roberts2019tensornetwork}, tntorch~\\cite{tntorch} , OSTD~\\cite{DBLP:conf\/iccvw\/SobralJJBZ15} and TensorTools~\\cite{williams2018unsupervised}.  OSTD is constructed for low-rank decomposition and implemented with MATLAB. TensorTools based on NumPy~\\cite{DBLP:journals\/cse\/WaltCV11} implements CP decomposition only, while T3F is explicitly designed for Tensor Train Decomposition on Tensorflow~\\cite{DBLP:conf\/osdi\/AbadiBCCDDDGIIK16}. Similarly based on Tensorflow, TensorD supports CP and Tucker decomposition. By contrast, TedNet implements five kinds of tensor decomposition with backend Pytorch~\\cite{NEURIPS2019_9015}.  TensorNetwork is built on Tensorflow and incorporates abundant tensor calculation tools. Nevertheless, TensorNetwork serves for tensor decomposition algorithms rather than TDNs. Tensorly supports with a variety of backends including CuPy, Pytorch, Tensorflow and MXNet~\\cite{chen2015mxnet}. Unfortunately, although Tensorly is powerful to process tensor algebra, tensor decomposition and tensor regressions, it still lacks support to Application Programming Interface (API) to build tensorial neural networks directly. Interestingly, Tensorly can assist to initialize TedNet network modules with its tensor decomposition operation. Compared with them, TedNet can \nset up a TDN layer quickly by calling API directly. In addition, we also provide three kinds of deep TDNs that are popular for researchers now. Due to the Dynamic Graph Mechanism of Pytorch, TedNet is also flexible to DEBUG for programmers.\n\n\n\n\n\n\n\n\n\n\\section*{Required Metadata}\n\n\n\n\n\\section*{Current code version}\n\\label{}\n\nAncillary data table required for subversion of the codebase. Kindly replace examples in right column with the correct information about your current code, and leave the left column as it is.\n\n\\begin{table}[!h]\n\\begin{tabular}{|l|p{6.5cm}|p{8.2cm}|}\n\\hline\n\\textbf{Nr.} & \\textbf{Code metadata description} & \\textbf{Please fill in this column} \\\\\n\\hline\nC1 & Current code version & 0.1.3 \\\\\n\\hline\nC2 & Permanent link to code\/repository used of this code version & $https:\/\/github.com\/tnbar\/tednet\/releases\/tag\/0.1.3$ \\\\\n\\hline\nC3 & Legal Code License   & MIT License \\\\\n\\hline\nC4 & Code versioning system used & git \\\\\n\\hline\nC5 & Software code languages, tools, and services used & Python, Pytorch \\\\\n\\hline\nC6 & Compilation requirements, operating environments \\& dependencies & Python3.X, NumPy\\\\\n\\hline\nC7 & If available Link to developer documentation\/manual & $https:\/\/tednet.readthedocs.io\/en\/latest\/index.html$ \\\\\n\\hline\nC8 & Support email for questions & iperryuu@gmail.com\\\\\n\\hline\n\\end{tabular}\n\\caption{Code metadata (mandatory)}\n\\end{table}\n\\section{Installation and Usage}\n\nThere are two ways to install TedNet. For the sake that the source code of TedNet is submitted to GitHub, it is feasible to install from the downloaded code by command \\textbf{python setup.py install}. Compared with aforementioned fussy way, another one, the recommended way is to install TedNet trough PyPI~\\footnote{https:\/\/pypi.org\/project\/tednet} by command \\textbf{pip install tednet}. After installation, all tensor decomposition models of TedNet can be used.\n\nAn example of Tensor Ring is shown in Listing~\\ref{ll:tr}. Tensor ring decomposition can be used by import module \\textbf{tednet.tnn.tensor\\_ring}. The usage of other decomposition is the same and more details can be found in the Document~\\footnote{https:\/\/tednet.readthedocs.io}.\n\n\n\n\\begin{lstlisting}[language={Python}, label={ll:tr}, caption={\\bf The Usage of Tensor Ring Models}]\n# import tensor ring module\nimport tednet.tnn.tensor_ring as tr\n\n# TR-LSRM\nmodel = tr.TRLSTM([16, 16], [32, 32], [5, 5, 5, 5])\n\n# TR-Lenet-5\nmodel = tr.TRLeNet5(10, [6, 6, 6, 6])\n\n# TR-ResNet-20\nmodel = tr.TRResNet20([7, 7, 7, 7, 7, 7, 7], 10)\n\\end{lstlisting}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGauge-Higgs unification \\cite{Manton, Fairlie, Hosotani} is one of the attractive scenarios \nsolving the hierarchy problem without invoking supersymmetry. \nIn this scenario, \nHiggs doublet in the Standard Model (SM) is identified with \nthe extra spatial components of the higher dimensional gauge fields. \nRemarkable feature is that the quantum correction to Higgs mass is insensitive \nto the cutoff scale of the theory and calculable \nregardless of the non-renormalizability of higher dimensional gauge theory. \nThe reason is that the Higgs mass term as a local operator is forbidden \nby the higher dimensional gauge invariance. \nThe finite mass term is generated radiatively and expressed by the Wilson line phase as a non-local operator. \nThis fact has opened up a new avenue to the solution of the hierarchy problem \\cite{HIL}. \nSince then, much attention has been paid to the gauge-Higgs unification and \nmany interesting works have been done from various points of view \n\\cite{KLY}-\\cite{LHC}. \n\nThe finiteness of Higgs mass has been studied and verified in various models \nand types of compactification at one-loop level\\footnote{For the case of gravity-gauge-Higgs unification, \nsee \\cite{HLM}} \\cite{ABQ}-\\cite{LMH} and even at two loop level \\cite{MY, HMTY}. \nIt is natural to ask whether any other finite physical observables exist in the gauge-Higgs unification. \nThe naive guess is that such observables are in the gauge-Higgs sector of the theory if they ever exist. \nTwo of the present authors (C.S.L. and N.M.) studied the structure of divergences for S and T parameters \nin the gauge-Higgs unification since such parameters are described \nby higher dimensional gauge invariant operators with respect to gauge and Higgs fields, \nand are expected to be finite by virtue of the higher dimensional gauge symmetry. \nThe result is that both parameters are divergent (convergent) more than (in) five dimensions \nas expected from the naive power counting argument.  \nHowever, a nontrivial prediction  we have found, specific to the gauge-Higgs unification, is \nthat some linear combination of S and T parameters is finite even in six dimensions \\cite{LM}. \n\nIn our previous paper \\cite{ALM} we have found a more striking fact:  \nwe have shown that the magnetic moment of fermion \nin the $(D+1)$ dimensional QED gauge-Higgs unification model \ncompactified on $S^1$ becomes finite for an arbitrary space-time dimension, \nregardless of the nonrenormalizability of the theory. \nActually, the reason is very simple. \nIn four dimensional space-time, \na dimension six gauge invariant local operator describes the magnetic moment: \n\\begin{eqnarray}\ni \\bar{\\psi}_L \\sigma^{\\mu\\nu} \\psi_R F_{\\mu\\nu} \\langle H \\rangle. \n\\label{MMM4}\n\\end{eqnarray}\nHowever, when included into the scheme of gauge-Higgs unification, \nthe Higgs doublet should be replaced by an extra space component of the higher dimensional gauge field $A_y$. \nThen the operator is forbidden by the higher dimensional gauge invariance, \nsince $A_y$ transforms inhomogeneously under the gauge transformation. \nThen, to preserve the gauge symmetry, \n$A_y$ should be further replaced by gauge covariant derivative $D_y$, \nand the relevant gauge invariant operator becomes \n\\begin{eqnarray}\ni \\bar{\\Psi} \\Gamma^{MN} D_L \\Gamma^L \\Psi F_{MN}  \n\\label{MMMD}\n\\end{eqnarray}\nwhere $L, M$ and $N$ denote $D+1$ dimensional Lorentz indices. \nThe key observation of our argument is that the operator (\\ref{MMMD}), \nwhen $D_L$ is replaced by $\\langle D_L \\rangle$ with the gauge field $A_L$ replaced by its VEV, \nvanishes because of the on-shell condition $i \\langle D_L \\rangle \\Gamma^L \\Psi = 0$. \nFrom this fact, we can expect that the magnetic moment is finite \nand have shown that it is indeed the case by explicit diagrammatical calculations \\cite{ALM}. \nThis is the specific prediction of the gauge-Higgs unification to be contrasted with \nthe case of Randall-Sundrum model \\cite{DHR} or the universal extra dimension scenario \\cite{ADW, AD}, \nin which the magnetic moment of fermion diverges in the models with more than five space-time dimensions. \n\nAlthough this result was quite remarkable, the above model is too simple and not realistic. \nIn particular, \nthe gauge group $U(1)$ is too small to incorporate the standard model. \nIn this paper, we study more about the cancellation mechanism of ultraviolet (UV) divergences \nin a realistic gauge-Higgs unification model. \nWe consider $(D+1)$ dimensional $SU(3)$ gauge-Higgs unification model compactified \non an orbifold $S^1\/Z_2$ with a massive bulk fermion in a fundamental representation. \nThe orbifolding is indispensable to obtain the SM Higgs $SU(2)_L$ doublet \nsince Higgs originally behaves as an adjoint representation of the gauge group in the gauge-Higgs unification. \nWe consider here a simple orbifold $S^1\/Z_2$. \nIn the case of $S^1\/Z_2$, the bulk mass parameter of fermion must have odd $Z_2$ parity \nsince the fermion bulk mass term connects fermions with different chiralities and opposite $Z_2$ parities. \nIt is well known that the zero mode wave functions take an exponential profile along a compactified space coordinate \nand $D$-dimensional effective Yukawa couplings obtained by an overlap integral of zero mode wave functions \nare exponentially suppressed. \nIn this way, we can freely obtain the light fermion masses, which are otherwise of ${\\cal O}(M_W)$, \nby tuning the bulk mass parameter. \nOne might worry if our argument for the finiteness in the previous paper \nstill holds in the present orbifold model since the on-shell condition for the fermion is changed to \n$i \\Gamma^M \\langle D_M \\rangle \\Psi = M \\varepsilon(y) \\Psi$ \n($\\varepsilon(y):$ the sign function of $y$, the extra space coordinate) and also the brane localized operator \n\\begin{eqnarray}\ni \\bar{\\psi}_L \\Gamma^{\\mu\\nu} A_y \\Gamma^y \\psi_R F_{\\mu\\nu} \n\\label{braneAMM}\n\\end{eqnarray}\nseems to be allowed. \nHowever, these two worries are not necessary. \nAs for the first one, we note that the fermion $\\Psi$ in the operator (\\ref{MMMD}) \nshould be understood as the zero mode fermion. \nThough the operator (\\ref{MMMD}) does not vanish even after imposing the on-shell condition, \nthe remaining operator $M \\varepsilon(y) \\bar{\\Psi} \\Gamma^{MN} \\Psi F_{MN}$ has no correspondence \nin the standard model (in the standard model $\\bar{\\Psi}_L \\gamma^{MN} \\Psi_R$ is not gauge invariant), \nand therefore vanishes automatically for the zero-mode fermion $\\Psi$. \nAs for the second one, note that the shift symmetry \n$A_y \\to A_y + {\\rm const}$ is operative as a remnant of higher dimensional gauge symmetry\neven at the branes \\cite{GIQ}. \nTherefore, the brane localized operator (\\ref{braneAMM}) is forbidden by the shift symmetry. \nFurthermore, the UV finiteness is independent of how we compactify the extra space, \nbecause the information about the compactification is an infrared property of the theory. \nFrom these observations, we can expect the magnetic moment still to be finite \neven for the orbifold compactification and the presence of bulk mass term. \nThe primary purpose of this paper is to clarify the cancellation mechanism of UV divergences \nin the calculation of the anomalous magnetic moment in the framework of the present orbifold model. \n\n\nThis paper is organized as follows. \nIn the next section, we introduce our model and discuss the mass eigenvalues and mode functions of \nfermions and gauge bosons. \nIn section 3, we derive various interaction vertices, which are needed in the calculation of \nthe anomalous magnetic moment and generally valid without any approximation. \nIn section 4, we provide the general formulae for the contributions of $A_\\mu$ ($D$-dimensional gauge field) \nand $A_y$ ($D$-dimensional scalar) exchange diagrams to the anomalous magnetic moment. \nThe mechanism of cancellation of divergence is clarified in the case of small bulk mass in section 5. \nOur conclusions are given in section 6.  \nThe detailed derivation of some useful properties \nconcerning the vertex functions are summarized in Appendices A and B. \n\n\n\\section{The Model}\n\nWe consider a $(D+1)$ dimensional $SU(3)$ gauge-Higgs unification model compactified on an orbifold $S^1\/Z_2$ \n($S^1$: a circle of radius $R$) with a massive bulk fermion in the fundamental representation of $SU(3)$ gauge group. \nThe Lagrangian is given by\n\\begin{eqnarray}\n{\\cal L} &=& -\\frac{1}{2} {\\rm Tr} (F_{MN} F^{MN}) + \\bar{\\Psi} (iD\\!\\!\\!\\!\/ - M \\varepsilon(y)) \\Psi\n\\label{Lagrangian1}\n\\end{eqnarray}\nwhere the indices $M,N = 0,1,2,3 \\cdots, D$, \nthe $(D+1)$ dimensional gamma matrices are $\\Gamma^M  = (\\gamma^\\mu, i \\gamma^{D+1})~(\\mu = 0,1,2,3, \\cdots, D-1)$, \n\\begin{eqnarray}\n&&F_{MN} = \\partial_M A_N - \\partial_N A_M - i g [A_M, A_N], \\\\\n&&D\\!\\!\\!\\!\/ = \\Gamma^M (\\partial_M -i g A_M), \\\\\n&&\\Psi = (\\psi_1, \\psi_2, \\psi_3)^T. \n\\end{eqnarray}\n$g$ denotes a gauge coupling constant in $(D+1)$ dimensional gauge theory. \n$M$ is a bulk mass of the fermion. \nHere we note that the bulk fermion mass must have an odd $Z_2$ parity \nto be consistent with an orbifold projection: we thus introduce here the bulk mass \nproportional to the sign function $\\varepsilon(y)$ of compactified extra space coordinate $y$. \n\nThe periodic boundary condition is imposed along $S^1$ \nand $Z_2$ parity assignments are taken as\n\\begin{eqnarray}\nA_\\mu = \n\\left(\n\\begin{array}{ccc}\n(+,+) & (+,+) & (-,-) \\\\\n(+,+) & (+,+) & (-,-) \\\\\n(-,-) & (-,-) & (+,+) \\\\\n\\end{array}\n\\right), \\quad \nA_y = \n\\left(\n\\begin{array}{ccc}\n(-,-) & (-,-) & (+,+) \\\\\n(-,-) & (-,-) & (+,+) \\\\\n(+,+) & (+,+) & (-,-) \\\\\n\\end{array}\n\\right), \n\\end{eqnarray}\n\\begin{eqnarray}\n\\Psi = \n\\left(\n\\begin{array}{c}\n\\psi_{1L}(+,+) + \\psi_{1R}(-,-) \\\\\n\\psi_{2L}(+,+) + \\psi_{2R}(-,-) \\\\\n\\psi_{3L}(-,-) + \\psi_{3R}(+,+) \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhere $\\mu = 0,1,2,3, \\cdots, D-1$  \nand $(+,+)$ means that $Z_2$ parities are even at the fixed points $y=0$ and $y=\\pi R$, for instance.\n$L, R$ on fermion denotes the chiral projection operator (for even $D$) defined as \n$L = \\frac{1+\\gamma^{D+1}}{2}, R = \\frac{1-\\gamma^{D+1}}{2}$. \nAs can be seen from the KK mode expansion consistent with the boundary conditions \nonly the fields with $(+,+)$ parities have massless modes as\n\\begin{eqnarray}\nA_\\mu^{(0)} = \\frac{1}{2}\n\\left(\n\\begin{array}{ccc}\nW_\\mu^3 +\\frac{B_\\mu}{\\sqrt{3}} & \\sqrt{2} W_\\mu^+ & 0 \\\\\n\\sqrt{2} W_\\mu^- & -W_\\mu^3 + \\frac{B_\\mu}{\\sqrt{3}} & 0 \\\\\n0 & 0 & -\\frac{2}{\\sqrt{3}}B_\\mu \\\\\n\\end{array}\n\\right), \\quad \nA_y^{(0)} = \\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{ccc}\n0 & 0 & \\phi^+ \\\\\n0 & 0 & \\frac{h -i \\phi^0}{\\sqrt{2}} \\\\\n\\phi^- & \\frac{h +i \\phi^0}{\\sqrt{2}} & 0 \\\\\n\\end{array}\n\\right), \n\\label{0modegauge}\n\\end{eqnarray}\nwhere $W_\\mu^3, W_\\mu^\\pm$ and $B_\\mu$ are the $SU(2)_L$ and $U(1)_Y$ gauge fields, respectively. \nFrom this expression, we see that the gauge symmetry $SU(3)$ is broken to $SU(2)_L \\times U(1)_Y$ \nby the boundary conditions. \nFurthermore, the SM Higgs doublet is just embedded into the off-diagonal elements of  $A_y^{(0)}$. \nAs for the fermion, we obtain massless modes \n\\begin{eqnarray}\n\\Psi^{(0)} = \n\\left(\n\\begin{array}{c}\n\\psi_{1L}^{(0)} \\\\\n\\psi_{2L}^{(0)} \\\\\n\\psi_{3R}^{(0)} \\\\\n\\end{array}\n\\right) \n= \n\\left(\n\\begin{array}{c}\nu_L \\\\\nd_L \\\\\nd_R \\\\\n\\end{array}\n\\right)\n\\end{eqnarray}\nwhich shows that chiral fermions are realized by orbifold projection. \n\n\nSome comments on this model are in order. \nFirst, the predicted Weinberg angle of this model is not realistic, $\\sin^2 \\theta_W = 3\/4$. \nAs was also discussed in \\cite{GW}, \nthe present $SU(3)$ model in five dimension ($D=4$) is inconsistent with \nthe experimental requirements $\\sin^2 \\theta_W \\simeq 1\/4$ \nand $\\rho = m_W^2\/(m_Z^2\\cos^2 \\theta_W) =1$. \nPossible way to cure the problem is to introduce an extra U(1) or \nthe brane localized gauge kinetic term \\cite{SSS}. \nSecond, the up quark remains massless and we have no up-type Yukawa coupling. \nA possible way out of this situation is to introduce second-rank symmetric tensors of $SU(3)$ \n(${\\bf 6}$ dimensional representation) \\cite{CCP}. \nThird point is that the fermion in the fundamental representation of $SU(3)$ has no lepton. \nIn order to incorporate the leptons, a third-rank symmetric tensor \n(${\\bf 10}$ dimensional representation) must be introduced. \nWhen such higher dimensional representations are added to the theory, there appear some massless exotic fermions and they should be removed from low energy sector \nby adding the brane localized fields to form brane localized mass terms with the exotic states. \n\nSince our primary purpose in this paper is to clarify \nthe cancellation mechanism of UV divergence for the magnetic moment, \nin this paper we calculate the $g-2$ of down quark belonging to the triplet, as an example of the anomalous magnetic moment of fermion. This makes our calculation greatly simplified since the fundamental representation has no massless exotic fermions and we does not need introducing additional brane localized fermions and mass terms. We hope that our results for UV finiteness remains  unchanged even for the case of well-discussed muon $g-2$, since the operator analysis given in the introduction is independent of the representation of fermion. We also hope that our calculation of $g-2$ of $d$ quark will be relevant for, $e.g.,$ the electric dipole moment of neutron. \n\nThroughout this paper, what we mean by ``realistic\" is in it's restricted sence, i.e. we mean that the gauge-Higgs model we consider contains the gauge group of the Standard Model and reproduces correct order of small Yukawa coupling relevant for the calculation of the magnetic moment. \n\n\n\\subsection{The mass eigenvalues and mode functions of fermions}\n\nIn order to derive $D$-dimensional effective Lagrangian and Feynman rules \nnecessary for the calculations of the magnetic moment, \nfirst we have to obtain the $D$-dimensional mass eigenvalues and corresponding mode functions of  fermions. \n\nWe first focus on the down quark sector $\\psi_d \\equiv (\\psi_2, \\psi_3)^t$ of the triplet fermion \n$(\\psi_1, \\psi_2, \\psi_3)^t$. \nThe $D$-dimensional mass term reads as \n\\begin{eqnarray}\n{\\cal L}_{{\\rm mass}} &=& \\bar{\\psi}_d [\\Gamma^y (i \\partial_y + g \\langle A_y \\rangle) -M \\varepsilon(y) ] \\psi_d, \n\\label{massfermion1} \\\\\ng \\langle A_y \\rangle &\\equiv& g_D v \n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n1 & 0 \\\\\n\\end{array}\n\\right) \\quad \n\\left( g_D \\equiv \\frac{g}{\\sqrt{2\\pi R}} \\right).\n\\label{massfermion2}\n\\end{eqnarray}\nWe diagonalize the matrix $g \\langle A_y \\rangle$ by an orthogonal transformation, \n\\begin{eqnarray}\n\\tilde{\\psi}_d = \n\\left(\n\\begin{array}{c}\n\\tilde{\\psi}_2 \\\\\n\\tilde{\\psi}_3 \\\\\n\\end{array}\n\\right) \n\\equiv  \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n-\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right) \\psi_d \n= \\frac{1}{\\sqrt{2}} \\left(\n\\begin{array}{c}\n\\psi_2+\\psi_3 \\\\\n-\\psi_2+\\psi_3 \\\\\n\\end{array}\n\\right). \n\\end{eqnarray}\nIn terms of $\\tilde{\\psi}_{2,3}$, (\\ref{massfermion1}) can be rewritten as\n\\begin{eqnarray}\n\\bar{\\tilde{\\psi}_2} [\\Gamma^y (i \\partial_y + g_D v) - M \\varepsilon(y) ] \\tilde{\\psi}_2 \n+ \\bar{\\tilde{\\psi}_3} [\\Gamma^y (i \\partial_y - g_D v) - M \\varepsilon(y) ] \\tilde{\\psi}_3. \n\\label{massfermion3}\n\\end{eqnarray}\nNow, we try to find mass eigenvalues and mode functions of $\\tilde{\\psi}_2$. \nExpanding the $\\tilde{\\psi}_2(x,y)$ as \n$\\tilde{\\psi}_2(x,y) = \\sum_n \\frac{1}{\\sqrt{2}} [ d_{L}^{(n)}(x) f_{d_L}^{(n)}(y) + d_{R}^{(n)}(x) f_{d_R}^{(n)}(y) ]$, \nwe found the equations of motion \n\\begin{eqnarray}\n&&[ (i \\partial_y + g_D v)^2 + M^2 -2M (\\delta(y) - \\delta (y-\\pi R) ) ] f_{d_L}^{(n)}(y) = m_n^2 f_L^{(n)}(y), \n\\label{modeeqL}\\\\\n&&[ (i \\partial_y + g_D v)^2 + M^2 +2M (\\delta(y) - \\delta (y-\\pi R) ) ] f_{d_R}^{(n)}(y) = m_n^2 f_R^{(n)}(y)\n\\label{modeeqR}\n\\end{eqnarray} \nwhere $m_n$ is the mass eigenvalue of $n$-th KK mode with $n$ being an arbitrary integer. \n\nLet us solve (\\ref{modeeqL}). \nBy use of the boundary condition that $f_{d_L}^{(n)}$ is continuous \nand $\\partial_y f_{d_L}^{(n)}$ has a discontinuity $-2M f_{d_L}^{(n)}$ \nat the fixed point $y=0$, \nthe solution is known to take a form \n\\begin{eqnarray}\nf_{d_L}^{(n)}(y) &=& e^{ig_D v y} \\left[ \nC_1^{(n)} \\left\\{ \\cos(\\sqrt{m_n^2-M^2}y) \n- \\frac{M}{\\sqrt{m_n^2-M^2}} \\varepsilon(y) \\sin(\\sqrt{m_n^2-M^2}y) \\right\\} \\right. \\nonumber \\\\\n&& \\left. + C_2^{(n)} \\sin(\\sqrt{m_n^2-M^2}y) \\right] \n\\label{generalsolL}\n\\end{eqnarray}\nwhere $C_{1,2}^{(n)}$ are integration constants. \n\nThen the similar boundary conditions \nat the fixed point $y=\\pi R$ can be written as\n\\begin{eqnarray}\n&&0 = (w-1) \\left(\\cos \\varphi_n -\\frac{M}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n \\right) C_1^{(n)} \n+ (w+1) \\sin \\varphi_n C_2^{(n)}, \n\\label{bc1} \\\\\n&&0 = -(w+1) \\frac{m_n^2}{m_n^2-M^2} \\sin \\varphi_n C_1^{(n)} \n+ (w-1) \\left(\\cos \\varphi_n + \\frac{M}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n \\right) C_2^{(n)} \\nonumber \n\\label{bc2} \\\\\n\\end{eqnarray}\nwhere $w$ is a Wilson loop $w \\equiv e^{2\\pi i g_D Rv}$ and $\\varphi_n \\equiv \\sqrt{m_n^2-M^2} \\pi R$. \nThese boundary conditions determine the mass eigenvalues $m_n$ through the condition\n\\begin{eqnarray}\n\\frac{m_n^2}{m_n^2-M^2} \\sin^2(\\sqrt{m_n^2-M^2}\\pi R) = \\sin^2(g_D v \\pi R), \n\\label{masscond}\n\\end{eqnarray}\nwhich cannot be solved analytically for $m_n$, in general.  \nAs a check, \nif we consider the case of $v=0$, we obtain\n\\begin{eqnarray}\nm_0 = 0, \\quad m_n^2 = \\left( \\frac{n}{R} \\right)^2 + M^2 \n\\end{eqnarray}\nwhich is a well known result. \nWe take a sign convention such that  \n\\begin{eqnarray}\nm_n = \\frac{n}{R} + g_D v \n\\end{eqnarray}\nin the case of $M=0$. \nFor $m_n$ satisfying (\\ref{masscond}), the ratio of $C_1^{(n)}$ and $C_2^{(n)}$ is known to be fixed as\n\\begin{eqnarray}\nC_1^{(n)} \/ C_2^{(n)} = \\sqrt{\\cos(\\varphi_n-\\alpha_n)} \/ -i \\frac{m_n}{\\sqrt{m_n^2 - M^2}} \\sqrt{\\cos(\\varphi_n + \\alpha_n)}\n\\end{eqnarray}\nwhere $\\tan \\alpha_n \\equiv \\frac{M}{\\sqrt{m_n^2-M^2}}$. \nThus, we have obtained the mode functions\n\\begin{eqnarray}\n&&f_{d_L}^{(n)}(y) = F_{M, M_W}^{(n)}(y), \\quad f_{d_R}^{(n)}(y) = F_{-M, M_W}^{(n)}(y), \\\\\n&&F_{M,M_W}^{(n)}(y) = e^{ig_D vy}C^{(n)} \n\\left[\n\\sqrt{\\cos(\\varphi_n -\\alpha_n)} \\cos(\\sqrt{m_n^2-M^2}|y| + \\alpha_n) \\right. \\nonumber \\\\\n&& \\left. \\hspace*{4cm} - \\varepsilon(n) i \\sqrt{\\cos(\\varphi_n + \\alpha_n)} \\sin(\\sqrt{m_n^2-M^2}y) \n\\right], \\label{modeL}\n\\label{modeR}\n\\end{eqnarray}\nwhere the normalization constant $C^{(n)}$ is given by\n\\begin{eqnarray}\nC^{(n)} &=& \\left[\n2\\pi R \\cos \\varphi_n \\cos \\alpha_n  \n- \\frac{2}{\\sqrt{m_n^2-M^2}} \\sin \\varphi_n  \\cos \\alpha_n \\sin^2 \\alpha_n \n\\right]^{-1\/2}. \n\\label{normalization}\n\\end{eqnarray}\nThe ``sign function\" $\\varepsilon(n)$ is defined as $1$ for $n \\ge 0$ and $-1$ for $n < 0$. \n\nAs a matter of fact when the mode functions are substituted in (\\ref{massfermion3}) \nwe get a mass $-i m_n$ for the Dirac fermion $d^{(n)} = d_L^{(n)} + d_R^{(n)}$. \nThus we perform a chiral transformation\n\\begin{eqnarray}\n\\hat{\\psi}_d \\equiv \n\\left(\n\\begin{array}{c}\n\\hat{\\psi}_2 \\\\\n\\hat{\\psi}_3 \\\\\n\\end{array}\n\\right)\n= e^{-i\\frac{\\pi}{4}\\gamma^{D+1}} \\tilde{\\psi}_2,\n\\end{eqnarray}\nso that $d^{(n)}$ has a mass $m_n$, with $\\hat{\\psi}_2$ still being mode-expanded as \n\\begin{eqnarray}\n\\hat{\\psi}_2(x,y) = \\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} \n\\left(\nf_{d_L}^{(n)} d_L^{(n)} + f_{d_R}^{(n)} d_R^{(n)}\n\\right). \n\\label{modeexpsi2}\n\\end{eqnarray}\n\nBecause of the orbifolding, \n$\\hat{\\psi}_3$ is not independent of $\\hat{\\psi}_2$. \nIn fact, the $Z_2$ parity assignment $\\psi_2(x,-y) = \\gamma^{D+1} \\psi_2(x,y)$ and \n$\\psi_3(x,-y) = -\\gamma^{D+1} \\psi_3(x,y)$ tells us\n\\begin{eqnarray}\n\\hat{\\psi}_3 (x,y) = -\\gamma^{D+1} \\hat{\\psi}_2(x,-y). \n\\end{eqnarray}\nSince $\\psi_1$ does not get a mass due to the VEV $v$ and the $Z_2$ parity assignment is \n$\\psi_1(x,-y) = \\gamma^{D+1} \\psi_1(x,y)$, $\\hat{\\psi}_1 \\equiv e^{-i\\frac{\\pi}{4}\\gamma^{D+1}} \\psi_1$ \nis mode-expanded as\n\\begin{eqnarray}\n\\hat{\\psi}_1(x,y) &=& \\sum_{n=1}^\\infty \n\\left\\{\n\\frac{1}{\\sqrt{2}} (f_{u_L}^{(n)}(y) + f_{u_L}^{(n)}(-y)) u_L^{(n)}(x) \n+ \\frac{1}{\\sqrt{2}} (f_{u_R}^{(n)}(y) - f_{u_R}^{(n)}(-y)) u_R^{(n)}(x)\n\\right\\} \\nonumber \\\\\n&& + f_{u_L}^{(0)}(y) u_L^{(0)}(x),  \n\\end{eqnarray}\nwhere the mode functions take relatively simple forms\n\\begin{eqnarray}\nf_{u_L}^{(n)}(y) &=& F_{M, 0}^{(n)}(y) = \\frac{1}{\\sqrt{2\\pi R}} \n\\left[\n\\cos \\left(\\frac{n}{R} |y| + \\alpha_n^{(0)} \\right) -i \\sin \\left(\\frac{n}{R} y \\right)\n\\right], \\\\\nf_{u_L}^{(0)}(y) &=& F_{M, 0}^{(0)}(y) = \\sqrt{\\frac{M}{1-e^{-2\\pi MR}}} e^{-M|y|}, \\\\ \nf_{u_R}^{(n)}(y) &=& F_{-M, 0}^{(n)}(y) = \\frac{1}{\\sqrt{2\\pi R}} \n\\left[\n\\cos \\left(\\frac{n}{R} |y| - \\alpha_n^{(0)} \\right) -i \\sin \\left(\\frac{n}{R} y \\right)\n\\right], \n\\end{eqnarray}\nwhere $\\alpha_n^{(0)}$ is defined by\n\\begin{eqnarray}\n\\tan \\alpha_n^{(0)} \\equiv \\frac{M}{n\/R}~(n \\ge 1). \n\\end{eqnarray}\nDirac fermion $u^{(n)}(x) \\equiv u_L^{(n)}(x) + u_R^{(n)}(x)$ has a mass\n\\begin{eqnarray}\n\\tilde{m}_n = \\sqrt{\\left(\\frac{n}{R} \\right)^2 + M^2}~(n \\ge 1), \n\\end{eqnarray}\nwhile $u_L^{(0)}(x)$ remains as a massless state: $\\tilde{m}_0 = 0$. \n\nWe thus have obtained the mass eigenstates of the fermion, \n\\begin{eqnarray}\n\\hat{\\Psi} &\\equiv& \n\\left(\n\\begin{array}{c}\n\\hat{\\psi}_1 \\\\\n\\hat{\\psi}_2 \\\\\n\\hat{\\psi}_3 \\\\\n\\end{array}\n\\right) \n= e^{-i \\frac{\\pi}{4}\\gamma_y} {\\cal O} \\Psi \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{c}\n\\sum_{n=1}^\\infty \\frac{1}{\\sqrt{2}} \n\\left\\{\n(f_{u_L}^{(n)}(y) + f_{u_L}^{(n)}(-y)) u_L^{(n)}(x) \n+ (f_{u_R}^{(n)}(y) - f_{u_R}^{(n)}(-y)) u_R^{(n)}(x)\n\\right\\}\n+ f_{u_L}^{(0)}(y) u_L^{(0)}(x) \\\\\n\\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} (f_{d_L}^{(n)}(y) d_L^{(n)}(x) + f_{d_R}^{(n)}(y) d_R^{(n)}(x)) \\\\\n\\sum_{n=-\\infty}^\\infty \\frac{1}{\\sqrt{2}} (-f_{d_L}^{(n)}(-y) d_L^{(n)}(x) + f_{d_R}^{(n)}(-y) d_R^{(n)}(x)) \\\\\n\\end{array}\n\\right), \\nonumber \\\\\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\cal O} &\\equiv& \n\\left(\n\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n0 & -\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right). \n\\label{rotation}\n\\end{eqnarray}\nLet us comment on the quantum mechanical supersymmetry (QMS) hidden in the mode functions of fermions. \nIt has been demonstrated that the mode functions of $A_\\mu$ and $A_y$ form a supermultiplet \nfor each non-zero KK modes, which reflects the Higgs-like mechanism to form massive gauge bosons \n$A_\\mu^{(n)}(n \\ne 0)$ \\cite{LNSS}. \nA similar thing is expected to take place in the fermion sector; \n$f_L^{(n)}$ and $f_R^{(n)}$ are expected to form a supermultiplet, \nas they are ``partner\" to form a massive Dirac fermion. \nIn fact, they are known to be related by a supercharge $Q$: \n\\begin{eqnarray}\n&&Q\n\\left(\n\\begin{array}{c}\nF_{M, M_W}^{(n)} \\\\\nF_{-M, M_W}^{(n)} \\\\\n\\end{array}\n\\right)\n= m_n \n\\left(\n\\begin{array}{c}\nF_{M, M_W}^{(n)} \\\\\nF_{-M, M_W}^{(n)} \\\\\n\\end{array}\n\\right), \n\\label{2.35}\\\\\n&&Q = \n\\left(\n\\begin{array}{cc}\n0 & i \\partial_y + M_W -i M \\varepsilon(y) \\\\\ni \\partial_y + M_W +i M \\varepsilon(y) & 0 \\\\\n\\end{array}\n\\right)\n\\label{2.36}\n\\end{eqnarray}\nwith $Q^2$ giving the differential operators in (\\ref{modeeqL}) and (\\ref{modeeqR}), \nnamely the Hamiltonian in QMS. \n\n\\subsection{The mass eigenvalues and mode functions of gauge bosons}\nNext, we turn to the mass eigenvalues and mode functions of $D$-dimensional gauge bosons and Higgs scalars, \n$i.e.~A_\\mu$ and $A_y$. \nFirst we explicitly write the gauge bosons and Higgs scalar fields as follows, \n\\begin{eqnarray}\nA_\\mu(x, y) &=& \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_\\mu}{\\sqrt{3}} & \\frac{W^+_\\mu}{\\sqrt{2}} & \\frac{\\phi^+_\\mu}{\\sqrt{2}} \\\\\n\\frac{W^-_\\mu}{\\sqrt{2}} & -\\frac{\\gamma_\\mu}{2\\sqrt{3}} -\\frac{Z_\\mu}{2} & \\frac{h_\\mu-i\\phi^0_\\mu}{2} \\\\\n\\frac{\\phi^-_\\mu}{\\sqrt{2}} & \\frac{h_\\mu +i \\phi^0_\\mu}{2} & -\\frac{\\gamma_\\mu}{2\\sqrt{3}} + \\frac{Z_\\mu}{2} \\\\\n\\end{array}\n\\right), \n\\label{gaugematrix} \\\\\nA_y(x, y) &=& \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_y}{\\sqrt{3}} & \\frac{W^+_y}{\\sqrt{2}} & \\frac{\\phi^+}{\\sqrt{2}} \\\\\n\\frac{W^-_y}{\\sqrt{2}} & -\\frac{\\gamma_y}{2\\sqrt{3}} -\\frac{Z_y}{2} & \\frac{h-i\\phi^0}{2} \\\\\n\\frac{\\phi^-}{\\sqrt{2}} & \\frac{h +i \\phi^0}{2} & -\\frac{\\gamma_y}{2\\sqrt{3}} + \\frac{Z_y}{2} \\\\\n\\end{array}\n\\right). \n\\label{Aymatrix}\n\\end{eqnarray}\nEach field has a mode expansion depending on its $Z_2$ parity, \n\\begin{eqnarray}\nA_{\\mu,y}(x,y) &=& \\frac{1}{\\sqrt{2\\pi R}} A_{\\mu,y}^{(0)}(x) \n+ \\frac{1}{\\sqrt{\\pi R}} \\sum_{n=1}^\\infty A_{\\mu,y}^{(n)}(x) \\cos \\left(\\frac{n}{R}y \\right)~({\\rm even}), \\\\\nA_{\\mu,y}(x,y) &=&  \n\\frac{1}{\\sqrt{\\pi R}} \\sum_{n=1}^\\infty A_{\\mu,y}^{(n)}(x) \\sin \\left(\\frac{n}{R}y \\right)~({\\rm odd})\n\\end{eqnarray}\nPutting these mode functions into the term ${\\rm Tr}(F_{\\mu y})^2$ and integrating over $y$ coordinate lead to \nthe necessary quadratic terms concerning nonzero KK modes;\n\\begin{eqnarray}\n{\\cal L}_{{\\rm quadratic}} &=& \\sum_{n=1}^\\infty \\left[ \n\\frac{1}{2} \\left(\\partial_\\mu \\gamma_y^{(n)} + \\frac{n}{R} \\gamma_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2} \\left(\\partial_\\mu h^{(n)} - \\frac{n}{R} h_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2}[(\\partial_\\mu Z_y^{(n)})^2 + (\\partial_\\mu \\phi^{0(n)})^2] \\right. \\nonumber \\\\\n&& \\left. + \\frac{1}{2}\\left[ \\left( \\frac{n}{R} \\right)^2 + (2g_D v)^2 \\right] [ (Z_\\mu^{(n)})^2 + (\\phi_\\mu^{0(n)})^2 ]\n+ 4\\frac{n}{R} g_D v Z_\\mu^{(n)} \\phi^{\\mu0(n)} \\right. \\nonumber \\\\\n&& \\left. + (\\partial^\\mu Z_y^{(n)}) \\left(\\frac{n}{R} Z_\\mu^{(n)} +2g_D v \\phi_\\mu^{0(n)} \\right) \n- (\\partial^\\mu \\phi^{0(n)}) \\left(\\frac{n}{R} \\phi_\\mu^{0(n)} +2g_D v Z_\\mu^{(n)} \\right) \\right. \\nonumber \\\\\n&& \\left. +|\\partial_\\mu W_y^{+(n)}|^2 + |\\partial_\\mu \\phi^{+(n)}|^2 \n+ \\left[\\left( \\frac{n}{R} \\right)^2 + (g_D v)^2 \\right](|W_\\mu^{+(n)}|^2 + |\\phi_\\mu^{+(n)}|^2) \\right. \\nonumber \\\\\n&& \\left. + \\left\\{\ni\\frac{n}{R} 2g_D v W_\\mu^{+(n)} \\phi^{-(n)\\mu} \n+(\\partial^\\mu W_y^{+(n)}) \\left(\\frac{n}{R} W_\\mu^{-(n)} +ig_D v \\phi_\\mu^{-(n)} \\right) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. \n+(\\partial^\\mu \\phi^{+(n)}) \\left(-\\frac{n}{R} \\phi_\\mu^{-(n)} +ig_D v W_\\mu^{-(n)} \\right) + {\\rm h.c.}\n\\right\\} \\right]\n\\end{eqnarray}\nThe mixings between $Z$ and $\\phi^0$, $W^\\pm$ and $\\phi^\\pm$ due to the VEV $v$ \nnecessitates the following diagonalization,\n\\begin{eqnarray}\n&&\\left(\n\\begin{array}{c}\nZ_{1\\mu}^{(n)} \\\\\nZ_{2\\mu}^{(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nZ_{\\mu}^{(n)} \\\\\n\\phi_{\\mu}^{0(n)} \\\\\n\\end{array}\n\\right), \\quad \n\\left(\n\\begin{array}{c}\nZ_{1y}^{(n)} \\\\\nZ_{2y}^{(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nZ_y^{(n)} \\\\\n\\phi^{0(n)} \\\\\n\\end{array}\n\\right), \n\\label{rotationZ}\\\\\n&&\\left(\n\\begin{array}{c}\nW_{1\\mu}^{+(n)} \\\\\nW_{2\\mu}^{+(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nW_{\\mu}^{+(n)} \\\\\ni \\phi_{\\mu}^{+(n)} \\\\\n\\end{array}\n\\right), \\quad \n\\left(\n\\begin{array}{c}\nW_{1y}^{+(n)} \\\\\nW_{2y}^{+(n)} \\\\\n\\end{array}\n\\right)\n= \\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\n\\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nW_y^{+(n)} \\\\\ni \\phi^{+(n)} \\\\\n\\end{array}\n\\right). \\nonumber \n\\label{rotationW}\\\\\n\\end{eqnarray}\nThus, the quadratic terms including the zero mode sector read as\n\\begin{eqnarray}\n{\\cal L}_{{\\rm quadratic}} &=& \\sum_{n=1}^\\infty \\left[ \n\\frac{1}{2} \\left(\\partial_\\mu \\gamma_y^{(n)} + \\frac{n}{R} \\gamma_\\mu^{(n)} \\right)^2 \n+ \\frac{1}{2} \\left(\\partial_\\mu h^{(n)} - \\frac{n}{R} h_\\mu^{(n)} \\right)^2 \\right. \\nonumber \\\\\n&& \\left. \n+ \\frac{1}{2} \\left[ \\partial_\\mu Z_{1y}^{(n)} + \\left(\\frac{n}{R} + M_Z \\right) Z_{1\\mu}^{(n)} \\right]^2 \n+ \\frac{1}{2} \\left[ \\partial_\\mu Z_{2y}^{(n)} + \\left(\\frac{n}{R} - M_Z \\right) Z_{2\\mu}^{(n)} \\right]^2 \\right. \\nonumber \\\\\n&& \\left. + \\left|\\partial_\\mu W_{1y}^{+(n)} + \\left(\\frac{n}{R} + M_W \\right) W_{1\\mu}^{+(n)} \\right|^2 \n+ \\left|\\partial_\\mu W_{2y}^{+(n)} + \\left(\\frac{n}{R} - M_W \\right) W_{2\\mu}^{+(n)} \\right|^2 \\right] \\nonumber \\\\\n&& \n+ \\frac{1}{2}(\\partial_\\mu h^{(0)})^2 + \\frac{1}{2}(\\partial_\\mu \\phi^{0(0)} + M_Z Z_\\mu^{(0)})^2 \n+|\\partial_\\mu (i\\phi^{+(0)}) +M_W W_\\mu^{+(0)}|^2\n\\label{gaugemass}\n\\end{eqnarray}\nwhere $M_Z = 2g_D v =2M_W$ (recall that $\\sin \\theta_W = \\frac{\\sqrt{3}}{2}$). \n\nWe note that the Higgs-like mechanism works between the partner of the pairs $(Z_{1,2\\mu}^{(n)}, Z_{1,2y}^{(n)})$ \nand $(W_{1,2\\mu}^{+(n)}, W_{1,2y}^{+(n)})$. \nAs we will see later, this structure is crucial for the cancellation of UV divergences \nin the calculation of the anomalous moment. \nThe mass spectrum for degenerate pairs of the gauge bosons and would-be N-G bosons \n(in the 't Hooft-Feynman gauge) are summarized as\n\\begin{eqnarray}\n(Z_{1\\mu}^{(n)}, Z_{1y}^{(n)}) &:& \\frac{n}{R} + M_Z, \\\\\n(Z_{2\\mu}^{(n)}, Z_{2y}^{(n)}) &:& \\frac{n}{R} - M_Z, \\\\\n(W_{1\\mu}^{+(n)}, W_{1y}^{+(n)}) &:& \\frac{n}{R} + M_W, \\\\\n(W_{2\\mu}^{+(n)}, W_{2y}^{+(n)}) &:& \\frac{n}{R} - M_W.  \n\\end{eqnarray}\n\nWe thus find that just as in the sector of fermions, the mass eigenstates are obtained \nby the orthogonal transformation due to ${\\cal O}$ in (\\ref{rotation}): \n\\begin{eqnarray}\n\\tilde{A}_\\mu &\\equiv& {\\cal O} A_\\mu {\\cal O}^t \n= \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_\\mu}{\\sqrt{3}} & \\frac{W_\\mu^+ + \\phi_\\mu^+}{2} & \\frac{-W_\\mu^+ + \\phi_\\mu^+}{2} \\\\\n\\frac{W_\\mu^- + \\phi_\\mu^-}{2} & -\\frac{\\gamma_\\mu}{2 \\sqrt{3}} + \\frac{h_\\mu}{2} & \\frac{Z_\\mu -i \\phi_\\mu^0}{2} \\\\\n\\frac{-W_\\mu^- + \\phi_\\mu^-}{2} & \\frac{Z_\\mu +i \\phi_\\mu^0}{2} & -\\frac{\\gamma_\\mu}{2 \\sqrt{3}} - \\frac{h_\\mu}{2} \\\\\n\\end{array}\n\\right) \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) & \n\\frac{1}{2} \\hat{W}_\\mu^{+(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2} \\hat{W}_\\mu^{-(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \n-\\frac{1}{2 \\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) + \\frac{1}{2}h_\\mu^{(n)}S_n \\\\\n-\\frac{1}{2} \\hat{W}_\\mu^{-(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \\frac{1}{2}\\hat{Z}_\\mu^{(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n&& \\left.\n\\begin{array}{c}\n-\\frac{1}{2} \\hat{W}_\\mu^{+(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2}\\hat{Z}_\\mu^{(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{1}{2 \\sqrt{3}} \\left(\\gamma^{(n)}_\\mu C_n + \\frac{\\gamma_\\mu^{(0)}}{\\sqrt{2\\pi R}} \\right) - \\frac{1}{2}h_\\mu^{(n)} S_n \\\\\n\\end{array}\n\\right), \n\\label{gaugefield1} \\\\\n\\tilde{A}_y &\\equiv& {\\cal O} A_y {\\cal O}^t \n= \n\\left(\n\\begin{array}{ccc}\n\\frac{\\gamma_y}{\\sqrt{3}} & \\frac{W_y^+ + \\phi^+}{2} & \\frac{-W_y^+ + \\phi^+}{2} \\\\\n\\frac{W_y^- + \\phi^-}{2} & -\\frac{\\gamma_y}{2 \\sqrt{3}} + \\frac{h}{2} & \\frac{Z_y -i \\phi^0}{2} \\\\\n\\frac{-W_y^- + \\phi^-}{2} & \\frac{Z_y +i \\phi^0}{2} & -\\frac{\\gamma_y}{2 \\sqrt{3}} - \\frac{h}{2} \\\\\n\\end{array}\n\\right) \\nonumber \\\\\n&=& \n\\left(\n\\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} \\gamma^{(n)}_y S_n & \n\\frac{1}{2} \\hat{\\phi}^{+(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\frac{1}{2} \\hat{\\phi}^{-(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \n-\\frac{1}{2 \\sqrt{3}} \\gamma^{(n)}_y S_n + \\frac{1}{2} \\left( h^{(n)} C_n + \\frac{h^{(0)}}{\\sqrt{2\\pi R}} \\right) \\\\\n\\frac{1}{2} \\hat{\\phi}^{-(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} & \\frac{i}{2} \\hat{\\phi}^{0(n)} \\frac{e^{i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n&& \\left.\n\\begin{array}{c}\n\\frac{1}{2} \\hat{\\phi}^{+(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{i}{2}\\hat{\\phi}^{0(n)} \\frac{e^{-i\\frac{n}{R}y}}{\\sqrt{2\\pi R}} \\\\\n-\\frac{1}{2 \\sqrt{3}} \\gamma^{(n)}_y S_n -\\frac{1}{2} \\left( h^{(n)} C_n + \\frac{h^{(0)}}{\\sqrt{2\\pi R}} \\right) \\\\\n\\end{array}\n\\right), \n\\end{eqnarray}\nwhere $C_n \\equiv \\frac{\\cos \\left(\\frac{n}{R}y \\right)}{\\sqrt{\\pi R}}, S_n \\equiv \\frac{\\sin \\left(\\frac{n}{R}y \\right)}{\\sqrt{\\pi R}}$, \nand the mode sum is for $n \\ge 1$ in the case of $C_n$ and $S_n$, while the sum is for all integer \nin the case of $e^{\\pm i \\frac{n}{R} y}\/\\sqrt{2\\pi R}$. \nWe have used the following notation, combining $Z_{1\\mu,y}^{(n)}, W_{1\\mu,y}^{+(n)}$ with $Z_{2\\mu,y}^{(n)}, W_{2\\mu,y}^{+(n)}$, \nrespectively: \n\\begin{eqnarray}\n&& \\hat{Z}_\\mu^{(n)} = Z_{1\\mu}^{(n)}, \\hat{Z}_\\mu^{(-n)} = Z_{2\\mu}^{(n)}~(n \\ge 1), \\hat{Z}_\\mu^{(0)} = Z_\\mu^{(0)} \\\\\n&& \\hat{\\phi}^{0(n)} = -Z_{1y}^{(n)}, \\hat{\\phi}^{0(-n)} = Z_{2y}^{(n)}~(n \\ge 1), \\hat{\\phi}^{0(0)} = \\phi^{0(0)} \\\\\n&& \\hat{W}_\\mu^{+(n)} = W_{1\\mu}^{+(n)}, \\hat{W}_\\mu^{+(-n)} = W_{2\\mu}^{+(n)}~(n \\ge 1), \\hat{W}_\\mu^{+(0)} = W_\\mu^{+(0)} \\\\\n&& \\hat{\\phi}^{+(n)} = -i W_{1y}^{+(n)}, \\hat{\\phi}^{+(-n)} = i W_{2y}^{+(n)}~(n \\ge 1), \\hat{\\phi}^{+(0)} = \\phi^{+(0)},\n\\end{eqnarray}\nso that the mass squared of $(\\hat{Z}_\\mu^{(n)}, \\hat{\\phi}^{0(n)})$ and $(\\hat{W}_\\mu^{+(n)}, \\hat{\\phi}^{+(n)})$ \nare $\\left(\\frac{n}{R} + M_Z \\right)^2$ and $\\left(\\frac{n}{R} + M_W \\right)^2$, respectively. \n\n\\section{Interaction vertices}\nHaving obtained the mode functions of the fermion $\\hat{\\Psi}$ and the ``gauge-Higgs\" sector $\\tilde{A}_\\mu, \\tilde{A}_y$, \nwe are ready to derive $D$-dimensional gauge and Yukawa interaction vertices of $\\hat{\\Psi}$ \nby the overlap integral of the relevant mode functions with respect to $y$, \nin the relevant part of the Lagrangian, \n\\begin{eqnarray}\ng \\bar{\\hat{\\Psi}} [\\tilde{A}_\\mu \\gamma^\\mu - \\tilde{A}_y]\\hat{\\Psi}. \n\\end{eqnarray}\n\nLet us start with the interaction vertex of $\\gamma_\\mu$, with $d$ quark, \nwhich is obtained by an integral\n\\begin{eqnarray}\n&&-\\frac{g}{2\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_\\mu(x,y) \n\\left[\n\\bar{\\hat{\\psi}}_2(x,y) \\gamma^\\mu \\hat{\\psi}_2(x,y) + \\bar{\\hat{\\psi}}_3(x,y) \\gamma^\\mu \\hat{\\psi}_3(x,y) \n\\right] \\nonumber \\\\\n&=& -\\frac{g}{\\sqrt{3}} \\int_{-\\pi R}^{\\pi R}dy \\gamma_\\mu(x,y) \\bar{\\hat{\\psi}}_2(x,y) \\gamma^\\mu \\hat{\\psi}_2(x,y) \n\\label{Intvtx}\n\\end{eqnarray}\nwhere $\\gamma_\\mu(x,y)$ denotes the photon field and \nthe properties $\\hat{\\psi}_3(x,y) = -\\gamma^{D+1} \\hat{\\psi}_2(x,-y)$ \nand $\\gamma_\\mu(x,-y) = \\gamma_\\mu(x,y)$ are used. \nSubstituting (\\ref{modeexpsi2}) and the $\\gamma_\\mu$ part of (\\ref{gaugefield1}) in (\\ref{Intvtx}), \nwe get the $D$-dimensional vertex function necessary for the calculation of $g-2$ of the zero-mode $d^{(0)}$, \n\\begin{eqnarray}\n-\\frac{g}{2\\sqrt{3}} \\gamma_\\mu^{(m)} \\bar{d}^{(n)} \\gamma^\\mu V_{\\gamma_\\mu}^{nm} d^{(0)},\n\\label{3.3}\n\\end{eqnarray}\nwhere the vertex function $V_{\\gamma_\\mu}^{nm}$ is defined as\n\\begin{eqnarray}\nV_{\\gamma_\\mu}^{nm} &=& I_c^{nm} (M, M_W,; M, M_W)L + I_c^{nm} (-M, M_W,; -M, M_W)R \\nonumber \\\\\n&=& (L + (-1)^{n+m}R) I_c^{nm}(M, M_W; M, M_W)~(m \\ge 1), \\label{3.4} \\\\\nV_{\\gamma_\\mu}^{n0} &=& (L + (-1)^{n}R) I^{n0}(M, M_W; M, M_W)\n\\end{eqnarray}\nin terms of functions\n\\begin{eqnarray}\n&&I_c^{nm}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \n\\cos \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y)~(m \\ge 1), \n\\label{3.6}\\\\\n&&I^{n0}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{2\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \nF_{M_1, M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y). \n\\label{3.7}\n\\end{eqnarray}\nHere we have used the fact that the vertices for the left-handed and right-handed $d$ quark are related \nwith the factor $(-1)^{m+n}$, as is shown in the Appendix A. \nFor the case of $M_1=M_3=M$ and $M_2=M_4=M_W$, the explicit forms of these functions are\n\\begin{eqnarray}\n&&I_c^{nm}(M, M_W; M, M_W) \n= \\frac{1}{\\sqrt{\\pi R}} C^{(n)}(\\varphi_n, \\alpha_n)^* C^{(0)}(\\varphi_0, \\alpha_0) \n\\left\\{\n\\sqrt{\\cos(\\varphi_n-\\alpha_n)} \\sqrt{\\cos(\\varphi_0 -\\alpha_0)} \\right. \\nonumber \\\\\n&& \\left. \\left[\n-\\frac{\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\alpha_n + \\varphi_0 + \\alpha_0) - \\sin(\\alpha_n + \\alpha_0)\n\\right) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left.  \n-\\frac{\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\alpha_n - \\varphi_0 - \\alpha_0) - \\sin(\\alpha_n - \\alpha_0)\n\\right)\n\\right] \\right. \\nonumber \\\\\n&& \\left. + \\varepsilon(n) \n\\sqrt{\\cos(\\varphi_n+\\alpha_n)} \\sqrt{\\cos(\\varphi_0 + \\alpha_0)} \\right. \\nonumber \\\\\n&& \\left. \\left[\n\\frac{\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} + \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n + \\varphi_0) \n\\right) \\right. \\right. \\nonumber \\\\\n&& \\left.  \\left. \n-\\frac{\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2}}{(\\frac{m}{R})^2-(\\sqrt{m_n^2-M^2} - \\sqrt{m_0^2-M^2})^2} \n\\left(\n(-1)^m \\sin(\\varphi_n - \\varphi_0) \n\\right)\n\\right]\n\\right\\}, \\label{Icnm}\\\\\n&& I^{n0}(M, M_W; M, M_W) = \\delta_{n0} \\frac{1}{\\sqrt{2\\pi R}}. \n\\label{In0} \n\\end{eqnarray}\nNote that (\\ref{In0}) is nothing but the orthonormality of the mode functions. \nWe therefore find that the $\\gamma_\\mu^{(0)}$ coupling is of ordinary form: \n\\begin{eqnarray}\n-\\frac{g}{2\\sqrt{3}} \\frac{1}{\\sqrt{2\\pi R}} \\gamma_\\mu^{(0)} \\bar{d}^{(0)} \\gamma^\\mu d^{(0)} \n= -\\frac{e_D}{3} \\gamma_\\mu^{(0)} \\bar{d}^{(0)} \\gamma^\\mu d^{(0)}, \n\\end{eqnarray}\nwhere $\\frac{g}{\\sqrt{2\\pi R}} =g_D, g_D \\sin \\theta_W = \\frac{\\sqrt{3}}{2} g_D = e_D$ are used. \n\nThe interaction vertex of $\\gamma_y$ with $d$ quark is derived by a similar step \nas in the case of $\\gamma_\\mu$ vertex: \n\\begin{eqnarray}\n&&\\frac{g}{2\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_y(x,y) \n\\left[\n\\bar{\\hat{\\psi}}_2(x,y) \\hat{\\psi}_2(x,y) + \\bar{\\hat{\\psi}}_3(x,y) \\hat{\\psi}_3(x,y)\n\\right] \\nonumber \\\\\n&=& \\frac{g}{\\sqrt{3}} \\int_{-\\pi R}^{\\pi R} dy \\gamma_y(x,y) \\bar{\\hat{\\psi}}_2(x,y) \\hat{\\psi}_2(x,y) \n\\to \\frac{g}{2\\sqrt{3}} \\gamma^{(m)}_y \\bar{d}^{(n)} V_{\\gamma_y}^{nm} d^{(0)}~(m \\ge 1), \n\\end{eqnarray}\nwhere the vertex function $V_{\\gamma_y}^{nm}$ is defined as \n\\begin{eqnarray}\nV_{\\gamma_y}^{nm} = (L + (-1)^{n+m}R) I_s^{nm}(-M, M_W; M, M_W)\n\\end{eqnarray}\nin terms of a function\n\\begin{eqnarray}\nI_s^{nm}(M_1, M_2; M_3, M_4) \\equiv \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \n\\sin \\left(\\frac{m}{R}y \\right) F_{M_1,M_2}^{(n)}(y)^* F_{M_3, M_4}^{(0)}(y)~(m \\ge 1). \n\\label{3.13}\n\\end{eqnarray}\n\nAs the matter of fact, \n$I_s^{nm}$ is not an independent function, and is related to $I_c^{nm}$. \nThis is because in the non-zero KK mode sector $\\gamma_y^{(m)}$ behaves as a would-be N-G boson \nto be ``eaten\" by $\\gamma_\\mu^{(m)}$; Higgs-like mechanism is operative and \nthe coupling of $\\gamma_y^{(m)}$ should be equivalent to that of the longitudinal component \nof $\\gamma_\\mu^{(m)}$ (``equivalence theorem\"). \nOr such relation may be attributed to the quantum mechanical supersymmetry, as is shown in the Appendix B. \nAnyway, we get a relation between $I_s^{nm}$ and $I_c^{nm}$, which means\n\\begin{eqnarray}\nV_{\\gamma_y}^{nm} = i \\frac{m_n-(-1)^{m+n}m_d}{\\frac{m}{R}} V_{\\gamma_\\mu}^{nm}~(m \\ge 1). \n\\end{eqnarray}\n\nThe interaction terms of $h_\\mu$ and $h$ are derived in a similar way, \nand we just give the result: \n\\begin{eqnarray}\n&&\\frac{g}{2} h_\\mu^{(n)} \\bar{d}^{(n)} \\gamma^\\mu V_{h_\\mu}^{nm} d^{(0)}, \\\\\n&&V_{h_\\mu}^{nm} = (L + (-1)^{m+n}R) I_s(M, M_W; M, M_W)~(m \\ge 1), \\\\\n&&-\\frac{g}{2} h^{(n)} \\bar{d}^{(n)} V_h^{nm} d^{(0)}, \\\\\n&&V_h^{nm} = \\left\\{\n\\begin{array}{c}\n(L + (-1)^{m+n} R) I_c^{nm}(-M, M_W; M, M_W)~(m \\ge 1) \\\\\n(L + (-1)^{n} R) I^{n0}(-M, M_W; M, M_W)~(m=0). \\\\\n\\end{array}\n\\right.\n\\end{eqnarray}\nAgain a relation holds:\n\\begin{eqnarray}\nV_h^{nm} = -i \\frac{m_n - (-1)^{m+n}m_d}{\\frac{m}{R}} V_{h_\\mu}^{nm}~(m \\ge 1). \n\\end{eqnarray}\n\n\nIn the case of interaction terms of $\\hat{Z}_\\mu$ and $\\hat{\\phi}^0$, \nthere appears a transition between $\\hat{\\psi}_2$ and $\\hat{\\psi}_3$ and \nthe vertex functions are described by new types of functions, $\\tilde{I}_c$ and $\\tilde{I}_s$.\nNamely, the interaction term of $\\hat{Z}_\\mu$ is (for an arbitrary integer $m$)\n\\begin{eqnarray}\n&&-\\frac{g}{2} \\hat{Z}_\\mu^{(m)} \\bar{d}^{(n)} \\gamma^\\mu V_{Z_\\mu}^{nm} d^{(0)}, \\\\\n&&V_{Z_\\mu}^{nm} = (L - (-1)^{m+n}R)\\frac{1}{\\sqrt{2}} \n\\left( \\tilde{I}_c^{nm}(M, M_W; M, M_W) + i \\tilde{I}_s^{nm}(M, M_W; M, M_W) \\right), \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\tilde{I}_c^{nm}(M_1, M_2; M_3, M_4) &\\equiv& \\frac{1}{\\sqrt{\\pi R}} \n\\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(-y)^* F_{M_3, M_4}^{(0)}(y), \\\\\n\\tilde{I}_s^{nm}(M_1, M_2; M_3, M_4) &\\equiv& \\frac{1}{\\sqrt{\\pi R}} \n\\int_{-\\pi R}^{\\pi R} dy \\sin \\left(\\frac{m}{R}y \\right) F_{M_1, M_2}^{(n)}(-y)^* F_{M_3, M_4}^{(0)}(y). \n\\end{eqnarray}\nThe interaction term of $\\hat{\\phi}^0$ reads as\n\\begin{eqnarray}\n&&-\\frac{g}{2} \\hat{\\phi}^{0(m)} \\bar{d}^{(n)} V_{\\phi^0}^{nm} d^{(0)}, \\nonumber \\\\\n&&V_{\\phi^0}^{nm} = (L - (-1)^{m+n}R)\\frac{i}{\\sqrt{2}} \n\\left( \\tilde{I}_c^{nm}(-M, M_W; M, M_W) + i \\tilde{I}_s^{nm}(-M, M_W; M, M_W) \\right). \\nonumber \\\\\n\\end{eqnarray}\nSome relations hold between $\\tilde{I}_s^{nm}$ and $\\tilde{I}_c^{nm}$ which mean, \n\\begin{eqnarray}\nV_{\\phi^0}^{nm} = i\\frac{m_n + (-1)^{m+n}m_d}{\\frac{m}{R} + M_Z}V_{Z_\\mu}^{nm}. \n\\end{eqnarray}\n\nFinally, in the case of interaction terms of $\\hat{W}_\\mu^+$ and $\\hat{\\phi}^+$ \nthe transition between $\\hat{\\psi}_1$ and $\\hat{\\psi}_2$ or $\\hat{\\psi}_3$ appears, \nwhich makes the vertex function a little complicated: \nfor $n \\ge 0$ and an arbitrary integer $m$, \n\\begin{eqnarray}\n&&\\frac{g}{2} \\hat{W}_\\mu^{+(m)} \\bar{u}^{(n)} \\gamma^\\mu V_{W_\\mu^+}^{nm} d^{(0)}, \\\\\n&&V_{W_\\mu^+}^{nm} = \n\\left\\{\n\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\left( I_c^{nm}(M, 0; M, M_W) +i I_s^{nm}(M, 0; M, M_W) \\right) (L + (-1)^{n+m}R) \\\\\n+ \\frac{1}{\\sqrt{2}} \\left( \\tilde{I}_c^{nm}(M, 0; M, M_W) +i \\tilde{I}_s^{nm}(M, 0; M, M_W) \\right) (L - (-1)^{n+m}R)~(n \\ge 1), \\\\\n\\left[I_c^{0m}(M, 0; M, M_W) +i  I_s^{0m}(M, 0; M, M_W) \\right] L~(n=0), \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n\\\\\n&&-\\frac{g}{2} \\hat{\\phi}^{+(m)} \\bar{u}^{(n)} V_{\\phi^+}^{nm} d^{(0)}, \\\\\n&&V_{\\phi^+}^{nm} = \n\\left\\{\n\\begin{array}{c}\n\\frac{1}{\\sqrt{2}} \\left( I_c^{nm}(-M, 0; M, M_W) +i I_s^{nm}(-M, 0; M, M_W) \\right) (L + (-1)^{n+m}R) \\\\\n-\\frac{1}{\\sqrt{2}} \\left( \\tilde{I}_c^{nm}(-M, 0; M, M_W) +i \\tilde{I}_s^{nm}(-M, 0; M, M_W) \\right) (L - (-1)^{n+m}R)~(n \\ge 1), \\\\\n\\left[I_c^{0m}(-M, 0; M, M_W) +i  I_s^{0m}(-M, 0; M, M_W) \\right] (-1)^m R~(n=0). \\\\\n\\end{array}\n\\right. \\nonumber \\\\\n\\end{eqnarray}\nWriting $V_{W_\\mu^+}^{nm}$ as\n\\begin{eqnarray}\nV_{W_\\mu^+}^{nm} = \\alpha^{nm} L + \\beta^{nm} R, \n\\end{eqnarray}\n$V_{\\phi^+}^{nm}$ is written as (noting $\\tilde{m}_0=0$), \n\\begin{eqnarray}\nV_{\\phi^+}^{nm} = -\\frac{\\tilde{m}_n \\alpha^{nm} - m_d \\beta^{nm}}{\\frac{m}{R} +M_W} L \n-\\frac{\\tilde{m}_n \\beta^{nm} - m_d \\alpha^{nm}}{\\frac{m}{R} +M_W} R. \n\\end{eqnarray}\n\nThe Feynman rules for the $D$-dimensional gauge and Yukawa interactions of $\\tilde{A}_\\mu, \\tilde{A}_y$ \nand $\\hat{\\psi}$ are readily read off from the results obtained above. \nFor instance, the Feynman rule for the $\\gamma_\\mu^{(n)}~(n \\ge 1)$ vertex is given by Fig. \\ref{photongaugeint}. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=8cm]{fig01.ps}\n \\end{center}\n\\caption{\nThe KK photon vertex between the down quark and KK fermion. \n} \n\\label{photongaugeint}\n\\end{figure}\n\nWe also need the Feynman rule for the three point self-interactions of the photon $\\gamma_\\mu^{(0)}$ \nwith charged gauge and Higgs bosons $\\hat{W}_\\mu^{\\pm(n)}, \\hat{\\phi}^{\\pm(n)}$ \nfor the calculation of $g-2$. \nWe skip all the detail of the derivation and just display the result: \nthe relevant terms of $D$-dimensional lagrangian are\n\\begin{eqnarray}\n&&i e_D \\left\\{\n(\\partial_\\mu \\gamma_\\nu^{(0)} - \\partial_\\nu \\gamma^{(0)}_\\mu ) \\hat{W}^{+(n)\\mu} \\hat{W}^{-(n)\\nu} \n- (\\partial_\\mu \\hat{W}_\\nu^{+(n)} - \\partial_\\nu \\hat{W}^{+(n)}_\\mu ) \\gamma^{(0)\\mu} \\hat{W}^{-(n)\\nu} \n\\right. \\nonumber \\\\\n&& \\left. \n+ (\\partial_\\mu \\hat{W}_\\nu^{-(n)} - \\partial_\\nu \\hat{W}^{-(n)}_\\mu ) \\gamma^{(0)\\mu} \\hat{W}^{+(n)\\nu} \n\\right\\} \\nonumber \\\\\n&&+ e_D \\gamma^{(0)\\mu} \n\\left\\{\n\\left( \\frac{n}{R} + M_W \\right) (\\hat{W}_\\mu^{+(n)} \\hat{\\phi}^{-(n)} + \\hat{W}_\\mu^{-(n)} \\hat{\\phi}^{+(n)}) \n+ i [(\\partial_\\mu \\hat{\\phi}^{+(n)}) \\hat{\\phi}^{-(n)} - (\\partial_\\mu \\hat{\\phi}^{-(n)}) \\hat{\\phi}^{+(n)} ]\n\\right\\}, \\nonumber \\\\\n\\end{eqnarray}\nwhere $n$ can be an arbitrary integer. \nA nice thing here is that if we regard $\\hat{W}_\\mu^{+(n)}$ as $W_\\mu^{+(0)}$, \nthe lagrangian is just the same as that in the standard model, \nexcept that now $M_W$ is replaced by $\\frac{n}{R} + M_W$. \nThus to obtain the Feynman rule is straightforward. \nWe get, {\\it e.g.}, \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=13cm]{fig_null.ps}\n \\end{center}\n\\end{figure}\n\n\n\\section{General formulae for the anomalous magnetic moment}\nIn this section, we derive general formulae for $A_\\mu$ and $A_y$-exchange diagrams \ncontributing to the anomalous magnetic moment, \nwhere $A_\\mu, A_y$ denote generic $D$-dimensional gauge and scalar bosons, respectively. \n\n\\subsection{$A_\\mu$-exchange diagram}\n\n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=7cm]{fig02.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA generic gauge interaction vertex of the $D$-dimensional gauge boson $A_\\mu^{(m)}$. \n} \n\\label{gaugeint}\n\\end{figure}\nWe first derive a general formula for the $A_\\mu^{(m)}$-exchange diagram due to the vertex in Fig. \\ref{gaugeint}, \nwith $A_\\mu^{(m)}$ and $\\psi^{(n)}$ being generic mass eigenstates of $A_\\mu$ and fermion ($d$ or $u$), respectively, \nwith masses $M_m$ and $m_n$. \nFor simplicity, hereafter, we indicate $d^{(0)}$ just as $d$.  \n\n\nThe diagram contributing to the anomalous magnetic moment is shown in Fig. \\ref{gaugeex}. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=5cm]{fig03.ps} \n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe contribution of $A_\\mu^{(m)}$-exchange to the anomalous magnetic moment. \n$p, p'$\nare external momenta of $d$ quark. \n} \n\\label{gaugeex}\n\\end{figure}\nThe $\\gamma_\\mu^{(0)}$ coupling at the tree level is modified \ndue to the quantum correction into\n\\begin{eqnarray}\n-\\frac{e_D}{3} \\bar{d}(\\gamma^\\mu + \\Gamma^\\mu) d. \n\\end{eqnarray}\nAmong a few term in $\\bar{d} \\Gamma^\\mu d$, we are interested in the term \nproportional to $p^\\mu + p'^\\mu$ with a form factor $F_2(0)$.\n\\begin{eqnarray}\n\\bar{d} \\Gamma^\\mu d \\to \\bar{d} \\left[-\\frac{1}{2m_d} (p^\\mu + p'^\\mu) F_2(0) \\right] d. \n\\label{gaugeex1}\n\\end{eqnarray}\nIt is the form factor $F_2(0)$ that gives the anomalous magnetic moment: \n$a=\\frac{g-2}{2} = F_2(0)$. \n\nWe obtain the contribution of Fig. \\ref{gaugeex} to the form factor $F_2(0)$ as \n\\begin{eqnarray}\nF_2^{A_\\mu^{(m)}}(0) &=& -4i \\frac{Q(\\psi)}{Q(d)}\n\\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\times \\nonumber \\\\ \n&& \\frac{-\\frac{a^*b+ab^*}{2} [4-DX] m_d m_n \n+ \\frac{|a|^2+|b|^2}{2}(1 - X)[4-(D-2)X] m_d^2}{[l^2 + X(1-X)m_d^2 - X m_n^2 - (1 - X) M_m^2]^3} \n\\label{g-2gauge}\n\\end{eqnarray}\nwhere $Q(\\psi)$ and $Q(d)$ are electric charges of $\\psi^{(n)}$ and $d$ quark, respectively. \n\n\n\n\\subsection{$A_y$-exchange diagram}\n\n\nLet us move to the calculation of $A_y^{(m)}$-exchange diagram due to the vertex shown in Fig. \\ref{Ayint} \nwhere $A_y^{(m)}$ is a generic mass eigenstate of $A_y$ with a mass $M_m$. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=6cm]{fig04.ps} \n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA generic interaction vertex of the $D$-dimensional scalar $A_y^{(m)}$. \n} \n\\label{Ayint}\n\\end{figure}\nThe diagram to evaluate is displayed in Fig. \\ref{Ayex}. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=5cm]{fig05.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\n$A_y^{(m)}$-exchange diagram contributing to the anomalous magnetic moment. \nThe external and internal momenta are defined in a similar way as in the $A_\\mu^{(m)}$-exchange diagram. \n} \n\\label{Ayex}\n\\end{figure}\nSimilarly to the case of the $A_\\mu^{(m)}$-exchange diagram, \nthe one-loop correction is given by\n\\begin{eqnarray}\nF_2^{A_y^{(m)}}(0) = 4i \\frac{Q(\\psi)}{Q(d)}\n\\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2  \n\\frac{\\frac{a'^*b'+a'b'^*}{2} m_d m_n + \\frac{|a'|^2+|b'|^2}{2} (1 - X) m_d^2}{[l^2 +X(1-X) m_d^2 - X m_n^2 - (1 - X) M_m^2]^3}. \n\\label{g-2Ay}\n\\end{eqnarray}\n\\subsection{Diagrams due to the three point self-interaction}\nWe also need a general formula for the diagrams due to the three point self-interactions of \n$\\gamma_\\mu^{(0)}$ with $A_\\mu^{(m)}$ and\/or $A_y^{(m)}$. \nWe first consider the contribution of a diagram (Fig. \\ref{3ptgauge})\n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=5cm]{fig06.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA diagram contributing to the anomalous magnetic moment due to a triple gauge boson vertex. \n} \n\\label{3ptgauge}\n\\end{figure}\ndue to the vertex shown in Fig. \\ref{gaugeint} and Fig. \\ref{3ptvtx}\n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=10cm]{fig07.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA triple gauge boson vertex. \n} \n\\label{3ptvtx}\n\\end{figure}\nwhere the factor $e_D$ is assigned assuming $A_\\mu^{(m)}$ is $\\hat{W}_\\mu^{-(m)}$, \nas is really the case. \nThe contribution of Fig. \\ref{3ptgauge} to $F_2(0)$ is given as\n\\begin{eqnarray}\nF_2(0)^{A_\\mu^{(m)} A_\\nu^{(m)}} &=& \n12i \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\frac{-\\frac{a^*b + ab^*}{2}[4-D+(D-1)X]m_d m_n \n+ \\frac{|a|^2+|b|^2}{2} X [5-D+(D-2)X] m_d^2}{[l^2 + X(1-X)m_d^2 -X M_m^2 -(1-X)m_n^2]^3}. \n\\nonumber \\\\\n\\end{eqnarray}\n\nNext we consider the contribution of combined two diagrams shown in Fig. \\ref{3ptmix}\n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=10cm]{fig08.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nDiagrams contributing to the anomalous magnetic moment \ndue to the $\\gamma_\\mu$-$A^\\mu$-$A_y$ coupling. \n} \n\\label{3ptmix}\n\\end{figure}\ndue to the vertices in Fig. \\ref{gaugeint}, Fig. \\ref{Ayint} and additional one, Fig. \\ref{mixed}. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=6cm]{fig09.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe 3-point vertex relevant for Fig. \\ref{3ptmix}.\n} \n\\label{mixed}\n\\end{figure}\nThe contribution to $F_2(0)$ from Fig. \\ref{3ptmix} is\n\\begin{eqnarray}\nF_2^{A_\\mu^{(m)}A_y^{(m)}}(0) = -12i M_m \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\frac{\\frac{1}{4}(b'^* a + a'^* b + b' a^* + a'b^*)X m_d}{[l^2 + X(1-X) m_d^2 - X M_m^2 - (1-X) m_n^2]^3}. \n\\nonumber \\\\\n\\end{eqnarray}\n\nFinally, we consider the contribution of Fig. \\ref{gyy}\n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=5.5cm]{fig10.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nA diagram contributing to the anomalous magnetic moment \ndue to the $\\gamma_\\mu$-$A_y$-$A_y$ coupling. \n} \n\\label{gyy}\n\\end{figure}\ndue to another additional vertex of Fig. \\ref{3ptgyy}. \n\\begin{figure}[ht]\n \\begin{center}\n  \\includegraphics[width=8cm]{fig11.ps}\n \\end{center}\n \\vspace*{-0.5cm}\n\\caption{\nThe 3-point vertex relevant for Fig. \\ref{gyy}. \n} \n\\label{3ptgyy}\n\\end{figure}\nThe contribution to $F_2(0)$ is\n\\begin{eqnarray}\nF_2^{A_y^{(m)}A_y^{(m)}}(0) &=& -12i \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\frac{\n\\frac{1}{2}(a'^* b' + a'b'^{*})(1-X) m_d m_n \n+ \\frac{1}{2}(|a'|^2 + |b'|^2) X(1-X) m_d^2}{[l^2 + X(1-X) m_d^2 - X M_m^2 - (1-X) m_n^2]^3}. \n\\end{eqnarray}\n\nHaving the general formulae, \nthe contribution of each type of $A_\\mu^{(m)}$ or $A_y^{(m)}$-exchange diagram is readily written down \nby use of the vertex functions derived in the previous section. \nFor instance, the contribution of $\\gamma_\\mu^{(m)}$-exchange diagram $(m \\ge 1)$ \nis obtained by setting (see (\\ref{3.3}), (\\ref{3.4}))\n\\begin{eqnarray}\na = (-1)^{m+n} b = -\\frac{g}{2\\sqrt{3}} I_c^{nm}(M, M_W; M, M_W)\n\\end{eqnarray}\nin (\\ref{g-2gauge}): \n\\begin{eqnarray}\nF_2^{\\gamma_\\mu^{(m \\ne 0)}}(0) &=& -\\frac{g^2}{3} i \\sum_{m=1}^\\infty \\sum_{n=-\\infty}^\\infty \n\\left[ I_c^{nm}(M, M_W; M, M_W) \\right]^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times  \\nonumber \\\\\n&& \\frac{-(-1)^{n+m} [4-DX] m_d m_n +(1-X) [4-(D-2)X] m_d^2}{[l^2 + X(1-X) m_d^2 -X m_n^2 -(1-X)(\\frac{m}{R})^2]^3}. \n\\end{eqnarray}\nTo get the other contributions is also straightforward and we do not write them down explicitly here. \nEven though in this way we can get the exact formulae for the contributions to the magnetic moment, \nto get the final analytic results is a hard task, since the vertex functions like $I_c(M, M_W; M, M_W)$ \nare rather complicated and $m_n$ to describe the function cannot be given analytically, \nunless some approximation is applied or some extreme cases are considered. \n\n\n\\section{The cancellation of divergences}\n\nWe are now ready to focus on the main issue of this paper, \n{\\em i.e.} the cancellation mechanism of UV divergence in the contribution to $g-2$. \nAs discussed in the introduction, \nthe anomalous magnetic moment in the present model is expected to be finite \nrelying on the operator analysis, similarly to the case of toy model, \n$(D+1)$ dimensional QED \\cite{ALM}. \nIn Ref. \\cite{ALM}, the cancellation seems to take place \nbetween photon-exchange and Higgs-exchange diagrams. \nIn the present model, however, these contributions behave differently. \nFor instance, the Yukawa coupling of Higgs to $d$ quark is suppressed compared to the gauge coupling. \nWe will find that in the present realistic model, \nthe cancellation takes place between the contributions of the partners $(A_\\mu^{(m)}, A_y^{(m)})$ \nhaving the same quantum number, such as $(\\hat{W}_\\mu^{+(m)}, \\hat{\\phi}^{+(m)})$, \namong which the Higgs(-like) mechanism is at work. \nIt is interesting to realize that they also form ``super-partners\" of the quantum mechanical SUSY \npresent in the higher dimensional gauge theories \\cite{LNSS}.\n\n\nAs has been discussed in the end of previous section, \nthe exact formulae for the contributions of each diagram are not easy to handle. \nSince the cancellation should occur for the arbitrary bulk mass $M$, \nbeing supported by the operator analysis, \nand our main purpose is to confirm the cancellation mechanism, \nwe simplify the analysis by considering the case of small $M$ in this section. \nWe will retain only the term up to ${\\cal O}(M)$ in the contribution of each diagram. \nAs the matter of fact, \nit turns out that the term linear in $M$ in each contribution vanishes. \nThis comes from the fact that the functions $I_c^{nm}$, {\\em etc.}, \nappearing in the fermion vertices satisfy a relation, as is shown in Appendix A: \n\\begin{eqnarray}\nI_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W). \n\\end{eqnarray}\nSince such functions appear twice in each diagram, \nwe conclude the contribution of each diagram is an even function of $M$. \nHence the term linear in $M$ actually disappears. \nWe thus take the limit of $M \\to 0$. \n\nIn the limit $M \\to 0$, $m_n$ reduces to $\\frac{n}{R}+M_W$ and the functions $I_c^{nm}$ etc. \nare greatly simplified, reflecting the (partial) recovery of momentum conservation \nalong the extra dimension: \n\\begin{eqnarray}\n&&I_c^{nm}(0, M_2; 0, M_4) = \\frac{1}{2\\sqrt{\\pi R}} (\\delta_{n,m} + \\delta_{-n,m})~(m \\ne 0), \\\\\n&&I_c^{n0}(0, M_2; 0, M_4) = \\frac{1}{\\sqrt{2\\pi R}} \\delta_{n,0}, \\\\\n&&I_s^{nm}(0, M_2; 0, M_4) = \\frac{i}{2\\sqrt{\\pi R}} (\\delta_{n,m} - \\delta_{-n,m})~(m \\ne 0), \\\\\n&&\\tilde{I}_c^{nm}(0, M_2; 0, M_4) = \\frac{1}{2\\sqrt{\\pi R}} (\\delta_{n,m} + \\delta_{-n,m})~(m \\ne 0), \\\\\n&&\\tilde{I}_s^{nm}(0, M_2; 0, M_4) = -\\frac{i}{2\\sqrt{\\pi R}} (\\delta_{n,m} - \\delta_{-n,m})~(m \\ne 0).\n\\end{eqnarray}\n  %\n\nThen the vertex functions such as $V_{\\gamma_\\mu}^{nm}$ in this limit \nalso take simple forms. \nFor instance, $V_{\\gamma_\\mu}^{nm}=\\frac{1}{2\\sqrt{\\pi R}}(\\delta_{n,m} + \\delta_{-n,m})~(m \\ge 1)$. \nHaving the Feynman rules for the interaction vertices of fermions, \nwe readily get the contribution to $g-2$, {\\em i.e.} $F_2(0)$, \nfrom each one-loop diagram, by use of the general formulae derived in section 4. \nWe display the results below, dividing them into the sectors of \ncharged current, neutral current, etc. \n\n\\subsection{The charged current sector}\n\nWe first display the contribution of $\\hat{W}_\\mu^+$-exchange \ndiagram (obtained with $A_\\mu^{(m)}$ being replaced by $\\hat{W}_\\mu^{+(\\pm n)}$ in Fig. \\ref{gaugeex}),\nwhich we denote by $F_2^{(W)}(0)$: \n\\begin{eqnarray}\nF_2^{(W)}(0) &=& -4i (-2) \\left( \\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1dX X \\nonumber \\\\\n&& \\times \\sum_{n=-\\infty}^\\infty \n\\frac{(4-DX)\\frac{n}{R}M_W +(1-X) [4-(D-2)X]M_W^2}{[l^2-(\\frac{n}{R}+(1-X)M_W)^2]^3}\n\\label{F2W}\n\\end{eqnarray}\nwhere the factor $(-2)$ comes from $Q(u)\/Q(d)$ and \nthe facts $m_d = M_W$, $\\tilde{m}_n =\\frac{n}{R}$, $M_{\\hat{W}_\\mu^{+(n)}} = \\frac{n}{R} + M_W$ \n(in the limit $M \\to 0$) have been used. \nA wisdom to treat the UV divergence is to invoke Poisson resummation \nand extract the ``zero-winding\" sector. \nHere, however, we just take the limit of ``de-compactification\", $R\\to \\infty$, \nas the zero-winding sector is easily known to correspond to the limit. \nIn this limit $\\frac{n}{R}$ may be replaced by the continuous extra space momentum $p_y$. \nThus, taking the limit $R \\to \\infty$, replacing $\\frac{n}{R}$ by $p_y$ \nand performing a shift of the momentum, $p_y +(1-X)M_W \\to p_y$, \nwe get the divergent part $F_2(0)_{{\\rm div.}}$ of the contribution, \n\\begin{eqnarray}\nF_2^{(W)}(0)_{{\\rm div.}} = i \\frac{2\\pi}{3} R g_D^2 \\int \\frac{d^Dl d p_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{[l^2-p_y^2]^3}. \n\\label{Wexdiv}\n\\end{eqnarray}\n\nWe next display the contribution of the exchange of the partner of $W_\\mu^+$, \n{\\em i.e.} $\\hat{\\phi}^+$, whose diagram is obtained with $A_y^{(m)}$ being replaced by $\\hat{\\phi}^{+(\\pm n)}$ in Fig. \\ref{Ayex}. \n\\begin{eqnarray}\nF_2^{(\\phi^+)}(0) = 4i (-2) \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \n\\sum_{n=-\\infty}^\\infty \\frac{-X \\frac{n}{R} M_W + X(1-X)M_W^2}{[l^2-(\\frac{n}{R} + (1-X) M_W)^2]^3}. \n\\label{F2phi}\n\\end{eqnarray}\nTaking the same step as we took above, \nwe obtain the divergent part of the contribution, \n\\begin{eqnarray}\nF_2^{(\\phi^+)}(0)_{{\\rm div.}} =-i \\frac{2\\pi}{3} Rg_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\frac{M_W^2}{[l^2-p_y^2]^3}. \n\\label{phiexdiv}\n\\end{eqnarray}\nWe realize , as we anticipated, the divergence exactly cancel out \nbetween the contributions of ``partners\" $(\\hat{W}_\\nu^{+(n)}, \\hat{\\phi}^{+(n)})$ \nshown in (\\ref{Wexdiv}) and (\\ref{phiexdiv}), \nthough the original forms of (\\ref{F2W}) and (\\ref{F2phi}) are quite different. \nLet us note that the Higgs(-like) mechanism is operative \nboth in the non-zero KK mode sector $(n \\ne 0)$ and the zero-mode sector $(n=0)$ \nfor the charged gauge-Higgs bosons, being triggered by the KK mass $\\frac{n}{R}$ for $n \\ne 0$ \nand by the VEV of $A_y$ for $n=0$ ({\\em i.e.} the ordinary Higgs mechanism).  \n\nIn addition to these diagrams, \nwe have to evaluate the contributions due to the three point self-interaction vertices, \nwhose diagrams are obtained with $A_\\mu^{(m)}$ and $A_y^{(m)}$ being replaced \nby $\\hat{W}_\\mu^{+(\\pm n)}$ and $\\hat{\\phi}^{+(\\pm n)}$ \nin Figs. \\ref{3ptgauge}, \\ref{3ptmix}, and \\ref{gyy}. \n  %\n  %\n  %\n  %\nThe contributions of each diagram are\n\\begin{eqnarray}\nF_2^{(WW)}(0) &=& 3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \n\\frac{-[D-4-(D-1)X] \\frac{n}{R} M_W + X[5-D+(D-2)X]M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}, \\\\\nF_2^{(W \\phi^+)}(0) &=& 3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\sum_{n=-\\infty}^\\infty \n\\frac{X \\frac{n}{R} M_W + X M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}, \\\\\nF_2^{(\\phi^+ \\phi^+)}(0) &=& -3ig_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\sum_{n=-\\infty}^\\infty \n\\frac{-(1-X) \\frac{n}{R} M_W + X (1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3}. \n\\end{eqnarray}\nIt is straightforward to show that \nthe divergent part just cancel out; \n\\begin{eqnarray}\nF_2^{(WW)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& 6\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \\\\\nF_2^{(W \\phi^+)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& 6\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \\\\\nF_2^{(\\phi^+ \\phi^+)}(0) &\\stackrel{R \\to \\infty}{\\longrightarrow}& -12\\pi i R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \n\\int_0^1 dX X \\frac{X(1-X)M_W^2}{(l^2-p_y^2)^3}, \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\left( F_2^{(WW)}(0) + F_2^{(W \\phi^+)}(0) + F_2^{\\phi^+ \\phi^+}(0) \\right)_{{\\rm div.}} = 0. \n\\end{eqnarray}\n\n\\subsection{The neutral current sector}\nTaking a similar step to the case of the charged current sector, \nthe contribution due to $\\hat{Z}_\\mu$-exchange diagram \nis known to be given as\n\\begin{eqnarray}\nF_2^{(Z)}(0) &=& -4i \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times  \\nonumber \\\\\n&& \n\\sum_{n=-\\infty}^\\infty \\frac{(4-DX)(\\frac{n}{R}+M_W)M_W +(1-X) [4 - (D-2)X] M_W^2}{[l^2-(\\frac{n}{R}+(2-X)M_W)^2]^3},\n\\end{eqnarray}\nwhere $m_n = \\frac{n}{R}+M_W, M_{\\hat{Z}_\\mu^{(n)}} = \\frac{n}{R} + 2M_W$ has been used.  \n\nThe $R\\to \\infty$ limit and the shift $p_y +(2-X) M_W \\to p_y$ gives\n\\begin{eqnarray}\nF_2^{(Z)}(0)_{{\\rm div.}} = -i \\frac{\\pi}{3}R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3}. \n\\end{eqnarray}\n\n\nThe contribution due to $\\hat{\\phi}^0$-exchange diagram \nis given as\n\\begin{eqnarray}\nF_2^{(\\phi^0)}(0) &=& 4i \\left(\\frac{g_D}{2} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X \\times  \\nonumber \\\\\n&& \n\\sum_{n=-\\infty}^\\infty \\frac{-X (\\frac{n}{R}+M_W) M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R}+(2-X)M_W)^2]^3}. \n\\end{eqnarray}\nThe divergent part reads as\n\\begin{eqnarray}\nF_2^{(\\phi^0)}(0)_{{\\rm div.}} = i \\frac{\\pi}{3}R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3}. \n\\end{eqnarray}\nWe thus confirm the cancellation of the divergence,\n\\begin{eqnarray}\nF_2^{(Z)}(0)_{{\\rm div.}} + F_2^{(\\phi^0)}(0)_{{\\rm div.}} = 0.  \n\\end{eqnarray}\n\n\\subsection{The photon sector}\nIn this subsection, we discuss the contributions of photon ($\\gamma_\\mu^{(n)}$)-exchange \nand the exchange of $\\gamma_y^{(n)}$, the partner of $\\gamma_\\mu^{(n)}$ for $n \\ge 1$. \nIn clear contrast to the previous cases, \nthe Higgs mechanism does not exist for the zero mode sector, $n=0$, \nand also $\\gamma_y^{(0)}$ is absent due to the orbifolding. \nThis suggests that the contribution of $n=0$ sector differs from \nwhat we obtain by formally setting $n=0$ in the formula valid for $n \\ne 0$. \nIn fact, the contribution of the $\\gamma_\\mu^{(n)}$-exchange ($n \\ge 0$)\nis given by \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0) &=& -4i \\left(-\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{-(4-DX)(\\frac{n}{R}+M_W)M_W + (1-X)[4-(D-2)X] M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&-4i \\left(-\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dXX^2 \\frac{[(D-2)X-2]M_W^2}{(l^2 - X^2 M_W^2)^3}. \n\\label{ptng-2}\n\\end{eqnarray}\nThe second term is to adjust the discrepancy mentioned above. \nThe divergent part reads as \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0)_{{\\rm div.}} &=& -i \\frac{2\\pi}{27} R e_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& -i \\frac{2}{9} e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2] M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{ptndiv}\n\\end{eqnarray}\nSimilarly, the contribution of the $\\gamma_y^{(n)}$-exchange ($n \\ge 1$) \nis given by \n\\begin{eqnarray}\nF_2^{(\\gamma_y)}(0) &=& 4i \\left(\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{X(\\frac{n}{R}+M_W)M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&-4i \\left(\\frac{e_D}{3\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dXX^2 \\frac{(2-X) M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{Ayg-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(\\gamma_y)}(0)_{{\\rm div.}} &=& i \\frac{2\\pi}{27} R e_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& -i \\frac{2}{9} e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{Aydiv}\n\\end{eqnarray}\nWe thus realize that although the cancellation of divergence is ``almost\" complete, \nthere remains a ``partial\" $D$-dimensional divergence originated from the $n=0$ sector, \n\\begin{eqnarray}\nF_2^{(\\gamma_\\mu)}(0)_{{\\rm div.}} + F_2^{(\\gamma_y)}(0)_{{\\rm div.}} \n= -i \\frac{2}{9} (D-3)e_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^3 \\frac{M_W^2}{(l^2-X^2 M_W^2)^3}. \n\\label{zeromodediv}\n\\end{eqnarray}\n\n\\subsection{The Higgs sector}\n\nFinally, in this subsection we discuss the contribution of the exchange \nof the partner of the Higgs $h_\\mu^{(n)}$, and the contribution of the Higgs ($h^{(n)}$)-exchange. \nThe situation concerning the UV divergence is similar to that in the photon sector, \nand we just summarize the result below. \n\nThe contribution of the $h_\\mu^{(n)}$-exchange ($n \\ge 1$)\n\nis given by \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0) &=& -4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{-(4-DX)(\\frac{n}{R}+M_W)M_W + (1-X)[4-(D-2)X] M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&+4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2]M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{hmug-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0)_{{\\rm div.}} &=& -i \\frac{\\pi}{6} R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& +i \\frac{1}{2} g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{[(D-2)X-2] M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{hmudiv}\n\\end{eqnarray}\n\nThe contribution of the $h^{(n)}$-exchange ($n \\ge 0$) \nis given by \n\\begin{eqnarray}\nF_2^{(h)}(0) &=& 4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X \\times \\nonumber \\\\\n&& \\sum_{n=-\\infty}^\\infty \\frac{X(\\frac{n}{R}+M_W) M_W + X(1-X) M_W^2}{[l^2-(\\frac{n}{R} + X M_W)^2]^3} \n\\nonumber \\\\\n&&+4i \\left(\\frac{g_D}{2\\sqrt{2}} \\right)^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2 - X^2 M_W^2)^3}, \n\\label{hg-2}\n\\end{eqnarray}\nwhose divergent part reads as \n\\begin{eqnarray}\nF_2^{(h)}(0)_{{\\rm div.}} &=& i \\frac{\\pi}{6} R g_D^2 \\int \\frac{d^Dl dp_y}{(2\\pi)^{D+1}} \\frac{M_W^2}{(l^2-p_y^2)^3} \n\\nonumber \\\\ \n&& +i \\frac{1}{2} g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^2 \\frac{(2-X) M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{Aydiv}\n\\end{eqnarray}\nAgain, the cancellation of UV divergence turns out to be not complete, \n\\begin{eqnarray}\nF_2^{(h_\\mu)}(0)_{{\\rm div.}} + F_2^{(h)}(0)_{{\\rm div.}} \n= i \\frac{1}{2} (D-3)g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \\int_0^1 dX X^3 \\frac{M_W^2}{(l^2-X^2M_W^2)^3}. \n\\label{zeromodehdiv}\n\\end{eqnarray}\n\nTo summarize this section, \nwe have found the cancellation mechanism of UV divergence \nbetween the contributions of the pairs of $D$-dimensional vector bosons and scalars \n$(A_\\mu, A_y)$, which form ``partners\" of Higgs(-like) mechanism and at the same time \nthe partners of quantum mechanical SUSY. \nThe cancellation is complete for the charged current and neutral current sectors, \n$(\\hat{W}_\\mu^{+(n)}, \\hat{\\phi}^{+(n)})$ and $(\\hat{Z}_\\mu^{(n)}, \\hat{\\phi}^{0(n)})$, \nwhile it is incomplete for the photon and Higgs sectors, \n$(\\gamma_\\mu^{(n)}, \\gamma_y^{(n)})$ and $(h_\\mu^{(n)}, h^{(n)})$. \nThe remaining ``partial\" and lower dimensional ($D$-dimensional) divergence \n({\\rm i.e.} the sum of Eqs. (\\ref{zeromodediv}) and (\\ref{zeromodehdiv}), by use of $e_D^2 =\\frac{3}{4} g_D^2$) \n\\begin{eqnarray}\nF_2(0)_{{\\rm div.}} = i \\frac{1}{3}(D-3)g_D^2 \\int \\frac{d^Dl}{(2\\pi)^D} \n\\int_0^1 dX X^3 \\frac{M_W^2}{(l^2 -X^2 M_W^2)^3}, \n\\label{remaindiv}\n\\end{eqnarray}\nseems to be attributed to the absence of Higgs mechanism for $n=0$ in the photon and Higgs sectors. \nAs the matter of fact, however, Eq.(\\ref{remaindiv}) is finite, so is the anomalous magnetic moment, \nfor 5 or 6 dimensional ($D=4~{\\rm or}~5$) space-time. \nLet us note the $g-2$ is divergent already at 6 dimensional space-time \nin other types of higher dimensional gauge theories, \n{\\em e.g.} in the scenario of universal extra dimension. \n\n\n\n\\section{Conclusions}\nIn this paper, \nwe have discussed the finiteness of the anomalous magnetic moment of fermion \nin a realistic model of gauge-Higgs unification. \nOur main purpose is to clarify the cancellation mechanism of UV divergences \nin various contributing Feynman diagrams. \nOur expectation that the anomalous moment should be finite and calculable, \nin spite of the fact that higher dimensional gauge theories are argued to be non-renormalizable, \nis based on an operator analysis: \nan operator corresponding to the dimension six gauge invariant operator \ndescribing the anomalous moment in the standard model \nis forbidden because of the higher dimensional gauge symmetry \npresent in the gauge-Higgs unification and the on-shell condition for the fermion, \nas was discussed in the introduction.   \n\nIn our previous paper, \nwe adopted a toy model, i.e. higher dimensional QED compactified on $S^{1}$, \nwhere $D$-dimensional gauge and scalar fields $(A_{\\mu}, A_{y})$ were identified with photon and Higgs fields, respectively, \nand showed by explicit calculation that the anomalous moment is in fact finite \nfor arbitrary space-time dimensions \\cite{ALM}. \nAlthough the result is quite remarkable, \nthis toy model is not realistic and has a few drawbacks: \nthe gauge group is too small to incorporate the $SU(2)$ doublet Higgs \nand the masses of light fermions or their small Yukawa couplings cannot be taken into account. \nIt cannot reproduce the famous result of Schwinger on the magnetic moment \\cite{Schwinger}, \nagain due to the unrealistic Yukawa coupling. \n \nThese unsatisfactory points are overcomed in this paper \nby considering a realistic $D+1$ dimensional ($D = 4,5,$ etc.) $SU(3)$ gauge-Higgs unification model \ncompactified on an orbifold $S^{1}\/Z_{2}$, with matter fermions of $SU(3)$ triplet. \nThe small Yukawa coupling is achieved by introducing $Z_{2}$-odd bulk mass $M$ for the fermions.\n\nAfter deriving various general formulae which are valid for arbitrary $M$ \nand are used to obtain the anomalous moment, \nwe have discussed the cancellation mechanism of the UV divergences \nfor the simplified case of small $M$. \nSince our operator analysis concerning the finiteness strongly \ndepends on the higher dimensional gauge symmetry, \nit will be natural to expect that the cancellation of UV divergence is realized \nbetween the contributions of the pair of $D$-dimensional gauge and scalar bosons \n$(A_{\\mu}, A_{y})$ with the same quantum number. $(A_{\\mu}, A_{y})$, at least for non-zero KK modes, \nplay the roles as the ``partners\" in the Higgs-like mechanism \nto form massive gauge bosons (from $D$-dimensional point of view), \nwhich is the manifestation of the higher dimensional gauge symmetry. \nWe also would like to point out that the pair $(A_{\\mu}, A_{y})$ is known to behave \nas a multiplet of quantum mechanical SUSY \\cite{LNSS}, \nso the cancellation may be regarded as the consequence of the supersymmetry. \nLet us note that in this model photon and Higgs have different quantum numbers and \nthe divergence cancellation does not take place between the contributions of these particles, \nin clear contrast to the case of our previous paper \\cite{ALM}.  \n\nWe have confirmed these expectations by explicit calculation of Feynman diagrams. \nConcerning the contributions of charged and neutral currents \ndue to the pairs $(\\hat{W}_{\\mu}^{\\pm (n)}, \\hat{\\phi}^{\\pm (n)})$ and $(\\hat{Z}_{\\mu}^{(n)}, \\hat{\\phi}^{0(n)})$, \nthe zero-mode sector also has ordinary Higgs mechanism. \nThus the UV divergence has been shown to be completely cancelled \nbetween the contributions of each partners. \nOn the other hand, \nconcerning the remaining contributions of photon and Higgs sector, \n$(\\gamma_{\\mu}^{(n)}, \\gamma_{y}^{(n)})$ and $(h_{\\mu}^{(n)}, h^{(n)})$, \nthe cancellation is not complete and there remains a UV divergence (but of lower degree) \nwhich originates from the zero modes. \nLet us recall that for these sectors the zero modes do not have ordinary Higgs mechanism and \none member of each partners is missing due to the orbifolding. \nSo far, we do not have any good reasoning why the cancellation mechanism does not perfectly work, \nwhile the operator analysis seems to be valid. \nThis issue remains to be settled. \nNevertheless, we would like to point out \nthat we have completely finite anomalous moments for 5 and 6 dimensional space-time, \nalthough the anomalous moment is divergent for the case of 6 dimensions \nin other higher dimensional gauge theories such as ``universal extra dimension\". \n\nIn our previous paper, \nwe could not reproduce the Schwinger's result in ordinary QED, \nsince the contribution of the Higgs-exchange was comparable to that of photon-exchange \nas the Yukawa coupling was of the order of the gauge coupling. \nThis drawback is overcomed in the present model \nby introducing the bulk mass which causes the localization of Weyl fermions \nwith different chiralities at two different fixed points. \nWe will report in the forthcoming paper \\cite{ALM2} \nthat the Schwinger's result is indeed reproduced. \nIn the paper, we will also discuss the constraint on the compactification scale \nby comparing our prediction on the anomalous magnetic moment \nincluding the contributions of non-zero KK modes \nwith the allowed deviation of the data from the prediction of the standard model. \n\n\n\\subsection*{Acknowledgment}\n\nThe work of the authors was supported \nin part by the Grant-in-Aid for Scientific Research \nof the Ministry of Education, Science and Culture, No.18204024 and No. 20025005.  \n\n\\begin{appendix}\n\n\\setcounter{equation}{0}\n\\section{The derivation of the property \\\\$I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$}\nIn this appendix, the relation between the vertex functions \ndue to the fermions with different chiralities is discussed. \nWe take a typical example of $I_c^{nm}(M, M_W; M, M_W)$ and \n$I_c^{nm}(-M, M_W; -M, M_W)$ defined in (\\ref{3.6}) and (\\ref{3.7}) \nto show the property \\\\$I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$. \nSince the interchange of the chirality $L \\leftrightarrow R$ corresponds \nto the sign flip of the bulk mass $M \\to -M$, therefore the exchange of the fixed points \n$y \\leftrightarrow y-\\pi R$, we expect that we can obtain $f_{d_R}$ by shifting $f_{d_L}$ by $\\pi R$ \nwith possible phase change. \nAs the shift does not mix the real and imaginary parts, \nwe expect that such property holds in each part of even and odd functions of $y$. \n\nLet us start with left-handed mode function,\n\\begin{eqnarray}\nf_{d_L}^{(n)}(y) &=& F_{M, M_W}^{(n)}(y) \\nonumber \\\\\n&=& e^{iM_W y} C^{(n)} \n\\left[\n\\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right. \\nonumber \\\\\n&& \\left. -\\varepsilon(n) i \\sqrt{\\cos(\\varphi_{n} +\\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y)\n\\right]  \\nonumber \\\\\n&=&\nC^{(n)} \n\\left[\n\\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right. \\nonumber \\\\\n&& \\left. + \\varepsilon(n) \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y) \n\\right. \\nonumber \\\\\n&& \\left. +i \\left(\n-\\varepsilon(n) \\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y) \n\\right. \\right. \\nonumber \\\\\n&& \\left. \\left. + \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos(\\sqrt{m_{n}^2-M^2}|y| + \\alpha_{n}) \n\\right) \n\\right], \n\\end{eqnarray}\nwhere $\\varepsilon(n)=1$ for $n \\ge 0$, $-1$ for $n < 0$.\n \nConcentrating on the real part of $f_{d_L}^{(n)}\/C^{(n)}$ ($C^{(n)}$ is invariant under $M \\to -M$), \nand making a shift $y \\to y-\\pi R$, we find\n\\begin{eqnarray}\n&&[f_{d_L}^{(n)}(y-\\pi R)\/C^{(n)}]_{{\\rm real}} \\nonumber \\\\\n&=& \n\\cos(M_W (y-\\pi R)) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \n\\cos(\\sqrt{m_{n}^2-M^2}(\\pi R -y) + \\alpha_{n}) \\nonumber \\\\\n&&+ \\varepsilon(n) \\sin(M_W (y-\\pi R)) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}(y-\\pi R)) \\nonumber \\\\\n&=& \\cos(M_W y) \n\\left[\n\\left(\n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos \\varphi_{n} \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. + \\varepsilon(n) \n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin (\\varphi_{n} - \\alpha_{n}) \n\\right) \\cos(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n}) \\right. \\nonumber \\\\\n&& \\left. + \\left(\n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin \\varphi_{n} \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. -\\varepsilon(n) \n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\cos (\\varphi_{n} - \\alpha_{n}) \n\\right) \\sin(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n})\n\\right] \\nonumber \\\\\n&& + \\sin(M_W y) \n\\left[\n\\left(\n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin (\\varphi_{n}+ \\alpha_{n}) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. +\\varepsilon(n) \n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\cos \\varphi_{n}  \n\\right) \\sin(\\sqrt{m_{n}^2-M^2}y) \\right. \\nonumber \\\\\n&& \\left. + \\left(\n\\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\cos (\\varphi_{n} + \\alpha_{n}) \\right. \\right. \\nonumber \\\\\n&& \\left. \\left. -\\varepsilon(n) \n\\cos(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})} \\sin \\varphi_{n}  \n\\right) \\cos(\\sqrt{m_{n\\pm}^2-M^2}y)\n\\right] \\nonumber \\\\\n&=& \n(-1)^n\n\\left[ \n\\cos(M_W y) \\sqrt{\\cos(\\varphi_{n} + \\alpha_{n})}  \n\\cos(\\sqrt{m_{n}^2-M^2}y-\\alpha_{n}) \\right. \\nonumber \\\\\n&& \\left. \n+ \\varepsilon(n) \\sin(M_W y) \\sqrt{\\cos(\\varphi_{n} - \\alpha_{n})} \\sin(\\sqrt{m_{n}^2-M^2}y)\n\\right] \\nonumber \\\\\n&=& (-1)^n [F_{-M, M_W}^{(n)}(y)\/C^{(n)}]_{{\\rm real}} \\nonumber \\\\\n&=& (-1)^n [f_{d_R}^{(n)}(y)\/C^{(n)}]_{{\\rm real}}\n\\end{eqnarray}\nwhere \nwe used the relations\n\\begin{eqnarray}\n\\sin(M_W \\pi R) \\cos \\alpha_{n} &=& \\varepsilon(n) (-1)^n \\sin \\varphi_{n}, \\\\\n\\cos(M_W \\pi R) \\sin \\varphi_{n} &=& \n\\varepsilon(n) \\sin(M_W \\pi R) \\sqrt{\\cos(\\varphi_{n} \\pm \\alpha_{n})} \n\\sqrt{\\cos(\\varphi_{n} \\mp \\alpha_{n})} \n\\end{eqnarray}\nwhich can be derived from\n\\begin{eqnarray}\n\\sin^2 \\varphi_n = \\sin^2(M_W \\pi R) \\cos^2 \\alpha_n \n= \\tan^2(M_W \\pi R) \\cos(\\varphi_n+\\alpha_n) \\cos(\\varphi_n-\\alpha_n). \n\\end{eqnarray}\nA similar relation for the imaginary part can be verified. \nThus, we obtain \n\\begin{eqnarray}\n&&\\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) f_{d_R}^{(n)}(y)^* f_{d_R}^{(0)}(y) \\nonumber \\\\\n&=& 2 \\int_0^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) \n\\left[\nf_{d_R}^{(n)}(y)_{{\\rm real}} f_{d_R}^{(0)}(y)_{{\\rm real}} + f_{d_R}^{(n)}(y)_{{\\rm imaginary}} f_{d_R}^{(0)}(y)_{{\\rm imaginary}} \n\\right] \\nonumber \\\\\n&=& (-1)^n 2\\int^{\\pi R}_{0} dy (-1)^m \\cos \\left(\\frac{m}{R}(y-\\pi R) \\right) \\times \\nonumber \\\\\n&& \\left[\nf_{d_L}^{(n)}(y-\\pi R)_{{\\rm real}} f_{d_L}^{(0)}(y-\\pi R)_{{\\rm real}} \n+ f_{d_L}^{(n)}(y-\\pi R)_{{\\rm imaginary}} f_{d_L}^{(0)}(y-\\pi R)_{{\\rm imaginary}} \n\\right] \\nonumber \\\\\n&=& (-1)^{n+m} \\int_{-\\pi R}^{\\pi R} dy \\cos \\left(\\frac{m}{R}y \\right) f_{d_L}^{(n)}(y)^* f_{d_L}^{(0)}(y),\n\\end{eqnarray}\nwhich concludes the fact $I_c^{nm}(-M, M_W; -M, M_W) = (-1)^{n+m} I_c^{nm}(M, M_W; M, M_W)$. \n\n\\section{The ``equivalence theorem\"}\n\nIn this appendix,  \nwe discuss the relation between the vertex functions of $D$-dimensional gauge boson and scalar $(A_{\\mu}, A_{y})$, \nwhich are the partners of the Higgs-like mechanism in the sector of non-zero K-K modes \nor form a multiplet of quantum mechanical SUSY. \nThe relation is what we naturally expect from the fact \nthat the interaction of the longitudinal component of the massive gauge boson is equivalent to \nthat of the would-be Nambu-Goldstone boson. \nAs the typical example, we will derive a relation between \n$I_{c}^{(nm)}$ and $I_{s}^{(nm)}$ defined in (\\ref{3.6}) and (\\ref{3.13}). \n\nA key ingredient is the fact that right- and left-handed Weyl fermions form a multiplet of quantum mechanical SUSY, \nwhose transformation is given as (see (\\ref{2.35}) and (\\ref{2.36})) \n\\begin{eqnarray} \n&& i(\\partial_{y} -iM_{W} -M \\epsilon (y)) f_{dR}^{(n)} (y) = m_{n} f_{dL}^{(n)} (y), \\\\  \n&& i(\\partial_{y} -iM_{W} +M \\epsilon (y)) f_{dL}^{(n)} (y) = m_{n} f_{dR}^{(n)} (y).  \n\\end{eqnarray}\nBy use of these relations we can verify (for $m \\geq 1$)\n\\begin{eqnarray} \n&& I_{s}^{(nm)}(-M, M_{W}; M, M_{W}) \n= \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\sin (\\frac{m}{R}y) f_{dR}^{(n)}(y)^{\\ast} \nf_{dL}^{(0)}(y) \\nonumber \\\\ \n&=& \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[(\\partial_{y} f_{dR}^{(n)}(y)^{\\ast})f_{dL}^{(0)}(y) + f_{dR}^{(n)}(y)^{\\ast}(\\partial_{y} f_{dL}^{(0)}(y))] \\nonumber \\\\ \n&=& \\frac{1}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[((\\partial_{y} +iM_{W} - M \\epsilon (y))f_{dR}^{(n)}(y)^{\\ast})f_{dL}^{(0)}(y) \\nonumber \\\\ \n&& \\hspace*{10mm} + f_{dR}^{(n)}(y)^{\\ast}((\\partial_{y} -iM_{W} + M \\epsilon (y))f_{dL}^{(0)}(y))] \\nonumber \\\\ \n&=& \\frac{i}{\\sqrt{\\pi R}} \\int_{-\\pi R}^{\\pi R} dy \\frac{1}{(\\frac{m}{R})} \\cos (\\frac{m}{R}y) \n[m_{n} f_{dL}^{(n)}(y)^{\\ast}f_{dL}^{(0)}(y) - m_{d} f_{dR}^{(n)}(y)^{\\ast}f_{dR}^{(0)}(y)] \\nonumber \\\\ \n&=& i \\frac{1}{(\\frac{m}{R})} [m_{n} - (-1)^{m+n} m_{d}] I_{c}^{(nm)} (M, M_{W}; M, M_{W}),   \n\\end{eqnarray}\nwhere the step to move to the second line is due to a partial integral. \nThis relation immediately leads to the relation between the vertex functions,  \n\\begin{eqnarray} \nV_{\\gamma_y}^{nm} = i \\frac{m_{n} - (-1)^{m+n} m_{d}}{(\\frac{m}{R})} V_{\\gamma_{\\mu}}^{nm} \\ \\ (m \\geq 1). \n\\end{eqnarray}\n\n\n\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nIn this paper the authors explicitly describe in terms of generators and relations and three families of polynomials, the universal central extension of an algebra appearing in work of Date, Jimbo, Kashiwara and Miwa (see \\cite{MR701334,MR823315}), where they study integrable systems arising from Landau-Lifshitz differential equation.  Two of these families of polynomials are described below in terms of elliptic integrals and the other family is a variant of certain ultraspherical polynomials.  The  authors Date, Jimbo, Kashiwara and Miwa solved the Landau-Lifshitz equation using methods developed in some of their previous work on affine Lie algebras.  The hierarchy of this equation is written in terms of free fermions on an elliptic curve. The  infinite-dimensional Lie algebra mentioned above is shown to act on solutions of the Landau-Lifshitz equation as infinitesimal B\\\"acklund transformations where they derive an $N$-soliton formula.These authors arrive at an algebra that is a one dimensional central extension of $\\mathfrak g\\otimes \\mathbb C[t,t^{-1},u|u^2=(t^2-b^2)(t^2-c^2)]$ where $b\\neq \\pm c$ are complex constants and $\\mathfrak g$ is a simple finite dimensional Lie algebra defined over the complex numbers.  Below we explicitly describe its four dimensional universal central extension.  Modulo the center this algebra is a particular example of a Krichever-Novikov current algebra (see (\\cite{MR902293}, \\cite{MR925072}, \\cite{MR998426}).  A fair amount of interesting and fundamental work has be done by Krichever, Novikov, Schlichenmaier, and Sheinman on the representation theory of certain one dimensional central extensions of these latter current algebras and of analogues of the Virasoso algebra. In particular Wess-Zumino-Witten-Novikov theory and analogues of the Knizhnik-Zamolodchikov equations are developed for  these algebras (see the survey article \\cite{MR2152962}, and for example \\cite{MR1706819}, \\cite{MR1706819},\\cite{MR2072650},\\cite{MR2058804},\\cite{MR1989644}, and \\cite{MR1666274}).\n\n\nM. Bremner on the other hand has explicitly described in terms of generators, relations and certain families of polynomials (ultraspherical and Pollacyk) the structure constants for the universal central extension of algebras of the form $\\mathfrak g\\otimes \\mathbb C[t,t^{-1},u|u^2=p(t)]$ where $p(t)=t^2-2bt+1$ and $p(t)=t^3-2bt^2+t$ (see \\cite{MR1249871,MR1303073}).  He determined more generally the dimension of the universal central extension for affine Lie algebras of the form $\\mathfrak g\\otimes R$ where $R$\nis the ring of regular functions defined on an algebraic curve with any number of points removed. He obtained this using C.  Kassel's result (\\cite{MR772062}) where one knows  that the center is  isomorphic  as a vector space to $\\Omega_R^1\/dR$ (the {\\it space of K\\\"ahler  differentials  of $R$ modulo  exact forms}).    We will review this material below as needed. \n\nIn previous work of the authors (see  \\cite{MR2373448,MR2541818}) we used Bremner's aforementioned description to obtain certain free field realizations of the four point and elliptic affine algebras depending on a parameter $r=0,1$ which correspond to two different normal orderings.  These later realizations are analogues of Wakimoto type realizations which have been used by Schechtman and Varchenko and various other authors in the affine setting to pin down integral solutions to the Knizhnik-Zamolodchikov differential equations (see for example \\cite{MR1136712}, \\cite{MR1138049}, \\cite{MR2001b:32028}, \\cite{MR1077959}).   Such realizations have also been used in the study of Drinfeld-Sokolov reduction in the setting of $W$-algebras and in E. Frenkel's and B. Feigin's description of the center of the completed enveloping algebra of an affine Lie algebra (see \\cite{MR1309540}, \\cite{MR2146349}, and \\cite{MR1187549}).   In future work we plan to use results of this paper to describe free field realizations of the universal central extension of the algebras of Date, Jimbo, Kashiwara and Miwa (which, since this is a mouth full, we will call DJKM algebras). \n\n\n\\section{Universal Central Extensions of Current Algebras}  Let $R$ be a commutative algebra defined over $\\mathbb C$.\nConsider the left $R$-module  with  action $f( g\\otimes h ) = f g\\otimes h$ for $f,g,h\\in R$ and let $K$  be the submodule generated by the elements $1\\otimes fg  -f \\otimes g -g\\otimes f$.   Then $\\Omega_R^1=F\/K$ is the module of K\\\"ahler differentials.  The element $f\\otimes g+K$ is traditionally denoted by $fdg$.  The canonical map $d:R\\to \\Omega_R^1$ by $df  = 1\\otimes f  + K$.  The {\\it exact differentials} are the elements of the subspace $dR$.  The coset  of $fdg$  modulo $dR$ is denoted by $\\overline{fdg}$.  As C. Kassel showed the universal central extension of the current algebra $\\mathfrak g\\otimes R$ where $\\mathfrak g$ is a simple finite dimensional Lie algebra defined over $\\mathbb C$, is the vector space $\\hat{\\mathfrak g}=(\\mathfrak g\\otimes R)\\oplus \\Omega_R^1\/dR$ with Lie bracket given by\n$$\n[x\\otimes f,Y\\otimes g]=[xy]\\otimes fg+(x,y)\\overline{fdg},  [x\\otimes f,\\omega]=0,  [\\omega,\\omega']=0,\n$$\n  where $x,y\\in\\mathfrak g$, and $\\omega,\\omega'\\in \\Omega_R^1\/dR$ and $(x,y)$  denotes the Killing  form  on $\\mathfrak g$.  \n  \n\nConsider the polynomial\n$$\np(t)=t^n+a_{n-1}t^{n-1}+\\cdots+a_0\n$$\nwhere $a_i\\in\\mathbb C$ and $a_n=1$. \nFundamental to the description of the universal central extension for $R=\\mathbb C[t,t^{-1},u|u^2=p(t)]$ is the following:\n\\begin{thm}[\\cite{MR1303073},Theorem 3.4]  Let $R$ be as above.  The set \n$$\n\\{\\overline{t^{-1}\\,dt},\\overline{t^{-1}u\\,dt},\\dots, \\overline{t^{-n}u\\,dt}\\}\n$$\n forms a basis of $\\Omega_R^1\/dR$ (omitting $\\overline{t^{-n}u\\ dt}$ if $a_0=0$).   \n\\end{thm}\nSet $u^m=p(t)$.  Then $u\\,d(u^m)=mu^mdu$ and\n$$\n\\sum_{j=1}^nja_jt^{j-1}u\\,dt-m\\left(\\sum_{j=0}^na_jt^j\\,du\\right)=0\n$$\nor \n$$\np'(t)udt-mp(t)du=0.\n$$\nMultiplying by $t^i$ we get \n\\begin{equation}\\label{lady1}\n\\sum_{j=1}^nja_jt^{i+j-1}u\\,dt-m\\left(\\sum_{j=0}^na_jt^{i+j}\\,du\\right)=0\n\\end{equation}\n\n\n\\begin{lem}  If $u^m=p(t)$ and $R=\\mathbb C[t,t^{-1},u|u^m=p(t)]$, then in $\\Omega_R^1\/dR$, one has\n\\begin{equation}\\label{recursionreln}\n((m+1)n+im)t^{n+i-1}u\\,dt \\equiv - \\sum_{j=0}^{n-1}((m+1)j+mi)a_jt^{i+j-1}u\\,dt\\mod dR\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe have expanding $d(t^{i+j}u)$\n$$\n(i+j)t^{i+j-1}u\\,dt\\equiv-t^{i+j}\\,du\\mod dR.\n$$\nso that \\eqnref{lady1} implies \n\\begin{equation}\n\\sum_{j=0}^nja_jt^{i+j-1}u\\,dt+m\\left(\\sum_{j=0}^n(i+j)a_jt^{i+j-1}u\\,dt\\right)=0 \\mod dR\n\\end{equation}\nor\n\\begin{equation}\n\\sum_{j=0}^n((m+1)j+mi)a_jt^{i+j-1}u\\,dt\\equiv 0\\mod dR\n\\end{equation}\nThis gives \\eqnref{recursionreln}.\n\n\n\\end{proof}\n\n\n\n\\section{Description of the universal central extension of Date-Jimbo-Miwa-Kashiwara algebras}\nIn the Date-Jimbo-Miwa-Kashiwara  setting one takes $m=2$ and $p(t)=(t^2-a^2)(t^2-b^2)=t^4-(a^2+b^2)t^2+(ab)^2$ with $a\\neq \\pm b$ and neither $a$ nor $b$ is zero.  We fix from here onward $R=\\mathbb C[t,t^{-1},u\\,|\\,u^2= (t^2-a^2)(t^2-b^2)]$.  As in this case $a_0=(ab)^2$, $a_1=0$, $a_2=-(a^2+b^2)$, $a_3=0$ and $a_4=1$, then letting $k=i-2$ the recursion relation in \\eqnref{recursionreln} looks like\n\\begin{align*}\n(6+2k)\\overline{t^{k}u\\,dt}   \n&=-2(k-3)(ab)^2\\overline{t^{k-4}u\\,dt} +2k(a^2+b^2)\\overline{t^{k-2}u\\,dt}.\n\\end{align*}\nAfter a change of variables we may assume that $a^2b^2=1$.   Then the recursion relation looks like\n\\begin{equation}\\label{recursionreln1}\n(6+2k)\\overline{t^{k}u\\,dt}   \n=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt},\n\\end{equation}\nafter setting $c=(a^2+b^2)\/2$, so that $p(t)=t^4-2ct^2+1$.  Let $P_k:=P_k(c)$ be the polynomial in $c$ satisfy the recursion relation \n$$\n(6+2k)P_k(c)   \n=4k cP_{k-2}(c)-2(k-3)P_{k-4}(c)\n$$\nfor $k\\geq 0$.\nThen set\n$$\nP(c,z):=\\sum_{k\\geq -4}P_k(c)z^{k+4}=\\sum_{k\\geq 0}P_{k-4}(c)z^{k}.\n$$\nso that after some straightforward rearrangement of terms we have\n\\begin{align*}\n0&=\\sum_{k\\geq 0}(6+2k)P_k(c)z^k \n-4c\\sum_{k\\geq 0}kP_{k-2}(c)z^{k} +2\\sum_{k\\geq 0}(k-3)P_{k-4}(c)z^{k}  \\\\\n&=(-2z^{-4} +8cz^{-2}-6)P(c,z) +(2z^{-3}-4cz^{-1}+2z)\\frac{d}{dz}P(c,z)  \\\\\n&\\quad+(2z^{-4}-8cz^{-2})P_{-4}(c)  -4cP_{-3}(c)  z^{-1} -2P_{-2}(c)z^{-2} -4P_{-1}(c)z^{-1}.\n\\end{align*}\nWe then multiply the above through by $z^{4}$ to get \n\\begin{align*}\n0&=(-2+8cz^{2}-6z^4)P(c,z) +(2z-4cz^{3}+2z^5)\\frac{d}{dz}P(c,z)  \\\\\n&\\quad+(2-8cz^{2})P_{-4}(c)  -4cP_{-3}(c)  z^{3} -2P_{-2}(c)z^{2} -4P_{-1}(c)z^{3}.\n\\end{align*}\n\nHence $P(c,z)$ must satisfy the differential equation\n\\begin{equation}\\label{funde}\n\\frac{d}{dz}P(c,z)-\\frac{3z^4-4c z^2+1}{z^5-2cz^3+z}P(c,z)=\\frac{2\\left(P_{-1}+cP_{-3} \\right)z^3 +P_{-2} z^2+(4cz^2-1)P_{-4} }{z^5-2cz^3+z}\n\\end{equation}\nThis has integrating factor\n\\begin{align*}\n\\mu(z)&\n=\\exp \\int\\left( \\frac{-2 \\left(z^3-cz\\right)}{1-2 c z^2+z^4 }-\\frac{1}{z}\\right)\\,dz  \\\\\n&=\\exp(-\\frac{1}{2} \\ln(1-2 c z^2+z^4)-\\ln (z))=\\frac{1}{z\\sqrt{1-2 c z^2+z^4}}.\n\\end{align*}\n\n\\subsection{Elliptic Case 1}\nIf we take initial conditions $P_{-3}(c)=P_{-2}(c)=P_{-1}(c)=0$ and $P_{-4}(c)=1$ then we arrive at a generating function \n$$\nP_{-4}(c,z):=\\sum_{k\\geq -4}P_{-4,k}(c)z^{k+4}=\\sum_{k\\geq 0}P_{-4,k-4}(c)z^{k},\n$$\ndefined in terms of an elliptic integral\n\\begin{align*}\nP_{-4}(c,z)&=z\\sqrt{1-2 c z^2+z^4}\\int \\frac{4cz^2-1}{z^2(z^4-2c z^2+1)^{3\/2}}\\, dz.\n\\end{align*}\n One way to interpret the right hand integral is to expand $(z^4-2c z^2+1)^{-3\/2}$ as a Talyor series about $z=0$ and then formally integrate term by term and multiply the result by the Taylor series of $z\\sqrt{1-2 c z^2+z^4}$.    More precisely one integrates formally with zero constant term\n $$\n \\int (4c-z^{-2})\\sum_{n=0}^\\infty Q_n^{(3\/2)}(c)z^{2n}\\,dz =\\sum_{n=0}^\\infty \\frac{4cQ_n^{(3\/2)}(c)}{2n+1}z^{2n+1} -\\sum_{n=0}^\\infty \\frac{Q_n^{(3\/2)}(c)}{2n-1}z^{2n-1}\n $$ \n where $Q_n^{(\\lambda)}(c)$ is the $n$-th Gegenbauer polynomial.\nAfter multiplying this by \n$$\nz\\sqrt{1-2cz^2+z^4}=\\sum_{n=0}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\n$$\none arrives at the series $P_{-4}(c,z)$.\n\n\n\\subsection{Elliptic Case 2}\nIf we take initial conditions $P_{-4}(c)=P_{-3}(c)=P_{-1}(c)=0$ and $P_{-2}(c)=1$ then we arrive at a generating function defined in terms of another elliptic integral:\n\\begin{align*}\nP_{-2}(c,z)&=z\\sqrt{1-2 c z^2+z^4}\\int \\frac{1}{ (z^4-2c z^2+1)^{3\/2}}\\, dz.\n\\end{align*}\n\n\n\n\\subsection{Gegenbauer  Case 3}\nIf we take $P_{-1}(c)=1$, and $P_{-2}(c)=P_{-3}(c)=P_{-4}(c)=0$ and set \n$$\nP_{-1}(c,z)=\\sum_{n\\geq 0}P_{-1,n-4}z^n,\n$$\nthen we get a solution which after solving for the integration constant can be turned into a power series solution \n\\begin{align*}\nP_{-1}(c,z)&=(z\\sqrt{1-2 c z^2+z^4})\\left(\\int \\frac{2cz^3}{t\\sqrt{1-2 c z^2+z^4}(z^5-2c z^3+z)}\\, dt+C\\right)  \\\\\n&=\\frac{ z(c-z^3)}{c^2-1}-\\frac{c}{c^2-1}z\\sqrt{z^4-2cz^2+1} \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-cz\\sqrt{z^4-2c z^2+1}\\right)  \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-\\sum_{k=0}^\\infty c Q_n^{(-1\/2)}(c)z^{2n+1}\\right)  \\\\\n&=\\frac{1}{c^2-1}\\left(cz-z^3-cz+c^2z^3-\\sum_{k=2}^\\infty c Q_n^{(-1\/2)}(c)z^{2n+1}\\right)  \n\\end{align*}\nwhere $Q^{(-1\/2)}_n(c)$ is the $n$-th Gegenbauer polynomial.   Hence\n\\begin{align*}\nP_{-1,-4}(c)&=P_{-1,-3}(c)=P_{-1,-2}(c) =P_{-1,2m}(c)=0, \\\\\nP_{-1,-1}(c)&=1,  \\\\\nP_{-1,2n-3}(c)&=\\frac{-cQ_{n}(c)}{c^2-1},\n\\end{align*}\nfor $m\\geq 0$ and $n\\geq 2$ .\nThe $Q^{(-1\/2)}_n(c)$ are known to satisfy the second order differential equation:\n\\begin{align*}\n(1-c^2)\\frac{d^2}{d^2 c}Q^{(-1\/2)}_n(c)+n(n-1)Q^{(-1\/2)}_{n}(c)=0\n\\end{align*}\nso that the $P_{-1,k}:=P_{-1,k}(c)$ satisfy the second order differential equation\n\\begin{align*}\n(c^4-c^2)\\frac{d^2}{d^2 c}P_{-1,2n-3}+2c(c^2+1)\\frac{d}{d  c}P_{-1,2n-3}+(-c^2n(n-1)-2)P_{-1,2n-3}=0\n\\end{align*}\nfor $n\\geq 2$.\n\n\\subsection{Gegenbauer Case 4}\nNext we consider the initial conditions $P_{-1}(c)=0=P_{-2}(c)=P_{-4}(c)=0$ with $P_{-3}(c)=1$ and set \n$$\nP_{-3}(c,z)=\\sum_{n\\geq 0}P_{-3,n-4}(c)z^n,\n$$\nthen we get a power series solution\n\\begin{align*}\nP_{-3}(c,z)&=(z\\sqrt{1-2 c z^2+z^4})\\left(\\int \\frac{2cz^3}{z\\sqrt{1-2 c z^2+z^4}(z^5-2c z^3+z)}\\, dz+C\\right)  \\\\\n&=\\frac{ cz(c-z^3)}{c^2-1}-\\frac{1}{c^2-1}z\\sqrt{z^4-2cz^2+1} \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-z\\sqrt{z^4-2c z^2+1}\\right)  \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-\\sum_{k=0}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\\right)  \\\\\n&=\\frac{1}{c^2-1}\\left(c^2z-cz^3-z+cz^3-\\sum_{k=2}^\\infty Q_n^{(-1\/2)}(c)z^{2n+1}\\right)  \\\\\n\\end{align*}\nwhere $Q^{(-1\/2)}_n(c)$ is the $n$-th Gegenbauer polynomial.  Hence\n\\begin{align*}\nP_{-3,-4}(c)&=P_{-3,-2}(c)=P_{-3,-1}(c) =P_{-1,2m}(c)=0, \\\\\nP_{-3,-3}(c)&=1,  \\\\\nP_{-3,2n-3}(c)&=\\frac{-Q_{n}(c)}{c^2-1},\n\\end{align*}\nfor $m\\geq 0$ and $n\\geq 2$ and hence \\begin{align*}\n(c^2-1)\\frac{d^2}{d^2 c}P_{-3,2n-3}+4c \\frac{d}{d c}P_{-3,2n-3} -(n+1)(n-2)P_{-3,2n-3} =0\n\\end{align*}\nfor $n\\geq 2$ and $P_{-1,2n-3}=cP_{-3,2n-3}$ for $n\\geq 2$.\n\\section{Main result}\nFirst we give an explicit description of the cocyles contributing to the {\\it even} part of the DJKM algebra. \n\n\\begin{prop}[cf. \\cite{MR1303073}, Prop. 4.2] \\label{cocyclecalc} Set $\\omega_0=\\overline{t^{-1}\\,dt}$. \n For $i,j\\in\\mathbb Z$ one has\n\\begin{equation}\nt^i\\,d(t^j)= j \\delta_{i+j,0}\\omega_0\n\\end{equation}\nand \n\\begin{equation}\nt^{i-1}u\\,d(t^{j-1}u)=\\left(\\delta_{i+j,-2}(j+1) -2cj\\delta_{i+j,0} +(j-1)\\delta_{i+j,2}\\right)\\omega_0.\n\\end{equation}\n\n\\end{prop}\n\\begin{proof}  First observe that \n$2u\\,du=d(u^2)=(4t^3-4c t)\\,dt$.\n The second congruence then follows from\n\\begin{align*}\nt^{i-1}u\\,d(t^{j-1}u)&=(j-1)t^{i+j-3}u^2\\,dt+t^{i+j-2}u\\,du \\\\\n&=(j-1)t^{i+j-3}(t^4-2ct^2+1)\\,dt+2t^{i+j-2}(t^3-c t)\\,dt \\\\\n&=(j-1)(t^{i+j+1}-2ct^{i+j-1}+t^{i+j-3})\\,dt+2(t^{i+j+1}-c t^{i+j-1})\\,dt \\\\\n&=(j+1)t^{i+j+1}\\,dt -2cjt^{i+j-1}\\,dt +(j-1)t^{i+j-3}\\,dt. \n\\end{align*}\n\\end{proof}\n\nThe map $\\sigma:R\\to R$ given by $\\sigma(t)=t^{-1}$, $\\sigma(u)=t^{-2}u$ is an algebra automorphism as \n$\\sigma(u^2)=t^{-4}u^2=1-2ct^{-2}+t^{-4}=\\sigma(1-2ct^{2}+t^{4})$.  This descends to a linear map $\\sigma:\\Omega_R^1\/dR$ where\n\\begin{align*}\n\\sigma(\\overline{t^{-1}\\,dt})&=-\\overline{t^{-1}\\,dt},\\\\\n\\sigma(\\overline{t^{-1}u\\,dt)}&=\\overline{t(t^{-2}u)d(t^{-1})}=-\\overline{t^{-3}u\\,dt},  \\\\\n\\sigma(\\overline{t^{-2}u\\,dt)}&=\\overline{t^2(t^{-2}u)d(t^{-1})}=-\\overline{t^{-2}u\\,dt}, \\\\\n\\sigma(\\overline{t^{-3}u\\,dt)}&=-\\overline{t^{-1}u\\,dt}, \\\\ \n\\sigma(\\overline{t^{-4}u\\,dt)}&=\\overline{t^4(t^{-2}u)d(t^{-1})}=-\\overline{u\\,dt}=-\\overline{t^{-4}u\\,dt },\n\\end{align*} \nwhereby the last identity follows from the recursion relation \\eqnref{recursionreln1} with $k=0$.  Setting $\\omega_{-k}=\\overline{t^{-k}u\\,dt}$, $k=1,2,3,4$, then $\\sigma(\\omega_{-1})=-\\omega_{-3}$,  and $\\sigma(\\omega_{-l})=-\\omega_{-l}$ for $l=2,4$.\n\n\\begin{thm} Let $\\mathfrak g$ be a simple finite dimensional Lie algebra over the complex numbers with   the Killing form $(\\,|\\,)$ and define $\\psi_{ij}(c)\\in\\Omega_R^1\/dR$ by\n\\begin{equation}\n\\psi_{ij}(c)=\\begin{cases} \n\\omega_{i+j}&\\quad \\text{ for }\\quad i+j=1,0,-1,-2 \\\\\nP_{-3,i+j-2}(c) (\\omega_{-3}+c\\omega_{-1})&\\quad \\text{for} \\quad i+j =2n-1\\geq 3,\\enspace n\\in\\mathbb Z, \\\\\nP_{-3,i+j-2}(c) (c\\omega_{-3}+\\omega_{-1})&\\quad \\text{for} \\quad i+j =-2n+1\\leq - 3, n\\in\\mathbb Z, \\\\\nP_{-4,|i+j|-2}(c) \\omega_{-4} +P_{-2,|i+j|-2}(c)\\omega_{-2}&\\quad\\text{for}\\quad |i+j| =2n \\geq 2, n\\in\\mathbb Z. \\\\\n\\end{cases}\n\\end{equation}\nThe universal central extension of the Date-Jimbo-Kashiwara-Miwa  algebra is the $\\mathbb Z_2$-graded Lie algebra \n$$\n\\widehat{\\mathfrak g}=\\widehat{\\mathfrak g}^0\\oplus \\widehat{\\mathfrak g}^1,\n$$\nwhere\n$$\n\\widehat{\\mathfrak g}^0=\\left(\\mathfrak g\\otimes \\mathbb C[t,t^{-1}]\\right)\\oplus \\mathbb C\\omega_{0},\\qquad \\widehat{\\mathfrak g}^1=\\left(\\mathfrak g\\otimes \\mathbb C[t,t^{-1}]u\\right)\\oplus \\mathbb C\\omega_{-4}\\oplus \\mathbb C\\omega_{-3}\\oplus \\mathbb C\\omega_{-2}\\oplus \\mathbb C\\omega_{-1}\n$$\nwith bracket\n\\begin{align*}\n[x\\otimes t^i,y\\otimes t^j]&=[x,y]\\otimes t^{i+j}+\\delta_{i+j,0}j(x,y)\\omega_0, \\\\ \\\\\n[x\\otimes t^{i-1}u,y\\otimes t^{j-1}u]&=[x,y]\\otimes (t^{i+j+2}-2ct^{i+j}+t^{i+j-2}) \\\\\n &\\hskip 40pt+\\left(\\delta_{i+j,-2}(j+1) -2cj\\delta_{i+j,0} +(j-1)\\delta_{i+j,2}\\right)(x,y)\\omega_0, \\\\ \\\\\n[x\\otimes t^{i-1}u,y\\otimes t^{j}]&=[x,y]u\\otimes t^{i+j-1}+ j(x,y)\\psi_{ij}(c).\n\\end{align*}\n\n\\end{thm}\n\n\n\\begin{proof}\nThe first two equalities follow from \\propref{cocyclecalc}.  For the last one we first observe that for $k=i+j-2\\neq -3$,\n\\begin{align*}\nj\\omega_{ij}(c)=\\overline{ t^{i-1}u\\,d( t^{j})}&=j\\overline{t^{i+j-2}u\\,dt} \\\\\n&=j\\left(\\frac{-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}}{6+2k}\\right),\n\\end{align*}\nwhere the last equality is derived from \\eqnref{recursionreln1}.\nThen by setting $k=0,1,2,3,4,5$ in \\eqnref{recursionreln1}\n\\begin{equation*}\n(6+2k)\\overline{t^{k}u\\,dt}   \n=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}.\n\\end{equation*}\ngives us \n\\begin{align*}\n6\\overline{u\\,dt}  &=6\\overline{t^{-4}u\\,dt}  , \\\\\n 8\\overline{tu\\,dt}   &=4\\overline{t^{-3}u\\,dt} +4c\\overline{t^{-1}u\\,dt}, \\\\\n  10\\overline{t^{2}u\\,dt}   \n&=2\\overline{t^{-2}u\\,dt} +8c\\overline{ u\\,dt}, \\\\ \n12\\overline{t^{3}u\\,dt}   \n&=12c\\overline{tu\\,dt} ,\\\\\n14\\overline{t^{4}u\\,dt}   \n&=-2\\overline{u\\,dt} +16c\\overline{t^{2}u\\,dt}, \\\\\n16\\overline{t^{5}u\\,dt}   \n&=-4\\overline{tu\\,dt} +20c\\overline{t^{3}u\\,dt}, \\\\\n(6+2k)\\overline{t^{k}u\\,dt}   \n&=-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt} .\n\\end{align*}\nHence when $i+j-2=k=0,1,2,3,4,5$\n\\begin{align*}\n\\overline{u\\,dt}  &=\\omega_{-4} , \\\\\n\\overline{tu\\,dt}   &=\\frac{1}{2}\\left(\\omega_{-3} +c\\omega_{-1}\\right), \\\\\n \\overline{t^{2}u\\,dt}   \n&=\\frac{1}{5} \\omega_{-2} +\\frac{4c}{5}\\omega_{-4}, \\\\ \n\\overline{t^{3}u\\,dt}   \n&=\\frac{c}{2}\\left(\\omega_{-3} +c\\omega_{-1}\\right), \\\\\n \\overline{t^{4}u\\,dt}   \n&=-\\frac{1}{7}\\overline{u\\,dt} +\\frac{8}{7}c\\overline{t^{2}u\\,dt}=-\\frac{1}{7}\\omega_{-4} +\\frac{8}{7}c\\left(\\frac{1}{5} \\omega_{-2} +\\frac{4c}{5}\\omega_{-4}\\right)\\\\\n&=\\left(\\frac{32c^2-5}{35}\\right)\\omega_{-4} +\\frac{8}{35}c \\omega_{-2} , \\\\ \n \\overline{t^{5}u\\,dt}   \n&=-\\frac{1}{8} \\left(\\omega_{-3} +c\\omega_{-1}\\right)+\\frac{5c^2}{8}\\left(\\omega_{-3} +c\\omega_{-1}\\right) \\\\\n&=\\frac{5c^2-1}{8} \\left(\\omega_{-3} +c\\omega_{-1}\\right),\n\\\\\n\\overline{t^{k}u\\,dt}    \n&=\\frac{-2(k-3)\\overline{t^{k-4}u\\,dt} +4kc\\overline{t^{k-2}u\\,dt}}{6+2k} .\n\\end{align*}\nThus by induction using the last equation above for $i+j-2=k=2n-3\\geq 1$, $n\\in\\mathbb Z$, we have \n\\begin{align}\\label{oddcase}\n\\omega_{ij}(c)\n&=P_{-3,i+j-2}(c)\\left(\\omega_{-3} +c\\omega_{-1}\\right),\n\\end{align}\nand for $i+j-2=k=2n-2\\geq 0$, $n\\in\\mathbb Z$, we have \n\\begin{align}\\label{evencase}\n\\omega_{ij}(c)\n&=P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}.\n\\end{align}\nApplying $\\sigma$ to \\eqnref{oddcase} for $i+j-2=k=2n-3\\geq 1$ to obtain \n\\begin{align*}\nj\\sigma(\\omega_{ij}(c))=\\overline{ t^{-i+1}u\\,d( t^{-j})}&=-j\\overline{t^{-i-j-2}u\\,dt} \\\\\n&=j\\sigma\\left(P_{-3,i+j-2}(c)\\left(\\omega_{-3} +c\\omega_{-1}\\right) \\right) \\\\ \n&=-jP_{-3,i+j-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\nHence for $i+j-2=2n-3\\geq 1$\n\\begin{align*}\n\\omega_{-i,-j}(c)=\\overline{t^{-i-j-2}u\\,dt} \n&=P_{-3,i+j-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\nSetting $i'=-i$ and $j'=-j$ we get for $i'+j'-2=-k-4=-2n+3\\leq -5$\n\\begin{align*}\n\\omega_{i'j'}(c)=\\overline{t^{i'+j'-2}u\\,dt} \n&=P_{-3,|i'+j'|-2}(c)\\left(\\omega_{-1} +c\\omega_{-3}\\right).\n\\end{align*}\n\nSimilarly if  we apply $\\sigma$ to \\eqnref{evencase} for $i+j=2n\\geq 2$, $n\\in\\mathbb Z$, we obtain \n\\begin{align*}\nj\\sigma(\\omega_{ij}(c))=\\overline{ t^{-i+1}u\\,d( t^{-j})}&=-j\\overline{t^{-i-j-2}u\\,dt} \\\\\n&=j\\sigma\\left(P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2} \\right) \\\\ \n&=-j\\left(P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}\\right)\n\\end{align*}\nHence for $i+j=2n\\geq 2$\n\\begin{align*}\n\\omega_{-i,-j}(c)=\\overline{t^{-i-j-2}u\\,dt} \n&=P_{-4,i+j-2}(c) \\omega_{-4} +P_{-2,i+j-2}(c)\\omega_{-2}.\n\\end{align*}\nSetting $i'=-i$ and $j'=-j$ we get for $i'+j'=-2n\\leq  -2$\n\\begin{align*}\n\\omega_{i'j'}(c)=\\overline{t^{i'+j'-2}u\\,dt} \n&=P_{-4,|i'+j'|-2}(c) \\omega_{-4} +P_{-2,|i'+j'|-2}(c)\\omega_{-2}.\n\\end{align*}\n\\end{proof}\nOne might want to compare the above theorem with the results that M.  Bremner obtained for the elliptic and four point affine Lie algebra cases (\\cite[Theorem 4.6]{MR1303073} and \\cite[Theorem 3.6]{MR1249871} respectively).\n\n\n\n\n\\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, an interesting effect was discovered by Ba\\~{n}ados, Silk\nand West \\cite{ban}, called usually the BSW effect after the names of the\nauthors: if two particles collide near the black hole horizon, the energy $%\nE_{c.m.}$ in their centre of mass frame can grow indefinitely large,\nprovided the parameters of one of the particles are fine-tuned. Immediately\nafter this observation, several considerations of theoretical nature were\nbrought forward suggesting that there must be restrictions that would\nprevent the realization of this effect. One of the basic objections is\nconnected with the force of gravitational radiation acting on particles. It\nwas pushed forward in \\cite{berti} and is mentioned from time to time in\nconsequent works starting from \\cite{ted}. There are also other similar effects which seem to\nrestrict the divergence of $E_{c.m.}$ -- say, synchrotron radiation by\ncharged particles near black holes \\cite{fr}.\n\nMeanwhile, the influence of the force of gravitational radiation (or any\nother force) on the BSW effect is not so obvious. First of all, the BSW\neffect is prepared from two main ingredients -- the presence of the horizon\nand the presence of special \"critical\" trajectories, (see below). It was\nshown in \\cite{gc}, with minimal assumptions, that even for neutral\nparticles and nongeodesic motion, such trajectories do exist. Therefore, the\nquestion is whether or not the force destroys these trajectories. If this\nhappens, the BSW effect is restricted. However, for a weak force, one can expect a large bound on $E_{c.m.}$. For instance, the analysis of\nparticle's motion on the innermost stable orbit near the Kerr black hole\nwith gravitational radiation taken into account showed that $E_{c.m.}$ can\nbe far beyond the Planck energy for collision of dark matter particles near\na stellar mass near-extremal black hole \\cite{insp}. The analysis suggested in \\cite{insp}, however, concerns special (although important for astrophysics) cases: it applies to\nnear-extremal Kerr black holes when fine-tuning required for the BSW effect\nis realized on circular orbits. It also remains incomplete since not all\nfactors responsible for the self-force are taken into account. Meanwhile, it\nis of interest to elucidate the issue under discussion in a\nmodel-independent way.\n\nIn this paper we develop such a general approach and analyze the BSW effect under the influence of a generic force near the horizon of a generic axially symmetric stationary ``dirty'' black hole (i.e. a black hole that is surrounded by matter, so its metric may deviate from the Kerr one). Here we only consider the case of an extremal horizon of a maximally rotating black hole. The approach used is applicable, with minimal modifications, to static or charged black holes, as shown explicitly for the case of Reissner-Nordstr\\\"{o}m metric.\n\nWe consider the conditions the force should satisfy for the effect to be either preserved in some form or not. The analysis is made in terms of tetrad components of the corresponding quantities in the frames attached both to an observer orbiting the black hole, and the one crossing the horizon. The nature of the force itself is not specified, we only assume that its tetrad components in the particle's proper frame are finite and restrict our consideration to equatorial motion. \nWe show that the BSW effect survives any force that satisfies the following assumptions: (i) it remains finite near the horizon, and (ii) its azimuthal component tends to zero fast enough (more detailed definition is given below). In case the above condition is not satisfied, e.g. the azimuthal force does not vanish in the horizon limit, the weaker version of the effect is realized whenever the acceleration's amplitude is small enough (as should be for e.g. radiation reaction). For the latter case, we find generic bounds on $E_{c.m.}$.\n\nIt is worth stressing that the BSW effect reveals itself not only for extremal black holes, but also for nonextremal ones. The mechanism in the latter case, however, is generally different, as it requires multiple scattering, which for extremal black holes is not necessary  \\cite{gp} (see also \\cite{prd}). Correspondingly, we postpone consideration of the BSW effect with a force near nonextremal horizons and, in the present paper, restrict ourselves to the extremal case. The effect for near-extremal horizons, considered in \\cite{insp}, occupies an intermediate position between the two. This problem contains some subtleties on its own related to the properties of near-circular orbits and in the general setting also needs separate treatment.\n\nThere are two aspects of the BSW effect --- the behavior of $E_{c.m.}$ near the horizon and the properties of energies of the collision outcome measured at infinity. The typical energies at infinity are quite modest even in the absence of force \\cite{inf1,inf2,inf3}, so taking the force into account can only change them slightly. It is the first aspect which is nontrivial and is being discussed in the present paper. \n\nThe paper is organized as follows. In Sec. II, we consider classification of\nparticles relevant for the BSW effect and discuss novel features that the\nforce brings into the system. In Sec. III, we consider behavior of\nacceleration near the horizon in different frames (attached to an observer orbiting the black hole or to one crossing the horizon). In Sec. IV, we illustrate\ngeneral relationships using the Reissner-Nordstr\\\"{o}m metric as an example.\nIn Sec. V, we consider generic motion in the equatorial plane under the action of finite forces and derive conditions on the force that allow or forbid critical trajectories.\nIn Sec. VI, we estimate the bounds on $E_{c.m.}$ for the case when the force is least favourable for the effect but small. In Sec. VII, we discuss pure kinematic restrictions on particle's trajectories (valid even in the absence of force) which can influence the properties of the BSW effect. Sec. VIII is devoted to conclusion.\n\n\\section{Particles' kinematics near extremal horizons}\n\n\\subsection{A particle in axially symmetric metric}\n\nWe consider the axially symmetric stationary metric written (at least in the\nvicinity of the horizon) in coordinates which are obtained from the Gaussian\nnormal ones by replacing the distance to the horizon $n$ with the\nradial coordinate $r$, defined so that\\footnote{Such as the quasiglobal coordinate of \\cite{BrRub}, ch.3.} $A(r)\\sim N^{2}$ in the\nhorizon limit, where $N^{2}\\rightarrow 0$ (hereafter $c=1$): \n\\begin{equation}\nds^{2}=-N^{2}dt^{2}+g_{\\phi }(d\\phi -\\omega dt)^{2}+\\frac{dr^{2}}{A}%\n+g_{z}dz^{2}.\n\\end{equation}\n\nLet there be some arbitrary, not necessarily geodesic, particle of mass $m$,\nfour-velocity $u^{\\mu }$ and four-momentum \n\\begin{equation*}\np^{\\mu }=mu^{\\mu }.\n\\end{equation*}%\nIt is convenient to represent the four-velocity, both with upper and lower\nindices, by the components of its four-momentum in the following way: \n\\begin{align}\nu^{\\mu }& =\\frac{1}{m}\\Big(\\frac{X}{N^{2}},\\frac{L}{g_{\\phi }}+\\frac{\\omega X%\n}{N^{2}},p^{r},p^{z}\\Big);  \\label{Uu} \\\\\nu_{\\mu }& =\\frac{1}{m}\\Big(-E,L,\\frac{1}{A}p^{r},g_{z}p^{z}\\Big),  \\label{Ud}\n\\end{align}%\nwhere $E=-mu_{0}$ is energy, $L=mu_{\\phi }$ angular momentum and \n\\begin{equation}\nX=E-\\omega L.  \\label{x}\n\\end{equation}\nDue to forward in time condition, $X$ is always positive.\n\nFor a free particle on a geodesic trajectory the energy $E$ and angular momentum \n$L$ are conserved; eqs. (\\ref{Uu}), (\\ref{Ud}) are nothing but the equations\nof motion with given fixed values of $E$ and $L$. In the general case, $E$\nand $L$ are not conserved and together with $u^{z}$ should be treated as\nfunctions of the particle's proper time. Nonetheless, we still write the\ncomponents of the four-velocity in the same form (\\ref{Uu}), (\\ref{Ud})\nwhich can be considered simply as useful parametrization.\n\nThe normalization condition $u^\\mu u_\\mu =-1$ can be written as \n\\begin{equation}\n\\frac{1}{A}(p^r)^2 +g_{z}(p^z)^2 =\\frac{X^2}{N^2}-\\frac{L^2}{g_\\phi}-m^2 .  \\label{norm}\n\\end{equation}\n\nThen $p^{r}$ is expressed through the three independent parameters $E$, $L$\nand $u^{z}$: \n\\begin{equation}\np^{r}=\\pm \\frac{\\sqrt{A}}{N}\\;Z,  \\label{pr}\n\\end{equation}\nwhere \n\\begin{align}\nZ^{2} \n& =X^{2}-N^{2}\\Big[\\frac{L^{2}}{g_{\\phi }}+g_{z}(p^{z})^{2}+m^{2}\\Big]. \\label{z}\n\\end{align}\n\nThe formulas in this section are applicable also to massless particles, with\nthe only difference that one has to set $m=0$. The four-momentum then, in\nthe appropriate parametrization of the worldline, is related to the wave\nvector $k^{\\mu }$ as $p^{\\mu }=\\hbar k^{\\mu }$.\n\n\\subsection{Two particles' collision near horizon}\n\nThe energy $E_{i\\;c.m.}$ of a particle $i$ with four-momentum $p_{i}^{\\mu }$\nin its center of mass (c.m.) frame is simply its rest mass, i.e. the norm of\nits four-momentum: \n\\begin{equation}\nE_{i\\;c.m.}^{2}=m_{i}^{2}=-p_i^{\\mu }p_{i\\,\\mu }.\n\\end{equation}%\nLikewise, for two particles with masses $m_{1}$ and $m_{2}$ and\nfour-velocities $u_{1}^{\\mu }$ and $u_{2}^{\\mu }$ the center of mass energy $%\nE_{c.m.}$ at the collision event is the norm of their total four-momentum \n\\begin{equation}\nE_{c.m.}^{2}=-(p_{1}^{\\mu }+p_{2}^{\\mu })(p_{1\\mu }+p_{2\\mu\n})=m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\\gamma _{c.m.},  \\label{cm}\n\\end{equation}%\nwhere \n\\begin{equation}\n\\gamma _{c.m.}=-u_{1\\mu }u_{2}^{\\mu }\n\\end{equation}%\nis the relative Lorentz factor.\n\nThe contraction can be written as \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{1}X_{2}-Z_{1}Z_{2}}{N^{2}}-\\frac{L_{1}L_{2}%\n}{g_{\\phi }}-g_{z}p_{1}^{z}p_{2}^{z}.  \\label{ga}\n\\end{equation}\n\nFor a collision of a massive particle of mass $m$ and a photon one obtains\nthat \n\\begin{equation}\nE_{c.m.}^{2}=m^{2}+2m\\hbar \\omega _{det},\n\\end{equation}%\nwhere $\\omega _{det}$ $=-k_{\\mu }u^{\\mu }$ is the photon's frequency as\ndetected in the frame of this massive particle.\n\n\\subsection{Usual and critical particles near extremal horizons}\n\nConsider a particle in the vicinity of a regular extremal horizon, for which \n\\cite{reg}\n\\begin{align}\nN^{2}(r)&\\sim (r-r_{H})^{2},  \\label{Extr}\\\\\n\\omega (r)&=\\omega _{H}-\\omega _{1}(z)N+O(N^{2}),\\qquad \\omega _{H}=const,\n\t\\label{OmExpansion}\n\\end{align}\nwhere $r=r_{H}$ is the horizon. The regularity of the horizon implies \\cite{reg} that,\nin particular, $\\omega_H$ is a constant, and that other metric functions can\nalso be expanded into series by $(r-r_{H})$ with positive powers. Note also\nthe sign by $\\omega_1$, defined so for consistency with earlier works.\n\nFrom the normalization condition (\\ref{norm}) then a particle's\nfour-velocity components can always be presented as series by $N$ (though\nthey can diverge at the horizon). Assuming $L$ and $E$ are finite, \n\\begin{equation}\nE=E_{H}+E_{1}N+O(N^{2}),\\quad L=L_{H}+L_{1}N+O(N^{2}),  \\label{el}\n\\end{equation}%\nand then \n\\begin{align}\nX=& X_{H}+O(N), \\\\\n& X_{H}=E_{H}-\\omega _{H}L_{H}.\n\\end{align}\n\nFor a usual (generic) particle $X_{H}\\neq 0$. The normalization (\\ref{norm})\nthen implies that \n\\begin{equation}\nu^{r}=O(1),  \\label{ru}\n\\end{equation}%\n$u^{z}=O(1\/N)$, so the particle reaches the horizon in finite proper time $%\n\\tau \\sim \\int dr<\\infty $.\n\nHowever, there are also worldlines of particles with angular momentum\nfine-tuned to energy in such a way that $X_{H}=0$, so that \n\\begin{equation}\nX=O(N).  \\label{xc}\n\\end{equation}%\nSuch particles are called critical.\n\nFor critical particles the right hand side of normalization condition(\\ref{norm}) is bounded, and as the left hand side there is a sum of squares, we obtain \n\\begin{equation}\nu^r =O(N),\\quad u^z =O(1).  \\label{Ubounded}\n\\end{equation}\n\nThen the equation of radial motion in the main order by $N$ is \n\\begin{equation}\n\\frac{dr}{d\\tau }=-\\frac{r-r_{H}}{\\tau _{0}},  \\label{rt}\n\\end{equation}%\nwhere $\\tau _{0}$ is a constant for motion in equatorial plane\\footnote{In general, the coordinate $z$ can oscillate between some limiting values, see \\cite{ne} for the Kerr metric and  \\cite{jh} for discussion of a more general case.}; its solution is \n\\begin{equation}\nr-r_{H}=r_{0}e^{-\\tau \/\\tau _{0}}  \\label{ur}\n\\end{equation}%\nand the proper time of reaching the horizon diverges as $\\ln (r-r_H )$. In case $u^{r}$ is of higher order that $N$, the divergence is stronger (i.e. if $u^{r}\\sim (r-r_{H})^2$,\nthen $\\tau $ diverges as $(r-r_{H})^{-1}$).\n\n\\subsubsection{Example: the Kerr metric}\n\nIt is instructive to look at the critical trajectory for the Kerr metric.\nLet us restrict ourselves to equatorial motion $\\theta =\\frac{\\pi }{2}$.\nThen, the metric coefficients near the horizon of the extremal Kerr black\nhole read \n\\begin{align}\n&N\\approx \\frac{r-r_{H}}{2r_{H}},\\qquad \\sqrt{A}\\approx \\frac{r-r_{H}}{r_{H}}%\n,\\qquad (g_{\\phi })_{H}=4r_{H}^{2}, \\\\\n&\\omega _{H}=\\frac{1}{2r_{H}},\\qquad \\omega -\\omega _{H}\\approx -\\frac{%\nr-r_{H}}{2r_H^2}.\n\\end{align}\nThen, it follows from eqs. (\\ref{pr}), (\\ref{z}) that the trajectory of the\nparticle with $E=\\omega _{H}L$ has exactly the form (\\ref{rt}) with \n\\begin{equation}\n\\tau _{0}=\\frac{r_{H}}{\\sqrt{3\\frac{E^{2}}{m^{2}}-1}}.\n\\end{equation}\n\n\\subsection{The BSW effect}\n\nConsider the collision of two particles. For a usual (generic) particle,\nassuming $E$, $L$ and $u^z$ are finite\\footnote{Those are natural assumptions, but for justification see the section on\ndynamics below.}, \n\\begin{equation}\nX=X_H +O(N),\\quad Z=X+O(N^2).\n\\end{equation}\n\nThen the relative Lorentz factor at the collision event of two usual\nparticles is \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{1}X_{2}-Z_{1}Z_{2}}{N^{2}}+O(1)=O(1).\n\\label{gacm}\n\\end{equation}\n\nHowever, for a critical particle \n\\begin{equation}\nX=X_{N}N+O(N^{2}),\\quad Z=Z_{N}N+O(N^{2}).\n\\end{equation}%\nThen for two critical particles $\\gamma _{c.m.}$ is also bounded, but the\nrelative Lorentz factor at the collision event of a critical (1) particle\nand a usual (2) particle is \n\\begin{equation}\nm_{1}m_{2}\\gamma _{c.m.}=\\frac{X_{H}^{(2)}(X_{N}^{(1)}-Z_{N}^{(1)})}{N}%\n+O(1)\\rightarrow \\infty .  \\label{div}\n\\end{equation}\n\nSo, the BSW effects occurs whenever one usual and one critical particle collide near the horizon. Geodesic particles can be\ncritical just due to the choice of initial conditions which fix $E$ and $L$,\nso one can always achieve $X_{H}=0.$ The question is how resilient is the\ncriticality attribute with respect to acceleration: whether a particle can\nremain critical under the action of finite forces, such as radiation\nreaction.\n\n\\subsection{Generalization: usual, critical and sub-critical particles}\n\nIn the absence of external forces acting on a particle, in the vicinity of a\nregular horizon, where all metric functions can be expanded into series by\nthe radial coordinate $r$, the geodesic equation induces the same\ntype of expansions for the parameters of a particle, such as $X$ and $E$.\nTherefore there are only two principally different types of particles: usual\nand critical ones. If we want to take into account forces acting on a\nparticle, however, we have to allow for more general setting. In particular,\nwe assume that acceleration components in the proper frame of a particle and \n$X$ can behave as $\\xi^{q}$ and $\\xi^{p}$ respectively, with some real $q$ and $p$, where\n\\begin{equation}\n \\xi \\equiv r-r_{H}. \\label{xi}\n\\end{equation}\nHereafter we consider this reasonably general while still relatively simple model.\n\nIf a particle reaches the horizon, $X$ must tend to zero more slowly than $N$, so that $Z^2$ remains positive. This is only possible for $p\\leq 1$. On the other hand, it is reasonable to\nrestrict our consideration to finite $E$ and $L$, and thus $X$, so $p\\geq 0$. Then there are three possible particle types, distinguished by $p$ in \n\\begin{equation}\nX\\sim \\xi^ p .\n\\end{equation}\n\n\\begin{enumerate}\n\\item $p=0$: \\textbf{usual particles}. \n\\begin{equation}\nX_u=x_H + x_1 \\xi +\\ldots, \\qquad Z_u=X+O(\\xi^2).\n\\end{equation}\n\n\\item $p=1$: \\textbf{critical particles}. \n\\begin{equation}\nX_{cr}=x_1 \\xi + x_2 \\xi^2 +\\ldots,\\qquad Z_{cr}=O(\\xi) .\n\\end{equation}\n\n\\item $p\\in (0,1)$: the intermediate case, which will be called \\textbf{sub-critical} particles hereafter: \n\\begin{equation}\nX_{sc}=\\alpha \\xi^{p}(1+x_1 \\xi+\\ldots),\\qquad Z_{sc}=X_{sc}+O(\\xi^{2-p}).\n\\end{equation}\nTheir proper time of reaching the horizon $\\sim \\int d\\xi \/Z_{sc}$ is finite.\n\\end{enumerate}\n\nAs shown above (\\ref{ga}), for collision of two particles the relative Lorentz factor\nis \n\\begin{equation}\n\\gamma_{c.m.} =\\frac{X_1 X_2 -Z_1 Z_2}{m_1 m_2 N^2}+O(1).  \\label{RelGamma}\n\\end{equation}\nFor collision of two usual or two critical particles near extremal horizon, for which $N^2 \\sim\n\\xi^2$ (\\ref{Extr}), we have $\\gamma_{c.m.} =O(1)$; for usual and critical $\\gamma_{c.m.} \\sim 1\/\\xi$.\nLikewise for usual and sub-critical one obtains \n\\begin{equation}\n\\gamma_{c.m.}\\sim \\xi^{-p} \\to \\infty ;\n\\end{equation}\nfor critical and sub-critical \n\\begin{equation}\n\\gamma_{c.m.} \\sim \\xi^{p-1}\\to \\infty .  \\label{SubGamma}\n\\end{equation}\nSo, the corresponding particles behave as\ncritical in collisions with usual ones and as usual in collisions with\ncritical ones.\n\nThis result can be derived in the general setting. First of all, let there\nbe a particle with \n\\begin{align}\n&X=\\alpha \\xi^p (1+O(\\xi)),\\qquad p\\in [0,1]; \\label{alpha} \\\\\n& \\frac{L^2}{g_{\\phi}}+m^2 =\\beta^2+O(\\xi),\\qquad \\alpha,\\beta\\sim 1. \\label{beta}\n\\end{align}\n\nThen \n\\begin{equation}\nZ-X =-C\\xi^{2-p}(1+O(\\xi)),  \\label{Z-X}\n\\end{equation}\nwhere \n\\begin{equation}\nC=\\left\\{\\begin{array}{ll}\n\t\\beta^2 \/ 2\\alpha & \\quad \\text{for}\\quad p< 1; \\\\ \n\t\\alpha-\\sqrt{\\alpha^2 -\\beta^2} & \\quad \\text{for}\\quad p=1 .\n\\end{array}\\right.\n\\end{equation}\nAs due to forward in time condition $\\alpha>0$ and $\\beta^2$ is also positive, $C$ is strictly positive as well.\n\nNow suppose we have two such particles, with $p_{1}$ and $p_{2}$, colliding\nnear the horizon. Then using (\\ref{RelGamma}) and (\\ref{Z-X}), the relative\nLorentz factor is reduced to \n\\begin{align}\nm_{1}m_{2}\\,\\gamma _{c.m.}& \n\t=O(1)+\\big(1+O(\\xi )\\big)\\Big[C_{1}\\alpha _{2}\\xi^{p_{2}-p_{1}}\n\t+C_{2}\\alpha _{1}\\xi ^{p_{1}-p_{2}}\\Big]  \\label{gamma12-1} \\\\\n& \\sim \\xi ^{-|p_{1}-p_{2}|}\\big[1+O(\\xi )+O(\\xi ^{2|p_{1}-p_{2}|})\\big],\n\\label{gamma12-2}\n\\end{align}%\nand thus \n\\begin{equation}\n\\gamma_{c.m.}\\sim \\xi ^{-|p_{1}-p_{2}|}.  \\label{p1-p2}\n\\end{equation}%\nHere, gamma becomes finite only if $p_{1}=p_{2}$. We see that consideration\nof sub-critical particles is convenient, as it allows to describe usual and\ncritical particles in a more coherent and unified way, while at the same\ntime providing greater generality, necessary when dealing with non-geodesic\nmotions.\n\n\\section{Dynamics}\n\n\\subsection{OZAMO and FZAMO frames}\n\nThere are two main qualitatively different frames of reference in the\nvicinity of a black hole horizon. The tetrad vectors and tetrad components\nof different quantities will be denoted by superscripts in parenthesis,\nwhile low case ``o'' or ``f'' in the subscript will denote which frame is used, i.e. $%\na_{o}^{(t)}$ is the $t$-component of acceleration in the OZAMO frame (see\nbelow).\n\n\\paragraph{OZAMO.}\n\nThe first kind of frame is attached to an observer who is orbiting the black hole with\nconstant $r$, having constant energy and zero angular momentum. We will call\nit OZAMO for orbital zero angular momentum observer\\footnote{This observer is usually called just ZAMO in textbooks, but we need to be more specific.}. It is the analogue of\nthe static observer in a static spacetime, and it becomes lightlike in the\nhorizon limit \\cite{72}.\n\nThe tetrad 1-forms of the OZAMO frame, denoted by small ``o'' subscripts,\nread \n\\begin{align}\ne^{(t)}_o& =-Ndt;  \\label{zamo} \\\\\ne^{(\\phi )}_o& =g_{\\phi }^{1\/2}\\;(d\\phi-\\omega dt); \\\\\ne^{(r)}_o& =A^{-1\/2}dr; \\\\\ne^{(z)}_o& =g_{z}^{1\/2}dz.  \\label{zamo-z}\n\\end{align}\n\nIf another particle's four-velocity is $u^{\\mu }$, then its Lorentz factor\nin this frame is \n\\begin{equation}\n\\gamma=-u^{\\mu }(e_{o }^{(t)})_\\mu=\\frac{X}{mN}.  \\label{gx}\n\\end{equation}\nThus for a particle with $X\\sim \\xi^p$\n\\begin{equation}\n\\gamma \\sim \\xi^{p-1}; \\label{gx1}\n\\end{equation}\nfor a usual particle it diverges in the horizon limit, while for a critical\none it stays finite.\n\nA particle's acceleration is \n\\begin{equation}\na^{\\mu }\\equiv u^{\\nu }\\nabla _{\\nu }u^{\\mu }.\n\\end{equation}\nIts tetrad components in the OZAMO frame \n\\begin{equation*}\na^{(i)}_o=a^\\mu (e^{(i)}_o)_{\\mu},\\qquad i=t,\\phi,r,z,\n\\end{equation*}\nare equal to \n\\begin{align}\na^{(t)}_o& =Na^{t};  \\label{a} \\\\\na^{(\\phi) }_o& =\\sqrt{g_{\\phi }}\\;(a^{\\phi }-\\omega a^{t}); \\label{a-phi} \\\\\na^{(r)}_o& =\\frac{1}{\\sqrt{A}}\\;a^{r}; \\label{a-r}\\\\\na^{(z)}_o& =\\sqrt{g_{z}}a^{z}; \\label{a-z}\n\\end{align}%\nthe acceleration scalar then can be presented as \n\\begin{equation}\na^{2}\\equiv a^{\\mu }a_{\\mu } =-(a^{(t)}_o)^{2} +(a^{(\\phi)}_o)^{2}\n+(a^{(r)}_o)^{2}+(a^{(z)}_o)^{2}.\n\\end{equation}\n\nThe OZAMO orbits the horizon at constant $r$ and does not cross it,\ntherefore it is not classified as either usual or critical particle, which\ndoes cross or approach the horizon in infinite proper time respectively. However, it\nis useful to note, that, as its Lorentz factor is finite with respect to a\ncritical particle, and vice versa, in the discussion that follows, OZAMO and\ncritical particles behave similarly.\n\nIt is well-known, that an OZAMO frame breaks down at the horizon, thus strictly speaking at the horizon it is not a valid frame and OZAMO is not an observer in the traditional sense. Hereafter, what we refer to as the values of some quantities measured in the OZAMO frame in the horizon limit are the limits of the corresponding quantities measured in successive different OZAMO frames, with different $r_{ZAMO}$, when $r_{ZAMO}\\to r_H$.\n\n\\paragraph{FZAMO.}\n\nThe other important frame of reference is realized by one of the usual\nparticles crossing the horizon. For simplicity, it is convenient to take for\nsuch an observer $L=0$ similarly to OZAMO and, additionally, $E=m$. Thus we\nwill call the corresponding observer FZAMO for falling zero angular\nmomentum observer. Its frame $\\{e_{f}^{(i)}\\}$, with $i=t,\\phi ,r,z$, is\nconstructed by making a local Lorentz transformation from the OZAMO in the\ndirection towards the horizon\\footnote{Note that this is the transformation for one-forms; vectors are\ntransformed by the inverse matrix, which differs by the sign of $v_{f}$.}: \n\\begin{align}\n\\begin{pmatrix}\n(e^{(t)}_{f})_{\\mu} \\\\ \n(e^{(r)}_{f})_{\\mu}%\n\\end{pmatrix}\n&=\\gamma_f\n\\begin{pmatrix}\n1 & v_f \\\\ \nv_f & 1%\n\\end{pmatrix}\n\\begin{pmatrix}\n(e^{(t)}_o)_{\\mu} \\\\ \n(e^{(r)}_o)_{\\mu}\n\\end{pmatrix}\n\\label{rrt}\n\\end{align}%\nThe FZAMO's Lorentz factor in the OZAMO frame is $\\gamma _{f}=-\\left( u^{\\mu\n}\\right) (e_{o}^{(t)})_{\\mu }$, where $u^{\\mu }$ is given by (\\ref{Uu}) with $L=0$ and $E=m$, and $v_{f}=\\sqrt{1-\\gamma_{f}^{-2}}$.\n\nThen, \n\\begin{equation}\n\\gamma _{f}=\\frac{1}{N},\\qquad v_{f}=\\sqrt{1-N^{2}}.  \\label{va}\n\\end{equation}\n\nThe corresponding tetrad components of acceleration \n\\begin{equation}\na_{f}^{(i)}=a^{\\mu }(e_{f}^{(i)})_{\\mu },\\qquad i=t,\\phi ,r,z  \\label{aa}\n\\end{equation}%\nare related to $a_{o}^{(i)}$ by the respective Lorentz transformation\nwhich becomes singular on the horizon, where $N\\rightarrow 0$, $\\gamma\n_{f}\\rightarrow \\infty $.\n\n\\paragraph{Proper frame.}\n\nFor non-critical particles with $p<1$ the Lorentz factor relative to the\nOZAMO frame diverges as $\\gamma \\sim \\xi ^{p-1}$ (\\ref{SubGamma}). Thus the\ncorrect reference frame for it will have the same behaviour of Lorentz\nfactor. We will construct it, analogously to FZAMO, by making the\ncorresponding boost in the radial direction, and call it for simplicity the\nproper frame for a particle, although it may not be exactly proper. What is\nimportant is that, in contrast to the OZAMO, the particle's velocity in it\nstays finite (does not tend to $c$).\n\nThus, given a particle's Lorentz factor in the OZAMO frame, $\\gamma$\n(without subscripts), the tetrad components of acceleration in the proper\nframe are \n\\begin{equation}\n\\begin{pmatrix}\na^{(t)}_{pr} \\\\ \na^{(r)}_{pr}%\n\\end{pmatrix}\n=\\gamma \n\\begin{pmatrix}\n1 & v \\\\ \nv & 1%\n\\end{pmatrix}\n\\begin{pmatrix}\na^{(t)}_o \\\\ \na^{(r)}_o%\n\\end{pmatrix}%\n.  \\label{AProper}\n\\end{equation}\nFor a usual or critical particle this reduces to the already considered\nOZAMO and FZAMO frames respectively, while for sub-critical particles the proper frame\ndoes not coincide with either one of those.\n\n\\subsection{Acceleration in different frames}\n\nWhen describing particles' motion near the horizon, we must restrict\nourselves to particles with finite acceleration. This necessarily means that\nthe acceleration scalar $a^{2}$ should be finite. It would seem that it is\nnatural to demand that tetrad components of acceleration are finite as well.\nHowever, as shown above, when we describe a particles' motion near the\nhorizon, we have different frames of reference, which are related to each\nother by singular Lorentz transformations. This means that finite tetrad\ncomponents of acceleration in one of the frames may correspond to diverging\ntetrad components in the other or vice versa.\n\nThe frame in which tetrad components of a particle's acceleration should be\nfinite is the instantly comoving frame, or equivalently, any frame which\nmoves with finite Lorentz factor with respect to that. For example, recall\nthe reasonably realistic problem of a charged particle in a uniform electric\nfield in Special Relativity. The tetrad components of acceleration in the\nlaboratory frame (with Minkowski metric) diverge proportionally to the\nLorentz factor, while those in the instantly comoving frame (with the\nRindler metric and the horizon) are constant (see, e.g. p. 403 of \\cite{car}%\n).\n\nFor a critical particle then acceleration is adequately and most easily\nmeasured in the OZAMO frame. For a usual particle we would have to attach\nthe tetrad also to one of the usual particles, for example to FZAMO. Due to\nnormalization of four-velocity $u^\\mu u_\\mu =-1$, which implies $a^\\mu u_\\mu\n=0$, in each case it is sufficient to show that three of the four\ntetrad components are finite.\n\n\\subsection{Energy and angular momentum}\n\nIf $\\xi ^{\\mu }$ is a Killing vector field, then \n\\begin{equation}\n\\frac{d}{d\\tau }(\\xi ^{\\mu }u_{\\mu })=\\xi ^{\\mu }a_{\\mu }.\n\\end{equation}%\nIn a stationary axisymmetric metric we have two Killing vectors $\\xi\n_{t}^{\\mu }=\\delta _{t}^{\\mu }$ and $\\xi _{\\phi }^{\\mu }=\\delta _{\\phi\n}^{\\mu }$, which give \n\\begin{align}\n\\frac{1}{m}\\frac{dE}{d\\tau }& =(N^{2}-\\omega ^{2}g_{\\phi })a^{t}+\\omega\ng_{\\phi }a^{\\phi }; \\\\\n\\frac{1}{m}\\frac{dL}{d\\tau }& =-\\omega g_{\\phi }a^{t}+g_{\\phi }a^{\\phi },\n\\end{align}%\nor through the tetrad components in the OZAMO frame (\\ref{a}, \\ref{a-phi})\n\\begin{align}\n\\frac{1}{m}\\frac{dE}{d\\tau }& =Na^{(t)}_o +\\omega \\sqrt{g_{\\phi }}%\n\\;a^{(\\phi)}_o;  \\label{dEdt} \\\\\n\\frac{1}{m}\\frac{dL}{d\\tau }& =\\sqrt{g_{\\phi }}\\;a^{(\\phi)}_o.  \\label{dLdt}\n\\end{align}%\n\nIt is clear, that if the proper time of crossing the horizon is finite, as\nis the case for the usual particles, then the finiteness of $Na^{(t)}_o$ and \n$a^{(\\phi)}_o$ implies that $E$ and $L$ are also bounded. However, this does\nnot seem to be necessarily so for critical particles, for which the proper\ntime of reaching the horizon diverges.\n\n\\subsection{Dynamic restrictions on a particle's velocity}\n\nLet us enumerate and classify all the possible variants of particle's type\nof asymptotic motion in the vicinity of the horizon, now in more detail than\nin the section on kinematics, so as to focus below only on those that are\nnot explicitly non-physical. \n\nFirst of all, diverging $L$, as seen from (\\ref{dLdt}), would correspond to\ncontinuous acceleration in the $\\phi$ direction, which would cost formally\ninfinitely large amounts of fuel per a unit mass particle. If one has the\nresources to make such experiments, he would not need the BSW effect in\norder to observe (formally) infinite energy in the center of mass frame. So\nthis variant is of not much interest.\n\nSecondly, one could imagine divergent $u^z$. Such a particle would have\nvelocity tending to $c$ and directed along the $z$ axis (or at finite angle\nwith respect to it) both in the OZAMO and FZAMO frames. This would mean that the\nparticle is ``accelerated'' (in the sense that its velocity increases) not\nonly in radial direction, but also along the horizon surface. This would be\nvery strange behaviour, and in the Kerr metric such particles are naturally\nabsent \\cite{ne}. We will not consider this variant here.\n\nGiven these two natural assumptions, from the normalizing condition \n\\begin{equation}\nZ^{2}=X^{2}-N^{2}\\tilde{\\beta}^{2},\n\\end{equation}%\nwhere $\\tilde{\\beta}$ in the horizon limit tends to a positive real number,\nfinite and separated from zero. Consequently, for a particle reaching the\nhorizon, where $Z^{2}$ must remain positive, and at the same time finite $X$ and $E$ (see discussion after Eq. (\\ref{xi})), we have\n\\begin{equation}\nX\\sim \\xi ^{p},\\quad \\text{with}\\quad p\\in[0,1],\n\\end{equation}%\nwhich corresponds to usual, sub-critical and critical particles as discussed\nabove.\n\n\\subsection{Usual particles}\n\nFor a usual particle $X_{H}\\neq 0$ by definition. As discussed above, the\ntetrad components of its acceleration in the FZAMO frame $a^{(i)}_f$ must be\nfinite. Then the components in the OZAMO frame $a^{(i)}_o$, related to them\nvia the singular Lorentz transform (\\ref{va}), with $\\gamma_f\n=1\/N$, can diverge as $1\/N$. Writing out explicitly the asymptotics for the $%\nt$ and $r$ components in both frames, we get \n\\begin{align}\na^{(t)}_f &= (a^{(t)}_f)_{0}+(a^{(t)}_f)_1 N+O(N^2);  \\label{af-t} \\\\\na^{(r)}_f &= (a^{(r)}_f)_{0}+(a^{(r)}_f)_1 N+O(N^2);  \\label{af-r} \\\\\na^{(t)}_o &=+\\frac{ (a^{(t)}_f)_{0}- (a^{(r)}_f)_{0}}{N} +\\big[ %\n(a^{(t)}_f)_{1}-(a^{(r)}_f)_{1}\\big]+O(N);  \\label{ao-t} \\\\\na^{(r)}_o &=-\\frac{ (a^{(t)}_f)_{0}- (a^{(r)}_f)_{0}}{N} -\\big[ %\n(a^{(t)}_f)_{1}-(a^{(r)}_f)_{1}\\big]+O(N),  \\label{ao-r}\n\\end{align}\n\nThe $\\phi $ and $z$ components are the same in the two frames and must be\nbounded: \n\\begin{align}\na^{(\\phi)}_f &=a^{(\\phi)}_o =O(1); \\\\\na^{(z)}_f &=a^{(z)}_o =O(1).\n\\end{align}\nThen, we see that if $a^{(i)}_f=O(1)$, the right hand side of (\\ref{dEdt}) is\nfinite. The left hand side is also finite, as for a usual particle, given $%\nu^{r}\\sim 1$ (\\ref{ru}), $dr\\sim d\\tau \\sim dN$.\n\nThe explicit expressions for $a_{o}^{(r)}$ and $a_o^{(z)}$ are \n\\begin{align}\na_o^{(r)}=& \\frac{1}{\\sqrt{A}}\\Big\\{(u^{r}\\partial _{r}+u^{z}\\partial\n_{z})u^{r} -\\frac{A^{\\prime }}{2A}(u^{r})^{2} -\\frac{A}{2}\\partial\n_{r}g_{z}(u^{z})^{2}+  \\notag \\\\\n& -\\frac{A}{2}\\Big[X^2 \\partial_r N^{-2}-L^2 \\partial_r g_\\phi^{-1}\n\t-2\\frac{XL}{N^2}\\partial_r \\omega\\Big]\\Big\\};  \\label{aR}\n\\\\\na_o^{z}=& \\sqrt{g_z}\\Big\\{ (u^{r}\\partial _{r}+u^{z}\\partial _{z})u^{z} +%\n\\frac{\\partial _{z}g_{z}}{2g_{z}}(u^{z})^{2} \n\t+\\frac{\\partial _{r}g_{z}}{g_{z}}u^{r}u^{z}+  \\notag \\\\\n& -\\frac{1}{2 g_z}\\Big[X^2 \\partial_z N^{-2}-L^2 \\partial_z g_\\phi^{-1}\n\t-2\\frac{XL}{N^2}\\partial_z \\omega\\Big]\\Big\\}.\n\\label{aZ}\n\\end{align}\n\nThe conditions $a^{(i)}_f=O(1)$ can be reformulated in the form of\nrestrictions on the coefficients $\\alpha_k$ and $\\beta_k$ in the expansions \n\\begin{align}\nu^{r}&=\\alpha _{0}(z)+\\alpha _{1}(z)(r-r_{H})+O((r-r_{H})^{2}),  \\label{u} \\\\\nu^{z}&=\\beta _{0}(z)+\\beta _{1}(z)(r-r_{H})+O((r-r_{H})^{2}).\n\\end{align}\n\n\\subsection{Critical particles}\n\nSuch particles approach the horizon but, in contrast to usual ones, the\nprocess takes infinite proper time. On the other hand, as seen from (\\ref{xc}) and (\\ref{gx}), their Lorentz factor in the OZAMO frame $\\gamma $ is\nfinite, and the velocity is $v<1$, so the tetrad components of acceleration\nin the OZAMO frame must be finite. As mentioned above, we consider only\nmotion with $E$ and $L$ bounded in the horizon limit $\\tau \\rightarrow\n\\infty $. This means that $a_{o}^{(\\phi )}$ should be not only bounded, but\nintegrable (\\ref{dLdt}): $\\int d\\tau a_{o}^{(\\phi )}<\\infty $. If we assume\nthat $a_{o}^{(\\phi )}$ is expandable in power series by $r$ with integer powers, this means \n\\begin{equation}\na_{o}^{(\\phi )}=O(N).\n\\end{equation}%\nWith this condition satisfied, and Eq. (\\ref{ao-t}) taken into account, the boundedness of $E$ from (\\ref{dEdt}) does\nnot give any more restrictions on $a_{o}^{(t)}$. Then, using that $E$, $L$,\nand $u^{z}$ are bounded (\\ref{el}), while $u^{r}$ and $X$ are $O(N)$, and $\\omega\n_{H}=const$ (which follows from regularity \\cite{reg}), it is easy to see\nthat all the terms in (\\ref{aR}) and (\\ref{aZ}) are automatically finite, so \n\\begin{equation}\na_{o}^{(r)},a_{o}^{(z)}=O(1).\n\\end{equation}\nThus all components of acceleration of a critical particle in the OZAMO\nframe, and therefore in the instantly comoving proper frame, are finite\nunconditionally. This is in contrast to usual particles, for which the\nconditions $a_{f}^{(i)}=O(1)$ impose some additional constraints on $\\alpha\n_{k}$ and $\\beta _{k}$ in (\\ref{u}).\n\nIn the FZAMO frame, and the frame of any usual particle, the picture looks\ndifferent, as the relative Lorentz factor of a usual and critical particle\ndiverges as $1\/N$. Using the Lorentz transformation (\\ref{rrt})\nbetween the OZAMO and FZAMO frames, with $\\gamma_f \\sim 1\/N$, we see that $%\na^{(t)}_f$ and $a^{(r)}_f$ can diverge as $1\/N$. Using (\\ref{rrt}), the asymptotics of these components of acceleration in the two\nframes can be brought to the form \n\\begin{align}\na^{(t)}_o &= (a^{(t)}_o)_{0}+(a^{(t)}_o)_1 N+O(N^2); \\\\\na^{(r)}_o &= (a^{(r)}_o)_{0}+(a^{(r)}_o)_1 N+O(N^2); \\\\\na^{(t)}_f &=\\frac{ (a^{(t)}_o)_{0}+ (a^{(r)}_o)_{0}}{N} +\\big[ %\n(a^{(t)}_o)_{1}+(a^{(r)}_o)_{1}\\big]+O(N); \\\\\na^{(r)}_f &=\\frac{ (a^{(t)}_o)_{0}+ (a^{(r)}_o)_{0}}{N} +\\big[ %\n(a^{(t)}_o)_{1}+(a^{(r)}_o)_{1}\\big]+O(N).\n\\end{align}\nThe $\\phi$ and $z$ components in the two frames are the same and therefore,\nas shown above, satisfy \n\\begin{align}\na^{(\\phi)}_o &=a^{(\\phi)}_f =O(N); \\\\\na^{(z)}_o &=a^{(z)}_f =O(1).\n\\end{align}\n\nThus we have two mutually complimentary cases. In the OZAMO frame $r$ and $t$\ncomponents of acceleration diverge for usual particles and stay finite for\nthe critical ones. In the FZAMO frame, the situation is opposite: $r$ and $t$\ncomponents of acceleration are finite for usual particles and diverge for\nthe critical ones. The $\\phi$ and $z$ components are the same in the two\nframes and are finite. For critical particles, additionally $%\na_{o}^{(\\phi)}=O(N)$ near the horizon for energy and angular momentum to\nremain bounded.\n\n\\section{Example: the Reissner-Nordstr\\\"{o}m metric}\n\nThe approach and results of the present paper are also valid in the case of\nthe electromagnetic interaction with minimal changes: in eq. (\\ref{x}) one\nshould make the replacement $X\\rightarrow X-q\\varphi $, where $\\varphi $ is\nthe electrostatic potential, and $q$ is the particle's charge. In order to\ndemonstrate this, it is instructive to consider as an example the extremal\nReissner-Nordstr\\\"{o}m metric. In this case the metric functions are \n\\begin{equation}\nN=\\sqrt{A}=1-\\frac{r}{r_{H}},\\qquad \\omega =0,\\qquad g_{\\phi }=r^{2},\n\\end{equation}%\nand the electromagnetic field potential is \n\\begin{equation}\nA_{\\mu }=-\\varphi \\delta _{\\mu }^{t},\\qquad \\varphi =\\frac{Q}{r},\n\\end{equation}%\nwhere \n\\begin{equation*}\nQ=r_{H}\n\\end{equation*}%\nis the extremal black hole's charge, so that the only nonvanishing\ncomponents of the electromagnetic field tensor are \n\\begin{equation*}\nF_{rt}=-F_{tr}=\\frac{Q}{r^{2}}.\n\\end{equation*}\n\nFor a particle of charge $q$ moving radially towards the horizon the\nfour-momentum can be parametrized as \n\\begin{equation}\np_\\mu=m u_\\mu=-(X,0,Z\/N^2 ,0) ,\n\\end{equation}\nthen the normalization condition implies \n\\begin{equation}  \\label{zz}\nZ=\\sqrt{X^2 -m^2 N^2}.\n\\end{equation}\nThe equation of motion \n\\begin{equation}\nma^{\\mu }=qF^{\\mu \\nu }u_{\\nu },\n\\end{equation}\nhas the integral of motion \n\\begin{equation}  \\label{RN-energy}\nE=X+q\\varphi =const.\n\\end{equation}\n\nFor usual particles, with $X_H \\neq 0$, \n\\begin{equation}\np^{r}=-X_{H}-\\frac{q}{r_H}(r-r_{H})+O((r-r_{H})^{2}),\n\\end{equation}%\nin agreement with (\\ref{u}).\n\nFor a critical particle\n\\begin{equation}\nE=q,\\quad X=qN,\\quad Z=N\\sqrt{q^{2}-m^{2}}.\\label{RNcrit}\n\\end{equation}\nThen, integrating the equation for radial motion \n\\begin{equation*}\n\\frac{dr}{d\\tau}=-\\frac{Z}{m} =-N(r)\\sqrt{q^2 \/m^2 -1},\n\\end{equation*}\nit is easy to obtain that in the horizon limit the same asymptotic as in eq.\n(\\ref{ur}) holds, with the characteristic time \n\\begin{equation}\n\\tau _{0}=r_H \\Big(\\frac{q^2}{m^2}-1\\Big)^{-1\/2}.\n\\end{equation}\n\nNow, we will consider the acceleration measured by the two types of\nobservers.\n\n\\subsection{Static observers}\n\nThe tetrad (\\ref{zamo})--(\\ref{zamo-z}) in this case turns into the tetrad\nof a static observer. Then, using (\\ref{zamo}) and (\\ref{a}), we obtain \n\\begin{align}\nm a^{(t)}_o&=-\\frac{qQ}{r^{2}}\\,\\frac{Z}{mN}, \\\\\nm a^{(r)}_o&=+\\frac{qQ}{r^{2}}\\,\\frac{X}{mN}, \\\\\n&m^2 a^{2}=\\Big(\\frac{qQ}{r^2}\\Big)^2 .\n\\end{align}\n\nFor the critical particle (\\ref{RNcrit}) both components of acceleration \n\\begin{align}\nm a^{(t)}_o&=-\\frac{qQ}{r^{2}}\\sqrt{q^{2}\/m^{2}-1}, \\\\\nm a^{(r)}_o&=\\frac{qQ}{r^{2}}\\frac{E}{m}\n\\end{align}\nare finite on the horizon, and can be expanded into a series by $(r-r_{H})$\nor $N$.\n\nHowever, for a usual particle, with $X_{H}\\neq 0$, near the horizon \n\\begin{equation*}\nX=X_{H}+O(N),\\quad Z=X+O(N^2),\n\\end{equation*}\nso \n\\begin{align}\n&a^{(r)}_o \\approx -a^{(t)}_o= \\frac{a_{-1}}{N}+O(1),\n\\end{align}\nwhere \n\\begin{equation}\na_{-1}=\\frac{q}{Q}\\frac{X_{H}}{m^{2}}.\n\\end{equation}\nThus both components diverge near the horizon, in accordance with (\\ref{ao-r}%\n,\\ref{ao-t}), while satisfying \n\\begin{equation}\na^{(r)}_o+a^{(t)}_o=O(N).  \\label{rt-sum}\n\\end{equation}\n\n\\subsection{Falling observers}\n\nThe falling frame $e^{(i)}_f$ is attached to a particle falling into the\nblack hole according to (\\ref{rrt})--(\\ref{va}), with the Lorentz factor $%\n\\gamma_{f}=1\/N$ and velocity $v_f =1-O(N^2)$ in the static frame.\n\nIn this frame the tetrad components of acceleration are equal to \n\\begin{align}\nm a^{(t)}_f&=-\\gamma_F \\;\\frac{qQ}{r^{2}}\\,\\frac{Z-v_F X}{mN}; \\\\\nm a^{(r)}_f&=+\\gamma_F \\;\\frac{qQ}{r^{2}}\\,\\frac{X-v_F Z}{mN}.\n\\end{align}\n\nFor usual particles, in the horizon limit $N\\rightarrow 0$, $X_{H}\\neq 0$.\nThen (\\ref{va}) and (\\ref{zz}) imply that $Z-X=O(N^{2})$, so $a^{(t)}_f$ and $a^{(r)}_f$\nare finite.\n\nIf the particle under consideration is critical, then $Z\\sim X\\sim N$, and\nboth components of acceleration diverge: \n\\begin{align}\na^{(t)}_f=&\\frac{\\tilde{a}_{-1}}{N}+O(N), \\\\\na^{(r)}_f=&\\frac{\\tilde{a}_{-1}}{N}+O(N), \\\\\n&\\tilde{a}_{-1} =r_H^{-1}\\,\\frac{q}{m} \\Big[\\frac{q}{m}-\\sqrt{%\n\\frac{q^2}{m^2}-1}\\Big].\n\\end{align}\nThus we see that, indeed, all the general properties (\\ref{af-t}), (\\ref{af-r}), (\\ref{ao-t}), (\\ref{ao-r}), described in the preceding section, are\nexplicitly verified in this exactly solvable case.\n\n\\section{BSW effect under finite forces: equatorial motion}\n\n\\subsection{Motion in equatorial plane}\n\nLet $m=1$, and let us consider motion in the equatorial plane so that $%\nu^{z}=0$ and $a_{o}^{(z)}=0$. Then for arbitrary motion we have 1) the\nnormalization condition for velocity \n\\begin{equation}\nu^{r}=-\\frac{\\sqrt{A}}{N}Z,\\qquad Z=\\sqrt{X^{2}-N^{2}\\Big[\\frac{L^{2}}{%\ng_{\\phi }}+1\\Big]},  \\label{zeq}\n\\end{equation}%\nand 2) orthogonality condition for acceleration, which can be written in terms of (\\ref{a}--\\ref{a-z}) as \n\\begin{align}\n0=u_{\\mu }a^{\\mu }& =+u_{t}a^{t}+u_{\\phi }a^{\\phi }+u_{r}a^{r} \\\\\n& =-Ea^{t}+La^{\\phi }+A^{-1}u^{r}a^{r} \\\\\n& =-\\frac{X}{N}a_{o}^{(t)}+\\frac{L}{\\sqrt{g_{\\phi }}}a_{o}^{(\\phi )}\n\t+\\frac{u^{r}}{\\sqrt{A}}a_{o}^{(r)}.  \\label{Orth3a}\n\\end{align}%\nGenerically, at least two of the three components of $a^{(i)}$ have to be\nnon-zero if there is acceleration. Also for simplicity we will assume\\footnote{This assumption is purely technical. In general, one should write $A=N^2 B$, where $B$ is some bounded function which does not vanish at the horizon. Its form does not affect the results qualitatively but leads to \nmore cumbersome expressions. Thus we put for simplicity $B=1$, which also\nfixes the time scale.} that $A=N^{2}$, so that $u^{r}=-Z$ and orthogonality condition takes form\n\\begin{equation}\n\\frac{X}{N}a_{o}^{(t)}-\\frac{L}{\\sqrt{g_{\\phi }}}a_{o}^{(\\phi )}\n\t+\\frac{Z}{N}a_{o}^{(r)}=0.\\label{Orth3}\n\\end{equation}\n\nOf the four components of the equation of motion \n\\begin{equation}\n(u^\\mu \\nabla_\\mu) u^\\nu=a^\\nu\n\\end{equation}\none is trivial\\footnote{In the equatorial plane derivatives of metric functions by $z$ in (\\ref{aZ}) must vanish due to symmetry.} $a^{z}=0$, and the other three are related through the\northogonality condition, so it is always sufficient to consider only two\ncomponents, for example (\\ref{dEdt}) and (\\ref{dLdt}), which can be written\nin terms of $X$ and $L$ as \n\\begin{align}\n\\frac{dX}{d\\tau}&=N a_o^{(t)}-L\\frac{d\\omega}{d\\tau};  \\label{Xeq} \\\\\n\\frac{dL}{d\\tau}&=\\sqrt{g_\\phi}\\; a_o^{(\\phi)} .  \\label{Leq}\n\\end{align}\nAs $dr\/d\\tau =u^r =-Z$, in terms of $X$ and derivatives by $\\xi\\equiv (r-r_H\n)$, which are denoted by primes, this can be written as \n\\begin{align}\n&X^{\\prime}+L\\omega^{\\prime}=-\\frac{N}{Z} a_o^{(t)};  \\label{Xeq2} \\\\\n&L^{\\prime}=-\\frac{\\sqrt{g_\\phi}}{Z}\\; a_o^{(\\phi)} .  \\label{Leq2}\n\\end{align}\nIt can be checked that, indeed, in case $u^z =0$, equations (\\ref{Xeq}) and (%\n\\ref{Leq}) together with (\\ref{Orth3}) give (\\ref{aR}).\n\n\n\n\\subsection{Acceleration in proper frame}\nExpressing acceleration components in the OZAMO frame through the particle's parameters $E$ and $L$ from (\\ref{Xeq2},\\ref{Leq2}) and the orthogonality condition (\\ref{Orth3}), one obtains\n\\begin{align}\na_o^{(\\phi)}&=-\\frac{Z}{\\sqrt{g_\\phi }}L' ;\\label{EQao-phi}\\\\\na_o^{(t)}&=-\\frac{Z}{N}(X'+L\\omega');\\label{EQao-t}\\\\\na_o^{(r)}&=-\\frac{X}{Z}a_o^{(t)}-N\\frac{LL'}{g_\\phi}.\\label{EQao-r}\n\\end{align}\nFor critical particles the OZAMO frame is the proper frame. For other types of particles the $r$ and $t$ components of acceleration in the proper frame are given by (\\ref{AProper}) with Lorentz factor (\\ref{gx}) \n\\begin{equation}\n\\gamma =\\frac{X}{N},  \\label{gx2}\n\\end{equation}\nwhile $a_{pr}^{(\\phi)}=a_{o}^{(\\phi)}$ for any type. Using (\\ref{EQao-phi}--\\ref{EQao-r}), this gives\n\\begin{equation}\n\\begin{pmatrix}\na_{pr}^{(t)}\\\\a_{pr}^{(r)}\n\\end{pmatrix}=\n\\frac{X}{N}\\left\\{\n\t\\frac{a_o^{(t)}}{Z}\n\t\\begin{pmatrix}\n\t\tZ-vX \\\\ Zv-X\n\t\\end{pmatrix}\n\t-N\\begin{pmatrix}\n\t\tv\\\\1\n\t\\end{pmatrix}\n\t\t\\frac{LL'}{g_\\phi}\\right\\}. \\label{ACCproper}\n\\end{equation}\n\nSuppose we have a particle with\n\\begin{align}\n\tX&=\\alpha \\xi^p (1+O(\\xi)),\\qquad \t\\gamma=\\frac{X}{\\xi},\\\\\n\tv&=\\sqrt{1-\\gamma^{-2}}=1-\\frac{1}{2\\alpha}\\xi^{2(1-p)}(1+O(\\xi)),\n\\end{align}\nwhere $p<1$. Using (\\ref{Z-X}), we get\n\\begin{align}\nZ-Xv&=(\\tfrac12 -C)\\xi^{2-p}(1+O(\\xi)), \\label{Z-Xv}\\\\\nX-Zv&=(\\tfrac12 +C)\\xi^{2-p}(1+O(\\xi)).\\label{X-Zv}\n\\end{align}\n\n\\subsubsection{Usual particles}\nFor a usual particle $p=0$, so assuming $L'$ is bounded, $a_o^{(\\phi,r,t)}=O(1)$, while $\\gamma \\sim 1\/\\xi$, and one can easily check term by term that \nacceleration in the proper frame (\\ref{ACCproper}) is always bounded: as expected, \\textbf{for usual particles there are no additional requirements}.\n\n\\subsubsection{Sub-critical particles}\nFor a sub-critical particle $p\\in(0,1)$. As $X\\sim Z$, the derivative $X^{\\prime}\\sim \\xi^{p-1}$ in (\\ref{EQao-t}) diverges, while $L\\omega^{\\prime}=O(1)$, so \n\\begin{equation}\n\\frac{a_o^{(t)}}{Z}\\sim \\xi^{p-2}.\n\\end{equation}\nThen taking into account (\\ref{Z-Xv}--\\ref{X-Zv}), the first term in the braces of (\\ref{ACCproper}) is $O(1)$, and different in the two rows, thus separated from zero. \n\nThe second term could only compensate the first one (in one of the two rows), if $L' \\sim \\xi^{-1}$, which would imply divergent $L\\sim \\ln \\xi$. Therefore the quantity in the braces is finite and separated from zero, so the proper acceleration diverges as (\\ref{gx2})\n\\begin{equation}\n\\gamma =\\frac{X}{N}\\sim \\xi^{p-1}\\to \\infty .\n\\end{equation}\nThis means that there are \\textbf{no sub-critical particles} with finite acceleration for motion in the equatorial plane.\n\n\\subsubsection{Critical particles}\nThe only remaining case to be considered is critical particles. Although (\\ref{ACCproper}) for them is unnecessary, one restores the acceleration in the OZAMO frame from it by setting $\\gamma=1$ and $v=0$. We see that\\footnote{Remember that components $a_o^{(i)}$ are related through the orthogonality condition (\\ref{Orth3}); if two of them are finite, then the third is bounded as well.}\n\\begin{align}\n\ta_o^{(\\phi)}&\\sim \\xi L' ;\\\\\n\ta_o^{(r,t)}&\\sim (X'+L\\omega '),\n\\end{align}\nso in order for such trajectory to be realized we need the azimuthal force to tend to zero fast enough:\n\\begin{equation}\na_o^{(\\phi)}=O(\\xi).\n\\end{equation}\nThere is no restriction on the radial component: it can be of the order of unity, as it will still be possible to fine-tune a critical particle by the appropriate choice of initial condition (this will be shown in more detail in the next Section). Thus the radial component does not affect or hinder the existence of critical trajectories and consequently the BSW effect.  This is in agreement with the already established fact that the radial force itself is the reason for the BSW effect near charged nonrotating black holes \\cite{jl}.\n\n\\subsection{Example: azimuthal dissipative force}\nLet us consider the particular case when the radial force, which does not hinder critical particles anyway, is absent:\n\\begin{equation}\na_{o}^{(r)}= 0,\\qquad a_{o}^{(t)}, a_{o}^{(\\phi)}\\neq 0 .\n\\end{equation}\nUsing orthogonality (\\ref{Orth3}), \n\\begin{equation}\na_o^{(t)}=\\frac{N}{X}\\frac{L}{\\sqrt{g_\\phi}} a_o^{(\\phi)} ,  \\label{OrthAD}\n\\end{equation}\nso in terms of derivatives with respect to $\\xi$ Eqs. (\\ref{Xeq2}) and (\\ref{Leq2}) can be rewritten as \n\\begin{align}\n&g_\\phi X(X^{\\prime}+L\\omega ^{\\prime})=N^2 LL^{\\prime};  \\label{AD-eq1} \\\\\n&a_o^{(t)}=-N \\frac{Z}{X}\\frac{LL^{\\prime}}{g_\\phi} ;  \\label{AD-eq2}\\\\\n&a_o^{(\\phi)}=-Z\\frac{L^{\\prime}}{\\sqrt{g_\\phi}}.\\label{AD-eq3}\n\\end{align}\n\n\\subsubsection{Tuning a critical particle}\nIn this section we show in more detail how one would tune the particle to be\ncritical $X\\sim \\xi$.\n\nAssuming expansions \n\\begin{align}\nN^2 &=\\nu_2 \\xi^2 +\\nu_3 \\xi^3 +\\ldots , \\\\\n\\omega&=\\omega_H -\\omega_1 \\xi +\\omega_2 \\xi^2 +\\ldots , \\\\\ng_\\phi &=g_H +g_1 \\xi +g_2 \\xi^2 +\\ldots , \\\\\nX&=x_1 \\xi +x_2 \\xi^2 +\\ldots , \\\\\nL&=l_H + l_1 \\xi +l_2 \\xi^2 +\\ldots ,\n\\end{align}\nfrom (\\ref{AD-eq1}) we obtain in consecutive orders \n\\begin{align}\nl_H &=\\frac{x_1}{\\omega_1}; \\\\\nl_{1}&=2\\frac{x_1 \\omega_2 +x_2 \\omega_1}{\\omega_1^2 +\\nu_2 \/g_H}; \\\\\nl_{2}&=l_{2} (x_1, x_2, x_3),\\quad \\ldots\n\\end{align}\n\nThen \n\\begin{equation}\n\\frac{Z^{2}}{\\xi^2}\n\t\\approx x_1^2 \\Big[1-\\frac{\\nu_2}{g_H}\\omega_1^{-2}\\Big]-\\nu_2 .\n\\end{equation}\nThere is a critical particle for \n\\begin{equation}\n|x_1| >x_{1\\min},\n\\end{equation}\nand there is a solution \n\\begin{equation}\nx_{1\\min}=\\frac{\\nu_2}{1-\\frac{\\nu_2}{g_H}\\omega_1^{-2}}\n\\end{equation}\nas long as \n\\begin{equation}\n\\omega_1^2 > \\frac{\\nu_2}{g_H}.  \\label{OmegaCond}\n\\end{equation}\n\nFrom (\\ref{AD-eq2}) we get \n\\begin{equation}\na_o^{(t)}=-\\frac{Z}{N}(X^{\\prime}+L\\omega ^{\\prime})\n\\end{equation}\nand after substitution of expansions for $X,L$ and $\\omega$, \n\\begin{equation}\na_o^{(t)}\\approx -\\frac{2Z}{\\omega_1 \\sqrt{\\nu_2}}\\frac{x_1 \\omega_2 +x_2\n\\omega_1}{1+g_H \\omega_1^2 \/\\nu_2}\\sim Z \\sim \\xi .\n\\end{equation}\nFurther terms are obtained straightforwardly but they are quite cumbersome.\n\nSo, if acceleration is expanded in a series by $\\xi$ \n\\begin{equation}\na_o^{(t)}=a_1 \\xi +a_2 \\xi^2 +\\ldots,\n\\end{equation}\nin the first order we obtain $a_{1}(x_1 ,x_2)$. As long as $a =O(\\xi)$, and the\nmetric coefficients satisfy (\\ref{OmegaCond}), we can take arbitrary $x_{1}$\nsuch that $|x_1|>x_{1\\min}$ (or equivalently $l_{H}=x_{1}\/\\omega_{1}$). Then\nfor the given $a_1$ in the first order we obtain $x_{2}(a_1)$, in the next\norder $x_3 (a_1, a_2)$ and so on. The set of critical trajectories\\footnote{For large enough $|x_1|$ the turning point, given by $Z=0$, will be at finite values of $(r-r_H )$ from the horizon. Thus it will be at the coordinate distance that does not have to be small in order to gain arbitrarily large $E_{c.m.}$ at the collision event near the horizon. This is in contrast to the case discussed in \\cite{BSW2}, which is realized near the turning point of a usual particle with small $X_H$, and the turning point itself must be close to the horizon: the nearer it is, the larger $E_{c.m.}$ is achieved.} is parametrized by one free parameter $x_1$ (or $l_H$).\n\n\\subsubsection{Other realizations of critical trajectories}\nSuppose now the azimuthal force tends to zero as $\\xi^s$ with some integer $s>1$. Then from (\\ref{AD-eq3}) we see that $L' \\sim \\xi^{s-1}$, and therefore expansion (\\ref{el}) for $L$ near the horizon takes the form\n\\begin{equation}\nL=L_H +L_{s}\\xi^{s}(1+o(1)).\\label{LS}\n\\end{equation}\nIt is perfectly consistent with the particle being critical, so that $X\\sim \\xi$: (\\ref{AD-eq1}) can be satisfied for any integer $s>1$ and solved for $X(L)$ (or, equivalently, $E(L)$) in each consecutive order by $\\xi$. Let us consider, for example, the case $s=2$. Assuming\n\\begin{equation}\nX=x_1 \\xi +x_2 \\xi^2 +x_3 \\xi^3 +O(\\xi^4),\n\\end{equation}\nfrom (\\ref{AD-eq1}) in consecutive orders one obtains\n\\begin{align}\nx_{1}&=L_{H}\\omega_{1};\\\\\nx_{2}&=-2\\omega_{2}L_{H};\\\\\nx_{3}&=-\\omega_3 L_H +\\frac{\\omega_1}{3}L_2 +\\frac{2}{3g_{\\phi H}}L_H L_2 ;\\\\\n\\ldots \\notag\n\\end{align}\nThis can be turned around to give $L_H (x_1)$ and $L_2 (x_3)$, but $x_2 \/ x_1$ is fixed to metric function coefficients; in terms of $E(L)$ this is\n\\begin{equation}\nE=\\omega_H L_H +(L_2 \\omega_H -L_H \\omega_2)\\xi^2 +\\ldots .\n\\end{equation}\nFor other integer $s$ the procedure is analogous. \n\n\n\\section{Energy bounds in collisions with near-critical particles}\nWe have seen in the previous section that as long as the azimuthal force is weak enough, critical particles exist and can be tuned via initial conditions. Then the BSW effect in its primary version \\cite{ban} manifests itself. However, what if this is not the case and azimuthal force is e.g. separated from zero on the horizon? The condition for critical particles is that\n\\begin{equation}\na^{(\\phi)} =O(x) \\label{Need},\n\\end{equation}\nwhere\n\\begin{equation}\nx=\\frac{\\xi}{r_H}  \\label{Xdimensionless}\n\\end{equation}\nis the dimensionless radial coordinate. Suppose that instead\n\\begin{equation}\na^{(\\phi)}(x)\\approx \\frac{a_{0}x ^{\\lambda }}{r_H},\\qquad \\lambda <1 ,  \\label{Asmall}\n\\end{equation}\nso that (\\ref{Need}) is violated, and we factored out the dimensional quantity $r_H^{-1}$, so that $a_0$ is dimensionless.\n\nDoes it mean that the BSW effect necessarily breaks down? Under no\nadditional assumptions -- yes. However, what if $a_0$ is small? Radiation\nreaction forces are usually considered very small (see e.g. \\cite{ne}, \\cite{Kesden}). In that case, the question is how high $E_{c.m.}$ can be\nachieved for the given small $a_{0}$ ? \n\nLet us reformulate the condition that \\emph{is} satisfied (\\ref{Asmall}) via\nanother small parameter: \n\\begin{equation}\na^{(\\phi)} (x)\\approx r_H^{-1} \\; x^{\\lambda}x_{m}^{1-\\lambda},\n\\end{equation}\nwhere \n\\begin{equation}\nx_{m}=a_0^{\\frac{1}{1-\\lambda}}\\ll 1 .\n\\end{equation}\nThen \n\\begin{equation}\na^{(\\phi)}(x_m) \\approx r_H^{-1} x_m\n\\end{equation}\nand for all $x\\gtrsim x_m$ the necessary condition for acceleration (\\ref{Need}) is\neffectively obeyed.\n\nBut then for $\\xi\\gtrsim r_H x_m$ the trajectory of a particle can be\neffectively tuned to be critical, (or sub-critical, for the chosen $p$), at\nwill. At the near-horizon end of this region, i.e. at $\\xi_{m}\\sim r_H x_m$,\nthe Lorentz factor with a usual particle with $p_2 =0$ will behave as\ndescribed (\\ref{p1-p2}) and can grow very large. Thus, for $p=1$ (the particle is tuned to be critical) from (\\ref{gamma12-1}) we get \n\\begin{equation}\n\\gamma_{12}^{(max)} \\approx \n\t\\gamma_{12}(x_m) \\approx \n\t\t\\frac{C_1 \\alpha_2}{r_H} \\cdot a_0^{-\\frac{1}{1-\\lambda}}.\n\\end{equation}\n\nWe see that as long as the amplitude $a_0$ of the azimuthal force acting on the particle is small enough, the BSW effect survives almost any\nkind of perturbation: one only has to calculate accurately the corresponding tuning parameters for the effectively critical trajectories.\n\n\\section{Kinematic restrictions on critical particles and two types of the\nBSW effect}\n\nIn the preceding Section, it was assumed that collision occurred not exactly\non the horizon but at some coordinate distance from it, its scale being tied\nto the amplitude of azimuthal acceleration, which is supposed to be small. The force, being too large, prevents\nthe critical particle from approaching the horizon. In this sense, the\nreason of it is dynamic. Meanwhile, even if the external force is small enough or absent at all, pure\nkinematic factors can also create an obstacle for reaching the horizon.\n\nLet us remind the situation with geodesic particles. If the horizon is\nnonextremal, the critical particle cannot reach the horizon at all.\nNonetheless, it was demonstrated in \\cite{gp} for the Kerr metric and in \n\\cite{prd} for generic dirty axially symmetric black holes, that $E_{c.m.}$\ncan be made as large as one likes provided (i) the critical particle is\nreplaced with a slightly noncritical one, (ii) the coordinate distance\nbetween the point of collision and horizon is adjusted to the small\ndeviation of the particle's parameters from the values corresponding to the\ncritical case.\n\nNow, we are dealing with an extremal horizon but there is a special situation \nwhen $X\\sim \\xi^p$ with $p>1$ (``supercritical'' particle). Such a particle cannot reach the horizon (in this sense it is similar to the case of the nonextremal horizon).\n\nThen, instead of taking a critical particle, we can choose a usual one with sufficiently\nsmall $X_{H}$. More precisely, let us consider expansion for $X$ of the form \n\\begin{equation}\nX=X_{H}+\\alpha_{s}\\xi ^{s}(1+x_{1}\\xi +\\ldots ),\\qquad p>1.  \\label{xpp}\n\\end{equation}\n\nThen, we look for the region in which both terms in $Z$ (\\ref{z}) are of the\nsame order of magnitude. This is achieved at $\\xi\\approx \\xi_c \\sim r_H X_H$. Then we can neglect the correction in (\\ref{xpp}), so that \n\\begin{equation}\nZ(\\xi_c)\\sim X_{H}\\sim \\xi_c \\sim N(\\xi_c ),\n\\end{equation}%\nand therefore (\\ref{gacm}) implies \n\\begin{equation}\n\\gamma _{c.m.}^{(max)}\n\t\\approx \\gamma_{c.m.}(\\xi_c)\n\t\t\\sim N^{-1}(\\xi_c) \n\t\t\\sim \\xi_{c} ^{-1} .  \\label{bsw}\n\\end{equation}\n\nThus one can distinguish between two main types of the BSW effect: BSW 1, in\nwhich the critical particle can approach the horizon, so that the\nhorizon limit can be taken, and BSW 2, for which the critical\nparticle does not reach the horizon. We see that, in general, the presence\nof the external force is compatible with both types of the BSW effect\\footnote{Another, more detailed classification of trajectories and corresponding\ntypes of the BSW effect can be found in Sec. IV of \\cite{ne} for the Kerr\nmetric and in \\cite{jh} for general dirty rotating axially symmetric black\nholes.}.\n\nIt is worth noting that even in the absence of force, the expansion for $X$\ncan take the form (\\ref{xpp}), if the linear terms cancel each other. Say,\nthis happens for solutions near the so-called ultraextremal horizon ($%\nN^{2}\\sim (r-r_{H})^{3}$) in special \"exotic\" metrics described in\nSec.\\,IV\\,B.\\,5 of \\cite{reg}, for which $\\partial_r \\omega \\big|_H\n=\\partial^2_r \\omega \\big|_H =0$, and thus $s$ can be equal to $2$ or $3$.\nCorrespondingly, the BSW-2 effect can be realized near such horizons.\n\n\\section{Conclusion}\n\nIn general, three main circumstances were considered as the factors which\nwere expected to restrict the indefinite growth of $E_{c.m.}$ and thus\ncreate obstacles to the manifestation of the BSW effect. These are (i)\nself-gravitation, (ii) deviation of a black hole from extremality \\cite{ted}%\n, and (iii) the force due to backreaction of gravitational or\nelectromagnetic radiation. As far as self-gravitation is concerned, it was\nshown in \\cite{shell} that for collisions of massive shells, either the BSW\neffect does not occur or it occurs but in the region inaccessible by a\nremote observer. However, in this case the shell does not approach the\nhorizon from the viewpoint of an external observer. As there is no horizon,\nthere is no BSW effect. Factor (ii) was analyzed in \\cite{gp} for the Kerr\nmetric where it was shown that for nonextremal black holes the BSW effect\ndoes exist (this conclusion was generalized in \\cite{prd} to generic dirty\naxially symmetric black holes). \n\nAnd, in the present work, we showed for extremal horizons that the BSW effect is compatible with a nonzero force under rather general assumptions: the radial force should be finite and the azimuthal force should tend to zero not too slowly. In terms of energy and angular momentum, the kinematic condition for the realization of the BSW effect is the same as for geodesic particles: $E=\\omega_H L$. In this sense, this condition by itself survives the action of the force (see also Sec. V of \\cite{insp} where, however, another physical situation was considered -- near-circular orbits around near-extremal black holes). Our approach is model-independent and is based on general properties of the horizons. \n\nFor the finite radial component of the force and the azimuthal one which tends to zero near the horizon as $r-r_H$, the BSW effect still exists. Otherwise, the effect is formally absent. The reason for the restriction on the azimuthal force seems to be clear: if azimuthal force was too large or did not tend to zero, in the infinite proper time that it takes for a critical particle to reach the horizon this force would accelerate the particle to infinite values of angular momentum. Obviously, one would not expect a force dissipative by nature, such as radiation reaction, to have such an effect. So, this only seeming restriction should be always obeyed. Even if it is not (so critical trajectories are absent), but the amplitude of the azimuthal force is small, the restrictions on $E_{c.m.}$ are shown to be inessential, and one can still attain very high energies. \n\nTo summarize, the BSW effect turned out to be more viable than one could expect.\n\nThe present work confirmed that the BSW effect relies on two main\nproperties: (i) the presence of the horizon and (ii) the existence of\nspecial types of trajectories. Thus it has geometric nature and reflects\ngeneral features of black holes irrespective of the details of the system.\nConcrete realization of the BSW effect certainly depends on particular\nproperties of a system but near the horizon these properties manifest\nthemselves in a universal way. We see that although dissipative forces in flat spacetime generically bound the values of energy peaks from above, in the strong gravitational field regime near the horizon the geometry dominates over the influence of dissipative forces on the system.\n\nThe present results refer to extremal horizons only. The nonextremal case\nand, especially, motion on circular orbits around near-extremal black holes,\nso important in astrophysical context, require separate treatment. Generalization to non-equatorial motion is also necessary. This will\nbe done elsewhere.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\n\nIn \\cite{LR2}, the authors have formulated boundary conformal field\ntheory in real time (Lorentzian signature) in the algebraic framework\nof quantum field theory. BCFT is a local M\\\"obius covariant QFT $B_+$\non the two-dimensional Minkowski halfspace $M_+$ (given by $x>0$), which\ncontains a (given) local chiral subtheory $A$, e.g., the stress-energy\ntensor. The reward of this approach was the surprisingly simple\nformula ((\\ref{relcomm}) below), expressing the von Neumann algebras\nof local observables $B_+(O)$ in a double cone $O\\subset M_+$ in terms\nof an (in general nonlocal) chiral conformal net $B$ of localized\nalgebras associated with intervals along the boundary (the time axis\n$x=0$). The net $B$ is M\\\"obius covariant and contains the local\nchiral observables $A$:   \n\\begin{eqnarray}\nA(I)\\subset B(I)\n\\end{eqnarray}\nfor each interval $I\\subset\\RR$. \n\nThe reduction to a single chiral net is responsible for a kinematical\nsimplification, explaining, e.g., Cardy's observation \\cite{C}\nthat in BCFT, bulk $n$-point correlation functions are linear\ncombinations of chiral $2n$-point conformal blocks.  \n\nThe algebra $B_+(O)$ is a relative commutant of $B(K)$ within $B(L)$, \n\\begin{eqnarray}\\label{relcomm}\nB_+(O) = B(K)'\\cap B(L),\n\\end{eqnarray}\nwhere $K\\subset L$ are a pair of open intervals on the boundary $\\RR$\nsuch that the disconnected complement $L\\setminus \\overline K = I\\cup\nJ$ is the set of advanced and retarded times $t\\pm x$ associated\nwith points in $(t,x)\\in O$ (see Fig.\\ 1). Although the chiral net $B$\nis not necessarily local, the intersections (\\ref{relcomm}) do commute\nwith each other when two double cones are spacelike separated. \n\nThe main result in \\cite{LR2} is that every BCFT is contained in a\nmaximal (Haag dual) BCFT of the form (\\ref{relcomm}).\n\n\\bigskip\n\n\\hskip40mm \\epsfig{file=intervals.eps,width=50mm} \n\\nopagebreak\n\n{\\bf Figure 1:} Intervals on the boundary and double cones in the halfspace.\n\n\\bigskip\n\nThis leads to a somewhat paradoxical conclusion: on the one\nhand, each local bulk observable is defined as a (special) observable\nfrom a chiral CFT. Thus, superficially, the ``degrees of freedom'' of\na BCFT are not more than those of a chiral CFT, containing only a\nsingle chiral component of the stress-energy tensor (Virasoro\nalgebra). One might argue that such a ``reduction of degrees of\nfreedom'' is a characteristic feature of QFT with a boundary. But this\npoint of view cannot be maintained, because on the other hand, the\nresulting BCFT $B_+$ is locally equivalent to another CFT $B_{2D}$ on\nthe full two-dimensional (2D) Minkowski spacetime, which has all the\ndegrees of freedom of a 2D QFT, and in particular contains a full 2D\nstress-energy tensor (two commuting copies of the Virasoro algebra). \nEven in the simplest case, when the chiral net $B$ on the boundary\ncoincides with $A$ (sometimes known as ``the Cardy case''), the\nassociated bulk QFT contains apart from the full 2D stress-energy \ntensor more (``non-chiral'') local fields that factorize into chiral\nfields with braid group statistics. Locally, also the BCFT contains\nthe same fields. \n\\medskip\n\nThis paradoxical situation is not a contradiction; it rather shows\nthat ``counting degrees of freedom'' of a QFT is an elusive\ntask. Trivially, there is no obstruction against a proper inclusion of\nthe form ${\\mathcal B}({\\mathcal H})\\otimes{\\mathcal B}({\\mathcal H})\\subset {\\mathcal B}({\\mathcal H})$ if ${\\mathcal H}$ is an\ninfinite-dimensional Hilbert space. But ``counting degrees of\nfreedoms'', e.g.\\ by entropy arguments, requires the specification of\nthe Hamiltonian. The BCFT shares the Hamiltonian and ground state\n(vacuum) of the chiral CFT, while the associated 2D CFT has a\ndifferent Hamiltonian and a different ground state. Thus, with respect\nto different Hamiltonians, the spacetime dimension (measured through\nsome power law behaviour of the entropy) may assume different values\n(1 or 2, in the present case). \n\nLooking at the issue from a different perspective, we may start from a\nvacuum representation of the Virasoro algebra. The latter integrates\nto a unitary projective representation of the diffeomorphism group of\nthe circle ${\\mathrm {Diff}\\;}(S^1)$, which contains the diffeomorphism group of an\ninterval ${\\mathrm {Diff}\\;}(I)$ as a subgroup. For two open intervals with\ndisjoint closures, there is a canonical identification between\n${\\mathrm {Diff}\\;}(I\\cup J)$ and ${\\mathrm {Diff}\\;}(I)\\times {\\mathrm {Diff}\\;}(J)$. In terms of the\nstress-energy tensor $T$, this amounts to an isomorphism between $\\exp\niT(f+g)$ and $\\exp iT(f)\\otimes \\exp iT(g)$, when $f$ and $g$ have\ndisjoint support. It would be hard to see this local isomorphism\ndirectly in terms of the Virasoro algebra.  \n\n\\medskip\n\nThe mathematical theorem underlying these facts is the well-known\nSplit Property \\cite{DL}, which can be derived in local QFT in any\ndimension under a suitable phase space assumption. In chiral local \nCFT, a sufficient assumption is the existence of the conformal\ncharacter $\\mathrm {Tr}\\; \\exp-\\beta L_0$. \n\nIn the algebraic framework, the {\\em chiral} observables of a BCFT\n(e.g., the stress-energy tensor) localized in a double cone $O$ are\noperators belonging to the von Neumann algebra $A_+(O) = A(I)\\vee\nA(J)$ where $I$ and $J$ are two open intervals of the time axis\n(``advanced and retarded times'') such that $ t+x\\in I$, $t-x\\in J$\nfor $(t,x)\\in O$ (this justifies the notation $O=I\\times J$), and\n$A(I)$ are the von Neumann algebras generated by the unitary\nexponentials of chiral fields smeared within $I$. In contrast, the\n{\\em chiral} observables in a 2D CFT are operators in the algebra\n$A_{2D}(O) = A_L(I)\\otimes A_R(J)$ where $I$ and $J$ are regarded as\ntwo open intervals of the lightcone axes, and $A_R(I)$ and $A_L(J)$\nare  generated by left and right chiral fields. Our present\nassociation between BCFT and 2D CFT applies to the case when $A_L(I) =\nA_R(I)= A(I)$, i.e., the left chiral observables $A_L(I)\\otimes 1$ are\nisomorphic with the right chiral observables $1\\otimes A_R(I)$, and\nboth are isomorphic with the chiral observables $A(I)$ of the BCFT.   \n\n\\medskip\n\nLet ${\\mathcal H}_0$ denote the vacuum Hilbert space for the chiral CFT\ndescribed by the algebras $A(I)$. The split property states that if\n$I$ and $J$ are two intervals with disjoint closures, there is a\ncanonical unitary ${\\mathcal{V}}:{\\mathcal H}_0\\to{\\mathcal H}_0\\otimes{\\mathcal H}_0$ implementing an\nisomorphism \n\\begin{eqnarray}\\label{splitiso}\n{\\mathcal{V}}\\big(A(I)\\vee A(J)\\big){\\mathcal{V}}^* = A(I)\\otimes A(J).\n\\end{eqnarray}\n\nThe split isomorphism does not preserve the vacuum vector, i.e.,\nthe canonical ``split vector'' $\\Xi={\\mathcal{V}}^*(\\Omega\\otimes\\Omega)$ is an\nexcited state in ${\\mathcal H}_0$. By construction, the split state\n$(\\Xi,\\cdot\\Xi)$ on $A(I)\\vee A(J)$ has the property that its\nexpectation values for either subalgebra $A(I)$ or $A(J)$ coincide\nwith those in the vacuum state, but the correlations between\nobservables $a_1\\in A(I)$ and $a_2\\in A(J)$ are suppressed: \n\\begin{eqnarray}\\label{splitstate}\n(\\Xi\\,,\\,a_1a_2\\,\\Xi) = (\\Xi\\,,\\,a_1\\,\\Xi)\\;(\\Xi\\,,\\,a_2\\,\\Xi) = (\\Omega\\,,\\,a_1\\,\\Omega)\\;(\\Omega\\,,\\,a_2\\,\\Omega).\n\\end{eqnarray}\nThe split isomorphism depends on the pair of intervals $I$ and \n$J$. It trivially restricts to algebras associated with subintervals,\nbut it does not, in general, extend to larger intervals. When the\nintervals touch or overlap, a split state and the split isomorphism\ncease to exist.  \n\nWhile the split isomorphism is well known, we discuss in this paper\nits extension to ``non-chiral'' local observables, which do {\\em not} \nbelong to  $A(I)\\vee A(J)$ in the BCFT, and to $A(I)\\otimes A(J)$ in\nthe 2D CFT. \n\n\\medskip\n\nAs a concrete demonstration for the resolution of the above ``paradox'', \nwe present two simple but nontrivial examples where the algebraic\nrelations outlined can be easily translated into the field-theoretic\nsetting, i.e., we characterize the local algebras of the various QFTs\nin terms of generating local Wightman fields. \n\nLet us translate (\\ref{relcomm}) into the field-theoretic language. \nThe intervals $I$ and $J$ shrink to the points $t\\pm x$ when\n$O=I\\times J$ shrinks to a point $(t,x)$. Thus, we have to approximate\na field $\\Phi(t,x)$ of the BCFT by observables in $A(L)$ (where the\ninterval $L$ approximates $(t-x,t+x)$ from the outside), that commute\nwith all fields localized in the interval $K$ (which approximates\n$(t-x,t+x)$ from the inside). This will be done in Sect.\\ 2. A crucial\npoint here is that generating the local algebra $A(L)$ involves\n``non-pointwise'' operations, e.g., typical observables may be\nexponentials of smeared field operators, so that an element of the\nrelative commutant is not necessarily localized in the disconnected\nset $L\\setminus \\overline K = I \\cup J$.   \n\n\\medskip\n\nA second, somewhat puzzling feature of the algebraic treatment of BCFT\nis the fact that the description of the local algebras $B_+(O)$ in\nterms of the chiral boundary net (Eq.\\ (\\ref{relcomm})) is much\nsimpler than that of the local algebras $B_{2D}(O)$ of the associated\n$2D$ conformal QFT without a boundary. The latter are (rather\nclumsily) defined as Jones extensions of the tensor products\n$A(I)\\otimes A(J)$ in terms of a Q-system constructed from the\nchiral extension $A\\subset B$ with the help of $\\alpha$-induction\n\\cite{KHR}.   \n\nOne purpose of this work is to present a more direct construction\nof the 2D CFT without boundary from the BCFT. The obvious idea is to\ntake a limit as the boundary is ``shifted to infinity''. But we shall\ndo more, and establish the {\\em covariant} local isomorphism between the\nsubnets $O\\mapsto B_+(O)$ and $O\\mapsto B_{2D}(O)$ as $O\\subset O_0$,\ni.e., the restriction of the AQFTs to any double cone $O_0$ within the\nhalfspace $x>0$, at finite distance from the boundary. \n\n\\medskip\n\nThe main problem here is, of course, the enhancement of the conformal\nsymmetry, i.e., the reconstruction of the unitary positive-energy\nrepresentation of the two-dimensional conformal group $\\Mb\\times \\Mb$\nfrom that of the chiral conformal group $\\Mb$. This is done by\na ``lift'' of the chiral M\\\"obius covariance of the local chiral net\n$A$, using the split property which allows to ``embed'' the 2D chiral\nalgebra $A(I)\\otimes A(J)$ into a local BCFT algebra $B_+(O)$. This\nwill be done in Sect.~3. The point is that only a single local\nalgebra of the BCFT is needed for this reconstruction of the 2D\nconformal group and the full 2D CFT.\n\n\\medskip \n\nIn Sect.~4, we show that the 2D CFT can also be obtained through a\nlimit where the boundary is ``shifted to the left'', or equivalently,\nthe BCFT observables are ``shifted to the right''. The translations in\nthe spatial direction ``away from the boundary'' do not belong to the\nchiral M\\\"obius group of the BCFT. But they are at our disposal by the\nprevious lifting of the 2D M\\\"obius group into the BCFT. Therefore, we\ncan study the behavior of correlation functions in the limit of\n``removing the boundary''. As we shift the boundary, the retarded and\nadvanced times are shifted apart from each other. The convergence of\nthe vacuum correlations of the BCFT to the vacuum correlations of the\n2D CFT is therefore a consequence of the cluster behavior of vacuum\ncorrelations of the chiral CFT $A$.  \n\n\\medskip\n\nWe add three appendices containing some related observations.\n\n\\section{Example}\\label{sec2}\n\\setcounter{equation}{0}\n\nThe purpose of this section is to illustrate the construction\n(\\ref{relcomm}) in a field-theoretic setting. It is convenient to\nassume the trivial chiral extension $B=A$ since even in this case\nthe construction (\\ref{relcomm}) is nontrivial, i.e., non-chiral local\nBCFT fields that factorize into nonlocal chiral fields can be\nconstructed from local chiral fields only. We exhibit local BCFT \nfields in a region $O=I\\times J\\subset M_+$ as ``neutral'' chiral\noperators, that behave like products of ``charged'' chiral operators\nlocalized in $I$ and $J$ in the limit of large distance from the\nboundary. The limit of pointlike localization is also discussed, and\nreproduces familiar vertex operators.\n\n\\medskip \n\nConsider the free $U(1)$ current $j$ with commutator\n$[j(x),j(y)]=2\\pi i\\delta'(x-y)$ and charge operator $Q=(2\\pi)^{-1}\\int\nj(x)dx$. The unitary Weyl operators $W(f)=e^{ij(f)}$ for real test\nfunctions $f$ satisfy the Weyl relation  \n\\begin{eqnarray}\\label{weyl}\nW(f)\\,W(g) = e^{-i\\pi\\sigma(f,g)}\\cdot W(f+g) = e^{-2\\pi i\\sigma(f,g)}\n\\cdot W(g)\\,W(f)\n\\end{eqnarray}\nand have the vacuum expectation value\n\\begin{eqnarray}\\label{ground}\n\\omega(W(f)) = e^{-i\\pi\\sigma(f_-,f_+)} = e^{-\\frac\n  12\\int_{\\RR_+}k\\,dk \\vert \\hat f(k)\\vert^2}\n\\end{eqnarray}\nwhere the symplectic form is\n\\begin{eqnarray}\\label{symp}\n\\sigma(f,g)= \\frac 12\\int_\\RR dx\\;\\big(f(x)\\;g'(x) - f'(x)\\;g(x)\\big)\n= \\frac 1{2\\pi i}\\int_\\RR k\\,dk\\;\\hat f(-k)\\;\\hat g(k),\n\\end{eqnarray}\nand $f_+$ ($f_-$) correspond to the restrictions to positive\n(negative) values of $k$ of the Fourier transform $\\hat f(k) =\n\\int_\\RR dx\\,e^{ikx}\\,f(x)$. With these conventions, $W(f)\\Omega$ is a \nstate with charge density $-f'(x)$.   \n\nThe vacuum correlations of Weyl operators are\n\\begin{eqnarray}\\label{corr}\n\\omega(W(f_1)\\cdots W(f_n)) =\ne^{-i\\pi\\big(\\sum_i\\sigma(f_{i-},f_{i+}) + 2\n  \\sum_{i<j}\\sigma(f_{i-},f_{j+})\\big)}\n\\end{eqnarray}\n\nThe Weyl operators $W(f)$ with $\\mathrm{supp}\\; f\\subset I$ define the local\nvon Neumann algebras of the chiral net $I\\mapsto A(I)$. We fix a\ndouble cone $O=I\\times J\\in M_+$. If $f$ is a test function that\nvanishes outside $L$ and is constant in $K$, then $W(f)$ belongs to\n$A(L)$ and commutes with $A(K)$ by (\\ref{weyl}) and (\\ref{symp}), hence\n\\begin{eqnarray}\\label{neut}\nW(f)\\subset B_+(O) = A(K)'\\cap A(L). \n\\end{eqnarray}\nThese are examples of operators that belong to $B_+(O)$ but (if\n$f\\vert_K\\neq 0$) not to $A_+(O)= A(I)\\vee A(J)$. \n\n\n\\medskip\n\nWeyl operators can also be defined for smooth functions\n$f$ such that $f'$ has compact support, and the relation\n(\\ref{weyl}) holds. Then $q= f(-\\infty)-f(\\infty)$ is called the\ncharge. However, $i\\sigma(f_-,f_+)$ diverges, and the vacuum\nexpectation value (\\ref{ground}) vanishes unless $q=0$ (see below).  \nThis implies that correlation functions (\\ref{corr}) of charged Weyl\noperators vanish whenever the total charge is non-zero (charge\nconservation), while the IR divergences in each term in the exponent\nof (\\ref{corr}) cancel for neutral\ncorrelations. The neutral Weyl operators (\\ref{neut}) in $B_+(O)$ are\n(up to a phase factor) products of charged Weyl operators with charge\ndensities localized in $J$ and in $I$. \n\nIn the limit of sharp step functions $G_u(x)=q\\cdot \\theta(x-u)$\n(requiring a regularization \\cite{CRW}), the regularized Weyl\noperators $W(G_u)$ become the well-known vertex operators of charge\n$-q$ and scaling dimension $\\frac 12q^2$ \\cite{SSV}, which are\nformally written as     \n\\begin{eqnarray} \nV_{-q}(u) = {:\\! \\exp\\Big(iq\\int_u^{\\infty} j(y)dy\\Big) \\!:}.\n\\end{eqnarray}  \nThus, as $O$ shrinks to a point $(t,x)\\in M_+$, and $I$ and $J$ shrink\nto the points $t+x$ and $t-x$, the (regularised) Weyl operators\n$W(G_{t-x}-G_{t+x})$ behave as \n\\begin{eqnarray}\n\\Phi_{q}(t,x) = V_{q}(t+x) V_{-q}(t-x).\n\\end{eqnarray}\nThe correlation functions of vertex operators are given by \n\\begin{eqnarray} \\langle\\ldots\\cdot V_{q_i}(u_i)\\cdot\\ldots\\rangle =\n\\lim_{\\varepsilon\\searrow 0}\\;  \\prod_{i<j}\n\\Big(\\frac{-i}{u_i-u_j-i\\varepsilon}\\Big)^{-q_iq_j} \\end{eqnarray}\nif $\\sum_i q_i=0$, and $=0$ otherwise, from which the well-known anyonic\ncommutation relations can be read off. It is then easily seen\nthat $\\Phi_{q_1}(t_1,x_1)$ commutes with $\\Phi_{q_2}(t_2,x_2)$ \nwhen either $t_1+x_1 > t_2+x_2 > t_2-x_2 > t_1-x_1$ or when $t_2+x_2 >\nt_1+x_1 > t_1-x_1 > t_2-x_2$, because in these cases the anyonic phase\nfactors cancel. It also commutes with $j(t_2\\pm x_2)$ if $t_2\\pm\nx_2\\neq t_1\\pm x_1$. These are precisely the requirements for locality\nof the fields $\\Phi_q(t,x)$ among each other, and relative to the\nconserved current \n\\begin{eqnarray}\nj_0(t,x) = j(t+x) + j(t-x),\\qquad j_1(t,x) = j(t+x) - j(t-x) \n\\end{eqnarray}\ndefined for $x>0$, i.e., $\\Phi_q$ and $j^\\mu$ are local fields on the\nhalfspace $M_+$. The correlation functions of $n$ fields\n$\\Phi_{q_i}(t_i,x_i)$ are correlations of $2n$ vertex operators\n($2n$-point conformal blocks).\n\n\\medskip \n\nAfter this digression to pointlike fields, let us resume the\nstudy of the correlation functions (\\ref{corr}) of the smooth\nWeyl operators $W(f_i)\\in B_+(O)$, and their behavior as $O$ is\nshifted away from the boundary. We choose $n$ test functions of the \nform \n\\begin{eqnarray}\\label{g-h}\nf_i=G_i-H_i\n\\end{eqnarray}\nwhere $G_i$, $H_i$ are smooth step functions with values $0$ at\n$-\\infty$ and $q_i$ at $+\\infty$, such that $G_i'=g_i$ is supported in\n$J$ and $H_i'=h_i$ is supported in $I$ (see Fig.\\ 2).\n\n\\bigskip\n\n\\hskip10mm \\epsfig{file=steps.eps,width=100mm} \n\\nopagebreak\n\n{\\bf Figure 2:} A test function $f$ such that $W(f)$ belongs to\n$B_+(O)$, but not to $A_+(O)$. $G$ and $H$ are smooth step functions,\n$\\mathrm{supp}\\; G' \\subset J$, $\\mathrm{supp}\\; H' \\subset I$.\n\n\\bigskip\n\nThe neutral states $W(f_i)\\Omega$ carry the charge $q_i$ in $I$ and the\ncharge $-q_i$ in $J$. \n\nThe neutrality\ncondition for each Weyl operator $W(f_i)$ can be written \n\\begin{eqnarray}\n\\label{charge}\n\\int_\\RR dx\\;g_i(x) - \\int_\\RR dx\\;h_i(x)= 0 \\qquad\\Leftrightarrow \\qquad\n\\hat g_i(0) - \\hat h_i(0) =0. \n\\end{eqnarray} \n\nThe exponent in (\\ref{corr}) is a linear combination of terms of the form \n(using $\\hat f_i = i(\\hat g_i-\\hat h_i)\/k$) \n\\begin{eqnarray}\n2\\pi i\\,\\sigma(f_{i-},f_{j_+}) = \\int_{\\RR_+} \\frac{dk}k\n\\int dx\\,\\big(g_i(x)-h_i(x)\\big)\\int dy\\, \\big(g_j(y)-h_j(y)\\big)  \n \\;e^{-ik(x-y)} \\quad\n\\end{eqnarray}\nwhich are IR finite because of (\\ref{charge}). The separate\ncontributions from $g_i$ and $h_i$, however, are IR divergent. \nTherefore, we first regularize at $k=0$ by the subtraction\n$e^{-ik(x-y)}\\to e^{-ik(x-y)}-e^{-k\/\\mu}$ ($\\mu>0$ arbitrary), \nwhich does not change the result because of (\\ref{charge}), and then\ncompute the contributions from $g$ and $h$ separately. \n\nWe are interested in the behavior of the correlation function\n(\\ref{corr}) as $O$ is shifted away from the boundary. This means that\nthe functions $g_i$ are shifted by a distance $a$ to the left, and\n$h_i$ are shifted by the same distance to the right. The $g$-$g$\ncontributions and the $h$-$h$-contributions to\n$\\sigma(f_{i-},f_{j_+})$ are obviously invariant under this shift,\nwhile in the mixed $h$-$g$ contributions $x-y$ is replaced by $x-y+2a$:\n\\begin{eqnarray}\n2\\pi i\\,\\sigma_{h_i,g_j}(a) := - \\int_I dx\\, h_i(x) \\int_J dy\\, g_j(y)  \n\\int_{\\RR_+} \\frac{dk}k \\Big[e^{-ik(x-y+2a)}-e^{-k\/\\mu}\\Big] \n\\end{eqnarray}\nand similar for the $g$-$h$ contributions. The last integrand\ncan be split into two parts:\n\\begin{eqnarray}\n\\big(e^{-ik(x-y+2a)}-1\\big)e^{-k\/\\mu} \\;+\\;\ne^{-ik(x-y+2a)}\\big(1-e^{-k\/\\mu}\\big) \n\\end{eqnarray}\nso that the first contribution to the momentum integral equals \n\\begin{eqnarray}\n-\\log\\big(1+i\\mu(x-y+2a)\\big)\n\\end{eqnarray}\nwhile the second (distributional) contribution is of order $O(a^{-1})$\nin the limit of large $a$. Because the remaining integrals have\ncompact support, we obtain    \n\\begin{eqnarray}\n\\lim_{a\\to\\infty} \\sigma_{h_i,g_j}(a) = q_iq_j\\cdot \\log(2i\\,a\\mu)\n+ O(a^{-1}).\n\\end{eqnarray}\nTogether with the $g$-$h$ contributions $q_iq_j\\cdot \\log(-2i\\,a\\mu)$,\nthese terms in the exponent of (\\ref{corr}) cumulate up to\nthe factor   \n\\begin{eqnarray}\n\\prod_{i} (2a\\mu)^{-q_i^2} \\prod_{i<j} (2a\\mu)^{-2q_iq_j} = (2a\\mu)^{-q^2},\n\\end{eqnarray}\nwhere $q=\\sum_i q_i$ is the total charge within $I$. Thus (\\ref{corr})\nvanishes in the limit $a\\to\\infty$ if $q\\neq 0$, enforcing ``chiral\ncharge conservation'' in the limit. If $q=0$, the remaining $g$-$g$\nand $h$-$h$ contributions yield\n\\begin{eqnarray}\n\\lim_{a\\to\\infty} \\omega\\big(W(f_1)\\cdots W(f_n)\\big) = \n\\omega\\big(W(G_1)\\cdots W(G_n)\\big)\\cdot \\omega\\big(W(-H_1)\\cdots\nW(-H_n)\\big) \\qquad \n\\end{eqnarray}\ninvolving charged Weyl operators. These expressions are well-defined\n(and independent of $\\mu$) because $\\sum_i G_i$ and $\\sum_i H_i$ are\nneutral precisely due to $q=0$.   \n\n\\medskip\n\nThe factorization of the vacuum correlations in the limit $a\\to\\infty$\nis the desired feature we wanted to illustrate by this example. In the\nlimit, $W(f_i)$ have the same correlations as $W(-H_i)\\otimes W(G_i)$,\nwhich are charged observables of the associated 2D CFT. \nNotice that in the limit of sharp test functions (see above), one\nobtains  \n\\begin{eqnarray}\nV_{q}(t+x) \\otimes V_{-q}(t-x)\n\\end{eqnarray}\nwhich are local fields in the entire two-dimensional Minkowski spacetime $M^2$.\n\n\\medskip\n\n{\\bf Remark:} The above construction can be generalized to the $SU(2)$\ncurrent algebra. The Frenkel-Kac representation of $SU(2)$ currents at\nlevel 1 is given by $j^3\\equiv j$ and $j^\\pm(x) = j^1(x) \\pm i j^2(x) =\nV_{\\pm \\sqrt2}(x)$. Then $V_{q}(x)\\cdot V_{-q}(y)$ commutes with\n$V_{q'}(w)$ at $w\\neq x,y$ provided $qq'\\in \\ZZ$. Hence the field \n\\begin{eqnarray}\n\\Phi_{\\frac 12\\sqrt2}(t,x) = V_{\\frac 12\\sqrt2}(t+x)\\cdot \nV_{-\\frac 12\\sqrt2}(t-x)\n\\end{eqnarray}\nis local (as before) and relatively local w.r.t.\\ the conserved currents \n$j^a$ ($a=1,2,3$)\n\\begin{eqnarray}\nj^a_0(t,x) = j^a(t+x) + j^a(t-x),\\quad j^a_1(t,x) = j^a(t+x) -\nj^a(t-x).\\qquad\n\\end{eqnarray}\n$\\Phi_{\\frac 12\\sqrt2}(t,x)$ is a neutral combination of charged\nprimary fields of dimension $\\frac 14$, transforming in the\nspin-$\\frac12$ representation of $SU(2)$, localized at $t+x$ and\n$t-x$. The description of this model in terms of\nsmooth Weyl operators is rather straightforward, see e.g., \\cite{BMT}:\nWeyl operators with integer multiples of the charge $\\sqrt 2$ belong to\n$A(I)$, while operators with half-integer multiples of the charge\n$\\sqrt 2$ in $I$ and in $J$ belong to $A(K)'\\cap A(L)$. \n\n\\medskip\n\nThe mechanism of ``charge separation'' described here for obtaining \nelements of $B_+(O)$ that do not belong to $A_+(O)$ is very general\n\\cite{LR2}, although in general it cannot be formulated in terms of\nWeyl operators. In Sect.\\ 4 we shall show that also the factorization\nbehavior far away from the boundary is a general feature, which allows\nto recover the 2D CFT from the BCFT. \n\n\\section{Reconstruction of the 2D symmetry}\\label{sec3}\n\\setcounter{equation}{0}\n\nWe work in this section with a fixed ``chiral extension'' $A\\subset B$. \nHere, $A$ is a Haag dual M\\\"obius covariant local net $\\RR\\supset\nI\\mapsto A(I)$ of von Neumann algebras on its vacuum Hilbert space\n${\\mathcal H}_0$, satisfying the split property and having finitely many\nirreducible DHR sectors of finite dimension (these properties together\nare called ``complete rationality'' \\cite{KLM}; in the case of\ndiffeomorphism covariant nets, Haag duality = strong additivity is a\nconsequence of the other properties \\cite{LX}. The fact that the $U(1)$\nWeyl algebra in Sect.~2 is {\\em not} completely rational, indicates\nthat the results to be reported in this section hold also in more\ngeneral situations).    \n\n$B$ is a M\\\"obius covariant net $\\RR\\supset I\\mapsto B(I)$ on its\nvacuum Hilbert space ${\\mathcal H}_0^B$ such that for each $I$ the inclusion\n$A(I)\\subset B(I)$ holds and is an irreducible subfactor, which has\nautomatically finite Jones index \\cite{KL} equal to the statistical\ndimension of the (reducible) representation of $A$ on ${\\mathcal H}_0^B$\n\\cite{LR1}. The net $B$ may be non-local, but is required to be\nrelatively local w.r.t.\\ $A$.  \n\nIf only $A$ is specified, the irreducible chiral extensions $B$ of $A$\ncan be classified in terms of Q-systems of $A$ \\cite{LR1}. The\ncomplete classification has been computed for $A$ the Virasoro nets\nwith central charge $c<1$ (and implicitly also for the $SU(2)$ current\nalgebras) in \\cite{KLPR}. \n\n\\medskip\n\nWith $A\\subset B$ one can associate a boundary CFT $B_+$ on the\nhalfspace $M_+$ and a two-dimensional CFT $B_{2D}$ on Minkowski spacetime\n$M^2$. To describe the former, we introduce a convenient notation (see\nFig.\\ 1). \nFor any quadruples of four real numbers such $a<b<c<d$ we define \n$I=(c,d)$, $J=(a,b)$, $K=(b,c)$, $L=(a,d)$, and $O =\n\\{(t,x):\\; t+x\\in I,\\, t-x \\in J\\}\\subset M_+$. Every double cone\n$O\\subset M_+$ is of this form and determines $I,J,K,L$, and similarly\nevery pair of open intervals $J<I$ (``$I$ is to the right = future of\n$J$'') determines $K,L$, and $O=I\\times J$. \n\nThen the BCFT associated with $A\\subset B$ is the net (\\ref{relcomm}),\ni.e., $O\\mapsto B_+(O)= B(K)'\\cap B(L)$. We have shown in \\cite{LR2}\nthat $B_+(O)$ contains $A_+(O) = A(I)\\vee A(J)$ as a subfactor with\nfinite index, $B_+$ is local and Haag dual on $M_+$, every Haag dual\nBCFT with chiral observables $A$ arises in this way (namely the chiral\nextension $B$ can be recovered from the BCFT), and every non-Haag-dual\nlocal BCFT net is intermediate between $A_+$ and $B_+$. If $B=A$,\n$B_+(O)$ equals the four-interval subfactor $A(E)\\subset A(E')'$ on\nthe circle \\cite{KLM} ($E=I\\cup J$).     \n\nThe 2D CFT $B_{2D}$ associated with $A\\subset B$ has been constructed\nin \\cite{KHR}. Its local algebras are extensions (with finite Jones\nindex) of the tensor products $A(I)\\otimes A(J)$, specified in terms\nof a Q-system constructed from the chiral extension $A\\subset B$\nwith the help of $\\alpha$-induction. \n\nWe know from \\cite{LR2} that $B_+$ and $B_{2D}$ are locally\nisomorphic, i.e., for each $O\\subset M_+$ there is an isomorphism\n$\\varphi^O: B_+(O)\\to B_{2D}(O)$ such that\n\\begin{eqnarray}\\label{lociso}\n \\varphi^O\\big(B_+(O_1)\\big) = B_{2D}(O_1)\\quad \\hbox{for all} \\;\nO_1\\subset O. \n\\end{eqnarray} \nHowever, the Hilbert space and the vacuum state for the two theories\nare very different. \n\nIn this section, we wish to understand the relation between these two\nnets, by giving an alternative construction of the 2D CFT directly from \nthe BCFT. The crucial point is the construction of the enhanced\nM\\\"obius symmetry of the 2D CFT, and its ground state (the 2D vacuum)\nwhich is different from the BCFT vacuum. \n\n\n\\medskip\n\nWe first construct the Hilbert space ${\\mathcal H}_{2D}$ for the 2D CFT. We\nchoose a fixed reference double cone $O_0=I_0\\times J_0\\subset M_+$. \nThe subfactor $A_+(O_0)=A(I_0)\\vee A(J_0) \\subset B_+(O_0)=B(K_0)'\\vee\nB(L_0)$ is irreducible with finite index \\cite{LR2}, and hence has a unique\nconditional expectation $\\mu:B_+(O_0)\\to A_+(O_0)$, which is automatically\nnormal and faithful. Let $\\Xi\\in{\\mathcal H}_0$ be the canonical split vector\nfor $A(I_0)\\vee A(J_0)$ as in (\\ref{splitstate}). The split state\n$\\xi=(\\Xi,\\cdot\\Xi)$ on $A_+(O_0)$ extends to the state\n$\\hat\\xi=\\xi\\circ\\mu$ on $B_+(O_0)$. Let $\\hat{\\mathcal H}$, $\\hat\\Xi$ and\n$\\hat\\pi$ denote the GNS Hilbert space, GNS vector and GNS\nrepresentation for $(B_+(O_0),\\hat\\xi)$. We also write $\\vert b\\rangle$\nfor $\\hat\\pi(b)\\hat\\Xi$. Let us analyze the structure of $\\hat{\\mathcal H}$. \n\n\\medskip\n\nThe structure of $B_+(O_0)$ has been described in \\cite{LR2}. \nBy complete rationality, $A$ has finitely many irreducible\nsuperselection sectors \\cite{KLM}. Choose for each irreducible sector \nof $A$ a representative DHR endomorphism \\cite{DHR} $\\sigma$ localized\nin $I_0$, and a representative $\\tau$ localized in $J_0$. (For the vacuum\nsector, $\\sigma=\\tau={\\mathrm {id}}$. $\\bar\\sigma$ and $\\bar\\tau$ are the\nrepresentatives of the conjugate sector.) Then the elements of\n$B_+(O_0)$ are (weak limits of) sums of operators of the form $\\iota(a_1\na_2) \\cdot \\psi$ where $\\iota$ is the injection $A\\to B$, $a_1\\in\nA(I_0)$, $a_2\\in A(J_0)$, and $\\psi \\in B(L_0)$ generalize the Weyl\noperators $W(f)$ (\\ref{g-h}) of Sect.~2: they are (for each pair\n$\\sigma,\\tau$) ``charged'' intertwiners in\n$\\mathrm{Hom}(\\iota,\\iota\\sigma\\bar\\tau)\\cap B(K_0)'$. We may express these\nintersections in a different way: Let $\\alpha_\\rho^\\pm$ denote the\nendomorphisms of $B$ extending the DHR endomorphisms $\\rho$ of $A$ \nby ``$\\alpha$-induction'' \\cite{LR1}, where $\\alpha^+_\\rho$\n($\\alpha^-_\\rho$) acts trivially on $b\\in B$ localized to the right =\nfuture (left = past) of the interval where $\\rho$ is localized. Thus\n$\\alpha_{\\sigma_1}^-\\alpha_{\\bar\\tau_1}^+$ acts trivially on $B(K_0)$, \nbecause $J_0<K_0<I_0$. Hence \n\\begin{eqnarray}\\label{chint}\n\\mathrm{Hom}(\\iota,\\iota\\sigma\\bar\\tau)\\cap B(K_0)' =\n\\mathrm{Hom}({\\mathrm {id}}_B,\\alpha_{\\sigma}^-\\alpha_{\\bar\\tau}^+).\n\\end{eqnarray}\n(For an alternative characterization of the charged intertwiners\nby means of an eigenvalue condition, see App.~\\ref{appeig}.) \nIf $O_1\\subset M_+$ is another double cone in the halfspace, the\nalgebra $B_+(O_1)$ is generated by\n$A(I_1)\\vee A(J_1)$ and charged intertwiners \n\\begin{eqnarray}\\label{chint1}\n\\psi_1 = \\iota(u\\times \\bar u)\\cdot\\psi \\in \\mathrm{Hom}({\\mathrm {id}}_B,\\alpha_{\\sigma_1}^-\\alpha_{\\bar\\tau_1}^+)\n\\end{eqnarray}\nwith\nunitary charge transporters $u\\in\\mathrm{Hom}(\\sigma,\\sigma_1)$ and $\\bar\nu\\in\\mathrm{Hom}(\\bar\\tau,\\bar\\tau_1)$, where $\\sigma_1$ is localized in $I_1$\nand $\\bar\\tau_1$ is localized in $J_1$. \n\n\nE.g., if $B=A$ (the ``Cardy case''), the charged intertwiners \n(generalizing the Weyl operators $W(f)$ in (\\ref{g-h}) of Sect.~2) are of\nthe form $\\psi\\in \\mathrm{Hom}({\\mathrm {id}},\\sigma\\bar\\tau)$. This implies that $\\tau$\nand $\\sigma$ are representatives of the same sector. \nThus, the charges of BCFT fields are in 1:1 correspondence\nwith the DHR sectors of $A$. \n\nIn the general case, when $\\psi$ and $\\psi'$ are two charged\nintertwiners, $\\mu(\\psi'\\psi^*)$ is an intertwiner  \n$\\in \\mathrm{Hom}(\\sigma'\\bar\\tau',\\sigma\\bar\\tau)\\cap (A(I_0)\\vee A(J_0))$. This\nspace is zero unless $\\sigma'=\\sigma$ and $\\tau'=\\tau$, and\n$\\mathrm{Hom}(\\sigma\\bar\\tau,\\sigma\\bar\\tau) \\cap (A(I_0)\\vee A(J_0)) = \\CC\\cdot 1$\n\\cite{L}. \nTherefore, we may choose (for each pair $\\sigma,\\tau$) a basis of\ncharged intertwiners $\\psi$  which is orthonormal w.r.t.\\ the inner\nproduct $\\mu(\\psi'\\psi^*)$.  \n\n\\medskip \n\n{\\bf Lemma 1:} {\\sl The subspaces $\\hat{\\mathcal H}_\\psi$ of $\\hat{\\mathcal H}$ spanned by\n$\\vert\\psi^*\\cdot \\iota(A(I_0) \\vee A(J_0))\\rangle$ are mutually\northogonal. Each subspace $\\hat{\\mathcal H}_\\psi$ factorizes as a\nrepresentation of $A_+(O_0)$ according to \n\\begin{eqnarray}\\label{hpsi}\n \\hat{\\mathcal H}_\\psi \\cong {\\mathcal H}_\\sigma\\otimes {\\mathcal H}_{\\bar\\tau} \n\\end{eqnarray}\nwhere ${\\mathcal H}_\\sigma$ and ${\\mathcal H}_{\\bar\\tau}$ carry the representations\n$\\sigma$ and $\\bar\\tau$ of $A(I_0)$ and $A(J_0)$, respectively. \n}\n\n\\medskip\n\n{\\it Proof:} The computation of matrix elements in a dense\nset of vectors\n\\begin{eqnarray}\n\\langle \\psi^*\\cdot\\iota(a''_1a''_2)\\vert \\,\\hat\\pi\\big(\\iota(a_1a_2)\\big)\\,\n\\vert \\psi^*\\cdot\\iota(a_1'a_2')\\rangle =\n\\big(\\Xi\\,,\\,a''_1{}^*a''_2{}^*\\,\\mu\\big(\\psi\\,\\iota(a_1a_2)\\,\\psi^*\\big)\\,a_1'a_2'\\,\\Xi\\big) =\n\\qquad \\\\\n= \\big(\\Xi\\,,\\,a''_1{}^*a''_2{}^*\\,\\sigma\\bar\\tau(a_1a_2)\\,a_1'a_2'\\,\\Xi\\big) =\n\\big(a''_1\\,\\Omega\\,,\\,\\sigma(a_1)\\,a_1'\\,\\Omega\\big)\\cdot\n\\big(a''_2\\,\\Omega\\,,\\,\\bar\\tau(a_2)\\,a_2'\\,\\Omega\\big)\n\\nonumber \\end{eqnarray}\nproves the claim. \\hspace*{\\fill}Q.E.D.\n\n\\medskip\n\nWe may therefore identify the vectors $\\vert\\psi^*\n\\iota(a_1'a_2')\\rangle$ with $a_1'\\Omega\\otimes\na_2'\\Omega\\in{\\mathcal H}_{\\sigma}\\otimes{\\mathcal H}_{\\bar\\tau}$ in \nthe representation $\\sigma\\otimes\\bar\\tau$ under the split\nisomorphism, such that in particular, the GNS vector\n$\\hat\\Xi=\\vert 1\\rangle\\in\\hat{\\mathcal H}$ corresponds to the 2D vacuum vector\n$\\Omega\\otimes\\Omega\\subset {\\mathcal H}_0\\otimes{\\mathcal H}_0$. We write\nthe extended Hilbert space $\\hat{\\mathcal H}$ in the form    \n\\begin{eqnarray}\\label{h2d} \n\\hat{\\mathcal H} \\equiv {\\mathcal H}_{2D} \\cong \\bigoplus\\nolimits_{\\sigma,\\tau} Z_{\\sigma,\\tau} \\;\n{\\mathcal H}_\\sigma\\otimes {\\mathcal H}_{\\bar\\tau} \n\\end{eqnarray} \n(the ``2D Hilbert space''). The nonnegative integer multiplicities are\n\\begin{eqnarray}\\label{dim}\nZ_{\\sigma,\\tau} = \\dim \\mathrm{Hom}(\\alpha_{\\tau}^+,\\alpha_{\\sigma}^-)\n\\end{eqnarray}\nby the above characterization (\\ref{chint}) of the spaces of charged\nintertwiners. \nThe chiral factorization (\\ref{h2d}) of the GNS construction from the\nextended state $\\xi\\circ\\mu$ may be viewed as the remnant of the original\n``splitting behavior'' of the split vector $\\Xi$. \n\nAs shown in \\cite{LR2} by comparison of the Q-system, the local\nsubfactor $\\hat\\pi(A_+(O_0)) \\subset \\hat\\pi(B_+(O_0))$ on $\\hat{\\mathcal H}$\nis isomorphic to $A(I_0)\\otimes A(J_0)\\subset B_{2D}(O_0)$ constructed\nin \\cite{KHR}. We may therefore consistently denote also the former by\n$A_{2D}(O_0)\\subset B_{2D}(O_0)$. \n\n\\medskip\n\nNext, we construct the action of the 2D M\\\"obius group on ${\\mathcal H}_{2D}$,\nby a ``lift'' of the M\\\"obius transformations of the chiral net $A$,\nusing the split isomorphism and the conditional expectation $\\mu$. \nThe action of $\\Mb\\times \\Mb$ on ${\\mathcal H}_{2D}$ will then be used to\ndefine $B_{2D}(O)$ as the images of the reference algebra\n$B_{2D}(O_0)$ under a 2D M\\\"obius transformation $g=(g_1,g_2)$ taking\n$O_0$ to $O$. \n\nThe 2D M\\\"obius group $\\Mb\\times \\Mb$ is unitarily represented in the\nvacuum Hilbert space ${\\mathcal H}_0$ of the chiral net $A$ by $U_+U_-$, the\npreimage of $U_0\\otimes U_0$ on ${\\mathcal H}_0\\otimes{\\mathcal H}_0$ under the split\nisomorphism. (See App.~\\ref{appmod}, how $U_+$ and $U_-$ can be obtained by\nmodular theory directly on the boundary Hilbert space.) We need to\nlift $U_+U_-$ to ${\\mathcal H}_{2D}$.\n\nLet $\\Sigma_I\\subset \\Mb$ denote the connected semigroup taking the\ninterval $I$ into itself, generated by the one-parameter\nsubgroup preserving $I$ and two one-parameter semigroups fixing either\nof its endpoints. Then $\\Sigma = \\Sigma_{I_0}\\times\\Sigma_{J_0}\\subset\n\\Mb\\times\\Mb$ is the connected semigroup taking the reference double\ncone $O_0$ into itself.  \n\nFor $g=(g_1,g_2)\\in\\Sigma$, the adjoint action of $U_+(g_1)U_-(g_2)$\non $a_1\\in A(I_0)$, $a_2\\in A(J_0)$ is given by the {\\em independent} \n(= product) action of the chiral M\\\"obius transformations given by\ngeometric automorphisms $\\alpha_g$ of the chiral net $A$:\n\\begin{eqnarray}\\label{alpha}\n\\alpha^+_{g_1}\\alpha^-_{g_2}(a_1\\cdot a_2) =\n\\alpha_{g_1}(a_1)\\cdot\\alpha_{g_2}(a_2),\n\\end{eqnarray}\nWe extend these endomorphisms of $A_+(O_0)$ to\nendomorphisms of $B_+(O_0)$ by\n\\begin{eqnarray}\\label{beta} \n\\beta^+_{g_1}\\beta^-_{g_2}\\big(\\iota(a_1a_2)\\cdot\\psi\\big) :=\n\\iota\\big(\\alpha_{g_1}(a_1)\\alpha_{g_2}(a_2)\n\\big)\\cdot\\iota\\big(z^\\sigma(g_1)z^{\\bar\\tau}(g_2)\\big)\\cdot \\psi.\n\\end{eqnarray}\nHere $z^\\rho(g)\\in \\mathrm{Hom}(\\rho,\\alpha_g\\rho\\alpha_g^{-1})$ are the\nunitary cocycles \\cite{FRS2,L} $z^\\rho(g) = U_0(g)U_\\rho(g)^*\\in A$\nwhere $U_0$ and $U_\\rho$ are the representations of the M\\\"obius group\nin the vacuum representation and in the DHR representation $\\rho$. \n\n\\medskip\n\n{\\bf Proposition 1:} {\\sl (i) The maps $\\beta^+_{g_1}\\beta^-_{g_2}$\n  defined by (\\ref{beta}) for $g\\in\\Sigma$ are homomorphisms from\n  $B_+(O_0)$ onto $B_+(g\\,O_0)\\subset B_+(O_0)$. \\\\[1mm] \n  (ii) For $O_1\\subset O_0$ we have\n  $\\beta^+_{g_1}\\beta^-_{g_2}\\big(B_+(O_1)\\big) = B_+(g\\,O_1)$, i.e.,\n  $\\beta^+_{g_1}\\beta^-_{g_2}$ ``act geometrically inside\n  $B_+(O_0)$''. \\\\[1mm] \n  (iii) $\\beta^+_{g_1}\\beta^-_{g_2}$ respect the group composition law\n  within the semigroup $\\Sigma$.   \\\\[1mm] \n  (iv) The conditional expectation $\\mu$ intertwines\n  $\\beta^+_{g_1}\\beta^-_{g_2}$ with $\\alpha^+_{g_1}\\alpha^-_{g_2}$. }\n\n\\medskip\n\n{\\it Proof:} (i) The homomorphism property follows from the\ncomposition and conjugation laws of charged intertwiners \\cite{LR2}\nand the intertwining and localization properties of the operators and\nendomorphisms involved. The statement about the range is just a\nspecial case of (ii). \\\\  \n(ii) It is sufficient to show that a charged intertwiner \n$\\psi_1\\in B_+(O_1)$ is mapped to a charged intertwiner in $B_+(g\\,O_1)$. By\nvirtue of (\\ref{chint1}), we compute \n\\begin{eqnarray} \\beta^+_{g_1}\\beta^-_{g_2}(\\psi_1) =\n\\iota\\big(\\alpha_{g_1}(u)z^\\sigma(g_1)\\alpha_{g_2}(\\bar\nu)z^{\\bar\\tau}(g_2)\\big)\\cdot \\psi.\n\\end{eqnarray}\nThen the claim follows, because $\\alpha_{g_1}(u)\\,z^\\sigma(g)\\in\n\\mathrm{Hom}(\\sigma,\\alpha_{g_1}\\sigma_1\\alpha_{g_1}^{-1})$, and \n$\\alpha_{g_1}\\sigma_1\\alpha_{g_1}^{-1}$ is localized in $g_1\\, I_1$, and\nsimilarly $\\alpha_{g_2}\\bar\\tau_1\\alpha_{g_2}^{-1}$ is localized in\n$g_2\\, J_1$.\\\\  \n(iii) The group composition law follows from the cocycle\nproperties \\cite{FRS2,L} of $z^\\rho$. \\\\ \n(iv) The intertwining property of $\\mu$ is due to the fact that $\\mu$ \nannihilates all charged intertwiners except the neutral one\n($\\sigma=\\bar\\tau={\\mathrm {id}}$).\\hspace*{\\fill}Q.E.D. \n\n\\medskip \n\nNext, we adapt a well-known Lemma about the implementation of\n(groups of) automorphisms to the case of (semigroups of) endomorphisms.\n\n\\medskip \n\n{\\bf Lemma 2:} {\\sl  Let $M$ be a von Neumann algebra on a Hilbert\n  space ${\\mathcal H}$ with a cyclic and separating vector $\\Psi$. Let $\\beta$\n  be an endomorphism of $M$, preserving the state $(\\Psi,\\cdot\\Psi)$. \n  Then the closure of the map $m\\Psi\\mapsto \\beta(m)\\Psi$ is an\n  isometry $U_\\beta$. If $\\Psi$ is cyclic also for $\\beta(M)$, then\n  $U_\\beta$ is unitary. For two endomorphisms $\\beta$, $\\beta'$ with\n  the same properties, such that $\\Psi$ is cyclic for $\\beta(M)$, one\n  has $U_{\\beta'\\beta} = U_{\\beta'}U_{\\beta}$. }  \n\n\\medskip\n\n{\\it Proof:} That $U_\\beta$ is an isometry is an obvious consequence\nof the invariance of the state. Since $\\beta(M)\\Psi$ is a dense\nsubset, the $U_\\beta$ is surjective, hence unitary. For the last\nstatement it is sufficient to notice that $U_{\\beta'}$ is densely\ndefined on $\\beta(M)\\Psi$. \n\\hspace*{\\fill}Q.E.D. \n\n\\medskip \n\nWe apply the Lemma to the endomorphisms $\\beta^+_{g_1}\\beta^-_{g_2}$\nof $B_+(O_0)$. Using (iv) of Prop.~1, we see that\n$\\beta^+_{g_1}\\beta^-_{g_2}$ leave the GNS state $(\\hat\\Xi,\\cdot\\hat\\Xi)$\ninvariant because the split state $(\\Xi,\\cdot\\Xi)$ on $A_+(O_0)$ is\ninvariant under $\\alpha^+_{g_1}\\alpha^-_{g_2}$. The vector $\\hat\\Xi$\nis cyclic and separating for each $\\hat\\pi\\big(B_+(O_1)\\big)$\n($O_1\\subset O_0$) because $\\mu$ is faithful and $\\Xi$ is cyclic and\nseparating for each $A_+(O)$, which in turn follows by the split\nisomorphism because $\\Omega$ is cyclic and separating for $A(I_1)$ and\nfor $A(J_1)$. Thus, Lemma~2 applies:\n\n\\medskip\n\n{\\bf Corollary 1:} {\\sl The homomorphisms \n$\\beta^+_{g_1}\\beta^-_{g_2}$ induce unitary\noperators on $\\hat{\\mathcal H}={\\mathcal H}_{2D}$, which satisfy the group composition\nlaw within the semigroup $\\Sigma$. Together with the inverse unitary\noperators, they generate a covering representation $\\hat U(g_1,g_2) =\n\\hat U_+(g_1)\\hat U_-(g_2)$ of $\\Mb\\times \\Mb$ on ${\\mathcal H}_{2D}$.}\n\n\\medskip  \n\nThe last statement is due to the fact that $\\Sigma$ and its inverse\ngenerate $\\Mb\\times\\Mb$, and the group law within $\\Sigma$ secures the\ncommutation relations of the Lie algebra. \n\nBy construction, for $g=(g_1,g_2)\\in\\Sigma$, $\\hat U(g_1,g_2)$ on the\nsubspace ${\\mathcal H}_\\psi$ is equivalent to $U_\\sigma(g_1)\\otimes U_{\\bar\\tau}(g_2)$ \non ${\\mathcal H}_\\sigma\\otimes{\\mathcal H}_{\\bar\\tau}$ under the isomorphism (\\ref{hpsi}). \nBy (ii) of Prop.~1, the adjoint action of $\\hat U(g_1,g_2)$ \ntakes $B_+(O_1)$ to $B_+(g\\,O_1)$ for $O_1\\subset O_0$.\n\nBy constructing $U_+U_-$, we have thus furnished the local subnet\n$O_0\\supset O_1\\mapsto B_+(O_1)$ of the BCFT with a covariant\n``two-dimensional re-interpretation''. In the representation $\\hat\\pi$\non $\\hat{\\mathcal H}={\\mathcal H}_{2D}$, this is precisely the local isomorphism\n$\\varphi^{O_0}$ referred to in (\\ref{lociso}). The present discussion\nshows that $\\varphi^{O_0}$ intertwines the global 2D M\\\"obius\ncovariance with a ``hidden'' symmetry of the BCFT, which is induced by the\nextended split state $\\hat\\xi$ and acts locally geometric. \n\n\\medskip\n\nWe now define for arbitrary double cones $O\\subset M^2$ the associated\nlocal algebras of the 2D conformal net on ${\\mathcal H}_{2D}$ by varying\n$g=(g_1,g_2)\\in \\Mb\\times\\Mb$ in the connected neighborhood of unity\nfor which $g\\,O_0\\subset M^2$, and putting\n\\begin{eqnarray}\\label{b2d}\nB_{2D}(O) := \\hat U(g_1,g_2)\\; \nB_{2D}(O_0)\\; \\hat U(g_1,g_2)^* \\quad\n\\hbox{if}\\quad O=g\\, O_0\\subset M^2.\\;\n\\end{eqnarray}\nFor $O\\subset O_0$, this coincides with\n$\\hat\\pi\\big(\\beta^+_{g_1}\\beta^-_{g_2}(B_+(O_0))\\big) \n= \\hat\\pi\\big(B_+(O)\\big)$ by virtue of (ii) of Prop.~1. \nNotice that $B_{2D}(g\\,O_0)$ is uniquely defined as long as \n$O= g\\,O_0\\subset M^2$ because in this case any two $g$ with the same\nimage $g\\,O_0$ differ by an element of $\\Sigma$, while it requires the\npassage to a covering space when $M^2$ is conformally completed. \n\n\\medskip \n\n{\\bf Theorem 1:} {\\sl The net of von Neumann algebras $O\\mapsto\n  B_{2D}(O)$ defined by (\\ref{b2d}) is covariant, isotonous, and\n  local.}\n\n\\medskip\n\n{\\it Proof:}  The covariance is by construction. Isotony and\nlocality of the 2D net follow from the geometric action inside $O_0$,\n(ii) of Prop.~1, and the fact that every pair of double cones in $M^2$\nsuch that either $O_1\\subset O_2$ are $O_1\\subset O_2'$ can be moved\ninside $O_0$ by a M\\\"obius transformation, where we know (from the\nboundary CFT) that isotony and locality hold. \\hspace*{\\fill}Q.E.D.\n\n\\medskip\n\n{\\bf Corollary 2:} {\\sl The extension $A_{2D}\\subset B_{2D}$ is\n  isomorphic to the extension constructed in \\cite{KHR}. } \n\n\\medskip\n\n{\\it Proof:}  Since the local subfactor $A_{2D}(O_0)\\subset B_{2D}(O_0)$\nconstructed in \\cite{KHR} is isomorphic to $A_+(O_0)\\subset B_+(O_0)$,\nand the isomorphism intertwines the representations of the 2D M\\\"obius\ngroup, the global isomorphism follows. \\hspace*{\\fill}Q.E.D.\n\n\\medskip\n\nWe have associated with the BCFT a 2D local CFT,\nthat is locally isomorphic. The association is intrinsic in the sense\nthat it requires only the subnet $O_0\\supset O_1\\mapsto B_+(O_1)$\ntogether with the covariance of the DHR sectors of the underlying\nchiral CFT $A$. \n\nIt should be noticed that the construction is up to unitary\nequivalence independent of the choice of the reference double cone\n$O_0\\subset M_+$. The reason is essentially that the charge structure\nof $B(K)'\\cap B(L)$ exhibited by the multiplicities $Z_{\\sigma,\\tau}$\nin (\\ref{h2d}) is independent of the pair $K\\subset L$. \n\n\\medskip\n\nWe conclude this section with an observation concerning diffeomorphism\ncovariance: \n\n\\medskip\n\n{\\bf Proposition 2:} {\\sl If $A\\subset B$ is a chiral extension of a\n  diffeomorphism covariant chiral net $A$, then the (possibly non-local) chiral net $B$, the BCFT\n  net $B_+$ defined by (\\ref{relcomm}), and the 2D net $B_{2D}$\n  associated with $B_+$ by Thm.~1 are also diffeomorphism covariant.}\n\n\\medskip\n\n{\\it Proof:} The chiral net $A$ is diffeomorphism covariant if for a\ndiffeomorphism $\\gamma$ of $S^1$ there is a unitary operator\n$w_\\gamma$ on ${\\mathcal H}_0$ such that $u_\\gamma A(I) u_\\gamma^* =\nA(\\gamma\\,I)$. Haag duality of $A$ implies that if $\\gamma$ is\nlocalized in an interval $I$ (i.e., acts trivially on the complement),\nthen $w_\\gamma$ is an observable in $A(I)$. \n\nFor a chiral extension $A\\subset B$ we claim that if $\\gamma$\nis localized in $I_0$, then for $I_1\\subset I_0$ one has\n$\\iota(w_\\gamma) B(I_1)\\iota(w_\\gamma^*) = B(\\gamma I_1)$, i.e.,\n$\\iota(w_\\gamma)$ implement the local diffeomorphisms. Namely,\n$B(I_1)$ is generated by \n$\\iota\\big(A(I_1)\\big)$  and $v_1=\\iota(u)\\cdot v$ where $v\\in B(I_0)$  \nis the canonical charged intertwiner $v\\in\\mathrm{Hom}(\\iota,\\iota\\theta)$ \nfor the canonical DHR endomorphism $\\theta$ localized in $I_0$ \n\\cite{LR1} (see also App.~\\ref{appeig}), and $\\theta_1$ is an\nequivalent DHR endomorphism localized \nin $I_1$. We find \n\\begin{eqnarray}\n\\iota(w_\\gamma)\\;v_1\\;\\iota(w_\\gamma^*) = \\iota(w_\\gamma u\n\\theta(w_\\gamma^*))\\cdot v.\n\\end{eqnarray}\nNow, $w_\\gamma u \\theta(w_\\gamma^*)\\in\n\\mathrm{Hom}(\\theta,\\gamma\\theta_1\\gamma^{-1})$, and $\\gamma\\theta_1\\gamma^{-1}$\nis localized in $\\gamma\\,I_1$. This proves the claim. The\ndiffeomorphism covariance of the chiral net $B$ follows because \nthe diffeomorphisms localized in $I_0$ together with the M\\\"obius group\ngenerate the diffeomorphism group of $S^1$. \n\nThe argument for the boundary CFT and for the 2D CFT are very\nsimilar: we first show that for diffeomorphisms\n$\\gamma=\\gamma_1\\gamma_2$ where $\\gamma_1$ is localized in $I_0$ and\n$\\gamma_2$ localized in $J_0$, the adjoint action with\n$\\iota(w_{\\gamma_1}w_{\\gamma_2})$ takes $B_+(O_1)$ to\n$B_+(\\gamma\\,O_1)$ if $O_1\\subset O_0$. Again, it is sufficient to\nverify the action on the charged intertwiners (\\ref{chint1}) of\n$B_+(O_1)$: \n\\begin{eqnarray}\\label{diff}\n\\iota(w_{\\gamma_1}w_{\\gamma_2}) \\cdot \\psi_1 \\cdot\n\\iota(w_{\\gamma_1}w_{\\gamma_2})^* = \\iota\\big((w_{\\gamma_1}u\\sigma(w_{\\gamma_1}^*))(w_{\\gamma_2}\\bar u\\bar\\tau(w_{\\gamma_2}^*))\\big)\\cdot \\psi\n\\end{eqnarray}\nwhere $w_{\\gamma_1}u\\sigma(w_{\\gamma_1}^*)\n\\in\\mathrm{Hom}(\\sigma,\\gamma_1\\sigma_1\\gamma_1^{-1})$ and $w_{\\gamma_2}\\bar\nu\\bar\\tau(w_{\\gamma_2}^*)\n\\in\\mathrm{Hom}(\\bar\\tau,\\gamma_2\\bar\\tau_1\\gamma_2^{-1})$, and\n$\\gamma_1\\sigma_1\\gamma_1^{-1}$ is localized in $\\gamma_1\\,I_1$ and\n$\\gamma_2\\bar\\tau_1\\gamma_2^{-1}$ is localized in\n$\\gamma_2\\,J_1$. Hence (\\ref{diff}) is a charged intertwiner of \n$B_+(\\gamma O_1)$. This proves the claim. Then the diffeomorphism\ncovariance of $B_+$ and $B_{2D}$ follow because \nthe diffeomorphisms localized in $O_0$ together with the M\\\"obius\ngroup generate all diffeomorphisms.  \\hspace*{\\fill}Q.E.D.\n\n\n\n\\section{Cluster limit}\\label{sec4}\n\\setcounter{equation}{0}\n\nLet $b_1,\\dots,b_n\\in B_+(O)$ be BCFT observables localized within any\nfixed double cone $O=I\\times J\\subset M_+$. We wish to consider the\nbehavior of a vacuum correlation  \n\\begin{eqnarray}\\label{b1bn} \n\\big(\\Omega\\,,\\, \\beta_x(b_1\\cdots b_n)\\,\\Omega\\big),\\end{eqnarray}\nwhere $\\beta_x=\\beta^+_x\\beta^-_{-x}$ is the one-parameter semigroup\nof ``right shifts'' ($x>0$, away from the boundary), that take $I$ to $I+x$\nand $J$ to $J-x$, represented as homomorphisms from $B_+(O)$\nto $B_+(I+x\\times J+x)$, see (\\ref{beta}).\n\nIn Sect.~3 (with $O$ as the fixed reference\ndouble cone) we have given the re-inter\\-pretation of $b_i$ in the GNS\nrepresentation $\\hat\\pi$ of the state $\\xi\\circ\\mu$ as observables of\nthe associated 2D CFT, with the 2D vacuum $\\Omega_{2D}$ given by the\nGNS vector. We shall show\n\n\\medskip \n\n{\\bf Theorem 2:} {\\sl Let each $b_i\\in B_+(O)$ ($i=1,\\dots,n)$ be of the\n  form $\\iota(a_1^{(i)} a_2^{(i)}) \\cdot \\psi^{(i)}$ with charged \n  intertwiners $\\psi^{(i)}$ and $a^{(i)}_1\\in A(I)$ and $a^{(i)}_2\\in A(J)$. \nAs $x$ goes to $+\\infty$, the BCFT vacuum correlations (\\ref{b1bn})\nconverge to the 2D vacuum correlations }\n\\begin{eqnarray}\\label{corr2d}\n\\big(\\Omega_{2D}\\,,\\,\\hat\\pi(b_1\\cdots\nb_n)\\,\\Omega_{2D}\\big) = \\xi\\circ\\mu(b_1\\cdots b_n). \n\\end{eqnarray}\n\n\\medskip\n\n{\\it Proof:} We compute the limit and the 2D vacuum expectation value\nseparately. \n\nUsing the decomposition of products $\\psi_1\\psi_2$ into\nfinite sums of operators of the form $\\iota(T_1T_2)\\cdot \\psi$\n\\cite{LR2}, where $T_i$ are intertwiners between DHR \nendomorphisms of $A$, we see that the product $b_1\\cdots b_n$ is a\nfinite sum of operators of the same form $\\iota(a_1a_2)\\cdot \\psi$.\n\nFor the present purpose, it is more convenient to write the charged\nintertwiners as $\\psi= t\\cdot\\iota(\\bar r)$ where $r\\in\n\\mathrm{Hom}({\\mathrm {id}},\\tau\\bar\\tau)\\subset A(J)$ and $t\\in\n\\mathrm{Hom}(\\alpha^+_\\tau,\\alpha^-_\\sigma)\\subset \\mathrm{Hom}(\\iota\\tau,\\iota\\sigma)$\n(Frobenius reciprocity). Then, because $a_2=\\sigma(a_2)$, we get\n$\\iota(a_2)\\cdot \\psi = t\\cdot \\iota(\\tau(a_2)\\bar r)$. Hence, the product\n$b_1\\dots b_n$ is a finite sum of operators\nof the form\n\\begin{eqnarray} \\label{ata}\n\\iota(a_1)\\cdot t\\cdot \\iota(a_2).\n\\end{eqnarray}\nThus, the above vacuum correlation function is a finite sum\nof expectation values\n\\begin{eqnarray} F(x) = \\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,\\Omega\\big) = \\qquad\\qquad\\qquad\\qquad \\nonumber  \n\\\\ = \\big(\\Omega\\,,\\,\\iota(\\alpha_x(a_1) z^\\sigma(x))\\cdot t\\cdot \n\\iota(z^\\tau(-x)^* \\alpha_{-x}(a_2))  \\,\\Omega\\big) =  \\\\\n= \\big(\\Omega\\,,\\,\\alpha_x(a_1) z^\\sigma(x)\\cdot \\varepsilon(t)\\cdot \nz^\\tau(-x)^* \\alpha_{-x}(a_2)\\,\\Omega\\big). \\nonumber \\end{eqnarray}\nHere, $\\varepsilon$ is the global conditional expectation $B\\to A$, which\npreserves the vacuum state \\cite{LR1}. In particular, $\\varepsilon(t)\\in\n\\mathrm{Hom}(\\tau,\\sigma)$. Therefore, the expression vanishes identically\nunless $\\sigma$ and $\\tau$ belong to the same sector.\n\nIn the latter case, we express the cocycles as\n$z^\\rho(g)=U_0(g)U_\\rho(g)^*$, and $\\alpha_g={\\mathrm {Ad}}_{U_0(g)}$, giving \n\\begin{eqnarray}\nF(x) = \\big(\\Omega\\,,\\,a_1 U_\\sigma(x)^* \\cdot \\varepsilon(t) \\cdot U_\\tau(x)^*\na_2\\,\\Omega\\big) =\\big(\\Omega\\,,\\,a_1 \\cdot U_\\sigma(-2x) \\cdot \\varepsilon(t) a_2\\,\\Omega\\big) ,\n\\end{eqnarray}\nbecause the intertwiners between DHR endomorphisms also intertwine the \nrepresentations of the M\\\"obius group \\cite{FRS2}. \nBy the spectrum condition, $F(x)$ has a bounded analytic\ncontinuation to the lower complex halfplane. $U_\\sigma(-z)$ weakly\nconverges in every direction $z=re^{i\\varphi}$ ($-\\pi < \\varphi <0$,\n$r\\to\\infty$) to the projection onto the zero eigenspace of the\ngenerator, and the latter projection is nonzero only if $\\sigma={\\mathrm {id}}$\nis the vacuum representation; in this case $t=\\varepsilon(t)=1$. Thus, $F(z)$\nconverges in these directions to the vacuum expectation value \n\\begin{eqnarray}\\label{limit}\n\\delta_{\\sigma,0}\\delta_{\\tau,0}\n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega).\n\\end{eqnarray}\n\nNext, we consider \n\\begin{eqnarray}\n\\overline{F(x)} = \\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(a_2^*)\\cdot t^*\\cdot\n\\iota(a_1^*)\\big)\\,\\Omega\\big).\n\\end{eqnarray}\nLet $r_\\sigma\\in \\mathrm{Hom}({\\mathrm {id}},\\bar\\sigma\\sigma)\\subset A(I)$\nand $r_\\tau\\in \\mathrm{Hom}({\\mathrm {id}},\\bar\\tau\\tau)\\subset A(J)$. Then we can write \n$t^* = \\iota(r_\\sigma^*)\\cdot \\bar t\\cdot \\iota(r_\\tau)$, where \n$\\bar t \\in \\mathrm{Hom}(\\alpha^+_{\\bar\\tau},\\alpha^-_{\\bar\\sigma})\\subset \n\\mathrm{Hom}(\\iota\\bar\\tau,\\iota\\bar\\sigma)$. Using the locality\nproperties of $a_1\\in A(I)$, $a_2\\in A(J)$, we can rewrite\n\\begin{eqnarray}\n\\overline{F(x)} =\n\\big(\\Omega\\,,\\,\\beta_x\\big(\\iota(r_\\sigma^*\\bar\\sigma(a_1^*))\\cdot \\bar t\\cdot\n\\iota(\\bar\\tau(a_2^*)r_\\tau)\\big)\\,\\Omega\\big). \n\\end{eqnarray}\nThis expression can be computed in the same way as $F(x)$ before,\ngiving\n\\begin{eqnarray}\n\\overline{F(x)} = \\big(\\Omega\\,,\\,\nr_\\sigma^*\\bar\\sigma(a_1^*))\\cdot U_{\\bar\\sigma}(-2x) \\cdot \\varepsilon(\\bar t)\n\\bar\\tau(a_2^*)r_\\tau\\,\\Omega\\big).\n\\end{eqnarray}\nThus $F(x)$ also has a bounded analytic continuation to the upper\ncomplex halfplane, and converges to the same limit (\\ref{limit}) also\nin the directions $z=re^{i\\varphi}$ ($0 < \\varphi < \\pi$, $r\\to\\infty$). \nFrom this, we may conclude the cluster limit  \n\\begin{eqnarray}\\label{cluster}\n\\lim_{x\\to\\infty}\\big(\\Omega\\,,\\,\\beta_{x}\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,\\Omega\\big) = \\delta_{\\sigma,0}\\delta_{\\tau,0}\n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega).\n\\end{eqnarray}\n\nOn the other hand, we now compute (\\ref{corr2d}) and show that it\ncoincides with the factorizing cluster limit of\n(\\ref{b1bn}). For each contribution of the form\n(\\ref{ata}), we have\n\\begin{eqnarray}\n(\\Omega_{2D}\\,,\\,\\hat\\pi\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\,,\\,\\Omega_{2D}) = \\xi\\circ\\mu\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big) = \\xi\\big(a_1\\cdot \\mu(t)\\cdot a_2\\big). \\qquad\n\\end{eqnarray}\nBut $\\mu(t)\\in A(I)\\vee A(J)$ is an intertwiner in $\\mathrm{Hom}(\\sigma,\\tau)$\nwhich vanishes unless $\\sigma={\\mathrm {id}}$ and $\\tau={\\mathrm {id}}$ both belong to the\nvacuum sector. In the latter case, $t=\\mu(t)=1$. Thus,\n\\begin{eqnarray}\\label{cblock}\n\\langle\\hat\\Xi\\vert \\hat\\pi\\big(\\iota(a_1)\\cdot t\\cdot\n\\iota(a_2)\\big)\\vert \\hat\\Xi\\rangle = \\delta_{\\sigma,0}\\delta_{\\tau,0} \\;\n\\xi(a_1a_2)  = \\delta_{\\sigma,0}\\delta_{\\tau,0} \n\\;(\\Omega,a_1\\Omega) \\cdot(\\Omega,a_2\\Omega). \\qquad\n\\end{eqnarray} \nThis coincides with the cluster limit (\\ref{cluster}) ``far away from\nthe boundary''. \\hspace*{\\fill}Q.E.D.\n\n\\medskip \n\nRecall that $a_1$ and $a_2$ in (\\ref{ata}) were obtained by\nmultiplying $b_1\\cdots b_n$ and successively decomposing the products\nof the charged intertwiners. Thus, the vacuum expectation values\n$(\\Omega,a_i\\Omega)$ in (\\ref{cblock}) are precisely the chiral\nconformal blocks of the corresponding 2D correlation functions. \n\nA variant of the conformal cluster theorem \\cite{FJ} should also give\na quantitative estimate for the rate of the convergence, depending on\nthe charges of the operators involved through the corresponding\nspectrum of $L_0$.\n\n\n\\section{Conclusion}\nWe have studied the passage from a local conformal quantum field\ntheory defined on the halfspace $x>0$ of two-dimensional Minkowski\nspacetime (boundary CFT, BCFT) to an associated local conformal\nquantum field defined on the full Minkowski spacetime (2D CFT). There\nare essentially two ways: the first is to consider BCFT vacuum\ncorrelations of observables localized far away from the boundary. In\nthe limit of infinite distance, these correlation factorize into\nchiral correlations (conformal blocks) of charged fields. We have\ntraced this effect back to the cluster property of the underlying\nlocal chiral subtheory. \n\nThe second method exploits the split property, i.e., the existence of\nstates of the underlying local chiral CFT in which correlations between\nobservables in two fixed intervals at a finite distance are suppressed. \nWith the help of the split property one can algebraically identify a\nfixed local algebra of the BCFT with a fixed local algebra of the 2D\nCFT, and one can generate a unitary representation of the 2D M\\\"obius\ngroup in the GNS Hilbert space of a suitable ``extended split state''\nof this algebra. Its ground state is different from the BCFT\nvacuum. Then, by acting with the 2D M\\\"obius group, one can obtain\n{\\em all} local algebras of the 2D CFT in the same Hilbert space. \n\nThe converse question: can one consistently ``add'' a boundary in any\n2D CFT (without affecting the algebraic structure away from the\nboundary), is not addressed here. However, there arises a necessary \ncondition from the discussion in App.~\\ref{apphmi}: the 2D partition\nfunction should be either modular invariant, or at least it should be\nintermediate between the vacuum partition function and some modular\ninvariant partition function. We hope to return to this problem, and\nfind also a sufficient condition. \n\n\\bigskip\n\n\\noindent\n{\\bf Acknowledgements:} KHR thanks the Dipartimento di Matematica of\nthe Universit\\`a di Roma ``Tor Vergata'' for hospitality and financial\nsupport, and M. Weiner and I. Runkel for discussions related to the subject.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nFungicide resistance is a prime example of adaptation of a population\nto an environmental change, also known as evolutionary rescue\n\\cite{bego11,hosg11}.\nWhile global climate change is expected to result in a loss of\nbiodiversity in natural ecosystems, evolutionary rescue is seen as a\nmechanism that may mitigate this loss. In the context of crop\nprotection the point of view is quite the opposite: reducing\nadaptation of crop pathogens to chemical disease control would help\nstabilize food production. Better understanding of the adaptive\nprocess may help slow or prevent it.\nThis requires a detailed quantitative understanding of the dynamics of\ninfection and the factors driving the emergence and development of\nfungicide resistance \\cite{bogi08}. \nDespite the global importance and\nurgency of fungicide resistance, this problem has received relatively\nlittle theoretical consideration (see\n\\cite{hopa+11a,hagu+07,sh06,pagi+05,sh89,mile+89} and \\cite{bogi08}\nfor a comprehensive review) as compared, for example, to antibiotic\nresistance \\cite{ozsh+12,boau+01,le01,auan99}.\n\nIn recent years, agrochemical companies have begun marketing mixtures\nthat contain fungicides with a low-risk of developing resistance with\nfungicides that have a high-risk developing of resistance. In extreme\ncases the high-risk fungicide is no longer effective against some\ncommon pathogens because resistance has become widespread. For\nexample, a large proportion of the European population of the\nimportant wheat pathogen \\emph{Mycosphaerella graminicola} (recently\nrenamed \\emph{Zymoseptoria tritici}) \\cite{orde+11,pask03} is\nresistant to strobilurin fungicides \\cite{tobr+09}.\n\n\nA number of previous modeling studies addressed the effect of\nfungicide mixtures on selection for fungicide resistance (for example,\n\\cite{kaje80,sk81,jodo85,sh93,hopa+11a,hopa+13}). Different studies\nused different definitions of ``independent action'' (also called\n``additivity'' or ``zero interaction'' in the literature) of\nfungicides in the mixtures \\cite{sh89a} and reported somewhat\ndifferent conclusions. One study \\cite{sh89a} critically reviewed the\noutcomes of these earlier studies and attempted to clarify the\nconsequences of using different definitions of ``independent\naction''. Some studies found that alternations are preferable to\nmixtures \\cite{kaje80}, while others found that mixtures are\npreferable to alternations \\cite{sk81}. A more recent study\n\\cite{hopa+13} addressed this question using a detailed population\ndynamics model and found that in all scenarios considered, mixtures to\nprovided the longest effective life of fungicides as compared to\nalternations or concurrent use (when each field receives a single\nfungicide, but the fungicides applied differ between the fields). This\nstudy used the Bliss' definition of ``independent action'' of the two\nfungicides \\cite{bl39} (also called Abbot's formula in the fungicide\nliterature \\cite{ab25}).\n\n\n\nWe addressed the question of whether mixtures of low-risk and high\nrisk fungicides can provide adequate disease control while minimizing\nfurther selection for resistance using a simple population dynamics\nmodel of host-pathogen interaction based on a system of ordinary\ndifferential equations. We found that the fitness cost associated with\nresistance mutations is a crucial parameter, which governs the outcome\nof the competition between the sensitive and resistant pathogen\nstrains.\n\n\n\nA single point mutation associated with fungicide resistance sometimes\nmakes the pathogen completely insensitive to a fungicide, as is the\ncase for the G143A mutation giving resistance to strobilurin\nfungicides in many fungal pathogens \\cite{feto+08,gisi+02}. In many\nother cases the resistance is partial, for example, resistance of\n\\emph{Z. tritici} and other fungi to azole fungicides\n\\cite{co08,zhst+06}.\nTherefore, we considered varying degrees of resistance in our model.\n\n\nIn contrast to our study, resistance in \\cite{hopa+11a} was assumed to\nbear no fitness costs for the pathogen. It was found that in the\nabsence of fitness costs the use of fungicide mixtures \\emph{delays}\nthe development of resistance \\cite{hopa+11a}. This conclusion is in\nagreement with our results (see Appendix\\,A.4).\nHere we focus on finding conditions under which the selection for the\nresistant pathogen strain is \\emph{prevented} by using fungicide\nmixtures.\n\n\n\\section{Theory and approaches}\n\\label{sec:model-assump}\n\nWe use a deterministic mathematical model of susceptible-infected\ndynamics (see \\fig{fig:model-scheme})\n\n\\begin{align\n\\frac{d H}{d t} &= r_H (K - H - I_\\mrm{s} - I_\\mrm{r} ) -  b \\left( \\left[ 1 - \\varepsilon_\\mrm{s}(C,\n  r_\\mrm{B}) \\right] I_\\mrm{s} + \\left[ 1 - \\varepsilon_\\mrm{r}(C, r_\\mrm{B}) \\right] (1-\\rho_\\mrm{r}) I_\\mrm{r} \\right) H, \\label{eq:1host2fung-gen-1}\\\\\n\\frac{d I_\\mrm{s}}{d t} & = b \\left[ 1 - \\varepsilon_\\mrm{s}(C,\n  r_\\mrm{B}) \\right] H I_\\mrm{s} - \\mu  I_\\mrm{s}, \\label{eq:1host2fung-gen-2}\\\\ \n\\frac{d I_\\mrm{r}}{d t} & = b \\left[ 1 - \\varepsilon_\\mrm{r}(C,\n  r_\\mrm{B}) \\right] (1-\\rho_\\mrm{r}) H I_\\mrm{r} - \\mu  I_\\mrm{r}. \\label{eq:1host2fung-gen-3}\n\\end{align}\nThe model has three compartments: healthy hosts $H$, hosts\ninfected by a sensitive pathogen strain $I_\\mrm{s}$, hosts infected by\na resistant pathogen strain $I_\\mrm{r}$; and is similar to the models\ndescribed in \\cite{bogi08,hagu+07}. The subscript ``s'' stands for the\nsensitive strain and the subscript ``r'' stands for the resistant\nstrain.\nThe quantities $H$, $I_\\mrm{s}$ and $I_\\mrm{r}$, represent the total amount of the\ncorresponding host tissue within one field, which could be leaves,\nstems or grain tissue, depending on the specific host-pathogen interaction.\nHealthy hosts $H$ grow with the rate $r_H$. Their growth is\nlimited by the ``carrying capacity'' $K$, which may imply limited\nspace or nutrients.\nFurthermore, healthy hosts may be infected by the sensitive\npathogen strain and transformed into infected hosts in the compartment\n$I_\\mrm{s}$ with the transmission rate $b$. This is a compound\nparameter given by the product of the sporulation rate of the infected\ntissue and the probability that a spore causes new infection.\nHealthy hosts may also be infected by the resistant pathogen strain\nand transformed into infected hosts in the compartment $I_\\mrm{r}$. In\nthis case, resistant mutants suffer a fitness cost $\\rho_\\mrm{r}$\nwhich affects their transmission rate such that it becomes equal to $b\n(1 - \\rho_\\mrm{r})$.\nThe corresponding terms in\nEqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}) are\nproportional to the amount of the available healthy tissue $H$ and\nto the amount of the infected tissue $I_\\mrm{s}$ or\n$I_\\mrm{r}$. Infected host tissue loses its infectivity at a rate\n$\\mu$, where $\\mu^{-1}$ is the characteristic infectious period.\n\nSince our description is deterministic we do not take into account the\nemergence of new resistance mutations but assume that the resistant\npathogen strain is already present in the\npopulation. Therefore, when ``selection for resistance'' is discussed\nbelow,\nwe refer to the process of winning the competition by this existing\nresistant strain due to its higher fitness with respect to the\nsensitive strain in the presence of fungicide treatment.\nEmergence of new resistance mutations is a different problem, which\ngoes beyond the scope of our study and requires stochastic simulation\nmethods. We do not consider the possibility of double resistance in\nthe model, but by preventing selection for single resistance\nas described here, one would also diminish the probability of the\nemergence of double resistance for both sexually and asexually\nreproducing pathogens (see Appendix\\,A.7).\n\nWe consider two fungicides A and B. Fungicide A is the high-risk\nfungicide, to which the resistant pathogen strain exhibits a variable\ndegree of resistance. However, the sensitive strain is fully sensitive to\nfungicide A. Fungicide B is the low-risk fungicide, i.\\,e. both pathogen\nstrains are fully sensitive to it.\nWe compare the effects of the fungicide A applied alone,\nfungicide B applied alone and the effect of their mixture in\ndifferent proportions.\n\nWe assume that the fungicides will decrease the pathogen transmission\nrate $b$ [see the\nexpression in square brackets in \\eq{eq:1host2fung-gen-2}, \\eq{eq:1host2fung-gen-3}]. \nFor example, application of a fungicide could result in production of\nspores that are deficient essential metabolic products such as ergosterol or\n$\\beta$-tubulin. Consequently, these spores would\nlikely have a lower success rate in causing new infections.\nSpores of sensitive strains of \\emph{Z. tritici} produced shorter germ\ntubes when exposed to azoles \\cite{leal+07}. Spores that produce\nshorter germ tubes are less likely to find and penetrate stomata,\nhence are less likely to give rise to new infections. Protectant\nactivity of fungicides will also reduce the transmission rate $b$\n\\cite{wowi01,pf06}. These studies \\cite{wowi01,pf06} also reported\nthat fungicide application leads to a reduction in the number of\nspores produced. This outcome can be attributed to the fungicide\ndecreasing the sporulation rate and thus affecting $b$ or decreasing\nthe infectious period and thus affecting $\\mu$, or both of these\neffects. More detailed measurements are often needed to distinguish\nbetween these different effects.\n\n\n\n\nWhen only one fungicide applied, the reduction of the transmission rate is described by\n\\begin{equation}\\label{eq:eps-1fungic}\n  \\varepsilon_\\mrm{A}(C_\\mrm{A}) = k_\\mrm{kA} \\frac{C_\\mrm{A}}{C_\\mrm{A} + C_\\mrm{50A}},\n\\end{equation}\nfor the fungicide A, and by\n\\begin{equation}\\label{eq:eps-1fungic}\n \\varepsilon_\\mrm{B}(C_\\mrm{B}) = k_\\mrm{kB} \\frac{C_\\mrm{B}}{C_\\mrm{B} + C_\\mrm{50B}},\n\\end{equation}\nfor the fungicide B.\nThese functions grow with the fungicide doses $C_\\mrm{A}$,\n$C_\\mrm{B}$ and saturate to values $k_\\mrm{kA}$, $k_\\mrm{kB}$,\nrespectively, which are the maximum reductions in the transmission\nrate (or efficacies). This functional form was used before in the fungicide\nresistance literature \\cite{hagu+07,gugi99}. We also performed the\nanalysis for the exponential fungicide action more common in plant\npathology and obtained qualitatively similar results. The reason for\nchoosing the function in \\eq{eq:eps-1fungic} was that it made possible\nto obtain all the results analytically.\nThe parameters $C_\\mrm{50A}$, $C_\\mrm{50B}$ represent the fungicide dose at\nwhich half of the maximum effect is achieved.\n These parameters can always be made equal by rescaling the\nconcentration axis for one of the fungicides. Hence, we set $C_\\mrm{50A}=C_\\mrm{50B}=C_\\mrm{50}$.\n\n\n\nWe next determine the effect of a mixture of two fungicides\naccording to the Loewe's definition of additivity (or non-interaction)\n\\cite{be89} (an equivalent graphic procedure is known as the Wadley\nmethod in the fungicide literature \\cite{lebe+86}). It is based on the\nnotion that a compound cannot interact pharmacologically with\nitself. A sham mixture of a compound A with itself can be created and\nits effect used as a reference point for assessing of whether the\ncomponents of a real mixture interact pharmacologically. When the two\ncompounds A and B have the same effect as the sham mixture of the\ncompound A with itself, they are said to have no interaction (or an\nadditive interaction). In this case, the isobologram equation\n\\begin{equation}\\label{eq:isobol}\nC_\\mrm{A}\/C_\\mrm{Ai} + C_\\mrm{B}\/C_\\mrm{Bi} = 1\n\\end{equation}\nholds (see Sec. VA of \\cite{be89} for the derivation).\nHere, $C_\\mrm{A}$ and $C_\\mrm{B}$ are the doses of the compounds A and B,\nrespectively, when applied in the mixture; $C_\\mrm{Ai}$ is the\nisoeffective dose of the compound A, that is the dose at which\ncompound A alone has the same effect as the mixture; and $C_\\mrm{Bi}$ is the\nisoeffective dose of the compound B.\nIf the mixture of A and B has a larger effect than the\nzero-interactive sham mixture, then $C_\\mrm{A}\/C_\\mrm{Ai} +\nC_\\mrm{B}\/C_\\mrm{Bi} < 1$ and the two compounds are said to interact\nsynergistically. On the contrary, when the mixture of A and B has a\nsmaller effect than the zero-interactive sham mixture,\n$C_\\mrm{A}\/C_\\mrm{Ai} + C_\\mrm{B}\/C_\\mrm{Bi} > 1$ and the two\ncompounds interact antagonistically.\n\nUsing the dose-response dependencies of each fungicide when applied\nalone, \\eq{eq:eps-1fungic} and \\eq{eq:isobol}, we derive the dose-response function\nfor the combined effect of the two fungicides on the sensitive\npathogen strain in the case of no pharmacological interaction (see Sec. VIB of \\cite{be89} for the derivation):\n\\begin{align}\\label{eq:veps-sens-addit}\n\\varepsilon_\\mrm{s}(C_\\mrm{A}, C_\\mrm{B}) =  \\frac{ k_\\mrm{kA} C_\\mrm{A} +\n k_\\mrm{kB} C_\\mrm{B}} { C_\\mrm{A} + C_\\mrm{B} + C_{50}}.\n\\end{align}\nSimilarly, we determine the combined effect of the two fungicides on the\nresistant pathogen strain still without pharmacological\ninteraction:\n\\begin{align}\\label{eq:veps-res-addit}\n\\varepsilon_\\mrm{r}(C_\\mrm{A}, C_\\mrm{B}) = \\frac{ k_\\mrm{kA} \\alpha C_\\mrm{A} +\nk_\\mrm{kB}  C_\\mrm{B}} { \\alpha C_\\mrm{A} + C_\\mrm{B} + C_{50}},\n\\end{align}\nwhere we introduced $\\alpha$, the degree of sensitivity of the resistant strain \nto the fungicide A (the high-risk fungicide). At $\\alpha=0$ the pathogen is fully resistant to\nfungicide A and the effect of the mixture $\\varepsilon_\\mrm{r}(C_\\mrm{A},\nC_\\mrm{B})$ in \\eq{eq:veps-res-addit} does not depend on its\ndose $C_\\mrm{A}$, while at $\\alpha = 1$ the pathogen is fully\nsensitive to fungicide A.\n\nThe expression in \\eq{eq:veps-sens-addit} and \\eq{eq:veps-res-addit}\nare only valid in the range of fungicide concentrations, over which\nisoeffective concentrations can be determined for both\nfungicides. Here, the isoeffective concentration is the concentration\nof a fungicide applied alone that has the same effect as the\nmixture. This requirement means that we are only able to consider the\neffect of the mixture at a sufficiently low total concentration: $C =\nC_\\mrm{A} + C_\\mrm{B}<k_\\mrm{kB} C_{50}\/(k_\\mrm{kA} - k_\\mrm{kB})\/(1 -\nr_\\mrm{B})$.\n\nNext, we introduce deviations from the additive pharmacological\ninteraction. There are several ways to do this, usually by adding an\ninteraction term to the isobologram equation \\cite{grbr95}. We chose a\nspecific form of the interaction term, which is proportional to the\nsquare root of the product of the concentrations of the two compounds\n[Eq.\\,(28) in \\cite{grbr95}]. Assuming\n$k_\\mrm{kA}=k_\\mrm{kB}=k_\\mrm{k}$, this form allows for a simple\nanalytical expression for the effect of the combination on the\nsensitive strain\n\\begin{align}\\label{eq:veps-mix-comp-sens}\n\\varepsilon_\\mrm{s}(C, r_\\mrm{B}) = k_\\mrm{k} \\frac{ C} { C + C_{50}\/ \\gamma_\\mrm{s}},\n\\end{align}\nand on the resistant strain\n\\begin{align}\\label{eq:veps-mix-comp-res}\n\\varepsilon_\\mrm{r}(C, r_\\mrm{B}) = k_\\mrm{k} \\frac{C} { C  + C_{50}\/ \\gamma_\\mrm{r}}.\n\\end{align}\nHere $C=C_\\mrm{A}+C_\\mrm{B}$, where $C_\\mrm{A}$ is the dose of the fungicide A\nand $C_\\mrm{B}$ is the dose of the fungicide B, $r_\\mrm{B} = C_\\mrm{B}\/C$\nis the proportion of the fungicide B in the mixture and\n\\begin{equation}\\label{eq:gammas}\n\\gamma_\\mrm{s} = 1 + u \\sqrt{r_\\mrm{B}(1-r_\\mrm{B})},\n\\end{equation}\n\\begin{equation}\\label{eq:gammar}\n\\gamma_\\mrm{r} = \\alpha (1 - r_\\mrm{B}) + r_\\mrm{B}  + u \\sqrt{\\alpha r_\\mrm{B}(1-r_\\mrm{B})}\n\\end{equation}\nare the parameters which modify $C_{50}$ due to pharmacological\ninteraction and partial\nresistance. Eqs.\\,(\\ref{eq:veps-mix-comp-sens}),\n(\\ref{eq:veps-mix-comp-res}) are obtained from \\eq{eq:isobol} with an\ninteraction term added and the dose-response functions of each\nfungicide when applied alone, \\eq{eq:eps-1fungic}.\nThe degree of pharmacological interaction is characterized by the\nparameter $u$. At $u=0$ the fungicides do not interact and\nEqs\\,(\\ref{eq:veps-mix-comp-sens}), (\\ref{eq:veps-mix-comp-res}) are\nthe same as Eqs\\,(\\ref{eq:veps-sens-addit}),\n(\\ref{eq:veps-res-addit}). The case when $u>0$ represents synergy: the\ninteraction term proportional to $u$ in \\eq{eq:veps-mix-comp-sens} and\n\\eq{eq:veps-mix-comp-res} is positive and it reduces the value of\n$C_{50}$, meaning that the same effect can be achieved at a lower dose\nthan at $u=0$. The case when $u<0$ corresponds to antagonism (see\nAppendix\\,A.1). Note, that the interaction term is proportional to\n$\\sqrt{r_\\mrm{B}(1-r_\\mrm{B})}$. This functional form guarantees that\nit vanishes, whenever only one of the compounds is used,\ni.\\,e. $r_\\mrm{B}=0$ or $r_\\mrm{B}=1$.\n\n\n\n In order to make clear the questions we ask and the assumptions we\n make, we consider the dynamics of the frequency of the resistant strain\n $p(t) = I_\\mrm{r}(t)\/\\left[ I_\\mrm{r}(t) + I_\\mrm{s}(t) \\right]$.\n The rate of its change is obtained from\n Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3})\n\\begin{equation}\\label{eq:resfreq-dyn}\n\\frac{d p}{d t} = s(t) p (1 - p),\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:sel-coeff}\n  s = b \\left[ (1 - \\varepsilon_\\mrm{r}(C, r_\\mrm{B})) (1 - \\rho_\\mrm{r} ) -\n    (1 -\\varepsilon_\\mrm{s}(C, r_\\mrm{B})) \\right] H(t)\n\\end{equation}\nis the selection coefficient [a similar expression was found in\n\\cite{gugi99}]. Here $\\varepsilon_\\mrm{s}(C, r_\\mrm{B})$ and\n$\\varepsilon_\\mrm{r}(C, r_\\mrm{B})$ are given by \\eq{eq:veps-mix-comp-sens}\nand \\eq{eq:veps-mix-comp-res}.\nIf $s>0$, then the resistant strain is favored by selection and will\neventually dominate the pathogen population ($p \\to 1$ at $t \\to\n\\infty$). Alternatively, if $s<0$, then the sensitive strain is\nselected and will dominate the population ($p \\to 0$ at $t \\to\n\\infty$). \n\nThe focus of this paper is to investigate the parameter range\nover which $s<0$, i.\\,e. the sensitive strain is favored by\nselection. Mathematically this corresponds to finding the range of\nstability of the equilibrium (fixed) point of the system\nEqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}),\ncorresponding to $H>0$, $I_\\mrm{s}>0$, $I_\\mrm{r}=0$. Our focus is\nmainly on the direction of selection. To address this point we\ndo not need to assume that the host-pathogen equilibrium is\nreached. However, we explicitly assume that the host-pathogen\nequilibrium is reached during one season in \\sec{sec:trben}, where we\nevaluate the benefit of fungicide treatment. The implications of this\nassumption are discussed at the beginning of \\sec{sec:trben}.\nFurthermore, we assume that the fungicide dose is constant over\ntime. (See Appendix A.\\,4 for the justification of this assumption.)\n\n\nA careful examination of the \\eq{eq:sel-coeff} reveals that the sign\nof the selection coefficient $s$, and therefore the direction of\nselection, is determined by the expression in square brackets, which\ncan be either positive or negative depending on the values of $C$,\n$r_\\mrm{B}$, $\\rho_\\mrm{r}$ and the shapes of the functions\n$\\varepsilon_\\mrm{s,r}$. The sign of the selection coefficient is unaffected\nby $b$ and $H(t)$ since both of them are non-negative. Consequently\nmost of the results of this paper do not depend on a particular shape\nof $H(t)$ and hence are independent of a particular form of the\ngrowth term (except for those in \\sec{sec:trben} concerned with the\nbenefit of fungicide treatment). This means that the main conclusions\nof the paper remain valid for both perennial crops, where the amount of\nhealthy host tissue steadily increases over many years, and for annual\ncrops, where the healthy host tissue changes cyclically during\neach growing season.\n\nWe also neglect the spatial dependencies of the variables $H$,\n$I_\\mrm{s}$ and $I_\\mrm{r}$ and all other parameters. The latent phase\nof infection, which can be considerable for some pathogens, is also\nneglected.\nSince we neglect mutation, migration and spatial heterogeneity, the\nresistant and sensitive pathogen strains cannot co-exist in the long\nterm (see Appendix\\,A.1). Only one of them eventually survives: the\none with a higher basic reproductive number.\n\nThe basic reproductive number, $R_0$, is often used in epidemiology as\na measure of transmission fitness of infectious pathogens\n\\cite{anma86}. It is defined as the expected number of secondary\ninfections resulting from a single infected individual introduced into\na susceptible (healthy) population. At $R_0>1$ the infection can spread over the\npopulation, while at $R_0<1$ the epidemic dies out.\n\nThe equilibrium stability analysis of the model\nEqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}) (see\nAppendix\\,A.1) shows that the relationship between the basic\nreproductive number of the sensitive strain $R_\\mrm{0s} = b \\left(1 -\n  \\varepsilon_\\mrm{s}(C, r_\\mrm{B}) \\right) K\/\\mu $ and the basic\nreproductive number of the resistant strain $R_\\mrm{0r} = b \\left(1 -\n  \\varepsilon_\\mrm{r}(C, r_\\mrm{B}) \\right) (1 - \\rho_\\mrm{r}) K \/ \\mu$\ndetermines the long-term outcome of the epidemic. The sensitive strain\nwins the competition and dominates the pathogen population if\n$R_\\mrm{0s}>1$, such that it can survive in the absence of the\nresistant strain, and $R_\\mrm{0s} > R_\\mrm{0r}$ (this is equivalent to\n$s<0$), such that it has a selective advantage over the resistant\nstrain.  The latter inequality is equivalent to\n\\begin{align}\n\\varepsilon_\\mrm{s}(C,r_\\mrm{B})   < \\rho_\\mrm{r} + \\varepsilon_\\mrm{r} (C, r_\\mrm{B}) (1 - \\rho_\\mrm{r}) \\label{eq:brn-ineq-gen0}\n\\end{align}\nSimilarly, the resistant strain wins the competition and dominates the\npopulation if $R_\\mrm{0r}>1$ and $R_\\mrm{0r} > R_\\mrm{0s}$ (this is equivalent to\n$s>0$).\n\nWe determined the range of the fungicide doses $C$ and fitness costs\n$\\rho_\\mrm{r}$, according to the inequality (\\ref{eq:brn-ineq-gen0})\nanalytically when (i) the high-risk fungicide has a higher efficacy\nthan the low-risk fungicide ($k_\\mrm{kA}>k_\\mrm{kB}$), but\npharmacological interaction is absent ($u=0$); and (ii) the two\nfungicides have the same efficacy ($k_\\mrm{kA}=k_\\mrm{kB}=k_\\mrm{k}$),\nbut may interact pharmacologically ($u \\ne 0$).\nIn case (i) the criterium (\\ref{eq:brn-ineq-gen0}) assumes the form\n\\begin{equation}\\label{eq:brn-ineq-gen-kka-kkb}\n\\frac{ C}{ C + C_{50}}   < \\rho_\\mrm{r}\/k_\\mrm{km} + k_\\mrm{kB}\/k_\\mrm{km} \\frac{ C}{ C +\n    C_{50}\/\\gamma_\\mrm{r}} (1 - \\rho_\\mrm{r}),\n\\end{equation}\nwhere $k_\\mrm{km} = k_\\mrm{kA}(1-r_\\mrm{B}) + k_\\mrm{kB} r_\\mrm{B}$, while in case (ii) the criterium (\\ref{eq:brn-ineq-gen0}) reads\n\\begin{equation}\\label{eq:brn-ineq-gen}\n\\frac{ C}{ C + C_{50}\/\\gamma_\\mrm{s}}   < \\rho_\\mrm{r}\/k_\\mrm{k} + \\frac{ C}{ C +\n    C_{50}\/\\gamma_\\mrm{r}} (1 - \\rho_\\mrm{r}),\n\\end{equation}\n\n\nTo keep the presentation concise, below we present the results\ncorresponding to case (ii), i.\\,e. solve the inequality\n(\\ref{eq:brn-ineq-gen}). However, we verified that all the conclusions remain the same in case (i). \nIn a more general case, when $k_\\mrm{kA}>k_\\mrm{kB}$ and $u \\ne 0$ the\nparameter ranges satisfying the inequality (\\ref{eq:brn-ineq-gen0})\ncan only be determined numerically.\n\n\nWe assume here that both the cost of resistance and fungicidal\nactivity decrease the transmission rate $b$. However, we performed the\nsame analysis when the effect of the resistance cost and the fungicide\nenter the model in other ways and obtained qualitatively similar\nresults (see Appendix\\,A.5)\n\nThe simplicity of the model allows us to obtain all of the results\nanalytically. We determined explicit mathematical relationships\nbetween the quantities of interest, which enabled us to study the\neffects over the whole range of parameters.\n\n\\section{Results}\n\\label{sec:results}\n\nWe first investigate the parameter ranges over which resistant or\nsensitive strains dominate the pathogen population for the case of\nfungicides A and B applied individually and in a mixture\n(\\sec{sec:selres}). Then, we consider the optimal proportion of\nfungicides to include in a mixture in \\sec{sec:optrat} and the benefit\nof fungicide treatment in \\sec{sec:trben}. Finally, we take into\naccount possible pharmacological interactions between fungicides and\nconsider the effect of partial resistance (\\sec{sec:pharmint},\n\\ref{sec:partres}).\n\n\\subsection{Selection for resistance}\n\\label{sec:selres}\n\nThe ranges of fungicide dose and cost of resistance at which the\nsensitive (white) or resistant (grey) pathogen strain is favored by\nselection are shown in \\fig{fig:phase-diagr}. In all scenarios\ncompetitive exclusion is observed: one of the strains takes over the\nwhole pathogen population and the other one is eliminated. If a low-risk\nfungicide is applied alone, the sensitive strain has a selective\nadvantage across the whole parameter range in\n\\fig{fig:phase-diagr}A. When only a high-risk fungicide is applied\n(\\fig{fig:phase-diagr}B) the resistance dominates if the fitness cost\nis lower than the maximum effect of the fungicide\n$\\rho_\\mrm{r}<k_\\mrm{k}$ and at a fungicide dose higher than a\nthreshold value which increases with the fitness cost (solid curve in\n\\fig{fig:phase-diagr}B). If the fitness cost exceeds $k_\\mrm{k}$\n(dotted line in \\fig{fig:phase-diagr}B), then the sensitive strain\ndominates at any fungicide dose.\nThe \\fig{fig:phase-diagr}C shows the outcome when the two fungicides\nare mixed at equal concentrations.\nHere the fitness cost at which the sensitive strain dominates is\nreduced (vertical dotted line is shifted to the left).\n\nAs expected, without a fitness cost ($\\rho_\\mrm{r} = 0$) the resistant\nstrain becomes favored by selection and will eventually dominate the\npopulation whenever the high-risk fungicide is applied, alone or in\ncombination with the low-risk fungicide (\\fig{fig:phase-diagr}B,C).\n\n\n\n\\subsection{Optimal proportion of fungicides in a mixture}\n\\label{sec:optrat}\n\nIt is highly desirable to keep existing fungicides effective for as\nlong as possible. \nFrom this point of view, an optimal mixture contains the largest proportion of the\nhigh-risk fungicide, at which (i) the resistant pathogen strain is not\nselected and (ii) an adequate level of disease control is achieved.\nIn order to fulfill both of these objectives, the fitness cost of\nresistance needs to be larger than a threshold value $\\rho_\\mrm{r} >\n\\rho_\\mrm{rb}$ [see Eq.\\,(A.16)]. The threshold\n$\\rho_\\mrm{rb}$ is shown by the dotted vertical line in \\fig{fig:phase-diagr}C.\n\nThe threshold $\\rho_\\mrm{rb}$ depends on the proportion of fungicides\nin the mixture.\nAdding more of the low-risk fungicide, while keeping the same total\ndose $C$, reduces the threshold. This diminishes the range of the\nvalues for fitness cost over which the resistant strain dominates. On\nthe other hand, adding less of the low-risk fungicide, while again\nkeeping $C$ the same, increases the threshold, which increases the\nparameter range over which the resistant strain is favored.\n\nTherefore, at a given fitness cost $\\rho_\\mrm{r}$, one can adjust the\nfungicide ratio $r_\\mrm{B}$ such that\n$\\rho_\\mrm{r}>\\rho_\\mrm{rb}$. This is shown in\n\\fig{fig:opt-fung-rat-comb}: the curve shows the critical proportion\nof the low-risk fungicide $r_\\mrm{Bc}$, above which no selection for\nresistance occurs at any total fungicide dose $C$. One can\nsee from \\fig{fig:opt-fung-rat-comb} that if the resistance cost is\nabsent ($\\rho_\\mrm{r} = 0$), then the high-risk fungicide should not\nbe added at all if one wants to prevent selection for resistance.\nAt larger fitness costs, the value of $r_\\mrm{Bc}$ decreases, giving\nthe possibility to use a larger proportion of the high-risk fungicide\nwithout selecting for resistance.\n\nFinding an optimum proportion of fungicides requires knowledge of both\nthe fitness cost $\\rho_\\mrm{r}$ and the maximum effect of the\nfungicide $k_\\mrm{k}$ [Eq.\\,(A.17)]. However, if the cost of\nresistance and fungicides affect the infectious period of the pathogen\n$\\mu^{-1}$ (see Sec.\\,A.5) and not the transmission rate $b$ as\nwe assumed above, then a simpler expression for the critical\nproportion of fungicides in the mixture is obtained\n[Eq.\\,(A.18)], which depends only on the ratio between the\nfitness cost and the maximum fungicide effect $\\rho_\\mrm{r}\/k_k$. In\nthis case, if the fitness cost is at least 5 percent of the maximum\nfungicide effect, then we predict that up to about 20 percent of the\nhigh-risk fungicide can be used in a mixture without selecting for\nresistance. \nAn example of the cost of fungicide resistance manifesting\nas a reduction in infectious period was in metalaxyl-resistant\nisolates of \\emph{Phytophthora infestans} \\cite{kaco89}. In this\nexperiment, the infectious period of the resistant isolates was\nreduced, on average, by 25\\,\\% compared to the susceptible isolates\n\\cite{kaco89}.\n\nSo far we have shown how choosing an optimal proportion of fungicides\nin the mixture prevents selection for resistance. Now, we will\nconsider in more detail how to achieve an adequate level of disease control.\n\n\\subsection{Treatment benefit}\n\\label{sec:trben}\n\nThe yield of cereal crops is usually assumed\nto be proportional to the healthy green leaf area, which corresponds\nin our model to the amount of healthy hosts $H(t)$. Accordingly,\nwe quantify the benefit of the fungicide treatment, $B(t)$, as the ratio\nbetween the amount of healthy hosts $H(t)$ when both the disease\nand treatment are present and its value $H_\\mrm{nd}(t)$ in the absence\nof disease: $B(t) = H(t)\/H_\\mrm{nd}(t)$. Hence, $B(t)=1$ corresponds\nto a perfect treatment, which eradicates the disease completely and\nthe treatment benefit of zero corresponds to a situation where all\nhealthy hosts are infected by disease.\nIn order to obtain analytical expressions for the treatment benefit\n$B(t)$, we consider one growing season and assume that the host-pathogen\nequilibrium is reached during the season (see Appendix\\,A.3 for a\ndiscussion of these assumptions).\n\nThe treatment benefit at equilibrium is shown in\n\\fig{fig:trben-vs-conc-cost-col} as a function of the fitness cost and\nthe fungicide dose (see Appendix\\,A.3 for equations).\nWhen a low-risk fungicide is applied alone\n[\\fig{fig:trben-vs-conc-cost-col}A], the sensitive strain is favored\nby selection over the whole range of parameters. Therefore, the\ntreatment benefit increases monotonically with the fungicide dose and\nis not affected by the cost of resistance.\nIn contrast, when a high-risk fungicide is applied alone\n[\\fig{fig:trben-vs-conc-cost-col}B], a region at low fitness costs appears [to\nthe left from the solid curve in \\fig{fig:trben-vs-conc-cost-col}B],\nwhere the resistant strain is favored. Here, the treatment\nbenefit does not depend on the fungicide dose, but increases with\nthe cost of resistance. Hence, if the fitness cost is too low to stop\nselection for resistance, then the fungicide treatment will fail.\n\nIn the case of a mixture of a high-risk and a low-risk fungicide, the\nparameter range over which the resistant strain is favored becomes\nsmaller [\\fig{fig:trben-vs-conc-cost-col}C, to the left from the solid\n  curve].\nIn this range the treatment benefit increases with the cost of resistance,\nsince larger costs reduce the impact of disease per se. Also, the\ntreatment benefit increases with the total fungicide dose in this\nrange, because the low-risk fungicide works against the resistant\nstrain.\n\n\nAs we have shown above in \\sec{sec:optrat}, in the presence of a\nsubstantial fitness cost, one can avoid selection for resistance by\nadjusting the proportion of the two fungicides in the mixture. Then,\nthe total fungicide dose such that the treatment benefit reaches a\nhigh enough value and an adequate disease control is achieved.\n\nIn the end of \\sec{sec:optrat} we estimated that up to about 20 percent of the\nhigh-risk fungicide can be used in a mixture without selecting for\nresistance if the fitness cost is at least 5 percent of the maximum\nfungicide effect on the infectious period $\\mu^{-1}$.\nBut how much extra control does one obtain by adding the high-risk\nfungicide to the mixture?\nWe estimate that adding 20 percent of the high-risk fungicide to the mixture\nincreases the treatment benefit by about 12 percent at $R_\\mrm{0s}(C=0)=b\nK\/\\mu=4$ and by about 9 percent at $R_\\mrm{0s}(C=0)=2$ (see Sec.\\,A.3 and\nFig.\\,A.1 for more details). In the case when the high-risk fungicide\nhas a larger maximum effect, i.\\,e. $k_\\mrm{kA}>k_\\mrm{kB}$, the\nbenefit of adding it to the mixture will increase. However, the largest\nproportion of the high-risk that can be added without selecting for\nresistance will decrease.\n\n\n\n\\subsection{The effect of pharmacological interaction between\n  fungicides}\n\\label{sec:pharmint}\nSynergistic interactions between fungicides make their combined effect\ngreater than expected with additive interactions.  The sensitive\npathogen strain is suppressed more by a synergistic mixture, while the\nresistant strain is not affected by the interaction (in case of full\nresistance $\\alpha=0$). This increases the range of fitness costs over\nwhich resistance has a selective advantage [the dashed line in\n\\fig{fig:phase-diagr}C shifts to the right]. Consequently, the\ncritical proportion of the low-risk fungicide in the mixture\n$r_\\mrm{Bc}$, above which the resistant mutants are eliminated\nincreases [dotted curve in\n\\fig{fig:rbc-vs-rhoa-interact-partres}A]. In contrast, an antagonistic\nmixture suppresses the sensitive strain less effectively than either\nfungicide used alone. In this case the range of fitness costs over\nwhich resistance dominates becomes smaller and the ratio $r_\\mrm{Bc}$\ndecreases [dashed curve in\n\\fig{fig:rbc-vs-rhoa-interact-partres}A]. Hence, reduced resistance\nevolution is achieved, however, at the expense of reduced disease\ncontrol.\nThis result is in agreement with studies on drug interactions in the\ncontext of antibiotic resistance, where antagonistic drug combinations\nwere found to select against resistant bacterial strains \\cite{chcr+07}.\n\n\\subsection{The effect of partial fungicide resistance}\n\\label{sec:partres}\nConsider the situation when the resistant pathogen strain is not fully\nprotected from the high-risk fungicide, but exhibits a partial\nresistance ($0 < \\alpha < 1$). In this case, the fungicide mixture is\nmore effective in suppressing the resistant strain than in the case of\nfull resistance ($\\alpha=0$) considered above.  Therefore, one needs\nless of the low-risk fungicide in the mixture to reach the conditions\nwhere resistance is eliminated by selection: the critical proportion\nof the low-risk fungicide in the mixture decreases with the degree of\nsensitivity $\\alpha$ in\n\\fig{fig:rbc-vs-rhoa-interact-partres}B. Also, in\n\\fig{fig:rbc-vs-rhoa-interact-partres}A the dependency of the\ncritical ratio of the fungicide B in the mixture for partial\nresistance (light grey curve) lies below the one at perfect resistance\nand reaches zero at a much smaller value.\nThus, knowledge of the degree of resistance is crucial for\ndetermining an appropriate proportion of fungicides in the mixture.\n\n\\section{Discussion}\n\\label{sec:conclusions}\n\nThe three main outcomes of our study are: (i) if fungicide resistance\ncomes without a fitness cost, application of fungicides prone to\nresistance (high-risk fungicides) in a mixture with fungicides still\nfree from resistance (low-risk fungicides) will select for resistance;\n(ii) if sufficiently high costs are found, then an optimal proportion\nof the high-risk fungicide in a mixture with the low-risk fungicide\nexists that does not select for resistance; (iii) this mixture can\npotentially be used for preventing de novo emergence of fungicide\nresistance, in which case the relevant fitness cost is the\n``inherent'' cost of fungicide resistance before the compensatory\nevolution occurs (see below).\n\nIn the absence of fitness costs application of a mixture of high-risk\nand low-risk fungicides will select for resistance.\nConsequently, the resistant strain will eventually dominate the pathogen\npopulation and the sensitive strain will be eliminated. Because of\nthis, the high-risk fungicide will not affect the amount of disease\nand only the low-risk fungicide component of the mixture will be\nacting against disease.\nHence, the high-risk fungicide becomes nonfunctional in the mixture\nand using the low-risk fungicide alone would have the same effect at a\nlower financial and environmental cost.\n\nIn contrast, if sufficiently high costs are found, then high-risk\nfungicides can be used effectively for an extended period of\ntime. According to our model, an optimal proportion of the high-risk\nfungicide in a mixture with the low-risk fungicide can be determined\nthat contains as much as possible of the high-risk fungicide, but\nstill does not select for resistance while providing adequate disease\ncontrol (see Box\\,\\ref{nbox:getrb-alg}). If a mixture with the optimum\nproportion is applied, then the rise of the resistant strain is\nprevented for an unlimited time. Thus, the scheme in\nBox\\,\\ref{nbox:getrb-alg} provides a framework for using our knowledge\nabout the evolutionary dynamics of plant pathogens and their\ninteraction with fungicides in devising practical strategies for\nmanagement of fungicide resistance.\n\nIn order to apply the scheme in Box\\,\\ref{nbox:getrb-alg}, one needs\nto know dose-response parameters of the fungicides $k_\\mrm{k}$ and\n$C_{50}$, the degree of fungicide sensitivity $\\alpha$ (or the\nresistance factor), the degree of pharmacological interaction $u$ and\nthe fitness cost of resistane mutations.\nFungicide dose-response curves are routinely determined empirically\n(for example, \\cite{locl05,pahi+98}) and can be used to estimate the\nmodel parameters $k_\\mrm{k}$ and $C_{50}$ \\cite{hopa+11a}. The\nfungicide sensitivity is known to be lost completely in some cases\n(for example, most cases of QoI resistance), i.\\,e. $\\alpha=0$, while\nin other cases with partial resistance the degree of sensitivity (or\nthe resistance factor) was\nmeasured (for example, \\cite{leal+07}). Pharmacological interaction between several different\nfungicides was also characterized empirically (see \\cite{gi96} and the\nreferences therein). Also, the fitness costs of resistance were\ncharacterized empirically in many cases (see below).\nIn the past these measurements were performed independently, but our\nstudy provides motivation to bring them together, since all these\nparameters need to be characterized for the same\nplant-pathogen-fungicide combination.\n\nThese measurements will allow one to predict the optimal proportion of the\ntwo fungicides in the mixture theoretically. This prediction needs to\nbe tested using field experimentation, in which the amount of disease\nand the frequency of resistance would be measured as functions of time\nat different proportions of the high- and low-risk fungicides in the\nmixture.\n From these measurements the optimal proportion of the fungicides can\n be obtained empirically. It is this empirically determined optimal\n proportion of fungicides that can be used for practical guidance on\n management of fungicide resistance. Moreover, from the comparison of\n the optimal proportion obtained theoretically and empirically, one\n can evaluate the performance of the model and identify the aspects of\n the model that need improvement.\n\n\n\n So far we considered the scenario where both the sensitive and the\n resistant pathogen strains increase from low numbers,\n i.\\,e. resistant mutants pre-exist in the pathogen population. In\n this scenario the strain with higher fitness (or basic reproductive\n number) eventually outcompetes the other strain. This competition may\n occur over a time scale of several growing seasons so that there is\n enough time for compensatory mutations that diminish fitness costs of\n resistance to emerge. This needs to be taken into account when\n determining the optimal proportion of fungicides in the\n mixture. However, an alternative scenario is possible when resistant\n mutants emerge de novo through mutation or migration and, in order to\n survive, they need to invade the host population already infected by\n the sensitive strain. The threshold of invasion in this case depends on the\n ``inherent'' fitness cost of resistance mutations, i.\\,e. their cost\n before the compensatory evolution occurs. In this case, one should\n measure the ``inherent'' cost of resistance mutations when performing\n step 3 in Box\\,\\ref{nbox:getrb-alg}.\n\n\nAs discussed above, it is crucial to know fitness costs of\nresistance mutations in order to determine whether the fungicide\nmixture will select for resistance. We extensively searched the\nliterature on fitness costs in different fungal pathogens of plants.\nA few studies inferred substantial fitness costs from field monitoring\n(see for example, \\cite{suya+10} and references in \\cite{pemi95}). But\nthese findings could result from other factors, including immigration\nof sensitive isolates, selection for other traits linked to resistance\nmutations or genetic drift \\cite{pemi95}. Though relatively few\ncarefully controlled experiments have been conducted, the majority\nindicate that fitness costs associated with fungicide resistance are\neither low (for example, \\cite{bifi+12,kixi11}) or absent (for\nexample, \\cite{codu+10,pemi94}). But in some cases fitness costs were\nfound to be substantial (for example,\n\\cite{iaba+08,kath+01,hoec95,we88}) both in laboratory measurements\nand in field experiments.\nAlthough measurements of fitness costs of resistant mutants performed\nunder laboratory conditions can be informative (as, for example in\n\\cite{bifi+12}), they do not necessarily reflect the costs connected\nwith resistant mutants selected in the field. This is because field\nmutants are likely to possess compensatory mutations improving\npathogen fitness \\cite{pemi95}. Moreover, a laboratory setting rarely\nreflects the balance of environmental and host conditions found\nthroughout the pathogen life-cycle, since the field environment is\nmuch more complex.\n\n\nHowever, the most relevant measure of pathogen fitness in the context\nof our study is the growth rate of the pathogen population at the very\nbeginning of an epidemic (often denoted as $r$). It is directly\nrelated to the basic reproductive number $R_0$.  To the best of our\nknowledge, the fitness costs of fungicide resistant strains were not\nmeasured with respect to $r$. In the studies cited above different\ncomponents of fitness were measured that may or may not be related to\n$r$.\nTherefore, we identified a major gap in our knowledge of fitness costs. We\nhope this study will stimulate further experimental investigations to\nbetter characterize fitness costs and expect that substantial costs\nwill be found in some cases.\n\n\n\n\n\n\n\n\n\n\n \n\nInteractions of plants with fungal pathogens, fungicide action and,\npossibly, pharmacological interaction can depend on environmental\nconditions. This means that the outcomes of measurements necessary for\napplying the scheme in Box\\,\\ref{nbox:getrb-alg}, may vary between\nseasons and geographical locations. Moreover, the outcomes may also be\ndifferent in different host cultivars. Therefore, the optimal\nproportion of fungicides in the mixture may vary between seasons,\ngeographical locations and host cultivars. Thus, to provide general\npractical guidance on management of fungicide resistance, one needs to\nmeasure the optimal proportion of fungicides over many seasons, in\ndifferent geographical locations and host cultivars. This difficulty\nis not a unique property of our study, but rather it is a general\nproblem in the field of mathematical modeling of fungicide resistance\nand plant diseases. For example, it is also relevant for choosing\nappropriate fungicide dose rate \\cite{locl05}.\n\n\nWhile it was previously discussed \\cite{sh06} that alternation of\nhigh-risk and low-risk fungicides might be a useful tactic for disease\ncontrol in the presence of a fitness cost, we have shown that a\nmixture of these fungicides in an appropriate proportion can provide\nadequate disease control without selecting for resistance. Mixtures\noffer an advantage compared to alternation because there is no need to\ndelay the application of the high-risk fungicide and the resistant\nstrain does not rise to high frequencies, which lowers the risk of its\nfurther spread (see Appendix\\,A.6)\n\n\\begin{nicebox}\nAccording to our model, one can avoid selection for resistance while providing adequate\ndisease control by choosing the fungicide ratio $r_\\mrm{B}$ and the\ntotal dose $C$ in the following way:\n\\begin{enumerate}\n\\item measure the pharmacological properties of both fungicides under field conditions to\n  determine $k_\\mrm{k}$, and $C_{50}$;\n\\item determine the degree of fungicide sensitivity $\\alpha$ under field conditions;\n\\item determine the degree of pharmacological interaction $u$ between\n  fungicides A and B under field conditions;\n\\item measure the fitness cost of resistance $\\rho_\\mrm{r}$ under field conditions;\n\\item choose the proportion of the fungicide B above the threshold:\n$r_\\mrm{B}>r_\\mrm{Bc}$, such that the resistance is not favored by selection at any\ntotal fungicide dose $C$;\n\\item choose the total fungicide dose, which should be large enough to\n  achieve an adequate  level of disease control (see \\fig{fig:trben-vs-conc-cost-col}C).\n\\end{enumerate}\n\\caption{How to determine an optimal mixture of fungicides theoretically.}\\label{nbox:getrb-alg}\n\\end{nicebox}\n\n\n\nThe problem of combining chemical biocides in order to delay or\nprevent the development of resistance also appears in other contexts,\nincluding resistance of agricultural weeds to herbicides \\cite{bere09} and\ninsect pests to insecticides \\cite{ro06}.\nThe fitness cost of resistance is also recognized as a crucial\nparameter for managing antibiotic resistance \\cite{anhu10}. \n\n\nDevelopment of mathematical models of fungicide resistance dynamics\nhas been influenced by theoretical insights from animal and human\nepidemiology \\cite{bogi08,hagu+04}. Similarly, we expect that lessons\nlearned from modeling fungicide combinations may well apply to the\nproblem of biocide resistance in the other contexts. In particular,\none can investigate the idea of adjusting the proportion of the\ncomponents in a mixture of drugs in order to prevent selection for\nresistance in a more general context of biocide resistance.\n\n\n\n\\section*{Acknowledgements}\n\nAM and SB gratefully acknowledge support by the European Research\nCouncil advanced grant PBDR 268540. The authors are grateful to\nMichael Milgroom and Michael Shaw for helpful comments concerning\nfitness costs of fungicide resistance and to two anonymous reviewers\nfor improving the manuscript.\n\n\n\\newpage\n\n\n\n\n\n\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=\\textwidth]{fig1_mdiagr_fl.eps}}\n \\caption{\\doublespacing Scheme of the model in\n     Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}).}\n\\label{fig:model-scheme}\n\\end{figure}%\n\\clearpage\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=\\textwidth]{fig2_pdiagr_pp_w.eps}} \\caption{\n    \\doublespacing \nOutcomes of the competition between the sensitive and resistant\n    pathogen strains depending on the fitness cost of resistance\n    $\\rho_\\mrm{r}$ and the fungicide dose $C$ when treated\n    with a single fungicide B at [$C_\\mrm{B} = C$, panel A], a\n    single fungicide A [$C_\\mrm{A}=C$, panel B] and the combination\n    of fungicides A and B [$C_\\mrm{A} = C_\\mrm{B} = C\/2$, panel\n    C]. The range of the total fungicide dose $C$ and the\n    fitness cost of resistance $\\rho_\\mrm{r}$, in which the resistant\n    strain is favored is shown in grey. The range where selection\n    favors the sensitive strain is shown in white. The dashed and the\n    solid curves in panel B are plotted according to Eq.\\,(A.22) and\n    Eq.\\,(A.23) in Appendix\\,A.2, respectively. The dashed and the\n    solid curves in panel C are plotted according to Eq.\\,(A.13) and\n    Eq.\\,(A.14) in Appendix\\,A.1, respectively, at $\\gamma_\\mrm{s}=1$,\n    $\\gamma_\\mrm{r}=1\/2$. Fungicides are assumed to have zero\n    interaction ($u = 0$) and the resistant strain is assumed to be\n    fully protected from fungicide A ($\\alpha=0$), the fungicide\n    dose-response parameters are $k_\\mrm{k}=0.5$, $C_{50}=1$.}\n\\label{fig:phase-diagr}\n\\end{figure}%\n\\clearpage\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=0.6\\textwidth]{fig3_rbc_vs_rhor_fl.eps}} \\caption{\n    \\doublespacing The critical proportion $r_\\mrm{Bc}$ of fungicide B\n    (low-risk fungicide) in the mixture, above which there is no\n    selection for the resistant strain at any total fungicide\n    dose $C$, plotted (black curve) according to Eq.\\,(A.17)\n    as a function of the resistance cost $\\rho_\\mrm{r}$, assuming no\n    pharmacological interaction ($u=0$), full resistance ($\\alpha=0$)\n    and the maximum fungicide effect $k_\\mrm{k}=0.5$.}\n\\label{fig:opt-fung-rat-comb}\n\\end{figure}%\n\\clearpage\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=0.6\\textwidth]{fig4_tb_gs_fl.eps}} \\caption{\n\\doublespacing \n    Treatment benefit as a function of fungicide dose $C$ and\n    fitness cost of resistance $\\rho_\\mrm{r}$, plotted according to\n    Eq.\\,(A.26) in panel A, according to Eq.\\,(A.27) in panel B\n    and according to Eq.\\,(A.28) in panel C. Treatment with\n    fungicide B is shown in panel A. Treatment with fungicide A is\n    shown in panel B. Treatment with a mixture of A and B at equal\n    concentrations ($r_\\mrm{B}=1\/2$) is shown in panel C. Solid and\n    dashed curves in panels B and C are the same as in\n    \\fig{fig:phase-diagr}.  Fungicides are assumed to have zero\n    interaction ($u = 0$) and the resistant strain is assumed to be\n    fully protected from fungicide A ($\\alpha=0$). The fungicide\n    dose-response parameters are $k_\\mrm{k}=0.5$, $C_{50}=1$, the basic\n    reproductive number of the sensitive strain without fungicide\n    treatment $R_\\mrm{0s}(C=0) =b K \/ \\mu=2$.}\n\\label{fig:trben-vs-conc-cost-col}\n\\end{figure}%\n\\clearpage\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=0.8\\textwidth]{fig5_pi_fl.eps}} \\caption{\n\\doublespacing \n    The effect of pharmacological interaction and partial resistance\n    on $r_\\mrm{Bc}$, the critical ratio of the fungicide\n    B. $r_\\mrm{Bc}$ is plotted as a function of the fitness cost of\n    resistance $\\rho_\\mrm{r}$ (left panel), according to\n    Eq.\\,(A.13) for the case of no interaction between the\n    fungicides $u=0$ (solid, the same as the curve in\n    \\fig{fig:opt-fung-rat-comb}), synergy $u=0.9$ (dotted), and\n    antagonism $u=-0.9$ (dashed) for the case of perfect resistance\n    $\\alpha=1$. The case of partial resistance at no interaction\n    ($\\alpha = 0.5$, $u=0$) is shown as a light grey curve.\n    $r_\\mrm{Bc}$ is shown as a function of the degree of fungicide\n    sensitivity $\\alpha$ at $\\rho_\\mrm{r}=0.05$ (solid) and\n    $\\rho_\\mrm{r}=0.1$ (dash-dotted) also according to\n    Eq.\\,(A.13) in the right panel.}\n\\label{fig:rbc-vs-rhoa-interact-partres}\n\\end{figure}%\n\\newpage\n\n\n\\section{Supplemental materials}\n\n\\subsection{Model equations}\n\\label{sec:ap-modeleq}\n\nIn order to explore the effect of the assumptions we made in\n\\sec{sec:model-assump}, we consider a more general system of\nequations, which describes the change in time of the same quantities\nas in Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}):\nthe amount of healthy host tissue $H$, the amount of host tissue\ninfected with the sensitive pathogen strain $I_\\mrm{s}$ and the amount\nof host tissue infected with the resistant pathogen strain $I_\\mrm{r}$\n\n\\begin{align}\n\\frac{d H}{d t} &= r_H (K - H - I_\\mrm{s} - I_\\mrm{r} ) -  b \\left[\n  \\left( 1 - \\varepsilon_\\mrm{s} \\right)\n(1-\\rho_\\mrm{s}) I_\\mrm{s} + \\left( 1 - \\varepsilon_\\mrm{r} \\right)  (1-\\rho_\\mrm{r}) I_\\mrm{r} \\right] H, \\label{eq:1host2fung-gen-app-1} \\\\\n\\frac{d I_\\mrm{s}}{d t} & = b \\left( 1 - \\varepsilon_\\mrm{s} \\right) (1-\\rho_\\mrm{s}) H I_\\mrm{s} - \\mu  I_\\mrm{s}, \\label{eq:1host2fung-gen-app-2}\\\\\n\\frac{d I_\\mrm{r}}{d t} & = b \\left( 1 - \\varepsilon_\\mrm{r} \\right) (1-\\rho_\\mrm{r}) H I_\\mrm{r} - \\mu  I_\\mrm{r}, \\label{eq:1host2fung-gen-app-3}\n\\end{align}\nwhere, the function $\\varepsilon_\\mrm{s} = \\varepsilon_\\mrm{s}(C_\\mrm{A}, C_\\mrm{B})$ describes the effect of the\napplication of the mixture fungicides A and B with doses\n$C_\\mrm{A}$ and $C_\\mrm{B}$ on the transmission rate of the sensitive pathogen strain\nand the function $\\varepsilon_\\mrm{r} = \\varepsilon_\\mrm{r}(C_\\mrm{A}, C_\\mrm{B})$ describes the effect of this\nmixture on the transmission rate of the resistant strain:\n\\begin{align}\n  \\varepsilon_\\mrm{s}(C_\\mrm{A}, C_\\mrm{B}) &= k_\\mrm{k} \\frac{\\alpha_\\mrm{s,A} C_\\mrm{A} + \\alpha_\\mrm{s,B} C_\\mrm{B}}{ \\alpha_\\mrm{s,A} C_\\mrm{A} + \\alpha_\\mrm{s,B} C_\\mrm{B} + C_{50}\/ \\left[1\n    + u \\sqrt{ \\alpha_\\mrm{s,A} \\alpha_\\mrm{s,B}  C_\\mrm{A} C_\\mrm{B}} \/\n    (\\alpha_\\mrm{s,A} C_\\mrm{A} + \\alpha_\\mrm{s,B} C_\\mrm{B})\\right]}, \\label{eq:veps-s-mix-gen-app}\\\\\n  \\varepsilon_\\mrm{r}(C_\\mrm{A}, C_\\mrm{B}) &= k_\\mrm{k} \\frac{\\alpha_\\mrm{r,A} C_\\mrm{A} + \\alpha_\\mrm{r,B} C_\\mrm{B}}{ \\alpha_\\mrm{r,A} C_\\mrm{A} + \\alpha_\\mrm{r,B} C_\\mrm{B} + C_{50}\/ \\left[1\n    + u \\sqrt{ \\alpha_\\mrm{r,A} \\alpha_\\mrm{r,B}  C_\\mrm{A} C_\\mrm{B}} \/ (\\alpha_\\mrm{r,A} C_\\mrm{A} + \\alpha_\\mrm{r,B} C_\\mrm{B})\\right]}. \\label{eq:veps-r-mix-gen-app}\n\\end{align}\nThe parameters $\\alpha_\\mrm{s,A}$, $\\alpha_\\mrm{s,B}$,\n$\\alpha_\\mrm{r,A}$ and $\\alpha_\\mrm{r,B}$ characterize the degree of\nsensitivity of each of the two pathogen strains (index \"s\" for\nthe sensitive strain, index ``r'' for the resistant strain) to each of the two\nfungicides A and B. Their values are between zero and one. In this\ngeneral case both pathogen strains are partially resistant to both\nfungicides. The maximum effect of the fungicide is characterized by\nthe parameter $k_k$ and assumed to be the same for both fungicides.\n\nThe parameter $C_{50}$ in Eqs.\\,(\\ref{eq:veps-s-mix-gen-app}),\n(\\ref{eq:veps-r-mix-gen-app}) is modified due to pharmacological\ninteraction between fungicides characterized by the degree of\ninteraction $u$. At $u=0$ fungicides do not interact, $u>0$ represents\nsynergy and $u<0$ corresponds to antagonism. [We restrict our\nconsideration to $u>-1$, since otherwise the term in the square\nbrackets of Eqs.\\,(\\ref{eq:veps-s-mix-gen-app}),\n(\\ref{eq:veps-r-mix-gen-app}) may become negative, which makes no\nsense.]  This way to define pharmacological interaction between\ncompounds is called ``Loewe additivity'' or ``concentration addition''\nin the literature \\cite{grbr95,be89}. In this approach an interaction\nof a compound with itself is set by definition to be additive (zero\ninteraction). For example, when the fungicide A is mixed with itself,\nthe resulting sham mixture is neither synergistic, nor antagonistic\nbut has zero interaction. An equivalent graphic procedure is known as\nthe Wadley method in the fungicide literature [second method described\nin \\cite{lebe+86}].\n\nAn alternative way to define pharmacological interaction assumes that\nthe two compounds have independent modes of action and is called\n``Bliss independence'' \\cite{bl39} or Abbott's formula\n\\cite{ab25}. However, in this definition a compound can have a\npharmacological interaction with itself, i.\\,e. be synergistic or\nantagonistic.\nThe study \\cite{sh89a} discusses the definition of ``independent action''\nof the two fungicides, according to which the two fungicides are\nindependent when one fungicide does not affect the evolution of\nresistance in the other. According to \\cite{sh89a,sh93}, this is only\npossible when each of the fungicides affects different stages of the\npathogen life cycle.\n\nThere are several ways to introduce a deviation from the zero\ninteraction regime, in which usually an interaction term is added to\nthe isobologram equation \\cite{grbr95}. We have chosen a specific form\nof the interaction term, which is proportional to the square root of\nthe product of the concentrations of the two compounds [Eq.\\,(28) in\n\\cite{grbr95}]. This form allows for a simple analytical expression of\nthe effect of the combination in Eqs.\\,(\\ref{eq:veps-s-mix-gen-app}), (\\ref{eq:veps-r-mix-gen-app}).\n\nWe assume that the cost of resistance decreases the transmission\nrate $b$ by a fixed amount $\\rho_\\mrm{s}$ for the sensitive strain and\nby $\\rho_\\mrm{r}$ for the resistant strain in\nEqs.\\,(\\ref{eq:1host2fung-gen-app-1})-(\\ref{eq:1host2fung-gen-app-3}).\nWe restrict our consideration here to the case when the ``sensitive''\npathogen strain is fully sensitive to both fungicides\n($\\alpha_\\mrm{s,A} = \\alpha_\\mrm{s,B} = 1$) and the ``resistant''\nstrain can have varying degrees of resistance to the fungicide A\n($\\alpha_\\mrm{r,A} \\equiv \\alpha$, $0 \\le \\alpha \\le 1$), but is fully\nsensitive to the fungicide B ($\\alpha_\\mrm{r,B} = 1$). Therefore, the\ncost of resistance for the sensitive strain is zero $\\rho_\\mrm{s}\n=0$. Then, the fungicide dose-response functions become simpler.\n\nIn order to determine the range of fitness costs $\\rho_\\mrm{r}$ and\nfungicide doses $C$, over which the sensitive or resistant strain is\nfavored by selection, we perform the linear stability analysis of the\nfixed points of the system\nEqs.\\,(\\ref{eq:1host2fung-gen-app-1})-(\\ref{eq:1host2fung-gen-app-3}). Fixed\npoints are the values of $H$, $I_\\mrm{s}$ and $I_\\mrm{r}$ at which the\nexpressions on the right-hand side of\nEqs.\\,(\\ref{eq:1host2fung-gen-app-1})-(\\ref{eq:1host2fung-gen-app-3})\nequal zero.\nThe system\nEqs.\\,(\\ref{eq:1host2fung-gen-app-1})-(\\ref{eq:1host2fung-gen-app-3})\nhas three fixed points: (i) $H^* = K$, $I_\\mrm{s}=I_\\mrm{r}=0$; (ii)\n$H^*= \\mu\/b_\\mrm{s}$, $I_\\mrm{s}=r_H (b_\\mrm{s} K - \\mu)\/\\left[ b_\\mrm{s}\n  (\\mu+r_H) \\right]$, $I_\\mrm{r} = 0$;\n(iii) $H^*= \\mu\/b_\\mrm{r}$, $I_\\mrm{s}= 0 $, $I_\\mrm{r} = r_H (b_\\mrm{r} K - \\mu)\/\\left[ b_\\mrm{r}\n  (\\mu+r_H) \\right]$. Here $b_\\mrm{s} = b \\left[ 1 -\n  \\varepsilon_\\mrm{s}(C_\\mrm{A}, C_\\mrm{B})\\right]$, $b_\\mrm{r} = \\left[ 1  - \\varepsilon_\\mrm{r}(C_\\mrm{A},\n    C_\\mrm{B}) \\right] (1 - \\rho_\\mrm{r})$.\nTo determine whether a fixed point is stable, we first linearize the system\nEqs.\\,(\\ref{eq:1host2fung-gen-app-1})-(\\ref{eq:1host2fung-gen-app-3})\nin its vicinity, then determine the Jacobian and its eigenvalues. A\nfixed point is stable if all the eigenvalues have negative real parts.\n\nThe results of this analysis can be conveniently expressed using the\nbasic reproductive number of the sensitive strain\n\\begin{equation}\\label{eq:ap-r0s}\nR_\\mrm{0s} = \\frac{b \\left[ 1 - \\varepsilon_\\mrm{s}(C_\\mrm{A}, C_\\mrm{B})\\right] K}{\\mu}\n\\end{equation}\nand the basic reproductive number of the resistant strain\n\\begin{equation}\\label{eq:ap-r0r}\nR_\\mrm{0r} = \\frac{b \\left[ 1  - \\varepsilon_\\mrm{r}(C_\\mrm{A},\n    C_\\mrm{B}) \\right] (1 - \\rho_\\mrm{r}) K}{ \\mu}.\n\\end{equation}\nThe sensitive strain is favored by selection [meaning that the fixed\npoint (ii) is stable and both fixed points (i) and (iii) are unstable]\nwhen both inequalities $R_\\mrm{0s}>1$, $R_\\mrm{0s} > R_\\mrm{0r}$ are\nfulfilled.\n\nWe consider then the inequality $R_\\mrm{0s} > R_\\mrm{0r}$, which\nis equivalent to\n\\begin{equation}\\label{eq:brn-ineq-gen-app}\n\\frac{ C}{ C + C_{50}\/\\gamma_\\mrm{s}}   < \\rho_\\mrm{r}\/k_\\mrm{k} + \\frac{ C}{ C +\n    C_{50}\/\\gamma_\\mrm{r}} (1 - \\rho_\\mrm{r}),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:app-gammas}\n\\gamma_\\mrm{s} = 1 + u \\sqrt{r_\\mrm{B}(1-r_\\mrm{B})},\n\\end{equation}\n\\begin{equation}\\label{eq:app-gammar}\n  \\gamma_\\mrm{r} = \\alpha (1 - r_\\mrm{B}) + r_\\mrm{B} + u\n  \\sqrt{\\alpha r_\\mrm{B}(1-r_\\mrm{B})}\n\\end{equation}\nand $r_\\mrm{B} = C_\\mrm{B}\/C$ is the proportion of the funcigide B in\nthe mixture, $C = C_\\mrm{A} + C_\\mrm{B}$. \n\nThe inequality (\\ref{eq:brn-ineq-gen-app}) holds at\n\\begin{align}\\label{eq:range-sens-gen1}\n\\rho_\\mrm{r} < \\rho_\\mrm{rb}, \\:\\: \\mrm{for} \\left( C<C_\\mrm{b1} \\: \\mrm{or} \\: C>C_\\mrm{b2}\n\\right)\n\\end{align}\nor at \n\\begin{align}\\label{eq:range-sens-gen2}\n\\rho_\\mrm{r} > \\rho_\\mrm{rb}, \\:\\: \\mrm{for\\:any\\:value\\:of}\\: C.\n\\end{align}\n Here,\n\\begin{equation}\\label{eq:rhorb-gen-app}\n\\rho_\\mrm{rb} = \\frac{k_\\mrm{k} (\\gamma_\\mrm{s} - \\gamma_\\mrm{r}) \\left( \\gamma_\\mrm{s} +\n    \\gamma_\\mrm{r} (1 -k_\\mrm{k} )  - 2 \\sqrt{\\gamma_\\mrm{r} \\gamma_\\mrm{s} (1 - k_\\mrm{k})}\n  \\right) } { \\left( \\gamma_\\mrm{s} - \\gamma_\\mrm{r} (1 - k_\\mrm{k}) \\right)^2},\n\\end{equation}\n\\begin{equation}\\label{eq:c12-gen-app}\nC_{b1,2} = \\frac{C_{50}}{2 \\gamma_\\mrm{s} \\gamma_\\mrm{r} \\rho_\\mrm{r} (1 - k_\\mrm{k})} \\left[\n  \\gamma_\\mrm{s} ( k_\\mrm{k} - \\rho_\\mrm{r}) - \\gamma_\\mrm{r}\n  (k_\\mrm{k} + \\rho_\\mrm{r} (1 -  k_\\mrm{k})) \\mp \\sqrt{D} \\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:d-gen-app}\nD = \\gamma_\\mrm{s}^2 \\left( k_\\mrm{k} - \\rho_\\mrm{r} \\right)^2 + \\gamma_\\mrm{r}^2\n(k_\\mrm{k} -  \\rho_\\mrm{r} - k_\\mrm{k} \\rho_\\mrm{r} )^2 - 2 \\gamma_\\mrm{s} \\gamma_\\mrm{r} \\left( k_\\mrm{k}^2 (1 - \\rho_\\mrm{r}) +\n  \\rho_\\mrm{r}^2(1 - k_\\mrm{k})  \\right).\n\\end{equation}\nAccording to the inequality (\\ref{eq:range-sens-gen2}), if the fitness\ncost of resistance is larger than a threshold value given by\n\\eq{eq:rhorb-gen-app}, the sensitive strain has a selective advantage\nand the resistant strain is eliminated from the population at any\nfungicide dose $C \\geq 0$.\n\n\nFor the case of no interaction between fungicides ($u=0$) and perfect\nresistance ($\\alpha=0$) we obtain from \\eq{eq:app-gammas} and\n\\eq{eq:app-gammar} $\\gamma_\\mrm{s} = 1$, $\\gamma_\\mrm{r} =\nr_\\mrm{B}$. Then, the \\eq{eq:rhorb-gen-app} is simplified:\n\\begin{equation}\\label{eq:rhorb-noint-app}\n\\rho_\\mrm{rb} = k_\\mrm{k} \\frac{(1 - r_\\mrm{B}) \\left[ 1 + r_\\mrm{B} (1-k_\\mrm{k})  - 2\n  \\sqrt{r_\\mrm{B} ( 1 -  k_\\mrm{k})} \\right]} { (1 -  r_\\mrm{B}  (1 -\n  k_\\mrm{k}) )^2 }.\n\\end{equation}\nWe then solve the inequality $\\rho_\\mrm{r} > \\rho_\\mrm{rb}$ with respect to\n$r_\\mrm{B}$ and find that it is fulfilled at $r_\\mrm{B}>r_\\mrm{Bc}$, where\n\\begin{equation}\\label{eq:rbc-fullres}\nr_\\mrm{Bc} = \\frac{ k_\\mrm{k}^2 (1 - \\rho_\\mrm{r}) + \\rho_\\mrm{r}^2 ( 1 -\n  k_\\mrm{k}) -  2 k_\\mrm{k} \\rho_\\mrm{r} \\sqrt{ (1 - k_\\mrm{k}) (1 - \\rho_\\mrm{r}) })} { (k_\\mrm{k}\n  + \\rho_\\mrm{r} - k_\\mrm{k} \\rho_\\mrm{r})^2}\n\\end{equation}\nIt represents the critical proportion of the fungicide B in the\nmixture above which the resistant strain is not favored by selection\n(\\fig{fig:opt-fung-rat-comb}). If the cost of resistance affects the\ndeath rate of the pathogen $\\mu$ (see \\sec{sec:gener}) and not the\ntransmission rate $b$ as considered above, then a simpler expression\nfor the critical proportion of the fungicide B is obtained\n\\begin{equation}\\label{eq:rbc-fullres1}\nr_\\mrm{Bc} = \\left( \\frac{1 - \\rho_\\mrm{r}\/k_k}{1 + \\rho_\\mrm{r}\/k_k} \\right)^2.\n\\end{equation}\nHere, $r_\\mrm{Bc}$ depends only on the ratio $\\rho_\\mrm{r}\/k_k$ of the\ncost of resistance to the maximum fungicide effect $k_k$, which allows to\nmake a more general prediction about the value of $r_\\mrm{Bc}$.\n\n\\subsection{Selection for resistance at no interaction between fungicides}\n\\label{sec:app-single-fung}\n\nWhen only the high risk fungicide (fungicide A) is applied with the\ndose $C_\\mrm{A}$, we set $r_\\mrm{B}=0$ in \\eq{eq:app-gammas}\nand \\eq{eq:app-gammar} to obtain $\\gamma_\\mrm{s} = 1$, $\\gamma_\\mrm{r}\n= \\alpha$. Then, the following expressions are obtained for the\nthreshold value of the resistance cost from \\eq{eq:rhorb-gen-app}\n\\begin{equation}\\label{eq:rhorb-fungica}\n\\rho_\\mrm{rb} =  k_\\mrm{k} \\frac{(1 - \\alpha) \\left[ 1 + \\alpha (1-k_\\mrm{k})  - 2\n  \\sqrt{\\alpha ( 1 -  k_\\mrm{k})} \\right]} { (1 -  \\alpha  (1 - k_\\mrm{k}) )^2 },\n\\end{equation}\nand the fungicide dose from \\eq{eq:c12-gen-app}\n\\begin{equation}\\label{eq:c12-fungica}\nC_{b1,2} = \\frac{C_{50}}{2 \\alpha \\rho_\\mrm{r} (1 - k_\\mrm{k})} \\left[\n  k_\\mrm{k} - \\rho_\\mrm{r} - \\alpha\n  (k_\\mrm{k} + \\rho_\\mrm{r} (1 -  k_\\mrm{k})) \\mp \\sqrt{D} \\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:d-fungica}\n  D = \\left( k_\\mrm{k} - \\rho_\\mrm{r} \\right)^2 + \\alpha^2\n(k_\\mrm{k} -  \\rho_\\mrm{r} - k_\\mrm{k} \\rho_\\mrm{r} )^2 - 2 \\alpha \\left( k_\\mrm{k}^2 (1 - \\rho_\\mrm{r}) +\n  \\rho_\\mrm{r}^2(1 - k_\\mrm{k})  \\right).\n\\end{equation}\nIn the simpler case of full resistance we take the limit $\\alpha \\to\n0$. Then, by taking this limit in \\eq{eq:rhorb-fungica},\n\\eq{eq:c12-fungica} and \\eq{eq:d-fungica} we obtain for the threshold\nvalues of the fitness cost and the fungicide dose\n\\begin{equation}\\label{eq:rhorb-fungica-perfres}\n\\rho_\\mrm{rb} = k_\\mrm{k},\n\\end{equation}\n\\begin{equation}\\label{eq:cb-fungica-perfres}\nC_\\mrm{b} = C_{50} \\frac{\\rho_\\mrm{r}}{k_\\mrm{k}- \\rho_\\mrm{r}}.\n\\end{equation}\nIn this case the sensitive strain dominates at $C<C_\\mrm{b}$ if $\\rho_\\mrm{r} <\n\\rho_\\mrm{rb}$ or at any positive values of $C$ if $\\rho_\\mrm{r}>\\rho_\\mrm{rb}$\n[white area in \\fig{fig:phase-diagr}B].\n\nWhen only the low risk fungicide (fungicide B) is applied, we set\n$r_\\mrm{B} = 1$ and, hence $\\gamma_\\mrm{s}=\\gamma_\\mrm{r}=1$ in the inequality (\\ref{eq:brn-ineq-gen-app}) and obtain\n\\begin{equation}\\label{eq:brn-ineq-fungica}\n\\frac{ C}{ C + C_{50}}   < \\rho_\\mrm{r}\/k_\\mrm{k} + \\frac{ C}{ C +\n    C_{50}} (1 - \\rho_\\mrm{r}),\n\\end{equation}\nThis inequality holds and the sensitive strain dominates for all positive values of $\\rho_\\mrm{r}$ and\n$C$ at which $R_\\mrm{0s}>1$.\n\nConsider the case when the two fungicides A and B are applied\ntogether at an arbitrary mixing ratio $r_\\mrm{B}$, assuming no\npharmacological interaction ($u=0$) and perfect resistance of the\nresistant strain to the fungicide A ($\\alpha=0$).\nIn this case, $\\gamma_\\mrm{s} = 1$ and $\\gamma_\\mrm{r} =\nr_\\mrm{B}$. Substituting these values in \\eq{eq:rhorb-gen-app},\n\\eq{eq:c12-gen-app} and \\eq{eq:d-gen-app} gives the same expressions\nas in \\eq{eq:rhorb-fungica}, \\eq{eq:c12-fungica} and\n\\eq{eq:d-fungica}, but with $\\alpha$ substituted by $r_\\mrm{B}$.\n \n\n\n\\subsection{Expressions for the treatment benefit}\n\\label{sec:app-trben}\n\nThe treatment benefit is defined as the ratio between the amount of\nhealthy hosts $H(t)$ when both the disease and treatment are\npresent and the amount of healthy hosts at no disease $B(t) =\nH(t)\/H_\\mrm{nd}(t)$ (see \\sec{sec:trben}). \n\nIn order to obtain analytical expressions for $B(t)$, we consider one growing season and assume the host-pathogen\nequilibrium is reached during the season.\n This corresponds to the time-dependent solution of\n Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3})\n reaching its stable fixed point (or steady state).  Fixed points of\n Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}) can be\n found by equating the right-hand sides of all equations to zero and\n solving the resulting algebraic equations with respect to $H(t)$,\n $I_\\mrm{s}(t)$ and $I_\\mrm{r}(t)$. Biologically this occurs when the\n first positive term in \\eq{eq:1host2fung-gen-1} corresponding to\n growth of healthy hosts, is compensated by the second, negative term\n that corresponds to the decrease in healthy hosts due to\n infection. In other words, equilibrium occurs when the rate of\n emergence of new healthy tissue as a result of plant growth is\n exactly offset by the rate of its decrease due to infection. The\n right-hand side of \\eq{eq:1host2fung-gen-2} goes to zero, when the\n rate of increase in $I_\\mrm{s}$ due to new infections is compensated\n by the loss of the infectious tissue due to the completion of the\n infectious period (similar reasoning applies\n for \\eq{eq:1host2fung-gen-3}).\n\nThen, the treatment benefit is given by\n\\begin{equation}\\label{trben-gen}\nB(t \\to \\infty) = B^* = \\frac{H^*}{K},\n\\end{equation}\nwhere $H^*$ is the equilibrium amount of healthy hosts and $K$ is the\nhost carrying capacity, and assume full resistance ($\\alpha=0$).\n\nWhen only the fungicide B is applied at a dose $C$\n[\\fig{fig:trben-vs-conc-cost-col}A], the basic reproductive number of\nthe sensitive pathogen strain always exceeds the one for the resistant\nstrain $R_\\mrm{0s}>R_\\mrm{0r}$. Therefore, the resistant mutants are\neliminated in the long run and the amount of the healthy host tissue is equal\nto $H^* = \\mu\/ (b \\left[ 1 - \\varepsilon(C) \\right] )$, where $\\varepsilon(C)$ is given\nby \\eq{eq:eps-1fungic}. Then, according to \\eq{trben-gen}, the\ntreatment benefit is\n\\begin{align}\\label{eq:trben-fungicb}\nB^*(C)=  \\frac{ \\mu } {b  \\left[ 1 - \\varepsilon(C) \\right] K}.\n\\end{align}\nIt grows with the fungicide dose and saturates, since the\nfunction $\\varepsilon(C)$ saturates.\n\nApplication of the fungicide A alone at a dose $C$ may favor\neither resistant or sensitive pathogen strain depending on the fitness\ncost of resistance $\\rho_\\mrm{r}$ and the fungicide dose $C$\n[see \\fig{fig:phase-diagr}B]. The treatment benefit in this case is\n\\begin{align}\\label{eq:trben-fungica}\nB^*(C, \\rho_\\mrm{r})= \n\\begin{cases} \n\\frac{ \\mu } { b \\left[ 1-\\varepsilon(C) \\right] K }, & \\mrm{for} \\:\n(\\rho_\\mrm{r} <k_k\\: \\mrm{and} \\:  C<C_\\mrm{b} ) \\\\ & \\mrm{or}\n\\: (\\rho_\\mrm{r} > k_k \\: \\mrm{and} \\: \\forall C),\\\\\n\\frac{\\mu} {  b ( 1-\\rho_\\mrm{r}) K }, &  \\mrm{for} \\:\n\\rho_\\mrm{r} < k_k \\: \\mrm{and} \\: C>C_\\mrm{b},\n\\end{cases}\n\\end{align}\nwhere the $C_\\mrm{b}$ is given by\n\\eq{eq:cb-fungica-perfres}.\n\nNow, consider application of both fungicides in a mixture at equal\nconcentrations ($r_\\mrm{B} = 1\/2$), assuming no interaction between\nfungicides ($u=0$). In this case, again either resistant or sensitive\npathogen strain will dominate the population depending on the fitness\ncost $\\rho_\\mrm{r}$ and the total fungicide dose $C$ [see\n\\fig{fig:phase-diagr}C]. The treatment benefit now has the following\nexpression\n\\begin{align}\\label{eq:trben-equalmix}\nB^*(C, \\rho_\\mrm{r})= \n\\begin{cases} \n   \\frac{\\mu} { b \\left[1 - \\varepsilon(C) \\right] K}, & \\mrm{for} \\: (\\rho_\\mrm{r}\n  <\\rho_\\mrm{rb} \\: \\mrm{and} \\: (C<C_\\mrm{b1} \\: \\text{or}\\:\n  C>C_\\mrm{b2})) \\\\ & \\mrm{or}\n  \\: (\\rho_\\mrm{r} > \\rho_\\mrm{rb} \\: \\mrm{and} \\: \\forall C),\\\\\n  \\frac{\\mu } { b \\left[1 - \\varepsilon(C\/2) \\right]  (1 - \\rho_r) K }, & \\mrm{for} \\:\n  \\rho_\\mrm{r} <\\rho_\\mrm{rb} \\: \\mrm{and} \\:\n  C_\\mrm{b1}<C<C_\\mrm{b2},\n\\end{cases}\n\\end{align}\nwhere $\\rho_\\mrm{rb}$, $C_\\mrm{b1}$ and $C_\\mrm{b2}$ and are given by\n\\eq{eq:rhorb-gen-app} and \\eq{eq:c12-gen-app} at $\\gamma_\\mrm{s} =1$,\n$\\gamma_\\mrm{r} = 1\/2$. \nThe treatment benefit\n$B^*(C, \\rho_\\mrm{r})$ is shown as a function of the fungicide\ndose $C$ and the fitness cost of resistance $\\rho_\\mrm{r}$ in\n\\fig{fig:trben-vs-conc-cost-col} for the three cases discussed above,\naccording to Eqs.\\,(\\ref{eq:trben-fungicb})-(\\ref{eq:trben-equalmix}).\n\n\nWhen a mixture of a high-risk and a low-risk fungicide is used with\nthe total dose $C$ and the proportion of the low-risk fungicide\n$r_\\mrm{B}$, a relevant question arises: ``How much additional control\ndoes the addition of the high-risk fungicide provide?''.\nIn order to quantify the degree of additional control due the\nhigh-risk fungicide component of the mixture, we first set the\nsufficient level of control that needs to be achieved in terms of the\ntreatment benefit $B^*=B_\\mrm{suf}$ (for example, we can set\n$B_\\mrm{suf}=0.9$). Then, we determine the dose $C_\\mrm{suf}$ of the\nfungicide mixture at which this level of control is achieved. We\nassume that the proportion of the two fungicides in the mixture was\nchosen such that $r_\\mrm{B}>r_\\mrm{Bc}$, i.\\,e. the sensitive pathogen\nstrain is favored by selection. Hence, the the treatment benefit is\ngiven by $B^*=\\mu\/(b K)$ [the upper expression in\n\\eq{eq:trben-equalmix}]. We set $B^*=B_\\mrm{suf}$ and substitute the\ndependence of the pathogen infectious period or the transmission rate\non the fungicide dose according to $\\mu \\to \\mu ( 1 + \\varepsilon(C))$ or\n$b \\to b ( 1 - \\varepsilon(C))$. Then, we obtain\n\\begin{align}\\label{eq:csuf}\n  C_\\mrm{suf} =\n\\begin{cases} \n   C_{50} \\frac {B_\\mrm{suf} R_0 - 1} {1 + k_\\mrm{k} - B_\\mrm{suf} R_0},\n   & \\mrm{if} \\: \\mrm{fungicides} \\: \\mrm{affect} \\: \\mu\\\\ \n   C_{50} \\frac {B_\\mrm{suf} R_0 - 1} {1 - B_\\mrm{suf} R_0 (1 - k_\\mrm{k}) }, &  \\mrm{if} \\: \\mrm{fungicides} \\: \\mrm{affect} \\: b.\n\\end{cases}\n\\end{align}\nThe ratio\n\\begin{align}\\label{eq:trben-rat}\n\\frac {B^*(C=C_\\mrm{suf})} {B^*(C=r_\\mrm{B} C_\\mrm{suf})} = \n\\begin{cases} \n   \\frac {1 + \\varepsilon(C_\\mrm{suf})} {1 + \\varepsilon(r_\\mrm{B} C_\\mrm{suf})},\n   & \\mrm{if} \\: \\mrm{fungicides} \\: \\mrm{affect} \\: \\mu\\\\ \n   \\frac {1 - \\varepsilon(r_\\mrm{B} C_\\mrm{suf}) } {1 - \\varepsilon(C_\\mrm{suf}) }, &  \\mrm{if} \\: \\mrm{fungicides} \\: \\mrm{affect} \\: b\n\\end{cases}\n\\end{align}\ncharacterizes the extra benefit due to addition of the high-risk\nfungicide, since $B^*(C=C_\\mrm{suf})$ is the treatment benefit when\nboth high-risk and low-risk fungicides are present and\n$B^*(C=r_\\mrm{B} C_\\mrm{suf})$ is the treatment benefit when the\nhigh-risk fungicide is absent (here, $r_\\mrm{B}$ is the proportion of\nthe low-risk fungicide in the mixture). The ratio $B^*(C=C_\\mrm{suf})\n\/ B^*(C=r_\\mrm{B} C_\\mrm{suf})$ is shown in \\fig{fig:benrat-vs-ra} as\na function of the proportion of the high-risk fungicide in the mixture\n$r_\\mrm{A} = 1 - r_\\mrm{B}$. One sees from \\fig{fig:benrat-vs-ra} that\nthe more high-risk fungicide is used in the mixture, the larger is the\nextra benefit from its application (provided that\n$r_\\mrm{B}>r_\\mrm{Bc}$, when the sensitive pathogen strain is favored\nby selection). However, the largest $r_\\mrm{A}$ which still does not favor the\nresistant strain is determined by the value of the fitness cost of\nfungicide resistance (see \\sec{sec:optrat}).\n\nInterestingly, the ratio $B^*(C=C_\\mrm{suf}) \/ B^*(C=r_\\mrm{B}\nC_\\mrm{suf})$ does not depend on where the fungicide acts, on the\ninfectious period $\\mu^{-1}$ or the transmission rate $b$, as long as\nthe maximum fungicide effects in these to cases $k_{\\mrm{k}b}$ and\n$k_{\\mrm{k}\\mu}$ are related by $k_{\\mrm{k}\\mu} = k_{\\mrm{k}b}\/(1 -\nk_{\\mrm{k}b})$ such that the basic reproductive number is reduced by\nthe same amount when the maximum effect is achieved.\n\n\n\\begin{figure}[!ht]\n  \\centerline{\\includegraphics[width=0.6\\textwidth]{fig_a1_benrat_vs_ra.eps}}\n  \\caption\n    Extra benefit of adding the high-risk fungicide to the mixture\n    plotted according to \\eq{eq:trben-rat} and \\eq{eq:csuf} versus the\n    proportion of the high-risk fungicide $r_\\mrm{A}$, provided the\n    sensitive pathogen strain is favored by selection. The basic\n    reproductive number of the sensitive strain in the absence of\n    fungicides $R_0 = b K \/ \\mu$ had the value $R_0=2$ (dashed curve)\n    and $R_0 =4$ (solid curve). Other parameters: $C_{50}=1$,\n   $B_\\mrm{suf}=0.9$, maximum fungicide effect on $b$ is\n   $k_{\\mrm{k}b}=0.9$ and the equivalent maximum fungicide effect on\n   $\\mu$ is $k_{\\mrm{k}\\mu}=k_{\\mrm{k}b}\/(1 - k_{\\mrm{k}b}) = 9$ (see\n   text for explanation).}\n\\label{fig:benrat-vs-ra}\n\\end{figure}%\n\n\n\\subsection{Dynamics of the frequency of the resistant pathogen strain}\n\\label{sec:app-delsel}\n\nIf the fungicide resistance is not associated with a fitness cost,\nthen the resistant strain is favored by selection and eventually\ndominates the population whenever the high risk fungicide is applied\nalone or in a mixture with the low risk fungicide [\\fig{fig:phase-diagr}B,C].\nHowever, for a given value of the total fungicide dose $C$,\nthe selection for resistance slows down when applying the fungicide\nmixture as compared to the treatment with the high risk fungicide\nalone [as seen from time-dependent numerical solutions of the model\nEqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3})] in\nagreement with the findings of \\cite{hopa+11a}.\n\nIn order to understand this result we consider the dynamics of the\nfrequency of the resistant pathogen strain $p(t) =\nI_\\mrm{r}\/(I_\\mrm{r} + I_\\mrm{s})$. The rate of its change is obtained\nfrom Eqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}) \\cite{mile+89}\n\\begin{equation}\\label{eq:resfreq-dyn}\n\\frac{d p}{d t} = s(t) p (1 - p),\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:sel-coeff-app}\n  s = b \\left[ (1 - \\varepsilon_\\mrm{r}(C, r_\\mrm{B})) (1 - \\rho_\\mrm{r} ) -\n    (1 -\\varepsilon_\\mrm{s}(C, r_\\mrm{B})) \\right] H(t)\n\\end{equation}\nis the selection coefficient [a similar expression was found in\n\\cite{gugi99}]. Here $\\varepsilon_\\mrm{s}(C, r_\\mrm{B}) = k_\\mrm{k} C \/ ( C\n+ C_{50}\/ \\gamma_\\mrm{s})$, $\\varepsilon_\\mrm{r}(C, r_\\mrm{B}) = k_\\mrm{k} C\n\/ ( C + C_{50}\/ \\gamma_\\mrm{r})$ and $r_\\mrm{B}$ is the proportion of\nthe fungicide B in the mixture. Here, $C=C_\\mrm{A} + C_\\mrm{B}$, where\nthe dose $C_\\mrm{A}$ of the fungicide A and the dose $C_\\mrm{B}$ of\nthe fungicide B may depend on time due to fungicide decay:\n\\begin{equation}\\label{eq:cab-t}\nC_\\mrm{A} = C_\\mrm{A0} \\exp \\left( -\\nu_\\mrm{A} t \\right), \\: C_\\mrm{B} = C_\\mrm{B0} \\exp \\left( -\\nu_\\mrm{B} t \\right)\n\\end{equation}\nwhere $C_\\mrm{A0}$, $C_\\mrm{B0}$ are the fungicide doses at the time\nof application, $\\nu_\\mrm{A}$ and $\\nu_\\mrm{B}$ are the fungicide\ndecay rates.\n\nThe expression (\\ref{eq:sel-coeff-app}) for the selection coefficient was\nobtained under the assumption that the fungicide decreases the\ntransmission rate $b$. In the case when the fungicide decreases the\ninfectious period $\\mu^{-1}$, the selection coefficient does not depend on the\namount of healthy hosts $H(t)$.\n\nVariables in \\eq{eq:resfreq-dyn} can be separated and a\nclosed-form solution is found\n\\begin{equation}\\label{eq:resfreq-dyn-sol}\n\\int \\frac{d p}{p (1 - p)} = \\int_0^{t_\\mrm{m}} s( t ) dt.\n\\end{equation}\n\nOne can see from \\eq{eq:resfreq-dyn-sol} that the overall selection\nover the time $t_\\mrm{m}$ is determined by the integral of the\nselection coefficient $s(t)$ over time $\\int_0^{t_\\mrm{m}} s(t)\ndt$. We are interested in the overall selection that occurs during the\ntime $t_\\mrm{m}$ which is longer than the time scale of change in the\nfungicide dose. In this case, an equivalent, constant over time\nfungicide dose can be determined, which gives rise to the same value\nof the integral $\\int_0^T s(t) dt$. This effective fungicide dose\nwould take into account the time-dependent effect of the amount of the\nhost tissue on the strength of selection.\n\nAssuming a zero fitness cost ($\\rho_\\mrm{r}=0$), no pharmacological\ninteraction ($u=0$) and full resistance ($\\alpha=0$), the selection\ncoefficient can be written as \n\\begin{equation}\\label{eq:sel-coeff1}\ns(t) = b \\left[ \\varepsilon(C) - \\varepsilon(r_\\mrm{B} C) \\right] H(t).\n\\end{equation}\nAssuming that $H(t)$ is a slowly varying function compared to the time\nscale of selection, the solution of \\eq{eq:resfreq-dyn} reads:\n\\begin{equation}\\label{eq:resfreq-logist}\np(t) = \\frac{p_0 \\exp [s(t) t]}{ 1 + p_0 \\left( \\exp [s(t) t] - 1 \\right)},\n\\end{equation}\nwhere $p_0 = p(t=0)$. At $s>0$, the function $p(t)$ grows monotonically\nand tends to one at large times. The rate, at which it grows is\ndetermined by the magnitude of the selection coefficient $s$.\n\nOne can see from \\eq{eq:sel-coeff1} that when the high risk fungicide is\napplied alone ($r_\\mrm{B} = 0$), the selection coefficient is larger\nthan when it is mixed with a low risk fungicide ($0<r_\\mrm{B}<1$) at the same\ntotal fungicide dose $C$. Hence, $s(r_\\mrm{B}=0,C) >\ns(r_\\mrm{B}>0,C)$. This is because the function $\\varepsilon(r_\\mrm{B}\nC)$ has positive values for any\n$r_\\mrm{B}>0$. Thus, the selection for the resistant strain (against the\nsensitive strain) is delayed when a mixture of high risk and low risk\nfungicides is applied compared to treatment with the high risk\nfungicide alone. A careful consideration of \\eq{eq:resfreq-dyn-sol}\nreveals that this conclusion holds also when $H(t)$ does not vary\nslowly over the time scale of selection. Lower fungicide dose will\ndecrease the selection coefficient under the integral on the right-hand side of\n\\eq{eq:resfreq-dyn-sol}. Hence, in order to achieve a given large value\nof the frequency of resistance $p$, one would need to integrate over a\nlonger time $t_\\mrm{m}$.\n\n\\subsection{Generalization of the model: effect of the fungicide and fitness cost of resistance\n  on the pathogen}\n\\label{sec:gener}\n\n\nSo far we assumed that both the resistance cost and fungicides affect\nthe transmission rate $b$. We performed the same analysis for the\nthree remaining cases possible in the model: When (i) both resistance\ncost and the fungicide affect the pathogen death rate according to\n$\\mu \\to \\mu (1 + \\rho_\\mrm{r} + \\varepsilon_\\mrm{r}(C, r_\\mrm{B}))$ for the\nresistant strain and $\\mu \\to \\mu (1 + \\varepsilon_\\mrm{s}(C, r_\\mrm{B}))$\nfor the sensitive strain; (ii) the resistance cost affects the\ntransmission rate $b \\to b (1 - \\rho_\\mrm{r})$ of the resistant strain\nand the fungicides affect the pathogen death rate $\\mu \\to\\mu (1 +\n\\varepsilon_\\mrm{s,r}(C, r_\\mrm{B}))$ ; (iii) resistance cost affects the\ndeath rate of the resistant pathogen strain $\\mu \\to \\mu (1 +\n\\rho_\\mrm{r})$, while the fungicide affects the infection rate of both\nresistant and sensitive strains $b \\to b (1 - \\varepsilon_\\mrm{r,s}(C, r_\\mrm{B}))$.\nWe have found that although the mathematical expressions for the\nresults have a different form in these cases and there is a slight\nquantitative difference, all the conclusions remain the same and do\nnot depend on whether the fungicide and the resistance cost manifest\nin the infection rate $b$ or in the pathogen death rate $\\mu$.\n\nMoreover, we have done the same analysis using a fungicide\ndose-response function different from \\eq{eq:eps-1fungic}, namely\nusing the function $\\varepsilon(C) = \\varepsilon_\\mrm{m} ( 1 - \\exp\n\\left[ - \\beta C \\right])$. If the two fungicides have the same values\nof $\\varepsilon_\\mrm{m}$ and $\\beta$ and are applied at doses $C_\\mrm{A}$\nand $C_\\mrm{B}$, then according to Loewe's additivity, their combined\naction has the form $\\varepsilon(C_\\mrm{A}, C_\\mrm{B}) = \\varepsilon_\\mrm{m} ( 1 -\n\\exp \\left[ - \\beta (C_\\mrm{A} + C_\\mrm{B}) \\right])$. We found again\nthat all the conclusions remain the same in this case.\n\nThis generalization applies to determination of the direction of\nselection (the sign of the selection coefficient in\n\\eq{eq:resfreq-dyn}) and to the outcomes for the treatment benefit at\nequilibrium obtained in \\sec{sec:trben}. However, the time-dependent\nsolutions of\nEqs.\\,(\\ref{eq:1host2fung-gen-1})-(\\ref{eq:1host2fung-gen-3}) may\nbehave differently depending on how the fungicide and the fitness cost\naffect the pathogen life cycle and the form of the fungicide\ndose-response function. This is an interesting topic for further\ninvestigations, but lies beyond the scope of this study.\n\n\n\n\\subsection{Fungicide mixture versus alternation}\n\\label{sec:app-mix-vs-altern}\n\nIt was previously discussed \\cite{sh06} that in the presence of a\nfitness cost the alternation of fungicides can be effective, but we\nhave shown here that fungicide mixtures can also be effective in this\ncase. When using an alternation strategy, the period of selection\nduring which the resistant strain is favored in the presence of the\nhigh risk fungicide is followed by a period during which selection\nfavors the sensitive strain in the absence of this fungicide. The\nlatter period is typically much longer because the selection pressure\ninduced by the high risk fungicide is much larger than that induced by\nthe fitness cost of resistance. Hence, one needs to wait for quite a\nlong time before the resistant strain disappears and the high risk\nfungicide can be used again. Moreover, there are times during which\nthe frequency of the resistant strain becomes large (at the end of the\nperiod of the application of the high risk fungicide), which increases\nthe risk that resistance will spread to other regions. Both of these\ndisadvantages are avoided by using a mixture where the proportion of\nthe low risk fungicide is above a critical value determined here\n(\\fig{fig:opt-fung-rat-comb}). In this case there is no need to delay\nthe application of the high risk fungicide and the frequency of the\nresistant strain does not rise above the mutation- or\nmigration-selection equilibrium because the mixture does not induce\nselection for resistance.\n\n\\subsection{The risk of double resistance}\n\\label{sec:dr-risk}\n\nAlthough we do not consider the possibility of double resistance in\nour model, by applying an optimal proportion of fungicides in the\nmixture as suggested here, one would prevent selection for resistance\nto the high risk fungicide. \nConsequently, the risk of development of double resistance would be\nreduced.\nFor both sexually and asexually reproducing pathogens, there are three\npathways for generating double resistance: (i) A-resistant\nmutants are produced first and then a proportion of them acquires also\nB-resistance by spontaneous mutation (ii) B-resistant mutants are\ngenerated first and subsequently acquire A-resistance and (iii) double\nresistance is generated directly from the wild-type. In this case, by\npreventing selection for A-resistance, one removes only the pathway (i)\nto double resistance. \nIf a pathogen is able to reproduce sexually, then a much more\nlikely scenario for the double resistance to emerge is through\nrecombination. For the recombination to occur, both singly resistant\nstrains (A-resistant and B-resistant) would need to be present in the\npopulation at significant frequencies. Hence, preventing selection for\nA-resistance would diminish the probability of the emergence of double\nresistance by recombination.\nThus, our findings would also help to significantly reduce the risk of\ndevelopment of double resistance, especially in sexually reproducing\npathogens.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nSeparation of the wave equation in prolate spheroidal coordinates leads to the\nprolate spheroidal wave equation (PSWE)\n\\begin{equation}\n\\left(  {1-z^{2}}\\right)  \\frac{d^{2}y}{dz^{2}}-2z\\frac{dy}{dz}+\\left(\n{\\lambda-\\frac{\\mu^{2}}{1-z^{2}}+\\gamma^{2}\\left(  {1-z^{2}}\\right)  }\\right)\ny=0, \\label{eq1}%\n\\end{equation}\nwhere $\\lambda$ and $\\mu$ are separation constants, and $\\gamma$ is\nproportional to the frequency (see [30] and [40]).\n\nSolutions of \\eqref{eq1}, the prolate spheroidal wave functions (PSWFs), are\nviewed as depending on the parameters $\\mu$ and $\\gamma$ from the equation, as\nwell as an implicitly defined parameter $\\nu$ (which describes the behavior of\nsolutions at infinity). This latter parameter is the so-called characteristic\nexponent, and for details see [1, \\S 8.1.1].\n\nThe parameter $\\lambda$ is usually regarded as an eigenvalue admitting an\neigensolution that is bounded at both $z=\\pm1$, which is equivalent to $\\mu=m$\nand $\\nu=n$ being integers (see [1] and [19]). Most of the literature focuses\non PSWFs with these parameters being integers, since this is the most useful\ncase in practical applications. We shall assume this as well throughout this paper.\n\nWe consider the important case of $\\gamma\\rightarrow\\infty$ (which\ncorresponds, for example, to high-frequency scattering in acoustics). In this\ncase it is known [1, p. 186] that $\\lambda\\rightarrow-\\infty$, and we shall\nassume this here. With the exception of \\S 5, our results will be uniformly\nvalid for $m$ bounded, $n$ small or large, and specifically\n\\begin{equation}\n0\\leq m\\leq n\\leq2\\pi^{-1}\\gamma\\left(  {1-\\delta}\\right)  , \\label{eq2}%\n\\end{equation}\nwhere (here and throughout) $\\delta\\in\\left(  {0,1}\\right)  $ is arbitrarily chosen.\n\nAlthough we will consider the case $z$ complex, our primary concern will be\nfor $z$ real (denoted by $x)$, and in particular the so-called angular\n($-1<x<1$) and the radial ($1<x<\\infty$) cases.\n\n\n\nWe shall apply several existing general asymptotic theories, each of which\ncontains explicit error bounds. The large range of validity \\eqref{eq2}, along\nwith error bounds, signifies a considerable improvement upon existing results.\nWe will primarily be concerned with $z$ real, but we shall use complex-valued\nargument results as needed to obtain our final results.\n\nOne of the principal difficulties in the asymptotic and numerical study of\nPSWFs is the determination of the eigenvalues, particularly for large value(s)\nof the other parameters. We mention that an extensive the theory of PSWFs with\narbitrary complex parameters $\\mu$ and $\\nu$ was developed in [19] and [20].\n\nPSWFs were first studied by Niven [24] in heat conduction in spheroidal\nbodies, and were subsequently investigated by a number of authors (see [3],\n[12], [17], [33], [34], [36]). For certain values of the parameters PSWFs are\neigenfunctions of the finite Fourier transform, and hence these functions play\nan important role in signal analysis. These band-limited functions are often\nencountered in physics, engineering, statistics; see [4], [32], [38], In [22]\nthe PSWE is also shown to play a fundamental role in Laplace's tidal\nequations. PSWFs also have important applications in fluid dynamics [13],\ngeophysics and theoretical cosmological models [8], atomic and molecular\nphysics [15], [28], and biophysics [6].\n\nDespite being studied extensively over the decades there are significant gaps\nin the literature on the rigorous analysis of their asymptotic behavior, and\ntheir computation is non-trivial, particularly for large values of the\nparameters. The literature contains many asymptotic approximations and\nexpansions (see [2], [7], [14], [21], [23], [31], [35], [40]), but most are\nheuristic, all parameters fixed except $\\gamma$, and with little or no error\nanalysis. For computational techniques see [11], [16], [18], [29], [37], [39].\n\nWe mention that in [9] rigorous results (with explicit error bounds) for more\nthan one large parameter were derived for PSWFs, using a theory of a\ncoalescing turning point and double pole [5], but not for the parameter range\nunder consideration in this paper. Specifically, in comparison to the current\npaper in which $\\lambda<0$, in [9] the case $\\lambda>0$ was assumed, which\ndoes not have many of the applications described above.\n\nThe PSWE (\\ref{eq1}) has regular singularities at $z=\\pm1$, each with\nexponents $\\pm{\\frac{1}{2}}m$. When $\\gamma=0$ the PSWE degenerates into the\nassociated Legendre equation (regular singularities at $z=\\pm1$ and $z=\\infty\n$), which for $-1<x<1$ has solutions the Ferrers functions $P_{\\nu}^{\\mu\n}\\left(  x\\right)  $, and for complex $z$ the associated Legendre functions\n$P_{\\nu}^{\\mu}\\left(  z\\right)  $.\n\nThe significant difference is that if $\\gamma\\neq0$ the PSWE has an irregular\nsingularity at infinity. In fact, one can show (from an algebraic form of\nFloquet's theorem [1, \\S 8.1.1]) that there exists a solution $S_{\\nu}%\n^{\\mu\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  $ (in the notation of\n[19]), with the property\n\\begin{equation}\nS_{\\nu}^{\\mu\\left(  1\\right)  }\\left(  {ze^{p\\pi i},\\gamma}\\right)\n=e^{p\\nu\\pi i}S_{\\nu}^{\\mu\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  ,\n\\label{eq3}%\n\\end{equation}\nfor any integer $p$. The LHS of (\\ref{eq3}) denotes the branch of the function\nafter completing $p$ negative half-circuits about $z=\\infty$ (equivalently,\n$p$ positive half-circuits about $z=\\pm1$).\n\n\nThis solution can be expressed as an infinite series involving Bessel\nfunctions of the first kind. Specifically, for integral $\\mu$ and $\\nu$ we\nhave from [1, \\S 8.3]\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nS_{n}^{m\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  =\\left(  {\\dfrac{\\pi\n}{2\\gamma z}}\\right)  ^{1\/2}\\dfrac{\\left(  {z^{2}-1}\\right)  ^{-m\/2}z^{m}%\n}{A_{n}^{m}\\left(  {\\gamma^{2}}\\right)  }\\\\\n\\times\\sum\\limits_{k=-k^{+ }}^{\\infty}{a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\nJ_{n+2k+\\left(  {1\/2}\\right)  }\\left(  {\\gamma z}\\right)  },\n\\end{array}\n\\label{eq4}%\n\\end{equation}\nwhere\n\\begin{equation}\nk^{\\pm }=\\left\\lfloor {{\\tfrac{1}{2}}\\left(  {n\\pm m}\\right)  }\\right\\rfloor .\n\\label{eq5}%\n\\end{equation}\n\nWe parenthetically note that related solutions $S_{n}^{m\\left(  j\\right)\n}\\left(  {z,\\gamma}\\right)  $ ($j=3,4$) are defined below\n\nIn (\\ref{eq4}) the coefficients are defined by the three-term recurrence\nrelation\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nA_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)  a_{n,k-1}^{m}\\left(  {\\gamma^{2}%\n}\\right)  +\\left\\{  {\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  +B_{n,k}%\n^{m}\\left(  {\\gamma^{2}}\\right)  }\\right\\}  a_{n,k}^{m}\\left(  {\\gamma^{2}%\n}\\right) \\\\\n+C_{n,k}^{m}a_{n,k+1}^{m}\\left(  {\\gamma^{2}}\\right)  =0,\n\\end{array}\n\\label{eq6}%\n\\end{equation}\nwhere\n\\begin{equation}\nA_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)  =\\frac{\\left(  {n-m+2k-1}\\right)\n\\left(  {n-m+2k}\\right)  }{\\left(  {2n+4k-3}\\right)  \\left(  {2n+4k-1}\\right)\n}\\gamma^{2}, \\label{eq7}%\n\\end{equation}%\n\\begin{equation}\nB_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)  =\\frac{2\\left[  {\\left(\n{n+2k}\\right)  \\left(  {n+2k+1}\\right)  +m^{2}-1}\\right]  }{\\left(\n{2n+4k-1}\\right)  \\left(  {2n+4k+3}\\right)  }\\gamma^{2}-\\left(  {n+2k}\\right)\n\\left(  {n+2k+1}\\right)  , \\label{eq8}%\n\\end{equation}\nand\n\\begin{equation}\nC_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)  =\\frac{\\left(  {n+m+2k+1}\\right)\n\\left(  {n+m+2k+2}\\right)  }{\\left(  {2n+4k+3}\\right)  \\left(  {2n+4k+5}%\n\\right)  }\\gamma^{2}. \\label{eq9}%\n\\end{equation}\nThe normalising constant $A_{n}^{m}\\left(  {\\gamma^{2}}\\right)  $ is defined\nby\n\\begin{equation}\nA_{n}^{m}\\left(  {\\gamma^{2}}\\right)  =\\sum\\limits_{k=-k^{+}}^{\\infty\n}{\\left(  {-1}\\right)  ^{k}a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)  }.\n\\label{eq10}%\n\\end{equation}\nWe also remark that the coefficients $a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\n$ vanish for $k\\leq-1-k^{+}$, where $k^{+}$ is defined by \\eqref{eq5}.\n\nFrom \\eqref{eq4}, \\eqref{eq10} and the well-known behavior of the Bessel\nfunction at infinity [25, Chap. 12, \\S 1.2] we observe that $S_{\\nu}%\n^{\\mu\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  $ has the important\nproperty\n\\begin{equation}\nS_{\\nu}^{\\mu\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  =\\frac{\\cos\\left\\{\n{\\gamma z-{\\frac{1}{2}}\\pi\\left(  {\\nu+1}\\right)  }\\right\\}  }{\\gamma\nz}\\left\\{  {1+{O}\\left(  {\\frac{1}{z}}\\right)  }\\right\\}  \\quad\\left(\n{z\\rightarrow\\infty}\\right)  , \\label{eq11}%\n\\end{equation}\nfor $\\left\\vert {\\arg\\left(  z\\right)  }\\right\\vert <\\pi$. With our assumption\nthat $\\nu=n=$ integer, all solutions of (\\eqref{eq1}) are single-valued in the\n$z$-plane having a cut along the interval $\\left[  {-1,1}\\right]  $.\n\nOther fundamental solutions at infinity are given by $S_{n}^{m\\left(\nj\\right)  }\\left(  {z,\\gamma}\\right)  $ ($j=3,4$). These are defined by\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nS_{n}^{m\\left(  j\\right)  }\\left(  {z,\\gamma}\\right)  =\\left(  {\\frac{\\displaystyle\\pi\n}{\\displaystyle 2\\gamma z}}\\right)  ^{1\/2}\\frac{\\left( \\displaystyle {z^{2}-1}\\right)  ^{-m\/2} \\displaystyle z^{m}%\n}{ \\displaystyle A_{n}^{m}\\left(  {\\gamma^{2}}\\right)  }\\\\\n\\times\\sum\\limits_{k=-k^{+}}^{\\infty}{a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\nH_{n+2k+\\left(  {1\/2}\\right)  }^{\\left(  {j-2}\\right)  }\\left(  {\\gamma\nz}\\right)  },\n\\end{array}\n\\label{eq12}%\n\\end{equation}\nwhere $H_{\\nu}^{\\left(  {1,2}\\right)  }\\left(  z\\right)  $ are the Hankel\nfunctions of the first and second kinds, respectively.\n\nThe solutions $S_{n}^{m\\left(  j\\right)  }\\left(  {z,\\gamma}\\right)  $ have\nthe fundamental properties\n\\begin{equation}\nS_{n}^{m\\left(  3\\right)  }\\left(  {z,\\gamma}\\right)  =i^{-n-1}\\frac\n{e^{i\\gamma z}}{\\gamma z}\\left\\{  {1+{O}\\left(  {\\frac{1}{z}}\\right)\n}\\right\\}  \\quad\\left(  {z\\rightarrow\\infty}\\right)  , \\label{eq13}%\n\\end{equation}\nand\n\\begin{equation}\nS_{n}^{m\\left(  4\\right)  }\\left(  {z,\\gamma}\\right)  =i^{n+1}\\frac\n{e^{-i\\gamma z}}{\\gamma z}\\left\\{  {1+{O}\\left(  {\\frac{1}{z}}\\right)\n}\\right\\}  \\quad\\left(  {z\\rightarrow\\infty}\\right)  , \\label{eq14}%\n\\end{equation}\nfor $\\left\\vert {\\arg\\left(  z\\right)  }\\right\\vert <\\pi$. In particular\n$S_{n}^{m\\left(  3\\right)  }\\left(  {z,\\gamma}\\right)  $ is the unique\nsolution that is recessive in the upper half plane, and $S_{n}^{m\\left(\n4\\right)  }\\left(  {z,\\gamma}\\right)  $ is the unique solution that is\nrecessive in the lower half plane.\n\nAn important connection formula, which comes directly from the corresponding\none relating the $J$ Bessel function to Hankel functions, is given by\n\\begin{equation}\nS_{n}^{m\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  ={\\tfrac{1}{2}}\\left\\{\n{S_{n}^{m\\left(  3\\right)  }\\left(  {z,\\gamma}\\right)  +S_{n}^{m\\left(\n4\\right)  }\\left(  {z,\\gamma}\\right)  }\\right\\}  . \\label{eq15}%\n\\end{equation}\nFor $n\\geq m\\geq0$ and $-1<x<1$ there is a solution $\\operatorname{Ps}_{n}%\n^{m}\\left(  {x,\\gamma^{2}}\\right)  $ defined in terms of Ferrers functions by\n(see [1, \\S 8.2])\n\\begin{equation}\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\sum\\limits_{k=-k^{-}%\n}^{\\infty}{\\left(  {-1}\\right)  ^{k}a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\n}\\operatorname{P}{_{n+2k}^{m}\\left(  x\\right)  }. \\label{eq16}%\n\\end{equation}\nThis is the unique solution having the property of being recessive at $x=1$,\nand in particular has the property\n\\begin{equation}\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =K_{n}^{m}\\left(\n{\\gamma^{2}}\\right)  \\left(  {1-x}\\right)  ^{m\/2}\\left\\{  {1+{O}%\n\\left(  {1-x}\\right)  }\\right\\}  \\quad\\left(  {x\\rightarrow1^{-}}\\right)  ,\n\\label{eq17}%\n\\end{equation}\nwhere $K_{n}^{m}\\left(  {\\gamma^{2}}\\right)  $ is a constant given by\n\\begin{equation}\nK_{n}^{m}\\left(  {\\gamma^{2}}\\right)  =\\frac{\\left(  {-1}\\right)  ^{m}%\n}{2^{m\/2}m!}\\sum\\limits_{k=-k^{-}}^{\\infty}{\\left(  {-1}\\right)  ^{k}%\n\\frac{\\left(  {n+2k+m}\\right)  !}{\\left(  {n+2k-m}\\right)  !}a_{n,k}%\n^{m}\\left(  {\\gamma^{2}}\\right)  }. \\label{eq18}%\n\\end{equation}\nFrom \\eqref{eq16} and [27, eq. 14.7.17 ] we remark that $\\operatorname{Ps}%\n_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $ also has the fundamental property of\nalso being bounded at $x=-1$; this is consequence of the characteristic\nexponent $\\nu=n$ being an integer. In this case the function satisfies the\nnormalisation condition\n\\begin{equation}\n\\int_{-1}^{1}{\\left\\{  {\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}%\n}\\right)  }\\right\\}  ^{2}dx}=\\frac{2\\left(  {n+m}\\right)  !}{\\left(\n{2n+1}\\right)  \\left(  {n-m}\\right)  !}. \\label{eq19}%\n\\end{equation}\nAs mentioned above, the separation constant $\\lambda=\\lambda_{n}^{m}\\left(\n{\\gamma^{2}}\\right)  $ is regarded as an eigenvalue for the case $m$ and $n$\nintegers, and hence admits the eigensolution ${\\operatorname{Ps}_{n}%\n^{m}\\left(  {x,\\gamma^{2}}\\right)  }$ that is bounded at $x=\\pm1$.\n\n\nFor our purposes the PSWE (\\ref{eq1}) therefore takes the form\n\\begin{equation}\n\\left(  {1-z^{2}}\\right)  \\frac{d^{2}y}{dz^{2}}-2z\\frac{dy}{dz}+\\left\\{\n{\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  -\\frac{m^{2}}{1-z^{2}}+\\gamma\n^{2}\\left(  {1-z^{2}}\\right)  }\\right\\}  y=0. \\label{eq20}%\n\\end{equation}\nFor $x$ real and lying in $\\left(  {1,\\infty}\\right)  $ (the radial case) we\nhave the following solution of (\\ref{eq20})\n\\begin{equation}\nPs_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\sum\\limits_{k=-k^{-}}^{\\infty\n}{\\left(  {-1}\\right)  ^{k}a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\nP_{n+2k}^{m}\\left(  x\\right)  }, \\label{eq21}%\n\\end{equation}\nwhich also defines $Ps_{n}^{m}\\left(  {z,\\gamma^{2}}\\right)  $ for complex\n$z$; in this case $Ps_{n}^{m}\\left(  {z,\\gamma^{2}}\\right)  $ is entire if $m$\nis even, and if $m$ odd $\\left(  {1-z^{2}}\\right)  ^{1\/2}Ps_{n}^{m}\\left(\n{z,\\gamma^{2}}\\right)  $ is entire.\n\nNow since $m$ and $n$ are integers, it is a straightforward to show from\n(\\ref{eq21}) that\n\\begin{equation}\nPs_{n}^{m}\\left(  {ze^{\\pi i},\\gamma^{2}}\\right)  =\\left(  {-1}\\right)\n^{n}Ps_{n}^{m}\\left(  {z,\\gamma^{2}}\\right)  , \\label{eq22}%\n\\end{equation}\nwhich is (unique) property of the Floquet solution $S_{n}^{m\\left(  1\\right)\n}\\left(  {z,\\gamma}\\right)  $. Hence\n\\begin{equation}\nS_{n}^{m\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  =\\left(  {-1}\\right)\n^{n}\\left(  {n-m}\\right)  !V_{n}^{m}\\left(  \\gamma\\right)  Ps_{n}^{m}\\left(\n{z,\\gamma^{2}}\\right)  , \\label{eq23}%\n\\end{equation}\nfor some constant $V_{n}^{m}\\left(  \\gamma\\right)  $, and hence from the known\nbehavior of $S_{n}^{m\\left(  1\\right)  }\\left(  {z,\\gamma}\\right)  $ at\ninfinity\n\\begin{equation}\nPs_{n}^{m}\\left(  {z,\\gamma^{2}}\\right)  =V_{n}^{m}\\left(  \\gamma\\right)\n\\frac{\\sin\\left\\{  {\\gamma z-{\\frac{1}{2}}\\pi n}\\right\\}  }{\\gamma z}\\left\\{\n{1+{O}\\left(  {\\frac{1}{z}}\\right)  }\\right\\}  \\quad\\left(\n{z\\rightarrow\\infty}\\right)  . \\label{eq24}%\n\\end{equation}\nAn explicit expression, in terms of $a_{n,k}^{m}\\left(  {\\gamma^{2}}\\right)\n$, for the constant $V_{n}^{m}\\left(  \\gamma\\right)  $ can be obtained from\n\\eqref{eq11}, \\eqref{eq21}, \\eqref{eq23}, and letting $z\\rightarrow\\infty$.\n\nFrom [1, p. 171] we also note the recessive behavior\n\\begin{equation}\nPs_{n}^{m}\\left(  {z,\\gamma^{2}}\\right)  =K_{n}^{m}\\left(  {\\gamma^{2}%\n}\\right)  \\left(  {z-1}\\right)  ^{m\/2}\\left\\{  {1+{O}\\left(\n{z-1}\\right)  }\\right\\}  \\quad\\left(  {z\\rightarrow1}\\right)  , \\label{eq25}%\n\\end{equation}\nwhere $K_{n}^{m}\\left(  {\\gamma^{2}}\\right)  $ is given by (\\ref{eq18}).\n\nThe plan of the paper is as follows. In \\S 2 we obtain Liouville-Green\napproximations for $S_{n}^{m\\left(  j\\right)  }\\left(  {z,\\gamma}\\right)  $\n($j=3,4)$ where $z$ is complex, and use these to obtain an asymptotic\napproximation for the radial PSWF $Ps_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $\nwhich is uniformly valid in the interval $1+\\delta\\leq x<\\infty$. In \\S 3 the\napproximation for $Ps_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $ is extended to\n$1<x<\\infty$ by applying the theory of differential equations having a simple\npole, which involves the Bessel function of the first kind. Also in this\nsection an asymptotic relationship involving $\\lambda_{n}^{m}\\left(\n{\\gamma^{2}}\\right)  _{\\,}$and the parameters $m$ and $n$ is obtained, by\nmatching the Liouville-Green and Bessel function approximations at infinity.\n\nIn \\S 4 the angular PSWF $\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}%\n}\\right)  $ is approximated, with the intervals $1-\\delta_{0}\\leq x<1$ and\n$0\\leq x\\leq1-\\delta_{0}$ considered separately (for some positive constant\n$\\delta_{0})$. In the former interval the asymptotic approximation involves\nthe modified Bessel function of the first kind, and in the latter interval the\nparabolic cylinder function is used. In \\S 5 the approximation involving the\nparabolic cylinder function is simplified under the assumption $n$ being\nbounded. Finally, in \\S 6 we summarise the main results of the paper.\n\n\\section{Liouville-Green asymptotics: the radial case}\n\nMaking the transformation $w=\\left(  {z^{2}-1}\\right)  ^{1\/2}y$ in\n(\\ref{eq20}) we remove the first derivative to obtain\n\\begin{equation}\n\\frac{d^{2}w}{dz^{2}}=\\left\\{  {-\\gamma^{2}+\\frac{\\lambda_{n}^{m}\\left(\n{\\gamma^{2}}\\right)  }{z^{2}-1}+\\frac{m^{2}-1}{\\left(  {z^{2}-1}\\right)  ^{2}%\n}}\\right\\}  w. \\label{eq26}%\n\\end{equation}\nNow, from [1, p. 186] it is known that for large $\\gamma$, with $m$ and $n$\nbounded, that\n\\begin{equation}\n\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  =-\\gamma^{2}+2\\left(\n{n-m+{\\tfrac{1}{2}}}\\right)  \\gamma+{O}\\left(  1\\right)  .\n\\label{eq27}%\n\\end{equation}\nWith this in mind we define a parameter $\\sigma$ by\n\\begin{equation}\n\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  =-\\gamma^{2}\\left(  {1-\\sigma^{2}%\n}\\right)  , \\label{eq28}%\n\\end{equation}\nand throughout we shall assume that\n\\begin{equation}\n0\\leq\\sigma=\\sqrt{1+\\gamma^{-2}\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)\n}\\leq\\sigma_{0}<1, \\label{eq29}%\n\\end{equation}\nwhere $\\sigma_{0}$ is an arbitrary positive constant.\n\nNext, from \\eqref{eq28} we can express (\\ref{eq26}) in the form\n\\begin{equation}\n\\label{eq30}\\frac{d^{2}w}{dz^{2}}=\\left[  {\\gamma^{2}f\\left(  {\\sigma,z}\n\\right)  +g\\left(  z \\right)  } \\right]  w,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq31}f\\left(  {\\sigma,z} \\right)  =\\frac{\\sigma^{2}-z^{2}}{z^{2}%\n-1},\\quad g\\left(  z \\right)  =\\frac{m^{2}-1}{\\left(  {z^{2}-1} \\right)  ^{2}%\n}.\n\\end{equation}\nWe observe for large $\\gamma$ the differential equation has turning points at\n$z=\\pm\\sigma$, and on account of our assumption (\\ref{eq29}) these turning\npoints lie in the interval $\\left(  {-1,1} \\right)  $, they may coalesce with\none another at $z=0$, but are bounded away from the poles $z=\\pm1$.\n\nWe shall construct Liouville-Green approximations for $Ps_{n}^{m} \\left(\n{z,\\gamma^{2}} \\right)  $, using the theory of [25, Chap. 10]. To this end, we\nintroduce a new independent variable\n\\begin{equation}\n\\label{eq32}\\xi=\\int_{1}^{z} {\\left\\{  {-f\\left(  {\\sigma,t} \\right)  }\n\\right\\}  ^{1\/2}dt} =\\int_{1}^{z} {\\left(  {\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\n\\right)  ^{1\/2}dt} .\n\\end{equation}\nBranch cuts are suitably chosen so that $0\\le\\xi<\\infty$ for $1\\le z<\\infty$.\n\nThe RHS of (\\ref{eq32}) can be expressed in terms of the elliptic integral of\nthe second kind [27, eq. 19.2.5]\n\\begin{equation}\nE\\left(  {a;b}\\right)  =\\int_{0}^{a}{\\left(  {\\frac{1-b^{2}t^{2}}{1-t^{2}}%\n}\\right)  ^{1\/2}dt}=b\\int_{0}^{a}{\\left(  {\\frac{b^{-2}-t^{2}}{1-t^{2}}%\n}\\right)  ^{1\/2}dt}. \\label{eq33}%\n\\end{equation}\nHere $b=\\sigma^{-1}>1$, and the branches of the square roots are such that\nintegrand is positive for $0\\leq t<b^{-1}$ and negative for $1<t<\\infty$, and\ncontinuous elsewhere in the complex $t$-plane having a cut along the interval\n$\\left[  {b^{-1},1}\\right]  $. We thus have\n\\begin{equation}\n\\xi=\\sigma E\\left(  {z;\\sigma^{-1}}\\right)  -\\sigma E\\left(  {1;\\sigma^{-1}%\n}\\right)  . \\label{eq34}%\n\\end{equation}\nThen with the new dependent variable $W=\\left\\{  {-f}\\right\\}  ^{1\/4}w$ we\nobtain\n\\begin{equation}\n\\frac{d^{2}W}{d\\xi^{2}}=\\left[  {-\\gamma^{2}+\\psi\\left(  \\xi\\right)  }\\right]\nW, \\label{eq35}%\n\\end{equation}\nwhere\n\\begin{equation}\n\\psi\\left(  \\xi\\right)  =\\frac{m^{2}-1}{\\left(  {z^{2}-1}\\right)  \\left(\n{z^{2}-\\sigma^{2}}\\right)  }+\\frac{\\left(  {1-\\sigma^{2}}\\right)  \\left(\n{6z^{4}-\\left(  {3+\\sigma^{2}}\\right)  z^{2}-2\\sigma^{2}}\\right)  }{4\\left(\n{z^{2}-1}\\right)  \\left(  {z^{2}-\\sigma^{2}}\\right)  ^{3}}. \\label{eq36}%\n\\end{equation}\nWe observe that $\\psi\\left(  \\xi\\right)  ={O}\\left(  {\\xi^{-2}%\n}\\right)  $ as $\\xi\\rightarrow\\infty$, but is unbounded at the singularities\n$z=\\pm1$, and also at the turning points $z=\\pm\\sigma$.\n\nFrom the definition of $\\xi$ we find that\n\\begin{equation}\n\\xi=z-J\\left(  \\sigma\\right)  +{O}\\left(  {z^{-1}}\\right)\n\\quad\\left(  {z\\rightarrow\\infty}\\right)  , \\label{eq37}%\n\\end{equation}\nwhere\n\\begin{equation}\nJ\\left(  \\sigma\\right)  =1-\\int_{1}^{\\infty}{\\left[  {\\left(  {\\frac\n{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)  ^{1\/2}-1}\\right]  dt}. \\label{eq38}%\n\\end{equation}\nNote $J\\left(  0\\right)  =0$ and $J\\left(  1\\right)  =1$. Now by Cauchy's\ntheorem\n\\begin{equation}\n0=\\operatorname{Re}\\int_{-\\infty}^{\\infty}{\\left[  {\\left(  {\\frac\n{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)  ^{1\/2}-1}\\right]  dt}=2\\operatorname{Re}%\n\\int_{0}^{\\infty}{\\left[  {\\left(  {\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)\n^{1\/2}-1}\\right]  dt}. \\label{eq39}%\n\\end{equation}\nHence\n\\begin{equation}\n\\int_{1}^{\\infty}{\\left[  {\\left(  {\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)\n^{1\/2}-1}\\right]  dt}=-\\operatorname{Re}\\int_{0}^{1}{\\left[  {\\left(\n{\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)  ^{1\/2}-1}\\right]  dt}, \\label{eq40}%\n\\end{equation}\nand consequently from (\\ref{eq38})\n\\begin{equation}\nJ\\left(  \\sigma\\right)  =1+\\operatorname{Re}\\int_{0}^{1}{\\left[  {\\left(\n{\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)  ^{1\/2}-1}\\right]  dt}=\\int\n_{0}^{\\sigma}{\\left(  {\\frac{\\sigma^{2}-t^{2}}{1-t^{2}}}\\right)  ^{1\/2}dt};\n\\label{eq41}%\n\\end{equation}\ni.e.\n\\begin{equation}\nJ\\left(  \\sigma\\right)  =\\sigma E\\left(  {\\sigma;\\sigma^{-1}}\\right)  ,\n\\label{eq42}%\n\\end{equation}\nfor $\\sigma>0$, in which $E$ is the Elliptic integral of the second kind given\nby (\\ref{eq33}). Thus\n\\begin{equation}\n\\xi=z-\\sigma E\\left(  {\\sigma;\\sigma^{-1}}\\right)  +{O}\\left(\n{z^{-1}}\\right)  \\quad\\left(  {z\\rightarrow\\infty}\\right)  . \\label{eq43}%\n\\end{equation}\nWe now apply Theorem 3.1 of [26], with $u$ replaced by $\\gamma$, and with\n$\\xi$ replaced by $i\\xi$. Then, by matching solutions that are recessive at\n$z=\\pm i\\infty$, we have from \\eqref{eq13}, (\\ref{eq14}) and (\\ref{eq43})\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nS_{n}^{m\\left(  3\\right)  }\\left(  {z,\\gamma}\\right)  =i^{-1-n}\\gamma\n^{-1}\\left[  {\\left(  {z^{2}-1}\\right)  \\left(  {z^{2}-\\sigma^{2}}\\right)\n}\\right]  ^{-1\/4}e^{i\\gamma J\\left(  \\sigma\\right)  }\\\\\n\\times\\left[  {e^{i\\gamma\\xi}\\sum\\limits_{s=0}^{p-1}{\\left(  {-i}\\right)\n^{s}\\frac{\\displaystyle A_{s}\\left(  \\xi\\right)  }{\\displaystyle\\gamma^{s}}}+\\varepsilon_{p,1}\\left(\n{\\gamma,\\xi}\\right)  }\\right]  ,\n\\end{array}\n\\label{eq44}%\n\\end{equation}\nand\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nS_{n}^{m\\left(  4\\right)  }\\left(  {z,\\gamma}\\right)  =i^{1+n}\\gamma\n^{-1}\\left[  {\\left(  {z^{2}-1}\\right)  \\left(  {z^{2}-\\sigma^{2}}\\right)\n}\\right]  ^{-1\/4}e^{-i\\gamma J\\left(  \\sigma\\right)  }\\\\\n\\times\\left[  {e^{-i\\gamma\\xi}\\sum\\limits_{s=0}^{p-1}{i^{s}\\frac{\\displaystyle A_{s}\\left(\n\\xi\\right)  }{\\displaystyle \\gamma^{s}}}+\\varepsilon_{p,2}\\left(  {\\gamma,\\xi}\\right)\n}\\right]  .\n\\end{array}\n\\label{eq45}%\n\\end{equation}\n\n\n\nThe error terms $\\varepsilon_{p,j}\\left(  {\\gamma,\\xi}\\right)  $ ($j=1,2$) are\nbounded by Olver's theorem, and are ${O}\\left(  {\\gamma^{-p}}\\right)\n$ in unbounded domains containing the real interval $1+\\delta\\leq z<\\infty$\n($\\delta>0$). Here the coefficients are defined recursively by $A_{0}\\left(\n\\xi\\right)  =1$ and\n\\begin{equation}\nA_{s+1}\\left(  \\xi\\right)  =-{\\tfrac{1}{2}A}_{s}^{\\prime}\\left(  \\xi\\right)\n+{\\tfrac{1}{2}}\\int{\\psi\\left(  \\xi\\right)  A_{s}\\left(  \\xi\\right)  d\\xi\n}\\quad\\left(  {s=0,1,2,\\cdots}\\right)  . \\label{eq46}%\n\\end{equation}\nThus, from (\\ref{eq15}), (\\ref{eq23}), (\\ref{eq42}), (\\ref{eq44}) and\n(\\ref{eq45}), we obtain the desired Liouville-Green expansion for $Ps_{n}%\n^{m}\\left(  {x,\\gamma^{2}}\\right)  $. In particular, to leading order, we\nhave\n\\begin{equation}\nPs_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\frac{\\left(  {-1}\\right)  ^{n}%\n\\sin\\left(  {\\gamma\\xi+\\gamma\\sigma E\\left(  {\\sigma;\\sigma^{-1}}\\right)\n-{\\frac{1}{2}}n\\pi}\\right)  +{O}\\left(  {\\gamma^{-1}}\\right)  }%\n{\\gamma\\left(  {n-m}\\right)  !V_{n}^{m}\\left(  \\gamma\\right)  \\left[  {\\left(\n{x^{2}-1}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)  }\\right]  ^{1\/4}},\n\\label{eq47}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, uniformly for $1+\\delta\\leq x<\\infty$. In order\nfor this approximation to be practicable, one requires an asymptotic\napproximation for $\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  $ as\n$\\gamma\\rightarrow\\infty$, and we shall discuss this in the next section. We\nalso remark that (\\ref{eq47}) breaks down at the simple pole $x=1$, and in the\nnext section we obtain asymptotic approximations that are valid at this pole.\n\n\\section{Bessel function approximations: the radial case}\n\n\nWe now obtain approximations valid at the simple pole of $f\\left(  {\\sigma\n,z}\\right)  $ at $z=1$, using the asymptotic theory of [25, Chap. 12]. We\nconsider $z=x$ real and positive. The appropriate Liouville transformation is\nnow given by\n\\begin{equation}\n\\eta=\\xi^{2}=\\left[  {\\int_{1}^{x}{\\left\\{  {-f\\left(  {\\sigma,t}\\right)\n}\\right\\}  ^{1\/2}dt}}\\right]  ^{2}, \\label{eq48}%\n\\end{equation}\nalong with\n\\begin{equation}\n\\hat{{W}}=\\left\\{  {\\frac{\\eta\\left(  {x^{2}-\\sigma^{2}}\\right)  }{x^{2}-1}%\n}\\right\\}  ^{1\/4}w, \\label{eq49}%\n\\end{equation}\nwhich yields the new equation\n\\begin{equation}\n\\frac{d^{2}\\hat{{W}}}{d\\eta^{2}}=\\left[  {-\\frac{\\gamma^{2}}{4\\eta}%\n+\\frac{m^{2}-1}{4\\eta^{2}}+\\frac{\\hat{{\\psi}}\\left(  \\eta\\right)  }{\\eta}%\n}\\right]  \\hat{{W}}. \\label{eq50}%\n\\end{equation}\nHere\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\hat{{\\psi}}\\left(  \\eta\\right)  =\\dfrac{1-4m^{2}}{16\\eta}+\\dfrac{m^{2}%\n-1}{4\\left(  {x^{2}-1}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)  }\\\\\n+\\dfrac{\\left(  {1-\\sigma^{2}}\\right)  \\left(  {6x^{4}-\\left(  {3+\\sigma^{2}%\n}\\right)  x^{2}-2\\sigma^{2}}\\right)  }{16\\left(  {x^{2}-1}\\right)  \\left(\n{x^{2}-\\sigma^{2}}\\right)  ^{3}}.\n\\end{array}\n\\label{eq51}%\n\\end{equation}\nThis has the same main features of (\\ref{eq30}), namely a simple pole for the\ndominant term (for large $\\gamma$) and a double pole in another term. We note\nthat $x=1$ corresponds to $\\eta=0$.\n\nThe difference here is that non-dominant term $\\hat{{\\psi}}\\left(\n\\eta\\right)  $ is now analytic at $\\eta=0$, i.e. $x=1$. Neglecting $\\hat\n{{\\psi}}\\left(  \\eta\\right)  $ in \\eqref{eq50} gives an equation solvable in\nterms of Bessel functions. We then find (by matching recessive solutions at\n$x=1$) and applying theorem 4.1 of [25, Chap. 12] (with $u$ replaced by\n$\\gamma$ and $\\zeta$ replaced by $\\eta$)\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nPs_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =c_{n}^{m}\\left(  \\gamma\\right)\n\\left\\{  {\\dfrac{\\eta}{\\left(  {x^{2}-1}\\right)  \\left(  {x^{2}-\\sigma^{2}%\n}\\right)  }}\\right\\}  ^{1\/4}\\\\\n\\times\\left[  {J_{m}\\left(  {\\gamma\\eta^{1\/2}}\\right)  +{O}\\left(\n{\\gamma^{-1}}\\right)  \\operatorname{env}J_{m}\\left(  {\\gamma\\eta^{1\/2}%\n}\\right)  }\\right]  ,\n\\end{array}\n\\label{eq52}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, uniformly for $1<x<\\infty$. Olver's theorem\nprovides an asymptotic expansion in inverse powers of $\\gamma$, but we present\njust the leading term here. The so-called envelope env of the $J$ Bessel\nfunction is defined by [27, \\S 2.8(iv)].\n\n\n\nThe constant of proportionality $c_{n}^{m}\\left(  \\gamma\\right)  $ can be\nfound by comparing both sides of (\\ref{eq52}) as $x\\rightarrow1$\n($\\eta\\rightarrow0$). Using\n\\begin{equation}\n\\eta=2\\left(  {1-\\sigma^{2}}\\right)  \\left(  {x-1}\\right)  +{O}%\n\\left\\{  {\\left(  {x-1}\\right)  ^{2}}\\right\\}  \\quad\\left(  {x\\rightarrow\n1}\\right)  , \\label{eq53}%\n\\end{equation}\nalong with (\\ref{eq25}), (\\ref{eq28}) and the behavior of the $J$ Bessel\nfunction at the origin (e.g. [25, Chap. 12, \\S 1]) we find that\n\\begin{equation}\nc_{n}^{m}\\left(  \\gamma\\right)  =\\left(  {-\\frac{2}{\\lambda_{n}^{m}\\left(\n{\\gamma^{2}}\\right)  }}\\right)  ^{m\/2}m!K_{n}^{m}\\left(  {\\gamma^{2}}\\right)\n. \\label{eq54}%\n\\end{equation}\nAn asymptotic approximation for this constant is given by (\\ref{eq105}) below.\n\nNext, from the well-known behavior of the $J$ Bessel function at infinity\n(e.g. see [25, Chap. 12, \\S 1]), we find from \\eqref{eq52} that\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nPs_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  \\sim\\text{constant}\\times\\left\\{\n{\\left(  {x^{2}-1}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)  }\\right\\}\n^{-1\/4}\\\\\n\\times\\left\\{  {\\cos\\left(  {\\gamma\\xi-\\frac{1}{2}m\\pi-\\frac{1}{4}\\pi}\\right)\n+{O}\\left(  \\xi^{-1}\\right)  }\\right\\}  \\quad\\left(  {\\eta=\\xi\n^{2}\\rightarrow\\infty}\\right)  .\n\\end{array}\n\\label{eq55}%\n\\end{equation}\nHowever, from (\\ref{eq47}) we observe an alternative expression of the\nbehavior of this function. On comparing both, we deduce that\n\\begin{equation}\n\\gamma\\sigma E\\left(  {\\sigma;\\sigma^{-1}}\\right)  =\\left(  {2N+{\\tfrac{1}{2}%\n}n-{\\tfrac{1}{2}}m+{\\tfrac{1}{4}}}\\right)  \\pi+{O}\\left(  {\\gamma\n^{-1}}\\right)  , \\label{eq56}%\n\\end{equation}\nfor some integer $N$, which we show is zero. Now, from (\\ref{eq27}) and\n(\\ref{eq28}) we have for \\textit{fixed} $m$ and $n$ that $\\sigma\n={O}\\left(  {\\gamma^{-1\/2}}\\right)  $ as $\\gamma\\rightarrow\\infty$,\nand more precisely,\n\\begin{equation}\n\\sigma^{2}=2\\left(  {n-m+{\\tfrac{1}{2}}}\\right)  \\gamma^{-1}+{O}%\n\\left(  {\\gamma^{-2}}\\right)  . \\label{eq57}%\n\\end{equation}\nThus in this case, using (\\ref{eq41}) and (\\ref{eq42}) in the LHS of\n(\\ref{eq56}), we have\n\\begin{equation}\n{\\tfrac{1}{4}}\\pi\\gamma\\sigma^{2}+{O}\\left(  {\\gamma^{-1}\\sigma^{4}%\n}\\right)  =\\left(  {2N+{\\tfrac{1}{2}}n-{\\tfrac{1}{2}}m+{\\tfrac{1}{4}}}\\right)\n\\pi+{O}\\left(  {\\gamma^{-1}}\\right)  . \\label{eq58}%\n\\end{equation}\nInserting (\\ref{eq57}) into (\\ref{eq58}) we deduce that $N=0$, at least for\nfixed $m$ and $n$; a continuity argument removes this restriction.\n\nIt is possible to extend (\\ref{eq47}) and (\\ref{eq55}) to asymptotic\nexpansions, and consequently from (\\ref{eq41}) and (\\ref{eq42}) we arrive at\n\\begin{equation}\n\\label{eq59}\\gamma\\int_{0}^{\\sigma}{\\left(  {\\frac{\\sigma^{2}-t^{2}}{1-t^{2}}}\n\\right)  ^{1\/2}dt} \\sim\\frac{1}{2}\\left(  {n-m+\\frac{1}{2}} \\right)  \\pi\n+\\sum\\limits_{s=0}^{\\infty}{\\frac{\\kappa_{s} }{\\gamma^{2s+1}}} ,\n\\end{equation}\nfor constants $\\kappa_{s} $ which can be determined in terms of the\ncoefficients appearing in (\\ref{eq44}) and (\\ref{eq45}). From (\\ref{eq29}) we\ncan invert this expansion to provide a means of computing the eigenvalue\n$\\lambda=\\lambda_{n}^{m} \\left(  {\\gamma^{2}} \\right)  $ asymptotically in\nterms of $m$ and $n$ as $\\gamma\\to\\infty$.\n\nNow, the elliptic integral on the LHS of (\\ref{eq59}) monotonically increases\nfrom 0 to 1 as $\\sigma$ increases from 0 to 1. We therefore see that the\ncondition (\\ref{eq29}) (along with our assumption that $m$ is bounded) is\nequivalent to (\\ref{eq2}).\n\n\n\\section{Bessel and parabolic cylinder function approximations: the angular\ncase}\n\nRecall $\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $ is the\nunique solution with the property of being recessive at $x=\\pm1$. It is also\nuniquely determined by the property\n\\begin{equation}\n\\operatorname{Ps}_{n}^{m}\\left(  {-x,\\gamma^{2}}\\right)  =\\left(  {-1}\\right)\n^{m+n}\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  . \\label{eq60}%\n\\end{equation}\nThus, it suffices to approximate $\\operatorname{Ps}_{n}^{m}\\left(\n{x,\\gamma^{2}}\\right)  $ in the interval $0\\leq x<1$. We consider the\nsubintervals $0\\leq x\\leq1-\\delta_{0}$ and $1-\\delta_{0}\\leq x<1$ separately,\nwhere $\\delta_{0}\\in\\left(  {0,1-\\sigma_{0}}\\right)  $ is arbitrary; recall\nthat $\\sigma_{0}$ is defined by (\\ref{eq29}). The significance of this choice\nis that the turning point $x=\\sigma$ is bounded away from the interval\n$\\left[  {1-\\delta_{0},1}\\right]  $.\n\nFor $1-\\delta_{0}\\leq x<1$ we apply theorem 3.1 of [25, Chap. 12]. It can then\nbe shown by utilising the recessive behavior at $x=1$ ($\\eta=0$) that\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =c_{n}^{m}\\left(\n\\gamma\\right)  \\left\\{  {\\dfrac{\\left\\vert \\eta\\right\\vert }{\\left(  {1-x^{2}%\n}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)  }}\\right\\}  ^{1\/4}\\\\\n\\times I_{m}\\left(  {\\gamma\\left\\vert \\eta\\right\\vert ^{1\/2}}\\right)  \\left[\n{1+{O}\\left(  {\\dfrac{1}{\\gamma}}\\right)  }\\right]  ,\n\\end{array}\n\\label{eq61}%\n\\end{equation}\nwhere $I_{m}\\left(  x\\right)  $ is the modified Bessel function, and\n$c_{n}^{m}\\left(  \\gamma\\right)  _{\\,}$is given by (\\ref{eq54}). Expansions\nand error bounds are obtainable from Olver's theorem.\n\nThe interval $0\\leq x\\leq1-\\delta_{0}$ is less straightforward. From\n(\\ref{eq30}) and (\\ref{eq31}) we observe that equation has the turning point\n$x=\\sigma$ in this interval, and this coalesces with the other turning point\n$x=-\\sigma$ when $\\sigma\\rightarrow0$. The appropriate asymptotic theory for\nthis situation is provided by [26], and from eq. (\\ref{eq28}) of this\nreference the appropriate transformation is given by\n\\begin{equation}\n\\frac{d\\zeta}{dx}=\\left(  {\\frac{\\sigma^{2}-x^{2}}{\\left(  {1-x^{2}}\\right)\n\\left(  {\\alpha^{2}-\\zeta^{2}}\\right)  }}\\right)  ^{1\/2}. \\label{eq62}%\n\\end{equation}\nUpon integration, this yields the implicit relationship\n\\begin{equation}\n\\int_{-\\alpha}^{\\zeta}{\\left(  {\\alpha^{2}-\\tau^{2}}\\right)  ^{1\/2}d\\tau}%\n=\\int_{-\\sigma}^{x}{\\left\\{  {-f\\left(  {\\sigma,t}\\right)  }\\right\\}\n^{1\/2}dt}=\\int_{-\\sigma}^{x}{\\left(  {\\frac{\\sigma^{2}-t^{2}}{1-t^{2}}%\n}\\right)  ^{1\/2}dt}. \\label{eq63}%\n\\end{equation}\nThe lower limits are selected to ensure that the turning point $x=-\\sigma$ is\nmapped to a new turning point at $\\zeta=-\\alpha$ (see \\eqref{eq69} below).\nFrom [26, eq. (\\ref{eq30})] we find that $\\alpha$ is given by\n\\begin{equation}\n\\alpha^{2}=\\dfrac{2}{\\pi}\\int_{-\\sigma}^{\\sigma}{\\left(  {\\frac{\\sigma\n^{2}-t^{2}}{1-t^{2}}}\\right)  ^{1\/2}dt}=\\frac{4}{\\pi}J\\left(  \\sigma\\right)  ,\n\\label{eq64}%\n\\end{equation}\nwhich ensures that the original turning point $x=\\sigma$ is mapped to the\nturning point at $\\zeta=\\alpha$ in the transformed equation.\n\n\nBy symmetry $x=0$ is mapped to $\\zeta=0$, and so the lower limits in the\nintegrals of (\\ref{eq63}) can be replaced by 0. Thus we have\n\\begin{equation}\n\\tfrac{1}{2}\\alpha^{2}\\arcsin\\left(  {\\frac{\\zeta}{\\alpha}}\\right)  +\\tfrac\n{1}{2}\\zeta\\left(  {\\alpha^{2}-\\zeta^{2}}\\right)  ^{1\/2}=\\sigma E\\left(\n{x;\\sigma^{-1}}\\right)  , \\label{eq65}%\n\\end{equation}\nfor $0\\leq x\\leq\\sigma$ ($0\\leq\\zeta\\leq\\alpha$).\n\nFor $\\sigma\\leq x\\leq1-\\delta_{0}$ we have\n\\begin{equation}\n\\int_{\\alpha}^{\\zeta}{\\left(  {\\tau^{2}-\\alpha^{2}}\\right)  ^{1\/2}d\\tau}%\n=\\int_{\\sigma}^{x}{\\left\\{  {f\\left(  {\\sigma,t}\\right)  }\\right\\}  ^{1\/2}%\ndt}=\\int_{\\sigma}^{x}{\\left(  {\\frac{t^{2}-\\sigma^{2}}{1-t^{2}}}\\right)\n^{1\/2}dt}. \\label{eq66}%\n\\end{equation}\nThus in this case\n\\begin{equation}\n-\\tfrac{1}{2}\\alpha^{2}\\operatorname{arccosh}\\left(  {\\frac{\\zeta}{\\alpha}%\n}\\right)  +\\tfrac{1}{2}\\zeta\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)\n^{1\/2}=\\left\\vert \\operatorname{Im}{\\left\\{  {\\sigma E\\left(  {x;\\sigma^{-1}%\n}\\right)  }\\right\\}  }\\right\\vert . \\label{eq67}%\n\\end{equation}\n\n\nWith\n\\begin{equation}\nW=\\left\\{  {\\frac{\\sigma^{2}-x^{2}}{\\left(  {\\alpha^{2}-\\zeta^{2}}\\right)\n\\left(  {1-x^{2}}\\right)  }}\\right\\}  ^{1\/4}w, \\label{eq68}%\n\\end{equation}\nwe transform (\\ref{eq30}) to the form\n\\begin{equation}\n\\frac{d^{2}W}{d\\zeta^{2}}=\\left\\{  {\\gamma^{2}\\left(  {\\zeta^{2}-\\alpha^{2}%\n}\\right)  +\\psi\\left(  {\\gamma,\\alpha,\\zeta}\\right)  }\\right\\}  W,\n\\label{eq69}%\n\\end{equation}\nwhere\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\psi\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =\\dfrac{\\left(  {1-m^{2}}\\right)\n\\left(  {\\alpha^{2}-\\zeta^{2}}\\right)  }{\\left(  {1-x^{2}}\\right)  \\left(\n{\\sigma^{2}-x^{2}}\\right)  }+\\dfrac{2\\alpha^{2}+3\\zeta^{2}}{4\\left(\n{\\alpha^{2}-\\zeta^{2}}\\right)  ^{2}}\\\\\n-\\dfrac{\\left(  {1-\\sigma^{2}}\\right)  \\left(  {\\alpha^{2}-\\zeta^{2}}\\right)\n\\left\\{  {6x^{4}-\\left(  {\\sigma^{2}+3}\\right)  x^{2}-2\\sigma^{2}}\\right\\}\n}{4\\left(  {1-x^{2}}\\right)  \\left(  {\\sigma^{2}-x^{2}}\\right)  ^{3}}.\n\\end{array}\n\\label{eq70}%\n\\end{equation}\nTo sharpen the subsequent error bounds it is possible to perturb the parameter\nby defining a new parameter $\\omega$ by $\\alpha^{2}=\\omega^{2}+\\psi\\left(\n{\\gamma,\\alpha,0}\\right)  \\gamma^{-2}$, but we shall not pursue this here.\n\n\nFrom theorem I of [26], with $u$ replaced by $\\gamma$, we obtain two\nindependent solutions of (\\ref{eq69}) given by\n\\begin{equation}\nw_{1}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =U\\left(  {-{\\tfrac{1}{2}}%\n\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  +\\varepsilon_{1}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  , \\label{eq71}%\n\\end{equation}\nand\n\\begin{equation}\nw_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =\\bar{{U}}\\left(  {-{\\tfrac{1}{2}%\n}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  +\\varepsilon_{2}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  . \\label{eq72}%\n\\end{equation}\nHere $U\\left(  {a,x}\\right)  $ and $\\bar{{U}}\\left(  {a,x}\\right)  $ are the\nparabolic cylinder functions defined in [26, \\S 5] and [27, \\S 12.2], and are\nlinearly independent for $a<0$. The approximants $U\\left(  {-{\\frac{1}{2}%\n}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  $ and $\\bar{{U}}\\left(\n{-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  $ satisfy the\ndifferential equation (\\ref{eq69}) with $\\psi\\left(  {\\gamma,\\alpha,\\zeta\n}\\right)  \\equiv0$.\n\nThe error terms are bounded by [26, \\S 6], and in particular these show that\n\\begin{equation}\n\\varepsilon_{1}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  ={O}\\left(\n{\\gamma^{-2\/3}\\ln\\left(  \\gamma\\right)  }\\right)  {\\operatorname{env}}U\\left(\n{-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  , \\label{eq73}%\n\\end{equation}\nand\n\\begin{equation}\n\\varepsilon_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  ={O}\\left(\n{\\gamma^{-2\/3}\\ln\\left(  \\gamma\\right)  }\\right)  {\\operatorname{env}}\\bar\n{{U}}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  ,\n\\label{eq74}%\n\\end{equation}\nuniformly for $0\\leq x\\leq1-\\delta_{0}$. Here the envelope function\n${\\operatorname{env}}$ is defined for the parabolic cylinder functions by [27,\neq. 14.15.23].\n\nThe parabolic cylinder function $U$ has the unique recessive property\n\\begin{equation}\nU\\left(  {-{\\tfrac{1}{2}}a,x}\\right)  \\sim x^{\\left(  {a-1}\\right)\n\/2}e^{-x^{2}\/4}\\quad\\left(  {x\\rightarrow\\infty}\\right)  , \\label{eq75}%\n\\end{equation}\nwhereas $\\bar{{U}}$ is dominant, with the behavior\n\\begin{equation}\n\\bar{{U}}\\left(  {-{\\tfrac{1}{2}}a,x}\\right)  \\sim\\left(  {2\/\\pi}\\right)\n^{1\/2}\\Gamma\\left(  {{\\tfrac{1}{2}}a+{\\tfrac{1}{2}}}\\right)  x^{-\\left(\n{a+1}\\right)  \/2}e^{x^{2}\/4}\\quad\\left(  {x\\rightarrow\\infty}\\right)  ;\n\\label{eq76}%\n\\end{equation}\nsee [26, \\S 5]. In addition, from [26, Eqs. (5.12) and (5.13)]\nand the definitions \\eqref{eq71} and \\eqref{eq72}, we note that $w_{1}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  $ and $w_{2}\\left(  {\\gamma,\\alpha,\\zeta\n}\\right)  $ are oscillatory in the $\\zeta$ interval $\\left[  {0,\\alpha\n}\\right]  $, with comparable amplitudes and complementary phases of the argument.\n\n\n\nNow, for negative $x$ and $\\zeta,$ we will also need the solution given in\n[26]\n\\begin{equation}\nw_{4}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =\\bar{{U}}\\left(  {-{\\tfrac{1}{2}%\n}\\gamma\\alpha^{2},-\\zeta\\sqrt{2\\gamma}}\\right)  +\\varepsilon_{4}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  . \\label{eq77}%\n\\end{equation}\nWe remark that\n\\begin{equation}\n\\varepsilon_{j}\\left(  {\\gamma,\\alpha,0}\\right)  =\\partial\\varepsilon\n_{j}\\left(  {\\gamma,\\alpha,0}\\right)  \/\\partial\\zeta=0\\quad\\left(\n{j=2,4}\\right)  , \\label{eq78}%\n\\end{equation}\nand hence\n\\begin{equation}\nw_{2}\\left(  {\\gamma,\\alpha,0}\\right)  =w_{4}\\left(  {\\gamma,\\alpha,0}\\right)\n=\\bar{{U}}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},0}\\right)  , \\label{eq79}%\n\\end{equation}\nas well as\n\\begin{equation}\n\\partial w_{2}\\left(  {\\gamma,\\alpha,0}\\right)  \/\\partial\\zeta=-\\partial\nw_{4}\\left(  {\\gamma,\\alpha,0}\\right)  \/\\partial\\zeta=\\sqrt{2\\gamma}\\bar\n{{{U}^{\\prime}}}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},0}\\right)  .\n\\label{eq80}%\n\\end{equation}\nThe error bounds for $\\varepsilon_{4}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  $\nonly apply for non-positive $\\zeta.$ In order to extend the solution to\npositive values of $\\zeta$ we use [26, Eqs. (6.23) and (6.24)] to obtain the\nconnection formula\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nw_{4}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =-\\left\\{  {\\sin\\left(  {{\\frac\n{1}{2}}\\pi\\gamma\\alpha^{2}}\\right)  +{O}\\left(  {\\gamma^{-2\/3}%\n}\\right)  }\\right\\}  w_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right) \\\\\n+\\left\\{  {\\cos\\left(  {{\\frac{1}{2}}\\pi\\gamma\\alpha^{2}}\\right)\n+{O}\\left(  {\\gamma^{-2\/3}}\\right)  }\\right\\}  w_{1}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  .\n\\end{array}\n\\label{eq81}%\n\\end{equation}\n\n\nNow from (\\ref{eq42}), (\\ref{eq56}) and (\\ref{eq64})\n\\begin{equation}\n{\\tfrac{1}{2}}\\pi\\gamma\\alpha^{2}=\\left(  {n-m+{\\tfrac{1}{2}}}\\right)\n\\pi+{O}\\left(  {\\gamma^{-1}}\\right)  . \\label{eq82}%\n\\end{equation}\nBearing in mind that $w_{1}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  $ is\nexponentially small compared to $w_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  $\nin $0\\leq x\\leq1-\\delta_{0}$ (except near its zeros) we deduce from\n\\eqref{eq81} and \\eqref{eq82} that\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nw_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  -\\left(  {-1}\\right)  ^{m+n}%\nw_{4}\\left(  {\\gamma,\\alpha,\\zeta}\\right)  =2w_{2}\\left(  {\\gamma,\\alpha\n,\\zeta}\\right) \\\\\n+{O}\\left(  {\\gamma^{-2\/3}}\\right)  \\left\\{  {w_{1}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  +w_{2}\\left(  {\\gamma,\\alpha,\\zeta}\\right)\n}\\right\\}  .\n\\end{array}\n\\label{eq83}%\n\\end{equation}\nWe next express\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\left\\{\n{\\dfrac{\\alpha^{2}-\\zeta^{2}}{\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(\n{1-x^{2}}\\right)  }}\\right\\}  ^{1\/4}\\\\\n\\times\\left[  {d_{n}^{m}\\left(  \\gamma\\right)  w_{1}\\left(  {\\gamma\n,\\alpha,\\zeta}\\right)  +e_{n}^{m}\\left(  \\gamma\\right)  \\left\\{  {w_{2}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  -\\left(  {-1}\\right)  ^{m+n}w_{4}\\left(\n{\\gamma,\\alpha,\\zeta}\\right)  }\\right\\}  }\\right]  ,\n\\end{array}\n\\label{eq84}%\n\\end{equation}\nand we shall determine the constant $d_{n}^{m}\\left(  \\gamma\\right)  $ (as\nwell as bounding $e_{n}^{m}\\left(  \\gamma\\right)  )$ by comparing both sides\nof this relationship at appropriate values of $x$.\n\n\nTo this end, firstly we assume that $\\operatorname{Ps}_{n}^{m}\\left(\n{x,\\gamma^{2}}\\right)  $ is even, so that $m+n$ is also even. Then, setting\n$x=\\zeta=0$ in (\\ref{eq84}), and invoking (\\ref{eq79}), immediately yields\n\\begin{equation}\nd_{n}^{m}\\left(  \\gamma\\right)  =\\left(  {\\frac{\\sigma}{\\alpha}}\\right)\n^{1\/2}\\frac{\\operatorname{Ps}_{n}^{m}\\left(  {0,\\gamma^{2}}\\right)  }%\n{w_{1}\\left(  {\\gamma,\\alpha,0}\\right)  }. \\label{eq85}%\n\\end{equation}\nAn asymptotic approximation for this constant, which does not involve\n$\\operatorname{Ps}_{n}^{m}\\left(  {0,\\gamma^{2}}\\right)  $, is given by\n(\\ref{eq102}) below.\n\nNext, if we differentiate both sides of (\\ref{eq84}) and again set $x=\\zeta=0$\nwe find from the property $\\operatorname{Ps}_{n}^{m}{}^{\\prime}\\left(\n{0,\\gamma^{2}}\\right)  =0$ that $e_{n}^{m}\\left(  \\gamma\\right)\n={O}\\left(  {\\gamma^{-1}}\\right)  $, which is not sharp enough.\nInstead we match the parabolic cylinder and Bessel function approximations,\nand their derivatives, at the fixed point $x=1-{\\frac{1}{2}}\\delta_{0}$ (at\nwhich both the parabolic cylinder function and modified Bessel function\napproximations are valid). Using (\\ref{eq61}) (\\ref{eq83}) and (\\ref{eq84}),\nwe therefore arrive at\n\\begin{equation}\ne_{n}^{m}\\left(  \\gamma\\right)  \\sim-d_{n}^{m}\\left(  \\gamma\\right)\n\\frac{\\mathcal{W}\\left\\{  {\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)\n^{1\/4}U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)\n,\\left\\vert \\eta\\right\\vert ^{1\/4}I_{m}\\left(  {\\gamma\\left\\vert\n\\eta\\right\\vert ^{1\/2}}\\right)  }\\right\\}  }{\\mathcal{W}\\left\\{  {\\left(\n{\\zeta^{2}-\\alpha^{2}}\\right)  ^{1\/4}\\overline{U}\\left(  {-{\\frac{1}{2}}%\n\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  ,\\left\\vert \\eta\\right\\vert\n^{1\/4}I_{m}\\left(  {\\gamma\\left\\vert \\eta\\right\\vert ^{1\/2}}\\right)\n}\\right\\}  }, \\label{eq86}%\n\\end{equation}\nand\n\\begin{equation}\nc_{n}^{m}\\left(  \\gamma\\right)  \\sim d_{n}^{m}\\left(  \\gamma\\right)\n\\frac{{\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)  ^{1\/2}}\\mathcal{W}\\left\\{\n{\\overline{U}\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  ,U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  }\\right\\}  }{\\mathcal{W}\\left\\{  {\\left(  {\\zeta^{2}-\\alpha^{2}%\n}\\right)  ^{1\/4}\\overline{U}\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\n\\sqrt{2\\gamma}}\\right)  ,\\left\\vert \\eta\\right\\vert ^{1\/4}I_{m}\\left(\n{\\gamma\\left\\vert \\eta\\right\\vert ^{1\/2}}\\right)  }\\right\\}  }. \\label{eq87}%\n\\end{equation}\nIn both of these the Wronskians $\\mathcal{W}$ are with respect to $x$, and\nevaluated at $x=1-{\\frac{1}{2}}\\delta_{0}$ (with $\\eta$ and $\\zeta$\ncorresponding to this value).\n\nNext, from (\\ref{eq66}) and [27, Eqs. 12.10.3 - 12.10.6], we have the\nasymptotic approximations for large $\\gamma$, fixed $\\zeta\\in\\left(\n{\\alpha,\\infty}\\right)  $ and fixed $\\alpha>0$\n\\begin{equation}\nU\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)\n\\sim\\left(  {\\frac{\\gamma\\alpha^{2}}{2e}}\\right)  ^{\\gamma\\alpha^{2}\/4}%\n\\frac{\\exp\\left\\{  {-\\gamma\\int_{\\sigma}^{x}{\\left\\{  {f\\left(  {\\sigma\n,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  }{\\left\\{  {2\\gamma\\left(\n{\\zeta^{2}-\\alpha^{2}}\\right)  }\\right\\}  ^{1\/4}}, \\label{eq88}%\n\\end{equation}%\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n{U}^{\\prime}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  \\sim-\\frac{1}{2}\\left(  {\\dfrac{\\gamma\\alpha^{2}}{2e}}\\right)\n^{\\gamma\\alpha^{2}\/4}\\\\\n\\times\\left\\{  {2\\gamma\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)  }\\right\\}\n^{1\/4}\\exp\\left\\{  {-\\gamma\\int_{\\sigma}^{x}{\\left\\{  {f\\left(  {\\sigma\n,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  ,\n\\end{array}\n\\label{eq89}%\n\\end{equation}%\n\\begin{equation}\n\\overline{U}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  \\sim2\\left(  {\\frac{\\gamma\\alpha^{2}}{2e}}\\right)  ^{\\gamma\n\\alpha^{2}\/4}\\frac{\\exp\\left\\{  {\\gamma\\int_{\\sigma}^{x}{\\left\\{  {f\\left(\n{\\sigma,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  }{\\left\\{  {2\\gamma\\left(\n{\\zeta^{2}-\\alpha^{2}}\\right)  }\\right\\}  ^{1\/4}}, \\label{eq90}%\n\\end{equation}\nand\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\overline{U}^{\\prime}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},\\zeta\n\\sqrt{2\\gamma}}\\right)  \\sim\\left(  {\\dfrac{\\gamma\\alpha^{2}}{2e}}\\right)\n^{\\gamma\\alpha^{2}\/4}\\left\\{  {2\\gamma\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)\n}\\right\\}  ^{1\/4}\\\\\n\\times\\exp\\left\\{  {\\gamma\\int_{\\sigma}^{x}{\\left\\{  {f\\left(  {\\sigma\n,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  .\n\\end{array}\n\\label{eq91}%\n\\end{equation}\n\n\n\n\nThese, along with\n\\begin{equation}\n\\left\\vert \\eta\\right\\vert ^{1\/4}I_{m}\\left(  {\\gamma\\left\\vert \\eta\n\\right\\vert ^{1\/2}}\\right)  \\sim\\left(  {2\\pi\\gamma}\\right)  ^{-1\/2}%\n\\exp\\left\\{  {\\gamma\\int_{x}^{1}{\\left\\{  {f\\left(  {\\sigma,t}\\right)\n}\\right\\}  ^{1\/2}dt}}\\right\\}  , \\label{eq92}%\n\\end{equation}%\n\\begin{equation}\n\\frac{d\\left\\{  {\\left\\vert \\eta\\right\\vert ^{1\/4}I_{m}\\left(  {\\gamma\n\\left\\vert \\eta\\right\\vert ^{1\/2}}\\right)  }\\right\\}  }{dx}\\sim-\\left(\n{\\frac{\\gamma}{2\\pi}}\\right)  ^{1\/2}\\left(  {\\frac{x^{2}-\\sigma^{2}}{1-x^{2}}%\n}\\right)  ^{1\/2}\\exp\\left\\{  {\\gamma\\int_{x}^{1}{\\left\\{  {f\\left(  {\\sigma\n,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  , \\label{eq93}%\n\\end{equation}\nand\n\\begin{equation}\n\\frac{d\\zeta}{dx}=\\left\\{  {\\frac{x^{2}-\\sigma^{2}}{\\left(  {1-x^{2}}\\right)\n\\left(  {\\zeta^{2}-\\alpha^{2}}\\right)  }}\\right\\}  ^{1\/2}, \\label{eq94}%\n\\end{equation}\ncan be used to simplify (\\ref{eq86}) and (\\ref{eq87}). In particular, we find\nthat\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\ne_{n}^{m}\\left(  \\gamma\\right)  \\left\\{  {w_{2}\\left(  {\\gamma,\\alpha,\\zeta\n}\\right)  -\\left(  {-1}\\right)  ^{m+n}w_{4}\\left(  {\\gamma,\\alpha,\\zeta\n}\\right)  }\\right\\} \\\\\n=o\\left(  1\\right)  A{\\operatorname{env}}U\\left(  {-{\\tfrac{1}{2}}\\gamma\n\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  ,\n\\end{array}\n\\label{eq95}%\n\\end{equation}\nwhere the $o\\left(  1\\right)  $ term is exponentially small as $\\gamma\n\\rightarrow\\infty_{\\,}$for $x\\in\\left[  {0,1-\\delta_{0}}\\right]  $. In\naddition, we obtain the useful result\n\\begin{equation}\nc_{n}^{m}\\left(  \\gamma\\right)  \\sim d_{n}^{m}\\left(  \\gamma\\right)  \\left(\n{\\frac{\\gamma\\alpha^{2}}{2e}}\\right)  ^{\\gamma\\alpha^{2}\/4}\\left(  {\\frac\n{\\pi^{2}}{2\\gamma}}\\right)  ^{1\/4}\\exp\\left\\{  {-\\gamma\\int_{\\sigma}%\n^{1}{\\left\\{  {f\\left(  {\\sigma,t}\\right)  }\\right\\}  ^{1\/2}dt}}\\right\\}  .\n\\label{eq96}%\n\\end{equation}\nFrom (\\ref{eq73}) and (\\ref{eq84}) - (\\ref{eq95}), for $m+n$ even, $m$ bounded\nand $n$ satisfying (\\ref{eq2}), we arrive at our desired result\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\dfrac\n{\\operatorname{Ps}\\left(  {0,\\gamma^{2}}\\right)  }{U\\left(  {-{\\frac{1}{2}%\n}\\gamma\\alpha^{2},0}\\right)  }\\left\\{  {\\dfrac{\\sigma^{2}\\left(  {\\alpha\n^{2}-\\zeta^{2}}\\right)  }{\\alpha^{2}\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(\n{1-x^{2}}\\right)  }}\\right\\}  ^{1\/4}\\\\\n\\times\\left\\{  {U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  +{O}\\left(  {\\gamma^{-2\/3}\\ln\\left(  \\gamma\\right)\n}\\right)  \\operatorname{env}U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2}%\n,\\zeta\\sqrt{2\\gamma}}\\right)  }\\right\\}  ,\n\\end{array}\n\\label{eq97}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, uniformly for $0\\leq x\\leq1-\\delta_{0}$.\n\nFrom [26, \\S 5] we note that\n\\begin{equation}\nU\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},0}\\right)  =\\pi^{-1\/2}2^{\\left(\n{\\gamma\\alpha^{2}-1}\\right)  \/4}\\Gamma\\left(  {{\\frac{1}{4}}\\gamma\\alpha\n^{2}+{\\frac{1}{4}}}\\right)  \\sin\\left(  {{\\frac{1}{4}}\\gamma\\alpha^{2}%\n\\pi+{\\frac{1}{4}}\\pi}\\right)  , \\label{eq98}%\n\\end{equation}\nas well as\n\\begin{equation}\n{U}^{\\prime}\\left(  {-{\\tfrac{1}{2}}\\gamma\\alpha^{2},0}\\right)  =-\\pi\n^{-1\/2}2^{\\left(  {\\gamma\\alpha^{2}+1}\\right)  \/4}\\Gamma\\left(  {{\\frac{1}{4}%\n}\\gamma\\alpha^{2}+{\\frac{3}{4}}}\\right)  \\sin\\left(  {{\\frac{1}{4}}%\n\\gamma\\alpha^{2}\\pi+{\\frac{3}{4}}\\pi}\\right)  . \\label{eq99}%\n\\end{equation}\nThus, on referring to (\\ref{eq82}), we observe that the RHS of (\\ref{eq98}) is\nbounded away from zero for large $\\gamma$ when $m+n$ is even, and likewise for\nthe RHS of \\eqref{eq99} when $m+n$ is odd (see \\eqref{eq101} below).\n\n\n\n\n\n\nFor the case $\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $ odd,\nequivalently $m+n$ odd, we differentiate both sides of \\eqref{eq84} with\nrespect to $\\zeta$, and then set $x=\\zeta=0$. As a result, using \\eqref{eq62}\nand \\eqref{eq80}, along with the fact that $\\operatorname{Ps}_{n}^{m}\\left(\n{0,\\gamma^{2}}\\right)  =0$, we obtain\n\\begin{equation}\nd_{n}^{m}\\left(  \\gamma\\right)  =\\left(  {\\frac{\\alpha}{\\sigma}}\\right)\n^{1\/2}\\frac{\\operatorname{Ps}_{n}^{m}{}^{\\prime}\\left(  {0,\\gamma^{2}}\\right)\n}{\\partial w_{1}\\left(  {\\gamma,\\alpha,0}\\right)  \/\\partial\\zeta}.\n\\label{eq100}%\n\\end{equation}\nThus, again from \\eqref{eq95}, we conclude for $m+n$ odd, $m$ bounded and $n$\nsatisfying \\eqref{eq2}, that\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\dfrac\n{\\operatorname{Ps}_{n}^{m}{}^{\\prime}\\left(  {0,\\gamma^{2}}\\right)  }%\n{{U}^{\\prime}\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},0}\\right)  }\\left\\{\n{\\dfrac{\\alpha^{2}\\left(  {\\alpha^{2}-\\zeta^{2}}\\right)  }{4\\gamma^{2}%\n\\sigma^{2}\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(  {1-x^{2}}\\right)  }%\n}\\right\\}  ^{1\/4}\\\\\n\\times\\left\\{  {U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  +{O}\\left(  {\\gamma^{-2\/3}\\ln\\left(  \\gamma\\right)\n}\\right)  \\operatorname{env}U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2}%\n,\\zeta\\sqrt{2\\gamma}}\\right)  }\\right\\}  ,\n\\end{array}\n\\label{eq101}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, uniformly for $0\\leq x\\leq1-\\delta$. In this\n${U}^{\\prime}\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},0}\\right)  $ is given by\n(\\ref{eq99}).\n\nWe now show that the proportionality constants in (\\ref{eq97}) and\n(\\ref{eq101}) can be replaced by one that does not involve $\\operatorname{Ps}%\n_{n}^{m}\\left(  {0,\\gamma^{2}}\\right)  $ or $\\operatorname{Ps}_{n}^{m}%\n{}^{\\prime}\\left(  {0,\\gamma^{2}}\\right)  $. Specifically, from (\\ref{eq19}),\n(\\ref{eq61}), (\\ref{eq84}), (\\ref{eq95}) and (\\ref{eq96}) we have (for both\nthe even and odd cases) that\n\\begin{equation}\nd_{n}^{m}\\left(  \\gamma\\right)  \\sim\\left\\{  {\\frac{\\left(  {n+m}\\right)\n!}{\\left(  {2n+1}\\right)  \\left(  {n-m}\\right)  !p_{n}^{m}\\left(\n\\gamma\\right)  }}\\right\\}  ^{1\/2}, \\label{eq102}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, again with $m$ bounded and $n$ satisfying\n(\\ref{eq2}). Here\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\np_{n}^{m}\\left(  \\gamma\\right)  =\\left[  {\\int_{0}^{1-\\delta_{0}}{\\left\\{\n{\\dfrac{\\alpha^{2}-\\zeta^{2}}{\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(\n{1-x^{2}}\\right)  }}\\right\\}  ^{1\/2}U^{2}\\left(  {-{\\frac{1}{2}}\\gamma\n\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  dx}}\\right. \\\\\n\\left.  {+q_{n}^{m}\\left(  \\gamma\\right)  \\int_{1-\\delta_{0}}^{1}{\\left\\{\n{\\dfrac{\\left\\vert \\eta\\right\\vert }{\\left(  {1-x^{2}}\\right)  \\left(\n{x^{2}-\\sigma^{2}}\\right)  }}\\right\\}  ^{1\/2}I_{m}^{2}\\left(  {\\gamma\n\\left\\vert \\eta\\right\\vert ^{1\/2}}\\right)  dx}}\\right]  ,\n\\end{array}\n\\label{eq103}%\n\\end{equation}\nin which\n\\begin{equation}\nq_{n}^{m}\\left(  \\gamma\\right)  =\\left(  {\\frac{\\gamma\\alpha^{2}}{2e}}\\right)\n^{\\gamma\\alpha^{2}\/2}\\left(  {\\frac{\\pi^{2}}{2\\gamma}}\\right)  ^{1\/2}%\n\\exp\\left\\{  {-2\\gamma\\int_{\\sigma}^{1}{\\left\\{  {f\\left(  {\\sigma,t}\\right)\n}\\right\\}  ^{1\/2}dt}}\\right\\}  . \\label{eq104}%\n\\end{equation}\nNote also, from (\\ref{eq96}), that under the same conditions\n\\begin{equation}\nc_{n}^{m}\\left(  \\gamma\\right)  \\sim\\left\\{  {\\frac{\\left(  {n+m}\\right)\n!q_{n}^{m}\\left(  \\gamma\\right)  }{\\left(  {2n+1}\\right)  \\left(\n{n-m}\\right)  !p_{n}^{m}\\left(  \\gamma\\right)  }}\\right\\}  ^{1\/2}.\n\\label{eq105}%\n\\end{equation}\n\n\n\\section{Fixed $m$ and $n$: the angular case}\n\n\nFor fixed $m$\\textit{ and }$n$ we can simplify the results of the previous\nsection, by applying the theory of [10]. To this end we observe that\n(\\ref{eq26}) can be expressed in the form\n\\begin{equation}\n\\frac{d^{2}w}{dx^{2}}=\\left[  {\\frac{\\gamma^{2}x^{2}}{1-x^{2}}-\\frac{a\\gamma\n}{1-x^{2}}+\\frac{m^{2}-1}{\\left(  {1-x^{2}}\\right)  ^{2}}}\\right]  w,\n\\label{eq106}%\n\\end{equation}\nwhere\n\\begin{equation}\na=\\lambda\\gamma^{-1}+\\gamma=2\\left(  {n-m+{\\tfrac{1}{2}}}\\right)\n+{O}\\left(  {\\gamma^{-1}}\\right)  , \\label{eq107}%\n\\end{equation}\nthe ${O}\\left(  {\\gamma^{-1}}\\right)  $ term being valid for fixed $m$\nand $n$ and $\\gamma\\rightarrow\\infty$. In particular, $a$ is bounded.\n\nEquation (\\ref{eq106}) is characterised as having a pair of almost coalescent\nturning points near $x=0$. The appropriate Liouville transformation in this\ncase is given by\n\\begin{equation}\n\\frac{1}{2}\\rho^{2}=\\int_{0}^{x}{\\frac{t}{\\left(  {1-t^{2}}\\right)  ^{1\/2}}%\ndt}=1-\\left(  {1-x^{2}}\\right)  ^{1\/2}. \\label{eq108}%\n\\end{equation}\nNote $x=0$ corresponds to $\\rho=0$, and $x=1$ corresponds to $\\rho=\\sqrt{2}$.\nThen with\n\\begin{equation}\nW=\\frac{x^{1\/2}}{\\rho^{1\/2}\\left(  {1-x^{2}}\\right)  ^{1\/4}}w, \\label{eq109}%\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\frac{d^{2}W}{d\\rho^{2}}=\\left[  {\\gamma^{2}\\rho^{2}-\\gamma a+\\gamma\\zeta\n\\phi\\left(  \\rho\\right)  +\\chi\\left(  \\rho\\right)  }\\right]  W, \\label{eq110}%\n\\end{equation}\nwhere\n\\begin{equation}\n\\phi\\left(  \\rho\\right)  =-\\frac{a\\rho}{4-\\zeta^{2}}, \\label{eq111}%\n\\end{equation}\nand\n\\begin{equation}\n\\chi\\left(  \\rho\\right)  =\\frac{\\rho^{2}\\left(  {4m^{2}-1}\\right)  }{\\left(\n{2-\\rho^{2}}\\right)  ^{2}}+\\frac{7\\rho^{2}-40}{4\\left(  {4-\\rho^{2}}\\right)\n^{2}}+\\frac{4m^{2}}{\\left(  {4-\\rho^{2}}\\right)  }. \\label{eq112}%\n\\end{equation}\nWe remark that $\\chi\\left(  \\rho\\right)  ={O}\\left(  1\\right)  _{\\,}%\n$as $\\gamma\\rightarrow\\infty$, and this function is analytic at $\\rho=0$\n($x=0$), but is not analytic at $\\rho=\\sqrt{2}\\ $(${x=1}$).\n\nOur approximants are again the parabolic cylinder functions $U\\left(\n{-{\\frac{1}{2}}a,\\rho\\sqrt{2\\gamma}}\\right)  $ and $\\bar{{U}}\\left(\n{-{\\frac{1}{2}}a,\\rho\\sqrt{2\\gamma}}\\right)  $ (c.f. (\\ref{eq71}) and\n(\\ref{eq72})). In this form they are solutions of\n\\begin{equation}\n\\frac{d^{2}W}{d\\rho^{2}}=\\left[  {\\gamma^{2}\\rho^{2}-\\gamma a}\\right]  W.\n\\label{eq113}%\n\\end{equation}\nOn comparing this equation with (\\ref{eq110}) we note the extra\n\\textquotedblleft large\\textquotedblright\\ term $\\gamma\\zeta\\phi\\left(\n\\rho\\right)  $. On account of this discrepancy we perturb the independent\nvariable, thus taking as approximants\n\\begin{equation}\nU_{1}=\\left\\{  {1+\\gamma^{-1}{\\Phi}^{\\prime}\\left(  \\rho\\right)  }\\right\\}\n^{-1\/2}U\\left(  {-{\\tfrac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)  ,\n\\label{eq114}%\n\\end{equation}\nand\n\\begin{equation}\nU_{2}=\\left\\{  {1+\\gamma^{-1}{\\Phi}^{\\prime}\\left(  \\rho\\right)  }\\right\\}\n^{-1\/2}\\bar{{U}}\\left(  {-{\\tfrac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)\n, \\label{eq115}%\n\\end{equation}\nwhere\n\\begin{equation}\n\\hat{{\\rho}}=\\rho+\\gamma^{-1}\\Phi\\left(  \\rho\\right)  , \\label{eq116}%\n\\end{equation}\nin which\n\\begin{equation}\n\\Phi\\left(  \\rho\\right)  =\\frac{1}{2\\rho}\\int_{0}^{\\rho}{\\phi\\left(  v\\right)\ndv}=\\frac{a\\ln\\left(  {1-{\\frac{1}{4}}\\rho^{2}}\\right)  }{4\\rho}.\n\\label{eq117}%\n\\end{equation}\nIn [10] it is shown that $U_{j}$ satisfy the differential equation\n\\begin{equation}\n\\frac{d^{2}U}{d\\rho^{2}}=\\left\\{  {\\gamma^{2}\\rho^{2}-\\gamma a+\\gamma\\rho\n\\phi\\left(  \\rho\\right)  +g\\left(  {\\gamma,\\rho}\\right)  }\\right\\}  U,\n\\label{eq118}%\n\\end{equation}\nwhere $g\\left(  {\\gamma,\\rho}\\right)  ={O}\\left(  1\\right)  _{\\,}$as\n$\\gamma\\rightarrow\\infty$, uniformly for $\\rho\\in\\left[  {0,\\sqrt{2}-\\delta\n}\\right]  $. Thus (\\ref{eq118}) is the appropriate comparison equation to\n(\\ref{eq110}).\n\n\n\nFollowing [10] we then define\n\\begin{equation}\n\\hat{{w}}_{j}\\left(  {\\gamma,\\rho}\\right)  =U_{j}\\left(  {\\gamma,\\rho}\\right)\n+\\hat{{\\varepsilon}}_{j}\\left(  {\\gamma,\\rho}\\right)  \\quad\\left(\n{j=1,2}\\right)  , \\label{eq119}%\n\\end{equation}\nas exact solutions of (\\ref{eq110}). Explicit error bounds are furnished in\n[10], and from these it follows that\n\\begin{equation}\n\\hat{{\\varepsilon}}_{1}\\left(  {\\gamma,\\rho}\\right)  ={O}\\left(\n{\\gamma^{-1}\\ln\\left(  \\gamma\\right)  }\\right)  {\\operatorname{env}}U\\left(\n{-{\\tfrac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)  , \\label{eq120}%\n\\end{equation}\nuniformly for $0\\leq x\\leq1-\\delta_{0}$, and similarly for $\\hat{{\\varepsilon\n}}_{2}\\left(  {\\gamma,\\rho}\\right)  $.\n\nLet us assume that $\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $\n(and hence $m+n)$ is even. Similarly to (\\ref{eq84}) we write\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\rho^{1\/2}%\nx^{-1\/2}\\left(  {1-x^{2}}\\right)  ^{-1\/4}\\\\\n\\times\\left[  {\\hat{{d}}_{n}^{m}\\left(  \\gamma\\right)  \\hat{{w}}_{1}\\left(\n{\\gamma,\\rho}\\right)  +\\hat{{e}}_{n}^{m}\\left(  \\gamma\\right)  \\left\\{\n{\\hat{{w}}_{2}\\left(  {\\gamma,\\rho}\\right)  -\\hat{{w}}_{4}\\left(  {\\gamma\n,\\rho}\\right)  }\\right\\}  }\\right]  ,\n\\end{array}\n\\label{eq121}%\n\\end{equation}\nwhere $\\hat{{w}}_{4}\\left(  {\\gamma,\\rho}\\right)  $ is the solution (involving\n$\\bar{{U}})$ given by eq. (110) of [10]. By matching at $x=\\rho=0$ we find\n\\begin{equation}\n\\hat{{d}}_{n}^{m}\\left(  \\gamma\\right)  =\\frac{Ps_{n}^{m}\\left(  {0,\\gamma\n^{2}}\\right)  }{\\hat{{w}}_{1}\\left(  {\\gamma,0}\\right)  }. \\label{eq122}%\n\\end{equation}\nAnalogously to the proof of (\\ref{eq95}) it can be shown that\n\\begin{equation}\n\\hat{{e}}_{n}^{m}\\left(  \\gamma\\right)  \\left\\{  {\\hat{{w}}_{2}\\left(\n{\\gamma,\\rho}\\right)  -\\hat{{w}}_{4}\\left(  {\\gamma,\\rho}\\right)  }\\right\\}\n=o\\left(  1\\right)  \\hat{{A}}{\\operatorname{env}}U\\left(  {-{\\tfrac{1}{2}%\n}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)  , \\label{eq123}%\n\\end{equation}\nwhere $o\\left(  1\\right)  $ is exponentially small for $0\\leq x\\leq\n1-\\delta_{0}$ as $\\gamma\\rightarrow\\infty$. Consequently, we arrive at our\ndesired result\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\dfrac\n{\\operatorname{Ps}_{n}^{m}\\left(  {0,\\gamma^{2}}\\right)  }{U\\left(\n{-{\\frac{1}{2}}a,0}\\right)  }\\left(  {\\dfrac{\\rho}{x}}\\right)  ^{1\/2}\\left(\n{1-x^{2}}\\right)  ^{-1\/4}\\\\\n\\times\\left[  {U\\left(  {-{\\frac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)\n+{O}\\left(  {\\gamma^{-1}\\ln\\left(  \\gamma\\right)  }\\right)\n\\operatorname{env}U\\left(  {-{\\frac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}%\n}\\right)  }\\right]  ,\n\\end{array}\n\\label{eq124}%\n\\end{equation}\nas $\\gamma\\rightarrow\\infty$, uniformly for $0\\leq x\\leq1-\\delta_{0}$.\n\nFor the case $\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  $ being\nodd we likewise obtain, under the same conditions,\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\dfrac\n{\\operatorname{Ps}_{n}^{m}{}^{\\prime}\\left(  {0,\\gamma^{2}}\\right)  }%\n{{U}^{\\prime}\\left(  {-{\\frac{1}{2}}a,0}\\right)  }\\left(  {\\dfrac{\\rho\n}{2\\gamma x}}\\right)  ^{1\/2}\\left(  {1-x^{2}}\\right)  ^{-1\/4}\\\\\n\\times\\left[  {U\\left(  {-{\\frac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}}\\right)\n+{O}\\left(  {\\gamma^{-1}\\ln\\left(  \\gamma\\right)  }\\right)\n\\operatorname{env}U\\left(  {-{\\frac{1}{2}}a,\\hat{{\\rho}}\\sqrt{2\\gamma}%\n}\\right)  }\\right]  .\n\\end{array}\n\\label{eq125}%\n\\end{equation}\n\n\n\\section{Summary}\n\n\n\nFor reference we collect the principal results of the paper. All results are\nuniformly valid for $\\gamma\\to\\infty$, $m$ and $n$ integers, $m$ bounded, and\n$n$ satisfying $0\\le m\\le n\\le2\\pi^{-1}\\gamma\\left(  {1-\\delta} \\right)  $\nwhere $\\delta\\in\\left(  {0,1} \\right)  $ is fixed.\n\nWe define $\\sigma=\\sqrt{1+\\gamma^{-2}\\lambda_{n}^{m}\\left(  {\\gamma^{2}%\n}\\right)  }$ and assume $0\\leq\\sigma\\leq\\sigma_{0}<1$ for an arbitrary fixed\npositive $\\sigma_{0}$. We further define variables $\\xi=\\xi\\left(  x\\right)  $\nand $\\zeta=\\zeta\\left(  x\\right)  $ by\n\\begin{equation}\n\\xi=\\int_{1}^{x}{\\left(  {\\frac{t^{2}-\\sigma^{2}}{t^{2}-1}}\\right)  ^{1\/2}dt},\n\\label{eq126}%\n\\end{equation}\nand\n\\begin{equation}\n\\int_{\\alpha}^{\\zeta}{\\left\\vert {\\tau^{2}-\\alpha^{2}}\\right\\vert ^{1\/2}d\\tau\n}=\\int_{\\sigma}^{x}{\\left(  {\\frac{\\left\\vert {t^{2}-\\sigma^{2}}\\right\\vert\n}{1-t^{2}}}\\right)  ^{1\/2}dt}, \\label{eq127}%\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha=2\\left\\{  {\\frac{1}{\\pi}\\int_{0}^{\\sigma}{\\left(  {\\frac{\\sigma\n^{2}-t^{2}}{1-t^{2}}}\\right)  ^{1\/2}dt}}\\right\\}  ^{1\/2}. \\label{eq128}%\n\\end{equation}\nThen, using the definition above for $\\sigma$, a uniform asymptotic\nrelationship between $\\lambda_{n}^{m}\\left(  {\\gamma^{2}}\\right)  $ and the\nparameters $m$, $n$ and $\\gamma$ is given implicitly by the relation\n\\begin{equation}\n\\gamma\\int_{0}^{\\sigma}{\\left(  {\\frac{\\sigma^{2}-t^{2}}{1-t^{2}}}\\right)\n^{1\/2}dt}=\\frac{1}{2}\\left(  {n-m+\\frac{1}{2}}\\right)  \\pi+{O}\\left(\n{\\frac{1}{\\gamma}}\\right)  . \\label{eq129}%\n\\end{equation}\n\n\nThe following approximation holds for the radial PSWF\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\nPs_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\left\\{  {\\dfrac{\\left(\n{n+m}\\right)  !q_{n}^{m}\\left(  \\gamma\\right)  }{\\left(  {2n+1}\\right)\n\\left(  {n-m}\\right)  !p_{n}^{m}\\left(  \\gamma\\right)  }}\\right\\}\n^{1\/2}\\left\\{  {\\left(  {x^{2}-1}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)\n}\\right\\}  ^{-1\/4}\\\\\n\\times\\xi^{1\/2}\\left[  {J_{m}\\left(  {\\gamma\\xi}\\right)  +{O}\\left(\n{\\gamma^{-1}}\\right)  \\operatorname{env}J_{m}\\left(  {\\gamma\\xi}\\right)\n}\\right]  ,\n\\end{array}\n\\label{eq130}%\n\\end{equation}\nthis being uniformly valid for $1<x<\\infty$. Here $J_{m}$ is the Bessel\nfunction of the first kind, ${\\operatorname{env}}J_{m}$ is defined by [27,\n\\S 2.8(iv)], and the constants $p_{n}^{m}\\left(  \\gamma\\right)  $ and\n$q_{n}^{m}\\left(  \\gamma\\right)  $ are given by\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\np_{n}^{m}\\left(  \\gamma\\right)  =\\left[  {\\int_{0}^{1-\\delta_{0}}{\\left\\{\n{\\dfrac{\\alpha^{2}-\\zeta^{2}}{\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(\n{1-x^{2}}\\right)  }}\\right\\}  ^{1\/2}U^{2}\\left(  {-{\\frac{1}{2}}\\gamma\n\\alpha^{2},\\zeta\\sqrt{2\\gamma}}\\right)  dx}}\\right. \\\\\n\\left.  {+q_{n}^{m}\\left(  \\gamma\\right)  \\int_{1-\\delta_{0}}^{1}{\\left\\{\n{\\dfrac{1}{\\left(  {1-x^{2}}\\right)  \\left(  {x^{2}-\\sigma^{2}}\\right)  }%\n}\\right\\}  ^{1\/2}\\left\\vert \\xi\\right\\vert I_{m}^{2}\\left(  {\\gamma\\left\\vert\n\\xi\\right\\vert }\\right)  dx}}\\right]  ,\n\\end{array}\n\\label{eq131}%\n\\end{equation}\nand\n\\begin{equation}\nq_{n}^{m}\\left(  \\gamma\\right)  =\\left(  {\\frac{\\gamma\\alpha^{2}}{2e}}\\right)\n^{\\gamma\\alpha^{2}\/2}\\left(  {\\frac{\\pi^{2}}{2\\gamma}}\\right)  ^{1\/2}%\n\\exp\\left\\{  {-2\\gamma\\int_{\\sigma}^{1}{\\left(  {\\frac{t^{2}-\\sigma^{2}%\n}{1-t^{2}}}\\right)  ^{1\/2}dt}}\\right\\}  . \\label{eq132}%\n\\end{equation}\nIn (\\ref{eq131}) $\\delta_{0}\\in\\left(  {0,1-\\sigma_{0}}\\right)  $ is\narbitrarily chosen, $I_{m}$ is the modified Bessel function of the first kind,\nand $U$ is the parabolic cylinder function (see [26, \\S 5]).\n\nIn terms of the modified Bessel function, we have for the radial PSWF\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\left\\{\n{\\dfrac{\\left(  {n+m}\\right)  !q_{n}^{m}\\left(  \\gamma\\right)  }{\\left(\n{2n+1}\\right)  \\left(  {n-m}\\right)  !p_{n}^{m}\\left(  \\gamma\\right)  }%\n}\\right\\}  ^{1\/2}\\left\\{  {\\left(  {1-x^{2}}\\right)  \\left(  {x^{2}-\\sigma\n^{2}}\\right)  }\\right\\}  ^{-1\/4}\\\\\n\\times\\left\\vert \\xi\\right\\vert ^{1\/2}I_{m}\\left(  {\\gamma\\left\\vert\n\\xi\\right\\vert }\\right)  \\left\\{  {1+{O}\\left(  {\\gamma^{-1}}\\right)\n}\\right\\}  ,\n\\end{array}\n\\label{eq133}%\n\\end{equation}\nuniformly for $1-\\delta_{0}\\leq x<1$.\n\nFinally, in terms of the parabolic cylinder function, the asymptotic\napproximation\n\\begin{equation}%\n\\begin{array}\n[c]{l}%\n\\operatorname{Ps}_{n}^{m}\\left(  {x,\\gamma^{2}}\\right)  =\\left\\{\n{\\dfrac{\\left(  {n+m}\\right)  !}{\\left(  {2n+1}\\right)  \\left(  {n-m}\\right)\n!p_{n}^{m}\\left(  \\gamma\\right)  }}\\right\\}  ^{1\/2}\\left\\{  {\\dfrac{\\alpha\n^{2}-\\zeta^{2}}{\\left(  {\\sigma^{2}-x^{2}}\\right)  \\left(  {1-x^{2}}\\right)\n}}\\right\\}  ^{1\/4}\\\\\n\\times\\left\\{  {U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2},\\zeta\\sqrt{2\\gamma}%\n}\\right)  +{O}\\left(  {\\gamma^{-2\/3}\\ln\\left(  \\gamma\\right)\n}\\right)  \\operatorname{env}U\\left(  {-{\\frac{1}{2}}\\gamma\\alpha^{2}%\n,\\zeta\\sqrt{2\\gamma}}\\right)  }\\right\\}  ,\n\\end{array}\n\\label{eq134}%\n\\end{equation}\nholds uniformly for $0\\leq x\\leq1-\\delta_{0}$, where ${\\operatorname{env}}U$\nis defined by [27, eq. 14.15.23].\n\n\n\\section*{Acknowledgement}\n        I thank the referee for a number helpful suggestions.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{Intro}\n\n\\vskip .1cm\n\nIn a previous paper~\\cite{Aziz} we focused on identities relating \nthe {\\em same} $\\, _2F_1$ hypergeometric function with \n{\\em two different\\footnote[2]{Beyond the \n$\\, x \\, \\rightarrow \\, 1-x, \\, 1\/x, ...$ known pullback symmetries of \nhypergeometric functions. The correspondence between the two pullbacks \nmust be an {\\em infinite order} rational or algebraic \ntransformation~\\cite{Aziz,Hindawi}.} algebraic pullback transformations}. \nThese identities correspond to {\\em modular forms}, \nthe algebraic transformations being solutions of a  \n(non-linear) {\\em differentially algebraic}~\\cite{Selected,IsTheFull} \n{\\em Schwarzian equation}, that also emerged in a paper \nby Casale on Galoisian envelopes~\\cite{Casale,Paul}. This covariance \nsymmetry of $\\, _2F_1$ hypergeometric functions could be regarded as\none of the simplest illustrations of the concept of symmetries \n (of the renormalization group type~\\cite{Hindawi,broglie}) in physics\nor enumerative combinatorics, \na {\\em univariate function being covariant (automorphic) with respect \nto an infinite set of rational or algebraic transformations}.\nThis paper~\\cite{Aziz} was essentially focused on $\\, _nF_{n-1}$ \n{\\em hypergeometric functions} and {\\em modular forms} \nactually represented as $\\, _2F_1$ hypergeometric \nfunction with two different algebraic pullback transformations\n({\\em modular correspondences}~\\cite{Aziz,Eichler}). \n\nThe applications of this Schwarzian equation~\\cite{Aziz} \nknown to be associated to a quite large mathematical \nframework\\footnote[1]{In Casale's paper~\\cite{Casale,Paul}\nthe Schwarzian equation is associated with meromorphic functions \ninstead of the rational functions of \nour paper~\\cite{Aziz}. See also~\\cite{Casale2,Casale3,Casale4}.} \n(Malgrange's pseudogroup, Galois \ngroupoid~\\cite{Casale2,Casale3,Casale4,Casale5,Casale6,Casale7,Malgrange}), \nextend well beyond hypergeometric functions in physics. We have \nseen, for instance in~\\cite{Aziz}, an example of identity relating \nthe same Heun function with two different \npullbacks\\footnote[9]{This Heun function being not, in general, \nreducible to a $\\, _2F_1$ pullbacked hypergeometric function~\\cite{maier-05}.}. This \nHeun example~\\cite{Aziz} could suggest that \nsuch Schwarzian differential equations emerge in physics with holonomic \nfunctions having a narrow set of singularities (three for hypergeometric \nfunctions, four for Heun functions, ...) like the Heun \nexample in~\\cite{Aziz}. \nGoing further we show, in this paper, that such \n{\\em differentially algebraic}~\\cite{Selected,IsTheFull} \nSchwarzian equations do emerge in a much more general holonomic framework. \n\nWe will show in section \\ref{order-two} \nthat the  covariance symmetry condition \nof {\\em general} order-two linear \ndifferential operators with polynomial coefficients automatically \nyields this Schwarzian differential equation. We will then show \nin sections \\ref{order-three} and  \\ref{order-four}\nthat the  covariance symmetry condition imposed on linear differential \noperators having  order three and order four \nwith respective orthogonal and symplectic  differential Galois groups,\nyield Schwarzian differential equations like the one examined in~\\cite{Aziz}.\nWhen their respective symmetric and exterior powers are of order {\\em five}\n(instead of six),  \none finds that these  order-three and order-four operators \n reduce to symmetric square and symmetric cube of an underlying \norder-two operator.\nIn section \\ref{order-N} we show that the Schwarzian condition \ncan be derived for linear differential operators \nof arbitrary order $\\, N$. The reduction of the solutions of this\nSchwarzian differential equation to only \n{\\em modular correspondences}~\\cite{Eichler} was an open question in~\\cite{Aziz}:\nin section \\ref{CalabiSchwResults} a necessary condition to have such\nmodular correspondences~\\cite{Eichler} is derived. \nIn section \\ref{pullbackhomomorphism} generalizations \nof modular forms provide  examples of   \n{\\em pullback invariance of an operator, up to operator homomorphism}.\n This invariance  should be important to describing the symmetries of \nlinear differential operators and thus, is of relevance to physics. \nFinally  in section \\ref{CalabiSchwatext}, we consider the more general \nproblem already addressed in~\\cite{Duco} where Schwarzian \ndifferential equations also occurred, of the \nequivalence of {\\em two different} order-four linear differential \n{\\em Calabi-Yau operators}~\\cite{Tables}\nup to pullbacks and conjugation, possibly yielding the \n{\\em same Yukawa couplings}~\\cite{Duco}, and \nwe will generalize it to linear differential operators of arbitrary orders. \n\n\\vskip .2cm \n\n\\section{Beyond hypergeometric and Heun functions: order-two linear differential operators} \n\\label{order-two}\n\nWe will show here that non-linear ODEs involving Schwarzian derivatives \n(cf. equation (\\ref{condition1}) below), that we will \ncall ``Schwarzian ODEs''\\footnote[2]{See~\\cite{Aziz,What} for \na definition. See also~\\cite{Schwarzian2,Schwarzian}.}, obtained \nin~\\cite{Aziz} for hypergeometric and Heun functions~\\cite{Vidunas,Belyi3}, \ncan be generalized to arbitrary globally nilpotent~\\cite{bo-bo-ha-ma-we-ze-09} \nlinear differential operators having an arbitrary numbers of \nsingularities (as opposed to three and four singularities\nfor hypergeometric and Heun functions).\n\nLet us consider a linear differential operator of order two\n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad  \n\\label{L_2}\nL_2 \\, \\, = \\, \\, \\, \\, \nD_x^2  \\, \\,\\,  + \\, p(x) \\cdot \\, D_x \\,\\,  + \\, \\, q(x), \n\\quad \\quad \\quad \\hbox{where:} \\quad  \\quad \\quad  \\quad\nD_x \\, = \\, \\, {{ d} \\over {dx}}, \n\\end{eqnarray}\nand let us also introduce two other linear differential operators \nof order two: the operator \n$\\, \\, L_2^{(c)} \\, = \\, \\, 1\/v(x) \\cdot \\, L_2 \\cdot \\, v(x)\\, $\nbeing the  conjugate of (\\ref{L_2}) by a function $\\, v(x)$, and \nthe pullbacked operator $\\, L_2^{(p)}$ which amounts to changing \n$\\, x \\, \\rightarrow \\, \\, y(x) \\, $ in (\\ref{L_2}), \nthe head coefficient being normalized\\footnote[1]{Throughout \nthe paper we consider, for clarity and simplicity, this \nnormalized form for the linear differential operators. The\n ``true'' pullbacked operator which amounts to  changing \n$\\, x \\, \\rightarrow \\, \\, y(x)$ (see the command ``dchange''\n in PDEtools in Maple) is in fact $\\, 1\/y'(x)^2 \\cdot \\,L_2^{(p)}$\nwhere $\\, L_2^{(p)}$ is given by (\\ref{L2p}).} to $\\,1$. These two \nlinear differential operators read respectively:\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{L2c} \\, \nL_2^{(c)} \\,  \\, \\, = \\, \\, \\, \\, \\, D_x^2  \\,\\, \\,  + \\, \n\\Bigl(p(x) \\, + \\, 2 \\cdot \\, {{v'(x)} \\over {v(x)}} \\Bigr) \\cdot \\, D_x \n\\, \\, \\, + \\, \\, q(x) \n \\, \\, + \\, p(x) \\cdot \\,  {{v'(x)} \\over {v(x)}} \n\\, \\,  + \\, {{v\"(x)} \\over {v(x)}},  \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{L2cwhere}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad  \\quad \\quad  \\quad \nv'(x) \\, \\, = \\, \\, \\, {{d v(x)} \\over {dx}},\n \\quad  \\quad \\quad \\quad \n v\"(x) \\, \\, = \\, \\, \\, {{d^2 v(x)} \\over {dx^2}},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \\quad \n\\label{L2p} \\, \\, \\, \\, \nL_2^{(p)} \\, \\, = \\, \\, \\, \\, \\, \\, D_x^2  \\, \\, \\, \n+ \\,  \\Bigl( p(y(x)) \\, \\cdot \\, y'(x) \\, - \\, {{y\"(x)} \\over {y'(x)}}  \\Bigr)\n  \\cdot \\, D_x \n\\,\\, \\,  + \\, \\, q(y(x)) \\cdot \\, y'(x)^2, \n\\end{eqnarray}\nwhere:\n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \\quad \\quad  \\quad \n\\label{L2pwhere}\ny'(x) \\, \\, = \\, \\, \\, {{d y(x)} \\over {dx}},\n \\quad  \\quad \\quad \\quad \n y\"(x) \\, \\, = \\, \\, \\, {{d^2 y(x)} \\over {dx^2}}.\n\\end{eqnarray}\nThe identification of these two  linear differential operators  \n$\\, L_2^{(c)} \\, = \\, \\, L_2^{(p)}$ gives two conditions:\n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\quad \n\\label{firstcond}\np(x) \\, \\, + \\, 2 \\cdot \\, {{v'(x)} \\over {v(x)}} \n \\, \\, \\, = \\, \\, \\,  \\,\\,  p(y(x)) \\, \\cdot \\, y'(x)\n \\, \\,\\, - \\, {{y\"(x)} \\over {y'(x)}} , \n\\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\quad \n\\label{seccond}\n q(x) \\, \\, + \\, p(x) \\cdot \\,  {{v'(x)} \\over {v(x)}} \n\\,  \\,+ \\, {{v\"(x)} \\over {v(x)}} \\, \\,  \\,= \\, \\,\\,  \\,  \\,\n q(y(x)) \\cdot \\, y'(x)^2.\n\\end{eqnarray}\nSince \n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\quad  \\quad  \\quad \n\\label{since}\n {{v\"(x)} \\over {v(x)}} \\, \\,  \\, = \\, \\,\\,  \\, \n{{d} \\over {dx}} \\Bigl({{v'(x)} \\over {v(x)}}\\Bigr)\n  \\, \\,  + \\,  \\Bigl({{v'(x)} \\over {v(x)}}\\Bigr)^2,  \n\\end{eqnarray}\none can eliminate the log-derivative $\\, v'(x)\/v(x)\\, $ between  (\\ref{firstcond}) \nand (\\ref{seccond}), and obtain the {\\em Schwarzian condition}\npreviously given in~\\cite{Aziz}\n\\begin{eqnarray}\n\\label{condition1}  \n\\hspace{-0.95in}&& \\quad   \\quad \\quad   \\quad \\quad  \\quad \\quad  \\quad \n W(x) \n\\, \\, \\,\\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\, \\, \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{wherecond}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad \\quad  \\quad \nW(x)  \\, \\, = \\, \\,  \\, \\, \\, {{ d p(x)} \\over { dx}} \n \\, \\,\\,  \\,  + \\, \\, \n   {{p(x)^2} \\over {2 }} \\, \\,  \\, \\,  -2 \\cdot \\, q(x),\n\\end{eqnarray}\nand where $\\,\\{ y(x), \\, x\\}\\, $ denotes the \n{\\em Schwarzian derivative}~\\cite{What}:\n\\begin{eqnarray}\n\\label{Schwa}\n\\hspace{-0.95in}&& \\quad \\, \\,     \\, \n\\{ y(x), \\, x\\}    \\, \\, = \\, \\,  \\,  \\,  \\,\n{{y'''(x) } \\over{ y'(x)}} \n \\,  \\,  - \\, \\, {{3} \\over {2}}\n \\cdot \\, \\Bigl({{y''(x)} \\over{y'(x)}}\\Bigr)^2\n \\, \\, = \\, \\,  \\,  \\,  \\,\n{{ d } \\over { dx  }} \\Bigl( {{y''(x) } \\over{ y'(x)}}   \\Bigr) \n\\, \\, - {{1} \\over {2}} \\cdot \\, \\Bigl( {{y''(x) } \\over{ y'(x)}}   \\Bigr)^2, \n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\, \\,     \\,  \\hbox{ where:}  \n\\, \\,   \\quad  \\quad \\quad \ny'''(x)   \\, \\, = \\, \\,  \\, {{d^3 y(x)} \\over{ dx^3}},  \\quad  \\,  \\,  \\,   \\,\ny''(x)   \\, \\, = \\, \\,  \\, {{d^2 y(x)} \\over{ dx^2}}, \\quad  \\,  \\,  \\,    \\,\n   y'(x)  \\, \\, = \\, \\,  \\, {{d y(x)} \\over{ dx}}.\n\\nonumber \n\\end{eqnarray}\nUnlike in~\\cite{Aziz}, the \n{\\em number of singularities of the second order operator} (\\ref{L_2}) \n{\\em  is arbitrary}: it does not need to be three or four like in the hypergeometric \nor Heun examples in~\\cite{Aziz}. The second order linear differential \noperator $\\, L_2$ is a {\\em general} order-two linear differential \noperator with polynomial coefficients. Introducing $\\, w(x)$ the wronskian of $\\, L_2$ \n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad  \\quad \n\\label{globnilp}\np(x) \\, \\,  = \\, \\,  \\,  \\,  -\\,{{w'(x)} \\over {w(x)}} \n\\quad \\quad  \\quad  \\,  \\, \\hbox{where:}  \\quad \\quad \\quad \\quad \n w'(x) \\, \\, = \\, \\,  \\, {{ d w(x)} \\over {dx}}, \n\\end{eqnarray}\nwe see that the LHS and  RHS of the first condition (\\ref{firstcond})\nare both log-derivatives. Thus one can immediately integrate  \nthe first condition (\\ref{firstcond}) and get (up to a multiplicative \nfactor $\\, \\mu$) the conjugation function\n$\\, v(x)$ in terms of\nthe wronskian $\\, w(x)$ and the pullback function $\\, y(x)$: \n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad   \\quad \\quad  \\quad \\quad \n\\label{integrate}\nv(x) \\, \\,  = \\, \\,  \\,  \\,  \\, \n\\mu \\cdot \\, \\Bigl({{w(x)} \\over {w(y(x)) \\cdot y'(x)}}\\Bigr)^{1\/2}.\n\\end{eqnarray}\n\n\\vskip .1cm\n\n{\\bf Remark 1:} When the wronskian $\\, w(x)$ is an $\\, N$-th root \nof a rational function, the exact expression (\\ref{integrate}) \nfor the conjugation function $\\, v(x)$, becomes an algebraic function \nwhen $\\, y(x)$ {\\em is an algebraic function}. This is actually \nthe case when the order-two linear \ndifferential operator  $\\, L_2$ is \n{\\em globally nilpotent}~\\cite{bo-bo-ha-ma-we-ze-09}.\nIn this case the linear differential operator is simply conjugated to its adjoint \nthrough its wronskian $\\, w(x)$ which is a $\\, N$-th root \nof a rational function:\n\\begin{eqnarray}\n\\label{simplyconj}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \n \\quad  \\quad  \\quad  \\quad \\, \\, \\,   \\,  \\,    \\, \nL_2 \\cdot \\, w(x) \\, \\, \\, = \\,  \\, \\,\\,  \\,   w(x) \\cdot \\, adjoint(L_2).\n\\end{eqnarray}\n\n\\vskip .1cm\n\n{\\bf Remark 2:} If the linear differential operator is {\\em not} \nglobally nilpotent~\\cite{bo-bo-ha-ma-we-ze-09}\nthe wronskian is {\\em not} necessarily an algebraic function. Introducing \n$\\, L_v(x)$, the log-derivative of the \nconjugation function $\\, v(x)$, one can rewrite the two \nconditions (\\ref{firstcond}) and (\\ref{seccond}) as:\n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad \\quad\n\\label{firstcondbis}\np(x) \\, \\, + \\, 2 \\cdot \\, L_v(x)\n \\, \\, \\, = \\, \\, \\,  \\,\\,  p(y(x)) \\, \\cdot \\, y'(x)\n \\, \\,\\, - \\, {{y\"(x)} \\over {y'(x)}} , \n\\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad\n\\label{seccondbis}\n q(x) \\, \\, + \\, p(x) \\cdot \\,  L_v(x)\n\\,  \\,+ \\, {{d  L_v(x)} \\over {dx}} \\, + L_v(x)^2 \n\\,  \\,= \\, \\,\\,  \\,  \\,\n q(y(x)) \\cdot \\, y'(x)^2.\n\\end{eqnarray}\nThe elimination of  $\\, L_v(x)$ in (\\ref{firstcondbis}) and (\\ref{seccondbis})\ngives the Schwarzian condition (\\ref{condition1}) \nwith (\\ref{wherecond}), however the conjugation function $\\, v(x)$ \n{\\em is no longer an algebraic function when $\\, y(x)$ is an algebraic function}\n(see (\\ref{integrate})): it is a transcendental function, and we certainly \nmove away from a modular correspondence~\\cite{Aziz,Eichler} \nframework\\footnote[1]{For modular correspondences  see also the concept \nof modular equations~\\cite{Morain,Yi,Chan,Yi2}.}.  \n\n\\vskip .2cm \n\\vskip .2cm \n\n\\section{Order-three linear differential operators} \n\\label{order-three}\n\n\\subsection{General order-three linear differential operators.\\\\} \n\\label{order-threegener}\n\n\\vskip .1cm \n\nConsidering an  {\\em irreducible} order-three linear differential operator\n\\begin{eqnarray}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad \\quad  \\quad  \\quad \n\\label{L_3}\nL_3 \\, \\, \\,  = \\, \\, \\,  \\, \n D_x^3  \\,  \\,\\,  + \\, p(x) \\cdot \\, D_x^2 \n\\, \\,  \\,  + \\, \\, q(x)\\cdot \\, D_x \\, \\,  + \\,  \\, r(x),\n\\end{eqnarray}\nwe introduce two other  linear differential operators of order three \ndefined as previously in section \\ref{order-two}:  \nthe operator $\\, L_3^{(c)}$ conjugated of (\\ref{L_3}) by a function $\\, v(x)$,\nnamely $\\, L_3^{(c)} \\, = \\, \\, 1\/v(x) \\cdot \\, L_3 \\cdot \\, v(x)$, and the \npullbacked\\footnote[1]{The $\\, D_x^3$ coefficient is normalized to $\\, 1$.  } \noperator $\\, L_3^{(p)}$ which amounts to changing \n$\\, x \\, \\rightarrow \\, \\, y(x)\\, $ in $\\, L_3$. These two linear differential \noperators read respectively\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{L3c} \\, \\,  \\quad  \\quad  \\quad  \\quad  \\quad \nL_3^{(c)} \\, \\, = \\, \\, \\, \\, \\, \nD_x^3  \\, \\, + \\, \n\\Bigl(p(x) \\, + \\, 3 \\cdot \\, {{v'(x)} \\over {v(x)}} \\Bigr) \\cdot \\, D_x^2 \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad \n\\, \\, \\, \\, \\, + \\, \\, \\Bigl(q(x) \\, + \\, 2 \\cdot \\,  p(x) \\cdot \\,  {{v'(x)} \\over {v(x)}} \n\\, + \\, 3 \\cdot \\, {{v\"(x)} \\over {v(x)}}\\Bigr) \\cdot  \\, D_x\n \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad \n\\, \\, \\,  + \\, \\,  r(x) \\, \\, + \\, q(x) \\cdot \\,  {{v'(x)} \\over {v(x)}} \n  \\, \\, + \\,  p(x) \\cdot \\,  {{v\"(x)} \\over {v(x)}} \\,  \\, + \\,  {{v^{(3)}(x)} \\over {v(x)}},\n  \\nonumber\n\\end{eqnarray}\nand: \n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{L3P} \\, \\, \\quad  \\quad  \\,  \\, \nL_3^{(p)} \\, \\, = \\, \\, \\, \\, \\, D_x^3  \\,\\, \\,   + \\, \n\\Bigl(p(y(x)) \\cdot \\, y'(x) \\, -\\,  3  \\, {{y\"(x)} \\over {y'(x)}} \\Bigr) \\cdot \\, D_x^2 \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad  \\quad   \n\\, \\, \\, + \\, \\, \\Bigl(  q(y(x)) \\cdot \\, y'(x)^2\\,  \\, -p(y(x)) \\cdot \\, y\"(x)\n\\, \\,\\,  -\\, {{y^{(3)}(x) } \\over {y'(x) }} \n \\,\\,  +3 \\cdot \\, \\Bigl({{y\"(x) } \\over {y'(x) }}  \\Bigr)^2  \\Bigr) \\cdot \\, D_x\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad  \\quad  \\quad \\quad \\quad  \\quad  \\quad   \n\\, + \\, r(y(x)) \\cdot \\, y'(x)^3.\n\\end{eqnarray}\n\nThe equality of these two order-three linear differential operators gives \nthree conditions $\\, {\\cal C}_n$, \nwith $\\, n= \\, 0, \\, 1, \\, 2$, corresponding, respectively, to the identification \nof the $\\, D_x^n$ coefficients of $\\, L_3^{(p)}$ and $\\, L_3^{(c)}$. Introducing \nthe wronskian $\\, w(x)$ of $\\, L_3$,  the LHS and RHS of condition \n$\\, {\\cal C}_2$ being, again, log-derivatives,\none can easily integrate  condition $\\, {\\cal C}_2$ and get the exact expression \nof the conjugation function $\\, v(x)$ in terms of the wronskian of $ \\, L_3$\nand of the pullback $\\, y(x)$:\n\\begin{eqnarray}\n\\label{v3} \n\\hspace{-0.95in}&&  \n \\, \\, \\quad  \\quad  \\quad  \\quad  \\quad  \\quad  \\quad \nv(x) \\, \\,  = \\, \\,  \\,  \\,  \\, \n\\mu \\cdot \\, \\Bigl( {{w(x)} \\over {w(y(x)) \\cdot \\, y'(x)^3 }} \\Bigr)^{1\/3}.\n\\end{eqnarray}\nSimilarly the elimination of the log-derivative $\\, v'(x)\/v(x)$ between condition \n$\\, {\\cal C}_2$ and condition $\\, {\\cal C}_1$ yields the Schwarzian condition\n\\begin{eqnarray}\n\\label{condition3}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad  \\quad  \\quad \n W(x) \n\\, \\, \\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\,  \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere this time $\\, W(x)$ reads:\n \\begin{eqnarray}\n \\label{wherecond3bis}\n \\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad  \\quad  \\,\\, \\, \\, \n W(x)  \\, \\, \\, = \\, \\,  \\, \\, \\,\n {{1} \\over {2}} \\cdot \\, {{d p(x)} \\over {dx }}\n \\,\\,   \\,+{{p(x)^2} \\over {6}} \\, \\,\\,  -{{q(x)} \\over {2}}.\n \\end{eqnarray} \n\n\\subsection{Symmetric Calabi-Yau condition.\\\\} \n\\label{SymCalab}\n\n\\vskip .1cm \n\nLet us consider the condition corresponding to imposing  \nthe {\\em symmetric square} of $\\, L_3$ to be of order {\\em five} \ninstead of the generic order six. This \n(``symmetric'' Calabi-Yau~\\cite{Diffalg}) \ncondition reads:\n\\begin{eqnarray}\n\\label{CalabiSymm}\n\\hspace{-0.95in}&& \\quad \\quad \\quad  \\quad  \nr(x) \\, \\, = \\, \\, \\,  \\, -{{2} \\over {27}} \\cdot \\, p(x)^3 \\, \n\\, \\, + \\, {{1} \\over {3}} \\cdot \\, p(x) \\cdot \\, q(x) \\,  \\, \\, \\,\n -{{1} \\over {3}} \\cdot \\,  p(x) \\cdot \\, {{ d p(x)} \\over { dx}}\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad   \\quad \\quad \\quad \\quad   \\quad  \\quad \n\\, \\, + {{1} \\over {2}} \\cdot \\, {{ d q(x)} \\over { dx}}\n \\, \\,\\,  -  {{1} \\over {6}} \\cdot \\,  {{ d^2 p(x)} \\over { dx^2}}.\n\\end{eqnarray}\nFor a  globally nilpotent~\\cite{bo-bo-ha-ma-we-ze-09} linear differential operator, \nthis  (symmetric Calabi-Yau) condition (\\ref{CalabiSymm}) together \nwith (\\ref{globnilp}) yields an order-three linear differential operator\n (\\ref{L_3}) simply conjugated to its adjoint:\n\\begin{eqnarray}\n\\label{simplyconj3}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad  \\quad \\quad \n \\quad \\, \\, \\, \\, \\,    \\,  \\,    \\, \nL_3 \\cdot \\, w(x)^{2\/3} \\, \\, \\, = \\,  \\, \\,\\,  \\,   w(x)^{2\/3} \\cdot \\, adjoint(L_3),\n\\end{eqnarray}\nwhere the wronskian $\\, w(x)$ is a $\\, N$-th root of a rational function.\n\nAgain for a  globally nilpotent~\\cite{bo-bo-ha-ma-we-ze-09}\n linear differential operator, the \nexact expression (\\ref{v3}) for the conjugation function $\\, v(x)$, \nbecomes an algebraic function {\\em when $\\, y(x)$ is an algebraic function}.\n\nThe symmetric square of an order-two linear differential operator \n$\\, L_2 \\, = \\, D_x^2 \\, + \\, A(x) \\cdot \\, D_x \\, + \\, B(x) \\, $ is an order-three\n linear differential operator (\\ref{L_3}) with the following coefficients:\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{param3} \\quad  \\quad   \\quad  \np(x) \\, \\, = \\, \\,\\,  \n3  \\cdot \\, A(x), \\quad  \\quad  \\, \\,   \\,   \nq(x) \\, \\, = \\, \\,\\, \n2 \\cdot \\, A(x)^2  \\, \\, + 4 \\cdot \\,B(x)  \\, \\, + {{ d A(x)} \\over { dx}}, \n \\\\\n\\label{param32}\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad  \\quad \\quad \\quad  \\quad  \\quad  \nr(x) \\, \\, = \\, \\,\\, \\,  \n4\\cdot \\, B(x) \\cdot \\, A(x) \n\\, \\, \\,  + 2  \\cdot \\, {{ d B(x)} \\over { dx}}.\n\\end{eqnarray}\nThese coefficients (\\ref{param3}), (\\ref{param32})\n{\\em automatically verify the (symmetric Calabi-Yau) condition}\n(\\ref{CalabiSymm}): the symmetric square of a symmetric square \nof  an order-two  linear differential operator is of order {\\em five}\ninstead of the generic order six.  Conversely, the (symmetric Calabi-Yau) \ncondition (\\ref{CalabiSymm}) can be parametrized\\footnote[1]{Note that \nrewriting the exact expression of $\\, W(x)$ given by (\\ref{wherecond3bis})\nin terms of $\\, A(x)$ and $\\, B(x)$ using  (\\ref{param3})\none recovers (\\ref{wherecond}), \n$\\, p(x)$ and $\\, q(x)$ in (\\ref{wherecond}) being now \n$\\, A(x)$ and $\\, B(x)$.} \nby (\\ref{param3}) and \n(\\ref{param32}) and amounts to imposing the order-three linear differential \noperator  (\\ref{L_3}) to be exactly the symmetric square of an order-two operator.\n\n\\vskip .2cm \n\n\\vskip .2cm \n\nThus our  calculations show that the pullback-compatibility \nof an order-three linear differential operator is equivalent to saying that this  \norder-three operator {\\em reduces to} (the symmetric square of) \n an underlying {\\em order-two linear differential operator}. The \nSchwarzian condition (\\ref{condition3}) \nwith $\\, W(x)$ given by (\\ref{wherecond3bis}), \nis {\\em thus inherited from the Schwarzian condition} \n(\\ref{condition1}) {\\em of the underlying order-two linear differential operator}.\n\n\\vskip .2cm \n\n\\section{Order-four linear differential operators} \n\\label{order-four}\n\nConsider the {\\em irreducible} order-four linear differential \noperator\n\\begin{eqnarray}\n\\label{L_4}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad \\,  \\,\nL_4 \\, \\, \\, = \\, \\, \\, \\,  \\,\n D_x^4  \\, \\,  \\, + \\, p(x) \\cdot \\, D_x^3 \\, \\, \\, + \\, \\, q(x)\\cdot \\, D_x^2\n \\, \\,  \\, + \\,  \\, r(x) \\cdot \\, D_x \\,  \\, + \\, \\, s(x),\n\\end{eqnarray}\nand  introduce two other  linear differential operators of order four\ndefined as previously \nin sections \\ref{order-two} and \\ref{order-threegener}: \nthe linear differential operator $\\, L_4^{(c)}$ conjugated of (\\ref{L_4}) \nby a function $\\, v(x)$ and the \n(normalized) pullbacked operator $\\, L_4^{(p)}$. These two \nlinear differential operators read respectively\n\\begin{eqnarray}\n\\label{L2c} \n\\hspace{-0.95in}&&  \\quad \\quad \\quad \n\\, \\, \nL_4^{(c)} \\, \\, = \\, \\, \\, \\, \\, D_x^4  \\, \\, \\, + \\, \n\\Bigl(p(x) \\, + \\, 4 \\cdot \\, {{v'(x)} \\over {v(x)}} \\Bigr) \\cdot \\, D_x^3 \n \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad \\quad \\,\n\\, \\, \\,  + \\, \\, \\Bigl(q(x) \\, + \\, 3 \\cdot \\, p(x) \\cdot \\,  {{v'(x)} \\over {v(x)}} \n\\, + \\,  6 \\cdot \\,  {{v\"(x)} \\over {v(x)}} \\Bigr) \\cdot \\, D_x^2\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad\\quad  \\quad  \\, \\,  \\, \n+ \\, \\, \\Bigl(r(x) \\, + \\, 2 \\cdot \\, q(x) \\cdot \\,  {{v'(x)} \\over {v(x)}}\n \\, + \\, 3 \\cdot \\, p(x) \\cdot \\,  {{v\"(x)} \\over {v(x)}}\n \\, + \\, 4 \\cdot \\,  {{v^{(3)}(x)} \\over {v(x)}}     \\Bigr) \\cdot \\, D_x  \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad   \\quad \\quad \\quad \\quad \n\\,\\, \\,  +  \\, \\, s(x) \\, \\, + \\,  \\, r(x) \\cdot \\,  {{v'(x)} \\over {v(x)}}\n \\, \\, + \\,  q(x) \\cdot \\,  {{v\"(x)} \\over {v(x)}}\n \\,\\,  + \\,   p(x) \\cdot \\,  {{v^{(3)}(x)} \\over {v(x)}} \n \\, \\, + \\,   {{v^{(4)}(x)} \\over {v(x)}}, \n \\nonumber\n\\end{eqnarray}\nand:\n\\begin{eqnarray}\n\\label{L2c}\n\\hspace{-0.95in}&&    \nL_4^{(p)} \\, \\, = \\, \\, \\, \\, \\, \nD_x^4  \\,  \\,  \\, + \\,  \\Bigl(p(y(x)) \\cdot \\, y'(x)  \\,\n - \\, 6 \\cdot \\,  {{y\"(x)} \\over {y'(x)}} \\Bigr) \\cdot \\, D_x^3\n \\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \n\\, +  \\Bigl( q(y(x)) \\cdot \\, y'(x)^2 \\, - \\, 3 \\cdot \\, p(y(x)) \\cdot \\, y\"(x) \\, \n-4 \\cdot \\, {{y^{(3)}(x)} \\over {y'(x)}} \n \\, + 15 \\cdot \\, \\Bigl( {{y\"(x)} \\over {y'(x)}} \\Bigr)^2 \\Bigr) \\cdot \\, D_x^2 \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad   \n\\, +  \\Bigl(r(y(x)) \\cdot \\,y'(x)^3 \\,\n -q(y(x)) \\cdot \\, y'(x)\\cdot \\, y\"(x)  \\, -p(y(x)) \\cdot \\, y^{(3)}(x) \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad   \\quad   \\quad \\quad   \n + 3 \\cdot \\, p(y(x))  \\cdot \\, {{y\"(x)^2} \\over {y'(x)}} \n \\, \\,  - {{y^{(4)} } \\over {y'(x)}} \\, \\, \n+10  \\cdot \\, {{y\"(x) \\cdot \\, y^{(3)} } \\over { y'(x)^2 }} \\, \\,\n  -15 \\cdot  \\, \\Bigl( {{y\"(x)  } \\over { y'(x) }} \\Bigr)^3\n \\Bigr) \\cdot \\, D_x\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad   \\,\\, \n\\, \\, + \\, s(y(x)) \\cdot y'(x)^4.\n\\end{eqnarray}\nThe identification of these two order-four linear differential \noperators $\\,L_4^{(p)}$ and $\\,L_4^{(c)}$ gives this time four conditions \n$\\, {\\cal C}_n$, $\\, n= \\, 0,  \\, 1, \\, 2, \\, 3$, corresponding, \nrespectively, to the identification  of the $\\, D_x^n$ coefficients \nof $\\, L_4^{(p)}$ and $\\, L_4^{(c)}$. \n\nEliminating once again the log-derivative $\\, v'(x)\/v(x)$ between  \n$\\, {\\cal C}_3$  and  $\\, {\\cal C}_2$ one deduces a Schwarzian condition\n\\begin{eqnarray}\n\\label{condition4}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad \n W(x) \n\\, \\, \\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\,  \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere this time:\n\\begin{eqnarray}\n\\label{wherecond4}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad \nW(x)  \\, \\, = \\, \\,  \\, \\, \\,\n{{3} \\over {10}} \\cdot \\,  {{ d p(x)} \\over { dx}}  \\, \\, \\,  + \\, \\, \n {{3} \\over {40}} \\cdot \\,   p(x)^2  \\, \\,  \\, \\,  - \\, {{q(x)} \\over {5}}.\n\\end{eqnarray}\n\nIntroducing the wronskian $\\, w(x)$ of the order-four linear differential\noperator $\\, L_4$ with (\\ref{globnilp}), the condition  $\\, {\\cal C}_3$ \njust corresponds to log-derivatives and can  be easily  integrated\ngiving the exact expression of the conjugation function \n$\\, v(x)$ as:\n\\begin{eqnarray}\n\\label{v4}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad  \\quad \\quad  \\quad \\quad  \\quad \nv(x) \\, \\, = \\, \\, \\,  \n\\Bigl({{ w(x) } \\over { w(y(x)) \\cdot \\, y'(x)^6 }}\\Bigr)^{1\/4}.\n\\end{eqnarray}\nThe next conditions  $\\, {\\cal C}_1$ and  $\\, {\\cal C}_0$ yield extremely \ninvolved non-linear differential conditions on the miscellaneous \nderivatives of the various coefficients. It turned out to be\n very difficult to proceed \nwith such huge expressions. Yet when the linear differential\noperator $\\, L_4$ has a selected (symplectic) differential \nGalois group one can go much further in the calculations, \nas we will see in the coming subsection.\n\n\\vskip .2cm  \n\n\\subsection{Calabi-Yau condition (exterior square).} \n\\label{CalabiYaucondi}\n\nImposing the \n{\\em Calabi-Yau condition}~\\cite{IsingCalabi,IsingCalabi2}\nin the case of an order-four  linear differential operator gives:\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{Calabi} \n\\, r(x) \\, \\, = \\, \\, \\,   \n {{p(x) \\cdot \\, q(x)} \\over {2}} \\, \\,  \\, \n -\\,  {{p(x)^3 } \\over {8}} \\, \\, \\,   + \\, {{d q(x)} \\over {dx}} \\,  \\,  \\, \n - {{3} \\over {4}} \\cdot \\,  p(x) \\cdot \\,  {{d p(x)} \\over {dx}} \n\\,  \\,  \\,  - {{1} \\over {2}} \\cdot \\,  {{d^2 p(x)} \\over {dx^2}}.\n\\end{eqnarray}\nIn this case the exterior square of the order-four operator $\\, L_4$ \nhas order {\\em five} instead of order six. \n\nWhen condition (\\ref{Calabi}) is verified,\nthe order-four linear differential operator $\\, L_4$ \nhas a {\\em symplectic} differential Galois group \n$\\, Sp(4, \\,  \\mathbb{C})$. Note that if condition (\\ref{Calabi}) is verified, \nthe Calabi-Yau conditions for the pullbacked and conjugated \noperators $\\, L_4^{(p)}$ and  $\\, L_4^{(c)}$ \n{\\em are automatically verified}: this is a consequence of the fact \nthat the Calabi-Yau condition  (\\ref{Calabi}) is left invariant by \nconjugation and pullback\\footnote[5]{To see that the  Calabi-Yau condition \nis preserved by conjugation is straightforward. However, \nas remarked in~\\cite{Duco}, to see that the\nCalabi-Yau condition is preserved by pullback transformations \nis very hard to see by direct computation, since one gets \nan enormous fourth-order nonlinear differential equation.\n}. In other words the {\\em following identification of the} $\\, D_x$ \n{\\em coefficients of} $\\, L_4^{(p)}$ and $\\, L_4^{(c)}$ \n{\\em is automatically verified when the Calabi-Yau condition} \n(\\ref{Calabi})  {\\em is verified}. \n\nRecall that the Calabi-Yau condition (\\ref{Calabi}) together \nwith the globally nilpotent condition~\\cite{bo-bo-ha-ma-we-ze-09} \nautomatically  yields $\\, L_4$ \nto be conjugated to its adjoint\n\\begin{eqnarray}\n\\label{simplyconj4}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad  \n\\quad \\quad  \\quad \\quad  \\quad \\, \\,   \\,  \\,    \\, \nL_4 \\cdot \\, w(x)^{1\/2} \\, \\, \\, = \\,  \\, \\,\\,  \\, \n  w(x)^{1\/2} \\cdot \\, adjoint(L_4),\n\\end{eqnarray}\nwhere $\\, w(x)$ is a $\\, N$-root of a rational function.\n\nAt the last step we consider the identification of the constant \nterms in $\\, D_x$ in $\\, L_4^{(p)}$ and $\\, L_4^{(c)}$. After injecting \nin this ``large'' non-linear differential equation, equation \n(\\ref{globnilp}), the Schwarzian condition (\\ref{condition4})\nwith $\\, W(x)$ given by (\\ref{wherecond4}), and the \nCalabi-Yau condition (\\ref{Calabi}), we eventually find that \nthis last ``large''  equation {\\em becomes independent of the pullback} \n$\\, y(x)$ and reduces to a quite simple condition {\\em giving} $\\, s(x)$ \n{\\em as a polynomial expression in the two coefficients \n$\\, p(x)$ and  $\\, q(x)$ and their derivatives}:\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{s(x)} \ns(x) \\, \\, = \\, \\, \\, \\,\n {{9} \\over {100}} \\cdot \\, q(x)^2 \n\\, \\,\\,   - \\, {{1} \\over {200}} \\cdot \\, q(x) \\cdot \\, p(x)^2 \\, \\,  \n+\\, {{1} \\over {4}} \\cdot \\, p(x) \\cdot \\, {{ d q(x)}  \\over {dx}}\n \\,\\,   - \\, {{1} \\over {50 }} \\cdot \\, q(x) \\cdot \\, {{ d p(x)}  \\over {dx}}\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad \\quad \n\\, + \\, {{3} \\over {10 }} \\cdot \\, {{ d^2 q(x)}  \\over {dx^2}} \\, \\,\n -  \\, {{11} \\over {1600}} \\cdot \\, p(x)^4 \\,  \\,\n -  \\, {{9} \\over {50}} \\cdot \\, p(x)^2 \\cdot \\,  {{ d p(x)}  \\over {dx}}\n \\,  \\, -  \\, {{21} \\over {100}} \\cdot \\, \\Bigl({{ d p(x)}  \\over {dx}}\\Bigr)^2 \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad \\quad  \\quad \\quad \n -  \\, {{1} \\over {5}} \\cdot \\,{{ d^3 p(x)}  \\over {dx^3}} \\, \n-  \\, {{7} \\over {20}} \\cdot \\, p(x) \\cdot \\, {{ d^2 p(x)}  \\over {dx^2}}.\n\\end{eqnarray}\nIn order to understand what this new condition (\\ref{s(x)}) \ncoming on top of the Calabi-Yau condition (\\ref{Calabi}) really means,\nwe calculated, for various MUM\\footnote[1]{Maximal unipotent \nmonodromy (MUM) linear operators~\\cite{bo-bo-ha-ma-we-ze-09,Christol}. } \norder-four linear differential operators\n$\\, L_4$ verifying (\\ref{Calabi}) and (\\ref{s(x)}), the corresponding nome \nand {\\em Yukawa couplings}~\\cite{Christol}. The corresponding Yukawa \ncouplings {\\em were actually found to be trivial}: $\\, K_q \\, = \\, 1$ !! \n\nThis amounts to saying that combining the two conditions (\\ref{Calabi}) \nand (\\ref{s(x)}) corresponds to a drastic reduction: \n the (irreducible) order-four linear differential operator $\\, L_4$\nis not a ``true'' order-four operator. Typically one can imagine that $\\, L_4$\nreduces to an order-two operator, being homomorphic to the {\\em symmetric cube} \nof an underlying order-two linear differential operator. In fact it is exactly\nthe symmetric cube of an order-two operator.\n\nLet us consider the {\\em symmetric cube} of an {\\em order-two} linear differential \noperator $\\, L_2 \\, = \\, D_x^2 \\, + \\, A(x) \\cdot \\, D_x \\, + \\, B(x) \\, $ which is an \norder-four linear differential (\\ref{L_4}) with the following coefficients:\n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \n\\label{param} \np(x) \\, = \\, \\,\\,  6  \\cdot \\, A(x), \\,\\,   \\quad   \\quad   \\quad \nq(x)  \\, = \\, \\,\\,  11\\cdot \\, A(x)^2 \\,\\, \n +4 \\cdot \\,  {{ d A(x)} \\over {dx}} \\, \\, +10 \\cdot \\, B(x),  \n \\nonumber  \\\\\n\\hspace{-0.95in}&&  \nr(x) \\, \\, =  \\, \\, \\, \n6\\cdot \\, A(x)^3 \\,\\, +7 \\cdot \\,A(x) \\cdot \\,  {{ d A(x)} \\over {dx}}\n \\,\\, +30 \\cdot \\,B(x) \\cdot \\,A(x) \\,\\,\n + {{ d^2 A(x)} \\over {dx^2}} \n\\,\\, +10 \\cdot \\,  {{ d B(x)} \\over {dx}},  \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \ns(x) \\, \\, = \\, \\, \\,\\,\\,  18 \\cdot \\, A(x)^2\\cdot \\,B(x)\\,\\,\\, \n  +6  \\cdot \\, B(x) \\cdot \\, {{ d A(x)} \\over {dx}}\\,\\,\\,\n +15 \\cdot \\,  {{ d B(x)} \\over {dx}} \\cdot \\, A(x)\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad   \\quad   \\quad \\quad   \\quad   \\quad \n\\,\\, +9 \\cdot\\, B(x)^2\\,\\,\\,  +3 \\cdot\\,  {{ d^2 B(x)} \\over {dx^2}}.\n\\end{eqnarray}\nOne finds straightforwardly that the coefficients given by (\\ref{param}) \nverify the Calabi-Yau condition (\\ref{Calabi}), \n{\\em as well as the new condition} \n(\\ref{s(x)}). In this case the differential Galois group  is no longer \nthe symplectic differential Galois group $\\, Sp(4,  \\,  \\mathbb{C})$, \nbut actually reduces\\footnote[2]{When an  order-four linear differential \noperator is the symmetric cube of an underling order-two operator \nits symmetric square is no longer of order $\\, 10$ \nbut reduces to order $\\, 7$.} to the\ndifferential Galois group of the underlying order-two linear differential \noperator, namely $\\, SL(2,  \\,  \\mathbb{C})$. The fact that the Calabi-Yau \ncondition (\\ref{Calabi}) is verified is not a surprise: the exterior \nsquare of a symmetric cube is naturally of order less than six. The \nfact that being the symmetric cube of an underlying order-two operator \nverifies automatically the new condition (\\ref{s(x)}) emerging \nfrom a compatibility condition of an order-four linear differential \noperator by pullback is far less obvious. The ``parametrization'' \n(\\ref{param}) necessarily yields the Calabi-Yau condition (\\ref{Calabi}) \nand the new condition (\\ref{s(x)}), and, conversely, (\\ref{Calabi})\nand (\\ref{s(x)}) can be  parametrized by (\\ref{param}). \n\n\\vskip .1cm\n\nOur large calculations thus show that the pullback-compatibility \nof an order-four linear differential operator which verifies the Calabi-Yau \ncondition (\\ref{Calabi}), amounts to saying that this order-four linear differential\noperator {\\em reduces to} (the symmetric cube of) an underlying \n{\\em order-two linear differential operator}. The Schwarzian \ncondition (\\ref{condition4}) with $\\, W(x)$ given by (\\ref{wherecond4}), \nis {\\em thus inherited from the Schwarzian condition} (\\ref{condition1})  \n{\\em of the underlying order-two linear differential operator}.\n\n\\vskip .1cm   \n\n\\subsection{Reducible operators} \n\\label{nonirreducible}\n\nThroughout the paper we make the assumption that the \nlinear differential operators are {\\em irreducible}. The reduciblility \nof the linear differential operators is not an academic\nscenario: it is  the situation {\\em we encounter} (almost)  \n{\\em systematically with the linear differential operators emerging in physics}, \ntypically in the case of the \n$\\, n$-fold integral $\\, \\chi^{(n)}$ of the two-dimensional \nIsing model~\\cite{High,bo-gu-ha-je-ma-ni-ze-08,ze-bo-ha-ma-05b}. \nWhen the linear differential operators are {\\em reducible}, \nit is clear that all the calculations of this paper must be \nrevisited, taking into account the miscellaneous factorization \nscenarios. \n\nSketching the kind of situation we may encounter,\nlet us consider an order-four linear differential operator\n$\\, L_4 \\, = \\, D_x^4 \\, + \\, p_r(x) \\cdot \\, D_x^3$\n$ \\, + \\,  q_r(x) \\cdot \\, D_x^2 \\, + \\, \\cdots \\, \\, $ \nwhich factorizes into the product of two  \norder-two linear differential operators: \n\\begin{eqnarray}\n\\label{p4q4}\n\\hspace{-0.95in}&& \\quad  \nL_4 \\, = \\,  \\, M_2 \\cdot \\, L_2, \\quad \\quad \\quad  \\quad  \\quad \n \\quad \\quad \\quad \\hbox{where:} \n\\nonumber \\\\\n \\hspace{-0.95in}&& \\quad  \n L_2 \\, = \\, \\, D_x^2 \\,  \\,+ p(x) \\cdot \\, D_x \\, +q(x), \\quad \\quad \\quad \n M_2 \\, = \\,  \\,D_x^2 \\, \\, + \\tilde{p}(x) \\cdot \\, D_x \\, +\\tilde{q}(x),\n \\\\\n \\hspace{-0.95in}&& \\quad   \n p_r(x) \\, = \\,  \\, p(x) \\, \\, + \\tilde{p}(x),  \\, \\quad\n q_r(x)  \\, = \\,  \\,   \\tilde{p}(x) \\cdot \\, p(x) \\,  \\, +\\tilde{q}(x) \\,  \\, \n+2 \\cdot \\, {{d p(x)} \\over {dx}} \\, \\,  +q(x), \n \\quad  \\, \\, \\cdots \n\\nonumber\n\\end{eqnarray}\nThe simple case where the two operators $\\, M_2$ and $\\, L_2$ are \nidentical is sketched in \\ref{reducibleapp}.   \nIn general the exterior square of $\\, L_4$ is an order-six linear differential \noperator which is the product of an order-one operator, of \nthe symmetric product of $\\, L_2$ and $\\, M_2$, and of the order-one\nlinear differential operator $\\, D_x \\, +p(x)$. Therefore, this reducible\norder-four  linear differential operator $\\, L_4$ \n{\\em does not verify in general}\nthe Calabi-Yau condition (\\ref{Calabi}). \n\nImposing the (normalized) pullback \nby $\\, y(x)$ of this reducible order-four  linear differential operator \n$\\, L_4 \\, = \\, M_2 \\cdot \\, L_2$ to be equal to a conjugation by a function\n$\\, v(x)$ of that operator, it is important to remember that a change \nof variable $\\, x \\, \\rightarrow \\, y(x)$ on a \nlinear differential operator which is the product of two  operators, \nis the product of these two linear differential operators \non which this change of variable has been performed. One gets a set \nof equations where one can disentangle two Schwarzian equations\n\\begin{eqnarray}\n\\label{disentang}\n\\hspace{-0.95in}&& \\quad  \\quad  \\quad  \\quad  \\quad \n W(x) \\, \\, \\,\\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\, \\, \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\\\\n\\label{disentang2}\n\\hspace{-0.95in}&& \\quad  \\quad  \\quad  \\quad  \\quad \n {\\tilde W}(x) \\, \\, \\,\\,  \\,-{\\tilde W}(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\, \\, \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere $\\, W(x)$ and $\\, {\\tilde W}(x)$ are the functions  (\\ref{wherecond})\nalready encountered in the analysis of order-two linear differential \noperators\n\\begin{eqnarray}\n\\label{disentangwhere} \n\\hspace{-0.95in}&& \\quad  \\quad  \\quad \\quad  \\quad  \\quad \\quad \nW(x)  \\, \\, = \\, \\,  \\, \\, \\, {{ d p(x)} \\over { dx}} \n \\, \\,\\,  \\,  + \\, \\, \n   {{p(x)^2} \\over {2 }} \\, \\,  \\, \\,  -2 \\cdot \\, q(x),\n\\\\\n\\hspace{-0.95in}&& \\quad  \\quad  \\quad \\quad  \\quad  \\quad \\quad \n\\label{disentangwhere2}\n {\\tilde W}(x)  \\, \\, = \\, \\,  \\, \\, \\, {{ d  {\\tilde p}(x)} \\over { dx}} \n \\, \\,\\,  \\,  + \\, \\, \n   {{ {\\tilde p}(x)^2} \\over {2 }} \\, \\,  \\, \\,  -2 \\cdot \\, {\\tilde  q}(x),\n\\end{eqnarray}\ncorresponding to  the Schwarzian conditions written separately \non $\\, L_2$ and $\\, M_2$, \ntogether with another relation which couples  $\\, L_2$ and $\\, M_2$:\n\\begin{eqnarray}\n\\label{couples}\n\\hspace{-0.95in}&& \\quad  \\quad  \\quad  \\quad \n4 \\cdot\\, {{  y''(x)} \\over { y'(x)}} \\,\\, \\,+  {\\tilde p}(x) \\, -p(x)\n \\, \\, = \\, \\, \\, \\Bigl({\\tilde p}(y(x)) \\, -p(y(x)) \\Bigr) \\cdot \\, y'(x).\n\\end{eqnarray}\nAmong the four solutions of the order-four operators $\\, L_4 \\, = \\, M_2 \\cdot \\, L_2$, \nthe two solutions of the order-two linear differential operator $\\, L_2$ transform \nnicely under the pullback $\\, \\, x \\, \\rightarrow \\, \\, y(x)$, provided the \nSchwarzian condition (\\ref{disentang}) is satisfied, but this just corresponds \nto a {\\em partial symmetry}. In general the set of equations (\\ref{disentang}),  \n(\\ref{disentang2}), (\\ref{couples}) seems to be too rigid to allow solutions\nother than trivial symmetries or partial symmetries.\n\n\\vskip .1cm\n\nIt is however worth mentioning a quite curious result. If one imposes \nthe reducible order-four  linear differential operator \n$\\, L_4 \\, = \\, M_2 \\cdot \\, L_2 \\, $ to verify the Calabi-Yau condition (\\ref{Calabi})\n(i.e. to be such that the exterior square of that operator is order five instead of order six),\none gets a condition that becomes remarkably simple when written in terms of \nthe functions  $\\, W(x)$ and $\\, {\\tilde W}(x)$ given by (\\ref{disentangwhere}) \nand  (\\ref{disentangwhere2}). Introducing the difference \n $\\, \\Delta W(x) \\,  = \\, \\,   W(x)\\, -   {\\tilde W}(x)$, \nthe Calabi-Yau condition (\\ref{Calabi}) simply reads:\n\\begin{eqnarray}\n\\label{verycurious}\n\\hspace{-0.95in}&& \\quad  \\quad  \\quad \\quad \\quad \\quad\n 2 \\cdot \\, {{ d \\Delta W(x)} \\over { dx}}\n \\, \\,\\,  = \\, \\,  \\, \\,\\, \n (p(x) \\, - \\,  {\\tilde p}(x))  \\cdot  \\, \\Delta W(x).\n\\end{eqnarray}\nTherefore, if one restricts oneself to $ \\,W(x)\\, = \\, \\,  {\\tilde W}(x)$ \nwhich identifies the two Schwarzian conditions\n(\\ref{disentang}) and (\\ref{disentang2}), one sees that condition (\\ref{verycurious})\nis automatically verified: condition  $ \\,W(x)\\, = \\, \\,  {\\tilde W}(x)$ \n{\\em is thus a sufficient condition for  the Calabi-Yau condition} (\\ref{Calabi}).\n\n\\vskip .1cm\n\nThe analysis of pullback symmetry on {\\em reducible} linear differential operators \nis clearly an interesting and challenging problem in physics. It will require \nmany more calculations to explore the arborescence of these various \nfactorization scenarios.\n\n\\vskip .1cm\n\n\\subsection{Symmetric Calabi-Yau condition} \n\\label{symmCalabiYau}\n\nThe condition, we called in~\\cite{Diffalg,Canonical} \n{\\em symmetric Calabi-Yau condition}\nfor the order-four linear differential operator $\\, L_4$ \n(which correspond to impose that its symmetric square is \nof order less than $\\, 10$), is a huge\\footnote[1]{This polynomial \nis the sum of $ \\, 3548$ monomials in the coefficients of $\\, L_4$ and \ntheir derivatives.} polynomial condition \non the coefficients of $\\, L_4$ and their derivatives.\nThis condition is invariant by pullback \nand conjugation. Provided the Schwarzian condition (\\ref{condition4}) \nwith $\\, W(x)$ given by (\\ref{wherecond4}) \nis satisfied, this symmetric Calabi-Yau \ncondition alone is not sufficient  to have $\\, L_4^{p} \\, = \\, \\, L_4^{c}$.\nSimilarly to what we saw with the \nCalabi-Yau condition (\\ref{Calabi}), would a supplementary \ncondition to the symmetric Calabi-Yau \ncondition be sufficient to have $\\, L_4^{p} \\, = \\, \\, L_4^{c}$ ?\nCould one also have, in this selected\nsubcase, a reduction of $\\, L_4$  to an underlying order-two operator? This \nscenario remains open.\n\n\\vskip .1cm \n\\vskip .1cm \n\nWorking with such huge polynomials will not get us far.  In order to advance, \nlet us introduce a parametrization based on the \nideas explained in~\\cite{Canonical}, namely that an order-four linear \ndifferential operator $\\, L_4$, with an orthogonal differential \nGalois group $\\, SO(4, \\,  \\mathbb{C})$ and such that its symmetric square  \nis of order less than $\\, 10$, is necessarily of the form\\footnote[2]{The differential \nGalois group $\\, SO(4, \\,  \\mathbb{C})$ with an order-$10$ symmetric square situation\ncorresponds to a decomposition \n$\\, L_4 \\, = \\, \\, (U_3 \\cdot U_1 \\, + \\, \\, 1) \\cdot \\, d(x)$, see~\\cite{Canonical}.} \n\\begin{eqnarray}\n\\label{L4symm}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad  \\quad \\quad  \\quad \\quad  \\quad \nL_4 \\, \\,\\, = \\, \\, \\,\\, (U_1 \\cdot U_3 \\, + \\, \\, 1) \\cdot \\, d(x), \n\\end{eqnarray}\nwhere $\\, U_1$ and $\\, U_3$ are order-one and  order-three {\\em self-adjoint}\nlinear differential operators:\n\\begin{eqnarray}\n\\label{U1U3}\n\\hspace{-0.95in}&& \nU_3 \\,  = \\, \\, a(x) \\cdot \\, D_x^3 \n\\,\\,  + \\,  {{3} \\over {2}} \\cdot \\, {{d a(x)} \\over {dx}}  \\cdot \\, D_x^2\n \\, \\, + \\,  b(x) \\cdot \\, D_x \\,\\,\n + \\,  {{1} \\over {2}} \\cdot \\,  {{d b(x)} \\over {dx}} \n\\,\\,  - \\,  {{1} \\over {4}} \\cdot \\,  {{d^3 a(x)} \\over {dx^3}}, \n\\quad \\quad \n\\\\\n\\label{U1U3U1}\n\\hspace{-0.95in}&& \nU_1 \\,\\, \\, = \\, \\, \\,\\, c(x) \\cdot \\, D_x\n\\,\\,\\,  + \\,  \\, \\, {{1} \\over {2}} \\cdot \\,  {{d c(x)} \\over {dx}}.  \n\\end{eqnarray}\nThis yields a parametrization of this huge polynomial differential \n(symmetric Calabi-Yau) condition:\n\\begin{eqnarray}\n\\label{U1U3param}\n\\hspace{-0.95in}&& \\quad \\quad \np(x) \\, \\, = \\, \\, \\, \\,   {{5} \\over { 2}} \\cdot \\, {{a'(x)} \\over {a(x)}}\n\\,\\,  + \\, \\,  {{1} \\over { 2}} \\cdot \\, {{c'(x)} \\over {c(x)}}\n\\,  \\, + \\, \\,  4 \\cdot \\, {{d'(x)} \\over {d(x)}},\n\\\\\n\\label{U1U3param1}\n\\hspace{-0.95in}&& \\quad \\quad \nq(x) \\, \\, = \\, \\, \\,\\,  {{b(x)} \\over {a(x)}}\n \\,\\,  + \\,  {{3} \\over { 2}} \\cdot \\, {{a''(x)} \\over {a(x)}}\n\\,\\,  + \\,  {{3} \\over { 4}} \\cdot \\, {{a'(x)} \\over {a(x)}}\n \\cdot \\, {{c'(x)} \\over {c(x)}} \n \\, \\, + \\, \\,  6 \\cdot \\, {{d''(x)} \\over {d(x)}} \n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad  \\quad \\quad \n \\, + \\,  {{15} \\over { 2}} \\cdot \\, {{a'(x)} \\over {a(x)}}\n \\cdot \\, {{d'(x)} \\over {d(x)}} \n\\, + \\,  {{3} \\over { 2}} \\cdot \\, {{c'(x)} \\over {c(x)}}\n \\cdot \\, {{d'(x)} \\over {d(x)}}, \n\\\\\n\\label{U1U3param2}\n\\hspace{-0.95in}&& \\quad \\quad \nr(x) \\, \\, = \\, \\, \\,\\,   {{1} \\over { 2}}\n \\cdot \\, {{c'(x)} \\over {c(x)}} \\cdot \\,  {{b(x)} \\over {a(x)}}\n\\, \\, + \\, \\,  4 \\cdot \\, {{d'''(x)} \\over {d(x)}}\n \\,\\,  + \\, \\,  4 \\cdot \\,  {{a'(x)} \\over {a(x)}} \n\\cdot \\, {{c'(x)} \\over {c(x)}} \\cdot \\, {{d'(x)} \\over {d(x)}}\n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \n+{{3} \\over { 2}} \\cdot \\, {{d''(x)} \\over {d(x)}} \\cdot \\, {{c'(x)} \\over {c(x)}} \n\\,\\,  - \\,  {{1} \\over { 4}} \\cdot \\, {{a'''(x)} \\over {a(x)}} \n\\,\\,  + \\,  {{3} \\over { 2}} \\cdot \\, {{b'(x)} \\over {a(x)}} \n\\,\\,   + \\, {{15} \\over { 2}} \\cdot \\, {{d''(x)} \\over {d(x)}}\n \\cdot \\, {{a'(x)} \\over {a(x)}}\n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \n+ 2 \\cdot \\, {{d'(x)} \\over {d(x)}} \\cdot \\,  {{b(x)} \\over {a(x)}} \n\\,\\,  + 3 \\cdot \\, {{d'(x)} \\over {d(x)}} \\cdot \\,  {{a''(x)} \\over {a(x)}}, \n\\\\\n\\label{U1U3param3}\n\\hspace{-0.95in}&& \\quad \\quad \ns(x) \\, \\, = \\, \\, \\, \\,  {{d^{(4)}} \\over {d(x)}} \\, \\, \\, \n+ \\,  {{1} \\over { 2}} \\cdot \\, {{c'(x)} \\over {c(x)}} \\cdot \\, {{d'''(x)} \\over {d(x)}}\n \\, \\, + \\,  {{1} \\over { 2}} \\cdot \\, {{b\"(x)} \\over {a(x)}}\n \\, \\, - \\,  {{1} \\over { 4}} \\cdot \\, {{a^{(4)}(x)} \\over {a(x)}}\n\\nonumber \\\\\n\\label{36}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\, \\,\n - \\,  {{1} \\over { 8}} \\cdot \\, {{a'''(x)} \\over {a(x)}} \\cdot \\,  {{c'(x)} \\over {c(x)}} \\,\\, \n+ \\,  {{1} \\over { 4}} \\cdot \\,  {{b'(x)} \\over {a(x)}}  \\cdot \\,  {{c'(x)} \\over {c(x)}}\n\\, + {{1} \\over {a(x) \\, c(x)}}\n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\, \\,  \n- \\,  {{1} \\over { 4}} \\cdot \\, {{a'''(x)} \\over {a(x)}}\\cdot \\, {{d'(x)} \\over {d(x)}} \\, \n+ \\,  {{3} \\over { 2}} \\cdot \\, {{b'(x)} \\over {a(x)}} \\cdot \\, {{d'(x)} \\over {d(x)}}\n\\,\\,  + \\,\\, {{b(x)} \\over {a(x)}} \\cdot \\, {{d\"(x)} \\over {d(x)}}\n \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\,\\,  \n+ \\,  {{3} \\over { 2}} \\cdot \\,  {{a''(x)} \\over {a(x)}} \\cdot \\, {{d''(x)} \\over {d(x)}}\n \\,\\,  \n+ \\,  {{5} \\over { 2}} \\cdot \\,  {{a'(x)} \\over {a(x)}} \\cdot \\, {{d'''(x)} \\over {d(x)}}\n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\,\\,  \n+ \\,  {{1} \\over { 2}} \\cdot \\,  {{c'(x)} \\over {c(x)}}\n \\cdot \\, {{d'(x)} \\over {d(x)}} \\cdot \\, {{b(x)} \\over {a(x)}}\n \\,\\,  + \\,  {{3} \\over { 4}} \\cdot \\, {{a'(x)} \\over {a(x)}} \n\\cdot \\, {{c'(x)} \\over {c(x)}} \\cdot \\, {{d''(x)} \\over {d(x)}}.\n\\nonumber \n\\end{eqnarray}\nOne easily verifies that this parametrization \n(\\ref{U1U3param}) ... (\\ref{36}) is \nsuch that the polynomial encoding the symmetric Calabi-Yau condition, \n{\\em  is identically equal to zero}. Moreover one verifies \nthat the order-four linear differential operator (\\ref{L4symm}), with \nparametrization (\\ref{U1U3param}), (\\ref{U1U3param1}),  (\\ref{U1U3param2}),  \n(\\ref{U1U3param3}), is, generically, such that its symmetric \nsquare has order $\\, 9$ (instead of $\\, 10$), its exterior \nsquare being of order $\\, 6$.\n\nImposing $\\, L_4^{(p)} \\, = \\, \\, L_4^{(c)}$ \nfor an order-four linear differential operator, \ncorresponding to this parametrization (such that it verifies \nthe symmetric Calabi-Yau condition, and such that its symmetric \nsquare is of order nine), one naturally finds the Schwarzian \ncondition (\\ref{condition4}) with (\\ref{wherecond4}), as well\nas the exact expression  (\\ref{v4}) for the conjugation \nfunction $\\, v(x)$. Taking into account the Schwarzian \ncondition (\\ref{condition4}), the identification of \nthe coefficients of $\\, D_x$  for $\\, L_4^{(p)}$ \nand  $ \\, L_4^{(c)}$ yields a relation of the form \n$\\, \\Phi(x) \\, \\,\\,  = \\, \\, \\, \\,  \\Phi(y(x)) \\cdot \\, y'(x)^3$,\nwhere $\\, \\Phi(x)$ is a rational function. \nTogether with the last condition, this gives an invariance  of the form \n $\\, \\Psi(x) \\,\\,  \\, = \\, \\,  \\, \\Psi(y(x)) \\, $ yielding only \ntrivial cases\\footnote[2]{See~\\cite{Aziz} for similar calculations.} \nfor $\\, L_4^{(p)} \\, = \\, \\, L_4^{(c)}$.\n\n\\vskip .1cm\n\nThis symmetric Calabi-Yau condition,  \n{\\em even if it is invariant by pullback and conjugation},\n is thus {\\em not sufficient to get} $\\, L_4^{(p)} \\, = \\, \\, L_4^{(c)}$. We \nhave here a situation similar to the one described in the previous \nsection \\ref{CalabiYaucondi}, with the emergence of the additional \ncondition (\\ref{s(x)}) on top of the Calabi-Yau condition \n(\\ref{Calabi}). However here the calculations are way too large: \nfinding the additional condition(s) together with the symmetric Calabi-Yau \ncondition yielding  $\\, L_4^{(p)} \\, = \\, \\, L_4^{(c)}$, is beyond \nour reach for now. The case, described in the previous \nsection \\ref{CalabiYaucondi}, where the order-four operator \n(\\ref{L4symm}) is the symmetric cube of an underlying order-two operator is \nalso such that the symmetric square of $\\, L_4$ is not of the generic \norder $\\, 10$, but, in fact, of order $\\, 7$: in this case the coefficients\nof  $\\, L_4$ {\\em verify}\\footnote[1]{This can be verified straightforwardly \nsubstituting (\\ref{param}) in the   $ \\, 3548$ monomials symmetric \nCalabi-Yau condition.} {\\em the symmetric Calabi-Yau condition}. \nSince the calculations are way too large, it is not possible for now to tell\nif the additional condition(s) to the symmetric Calabi-Yau condition, also \ngives eventually a linear differential operator that is the \nsymmetric cube of an order-two \noperator, as described in the previous section \\ref{CalabiYaucondi}, \nor whether it would give something else. This would mean the emergence\nof the ``classic'' Calabi-Yau condition (\\ref{Calabi}) \ncombined with the condition (\\ref{s(x)}). This remains an open question.\n\n\\vskip .2cm \n\n\\section{Order-$N$ linear differential operators} \n\\label{order-N}\n\nThe analysis of {\\em irreducible} order-five operators is sketched \nin \\ref{order-five}. Let us now consider an  {\\em irreducible} \norder-$N$ linear differential operator\n\\begin{eqnarray}\n\\label{L_N}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad  \\quad \\quad  \nL_N \\, \\, \\, = \\, \\, \\, \\, \n D_x^N  \\, \\, \\, + \\, p(x) \\cdot \\, D_x^{N-1} \\, \\,\\,  + \\, \\, q(x)\\cdot \\, D_x^{N-2}\n \\, \\,\\, \\,   + \\, \\,  \\, \\cdots\n\\end{eqnarray}\nand let us also introduce two other  linear differential operators of order $N$:\nthe operator $\\, L_N^{(c)}$ conjugated of (\\ref{L_N}) by a function $\\, v(x)$,\nnamely $\\, L_N^{(c)} \\, = \\, \\, 1\/v(x) \\cdot \\, L_N \\cdot \\, v(x)$, and the \n(normalized) pullbacked operator $\\, L_N^{(p)}$ which amounts to changing \n$\\, x \\, \\rightarrow \\, \\, y(x)$ in $\\, L_N$.  \nThe pullbacked operator $\\, L_N^{(p)}$ reads\n\\begin{eqnarray}\n\\label{L_Np}\n\\hspace{-0.95in}&& \\quad   \\quad  \nL_N^{(p)} \\, \\, \\, = \\, \\, \\, \\,\\,  D_x^N  \\, \\,\\,  \\, + \\, \n\\Bigl(p(y(x)) \\cdot \\, y'(x) \\,\n - {{N\\cdot \\, (N-1)} \\over {2}} \\cdot \\,  {{y\"(x)} \\over {y'(x)}} \\Bigr) \n\\cdot \\, D_x^{N-1}  \n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad  \\quad  \\quad  \n+ \\, \\Bigl(  q(y(x)) \\cdot \\, y'(x)^2 \\, \n - {{ (N-2) \\cdot \\, (N-1)} \\over {2}} \\cdot \\, p(y(x)) \\cdot \\, y\"(x) \n\\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\quad  \\quad  \n\\,  - {{ N \\cdot \\, (N-1) \\cdot \\, (N-2)} \\over {6}} \n\\cdot \\, {{y^{(3)} } \\over { y'(x)}}\n \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\quad  \\quad  \n\\,  - {{ (N+1) \\cdot \\, N \\cdot \\, (N-1) \\cdot \\, (N-2)} \\over {8}}\n \\cdot \\, \\Bigl({{y^{(2)} } \\over { y'(x)}}\\Bigr)^2\n \\Bigr)  \\cdot \\, D_x^{N-2} \n\\, \\,\\,  \\, \\, + \\, \\,\\, \\cdots\n \\nonumber \n\\end{eqnarray}\nand the conjugate of (\\ref{L_N}) reads:\n\\begin{eqnarray}\n\\label{L_Nc}\n\\hspace{-0.95in}&& \\quad \\, \\, \nL_N^{(c)} \\, \\, \\, = \\, \\, \\, \\, \\, D_x^N  \n\\, \\, \\, \\, + \\, \n\\Bigl(p(x) \\, +N \\cdot \\, {{v'(x)} \\over {v(x)}}\\Bigr) \\cdot \\, D_x^{N-1} \n \\\\\n\\hspace{-0.95in}&& \\quad \\quad \n \\, \\, \\,\\,\n + \\,  \\Bigl( q(x) \\,\n \\, +(N-1) \\cdot \\,  {{v'(x) } \\over {v(x) }}\n  \\cdot \\, p(x)  \n \\,  \\,  + {{N\\cdot \\, (N-1)} \\over {2}} \\cdot \\, {{v\"(x) } \\over {v(x) }} \n    \\Bigr)  \\cdot \\, D_x^{N-2} \n\\,\\, \\, \\, \\, \\,  + \\, \\, \\, \\cdots\n \\nonumber \n\\end{eqnarray}\nWe impose the identification of these two order-$N$ linear differential operators:\n\\begin{eqnarray}\n\\label{vLNgenerggfirst}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad   \\quad   \\quad   \\quad  \n {{1} \\over {v(x)}} \\cdot \\,  L_N  \\cdot \\, v(x) \n\\, \\,\\,  = \\, \\, \\,  \\, pullback\\Bigl(L_N, \\,  y(x)  \\Bigr).\n\\end{eqnarray}\nThe identification of the $\\, D_x^{N-1} $ coefficients gives the \nexact expression of $\\, v(x)$ in terms of the wronskian $\\, w(x)$ \nand of the pullback $\\, y(x)$:\n\\begin{eqnarray}\n\\label{vN}\n\\hspace{-0.95in}&& \\quad  \nv(x) \\, \\, = \\, \\, \\,  \n y'(x)^{ -(N-1)\/2} \\cdot \\,   \\Bigl({{w(x) } \\over {w(y(x)) }}\\Bigr)^{1\/N}\n \\quad  \\,   \\, \\hbox{where:}   \\,  \\,   \\,   \\,   \\,   \\quad \np(x) \\, \\, = \\, \\, \\, - \\, {{w'(x)} \\over {w(x)}}.\n\\end{eqnarray}\nInjecting this exact expression in (\\ref{L_Nc}), or eliminating the \nlog-derivative $\\, v'(x)\/v(x)$, the identification of the\n$\\, D_x^{N-2} $ coefficients gives the following Schwarzian equation\n\\begin{eqnarray}\n\\label{SchwarN}\n\\hspace{-0.95in}&& \\quad \\quad   \\quad \\quad  \\quad \\quad  \nW(x) \n\\, \\, \\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\,  \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\label{wherecondN}\n\\hspace{-0.95in}&&  \nW(x)  \\, \\, = \\, \\,  \\,  \\,\n{{6} \\over { (N+1) \\cdot \\, N }} \\cdot \\,  {{ d p(x)} \\over { dx}} \n \\, \\,  \\,   + \\, \\,  {{6 \\cdot \\,   p(x)^2 } \\over {(N+1) \\cdot \\, N^2}} \n \\,  \\,  \\, \n - \\, {{12 \\cdot \\,  q(x)} \\over {(N+1) \\cdot \\, N \\cdot \\, (N-1) }},\n\\end{eqnarray}\ni.e.\n\\begin{eqnarray}\n\\label{wherecondNmore}\n\\hspace{-0.95in}&&  \\quad \\quad \nW(x)  \\, \\, = \\, \\,  \\,  \\, \n {{6} \\over { (N+1) \\cdot \\, N }} \\cdot \\, {\\cal W}(x)\n \\quad \\quad \\quad\\quad \\quad \\quad\n\\hbox{where:} \n \\\\\n\\hspace{-0.95in}&&  \\quad \\quad \n{\\cal W}(x)  \\, \\, = \\, \\,  \\,  \\,  \n {{ d p(x)} \\over { dx}} \n \\, \\,   \\,  + \\, \\,  {{  p(x)^2 } \\over { N}} \n \\,  \\,  \\,  - \\, {{2 \\cdot \\,  q(x)} \\over {N-1 }}\n \\, \\, \\,  = \\, \\,  \\,  \\,  \\,  \n  N \\cdot \\,  {{z\"(x)} \\over  {z(x)}} \n \\, \\,  \\,  - \\, {{2 \\cdot \\,  q(x)} \\over {N-1 }},  \n\\end{eqnarray}\nwhere: \n\\begin{eqnarray}\n\\label{wherecondNbis}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad  \\, \\, \\, \\,  \\quad \\quad \\quad \n z(x) \\, \\,  = \\, \\,  \\, w(x)^{-1\/N}, \\quad \\quad \\quad \\,\\, \n p(x) \\, \\, = \\, \\, \\, - {{w'(x)} \\over {w(x)}}.\n\\end{eqnarray}\n\n\\vskip .1cm\n\nThis is in agreement with the fact that the symmetric $\\, (N-1)$-th \npower of an order-two linear differential operator \n$\\, L_2 \\, = \\, D_x^2 \\, +A(x) \\cdot \\, D_x \\, + \\, B(x)\\, $ \ngives an order-$N$ linear differential operator \n$\\, L_N\\, = \\,\\, D_x^N \\,\\, +p(x) \\cdot \\, D_x^{N-1} \n\\,\\, + \\, q(x)\\cdot \\, D_x^{N-2} \\,\\, + \\, \\, \\cdots \\, \\,\\,\\, $ \nsuch that\n\\begin{eqnarray}\n\\label{suchthat}\n\\hspace{-0.95in}&&  \np(x) \\, \\, = \\, \\, \\,\n {{ N \\cdot \\, (N-1)} \\over {2}} \\cdot A(x), \\quad \n\\nonumber \\\\ \n\\hspace{-0.95in}&&  \nq(x) \\, \\, = \\, \\, \\, \\,\n {{ (3\\,N-1) \\cdot \\, N \\cdot \\, (N-1)  \\cdot \\, (N-2) } \\over {24}} \\cdot \\,  A(x)^{2}\n\\, \\, +\\,  {{ N \\cdot \\, (N-1)  \\cdot \\, (N+1)} \\over {6}} \\cdot \\,  B(x) \n\\nonumber \\\\ \n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad \\quad  \\quad \\quad  \\quad \n+ \\, {{ N \\cdot \\, (N-1)  \\cdot \\, (N-2)} \\over  {6}} \n \\cdot \\,  {{ d A(x)} \\over {dx}}, \n\\end{eqnarray}\nand thus conversely:\n\\begin{eqnarray}\n\\label{suchthatconverse}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \nA(x) \\,\\, = \\, \\,\\, {{2 } \\over { N \\cdot \\, (N-1)}} \\cdot \\, p(x), \\quad \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \nB(x) \\,\\, = \\, \\,\\, \\,\n {{6 \\cdot \\, q(x)} \\over { (N+1) \\cdot \\, N \\cdot \\, (N-1)}}\n \\,  \\, \\,  \\, - {{(3\\, N \\, -1) \\cdot \\, (N -2)  \\cdot \\, p(x)^2 } \\over {\n   (N+1) \\cdot \\, N^2 \\cdot \\, (N-1)^2 }} \n\\nonumber  \\\\ \n\\hspace{-0.95in}&& \\quad  \\quad  \\quad \\quad \\quad  \\quad \\quad \\quad \\quad \n\\,  -\\, {{2 \\cdot \\, (N-2) } \\over {(N+1) \\cdot \\, N \\cdot \\, (N-1) }} \n \\cdot \\,  {{ d p(x)} \\over { dx}}. \n\\end{eqnarray}\nInjecting (\\ref{suchthatconverse}) in the expression of $\\, W(x)$ for \nan order-two linear differential operator $\\, L_2$ (see (\\ref{wherecond}))\n\\begin{eqnarray}\n\\label{WL2}\n\\hspace{-0.95in}&&   \\quad \\quad  \\quad \\quad  \\quad \\quad \nW(x)  \\, \\, = \\, \\,  \\,  \\, {{ d A(x)} \\over { dx}} \n \\, \\, \\,   + \\, \\,  {{ A(x)^2 } \\over {2}} \n \\,  \\, \\,  - \\, 2 \\cdot \\, B(x),\n\\end{eqnarray}\none gets again the expression (\\ref{wherecondN}) \nfor $\\, W(x)$ for an order-$N$ linear differential operator \n$\\,\\,\\, L_N\\, = \\,\\, $\n$\\,\\,D_x^N \\, \\, +p(x) \\cdot \\, D_x^{N-1} $\n$\\,\\,  + \\, q(x)\\cdot \\, D_x^{N-2} \\,\\,  + \\, \\, \\cdots $ \n\n\\vskip .1cm \n\\vskip .1cm \n\n{\\bf Remark:} the Schwarzian condition (\\ref{SchwarN}) and the \nassociated function $\\, W(x)$ given by (\\ref{wherecondN}), \ncorrespond to an elimination of the conjugation function \n$\\, v(x)$ in (\\ref{vLNgenerggfirst}). If one changes \nthe order-$N$ linear differential operator $\\, L_N$ by conjugation, \n$\\,  L_N \\, \\rightarrow \\, \\, $\n$\\tilde{L}_N \\, = \\, \\, 1\/\\rho(x) \\cdot \\,  L_N   \\cdot \\, \\rho(x)$,\none gets again (\\ref{vLNgenerggfirst}),  $\\, L_N$ being replaced \nby $ \\, \\tilde{L}_N$ and $\\, v(x)$ being replaced by  $ \\, \\tilde{v}(x)$:\n\\begin{eqnarray}\n\\label{new}\n\\hspace{-0.95in}&&  \\,   \\,   \\quad \\quad  \\quad \\quad  \\quad \\, \\,    \nv(x) \\quad \\quad  \\longrightarrow \\quad \\, \\,   \\quad \n \\tilde{v}(x) \\, \\, = \\, \\,  {{ v(x) \\cdot \\,  \\rho(y(x)) } \\over {  \\rho(x)  }}.  \n\\end{eqnarray}\nConsequently one gets again the same Schwarzian condition (\\ref{SchwarN}) \nwith the function $\\, W(x)$ given by (\\ref{wherecondN}), \nsince they are obtained by elimination of the conjugation functions \n$\\, v(x)$ or $ \\, \\tilde{v}(x)$. Therefore $\\, W(L_N, \\,x)$ given \nby (\\ref{wherecondN}), which is invariant by the conjugation \n$\\,  L_N \\, \\rightarrow \\, \\, 1\/\\rho(x) \\cdot \\,  L_N   \\cdot \\, \\rho(x)$, \nis left invariant by:\n\\begin{eqnarray}\n\\label{accordingxtext}\n\\hspace{-0.95in}&&  \\,   \\,   \n p(L_N, \\, x) \\, \\, \\,   \\,  \\longrightarrow \\, \\, \n \\quad   \\, \\,  \\, \\,   p(L_N, \\, x)\n\\, \\, \\,\\, + \\, N \\cdot \\, {{\\rho'(x)} \\over {\\rho(x)}},\n\\\\\n\\label{accordingx2text}\n\\hspace{-0.95in}&&  \\,  \\,   \n  q(L_N, \\, x) \\, \\,  \\,   \\, \\longrightarrow \\, \\,  \\, \\, \n \\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad   \\quad  \\quad    \\quad  \n q(L_N, \\, x) \\,\\,  \\, \\,\n + \\, (N-1) \\cdot \\, {{\\rho'(x)} \\over {\\rho(x)}} \\cdot \\,  p(L_N, \\, x)\n\\, \\, \\, \\, \n+ \\, {{N \\cdot \\, (N-1)} \\over {2}} \\cdot \\, {{\\rho\"(x)} \\over {\\rho(x)}}. \n\\end{eqnarray}\nConversely imposing this invariance by conjugation (\\ref{accordingxtext}),  \n(\\ref{accordingx2text}), on a function of the form \n$\\,\\,  W(x)  =   \\alpha_N \\cdot \\, p'(x) \n + \\beta_N \\cdot \\, p(x)^2  + \\gamma \\cdot \\, q(x)\\, $\ngives  (\\ref{wherecondN}) {\\em up to an overall constant factor}.\n\n\\vskip .2cm \n\n\\section{Solutions of the Schwarzian conditions} \n\\label{CalabiSchwResults}\n\nLet us study the solutions $\\, y(x)$ of the  Schwarzian equation (\\ref{SchwarN})\nthat  emerge for any pullback-symmetry condition of linear differential \noperators of arbitrary order $\\, N$. This should provide valuable\ninformation on the pullbacks that are symmetries of linear \ndifferential operators.\n\n\\vskip .2cm\n\n\\subsection{Solutions of the Schwarzian equation that are \ndiffeomorphisms of the identity: a condition on $\\, W(x)$} \n\\label{around}\n\nThe Schwarzian condition (\\ref{condition1}) has been shown in~\\cite{Aziz} \nto be compatible under the {\\em composition of the pullback-functions} \n$\\, y(x)$ verifying (\\ref{condition1}). \nThe fact that the composition of two solutions  $\\, y(x)$ \nof the  Schwarzian condition (\\ref{condition1})\nis also a solution\\footnote[2]{See Appendix D in~\\cite{Aziz}.} of the  \nSchwarzian condition (\\ref{condition1}), is crucial to describe the set \nof solutions $\\, y(x)$ of (\\ref{condition1}). Once a solution \n$\\, y(x)$ of the  Schwarzian condition (\\ref{condition1}) is known, \nthe $\\, n$-th composition \n$\\, y^{(n)}(x) \\, = \\, \\, y(y( \\, \\cdots \\, y(x) \\, \\cdots \\, )))$, \nyields automatically a commuting set of solutions\\footnote[5]{Cum grano salis:\nwhen the pullbacks $\\, y(x)$ are algebraic functions, they are \n{\\em multivalued functions}. The composition \nof multivalued functions is limited to their analytic series expansions \n(setting aside Puiseux series).} \nof (\\ref{condition1}). \nBy obtaining the series expansions of these solutions, one \ncan extend to non integer complex values \nof $\\, n$, and in order to build a one-parameter family of {\\em commuting} solution \nseries, consider the infinitesimal composition~\\cite{Hindawi}:\n\\begin{eqnarray}\n\\label{yinfini}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \n \\quad  \\quad  \\quad \\, \\,   \\,  \\,    \\, \ny_{\\epsilon}(x) \\, \\, \\, = \\,  \\, \\,\\,  \\,\n  x \\, \\, \\, \\,  + \\epsilon \\cdot \\, F(x) \\,  \\,\\, +  \\, \\, \\, \\cdots \n\\end{eqnarray}\nThe  one-parameter family of commuting solution series $\\, y^{(n)}(x)$ \ncommutes with (\\ref{yinfini}) yielding the functional \nequations~\\cite{Hindawi}:\n\\begin{eqnarray}\n\\label{functional}\n\\hspace{-0.95in}&& \\, \\,\\, \\quad  \\quad   \\, \\, \nF(x)  \\cdot \\,  {{d y^{(n)}(x)} \\over {dx}}  \\, \\, = \\, \\, \\,  F(y^{(n)}(x)),\n\\quad \\quad  \\, \\,  \n F(x)  \\cdot \\,  {{d y_{\\epsilon}(x)} \\over {dx}}  \\, \\, = \\, \\, \\,  F(y_{\\epsilon}(x)).\n\\end{eqnarray}\nInserting (\\ref{yinfini}) in the Schwarzian condition (\\ref{condition1}), \none sees that $\\, F(x)$ is actually {\\em holonomic} being solution of \nthe linear differential equation of {\\em order-three}:\n\\begin{eqnarray}\n\\label{equaF}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad  \\quad \\quad  \\,\\,\n{{d^3 F(x)} \\over { dx^3}}  \\, \\,  \\,\n - \\, 2 \\cdot \\,  W(x) \\cdot \\,  {{d F(x)} \\over { dx}}\n \\, \\, \\,  - \\, {{d W(x)} \\over { dx}}     \\cdot \\,   F(x) \n\\,\\, \\, \\,  = \\, \\,   \\, \\,\\, \\, 0,\n\\end{eqnarray}\nwhose corresponding order-three linear differential operator \n$\\, {\\cal L}_3$ is {\\em exactly the symmetric square of an \nunderlying order-two linear differential operator}\\footnote[3]{The \nreduction of $\\, {\\cal L}_3$ to a symmetric square  (\\ref{equaFop}) \n{\\em does not mean that} $\\, F(x)$ is solution of a second order \nlinear differential (Liouvillian) equation \n$\\, F\"(x)\/F(x) \\, \\, \\, = \\,  \\, \\,\\, \\,  W(x)\/2$.} \n$\\, {\\cal L}_2$:\n\\begin{eqnarray}\n\\label{equaFop}\n\\hspace{-0.95in}&& \\quad  \\quad  \n {\\cal L}_3  \\,  \\, = \\,  \\, \n D_x^3  \\, \\, - \\, 2 \\cdot \\,  W(x) \\cdot \\, D_x \\, \\, - \\, {{d W(x)} \\over { dx}} \n \\, \\,  \\,  \\, = \\,  \\,  \\, \\,\\,  Sym^2\\Bigl( D_x^2 \\, -\\, {{W(x)} \\over {2}}  \\Bigr).\n\\end{eqnarray}\nConversely $\\, W(x)$ can be expressed in terms of $\\, F(x)$ as follows:\n\\begin{eqnarray}\n\\label{equaFthus}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \\quad \\quad \nW(x)\\, \\, \\, = \\,  \\, \\,\\, \\, \\,   \n{{F\"(x) } \\over { F(x)}} \\,  \\,\n -{{1} \\over {2}} \\cdot \\,\\Bigl( {{F'(x) } \\over { F(x)}} \\Bigr)^2 \n\\, \\, \\,  + \\, \\, {{ \\lambda} \\over {F(x)^2}}\n\\\\\n\\label{equaFthusricatti}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \\quad \\quad \\quad \\quad \n\\, \\, \\, = \\,  \\, \\,\\, \\, \\,  \n{{d } \\over {dx}} \\Bigl(  {{F'(x) } \\over { F(x)}} \\Bigr)  \\, \n+ \\, \\, {{1} \\over {2}} \\cdot \\,\\Bigl( {{F'(x) } \\over { F(x)}} \\Bigr)^2 \n\\, \\, \\,  + \\, \\, {{ \\lambda} \\over {F(x)^2}}.\n\\end{eqnarray}\nThis last result (\\ref{equaFthus}) is easily obtained \nby multiplying the LHS of (\\ref{equaF}) by $\\, F(x)$ and integrating \nthe result. One gets this way\\footnote[1]{This ``gauge'' \n$\\,  W(x) \\, \\rightarrow \\, W(x) \\,+ \\, \\lambda\/F(x)^2 \\,  $ \nin (\\ref{equaFthus}) corresponds to the fact that because \nof (\\ref{functional}) one has \n$\\, \\lambda\/F(x)^2 \\, - \\lambda\/F(y(x))^2 \\cdot \\, y'(x)^2 \\, = \\, 0$ \nwhich allows to change  \n$\\,  W(x) \\, \\rightarrow \\, \\, W(x) \\,+ \\, \\lambda\/F(x)^2\\, $ \nin the Schwarzian equation  (\\ref{condition1}), as well as in the \nthird order linear differential ODE (\\ref{equaF}). One easily verifies \nthat inserting (\\ref{equaFthus}) in (\\ref{equaF}) {\\em gives an identity}. \n}:\n\\begin{eqnarray}\n\\label{Casimir}\n\\hspace{-0.95in}&& \\quad   \\quad \\quad \nF(x) \\cdot \\, {{d^2 F(x)} \\over {dx^2}} \\, \\, \\, \n -{{1} \\over {2}} \\cdot \\,  \\Bigl({{d F(x)} \\over {dx}}\\Bigr)^2\n \\, \\,  + \\, \\lambda \\, \\,  \\,  -\\, F(x)^2 \\cdot \\, W(x) \\,  \\,\\, = \\, \\, \\, 0,\n\\end{eqnarray}\nwhich is (\\ref{equaFthus}). \nThus, for a given pullback $\\, y(x)$, or for\na given {\\em one-parameter} family of commuting solution series (\\ref{yinfini}), \nor for a given $\\, F(x)$, one has a one-parameter family (\\ref{equaFthus})\nof $\\, W(x)$ in the Schwarzian equation  (\\ref{condition1}). Conversely, \nfor a given  $\\, W(x)$, one has {\\em at least} a one-parameter family of \ncommuting solution series (\\ref{yinfini}). \n\n\\vskip .1cm\n\n\\subsubsection{Selected subcase of the Schwarzian equation. \\\\} \n\\label{subcase}\n\n\\vskip .1cm\n\nLet us consider an order-two linear differential operator\n $\\, L_2\\, = \\, \\, D_x^2 \\, + \\, A(x) \\cdot \\, D_x  \\, + \\, B(x)$ \n(where $A(x)$ and $\\, B(x)$ are rational functions), \nsuch that its corresponding function\n$\\, W(x) \\, = \\, \\,\\, A'(x)\\, +A(x)^2\/2 \\, -2 \\, B(x) \\, \\, $ \n(see (\\ref{wherecond})) in the  \nSchwarzian equation (\\ref{condition1}), is of the form \n(see subsection 6.2 of~\\cite{Aziz}) \n\\begin{eqnarray}\n\\label{WAR}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \\quad \\quad \\quad \n W(x) \\,  \\, = \\, \\,  \\, \\,  \n {{d A_R(x)} \\over {dx}} \\,\\, + \\, {{ A_R(x)^2 }  \\over {2}}, \n\\end{eqnarray}\nwhere $\\, A_R(x)$ is a {\\em rational function}.\nIntroducing the rational function $\\, C(x) \\, = \\, (A(x) \\, - A_R(x))\/2$,  \nthe identification of the expression of $\\, W(x)$, namely \n$\\, W(x) \\, = \\, \\, A'(x)\\, +A(x)^2\/2 \\, -2 \\, B(x)$\nwith (\\ref{WAR}), gives $\\, B(x)$ in terms of $\\, A_R(x)$ and $\\, C(x)$\n\\begin{eqnarray}\n\\label{Bx}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \\quad \\quad \\quad \n B(x) \\,  \\, = \\, \\,  \\, \\,  \n  \\, {{d C(x)} \\over {dx}} \n\\,\\,   +  C(x) \\cdot \\, ( C(x) \\,  \\, + \\, A_R(x)),    \n\\end{eqnarray}\nwhich is the condition for the order-two linear differential operator \n$\\, L_2$ {\\em to factorize into two order-one linear differential operators}:\n\\begin{eqnarray}\n\\label{factoL2}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \\quad \\, \\, \n L_2\\,\\, = \\, \\, \\,\n \\Bigl(D_x \\, + \\, A_R(x) \\, + \\,C(x)\\Bigr) \\cdot \\, \\Bigl(D_x \\, \\, + \\,C(x)\\Bigr).\n\\end{eqnarray}\nIn other words, condition (\\ref{WAR}) with $\\, A_R(x)$ a rational function, is \nthe condition of factorization of the order-two linear differential operator \n$\\, L_2$. In this case, the Schwarzian equation (\\ref{condition1}) \nreduces to a simpler second order {\\em non-linear} differential \nequation (that was studied extensively in~\\cite{Aziz,Hindawi}):\n\\begin{eqnarray}\n\\label{WAR2}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\quad \\quad \n {{d^2 y(x)} \\over {dx^2}}  \\,  \\, = \\, \\,  \\, \\,  \n A_R(y(x)) \\cdot \\, \\Bigl({{d y(x)} \\over {dx}}\\Bigr)^2 \n\\,\\, - \\,  A_R(x) \\cdot \\, {{d y(x)} \\over {dx}}. \n\\end{eqnarray}\nSeeking the following one-parameter solutions (\\ref{yinfini}), \n$\\, y_{\\epsilon}(x) \\, = \\,   \nx \\, + \\,  \\epsilon \\cdot \\, F(x) \\, +  \\, \\cdots$,  \none finds that $\\, F(x)$ verifies a linear \ndifferential equation of order two~\\cite{Hindawi} \n\\begin{eqnarray}\n\\label{FepsAR}\n\\hspace{-0.95in}&& \\quad  \\,  \n \\quad  \\quad  \\quad \\, \\,   \\,  \\,    \\, \n{{d^2 F(x)} \\over { dx^2}}  \\, \\,  \\,\n -  \\,  A_R(x) \\cdot \\,  {{d F(x)} \\over { dx}}\n \\, \\, \\,  - \\, {{d A_R(x)} \\over { dx}}     \\cdot \\,   F(x) \n\\,\\, \\, \\,  = \\, \\,   \\, \\,\\, \\, 0,\n\\end{eqnarray}\ncorresponding to the linear differential \noperator of order two\\footnote[1]{In fact the order-two operator \n$\\, {\\cal L}_F$  is the adjoint of the operator \n$\\, \\, \\Omega = \\,(D_x \\, +A_R(x)) \\cdot \\, D_x$ \n(see~\\cite{Hindawi}). When  $\\, A_R(x) \\, = \\, \\, -w'(x)\/w(x)$ \nthe linear differential operator $\\, {\\cal L}_F$ is conjugated by the \nwronskian $\\, w(x)$ to the linear differential operator $\\, \\Omega$, \nnamely $\\, \\Omega \\cdot \\, w(x) = \\, w(x) \\cdot \\,{\\cal L}_F$.}:\n\\begin{eqnarray}\n\\label{FepsARop}\n\\hspace{-0.95in}&& \\quad  \\,  \n \\quad  \\, \\,  \\,  \\,  \\, \n{\\cal L}_F  \\,  \\,  \\, = \\, \\,  \\,  \\, \nD_x^2  \\, \\, \n -  \\,  A_R(x) \\cdot \\, D_x\n \\, \\, \\,  - \\, {{d A_R(x)} \\over { dx}}\n  \\,  \\, \\, = \\,  \\, \\,  \\, D_x \\cdot \\, \n\\Bigl(D_x\\, -A_R(x)\\Bigr).\n\\end{eqnarray}\nIntroducing the wronskian $\\, w(x)$, $\\, A_R(x)$ reads\n $\\, A_R(x) \\, = \\, \\, -w'(x)\/w(x)$. Thus  the linear differential \noperator (\\ref{FepsARop}) has two solutions:  $\\, 1\/w(x)$ which is \nthe solution of the right factor $\\, D_x\\, -A_R(x)$, and another (transcendental) \nsolution that we denote $\\, {\\cal S}_F$. The function $\\, F(x)$ \ncorresponds to this last (transcendental) solution, and \n{\\em not the} $\\, 1\/w(x)$ \n{\\em solution}. Conversely $\\, A_R(x)$ can be \nexpressed\\footnote[2]{Just integrate the LHS of (\\ref{FepsAR}).} \nin terms of $\\, F(x)$ as follows:\n\\begin{eqnarray}\n\\label{conversFepsAR}\n\\hspace{-0.95in}&& \\quad \\quad \\quad  \\quad  \\quad  \\quad \\quad  \\,  \nA_R(x) \\,  \\,  = \\, \\, \\,\\, {{ F'(x)} \\over { F(x)}} \n\\, \\, + \\, \\, {{\\mu} \\over { F(x)}}.\n\\end{eqnarray}\nOne easily verifies that by inserting \n(\\ref{conversFepsAR}) in (\\ref{FepsAR}) ones gets\nan identity, and that by inserting (\\ref{conversFepsAR}) \nin (\\ref{WAR}) one recovers (\\ref{equaFthusricatti}) with\n$\\, \\lambda \\, = \\, \\mu^2\/2$. Here the $\\, \\mu\/F(x)$ \n{\\em term is crucial}, because when\n$\\,  \\mu \\, = \\, 0$ condition (\\ref{conversFepsAR})\n with $\\, A_R(x) \\, = \\, \\, -w'(x)\/w(x)$  yield \nthe trivial result,  $\\,\\, F(x) \\, = \\, 1\/w(x)$ which is\ndifferent from the transcendental (holonomic) function \nwe are looking for. For instance in the example detailed \nin~\\cite{Hindawi}, we \nhad the condition (\\ref{conversFepsAR})  verified \nwith $\\, \\mu \\, \\ne \\, 0$, namely $\\,\\mu \\, = 1\/4$:\n \\begin{eqnarray}\n\\label{Hindwaiexample}\n\\hspace{-0.95in}&&  \\quad  \\quad \nF(x) \\, = \\,   \\, \n x \\cdot \\, (1-x)^{1\/2} \\cdot \\, \n_2F_1\\Bigl([{{1} \\over {2}}, \\, {{1} \\over {4}}],[{{5} \\over {4}}], \\, x\\Bigr), \n\\, \\, \\quad  \\,\\,\\, A_R(x) \\, = \\, \\, {{ 3 \\,-5\\,x } \\over {4 \\, x \\, (1\\, -x) }}.\n\\end{eqnarray}\n\n\\vskip .2cm  \n  \nAt first sight one expects the order-two linear differential \nequation (\\ref{FepsAR}) on $\\, F(x)$ to be a simple limit \nof the order-three linear differential \nequation (\\ref{equaF}) when the condition \n(\\ref{WAR}) is imposed.\nThis reduction is not obvious however and the interested reader can\nfind it explained in \\ref{Reduction32}. \n\n\\vskip .2cm    \n\n{\\bf Remark:} the global nilpotence of the linear differential \noperators gives an $\\, A_R(x)$ of the form  \n$\\, A_R(x) \\, = \\, \\, -w'(x)\/w(x)$, \nwhere the wronskian $\\, w(x)$ is an $\\, N$-th root of a \nrational function~\\cite{bo-bo-ha-ma-we-ze-09}. Using \n$\\, A_R(x) \\, = \\, \\, -w'(x)\/w(x)$, \ncondition (\\ref{WAR2}) can be  easily integrated into\n\\begin{eqnarray}\n\\label{WAR21}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \n{{ d y(x)} \\over { dx}} \\, \\, = \\, \\, \\,  \\, \nc_1 \\cdot \\, {{ w(x)} \\over {w(y(x))}}\n \\quad   \\quad  \\quad   \\quad   \\quad  \\hbox{or:}  \n\\\\\n\\label{WAR22}\n\\hspace{-0.95in}&& \\quad \\quad  \n\\Theta(y(x)) \\, \\, = \\, \\, \\,  \\, c_1 \\cdot \\,  \\Theta(x) \\, + \\, c_2\n \\quad  \\quad   \\quad\\hbox{with:}  \\quad  \\quad  \\quad \n\\Theta(x) \\, = \\, \\, \\int^x \\, w(x) \\, dx \n\\end{eqnarray}\nwhere $\\, c_1$ and  $\\, c_2$ are constants of integration.\n\n\\vskip .1cm\n\n\\vskip .1cm\n\n\\vskip .1cm\n\n Now let us describe \nthis one-parameter family of commuting solution series (\\ref{yinfini}) \nof the  Schwarzian equation  (\\ref{condition1}).\n\n\\vskip .1cm\n\n\\subsection{Solutions of the Schwarzian equation that are diffeomorphisms \nof the identity: the general formal solution} \n\\label{formal}\n\nLet us consider (\\ref{yinfini}) as a series in $\\, \\epsilon$:\n\\begin{eqnarray}\n\\label{expansion}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\, \\,  \\quad \\quad \ny_{\\epsilon}(x)  \\, \\, = \\, \\, \\, \\,  \\,\n x \\, \\, \\, \\, +  \\, \\epsilon \\cdot \\, F(x) \\, \\, \\, \n+ \\, \\sum_{n=2}^{\\infty} \\,  {{\\epsilon^n} \\over {n!}} \n \\cdot \\,  F(x)  \\cdot \\, Q_n(x),  \n\\end{eqnarray}\n solution of the functional equation (\\ref{functional}). This \n is sufficient to find, order by order in $\\, \\epsilon$, \nthe solution (\\ref{expansion})  of (\\ref{functional}) \nwhere the $\\, Q_n(x)$ are given by\n\\begin{eqnarray}\n\\label{recursion}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad \nQ_1(x) \\, = \\, \\, F(x), \\quad \\quad \\quad  \\quad  \\, \\,  \\, \\,  \\,\nQ_2(x)  \\, = \\,\\, \\,  F(x) \\cdot \\, {{ d Q_1(x) } \\over {dx}} \n\\, = \\, \\, \\,  F(x) \\cdot \\, {{ d F(x)  } \\over {dx}} , \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad \\quad   \\quad \nQ_3(x)  \\, = \\, \\,  F(x)  \\cdot \\, {{ d } \\over {dx}} \\, Q_2(x) \n \\,\\,  = \\, \\, \\,   F(x) \\cdot \\, \\Bigl(F(x) \\cdot \\, F\"(x)\n \\,\\,  + \\, F'(x)^2\\Bigr),\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad  \nQ_4(x)  \\, = \\, \\,  F(x)  \\cdot \\,  {{ d } \\over {dx}}  \\, Q_3(x), \n\\quad \\quad  \\quad  \\, \\,  \nQ_5(x)  \\, = \\, \\,  F(x)  \\cdot \\, {{ d } \\over {dx}} \\, Q_4(x), \n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad   \\quad \\quad \n\\, \\,   \\, \\, \\,  \\cdots\n \\quad \\quad  \\quad  \\quad \nQ_{n+1}(x)  \\, = \\, \\, \\,  F(x)  \\cdot \\, {{ d } \\over {dx}}  \\, Q_n(x),\n\\end{eqnarray}\nthe most general solution (\\ref{expansion}) of (\\ref{functional}) \ncorresponding to linear combinations of the $\\, Q_n$'s \nwhich amounts to changing \n$\\, \\epsilon$ in (\\ref{expansion}) into:\n\\begin{eqnarray}\n\\label{change}\n\\hspace{-0.95in}&&  \\quad  \\quad  \\quad \\quad  \\quad \\quad \n\\epsilon \\, \\, \\,\\, \\, \\, \\longrightarrow \\, \\, \\,  \\, \\,  \\quad \n \\epsilon \\cdot \\, \n(1 \\, + \\, \\lambda_1 \\cdot \\epsilon \\, \n +\\,  \\lambda_2 \\cdot \\epsilon^2\n \\, +\\,  \\lambda_3 \\cdot \\epsilon^3 \\, +\\, \\, \\cdots ).\n\\end{eqnarray}\nNote that all the $\\, Q_n$'s are {\\em polynomial expressions of} \n$\\, F(x)$ {\\em and its derivatives}.\n\nThe functional equation (\\ref{functional}) corresponds to the one-form \n $\\,d \\Theta   \\,  = \\, \\, dx\/F(x)  \\,  = \\, \\, dy\/F(y)$ giving:\n\\begin{eqnarray}\n\\label{deriv}\n\\hspace{-0.95in}&&  \\quad \\quad   \\quad \\quad  \\quad    \\quad \\quad\n\\Theta(x)  \\, \\, = \\, \\, \\, \\int^{x} \\, {{ dx} \\over {F(x)}}, \n \\quad \\quad \\quad \\, \\,  \\,  \n {{d} \\over { d\\Theta}} \\, \\,  = \\, \\, \\,\\,  F(x) \\cdot \\, {{ d  } \\over {dx}}.   \n\\end{eqnarray}\nSeeing $\\, x$ as a function \nof $\\, \\Theta$, one finds that the series (\\ref{expansion}) \ntogether with the recursion (\\ref{recursion}),\ngives the well-known Taylor expansion\n\\begin{eqnarray}\n\\label{Taylor}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad  \\quad  \\quad \n y_{\\epsilon}(x(\\Theta)) \\, \\, = \\, \\, \\,\\, \\, \n x(\\Theta) \\,   \\,  \\, \n+ \\, \\sum_{n=1}^{\\infty} \\,  {{\\epsilon^n} \\over {n!}}\n  \\cdot \\,   \\,  {{d^n  \\, x(\\Theta)} \\over { d\\Theta^n}}\n\\, \\, = \\, \\, \\,\\, x(\\Theta \\, + \\, \\epsilon),\n\\end{eqnarray}\nmeaning that  $\\, x \\, \\rightarrow \\, \\, y_{\\epsilon}(x)$ \n{\\em is just a shift in} $\\, \\Theta$\n\\begin{eqnarray}\n\\label{justashift}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad  \\quad  \\quad  \\quad \\quad  \\quad    \n\\, \\Theta_x   \\,\\quad   \\longrightarrow \\, \\quad   \\, \n \\Theta_y   \\,  \\, = \\, \\,  \\, \\,  \\Theta_x \\,  + \\, \\, \\epsilon, \n\\end{eqnarray}\ncorresponding to the integration of the one-form\n$\\,d \\Theta   \\,  = \\, \\, dx\/F(x)  \\,  = \\, \\, dy\/F(y)$.\nThe two transformations \n$\\, y_{\\epsilon_1}(x)$ and  $\\, y_{\\epsilon_2}(x)$ \nof the one-parameter family clearly commute\\footnote[2]{This can \nalso be checked directly using (\\ref{expansion}) with \n(\\ref{recursion}) for any rational function $\\, F(x)$.}:\n\\begin{eqnarray}\n\\label{deriv2}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad  \\quad  \\quad \ny_{\\epsilon_1}(y_{\\epsilon_2}(x(\\Theta))) \n\\, \\, = \\, \\, \\,\\, y_{\\epsilon_1}(x(\\Theta\\, + \\, \\epsilon_2))\n \\, \\, = \\, \\, \\,\\, x(\\Theta\\, + \\, \\epsilon_1 \\, + \\, \\epsilon_2).\n\\end{eqnarray}\n\nOne verifies order by order in $\\, \\epsilon$,  that the one-parameter \nfamily of commuting series (\\ref{expansion}) with (\\ref{recursion})\n{\\em is solution of the Schwarzian equation}\n\\begin{eqnarray}\n\\label{condition1ter}  \n\\hspace{-0.95in}&& \\quad   \\quad \\quad   \\quad  \\quad   \\quad \n W(x) \n\\, \\, \\,\\,  \\,-W(y_{\\epsilon}(x)) \\cdot  \\, y_{\\epsilon}'(x)^2\n\\, \\, \\, \\, \\,+ \\,  \\{ y_{\\epsilon}(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere $\\, W(x)$ is given by (\\ref{equaFthus}). In terms of $\\, \\Theta$, \nthe expression (\\ref{equaFthus}) for $\\, W(x)$ \ncan be  written using the Schwarzian derivative:\n\\begin{eqnarray}\n\\label{identityQbis}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad  \\quad   \\quad  \\quad  \nW(x) \\,\\,  + \\,  \\{\\Theta(x), \\, x\\}  \\,\\,\\, \n -\\, \\lambda \\cdot \\, \\Bigl( {{d \\Theta(x)} \\over {dx}}\\Bigr)^2\n\\, \\, = \\, \\, \\, 0. \n\\end{eqnarray}\nRecalling the chain rule for the Schwarzian derivative of a \ncomposition of functions\\footnote[9]{Namely \n$ \\, \\, \\, \\{\\Theta(y(x)), \\, x\\} \\, \\,\\,  = \\,\\,  \\,$ \n$\\,\\{\\Theta(y(x)), \\, y(x)\\}  \\cdot  \\, y'(x)^2 \\,\\,\\, + \\,  \\{ y(x), \\, x\\}$. } \nand the fact that $\\, d \\Theta(y(x))\/dx \\, = \\, \\, d \\Theta(x)\/dx$, \none finds that the Schwarzian condition (\\ref{condition1ter}) \ncorresponds to the equality of the two Schwarzian derivatives:\n\\begin{eqnarray}\n\\label{identityQbis}\n\\hspace{-0.95in}&&  \\quad  \\quad   \\quad \n \\quad    \\quad  \\quad    \\quad   \\quad    \\quad  \n\\{\\Theta(y(x)), \\, x\\}  \\, \\, \\, = \\, \\, \\, \\,   \\{\\Theta(x), \\, x\\},  \n\\nonumber \n\\end{eqnarray}\nwhich is verified since \n$\\, d \\Theta(y(x))\/dx \\, = \\, \\, d \\Theta(x)\/dx$.\nThis is another way\nto see that the one-parameter \nfamily of commuting series (\\ref{expansion}) \n(with the $\\, Q_n$'s given by (\\ref{recursion}))\n is  solution of the Schwarzian equation.\n\n\\vskip .2cm \n\n\\subsection{A simple modular form example. } \n\\label{example}\n\n We have considered in~\\cite{Aziz,IsingCalabi,IsingCalabi2,Christol,perimeter} \nmany examples of {\\em modular forms}  \nrepresented as pullbacked $\\, _2F_1$ hypergeometric functions. Each time the \none-parameter commuting series combined with the \nmodular correspondences~\\cite{Eichler} series yields one-parameter series\nof the form $\\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots, \\, $\n $\\, n \\, = \\, 2, \\, 3, \\, 4, \\,  \\cdots \\, \\, $ that are solutions of the\n Schwarzian equation (\\ref{condition1ter}). \n\nIn~\\cite{Aziz} the pullback symmetry of the order-two linear \ndifferential operator was given as a covariance of its  \nsolution, namely a hypergeometric function\nwith {\\em two different\\footnote[5]{We exclude the trivial well-known changes \nof variables on hypergeometric functions \n$\\, x \\, \\rightarrow $\n$ \\, \\, 1\\, -x, \\, 1\/x, ...$ The transformation \n$\\, x \\, \\rightarrow \\, \\, y(x)$ must be an {\\em infinite order} \ntransformation symmetry.} pullbacks} related by modular \nequations\\footnote[3]{\nThe emergence of a {\\em modular form}~\\cite{IsingCalabi,IsingCalabi2,RatMaier} \ncorresponds to the emergence of a selected hypergeometric function \nhaving an exact covariance property~\\cite{Stiller,Zudilin} \nwith respect to an {\\em infinite order algebraic transformation} \n(the modular correspondences).\n}\n\\begin{eqnarray}\n\\label{modularform2explicit}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad  \\quad  \\quad \n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\, y(x)  \\Bigr)\n\\,\\, = \\, \\, \\, \n {\\cal A}(x) \\cdot \\,\n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\,  x  \\Bigr),\n\\end{eqnarray}\nthe pullback $\\, y(x)$ being solution of the Schwarzian \ncondition (\\ref{condition1ter}).\n\nIn this example, the pullback $\\, y_{\\epsilon}(x)$ is solution of \nthe Schwarzian solution (\\ref{condition1ter}) with $\\, w(x)$ and $\\, F(x)$ \ngiven by\\footnote[1]{ One can easily check that these \nexpressions (\\ref{firstbister}) \nfor $\\, W(x)$ and $\\, F(x)$ verify  (\\ref{equaF}).}:\n\\begin{eqnarray}\n\\label{firstbister}\n\\hspace{-0.96in}&&  \nW(x)  = \\, \\,\n -{\\frac {32\\,{x}^{2}-41\\,x+36}{72\\,{x}^{2} \\cdot \\, (x-1)^{2}}}, \n\\, \\,  \\, \\,  \\,   \\, \nF(x) = \\, \n x  \\cdot \\, (1-x)^{1\/2} \\cdot \\,\n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\, x\\Bigr)^2.\n\\end{eqnarray}\nOne can also check that these expressions (\\ref{firstbister})  \n verify  (\\ref{equaFthus}) with\\footnote[2]{This selected value of $\\, \\lambda$ \nhas to be compared with the value $\\, \\mu = \\, 1\/4$ in (\\ref{Hindwaiexample}).} \n$\\, \\lambda \\, = \\, \\, 0$,\n{\\em thus providing a quite non-trivial (non-linear differential)\nidentity between the rational function} $\\, W(x)$ {\\em and the holonomic \nfunction} $\\, F(x)$. \n\nThe one-parameter commuting \nfamily (\\ref{yinfini}) solution of the Schwarzian equation \n(\\ref{condition1ter}) can be expressed using the two (mirror maps) \n{\\em differentially algebraic}~\\cite{Selected,IsTheFull}\nfunctions $\\, P(x)$ and $\\, Q(x)$\ndescribed in~\\cite{Aziz} and in \\ref{Mirror},  \nas $\\,\\,\\, y_1(a_1, \\, x)  =  P(a_1 \\cdot \\, Q(x))$:\n\\begin{eqnarray}\n\\label{Qseries}\n\\hspace{-0.95in}&&   \ny_1(a_1, \\, x) \\,  = \\, \\, a_1 \\cdot \\, x \\, \\, \n-{\\frac {31\\,a_1 \\cdot \\, (a_1 \\,-1) }{72}} \\cdot \\, x^2  \\, \n +{\\frac {a_1 \\cdot \\, (9907\\,{a_1}^{2} -30752\\,a_1 +20845) }{82944}} \\cdot \\, x^3 \n\\nonumber  \\\\\n\\hspace{-0.95in}&&  \\,     \\quad    \\quad      \\quad  \n\\, -{\\frac {a_1 \\cdot \\, (a_1 -1) \\cdot \\,\n (4386286\\,{a_1}^{2} -20490191\\,a_1 +27274051) }{161243136}} \\cdot \\, x^4\n\\, \\, \\,\\, + \\, \\cdots \n\\end{eqnarray}\nwhere $\\,\\,\\, a_1 \\, = \\, \\, \\exp(\\epsilon)$.  \n\nBesides this one-parameter commuting family (\\ref{yinfini}), the Schwarzian \nequation  (\\ref{condition1ter}) has a remarkable (infinite) set of algebraic \nfunctions solutions~\\cite{Aziz}  $\\, y(x)$ defined by the corresponding \n{\\em modular equations}~\\cite{Morain,Andrews,Atkin,Hermite,Hanna,Weisstein}. Their \nseries expansions near \n$\\, x \\, = \\, \\, 0$ read:\n\\begin{eqnarray}\n\\label{Qseriesn}\n\\hspace{-0.95in}&&    \\quad \\quad       \\quad \\quad  \\quad  \ny_n(x) \\, \\,\\, = \\, \\, \\, P(Q^n(x)) \\,  \\, = \\, \\,\\, \\, \\,\n  1728 \\cdot \\, \\Bigl({{ x} \\over {1728}}\\Bigr)^n  \\, \\, + \\,   \\, \\cdots \n\\end{eqnarray}\nwhere $\\, n$ is an integer $\\, n \\, = \\, 2, \\, 3, \\, 4, \\cdots \\, \\, $ These \nseries expansions {\\em commute for different values of the integer} \n$\\, n$. This is a consequence of the fact that, up to the \nprevious change of variables $\\, P(x)$, $\\, Q(x)$, these \n{\\em modular correspondences} (\\ref{Qseriesn}) correspond to taking \nthe $\\, n$-th power of the nome: \n$\\, q \\, \\rightarrow \\, q^n$ (see~\\cite{Aziz} for more details).\n\n\\vskip .2cm \n\n\\subsubsection{A pre-modular concept. \\\\} \n\\label{premodul}\n\nThe composition of the one-parameter series (\\ref{yinfini}) \n(which corresponds to \n$\\, q \\, \\rightarrow \\, a_1 \\cdot \\, q$) and of the modular correspondences \n(\\ref{Qseriesn}), yields an {\\em infinite set of one-parameter series} \n$\\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots, \\, \\, $\n $ n \\, = \\, 2, \\, 3, \\, 4, \\,\\, \\cdots \\, \\,  \\, $\nfor instance~\\cite{Aziz}:\n\\begin{eqnarray}\n\\label{seriesmodcurve3a}\n\\hspace{-0.95in}&& \ny_3  = \\,  \na_3 \\cdot \\, {x}^{3} \\, \n+{\\frac {31\\, a_3  }{24}} \\cdot \\, {x}^{4}\\, \n+{\\frac {36221\\, a_3  }{27648}} \\cdot \\, {x}^{5}  \\,  \n-{\\frac {a_3   \\cdot \\, (23141376\\, a_3 -66458485)\n}{53747712}} \\cdot \\,  {x}^{6}  + \\, \\, \\cdots \n\\nonumber \n\\end{eqnarray}\nThese  one-parameter series do not commute but verify~\\cite{Aziz} \nthe simple composition \nformulae\\footnote[5]{Consequence of the fact, in the nome, they \ncorrespond to the composition of transformations like \n$\\,  q \\, \\rightarrow \\, a_n \\cdot \\, q^n$.}:\n\\begin{eqnarray}\n\\label{curious5}\n\\hspace{-0.95in}&& \\quad   \\quad  \\quad   \\,  \\,  \\, \ny_n(a_n, \\, \\, y_m(a_m,  \\,\\, x)) \\, \\, = \\, \\,  \\,\\, y_{n m}(a_n a_m^n, \\, \\, x),\n \\quad \\, \\,   \\, \\, \\,\\, \\,\\, \\,\nn, \\, m \\, \\, = \\, \\, \\, 1, \\, 2, \\, 3, \\,  \\, \\,  \\, \\cdots \n\\end{eqnarray}\nWhen the $\\, a_n$ are arbitrary rational numbers the corresponding series \n$\\, y_n(a_n, \\, x)$ are {\\em not globally bounded series}~\\cite{Christol} in general. \nTherefore, they {\\em cannot be the series expansion of an algebraic function}: \nthey are {\\em differentially algebraic}~\\cite{Selected,IsTheFull} since \nthey are solutions of the Schwarzian equation (\\ref{condition1ter}).\n\n\\vskip .2cm \n\nIn general, finding the Schwarzian equation (\\ref{condition1ter}) is easy, and\ngetting solutions order by order as series expansions is also easy. \nHowever finding the selected values of the rational numbers $\\, a_n$ \nsuch that the \n{\\em differentially algebraic}~\\cite{Selected,IsTheFull} series\n$\\, y_n(a_n, \\, x)$ are globally bounded and \n{\\em thus can be algebraic functions}, and, possibly, modular \ncorrespondences, is a quite difficult \ntask\\footnote[2]{Similar to finding the selected values\nof the parameters so that a quantum Hamiltonian becomes  integrable,\nor finding modular forms among Beukers' second order differential equations\ndepending on three parameters~\\cite{Zagier} (36 cases emerging\nof a numerical exploration of 10 millions triples). }.\n\nWe will call ``pre-modular\\footnote[8]{Of course, this ``pre-modular'' term\nshould not be confused with the term premodular in premodular \ncategories, (ribbon fusion categories). Here we mean prerequisites\nfor the emergence of modular forms.}'' the existence of an infinite set of \none-parameter differentially algebraic series (solution of the \nSchwarzian equation) of the form \n$\\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots \\, $ which \nverify (\\ref{curious5}), but for which \n{\\em one does not know if there exist some selected values of the parameter} \n$\\, a_n$ such that these {\\em differentially algebraic} \nseries~\\cite{Selected,IsTheFull} become {\\em algebraic functions}.\n\n\\vskip .1cm \n\n\nIn the next section, we will characterize the  Schwarzian \nequations corresponding to these ``pre-modular'' structure, thus finding \n{\\em conditions that are necessary} for the emergence of modular forms.\n\n\\vskip .2cm \n\n\\subsection{Schwarzian equation: conditions for modular correspondence } \n\\label{minus1over2}\n\nIn the previous sections it was shown that the pullback symmetry condition \nof {\\em arbitrary} order-two linear differential operators\nyields Schwarzian equation (\\ref{condition1ter}). The solutions of these\norder-two linear differential operators {\\em are much more general than \nhypergeometric functions and Heun functions}~\\cite{Aziz}: they can \nhave an {\\em arbitrary number of singularities}. Let us see which  \nSchwarzian equation (\\ref{condition1ter}), or equivalently, which \nfunction $\\, W(x)$ gives relations (\\ref{curious5}) corresponding to\n{\\em  rigid constraints necessary to have modular correspondences}~\\cite{Aziz}.\n\nSeries calculations give the conditions on $\\, W(x)$ \nsuch that series solutions of the form \n$\\,  \\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots \\, $ are \nsolutions of the Schwarzian equation with these $\\,  \\, y_n(x)$'s verifying \nrelations (\\ref{curious5}). These constraints are  conditions on the  \n{\\em Laurent series} of $\\, W(x)$. \nFor the solution series of the Schwarzian equation to have the \npre-modular structure (\\ref{curious5}), i.e. the same structure \nas modular correspondences, \nthe  Laurent series of $\\, W(x)$ must be of the form:\n\\begin{eqnarray}\n\\label{Laurent}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad  \\quad \\quad \nW(x)  \\, \\, \\, = \\,\\, \\,\\,\n  \\, -\\, {{1} \\over {2 \\, x^2}}\\, \\, \\,  + \\,  {{b_1} \\over {x}}\n \\,\\,\\, + \\, \\sum_{m=0}^{\\infty} \\, a_m \\cdot \\, x^m. \n\\end{eqnarray}\nOne easily verifies that this is the case for the previous modular form \nexample where $\\, W(x)$ reads (\\ref{firstbister}), as well as for all the other \nmodular forms emerging in physics or enumerative combinatorics we mentioned \nin previous papers~\\cite{IsingCalabi,IsingCalabi2,Christol,Diffalg,perimeter}.\n\n\\vskip .2cm\n\nCondition (\\ref{Laurent}) is a result whose scope transcends the\nhypergeometric functions framework. \nIn order to show this, let us apply this result on the open\nproblem of finding Heun functions\\footnote[1]{Finding the selected values \nof the parameters of a Heun function~\\cite{Heun1} (in particular the \n{\\em accessory parameter}~\\cite{Katz}) such that its series\nexpansion is a series with {\\em integer coefficients} (or more generally is globally \nbounded~\\cite{Christol}), or such that the corresponding order-two linear differential \noperator is {\\em globally nilpotent}~\\cite{bo-bo-ha-ma-we-ze-09} is a difficult problem. \nThese classification problems are closely related to finding the Heun functions \nreducible to pullbacked hypergeometric functions~\\cite{VidunasHoeij}, \nand to modular forms~\\cite{Zagier}.} that  \ncould be modular forms~\\cite{RatMaier}, or pullbacked $\\, _2F_1$ \nfunctions~\\cite{maier-05,BelyiMaier}.\n The Heun function \n$\\, {\\it HeunG} \\left( a, \\,q, \\, \\alpha,\\, \\beta,\\, \\gamma,\\, \\delta, \\, x \\right)$ \nis solution of a linear differential operator of order two \n$\\, L_2 \\, = \\, \\, D_x^2 \\, +A(x) \\cdot \\, D_x \\, +B(x)\\,$ \nwhere $\\, \\,A(x)$ and  $\\, B(x)$\nread:\n\\begin{eqnarray}\n\\label{Heun2F1with}\n\\hspace{-0.96in}&&   \\quad    \nA(x)  \\,\\, = \\, \\, \\,  \\,  \\, \n{\\frac { (\\alpha+\\beta \\, +1) \\cdot \\,  {x}^{2} \\, \n- \\, ( ( \\delta +\\gamma) \\cdot \\,  a \n\\, +\\alpha \\, -\\delta\\,  +\\beta\\, +1) \\cdot \\,  x \n\\,\\, +\\gamma \\cdot \\,a}{ x \\cdot \\, (x-1)  \\cdot \\, (x \\, -a) }},\n\\\\\n\\label{Heun2F1withB}\n\\hspace{-0.96in}&& \n\\quad \\quad \\quad \\quad \\quad \\quad \\quad \nB(x) \\, \\, = \\, \\, \\, \\,  \\, \n {\\frac {\\alpha \\, \\,\\beta \\cdot \\,x \\, \\, -q}{\nx \\cdot \\, (x-1)  \\cdot \\, (x \\, -a) }}.\n\\end{eqnarray}\nThe corresponding function $\\, W(x)$ is easily deduced\nfrom the formula (\\ref{wherecond}) given by \n$\\, W(x) \\, = \\, A'(x) \\, A^2(x)\/2 \\, -2 \\, B(x)$. It\nhas the following {\\em Laurent series} expansion:\n \\begin{eqnarray}\n\\label{Heunexpansion}\n\\hspace{-0.96in}&&    \\quad     \\,    \nW(x)\\, \\, = \\, \\, \\, \\,  \\, \n\\,{\\frac {\\gamma \\cdot \\, (\\gamma  -2) }{{2 \\, \\, x}^{2}}}  \\, \\, \\, \\,   \n-{\\frac {a \\, \\delta\\,\\gamma \\, +\\alpha\\,\\gamma+\\beta\\,\\gamma \\,\n -\\delta\\,\\gamma-{\\gamma}^{2}+\\gamma-2\\,q}{ a \\, \\, x}}\n \\,\\,  \\,  \\, \\, + \\, \\, \\, \\cdots,  \n\\end{eqnarray}\nand has the form (\\ref{Laurent}) given by\n$\\, -1\/2\/x^2 \\, + \\, \\, \\cdots \\, \\,\\, $\n {\\em only when} $\\, \\gamma \\, = \\, \\, 1$. \nThus a general analytical constraint like (\\ref{Laurent}) yields a\nsimple exact constraint on the intriguing problem of the classification \nof the Heun functions that can be modular forms, and more specifically \non the necessary conditions for the Heun functions  to have a ``pre-modular'' structure.\n\n\\vskip .2cm\n\n\\subsubsection{Schwarzian equation for $\\, \\, \\, W(x) \\, = \\, -1\/2\/x^2$. \\\\} \n\\label{limit}\n\nIn order to understand the Laurent series condition (\\ref{Laurent}), let \nus try to see what is so ``special'' in the case where\n $\\, W(x) \\, = \\, -1\/2\/x^2$. For\n\\begin{eqnarray}\n\\label{first}\n\\hspace{-0.95in}&& \\quad \\quad   \\quad \\quad \\quad \\quad   \\, \nW(x)  \\, \\, \\, = \\,\\, \\,  \n-\\, {{1} \\over {2 \\, x^2}} \\, \\, = \\,\\, \\,  \\,  \\, - \\,  \\{ \\ln(x), \\, x\\},\n\\end{eqnarray}\nthe most general solutions of corresponding Schwarzian equation read:\n\\begin{eqnarray}\n\\label{solSchwarrec}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad \\quad  \\quad  \\quad \\quad  \\quad \\quad\ny(x) \\, \\, \\,  = \\, \\,  \\, \\,\n \\exp\\Bigl({{a \\cdot \\ln(x) \\, + \\, b } \\over { \nc \\cdot \\ln(x) \\, + \\, d }}   \\Bigr),\n\\end{eqnarray}\nwhich just amounts to a simple transformation on $\\, \\ln(x)$:\n\\begin{eqnarray}\n\\label{justamounts}\n\\hspace{-0.95in}&& \\quad \\quad    \\quad \\quad  \\quad \\quad\n\\ln(x) \\quad \\longrightarrow \\quad \\quad \n\\ln(y(x))  \\, \\, \\,  = \\, \\,  \\, \\, \\,\n {{a \\cdot \\ln(x) \\, + \\, b } \\over { c \\cdot \\ln(x) \\, + \\, d }}.\n\\end{eqnarray}\nThe solutions of the form\n$\\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots \\, \\, $ \nare given by $\\, \\, \\, y_n(x) \\, = \\, \\, \\, a_n \\cdot \\, x^n$ and are thus ``trivial'': \nthis is the case because the nome\\footnote[2]{Such that the transformations \n$ x \\, \\rightarrow \\,  y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots \\, \\, $ \nsimply reduce to $\\, q \\, \\rightarrow \\,  a_n \\cdot \\, q^n$, see \nthe concept of mirror maps~\\cite{Aziz}. } $\\, q$ {\\em is nothing but \nthe} $\\, x$ {\\em variable!} Similarly, the ratio of periods $\\, \\tau$ is \njust $\\, \\ln(x)$, and thus the condition $\\, W(x) \\, = \\, -1\/2\/x^2$ \nis a ``trivialization'' of the mirror map.\n\n\\vskip .1cm\n\n\\subsubsection{Rank-two condition (\\ref{WAR2})  and pre-modular  structures. \\\\} \n\\label{rank}\n\nThe factorization of the order-two linear differential \noperator which corresponds to $\\, W(x)$ of the form (\\ref{WAR}), yields the \nrank-two {\\em non-linear} differential equation (\\ref{WAR2}) \n(see section \\ref{subcase}).\nWe would like to know when the modular correspondences structures  (existence of \nsolutions series $\\,  \\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots, \\, $\n$n \\, = \\, 2, \\, 3, \\, 4, \\cdots \\, \\, $ such that (\\ref{curious5}), thus \nrequiring $\\, W(x)\\, = \\, -1\/2\/x^2 \\, + \\, \\cdots$) \nare compatible with a factorization of the order-two linear \ndifferential operator and thus with condition (\\ref{WAR}). Imposing \n\\begin{eqnarray}\n\\label{WARminus1over2}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad  \\quad \\quad \n W(x) \\,  \\, = \\, \\,  \\, \\,  \n {{d A_R(x)} \\over {dx}} \\,\\, + \\, {{ A_R(x)^2 }  \\over {2}}\n \\,  \\, = \\, \\,  \\, \\,  \n- \\, {{1} \\over {2 \\, x^2}} \\,  \\, \\, \\, + \\, \\, \\cdots  \n\\end{eqnarray}\nwhere $\\, A_R(x)$ is a rational function, one finds that $\\, A_R(x)$  \nmust have the following Laurent series expansion:\n\\begin{eqnarray}\n\\label{LaurentAR}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad  \\quad \\quad \n A_R(x)  \\, \\,\\, \\, = \\,\\, \\,\\,\n  \\, \\, {{1} \\over { x}} \n\\,\\,\\, \\, + \\, \\sum_{m=0}^{\\infty} \\, r_m \\cdot \\, x^m. \n\\end{eqnarray}\nThis result (\\ref{LaurentAR}) can be directly obtained \nby looking for the {\\em Laurent series} \nfor $\\, A_R(x)$ with a pre-modular structure, i.e. such that the series \n$\\,  \\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots, \\, $\n$\\, n \\, = \\, 2, \\, 3, \\, 4, \\cdots \\, \\, $ are \nsolutions of \n condition (\\ref{WAR2}). As a byproduct, one finds that \nin the case (\\ref{LaurentAR}) the \nsolutions $\\,  \\, y_n(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots \\, $\nare such that (\\ref{curious5}). In particular the \nsolution $\\,  \\, y_1(x) \\, = \\, \\, a_1 \\cdot \\, x \\, + \\, \\, \\cdots \\, $\nis a one-parameter family of commuting series. The case $\\, W(x)\\, = \\, -1\/2\/x^2$, \nor $\\, A_R(x)  \\, \\, \\, = \\,\\, 1\/x$,\ncorresponds to the simple order-two linear differential \noperator $\\, \\theta^2$ where $\\, \\theta$ is \nthe homogeneous derivative $\\, \\theta\\, = \\, \\, x \\cdot \\, D_x$.\n\n\\vskip .2cm\n\nMore specifically, if one revisits our Heun classification problems,\nimposing the {\\em factorization condition} (see the analysis sketched \nin \\ref{subcaseHeun}) {\\em together} with the condition \n(\\ref{Laurent}) required for the \n{\\em emergence of modular correspondence structure} \n(\\ref{curious5}), one gets the following Laurent series expansion\n(see (\\ref{mustbe}) for the definition of the $\\, u, \\, v, \\, w$ parameters): \n\\begin{eqnarray}\n\\label{LaurentARHeun}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \nW(x) \\,\\, = \\, \\, \\,  \\,\n{\\frac {v \\cdot \\, (v \\, -2) }{ 2 \\cdot \\, x^{2}}}\n \\, \\,\\,\\, -{\\frac {v \\cdot \\, (a\\,w\\,+u) }{a \\,\\cdot  \\, x}}\n\\, \\,\\,\\,\\, + \\, \\,\\, \\cdots \n\\end{eqnarray}\nThis gives the condition $\\, v\\, = \\, \\, 1$ (in agreement \nwith condition (\\ref{LaurentAR})) and four other conditions.\nExcluding the case $\\, a \\, =\\, 0$ corresponding to the reduction \nfrom the four singularities of the Heun function \nto three singularities, one gets \n$\\, \\gamma \\, = \\, \\, v\\, = \\, \\, 1$. \nThe Heun function\n$\\, {\\it HeunG} \\left( a, \\,0, \\, 0,\\, \\beta,\\, 1,\\, \\delta, \\, x \\right)$ \nis a (Liouvillian) solution of a reducible linear differential operator \nof order two \n$\\, L_2 \\, = \\, \\, (D_x \\, +A_R(x)) \\cdot \\, D_x$,  where $\\, A_R(x)$\nthen reads:\n\\begin{eqnarray}\n\\label{ARgood}\n\\hspace{-0.96in}&&   \\quad  \\quad \\quad \n   \\quad  \\quad   \\quad  \\quad \\quad    \nA_R(x)  \\,\\, = \\, \\, \\,  \\,  \\, \n{{1} \\over {x}} \\,\\, \\, + \\, {{\\delta } \\over {x \\, -1 }}\n \\,\\,  \\,+ \\, {{ \\beta \\, -\\delta} \\over {x\\, -a }}.\n\\end{eqnarray}\nThe pullbacks $\\, y(x)$ are solutions of the rank-two non-linear \ndifferential equation (\\ref{WAR2}) which can easily be \nintegrated into (see (\\ref{WAR21}), (\\ref{WAR22})):\n\\begin{eqnarray}\n\\label{ARgood1}\n\\hspace{-0.96in}&&   \\quad   \\quad  \n\\quad \\quad  \\quad  \\quad  \\quad \nx \\cdot \\, {{ y'(x)} \\over {y(x)}} \n  \\, \\,\\, = \\, \\, \\, \\,  \\,  c_1 \\cdot \\, \n {{ (y(x)-1)^{\\delta} \\cdot \\, (y(x)-a)^{\\beta \\, -\\delta}} \\over { \n(x-1)^{\\delta} \\cdot \\, (x-a)^{\\beta \\, -\\delta}} }, \n\\end{eqnarray}\ngiving a functional equation on the pullbacks \n$\\, y(x)$ with an Abel integral $\\, \\Theta(x)$:\n\\begin{eqnarray}\n\\label{WAR2good}\n\\hspace{-0.96in}&&  \n\\Theta(y(x) \\, = \\, \\, c_1 \\cdot \\Theta(x) \\, + \\, c_2 \n\\quad \\hbox{where:} \\, \\,  \\quad  \n\\Theta(x)  \\,\\, = \\, \\, \\, \n\\int^x \\, {{ dx } \\over {\nx \\cdot \\, (x\\, -1)^{\\delta} \\cdot \\, (x-a)^{\\beta \\, -\\delta}  }}. \n\\end{eqnarray}\nOne has for instance the following one-parameter series solutions \nfor the pullback $\\, y(x)$, which verify (\\ref{curious5}):\n\\begin{eqnarray}\n\\label{WAR2goody1}\n\\hspace{-0.96in}&& \\quad  \\quad  \\quad   \\quad  \\quad  \ny_1 \\,\\,\\, = \\,\\, \\, \\,\\, a_1 \\cdot \\, x \\,\\,\\, \\, \n - \\, a_1 \\cdot \\, (a_1 \\, -1)\\cdot \\, \n{{ a\\, \\delta \\, + \\beta \\, -\\delta} \\over {a }} \\cdot \\, x^2\n \\, \\,\\,\\, + \\, \\, \\, \\cdots \n\\\\\n\\hspace{-0.96in}&& \\quad  \\quad  \\quad   \\quad  \\quad  \ny_2 \\,\\,\\, = \\,\\, \\, \\,\\, a_2 \\cdot \\, x^2 \\,\\, \n\\, \\, + \\, \\,  2 \\cdot \\,\n {{ a\\, \\delta \\, +\\beta \\, -\\delta } \\over {a }}\n \\cdot \\, a_2 \\cdot \\, x^3\n \\,\\,\\,\\, + \\, \\,  \\cdots \n\\end{eqnarray}\nThe fact that solutions of the form \n$\\, y(x) \\, = \\, \\, a_n \\cdot \\, x^n \\, + \\, \\, \\cdots\\,\\,$\noccur can be clearly seen on equation (\\ref{ARgood1}). Even if \nthe ``pre-modular'' conditions (\\ref{curious5}) \nare verified for this example, this Heun function\n$\\, {\\it HeunG} \\left( a, \\,0, \\, 0,\\, \\beta,\\, 1,\\, \\delta, \\, x \\right)$ \nwill not be necessarily a modular form represented as a pullbacked \n$\\, _2F_1$ hypergeometric function with more than one pullback\nfor generic parameters\\footnote[1]{The exponent-differences \nat the four singularities are: \n$\\,0, 1\\,-\\delta, \\,1 \\,+\\delta \\,-\\beta, \\beta$. Introducing \n$\\, e_1$,$\\, e_2$,$\\, e_3$ the exponents difference of the three \nsingular points of the $\\, _2F_1$ hypergeometric function each the\nprevious exponent-differences must be a multiple of the $\\, e_i$'s.}. \n\n\\vskip .2cm\n\n\\section{Pullback symmetry of an operator up to equivalence of operators} \n\\label{pullbackhomomorphism}\n\nWith the aim of generalizing \n covariance  (\\ref{modularform2explicit}), we introduce \nthe derivative of $\\, _2F_1([1\/12,5\/12],[1],x)$ \n\\begin{eqnarray}\n\\label{modularform2explicitPhi}\n\\hspace{-0.95in}&& \\quad \n \\Phi(x) \\,  \\, = \\, \\, \\, \n {{d } \\over { dx}} \n\\Bigl( \\, _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\, x  \\Bigr) \\Bigr)\n\\,\\, = \\, \\, \\, \n {{5} \\over { 144}} \\cdot \\,\n  _2F_1\\Bigl([{{13} \\over {12}}, \\, {{17} \\over {12}}], \\, [2], \\, x  \\Bigr), \n\\end{eqnarray}\nwhich {\\em does not} correspond to a modular form, \n{\\em since the derivative of a modular form is not a modular form}.  A derivative \nof the simple covariance identity (\\ref{modularform2explicit}) gives\n\\begin{eqnarray}\n\\label{modularform2explicitder}\n\\hspace{-0.95in}&& \\quad  \\quad  \\, \\,      \n \\Phi(y(x)) \\cdot \\, y'(x) \n\\, = \\, \\, \\, \\,\\,  {\\cal A}(x) \\cdot \\, \\Phi(x)\n\\, \\, \\,  + \\, \n  {\\cal A}'(x) \\cdot \\,\n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\,  x  \\Bigr).\n\\end{eqnarray}\nUsing the order-two linear differential equation\nverified by  $\\, _2F_1([1\/12,5\/12],[1],x)$, one can rewrite the \n $\\, _2F_1([1\/12,5\/12],[1],x)$ in the RHS of (\\ref{modularform2explicitder}),\nas a linear combination of $\\, \\Phi(x)$\nand its derivative $\\, \\Phi'(x)$. One then deduces from \nrelation (\\ref{modularform2explicitder}) a slightly \nmore general relation\nthan the initial simple covariance (\\ref{modularform2explicit})\n\\begin{eqnarray}\n\\label{moregeneralPhi}\n\\hspace{-0.95in}&& \\quad \\quad \\quad  \\quad  \\quad \n \\Phi(y(x)) \\,  \\,  \\, = \\, \\,  \\, \\, \\Bigl(   {\\cal A}_{\\Phi}(x) \\cdot \\, {{d} \\over { dx}}\n\\, \\, + {\\cal B}_{\\Phi}(x)   \\Bigr)\n\\, \\cdot \\, \\Phi(x),\n\\end{eqnarray}\nwhere $\\, {\\cal A}_{\\Phi}(x)$ and $\\,  {\\cal B}_{\\Phi}(x)$ read in this particular \nexample\\footnote[2]{If instead of the simple derivative \n(\\ref{modularform2explicitPhi}) we had introduced \n $\\, \\Phi(x) \\, = \\,\\, $$L_1( \\,_2F_1([1\/12,5\/12],[1],x))$ where  $\\, L_1$ is an \narbitrary order-one linear differential operator, we would have also obtained \na relation of the form (\\ref{moregeneralPhi}) but where  $\\, {\\cal A}_{\\Phi}(x)$ \nand $\\,  {\\cal B}_{\\Phi}(x)$ are much more involved expressions.   }:\n\\begin{eqnarray}\n\\label{particularmoregeneralPhi}\n\\hspace{-0.95in}&& \n{\\cal A}_{\\Phi}(x)  \\,   = \\, \\,\n  {{144  \\cdot \\, x \\cdot \\, (x-1)  \\cdot \\, {\\cal A}(x) } \\over { 5 \\cdot \\, y'(x)}}, \n\\quad \\quad \n {\\cal B}_{\\Phi}(x) \\,   = \\, \\, \n  {{ 5 \\cdot \\,  {\\cal A}(x) \\, +\\, 72 \\cdot \\, (2 \\,-3\\,x) \n\\cdot \\,  {\\cal A}'(x) } \\over { 5 \\cdot \\,  y'(x)}}.\n\\nonumber \n\\end{eqnarray}\nRecalling two Hauptmoduls $\\, p_1(x)$ and $\\, p_2(x)$\n\\begin{eqnarray}\n\\label{recallHaupt}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad  \\quad \\,\np_1(x) \\,\\, = \\, \\, \\,{{ 1728 \\cdot \\, x } \\over {(x\\, +16)^3 }}, \n\\quad \\quad  \\quad \\quad \n p_2(x) \\, \\,= \\, \\,\\, {{ 1728 \\cdot \\, x^2 } \\over {(x\\, +256)^3 }}, \n\\end{eqnarray}\none can also write relation (\\ref{moregeneralPhi}) in a more ``balanced''  form  \n(see equation (7) in~\\cite{Hindawi}).\nIntroducing the two algebraic functions $\\, A_1(x)$ and  $\\, A_2(x)$\n\\begin{eqnarray}\n\\label{recallHaupt2}\n\\hspace{-0.95in}&& \\quad \\quad  \\quad  \\quad \nA_1(x) \\,\\, = \\, \\, \\,\\Bigl(1 \\, +\\, {{x} \\over {16}} \\Bigr)^{-1\/4}, \n \\quad \\quad  \\quad \nA_2(x) \\, \\,= \\, \\, \\,\\Bigl(1 \\, +\\, {{x} \\over {256}} \\Bigr)^{-1\/4}, \n\\end{eqnarray}\none has the (modular form) hypergeometric identity: \n\\begin{eqnarray}\n\\label{modularformident}\n\\hspace{-0.95in}&& \n\\, \\, \\,  \\, \\,  \\, \\, \nA_1(x) \\cdot \\,\n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\, p_1(x) \\Bigr)\n\\,\\,  = \\, \\, \\, \nA_1(x) \\cdot \\,\n _2F_1\\Bigl([{{1} \\over {12}}, \\, {{5} \\over {12}}], \\, [1], \\, p_2(x)  \\Bigr). \n\\end{eqnarray}\nAfter performing calculations of a similar nature of the ones previously seen,\none deduces\nthe $\\, 1 \\, \\leftrightarrow \\, 2$ balanced relation on $\\, \\Phi(x)$:\n\\begin{eqnarray}\n\\label{modularformidentder}\n\\hspace{-0.95in}&&  \\quad \\quad \\quad \\, \\, \\, \n144 \\cdot \\, p_1(x) \\cdot \\, (p_1(x) \\, -1) \n\\cdot \\, {{d A_1(x)} \\over { dx}} \\cdot \\,  \\Phi'(p_1(x))\n \\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad\n\\, + \\, \\Bigl(72 \\cdot \\, (3\\,p_1(x) \\, -2) \\cdot \\,  {{d A_1(x)} \\over { dx}}\n \\, - 5 \\cdot \\,  A_1(x) \\cdot \\,  \n {{d p_1(x)} \\over { dx}} \\Bigr) \\cdot \\, \\Phi(p_1(x))\n \\nonumber \\\\\n\\hspace{-0.95in}&& \\quad \\, \\, \\, \n\\, = \\, \\,  \\,  \\, 144 \\cdot \\, p_2(x) \\cdot \\, (p_2(x) \\, -1) \n\\cdot \\, {{d A_2(x)} \\over { dx}} \\cdot \\,  \\Phi'(p_2(x))\n \\\\\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad\n\\, + \\, \\Bigl(72 \\cdot \\, (3\\,p_2(x) \\, -2) \\cdot \\,  {{d A_2(x)} \\over { dx}}\n \\, - 5 \\cdot \\,  A_1(x) \\cdot \\, \n  {{d p_2(x)} \\over { dx}} \\Bigr) \\cdot \\, \\Phi(p_2(x)),\n\\nonumber \n\\end{eqnarray}\nwhich should be viewed as a (rational) parametrization of the relation having \nthe form (\\ref{moregeneralPhi}).\n\nThe interested reader shall find in \\ref{Asimpler} \na detailed (and we hope pedagogical) analysis of the \nmore general relation (\\ref{moregeneralPhi}) given\nfor a selected  hypergeometric function\\footnote[5]{We thank A.J. Guttmann for \nshowing us this remarkable hypergeometric function emerging in a \ndual context of combinatorics and random-matrix theory, counting \nthe number of avoiding permutations~\\cite{Conway,Bona}.} solution \n$\\, \\,  _2F_1([-1\/4, 3\/4],[1],x)$.\n\n\\vskip .1cm\n\nLet us provide an example of the relevance of the relation (\\ref{moregeneralPhi}) \nin the context of integrable models in physics.\nIn the case of the two-dimensional Ising model, the covariance (\\ref{moregeneralPhi})\nis instantiated on  $\\, \\tilde{\\chi}^{(2)}$, the simplest of the low-temperature\n$\\, n$-fold integrals\n $\\, \\tilde{\\chi}^{(n)}$ occurring in the decomposition \nof the susceptibility of the square \nIsing model~\\cite{High,bo-gu-ha-je-ma-ni-ze-08,ze-bo-ha-ma-05b} \n(see subsection 5.1 in~\\cite{Automaton}). When applied to $\\, \\tilde{\\chi}^{(2)}$,\nthe {\\em Landen transformation}  $\\, k \\, \\rightarrow \\, \\, $\n$k_L \\, = \\, \\,  {{2 \\sqrt{k}} \\over {1 \\, + \\, k}}$, which provides \nan exact representation of a generator of the renormalization \ngroup~\\cite{Hindawi,broglie,Heegner}, gives the following covariance relation\n(see equation\\footnote[2]{Note a misprint in the\n expression of the Landen transformation in the unlabelled equation\n above equation (62) in~\\cite{Automaton}.} (64) in~\\cite{Automaton}):\n\\begin{eqnarray}\n\\label{Automat}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad \\,  \\,  \\, \n\\tilde{\\chi}^{(2)}\\Bigl( {{2 \\sqrt{k}} \\over {1 \\, + \\, k}}  \\Bigr) \n  \\,\\,  = \\, \\,  \\,  \\, \n 4 \\cdot \\, {{1\\, + \\, k} \\over { k}} \\cdot \\, \n{{ d \\, \\tilde{\\chi}^{(2)}(k)} \\over { d k}}, \n \\\\\n\\label{Automat2}\n\\hspace{-0.95in}&& \\quad \\quad \\,  \\,  \\, \n\\hbox{where:} \\quad \\quad \\quad \\quad \\quad\n \\tilde{\\chi}^{(2)}(k)  \\,\\, \\,  = \\, \\,  \\,  \\,  \n{{k^4} \\over {4^3}} \\cdot \\,\n _2F_1\\Bigl( [{{3} \\over {2}}, \\,  {{5} \\over {2}}], \\,\\, [3],\\, \\, k^2\\Bigr). \n\\end{eqnarray}\nThis relation (\\ref{Automat}) can also be written as \n\\begin{eqnarray}\n\\label{Automat2}\n\\hspace{-0.95in}&& \\quad \\quad \\,  \\,  \\, \n\\tilde{\\chi}^{(2)}(k)  \\,\\,  = \\, \\,  \\,  \\, \n{{1} \\over {4}} \\cdot \\, \\Bigl( k \\cdot \\, (k-1)  \\cdot \\, {{d} \\over {dk}} \n\\, + \\, {{k^2\\, +k\\, +2 } \\over {k\\, +1}}   \\Bigr) \n\\,  \\,  \\tilde{\\chi}^{(2)}\\Bigl( {{2 \\sqrt{k}} \\over {1 \\, + \\, k}}  \\Bigr), \n\\end{eqnarray}\nor introducing the {\\em inverse Landen transformation} (descending Landen transformation):\n\\begin{eqnarray}\n\\label{inverseLanden}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad \n{{ 1 \\, -(1\\, -k^2)^{1\/2} } \\over {  1 \\, +(1\\, -k^2)^{1\/2} }} \n\\, \\,   \\,   \\,   = \\, \\,  \\,  \\, {\\frac{k^2}{4}}  \\,   \\, +{\\frac{k^4}{8}} \\,   \\,  \n+{\\frac{5}{64}}{k}^{6} \\,   \\,  +{\\frac{7}{128}}{k}^{8} \\,  \\,   +{\\frac{21}{512}}{k}^{10} \\,   \\,  \n \\, + \\, \\, \\cdots,  \n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{Automat3}\n\\hspace{-0.95in}&& \\quad  \\quad   \n\\tilde{\\chi}^{(2)}\\Bigl( {{ 1 \\, -(1\\, -k^2)^{1\/2} } \\over {  1 \\, +(1\\, -k^2)^{1\/2} }}  \\Bigr)\n \\,  \\,  \\,  \\,   = \\,  \\,   \\,  \\,   \n\\Bigl(  {{(k^2 \\, -2)  \\cdot \\, (1\\, -k^2)^{1\/2} \\, + \\, 2 } \\over { \n4 \\, k^2}} \\Bigr)  \\cdot \\,  \\tilde{\\chi}^{(2)}(k)\n\\nonumber \\\\\n\\hspace{-0.95in}&&  \\quad  \\quad \\quad \\quad  \\quad  \\quad  \\quad  \\quad \n \\,   + \\,   \\,  \\, \n {{k^2 \\, -1 } \\over { 4 \\, k}} \\cdot \\, \n\\Bigl(1 \\, - \\, (1\\, -k^2)^{1\/2}\\Bigr)  \\cdot \\, {{d \\tilde{\\chi}^{(2)}(k)} \\over {d k}}. \\,  \\, \n\\end{eqnarray}\n\n\\vskip .1cm\n\n{\\bf Remark:} Note that the premodular condition (\\ref{Laurent}), \n$\\, W(x) \\, = -1\/2\/x^2 \\, + \\, \\, \\cdots, \\, $\nhas no reason to be verified for such generalizations of modular \nforms (\\ref{modularform2explicitPhi}), (\\ref{moregeneralPhi}).  For instance \nfor $\\, \\tilde{\\chi}^{(2)}$ given by (\\ref{Automat2}), the function \n$\\, W(x) \\, = \\, p'(x)\\, +p(x)^2\/2 \\, -2\\, q(x)$ (see (\\ref{wherecond}))  \n has the following Laurent series \nexpansion (here $x\\, = \\, k$):\n\\begin{eqnarray}\n\\label{Laurentdonotmatch}\n\\hspace{-0.95in}&& \\quad \\quad  \nW(x) \\, \\,  = \\, \\,  \\,\n  {{3} \\over {2}} \\cdot \\,  {{x^2 \\, -5 } \\over {x^2 \\cdot \\, (x^2 \\, -1) }} \n \\, \\,  = \\, \\,  \\,  \\,\n{{15} \\over {2}} \\cdot \\, {{1} \\over {x^2}}\n \\, \\, \\, + 6 \\, \\,  + \\, 6 \\, x^2 \\,  \\, + \\, \\, 6 \\, x^4 \\,  \\,\\, + \\, \\, \\cdots \n\\end{eqnarray}\n\n\\vskip .2cm\n\nMore generally these  (hypergeometric) examples provide  simple illustrations of \na more general pullback symmetry, where one imposes  the pullback of an  \norder $\\, N$ linear differential operator to be {\\em homomorphic to that operator}.\nIn this case there exists two intertwiners (of order $\\, N-1$ in general) \n$\\, L_{N-1}$ and $\\, \\, M_{N-1}$, such that:  \n\\begin{eqnarray}\n\\label{onepullbackhomo}\n\\hspace{-0.95in}&&  \\quad   \\quad   \\quad   \\quad   \\quad    \\quad  \n M_{N-1} \\cdot \\,  L_N \\, \\,\\,  = \\, \\, \\,  \\, \npullback\\Bigl(L_N, \\,  y(x)  \\Bigr) \\cdot L_{N-1}.\n\\end{eqnarray}\nThe pullback symmetry up to conjugation studied in sections \n\\ref{order-two}, \\ref{order-three}, \\ref{order-four}, \n\\ref{order-N}, \\ref{CalabiSchwResults}  is  appropriate for \nmodular forms~\\cite{IsingCalabi,IsingCalabi2,Christol,perimeter},\nbut {\\em not for derivatives of modular forms} that \n{\\em also occur in physics} (see for instance the previous \nrelation (\\ref{Automat}) on the \nsquare Ising model). The emergence of such generalized covariance \n(\\ref{onepullbackhomo}) for the representation of the Landen transformation\n(and more generally the modular correspondences providing exact \nrepresentations of the generators of the renormalization group) on the \nother $n$-fold integrals $\\, \\tilde{\\chi}^{(n)}$'s of the susceptibility \nof the Ising model~\\cite{High,bo-gu-ha-je-ma-ni-ze-08,ze-bo-ha-ma-05b}\nis a {\\em challenging open problem}, that will require one to consider \n {\\em reducible operators} (see subsection \\ref{nonirreducible}).\n\nAnalyzing these more general constraints (\\ref{onepullbackhomo})\nwill require many additional assumptions (beyond the \none of having selected differential Galois group) \non the linear differential operator  $\\, L_N$ to be able \nto perform more calculations. \n\n\\vskip .2cm\n\n\\section{Schwarzian conditions for different Calabi-Yau operators with the same Yukawa couplings} \n\\label{CalabiSchwatext}\n\nIn the previous sections we have analyzed the question of the covariance under algebraic \npullbacks of a linear differential operator of arbitrary order $\\, N$, i.e. the \nquestion of linear  differential operators with algebraic pullback symmetries. Let \nus consider here the more general problem  of the equivalence under \npullbacks up to conjugations of {\\em two different linear  differential operators}, \nwhich is an enlightening\n sieve when one tries to classify selected \nlinear differential operators in theoretical physics (Calabi-Yau linear  \ndifferential operators~\\cite{Duco,Tables}). The interested reader \nwill find in \\ref{CalabiSchwa} an illustration of this \nimportant question where we revisit in detail some calculations of a paper by  Almkvist, \nvan Straten and Zudilin~\\cite{Duco}. This calculation reexamines the\nquestion of pullback equivalence up to \nconjugation, of two selected order-four operators $\\, L_4$ and ${\\cal L}_4$ verifying \nthe Calabi-Yau condition:\n\\begin{eqnarray}\n\\label{vL4text}\n\\hspace{-0.95in}&&   \\quad \\quad \\quad \\quad  \\quad  \\quad \n v(x) \\cdot \\,  {\\cal L}_4   \\cdot \\, {{1} \\over {v(x)}}\n\\, \\,   = \\, \\, \\, \npullback\\Bigl(L_4, \\,  {\\frac {-4 \\, x}{ (1 \\, -\\,x)^{2}}}  \\Bigr), \n\\\\\n\\hspace{-0.95in}&&   \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \n\\, \\, \\, \\,  \\hbox{with:} \\, \\, \\,  \\quad  \\quad \\quad \\quad \nv(x) \\, = \\, \\,  \\Bigl({{ x \\cdot (1\\, +x)} \\over { 1\\, -x}} \\Bigr)^{1\/2}.\n\\end{eqnarray}\nOne finds that a  Schwarzian equation  verified \nby these two order-four linear differential operators\n $\\, L_4$ and ${\\cal L}_4$ reads:\n\\begin{eqnarray}\n\\label{SchwarUUhatAlmkvistext}\n\\hspace{-0.95in}&& \\quad  \\quad \\quad  \\quad  \\quad \n\\hat{U}_R(x)  \\,\\, \\,  \\, - \\, U_M(y(x)) \\cdot \\, y'(x)^2 \n\\, \\,\\,  + \\,  \\{y(x), \\, x\\} \n \\,\\,\\, \\, = \\, \\,\\, \\,   \\, 0, \n\\end{eqnarray}\nwhere $\\, U_M(x)$ and  $\\, \\hat{U}_R(x)$ are given by (\\ref{wherecond4}),\nand where $\\, p(x)$ and $\\, q(x)$ are the coefficients of \n$\\, D_x^3$ and  $\\, D_x^2$ for respectively $\\, L_4$ and $\\, {\\cal L}_4$, \n (see (\\ref{U}) and (\\ref{Uhat}) in \\ref{CalabiSchwa}).\n\n\\vskip .1cm\n\nOne sees on this example that the nome and Yukawa couplings, expressed \nin terms of the $\\, x$ variable, are related \n(see (\\ref{nomeL4}), (\\ref{YukcalL2})) by the pullback \ntransformation. Yet, the Yukawa couplings of the two linear differential \noperators {\\em expressed in term of the nome}, \nare related  in an even simpler and ``universal'' way:\n $\\, K_q({\\cal L}_4) \\, \\, = \\, \\, \\, K_q(L_4)(-4 \\cdot \\, q)$,\nas shown in Appendix E of~\\cite{Christol}. For \na pullback $\\, y(x)$ with a series expansion of the form \n\\begin{eqnarray}\n\\label{pullform}\n\\hspace{-0.95in}&&    \\quad \\quad \n\\quad \\quad \\quad \\quad \\quad  \\quad \\quad \ny(x) \\,\\, \\,\\, = \\, \\, \\, \\,\\,\\lambda \\cdot x^n \\,\\,\\, \\, + \\, \\, \\cdots \n\\end{eqnarray} \nthe nome and Yukawa couplings expressed in terms of the $\\, x$ variable,\nof two order-four operators such that \n\\begin{eqnarray}\n\\label{vL4suchthat}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad \\quad  \\quad   \\quad   \\quad   \\quad  \n v(x) \\cdot \\,  {\\cal L}_4   \\cdot \\, {{1} \\over {v(x)}}\n\\, \\,\\,  = \\, \\, \\,  \\, pullback\\Bigl(L_4, \\,  y(x) \\Bigr),\n\\end{eqnarray}\nare simply related through the relations\n\\begin{eqnarray}\n\\label{YukcalL2gen}\n\\hspace{-0.95in}&&    \\quad  \\quad  \\quad  \\quad\n q_x({\\cal L}_4)^n \\, \\, = \\, \\, \\,   \n \\, {{1} \\over {\\lambda}} \\cdot \\, q_x(L_4)\\Bigl( y(x) \\Bigr), \n \\quad  \\quad \nK_x({\\cal L}_4) \\, \\, = \\, \\, \\,K_x(L_4)\\Bigl( y(x) \\Bigr). \n\\end{eqnarray}\nThe Yukawa couplings \n{\\em expressed in terms of the nome}\\footnote[2]{This function \nis often viewed as a function of the nome $\\, q \\, = \\, e^{\\tau}$, since \nits $ \\,q$-expansion in the case of degenerating family of Calabi-Yau 3-folds is \nsupposed to encode the counting of rational curves of various degrees on a mirror \nmanifold.}, {\\em are related in an even simpler ``universal'' way} as so: \n\\begin{eqnarray}\n\\label{YukcalL2nomegenertext}\n\\hspace{-0.95in}&&    \\quad  \\quad  \\quad \n \\quad    \\quad  \\quad  \\quad \\quad  \\quad  \\quad \nK_q({\\cal L}_4) \\, \\, = \\, \\, \\, K_q(L_4)(\\lambda  \\cdot \\, q^n).\n\\end{eqnarray} \nThe previous example (\\ref{vL4text}) corresponds to $\\, n\\, = \\, 1$ \nand $\\, \\lambda \\, = -4$.\nIn the case  $\\, n\\, = \\, 1$ and $\\, \\lambda \\, = \\, 1$, the pullback \nis a deformation of the identity $\\, y(x) \\, = \\, \\, x \\, + \\, \\, \\cdots \\,$ \nand the  Yukawa couplings expressed in terms of the nome, of the two \nlinear differential operators are equal. Thus one recovers \nProposition (6.2) of Almkvist et al. paper~\\cite{Duco}\nwhere the Yukawa couplings coincide.\n                                    \nSince the Schwarzian equation (\\ref{SchwarUUhatAlmkvistext}) corresponds to the \nequivalence of two linear differential operators by pullback with remarkably \nsimple relations (\\ref{YukcalL2nomegenertext}) on their Yukawa couplings\nexpressed in terms of the nome,  the Schwarzian equation (\\ref{SchwarUUhatAlmkvistext})\ncan be seen as a condition to have simply related Yukawa couplings. In the case \nof {\\em  deformation of the identity} $\\, y(x) \\, = \\, \\, x \\, + \\, \\, \\cdots \\,\\,$ \npullbacks, it can be seen  as a condition of preservation of the Yukawa \ncouplings ({\\em seen as functions of the nome}). These results {\\em are not restricted \nto order-four linear differential operators} (see Appendix E of~\\cite{Christol} \nand \\ref{CalabiSchwa}). For instance, one can impose that \n{\\em two different pullbacks of the same order-$N$ \nlinear differential operator $\\, L_N$  are homomorphic}, i.e. there exist two\nintertwiners (of order $\\, N-1$ in general) $\\, L_{N-1}$ and $\\, \\, M_{N-1}$\nsuch that:    \n\\begin{eqnarray}\n\\label{twopullbacks}\n\\hspace{-0.95in}&&  \\quad   \\quad   \\quad  \npullback\\Bigl(L_N, \\,  p_1(x)  \\Bigr) \\cdot L_{N-1} \n\\, \\,\\,  = \\, \\, \\,  \\, \n M_{N-1} \\cdot \\,  pullback\\Bigl(L_N, \\,  p_2(x)  \\Bigr).\n\\end{eqnarray}\nThis last generalization turns out to be instructive for physics and \nenumerative combinatorics.\n\n\\vskip .2cm\n\n\\section{Conclusion} \n\\label{Conclu}\n\nIn a previous paper~\\cite{Aziz} we focused on identities relating \nthe {\\em same} $\\, _2F_1$ hypergeometric function with \n{\\em two different algebraic pullback transformations} \n\\begin{eqnarray}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad   \\quad  \\quad  \\quad     \n\\label{prevsymmetry}\n{\\cal A}(x) \\cdot \\,   _2F_1 \\Bigl( [a,b],[c],x \\Bigr) \n\\,   \\,   = \\,  \\,    \\,   _2F_1 \\Bigl( [a,b],[c],y(x) \\Bigr), \n\\end{eqnarray}\nalong with the existence of $\\, _nF_{n-1}$ analogues of \nthe previous relation. \nSuch remarkable identities correspond to {\\em modular forms} \nthat emerged in the analysis of multiple integrals related to\nthe square Ising model~\\cite{IsingCalabi,IsingCalabi2,Christol,Diffalg}\nor in other enumerative combinatorics context~\\cite{perimeter}. \nThey can be seen as a simple occurence of \n{\\em infinite order}\\footnote[9]{We have for instance in mind \nto provide exact representations of the renormalization \ngroup~\\cite{Hindawi,broglie,Heegner}.}\ncovariance symmetries \nin physics~\\cite{Hindawi} or enumerative combinatorics. \n\nThe current paper generalizes these previous results \n beyond hypergeometric \nfunctions\\footnote[5]{Or even Heun functions, see~\\cite{Aziz}.},\nanalyzing the conditions for  order-$N$ linear differential \noperators with an arbitrary number of \nsingularities\\footnote[1]{Far beyond operators with\nhypergeometric solutions, or pullbacked  hypergeometric solutions.}  \nto be {\\em pullback invariant up to conjugations}: \n\\begin{eqnarray}\n\\label{vLNgenergg}\n\\hspace{-0.95in}&&  \\quad \\quad  \\quad   \\quad   \\quad   \\quad  \n {{1} \\over {v(x)}} \\cdot \\,  L_N  \\cdot \\, v(x) \n\\, \\,\\,  = \\, \\, \\,  \\, pullback\\Bigl(L_N, \\,  y(x)  \\Bigr).\n\\end{eqnarray}\nOne finds that the pullbacks $\\, y(x)$ are\n{\\em differentially algebraic}~\\cite{Selected,IsTheFull}, \nbeing {\\em necessarily solutions of \nthe same Schwarzian equations} as in~\\cite{Aziz}\n\\begin{eqnarray}\n\\label{SchwarNconcl}\n\\hspace{-0.95in}&& \\quad \\quad   \\quad \\quad  \\quad \\quad  \nW(x) \n\\, \\, \\,  \\,-W(y(x)) \\cdot  \\, y'(x)^2\n\\, \\, \\,  \\,+ \\,  \\{ y(x), \\, x\\} \n\\, \\,\\, \\, = \\,\\, \\, \\,  \\, 0, \n\\end{eqnarray}\nwhere the function $\\, W(x)$ encoding the Schwarzian equation (\\ref{SchwarNconcl})\nis a simple expression of the first two coefficients of the linear \ndifferential operator (see (\\ref{wherecondN})). For order-two linear differential \noperators this Schwarzian condition turns out to be sufficient. In the case \nof linear differential operators with\nselected differential Galois groups however, we showed, for orders three \nand four, that the  ``Calabi-Yau'' conditions\n(see sections \\ref{CalabiYaucondi})\nare rigid enough to force the pullbacked-invariant (up to conjugation) \noperators (see (\\ref{vLNgenergg})) to reduce to symmetric powers of an \norder-two linear differential operator.  \n\nThe reduction of the solutions \nof this Schwarzian differential equation to {\\em modular correspondences}\nwas an open question in~\\cite{Aziz}. Modular correspondences require \nthe existence, for any integer $\\, n$, of solutions of the Schwarzian \nequation (\\ref{SchwarNconcl})\nof the form $ \\, y_n (x)=  \\, a_n \\cdot \\,  x^n \\,  + \\cdots \\, \\,$\nsuch that, for any integer $\\, m$ and $\\, n$, the following \n``pre-modular'' condition is satisfied:\n\\begin{eqnarray}\n\\label{premodular}\n\\hspace{-0.95in}&& \\quad \\quad   \\quad \\quad  \\quad \\quad  \\,  \\,  \\, \ny_n(a_n, \\, \\, y_m(a_m,  \\,\\, x)) \\, \\,   = \\, \\,   \\,  y_{n m}(a_n a_m^n, \\, \\, x).\n\\end{eqnarray}\nWe derived in this paper a necessary and sufficient condition to obtain \nsuch ``pre-modular'' \nsolutions for the ``Schwarzian condition''  (\\ref{SchwarNconcl}).\nThis condition turns out to be a simple condition on the Laurent series \nof $ \\,W(x)$ encoding the Schwarzian condition:\n\\begin{eqnarray}\n\\label{adB}\n\\hspace{-0.95in}&& \\quad \\quad \\quad \\quad \\quad  \\quad \\quad \nW(x) \\,  \\, = \\, \\,   \\, \n -\\frac{1}{2\\cdot x^2} \\, \\,  \\, + \\frac{b}{x}\n  \\,  \\, \\, + \\sum_{m=0}^{\\infty} \\, a_m \\cdot x^m.\n\\end{eqnarray}\n\nIn light of what we have discussed so far, the current paper \ngenerates more questions than answers \nthat give directions for further research. We have seen \nfor example that (\\ref{adB})\nis a necessary and sufficient condition for \nobtaining ``pre-modular'' solutions for the ``Schwarzian condition'', \ncorresponding, in general, to a {\\em transcendental\\footnote[2]{The series \n$\\, y_n(x)$ (see (\\ref{premodular}))  are {\\em differentially algebraic}, \nbut, {\\em not necessarily algebraic functions}.} \ndeclination of modular correspondences}.\nTo have modular correspondences one needs the existence of\n{\\em selected values}  \nof the parameters such that the solution series \n$ \\, y_n(x)= \\,  a_n \\cdot  \\, x^n \\,  + \\cdots \\, $ (see (\\ref{curious5})) \nactually {\\em reduce to algebraic functions}. Is it only in the case of \nmodular correspondences that such algebraic reductions for \nselected values take place ? \n\nThen we showed that an order-two linear differential operator \nemerging in the context of avoiding \npermutations counting~\\cite{Conway,Bona}, provides a good illustration \nof a generalization of the pullback-covariance (\\ref{prevsymmetry})\nor of the pullback invariance up to conjugation (\\ref{vLNgenergg}):\nthe $ \\, _2F_1 \\Bigl( [-1\/4,3\/4],[1],x \\Bigr)$ \nthat comes up in the context of avoiding \npermutations counting~\\cite{Conway,Bona}, verify\na relation (see (\\ref{followingidentity}), (\\ref{followingidentity2})), \nwhose general form is given by   \n\\begin{eqnarray}\n\\label{adB2F1}\n\\hspace{-0.95in}&&\n\\quad \\quad \\quad \\quad \\quad \\quad \n\\Phi(y(x)) \\, \\,\\,  \\,\n\\, \\,\\, =  \\, \\, \\,\n\\Bigl(   {\\cal A}(x) \\cdot \\, {{d} \\over { dx}}\n\\, \\, + {\\cal B}(x)   \\Bigr)\n\\, \\cdot \\,  \\Phi(x),\n\\end{eqnarray}\ngiving a non-trivial explicit example of a \n{\\em pullback invariance of an operator up to operator homomorphisms} \n(see (\\ref{onepullbackhomo}))\n\\begin{eqnarray}\n\\label{onepullbackhomoconcl}\n\\hspace{-0.95in}&&  \\quad   \\quad   \\quad   \\quad   \\quad  \\quad \\quad \n M_{N-1} \\cdot \\,  L_N \\, \\,\\,  = \\, \\, \\,  \\, \npullback\\Bigl(L_N, \\,  y(x)  \\Bigr) \\cdot L_{N-1}.\n\\end{eqnarray}\nEquation (\\ref{Automat}) providing an exact representation \nof the {\\em Landen} transformation \n(generator of the renormalization group) on $\\, \\tilde{\\chi}^{(2)}$,\ntogether with the explicit calculations of section \\ref{pullbackhomomorphism},\nmake quite clear that conditions like (\\ref{adB2F1}) \nprovide a natural and interesting generalization \nof {\\em modular forms}, going beyond the Schwarzian \nequation (\\ref{SchwarNconcl}).\n\nAt last, we examined the equivalence of two different linear differential \noperators, under pullback and conjugation, yielding again some Schwarzian \ncondition relating these two linear differential operators \n(see relation (\\ref{SchwarUUhatAlmkvistgener})), and we discussed \nthe consequence of such equivalence on the corresponding \nYukawa couplings. These results revisiting and complementing \nthe results of~\\cite{Duco}, provide \npowerful tools to analyze various \nsymmetry and classification problems of selected linear \ndifferential operators, in particular  linear differential \noperators of the Calabi-Yau type~\\cite{Tables} \n({\\em not necessarily of order four}~\\cite{Christol}). \n\nWhen dealing with {\\em linear} differential operators, \nwe have seen the emergence of Schwarzian derivatives, \nconsequence of the fact that\n{\\em the Schwarzian derivative is appropriate for the composition of functions}~\\cite{What} \n(see the chain rule of the  Schwarzian derivative\nof the composition of function).\nDo {\\em higher order Schwarzian derivatives}~\\cite{Matone,Matone2,HigherSchw,HigherSchw2} \noccur for pullback-symmetries of {\\em non-linear} ODE's, \nor, more generally, for {\\em functional} equations? \n\nRestraining oneself to the univariate {\\em linear} differential \noperators case, let us remark that \nif condition (\\ref{prevsymmetry}), or (\\ref{vLNgenergg}),\ndescribe effectively all the modular forms that \noften occur in physics~\\cite{IsingCalabi,IsingCalabi2,Diffalg},  \nor enumerative combinatorics~\\cite{perimeter}, a\npullback symmetry up to conjugation constraint like (\\ref{vLNgenergg}) \ncould be restrictive in some sense since it seems to \nyield systematic reduction\\footnote[5]{At least in the case where the\noperators verify Calabi-Yau conditions and thus \nhave selected differential Galois groups.} to order-two linear differential \noperators. In contrast the simple hypergeometric \nexample of section \\ref{pullbackhomomorphism} seems to provide a natural \ngeneralization of {\\em modular forms}:\n the {\\em pullback invariance of an operator up to operator homomorphisms} \ncondition (\\ref{onepullbackhomoconcl}) promises to cover a larger ensemble of\nexact representations of \nsymmetries in physics or enumerative combinatorics. In particular \nthe emergence of conditions like (\\ref{adB2F1})\nof higher order, namely generalized covariance \n(\\ref{onepullbackhomoconcl}) for the representation of the Landen \ntransformation\\footnote[1]{And more generally the modular correspondences \nproviding exact representations of the generators of the renormalization \ngroup~\\cite{Hindawi,Heegner}.} \non the other $\\, n$-fold $\\, \\tilde{\\chi}^{(n)}$'s of the Ising susceptibility \n(see~\\cite{High,bo-gu-ha-je-ma-ni-ze-08,ze-bo-ha-ma-05b}), together with \ntheir corresponding large order {\\em reducible} linear differential operators, \nis a challenging open problem.\n\n\\vskip .3cm\n\\vskip .3cm\n\n{\\bf Acknowledgments:} \nWe would like to thank\n S. Boukraa,  M. van Hoeij and J-A. Weil for very fruitful \ndiscussions   on differential systems. We thank A.J. Guttmann for\nproviding an interesting pullbacked hypergeometric example.\nThis work has been performed \nwithout any ERC, ANR or MAE financial support. \n\n\n\\vskip .5cm\n\\vskip .5cm\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe brain has been a productive source of inspiration for AI, from the perceptron and the neocognitron to deep neural nets. Machine learning has since advanced to dizzying heights of analytical understanding and practical success, but the study of the brain has lagged behind in one important dimension: After half a century of intensive effort by neuroscientists (both computational and experimental), and despite great advances in our understanding of the brain at the level of neurons, synapses, and neural circuits, we still have no plausible mechanism for explaining intelligence, that is, the brain's performance in planning, decision-making, language, etc. As Nobel laureate Richard Axel put it, ``we have no logic for translating neural activity into thought and action'' \\citep{AxelNeuron2018}.\n\nRecently, a high-level computational framework was developed with the explicit goal to fill this gap: the Assembly Calculus (AC)  \\citep{papadimitriou2020brain}, a computational model whose basic data type is the {\\em assembly of neurons}.  Assemblies, called ``the alphabet of the brain'' \\citep{buzsaki2019brain}, are large sets of neurons whose simultaneous excitation is tantamount to the subject's thinking of an object, idea, episode, or word (see \\citet{Josh2016}). Dating back to the birth of neuroscience, the ``million-fold democracy'' by which groups of neurons act collectively without central control was first proposed by \\citet{sherrington1906integrative} and was the empirical phenomenon that Hebb attempted to explain with his theory of plasticity \\citep{hebb1949}. Assemblies are initially created to record memories of external stimuli \\citep{quiroga2016neuronal}, and are believed to be subsequently recalled, copied, altered, and manipulated in the non-sensory brain \\citep{Josh2012, Buzsaki:10}.  The Assembly Calculus provides a repertoire of operations for such manipulation, namely {\\tt project, reciprocal-project, associate, pattern-complete,} and {\\tt merge} encompassing a complete computational system.  Since the Assembly Calculus is, to our knowledge, the only extant computational system whose purpose is to bridge the gap identified by Axel in the above quote \\citep{AxelNeuron2018}, it is of great interest to establish that complex cognitive functions can be plausibly expressed in it.  Indeed, significance progress has been made over the past year, see for example \\citet{mitropolsky2021biologically} for a parser of English and  \\citet{damore2021planning} for a program mediating planning in the blocks world, both written in the AC programming system.  Yet despite these recent advances, one fundamental question is left unanswered: If the Assembly Calculus is a meaningful abstraction of cognition and intelligence, why does it not have a {\\tt learn} command?  {\\em How can the brain learn through  assembly representations?}  \n\nThis is the question addressed and answered in this paper. As assembly operations are a new learning framework and device, one has to start from the most basic questions: Can this model classify assembly-encoded stimuli that are separated through clustering, or by half spaces? Recall that learning linear thresholds is a theoretical cornerstone of supervised learning, leading to a legion of fundamental algorithms: Perceptron, Winnow, multiplicative weights, isotron, kernels and SVMs, and many variants of gradient descent. \n\nFollowing \\citet{papadimitriou2020brain}, we model the brain as a directed graph of excitatory neurons with dynamic edge weights (due to plasticity).  The brain is subdivided into {\\em areas}, for simplicity each containing $n$ neurons connected through a $G_{n,p}$ random directed graph \\citep{erdHos1960evolution}. Certain ordered pairs of areas are also connected, through random bipartite graphs.  We assume that neurons fire in discrete time steps. At each time step, each neuron in a brain area will fire \nif its synaptic input from the firings of the previous step is among the top $k$ highest out of the $n$ neurons in its brain area.  This selection process is called {\\em $k$-cap}, and is an abstraction of the process of {\\em inhibition} in the brain, in which a separate population of inhibitory neurons is induced to fire by the firing of excitatory neurons in the area, and through negatively-weighted connections prevents all but the most stimulated excitatory neurons from firing. Synaptic weights are altered via Hebbian plasticity and homeostasis (see Section 2 for a full description).  In this stylized mathematical model, reasonably consistent with what is known about the brain, it has been shown that the operations of the Assembly Calculus converge and work as specified (with high probability relative to the underlying random graphs). These results have also been replicated by simulations in the model above, and also in more biologically realistic networks of spiking neurons (see \\citet{LMPV,papadimitriou2020brain}). {\\em In this paper  we develop, in the same model, mechanisms for learning to classify well-separated classes of stimuli}, including clustered distributions and linear threshold functions with margin. Moreover, considering that the ability to learn from few examples, and with mild supervision, are crucial characteristics of any brain-like learning algorithm, we show that learning with assemblies does both quite naturally.\n\n\\section{A mathematical model of the brain} \nHere we outline the basics of the model in \\citet{papadimitriou2020brain}.  There are a finite number $a$ of brain areas, denoted $X, Y, \\ldots$ (but in this paper, we will only need one brain area where learning happens, plus another area where the stimuli are presented).  Each area is a random directed graph with $n$ nodes called {\\em neurons} with each directed edge present independently with probability $p$ (for simplicity, we take $n$ and $p$ to be the same across areas).  Some ordered pairs of brain areas are also connected by random bipartite graphs, with the same connection probability $p$.  Importantly, each area may be {\\em inhibited}, which means that its neurons cannot fire; the status of the areas is determined by explicit {\\tt inhibit\/disinhibit} commands of the AC. \\footnote{The brain's neuromodulatory systems \\citep{jones2003arousal, harris2011cortical} are plausible candidates to implement these mechanisms.} This defines a large random graph $G=(N,E)$ with $|N|=an$ nodes and a number $|E|$ of directed edges which is in expectation $(a+b)pn^2-apn$, where $b$ is the number of pairs of areas that are connected. Each edge $(i,j)\\in E$ in this graph, called a {\\em synapse}, has a dynamic non-negative weight $w_{ij}(t)$, initially $1$. \n\nThis framework gives rise to a discrete-time dynamical system, as follows: The {\\em state} of the system at any time step $t$ consists of (a) a bit for each area $X$, inh$(X,t)$, initially $0$, denoting whether the area is inhibited; (b) a bit for each neuron $i$, fires$(i,t)$, denoting whether $i$ spikes at time $t$ (zero if the area $X$ of $i$ has inh$(X,t)=1$); and (c) the weights of all synapses $w_{ij}(t)$, initially one.  \n\nThe state transition of the dynamical system is as follows: For each neuron $i$ in area $A$ with inh$(X,t+1)=0$ (see the next paragraph for how inh$(X,t+1)$ is determined), define its {\\em synaptic input} at time $t+1$, \n\\[\n\\mbox{SI}(i,t+1) = \\sum_{(j,i)\\in E} \\hbox{\\rm fires}(j,t)w_{ji}(t).\n\\]\nFor each $i$ in area $X$ with inh$(X,t+1)=0$, we set fires$(i,t)=1$ iff $i$ is among the $k$ neurons in its area that have highest SI$(i,t+1)$ (breaking ties arbitrarily).  This is the {\\em $k$-cap} operation, a basic ingredient of the AC framework, modeling the inhibitory\/excitatory balance of a brain area.\\footnote{\\citet{binas2014learning} showed rigorously how a $k$-cap dynamic could be emerge in a network of excitatory and inhibitory neurons.}  As for the synaptic weights, \n\\[\nw_{ji}(t+1)=w_{ji}(t)(1+\\beta\\cdot\\hbox{\\rm fires}(j,t)\\hbox{\\rm fires}(i,t+1)).\n\\]\nThat is, if $j$ fires at time $t$ and $i$ fires at time $t+1$, Hebbian plasticity dictates that $w_{ji}$ be increased by a factor of $1+\\beta$ at time $t+1$. So that the weights do not grow unlimited, a {\\em homeostasis} process renormalizes, at a slower time scale, the sum of weights along the incoming synapses of each neuron (see \\citet{davis2006homeostatic} and \\citet{turrigiano2011too} for reviews of this mechanism in the brain).\n\nFinally, the AC is a computational system {\\em driving} the dynamical system by executing commands at each time step $t$ (like a programming language driving the physical system that is the computer's hardware).  The AC commands {\\tt (dis)inhibit$(X)$} change the inhibition status of an area at time $t$; and the command {\\tt fire$(x)$,} where $x$ is the name of an assembly (defined next) in a disinhibited area, overrides the selection by $k$-cap, and causes the $k$ neurons of assembly $x$ to fire at time $t$.  \n\nAn {\\em assembly} is a highly interconnected (in terms of both number of synapses and their weights) set of $k$ neurons {\\em in an area} encoding a real world entity. Initially, assembly-like representations exist only in a special {\\em sensory area,} as representations of perceived real-world entities such as a heard (or read) word. Assemblies in the remaining, non-sensory areas are {\\em an emergent behavior} of the system, copied and re-copied, merged, associated, etc.,~through further commands of the AC.  This is how the model is able to simulate arbitrary $\\frac{n}{k}$ space bounded computations \\citep{papadimitriou2020brain}. The most basic such command is {\\tt project$(x,Y,y)$,} which, starting from an assembly $x$ in area $X$, creates in area $Y$ (where there is connectivity from $X$ to $Y$) a new assembly $y$, which has strong synaptic connectivity from $x$ and which will henceforth fire every time $x$ fires in the previous step, and $Y$ is not inhibited.  This command entails disinhibiting the areas $X,Y$, and then {\\tt firing} (the neurons in) assembly $x$ for the next $T$ time steps.  It was shown by \\citet{papadimitriou2019random} that, with high probability, after a small number of steps, a stable assembly $y$ in $Y$ will emerge, which is densely intraconnected and has high connectivity from $x$. The mechanism achieving this convergence involves synaptic input from $x$, which creates an initial set $y^1$ of firing neurons in $Y$, which then evolves to $y^t, t=2,\\ldots$ through sustained synaptic input from $x$ {\\em and recurrent input from $y^{t-1}$}, while these two effects are further enhanced by plasticity.  \n\nIncidentally, this convergence proof (see \\cite{legenstein2018long, papadimitriou2019random, papadimitriou2020brain} for a sequence of sharpened versions of this proof over the past years) is the most mathematically sophisticated contribution of this theory to date.  The theorems of the present paper can be seen as substantial {\\em generalizations} of that result:  Whereas in previous work an assembly is formed as a copy of one stimulus firing repeatedly (memorization), so that this new assembly will henceforth fire whenever the same stimulus us presented again, in this paper we show rigorously that an assembly will be formed in response to the sequential firing of many stimuli, all drown from the same distribution (generalization), and the formed assembly will fire reliably every time another stimulus from the same distribution is presented. \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{images\/schematic.png}\n    \\caption{A mathematical model of learning in the brain. Our model (left) has a sensory area (column) connected to a brain area (square), both made up of spiking neurons. Two different stimuli classes (with their core sets in blue\/green on the left) project from the sensory area via synaptic connections (arrows). Assemblies in the brain area (with core sets in the corresponding colors) form in response to these stimuli classes, each of which consistently fires when a constant fraction of the associated stimuli class's core set does. Our learning algorithm (right) consists of presenting a stream of stimuli from each class\n    }\n    \\label{fig:schematic}\n\\end{figure}\n\nThe key parameters of our model are $n, k, p$, and $\\beta$. Intended values for the brain are $n = 10^7, k= 10^4, p = 10^{-3}, \\beta=0.1$, but in our simulations we have also had success on a much smaller scale, with $n=10^3, k=10^2, p=10^{-1}, \\beta =0.1$. $\\beta$ is an important parameter, in that adequately large values of $\\beta$ guarantee the convergence of the AC operations.\nFor a publicly available simulator of the Assembly Calculus (in which the Learning System below can be readily implemented) see \\url{http:\/\/brain.cc.gatech.edu}.\n\n\\paragraph{The learning mechanism.} \nFor the purpose of demonstrating learning within the framework of AC, we consider the specific setting described below. First, there is a special area, called the {\\em sensory area,} in which training and testing data are encoded as assembly-like representations called {\\em stimuli}. There is only one other brain area (besides the sensory area), and that is where learning happens, through the formation of assemblies in response to sequences of stimuli.\n\nA {\\em stimulus} is a set of about $k$ neurons firing simultaneously (``presented'') in the sensory area. Note that, exceptionally in the sensory area, a number of neurons that is a little different from $k$ may fire at a step.  A {\\em stimulus class} $A$ is a distribution over stimuli, defined by three parameters: two scalars $r,q \\in [0,1], r > q$, and a set of $k$ neurons $\\stimcore{A}$ in the sensory area.  To generate a stimulus $x \\in \\{0,1\\}^n$ in the class $A$, each neuron $i \\in \\stimcore{A}$ is chosen with probability $r$, while for each $i \\not\\in \\stimcore{A}$, the probability of choosing neuron $i$ is $qk\/n$. It follows immediately that, in expectation, an $r$ fraction of the neurons in the stimulus core are set to $1$ and the number of neurons outside the core that are set to $1$ is also $O(k)$. \n\nThe presentation of a sequence of stimuli from a class $A$ in the sensory area evokes in the learning system a {\\em response} $R$, a {\\em distribution over assemblies} in the brain area. \nWe show that, as a consequence of plasticity and $k$-cap, this distribution $R$ will be highly concentrated, in the following sense: Consider the set $S_R$ of all assemblies $x$ that have positive probability in $R$.  Then the numbers of neurons in both the intersection $\\assmcore{R}=\\bigcap_{x\\in S_R} x$, called the {\\em core} of $R$ and the union $\\support{R}=\\bigcup_{x\\in S_R} x$ are close to $k$, in particular $k-o(k)$ and $k+o(k)$ respectively.\\footnote{The larger the plasticity, the closer these two values are (see \\citet{papadimitriou2019random}, Fig. 2).} \nIn other words, neurons in $\\assmcore{R}$ fire far more often on average than neurons in $\\support{R} \\setminus \\assmcore{R}$.\n\nFinally, our learning protocol is this: Beginning with the brain area at rest, stimuli are repeatedly sampled from the class, and made to fire.\nAfter a small number of training samples, the brain area returns to rest, and then the same procedure is repeated for the next stimulus class, and so on.  Then testing stimuli are presented in random order to test the extent of learning.\n(see Algorithm 1 in an AC-derived programming language.)\n\\if{false}\nAt a particular moment in time, the state of the network can be described as $\\mathcal B = (x, y, A, W)$, where $x \\in \\{0, 1\\}^n$ is the activations of neurons in the sensory area, $y \\in \\{0, 1\\}^n$ is the activations of neurons in the learning area (with $\\sum_{i} y_i \\le k$), $A$ is a set of weighted directed edges from  neurons in $X$ to neurons in $Y$, and $W$ is a set of weighted directed edges from neurons in $Y$ to other neurons in $Y$, excluding loops. To interact with the brain area, we provide a few basic commands: \n\n\\begin{itemize}\n    \\item $\\texttt{input}(\\mathcal B, x)$ updates the current value of the sensory area activations (the input) to $x$\n    \\item $\\texttt{step}(\\mathcal B)$ progresses to the next time step: $\\mathcal B$ computes the new value $y'$ of $y$ based on the current value of $(x, y, A, W)$, and updates $A$ and $W$ using Hebbian plasticity.\n    \\item $\\texttt{read}(\\mathcal B)$ returns $y$, i.e. the indicator vector of the neurons in the learning area which are currently firing\n    \\item $\\texttt{inhibit}(\\mathcal B)$ forces $y$ to zero, silencing all activity in the learning area\n\\end{itemize}\n\\fi\n\n\\begin{algorithm} \\label{alg:mechanism}\n\\caption{The learning mechanism. ($B$ denotes the brain area.)}\n\\KwIn{a set of stimulus classes $A_1, \\ldots, A_c$; $T \\ge 1$}\n\\KwOut{A set of assemblies $y_1, \\ldots, y_c$ in the brain area encoding these classes}\n\\ForEach{ stimulus class $i$}{\n    inh$(B) \\gets 0$\\;\\\\\n    \\ForEach{ time step $1 \\le t \\le T$}{\n        Sample $x \\sim A_i$ and fire $x$\\; \n        }\n    $y_i \\gets \\texttt{read}(B)$\\;\\\\\n    inh$(B) \\gets 1$\\;\n   \n    }\n\\end{algorithm}\n\nThat is, we sample $T$ stimuli $x$ from each class, fire each $x$ to cause synaptic input in the brain area, and after the $T$th sample has fired we record the assembly which has been formed in the brain area.  This is the representation for this class.\n\n\\subsection*{Related work}\nThere are numerous learning models in the neuroscience literature. \nIn a variation of the model we consider here, \\citet{RangamaniGandhi2020} have considered supervised learning of Boolean functions using assemblies of neurons, by setting up separate brain areas for each label value.\nAmongst other systems with rigorous guarantees, assemblies are superficially similar to the ``items'' of Valiant's neuroidal model \\citep{Valiant94}, in which supervised learning experiments have been conducted \\citep{Valiant00, FeldmanV09}, where an output neuron is clamped to the correct label value, while the network weights are updated under the model. The neuroidal model is considerably more powerful than ours, allowing for arbitrary state changes of neurons and synapses; in contrast, our assemblies rely on only two biologically sound mechanisms, plasticity and inhibition. \n\nHopfield nets \\citep{hopfield1982neural} are recurrent networks of neurons with symmetric connection weights which will converge to a memorized state from a sufficiently similar one, when properly trained using a local and incremental update rule.  In contrast, the memorized states our model produces (which we call assemblies) emerge through plasticity and randomization from the structure of a random directed network, whose weights are asymmetric and nonnegative, and in which inhibition --- not the sign of total input --- selects which neurons will fire. \n\nStronger learning mechanisms have recently been proposed. Inspired by the success of deep learning, a large body of work has shown that cleverly laid-out microcircuits of neurons can approximate backpropagation to perform gradient descent \\citep{Lillicrap2016RandomSF, sacramento2017dendritic, guerguiev2017towards, sacramento2018dendritic, whittington2019theories, lillicrap2020}. These models rely crucially on novel types of neural circuits which, although biologically possible, are not presently known or hypothesized in neurobiology, nor are they proposed as a theory of the way the brain works. These models are capable of matching the performance of deep networks on many tasks, which are more complex than the simple, classical learning problems we consider here. The difference between this work and ours is, again, that here we are showing that learning arises naturally from well-understood mechanisms in the brain, in the context of the assembly calculus.\n\n\\section{Results} \\label{results}\nVery few stimuli sampled from an input distribution are activated sequentially at the sensory area.  \nThe only form of supervision required is that all training samples from a given class are presented consecutively.\nPlasticity and inhibition alone ensure that, in response to this activation, an assembly will be formed for each class, and that this same assembly will be recalled at testing upon presentation of other samples from the same distribution. In other words, learning happens. \nAnd in fact, despite all these limitations, we show that the device is an efficient learner of interesting concept classes.\n\nOur first theorem is about the creation of an assembly in response to inputs from a stimulus class. This is a generalization of a theorem from \\citet{papadimitriou2019random}, where the input stimulus was held constant; here the input is a stream of random samples from the same stimulus class. Like all our results, it is a statement holding with high probability (WHP), where the underlying random event is the random graph and the random samples. When sampled stimuli fire, the assembly in the brain area changes. The neurons participating in the current assembly (those whose synaptic input from the previous step is among the $k$ highest) are called the current {\\em winners.}\nA {\\em first-time winner} is a current winner that participated in no previous assembly (for the current stimulus class).\n\n\\begin{theorem}[Creation] \\label{theorem:creation} \nConsider a stimulus class $A$ projected to a brain area. Assume that \n\\[\\beta \\geq \\beta_0 = \\frac{1}{r^2}\\frac{\\left(\\sqrt{2} - r^2\\right)\\sqrt{2\\ln\\left(\\frac{n}{k}\\right)} + \\sqrt{6}}{\\sqrt{kp} + \\sqrt{2\\ln\\left(\\frac{n}{k}\\right)}}\\] Then WHP no first-time winners will enter the cap after $O(\\log k)$ rounds, and moreover the total number of winners $\\support{A}$  can be bounded as \\[|\\support{A}| \\leq \\frac{k}{1-\\exp(-(\\frac{\\beta}{\\beta_0})^2)} \\leq k + O\\left(\\frac{\\log n}{r^3p\\beta^2}\\right)\\]  \\end{theorem}\n\\begin{remark}\nThe theorem implies that for a small constant $c$, it suffices to have plasticity parameter \n\\[\n\\beta \\ge \\frac{1}{r^2}\\frac{c}{\\sqrt{kp\/(2\\ln(n\/k))}+1}.\n\\]\n\\end{remark}\n\n\\noindent Our second theorem is about {\\em recall} for a single assembly, when a new stimulus from the same class is presented.  We assume that examples from an an assembly class $A$ have been presented, and a response assembly $A^*$ encoding this class has been created, by the previous theorem.  \n\n\\begin{theorem}[Recall] \\label{theorem:recall}\nWHP over the stimulus class, the set $C_1$ firing in response to a test assembly from the class $A$ will overlap $\\assmcore{A}$ by a fraction of at least $1 - e^{-kpr}$, i.e. \\[\\frac{|C_1 \\setminus \\assmcore{A}|}{k} \\leq e^{-kpr}\\]\n\\end{theorem}\n\\noindent  The proof entails showing that the average weight of incoming connections to a neuron in $\\assmcore{A}$ from neurons in $\\stimcore{A}$ is at least \\[1 + \\frac{1}{\\sqrt{r}}\\left(\\sqrt{2} + \\sqrt{\\frac{2}{kpr}\\ln\\left(\\frac{n}{k}\\right) + 2}\\right)\\]\n\n\\noindent Our third theorem is about the creation of a second assembly corresponding to a second stimulus class. This can easily be extended to many classes and assemblies.  As in the previous theorem, we assume that $O(\\log k)$ examples from assembly class $A$ have been presented, and $\\support{A}$ has been created.  Then we introduce $B$, a second stimulus class, with $|\\stimcore{A} \\cap \\stimcore{B}| = \\alpha k$, and present $O(\\log k)$ samples to induce a series of caps, $B_1, B_2, \\ldots$, with $B^*$ as their union.\n\n\\begin{theorem}[Multiple Assemblies] \\label{theorem:multiple}\nThe total support of $B^*$ can be bounded WHP as \n\\[|B^*| \\leq \\frac{k}{1-\\exp(-(\\frac{\\beta}{\\beta_0})^2)} \\leq k + O\\left(\\frac{\\log n}{r^3p\\beta^2}\\right)\\]\nMoreover, WHP, the overlap in the core sets $\\assmcore{A}$ and $\\assmcore{B}$ will preserve the overlap of the stimulus classes, so that $|\\assmcore{A} \\cap \\assmcore{B}| \\leq \\alpha k$.\n\\end{theorem}\nThis time the proof relies on the fact that the average weight of incoming connections to a neuron in $\\assmcore{A}$ is {\\em upper-bounded} by \\[\\gamma \\leq 1 + \\frac{\\sqrt{2\\ln\\left(\\frac{n}{k}\\right)} - \\sqrt{2\\ln((1+r)\/r\\alpha)}}{\\alpha r \\sqrt{kp}}\\] \n\n\\noindent Our fourth theorem is about classification after the creation of multiple assemblies, and shows that random stimuli from any class are mapped to their corresponding assembly. We state it here for two stimuli classes, but again it is extended to several. We assume that stimulus classes $A$ and $B$ overlap in their core sets by a fraction of $\\alpha$, and that they have been projected to form a distribution of assemblies $\\assmcore{A}$ and $\\assmcore{B}$, respectively.\n\n\\begin{theorem}[Classification] \\label{theorem:classify}\nIf a random stimulus chosen from a particular class (WLOG, say $B$) fires to cause a set $C_1$ of learning area neurons to fire, then WHP over the stimulus class the fraction of neurons in the cap $C_1$ and in $\\assmcore{B}$ will be at least \n\\[\\frac{|C_1 \\cap \\assmcore{B}|}{k} \\geq 1 - 2\\exp\\left(-\\frac{1}{2}(\\gamma \\alpha - 1)^2 kpr \\right)\\]\nwhere $\\gamma$ is a lower bound on the average weight of incoming connections to a neuron in $\\assmcore{A}$ (resp. $\\assmcore{B}$) from neurons in $\\stimcore{A}$ (resp. $\\stimcore{B}$).\n\\end{theorem}\n\n\n\n\\noindent Taken together, the above results guarantee that this mechanism can learn to classify well-separated distributions, where each distribution has a constant fraction of its nonzero coordinates in a subset of $k$ input coordinates. The process is {\\em naturally interpretable:} an assembly is created for each  distribution, so that random stimuli are mapped to their corresponding assemblies, and the assemblies for different distributions overlap in no more than the core subsets of their corresponding distributions. \n\n\nFinally, we consider the setting where the labeling function is a linear threshold function, parameterized by an arbitrary nonnegative vector $v$ and margin $\\Delta$. We will create a single assembly to represent examples on one side of the threshold, i.e. those for which $v \\cdot X \\ge  \\|v\\|_1 k \/ n$. We define $\\mathcal D_+$ denote the distribution of these examples, where each coordinate is an independent Bernoulli variable with mean $\\mathbb{E}(X_i) = k\/n + \\Delta v_i$, and define $\\mathcal D_-$ to be the distribution of negative examples, where each coordinate is again an independent Bernoulli variable yet now all identically distributed with mean $k\/n$. (Note that the support of the positive and negative distributions is the same; there is a small probability of drawing a positive example from the negative distribution, or vice versa.) To serve as a classifier, a fraction $1- \\epsilon_+$ of neurons in the assembly must be guaranteed to fire for a positive example, and a fraction $\\epsilon_- < 1 - \\epsilon_+$ guaranteed \\emph{not} to fire for a negative one. A test example is then classified as positive if at least a $1 - \\epsilon$ fraction of neurons in the assembly fire (for $\\epsilon \\in [\\epsilon_-, 1 - \\epsilon_+]$), and negative otherwise. The last theorem shows that this can in fact be done with high probability, as long as the normal vector $v$ of the linear threshold is neither too dense nor too sparse. Additionally, we assume synapses are subject to homeostasis in between training and evaluation; that is, all of the incoming weights to a neuron are normalized to sum to 1.\n\\begin{theorem}[Learning Linear Thresholds]\\label{theorem:halfspace}\nLet $v$ be a nonnegative vector normalized to be of unit Euclidean length ($\\|v\\|_2 = 1$). Assume that  $\\Omega(k) = \\|v\\|_1 \\le \\sqrt{n}\/2$ and \n\\[\\Delta^2\\beta \\ge \\sqrt{\\frac{2k}{p}}(\\sqrt{2\\ln(n\/k)+2)} + 1).\\] \nThen,\nsequentially presenting $\\Omega(\\log k)$ samples drawn at random from $\\mathcal D^+$ forms an assembly $\\assmcore{A}$ that correctly separates $D^+$ from $D^-$: with probability $1-o(1)$ a randomly drawn example from $\\mathcal D^+$ will result in a cap which overlaps at least $3k\/4$ neurons in $\\assmcore{A}$, and an example from $\\mathcal D^-$ will create a cap which overlaps no more than $k\/4$ neurons in $\\assmcore{A}$.\n\\end{theorem}\n\n\\begin{remark}\nThe bound on $\\Delta^2 \\beta$ leads to two regimes of particular interest: In the first, \\[\\beta \\ge \\frac{\\sqrt{2\\ln(n\/k) + 2} + 1}{\\sqrt{kp}}\\] and $\\Delta \\ge \\sqrt{k}$, which is similar to the plasticity parameter required for a fixed stimulus \\citep{papadimitriou2019random} or stimulus classes; in the second, $\\beta$ is a constant, and \\[\\Delta \\ge \\left(\\frac{2k}{\\beta^2 p}\\right)^{1\/4}\\left(\\sqrt{2\\ln(n\/k)+2} +1\\right)^{1\/2}.\\]\n\\end{remark}\n\n\\begin{remark}\nWe can ensure that the number of neurons outside of $\\assmcore{A}$ for a positive example or in $\\assmcore{A}$ for a negative example are both $o(k)$ with small overhead\\footnote{i.e. increasing the plasticity constant $\\beta$ by a factor of $1 + o(1)$}, so that plasticity can be active during the classification phase. \n\\end{remark}\n\nSince our focus in this paper is on highlighting the brain-like aspects of this learning mechanism, we emphasize stimulus classes as a case of particular interest, as they are a probabilistic generalization of the single stimuli considered in \\citet{papadimitriou2019random}.\nLinear threshold functions are an equally natural way to generalize a single $k$-sparse stimulus, say $v$; all the 0\/1 points on the positive side of the threshold $v^\\top x \\ge \\alpha k$ have at least an $\\alpha$ fraction of the $k$ neurons of the stimulus active.\n\n\nFinally, reading the output of the device by the Assembly Calculus is simple: Add a {\\em readout area} to the two areas so far (stimulus and learning), and project to this area one of the assemblies formed in the learning area for each stimulus class.  The assembly in the learning area that fires in response to a test sample will cause the assembly in the readout area  corresponding to the class to fire, and this can be sensed through the {\\tt readout} operation of the AC.\n\n\\paragraph{Proof overview.}\nThe proofs of all five theorems can be found in the Appendix.  The proofs hinge on showing that large numbers of certain neurons of interest will be included in the cap on a particular round --- or excluded from it. More specifically: \\begin{itemize}\n    \\item To create an assembly, the sequence of caps should converge to the assembly's core set. In other words, WHP an increasing fraction of the neurons selected by the cap in a particular step will also be selected at the next one.\n    \\item For recall, a large fraction of the assembly should fire (i.e. be included in the cap) when presented with an example from the class.\n    \\item To differentiate stimuli (i.e. classify), we need to ensure that a large fraction of the correct assembly will fire, while no more than a small fraction of the other assemblies do.\n\\end{itemize} Following \\citet{papadimitriou2019random}, we observe that if the probability of a neuron having input at least $t$ is no more than $\\epsilon$, then no more than an $\\epsilon$ fraction of the cohort of neurons will have input exceeding $t$ (with constant probability). By approximating the total input to a neuron as Gaussian and using well-known bounds on Gaussian tail probabilities, we can solve for $t$, which gives an explicit input threshold neurons must surpass to make a particular cap. Then, we argue that the advantage conferred by plasticity, combined with the similarity of examples from the same class, gives the neurons of interest enough of an advantage that the input to all but a small constant fraction will exceed the threshold.\n\n\n\\section{Experiments}\nThe learning algorithm has been run on both synthetic and real-world datasets, as illustrated in the figures below. Code for experiments is available at \n\\url{https:\/\/github.com\/mdabagia\/learning-with-assemblies}.\n\nBeyond the basic method of presenting a few examples from the same class and allowing plasticity to alter synaptic weights, the training procedure is slightly different for each of the concept classes (stimulus classes, linearly-separated, and MNIST digits). In each case, we renormalize the incoming weights of each neuron to sum to one after concluding the presentation of each class, and classification is performed on top of the learned assemblies by predicting the class corresponding to the assembly with the most neurons on. \\begin{itemize}\n    \\item For stimulus classes, we estimate the assembly for each class as composed of the neurons which fired in response to the last training example, which in practice are the same as those most likely to fire for a random test example. \n    \\item For a linear threshold, we only present positive examples, and thus only form an assembly for one class. As with stimulus classes, the neurons in the assembly can be estimated by the last training cap or by averaging over test examples. We classify by comparing against a fixed threshold, generally half the cap size.\n   \n\\end{itemize}\nAdditionally, it is important to set the plasticity parameter ($\\beta$) large enough that assemblies are reliably formed. We had success with $\\beta = 0.1$ for stimulus classes and $\\beta = 1.0$ for linear thresholds.\n\n\nIn Figure \\ref{figure:experiments} (a) \\& (b), we demonstrate learning of two stimulus classes, while in\nFigure \\ref{figure:experiments} (c) \\& (d), we demonstrate the result of learning a well-separated linear threshold function with assemblies.  Both had perfect accuracy. Additionally, assemblies readily generalize to a larger number of classes (see Figure \\ref{figure:fourclasses} in the appendix). We also recorded sharp threshold transitions in classification performance as the key parameters of the model are varied (see Figures \\ref{fig:accuracies} \\& \\ref{fig:transition}).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{images\/overlap.png}\n    \\includegraphics[width=0.7\\linewidth]{images\/overlap_hs.png}\n    \\caption{Assemblies learned for various concept classes. On the top two lines, we show assemblies learned for stimulus classes, and on the bottom two lines, for a linear threshold with margin. In (a) \\& (c) we exhibit the distribution of firing probabilities over neurons of the learning area. In (b) \\& (d) we show the average overlap of different input samples (red square) and the overlaps of the corresponding representations in the assemblies (blue square). Using a simple sum readout over assembly neurons, both stimulus classes and linear thresholds are classified with 100\\% accuracy. Here, $n=10^3, k=10^2, p=0.1, r=0.9, q=0.1, \\Delta = 1.0$, with 5 samples per class, and $\\beta = 0.01$ (stimulus classes) and $\\beta=1.0$ (linear threshold).}\n    \\label{figure:experiments}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.45\\linewidth]{images\/stimulus_acc.png}\n    \\includegraphics[width=0.45\\linewidth]{images\/halfspace_acc.png}\n    \\caption{Mean (dark line) and range (shaded area) of classification accuracy for two stimulus classes (left) and a fixed linear threshold (right) over 20 trials, as the classes become more separable. For stimulus classes, we vary the firing probability of neurons in the stimulus core while fixing the probability for the rest at $k\/n$, while for the linear threshold, we vary the margin. \n   \n   \n    For both we used 5 training examples with $n=1000, k=100, p=0.1$, and $\\beta = 0.1$ (stimulus classes), $\\beta = 1.0$ (linear threshold).}\n    \\label{fig:accuracies}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.45\\linewidth]{images\/acc_vs_n.png}\n    \\includegraphics[width=0.45\\linewidth]{images\/acc_vs_k.png}\n    \\caption{Mean (dark line) and range (shaded area) of classification accuracy of two stimulus classes for various values of the number of neurons ($n$, left) and the cap size ($k$, right). For variable $n$, we let $k = n \/ 10$; for variable $k$, we fix $n = 1000$. Other parameters are fixed, as $p = 0.1, r = 0.9, q = k\/n$, and $\\beta = 0.1$.}\n    \\label{fig:transition}\n\\end{figure}\n\nThere are a number of possible extensions to the simplest strategy, where within a single brain region we learn an assembly for each concept class and classify based on which assembly is most activated in response to an example. \nWe compared the performance of various classification models on MNIST as the number of features increases. The high-level model is to extract a certain number of features using one of the five different methods, and then find the best linear classifier (of the training data) on these features to measure performance (on the test data). The five different feature extractors are:\n\\begin{itemize}\n    \\item Linear features. Each feature's weights are sampled i.i.d. from a Gaussian with standard deviation $0.1$.\n    \\item Nonlinear features. Each feature is a binary neuron: it has $784$ i.i.d. Bernoulli$(0.2)$ weights, and `fires' (has output $1$, otherwise $0$) if its total input exceeds the expected input ($70 \\times 0.2$).\n    \\item Large area assembly features. In a single brain area of size $m$ with cap size $m \/ 10$, we attempt to form an assembly for each class. The area sees a sequence of $5$ examples from each class, with homeostasis applied after each class. Weights are updated according to Hebbian plasticity with $\\beta = 1.0$. Additionally, we apply a negative bias: A neuron which has fired for a given class is heavily penalized against firing for subsequent classes.\n    \\item 'Random' assembly features. For a total of $m$ features, we create $m \/ 100$ different areas of $100$ neurons each, with cap size $10$. We then repeat the large area training procedure above in each area, with the order of the presentation of classes randomized for each area.\n    \\item 'Split' assembly features: For a total of $m$ features, we create $10$ different areas of $m \/ 10$ neurons each, with cap size $m \/ 100$. Area $i$ sees a sequence of $5$ examples from class $i$. Weights are updated according to Hebbian plasticity, and homeostasis is applied after training.\n\\end{itemize}\nAfter extracting features, we train the linear classification layer to minimize cross-entropy loss on the standard MNIST training set ($60000$ images) and finally test on the full test set ($10000$ images). \n\n\\begin{figure}[b!]\\label{fig:mnistcompare}\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{images\/acc_v_neurons.png}\n    \\caption{MNIST test accuracy as the number of features increases, for various classification models. 'Split' assembly features, which forms an assembly for class $i$ in area $i$, achieves the highest accuracy with the largest number of features.}\n\\end{figure}\n\nThe results as the total number of features ranges from $1000$ to $10000$ is shown in Fig. \\ref{fig:mnistcompare}. 'Split' assembly features are ultimately the best of the five, with 'split' features achieving $96\\%$ accuracy with $10000$ features. However, nonlinear features outperform 'split' and large-area features and match 'random' assembly features when the number of features is less than $8000$. \nFor reference, the linear classifier gets to $89\\%$, while a two-layer neural network with width $800$ trained end-to-end gets to $98.4\\%$.\n\nGoing further, one could even create a hierarchy of brain areas, so that the areas in the first ``layer'' all project to a higher-level area, in hopes of forming assemblies for each digit in the higher-level area which are more robust. In this paper, our goal was to highlight the potential to form useful representations of a classification dataset using assemblies, and so we concentrated on a single layer of brain areas with a very simple classification layer on top. It will be interesting to explore what is possible with more complex architectures. \n\n\\section{Discussion}\nAssemblies are widely believed to be involved in cognitive phenomena, and the AC provides evidence of their computational aptitude.  Here we have made the first steps towards understanding how {\\em learning} can happen in assemblies. Normally, an assembly is associated with a stimulus, such as Grandma. We have shown that this can be extended to {\\em a distribution over stimuli.} Furthermore, for a wide range of model parameters, distinct assemblies can be formed for multiple stimulus classes in a single brain area, so long as the classes are reasonably differentiated.\n\nA model of the brain at this level of abstraction should allow for the kind of classification that the brain does effortlessly --- e.g., the mechanism that enables us to understand that individual frames in a video of an object depict the same object. With this in mind, the learning algorithm we present is remarkably parsimonious: it generalizes from a handful of examples which are seen only once, and requires no outside control or supervision other than ensuring multiple samples from the same concept class are presented in succession (and this latter requirement could be relaxed in a more complex architecture which channels stimuli from different classes).  Finally, even though our results are framed within the Assembly Calculus and the underlying brain model, we note that they have implications far beyond this realm. In particular, they suggest that \\emph{any} recurrent neural network, equipped with the mechanisms of plasticity and inhibition, will naturally form an assembly-like group of neurons to represent similar patterns of stimuli.\n\nBut of course, many questions remain.  In this first step we considered a single brain area --- whereas it is known that assemblies draw their computational power from the interaction, through the AC, among many areas.  \nWe believe that a more general architecture encompassing a hierarchy of interconnected brain areas, where the assemblies in one area act like stimulus classes for others, can succeed in learning more complex tasks --- and even within a single brain area improvements can result from optimizing the various parameters, something that we have not tried yet.   \n\nIn another direction, here we only considered Hebbian plasticity, the simplest and most well-understood mechanism for synaptic changes. Evidence is mounting in experimental neuroscience that the range of plasticity mechanisms is far more diverse \\citep{magee2020synaptic}, and in fact it has been demonstrated recently \\citep{payeur2021burst} that more complex rules are sufficient to learn harder tasks. Which plasticity rules make learning by assemblies more powerful?\n\nWe showed that assemblies can learn nonnegative linear threshold functions with sufficiently large margins. Experimental results suggest that the requirement of nonnegativity is a limitation of our proof technique, as empirically assemblies readily learn arbitrary linear threshold functions (with margin). What other concept classes can assemblies provably learn?  We know from support vector machines that linear threshold functions can be the basis of far more sophisticated learning when their input is pre-processed in specific ways, while the celebrated results of \\citet{rahimi2007random} demonstrated that certain families of random nonlinear features can approximate sophisticated kernels quite well. What would constitute {\\em a kernel} in the context of assemblies? The sensory areas of the cortex (of which the visual cortex is the best studied example) do pre-process sensory inputs extracting features such as edges, colors, and motions. Presumably learning by the non-sensory brain --- which is our focus here --- operates on the output of such pre-processing. We believe that studying the implementation of kernels in cortex is a very promising direction for discovering powerful learning mechanisms in the brain based on assemblies.\n\n\n\n\\acks{We thank Shivam Garg, Chris Jung, and Mirabel Reid for helpful discussions. MD is supported by an NSF Graduate Research Fellowship. SV is supported in part by NSF awards CCF-1909756, CCF-2007443 and CCF-2134105. CP is supported by NSF Awards CCF-1763970 and CCF-1910700, and by a research contract with Softbank.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Local Description} \n\nAmong the singular Riemannian metrics on surfaces, the simplest ones are those with isolated singularities. Away from a discrete set, these metrics are then  smooth (say of class $C^2$). We will make two additional  hypotheses which are natural from a geometric point of view.\nThe first concerns the conformal structure.  If $g$ is a Riemannian metric on a surface $S$ having an isolated singularity at $p$, and if $U$ is a neighborhood of $p$ homeomorphic to the disk, then $U' = U \\setminus \\{0\\}$  has a well-defined conformal structure (since the metric $g$ is smooth on $U'$,  one may  apply Korn-Lichtenstein's Theorem). From  the classification of conformal structures on the annulus, we know that $U'$  is conformally equivalent to a standard annulus \n$$\n  A_{\\rho} = \\{z \\in \\mathbb{C} \\mid \\rho < |z| < 1\\},\n$$\nfor some fixed parameter $\\rho \\in [0, 1)$.\nIn this paper, we will always assume  $U'$  to be conformally equivalent to the punctured disk $A_0$. In other words, we are assuming that the conformal structure of $U'$ extends to $U$ (i.e. the point $p$ is a removable singularity from the conformal viewpoint). \n\nIn particular, every point of $S$ (singular or not) has a coordinate neighborhood  in which the metric  $g$  can be written as \n$$\n  g = \\rho(x,y)(dx^2 + dy^2) = \\rho(z) |dz|^2,\n$$\nwhere $z = x+iy$ and $\\rho$ is a positive function that is of class $C^2$ outside the singularities. Such coordinates are called \\textit{isothermal coordinates}.\n\n\\smallskip\n\n{\\small As a counterexample, one may consider a non-negative function $\\phi$ on $\\mathbb{C}$ that vanishes exactly on a contractible compact subset $Q \\subset \\mathbb{C}$. Let us denote by $S = \\mathbb{C}\/Q$ the space obtained by identifying all the points of $Q$ and $g = \\phi(z) |dz|^2$. If $Q$ contains more than one point, then $(S,g)$ has a singular point $q = [Q]$ that does not satisfy the above condition.\n}\n\n\\medskip\n\nOur  second assumption concerns the curvature. It says that if $K$ denotes the curvature   and $dA$ the area element of $g$, then\n$$\n \\int_{U'} |K| dA < \\infty.\n$$\n\nAn important class of singularities satisfying the above conditions is given by the simple singularities:\n\n\\medskip\n\n\\textbf{Definition.}    \\index{Simple singularity}\nA conformal metric $g$ on a Riemann surface $S$ is said to have a \\textit{simple singularity \nof order} $\\beta$ at $p\\in S$ if it can be locally written as \n$$\n  g = e^{2u(z)} |z|^{2\\beta} |dz|^2,\n$$\nwhere $\\beta$ is a real number and $u$ is a function satisfying\n$$\n  u  \\in L^1  \\quad  \\text{and}  \\quad   \\Delta u  \\in L^1.\n$$\nIn this definition, $z = x + iy$ is a local coordinate on $S$ defined in a neighborhood $U$ of $p$ and such that $z (p) = 0$. The Lebesgue space $L^1$  is defined  with respect to the  Lebesgue measure $dx dy$ on $U$ and the Laplacian of $u$  is defined in the sense of distributions by\n$\\Delta u= - \\frac{\\partial^2}{\\partial x^2}   - \\frac{\\partial^2}{\\partial y^2}$. \n\n\n\\medskip \n\n\nSimple singularities naturally appear  in several  contexts as we shall soon see. A first class of examples \nis given by the following result due to MacOwen \\cite[Appendix B]{McO}:\n\n\\begin{theorem}\n Let $g = e^{2u} |dz|^2$  be a conformal metric on the unit disk $D = \\{z \\in \\mathbb{C}\u00a0\\mid   |z| <1\\}$ having a singularity at the origin. Suppose that $g$ is smooth on the punctured disk  $D' =  D \\setminus \\{0\\}$.\nIf there exist $\\ell \\in \\r$ and $a, b> 0$ such that the curvature $K$ of $g$ satisfies\n$$\n - b|z|^{\\ell} \\leq K(z) \\leq  - a|z|^{\\ell}, \n$$\nthen $0$ is a simple singularity of $g$.\n\\end{theorem}\n\nA simple singularity of order $\\beta < -1$  is always at infinite distance while a simple singularity of order $\\beta > -1$ is always at a finite distance of ordinary points. For a singularity of order $- $1, both cases can occur, see \\S 2.2 in  \\cite{HT}. A \\textit{cusp} \\index{Cusp} is a simple singularity  of order  $\\beta = -1$ admitting a neighborhood of finite area.\n\n\\medskip\n\nA simple singularity of order  $\\beta > -1$  is also called a \\textit{conical singularity}   \\index{Conical Singularity}  \nof (total)  angle $\\theta = 2\\pi  (\\beta+ 1)$. Such a singularity can indeed be approximated by a Euclidean cone of total angle $\\theta$. In particular, if the curvature of $g$ is  bounded in some neighborhood of a conical singularity, there exists an ``exponential map'' making it possible to parametrize a neighborhood of the conical singularity by a neighborhood of the vertex of its tangent cone. In other words, one can introduce polar coordinates near a conical singularity. Moreover, if the curvature is continuous, then these polar coordinates are of class $C^1$ with respect to the isothermal coordinates, see \\cite{Troyanov1990b} for proofs of these facts. \n\n\\section{Global   description}\n\nTo investigate singular surfaces  having several simple singularities, it is convenient to introduce the following notion:\n\\medskip \n\n\\textbf{Definition.}  \\index{Divisor}\nA   \\textit{divisor}  on a Riemann surface $S$ is a formal sum \n$\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$.\nThe \\textit{support} of this divisor is the set $\\mathrm{supp} (\\pmb{\\beta}) = \\{p_1, \\dots, p_n\\}$. A conformal metric $g$ on $S$ \\textit{represents the divisor} $\\pmb{\\beta}$ if it is smooth on the complement of $\\mathrm{supp} (\\pmb{\\beta})$ and if $g$ has a simple singularity of order $\\beta_i$ at $p_i$  for $i = 1, \\dots, n$.\n\n\\medskip \n\n\\textbf{Examples.} \\  (1) The metric $g = |dz|^2$ on the Riemann sphere $\\mathbb{C} \\cup \\{\\infty\\}$ represents the divisor $\\pmb{\\beta} = (-2)\\cdot \\infty$. \n\n\\medskip\n\n(2) More generaly, the metric $g = |z|^{2\\alpha}|dz|^2$ on $\\mathbb{C} \\cup \\{\\infty\\}$ represents the divisor $\\pmb{\\beta} = \\alpha \\cdot 0 +  (-2- \\alpha)\\cdot \\infty$. \n\n\\medskip\n\n (3) If $\\omega = \\varphi (z) dz$ is a meromorphic differential on the Riemann surface $S$, then $g = |\\omega|^2$ is a flat Riemannian metric with simple singularities representing the divisor $\\pmb{\\beta} = \\mathrm{div} (\\omega)$. \n\n\\medskip\n\n(4)  If $(S_1,g_1)$ is a smooth Riemannian surface and $f : S \\to S_1$ is a branched  covering, then $g = f^*(g_1)$ is a Riemannian metric on $S$ representing the ramification divisor of $f$, that is $\\pmb{\\beta} = \\sum_p O_p(f) \\cdot p$, where $O_p(f)$ is the ramification order of $f$ at $p$  (i.e. the local degree minus $1$).\n\n\\medskip\n\n(5) If $S$ is a two-dimensional polyhedron (euclidean, spherical or hyperbolic) with vertices $p_1, \\dots, p_n$, then the metric induced by the geometric realization of that polyhedron represents the divisor $\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$, where \n$2\\pi(\\beta_i+1) = \\theta_i$ is the sum of the angles at $p_i$ of all faces incident with $p_i$.\n\n\\medskip\n\n(6) Let $(\\tilde{S},\\tilde{g})$ be a smooth Riemannian surface on which a finite group $\\Gamma$ acts by isometries. If\n$S = \\tilde{S}\/\\Gamma$ is a surface without boundary, then it inherits a Riemannian metric with simple singularities representing the divisor\n$\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$, where $\\beta_i = (\\frac{1}{n_i}-1)$. Here, the point $p_i$ is the image of a point\n$\\tilde{p}_i \\in \\tilde{S}$ such that $\\Gamma$ has a stabiliser of order $n_i$ at $\\tilde{p}_i $. This examples generalizes to two-dimensional orbifolds.\n\n\\medskip \n\nIn examples 4 to 6, all the singular points are conical singularities. An important source of examples, where no singular point is conical, is given by the following Theorem:\n\\begin{theorem}\\label{th.Huber} \n Let $(S', g ')$ be a complete Riemannian surface of class $C^2$ with finite total curvature: $\\int_{S'}|K|dA < \\infty$. Then there exists a compact Riemann surface $S$, a divisor $\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$ on $S$  such that $\\beta_i \\leq -1$ for all $i$,  and a conformal metric $g$ on $S$ representing this divisor such that   $(S', g ')$ is isometric to\n$(S \\setminus \\mathrm{supp}(\\pmb{\\beta}), g)$.\n\\end{theorem}\nThis result is essentially due to A. Huber, we  refer to \\S 1.1 and  \\S 2.9 in  \\cite{HT} for a discussion and a proof  of Huber's theorem in the above formulation.  \nIt is not difficult to see that if the surface  $(S', g ')$  has finite area, then $\\beta_i = -1$ for all $i$.\n\n\\section{Some global geometry}\n\nFor compact Riemannian surfaces with simple singularities, there is a well known Gauss-Bonnet Formula (see e.g. \\cite{Finn} for the case where all orders satisfy $\\beta_i \\leq -1$). To state the Formula we define  the \\textit{Euler characteristic of a surface $S$ with divisor} as  $\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$ as $\\chi(S, \\pmb{\\beta}) = \\chi(S) + \\sum_i \\beta_i$. \n\n\n\\begin{theorem}[The Gauss-Bonnet Formula]  \\index{Gauss-Bonnet Formula} \\label{GB}\nLet $(S, g)$ be a compact Riemannian surface whose metric represents a divisor $\\pmb{\\beta}$. Then the total curvature of $(S, g)$ is finite and \nwe have\n$$\n \\frac{1}{2\\pi}\\int_S K dA = \\chi(S, \\pmb{\\beta}).\n$$\n\\end{theorem}\n\nSee e.g.  \\cite[Theorem 2.8]{HT} for a proof.\n\n\\medskip \n\nFor example, if $\\omega$ is a meromorphic differential on the closed Riemann surface of genus $\\gamma$, then the Gauss-Bonnet Formula implies that the degree of $\\omega$ (i.e. the  number of zeroes minus the number of poles), is equal to $2\\gamma  - 2$. Indeed, $g = |\\omega|^2$ is a flat metric representing the divisor $\\mathrm{div}(\\omega)$. Another application of the Gauss-Bonnet formula is the Riemann-Hurwitz formula:\n\n\n\\begin{proposition}[The Riemann-Hurwitz Formula]   \\index{Riemann-Hurwitz Formula}\nLet $f : S \\to S_1$ be a branched cover of degree $d$ between two closed surfaces, then \n$$\n  \\chi(S) + \\sum_{p\\in S} O_p(f)  = d \\chi(S_1),\n$$\nwhere $O_p(f)$ is the branching order of $f$ at $p$. \n\\end{proposition} \n\n\\begin{proof}\nPick an arbitrary smooth metric $g_1$ on $S_1$ and set $g = f^*(g_1)$. Then $g$ is a Riemannian metric with simple singularities on\n  $S$ representing the ramification  divisor of $f$. The above formula follows now from Theorem \\ref{GB},  since we obviously have \n$$\n\\int_S K dA = d \\cdot \\int_{S_1} K_1 dA_1. \n$$\n\\end{proof}\n\n\\medskip\n\n  \nHuber's theorem, together  with the Gauss-Bonnet Formula, is a refinement of the Cohn-Vossen inequality:\n\n\\begin{proposition}[The Cohn-Vossen inequality]  \\index{Cohn-Vossen inequality}\nLet $(S', g')$ be a compact Riemannian surface of class $C^2$ with finite total curvature: $\\int_{S'} |K'| dA' < \\infty$, then we have\n$$\n \\frac{1}{2\\pi}\\int_{S'} K' dA' \\leq  \\chi(S').\n$$\nMoreover  we have equality if  $(S', g')$ has finite area.\n\\end{proposition}\n\n\\begin{proof}\nHuber's Theorem tells us that $(S', g')$ admits a compactification $(S, g)$ where $g$ is a metric representing a divisor\n$\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$ such that $\\beta_i \\leq -1$ for all $i$. We then have from the Gauss-Bonnet Formula\n$$\n   \\frac{1}{2\\pi}\\int_{S'} K' dA' =  \\frac{1}{2\\pi}\\int_{S} K dA = \\chi(S) +  \\sum_{i=1}^n \\beta_i  \\leq \\chi(S) -n = \\chi(S').\n$$\nIf $S'$ has finite area, then we have  $\\beta_i =-1$ for all $i$ and the above inequality is in fact an equality. \n\\end{proof}\n{\\small Note that although the Cohn-Vossen inequality is  a consequence of Huber's Theorem  \\ref{th.Huber} and the Gauss-Bonnet Formula, \none should not  consider it to be a corollary of these results. The reason is that the proof of Huber's theorem is in part based on the Cohn-Vossen inequality.}\n\n\\medskip\n\n\nThe difference between $\\chi(S')$ and $\\chi(S, \\pmb{\\beta})$ in the Cohn-Vossen inequality is an isoperimetric constant: \n\n\\begin{theorem}\nLet $(S, g)$ be a compact Riemannian surface whose metric $g$ represents a divisor $\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$ such that $\\beta_i \\leq -1$ for all $i$.  \nFix a point $q$ on $S' = S\\setminus \\{p_1, \\dots, p_n\\}$ and denote by  $A(q, r)$  the area of $B_q(r) := \\{x \\in S' \\mid d(q,x) \\leq r\\}$ and $L(q, r)$ the length of $\\partial B_q(r)$. Then\n$$\n \\lim_{r \\to \\infty} \\frac{L^2(q,r)}{4\\pi A(q,r)}  = - \\sum_{i=1}^n (\\beta_i +1) = \\chi(S') -  \\chi(S, \\pmb{\\beta}).\n$$\n\\end{theorem}\n\nThis result is due to K. Shiohama \\cite{Shiohama}, but R. Finn obtained a partial result in this direction \\cite[Theorem 10]{Finn}. \n\n\n\n\\section{Classifying Flat Metrics}\n\nLet us now formulate a classification theorem for flat metrics with simple singularities on a compact surface.\n\n\\begin{theorem}\n Let $S$ be a compact Riemann surface with divisor $\\pmb{\\beta} = \\sum \\beta_ip_i$. Then there exists a conformal flat metric representing $\\pmb{\\beta}$ on $S$ if and only if $\\chi(S,\\pmb{\\beta}) = 0$. Moreover this metric is unique up to homothety.\n\\end{theorem}\n\nThis theorem has several proofs, see e.g. \\cite{Troyanov1986} and \\cite[\\S 7]{HT}. We  give  here the proof  from \\cite{HT}.\n\n\\begin{proof}\nIntroduce in the neighborhood  of each $p_i$  a coordinate $z_i$  such that $z_i(p_i) = 0$ and choose an arbitrary conformal metric $g_0$ on $S$ such that  $g_0 = |dz_i|^2$ in the neighborhood of  each $p_i$. Let us now choose a positive function $\\rho : S \\to \\r$ which is of class $C^2$ on  $S\\setminus \\mathrm{supp}(\\pmb{\\beta})$  and such that $\\rho = |z_i|^{2\\beta_i}$ in the neighborhood of each $p_i$.\nThe metric $g_1 = \\rho g_0$ is then a conformal metric representing the divisor $\\pmb{\\beta}$. \n\nSince the desired metric must be conformal on $S$, it can be written as $g = e^{2u} g_1$. Note that if $u$ is a function of class $C^2$ on $S$ such that \n\\begin{equation}\\label{LaplK1}\n  \\Delta_1 u = -K_1,\n\\end{equation}\nwhere $\\Delta_1$ and $K_1$ denote   the Laplacian and the curvature of $g_1$, then  $g = e^{2u} g_1$  is a flat conformal metric representing $\\pmb{\\beta}$ on $S$. \nBecause $\\Delta_1$ is a singular operator,  it is more convenient  to write the previous equation as\n\\begin{equation}\\label{LaplK2}\n  \\Delta_0 u = -\\rho K_1,\n\\end{equation}\nwhere $\\Delta_0$ is the Laplacian of the smooth metric  $g_0$.\n\nNote that \\eqref{LaplK1} and  \\eqref{LaplK2} are equivalent equations, but since $K_1$ vanishes in a  neighborhood of the points $p_i$ and the functions $\\rho$ and $K_1$  are of class $C^2$ on  $S\\setminus \\mathrm{supp}(\\pmb{\\beta})$, the right hand side  of \\eqref{LaplK2} is  of class $C^2$ on the whole surface $S$.\n\nIt is well known that the partial differential equation \\eqref{LaplK2} has a solution if and only if the integral of the right hand side vanishes, which  follows from the Gauss-Bonnet formula:\n$$\n \\int_S K_1 \\rho dA_0 = \\int_S K_1 dA_1 = 2\\pi \\chi(S,\\pmb{\\beta}) = 0.\n$$\nWe have thus proved the existence of a flat conformal metric on $S$ representing the divisor $\\pmb{\\beta}$. \nThe uniqueness follows from the fact that if $g_1$ and $g_2$ are two such metrics, then $g_2 = e^{2v} g_1$ for a harmonic function $v$ on $S$. This function is constant (because $S$ is a closed surface) and the two metrics are therefore homothetic.\n\\end{proof}\n \n\\medskip\n\nThe above Theorem gives us a short proof of the Uniformization Theorem for the sphere:\n\n\\begin{theorem}  \\index{Uniformization Theorem (for the 2-sphere)}\n  Any  Riemann surface homeomorphic to the two-sphere  is conformally equivalent to $\\mathbb{C} \\cup \\{\\infty\\}$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet us choose a point $p$ in $S$ and consider the divisor $\\pmb{\\beta} = (-2)\\cdot p$. Observe that $\\chi(S, \\pmb{\\beta}) = 2 - 2 = 0$.\nThe previous theorem tells us   that there is a conformal  flat metric $g$ on $S$ representing this divisor. It is clear that $(S, g)$ is isometric (and thus conformally equivalent) to   $\\left(\\mathbb{C} \\cup \\{\\infty\\}, |dz|^2\\right) $.\n\\end{proof}\n \n\\medskip\n\n{\\small  Of course the Uniformization  Theorem also  means   that  there exists a smooth  conformal  metric of constant positive curvature on $S$. However, it is hard to prove this result by directly solving the corresponding Berger-Nirenberg problem, that is by  directly constructing a conformal metric of curvature $+1$. The above proof on the other hand is almost trivial.}\n\n\n\\section{The  Berger--Nirenberg Problem  on Surfaces with Divisors} \\index{The  Berger--Nirenberg Problem}\n\nThe classical  Berger-Nirenberg problem is the following:\n\n\\medskip \n\n\\textbf{Problem 1.} Let $S$ be a Riemann surface and  $K : S \\to  \\mathbb{R}$ a function on this surface. Is there a conformal metric on $S$ whose curvature is the function $K$? If it exists, is such a metric unique?\n\n\\medskip \n\nThis problem is clearly not well posed  for open surfaces. One could hope that the   problem is well posed for  complete Riemannian metrics, however, it is not difficult to construct families $\\{g_{\\lambda}\\}$ of conformal metrics on a Riemann surface which are complete, conformal, of the same curvature and whose geometry at infinity drastically varies with $\\lambda$,  in the sense that they are not mutually bilipschitz.   An example is given in \\cite{HT1990}.\nThe previous discussion, in particular Huber's Theorem \\ref{th.Huber}, suggests to replace the Berger-Nirenberg Problem on open  surfaces by a version of the  problem on compact surfaces with a divisor.\n \n\n\\medskip \n\n\\textbf{Problem 2.} Let $(S,\\beta)$ be a compact  Riemann surface  with divisor, and  $K : S \\to  \\mathbb{R}$ be a smooth function. Is there a conformal metric $g$ on $S$ that  represents $\\pmb{\\beta}$ and whose curvature is the function $K$? If it exists, is such a metric unique?\n\n\\medskip \n\nWe have already answered this question when $K$ vanishes everywhere.\n\n\\smallskip \n\nProblem 2 is studied in the papers \\cite{Troyanov1991} (in the case of conic singularity) and \\cite{HT} in the general case. The results can be summarized in a form which is similar to the classical theory in the smooth case as it is exposed   in the foundational article  \\cite{KW} by Jerry Kazdan and Frank Warner.\n\n\\begin{theorem}\\label{prescrK}\nLet $(S, \\pmb{\\beta})$ be a compact Riemann surface with a divisor $\\pmb{\\beta} = \\sum \\beta_ip_i$ , and $K : S \\to \\r$ be  a smooth function. Suppose that there exists $p>1$ such that   $h_i(z) = |z-p_i|^{2\\beta_i}K(z)$ is a function of class $L^p$ in a neighborhood of each $p_i$. Moreover\n\\begin{enumerate}[(a)]\n\\item If $\\chi(S,\\pmb{\\beta}) >0$, we assume  $\\sup(K) >0$ and $q\\chi(S,\\pmb{\\beta})<2$, where $q=p\/(p-1)$.\n\\item If $\\chi(S,\\pmb{\\beta}) =0$, we assume either that $K \\equiv 0$  or  $\\sup(K) >0$  and $\\int_S KdA_0 < 0$, where $dA_0$\nis the area element of a flat conformal metric representing $\\pmb{\\beta}$.\n\\item If $\\chi(S,\\pmb{\\beta}) <0$, we assume $K \\leq 0$ and $K \\not\\equiv 0$.\n\\end{enumerate}\nThen there exists a conformal metric $g$ on $S $ which represents the divisor $\\pmb{\\beta}$\nand whose curvature is $K$. In case (c), this metric is unique.\n\\end{theorem}\n\n\nA very brief idea of the proof is presented in \\cite{HT1990} (see \\cite{Troyanov1991} and \\cite{HT} for details).\nSome particular cases of this theorem have been obtained previously by W. M. Ni, R. MacOwen and P. Aviles. At the beginning of the twentieth century, Emile Picard had already studied the case of curvature $- 1$ in \\cite{Picard}.\nThe hypotheses of the previous theorem impose a decay of the curvature  when approaching the singularities of order $< - 1$. The next result, only valid for non-positive curvature,  does not impose such a behavior.\n\n\\begin{theorem}\nLet $S$ be a compact Riemann surface and $g_1$ be a conformal metric representing a divisor $\\pmb{\\beta} = \\sum_{i=1}^n  \\beta_ip_i$\nsuch that $n \\geq 1$ and $\\chi(S,\\pmb{\\beta}) < 0$. Let  $K : S \\to \\r$  be a smooth nonpositive function  such that \n$$ bK\\leq K_1\\leq aK<0$$\non the complement of  a compact subset of $S' = S \\setminus \\{p_1, \\dots, p_n\\}$, where $K_1$  is the curvature of $g_1$ and $a,b$  are positive constants. Then there exists a unique conformal metric $g$ on $S$ which represents $\\pmb{\\beta}$, has curvature $K$ and is \nconformally quasi-isometric to $g_1$.\n\\end{theorem}\n\nSee \\cite[Theorem 8.1, 8.4]{HT}  or  \\cite{McO} for the proof.  As an application, using this  Theorem,  one can  construct metrics with prescribed (negative) curvature having cusps. The previous Theorem  also admits a generalization to non-compact Riemann surfaces of finite type having hyperbolic ends.\n\n\\medskip\n\nWe end this section  with two  results on the Berger-Nirenberg Problem on closed Riemann surfaces with divisor. The first result, proved in \\cite{Tang} by Junjie Tang,  allows us to solve the Berger-Nirenberg Problem  when $\\chi (S, \\pmb{\\beta})$ is small enough. \n\n \n\n\\begin{theorem}\nLet $S$ be a closed Riemann surface with a divisor  ${\\pmb{{\\pmb{\\alpha}}}} = \\sum_{i=1}^n \\alpha_i p_i$ such that $\\chi(S, {\\pmb{\\alpha}}) = 0$, and let\nus pick a conformal metric $g_0$ representing ${\\pmb{\\alpha}}$. \n\nSuppose another divisor $\\pmb{\\beta}= \\sum_{i=1}^n \\beta_i p_i$ with same support is given on $S$,  such that  $\\chi(S, \\pmb{\\beta}) < 0$ and consider a  function $K : S\\to \\r$  such that  $K = O(|z - p_i|^{\\ell_i})$ (in the neighborhood of each $p_i$), where $\\ell_i > -2(1+\\alpha_i)$.\n\n\\smallskip \n\n\\emph{(A)}  If $\\beta_i \\leq \\alpha_i$ for all $i$, then a necessary condition for the existence of a conformal metric $g$ on $S$ representing \n$\\pmb{\\beta}$ is\n\\begin{equation}\\label{kaneg}\n  \\int_S KdA_0 < 0,\n\\end{equation}\nwhere $dA_0$ is the area element of $g_0$.\n\n\\smallskip \n\n\\emph{(B)}  There exists $\\varepsilon >0$  (depending on $S$, $\\alpha$ and $K$) such that if $\\max_i |\\beta_i - \\alpha_i| \\leq \\varepsilon$,\nthen \\eqref{kaneg} is a sufficient condition for the existence of a conformal metric $g$ representing $\\beta$.\n\\end{theorem}\n\n\n\\medskip\n\nThe second result is due to Dominique Hulin \\cite{Hulin1993}. It  allows us to solve the problem when $K$ is close enough  to a given non-positive function: \n\\begin{theorem} \nLet $S$ be a compact Riemann surface with a divisor $\\pmb{\\beta}= \\sum_{i=1}^n \\beta_i p_i$  such that $\\chi(S, \\pmb{\\beta}) < 0$.\nLet $K_1, k : S \\to \\mathbb{R}$ be two smooth functions on $S$ such that $K_1 \\leq 0 \\leq k$ everywhere on $S$, and $K_1 \\not\\equiv 0$.\nSuppose that $|z-p_i|^{2\\beta_i}(k(z) - K_1(z)) \\in L^p$ for some $p>1$ in the neighborhood of the points  $p_i$. \n\\  Then there exists a constant $C > 0$ (depending on $S$, $\\pmb{\\beta}$, $K_1$ and $k$) such that if\n$$\nK_1\\leq K \\leq K_1 + Ck\n$$\non $S$, then there exists a conformal metric $g$ of curvature $K$ on $S$ which represents $\\pmb{\\beta}$. \n\\end{theorem}\nThe dependence of the constant $C$ on $S$, $\\pmb{\\beta}$, $k$ and $K_1$ is explicitly  given in  \\cite[Theorem 6.1]{Hulin1993}\n\n\\newpage\n\n\\section{Spherical Polyhedra}\n\nThe following result classifies spherical surfaces with less than three conical points.\n\\begin{theorem}\nLet $g$ be a metric on the sphere $S^2$ representing a divisor $\\pmb{\\beta} = \\beta_1p_1 + \\beta_2p_2$   and whose curvature is constant $K = + 1$. Then $\\beta_1 = \\beta_2$,  moreover:\n\\begin{enumerate}[(a)]\n\\item  If $\\beta_i$ is not an integer, then $p_1$ and $p_2$ are conjugate points (that is $d(p_1,p_2) = \\pi$).\n\\item  If $\\beta = m \\in \\mathbb{N}$, then   $(S, g)$ is isometric to a branched covering of degree $m+1$ of the standard sphere  $\\mathbb{S}^2$, \n(with its canonical metric) branched over two points, with  ramification order equal to $m$.\nMoreover, two such metrics are isometric if and only if their singularities are of the same order and separated by the same distance.\n\\end{enumerate}\n\\end{theorem}\n\nLet us call  \\emph{spherical polyhedra}    \\index{Spherical polyhedra} a Riemannian surface homeomorphic to the sphere, with conical singularities of order \n$\\beta_i  \\in (-1,0)$ and whose  Gauss curvature is constant $K = +1$. \nA fundamental Theorem by A. D.  Aleksandrov  states that a Riemannian surface, homeomorphic to the sphere, with   constant curvature  $K = +1$ and\n conical singularities of order $\\beta_i \\in (-1,0)$  can be realized as the boundary  of a convex polytope \nin the standard three-sphere $\\mathbb{S}^3$ (see  \\cite[Chapter XII, p. 400]{Alexandrov} ).  Note that from the previous Theorem,  a spherical polyhedron cannot have exactly one singularity.\n\n\\medskip \n\nThe last result  classifies the divisors that can be represented by a spherical polyhedron having at least three singularities:\n\\begin{theorem}\nLet $\\pmb{\\beta} = \\sum_{i=1}^n \\beta_i p_i$ be a divisor on $S^2 = \\mathbb{C} \\cup \\infty$ such  $n\\geq 3$ and that $- 1 < \\beta_i < 0$ for all $i$.\nThen there exists a unique conformal metric $g$ with constant curvature $K = +1$ representing $\\pmb{\\beta}$ if and only if\n\\begin{equation}\\label{CondLT}\n  0 < 2 + \\sum_{i=1}^n \\beta_i < 2(1 + \\min_i \\{\\beta_i\\}).\n\\end{equation}\n\\end{theorem}\nExpressed with the cone angles $\\theta_i = 2\\pi(1+\\beta_i)$, the hypothesis can also be written as $\\theta_i < 2\\pi$ and \n\\begin{equation}\\label{CondLT2}\n 0 < 4\\pi +  \\sum_{i=1}^n (\\theta_i - 2\\pi)  < 2  \\min_i \\{\\theta_i\\}.\n\\end{equation}\nThe first inequality in this condition is none other than the Gauss-Bonnet formula. \n\n\nLet us observe that the condition \\eqref{CondLT2} is similar to the condition satisfied by the angles \n$\\varphi_1, \\dots, \\varphi_n$   of a spherical convex polygon:\n$$\n 0 < 2\\pi +  \\sum_{i=1}^n (\\varphi_i - \\pi)  < 2  \\min_i \\{\\varphi_i\\}.\n$$\nThe existence of a spherical metric representing $\\pmb{\\beta}$ follows from Theorem \\ref{prescrK} above (see also \\cite{Troyanov1991}). The necessity of \\eqref{CondLT}, as well as the uniqueness of the metric have been proved by Feng Luo and Gang Tian  in \\cite{LT}. \n\n\\bigskip \n\n\n\\textit{Note added in 2021.} \nThe subject of this survey has grown considerably over the past 30 years. A nice reference covering more recent aspects of the theory is the article \\cite{Lai} by M. Lai\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMagnetometers based on the nitrogen-vacancy (NV) center in diamond~\\cite{doherty_nitrogen-vacancy_2013, schirhagl_nitrogen-vacancy_2014} provide $\\upmu$T\\,- nT sensitivity for single centers at ambient temperature and mm-to-sub-$\\upmu$m length scales, making them attractive resources for studying biomagnetism~\\cite{gille_quantum_2021}, solid state systems~\\cite{tetienne_quantum_2017, casola_probing_2018} and nanoscale NMR~\\cite{mamin_nanoscale_2013} in challenging real-world sensing environments~\\cite{fu_sensitive_2020}. Since many magnetic phenomena of importance in navigation and biomagnetism manifest as slowly varying or static magnetic fields, intense effort has been devoted in particular to improving dc sensitivity~\\cite{barry_sensitivity_2020}, focusing on the diamond material~\\cite{balasubramanian_ultralong_2009, herbschleb_ultra-long_2019}, photon collection efficiency~\\cite{clevenson_broadband_2015}, quantum control sequences to eliminate decoherence~\\cite{lange_controlling_2012, mamin_multipulse_2014, bauch_ultralong_2018} and more recently, the addition of ferrite flux-concentrators~\\cite{fescenko_diamond_2020, zhang_diamond_2021}. Many approaches to improving the measurement signal are frustrated by a commensurate increase in noise, limiting the attainable sensitivity.   \n\nTo date, Ramsey-type interferometry~\\cite{ramsey_molecular_1950} is the optimum dc measurement sequence~\\cite{rondin_magnetometry_2014, barry_sensitivity_2020}, and the sensitivity of Ramsey magnetometry is limited by the ensemble dephasing time $T_2^\\ast$, which reflects the magnitude of low frequency noise in the system. In diamond, impurities such as $^{13}$C~\\cite{childress_coherent_2006} or paramagnetic nitrogen~\\cite{bauch_decoherence_2020} are the dominant contributions to $T_2^\\ast$, resulting in $T_2^\\ast<1\\,\\upmu$s for readily-available CVD diamond samples. Additionally, $T_2^\\ast$ varies considerably between diamond samples, and often significantly \\emph{within} a single sample due to gradients or spatial variations of crystal strain or varied dopant density. This decoherence can be largely eliminated by employing time-reversal dynamical decoupling measurement schemes, such as Hahn spin-echo~\\cite{hahn_spin_1950}, but at the cost of insensitivity to slowly varying or static dc fields~\\cite{taylor_high-sensitivity_2008}. Alternative NV-magnetometry techniques~\\cite{acosta_broadband_2010, jeske_laser_2016, wickenbrock_microwave-free_2016} that eschew conventional quantum sensing protocols have also been proposed and demonstrated, though to date yielded comparable sensitivities to standard methods.  \n\nIn this work, we use physical rotation to realize a significant improvement in the sensitivity of a dc quantum magnetometer. Mechanically rotating a diamond at rates comparable to the spin coherence time $T_2\\sim 0.1-1$\\,ms up-converts an external dc magnetic field to the rotation frequency, which is then detected using NV spin-echo magnetometry~\\cite{wood_t_2-limited_2018}. The quantum sensing time is increased from $T_2^\\ast$ up to $T_2$, with a potential $\\sqrt{T_2\/T_2^\\ast}$ sensitivity gain, typically an order of magnitude~\\cite{taylor_high-sensitivity_2008}. We demonstrate a 30-fold improvement in sensitivity over comparable Ramsey magnetometry. Furthermore, our demonstrated sensitivity exceeds, by a factor of 4.5, the theoretical shot-noise-limited, unity duty cycle sensitivity of $T_2^\\ast$-limited Ramsey magnetometry with this diamond. \n\n\\begin{figure*}\n\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{fig1.pdf}\n\t\\caption{Diamond rotation up-conversion magnetometry (DRUM) schematic and key results. a) Schematic model of NV center, axis tilted from $z$ by $\\theta_\\text{NV} = 30.2^\\circ$, rotating at $\\omega_\\text{rot}$ with external magnetic field components $B_z$ and $B_x$. b) Rotation of the diamond modulates the Zeeman shift in proportion to $B_x$, at a rate set by the rotation frequency. c) Typical stationary spin-echo (blue circles) and Ramsey (orange squares, detail inset) signals at $B_z = 2.3\\,$mT for the diamond sample used in this work. d) Comparing magnetometry signals: varying an applied dc $B_x$ field (DRUM) or effective $z$-field (Ramsey) at the optimum sensing times traces out fringes. We used 10\\,s and 30\\,s measurement times per point, for Ramsey and DRUM, respectively. (e, zoom-in of d) The DRUM fringes are far faster, as considerably more phase accumulates within the $500\\times$ longer sensing time. f) Allan deviation of DRUM and Ramsey when sensitive (blue) and insensitive (orange, microwaves detuned) to magnetic fields. Both measurements follow $T^{-1\/2}$ scaling for a few hundred seconds, before drifts become dominant.}\n\t\\label{fig:fig1}\n\\end{figure*}\n\nOur method, depicted in Fig. \\ref{fig:fig1}(a,b), is called diamond rotation up-conversion magnetometry (``DRUM\") and was introduced in Ref.~\\cite{wood_t_2-limited_2018}. The NV Hamiltonian in the presence of a magnetic field $\\boldsymbol{B}$ is $H = D_\\text{zfs} S_{z'}^2 + \\gamma_e \\boldsymbol{B}\\cdot\\boldsymbol{S}$, with $D_\\text{zfs}\/2\\pi = 2870\\,$MHz at room temperature, $\\gamma_e\/2\\pi = 2.8\\,$kHz\\,$\\upmu$T$^{-1}$ and $\\boldsymbol{S} = (S_{x'}, S_{y'}, S_{z'})$ the vector of spin-1 Pauli matrices. We have ignored strain, electric fields and coupling to temperature. The quantization axis $z'$ is taken as the nitrogen-vacancy axis, lying along one of the four $[111]$ crystallographic axes of the crystal and therefore at an angle $\\theta_\\text{NV}$ to $z$. We use a single-crystal diamond containing an ensemble of NV centers, with a surface normal parallel to the rotation axis. A bias magnetic field is applied along the rotation axis $z$ to spectrally select one NV class and isolate $m_S$ transitions. The energies of the NV $m_S = 0, \\pm1$ spin states for weak applied fields ($\\hbar = 1$) are $\\omega(m_S) \\approx D_\\text{zfs}+m_S \\gamma_e B\\cos(\\theta_\\text{NV})$, with $B = |\\boldsymbol{B}|$. A weak magnetic field $B_x$ is applied along the lab-frame $x$-axis, making the Zeeman shift time-dependent during rotation at an angular frequency $\\omega_\\text{rot}$. Considering the $|m_S = 0\\rangle-\\leftrightarrow|m_S = -1\\rangle$ transition, the time dependent component is \n\\begin{equation}\n\\omega_{-1,0}(t) \\approx \\gamma_e B_x\\sin\\theta_\\text{NV}\\cos(\\omega_\\text{rot}t - \\phi_0),\n\\label{eq:zeemant}\n\\end{equation}\nwith $\\phi_0$ set by the initial orientation of the diamond, and adjusted to maximize sensitivity to either $x$ or $y$-oriented fields. The dc field in the lab frame is now effectively an ac field in the  NV frame, with amplitude $\\delta B_x\\sin\\theta_\\text{NV}$ and frequency $\\omega_\\text{rot}$. \n\nOur previous realization of DRUM used an NV tilt angle $\\theta_\\text{NV} =4^\\circ$, resulting in up-conversion of only a small fraction of the dc field. In this work, we use a $\\langle 110\\rangle$-cut CVD-grown type IIa diamond with a natural abundance of $^{13}$C and approximately [N] = 1\\,ppm, [NV] = 0.01\\,ppm, mounted on an electric motor that can spin at up to 5.83\\,kHz. We choose an NV orientation class making an angle of $30.2^\\circ$ to the $z$-axis for our measurements, yielding $\\sin\\theta_\\text{NV} = 0.5$, which is easily resolved and still sufficient to attain sensitivities exceeding that of optimized Ramsey sensing. Typical stationary spin-echo and Ramsey signals are shown in Fig. \\ref{fig:fig1}(c) for $B_z$ = 2.3\\,mT. The $^{13}$C spin bath is the dominant source of decoherence~\\cite{barry_sensitivity_2020}, limiting $T_2^\\ast$ to less than half a microsecond, but the sample exhibits a much longer $T_2 = 250(9)\\,\\upmu$s, with interferometric visibility restricted to revivals spaced at twice the $^{13}$C Larmor period~\\cite{childress_coherent_2006}. Such a wide gulf separating $T_2$ and $T_2^\\ast$ is not unusual, reflecting the much greater sensitivity of $T_2^\\ast$ to the presence of magnetic impurities and sample-specific imperfections~\\cite{bauch_ultralong_2018, bauch_decoherence_2020}.   \n\nThe remaining experimental details are similar to that described in Refs.~\\cite{wood_quantum_2018, wood_t_2-limited_2018, wood_anisotropic_2021}, and further details are provided in the Supplementary Material. Briefly, a scanning confocal microscope optically polarizes (1\\,mW 532\\,nm) and reads (600-800\\,nm, 9$\\times10^6$ cts\/s) the NV fluorescence. Microwave fields are applied along the $z$-axis with a coil antenna to ensure rotational symmetry. Further coils supply the $z$-oriented bias field and create the transverse dc test fields. A fast pulse generator controls the timing of laser and microwave pulses, and is triggered synchronously with the rotation. The laser is pulsed for $3\\,\\upmu$s to optically pump NVs to the $m_S = 0$ state, and the subsequent microwave spin-echo sequence is timed so that the $\\pi$-pulse is applied at the zero-crossing of the up-converted field (Fig. \\ref{fig:fig1}(b)), conferring maximum sensitivity. After the microwave pulses, optical readout is performed after a shuttling time so that the whole sequence takes one rotation period. A second sequence is applied back-to-back, with the final $\\pi\/2$-pulse phase shifted by $180^\\circ$. The detected photoluminescence from each trace is then normalized and the difference computed to extract the contrast $\\mathcal{S}$, which constitutes the DRUM signal.\n\nThe concept of rotational dc up-conversion should be applicable to a wide range of competing magnetometry architectures, including laser and absorption based readout schemes. The point of this paper is to demonstrate the sensitivity advantage of the DRUM technique over standard $T_2^\\ast$-limited Ramsey magnetometry. Our priorities are therefore not to improve the ultimate sensitivity of the measurement, but rather ensure that the two techniques are compared on an equal basis. Using isotopically-purified diamonds with very high NV densities, wide-area collection optics and powerful excitation beams has been shown to be the most effective means to achieve high sensitivities~\\cite{wolf_subpicotesla_2015}, and these techniques should be compatible with diamond rotation, with sufficient engineering application. The shorter coherence times intrinsic to these NV-dense configurations place additional requirements on rotation speed, and these will be discussed later in this work.\n\nDue to strain and impurity inhomogeneity, $T_2^\\ast$ exhibits considerable spatial variation\\footnote{See Supplementary Material}. To assess the peak Ramsey sensitivity, we locate a region with a comparatively high $T_2^\\ast = 360\\,$ns and determine the sensitivity, $\\delta B = \\left(\\frac{d\\mathcal{S}}{dB}\\right)^{-1} \\sigma(\\mathcal{S}) \\sqrt{T}$, with $d\\mathcal{S}\/dB$ the mid-fringe signal slope, $T$ the total integration time, and $\\sigma(\\mathcal{S})$ the standard deviation of the Ramsey signal. For Ramsey, changing the microwave frequency is equivalent to varying a magnetic field exactly parallel to the NV axis. Ramsey fringes as a function of effective magnetic field are shown in Fig. \\ref{fig:fig1}(d). The sensitivity of Ramsey magnetometry was found by detuning to the mid-fringe point and repeating the same 10\\,s averaging interval 10 times. The best we achieved was a standard deviation of this data yielding $\\delta B =0.86\\upmu$T\\,$\\text{Hz}^{-1\/2}$.\n\nNext, we rotated the diamond at 3.75\\,kHz and performed DRUM with a spin-echo time of $\\tau = 180\\,\\upmu$s ($B_z = 0.7\\,$mT), yielding fringes as an applied $x$-field is varied as shown in Fig.\\ref{fig:fig1}(d,e), with each point averaged for 30\\,s. We calculate the operational sensitivity to be $28\\,\\text{nT}\\,\\text{Hz}^{-1\/2}$, about 30 times better than that of Ramsey in the same diamond sample. DRUM exhibits similar long-time averaging behavior to Ramsey, but towards a much lower minimum detectable field. Figure \\ref{fig:fig1}(f) shows the Allan deviation~\\footnote{A comprehensive description of Allan deviation as it applies to our work is provided in the Supplementary Material.} of DRUM and Ramsey as a function of averaging time. To assess the relative magnitudes of magnetic drifts and intrinsic noise in each technique, we measure while sensitive to magnetic fields, and then again with the microwaves detuned by 15\\,MHz, so that only intrinsic measurement noise is present. Ramsey, sensitive to drifts in temperature, local strain as well as magnetic drifts (common also to DRUM) exhibits a significantly higher Allan deviation compared to DRUM. No amount of averaging time with Ramsey can exceed the performance of DRUM. This demonstration constitutes a significant achievement, showing that $T_2^\\ast$ does not intrinsically limit dc magnetic sensitivity for a spin-based quantum magnetometer. In what follows, we describe how the optimum parameters were deduced, and how the ultimate sensitivity depends on parameters we can control.\n\nThe shot-noise limited dc sensitivity of the DRUM measurement is given by~\\footnote{See Supplementary Material}\n\\begin{equation}\n\\delta B = \\frac{\\pi e^{\\left(\\frac{\\tau}{T_2}\\right)^n}}{4 C \\gamma_e \\sin\\theta_\\text{NV}\\sin\\left(\\frac{\\pi\\tau}{2t_\\text{rot}}\\right)}\\frac{1}{\\sqrt{t_\\text{rot}}},\n\\label{eq:sensitivity}\n\\end{equation}\nwith $n\\approx 3$ reflecting sample-specific decoherence processes, $C$ the readout efficiency~\\cite{degen_quantum_2017} and $t_\\text{rot}$ the rotation period. Extracting the optimum performance for DRUM amounts to increasing the slope $d\\mathcal{S}\/dB$, which depends on the rotation speed and spin-echo measurement time, and increasing the measurement signal-to-noise, largely via well-understood means~\\cite{barry_sensitivity_2020} that benefit both Ramsey and DRUM.\n\nThe rotation speed can be chosen to optimize sensitivity. The intrinsic amplitude of the up-converted field is set by the lab-frame dc field, but the phase accumulated by the NV centers depends on the rotation speed. Faster rotation speeds enable the same integration time to sample a greater fraction of the modulated field, and more measurements are possible within a given averaging time, increasing the number of photons collected. However, without recourse to higher-order dynamical decoupling sequences, which supplant spin-echo as the optimum measurement only when $T_2\\gg t_\\text{rot}$, increasing the rotation speed eventually reduces the accumulated phase due to the smaller integrated area under $\\omega_{-1,0}(t)$. \n\nMaximizing the slope also requires minimizing decoherence sources. For a natural abundance diamond, the NV-$^{13}$C bath interaction depends on the strength~\\cite{zhao_decoherence_2012, hall_analytic_2014} and direction~\\cite{stanwix_coherence_2010, wood_anisotropic_2021} of the magnetic field. Consequently, the $z$-bias field tunes the spin-echo time $\\tau$, and must be set so that $\\tau = n_c \/ (B\\times 10.71\\,\\text{kHz\\,mT}^{-1}\\pm\\omega_\\text{rot}\/2\\pi)$, $n_c\\in\\{1, 2, ...\\}$. Stronger $z$-fields result in faster decoherence due to the anisotropic NV-$^\\text{13}$C hyperfine coupling~\\cite{stanwix_coherence_2010,wood_anisotropic_2021}. We therefore minimize the requisite $z$-field by ensuring just the \\emph{first} $^{13}$C revival coincides with the desired measurement time, and choose the rotation direction so that the induced magnetic pseudo-field~\\cite{wood_magnetic_2017} adds to the $z$-bias field, \\emph{i.e.} $\\omega_\\text{rot} >0$.\n\nThe slope $d\\mathcal{S}\/dB$ is given by \n\\begin{equation}\n\\frac{d\\mathcal{S}}{dB} = \\frac{4A}{\\omega_\\text{rot}}e^{-\\left(\\frac{\\tau}{T_2}\\right)^n}\\gamma_e \\sin\\theta_\\text{NV}\\sin\\left(\\frac{\\omega_\\text{rot}\\tau}{4}\\right).\n\\label{eq:slope}\n\\end{equation}\nFor comparison with data, we leave $A$ and $T_2$ as free parameters, possibly dependent on rotation speed, and fix $n = 3$, which best describes our observed relaxation and is consistent with other experiments~\\cite{stanwix_coherence_2010, hall_analytic_2014}. We measured the mid-fringe slope of DRUM fringes as a function of $\\tau$ and $\\omega_\\text{rot}$. Fig.~\\ref{fig:fig2}(a, inset) shows DRUM fringes as the spin-echo time is varied for $\\omega_\\text{rot}\/2\\pi = 3.75\\,$kHz, and the slope extracted from fits of Eq. \\ref{eq:slope} to such data is plotted versus $\\tau$ for rotation speeds from $1-6\\,$kHz (Fig.~\\ref{fig:fig2}(a)). We find that as the rotation speed is increased, an optimal sensing time appears around 3 to 4\\,kHz. \n\n\\begin{figure}[t!]\n\t\\centering\n\t\t\\includegraphics[width = \\columnwidth]{fig2.pdf}\n\t\\caption{Optimization of DRUM. (a) Peak slope $d\\mathcal{S}\/dB$ as a function of r$\\omega_\\text{rot}$, deduced from fits of Eq. \\ref{eq:slope} to slope-vs-$\\tau$ data gathered for DRUM fringes at different rotation speeds (inset, 3.75\\,kHz) (b) Measured $T_2$ (gray dashed line is linear fit) and spin-echo time $\\tau_\\text{opt}$ where the peak slope occurs: decoherence results in $\\tau_\\text{opt} < T_2$. Dashed lines in (a) and (b) correspond to theoretical predictions from measured $T_2$ data. (c) Increasing photon number per measurement bin (blue circles), and reducing measurement noise (orange squares). (d) Sensitivity $\\delta B$ as a function of rotation speed. Shaded regions denote maximum and minimum ranges from 5 repetitions.}\n\t\\label{fig:fig2}\n\\end{figure}\n\nWe also monitored the coherence time $T_2$ and time at which the peak slope occurs $\\tau_\\text{opt}$ as a function of rotation speed (Fig. \\ref{fig:fig2}(b)). The measured $T_2$ is higher than that in Fig. \\ref{fig:fig1}(c) due to the overall lower magnetic field strengths, and drops slightly as a function of rotation speed, which we believe is due to imperfectly canceled $y$-fields or the test field itself inducing weak anisotropy in the NV-$^{13}$C interaction~\\footnote{See Supplementary Material}. Numerically maximizing Eq. \\ref{eq:slope} for the measured $T_2$ yields the dashed theoretical predictions in Fig.~\\ref{fig:fig2}(a, b), confirming the key role $T_2$ plays. While the theory accurately reproduces the peak slope, the optimum time $\\tau_\\text{opt}$ where the slope is maximized is lower than predicted; we attribute this to the particular choice of $n = 3$ in Eq. \\ref{eq:slope}.\n\nThe variation in slope is tempered by increasing photon collection rates (and hence reduction in shot noise) due to increased duty cycle at higher rotation speeds, as shown in Fig. \\ref{fig:fig2}(c). This leads to an almost flat dependence on sensitivity with rotation speed, as shown in Fig. \\ref{fig:fig2}(d). DRUM operates close to the shot-noise predicted sensitivity limit, 30 times better than the Ramsey sensitivity. This is actually greater than $\\sqrt{T_2\/T_2^\\ast} = 26$ due to the $10\\%$ duty cycle of Ramsey measurements (including laser preparation and readout), and the trigonometric factor and integration for less than a full period in DRUM. However, computing the idealized Ramsey sensitivity with unity duty cycle yields $122$\\,nT\\,Hz$^{-1\/2}$, still a factor of 4.5 worse than experimentally demonstrated DRUM with $\\tau < t_\\text{rot}$ and $\\theta_\\text{NV} = 30^\\circ$.\n\n\\emph{Discussion.} We have shown in this work that dc up-conversion magnetometry with a rotating diamond can significantly exceed the sensitivity of conventional Ramsey magnetometry. Unlike previous proposals~\\cite{ajoy_DC_2016}, our scheme up-converts just the magnetic fields of interest and not the deleterious noise that limits quantum coherence, thus definitively improving sensitivity to dc fields. We also retain the vector sensitivity of the NV center, with a timing adjustment to the synchronization allowing for exclusive $y$-field detection. In principle, our scheme is equally applicable to any quantum system where the coupling between qubit and parameter of interest can be modulated in time, and this paper shows that sensitivity exceeding the $T_2^\\ast$ limit is thus possible. \n\nFor up-conversion to be worthwhile, $T_2\\gg T_2^\\ast$ and $t_\\text{rot} \\sim T_2$. More practically, however, the $\\sqrt{\\omega_\\text{rot}}$ scaling of sensitivity in Eq. \\ref{eq:sensitivity} requires a commensurate increase in $C$, \\emph{i.e.} by increasing NV density, which in turn reduces $T_2$~\\cite{bauch_decoherence_2020}. However, with the simplified assumption that $\\tau = t_\\text{rot} = T_2$ and $\\theta_\\text{NV} = 90^\\circ$, DRUM can confer a sensitivity gain over Ramsey in any diamond sample of $\\delta B_\\text{DRUM}\/\\delta B_R  = \\frac{\\pi}{4}\\sqrt{T_2^\\ast\/T_2}$. For DRUM to have a factor of 10 better sensitivity than unity-duty-cycle Ramsey, $T_2 =62\\,T_2^\\ast$.\n\nMechanical rotation can be challenging, though not impossible, to achieve for the short $T_2\\sim 5-10\\,\\upmu$s exhibited by NV-dense diamond samples. Commercial NMR magic-angle spinning devices can achieve rotation rates of mm$^3$-scale samples now up to 150\\,kHz~\\cite{schledorn_protein_2020}, and demonstrations of ultrafast rotation of optically-trapped microscale structures~\\cite{reimann_GHz_2018, ahn_ultrasensitive_2020} yield GHz rotation frequencies. Since rapid libration can be substituted for rotation, alternative approaches could potentially leverage fast piezoelectric tip-tilt transducers~\\cite{csencsics_fast_2019} for larger samples or micro-to-nano structures for smaller length scales, for instance in optically or electrically-trapped~\\cite{perdriat_spin-mechanics_2021} micro-to-nano diamonds. Modulation can also be achieved by position displacement in a spatially-varying field, and recent work has demonstrated up-conversion of dc fields to ac using scanning single NV magnetometry by rapidly modulating the distance between tip and sample~\\cite{huxter_scanning_2022}. Another option could be modulation, \\emph{i.e.} via position displacement, of the sensitivity enhancement conferred by ferrite flux concentrators~\\cite{fescenko_diamond_2020}.   \n\nIn conclusion, we have demonstrated that $T_2$-limited dc magnetometry can exceed the sensitivity of $T_2^\\ast$-limited Ramsey magnetometry for diamond-based quantum magnetometers. We anticipate our work will stimulate other approaches, not necessarily based on sample rotation, to combine up-conversion with existing schemes to improve magnetic sensitivity. Augmented with fast rotation, improvements such as engineered diamonds with higher NV densities and larger optical excitation and collection areas may be sufficient to achieve the much sought-after fT\\,Hz$^{-1\/2}$ regime of dc magnetic sensitivity with room-temperature, microscale diamond sensors, where applications such as magnetoencephalography in unshielded environments become possible~\\cite{boto_moving_2018}.\n\nThis work was supported by the Australian Research Council (DE210101093, DE190100336). We thank R. E Scholten for insightful discussions and a careful review of the manuscript. The authors (AAW, AS, AMM) are inventors on a United States Patent, App. 16\/533,167 which is based on this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEvolutionary game theory is used on different levels of biological systems, ranging from the genetic level to ecological systems. The language of game theory allows to address basic questions of ecology, related to the emergence of cooperation and biodiversity, as well as the formation of coalitions with applications to social systems. Darwinian dynamics, in particular frequency-dependent selection, can be formulated in terms of game-theoretic arguments \\cite{nowak}. The formation of dynamical patterns is considered as one of the most important promoters of biodiversity \\cite{may,levin,durret,hassel}. Here we consider games of competition, where the competition is realized as predation among $N$ species, where each species preys on $r$ others in a cyclic way. A subclass of these $(N,r)$-games are cyclic games, that is $(N,1)$ with $(3,1)$ being the famous rock-paper-scissors game. An extensive overview on cyclic games is given in \\cite{szabo,perc}. The $(3,1)$-game has been studied in various extensions (spatial, reproduction and deletion, swapping or diffusion, mutation). One of the first studies of a $(3,1)$-game without spatial assignment, but in a deterministic and stochastic realization revealed that fluctuations due to a finite number of agents  can drastically alter the mean-field predictions, including an estimate of the extinction probabilities at a given time \\cite{reichen1}. This model was extended to include a spatial grid in \\cite{reichen2}, where the role of stochastic fluctuations and spatial diffusion was analyzed both numerically and analytically. The influence of species mobility on species diversity was studied in \\cite{reichen3}, pattern formation close to a bifurcation point was the topic of \\cite{reichen4}, see also \\cite{reichen5} and the impact of asymmetric interactions was considered in \\cite{reichen6}.\n\nAn extension to four species, first without spatial assignment, shows interesting new features as compared to $(3,1)$: Already in the deterministic limit the trajectories show a variety of possible orbits, and from a certain conserved quantity the late-time behavior can be extrapolated \\cite{durney}. The four species can form alliance pairs similarly to the Game of Bridge \\cite{case}. Under stochastic evolution various extinction scenarios and the competition of the surviving set of species can be analyzed \\cite{28}. Domains and their separating interfaces were studied in \\cite{arXiv:1205.4914}. $(4,1)$ cyclic games on a spatial grid were the topic in \\cite{szabosznaider,luetz}. A phase transition as a function of the concentration of vacant sites is identified between a phase of four coexisting species and a phase with two neutral species that protect each other and extend their domain over the grid. For an extension of this model to long-range selection see \\cite{hua}.\n\nIn this paper we focus on  the $(6,3)$-game, including both spiral formation inside domains and domain formation. It is a special case of $(N,r)$-games, which were considered for $N\\ge3$ and $r\\ge1$ by \\cite{m1,m2} and more recently by \\cite{m3}. The authors of \\cite{m1,m2} were the first to notice that for certain combinations of N and r one observes the coexistence of both spiral formation and domain formation. However, it should be noticed that our set of reactions, even if we specialize $(N,r)$ to the $(3,1)$-game, is similar, but not identical with the versions, considered in \\cite{m1,m2,m3}  or in \\cite{reichen1}-\\cite{reichen6}. The seemingly minor difference refers to the implementation of an upper threshold to the occupation number of single sites (set to 1 or a finite fixed number), while we use a ``bosonic\" version. We introduce a dynamical threshold, realized via deletion reactions, so that we need not explicitly restrict the occupation number per site. Due to this difference, the bifurcation structure of the mean-field equations is changed.\n\nThe reason why we are interested in the particular combination of $N=6$ and $r=3$ is primarily motivated by two theoretical aspects rather than by concrete applications. As to the first aspect, this game is one of the simplest examples of ``games within games\" in the sense that the domains effectively play a $(2,1)$-game as transient dynamics on a coarse scale (the scale of the domain diameter), while the actors inside the domains  play a $(3,1)$-game on the grid scale. Finally, one of the domains gets extinct along with all its actors. As such, this game provides a simple, yet non-trivial example for a mechanism that may be relevant for evolution: In our case, due to the spatial segregation of species, the structural complexity of the system increases in the form of patterns of who is chasing whom, appearing as long-living transients, along with a seemingly change of the rules of the game that is played between the competing domains on the coarse scale, while the rules, which individuals use on the elementary grid sites, are not changed at all. As outlined by Goldenfeld and Woese \\cite{goldenfeldwoese}, it is typical for processes in evolution, in particular  in ecology, that ``the governing rules are themselves changed\", as the system evolves in time and the rules depend on the state. In our example it is spatial segregation, which allows for a change of rules from a coarse perspective, as we shall see.\n\nAs to the second aspect, an interesting feature of such an arrangement is the multitude of time and spatial scales that are dynamically generated. Concretely in the $(6,3)$-game the largest of the reaction\/diffusion rates sets the basic time unit. When the species segregate and form domains, the next scale is generated: it is the time it takes the two domains to form until both cover the two-dimensional grid or the one-dimensional chain. The domains are not static, but play the $(2,1)$-game that has a winner in the end. So the extinction time of one of the domains sets the third scale. A single domain then survives, including the moving spirals from the remaining $(3,1)$-game inside the domain. The transients can last very long, depending on the interaction rates and the system size. In the very end, however, in a stochastic realization as well as due to the finite accuracy in the numerical solutions even in the mean-field description, only one out of the three species will survive, and the extinction of the other two species sets the fourth scale. Along with these events, spatial scales emerge, ranging from the basic lattice constant to the radii of spirals and the extension of the domains.\n\nOne of the challenges is to explore which of the observed features in the Gillespie simulations can be predicted analytically. We shall study the predictions on the mean-field level, which is rather conclusive in our ultralocal implementation of reactions and reproduces the results of the Gillespie simulations quite well, since fluctuations turn out to play a minor role for pattern formation. The deterministic equations are derived as the lowest order of a van Kampen expansion. The eigenvalues of the Jacobian are conclusive for the number of surviving species in a stable state, the composition of the domains, and transient behavior, which is observed in the Gillespie simulations. The mean-field equations, including the diffusion term, will be integrated numerically and compared to the results of the Gillespie simulations.\n\nThe paper is organized as follows. In section~\\ref{sec_reactions} we present the model in terms of basic reactions and the corresponding master equation. For generic $(N,r)$ games we summarize in section~\\ref{sec_vankampen} the derivation of the mean-field equations from a van Kampen expansion, followed by a stability analysis via the Jacobian  with and without spatial dependence for the specific $(6,3)$ game, and a derivation of the numerical solutions of the mean-field equations in section~\\ref{sec_Jacobian}. In section~\\ref{sec_numerical} we present our results from the Gillespie simulations in comparison to the mean-field results. Section~\\ref{sec_conclusions} summarizes our conclusions and gives an outlook to further challenges related to this class of games. For comparison, the supplementary material contains a detailed stability analysis for the $(3,1)$-game with spiral formation and the $(3,2)$-game with domain formation, as well as the numerical solutions of the mean-field equations and the corresponding Gillespie simulations.\n\n\n\\section{Reactions and Master Equation}\\label{sec_reactions}\nWe start with the simplest set of reactions that represent predation between individuals of different species, followed by reproduction, deletion\n\\begin{eqnarray}\\label{eq:rec_sys}\n\tX_{\\alpha,i}\\, +\\, X_{\\beta,i} & \\overset{k_{\\alpha\\beta}\/V}{\\longrightarrow} & X_{\\alpha,i}  \\label{pred}\\\\\n\tX_{\\alpha,i}\\,  & \\overset{r_{\n\\alpha,i}}{\\longrightarrow} & 2 X_{\\alpha,i} \\label{repr} \\\\\n\t2X_{\\alpha,i}\\,& \\overset{p_{\\alpha}\/V}{\\longrightarrow} & X_{\\alpha,i}  \\label{anih}\n\\end{eqnarray}\t\nand finally diffusion\n\\begin{eqnarray}\\label{diff}\n\tX_{\\alpha,i}\\, & \\overset{D_{\\alpha}\/h^2}{\\longrightarrow} & X_{\\alpha,j}.\n\\end{eqnarray}\n$X_{\\alpha,i}$ represents an individual of species $\\alpha$ at lattice site $i$, while the total number of individuals of species $\\alpha$ at site $i$ will be denoted with $n_{\\alpha,i}$. (We use small characters $n$ for convenience, although the meaning of $n$ is not a density, but the actual occupation number of a certain species at a certain site.) In view of applications to ecological systems, each lattice site stands  for a patch housing a subpopulation of a metapopulation, where the patch is not further spatially resolved.  Eq.~(\\ref{pred}) represents the predation of species $\\alpha$ on species $\\beta$ with rate $k_{\\alpha\\beta}\/V$, where the parameter $V$ does not stand for the physical volume, but parameterizes the distance from the deterministic limit in the following way: According to our set of reactions, larger values of $V$ lead to higher occupation numbers $n_{\\alpha,i}$ of species $\\alpha$ at sites $i$, since predation and deletion events are rescaled with a factor $1\/V$, and therefore to a larger total rate. The fluctuations in occupation numbers, realized via the Gillespie algorithm, are independent of $V$ or the occupation numbers of sites, since only relative rates enter the probabilities for a certain reaction to happen. Therefore the size of the fluctuations relative to the absolute occupation numbers or to the overall $V$ gets reduced for large V, that is, in the deterministic limit. Predation is schematically described in figure~\\ref{(6,3)}.\n\nEq.~(\\ref{repr}) represents reproduction events with rate $r_{\\alpha}$, and Eq.~(\\ref{anih}) stands for death processes of species $\\alpha$ with rate $p_{\\alpha}\/V$. Death processes are needed to compensate for the reproduction events, since we do not impose any restriction on the number of individuals that can occupy lattice sites. Here we should remark why we implement death processes in the form of Eq.~\\ref{anih} rather than simpler as $X_{\\alpha,i} \\overset{p_{\\alpha}}{\\longrightarrow} \\oslash$. The latter choice could be absorbed in a term $(\\rho-\\gamma)\\phi_i\\equiv \\tilde{\\rho}\\phi_i$ in the mean-field equation (\\ref{eq:pde}) below with uniform couplings $\\rho$ and $\\gamma$. This choice would not lead to a stable coexistence-fixed point \\cite{josef} and therefore not to the desired feature of games within games\\footnote{For the $(6,3)$-game we would have 40 fixed points, the sign of the eigenvalues would then only depend on the sign of the parameter $\\tilde{\\rho}$. At $\\tilde{\\rho}=0$ all fixed points collide and exchange stability through a multiple transcritical bifurcation. For $\\tilde{\\rho}>0$ (the only case of interest), the system has no stable fixed points, and the numerical integration of the differential equations diverges. (Similarly for the (3,1)-game, for $\\tilde{\\rho}>0$, the trivial fixed point with zero species is always an unstable node, while the coexistence fixed point is always a saddle.)}.\n\nThe species diffuse within  a two-dimensional lattice, which we reduce to one dimension for simplicity if we analyze the behavior in more detail. We assume that there can be more than one individual of one or more species at each lattice site. Individuals perform a random walk on the lattice with rate $D_{\\alpha}\/h^d$, where $D_{\\alpha}$ is the diffusion constant, $h$ the lattice constant and $d$ the dimension of the grid. Diffusion is described by Eq.~(\\ref{diff}), where $i$ represents the site from which an individual hops, and $j$ is one of the neighboring sites to which it hops. It should be noticed that diffusion is the only place, which leads to a spatial dependence of the results, since apart from diffusion, species interact on-site, that is, within their patch.\n\nIn summary, the main differences to other related work such as references \\cite{reichen1,reichen2,reichen3,reichen4,reichen5,reichen6,durney,case, 28,m1,m2,m3} are the ultralocal implementation of prey and predation, no swapping, no mutations as considered in \\cite{mobilia1,mobilia2}, and a bosonic version with a dynamically ensured finite occupation number of sites. Even if qualitatively similar patterns like spirals or domains are generated in all these versions, the bifurcation diagram, that is, the stability properties and the mode of transition from one to another regime depend on the specific implementation.\n\nWe can now write a master equation for the probability of finding $\\{n\\}$ particles at time $t$ in the system for reaction and diffusion processes, where $\\{n\\}$ stands for $(n_{1,1},...,n_{N,L^d})$  and $N$ is the number of species, $L^d$ the number of sites.\n\\begin{eqnarray}\\label{eq:me_reac}\n\n\\frac{\\partial P^{reac} \\left( \\left\\{ n \\right\\};t \\right)}{\\partial t} &=&\n\t \\underset{i}{\\sum} \\left\\{\n\t %\n\t \\underset{\\alpha,\\beta}{\\sum} \\frac{k_{\\alpha\\beta}}{V} \\left[\n\t  n_{\\alpha,i}\\left(n_{\\beta,i}+1\\right)P\\left(n_{\\alpha,i},n_{\\beta,i}+1,...;t \\right)\n\t   - n_{\\alpha,i}n_{\\beta,i}P\\left(\\{n\\};t \\right) \\right] \\right.\\nonumber \\\\\n\t  \n\t  &+&\\left. \\underset{\\alpha}{\\sum}  \\frac{p_{\\alpha}}{V} \\left[\n\t  \\left( n_{\\alpha,i}+1 \\right) n_{\\alpha,i} P\\left( ...,n_{\\alpha,i}+1,...;t \\right)\n\t  - n_{\\alpha,i}\\left( n_{\\alpha,i}-1 \\right) P(\\{n\\};t) \\right] \\right. \\nonumber \\\\\n\t  %\n       &+& \\left. \\underset{\\alpha}{\\sum}  r_{\\alpha} \\left[\n       (n_{\\alpha,i}-1)P(n_{\\alpha,i}-1,...;t) - n_{\\alpha,i}P(\\{n\\};t) \\right]\n\t  \\right\\}\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:me_diff}\n\t\\frac{\\partial P^{diff} \\left( \\left\\{ n \\right\\};t \\right)}{\\partial t} &=& \\underset{\\alpha}{\\sum} \\frac{D_\\alpha}{h^2} \\underset{\\left\\langle i,j \\right\\rangle}{\\sum} \\left[ (n_{\\alpha,i}+1)P(...,n_{\\alpha,i}+1,n_{\\alpha,j}-1,...;t)-n_{\\alpha,i}P(\\{n\\};t)\\right. \\nonumber\\\\\n\t&+&\\left. (n_{\\alpha,j}+1)P(...,n_{\\alpha,i}-1,n_{\\alpha,j}+1,...;t)-n_{\\alpha,j}P(\\{n\\};t)\\right]\n\\end{eqnarray}\nwith $n_{\\alpha,i}\\ge 1$ for all $\\alpha,i$, and\n\\begin{equation}\\label{eq7}\n\t\\partial_tP = \\partial_tP^{reac}+\\partial_tP^{diff}.\n\\end{equation}\n\nAs uniform (with respect to the grid) random initial conditions we assume a Poissonian distribution on each site $i$\n\\begin{equation}\n\tP \\left( \\{n\\} ;0 \\right)=\\underset{\\alpha,i}{\\prod}\\left( \\frac{\\overline{n}^{n_{\\alpha,i}}_{\\alpha,0}}{n_{\\alpha,i}!} e^{-\\overline{n}_{\\alpha,0}} \\right),\n\\end{equation}\nwhere $\\overline{n}_{\\alpha,0}$ is the mean initial number of individuals of species $\\alpha$ per site.\n\n\n\\section{Derivation of the mean-field equations}\\label{sec_vankampen}\nThe master equation is continuous in time and discrete in space.  The diffusion term is included as a random walk. Next one takes the continuum limit in space, in which the random walk part leads to the usual diffusion term in the partial differential equation (pde) for the concentrations $\\varphi(\\vec{x},t)\\equiv n_\\alpha(\\vec{x},t)\/V$. The mean-field equations can then be derived by calculating the equations of motion for the first moments $\\langle n_\\alpha(\\vec{x},t)\\rangle$ from the master equation, where the average is defined as $\\langle n_\\alpha(\\vec{x},t)\\rangle=\\sum n_\\alpha(\\vec{x},t)P(\\{n_\\alpha(\\vec{x},t)\\})$ with $P(\\{n_\\alpha(\\vec{x},t)\\})$ being a solution of the master equation, and factorizing higher moments in terms of first-order moments.\n\nAlternatively,  we insert the ansatz for the van Kampen expansion according to $n_\\alpha=V\\varphi_\\alpha + \\sqrt{V}\\eta_\\alpha$ in the reaction part. To leading order in $V$ we obtain the deterministic pde for the concentrations of the reaction part. Combined with the diffusion part this leads to the full pde that is given as Eq.~\\ref{eq:pde} in the next section. While this leading order then corresponds to the mean-field level, the next-to-leading order leads a Fokker-Planck equation with associated Langevin equation, from which one can determine the power spectrum of fluctuations. In our realization, the visible patterns are not fluctuation-induced, differently from noise-induced fluctuations as considered in \\cite{goldenbutler}. Therefore our power spectrum of fluctuations is buried under the dominating spectrum that corresponds to patterns from the mean-field level. Therefore we do not further pursue the van Kampen expansion here\\footnote{For details of a possible derivation of the mean-field equations we refer to \\cite{darkathesis}; however, there we derived the mean-field equations via a longer detour towards a field theoretic formulation, where we read off the mean-field equations as leading order of a van Kampen expansion, not applied to the master equation, but to a Lagrangian that appears in the path integral derived from the master equations, in analogy to the derivation in \\cite{goldenbutler}.}.\n\n\n\n\\section{Stability analysis of the mean field equations and their solutions}\\label{sec_Jacobian}\nWe perform a linear stability analysis of the mean field equations by finding the fixed points of the system of partial differential equations\n\\begin{equation}\\label{eq:pde}\n\t\\partial_t\\varphi_\\alpha =\n\tD_\\alpha\\nabla^2\\varphi_\\alpha+\n\tr_\\alpha\\varphi_\\alpha-\n\tp_\\alpha\\varphi_\\alpha^2 -\n\t\\underset{\\beta}{\\sum}k_{\\beta\\alpha}\\varphi_\\alpha\\varphi_\\beta,\n\\end{equation}\nwith $\\phi_\\alpha$ the concentration of species $\\alpha$,\nby setting $\\partial_t\\varphi_\\alpha =D_\\alpha\\nabla^2\\varphi_\\alpha=0$. We will focus on the system with homogeneous parameters $r_\\alpha=\\rho$, $k_{\\alpha\\beta}=\\kappa$ if species $\\alpha$ preys on $\\beta$ and 0 otherwise, $p_\\alpha=\\gamma$, and $D_\\alpha=\\delta$, $\\forall\\alpha,\\beta\\in\\{1,...,N\\}$ and consider the special case of the (6,3)-game. After finding the fixed points, we look at the eigenvalues of the Jacobian $J$ of the system~(\\ref{eq:pde}) to determine the stability of the fixed points. We then extend our analysis to a spatial component by analyzing a linearized system in Fourier space, with Jacobian~\\cite{cianci}\n\\begin{equation}\nJ^{SP}=J+\\underline{D}\\tilde{\\Delta},\n\\end{equation}\nwhere $\\tilde{\\Delta}=-k^2$ is the Fourier transform of the Laplacian and $\\underline{D}$ is the diffusion matrix evaluated at a given fixed point. In our case, the diffusion matrix is a diagonal matrix $\\underline{D}=\\delta\\mathbb{1}$. This leads to a dependence of the stability of the fixed points on diffusion. In the following we focus on the special case to be considered.\n\n\\subsection{Stability analysis and numerical integration for the (6,3)-game}\n{\\bf Stability analysis of the (6,3)-game.}\nThe (6,3)-game is given by the system of mean field equations:\n\\begin{eqnarray}\\label{eq:MF(6,3)}\n\\frac{\\partial\\varphi_1}{\\partial t} & = & \\delta\\nabla^2\\varphi_1 + \\rho\\varphi_1 - \\gamma\\varphi_1^2 - \\kappa\\varphi_1(\\varphi_4+\\varphi_5+\\varphi_6) \\nonumber \\\\\n\\frac{\\partial\\varphi_2}{\\partial t} & = & \\delta\\nabla^2\\varphi_2 + \\rho\\varphi_2 - \\gamma\\varphi_2^2 - \\kappa\\varphi_2(\\varphi_5+\\varphi_6+\\varphi_1) \\nonumber \\\\\n\\frac{\\partial\\varphi_3}{\\partial t} & = & \\delta\\nabla^2\\varphi_3 + \\rho\\varphi_3 - \\gamma\\varphi_3^2 - \\kappa\\varphi_3(\\varphi_6+\\varphi_1+\\varphi_2) \\nonumber\\\\\n\\frac{\\partial\\varphi_4}{\\partial t} & = & \\delta\\nabla^2\\varphi_4 + \\rho\\varphi_4 - \\gamma\\varphi_4^2 - \\kappa\\varphi_4(\\varphi_1+\\varphi_2+\\varphi_3) \\nonumber\\\\\n\\frac{\\partial\\varphi_5}{\\partial t} & = & \\delta\\nabla^2\\varphi_5 + \\rho\\varphi_5 - \\gamma\\varphi_5^2 - \\kappa\\varphi_5(\\varphi_2+\\varphi_3+\\varphi_4) \\nonumber\\\\\n\\frac{\\partial\\varphi_6}{\\partial t} & = & \\delta\\nabla^2\\varphi_6 + \\rho\\varphi_6 - \\gamma\\varphi_6^2 - \\kappa\\varphi_6(\\varphi_3+\\varphi_4+\\varphi_5). \\nonumber\\\\\n\\end{eqnarray}\nIn total there are 64 different fixed points $FP_1$ to $FP_{64}$, of which some have the same set of eigenvalues and differ only by a permutation of the fixed-point coordinates of the eigenvalues,\nso that we can sort all fixed points in 12 groups $FP^1$-$FP^{12}$: for example, the  fixed points $(\\rho\/(\\gamma+\\kappa),0,\\rho\/(\\gamma+\\kappa),0,\\rho\/(\\gamma+\\kappa),0)$ and $(0,\\rho\/(\\gamma+\\kappa),0,\\rho\/(\\gamma+\\kappa),0,\\rho\/(\\gamma+\\kappa))$ are in the same group $FP^3$. We will refer to all fixed points by the number of the group they belong to, that is to $FP^1$ to $FP^{12}$,  instead of $FP_1$ to $FP_{64}$.\\\\\nThe zero-fixed point $FP^1$, where all components are equal to zero, with all eigenvalues equal to $\\rho$ for $\\delta=0$, and equal to $\\rho-\\delta k^2$ for $\\delta\\neq0$, is unstable for a system without spatial assignment, while it can become stable for a spatial system if $\\rho<\\delta k^2$, as in the cases of the (3,1) and (3,2) games, which are discussed in detail in the supplementary material. In the coexistence-fixed point $FP^2$, all components are equal to $\\rho\/(\\gamma+3 \\kappa)$. It is stable for $\\kappa\/\\gamma<1$, three of the eigenvalues are always negative, the first one being $-\\rho$ and an the second and third one equal to $-\\rho\\gamma\/(\\gamma+\\kappa)$, two are complex conjugates $-\\rho(\\gamma-\\kappa\\pm i \\sqrt{3}\\kappa)\/(\\gamma+3\\kappa)$, and the last one is real $-\\rho(\\gamma-\\kappa)\/(\\gamma+3\\kappa)$. At $\\kappa\/\\gamma=1$, $FP^2$ becomes a saddle, three of the six eigenvalues change sign, complex conjugates change sign of their real part, so a Hopf bifurcation occurs, and the direction corresponding to the last eigenvalue becomes unstable.\\\\\nOther fixed points include the survival of one species ($FP^4$), two species (for both $FP^5$ and $FP^6$), three (for $FP^7$ and $FP^8$), four ( for $FP^9$, $FP^{10}$ and $FP^{11}$), and five species (for $FP^{12}$). All fixed points $FP^4$ to $FP^{12}$ are always saddles in the case of $\\delta=0$.\\\\\nFor $\\delta\\neq0$ all eigenvalues get a $(-\\delta k^2)$-term, which can extend the stability regime in the parameter space, as long as $k\\neq0$, and lead to the coexistence of stable fixed points, which cannot be found for $\\delta=0$.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\includegraphics[width=5cm]{fig1.pdf}\n\t\t\\end{center}\n\t\\caption{Diagram of a (6,3)-game. Colors represent species, each preys on three other species in clockwise direction, shown only for the red species by black arrows. Red lines connect species, which form one domain (red, green, and blue), the other three species (cyan, magenta, and yellow) form the second domain. Each species (like the red one) preys on only one species from its own domain (green), and on two species from the other domain (cyan and magenta), this way eliminating all predators of the third species (blue) from the domains. These rules are characteristic for all games, in which a domain forms of three species playing the (3,1)-game. Colors in this scheme will be used throughout the paper to represent species one (red) to six (yellow). }\\label{(6,3)}\n\\end{figure}\n\nIn view of pattern formation we shall distinguish three regimes. Before we go into detail, let us first give an overview of the sequence of events, if we vary the bifurcation parameters $\\kappa$ and $\\gamma$ as $\\kappa\/\\gamma$. These are events, which we see both in the Gillespie simulations and the numerical integration of the mean-field equations in space, described as a finite grid upon integration.\n\\begin{itemize}\n\\item  $\\kappa\/\\gamma<1$: the first regime with $\\kappa\/\\gamma$ smaller than its value at the first Hopf bifurcation at $\\kappa\/\\gamma=1$, where the 6-species coexistence fixed point becomes unstable. As long as this fixed point is stable, we see no patterns, as the system converges at each site of the grid to the 6-species fixed point without dominance of any species, so that the uniform color is gray.\n\\item $1<\\kappa\/\\gamma<2$: the second regime with $\\kappa\/\\gamma$ chosen between the first and second Hopf bifurcations, where the second one happens at $\\kappa\/\\gamma=2$ for the $FP^3$ fixed points. When  $\\kappa\/\\gamma=1$ is approached from below, that is from $\\kappa\/\\gamma<1$, two fixed points, belonging to the $FP^3$-group, become stable through a transcritical bifurcation until $\\kappa\/\\gamma=2$, where they become unstable through the second Hopf bifurcation. Each of the two predicts the survival of three species, the ones, which are found inside the domains. Each of these fixed points is, of course, a single-site fixed point, so in principle a subset of the nodes of the grid can individually approach one of the two fixed points, while the complementary set of the nodes would approach the other fixed point. However, as a transient we see two well separated domains with either even or odd species. At the interfaces between them all six species are present and oscillate with small amplitude oscillations, caused by the first Hopf bifurcation of the six-species coexistence fixed-point, where it became a saddle. Which one of the domains wins the effective (2,1)-game in the end, where a single domain with all its three species survives, depends on the initial conditions and on the fact that diffusion is included; the mere stability analysis only suggests that six species at a site destabilize the interface between domains with either even or odd species.  In fact, the numerical integration and the Gillespie simulations both show that one domain gets extinct if the lattice size is small enough and\/or the diffusion fast enough. As long as the two fixed points are stable, the (3,1)-game is played at each site of a domain in the sense of coexisting three species, which are not chasing each other, related to the neighboring sites only via diffusion, without forming any patterns. Patterns are only visible at the interface of the domains as a remnant of the unstable six-species coexistence fixed point.\n\\item  $\\kappa\/\\gamma>2$: the third regime, which is of most interest for pattern formation.\nStarting from random initial conditions, the species segregate first into two domains, each consisting of three species, one with species 1,3, and 5, the second one with species 2, 4, and 6, and inside both domains the three species play a rock-paper-scissors game, chasing each other, since the two fixed points of the $FP^3$ group became unstable at the second Hopf bifurcation. Due to the interactions according to an effective $(2,1)$-game at the interfaces of the domains (here with either two or four species coexisting), one of the domains will also here get extinct, including the involved three species, while the remaining three survive. Which domain survives  depends also here on the initial conditions. As we shall see, the temporal trajectories of the concentrations of the three species in the surviving domain show that they still explore  the vicinity of the second Hopf bifurcation from time to time, while they otherwise are attracted by the heteroclinic cycle. The three species in the surviving domain live the longer, the larger the grid size is, in which the species continue playing (3,1).  In contrast to the second regime, however, two of the three species in the surviving domain will get extinct as well, and a single one remains in the end. This extinction is caused by fluctuations in the finite population in the stochastic simulation or by the numerical integration on a spatial grid with finite numerical accuracy, respectively.\n\\end{itemize}\nSo the linear stability analysis indicates options for when we can expect oscillatory trajectories: it is  the Hopf bifurcations in the (6,3)-game for the $FP^2$ and $FP^3$ fixed points that induce the creation of limit cycles, which here lead to the {\\it formation of spirals} in space in the third regime and only temporary patterns at the interfaces in the second regime, before the system converges to one of the  $FP^3$ fixed points.\n\nMoreover, it is the two $FP^3$ fixed points in the (6,3)-game that correspond to the {\\it formation of two domains}. In both the (3,2) and the (6,3)-games, one of these fixed points will be approached as a collective fixed point (shared by all sites of the grid), while the domain corresponding to the other one gets extinct, and patterns are seen if this fixed point is unstable. So in the (6,3)-game the existence of domains including their very composition is due to two stable (second regime) or unstable (third regime) fixed points. Their coexistence is in both regimes transient. In the second regime three species will survive in the end, because the three-species coexistence-fixed point is stable, and it would need a large fluctuation to kick it towards a 1-species unstable fixed point. In contrast, only one species will survive in the third regime, where the same fixed point is unstable. Obviously here it does not need a rare, large fluctuation to kick the system towards the 1-species unstable fixed point, as we always observed a single species to survive in the end, both in the Gillespie simulations and the numerical integration in a relatively short time.\n\nWe should mention, however, that from our Gillespie simulations we cannot exclude that after all, a large fluctuation  would also kick the system in the second regime from its metastable state towards one of the unstable 1-species fixed points  as well as in the first regime  to either one of the two three-species fixed points, or to one of the six 1-species fixed points, when the six-species fixed point is stable in the deterministic limit. So far we have not searched for these rare events, in which two, three or five species would get extinct, respectively.\n\\vskip5pt\n\\textbf{Numerical solutions of the (6,3)-game.}\n\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig2.pdf}\n\t\\end{center}\n\t\\caption{Evolution of the (6,3)-game in the second regime in one dimension. The parameters are $ \\gamma = \\kappa = 1 $, $ \\delta\/dx = 0.1 $, and $ \\kappa = 1.5 $. The right and middle columns show the species of each domain separately.\nFor further explanations see the text.}\\label{(6,3)_array_DOM}\n\\end{figure}\n\nIn the following we show evolutions of species concentrations in space and time for parameters, chosen from the second and third regime of the (6,3)-game. These solutions are obtained from the numerical integration of Eq.~\\ref{eq:MF(6,3)}.\nFor the representation on a lattice we will use the following procedure to visualize site occupation: Odd species are represented by the rgb-(red, green, blue) color scheme, while even species are represented by  cmy-colors (cyan, magenta, yellow). The three numbers of species $(r,g,b)$, or $(c,m,y)$, divided by the total sum of all species at the site, give a color in the rgb-, or cmy-spectrum that results from a weighted superposition of individual colors, where the weights (color intensities) depend only on the ratios of occupation numbers, rather than on absolute numbers. Moreover, we display the rgb-color scheme if odd species make up the majority at a site and the cmy-scheme otherwise. We should note that a well mixed occupation of  odd (even) species leads to a dark (light) gray color in these color schemes.\nFigure~\\ref{(6,3)_array_DOM} shows coexisting domains with oscillations at the interfaces in the second regime.  To justify the visualization of data according to the ``majority rule\", we show even and odd species also separately in the two right panels. This way we can see the transitions at the interfaces  of the domains between even and odd species more clearly. The light (dark) gray domain corresponds to a well mixed occupancy with even (odd) species, respectively. On the boundaries of the domains all six species are present, and if we zoom into the boundary, we can see small amplitude oscillations caused by the Hopf bifurcation of the 6-species coexistence-fixed point, see figure~\\ref{(6,3)_tx_DOM}. Figures~\\ref{(6,3)_array_DOM} (a)-(c) show the evolution of the system at the first 100 t.u. at which time the domains are already starting to form. In panel (a) it is seen how the transient patterns, generated by transient small domains, shortly after the domains disappear also fade away, so that the transient patterns are generated by oscillations at the interfaces. The figure also reminds to the early time evolution of condensate formation in a zero-range process, where initially many small condensates form, which finally get absorbed in a condensate that is located at a single site with macroscopic occupation in the thermodynamic limit. Here initially many small and short-lived domains form, which get first absorbed into four domains as seen in the figure, but later end up in a single domain  with three surviving species. So we see a ``condensation\" in species space, where three out of six species get macroscopically occupied as a result of the interaction, diffusion and an unstable interface, while the remaining three species get extinct,  so that the symmetry between the species in the cyclic interactions with identical rates gets dynamically broken.\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig3.pdf}\\\\\n\t\\end{center}\n\t\\caption{Trajectories of all six species corresponding to figure~\\ref{(6,3)_array_DOM} at an interface. Red (1), green (3), and blue (5) represent odd species, cyan (2), magenta (4), and yellow (6) even species, from smaller to larger labels, respectively. (a) and (b) show temporal and spatial trajectories, respectively, at the beginning of the integration, corresponding to  (a)-(c) in figure~\\ref{(6,3)_array_DOM}, while (c) and (d) refer to late times. For further explanations see the text.}\\label{(6,3)_tx_DOM}\n\\end{figure}\n\nPanels (d)-(f) show the evolution from 10000-10100 time units (t.u.). The displayed domains were checked to coexist numerically stable up to $10^6$ t.u., while for smaller lattices and faster diffusion one domain gets extinct. Figures~\\ref{(6,3)_array_DOM} (a) and (d) should be compared with figures~\\ref{(6,3)1D_2} (a) and (b) of the Gillespie simulations, respectively.\n\nFigure~\\ref{(6,3)_tx_DOM} shows the corresponding oscillating concentration trajectories at early (a) and late (c) times at a site of an interface (x=124), where all six species oscillate around the coexistence-saddle fixed point, as indicated by the horizontal black line in (c), while the spatial dependence at (b) (early) and (d) (late) times displays the domain formation due to two stable fixed points, corresponding to figures~\\ref{(6,3)_array_DOM} (a) and (d), respectively, so that the oscillations are restricted to the interfaces.\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig4.pdf}\n\t\\end{center}\n\t\\caption{Evolution of the (6,3)-game in the third regime in one-dimension. The parameters are $ \\gamma = \\kappa = 1 $, $ \\delta\/dx = 0.1 $, and $ \\kappa = 4.0 $. The middle and right columns show the species of each domain separately. For further explanations see the text. }\\label{(6,3)_array_OSC}\n\t%\n\\end{figure}\n\nThe evolution of the (6,3)-game in the third, oscillatory regime in one dimension is shown in figure~\\ref{(6,3)_array_OSC}. Species are represented in the same way as in figure~\\ref{(6,3)_array_DOM}. Panel (a)-(c) show the evolution of the system in the first 100 t.u.,  (d)-(f) in the first 10000 t.u. The two stable fixed points from the second regime became unstable (saddles) through the second Hopf bifurcation. As in the second regime, at the beginning of the integration there is a separation of odd and even species, but at the same time they start to chase  each other, resulting in oscillatory behavior in space and time. Here we see no longer traces of the limit cycle around the six-species coexistence fixed point as in the second regime, since no sites have six species coexisting, even not for a short period of time. At the interfaces between even and odd species usually three species coexist, either two odd and one even, or vice versa, two even and one odd, but these mixtures are not stable, as these 3-species coexistence-fixed points in the deterministic limit are saddles. It also happens that just two or four species coexist at the interface, but also their coexistence-fixed points are saddles. Therefore also here the coexistence of domains is not stable,  only one of them survives, and which one depends on the initial conditions, resulting in the extinction of three either odd or even species. In view of Gillespie simulations, figures~\\ref{(6,3)_array_OSC} (a) (early times) and (b) (late times) should be compared with figures~\\ref{(6,3)1D_1} (a) (early) and (b) (late), respectively.\n\nFigure~\\ref{(6,3)_tx_OSC} (a) shows the evolution in time at late times, when only one domain survives.  All three species oscillate between zero and one, corresponding to the heteroclinic cycle. From time to time the trajectories are also attracted by the saddle-limit cycle, which is created by the second Hopf bifurcation of the three species-fixed point (black line) as indicated by the small amplitude oscillations. Apart from the amplitude, the heteroclinic and saddle-limit cycles differ  in their frequency: the saddle-limit cycle has a higher frequency than the heteroclinic cycle. Panel (b) shows the spatial trajectories at the beginning of the integration when both domains still coexist. Yet we see no mixing of all six species at a single site, the 6-species coexistence-fixed point is no longer felt in this regime.\n\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig5.pdf}\\\\\n\t\\end{center}\n\t\\caption{In correspondence to figure~\\ref{(6,3)_array_OSC} (a) temporal trajectories of the surviving domain at late times in the interval 10200-10600 t.u. when only even species exist, and (b) spatial trajectories  at early times when still both domains exist. For further explanations see the text.}\\label{(6,3)_tx_OSC}\n\t%\n\\end{figure}\n\nAs we see from figure~\\ref{(6,3)_array_OSCend}, the numerical integration evolves to one of the saddles  after having spent a finite time on the heteroclinic cycle and not according to the analytical prediction, where it were only in the infinite-time limit that the trajectory would get stuck in one of the saddles, which are connected by the heteroclinic cycle. According to  figure~\\ref{(6,3)_array_OSCend}(a) all trajectories get absorbed in one (the pink one) of the 1-species saddles already at finite time as a result of the finite accuracy of the numerical integration. Yet figure~\\ref{(6,3)_array_OSCend} (b) shows the characteristics of a heteroclinic cycle at finite time: The dwell time of the trajectory in the vicinity of the 1-species saddles gets longer and longer in each cycle, before it fast moves towards the next saddle in the cycle. This escape stops after a finite number of cycles, when the concentration of two of the three  species are zero within the numerical accuracy, and therefore no ``resurrection\" is possible.\n\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig6.pdf}\n\t\\end{center}\n\t\\caption{Extinction of all but one species upon approaching the heteroclinic cycle, (a) (x,t)-diagram, (b) species concentrations as a function of time. For further explanations see the text.}\\label{(6,3)_array_OSCend}\n\t%\n\\end{figure}\n\n\n\n\\section{Numerical Methods and Results}\\label{sec_numerical}\nGoing back to the set of reactions, in this section we describe their Gillespie simulations.\nWe solve the system (\\ref{eq:rec_sys})\nby stochastic simulations on a regular square lattice as well as on a one-dimensional ring, using the Gillespie algorithm~\\cite{gillespie}, combined with the so-called next-subvolume method~\\cite{elf}. This method is one option to generalize Gillespie simulations to spatial grids. We choose periodic boundary conditions on a square $L\\times L$ lattice or on a ring with $L$ nodes. In our case nodes, or synonymously sites, represent subvolumes. All reactions except of the diffusion happen between individuals on the same site (in the same subvolume), and a diffusion reaction is a jump of one individual to a neighboring site. One event can change the state of the system of only one (if a reaction happens) or two neighboring (if  a diffusion event happens) subvolumes. At each site the initial number of individuals of each species is chosen from a Poisson distribution $P \\left( \\{n\\} ;0 \\right)=\\underset{\\alpha,i}{\\prod}\\left( \\frac{\\overline{n}^{n_{\\alpha,i}}_{\\alpha,0}}{n_{\\alpha,i}!} e^{-\\overline{n}_{\\alpha,0}} \\right)$, with a mean $\\overline{n}_{\\alpha,0}$, which is randomly chosen for each species $\\alpha$.\n\nIn the next-subvolume method we assign the random times of the Gillespie algorithm to subvolumes rather than to a specific reaction. To each subvolume, or site, we assign  a time $\\tau$, at which one of the possible events, in our case reactions (predation, birth or death), or diffusion, will happen. The time $\\tau$ is calculated as $\\tau = -\\ln(rn)\/r_{total}$, where $rn$ is a random number generated from a uniform distribution between 0 and 1. The total rate $r_{total}$ depends on the reaction rates and the number of individuals which participate in the event. Events happen at sites in the order of the assigned times $\\tau$. Once it is known at which subvolume the next event happens, the event (reaction or diffusion) is chosen randomly according to the specified reaction rates.\n\nWe start the simulations with initial conditions from a Poisson distribution such that each site of the entire lattice is well mixed with all species. We want to study the dynamics of the system in a parameter regime, where we expect pattern formation. From the linear stability analysis of the mean-field system we expect stable patterns in the regime without stable fixed points in the (6,3)-game, as long as three species are alive, and transient patterns in the (6,3)-game for  coexisting stable fixed points.\nWe use the same color scheme as we used before to visualize the numerical solutions of the mean-field equations.\nAs one Gillespie step (GS) we count one integration step here.\n\nOur results confirm the predictions from the mean-field analysis: there are on-site oscillations in time in a limit-cycle regime. There are also oscillations in space, which form spirals on two-dimensional lattices. If the evolution approaches the stochastic counterpart of a stable fixed point in the deterministic limit, we call it shortly  ``noisy fixed point\",  where the trajectories fluctuate around a value that is the mean field-fixed point value multiplied by the parameter $V$ as defined before. Of particular interest is the influence of the diffusion in relation to the ratio $\\kappa\/\\gamma$ on the patterns. It was the ratio of $\\kappa\/\\gamma$ that determines the stability of the fixed points. As mentioned earlier, the value of $\\delta k^2$, which enters the stability analysis, can extend the stability regime. So in the Gillespie simulations it is intrinsically hard to disentangle the following two reasons for the absence of patterns in the case of fast diffusion: either the stability regime of a fixed point with only one surviving species is extended, or the diffusion is so fast, that the extension of visible patterns is larger than the system size, so that a uniform color may just reflect the homogeneous part within a large pattern.\nAll the mean-field-fixed points are proportional to the value of the parameter $\\rho$. If this value is much larger than $\\kappa$ and $\\gamma$, the fixed-point value is very large. This leads to a large occupation on the sites, which slows down the formation of patterns. The reason is that the number of reactions, which are needed for the system to evolve to stable trajectories, either to oscillations, or to fixed points, increases with the number of individuals in the system.\n\n\\vskip5pt\n\nWe study the stochastic dynamics of a (6,3)-game in regimes, for which we expect pattern formation, i.e. for $\\gamma<\\kappa$. When the coexistence-fixed point $FP^2$ becomes unstable at $\\gamma=\\kappa$, we find the formation of two domains, each consisting of three species, one domain containing odd species, in the figures represented by shared colors red, green, and blue in the rgb-color scheme. The other domain consists of even species, represented by shared colors of cyan, magenta, and yellow in the cmy-color representation, see figure~\\ref{(6,3)2D}. Inside the domains the three species play the (3,1)-game and form spiral patterns. We have checked that the domains in figure~\\ref{(6,3)2D} are not an artefact of the visualization, and determined, for example, the occupancy on a middle column of the lattice (not displayed here).\nOn sites with oscillations of species from one domain, there is a very small or no occupation of species of the second domain, confirming the very existence of the domains.\nThe time evolution of the six species on two sites, chosen, for example, from the middle column of the lattice confirm that the species' trajectories  oscillate in time, reflecting the stable limit cycles in the deterministic limit.\n\nHere a remark is in order as to whether radii, propagation velocity or other features of the observed spiral patterns can be predicted analytically. While  spiral patterns in spatial rock-paper-scissors games were very well predicted via a multi-scale expansion in the work of \\cite{mobilia1,mobilia2}, we performed a multi-scale expansion (see, for example \\cite{bookkuramoto}) to derive amplitude equations for the time evolution of deviations from the two unstable fixed points, which lose their stability at the two Hopf bifurcations. However, the resulting amplitude equations differ from Ginzburg-Landau equations by a missing imaginary part, which can be traced back to the absence of an explicit constraint to the occupation numbers on sites and the absence of a conserved total number of individuals. As a result, the amplitude equations only predict the transient evolution as long as the trajectory is in the very vicinity of the unstable fixed point, but cannot capture the long-time behavior, which here is determined by an attraction towards the heteroclinic cycle that is responsible for the spiral patterns in our case. So it seems to be this non-local feature in phase space that the multi-scale expansion about the Hopf bifurcation misses.\n\nFor a further discussion of how the patterns depend on the choice of parameters  we shall focus on the results on a one-dimensional lattice, since the simulation times are much longer for two dimensions. (In two dimensions, the period of oscillations is as long as about one fifth of the $2^{30}$ Gillespie steps.)\n\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.9]{fig7.pdf}\n\t\t\\end{center}\n\t\\caption{Pattern formation for a (6,3)-game on a two-dimensional $(64\\times 64)$-lattice for weak diffusion and far from the bifurcation point. Snap shots are taken at  $1000\\cdot2^{15}$ (a), $10000\\cdot2^{15}$ (b), and $32000\\cdot2^{15}$ (c) GS. Two domains are formed, each containing three species, indicated by the different color groups. These species play a (3,1)-game inside the domains and evolve spiral patterns. The parameters are $\\rho=1$, $\\kappa=1$, $\\delta=1$, and $\\gamma=0.2$.\n}\\label{(6,3)2D}\n\\end{figure}\n\n\n\\begin{figure}[tp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig8.pdf}\n\t\t\\end{center}\n\t\\caption{Pattern formation in the (6,3)-game on a one-dimensional lattice of 64 sites for $\\kappa\/2<\\gamma<\\kappa$, that is in the second regime, where the coexistence-fixed points $FP^3$ are stable, for weak diffusion $\\delta=0.1$ ((a) and (b)), and strong diffusion $\\delta=1$ ((c) and (d)). For both strengths of the diffusion domains form. In the case of weak diffusion no extinction of domains is observed within the simulation time of $2^{30}$ GS. For strong diffusion, one domain goes extinct after $800\\cdot2^{15}$ GS. Initially,  oscillatory patterns appear as remnants of many interfaces between small domains, where within the interfaces six species oscillate due to the unstable 6-species coexistence-fixed point, which fade away with time. This confirms the analytical results that in this parameter regime the $FP^3$-fixed points with $2\\times 3$ coexisting species  are both stable, leading to the black color in (c) and (d) for the one surviving domain. Panel (a) shows the time evolution on a lattice for the time interval $(0-1000)\\cdot2^{15}$, (b) for $(30000-31000)\\cdot2^{15}$, (c) for $(0-1000)\\cdot2^{15}$, and (d) for $(10000-11000)\\cdot2^{15}$ GS. The parameters are $\\gamma=\\rho=0.5$, $\\kappa=0.6$.}\\label{(6,3)1D_2}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[scale=1]{fig9.pdf}\n\t\t\\end{center}\n\t\\caption{Pattern formation in the (6,3)-game on a one-dimensional lattice of size 64, shown  on a space-time grid, for $\\gamma<\\kappa\/2$, that is, in the third regime, where the coexistence-fixed points $FP^3$ are unstable, weak diffusion $\\delta=0.1$ ((a) and (b)), and strong diffusion $\\delta=0.1$ ((c) and (d)). The parameters are $\\gamma=\\rho=0.5$ and $\\kappa=1.2$. Two domains form, of which one goes extinct after $600\\cdot2^{15}$ GS in the case of weak diffusion and $150\\cdot2^{15}$ GS in the case of strong diffusion. The surviving domain keeps playing the (3,1)-game. For weak diffusion no further extinction is observed for the simulation time of $2^{30}$ GS, while for strong diffusion, an extinction of all but one species, here the red one, happens after $1470\\cdot2^{15}$ GS. Panel (a) shows patterns in a time interval of $(0-700)\\cdot2^{15}$, (b) for $(30000-30100)\\cdot2^{15}$, (c) for $(0-200)\\cdot2^{15}$, and (d) for $(0-1470)\\cdot2^{15}$ GS.}\\label{(6,3)1D_1}\n\\end{figure}\n\nAs to diffusion, for stronger diffusion patterns are more homogeneous, extinction events happen faster,  sometimes they happen only for sufficiently strong diffusion, see figure~ \\ref{(6,3)1D_2}. The extinction time depends also on the $\\kappa\/\\gamma$-ratio, i.e. if the ratio is from the interval (i) $1<\\kappa\/\\gamma<2$, where $FP^3$ is stable and $FP^2$ is unstable, or, (ii) from $\\kappa\/\\gamma>2$, where both fixed points are unstable.\n\nFor case (i), the $FP^3$-fixed points are stable, yet at the beginning of the simulations the dynamics shows oscillatory behavior, caused by the interfaces between the small domains, where six species feel the unstable coexistence-fixed point at $\\gamma=\\kappa$; but after about $500\\cdot2^{15}$ Gillespie steps for weak diffusion, and $1000\\cdot2^{15}$ Gillespie steps for strong diffusion, and for our choice of parameters, the patterns fade away and the system evolves to a homogeneous state in both domains as long as they coexist, see figures~\\ref{(6,3)1D_2} (b) and \\ref{(6,3)1D_2} (d). The closer the system is to the bifurcation point $\\gamma=\\kappa$, the longer live the oscillatory patterns, the stronger feels the system the unstable 6-species fixed point.\n\nFigure~\\ref{(6,3)1D_2} (a) should be compared with the corresponding mean-field solution of figure~\\ref{(6,3)_array_DOM} (a) at early times and figure~\\ref{(6,3)1D_2} (b) with figure~\\ref{(6,3)_array_DOM} (d) at late times, from which we see that the mean-field solutions reproduce the qualitative features including the transient patterns.\n\nIn case (ii), the third regime, domains get faster extinct, both domains play a (3,1)-game inside the domains. After one domain gets extinct, the surviving one keeps on playing the (3,1)-game, until only one species survives. Here one should compare figure~\\ref{(6,3)1D_1} (a) with the corresponding mean-field solution of figure~\\ref{(6,3)_array_OSC} (a) at early times and figure~\\ref{(6,3)1D_1} (b) with figure~\\ref{(6,3)_array_OSC} (d) at later times, where one is left with one domain and three species chasing each other; the final extinction of two further species is not visible in this figure due to the weak diffusion and the larger extinction time.\n\n\n\\section{Conclusions and Outlook}\\label{sec_conclusions}\nBeyond the emerging dynamically generated time-and spatial scales, the most interesting feature of the (6,3)-game is the fact that the rules of the game, specified initially as (6,3), dynamically change to effectively (2,1) and (3,1) as a result of spatial segregation. In view of evolution, here the rules of the game change while being played. They change as a function of the state of the system if the state corresponds to the spatial distribution of coexisting species over the grid.\n\nIn preliminary studies, we investigated the $(27,17)$-game with the following set of coexisting  games in a transient period: From a random start we observe segregation towards nine domains playing $(9,7)$ with each other and inside the domains again the $(3,1)$ game. From the superficial visualization of Gillespie simulations, this system looks like a fluid with whirling ``vortices\", where the (3,1)-game is played inside the domains. We expect a rich variety of games with new, emerging, effective rules on a coarser scale, if we not only increase the number $N$ of species, or release the restriction to cyclic predation, but allow for different time scales, defined via the interaction rates. So far we chose the same rates for all interactions and always started from random initial conditions.\n\nWe performed a detailed linear stability analysis, which together with the numerical integration of the mean-field equations reproduced all qualitative features of the Gillespie simulations, even extinction events. That the mean-field analysis worked so well is due to the ultralocal implementation of the interactions, so that the spatial dependence enters only via diffusion. The  stability analysis revealed already a rather rich structure with 12 groups of in total 64 fixed points for the (6,3)-game. We focussed on coexistence-fixed points of six, three or one species.\n\nAlong with the fixed points' repulsion or attraction properties we observed three types of extinction, whose microscopic realization is different and deserves further studies:\n\n(i) In the second regime of the (6,3)-game, both domains with either even or odd species are in principle stable, as long as they are not forced to coexist. We have seen a spatial segregation towards a domain with only even and one with only odd species, occupying the sites. At the interface between both domains, six species cannot escape from playing the (6,3)-game. Since the 6-species coexistence-fixed point is unstable, the unstable interface seems to be the driving force to initiate the extinction of one of the two domains including its three species, since interface areas should be reduced to a minimal size. From the coarse-grained perspective, one domain preys on the other domain, which is a (2,1)-game.\n\n(ii) In the third regime of the (6,3)-game, the domain structure in odd and even domains is kept, but in the interior of the domains the species follow heteroclinic cycles, which explain the patterns of three species, chasing each other, inside each domain.\nAt the interface between the domains, two to four species coexist at a site, but for small enough diffusion, coexistence-fixed points of the respective species are always saddles, so also here the instability of the interfaces seems to induce their avoidance, leading\nagain to the extinction of one of the two domains. So from the coarse-grained perspective, again a (2,1)-game is played between the domains.\\\\\nIt should be noticed that in contrast to systems, where the fate of interfaces between domains is explained in terms of the competition between free energy and interface tension, here the growth of domains and the reduction of interfaces are traced back to the linear stability analysis of the system in the deterministic limit, which is conclusive for the dynamics.\n\n(iii) The third type of extinction event was the extinction of two species, when the individual trajectories move either in the vicinity of, or  along a heteroclinic cycle, and either a fluctuation from the Gillespie simulations, or the finite numerical accuracy on the grid (used for integration) captured the trajectory in one of the 1-species saddles.\n\n\nWe have not studied rare large fluctuations, which could induce other extinction events and kick the system out of the basin of attraction from the 6- or 3-species stable coexistence-fixed points when stochastic fluctuations are included. Neither have we measured any scaling of the extinction times with the system size or of the domain growth with the system size. This is left for future work.\\\\\nFurthermore, for future work it would be challenging to derive and predict the domain formation on the coarse scale from the underlying $(6,3)$-game on the basic lattice scale in the spirit of the renormalization group, here, however, applied to differential equations rather than to an action.\n\n\n\\section{Acknowledgments}\nOne of us (D.L.) is grateful to the German Research Foundation (DFG)(ME-1332\/25-1) for financial support during this project. We are also indebted to the German Exchange Foundation (DAAD)(ID 57129624)for financial support of our visit at Virginia Tech Blacksburg University, where we would like to thank Michel Pleimling for valuable discussions. We are also indebted to Michael Zaks (Potsdam University) for useful discussions.\\\\\n\n\\section{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Insights and Conclusions}\n\\label{section:conclusions}\n\\emph{Dimensional circuit synthesis} is a new method for generating\ndigital logic circuits that improve the efficiency of training and\ninference of machine learning models from sensor data. The method\ncomplements prior work on\\emph{ dimensional function synthesis}, a\nnew method for learning models from sensor data that enables orders\nof magnitude improvements in training and inference on physics-constrained\nsignal data.  Dimensional circuit synthesis, which we present in\nthis paper, implements preprocessing steps required by dimensional\nfunction synthesis in hardware.  This article presented the principle\nbehind the methods and the design and implementation of a compiler\nbackend that implements dimensional circuit synthesis for the iCE40,\na low-power miniature FPGA in a miniature wafer-scale\n2.15\\,mm$\\times$2.50\\,mm WLCSP package which is targeted at sensor\ninterfacing tasks and at on-device machine learning.  The hardware\naccelerators that the method generates are compact (fewer than four\nthousand gates for all the examples investigated) and low power\n(dissipating less than 6\\,mW even on a non-optimum FPGA).  These\nresults show, for the first time, that it could be feasible to\nintegrate physics-inspired machine learning methods within low-cost\nminiaturized sensor integrated circuits, right next to the sensor\ntransducer.\n\n\\section{Introduction}\nSensor integrated circuits are at the forefront of the data pipeline\nfeeding the recent revolution in machine learning systems. Sensors\ntransduce a physical signal such as acceleration, temperature, or\nlight, into a voltage which is then converted by analog-to-digital\nconverters (ADCs) into a numeric representation, for input to\ncomputation. Digital preprocessing within sensor integrated circuits,\nor the software that consume their output, then apply appropriate\ncalibration constants and scaling to convert these digitized voltages\ninto a scaled and dimensionally-meaningful representation of the\nsignal (e.g., acceleration in $m\/s^2$).\n\nFigure~\\ref{fig:introduction:sensors-in-systems} shows how contemporary\nsensor-driven computing systems move the digitized data at the\noutput of signal conversion circuits through many transmission and\nstorage steps before the data are used in training a model or in\ndriving an inference, typically on server far removed from the\nsensing process. This data movement costs time and energy.  When\never-greater volumes of data will enable new applications and\ninference models, it will be valuable to perform the necessary\ncomputations as close to the signal acquisition and transduction\nprocess as possible: ideally, in the sensor integrated circuit\nitself (labeled ``\\ding{202}'' in\nFigure~\\ref{fig:introduction:sensors-in-systems}).\n\nHowever, since these sensor integrated circuits are typically\nrequired be low cost (often under 10\\,USD), have small die area\n(often less than 4\\,mm$^2$), and use minimal power (typically under\n1\\,mW), it is challenging to integrate even the most efficient and\ncompact traditional learning and inference methods into these devices\nthemselves.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, angle=0, width=0.485\\textwidth]{Illustrations\/sensors-in-systems.pdf}\n\\caption{Existing sensing systems typically send data to servers\nfor training models and generating inferences.  Moving data both\nwithin a system and over networks adds latency and costs energy.}\n\\label{fig:introduction:sensors-in-systems}\n\\end{figure}\n\n\n\n\\subsection{Physics constrains signals from sensors}\nThe values taken on by data from sensors are constrained by the\nlaws of physics and by the dynamics of the structures to which\nsensors are attached. Most physical laws and the governing equations\nfor most system dynamics take the form of sums of product terms\nwith each product term comprising powers of the system's\nvariables~\\cite{feynman1967character, feynman1965feynman,\nBuckingham1914}.  Because there are a bounded number of irreducible\nrelative powers in these product terms whose units of measure result\nin meaningful units for the whole expression, it is common in many\nengineering disciplines to use information on units of measure\n(\\textit{dimensional analysis}) to derive candidate relations for\nexperimentally-observed phenomena~\\cite{mahajan2010street,\nBuckingham1914}. Recent work~\\cite{10.1145\/3358218} has used this\nobservation to prune the hypothesis set of functions considered\nduring machine learning to achieve significant improvements in both\ntraining and inference, improving training latency by 8660$\\times$\nand reducing the arithmetic operations in inference over 34$\\times$.\n\nIn this work, we build on these results to develop a new backend\nfor the Newton compiler~\\cite{lim2018newton}. The backend generates\nregister transfer level (RTL) hardware designs for accelerating the\nexecution of the required pre-inference parts of analytic models\nrelating the signals in a multi-sensor system according to the\nspecifications of the physics of a system. Because our method uses\ninformation from dimensional analysis and units of measure to\nsynthesize hardware to accelerate inference from sensors, we call\nthe method \\emph{dimensional circuit synthesis}.\n\n\\subsection{Dimensional circuit synthesis: physics-derived pre-inference processing in sensors}\nDimensional circuit synthesis is a compile-time method to generate\ndigital logic circuits for performing pre-inference processing on\nsensor signals. Dimensional circuit synthesis takes as its input a\nspecification of the signals that can be obtained from the sensors\nin a system and their units of measure.  Using these specifications,\ndimensional circuit synthesis generates hardware to compute a set\nof physically plausible expressions relating the signals in the\nsystem.  The logic circuits which the method generates represent\nsets of monomial expressions which form dimensionless groupings of\nsensor signals (i.e., products whose units cancel out).  The\nsynthesized hardware takes as input digital representations of\nsensor readings and generates the computed value of the dimensionless\nexpressions as its output. A machine learning training or inference\nprocess then uses these dimensionless products as its inputs and\nprior work has shown that this preprocessing can significantly\nimprove both the latency and accuracy of inference. An on-device\n(in-sensor) inference engine will integrate the generated RTL that\nperforms pre-processing with either custom RTL or a programmable\ncore implementing the inference using, e.g., a neural network.\nWe evaluate the generated RTL on the Lattice iCE40, a\nstate-of-the-art, ultra-miniature FPGA that meets the size\nand power consumption constraints of in-sensor processing.\n\n\\subsection{Contributions}\nThis article makes two main contributions to on-device and in-sensor \ninference:\n\\begin{itemize}\n\\item We present dimensional circuit synthesis\n(Section~\\ref{sec:methodology}), a new method to generate RTL\nhardware for pre-processing sensor data prior to inference, thereby\nimproving latency and reducing overhead.\n\n\\item We evaluate the generated RTL on the Lattice\niCE40 ultra-miniature FPGA (Section~\\ref{sec:results}) and show\nthat the generated RTL is fast enough to allow real-time processing,\nwhile consuming minimal power.\n\\end{itemize}\n\n\n\n\n\n\\section{Background and Methodology}\n\\label{sec:methodology}\n\\label{subsec:dfs}\nDimensional circuit synthesis takes as input descriptions of the\nunits of measure of the sensor signals in a system.\nFigure~\\ref{figure:methodology:NewtonExample} shows an example\nNewton description for an unpowered UAV (i.e., a glider).  Let a\nphysical system for which we want to construct an efficient predictive\nmodel have $k$ symbols corresponding to physical constants or sensor\nsignals.  From the Buckingham $\\Pi$-theorem~\\cite{Buckingham1914},\nwe can form $N \\le k$ dimensionless products, $\\Pi_1 \\ldots \\Pi_N$\nand these dimensionless products are the roots of some function\n$\\Phi$, where\n\\begin{align}\n\\label{equation:Phi}\n\\Phi (\\Pi_1, \\Pi_2, \\dots, \\Pi_i, \\ldots, \\Pi_{N}) = 0 .\n\\end{align}\nWang \\textit{et al.}~\\citep{10.1145\/3358218} use\nEquation~\\ref{equation:Phi} as the basis for generating a preprocessing\nstep of offline training and inference of models of physical systems\nand propose an automated framework for generating these dimensionless\nproducts.  In a subsequent calibration step, they learn a model for\nthe function $\\Phi$ and demonstrate that learning $\\Phi$ from the\n$\\Pi_1 \\ldots \\Pi_N$ can be both significantly more efficient and\nmore accurate than learning a function from the original $k$ sensor\nsignals directly. The dimensionless products $\\Pi_1 \\ldots \\Pi_N$\nare essential to both training and inference and to achieving the\norders-of-magnitude speedup.  In this work, we present  a method\nfor generating hardware to efficiently compute these $\\Pi$s. Doing\nso close to the sensor transducer also reduces the data sensing\nsystems must transmit from the sensor transducer to either a sensor\nhub, microcontroller, or other component performing on-device\ntraining and inference, potentially improving system efficiency and\nperformance.  Figure~\\ref{fig:introduction:dfsRTL} shows how the\ngenerated hardware for $\\Pi$ computation fits within an on-device\ninference system.\n\n\n\n\\begin{figure}\n\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, angle=0, width=1.0\\columnwidth]{Illustrations\/DFS-RTL.pdf}\n\\caption{The hardware generated by dimensional circuit synthesis\npreprocesses $k$ sensor signals to obtain $N < k$ \\emph{dimensionless\nproducts} $\\Pi_1 \\ldots \\Pi_k$.  A predictive model takes these\nproducts as input and generates an inference output.}\n\\label{fig:introduction:dfsRTL}\n\\end{figure}\n\n\n\n\\subsection{Dimensional Circuit Synthesis}\nFigure~\\ref{fig:introduction:dfsRTL} shows how hardware blocks\ngenerated by dimensional circuit synthesis calculate the values of\nthe $\\Pi$ products.  The input of these modules are the sensor\nsignals corresponding to the physical parameters specified as the\ninput to the dimensional circuit synthesis analysis in the Newton\nspecification language (see, e.g.,\nFigure~\\ref{figure:methodology:NewtonExample}).  The calculated\n$\\Pi$ product values correspond to the output of the pre-processing\nstep of the inference function and they feed into any existing\nmethod for classification or regression.  This final step could be\na programmable low-power core such as the 32-bit RISC cores now\nintegrated into some state-of-the-art sensor integrated circuits\nor a low-power machine learning accelerator such as\nMarlann~\\cite{SymbioticEDA:Marlann}, implemented in either RTL or\nin a miniature FPGA like we use in our evaluation in\nSection~\\ref{sec:results}.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, angle=0, width=2.0\\columnwidth]{Illustrations\/DFS-compilation-process.pdf}\n\\caption{Proposed dimensional circuit synthesis framework. In\nStep~\\ding{202}, the users provide the specifications and target\ninference parameter of the examined physical system. Newton compiler\nincluding our implemented backend is executed in Step~\\ding{203}.\nIn Step~\\ding{204} the framework translates the generated RTL modules\nto an FPGA bitstream and in parallel we execute a manual calibration\nof dimensional functions~\\cite{10.1145\/3358218} (box with dashed\nborder).  In Step~\\ding{205} the generated SW and HW modules are\ndownloaded to the in-sensor inference engine.}\n\\label{fig:introduction:dfsCompilationProcess}\n\\end{figure*}\n\n\n\n\n\\subsubsection{Approach and implementation}\nFigure~\\ref{fig:introduction:dfsCompilationProcess} shows the four\nsteps which make up our implementation of dimensional circuit\nsynthesis.  We implemented these steps as a new backend of the\nNewton compiler~\\cite{lim2018newton}, but the techniques in\nFigure~\\ref{fig:introduction:dfsCompilationProcess} could in principle\nbe applied to any specification for physical systems that contains\ninformation on units of measure.\n\nIn Step~\\ding{202}, a user of dimensional circuit synthesis creates\na Newton language description such at that in\nFigure~\\ref{figure:methodology:NewtonExample}, specifying the\nphysical signals that describe the target physical system and from\nwhich a machine learning model will eventually be trained.\n\nNext, in Step~\\ding{203}, the user \ninvokes the Newton compiler with our new dimensional circuit synthesis\nbackend activated.  Because the method of constructing dimensionless\ngroups can result in multiple dimensionless products ($\\Pi_i$ in\nEquation~\\ref{equation:Phi}), the user specifies which of the physical\nsignals in the input physical system description will be the target\nvariable of a machine learning model for the function $\\Phi$ from\nEquation~\\ref{equation:Phi}.  Our new dimensional circuit synthesis\nbackend identifies the group of dimensionless products where the\ntarget parameter appears in only one of the dimensionless products.\nThe outputs of Step~\\ding{203} are: (i) a function $\\Phi$, defined\nin terms of the dimensionless products $\\Pi_i$, but whose form has\nnot yet been fully defined; (ii) RTL descriptions for hardware to\ncompute the dimensionless products $\\Pi_i$, including RTL descriptions\nof the functional units (multipliers and dividers) that will perform\nthe arithmetic operations of the dimensionless product monomials.\nBecause floating-point operations can be expensive in both resources\nand execution latency on energy-constrained on-device training and\ninference systems, we use a signed fixed-point approximate real\nnumber representation~\\cite{behrooz2000computer} for the signals\nin the dimensionless products computed in the synthesized hardware.\nEach real number is represented by 32 bits, using 1 bit for the\nsign, 16 bits for the decimal part and 15 bits for the fractional\npart (i.e., a Q16.15 fixed-point representation).  This choice leads\nto fast and lightweight multiplication and division units by\nsacrificing the ability to use an arbitrary precision floating point\nrepresentation. The compiler backend is fully parametric with respect\nto the length of the fixed point representation as well the precision\nof the fractional part and can generate hardware with arbitrary\nfixed-point representation sizes. This will allow future designs\nto tailor the precision of the compute modules to the requirements\nof the inference algorithms~\\cite{micikevicius2017mixed}.\n\nIn Step~\\ding{204}, we can train the uncalibrated dimensional\nfunction offline on values of the dimensionless groups $\\Pi_i$\ncomputed offline as done in prior work~\\cite{10.1145\/3358218}.\n\n\nFinally, in Step~\\ding{205} the outputs of the hardware blocks\ncomputing the dimensionless products feed the models trained offline\nto generate inferences.  Alternatively a system could also use the\nvalues of the dimensionless products to feed in situ training of\nmodels implemented in a processor core, or to feed training in situ\nof a hardware neural network accelerator~\\cite{SymbioticEDA:Marlann}.\n\n\n\n\n\n\n\n\\section{Experimental Evaluation}\n\\label{sec:results}\n\\label{subsec:experimental-setup}\nWe evaluated the hardware generated by the dimensional circuit\nsynthesis backend using a Lattice Semiconductor iCE40 FPGA. The\niCE40 is a low-power miniature FPGA in a miniature wafer-scale\n2.15\\,mm$\\times$2.50\\,mm WLCSP package and is targeted at sensor\ninterfacing tasks and at on-device machine learning.  We used the\nfully open-source FPGA design flow, comprising the\nYoSys~\\cite{shah2019yosys+} synthesis tool (version 0.8+456) for\nsynthesis and NextPNR ~\\cite{shah2019yosys+} (version git sha1\n5344bc3) for placing, routing, and timing analysis.\n\nWe performed our measurements on an iCE40 Mobile Development Kit (MDK)\nwhich includes a 1$\\Omega$ sense resistor in series with each of\nthe supply rails of the FPGA (core, PLL, I\/O banks).  We measure the current\ndrawn by the FPGA core by measuring the voltage drop across the\nFPGA core supply rail (1.2\\,V) resistor using a Keithley DM7510, a\nlaboratory-grade 7-\\sfrac{1}{2} digital multimeter that can measure\nvoltages down to 10\\,nV and we thereby computed the power dissipated\nby the FPGA core for each configured RTL design.\nWe used a pseudorandom number generator to feed the $\\Pi$ computation\ncircuit modules under evaluation with random input data.\n\n\\begin{table*}\n        \\centering\n        \\caption{Experimental evaluation on iCE40 FPGA of dimensional circuit modules generated from the description of 7 physical systems.} \\label{table:results}\n\t\t%\n        \\tiny\n        \\begin{tabular}{lm{3.0cm}lllllll}\n                \\toprule\n                \\textbf{Name}\t\t\t& \\textbf{Description}\t& \\textbf{Target}\t\t&\\textbf{LUT4}\t\t&\\textbf{Gate}\t&\\textbf{Maximum}\t&\\textbf{Execution}\t&\\textbf{Avg. Power}\t\t&\\textbf{Avg. Power}\\\\\n                                       &\t\t\t\t\t\t& \\textbf{Parameter}\t&\\textbf{Cells} \t&\\textbf{Count}\t&\\textbf{Frequency}\t&\\textbf{Latency}\t&\\textbf{at 12 MHz}\t\t&\\textbf{at 6 MHz}\\\\\n                \\hline\n\\textbf{Beam}\t\t\t\t\t& Cantilevered beam model, excluding mass of beam\t\t\t\t\t\t& Beam deflection\t& 2958 & 2590 & 16.88\\,Mhz & 115 cycles & 3.5\\,mW & 1.8\\,mW\\\\\n\\textbf{Pendulum, static} \t\t& Simple pendulum excluding dynamics and friction\t\t\t\t\t\t& Osc. period \t\t& 1402 & 1239 & 17.07\\,Mhz & 115 cycles & 2.0\\,mW & 1.1\\,mW\\\\\n\\textbf{Fluid in Pipe} \t\t\t& Pressure drop of a fluid through a pipe\t\t\t\t\t\t\t\t& Fluid velocity \t& 4258 & 3752 & 15.65\\,Mhz & 188 cycles & 5.8\\,mW & 3.0\\,mW\\\\\n\\textbf{Unpowered flight} \t\t& Unpowered flight (e.g., catapulted drone)\t\t\t\t\t\t\t& Position (height)\t& 1930 & 1865 & 16.44\\,Mhz & 81 cycles & 2.3\\,mW & 1.2\\,mW\\\\\n\\textbf{Vibrating string}\t\t& Vibrating string\t\t\t\t\t\t\t\t\t\t\t\t\t\t& Osc. frequency \t& 2183 & 1787 & 16.67\\,Mhz & 183 cycles & 2.5\\,mW & 1.3\\,mW\\\\\n\\textbf{Warm vibrating string}\t& Vibrating string\twith temperature dependence\t\t\t\t\t\t\t& Osc. frequency \t& 3137 & 2718 & 16.77\\,Mhz & 269 cycles & 1.9\\,mW & 1.0\\,mW\\\\\n\\textbf{Spring-mass system}\t\t& Vertical spring with attached mass \t\t\t\t\t\t\t\t\t& Spring constant \t& 1419 & 1240 & 16.67\\,Mhz & 115 cycles & 3.4\\,mW & 1.8\\,mW\\\\\n                \\bottomrule\n        \\end{tabular}\n\\end{table*}\n\n\\subsection{Results}\nWe evaluated dimensional circuit synthesis on seven different\nphysical systems described in the Newton specification language.\nTable~\\ref{table:results} provides a brief description of the inputs\nas well as a summary of the measurement results.  The table also\nincludes the target parameter for each respective execution of the\nNewton compiler.  For example, for the physical description of the\npendulum the target parameter was its oscillation period, while for\nthe physical descriptions of a fluid in pipe the target parameter\nwas the velocity of the fluid.  This value of this parameter is\ninferred at run-time by the machine learning model that is fed with\nthe output of the $\\Pi$ computation.  The results in\nTable~\\ref{table:results} show the FPGA resource utilization as\nwell as resource utilization when mapped to CMOS gates, of each\ngenerated $\\Pi$ computation module, including the fixed point\narithmetic modules that implement the required arithmetic operations.\n\nThe execution latency column lists the cycles required for\ncompleting the calculation of each of the generated RTL modules.\nWe obtained the number of cycles by simulating the execution of the\nRTL modules for pseudorandom inputs generated by an LFSR. In each\nRTL module, the calculation of different $\\Pi$ products is parallelized\nbut the required operations per $\\Pi$ product are executed serially.\nAs a result, designs with larger resource usage in the table, such\nas the hardware for the unpowered flight model, conclude faster\nthan smaller designs such as the static pendulum. All modules require\nless than 300 cycles. As a result, for both 6 and 12\\,Mhz clocks,\nthe generated hardware can handle sample rates of over 10k\nsamples\/second, permitting real-time operation.\n\nThe last column of Table~\\ref{table:results} shows the measured\npower dissipation of each design running in the iCE40 FPGA. In all\ncases, the power dissipation is less than 6\\,mW and as low as 1\\,mW,\ndemonstrating the suitability of our method for small-factor,\nbattery-operated on-device inference at the edge.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany different techniques have been proposed\nfor the detection of dark matter particles which could \nmake up the halo of our Galaxy. Among the different\npossibilities, the detection of a neutrino flux by\nmeans of neutrino telescopes represents certainly \nan interesting tool, which is already at the level of\nimposing some (mild) constraint on the particle physics\nproperties of the neutralino, the most interesting and\nstudied dark matter candidate \\cite{refer}. This\nparticle is present in all the supersymmetric extensions\nof the standard model as a linear combination\nof the superpartners of the neutral gauge and higgs fields.\nIn the present paper we will perform our calculations\nin the minimal supersymmetric extension of the standard\nmodel (MSSM), for a definition of which we refer to Ref.\\cite{ICTP}\nand to the references therein quoted. \n\n\n\\section{Up--going muons from neutralino annihilation in the Earth}\nThe neutrino flux has origin from neutralino pair annihilation inside \nthe Earth where these dark matter particles \ncan be accumulated after having been captured by gravitational \ntrapping. The differential flux is \n\\begin{equation}\n\\Phi^0_{\\stackrel{(-)}{\\nu_\\mu}} (E_\\nu) \\equiv \n\\frac{dN_{\\stackrel{(-)}{\\nu_\\mu}}}{dE_\\nu} =\n\\frac{\\Gamma_A}{4\\pi d^2} \\sum_{F,f}\nB^{(F)}_{\\chi f}\\frac{dN_{f {\\stackrel{(-)}{\\nu_\\mu}}}}{dE_\\nu} \\, ,  \n\\label{eq:fluxnu}\n\\end{equation}\nwhere $\\Gamma_A$ denotes the annihilation rate,\n$d$ is the distance of the detector from the source (i.e. the\ncenter of the Earth), $F$ lists the \nneutralino pair annihilation final states,\n$B^{(F)}_{\\chi f}$ denotes the branching ratios into\nheavy quarks, $\\tau$ lepton and gluons in the channel $F$.\nThe spectra $dN_{f {\\stackrel{(-)}{\\nu_\\mu}} }\/dE_{\\nu}$ \nare the differential distributions of the (anti) neutrinos generated \nby the $\\tau$ and by hadronization of quarks\nand gluons and the subsequent semileptonic decays of the\nproduced hadrons. For details, see for instance\nRefs. \\cite{ICTP,noi_nuflux,altri_nuflux}. Here we only\nrecall that the annihilation rate depends, through\nits relation with the capture rate of neutralinos in the Earth,\non some astrophysical parameters, the most relevant of which\nis the local density $\\rho_l$.\n\nThe neutrino flux is produced in the inner part of the Earth \\cite{ICTP}\nand propagates toward a detector\nwhere it can be detected as a flux of up--going muons,\nas a consequence of neutrino--muon conversion inside\nthe rock that surrounds the detector. A double differential\nmuon flux can be defined as\n\\begin{eqnarray}\n& & \\frac{d^2 N_\\mu}{d E_\\mu d E_\\nu} = \\\\\n& & N_A \\int_0^\\infty dX \\int_{E_\\mu}^{E_\\nu} d E'_\\mu \ng(E_\\mu, E'_\\mu; X) \\; S(E_\\nu, E_\\mu) \\nonumber \\, ,\n\\end{eqnarray}\nwhere $N_A$ is the Avogadro's number, \n$g(E_\\mu,E'_\\mu; X)$ is the survival\nprobability that a muon of initial energy $E'_\\mu$\nwill have a final energy $E_\\mu$ after propagating\nalong a distance $X$ inside the rock  and\n\\begin{equation}\nS(E_\\nu, E_\\mu) = \\sum_i\n      \\Phi_i(E_\\nu) \\frac{d \\sigma_i(E_\\nu,E'_\\mu)}{d E'_\\mu}\n\\end{equation}\nwhere $i =  \\nu_\\mu, \\bar\\nu_\\mu$\nand $d \\sigma_{{\\stackrel{(-)}{\\nu_\\mu}}} (E_\\nu,E'_\\mu) \/ d E'_\\mu$ is\nthe charged current cross--section for the\nproduction of a muon of energy $ E'_\\mu$ from \na neutrino (antineutrino) of energy $E_\\nu$.\n\n\\FIGURE[t]{\n\\epsfig{figure=yieldx.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{Muon response function $d N_\\mu \/ d\\log x$ vs.\nthe parent neutrino fractional energy $x = E_\\nu \/ m_\\chi$ for\nneutralino annihilation in the Earth. Different curves refer to\ndifferent neutralino masses : $m_\\chi = 50$ GeV (solid), \n$m_\\chi = 80$ GeV (dotted), $m_\\chi = 120$ GeV (shot--dashed), \n$m_\\chi = 200$ GeV (long--dash), $m_\\chi = 500$ GeV (dot--dashed).}\n}\n\n\n\nA useful quantity for the discussion in the following Sections is\nthe muon response function\n\\begin{equation}\n\\frac{d N_\\mu}{d E_\\nu} = \\int_{E^{\\mathrm{th}}}^{E_\\nu}\nd E_\\mu \\; \\frac{d^2 N_\\mu}{d E_\\mu d E_\\nu}\n\\end{equation}\nwhere $E^{\\mathrm{th}}$ is minimal energy for detection\nof up--going muons. For SuperKamiokande and MACRO, \n$E^{\\mathrm{th}} \\simeq 1.5$ GeV \\cite{oscill_exp}. \nThe muon response\nfunction indicates the neutrino energy range\nthat is mostly responsile for the up--going muon signal.\nFig. 1 shows a few examples of it, plotted as \nfunctions of the variable $x = E_\\nu\/m_\\chi$, where\n$m_\\chi$ denotes the neutralino mass. Fig. 1 shows that\nthe maximum of the muon reponse happens for neutrino\nenergies of about $E_\\nu \\simeq (0.4 - 0.6) \\; m_\\chi$, with\na half width which extends from $E_\\nu \\simeq 0.1\\; m_\\chi$\nto $E_\\nu \\simeq 0.8 \\; m_\\chi$.  \n\nFinally, the total flux of up--going muons is defined as\n\\begin{equation}\n\\Phi_\\mu = \\int_{E^{\\mathrm th}}^{m_\\chi}\nd E_\\nu \\; \\frac{d N_\\mu}{d E_\\nu}\n\\end{equation}\n\n\n\\FIGURE[t]{\n\\epsfig{figure=flux_earth.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{Flux of up--going muons $\\Phi_\\mu^{\\mathrm{Earth}}$ from\nneutralino annihilation in the Earth, plotted as a function of\n$m_\\chi$. The solid line denotes the present upper limit \\cite{MACRO}.\nDifferent neutralino compositions are shown with different symbols:\ncrosses for gauginos, open circles for higgsinos and dots for mixed\nneutralinos.}\n}\n\n\nThe natural background for these kind of\nsearches is represented by the flux of up--going\nmuons originated by the atmospheric neutrino flux.\nExperimentally one searches, inside a small angular \ncone around the center of the Earth,\nfor a statistically significant up--going\nmuon excess over the muons of atmospheric $\\nu_\\mu$ origin.\nNo excess has been found so far and therefore, an upper\nlimit on $\\Phi_\\mu$ can be derived. Fig. 2 shows the\npresent most stringent upper limit obtained by the \nMACRO Collaboration \\cite{MACRO}. In the same figure \nthe theoretical calculations of $\\nu_\\mu$ for a scan\nof the supersymmetric parameter space are also displayed.\nThe plot refers to  $\\rho_l = 0.3$ GeV cm$^{-3}$\nand is obtained by a variation of the MSSM parameters  \nin the following ranges:\n$20\\;\\mbox{GeV} \\leq M_2 \\leq  500\\;\\mbox{GeV}$,  \n$20\\;\\mbox{GeV} \\leq |\\mu| \\leq  500\\;\\mbox{GeV}$,\n$80\\;\\mbox{GeV} \\leq m_A \\leq  1000\\;\\mbox{GeV}$,\n$100\\;\\mbox{GeV} \\leq m_0 \\leq  1000\\;\\mbox{GeV}$,\n$-3 \\leq {\\rm A} \\leq +3,\\; 1 \\leq \\tan \\beta \\leq 50$.\nFor further details\nof the calculation, we refer to Ref. \\cite{ICTP}. The comparison\nof the scatter plot with the experimental upper limit would\nimply that a fraction of the supersymmetric configuration\ncould be excluded. However, a variation of the value\nof $\\rho_l$ inside its range of uncertainty can lower the\ntheoretical prediction by about a factor of 3 \\cite{ICTP}. As a consequence,\nwe can conservatively consider that only a small fraction of the susy \nconfigurations can be potentially in conflict with the experimental\nupper limit, when no oscillation effect on the neutrino signal\nis assumed.\n\n\n\n\n\\section{Neutrino oscillation effect on the up--going muon signal}\n\nThe recent data on the atmospheric neutrino deficit indicate that\nthe $\\nu_\\mu$ may oscillate, either into $\\nu_\\tau$ or into\na sterire neutrino $\\nu_s$ \\cite{oscill_exp,oscill_the}. If this\nis the case, also the  $\\nu_\\mu$ produced by neutralino annihilations\nwould undergo an oscillation process. The energies involved in both\natmospheric and neutralino--produced neutrinos are the same. \nThe baseline of oscillation of the two neutrino components is\ndifferent, since atmospheric neutrinos cross the entire Earth,\nwhile neutrinos produced by  neutralino annihilation travel\nfrom the central part of the Earth to the detector\n(we recall once more that neutralinos annihilate in the core of the\nEarth). On the basis of the features of the $\\nu_\\mu$  oscillation\nwhich are required to fit the experimental data on atmospheric\nneutrinos \\cite{oscill_exp,oscill_the}, we expect that also\nthe neutrino flux from dark matter annihilation would be\naffected. In the next Sections we will explicitely discuss\nthe  $\\nu_\\mu \\rightarrow \\nu_\\tau$ and the\n$\\nu_\\mu \\rightarrow \\nu_s$ cases, in a two neutrino mixing \nscenario \\cite{Ellis}.\n\n\n\n\\subsection{$\\nu_\\mu \\rightarrow \\nu_\\tau$ vacuum oscillation}\n\\FIGURE[t]{\n\\epsfig{figure=psurv_vacuum.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{$\\nu_\\mu$ survival probability in the case of\n$\\nu_\\mu \\rightarrow \\nu_\\tau$ oscillation. The solid line refers to\n$\\sin^2 (2\\theta_v) = 1$, the dashed line is for $\\sin^2 (2\\theta_v) = 0.8$.\nIn both cases, $\\Delta m^2 = 5\\cdot 10^{-3}$ eV $^{-2}$.}\n}\n\n\\FIGURE[t]{\n\\epsfig{figure=flux_vacuum.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{Scatter plot of the ratio \n$(\\Phi_\\mu)^{\\rm VAC}_{\\rm oscill}\/\\Phi_\\mu$ vs. the neutralino\nmass $m_\\chi$. $(\\Phi_\\mu)^{\\rm VAC}_{\\rm oscill}$ is the up--going\nmuon flux in the case of $\\nu_\\mu \\rightarrow \\nu_\\tau$ oscillation,\nwhile $\\Phi_\\mu$ is the corresponding flux in the case of no oscillation.}\n}\n\nIn the case of $\\nu_\\mu \\rightarrow \\nu_\\tau$ oscillation,\nthe  $\\nu_\\mu$ flux is reduced because of oscillation, but \nwe have to take into account also that neutralino annihilation \ncan produce $\\nu_\\tau$ which in turn can oscillate into\n $\\nu_\\mu$ and contribute to the up--going muon flux.\nThe $\\nu_\\tau$ flux can be calculated as discussed in Sect. 1\nfor the $\\nu_\\tau$ flux, and it turns out to be always\na relatively small fraction of the  $\\nu_\\mu$ flux.\nThe muon neutrino flux at the conversion region can therefore\nbe expressed as\n\\begin{eqnarray}\n\\Phi_{{\\stackrel{(-)}{\\nu_\\mu}}} (E_\\nu) &=& \n\\Phi^0_{{\\stackrel{(-)}{\\nu_\\mu}}}\\; \nP^{\\mathrm{vac}} ({{\\stackrel{(-)}{\\nu_\\mu}}} \\rightarrow\n{{\\stackrel{(-)}{\\nu_\\mu}}}) \\nonumber \\\\\n& + & \n\\Phi^0_{{\\stackrel{(-)}{\\nu_\\tau}}}\\; \nP^{\\mathrm{vac}} ({{\\stackrel{(-)}{\\nu_\\tau}}} \\rightarrow\n{{\\stackrel{(-)}{\\nu_\\mu}}})\n\\end{eqnarray}\nwhere the vacuum survival probability is\n\\begin{eqnarray}\n& & P^{\\mathrm{vac}} ({{\\stackrel{(-)}{\\nu_\\mu}}} \\rightarrow\n{{\\stackrel{(-)}{\\nu_\\mu}}})\n\\;\\; = \\;\\; \\\\  \n& & 1 - \\sin^2(2\\theta_v)\\sin^2\n\\left (\n\\frac{1.27 \\Delta m^2 (\\mathrm{eV}^2) R(\\mathrm{Km})}\n{E_\\nu (\\mathrm{GeV})} \\nonumber\n\\right )\n\\label{vac}\n\\end{eqnarray}\nwhere $\\Delta m^2$ is the mass square difference of the\ntwo neutrino mass eigenstates, $\\theta_v$ is the mixing angle\nin vacuum and $R$ is the Earth's radius. \nFig. 3 shows the survival probability for two different values of\nthe neutrino oscillation parameters. Smaller (larger) values of \n$\\Delta m^2$ have the effect of shifting the curves to the left (right).\nComparing Fig. 1 with Fig. 3, we notice that the reduction of the\nup--going muon flux is stronger when there is matching between the\nthe energy \n$E_\\nu^1 \\simeq 5.2 \\cdot 10^{-3} \\Delta m^2 (\\mathrm{eV}^2)$ \nof the first (from the right) minimum of the \nsurvival probability and the energy \n$E_\\nu \\simeq 0.5 m_\\chi$ \nwhich is responsible for most of the muon response in the detector.\nThis implies that a maximum reduction of the signal could\noccur for neutralino masses of the order of \n$m_\\chi (\\mathrm{GeV}) \\simeq 10^4 \\Delta m^2 (\\mathrm{eV}^2)$.\nThe $\\nu_\\tau \\rightarrow \\nu_\\mu$ oscillation makes the\nreduction of the muon flux less severe, but it is not able\nto completely balance the reduction effect because the\noriginal $\\nu_\\tau$ flux at the source is sizeably smaller than\nthe $\\nu_\\tau$ flux. Therefore, the overall effect of\nthe neutrino oscillation is to reduce the up--going muon\nsignal. This effect is summarized in Fig. 4, where the\nratio between the up--going muon signals in the presence\nand in the absence of oscillation are plotted as a function\nof the neutralino mass. The susy parameter space has been\nvaried in the same ranges quoted for Fig. 2. We notice\nthat the strongest effect is present for light neutralinos,\nsince in this case the muon flux is mostly produced from\nneutrinos whose energy is in the range of maximal suppression\nfor the oscillation phenomenon. The effect is between\n0.5 and 0.8 for $m_\\chi \\lsim 100$ GeV.\nOn the contrary, the fluxes for larger masses are less \naffected, and the reduction is less than about \n20\\% for  $m_\\chi \\gsim 200$ GeV.\n\n\n\\subsection{$\\nu_\\mu \\rightarrow \\nu_s$ matter oscillation}\n\\FIGURE[t]{\n\\epsfig{figure=psurv_matter.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{$\\nu_\\mu$ survival probability in the case of\n$\\nu_\\mu \\rightarrow \\nu_s$ oscillation, for $\\sin^2 (2\\theta_v) = 0.8$\nand $\\Delta m^2 = 5\\cdot 10^{-3}$ eV $^{-2}$. The solid line refers to\nneutrinos, the dashed line is for antineutrinos.}\n}\n\n\n\\FIGURE[t]{\n\\epsfig{figure=flux_matter.ps,width=6.5cm,bbllx=50bp,bblly=200bp,bburx=520bp,bbury=650bp,clip=}\n\\caption{Scatter plot of the ratio \n$(\\Phi_\\mu)^{\\rm MAT}_{\\rm oscill}\/\\Phi_\\mu$ vs. the neutralino\nmass $m_\\chi$. $(\\Phi_\\mu)^{\\rm MAT}_{\\rm oscill}$ is the up--going\nmuon flux in the case of $\\nu_\\mu \\rightarrow \\nu_s$ oscillation,\nwhile $\\Phi_\\mu$ is the corresponding flux in the case of no oscillation.}\n}\n\nIn the case of $\\nu_\\mu \\rightarrow \\nu_s$ oscillation, \nthe neutrino flux is simply\n\\begin{equation}\n\\Phi_{{\\stackrel{(-)}{\\nu_\\mu}}} (E_\\nu) = \n\\Phi^0_{{\\stackrel{(-)}{\\nu_\\mu}}}\\; \nP^{\\mathrm{mat}} ({{\\stackrel{(-)}{\\nu_\\mu}}} \\rightarrow\n{{\\stackrel{(-)}{\\nu_\\mu}}})\n\\end{equation}\nand no $\\nu_\\mu$ regeneration is possible from the\nsterile neutrino. In this case, the effective potential\nof $\\nu_\\mu$ and $\\nu_s$ inside the Earth are different\nand we have to solve the evolution equation for propagation\nin the core and in the mantle. Neutrinos\n(produced in the center of the Earth) cross once half\nof the core and once the mantle. By considering both\ncore and mantle as of constant density, we can express the\nsurvival probability as \\cite{akhmedov, Kim}\n\\begin{eqnarray}\n& & P^{\\mathrm{mat}} ({{\\stackrel{(-)}{\\nu_\\mu}}} \\rightarrow\n{{\\stackrel{(-)}{\\nu_\\mu}}}) \\,\\,=\\,\\,  \\\\\n& & \\left [\nU(\\theta_c) D(\\phi_c) U^\\dagger (\\theta_c - \\theta_m)\n D(\\phi_m) U^\\dagger (\\theta_m)\n\\right ]_{\\mu\\mu} \\nonumber\n\\label{mat}\n\\end{eqnarray}\nwhere $U$ is the $2 \\times 2$\nneutrino mixing matrix, $\\theta_a$ \n($a=c,m$ for core and mantle, respectively) are \nthe effective mixing angles in matter and they are related\nto the vacuum mixing angle $\\theta_v$ as\n\\begin{equation}\n\\sin^2 (2\\theta_a) = \\frac{\\sin^2 (2\\theta_v) \\xi_a^2}\n{\\left [\n       {(\\xi_a \\cos(2\\theta_v) + 1)^{2} +\n\\xi_a^2 \\sin^2 (2\\theta_v)}\n\\right ]}\n\\end{equation}\nwith $\\xi_a = \\Delta m^2 \/ (2E_\\nu V_a)$;\n$V_a = \\pm G_F N_n^a \/ \\sqrt{2}$ is the matter \npotential in a medium of number density $N_n^a$\nfor neutrinos ($+$) and antineutrinos ($-$).\nIn Eq.(3.4), $D$ is the evolution\nmatrix $D_{ij}(\\phi_a) = \\delta_{ij} d_j^a$,\nwhere $d_1^a = 1$, $d_2^a = \\exp(i \\phi_a)$ and\n\\begin{equation}\n\\phi_a = V_a R_a \n\\left [\n(\\xi_a \\cos(2\\theta_v) + 1)^2 +\n\\xi_a^2 \\sin^2 (2\\theta_v)\n\\right ]^{1\/2}\n\\end{equation}\n\nIn Fig. 5 an example of the $\\nu_\\mu$ and $\\bar \\nu_\\mu$  \nsurvival probability is given for representative\nvalues in the range allowed by the fits on the\natmospheric neutrino data \\cite{oscill_exp,oscill_the}:\n$\\Delta m^2 = 5\\cdot 10^{-3}$ eV$^2$\nand  $\\sin^2 (2\\theta_v) = 0.8$. {}From Fig. 5\nand the previous discussion relative to Fig. 3, we \nexpect that in the case of  $\\nu_\\mu \\rightarrow \\nu_s$\nthe reduction of the muon signal is significantly\nless severe than in the case of   $\\nu_\\mu \\rightarrow \\nu_\\tau$.\nIn fact, in these case the minima of the survival probability\noccur for lower neutrino energies, and threfore the\noscillation can affect only muon fluxes originated by very\nlight neutralinos. This is manifest in Fig. 6, where\nthe ratio of the up--going muon fluxes in presence and\nabsence of oscillation are shown. In this case, the reduction \nof the signal is always less than 30\\%. This maximal reduction\noccurs for neutralino masses lower that about 80 GeV. For\nlarger masses, the up--going muon flux is almost unaffected.\n\n\n\\section{Conclusions}\nWe have discussed the effect on the up--going muon\nsignal from neutralino annihilation in the Earth, in the\ncase that the $\\nu_\\mu$ flux produced by neutralinos would oscillate\nas indicated by the data on the atmospherice neutrino deficit.\nWhile the experimental upper limit is, at present, practically\nnot affected by the possibility of neutrino \noscillation \\cite{MACRO}, the\ntheoretical predictions are reduced in the presence of\noscillation. With the oscillation parameters deduced\n{}from the fits on the atmospheric neutrino data,\nthe effect is always larger for lighter neutralinos.\nIn the case of $\\nu_\\mu \\rightarrow \\nu_\\tau$\nthe reduction is between 0.5 and 0.8 for \n$m_\\chi \\lsim 100$ GeV and less than about\n20\\% for  $m_\\chi \\gsim 200$ GeV.\nIn the case of  $\\nu_\\mu \\rightarrow \\nu_s$,\nthe reduction of the signal is up to 30\\% for neutralino \nmasses lower that about 80 GeV and smaller than 10\\% for\nheavier neutralinos.\n\n\n\\acknowledgments\nI wish to thank Sandro Bottino for very stimulating and\ninteresting discussions about the topic of this paper.\nThis work was supported by DGICYT under grant number \nPB95--1077 and by the TMR network grant ERBFMRXCT960090 of \nthe European Union.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nOur understanding of star formation has grown primarily from the study of local regions forming low-mass stars in relative isolation.  Nearby regions such as Taurus \\citep[e.g.][]{Goldsmith_Taurus}, $\\rho$ Ophiuchus \\citep[e.g.][]{Young_oph, Johnstone_Oph}, Perseus \\citep[e.g.][]{Enoch2008, Jorgensen_perseus,Kirk_perseus}, Serpens \\citep[e.g.][]{ts98, Harvey_serpensIRAC, Enoch2008}, the Pipe nebula \\citep[e.g.][]{Lombardi_pipe, Muench_pipe}, and Orion \\citep[e.g.][]{Li_orion, Johnstone_OrionB} have been studied to great lengths using a variety of techniques including dust emission, extinction mapping, and molecular line emission.  This past decade of research has shown that star formation regions are assembled hierarchically.  Within molecular clouds (tens of parsecs in size, containing 10$^4$-10$^5~M_{\\sun}$), we adopt the nomenclature used by \\citet{BerginTafalla_ARAA2007} distinguishing ``clouds'' (10$^3-10^4~M_{\\sun}$, 10$^0-10^1$~pc), ``clumps'' (10-10$^3~M_{\\sun}$, 10$^{-1}$-10$^0$~pc), and ``cores'' (10$^{-1}$-10$^1$~$M_{\\sun}$, 10$^{-2}$-10$^{-1}$~pc).\n\nIn studies of nearby regions, it is possible to resolve pre-stellar cores with single dish observations \\citep[e.g.][]{Johnstone_Oph}.  This permits the examination of the properties of the fragmentation of the natal molecular clouds into smaller components.  It is then straightforward to construct a mass function of cores and a mass function for individual cores can then be constructed.  The core mass distributions typically derived from dust emission studies are found to be strikingly similar to the mass spectra of stars, implying that the masses of stars are a direct result of the way in which the natal molecular cloud fragments.  In contrast, when CO line emission is used as a mass probe for cores \\citep{Kramer_CMF}, a more top-heavy distribution results.  \n\nWhile these studies have brought us a deep understanding of isolated low-mass star formation, this is not the complete picture for star formation in the Galaxy.  The necessary ingredient for star formation is dense molecular gas.  However, the H$_2$ distribution in the Milky Way is not uniform.  The primary reservoir is in the Molecular Ring, which resides at 4~kpc from the Galactic Center and contains $\\sim$70\\% of the molecular gas inside the solar circle \\citep{Jackson_GRS}.  Thus, the Molecular Ring is the heart of Galactic star formation.  Indeed, as \\citet{Robinson1984} show, the peak of Galactic far-infrared emission originates from this region.\n\nIn local clouds, most of the recent progress in our understanding of the early stages of star formation have come from studies of pre-stellar objects.  Populations of starless cores have been identified in numerous local regions, and they are universally cold and quiescent, exhibiting thermal line widths \\citep{BerginTafalla_ARAA2007}.  Recently, a population of cold, dense molecular clouds within the Molecular Ring were detected against the bright Galactic mid-infrared background \\citep[from 7 to 25~$\\mu$m;][]{egan_msx, carey_msx}.  These clouds are opaque to mid-infrared radiation and show little or no typical signs of star formation, such as association with IRAS point sources.  Initial studies demonstrated that these objects, termed infrared-dark clouds (IRDCs) are dense (n(H$_2$) $> 10^5$ cm$^{-3}$), cold (T $<$ 20K) concentrations of 10$^3$ - 10$^5~M_{\\sun}$ of molecular gas.  Based upon the available mass for star formation, infrared-dark clouds are likely the sites of massive star formation.\n\nSince their discovery, further studies of infrared-dark clouds have established their place as the precursors to clusters.  A number of studies have detected the presence of deeply embedded massive protostars using sub-millimeter probes \\citep{Beuther_protostars_IRDC, Rathborne_2007_protostars, Pillai_G11}, which confirms that IRDCs are the birth-sites of massive stars.  Detailed molecular surveys show that molecules such as NH$_3$ and N$_2$H$^+$ trace the dense gas extremely well \\citep{ragan_msxsurv, Pillai_ammonia}, as seen in local dense prestellar cores \\citep{Bergin2002}.  Furthermore, the molecular emission corresponding to the absorbing structure of infrared-dark clouds universally exhibit non-thermal linewidths on par with massive star formation regions.  Other studies have uncovered the presence of masers \\citep{Beuther2002_masers, Wang_ammonia} and outflows \\citep{Beuther_IRDCoutflow}, known indicators of ongoing embedded star formation.  Already, the evidence shows that these are the sites where massive stars and star clusters will form or are already forming.  In order to understand massive star formation, and thus Galactic star formation, it is crucial to understand the structure and evolution of IRDCs.\n\nStudies of infrared-dark clouds to date have left the fundamental properties of cloud fragmentation go relatively unexplored.  \\citet{rathborne2006} showed that IRDCs exhibit structure with median size of $\\sim$0.5~pc, but observations of IRDCs with the Spitzer Space Telescope, which we describe in $\\S$\\ref{obs}, reveal that there exists structure well below this level.  We characterize the environment in $\\S$\\ref{env} and highly structured nature of infrared-dark clouds in $\\S$\\ref{clumps} by utilizing the high-resolution imaging capabilities of the Spitzer.  In $\\S$\\ref{mf}, we analyze the IRDC absorbing structure, derive the clump mass function, and put the results in the context of previous studies.  We find the mass function to be shallower than Salpeter initial mass function (IMF) \\citep{Salpeter_imf} and more closely aligned with that observed using CO in massive star forming regions.  Given the strong evidence for fragmentation and star formation characteristics of these objects, we suggest they are in the initial stages of fragmentation.  The conclusions as well as the broad impact of these results are discussed in $\\S$\\ref{conclusion}. The results of this study provide an important foundation for further studies of IRDCs with the instruments of the future, allowing us to probe the dominant mode of star formation in the Galaxy, which may be fundamentally different from the processes that govern local star formation.\n\n\n\\section{Observations \\& Data Reduction}\n\\label{obs}\n\n\\subsection{Targets}\n\nSearching in the vicinity of ultra-compact HII (UCHII) regions \\citep{wc89} for infrared-dark cloud candidates, \\citet{ragan_msxsurv} performed a survey of 114 candidates in N$_2$H$^+$(1-0), CS(2-1), and C$^{18}$O(1-0) with the FCRAO.  In order to study substructure with Spitzer, we have selected a sample of targets from the \\citet{ragan_msxsurv} sample  which are compact, typically $2\\arcmin \\times 2\\arcmin$ (or $2 \\times 2$~pc at 4~kpc), and opaque, providing the starkest contrast at 8~$\\mu$m (MSX Band A) with which to examine the absorbing structure.  The selected objects also exhibit significant emission in transitions of CS and N$_2$H$^+$ that are known to trace high-density gas, based on their high critical densities.  By selecting objects with strong emission in these lines, we ensure that their densities are  $>$10$^4$ cm$^{-3}$ and their temperatures are less than 20~K.  Under these conditions in local clouds, N$_2$H$^+$ is strongest when CO is depleted in the pre-stellar phase \\citep{bl97}, hence a high N$_2$H$^+$\/CO ratio guided our attempt to select the truly ``starless'' dark clouds in the IRDC sample.  Our selection criteria are aimed to isolate earliest stages of star formation in local clouds and give us the best hope of detecting massive starless objects. The eleven IRDCs observed are listed in Table~1 with the distances derived in \\citet{ragan_msxsurv} using a Milky Way rotation curve model \\citep{Fich:1989} assuming the ``near'' kinematic distance.  The listed uncertainties in Table~1 arise from the $\\pm$14\\% maximal deviation inherent in the rotation curve model. \n\n\\vspace{0.5in}\n\n\\subsection{Spitzer Observations \\& Data Processing}\n\nObservations of this sample of objects were made on 2005 May 7 -- 9 and September 15 -- 18 with IRAC centered on the coordinates listed in Table~1.  Each region was observed 10 times with slightly offset single points in the 12s high dynamic range mode.  All four IRAC bands were observed over 7$' \\times$ 7$'$ common field-of-view.  MIPS observations were obtained on 2005 April 7 -- 10 of the objects in this sample.  Using the ``large\" field size, each region was observed in 3 cycles for 3s at 24~$\\mu$m.  MIPS observations cover smaller 5.5$' \\times$ 5.5$'$ fields-of-view but big enough to contain the entire IRDC.  Figures~\\ref{fig:g0585}$-$\\ref{fig:g3744} show each IRDC field in all observed wavebands.  The absorbing structures of the IRDCs are most prominent at 8~$\\mu$m and 24~$\\mu$m.\n\nWe used IRAC images processed by the Spitzer Science Center (SSC) using pipeline version S14.0.0 to create basic calibrated data (BCD) images.  These calibrated data were corrected for bright source artifacts (``banding'', ``pulldown'', and ``muxbleed''), cleaned of cosmic ray hits, and made into mosaics using Gutermuth's WCS-based IRAC post-processing and mosaicking package \\citep[see][for further details]{Gutermuth_ngc1333}.  \n\nSource finding and aperture photometry were performed using Gutermuth's PhotVis version 1.10 \\citep{Gutermuth_ngc1333}.  We used a 2.4$\\arcsec$ aperture radius and a sky annulus from 2.4$\\arcsec$ to 6$\\arcsec$ for the IRAC photometry.  The photometric zero points for the [3.6], [4.5], [5.8], and [8.0] bands were 22.750, 21.995, 19.793, and 20.187 magnitudes, respectively.  For the MIPS 24~$\\mu$m photometry, we use a 7.6$\\arcsec$ aperture with 7.6$\\arcsec$ to 17.8$\\arcsec$ sky annuli radii and a photometric zero point of 15.646 magnitude.  All photometric zero points are calibrated for image units of DN and are corrected for the adopted apertures.\n\nTo supplement the Spitzer photometry, we incorporate the source photometry from the Two-Micron All Sky Survey (2MASS) Point Source Catalog (PSC).  Source lists are matched for a final catalog by first matching the four IRAC band catalogs using Gutermuth's WCSphotmatch utility, enforcing a 1$\\arcsec$ maximal tolerance for positive matches.  Then, the 2MASS sources are matched with tolerance 1$\\arcsec$ to the mean positions from the first catalog using the same WCS-based utility.  Finally, the MIPS 24~$\\mu$m catalog is integrated with matching tolerance 1.5$\\arcsec$.\n\n\\section{Stellar Content \\& IRDC Environment}\n\\label{env}\n\nThe tremendous sensitivity of Spitzer has given us the first abilty to characterize young stellar populations in detail.  Before the Spitzer era, IRAS led the effort in identifying the brightest infrared point sources in the Galaxy.  Only one object in this sample, G034.74$-$0.12 (Figure~\\ref{fig:g3474}) has an IRAS point source (18526+0130) in the vicinity.  Here, with Spitzer, we have identified tens of young stellar objects (YSOs) in the field of each IRDC.\n\n\\vspace{-0.05in}\n\n\\subsection{Young Stellar Object Identification \\& Classification}\n\nWith this broad spectral coverage from 2MASS to IRAC to MIPS, we apply the robust critieria described in \\citet{Gutermuth_ngc1333} to identify young stellar objects (YSOs) and classify them.  Table~2 lists the J, H, K$_s$, 3.6, 4.5, 5.8, 8.0 and 24~$\\mu$m photometry for all stars that met the YSO criteria, and we note the classification as Class I (CI), Class II (CII), embedded protostars (EP), or transition disk objects (TD).  A color-color diagram displaying these various classes of YSOs \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g0585}\n\\end{center}\n\\caption{G005.85$-$0.23: {\\it Top Row Right: 3.6$\\micron$.  Middle Row Left: 4.5$\\micron$.  Middle Row Right: 5.8$\\micron$.  Bottom Row Left: 8$\\micron$.  Bottom Row Right: 24$\\micron$.}}\n\\label{fig:g0585}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g0626}\n\\end{center}\n\\caption{G006.26$-$0.51: Wavelengths as noted in Figure~1.}\n\\label{fig:g0626}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g0916}\n\\end{center}\n\\caption{G009.16$+$0.06: Wavelengths as noted in Figure~1.}\n\\label{fig:g0916}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g0928}\n\\end{center}\n\\caption{G009.28$-$0.15: Wavelengths as noted in Figure~1.}\n\\label{fig:g0928}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g0986}\n\\end{center}\n\\caption{G009.86$-$0.04: Wavelengths as noted in Figure~1. - Embedded Objects (indices 6 and 7 in Table~2 under source G009.86$-$0.04) are labeled.  Source G009.86$-$0.04 index 6 is only detectable at 24~$\\mu$m and lies right at the heart of the dust absorption.}\n\\label{fig:g0986}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g1250}\n\\end{center}\n\\caption{G012.50$-$0.22: Wavelengths as noted in Figure~1. - Embedded Object (index 5 in Table~2 under source G012.50-0.22) is labeled.}\n\\label{fig:g1250}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g2337}\n\\end{center}\n\\caption{G023.37$-$0.29: Wavelengths as noted in Figure~1. - Embedded Objects (indices 9 and 10 in Table~2 under source G023.37$-$0.29) are labeled.}\n\\label{fig:g2337}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g2348}\n\\end{center}\n\\caption{G023.48$-$0.53: Wavelengths as noted in Figure~1.}\n\\label{fig:g2348}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g2405}\n\\end{center}\n\\caption{G024.0$5-$0.22: Wavelengths as noted in Figure~1. - Embedded Object (index 1 in Table~2 under source G24.05$-$0.22) is labeled.}\n\\label{fig:g2405}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g3474}\n\\end{center}\n\\caption{G034.74$-$0.12: Wavelengths as noted in Figure~1. - Embedded Object (index 5 in Table~2 under source G034.74$-$0.12.) is labeled.}\n\\label{fig:g3474}\n\\end{figure}\n\n\\clearpage\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{g3744}\n\\end{center}\n\\caption{G037.44$+$0.14: Wavelengths as noted in Figure~1.}\n\\label{fig:g3744}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\hspace{-0.85in}\n\\includegraphics[scale=0.8]{f12}\n\\caption{\\footnotesize{IRAC four color plot for all objects in the IRDC sample for all objects with photometry in all four bands that had errors less than 0.2 magnitudes.  Class I protostars are marked with red squares, green circles mark the more-evolved Class II sources, and transition\/debris disk objects are marked with purple circles.  The deeply embedded objects identified with this analysis did not have sufficient detections in IRAC bands to appear on the color-color plots.  The extinction law from \\citet{Flaherty2007} indicated by the black arrow, and the extinction law from \\citet{Indebetouw2005} is plotted as the blue arrow.}}\n\\label{fig:colorcolor}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=270,scale=0.65]{f13}\n\\caption{\\footnotesize{FCRAO molecular line contours of N$_2$H$^+$~(1-0) (left), C$^{18}$~(1-0) (center) and $^{12}$CO~(1-0) (right) plotted over the {\\em Spitzer} 8~$\\mu$m image of G012.50$-$0.22.  The critical density of the molecular transition decreases from left to right.}}\n\\label{fig:g1250_env}\n\\end{center}\n\\end{figure*}\n\n\n\\noindent in the entire sample is shown in Figure~\\ref{fig:colorcolor}\\footnote{No embedded protostar was detected in all four IRAC bands, so none are plotted in Figure~\\ref{fig:colorcolor}.}.  The extinction laws from both \\citet{Flaherty2007} and \\citet{Indebetouw2005} are plotted to show the effect of five magnitudes of visual extinction.  The objects associated with these IRDCs are a great distance from us and in the plane of the Galaxy, so they naturally suffer from a great deal of extinction, reddening, and foreground contamination.  Furthermore, the reddening law used in this classification scheme and the measures taken to extricate extragalactic contaminants may be inaccurate due to the great distance to IRDCs, as the criteria were originally designed to suit local regions.  This may result in misclassification of sources. For example, a highly reddened Class I object might appear as an embedded protostar.  Nonetheless, if these objects are indeed protostars, it is likely that they are associated with the IRDC.  \n\n\nIn Table~3, we summarize the number of each class of YSO in each IRDC field.  We note the number of these YSOs that are spatially coincident with the absorbing IRDC clumps (see $\\S$\\ref{structure}).  Only $\\sim$10\\% of the YSOs are associated with the dense gas.  The rest appear to be a distributed population of stars surrounding the IRDC.  This may be because any star directly associated with the IRDC is too heavily obscured to be detected even with the deep Spitzer observations we undertook.  Our observations are sensitive to 1-3~$M_{\\sun}$, 1Myr-old pre-main sequence stars \\citep{Baraffe1998}, or 1~$L_{\\sun}$ Class 0 protostar at 4~kpc with no extinction \\citep{whitney_protostar}.  With extinction, which can reach 1-2 magnitudes in the {\\em Spitzer} bands, embedded YSOs up to 3-4~$M_{\\sun}$ might be present, but hidden from our view.  Another possible reason for the lack of YSOs detected coincident with the dense gas is that the IRDC itself could be in a stage prior to the onset of star formation, and the surrounding stars that are observed have disrupted their natal molecular gas.\n\nTable~4 lists all of the objects identified as embedded objects that are spatially coincident with an IRDC.  We list the flux density at each Spitzer wavelength and an estimate of the mid-infrared luminosity derived from integrating the spectral energy distribution, which is dominated by emission at 24~$\\mu$m.  In the likely event that the embedded objects are extincted, these mid-infrared luminosities will be underestimated.  Taking the average extinction estimations, which can be derived most reliably from the measurements of Class II objects, A$_K$ ranges from 1 to 3, which, if the extinction law \\citet{Flaherty2007} is applied, corresponds to A$_{24}$ of 0.5 to 1.6.  As a check, we use a second method to estimate the extinction: based on average values of the optical depth we measure in the IRDCs, we confirm that A$_{24}\\sim$1 is typical in these objects.  Given the uncertain extinction properties, and the fact that a large portion of these embedded sources' luminosity will emerge at longer wavelengths not observed here, the luminosities presented in Table~4 are lower limits.  Stars with luminosities in this range, according to \\citet{Robitaille2006}, arise from stars ranging from 0.1 to 2~$M_{\\sun}$, but are likely much greater.\n\n\\subsection{Nebulosity at 8 and 24~$\\mu$m}\n\nFour IRDCs in our sample (G006.26$-$0.51, Figure~\\ref{fig:g0626}; G009.16$+$0.06, Figure~\\ref{fig:g0916}; G023.37$-$0.29 Figure~\\ref{fig:g2337}; G034.74$-$0.12, Figure~\\ref{fig:g3474}) exhibit bright emission nebulosity in the IRDC field at 8 and 24~$\\mu$m.  These regions tend to be brightest in the thermal infrared (e.g. 24~$\\mu$m) but show some emission at 8~$\\mu$m, which suggests they are sites of high mass star cluster formation.  To test whether the apparent active star formation is associated with the IRDC in question, or if it is in the vicinity, we correlate each instance of a bright emission with the molecular observations of the object obtained by \\citet{ragan_msxsurv}.  The molecular observations provide velocity information which, due to Galactic rotation, aid in estimating the distance to the mid-infrared emission \\citep{Fich:1989}.  This distance compared with the distance to the IRDC enables us to discern whether the IRDC and young cluster are at the same distance or one is in the foreground or background.\n\nIn the case of G006.26$-$0.51 (Figure~\\ref{fig:g0626}), we detect infrared emission at 24~$\\mu$m east of the IRDC.  This is spatially coincident and has similar morphology to C$^{18}$O~(1-0) emission emitting at a characteristic velocity of 17~km~s$^{-1}$ \\citep{ragan_msxsurv}, corresponding to a distance of about 3$\\pm$0.5~kpc.  The IRDC has a velocity of 23~km~s$^{-1}$, which gives a distance of 3.8~kpc, but with an uncertainty of over 500~pc (see Table~1 and \\citet{ragan_msxsurv}).  Given the errors inherent in the distance derivation from the Galactic rotation curve, we cannot conclusively confirm or rule out association.  G009.16$+$0.06 (Figure~\\ref{fig:g0916}), has neither distinct velocity component evident in the molecular observations nor does the molecular emission associated with the IRDC overlap with the 24~$\\mu$m emission.  Embedded clusters should be associated with molecular emission especially C$^{18}$O which is included in the FCRAO survey.  Associated emission for this object likely lies outside the bandpass of the FCRAO observations and is at a greater or lesser distance than IRDC.  The 24~$\\mu$m image of G023.37$-$0.29 (Figure~\\ref{fig:g2337}) shows bright emission to the south of the IRDC and another region slightly south and west of the IRDC.  This emission is not prominent in the IRAC images, suggesting that this is potentially an embedded star cluster.  Molecular observations show strong emission peaks in both CS~(2-1) and N$_2$H$^+$~(1-0) in the vicinity of the IRAC 8~$\\mu$m and MIPS 24~$\\mu$m emission.  However, there are three distinct velocity components evident in the observed bandpass, none of which is more spatially coincident with the 24~$\\mu$m emission than the others.  Unfortunately, the spatial resolution of the FCRAO survey is insufficient for definitive correlation.  Finally, in G34.74$-$0.12 (Figure~\\ref{fig:g3744}), no molecular emission is distinctly associated with the nebulosity; the most likely scenario for this object is that the associated molecular emission lies outside the bandpass of the FCRAO observation and, therefore, is not associated.  \n\n\\subsection{Summary of Stellar Content and IRDC Environment}\n\nWe have characterized the star formation that is {\\em possibly} associated with the IRDCs to the extent that the Spitzer and millimeter data allow.  The YSO population is distributed, and only a handful of objects identified are directly spatially associated with the IRDC.  More explicitly, in this sample, half (5\/11) of the sample shows no clear evidence for {\\em embedded} sources in the dense absorbing gas, and instead appear populated sparsely with young protostars, the photometric properties of which are given in Table~2, and the overall IRDC star content is summarized in Table~3.  Among those embedded objects correlated with the absorbing structure at 8~$\\mu$m, which are summarized in Table~4, we find a marked lack of luminous sources ($>$5~$L_{\\sun}$) at these wavelengths.  There may be significant extinction at 24~$\\mu$m, in which case we would underestimate their luminosity.  Further, even in IRDCs with embedded protostars, most of the cloud core mass is not associated with an embedded source.  It is our contention that most of the IRDC mass does not harbor significant massive star formation, and, hence IRDCs are in an early phase of cloud evolution.  \n\nBright emission nebulosity is evident at 8~$\\mu$m and 24~$\\mu$m in four fields, presumably due to the presence of high mass stars or a cluster.  If the IRDC were associated with the nebulosity, it would be a strong indication that the IRDCs have massive star formation occurring already in the vicinity.  Molecular data give no definitive clues that these regions are associated with the IRDCs.  \n\nMost studies including this one focus primarily on the dense structures that comprise infrared-dark clouds, yet their connection to the surrounding environment has not yet been discussed in the literature.  While it is clear that some star formation is directly associated with the dense material, star formation is also occurring beyond the extent of the IRDC as it appears in absorption.  Figure~\\ref{fig:g1250_env} shows molecular line contours from \\citet{ragan_msxsurv} over the {\\em Spitzer} 8~$\\mu$m image.  N$_2$H$^+$, a molecule known to trace very dense gas, corresponds exclusively to the dark cloud.  On the other hand, C$^{18}$O and, to a greater extent $^{12}$CO, show a much more extended structure, which suggests that the infrared-dark cloud resides within a greater molecular cloud complex.  For all of the objects in our sample, the $^{12}$CO emission was present at the edge of the map (up to 2$'$ away from the central position), so it is likely that the emission, and therefore the more diffuse cloud that it probes, extends beyond the mapped area.  Thus, the full extent of the surrounding cloud is not probed by our data.\n\n\\section{Tracing mass with dust absorption at 8~$\\micron$}\n\\label{clumps}\n\nEach infrared-dark cloud features distinct absorbing structures evident at all Spitzer wavelengths, but they are most pronounced at 8~$\\mu$m and 24~$\\mu$m due to strong background emission from polycyclic aeromatic hydrocarbons (PAHs) and small dust grains in the respective bandpasses \\citep{Draine_dustreview}.  The IRDCs in this sample exhibit a range of morphologies and surrounding environments.  Figures~\\ref{fig:g0585}-\\ref{fig:g3744} shows a morphological mix of filamentary dark clouds (e.g. G037.44$+$0.14, Figure~\\ref{fig:g3744}) and large ``round'' concentrations (e.g. G006.26$-$0.51, Figure~\\ref{fig:g0626}). Remarkably, these detailed structures correspond almost identically between the 8~$\\mu$m and 24~$\\mu$m bands, despite the fact that the source of the background radiation arises from separate mechanisms.  At 8~$\\mu$m emission from PAHs dominate on average, and at 24~$\\mu$m, the bright background is due to the thermal emission of dust in the Galactic plane.  Considering this scenario, it is unlikely that we are mistaking random background fluctuations for dense, absorbing gas with the appropriate characteristics to give rise to massive star and cluster formation.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=270,scale=0.6]{f14}\n\\end{center}\n\\caption{\\footnotesize{{\\it Upper left:} Original IRAC 8~$\\mu$m image of G024.05-0.22.  {\\it Upper right:} Background model using the spatial median filtering technique with a 3$\\arcmin$ radius.  The dark cloud is virtually eliminated from the background, but still accounts for the large-scale variations. {\\it Lower left:}  Same as upper right panel, except that a 1$\\arcmin$ radius is used, which models the dark cloud as part of the background.  {\\it Lower right:} Same as upper right and lower left panels, except that a 5$\\arcmin$ radius is used, which misses the background variation and is almost a constant value.}}\n\\label{fig:bgmethod}\n\\end{figure*}\n\n\n\\subsection{Modeling the Foreground and Background}\n\\label{bg}\n\nIn the Galactic plane, the 8~$\\mu$m background emission varies on scales of a few arcminutes.  To accurately estimate structures seen in absorption, we account for these variations using a spatial median filtering technique, motivated by the methods used in \\citet{Simon2006}.  For each pixel in the IRAC image, we compute the median value of all pixels within a variable radius and assign that value to the corresponding pixel in the background model.  Figure~\\ref{fig:bgmethod} illustrates an example of several trials of this method, including models with 1$\\arcmin$, 3$\\arcmin$, and 5$\\arcmin$ radius of pixels included in a given pixel's median calculation.  We select the size of the filter to be as small as possible such that the resulting map shows no absorption as background features.  If the radius is too small, most of the included pixels will have low values with few representing the true background in the areas where absorption is concentrated (lower left panel of Figure~\\ref{fig:bgmethod}).  The background variations are also not well-represented if we select a radius too large (lower right panel of Figure~\\ref{fig:bgmethod}).  Based on our analysis, the best size for the filter is 3$\\arcmin$.  The observed 8~$\\mu$m emission is a combination of both background and foreground contributions.  \n\n\\begin{equation}\n\\int I^{estimate} d\\lambda = \\int I_{BG}^{true} d\\lambda + \\int I_{FG} d\\lambda\n\\label{eq1}\n\\end{equation}\n\n\\noindent where $\\int I^{estimate}~d\\lambda$ is the intensity that we measure from the method described above, $\\int I_{BG}^{true}~d\\lambda$ is the true background intensity, which can only be observed in conjunction with $\\int I_{FG}~d\\lambda$, the foreground intensity, all at 8~$\\mu$m.  The relative importance of the foreground emission is not well-known. For simplicity, we assume the foreground can be approximated by constant fraction, $x$, of the emission across each field.\n\n\\begin{equation}\n\\int I_{FG}~d\\lambda = x \\int~I_{BG}^{true}~d\\lambda\n\\end{equation}\n\nOne way to estimate the foreground contribution has already been demonstrated by \\citet{Johnstone_G11}.  The authors compare observations of IRDC G011.11$-$0.12 with the {\\em Midcourse Science Experiment (MSX)} at 8~$\\mu$m and the Submillimeter Common-User Bolometer Array (SCUBA) on the James Clerk Maxwell Telescope (JCMT) at 850~$\\mu$m (see their Figure~3) and use the point at which the 8~$\\mu$m integrated flux is at its lowest at high values of 850~$\\mu$m flux for the foreground estimate.  The top panel Figure~\\ref{fig:g1111_fluxplot} shows a similar plot to Figure~3 in \\citet{Johnstone_G11}, except our integrated 8~$\\mu$m flux is measured with {\\em Spitzer} and presented here in units of MJy\/sr.  SCUBA 850$\\mu$m data for two of the IRDCs in this sample (G009.86$-$0.04 and G012.50$-$0.22) are available as part of the legacy data release \\citep{SCUBA_legacy} and are included in this plot.  Just as \\citet{Johnstone_G11} point out, we see a clear trend: where 8~$\\mu$m emission is low along the filament, the 850$\\mu$m flux is at its highest.  In the case of G011.11$-$0.12, where the SCUBA data are of the highest quality, we take the minimum 8~$\\mu$m flux density to be an estimate of the foreground contribution.  Assuming this trend is valid for our sample of IRDCs, we use the 8~$\\mu$m emission value measured at the dust opacity peak in each source as our estimation of the foreground level for that object (for the remainder of this paper, we will refer to this method as foreground estimation method ``A'').  Given these considerations, we find values for $x$ to range between 2 and 5.  Up to 20\\% of this foreground contamination is likely due to scattered light in the detector (S.T. Megeath, private communication).  We assume constant foreground flux at this level.  As an alternative foreground estimate, we also test a case in which we attribute half of the model flux to the background and half to the foreground.  This is equivalent to choosing a value of $x$ of 1, and based on Figure~\\ref{fig:g1111_fluxplot}, is also a reasonable estimate.  This method will be referred to as foreground estimation method ``B.''  For most of the following figures and discussion, we use estimation method A and refer the results from method B in the text when applicable.\n\nWith an estimation of the foreground contribution, the absorption can be quantitatively linked to the optical depth of the cloud.  The measured integrated flux, $\\int I_m d\\lambda$ at any point in the image, including contributions from both the foreground and background, can then be expressed as \n\n\\begin{equation}\n\\int I_m d\\lambda =\\int I_{BG}^{true}e^{-\\tau_8} d\\lambda+ \\int I_{FG} d\\lambda\n\\end{equation}\n\n\\noindent where $\\tau_8$ is the optical depth of the absorbing material.  For the subsequent calculations, we use the average intensity, assuming uniform transmission over the IRAC channel 4 passband, and average over the extinction law \\citep[][see Section~\\ref{structure}]{weingartner_draine01} in this wavelength region in order to convert the optical depth into a column density (see discussion in next section).  We note that we make no attempt to correct for the spectral shape of the the dominant PAH emission feature in the 8~$\\mu$m Spitzer bandpass, which we assume dominates the background radiation.  In addition, clumpy material that may be optically thick and is not resolved by these observations will cause us to underestimate the column density.  These factors could introduce an uncertainty in the conversion of optical depth to column density.  Still, we will show in Section~\\ref{columnprobe} that dust models compare favorably to our estimation of the dust absorption cross section, lending credence to our use of $\\tau$ as a tracer of column density.\n\n\\begin{figure}\n\\hspace{-0.3in}\n\\includegraphics[scale=0.55]{f15}\n\\caption{\\footnotesize{Spitzer 8~$\\mu$m vs. SCUBA 850$\\mu$m flux for IRDC G011.11$-$0.12, G009.86$-$0.04, and G012.50$-$0.22.  The horizontal dashed line marks where the 8~$\\mu$m flux density reaches a minimum in G011.11$-$0.12, which is also indicated for the two other IRDCs with available SCUBA data.  This flux density serves as an estimate of the foreground emission at 8~$\\mu$m.  The dash-dotted line indicates the mean 8~$\\mu$m emission.}}\n\\label{fig:g1111_fluxplot}\n\\end{figure}\n\n\n\\subsection{Identification of Structure}\n\\label{structure}\n\nFigure~\\ref{fig:opaccontours} shows a map of optical depth G024.05$-$0.22.  This provides an example of the the absorbing substructure in one of the IRDCs in our sample.  Owing to the high spatial resolution of Spitzer at 8~$\\mu$m (1 pixel = 0.01~pc at 4~kpc, accounting for oversampling), we see substructures down to very small scales ($\\sim$0.03~pc) in {\\em all} IRDCs in our sample.  \n\nIn order to identify independent absorbing structures in the 8~$\\mu$m optical depth map, we employed the {\\tt clumpfind} algorithm \\citep{williams_clumpfind}.  In the two-dimensional version, {\\tt clfind2d}, the algorithm calculates the location, size, and the peak and total flux of structures based on specified contour levels.  We use the Spitzer PET\\footnote{http:\/\/ssc.spitzer.caltech.edu\/tools\/senspet\/} to calculate the sensitivity of the observations, i.e. to what level the data permit us to discern true variations from noise fluctuations.  At 8~$\\mu$m, the observations are sensitive to 0.0934 MJy\/sr which, on average, corresponds to an optical depth sensitivity (10-$\\sigma$) of $\\sim$0.02.  While the clumps take on a variety of morphologies, since {\\tt clumpfind} makes no assumptions about the clump shapes, we approximate the clump ``size'' by its effective radius, \n\n\\begin{equation}\nr_{eff}=\\sqrt{\\frac{n_{pix}~A_{pix}}{\\pi~f_{os}}}\n\\end{equation}\n\n\\noindent where n$_{pix}$ is the number of pixels assigned to the clump by {\\tt clumpfind}, and A$_{pix}$ is the area subtended by a single pixel.  The correction factor for oversampling, $f_{os}$ accounts for the fact that the {\\em Spitzer Space Telescope} has an angular resolution of 2.4$''$ at 8~$\\mu$m, while the pixel scale on the IRAC chip is 1.2$''$, resulting in oversampling by a factor of 4.  \n\nThe number and size of structures identified with {\\tt clumpfind} varies depending on the number of contouring levels between the fixed lower threshold, which is set by the sensitivity of the observations, and the highest level set by the deepest absorption.  We set the lowest contour level to 10$\\sigma$ above the average background level.  In general, increasing the number of contour levels serves to increase the number of clumps found.   In all cases, we reach a number of levels where the addition of further contouring levels results in no additional structures.  We therefore select the number of contour levels at which the number of clumps levels off, i.e. when the addition of more contour levels reveals no new clumps.  We also remove those clumps found at the image edge or bordering a star, as the background estimation is likely inaccurate and\/or at least a portion of the clump is probably obscured by the star, rendering any estimation of the optical depth inaccurate.\n\nUsing {\\tt clumpfind}, each IRDC broke down into tens of clumps, ranging in size from tens to hundreds of pixels per clump.  The average clump size is 0.04~pc.  Typically, there is one or two central most-massive clumps and multiple smaller clumps in close proximity.  In some instances, clumps are strung along a filamentary structure, while in other cases, clumps are radially distributed about a highly-concentrated center.  Figure~\\ref{fig:numberedclumps} shows an example of how the clumps are distributed spatially in G024.05$-$0.22 as {\\tt clumpfind} identifies them.   \n\nWith reliable identification of clumps, we next calculate individual clump masses.  As described, {\\tt clumpfind} gives total optical depth measured at 8~$\\mu$m, $\\tau_{8,tot}$, within the clump boundary, its size and position.  This can be directly transformed into $N(H)_{tot}$ via the relationship \n\n\\begin{equation}\n\\label{colequation}\nN(H)_{tot} = {\\frac{\\tau_{8,tot}}{\\sigma_8~f_{os}}}\n\\end{equation}\n\n\\noindent where $\\sigma_8$ is the dust absorption cross section at 8~$\\mu$m.  We derive an average value of $\\sigma_8$ over the IRAC channel 4 bandpass using dust models that take into account higher values of R$_V$ corresponding to dense regions in the ISM.  Using \\citet{weingartner_draine01}, we use $R_V$ = 5.5, case B values, which agree with recent results from \\citet{Indebetouw2005}.  We find the value of $\\sigma_8$ to be 2.3$\\times$10$^{-23}$cm$^2$.  \n\nThe column density can then be used with the average clump size and the known distance to the IRDC, assuming all clumps are at approximately the IRDC distance, to find the clump mass.  The mass of a clump is given as \n\n\\begin{equation}\nM_{clump} = 1.16 m_H N(H)_{tot} A_{clump} \n\\end{equation}\n\n\\noindent where m$_H$ is the mass of hydrogen, N(H)$_{tot}$ is the total column density of hydrogen, the factor 1.16 is the correction for helium and A$_{clump}$ is the area of the clump.  Table~6 gives the location, calculated mass and size of all the clumps identified with {\\tt clumpfind}.  We also note which clumps are in the vicinity of candidate young stellar objects (Table~2) or foreground stars, thereby subjecting the given clump properties to greater uncertainty.  On average (for foreground estimation method A), 25\\% of clumps border a field star, and these clumps are flagged and not used in the further analysis.  In each infrared-dark cloud, we find between 3000$M_{\\sun}$ and 10$^4M_{\\sun}$ total mass in clumps, and typically $\\sim$15\\% of that mass is found in the most massive clump.  \n\nWe perform the same analysis on the maps produced with foreground estimation method B. The foreground assumption in this case leads to lower optical depths across the map.  Due to the different dynamic range in the optical depth map, {\\tt clumpfind} does not reproduce the clumps that are found with method A exactly.  The discrepancy arises in how {\\tt clumpfind} assigns pixels in crowded regions of the optical depth map, so while at large the same material is counted as a clump, the exact assignment of pixels to specific clumps varies somewhat.  On average, the clumps found in the ``method B'' maps tend to have lower masses by a factor of 2, though the sizes do not differ appreciably from those found with foreground estimation method A.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{f16}\n\\end{center}\n\\caption{\\footnotesize{G024.05$-$0.22. 8~$\\mu$m optical depth with contours highlighting the structures.}}\n\\label{fig:opaccontours}\n\\medskip\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=270, scale=0.4]{f17}\n\\end{center}\n\\caption{\\footnotesize{G024.05$-$0.22. Results of the {\\tt clumpfind} algorithm plotted over Spitzer 8~$\\mu$m image.  Absorption identified as a ``clump'' is denoted by a number.}\n\\label{fig:numberedclumps}}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=270,scale=0.6]{f18}\n\\end{center}\n\\caption{\\footnotesize{Original optical depth image of G024.05$-$0.22 (left) and the wavelet subtracted image (right) of the same region.}}\n\\label{fig:wavelet}\n\\end{figure*}\n\n\n\n\\subsection{Resolving Inaccuracy in Clump Mass Calculation}\n\\label{masserr}\n\nThe clumps identified in this fashion include a contribution from the material in the surrounding envelope.  As a result, a portion of the low-mass clump population may not be detected, and the amount of material in a given clump may be overestimated.  To examine this effect, we use the {\\tt gaussclumps} algorithm \\citep{StutzkiGusten1990} to identify clumps while accounting for the contribution from the cloud envelope.  This method was designed to decompose three-dimensional molecular line observations by deconvoloving the data into clumps fit by Gaussians.  To use the algorithm here without altering the code, we fabricated a data cube by essentially by mimicking a third (velocity) dimension, thus simulating three-dimensional clumps that were all centered in velocity on a single central plane.  \\citet{mookerjea2004} and \\citet{Motte2003} have used similar techniques to simulate a third dimension to their dust continuum data sets.  The {\\tt gaussclumps} algorithm inherently accounts for an elevated baseline level, which can be used to approximate the envelope.  Applied to our data set, {\\tt gaussclumps} finds that 15-50\\% of the material is in the envelope.   Further discussion of the envelope contribution, including its effect on the mass function, is given in $\\S$\\ref{envelope}.\n\nThe {\\tt clumpfind} and {\\tt gaussclumps} methods result in nearly one-to-one clump identification in the central region of the IRDC.  However, because the contribution from the cloud envelope falls off further away from the central concentration of mass in the IRDC, {\\tt gaussclumps} fails to find low-mass clumps on the outskirts of IRDCs as successfully as {\\tt clumpfind}, despite being statistically valid relative to their local background.  We conclude that {\\tt gaussclumps} is not suitable to identify of structure in the outskirts of the IRDCs where the envelope is below the central level.  \n\nAnother method commonly employed in the literature to account for the extended structures in which dense cores reside is a ``wavelet subtraction'' technique, which is described in \\citet{alves_cmfimf}.  To address the varying levels of background across the optical depth map, we use the wavelet transform of the image to extract the dense cores.  For one IRDC in our sample, G024.05$-$0.22, we (with the help of J. Alves, private communication) perform the wavelet analysis on the optical depth map.  Figure~\\ref{fig:wavelet} shows a comparison between the original optical depth map and the wavelet-subtracted map.  With the removal of the ``envelope'' contribution in this fashion, the clumps are up to 90\\% less massive on average, and their average size decreases by 25\\%, or $\\sim$0.02~pc.  \n\nBoth using {\\tt gaussclumps} and applying wavelet subtraction methods to extract clumps show that the contribution of the cloud envelope is not yet well-constrained quantitatively.  Not only is the cloud envelope more difficult to detect, its structure is likely not as simple as these first order techniques have assumed in modeling it.  As such, for the remainder of the paper, we will not attempt to correct the clump masses on an individual basis, but rather focus our attention on the clump population properties as a whole.  In $\\S$\\ref{validate}, we employ several techniques to calibrate our mass estimation methods.  We will show in $\\S$\\ref{mf} that the effect of the envelope is systematic and does not skew the derived relationships, such as the slope of the mass function.  \n\n\\subsection{Validating 8~$\\micron$ absorption as a Tracer of Mass}\n\\label{validate}\n\nIn previous studies, molecular clouds have been predominantly probed with using the emission of warm dust at sub-millimeter wavelengths.  While there are inherent uncertainties in the conversion of flux density to mass, the emission mechanism is well-understood.  The method described above is a powerful way to trace mass in molecular clouds.  To understand the extent of its usefulness, here we validate dust absorption as a mass tracer by drawing comparisons between it and results using more established techniques.  First, we relate the dust absorption to dust emission as probes of column density.  Second, we use observations of molecular tracers of dense gas not only to further cement the validity of the absorbing structures, but also to place the IRDCs in context with their surroundings.  Finally, we show that the sensitivity of the technique does not have a strong dependence on distance.  \n\n\\subsubsection{Probing Column Density at Various Wavelengths}\n\\label{columnprobe}\n\nAs we discussed in $\\S$\\ref{bg}, there is an excellent correlation between the 8~$\\mu$m and 850~$\\mu$m flux densities in IRDC G011.11-0.12.  Figure~\\ref{fig:g1111_fluxplot} shows the point-to-point correlation between the SCUBA 850~$\\mu$m flux density and Spitzer 8~$\\mu$m flux density.  This correspondence itself corroborates the use of absorption as a dust tracer.  In addition, the fit to the correlation can confirm that the opacity ratio, $\\kappa_8~$\/$\\kappa_{850}$, is consistent with dust behavior in high density environments.  Relating the 8~$\\mu$m flux density \n\n\\begin{equation}\nf_{8}=f_{bg}e^{-\\kappa_{8}\\Sigma(x)}+f_{fg}\n\\end{equation}\n\n\\noindent where $\\kappa_8~$ is the 8~$\\mu$m dust opacity, $\\Sigma(x)$ is the mass column density of emitting material, and $f_{bg}$ and $f_{fg}$ are the background and foreground flux density estimates, respectively (from $\\S$\\ref{bg}), and the 850~$\\mu$m flux density \n\n\\begin{equation}\nf_{850}=B_{850}(T_d=13K) \\kappa_{850} \\Sigma(x) \\Omega\n\\end{equation}\n\n\\noindent where $B_{850}$ is the Planck function at 850~$\\mu$m evaluated for a dust temperature of 13~K, $\\kappa_{850}$ is the dust opacity at 850~$\\mu$m and $\\Omega$ is the solid angle subtended by the JCMT beam at 850~$\\mu$m, one can find a simple relation between the two by solving each for $\\Sigma(x)$ and equating them. The opacity ratio, put in terms of the flux density measurements is as follows:\n\n\\begin{equation}\n\\frac{\\kappa_{8}}{\\kappa_{850}}=\\frac{B_{850}~\\Omega}{f_{850}} ln \\left( \\frac{f_{bg}}{f_{8}-f_{fg}}\\right)\n\\end{equation}\n\n\\noindent From our data, we confirm this ratio is considerably lower ($\\sim$500) in cold, high density environments than in the diffuse interstellar dust as found by \\citet{Johnstone_G11}.\n\nWe perform another consistency check between our data and dust models.  With maps at both 8 and 24~$\\mu$m, both showing significant absorbing structure against the bright Galactic background (albeit at lower resolution at 24~$\\mu$m), we can calculate the optical depth of at 24~$\\mu$m in the same way we did in Section~\\ref{bg}.  The optical depth scales with the dust opacity by the inverse of the column density ($\\tau_{\\lambda} \\propto \\kappa_{\\lambda} \/ N(H)$), so the ratio of optical depths is equal to the dust opacity ratio.  We find that the typical ratio as measured by Spitzer in IRDCs is\n\n\\begin{equation}\n\\frac{\\kappa_{8}}{\\kappa_{24}} = \\frac{\\tau_8}{\\tau_{24}} \\sim 1.2\n\\end{equation}\n\n\\noindent which is comparable to 1.6, the \\citet{weingartner_draine01} prediction (for R$_V$ = 5.5, case B) and 1-1.2 predicted by \\citet{ossenkopf_henning} in the high-density case.  We conclude that the dust properties we derive are consistent with the trends that emerge from models of dense environments typical of infrared-dark clouds.\n\n\\begin{figure*}\n\\begin{center}\n\\hbox{\n\\vspace{1.0cm}\n\\hspace{1.5cm}\n\\psfig{figure=f19a.ps,angle=270,height=7.0cm}\n\\hspace{0.4cm}\n\\psfig{figure=f19b.ps,angle=90,height=7.0cm,width=7.0cm}\n}\n\\vspace{-1.0cm}\n\\end{center}\n\\caption{\\footnotesize{Left: Contours of integrated intensity of N$_2$H$^+$ plotted over the IRAC 8~$\\mu$m image of IRDC G012.50$-$0.22.  Right: Point-to-point correlation of the N$_2$H$^+$ integrated intensity and the 8~$\\mu$m optical depth.  Points with high integrated intensity but low optical depth correspond to stars, whose presence leads to the underestimation of optical depth in the vicinity.}}\n\\label{fig:g1250_bimaplot}\n\\end{figure*}\n\n\n\\begin{figure}\n\\hspace{-1in}\n\\includegraphics[scale=0.8]{f20}\n\\caption{\\footnotesize{Comparison of the total mass derived from N$_2$H$^+$ maps from \\citet{ragan_msxsurv} and total clump mass as derived from dust absorption at 8~$\\micron$, where the black diamonds represent the mass using foreground estimation method A and the grey squares show the masses derived using foreground estimation method B (see $\\S$\\ref{bg}).  Three of the IRDCs in the sample did not have adequate N$_2$H$^+$ detections.  Error bars for 30\\% systematic errors in the mass are plotted for the clump mass estimates, and a factor of 5 uncertainty is plotted for the N$_2$H$^+$ mass estimates.  The dashed line shows a one-to-one correspondence for reference.}\n\\label{fig:n2hpclumps}}\n\\end{figure}\n\n\\subsubsection{Molecular Line Tracers}\n\nMolecular lines are useful probes of dense clouds, with particular molecules being suited for specific density ranges.  For instance, chemical models show that N$_2$H$^+$ is an excellent tracer of dense gas in pre-stellar objects \\citep{bl97}.  In support of these models, observations of low-mass dense cores \\citep{tafalla_dep, Bergin2002} demonstrate that N$_2$H$^+$ highlights regions of high central density (n$\\sim$10$^6$~cm$^{-3}$), while CO readily freezes out onto cold grains (when n~$>~10^4$~cm$^{-3}$), rendering it undetectable in the central denser regions of the cores.  CO is a major destroyer of N$_2$H$^+$, and its freeze-out leads to the rapid rise in N$_2$H$^+$ abundance in cold gas.  When a star is born, the CO evaporates from grains and N$_2$H$^+$ is destroyed in the proximate gas \\citep{lbe04}.  Thus, N$_2$H$^+$ is a preferential tracer of the densest gas that has not yet collapsed to form a star in low-mass pre-stellar cores.\n\nWhile N$_2$H$^+$ has been used extensively as a probe of the innermost regions of local cores, where densities can reach 10$^6$cm$^{-3}$ \\citep[e.g.][]{taf04}, this chemical sequence has not yet been observationally proven in more massive star forming regions.  Nonetheless recent surveys \\citep[e.g.][]{Sakai2008, ragan_msxsurv} confirm that N$_2$H$^+$ is prevalent in IRDCs, and mapping by \\citet{ragan_msxsurv} shows that N$_2$H$^+$ more closely follows the absorbing gas than CS or C$^{18}$O, which affirms that the density is sufficient for appreciable N$_2$H$^+$ emission.  These single dish surveys do not have sufficient resolution to confirm the tracer's reliability on the clump or pre-stellar core scales in IRDCs.  Interferometric observations will be needed to validate N$_2$H$^+$ as a probe of the chemistry and dynamics of individual clumps (Ragan et al., in prep.).  \n\nFor one of the objects in our sample, G012.50$-$0.22, we had previous BIMA observations of N$_2$H$^+$ emission with 8$'' \\times 4.8''$ spatial resolution.  The BIMA data were reduced using the standard MIRIAD pipeline reduction methods \\citep{MIRIAD}.  As in nearby clouds, such as \\citet{wmb04}, the integrated intensity of N$_2$H$^+$ relates directly to the dust (measured here in absorption) in this infrared-dark cloud.  Figure~\\ref{fig:g1250_bimaplot} illustrates the quality of N$_2$H$^+$ as a tracer of dense gas, both in the N$_2$H$^+$ contours plotted over the 8~$\\mu$m {\\em Spitzer} image and the point-to-point correlation between the 8~$\\mu$m optical depth and the integrated intensity of N$_2$H$^+$.  The points that lie above the average line, with high integrated intensities but low optical depth, are all in the vicinity of a foreground star in the 8~$\\mu$m image, which lowers our estimate for optical depth.  In the sample, however, we have shown that the foreground and young stellar population is largely unassociated with the absorption.  \n\nTwo trends are apparent in Figure~\\ref{fig:g1250_bimaplot}.  First, below $\\tau~<~0.25$ there is a lack of N$_2$H$^+$ emission.  This suggests that the absorption may be picking up a contribution from a lower density extended envelope that is incapable of producing significant N$_2$H$^+$ emission.  This issue is discussed in greater detail in $\\S$\\ref{envelope}.  Alternatively, the interferometer may filter out extended N$_2$H$^+$ emission.  The second trend evident in Figure~\\ref{fig:g1250_bimaplot} is that for $\\tau~>~0.25$, there is an excellent overall correlation, confirming that mid-infrared absorption in clouds at distances of 2 to 5~kpc is indeed tracing the column density of the {\\em dense} gas likely dominated by pre-stellar clumps.\n\nIn addition to directly tracing the dense gas in IRDCs, molecular observations can be brought to bear on critical questions regarding the use of absorption against the Galactic mid-infrared background and how best to calibrate the level of foreground emission.  One way to approach this is to use the molecular emission as a tracer of the total core mass and compare this to the total mass estimated from 8~$\\mu$m absorption with differing assumptions regarding the contributions of foreground and background (see $\\S$~\\ref{bg}).  In \\citet{ragan_msxsurv} we demonstrated that the distribution of N$_2$H$^+$ emission closely matches that of the mid-infrared absorption (see also $\\S$~\\ref{clumps}).  This is similar to the close similarity of N$_2$H$^+$ and dust continuum emission in local pre-stellar cores \\citep[e.g.][]{BerginTafalla_ARAA2007}.  Thus we can use the mass estimated from the rotational emission of N$_2$H$^+$ to set limits on viable models of the foreground.  In \\citet{ragan_msxsurv} we directly computed a mass using an N$_2$H$^+$ abundance assuming local thermodynamic equilibrium (LTE) and using the H$_2$ column density derived from the MSX 8~$\\mu$m optical depth.  However, this estimate is highly uncertain as the optical depth was derived assuming no foreground emission, and the N$_2$H$^+$ emission may not be in LTE.  Instead, here, we will use chemical theory and observations of clouds to set limits.\n\nN$_2$H$^+$ appears strong in emission in dense pre-stellar gas due to the freeze-out of CO, its primary destruction route.  Detailed theoretical models of this process in gas with densities in excess of 10$^5$~cm$^{-3}$ \\citep{aikawa_be}, as expected for IRDCs, suggest a typical abundance should be $\\sim$10$^{-10}$ with respect to H$_2$ \\citep{maret_n2, aikawa_be, Pagani2007}.  This value is consistent with that measured in dense gas in several starless cores \\citep{tafalla_dep, maret_n2}.  Using this value we now have a rough test of our foreground and background estimates.  For example, in G024.05$-$0.22 we find a total mass of 4100~$M_{\\sun}$ (foreground estimation method A).  Using the data in \\citet{ragan_msxsurv}, we find that the total mass traced by N$_2$H$^+$ is 4400~$M_{\\sun}$, providing support for our assumptions.  Figure~\\ref{fig:n2hpclumps} shows the relationship between the total clump mass derived from absorption and the total mass derived from our low-resolution maps of N$_2$H$^+$ for the eight IRDCs in our sample that were detected in N$_2$H$^+$.  In general, there is good agreement.  We plot a 30\\% systematic error in the total clump masses (abscissa) and a factor of 5 in for the total N$_2$H$^+$ mass estimate (ordinate).  In the cases where the estimates differ, the N$_2$H$^+$ mass estimate tends to be greater than the total mass derived from the dust absorption clumps.  This discrepancy likely arises in large part from an under-estimation of N$_2$H$^+$ abundance and\/or non-LTE conditions.  All the same, the consistency of the mass estimates, together with the morphological correspondence, reaffirms that the we are probing the dense clumps in IRDCs and that our mass probe is reasonably calibrated.\n\nWe find no discernible difference between methods A and B of foreground estimation.  However, we note that both are substantially better than assuming no foreground contribution.  We therefore believe that method A is an appropriate estimate of the foreground contribution (see $\\S$\\ref{bg}).  \n\n\\subsubsection{Effects of Distance on Sensitivity}\n\\label{sens}\n\nInfrared-dark clouds are much more distant than the local, well-studied clouds such as Taurus or $\\rho$ Ophiuchus.  As such, a clear concern is that the distance to IRDCs may preclude a well-defined census of the clump population.  The most likely way in which the our survey is incomplete is the under-representation of low-mass objects due to their relatively small size, blending of clumps along the line of sight, or insensitivity to their absorption against the background.  One observable consequence of this effect, assuming IRDCs are a structurally homogeneous class of objects, might be that more distant IRDCs should exhibit a greater number of massive clumps at the expense of the combination of multiple smaller clumps.  Another possible effect is the greater the distance to the IRDC, the less sensitive we become to small clumps, and clumps should appear to blend together (i.e. neighboring clumps will appear as one giant clump).  Due to this effect, we expect that the most massive clumps of the population will be over-represented.  As a test, we examine the distribution of masses and sizes of clumps as a function of IRDC distance, which is shown in Figure~\\ref{fig:masssens}.  This sample, with IRDCs ranging in distance from 2.4 to 4.9~kpc away, does not show a strong trend of this nature.  We show the detection limit for clumps to illustrate the very good sensitivity of this technique and that while it does impose a lower boundary on clump detectability, most clumps are not close to this value.  We found no strong dependence of clump mass or size on the distance to the IRDC and conclude that blending of clumps does not have a great effect on the mass sensitivity.\n\nTypical low-mass star forming cores range in size from 0.03 to 0.1~pc \\citep{BerginTafalla_ARAA2007}.  If one were to observe such objects 4~kpc, they would only subtend a few arcseconds.  For example, if L1544, a prototypical pre-stellar core, resided at the typical distance to the IRDCs in the sample, it would show sufficient absorption \\citep[based on reported column density measurements by][]{bacmann_iso} against the Galactic background, but according to \\citet{Williams_2006}, would subtend 3$''$ in diameter at our fiducial 4~kpc distance, which is very close to our detection limit.  In addition, very low mass clumps could blended into any extended low-density material that is included in our absorption measurement.  These effects should limit our sensitivity to the very low-mass end of our clump mass function. \n\nTo first order, we have shown distance is not a major factor because the high-resolution offered by {\\em Spitzer} improves our sensitivity to small structures.  However, infrared-dark clouds are forming star clusters and by nature are highly structured and clustered.  As such, we can not rule out significant line-of-sight structure.  Since independent clumps along the line-of-sight might have distinct characteristic velocities, the addition of kinematical information from high-resolution molecular data (Ragan et al., in prep.) will help the disentanglement.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=90,scale=0.4]{f21}\n\\end{center}\n\\caption{\\footnotesize{Top: The range in clump mass as a function of distance.  The median clump mass for each IRDC in the sample is indicated with a diamond.  Bottom: The range in clump size as a function of distance.  The median clump size for each IRDC in the sample is indicated with a diamond.  The resolution limit is plotted as a solid line, and it shows the boundary at which {\\tt clumpfind} defines a ``clump'' for an object at the distance of the indicated host IRDC.}}\n\\vspace{0.2in}\n\\label{fig:masssens}\n\\end{figure}\n\n\\section{Mass Function}\n\\label{mf}\n\nA primary goal of this study is to explore the mass function of clumps in infrared-dark clouds and compare it to that of massive star formation regions, local star formation regions, and the stellar IMF.  We note that there is some ambiguity in the literature about the ``clump'' versus the ``core'' mass functions.  In the following description, a ``core'' mass function refers to the mass spectrum objects with masses in the ``core'' regime (10$^{-1}$-10$^1$~$M_{\\sun}$, 10$^{-2}$-10$^{-1}$~pc), and a ``clump'' mass function for objects in the ``clump'' regime (10$^{-1}$-10$^1$~$M_{\\sun}$, 10$^{-2}$-10$^{-1}$~pc), as summarized in \\citet{BerginTafalla_ARAA2007}.  Here we present the infrared-dark cloud clump mass function.  We describe the relevance of this result in the context of Galactic star formation and discuss several methods we use to test its validity.\n\n\\subsection{Mass Function in Context}\n\\label{context}\n\nA fundamental property of the star formation process is the mass spectrum of stars, and, more recently, the mass function of pre-stellar objects.  The mass spectrum in either case is most typically characterized by a power law, taking the form $dN\/dM \\propto M^{-\\alpha}$, known as the differential mass function (DMF).  In other contexts, the mass function can be described as a function of the logarithm of mass, which is conventionally presented as $dN\/d(log~m) \\propto M^{\\Gamma}$, in which case $\\Gamma=-(\\alpha-1)$.  In the results that follow, we present the slope of the clump mass function in terms of $\\alpha$.\n\nA commonly used method for studying mass functions of pre-stellar cores is observation of dust thermal continuum emission in nearby star-forming clouds.  Cold dust emission is optically thin at millimeter and sub-millimeter wavelengths, and can therefore be used as a direct tracer of mass.  A number of surveys of local clouds \\citep[e.g.][]{Johnstone_Oph, motte_rhooph} have been performed with single-dish telescopes covering large regions in an effort to get a complete picture of the mass distribution of low-mass clouds.  This is an extremely powerful technique, but as \\citet{Goodman_col} demonstrate, this technique suffers from some limitations, chief among them poor spatial resolution (in single-dish studies), required knowledge of dust temperatures \\citep{Pavlyuchenkov_2007}, and the insensitivity to diffuse extended structures.  \n\nAnother technique that has been employed to map dust employs near-infrared extinction mapping \\citep{alves_cmfimf, Lombardi_pipe}, which is a way of measuring $A_V$ due to dark clouds by probing the color excesses of background stars \\citep{Lombardi_NICER}.  This method is restricted to nearby regions of the Galaxy because of sensitivity limitations and the intervention of foreground stars, both of which worsen with greater distance. Also, the dynamic range of $A_V$ in such studies is limited to $\\sim$1-60 \\citep{Lombardi_NICER}, while our technique probes from $A_V$ of a few to $\\sim$100.\n\nThe dust-probing methods mentioned above, both thermal emission from the grains and extinction measures using background stars, often find a core mass function (CMF) that is similar in shape to the stellar initial mass function (IMF), as described by \\citet{Salpeter_imf}, where $\\alpha=2.35$ ($\\Gamma=-1.35$), or \\citet{Kroupa_imf}.  This potentially suggests a one-to-one mapping between the CMF and IMF, perhaps scaled by a constant ``efficiency'' factor \\citep[e.g.][]{alves_cmfimf}.  Also, both techniques are difficult to apply to regions such as infrared-dark clouds due to their much greater distance.  As we show in $\\S$\\ref{structure}, absorbing structure exists below the spatial resolution limit of single-dish surveys.  Sensitivity limitations and foreground contamination preclude use of extinction mapping to probe IRDCs.\n\nStructural analysis using emission from CO isotopologues find a somewhat different character to the distribution of mass in molecular clouds.  \\citet{Kramer_CMF} determined that the clump mass function in molecular clouds follows a power law with $\\alpha$ between 1.4 and 1.8 ($-0.8 < \\Gamma < -0.4 $).  This is significantly shallower than the Salpeter-like slope for clumps found in works using dust as a mass probe.  This disagreement may be due to an erroneous assumption about one technique or the other, or it may be that the techniques are finding information about how the fragmentation process takes place from large scale, probed by CO, to small scales, probed by dust.  Another possible explanation is that most of the objects in  \\citet{Kramer_CMF} are massive star forming regions, and star formation in these regions may be intrinsically different than tyical regions studied in the local neighborhood (e.g. Taurus, Serpens).\n\nSub-millimeter observations of more distant, massive star-formation regions have been undertaken \\citep[e.g.][]{reid_2, Li_orion, mookerjea2004, rathborne2006} with a mixture of results regarding the mass function shape.  \\citet{rathborne2006}, for example, performed IRAM observations of a large sample of infrared-dark clouds.  Each cloud in that sample is comprised of anywhere from 2 to 18 cores with masses ranging from 8 to 2000$M_{\\sun}$.  They find a Salpeter-like ($\\alpha$~$\\sim$2.35) mass function for IRDC cores.  However, our Spitzer observations reveal significant structure below the spatial resolution scales of \\citet{rathborne2006}.  As we will show (see Section~\\ref{mf}), the mass function within a fragmenting IRDC is shallower than Salpeter and closer to the mass function derived from CO emission.\n\nGiven the strong evidence for fragmentation, it is clear that IRDCs are the precursors to massive clusters.  We then naturally draw comparisons between the characteristics of fragmenting IRDCs and the nearest region forming massive stars, Orion.  At $\\sim$500~pc, it is possible to resolve what are likely to be pre-stellar objects in Orion individually with current observational capabilities.  With the high-resolution of our study, we can examine star formation regions (IRDCs) at a similar level of detail as single-dish telescopes can survey Orion.  For example, we detect structures on the same size scale ($\\sim$0.03~pc) as the quiescent cores found by \\citet{Li_orion} in the Orion Molecular Cloud, however the most massive core in their study is $\\sim$50~$M_{\\sun}$.  These cores account for only a small fraction of the total mass in Orion.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{f22}\n\\end{center}\n\\caption{\\footnotesize{\nDifferential mass function of ensemble IRDC sample.  Black filled circles indicate results of the {\\tt clumpfind} technique, and the green open triangles denote the results of the {\\tt gaussclumps} clump-finding method.  The fits are broken power laws.  On the high-mass end, the slope of the {\\tt gaussclumps} method mass function ($\\alpha=1.15\\pm0.04$) is shallower than the slope of hte {\\tt clumpfind} mass function ($\\alpha=1.76\\pm0.05$).}}\n\\label{fig:cfgccompareMF}\n\\end{figure}\n\n\n\\subsection{Results: Differential Mass Function}\n\\label{dmf}\n\nWe use the IRDC clump masses calculated in $\\S$\\ref{structure} (using {\\tt clumpfind} and foreground estimation method A) to construct an ensemble mass function in Figure~\\ref{fig:cfgccompareMF}.  The mass function that results from using foreground estimation method B is shifted to lower masses by a factor of 2, but the shape is identical.  Because IRDCs appear to be in a roughly uniform evolutionary state over the sample (i.e. they are all likely associated with the Molecular Ring, and they possess similar densities and temperatures), we merge all the clumps listed in Table~6 as ensemble and present a single mass function for all the objects at a range of distances.  This assumes that the character of the mass function is independent of the distance to a given IRDC.  Recall that we see no evidence (see Figure~\\ref{fig:masssens}) for the mass distributions to vary significantly with distance.\n\nFor the calculation of the errors in the DMF we have separately accounted for the error in the mass calculation and the counting statistics.  We used a method motivated by \\citet{reid_1} to calculate the mass error.   We have assumed that the clump mass error is dominated by the systematic uncertainty of 30\\% in the optical depth to mass correction.  For each clump we have randomly sampled a Guassian probability function within the 1$\\sigma$ envelope defined by the percentage error.  With these new clump masses we have re-determined the differential mass function.   This process is repeated 10$^4$ times, and the standard deviation of the DMF induced by the error in the mass is calculated from the original DMF.  This error is added in quadrature to the error introduced by counting statistics.  The provided errors are 1$\\sigma$, with the caveat that the value assumed for the systematic uncertainty is open to debate.  As a result, when there are large numbers in a given mass bin, the error is dominated by the mass uncertainty.  Conversely, when there are few objects in a mass bin, the error is dominated by counting.\n\nThe IRDC clump mass function for this sample spans nearly four orders of magnitude in mass.  We fit the mass function with a broken power law weighted by the uncertainties.  At masses greater than $\\sim$40$M_{\\sun}$, the mass function is fit with a power law of slope $\\alpha$=1.76$\\pm$0.05.  Below $\\sim$40$M_{\\sun}$, the slope becomes much shallower, $\\alpha$=0.52$\\pm$0.04.  We also include in Figure~\\ref{fig:cfgccompareMF} the mass function of clumps found with the {\\tt gaussclumps} algorithm, with errors calculated in the identical fashion.  Performing fits in the equivalent mass regimes results in a shallower slope for masses greater than 40$M_{\\sun}$ ($\\alpha$=1.15$\\pm$0.04), while the behavior at low masses is similar.  As discussed in $\\S$\\ref{structure}, the clumps found with {\\tt clumpfind} and {\\tt gaussclumps} are in good agreement in the central region of each IRDC, but tend to disagree on the outskirts.  This is a consequence of the failure of {\\tt gaussclumps} to model the varying background.  Examination of the images reveals that the contribution of the diffuse material varies across the image, thereby setting the background level too high for outer clumps (where the envelope contributes less) to be detected.  In fact, these clumps appear to be preferentially in the 30 to 500$M_{\\sun}$ range, and a mass function constructed with the {\\tt gaussclumps} result is significantly shallower than derived with {\\tt clumpfind} (see Figure~\\ref{fig:cfgccompareMF}).  We conclude that {\\tt gaussclumps} is not suitable to identify structure away from the central region of the IRDC where the envelope level is below the central level.  This is further supported by the wavelet analysis which is capable of accounting for a variable envelope contribution.  It is worth noting that for the one IRDC for which we have the wavelet analysis, that the slope of the derived mass function shows little appreciable change and agrees with the {\\tt clumpfind} result.   \n\nTo put the mass function into context with known Galactic star formation, we plot the clump mass function of all clumps in our sample in Figure~\\ref{fig:literature_MF} along with the core\/clump mass function of a number of other studies probing various mass ranges.  We select four studies, each probing massive star forming regions at different wavelengths and resolutions including quiescent cores in Orion \\citep{Li_orion}, clumps in M17 \\citep{reid_2}, clumps in RCW 106 \\citep{mookerjea2004}, and clumps in massive star formation region NGC 6334 \\citep{munoz_ngc6334}.  In their papers, each author presents the mass function in a different way, making it difficult to compare the results directly to one another.  Here, we recompute the mass function for the published masses in each work uniformly (including the treatment of errors, see above).  Each of the mass functions is fit with a power law.  Figure~\\ref{fig:literature_MF} highlights the uniqueness of our study in that it spans over a much larger range in masses than any other study to date.  \n\nAt the high-mass end, the mass function agrees well with the \\citet{mookerjea2004} and \\citet{munoz_ngc6334} studies, which probed to lower mass limits of 30$M_{\\sun}$ and 4$M_{\\sun}$, respectively.  The fall-off from the steep slope at the high mass end to a shallower slope at the low mass end immediately suggests that completeness, enhanced contribution from the envelope and\/or clump blending become an issue.  However, the slope at the low mass end compares favorably with \\citet{Li_orion} and \\citet{reid_2} which probe mass ranges 0.1 to 46$M_{\\sun}$ and 0.3 to 200$M_{\\sun}$, respectively.  In addition to the general DMF shape at both the high mass and low mass end, the ``break'' in the mass function falls in the 10$M_{\\sun}$ to 50$M_{\\sun}$ range for the ensemble of studies, including ours.  If this is a real feature of the evolving mass spectrum, this can shed some light on the progression of the fragmentation process from large, massive objects to the numerous low-mass objects like we see in the local neighborhood.  The characteristic ``break'' mass can also be a superficial artifact of differences in binning, mass determination technique, and observational sensitivity.  Our study is the only one that spans both mass regimes, and further such work is needed to explore the authenticity of this feature.  However, in $\\S$\\ref{conclusion} we speculate that this may be an intrinsic feature. \n\nIt is possible that the slope of the IRDC clump mass function might be an artifact of a limitation in our technique.  With the great distances to these clouds, one would expect the effect of clump blending to play a role in the shape of their mass spectrum.  We have shown in $\\S$~\\ref{sens} that distance does not dramatically hinder the detection of small clumps.  Our study samples infrared-dark clouds from 2.4~kpc to 4.9~kpc, and we find that the number of clumps does not decrease with greater distance, nor does the median mass tend to be be significantly greater with distance.  Furthermore, with the present analysis, we see no evidence that including clumps from IRDCs at various distances affects the shape of the mass function.\n\nFrom past studies of local clouds there has been a disparity between mass function slope derived with dust emission and CO \\citep[e.g. compare][]{Johnstone_OrionB, Kramer_CMF}.  Our result suggests that massive star forming regions have mass functions with slope in good agreement with CO isotopologues, e.g. $\\alpha$=1.8.  This is crucial because CO observations contain velocity information, which allow for the clumps to be decomposed along the line-of-sight.  Still, the authors find a shallow slope in agreement with ours.  We conclude that clump blending, while unavoidable to some extent, does not skew the shape of the mass function as derived from dust emission or absorption.  A close look at \\citet{Kramer_CMF} results finds that the majority of objects studied are massive star formation regions.  Given the general agreement of the clump mass function of this sample of IRDCs with other studies of massive star formation regions, we believe this result represents the true character of these objects, not an artifact of the observing technique.  \n\nSeveral studies of pre-stellar cores in the local neighborhood show a mass distribution that mimics the shape of the stellar IMF.  That the slope of the mass function in infrared-dark clouds is considerably shallower than the stellar IMF should not be surprising.  The masses we estimate for these clumps are unlikely to give rise to single stars.  Instead, the clumps themselves must fragment further and eventually form a star cluster, likely containing multiple massive stars.  Unlike Orion A, for example, which contains $\\sim$10$^4$~$M_{\\sun}$ distributed over a 380 square parsec (6.2 square degrees at 450~pc) region \\citep{Carpenter2000}, in IRDCs, a similar amount of mass is concentrated in clumps extending only a 1.5 square parsec area.  Therefore, we posit that IRDCs are not distant analogues to Orion, but more compact complexes capable of star formation on a more massive scale.\n\nGiven the high masses estimated for infrared-dark clouds, yet the lack evidence for the massive stars they must form is perhaps indicating that we see them necessarily {\\em because} we are capturing them just before the onset of star formation.  Such a selection effect would mean that we preferentially observe these dark objects because massive stars have yet to disrupt their natal cloud drastically in the process of protostar formation.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{f23}\n\\end{center}\n\\caption{\n\\footnotesize{ \nDifferential mass function of this IRDC sample (black filled circles) fit with a single power-law for M$_{clump}>30M_{\\sun}$ ($\\alpha = 1.76\\pm0.05$) compared with various star formation regions in the high mass regime and their respective single power-law fit slopes.  At the high mass end, our fit agrees well with that of other studies: {\\it Open purple diamonds} from \\citet{munoz_ngc6334} ($\\alpha = 1.64\\pm0.06$); {\\it Open green inverted triangles} from \\citet{mookerjea2004} ($\\alpha = 1.59\\pm0.10$).  At the low mass end, we fit a second power law for the bins with M$_{clump}<30M_{\\sun}$ ($\\alpha = 0.52\\pm0.04$), which agrees well with other studies in this mass regime: {\\it Open blue diamonds} from \\citet{reid_2} ($\\alpha = 0.80\\pm0.07$); {\\it Open red circles} from quiescent Orion cores from \\citet{Li_orion}($\\alpha = 0.82\\pm0.09$).  Note that only this study spans the entire range of masses, so the reality of the apparent break at $\\sim$30$M_{\\sun}$ is in question. }}\n\\label{fig:literature_MF}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\hbox{\n\\vspace{1.0cm}\n\\hspace{-1cm}\n\\psfig{figure=f24a.eps,height=9cm}\n\\hspace{-4cm}\n\\psfig{figure=f24b.eps,height=9cm}\n}\n\\vspace{-1.0cm}\n\\end{center}\n\\caption{\\footnotesize{Left: The mass-radius relationship for {\\tt clumpfind} clumps (foreground method A) in the entire sample of IRDCs (gray), with the clumps found only in G024.05$-$0.22 highlighted in black, and the clumps found in the wavelet subtracted image (red).  The solid line denotes the critical Bonnor-Ebert mass-radius relation for T$_{internal}$=15~K.  The dashed line is the M~$\\propto$~R$^{2.2}$ from the \\citet{Kramer1996} CO multi-line study of Orion.  The dash-dotted line is taken from \\citet{williams_clumpfind}, which finds M~$\\propto$~R$^{2.7}$.  Right: The mass-radius relationship for IRDC clumps, including a comparison to all the studies of massive star forming regions included in Figure~\\ref{fig:literature_MF}.}}\n\\label{fig:massrad}\n\\end{figure*}\n\n\n\\subsection{The Contribution from the IRDC Envelope}\n\\label{envelope}\n\nLike nearby clouds, infrared-dark clouds are structured hierarchically, consisting of dense condensations embedded in a more diffuse envelope.  Here we present various attempts to estimate the fraction of the total cloud mass resides in dense clumps compared to the extended clouds.  First, we use archival $^{13}$CO data to probe the diffuse gas and use it to estimate the envelope mass.  To further explore the contribution of the envelope, we demonstrate that a wavelet analysis, a technique designed to remove extended structures from emission maps, gives a similar relationship between envelope and dense clump mass.  Alternatively, applying the {\\tt gaussclumps} algorithm to the data provides an average threshold that describes the diffuse structure.  \n\nWe use $^{13}$CO~(1-0) molecular line data from the Galactic Ring Survey \\citep{Jackson_GRS} in the area covered by our {\\em Spitzer} observations of G024.05$-$0.22 to probe the diffuse material in the field.  The $^{13}$CO emission is widespread, covering the entire area in the IRAC field, thus we are not probing the entire cloud.  Assuming local thermodynamic equilibrium (LTE) at a temperature of 15~K and a $^{13}$CO abundance relative to H$_2$ of $4\\times10^{-6}$ \\citep{Goldsmith_Taurus}, we find that the the clump mass is $\\sim$20\\% of the total cloud mass. This demonstrates that IRDCs are the densest regions of much larger molecular cloud complexes;  however, the fraction of mass that we estimate the clumps comprise relative to the cloud is an upper limit because the full extent of the cloud is not probed with these data.\n\nIn $\\S$\\ref{masserr}, we discuss two ways in which we account for the envelope in the clump-finding process.  First, the {\\tt gaussclumps} algorithm is an alternative method of identifying clumps, and in $\\S$\\ref{dmf} we examine the effect this method has on the clump mass function.  The algorithm is insensitive to clumps on the outskirts of the IRDC, thereby flattening the mass function.  While {\\tt gaussclumps} may oversimplify the structure of the envelope for the purposes of identifying clumps, it does provide a envelope {\\em threshold}, above which optical depth peaks fit as clumps and below which emission is subtracted.  This threshold approximates the level of the envelope, and as a result, {\\tt gaussclumps} finds 15-50\\% of the optical depth level is from the diffuse envelope.  The wavelet subtraction technique results in clumps that are on average 90\\% less massive and smaller in size by 25\\% ($\\sim$0.02~pc) than those extracted from the unaltered map.\n\nThese analyses of the IRDC envelope show us that our technique is only sampling 20-40\\% of the clouds total mass and, at the same time, the clump masses themselves include a contribution from the surrounding envelope.  Because of these factors, the different methods for isolating ``clumps'' have varying levels of success.  For example, using {\\tt gaussclumps} equips us to parametrically remove the envelope component to the clump, but due to the underlying assumption of the baseline level, it misses many clumps that {\\tt clumpfind} identifies successfully.  The mass function that results from using the {\\tt gaussclumps} method is shallower than that from {\\tt clumpfind}, as {\\tt gaussclumps} fails to find clumps on the periphery of the dominant (often central) concentration of clumps, where the envelope level is lower.  \n\nWhile both the {\\tt clumpfind} and {\\tt gaussclumps} methods have their drawbacks, it is clear that IRDCs have significant structure on a large range of scales.  The relatively shallow mass function for IRDC clumps and other massive star forming regions shows that there is a great deal of mass in large objects, and future work is needed to understand the detailed relationship between the dense clumps and their surroundings.\n\n\n\n\\section{Mass-Radius Relation}\n\nNext we investigate the relationship between the mass and size of the clumps found in IRDCs, which informs us of the overall stability of the clump structures.  Figure~\\ref{fig:massrad} shows the mass-radius relationship of the {\\tt clumpfind}-identified clumps, highlighting the results for G024.05$-$0.22 and the wavelet-subtracted case.  Indeed, the clumps extracted from the wavelet-subtracted map are shifted down in mass by 90\\% and down in size by 25\\%, but the relationship between the quantities does not change.  We plot the relation of simple self-gravitating Bonnor-Ebert spheres ($M~(R)~=~2.4 R a^2 \/ G$, where $a$ is the sound speed and set to 0.2~km~s$^{-1}$, solid line) and also the mass-radius relationship observed in a multi-line CO survey of Orion \\citep[M~$\\propto$~R$^{2.2}$,][(dashed line)]{Kramer1996}.  For comparison, Figure~\\ref{fig:massrad} also shows these properties from the other studies of massive star formation regions.  We note that the spatial resolution of the comparison studies is larger than the resolution of this study.  The relationship for Orion \\citep{Li_orion}, M17 \\citep{reid_2}, NGC 6334 \\citep{munoz_ngc6334} and RCW 106 \\citep{mookerjea2004} all agree with the \\citet{Kramer1996} relationship, which is consistent with the mass function agreement to CO studies (see $\\S$\\ref{dmf}).  \n\nThe IRDC clumps are clearly gravitationally unstable, showing higher densities than their local Bonnor-Ebert sphere counterparts.  The relationship for clumps in IRDCs shows a steeper trend, one closer to the \\citet{williams_clumpfind} relationship, $M~\\propto~R^{2.7}$.  Also, dust extinction at 8~$\\mu$m has greater sensitivity to high-densities than CO, which is known to freeze out at extreme densities.  Hence, while the IRDC clumps are clearly Jeans unstable, the slope of the relation may be simply a reflection of the different mass probe used here.\n\n\\section{Discussion \\& Conclusion} \n\\label{conclusion}\n\nThe {\\em Spitzer Space Telescope} affords us the ability to probe a spatial regime of massive clouds in the Galactic Ring at comparable resolution as has been applied to the numerous studies of local, low-mass star formation.  In this way, we can extend the frontier of detailed star formation studies to include regions the likes of which are not available in the solar neighborhood.  This study demonstrates a powerful method for characterizing infrared-dark clouds, the precursors to massive stars and star clusters.  These objects provide a unique look at the initial conditions of star formation in the Galactic Ring, the dominant mode of star formation in the Galaxy.\n\nWe present new {\\em Spitzer} IRAC and MIPS 24~$\\mu$m photometric measurements supplemented with 2MASS J, H, K$_s$ photometry of the distributed young stellar population observed in the Spitzer fields.  Rigid color criteria are applied to identify candidate young stellar objects that are potentially associated with the infrared-dark clouds.  In all, 308 young stellar objects were identified (see Table~2), seven of which are classified as embedded protostars.  For those objects, we set lower limits on the infrared luminosities.  One IRDC has an IRAS source in the field, which is the best candidate for an associated massive star.  Otherwise, our observations provide no evidence for massive star formation in IRDCs, though sensitivity limitations do not rule out the presence of low mass stars and heavily extincted stars.  Nebulosity at 8 and 24~$\\mu$m was detected in four of the fields, but when these regions were correlated with molecular data, they do not appear to be associated with the IRDCs.  On average, 25\\% of clumps are in the vicinity of stars and $\\sim$10\\% are in near YSOs, which are the most likely sources to be associated with the infrared-dark cloud.  Since most of the mass is not associated with any indicator of star formation.  This leads us to conclude that IRDCs are at in earlier stage than, say, the nearest example of massive star formation, the Orion Nebula, and these results are powerful clues to the initial conditions of star cluster formation.  \n\nWe detail our method of probing mass in IRDCs using dust absorption as a direct tracer of column density.  We perform the analysis using two different assumptions (methods A and B) for the foreground contribution to the 8~$\\mu$m flux.  The IRDC envelope contribution to the  To validate our method in the context of others, we compare and find good agreement between the 8~$\\mu$m absorption and other tracers of dust, such as sub-millimeter emission from dust grains measured with SCUBA and N$_2$H$^+$ molecular line emission measured with FCRAO and BIMA.  We show that distance does not play a role in the effectiveness of the technique.  The high resolution {\\em Spitzer} observations allows us to probe the absorbing structures in infrared-dark clouds at sub-parsec spatial scales.  We apply the {\\tt clumpfind} algorithm to identify independent absorbing structures and use the output to derive the mass and size of the clumps.  Tens of clumps are detected in each IRDC, ranging in mass from 0.5 to a few $\\times$ 10$^3~M_{\\sun}$ with sizes from 0.02 to 0.3~pc in diameter.  We also apply the {\\tt gaussclumps} algorithm to identify clumps.  The structures in the central region of the IRDC correspond almost perfectly to the {\\tt clumpfind} result, but {\\tt gaussclumps} misses clumps on the outskirts because it fails to account for a spatially variable background level.  \n\nThe existence of substructure -- from 10$^3$~$M_{\\sun}$ clumps down to 0.5~$M_{\\sun}$ ``cores'' -- indicates that IRDCs are undergoing fragmentation and will ultimately form star clusters.  The typical densities (n $>$ 10$^5$ cm$^{-3}$) and temperatures (T~$<$~20 K) of IRDCs are consistent with massive star forming regions, but they lack the stellar content seen in more active massive star formation regions, such as the Orion molecular cloud or W49, for example.  The mass available in the most massive clumps, however, leads us to conclude that IRDCs will eventually form multiple massive stars.  \n\nThe IRDC clump mass function, with slope $\\alpha = 1.76\\pm0.05$ for masses greater than $\\sim$40$M_{\\sun}$, agrees with the mass function we calculate based on data from other studies of massive objects.  The mass function for both IRDCs and these massive clump distributions is shallower than the Salpeter-like core mass function reported in local regions.  In fact, the IRDC clump mass function is more consistent with that found when probing molecular cloud structure using CO line emission ($\\alpha = 1.6 - 1.8~$), again supporting the assertion that these objects are at an earlier phase of fragmentation.  At the low-mass end ($M < 40M_{\\sun}$), we find a much shallower slope, $\\alpha = 0.52\\pm0.04$, which is somewhat flatter than other studies that cover the same range in masses.  This could be due in part to incomplete sampling of the fields.  Alternatively, the apparent flattening of the clumps mass function around 40~$M_{\\sun}$ could indicate a transition between objects that will generate clustered star formation and those that give rise to more distributed star formation \\citep{AdamsMyers2001}.\n\nIRDC clumps are generally not in thermodynamic equilibrium, but rather are undergoing turbulent fragmentation.  The mass spectrum is consistent with the predictions of gravoturbulent fragmentation of molecular clouds \\citep{Klessen2001}.  The dynamic Molecular Ring environment could naturally be conducive for producing concentrated cluster-forming regions.\n\nJust as in all surveys of IRDCs to date, this study is subject to the blending of clumps, which could alter the shape of the mass function to over-represent the most massive clumps at the expense of clumps of all masses and sizes.  To the extent that this sample allows, we find that this does not drastically effect the shape of the mass function.  Other studies of cloud fragmentation that have the advantage of a third dimension of information also find a shallower clump mass function slope \\citep{Kramer_CMF}.  We therefore conclude that this result is a true reflection of the structure in IRDCs and nature of massive star formation. \n\nInfrared-dark clouds are already well-established candidates for the precursors to stellar clusters and exhibit significant structures down to 0.02~pc scales.  The properties of IRDCs provide powerful constraints on the initial conditions of massive and clustered star formation.  We suggest that the mass function is an evolving entity, with infrared-dark clouds marking one of the earliest stages of cluster formation.  The mass distribution is top-heavy, with most of the mass in the largest structures.  As the massive clumps fragment further, the mass function will evolve and become steeper.  The clumps will ultimately fragment to the stellar scale and {\\em then} take on the Salpeter core mass function that has been observed so prevalently in local clouds.  For example, following the (mostly) starless IRDC phase of cluster evolution, the mass spectrum will evolve into its steeper form, aligning with the mass function of local embedded clusters \\citep{Lada_araa03} or star clusters in the Large and Small Magellanic Clouds \\citep{Hunter2003}, with a slope $\\alpha\\sim$~2.  As fragmentation proceeds on smaller scales, the mass function would take on yet a steeper character observed in core mass functions \\citep[e.g.][]{alves_cmfimf} and, ultimately, stars \\citep[e.g.][]{Kroupa_imf}.  \n\n\\acknowledgements\nSR is indebted to Doug Johnstone, Fabian Heitsch, Lori Allen and Lee Hartmann for useful suggestions on this work.  SR and EB thank Darek Lis, Carsten Kramer and Joseph Weingartner for their invaluable assistance in the analysis.  This research was supported by {\\em Spitzer} under program ID 3434.  This work was also supported by the National Science Foundation under Grant 0707777.\n\n\\bibliographystyle{apj}  \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{I. Introduction}\nSince the early description of quantum mechanics, the effect of quantum interference led to fascinating and peculiar predictions, one being the appearance of quantum beats caused by interference of emission paths from excited atoms \\cite{Breit1933}. \nNowadays, quantum interference is an essential feature in a quantum network based on quantum optical systems with single atoms as nodes and single photons transferring information between them \\cite{Cirac1997,Kimble2008}. It allows for entanglement between two nodes \\cite{Moehring2007,Slodicka2013,Hofmann2012} and employing coherent phenomena at atom-photon interfaces to faithfully convert quantum information between photonic communication channels and atomic quantum processors \\cite{Ritter2012}. In this context, the controlled emission \\cite{Blinov2004,Maunz2007,Almendros2009,Kurz2013} and absorption \\cite{Piro2011,Huwer2013} of single photons by a single atom is crucial for schemes proposing the heralded mapping of a photonic polarization state into an atomic quantum memory \\cite{Muller2013}. Here we show the quantum coherent character of the absorption and emission of single photons through the interference between indistinguishable quantum channels in a single atom.\n \nFirst experimental observations of quantum beats were induced by pulsed optical excitation of atoms with two upper states which decay to the same ground state \\cite{Dodd1967, Aleksandrov1964}. In continuously excited atomic ensembles, the observation of quantum beats in the correlation of two photons was demonstrated for a coherent superposition state with short lifetime in a calcium cascade \\cite{Aspect1984} and in cavity mediated systems with coherent ground states \\cite{Norris2010}. For a single ion \\cite{Schubert1995}, transient effects of the internal dynamics showed photonic oscillations by interference in absorption. \n\nHere we generate controlled, high-contrast quantum beats in a spontaneous Raman scattering process which consists of the emission of single 393-nm photons after absorption of 854-nm laser photons in a single trapped $^{40}\\text{Ca}^+$ ion. We consider two distinct excitation schemes, called $\\Lambda$ and V. For both schemes we employ the controlled generation of an initial coherent superposition state of two Zeeman sublevels in the metastable D$_{5\/2}$ manifold with phase-coherent laser pulses. Quantum beats are then observed in the arrival-time distribution of the detected 393-nm photons after onset of the 854-nm laser pulse that induces their emission. We show that the observed quantum beats are fully controlled through changes of the phase in the atomic superposition and the photonic polarization input states. In particular, we highlight two distinct physical origins of the quantum beats, namely, quantum interference of two 854-nm absorption amplitudes and two 393-nm emission amplitudes, respectively. The two absorption-emission pathways resemble a which-way experiment where indistinguishability is afforded by a quantum eraser \\cite{Scully1982}.    \n\n\\section{II. Experimental sequence}\n\nThe experimental setup is illustrated in Fig.~\\ref{setup}(a). A single $^{40}$Ca$^+$ ion is trapped in a linear Paul trap and excited by various laser beams. A magnetic field $B$ parallel to the trap axis defines the quantization axis and lifts the degeneracy of the atomic levels [Fig.~\\ref{setup}(b)]. The experimental sequence starts with Doppler cooling on the S$_{1\/2}$-P$_{1\/2}$ transition with a 397-nm laser, aided by an 866-nm laser that repumps the population from the metastable D$_{3\/2}$ state. After cooling, a circularly polarized laser pulse optically pumps the ion to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$. Then it is excited to two selected Zeeman sublevels in the metastable D$_{5\/2}$ manifold, employing a laser at 729~nm which is locked to an ultra-stable high-finesse cavity. Efficient population transfer up to 99.6(3)$\\%$ on each of the two transitions is achieved by frequency-selective coherent pulses. From the difference of the two frequencies we extract directly the Larmor frequency $\\nu_{L}$ in D$_{5\/2}$ [see Eq.~(\\ref{Zeeman-splitting}) below]. The superposition state is excited to the $\\text{P}_{3\/2}$ level by a laser at 854-nm wavelength with controlled polarization and frequency. Its detuning $\\Delta$ from the line center is previously calibrated by a spectroscopic measurement and it is adjusted to provide identical pumping rates for the two initial Zeeman sublevels. The 854-nm beam enters parallel to the quantization axis and its polarization is adjustable to any defined linear or circular state by a combination of $\\lambda\/4$ and $\\lambda\/2$ waveplates. The absorption of the 854-nm laser photons leads to the emission of a 393-nm Raman-scattered photon that is collected perpendicular to the quantization axis by an in-vacuum high-numerical-aperture laser objective (HALO) with a numerical aperture of 0.4 \\cite{Gerber2009}. The photons are selected by their polarization with a polarizing beam splitter (PBS) cube and detected by a photomultiplier tube (PMT). The PMT pulses are fed into a time-correlated single-photon counting module (Pico Harp 300) for temporally correlating them with the sequence trigger, i.e., with the onset of the 854-nm pulse. \n\nAn inherent requirement for quantum interference between two scattering paths is to keep indistinguishability in all degrees of freedom of the involved quantum channels. In the $\\Lambda$- and V-type level configurations, the two absorption channels exhibit a frequency difference originating from the differential Zeeman shift of the two D$_{5\/2}$ sublevels, up to $\\sim$20~MHz for typical magnetic fields of $\\sim$3~G. The PMT time resolution of 300~ps sets a much lower frequency resolution and thus erases the information of this frequency splitting \\cite{Togan2010}. The indistinguishability concerning the polarization is attained by the detection perpendicular to the magnetic field of only $\\left\\lvert\\rm{H}\\right\\rangle$-polarized or only $\\left\\lvert\\rm{V}\\right\\rangle$-polarized 393-nm photons (i.e.,\\ polarization parallel or orthogonal to the quantization axis, respectively).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.6cm]{setup}\n\\caption{(Color online) (a) Schematic of the experimental setup: excitation of a single trapped ion with an 854-nm laser beam parallel to the magnetic field axis and polarization-selective detection of 393-nm photons perpendicular to it. HALO stands for high-numerical-aperture laser objective \\cite{Gerber2009}, PBS for polarizing beam splitter, and PMT for photomultiplier tube. (b) Level scheme and relevant transitions of the $^{40}$Ca$^+$ ion.}\n\\label{setup} \n\\end{figure}\n\n\n\n\\section{III. Theoretical analysis}\n\nA brief and simplified theoretical description of the population transfer from D$_{5\/2}$ to S$_{1\/2}$ is presented which emphasizes the coherent evolution of the internal states for the two different level configurations.\n\nFigure~\\ref{levelscheme_both} shows the $\\Lambda$- and V-type level configurations with the relevant transitions and the squared Clebsch-Gordan coefficients (CGCs).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.4cm]{levelscheme_both}\n\\caption{(Color online) (a) The $\\Lambda$-shaped three-level system consisting of\n$\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$, $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$ and $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$ including the relevant transitions with their squared Clebsch-Gordan coefficients. (b) The V-shaped three-level system consisting of\n$\\left\\lvert\\rm{P}_{3\/2},-\\frac{3}{2}\\right\\rangle$, $\\left\\lvert\\rm{P}_{3\/2},+\\frac{1}{2}\\right\\rangle$ and $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ including the relevant transitions with their squared Clebsch-Gordan coefficients. Red arrows, absorption of 854-nm photons ($\\sigma^+$ and $\\sigma^-$); blue wavy arrows, emission of 393-nm photons; gray arrows, parasitic absorption (854~nm) and emission channels (393~nm).}\n\\label{levelscheme_both}\n\\end{figure}\n\nBoth schemes are initialized in a coherent superposition state,  \n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle = \\sqrt{\\rho_1} \\left\\lvert \\text{D}_{5\/2},m_\\text{D}\\right\\rangle + \\text{e}^{i\\Phi_{\\text{D}}(t)} \\sqrt{\\rho_2}\\left\\lvert \\text{D}_{5\/2},m_{\\text{D'}}\\right\\rangle, \n\\label{initial_state}\n\\end{equation}\nwith populations $\\rho_1$ and $\\rho_2$ adjusted by two consecutive Rabi pulses from $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$. The phase $\\Phi_{\\rm{D}}(t)=\\Phi_{\\rm{D}}(0)+2\\pi\\nu_{L}t$ is composed of a starting phase $\\Phi_{\\rm{D}}(0)$ which is set by the sequence control as the relative phase between the two 729-nm pulses and a part oscillating with the Larmor frequency,\n\\begin{equation}\n \\nu_{L}=\\frac{\\mu_B}{h}\\Delta m_j \\,g_j\\,B,\n \\label{Zeeman-splitting}\n\\end{equation} \nincluding the magnetic field $B$, the Land$\\acute{\\rm{e}}$ factor $g_j=\\frac{6}{5}$, and the Bohr magneton $\\mu_B$.\n\nThe polarization state of the laser photons at 854~nm is defined as a superposition of two orthogonal states, namely, right ($\\left\\lvert\\rm{R}\\right\\rangle$) and left ($\\left\\lvert\\rm{L}\\right\\rangle$) circularly polarized light \n\\begin{equation}\n\\left\\lvert\\psi_{854}\\right\\rangle=\\rm{cos}\\tfrac{\\vartheta}{2}\\left\\lvert\\rm{R}\\right\\rangle+\\rm{sin}\\tfrac{\\vartheta}{2}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{L}\\right\\rangle.\n\\label{state854}\n\\end{equation}\nAny linear photonic polarization state, such as horizontal $\\left\\lvert\\rm{H}\\right\\rangle$, vertical $\\left\\lvert\\rm{V}\\right\\rangle$, diagonal $\\left\\lvert\\rm{D}\\right\\rangle$, and antidiagonal $\\left\\lvert\\rm{A}\\right\\rangle$ is adjusted by rotating the two wave plates [see Fig.~\\ref{setup}(a)] to change $\\Phi_{854}$, while $\\vartheta$ is fixed to $\\pi\/2$. As the propagation direction of the incoming laser photons is chosen parallel to the applied magnetic field, the photon polarization translates to the reference frame of the atom according to \\begin{equation}\n\\left\\lvert\\psi_{854}\\right\\rangle = \\rm{cos}\\tfrac{\\vartheta}{2}\\left\\lvert+1\\right\\rangle + \\rm{sin}\\tfrac{\\vartheta}{2}\\rm{e}^{i\\Phi_{854}}\\left\\lvert-1\\right\\rangle,\n\\end{equation}\nwhereby $\\left\\lvert m_{854} \\right\\rangle = \\left\\lvert \\pm1 \\right\\rangle$ stand for the photon polarizations that effect $\\Delta m = \\pm1$ (i.e., $\\sigma^{\\pm}$) transitions, respectively, between the Zeeman sublevels of D$_{5\/2}$ and P$_{3\/2}$. \n\nThe coupled quantum system is represented by a joint state between photon and atom,\n\\begin{equation}\n\\left\\vert\\psi(t)\\right\\rangle=\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle\\otimes \\left\\lvert\\psi_{854}\\right\\rangle. \n\\label{joint}\n\\end{equation}\nBased on \\cite{Muller2013} the absorption process is described by\n\\begin{equation}\n\\hat{A}=\\sum_{m_{\\rm{D}},m_{\\rm{P}}}C_{m_{\\rm{D}},m_{854},m_{\\rm{P}}} ~ \\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta) \\left\\lvert m_{\\rm{P}}\\right\\rangle\\left\\langle m_{\\rm{D}}\\right\\lvert\\left\\langle m_{854}\\right\\lvert\n\\label{absop}\n\\end{equation}\nwhere $m_{854}=0,\\pm1$, and $C_{m_{\\rm{D}},m_{854},m_{\\rm{P}}}$ are the CGCs \\cite{footnoteCGC}. The detuning-dependent atomic response $\\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta)=\\lvert \\tilde{c}(\\Delta)\\rvert\\rm{e}^{i\\phi(\\Delta)}$ has a complex Lorentzian lineshape with linewidth $\\Gamma_{\\rm{P}_{3\/2}}$. We combine these coefficients according to \n\\begin{equation}\nC_{m_{\\rm{D}},m_{854},m_{\\rm{P}}} ~ \\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta) = c_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta)\n\\end{equation}\nusing $m_{\\rm{P}}=m_{\\rm{D}}+m_{854}$. The detuning $\\Delta=\\omega_{\\rm{L}}-\\omega_{0}$ between the laser frequency $\\omega_{\\rm{L}}$ and the D$_{5\/2}$ to P$_{3\/2}$ line center $\\omega_{0}$ will be used as a control knob to compensate for the different CGCs in the absorption channels.\n\nAfter the absorption, the ion decays in a spontaneous (Raman) emission process to the S$_{1\/2}$ ground state, described by an emission operator,\n\\begin{equation}\n\\hat{E}=\\sum_{m_{\\rm{S}},m_{\\rm{P}}}C_{m_{\\rm{P}},m_{393},m_{\\rm{S}}}\\left\\lvert m_{393}\\right\\rangle\\left\\lvert m_{\\rm{S}}\\right\\rangle\\left\\langle m_{\\rm{P}}\\right\\lvert,\n\\label{emop}\n\\end{equation}\nwith $m_{393}=0,\\pm1$. Applying the absorption and emission operator to the joint state of Eq.~(\\ref{joint}) gives $\\hat{E}\\hat{A}\\left\\vert\\psi(t)\\right\\rangle$, a new joint state between an atom in S$_{1\/2}$ and a single photon in the 393-nm mode, which is projected onto $\\left\\lvert \\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$ conditioned on the detection of a 393-nm photon with certain polarization. \n\n\\subsection{A. The \\texorpdfstring{$\\Lambda$}{Lambda} system\n\nIn the case of the $\\Lambda$ system, the ion is prepared in the coherent superposition state \n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle=\\sqrt{\\tfrac{1}{2}}\\left(\\left\\lvert\\rm{D}_{5\/2},-\\tfrac{3}{2}\\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{D}_{5\/2},+\\tfrac{1}{2}\\right\\rangle\\right),\n\\label{L-superposition}\n\\end{equation}\nsuch that two absorption paths share the same emission channel [Fig.~\\ref{levelscheme_both}(a)]. Applying the absorption operator to the joint state gives\n\\begin{equation}\n\\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\sqrt{\\tfrac{1}{2}}\\cos\\tfrac{\\vartheta}{2}~c_{-\\nicefrac{3}{2},-\\nicefrac{1}{2}}(\\Delta) \\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{2}}\\sin\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{1}{2},-\\nicefrac{1}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{2}}\\cos\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{1}{2},+\\nicefrac{3}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{3}{2}\\right\\rangle.\n\\end{split}\n\\end{equation}\nHighest visibility of the quantum beats is expected when the interfering transitions have equal weights. The amplitudes of the two absorbing paths are determined by the detuning $\\Delta$, which is adjusted in order to compensate for the two different CGCs, i.e., such that $|c_{-\\nicefrac{3}{2},-\\nicefrac{1}{2}}(\\Delta)| = |c_{+\\nicefrac{1}{2},-\\nicefrac{1}{2}}(\\Delta)| = c$.  \nFor a linear photonic polarization state ($\\vartheta=\\tfrac{\\pi}{2}$) it follows that \n\\begin{equation}\n\\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\tfrac{1}{2}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n& \\quad\n+\\tfrac{1}{2}c'\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{3}{2}\\right\\rangle.\n\\end{split}\n\\label{absorbH}\n\\end{equation}\nwith $c' = c_{+\\nicefrac{1}{2},+\\nicefrac{3}{2}}(\\Delta)$. Here the term $\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)$ already shows the interference of the two absorption paths in the amplitude of $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$.\nThe transfer of this oscillation to the emitted 393-nm photons is obtained by applying the emission operator $\\hat{E}$,\n\\begin{equation}\n\\begin{split}\n\\hat{E}\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\sqrt{\\tfrac{1}{6}}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert 0 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{12}}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert -1 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{8}}c'\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert +1 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle.\n\\end{split}\n\\label{state1}\n\\end{equation}\nOf these three different decay channels, selection of $\\Delta m = 0$, i.e., of $\\pi$ photons, is favorable for keeping indistinguishability between the two interfering scattering channels. A $\\pi$ photon detected perpendicular to the magnetic field transforms to an $\\left\\lvert\\rm{H}\\right\\rangle$-polarized photon in the photonic reference frame. The detection of these photons projects the joint state of Eq.~(\\ref{state1}) onto $\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$. The intensity of the emitted light is proportional to \n\\begin{equation}\nI \\propto \\tfrac{1}{3} c^2(1+\\text{cos}\\,(\\Phi_{\\text{D}}(t)+\\Phi_{854}))\n\\label{intensity1}\n\\end{equation}\nand shows the possibility to be controlled by changing the photonic input phase $\\Phi_{854}$ or the atomic superposition phase $\\Phi_{\\rm{D}}(t)$ through the offset phase $\\Phi_{\\rm{D}}(0)$. We note again that interference happens in the absorption process, since two pathways lead to the same excited intermediate state before the emission process takes place.\n\n\\subsection{B. The V system}\nThe V-type configuration and the corresponding excitation scheme for the generation of 393-nm photons is shown in Fig.~\\ref{levelscheme_both}(b). Starting in the eigenstate $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$, the first resonant 729-nm pulse transfers 75$\\%$ of the population to $\\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$. The remaining population is subsequently transferred to $\\left\\lvert\\rm{D}_{5\/2},-\\frac{5}{2}\\right\\rangle$  with a $\\pi$ pulse, resulting in the coherent superposition state\n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle=\\sqrt{\\tfrac{1}{4}}\\left\\lvert\\rm{D}_{5\/2},-\\tfrac{5}{2}\\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\sqrt{\\tfrac{3}{4}}\\left\\lvert\\rm{D}_{5\/2},+\\tfrac{3}{2}\\right\\rangle. \n\\label{V-superposition}\n\\end{equation}\nThe unequal squared CGCs of the two 393-nm decay channels, i.e., the spurious decay into unwanted channels [see Fig.~\\ref{levelscheme_both}(b)], is compensated for by the unequal initial population distribution in D$_{5\/2}$. Applying the absorption operator (\\ref{absop}) to the joint state gives\n\\begin{equation}\n \\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle\n &=\\sqrt{\\tfrac{1}{4}}\\cos\\tfrac{\\vartheta}{2}~c_{-\\nicefrac{5}{2},-\\nicefrac{3}{2}}(\\Delta)\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{3}{2}\\right\\rangle\\\\\n& +\\sqrt{\\tfrac{3}{4}}\\sin\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{3}{2},+\\nicefrac{1}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)} \\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{1}{2}\\right\\rangle.\n \\end{split}\n\\end{equation}\nThe two different CGCs on the 854-nm absorption channels are again compensated by two different 854-nm repumping rates, set by adjusting the detuning $\\Delta$ such that $|c_{-\\nicefrac{5}{2},-\\nicefrac{3}{2}}(\\Delta)| = |c_{+\\nicefrac{3}{2},+\\nicefrac{1}{2}}(\\Delta)| = c$. For a linear 854-nm polarization with phase $\\Phi_{854}$ we get \n\\begin{equation}\n \\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle&=\\sqrt{\\tfrac{1}{8}}c\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{3}{2}\\right\\rangle+\\sqrt{\\tfrac{3}{8}}c~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{1}{2}\\right\\rangle.\\\\\n \\end{split}\n\\end{equation}\n\nApplying the emission operator (\\ref{emop}) results in\n\\begin{equation}\n \\begin{split}\n\\hat{E}\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\left(\\left\\lvert -1 \\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert +1 \\right\\rangle\\right)\\sqrt{\\tfrac{1}{8}}c\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+ \\tfrac{1}{2}c~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert 0 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle.\n\\label{superpos}\n \\end{split}\n\\end{equation}\nThe first term in this state shows the oscillating phase between two atomic ($\\sigma^{\\pm}$) transitions leading to interference of two emission channels when they are projected on the same axis. More precisely, the superposition of $\\sigma^-$ and $\\sigma^+$ emission causes the dipolar emission pattern to rotate with the Larmor frequency about the quantization axis, which leads to a temporal modulation of the detected photons in the direction perpendicular to that axis. Measurement of the photonic superposition $\\left\\lvert\\rm{V}\\right\\rangle=\\sqrt{\\frac{1}{2}}\\left(\\left\\lvert\\sigma^+\\right\\rangle-\\left\\lvert\\sigma^-\\right\\rangle\\right)$ projects the joint state of Eq.~(\\ref{superpos}) onto $\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$, which results in an intensity  \n\\begin{equation}\nI \\propto \\tfrac{1}{8}c^2(1-\\rm{cos}\\,(\\Phi_{\\rm{D}}(t)+\\Phi_{854})),\n\\label{intensity}\n\\end{equation}\nshowing again oscillations with the Larmor frequency.\nThe second term in (\\ref{superpos}) describes the emission of parasitic $\\pi$ photons which transforms in the photonic reference frame to $\\left\\lvert\\rm{H}\\right\\rangle$. They are suppressed by rotating the PBS by $90^{\\circ}$ with respect to the $\\Lambda$ case.\n\n\\section{IV. Experimental Results}\n\\subsection{A. Quantum beats in arrival-time distributions}\n\nWe first discuss the experimental results for the $\\Lambda$ scheme. The blue circles in Fig.~$\\ref{L-incoherent}$ show the arrival-time distribution of the 393-nm Raman-scattered photons for the case of an initial coherent superposition between $\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$ and $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$. The exponential decay of the photon wave packet is modulated with a period of 106 ns, corresponding to a Larmor frequency in D$_{5\/2}$ of 9.4 MHz, which is in agreement with the frequency difference of the initially populated Zeeman sublevels, as determined from 729-nm spectroscopy. The data points are fitted by numerically solving the optical Bloch equations, including all relevant Zeeman sublevels and the projection of the final joint state [Eq.~(\\ref{state1})] according to the detection of an $\\left\\lvert\\rm{H}\\right\\rangle$-polarized photon. The visibility of the oscillation, determined from the envelope of the fit (gray dotted line), is 93.1(6)$\\%$. For comparison, we also show the photon arrival-time distribution for the case of an initial statistical mixture in D$_{5\/2}$ (black dots). The data set is generated by averaging the two individually recorded arrival-time distributions for the $\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$ to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ and  $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$ to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ scattering processes. From the fit of the exponential decay (green line) we get an 1\/e decay time of 461(2) ns. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.4cm]{L-incoherent}\n\\caption{(Color online) Arrival-time distribution of single 393 nm photons from an initial coherent superposition (blue circles) and statistical mixture (black dots) in D$_{5\/2}$. Red solid line, fit to the data by numerically solving the 18-level optical Bloch equations; green dash-dotted line, exponential fit to the data for the mixture; gray dotted line, exponential fit to the envelope of the oscillation, from which the visibility is determined.}\n\\label{L-incoherent}\n\\end{figure}\n\nIn the following we show that we have full control over the quantum phases that enter into the quantum beats of Fig.~\\ref{L-incoherent}. First we study their dependence on the linear polarization of the absorbed 854-nm photons. Rotation of the wave plates allows adjustment of the incoming linear polarization, i.e., of the photonic phase $\\Phi_{854}$ in Eq.~(\\ref{intensity1}), to the canonical basis states $\\left\\lvert\\rm{H}\\right\\rangle$, $\\left\\lvert\\rm{V}\\right\\rangle$, $\\left\\lvert\\rm{D}\\right\\rangle$, and $\\left\\lvert\\rm{A}\\right\\rangle$ (and any value in between). Figure~\\ref{L-osc-pol+phase}(a) shows arrival-time distributions for the polarization input states $\\left\\lvert\\text{D}\\right\\rangle$ and $\\left\\lvert\\text{A}\\right\\rangle$ (for the sake of clarity, we only show these two). The phase difference of $\\Delta \\Phi_{854}=180^{\\circ}$ set by the waveplates is revealed in the two oscillations; the value derived from fitting the Bloch equations is $178.2(1.6)^{\\circ}$. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{L-osc-pol+phase}\n\\caption{(Color online) $\\Lambda$ scheme. (a) Arrival-time distributions of the 393-nm photons showing quantum beats for two different 854-nm input polarization states, $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. (b) Arrival-time distributions showing quantum beats for two values, 0 and $\\pi$, of the phase $\\Phi_{\\rm D}(0)$ of the initial atomic superposition. In (a) and (b), the bin size is 2~ns for 6~min measurement time. }\n\\label{L-osc-pol+phase}\n\\end{figure}\n\nSimilarly, according to Eq.~(\\ref{intensity1}), also the phase of the initial atomic superposition state $\\Phi_{\\rm D}(0)$ enters directly into the quantum beats. We control this phase via the radio-frequency source that drives the acousto-optic modulator setting the amplitude of the 729-nm laser. Figure~\\ref{L-osc-pol+phase}(b) displays the change in the phase of the quantum beats effected by changing $\\Phi_{\\rm D}(0)$ by $180^{\\circ}$, while keeping $\\Phi_{854}$ constant; the Bloch equation fits yield a phase difference of $181.1(1.1)^{\\circ}$. The $<1$\\% deviation between the set values and the fitted values highlights the precise control that we have over the quantum beats. \nWith the V scheme, we achieve analogous control through changes of both the photonic phase, i.e.\\ the polarization of the incoming 854-nm photons, and the atomic phase, i.e.\\ the difference phase of the preparing 729-nm pulses. Figure~\\ref{V-osc-pol+phase}(a) displays the arrival-time distributions when the 854-nm polarization is adjusted to the orthogonal basis states $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. The Bloch equation fits yield a phase difference of 176.5(1.7)$^{\\circ}$. In Fig.~\\ref{V-osc-pol+phase}(b) the offset phase $\\Phi_{\\rm D}(0)$ of the coherent superposition in D$_{5\/2}$ is set to 0 and $\\pi$. Here the phase difference from the fits is $181.4(1.5)^{\\circ}$. The deviation by 1-2\\% from the ideal values confirms the precise control over the quantum-beat phase also for the case of the V scheme. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-osc-pol+phase}\n\\caption{(Color online) V scheme. (a) Quantum beats for two different 854-nm input polarization states, $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. (b) Quantum beats for two values, 0 and $\\pi$, of the phase $\\Phi_{\\rm D}(0)$ of the initial atomic superposition. The bin size is 2~ns for 10~min measurement time [6~min in (b)]. }\n\\label{V-osc-pol+phase}\n\\end{figure}\n\nDue to the spurious decay from $|\\rm{P}_{3\/2},+\\frac{1}{2}\\rangle$ to $|\\rm{S}_{1\/2},+\\frac{1}{2}\\rangle$ [see Fig.~\\ref{levelscheme_both}(b)] in the V scheme, maximum visibility of the quantum beats requires an unequal population ratio in the initial superposition state, as can be seen from Eq.~(\\ref{V-superposition}). The data of Fig.~\\ref{V-osc-pol+phase} are acquired with the optimum population ratio, which is experimentally verified by the following procedure: From the two pulses which prepare the coherent superposition, the duration of the first $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle \\to \\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$ pulse is varied, thereby adjusting the amount of transferred population. The subsequent $\\pi$ pulse to $\\left\\lvert\\rm{D}_{5\/2},-\\frac{5}{2}\\right\\rangle$ transfers the remaining population. In Fig.~\\ref{best-pop} the quantum-beat visibility is shown as a function of the population in $\\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$. The highest visibility of 78.2(9)$\\%$ is obtained for 75$\\%$ population, in agreement with the ratio of the two involved CGCs. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-best-pop}\n\\caption{(Color online) Quantum-beat visibility in the V scheme as a function of the initial population in $|\\rm{D}_{5\/2},+\\frac{3}{2}\\rangle$. The data points are calculated from exponential fits to the envelopes of the observed quantum beats.}\n\\label{best-pop}\n\\end{figure}\n\nThe reduced visibility of the quantum beats in the V scheme (Fig.~\\ref{V-osc-pol+phase}) compared to the $\\Lambda$ scheme (Fig.~\\ref{L-osc-pol+phase}) results from the nature of the respective interference phenomena. In the $\\Lambda$ scheme the excitation amplitudes to the P level interfere constructively or destructively; hence, the overall probability for 393-nm photon emission is modulated. In contrast, in the V scheme the photons are emitted with a spatially rotating pattern. Therefore, the HALO that collects the 393-nm photons in $\\sim 4\\%$ solid angle will also pick up a small fraction of the dipolar emission when the preferred emission direction is perpendicular to the HALO axis. \n \n\n\\subsection{B. Phase-dependent photon-scattering probability}\nThe quantum beats in Figs.~\\ref{L-osc-pol+phase} and ~\\ref{V-osc-pol+phase} extend over many periods of the underlying Larmor precession; therefore, the total probability to detect a photon (i.e., time-integrated over the whole wave packet) is nearly independent of the control phases $\\Phi_{854}$ and $\\Phi_{\\rm D}(0)$. This behavior changes, however, when the exciting laser pulse is shorter than the quantum-beat period (or at least of comparable duration). In this case the time-integrated probability of detecting a photon may be significantly enhanced or suppressed by the quantum interference. In order to emphasize this effect, we created quantum beats with high 854-nm laser power, such that the total duration of the generated photon wave packet was as short as $\\sim$70~ns (including about 60~ns of broadening by the acousto-optic modulator rise time); at the same time, we reduced the magnetic field to extend the Larmor period. In Fig.~\\ref{det-eff-both}, the time-integrated photon detection probability for the case of a short excitation pulse is plotted against the atomic control phase $\\Phi_{\\rm D}(0)$ (dots). With the $\\Lambda$-type level scheme [Fig.~\\ref{det-eff-both}(a)] we find a variation by a factor of 7 ($\\sim76\\%$ visibility), while for the V-scheme [Fig.~\\ref{det-eff-both}(b)] the ratio is about 3 ($\\sim48\\%$ visibility). For comparison, the phase-dependent photon detection probability for a long photon, covering many quantum-beat periods, is also shown (crosses). Here the modulation is hardly visible; it disappears asymptotically. Hence, in the regime of short excitation pulses compared to the Larmor-precession period, the atomic control phase serves as a knob to determine the probability with which a photon is scattered into the detector. With faster modulation of the exciting laser than what could be attained in our setup, even much larger enhancement-suppression ratios may be reached. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{det-eff-both}\n\\caption{(Color online) Suppression and enhancement of the generation efficiency of 393~nm photons controlled via the atomic phase $\\Phi_{\\rm D}(0)$. The dots and solid curves (sinusoidal fits) are for short 854~nm excitation pulses (duration $<$ quantum-beat period), while the crosses and dashed curves are for long pulses (duration $\\gg$ quantum-beat period). (a) $\\Lambda$ scheme: The short photon has about 70~ns duration and 553~ns quantum-beat period, the long photon about 850~ns duration and 106.5~ns quantum-beat period. (b) V scheme: The short photon extends over $\\sim$120~ns at 277.4~ns beat period, for the long photon the values are 750 and 53.2~ns. }\n\\label{det-eff-both}\n\\end{figure}\n\n\n\\section{V. PHYSICAL MECHANISM}\n\nIn this final section we highlight the fundamental difference between the interference processes involved in the two schemes. This is done by analyzing the time dependence of the population in D$_{5\/2}$ during the excitation with 854~nm light: In the experiment, the excitation is interrupted after a certain pulse duration and the remaining population in the D level is determined through state-selective fluorescence \\cite{Roos1999}. For the $\\Lambda$ scheme, the result is displayed in Fig.~\\ref{L-stairs}. Here the input polarization is set to $\\left\\lvert\\rm{V}\\right\\rangle$, and the magnetic field has a value of 0.987~G, which results in a Larmor period between the Zeeman sublevels of 302 ns. The pulse length of the 854-nm excitation is varied in steps of 12.5 ns. The figure shows that the depopulation of the D$_{5\/2}$ manifold exhibits a \"stairs\"-like modulation with the Larmor frequency. The population dynamics are very well fitted by 18-level Bloch equations (solid line in Fig.~\\ref{L-stairs}). The inset in Fig.~$\\ref{L-stairs}$ shows the time derivative of the population in D$_{5\/2}$, which is proportional to the population in P$_{3\/2}$. Comparison with Fig.~\\ref{L-osc-pol+phase} confirms that the observed quantum beats in the case of the $\\Lambda$ scheme are due to a corresponding oscillation of the population of the excited P level. This manifests that the interference happens indeed between the two absorption pathways from D$_{5\/2}$ to P$_{3\/2}$: As these require $\\sigma^+$- and $\\sigma^-$-polarized light, respectively, the Larmor-precessing phase of the initial superposition state creates an oscillatory excitation probability for any fixed linear incoming polarization.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{L-stairs}\n\\caption{(Color online) Change of the D$_{5\/2}$ population for different 854-nm pulse lengths in the $\\Lambda$ scheme. The stairs-like behavior results from interference of two excitation amplitudes to $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$. The pulse length is varied in steps of 12.5~ns. The solid line is calculated with the 18-level Bloch equations including experimental parameters. (Inset) Time derivative of the calculated solid line showing the oscillation of the population in P$_{3\/2}$, which leads to suppression and enhancement of the photon emission at 393~nm.}\n\\label{L-stairs}\n\\end{figure} \n\nIn contrast, a similar measurement for the case of the V scheme, shown in Fig.~\\ref{V-nostairs}, does not exhibit any modulation on the exponential depopulation curve of the D$_{5\/2}$ level. The Bloch equation fit describes excitation from D$_{5\/2}$ to P$_{3\/2}$ at a constant rate, i.e., with no interference in the excitation pathways. The observed quantum beats in the emitted 393-nm photon wave packet are therefore due to interference of emission amplitudes: The Larmor precession of the initial state enters into the emitted superposition of $\\sigma^+$- and $\\sigma^-$-polarized components and causes a spatially rotating emission pattern, which leads to an oscillatory detection probability. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-nostairs}\n\\caption{(Color online) Depopulation of D$_{5\/2}$ for different 854-nm pulse lengths in the V scheme, showing simple exponential decay without modulation. The points are measured data; the line is a fit by Bloch equation dynamics. (Inset) Time derivative of the calculated curve \\cite{footnoteV}.}\n\\label{V-nostairs}\n\\end{figure} \n\n \nFurther evidence for our explanation of the physical mechanism is provided by measuring the depletion of the D$_{5\/2}$ level as a function of the initial phase, $\\Phi_{\\rm D}(0)$, after an excitation pulse of fixed duration. A comparison of the results for the two distinct level schemes, under otherwise equal conditions, is displayed in Fig.~\\ref{depletion}. The phase enters into the depletion in the $\\Lambda$ case, while the V case is practically insensitive to it. \n \nThe two types of quantum beats of Figs.~\\ref{L-osc-pol+phase} and \\ref{V-osc-pol+phase}, while looking very similar, are hence identified to have manifestly different physical origins. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{depletion}\n\\caption{(Color online) Depopulation of D$_{5\/2}$ for different intial phases $\\Phi_{\\rm D}(0)$ of the initial superposition, after an excitation pulse of 12.5~ns. The blue dots (solid curve) are measured (calculated) for the $\\Lambda$ scheme, the red crosses (dashed curve) for the V scheme. The Larmor period was 300~ns ($\\Lambda$) and 150~ns (V).}\n\\label{depletion}\n\\end{figure} \n\n\\section{V. Summary}\nWe investigated experimentally two cases of single-photon quantum beats originating from interference of photon-scattering amplitudes. A single trapped Ca$^+$ ion is prepared in a coherent superposition state in its D$_{5\/2}$ manifold and then excited on the D$_{5\/2}$ to P$_{3\/2}$ transition (854~nm), whereby it releases a single photon on the P$_{3\/2}$ to S$_{1\/2}$ transition (393~nm). Quantum beats with the frequency of the atomic Larmor precession are manifested as oscillations in the photon arrival-time distribution with $>93\\%$ visibility. The phase of the quantum beats is controlled through setting the phase of the initial atomic superposition and through the polarization of the exciting 854-nm light. For a $\\Lambda$- and a V-type atomic level scheme, we identified interference of absorption and emission amplitudes, respectively, to be the physical mechanisms behind the quantum beats. The two mechanisms are fundamentally different in that the remaining population in the D$_{5\/2}$ level reveals the interference in the $\\Lambda$ case but not in the V case. As a consequence, for small ratios between the excitation pulse length and the Larmor period, excitation out of D$_{5\/2}$ can be efficiently suppressed and enhanced in the $\\Lambda$ scheme through the choice of the control phases, while in the V case the excitation probability is phase insensitive. The presented experimental techniques and the physical mechanisms are essential ingredients for mapping arbitrary polarization states of photons onto a single ion \\cite{Kurz2014}.\n\n\n\\vspace{0.2cm}\n\\begin{acknowledgments}\nWe acknowledge support by the BMBF (Verbundprojekt QuOReP, CHIST-ERA project QScale), the German Scholars Organization \/ Alfried Krupp von Bohlen und Halbach-Stiftung, and the ESF (IOTA COST Action).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n Since the first observations of baryonic decays of $B$ mesons\nby ARGUS~\\cite{1} and CLEO~\\cite{2}, many three-body baryonic $B$ \ndecays have been found~\\cite{3}. \nAlthough the general pattern of these decays \ncan be understood intuitively from heavy $b$ quark decays~\\cite{4}, \nmany specific details cannot be explained by this simple picture. \n\nUsing a generalized factorization \napproach, Ref.~\\cite{5} predicts rather large branching fractions \n($\\sim$$10^{-5}$) for the Cabibbo-suppressed processes\n$B \\to \\overline{p} \\Lambda  D^{(*)} $. \nThe branching fractions of other related \nbaryonic decays such as $B^0 \\to p\\overline{p}D^0$~\\cite{6,7},\n~$B^0 \\to p\\overline{p}K^{*0}$~\\cite{8}, $B^- \\to p\\overline{p}K^{*-}$~\\cite{9,10} and \n$B^- \\to p\\overline{p}\\pi^-$~\\cite{9} are used as\ninputs in such estimates because  baryon form factors entering\nthe decay amplitudes are difficult to calculate from first principles. \nThe expected values of the branching fractions for  $B^- \\to \\overline{p}\\Lambda D^0$ and \n$B^- \\to \\overline{p}\\Lambda D^{*0}$\nare already within reach with the data sample accumulated at Belle.\n\nNearly all  baryonic $B$ decays into three- and\nfour-body final states possess a common feature: baryon-antibaryon \ninvariant masses that peak near threshold. This \nthreshold enhancement is found both in charmed and charmless \ncases~\\cite{3}. A similar effect has been observed \nin $J\/\\psi \\to p\\overline{p}\\gamma$ decays by BES~\\cite{bes1,bes2} \nand CLEO~\\cite{cleo1}, but is not seen in \n$J\/\\psi \\to p \\overline{p} \\pi^0 $~\\cite{bes1} and \n$\\Upsilon(1S) \\to  p\\overline{p}\\gamma$~\\cite{cleo2}. One of the possible \nexplanations of this phenomenon suggested in the literature is a final state \n$N\\overline{N}$ interaction~\\cite{int}.\n\nIn this paper, we present results on the $B^- \\to \\overline{p}\\Lambda D^{(*)0}$ decays in order to test the  factorization hypothesis\nand study the $\\overline{p}\\Lambda$ threshold enhancement effect.\n\n\n\nThe data sample used in the study corresponds to an integrated luminosity of 605 \nfb$^{-1}$, containing 657 $ \\times 10^6~B\\overline{B}$ pairs, collected at the $\\Upsilon(4S)$ resonance\nwith the Belle detector at the KEKB asymmetric-energy\n$e^+e^-$ (3.5~GeV and 8~GeV) collider~\\cite{KEKB}.\nThe Belle detector~\\cite{Belle} is a large-solid-angle magnetic spectrometer that consists of\n a silicon vertex detector (SVD),\na 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov\ncounters (ACC), a barrel-like arrangement of time-of-flight scintillation counters \n(TOF), and an electromagnetic calorimeter (ECL) composed of CsI(Tl) crystals \nlocated inside a superconducting solenoid coil that provides a 1.5~T magnetic field.\n\n\nThe selection criteria for the final state charged particles in $B^{-} \\to \\pld$ and $B^{-} \\to \\pldst$ are \nbased on information obtained from the tracking system (SVD and CDC) and \nthe hadron identification system (CDC, ACC, and TOF). \nThe primary and $D^0$ daughter charged tracks\nare required to have a point of\nclosest approach to the interaction point (IP) that is within\n$\\pm$0.3 cm in the transverse ($x$--$y$) plane, and within $\\pm$3.0 cm\nin the $z$ direction, where the $+z$ axis is opposite to the\npositron beam direction.\nFor each track, the likelihood values $L_p$, $L_K$, or $L_\\pi$ that it is a proton, kaon, or pion,\nrespectively, are determined from the information provided by the hadron identification system. \nA track is identified as a proton if $L_{p}\/(L_p +L_{K}) > 0.6$ and $L_{p}\/(L_p +L_{\\pi}) > 0.6$, as a kaon\nif $L_{K}\/(L_K +L_{\\pi}) > 0.6$, or as a pion if $L_{\\pi}\/(L_K +L_{\\pi}) > 0.6$.\n The efficiency for identifying a kaon (pion) is 85$-$95\\% depending on the momentum of the track, while the probability for a pion\n(kaon) to be misidentified as a kaon (pion) is 10$-$20\\%.\nThe proton identification efficiency is 84\\% while the probability for a kaon or a pion to be misidentified as a proton is less than 10\\%.\n\nWe reconstruct $\\Lambda$'s from their decays to $p \\pi^-$. \nEach $\\Lambda$ candidate must have a displaced vertex and the direction of its momentum vector must be\nconsistent with an origin at the IP.\nThe proton-like daughter is required to satisfy the proton criteria described above, and no further selections are applied\nto the daughter tracks. \nThe reconstructed $\\Lambda$ mass is required to be in the range 1.111 GeV\/$c^2 < M_{p\\pi^-} <1.121 $ GeV\/$c^2$~\\cite{3}.\n\nCandidate $D^0$ mesons are reconstructed in the following two sub-decay channels: \n$D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$, $\\pi^0 \\to \\gamma\\gamma$. \nThe $\\gamma$'s that constitute $\\pi^0$ candidates are required to have \nenergies greater than 50 MeV if the $\\gamma$ is reconstructed from the barrel ECL\nand greater than 100 MeV for the endcap ECL, and not be associated with any charged\ntracks in CDC.\nThe energy asymmetry of $\\gamma$'s from a $\\pi^0 $, $\\frac{|E_{\\gamma1}-E_{\\gamma2}|}{E_{\\gamma1}+E_{\\gamma2}}$, \nis required to be less than 0.9.\nThe mass of a $\\pi^0$ candidate is required to be within the range \n$0.118$ GeV\/$c^2 < M_{\\gamma\\gamma}<0.150$ GeV\/$c^2$ before a mass-constrained fit is applied to improve the $\\pi^0$ momentum resolution. We impose a cut on the invariant masses of the $D^0$ candidates, $|M_{K^-\\pi^+}-1.865$ GeV\/$c^2|<0.01$ GeV\/$c^2$ and $1.837$ \nGeV\/$c^2 <M_{K^-\\pi^+\\pi^0}<1.885$ GeV\/$c^2$ for $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$, respectively, which retains about 87\\% of the signal.\n\nWe reconstruct $D^{*0}$ mesons in the decay mode $D^{*0} \\to D^0 \\pi^0$ \nwith $D^0 \\to K^- \\pi^+$ only. \nSince the $\\pi^0$ coming from the $D^{*0}$ decay is expected to have low energy, we adjust the\nphoton selection criteria accordingly.\nThe energy of $\\gamma$'s that constitute $\\pi^0$ candidates from a $D^{*0}$ must be greater than 50 MeV.\nThe energy asymmetry of the two $\\gamma$'s is required to be less than 0.6\nand the di-photon invariant mass should be in the range $0.120$ GeV\/$c^2< M_{\\gamma\\gamma}<0.158$ GeV\/$c^2$. \nFor the $D^{*0}$ candidates, we require $0.139$ GeV\/$c^2< \\Delta M < 0.145$ GeV\/$c^2$, where $\\Delta M$ denotes the mass difference between $D^{*0}$ and $D^{0}$. \n\nCandidate $B$ mesons are identified with two kinematic variables calculated \nin the center-of-mass (CM) frame: the beam-energy-constrained mass \n$M_{\\rm bc} = \\sqrt{E^2_{\\rm beam}-p^2_B}$, and the energy difference ${\\Delta{E}} = E_B - E_{\\rm beam}$, \nwhere $E_{\\rm beam}$ is the beam energy, and $p_B$ and $E_B$ are the momentum and energy, \nrespectively, of the reconstructed $B$ meson. \nIn order to reduce the contribution from combinatoric backgrounds, we define the candidate region for $B^{-} \\to \\pld$ ($\\pldst$) as $5.2$ GeV\/$c^2 < M_{\\rm bc} < 5.3$ GeV\/$c^2$, $-0.1$ GeV $ < {\\Delta{E}}< 0.4$ GeV and ${M_{\\overline{p}\\Lambda}}< 3.4$ ($3.3$) GeV\/$c^2$, where ${M_{\\overline{p}\\Lambda}}$ denotes the invariant mass of the baryon pair.\nThe lower bound in ${\\Delta{E}}$ is chosen to exclude backgrounds from multibody baryonic $B$ decays.\nFrom Monte Carlo (MC) simulations based on GEANT~\\cite{geant},\nwe define the signal region as $5.27$ GeV\/$c^2 < M_{\\rm bc} < 5.29$ GeV\/$c^2$ \nand $|{\\Delta{E}}|< 0.05$ GeV.\n\n\nThe dominant background for $B^{-} \\to \\pld$ in the candidate region is from continuum $e^+e^-\n\\to q\\overline{q}$ ($q = u,\\ d,\\ s,\\ c$) processes. \nWe suppress the jet-like continuum background relative to the more\nspherical $B\\overline{B}$ signal using a Fisher discriminant \nthat combines seven event-shape variables derived from modified Fox-Wolfram moments~\\cite{rr-104} as described in Ref.~\\cite{rr-551}.\nThe Fisher discriminant is a linear combination of several variables with coefficients that are optimized to separate signal and background.\nIn addition to the Fisher discriminant, two variables $\\cos\\theta_{B}$ and $\\Delta z$ are used to form signal and background probability density functions (PDFs). \nThe variable $\\theta_{B}$ is the angle between the reconstructed $B$ direction and\nthe beam axis in the CM frame,\nand $\\Delta z$ is the difference between the $z$ positions of the candidate $B$ vertex \nand the vertex of the rest of the final state particles, presumably, from the other $B$ in the $\\Upsilon({\\rm 4S})$ decay. \nThe products of the above PDFs, obtained from signal and continuum MC simulations, \ngive the event-by-event signal and background likelihoods, ${\\mathcal L}_S$ and ${\\mathcal L}_B$. \nWe apply a selection on the likelihood ratio, ${\\lr} = {\\mathcal L}_S\/({\\mathcal L}_S+{\\mathcal L}_B)$ to suppress background.\nInformation associated with the accompanying B meson can also be used to distinguish $B$ events from continuum events. The\nvariables used are ``$q$'' and ``$r$'' from a $B$ flavor-tagging algorithm~\\cite{flavor_tag}. \nThe value of the preferred flavor $q$ equals $+1$ for $B^0 \/ B^+$ \nand $-1$ for $\\overline{B}^0 \/ B^-$. \nThe $B$ tagging quality factor $r$ ranges from zero\nfor no flavor information to unity for unambiguous flavor assignment.\nSets of $q \\times r$-dependent $\\lr$  selection requirements are optimized by maximizing a figure of merit defined as $N_S \/ \\sqrt{N_S+N_B}$, where $N_{S}$ denotes the expected number of signal events based on MC simulation and the predicted branching fraction, and $N_{B}$ denotes the expected number of background events from the continuum MC.\nThe requirements on $\\lr$ remove 75\\% (89\\%) of continuum background\nwhile retaining 88\\% (69\\%) of the signal for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$ ($ D^0 \\to K^-\\pi^+\\pi^0$). \nThe continuum background suppression is not applied for $B^{-} \\to \\pldst$ since the optimal $\\lr$ requirement is close to zero.\n\nIn order to avoid multiple counting, \nin cases where more than one $B$ candidate is found in a single event,\nwe choose the one with the smallest $\\chi^2 = \\chi^2_{B} + \\chi^2_{\\Lambda} (+\\chi^2_{\\pi^0})$, where $\\chi^2$ \nis calculated from the vertex fit to the $B$ using $\\overline{p}$ and $D^{0}$ measurements, the vertex fit to $\\Lambda$ using $p$ and $\\pi^-$ tracks, and the mass-constrained $\\pi^0$ fit if applicable. \nThe fraction of events with multiple candidates are 2.3\\% (14.1\\%) of the sample according to \nMC simulations for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$ ($D^0 \\to K^-\\pi^+ \\pi^0$), \nand 17.8\\% for $B^{-} \\to \\pldst$. \nThe dominant background for $B^{-} \\to \\pldst$ is \nfrom $\\overline{ B^0} \\to \\pldstp$ cross-feed and $B^{-} \\to \\pldst$ self cross-feed (both referred to as CF) events according to a MC simulation based on \\textsc{Pythia}~\\cite{Pythia}.\nIn CF events,  two low-energy $\\gamma$'s can form a $\\pi^0$ candidate that is combined with a correctly reconstructed  $D^0$, $\\overline{p}$\nand $\\Lambda$ from a $B$ decay to form a candidate event in the signal region.\nThese backgrounds cannot be distinguished from the signal in the $M_{\\rm bc} - {\\Delta{E}}$ two-dimensional fit alone, \nalthough their distributions in $M_{\\rm bc}$ and ${\\Delta{E}}$ have a slightly wider spread than the signal.  \nWe can, however, estimate this background contribution by analyzing the $\\Delta M$ \ndistribution in which signal events have a Gaussian\nshape and background events have a threshold function shape as shown in Fig.~\\ref{fg:fit_dm_scf3}.\n\n\n\n\n\n\n\\begin{figure}[htb]\n\\vskip -3.1cm\n\\begin{sideways}  ~~~~~~~~~~~~~~~~~~~~~~Events \/ (0.8 MeV\/$c^2$)  \\end{sideways}\n\\hskip -0.5cm\n\\includegraphics[width=0.32\\textwidth]{.\/Fig1.eps}\\\\\n\\vskip -0.3cm\n\\vskip -4.cm \\hskip 4.cm{MC}\n\\vskip 3.6cm \n\\hskip 3.5cm {$\\Delta M $ (GeV\/$c^2$)}\n\\caption{\nThe $\\Delta M$ distribution of the $B^{-} \\to \\pldst$ MC sample with fit curves overlaid,  where $\\Delta M$ denotes the mass difference between $D^{*0}$ and $D^{0}$. \nThe solid curve is the overall fit result, the dashed curve shows the CF background and the black filled squares are the MC events. }\n\\label{fg:fit_dm_scf3}\n\\end{figure}\n\nThe signal yields the of $B^{-} \\to \\pldstzp$ modes are extracted from \na two-dimensional extended unbinned maximum likelihood fit with the likelihood defined as \n\\begin{equation}\nL = \\frac{e^{-\\sum_{j}N_j}}{ N!}\\prod_{i=1}^{N}(\\sum_{j}N_jP_i^j),\n\\end{equation}\nwhere $N$ is the total number of candidate events, $N_j $ denotes the number of corresponding category events and $P_i^j$ represents the corresponding\ntwo-dimensional PDF in $M_{\\rm bc}$ and ${\\Delta{E}}$; $i$ denotes the $i$-th event,\nand $j$ indicates the index of different event categories in the fit.\nThus, $j$ could either indicate signal or combinatorial background for the $\\pld$ case and includes one more category (CF) for the $\\pldst$ case. \nWe use a Gaussian function to represent the $M_{\\rm bc}$ signal and a double Gaussian function for the \n ${\\Delta{E}}$ signal with parameters determined using MC simulations.\nCombinatorial background is described by an ARGUS function~\\cite{Argus} and \na second-order Chebyshev polynomial in the $M_{\\rm bc}$ and ${\\Delta{E}}$ distributions, respectively.\n\nSince it is difficult to separate the $B^{-} \\to \\pldst$ signal and CF events in the fit, \nwe estimate the number of CF events in the $\\Delta M$ signal region \n($0.139$ GeV\/$c^2 < \\Delta M <$ 0.145 GeV$\/c^2$) from the fitted CF yield in the $\\Delta M$ sideband \nregion ($0.15$ GeV\/$c^2 < \\Delta M < 0.17$ GeV$\/c^2$). \nThe ratio of the area of the CF in the $\\Delta M$ signal region to that in the sideband region is $26.0\\pm0.9\\%$, which is determined from MC samples of $B^{-} \\to \\pldst$ (Fig.~1) and $\\overline{B^{0}} \\to \\pldstp$.\nThe PDF used for CF events is a product of a Gaussian-like smoothed histogram for $M_{\\rm bc}$ and a \ndouble Gaussian function for ${\\Delta{E}}$ with parameters determined using MC simulations.   \nWe fix the number of CF events in the $M_{\\rm bc}-{\\Delta{E}}$ fit to determine the \nsignal yield within the $\\Delta M$ signal region.\n\nFigure \\ref{fg:data_fit1} shows the result of the fit for $B^{-} \\to \\pld$.\nThe fitted signal yields in the data sample are $26.5^{+6.3}_{-5.6}$ and $35.6^{+11.7}_{-10.7}$ events with statistical significances of 7.6 and 3.6 standard deviations ($\\sigma$) for $B^{-} \\to \\pld, ~D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$, respectively.\nThe significance is defined as $\\sqrt{-2{\\rm ln}(L_0\/L_{\\rm max})}$,\nwhere $L_0$ and $L_{\\rm max}$ are the likelihood values returned by the fit with a signal yield fixed to zero and  the nominal fit, respectively. \nThe branching fractions are calculated using the formula\n\\begin{eqnarray}\n\\nonumber \\mathcal{B} =\\frac{N_{signal}}{\\epsilon \\times f \\times N_{B\\overline{B}} },\n\\end{eqnarray}\nwhere $N_{signal}$, $N_{B\\overline{B}}$, $\\epsilon$, and $f$ are the fitted number of signal events, the number of $B\\overline{B}$ pairs, the reconstruction efficiency, and the relevant sub-decay branching fractions:\n$\\mathcal{B}(\\Lambda \\to p\\pi^-) ~= 63.9 \\pm 0.5 \\%$,\n$\\mathcal{B}(D^0 \\to K^-\\pi^+)  ~= 3.89 \\pm 0.05\\% $,\n$\\mathcal{B}(D^0 \\to K^-\\pi^+\\pi^0)  ~= 13.9 \\pm 0.5\\% $, and\n$\\mathcal{B}(D^{*0} \\to D^0\\pi^0) ~= 61.9 \\pm 2.9\\%$~\\cite{3}. \nWe assume that charged and neutral $B\\overline{B}$ pairs are equally produced at the $\\Upsilon(4S)$.\n\nTo investigate the threshold enhancement feature, we\ndetermine the differential branching fractions in bins of $M_{\\overline{p}{\\Lambda}}$; \nthe results obtained from the weighted averages of the fits to $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$ separately are shown in Fig.~\\ref{fg:pld0_mpl} where an enhancement near threshold is evident.\nWe fit the $\\overline{p}{\\Lambda}$ mass spectrum with a threshold function and then reweight MC events to match the fitted threshold\nfunction in order to obtain a proper estimate of the reconstruction efficiency \nfor signal events. \nThe observed branching fractions are \n$ (1.39^{+0.33}_{-0.29} \\pm 0.16)\\times 10^{-5}$ for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$ and\n$ (1.54^{+0.50}_{-0.46} \\pm 0.26)\\times 10^{-5}$ for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+\\pi^0$. \nThe weighted average of the branching fractions is $(1.43^{+0.28}_{-0.25} \\pm 0.18)\\times 10^{-5}$ with a significance of 8.1$\\sigma$, where the\nsystematic uncertainties (described below) on the signal yield are also included in the significance evaluation.\n\nThe fit results for $B^{-} \\to \\pldst$ in the $\\Delta M$ sideband region are shown in Fig.~\\ref{fg:data_fit2} (a, b). \nThe number of CF events in the sideband  is $11.6 \\pm 5.4$, which is used to estimate the number of CF events in the $\\Delta M$ signal region, $3.0 \\pm 1.4$, after scaling by the \n area ratio of CF (26.0$\\pm$ 0.9\\%). \nWe then fix the normalization of the CF component \nin the fit to the $\\Delta M$ signal region [fit results are shown in Fig.~\\ref{fg:data_fit2} (c, d)], and \nobtain a signal yield of $4.3^{+3.2}_{-2.4}$ with a statistical significance of 2.2$\\sigma$.\nAssuming $B^{-} \\to \\pldst$ and $B^{-} \\to \\pld$ have the same $\\overline{p}{\\Lambda}$ spectrum,\nwe determine $\\mathcal{B}(B^{-} \\to \\pldst)$ to be $(1.53^{+1.12}_{-0.85} \\pm 0.47)\\times 10^{-5}$. \nIn the absence of a statistically compelling signal yield,\n we set an upper limit $\\mathcal{B}(B^{-} \\to \\pldst) < 4.8 \\times 10^{-5} $ at the 90\\% confidence level\nusing the Feldman-Cousins method~\\cite{FC, pole}. \nThe information used to obtain the upper limit includes the number of events in the signal region (13) and $8.1  \\pm 1.4$ background events.\nHere, the background that is integrated in the signal region, consists of 5.3$\\pm$0.5 continuum events and 2.9$\\pm$1.3 CF events from the fit to the $\\Delta M$ sideband. \nThe 11.7\\% additive systematic uncertainty due to the selection criteria is included in the determination  of the upper limit on the branching fraction.\\\\\n\n\n\n\\begin{figure}[htb]\n \\hskip  2.35 cm  {\\bf{(a)}} \\hskip  3.15 cm  {\\bf{(b)}}\n \\vskip -1.2cm\n \\includegraphics[width=0.45\\textwidth]{.\/Fig2ab.eps}\\\\\n \\hskip  2.35 cm  {\\bf{(c)}} \\hskip  3.15 cm  {\\bf{(d)}}\n  \\vskip -1.2cm\n\\includegraphics[width=0.45\\textwidth]{.\/Fig2cd.eps}\\\\\n\\caption{ \nDistributions of ${\\Delta{E}}$ (a, c) for $M_{\\rm bc} > 5.27$ GeV\/$c^2$ and of $M_{\\rm bc}$ (b, d) for $|{\\Delta{E}}| <0.05 $ GeV; \nthe top row is the fit result for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ $ (a, b) and the bottom row for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ \\pi^0$ (c, d). \nThe points with error bars are data; the solid\ncurve shows the fit; the dashed curve represents the signal, \nand the dotted curve indicates continuum background.}\n\\label{fg:data_fit1}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{sideways}   {~~~~~~~~~~~~~~~~~~~~$d\\mathcal{B} \/ dM_{\\overline{p}{\\Lambda}}$ $(10^{-6})$ \/ ( 0.2 GeV \/$c^2)$ }  \\end{sideways}\n \\hskip  -0.5 cm \n{\\includegraphics[width=0.35\\textwidth]{.\/Fig3.eps}}\\\\\n\\vskip -0.9cm  \n\\hskip  3.5 cm  \n {\\large $M_{\\overline{p}{\\Lambda}} $ (GeV\/$c^2$)}\n\\caption{ Differential branching fraction ($d\\mathcal{B} \/ dM_{\\overline{p}{\\Lambda}}$) as a function of the $\\overline{p}{\\Lambda}$ mass for $B^{-} \\to \\pld$.\nNote that the last bin with the central value of 3 GeV\/$c^2$ has a bin width of 0.8 GeV\/$c^2$. The solid curve is a fit with a threshold function.\n}\n\\label{fg:pld0_mpl}\n\\end{figure}\n\n\\begin{figure}[htb]\n \\vskip  0.0cm\n \\hskip  2.cm  {\\bf{(a)}} \\hskip  3.15 cm  {\\bf{(b)}}\n \\vskip -1.2cm\n\\includegraphics[width=0.43\\textwidth]{.\/Fig4ab.eps}\\\\\n \\hskip  2. cm  {\\bf{(c)}} \\hskip  3.15 cm  {\\bf{(d)}}\n  \\vskip -1.2cm\n\\includegraphics[width=0.43\\textwidth]{.\/Fig4cd.eps}\n\\caption{\nDistributions of ${\\Delta{E}}$ (a, c) for $M_{\\rm bc} > 5.27$ GeV\/$c^2$ and of $M_{\\rm bc}$ (b,d) for $|{\\Delta{E}}| <0.05 $ GeV; \nthe top row is the fit result for $B^{-} \\to \\pldst$ in the $\\Delta M$ sideband region (a,b) and the bottom row for $B^{-} \\to \\pldst$ in the $\\Delta M$ signal region (c, d). \nThe points with error bars are data; the solid curve shows the result of the fit; \nthe dot-dashed and dotted curve indicates the CF and continuum background; \nthe dashed curve represents the signal.}\n\\label{fg:data_fit2}\n\\end{figure}\n\n\\begin{table}\n\\caption{ Summary of the results: event yield, significance, efficiency, and branching\nfraction.}\n\\tabcolsep= 6pt\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{clcccc}\n\\hline\\hline\n Mode        & $N_{signal}$ &\t$\\mathcal{S}$        &   $\\epsilon$(\\%) &\t    $\\mathcal{B}( 10^{-5})$  \\\\\n\\hline\\hline \n$\\overline{p} \\Lambda D^0_{K^-\\pi^+}$ \t         & $26.5^{+6.3}_{-5.6}$    & 7.4  &\t 11.7 & \t $1.39^{+0.33}_{-0.29} \\pm 0.16$\\\\\n$\\overline{p} \\Lambda D^0_{K^-\\pi^+ \\pi^0}$     & $35.6^{+11.7}_{-10.7}$    & 3.4  &\t 4.0  &   $1.54^{+0.50}_{-0.46} \\pm 0.26$\t\\\\    \n\\hline\n$B^{-} \\to \\pld$\t \t\t\t\t\t\t    \t &               & 8.1  &\t \t\t   &   $1.43^{+0.28}_{-0.25} \\pm 0.18$\t\\\\ \n\\hline\\hline \n$B^{-} \\to \\pldst$      \t\t\t & $4.3^{+3.2}_{-2.4}$    & 2.1  &\t 2.8 & \t $1.53^{+1.12}_{-0.85} \\pm 0.47$\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\label{data_yields}\n\\end{table}\t\t\t\n\n\nSystematic uncertainties are estimated using high-statistics control samples.\nA track reconstruction efficiency uncertainty of 1.2\\% is assigned for each track.\nFor the proton identification efficiency uncertainty, we use a $\\Lambda \\to p \\pi^-$ sample, and for\n$K-\\pi$ identification uncertainty we use a sample of kinematically identified  $D^{*+} \\to D^0\\pi^+$,\n $D^0 \\to K^-\\pi^+$ decays. \nThe average efficiency discrepancy due to hadron identification differences \nbetween data and MC simulations has been corrected for the final branching fraction measurements. \nThe corrections due to the  hadron identification are 10.7\\% and 10.6\\% for $B^{-} \\to \\pld$ and $B^{-} \\to \\pldst$, respectively.\nThe uncertainties associated with the  hadron identification corrections are 4.2\\% for \ntwo protons (one from $\\Lambda$ decay), 0.5\\% for a charged pion, and 1.0\\% for a charged kaon.\n\nThe $\\pi^0$ selection uncertainty is found to be 5.0\\% by comparing the ratios of efficiencies between $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$ for data and MC samples.\nIn the $\\Lambda$ reconstruction, we find an uncertainty of $4.1\\%$ from the differences between data and MC for the efficiencies of tracks displaced from the interaction point, the $\\Lambda$ proper time distributions, and the $\\Lambda$ mass spectrum.\nThe uncertainty due to the $\\mathcal R$ selection for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+$ is estimated \nfrom the control sample $B^{-} \\to {D}^0 \\pi^-$, $D^0 \\to K^-\\pi^+\\pi^-\\pi^+$ and is determined to be 1.3\\%.\nThe $\\mathcal R$ related uncertainty for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ \\pi^0$ is 3.0\\% estimated \nfrom $B^{-} \\to {D}^0 \\pi^-$, $D^0 \\to K^-\\pi^+\\pi^0$.\nThe uncertainties due to the $D^0$ mass selection for $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$ are 1.9\\% and 1.6\\%, respectively.\n\nThe dominant systematic uncertainty for $B^{-} \\to \\pld$ is due to the modeling of PDFs, estimated by including a $B \\to \\pld\\pi$ or a nonresonant $B^{-} \\to \\overline{p} \\Lambda K^-\\pi^+ (K^-\\pi^+ \\pi^0)$ component in the fit, \nmodifying the efficiency after changing the signal ${M_{\\overline{p}\\Lambda}}$ distribution, and varying the parameters of the signal and background PDFs by one standard deviation using MC samples.\nThe modeling uncertainties are 7.5\\% and 12.9\\% for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+ \\pi^0$, respectively. \nThe overall modeling uncertainty for $B^{-} \\to \\pldst$ of 28.6\\% is obtained from two kinds of PDF modifications.\nThe parameters of the fixed CF component are varied by their $\\pm 1\\sigma$ statistical uncertainties, which were obtained from the fit to the $\\Delta M$ sideband region.\nWe also include an additional PDF for the \ncombinatorial background based on the \\textsc{Pythia}~\\cite{Pythia} $b$ quark fragmentation process, e.g.,\n$B^{-} \\to \\overline{p} \\Lambda {D}^{0}$, $B^{+} \\to \\overline{p} \\Delta^{++} {D}^{*0}$, $B^{-} \\to \\overline{p} \\Delta^{0} {D}^{*0}$, $B^{-} \\to \\overline{p} \\Sigma^0 {D}^{*0}$, etc.\n\nThe systematic uncertainties from the sub-decay branching fractions are calculated from the corresponding \nbranching uncertainties in~\\cite{3}; they are 1.5\\% (3.7\\%) and 6.0\\% \nfor $B^{-} \\to \\pld,~D^0 \\to K^-\\pi^+$ ($D^0 \\to K^-\\pi^+\\pi^0$) and $B^{-} \\to \\pldst$, respectively.\nThe uncertainty in the number of $B\\overline{B}$ pairs is 1.4\\%.\nThe total systematic uncertainties are 11.6\\% (17.1\\%) and 30.9\\% for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$\n ($D^0 \\to K^-\\pi^+ \\pi^0$) and $B^{-} \\to \\pldst$, respectively.\nThe final results are listed in Table~\\ref{data_yields}, where the significance values are\nmodified and include the systematic uncertainty related to PDF modeling.\n\nIn summary, using a sample of $657 \\times 10^6~B\\overline{B}$ events, we report the first \nobservation of $B^{-} \\to \\pld$ with a \nbranching fraction of $(1.43^{+0.28}_{-0.25} \\pm 0.18)\\times 10^{-5}$ and a significance of 8.1$\\sigma$.\nNo significant signal is found for $B^{-} \\to \\pldst$ and the corresponding upper limit is $4.8 \\times 10^{-5}$ at the 90\\% confidence level.\nWe also observe a $\\overline{p}{\\Lambda}$ enhancement near\nthreshold for $B^{-} \\to \\pld$, which is similar to a common feature found in charmless three-body \nbaryonic $B$ decays~\\cite{3}. \nThe measured $B^{-} \\to \\pld$ branching fraction agrees with the theoretical prediction \nof $(1.14 \\pm 0.26) \\times 10 ^{-5} $~\\cite{5}. This indicates that the generalized factorization approach with parameters determined from experimental data gives reasonable estimates for $b \\to c$ decays. \nThis information can be helpful for future theoretical studies of the angular distribution puzzle in the penguin-dominated processes, $B^- \\to p\\overline{p}K^{-}$ and $B^{0} \\to {p\\overline{\\Lambda}\\pi^-}$~\\cite{5}.\nThe measured branching fraction for $B^{-} \\to \\pld$ can also be used to tune the parameters in the event generator, e.g., \n\\textsc{Pythia}, for fragmentation processes involving $b$ quarks. \nAlthough the current statistics for $B^{-} \\to \\pld$ are still too low to perform\nan angular analysis of the baryon-antibaryon system, the proposed super flavor factories~\\cite{SFF1,SFF2}\n offer promising venues for such studies. \n\\\\\n\n\nWe thank the KEKB group for the excellent operation of the\naccelerator, the KEK cryogenics group for the efficient\noperation of the solenoid, and the KEK computer group and\nthe National Institute of Informatics for valuable computing\nand SINET4 network support.  We acknowledge support from\nthe Ministry of Education, Culture, Sports, Science, and\nTechnology (MEXT) of Japan, the Japan Society for the \nPromotion of Science (JSPS), and the Tau-Lepton Physics \nResearch Center of Nagoya University; \nthe Australian Research Council and the Australian \nDepartment of Industry, Innovation, Science and Research;\nthe National Natural Science Foundation of China under\ncontract No.~10575109, 10775142, 10875115 and 10825524; \nthe Ministry of Education, Youth and Sports of the Czech \nRepublic under contract No.~LA10033 and MSM0021620859;\nthe Department of Science and Technology of India; \nthe BK21 and WCU program of the Ministry Education Science and\nTechnology, National Research Foundation of Korea,\nand NSDC of the Korea Institute of Science and Technology Information;\nthe Polish Ministry of Science and Higher Education;\nthe Ministry of Education and Science of the Russian\nFederation and the Russian Federal Agency for Atomic Energy;\nthe Slovenian Research Agency;  the Swiss\nNational Science Foundation; the National Science Council\nand the Ministry of Education of Taiwan; and the U.S.\\\nDepartment of Energy.\nThis work is supported by a Grant-in-Aid from MEXT for \nScience Research in a Priority Area (``New Development of \nFlavor Physics''), and from JSPS for Creative Scientific \nResearch (``Evolution of Tau-lepton Physics'').\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nIt is generally believed that most of the stars and galaxies can be described \nin good approximation as fluid bodies in thermodynamical equilibrium. In \nthe framework of general relativity, this implies (see e.g.\\ \n\\cite{hartle,lindblom}) that the corresponding spacetimes are stationary and \naxisymmetric. Moreover it is usually assumed (though there is no proof \nknown to us) that they are equatorially symmetric.  \nThis stresses the importance of the study of stationary axisymmetric \nspacetimes. A relativistic treatment is necessary for rapidly \nrotating and massive compact objects like pulsars, neutron stars and \nblack-holes. \n\nThough the importance of global solutions describing stationary \naxisymmetric fluid bodies is generally accepted, the complicated structure \nof the Einstein equations with matter gives little hope that such solutions \ncan be found in the near future. Only for special and somewhat unphysical \nequations of state \n\\cite{wahl,kramer,seno}, it was possible to give solutions in the \nmatter region which are discussed as candidates for an interior solution. \nIn the exterior vacuum region, however, powerful solution generating techniques \nare at hand. Since the surface $\\Gamma_z$\nof a compact astrophysical object constitutes a \nnatural boundary at which the metric functions are not continuously \ndifferentiable, one is looking \nfor solutions to the vacuum equations\nthat are analytic outside this contour and can be at least \ncontinuously extended to $\\Gamma_z$. This means that the typical problem \none has to consider for the vacuum Einstein equations is a boundary value \nproblem of Dirichlet, von Neumann or mixed type, see \\cite{hp2}.  The \nmatter then enters only in form of boundary conditions for the vacuum \nequations.  This is possible if an interior solution is known or if only \ntwo--dimensionally extended bodies like disks or shells are considered. \nIn the latter case,\nthe surfacelike distribution of the matter implies \nthat the matter equations reduce to ordinary differential equations. \nNotice that disks are important models in astrophysics for certain types \nof galaxies.\n\nThe reason why it is much more promising to treat only the vacuum case \nis the equivalence of the stationary axisymmetric Einstein equations \nto a single nonlinear differential equation for a complex \npotential, the so called \nErnst equation \\cite{ernst}. The latter belongs to a family of \ncompletely integrable nonlinear equations that are studied as\nthe integrability conditions for associated linear differential systems. The \ncommon feature of these linear systems is that they contain an additional \nvariable, the so called spectral parameter, which reflects an underlying \nsymmetry of the differential equations under investigation, in the case of \nthe Ernst equation the Geroch group \\cite{geroch}. \nAssociated linear systems for the Ernst equations \nwere given in \\cite{belzak,maison,neuglinear}. \nThe existence of this parameter can be used to construct solutions by\nprescribing the singular structure of the matrix of the linear system with\nrespect to the spectral parameter.\n\nOne of the most successful solution techniques for nonlinear differential\nequations rests on methods of\nalgebraic geometry and leads to the so called finite-gap solutions that can be\nexpressed elegantly in terms of theta functions. Such methods were first used\nto construct periodic and quasiperiodic\nsolutions to nonlinear evolution equations like the Korteweg--de~Vries \n(KdV) and the Sine--Gordon (SG) equation.  For a\nsurvey of this subject we refer the reader to~\\cite{soliton1,algebro}. \nHowever, it was only recently that \nalgebro-geometrical methods were applied to the Ernst equation,\nsee~\\cite{korot1}. The found solutions differ from similar \nsolutions of other equations in several aspects, e.g.\\ they are in general \nnot  periodic or quasi-periodic. The main difference is that \nthis class is much richer than previously obtained ones.  \n\nThe development of solution techniques yields a deeper insight into the\nstructure of nonlinear differential equations. However, from a practical point\nof view, it would also be desirable to solve initial value problems or, for the\nErnst equation, boundary value problems.\nOne approach to solve boundary value problems of the above mentioned\ntype with the help of the linear system is to translate the \nphysical boundary conditions into a Riemann-Hilbert problem which is\nequivalent to a linear integral equation, see~\\cite{soliton1}.\nNeugebauer and Meinel~\\cite{ngbml1}\nsucceeded in doing this in the case of the rigidly rotating dust disk.  They\nwere able to reduce the matrix problem on a sphere to a scalar\nRiemann-Hilbert problem\non a hyperelliptic Riemann surface which can be solved explicitly via \nquadratures. By making use of the gauge transformations of the linear system\nwe were able to show~\\cite{prd} that this is possible in\ngeneral if the boundary value problem leads to a Riemann-Hilbert\nproblem with rational jump data. Up to now there is however no direct \nway to infer the jump data from the boundary value problem one wants to solve.\nThe explicit form of the hyperelliptic solutions possibly offers a different\napproach to boundary value problems: one can try to identify the free \nparameters in the solutions, a real valued function and a set of complex\nparameters, the branch points of the hyperelliptic Riemann surface, from \nthe problem one wants to solve. \n\nTo this end we study a class of solutions -- which is essentially\nequivalent to \\cite{korot1} and \\cite{meinelneugebauer} --\nthat is constructed via a  \ngeneralized Riemann--Hilbert problem on a hyperelliptic Riemann surface. \nWe present a complete  discussion of the singularity\nstructure  of these Ernst potentials. It is possible to \nidentify a subclass of solutions that are everywhere regular except  \nat some contour, which can possibly be related to the surface of an\nisolated body, where the Ernst potential is bounded. These solutions \nare asymptotically flat and\nequatorially symmetric, and thus show all the features one might expect \nfrom the exterior solution for an isolated relativistic ideal fluid. \nThey can have a Minkowskian and an extreme relativistic limit in \nwhich the body is `hidden' behind a horizon, and the exterior solution \nbecomes the extreme Kerr solution. This provides the hope that further \nsolutions to physically interesting boundary value problems to the Ernst\nequation, besides the rigidly rotating dust disk, can be identified within this\nclass. First results on this subclass where published in \\cite{prl}.\n\nThe paper is organized as follows. In section~\\ref{sec2} we introduce the \nlinear system associated to the Ernst equation and  discuss how the matrix \nof the system has to be constructed in order to end up with new solutions \nto the Ernst equation. Using the results of~\\cite{krichever1}, we show how\nRiemann surfaces arise naturally in the context of linear systems with a\nspectral parameter. In the case of the Ernst equation, these are hyperelliptic\nRiemann surfaces with a special structure of the branch points. We will\nrestrict ourselves to regular compact Riemann surfaces and are eventually led\nto consider families of hyperelliptic Riemann surfaces of arbitrary genus,\nparametrized by the physical coordinates.\n\nIn section 3 we recall some basic notions of the theory of Riemann surfaces,\ntheta functions and the solution of Riemann--Hilbert problems on\nRiemann surfaces due to Zverovich, and present the class of solutions. It \nis shown that the solution of the axisymmetric Laplace equation which can \nbe freely prescribed in \\cite{meinelneugebauer}\nis a period of the Abelian integrals \nwhich determine the singularity structure of the matrix of the linear \nsystem. The differential relations between these periods are a subset of \nthe so called Picard--Fuchs equations which we write down for the Ernst \nequation.\nIn section 4 we discuss the singularity structure of these\nsolutions. It is shown that the solutions can have a regular axis and \nare in general asymptotically flat. Using an identity for \ntheta functions, we are able to give in section 5 \ncompact formulas for two metric \nfunctions and a simple condition for the occurrence of ergospheres.\nA subclass of solutions with equatorial symmetry is presented in section 6. \nThe common physical features of this subclass like  the \nextreme relativistic limit are discussed.  \nIn section 7, we use the equatorial symmetry to give simplified formulae \nfor the potential in the equatorial plane  and on the axis. Since the rigidly \nrotating dust disk belongs to the simplest non-static solutions which are\nof genus 2, we consider this case in detail in section 8. In section 9, we \nsummarize the results and add some concluding remarks. \n\n\n\\section{Linear System for the Ernst equation and \nMonodromy matrix}\\label{sec2}\n\\setcounter{equation}{0}\n\nIt is well known (see \\cite{exac}) that the metric of stationary axisymmetric \nvacuum spacetimes can be written in the  Weyl--Lewis--Papapetrou form\n\\begin{equation}\\label{3.1}\n\t{\\rm d} s^2 =-{\\rm e}^{2U}({\\rm d} t+a{\\rm d} \\phi)^2+{\\rm e}^{-2U}\n\t\\left({\\rm e}^{2k}({\\rm d} \\rho^2+{\\rm d} \\zeta^2)+\n\t\\rho^2{\\rm d} \\phi^2\\right)\n\t\\label{vac1}\n\\end{equation}\nwhere $\\rho$ and $\\zeta$ are Weyl's canonical coordinates and \n$\\partial_{t}$ and $\\partial_{\\phi}$ are the two commuting asymptotically\ntimelike respectively spacelike Killing vectors. \n\nIn this case the vacuum field equations are equivalent \nto the Ernst equation for the \ncomplex potential $f$ where $f={\\rm e}^{2U}+{\\rm i}b$, and where\nthe real function $b$ \nis related to the metric functions via\n\\begin{equation}\\label{3.2}\n\tb_{,z}=-\\frac{{\\rm i}}{\\rho}{\\rm e}^{4U}a_{,z}\n\t\\label{vac9}.\n\\end{equation}\nHere the complex variable $z$ stands for $z=\\rho+{\\rm i}\\zeta$. With these\nsettings, the Ernst equation reads\n\\begin{equation}\\label{3.3}\n\tf_{z\\bar{z}}+\\frac{1}{2(z+\\bar{z})}(f_{\\bar{z}}+f_z)=\\frac{2 }{f+\\bar{f}}\n\tf_z f_{\\bar{z}}\n        \\label{vac10}\\enspace,\n\\end{equation}\nwhere a bar denotes complex conjugation in $\\bigBbb{C}$. With a solution $f$,\nthe metric function $U$ follows directly from the definition of the Ernst \npotential whereas $a$ can be obtained from (\\ref{vac9}) via quadratures. \nThe metric function $k$ can be calculated from the relation\n\\begin{equation}\\label{3.4}\n\t\tk_{,z}  =  2\\rho \\left(U_{,z}\\right)^2-\\frac{1}{2\\rho}{\\rm e}^{4U}\n                \\left(a_{,z}\\right)^2.\n\t\\label{vac8}\n\\end{equation}\nThe integrability condition of (\\ref{vac9}) and  (\\ref{vac8}) is the \nErnst equation.\n\nThe remarkable feature of the Ernst equation is that it is completely \nintegrable. This means that it can be considered as the integrability \ncondition of an overdetermined linear differential system for a \nmatrix valued function $\\Phi$ that contains an additional variable, the so \ncalled spectral parameter $K$. The occurrence of the linear system with a \nspectral parameter is a consequence of the symmetry group of the Ernst \nequation, the Geroch group \\cite{geroch}. Several forms of the linear system \nare known \nin the literature (\\cite{belzak,maison,neuglinear}). They are related \nthrough gauge transformations (see \\cite{cosgrove}). The choice of a \nspecific form of the linear system is equivalent to a gauge fixing. We \nwill use the form of \\cite{neuglinear},\n\\alpheqn\n\\begin{eqnarray}\n\t\\Phi_{,z}(K,\\mu_0;z,\\bar{z})& = & \\left\\{\\left(\n\t\\begin{array}{cc}\n\t\tN & 0  \\\\\n\t\t0 & M\n\t\\end{array}\n\t\\right)\n        +\\frac{K-{\\rm i}\\bar{z}}{\\mu_0(K)}\\left(\n\t\\begin{array}{cc}\n\t\t0 & N  \\\\\n\t\tM & 0\n\t\\end{array}\n        \\right)\\right\\}\\Phi(K,\\mu_0;z,\\bar{z})\\doteq W\\Phi\\enspace,\n\t\\label{lin1} \\\\\n\t\\Phi_{,\\bar{z}}(K,\\mu_0;z,\\bar{z}) & = & \\left\\{\\left(\n\t\\begin{array}{cc}\n\t\t\\bar{M} & 0  \\\\\n\t\t0 & \\bar{N}\n\t\\end{array}\n\t\\right)\n        +\\frac{K+{\\rm i}z}{\\mu_0(K)}\\left(\n\t\\begin{array}{cc}\n\t\t0 & \\bar{M}  \\\\\n\t\t\\bar{N} & 0\n\t\\end{array}\n        \\right)\\right\\}\\Phi(K,\\mu_0;z,\\bar{z})\\doteq V\\Phi\n\t\\label{lin2}\n\\end{eqnarray}\n\\reseteqn\nwhere\n\\begin{equation}\\label{3.6}\n\tM  =  \\frac{f_z}{f+\\bar{f}}, \\quad\n\tN  =  \\frac{\\bar{f}_z}{f+\\bar{f}}.\n\\end{equation}\nObviously $M$ and $N$ depend only on the coordinates $z$ and $\\bar{z}$ \nand not on the spectral parameter $K$ that lives on the Riemann surface\n${\\cal L}(z,\\bar{z})={\\cal L}$  given by $\\mu_0^2(K)=(K-{\\rm i}\n\\bar{z})(K+{\\rm i}z)$. Notice that\n${\\cal L}$ is a Riemann surface of genus zero with coordinate dependent \nbranch points. This is a special feature of the family of chiral field \nequations to which the Ernst equation belongs that has no counterpart among \nthe completely integrable nonlinear evolution equations for which \nalgebro-geometric solutions have been constructed first.\n\nOn $\\cal L$ we have an involutive map $\\sigma$, defined by\n\\begin{equation}\\label{3.61}\n{\\cal L}\\ni P=(K,\\pm\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\to\\sigma(P)\\equiv\nP^{\\sigma}=(K,\\mp\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\in{\\cal L}\\enspace,\n\\end{equation}\nand an anti--holomorphic involution $\\tau$, defined by\n\\begin{equation}\\label{3.62}\n{\\cal L}\\ni P=(K,\\pm\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\to\\tau(P)\\equiv\n\\bar{P}=(\\bar{K},\\pm\\sqrt{(\\bar{K}-{\\rm i}\\bar{z})(\\bar{K}+\n{\\rm i}z)})\\in{\\cal L}\\enspace.\n\\end{equation}\n\nIt is possible to use the existence of the above linear system for the \nconstruction of solutions to the Ernst equation. To this end one \ninvestigates the singularity structure of the matrices $\\Phi_{z}\\Phi^{-1}$ and \n$\\Phi_{\\bar{z}}\\Phi^{-1}$ with respect to the spectral parameter and infers \na set of conditions for the  matrix $\\Phi$ \n(at least twice differentiable with respect \nto $z$, $\\bar{z}$) that satisfies the linear system \n(\\ref{lin1}) and (\\ref{lin2}). \nThis is done (see e.g.~\\cite{korot1}) in\n\\begin{theorem}\\label{theorem2.1}\n Let $\\Phi(P)$ ($P\\in{\\cal L}$) be a $2\\times 2$--matrix with the following\nproperties:\\\\\nI. $\\Phi(P)$  is holomorphic and invertible at the branch\npoints $P_0=-{\\rm i}z$ and $\\bar{P}_0$ such that the logarithmic derivative\n$\\Phi_{z}\\Phi^{-1}$ diverges as $(K+{\\rm i}z)^{\\frac{1}{2}}$ at $P_0$ and \n$\\Phi_{\\bar{z}}\\Phi^{-1}$ as $(K-{\\rm i}\\bar{z})^{\\frac{1}{2}}$ at\n$\\bar{P}_0$.\\\\\nII. All singularities of $\\Phi$ on ${\\cal L}$ (poles, essential\nsingularities, zeros of the determinant of $\\Phi$, branch cuts and branch\n\tpoints) are regular which means that the logarithmic derivatives \n\t$\\Phi_{z}\\Phi^{-1}$ and $\\Phi_{\\bar{z}}\\Phi^{-1}$ are holomorphic in the \n        neigbourhood of the singular points (this implies they have to be\n        independent\n        of $z$, $\\bar{z}$). In particular $\\Phi(P)$ should have\\\\\n\ta) regular singularities at the points $A_i\\in {\\cal L}$ ($i=1,\\dots,n$)\n        which do not depend on $z$, $\\bar{z}$,\\\\\n\tb)  regular essential singularities at the points $S_i$ \n        ($i=1,\\dots,m$) which do not depend on $z$, $\\bar{z}$,\\\\\n        c)  boundary values at a set of (orientable, piecewise smooth)\n\tcontours $\\Gamma_i \\subset {\\cal L}$ ($i=1,\\dots,l$)  \n        independent of $z$, $\\bar{z}$, which are related on both sides of the\n        contours via\n\t\\begin{equation}\\label{3.9}\n\\left.\\Phi_- (P)=\\Phi_+ (P) {\\cal G}_i(P)\\right|_{P\\in\\Gamma_i}\\enspace.\n          \\label{lin6}\n\t\\end{equation}\n\twhere  ${\\cal G}_i(P)$ are matrices independent of $z$, $\\bar{z}$ with \n        H\\\"older--continuous components and non--vanishing determinant.\\\\\nIII. $\\Phi$ satisfies the reduction condition\n\t\\begin{equation}\\label{3.10}\n\t        \\Phi(P^{\\sigma}) = \\sigma_3 \\Phi(P) \\gamma\\enspace,\n\t\t\\label{lin7}\n\t\\end{equation}\n\twhere $\\sigma_3$ is the third Pauli matrix, and where $\\gamma$ is an \n        invertible matrix independent of $z$ and $\\bar{z}$.\\\\\n\tIV. The normalization and reality condition\n\\begin{equation}\n\t\\Phi(P=\\infty^+)=\\left(\n\t\\begin{array}{rr}\n\t\t\\bar{f} & 1  \\\\\n\t\tf & -1\n\t\\end{array}\n\t\\right)\n\t\\label{lin9}.\n\\end{equation}\n\nThen the function $f$ in (\\ref{lin9}) is a solution to the Ernst equation.\n\\end{theorem}\nA proof of this Theorem may be obtained by comparing the above matrix\n$\\Phi$ with the linear system (\\ref{lin1}) and (\\ref{lin2}).\n\\begin{proof}\nBecause of I, $\\Phi$ and $\\Phi^{-1}$ can be expanded in a series in\n$t=\\sqrt{K+iz}$ and $t'=\\sqrt{K-i\\bar{z}}$ in a neighbourhood of $P=P_0$\nand $P=\\bar{P}_0\\neq P_0$ respectively at all points $P_0$, $\\bar{P}_0$ which \ndo not belong to the singularities given in II. This implies that $\\Phi_z\n\\Phi^{-1}=\\alpha_0\/t+\\alpha_1+\\alpha_2t +\\cdots$. We recognize that, because of\nI and II, $\\Phi_z \\Phi^{-1}-\\alpha_0\/t$ is a holomorphic function. The\nnormalization condition IV implies that this quantity is bounded at \ninfinity. According to Liouvilles theorem, it is a constant. \nSince $\\Phi$, $\\Phi^{-1}$ and $\\Phi_z$ are single valued functions on ${\\cal \nL}$, they must be functions of $K$ and $\\mu_0$. Therefore we have $\\Phi_z\n\\Phi^{-1}=\\beta_0 \\sqrt{\\frac{K-\\bar{P}_0}{K-P_0}}+\\beta_1$. The matrix \n$\\beta_0$ must be \nindependent of $K$ and $\\mu$ since $\\Phi_z \\Phi^{-1}$ must have the same \nnumber of zeros and poles on ${\\cal L}$. The structure of the matrices \n$\\beta_0$ and $\\beta_1$ follows from III. From the normalization condition IV,\nit follows that $\\Phi_z \\Phi^{-1}$ has the structure of (\\ref{lin1}).\nThe corresponding equation for $\\Phi_{\\bar{z}}\\Phi^{-1}$ can be obtained \nin the same way.\n\\end{proof}\n\nFor a given Ernst potential $f$, the matrix $\\Phi$ in the above theorem is \nnot uniquely determined. This reflects the fact that the gauge is not uniquely \nfixed in the linear system (\\ref{lin1}) and (\\ref{lin2}). \nIf we choose without loss of generality \n$\\gamma=\\sigma_1$ (the first Pauli matrix), \nthe remaining gauge freedom can be seen from \n\\begin{corollary}\n        Let $\\Phi(P)$ be a matrix subject to the conditions of\nTheorem~\\ref{theorem2.1},\nand $C(K)$ be a $2\\times2$-matrix that only depends on $K\\in \\bigBbb{C}$\nwith the properties\n\\begin{eqnarray}\n\tC(K) & = & \\alpha_1(K) \\hat{1}+\\alpha_2(K) \\sigma_1,\n\t\\nonumber \\\\\n\t\\alpha_1(\\infty) &=  &1,\\quad \\alpha_2(\\infty)=0.\n\t\\label{c}\n\\end{eqnarray}\nThen the matrix $\\Phi'(P)=\\Phi(P) C(K)$ also satisfies the conditions of\nTheorem~\\ref{theorem2.1} and $\\Phi'(\\infty^+)=\\Phi(\\infty^+)$.\n\\end{corollary}\nIt is this gauge freedom to which we refer when we speak of the gauge \nfreedom of the linear system in the following.\n\nIt is interesting to note that the metric function $a$ can be obtained \nfrom a given matrix $\\Phi$ without solving the equation (\\ref{vac9}), see \n\\cite{korot1}. We get  \n\\begin{proposition}\n        Let $\\delta$ be a local parameter in the vicinity of $\\infty^-$. Then\n\t\\begin{equation}\n                (a-a_0)e^{2U}={\\rm i}(\\Phi_{11}-\\Phi_{12})_{,\\delta}\\enspace,\n\t\t\\label{a}\n\t\\end{equation}\n\twhere $a_0$ is a constant that is fixed by the condition that $a=0$ on \n\tthe regular part of the axis and at spatial infinity, and where \n\t$\\Phi_{,\\delta}$ denotes the linear term in the expansion of $\\Phi$ in \n\t$\\delta$ divided by $\\delta$. \n\\end{proposition}\nThe proof follows from the linear system (\\ref{lin1}) and (\\ref{lin2}).\n\\begin{proof}\n\tIt is straightforward to check the relation \n\t\\begin{equation}\n\t\t(\\Phi^{-1}\\Phi_{,\\delta})_{,z}=\\Phi^{-1}(\\Phi_{z}\\Phi^{-1})_{,\\delta} \n\\Phi\n                \\label{a1}\\enspace.\n\t\\end{equation}\nWith  (\\ref{lin1}), we get \n\\begin{equation}\n\\left(\\Phi^{-1}\\Phi_{\\delta}\\right)_{21,z}=\\frac{{\\rm i}\\rho}{(f+\\bar{f})^2}\n        (\\bar{f}-f)_z\\enspace,\n\t\\label{a2}\n\\end{equation}\nfrom which, together with (\\ref{vac9}), (\\ref{a}) follows.\n\\end{proof}\nNotice that $a_0$ is not gauge independent (in the sense of the above \ncorollary) whereas $a$ is.\n\nTheorem~\\ref{theorem2.1} can be used to construct solutions to the\nErnst equation by determining the structure and the singularities of \n$\\Phi$ in accordance with the conditions I--IV. For nonlinear evolution\nequations, large classes of solutions were constructed with the help of\nalgebro-geometric methods, in particular Riemann surface techniques. A \nkeypoint in this context is the occurrence of Riemann surfaces\nwhich are related to the linear system of the integrable equation under\nconsideration. In this paper we want to show how solutions for the Ernst \nequation\ncan be constructed by making use of the so called monodromy matrix\nof the Ernst system, which -- following~\\cite{krichever1} --  can be introduced\nas follows.\n\nFor a given linear system (\\ref{lin1}) and (\\ref{lin2}), we define the\nmonodromy matrix $L$ as a solution to the system\n\\begin{equation}\\label{3.19}\n        L_z=[W,L],\\quad L_{\\bar{z}}=[V,L]\n        \\label{mono3}\\enspace.\n\\end{equation}\nFor a known solution $\\Phi$ of (\\ref{lin1}) and (\\ref{lin2}), \n$L$ can be directly constructed in the form\n\\begin{equation}\\label{3.20}\n        L(K)=-\\hat{\\mu}(K) \\Phi{\\cal C}\\Phi^{-1}\n        \\label{mono5}\n\\end{equation}\nwhere ${\\cal C}$ is an arbitrary constant matrix with $\\det{\\cal C}=-1$ and\n$\\hat{\\mu}$ does not depend on the physical coordinates. Since $\\Phi$\nis analytic in $K$, there is a solution to (\\ref{mono3}) with the same \nproperties.\n\nIt follows from (\\ref{mono3}) that the coefficients of the characteristic \npolynomial $Q(\\mu, K)=\\det (L(K)-\\hat{\\mu}\\hat{1})$ are independent of the\ncoordinates. Without loss of generality we may assume $\\mbox{Tr}L(K)=0$. Then\n$L$ has the structure\n\\begin{equation}\\label{3.21}\n        L=\\left(\n        \\begin{array}{rr}\n                A(K) & B(K)  \\\\\n                C(K) & -A(K)\n        \\end{array}\n        \\right)\n        \\label{mono4}\\enspace.\n\\end{equation}\nThe equation $Q(\\hat{\\mu},K)=0$, i.e. \n\\begin{equation}\\label{3.21.1}\n        \\hat{\\mu}^2=A^2+BC\\enspace,\n        \\label{mono6}\n\\end{equation}\nis then the equation of an algebraic curve which \nin general will have infinite genus. We will restrict the analysis in the \nfollowing to the case of a regular curve with finite genus.\n\nIn this case, the Riemann surface $\\hat{{\\cal L}}$ is given by an \nequation of the form  \n\\begin{equation}\\label{3.22}\n        \\hat{\\mu}^2=\\prod_{i=1}^{g}(K-E_i)(K-F_i)\n        \\label{mono20}\n\\end{equation}\nwhere $E_i$ and $F_i$\nare obviously independent of the physical coordinates. This equation represents\na two sheeted covering of the Riemann sphere and thus a four sheeted covering\nof the complex plane. A point $\\hat{P}\\in\\hat{\\cal L}$ can be given by\n$\\hat{P}=(K,\\mu_0(K),\\hat{\\mu}(K))$. The Hurwitz diagram of $\\hat{{\\cal L}}$ \nis shown in figure 1.\n\\begin{figure}[ht]\n\\begin{center}\n\\unitlength1cm\n\\begin{picture}(7,4)\n\\thicklines\n\\put(0.4,4){\\line(1,0){6.6}}\n\\put(0.4,3){\\line(1,0){6.6}}\n\\put(0.4,2){\\line(1,0){6.6}}\n\\put(0.4,1){\\line(1,0){6.6}}\n\\put(0.9,4){\\line(0,-1){2}}\n\\put(1.2,3){\\line(0,-1){2}}\n\\put(2.2,4){\\line(0,-1){2}}\n\\put(2.5,3){\\line(0,-1){2}}\n\\multiput(3.7,4)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(3.7,2)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(5.7,4)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(5.7,2)(0.8,0){2}{\\line(0,-1){1}}\n\\put(0.9,4){\\circle*{0.15}}\n\\put(0.9,2){\\circle*{0.15}}\n\\put(1.2,3){\\circle*{0.15}}\n\\put(1.2,1){\\circle*{0.15}}\n\\put(2.2,4){\\circle*{0.15}}\n\\put(2.2,2){\\circle*{0.15}}\n\\put(2.5,3){\\circle*{0.15}}\n\\put(2.5,1){\\circle*{0.15}}\n\\multiput(3.7,4)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,3)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,2)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,1)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,4)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,3)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,2)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,1)(0.8,0){2}{\\circle*{0.15}}\n\\put(0,3.9){4}\n\\put(0,2.9){3}\n\\put(0,1.9){2}\n\\put(0,0.9){1}\n\\put(0.9,0.5){${\\rm i}\\bar{z}$}\n\\put(2,0.5){$-{\\rm i}z$}\n\\put(3.5,0.5){$E_1$}\n\\put(4.3,0.5){$F_1$}\n\\put(4.9,0.5){$\\cdots$}\n\\put(5.5,0.5){$E_g$}\n\\put(6.3,0.5){$F_g$}\n\\put(3.5,2.5){$E_1^\\sigma$}\n\\put(4.3,2.5){$F_1^\\sigma$}\n\\put(4.9,2.5){$\\cdots$}\n\\put(5.5,2.5){$E_g^\\sigma$}\n\\put(6.3,2.5){$F_g^\\sigma$}\n\\end{picture}\n\\end{center}\n\\caption{The Hurwitz diagram of $\\hat{\\cal L}$.}\n\\end{figure}\n\nThere is an automorphism $\\sigma$ of $\\hat{\\cal L}$ inherited from ${\\cal L}$\nwhich ensures $E_i^{\\sigma}=E_i$ and $F_i^{\\sigma}=F_i$. The orbit space\n${\\cal L}_H=\\hat{\\cal L}\/\\sigma$ is then, see~\\cite{algebro}, again a Riemann\nsurface, namely a hyperelliptic surface given by\n\\begin{equation}\\label{3.22a}\n\\mu_H^2=(K-{\\rm i}\\bar{z})(K+{\\rm i}z)\\prod_{i=1}^{g}(K-E_i)(K-F_i)\\enspace.\n\\end{equation}\nThus it is possible to construct components of the matrix $\\Phi$ on ${\\cal \nL}_H$ which makes it possible to use the powerful calculus of hyperelliptic \nRiemann surfaces. These functions may be lifted to $\\hat{{\\cal L}}$. \nAs we will show in the following, it is possible to construct a \nmatrix $\\Phi$ on ${\\cal L}$ in accordance with the conditions of Theorem 2.1 \nby projecting onto this surface. \n\n\n\\section{Hyperelliptic solutions of the Ernst equation}\n\\setcounter{equation}{0}\n\n\\subsection{Theta functions asscociated with a Riemann surface and the\nRiemann--Hilbert problem}\n\nIn this section, we want to give an explicit construction of the matrix \n$\\Phi$ in accordance with Theorem 2.1. Condition II can be used to \nconstruct solutions by \nprescribing the poles, essential singularities and cuts of $\\Phi$ \nwhich is equivalent to the solution of \na generalized Riemann--Hilbert problem for the matrix $\\Phi$. The \ninvestigation of such matrix Riemann--Hilbert problem  turns out to be \nrather difficult and is not yet fully done (in general it can be merely \nreduced to the solution of a linear integral equation, see e.g.\\ \n\\cite{musk}). Therefore we will use here a different approach. The \noccurrence of the monodromy matrix suggests that it might be possible to \nconstruct a matrix $\\Phi$ on the Riemann surface $\\hat{{\\cal L}}$ of the \nprevious section. The additional freedom we thus gain is used to restrict \nthe problem to a scalar one, namely to a Riemann--Hilbert problem for one \ncomponent of $\\Phi$ on the hyperelliptic surface ${\\cal L}_H$ obtained \nfrom $\\hat{{\\cal L}}$ by factorizing with respect to the involution \n$\\sigma$.  We impose the reality condition $E_i, F_i\\in \\bigBbb{R}$ or \n$E_i=\\bar{F}_i$ on the branch points in order to satisfy the reality \ncondition of Theorem 2.1. \nThen we construct the whole matrix $\\Phi$ in accordance with \nthis theorem. In fact it was shown in \\cite{prd} that all matrix \nRiemann--Hilbert problems with rational jump data are gauge equivalent \nto scalar problems on a suitably chosen hyperelliptic surface. Thus the \nlimitation to the scalar case is only a comparatively weak restriction \nwhich allows, as we will show below, for an explicit solution of the \nproblem in terms of theta functions. \n\nFor the moment, we fix the physical coordinates $z$ and \n$\\bar{z}$ in a way that $\\rho\\neq0$ and that $-{\\rm i}z$ and ${\\rm i}\\bar{z}$\ndo not coincide with the singular points of \n$\\Phi$ in order to ensure that the first condition of Theorem 2.1 is valid.\nIn the next section we study the dependence of the found solution on\n$z$ and $\\bar{z}$.\nIn order to give the solution to this special case of the generalized\nRiemann--Hilbert problem, we use the theory of theta functions associated to\na Riemann surface (see \\cite{dubrovin}) and the solution of the Riemann--Hilbert problem on a Riemann\nsurface, as given in~\\cite{zverovich1}.\nAs we will need only hyperelliptic Riemann surfaces\nof the form (\\ref{3.22a}), we restrict ourselves to this case.\n\nLet us denote by $(a_1,\\dots,a_g,b_1,\\dots,b_g)$ a basis of the\nfirst (integral) homology group $H_1({{\\cal L}_H})$ of ${{\\cal L}_H}$ (see \nthe picture below) where  the\ncuts are either between real branch points (which are ordered \n$E_{k+1}<F_{k+1}<\\ldots $) or between $E_i$ and $\\bar{E}_i$ (for the moment \nwe ignore the case that more than two branch points may have the same real \npart).\\\\\n\\begin{figure}[ht]\n\\setlength{\\unitlength}{.5cm}\n\\begin{center}\n\\psset{unit=.5cm}\n\\pspicture[](-2,0)(14,6)\n\\psline[linewidth=1.5pt]{*-*}(0,-1)(0,1)\n\\rput[bl](.3,-1.05){$\\scriptstyle P_0$}\n\\rput[b](0,1.2){$\\scriptstyle\\bar{P}_0$}\n\\psellipse[](2,0)(.7,1.5)\n\\psdots*[dotstyle=triangle*,dotscale=1.2 3](1.3,0)\n\\rput[b](2.8,1){$\\scriptstyle a_1$}\n\\psline[linewidth=1.5pt]{*-*}(2,-1)(2,1)\n\\rput(3.5,0){$\\dots$}\n\\psellipse[](5,0)(.7,1.5)\n\\psdots*[dotstyle=triangle*,dotscale=1.2 3](4.3,0)\n\\rput[b](5.8,1){$\\scriptstyle a_k$}\n\\psline[linewidth=1.5pt]{*-*}(5,-1)(5,1)\n\\psellipse[](8.5,0)(1.5,0.7)\n\\psdots*[dotstyle=triangle*,dotangle=-90,dotscale=1.2 3](8.5,.7)\n\\rput[b](9.5,.7){$\\scriptstyle a_{k+1}$}\n\\psline[linewidth=1.5pt]{*-*}(7.5,0)(9.5,0)\n\\rput(10.5,0){$\\dots$}\n\\psellipse[](12.5,0)(1.5,0.7)\n\\psdots*[dotstyle=triangle*,dotangle=-90,dotscale=1.2 3](12.5,.7)\n\\rput[b](13.5,.7){$\\scriptstyle a_g$}\n\\psline[linewidth=1.5pt]{*-*}(11.5,0)(13.5,0)\n\\pscurve[showpoints=false,linewidth=1pt]{-}(0,.5)(1,.5)(2,.3)\n\\psdots*[dotstyle=triangle*,dotangle=85,dotscale=1.2 3](1,.5)\n\\rput[bl](.9,.6){$\\scriptstyle b_1$}\n\\pscurve[showpoints=false,linewidth=1pt,\nlinestyle=dashed]{-}(2,.3)(2.5,.2)(3,-1)(1,-2)(-1,-1)(-.5,.4)(0,.5)\n\\pscurve[showpoints=false,linewidth=1pt]{-}(0,.8)(2,1.8)(5,.7)\n\\psdots*[dotstyle=triangle*,dotangle=85,dotscale=1.2 3](2,1.8)\n\\rput[b](3.4,1.6){$\\scriptstyle b_k$}\n\\pscurve[showpoints=false,linewidth=1pt,\nlinestyle=dashed]{-}(5,.7)(5.5,.3)(6,-1.3)(1,-2.5)(-1,-1.5)(-.6,.6)(0,.8)\n\\pscurve[showpoints=false,linewidth=1pt]{-}(0,.9)(3,2.5)(6,2.2)(8.5,0)\n\\rput[b](6.2,2.3){$\\scriptstyle b_{k+1}$}\n\\psdots*[dotstyle=triangle*,dotangle=75,dotscale=1.2 3](6,2.2)\n\\pscurve[showpoints=false,linewidth=1pt,\nlinestyle=dashed]{-}(8.5,0)(8,-3)(-1.5,-2.7)(-1.5,.7)(0,.9)\n\\pscurve[showpoints=false,linewidth=1pt]{-}(0,.95)(4,3.6)(10,3.2)(12,0)\n\\rput[b](10.2,3.3){$\\scriptstyle b_g$}\n\\psdots*[dotstyle=triangle*,dotangle=73,dotscale=1.2 3](10,3.2)\n\\pscurve[showpoints=false,linewidth=1pt,\nlinestyle=dashed]{-}(12,0)(11.5,-3.5)(-1.9,-3.4)(-1.7,.75)(0,.95)\n\\endpspicture\n\\vspace*{2.5cm}\n\\end{center}\n\\caption{The homology basis for ${\\cal L}_H$.}\\label{fig1}\n\\end{figure}\n\n\\noindent\nLet $\\{{\\rm d}\\omega_i\\}$ denote a basis of $H^1({{\\cal L}_H})$ such that\n$\\oint_{a_i}{\\rm d}\\omega_j=2\\pi{\\rm i}\\delta_{ij}$. Using these \nnormalized differentials, we define the Abel--Jacobi map of $P_H\\in{\\cal L}_H$\nby\n$\\omega(P_H)=(\\int_{P_0}^{P_H}{\\rm d}\\omega_1,\\dots,\\int_{P_0}^{P_H}{\\rm d}\\omega_g)$ with\n$P_0\\in{\\cal L}_H$ fixed. In the following, we will always choose \n$P_0=-{\\rm i}z$.\n\nWe define a $g\\times g$ matrix ${\\mit\\Pi}$ -- the Riemann matrix -- with \nelements $\n\\pi_{ij} \\doteq\\oint\\limits_{b_i}{\\rm d}\\omega_j$.\nThis matrix is symmetric and has a negative definite real part. These\nproperties ensure that the theta function with integer characteristic\n$\\left [\\alpha\\atop\\beta\\right]$ defined by\n\\begin{equation}\\label{4.2}\n\\Theta\\left [\\alpha\\atop\\beta\\right](x,{\\mit\\Pi})=\n\\sum\\limits_{N\\in\\SBbb{Z}^g}\\exp\\left\\{\n\\frac{1}{2}\\left\\langle{\\mit\\Pi}\\left(N+\\frac{\\alpha}{2}\\right),N+\\frac{\\alpha}{2}\n\\right\\rangle+\n\\left\\langle x+\\pi{\\rm i}\\beta,N+\\frac{\\alpha}{2}\n\\right\\rangle\\right\\}\n\\enspace,\n\\end{equation}\nwith $x\\in{\\bigBbb{C}}^g$ and $\\alpha$, $\\beta\\in{\\bigBbb{Z}}^g$, where $\\left\n\\langle\\cdot,\\cdot\\right\\rangle$ denotes the Euclidean scalar product\n$\\left\\langle N,x\\right\\rangle=\\sum_{i=1}^gN_ix_i$, is an analytic function on\n$\\bigBbb{C}^g$. A characteristic is called \nodd if $\\langle \\alpha,\\beta\\rangle \\neq 0 \\mbox{ mod } 2$. The Riemann \nvector is denoted by $K_R$.\n\nThe reality condition on the branch points implies for the theta function \n$\\Theta$ with characteristic $\\left[0\\atop0\\right]$, the Riemann theta \nfunction, \n\\begin{equation}\n        \\bar{\\Theta}(x)=\\Theta(\\bar{x}+\n        {\\rm i}\\pi\\Delta)\\enspace,\n\t\\label{real}\n\\end{equation} \nwhere $\\Delta_i=1$ if $E_i$ and $F_i$ are real and $\\Delta_i=0$ otherwise.\n\nWe recall that a divisor $\\eufm{A}$ on a general Riemann surface $\\Sigma_g$ is a\nformal\nsymbol $\\eufm{A}=n_1P_1+\\cdots n_kP_k$ with $P_i\\in\\Sigma_g$ and $n_i\\in\\bigBbb{Z}$.\nThe set of divisors admits a partial ordering and we may associate to a\nmeromorphic\nfunction $f$ a divisor $(f)$, a principal divisor. Using this partial ordering\nwe define for a divisor $\\eufm{A}$ a vector space $L(\\eufm{A})$ consisting of all principal\ndivisors not less than $\\eufm{A}$.\n\nA Riemann--Hilbert problem can be stated as follows:\nlet $\\Gamma$ be a piecewise smooth contour on ${{\\cal L}_H}$.\nLet $\\Lambda=t_1+\\cdots+t_r$ be a divisor\non $\\Gamma$ consisting of a finite number of pairwise different points\nsubject to the following condition: \n$\\Gamma\\setminus\\Lambda$ decomposes into a finite set of connected components\n$\\{\\Gamma_j\\}$ ($j=1,\\dots,N$), each of which is homeomorphic to the interval\n$(0,1)$. We call $\\Gamma_j$ a {\\em curve of the contour} $\\Gamma$.\nEach $\\Gamma_j$ has a starting and an end point, given by two points of $\\Lambda$,\nwhere the starting respectively end points may also coincide. We define the\nfunction $\\alpha(t,\\Gamma_j)$ on $\\Gamma$ by\n\\begin{equation}\\label{4.3}\n\\alpha(t,\\Gamma_j)=\\left\\{\n\\begin{array}{ll}\n1 & \\mbox{if $t\\in\\Gamma_j$}\\\\\n0 & \\mbox{otherwise}\n\\end{array}\\enspace,\n\\right.\n\\end{equation}\n($j=1,\\dots,N$). On each curve $\\Gamma_j$ let there be defined a\nH\\\"older-continuous function $G_j(t)$, which is finite and nonzero. We denote\n\\begin{equation}\\label{4.4}\nG(t)=\\sum\\limits_{j=1}^N\\,\\alpha(t,\\Gamma_j)G_j(t)\\enspace, \\quad\nt\\in\\Gamma\\setminus\\Lambda.\n\\end{equation}\n\nLet there be given a divisor $\\eufm{A}$ of degree $m$, consisting of points of the\ndivisor $\\Lambda$, taken at arbitrary degree. Let on ${{\\cal L}_H}\\setminus\n\\Gamma$ be given another divisor $\\eufm{B}$ of degree $n$. Now we can\nformulate the\\\\ \n{\\bf Homogeneous scalar Riemann--Hilbert problem:}\\\\ Give a function $\\psi$ \nwith the properties\n\\begin{equation}\\label{4.5}\n\\psi^+(t)=G(t)\\psi^-(t)\\enspace,\n\\end{equation}\nwith $(\\psi)\\in L(\\eufm{A}^{-1}\\eufm{B}^{-1})$.\n\nThe solution of this problem was given by Zverovich~\\cite{zverovich1}. A key\npoint in the construction is the introduction of an analogue to the usual\nCauchy kernel. This Cauchy analogue is given by a normalized (i.e.~all\n$a$-periods are zero)\ndifferential of the third kind with poles at $P_H$ and $P_0$ and is denoted by\n${\\rm d}\\omega_{P_HP_0}(\\tau)$. With the Cauchy analogue at hand the\nsolution to the above Riemann-Hilbert problem is given by\n\\begin{equation}\\label{4.10}\n\\psi(P_H)={\\rm e}^{\\Psi(P_H)}\\enspace,\n\\end{equation}\nwith\n\\begin{equation}\\label{4.10a}\n\\Psi(P_H)=\\frac{1}{2\\pi{\\rm i}}\\int\\limits_{\\Gamma}\\,\\ln G(\\tau)\\,{\\rm d}\n\\omega_{P_HP_0}(\\tau)\\enspace\n\\end{equation}\nwhere $P_0\\notin\\Gamma$, and the integration goes over all curves\n$\\Gamma_j$ of the contour $\\Gamma$, and where we have put \n$\\ln G(t)=\\sum\\limits_{j=1}^N\\alpha(t,\\Gamma_j)\\ln G_j(t)$.\nThe $b$-periods $u_i$ of $\\psi$ are given by\n\\begin{equation}\nu_i=\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\ln G\\,{\\rm d}\\omega_i\n\\label{bperiod2}\\enspace.\n\\end{equation}\nApplying the Plemelj formulae to the function $\\Psi(P_H)$, we find\n\\begin{equation}\\label{4.12}\n\\psi^+(t) =  G(t)\\psi^-(t)\\enspace,\\enspace\\enspace t\\in\\Gamma\\setminus\\Lambda\n\\setminus\\bigcup\\limits_{i=1}^g a_i.\n\\end{equation}\nFor more details on the solution to the scalar Riemann--Hilbert problem, \nthe reader is referred to \\cite{zverovich1}, \\cite{rodin} or \\cite{olaf}.\n\n\n\\subsection{Solutions to the Ernst equation via the scalar Riemann--Hilbert \nproblem}\n\nWith the help of the results of the previous section,\nwe are now able to state the following\ntheorem, which gives the solution to the generalized Riemann--Hilbert problem\non the hyperelliptic Riemann surface ${\\cal L}_H$.\n\\begin{theorem}\nLet $P_0$  be a fixed complex constant ($\\rho\\neq0$) \nnot coinciding with the \nsingularities of $\\psi$ or the branch points $E_i$ or $F_i$. \nLet $\\Omega(P_H)$ be a linear combination of \nnormalized  Abelian integrals of the \nsecond kind (with singularities $p\\neq E_i$ and $p\\neq F_i$, independent of $z$ \nand $\\bar{z}$) and third kind (with in addition singularities at all real \nbranch points with residues $\\pm \\frac{1}{2}$), satisfying \n$\\bar{\\Omega}(P_H)=\\Omega(\\bar{P}_H)$. Then the solution to the generalized\nRiemann--Hilbert problem  on the real hyperelliptic Riemann surface ${\\cal\nL}_H$ is given by\n\\begin{equation}\\label{8.9}\n\\psi(P_H)=\\psi_0\\,\\frac{\\Theta(\\omega(P_H)-\n\\omega(D)+u+b-K_R)}{\\Theta(\\omega(P_H)-\\omega(D)-K_R)}\n\\exp\\left\\{\\Omega(P_H)+\\frac{1}{2\\pi{\\rm i}}\\int\\limits_\\Gamma\n\\ln G(\\tau){\\rm d}\\omega_{P_HP_0}(\\tau)\\right\\}\\enspace,\n\\end{equation}\nwhere $D=P_1+\\cdots+P_g$ is a fixed non--special divisor on ${\\cal L}_H$ \nwhich is subject to the reality condition: \neither $P_i\\in \\bigBbb{R}$ or with $P_i\\in D$ we have $\\bar{P}_i\\in D$ or $P_i$ is a \nbranch point $E_i$ or $F_i$. $b$ is the vector of $b$-periods of $\\Omega$ with\ncomponents\n\\begin{equation}\nb_i=\\oint\\limits_{b_i}\\,\\Omega\\label{8.10a}\\enspace,\n\\end{equation}\n$i=1,\\dots,g$, the $u_i$ are given by (\\ref{bperiod2}), \nand $\\psi_0$ is a normalization constant. The paths of integration \nhave to be the same for all integrals.\n\\end{theorem}\n\\begin{proof}\nWe want to prove that $\\psi(P_H)$ is a single valued function on ${\\cal L}_H$.\nIf we choose a different path of integration for the integrals in the\nexponent and the map $\\omega(P_H)$ and denote the corresponding integrals by a \nprime, the primed and unprimed integrals are connected via \n\\begin{equation}\\label{8.11}\n\\Omega'(P_H)=\\Omega(P_H)+\\oint\\limits_{{\\cal E}}\\,{\\rm d}\\Omega\\enspace,\n\\end{equation}\n(similarly for the other integrals) where ${\\cal E}$ is a closed contour on \n${\\cal L}_H$ which may be decomposed in the homology basis as follows\n\\begin{equation}\\label{8.12}\n{\\cal E}=\\sum_{i=1}^g m_i\\,a_i+\\sum_{i=1}^g n_i\\,b_i\\enspace,\n\\end{equation}\nwith $m_i$, $n_i\\in\\bigBbb{Z}$. Then we have, e.g.~for $\\Omega$ and $\\omega$\n\\begin{eqnarray}\\label{8.13}\n\\Omega(P_H) & \\to & \\Omega(P_H)+\\sum_{i=1}^gn_i b_i=\\Omega(P_H)+\n\\left\\langle N,b\\right\\rangle\\enspace,\\nonumber\\\\\n\\omega(P_H) & \\to & \\omega(P_H)+2\\pi{\\rm i}M +{\\mit\\Pi} N\\enspace,\n\\end{eqnarray}\nwhere $M=(m_1,\\dots,m_g)$, $N=(n_1,\\dots,n_g)\\in{\\bigBbb{Z}}^g$. Under this \ntransformation, the original quotient of theta functions in (\\ref{8.9}) \nwill be multiplied by \n\\begin{equation}\\label{8.14}\n\\exp\\left (-\\left\\langle N,b\\right\\rangle\\right )\\enspace,\n\\end{equation}\nbut this term is just compensated by the contour integral over ${\\cal E}$ in the\nexponent.\nThe same argument holds for the line integral over the contour $\\Gamma$ \nsince the $u_i$ are its $b$-periods.\nThis shows that $\\psi(P_H)$ is a single valued function on ${\\cal L}_H$. \n\n>From the properties of the theta function, we also find that $\\psi(P)$ has\n$g$ simple poles at the points $P_1,\\dots,P_g$ and $g$ simple \nzeros. Additional poles, zeros and essential singularities can be obtained by a\nsuitable choice of\nAbelian integrals of the second kind (essential singularities) \nand third kind (zeros\nand poles). We remark that the assumption $\\bar{\\Omega}(P)=\\Omega(\\bar{P})$ had\nto be introduced in order to satisfy the reality condition of Theorem 2.1. \n\\end{proof}\n\\begin{remark}\nWithout loss of generality we can choose $D$ to consist only of branch\npoints since  $D$ gives the poles of $\\Psi$ due to the zeros of the theta\nfunction in the denominator. This can always\nbe compensated by a suitable choice of the zeros and poles of $\\Psi$\nwhich arise from the integrals of the third kind in\n$\\Omega$. All $P_i\\in D$ shall have multiplicity 1 and be chosen in a way\nthat $\\Theta\\left[\\alpha\\atop\\beta\\right](x)$ with\n$\\left[\\alpha\\atop\\beta\\right] =\\omega(D)+\\omega(\\bar{P}_0)+K_R$ \nhas the same reality\nproperties as the Riemann theta function $\\Theta(x)$.\n\\end{remark}\n\nOur next aim is to define a matrix\nvalued function $\\Phi(P)$ on ${\\cal L}$, satisfying the conditions of\ntheorem~\\ref{theorem2.1}, with the help of the above solution to the scalar\nRiemann--Hilbert problem on the hyperelliptic surface ${\\cal L}_H$. \nTo this end we define a further function on ${\\cal L}_H$ by\n\\begin{equation}\n\\chi(P_H)=\\chi_0\\frac{\\Theta(\\omega(P_H)+u-\\omega(\\bar{P}_0)\n-\\omega(D)-K_R)}{\\Theta(\\omega(P_H)-\\omega(D)-K_R)}\\exp\\left(\n\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\ln G{\\rm d}\\omega_{P_H P_0}\\right)\n\t\\label{gauge17},\n\\end{equation}\nwhere $\\chi_0$ is again a normalization constant. \nIt can be easily seen that the analytic behaviour of $\\chi(P_H)$ is\nidentical to that of $\\psi(P_H)$, except that it changes the sign at every\n$a$-cut. $\\chi$ is thus not a single valued function on ${\\cal L}_H$.\nHowever, it is single valued on $\\hat{{\\cal L}}$ which can be\nviewed as two copies of ${\\cal L}_H$ cut along \n$\\left[P_0,\\bar{P}_0\\right]$ and glued together along this cut. We define \nthe vector $X$  on $\\hat{{\\cal L}}$ by fixing the sign in front of $\\chi$ in the \nvicinity of the points $P_0^{\\pm}=(K_0,0,\\pm\\hat{\\mu}(K_0))\\in \\hat{{\\cal L}}$,\n\\begin{equation}\n\tX(\\hat{P})=\\left(\n\t\\begin{array}{c}\n\t\t\\psi(\\hat{P})  \\\\\n\t\t\\pm \\chi(\\hat{P})\n\t\\end{array}\\right), \\quad \\hat{P}\\sim P_0^{\\pm}.\n\t\\label{gauge18}\n\\end{equation}\nWith the help of this vector, we can construct the matrix $\\Phi$ on ${\\cal \nL}$ via\n\\begin{equation}\n\t\\Phi(P)=(X(K,\\mu_0(K),+\\hat{\\mu}(K)),X(K,\\mu_0(K),-\\hat{\\mu}(K)))\n\t\\label{gauge19}\n\\end{equation}\nwhere the signs are again fixed in the vicinity of $P_0^{\\pm}$. Notice \nthat this matrix consists of eigenvectors of the monodromy matrix, \n$LX(K,\\mu_0(K),\\pm\\hat{\\mu}(K))=\\hat{\\mu} X(K,\\mu_0(K),\\pm\\hat{\\mu}(K))$ \nif $L$ is written as $L=\\hat{\\mu}\\Phi \\gamma \n\\Phi^{-1}$ where $\\gamma$ is the matrix from~\\ref{theorem2.1}.\n\nIt may be readily checked that this ansatz is in accordance with the reduction\ncondition (\\ref{lin7}) (this is in fact the reason why one has to define \nthe function $\\chi$ in the way (\\ref{gauge17})). The behaviour at the\nsingularities is as required in condition II: For the contour $\\Gamma$ and the\nsingularities of the Abelian integrals $\\Omega$, this is obvious. At the\nbranch points $E_i$ and $F_i$, one gets the following behaviour: at  \npoints $P_i$ of the divisor $D$, the components of $\\Phi$ have a simple pole, and \nthe determinant diverges as $(K-P_i)^{-\\frac{1}{2}}$, if this branch point \nis not a singularity of an integral of the third kind in $\\Omega$ or lies \non the contour $\\Gamma$. If the same condition holds  at \nthe remaining branch points, the components are regular there but the \ndeterminant vanishes as $(K-P_i)^{\\frac{1}{2}}$. If the branch points \ncoincide with one of the singularities of the integrals in the exponent in\n(\\ref{8.9}), this merely changes the singular behaviour of $\\Phi$ and its \ndeterminant there. Condition II of theorem~\\ref{theorem2.1} is however \nobviously satisfied.\n\nSince $\\Phi$ in (\\ref{gauge19}) is only a function of $P$, it will not be\nregular at the cuts $\\left[E_i,F_i\\right]$. At the $a$-cuts around non-real\nbranch points, we get $\\Phi^-=\\Phi^+\\sigma_1|_{a_i}$, whereas we have\n$\\Phi^-|_{a_i}=-\\Phi^+|_{a_i}\\sigma_2$ at\nthe $a$-cuts around real branch points. The logarithmic derivatives of $\\Phi$\nwith respect to $z$ and \n$\\bar{z}$ are however holomorphic at all these points. One can recognize that \nthe behaviour at the non-real branch\npoints is related to a gauge transformation of the form (\\ref{c}). This \nmeans that one can find a gauge transformed matrix $\\Phi'$ that is \ncompletely regular at these points if the integrals in the exponent are \nregular there. With \n\\begin{equation}\n\t\\alpha_1=\\frac{1}{2}(1+\\lambda), \\quad \\alpha_2=\\frac{1}{2}(1-\\lambda)\n\t\\label{cc}\n\\end{equation} \nand $\\lambda=\\prod_{i=1}^{g}\\sqrt{\\frac{K-\\bar{P}_i}{K-P_i}}$ where\n$D=\\sum_{i=1}^{g}P_i$, this may be checked by direct calculation. The real\nbranch points, however, cannot be related to gauge transformations.\n\nNormalizing $\\psi$ and $\\chi$ (if possible)\nin a way that $\\psi(\\infty^-_H)=1$ and $\\chi(\\infty^-_H)=-1$, one can see \nthat $\\Phi$ is then in accordance with all conditions of Theorem 1 since \nthe reality condition follows from the reality properties of the theta \nfunctions and the Riemann--Hilbert problem. The fact that $\\Phi$ is at least\ndifferentiable with respect to $z$ and $\\bar{z}$ at points where $P_0$ \ndoes not coincide with the singularities of the integrals in the exponent \nor the remaining branch points of ${\\cal L}_H$ follows from the modular \nproperties of the theta function. Let the paths between \n$\\left[P_0,\\infty^-\\right]$ and \n$\\left[P_0,\\infty^+\\right]$ be the same in \nall integrals and let them have the same projection into the complex plane \n(i.e.\\ one is the involuted of the other).\nThen the results may be summarized in\n\\begin{theorem}\\label{theorem3.2}\nLet $\\Theta\\left[\\alpha\\atop\\beta\\right](\\omega(\\infty^-)+u)\\neq0$. Then the function\t\n\t\\begin{equation}\\label{8.16}\n\tf(z,\\bar{z})=\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](\\omega(\\infty^{+})+u+b)}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\n\t(\\omega(\\infty^{-})+u+b)}\n\t\\exp\\left\\{\\Omega(\\infty^{+})-\\Omega(\\infty^{-})+\n\t\\frac{1}{2\\pi{\\rm i}}\\int\\limits_\\Gamma\n\t\\ln G(\\tau){\\rm d}\\omega_{\\infty^{+}\\infty^-}(\\tau)\n\t\\right\\}\n\t\\end{equation}\n\tis a solution to the Ernst equation. \n\\end{theorem}\n\\begin{remarks}\n\\item In the case $g=0$ the Ernst potential (\\ref{8.16}) is real, $f={\\rm\ne}^{2U}$. This means\nthat $U$ is a solution to the axisymmetric Laplace equation and belongs \ntherefore to the\nWeyl-class. For $g>0$, there are no real solutions other than $f=1$ which\ndescribes Minkowski space.\n\\item The multi-black-hole solutions which can be obtained via B\\\"acklund\ntransformations (see e.g.\\ \\cite{hkx}) are contained in the class \n(\\ref{8.16}) as the limiting case that the branch points $E_i$ and $F_i$ \ncoincide pairwise. In this limit, all branch points become double points \nand the theta functions break down to purely algebraic functions. Notice \nthat the analysis of $f$ at the branch points in the following section \nalways assumes a regular surface. The obtained results for the regularity \nof $f$ do not hold in this limit. \n\\end{remarks}\nThe above explicit construction of the solutions makes it possible to\nderive useful formulae for the metric function $a$ and the derivatives of the\nErnst potential. Let $\\int_{P_H}^{P_H+\\delta}{\\rm d}\\omega_i= \ng_i\\delta+o(\\delta)$ where\n$\\delta$ is the local parameter in the vicinity of $P_H\\in {\\cal L}_H$. We\ndefine the derivative\n\\begin{equation}\n\tD_{P_H}\\Theta(x)=\\sum_{i=1}^{g} g_i\\partial_{x_i}\\Theta(x).\n\t\\label{local}\n\\end{equation}\nUsing (\\ref{a}) and (\\ref{8.9}), (\\ref{gauge17}), we get \n\\begin{equation}\n\t(a-a_0) {\\rm e}^{2U}={\\rm i}D_{\\infty^-}\\ln \n\t\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\\left(\\int_{\\bar{P}_0}^{\\infty^- }d\\omega+u+b\\right)}{\n\t\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\\left(\\int_{P_0}^{\\infty^-}d\\omega+u+b\\right)}\n\t\\label{fay6}.\n\\end{equation}\n>From the linear system (\\ref{lin1}) and (\\ref{lin2}), we obtain with\n(\\ref{8.9}) and (\\ref{gauge17})\n\\begin{eqnarray}\n\t\\frac{\\bar{f}_z}{f+\\bar{f}}&=&\\frac{{\\rm i}}{2\\sqrt{P_0-\\bar{P}_0}}\n\t\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b-\\omega(\\bar{P}_0))\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-))}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-)-\\omega(\\bar{P}_0))}\\times\\nonumber\\\\\n\t&&\\left(D_{P_0}\\ln \\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u-\\omega(\\bar{P}_0))+I_{P_0}\\right)\n\t\\label{fay7}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\t\\frac{f_z}{f+\\bar{f}}&=&\\frac{{\\rm i}}{2\\sqrt{P_0-\\bar{P}_0}}\n        \\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-)-\\omega(\\bar{P}_0))}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b-\\omega(\\bar{P}_0))\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-))}\\times\\nonumber\\\\\n\t&&\\left(D_{P_0}\\ln \\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u-\\omega(\\bar{P}_0))+I_{P_0}\\right)\n\t\\label{fay8},\n\\end{eqnarray}\nwhere $I_{P_0}$ is the linear term of the expansion of the integrals \nin the exponent of (\\ref{8.9}) in the local parameter around $P_0$.\n\n\n\\subsection{Finite gap solutions and Picard--Fuchs equations}\n\nThe original finite gap solutions of~\\cite{korot1} are those among\n(\\ref{8.16}) without the contour integral (in our notation\nonly an arbitrary linear combination of Abelian integrals of the second and\nthird kind $\\Omega$). They just correspond to the so called\nBaker--Akhiezer function (see \\cite{algebro}) for the Ernst system. This\nfunction that has essential singularities and poles gives the periodic or\nquasiperiodic solutions to the integrable nonlinear evolution equations. \nThere the essential singularity is uniquely determined by the structure of \nthe differential equation. In contrast to these equations, the solutions \n(\\ref{8.16}) are in general neither periodic nor quasiperiodic, and the \nessential singularity can be nearly arbitrarily chosen. The form of the \nsolution to the Riemann--Hilbert problem shows that one might even think\nof ``putting the singularities densely on a line and integrate over the\nintegrals with some measure\": an Abelian integral $\\Omega_p$ of the second\nkind with a pole of first order at $p$ can be used as an analogue to the Cauchy\nkernel. A contour integral over this kernel with some measure,\n$\\int_{\\Gamma}^{}\\ln G \\Omega_p{\\rm d} p$, is thus\njust another way to write down the solution to a Riemann--Hilbert problem \non a Riemann surface. \n\nIn studying the boundary value problem for the rigidly rotating disk of dust,\nMeinel and Neugebauer~\\cite{meinelneugebauer} observed that it is possible to\nobtain solutions to the Ernst equations via\n\\begin{equation}\nf=\\exp\\left(\\sum_{m=1}^{g}\\int_{E_m}^{C_m}\\frac{K^g{\\rm d} K}{\\mu_H}-I_g\\right)\n\\label{mn1}\n\\end{equation}\nwhere the divisor $C=\\sum_{m=1}^{g}C_m$ is determined by\n\\begin{equation}\n\t\\sum_{m=1}^{g}\\int_{E_m}^{C_m}\\frac{K^i {\\rm d} K}{\\mu_H}=I_i\n\t\\label{mn2}\n\\end{equation}\n($i=0,1,\\dots,g-1$), i.e.\\ as the solution of a Jacobi inversion problem. The\n$I_i$ are (in the absence of real branch points)\nreal solutions to the axisymmetric Laplace equation which satisfy the \nrecursive condition,\n\\begin{equation}\n        {\\rm i} I_{n+1,z}=zI_{n,z}+\\frac{1}{2}I_n\\enspace.\n        \\label{8.18}\n\\end{equation}\nThe relation to the class obtained in theorem~\\ref{theorem3.2} is the following:\nThe integral of the third kind in\n(\\ref{mn1}) can be expressed by the help of a formula in~\\cite{stahl}\nvia theta functions. Equation (\\ref{mn2}) ensures that the resulting \nexpression is independent of the chosen integration path which is shown in \nthe proof of the theorem. Thus the $u_i$ (obtained from the $I_i$ by \nnormalization) are as in our case the $b$-periods\nof the integral $I_g$ in the exponent. In fact, it was shown in\n\\cite{meinelneugebauer} that one of these periods, say $I_1$, can be chosen \nas an arbitrary solution to the axisymmetric Laplace equation. The other \nperiods as well as the integral in the exponent then follow from \ndifferential identities plus boundary conditions.\n\nThe underlying reason for this fact is that the \nErnst potential $f$ is studied on a family of Riemann surfaces parametrized\nby the moving branch points $-{\\rm i}z$ and ${\\rm i}\\bar{z}$. The periods \non this surface (i.e.~integrals along closed curves) are\nsubject to differential identities, the so called Picard--Fuchs equations. \nIt is a general feature of the periods of rational functions \n\\cite{griffiths,morrisson,foucault} that \nthey satisfy a differential system of finite order with Fuchsian \nsingularities. An elegant way to find the Picard--Fuchs system explicitly is\nvia the notion of the Manin connection in the bundle $H^1_{\\rm\nDR}(\\Sigma_g)\\to\\Sigma_g$, see~\\cite{manin1}. The investigation turns out to\nbe particularly simple if one uses the following standard form of the\n(hyperelliptic) Riemann surface $\\Sigma_g$ (all hyperelliptic surfaces of \ngenus $g$ are conformally equivalent to this standard form)\n\\begin{equation\ny^2=(x-z)\\prod\\limits_{i=1}^{2g} (x-E_i)\\doteq (x-z)P(x)\n=(x-z)\\sum_{j=0}^{2g}a_jx^j\\enspace,\n\\end{equation}\nwhere the $E_i$ do not depend on $z$. Using $j_0={\\rm d} x\/y$, \n$j_1=xj_0,\\dots,j_{2g-1}=x^{2g-1}j_0$ as the basis for the de Rham \ncohomology $H^1_{\\rm DR}(\\Sigma_g)$ we\nobtain for the matrix $M_n^m$ ($m,n=0,\\dots,2g-1$) of the Manin connection\n(defined by $\\frac{\\partial j_n}{\\partial z}=M^m_nj_m$)\n\\begin{equation\nM_n^m=\n\\left\\{\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{array}{cl}\n\\displaystyle \\frac{z^n}{2P(z)}\\left((m+1)a_{m+1}+\nz^{-m-1}\\,\\sum_{j=0}^ma_j z^j\\right)& \\mbox{for $0\\leq m<n$,} \\\\\n\\displaystyle \\frac{z^n}{2P(z)}\\left((m+1)a_{m+1}-\n\\sum_{j=0}^{2g-1-m}a_{m+1+j}z^j\\right) & \\mbox{for $n\\leq m\\leq 2g-1$} \\\\\n\\end{array}\\right.\\enspace.\n\\end{equation}\nOne finds that the periods satisfy a similar recursive condition as\n(\\ref{8.18}).\nAn analogous consideration can be performed for the $\\bar{z}$-dependence of\nthe periods. One finds that the integrability condition of the Picard--Fuchs\nsystems is just the axisymmetric Laplace equation.\n\nOn the other hand, with the help of some boundary conditions (for instance at\n$|z|\\to\\infty$), the $I_n$ can be uniquely determined from the above system\n(\\ref{8.18}).\nThus the class of solutions discussed by Meinel and Neugebauer may be phrased\nin the following form: if an arbitrary solution of the Laplace equation is\ngiven, one can calculate the functions $I_n$ with (\\ref{8.18}) and the boundary\ncondition, and ends up with a solution to the Ernst equation of the form \n(\\ref{8.16}).\n\n\n\\section{The singular structure of the Ernst potential}\n\\setcounter{equation}{0}\n\nThe construction of the solutions  in the previous sections with \nthe help of Theorem~\\ref{theorem2.1} also indicates where the resulting Ernst\npotential\n(\\ref{8.16}) may be singular: only at points $P_0=-{\\rm i}z$ where the \nconditions of Theorem~\\ref{theorem2.1} do not hold. Notice that these\nconditions are sufficient for the regularity of $f$ at all other points. It may\nturn out though that the Ernst potential is perfectly regular at points where\nTheorem~\\ref{theorem2.1} is not fulfilled, e.g.\\ in the case of singularities\nthat are pure gauge. We will therefore discuss all possible singular points of\nthe solutions (\\ref{8.16}).\n\nIt is very helpful that this whole discussion can be  performed on the \nRiemann surface ${\\cal L}_H$ where one can use the powerful calculus on\nhyperelliptic surfaces.\nOne does not have to work on the four sheeted surface $\\hat{{\\cal L}}$ \nwhose introduction was necessary for the\nconstruction of the solutions, and which provides an understanding\nof the mathematical properties of the Ernst equation. Since we will work \nfrom now on on ${\\cal L}_H$ only, unless otherwise noted, we will drop the \nindex $H$ at points $P\\in {\\cal L}_H$.\n\nThe possible singularities of $f$ can be directly inferred \nfrom the potential in the form (\\ref{8.16}). The Ernst potential will be \nsingular at the zeros of the denominator.\nIt is possibly not regular at the points \nwhere $P_0$  is identical to the singularities of $\\Omega$ or\nis on $\\Gamma$. Critical points of a \ndifferent kind are the branch points $E_i$ and $F_i$. If $P_0$  \ncoincides with these points, the Riemann surface ${\\cal L}_H$\nbecomes singular. Something similar happens at the axis where the \nbranch points $P_0$ and $\\bar{P}_0$ coincide. This is a reminiscent of the\nsingular behaviour of the three--dimensional Laplace operator on the axis \nin the axisymmetric case. The\nmain aim of the following analysis is to single out a class of solutions that \nmay be interesting in the context of boundary value problems for the Ernst \nequation that describe e.g.\\ the exterior of a body of revolution. Thus we \nwill not study the nature of the singularities (e.g.\\ curvature \nsingularities) but single out a large class of solutions where the Ernst \npotential is only discontinuous at a (closed) contour that could be identified \nwith the surface of a body.\n\n\n\\paragraph{Zeros of the denominator}\n\nZeros of the denominator of (\\ref{8.16}) will lead to singularities in the \nspacetime. From condition IV of Theorem 2.1 it follows that these are just \nthe points at which the matrix $\\Phi$ cannot be normalized in the required \nway. This leads to the transcendental condition\n\\begin{equation}\n\t\\Theta\\left[\\alpha\\atop\\beta\\right](\\omega(\\infty^-)+u+b)\\neq 0,\n\t\\label{reg}\n\\end{equation}\nif one wants to exclude these zeros of the denominator.\nWe will show in the next section \nhow the points at which \n$\\Theta\\left[\\alpha\\atop\\beta\\right](\\omega(\\infty^-)+u+b) =0$  \ncan be found as the solution of a set of algebraic equations.\n\n\n\\paragraph{Essential singularities}\n\nThe integrals of the third kind occuring in $\\Omega$ are nothing but\na particular case of line integrals over contours with constant jump function\n$G(t)$. Therefore, we are left with the investigation of the integrals of the\nsecond kind at this point. Since the theta functions in (\\ref{8.16}) are \nregular as long as the Riemann surface ${\\cal L}_H$ is, we are left with \nthe exponent if (\\ref{reg}) holds. \nFor the behaviour of the exponent, we get the following\n\\begin{proposition}\\label{prop4.1}\nThe Ernst potential (\\ref{8.16}) has an essential singularity at the points\nwhere $P_0$  coincides with the singularities of the integrals of\nthe second kind on ${\\cal L}_H$.\n\\end{proposition}\n\\begin{proof}\nThe exponent has by construction an algebraic\npole there and, therefore, the Ernst potential has an essential\nsingularity.\n\\end{proof}\nAn essential singularity of the real part of the Ernst potential \ncorresponds to a line singularity of the metric function $U$. \nIn the context of exterior solutions for bodies of revolution we are \ninterested in, there seems to be no \nsituation where such a line singularity in a spacetime might be interesting.\n\n\n\\paragraph{Contours}\n\nIn the case that $P_0$ lies on a contour $\\Gamma_i$ but\nnot on an endpoint of $\\Gamma_i$,\non the axis, or on one of the branch points $E_i$ or $F_i$, it\ncan be easily seen that the\nintegral in the exponent as well as the $b$--periods $u_i$ are bounded \nsince $G$ is H\\\"older--continuous, finite and non--zero on $\\Gamma$. At \nthe endpoints of the contours $\\Gamma_i$, singularities may occur.\nThe value of these integrals at the remaining points will \nhowever not be the same in general if the contour is approached from one or \nthe other side. This can be seen from the following fact: The point \n$P_0$ is a branch point of ${\\cal L}_H$. If it lies on the contour\n$\\Gamma$, care has to be taken of the sign of the root whilst evaluating \nthe integrals of the form $J_n=\\int_{\\Gamma}^{}\\ln G(\\tau)\\frac{ \\tau^n \n{\\rm d} \\tau}{\\mu}$. The decisive factor is $\\mu_0^2=(K-P_0)(K-\\bar{P}_0)$.  \nWe get for $K\\in\\Gamma$ with $K=K_1+{\\rm i}K_2$,  and $K_1,K_2\\in \\bigBbb{R}$ \nfor the imaginary part of $\\mu_0$,\n\\begin{equation}\n\t\\Im\\mu_0=\\pm\\frac{1}{\\sqrt{2}}\\mbox{sgn}\\left((K_1-\\zeta)K_2\\right)\n\t\\sqrt{|\\mu_0^2|-\\Re\\left(\\mu_0^2\\right)}\n\t\\label{4.2a},\n\\end{equation}\ni.e.\\ the sign of the imaginary part of \n$\\mu$ depends on the sign of $K_1-\\zeta$. Thus the value of the integrals \nwill in general not be the same whether the contour is approached from the \ninterior or the exterior region.\nThis reasoning does not work for points $K=K_0$ \nnot on the axis ($K_2\\neq 0$) with $K_1=0$. There the imaginary part \nof $\\mu_0^2$ is zero which means that $\\mu_0$ is either purely \nimaginary or real in the vicinity of $P_0=K_0$ depending  on the sign of\n$K_2-\\zeta$. \nWe conclude that the integrals over $\\Gamma$ with $P_0\\in \\Gamma$ have the form \n$J=J^1+\\mbox{sgn}(\\epsilon) J^2$ (where the $J^i$ are independent of \n$\\epsilon$ which indicates if the contour is approached from the interior \nor the exterior) which implies that the limiting value of the \nErnst potential calculated via (\\ref{8.16}) exists but depends on\n$\\epsilon$. Therefore, we have proven the following\n\\begin{proposition}\nLet $P_0$ lie on the contour $\\Gamma$ but not on the\naxis, on one of the branch points $E_i$ or $F_i$, or at an endpoint \nof $\\Gamma_i$. Then $f$  will, in general, have a jump at\n$\\Gamma$. The limiting value of $f$ will exist and be H\\\"older--continuous \nthere. $f$ may be singular at the endpoints of the $\\Gamma_i$.\n\\end{proposition}\nThus the Ernst potential will be finite but discontinuous at a \ncontour $\\Gamma_z$ in the $(\\rho,\\zeta)$--plane given by $P_0\\in \\Gamma$ \nwhich means that the \nsolution to the vacuum equations will not be regular at a surface in the \n$(\\rho$, $\\zeta$, $\\phi$)--space. If this surface is closed, it can \npossibly be \nidentified with the surface of a body of revolution. The interior of the \nbody is supposed to be filled with matter. Therefore the vacuum solution is \nonly considered in the exterior (that contains $z=\\infty$); it is not \nregular at the boundary to the matter region. \n\n\n\\paragraph{The axis}\n\nThe axis is a double point on the Riemann surface ${\\cal L}_H$ since two branch\npoints coincide. In this case, all quantities may be considered on\nthe Riemann surface $\\Sigma'$ given by \n$\\mu'{}^2=\\prod_{i=1}^{g}(\\tau-E_i)(\\tau-F_i)$, as was shown by Fay\n\\cite{fay}. Let a prime denote here and in the following that the primed \nquantity is taken on $\\Sigma'$. This surface is obtained from ${\\cal L}_H$ \nby removing the cut $\\left[P_0,\\bar{P}_0\\right]$. For the \nanalysis of the axis, we will use a slightly different cut--system than the \none introduced in the previous section: we take a closed curve around \n$\\left[P_0,\\bar{P}_0\\right]$ in the $+$--sheet as the cut $a_g$. All $b$--cuts \nshall begin at the cut $\\left[E_1,F_1\\right]$. The rest is unchanged.\nThis implies for the characteristic of the theta function that it has the form\n\\begin{equation}\n\t\\left[\n\t\\begin{array}{ll}\n\t\t\\alpha' & 1  \\\\\n\t\t\\beta' & \\varepsilon\n\t\\end{array}\n\t\\right]\n\t\\label{chara}\n\\end{equation}\nwhere $\\varepsilon=0,1$ and $\\alpha_i'=0$. \n\nSince the expansions of all characteristic quantities of the Riemann surface\nare smooth in $\\rho$ except $\\pi_{gg}$ which is divergent as $\\ln \\rho$ for\n$\\rho\\to 0$, it follows that the Ernst potential has a regular expansion \nin $\\rho$. For points $P_0$ not coinciding with real branch points or \nsingularities of the exponent in (\\ref{8.16}), the Ernst potential is thus \nat least $C^3$. It follows from a theorem of M\\\"uller zum Hagen \\cite{mzh} that \nit is therefore analytic. Consequently it is sufficient \nto calculate the limiting case. If this is well defined, the \nErnst potential is regular at these points of the axis. \nThe differentials of the first kind for $\\rho=0$ turn out to be\n\\begin{equation}\n\t{\\rm d}\\omega_i={\\rm d}\\omega_i', \\quad i=1,\\ldots, g-1, \\quad\n\t{\\rm d}\\omega_g=-{\\rm d}\\omega'_{\\zeta^+\\zeta^-},\n\t\\label{sing1}\n\\end{equation}\nwhere ${\\rm d}\\omega'_{\\zeta^+\\zeta^-}$ is the normalized differential of \nthe third kind on $\\Sigma'$ with poles in $\\zeta^+$ and $\\zeta^-$. \nThis implies for the $b$-periods\n\\begin{eqnarray}\n\t\\pi_{ij} & = & \\pi_{ij}',\\quad i,j=1,\\ldots, g-1,\n\t\\label{sing2} \\\\\n\t\\pi_{ig} & = & -\\int_{\\zeta^-}^{\\zeta^+}{\\rm d} \\omega_i', \\quad i=1,\\ldots, \n\tg-1,\n\t\\label{sing3} \\\\\n        \\pi_{gg} & = & 2\\ln\\rho+\\mbox{reg. terms}\\enspace.\n\t\\label{sing4}\n\\end{eqnarray}\nSince $\\pi_{gg}$ diverges, the theta function will break down to a sum of \ntwo theta series on $\\Sigma'$ (in the case of genus $g=1$, the surface \n$\\Sigma'$ has genus 0; the formula below can however be used if one \nreplaces the theta function $\\Theta'$ simply by a factor 1 which means \nthat the axis potential can be expressed in terms of elementary functions \nin this case). We introduce integrals $\\omega'(P)$ of the first kind with the\nproperty $\\omega'(E_1)=0$. \nThe differential ${\\rm d}\\omega_{\\infty^+\\infty^-}$ on the axis becomes\n${\\rm d}\\omega'_{\\infty^+\\infty^-}$. In the case of the contour integrals \none has to observe that an additional factor $\\mbox{sgn}(K_1-\\zeta)$ in the \nnotation of (\\ref{4.2a}) occurs for the same reasons as there. Since the \nAbelian integrals of the second kind can be obtained from the integrals of \nthe third kind by a limiting procedure, the same holds for these integrals \nand their $b$-periods. With the above settings, we obtain for (\\ref{8.16})\n\\begin{eqnarray}\n        f&=&\\frac{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n        \\left(\\omega'|^{\\infty^+}_{\\zeta^+}+u'+b'\\right)\n\t+(-1)^{\\varepsilon}\\exp(-(\\omega'_g(\\infty^+)+u_g+b_g))\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\\left(\\omega'|^{\\infty^+}_{\\zeta^-}\n\t+u'+b'\\right)}{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n        \\left(\\omega'|^{\\infty^+}_{\\zeta^+}-u'-b'\\right)\n\t+(-1)^{\\varepsilon}\\exp(-(\\omega'_g(\\infty^+)-u_g-b_g))\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\\left(\\omega'|^{\\infty^+}_{\\zeta^-}\n\t-u'-b'\\right)}\n        \\nonumber\\\\\n        &&\\exp\\left\\{\\Omega'|_{\\infty^-}^{\\infty^+}+\\frac{1}{2\\pi i}\n        \\int\\limits_\\Gamma\n\\ln G(\\tau){\\rm d}\\omega'_{\\infty^{+}\\infty^-}(\\tau)+b_g+u_g\\right\\}.\n\t\\label{sing7}\n\\end{eqnarray}\nIt can be seen from the above formula that the limiting value of $f$ exists \neven if $u_g$ diverges, provided (\\ref{reg}) holds ($f$ will be\nH\\\"older-continuous if $u_g$ diverges). The Ernst potential will however have\nan essential singularity at the real singularities of $\\Omega$. We can\nsummarize the above results.\n\\begin{proposition}\nLet condition (\\ref{reg}) hold. Then the\nErnst potential is regular on the axis except at the points where $P_0$\ncoincides with singularities\nof $\\Omega$, points of $\\Gamma$, and branch points $E_i$, $F_i$.\n\\end{proposition}\n\\begin{remark}\n        Though $f$ is H\\\"older-continuous even if $u_g$ diverges, it is\n        interesting to note for the following when this will be the case.\n\tObviously this can only happen at the \n\treal points of $\\Gamma$. It can be seen however that $u_g$ is always \n\tbounded at these points due to the reality condition, unless they are \n\tendpoints of $\\Gamma_i$ (this would lead to a conic singularity on the \n\taxis).\n\\end{remark}\n\n\n\\paragraph{Real branch points}\n\nIf $P_0$ coincides with a real branch point $E_i$ or $F_i$, this will be a \ntriple point on ${\\cal L}_H$. We get the following.\n\\begin{proposition}\n\tAt points where $P_0$ coincides with the real branch point $E_g$, the \n\tlimiting value of $f$ exists. The Ernst potential \n\tis in general not differentiable there. \n\\end{proposition}\n\\begin{proof}\n\tWe use the same cut system and the same notation as on the axis. Put \n\t$P_0=E_g+x$ with $x=\\delta {\\rm e}^{{\\rm i}\\phi}$ and $\\phi\\in\\bigBbb{R}$, \n\t$\\delta\\in\\bigBbb{R}^+$. In order to expand $f$ in powers of $x$ and $\\bar{x}$, \n        one has to consider the $a$-periods, in particular\n\t\\begin{eqnarray}\n\\oint_{a_g} \\frac{{\\rm d}\\tau}{\\mu} &=& \\frac{4}{\\sqrt{\\bar{x}(F_g-E_g)}}\n\\int_{1}^{\\frac{1}{k}}\\frac{{\\rm d} t}{\\sqrt{(1-t^2)(1-k^2 t^2)(F_g-E_g+xt^2)}\n\\mu''(F_g+xt^2)}\\nonumber\\\\\n&=&\\frac{4}{\\sqrt{\\bar{x}(F_g-E_g)}\\mu''(F_g)}\n({\\rm i}\\tilde{K}(k)+O(\\delta))\n\t\t\\label{reala} \n\\end{eqnarray}\nwhere $k={\\rm e}^{{\\rm i}\\phi}$, where\n$\\tilde{K}(k)=K(\\sqrt{1-k^2})$ and $K(k)$ are the complete\nelliptic integrals of the first kind, and where\n$\\mu''{}^2(\\tau)=\\prod_{i=1}^{g-1}(\\tau-E_i) (\\tau-F_i)$. It can be seen\nfrom (\\ref{reala}) that the $a$--period has an expansion in powers of\n$\\sqrt{\\delta}$. The coefficients of the expansion in $\\sqrt{x}$,\nand $\\sqrt{\\bar{x}}$ are $\\phi$--dependent, since the modul of the elliptic\nintegrals is just $k= {\\rm e}^{{\\rm i}\\phi}$.\nThis implies for the differentials of the first kind\n${\\rm d}\\omega_i={\\rm d}\\omega_i'+O(\\sqrt{\\delta})$ for\n$i=1,...,g-1$, and ${\\rm d}\\omega_g={\\rm d}\\omega_{g-1}$. Similarly\n${\\rm d}\\omega_{\\infty^+\\infty^-}={\\rm d}\\omega'_{\\infty^+\n\\infty^-}+O\\left(\\sqrt{\\delta}\\right)$. We get for the\n$b$-periods,\n\\begin{equation}\n\\pi_{gg}=-2\\pi\\frac{K(k)}{K_1(k)}\\left(1+O\\left(\\sqrt{\\delta}\\right)\\right)\n\\label{realb}\\enspace,\n\\end{equation}\nwhereas $\\pi_{(g-1)g}=O(\\sqrt{\\delta})$ and $\\pi_{ij}=\\pi_{ij}'$ for\n$i,j=1,\\dots,g-1$ in the limit. Thus $f$ can be expanded in $\\sqrt{x}$ and $\n\\sqrt{\\bar{x}}$. Even in case that only integer powers in the expansion\noccur, the coefficients will be in general $\\phi$-dependent. Though the\nlimiting value of $f$ at $P_0=E_g$ exists, $f$ will in general not be\ndifferentiable at this point.\n\\end{proof}\nThis implies that the real branch points are singular points on the axis,\npossibly topological defects in the spacetime, see \\cite{korot1}. They should not occur in \nthe context of exterior solutions for bodies of revolution we are \ninterested in.\n\n\n\\paragraph{Non-real branch points}\n\nIf $P_0$ coincides with a branch point $E_g=\\bar{F}_g$, the points $E_g$ \nand $F_g$ will be double points on ${\\cal L}_H$. Thus the situation is \nsimilar to the one on the axis with the only exception that one ends up \nhere with two double points.  \nAs on the axis, it is convenient to consider all quantities on \na Riemann surface $\\Sigma''$ given by \n$\\mu''{}^2=\\prod_{i=1}^{g-1}(K-E_i)(K-F_i)$ where the double points are\nremoved. All quantities with two primes \nare understood to be taken on this surface. We use the following cut \nsystem: let $a_{g-1}$ be the circle around $\\left[P_0,E_g\\right]$, and $a_g$ \nthe circle around $\\left[\\bar{P}_0,F_g\\right]$, both in the plus \nsheet. The remaining cuts are as on the axis, i.e.\\ all $b$--cuts start at \n$\\left[E_1,F_1\\right]$.  As on the axis, we get\n\\begin{proposition}\nLet (\\ref{reg}) hold, and let $E_g=\\bar{F}_g\\notin\\Gamma$.\n\tThen the Ernst potential is regular at the point $P_0=E_g$. \n\tFor $E_g\\in\\Gamma$, $f$ is in general H\\\"older--continuous at $P_0=E_g$.\n\\end{proposition}\nThe proof is similar to the one on the axis and basically uses again results of \nFay \\cite{fay}.\n\\begin{proof}\n\tThe case $g=1$ may be checked directly with the help of the standard \n        theory of elliptic theta functions (see e.g.\\ \\cite{erdelyi}).\n\tFor $g>1$ with the cut system in use and $P_0=E_g+x$, \n\twhere $x$ is chosen as in the \n\tcase of the real branch points, the differentials of the \n\tfirst kind have a smooth expansion in $x$ and $\\bar{x}$. In contrast to \n\tthe case of real branch points, the coefficients in the expansion are \n\t$\\phi$--independent. The differentials\n\t${\\rm d}\\omega_{i}$ become in leading order the differentials of the first kind \n\t${\\rm d}\\omega''_i$ on $\\Sigma''$.  The differential \n\t${\\rm d}\\omega_{g-1}$ becomes in the limit the differential \n\t$-{\\rm d}\\omega''_{E_g^+E_g^-}$, and similar for ${\\rm d}\\omega_g$ at $F_g$. \n\tThe differential of the third kind becomes \n\t${\\rm d}\\omega_{\\infty^+\\infty^-}=\n\t{\\rm d}\\omega_{\\infty^+\\infty^-}''$. All these differentials have \n\tcoefficients in the $x$ and $\\bar{x}$ expansion that contain Abelian \n\tintegrals of the second kind with poles in $E_g^{\\pm}$ and $F_g^{\\pm}$ as \n\tmay be checked by direct calculation. This implies for the \n\t$b$--periods that $\\pi_{ij}=\\pi''_{ij}$ for $i,j=1,...,g-2$ and \n\t\\begin{eqnarray}\n\t\t\\pi_{(g-1)(g-1)} & = & \\pi_{gg}=2\\ln\\delta +...,\n\t\t\\nonumber \\\\\n\t\t\\pi_{i(g-1)} & = & -2\\omega''(E_g^+),\n\t\t\\nonumber \\\\\n\t\t\\pi_{ig} & = & -2\\omega''(F_g^+),\n\t\t\\label{com1}\n\t\\end{eqnarray} \n\twhereas $\\pi_{(g-1)g}$ is finite in the limit $\\delta\\to0$.\n\tIf $E_g\\notin \\Gamma$, the $u_i$ as well as the Cauchy integral in the \n\texponent have a smooth expansion in $x$ and $\\bar{x}$ with finite \n\tcoefficients.  The theorem of \\cite{mzh} then guarantees regularity if the\n\tlimiting value that may be calculated as on the axis exists. \n\tThe theta function \n\ton ${\\cal L}_H$ breaks down to a sum of four theta functions on \n\t$\\Sigma''$ times a multiplicative factor. If $E_g\\in\\Gamma$, \n\tthe coefficients in the expansion of $f$ in $x$ and $\\bar{x}$ will diverge \n\twhich implies that $f$ is possibly \n\tnot differentiable there though the limiting \n\tvalue exists if (\\ref{reg}) holds.\n\\end{proof}\n\n\n\\section{Metric functions and ergospheres}\n\nIn the previous sections we have made extensive use of the \ncomplete integrability of the Ernst equation to \nconstruct a large class of solutions. To discuss physical \nfeatures of the resulting spacetimes however, it would be helpful to have \nexpressions in closed form not only for the Ernst potential but for the \nmetric functions, at least for the functions ${\\rm e}^{2U}$ and $a$ that \ncan be expressed invariantly via the Killing vectors. It is a remarkable \nfact already noticed by Korotkin \\cite{korot1} that the metric function \n$a$ can be related to derivatives of the matrix $\\Phi$ without solving \nthe differential equation (\\ref{vac9}). In the following we will show that \na theta identity of Fay \\cite{fay} can be used to go one step \nfurther to obtain a formula for $a$ that is free of derivatives. The same \nidentity leads to a simplified expression for the metric function ${\\rm e}^{2U}$ \nthat can be directly used to identify ergospheres in the spacetime.\n\nFay's trisecant identity establishes a relation between four points\n$A_1,\\dots,A_4$ on a\nRiemann surface, in our case ${\\cal L}_H$, in arbitrary position (see e.g.\\ \n\\cite{mumford}, \\cite{taimanov}). Let $x$ be an arbitrary $g$-dimensional\nvector. Then the following identity holds,\n\\begin{eqnarray}\n        &  &\\Theta(x)\\Theta\\left(x+\\int_{A_1}^{A_3}{\\rm d}\\omega+\n        \\int_{A_2}^{A_4}{\\rm d}\\omega\\right)\n        -\\exp \\left(\\Omega_{A_1A_4}|^{A_3}_{A_2}\\right)\n        \\Theta\\left(x+\\int_{A_2}^{A_3}{\\rm d}\\omega\\right)\n        \\Theta\\left(x+\\int_{A_1}^{A_4}{\\rm d}\\omega\\right)\n\t\\nonumber \\\\\n        &&- \\exp\\left(\\Omega_{A_2A_4}|^{A_3}_{A_1}\\right)\n        \\Theta\\left(x+\\int_{A_1}^{A_3}{\\rm d}\\omega\\right)\n        \\Theta\\left(x+\\int_{A_2}^{A_4}{\\rm d}\\omega\\right) =  0\\enspace,\n\t\\label{fay1a}\n\\end{eqnarray}\nwhere e.g.~$\\Omega_{A_1A_4}|^{A_3}_{A_2}$ denotes the integral of a normalized\ndifferential of the third kind with simple poles at $A_1$ respectively $A_4$\nwith residues $+1$ respectively $-1$ along a path from $A_2$ to $A_3$. For a\ngeometric interpretation of this identity in terms of the Kummer variety see\n\\cite{mumford}, \\cite{taimanov}, for an interpretation via generalized cross\nratio functions see \\cite{farkas}. The strength of the above identity arises\nfrom the fact that it holds for points $A_i$ in general position. By a \nsuitable choice of these points, we obtain for the metric function ${\\rm \ne}^{2U}$, the real part of the Ernst potential, \n\\begin{equation}\n\te^{2U}=\\frac{1}{2}\\exp \\left(\\Omega_{\\bar{P}_0\\infty^-}|^{P_0}_{\\infty^+}\n\t\\right)\n\t\\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\bar{P}_0))}{\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\infty^-)+\\omega(\\bar{P}_0))\n\t\\Theta\\left[\\alpha\\atop\\beta\\right](u+b+\\omega(\\infty^-))} e^{I}\n\t\\label{fay14}\n\\end{equation}\nwhere $I$ denotes the integral in the exponent of (\\ref{8.16}).\nThis formula makes it possible to identify directly the zeros of ${\\rm \ne}^{2U}$ which give the ergospheres, the limiting surfaces of stationarity \n(inside these surfaces there can be no observer at rest with respect to \nspatial infinity). Since the exponent of \nthe integral of the third kind in (\\ref{fay14}) in front of the fraction \ncannot vanish, the necessary condition for ergospheres is \n\\begin{equation}\n\t\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\bar{P}_0))=0\n\t\\label{ergo}.\n\\end{equation}\nDefining the divisor $A$ as the solution of the Jacobi inversion problem\n\\begin{equation}\n\t\\omega(A)-\\omega(D)=u+b\n\t\\label{ergo1},\n\\end{equation}\nwe find that an ergosphere can occur if $P_0$ or $\\bar{P}_0$ are in $A$. It\nis however possible that the denominator of (\\ref{fay14}) vanishes at the \nsame time which would imply a violation of (\\ref{reg}) (and thus a \nsingularity of the spacetime). Summing up we get\n\\begin{proposition}\n\tI. Let $P_0$ or $\\bar{P}_0$ and $\\infty^-$ be in $A$ for some $P_0$, \n\tthen condition \n\t(\\ref{reg}) is violated and the Ernst potential is singular\n\tat these points.\\\\\n\tII. Let $P_0$ or $\\bar{P}_0$ but not $\\infty^-$ be in $A$ for some $P_0$, \n\tthen the real part of the Ernst potential vanishes at these points which \n\t describe an ergosphere.\n\\end{proposition}\n\nThe same formula can be used on the axis where we obtain for the metric \nfunction ${\\rm e}^{2U}$ in the notation of (\\ref{sing7})\n\\begin{eqnarray}\n\t{\\rm \n        e}^{2U}&=&\\frac{1}{2}\\exp\\left(\n        \\Omega_{\\infty^-\\zeta^+}|^{\\infty^+}_{\\zeta^-}\n\t\\right)\\nonumber\\\\\n        &&\\frac{\\Theta'\\left[\\alpha' \\atop \\beta'\\right]^2(u'+b')}{\\Theta'\n        \\left\n        [\\alpha' \\atop \\beta'\\right]^2(\\omega'|^{\\infty^-}_{\\zeta^-}+u'+b')-\n        \\exp(2(\\omega_g(\\infty^-)+u_g+b_g))\\Theta'\\left[\\alpha' \\atop\n        \\beta'\\right]^2\n        (\\omega'|^{\\infty^-}_{\\zeta^+}+u'+b')}\\nonumber\\\\\n\t\\label{ergo2}.\n\\end{eqnarray}\nThe condition for an ergosphere to hit the axis is then\n\\begin{equation}\n\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u'+b')=0\\enspace,\n\t\\label{ergo3}\n\\end{equation}\nsince the integral of the third kind in the exponent in front of the fraction\ncannot diverge for finite values of $\\zeta$. The interesting feature of \nthis relation is that it is completely independent of the physical \ncoordinates. This implies that if an ergosphere extends to the axis, this \nwill be only possible if the metric function ${\\rm e}^{2U}$ vanishes on the\nwhole axis. In the case of the Kerr solution, the ergosphere touches the \naxis at the horizon. An interpretation of the fact \nthat the whole axis would be singular in the present case is given in the \nnext section where the above case is related to the ultrarelativistic limit \nin which the source of the gravitational field becomes so strong that it\nvanishes behind the horizon of the extreme Kerr metric.\n\nThe metric function $a$ can be calculated from (\\ref{a}) if one uses the \ntrisecant identity (\\ref{fay1a}) in the limit that two points coincide. \nUsing the trisecant identity several times, we get\n\\begin{eqnarray}\n        (a-a_0){\\rm e}^{2U}&=&-\\rho\\left(\n        \\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](0)\n        \\Theta\\left[\\alpha\\atop\\beta\\right]\n        \\left(\\int_{\\bar{P}_0}^{P_0}{\\rm d}\\omega\\right)}{\n        \\Theta\\left[\\alpha\\atop\\beta\\right]\\left(\n        \\int_{\\infty^-}^{P_0}{\\rm d}\\omega\\right)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n        \\left(\\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)}\\times\n        \\right.\\nonumber\\\\\n\t&&\\left.\\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n        \\left(u+b+\\int_{P_0}^{\\infty^-}{\\rm d}\\omega+\n        \\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)}{\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n        \\left(u+b+\\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n        \\left(u+b+\\int_{P_0}^{\\infty^-}{\\rm d}\\omega\\right)}-1\\right)\n        \\label{fay7a}\\enspace.\n\\end{eqnarray}\nThe constant $a_0$ can be obtained in a similar way from the condition \nthat $a=0$ on the regular part of the axis (we assume here that the \nsingularities in the exponent of (\\ref{8.16}) are situated in a compact \nregion of the $(\\rho,\\zeta)$--plane). Care has to be taken in the above \nformula that some of the terms in brackets explode as $1\/\\rho$ in the \nlimit $\\rho\\to0$. We get\n\\begin{equation}\n\ta_0=\\frac{{\\rm i}}{2}(D-\\bar{D})\\frac{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n\t(u'+b'+\\int_{\\infty^+}^{\\infty^-}d\\omega')\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right](0)}{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n\t(\\int_{\\infty^+}^{\\infty^-}d\\omega')\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u'+b')}e^{-I'}\n\t\\label{fay18b}.\n\\end{equation}\nIt can be seen from this formula that $a_0$ does not vanish if there \nare no singularities in the exponent ($I=u=b=0$) in which case $f=1$ which \ndescribes Minkowski spacetime. This reflects, as already noted, the fact \nthat $a_0$ is a gauge dependent quantity. The metric function $a$ however \nis gauge independent. In the above example of Minkowski spacetime, it will \nof course vanish in the used asymptotically non-rotating coordinates.\n\n\n\\section{Asymptotic behaviour and equatorial symmetry}\n\nSince we are mainly interested in solutions to the Ernst equation that could\ndescribe the gravitational field outside a compact matter source, we will study\nthe asymptotic behaviour (near spatial infinity) of the solutions (\\ref{8.16}).\nIt is generally believed\nthat the Ernst potentials of the corresponding spacetimes are regular except\nat the contour $\\Gamma_z$ in \nthe $(\\rho,\\zeta)$--plane which corresponds to the surface of the body, \nasymptotically flat and equatorially symmetric. We will investigate in the \nfollowing whether it is possible to identify solutions with these properties in \nthe class (\\ref{8.16}).\n\n\n\\paragraph{Asymptotic behaviour}\n\nAsymptotic flatness implies that the Ernst potential is of the form\n$f=1-2m\/|z|+o(1\/|z|)$ for $|z| \\to \\infty$ where $m$ is a positive real\nconstant. A complex $m$ is related to a so called NUT-parameter that is\ncomparable to a magnetic monopole.\n\nThe asymptotic properties of the solutions (\\ref{8.16}) can be read off at the \naxis. Notice that the ${\\rm d}\\omega_i'$ are independent of $\\zeta$. \nFor ${\\rm d}\\omega_g$, we get\n\\begin{equation}\n\t{\\rm d}\\omega_g={\\rm d}\\omega'_{\\infty^+\\infty^-}\\left(1-\\frac{1}{2\\zeta}\n\t\\sum_{i=1}^{g}(E_i+F_i)\\right)+\\frac{1}{\\zeta}{\\rm d}\\omega'_{\\infty^+,1}\n\t+o(1\/\\zeta)\n\t\\label{sing9}\n\\end{equation}\nwhere ${\\rm d}\\omega'_{\\infty^+,1}$ is the differential of the second kind \nwith a pole of second order at $\\infty^+$. Furthermore it can be seen \nthat $\\exp(-\\omega_g(\\infty^+))$ is proportional to $1\/\\zeta$ for $\\zeta\\to\n\\infty$. Thus we get\n\\begin{proposition}\n        Let $\\lim_{\\tau\\to \\infty}\\tau\\ln G(\\tau)=0$ \n        on all contours that go through\n\t$\\infty^+$ or $\\infty^-$ and let $\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n        \\left(u'+b'\\right)\\neq0$. Then $f$ has the form $f=1-2m\/\\zeta$ for \n\t$\\zeta\\to\\infty$ where $m$ is a complex constant. \n\\end{proposition}\nThe proof of this proposition follows from (\\ref{sing9}) and (\\ref{sing7}).\n\n\n\\paragraph{Equatorial symmetry}\n\nThe fact that the mass is in general complex implies that the class we are \nconsidering here is too large if one wants to study only solutions that are\nasymptotically flat in the strong sense ($m$ real). There is the belief that\nstationary axisymmetric spactimes describing isolated bodies in thermodynamical\nequilibrium are equatorially symmetric. This implies for the Ernst potential\n$f(-\\zeta)=\\bar{f}(\\zeta)$. Solutions with this property always have a real\nmass due to the symmetry. It is therefore of special interest to single out\nequatorially symmetric solutions among those in (\\ref{8.16}). We get\n\\begin{theorem}\nLet ${\\cal L}_H$ be a hyperelliptic surface of the form (\\ref{3.22a}) with \neven genus $g=2s$ and the property $\\mu(-K,-\\zeta)=\\mu(K,\\zeta)$.\t\nLet $\\Gamma$ be a piecewise smooth contour on ${\\cal L}_H$ such that \nwith $P=(K,\\mu(K))\\in \\Gamma$ also $\\bar{P}\\in\n\\Gamma$ and $(-K,\\mu(K))\\in \\Gamma$. Let there be \ngiven a finite nonzero function $G$  on \n$\\Gamma$ subject to $G(\\bar{P})=\\bar{G}(P)=G((-K,\\mu(K)))$. If $(p,\\mu(p))$\nis a singularity of $\\Omega$, the same should hold for $(-p,\\mu(-p))$. \nChoose a cut sytem in a way that the cuts $a_i^1$ ($i=1,\\ldots ,s$) encircle \n$\\left[-F_i,-E_i\\right]$ and $a_i^2$ encircle\n$\\left[E_i,F_i\\right]$  in the $+$--sheet (in the case of real branch \npoints, the points are ordered in the way $E_i<F_i<E_{i+1}<\\ldots $; points \nwith the same real part are ordered in the way $\\Im (E_i)<\\Im \n(F_i)<\\Im(E_{i+1})<\\ldots $ which implies that $E_i\\neq \\bar{F}_i$ in this \nspecial case).  \\\\\nThen  $f$ is equatorially symmetric if the characteristics \nin the $i$--th position  (any combination of the two cases is \nallowed) have the form\n\\begin{equation}\n\t\\left[\n\t\\begin{array}{cc}\n\t\t0 & 0  \\\\\n\t\t1 & 1\n\t\\end{array}\n\t\\right], \\quad \n\t\\left[\n\t\\begin{array}{cc}\n\t\t0 & 0  \\\\\n\t\t0 & 0\n\t\\end{array}\n\t\\right]\n\t\\label{equ1}.\n\\end{equation}\n\n\\end{theorem}\n\\begin{proof}\n\tThe property $\\mu(\\zeta,K)=\\mu(-\\zeta,-K)$ on ${\\cal L}_H$ makes it \n\tpossible to express quantities on a surface with $\\zeta=-\\zeta_0$ in terms \n\tof  the \n\tcorresponding quantities on the surface with $\\zeta=\\zeta_0$. We have \n\t$a_i^1(-\\zeta)=\\tau a_i^2(\\zeta)$ and $b_i^1(-\\zeta)=\\tau \n\tb_i^2(\\zeta)$ where $\\zeta$ and $-\\zeta$ denote the surface on which the \n\tquantity is considered, and where $\\tau$ is the anti--holomorphic \n\tinvolution on ${\\cal L}_H$. Together with the symmetry properties of the \n\tAbelian integrals in the exponent of (\\ref{8.16}), \n\tthis implies that the transformation \n\t$\\zeta\\to-\\zeta$ acts as the complex conjugation together with a change \n\tof the upper index. Thus we have for the \n\tcharacteristics in (\\ref{equ1}) $f(-\\zeta)=\\bar{f}(\\zeta)$.\n\t\\end{proof}\n\\begin{remark}\nIf the theta function contains only blocks of the form\n\t $\\left[\\begin{array}{cc}\n\t\t0 & 0  \\\\\n\t\t1 & 1\n\t\\end{array}\\right]$, the resulting $f$ is just the complex conjugate of \n\tthe Ernst potential built with the Riemann theta function. This means \n\tthat the two cases in (\\ref{equ1}) are related through  complex \n\tconjugation. It is however possible to combine any number of these \n\ttwo blocks in which case the Ernst potential cannot be simply reduced to \n\tthe case of the Riemann theta function or its complex conjugate. In the \n\tcase of a rotating body, complex conjugation of the Ernst potential \n\tonly implies that the angular velocity of the body changes its \n\tsign.\n\\end{remark}\n\n The above results suggest that it is \n possible to identify a whole subclass of solutions among (\\ref{8.16}) that are \n asymptotically flat, regular except at a closed contour and equatorially \n symmetric, i.e.\\ solutions that might describe the \n exterior of a rotating body and might be helpful in the construction of \n solutions to boundary value problems for the Ernst equation. \n We get\n\\begin{theorem}\n\tLet ${\\cal L}_H$ be a regular hyperelliptic surface of even genus $g=2s$ \n\tof the form (\\ref{3.22a})  without real branch points.  \n\tLet $\\Gamma$ be a closed, \nsmooth contour on ${\\cal L}_H$ such that \nwith $P=(K,\\mu(K))\\in \\Gamma$ also $\\bar{P}=(\\bar{K},\\mu(\\bar{K}))\\in\n\\Gamma$ and $(-K,\\mu(K))\\in \\Gamma$ and $E_i\\notin\\Gamma$. Let there be \ngiven a finite nonzero function $G$  on \n$\\Gamma$ subject to $G(\\bar{P})=\\bar{G}(P)=G((-K,\\mu(K))$. \nChoose the characteristic $\\left[\\alpha\\atop\\beta\\right]$ such \nthat it consists of blocks of the form $\\left[\n\\begin{array}{cc}\n\t0 & 0  \\\\\n\t1 & 1\n\\end{array}\n\\right]$ and $\\left[\n\\begin{array}{cc}\n\t0 & 0  \\\\\n\t0 & 0\n\\end{array}\n\\right]$ as in theorem 6.2.\n\nThen\n\\begin{equation}\\label{2}\nf(\\rho,\\zeta)=\\frac{\\Theta\\left[\\alpha\\atop\n\\beta\\right](\\omega(\\infty^{+})+u)}{\\Theta\\left[\\alpha\\atop\n\\beta\\right](\\omega(\\infty^{+})-u)}\n\\exp\\left\\{\n\\frac{1}{2\\pi{\\rm i}}\\int\\limits_\\Gamma\n\\ln G(\\tau){\\rm d}\\omega_{\\infty^{+}\\infty^-}(\\tau)\n\\right\\}\\enspace,\n\\end{equation}\nis\\\\\n1. a regular solution to the Ernst equation for $P_0\\notin \\Gamma$ if \ncondition (\\ref{reg}) holds.\\\\\n2. in general discontinuous at \n$\\Gamma_z$ given by $P_0\\in \\Gamma$,\\\\\n3. asymptotically ($|z|\\to \\infty$) given by $f=1-2m\/|z|$ where $m$ is a \nfinite real constant if $\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u')\\neq0$,\\\\\n4. equatorially symmetric.\n\\end{theorem}\n\\begin{proof}\n\t From \n\t(\\ref{8.16}) it can be seen that $f$ is a solution to the Ernst equation.  \n\tThe regularity properties  follow from the previous section. \n\t Asymptotic behaviour and \n\tequatorial symmetry follow from above.\n\\end{proof}\n\n\\begin{remark}\n\tThe choice of this class is mainly due to regularity requirements. \n\tIf all singularities like real branch points or the singularities of \n\tthe Abelian integrals of the second kind lie within the contour $\\Gamma$ \n\twhere the solution is not considered since the region is \n\tassumed to be filled with matter, \n\tthey would not affect the vacuum region. However this would not enlarge \n\tthe degrees of freedom (one real--valued function and a set of complex \n\tparameters) if one wants to solve boundary value problems.\n\t\\end{remark}\nWe will discuss the common properties of the solutions in this subclass in \nthe following.\n\\paragraph{Mass and Angular Momentum}\nThe asymptotic behaviour of the Ernst potential, $f=1-2m\/|z|-\n2{\\rm i}j\/|z|^2+\\cdots$,\nfollows already from the axis potential (\\ref{sing7}). The equatorial \nsymmetry implies that $m$ and $j$ are real constants. In the following it \nis convenient to introduce rescaled coordinates $\\tilde{z}=z\/R$ where $R$ \nis the radius of the smallest sphere that totally contains the contour \n$\\Gamma_z$, and the dimensionless quantities $M=m\/R$, $J=j\/R^2$. \nThis implies that the contour shrinks to a point in the limit \n$R\\to0$. If we use the differential operator $D_{\\infty^+}$ we get for the \nADM-mass with (\\ref{sing7})\n\\begin{equation}\n\tM=\\frac{D_{\\infty^+}\\Theta\\left[\\alpha'\\atop\\beta'\\right](u')}{\n\t\\Theta\\left[\\alpha'\\atop\\beta'\\right](u')}+\\frac{1}{2p {\\rm i}}\n\t\\int_{\\Gamma}^{}\\ln G {\\rm d}\\omega'_{1,\\infty^+}\n\t\\label{d1}.\n\\end{equation}\nA solution is of course \nonly physically acceptable if the mass is positive. \nSimilarly one can see that the angular momentum is proportional to $1\/\n\\Theta^2\\left[\\alpha'\\atop\\beta'\\right](u')$.\n\n\n\\paragraph{Minkowskian limit}\n\nIt is possible to parametrize a solution by the mass and the angular \nmomentum. For $M<<1$ and $J<<1$, the solution is nearly Minkowskian. \nIt can be directly seen that this limit is obtained for $G\\to1$. This \nimplies that the solutions from above for $|G|\\sim 1$ are in the regime of \nsmall gravitational fields. This is also the regime of the Newtonian limit \nif the solution has one.  \n\n\n\\paragraph{Ultrarelativistic limit}\n\nIt can be seen from (\\ref{d1}) that both the mass and the angular momentum\ndiverge if $\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u')=0$ but that $M^2\/J$\nremains finite in this case. This suggests that this divergence is best\nunderstood as the limit $R\\to0$ as was already done in \\cite{bawa} for \nthe case of the rigidly rotating dust disk: in this limit, the \ngravitational fields at the surface $\\Gamma_z$ of the matter source become \nso strong that it is hidden behind a horizon, i.e.\\ its coordinate radius \ntends to zero. It is suggestive to consider this limit as the \nultrarelativistic limit of the solution. For finite $R$, the Ernst \npotential is purely imaginary on the axis which means that the solution \nis no longer asymptotically flat. If one takes the limit $R\\to0$, one ends \nup with the Kerr solution in this case which gives further support to the\ninterpretation of this limit as the ultrarelativistic limit (this\ninterpretation is of course only consistent if the limit gives the extreme Kerr\nsolution for which the horizon is given in the used coordinates by\n$\\rho=\\zeta=0$). If the limit $R\\to0$ is taken for $\\rho=\\zeta=0$ with\n$\\rho\/\\rho_0$ and $\\zeta\/\\rho_0$ finite (this \ncorresponds to an observer on the contour that vanishes behind the horizon),\nthe resulting solution will not be asymptotically flat. \nFor an observer on $\\Gamma_z$, the exterior region is in infinite geodesic \ndistance, and thus completely decouples from the exterior. \nIn the context of a boundary value problem this limit corresponds to a stability\nlimit for the solution: if the boundary data reach a certain critical value,\nthe solution will no longer be regular.\n\nThus solutions in (\\ref{2}) should be physically interesting in the \nparameter range $0<M\/R<\\infty$. Notice that the solutions have an analytic \ncontinuation beyond the upper limit. It can be seen however from (\\ref{d1}) \nthat the mass changes its sign in this case since \n$\\Theta\\left[\\alpha'\\atop\\beta'\\right](u')$ has zeros of first order. \nConsequently these `overextreme' solutions, that do not have a Newtonian\nlimit, are probably not physically interesting, at least they are\nunacceptable in the region with negative mass. \n\n\n\\section{Reduction of the Ernst potential}\n\nThe explicit form of the solutions (\\ref{2}) in terms of theta functions \nhas a number of advantages in contrast to the linear integral equations to \nwhich the solution of boundary or initial value problems can be reduced in \nthe case of integrable non--linear evolution equations\\footnote{There are \nsolutions in terms of theta functions for these equations, too. But as we \nhave pointed out already, these solutions are always periodic or \nquasiperiodic. The solutions to the Ernst equation discussed here are \nhowever gauge equivalent to solutions to a linear integral equation as was \nshown in \\cite{prd}.}: it is possible to identify physically interesting \nfeatures as ergospheres or the ultrarelativistic limit explicitly. Since \nthe theta functions are transcendental, final results normally \ncan only be obtained \nnumerically. It is however possible to address most features like the \ncondition for the occurrence of ergospheres (\\ref{ergo}) \ndirectly without having to determine the Ernst potential numerically in the \nwhole spacetime.\n\nThe numerical treatment of theta functions is comparatively simple since \nthe exponential series converges rapidly due to the factor \n$\\exp\\left(\\frac{1}{2}\\pi_{ij}n_in_j\\right)$ where $\\Re(\\pi_{ij})<0$. It is\nhowever obvious that the numerics become more and more tedious the larger \nthe genus $g$ of the Riemann surface ${\\cal L}_H$ is. Therefore it is an \nimportant question whether the Riemann surface can be reduced in \nphysically interesting cases to surfaces of lower genus. Loosely speaking \nthis is possible if there exists a special relation between the branch \npoints (see Weierstrass' discussion of the case $g=2$ which is referred to \nin \\cite{algebro} \nand references given therein). Since the branch points $P_0$, \n$\\bar{P}_0$ are parametrized by the physical coordinates and can thus take \non arbitrary complex values, such a reduction will only be possible at \nspecial points of the spacetime which will in general not be of special \nphysical interest. A general reduction of the Riemann surface is possible \nif there exist non--trivial automorphisms on the surface. For the class of \nequatorially symmetric solutions discussed here, this is the case in the \nequatorial plane and on the axis. There the surfaces ${\\cal L}_H$  and \n$\\Sigma'$ have defining equations $\\mu(K)$ and $\\mu'(K)$ which both depend \nonly on $K^2$. Thus on both surfaces there is the involution $T$ defined \nby $(K,\\mu(K))\\to (-K,\\mu(-K))$. \n\nFor the sake of simplicity, we will only discuss the characteristic \n$\\alpha_i=\\beta_i=0$ and the case $E^1_i=-\\bar{E}^2_i$ (the general case \ncan be inferred from the resulting relations without problems). \nWe will concentrate on disks of radius $\\rho_0$ since they \nare an interesting model for galaxies. The Ernst potential simplifies \nin the equatorially \nsymmetric case at the disk where the boundary data are prescribed, what\nmakes disks the most promising objects in the search for solutions to \nboundary value problems to the Ernst equation in closed form. We recall \nthat the first solution of such a problem was found for the \nrigidly rotating dust disk \\cite{ngbml1}.\n\nIn the equatorial plane ($\\zeta=0$), the surface ${\\cal L}_H$ is then given \nby $\\mu^2(K)=(K^2+\\rho^2)\\prod_{i=1}^{s}(K^2-E_i^2)(K^2-\\bar{E}_i^2)$. We cut \nthe surface as before which implies $Ta_i^1=a_1^2$, $Tb_i^1=b_i^2$ and \n${\\rm d}\\omega_i^1(TP)=-{\\rm d}\\omega_i^2(P)$ with $P\\in {\\cal L}_H$. The \nRiemann surface $\\Sigma_1={\\cal L}_H\/T$ of genus $s$ \nis then given by \n\\begin{equation}\n\t\\mu_1^2(x)=x(x+\\rho^2)\\prod_{i=1}^{s}(x-E_i^2)(x-\\bar{E}_i^2)\n        \\label{prym1}\\enspace.\n\\end{equation}\nThe holomorphic differentials $dv_i$ in $\\Sigma_1$ dual to $(a_i,b_i)$ (the\nprojection of the cuts on ${\\cal L}_H$ onto $\\Sigma_1$) follow from ${\\rm d}v_i\n={\\rm d}\\omega_i^1-{\\rm d}\\omega_i^2$. The so called Prym differentials ${\\rm \nd}w_i$ which change the sign under $T$ are given  by ${\\rm d}w_i=\n{\\rm d}\\omega_i^1+{\\rm d}\\omega_i^2$. They are holomorphic differentials \non the Riemann surface $\\Sigma_2$ of genus $s$ with \n\\begin{equation}\n\t\\mu_2^2(y)=(y+\\rho^2)\\prod_{i=1}^{s}(y-E_i^2)(y-\\bar{E}_i^2)\n        \\label{prym2}\\enspace,\n\\end{equation}\nwhich implies that the Prym variety is a Jacobi variety in this case. The \nRiemann matrix on ${\\cal L}_H$ has the form \n\\begin{equation}\n\t\\Pi=\\frac{1}{2}\\left(\n\t\\begin{array}{cc}\n\t\t\\Pi^1+\\Pi^2 & \\Pi^2-\\Pi^1  \\\\\n\t\t\\Pi^2-\\Pi^1 & \\Pi^1+\\Pi^2\n\t\\end{array}\n        \\right)\\enspace,\n\t\\label{red3}\n\\end{equation}\nwhere the $\\Pi^i$ are the Riemann matrices on $\\Sigma^i$ respectively. The \ntheta function on ${\\cal L}_H$ thus factorizes into products of theta \nfunctions on the $\\Sigma_i$,\n\\begin{equation}\n\t\\Theta(x_1|x_2,\\Pi)=\\sum_{\\delta}^{}\\Theta\\left[\\delta\\atop0\\right]\n\t(x_1+x_2;2\\Pi^2)\\Theta\\left[\\delta\\atop0\\right]\n        (x_1-x_2;2\\Pi^1)\\enspace,\n\t\\label{red4}\n\\end{equation}\nwhere each component of the $s$-dimensional vector $\\delta$ takes the\nvalues $0,1$. Thus the theta function on the surface of genus $2s$ can be \nexpressed via theta functions on surfaces of genus $s$.\n\nIn the case of the Ernst potential (\\ref{2}), further simplifications \nfollow from the fact that $\\infty$ is a branch point of $\\Sigma_2$. For the \ncontour integrals $u$, we obtain for disks\n\\begin{equation}\n        u_v=\\frac{1}{\\pi {\\rm i}}\\int_{\\Gamma_v}^{}\\ln G{\\rm d} v,\\quad\n        u_w=\\mbox{sgn}\\zeta\\frac{1}{\\pi {\\rm i}}\\int_{\\Gamma_w}^{}\\ln G{\\rm d} w\n        \\enspace,  \\label{red11}\n\\end{equation}\nwhere $\\Gamma_v$ is the contour in the $+$-sheet of $\\Sigma_1$ between $0$\nand $-\\rho^2$  along the real axis, and $\\Gamma_w$ is the part of the real \naxis in the upper sheet of $\\Sigma_2$ between $-\\infty$ and $-\\rho^2$. The \nformula for $u_w$ shows that it does matter whether the equatorial plane is \napproached from the upper or the lower side (the Ernst potential is not \nregular at the disk). For $\\rho>\\rho_0$, we have $u_w=0$ ($G=1$ in the \nexterior of the disk). Similarly we get for the integral in the exponent\nof (\\ref{2})\n\\begin{equation}\n        I_v\\doteq\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\ln G{\\rm d}\n        \\omega_{\\infty^+\\infty^-}=\n        \\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma_v}^{}\\ln G{\\rm d} v_{\\infty^+\\infty^-}\n        \\label{red12}\\enspace.\n\\end{equation}\nSumming up we can write the Ernst potential in the equatorial plane in the \nform \n\\begin{equation}\n\tf=\\frac{\\sum_{\\delta}^{}\\Theta\\left[\\delta\\atop0\\right]\n\t(v(\\infty^+)+u_v;2\\Pi^1)\\Theta\\left[\\delta\\atop\\beta\\right]\n\t(u_w;2\\Pi^2)}{\\sum_{\\delta}^{}\\Theta\\left[\\delta\\atop0\\right]\n\t(v(\\infty^-)+u_v;2\\Pi^1)\\Theta\\left[\\delta\\atop\\beta\\right]\n        (u_w;2\\Pi^2)}{\\rm e}^{I_v}\\enspace,\n\t\\label{prym3}\n\\end{equation}\nwhere $\\beta_i=1$, and where $v(P)=\\int_{-\\rho^2}^{P}dv$. \nThe reality properties of the above theta functions \nimply together with (\\ref{red11}) the condition for equatorial symmetry \n$f(-\\zeta)=\\bar{f}(\\zeta)$. Thus the imaginary part of $f$ jumps at the \ndisk. For $\\rho>\\rho_0$ (where $u_w=0$), only the terms with even  \ncharacteristics in \n(\\ref{prym3}) will survive which leads to a real Ernst potential. This \nimplies that the Ernst potential is regular outside the disk as it should \nbe. The formula (\\ref{prym3}) can also be used to determine asymptotic \nquantities as angular momentum and ADM-mass in the limit of\n$\\rho\\to\\infty$ as was done previously on the axis.\n\nA similar reduction as in the equatorial plane is possible on the axis. \nThere the Riemann surface $\\Sigma'$ also has the involution $T$ which \nmakes it possible to factorize the surface into the surfaces $\\Sigma_1'$ \nand $\\Sigma_2$ where the $\\Sigma_i$ are as above and where $\\Sigma_1'$ \nis $\\Sigma_1$  with the cut \n$\\left[0,-\\rho^2\\right]$ removed. Thus the theta function $\\Theta'$ \non the surface  $\\Sigma'$ of \ngenus $2s-1$ can be expressed via theta functions on surfaces of genus \n$s-1$ and $s$ respectively. In the case $g=2$, this will not lower the \ngenus of the Riemann surfaces under consideration on the axis.\nWe will not give the Ernst potential on the axis\nsince it will not be used here (the formula is helpful if one wants to \ncalculate the multipole moments on the axis for higher genus).\n\n\n\\section{The case $g=2$}\n\nThe simplest non-static solutions within the class (\\ref{2}) are of genus\n2 since solutions of genus 0 belong to the Weyl-class. Interestingly the\nfirst solution to a physically relevant boundary value problem, \nthe rigidly rotating dust disk \\cite{ngbml1} with dust parameter $\\nu=\n2\\Omega^2\\rho_0^2{\\rm e}^{-2V_0}$ where $\\Omega$ is the angular velocity \nin the disk, and  ${\\rm e}^{-V_0}-1$ is the central redshift and radius \n$\\rho_0$,  belongs to\nthis subclass. There the characteristic is $\\left[\n\\begin{array}{cc}\n\t0 & 0  \\\\\n\t1 & 1\n\\end{array}\n\\right]$, the branch points are given by \n$E=\\sqrt{{\\rm i}\/\\nu-1}$, and the function $G$ has the form $G=\\left(\n\\sqrt{1+\\nu^2(K^2+1)^2}+\\nu(K^2+1)\\right)^2$ where we have used \ndimensionless coordinates $\\rho\/\\rho_0$ and $\\zeta\/\\rho_0$.  \nTherefore we will discuss the case $g=2$ as an example in more detail.\n\nSolutions (\\ref{2}) of genus 2 will be regular except at the contour \n$\\Gamma_z$ if $\\Theta(\\omega(\\infty^-)+u)\\neq 0$.  This is \nequivalent to the condition that the divisor $A$, defined by the Jacobi \ninversion problem $\\omega(A)-\\omega(D)=u$, does not contain both $\\infty^-$ \nand $X$ where $X=P_0$ or $X=\\bar{P}_0$. This implies that the equations\n\\begin{eqnarray}\n\t\\int_{-E}^{\\infty^-}\\frac{{\\rm d}\\tau}{\\mu(\\tau)}\n\t+ \\int_{\\bar{E}}^{X}\\frac{{\\rm d}\\tau}{\\mu(\\tau)}& = & \n\t\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\frac{\\ln G{\\rm d}\\tau}{\\mu(\\tau)}\n\t\\label{g2.1} \\\\\n\t\\int_{-E}^{\\infty^-}\\frac{\\tau{\\rm d}\\tau}{\\mu(\\tau)}\n\t+ \\int_{\\bar{E}}^{X}\\frac{\\tau{\\rm d}\\tau}{\\mu(\\tau)}& = & \n\t\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\frac{\\ln G\\tau{\\rm d}\\tau}{\\mu(\\tau)}\n\t\\label{g2.2}\n\\end{eqnarray}\nmust not hold simultaneously for all $\\rho$ and $\\zeta$ in the vacuum \nregion. Since the limits of the integration are fixed, this inversion \nproblem will in general not have a solution. The equations constitute a\nrelation between the physical parameters that characterize the solution. In \nthe case of the dust disk, this is the parameter $\\nu$ and the radius \n$\\rho_0$. This implies that \nthe above condition will determine the allowed parameter range for $\\nu$ \nfor which there are no further singularities in the whole spacetime except the \ndisk. \n\nThe condition for ergospheres can be obtained from a similar system of \nequations as above: simply replace $\\infty^-$ in (\\ref{g2.1}) and \n(\\ref{g2.2}) by a point on ${\\cal L}_H$ which must not be \n$\\infty^-$ or $X$ but is otherwise arbitrary. Then an ergosphere occurs if both \nequations hold simultaneously which gives in the above example the values \nfor $\\nu$ for which there exists a non--empty set of points $\\rho$ and \n$\\zeta$, the ergosphere. Since one of the points in the divisor $A$ is in \nthis case essentially arbitrary, the conditions for ergospheres will be \nsatisfied much more frequently than the condition for a singularity where \nboth points of $A$ are prescribed.\n\nThe ergosphere has only common points with the axis \nin the ultrarelativistic limit which is \ngiven by $\\vartheta_4(u_1')=0$ (we use the notation for elliptic theta \nfunctions of \\cite{erdelyi}). This condition is equivalent to\n\\begin{equation}\n\t\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\frac{\\ln G {\\rm d}\\tau}{\\mu'(\\tau)}\n\t=(2n+1)\\int_{-E}^{E}\\frac{{\\rm d}\\tau}{\\mu'(\\tau)}\n\t\\label{g2.3}\n\\end{equation}\nwhere $n=0,1,2,\\ldots$. Since this relation is independent of the physical \ncoordinates, it determines the values for $\\nu$ at which the \nADM-mass diverges. Together with the conditions (\\ref{g2.1}) and\n(\\ref{g2.2}), this determines the allowed parameter range for $\\nu$: the \nabsence of singularities except the disk and the fact that the mass shall \nvary between 0 and $\\infty$ (the ultrarelativistic limit). \n\nIn the case $g=2$, the potential in the equatorial plane and on the axis \ncan be expressed in terms of elliptic theta functions. The formula for the \naxis reads \n\\begin{eqnarray}\n\tf(\\rho=0,\\zeta)&=&\\frac{\\vartheta_4(\\omega_1'|_{\\zeta^+}^{\\infty^+}+u_1')\n\t\\vartheta_1(\\omega_1'|_{\\zeta^-}^{\\infty^+})+\n\t\\exp(-u_2)\\vartheta_4(\\omega_1'|_{\\zeta^-}^{\\infty^+}+u_1')\n\t\\vartheta_1(\\omega_1'|_{\\zeta^+}^{\\infty^+})}{\n\t\\vartheta_4(\\omega_1'|_{\\zeta^+}^{\\infty^+}-u_1')\n\t\\vartheta_1(\\omega_1'|_{\\zeta^-}^{\\infty^+})+\n\t\\exp(u_2)\\vartheta_4(\\omega_1'|_{\\zeta^-}^{\\infty^+}-u_1')\n\t\\vartheta_1(\\omega_1'|_{\\zeta^+}^{\\infty^+})}\n\t\\nonumber\\\\\n\t&&\\exp\\left\\{\\frac{1}{2\\pi {\\rm i}}\n        \\int\\limits_\\Gamma\n\\ln G(\\tau){\\rm d}\\omega'_{\\infty^{+}\\infty^-}(\\tau)+u_2\\right\\}\n\t\\label{7}.\n\\end{eqnarray}\nIn the equatorial plane we have\n\\begin{equation}\n\t\\bar{f}=\\frac{\\vartheta_3(v+u_v;2\\Pi^1)\\vartheta_4(u_w;2\\Pi^2)\n\t-\\vartheta_2(v+u_v;2\\Pi^1)\\vartheta_1(u_w;2\\Pi^2)}{\n\t\\vartheta_3(u_v-v;2\\Pi^1)\\vartheta_4(u_w;2\\Pi^2)\n\t+\\vartheta_2(u_v-v;2\\Pi^1)\\vartheta_1(u_w;2\\Pi^2)}e^{I_v}\n\t\\label{red13}\n\\end{equation}\nwhere $v=\\int_{-\\rho^2}^{\\infty^+}{\\rm d} v$. For $\\rho>\\rho_0$, the exterior of\nthe disk, we get\n\\begin{equation}\n\tf=\\bar{f}=\\frac{\\vartheta_3(v+u_v;2\\Pi^1)}{\\vartheta_3(\n\tu_v-v;2\\Pi^1)}e^{I_v}.\n\t\\label{red14}\n\\end{equation}\nIn both cases the formulae for a different characteristic are obtained by \ncomplex conjugation.\n\nThe above relations illustrate that important features of the solutions of \ngenus 2 can be discussed with the help of the standard elliptic theory. It \nis thus an interesting question which boundary value problems lead to \nsolutions within this subclass.\n\n\n\\section{Outlook}\n\nIn this paper, it was shown that it is possible to identify within a class\nof hyperelliptic solutions to the Ernst equation a subclass whose solutions \ncould describe the exterior of a body of revolution: they are asymptotically \nflat, equatorially symmetric and regular except at the surface of the \nbody. This subclass could consequently be interesting in the context of \nboundary value problems for the Ernst equation. There the boundary data are \neither induced by an interior solution or a surfacelike matter distribution \nas in the case of the dust disk: in the general case, one would have to \nfix two real functions at the boundary in order to satisfy the boundary \nconditions. Within the class considered here, \none has the freedom to choose one real valued function \n($G$) and a set of free parameters $E_i$, the branch points of the Riemann \nsurface. Whether the subclass discussed here can actually be used to  \nsolve boundary value problems, and when these degrees of freedom will be \nindeed sufficient is an open question. However it is remarkable \nthat it is possible to identify a whole generic subclass of regular \nequatorially symmetric solutions. This gives reasonable hope that further \nsolutions to physically interesting boundary value problems may be found \nwithin the class of hyperelliptic solutions to the Ernst equation.\\\\[1ex]\n{\\bf Acknowledgement:}\\\\\nWe thank H.~Farkas, J.~Frauendiener and H.~Pfister for helpful discussions \nand hints. One of us (CK) acknowledges support by the DFG.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nDwarf galaxies (DGs) present a variety of morphological types.\nTheir structural and chemical properties differ from those of \ngiant galaxies. In addition, low-mass galaxies seem to \nform at all cosmological epochs and by different processes.\nDwarf elliptical galaxies (dEs) are an extremely common and \nastrophysically interesting class of galaxy.  Most known dEs \nare found in regions with high galaxy densities, and they are \nthe most numerous of all galaxy types in the cores of nearby \ngalaxy clusters (see e.g., \\cite{WG84, Bin94}). \nSince the bulk of their star formation (SF) occurred in the past \n(see review by \\cite{FB94}), dEs thus are frequently \nconsidered to be ``stellar fossil'' systems. \nYet many dwarf spheroidals (dSphs) which \nrepresent the low-mass end of DGs show not only a significant \nintermediate-age stellar population \\cite{Hod89,Gre97}, \nbut also more recent SF events \\cite{Seta94, Heta97} with \nincreasing metallicity, indicating that gas was kept in the system.\nIn galaxies where gas is depleted by astration stellar abundances \nare predicted to be near the solar value. On the other hand,\nthe moderate-to-low stellar metallicities in dE and the related \ndSph galaxies (about 0.1 of solar or less) suggest that \nextensive gas loss occurred during their evolution by means of\na supernova typeII (SNII) driven galactic winds (\\cite{Lar74,DS86}). \n\nAs another DG type, that consists of the same and higher \ngas fraction as giant spiral \ngalaxies (gSs), dwarf irregular galaxies (dIrrs) appear \nwith a wide range but lower $Z$ as gSs. \nAgain this implies that the metal-enriched \ngas from SNeII was lost from the galaxy. dIrrs show a large \nvariety of SF rates from moderate to extraordinarily high values\nin starbursts (SBs) and are forming very \ncompact star clusters which are embedded in at least one \nintermediate-age to old underlying stellar population. \nBecause of their low binding energy SBDGs are characterized by\nsuperwinds \\cite{Meta95} or large expanding X-ray plumes\nwhich are driven by SNeII. Their existence\ndemonstrates the occurrence of metal loss by means of \nlarge-scale galactic winds and allows to study this phenomenon \nstill today. \n\nThe faint blue galaxies at medium redshifts have experienced \nstrong SBs and show signatures of present H{\\sc ii}\\ galaxies \n\\cite{Kota95} what might represent the formation epoch\nof the older stellar populations in SBDGs. \nNevertheless, very metal-poor dIrrs like e.g.\\ I~Zw~18 \n\\cite{Keta95} or SBS~0335-52 \\cite{Teta97} seem to form \ntheir first generation of stars today. In addition, a strange\nbut perhaps common mode of DG formation exists in tidal tails \nof merging galaxies \\cite{Meta92,DM98}. \n\n\nWhile the low metal content in DGs can be attributed to\nthis metal-enhanced mass loss, another puzzling fact needs \nexplanation: Why do DGs, though with O abundances below 1\/10 \nsolar, show also low N\/O ratios of almost 0.7 dex smaller than \nin gSs with a large scatter and no significant correlation with \nO\/H (see fig.1)?\n\n\n\\begin{figure}\n\\psfig{figure=garnett90_new.eps,height=8.5cm,width=15.5cm,angle=-90}\n\\caption[]{Evolutionary tracks of 2d chemodynamical models for 10$^9 \\, {\\rm M}_{\\odot}$\n         galaxies with (long curve with diamonds) and without DM halos \n         (open circles) in comparison with \n         N\/O vs.\\ O\/H measurements of irregular galaxies (stars and full\n\t dots) and two chemical evolutionary models by \n\t \\cite{MT85} (upper lines) \n         and a simple model track (arrows; see \\cite{Gar90}). }\n\\end{figure}\n\n\nWhile C and N are mainly contributed by planetary nebulae (PNe) from\nintermediate-mass stars (IMS) to the warm cloudy gas phase (CM)\nof the the interstellar medium (ISM), \nO and Fe are the dominant tracers of SNII and SNIa ejecta, \nrespectively, and are therefore initially encorporated in \nthe hottest phase of the ISM, the intercloud medium (ICM). \nVarious authors have successfully reached the observed \nN\/O-O regime of DGs in different chemical models. \n\\cite{MT85} and \\cite{Meta94} took galactic \nwinds into account, which are driven by numerous SB episodes, \nin order to yield the necessary reduction of O. Similarly,\n\\cite{Gar90} and \\cite{Pil92} report models that follow \nthe assumption of abundance self-enrichment within the observed \nH{\\sc ii}\\ regions \\cite{KS86}. All these models differentiate the \nO and N release in accordance to their progenitor lifetimes, \nby this, leading to zick-zack evolutionary tracks in the \nlog(N\/O)-log(O\/H) diagram (\\cite{Pil92} and fig.1).\nParameters like wind efficiency, IMF, and gas mass fraction make \nsufficiently good model approachs to the observations possible. \nThe yields of these models are in principle based on almost \nthe same stellar evolution calculations \n(N by \\cite{RV81} and O by \\cite{WW86} or by \\cite{Mae92} \nand \\cite{MM89}). Because of the assumption of an already existing \nelement enrichment by ancient stellar populations, the\nchemical evolutionary models start \nalready from high N\/O values at low O abundance.\n\nSince N and O are polluting the different phases of the ISM separately,\ntheir simultaneous existence in H{\\sc ii}\\ regions, which represent the ionized\nCM, can only result from phase transitions between CM and ICM. \nIn models where these processes are taken into account, the \nobserved N\/O ratio should, therefore, permit qualitative studies \nabout both the mixing direction and efficiency. Additionally, its radial \ndistribution in a DG provides an insight into dynamical effects of \nthe ISM. The assumption of an individual self-enrichment of H{\\sc ii}\\ \nregions during their ionization-caused observability \\cite{Pil92}\nwould plausibly involve both, abundance variations during their visibility\nepoch and, by this, differences between H{\\sc ii}\\ regions within\nthe same dIrr. In contrast, \\cite{Kob98} reports the non-detection\nof any sizable O, N, and He anomalies from H{\\sc ii}\\ regions in the \nvicinity of young star clusters in SBDGs \nwith one exception, NGC~5253, which reveals a central N overabundance. \nIn NGC~1569, that has recently formed two super-star clusters, \n\\cite{Kob98} finds constant O and N\/O values over a radial extent \nof more than \n400 pc with a scatter of N\/O by only 0.2 dex, while self-enrichment\ntracks extent over one dex in N\/O \\cite{Pil92} and dispersal\ndistances are much less. The same fact holds in I~Zw~18 \n(Izotov, this conference).\n\n\n\\section{Chemodynamical Models of Dwarf Irregulars}\n\nNot all observed dIrrs that cover the same N\/O-O regime \nhave passed or experience at present strong SBs, but form\nstars at an almost moderate and continuous rate. \nEmpirical studies and theoretical investigations \nof systems at low potential energies have decovered that \ntheir ISM should be balanced by counteracting \nprocesses like heating and cooling, turbulence and dissipation \n\\cite{BH89}. The SF is self-regulated under various conditions \n(see e.g.\\ \\cite{FC83,Keta95,Keta98}). \nIf the evolution of galaxies is sensitive to the energetical\nimpact of different processes, an adequate treatment of the \ndynamics of stellar and gaseous components and of their mutual \nenergetical and materialistic interactions is essential. \nThis modelling of galactic evolution is properly performed by the \n{\\bf chemodynamical} ({\\it cd} ) prescription (for its formulation see \ne.g.\\ \\cite{TBH92,SHT97}).\nThe evolution of DGs can proceed in self-regulated ways\nboth, globally by large-scale flows of unbound gas but also \nlocally in the SF sites. {\\it cd}\\ models of non-rotating DG systems \nachieve SBs from the initial collapse and after mass-loss \ninduced expansion in the recollapse phase of its bound \nfraction of the ISM \\cite{Heta93,Heta98}. Particular models\nare evolving also by oscillatory SF episodes. \nThe stellar populations represent dSphs as well as more massive \n{\\it blue compact} DGs and reveal a central\nconcentration of the recent SF region embedded in older elliptical \npopulations. External effects \\cite{Vil95} like extended \ndark matter (DM) halos, \nthe intergalactic gas pressure, etc., could cause further \nmorphological differences of DGs e.g.\\ by regulating the outflow\nand allowing for a recollapse that fuels subsequent SF.\n\nFor rotating dIrrs 2d {\\it cd}\\ models are necessary. \nA gaseous protogalactic cloud starts with or\nwithout a DM halo \\cite{Bur95} and with a Plummer-Kuzmin-type \ngas distribution. In the following we wish to discuss only\n10$^9 \\, {\\rm M}_{\\odot}$ DG models which start with a 2 kpc \nbaryonic density scalelength. \nAs in former 2d {\\it cd}\\ models for massive gSs \\cite{SHT97}\nwe have, at first, divided the yields of N and O between IMS and \nmassive stars (HMS) according to \\cite{MM89,Seta92,Seta93} and to\n\\cite{WW86}, respectively. Surprisingly, because the {\\it cd}\\ prescription \nalso differentiates the N and O pollution to different gas phases \nand distinguishs between their progenitors' lifetimes, but \nin agreement with the above-mentioned purely chemical models of \ndifferent authors, our {\\it cd}\\ models reach very rapidly a  \ntoo high log(N\/O) ratio between -1.3 to -1.4 at low O \nabundance of almost 7.2 in 12+log(O\/H) (see fig.1). Moreover,\nwithout any burst behaviour the further evolutionary track \nsurpasses the regime of the observed N\/O-O values for dIrrs \n\\cite{RH98}. In contrast to the expectation that only the \nICM is expelled from the dIrrs carrying away the O yield, \na substatial mixing of CM and ICM determines the \nchemical evolution. Since the hot material cannot completely \ncondense onto the clouds, nearly total evaporation of \nthe CM in the vicinity of the SF and SNII explosion sites \nmust be responsible. This is displayed in the left panel\nof fig.2. \nReasonably, in this case the N mixes perfectly  with O in \nthe ICM to an almost constant abundance ratio and can spread \nover larger distances within the DG (right panel of fig.2). \nDue to cooling and dynamical shocks the ICM forms new \ncondensations of CM on all galactic scales. The right \npanel of fig.2 seems to reveal a strong outflow, however, \nthe greayscale reflects the pure N abundance only, \nbut not the absolute N density, which is low and would not \nrepresent a perceivible  mass loss \\cite{RH98}.\nBut even with a galactic wind the N\/O ratio would remain constant\nbecause both elements get lost by the ICM. This means that\nthe dIrr abundance problem cannot be solved by the selective \nexpulsion of O.\n\n\n \\begin{figure}\n\\psfig{figure=evap_nicm.eps,height=8.0cm,width=15.5cm,angle=-90}\n\\caption[]{Physical condidions of a 2d chemodynamical 10$^9 \\, {\\rm M}_{\\odot}$\n         dwarf galaxy model with a DM halo after 2 Gyr. \n         The size of the figures is 6.8 kpc. \\\\\n         left: grayscale of the evaporation rate of warm \n\t interstellar clouds by the hot intercloud medium (ICM); \n\t the maximum lies at the lower left corner. \\\\\n\t right: greyscale of N abundance in the ICM; the maximum \n         lies at the lower left corner. \\\\ }\n\\end{figure}\n\nRecent stellar evolution models by \\cite{WW95}, however, \nprovide a secondary N production also by HMS.\nTheir new yields combined with the most recent models \nfor IMS by \\cite{vdHG97} have been successfully fitted by \n\\cite{Sam98} to the abundance distributions in our {\\it cd}\\\nMilky Way model \\cite{SHT97}. \nIn contrast to the cited chemical and first 2d {\\it cd}\\ models of \nDGs which reach too large N\/O ratios or even begin in that range,\nboth our 10$^9 \\, {\\rm M}_{\\odot}$ models with new yield \nprescriptions commence at very low N\/O and O values \ndue to the delayed N release by PNe and the lower \nN production in massive stars at low metallicities.\nThe N\/O-O track of the model with DM halo then rises rapidly,\nboth N\/O and O, and reaches log(N\/O) of -1.8 after 1 Gyr \nand -1.6 after 2 Gyrs, respectively (see fig.1). \nThese values are only somewhat smaller than the numerical \nyield ratios and are based on an almost perfect \nmixing of CM and ICM. Due to the large-scale hot gas streaming\nand subsequent condensation the abundances observed in \nH{\\sc ii}\\ regions show a constant value \\cite{Kob98}. \nWith the same differential mixing processes the \nobserved C\/O tendency of DGs \\cite{Geta95} could also be \nexplained \\cite{RH98}. \n\n\n\\section{Conclusion}\n\nHere we have shown that in {\\it cd}\\ models of dIrrs a strong \nmixing of the gas phases prevents a selective element depletion. \nThe self-consistent {\\it cd}\\ models \nevolve with local variations but globally moderate SF \nactivity \\cite{RH98} and therefore typically for a large\nfraction of dIrrs.\nThe observed N and O abundances in the H{\\sc ii}\\ regions\ncan be strikingly attained, if N is\nproduced as a secondary element also in HMS. Their\nevolutionary tracks of log(N\/O)-log(O\/H) rise from low values,\nreach the regime of observations after almost 3 Gyr and\ncover the region for another 3 Gyr depending on the DM halo\ncontribution. Only for a much older stellar populations \nthe existing ISM could be more metal-enriched, so\nthat it must be partly rejuvenated by infall of \nintergalactic gas or from a bound gas reservoir, both with almost \npristine abundances. \n\nThe results change, if condensation would dominate\nthe phase transitions and parts of the ICM \n(and therefore of the O content) would be adapted by the CM. \nReasonably, this would lead to higher N\/O ratios \nand, in addition, to inhomogeneous N\/O distributions.\nStrong operation of condensation in the SF body of dIrrs\ncan therefore be excluded but works preferably at higher \ngas densities. \nBecause larger gravitational potentials and resulting mass \ndensities in more massive galaxies lead to a faster evolution\nby a more intense cooling of the ICM and significantly higher condensation \nand SF rates, the N\/O increase is stronger, so that discrepancies between\nthe different yield assumptions (with and without secondary N\nproduction in HMS) cannot be noticed in {\\it cd}\\ models of\ngSs like the Milky Way \\cite{SHT97,Sam98}. \\\\\n\n{\\small\nAcknowledgments: We are grateful to J.\\ K\\\"oppen, M.\\ Samland \nand C.\\ Theis for discussions. \nThis work is supported by the {\\it Deutsche \nForschungsgemeinschaft} (DFG) under grant no.\\ He 1487\/5-3 (A.R.).\n\n\n\\begin{moriondbib}\n\n\\bibitem{Bin94} Binggeli B., 1994, in {\\it ``Panchromatic View of Galaxies -- \ntheir Evolutionary Puzzle''}, eds.\\ G.\\ Hensler, J.S.\\ Gallagher, \nC.\\ Theis, Editions Fronti\\`{e}res, Gif-sur-Yvettes, p.\\ 173\n\\bibitem{Bur95} Burkert A., \\apj {447} {L25}\n\\bibitem{BH89} Burkert A., Hensler G., 1989b, {\\it ``Evolutionary \nPhenomena in Galaxies''}, Tenerife, \neds.\\ J.E.\\ Beckmann \\& B.E.J.\\ Pagel, Cambridge University Press, p.\\ 230\n\\bibitem{DS86} Dekel A., Silk J., 1986, \\apj {303} {39}\n\\bibitem{DM98} Duc P.-A., Mirabel I.F., 1998, \\aa {333} {813}\n\\bibitem{FB94} Ferguson H.C., Binggeli B., 1994, \n{\\it Astr.\\ Astrophys.\\ Reviews} {\\bf 6}, 67\n\\bibitem{FC83} Franco J., Cox D.P., 1983, \\apj {273} {243} \n\\bibitem{GW95} Gallagher J.S., Wyse R.F.G., 1995, {\\it PASP} {\\bf 106},1225\n\\bibitem{Gar90} Garnett D.R., 1990, \\apj {360} {142}\n\\bibitem{Geta95} Garnett D.R., Skillman E.D., Dufour R.J., et al., 1995, \n\\apj {443} {64}\n\\bibitem{Gre97} Grebel E., 1997, {\\it Rev.\\ Modern Astron.} {\\bf 10}, 29 \n\\bibitem{Geta98} Guzman R., Jangren A., Koo D.C., et al., 1998, \n\\apj {495} {L13}\n\\bibitem{Heta97} Han M., Hoessel J.G., Gallagher J.S., et al., 1997, \n\\aj {113} {1001}\n\\bibitem{Heta93} Hensler G., Theis C., Burkert A., 1993, in \n{\\it Proc.\\ 3$^{rd}$ DAEC Meeting\n``The Feedback of Chemical Evolution on the Stellar Content in\nGalaxies''}, eds.\\ D.\\ Alloin \\& G.\\ Stasinska, p.\\ 229\n\\bibitem{Heta98} Hensler G., Theis C., Gallagher J.S., 1998, \n{\\it in preparation}\n\\bibitem{Hod89} Hodge P.W., 1989, {\\it Ann.\\ Rev.} \\aa {27} {139}\n\\bibitem{Kob98} Kobulnicky H.A., 1998, {\\it  Proc. Quebec Conference: \n``Abundance Profiles: Diagnostic Tools for Galaxy History''},\neds. D. Friedli, M. Edmunds, C. Robert, \\& L. Drissen \nSan Francisco: ASP, in press\n\\bibitem{Keta95} K\\\"oppen J., Theis C., Hensler G., 1995, \\aa {296} {99}\n\\bibitem{Keta98} K\\\"oppen J., Theis C., Hensler G., 1998, \\aa {328} {121}\n\\bibitem{Kota95} Koo D.C., Guzman R., Faber S.M., et al., 1995, \\apj {440} {L49}\n\\bibitem{Kuta95} Kunth D., Matteucci F., Maroni G., 1995, \\aa {297} {634}\n\\bibitem{KS86} Kunth D., Sargent W.L., 1986, \\apj {300} {496}\n\\bibitem{Lar74} Larson  R.B., 1974, \\mnras {169} {229}\n\\bibitem{Mae92} Maeder A., 1992, \\aa {264} {105}\n\\bibitem{MM89} Maeder A., Meynet G., 1989, \\aa {210} {155}\n\\bibitem{Meta94} Marconi G., Matteucci F., Tosi M., 1994, \\mnras {270}{35}\n\\bibitem{Meta95} Marlowe A.T., Heckman T.M., Wyse R.F.G., Schommer R., \n1995, \\apj {438} {563}\n\\bibitem{MT85} Matteucci F., Tosi M., 1985, \\mnras {217} {391}\n\\bibitem{Meta92} Mirabel I.F., Dottori H., Lutz D., 1992, \\aa {256} {L19}\n\\bibitem{Pil92} Pilyugin L.S., 1992, \\aa {260} {58}\n\\bibitem{RV81} Renzini A., Voli M., 1981, \\aa {94} {175} \n\\bibitem{RH98} Rieschick A., Hensler G., 1998, in preparation\n\\bibitem{Sam98} Samland M., 1998, \\apj {496} {155}\n\\bibitem{SHT97} Samland M., Hensler G., Theis C., 1997, \\apj {476} {277} \n\\bibitem{Seta93} Schaerer D., Meynet G., Maeder A., Schaller G., 1993, \n\\aas {98} {523}\n\\bibitem{Seta92} Schaller G., Schaerer D., Meynet G., Maeder A., 1992, \n\\aas {96} {269}\n\\bibitem{Seta94} Smecker-Hane T.A., Stetson P.B., Hesser J.E., et al., 1994, \n\\aj {108} {507} \n\\bibitem{TBH92} Theis C., Burkert A., Hensler G., 1992, \\aa {265} {465}\n\\bibitem{TI97} Thuan T.X., Izotov Y.I., 1997, \\apj {489} {623}\n\\bibitem{Teta97} Thuan T.X., Izotov Y.I., Lipovetsky V.A., 1997, \n\\apj {477} {661}\n\\bibitem{vdHG97} van der Hoek L.B., Groenewegen M.A.T., 1997, \\aas {123} {305}\n\\bibitem{Vil95} Vilchez J.M., 1995, \\aj {110} {1090}\n\\bibitem{WG84} Wirth A., Gallagher J.S., 1984, \\apj {282} {85}\n\\bibitem{WW86} Woosley S.E., Weaver T.A., 1986, in {\\it ``Radiation\nHydrodynamics in Stars and Compact Objects''}, \neds.\\ D.\\ Mihalas \\& H.A.\\ Winkler, Springer, Berlin, p.\\ 91\n\\bibitem{WW95} Woosley S.E., Weaver T.A., 1995, \\apjs {101} {181} \n\n\\end{moriondbib}\n}\n\\vfill\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{I}\n   Photo-production of iso-scalar vector mesons ($V$) at the proton ($p$),\n   $\\gamma p \\to V p $,\n   plays an important role\n   in understanding phenomena of hadronic physics\n   which are far beyond the scope of perturbative QCD.\n   Thereby, reactions with polarized photons ($\\vec \\gamma$)\n   at relatively low energies are particularly interesting.\n   For example, photo-production of $\\phi$ mesons\n   at forward angles can provide some information on the Pomeron\n   exchange channel.\n   A recent analysis of the LEPS collaboration\n   shows a sizeable deviation\n   from predictions based on conventional approaches~\\cite{TL03},\n   in particular, as the data show a bump structure\n   at photon energies $E_\\gamma\\sim 2$~GeV~\\cite{Mibe05}.\n   Another peculiarity of the LEPS data is a strong deviation\n   of the spin-density matrix element $\\rho^1_{1-1}$ from 0.5,\n   which points to a sizable contribution of un-natural parity\n   exchange processes, contrary to (conventional) expectations based on the\n   dominance of the Pomeron exchange channel.\n\n   $\\omega$ photo-production near the threshold provides further\n   unique information on the mechanism of nucleon\n   resonance ($N^*$) excitation~\\cite{Ajaka06,TL02,Zhao01,Shklyar04}\n   and the strength of the $\\omega NN^*$ coupling which are\n   crucial for understanding the structure of baryon\n   resonances~\\cite{Capstick00}.\n\n   Most experiments of $\\phi$ and $\\omega$ photo-production\n   are aimed at analyzing the angular distribution of the exclusively outgoing\n   hadrons in reactions with polarized photons.\n   The corresponding formalism for such analysis,\n   expressing the angular distributions of the outgoing hadrons\n   in terms of the spin-density matrix elements, is developed in\n   Ref.~\\cite{Schilling70}. However, there is also an opportunity\n   to extend the study of vector meson photo-production\n   by measuring the $V \\to \\pi^0\\gamma$ decay channel.\n  \n   The Crystal Barrel detector~\\cite{Aker}, which is now operated\n   at the electron accelerator ELSA at Bonn and\n   delivers a new data complementary to the previous ones,\n   providing independent information about\n   various aspects of $\\phi$ and $\\omega$ photo-production~\\cite{Schmiden}.\n  \n  \n  \n\n   Given this motivation,\n   the aim of our Note is to extend the formalism of\n   Ref.~\\cite{Schilling70} to the case of the $V\\to\\pi^0\\gamma$\n   decay distribution.\n   We are going to present useful formulas for corresponding distributions\n   and asymmetries. Our focus is on iso-scalar vector meson photo-production\n   off the proton.\n\nTaking into account the small total decay width of $\\omega$ and\n$\\phi$ mesons, one can express the cross section of the reaction\n$\\vec \\gamma p\\to Vp\\to \\pi^0\\gamma p$ with linearly polarized\nphotons as\n  \\begin{eqnarray}\n\\frac{d\\sigma^{\\pi^0\\gamma}(\\Omega_\\pi,\\Psi)}{dt d\\Omega_\\pi} =\nB_{\\pi^0\\gamma}\\frac{d\\sigma_0}{dt}\\,\nW^{\\pi^0\\gamma}(\\Omega_\\pi,\\Psi)~, \\label{E1}\n\\end{eqnarray}\nwhere $\\Psi$ is the angle between the polarization of the incoming\nphoton, ${\\bm \\varepsilon}=(\\cos\\Psi,\\sin\\Psi, 0)$, and the\nproduction plane; $\\Omega_\\pi$ denotes the solid angle of the\nmomentum of the outgoing pion in the vector meson rest frame,\n$W^{\\pi^0\\gamma}(\\Omega_\\pi,\\Psi)$ is the decay distribution; $\nB_{\\pi^0\\gamma}={\\Gamma_{\\pi^0\\gamma}}\/{\\Gamma_{\\rm tot}}$ stands\nfor the branching ratio of the $V\\to\\pi^0\\gamma$ decay, and\n${d\\sigma_0}\/{dt}$ is the differential cross section of the vector\nmeson production for an unpolarized photon beam. Note, that our\nconsideration is valid for $V\\to\\eta\\gamma$ decay as well, and in\nparticularly for $\\phi\\to\\eta\\gamma$ decay mode, which branching\nratio exceeds the branching ratio of $\\phi\\to\\pi^0\\gamma$ decay by\norder of magnitude. But for simplicity, we will not distinguish\nbetween these two modes, assuming that the result for\n$V\\to\\eta\\gamma$ decay is obtained by the evident substitution\n$\\pi^0\\to\\eta$.\n\nFor calculating the decay distribution we cast the decay amplitude\nin the form\n\\begin{eqnarray}\nT^{V\\to\\pi^0\\gamma} =\\frac{g_{V\\pi^0\\gamma}}{M_{V}}\n\\epsilon^{\\mu\\nu\\alpha\\beta} V_\\mu \\,\n\\varepsilon^{*\\,(\\gamma)}_\\alpha(\\lambda_{(\\gamma)}) \\, q_\\nu \\,\nk_{\\beta}~, \\label{E2}\n\\end{eqnarray}\nwhere $V_\\mu$ and $\\varepsilon^{*\\,(\\gamma)}_\\alpha$ are the\npolarization four-vectors of the vector meson $V$ with momentum\n$q$ and mass $M_V$ and the outgoing photon $\\gamma$ with momentum\n$k$, respectively; $g_{V\\pi^0\\gamma}$ is the coupling strength of\nthe $V\\pi^0\\gamma$ interaction. By summing up the polarization\nstates $\\lambda_{(\\gamma)}$ of the outgoing photon in the total\nprobability for the reaction $\\vec \\gamma p\\to \\pi^0\\gamma p$, one\ngets the following expression for the decay distribution in the\nvector meson's rest frame\n\\begin{eqnarray}\nW^{\\pi^0\\gamma}(\\Omega_\\pi,\\Psi) =\\frac{3}{8\\pi}\\frac{1}{N} \\left(\n{\\vec {\\cal M}}{\\vec {\\cal M}}^* -({\\vec {\\cal M}}\\cdot{\\vec\nn}_\\pi) ({\\vec {\\cal M}}\\cdot{\\vec n}_\\pi)^* \\right), \\label{E3}\n\\end{eqnarray}\nwhere ${\\vec n}_\\pi$ is the unit three-vector along direction of\nflight of the outgoing $\\pi$ meson,  $\\vec {\\cal M}$ is the\namplitude for the vector meson photo-production with components\n${\\cal M}_\\lambda$ referring to the three polarization states\n$\\lambda=\\pm1,0$. Thus,\n\\begin{eqnarray}\n{\\vec {\\cal M}}\\cdot{\\vec n}_\\pi=\\sqrt{\\frac{4\\pi}{3}}\n\\sum\\limits_{\\lambda} {\\cal M}_{\\lambda}\nY_{1\\lambda}(\\Omega_\\pi)~. \\label{E4}\n\\end{eqnarray}\nThe normalization factor $N$ in Eq.~(\\ref{E3}) is related to the\ndifferential cross section of the vector meson photo-production by\nan unpolarized photon beam\n\\begin{eqnarray}\n\\frac{d\\sigma_{0}}{dt} =\\frac{1}{16\\pi(s-M_p^2)^2}\\,|N|^2~.\n\\label{E5}\n\\end{eqnarray}\nHere, $s$ is the Mandelstam variable for the entrance channel\n$\\gamma p$, and $M_p$ denotes the proton mass. It is worth noting\nthat the distribution for purely hadronic decays, say $\\phi\\to\nK^+K^-$ or $\\omega\\to\\pi^+\\pi^-\\pi^0$, differs noticeably from\nEq.~(\\ref{E3}):\n\\begin{eqnarray}\nW^{h}(\\Omega_\\pi,\\Psi) =\\frac{3}{4\\pi}\\frac{1}{N} ({\\vec {\\cal\nM}}\\cdot{\\vec n}_h) ({\\vec {\\cal M}}\\cdot{\\vec n}_h)^*~,\n\\label{E6}\n\\end{eqnarray}\nwhere ${\\vec n}_h$ is the unit three-vector along direction of\nflight of a kaon in case of the $\\phi$ meson decay; for the\n$\\omega$ meson decay, ${\\vec n}_h$ defines the direction of the\nvector ${\\bf p}_{\\pi^0}\\times({\\bf p}_{\\pi^+}-{\\bf p}_{\\pi^-})$ .\n\nThe evaluation of Eq.~(\\ref{E3}) for polarized photons is\nperformed with the photon density matrix~\\cite{Schilling70}\n\\begin{eqnarray}\n\\rho(\\gamma)=\\frac12 I +\\frac12{\\bf P}_\\gamma\\cdot{\\bm \\sigma}~,\n\\label{E7}\n\\end{eqnarray}\nwhere $I$ is the $2\\times2$ unity matrix, and $\\sigma_i$ stand for\nthe Pauli matrices. The three components of ${\\bf P}_\\gamma$ read\nfor linear and circular polarizations\n\\begin{equation}\n{\\bf P}_\\gamma = \\left\\{\n\\begin{array}{ll}\nP_\\gamma\\left(-\\cos2\\Psi,-\\sin2\\Psi,0\\right)& ({\\rm lin. \\, pol.}),\\\\[1mm]\nP_\\gamma\\left(0,0,\\pm 1\\right) & ({\\rm circ. \\,pol.}),\n\\end{array}\n\\right.  \\label{E8}\n\\end{equation}\nwhere $P_\\gamma$ is the degree of the polarization. The final\nresult for linearly polarized photons, expressed in terms of the\nspin-density matrix elements, reads\n\\begin{eqnarray}\n  W^{\\pi^0\\gamma}(\\Omega_\\pi,\\Psi) &=&\n  W^{\\pi\\gamma}_0(\\Omega_\\pi)\n  -P_\\gamma\\,W^{\\pi^0\\gamma}_1(\\Omega_\\pi)\\cos2\\Psi \\nonumber\\\\\n  &-&P_\\gamma\\,W^{\\pi^0\\gamma}_2(\\Omega_\\pi)\\sin2\\Psi~, \\label{E9}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&W^{\\pi^0\\gamma}_0(\\Omega_\\pi) = \\frac{3}{8\\pi} \\left(\n1-\\rho^0_{11}\\sin^2\\Theta-\\rho^0_{00}\\cos^2\\Theta\\right. \\label{E101}\\\\\n&&\\quad\\quad + \\left.\\rho^0_{1-1}\\sin^2\\Theta\\cos2\\Phi\n+\\sqrt{2}{\\rm Re}\\rho^0_{10}\\sin2\\Theta\\cos\\Phi\n\\right), \\nonumber\\\\\n&&W^{\\pi^0\\gamma}_1(\\Omega_\\pi) = \\frac{3}{8\\pi} \\left(\n2\\rho^1_{11}+(\\rho^1_{00}-\\rho^1_{11})\\sin^2\\Theta\\right. \\label{E102}\\\\\n&&\\quad\\quad + \\left. \\rho^1_{1-1}\\sin^2\\Theta\\cos2\\Phi\n+\\sqrt{2}{\\rm Re}\\rho^1_{10}\\sin2\\Theta\\cos\\Phi\n\\right), \\nonumber\\\\\n&&W^{\\pi^0\\gamma}_2(\\Omega_\\pi) = -\\frac{3}{8\\pi} \\left( {\\rm\nIm}\\rho^2_{1-1}\\sin^2\\Theta\\cos2\\Phi\\right.  \\label{E103}\\\\\n&&\\quad\\quad + \\left. \\sqrt{2}{\\rm {\\rm\nIm}}\\rho^2_{10}\\sin2\\Theta\\cos\\Phi \\right)~, \\nonumber\n\\end{eqnarray}\nwhile the decay distribution for circular polarized photons with\npolarization $\\lambda_\\gamma=\\pm1$ is\n\\begin{eqnarray}\nW^{\\pi^0\\gamma}_{\\lambda_\\gamma}(\\Omega_\\pi) &=& W^{\\pi^0\\gamma}_0\n- P_\\gamma \\frac{3\\lambda_\\gamma}{8\\pi} \\left(\n  {\\rm Im}\\rho^3_{1-1}\\sin^2\\Theta\\cos2\\Phi\\right.\n  \\nonumber\\\\\n  &+&\\,\\left.\n\\sqrt{2}{\\rm Im}\\rho^3_{10}\\sin2\\Theta\\cos\\Phi \\right)~.\n\\label{E11}\n\\end{eqnarray}\nThe spin-density matrix elements are defined by the components of\n$\\vec {\\cal M}$; their explicit form may be found in\nRef.~\\cite{Schilling70}.\n\n\\begin{figure}[hb!]\n    \\includegraphics[width=0.45\\columnwidth]{Fig1a.eps}\\qquad\n    \\includegraphics[width=0.45\\columnwidth]{Fig1b.eps}\n    \\caption{\\small{\n    The polar angular distribution (left panel) and\n    the azimuthal angle decay distribution (right panel)\n    for the $\\vec \\gamma p\\to \\phi p\\to \\pi^0\\gamma p$ reaction\n    (solid curve) and for the $\\vec \\gamma p\\to \\phi p\\to K^+K^- p$ reaction\n   (dashed curve).\n    in the Gottfried-Jackson frame for $E_\\gamma=1.97-2.17$~GeV and\n    $t=t_{\\rm max}-0.2$~GeV$^2$.\n    The experimental data are taken from~\\protect\\cite{Mibe05}.\n     \\label{Fig:1}}}\n    \\end{figure}\n\nEqs.~(\\ref{E101} - \\ref{E11}) show that the angular distributions\nfor the $V\\to\\pi^0\\gamma$ decay differ from the distributions of a\npurely hadronic decay emerging from (\\ref{E6})\n\\cite{TK07,TL03,Schilling70}. Thus, the polar angular\ndistributions in reaction with unpolarized photons, integrated\nover the azimuthal angle $\\Phi$, read for the $\\pi^0 \\gamma$ and\nexclusively hadronic decays\n\\begin{eqnarray}\nW^{\\pi^0\\gamma}(\\Theta) &=& \\frac{3}{8} \\left(1+\\cos^2\\Theta +\n\\rho^0_{00}(1-3\\cos^2\\Theta) \\right), \\label{E12}\\\\\nW^{h}(\\Theta) &=& \\frac{3}{4} \\left(\\sin^2\\Theta -\n\\rho^0_{00}(1-3\\cos^2\\Theta) \\right)~. \\label{E13}\n\\end{eqnarray}\nWe emphasize the difference between the two distributions.\n\nThe azimuthal angle distribution, integrated over the polar angle\n$\\Theta$, for the two considered cases may be written in the\nuniversal form\n\\begin{eqnarray}\n2\\pi W^{f}(\\Phi,\\Psi)&=& 1 -\\Sigma^f_{\\Phi}\\cos2\\Phi\n-P_\\gamma\\Sigma^f_b\\cos2\\Psi\\nonumber\\\\\n  &+&P_\\gamma\\Sigma^f_d\\cos2(\\Phi-\\Psi)~,\n  \\label{E14}\n\\end{eqnarray}\nwhere the symbol $f$ labels either the $\\pi^0\\gamma$ or the pure\nhadronic ($h$) decay; $\\Sigma^f_{\\Phi}$ denotes the azimuthal\ndecay asymmetry for unpolarized photon beam, $\\Sigma^f_{b}$\ndenotes the beam asymmetry and $\\Sigma^f_d$ is the azimuthal decay\nasymmetry relative to the photon polarization plane. For\ncircularly polarized photons, the azimuthal angle distribution\nreads\n\\begin{eqnarray}\n2\\pi W^{f}_{\\lambda_\\gamma}(\\Phi,\\Psi)= 1 + \\lambda_\\gamma\nP_\\gamma\\Sigma^f_C\\cos2\\Phi~,\n  \\label{E15}~\n\\end{eqnarray}\nwhere $\\Sigma^f_C$ is the azimuth angle asymmetry for circularly\npolarized photons with polarization $\\lambda_\\gamma=\\pm1$. The\nasymmetries are expressed through the spin-density matrices\n  \\begin{eqnarray}\n\\Sigma^{\\pi^0\\gamma }_{\\Phi}&=&\n-\\frac12\\Sigma^{h}_{\\Phi}=-\\rho^0_{1-1}~,\\label{16}\\\\\n\\Sigma^{\\pi^0\\gamma }_{b}&=&\n\\Sigma^{h}_{b}=2\\rho^1_{11}+\\rho^1_{00}~,\\label{17}\\\\\n\\Sigma^{\\pi^0\\gamma }_{d}&=&\n-\\frac12\\Sigma^{h}_{d}=-\\rho^1_{1-1}~.\\label{18}\\\\\n\\Sigma^{\\pi^0\\gamma }_{C}&=& -\\frac12\\Sigma^{h}_{c}=-{\\rm Im\n}\\rho^3_{1-1}~,\\label{19}\n\\end{eqnarray}\nwhere we assume $\\rho^1_{1-1} \\approx -{\\rm Im}\\rho^2_{1-1}$\naccording to \\cite{TL03}. It is clear that the beam asymmetry does\nnot depend on the decay mode. However, the other asymmetries\ndepend on the decay mode: (i) they have opposite signs and (ii)\nthe absolute values for the purely hadronic decay is two times\nlarger.\n\\begin{figure}[ht!]\n    \\includegraphics[width=0.45\\columnwidth]{Fig2a.eps}\\qquad\n    \\includegraphics[width=0.45\\columnwidth]{Fig2b.eps}\n    \\caption{\\small{The same as in Fig.~\\protect\\ref{Fig:1}\n    but for $\\omega$ meson photo-production.\n    Our prediction is for $E_\\gamma=1.6$~GeV and\n    $t=t_{\\rm max}-0.2$~GeV$^2$.\n     \\label{Fig:2}}}\n    \\end{figure}\n\nThe different distributions, for the case of $\\phi$ meson\nphoto-production, are exhibited in Fig.~\\ref{Fig:1}.  The left\npanel shows the polar angular distribution according to\nEq.~(\\ref{E13}). The spin-density matrix elements are calculated\nin the Gottfried-Jackson system using the model of\nRef.~\\cite{TK07}. The photon energy is integrated over the\ninterval $E_\\gamma=1.97-2.17$~GeV, and the momentum transfer is\n$t=t_{\\rm max}-0.2$~GeV$^2$, where $t_{\\max}$ corresponds to the\n$\\phi$ meson production at forward angle, i.e., $\\theta=0$. The\nexperimental data are from~Ref.~\\cite{Mibe05}. The azimuthal angle\ndistribution relative to the photon polarization plane is\nexhibited in the right panel. One can see the striking difference\nof the angular distributions for $K^+K^-$ and $\\pi^0\\gamma$\ndecays.\n\nA similar comparison for $\\omega$ meson photo-production is\npresented in Fig.~\\ref{Fig:2}. Here, the spin-density matrix\nelements are calculated using the model of Ref.~\\cite{TL02}. The\ncalculation is for $E_\\gamma=1.6$~GeV and $t=t_{\\rm\nmax}-0.2$~GeV$^2$. Again, there is a striking difference in the\nangular distributions for the two considered decay modes.\n\nIn summary, we derived suitable formulas of the angular decay\ndistributions for $\\phi$ and $\\omega$ meson photo-production and\nsubsequent decay to $\\pi^0\\gamma$ ($\\eta\\gamma$). We found that\nthe distributions differ from corresponding distributions of\npurely hadronic decay channels on a qualitative level. Thus,\nasymmetries, with the exception of the beam asymmetry, are\ndifferent in signs and absolute values.\nFinally, we note that the spin-density matrices depend on the\nchoice of the reference frame (Gottfried-Jackson, or Helicity, or\nAdair, cf.~\\cite{TK07,Schilling70,TL03}). This circumstance must\nbe taken\ninto account in the analysis and interpretation of data.\\\\\n\n\n  We thank H.~Schmieden for stimulating discussions.\n  One of the authors (A.I.T.) appreciates the hospitality in FZD.\n  This work was supported by BMBF grant 06DR136 and GSI-FE.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{sec:level1}Introduction}\nResearch in the area of nanoscale thermoelectrics is primarily carried out along two broad directions: (i) materials and devices for practical energy conversion, and (ii) fundamental studies on heat flow in the nanoscale. The former concerns materials and interface design, device optimization and integration at the systems level \\cite{shakouri}. The physics that is actively considered at the materials level is that of maximizing the thermoelectric figure of merit, $zT$ \\cite{Hicks1993,Hickss1993,Dresselhaus2007,Poudel2008,Snyder2008,Heremans2008,Murphy2008}, via detailed electronic structure and interface considerations. The latter is exploratory in nature and concerns fundamental transport studies on the physics of heat flow at the nanoscale \\cite{Andreev2001,Humphrey2002,Kubala2006,Kubala2008,nak,Kim2014,Reddy1568,jordan1,jordan2,Agarwal2014,Sothmann2014}. It is also well known from the thermodynamic interpretation of thermoelectric processes that $zT$ is conceptually meaningful only in the linear response regime \\cite{ioffe} and does not provide the full picture of heat flow at the nanoscale\\cite{Muralidharan2012,jordan1,Sothmann2014,whitney}. Therefore, themoelectric analysis based on power and efficiency considerations \\cite{whitney,Muralidharan2012,Sothmann2014} have gained precedence when it comes to fundamental studies \\cite{nak,jordan2,Muralidharan2012,Agarwal2014,Zimb2016,Leijnse2010}.\\\\\n\\indent Zero-dimensional systems such as molecules or quantum dots are known to possess unique thermoelectric properties \\cite{Mahan1996,Zimb2016} owing to their highly distorted electronic density of states (DOS). From a fundamental stand point, nanoscale heat engines are built using quantum dots or molecules sandwiched between two contacts. Thermoelectric transport measurements are performed by subjecting electrochemical potential and thermal gradients across them \\cite{Kim2014,Reddy1568}.  Specific to short molecules and quantum dots, a lot of recent research work has focused on how strong correlation effects related to Coulomb charging may influence the thermoelectric performance \\cite{jordan1,jordan2,Sothmann2014,Muralidharan2012,Zimb2016} under various bias situations.\\\\\n\\begin{figure*}[]\n\t\\begin{center}\n\t\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth, height=0.25\\textwidth]{image1a.png}\\label{1a}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth, height=0.25\\textwidth]{image1b.PNG}\\label{1b}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth, height=0.25\\textwidth]{image1c.png}\\label{1c}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth, height=0.25\\textwidth]{image1d.png}\\label{1d}}\n\t\t\n\t\\end{center}\n\t\n\t\\caption{Two operating limits of the dissipative heat engine set up studied here: (a) the dot phonon degrees of freedom (DOF) are out of equilibrium, and (b) the dot phonon degrees of freedom (DOF) relax via coupling to a heat bath with a rate $\\beta$. In this case, the dot temperature is estimated by coupling a {\\it{thermometer bath}} weakly, with an associated rate $\\beta_{th}$. In both cases, the quantum dot is  coupled weakly to the contacts with electronic rates, $\\gamma_H$, $\\gamma_C$ and phonon rates $\\beta_H$, $\\beta_C$. (c) State transition diagram in the quantum dot electronic Fock space. Electron transitions take place between the states where electron numbers differ by $\\pm$ 1. (d) The state transition diagram in the electron-phonon Fock space.The black solid and black dash arrows represent electronic tunneling processes. The gray solid arrows and gray dotted arrows represent the phonon-assisted tunneling processes. The black double-sided arrows represent the heat bath assisted phonon transitions. In both cases, the currents are driven by a temperature gradient applied between the two contacts, $H$ and $C$, in a voltage controlled set up (see text). }\n\\end{figure*}\n\\indent However, the charging of the system due to electronic transport processes changes the nuclear geometry and couples with various vibrational modes \\cite{Hartle2011,Zazunov2006}. The interplay between electronic and vibrational degrees of freedom have found numerous signatures in a multitude of charge transport experiments \\cite{Park2000,Leroy2004,LeRoy2005,Yu2004,Sapmaz2006,Zhitenev2002}.  From the point of view of thermoelectrics, the electron-phonon interactions are important since they modify both charge and heat currents at the nanoscale. While charge and heat transport in the dissipative quantum dot set up described using the Anderson-Holstein model have been the subject of many works \\cite{Siddiqui2006,Entin-Wohlman2010,Entin-Wohlman2012,Entin-Wohlman2014,Hartle2011,Zazunov2006,Segal2006,Segal2005,Segall2005}, analysis of thermoelectric transport in this regime has not received much attention \\cite{Leijnse2010}. \\\\\n\\indent The object of this paper is to advance the basic understanding developed in an earlier work \\cite{Leijnse2010} on dissipative quantum dot heat engines by elucidating some relevant and novel physics that arises under two experimentally relevant operating limits. The first limit, being when the dot phonon modes are in non-equilibrium. Such a situation occurs in quantum dot systems that are suspended \\cite{LeRoy2005} over metallic contacts and can hence be driven out of equilibrium, giving rise to interesting charge transport signatures \\cite{Siddiqui2006}. The second limit occurs when the dot phonons relax via coupling to a heat bath held at a fixed temperature \\cite{Siddiqui2006,Entin-Wohlman2010,Entin-Wohlman2012,Entin-Wohlman2014}. In the first case, a detailed analysis of the physics related to the interplay between the quantum dot level quantization, the on-site Coulomb interaction and the electron phonon coupling on the thermoelectric performance is carried out. Importantly, it is demonstrated that due to such an interplay, an n-type heat engine performs better than a p-type heat engine. In the second case, with the aid of the dot temperature estimated by incorporating a {\\it{thermometer bath}}, it is shown that the dot temperature deviates strongly from the bath temperature as electron-phonon interaction in the dot becomes stronger. An interesting consequence of which is that the dot temperature intimately controls the direction of phonon heat current thereby influencing the thermoelectric performance. Finally, we evaluate the trend of the maximum efficiency as the phonon couplings between the dot and all the other macroscopic bodies vary.  \\\\\n\\indent This paper is organized as follows: Sec. II introduces the Anderson-Holstein based dissipative heat engine and formulates the transport equations related to the two operating limits we consider. In Sec. IIIA and Sec. IIIB we analyze the performance of heat engines when phonons are out of equilibrium. In Sec. IIIC and Sec. IIID, we elucidate the physics related to heat engines coupled to heat baths with the aid of the dot temperature. In Sec. IV, we summarize our result and conclude.\n\n\n\n\\section{\\label{sec:level2}Physics and Formulation}\n\\indent A schematic of the heat engine setups studied here is presented in Fig.~\\ref{1a} and Fig.~\\ref{1b}. Both setups constitute a quantum dot described by the dissipative Anderson Holstein Hamiltonian coupled weakly to two macroscopic contacts denoted as $H$ and $C$, which drive charge and heat currents through the dot. Additionally, the dot can be coupled strongly to a heat bath $B$, and be weakly coupled to the thermometer bath \\cite{Segall2005} as presented in Fig.~\\ref{1b}. We note that the set ups described here are typically {\\it{voltage controlled}}, primarily driven by  the application of a temperature gradient accompanied by the control of the voltage drop via a variable load resistor.\n\\subsubsection{Model Hamiltonian}\n\\indent The composite Hamiltonian of the set up is given as $\\hat{H}=\\hat{H}_{D}+\\hat{H}_{C}+\\hat{H}_{B}+\\hat{H}_{CD}+\\hat{H}_{BD}$, where $\\hat{H}_{D}$, $\\hat{H}_{C}$, and $\\hat{H}_{B}$ are the respective Hamiltonians of the dot, the contacts, and the bath, while $\\hat{H}_{CD}$ and $\\hat{H}_{BD}$ represent the coupling Hamiltonians between the dot and the contacts and between the dot and the bath respectively. The dot Hamiltonian is described via the Anderson-Holstein model given by\n\\begin{equation}\n\t\\begin{split}\n\t\\hat{H}_{D}=(\\sum_{\\sigma}^{} \\epsilon_{\\sigma} \\hat{n}_{\\sigma}+U \\hat{n}_{\\uparrow}\\hat{n}_{\\downarrow}) \n\t+\\hbar \\omega_{\\nu}\\hat{n}_{\\nu} \\\\+\\sum_{\\sigma}^{} \\lambda_{\\nu} \\hbar \\omega_{\\nu} \\hat{n}_{\\sigma} (\\hat{b}_{\\nu}^{\\dagger}+\\hat{b}_{\\nu}),\n\t\\end{split}\n\t\\label{Anderson}\n\\end{equation}\nwhere the dot comprises a single spin degenerate energy level with an on-site energy, $\\epsilon_{\\sigma}$, and a Coulomb interaction energy, $U$. The phonon degree of freedom is described via a single phonon mode of angular frequency $\\omega_{\\nu}$. Inside the dot, the electrons and phonons interact via the electron-phonon coupling, $\\lambda_{\\nu}$. Here, $\\hat{n}_{\\sigma}$=$\\hat{d}_{\\sigma}^{\\dagger} \\hat{d}_{\\sigma}$ and $\\hat{n}_{\\nu}$=$\\hat{b}_{\\nu}^{\\dagger} \\hat{b}_{\\nu}$ are the dot electron and dot phonon number operators respectively, given that $\\hat{d}_{\\sigma}^{\\dagger} (\\hat{d}_{\\sigma})$ and $\\hat{b}_{\\nu}^{\\dagger} (\\hat{b}_{\\nu})$ represent the creation (annihilation) operator for the electrons and phonons in the dot respectively. \\\\\n\nThe contact and heat bath Hamiltonians and their respective coupling Hamiltonians with the dot are defined as\n\\begin{equation}\n\t\\begin{split}\n\t\t\\hat{H}_{C}=\\sum_{\\alpha\\in H,C}^{} \\sum_{k\\sigma'}^{}\\epsilon_{\\alpha k\\sigma'} \\hat{n}_{\\alpha k\\sigma'}+\\sum_{\\alpha\\in H,C}^{} \\sum_{p}^{}\\hbar \\omega_{\\alpha p} \\hat{n}_{\\alpha p}\n\t\\end{split}\n\\end{equation}\n\\begin{equation}\n\t\\begin{split}\n\t\t\\hat{H}_{B}=\\sum_{t}^{} \\hbar \\omega_{t}\\hat{B}_{t}^{\\dagger} \\hat{B}_{t}\n\t\\end{split}\n\\end{equation}\n\\begin{equation}\n\t\\begin{split}\n\t\t\\hat{H}_{CD}=\\sum_{k\\sigma', \\sigma}^{} [\\tau_{ \\alpha k\\sigma'\\sigma}^{el} \\hat{c}_{\\alpha k \\sigma'}^{\\dagger}\\hat{d}_{\\sigma}+h.c] \\\\ +\\sum_{\\nu,p}^{}[\\tau_{ \\alpha p\\nu}^{ph} (\\hat{a}_{\\alpha p}^{\\dagger}+\\hat{a}_{\\alpha p})(\\hat{b}_{\\nu}^{\\dagger}+\\hat{b}_{\\nu})+h.c]\n\t\\end{split}\n\\end{equation} \n\\begin{equation}\n\t\\begin{split}\n\t\t\\hat{H}_{BD}=\\sum_{\\nu,t}^{}\\tau^{ph}_{ t\\nu}(\\hat{B}_{t}^{\\dagger}+\\hat{B}_{t})(\\hat{b}_{\\nu}^{\\dagger}+\\hat{b}_{\\nu}).\n\t\\end{split}\n\\end{equation}\nHere, $\\hat{n}_{\\alpha k \\sigma'}$=$\\hat{c}_{\\alpha k\\sigma'}^{\\dagger} \\hat{c}_{\\alpha k\\sigma'}$ and  $\\hat{n}_{\\alpha p}$=$\\hat{a}_{\\alpha p}^{\\dagger} \\hat{a}_{\\alpha p}$ are the electron and phonon number operators in the contacts. The contacts are assumed to be in the eigen-basis with wave vectors $k$ and spin orientation ${\\sigma'}$. An electron in the dot with a spin orientation $\\sigma$ is coupled to an electron in contact $\\alpha$ ($\\alpha \\in H,C$) through $\\tau_{ \\alpha k\\sigma'\\sigma}^{el}$. Similarly $\\nu$-th phonon mode of the dot is coupled to the $t$-th phonon mode of the contact through $\\tau_{ t\\nu}^{ph}$.\\\\\n\\indent The dot Hamiltonian $\\hat{H}_{D}$ is diagonalized by the polaron transformation \\cite{Braig2003,Siddiqui2006} leading to the renormalization of the on-site and Coulomb interaction energies given by\n\\begin{equation}\n\\tilde{\\epsilon}_{\\sigma}=\\epsilon_{\\sigma}-(\\lambda^2 \\hbar \\omega_{\\nu}),\n\\end{equation}\n\\begin{equation}\n\\tilde{U}=U-(2\\lambda^2 \\hbar \\omega_{\\nu}).\n\\end{equation}\nThe renormalized dot many-particle energies are given by $E=\\tilde{E}_{\\sigma}+m \\hbar \\omega_{\\nu}$, where $m=(0,1,2,3,...)$ and $\\tilde{E}_{\\sigma}$=$(0,\\tilde{\\epsilon}_{\\uparrow},\\tilde{\\epsilon}_{\\downarrow},\\tilde{\\epsilon}_{\\uparrow}+\\tilde{\\epsilon}_{\\downarrow}+\\tilde{U})$.\\\\\n \\indent Both $\\hat{H}_C$ and  $\\hat{H}_B$ remain unchanged due to the renormalization since they are independent of the dot operators. The transformation of the electron tunneling part of the Hamiltonian $\\hat{H_{CD}}$ leads to a modification of the electron coupling factor, $\\tilde{\\tau}_{ \\alpha k\\sigma'\\sigma}^{el}=\\tau_{ \\alpha k\\sigma'\\sigma}^{el}\\exp[-\\lambda_{\\nu}(\\hat{b_{\\nu}}-\\hat{b_{\\nu}}^{\\dagger})]$. We can neglect the renormalization of phonon coupling factors $\\tau_{ \\alpha p\\nu}^{ph}$ and $\\tau^{ph}_{ t\\nu}$,  considering that both of them are very small, which is an essential condition to get optimized thermoelectric efficiency \\cite{Leijnse2010}. \\\\\n \\indent With the above definitions, in the calculations to follow, it is also customary to define various tunneling rates under the assumption of dispersionless contacts as follows: The electronic tunneling rate between the dot and the contact $\\alpha$ with density of states $\\rho_{\\alpha}$ is derived from the Fermi's golden Rule as  $\\gamma_{\\alpha}=\\frac{2\\pi}{\\hbar}\\sum_{\\sigma} |\\tilde{\\tau}^{el}_{\\alpha k\\sigma}|^2 \\rho_{\\alpha \\sigma}$. Similarly, the phonon relaxation rates between the dot and other macroscopic bodies $r'$( where, $r' \\in H,C,B$) with phonon density of states $D_{r'}$ is expressed as $\\beta_{r'}=\\frac{2\\pi}{\\hbar}|\\tau^{ph}_{r'\\nu}|^2D_{r'}$. \n\\subsubsection{Transport formulation}\nWe first state the important assumptions made in the set ups that we consider. First, we work in a regime where the dot-bath phonon relaxation rate is smaller compared to the tunneling rates between the dot and the contacts. Larger phonon couplings may cause further energy shift in the dot phonon modes leading to a non-separable terminal phonon currents \\cite{Segal2006,Segal2005,Segall2005}. The assumption of small phonon coupling also allows us to exclude system damping \\cite{Braig2003}. Second, we perform all calculations within the sequential tunneling limit \\cite{Timm,Leijnse2010}, where, the associated tunneling energies, $\\hbar \\gamma_{\\alpha}, \\hbar \\beta_{r} <<  k_BT$. The mathematical expressions for the electron tunneling rate $\\gamma$ and the phonon relaxation rate $\\beta$ are to be defined shortly. The sequential tunneling limit is the relevant regime when describing quantum dot transport as most experiments are performed in this regime \\cite{tarucha1,Timm}. Under this approximation, given a spin degenerate level coupled to non magnetic contacts, transport is described via rate equations \\cite{Beenakker,Basky_Beenakker,Basky_Datta,Timm} in the diagonal subspace of the quantum dot reduced density matrix \\cite{Koenig_1,Koenig_2,Brouw,Milena_noncoll,Basky_Milena}. \\\\\n\\indent The use of the diagonal subspace is justified in the absence of coherences. In the current context, we are faced with two types of coherences, (a) coherence between the degenerate up-spin and down-spin levels and (b) coherences between various phonon induced side band energies. The first type can be neglected simply because electron-phonon interaction is described via a coupling factor $\\lambda_{\\nu}$, which is spin independent. Such a coherence between up-spin and down-spin levels is characteristic of systems with non-collinear magnetic contacts \\cite{Koenig_1,Koenig_2,Basky_Milena}, or in systems which have orbital degeneracies \\cite{Brouw,Milena_noncoll}. The second type, namely, the coherence between two phonon induced side bands can be safely neglected by assuming that the energy spacing between two adjacent side bands is larger than the tunneling induced broadening of energy levels, i.e., $\\hbar\\omega_{\\nu}>>\\hbar\\gamma$. In this limit electron-phonon interactions that occur at two consecutive times are completely uncorrelated \\cite{Theses1}, and a Markovian approximation is also justified \\cite{Timm}. This allows us to neglect bath memory also. Hence, the secular terms in density matrix get decoupled from the off-diagonal terms, which ultimately implies that such coherences may be safely ignored.  \\\\\n\\indent The electronic tunneling rate between two electron-phonon Fock states, $\\mid n,q \\rangle$ and $ \\mid n \\pm 1,q \\rangle$, with $n$ and $q$ representing the electronic and phonon state label respectively, is given by\n\\begin{equation}\n\\begin{gathered}\nR_{(n,q)\\rightarrow(n+1,q')}^{el}=\\\\\\sum\\limits_{\\alpha \\in H,C}^{}\\gamma_{\\alpha}|\\bra{n,q}\\hat{\\tilde{{d}}}_{\\sigma}\\ket{n+1,q'}|^2\\\\ \\times f_{\\alpha}\\bigg(\\frac{E_{(n+1,q')}-E_{(n,q)}-\\mu_{\\alpha}}{k_B T_{\\alpha}}\\bigg) \n\\end{gathered}\n\\end{equation}\n\\begin{equation}\n\\begin{gathered}\nR_{(n,q)\\rightarrow(n -1,q')}^{el}=\\\\\\sum\\limits_{\\alpha \\in H,C}^{} \\gamma_{\\alpha}|\\bra{n,q}\\hat{\\tilde{{d}}}_{\\sigma}^{\\dagger}\\ket{n-1,q'}|^2\\\\ \\times \\bigg[1-f_{\\alpha}\\bigg(\\frac{E_{(n,q)}-E_{(n-1,q')}-\\mu_{\\alpha}}{k_B T_{\\alpha}}\\bigg)\\bigg]\n\\end{gathered}\n\\end{equation}\nThe relaxation of the dot phonons to the contacts and the heat bath cause transition between the states $(n,q)$ and $(n,q\\pm 1)$ follow the Boltzmann ratio:\n\\begin{equation}\n\\begin{split}\nR_{(n,q)\\rightarrow(n,q+1)}^{ph}=\\sum\\limits_{r'\\in H,C,B}^{}\\beta_{r'} (q+1) exp(-\\frac{\\hbar \\omega_{r}}{k_B T_{r'}})\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\nR_{(n,q)\\rightarrow(n,q-1)}^{ph}=\\sum\\limits_{r'\\in H,C,B}^{}\\beta_{r'} (q+1).\n\\end{split}\n\\end{equation} \nWith various rates defined above, the master equation for the probabilities, $P_{n,q}$ of the many particle states $\\mid n,q \\rangle$, then reads:\n\\begin{equation}\n\\begin{split}\n\\frac{d P_{(n,q)}}{dt}=\\sum_{k'=0}^{N_q'-1}\\bigg[R_{(n\\pm 1,q')\\rightarrow(n,q)}^{el}P_{(n\\pm 1,q')} \\\\ - R_{(n,q)\\rightarrow(n\\pm 1,q')}^{el}P_{(n,q)} \\bigg]\\\\+\\bigg[R_{(n ,q\\pm 1)\\rightarrow(n,q)}^{ph}P_{(n,q\\pm 1)} \\\\ -R_{(n,q)\\rightarrow(n,q\\pm 1)}^{ph} P_{(n,q)} \\bigg].\n\\end{split}\n\\end{equation}\nIn steady state, we set $\\frac{d P_{(n,q)}}{dt}=0$, and find the null space of the rate matrix to evaluate the steady state probabilities. Using the steady state probabilities, we can get the expressions for the terminal electronic charge currents $J$ and heat currents $J_{el_{\\alpha}}^{Q}$,$J_{ph_{r'}}^{Q}$  as\n\\begin{equation}\n\\begin{split}\nJ_{\\alpha}=\\sum_{n,q=0}^{N_e,N_p-1}\\sum_{q'=0}^{N'_q-1} -q\\bigg[R_{(n \\pm 1,q') \\rightarrow(n,q)}^{el_{\\alpha}}P_{(n \\pm 1,q')}\\\\ -R_{(n,q) \\rightarrow(n \\pm 1,q')}^{el_{\\alpha}}P_{(n,q)}\\bigg]\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{gathered}\nJ_{el_{\\alpha}}^{Q}=\\sum_{n,q=0}^{N_e,N_p-1}\\sum_{k'}^{N'_p-1} \\bigg[E_{(n\\pm1,q')}-\\mu_{\\alpha}\\bigg] \\\\ \\times R_{(n\\pm 1,q') \\rightarrow(n,q)}^{el_{\\alpha}}P_{(n \\pm 1,q')}\\\\ - \\bigg[E_{(N_e,N_p)}-\\mu_{\\alpha}\\bigg]\\\\ \\times R_{(n,q) \\rightarrow(n \\pm1,q)}^{el_{\\alpha}}P_{(n,q)}\n\\end{gathered}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\nJ_{ph_{r'}}^{Q}=\\sum_{n,q=0}^{N_e,N_p-1} \\hbar \\omega_{r} \\bigg[R_{(n,q) \\rightarrow(n,q \\pm 1)}^{ph_{\\alpha}}P_{(n,q)}\\\\- R_{(n,q \\pm 1) \\rightarrow(n,q)}^{ph_{\\alpha}}P_{(n,q +1)}\\bigg]\n\\end{split}\n\\end{equation}\nThe charge or electronic heat currents associated with contacts $\\alpha=H(C)$ involve only the rates associated with the respective contact. However, the phonon heat current is associated with both the contacts as well as the heat bath. \n\\subsubsection{Calculation of power and efficiency}\nA thermal bias applied across the contacts, $T_H$, at the hot contact $H$, and $T_C$, at the cold contact $C$, can result in charge and heat currents. In the voltage controlled setup, a variable resistor controls the back flow charge current. At a voltage $V_S$, the back flow current completely cancels the charge current set up by the temperature gradient. This is referred to as the built-in potential or Seebeck voltage. The set up hence functions as a heat-to-charge-current converter or a heat engine in the voltage range $[0,V_S]$, which we term as the operating region. The electrical power generated in the circuit is given b $P=-J_{\\alpha}\\times V_{app}$. The thermoelectric efficiency is then expressed as\n\\begin{equation}\n\\eta=\\frac{P}{J^Q_{in}},\n\\end{equation}\nwhere, the input heat current includes both the electron and phonon heat currents such that the net heat input is $J^Q_{in}= J_{{el}_{H}}^{Q} + J_{{ph}_{in}}^{Q}$. It must be noted that while the input electronic heat current can be supplied only from the hot contact, the phonon heat current can be supplied from contacts or the heat bath depending on the dot temperature $T_M$. This aspect will be studied in detail in a later section.\n\\section{Results}\n\\subsection{Non-equilibrium phonons}\nWe first elaborate on the effect of the electron-phonon interaction parameter, $\\lambda$, on the delivered electronic power, $P$, and the efficiency, $\\eta$. Each operating point is signified by a constant applied thermal gradient ($T_H$=10K,$T_C$=5K) and a variable voltage bias $V$ between the two contacts. In the current section we assume that the Coulomb interaction is kept much larger, i.e., $U>>k_BT$, the electronic contact coupling and the phonon couplings, $\\gamma_H,\\gamma_C,\\beta_H,\\beta_C$ are small enough ($\\hbar\\gamma_{H(C)}=5\\times10^{-6} eV,\\hbar\\beta_{H(C)}=2\\times10^{-12} eV$) to keep the tunneling induced broadening of the states in the quantum dot small and ensure that the dot phonons are out of equilibrium. Transport under the sequential tunneling limit where the rate equation formalism is applicable is also ensured under these conditions.  Additionally, in this part, we consider that the dot functions as an n-type, i.e., when $\\epsilon-\\mu>0$.\\\\\n\\begin{center}\n\t\t\t\\begin{figure}[!htb]\n\t\t\t\t\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2a.png}\\label{2a}}\n\t\t\t\t\\quad\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2b.png}\\label{2b}}\n\t\t\t\t\\quad\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2c.png}\\label{2c}}\n\t\t\t\t\\quad\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2d.png}\\label{2d}}\n\t\t\t\t\\quad\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2e.png}\\label{2e}}\n\t\t\t\t\\quad\n\t\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic2f.png}\\label{2f}}\n\t\t\t\t\\quad\t\t\n\\caption{Thermoelectric performance with non-equilibrium phonons. The Carnot efficiency is set to $\\eta_C=0.5$ at $\\epsilon-\\mu<2KT$. (a) Variation of the electronic power as a function of voltage as  $\\lambda$ is varied. (b) Variation of $\\eta$ as a function of voltage as  $\\lambda$ is varied. (c) and (d) 3-D color plots of $\\gamma_{eff}\/\\gamma$ pertaining to the tunneling between two states with phonon numbers, $q_1$ and $q_2$, for $\\lambda=0$ and $\\lambda=2$ respectively. As $\\lambda$ is increased, we notice a larger off-diagonal contribution in the associated phonon numbers between the two states. (e) Electronic power delivered at $\\eta_{max}$ as $\\lambda$ increases, and (f) Variation of $\\eta$  at $P_{max}$, representing the efficiency at maximum power as $\\lambda$ is increased. We infer that an increase in the electron-phonon coupling leads to a better performance in terms of the power at maximum efficiency while maintaining the efficiency at maximum power. }\n\\end{figure}\n\\end{center}\n\\begin{center}\n\t\\begin{figure}[!htb]\n\t\t\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.222\\textwidth]{pic4a.png}\\label{3a}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.222\\textwidth]{pic4b.png}\\label{3b}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.222\\textwidth]{pic4c.png}\\label{3c}\n\t\t}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.222\\textwidth]{pic4d.png}\\label{3d}\n\t\t}\n\t\t\\quad\n\t\t\\caption{A performance comparison between n-type and p-type heat engines. Variation of $\\eta_{max}$ with Coulomb interaction and electron-phonon interaction for (a) an n-type ($\\epsilon-\\mu=1 meV$) and (b) a p-type setup ($\\epsilon-\\mu=-1 meV$). For both set ups, $\\eta_{max}$ maximizes at $\\tilde{U}=0$. The p-type setup can have a vanishing $\\eta_{max}$ at the particle-hole symmetry point. (c) Maxima and minima points of $\\eta_{max}$ for an n-type engine with $\\lambda=1.5$, with the grey region depicting the negative $U$ regime. (d) The region around the particle-hole symmetry point for a p-type setup where $\\eta_{max}$ vanishes. We note that an n-type setup can provide a better $\\eta_{max}$ in the presence of a finite $U$ since it can avoid particle-hole symmetry.}\n\t\\end{figure}\n\\end{center}\n\\indent  A finite electron-phonon coupling causes a displacement of the potential profile of the dot and alters the electron tunneling rate between two electron-phonon states, $\\ket{n,q_1}$ and $\\ket{n\\pm 1,q_2}$ to $\\gamma_{eff}$\\cite{Theses} defined as\n\\begin{equation}\n\\begin{gathered}\n\\gamma_{eff}=\\gamma\\times |C_{q_1q_2}|^2\\\\\n=\\gamma\\times exp(-\\frac{\\lambda^2}{2})\\times {\\bigg(\\frac{q!}{Q!}\\bigg)}^2\\\\\\times\\lambda^{Q-q}\\times L_{q}^{Q-q}(\\lambda^2)\\times[sgn(q_1-q_2)]^{(q_1-q_2)},\n\\end{gathered}\n\\end{equation}\nwhere $q=min(q_1,q_2)$ and $Q=max(q_1,q_2)$, $C_{q_1q_2}$ is a measure of the overlap between two many body electron-phonon states with phonon numbers, $q_1$ and $q_2$, arising from the electron-phonon interaction. Referring to Fig.~\\ref{2a}, we see that the peak power, as well as the Seebeck voltage, $V_S$, drops as the electron-phonon coupling parameter $\\lambda$ is increased.  As $\\lambda$ is increased, the charge current as well as the peak power falls, since $\\gamma_{eff}$ between two states become smaller. In Fig.~\\ref{2c}, we see that for $\\lambda=0$, $\\gamma_{eff}\\sim\\gamma$ is only non-zero between two states with equal phonon number. Hence, in the non-interacting case, only direct tunneling is feasible. As $\\lambda$ is increased, strong electron-phonon interaction leads to the  suppression of  direct tunneling and the facilitation of phonon-assisted tunneling. \\\\ \n\\indent We see in Fig.~\\ref{2d} that a nonzero $\\lambda$ results in phonon assisted tunneling between two states with unequal phonon numbers. Evidence of this phenomenon has been found in various transport experiments \\cite{Steele1103,Park2000,Leturcq2008} and also has been demonstrated theoretically \\cite{Koch2005,Kochh2006}. Notice that $\\gamma_{eff}$ for non-zero $\\lambda$ is always less than $\\gamma$, due to electron-phonon coupling making the set up dissipative. For this reason, the charge current decreases and the open-circuit point is reached at a smaller voltage, leading to a fall in $V_S$ as $\\lambda$ is increased. Turning to the analysis of the efficiency $\\eta$, first we note the well known result \\cite{Muralidharan2012} that for $\\lambda=0$, $\\eta$ attains a maximum of $\\eta_C$ at $V_S$. But as $\\lambda$ is increased, due to phonon assisted tunneling, $J^{Q}_{el}$ is always greater than $P$ within the operating range. Hence according to (16), $\\eta_{max}$ falls below $\\eta_C$.\\\\\n\\indent One must appreciate the fact that although a non-interacting system gives the maximum $\\eta$, the power it delivers at that point is identically zero. With the inclusion of electron-phonon interaction, we evaluate two different trends: (a) Power at maximum efficiency $P_{\\eta_{max}}$, and (b) Efficiency at maximum power $\\eta_{P_{max}}$. In Fig.~\\ref{2e} we see that the electronic power delivered at $\\eta_{max}$ increases monotonically as $\\lambda$ increases. On the other hand, $\\eta$ at maximum power keeps almost constant as shown in Fig.~\\ref{2f}. Hence, although there is a fall in the peak power as $\\lambda$ increases,  $\\eta_{P_{max}}$ increases slightly. This may be counted as an advantage of having stronger electron-phonon interaction. Thus, so far, we can conclude from here that when phonons are out of equilibrium, $\\lambda$ is the deciding factor for the thermoelectric performance. \n\\subsection{Comparison between n-type and p-type heat engines}\n\\begin{center}\n\t\t\n\t\t\n\t\t\\begin{figure}[!htb]\n\t\t\t\n\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic3a.png}\\label{4a}}\n\t\t\t\\quad\n\t\t\t\\subfigure[]{\\includegraphics[height=0.225\\textwidth,width=0.225\\textwidth]{pic3b.png}\\label{4b}}\n\t\t\t\n\t\t\t\\caption{Peak power characteristics of the heat engine as a function of $U$ and on-site energy position, keeping $\\lambda$ constant. (a) Variation of  $V_S$ as a function of $(\\epsilon-\\mu)\/k_{B}T$ and $U\/k_{B}T$. (b) Variation in peak electronic power as a function of $(\\epsilon-\\mu)\/k_{B}T$ and  $U\/k_{B}T$. We note that $V_S$ and $P_M$ remain independent of $U$ for an n-type setup but decrease significantly for a p-type setup as the particle-hole symmetry condition is approached.}\n\t\t\t\n\t\t\\end{figure}\n\t\t\n\t\\end{center}\n\\indent We now turn our attention to an analysis with the inclusion of Coulomb interaction $U$, which brings to fore the difference between an n-type set up and a p-type set up. If $\\epsilon>\\mu$, then the thermal bias induced current flows from cold to hot contact, whereas the current direction is just reverse for $\\epsilon<\\mu$. The nomenclature due to the sense of particle flow being identical to that noted in the thermoelectric transport of n-type and p-type semiconductors . However, turning on $U$ for a p-type set up may give rise to a situation where transport channels $\\epsilon$ and $\\epsilon+U$ in conjunction with the phonon sidebands may give rise to a particle-hole symmetry, which will not be possible for an n-type setup. \\\\\n\\indent A schematic of the variation in $\\eta_{max}$ with respect to $U$ for an n-type set up shown in Fig.~\\ref{3a}. We notice that for a non-zero $\\lambda$, $\\eta_{max}$ never equals the Carnot efficiency and maximizes for a non-zero Coulomb interaction. In Fig.~\\ref{3c}, we present a zoomed-in view for $\\lambda=1.5$, which clearly shows the maxima and minima of $\\eta_{max}$. We notice that $\\eta_{max}$ reaches a maximum when $\\tilde{U}$ disappears. Hence, electron-phonon interaction results in a region of increasing $\\eta_{max}$ in the negative $\\tilde{U}$ regime  \\cite{Andergassen2011,Alexandrov2002}, shown as a gray shaded area in  Fig.~\\ref{3c}. In Fig.~\\ref{3b} we depict the $\\eta_{max}$ variation for a p-type heat engine, which follows a similar trend except that it vanishes when the particle-hole symmetry point is reached \\cite{Wang2010,Dubi2009,Rejec2012,Buddhiraju2015}, as shown schematically in Fig.~\\ref{3d}. \\\\\n\\indent Next we compare the power generation of the n-type and the  p-type setup. In Fig.~\\ref{4a} and Fig.~\\ref{4b}, we detail the variation in the Seebeck voltage $V_S$ and the peak power $P_M$ as a function of  $U$ and the relative onsite-energy $\\epsilon-\\mu$. Both $V_S$ and $P_M$ rise with increasing $|\\epsilon-\\mu|$, since more voltage bias is needed to reach the open-circuit point, justifying the increase of $|V_S|$. As the operating region $[0, V_S]$, of the heat engine broadens, the peak power also increases. We see that for the n-type engine, where $\\epsilon-\\mu>0$, the variation of both $V_S$ and $P_M$ remains almost constant with $U$ while for a  p-type engine, where $\\epsilon-\\mu<0$, this is not so . \n\\begin{center}\n\t\t\\begin{figure}[!htb]\n\t\t\t\n\t\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic5a.png}\\label{5a}}\n\t\t\t\\quad\n\t\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic5b.png}\\label{5b}}\n\t\t\t\n\t\t\t\\caption{Study of $\\eta_{max}$ as cumulative function of $U$ and the phonon angular frequency $\\omega$, keeping $\\lambda$ constant ($\\lambda=0.5$) for (a) an n-type and (b) a p-type set up. (b) The region $\\textbf{P1}$ depicts the locus along which $\\eta_{max}$ vanishes for the p-type setup. Here both $U$ and $\\omega$ contribute to achieve the particle-hole symmetry. The region $\\textbf{P3}$ depicts where the p-type setup can perform as a heat engine. The region $\\textbf{P2}$ represents the out of the operating limit region. In (a), an n-type setup performs like a p-type setup for large $\\omega$ with the regions $\\textbf{N1}$ and $\\textbf{N2}$ being similar to $\\textbf{P1}$, $\\textbf{P2}$. The region $\\textbf{N3}$ has a low efficiency since $\\epsilon\\sim\\mu$. For small $\\omega$, the n-type setup achieves a high efficiency which is independent of $\\omega$. Hence in the limit of low $\\omega$, an n-type setup is a better heat engine. }\n\t\t\t\n\t\t\\end{figure}\n\t\t\n\t\\end{center}\nWe see that for a p-type heat engine, significant power is delivered at small values of $U$. If we increase $U$, first both $|V_S|$ and $P_M$ drops to zero before increasing again to the previous value. \nThis is again due to the particle-hole symmetry condition. The analysis in Fig. 3 and Fig. 4 thus clearly indicates that an n-type engine can avoid particle-hole symmetry condition and hence performs better than the p-type engine. \\\\\t\n\\indent An important aspect to be noticed is that particle-hole symmetry can be reached in two ways, either by changing $U$ or by tuning $\\omega$. In Fig.~\\ref{5a} and Fig.~\\ref{5b} we produce a 3-D plots for the variation of $\\eta_{max}$ for the n-type dot and p-type dot respectively. According to (2) and (3), for large values of $\\omega$, an n-type setup performs like a p-type setup. So the upper half of Fig.~\\ref{5a} (where $\\omega$ is high) resembles that of Fig.~\\ref{5b}. For low frequencies, we see that $\\eta_{max}$ does nullify along the black branch $\\textbf{N3}$, where $\\tilde{\\epsilon}$ almost merges with $\\mu$. In the region $\\textbf{N4}$, we get a high $\\eta_{max}$, which is almost independent of $U$. \\\\\n\\begin{center}\n\t\\begin{figure}[!htb]\n\t\t\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic9a.png}\\label{6a}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic9b.png}\\label{6b}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic9c.png}\\label{6c}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic9d.png}\\label{6d}}\n\t\t\\caption{Trends in the dot temperature $T_M$. (a),(b) Variation of $T_M$ as a function of $V_S$ and $\\gamma\/\\beta$ for (a) $T_B>>T_H,T_C$, and (b) $T_B<<T_H, T_C$. It shows that $T_M$ is almost independent of $V_S$. (c),(d) Variation of $T_M$ as a function of $\\gamma\/\\beta$ for different $\\lambda$, for the hot and cold bath conditions respectively. In the limit of strong dot to bath coupling, $T_M$ just follows $T_B$ for $\\lambda=0$, but deviates considerably as $\\lambda$ increases. This implies that a bath-to-dot heat current is also feasible and will control the calculation of efficiency.}\n\t\t\\end{figure}\n\\end{center}\n\n\\indent Switching our attention to Fig.~\\ref{5b}, for the case of the p-type heat engine, we see that $\\eta_{max}$ vanishes along the black branch (region marked as $\\textbf{P1}$), which represents the locus of the particle-hole symmetry points. In the low frequency region marked $\\textbf{P3}$, we get a comparatively high value of $\\eta_{max}$ as described in Fig. 3. On the other hand, in the high frequency region (marked $\\textbf{P2}$), both the polaronic shifted energy channels and their corresponding phonon sidebands go out of the transport window. Theoretically we get high efficiency in this region but it is of no use since this region lies outside the operating region.\\\\\n\\indent In general, hence, we should be interested in the low-frequency range since it serves as the power generating region, where $\\eta_{max}$ is more for the n-type setup. Hence the overall study confirms that n-type engine is optimal compared to the p-type engine. One fact must be noted that we have chosen $\\omega$ so that contact phonon heat current $J_{ph}^{Q}$ is low enough to control $Q^{in}$ or $\\eta$. However $J_{ph}^{Q}$ becomes significant as the dot is strongly coupled to phonon modes of macroscopic bodies which we discuss in the subsequent sections. \\\\\n\\begin{center}\n\t\\begin{figure}[!htb]\n\t\t\n\t\t\\subfigure[]{\\includegraphics[height=0.22\\textwidth,width=0.22\\textwidth]{image9a.png}\\label{7a}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.22\\textwidth,width=0.22\\textwidth]{image9c.png}\\label{7b}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic7a.png}\\label{7c}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic7c.png}\\label{7d}}\n\t\t\\quad\n\t\t\\caption{Evaluation of the sense of contact and bath phonon heat currents when the bath is strongly coupled to the dot for (a) Hot bath, $T_B$$>$$T_H$,$T_C$ and (b) Cold bath, $T_B$$<$$T_H$,$T_C$. (c),(d) Contact phonon heat currents for hot and cold bath respectively. (c) Note that the hot bath forces that the contact phonon currents flow away from the dot, and (d) that cold bath just reverses the direction.}\n\t\\end{figure}\n\\end{center}\n\\subsection{Dot phonons coupled to a heat bath}\n\\indent In this section, we discuss exclusively the role of a heat bath in determining the thermoelectric performance of a heat engine and how the temperature of the bath controls the thermoelectric efficiency. In the limit of non-equilibrium phonons, the electronic heat current is much greater than the phonon heat current and takes the major role in determining $\\eta$. But as the phonons of the dot become strongly coupled to the bulk phonon mode of any macroscopic body, the phonon heat current also becomes a relevant quantity. \n\\indent We start by estimating the temperature of the quantum dot by coupling the central system with a thermometer phonon bath as described in earlier works \\cite{Galperin2007, Galperin2006, Galperin2004}. This is based on the principle that the phonon heat current between the thermometer and the dot vanish when the temperature of thermometer equals the temperature of the dot, $T_M$. In Fig.~\\ref{6a} and ~\\ref{6b}, we present the trends of the molecular temperature as a function of $\\gamma$\/$\\beta$ and applied voltage. We see that dot temperature, $T_M$, is a  weak function of the bias voltage. In Fig.~\\ref{6c} and ~\\ref{6d}, we have shown the dependence of $T_M$ with the electron-phonon coupling parameter $\\lambda$. In the strong coupling limit, for $\\lambda$=0, $T_M$ just follows $T_B$ and the bath phonon current cancels out. But as $\\lambda$ increases, $T_M$ deviates from $T_B$ and gives rise to a bath phonon heat current.\n\\begin{center}\n\\begin{figure}[!htb]\n\t\t\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic6a.png}\\label{8a}}\n\t\t\\quad\n\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic6b.png}\\label{8b}}\n\t\t\\quad\n\t\t\\caption{Variation of $\\eta_{max}$ as a function of $U$ in the limit of strong coupling to the phonon bath. (a) Variation of efficiency as a function of $U$ when $T_B<T_H,T_C$ (b) Same plot for $T_B>T_H,T_C$. This figure establishes the fact that a cold bath does not affect $\\eta$ much, while a hot bath does.}\n\t\t\t\n\t\t\\end{figure}\n\t\t\n\t\\end{center}\n\\indent To evaluate how $T_M$ controls the sense of contact phonon heat currents when the dot is strongly coupled to an external heat bath, We illustrate a schematic in Fig.~\\ref{7a} and Fig.~\\ref{7b}. For a non-zero $\\lambda$, a hot bath always keeps $T_M$ greater than both $T_H$ and $T_C$, compelling the contact phonon heat currents to flow away from the dot. On the other hand, the hot bath itself pumps a phonon heat current into the dot. A cold bath does just the opposite and extracts phonon heat currents out of the dot which, in turn, compels phonon heat currents to flow from the contacts. In Fig.~\\ref{7c} and Fig.~\\ref{7d}, we show the plot of contact phonon heat currents considering that the dot is strongly coupled to the hot and the cold bath respectively. By convention, phonon currents from the contact to the dot are taken to be positive. We notice that for the hot bath, phonon heat currents flow away from the dot leading to a cooling of the dot by the contacts. A strongly coupled cold bath just does the reverse. If the bath is kept at an intermediate temperature, then the contact phonon heat currents will maintain the same direction, i.e., the hot contact will push phonons into the dot and the cold contact will extract phonons out of the dot. \\\\\n\\indent In Fig.~\\ref{8a} and Fig.~\\ref{8b}, we repeat the same plot as Fig.~\\ref{4a} and Fig.~\\ref{4b} for a fixed $\\lambda$ as the temperature of the heat bath varies. We see that when the bath is hot ($T_B >>T_H, T_C$), the maximum efficiency reduces the most. The maximum efficiency $\\eta_{M}$ improves monotonically as we reduce the bath temperature. The expression for the efficiency under the hot and cold bath conditions may be written as\n\t\\begin{equation}\n\t\t\\eta_{Hot}=\\frac{P}{J_{el}^{Q}+J_{phB}^{Q}}\n\t\\end{equation}\n\t\\begin{equation}\n\t\t\\eta_{cold}=\\frac{P}{J_{el}^{Q}+J_{Q}^{phH}+J_{Q}^{phC}}.\n\t\\end{equation}\n\t\\begin{center}\n\t\t\\begin{figure}[!htb]\n\t\t\t\n\t\t\t\\subfigure[]{\\includegraphics[height=0.218\\textwidth,width=0.229\\textwidth]{pic12a.png}\\label{9a}}\n\t\t\t\\quad\n\t\t\t\\subfigure[]{\\includegraphics[height=0.218\\textwidth,width=0.229\\textwidth]{pic12b.png}\\label{9b}}\n\t\t\t\\quad\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic12c.png}\\label{9c}}\n\t\t\t\\quad\n\t\t\t\\subfigure[]{\\includegraphics[height=0.221\\textwidth,width=0.229\\textwidth]{pic12d.png}\\label{9d}}\n\\caption{A trade-off between $\\beta$ and $\\beta_{H(C)}$ to optimize $\\eta_{max}$. We consider that the heat bath is cold, i.e $T_B<T_H,T_C$. (a) 3-D plot showing the variation of $\\eta_{max}$(normalized by $\\eta_C$) as a function of $\\beta$ and $\\beta_{H(C)}$, when the dot is (a) symmetrically and (b) asymmetrically phonon coupled to the contacts. Two blank circles represent the regions which are very weakly and very strongly coupled to both the bath and the contacts. The region \\textbf{SB} indicates where the dot is strongly coupled to the bath but weakly to the contacts. The region \\textbf{SC} indicates the reverse case. (c) and (d), Schematic of the directions of the phonon heat currents in all cases considered. The thick (thin) arrows represent the sense of phonon heat currents between the dot and the strongly (weakly) coupled macroscopic bodies respectively. }\n\t\t\t\n\t\t\\end{figure}\n\t\t\n\t\\end{center}\nThe above expressions are based on the sense of the phonon heat currents. When the dot is strongly coupled to the hot bath, the input phonon heat current is supplied by the heat bath only. Whereas, when the dot is strongly coupled to a cold bath, the contacts supply phonon heat currents into the dot. Since $\\beta>>\\beta_{H},\\beta_{C}$, a hot bath deteriorates $\\eta$ much more in comparison with the cold bath. A heat bath with an intermediate temperature keeps $\\eta$ in between. Thus, a quantum dot coupled to a cold environment ensures a better thermoelectric performance and merits a greater efficiency. \n\\subsection{Trade-off between different phonon couplings and efficiency optimization }\n \\indent In the earlier section, we have discussed the effect of a strong coupling to a heat bath and established that the efficiency, $\\eta$, is controlled by the bath temperature. We must also note that the degree of phonon coupling between the dot and contacts is an important factor in determining $\\eta_{max}$. In this section we investigate how $\\eta_{max}$ is influenced as a function of $\\beta$, $\\beta_H$ and $\\beta_C$ and present conditions on the optimization of $\\eta_{max}$. The preceding section established that the efficiency can be improved by coupling the dot to a cold environment which drives the phonons out of the dot. Hence from now on we will focus on the situation where the heat bath is a cold one.\\\\\t\n\\indent In Fig.~\\ref{9a} we present the variation of $\\eta_{max}$ as a function of the dot to bath and the dot to contact phonon couplings when $\\beta_H=\\beta_C$. When the dot is coupled strongly to both the contacts and the bath, $\\eta_{max}$ is low. When the dot is weakly connected to both of them, phonons remain in non-equilibrium and hence this results in a high $\\eta_{max}$. These two regions are represented by blank circles. In the regime strong coupling to the contact $\\textbf{SC}$, $T_M$ remains close to the average temperature and the $H$ contact pushes large phonon currents to decrease $\\eta_{max}$. On the other hand, in the regime of strong coupling to the bath, i.e., in the region marked as $\\textbf{SB}$, $T_M$ remains close to the bath temperature and the cold bath extracts heat currents from the dot to increase $\\eta_{max}$. The thermodynamics of phonon heat flow is shown in Fig.~\\ref{9c}. Hence, when the dot is equally coupled to the contacts, strong bath coupling is better than strong contact coupling. \\\\\n\\indent  We now turn our attention to the case when the dot is asymmetrically coupled to both the contacts. In Fig.~\\ref{9b}, we present a plot similar to that in Fig.~\\ref{9a}. We are only interested in the case $\\beta_H<<\\beta_C$, where there is a chance of getting high $\\eta_{max}$. Here we see just the opposite case. The $\\textbf{SB}$ regime gives a similar performance like the earlier case. But $\\eta_{max}$ is maximized in the regime $\\textbf{SC}$ since phonon currents pushed in by the hot contact are smaller. Even in this case, we can ensure $\\eta_{max}$ to be the same as the non-equilibrium case. Hence, this is the region where the efficiency is optimized.\\\\\n\\section{\\label{sec:level4} Conclusion} This paper examined the thermoelectric response of a dissipative quantum dot heat engine based on the Anderson-Holstein model in two relevant operating limits (i) when the dot phonon modes are out of equilibrium, and (ii) when the dot phonon modes are strongly coupled to an external heat bath. In the first case, a detailed analysis of the related physics was elucidated and it was  conclusively demonstrated that an n-type heat engine performs better than a p-type as a result of an interplay between the on-site Coulomb interaction and the coupling to dot phonons. In the second case, with the aid of the dot temperature estimated by incorporating a {\\it{thermometer bath}}, it was shown that the dot temperature deviates from the bath temperature as electron-phonon interaction becomes stronger. Consequently, we showed that the dot temperature intimately controls the direction of phonon heat current thereby influencing the thermoelectric performance. Our simulations highlight two crucial aspects: (a) a cold bath strongly coupled to the dot does not affect the efficiency that much but a hot bath does. (b) When the dot is phonon coupled with contacts, $H$ and $C$ and the cold bath $B$, it is better to couple it strongly to $B$ provided the phonon couplings with $H$ and $C$ are symmetric, whereas it is better to couple it strongly to $C$ if the phonon couplings with $H$ and $C$ are asymmetric. While the current work explored many aspects related to the functioning of a dissipative quantum dot heat engine, we believe some of the latter ideas developed here might merit a separate investigation by examining separately, the aspect of molecular Peltier cooling and refrigeration. \\\\ \\\\\n{\\it{Acknowledgements:}} Financial support from the Center of Excellence in Nanoelectronics (CEN) is acknowledged. We would like to thank Dr. S. D. Mahanti and Dr. R. H\\\"{a}rtle for illuminating discussions. \n\t\n\\bibliographystyle{apsrev}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Acknowledgements}\n\\label{sec:acknowledgements}\nThis work was supported by Shanghai AI Laboratory, \nthe National Natural Science Foundation of China No. 61976206 and No. 61832017, \nBeijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098, \nBeijing Academy of Artificial Intelligence (BAAI), \nthe Fundamental Research Funds for the Central Universities, \nthe Research Funds of Renmin University of China 21XNLG05, \nand Public Computing Cloud, Renmin University of China. \n\n\\section{Appendix}\n\nThe supplementary materials include\nsome additional ablation studies on the different \nnumber of masked joints in CSM module and different $K$ in MATM module. \n\n{\\noindent\\bf Effects of the number of masked joints in CSM.}\nTab.~\\ref{tab:jointnum} illustrates the effects of the number of masked joints on our PSTL.\nOn NTU-60 xsub datasets, PSTL can acquire a good \nperformance (77.3\\%) on a single stream with 9 masked joints,\nso we choose to report these results in our paper.\nNote that, the same number of masked joints may not suitable for all\nthe three streams, and when the number is 11, \nPSTL can acquire a better fusion result (79.6\\%).\n\n{\\noindent\\bf Effects of top-$K$ in MATM.}\nTab.~\\ref{tab:topk} shows the effects of different \nchoice of $K$ on NTU-60 xsub datasets. \nThe number of masked joints is fixed to 9.\nWhen the top 10 keyframes are masked, \nPSTL can achieve the best result (77.3\\%). \n\n{\\noindent\\bf Effects of the dimension of the embeddings.}\nIn our proposed PSTL, the structure of the projector $g$ \nis a 3-layer MLP.\nThe first two layers are followed by a\nbatch normalization layer and ReLU nonlinearity. \nThe output of the encoder is called the 'representations' and the output of the projector is the 'embeddings'.\n\nTo figure out the effects of the embedding dimension, \nwe conduct an ablation study with different dimensions on NTU-60 dataset. \nAs shown in Tab.~\\ref{tab:dim}, \n6144-dim can achieve a better performance than others \non both xsub and xview splits. \n\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n\\toprule\n\\# of masked joints & 5 & 7 & 9 & 11 \\\\ \\midrule\nJ     & 76.7  & 76.0  & \\textbf{77.3}  & 75.4   \\\\\nM     & 74.8  & 75.1  & \\textbf{75.4}  & 75.2   \\\\\nB     & 69.1  & 69.6  & 69.9  & \\textbf{70.5}   \\\\\nJ+M+B & 79.4  & 78.9  & 79.1  & \\textbf{79.6}   \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of the number of masked joints in CSM on NTU-60 xsub dataset.}\n\\label{tab:jointnum}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n\\toprule\n$K$   & 6 & 8 & 10 & 12 \\\\ \\midrule\nJ     & 75.3  & 76.8  & \\textbf{77.3}  & 76.1   \\\\\nM     & 73.9  & 74.5  & \\textbf{75.4}  & 74.9   \\\\\nB     & 68.5  & 68.8  & 69.9  & \\textbf{70.0}   \\\\\nJ+M+B & 77.7  & 78.8  & \\textbf{79.1}  & 78.9   \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of top-$K$ in MATM on NTU-60 xsub dataset.}\n\\label{tab:topk}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cc}\n\\toprule\n\\multirow{2}{*}{dimension} & \\multicolumn{2}{l}{NTU-60(\\%)} \\\\\n                           & xsub  & xview          \\\\ \\midrule\n4096                       &    76.7 &  81.3        \\\\\n6144                       &    \\textbf{77.3} &  \\textbf{81.8}         \\\\\n8196                       &    75.8 &  81.3        \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of the embedding dimension on NTU-60 dataset.}\n\\label{tab:dim}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we introduce a triple stream network named\nPartial Spatio-Temporal skeleton sequences Learning framework\nfor self-supervised skeleton-based action recognition. \nFirstly, all streams apply an ordinary augmentation for producing a variety of skeleton views.\nSecondly, one stream introduces a Central Spatial Masking strategy to force the selected \njoints out of the calculation of GCN backbone. \nThe third  stream utilizes a Motion Attention Temporal Masking module to find \nthe keyframes and mask them. \nAt last, the feature cross-correlation matrix\nis measured to acquire the correlation among each stream. \nExperiments show that such a masking strategy can boost the robustness of \nrepresentations of skeleton joints, \nwhich is essential to downstream tasks. \nour PSTL achieves remarkable performance \nin three large skeleton datasets under various evaluation settings.\n\\section{Experiments}\n\\label{sec:experiment}\n\n\\subsection{Datasets}\n\n\\noindent\\textbf{NTU-RGB+D 60.}\nNTU-60 dataset~\\cite{shahroudy2016ntu} is collected by Microsoft Kinect sensors. \nIt contains $56,578$ skeleton sequences of $60$ action categories,\nperformed by $40$ volunteers.\nThe skeleton of each subject is represented by $25$ joints, \nwhile each video has no more than two subjects. \nThere are two official dataset splits: \n1) Cross-Subject (xsub): half of the subjects belong to the training set, \nand the rest make up the testing sets;\n2) Cross-View (xview): training and testing sets are captured by cameras with different views. \n\n\\noindent\\textbf{NTU-RGB+D 120.}\nNTU-120 dataset~\\cite{liu2019ntu} is the extended version of the NTU-60,\nwhich contains $113,945$ skeleton sequences of $120$ action categories, \nperformed by $106$ volunteers. \nThere are two official dataset splits:\n1) Cross-Subject (xsub): the training set is performed by $53$ subjects,\nand testing data is performed by the rest volunteers;\n2) Cross-Setup (xset): the training and testing sets are captured by cameras with different setups. \n\n\\noindent\\textbf{PKU-MMD.}\nPKU-MMD dataset~\\cite{liu2020benchmark} is captured via \nthe Kinect v2 sensors from multiple viewpoints.\nIt has almost $20000$ action instances in $51$ categories collected by $66$ subjects. \nThere are two parts of the dataset:\n1) Part I contains $21539$ sequences;\n2) Part II contains $6904$ sequences.\nPart II is more challenging due to the large view variation with more skeleton noises.\n\n\\subsection{Implementation Details}\n\nTo perform an apple-to-apple comparison with other methods, \nwe follow the same pre-processing methods of CrossCLR~\\cite{Li_2021_CVPR} and AimCLR~\\cite{guo2022contrastive}, \nwhich resize the skeleton sequences to 50 frames. \nSimilarly, ST-GCN~\\cite{yan2018spatial} with 16 hidden channels is used as the backbone.\nFor all experiments (both representation learning and downstream tasks), we adopt Adam optimizer and\nthe CosineAnnealing scheduler with 150 epochs. \nThe mini batch size is 128. \n\n\\noindent\\textbf{Data Augmentation.}\nBefore applying the joint and temporal masking strategy proposed, \nsome ordinary augmentations are used to generate different views\nof the input skeleton sequences.\nThe ordinary augmentations include 3 spatial augmentations and 1 temporal augmentation. \n\n(1) \\textit{Shear.}\nIt is a linear transformation for skeleton 3D coordinates.\nA shear matrix is\n$$\n\\mathbf{S}=\\left[\\begin{array}{ccc}\n1 & s_{12} & s_{13} \\\\\ns_{21} & 1 & s_{23} \\\\\ns_{31} & s_{32} & 1\n\\end{array}\\right]\n,\n$$ \nwhere $s_{12}, s_{13}, s_{21}, s_{23}, s_{31}, s_{32}$ are the shear factors and are\nrandomly sampled from $[-\\beta \\sim \\beta]$. $ \\beta $ is the amplitude to adjust \nthe augmentation strength. \nHere we set $ \\beta = 1$.  \nBy directly multiplying the shear matrix with the 3D coordinates, the human pose slants\nin a random angle. \n\n(2) \\textit{Crop.}\nAs a temporal augmentation, Crop aims to improve data diversity \nby first padding part of frames to the original sequence and then \nrandomly cropping it to the original length. \nThe padding length is $\\gamma\\, T$, $T$ is the original sequence length, and $\\gamma$ is the padding ratio (we set $\\gamma = 1 \/ 6$). \n\n(3) \\textit{Rotate.}\nThis is an efficient spatial augmentation which \nhelps the representation adapt to the variation of spatial perspective. \nSpecifically, it first randomly selects an axis $M \\in \\{X, Y, Z\\} $ as main axis and then a random angle in $[0 \\sim \\pi\/6]$ is selected for it to rotate. \nNext, the rest two axis both randomly choose the angle in $[0 \\sim \\pi\/180]$.\n\n(4) \\textit{Spatial Flip.}\nIt is another spatial augmentation,\nwhich swaps the left and right sides of the skeleton data with the probability $p=0.5$. \n\n\\subsection{Experiment Setting}\n\n\\noindent\\textbf{Self-supervised Pre-training.}\nWe use ST-GCN~\\cite{yan2018spatial} as the encoder,\nwhich extracts $256$-dim features from skeleton sequences.\nThen we project features to $6144$-dim embeddings. \n$\\lambda$ in the loss of each stream is set to 2e-4. \nA 10-epoch warmup is used for stabilizing the training process. \nThe weight decay is set to 1e-5.\nNote that for all datasets and evaluation settings, \nthe number of mask joints is 9 in CSM and $K=10$ in MATM.\nTo compare with the state-of-the-art, we use multiple skeleton modalities,\nincluding joint (\\textbf{J}), bone (\\textbf{B}), and motion (\\textbf{M}).\nWe evaluate the learned representations with various evaluation protocols.\n\n\\noindent\\textbf{Linear Evaluation.}\nWe add a linear classifier on top of the frozen pre-trained encoder,\nand then train the recognizer on the target skeleton action recognition dataset with initial learning rate $0.01$.\n\n\\noindent\\textbf{Partial Body Evaluation.}\nTo encourage the learned representation to adapt to real-life scenarios \nwith skeleton shaded or partial body information lost,\nwe introduce the Partial Body Evaluation protocol.\nBasically, we follow the settings of Linear Evaluation. \nHowever, some joints or body parts may be masked to simulate the joint\/part missing in extracted skeletons.\nWe consider two settings:\n1) Joint shaded: some joints are masked randomly during evaluation;\n2) Body part shaded: we divide skeletons into 5 major parts\\footnote{left\/right arms, left\/right legs, torso} and randomly drop parts during evaluation.\n\n\n\\noindent\\textbf{Finetuning.}\nWe append a linear classifier on top of the encoder and then finetune the entire network\non the target skeleton action recognition dataset with an initial learning rate $0.005$. \n\n\\noindent\\textbf{Semi-Supervised Learning.}\nThis protocol shares the same setting as ``Finetuning\",\nbut only uses $1\\%$ or $10\\%$ training labeled data that is randomly selected. \n\n\n\\begin{table*}[!ht]\n\\begin{center}\n\\begin{tabular}{l|c|cc|cc|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{1}{l|}{\\multirow{3}{*}{Method}} &\n  \\multicolumn{1}{c|}{\\multirow{3}{*}{Stream}} &\n  \\multicolumn{4}{c|}{NTU-60(\\%)} &\n  \\multicolumn{4}{c|}{PKU(\\%)} &\n  \\multicolumn{4}{c}{NTU-120(\\%)} \\\\ \\cline{3-14} \n\\multicolumn{1}{c|}{} &\n  \\multicolumn{1}{c|}{} &\n  \\multicolumn{2}{c|}{xsub} &\n  \\multicolumn{2}{c|}{xview} &\n  \\multicolumn{2}{c|}{part I} &\n  \\multicolumn{2}{c|}{part II} &\n  \\multicolumn{2}{c|}{xsub} &\n  \\multicolumn{2}{c}{xset} \\\\\n\\multicolumn{1}{c|}{} &\n  \\multicolumn{1}{c|}{} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &  $\\Delta$ \\\\\\midrule\nSkeletonBT    & \\textbf{J} & 68.5 &       & 71.6 &       & 84.7 &       & 38.2 &       & 53.9 &       & 52.7 &  \\\\\n\\textbf{PSTL}          & \\textbf{J} & \\textbf{77.3} & +8.8  & \\textbf{81.8} & +10.2  & \\textbf{88.4} & +3.9  &\\textbf{ 49.3} & +11.1 & \\textbf{66.2} & +12.3 & \\textbf{67.7} & +15.0 \\\\\\midrule\nSkeletonBT    & \\textbf{M} & 58.9 &       & 58.2 &       & 63.2 &       & 30.3 &       & 43.5 &       & 44.7 &   \\\\\n\\textbf{PSTL}          & \\textbf{M} & \\textbf{69.9} & +11.0 & \\textbf{72.6} & +14.4 & \\textbf{80.4} & +17.2 & \\textbf{34.7} & +4.4  & \\textbf{65.4} & +21.9 & \\textbf{57.7} & +13.0 \\\\\\midrule\nSkeletonBT    & \\textbf{B} & 62.4 &       & 64.7 &       & 79.1 &       & 29.8 &       & 52.4 &       & 53.6 &   \\\\\n\\textbf{PSTL }         & \\textbf{B} & \\textbf{75.4} & +13.0 & \\textbf{79.2} & +14.5 & \\textbf{84.8} & +5.7  & \\textbf{41.3} & +11.5 & \\textbf{58.7} & +6.3  & \\textbf{66.6} & +13.0 \\\\\\midrule\n3s-SkeletonBT & \\textbf{J+M+B} & 69.3 &      & 72.2 &       & 84.8 &      & 44.1 &      & 56.3 &       & 56.8     &   \\\\\n\\textbf{3s-PSTL}       & \\textbf{J+M+B} & \\textbf{79.1} & +9.8 & \\textbf{82.6} & +10.4 & \\textbf{89.2} & +4.4 & \\textbf{52.3} & +8.2 & \\textbf{69.2} & +12.9 & \\textbf{70.3} & +13.5  \\\\\\bottomrule\n\\end{tabular}\n\\vspace{-5pt}\n\\caption{Linear evaluation results compared with SkeletonBT on NTU-60, PKU-MMD, and NTU-120 datasets.\n\"$\\Delta$\" represents the gain compared to SkeletonBT using the same stream data. \n\\textbf{J}, \\textbf{M} and \\textbf{B} indicate joint stream, motion stream, and bone stream.}\n\\label{tab:compBT}\n\\vspace{-8pt}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.9\\linewidth}{!}{\n\\begin{tabular}{ccc|cc}\n\\toprule\n\\multirow{2}{*}{SkeletonBT} & \\multirow{2}{*}{w\/ CSM} & \\multirow{2}{*}{w\/ MATM} & \\multicolumn{2}{l}{NTU-60(\\%)} \\\\\n  &   &   & xsub & xview \\\\ \\midrule\n\\checkmark &            &            & 68.5 & 71.6  \\\\\n\\checkmark & \\checkmark &            & 71.8 & 75.4  \\\\\n\\checkmark &            & \\checkmark & 73.7 & 78.0  \\\\\n\\checkmark & \\checkmark & \\checkmark & \\textbf{77.3} & \\textbf{81.8}  \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Ablation studies of PSTL on NTU-60 dataset.}\n\\label{tab:overall}\n\\vspace{-9pt}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Ablation Studies of PSTL}\nWe ablate the proposed CSM and MATM strategies on our baseline SkeletonBT respectively to \nshow their effectiveness. \nThe linear evaluation results are shown in Tab.~\\ref{tab:overall}, \nSkeletonBT achieves 68.5\\% and 71.6\\% accuracy on xsub and xview.\nWe find that the CSM module improves about 3.3\\%, 3.8\\% Top-1 accuracies\nand the MATM module brings about 5.2\\%, 6.4\\% performance gain on\nxsub and xview, respectively. \nWith the help of both modules, \nour PSTL achieves 8.8\\% and 9.4\\% increments. \nThese results show that our proposed CSM and MATM strategies can \nencourage the encoder to acquire stronger features\nwhich are more suitable for downstream tasks.\n\n\n\\noindent\\textbf{Comparisons with SkeletonBT.}\nTo further verify the effectiveness of our proposed method, \nwe conduct experiments on three datasets to \ncompare our PSTL with the baseline method SkeletonBT in detail. \nAs shown in Tab.~\\ref{tab:compBT}, PSTL performs much better than SkeletonBT \non all three different streams. \nFor the fusion results, 3s-PSTL substantially improves\nSkeletonBT, especially for the NTU-120 datasets. \n\n\\noindent\\textbf{Comparisons with Random Masking Strategy.}\nTo demonstrate the effectiveness of our masking strategy, \nwe compare our CSM and MATM modules with random spatial masking (RSM) strategy \nand random temporal masking (RTM) strategy, which selects the mask position randomly. \nThe linear evaluation results are shown in Tab.~\\ref{tab:random}, \nwith the help of our triple stream network, \nthe random masking can capture the relationship between \njoints and frames and get a good results on downstream tasks. \nAnd then, our CSM and MATM can further strengthen this capacity \nto get a better accuracy.\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.56\\linewidth}{!}{\n\\begin{tabular}{l|cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{NTU-60(\\%)} \\\\\n                        & xsub           & xview          \\\\ \\midrule\nRSM + RTM                 & 75.6           & 79.7           \\\\\nRSM + MATM                & 76.4           & 81.2               \\\\\nCSM + RTM                 & 76.1           & 80.4                \\\\\nCSM + MATM                & \\textbf{77.3} & \\textbf{81.8}          \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Compared with random masking strategy on NTU-60 dataset. \nRSM means random spatial masking and RTM means random temporal masking.}\n\\label{tab:random}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Comparison with State-of-the-art}\n\n\\noindent\\textbf{Partial Body Evaluation.}\nTo verify the \\emph{joint robustness} of extracted features,\nwe conduct the partial body evaluation on NTU-60 xsub datasets.\nThe first level of the evaluation protocols is \n\\textbf{joint shaded}, \nwhich applies random masks on joints to simulate the situation of \njoints shade in realistic scenarios.\nTab.~\\ref{tab:jointshad} shows that, without CSM and MATM, the baseline  SkeletonBT suffers\nserious performance loss. \nWith the number of mask joints increasing to 10, \nthe SOTA method AimCLR~\\cite{guo2022contrastive} \nand its baseline methods SkeletonCLR~\\cite{Li_2021_CVPR} also \nsuffer about 20.0\\% and 17.6\\% performance decrease. \nNote that the feature extracted by our PSTL acquires strong joint robustness,\nand only 3.5\\% performance decrease under 10 masked joints. \n\nThe second level evaluation protocols \\textbf{body part shaded}, as shown in Tab.~\\ref{tab:partshad}, \nis much harder since we mask the total parts of the body.\nBut our PSTL still achieves better results than other methods.\nTo sum up, more skeleton sequence information is lost, \nand the advantage of our method becomes more obvious. \nAnd the results tell that with the help of\nlocal relationship between joints and frames, our CSM and MATM strategy \ncan improve the robustness of learned features, \nwhich is essential to downstream tasks.\n\n\\begin{table}[!t]\n\\begin{center}\n\t\t\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|cccccc}\n\\toprule\n\\# of masked joints      & 0 & 2 & 4 & 6 & 8 & 10 \\\\ \\midrule\nSkeletonCLR  & 68.3 & 65.6  & 61.9  & 58.6  & 54.5  & 50.7   \\\\\nSkeletonBT     & 68.5 & 65.0  & 61.7  & 58.6  & 55.0  & 50.6   \\\\\nAimCLR       & 74.3 & 70.4  & 67.2  & 63.0  & 58.5  & 54.3  \\\\\n\\textbf{PSTL(ours)}  & \\textbf{77.3} & \\textbf{76.9}  & \\textbf{76.2} & \\textbf{75.7}  & \\textbf{74.6}  & \\textbf{73.8}  \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results of \\textbf{joints shaded} on NTU-60 xsub dataset.}\n\\label{tab:jointshad}\n\\vspace{-5pt}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|ccccc}\n\\hline\n\\# of masked body parts       & 0 & 1 & 2 & 3 & 4 \\\\ \\hline\nSkeletonCLR  & 68.3 & 63.6 & 57.7  & 47.5  & 33.5 \\\\\nSkeletonBT   & 68.5 & 64.0 & 58.5 & 50.2 & 38.4  \\\\\nAimCLR       & 74.3 & 69.3 & 63.9 & 53.7 & 37.9 \\\\\n\\textbf{PSTL (ours)}  & \\textbf{77.3} & \\textbf{73.6} & \\textbf{68.3} & \\textbf{60.0} & \\textbf{47.3} \\\\ \\hline\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results of \\textbf{body part shaded} on NTU-60 xsub dataset.}\n\\label{tab:partshad}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\noindent\\textbf{Linear Evaluation.}\nAs shown in Tab.~\\ref{tab:ntu60&120lp}, for a single stream, \nPSTL achieves considerable gains on all other methods on NTU60 and NTU120.\nCompared with the current SOTA method AimCLR, \nour PSTL improves \n3.0\\% and 2.1\\% on Xsub and Xview respectively. \nThe results indicate that the local relationship between joints and frames can \nhelp extract suitable features for downstream tasks. \nFor the performance of the 3-streams, PSTL still \nachieves comparable results to current methods. \nOur method also achieves the best results on NTU-120 datasets. \nFor single stream, PSTL outperforms the current SOTA method AimCLR \n2.8\\% and 4.3\\% on Xsub and Xset respectively. \nAnd the fusion can also achieve 1.0\\% and 1.5\\% increment. \n\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|cc|cc}\n\\toprule\n\\multirow{2}{*}{Method}          & \\multicolumn{2}{c|}{NTU-60(\\%)} & \\multicolumn{2}{c}{NTU-120(\\%)} \\\\\n                                 & xsub          & xview          & xsub           & xset           \\\\\\midrule\n\\textit{Single-stream:}         &               &               &                &                \\\\\nLongT GAN (AAAI 18)              & 39.1          & 48.1         &  -             &  -              \\\\\nMS2L (ACM MM 20)                 & 52.6         & -             &  -             &  -              \\\\\nAS-CAL (Inf Sci 21)             &  58.5         & 64.8          &  48.6          & 49.2           \\\\\nP\\&C (CVPR 20)                   & 50.7         & 76.3          &  42.7          & 41.7             \\\\\nSeBiReNet (ECCV 20)              &  -            & 79.7         &   -            & -               \\\\\nAimCLR (AAAI 22)                 & 74.3         & 79.7          &  63.4          & 63.4           \\\\\n\\textbf{PSTL (ours)}            & \\textbf{77.3} & \\textbf{81.8} & \\textbf{66.2}& \\textbf{67.7} \\\\\\midrule\n\\textit{Three-stream:}          &               &               &                &                \\\\\n3s-SkeletonBT (ours)            &   69.3        & 72.2          &  56.3          & 56.8          \\\\\n3s-SkeletonCLR (CVPR 21)        &    75.0        & 79.8          &  60.7          &  62.6             \\\\\n3s-Colorization (ICCV 21)       &   75.2        & 83.1          &  -             &   -             \\\\\n3s-CrosSCLR (CVPR 21)           &  77.8        & 83.4          &  67.9          &  67.1              \\\\\n3s-AimCLR (AAAI 22)             &   78.9        & 83.8          &  68.2          &  68.8               \\\\\n\\textbf{3s-PSTL (ours) }        &   \\textbf{79.1}        &  \\textbf{83.8}        & \\textbf{69.2}         &  \\textbf{70.3} \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results on NTU-60 and NTU-120 datasets.}\n\\label{tab:ntu60&120lp}\n\\vspace{-10pt}\n\\end{center}\n\\end{table}\n\n\n\nNoting that the PKU-MMD dataset has about half the data size of NTU-60,\nTab.~\\ref{tab:pkulp} shows that the advantage of our method is more obvious in a small dataset. \nIt is worth mentioning that our baseline method SkeletonBT can achieve better \nresults than AimCLR on the single stream, \nwhich indicates that our negative-sample-free methods without a memory bank\nis more efficient for small datasets.\nSimilarly, our PSTL achieves remarkable results on PKU-MMD datasets.\nFor single stream, PSTL outperforms AimCLR 5.0\\% and 12.5\\% on Part I and Part II.\nFor 3 stream fusion, our 3s-PSTL outperforms 3s-AimCLR 1.4\\% and 13.8\\% on both parts. \n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.74\\linewidth}{!}{\n\\begin{tabular}{l|c|c}\n\\toprule\nMethod                & Part I & Part II \\\\\\midrule\n\n\\textit{Single-stream:}         &        &         \\\\\nLongT GAN (AAAI 18)         &  67.7      & 26.0        \\\\\nMS2L (ACM MM 20)             & 64.9      & 27.6        \\\\\nSkeletonCLR (CVPR 21)       &  80.9      & 35.2        \\\\\nISC (ACM MM 20)             &  80.9     &  36.0        \\\\\nAimCLR (AAAI 22)                & 83.4  & 36.8          \\\\\nSkeletonBT (ours)               & 84.7  & 38.2         \\\\\n\\textbf{PSTL(ours) }            & \\textbf{88.4}  & \\textbf{49.3}    \\\\\\midrule\n\\textit{Three-stream:}          &        &         \\\\\n3s-CrosSCLR (CVPR 21)           & 84.9   & 21.2    \\\\\n3s-AimCLR (AAAI 22)             & 87.8   & 38.5   \\\\\n3s-SkeletonBT (ours)            & 84.8  & 44.1    \\\\\n\\textbf{3s-PSTL (ours)}         & \\textbf{89.2}   & \\textbf{52.3}   \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results on PKU-MMD dataset.}\n\\label{tab:pkulp}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\noindent\\textbf{Finetune Evaluation.}\nWe compare PSTL with other self-supervised methods and\nthe supervised method ST-GCN using the same structure and parameters as ours. \nAs shown in Tab.~\\ref{tab:ntu60&120ft}, \nfor the single steam, \nall methods use bone steam since \nthe results are better than the joint stream. \nOur PSTL achieves better performance than others on \nNTU-60 and NTU-120 datasets.\nNotably, only single steam PSTL outperforms supervised 3s-ST-GCN on NTU-120 datasets,\nwhich indicates the effectiveness of our method. \nSimilarly, 3s-PSTL outperforms 3s-CrossCLR and 3s-AimCLR \nfor the fusion results on both datasets.  \n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|cc|cc}\n\\toprule\n\\multirow{2}{*}{Method}          & \\multicolumn{2}{c|}{NTU-60(\\%)} & \\multicolumn{2}{c}{NTU-120(\\%)} \\\\\n                                 & xsub          & xview          & xsub           & xset           \\\\\\midrule\n\\textit{Single-stream:}         &               &                &                &                \\\\\nSkeletonCLR (CVPR 21)           &  82.2         & 88.9          & 73.6              & 75.3            \\\\\nAimCLR (AAAI 22)                 & 83.0         & 89.2         &  77.2           & 76.0               \\\\\n\\textbf{PSTL (ours)}            &  \\textbf{84.5}& \\textbf{92.0 }& \\textbf{78.6}   &  \\textbf{78.9}     \\\\\\midrule\n\\textit{Three-stream:}          &               &                &                &                \\\\\n3s-ST-GCN (AAAI 18) ${}^{\\S}$   &   85.2        & 91.4          &  77.2         & 77.1              \\\\\n3s-CrosSCLR (CVPR 21)            &  86.2        & 92.5          & 80.5          & 80.4               \\\\\n3s-AimCLR (AAAI 22)              &  86.9        & 92.8          & 80.1          & 80.9               \\\\\n\\textbf{3s-PSTL (ours) }         &  \\textbf{87.1} &\\textbf{93.9} & \\textbf{81.3} & \\textbf{82.6}   \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Finetune evaluation results on NTU-60 and NTU-120 datasets.\n\"\u00a7\" means the model is trained\nwith a fully-supervised manner.}\n\\label{tab:ntu60&120ft}\n\\vspace{-5pt}\n\\end{center}\n\\end{table}\n\n\\noindent\\textbf{Semi-Supervised Evaluation.}\nTo show the superiority of our approach,\nwe test all methods under the semi-supervised setting, where only 1\\% or 10\\% labeled data can be used such that models are more likely to overfit.\nAs shown in Tab.~\\ref{tab:semi}, our PSTL far exceeds the current SOTA method AimCLR on PKU-MMD datasets.\nAnd for fusion results, 3s-PSTL outperforms 3s-AimCLR 8.6\\% on Part II. \nIt demonstrates that such contrastive methods which need a large memory bank \nare subjected to serious performance loss with insufficient samples. \nBut our negative-sample-free PSTL can well alleviate this problem and \nachieve better results. \n\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.75\\linewidth}{!}{\n\\begin{tabular}{l|c|c}\n\\toprule\nMethod                & Part I & Part II \\\\\\midrule\n\\textit{1\\% labeled data:}     &        &         \\\\\nLongT GAN (AAAI 18)   & 35.8   & 12.4     \\\\\nMS2L (ACM MM 20)      & 36.4  & 13.0        \\\\\nISC (ACM MM 21)       & 37.7  & -        \\\\\n3s-CrosSCLR (CVPR 21) & 49.7  & 10.2        \\\\\n3s-AimCLR (AAAI 22)   & 57.5  & 15.1        \\\\\n\\textbf{3s-PSTL(ours)} & \\textbf{62.5}  & \\textbf{16.9} \\\\\\midrule\n\\textit{10\\% labeled data:}    &        &         \\\\\nLongT GAN (AAAI 18)   &  69.5  & 25.7       \\\\\nMS2L (ACM MM 20)      & 70.3  & 26.1        \\\\\nISC (ACM MM 21)       & 72.1  & -        \\\\\n3s-CrosSCLR (CVPR 21) & 82.9  & 28.6    \\\\\n3s-AimCLR (AAAI 22)   & 86.1  & 33.4        \\\\\n\\textbf{3s-PSTL (ours)} & \\textbf{86.9} & \\textbf{42.0}      \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Semi-supervised evaluation results on PKU-MMD dataset.}\n\\label{tab:semi}\n\\vspace{-7pt}\n\\end{center}\n\\end{table}\n\\section{Introduction}\n\\label{sec:introduction}\n\n\n\nIn recent years, human action recognition has been extensively studied for various applications in the real world, \nsuch as security, human-robot interaction, and virtual reality.\nWith the development of sensor technology~\\cite{zhang2012microsoft,shotton2011real} and human pose estimation methods~\\cite{Cao_2017_CVPR,hua2022weakly}, \nskeleton data becomes easier to obtain.  \nIts robustness against the variation of appearance and backgrounds \nhas attracted many researchers to explore fully supervised methods \nfor action recognition, \nsuch as applying CNN~\\cite{ke2017a}, \nRNN~\\cite{du2015hierarchical} or GCN~\\cite{yan2018spatial,shi2019two} on skeleton data. \n\n\nHowever, fully supervised learning relies on abundant manually annotated labels, \nwhich is time-consuming and expensive. \nTherefore, self-supervised frameworks, \\eg, MoCo~\\cite{he2020momentum}, \nSimCLR~\\cite{chen2020simple}, Barlow Twins~\\cite{zbontar2021barlow}, \nand MAE~\\cite{he2022masked} have received a lot of attention. \nFor self-supervised learning on skeleton data, \nearly methods also apply pretext tasks~\\cite{lin2020ms2l,su2020predict,zheng2018unsupervised}.\nWhile nowadays, most researchers focus on contrastive learning and apply MoCo structure \nto the skeleton squences~\\cite{thoker2021skeleton,Li_2021_CVPR,rao2021augmented,guo2022contrastive}.\n\n\n\n\nThe key of current contrastive learning-based methods is finding effective global data augmentation and using them to create various views of the skeleton carefully. \nHowever, due to the rich action clues stored in the skeleton sequence, this paradigm can readily discriminate different views from a global perspective without thoroughly leveraging the local relationship between different skeleton joints and video frames. \nAs a result, it limits the performance and robustness of the model in real-world scenarios. Moreover, a large batch size or a memory bank is necessary for contrastive learning, which is unsuitable for a small amount of skeleton data. \nTo evaluate the robustness of the current methods in real-world scenarios (such as video surveillance), \nwe formulate a new evaluating setting where only a partial skeleton is available for downstream tasks to simulate that the skeleton data captured by the camera are shaded. \nIn this challenging setting, previous works suffer severe performance drops. \nWe conjecture that if the local spatial and temporal relationship can be explored and captured, it will boost the representation ability and robustness of the model on downstream tasks.\n\n\n\n\nTherefore, we propose a Partial Spatio-Temporal Learning (PSTL) framework to exploit the local relationship from a partial skeleton sequence built by a unique spatio-temporal masking strategy. \nSpecifically, we construct a negative-sample-free triplet steam structure that is composed of an anchor stream, a spatial masking stream with Central Spatial Masking (CSM), and a temporal masking stream with Motion Attention Temporal Masking (MATM). \n\nIn each stream, an ordinary augmentation is first applied to improve the diversity of the input samples. \n(1) In the spatial masking stream, Central Spatial Masking discards selected joints from the feature calculation process. Notably, it is not simply set the selected joint position to zeros, which is unreasonable for skeleton data. \nConsidering that the topology of the human skeleton is a predefined graph, the joints with a higher degree of centrality get stronger connectivity. By filtering out such joints with higher probabilities, the encoder can focus more on these skeletons that are less explored. \n(2) In the temporal masking stream, considering the frames that change quickly always contain more semantic information about the actions, \nwe propose a Motion Attention Temporal Masking strategy, where\nframes moves quickly are more likely to be masked.\n(3) In the anchor stream, the masking policy is not introduced to keep the original semantic information. A shared encoder extracts \nthe triple stream features simultaneously.\nInspired by the loss of Barlow Twins~\\cite{zbontar2021barlow},\nwe adopt the cross-correlation matrix to \ncapture the correlation between the anchor stream and the other two masking streams, respectively.\n\nAs a negative-sample-free self-supervised method, \nPSTL encourages the empirical cross-correlation matrix \nbetween learned representations to be an identity matrix, \nand minimizes the redundancy within the \nrepresentation vector to alleviate feature collapse.\nIn this way, the spatial and temporal relationship can be well captured \nand the joint robustness can be boosted for downstream tasks. \nExtensive results on three large benchmark\ndemonstrate the effectiveness of our proposed masking strategy. \nOur contributions:\n\n\\vspace{2pt}\n\\begin{itemize}\n\n\\item We propose Partial Spatio-Temporal skeleton representation Learning (PSTL), a negative-sample-free self-supervised framework for skeleton-based action representation.\n\n\\item We are the first to introduce the masking strategy \nto skeleton-based self-supervised learning and \nusing Central Spatial Masking (CSM) strategy during pre-training to \nimprove the joint robustness of learned representation for downstream tasks.\n\n\\item A unique Motion Attention Temporal Masking strategy (MATM) \nis proposed to use the motion of skeleton data to guild \ntemporal masking modules to find and mask key frames of the sequences. \n\n\\item Extensive experiment results on NTU-60, NTU120, and PKU-MMD datasets \nshow the proposed PSTL achieves the-state-of-art performance under diverse downstream tasks,\nsuch as linear evaluation, finetune evaluation, and semi-supervised evaluation protocol.\n\n\\end{itemize}\n\\section{Method}\n\\label{sec:method}\n\nAs shown in Fig~\\ref{fig:PSTL},\nPartial Spatio-Temporal skeleton representation Learning (PSTL) \ntakes a triple stream architecture, which is based on our Skeleton Barlow Twins (SkeletonBT),\nand applies Central Spatial Masking (CSM) on its spatial masking stream and Motion Attention Temporal Masking (MATM) on its temporal masking stream together with an extra anchor stream to keep the original semantic information. \nSpecifically,\n1) \\textbf{SkeletonBT} is a basic negative-sample-free double stream network\nwith ordinary augmentation for skeleton self-supervised learning. \n2) \\textbf{CSM} first adjusts the computing mode of the GCN backbone\nto ensure the information of masked joints out of the feature calculation process. \nThen, the joints with higher degree centrality are assigned higher probability to be masked. \n3) \\textbf{MATM} calculates the corresponding motion value for each frame of the action, \nwhich serves as the attention weight to select the masked frames. \n\n\n\\subsection{Skeleton Barlow Twins}\n\\label{subsec:Bt}\n\nInspired by the recent advance on negative-sample-free self-supervised methods~\\cite{zbontar2021barlow} that avoid negative sample \nand do not need a large batch size\/memory bank, \nwe develop Skeleton Barlow Twins (SkeletonBT) to learn the skeleton-based action \nrepresentation.\nIt is a symmetric network, \nwhere a shared encoder is used to generate features.\n\nSpecifically, a 3D human skeleton sequence can be denoted as\n$s \\in \\mathbb{R}^{C \\times T \\times V}$ ,\nwhich has $T$ frames and $V$ joints. \n$C$ is the channel dimension representing the 3D position. \nAs shown in Fig.~\\ref{fig:SkeletonBT},\nthe input skeleton $s$ is first applied with an \nordinary augmentation $\\mathcal{T}$\nto get diverse views $x$ and $x^{'}$. \nThe encoder $f$ is used to extract features: \n$h = f(x;\\theta)$ and $h^{'} = f(x^{'};\\theta)$, \nwhere $h, h^{'} \\in \\mathbb{R}^{c_{h}}$. \nThen a projector $g$ maps each feature \nto the higher dimension space and acquire the embeddings: \n$z = g(h)$ and $z^{'} = g(h^{'})$, \nwhere $z, z^{'} \\in \\mathbb{R}^{c_{z}}$. \nAt last, by encouraging the empirical cross-correlation matrix \nbetween embeddings of the distorted variations be an identity matrix,\nthe encoder can capture the relationship between two streams. \nThe loss can be formulated as: \n\\begin{equation}\n\\mathcal{L} = \\sum_{i}\\left(1-\\mathcal{C}_{i i}\\right)^{2}+\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{C}_{i j}^{2}.\n\\label{eq:BT}\n\\end{equation}\nHere $\\mathcal{C}$ is the cross-correlation matrix \ncomputed between embeddings $z$ and $z^{'}$ \nalong the batch dimension $b$: \n\n\\begin{equation}\n\\label{eq:cc}\n\\mathcal{C}_{i j} = \\frac{\\sum_{b} z_{b, i} z_{b, j}^{'}}{\\sqrt{\\sum_{b}\\left(z_{b, i}\\right)^{2}} \\sqrt{\\sum_{b}\\left(z_{b, j}^{'}\\right)^{2}}},\n\\end{equation}\nwhere $i,j$ represent the embedding dimension. \n\nSpecifically, in Eq.~\\eqref{eq:BT} the first term encourages the diagonal elements of $\\mathcal{C}$ to 1,\nwhich makes the embedding invariant to the variety augmentation applied. \nThe second term equates the off-diagonal elements of $\\mathcal{C}$ to 0, \nhelping the different embedding components to be decoupled with each other, \nwhich can minimize the redundancy within the representation.\n$\\lambda$ serves as a trade-off parameter to balance two terms.\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{AAAI 2023 skeleton self_supervised\/SkeletonBT.pdf}  \n  \\vspace{-5pt}\n  \\caption{The framework of Skeleton Barlow Twins.}\n  \\label{fig:SkeletonBT}\n  \\vspace{-5pt}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{AAAI 2023 skeleton self_supervised\/PSTL.pdf} \n\\vspace{-6pt}\n\\caption{The overall framework of Partial Spatio-Temporal skeleton representation Learning. \nIn Central Spatial Masking \\textbf{(CSM)},  the degree is red (4), blue (3), dark grey (2) and light grey (1).\nIn Motion Attention Temporal Mask \\textbf{(MATM)}, \n$m$ denotes motion density while $t$ is the time code.\n}\n\\label{fig:PSTL}\n\\end{center}\n\\vspace{-10pt}\n\\end{figure*}\n\n\n\n\n\\subsection{Central Spatial Masking}\n\\label{subsec:CSM}\nTo boost the joint robustness of learned representation,\nwe propose a Central Spatial Masking (CSM) module that \nencourages partial skeleton data to generate similar features as the whole skeleton data. \nThe relationship between masked and unmasked joints can be captured by the encoder, \nwhich is essential for the hard downstream task.\n\nAs discussed above, it is critical to construct partial skeleton data for the encoder. \nEarly methods~\\cite{rao2021augmented} try to directly set the values of masked joints to zero.\nBut it is unreasonable for skeleton data since\nthe joint semantic information is the 3D position that is changed. \nFor CSM, we first adjust the computing mode of the backbone ST-GCN~\\cite{yan2018spatial}.\nSpecifically, for a skeleton sequence $s$, the masked joints are abandoned and then\ndrop the corresponding rows and columns in the adjacency matrix, \nforcing masked joints out of the calculation process and the learned feature \nis only generated by the unmasked ones. \n\nNext, we take the advantage of the degree centrality \nof the human skeleton graph topology, \nand observe that the joints with more degrees can \nacquire richer neighborhood information. \nThus, we assign higher probabilities to mask \nsuch connective joints which help the encoder capture the relationship between \na wider range.\nAs shown in the green box of Fig~\\ref{fig:PSTL}, the human skeleton joint is $V_i, i \\in (1,2,...,n)$,\nwhere $n$ is the total number of joints. \nNote that there are four types of joints.\nThe degree of light gray joints located at the margin of the graph is 1. \nThe degree of dark gray joints is 2, which are more connective \nand their number is the majority in the total graph. \nThe blue one and red one hold degree of 3 and 4, \nwhich are the center of the upper half and lower half of the body. \nMasking such center joints can help the encoder to\ncapture the relationship between a large range of joint information.\nTo this end, our CSM strategy first calculates the degree $d_i$ of each joint $V_i$, \nand sets their masked probability:\n\\begin{equation}\np_i = \\frac{d_i}{\\sum_{j=1}^{n} d_j}.\n\\end{equation}\n\n\n\n\n\\subsection{Motion Attention Temporal Masking}\n\\label{subsec:MATM}\nCompared with the spatial joints, the temporal information is more redundant and\nthe semantic of action is more concentrated. \nSome key-frames play an pivotal role in representing the semantics of an action. \nTo utilize this feature, \nour MATM aims to find such key-frames and mask them to \ngenerate a hard pair for anchor stream and temporal masking stream in our proposed PSTL. \nBy encouraging the empirical cross-correlation matrix between embeddings of two stream\nbe an identity matrix, the encoder can capture the relationship between \nredundant frames and key-frames effectively. \n\nSpecifically, the motion $m \\in \\mathbb{R}^{C \\times T \\times V}$ of the sequence $s$\nis computed by the temporal displacement between frames:\n$m_{:, t, :} = x_{:, t+1,:}-x_{:, t, :}$. \nThen we acquire the overall motion rate of a frame which serves as the attention weight:\n\\begin{equation}\na_t = \\frac{m_t^2} {\\sum_{i=1}^{T} {m_i^2}}.\n\\end{equation}\nThen, the top-$K$ attention weights $a_{t_1},...,a_{t_K}$ are selected \nand the sequence $x_{t_1},...,x_{t_K}$ serves as the key-frames which\ncontain more semantic information about the actions. \nAs shown in the blue box of Fig~\\ref{fig:PSTL}, the bottom one is a skeleton sequence of \\textit{put on glasses}, \nthe top one shows the motion density. \nWe can find that some frames in the middle of the sequence acquire higher motion values, \nwhich are the key-frames for this action. \nThus, we mask these key-frames and encourage the feature from masked sequence close to \nthe anchor feature which contains total semantic information. \n\n\\subsection{Loss Function}\n\\label{subsec:PSTL}\nRecall the pipeline of the proposed triple stream PSTL shown in Fig.~\\ref{fig:PSTL}.\nGiven a skeleton sequence $s$, an ordinary augmentation is applied \nto obtain different views $x,x^{'},\\hat{x}$. \nFor sequence $x^{'}$, a Central Spatial Masking strategy is used to \ngenerate partial skeleton data $x_c^{'}$. \nConsidering the degree centrality of the skeleton graph, \nthe joints with more degrees are assigned higher masked probability in this module.   \nFor sequence $\\hat{x}$, a Motion Attention Temporal Masking strategy is applied.\nWe first compute the motion value of each frame which represents the key \nfragment in a skeleton action. \nThen the top-$K$ frames are selected to be masked \nand to further improve diversity,\n$K$ random frames are still masked from remain sequence to \ngenerate $\\hat{x}_m$. \nThe sequence $x$ in the anchor stream does not introduce any mask\nstrategy to keep the original semantic information. \n\nAfter that, the shared encoder $f$ is used to extract the features simultaneously.\n$h = f(x;\\theta), h_c^{'} = f(x^{'};\\theta), \\hat{h}_m = f(\\hat{x}_m;\\theta)$,\nwhere $\\theta$ is the parameters learned by the encoder\nwhich is also used in downstream tasks. \nThen a projector $g$ is applied and the features are projected to\nthe embeddings $z, z_c^{'}, \\hat{z}_m$ in a high dimension space. \n\\ie $z = g(h), z_c^{'} = g(h_c^{'}), \\hat{z}_m = g(\\hat{h}_m)$. \nTo capture the relationship between masked joints and unmasked ones, \nwe compute the cross-correlation matrix $\\mathcal{C^{'}}$ between embeddings $z$ and $z_c^{'}$. \nThe calculation process refers to Eq.~\\eqref{eq:cc}.\nAnd then $\\mathcal{L}_1$ is applied:\n\n\\begin{equation}\n\\label{eq:BT1}\n\\mathcal{L}_1 = \\sum_{i}\\left(1-\\mathcal{C^{'}}_{i i}\\right)^{2}+\n\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{C^{'}}_{i j}^{2},\n\\end{equation}\nwhere the first term encourages the diagonal element of $\\mathcal{C^{'}}$ to 1.\nIt forces the representation of partial data to be similar to the total one. \nThe second term is designed to prompt the different embedding components to be decoupled with each other,\nwhich helps to minimize the redundancy within the representation and avoid it being a constant. \n\nSimilarly, the cross-correlation matrix $\\mathcal{\\widehat{C}}$ between $z$ and $\\hat{z}_m$\nis acquired for the $\\mathcal{L}_2$:\n\\begin{equation}\n\\label{eq:BT2}\n\\mathcal{L}_2 = \\sum_{i}\\left(1-\\mathcal{\\widehat{C}}_{i i}\\right)^{2}+\n\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{\\widehat{C}}_{i j}^{2},\n\\end{equation}\nwhich is utilized to capture the relationship between masked and unmasked frames.\nNote that the trade-off parameter $\\lambda$ is used to balance the dimension difference \nbetween first and second terms, so it keep same on Eq.~\\eqref{eq:BT1} and Eq.~\\eqref{eq:BT2}. \n\nThe total loss $\\mathcal{L}_p$ for our PSTL can be formulated as:\n\n\\begin{equation}\n\\mathcal{L}_p = \\mathcal{L}_1 + \\mathcal{L}_2.\n\\end{equation}\n\\section{Related Work}\n\\label{sec:related}\n\n{\\noindent\\bf Skeleton-based action recognition.}\nTo tackle skeleton-based action recognition,\nearly methods mainly focus on hand-crafted features~\\cite{vemulapalli2014human,xia2012view}.\nWith the development of the deep neural network, \nskeleton data is processed as pseudo-images with CNNs~\\cite{ke2017a,kim2017interpretable}.\nOthers structure it as a long-term sequence\nand feed it into RNNs~\\cite{du2015hierarchical,zhang2017view}.\nLater on, ST-GCN~\\cite{yan2018spatial} structures the skeleton data as a predefined spatial graph and \nutilize GCN to integrate the skeleton joint information. \nAfter that, many GCN-based methods are proposed, \nwhich contain more data stream or add attention mechanisms~\\cite{shi2019two,song2020stronger,chen2021multi}. \nIn this paper, we also utilize the widely used ST-GCN as the backbone. \n\n\n{\\noindent\\bf Self-supervised representation learning.}\nSelf-supervised learning aims to acquire a high-level representation from a large number of unlabeled samples. \nEarly methods rely on pretext tasks to generate supervision, \nsuch as jigsaw puzzles~\\cite{noroozi2016unsupervised,wei2019iterative}, \ncolorizing~\\cite{larsson2016learning, zhang2016colorful}, \nor rotation prediction~\\cite{gidaris2018unsupervised}. %\nThen, discriminative methods like MoCo~\\cite{he2020momentum} and SimCLR~\\cite{chen2020simple} achieve \nremarkable performance, \nwhich are comparable to supervised learning. \nBy distinguishing the positive pairs from the negative ones, \nthe backbone network can acquire the representation with rich discriminant information. \nBut it requires a large batch size or memory bank to store the negative samples. \nTo tackle this issue, negative-sample-free methods are introduced \nsuch as BYOL~\\cite{grill2020bootstrap}, SimSiam~\\cite{chen2021exploring}, and Barlow Twins~\\cite{zbontar2021barlow}. \nThey do not require explicit construction of negative sample pairs,\nbut most of them utilize an asymmetric network to avoid feature collapse.\nSpecially, \nBarlow twins avoids the reliance on the symmetry-breaking network \nbut minimizes the redundancy within the representation vector to alleviate collapse. \nMore recently, mask auto-encoder~\\cite{he2022masked} is introduced to learn effective visual representations\nfrom local complementary information.\n\n{\\noindent\\bf Self-supervised skeleton-based action recognition.}\nTo apply self-supervised learning to skeleton data,\nLongT GAN~\\cite{zheng2018unsupervised} uses a recurrent encoder-decoder GAN to reconstruct the input sequence.\nBased on this, Predict\\&Cluster~\\cite{su2020predict} proposes\na decoder to improve the representation ability of the encoder. \nMS2L~\\cite{lin2020ms2l} proposes a multi-task self-supervised learning framework,\nincluding motion prediction and jigsaw puzzle. \nAS-CAL~\\cite{rao2021augmented} uses momentum LSTM to regularize the feature space together with several skeleton augmentation strategy.\nISC~\\cite{thoker2021skeleton} proposes inter-skeleton contrastive method that combines sequence-based and graph-based representation.\nCrossCLR~\\cite{Li_2021_CVPR} introduces\nSkeletonCLR that applies memory bank to store the negative samples and \na cross-view knowledge mining strategy to capture more comprehensive representation. \nAimCLR~\\cite{guo2022contrastive} aims to utilize plenty of augmentation to introduce movement patterns and force \nthe encoder to learn more general representations. \nHowever, these contrastive-based methods heavily rely on strong data augmentation strategies, \nbut the redundancy within the spatial joints and temporal frames have not been considered, \nwhich can improve the robustness of 3D visual representation and essential for downstream task.\nHence, we introduce PSTL and propose a spatial-temporal mask strategy to find a more general representation. \n\n\n\\section{Acknowledgements}\n\\label{sec:acknowledgements}\nThis work was supported by Shanghai AI Laboratory, \nthe National Natural Science Foundation of China No. 61976206 and No. 61832017, \nBeijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098, \nBeijing Academy of Artificial Intelligence (BAAI), \nthe Fundamental Research Funds for the Central Universities, \nthe Research Funds of Renmin University of China 21XNLG05, \nand Public Computing Cloud, Renmin University of China. \n\n\\section{Appendix}\n\nThe supplementary materials include\nsome additional ablation studies on the different \nnumber of masked joints in CSM module and different $K$ in MATM module. \n\n{\\noindent\\bf Effects of the number of masked joints in CSM.}\nTab.~\\ref{tab:jointnum} illustrates the effects of the number of masked joints on our PSTL.\nOn NTU-60 xsub datasets, PSTL can acquire a good \nperformance (77.3\\%) on a single stream with 9 masked joints,\nso we choose to report these results in our paper.\nNote that, the same number of masked joints may not suitable for all\nthe three streams, and when the number is 11, \nPSTL can acquire a better fusion result (79.6\\%).\n\n{\\noindent\\bf Effects of top-$K$ in MATM.}\nTab.~\\ref{tab:topk} shows the effects of different \nchoice of $K$ on NTU-60 xsub datasets. \nThe number of masked joints is fixed to 9.\nWhen the top 10 keyframes are masked, \nPSTL can achieve the best result (77.3\\%). \n\n{\\noindent\\bf Effects of the dimension of the embeddings.}\nIn our proposed PSTL, the structure of the projector $g$ \nis a 3-layer MLP.\nThe first two layers are followed by a\nbatch normalization layer and ReLU nonlinearity. \nThe output of the encoder is called the 'representations' and the output of the projector is the 'embeddings'.\n\nTo figure out the effects of the embedding dimension, \nwe conduct an ablation study with different dimensions on NTU-60 dataset. \nAs shown in Tab.~\\ref{tab:dim}, \n6144-dim can achieve a better performance than others \non both xsub and xview splits. \n\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n\\toprule\n\\# of masked joints & 5 & 7 & 9 & 11 \\\\ \\midrule\nJ     & 76.7  & 76.0  & \\textbf{77.3}  & 75.4   \\\\\nM     & 74.8  & 75.1  & \\textbf{75.4}  & 75.2   \\\\\nB     & 69.1  & 69.6  & 69.9  & \\textbf{70.5}   \\\\\nJ+M+B & 79.4  & 78.9  & 79.1  & \\textbf{79.6}   \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of the number of masked joints in CSM on NTU-60 xsub dataset.}\n\\label{tab:jointnum}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n\\toprule\n$K$   & 6 & 8 & 10 & 12 \\\\ \\midrule\nJ     & 75.3  & 76.8  & \\textbf{77.3}  & 76.1   \\\\\nM     & 73.9  & 74.5  & \\textbf{75.4}  & 74.9   \\\\\nB     & 68.5  & 68.8  & 69.9  & \\textbf{70.0}   \\\\\nJ+M+B & 77.7  & 78.8  & \\textbf{79.1}  & 78.9   \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of top-$K$ in MATM on NTU-60 xsub dataset.}\n\\label{tab:topk}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\begin{tabular}{l|cc}\n\\toprule\n\\multirow{2}{*}{dimension} & \\multicolumn{2}{l}{NTU-60(\\%)} \\\\\n                           & xsub  & xview          \\\\ \\midrule\n4096                       &    76.7 &  81.3        \\\\\n6144                       &    \\textbf{77.3} &  \\textbf{81.8}         \\\\\n8196                       &    75.8 &  81.3        \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-4pt}\n\\caption{Ablation studies of the embedding dimension on NTU-60 dataset.}\n\\label{tab:dim}\n\\end{center}\n\\vspace{-4pt}\n\\end{table}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we introduce a triple stream network named\nPartial Spatio-Temporal skeleton sequences Learning framework\nfor self-supervised skeleton-based action recognition. \nFirstly, all streams apply an ordinary augmentation for producing a variety of skeleton views.\nSecondly, one stream introduces a Central Spatial Masking strategy to force the selected \njoints out of the calculation of GCN backbone. \nThe third  stream utilizes a Motion Attention Temporal Masking module to find \nthe keyframes and mask them. \nAt last, the feature cross-correlation matrix\nis measured to acquire the correlation among each stream. \nExperiments show that such a masking strategy can boost the robustness of \nrepresentations of skeleton joints, \nwhich is essential to downstream tasks. \nour PSTL achieves remarkable performance \nin three large skeleton datasets under various evaluation settings.\n\\section{Experiments}\n\\label{sec:experiment}\n\n\\subsection{Datasets}\n\n\\noindent\\textbf{NTU-RGB+D 60.}\nNTU-60 dataset~\\cite{shahroudy2016ntu} is collected by Microsoft Kinect sensors. \nIt contains $56,578$ skeleton sequences of $60$ action categories,\nperformed by $40$ volunteers.\nThe skeleton of each subject is represented by $25$ joints, \nwhile each video has no more than two subjects. \nThere are two official dataset splits: \n1) Cross-Subject (xsub): half of the subjects belong to the training set, \nand the rest make up the testing sets;\n2) Cross-View (xview): training and testing sets are captured by cameras with different views. \n\n\\noindent\\textbf{NTU-RGB+D 120.}\nNTU-120 dataset~\\cite{liu2019ntu} is the extended version of the NTU-60,\nwhich contains $113,945$ skeleton sequences of $120$ action categories, \nperformed by $106$ volunteers. \nThere are two official dataset splits:\n1) Cross-Subject (xsub): the training set is performed by $53$ subjects,\nand testing data is performed by the rest volunteers;\n2) Cross-Setup (xset): the training and testing sets are captured by cameras with different setups. \n\n\\noindent\\textbf{PKU-MMD.}\nPKU-MMD dataset~\\cite{liu2020benchmark} is captured via \nthe Kinect v2 sensors from multiple viewpoints.\nIt has almost $20000$ action instances in $51$ categories collected by $66$ subjects. \nThere are two parts of the dataset:\n1) Part I contains $21539$ sequences;\n2) Part II contains $6904$ sequences.\nPart II is more challenging due to the large view variation with more skeleton noises.\n\n\\subsection{Implementation Details}\n\nTo perform an apple-to-apple comparison with other methods, \nwe follow the same pre-processing methods of CrossCLR~\\cite{Li_2021_CVPR} and AimCLR~\\cite{guo2022contrastive}, \nwhich resize the skeleton sequences to 50 frames. \nSimilarly, ST-GCN~\\cite{yan2018spatial} with 16 hidden channels is used as the backbone.\nFor all experiments (both representation learning and downstream tasks), we adopt Adam optimizer and\nthe CosineAnnealing scheduler with 150 epochs. \nThe mini batch size is 128. \n\n\\noindent\\textbf{Data Augmentation.}\nBefore applying the joint and temporal masking strategy proposed, \nsome ordinary augmentations are used to generate different views\nof the input skeleton sequences.\nThe ordinary augmentations include 3 spatial augmentations and 1 temporal augmentation. \n\n(1) \\textit{Shear.}\nIt is a linear transformation for skeleton 3D coordinates.\nA shear matrix is\n$$\n\\mathbf{S}=\\left[\\begin{array}{ccc}\n1 & s_{12} & s_{13} \\\\\ns_{21} & 1 & s_{23} \\\\\ns_{31} & s_{32} & 1\n\\end{array}\\right]\n,\n$$ \nwhere $s_{12}, s_{13}, s_{21}, s_{23}, s_{31}, s_{32}$ are the shear factors and are\nrandomly sampled from $[-\\beta \\sim \\beta]$. $ \\beta $ is the amplitude to adjust \nthe augmentation strength. \nHere we set $ \\beta = 1$.  \nBy directly multiplying the shear matrix with the 3D coordinates, the human pose slants\nin a random angle. \n\n(2) \\textit{Crop.}\nAs a temporal augmentation, Crop aims to improve data diversity \nby first padding part of frames to the original sequence and then \nrandomly cropping it to the original length. \nThe padding length is $\\gamma\\, T$, $T$ is the original sequence length, and $\\gamma$ is the padding ratio (we set $\\gamma = 1 \/ 6$). \n\n(3) \\textit{Rotate.}\nThis is an efficient spatial augmentation which \nhelps the representation adapt to the variation of spatial perspective. \nSpecifically, it first randomly selects an axis $M \\in \\{X, Y, Z\\} $ as main axis and then a random angle in $[0 \\sim \\pi\/6]$ is selected for it to rotate. \nNext, the rest two axis both randomly choose the angle in $[0 \\sim \\pi\/180]$.\n\n(4) \\textit{Spatial Flip.}\nIt is another spatial augmentation,\nwhich swaps the left and right sides of the skeleton data with the probability $p=0.5$. \n\n\\subsection{Experiment Setting}\n\n\\noindent\\textbf{Self-supervised Pre-training.}\nWe use ST-GCN~\\cite{yan2018spatial} as the encoder,\nwhich extracts $256$-dim features from skeleton sequences.\nThen we project features to $6144$-dim embeddings. \n$\\lambda$ in the loss of each stream is set to 2e-4. \nA 10-epoch warmup is used for stabilizing the training process. \nThe weight decay is set to 1e-5.\nNote that for all datasets and evaluation settings, \nthe number of mask joints is 9 in CSM and $K=10$ in MATM.\nTo compare with the state-of-the-art, we use multiple skeleton modalities,\nincluding joint (\\textbf{J}), bone (\\textbf{B}), and motion (\\textbf{M}).\nWe evaluate the learned representations with various evaluation protocols.\n\n\\noindent\\textbf{Linear Evaluation.}\nWe add a linear classifier on top of the frozen pre-trained encoder,\nand then train the recognizer on the target skeleton action recognition dataset with initial learning rate $0.01$.\n\n\\noindent\\textbf{Partial Body Evaluation.}\nTo encourage the learned representation to adapt to real-life scenarios \nwith skeleton shaded or partial body information lost,\nwe introduce the Partial Body Evaluation protocol.\nBasically, we follow the settings of Linear Evaluation. \nHowever, some joints or body parts may be masked to simulate the joint\/part missing in extracted skeletons.\nWe consider two settings:\n1) Joint shaded: some joints are masked randomly during evaluation;\n2) Body part shaded: we divide skeletons into 5 major parts\\footnote{left\/right arms, left\/right legs, torso} and randomly drop parts during evaluation.\n\n\n\\noindent\\textbf{Finetuning.}\nWe append a linear classifier on top of the encoder and then finetune the entire network\non the target skeleton action recognition dataset with an initial learning rate $0.005$. \n\n\\noindent\\textbf{Semi-Supervised Learning.}\nThis protocol shares the same setting as ``Finetuning\",\nbut only uses $1\\%$ or $10\\%$ training labeled data that is randomly selected. \n\n\n\\begin{table*}[!ht]\n\\begin{center}\n\\begin{tabular}{l|c|cc|cc|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{1}{l|}{\\multirow{3}{*}{Method}} &\n  \\multicolumn{1}{c|}{\\multirow{3}{*}{Stream}} &\n  \\multicolumn{4}{c|}{NTU-60(\\%)} &\n  \\multicolumn{4}{c|}{PKU(\\%)} &\n  \\multicolumn{4}{c}{NTU-120(\\%)} \\\\ \\cline{3-14} \n\\multicolumn{1}{c|}{} &\n  \\multicolumn{1}{c|}{} &\n  \\multicolumn{2}{c|}{xsub} &\n  \\multicolumn{2}{c|}{xview} &\n  \\multicolumn{2}{c|}{part I} &\n  \\multicolumn{2}{c|}{part II} &\n  \\multicolumn{2}{c|}{xsub} &\n  \\multicolumn{2}{c}{xset} \\\\\n\\multicolumn{1}{c|}{} &\n  \\multicolumn{1}{c|}{} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &\n  \\multicolumn{1}{c|}{$\\Delta$} &\n  acc. &  $\\Delta$ \\\\\\midrule\nSkeletonBT    & \\textbf{J} & 68.5 &       & 71.6 &       & 84.7 &       & 38.2 &       & 53.9 &       & 52.7 &  \\\\\n\\textbf{PSTL}          & \\textbf{J} & \\textbf{77.3} & +8.8  & \\textbf{81.8} & +10.2  & \\textbf{88.4} & +3.9  &\\textbf{ 49.3} & +11.1 & \\textbf{66.2} & +12.3 & \\textbf{67.7} & +15.0 \\\\\\midrule\nSkeletonBT    & \\textbf{M} & 58.9 &       & 58.2 &       & 63.2 &       & 30.3 &       & 43.5 &       & 44.7 &   \\\\\n\\textbf{PSTL}          & \\textbf{M} & \\textbf{69.9} & +11.0 & \\textbf{72.6} & +14.4 & \\textbf{80.4} & +17.2 & \\textbf{34.7} & +4.4  & \\textbf{65.4} & +21.9 & \\textbf{57.7} & +13.0 \\\\\\midrule\nSkeletonBT    & \\textbf{B} & 62.4 &       & 64.7 &       & 79.1 &       & 29.8 &       & 52.4 &       & 53.6 &   \\\\\n\\textbf{PSTL }         & \\textbf{B} & \\textbf{75.4} & +13.0 & \\textbf{79.2} & +14.5 & \\textbf{84.8} & +5.7  & \\textbf{41.3} & +11.5 & \\textbf{58.7} & +6.3  & \\textbf{66.6} & +13.0 \\\\\\midrule\n3s-SkeletonBT & \\textbf{J+M+B} & 69.3 &      & 72.2 &       & 84.8 &      & 44.1 &      & 56.3 &       & 56.8     &   \\\\\n\\textbf{3s-PSTL}       & \\textbf{J+M+B} & \\textbf{79.1} & +9.8 & \\textbf{82.6} & +10.4 & \\textbf{89.2} & +4.4 & \\textbf{52.3} & +8.2 & \\textbf{69.2} & +12.9 & \\textbf{70.3} & +13.5  \\\\\\bottomrule\n\\end{tabular}\n\\vspace{-5pt}\n\\caption{Linear evaluation results compared with SkeletonBT on NTU-60, PKU-MMD, and NTU-120 datasets.\n\"$\\Delta$\" represents the gain compared to SkeletonBT using the same stream data. \n\\textbf{J}, \\textbf{M} and \\textbf{B} indicate joint stream, motion stream, and bone stream.}\n\\label{tab:compBT}\n\\vspace{-8pt}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.9\\linewidth}{!}{\n\\begin{tabular}{ccc|cc}\n\\toprule\n\\multirow{2}{*}{SkeletonBT} & \\multirow{2}{*}{w\/ CSM} & \\multirow{2}{*}{w\/ MATM} & \\multicolumn{2}{l}{NTU-60(\\%)} \\\\\n  &   &   & xsub & xview \\\\ \\midrule\n\\checkmark &            &            & 68.5 & 71.6  \\\\\n\\checkmark & \\checkmark &            & 71.8 & 75.4  \\\\\n\\checkmark &            & \\checkmark & 73.7 & 78.0  \\\\\n\\checkmark & \\checkmark & \\checkmark & \\textbf{77.3} & \\textbf{81.8}  \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Ablation studies of PSTL on NTU-60 dataset.}\n\\label{tab:overall}\n\\vspace{-9pt}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Ablation Studies of PSTL}\nWe ablate the proposed CSM and MATM strategies on our baseline SkeletonBT respectively to \nshow their effectiveness. \nThe linear evaluation results are shown in Tab.~\\ref{tab:overall}, \nSkeletonBT achieves 68.5\\% and 71.6\\% accuracy on xsub and xview.\nWe find that the CSM module improves about 3.3\\%, 3.8\\% Top-1 accuracies\nand the MATM module brings about 5.2\\%, 6.4\\% performance gain on\nxsub and xview, respectively. \nWith the help of both modules, \nour PSTL achieves 8.8\\% and 9.4\\% increments. \nThese results show that our proposed CSM and MATM strategies can \nencourage the encoder to acquire stronger features\nwhich are more suitable for downstream tasks.\n\n\n\\noindent\\textbf{Comparisons with SkeletonBT.}\nTo further verify the effectiveness of our proposed method, \nwe conduct experiments on three datasets to \ncompare our PSTL with the baseline method SkeletonBT in detail. \nAs shown in Tab.~\\ref{tab:compBT}, PSTL performs much better than SkeletonBT \non all three different streams. \nFor the fusion results, 3s-PSTL substantially improves\nSkeletonBT, especially for the NTU-120 datasets. \n\n\\noindent\\textbf{Comparisons with Random Masking Strategy.}\nTo demonstrate the effectiveness of our masking strategy, \nwe compare our CSM and MATM modules with random spatial masking (RSM) strategy \nand random temporal masking (RTM) strategy, which selects the mask position randomly. \nThe linear evaluation results are shown in Tab.~\\ref{tab:random}, \nwith the help of our triple stream network, \nthe random masking can capture the relationship between \njoints and frames and get a good results on downstream tasks. \nAnd then, our CSM and MATM can further strengthen this capacity \nto get a better accuracy.\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.56\\linewidth}{!}{\n\\begin{tabular}{l|cc}\n\\toprule\n\\multirow{2}{*}{Method} & \\multicolumn{2}{c}{NTU-60(\\%)} \\\\\n                        & xsub           & xview          \\\\ \\midrule\nRSM + RTM                 & 75.6           & 79.7           \\\\\nRSM + MATM                & 76.4           & 81.2               \\\\\nCSM + RTM                 & 76.1           & 80.4                \\\\\nCSM + MATM                & \\textbf{77.3} & \\textbf{81.8}          \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Compared with random masking strategy on NTU-60 dataset. \nRSM means random spatial masking and RTM means random temporal masking.}\n\\label{tab:random}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Comparison with State-of-the-art}\n\n\\noindent\\textbf{Partial Body Evaluation.}\nTo verify the \\emph{joint robustness} of extracted features,\nwe conduct the partial body evaluation on NTU-60 xsub datasets.\nThe first level of the evaluation protocols is \n\\textbf{joint shaded}, \nwhich applies random masks on joints to simulate the situation of \njoints shade in realistic scenarios.\nTab.~\\ref{tab:jointshad} shows that, without CSM and MATM, the baseline  SkeletonBT suffers\nserious performance loss. \nWith the number of mask joints increasing to 10, \nthe SOTA method AimCLR~\\cite{guo2022contrastive} \nand its baseline methods SkeletonCLR~\\cite{Li_2021_CVPR} also \nsuffer about 20.0\\% and 17.6\\% performance decrease. \nNote that the feature extracted by our PSTL acquires strong joint robustness,\nand only 3.5\\% performance decrease under 10 masked joints. \n\nThe second level evaluation protocols \\textbf{body part shaded}, as shown in Tab.~\\ref{tab:partshad}, \nis much harder since we mask the total parts of the body.\nBut our PSTL still achieves better results than other methods.\nTo sum up, more skeleton sequence information is lost, \nand the advantage of our method becomes more obvious. \nAnd the results tell that with the help of\nlocal relationship between joints and frames, our CSM and MATM strategy \ncan improve the robustness of learned features, \nwhich is essential to downstream tasks.\n\n\\begin{table}[!t]\n\\begin{center}\n\t\t\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{l|cccccc}\n\\toprule\n\\# of masked joints      & 0 & 2 & 4 & 6 & 8 & 10 \\\\ \\midrule\nSkeletonCLR  & 68.3 & 65.6  & 61.9  & 58.6  & 54.5  & 50.7   \\\\\nSkeletonBT     & 68.5 & 65.0  & 61.7  & 58.6  & 55.0  & 50.6   \\\\\nAimCLR       & 74.3 & 70.4  & 67.2  & 63.0  & 58.5  & 54.3  \\\\\n\\textbf{PSTL(ours)}  & \\textbf{77.3} & \\textbf{76.9}  & \\textbf{76.2} & \\textbf{75.7}  & \\textbf{74.6}  & \\textbf{73.8}  \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results of \\textbf{joints shaded} on NTU-60 xsub dataset.}\n\\label{tab:jointshad}\n\\vspace{-5pt}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|ccccc}\n\\hline\n\\# of masked body parts       & 0 & 1 & 2 & 3 & 4 \\\\ \\hline\nSkeletonCLR  & 68.3 & 63.6 & 57.7  & 47.5  & 33.5 \\\\\nSkeletonBT   & 68.5 & 64.0 & 58.5 & 50.2 & 38.4  \\\\\nAimCLR       & 74.3 & 69.3 & 63.9 & 53.7 & 37.9 \\\\\n\\textbf{PSTL (ours)}  & \\textbf{77.3} & \\textbf{73.6} & \\textbf{68.3} & \\textbf{60.0} & \\textbf{47.3} \\\\ \\hline\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results of \\textbf{body part shaded} on NTU-60 xsub dataset.}\n\\label{tab:partshad}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\noindent\\textbf{Linear Evaluation.}\nAs shown in Tab.~\\ref{tab:ntu60&120lp}, for a single stream, \nPSTL achieves considerable gains on all other methods on NTU60 and NTU120.\nCompared with the current SOTA method AimCLR, \nour PSTL improves \n3.0\\% and 2.1\\% on Xsub and Xview respectively. \nThe results indicate that the local relationship between joints and frames can \nhelp extract suitable features for downstream tasks. \nFor the performance of the 3-streams, PSTL still \nachieves comparable results to current methods. \nOur method also achieves the best results on NTU-120 datasets. \nFor single stream, PSTL outperforms the current SOTA method AimCLR \n2.8\\% and 4.3\\% on Xsub and Xset respectively. \nAnd the fusion can also achieve 1.0\\% and 1.5\\% increment. \n\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|cc|cc}\n\\toprule\n\\multirow{2}{*}{Method}          & \\multicolumn{2}{c|}{NTU-60(\\%)} & \\multicolumn{2}{c}{NTU-120(\\%)} \\\\\n                                 & xsub          & xview          & xsub           & xset           \\\\\\midrule\n\\textit{Single-stream:}         &               &               &                &                \\\\\nLongT GAN (AAAI 18)              & 39.1          & 48.1         &  -             &  -              \\\\\nMS2L (ACM MM 20)                 & 52.6         & -             &  -             &  -              \\\\\nAS-CAL (Inf Sci 21)             &  58.5         & 64.8          &  48.6          & 49.2           \\\\\nP\\&C (CVPR 20)                   & 50.7         & 76.3          &  42.7          & 41.7             \\\\\nSeBiReNet (ECCV 20)              &  -            & 79.7         &   -            & -               \\\\\nAimCLR (AAAI 22)                 & 74.3         & 79.7          &  63.4          & 63.4           \\\\\n\\textbf{PSTL (ours)}            & \\textbf{77.3} & \\textbf{81.8} & \\textbf{66.2}& \\textbf{67.7} \\\\\\midrule\n\\textit{Three-stream:}          &               &               &                &                \\\\\n3s-SkeletonBT (ours)            &   69.3        & 72.2          &  56.3          & 56.8          \\\\\n3s-SkeletonCLR (CVPR 21)        &    75.0        & 79.8          &  60.7          &  62.6             \\\\\n3s-Colorization (ICCV 21)       &   75.2        & 83.1          &  -             &   -             \\\\\n3s-CrosSCLR (CVPR 21)           &  77.8        & 83.4          &  67.9          &  67.1              \\\\\n3s-AimCLR (AAAI 22)             &   78.9        & 83.8          &  68.2          &  68.8               \\\\\n\\textbf{3s-PSTL (ours) }        &   \\textbf{79.1}        &  \\textbf{83.8}        & \\textbf{69.2}         &  \\textbf{70.3} \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results on NTU-60 and NTU-120 datasets.}\n\\label{tab:ntu60&120lp}\n\\vspace{-10pt}\n\\end{center}\n\\end{table}\n\n\n\nNoting that the PKU-MMD dataset has about half the data size of NTU-60,\nTab.~\\ref{tab:pkulp} shows that the advantage of our method is more obvious in a small dataset. \nIt is worth mentioning that our baseline method SkeletonBT can achieve better \nresults than AimCLR on the single stream, \nwhich indicates that our negative-sample-free methods without a memory bank\nis more efficient for small datasets.\nSimilarly, our PSTL achieves remarkable results on PKU-MMD datasets.\nFor single stream, PSTL outperforms AimCLR 5.0\\% and 12.5\\% on Part I and Part II.\nFor 3 stream fusion, our 3s-PSTL outperforms 3s-AimCLR 1.4\\% and 13.8\\% on both parts. \n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.74\\linewidth}{!}{\n\\begin{tabular}{l|c|c}\n\\toprule\nMethod                & Part I & Part II \\\\\\midrule\n\n\\textit{Single-stream:}         &        &         \\\\\nLongT GAN (AAAI 18)         &  67.7      & 26.0        \\\\\nMS2L (ACM MM 20)             & 64.9      & 27.6        \\\\\nSkeletonCLR (CVPR 21)       &  80.9      & 35.2        \\\\\nISC (ACM MM 20)             &  80.9     &  36.0        \\\\\nAimCLR (AAAI 22)                & 83.4  & 36.8          \\\\\nSkeletonBT (ours)               & 84.7  & 38.2         \\\\\n\\textbf{PSTL(ours) }            & \\textbf{88.4}  & \\textbf{49.3}    \\\\\\midrule\n\\textit{Three-stream:}          &        &         \\\\\n3s-CrosSCLR (CVPR 21)           & 84.9   & 21.2    \\\\\n3s-AimCLR (AAAI 22)             & 87.8   & 38.5   \\\\\n3s-SkeletonBT (ours)            & 84.8  & 44.1    \\\\\n\\textbf{3s-PSTL (ours)}         & \\textbf{89.2}   & \\textbf{52.3}   \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Linear evaluation results on PKU-MMD dataset.}\n\\label{tab:pkulp}\n\\vspace{-15pt}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\noindent\\textbf{Finetune Evaluation.}\nWe compare PSTL with other self-supervised methods and\nthe supervised method ST-GCN using the same structure and parameters as ours. \nAs shown in Tab.~\\ref{tab:ntu60&120ft}, \nfor the single steam, \nall methods use bone steam since \nthe results are better than the joint stream. \nOur PSTL achieves better performance than others on \nNTU-60 and NTU-120 datasets.\nNotably, only single steam PSTL outperforms supervised 3s-ST-GCN on NTU-120 datasets,\nwhich indicates the effectiveness of our method. \nSimilarly, 3s-PSTL outperforms 3s-CrossCLR and 3s-AimCLR \nfor the fusion results on both datasets.  \n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.95\\linewidth}{!}{\n\\begin{tabular}{l|cc|cc}\n\\toprule\n\\multirow{2}{*}{Method}          & \\multicolumn{2}{c|}{NTU-60(\\%)} & \\multicolumn{2}{c}{NTU-120(\\%)} \\\\\n                                 & xsub          & xview          & xsub           & xset           \\\\\\midrule\n\\textit{Single-stream:}         &               &                &                &                \\\\\nSkeletonCLR (CVPR 21)           &  82.2         & 88.9          & 73.6              & 75.3            \\\\\nAimCLR (AAAI 22)                 & 83.0         & 89.2         &  77.2           & 76.0               \\\\\n\\textbf{PSTL (ours)}            &  \\textbf{84.5}& \\textbf{92.0 }& \\textbf{78.6}   &  \\textbf{78.9}     \\\\\\midrule\n\\textit{Three-stream:}          &               &                &                &                \\\\\n3s-ST-GCN (AAAI 18) ${}^{\\S}$   &   85.2        & 91.4          &  77.2         & 77.1              \\\\\n3s-CrosSCLR (CVPR 21)            &  86.2        & 92.5          & 80.5          & 80.4               \\\\\n3s-AimCLR (AAAI 22)              &  86.9        & 92.8          & 80.1          & 80.9               \\\\\n\\textbf{3s-PSTL (ours) }         &  \\textbf{87.1} &\\textbf{93.9} & \\textbf{81.3} & \\textbf{82.6}   \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Finetune evaluation results on NTU-60 and NTU-120 datasets.\n\"\u00a7\" means the model is trained\nwith a fully-supervised manner.}\n\\label{tab:ntu60&120ft}\n\\vspace{-5pt}\n\\end{center}\n\\end{table}\n\n\\noindent\\textbf{Semi-Supervised Evaluation.}\nTo show the superiority of our approach,\nwe test all methods under the semi-supervised setting, where only 1\\% or 10\\% labeled data can be used such that models are more likely to overfit.\nAs shown in Tab.~\\ref{tab:semi}, our PSTL far exceeds the current SOTA method AimCLR on PKU-MMD datasets.\nAnd for fusion results, 3s-PSTL outperforms 3s-AimCLR 8.6\\% on Part II. \nIt demonstrates that such contrastive methods which need a large memory bank \nare subjected to serious performance loss with insufficient samples. \nBut our negative-sample-free PSTL can well alleviate this problem and \nachieve better results. \n\n\n\n\\begin{table}[!t]\n\\begin{center}\n\\resizebox{0.75\\linewidth}{!}{\n\\begin{tabular}{l|c|c}\n\\toprule\nMethod                & Part I & Part II \\\\\\midrule\n\\textit{1\\% labeled data:}     &        &         \\\\\nLongT GAN (AAAI 18)   & 35.8   & 12.4     \\\\\nMS2L (ACM MM 20)      & 36.4  & 13.0        \\\\\nISC (ACM MM 21)       & 37.7  & -        \\\\\n3s-CrosSCLR (CVPR 21) & 49.7  & 10.2        \\\\\n3s-AimCLR (AAAI 22)   & 57.5  & 15.1        \\\\\n\\textbf{3s-PSTL(ours)} & \\textbf{62.5}  & \\textbf{16.9} \\\\\\midrule\n\\textit{10\\% labeled data:}    &        &         \\\\\nLongT GAN (AAAI 18)   &  69.5  & 25.7       \\\\\nMS2L (ACM MM 20)      & 70.3  & 26.1        \\\\\nISC (ACM MM 21)       & 72.1  & -        \\\\\n3s-CrosSCLR (CVPR 21) & 82.9  & 28.6    \\\\\n3s-AimCLR (AAAI 22)   & 86.1  & 33.4        \\\\\n\\textbf{3s-PSTL (ours)} & \\textbf{86.9} & \\textbf{42.0}      \\\\\\bottomrule\n\\end{tabular}\n}\n\\vspace{-5pt}\n\\caption{Semi-supervised evaluation results on PKU-MMD dataset.}\n\\label{tab:semi}\n\\vspace{-7pt}\n\\end{center}\n\\end{table}\n\\section{Introduction}\n\\label{sec:introduction}\n\n\n\nIn recent years, human action recognition has been extensively studied for various applications in the real world, \nsuch as security, human-robot interaction, and virtual reality.\nWith the development of sensor technology~\\cite{zhang2012microsoft,shotton2011real} and human pose estimation methods~\\cite{Cao_2017_CVPR,hua2022weakly}, \nskeleton data becomes easier to obtain.  \nIts robustness against the variation of appearance and backgrounds \nhas attracted many researchers to explore fully supervised methods \nfor action recognition, \nsuch as applying CNN~\\cite{ke2017a}, \nRNN~\\cite{du2015hierarchical} or GCN~\\cite{yan2018spatial,shi2019two} on skeleton data. \n\n\nHowever, fully supervised learning relies on abundant manually annotated labels, \nwhich is time-consuming and expensive. \nTherefore, self-supervised frameworks, \\eg, MoCo~\\cite{he2020momentum}, \nSimCLR~\\cite{chen2020simple}, Barlow Twins~\\cite{zbontar2021barlow}, \nand MAE~\\cite{he2022masked} have received a lot of attention. \nFor self-supervised learning on skeleton data, \nearly methods also apply pretext tasks~\\cite{lin2020ms2l,su2020predict,zheng2018unsupervised}.\nWhile nowadays, most researchers focus on contrastive learning and apply MoCo structure \nto the skeleton squences~\\cite{thoker2021skeleton,Li_2021_CVPR,rao2021augmented,guo2022contrastive}.\n\n\n\n\nThe key of current contrastive learning-based methods is finding effective global data augmentation and using them to create various views of the skeleton carefully. \nHowever, due to the rich action clues stored in the skeleton sequence, this paradigm can readily discriminate different views from a global perspective without thoroughly leveraging the local relationship between different skeleton joints and video frames. \nAs a result, it limits the performance and robustness of the model in real-world scenarios. Moreover, a large batch size or a memory bank is necessary for contrastive learning, which is unsuitable for a small amount of skeleton data. \nTo evaluate the robustness of the current methods in real-world scenarios (such as video surveillance), \nwe formulate a new evaluating setting where only a partial skeleton is available for downstream tasks to simulate that the skeleton data captured by the camera are shaded. \nIn this challenging setting, previous works suffer severe performance drops. \nWe conjecture that if the local spatial and temporal relationship can be explored and captured, it will boost the representation ability and robustness of the model on downstream tasks.\n\n\n\n\nTherefore, we propose a Partial Spatio-Temporal Learning (PSTL) framework to exploit the local relationship from a partial skeleton sequence built by a unique spatio-temporal masking strategy. \nSpecifically, we construct a negative-sample-free triplet steam structure that is composed of an anchor stream, a spatial masking stream with Central Spatial Masking (CSM), and a temporal masking stream with Motion Attention Temporal Masking (MATM). \n\nIn each stream, an ordinary augmentation is first applied to improve the diversity of the input samples. \n(1) In the spatial masking stream, Central Spatial Masking discards selected joints from the feature calculation process. Notably, it is not simply set the selected joint position to zeros, which is unreasonable for skeleton data. \nConsidering that the topology of the human skeleton is a predefined graph, the joints with a higher degree of centrality get stronger connectivity. By filtering out such joints with higher probabilities, the encoder can focus more on these skeletons that are less explored. \n(2) In the temporal masking stream, considering the frames that change quickly always contain more semantic information about the actions, \nwe propose a Motion Attention Temporal Masking strategy, where\nframes moves quickly are more likely to be masked.\n(3) In the anchor stream, the masking policy is not introduced to keep the original semantic information. A shared encoder extracts \nthe triple stream features simultaneously.\nInspired by the loss of Barlow Twins~\\cite{zbontar2021barlow},\nwe adopt the cross-correlation matrix to \ncapture the correlation between the anchor stream and the other two masking streams, respectively.\n\nAs a negative-sample-free self-supervised method, \nPSTL encourages the empirical cross-correlation matrix \nbetween learned representations to be an identity matrix, \nand minimizes the redundancy within the \nrepresentation vector to alleviate feature collapse.\nIn this way, the spatial and temporal relationship can be well captured \nand the joint robustness can be boosted for downstream tasks. \nExtensive results on three large benchmark\ndemonstrate the effectiveness of our proposed masking strategy. \nOur contributions:\n\n\\vspace{2pt}\n\\begin{itemize}\n\n\\item We propose Partial Spatio-Temporal skeleton representation Learning (PSTL), a negative-sample-free self-supervised framework for skeleton-based action representation.\n\n\\item We are the first to introduce the masking strategy \nto skeleton-based self-supervised learning and \nusing Central Spatial Masking (CSM) strategy during pre-training to \nimprove the joint robustness of learned representation for downstream tasks.\n\n\\item A unique Motion Attention Temporal Masking strategy (MATM) \nis proposed to use the motion of skeleton data to guild \ntemporal masking modules to find and mask key frames of the sequences. \n\n\\item Extensive experiment results on NTU-60, NTU120, and PKU-MMD datasets \nshow the proposed PSTL achieves the-state-of-art performance under diverse downstream tasks,\nsuch as linear evaluation, finetune evaluation, and semi-supervised evaluation protocol.\n\n\\end{itemize}\n\\section{Method}\n\\label{sec:method}\n\nAs shown in Fig~\\ref{fig:PSTL},\nPartial Spatio-Temporal skeleton representation Learning (PSTL) \ntakes a triple stream architecture, which is based on our Skeleton Barlow Twins (SkeletonBT),\nand applies Central Spatial Masking (CSM) on its spatial masking stream and Motion Attention Temporal Masking (MATM) on its temporal masking stream together with an extra anchor stream to keep the original semantic information. \nSpecifically,\n1) \\textbf{SkeletonBT} is a basic negative-sample-free double stream network\nwith ordinary augmentation for skeleton self-supervised learning. \n2) \\textbf{CSM} first adjusts the computing mode of the GCN backbone\nto ensure the information of masked joints out of the feature calculation process. \nThen, the joints with higher degree centrality are assigned higher probability to be masked. \n3) \\textbf{MATM} calculates the corresponding motion value for each frame of the action, \nwhich serves as the attention weight to select the masked frames. \n\n\n\\subsection{Skeleton Barlow Twins}\n\\label{subsec:Bt}\n\nInspired by the recent advance on negative-sample-free self-supervised methods~\\cite{zbontar2021barlow} that avoid negative sample \nand do not need a large batch size\/memory bank, \nwe develop Skeleton Barlow Twins (SkeletonBT) to learn the skeleton-based action \nrepresentation.\nIt is a symmetric network, \nwhere a shared encoder is used to generate features.\n\nSpecifically, a 3D human skeleton sequence can be denoted as\n$s \\in \\mathbb{R}^{C \\times T \\times V}$ ,\nwhich has $T$ frames and $V$ joints. \n$C$ is the channel dimension representing the 3D position. \nAs shown in Fig.~\\ref{fig:SkeletonBT},\nthe input skeleton $s$ is first applied with an \nordinary augmentation $\\mathcal{T}$\nto get diverse views $x$ and $x^{'}$. \nThe encoder $f$ is used to extract features: \n$h = f(x;\\theta)$ and $h^{'} = f(x^{'};\\theta)$, \nwhere $h, h^{'} \\in \\mathbb{R}^{c_{h}}$. \nThen a projector $g$ maps each feature \nto the higher dimension space and acquire the embeddings: \n$z = g(h)$ and $z^{'} = g(h^{'})$, \nwhere $z, z^{'} \\in \\mathbb{R}^{c_{z}}$. \nAt last, by encouraging the empirical cross-correlation matrix \nbetween embeddings of the distorted variations be an identity matrix,\nthe encoder can capture the relationship between two streams. \nThe loss can be formulated as: \n\\begin{equation}\n\\mathcal{L} = \\sum_{i}\\left(1-\\mathcal{C}_{i i}\\right)^{2}+\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{C}_{i j}^{2}.\n\\label{eq:BT}\n\\end{equation}\nHere $\\mathcal{C}$ is the cross-correlation matrix \ncomputed between embeddings $z$ and $z^{'}$ \nalong the batch dimension $b$: \n\n\\begin{equation}\n\\label{eq:cc}\n\\mathcal{C}_{i j} = \\frac{\\sum_{b} z_{b, i} z_{b, j}^{'}}{\\sqrt{\\sum_{b}\\left(z_{b, i}\\right)^{2}} \\sqrt{\\sum_{b}\\left(z_{b, j}^{'}\\right)^{2}}},\n\\end{equation}\nwhere $i,j$ represent the embedding dimension. \n\nSpecifically, in Eq.~\\eqref{eq:BT} the first term encourages the diagonal elements of $\\mathcal{C}$ to 1,\nwhich makes the embedding invariant to the variety augmentation applied. \nThe second term equates the off-diagonal elements of $\\mathcal{C}$ to 0, \nhelping the different embedding components to be decoupled with each other, \nwhich can minimize the redundancy within the representation.\n$\\lambda$ serves as a trade-off parameter to balance two terms.\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{AAAI 2023 skeleton self_supervised\/SkeletonBT.pdf}  \n  \\vspace{-5pt}\n  \\caption{The framework of Skeleton Barlow Twins.}\n  \\label{fig:SkeletonBT}\n  \\vspace{-5pt}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{AAAI 2023 skeleton self_supervised\/PSTL.pdf} \n\\vspace{-6pt}\n\\caption{The overall framework of Partial Spatio-Temporal skeleton representation Learning. \nIn Central Spatial Masking \\textbf{(CSM)},  the degree is red (4), blue (3), dark grey (2) and light grey (1).\nIn Motion Attention Temporal Mask \\textbf{(MATM)}, \n$m$ denotes motion density while $t$ is the time code.\n}\n\\label{fig:PSTL}\n\\end{center}\n\\vspace{-10pt}\n\\end{figure*}\n\n\n\n\n\\subsection{Central Spatial Masking}\n\\label{subsec:CSM}\nTo boost the joint robustness of learned representation,\nwe propose a Central Spatial Masking (CSM) module that \nencourages partial skeleton data to generate similar features as the whole skeleton data. \nThe relationship between masked and unmasked joints can be captured by the encoder, \nwhich is essential for the hard downstream task.\n\nAs discussed above, it is critical to construct partial skeleton data for the encoder. \nEarly methods~\\cite{rao2021augmented} try to directly set the values of masked joints to zero.\nBut it is unreasonable for skeleton data since\nthe joint semantic information is the 3D position that is changed. \nFor CSM, we first adjust the computing mode of the backbone ST-GCN~\\cite{yan2018spatial}.\nSpecifically, for a skeleton sequence $s$, the masked joints are abandoned and then\ndrop the corresponding rows and columns in the adjacency matrix, \nforcing masked joints out of the calculation process and the learned feature \nis only generated by the unmasked ones. \n\nNext, we take the advantage of the degree centrality \nof the human skeleton graph topology, \nand observe that the joints with more degrees can \nacquire richer neighborhood information. \nThus, we assign higher probabilities to mask \nsuch connective joints which help the encoder capture the relationship between \na wider range.\nAs shown in the green box of Fig~\\ref{fig:PSTL}, the human skeleton joint is $V_i, i \\in (1,2,...,n)$,\nwhere $n$ is the total number of joints. \nNote that there are four types of joints.\nThe degree of light gray joints located at the margin of the graph is 1. \nThe degree of dark gray joints is 2, which are more connective \nand their number is the majority in the total graph. \nThe blue one and red one hold degree of 3 and 4, \nwhich are the center of the upper half and lower half of the body. \nMasking such center joints can help the encoder to\ncapture the relationship between a large range of joint information.\nTo this end, our CSM strategy first calculates the degree $d_i$ of each joint $V_i$, \nand sets their masked probability:\n\\begin{equation}\np_i = \\frac{d_i}{\\sum_{j=1}^{n} d_j}.\n\\end{equation}\n\n\n\n\n\\subsection{Motion Attention Temporal Masking}\n\\label{subsec:MATM}\nCompared with the spatial joints, the temporal information is more redundant and\nthe semantic of action is more concentrated. \nSome key-frames play an pivotal role in representing the semantics of an action. \nTo utilize this feature, \nour MATM aims to find such key-frames and mask them to \ngenerate a hard pair for anchor stream and temporal masking stream in our proposed PSTL. \nBy encouraging the empirical cross-correlation matrix between embeddings of two stream\nbe an identity matrix, the encoder can capture the relationship between \nredundant frames and key-frames effectively. \n\nSpecifically, the motion $m \\in \\mathbb{R}^{C \\times T \\times V}$ of the sequence $s$\nis computed by the temporal displacement between frames:\n$m_{:, t, :} = x_{:, t+1,:}-x_{:, t, :}$. \nThen we acquire the overall motion rate of a frame which serves as the attention weight:\n\\begin{equation}\na_t = \\frac{m_t^2} {\\sum_{i=1}^{T} {m_i^2}}.\n\\end{equation}\nThen, the top-$K$ attention weights $a_{t_1},...,a_{t_K}$ are selected \nand the sequence $x_{t_1},...,x_{t_K}$ serves as the key-frames which\ncontain more semantic information about the actions. \nAs shown in the blue box of Fig~\\ref{fig:PSTL}, the bottom one is a skeleton sequence of \\textit{put on glasses}, \nthe top one shows the motion density. \nWe can find that some frames in the middle of the sequence acquire higher motion values, \nwhich are the key-frames for this action. \nThus, we mask these key-frames and encourage the feature from masked sequence close to \nthe anchor feature which contains total semantic information. \n\n\\subsection{Loss Function}\n\\label{subsec:PSTL}\nRecall the pipeline of the proposed triple stream PSTL shown in Fig.~\\ref{fig:PSTL}.\nGiven a skeleton sequence $s$, an ordinary augmentation is applied \nto obtain different views $x,x^{'},\\hat{x}$. \nFor sequence $x^{'}$, a Central Spatial Masking strategy is used to \ngenerate partial skeleton data $x_c^{'}$. \nConsidering the degree centrality of the skeleton graph, \nthe joints with more degrees are assigned higher masked probability in this module.   \nFor sequence $\\hat{x}$, a Motion Attention Temporal Masking strategy is applied.\nWe first compute the motion value of each frame which represents the key \nfragment in a skeleton action. \nThen the top-$K$ frames are selected to be masked \nand to further improve diversity,\n$K$ random frames are still masked from remain sequence to \ngenerate $\\hat{x}_m$. \nThe sequence $x$ in the anchor stream does not introduce any mask\nstrategy to keep the original semantic information. \n\nAfter that, the shared encoder $f$ is used to extract the features simultaneously.\n$h = f(x;\\theta), h_c^{'} = f(x^{'};\\theta), \\hat{h}_m = f(\\hat{x}_m;\\theta)$,\nwhere $\\theta$ is the parameters learned by the encoder\nwhich is also used in downstream tasks. \nThen a projector $g$ is applied and the features are projected to\nthe embeddings $z, z_c^{'}, \\hat{z}_m$ in a high dimension space. \n\\ie $z = g(h), z_c^{'} = g(h_c^{'}), \\hat{z}_m = g(\\hat{h}_m)$. \nTo capture the relationship between masked joints and unmasked ones, \nwe compute the cross-correlation matrix $\\mathcal{C^{'}}$ between embeddings $z$ and $z_c^{'}$. \nThe calculation process refers to Eq.~\\eqref{eq:cc}.\nAnd then $\\mathcal{L}_1$ is applied:\n\n\\begin{equation}\n\\label{eq:BT1}\n\\mathcal{L}_1 = \\sum_{i}\\left(1-\\mathcal{C^{'}}_{i i}\\right)^{2}+\n\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{C^{'}}_{i j}^{2},\n\\end{equation}\nwhere the first term encourages the diagonal element of $\\mathcal{C^{'}}$ to 1.\nIt forces the representation of partial data to be similar to the total one. \nThe second term is designed to prompt the different embedding components to be decoupled with each other,\nwhich helps to minimize the redundancy within the representation and avoid it being a constant. \n\nSimilarly, the cross-correlation matrix $\\mathcal{\\widehat{C}}$ between $z$ and $\\hat{z}_m$\nis acquired for the $\\mathcal{L}_2$:\n\\begin{equation}\n\\label{eq:BT2}\n\\mathcal{L}_2 = \\sum_{i}\\left(1-\\mathcal{\\widehat{C}}_{i i}\\right)^{2}+\n\\lambda \\sum_{i} \\sum_{j \\neq i} \\mathcal{\\widehat{C}}_{i j}^{2},\n\\end{equation}\nwhich is utilized to capture the relationship between masked and unmasked frames.\nNote that the trade-off parameter $\\lambda$ is used to balance the dimension difference \nbetween first and second terms, so it keep same on Eq.~\\eqref{eq:BT1} and Eq.~\\eqref{eq:BT2}. \n\nThe total loss $\\mathcal{L}_p$ for our PSTL can be formulated as:\n\n\\begin{equation}\n\\mathcal{L}_p = \\mathcal{L}_1 + \\mathcal{L}_2.\n\\end{equation}\n\\section{Related Work}\n\\label{sec:related}\n\n{\\noindent\\bf Skeleton-based action recognition.}\nTo tackle skeleton-based action recognition,\nearly methods mainly focus on hand-crafted features~\\cite{vemulapalli2014human,xia2012view}.\nWith the development of the deep neural network, \nskeleton data is processed as pseudo-images with CNNs~\\cite{ke2017a,kim2017interpretable}.\nOthers structure it as a long-term sequence\nand feed it into RNNs~\\cite{du2015hierarchical,zhang2017view}.\nLater on, ST-GCN~\\cite{yan2018spatial} structures the skeleton data as a predefined spatial graph and \nutilize GCN to integrate the skeleton joint information. \nAfter that, many GCN-based methods are proposed, \nwhich contain more data stream or add attention mechanisms~\\cite{shi2019two,song2020stronger,chen2021multi}. \nIn this paper, we also utilize the widely used ST-GCN as the backbone. \n\n\n{\\noindent\\bf Self-supervised representation learning.}\nSelf-supervised learning aims to acquire a high-level representation from a large number of unlabeled samples. \nEarly methods rely on pretext tasks to generate supervision, \nsuch as jigsaw puzzles~\\cite{noroozi2016unsupervised,wei2019iterative}, \ncolorizing~\\cite{larsson2016learning, zhang2016colorful}, \nor rotation prediction~\\cite{gidaris2018unsupervised}. %\nThen, discriminative methods like MoCo~\\cite{he2020momentum} and SimCLR~\\cite{chen2020simple} achieve \nremarkable performance, \nwhich are comparable to supervised learning. \nBy distinguishing the positive pairs from the negative ones, \nthe backbone network can acquire the representation with rich discriminant information. \nBut it requires a large batch size or memory bank to store the negative samples. \nTo tackle this issue, negative-sample-free methods are introduced \nsuch as BYOL~\\cite{grill2020bootstrap}, SimSiam~\\cite{chen2021exploring}, and Barlow Twins~\\cite{zbontar2021barlow}. \nThey do not require explicit construction of negative sample pairs,\nbut most of them utilize an asymmetric network to avoid feature collapse.\nSpecially, \nBarlow twins avoids the reliance on the symmetry-breaking network \nbut minimizes the redundancy within the representation vector to alleviate collapse. \nMore recently, mask auto-encoder~\\cite{he2022masked} is introduced to learn effective visual representations\nfrom local complementary information.\n\n{\\noindent\\bf Self-supervised skeleton-based action recognition.}\nTo apply self-supervised learning to skeleton data,\nLongT GAN~\\cite{zheng2018unsupervised} uses a recurrent encoder-decoder GAN to reconstruct the input sequence.\nBased on this, Predict\\&Cluster~\\cite{su2020predict} proposes\na decoder to improve the representation ability of the encoder. \nMS2L~\\cite{lin2020ms2l} proposes a multi-task self-supervised learning framework,\nincluding motion prediction and jigsaw puzzle. \nAS-CAL~\\cite{rao2021augmented} uses momentum LSTM to regularize the feature space together with several skeleton augmentation strategy.\nISC~\\cite{thoker2021skeleton} proposes inter-skeleton contrastive method that combines sequence-based and graph-based representation.\nCrossCLR~\\cite{Li_2021_CVPR} introduces\nSkeletonCLR that applies memory bank to store the negative samples and \na cross-view knowledge mining strategy to capture more comprehensive representation. \nAimCLR~\\cite{guo2022contrastive} aims to utilize plenty of augmentation to introduce movement patterns and force \nthe encoder to learn more general representations. \nHowever, these contrastive-based methods heavily rely on strong data augmentation strategies, \nbut the redundancy within the spatial joints and temporal frames have not been considered, \nwhich can improve the robustness of 3D visual representation and essential for downstream task.\nHence, we introduce PSTL and propose a spatial-temporal mask strategy to find a more general representation. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the search for carriers of unexplained interstellar absorption spectra, the\nelectronic properties of rarely studied systems offer a viable new avenue of\nresearch to explore.  The set of as-of-yet unexplained visible to near-infrared\npeaks known as the diffuse interstellar bands (DIBs), for example, has remained\na mystery for nearly a century \\citep{Sarre06}. However, it has been suggested\nthat anions may be responsible for some of the DIB features \\citep{Sarre00,\nCordiner07}.  Recent work has shown a relative richness of electronic absorption\nfeatures below the electronic binding energy (eBE) \\citep{Simons08, Simons11} for\nsome anions that exhibit singlet dipole-bound excited states and even some\nspecies that appear to possess additional valence\nexcitations \\citep{Fortenberry11dbs, Fortenberry112dbs} For any dipole-bound\nanion, the corresponding radical must possess large a dipole\nmoment \\citep{Fermi47, Turner66, Coulson67, Crawford67, Jordan76, Turner77,\nCrawford77, Gutowski96, Wang02, Jordan03} in order for the electron to remain\nbound within the system.  In an ideal situation the dipole moment must be at\nleast greater than 1.625 D \\citep{Fermi47, Turner66, Coulson67} and probably\ncloser to 2.5 D \\citep{Cordiner07, Gutsev95a, Ard09} typically in a highly\ndiffuse orbital \\citep{Mead84}.  However, valence ground and excited states may\nexist below the single dipole-bound state which functions as the upper\nexcitation energy limit since the dipole-bound state excitation energy must be\nnearly coincident with the eBE \\citep{Simons08}.\n\n\n``Bound excited valence states in molecular anions are rare\" \\citep{Brinkman93}.\nHowever, some anions are known to exhibit valence electronic excitations, but\nmost of these are open-shell anions where excitations do not have to overcome\nthe additional energy barrier of sorts involved in splitting an electron pair\nwhile still remaining below the eBE.  Negatively charged small fragments or\natoms \\citep{Brinkman93} and, especially, carbon chains \\citep{Maier98,\nJochnowitz08, Zhao09} have been documented to possess bound valence excited\nstates.  Additionally, some anions, including many with second row atoms, are\nknown to possess triplet excited states above the ground state \\citep{Meloni04,\nPino04, Sheehan08, Inostroza10} where the energy splitting between the ground\nstate (which may either be singlet or triplet) and the triplet excited state is\nrelatively small.  Much larger anions are also known to possess excited\nstates, \\citep{Brinkman93, Skurski00TCNQ, Sobczyk03} and such behavior is\nrelevant to the study of many biological species \\citep{Chen01}. However, a new\nset of small, closed-shell singlet anions have been studied previously for\ntheir electronic properties \\citep{Fortenberry11dbs, Fortenberry112dbs}.  They\nrepresent an emerging class of anions that are not: 1) large, long, or exotic,\n2) atomic, elemental, or diatomic, or 3) radical, diradical, or open-shell.\nThey are closed-shell anions of three to seven atoms (with no more than four\nheavy atoms) that are composed of common substituent groups.\n\nSilicon is present in each of the small singlet systems where previous study\nindicates the existence of valence excited states \\citep{Fortenberry11dbs,\nFortenberry112dbs}. Silicon and other higher row main group elements are\nwell-known to have very different bonding environments than their more common\nC, O, and N (first row) analogues \\citep{Kutzelnigg84, Driess96, Power99,\nBrown00, Owens06, Woon09SFn, Woon10PFn, Woon11}. Silicon provides many\nunexpected differences in the molecular structures as compared to carbon\nmolecules of the analogous chemical formula including the butterfly conformer\nof Si$_2$H$_2$ and cyclic Si$_3$ \\citep{Li95, Chesnut02, Reilly12}. Steric\neffects from the presence of the larger $n=3$ orbitals in the higher row atoms\ninhibit standard valence bond $sp^3$\/$sp^2$ hybridization or electron pair\nrecoupling \\citep{Woon11} making molecules containing these atoms ``reluctant to\nhybridize\" \\citep{Kutzelnigg84, Driess96, Owens06}. The lack of hybridization is\nmost extreme in second row atoms, most notably for silicon, which forms strong\n$\\sigma$ bonds due to an increase in its inclusion of the $s$-type atomic\norbitals \\citep{Power99, Brown00, Owens06}.  Hence, silicon is even less likely\nto create bonds of significant hybridized character than its rowmates\nphosphorus and sulfur.  The increase in $s$ character is most straightforwardly\nobserved in $^1A_1$ SiH$_2$ where the optimal bond angle is\n92.1$^{\\circ}$, \\citep{Dubois68, Driess96} much closer to 90$^{\\circ}$ than\n102$^{\\circ}$, which the bond angle for singlet carbene \\citep{Bourissou00}. Even\nso, the carbene carbon here exibits similar effects since the bond angles in\nboth $^1A_1$ CH$_2$ and SiH$_2$ are much less than the 120$^{\\circ}$ bond angle\nexpected for standard $sp^2$ hybridized atoms \\citep{Woon11}.\n\nThe previous work examining the singlet excited states of the aforementioned\nsmall, closed-shell anions \\citep{Fortenberry11dbs} employed coupled cluster\nmethods to reproduce the experimental eBEs and the lone dipole-bound state\nexcitation energies of CH$_2$CN$^-$ \\citep{Sarre00, Cordiner07, Lykke87} and\nCH$_2$CHO$^-$ \\citep{Mead84, Mullin92, Mullin93} to within 0.06 eV or better.\nThe same accurate methodology was extended to provide likely dipole-bound and\nvalence excited states of other anions for which experiments have not yet been\nperformed.  The present study employs the same methodology and is an\nexamination of the excited state properties of similar molecular anions\ncontaining the second row elements aluminum, silicon, phosphorus, and sulfur\nand how they relate to their first row analogues.  Some systems analyzed have\nbeen constructed to produce structures related to those where the previous\ncomputational studies indicate the presence of dipole-bound and valence excited\nstates for silicon-containing species \\citep{Fortenberry11dbs,\nFortenberry112dbs}.  Other anions of interest to this work are unique attempts\nto find if the larger atoms play a role in the stabilization of valence\nexcitations below the eBE.  The search for dipole-bound states provides\nmotivation for analysis of these types of systems, but the search for even\nrarer valence states offers a more fascinating result.  It is currently unknown\nhow anions may affect attribution of the DIBs, but the probable existence of \nexcited states for previously unexamined molecules may hold some new insight\ninto this near-century-old problem.\n\n\n\\section{Computational Details}\n\nCoupled cluster theory is one of the most accurate quantum chemical methods\ndeveloped to date, \\citep{Helgaker04, Lee95Accu, Shavitt09} and our\npast \\citep{Fortenberry11dbs, Fortenberry112dbs} and current computations of\nexcited states of anions exclusively utilize this method extensively.  The\ngeometries are optimized at the coupled cluster singles, doubles, and\nperturbative triples level [CCSD(T)] \\citep{Ragh89} in conjunction with Dunning's\nsingly augmented correlation consistent triple-zeta basis set, aug-cc-pVTZ\n\\citep{Dunning89, cc-pVXZ} and aug-cc-pV(T+d)Z for the higher row\natoms \\citep{Dunning01}.  Spin-restricted (RHF) \\citep{ScheinerRHF87, Lee91}\nreference wavefunctions are used in computations involving the closed-shell\nanions while spin-unrestricted (UHF) \\citep{Gauss-UHF91, Watts-UHF92}\nwavefunctions are chosen for the open-shell radical computations.  Dipole\nmoments are computed at the UHF-CCSD\/aug-cc-pVTZ level of theory from the\nUHF-CCSD(T)\/aug-cc-pVTZ optimized geometry of the radical.\n\nThe computation of vertically excited states from the RHF-CCSD(T)\/aug-cc-pVTZ\n{\\footnote{The aug-, n-aug-, and cc-pV(X+d)Z basis sets for the higher row\natoms are simply referred to as the aug-, n-aug-, and cc-pVXZ basis sets beyond\nthe Computational Details section for ease of discussion.}} reference geometry\nmakes use of equation of motion coupled cluster (EOM-CC) theory\n\\citep{Stanton93EOM, Monkhorst77, Mukherjee79} with increasingly diffuse basis\nsets constructed in an even-tempered fashion.  Inclusion of this series of\nbasis sets provides evidence as to the classification of whether a state is\nvalence or dipole-bound.  A precipitous decrease in excitation energy from a\nstandard cc-pVXZ (where X=D or T) basis set \\citep{Dunning89, cc-pVXZ,\nDunning01} to t-aug-cc-pVXZ indicates the presence of a dipole-bound state.\nHowever, such is only the case if the excitation energy is below or within a\ncomputational limit (usually 0.1 eV) of the vertical eBE\n\\citep{Fortenberry112dbs} computed with the same level of theory and basis set\nfor the reference geometry.  These vertical eBEs are computed with the equation\nof motion coupled cluster theory for ionization potentials (EOMIP)\n\\citep{Stanton94:EOMIP} approach where an electron is removed from the HOMO of\nthe anion.  A valence state must also be less energetic than the vertical eBE,\nbut its transition energy is typically much less than this limit and certainly\nbelow that of the dipole-bound state excitation energy \\citep{Fortenberry11dbs,\nSimons08}. Additionally, the decrease in vertical excitation energy with more\ndiffuse basis sets is much less striking for a valence state where nearly all\nof the necessary orbital extent is recovered with the use of the aug-cc-pVXZ\nbasis \\citep{Skurski00}.  Convergence is thus clearly achieved for computations\ninvolving the other, more highly diffuse bases.\n\nAdiabatic computations of the eBEs and excitation energies further provide\ninsight into the potential existence of either type of excited state since they\nallow for a more direct comparison to experiment \\citep{Fortenberry11dbs,\nFortenberry112dbs}.  Adiabatic eBEs represent the energy difference between\nCCSD(T)\/aug-cc-pVTZ optimized geometries of the neutral and the\nanion \\citep{Simons11}.  Similarly, the adiabatic excitation energies are the\nenergy difference between the CCSD\/d-aug-cc-pVDZ optimized ground state and the\nEOM-CCSD\/d-aug-cc-pVDZ optimized excited state.  Also, the optimized geometries of\nthe ground state radicals and dipole-bound excited states should be very\nsimilar, which gives a further means by which excited states can be classified.\n\nCore orbitals are frozen in all computations: $1s^2$ for B, C, N, and O with\n$1s^2 2s^2 2p^6$ frozen for Al, Si, P, and S.  All geometry optimizations are\nminima according to harmonic vibrational frequency analyses unless otherwise\nnoted and discussed.  The EOMIP-CC computations make use of the\n{\\footnotesize{CFOUR}} \\citep{CFOUR} quantum chemical program package, while all\nother computational results were obtained using the\n{\\footnotesize{PSI}}{\\small{3}} suite of computational chemistry\nprograms \\citep{Psi3}.\n\n\n\\section{Results and Discussion}\n\nThe presence of valence excited states in silicon-containing anions from our\nprevious studies \\citep{Fortenberry11dbs, Fortenberry112dbs} has led us to to\nexplore if silicon's properties are unique in the retention of anionic excited\nstates, especially valence states, or if the general properties of higher row\natoms allow for behavior similar to that previously computed.  The systems\nchosen for this study all have closed-shell ground states that are valence in\nnature.  In order to meet this requirement, the first set of systems considered\nhave a methylene group on one end of the structure and a single ``dipole moment\ninducing\" atom (labeled as M) on the other where the two are bridged by our\natom of interest, X.  These anions have the form CH$_2$XM$^-$ since\nCH$_2$SiN$^-$ appears to possess both valence and dipole-bound excited states.\nIn the CH$_2$XM$^-$ systems, X=N, P, O, and S which compare with X=C and Si\nfrom \\citep{Fortenberry11dbs}.  M equals N when X is either O or S, but\nM=O where X=N and P.  The choice of M is limited to first row atoms of groups\n15 or 16 due to electronegativity effects, which influence dipole moment\nstrength, and the lack of dipole-bound states previously computed for fluorine\ncontaining anions \\citep{Fortenberry11dbs}.\n\nThe second set of systems studied are carbide groups paired with XH$_n$ groups.\nHere the X atoms studied are C, N, O, Si, P, and S.  The $n$ values are 2 when\nX=N and P, and $n=1$ for all other systems.  This construction gives six\ndifferent anions including C$_3$H$^-$ and CCSiH$^-$.  These are chosen since\nthe carbide containing CCSiN$^-$ has been shown to possess two valence excited\nstates, \\citep{Fortenberry112dbs} and we are exploring how the carbide moiety can\naffect atoms in both rows without the inclusion of further influence from the\nadditional N atom present in CCSiN$^-$.\n\nThe third class of systems in this study is the $^1A'$ XH$_3$YH$_2^-$ systems\nwhere X=B or Al and Y=N or P.  Related systems, with BH$_3$NH$_3$ being the\nmost notable, have been the subject of much computational study, especially\nwith regards to the nature of the X$-$Y bond length, stretching frequency, and\ndissociation \\citep{Thorne83, Binkley83, Marsh92, Leboeuf95, Reinemann11,\nSams12}. Presently, this system allows us to probe anions with group 13 elements\nand further explore group 15 elements.  The group 15 atoms of the\nXH$_3$YH$_2^-$ systems are chosen to be the Y atoms in the YH$_2$ groups.  This\narrangement creates dipole moments in the radicals large enough to allow\npotentially for dipole-bound states.  Additionally, some anionic and\ncorresponding radical systems with the group 13 atoms in YH$_2$ are not stable\nstructures.\n\n\\subsection{Excited States of CH$_2$XM$^-$ Anions}\n\n\nTable \\ref{ebe} and Table \\ref{basis} provide a summary of the data for each of\nthe systems of interest for this work.  The first system similar to\nCH$_2$CN$^-$ and CH$_2$SiN$^-$ listed in Table \\ref{ebe} is CH$_2$ON$^-$, which\nis also shown in Figure \\ref{ebeFig}a \\& Figure \\ref{AdFig}a.  CH$_2$ON$^-$ is\nsimply a replacement of the cyano carbon with an oxygen to create an anion\nsimilar to CH$_2$CN$^-$.  Both the neutral radical and the anionic forms of\nCH$_2$ON are asymmetric molecules as has been shown previously for the\nradical \\citep{Shapley98}.  The adiabatic eBE for CH$_2$ON$^-$ cannot actually be\ndetermined here since the $C_1$ computations are not well-behaved in the SCF\nstep.  However, it can clearly be determined that the eBE will be lower than\n0.46 eV, which is the adiabatic eBE from consideration of the planar CH$_2$ON\nradical and the optimized asymmetric anion.  This upper bound negates the\nexistence of any stable excited states for CH$_2$ON$^-$ and is much lower in\nenergy than the vertical eBE. Even though two vertically excited states are\ncomputed to lie below the vertical eBE, only one of these dipole-bound states\ncould exist if the adiabatic eBE was much higher \\citep{Fermi47, Simons08}.\n\n\n\nIsomerization of CH$_2$ON$^-$ to CH$_2$NO$^-$ results in a species that is more\nenergetically favorable.  The computations report that CH$_2$NO$^-$ is 75.0\nkcal\/mol more stable than the CH$_2$ON$^-$ isomer.  The corresponding radicals\nare similarly separated by around 50 kcal\/mol, which is in agreement with\nprevious computations at various levels of theory \\citep{Shapley98, Shapley99}.\nThese two anions have comparable dipole moments of $\\sim 2.3$ D as shown in\nTable \\ref{ebe}.  CH$_2$NO$^-$ and its excited state properties have been\npreviously examined by \\cite{Fortenberry11dbs}, but they are relevant for\nthis discussion.  The 1.55 eV adiabatic excitation energy to the lowest excited\nstate (2 $^1A'$) is more than 0.1 eV higher than the 1.42 eV adiabatic eBE,\nwhich indicates that this state does not exist.  However, vertical excitation\nenergies from the RHF-CCSD(T)\/aug-cc-pVTZ optimized geometry reported in Table\n\\ref{basis} show a clear pattern of energy decrease with the inclusion of more\ndiffuse functions indicative of a highly diffuse dipole-bound state.\nInterestingly, the vertical CCSD\/t-aug-cc-pVTZ eBE computed via EOMIP is much\nhigher than the vertical EOM-CCSD\/t-aug-cc-pVTZ 2 $^1A'$ state energy.\nHowever, the adiabatic results better model physical behavior and this leads to\nthe conclusion that, again, CH$_2$NO$^-$ probably does not possess any bound\nsinglet excited states.\n\nPhosphorus substitution for the nitrogen atom in CH$_2$NO$^-$ to give\nCH$_2$PO$^-$ and sulfur substitution of the oxygen atom in CH$_2$SN$^-$ both\nappear to give dipole-bound states.  CH$_2$PO radical's dipole moment of 2.477\nD is well above Fermi and Teller's \\citep{Fermi47} theoretical 1.625 D necessary\ndipole moment for a dipole-bound state in the corresponding anion and right at\nGutsev and Adamowicz's \\citep{Gutsev95a} more practical 2.5 D limit.  The 2.477 D\ndipole moment for CH$_2$PO is higher than either of the previous first row\nanalogues, indicating that presence of the second row atom has some effect on\nthe system.\n\nThe adiabatic transition energy to the first excited state of CH$_2$PO$^-$, 1\n$^1A''$, is computed to be 2.89 eV, while the eBE is 2.82 eV.  This is within\nthe 0.1 eV limit of computational accuracy.  Vertically, the eBE is 2.79 eV,\nand the vertical excited state energy for the t-aug-cc-pVDZ basis set is the\nsame.  Coupling these factors with a steep decline in the vertical excitation\nenergy for increasingly more diffuse basis sets shown in Table \\ref{basis}\ngives classic behavior of a dipole-bound excited state.  However, the\nexcitation into a highly diffuse $s$-type (totally symmetric) orbital\nnecessitates that the term for the dipole-bound excited state must be directly\nrelated to the term of the radical, $^2A'$ in this case.  There is a second\nexcited state of CH$_2$PO$^-$, 2 $^1A'$, that has an adiabatic excitation\nenergy (3.08 eV) much higher than 2.82 adiabatic eBE.  Its t-aug-cc-pVDZ\nvertical excitation energy of 2.81 eV is only 0.02 eV higher than the vertical\neBE.  Another key hallmark of a dipole bound state is that due to the\ndiffuseness of the orbital in which the bound electron is held, the excited\nanion will take on not only the similar term but also the geometry of the\nradical whose dipole moment is binding the electron.  Figure \\ref{AdFig}d shows\nthat the 2 $^1A'$ CH$_2$PO$^-$ geometry is very much like the Figure\n\\ref{ebeFig}d CH$_2$PO geometry.  Additionally, only one dipole bound state can\nexist for any given anion \\citep{Fermi47, Simons08}.  Hence, the 2 $^1A'$ state\nis actually the dipole-bound state, but the adiabatic results for this\ndipole-bound excited state do not suggest that it will be stable since it is\nmore than 0.1 eV higher than the eBE.\n\n\nThe 1 $^1A''$ state of CH$_2$SN$^-$ behaves somewhat differently than the\nanalogous 3 $^1A$ state of CH$_2$ON$^-$.  There is relatively little change in\nthe vertical excitation energy (0.72 eV) from cc-pVDZ to t-aug-cc-pVDZ for the\nCH$_2$SN$^-$ 1 $^1A''$ state, and the vertical eBE at 2.24 eV is higher than\nthe t-aug-cc-pVDZ excitation energy at 2.04 eV.  However, the change from d- to\nt-aug-cc-pVDZ excitation energy is fairly substantial at 0.39 eV.  This is over\nhalf of the total decrease in excitation energy across the range of diffuse\nbasis sets indicating that there is a mixture of valence and dipole-bound\ncharacter.  Even so, there is not enough pure valence character in the\nexcitation in order for this state to be predominantly valence in nature.\nSimilar vertical CC3 excited state computations also do not give any further\nevidence to show that the 1 $^1A''$ state is a valence state.  Additionally,\nthe adiabatic excitation energy for this state is more than 0.1 eV higher than\nthe adiabatic eBE.  As a result, we must conclude that the 1 $^1A''$ state of\nCH$_2$SN$^-$ is not accessible.  Differently, the lower $2\\ ^1A'$ state is the\ndipole-bound state since the adiabatic eBE in Table \\ref{ebe} is 0.03 eV above\nthe excitation energy, the basis set convergence in Table \\ref{basis} is\nconsistent with dipole-bound states, and the geometry of the radical in Table\n\\ref{ebe} is comparable to that of the optimized $2\\ ^1A'$ state.\n\nFor CH$_2$SN, both the radical and the anion are planar while its first row\nanalogue, CH$_2$ON, is asymmetric in both electronic occupations.  The\n``reluctance to hybridize\" \\citep{Kutzelnigg84, Driess96} in the sulfur atom\nallows CH$_2$SN$^-$ to remain planar, whereas the MOs in CH$_2$ON$^-$ force the\nhydrogens in the methylene group to pyramidalize.  This same behavior is\npresent in CH$_2$CN$^-$.  Hence, the geometrical differences and the subsequent\nsubtle changes to the orbitals for the inclusion of larger atoms from the\nsecond row alone do not bring about valence states.  Otherwise, the\nsulfur-containing anion, CH$_2$SN$^-$, would have at least one valence state.\nWhile it is fascinating that these anions containing higher row atoms possess\ndipole-bound excited states and the analogous first row anions do not, it is\nthe search for valence states that is driving this study.  Even though the\nCH$_2$XM$^-$ anions are as close as possible to the structure of CH$_2$SiN$^-$,\nthese anions with second row elements do not give similar valence excitations.\nHowever, the inclusion of a single second row atom systematically increases the\neBEs as compared to the first row only anions.  This increase appears to open\nup the possibility for excited states to be present in the spectra of anions\ncontaining second row atoms, but it does not guarantee the existence of\nstable valence states. \n\n\n\n\n\n\\subsection{Excited States of CCXH$_n^-$ Anions}\n\nMolecules containing a carbide group have gained recent interest in\nrelation to the diffuse interstellar bands, \\citep{Linnartz10, Maier11} and this\nmoiety is present in CCSiN$^-$, which has been shown to possess two valence\nstates \\citep{Fortenberry112dbs}. As a result, simple anions of the CCXH$_n^-$\nfamily where X=C, N, and O, as well as their second row analogues Si, S, and P,\nare examined here.  An interesting feature of the entire CCXH$_n^-$ family of\nanions is that the corresponding radicals all have very large dipole moments\nfrom 3.409 D for C$_3$H to nearly 6 D for $C_{2v}$ CCNH$_2$.  This\nbehavior does not vary for the inclusion of the first or second row atoms; the\ncarbide moiety is responsible for the large dipole moments.\nPrevious computations of the dipole moment for CCOH are\nconsistent, \\citep{Yamaguchi98, Fortenberry11dbs} and the 3.409 D dipole moment\nof C$_3$H is also in line with previous CASSCF computations of its dipole\nmoment, 3.163 D, by Takahashi and Yamashita \\citep{Takahashi96}.\n\nThe C$_3$H$^-$ system has received much attention over the past two\ndecades \\citep{Takahashi96, Ochsenfeld97, Pachkov01, Pino04, Sheehan08} mainly\nexamining the anion to neutral (triplet to doublet) photochemistry.  Even\nthough it has been reported that the cyclic singlet is the most stable isomer\nof C$_3$H$^-$, \\citep{Sheehan08} consistency for the scope of this study and how\nC$_3$H relates to the other anions from groups 15 and 16 in these CCXH$_n^-$\nmodel systems mandates exploration of the excited state properties of the $C_s$\npseudo-linear $^1A'$ C$_3$H$^-$ and the silicon analogue, CCSiH$^-$.\nCCSD(T)\/aug-cc-pVTZ computations put the $^1A_1$ $c$-C$_3$H$^-$ isomer only 6.6\nkcal\/mol lower in energy than the $^1A'$ C$_3$H$^-$ isomer.  The pseudo-linear\nsinglet isomer is 6.9 kcal\/mol lower in energy than $^3A'$ C$_3$H$^-$.\n\nFascinatingly, pseudo-linear C$_3$H$^-$ appears to possess a valence state.\nTable \\ref{ebe} shows that the adiabatic excitation energy of the $1\\ ^1A''\n\\leftarrow 1\\ ^1A'$ transition at 0.93 eV is much less than the 1.83 eV\nadiabatic eBE.  The vertical excitation energies for the increasingly diffuse\nbasis sets given in Table \\ref{basis} decrease from 1.21 eV for cc-pVDZ merely\nto 1.15 eV for aug-cc-pVDZ and to 1.14 eV for the converged excitation energy\nfor d- and t-aug-cc-pVDZ.  The converged vertical excitation energy is well\nbelow the 2.34 vertical eBE.  Finally, the excited state geometry given in\nFigure \\ref{AdFig}k is markedly different from the radical geometry in Figure\n\\ref{ebeFig}k, especially for the H$-$C$_3-$C$_2$ bond angle.  Differently, the\nvertical excitation energy of the $2\\ ^1A' \\leftarrow 1\\ ^1A'$ transition at\n2.37 eV for t-aug-cc-pVDZ is just above (but within computational accuracy of)\nthe vertical eBE.  However, the adiabatic computations put this state at 2.05\neV, 0.22 eV above the adiabatic eBE.  Our analysis strongly indicates that a\nvalence state is present in this anion.  It may yet possess a dipole-bound\nstate in addition to its valence excited state, but our computations do not\nstrongly support this interpretation.\n\nThe silicon analogue, CCSiH$^-$, exhibits similar behavior.  The 2.06 eV $1\\\n^1A'' \\leftarrow 1\\ ^1A'$ excitation energy is also well under the 3.11 eV\nadiabatic eBE.  Table \\ref{basis} also confirms that the $1\\ ^1A''$ state is\nvalence in nature since the more diffuse basis sets lower the vertical\nexcitation energy for this state by only 0.04 eV.  The vertical eBE at 3.27 eV\nis, again, well above the vertical excitation energy.  Differently, the $2\\\n^1A'$ state appears to be the dipole-bound state of CCSiH$^-$.  Like the\nanalogous $2\\ ^1A'$ state of C$_3$H$^-$, this state is also more than 0.1 eV\nabove the eBE adiabatically,  but it is only 0.12 eV higher than the eBE here.\nThe EOM-CCSD\/t-aug-cc-pVDZ vertical excitation energy of 3.26 eV is actually\n0.01 eV less than the vertical eBE.  Hence, the dipole-bound state is more\nlikely to be present in the spectrum of CCSiH$^-$ than that of C$_3$H$^-$, but\nthe presence of dipole-bound states for these systems cannot be confirmed.\nRegardless, both C$_3$H$^-$ and CCSiH$^-$ show clear indications of possessing\n$1\\ ^1A''$ valence states, the first time our methodology is reporting a\nvalence state for a small anion without silicon.\n\n\nThe valence state for C$_3$H$^-$ is probably the result of a more favorable\norbital arrangement arising from the presence of the carbene.  This allows the\nvalence $a''$ virtual orbital involved in the excitation to be more easily\naccessed photochemically.  Figure \\ref{ebeFig}k \\& Figure \\ref{ebeFig}l show\nthat the ground state structures of C$_3$H$^-$ and CCSiH$^-$ are both\nsubstantially bent at the C$-$X$-$H (X=C, Si) bond angle.  The C$_3$H$^-$\nground state C$-$C$-$H bond angle is 109.1$^{\\circ}$, while this same angle for\nthe radical in Figure \\ref{ebeFig}k is 162.7$^{\\circ}$.  The ground state\nanionic C$-$Si$-$H bond angle is much closer to perpendicular at 99.8$^{\\circ}$\nas shown in Figure \\ref{ebeFig}l.  The similar C$-$C$-$H bend in C$_3$H$^-$\nallows its orbitals to behave more like those in CCSiH$^-$ than in previous\nfirst-row-only anions studied.  One interpretation of this result is that the\ncentral carbene carbon in the C$-$C$-$H bond angle in C$_3$H$^-$ has less\n``hybridized\" character since the C$-$C$-$H bond angle is noticeably less than\nthe desired 120$^{\\circ}$ typically associated with an sp$^2$ carbon.  Such a\nresult is, again, a well-known property of singlet carbenes \\citep{Bourissou00}.\nConsequently, the carbene carbon is behaving more like silicon, which is much\nless likely to hybridize \\citep{Kutzelnigg84, Driess96} as is evidenced by its\neven smaller C$-$Si$-$H bond angle in CCSiH$^-$ (99.8$^{\\circ}$ in the ground\nstate) consistent with the 92.1$^{\\circ}$ H$-$Si$-$H bond angle in singlet\nSiH$_2$ \\citep{Driess96, Dubois68}.  Further interpretation of the presence of\nvalence states and the C$-$X$-$H bond angles of less than 120$^{\\circ}$ in both\nC$_3$H$^-$ and CCSiH$^-$ is that the the amount of recoupled pair bonding in\nC$_3$H$^-$ is much less than that present in typical carbon systems and is more\nlike that from higher row, early $p$-block atoms such as silicon \\citep{Woon11,\nShaik12}. \n\nThe presence of the CCOH$^-$ dipole-bound 1 $^1A''$ excited state is\nunclear \\citep{Fortenberry11dbs}.  However, the existence of the dipole-bound 1\n$^1A''$ excited state of the sulfur analogue analyzed in this study, CCSH$^-$,\nis more promising.  This corresponding radical was first analyzed\nphotochemically to compare its chemistry to CCOH and HCCO \\citep{Venk91}. CCSH\nshares similarities with CCOH in that the 4.492 D dipole moment is similar to\nthe 4.401 D dipole moment of CCOH.  The adiabatic excitation energy for the 1\n$^1A''$ state of CCSH$^-$ is approximated to be 2.82 eV since the geometry\noptimizations could not converge the root mean squared force to better than\n$10^{-3} E_h\/a_0$.  However, this approximate excitation energy is still 0.04\neV lower than the adiabatic eBE.  Additionally, the trend of vertical\nexcitations places this excitation beneath the vertical eBE, giving strong\nevidence that this state is present in the electronic spectrum of CCSH$^-$.\nCombining these factors with the similarities in the optimized geometry of the\nradical and the approximate geometry of the anion, especially for the\nH$-$S$-$C$_2$ bond angle from Figure \\ref{ebeFig}m \\& Figure \\ref{AdFig}m, the\n1 $^1A''$ state of CCSH$^-$ is a dipole-bound state.  However, no evidence is\npresent for a valence state of CCSH$^-$.\n\nIt was hoped that the large 5.903 D dipole moment of the stable \\citep{Tao05}\nCCNH$_2$ radical would increase the anionic eBE enough so that a valence state\ncould exist, but our computations show that no valence state is present for\nthis anion.  A dipole-bound state does appear to be present for CCNH$_2^-$, an\nisomer of CH$_2$CN$^-$, which is the anion hypothesized to be a DIB\ncarrier \\citep{Cordiner07, Fortenberry13CH2CN-}. The adiabatic excitation energy\nof the 1 $^1B_1$ state (2 $^1A'$ in the vertical computations) of CCNH$_2^-$ is\n2.00 eV, while the adiabatic eBE is nearly coincident at 1.97 eV, which is\nwithin the computational limit for this state to exist.  The vertical\ncomputations fully place the eBE higher than the excitation energy.\nAdditionally, the optimized bond angles and bond lengths of the anion excited\nstate and those of the radical given in Figure \\ref{ebeFig}e \\& Figure\n\\ref{AdFig}e show a strong correspondence, and the excited state is of $C_{2v}$\nsymmetry like that of the radical.  Hence, the first excited state of\nCCNH$_2^-$ is the dipole-bound state.  No valence states exist.\n\nThe second row analogue, CCPH$_2^-$, exhibits similar dipole-bound behavior in\nits 2 $^1A'$ state.  The dipole moment of the radical is smaller than the N\nanalogue, but it is still quite large at 4.759 D.  The 2 $^1A'$ state excites\nat 3.16 eV from adiabatic computations.  The adiabatic eBE is slightly more\nthan this at 3.21 eV, which is strong evidence that this state is bound upon\nexcitation.  Additionally, the energy progression with increased diffuse\ncharacter of the basis set for the vertical excitation energy of this\nCCPH$_2^-$ state clearly shows that it is indeed dipole-bound in nature and\nalso underneath the vertical eBE.  Comparison of the geometries in Figure\n\\ref{ebeFig}f \\& Figure \\ref{AdFig}f gives further evidence for the 2 $^1A'$\nstate to be dipole-bound.  Like its first row analogue, CCPH$_2^-$ possesses\none state that exhibits dipole-bound character, but no valence states appear to\nbe present even though the adiabatic eBE for CCPH$_2^-$ is 1.24 eV greater than\nthat of CCNH$_2^-$.\n\n\n\\subsection{Excited States of XH$_3$YH$_2^-$ Anions}\n\nThe XH$_3$YH$_2$ systems are all depicted in Figure \\ref{ebeFig} and Figure\n\\ref{AdFig} in items g-j.  These systems are based on the dative bonding\nmolecule ammonia borane, BH$_3$NH$_3$, but the removal of a hydrogen from the\nnitrogen changes the bonding chemistry.  For instance, experiment and theory\nboth agree that the B$-$N bond length in standard ammonia borane is 1.657\n\\AA\\ \\citep{Thorne83, Sams12}.  The lone pair on the ammonia molecule fills the\nempty out-of-plane $p$ orbital in the borane.  Hence, the bonding environment\nis not truly covalent but dative as evidenced by the long B$-$N bond length.\nHowever, our systems have shorter B$-$N bond lengths than other dative\nstructures.  As shown in Figure \\ref{ebeFig}g, the 1.448 \\AA\\ B$-$N bond length\nin BH$_3$NH$_2$ is much less than the standard ammonia borane dative bond.\nEven though the anion has a longer B$-$N bond than the radical at 1.584 \\AA,\nthis bond is still much shorter than dative bonds in similar\nsystems \\citep{Reinemann11}.\n\nThe molecules examined for this study appear to have a combination of dative\nand covalent bonding due to the presence of bonds longer than typical covalent\nbonds but not as long as dative bonds.  For instance, BH$_3$NH$_2^-$ can \nbe thought of as the aminoborane anion, a $C_s$ molecule with the plane of\nsymmetry contained in the B$-$N bond bisecting the H$-$N$-$H bond angle.  The\nborane still has its empty $p$ orbital which is filled not by the nitrogen's\nlone pair but by another pair of electrons from the nitrogen giving a standard\ncovalent interaction while nitrogen retains its lone pair.  Hence, the bonding\nis mixed from the covalency of the amino group and the dative interaction\noriginating from the borane.  The ground state of the BH$_3$NH$_2$ radical\nactually has a different geometrical conformation than the anion.  Its geometry\nis rotated so that the plane of symmetry in this $C_s$ radical contains both\nthe B$-$N bond and the H$-$N$-$H bond angle.\n\nThe aminoborane anion, BH$_3$NH$_2^-$, gives some evidence for the possession\nof a dipole-bound excited state.  The radical has a dipole moment (2.524 D)\nlarge enough for one to exist, but the 2.28 eV adiabatic eBE given in Table\n\\ref{ebe} is much lower than the adiabatic excitation energy to the 2 $^1A'$\nstate at 2.53 eV.  Differently, the vertical eBE of 2.88 eV given in Table\n\\ref{basis} is coincident with the converged vertical excitation energy for\nthis state.  Even though the energy difference between the eBE and excitation\nenergy adiabatically appears to be too great to support a dipole-bound state,\nthe vertical computations show that if this excited state does exist it is\nprobably dipole-bound.  The 2 $^1A'$ state of BH$_3$NH$_2^-$ has a geometry\nsimilar in many respects to that of the radical from the the bottom values of\nFigure \\ref{ebeFig}g \\& Figure \\ref{AdFig}g, but the H$-$B$-$H bond angle\ndiscrepancies highlight that these are geometrically different enough for the 2\n$^1A'$ state not to be purely dipole-bound.  There is mixing in the character\nof the excited state wavefunction, and it is great enough to stabilize this\nstate before it assumes an optimized geometry similar to that of the radical.\nRegardless, our adiabatic energy difference in the eBE and excitation energy of\n0.25 eV is probably too great for this excited state to exist.\n\n\nNot surprisingly, both the BH$_3$PH$_2$ radical and anion have longer central\nbonds than the analogous aminoborane compounds due to the larger valence\norbitals present in the phosphorus atom.  Figure \\ref{ebeFig}h shows this\nincrease to be on the order of 0.4 \\AA\\ for the anion and 0.5 \\AA\\ for the\nradical.  Also, the B$-$P$-$H bond angles are smaller in the BH$_3$PH$_2$\nsystems than the B$-$N$-$H bond angles in the BH$_3$NH$_2$ systems.  This is,\nagain, typical of second row atoms \\citep{Driess96}.  Additionally, the $C_s$\nBH$_3$PH$_2$ radical has its plane of symmetry bisecting the H$-$P$-$H bond\nangle.  This is consistent with the corresponding anion but different from the\nBH$_3$NH$_2$ radical.  The dipole moment of BH$_3$PH$_2$ is slightly larger\nthan its first row analogue at 2.889 D.  This makes it large enough for the\ncorresponding anion to possess a dipole-bound state, which it appears to have.\nTable \\ref{ebe} shows the adiabatic excitation to the 2 $^1A'$ state of\nBH$_3$PH$_2^-$ to be 2.78 eV, while the adiabatic eBE is within our 0.1 eV\ncutoff at 2.74 eV.  The adiabatic eBE is also nearly 0.5 eV greater than the\nBH$_3$NH$_2^-$ adiabatic eBE.  The vertical excitation energy converges to the\nsame value as the vertical eBE, 3.26 eV, for the double-zeta basis sets.\nHence, the lone stable excited state, the 2 $^1A'$ state, of BH$_3$PH$_2^-$\nappears to dipole-bound.\n\n\nThe insertion of aluminum into the XH$_3$YH$_2^-$ systems brings about some\ninteresting chemistry.  From Figure \\ref{ebeFig}, the Al$-$N bond (1.887 \\AA)\nin AlH$_3$NH$_2^-$ is longer than the original B$-$N bond (1.584 \\AA), but it\nis not quite as long the B$-$P bond in BH$_3$PH$_2^-$ (2.013 \\AA).\nAdditionally, the Al$-$N bonds in both the AlH$_3$NH$_2$ radical and anion are\nshorter than the previously computed Al$-$N bond lengths in AlH$_3$NH$_3$,\nwhich are more than 2.0 \\AA, irrespective of the level of theory\nemployed \\citep{Marsh92, Leboeuf95}. This is similar to the mixed covalent\/dative\nbonding in the boranes above.  However, the N$-$Al$-$H bond angle contained\nwithin the plane of symmetry in these $C_s$ molecules is beyond perpendicular\nfor both the AlH$_3$NH$_2$ (68.2$^{\\circ}$) and AlH$_3$PH$_2$ (69.7$^{\\circ}$)\nradicals.  The anions revert back to more typical bond angles \\citep{Marsh92,\nLeboeuf95} once the out of plane $p$ orbital in the N or P atom is filled to\ngive the standard lone pair.\n\nBoth alane species, AlH$_3$NH$_2^-$ and AlH$_3$PH$_2^-$, have two excited\nstates adiabatically either below or within 0.1 eV of their respective eBEs as\ngiven in Table \\ref{ebe}.  The inclusion of aluminum significantly increases\nthe eBEs as compared to the aminoborane or phosphinoborane anions, especially\nwhen phosphorus is also included.  The corresponding radical to AlH$_3$NH$_2^-$\nhas the stronger dipole moment at 2.579 D whereas the dipole moment of\nAlH$_3$PH$_2$ is somewhat below Gutsev and Adamowicz's \\citep{Gutsev95a} 2.5 D\nlimit at 2.167 D.  This ensures that neither dipole moment is strong enough to\nbind a second dipole-bound state and maybe not a first \\citep{Fermi47,\nGutsev95a, Simons08}.  The only way that both states in each alane anion may be\nretained is if one is valence.  Table \\ref{basis} gives the vertical excitation\nenergies and eBEs for both the 2 $^1A'$ and 1 $^1A''$ states of both alanes.\nIn each of the two cases, the higher 1 $^1A''$ state excitation energy (3.85 eV\nand 3.86 eV, respective of AlH$_3$NH$_2^-$ and AlH$_3$PH$_2^-$) is in the range\nof what we expect for a classic dipole-bound state since the eBEs are 3.80 eV\nand 3.85 eV, respectively.  The basis set convergence confirms the dipole-bound\ncharacter for both states of both anions.  However, if the lower energy excited\nstates are dipole-bound, only the lower energy states can exist since only one\ndipole-bound state is allowed for these dipole moment magnitudes \\citep{Fermi47}.\n\nThe classification of the lower 2 $^1A'$ states for each alane anion is a bit\nmore nebulous.  Due to the orbitals involved in the excitation with the highly\ndiffuse basis sets, no vertical excitation energies for the cc-pVDZ or cc-pVTZ\nbasis sets could be conclusively linked to comparable excitations utilizing the\nmore diffuse basis sets.  This is marked in Table \\ref{basis}.  The lack of\ncorrelation in vertical excitation energies does not indicate valence\nexcitation since all previous valence excitations characterized with this\nmethodology have had little change in the excitation energy from the cc-pVDZ\nvertical excitation energy out to those computed with more diffuse basis sets.\nAdditionally, the vertical excitation energy convergence for the double zeta\nbasis sets from the aug-cc-pVDZ to the t-aug-cc-pVDZ level has previously been\nless than 0.1 eV for valence states.  The change in vertical excitation\nenergies due to basis set augmentation for the 2 $^1A'$ state of\nAlH$_3$NH$_2^-$ is 0.30 eV and 0.42 eV for the 2 $^1A'$ state of\nAlH$_3$PH$_2^-$.  However, the change from d-aug-cc-pVDZ to t-aug-cc-pVDZ in\nthe aminoalane anion is only 0.01 eV and 0.04 eV in the phosphinoalane.  To\nfurther complicate the interpretation, the optimized geometries of the 2 $^1A'$\nstates of both anions (given in Figures \\ref{AdFig}i \\& j) are not as similar\nto the optimized geometries of the 1 $^2A'$ states of the radicals as one would\nexpect for dipole-bound states.  For instance, the Al$-$Y (Y=N \\& P) bond\nlengths are on the order of 0.1 \\AA\\ longer in the anion excited states than\nthey are in the corresponding radicals.  The Al$-$Y$-$H bond angles are larger\nin the excited 2 $^1A'$ states of the anion than they are in the radical:\n121.8$^{\\circ}$ in the AlH$_3$NH$_2$ radical as opposed to 125.7$^{\\circ}$ for\nthe 2 $^1A'$ state of the anion and 108.0$^{\\circ}$ for the AlH$_3$PH$_2$\nradical while the excited state of the anion has this bond angle at\n126.6$^{\\circ}$.  Similar to BH$_3$NH$_2$, these two closely related cases\nappear to have mixed valence\/dipole-bound character, the most of which for\nany anionic excited states examined so far with this methodology.  It is\nunclear if the valence character mixed into these excited states is large\nenough to allow the higher, clearly dipole-bound states to exist above them.\nRegardless, the presence of at least one excited state, the 2 $^1A'$, in the\nelectronic spectra is possible for both AlH$_3$NH$_2^-$ and AlH$_3$PH$_2^-$.\n\n\n\n\n\n\\section{Conclusions}\n\nIn the further examination of singlet excited states of small, closed-shell\nmolecular anions, specifically those that may possess valence excited states,\nwe have examined new species containing the second row atoms Al, Si, S, and P.\nOf the twelve new anions examined here with further inclusion of CH$_2$NO$^-$\nand CCOH$^-$ previously studied by \\cite{Fortenberry11dbs}, nine exhibit\ndefinite bound excited state character: CH$_2$SN$^-$, C$_3$H$^-$, CCSiH$^-$,\nCCSH$^-$, CCNH$_2^-$, CCPH$_2^-$, BH$_3$PH$_2^-$, AlH$_3$NH$_2^-$, and\nAlH$_3$PH$_2^-$.  Only two of these, C$_3$H$^-$ and CCNH$_2^-$, do not contain\nsecond row atoms.  Adiabatic computations indicate that CH$_2$ON$^-$,\nCH$_2$NO$^-$, CCOH$^-$, and BH$_3$NH$_2^-$ do not possess excited states,\ndipole-bound or otherwise, and CH$_2$PO$^-$ likely does not either.\n\nThe only silicon system examined in this study, CCSiH$^-$, possesses two\nexcited states: one valence and one dipole-bound.  Interestingly, the carbon\nanalogue, C$_3$H$^-$, also exhibits both a single valence excited state and a\ndipole-bound excited state.  This is possibly the result of similar orbital\narrangements present in both systems that was not present in those first and\nsecond row analogues previously studied.  In this regard, silicon does not\nappear unique in its ability to foster valence singlet excited states in\nanions.  Even so, when varying the atoms at all non-hydrogen and\nnon-dipole-inducing positions, only the inclusion of group 14 atoms has been\nshown to contribute to stable valence excitations for closed-shell anions.\nThis could also be true for inclusion of a directed group 15 and 16 atom since\nthe presence of these second row atoms increases the eBEs noticeably as\ncompared to their first row analogues.  However, the properties of anions with\ngroup 15 and 16 atoms do not appear as favorable for the existence of valence\nexcited states of closed-shell anions based on the systems analyzed both here\nand previously \\citep{Fortenberry11dbs, Fortenberry112dbs}.\n\nMixing of valence and dipole-bound character in the excited state wavefunctions\ncan be more substantial than originally thought.  This is present in some of\nthe carbide anions, but the aminoborane anion and its analogues, especially the\nalanes, have substantial mixing of valence and dipole-bound excited state\ncharacter in their first excited states.  The vertical excitation energies for\nAlH$_3$NH$_2^-$ and AlH$_3$PH$_2^-$, for example, shift noticeably for more\ndiffuse basis sets unlike typical valence excitations but not as much as the\nshifts observed for established dipole-bound anions.  The mixing is also\nshowcased in the lack of correspondence between the optimized radical and\nexcited 2 $^1A'$ state geometries.  It is unclear if the amount of mixing in\nthese first excited states gives enough valence character for the second state\nto play the role of the ``threshold resonance,\" but the order of magnitude\nvariance in the oscillator strengths (Table \\ref{basis}) between the lower and\nstronger 2 $^1A'$ states and the higher and less intense 1 $^1A''$ states of\neach alane anion would make this experimentally measurable.\n\n\nThe presence of second row atoms in the types of closed-shell anions examined\nthus far increases the likelihood of dipole-bound excited states since the eBE\nand dipole moments are both increased with the larger atoms.  Combining these\nfacts with the increased prevalence of mixing for valence and dipole-bound\ncharacter in the excited state wavefunction indicates that the larger atoms\ncreate environments more favorable for excitation of electrons in closed-shell\nanions, but the anions built around group 14 atoms, especially silicon, are the\nmost favorable for anion excitation.\n\nFinally, direct attribution of known DIBs to computed anionic excitations is\nimpractical for the present level of computational accuracy.  However, the\naccuracy range combined with the density of the DIBs (with a band present every\nnm or so between 400 nm and 880 nm) gives many options for correlation between\nthe computed transitions of our set of anions and the various\nDIBs \\citep{Joblin90, Jenniskens94, Jenn}.  In fact, one of the longest\nwavelength DIBs at 1317.5 nm \\citep{Joblin90} is in the proper range to be\nrelated potentially to the 1328 nm dipole-bound $1\\ ^1A'' \\leftarrow 1\\ ^1A'$\ntransition of C$_3$H$^-$, while the 604 nm valence $2\\ ^1A'' \\leftarrow 1\\\n^1A'$ transition is within 10 nm of more than a dozen known\nDIBs \\citep{Jenniskens94, Jenn}.\n\n\n\\section{Acknowledgements}\n\nRCF was funded for this work partly by the NASA Postdoctoral Program\nadministered through Oak Ridge Associated Universities.  Funding for this work\nhas also come from the U.S. National Science Foundation: award CHE-1058420 and\na Multi-User Chemistry Research Instrumentation and Facility (CRIF:MU) award\nCHE-0741927 and by the Virginia Space Grant Consortium in the form of a\nGraduate Research Fellowship for RCF.  The figures were generated in part with\nthe CheMVP program made available by Dr.\\ Andrew Simmonett of the University of\nGeorgia.  RCF would like to thank Dr.~David Woon of the University of Illinois\nfor discussions regarding the bonding of the higher row atoms and Dr.~Fabio\nCarelli of Spienza$-$University of Rome for insights into the nature of\nelectronic excitations and binding in anions.  Dr.~Timothy J.~Lee of the NASA\nAmes Research Center is also due thanks for his encouragement on finalizing\nthis project.  A tremendous debt of gratitude belongs to Prof.~T.~Danial\nCrawford of Virginia Tech for providing the computer resources necessary to\nexecute the computations as well as for guiding many aspects of this research\nand for assistance in editing the manuscript.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum mechanics allows for the existence of states without any classical correspondence. Examples of such states are entangled states, that may appear when describing the total state of a many particle  system, or when describing different degrees of freedom of a single particle. Entangled states cause debate because they can present properties which contradict our classical intuition. In addition,  some of these properties can be use to increase the security of quantum communication and the efficiency of  algorithmic protocols when compared with  classical techniques. This is why so much attention has been payed to problems belonging to the foundations of quantum mechanics, and in particular to quantum entanglement, which is believed to be one of its main traits \\cite{SCHRO}. One of the most important discussions on entanglement and its conflicts with classical physics concerns the realism and (non-)locality of quantum physics. As pointed out by Einstein, Podolski and Rosen in the so called EPR \\cite{EPR} paradox, entangled states are closely connected to apparent contradictions between quantum mechanics and fundamental physical assumptions. Even if experimental evidence has been obtained to support quantum mechanics against the EPR criticism \\cite{ASPECT}, it is still a matter of debate if such experiments were realised in the ideal conditions, closing all the loopholes, so as a definitive conclusion can be reached. At the same time, from the fundamental point of view, identifying precisely which essential quantum properties are involved in local realism violation is still an open problem. This is why extending local realism tests to different physical systems and different physical scenarii still presents so much interest.  \n\n\nA common property of non-local states is entanglement.  In spite of its importance and  consequences in different physical contexts, it remains an open question to determine whether a general quantum system is entangled or not and to quantify the degree if entanglement of a given state. The problem has been solved for some particular cases, as for a general bipartite system of dimension  ${\\cal H}_{2}\\otimes {\\cal H}_{2}$ and ${\\cal H}_{2}\\otimes {\\cal H}_{3}$ \\cite{PERES}, where ${\\cal H}_d$ is the one particle Hilbert space of dimension $d$. In such cases,  necessary and sufficient conditions for telling whether a given state is entangled or not exists. In other particular cases, or subspaces,   one can also find necessary and sufficient conditions. For instance,  entanglement in bipartite pure states can always be recognised and quantified irrespectively of each parties' dimension. However, when  dealing  with arbitrary states, including the more realistic mixed ones, only necessary conditions for separability (non-entanglement) can be provided. A notion that will be useful in the following of this paper is the one of {\\it entanglement witnesses}, defined as an operator $\\hat W$ for which the expectation value $\\langle \\hat W \\rangle \\leq S$ for all separable states. This ensures that the state is entangled if  $\\langle \\hat W \\rangle > S$. On the contrary, the case $\\langle \\hat W \\rangle \\leq S$ \\cite{WITNESS} is not conclusive. Examples of entanglement witnesses that are also useful for fundamental tests of quantum mechanics are Bell-type inequalities \\cite{WITBELL}, which are the main scope of this paper.\n\nBell-type inequalities are composed of combinations of observables that, when measured, allow for setting a board between crucial aspects of quantum theory and the classical one. They were formulated by J. S. Bell \\cite{BELL} as a late reply to the EPR criticism to quantum physics \\cite{EPR}. Bell inequalities aim at answering the question: is quantum mechanics a local and realistic theory?  In order to do so, they combine correlations between local measurements realised in a multiparty system. Since their first formulation, several other inequalities have been proposed, some studying the same type of problem as Bell, other generalising locality and realism tests to many observers and many possible observables. There are a number of Bell type inequalities, and classifying all of them is a work by itself \\cite{GISIN2}. \n \n \n The inequalities derived here follow the original formulation of Clauser, Horne, Shimony and Holt (CHSH)  \\cite{CHSH}, and deal with the following scenario: two observers $A$ and $B$ perform local measurements on a bipartite system. Each observer can chose among two experimental set-ups ($a$ and $a'$ for $A$ and $b$ and $b'$ for $B$). Each measurement performed by $A$ and $B$ can give only two outcomes. At its origin, CHSH inequalities have been formulated for a pair of spin $1\/2$ particles or equivalent two--level systems. In the framework of local hidden variable (LHV) theories the measurement outcomes correlation statistics which must fulfil:\n\\begin{equation}\\label{chsh}\n|\\langle \\sigma_a \\sigma_b\\rangle+\\langle \\sigma_a \\sigma_{b'}\\rangle+\\langle \\sigma_{a'} \\sigma_b\\rangle-\\langle \\sigma_{a'} \\sigma_{b'}\\rangle|\\leq 2,\n\\end{equation}\nwhere $\\sigma_{\\alpha}$ is the Pauli matrix in the $\\alpha$\ndirection.  Briefly, in a LHV theory,  one  assumes that measurements performed by each observer are independent and their outcomes have a probability distribution which is a product of independent probabilities for each subsystem. Such probabilities can also depend on some random local variable. Details are discussed in many works, as \\cite{GISIN}, for instance. It can be shown that some entangled states can  violate (\\ref{chsh}), and this experimental violation was observed using photon pairs entangled in polarization~\\cite{ASPECT}. In this case, directions $a$, $a'$, $b$ and $b'$ refer to different \norientations of polarizers placed before the detectors, determining the direction of the Pauli matrix that is measured. It was shown by Cirelson \\cite{CIRELSON} that the maximum value of (\\ref{chsh}) is $2\\sqrt{2}$, and it is easy to verify that this maximal violation can be obtained with maximally entangled states. \n\nInequalities as (\\ref{chsh}) have proved to hold for two-level systems or equivalent ones. By ``equivalent ones\", we also include Bell type inequalities involving  a continuum of possible measurement outcomes   that are dichotomised and transformed into a two outcomes measurement set-up. Dichotomization works as follows: one splits in two classes the range of possible measurement  outcomes. All results obtained lying in one of the  classes is identified to a given value ($+$ or $-$) and the results obtained in the complementary space are associated to the opposite sign. Some examples of systems where this can be sucesfully done are optical fields \\cite{BUZEK,EPJD}.  An interesting problem is to derive Bell-type inequalities for continuous variables without appealing to dichotomisation.  Cavalcanti {\\it et al.} found a way out by using second moment correlations instead of first moment ones, as done in CHSH-type inequalities as (1) \\cite{REID}. Here, we deal with this problem in a different way: by using bounded observables, one can still use CHSH-type inequalities to detect non-local properties.  In this case, a Cirelson bound depending on the norm of the measured observable can also be derived, even if, at least for the specific case treated in this paper, we have not shown yet that it can be attained. An interesting property of the inequality discussed in this work is that it can be used not only for infinite  dimensional systems, but also in $N$ levels systems, where $N$ is a finite number. In this case, the numerical value of the bound  splitting between a local theory and non-local one depends on the maximal eigenvalue of the measured observable.\n\n\n Up to now, several studies have been made on Bell-type inequalities in different contexts. The general conclusion is that the subject still presents several open questions and no general theory is available. In particular, a number of intriguing features coming out from such studies somewhat contradict our acquired quantum ``intuition\":  in  \\cite{ZUKOWSKI} it is  numerically shown that bipartite multidimensional states may violate locality tests more than two qubits. For two qubits, it has been proven that  a maximal  violation exists, and it is  given by the Cirelson bound\\cite{CIRELSON}, as will be discussed hereafter. Acin, Gill and Gisin showed, some years later, also using numerical tools, that the maximal violation for pure bipartite multidimensional systems is not obtained for maximally entangled sates \\cite{ANTONIO}. General rules relating local realism violation and entanglement have not yet been found, even if it has been proven that all non-local states are entangled in some way. This fact can be easily understood with the help of entanglement witnesses. It has been shown in \\cite{WITBELL} that Bell-type inequalities are entanglement witnesses, and all non-local states are entangled.  \n\nThe inequalities studied in this paper are based on molecular  spatial orientation correlations measurements \\cite{PRL} instead of spin-like observables. However, the same type of idea can be generalised to other continuous bounded observable. We have shown that they can be implemented using time delayed measurements of correlations between the spatial orientations of two molecules. As in usual Bell inequalities scenarios, the proposed non-locality tests rely on measurements performed independently on each molecule by  observers placed far apart enough, so that no communication between them is possible during the realization of the protocol. We have shown that the proposed inequalities can be violated by a set of entangled states. An interesting point is that the inequalities derived in the present paper can also be used as entanglement witnesses, as shown, for general CHSH-tye inequalities in  \\cite{WITNESS}. In this situation, one can loosen measurement conditions, since we are interested only in detecting a particular quantum correlation, and not a fundamental aspect of quantum physics. \n\nFrom the experimental point of view, motivations for this work are the recent advances in single molecule manipulation and detection with entanglement creation \\cite{Hettich2002} and quantum information purposes \\cite{ZOLLER, Charron2006, Yelin2006, ORIENT}.  In particular, trapped cold polar molecules are  promissing candidates for quantum information processing based on the manipulation of their rotational levels \\cite{DEMILLE}. Rotational states, which are also involved in the Bell-type inequalities studied in the present paper, are relatively long lived, allowing for short quantum gate implementation times: one can hope to perform about $10^4$ gate operations before decoherence takes place. This excellent performance when compared to cold collision based quantum gates \\cite{COLLI} is  due to the strength of  molecular interactions, based on dipolar forces. \n\nThe present paper is organised as follows: in section \\ref{section 1} we discuss some general properties of Bell-type inequalities that are useful in the context of the non-locality tests we propose. In section \\ref{section2} we explicit the inequalities based on molecular orientation and show how it can be used for non-locality tests. We exploit its performance and study some entangled states that violate it. We then discuss some constraints involved in an eventual physical implementation of our ideas in section \\ref{section3}, ending up with a concluding note in section \\ref{conclusion} \n\n\n\n\n\\section{General Properties}\\label{section 1}\n\nWe describe now some general properties of the inequalities studied in the present paper.  Our inequalities involve four observables that can be combined as follows:  \n\n\\begin{eqnarray}\\label{genchsh}\n&{\\cal B}=&O_1(\\phi_1) \\otimes O_2(\\phi_2)+O_1(\\phi_1) \\otimes O_2(\\phi'_2)+\\\\ \\nonumber\n&&O_1(\\phi'_1) \\otimes O_2(\\phi_2)-O_1(\\phi'_1) \\otimes O_2(\\phi'_2).\n\\end{eqnarray}\nThe operators appearing in the equation above are defined as follows:  operators $O_i(\\phi_i)$, $i=1,2$ are  local observables chosen by observers $1$ and $2$. The variables $\\phi_i$ also depend on local properties only, and $O_i(\\phi_i)=U(\\phi_i)O(0)U^{\\dagger}(\\phi_i)$, with $U(\\phi_i)$ a one-parameter group of unitary operators with periodicity $2\\pi$. If the operator $O_i(\\phi_i)$ is bounded, we can show that, under the assumption of local realism, (\\ref{genchsh}) satisfies:\n\n\\begin{equation}\\label{genuine}\n|\\langle {\\cal B} \\rangle| \\leq S,\n\\end{equation}\nwhere $S$ is a number, representing the maximal allowed value for a local theory to hold whenever measurements of correlations between observables are compared as in (\\ref{genchsh}). The basic assumption to derive (\\ref{genuine}) is that correlations can be described by probability distributions which are independent for each party ($1$ and $2$) and depend only on local parameters and that the spectrum of observables $O_i(\\phi_i)$ is bounded. \nThe same inequality can be derived by assuming that the average of ${\\cal B}$ is taken with respect to a separable (non-entangled) states. \n\nFor simplifying reasons, we  focus on the specific case in which  the norm of  $O_i(\\phi_i)$ is the same for each subsystem and is equal to $\\lambda_{max}$. In this case,   the numerical value of $S$ is $S=2\\lambda_{max}^2$. \n\n\n Quantum mechanics can violate inequality (\\ref{genuine}), but in order to do so,  observables $O_i$ and the state considered should be judiciously chosen. In order to check \n whether an inequality of the type of (\\ref{genuine}) allows for a  non-locality test, one should maximize the left side of (\\ref{genuine}) for an arbitrary bipartite quantum state. If  the maximum obtained value is greater than $S$, all states violating (\\ref{genuine}) are non-local. \n \n\nBefore proceeding on testing the power of inequalities of the type of (\\ref{genuine}) for non-locality tests using the proposed molecular orientation-related observables, we demonstrate some simple general properties of  (\\ref{genchsh}) which are independent of the observables $O_i$. Such properties are useful  since they help to simplify the numerical optimization while giving some physical insight. We show that, thanks to these properties, the number of  degrees of freedom to be considered in order to evaluate the maximal value of the violation  is decreased. We start by decomposing operators $O_i(\\phi_i)$ in terms of unitary transformations. Using the group property  $U(\\phi_i+\\phi_j)=U(\\phi_i)U(\\phi_j)$ we can write:\n\\begin{eqnarray}\\label{decomp1}\n&{\\cal B}=&(U_1(\\phi_1)\\otimes U_2(\\phi_2))(O_1(0)\\otimes O_2(0))+\\\\ \\nonumber\n&&O_1(0)\\otimes O_2(\\phi'_2-\\phi_2)+ \\\\ \\nonumber\n&&O_1(\\phi'_1-\\phi_1) \\otimes O_2(0)-\\\\\n&&O_1(\\phi'_1-\\phi_1)\\otimes O_2(\\phi'_2-\\phi_2)(U_1^{\\dagger}(\\phi_1)\\otimes U_2^{\\dagger}(\\phi_2)),\\nonumber\n\\end{eqnarray}\nor, more succinctly,  \n\\begin{eqnarray}\\label{decomp2}\n&{\\cal B}=&(U_1(\\phi_1)\\otimes U_2(\\phi_2))\\\\\n&&{\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2)(U_1^{\\dagger}(\\phi_1)\\otimes U_2^{\\dagger}(\\phi_2)), \\nonumber\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\label{decomp1}\n&&{\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2) \\equiv O_1(0)\\otimes O_2(0)+\\\\ \\nonumber\n&&O_1(0) \\otimes O_2(\\phi'_2-\\phi_2)+ \\\\ \\nonumber\n&&O_1(\\phi'_1-\\phi_1) \\otimes O_2(0)-\\\\\n&&O_1(\\phi'_1-\\phi_1)\\otimes O_2(\\phi'_2-\\phi_2),\\nonumber\n\\end{eqnarray}\nThe interest of such decomposition is that operator ${\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2)$ depends only on two variables and possesses the same spectrum as ${\\cal B}$. Thus, ${\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2)$  and   ${\\cal B}$ share the same entanglement witnessing properties. Of course,  ${\\cal B}$ and  ${\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2)$ do not detect the same entangled states. Nevertheless, they are connected by a local unitary transformation. \n\nThe first question we wish to answer concerns the usefulness of orientation correlation measurement for non-locality tests. In order to answer that, it is enough to determine the norm of  ${\\cal B}(0,0,\\phi_1-\\phi'_1,\\phi_2-\\phi'_2)$.  \n\nIn the next section, we will study a specific example of an operator $O_{1,2}(\\phi_{1,2})$, and the physical meaning of the previous results will be made explicit. \n\n\\section{Orientation based non-locality test}\\label{section2}\n\n\\subsection{Molecular orientation and correlation}\n\nWe introduce now an essential ingredient for the inequalities studied in the present paper, which is  molecular spatial orientation. Molecular orientation is an observable, and its classical correspondent is the spatial orientation of the molecular inter-atomic axis with respect to some reference frame. By supposing that the molecules are addressed and manipulated by a linearly polarized laser field, we can define the laser's polarization axis as $z$, and take it as a reference for molecular orientation. In this case, the orientation of a given molecular state can be defined as the average value of operator $\\cos{\\hat \\theta}$, where $\\hat \\theta $ is the angle relative to the $z$ axis.  Note that here, $\\cos(\\hat \\theta)$ is taken as an operator, in perfect analogy to the position operator and related functions.  Orientation as defined above can be experimentally measured, as discussed below. \n\n\nSince we are interested on probing properties related to a two party system, our system is composed by two molecules. We suppose that they behave like rigid rotors that can freely evolve. Their state depends thus on the  time $t$. The Hamiltonian describing each individual molecule's free evolution is $H_i=J_i^{2}\/\\hbar^{2}$, where $i=1,2$. It is expressed in units of the rotational energy. $J_i$ is the angular momentum \noperator and therefore the associated evolution operator is given by $U_i(t)= e^{-i\\pi H_i t}$, where time\n$t$ is written in units of the rotational period. $U_i(t)$ is therefore time-periodic with period 1. \n\nFor each molecule of the bipartite set, the orientation at time $t$ is defined as the expectation value of \nthe $\\hat O_i(t)=U_2^{-1}(t)\\otimes U_1^{-1}(t)\\cos(\\hat \\theta_i)U_1(t)\\otimes U_2(t)$ operator, $\\moy  {\\hat O_i(t)} = \\bra{\\psi_o}\\hat O_i(t)\\ket{\\psi_o}$, where $\\ket{\\psi_o}$ is the initial state of the system. Orientation correlations between two particles are given by\nthe average value of \n$\\moy{C(t_1,t_2)} = \\moy{O_1(t_1)\\otimes O_2(t_2)}$, and this quantity can be measured at different times $t_1$ and $t_2$ for each molecule. \nOperator $\\cos{\\hat \\theta}$  is useful for entanglement detection since it ``mixes\"\ndifferent values of $j$, without affecting their projection $m$. Previous works have considered correlations between different values of the projection $m$ of\na given (fixed) value of $j$ ~\\cite{HD,agarwal}.  \n\n\n\n\nWith an arbitrary accuracy, each molecule's state (subscripts  have been omitted) can  be considered to\nreside in a finite dimensional Hilbert space\n$\\mathcal{H}$ generated by the basis set \n$\\{\\ket{j,m}; 0\\le j\\lej_{\\text{max}},\\abs{m}\\le j\\}$,  where $\\ket{j,m}$ are the eigenstates of\n$J^2$ and\n$J_{z}$. Note that $\\mathcal{H}$ has dimension $(j_{\\text{max}} +1)^2$. The corresponding wavefunctions are  $\\braket{\\theta,\\varphi}{j,m}=Y_{jm}(\\theta,\\varphi)$, the spherical harmonics.\nIn the finite space $\\mathcal{H}^{(j_{\\text{max}})}$, the $\\cos\\theta$ operator is characterized by a discrete, non\ndegenerate spectrum of eigenvalues $\\lambda_n$, with corresponding\neigenvectors $\\ket{\\lambda_n}$, also called {\\it orientation eigenstates}. The two maximally\noriented states $\\ket{+}$ and $\\ket{-}$ are the two eigenstates\ncorresponding to the extreme eigenvalues $\\pm \\lambda_{max}$, where\n$\\lambda_{max} \\equiv \\text{Max}_n (\\lambda_n)$. In the particular case of $j_{\\text{max}} = 1$ and $m=0$, maximally oriented states can be written in the basis of the angular momentum eigenstates $\\ket{0,0}$ and $\\ket{1,0}$ as  $\\ket{+}= \\sqrt{1\/2}(\\ket{1,0}+\\ket{0,0})$ and   $\\ket{-}= \\sqrt{1\/2}(\\ket{1,0}-\\ket{0,0})$. In this particular case, $\\cos{\\theta}=  \\sqrt{1\/3}\\hat \\sigma_x$ and we recover the results of a two-level system. In this context,  the free evolution operator $U_i(t)$ changes the orientation of a state, since orientation eigenstates  are not eigenstates of the free Hamiltonian. Time evolution creates  superpositions of  orientation eigenstates, exactly as it happens when one projects photon's polarization with polarizers. \n\n\\subsection{The inequalities}\n\nWe can now combine all the ingredients to build the orientation based Bell-type inequalities. We start with a system composed by two molecules, and measurements independently performed in each one of them should be combined in order to tell whether the total molecular state violates or not local realism. The two molecule bipartite state $\\ket{\\psi_o}$ has been created at a given time $t_o$ after which it freely evolves.  We suppose that, after $t_o$, there is no interaction between molecules. \nThe total molecular state is thus given by the wavefunction $\\psi(\\theta_1,\\theta_2,\\varphi_1,\\varphi_2,t)\\equiv \\braket{\\theta_1,\\theta_2,\\varphi_1,\\varphi_2}{\\psi(t)}$, where $\\theta_i$ and $\\varphi_i$ \ndenote here the polar and azimuthal spherical coordinates in the laboratory frame. \n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{B12_l1_m0.eps}\n\n\\includegraphics[width=0.5\\textwidth]{B1_l5_m0.eps}\n\\caption{(Color online) Maximal value of $\\moy{{\\cal B}_1}$ as a function of $t_1$ and $t_2$ in units of\nthe rotational period.\nLeft $z$--axis: highest eigenvalue $\\beta_1$.\nRight $z$--axis: relative violation $b_1$ defined by Eq.~(\\ref{eq:relatviolat}).\n(a): $j_{\\text{max}}=1, m=0$, (b): $j_{\\text{max}}=5, m=0$.\n\\label{fig1}}\n\\end{figure}\n\n\n\n\nThis state is the one whose non-local properties are to be checked. We can use it to compute the average values of $\\moy  {\\hat C(t_1,t_2)}$ and combine such correlations taken at different times in a way analog to  Eq.(~\\ref{chsh}) and (\\ref{genchsh}), defining the operator\n\\begin{eqnarray}\n\\label{cos}\n{\\cal B}_1(t_1,t_2,t'_1,t'_2) \\equiv&& \\\\\n&&C(t_1,t_2) + C(t_1,t'_2) + C(t'_1,t_2) - C(t'_1,t'_2).\\nonumber\n\\end{eqnarray}\nFor a local theory (LT), it  obeys an  inequality similar to Eq.~(\\ref{genuine}):\n\\begin{equation}\n\\label{eq:threshold}\n\\abs{\\moy{{\\cal B}_1}_{\\text{LT}}} \\leq 2 (\\lambda_{max})^2;\\quad \\forall (t_i,t'_i) \\in \\mathbb{R}^2.  \t\n  \\end{equation}\nWithout loss of generality, we have assumed that each particle state resides in the same finite\ndimensional Hilbert space $\\mathcal{H}^{(j_{\\text{max}})}$. Notice that, in Equation (\\ref{cos}), time plays the role of polarisers in Bell inequalities based in the photonic polarisation. Other CHSH inequalities using the time evolution instead of polarizers\nwere studied in the literature in very different contexts: in~\\cite{KAONS,GO,GISIN} they allow the detection of entanglement between products \nof decaying mesons. In~\\cite{LEGGET,TEMP_DISC}, they reveal quantum properties of single particles. \n\n\n\\begin{figure}\n\\includegraphics[width=.5\\textwidth]{relatviol.eps}\n\\caption{Relative violation of the inequality (\\ref{eq:threshold}) as a function of the dimension of the subspace for $m=0$. Blue: relative violation with respect to the locality threshold when only the finite dimension subspace is considered. Red: relative violation when the infinite dimensional threshold of $2$ is considered.}\n\\label{relative}\n\n\\end{figure}\n\n\nWe note that ~(\\ref{eq:threshold}) is valid for all possible values of $j_{\\text{max}}$, \nand that it can, in particular, be extended to the limit $j_{\\text{max}} \\rightarrow + \\infty$, in which case the spectrum of $\\cos{\\hat \\theta}$  forms a continuum. An interesting characteristic of the separability threshold ~(\\ref{eq:threshold}) is its dependence on $\\lambda_{max}$.  \n\n\nWe will see in the next subsection how the general properties of CHSH-type inequalities can be used in the specific case discussed in this paper to show that the studied inequalities can be violated. \n\n\\subsection{Reference Frame and Temporal Origin}\n\nA straightforward application of the results derived in section \\ref{section2}  consists of showing that, in order to study the spectrum of  ${\\cal B}_1$, we can, without loss of generality, \nnumerically diagonalize it in the specific case of $t'_1=t'_2=0$.  This result is due to the fact that both operators are  related  by a local unitary transformation. Using the notation of Section II, we have here that  $t_1=\\phi'_A-\\phi_A$ and $t_2=\\phi'_B-\\phi_B$. Another interesting application of the general results of Section II consists of showing that the inequalities discussed here allow for local realism violation even in the case where observers $A$ and $B$ have different time origins. The same happens for the spatial reference frame: violation is independent of any previous agreement between observers.  However, different temporal origins and reference frames correspond to different non-local detected states, that are related to each other by  local unitary transformations.\n\n We start by discussing in more details the time origin chosen by both observers. Usually, the orientation  Bell-type inequalities depend on four times of measurements, as defined in Eq. (\\ref{cos}).   However, as pointed out previously, different times correspond to the application of different unitary transformations. We can thus apply the results of section \\ref{section2}, identifying  the general operator $\\hat O$ to the operator $\\cos{\\hat \\theta}$.  This leads to the inequalities:\n\\begin{equation}\n\\moy { {\\cal B}_1(t_1,t_2,t'_1,t'_2)}=\\moy { {\\cal B}_1(0,0,t'_1-t_1,t'_2-t_2)} \\leq 2\\lambda_{max}^2,\n  \\end{equation} \nThe first identity shows that states maximally violating the Bell-type inequality for $t_1,t_2,t'_1,t'_2$, defined as $\\ket{s_{max}(t_1,t_2,t'_1,t'_2)}$ can be obtained by the one maximally violating it for $0,0,t'_1-t_1,t'_2-t_2$, that we will call $\\ket{s_{max}(0,0,t'_1-t_1,t'_2-t_2)}$ by the application of the transformation $U(t_1) \\otimes U(t_2)\\ket{s_{max}(0,0,t'_1-t_1,t'_2-t_2)}$. Temporal uncertainties of $\\tau_1, \\tau_2$ for each one of the molecules  can always be translated as the application of the unitary operator $U(\\tau_1) \\otimes U(\\tau_2)$, so that their only effect is to change the eigenstates of operators ${\\cal B}_i$ by the same transformation. Violation can thus still be observed, and the subspace of states violating local realism are obtained by a simple unitary transformation on the original subspace. \n\n\nThe same type of argument can be used for uncertainties of the reference frame for each observer. All reference frames are connected by local unitary transformations describing rotations about some direction of space, and results for different references frames are connected by these same unitary transformations.  \n\nWe apply these results to simplify the investigation of possible violations of inequality (\\ref{eq:threshold}). \n\n\\subsection{Results}\n\n For a given value of $j_{\\text{max}}$, we have numerically diagonalized operator ${\\cal B}_1(0,0,t_1,t_2)$, and obtained for each $(t_1,t_2)$,  \n its highest eigenvalue $\\beta_1(t_1,t_2)$, which gives the maximal value of\n$\\moy{{\\cal B}_1}$  (maximal violation of (\\ref{eq:threshold})). This quantity depends on the dimensionality of the system, and we compare the amplitude of the violation when different values of  \n$j_{\\text{max}}$ are used by defining the relative violation\n\\begin{equation}\n\\label{eq:relatviolat}\nb_1(t_1,t_2) \\equiv \\frac{\\beta_1(t_1,t_2) - 2 (\\lambda_{max})^2}{2 (\\lambda_{max})^2}.\n  \\end{equation}\nResults for different values of $j_{\\text{max}}$ and $m=0$ are shown in Figure \\ref{fig1}. We can see that the proposed inequalities are violated for a significant region of parameters $t_1$ and $t_2$. Figure \\ref{fig1} also calls one's attention because of its symmetries. The central symmetry with respect to the point $t_1=t_2=0.5$ corresponds to the time reversal symmetry. One can also easily understand the mirror symmetry about the $t_1=t_2$ line with the help of the particle  exchange symmetry of operator ${\\cal B}_1$. \n\nWe have shown in \\cite{PRL} that Eq. (\\ref{eq:threshold}) can be violated by a number of pure states, and numerically studied the relative violation (\\ref{eq:relatviolat}) with increasing dimension. The results, shown in Figure (\\ref{relative}) were obtained in a particular case, where both molecules had a vanishing angular momentum $z$ axis projection ($m=0$ for both molecules).  The effects of considering different values of $m$ will be discussed in the following. We focus here in the $m=0$ case in order to discuss the behaviour of violation of (\\ref{eq:threshold}) with respect to the dimensionality of the system. We see that the maximal relative violation $b_1(t_1,t_2)$ decreases with the dimensionality of the system for low dimensions and then starts smoothly growing starting from  $j_{\\text{max}}=4$ (blue bars, Figure (\\ref{relative})). Up to now, we have not found an asymtoptical numerical behaviour for $j_{\\text{max}} \\rightarrow \\infty$. We have shown that the maximal violation is bounded by $3$. However, it is still unknown if this value can be reached. Since the exact behaviour of the violation of inequality (\\ref{eq:threshold}) in infinite dimension is still not known, one pertinent  question is   whether the proposed inequality is still violated in the limit $j_{max}\\rightarrow \\infty$  for physically acceptable states. We can see that it is indeed the case by calculating the violation relative to the classical bound obtained for an infinite dimensional system. In this case, the classical threshold separating local theory from non-local ones is $2$, since $\\lambda_{max}=1$. Figure (\\ref{relative}) shows the results of such violation (pink bars). We see that   states violating the infinite dimensional classical threshold can be  found when one considers  subspaces with dimensions higher than $N=j_{max}+1 \\geq 5$ for $m=0$. In this subspace, we can see that the violation relative to the infinite dimensional threshold is still small. However, by increasing the size of the subspace and going to $j_{max}=10$, we can see that this relative violation increases, and it is not negligible when compared to the two dimensional case, for instance. Notice that the relative violation with respect to the infinite dimensional subspace and the one relative to the restricted subspace approach with increasing dimension, as one should expect. Violating the inequalities (\\ref{eq:threshold}) in the infinite dimensional limit with entangled states of low dimension is a surprising result that proves that (\\ref{eq:threshold}) can be violated for all possible values of $\\lambda_{max}$. \n\nA natural question is what states maximally violate  (\\ref{eq:threshold}). Since we are dealing here with correlations in orientation, one could expect that maximally oriented  entangled states of the form $\\frac{1}{\\sqrt{2}}(\\ket{\\lambda_{max},\\lambda_{max}}\\pm\\ket{-\\lambda_{max},-\\lambda_{max}})$ are those that maximally violate them. However, this is not the case, and such state, except for the case $j_{max}=1$ and $m=0$ do not violate  (\\ref{eq:threshold}) at all. Nevertheless, states maximally violating  (\\ref{eq:threshold}) have  most of their population in highly oriented states. For  $j_{max}=5$, states $\\ket{\\lambda_{max},\\lambda_{max}}$ and $\\ket{-\\lambda_{max},-\\lambda_{max}}$ carry, each one of them, $36\\%$ of the population.  In addition, states maximally violating (\\ref{eq:threshold}) are not maximally entangled (except in the case $j_{max}=1$, a result which is not surprising and has been observed for other Bell-type inequalities, as in \\ref{ANTONIO}, for example. Using the maximally violating state, we have computed the reduced density matrix entropy $S=-Tr[\\rho_i\\log{\\rho_i}]$, where $\\rho_i$ is the reduced density matrix with respect to one of the two entangled molecules. In order to compare $S$ for different values of $j_{\\text{max}}$, we have normalised it with respect to $\\log{(j_{max}+1)}$, which is the maximal value of the entropy in a subspace of dimension $j_{\\text{max}}+1$. The  reduced density matrix entropy is a measure of entanglement for pure states, as the ones we are considering here.The results are shown in Figure(\\ref{entropy}), together with a plot of $\\log{2}\/\\log{(j_{\\text{max}}+1)}$.  We see that the two plots almost completely match, showing that maximally violating states are very close to entangled states involving only two orthogonal states of each molecule. We have checked that the two states involved are the maximally oriented states.  \n\n\\begin{figure}\n\n\\includegraphics[width=0.5\\textwidth]{entropy_norm.eps}\n\\caption{Normalised entropy of the maximally violating state as a function of the subspace dimensionalty $j_{\\text{max}}$ (dots). Continuous (green) line represents $\\log{2}\/\\log{(j_{\\text{max}}+1)}$.}\n\\label{entropy}\n\n\\end{figure}\n\n\n\n\nThe results presented above were obtained for the specific case of $m=0$ for both molecules. Allowing $m$ to take a value different from $m=0$ does not bring qualitative changes to our results: the classical threshold will still be given by the case $m=0$, since it corresponds to the maximal value of   $\\moy{{\\cal B}_1}$ for a local state.   However,  the dimension of the subspace considered depends on $m$. For a given fixed $m$, it is given by $j_{max}-|m|+1$. The value of the maximal orientation is thus determined not only by the dimension  of the subspace, but also on the specific value of $m$. As a consequence, the maximal value of  $\\moy{{\\cal B}_1}$ is determined by the same parameters. For instance, if  the considered state is a superposition state of different $m$'s, each one of this subspaces will lead to different contributions when computing $\\moy{{\\cal B}_1}$. In particular, those for which $j_{max}=|m|$ give a null contribution, and the maximal possible contribution decreases as $|m|$ increases for each molecule.  Physically, this is related to the fact that states with high values of the angular momentum projection $|m|$ are less oriented than those for which $j \\gg |m|$. As a conclusion, by considering different $m$'s, it is still possible to violate (\\ref{eq:threshold}), even if  fulfilling  the required conditions for it becomes harder. \n\n\n\\subsection{Non-pure states and the effect of noise}\nWe now formalize the conditions a non-pure state should satisfy in order to violate Eq. (\\ref{eq:threshold}). This is useful to estimate the effect of noise  in our system.  A non-pure bipartite state is described by the density matrix $\\rho_{1,2}$, where the subscripts $1$ and $2$ refer to each one of the observers. The average values of  (\\ref{cos}) can be obtained from $\\rho_{1,2}$ by using $\\moy{C(t_1,t_2)} = \\moy{O_1(t_1)\\otimes O_2(t_2)}= {\\rm Tr}[\\rho_{1,2} O_1(t_1)\\otimes O_2(t_2)]$. Defining the  ${\\cal B}_1$ eigenstates,  by $\\{\\ket{s_i}\\}$, with $i$ ranging form $0$ to $(j_{max}+1)^2$ if one assumes that each particle's  subspace have the same dimension and $m=0$, it is clear that the density matrix $\\rho_{1,2}$ can be expressed in this basis. By doing so,  we have that\n\\begin{equation}\\label{rho}\n{\\rm Tr}[{\\cal B}_1\\rho_{1,2}]=\\sum_i^{N^2} p_i s_i, \n\\end{equation}\nwhere $s_i$ is the eigenvalue associated to the eigenstate $\\ket{s_i}$ of ${\\cal B}_1$, $p_i=\\bra{s_i}\\rho_{1,2} \\ket{s_i}$ the statistical weight of each one of such eigenstates and $N=j_{max}+1$ is the dimensionality of each molecule's subspace. This means, that for a local realistic theory, one should obey\n\\begin{equation}\\label{rho}\n\\sum_{i=1}^{N^2} p_i s_i \\leq 2\\lambda_{max}^2. \n\\end{equation}\nAn  example of non-pure state is given by\n\\begin{equation}\\label{zz}\n\\rho_{1,2}=P_N \\frac{\\openone}{N^2}+(1-P_N)\\ket{s_{max}}\\bra{s_{max}},\n\\end{equation}\nwhere $\\openone$ is the $N^2 \\times N^2$ identity matrix, $\\ket{s_{max}}$ is the eigenstate of ${\\cal B}_1$ with maximal eigenvalue $s_{max}$ and $P_N$ is the probability of the state to be in a complete mixture. This type of state has been studied  in  \\cite{ZUKOWSKI} and can illustrate the presence of noise in the preparation of a state $\\ket{s_{max}}$maximally violating a Bell-type inequality. They defined that the robustness of a Bell-type test with respect to noise is measured by the maximal allowed value of $P_N$ still leading to locality violation.   In the notation of Eq. (\\ref{rho}), $\\bra{s_{max}}\\rho \\ket{s_{max}}=p_{max}=P_N\/N^2+(1-P_N)$ and $\\bra{s_{i} \\neq s_{max}}\\rho \\ket{s_{i} \\neq s_{max}}=p_{i}=P_N\/N^2$. By calculating the expectation value of ${\\cal B}_1$ using (\\ref{zz}) one gets that local realism should obey:\n\\begin{equation}\\label{zz2}\nP_N \\sum_{i=1}^{N^2} \\frac{s_i}{N^2}+(1-P_N)s_{max} \\leq 2\\lambda_{max}^2.\n\\end{equation}\n\nSince ${\\cal B}_1$ is traceless, $\\sum_i s_i=0$, so the inequality is violated for $P_N < 1-2\\lambda_{max}^2\/s_{max}$. A plot of $P_N$ as a function of the considered subspace is given in Figure (\\ref{noise}). Notice that, when a two-level system is considered, the known result of $P_2=1-\\frac{1}{\\sqrt{2}}$ is recovered. Contrary to other Bell-type inequalities for which $P_N$ was optimised \\cite{ZUKOWSKI}, the maximal allowed value of noise in our system decreases with dimensionality up to $j_{max}=5$ and then starts increasing again, but doesn't change significantly in the range of dimensions that we have calculated. This result shows that the non-local properties of the maximally violating state are quite robust with respect to noise.   \n\n\n\n\n\\begin{figure}\n\n\\includegraphics[width=0.5\\textwidth]{noise.eps}\n\\caption{Maximal value of $P_N$ for which our inequalities are violated as a function of $j_{\\text{max}}=N-1$}\n\\label{noise}\n\n\\end{figure}\n\n\n\\subsection{Dichotomisation}\\label{secdicho}\n\nWe have shown that  the operator defined by Eq.~(\\ref{cos})  allows not only for the realization of Bell-type tests in finite angular momentum subspaces, but also when the size of the subspace is not a priori known. This proves that, even in the case where we consider an infinite dimensional space, our inequalities can be violated for some states. However, as shown previously, the contrast of the maximal violation depends on the value of $j_{max}$. With dichotomizing procedure,  a high dimensional system is transformed into an effective two level one by splitting in two the set of measurement outcomes, and the contrast is kept constant, equal to the one for a two-level system, irrespectively of $j_{max}$. In our case, we can dichotomise  as follows: \nstates $\\ket{\\varphi}$, for which $\\moy{\\cos\\theta}_{\\varphi}>0$, are said to be positively oriented, while\nthose for which $\\moy{\\cos\\theta}_{\\varphi}\\le 0$ are considered as negatively oriented.\nWe  define the associated projectors: \n$\\Pi_{\\pm}=\\sum_{\\lambda_{\\pm}}\\ket{\\lambda_{\\pm}}\\bra{\\lambda_{\\pm}}$ which project states\nin the subspace of positive (negative) orientation.\n$\\lambda_{\\pm}$ are the positive (negative) $\\cos{\\theta}$ eigenvalues in the given\nfinite dimensional subspace and $\\ket{\\lambda_{\\pm}}$ are the corresponding eigenstates. The\nmeasured observable for each molecule $i$ is then $\\Pi_i=\\Pi_+-\\Pi_-$. \nFor a given single-molecule state\n$\\ket{\\varphi}=\\sum_{\\lambda_+}c_{\\lambda_+}\\ket{\\lambda_+}+\\sum_{\\lambda_-}c_{\\lambda_-}\\ket{\\lambda_-}$,\n$\\langle \\Pi \\rangle_{\\varphi}$ can take any value in the interval $[+1,-1]$. The total two molecule observable is  $\\Pi=\\Pi_1\\otimes \\Pi_2$. \nWe refer, as before, to two molecule correlation measurements realized at two different \ntimes, using $\\Pi(t_1,t_2)=\\Pi_1(t_1)\\otimes\\Pi_2(t_2)$ where $\\Pi_i(t_i)=U_i^{-1}(t_i)\\Pi_iU_i(t_i)$.\nIn analogy with Eq.~\\ref{cos}, we now define the operator\n\\begin{equation}\n\\label{eq:B2}\n{\\cal B}_2=\\Pi(t_1,t_2)+ \\Pi(t_1,t'_2)+\\Pi(t'_1,t_2)- \\Pi(t'_1,t'_2).\n\\end{equation} \nSince $\\Pi(t_i,t_j)^2=1$, one can show that the highest value $\\moy {{\\cal B}_2}$ can reach is given by the Cirel'son bound $2\\sqrt{2}$~\\cite{CIRELSON}. Also, for a local theory, we have\n\\begin{equation}\n\\label{dico} \n\\abs{\\moy{{\\cal B}_2}_{\\text{LT}}} \\leq 2;\\quad \\forall (t_1,t_2) \\in \\mathbb{R}^{2}. \n  \\end{equation}\nAs in the case of ${\\cal B}_1$, we can define the relative violation $b_2(t_1,t_2)$ as a function of the maximal eigenvalue $\\beta_2(t_1,t_2)$ of ${\\cal B}_2(t_1,t_2)$ (see Equation (\\ref{eq:relatviolat})).  The interest of dichotomisation is that the maximal value of  the relative violation is always given by $b_2(t_1,t_2)=\\sqrt{2}-1$,  independently of the dimension  of the subspace. Fig.~\\ref{dicho} shows the maximum value $\\beta_2(t_1,t_2)$ of $\\moy{{\\cal B}_2(t_1,t_2)}$ \nas a function of $t_1$ and $t_2$ for $j_{\\text{max}}=5$, while for  $j_{\\text{max}}=1$ we obtain trivially \nthe same result as with $b_1(t_1,t_2)$ (Fig.~\\ref{fig1}.a right z--axis). We see in Fig. \\ref{dicho} that the dichotomisation procedure not only keeps the contrast of the maximal violation constant, but also allows for high violation for a wider range of values of $t_1$ and $t_2$. \n\n\n\\begin{figure}\n\n\\includegraphics[width=0.5\\textwidth]{B2_l5_m0.eps}\n\\caption{(Color online) Same as Fig.~\\ref{fig1}.b, but for $\\moy{{\\cal B}_2}$}\n\\label{dicho}\n\n\\end{figure}\n\n\n\\section{Discussion on possible experimental implementations}\\label{section3}\nOrientation entanglement between two molecules can be created, for instance, with the help of the dipolar interaction \\cite{Charron2006}. A possible experimental scenario consists of two trapped molecules that can be submitted to spatial displacements, as it is currently done with atoms \\cite{BLOCH}. By putting molecules close enough to each other, and with the help of laser manipulation, one can tailor rotational entangled states. In order to realise Bell-type tests, molecules would have to be separated and measurements performed independently in each one of them. The proposed scenario is  realistic and there has been rapid progress on cooling, trapping and manipulating polar molecules, especially those formed with two alcali atoms. We believe that such manipulations are going to be shown in laboratories in a near future.  With respect to the tests we wish to perform in the present paper,  there are some crucial points to analyse for the  experimental implementation of the proposed Bell-type inequality test. One of them  concerns trapping conditions. Traps should allow molecules to make spatial displacements so that they are sufficiently far apart and so that signaling of one observer's results to the other is not possible during at least one molecular rotational period. This condition is attained for an inter-molecular distance $L > cT$, where $c$ is the speed of light and $T$ is the molecular rotational period. To put some numbers on it, the molecular period is usually of the order of $10^{-12}$s, giving a lower bound to $L \\approx 10^{-6}$m. Even if at this distance dipolar interaction between molecules is not completely negligible, measurements on both molecules are performed during the same rotational period, which is much faster than the characteristic interaction time related to dipolar coupling at this distance (see \\cite{Charron2006}, for example, for a detailed discussion on the relevant time scales). Optical traps seem  promising candidates to trap and to displace molecules. Optical tweezers \\cite{GRANGIER1}, optical lattices and optical conveyor belts \\cite{ARNO1} are some examples of dipole force based optical traps allowing for atomic trapping and coherence preserving displacements \\cite{GRANGIER2, ARNO1, BLOCH}. In the case of optical tweezers, highly focused beams are used to trap individual atoms \\cite{GRANGIER1}, and we can, in principle,  load two atoms, each one of them in an individual trap \\cite{GRANGIER3}. In optical conveyor belts, a stationary field made of two counter propagating beams is used to trap atoms, that can be displaced over distances of $\\approx 10$nm by changing the relative detuning between the laser beams \\cite{ARNO1}. Going from atoms to molecules in this type of experimental system could in principle be done by the usual photo association techniques, already demonstrated in the context of optical lattices \\cite{MOLECULES}, where diatomic molecules of two alcali atoms of the same specie have been produced. Once  molecules are created and trapped, they interact with each other by dipole interaction and a non-local state may be created, as discussed in the previous section. Molecules are thus separated until dipole interaction is negligible, and measurements can be performed. One natural question concerns the effect of the trapping potential itself to the orientation of the molecular state. Optical traps are based on an applied non resonant electric field that interacts with rotational levels. We suppose that molecules in our setup are created in their ground electronic level. The optical trap is created by coupling non-resonantly this electronic level to the first excited one. The detuning between laser and the electronic transition is $\\delta$. In usual experiments on optical tweezers, for instance,, $\\delta$ can vary vary in the range  $\\approx 10-10^4$GHz, providing photon emission rates from the excited electronic state in the range of $0.1-100$ MHz. These figures are of the same order of magnitude of coherence lifetime for rotational levels, while the range of variation of $\\delta$ is also of the same order of magnitude of the frequency difference between neighboring rotational levels. In order to estimate various effects of trapping lasers, we will put ourselves in a close to realistic configuration in which the rotational frequency is $\\approx 10$GHz and $\\delta \\approx 10^4$GHz. Each rotational level has a different energy, making the effective detuning $j$ dependent. This corresponds to adding a factor $\\delta_j=B j(j+1)\/\\hbar$ to the electronic detuning $\\delta$, taken with respect to the rotational ground state. Because each rotational level has a detuning which depends on the value of $j$,  there will be a phase difference between each rotational level due to the non resonant coupling to light. In the case of a harmonic trapping potential, this phase is given by $e^{-i 2\\Omega^2\/(\\delta+\\delta_j) t}$ if only one electronic transition is considered. This corresponds to a local action on each molecule that does not play a significant role on non-locality tests, since it maps one state to another one with the same degree of entanglement. As shown before, by properly choosing the time where measurements are performed, such effect can be compensated. This means that  the parameters for which maximal violation occurs may be modified.  As it can be shown, if a given state violates the inequalities of the type of (\\ref{eq:threshold}) and (\\ref{dico}), a collection  of states  connected to it by local unitary transformations will also violate the same inequalities, even if for different parameters. Also, it is important to notice that such dephasing effects can be completely neglected depending on the precise circumstances under which the experiment is performed. This happens because dephasing occurs on a time scale much longer than the rotational period, so that if measurements are performed in this interval, it will not affect the expected results.\n\n \nFinally, we address the question of orientation measurement. A possibility is to use Coulomb explosion, a destructive techniques already employed for molecular orientation measurements \\cite{SAKAI}. After a quasi-instantaneous  (when compared to the molecular rotational period), laser induced dissociative multi-ionisation, the molecular fragments are recorded at different directions of space, in the case of the orientation Bell-type inequalities, or at different hemispheres, in the case of the dichotomized inequalities. This technique is still experimentally challenging since it demands highly efficient single atom detectors. Usually this technique is employed for molecular ensembles, and the detection efficiency is less determinant than in our case. Another possibility is to detect the orientation of single molecules optically, as realized in \\cite{NIST}. Fluorescence intensity when a single molecule is excited by a linearly polarized laser beam that can change polarization in time, can reveal molecular orientation since the scalar product between the field polarization and the molecular dipole depends on its orientation. The phase of the fluorescence intensity with respect to the exciting laser beam when its polarization is turned in time, depends on the molecular orientation, as observed in \\cite{NIST}. \n \n\n\n\n\n\n \\section{Conclusion}\\label{conclusion}\n \nWe have extensively discussed some important properties of a recently proposed Bell-type inequality based on the measurement of a continuous, bounded, observable, which is molecular orientation. We have discussed some important properties of it, as its symmetries and its non-dependency on a specific choice of common temporal origin or reference frame. The role of noise in our system, that would transform pure states into statistical mixtures was also analysed, as well as how violation depends on the amount of noise. We have also discussed some important conditions that an eventual experimental set-up should satisfy and the influence of physical parameters as light forces and residual interaction between molecules on the proposed measurements.  It seems, from our analysis, that once the necessary degree of advancement is attained by experimental set-ups, our proposal is realistic and should not present major difficulties.  Our results open the perspective \nof entanglement detection and non-locality tests for high angular momentum systems in atomic and molecular physics. \n\n\nOne interesting aspect of the derived inequalities is the fact that they assume the simple form of CHSH inequalities and deal with continuous variables at the same time. This happens because  the observable that is measured for each particle has a bounded spectrum, naturally limiting the locality threshold and the norm of the Bell operator. Notice that, in spite of having discussed here the specific case of molecular orientation, the same type of inequalities could be derived for other continuous bounded observables, easily measurable in other physical contexts. \n\n\\begin{acknowledgments}\n\nThis work was partially supported by  ANR  (Agence Nationale de la Recherche), project ImageFemto number ANR-07-BLAN-0162-02 \\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA group $G$ possesses the Magnus property if for every two elements $u,v \\in G$ with the same normal closure, $u$ is conjugate in $G$ to $v$ or $v^{-1}$.\nThe Magnus property was named after W. Magnus who proved the so-called Freiheitssatz (see Theorem \\ref{Magnus}) and the Magnus property for free groups \\cite{ArtMagnus}. Since then, many mathematicians proved or disproved the Freiheitssatz and the Magnus property for certain classes of groups (see e.g. \\cite{ArtSurGr}, \\cite{ArtOneRel}, \\cite{ArtEdjvet}, \\cite{ArtHowie81}, \\cite{ArtHowSurGr}).\n\nLet $\\pi_{1}(S_{g}^{+})$, respectively $\\pi_{1}(S_{g}^{-})$, be the fundamental group of the compact orientable, respectively non-orientable, surface of genus~$g$.\nThe Magnus property of $\\pi_{1}(S_{g}^{+})$ for all $g$ was proved independently by O. Bogopolski \\cite{ArtSurGr} (by using algebraic methods) and by J.~Howie \\cite{ArtHowSurGr}\n(by using topological methods). As observed in \\cite{ArtSurGr}, there is a third, model theoretic method: two groups $G_1$, $G_2$\nare called elementarily equivalent if their elementary theories coincide: $Elem(G_1)=Elem(G_2)$, see \\cite{ArtTheorie}.\nIt is easy to show that elementarily equivalent groups either both possess\nthe Magnus property, or both do not possess it.\nSince the groups $\\pi_1(S_g^+)$ for $g\\geqslant 2$ and $\\pi_1(S_g^-)$ for $g\\geqslant 4$ are elementarily equivalent\nto the free group on two generators, all these groups possess the Magnus property.\n\nIn \\cite{ArtOneRel}, O. Bogopolski and K. Sviridov proved the following theorem:\n\n\\begin{theorem} \\label{ArtikelBSMain} \\emph{\\textbf{\\cite[Main Theorem]{ArtOneRel}}}\nLet $G=\\langle a,b,y_{1}, \\dots , y_{e} \\mid [a,b]uv \\rangle$, where $e \\geqslant 2$, $u,v$ are non-trivial reduced words in letters $y_{1}, \\dots, y_{e}$, and $u,v$ have no common letters. Let $r,s \\in G$ be two elements with the same normal closures. Then $r$ is conjugate to $s$ or $s^{-1}$.\n\\end{theorem}\n\nAs a corollary of that theorem (\\!\\!\\cite[Corollary 1.3]{ArtOneRel}), they showed the Magnus property of $\\pi_{1}(S_{g}^{-})$ for  $g \\geqslant 4$. Since the Magnus property trivially holds for genus $1$ and $2$, the authors asked, whether it also holds for the fundamental group of the non-orientable surface of genus 3. With our Main Theorem, that proves the Magnus property for a slightly larger subclass of one-relator groups than in \\mbox{Theorem \\ref{ArtikelBSMain}}, we answer this question positively. The difficulty with genus $g=3$\nis essential since it is well known that the group $\\pi_1(S_3^-)=\\langle x,y,z\\,|\\, x^2y^2z^2\\rangle$ is not even existentially equivalent to a free group $F_{n}$ on $n$ generators. However, in large parts, our proof follows the proof of the Main Theorem in \\cite{ArtOneRel}.\n\n\n\\begin{main} \\label{Main}\nLet $G=\\langle a,b,y_{1},\\dots,y_{n} \\mid [a,b]u \\rangle$, where $n\\in \\mathbb{N}$ and $u$ is a non-trivial reduced word in the letters $y_{1},\\dots,y_{n}$. Then $G$ possesses the Magnus property.\n\\end{main}\n\nUsing the isomorphism $\\varphi$ between $\\pi_1(S_3^-) = \\langle x,y,z \\, | \\, x^2y^2z^2 \\rangle$ and $\\widetilde{H}=\\langle a,b,c \\mid [a,b]c^{2} \\rangle$ defined by $\\varphi(x)=ca^{-1}, \\varphi(y)=b^{-1}c^{-1}$ and $\\varphi(z)=cbcac^{-1}$, we get the following corollary.\n\n\\begin{corollary} \\label{Corgen3}\nThe group $\\pi_{1}(S_{3}^{-})$ possesses the Magnus property.\n\\end{corollary}\n\nIn the proof of Theorem \\ref{ArtikelBSMain}, the authors of \\cite{ArtOneRel} used an automorphism of $G$ with certain convenient properties. This automorphism is in general absent for the group given in our Main Theorem. For example, it is absent for the group $\\pi_1(S_3^-)$. So we introduce an additional tool which we call $\\alpha$- and $\\omega$-limits (see Section \\ref{secalphaomega}).\nTogether with the results of \\cite{ArtOneRel,ArtSurGr,ArtHowSurGr} Corollary \\ref{Corgen3} implies the following.\n\n\\begin{corollary} The fundamental groups of all compact surfaces possess the Magnus property.\\end{corollary}\n\n\nWe will start the proof of our main theorem by recapitulating the notation of \\cite[Section 3]{ArtOneRel} with some small alterations. In Section 3 we introduce $\\alpha$- and $\\omega$-limits, $\\alpha$-$\\omega$-length $|| \\cdot ||_{\\alpha,\\omega}$ and suitable elements. In Section 4 we give the proof for the case, where the $\\alpha$-$\\omega$-length of suitable elements is positive (the consideration of this case is close to that in \\cite{ArtOneRel}). In Section 5 we complete the proof in the remaining case.\n\n\\section{Reduction to a new group and left\/right bases} \\label{leftright}\n\nWe denote the normal closure of an element $g$ in a group $G$ by $\\langle \\! \\langle g \\rangle \\! \\rangle_{G}$ and the exponent sum of an element $g \\in G$ in a letter $x$ by $g_{x}$. Note that this sum is well-defined if all relations of $G$ considered as words of a free group have exponent sum $0$ in $x$. Our main theorem can be deduced from the following proposition in the same way as in the Main Theorem of \\cite{ArtOneRel} from \\cite[Proposition 2.1]{ArtOneRel}. Therefore we leave this argumentation out.\n\n\\begin{proposition} \\label{prop1} Let $H=\\langle x,b,y_{1},\\dots,y_{n} \\ | \\ [x^{k},b]u \\rangle$, where $k,n \\ge 1$ and $u$ is a non-trivial reduced word in the letters $y_{1},\\dots,y_{n}$. Further, let $r,s \\in H \\backslash \\{1\\}$ with $r_{x}=0$. Then $\\langle \\! \\langle r \\rangle \\! \\rangle_{H} = \\langle \\! \\langle s \\rangle \\! \\rangle_{H}$ implies that $r$ is conjugate to $s$ or $s^{-1}$.\n\\end{proposition}\n\nWe briefly summarise the concept of left and right bases from \\cite{ArtOneRel}. For a given group $G$ and an element $g \\in G$ we denote by $g_{i}$ the element $x^{-i}gx^{i}$ for $i \\in \\mathbb{Z}$.\n\nLet $H$ be as in Proposition \\ref{prop1}. We consider the homomorphism $\\varphi: H \\rightarrow \\mathbb{Z}$ which sends $x$ to $1$ and $b,y_{1},y_{2},\\dots, y_{n}$ to $0$. For each $i \\in \\mathbb{Z}$ let $Y_{i}=\\{ b_{i}, y_{1,i},y_{2,i},\\dots, y_{n,i} \\}$, where $y_{j,i}:=x^{-i}y_{j}x^{i}$. Using the rewriting process of Reidemeister-Schreier, we obtain that the kernel $N$ of $\\varphi$ has the presentation\n\\begin{eqnarray*}\nN = \\langle \\, \\underset{i \\in \\mathbb{Z}}{\\bigcup} Y_{i} \\ | \\ b_{i}u_{i}=b_{i+k} \\ (i \\in \\mathbb{Z}) \\, \\rangle.\n\\end{eqnarray*}\n\nThe group $N$ is the free product of the free groups $G_{i}=\\langle Y_{i} \\ | \\ \\rangle \\ (i \\in \\mathbb{Z})$ with amalgamation, where $G_{i}$ and $G_{i+k}$ are amalgamated over a cyclic group $Z_{i+k}$ that is generated by $b_{i}u_{i}$ in $G_{i}$ and by $b_{i+k}$ in $G_{i+k}$. This gives $N=N_{t} \\ast \\dots \\ast N_{t+k-1}$ for all $t \\in \\mathbb{Z}$, where \\[ N_{l}=\\dots \\ \\underset{ Z_{l-k}}{\\ast} \\ G_{l-k} \\ \\underset{Z_{l}}{\\ast} \\ G_{l} \\underset{Z_{l+k}}{\\ast} G_{l+k} \\ \\underset{Z_{l+2k}}{\\ast} \\dots \\] for $l \\in \\mathbb{Z}$.\n\n\\begin{proposition} \\label{Nfree}\nFor every $i \\in \\mathbb{Z}$, the group $N$ is free with basis\n$$\\mathcal{B}(i) =\\{b_{i},b_{i+1}, \\dots,b_{i+k-1}\\} \\, \\cup \\{y_{m,j} \\mid 1 \\leqslant m \\leqslant n, j \\in \\mathbb{Z}\\}.$$\n\\end{proposition}\n\n\n\\begin{proof} Since $N=N_{i} \\ast \\dots \\ast N_{i+k-1}$, it suffices to show that each $N_l$ has the basis\n$$B_l:=\\{b_l\\}\\cup \\{y_{m,l+tk}\\,|\\, 1 \\leqslant m \\leqslant n, t\\in \\mathbb{Z}\\}.$$\nWe have $N_{l}= \\underset{p \\in \\mathbb{N}}{\\bigcup} N_{l,p}$, where $N_{l,p}= G_{l-pk} \\underset{Z_{l-(p-1)k}}{\\ast} \\dots \\ \\underset{Z_{l}}{\\ast} G_{l}  \\underset{Z_{l+k}}{\\ast}   \\dots  \\underset{Z_{l+pk}}{\\ast} G_{l+pk}$. Using Tietze transformations, one can show that $N_{l,p}$ is free with basis $B_{l,p}:=\\{b_{l}\\} \\cup \\{y_{m,l+tk} \\mid 1 \\leqslant m \\leqslant n, -p \\leqslant t \\leqslant p \\}$. Since $B_{l,p} \\subset B_{l,p+1}$, the group $N_{l}$ is free with basis $B_l= \\underset{p \\in \\mathbb{N}}{\\bigcup} B_{l,p}$.\n\\end{proof}\n\n\\noindent\n{\\bf Notation.} Let $G_{i,j} = \\langle G_{i},G_{i+1} \\dots,G_{j} \\rangle$, $G_{i,\\infty}= \\langle G_{l} \\mid l \\geqslant i \\rangle$ and $G_{-\\infty,i}=\\langle G_{l} \\mid l \\leqslant i \\rangle$ for all $i,j \\in \\mathbb{Z}$ with $i \\leqslant j$.\nFurther, we use the\n$$\\begin{array}{llllllll}\n\\emph{b-left basis} & \\mathcal{B}^{+}(i) & := & \\{b_{i},b_{i+1},\\dots,b_{i+k-1} \\} & \\cup & \\{y_{m,j} \\mid 1 \\leqslant m \\leqslant n,j \\geqslant i \\} & \\text{of} & G_{i,\\infty} \\ \\ \\text{and the} \\\\\n\\emph{b-right basis} & \\mathcal{B}^{-}(i) & := &  \\{b_{i-k+1},b_{i-k+2},\\dots,b_{i} \\}  & \\cup & \\{y_{m,j} \\mid 1 \\leqslant m \\leqslant n,j \\leqslant i \\} & \\text{of} & G_{-\\infty,i}. \\\\\n\\end{array}$$\n\n\\begin{remark} \\label{remchange}\nLet $r \\in N$ be written as a word in the letters $b_{j},y_{m,j} \\ (1 \\leqslant m \\leqslant n,j \\in \\mathbb{Z})$. For an arbitrary $i \\in \\mathbb{Z}$ we describe how to rewrite $r$ in the basis $\\mathcal{B}^{+}(i)$. First, replace each letter $b_{j}$ of $r$ by $b_{j-k}u_{j-k}$, if $j > i+k-1$, and by $b_{j+k}u_{j}^{-1}$, if $j< i$, and reduce. If the resulting word contains $b_{j}$ that do not belong to $\\mathcal{B}^{+}(i)$, we repeat this procedure. After finitely many steps, we will obtain the desired form of~$r$. Analogously, we can rewrite $r$ in the basis $\\mathcal{B}^{-}(i)$.\n\\end{remark}\n\n\\section{\\boldmath{$\\alpha$}- and \\boldmath{$\\omega$}-limits} \\label{secalphaomega}\n\nWe keep using the notation introduced in Section $2$. An arbitrary element $r\\in N\\setminus \\{1\\}$\ncan be written in many ways as a reduced word in letters $b_i,y_{m,i}$. For example, $b_0u_0u_1^{-1}b_1^{-1}=b_kb_{k+1}^{-1}=\nb_{2k}u_k^{-1}u_{k+1}b_{2k+1}^{-1}$.\nWe give an algorithm which finds a word $r^{*}$ representing $r$\nsuch that the smallest index of letters used in $r^{*}$ is maximum possible. We call such index the \\fbox{{\\it $\\alpha$-limit} of $r$} and denote it by $\\alpha_{r}$. In other words, $\\alpha_{r}$ is the largest index such that $r$ is an\nelement of $G_{\\alpha_{r}, \\infty}$. The following algorithm rewrites an arbitrary word $r \\in N\\setminus \\{1\\}$ into the presentation $r^{*}$ of $r$ written in basis $\\mathcal{B}^{+}(\\alpha_{r})$ (see Lemma \\ref{Lemefficiency}).\nFor a word $r$ in the alphabet $\\{b_i\\,|\\, i\\in \\mathbb{Z}\\}\\cup \\{y_{m,i}\\,|\\, 1 \\leqslant m \\leqslant n,i\\in \\mathbb{Z}\\}$, let $\\min(r)$\ndenote the minimal index of letters of $r$.\n\n\\begin{algorithm} \\label{algorithm}\nLet $r \\in N \\backslash \\{1\\}$. Suppose that $r$ is given as a finite word in letters $b_i,b_{i+1},\\dots $ and $y_{m,i},y_{m,i+1},\\dots$ ($1 \\leqslant m \\leqslant n$). In particular, $r\\in G_{i,\\infty}$.\nLet $r[0]$ be the reduced word representing $r$ in basis $\\mathcal{B}^{+}(i)$. Increasing~$i$ if necessary, we may assume that $i=\\min(r[0])$.\n\n\\begin{itemize}\n\\item[(1)] Let $r[1]$ be the word obtained from $r[0]$ by replacement of each occurrence of the letter $b_i$ by $b_{i+k}u_{i}^{-1}$ followed by free reduction.\nThen $r[1]$ presents $r$ in the following basis of $G_{i,\\infty}$:\n\\begin{align} \\label{Basisplus1}\n\\{b_{i+1}, b_{i+2}, \\dots, b_{i+k} \\} \\cup \\{y_{m,l} \\mid 1 \\leqslant m \\leqslant n,i \\leqslant l \\}.\\tag{$\\star$}\n\\end{align}\n\n\\begin{enumerate}\n\\item[(1a)] If $r[1]$ does not contain a letter $y_{m,i}$ ($1 \\leqslant m \\leqslant n$), we reset $r[0]:=r[1]$, $i:=\\min(r[1])$, and go back to $(1)$. Clearly, the new $i$ is larger than the old one.\n\\item[(1b)] If $r[1]$ contains a letter $y_{m,i}$ ($1 \\leqslant m \\leqslant n$),  the algorithm ends with $r^{*}=r[0]$ and $\\alpha_r=i$.\n\\end{enumerate}\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{lemma} \\label{Lemefficiency}\nThe output $r^{*}$ of Algorithm \\ref{algorithm} coincides with the word representing $r$ in Basis $\\mathcal{B}^{+}(\\alpha_{r})$, where $\\alpha_{r}$ is the $\\alpha$-limit of $r$.\n\\end{lemma}\n\n\\begin{proof}\nWe shall prove that the algorithm ends and that the integer assigned to $\\alpha_{r}$ by the algorithm is really the $\\alpha$-limit of $r$.\n\nThe algorithm ends since the length of $r[0]$ decreases with each iteration of (1) which does not terminate the algorithm.\nThis can be verified in the following way:\nIf we arrive in (1a), then the word $r[1]$ does not contain a letter $y_{m,i}$ ($1 \\leqslant m \\leqslant n$).\nRecall that $r[1]$ was obtained from $r[0]$ by replacing all $b_{i}$ with $b_{i+k}u_{i}^{-1}$ and reducing the resulting word. The fact that there is no letter $y_{m,i}$ ($1 \\leqslant m \\leqslant n$) left in $r[1]$\nmeans that each $b_{i}$ in $r[0]$ occurred in a subword of the form $b_{i}u_{i}$.\nTherefore $r[1]$ is shorter than $r[0]$.\n\nFinally, we show that $r$ is not an element of $G_{i+1,\\infty}$, where $i$ as in (1b).\nObserve that $$G_{i,\\infty}=G_{i+1,\\infty}\\ast \\langle y_{m,i} \\mid 1 \\leqslant m \\leqslant n \\rangle.$$ By (1b), $r$, written in basis \\eqref{Basisplus1},\nuses a letter $y_{m,i}$ ($1 \\leqslant m \\leqslant n$). Therefore $r\\notin G_{i+1,\\infty}$ and $\\alpha_r=i$.\n\\end{proof}\n\n\\begin{corollary} \\label{Corfinite}\nLet $r$ be an element from $N \\backslash \\{1\\}$ given as a word in some $b$-left basis. Starting \\mbox{Algorithm \\ref{algorithm}} with $r$, we either get the same presentation or a presentation of shorter length.\n\\end{corollary}\n\n\\begin{proof}\nIn the proof of Lemma~\\ref{Lemefficiency}, we already showed that the length of $r[0]$ in Algorithm \\ref{algorithm} decreases with each iteration of (1) which does not terminate the algorithm. Moreover, the last iteration of \\mbox{Algorithm~\\ref{algorithm}} does not change the current presentation $r[0]$.\n\\end{proof}\n\nAnalogously, we can find a presentation $r_{*}$ of a word $r \\in N \\backslash \\{1\\}$ such that the largest used index of letters in $r_{*}$ is minimum possible. We call this index \\fbox{{\\it $\\omega$-limit} of $r$} and denote it by $\\omega_{r}$. In\nother words, $\\omega_{r}$ is the smallest index such that $r$ is an element of $G_{-\\infty, \\omega_{r}}$. The algorithm to find $r_{*}$ can be received from Algorithm~\\ref{algorithm} by ``mirroring'' this algorithm using the $b$-right basis of $G_{-\\infty,i}$  and replacements of $b_i$ by $b_{i-k}u_{i-k}$. We define the \\fbox{\\emph{$\\alpha$-$\\omega$-length} of $r$} by $||r||_{\\alpha,\\omega}:=\\omega_{r}-\\alpha_{r}+1$. Note that the $\\alpha$-$\\omega$-length of a non-trivial element can be non-positive.\n\n\\noindent\n\\textbf{Examples.}\n\\begin{itemize}\n\\item[(i)] Let $r=b_{-2}u_{-2}y_{1,0}b_{4}u_{1}^{-1}$ and $k=3$. To determine $\\alpha_{r}$, we write $r=b_{-2}u_{-2}y_{1,0}b_{4}u_{1}^{-1} = b_{1}y_{1,0}b_{4}u_{1}^{-1}$\nand get $\\alpha_{r}=0$. For $\\omega_{r}$ we have $r=b_{-2}u_{-2}y_{1,0}b_{4}u_{1}^{-1} =b_{-2}u_{-2}y_{1,0}b_{1}u_{1}u_{1}^{-1}=b_{-2}u_{-2}y_{1,0}b_{1}$ \\mbox{$=b_{-2}u_{-2}y_{1,0}b_{-2}u_{-2}.$}\nThus, $\\omega_{r}=0$ and $||r||_{\\alpha,\\omega}=1$. Note that $r \\notin G_{\\alpha_{r},\\omega_{r}}=G_{0}$.\n\\item[(ii)] Let $r=b_{5}b_{6}^{-1}$ and $k=4$. Clearly, $\\alpha_{r}=5$. To determine $\\omega_{r}$, we write $r=b_{5}b_{6}^{-1}= b_{5}u_{2}^{-1}b_{2}^{-1}$ $= b_{1}u_{1}u_{2}^{-1}b_{2}^{-1}=b_{1}u_{1}u_{2}^{-1}u_{-2}^{-1}b_{-2}^{-1}$. Thus, $\\omega_{r}=2$ and $||r||_{\\alpha,\\omega}=-2$.\n\\end{itemize}\n\n\\begin{lemma} \\label{LemEig}\nLet $r \\in N$.\nThen $\\alpha_{r_{i+j}}=\\alpha_{r_{i}}+j$ and $\\omega_{r_{i+j}}=\\omega_{r_{i}}+j$ for all $i,j \\in \\mathbb{Z}$. In particular, $||r_{i}||_{\\alpha,\\omega}=||r_{j}||_{\\alpha,\\omega}$ for all $i,j \\in \\mathbb{Z}$.\n\\end{lemma}\n\n\\begin{proof}\nIndeed, $r_{i+j}$ can be obtained from $r_i$ by increasing the (second) indices of all letters by $j$.\n\\end{proof}\n\n\\noindent\n\\textbf{Notation.} Let $r \\in N$. For the following two lemmata, let $r(i)$\nbe the presentation of $r$ written in basis $\\mathcal{B}(i)$.\n\n\\begin{lemma} \\label{Lemcyclical}\nLet $r \\in N \\backslash \\{1\\}$. The following statements are equivalent:\n\\begin{itemize}\n\\item[{\\rm (1)}] For some $i\\in \\mathbb{Z}$, the word $r(i)$ begins with a positive power of a $b$-letter.\n\\item[{\\rm (2)}] For all $i\\in \\mathbb{Z}$, the word $r(i)$ begins with a positive power of a $b$-letter.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} The word $r(i+1)$ can be obtained from $r(i)$, by replacement of each occurrence of $b_i$ in $r(i)$ by $b_{i+k}u_i^{-1}$\nfollowed by free reduction. The new letter $b_{i+k}$ does not lie in $\\{b_i,\\dots,b_{i+k-1}\\}$. That prevents cancellation between $b$-letters in $r(i+1)$. \nTherefore $r(i)$ starts with a positive exponent of a $b$-letter if and only if $r(i+1)$ starts with a positive exponent of a $b$-letter. This proves the equivalence $(1)\\Leftrightarrow (2)$.\n\\end{proof}\n\n\\begin{corollary} \\label{lemdef}\nFor every $r \\in N \\backslash \\{1\\}$ there exists a conjugate $\\widetilde{r}$ of $r$ \nsuch that $\\widetilde{r}(i)$ is cyclically reduced for each $i\\in \\mathbb{Z}$.\n\\end{corollary}\n\n\\begin{proof}\nUsing conjugation, we may assume that $r(0)$ is cyclically reduced.\nIf $r(0)$ contains only $y$-letters, we are done with the element $\\widetilde{r}$ represented by $r(0)$.\nSuppose that $r(0)$ contains a $b$-letter. Let $\\widetilde{r}$ be the element represented by a cyclic permutation of $r(0)$ which either starts with a positive power of a $b$-letter, or ends with a negative power of a $b$-letter (but not both).\nBy Lemma~\\ref{Lemcyclical}, this $\\widetilde{r}$ has the desired property.\n\\end{proof}\n\n\\begin{definition} \\label{defsc}\nLet $r \\in N\\backslash \\{1\\}$. Any element $\\widetilde{r}$ as in Lemma~\\ref{lemdef} is called \\emph{suitable conjugate for $r$}.\n\\end{definition}\n\n\\begin{remark} {\\bf (dual structure of $N$)} \\label{Remdual}\nDenote $b'_i:=b_{-i}u_{-i}$, $y_{m,i}':=y_{m,-i}^{-1}$ ($1 \\leqslant m \\leqslant n$) and $G_i':=G_{-i}$.\nWe call the elements $b_i',y_{m,i}'$ ($1 \\leqslant m \\leqslant n$) {\\it dual} to $b_i,y_{m,i}$ ($1 \\leqslant m \\leqslant n$) and the subgroup $G_i'$ dual to $G_i$.\nExpressing $b$-letters and $y$-letters via their dual, we obtain $b_i=b_{-i}'u_{-i}'$, $y_{m,i}=y_{m,-i}'^{-1}$ ($1 \\leqslant m \\leqslant n$), where $u'_{-j}$ ($j \\in \\mathbb{Z}$) is the word obtained from $u_{j}$ by replacing each letter $y_{m,j}$ with $y_{m,-j}'^{-1}$. That justifies the terminology. \n\nObserve that the old relations $b_iu_i=b_{i+k}$ $(i\\in \\mathbb{Z})$ can be rewritten in dual letters as $b_i'u_i'=b_{i+k}'$ $(i\\in \\mathbb{Z})$. Thus, the relations preserve their form. \nMoreover, we have $G_i'=\\langle b_i',y_{m,i}' \\mid 1 \\leqslant m \\leqslant n \\rangle$ that repeats the form $G_i=\\langle b_i,y_{m,i} \\mid 1 \\leqslant m \\leqslant n \\rangle$.\nOther dual objects can be defined analogously: \nFor example, $G_{i,\\infty}':=\\langle G_i',G_{i+1}'\\dots \\rangle=G_{-\\infty,-i}$ \nand $G_{-\\infty,i}':=\\langle \\dots ,G_{i-1}',G_i'\\rangle =G_{-i,\\infty}$. We use the following bases of $N$\\,\n($i\\in \\mathbb{Z}$):\n$$\\mathcal{B}(i)' =\\{b'_{i},b'_{i+1}, \\dots,b'_{i+k-1}\\} \\, \\cup \\{y'_{m,j} \\mid 1 \\leqslant m \\leqslant n, j \\in \\mathbb{Z}\\}.$$\nalong with the\n$$\\begin{array}{llllllll}\n\\emph{$b'$-left basis} & \\mathcal{B}^{+}(i)' & := & \\{b_{i}',b_{i+1}',\\dots,b_{i+k-1}' \\} & \\cup & \\{y_{m,j}' \\mid 1 \\leqslant m \\leqslant n, j \\geqslant i \\} & \\text{of} & G_{i,\\infty}'\\ \\text{and the} \\\\\n\\emph{$b'$-right basis} & \\mathcal{B}^{-}(i)' & := &  \\{b_{i-k+1}',b_{i-k+2}',\\dots,b_{i}' \\}  & \\cup & \\{y_{m,j}' \\mid 1 \\leqslant m \\leqslant n, j \\leqslant i \\} & \\text{of} & G_{-\\infty,i}'. \\\\\n\\end{array}$$\n\n\nFor $r\\in N\\backslash \\,\\{1\\}$, let $\\alpha'_r$ be the largest index $i$ such that $r$ is an element of $G'_{i,\\infty}$ and let  $\\omega'_r$ be the smallest index $i$ such that $r$ is an element of $G'_{-\\infty,i}$. We denote $||r||_{\\alpha',\\omega'}:=\\omega_r'-\\alpha_r'+1$.\n\\end{remark}\n\n\\begin{lemma} \\label{LemDual} For any non-trivial $r\\in N$ the following statements are valid.  \n\\begin{enumerate}\n\\item[\\rm (1)] We have $\\alpha_r'=-\\omega_{r}$, $\\omega_r'=-\\alpha_{r}$, and $||r||_{\\alpha',\\omega'}=||r||_{\\alpha,\\omega}$.\n\n\\item[\\rm (2)] Suppose that $r$ written in a basis $\\mathcal{B}(i)$ is cyclically reduced and begins with a positive power of a $b$-letter. Then $r$ written in the basis $\\mathcal{B}(-i-k+1)'$ is cyclically reduced and begins with a positive power of a $b'$-letter.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nBoth statements can be verified straightforward by using the relations $b_i=b_{-i}'u_{-i}'$, $y_{m,i}=y_{m,-i}'^{-1}$ ($1 \\leqslant m \\leqslant n$).\n\\end{proof}\n\n\\section{Proof of Proposition \\ref{prop1} for \\boldmath{$\\widetilde{r}$} with positive  {\\boldmath $\\alpha$-$\\omega$}-length }\n\n\\subsection{Properties of \\boldmath{$\\widetilde{r}$} with positive \\boldmath{$\\alpha$-$\\omega$}-length}\n\n\nWe use the following version of Magnus' Freiheitssatz:\n\n\\begin{theorem}[\\textbf{Magnus' Freiheitssatz (cf. \\cite{ArtMagnus})}] \\label{Magnus} Let $F$ be a free group on a basis $X$, and let $g$ be a cyclically reduced word in $F$ with respect to $X$, containing a letter $x \\in X$. Then the subgroup generated by $X \\backslash \\{x\\}$ is naturally embedded into the group $F \/ \\langle \\! \\langle g \\rangle \\! \\rangle_{F}$.\n\\end{theorem}\n\nBy abuse of notation, we write $A \/ \\langle \\! \\langle a \\rangle \\! \\rangle$ instead of $A \/ \\langle \\! \\langle a \\rangle \\! \\rangle_{A}$, where $A$ is a group and $a \\in A$.\n\n\\begin{corollary} \\label{KorEinb} Let $\\widetilde{r} \\in N$ be a suitable element. Then we get the embeddings $G_{\\alpha_{\\widetilde{r}}+1,\\infty} \\hookrightarrow N \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$ and $G_{-\\infty, \\omega_{\\widetilde{r}}-1} \\hookrightarrow N \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$.\n\\end{corollary}\n\n\\begin{proof}\nBy the definition of $\\alpha$-limits, $\\widetilde{r}$ written in basis $\\mathcal{B}^{+}(\\alpha_{\\widetilde{r}})$ contains at least one letter in $Y_{\\alpha_{\\widetilde{r}}}$, and by Definition~\\ref{defsc}, the suitable element $\\widetilde{r}$ is cyclically reduced in this basis. We extend $\\mathcal{B}^{+}(\\alpha_{\\widetilde{r}})$ to a free basis of $N$ by adding the letters $\\{y_{m,i} \\mid 1 \\leqslant m \\leqslant n, i<\\alpha_{\\widetilde{r}}\\}$. The presentation of $\\widetilde{r}$ will not change by that. Now, $G_{\\alpha_{\\widetilde{r}}+1,\\infty} \\hookrightarrow N \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$ follows immediately from Theorem~\\ref{Magnus}. The other embedding follows analogously.\n\\end{proof}\n\n\\begin{lemma} \\label{Lembw}\nSuppose that $r \\in N$ satisfies $||r||_{\\alpha,\\omega} \\geqslant 1$.\nThen $r$, written in the basis $\\mathcal{B}^{+}(\\alpha_{r})$ of $G_{\\alpha_{r},\\infty}$, contains at least one letter $y_{m,l}$ with $l \\geqslant \\omega_{r}$.\n\\end{lemma}\n\n\\begin{proof}\nBy assumption, $\\alpha_r\\leqslant \\omega_r$.\nTo the contrary, suppose that $$r\\in \\langle b_{\\alpha_r},b_{\\alpha_r+1}, \\dots ,b_{\\alpha_r+k-1},y_{m,\\alpha_r},y_{m,\\alpha_r+1},\\dots ,y_{m,\\omega_r-1} \\mid 1 \\leqslant m \\leqslant n\\rangle.$$ Then, using relations $b_{i}=b_{i-k}u_{i-k}$, we obtain\n$$r\\in \\langle b_{\\alpha_r-k},\\dots ,b_{\\alpha_r-1},y_{m,\\alpha_r-k},y_{m,\\alpha_r-k+1}, \\dots ,y_{m,\\omega_r-1}\\mid 1 \\leqslant m \\leqslant n \\rangle.$$ Hence $r\\in G_{-\\infty, \\omega_r-1}$. A contradiction.\n\\end{proof}\n\n\\subsection{The structure of some quotients of \\boldmath{$N$}}\n\nThis section and the next one are very similar to \\cite[Section 4 and 5]{ArtOneRel}, but due to some important changes we cannot skip them. Our aim in this subsection is to present $N \/  \\langle \\! \\langle \\widetilde{r_{i}}, \\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$ as an amalgamated product. We denote $w_i=b_iu_i$.\n\n\\begin{lemma} \\label{LemH}\nLet $\\widetilde{r} \\in N$ be a suitable element with $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$, and let $i,j$ be two integers with $i \\leqslant j$. We denote $s=\\alpha_{\\widetilde{r_j}}$ and $t=\\omega_{\\widetilde{r}_{j}}-1$. Then we have:\n\\begin{itemize}\n\\item[{\\rm (1)}] $N \/  \\langle \\! \\langle \\widetilde{r_{i}},\\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,t} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j-1} \\rangle \\! \\rangle \\underset{\n    \\begin{array}{ll}\n    w_{t-k+1}\\!\\! & =b_{t+1},\\vspace*{-3mm}\\\\\n    \\dots & \\vspace*{-3mm}\\\\\n    w_{t} \\!\\!& =b_{t+k}\\\\\n    \\end{array}\n    }{\\underset{G_{s,t}}{\\ast}} G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\\item[{\\rm (2)}] $G_{s+1,\\infty}$ naturally embeds into $N\/\\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots,\\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\\end{itemize}\n\\end{lemma}\n\nBefore we give a formal proof, we consider an illustrated example. This will help\nto visualise a lot of technical details in the formulation of lemma.\n\n\\noindent\n{\\bf  Example.} Let $k=4$. We consider the element $\\widetilde{r}=\\widetilde{r}_0=b_4y_{2,1}y_{1,3}b_0u_0$ and the integers $i=-1$ and $j=2$.\n\nAlgorithm~\\ref{algorithm} applied to $\\widetilde{r}$ gives $\\alpha_{\\widetilde{r}_0}=1$. Its ``mirrored'' version gives $\\omega_{\\widetilde{r}_0}=3$.\nIn particular, $||\\widetilde{r}||_{\\alpha,\\omega}=3$. Furthermore, we have $s=\\alpha_{\\widetilde{r}_2}=3$ and\n$t=\\omega_{\\widetilde{r}_2}-1=4$.\nThen Lemma~\\ref{LemH} (1) says that\n$$N \/  \\langle \\! \\langle \\widetilde{r}_{-1},\\widetilde{r}_0, \\widetilde{r}_1 ,\\widetilde{r}_2\\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,4} \/  \\langle \\! \\langle \\widetilde{r}_{-1},\\widetilde{r}_0, \\widetilde{r}_1  \\rangle \\! \\rangle \\underset{\n    \\begin{array}{rl}\n    w_1\\!\\!\\!\\! & =\\, b_5,\\vspace*{-3mm}\\\\\n    w_2\\!\\!\\!\\! & =\\, b_6\\vspace*{-3mm}\\\\\n    w_3\\!\\!\\!\\! & =\\, b_7,\\vspace*{-3mm}\\\\\n    w_4\\!\\!\\!\\! & =\\, b_8\n    \\end{array}\n    }{\\underset{G_{3,4}}{\\ast}} G_{3,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_2 \\rangle \\! \\rangle.$$\n\n\n\nIn Figure 1, the word $\\widetilde{r}_0=\\widetilde{r}$ is pictured by the (partially dashed) line crossing the blocks $G_0,\\dots,G_4$ since $\\widetilde{r}$\nuses letters with indices from the segment $[0,4]$.\nWe can represent $\\widetilde{r}$ by the word $b_4y_{2,1}y_{1,3}b_4$ that uses letters with indices from the segment $[1,4]$,\nand we can represent $\\widetilde{r}$ by the word $b_0u_0y_{2,1}y_{1,3}b_0u_0$ that uses letters with indices from the segment $[0,3]$.\nTo visualise the fact that $\\alpha_{\\widetilde{r}_0}=1$ and $\\omega_{\\widetilde{r}_0}=3$, we draw a continuous line crossing the blocks $G_1,G_2,G_3$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.7]{Abb1.eps}\n\\caption{Illustration to Lemma~\\ref{LemH} for $k=4$, $\\widetilde{r}=b_{4}y_{2,1}y_{1,3}b_{0}u_{0}$, $i=-1$, and $j=2$}\n\\label{Abb1}\n\\end{figure}\n\n\n\n\\begin{proof}\nFirst, we prove that (1) implies (2). By Corollary~\\ref{KorEinb}, we have $G_{s+1,\\infty} \\hookrightarrow G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$, and if (1) holds, then we have $G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle \\hookrightarrow N \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$. The composition of these two embeddings gives (2). Now we prove (1) for fixed $i$ by induction on $j$.  \\\\\n\\textbf{Base of induction.} For $j=i$ we shall show\n\n$$\nN\/  \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,t}\n\\underset{\n\\begin{array}{ll}\n    w_{t-k+1}\\!\\! & =b_{t+1},\\vspace*{-3mm}\\\\\n    \\dots & \\vspace*{-3mm}\\\\\n    w_{t} \\!\\!& =b_{t+k}\\\\\n    \\end{array}\n}\n{\\underset{G_{s,t}}{\\ast}} G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle.\\eqno{(4.1)}\n$$\nIt suffices to show the following claim.\n\n\\emph{Claim.} Let $P$ be the subgroup of $N$ generated by $G_{s,t}\\cup \\{b_{t+1},\\dots ,b_{t+k}\\}$. Then\n\\vspace{-1mm}\n\\begin{enumerate}\n\\item[(a)] $P$ embeds into $G_{-\\infty,t}$ and $ G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\\vspace{-1mm}\n\\item[(b)] The abstract amalgamated product in the right side of (4.1) is canonically isomorphic to $N\/  \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\\end{enumerate}\n\n\\emph{Proof of the claim.} (a) Clearly, $P$ embeds into $G_{-\\infty,t}$. So, we show that $P$ embeds into $G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$. Note that $P = \\langle Y_{s} \\cup \\dots \\cup Y_{t} \\cup \\{ b_{t+1},\\dots ,b_{t+k} \\} \\rangle = \\langle b_{s},b_{s+1}, \\dots , b_{s+k-1}, y_{m,s},y_{m,s+1}, \\dots , y_{m,t} \\mid 1 \\leqslant m \\leqslant n \\rangle$. Thus, the group $P$ is generated by the set\n$$\\{ b_{s},b_{s+1}, \\dots , b_{s+k-1}, y_{m,s},y_{m,s+1}, \\dots , y_{m,t} \\mid 1 \\leqslant m \\leqslant n \\}. \\eqno{(4.2)}$$\nThis set is a part of the free basis $\\mathcal{B}^{+}(s)$ of $G_{s,\\infty}$. By Definition~\\ref{defsc}, $\\widetilde{r}_{j}$ written in $\\mathcal{B}^{+}(s)$ is cyclically reduced. Further, by statement (1) of Lemma~\\ref{Lembw}, the element $\\widetilde{r}_{j}$ written in $\\mathcal{B}^{+}(s)$ contains at least one letter $y_{m,\\ell}$ with $1 \\leqslant m \\leqslant n$ and $\\ell \\geqslant t+1$. In particular, this letter does not lie in\nthe set~(4.2). By Magnus' Freiheitssatz \\mbox{(Theorem~\\ref{Magnus})}, $P$ embeds into $G_{s,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\nNow, the amalgamated product in (4.1) is well defined. It is easy to check that the groups written in (4.1) are isomorphic by finding a common presentation. This completes the base of induction. \\\\\n\\textbf{Inductive step} $i,j \\ \\rightarrow i,j+1$. We need to show the formula,\nwhere  $s=\\alpha_{\\widetilde{r_j}}$ and $t=\\omega_{\\widetilde{r}_{j}}-1$:\n$$\nN \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1}\\dots, \\widetilde{r}_{j+1} \\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,t+1} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\underset{\n\\begin{array}{ll}\n    w_{t-k+2}\\!\\! & =b_{t+2},\\vspace*{-3mm}\\\\\n    \\dots & \\vspace*{-3mm}\\\\\n    w_{t+1} \\!\\!& =b_{t+k+1}\\\\\n    \\end{array}\n}{\\underset{G_{s+1,t+1}}{\\ast}} G_{s+1,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j+1} \\rangle \\! \\rangle.\n\\eqno{(4.3)}\n$$\n\nLet $P$ be the subgroup of $N$ generated by $G_{s+1,t+1}$ and the set $\\{b_{t+2},b_{t+3},\\dots,b_{t+k+1} \\}$. First, we prove that $P$ canonically embeds into both factors. As above $P$ embeds into $G_{s+1,\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j+1} \\rangle \\! \\rangle$. So we show that $P$ embeds into $G_{-\\infty,t+1} \/  \\langle \\! \\langle \\widetilde{r}_{i}, \\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$ using the following commutative diagram:\n\\begin{displaymath}\n\t\\xymatrix@C=2cm@R=2cm{\n        P \\ \\ar@{^{(}->}[r] \\ \\ar@{^{(}->}[d] \\ar@{-->}[rrd]^{\\varphi} & G_{s+1,\\infty} \\ \\ar@{^{(}->}[r]^{\\hspace{-0.9cm} by \\ (2)} &  \\ N \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\\\\n        G_{-\\infty,t+1} \\ \\ar[rr]_{}  && \\ G_{-\\infty,t+1} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\ar@{^{(}->}[u] }\n\\vspace{0.3cm}\n\\end{displaymath}\nLet $\\varphi$ be the composition of the canonical embedding of the subgroup $P$ into $G_{-\\infty,t+1}$ and the canonical homomorphism from $G_{-\\infty,t+1}$ to the factor group $G_{-\\infty,t+1} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$. It remains to prove that $\\varphi$ is an embedding. Considering $P$ as a subgroup of $G_{s+1,\\infty}$ and using statement (2) for $i,j$ (recall that (1) implies (2)), we have an embedding of $P$ into $N\/\\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots,\\widetilde{r}_{j} \\rangle \\! \\rangle$. Since the diagram is commutative, $\\varphi: P \\rightarrow G_{-\\infty,t+1} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$ is an embedding. Again, it is easy to check that the groups in (4.3) are isomorphic.\n\\end{proof}\n\nFinally, we need a ``mirrored'' version of Lemma~\\ref{LemH}:\n\n\\begin{lemma} \\label{LemS}\nLet $\\widetilde{r} \\in N$ be a suitable element with $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$, and let $i,j$ be two integers with $i \\leqslant j$. We denote $s=\\alpha_{\\widetilde{r}_{i}}+1$ and $t=\\omega_{\\widetilde{r}_{i}}$. Then we have:\n\\begin{itemize}\n\\item[{\\rm (1)}] $N \/  \\langle \\! \\langle \\widetilde{r_{i}},\\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,t} \/ \\langle \\! \\langle \\widetilde{r}_{i} \\rangle \\! \\rangle \\underset{\n    \\begin{array}{ll}\n    w_{s-k}\\!\\! & =b_{s},\\vspace*{-3mm}\\\\\n    \\dots & \\vspace*{-3mm}\\\\\n    w_{s-1} \\!\\!& =b_{s+k-1}\\\\\n    \\end{array}\n    }{\\underset{G_{s,t}}{\\ast}} G_{s,\\infty} \/  \\langle \\! \\langle \\widetilde{r}_{i+1}, \\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle $.\n\\item[{\\rm (2)}] $G_{-\\infty,t-1}$ naturally embeds into $N\/\\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots,\\widetilde{r}_{j} \\rangle \\! \\rangle$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} \nDue to the dual structure noticed in Remark \\ref{Remdual}, the statements of Corollary \\ref{KorEinb}, Lemma \\ref{Lembw} and Lemma \\ref{LemH} also hold for the dual objects. By rewriting the dual version of Lemma \\ref{LemH} with the help of Lemma \\ref{LemDual}, we get the desired statement.\n\\end{proof}\n\n\\subsection{Conclusion of the proof for \\boldmath{$\\widetilde{r}$} with positive \\boldmath{$\\alpha$-$\\omega$}-length} \\label{conclusion1}\n\nBy assumption of Proposition~\\ref{prop1}, we have two elements $r,s \\in H$ with the same normal closure and $r_{x}=0$. Thus, $r,s$ are elements of $N$. By Lemma~\\ref{lemdef}, we can choose conjugates $\\widetilde{r},\\widetilde{s}$ of $r,s$ such that $\\widetilde{r}$ and $\\widetilde{s}$ are suitable elements. Normal closures are invariant under conjugation. So, without loss off generality, we can replace $r,s$ by $\\widetilde{r},\\widetilde{s}$. In this section we assume $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$. Since the normal closures of $\\widetilde{r}$ and $\\widetilde{s}$ in $H$ are equal, the normal closures of $\\mathcal{R}:=\\{ \\widetilde{r}_{i} \\ | \\ i \\in \\mathbb{Z} \\}$ and $\\mathcal{S}:=\\{ \\widetilde{s}_{i} \\ | \\ i \\in \\mathbb{Z} \\}$ in $N$ are equal. In particular, $\\widetilde{s}_{0}$ is trivial in $N \/ \\langle \\! \\langle \\mathcal{R} \\rangle \\! \\rangle$. Thus, there are indices $i,j \\in \\mathbb{Z}$ such that $\\widetilde{s}_{0}$ is trivial in $N \/ \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots,\\widetilde{r}_{j} \\rangle \\! \\rangle$. We choose a pair $i,j$ with this property and $j-i$ minimal. By Lemma~\\ref{LemH}~(1), we have\n$$\nN \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle \\ \\cong \\ G_{-\\infty,\\omega_{\\widetilde{r}_{j}}-1} \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j-1} \\rangle \\! \\rangle \\ \\underset{A}{\\ast} \\ G_{\\alpha_{\\widetilde{r}_{j}},\\infty} \/ \\langle \\! \\langle \\widetilde{r}_{j} \\rangle \\! \\rangle\\eqno{(4.4)}\n$$\nfor some subgroup $A$.\n\\begin{lemma}\\label{new} Suppose that $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$. Then\n\\vspace*{-2mm}\n\\begin{enumerate}\n\\item[{\\rm (1)}] $\\omega_{\\widetilde{s}_{0}} \\geqslant \\omega_{\\widetilde{r}_{j}}\\geqslant \\omega_{\\widetilde{r}_{i}}$,\n\\vspace*{-1mm}\n\\item[{\\rm (2)}] $\\alpha_{\\widetilde{s}_0}\\leqslant \\alpha_{\\widetilde{r}_i}\\leqslant \\alpha_{\\widetilde{r}_j}$,\n\\vspace*{-1mm}\n\\item[{\\rm (3)}] $\n||\\widetilde{s}||_{\\alpha,\\omega} \\geqslant ||\\widetilde{r}||_{\\alpha,\\omega}.\n$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} (1) We prove $\\omega_{\\widetilde{s}_{0}} \\geqslant \\omega_{\\widetilde{r}_{j}}$. Suppose the contrary. Then $\\widetilde{s}_{0} \\in G_{-\\infty,\\omega_{\\widetilde{r}_{j}}-1}$. Since $\\widetilde{s}_{0}$ is trivial in $N \/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1},\\dots, \\widetilde{r}_{j} \\rangle \\! \\rangle$, it is also trivial in the left factor of the amalgamated product (4.4). Therefore, $\\widetilde{s}_0$ is trivial in $N\/  \\langle \\! \\langle \\widetilde{r}_{i},\\widetilde{r}_{i+1}, \\dots, \\widetilde{r}_{j-1} \\rangle \\! \\rangle$. This contradicts the minimality of $j-i$. The inequality \\mbox{$\\omega_{\\widetilde{r}_{j}}\\geqslant \\omega_{\\widetilde{r}_{i}}$} follows by Lemma \\ref{LemEig} since $j \\geqslant i$. Inequalities (2) can be proved analogously with the help of Lemma~\\ref{LemS}~(1). Inequality (3) follows straightforward from (1) and (2).\n\\end{proof}\n\nSince $||\\widetilde{s}||_{\\alpha,\\omega} \\geqslant ||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$, we have $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant ||\\widetilde{s}||_{\\alpha,\\omega}$ by symmetry and therefore $||\\widetilde{s}||_{\\alpha,\\omega} = ||\\widetilde{r}||_{\\alpha,\\omega}$. By Lemma~\\ref{LemEig}, this is equivalent to $||\\widetilde{s}_{0}||_{\\alpha,\\omega} = ||\\widetilde{r}_{i}||_{\\alpha,\\omega}$.\nThis and the first two statements of Lemma~\\ref{new} imply that\n$\\alpha_{\\widetilde{s}_{0}} = \\alpha_{\\widetilde{r}_{i}} = \\alpha_{\\widetilde{r}_{j}}$, and $\\omega_{\\widetilde{r}_{j}}=\\omega_{\\widetilde{r}_{i}} = \\omega_{\\widetilde{s}_{0}}$, in particular, $j=i$. Therefore, $\\widetilde{s}_{0}$ is trivial in $N \/ \\langle \\! \\langle \\widetilde{r}_{i} \\rangle \\! \\rangle$, and the index $i$ is determined by $\\alpha_{\\widetilde{s}_{0}} = \\alpha_{\\widetilde{r}_{i}}$. One can prove in the same way that $\\widetilde{r}_{i}$ is trivial in $N \/ \\langle \\! \\langle \\widetilde{s}_{0} \\rangle \\! \\rangle$. Thus, $\\langle \\! \\langle \\widetilde{s}_{0} \\rangle \\! \\rangle_{N} =  \\langle \\! \\langle \\widetilde{r}_{i} \\rangle \\! \\rangle_{N}$. From Section~\\ref{leftright} we know that $N$ is a free group. So by \\mbox{Theorem~\\ref{Magnus}}, $\\widetilde{s}_{0}$ is conjugate to $\\widetilde{r}_{i}^{\\pm1}$ in $N$. Finally, $\\widetilde{r}$ is conjugate to $\\widetilde{s}^{\\pm1}$ in $H$, and the proof of Proposition~\\ref{prop1} in the case $||\\widetilde{r}||_{\\alpha,\\omega} \\geqslant 1$ is completed. \\hfill $\\qed$\n\n\\section{Proof of Proposition \\ref{prop1} for \\boldmath{$\\widetilde{r}$} with non-positive  {\\boldmath $\\alpha$-$\\omega$}-length}\n\n\\subsection{Properties of \\boldmath{$\\widetilde{r}$} with non-positive \\boldmath{$\\alpha$-$\\omega$}-length}\n\n\\begin{lemma} \\label{LemEig2}\nLet $r$ be an element in $N$ with $||r||_{\\alpha,\\omega} < 1$.\nThen the reduced word representing $r$ in basis $\\mathcal{B}^{+}(\\alpha_{r})$ contains only $b_{i}$'s.\n\\end{lemma}\n\n\\begin{proof} We have $\\omega_r<\\alpha_r$. By definition,  $r\\in G_{-\\infty,\\omega_r}$ and $r\\in G_{\\alpha_r,\\infty}$.\\\\\n$\\bullet$ Let $r^{-}$ be the reduced word representing $r$ in the basis $\\mathcal{B}^{-}(\\omega_r)$ of $G_{-\\infty,\\omega_r}$. Recall that\n$$\\mathcal{B}^{-}(\\omega_r)=\\{b_i\\,|\\, i\\in I \\}  \\cup \\{ y_{m,\\ell} \\mid 1 \\leqslant m \\leqslant n, \\ell \\leqslant \\omega_r \\},$$\nwhere $I=\\{\\omega_r-k+1,\\dots ,\\omega_r\\}$.\\\\\n\\noindent\n$\\bullet$ Let $r^{+}$ be the reduced word representing $r$ in the basis $\\mathcal{B}^{+}(\\alpha_r)$ of $G_{\\alpha_r,\\infty}$. Recall that\n$$\\mathcal{B}^{+}(\\alpha_r)=\\{b_j\\,|\\, j\\in J\\}  \\cup \\{y_{m,\\ell} \\mid 1 \\leqslant m \\leqslant n, \\ell \\geqslant \\alpha_r \\},$$\nwhere $J=\\{\\alpha_r,\\dots,\\alpha_r+k-1\\}$.\n\nFor each $i\\in I$, there is a unique $j\\in J$ such that $j\\equiv i \\mod k$.\nThen $r^{+}$ can be obtained from $r^{-}$ by replacements of all letters $b_i$ by\n$b_ju_{j-k}^{-1}u_{j-2k}^{-1}\\dots u_i^{-1}$ followed by reduction (see Remark~\\ref{remchange}). The second indices of $y$-letters appearing in these\nreplacements lie in\nthe interval $(-\\infty, \\max J -k]= (-\\infty, \\alpha_r-1]$.\nThe second indices of $y$-letters of $r^{-}$ lie in the interval $(-\\infty, \\omega_r]\\subseteq (-\\infty, \\alpha_r-1]$.\nTherefore, the second indices of $y$-letters of $r^{+}$ (if exist) lie in $(-\\infty, \\alpha_r-1]$.\nHence, $r^{+}$ does not have $y$-letters.\n\\end{proof}\n\n\\subsection{Conclusion of the proof for \\boldmath{$\\widetilde{r}$} with non-positive \\boldmath{$\\alpha$-$\\omega$}-length} \\label{conclusion2}\n\nAs in Section \\ref{conclusion1}, we can replace the elements $r,s \\in N$ from Proposition~\\ref{prop1} by suitable elements~$\\widetilde{r}, \\widetilde{s}$.\nIn this section, we consider the case $||\\widetilde{r}||_{\\alpha,\\omega} < 1$. Assuming $||\\widetilde{s}||_{\\alpha,\\omega} \\geqslant 1$, we immediately get a contradiction by applying Lemma~\\ref{new} with reversed roles of $\\widetilde{r}$ and $\\widetilde{s}$. Thus, we have $||\\widetilde{s}||_{\\alpha,\\omega} < 1$.\n\nBy Lemma~\\ref{LemEig2}, there are presentations of $\\widetilde{r}$ and $\\widetilde{s}$ in $N$ which use only $b_i$, $i\\in \\mathbb{Z}$. So by means of the relations $b_{i}=x^{-i}bx^{i}$, we get presentations of $\\widetilde{r}$ and $\\widetilde{s}$ as elements in $$H=\\langle x,b,y_{1}, \\dots , y_{n} \\ | \\ [x^{k},b]u \\rangle$$ that only use $x$ and $b$.\n\nThus, $\\widetilde{r},\\widetilde{s}\\in \\widetilde{H}$, where $\\widetilde{H}$ is the subgroup of $H$ generated by $x$ and $b$. We have the presentations\n\n$$\n\\widetilde{H}=\\langle x,b \\rangle \\cong F_{2} \\ \\ \\text{and} \\ \\ H=\\widetilde{H} \\underset{[x^{k},b]=u^{-1}}{\\ast} \\langle y_{m} \\mid 1 \\leqslant m \\leqslant n \\rangle, \\eqno{(5.1)}\n$$\n\nwhere $F_{2}$ is the free group in two generators and the isomorphism to $F_{2}$ follows from the Magnus' \\mbox{Freiheitssatz}.\n\n{\\bf Convention.} For $h\\in \\widetilde{H}$, let $(\\!(h)\\!)$ be the normal closure of $h$ in $\\widetilde{H}$ and\nlet $\\langle\\! \\langle h\\rangle\\! \\rangle$ be the normal closure of $h$ in $H$.\n\nConsider the canonical homomorphism $\\iota: \\widetilde{H} \\rightarrow \\widetilde{H} \/ (\\! ( \\widetilde{r})\\! )$.\nThere are two cases:\n\n{\\bf Case 1.} $\\ [x^{k},b]^{p} \\notin (\\! ( \\widetilde{r} )\\! )$\nfor all $p \\geqslant 1$. \\\\\nIn this case $\\iota \\mid_{\\langle [x^{k},b] \\rangle}: \\widetilde{H} \\rightarrow \\widetilde{H} \/\n(\\! (\\widetilde{r} ) \\!)$ is an embedding. We identify the subgroup $\\langle [x^{k},b] \\rangle$ of $\\widetilde{H}$ with its image in $\\widetilde{H}\/(\\! ( \\widetilde{r} ) \\! )$. Then\n\\begin{eqnarray*}\nH \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle &=& \\langle x,b,y_{1},\\dots,y_{n} \\ | \\ [x^{k},b]u, \\ \\widetilde{r} \\rangle \\ \\ = \\ \\ \\widetilde{H} \/ (\\!( \\widetilde{r} )\\!) \\underset{[x^{k},b]=u^{-1}}{\\ast} \\langle y_{m} \\mid 1 \\leqslant m \\leqslant n \\rangle \\ \\ = \\ \\ \\widetilde{H} \/ ( \\!( \\widetilde{r} )\\!) \\underset{\\mathbb{Z}}{\\ast} \\mathbb{F}_{n}.\n\\end{eqnarray*}\n\n{\\bf Case 2.} $[x^{k},b]^{p} \\in ( \\! ( \\widetilde{r} ) \\! )$ for some $p\\geqslant 1$. \\\\\nWe choose the smallest $p$ with this property.\nUsing Tietze transformations, we write:\n\\begin{eqnarray*}\nH \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle &=& \\langle x,b,y_{1},\\dots,y_{n} \\ | \\ [x^{k},b]u, \\ \\widetilde{r} \\, \\rangle \\ = \\ \\langle x,b,y_{1},\\dots , y_{n} \\ | \\ [x^{k},b]u, \\ \\widetilde{r}, \\ [x^{k},b]^{-p} \\, \\rangle \\\\\n&=& \\langle x,b,y_{1}, \\dots , y_{n} \\ | \\ [x^{k},b]u, \\ \\widetilde{r}, \\ u^{p} \\, \\rangle \\ = \\ \\widetilde{H} \/ (\\! ( \\widetilde{r} )\\! ) \\underset{[x^{k},b]=u^{-1}}{\\ast} \\langle y_{1}, \\dots , y_{n} \\ | \\ u^{p} \\, \\rangle.\n\\end{eqnarray*}\nThus, in both cases, we have the embedding $\\widetilde{H} \/ ( \\! (\\widetilde{r} )\\! ) \\hookrightarrow H \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$. \\\\\n\nWe already know that $\\widetilde{s}$ is an element of the subgroup $\\widetilde{H}$ of $H$. Further, $\\widetilde{s}$ is trivial in $H \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$ since $\\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle = \\langle \\! \\langle \\widetilde{s} \\rangle \\! \\rangle$. Now, we use the embedding $\\widetilde{H} \/ (\\! ( \\widetilde{r} )\\! ) \\hookrightarrow H \/ \\langle \\! \\langle \\widetilde{r} \\rangle \\! \\rangle$ to conclude that $\\widetilde{s}$ is trivial in $\\widetilde{H} \/ ( \\! ( \\widetilde{r} ) \\! )$. By symmetry, $\\widetilde{r}$ is trivial in $\\widetilde{H} \/ ( \\! ( \\widetilde{s} )\\! )$, and we get $(\\! ( \\widetilde{r} ) \\! ) = (\\! ( \\widetilde{s} )\\! )$.\n\nAs we have seen in (5.1), the group $\\widetilde{H}$ is isomorphic to the free group of rank 2, and this group possesses the Magnus property (cf. \\cite{ArtMagnus}). Hence, $\\widetilde{r}$ is conjugate to $\\widetilde{s}^{\\pm1}$ in $\\widetilde{H}$, and since $\\widetilde{H} \\hookrightarrow H$, the element $\\widetilde{r}$ is conjugate to $\\widetilde{s}^{\\pm1}$ in $H$. \\hfill $\\qed$ \\\\\n\n\\textbf{Acknowledgements.} This work is based on my M.Sc. thesis, Heinrich Heine Universit\\\"at D\\\"usseldorf, 2015. I would like to express my special thanks to my supervisor Professor Oleg Bogopolski for the subject proposal and the support during and after the preparation of my M.Sc. thesis. I also want to thank Professor Benjamin Klopsch for his editorial help.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n$\\indent$ We consider the Cauchy problem for the nonlinear Schr\u00f6dinger equation\n\\ben\\label{cauchy}\n\\left\\{\\begin{array}{llr}\niu_t + u_{xx} + |u|^{4}u +iau =0,& (t,x)\\in[0,\\infty)\\times{\\mathbb R}& \\mbox{(NLS}_a\\mbox{)}\\\\\nu(0,x)=u_0(x),\\ u_0:{\\mathbb R}\\to{\\mathbb C}\n\\end{array}\\right.\n\\een\nwith a real inhomogeneous damping term $a\\in C^1({\\mathbb R};{\\mathbb R})\\cap W^{1,\\infty}({\\mathbb R};{\\mathbb R})$. This is the one-dimensional $L^2$-critical case of the equation\n\\ben\\label{geral}\niu_t + \\Delta u + |u|^{p-1}u +iau =0,\n\\een\nwith $1<p<1+4\/(N-2)$ if $N\\ge3$ and $1<p<\\infty$ if $N=1,2$. The equation (\\ref{geral}) arises in several areas of nonlinear optics and plasma physics. The inhomogenous damping term corresponds to an electromagnetic wave absorved by an inhomogenous medium (cf. \\cite{b1}, \\cite{b2}).\n\nIt is known that, if $u_0\\in H^1({\\mathbb R}^N)$, the Cauchy problem for (\\ref{geral}) is locally well-posed (see Cazenave \\cite{b3}, theorem 4.4.6). Moreover, if $T_a(u_0)$ is the maximal time of existence for the solution $u(t)$, one has the blowup alternative: if $T_a(u_0)<\\infty$, then $\\|\\nabla u(t)\\|_2\\to\\infty$ when $t\\to T$.\n\nThe case where $a$ is constant was studied in \\cite{darwich}, \\cite{ohtatodorova}, \\cite{tsutsumi}. In the supercritical case ($1+4\/N<p <1 + 4\/(N-2)$), for sufficiently small damping and special initial data with negative energy, the blowup of the solution is proved. The proof of this result is based on the variance method introduced in \\cite{glassey} and \\cite{zakharov}. Such method does not seem to work on the $L^2$-critical case for $a>0$. Also, for $1<p<1+4\/(N-2)$ and for all initial data in $H^1({\\mathbb R}^N)$, one proves the global existence of the solutions for sufficiently large damping. The critical case ($p=1+4\/N$) was studied in \\cite{darwich}, where one proves, for small dimensions, the existence of blowup phenomena for small damping. The technique used therein is strongly based on the works of Merle and Rapha\u00ebl (\\cite{merleraphael1}, \\cite{merleraphael2}).\n\nRegarding the equation with inhomogeneous damping, it has been recently proved in \\cite{diasfigueira} the existence of blow-up phenomena in the supercritical case, under similar conditions to those of the homogeneous case. Here, we shall consider the critical exponent $p=1+4\/N$ and we prove the following result:\n\\begin{teo}\nThere exists $\\delta>0$ such that, for $\\|a\\|_{W^{1,\\infty}}<\\delta$, there exists $u_0\\in H^1({\\mathbb R})$ such that the solution of (\\ref{cauchy}) blows up in finite time.\n\\end{teo}\n\\begin{nota}\nThe result is stated in dimension one. We conjecture that it can be extended to higher dimensions (see \\cite{merleraphael1}, \\cite{darwich}).\n\\end{nota}\nAs a consequence of the technique used to prove the existence of blowup, we can prove an upper bound on the blow-up rate:\n\\begin{cor}\nThe explosive solution $u$ constructed in theorem 1 satisfies\n$$\n\\|u_x\\|_2\\le C^*\\frac{|\\log(T-t)|^{1\/4}}{\\sqrt{T-t}}, \\mbox{ for } t \\mbox{ close to } T,\n$$\nwhere $C^*$ is an universal constant.\n\\end{cor}\n\nAs it was said before, the variance method does not seem to work in the critical case for the damped equation.  However, another method to prove the blow-up of\ncertain solutions of equation (1) in the case $a=0$ was introduced by\nMerle and Raphael in \\cite{merleraphael1}, based on the so-called {\\it geometric\ndecomposition}\ntechnique. The main goal of this method was to obtain an\nupper bound on the blow-up rate, similar to the one presented in Corollary 2,\nwhich was improved in \\cite{merleraphael2} with a sharp upper bound estimate\n(the log log upper bound). In \\cite{darwich}, an extension of such a technique was made to the case where $a$ is a positive constant function, thus obtaining the first blow-up result for this critical case. Here, despite an inspiration on the arguments presented in \\cite{darwich}, we do not follow the same steps. A simplification is made, to make the method used clearer. For example, in this proof, we shall not use Strichartz estimates, which were of particular importance in \\cite{darwich}. One advantage is that the proof of blowup using this technique is done in a simpler way, which also implies a simplification of the proof of the upper bound on the blow-up rate. The disadvantage is that, while in \\cite{darwich} one proves the log-log upper bound for the particular solution previously constructed, here we shall only prove the log upper bound.\n\nWe now recall some important invariances in the energy space $H^1({\\mathbb R})$ for the nonlinear Schr\u00f6dinger equation\n\\ben\niu_t + \\Delta u + |u|^{4\/N}u=0\n\\een\nnamely:\n\\bi\n\\item Mass (or charge): $C(u)=\\|u\\|^2_2=C(u_0)$;\n\\item Energy: $E(u)=\\frac{1}{2}\\|\\nabla u\\|_2^2 - \\frac{1}{p+1}\\|u\\|_{p+1}^{p+1}=E(u_0)$;\n\\item Linear momentum: $M(u)=\\parteim \\int_{{\\mathbb R}^N} {\\bar{u}}\\nabla u=M(u_0)$;\n\\ei\nFor the (NLS$_a$) equation, these quantities are no longer conserved and one obtains the following evolution laws:\n\\bi\n\\item Mass evolution law:\n\\ben\n\\frac{d}{dt}C(u(t)) + 2\\int_{{\\mathbb R}^N} a(x)|u(t,x)|^2dx = 0;\n\\een\n\\item Energy evolution law:\n\\begin{align}\n\\frac{d}{dt}E(u(t))&= -\\int_{{\\mathbb R}^N}a(x)|\\nabla u(t,x)|^2dx + \\int_{{\\mathbb R}^N}a(x)|u(t,x)|^{p+1}dx \\\\&\\quad- \\partere \\int_{{\\mathbb R}^N} \\left(\\nabla u(t,x)\\cdot \\nabla a(x)\\right){\\bar{u}}(t,x)dx;\\nonumber\n\\end{align}\n\\item Linear momentum evolution law:\n\\ben\n\\frac{d}{dt}M(u(t)) + 2\\int_{{\\mathbb R}^N} a(x)\\parteim\\nabla u(t,x){\\bar{u}}(t,x) dx = 0.\n\\een\n\\ei\nNote that, from the mass evolution law, one has\n\\ben\\label{controlomassa}\n\\|u_0\\|_2\\ e^{-\\|a\\|_\\infty t}\\le \\|u(t)\\|_2 \\le \\|u_0\\|_2\\ e^{\\|a\\|_\\infty t}.\n\\een\n\nThe rest of this paper in organized as follows: in section 2, we will make a brief presentation of the technique used in \\cite{merleraphael1}, highlighting the main steps. In section 3, a general idea of the proof is given, followed by its demonstration and, at the end, the log upper bound will be proved.\n\n\\begin{section}{The geometric decompositon method}\n$\\indent$ In this section, we shall consider the case where $a\\equiv0$ and $N=1$,\n\\ben\\tag{NLS}\niu_t+u_{xx} + |u|^4u=0.\n\\een\n\nIn this context, one may look for time-periodic solutions of the form $u(t,x)=e^{it}\\phi(x)$. Inserting this expression in (NLS), we obtain the equation satisfied by $\\phi$:\n$$\n-\\phi_{xx}+\\phi=|\\phi|^4\\phi.\n$$\nAs proved in \\cite{b3}, section 8.1, the above equation has non-trivial solutions in the energy space $H^1({\\mathbb R})$. Futhermore, all the solutions are of the form \n$$\n\\phi(x)=e^{i\\omega}Q(x-y),\\ \\omega,y\\in{\\mathbb R},\n$$\nwhere $Q:{\\mathbb R}\\to{\\mathbb R}$ is a positive decreasing radial function with exponential decay at infinity called the ground-state associated with (NLS). One may also prove (see \\cite{b3}, section 8.4) that the ground-state is the only function (modulo translations and multiplication by a complex exponential) which minimizes the functional\n$$\nG(u)=\\frac{\\|\\nabla u\\|_2^2 \\|u\\|_2^{4}}{\\|u\\|_{6}^{6}}, \\ u\\neq 0.\n$$\nWe define $Q_d=\\frac{1}{2}Q+yQ_y$ and $Q_{dd}=\\frac{1}{2}Q_d+y\\left( Q_d\\right)_y$. Moreover, we write the inner product in $L^2({\\mathbb R})$ as $(\\cdot,\\cdot)$.\n\nConsider a continuous function $u:[0,T]\\to H^1({\\mathbb R})$. From the variational characterization of the ground-state, it is proved in \\cite{merleraphael1}, lemma 1, that, for small $\\alpha$, if $0<\\|u(t)\\|_2^2-\\|Q\\|_2^2<\\alpha$ and $E(u(t))\\le \\alpha\\|u_x(t)\\|_2^2$, there exist $C^1$ functions $x,\\theta:[0,T]\\to{\\mathbb R}$ and $\\lambda:[0,T]\\to{\\mathbb R}^+$ such that\n\\ben\n\\left| 1-\\lambda(t)\\frac{\\| u_x(t)\\|_2}{\\|Q\\|_2}\\right| < \\delta(\\alpha)\n\\een\nand\n\\ben\n\\|\\lambda(t)^{1\/2}e^{i\\theta(t)}u(\\lambda(t)(\\cdot-x(t)),t)-Q\\|_{H^1({\\mathbb R})} < \\delta(\\alpha),\n\\een\nwhere $\\delta(\\alpha)>0$ satisfies $\\delta(\\alpha)\\to 0$ when $\\alpha\\to 0$. We define \n\\ben\\label{epsilon}\n\\epsilon(t)=\\lambda(t)^{1\/2}e^{i\\theta(t)}u(\\lambda(t)(\\cdot-x(t)),t)-Q.\n\\een\nThe set of the functions $x,\\theta,\\lambda$ and $\\epsilon$ is called a \\textit{geometric decomposition} of $u$.\n\nIf $u_0\\in H^1({\\mathbb R})$ satisfies $0<\\|u_0\\|_2^2-\\|Q\\|_2^2<\\alpha$ and $E(u_0)<0$, then, by the conservation of charge and energy, the corresponding solution  of (NLS) satisfies the conditions for the geometric decomposition. Therefore, one may write (NLS) using $\\epsilon$ and the change  temporal variable\n\\be\ns(t)=\\int_0^t\\frac{1}{\\lambda^2(\\tau)}d\\tau,\n\\ee\nthus obtaining the following system:\n\\begin{align}\\label{eqe}\n\\partial_s\\epsilon_1 - L_-\\epsilon_2 &= \\frac{\\lambda_s}{\\lambda}Q_d + \\frac{x_s}{\\lambda}Q_y + \\frac{\\lambda_s}{\\lambda}(\\epsilon_1)_d + \\frac{x_s}{\\lambda}(\\epsilon_1)_y + \\tilde{\\theta_s}\\epsilon_2 - R_2(\\epsilon)\\\\\\label{eqe2}\n\\partial_s\\epsilon_2 + L_+\\epsilon_1 &= -\\tilde{\\theta_s}Q - \\tilde{\\theta_s}\\epsilon_1 + \\frac{\\lambda_s}{\\lambda}(\\epsilon_2)_d + \\frac{x_s}{\\lambda}(\\epsilon_2)_y + R_1(\\epsilon).\n\\end{align}\nwhere $\\tilde{\\theta}_s=-1-\\theta_s$, $\\epsilon=\\epsilon_1+i\\epsilon_2$,  $L_+=-\\Delta + 1 - 5Q^4$, $L_-=-\\Delta + 1 - Q^4$ and $R_1(\\epsilon), R_2(\\epsilon)$ are formally quadratic in $\\epsilon$. The operator $L=(L_+,L_-)$ is the linear operator close to the ground-state, which has been studied in \\cite{weinstein}. Therein, there are proved the following identities:\n$$\nL_+(Q_d)=-2Q; \\quad L_+(Q_y)=0; \\quad L_-(Q)=0;\n$$\n$$\n L_-(yQ)=-2Q_y; \\quad L_-(y^2Q)=-4Q_d.\n$$\nThese properties are essential to make the estimates on the parameters of the geometric decompositon.\n\nOne can also write the linear momentum of $u$ as a function of $\\epsilon$:\n\\ben\\label{momento}\nM(u(t)) = \\frac{1}{\\lambda}\\left[\\parteim\\left(\\int_{\\mathbb R} \\epsilon_y(t)\\bar{\\epsilon}(t)dx\\right) - 2(\\epsilon_2,Q_y)\\right].\n\\een\nBy choosing $u_0\\in H^1({\\mathbb R})$ such that $M(u_0)=0$, then $M(u(t))\\equiv 0$ and one has easily \n\\ben\n|(\\epsilon_2, Q_y)(t)|\\le \\delta(\\alpha)\\|\\epsilon(t)\\|,\n\\een\n where $\\|\\epsilon\\|^2=\\int_{\\mathbb R} |\\epsilon_y|^2 + \\int_{\\mathbb R} |\\epsilon|^2e^{-2^-|y|}$ and $2^-$ is a positive constant smaller than $2$ related with the properties of $Q$. This inequality is essential in the following results.\n\nNow, if we choose a geometric decompositon (this choice can be made using the implicit function theorem) such that $\\epsilon$ satisfies $(\\epsilon_1,Q_d)=0;\\ (\\epsilon_2,Q_{dd})=0;\\ (\\epsilon_1,yQ)=0$ (the so-called ortogonality conditions), then it is possible to obtain\n\\ben\\label{virialtransformado}\n\\left[ \\left(1+\\frac{1}{4\\delta_0}(\\epsilon_1,Q)\\right)(\\epsilon_2,Q_d)\\right]_s\\ge \\delta_0\\|\\epsilon\\|^2 + 2\\lambda^2|E_0| - \\frac{1}{\\delta_0}(\\epsilon_2,Q_d)^2.\n\\een\nwhere $\\delta_0$ is a positive constant.\n\nBy using the equations \\eqref{eqe}, \\eqref{eqe2} and the inequality (\\ref{virialtransformado}), we derive the following result:\n\\begin{lema}\nFor $\\alpha_1>0$ small, there exists $s_0>0$ such that\n\\be\n(\\epsilon_2,Q_d)(s)>0,\\ \\forall s>s_0.\n\\ee\nMoreover, if $s_2>s_1>s_0$,\n\\ben\\label{desigualdadeintegral}\n3\\int_{s_1}^{s_2}(\\epsilon_2,Q_d) ds - C(\\delta_0)\\delta(\\alpha)\\le -\\|yQ\\|_2^2\\log \\frac{\\lambda(s_2)}{\\lambda(s_1)} \\le 5\\int_{s_1}^{s_2}(\\epsilon_2,Q_d) ds + C(\\delta_0)\\delta(\\alpha)\n\\een\nand one has the quasi-monotony property\n\\ben\\label{quasemonotonia}\n\\lambda(s_2)<2\\lambda(s_1).\n\\een\n\\end{lema}\nFrom the inequalities (\\ref{virialtransformado}), (\\ref{desigualdadeintegral}) and (\\ref{quasemonotonia}), one proves that  $\\lim_{t\\to T_{max}} \\|\\nabla u(t)\\|_2=\\infty$ (or, equivalently, $\\lim_{t\\to T_{max}} \\lambda(t)=0$), where $T_{max}$ is the maximal time of existence of the solution of (NLS). By using a refinement of the geometric decomposition, in which one introduces  $\\tilde{\\epsilon}=\\epsilon+i\\frac{(\\epsilon_2,Q_d)}{\\|yQ\\|_2^2}W$, where $W=y^2Q+\\nu Q$ and $\\nu$ is such that $(W,Q_{dd})=0$, one obtains the following:\n\\begin{lema}\\label{refinada}\n\n1. There exist universal constants $\\tilde{\\delta}_0>0$ and $C>0$ such that, for small $\\alpha>0$, there exists $\\tilde{s}_1$ verifying\n\\ben\\label{refinado}\n\\left[\\left(1+\\frac{(\\epsilon_1, W_d)}{\\|yQ\\|_2^2}\\right)(\\epsilon_2,Q_d)\\right]_s + C(\\epsilon_2,Q_d)^4 \\ge \\tilde{\\delta}_0\\|\\tilde{\\epsilon}\\|^2 + \\lambda^2|E_0|, \\ \\forall \\ s>\\tilde{s}_1\n\\een\n2. There exists a universal constant $B>0$ such that, for small $\\alpha$, there exists $\\tilde{s}_2>0$ such that\n\\ben\\label{controloexp}\n\\lambda(s)^2\\le \\exp\\left(-\\frac{B}{(\\epsilon_2,Q_d)^2(s)}\\right), \\forall \\ s>\\tilde{s}_2.\n\\een\n\n\\end{lema}\n\nFinally, defining $t_k$ such that $\\lambda(t_k)=2^{-k}$, it follows from the inequalities (\\ref{desigualdadeintegral}), (\\ref{quasemonotonia}) and (\\ref{controloexp}) that $t_{k+1}-t_k\\le C\\lambda^2(t_k)|\\log\\lambda(t_k)|^{1\/2}$, for large $k$. By summing in $k$, we deduce the finiteness of $T_{max}$ and the blowup is proved. The log upper bound is then a simple consequence of the above considerations.\n\n\\end{section}\n\n\\begin{section}{Proof of the main theorem}\n\n$\\indent$ The technique presented in the previous section works as long as it is possible to obtain the geometric decomposition for the solution of the equation one is working with. Unlike the (NLS) setting, since the mass and the energy are no longer conserved, one cannot guarantee \\textit{a priori} that the solution of (NLS$_a$) is decomposable, even if the initial data satisfies the same conditions as before. Therefore, we shall work over certain uniformly bounded intervals contained in the maximal interval of existence of the solution of (NLS$_a$), where we know that it is possible to obtain the decomposition. The goal will be to prove that, by conveniently choosing the initial data and assuming $\\|a\\|_{W^{1,\\infty}}$ small, then the largest of those intervals is actually the maximal interval of existence. Since those intervals are bounded uniformly, one has $T_a(u_0)<\\infty$ and the blowup phenomenon is proved.\n\nLet $u_0\\in H^1({\\mathbb R})$ with $E(u_0)<0$ and $M(u_0)=0$. Set $\\alpha=2\\left(\\|u_0\\|_2^2-\\|Q\\|_2^2\\right)$, and assume that $\\alpha>0$ is small. Therefore, on a small interval $[0,T_0]$, it is possible to decompose the solution geometrically. We denote $m=\\lambda(0)$, parameter that has to be small for the following calculations. Futhermore, we suppose that $(\\epsilon_2,Q_d)(0)>0$ and that\n\\ben\n\\lambda(0)^2\\le e^{-\\frac{B}{(\\epsilon_2,Q_d)^2(0)}} \\le \\|\\epsilon(0)\\|^8.\n\\een\nNotice that these conditions can be fulfilled: given $\\tilde{u}_0\\in H^1({\\mathbb R})$ with negative energy and mass just above the critical mass $\\|Q\\|_2^2$, we consider the respective solution $\\tilde{u}$ of (NLS). By the previous section, we know that the solution blows-up and that, for $t$ close to $T_{max}$, $(\\epsilon_2, Q_d)(t)>0$ and $\\lambda(t)^2\\le \\exp\\left(-\\frac{B}{(\\epsilon_2,Q_d)^2(t)}\\right)$ (cf. Lemmas 3,4). Now it is enough to consider $u_0=\\tilde{u}(t)$, for a large fixed $t$.\n\nIn the following, we write $E(t):=E(u(t))$. Fixed $\\alpha, m$ and $\\|a\\|_{W^{1,\\infty}}$ small, we define the set $X$ as the set of all $T\\ge 0$ such that\\newline\n\n(H1) $\\quad T\\le\\frac{1}{2\\|a\\|_\\infty}\\log\\frac{\\|Q\\|_2^2+\\alpha}{\\|Q\\|_2^2+\\frac{\\alpha}{2}}$;\\newline\n\n(H2) $\\quad E(t)\\le\\alpha\\|u_x(t)\\|^2_2,\\ 0\\le t\\le T$;\\newline\n\nThese two conditions and \\eqref{controlomassa} allow us to obtain the geometric decomposition on the interval $[0,T]$. Now we define $k_0$ as the positive integer such that $\\frac{1}{2^{k_0}}\\ge\\lambda(0)>\\frac{1}{2^{k_0+1}}$ and $k_T$ as the integer such that $k_T\\ge k_0$ and $\\frac{1}{2^{k_T}}\\ge\\lambda(T)>\\frac{1}{2^{k_T+1}}$.\\newline\n\n(H3) For each $k_T\\ge k> k_0$, choose $t_k$ (taken in increasing order) such that\n$\\lambda(t_k)=2^{-k}$.\nWe also write $T=t_{k_T+1}$ and $0=t_{k_0}$. Then we require $t_{k+1}-t_k\\le \\lambda^{3\/2}(t_k)$, $ k_T\\ge k\\ge k_0$;\\newline\n\n(H4) $\\quad\\lambda(\\tilde{t})\\le2\\lambda(t)$, $\\forall \\tilde{t}, t:\\ T\\ge \\tilde{t}\\ge t\\ge 0$;\\newline\n\n(H5) $\\quad\\lambda^{1\/2}(t)\\le \\|\\epsilon(t)\\|^2$, $0\\le t\\le T$.\\newline\n\nIt is important to notice that the hypothesis placed over the interval $[0,T]$ have a direct analogy with the lemmas from the previous section.\n\nAs a consequence of the broad inequalities and the continuity of the functions involved in the conditions (H1)-(H5), the set $X$ is closed in $[0,T_a(u_0))$. Since $0\\in X$, $X$ is nonempty. If one proves that $X$ is open in $[0, T_a(u_0))$, then one obtains $X=[0,T_a(u_0))$. Since $X$ is bounded (by (H1)), this proves finite-time blowup. To show that $X$ is open in $[0, T_a(u_0))$, we shall prove that, if $T\\in X$, then, on the interval $[0,T]$, one verifies stronger conditions than those that define the set $X$. By continuity, this implies that, for small $\\delta>0$, $T+\\delta\\in X$, and $X$ is open.\n\n\\begin{nota}\nFor small $\\|a\\|_\\infty$,\n\\be\n T< 2 \\le\\frac{1}{2\\|a\\|_\\infty}\\log\\frac{\\|Q\\|_2^2+\\alpha}{\\|Q\\|_2^2+\\frac{\\alpha}{2}}.\n\\ee\nIn fact, using the lenght hypothesis for the intervals $[t_k,t_{k+1}]$,\n\\be\nT=\\sum_{k_0}^{k_T+1} t_{k+1}-t_k \\le \\sum_{k_0}^{k_T+1}\\left(\\frac{1}{2\\sqrt{2}}\\right)^k< 2\n\\ee\n\\end{nota}\n\n\\begin{lema}\nFor any $\\delta>0$, there exists $a_0>0$ such that, for $0<\\|a\\|_{W^{1,\\infty}}<a_0$, one has, over the interval $[0,T]$,\n\\be\nE(t)\\lambda^{3\/2}(t)<\\delta.\n\\ee\n\\end{lema}\n\\begin{proof}\nWe shall prove that, for each $i\\ge k_0$, if $E(t_j)\\lambda^{3\/2}(t_j)<\\delta\\left(1-\\frac{1}{j}\\right),\\ \\forall j\\le i$, then\n\\be\nE(t)\\lambda^{3\/2}(t) < \\delta\\left(1-\\frac{1}{i+1}\\right), \\ \\forall \\ t\\in[t_i,t_{i+1}].\n\\ee\nThe result then follows by induction.\nSuppose that\n\\be\nE(t_j)\\lambda^{3\/2}(t_j)<\\delta\\left(1-\\frac{1}{j}\\right), \\forall j\\le i\n\\ee\nRecall that, over the interval $[t_i,t_{i+1}]$, $\\lambda(t)>2^{-i+2}$ and, by the geometric decomposition, $\\lambda(t)$ is approximately $\\|Q\\|_2\\|u_x(t)\\|_2^{-1}$.\nUsing the energy evolution law,\n$$\n\\frac{dE(t)}{dt}\\le -\\|a\\|_\\infty(p+1)E(t) + \\|a\\|_\\infty\\frac{p+3}{2}\\| u_x(t)\\|_2^2 + \\|a_x\\|_\\infty\\|u_x(t)\\|_2e^{\\|a\\|_\\infty t}\\|u_0\\|_2\n$$\nand so\n\\begin{align}\\label{diferencial}\n\\frac{d}{dt}\\left(e^{\\|a\\|_\\infty(p+1)t}E(t)\\right) &\\le \\|a\\|_\\infty\\frac{p+3}{2}e^{\\|a\\|_\\infty(p+1)t}\\| u_x(t)\\|_2^2 \\nonumber\\\\&\\quad+ \\| a_x\\|_\\infty\\|u_x(t)\\|_2e^{\\|a\\|_\\infty(p+2) t}\\|u_0\\|_2.\n\\end{align}\nIf $i\\neq k_0+1$, integrating over the interval $[t_{i-1},t]$ with $t_i<t\\le t_{i+1}$,\n\\begin{align*}\ne^{\\|a\\|_\\infty(p+1)t}E(t) - e^{\\|a\\|_\\infty(p+1)t_{i-1}}E(t_{i-1})&\\le \\frac{p+3}{2(p+1)}2^{2(i+1)}\\left(e^{\\|a\\|_\\infty(p+1)t}-e^{\\|a\\|_\\infty(p+1)t_{i-1}}\\right) \\\\\n&+ \\frac{\\| a_x\\|_\\infty}{\\|a\\|_\\infty(p+2)}2^{i+1}\\left(e^{\\|a\\|_\\infty(p+2)t}-e^{\\|a\\|_\\infty(p+2)t_{i-1}}\\right),\n\\end{align*}\nwhich implies\n\\begin{align*}\nE(t)&\\le E(t_{i-1})e^{-\\|a\\|_\\infty(p+1)(t-t_{i-1})} +  \\frac{p+3}{2(p+1)}2^{2(i+1)}\\left(1-e^{-\\|a\\|_\\infty(p+1)(t-t_{i-1})}\\right)\\\\\n&\\quad+ \\frac{\\| a_x\\|_\\infty}{\\|a\\|_\\infty(p+2)}2^{i+1}\\left(e^{\\|a\\|_\\infty t} - e^{-\\|a\\|_\\infty(p+1)(t-t_{i-1})}e^{\\|a\\|_\\infty t_{i-1}}\\right).\n\\end{align*}\nMultiplying by $\\lambda^{3\/2}(t)<\\lambda^{3\/2}(t_{i-1})=2^{-\\frac{3}{2}(i-1)}$ and using the induction hypothesis,\n\\begin{align*}\nE(t)\\lambda^{3\/2}(t)&\\le \\delta\\left(1-\\frac{1}{i-1}\\right) +  C2^{i\/2}\\left(1-e^{-\\|a\\|_\\infty(p+1)(t-t_{i-1})}\\right) \\\\&\\quad+ \\frac{\\| a_x\\|_\\infty}{\\|a\\|_\\infty(p+2)}2^{-i\/2}\\left(e^{\\|a\\|_\\infty t} - e^{-\\|a\\|_\\infty(p+1)(t-t_{i-1})}e^{\\|a\\|_\\infty t_{i-1}}\\right).\n\\end{align*}\nFrom the interval lenght hypothesis, one has $t_{i+1}-t_{i-1}\\le  2^{-i+2}$. Since\n\\begin{align*}\n&\\frac{1}{\\|a\\|_\\infty}2^{-i\/2}\\left(e^{\\|a\\|_\\infty t} - e^{-\\|a\\|_\\infty(p+1)(t-t_i)}e^{\\|a\\|_\\infty t_i}\\right)\\\\\\le\\ & e^{\\|a\\|_\\infty T}2^{-3i\/2}\\left(\\frac{e^{\\|a\\|_\\infty 2^{-i+2}} - e^{-\\|a\\|_\\infty(p+1)2^{-i}+2}}{\\|a\\|_\\infty 2^{-i}}\\right) \\le K2^{-3i\/2},\n\\end{align*}\nwith $K$ independent of $i$ and $\\|a\\|_\\infty$, we deduce\n\\begin{align*}\nE(t)\\lambda^{3\/2}(t)\\le \\delta\\left(1-\\frac{1}{i}\\right) +  C2^{i\/2}\\left(1-e^{-\\|a\\|_\\infty(p+1)2^{-i+2}}\\right) + \\frac{\\| a_x\\|_\\infty}{(p+2)}K2^{-3i\/2}.\n\\end{align*}\nIt now suffices to check that, independently of $i$, for small $\\|a\\|_{W^{1,\\infty}}$, \n$$\nC2^{i\/2}\\left(1-e^{-\\|a\\|_\\infty(p+1)2^{-i}}\\right) + \\frac{\\| a_x\\|_\\infty}{(p+2)}K2^{-3i\/2}\\le \\delta\\left(\\frac{1}{i}-\\frac{1}{i+1}\\right).\n$$\nFor the case $i=k_0+1$, we integrate (\\ref{diferencial}) over the interval $[t_{k_0}, t]$, with $t_{k_0}<t\\le t_{k_0+1}$ and we use the fact that $E(t_{k_0})<0$.\n\\end{proof}\n\nThe following lemma solves the problem of the non-conservation of the linear momentum:\n\\begin{lema}\nFor small $\\|a\\|_\\infty$ and $\\alpha$, one has $|(\\epsilon_2,Q_y)(t)|\\le 2\\delta(\\alpha)\\|\\epsilon(t)\\|$, $\\forall t\\in [0,T]$.\n\\end{lema}\n\\begin{proof}\nFor each $k_0\\le k\\le k_T$ and $t\\in[t_k,t_{k+1}]$,\n\\begin{align*}\n&\\left|\\int_{\\mathbb R} \\parteim u_x(t){\\bar{u}}(t) dx - \\int_{\\mathbb R} \\parteim u_x(t_k){\\bar{u}}(t_k) dx\\right|=2\\left|\\int_{t_k}^{t}\\int_{\\mathbb R} a \\parteim u_x(s,x){\\bar{u}}(s,x)dx ds\\right|  \\\\\n\\le&\\ 2\\|a\\|_\\infty\\|u_0\\|_2e^{\\|a\\|_\\infty T}\\int_{t_k}^{t} \\|\\nabla u(s)\\|_2ds \\le\n C\\|a\\|_\\infty\\|u_0\\|_2  \\int_{t_k}^{t_{k+1}} \\frac{1}{\\lambda(s)} ds\\\\\\le&\\ C\\|a\\|_\\infty\\|u_0\\|_2 \\frac{2}{\\lambda(t_{k+1})} (t_{k+1}-t_k)\\\\\n\\le&\\ C'\\|a\\|_\\infty\\|u_0\\|_2\\lambda^{1\/2}(t_k).\n\\end{align*}\nGiven $t\\in[0,T]$, let $k_t$ be such that $t\\in [t_{k_t}, t_{k_t+1}]$. Then\n\\begin{align*}\n\\left|\\int_{\\mathbb R} \\parteim u_x(t){\\bar{u}}(t) dx\\right| &\\le \\left|\\int_{\\mathbb R} \\parteim u_x(t){\\bar{u}}(t) dx - \\int_{\\mathbb R} \\parteim u_x(t_k){\\bar{u}}(t_k) dx\\right|\\\\&\\quad+ \\sum_{i=0}^{k_t-1}\\left|  \\int_{\\mathbb R} \\parteim u_x(t_{i+1}){\\bar{u}}(t_{i+1}) dx - \\int_{\\mathbb R} \\parteim u_x(t_i){\\bar{u}}(t_i) \\right|\\\\&\\le C'\\|a\\|_\\infty\\|u_0\\|_2\\sum_{i=0}^{k_t}\\lambda^{1\/2}(t_i)\\le C'\\|a\\|_\\infty\\|u_0\\|_2\\sum_{i=0}^\\infty \\left(\\frac{1}{\\sqrt{2}}\\right)^i.\n\\end{align*}\nRecalling the last property of the interval $[0,T]$, (H5), we obtain\n$$\n\\left|\\lambda(t)\\int_{\\mathbb R} \\parteim u_x(t){\\bar{u}}(t) dx\\right|\\le \\|\\epsilon(t)\\|^4\\left(C'\\|a\\|_\\infty\\|u_0\\|_2\\sum_{i=0}^\\infty \\left(\\frac{1}{\\sqrt{2}}\\right)^i\\right)\\le \\delta(\\alpha)\\|\\epsilon(t)\\|,\n$$\nfor small $\\|a\\|_\\infty$ and $\\alpha$.\nUsing \\eqref{momento}, we deduce finally\n\\begin{align*}\n|(\\epsilon_2,Q_y)(t)|&\\le \\left|\\parteim\\left(\\int_{\\mathbb R} \\epsilon_y(t)\\bar{\\epsilon}(t)dx\\right)\\right| + \\left|\\lambda(t)M(u(t))\\right|\\le \\|\\epsilon(t)\\|_2\\|\\epsilon_y(t)\\|_2+\\delta(\\alpha)\\|\\epsilon(t)\\| \\\\&\\le 2\\delta(\\alpha)\\|\\epsilon(t)\\|.\n\\end{align*}\n\n\\end{proof}\nLet us introduce a new time variable\n\\be\ns(t)=\\int_0^t \\frac{1}{\\lambda^2(\\tau)}d\\tau\n\\ee\nand define $S=s(T)$ and $s_k=s(t_k)$, $ k_0\\le k \\le k_T$. Then, from the expression of $\\epsilon$, \\eqref{epsilon}, we may write (NLS$_a$) in terms of $\\epsilon=\\epsilon_1 + i\\epsilon_2$ over the interval $[0,S]$:\n\n\\begin{align}\\label{aeqeum}\n\\partial_s\\epsilon_1 - L_-\\epsilon_2 &= \\frac{\\lambda_s}{\\lambda}Q_d + \\frac{x_s}{\\lambda}Q_y + \\frac{\\lambda_s}{\\lambda}(\\epsilon_1)_d + \\frac{x_2}{\\lambda}(\\epsilon_1)_y + \\tilde{\\theta_s}\\epsilon_2 - R_2(\\epsilon) -a\\lambda^2\\epsilon_1\n\\\\\n\\label{aeqedois}\n\\partial_s\\epsilon_2 + L_+\\epsilon_1 &= -\\tilde{\\theta_s}Q - \\tilde{\\theta_s}\\epsilon_1 + \\frac{\\lambda_s}{\\lambda}(\\epsilon_2)_d + \\frac{x_s}{\\lambda}(\\epsilon_2)_y + R_1(\\epsilon)-a\\lambda^2\\epsilon_2.\n\\end{align}\n\nThrough the control of $|(\\epsilon_2,Q_y)|$ given by the previous lemma and the same ortogonality conditions as the last section, one has the following (see \\cite{merleraphael1}, proposition 1):\n\n\\begin{lema}\\label{asem(e1,Q)}\nThere exists an universal constant $\\delta_0>0$ such that, for $~\\alpha$ and $\\|a\\|_{W^{1,\\infty}}$ small,\n\\begin{align*}\n\\left[\\left(1+\\frac{1}{4\\delta_0}(\\epsilon_1,Q)\\right)(\\epsilon_2,Q_d)\\right]_s\\ge&\\ \\delta_0\\|\\epsilon\\|^2 - 2 \\lambda^2E - \\frac{1}{\\delta_0}(\\epsilon_2,Q_d)^2 \\\\&- \\|a\\|_\\infty\\lambda^2|(\\epsilon_2,Q_d)\\left((\\epsilon_2,Q_d)+2(\\epsilon_1,Q)\\right)|\n\\end{align*}\n\\end{lema}\n\nTo prove such a result, several steps are needed: first, one calculates $(\\epsilon_2, Q_d)_s$, use the first two ortogonality conditions and the energy expression in terms of $\\epsilon$ to obtain\n$$\n(\\epsilon_2, Q_d)_s \\ge H(\\epsilon, \\epsilon) -2\\lambda^2E - \\frac{x_s}{\\lambda}(\\epsilon_2, (Q_d)_y) + G(\\epsilon) - \\lambda^2(a\\epsilon_2, Q_d),\n$$\nwhere $H$ is some bilinear form related to $(L_-, L_+)$ and $G$ is a higher-order remainder. The last ortogonality condition guarantees that $|x_s\/\\lambda|\\le C\\delta(\\alpha)\\|\\epsilon\\|$. A precise study of the bilinear form $H$ insures that, in the subspace where $(\\epsilon_2,Q_d)=(\\epsilon_1,Q)=(\\epsilon_1, yQ)=(\\epsilon_2, Q_{dd})=0$, the form is coercive. Using this information, one obtains the following intermediate inequality\n$$\n(\\epsilon_2, Q_d)_s\\ge \\tilde{\\delta}_0\\|\\epsilon\\|^2-2\\lambda^2E - \\frac{4}{\\tilde{\\delta}_0}((\\epsilon_1,Q)^2 + (\\epsilon_2,Q_d)^2) - \\|a\\|_\\infty\\lambda^2|(\\epsilon_2, Q_d)|.\n$$\nFinally, one proves that $(\\epsilon_1,Q)^2$ is controled by $((\\epsilon_1, Q)(\\epsilon_2, Q_d))_s$ and obtains the final inequality.\n\\begin{nota}\\label{nota}\nDue to the hypothesis over the interval $[0,T]$, it is possible to simplify the previous inequality:\n\\begin{enumerate}\n\\item Since $2E\\lambda^2\\le 2E\\lambda^{3\/2}\\lambda^{1\/2}\\le \\frac{\\delta_0}{2}\\|\\epsilon\\|^2$ over $[0,T]$, we obtain\n$$\n\\delta_0\\|\\epsilon\\|^2 - 2\\lambda^2E\\ge \\frac{\\delta_0}{2}\\|\\epsilon\\|^2;\n$$\n\\item On the other hand, using (H4), $\\lambda(t)\\le 2\\lambda(0)=2m, \\forall\\ t\\in[0,T]$. Therefore $\\lambda$ is bounded on $[0,T]$ by a constant $L$ that only depends on $m$ and, for small $\\|a\\|_\\infty$,\n$$\n\\left|\\|a\\|_\\infty\\lambda^2(\\epsilon_2,Q_d)((\\epsilon_2,Q_d)+2(\\epsilon_1,Q))\\right|\\le \\|a\\|_\\infty L^2C\\|\\epsilon\\|^2\\le \\frac{\\delta_0}{4}\\|\\epsilon\\|^2.\n$$\n\\end{enumerate}\nIn this way, one obtains the following inequality:\n\\ben\\label{semenergia}\n\\left[\\left(1+\\frac{1}{4\\delta_0}(\\epsilon_1,Q)\\right)(\\epsilon_2,Q_d)\\right]_s\\ge \\frac{\\delta_0}{4}\\|\\epsilon\\|^2 - \\frac{1}{\\delta_0}(\\epsilon_2,Q_d)^2.\n\\een\n\\end{nota}\n\nWe now turn to the inequality analogous to (\\ref{desigualdadeintegral}). The terms associated to the damping parameter turn out to be irrelevant, since their integral over the set $[0,S]$ is bounded by a function of $\\|a\\|_\\infty$ which converges to 0 when $\\|a\\|_\\infty\\to 0$.\n\n\\begin{lema}\nFor small $a$ and $\\alpha$, one has, over the interval $[0,S]$, $(\\epsilon_2, Q_d)>0$ and \n\\ben\\label{desigualdadeintegralamort}\n3\\int_{s_1}^{s_2}(\\epsilon_2,Q_d) ds - C(\\delta_0)\\delta(\\alpha)\\le -\\|yQ\\|_2^2\\log \\frac{\\lambda(s_2)}{\\lambda(s_1)} \\le 5\\int_{s_1}^{s_2}(\\epsilon_2,Q_d) ds + C(\\delta_0)\\delta(\\alpha).\n\\een\n\\end{lema}\n\\begin{proof}\nSince $(\\epsilon_2,Q_d)(0)>0$, it is enough to check that, if $(\\epsilon_2,Q_d)=0$, then $(\\epsilon_2,Q_d)_s>0$. If there exists $s\\ge0$ such that $(\\epsilon_2,Q_d)=0$ and $(\\epsilon_2,Q_d)_s\\le0$, then, by (\\ref{semenergia}), \n\\be\n\\|\\epsilon\\|^2\\le 0,\n\\ee\nwhich is absurd. Therefore $(\\epsilon_2,Q_d)>0$ on $[0,S]$.\nTo obtain the integral inequality, we proceed as in \\cite{merleraphael1}. The problem is controlling the terms associated with $a$. For example, by taking the $L^2$ inner product of (\\ref{aeqeum}) with $y^2Q$ and integrating, one obtains the term\n\\be\n\\int_{s_1}^{s_2} \\|a\\|_\\infty \\lambda^2(\\epsilon_1,y^2Q) ds.\n\\ee\nHowever, simply notice that\n\\be\n\\left|\\int_{s_1}^{s_2} \\|a\\|_\\infty \\lambda^2(\\epsilon_1,y^2Q) ds\\right| \\le \\delta(\\alpha)\\|a\\|_\\infty\\int_0^S \\lambda^2(s)ds = \\delta(\\alpha)\\|a\\|_\\infty T\\le 2\\delta(\\alpha)\\|a\\|_\\infty.\n\\ee\nTherefore, for small $\\|a\\|_\\infty$, we deduce $\\left|\\int_{s_1}^{s_2} a \\lambda^2(\\epsilon_1,y^2Q) ds\\right|\\le \\delta(\\alpha)$. The remainder terms are controlled in a similar way.\n\\end{proof}\n\nUsing the previous result, we prove a stronger quasi-monotony property than the one in the definition of $X$:\n\\begin{lema}\nFor small $\\alpha$,\n\\be\n\\lambda(\\tilde{t})<\\frac{3}{2}\\lambda(t), \\quad T\\ge\\tilde{t}\\ge t\\ge0.\n\\ee\n\\end{lema}\n\\begin{proof}\nIf such an inequality was not true for some $t_1<t_2$, then, by (\\ref{desigualdadeintegralamort})\n$$\n\\|yQ\\|_2^2\\log\\frac{3}{2} - C(\\delta_0)\\delta(\\alpha)\\le \\|yQ\\|_2^2\\log\\frac{\\lambda(s_2)}{\\lambda(s_1)} - C(\\delta_0)\\delta(\\alpha) \\le -3\\int_{s_1}^{s_2}(\\epsilon_2, Q_d) ds <0, \n$$\nwhich is absurd, for small enough $\\alpha$.\n\\end{proof}\n\nSince the term $a\\lambda^2$ is bounded by a small constant, one may apply a reasoning similar to remark 3.2 to prove a result completely analogous to the first part of lemma \\ref{refinada}:\n\\begin{lema}\nThere exist universal constants $\\tilde{\\delta}_0>0$ e $C>0$ such that, for $m$ and $\\alpha$ small,\n\\ben\\label{refinadoamort}\n\\left[\\left(1+\\frac{(\\epsilon_1, W_d)}{\\|yQ\\|_2^2}\\right)(\\epsilon_2,Q_d)\\right]_s + C(\\epsilon_2,Q_d)^4 \\ge \\tilde{\\delta}_0\\|\\tilde{\\epsilon}\\|^2 - \\lambda^2E, \\ \\forall \\ s>0\n\\een\n\\end{lema}\n\nNow we prove the following\n\\begin{prop}\nThere exist universal constants $B', \\sigma>0$ such that, for small $\\alpha$ and $m$, \n\\ben\\label{controloexpamort}\n\\lambda(0)^{2\\sigma}\\lambda(s)^2\\le \\exp\\left(-\\frac{B}{(\\epsilon_2,Q_d)^2(s)}\\right), \\  \\ 0\\le s\\le S.\n\\een\n\\end{prop}\n\\begin{nota}\nThe above inequality is equivalent to\n\\ben\\label{maiorlog}\n(\\epsilon_2,Q_d)(s)\\ge \\frac{B^*}{|\\log \\left(\\lambda(0)^\\sigma\\lambda(s)\\right)|^{1\/2}}.\n\\een\n\\end{nota}\n\\begin{proof}\nWhat follows is an adaptation of the proof for the proposition 8 in \\cite{merleraphael1}.\nDefine \n\\ben\\label{expressaof}\nf(s)=\\left(1+\\frac{(\\epsilon_1,W_d)}{\\|yQ\\|_2^2}\\right)(\\epsilon_2,Q_d). \n\\een\nFor small $\\alpha>0$,\n\\be\n\\frac{1}{2}(\\epsilon_2,Q_d)\\le f \\le 2(\\epsilon_2,Q_d).\n\\ee\nThen $f>0$ for $s\\in[0,S]$ and, using (\\ref{refinadoamort}), there exists a universal constant $C'>0$ such that (see remark \\ref{nota}.1)\n\\be\nf_s+C'f^4\\ge0.\n\\ee\nIntegrating this inequality over $[0,s]$, we obtain\n\\be\n\\frac{1}{f^3(s)}\\le C's+\\frac{1}{f^3(0)},\n\\ee\nand from \\eqref{expressaof}, we deduce\n\\ben\\label{scuboamort}\n(\\epsilon_2,Q_d)(s)\\ge \\frac{1}{2\\left(C's+\\frac{1}{f^3(0)}\\right)^{1\/3}},\\ \\forall s>0.\n\\een\nNow, from (\\ref{desigualdadeintegralamort}) and \\eqref{scuboamort}, we obtain\n\\be\n3\\int_0^s (\\epsilon_2,Q_d)ds \\le -\\|yQ\\|_2^2\\log\\frac{\\lambda(s)}{\\lambda(0)} + C(\\delta_0)\\delta(\\alpha)\\le -\\frac{\\|yQ\\|_2^2}{2}\\log\\frac{\\lambda(s)}{\\lambda(0)}\n\\ee\nand\n\\be\nC''\\left(\\left(C's+\\frac{1}{f^3(0)}\\right)^{2\/3}-\\frac{1}{f^2(0)}\\right)\\le -\\log\\frac{\\lambda(s)}{\\lambda(0)}.\n\\ee\nHence, \n\\ben\\label{loglambda}\n\\frac{C''}{4(\\epsilon_2,Q_d)^2}\\le -\\log \\lambda(s) + \\log \\lambda(0) + \\frac{C''}{f^2(0)}.\n\\een\nSince $\\lambda(0)\\le e^{-\\frac{B}{(\\epsilon_2,Q_d)^2(0)}}$, there exists $\\sigma>0$ universal constant such that\n$$\n \\frac{C''}{f^2(0)}\\le -(\\sigma+1)\\log\\lambda(0)\n$$\nand from \\eqref{loglambda},\n$$\n\\frac{C''}{4(\\epsilon_2,Q_d)^2}\\le -\\log \\left(\\lambda(0)^\\sigma\\lambda(s)\\right).\n$$\nTherefore, there exists an universal constant $B'>0$ such that\n\\be\n- \\log \\left(\\lambda(0)^{2\\sigma}\\lambda^2(s)\\right)\\ge -2\\log \\left(\\lambda(0)^\\sigma\\lambda(s)\\right) \\ge \\frac{B'}{(\\epsilon_2,Q_d)^2(s)}, \n\\ee\nor, equivalently,\n$$\n\\lambda(0)^{2\\sigma}\\lambda(s)^2\\le \\exp\\left(-\\frac{B'}{(\\epsilon_2,Q_d)^2(s)}\\right), \\forall\\ s\\in [0,S].\n$$\n\\end{proof}\n\\begin{lema}\nThere exists an universal constant $D$ such that, for each $k_0\\le k\\le k_T$,\n$$\nt_{k+1}-t_k\\le D\\left|\\log\\left(\\lambda(0)^\\sigma\\lambda(t_k)\\right)\\right|^{1\/2}\\lambda^2(t_k)\n$$\n\\end{lema}\n\\begin{proof}\nFirst, using the quasi-monotonicity property and (\\ref{desigualdadeintegralamort}),\n$$2\\|yQ\\|_2^2\\log 2\\ge \\|yQ\\|_2^2 + C(\\delta_0)\\delta(\\alpha) \\ge 3\\int_{s_k}^{s_{k+1}} (\\epsilon_2, Q_d)(s)ds.\n$$\nNow, from (\\ref{maiorlog}),\n\\begin{align*}\n\\int_{s_k}^{s_{k+1}} (\\epsilon_2, Q_d)(s)ds &\\ge B^*\\int_{s_k}^{s_{k+1}} \\frac{1}{|\\log(\\lambda(0)^{\\sigma}\\lambda(s)|^{1\/2}}ds  \\ge  B^*\\int_{t_k}^{t_{k+1}} \\frac{1}{\\lambda^2(t)|\\log(\\lambda(0)^{\\sigma}\\lambda(t)|^{1\/2}}dt \\\\ &\\ge \\frac{t_{k+1}-t_k}{4\\lambda^2(t_k)|\\log(\\lambda(0)^{\\sigma}\\lambda(t)|^{1\/2}}.\n\\end{align*}\nThe result follows from combining the two above inequalities.\n\\end{proof}\n\n\\textit{Proof of theorem 1}.\nSince $X$ is nonempty and closed in $[0,T_a(u_0))$, $X=[0,T_a(u_0))$ iff $X$ open in $[0,T_a(u_0))$. Let $T\\in X$ be arbitrary. Joining the conclusions of remark 2 and lemmas 5, 6, 9, 12 and proposition 11, one has, for small $\\alpha, m$ and $\\|a\\|_{W^{1,\\infty}}$  (chosen by this order) \\footnote{Note that this choice is independent of $T$.}\\newline\n\n(\\~{H}1) $T<2<\\frac{1}{2\\|a\\|_\\infty}\\log\\frac{\\|Q\\|_2^2+\\alpha}{\\|Q\\|_2^2+\\frac{\\alpha}{2}}$; \\newline\n\n(\\~{H}2) $E(t)<\\delta(\\alpha)\\|u_x(t)\\|^{3\/2}_2< \\delta(\\alpha)\\|u_x(t)\\|^{2}_2$; \\newline\n\n(\\~{H}3) For each $k_T\\ge k\\ge k_0$, $\\ t_{k+1}-t_k\\le D\\left|\\log\\left(\\lambda(0)^\\sigma \\lambda(t_k)\\right)\\right|^{1\/2}\\lambda^2(t_k)< \\lambda^{7\/4}(t_k)$; \\newline\n\n(\\~{H}4) $\\lambda(\\tilde{t})\\le \\frac{3}{2}\\lambda(t)<2\\lambda(t)$, $\\forall  \\tilde{t}, t: \\ T\\ge \\tilde{t}\\ge t\\ge 0$; \\newline\n\n(\\~{H}5) $\\lambda^{1\/2}(t)\\le\\exp\\left(-\\frac{B}{4(\\epsilon_2,Q_d)^2(t)}\\right) < \\frac{1}{2}\\|\\epsilon(t)\\|^2$.\\newline\n\nOne now applies a standart bootstrap argument since, in a neighbourhood of $T$, one has stronger conditions than those defining the set $X$. Then $X$ is open and $X=[0,T_a(u_0))$. From the definition of $X$, $T_a(u_0)\\le \\frac{1}{2\\|a\\|_\\infty}\\log\\frac{\\|Q\\|_2^2+\\alpha}{\\|Q\\|_2^2+\\frac{\\alpha}{2}}$, which proves finite-time blowup.\n\\hfill$\\qed$\n\\vskip10pt\n\\textit{Proof of corollary 2.}\nFor the sake of simplicity, we write $T=T_a(u_0)$. Since the solution blows-up in finite time, we may define, for each $k\\ge k_0$ $t_k\\in [0,T)$ such that $t_k\\to T$ and $\\lambda(t_k)=2^{-k}$. By the previous proof, $X=[0,T)$, and so, by (\\~H3),\n$$\n t_{k+1}-t_k\\le D\\left|\\log\\left(\\lambda(0)^\\sigma \\lambda(t_k)\\right)\\right|^{1\/2}\\lambda^2(t_k), \\ k\\ge k_0.\n$$\nThen, for $k$ large,\n $$\n t_{k+1}-t_k\\le C\\left|\\log\\left(\\lambda(t_k)\\right)\\right|^{1\/2}\\lambda^2(t_k).\n$$\nFix $n$ large. Summing in $k\\ge n$,\n\\begin{align*}\nT-t_n &\\le C\\sum_{k\\ge n} 2^{-2k}\\sqrt{k}=C \\sum_{n\\le k < 2n} 2^{-2k}\\sqrt{k} + C\\sum_{k\\ge 2n} 2^{-2k}\\sqrt{k}&\\\\\n&\\le C2^{-2n}\\sqrt{n} + C\\sum_{j\\ge0} 2^{-2(j+2n)}\\sqrt{2n+j} &\\\\&\\le C2^{-2n}\\sqrt{n} + C2^{-4n}\\sqrt{n}\\sum_{j\\ge0} 2^{-2j}\\sqrt{2+\\frac{j}{n}}&\\\\\n&\\le C2^{-2n}\\sqrt{n} +C2^{-4n}\\sqrt{n}\\le C2^{-2n}\\sqrt{n} = C\\lambda^2(t_n)|\\log\\lambda(t_n)|^{1\/2}.\n\\end{align*}\nGiven $t$ close to $T_a(u_0)$, $t\\in[t_n,t_{n+1}]$ for some large $n$. Therefore, by (H4),\n\\ben\\label{log1}\n\\lambda^2(t)\\left|\\log\\frac{\\lambda(t)}{2}\\right|^{1\/2} \\ge C\\lambda^2(t_n)|\\log\\lambda(t_n)|^{1\/2}\\ge C(T-t_n).\n\\een\nSet $g(x)=x^2|\\log\\frac{x}{2}|^{1\/2}$. For $t$ close to $T$ and $C^*=\\sqrt{C}$,\n\\begin{align}\\label{log2}\ng\\left(\\frac{C^*\\sqrt{T-t}}{|\\log(T-t)|^{1\/4}}\\right)&=\\frac{C(T-t)}{|\\log(T-t)|^{1\/2}}\\left|\\log\\frac{C^*\\sqrt{T-t}}{2|\\log (T-t)|^{1\/4}}\\right|^{1\/2}\\nonumber\\\\&\n=\\frac{C(T-t)}{|\\log(T-t)|^{1\/2}}\\frac{1}{\\sqrt{2}}\\left|\\log(T-t)-\\log\\left(\\frac{4}{C}|\\log (T-t)|^{1\/2}\\right)\\right|^{1\/2}\\nonumber\\\\\n&=\\frac{C}{\\sqrt{2}}(T-t)\\left|1-\\frac{\\log\\left(\\frac{4}{C}|\\log (T-t)|^{1\/2}\\right)}{\\log(T-t)}\\right|^{1\/2}\n\\le C(T-t).\n\\end{align}\nSince $g$ is nondecreasing in a neighbourhood of $0$, by \\eqref{log1} and \\eqref{log2}, one has, for $t$ close to $T$,\n\\be\n\\lambda(t)\\ge \\frac{C^*\\sqrt{T-t}}{|\\log(T-t)|^{1\/4}},\n\\ee\nwhich concludes this proof.\n\n\\end{section}\n\\begin{section}{Acknowledgements}\n\n$\\indent$ I thank the Funda\u00e7\u00e3o Calouste Gulbenkian for the financial support, Darwich Mohamad for his availability to discuss his work and Thierry Cazenave for important improvements. Finally, I am grateful to M\\'{a}rio Figueira, who suggested this problem and offered multiple and interesting points of view.\n\\end{section}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we look at the problem of obtaining lower bounds on the index of free boundary minimal surfaces of convex bodies in terms of their topology. Index estimates for minimal surfaces are generally difficult to obtain, and there are few minimal surfaces for which the index is explicitly known. However, index bounds can help in the classification of minimal surfaces, especially when the topology is explicitly represented in the bounds, and have applications in understanding the relationships between the curvature and topology of Riemannian manifolds. Moreover, minimal surfaces whose index is known have proven to be useful in other problems; Urbano's \\cite{Ur90} index characterization of the Clifford torus as being the unique minimal surface in $\\mathbb{S}^3$ of index 5 was recently used by Marques and Neves \\cite{Ma12} in their celebrated proof of the longstanding Willmore Conjecture. In \\cite{Sav10}, Savo was able to obtain index bounds for minimal hypersurfaces in $\\mathbb{S}^n$ in terms of their topology making use of the Laplacian on 1-forms. His work has inspired the approach taken in this paper.\n\n\\subsection{Free Boundary Minimal Hypersurfaces in Convex Bodies}\nA submanifold $M$ of a compact Riemannian manifold $\\overline{M}$ with boundary $\\partial M\\subset \\partial \\overline{M}$ is said to be a \\emph{free boundary minimal submanifold} in $\\overline{M}$ if it is a critical point for the volume functional among submanifolds with boundary in $\\partial \\overline{M}$. That is, $M$ is a free boundary minimal submanifold of $\\overline{M}$ if for every admissible variation $M_t$ of $M$, $\\frac{d}{dt}\\text{Vol}(M_t)\\big|_{t=0} =0$. \nThe first variation formula for a variation $M_t$ of $M$ with variation field $V$ is given by,\n\\begin{equation*}\n\\frac{d}{dt}\\text{Vol}(M_t)\\big|_{t=0} = -\\int_M\\la{V, H}\\ra dV +\\int_{\\partial M}\\la{V, \\eta}\\ra dA,\n\\end{equation*}\nwhere $\\eta$ is the outward unit conormal vector field.\nIt follows that $M$ is a free boundary minimal submanifold of $\\overline{M}$ if and only if $H\\equiv 0$ and $\\eta$ is orthogonal to $T(\\partial \\overline{M})$, \\textit{i.e.}, $M$ meets $\\partial \\overline{M}$ orthogonally. \n\n\nThroughout, we will focus our attention on free boundary minimal hypersurfaces $M^n$ properly immersed in convex bodies $B^{n+1}$. Here, a \\emph{convex body} is a smooth, compact, connected $(n+1)$-dimensional submanifold of $\\mathbb{R}^{n+1}$ for which the scalar second fundamental form of the boundary satisfies $h^{\\partial B}(U,U)<0$ (with respect to the outward pointing normal vector) for all vectors $U$ tangent to $\\partial B$. \n\n\n\nWe will also place some attention on the special case when $B=\\mathbb{B}$, the Euclidean ball, as there are more existence results for free boundary minimal hypersurfaces of Euclidean balls. Free boundary minimal hypersurfaces of Euclidean balls have also been shown to have an alternative characterization: in \\cite{FS11}, Fraser and Schoen showed that if $\\Sigma^k$ is a properly immersed submanifold of the Euclidean unit ball $\\mathbb{B}^{n+1}$, then $\\Sigma$ is a free boundary minimal submanifold if and only if the coordinate functions of the immersion are (Steklov) eigenfunctions of the Dirichlet-to-Neumann map with (Steklov) eigenvalue 1. Furthermore, free boundary minimal surfaces in $\\mathbb{B}^{n+1}$ are extremal surfaces for the Steklov eigenvalue problem.\n\n\n\\subsection{The Index of a Minimal Hypersurface}\nSuppose that $M^n\\subset B^{n+1}$ is a free boundary minimal hypersurface and that $N$ is a smooth unit normal vector field. Then, for a normal variation with variation field $uN$, the second variation formula is\n\\begin{equation*}\n\\frac{d^2}{dt^2}\\text{Vol}(M_t)\\big|_{t=0} = \\int_M \\left(\\|\\nabla u\\|^2 - \\|A\\|^2u^2\\right) dV + \\int_{\\partial M}h^{\\partial B}(N,N) u^2 dA.\n\\end{equation*}\n\n\nLet $J$ denote the \\emph{Jacobi operator} (also called the \\emph{stability operator}),\n\\begin{equation*}\nJ = \\Delta - \\|A\\|^2,\n\\end{equation*}\nand let $Q$ denote the associated symmetric bilinear form,\n\\begin{equation*}\n\\begin{split}\nQ(u) &= \\int_M \\left[ \\|\\nabla u\\|^2- \\|A\\|^2u^2\\right]dV+\\int_{\\partial M}h^{\\partial B}(N,N) u^2\\, dA\\\\\n& = -\\int_M u\\cdot Ju\\, dV + \\int_{\\partial M} \\left(\\frac{\\partial u}{\\partial \\eta}+h^{\\partial B}(N,N)u\\right)u\\, dA.\n\\end{split}\n\\end{equation*}\nWe say that $\\lambda(J)$ is an eigenvalue of $J$ with eigenfunction $u\\in C^{\\infty}(M)$ if \n\\begin{equation*}\n\\begin{cases}\nJu = \\lambda u \\ \\ \\text{on } M,\\\\\n\\frac{\\partial u}{\\partial \\eta} +h^{\\partial B}(N,N) u =0 \\ \\ \\text{on } \\partial M.\n\\end{cases}\n\\end{equation*}\n\n\nThe (\\emph{Morse}) \\emph{index} of a minimal hypersurface is the maximal dimension of a subspace of $C^{\\infty}(M)$ on which the second variation is negative. \n\n\nA free boundary minimal hypersurface is said to be \\emph{stable} if it has index 0. For free boundary minimal hypersurfaces in $B^{n+1}$, there are none which are stable. This is easy to see since if we use the variation with variation field $1\\cdot N$, then we get that\n\\begin{equation*}\nQ(1) = -\\int_M \\|A\\|^2\\, dV + \\int_{\\partial M}\\left(0+h^{\\partial B}(N,N)\\right)\\cdot 1 \\,dA <0.\n\\end{equation*}\nHence, any free boundary minimal hypersurface in $B^{n+1}$ has index at least 1.\n\n\\subsection{Examples and Existence Results} For general convex bodies, little is known about the existence of free boundary minimal submanifolds.  In \\cite{St84}, Struwe showed the existence of a (possibly branched) immersed free boundary minimal disk in any domain in $\\mathbb{R}^3$ diffeomorphic to $\\mathbb{B}^3$, and Gr\\\"uter and Jost \\cite{GJ86} showed that there is an embedded free boundary minimal disk in any convex body in $\\mathbb{R}^3$. Jost \\cite{Jo89} was also able to show that any convex body in $\\mathbb{R}^3$ actually contains at least three embedded free boundary minimal disks.  More recently, Maximo, Nunes and Smith \\cite{MNS} showed that any convex body in $\\mathbb{R}^3$ contains a minimal annulus. By the above argument, we know that any free boundary minimal hypersurface of a convex body has index at least one. Currently, there are no other existence results or index results for minimal surfaces of greater topological complexity, nor are there any existence results in higher dimensions.  \n\nIf we focus on minimal submanifolds of Euclidean balls, then more is known. \nThe simplest examples of free boundary minimal submanifolds in $\\mathbb{B}^{n+1}$ are the equatorial $k$-planes $D^k \\subset \\mathbb{B}^{n+1}$. By \\cite{Ni85} and \\cite{FS15b}, any simply connected free boundary minimal surface in $\\mathbb{B}^n$ must be a flat equatorial disk,  and it is well known that the equatorial disk has index 1 (see p.\\ 3741 in \\cite{Fr07}). In fact, it is the only free boundary minimal surface of $\\mathbb{B}^{3}$ to have index 1. However, there are many examples of free boundary minimal surfaces of different topological type. The critical catenoid, a minimal surface with genus 0 and 2 boundary components, is an explicit example of such a surface. In \\cite{FS15}, Fraser and Schoen prove existence of free boundary minimal surfaces in $\\mathbb{B}^3$ with genus 0 and $k>0$ boundary components. Using gluing techniques, in \\cite{FPZ15} Folha, Pacard and Zolotareva give an independent construction of free boundary minimal surfaces in $\\mathbb{B}^3$ of genus 0 with $k$ boundary components for $k$ large. They are also able to use the same techniques to construct a genus 1 free boundary minimal surface with $k$ boundary components for $k$ large. Kapouleas and M.\\ Li have constructed free boundary minimal surfaces in $\\mathbb{B}^3$ with any sufficiently large genus and 3 boundary components by gluing an equatorial disk to a critical catenoid, though this has not yet been published. If $M$ is not an equatorial disk, then all that is known about its index is that $\\Ind(M)\\geq 3$ (see Theorem 3.1 in \\cite{FS15}).   \n\n\n\n\nIn this paper, we give a relationship between the eigenvalues of the Jacobi operator and the eigenvalues of the Laplacian on 1-forms and, as a corollary, obtain new index bounds for orientable free boundary minimal hypersurfaces of convex bodies. More specifically, our first main result is:\n\\begin{thm}\\label{ER} Let $M^n$ be an orientable free boundary minimal hypersurface of a convex body in $\\mathbb{R}^{n+1}$ with Jacobi operator $J$. Then, for all positive integers $j$, one has that\n\\begin{equation*}\n\\lambda_j(J)\\leq\\lambda_{m(j)}(\\Delta_1),\n\\end{equation*}\nwhere $m(j) = {n+1\\choose 2}(j-1)+1$ and $\\lambda_{m(j)}(\\Delta_1)$ is the $m(j)$th eigenvalue of the Laplacian eigenvalue problem with absolute boundary conditions.\n\\end{thm}\n\nLet $\\beta^1_a=\\dim H_a^1(M)$ be the first absolute Betti number of $M$. Our second main result is:\n\\begin{thm}{\\upshape (Index Bound)}\\label{IB} If $M$ is an orientable free boundary minimal hypersurface of a convex body in $\\mathbb{R}^{n+1}$, then\n{\\upshape\\begin{equation*}\n\\Ind(M) \\geq \\left \\lfloor{\\frac{\\beta_a^1+{n+1\\choose 2}-1}{{n+1\\choose 2}}}\\right \\rfloor.\n\\end{equation*}}\n\\end{thm}\n\n\\begin{cor}\\label{ICor} If $M$ is an orientable free boundary minimal surface of a convex body in $\\mathbb{R}^3$ with genus $g$ and $k$ boundary components, then\n{\\upshape\\begin{equation*}\n\\Ind(M)\\geq \\left \\lfloor{\\frac{2g+k+1}{3}}\\right \\rfloor.\n\\end{equation*}} \n\\end{cor}\n\nCorollary \\ref{ICor} provides new index bounds for free boundary minimal surfaces of $\\mathbb{B}^3$ with large topology. In particular, it shows that $\\text{Ind}(M)>3$ when $2g+k\\geq 11$ and $\\text{Ind}(M)$ tends to infinity as the genus or the number of boundary components tend to infinity. \n\n\nThe paper is structured as follows: In the second section, we outline the basic notation and conventions that we will use throughout the paper, and  give a brief introduction to the Hodge Laplacian on $p$-forms. Here, we define the Hodge Laplacian on $p$-forms and then focus on the special case when $p=1$. \nWe also introduce the two main boundary conditions for the eigenvalue problem of the Laplacian for 1-forms on a manifold with boundary. \nIn the third section, we provide several preliminary calculations that will ultimately allow us to see how the Jacobi operator acts on specific test functions, which will be needed to prove our main results.\nWe give the proofs of our two main results in the fourth section.  \n\n\n\n\n\n\\vspace{1cm}\n\n\\section{Notation and Conventions}\nLet $M^n$ be an orientable free boundary minimally immersed hypersurface in $B^{n+1}$ ($\\partial M\\neq \\emptyset$). Throughout, we will let $N$ be a unit normal vector field on $M$. \n\nLet $D$ denote the Levi-Civita connection on $\\mathbb{R}^{n+1}$ and $\\nabla$ the Levi-Civita connection on $M$. We will let $A$ denote the second fundamental form of $M\\subset B$, and $S$ the associated shape operator. That is, for $X,Y\\in \\Gamma(TM)$,\n\\begin{equation*}\n\\begin{split}\nA(X,Y) &= \\left(D_X Y\\right)^N = \\la{D_X Y,N}\\ra\\cdot N\\\\\nS(X) & = -\\left(D_XN\\right)^T,\n\\end{split}\n\\end{equation*}\nso that $\\la{A(X,Y),N}\\ra = \\la{S(X),Y}\\ra$\n\n\nFor any parallel vector field $\\overline{V}$ in $\\mathbb{R}^{n+1}$, we have the orthogonal decomposition\n\\begin{equation*}\n\\overline{V} = V+V^{N},\n\\end{equation*}\nwhere $V\\in TM$ is the orthogonal projection of $\\overline{V}$ onto M and $V^{N} = \\la{\\overline{V},N}\\ra\\cdot N\\in NM$. Since parallel vector fields on $\\mathbb{R}^{n+1}$ and their orthogonal projections onto $M$ will be used throughout, we introduce the following vector spaces:\n\\begin{equation*}\n\\begin{split}\n\\overline{\\mathcal{P}} &= \\{\\text{parallel vector fields on } \\mathbb{R}^{n+1}\\},\\\\\n\\mathcal{P} & = \\{\\text{vector fields on } M \\text{ which are orthogonal projections of elements of } \\overline{\\mathcal{P}}\\}.\n\\end{split}\n\\end{equation*}\n\n\nThroughout, we will let $\\Delta_p$ denote the Hodge Laplacian on $p$-forms (though the $p$ will usually be dropped for convenience) and we will  let $\\nabla^*\\nabla$ denote the rough Laplacian on vector fields. So, if $\\omega$ is a $p$-form on $M$ and $\\xi$ is a vector field on $M$, then\n\\begin{equation*}\n\\begin{split}\n\\Delta_p \\omega & = (d\\delta + \\delta d)\\omega\\\\\n\\nabla^*\\nabla\\xi & =-\\sum_{j=1}^n\\left(\\nabla_{e_j}\\nabla_{e_j}\\xi-\\nabla_{\\nabla_{e_j}e_j}\\xi\\right),\n\\end{split}\n\\end{equation*}\nwhere $d$ is the exterior derivative, $\\delta$ is the codifferential, and $\\{e_1,\\ldots,e_n\\}$ is any local orthonormal frame of $TM$.  Recall that a vector field $X$ is dual to a 1-form $\\theta$ if and only if $\\la{X,Y}\\ra = \\theta(Y)$ for all $Y\\in\\Gamma(TM)$. \nIf $\\xi$ is the vector field dual to $\\omega$, then one can also define the Hodge Laplacian of $\\xi$, denoted $\\Delta\\xi$, to be the vector field dual to the 1-form $\\Delta_1\\omega$.\nThe Bochner formula relates the two Laplacians:\n\\begin{equation*}\n\\Delta\\xi = \\nabla^*\\nabla\\xi +\\Ric(\\xi),\n\\end{equation*}\nwhere $\\Ric$ is seen as a symmetric endomorphism of $TM$.\n\nTo get a bound on the index of $M$, we will consider the following eigenvalue problem defined by the \\emph{absolute} boundary conditions:\n\\begin{equation*}\\label{AbsCond}\n\\begin{cases}\nJ_1\\omega & =\\lambda\\omega,\\\\\ni^*\\iota_{\\eta}\\omega &= i^*\\iota_{\\eta}d\\omega =0,\n\\end{cases}\n\\end{equation*}\nwhere $i$ is the inclusion $\\partial M\\hookrightarrow M$,  $\\iota_{\\eta}$ denotes interior multiplication by $\\eta$ and $J_1$ is the Jacobi operator on 1-forms defined by $J_1 = \\Delta_1-\\|A\\|^2$. We will often drop the subscripts for convenience. These absolute boundary conditions are a generalization of Neumann boundary conditions for functions.\n We say that $\\omega$ is \\emph{tangential} on $\\partial M$ if $i^*\\iota_{\\eta}\\omega =0$, \\textit{i.e.}, $\\omega$ vanishes whenever its argument is normal to the boundary of $M$. So, if $\\omega$ satisfies the absolute boundary conditions, then both $\\omega$ and $d\\omega$ are tangential ($d\\omega$ is tangential whenever one of its arguments is normal to $\\partial M$).\n\nWe define the following space of harmonic 1-forms\n\\begin{equation*}\n\\begin{split}\n\\mathcal{H}_N^1(M)& = \\{\\omega\\in\\Omega^1(M) \\ | \\ \\Delta\\omega =0, \\ \\omega \\ \\text{satisfies the absolute boundary conditions}\\},\\\\\n\\end{split}\n\\end{equation*}\nand note that $\\beta_a^1 = \\dim H^1_a(M) = \\dim \\mathcal{H}_N^1(M)$, where $H^1_a(M)$ is the first absolute cohomology group of $M$.\n\n\n\n\n\n\\vspace{1cm}\n\n\\section{Preliminary Calculations}\nThe calculations done here are analogous to those done by Savo in \\cite{Sav10} for the case of a minimal hypersurface in $\\mathbb{S}^{n+1}$. In $\\mathbb{S}^{n+1}$, a hypersurface has two normal vectors (one tangent to the sphere and one normal to both the sphere and the hypersurface) whereas a free-boundary minimal hypersurface of a convex body $B^n$ just has one. The absence of a second normal vector simplifies many of the preliminary calculations. \n\\begin{lem}\\label{PPC} Let $\\overline{V}\\in\\overline{\\mathcal{P}}$ and let $V\\in\\mathcal{P}$ be its orthogonal projection onto $M$. Let $A$ and $S$ be the second fundamental form and shape operator (respectively) of the immersion $\\phi:M\\rightarrow B^n$. Then{\\upshape\n\\begin{enumerate}[(a)]\n\\item $\\nabla\\la{\\overline{V},N}\\ra = -S(V)$.\n\\item $\\Delta\\la{\\overline{V},N}\\ra = |S|^2\\la{\\overline{V},N}\\ra$.\n\\end{enumerate}}\n\\end{lem}\n\\begin{proof}\nTo show (a), take any $X\\in\\Gamma(TM)$. Then we have that\n\\begin{equation*}\n\\la{\\nabla\\la{\\overline{V},N}\\ra,X}\\ra  = \\la{D_X\\overline{V}, N}\\ra + \\la{\\overline{V}, D_X N}\\ra = \\la{\\overline{V}, D_X N}\\ra,\n\\end{equation*}\nsince $\\overline{V}$ is parallel. Now, since $\\la{N, D_X N}\\ra = \\frac{1}{2}X\\left(\\|N\\|^2\\right) \\equiv 0$ and $[X,V]$ is tangent to $M$, we have that\n\\begin{equation*}\n\\la{\\overline{V}, D_X N}\\ra  = -\\la{D_X V, N}\\ra = -\\la{D_V X, N}\\ra = \\la{X, \\left(D_V N\\right)^T}\\ra.\n\\end{equation*}\nHence, $\\nabla\\la{\\overline{V}, N}\\ra = -S(V)$.\n\nFor (b), let $\\{e_1, \\ldots e_n\\}$ denote normal coordinate vector fields centred at a point $p\\in M$. Then (at $p$),\n\\begin{equation*}\n\\begin{split}\n-\\Delta\\la{\\overline{V}, N}\\ra & = \\sum_{i=1}^n \\la{\\nabla_{e_i}\\nabla\\la{\\overline{V}, N}\\ra, e_i}\\ra = \\sum_{i=1}^n \\la{\\nabla_{e_i}\\left(D_V N\\right)^T, e_i}\\ra = -\\sum_{i=1}^n e_i\\la{N, D_V e_i}\\ra\\\\\n& = -\\sum_{i=1}^n \\la{D_{e_i} N, A(V, e_i)}\\ra + \\la{N, D_{e_i}A(V, e_i)}\\ra.\n\\end{split}\n\\end{equation*}\nSince $D_{e_i} N$ has no normal component, and $A$ is symmetric, we have that\n\\begin{equation*}\n\\Delta\\la{\\overline{V}, N}\\ra = \\sum_{i=1}^n\\la{N, \\left(D_{e_i}A(e_i, V)\\right)^N}\\ra.\n\\end{equation*}\nNow, it follows from the Codazzi Equation that\n\\begin{equation*}\n\\left(D_{e_i}A(e_i, V)\\right)^N = \\left(D_VA(e_i, e_i)\\right)^N-2A(e_i, \\nabla_V e_i) +A(\\nabla_{e_i}V, e_i).\n\\end{equation*}\nHowever\n$A(e_i, \\nabla_V{e_i}) = \\left(D_{e_i}\\nabla_V e_i\\right)^N$,\nand, at $p$,\n$\\la{D_{e_i}\\nabla_V e_i, N}\\ra  = -\\la{\\nabla_V e_i, D_{e_i} N}\\ra= 0$.\nMoreover, since $M$ is minimal,\n$\\sum_{i=1}^n\\left(D_VA(e_i, e_i)\\right)^N = 0$.\nTherefore,\n\\begin{equation*}\n\\Delta\\la{\\overline{V}, N}\\ra  = \\sum_{i=1}^n\\la{N, A(\\nabla_{e_i}V, e_i)}\\ra = -\\sum_{i=1}^n\\la{\\left(D_{e_i}N\\right)^T, \\nabla_{e_i}V}\\ra.\n\\end{equation*}\n\\end{proof}\n\n\n\n\\begin{lem}\\label{PC1} For any vector field $\\xi\\in\\Gamma(TM)$ and any $\\overline{V}\\in\\overline{P}$ with orthogonal projection $V$,{\\upshape\n\\begin{enumerate}[(a)]\n\\item $\\Delta\\xi = \\nabla^*\\nabla\\xi-S^2(\\xi)$.\n\\item $\\nabla^*\\nabla V = S^2(V), \\ \\Delta V =0$.\n\\end{enumerate}\n}\n\\end{lem}\n\n\\begin{proof} Let $\\{e_1,\\ldots, e_n\\}$ be local normal coordinate vector fields centred at a point $p\\in M$. Then,\nusing the minimality of $M$ and the Gauss equation, we have that (at $p$)\n\\begin{equation*}\n\\text{Ric}(\\xi) =\\sum_{i,k=1}^nR_M(e_k,e_i, \\xi, e_k)e_i  = -\\sum_{i,k=1}^n \\la{A(e_k, \\xi), A(e_i, e_k)}\\ra e_i.\n\\end{equation*}\nNow \n\\begin{equation*}\n\\begin{split}\n-\\sum_{k=1}^n \\la{A(e_k, \\xi), A(e_i, e_k)}\\ra = -\\sum_{k=1}^n \\la{e_k, D_{\\xi}N}\\ra\\la{e_k, D_{e_i}N}\\ra & = - \\la{(D_{\\xi}N)^T, D_{e_i}N}\\ra\\\\\n& = \\la{D_{(D_{\\xi}N)^T}e_i+[e_i, (D_{\\xi}N)^T], N}\\ra\\\\\n& = -\\la{e_i, S^2(\\xi)}\\ra.\n\\end{split}\n\\end{equation*}\nTherefore, $\\text{Ric}(\\xi) = -S^2(\\xi)$, and (a) follows from the Bochner formula. \n\nTo see that $\\nabla^*\\nabla V=S^2(V)$, we'll first show that $\\nabla^*\\nabla N=0$ in the sense that if $\\{e_1,\\ldots, e_n\\}$ are again local normal coordinate vector fields centred at $p\\in M$, then, at $p$,\n\\begin{equation*}\n\\sum_{i=1}^n\\left(D_{e_i}(D_{e_i}N)^T\\right)^T=0.\\label{DN0}\n\\end{equation*}\nSince, $D_{e_i}N$ is tangential,\n\\begin{equation*}\n\\begin{split}\n\\sum_{i=1}^n\\left(D_{e_i}(D_{e_i}N)^T\\right)^T  = \\sum_{i,j=1}^n\\la{D_{e_i}D_{e_i}N, e_j}\\ra e_j\n& = -\\sum_{i,j=1}^ne_i\\la{N, D_{e_i}e_j}\\ra e_j\\\\\n& = -\\sum_{i,j=1}^n\\la{N, D_{e_i}(A(e_j,e_i))}\\ra e_j.\n\\end{split}\n\\end{equation*}\nNow, it follows from the Codazzi Equation that $\\left(D_{e_i}(A(e_j,e_i))\\right)^N = \\left(D_{e_j}(A(e_i,e_i))\\right)^N$.\nSo, again using the minimality of $M$, we have that\n\\begin{equation*}\n\\sum_{i=1}^n\\left(D_{e_i}(D_{e_i}N)^T\\right)^T =-\\sum_{i,j=1}^n\\la{N, D_{e_i}(A(e_j,e_i))}\\ra e_j = -\\sum_{i,j=1}^n\\la{N, D_{e_j}(A(e_i,e_i))}\\ra e_j = 0.\n\\end{equation*}\n\nNow, if we write $V=\\overline{V} - \\la{\\overline{V}, N}\\ra N$, then we can use this calculation and the fact that $\\overline{V}$ is parallel to help us calculate $\\nabla^*\\nabla V$. \n\\begin{equation*}\n\\begin{split}\n\\nabla^*\\nabla V &= \\sum_{i=1}^n \\left(D_{e_i}(D_{e_i}\\overline V)^T\\right)^T - \\left(D_{e_i}(D_{e_i}(\\la{\\overline{V}, N}\\ra N))^T\\right)^T\\\\\n& = -\\sum_{i=1}^n\\left(D_{e_i}(\\la{\\overline{V}, N}\\ra D_{e_i}N)\\right)^T = -\\left(\\sum_{i=1}^ne_i(\\la{\\overline{V}, N}\\ra) D_{e_i}N\\right)-\\la{\\overline{V},N}\\ra \\nabla^*\\nabla N\\\\\n& = -\\sum_{i=1}^n\\la{V, D_{e_i}N}\\ra D_{e_i}N = -\\sum_{i=1}^n\\la{e_i,D_V N}\\ra D_{e_i}N = S^2(V).\n\\end{split}\n\\end{equation*}\nThe fact that $\\Delta V = 0$ now follows from (a).\n\\end{proof}\n\n\n\n\\begin{lem}\\label{LapIP} Let $\\overline{V},\\overline{W}\\in\\overline{\\mathcal{P}}$ and let $V, W\\in\\mathcal{P}$ be their orthogonal projections onto $M$. Then, for any $\\xi\\in\\Gamma(TM)$,{\\upshape\n\\begin{enumerate}[(a)]\n\\item $\\Delta\\la{V,\\xi}\\ra = 2\\la{S(V),S(\\xi)}\\ra + \\la{V,\\Delta\\xi}\\ra -2\\la{\\overline{V}, N}\\ra\\la{S,\\nabla\\xi}\\ra$.\n\\item $\\la{\\nabla\\la{\\overline{V},N}\\ra,\\nabla\\la{W,\\xi}\\ra}\\ra = -\\la{\\overline{W},N}\\ra\\la{S(V),S(\\xi)}\\ra-\\la{W,\\nabla_{S(V)}\\xi}\\ra$.\n\\item $\\Delta(\\la{\\overline{V},N}\\ra\\la{W, \\xi}\\ra) = |S|^2 \\la{\\overline{V}, N}\\ra\\la{W, \\xi}\\ra +2(\\la{\\overline{W}, N}\\ra\\la{S(V), S(\\xi)}\\ra+\\la{W, \\nabla_{S(V)}\\xi}\\ra)\\\\\n\\textcolor{white}{.} \\qquad  \\qquad\\qquad  \\qquad + \\la{\\overline{V}, N}\\ra(2\\la{S(W), S(\\xi)}\\ra+\\la{W, \\Delta\\xi}\\ra-2\\la{\\overline{W}, N}\\ra\\la{S, \\nabla\\xi}\\ra)$.\n\\end{enumerate}\n}\n\\end{lem}\n\n\\begin{proof}\nLet $\\{e_1,\\ldots, e_n\\}$ be local normal coordinate vector fields centred at a point $p\\in M$. Then, at $p$,\n\\begin{equation*}\n\\Delta\\la{V, \\xi}\\ra = \\la{\\nabla^*\\nabla V, \\xi}\\ra -2\\la{\\nabla V, \\nabla \\xi}\\ra + \\la{V, \\nabla^*\\nabla \\xi}\\ra,\n\\end{equation*}\nwhere $\\la{\\nabla V, \\nabla \\xi}\\ra = \\sum_{i=1}^n\\la{\\nabla_{e_i}V, \\nabla_{e_i}\\xi}\\ra$.\nFrom Lemma \\ref{PC1} we have that $\\nabla^*\\nabla V = S^2(V)$ and $\\nabla^*\\nabla\\xi = \\Delta\\xi+S^2(\\xi)$. We also have that\n\\begin{equation*}\n\\la{S^2(V),\\xi}\\ra  = -\\la{N, D_{D_V N}\\xi}\\ra = \\la{D_{\\xi}N, D_V N}\\ra = \\la{S(\\xi), S(V)}\\ra,\n\\end{equation*}\nand, similarly, $\\la{V, S^2(\\xi)}\\ra = \\la{S(V), S(\\xi)}\\ra$. Therefore,\n\\begin{equation*}\n\\Delta\\la{V, \\xi}\\ra = 2\\la{S(V), S(\\xi)}\\ra+\\la{V, \\Delta\\xi}\\ra -2\\la{\\nabla V, \\nabla \\xi}\\ra.\n\\end{equation*}\nFinally,\n\\begin{equation*}\n\\la{\\nabla_{e_i} V, \\nabla_{e_i}\\xi}\\ra = \\la{D_{e_i}(\\overline{V}-\\la{\\overline{V}, N}\\ra N), \\nabla_{e_i}\\xi}\\ra = \\la{\\overline{V}, N}\\ra\\la{S(e_i), \\nabla_{e_i}\\xi}\\ra.\n\\end{equation*}\nHence, summing over $i$ gives us that\n\\begin{equation*}\n\\Delta\\la{V, \\xi}\\ra = 2\\la{S(V), S(\\xi)}\\ra+\\la{V, \\Delta\\xi}\\ra -2\\la{\\overline{V}, N}\\ra\\la{S, \\nabla\\xi}\\ra.\n\\end{equation*}\n\nFrom Lemma \\ref{PPC}(a) we know that $\\nabla\\la{\\overline{V}, N}\\ra = -S(V)$, so we just need to calculate $\\nabla\\la{W, \\xi}\\ra$. First, notice that for any vector field $X$ on $M$, since $\\overline{W}$ is parallel,\n\\begin{equation*}\n\\nabla_X W  = \\left(D_X(\\overline{W} - \\la{\\overline{W}, N}\\ra N)\\right)^T\\ = \\la{\\overline{W}, N}\\ra S(X)\n\\end{equation*}\nHence, \n\\begin{equation*}\n\\la{\\nabla\\la{W, \\xi}\\ra, X}\\ra  =X(\\la{W, \\xi}\\ra) = \\la{\\overline{W}, N}\\ra\\la{S(X), \\xi}\\ra + \\la{W, \\nabla_X\\xi}\\ra.\n\\end{equation*}\nSo, for $X=-S(V) \\ (=\\nabla\\la{\\overline{V}, N}\\ra)$, we have that\n\\begin{equation*}\n\\begin{split}\n\\la{\\nabla\\la{\\overline{V}, N}\\ra, \\nabla\\la{W, \\xi}\\ra}\\ra & = -\\la{\\overline{W}, N}\\ra\\la{S^2(V), \\xi}\\ra  -\\la{W, \\nabla_{S(V)}\\xi}\\ra\\\\\n& = -\\la{\\overline{W}, N}\\ra\\la{S(V), S(\\xi)}\\ra  -\\la{W, \\nabla_{S(V)}\\xi}\\ra.\\\\\n\\end{split}\n\\end{equation*}\nNow (c) follows from (a) and (b) and Lemma \\ref{PPC}(b).\n\\end{proof}\n\n\nLet $\\overline{\\mathcal{U}} = \\{\\overline{V}\\in\\overline{\\mathcal{P}} \\ | \\ \\|V\\|\\equiv 1\\}$. Then $\\mathcal{U}$ can naturally be identified with $S^n$ if we endow it with the measure $\\mu = \\frac{n+1}{\\text{Vol}(S^n)}dv_{S^n}$. \n\\begin{lem}\\label{IC} For any $\\overline{X}, \\overline{Y}\\in\\mathbb{R}^{n+1}$, \n\\begin{equation*}\n\\int_{\\overline{\\mathcal{U}}} \\la{\\overline{V}, \\overline{X}}\\ra\\la{\\overline{V}, \\overline{Y}}\\ra\\, d\\overline{V} = \\la{\\overline{X}, \\overline{Y}}\\ra.\n\\end{equation*}\n\\end{lem}\nThe proof of Lemma \\ref{IC} follows from a direct, but tedious, calculation after changing to spherical coordinates and repeatedly applying the integral identity\n\\begin{equation*}\n\\int \\sin^mx\\, dx = -\\frac{1}{m}\\sin^{m-1}x\\cos x + \\frac{m-1}{m}\\int \\sin^{m-2}x\\, dx.\n\\end{equation*}\n\n\n\\bigskip\n\nThe following lemma was originally proved by Ros \\cite{Ro09} for free boundary minimal surfaces in a smooth domain in $\\mathbb{R}^3$. Here, we extend his proof to obtain the analogous result for free boundary minimal hypersurfaces in smooth domains in $\\mathbb{R}^n$.\n\\begin{lem}\\label{BC}\nSuppose $\\xi$ is a vector field on $M$ dual to a 1-form $\\omega$ which satisfies the absolute boundary conditions. Then, at a point $p\\in\\partial M$,\n\\begin{equation*}\n\\la{\\nabla_{\\eta}\\xi, \\xi}\\ra =h^{\\partial B}(N,N)\\|\\xi\\|^2.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $\\eta$ be the (outward pointing) conormal vector along $\\partial M$. Then, since $\\omega$ satisfies the absolute boundary conditions on $\\partial M$, at $p$ we have that\n\\begin{equation*}\n\\begin{split}\n\\omega(\\eta) &= 0,\\\\\nd\\omega(\\eta,t) & = \\eta(\\omega(t))-t(\\omega(\\eta))-\\omega([\\eta,t]) = 0,\n\\end{split}\n\\end{equation*}\nfor any vector $t\\in T_p(\\partial M)$. In particular, if $\\xi$ is the vector field dual to $\\omega$, the the first condition implies that $\\xi_p\\in T_p(\\partial M)$, and so the second condition implies that $d\\omega (\\eta, \\xi)=0$ at $p$. Now,\n\\begin{equation*}\n\\la{\\xi, \\nabla_\\eta\\xi}\\ra = \\eta\\la{\\xi, \\xi}\\ra - \\la{\\xi, \\nabla_\\eta\\xi}\\ra = (\\nabla_{\\eta}\\omega)(\\xi),\n\\end{equation*}\nand we claim that $(\\nabla_\\eta\\omega)(\\xi) = (\\nabla_{\\xi}\\omega)(\\eta)$. To see this, note that, by definition,\n\\begin{equation*}\n(\\nabla_{\\xi}\\omega)(\\eta) - (\\nabla_{\\eta}\\omega)({\\xi}) = {\\xi}(\\omega(\\eta))-\\omega(\\nabla_{\\xi} \\eta) -{\\eta}(\\omega(\\xi))+\\omega(\\nabla_{\\eta}\\xi).\n\\end{equation*}\nHowever,\n\\begin{equation*}\n\\omega(\\nabla_{\\xi}\\eta) - \\omega(\\nabla_{\\eta}{\\xi}) = \\omega(\\nabla_{\\xi}\\eta -\\nabla_{\\eta}{\\xi}) = \\omega([{\\xi},\\eta]),\n\\end{equation*}\nand, since $d\\omega(\\eta, {\\xi}) = 0$, $\\omega([\\eta, {\\xi}]) = \\eta(\\omega({\\xi})) -{\\xi}(\\omega(\\eta))$. Therefore \n\\begin{equation*}\n(\\nabla_{\\xi}\\omega)(\\eta) - (\\nabla_{\\eta}\\omega)({\\xi}) = {\\xi}(\\omega(\\eta))-{\\eta}(\\omega(\\xi))+\\omega([\\eta, {\\xi}])=0.\n\\end{equation*}\nSo, \n\\begin{equation*}\n\\la{\\xi, \\nabla_{\\eta} \\xi}\\ra = (\\nabla_{\\eta}\\omega)(\\xi) = (\\nabla_{\\xi}\\omega)(\\eta).\n\\end{equation*}\nNow, since $\\xi$ is tangent to $\\partial M$ and $\\omega(\\eta)=0$ on $\\partial M$,\n\\begin{equation*}\n(\\nabla_{\\xi}\\omega)(\\eta) = \\xi(\\omega(\\eta))-\\omega(\\nabla_{\\xi}\\eta) = \\la{\\nabla_{\\xi}\\xi, \\eta}\\ra = h^{\\partial B}(\\xi,\\xi).\n\\end{equation*}\nHence,\n\\begin{equation*}\n\\la{\\nabla_{\\eta}\\xi,\\xi}\\ra = \\la{\\xi, \\nabla_{\\eta} \\xi}\\ra =h^{\\partial B}(\\xi,\\xi)=h^{\\partial B}(N,N)\\|\\xi\\|^2.\n\\end{equation*}\n\\end{proof}\n\n\\section{Proofs of Main Theorems}\n\n\\subsection{Eigenvalue Relationship}\n\\begin{thmnonum}{\\upshape \\textbf{\\ref{ER}}} Let $M^n$ be an orientable free boundary minimal hypersurface of a convex body in $\\mathbb{R}^{n+1}$ with Jacobi operator $J$. Then, for all positive integers $j$, one has that\n\\begin{equation*}\n\\lambda_j(J)\\leq\\lambda_{m(j)}(\\Delta_1),\n\\end{equation*}\nwhere $m(j) = {n+1\\choose 2}(j-1)+1$ and $\\lambda_{m(j)}(\\Delta_1)$ is the $m(j)$th eigenvalue of the Laplacian eigenvalue problem with absolute boundary conditions.\n\\end{thmnonum}\n\\medskip\n\n\\begin{lem}\\label{JC} For $\\overline{V}, \\overline{W}\\in\\overline{\\mathcal{P}}$,  let \n\\begin{equation*}\nX_{V,W} = \\la{\\overline{V},N}\\ra W - \\la{\\overline{W}, N}\\ra V.\n\\end{equation*}\nLet $\\xi$ be any vector field on $M$ and consider the function $u=\\la{X_{V,W}, \\xi}\\ra$. Then\n\\begin{equation*}\nJu = \\la{X_{V,W},\\Delta\\xi}\\ra +2v,\n\\end{equation*}\nwhere $v$ is the smooth function\n\\begin{equation*}\nv = \\la{\\nabla_{S(V)}\\xi, W}\\ra - \\la{\\nabla_{S(W)}\\xi, V}\\ra.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}[Proof of Lemma \\ref{JC}]\nSince $u = \\la{X_{V,W}, \\xi}\\ra = \\la{\\overline{V}, N}\\ra\\la{W, \\xi}\\ra - \\la{\\overline{W}, N}\\ra\\la{V, \\xi}\\ra$, from part (c) of Lemma \\ref{LapIP}, (after some cancellations) we get that\n\\begin{equation*}\n\\Delta u = |S|^2u+\\la{X_{V,W},\\Delta\\xi}\\ra + 2v,\n\\end{equation*}\nand so $Ju = \\la{X_{V, W}, \\Delta\\xi}\\ra+ 2v$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{ER}]\nLet $\\{\\phi_1, \\phi_2,\\ldots, \\}$ be an orthonormal basis for $L^2(M)$ given by eigenfunctions of $J$, where $\\phi_i$ is an eigenfunction associated to $\\lambda_i(J)$. Let $V^m(\\Delta_1) = \\bigoplus_{i=1}^m E^N_{\\lambda_i(\\Delta_1)}$, where $E^N_{\\lambda_i(\\Delta_1)}$ is the space of eigenforms of $\\Delta_1$ associated with $\\lambda_1(\\Delta_1)$ with absolute boundary conditions. We want to find $\\omega\\in V^m(\\Delta_1)$, $\\omega\\not\\equiv 0$, for which\n\\begin{equation}\n\\int_{M}\\la{X_{V,W},\\xi}\\ra\\phi_i dV = 0,\\label{SOE}\n\\end{equation}\nfor $i=1,\\ldots, j-1$ and for all $\\overline{V}, \\overline{W}\\in\\overline{\\mathcal{P}}$, where $\\xi$ is the vector field dual to $\\omega$. Since $X_{V,W}$ is a skew-symmetric bilinear function of $\\overline{V}, \\overline{W}$, and since $\\text{dim}\\overline{\\mathcal{P}} = \\dim\\mathbb{R}^{n+1}=n+1$,  there are $\\binom{n+1}{2}$ equations that need to be satisfied in (\\ref{SOE}) for each $i$, and therefore $\\binom{n+1}{2} (j-1)$ homogeneous linear equations in total. So, if $m(j) = \\binom{n+1}{2} (j-1)+1$, then we're guaranteed that there is a $\\omega\\in V^{m(j)}(\\Delta_1)$, $\\omega\\not\\equiv 0$, whose dual vector field satisfies (\\ref{SOE}) for all $V, W$ and for $i=1,\\ldots j-1$.\nFrom the min-max principle and Lemma \\ref{JC} we have that,\n\\begin{equation}\\label{MMI}\n\\begin{split}\n\\lambda_j(J)\\int_M u^2\\,dV & \\leq \\int_M uJu\\,dV + \\int_{\\partial M}\\left(\\frac{\\partial u}{\\partial \\eta} +h^{\\partial B}(N,N)u\\right)u\\,dA\\\\\n& = \\int_Mu\\la{X_{V, W}, \\Delta\\xi}\\ra\\, dV+ 2\\int_Muv\\, dV +\\int_{\\partial M}\\left(\\frac{\\partial u}{\\partial \\eta} +h^{\\partial B}(N,N)u\\right)u\\,dA.\\\\\n\\end{split}\n\\end{equation}\nIn addition, \n\\begin{equation*}\n\\begin{split}\n\\frac{\\partial u}{\\partial \\eta} &= \\eta\\left(\\la{\\overline{V}, N}\\ra\\la{W, \\xi}\\ra - \\la{\\overline{W}, N}\\ra\\la{\\overline{V}, \\xi}\\ra\\right)\\\\\n& = \\la{\\overline{V}, D_{\\eta}N}\\ra\\la{W, \\xi}\\ra + \\la{\\overline{V}, N}\\ra\\left(\\la{D_{\\eta}\\overline{W}, \\xi}\\ra + \\la{\\overline{W}, D_{\\eta}\\xi}\\ra\\right)\\\\\n&\\quad  - \\la{\\overline{W}, D_{\\eta}N}\\ra\\la{V, \\xi}\\ra + \\la{\\overline{W}, N}\\ra\\left(\\la{D_{\\eta}\\overline{V}, \\xi}\\ra + \\la{\\overline{V}, D_{\\eta}\\xi}\\ra\\right).\\\\\n\\end{split}\n\\end{equation*}\n\n\nWe'll now use an integration technique that exploits Lemma \\ref{IC} to help us simplify (\\ref{MMI}). We'll then apply Lemma \\ref{BC} to get the claimed eigenvalue relationship.  \n\nUsing the product metric on $\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}$, Lemma \\ref{IC} implies that (pointwise)\n\\begin{equation*}\n\\begin{split}\n&\\int_{\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}} u^2\\,d\\overline{V}d\\overline{W}  = 2\\|\\xi\\|^2,\\\\\n&\\int_{\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}} u\\la{X_{V,W}, \\Delta\\xi}\\ra\\, d\\overline{V}d\\overline{W}  = 2\\la{\\xi, \\Delta\\xi}\\ra,\\\\\n&\\int_{\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}} uv\\, d\\overline{V}d\\overline{W}  = 0,\\\\\n& \\int_{\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}} u\\la{\\overline{V}, D_{\\eta}N}\\ra\\la{\\overline{W}, \\xi}\\ra \\,d\\overline{V}d\\overline{W} = 0,\\\\\n& \\int_{\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}} u\\la{\\overline{V}, N}\\ra \\la{\\overline{W}, D_{\\eta} \\xi}\\ra\\, d\\overline{V}d\\overline{W} = \\la{\\xi, D_{\\eta}\\xi}\\ra = \\frac{1}{2}\\eta(\\|\\xi\\|^2). \n\\end{split}\n\\end{equation*}\nTherefore, integrating (\\ref{MMI}) over $\\overline{\\mathcal{U}}\\times \\overline{\\mathcal{U}}$ yields\n\\begin{equation*}\n2\\lambda_j(J) \\int_M\\|\\xi\\|^2\\, dV\\leq 2\\int_M\\la{\\xi, \\Delta\\xi}\\ra\\, dV + \\int_{\\partial M}\\left( \\eta(\\|\\xi\\|^2) +2h^{\\partial B}(N,N)\\|\\xi\\|^2\\right) dA.\n\\end{equation*}\nFrom Lemma \\ref{BC} we know that $\\eta(\\|\\xi\\|^2) =2h^{\\partial B}(N,N)\\|\\xi\\|^2$ on $\\partial M$, since $\\xi$ is the dual vector field of a 1-form satisfying the absolute boundary conditions. Moreover, since $\\xi$ is the dual vector field to a linear combination of eigenforms of $\\Delta_1$, it now follows that\n\\begin{equation*}\n2\\lambda_j(J)\\int_M\\|\\xi\\|^2\\, dV\\leq 2\\lambda_{m(j)}(\\Delta_1)\\int_{M}\\|\\xi\\|^2\\, dV +4 \\int_{\\partial M}h^{\\partial B}(N,N)\\|\\xi\\|^2 \\, dA.\n\\end{equation*}\nSince $h^{\\partial B}(U,U)<0$ for any vector tangent to $\\partial B$, we get that\n\\begin{equation*}\n2\\lambda_j(J)\\int_M\\|\\xi\\|^2\\, dV\\leq 2\\lambda_{m(j)}(\\Delta_1)\\int_{M}\\|\\xi\\|^2\\, dV.\n\\end{equation*}\nNow, since $\\omega\\not\\equiv0$, we can divide both sides by the $L^2(M)$-norm of $\\xi$ to get \n\\begin{equation*}\n\\lambda_j(J) \\leq \\lambda_{m(j)}(\\Delta_1).\n\\end{equation*}\n\\end{proof}\n\n\\begin{rmk} We note that when $m(j)\\leq\\dim\\mathcal{H}_N^1(M)$, \\emph{i.e.} when $\\omega$ is a linear combination of harmonic forms and therefore a harmonic form itself, we actually get the strict inequality $\\lambda_j(J)<\\lambda_{m(j)}(\\Delta_1)=0$. This follows from the fact that $\\omega\\not\\equiv 0$ implies that $\\omega|_{\\partial M}\\not\\equiv 0$ (see Theorem 3.4.4 on p.131 of \\cite{Sch95}), and so we get the strict inequality $4\\int_{\\partial M} h^{\\partial M}(N,N)\\|\\xi\\|^2\\,dA<0$.\n\\end{rmk}\n\n\n\\subsection{Index Bound} \n\n\\medskip\n\n\n\\begin{thmnonum}{\\upshape\\textbf{\\ref{IB}} (Index Bound)}  If $M$ is an orientable free boundary minimal hypersurface of a convex body in $\\mathbb{R}^{n+1}$, then\n{\\upshape\\begin{equation*}\n\\Ind(M) \\geq \\left \\lfloor{\\frac{\\beta_a^1+{n+1\\choose 2}-1}{{n+1\\choose 2}}}\\right \\rfloor.\n\\end{equation*}}\n\\end{thmnonum}\n\n\\medskip\n\n\\begin{proof} Suppose $j$ is such that $m(j) \\leq \\dim\\mathcal{H}_N^1(M):=\\beta_a^1$. Then $\\lambda_j(J)<\\lambda_{m(j)}(\\Delta)= 0$, so $\\Ind(M)\\geq j$. Now, $m(j) = {n+1\\choose 2}(j-1)+1\\leq \\beta_a^1$, so $j\\leq \\left \\lfloor{\\frac{\\beta_a^1+{n+1\\choose 2}-1}{{n+1\\choose 2}}}\\right \\rfloor$. Hence, $\\Ind(M)\\geq \\left \\lfloor{\\frac{\\beta_a^1+{n+1\\choose 2}-1}{{n+1\\choose 2}}}\\right \\rfloor$.\n\\end{proof}\n\n\\begin{cornonum}{\\upshape\\textbf{\\ref{ICor}}}  If $M$ is an orientable free boundary minimal surface in a convex body in $\\mathbb{R}^3$ with genus $g$ and $k$ boundary components, then\n{\\upshape\\begin{equation*}\n\\Ind(M)\\geq \\left \\lfloor{\\frac{2g+k+1}{3}}\\right \\rfloor.\n\\end{equation*}} \n\\end{cornonum}\n\\begin{proof} Since $\\beta_a^1 = 2g+k-1$ for a surface (see Appendix \\ref{DimCalc}), this follows directly from Theorem \\ref{IB}.\n\\end{proof}\n\n\n\\begin{rmk} We note that Corollary \\ref{ICor} can also be obtained by using the work of Ros. In \\cite{Ro09}, Ros shows that if $\\omega$ is a harmonic 1-form and $\\xi$ is its dual vector field, then \n\\begin{equation*}\n\\Delta\\xi + \\|A\\|^2\\xi = 2\\la{\\nabla\\omega, A}\\ra N,\n\\end{equation*}\nand, if $\\omega$ satisfies the absolute boundary conditions, then\n\\begin{equation*}\n\\la{\\nabla_{\\eta}\\xi, \\xi}\\ra = h^{\\partial B}(N,N)\\|\\xi\\|^2.\n\\end{equation*}\n\n So, for $\\xi = (\\xi_1, \\xi_2, \\xi_3)$, if we use the notation $Q(\\xi, \\xi) = \\sum_{i=1}^3Q(\\xi_i, \\xi_i)$. \nNow, assuming $\\omega\\not\\equiv 0$,  \n\\begin{equation*}\n\\begin{split}\nQ(\\xi, \\xi) & = -\\int_M\\la{\\Delta\\xi+\\|A\\|^2\\xi, \\xi}\\ra dV + \\int_{\\partial M}(\\la{\\nabla_{\\eta}\\xi, \\xi}\\ra +h^{\\partial B}(N,N)\\|\\xi\\|^2) dA\\\\\n& = 2\\int_{\\partial M}h^{\\partial B}(N,N)\\|\\xi\\|^2dA<0.\n\\end{split}\n\\end{equation*}\nHence $Q(X, X)<0$, and we get that $\\dim\\mathcal{H}_N^1(M)-3\\cdot\\Ind(M) = (2g+k-1)-3\\cdot \\Ind(M)\\leq 0$, or $\\Ind(M)\\geq \\lceil{\\frac{(2g+k-1)}{3}}\\rceil = \\lfloor{\\frac{2g+k+1}{3}}\\rfloor$. \n\\end{rmk}\n\n\n\\vspace{1cm}\n\n\\begin{appendix}\n\\section{The Dimension of the Space of Harmonic 1-Forms with Dirichlet Boundary Condition}\\label{DimCalc}\n\n\nIt is well-known, we believe, that if $M$ is a surface with boundary $\\partial M\\neq\\emptyset$, genus $g$ and $k$ boundary components, then $\\dim \\mathcal{H}_N^1(M ) = 2g + k -1$, but this result seems difficult to find in the literature. We give a proof here for completeness. When $M$ is a surface, it follows from Lefschetz duality that $\\dim\\mathcal{H}^1_N(M) = \\dim\\mathcal{H}^{2-1}_D(M)$, where $\\mathcal{H}_D^1(M)$ is the space of harmonic 1-forms on $M$ which satisfy the \\emph{relative} boundary conditions:\n\\begin{equation*}\ni^*\\omega = i^*\\delta\\omega =0,\n\\end{equation*}\nwhere $i:\\partial M \\hookrightarrow M$ is the inclusion. So, to prove that $\\dim \\mathcal{H}_N^1(M ) = 2g + k -1$, we will show that $\\dim \\mathcal{H}_D^1(M ) = 2g + k -1$.\n\n\\begin{lem} Let $M$ be an orientable surface of genus $g$ with $k$ boundary components. Then $\\dim\\mathcal{H}^1_D(M) = 2g+k-1$.\n\\end{lem}\n\n\\begin{proof} Let $\\mathcal{EH}_D^1(M)$ denote the subspace of harmonic fields with Dirichlet boundary conditions which are exact. Then,\n\\begin{equation*}\n \\mathcal{H}_D^1(M) = \\mathcal{EH}_D^1(M)\\oplus \\left(\\mathcal{EH}_D^1(M)\\right)^{\\perp},\n \\end{equation*}\n and $\\dim\\mathcal{H}^1_D(M) = \\dim\\mathcal{EH}_D^1(M) +\\dim\\left(\\mathcal{EH}^1_D(M)\\right)^{\\perp}$. We claim that $\\dim\\mathcal{EH}_D^1(M) = k-1$ and $\\dim\\left(\\mathcal{EH}^1_D(M)\\right)^{\\perp}=2g$.\n \n For the first claim, if $\\omega \\in \\mathcal{EH}_D^1(M)$, then there is a function $u\\in C^{\\infty}(M)$ for which $\\omega= du$. Since $\\omega$ is a harmonic field with Dirichlet boundary conditions, it follows that $u$ is a harmonic function and is constant on the boundary. If we write the boundary as a disjoint union of $k$ curves, $\\partial M = \\Gamma_1\\cup\\dots\\cup \\Gamma_k$, then we get that $u|_{\\Gamma_i}=c_i$, for some constant $c_i$, $i=1,\\ldots k$. Now, the Dirichlet problem\n \\begin{equation*}\n\\begin{cases}\n\\Delta u = 0\\\\\nu|_{\\Gamma_i} = c_i,\n\\end{cases}\n\\end{equation*}\nhas a unique solution for each choice of $(c_1,\\ldots, c_k)$ (see pg. 307 of \\cite{Ta96}). Let \n\\begin{equation*}\n\\mathcal{F} = \\left\\{u\\in C^{\\infty}(M)\\ \\bigg| \\ \\Delta u = 0, u|_{\\Gamma_i}=c_i, i=1\\ldots k, \\sum_{i=1}^k c_i = 0\\right\\}.\n\\end{equation*}\nIt easy to see that the differential $d|_{\\mathcal{F}}:\\mathcal{F}\\rightarrow \\mathcal{E}\\mathcal{H}_D^1(M)$ is linear and bijective,  and so $\\dim\\mathcal{E}\\mathcal{H}_D^1(M) = \\dim\\mathcal{F} = k-1$. \n\n \nLet $\\overline{M}$ be a smooth Riemannian manifold obtained from $M$ by gluing a disk into each of its boundary curves $\\Gamma_i$. To prove the second claim, we will construct an isomorphism between $\\left(\\mathcal{EH}^1_D(M)\\right)^{\\perp}$ and  $H^1(\\overline{M})$. The result will then follow from the fact that there are $2g$ cohomology classes of closed forms on $\\overline{M}$.\n\nLet $\\theta \\in \\Omega(\\overline{M})$ be a closed form. We'll first show that there is a closed form $\\tilde{\\omega}\\in \\Omega(\\overline{M})$ supported on $M$ which is cohomologous to $\\theta$. To see this, let $\\tilde{D_i}$, $i=1,\\ldots, k$, be a disk slightly larger than and containing $D_i$, and let $\\phi_i$ be a smooth cut-off function for which $\\phi_i|_{D_i}\\equiv 1$ and $\\phi_i|_{\\overline{M}\\setminus\\tilde{D_i}}\\equiv 0$. Since $\\tilde{D_i}$ is simply-connected, $\\theta|_{\\tilde{D_i}} = df_i$ for some smooth functions $f_i$. Let $\\tilde{\\omega} = \\theta - \\sum_{i=1}^k d(\\phi_i f_i)$. Then $\\tilde{\\omega}|_{D_i}\\equiv 0$ and $d\\tilde{\\omega} =0$, so $\\tilde{\\omega}$ is a closed form in $\\Omega(\\overline{M})$ with compact support. Since $\\sum_{i=1}^kd(\\phi_i f_i)$ is exact, it follows that $\\theta$ and $\\tilde{\\omega}$ are cohomologous. For simplicity, we will suppress the restriction notation and write $\\tilde{\\omega}|_M$ by $\\tilde{\\omega}$. Now, we claim that any closed form $\\tilde{\\omega}\\in\\Omega(M)$ with compact support is cohomologous to a form $\\omega_0 \\in (\\mathcal{EH}_D(M))^{\\perp}$. To see this, let $u$ be a solution to the Poisson problem\n\\begin{equation*}\n\\begin{cases}\n\\Delta u = -\\delta \\tilde{\\omega}\\\\\nu|_{\\Gamma_i} = 0\n\\end{cases},\n\\end{equation*}\nand define $\\omega = \\tilde{\\omega}+du$. Then, $\\omega$ is harmonic, since $\\Delta\\omega = \\Delta\\tilde{\\omega}+\\Delta du = d\\delta\\tilde{\\omega} + 0 -d\\Delta u=0$. Moreover, $i^*\\omega = i^*\\tilde{\\omega}+d(i^*u) = 0$, so $\\omega$ satisfies the Dirichlet boundary condition. Now, $\\omega = \\omega_0 + dv$ for some $\\omega_0\\in (\\mathcal{EH}^1_D(M))^{\\perp}$ and $dv \\in\\mathcal{EH}^1_D(M)$. Hence, $\\omega_0$ is cohomologous to $\\omega$, and therefore $\\tilde{\\omega}$ and $\\theta$. Note that $\\omega_0$ is unique, \\textit{i.e.}, for any closed form $\\theta\\in \\Omega(\\overline{M})$, there is a unique $\\omega_0\\in(\\mathcal{EH}_D^1(M))^{\\perp}$ for which $\\omega_0\\sim\\theta$. If $\\omega_0^1,\\omega_0^2\\in(\\mathcal{EH}_D^1(M))^{\\perp}$ are two such forms, then $\\omega_0^1\\sim\\theta\\sim\\omega_0^2$. Hence, $\\omega_0^1-\\omega_0^2 = d\\zeta$, for some smooth function $\\zeta$. However, $\\omega_0^1-\\omega_0^2\\in(\\mathcal{EH}_D^1(M))^{\\perp}\\subset(\\mathcal{E}\\Omega(M))^{\\perp}$ and $d\\zeta\\in\\mathcal{E}\\Omega(M)$, so it follows that $\\omega_0^1=\\omega_0^2$.\n\nLet $\\mathcal{L}:H^1(\\overline{M})\\rightarrow (\\mathcal{EH}_D^1(M))^{\\perp}$ be the map $[\\theta]\\mapsto \\omega_0$ (as above). Note that it follows from the uniqueness of $\\omega_0$ that $\\mathcal{L}$ is well-defined and linear. \n\n\nNow, $\\mathcal{L}$ is also injective. If $\\mathcal{L}([\\theta_1]) = \\mathcal{L}([\\theta_2])$, then $\\theta_1+du_1=\\theta_2+du_2$, for some smooth functions $u_1, u_2$, which yields $\\theta_1\\sim\\theta_2$.\n\n\n\nFinally, $\\mathcal{L}$ is surjective. Suppose $\\omega_0\\in\\left(\\mathcal{EH}_D^1(M)\\right)^{\\perp}$. Then, since $i^*\\omega_0\\equiv0$, \n\\begin{equation*}\n\\int_{\\partial M}\\omega_0 = 0,\n\\end{equation*}\nand it follows that $\\omega_0$ is exact in a neighbourhood of each boundary curve, \\textit{i.e.}, $\\omega_0 = d\\psi_i$ in a neighbourhood of $\\Gamma_i$. Since we can extend each $\\psi_i$ smoothly over $D_i$, we can extend $\\omega_0$ to a closed form $\\theta\\in\\overline{M}$. It follows from the well-definedness of $\\mathcal{L}$ that $\\mathcal{L}$ does not depend on the choice of $\\tilde{D}_i$ or $\\phi_i$, $i=1,\\ldots k$. Hence, $\\mathcal{L}([\\theta]) = \\omega_0$. \n\\end{proof}\n\n\n\\end{appendix}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\nThe analysis of time series from complex systems calls for numerical methods that capture the \nmost relevant features in the observed variability. At the same time, the impact of various \nfrequently encountered data-related intricacies such as low signal-to-noise ratio, \nnonstationarity, and limited time series length must be accounted for. A major challenge is \nposed by irregular sampling, i.e., variations in the interval $\\Delta_i = t_i - t_{i-1}$ \nbetween consecutive measurement times\n$t_{i-1}$ and $t_i$. Irregular sampling is observed in many complex real-world systems. The underlying mechanisms that render the temporal sampling irregular may differ: sampling can be inherently \nirregular due to an additional process that controls the sampling interval (e.g., financial or \ncardiac time series \\cite{reno2003closer, fedotov2020methods}); a mixture of various external \nprocesses can result in `missing values', i.e., multiple interacting processes result in the \nnon-availability of measurements (e.g., sociological or psychological survey data \n\\cite{enders2010applied}) or cause failures of the system (e.g., mechanical\/electronical \nsystems \\cite{khayati2020mind}); finally, the measurement process often results in irregularly \nsampled time series (e.g., astronomical \\cite{scargle1982studies} or geophysical systems \n\\cite{rehfeld2011comparison}). Proxy time series obtained from \npalaeoclimate archives are a particularly challenging example since irregularity in the \ntemporal sampling can itself contain valuable information on the processes of interest \\cite{baldini2021detecting}. The \ngrowth rate of a stalagmite for example depends on variable environmental factors, including temperature in \nthe cave and drip rate \\cite{muhlinghaus2007modelling}, among others. Since these factors and their variability are strongly coupled to the environmental conditions outside the cave, growth rate must be regarded as a dynamical indicator for example, hydrological conditions which in turn determine variations in \nthe temporal sampling of the proxy time series.\n\nAcross many research communities, resampling based on interpolation techniques and imputation \napproaches are popular methods for making irregularly sampled time series compatible with standard time \nseries analysis tools \\cite{garland2018anomaly, trauth2021spectral}. Artefacts and statistical biases caused by interpolation techniques are well-known and may result in misinterpretation of the extracted time series properties, an issue further aggravated by the fact that biases introduced by interpolation may vary among different \nsystems \\cite{rehfeld2013late}. The robustness of results arising from different \ninterpolation techniques for the same data set is rarely examined. For instance, linear \ninterpolation will not compensate for the effect of lower variability during sparesely sampled \nepisodes in a time series compared to more densely sampled periods. In fact, linear \ninterpolation and mean imputation decrease variance to a hardly quantifiable, data-related degree \\cite{schulz1997spectrum}.\nFinally, more complex imputation models may account for such finite (sampling) size effects \nbut may not represent the `natural' variability of a time series adequately. For data \n\\textit{not-missing-at-random}, the assignment of a sufficient imputation model can be challenging and \nmust account for nonstationarity in the underlying non-random effects (e.g., for the palaeoclimate example mentioned above). Similar biases are known from the problem of imbalanced data, i.e., given two populations that should be compared based on a statistical model, a majority class exists that contains significantly more samples than the minority class and thus, oversampling techniques are applied to compensate for the resulting bias \\cite{chawla2002smote, barua2012mwmote}.\n\n\nGeophysical time series frequently exhibit nonlinear features such as nonlinear oscillations \nand critical regime transitions, e.g., tipping points \\cite{lenton2019climate}. Dynamical \nsystem theory regards observations from such systems as embedded in a higher-dimensional phase \nspace and offers a range of tools to quantify gradual or abrupt changes in these \ndynamics  \\cite{bradley2015nonlinear, marwan2021nonlinear}. The power of these methods relies on their ability to \nuncover features that regular techniques, such as autocorrelations or \nvariance estimation, fail to uncover \\cite{scheffer2009early}.\nAiming for higher applicability of nonlinear time series analysis methods in the Earth \nsciences, irregular sampling approaches have been proposed \n\\cite{lekscha2018phase, mccullough2016counting, sakellariou2016counting}. One of these approaches is based on the idea of transforming sub-sequences of unequal lengths in a time series into each other and comparing the costs of these transformations for all sub-sequences \n\\cite{suzuki2010definition}.\nMore generally, the definition of a metric distance between states at different instances of \ntime can entail dynamical information on the evolution of the phase space trajectory of the \nstudied system.\nWhile standard metrics (such as Euclidean distance) fail to account for irregular sampling, \nthe TrAnsformation-Cost Time-Series (TACTS)  \\cite{ozken2015transformation} includes the \ntemporal information for distinct time series segments. Similar approaches based on the edit \nor Levenshtein distance have been used in natural language processing \n\\cite{ukkonen1985algorithms} and metric analyses of point processes \n\\cite{banerjee2020recurrence}, among many others.\n\n\nIn this work, we focus on the application of the (m)Edit-distance \\cite{ozken2018recurrence} as a base for the \ncomputation of recurrence plots (RPs) \\cite{marwan2007recurrence}. The (m)Edit-distance \napproach can potentially be employed in any methodological framework that includes computation \nof a distance (or similarity) measure. The RP technique represents one particular application that has proven to be a powerful \napproach, tackling many of the fundamental problems in time series analysis, such as time \nseries classification \\cite{garcia2018classification}, the study of synchronization between \nmultiple time series \\cite{romano2004multivariate}, and detection of regime transitions \n\\cite{marwan2013recurrence}. Recurrence quantification analysis (RQA) provides a means of \nquantifying the tendency of a time series to revisit previously visited states and has grown \nin its scope from basic predictability quantification towards more ambitious measures that, \ne.g., capture the multiscale nature of transitions \\cite{corso2018quantifying, \nschinkel2007order, braun2021detection}.\nThe identification of shifts stands out as a \nparticularly interesting application since critical transitions can often be linked to the \nvulnerability of the respective regional climate system towards external shocks or feedback \nmechanisms. The combination of the (m)Edit-distance approach and RPs offers a promising \napproach to identify regime transitions in irregularly sampled records, which may otherwise be impeded \nwithout an adequate technique designed to account for sampling variations \n\\cite{ozken2018recurrence, marwan2018regime, stegner2019inferring}. In following this approach, \nspecial care must be taken if irregular sampling intervals undergo strong variations, i.e., where the process(es) that control the sampling rate are rendered non-stationary. In some \napplications, segments can be chosen such that they do not cover the same time period but the same number of values on average. Other applications \nrequire fixing a particular time period to be covered by each segment since this time period corresponds to the time scale under investigation, e.g., a year for seasonal time series. \nEven if such an approach is not motivated by the research question, splitting the time series \ninto segments that correspond to non-equal time periods will result in mixing of time scales \nin the resulting distance matrix if the sampling rate is highly non-stationary. \nHere, we focus on segments that cover equal time periods but varying numbers of values, refered \nto as \\textit{segment size}. We will show that in such cases, the resulting strong variations in segment size entail a \nnon-trivial sampling bias of the (m)Edit-distance.\n\n\n\nWe introduce the (m)Edit-distance methodology in \nSect.~\\ref{sec2.1} followed by a short summary of recurrence analysis in Sect.~\\ref{sec2.2}. \nSect.~\\ref{sec3} illustrates the problem of strong variations in the sampling rate whereas model time series are studied to elucidate the sample size effects. A correction \nscheme based on constrained randomization is proposed in Sect.~\\ref{sec4}. \nIn Sect.~\\ref{sec5}, we demonstrate the importance to correct for the identified sample-size dependence in an application to a palaeoclimate record from Niue island in the central Pacific where we identify variations in seasonal predictability. We conclude our findings \nin Sect.\\ref{sec6}.\n\n\n\n\n\\section{Methodology}\n\\label{sec2}\n\\subsection{The (m)Edit-distance measure}\n\\label{sec2.1}\nMany approaches in nonlinear time series analysis are based on some notion of a \n(dis)similarity measure. For deterministic systems, embedding the univariate time series into \nan $m$-dimensional phase space offers a multitude of quantitative approaches to analyse the \nvariability of its trajectory \\cite{kraemer2021unified}. Yet appropriate techniques to extract the embedding dimension and delay from empirical data are needed. These approaches can be cumbersome. In this work we focus \non univariate time series wherein the most widespread dissimilarity measure \nbetween distinct segments $\\mathcal{S}_{a},\\, \\mathcal{S}_{b}$ is the Euclidean distance. It \nis a metric distance, i.e., its value is always positive $D(\\mathcal{S}_{a},\\, \n\\mathcal{S}_{b}) \n\\, \\geq \\, 0$, it is symmetric $D(\\mathcal{S}_{a},\\, \\mathcal{S}_{b}) \\, = \\, \nD(\\mathcal{S}_{b},\\, \\mathcal{S}_{a})$, and the triangle inequality holds \n$D(\\mathcal{S}_{a},\\, \n\\mathcal{S}_{\\mathrm{c}}) \\, \\leq D(\\mathcal{S}_{a},\\, \\mathcal{S}_{b}) + \nD(\\mathcal{S}_{b},\\, \n\\mathcal{S}_{\\mathrm{c}})$. If the time series is characterized by missing values or the \nsampling interval $\\Delta_i$ is irregular (e.g., due to irregularities in the measurement process), no straight-forward application of Euclidean distance or comparable metrics is possible: dissimilarity of values at unequal time scales would be computed without accounting for their non-equality.\nLinear interpolation as a means of resampling the time series values onto a regular time axis is among the most popular \napproaches to regularize sampling \\cite{rehfeld2014similarity}. Yet, hardly controllable artefacts arise from linear interpolation, ranging from difficulties related to altered absolute timing to underestimation of variance or overestimation of persistence \\cite{rehfeld2013late, \nmudelsee2013climate}. \n\n\nOriginally proposed for natural language processing, the edit distance measure \n\\cite{masek1980faster} is designed to compare sequences of variable length. \nShifting and adding $\\&$ deleting of strings were proposed as two elementary operations to \nquantify dissimilarities between words, an objective also pursued by other methods such as \ndynamic time warping \\cite{rabiner1978considerations}. The resulting costs are calculated by \nidentifying a minimum cost path to transform one sequence into the other. Taking the next step \ntowards an application to empirical time series, the edit distance was applied to point \nprocess data whereby cost parameters for the elementary operations remained arbitrary \n\\cite{suzuki2010definition, victor1997metric}. By equipping the technique with data-driven \ncost parameter estimates, it was then applied to irregularly sampled palaeoclimate time series \n\\cite{ozken2015transformation}. A further modification ((m)Edit distance) with an application \nto extreme events was proposed to consider the saturation of shifting costs when a \ncertain time scale $\\tau$, separating the two compared segments, is exceeded \n\\cite{banerjee2020recurrence}. The main difference between applying the edit distance to \nseries of events\/spike trains and irregularly sampled time series is that for the latter, \namplitudes of time series values must be considered. In the following, whenever no assumptions are \nmade about the amplitudes of a signal, we refer to `events'.\nThe edit distance between two segments $\\mathcal{S}_{a},\\, \\mathcal{S}_{b}$ of an irregularly \nsampled time series is computed by minimizing the transformation costs by:\n\\begin{widetext}\n\\begin{align}\nD(\\mathcal{S}_{a},\\, \\mathcal{S}_{b}) \\, = \\, {\\mathrm{min}}\n\\left\\{\n\\sum_{\\alpha,\\beta \\in \\mathcal{C}} \n\\left[\n\\underbrace{f_{\\Lambda_0}\\left(t(\\alpha), t(\\beta);\\,\\tau\\right)}_\\text{shifting}\n\\, + \\, \n\\underbrace{\\Lambda_{\\mathrm{k}}\\bigl\\|L_{a}(\\alpha) - L_{b}(\\beta)\\bigr\\|}_\\text{amplitude change}\n\\right]\n\\ + \\ \n\\underbrace{\\Lambda_{\\mathrm{S}}\\left(|I| + |J| - 2|\\mathcal{C}|\\right)}_\\text{adding and deleting}\n\\right\\}\n\\label{meq1}\n\\end{align}\n\\end{widetext}\nwith a norm $\\|\\cdot\\|$ (e.g., the Euclidean \nnorm), the $\\alpha\\,$-th\/$\\beta\\,$-th amplitudes $L_{a}(\\alpha),\\, L_{b}(\\beta)$ of the \nsegments $\\mathcal{S}_{a},\\, \\mathcal{S}_{b}$ and the cardinalities $|\\cdot|$ of the sets $I,\\, J \n\\,$ and $\\mathcal{C}$. While the latter are a  set  of  indices  of  the time series values, $\\mathcal{C}$ denotes the values that are shifted.\n$D(\\mathcal{S}_{a},\\, \\mathcal{S}_{b})$ is a metric distance. The cost \nparameters $\\Lambda_0,\\,\\Lambda_{\\mathrm{k}}\\, $, and $\\Lambda_{\\mathrm{S}}$ need to be fixed \nprior to cost optimization.\nWe choose the cost parameter for amplitudes changes $\\Lambda_{\\mathrm{k}}$ as suggested in \\cite{ozken2015transformation}:\n\\begin{align}\n\\Lambda_{\\mathrm{k}} \\, = \\, \\frac{M-1}{\\sum_{i=1}^{M-1}\\|x_i - x_{i+1}\\|}\n\\label{meq2}\n\\end{align}\nThe cost parameter $\\Lambda_{\\mathrm{S}}$ for deleting $\\&$ adding has to be chosen such that \ndeletions are neither `too cheap' nor `too expensive'. For a set of time series values with a \nlarge temporal distance or very distinct amplitudes, a deletion and addition should be favorable while a \ntoo low value of $\\Lambda_{\\mathrm{S}}$ will result in a transformation of sequences solely by \ndeletion and adding operations even for very close time series values. We follow the scheme proposed in \\cite{ozken2018recurrence} by \nassuming normality for the distance values between all segments of the time series and \noptimize $\\Lambda_{\\mathrm{S}}$ within a specified range using a Kolmogorov-Smirnov (KS)-test to ensure that the normality assumption holds as close as possible.\nFollowing the modification proposed in \\cite{banerjee2020recurrence}, costs associated with \nshifting of time instances between two time series values are controlled by the logistic \nfunction\n\\begin{align}\nf_{\\Lambda_0}\\left(t(\\alpha), t(\\beta);\\,\\tau\\right) \\, = \\, \\frac{\\Lambda_0}{1 + \\mathrm{e}^{-\\left(\\|t_{a}(\\alpha) - t_{b}(\\beta)\\| - \\tau\\right)}}\n\\label{meq3}\n\\end{align}\nwhere $\\tau$ is the location parameter of the logistic function, reflecting a \ncharacteristic time scale that separates exponentially increasing from saturating\/bounded \nexponentially increasing costs for shifting. We choose $\\tau$ as \nthe average sampling interval of the time series; $\\tau \\, = \\, \\left. T \\middle\/ M \\right.$  \nwith the total time period $T$ and the number of samples $M$. Interpreting $\\tau$ as a `temporal \ntolerance', this choice ensures that shifting exponentially fast becomes less favorable if \ntime instances are separated by several standard deviations of the sampling interval \ndistribution. Finally, a value for the maximum costs associated with shifting $\\Lambda_0$ needs to be \nset. The ratio $\\left. \\Lambda_{\\mathrm{K}}\\middle\/\\Lambda_0 \\right.$ reflects the relative \nimportance of temporal and magnitudinal separation; in the limiting case $\\left. \n\\Lambda_{\\mathrm{K}}\\middle\/\\Lambda_0 \\right. \\,\\gg\\, 1\\,$, irregular sampling is no longer \naccounted for and the resulting distance between two segments solely \nreflects the norm $||L_{a}(\\alpha) - L_{b}(\\beta)||$\nfor all amplitudes $L_{a}(\\alpha),\\, L_{b}(\\beta)$ of both segments $\\mathcal{S}_{a},\\, \n\\mathcal{S}_{b}$.\nIn the opposite case $\\left. \\Lambda_{\\mathrm{K}}\\middle\/\\Lambda_0 \\right. \\,\\ll\\, 1\\,$, the \ntime series can be regarded as a series of events since cost optimization is independent of \ntheir amplitudes. We choose $\\Lambda_{\\mathrm{K}} = \\Lambda_0 = 1$. It must be stressed that this rate depends on the research question and the data under study.\n\nIn the following, we discuss the finite-sample effects bias (m)Edit-distance values \n$D(\\mathcal{S}_{a},\\, \\mathcal{S}_{b})$ and give a summary of the RP methodology. This facilitates the presentation of finite-sample effects discussed in \nSect.~\\ref{sec3} alongside an illustration of the (m)Edit-distance \nmethodology (Fig.~\\ref{fig1}).\n\n\n\n\\subsection{Recurrence analysis}\n\\label{sec2.2}\n\nThe tendency to recur to previously visited states is a ubiquitous feature shared by time \nseries from many different complex systems. Recurrence plots encode this information in a \n2-dimensional \nbinary matrix, indicating a recurrence between two states $\\vec{x}_i$ and \n$\\vec{x}_j$ at times $i$ and $j$ if the respective states are similar with respect to a given \nnorm ${D}(\\vec{x}_i, \\vec{x}_j)$ \\cite{jp1987recurrence}:\n\\begin{align}\n{R}_{ij} \\, = \\, \\left\\{\n\\begin{array}{ll}\n1 & \\mathrm{if} \\ {D}\\left(\\vec{x}_i, \\vec{x}_j\\right) \\leq \\varepsilon \\\\\n0 & \\mathrm{if} \\ {D}\\left(\\vec{x}_i, \\vec{x}_j\\right) > \\varepsilon.\n\\end{array}\n\\right.\n\\label{meq4}\n\\end{align}\nThe norm ${D}(\\vec{x}_i, \\vec{x}_j)$ yields a symmetric, real-valued distance \nmatrix $\\mathbf{D}$ between states at all time instances $i,\\,j$. By thresholding $\\mathbf{D}$ with the vicinity \nthreshold $\\varepsilon$, a notion of similar and dissimilar states is implemented and defines the \nrecurrence between each pair of states. The underlying idea is based on the Poincar\u00e9 \nrecurrence theorem that states the recurrence of a dynamical system's trajectory \n$\\vec{x}(t)$ \nto an $\\varepsilon$-neighborhood of any perviously visited state after sufficiently long \ntime \\cite{poincare1890probleme}. For the main diagonal of the RP, it always holds that \n${R}_{ij} \n\\equiv 1$.  If no phase space reconstruction is applied, states \n$\\vec{x}_i$ and $\\vec{x}_j$ correspond to time series amplitudes ${x}_i$ and ${x}_j$. \nThe threshold $\\varepsilon$ can be chosen \nbased on different data-dependent criteria. In many applications, the recurrence rate is \nfixed to a certain percentage (e.g., $10\\,\\%$ recurrences \\cite{kraemer2018recurrence}) or set to a multiple of the standard \ndeviation of the distance matrix $\\mathbf{D}$ \\cite{schinkel2008}.\nThe geometric recurrence patterns encoded in a RP can be exploited to distinguish between \nstochastic and deterministic systems \\cite{marwan2007recurrence}; while a purely random white \nnoise process will result in isolated dots in the recurrence matrix, time series from \ndeterministic systems are known to yield diagonal line structures \\cite{marwan2007recurrence}. Long diagonal lines are \ncharacteristic for periodic systems; interrupted diagonal lines indicate chaotic dynamics. \nRecurrence quantification analysis (RQA) which evaluates the statistical properties of a RP has proven a versatile tool for diverse real-world applications, such as time series classification \\cite{hirata2021recurrence}, study of causal relations \\cite{ramos2017recurrence} or regime shift detection \\cite{westerhold2020astronomically}.\n\n\nRecurrence analysis overcomes some of the flaws of other statistical analysis tools \nwhen applied to geophysical time series, such as the Lyapunov exponent or correlation dimension \n\\cite{kantz1994robust, donner2008nonlinear}. It is less sensitive to noise and can be \napplied to short time series. In combination with the (m)Edit-distance approach, first \napplications demonstrated its ability to detect regime transitions in palaeoclimate proxy \nrecords \\cite{marwan2018regime}. In order to compute a RP for irregularly sampled time series, ${D}(\\vec{x}_i, \\vec{x}_j)$ in Eq.~(\\ref{meq4}) is identified with the \nmodified edit distance from Eq.~(\\ref{meq1}). In contrast to regular computation of metric \ndistances, segments of the time series are required to obtain a distance value between two \nstates. Genereally speaking, segment size should be chosen sufficiently small to ensure that \nno aliasing effects arise due to interference between the segment width and the characteristic \ntime scale of a time series (e.g., characteristic period of a periodic time series).\nFor some applications the segments can be chosen such that all are equally sized \n$|\\mathcal{S}_{a}| \n= |\\mathcal{S}_{b}| = \\dots = N$. If this is not possible, the variance of \nsegment widths can still be minimized and for each pair of segments with differing widths; \ndeletion $\\&$ adding operations will contribute to the resulting transformation cost. If time \nseries are short, we can allow for an overlap between segments, although caution is advised \nsince this introduces a serial dependence in the resulting edit distances of overlapping \nsegments and violates the normality assumption used in the estimation of \n$\\Lambda_{\\mathrm{S}}$.\nHere, we focus on the most general case of unequal segment sizes. Apart from cases where \nsegment size deviations can hardly be minimized, this is relevant in some real-world \napplications where we are interested in the recurrences between segments that correspond \nto a particular time scale, or where sampling rate is highly non-stationary and selecting a \nconstant segment size would result in mixing of distinct time scales. The application to \npalaeoclimate data (Sect.~\\ref{sec5}) will illustrate such a case. There, the focus lies on \nthe comparison of seasonal sequences in an irregularly sampled proxy time series.\n\nPredictability is a feature of time series that can help to identify and classify different dynamical regimes in the evolution of the studied system. Since the lengths of diagonal lines in a RP reflect the predictability of a system, the number of diagonal lines which exceed a specified minimum line length $l_{\\mathrm{min}}$ can be used as a predictiability measure:\n\\begin{align}\n\\mathrm{DET} \\, = \\, \\frac{\\sum_{l=l_{\\mathrm{min}}}^{N}P(l)}{\\sum_{l=1}^{N}P(l)}\n\\label{meq5}\n\\end{align}\nwith the number $P(l)$ lines of length $l$.\nDeterminism (DET) can be linked to the correlation dimension of a dynamical system \n\\cite{march2005recurrence} and has successfully been used in diverse empirical analyses \n\\cite{ozken2018recurrence,marwan2018regime,westerhold2020astronomically} to detect transitions \nbetween regimes of varying predictability. We use DET as a recurrence quantifier to test the impact of the sampling-based correction scheme introduced below.\n\n\n\n\\section{Segment size dependence}\n\\label{sec3}\nFinite-sample effects are known to entail statistical biases in various time series analysis methods. Linear or spline interpolation is often employed as a pre-processing technique to enable the application of standard time series analysis tools to irregularly sampled time series. Interpolation techniques do not account for basic finite-sample biases. \nFor instance, statistical location and scale measures (such as the median or volatility \nindicators) are known to be biased for small sample sizes \\cite{williams2011finite, \npark2019investigation}. Given two segments $\\mathcal{S}_{a},\\, \\mathcal{S}_{b}$ with \n$|\\mathcal{S}_{a}| \n\\gg |\\mathcal{S}_{b}|$, estimating their variance (e.g., as a volatility \nindicator or in order to compute a continuous wavelet spectrum) can result in underestimation of the variance for the shorter segment. \nSimilarly, persistence estimators are generally biased due to finite-sample effects, even for \nMarkovian stationary stochastic processes \\cite{trenberth1984some}. Whenever a sliding-window \nanalysis for nonstationary, irregularly sampled time series is carried out, variations in the \nsampling rate will inevitably result in a mixture between the actual variability of the statistical \nindicator and purely sampling-related variations. As interpolation techniques are usually \nlimited to resampling values such that sampling intervals are equal, this effect is not \ncompensated. Similar intricacies need to be considered in short time series, e.g., \nwhen computing correlations between multiple time series (of varying length) \\cite{goswami2017inferring}.\n\nWhile not designed to compensate such effects, the (m)Edit-distance methodology does not introduce any known additional biases. The computation of \ntransformation costs is demonstrated with two exemplary pairs of segments $\\mathcal{S}_a,\\, \n\\mathcal{S}_b$ and $\\mathcal{S}_c,\\, \\mathcal{S}_d$ (Fig.\\ref{fig1}). The \nsegments $\\mathcal{S}_a,\\, \\mathcal{S}_b$ all display distinct operations for transforming a \nsegment into another: in the first step, a shift of amplitude and time are applied to transform the time instance \n$t_a(1)$ and amplitude $L_a(1)$ of the first segment into time instance $t_b(2)$ \nand amplitude $L_b(2)$ of the second segment. The cost $C_1$ associated with this operation is \nthe sum of shifting both time and amplitude. After shifting the third value of $\\mathcal{S}_a$ \nto match the third value of $\\mathcal{S}_b$, both a deletion and an adding operation are \nperformed in step 3 with twice the cost $\\Lambda_{\\mathrm{S}}$ for a adding\/deleting \noperation. The same transformation could have been achieved with an additional shifting \noperation. The prefered operation is determined by the particular choice of cost parameters. \nAs $|\\mathcal{S}_a|=3$ and $|\\mathcal{S}_b|=4$, the first value of $\\mathcal{S}_b$ is added in \nstep 4. The resulting cost is the sum of all costs for each step. While different \ntransformation paths are possible, the algorithmic implementation ensures that $C$ is \nminimized with respect to all possible combinations. Another example is displayed in the right \ncolumn of Fig.\\ref{fig1}. The setup differs in that the indicated segments \n$\\mathcal{S}_c,\\, \n\\mathcal{S}_d$ are longer than $\\mathcal{S}_a,\\, \\mathcal{S}_b$ ($|\\mathcal{S}_c|=8,\\, \n|\\mathcal{S}_d|=7$). \nDespite a similar set of transformations, the resulting costs \n$\\tilde{C}$ \nare significantly higher for the exemplary choice of parameters. \n\n\nA systematic derivation of transformation costs on segment size\/sampling rate for exponentially distributed sampling intervals is given in appendix A.\nNote that the identified effect is not due \nto an immanent misconception in the edit distance computation. It solely arises from the \nfact that the edit distance is applied in a setting where the time axis is not only irriations in its sampling rate.\nIn particular, abrupt transitions in the sampling rate between a time period $T_1$ with low \nsampling rate $\\lambda_1$ and $T_2$ with high sampling rate $\\lambda_2$ will imprint a \nnon-trivial \n$\\lambda_1,\\,\\lambda_2$-dependence on the transformation cost  \n$D(\\mathcal{S}_a\\,\\mathcal{S}_b)$ \nbetween any two segments.\nIn a recurrence analysis of time series, the focus lies on the similarity of states based on egular but undergoes \nsignificant var\nthe amplitudes of the time series. Hence, we argue that the identified dependencies counteract \nthe goal of recurrence analysis of irregularly sampled time series and thus need to be \ncorrected such that recurrence quantification measures reflect the dynamical behaviour of the \nunderlying system rather than mere shifts in the sampling rate.\n\n\nWe numerically examine the dependence of transformation costs between segments $\\mathcal{S}_{a},\\, \\mathcal{S}_{b}$ on their sizes $N_a, N_b$ for simple synthetic time series.\nWe test irregularly sampled time series from three different model systems: uncorrelated \nuniform noise, an AR(1)-process ($\\tau=5$), and a sinusoidal ($\\nu =\\left.1 \\middle\/ 25\\right.$) \nwith superimposed low-amplitude white noise. Segments of specified sizes from each of these \nsystems are drawn to compute segment size-specific costs.\n\\begin{figure*}[]\n\\centering\n\\includegraphics[width=.9\\textwidth]{fig1}\\\\\n\\caption{Schematic illustration of how irregularly sampled segments of varying lengths are \ntransformed with the (m)Edit-distance method. Two exemplary pairs of segments \n$\\mathcal{S}_a,\\, \n\\mathcal{S}_b$ (\\textbf{A}: red, blue) and $\\mathcal{S}_c,\\, \\mathcal{S}_d$ (\\textbf{B}: green, \norange) of an irregularly sampled synthetic AR(1)-time series are displayed. Each row shows an \noperation applied to the respective segment (shift: purple, deletion\/adding: cyan). Final \ncosts $C$ and $\\tilde{C}$ result from a specific choice of cost parameters as described in \nSect.~\\ref{sec2.1}. Please note that the higher total cost in \\textbf{B} showcase the dependence on segment length.}\n\\label{fig1}\n\\end{figure*}\n\\FloatBarrier\nIrregular time axes are generated from a $\\gamma(\\Delta; k,\\Theta)$-distribution with scale \n$\\Theta$ and shape $k = \\sqrt{\\left. 2 \\middle\/ \\Gamma \\right.}\\,$, where $\\Gamma$ denotes the \nskewness of the distribution.\nThis choice is motivated by the observation that sampling \nintervals in palaeoclimate proxy time series are often $\\gamma$- rather than \nexponentially-distributed. \nFor each system, we generate a `superpopulation' $(K=100)$ of time series and \ntime axes. Fixing a different skewness $\\Gamma$ of the $\\gamma\\,$-distribution of each \nof the time axes between $\\Gamma\\in[1,8]$ ensures that for $T=10,000$, segment sizes range \nbetween $N\\in[1, 20]$. The modified edit-distance is used, Eq.~(\\ref{meq1}) and deletions are included as a competing operation to shifting.\nThe optimal $\\Lambda_{\\mathrm{S}}$ is estimated for each system according to the procedure \noutlined in Sect.~\\ref{sec2.1}: the KS-statistic is minimized for each systems, yielding\n$\\Lambda^{\\mathsmaller{(\\mathrm{unif})}}_{\\mathrm{S}} = 1.5,\\, \\Lambda^{\\mathsmaller{(\\mathrm{AR1})}}_{\\mathrm{S}} = 1.5,\\, \\Lambda^{\\mathsmaller{(\\mathrm{sin})}}_{\\mathrm{S}} = 3.5$.\n\n\nFigure \\ref{fig2a} displays the obtained transformation costs in the cost matrices \n${{C}}(N_a,N_b)$ \nand ${{\\tilde{C}}}_{\\mathrm{shift}}(N_a,N_b)$ \nafter averaging over $K=100$ different realizations. Regardless of the irregularity of the \ntime axis and the respective system, a tendency of increasing total costs for larger segment \nsizes is observed (upper row). For the AR(1)-system, this increase is slower for fixed \n$N_b$ and increasing $N_a$ composed to the uncorrelated noise and the sinusoidal examples. More generally, \nthe rate of increase differs between the considered systems but follows the same trend.\nIn total, $|N_a-N_b|$ `basic deletions' (or adding operations) need to be carried out for each pair of segments with $N_a\\neq N_b$.\nIf costs for these basic deletions are subtracted and computed per shifting step, a similar \ndependency on $N_a,\\,N_b$ as observed in Fig.~\\ref{figA1}c for the more simple case can be \nobserved in the cost matrices ${{\\tilde{C}}}_{\\mathrm{shift}}(N_a,N_b)$ in \nFig.~\\ref{fig2b}\\,: the cost of an average shift from a segment with $N=N_a$ increases towards \n$N_b=N_a$ and decays if segment size increases further.\nConsequently, the leading effect results from the basic deletions that are directly linked to the difference in segment sizes $|N_b-N_a|$. Yet, transformation costs still depend on segment size even after aligning both segment sizes by means of basic deletions; this effect likely results from having a higher probability of finding closely spaced values on the time axis as the sampling rate of one segment increases, yielding an increasing trend for average costs per operation (in \nfig.~\\ref{fig2b}).\n\\section{Sampling rate constrained surrogates}\n\\label{sec4}\nIrregularly sampled time series with constant sampling rate can be studied with the \n(m)Edit-distance to obtain dissimilarity estimates between different time series \nsegments. The resulting distance matrix can be used to perform a recurrence analysis. \nMoreover, other analysis techniques such as complex networks, clustering, or correlation analysis are based on (dis)similarity measures and could use the (m)Edit-distance as a metric to account for irregular sampling or to characterize event-like \ndata. In section \\ref{sec3} we showed that in case of a non-constant sampling rate, an \nestimation of the (m)Edit-distance matrix is biased by significant differences in the \nsegment sizes.\n\nIn the following, we propose a numerical correction-technique for recurrence analysis. We generate an ensemble of time series and time axis surrogates that reproduces the sampling properties of the real irregularly sampled time series. This surrogate ensemble is used for \nbias-correction of recurrence quantification measures, exemplified by the determinism DET.\n\\subsection{Constrained randomization}\n\\label{sec4.1}\nWhen studying a system's dynamics with time series analysis tools, a null-hypothesis \nis formulated which can be be tested. In case of recurrence analysis, \nthis hypothesis could for example be non-stationarity of a dynamical property of the \nsystem (predictability, serial\/cross-dependence, \\ldots) expressed by a particular \nrecurrence quantification measure. In the used example, the \nnull-hypothesis \ntests whether the observed dynamics could be solely caused by variations in the sampling rate. \n\nParametric hypothesis testing for time series analysis often poses severe constraints \non the statistical properties of the underlying probability distribution, e.g., \nnormality. Surrogate tests represent a non-parametric and flexible method to test for \na range of properties in a system, including nonlinearity or periodicity, among \nothers  \\cite{schreiber2000surrogate}. Time series surrogates are altered copies of a \nreal, underlying time series that only preserve a specified set of properties of the \nreal time series. The general technique to generate surrogate realizations of a time \nseries is constrained randomization. After defining a set of constraints that state \nwhich properties of the real time series should be preserved, the time series is \nrandomized such that these constraints are still fullfilled. Here, randomization will be carried out on the sampling interval $\\Delta_i$ with \nthe constraint that for each segment $\\mathcal{S}_i$ of the real time series, segment \nsize $N_i$ is preserved. This is achieved by drawing sampling intervals $\\Delta_i$ \n(with replacement) from the empirical sampling interval distribution $p(\\Delta, \n\\lambda (t))$. For a given segment $S_i$ with size $N_i$, $N_i$ \nsampling intervals are drawn from $p(\\Delta, \\lambda (t))$ and cumulated to generate a \nsurrogate realization of the particular time axis segment: \n\\begin{align}\n\\tilde{t}_{\\mathcal{S}_i}^{(0)} = t_{\\mathcal{S}_i}^{(0)} \\, , \\quad  \\tilde{t}_{\\mathcal{S}_i}^{(j+1)} = \\tilde{t}_{\\mathcal{S}_i}^{(0)} + \\sum_{m=0}^j \\Delta_{i}^{(m)}\n\\label{meq6}\n\\end{align}\nLet $w$ be the time period covered by each segment. For any randomly sampled set of \nsampling intervals, the constraint of preserved segment size requires that\n\\begin{equation}\\label{condition_t}\n\\tilde{t}_{\\mathcal{S}_i}^{(N_i)} \\stackrel{\\text{!}}{\\leq} w, \n\\end{equation}\notherwise the random sampling of sampling intervals $\\Delta_i$ has to be repeated. \nIf the distribution of segment sizes is short-tailed, i.e., no segments with size \n$N\\gg \\mathbb{E}[k]$ exist, this simple randomization procedure converges rapidly for \neach segment. If segments of relatively large size are present -- which is likely the \ncase for non-stationary sampling rates -- only a small subset of sampling intervals from \nthe left tail of $p(\\Delta, \\lambda (t))$ will fulfill the condition \n(\\ref{condition_t}). In order to ensure convergence of the algorithm for large \nsegments, a weight-function can be introduced for all sampling intervals to increase \nthe likelihood of drawing short sampling intervals when a segment with large size is \ngenerated. We suggest the use of $\\beta$-distributed weights $\\omega$:\n\\begin{align}\n\\omega(X;\\,\\alpha, \\beta) \\, = \\, \\frac{1}{B(\\alpha, \\beta)}x^{\\alpha-1}(1-x)^{\\beta-1}\n\\label{meq7}\n\\end{align}\nwith the $\\beta$-function $B(\\alpha, \\beta)$. This choice is motivated by the fact \nthat for $\\alpha = \\beta = 1$, $\\omega(X;\\,\\alpha, \\beta)$ becomes a uniform \ndistribution. In our application, we choose $\\alpha = \\beta = 1$ when the first \niteration of sampling $N_i$ sampling intervals $\\Delta_i$ is carried out. The \npopulation of sampling intervals is ordered from shortest to largest and each \n$x_i\\leftrightarrow \\Delta_i$ is assigned a $\\beta$-distributed weight $\\omega_i$, \ni.e., for the first iteration, every sampling interval is drawn with equal \nprobability. If the iteration fails $\\left(\\tilde{t}_{\\mathcal{S}_i}^{(N_i)} > \nw\\right)$, $\\alpha$ is increased by a small number $\\Delta\\alpha$, reshaping the \nbeta-distribution \nand increasing the probability of drawing small sampling intervals. Thus, \nwe perform a weighted sampling from the empirical distribution $p(\\Delta, \\lambda \n(t))$ \nof sampling intervals with $\\beta$-distributed weights. In the $l$-th \niteration, we use $\\omega(X;\\,\\alpha_l, \\beta=1),\\, \\alpha_l = 1 + l\\Delta\\alpha$ as \nthe weight function for each segment. Finally, we can identify an amplitude difference \n$\\Delta y_i$ of the time series with each sampling interval $\\Delta_i$. This \ncorrespondence is exploited by also drawing the respective amplitude difference for \neach drawn sampling interval. After the procedure is finalized and a surrogate has \nbeen generated, amplitude differences are cumulated, that yields both a time axis and \ntime series surrogate. Both are denoted as sampling rate constrained \nsurrogate (SRC-surrogates).\nThe full randomization procedure thus preserves segment sizes in the correct temporal \norder and by definition approximately reproduces the distribution of amplitude differences and \nsampling intervals.\n\\begin{figure*}[]\n\\centering\n\\begin{subfigure}[t]{1\\textwidth}\n\\centering\n\\includegraphics[width=.32\\textwidth]{fig2a}\n\\includegraphics[width=.32\\textwidth]{fig2b}\n\\includegraphics[width=.32\\textwidth]{fig2c}\n\\caption{Cost matrices ${{C}}(N_a,N_b)$ (including basic deletions, total costs).}\n\\label{fig2a}\n\\end{subfigure}\n\\begin{subfigure}[t]{1\\textwidth}\n\\centering\n\\includegraphics[width=.32\\textwidth]{fig2d}\n\\includegraphics[width=.32\\textwidth]{fig2e}\n\\includegraphics[width=.32\\textwidth]{fig2f}\n\\caption{Cost matrices ${{\\tilde{C}}}_{\\mathrm{shift}}(N_a,N_b)$ (excluding basic deletions, average costs per operation).}\n\\label{fig2b}\n\\end{subfigure}\n\\caption{Cost matrices ${{C}}(N_a,N_b)$ for the transformation of segments with different lengths, including basic deletions (a) and excluding basic deletions (b). Costs are shown for uncorrelated uniform-distributed noise (left), an AR(1)-process (center) and a sinusoidal with superimposed white noise (right). Sampling intervals are $\\gamma\\,$-distributed.}\n\\label{fig2}\n\\end{figure*}\n\\FloatBarrier\n It also preserves the correspondance between sampling \nintervals and amplitude differences, ensuring that if periods with high local sampling \nrate entail larger variance\/strong amplitude changes in a real time series, this \nproperty is also included in the SRC-surrogates. The full procedure is outlined in \nFig.~\\ref{fig3} for an exemplary time series. \nOther randomization schemes are conceivable, e.g., varying the sampling weights after \ndrawing each single sampling interval based on the size of the latter, or stratified \nrandomization, i.e., performing the randomization differently for strata that correspond to the \ndifferent segment sizes. However, the proposed scheme has proven to be effective within the scope of this work.\n\nWith the presented scheme of generating SRC-surrogates, an ensemble of surrogates can be \ngenerated and (m)Edit-distance matrices $\\mathbf{D}$ computed for each SRC-surrogate. \nAny measure that is based on $\\mathbf{D}$ can consequently be computed for each surrogate \nseparately, yielding a distribution that can be used for testing the null-hypothesis \nformulated above based on the desired $\\alpha$-confidence level.\n\\subsection{Recurrence analysis of an AR(1)-process}\n\\label{sec4.2}\n\nIn the example below, the proposed correction scheme is applied to an irregularly sampled AR(1)-process (Fig.~\\ref{fig5}a).\nWe consider an autocorrelation increasing with time, visible by autocorrelation time $\\tau$ (Fig.~\\ref{fig5}b).\nA recurrence analysis is used to characterize the predictability of the time series in a sliding window analysis. \nPredictability is computed by means of determinism, DET, as defined in Eq.~(\\ref{meq5}).\nIn particular, we study how an abrupt shift of the sampling rate (represented by the skewness of $\\gamma$-distributed \nsampling intervals) affects DET and if a continuous increase of predictability can be recovered despite this shift by using \nthe proposed SRC-surrogate method. The shift appears at $t=1250$ (visible by variation of the segment size, Fig.~\\ref{fig5}b).\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=1\\textwidth]{fig3.pdf}\n\\caption{Schematic illustration of the constrained randomization procedure that generates SRC-surrogates for an exemplary irregularly sampled time series with non-stationary sampling rate. The left column shows the segmentation of the time series into segments of constant time period $s$ but of variable size $N_i$. The center column illustrates the weighted sampling of sampling intervals and amplitude differences. Each sampling interval is assigned a $\\beta\\,$-distributed weight whereby the $\\alpha$-parameter of the weight distribution is increased with each $l$-th failed iteration to favor short sampling intervals. The resulting surrogates preserve the empirical distributions and segment sizes. Since amplitude differences are sampled jointly with the respective sampling intervals, increased volatility simply due to a higher local \nsampling rate is reproduced by the SRC-surrogates.}\n\\label{fig3}\n\\end{figure*}\n\\FloatBarrier\nWe expect DET to reproduce the linear increase in autocorrelation time, because increased serial dependence implies longer \nand more diagonal lines in the RP.\nFor the computation of the (m)Edit-distance measure, segments are picked such that each covers a constant time interval of \n$w = 1$ which could correspond to a year in a real-world application. 200 SRC-surrogates are generated (see appendix B) with $\\alpha_0=1, \\beta=1$ and a step size for the shape parameter \n$\\alpha$ of the beta-distribution $\\Delta\\alpha=0.15$.\nWe set an upper limit of $N_{\\mathrm{it}}^{(\\mathrm{max})}=1000$ for \nthe number of iterations in the generation of each segment which is never exceeded in the performed simulations. \nThe deletion\/adding cost parameter $\\Lambda_S$ is estimated separately for the real time series and the surrogate realizations, yielding $\\Lambda_S^{(\\mathrm{real})} = 5.3$ and $\\Lambda_S^{(\\mathrm{SRC})} = 2.6$. Recurrence plots are \ncomputed on sliding windows of size $s=200\\Delta$ with $75\\%$ overlap (time series length: $T=5000$).\nWe fix a recurrence rate of $15\\%$ and do not apply any time-delay embedding.\nFor each window, two DET values are obtained (Fig.~\\ref{fig5}c): the DET value of the real time series and the \n$\\alpha(=95\\%)$-confidence \nlevel of DET values calculated from the SRC-surrogate ensemble. The DET measure indicates a \nspurious transition of predictability induced by the abrupt shift in sampling rate (Fig.~\\ref{fig5}c, grey shading). Both the real time series \nand the surrogate ensemble indicate this shift, demonstrating that the proposed SRC-surrogates effectively reproduce the \nsampling bias. \n\\begin{figure*}[]\n\\centering\n\\includegraphics[width=.7\\textwidth]{fig4.pdf}\n\\caption{Application of SRC-surrogate correction method to \\textbf{(a)} an irregularly sampled AR(1)-process with \n\\textbf{(b)} non-stationary sampling rate (blue) and linearly increasing autocorrelation time (red). Gray shading indicates \nthe abrupt shift in sampling rate. \\textbf{(c)} A sliding window RQA using determinism (DET) as a predictability measure is \ncarried out. Real DET values are displayed in dark blue, the $95\\%$-confidence level computed from 200 SRC-surrogates is \nshown in yellow. \\textbf{(d)} The ratio $\\mathrm{DET}_{\\mathrm{real}}$ by $\\mathrm{DET}_{\\mathrm{surr}}$ provides a \nsampling-bias corrected predictability measure that reproduces the linear increase in serial dependence.}\n\\label{fig4}\n\\end{figure*}\nThe SRC-based correction is applied to DET values by dividing the real \nDET-series \nby the $95\\%$-confidence level for each window (Fig.~\\ref{fig5}d). The resulting predictability estimates reproduce the expected linear \nincrease in serial dependence whilst eliminating the spurious shift due to the jump of the sampling rate.\n\\section{Real-world application: rainfall seasonality in the central Pacific}\n\\label{sec5}\nMany real-world proxy time series are characterized by irregular sampling or missing data and stationarity of the underlying process that controls the \nsampling rate cannot be guaranteed. This perspective even goes beyond uneven time axes as for some systems, it might be \ndesirable to apply an adaptive windowing in order to obtain segments with segment sizes depending on specific parameters of \nthe system. For instance, when analyzing cardiac time series it might be reasonable to choose the segment size adaptively to capture one heart-beat cycle within each segment. The length of every cycle is controlled by a variety of other physical, non-stationary parameters.\nBelow, we focus on an irregularly sampled palaeoclimate proxy time series with a non-stationary temporal sampling rate.\nWe demonstrate the effectiveness of the proposed approach by carrying out a sliding window recurrence analysis.\n\nThe palaeoclimate record analysed here is a seasonally resolved stalagmite proxy record from Niue Island in the southwestern Pacific (19$^{\\circ}$S, 169$^{\\circ}$W). It covers 1000 years in the mid-Holocene (6.4-5.4 thousand years before present (ka BP)). \nNiue island has a tropical climate, receiving an average of 2000 mm of precipitation annually with a pronounced wet season from November to April. Rainfall is most strongly controlled by seasonal displacement of the South Pacific Convergence Zone but also reacts sensitively to atmospheric circulation changes associated with the El Ni\u00f1o-Southern Oscillation. \nHere, we analyse seasonal rainfall variability on Niue recorded in greyscale changes that arise from crystallographic variations caused by changes in the stalagmite growth rate (Fig.~\\ref{fig6}). Greyscale values are obtained from high resolution scans of the stalagmite surface along its growth axis subsequently extracted with ImageJ \\cite{abramoff2004image}. \nDuring the dry season, low drip rates promote the deposition of layers with compact and dark crystals, yielding low greyscale values. In the wet season, the drip rates are higher and crystal growth is enhanced as dissolved inorganic carbon is supplied to a greater extent. (see Fig.~\\ref{fig6}). The inferred link between dark layers and dry season is supported by earlier studies \\cite{aharon2006caves} \\citep{CINTHYA}.\n\nPrior to the recurrence analysis of the greyscale time series, we \nsubtracted a centennial-scale trend using a Gaussian kernel filter in order tp focus on the high-frequency variability in the record (Fig.~\\ref{fig6}a, black line). Next, we downsampled the time series uniformly by only storing every 10$^\\text{th}$ value due to computational constraints. \nThis downsampling does not alter the relative changes in the sampling rate (Fig.~\\ref{fig6}b). The number of samples per year (i.e., the segment \nsize) undergoes an abrupt shift at $\\approx\\,$6.15~ka~BP. The period with the highest average segment size ($\\approx\\,$ 6.4 to 6.15~ka BP) coincides with the wettest period covered by the record, indicated by high greyscale values. \nThis suggests that during this wet period, stalagmite growth was enhanced which resulted in thicker crystal laminae and a higher number of samples per layer. This observation reflects the complex nature of irregular sampling of palaeoclimate-proxy data. If spatial \nsampling on the stalagmite is performed such that the number of samples is as high as possible, it will inevitably be linked \nto its growth rate and thus to other environmental parameters and their non-stationary characteristics.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.7\\textwidth]{fig5}\n\\caption{Greyscale record (in black) extracted from a high resolution scan of the surface of the stalagmite C132 from Niue island. Lower grey values are associated with dense microcrystalline calcite layers that form during drier periods.}\n\\label{fig5}\n\\end{figure}\nFinally, we perform the recurrence analysis (Fig.~\\ref{fig6}c). In order to characterize seasonal features, the period covered by one segment is fixed as one year. Optimization of deletion\/adding costs yields $\\Lambda_s = 2$. A window size of $s = \n200\\,\\mathrm{years}$ is chosen with $90\\%$ overlap. A recurrence plot with fixed recurrence rate of $15\\%\\,$ is obtained for each window \nand analyzed with DET. DET reveals variations in seasonal-scale predictability for the \nreal greyscale record (Fig.~\\ref{fig6}c, blue line). The effect of the varying sampling size is obtained by\nthe $95\\%\\,$-quantile of the DET distribution from 200 SRC-surrogates (Fig.~\\ref{fig6}c, yellow line). Five exemplar SRC-surrogate realizations are shown in appendix B.\nBoth DET time series indicate an increase of seasonal-scale predictability during the wet period between 6.35 to 6.2 ka BP, potentially \ncaused by the simultaneously increased sampling rate. The predictability estimate is corrected for the identified sampling bias by considering\nthe ratio $\\mathrm{DET}_\\mathrm{r} \/ \\mathrm{DET}_\\mathrm{surr}$ (Fig.~\\ref{fig6}d).\nTwo periods (6.4 and 6.2 ka BP, and between 5.9 and 5.72 ka BP) show \nrelatively low segment size-corrected seasonal \npredictability $\\left. \\mathrm{DET}_{\\mathrm{r}} \\middle\/ \\mathrm{DET}_{\\mathrm{surr}} \\right. < 1$.\nWhile the latter is not significantly affected by the correction, the former can only be \nidentified as less predictable when the variations in sampling rate are taken into account. This result corroborates \nprevious findings that suggested that both of these identified periods were more irregular, i.e. showing less steady seasonal fluctuations \\citep{CINTHYA}. \nHowever, it was not possible to characterize all sub-annual values as a proxy for sub-annual rainfall \ndistribution rather extracting only the contrast between wet and dry season. The (m)Edit-distance approach employed here in combination \nwith the proposed correction technique allows for a more reliable interpretation of mid-Holocene seasonal variations in the \nwest Pacific. \n\nIn particular, an enhanced control of the seasonal cycle by ENSO-scale variability was found in \\citep{CINTHYA} for the periods of reduced predictability (6.4 and 6.2 ka BP, and between 5.9 and 5.72 ka BP). High tropical cyclone activity between 6.4-6.2 ka BP could have been triggered by increased ENSO activity, yielding a more irregular sub-annual distribution of rainfall. Our results indicate that not only contrast between both seasons is rendered less predictable during this period but also the seasonal rainfall distribution appears less stable from one year to another. Reconstructing past climate variability at seasonal scale plays a critical role in the context of human adaption to continuous and abrupt climate variations and therefore our approach has direct relevance for teasing out the seasonal-scale signals.\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=.75\\textwidth]{fig6}\n\\caption{Application of SRC-surrogate correction method to an irregularly sampled greyscale proxy-record from the central \nPacific \\textbf{(a)} with non-stationary sampling rate (blue) \\textbf{(b)}. A sliding window RQA using determinism (DET) as \na predictability measure is carried out \\textbf{(c)}. Real DET values are displayed in dark blue, the $95\\%\\,$-confidence \nlevel computed from 200 SRC-surrogates is shown in yellow. Dividing $\\mathrm{DET}_{\\mathrm{real}}$ by \n$\\mathrm{DET}_{\\mathrm{surr}}$ \nyields a sampling-bias corrected predictability series \\textbf{(d)} that indicate mid-Holocene variations \nin seasonal-scale predictability. Gray shading indicates two periods with low seasonal predictability.}\n\\label{fig6}\n\\end{figure*}\n\n\n\\section{Conclusion}\n\\label{sec6}\nThe characterization of time series from complex nonlinear systems is a challenging task. Irregular sampling, \ni.e., variations in the sampling interval between consecutive values, additionally impedes typical research objectives such \nas spectral analysis, persistence estimation or quantifying the predictability of a system. Even though interpolation \ntechniques offer a seemingly efficient way of pre-processing a time series to allow application of standard \ntime series analysis tools, these entail various biases which are difficult to control. A different perspective is pursued by \nthe (m)Edit-distance method. Many analysis methods are based on an estimate of (dis)similarity. With the \n(m)Edit-distance, \na suitable distance measure between states of a system at different times $i$ and $j$ is defined by computing the \ntransformation cost of segments centered at these time instances. First analyses demonstrated  its scope in the context of \nrecurrence analysis, enabling researchers to examine predictability variations of irregularly sampled palaeoclimate time \nseries. Applications to other complex systems (also for time series with `missing values') and other methodological \nframeworks (e.g., complex networks, clustering, correlation analysis, \\ldots) are possible and should \nbe attempted in the near future.\n\nFor some real-world systems, it is instructive to quantitatively compare sequences corresponding to a specific time scale in \norder to analyse the scale-specific variability. In such cases, segment sizes will vary in the presence of irregular sampling. \nFurthermore, splitting time series with a non-stationary sampling rate into segments that do not cover the same time period \nwill result in a mixing of time scales.\nWe have shown that (m)Edit-distance-based recurrence analyses are affected by variations in segment sizes, resulting in a \nnon-trivial sampling bias if episodes with variable sampling rate are included in a single RP. The (m)Edit distance regards \npairs of longer segments to be generally more dissimilar than shorter segments due to higher deletion costs. Shifting costs \nconversely decrease for increasing segment size, resulting in a non-trivial dependence of costs on local sampling rates. \nWhen including amplitudes of a signal into the (m)Edit-distance computation, similar general tendencies \nwere observed but the strength of the segment size-dependencies varied for different systems. A more detailed examination of how \ndissimilar amplitude segments of different paradigmatic systems depend on their time scale will be investigated in \na future study.\n\nWe introduced a numerical technique based on constrained randomization to address and correct the issue of segment-size dependence in \nrecurrence analysis.\nThis method involves generating an ensemble of sampling rate constrained surrogate realizations (SRC-surrogates). Each SRC-surrogate \nreproduces the real variations of sampling rate and its assignment to the corresponding amplitude differences, allowing the \nensemble to be used for correcting the undesired segment size-dependence. The effectiveness of the proposed correction was \napplied to a synthetic AR(1)-time series and real palaeo-proxy data. In both applications, a recurrence analysis \nsuccessfully recovered variations in the scale-specific predictability of the system whilst discarding spurious effects \nimprinted by sampling rate variations. We found that seasonal-scale predictability varied significantly during the mid-Holocene in the \nWest Pacific, corroborating and extending the results from a recent study.\nThe reasons for these changes in predictability warrant further investigation.\n\nThe identified sampling bias is a specific case of a more general problem in time series analysis; \nsliding window analyses (or the study of short time series) often suffer from finite-sampling biases which may introduce \nartificial variability into any statistical indicator that is computed. As pointed out in Section \\ref{sec5}, finite-sampling biases are also not limited to irregular temporal sampling but are likely to also occur in settings where other parameter axes \ndetermine suitable window sizes or adaptive windowing is required. In future, the proposed method could also be \nappliedd in such settings to test its effectiveness beyond the examples considered in this study.\n\\section*{Code availability}\nPython code for the generation of SRC-surrogates is available at \\url{https:\/\/github.com\/ToBraun\/SRC-surrogates}. \n\\section*{Acknowledgements}\nThis research was supported by the\nDeutsche Forschungsgemeinschaft in the context of the DFG\nproject MA4759\/11-1 'Nonlinear empirical mode analysis of\ncomplex systems: Development of general approach and appli-\ncation in climate'. It also received financial support from the European Union's Horizon 2020 Research and Innovation programme (Marie Sklodowska-Curie grant agreement No. 691037).\nDeniz Eroglu acknowledges funding by T\\\"UB\\.ITAK (Grant No.~118C236) and the BAGEP Award of the Science Academy.\nCinthya Nava-Fernandez acknowledges financial support from the German Academic Exchange Service (DAAD). AH also acknowledges support from the Royal Society of New Zealand (grant no. RIS-UOW1501), and the Rutherford Discovery Fellowship programme (grant no. RDF-UOW1601).\n\n\\section*{Conflict of interest}\nThe authors declare that they have no conflict of interest.\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nIt is widely known that the BPS (self-dual) solitons in (1+1) dimensions are \nintimately related to the existence of a $\\mathcal{N}=1$ supersymmetric extension \nof the bosonic model. Here, by a BPS solution we understand a static solution of the so-called Bogomolnyi equations (which are of lower order than the Euler-Lagrange (EL) equations), which\nsaturates a pertinent topological energy bound. This guarantees the topological stability of the\nsolution for topologically nontrivial configurations (kinks). Furthermore, the BPS sector is equivalent to the zero pressure sector, as the Bogomolnyi equation is just the zero pressure condition. \n\nSupersymmetry provides a systematic tool for the derivation and analysis of such Bogomolnyi\nequations. Indeed, they can be obtained from the $\\mathcal{N}=1$ supersymmetric transformations of the fermions. Then, the self-dual sector (supporting kinks with topological charge $Q=1$) is invariant under one-half of the supersymmetry. Of course, the same happens with the anti self-dual sector. \n\nFurthermore, the model allows for a central extension,  where the central charge is the difference of the values of the superpotential at asymptotical values of the fields (vacua). The superpotential is related to the potential of the bosonic part of the theory by a target space differential equation.\nHence, for topologically nontrivial solutions (BPS solitons), the central charge takes a nonzero value.  \n\nAll these properties concern not only a scalar field theory with the standard kinetic term and an arbitrary (at least two vacua) potential but can be generalized to a quite arbitrary target space \\cite{adam-santamaria} (multi-field and curved target space) as well as to models with nonstandard derivative terms \\cite{k-model-1}-\\cite{k-susy-7}. In fact, very recently it has been proved that the BPS property is shared even by theories with higher derivatives and, therefore, is a generic feature of {\\it all} translational invariant scalar field Lagrangians in (1+1) dimensions \\cite{higher-der}. Of course, this is at odds with higher dimensional models, where only very few models enjoy the BPS property. \n\nOne could ask the obvious question of how to break the BPS-ness in (1+1) dimensions. This is possible but requires a quite drastic modification of the action, i.e., the addition of a term which breaks the translational invariance of the model, that is, an impurity (defect). Then, typically, no Bogomolnyi equations exist and solitons are solutions of the full EL equations. However, it has been found that there is a very special coupling of the defect which preserves  {\\it one-half} of the BPS-ness \\cite{higher-der}. This means that a kink (or antikink) is a BPS solution (solving a Bogomolnyi equation and saturating the pertinent bound) while its topological charge conjugated partner, i.e., the antikink (kink) does not have this property. \n \n In the present work, we want to understand these half-BPS soliton-impurity standard scalar theories in (1+1) dimensions from a supersymmetric point of view. In particular, we will show that the Bogomolnyi equation again emerges via a supersymmetry transformation of fermions, leading to an invariance of the BPS sector under one-half of the SUSY. Similarly, we will obtain a central charge extension which, however, possesses only one nontrivial supercharge. Also the fermionic and bosonic zero modes coincide. This further explains the existence of the generalized translational symmetry of the BPS soliton. Finally, our approach allows for a derivation of a whole family of impurity deformed models which preserve one-half of the BPS-ness (and in the limit of the vanishing impurity reduce to the original scalar soliton BPS model). All such extensions preserve $1\/2$ of the original $\\mathcal{N}=1$ supersymmetry.\n \n \\hspace*{0.2cm}\n \nAs $\\mathcal{N}=1$ supersymmetry in (1+1) and (2+1) dimensions have basically the same structure, all our findings can be generalized to the (2+1) case. This gives us a chance to understand half-BPS preserving impurities in a unified way. Using these results, we construct half-BPS impurity extensions for the baby BPS Skyrme model. The Lagrangian with the impurity presents the same type of couplings as the scalar model in (1+1) dimensions. The only difference is the particular form of the topological current used in the construction, which for (1+1) gives the usual kinetic term while for the baby BPS Skyrme model is just the topological degree current. As a result, we get a model which preserves $1\/4$ of $\\mathcal{N}=2$ sypersymmetry. \n\nInterestingly, such a partially BPS preserving coupling of the impurity to a BPS model resembles in many aspects the partially BPS Abelian Higgs model at critical coupling with a magnetic impurity \\cite{vortices}. In both theories, only half of the solitons enjoy the BPS property (are solutions of the pertinent Bogomolnyi equations) while the other half obey the full EL equations. Hence, only the former ones saturate the topological bound. Moreover, the impurity enters the Bogomolnyi equations of the original (no impurity) model as an inhomogenous term. Finally, the action requires the appearance of a coupling between a 'topological object' (the topological density or the magnetic field, respectively) and the impurity. \n \nEven more interestingly, we find the half-BPS preserving coupling of the impurity to the $CP^1$ model. In this case, the original Bogomolnyi equations, i.e., the Cauchy-Riemann (or anti Cauchy-Riemann) equations, are replaced by their non-homogeneous versions where the inhomogeneity is just the impurity. \nThis implies the complete solvability in the BPS sector (which hosts half of the solitons of the original model).  \nThis result will enable us to introduce another impurity-Abelian Higgs model where  one-half of the BPS-ness is preserved. This is a different construction than the original one presented in \\cite{vortices}. \n \n  \\hspace*{0.2cm}\n\nThe last comment concerns our terminology. All  impurity models presented here are theories where one-half of the solitons are still BPS objects, so frequently we call them half-BPS soliton-impurity models. However, for the sake of simplicity sometimes we call them just BPS soliton-impurity models.\nA related but different issue is the amount of SUSY preserved by the SUSY extensions of the impurity models.  Concretely, in the case of $\\mathcal{N}=1$ SUSY always $1\/2$ of the SUSY is preserved by the impurity BPS models, whereas in the $\\mathcal{N}=2$ case both the preservation of $1\/4$ or of $1\/2$ of the supersymmetry are possible. We shall always denote these by $1\/4$ SUSY and $1\/2$ SUSY, respectively.\n\n\n\n\\section{The BPS preserving impurity in the scalar model in (1+1) dim}\n\n\\subsection{The BPS property from supersymmetry}\n\\label{BPS}\n\nWe will focus on $\\mathcal{N}=1$ SUSY in $d=1+1$ dimensions. The modified Lagrangian with impurity $\\sigma$ preserving one-half of the BPS property of the original model (without impurity) has the following form \\cite{higher-der}\n\\begin{equation}\n\\mathcal{L}=\\frac{1}{2}\\partial_\\mu\\phi\\partial^\\mu\\phi-U-2 \\sigma \\sqrt{U}-\\sqrt{2}\\sigma \\phi_x-\\sigma^2. \\label{sec:intro:imp}\n\\end{equation}\nHere $U$ is a (at least) two vacuum potential and $\\phi_x\\equiv \\partial_x \\phi$. \nWe will give an explanation, based on a possible SUSY breaking, for the preservation of the BPS property. The first two terms in (\\ref{sec:intro:imp}) have a simple SUSY extension\n\\begin{equation}\n\\mathcal{L}_{0}=\\frac{1}{4}\\int d^2 \\theta D^\\alpha \\Phi D_\\alpha \\Phi+\\int d^2\\theta \\,W(\\Phi) \\vert= \\frac{1}{2}\\partial_\\mu \\phi\\partial^\\mu\\phi+\\frac{1}{2}F^2+W_\\phi(\\phi)F\n\\end{equation}\nwhere $\\vert$ means setting $\\psi_\\alpha=0$. $\\Phi$ is a scalar superfield and $D_\\alpha$ is a superderivative, whose components are\n\\begin{eqnarray}\n\\Phi&=&\\phi+\\theta^\\alpha \\psi_\\alpha-\\theta^2 F,\\\\\nD_\\alpha&=&\\frac{\\partial}{\\partial \\theta^\\alpha}+i\\theta^\\beta \\partial_{\\alpha\\beta}.\n\\end{eqnarray}\nSince for (\\ref{sec:intro:imp}) only the kink (or the antikink) are BPS solutions of the model, one should not expect to have a superfield formulation of the full model with impurity. If this were the case, then the existence of a BPS kink solution would imply the existence of the antikink and vice versa. Let us consider the following term\n\\begin{equation}\n\\mathcal{L}_{impurity}^1=\\sqrt{2}\\phi_x \\sigma.\\label{sec:intro:imps}\n\\end{equation}\nTaking into account the supersymmetric transformations of the fields\n\\begin{eqnarray}\n\\delta\\phi&=&-\\epsilon^\\alpha \\psi_\\alpha,\\\\\n\\delta\\psi_\\beta&=&-\\epsilon^\\beta\\left(C_{\\alpha\\beta} F+i\\partial_{\\alpha\\beta}\\phi\\right),\\label{sec:intro:tpsi}\\\\\n\\delta F&=&-\\epsilon^\\alpha i\\partial_\\alpha^{\\,\\,\\beta}\\psi_\\beta, \\label{sec:intro:tF}\n\\end{eqnarray}\nwe have\n \\begin{equation}\n \\delta \\mathcal{L}_{impurity}^1=-\\sqrt{2}\\sigma (\\epsilon^\\alpha \\psi_{\\alpha,x}).\\label{sec:intro:sigma1}\n \\end{equation}\n If the impurity $\\sigma$ is trivial ($\\sigma_x=0$), then (\\ref{sec:intro:sigma1}) is a total derivative and SUSY is preserved. If $\\sigma_x\\neq0$, this term has to be compensated in order to preserve (a part of) supersymmetry. The transformation (\\ref{sec:intro:tF}) suggests the addition of the following term\n  \\begin{equation}\n\\mathcal{L}_{impurity}^2=\\sqrt{2}\\sigma F.\\label{sec:intro:impF}\n \\end{equation}\n The combination of (\\ref{sec:intro:imps}) and (\\ref{sec:intro:impF}) gives the following SUSY (static) transformations\n \\begin{eqnarray}\n\\frac{1}{\\sqrt{2}} \\delta ( \\mathcal{L}_{impurity}^1+ \\mathcal{L}_{impurity}^2)&=&-2\\sigma \\epsilon^2\\psi_{2,x}-\\sigma\\epsilon^1\\psi_{2,t}+\\sigma\\epsilon^2\\psi_{1,t}, \\label{sec:intro:varp}\\\\\n\\frac{1}{\\sqrt{2}} \\delta ( \\mathcal{L}_{impurity}^1- \\mathcal{L}_{impurity}^2)&=&- 2\\sigma \\epsilon^1\\psi_{1,x}-\\sigma\\epsilon^2\\psi_{1,t}+\\sigma\\epsilon^1\\psi_{2,t}.\\label{sec:intro:varm}\n \\end{eqnarray}\n \n The conclusion is as follows: if we add $\\mathcal{L}_{impurity}^1+ \\mathcal{L}_{impurity}^2$ to the Lagrangian then $1\/2$ SUSY is preserved (provided that $\\epsilon^2=0$), while the combination $ \\mathcal{L}_{impurity}^1- \\mathcal{L}_{impurity}^2$ preserves $1\/2$ SUSY (if $\\epsilon^1=0$). Phrased differently, the addition of the impurity breaks explicitly one-half of the supersymmetry generators (one real Grassmann degree for $\\mathcal{N}=1$ in $d=1+1$). We have therefore\n \\begin{eqnarray}\n  \\delta ( \\mathcal{L}_{impurity}^1+ \\mathcal{L}_{impurity}^2)\\vert_{\\epsilon^2=0}&=&-\\sqrt{2}\\epsilon^1\\partial_t (\\sigma \\psi_2)\\label{sec:intro:cur}\\\\\n   \\delta ( \\mathcal{L}_{impurity}^1- \\mathcal{L}_{impurity}^2)\\vert_{\\epsilon^1=0}&=&-\\sqrt{2}\\epsilon^2\\partial_t (\\sigma \\psi_1)\\label{sec:intro:cur2}\n \\end{eqnarray}\n The total Lagrangian $\\mathcal{L}_{tot}=\\mathcal{L}_{0}-\\mathcal{L}_{impurity}^1+ \\mathcal{L}_{impurity}^2$ has the following form in components\n \\begin{equation}\n \\mathcal{L}_{tot}\\vert=\\frac{1}{2}\\partial_\\mu \\phi\\partial^\\mu\\phi+\\frac{1}{2}F^2+W_\\phi(\\phi)F-\\sqrt{2}\\phi_x \\sigma+\\sqrt{2}\\sigma F.\n \\end{equation}\n  After eliminating $F$ ($F=-W_\\phi(\\phi)-\\sqrt{2}\\sigma$) and for $\\phi_t=0$ the on-shell Lagrangian reads\n \\begin{equation}\n  \\mathcal{L}_{tot,on-shell}\\vert=-\\frac{1}{2}\\phi_x^2-\\frac{1}{2}W_\\phi^2(\\phi)-\\sqrt{2}W_\\phi(\\phi)\\sigma-\\sigma^2-\\sqrt{2}\\phi_x\\sigma \\label{sec:intro:fullL}\n \\end{equation}\n We recall that (\\ref{sec:intro:fullL}) is invariant under one-half of the SUSY transformations (for $\\epsilon^1=0$). As usual, the BPS equation can be obtained from (\\ref{sec:intro:tpsi})\n \\begin{eqnarray}\n \\delta\\psi_1&=&i\\epsilon^2(F-\\phi_x)\\label{sec:intro:varpsi1},\\\\\n  \\delta\\psi_2&=&i\\epsilon^1(-F-\\phi_x)\\label{sec:intro:varpsi2}.\n \\end{eqnarray}\n Since in order to preserve one-half of SUSY at the level of the Lagrangian we have to impose $\\epsilon^1=0$, the condition $ \\delta\\psi_2=0$ is automatically satisfied. From (\\ref{sec:intro:varpsi1}) we obtain the condition\n \\begin{equation}\n F-\\phi_x=0, \\,\\,\\,\\text{or}\\,\\,\\, \\phi_x=-W_\\phi(\\phi)-\\sqrt{2}\\sigma \\label{sec:intro:BPS}.\n \\end{equation}\n It is easy to check that (\\ref{sec:intro:BPS}) implies the Euler-Lagrange equations for (\\ref{sec:intro:fullL}). Also, under the replacement\n \\begin{equation}\n W_\\phi(\\phi)\\rightarrow \\sqrt{2 U}\n \\end{equation}\nthe Lagrangian (\\ref{sec:intro:fullL}) corresponds to (\\ref{sec:intro:imp}).\n\n\nThe specific form of introducing the impurity preserving the BPS property gives us some hints on how to generalize this result to other models. Since the impurity breaks the translation invariance, one should expect that, if the model preserves part of the SUSY, the superalgebra contains only time translations (see Sec. \\ref{central}). Now let us assume that the original model is BPS. From the SUSY point of view, this implies that the fermionic transformations (after a proper reduction of the parameter space) are either time derivatives or proportional to the BPS equations. In the previous case we had\n\\begin{eqnarray}\n\\delta_\\epsilon\\psi_1&=&i\\epsilon^2(F-\\phi_x), \\label{sec:intro:fert1}\\\\\n\\delta_\\epsilon\\psi_2&=&-i\\epsilon^2\\phi_t.\\label{sec:intro:fert2}\n\\end{eqnarray}\nOn the other hand we have, in general \n\\begin{equation}\n[\\delta_\\epsilon,\\delta_\\eta]X=-2i \\epsilon^\\alpha\\eta^\\beta\\partial_{\\alpha\\beta}X.\n\\end{equation}\nTaking into account (\\ref{sec:intro:fert1}) we have\n\\begin{equation}\n[\\delta_\\epsilon,\\delta_\\eta]\\psi_1=2\\delta_\\epsilon\\delta_\\eta\\psi_1=2i \\eta^2 \\delta_\\epsilon(F-\\phi_x)=-2i\\eta^2\\epsilon^2\\psi_{1,t}\n\\end{equation}\nTherefore, after the reduction to the BPS space ($\\epsilon^1=0$), the transformation of the off-shell BPS equation is a time derivative. This property can be stated as follows: the addition of a term of the form $\\sigma(x)\\Sigma$ (where $\\Sigma=0$ is the off-shell BPS equation) to a SUSY BPS Lagrangian preserves the amount of supersymmetry preserved by the BPS solutions of the original model and, as a consequence, the BPS property.\n\n\n\\subsection{The central charge}\n\\label{central}\n\n\nLet us consider again the model (\\ref{sec:intro:imp}). As we discussed above, this model breaks explicitly one real supersymmetric generator. The supercharge is defined as follows\n\\begin{equation}\nQ_\\alpha=\\int dx\\, J_\\alpha^0,\n\\end{equation}\nwhere $J_\\alpha^\\mu$ is the supercurrent. The model (\\ref{sec:intro:imp}) is $1\/2$-supersymmetric provided that $\\epsilon^1=0$ and this implies that $J_1^0=0\\rightarrow Q_1=0$. Thus we have\n\\begin{eqnarray}\n&&\\lbrace Q_2,Q_2\\rbrace=2 P_0-2Z,\\label{sec:central:alge}\\\\\n&&\\lbrace Q_1,Q_1\\rbrace=0,\\\\\n&&\\lbrace Q_1,Q_2\\rbrace=\\lbrace Q_2,Q_1\\rbrace=0.\n\\end{eqnarray}\nNow, the impurity only adds an extra term proportional to the supercurrent (\\ref{sec:intro:cur2}). A direct computation shows that\n\\begin{equation}\nQ_2=\\int dx\\left(\\partial_t\\phi\\,\\psi_1+\\partial_x\\phi\\,\\psi_2+\\frac{1}{2}(F+W_\\phi+\\sqrt{2}\\sigma)\\,\\psi_2\\right).\\label{sec:central:charge}\n\\end{equation}\nWith the (anti)commutation relations\n\\begin{equation}\n[p(x),\\phi(y)]=i\\delta(x-y),\\,\\,\\, \\lbrace \\psi_\\alpha(x),\\psi_\\beta(y)\\rbrace=\\delta_{\\alpha\\beta}\\delta(x-y).\n\\end{equation}\nExplicitly\n\\begin{eqnarray}\nP_0&=&\\int dx\\left(\\frac{1}{2}\\phi_t^2+\\frac{1}{2}\\phi_x^2+i\\psi_{2,x}\\psi_1+\\frac{1}{2}W_\\phi^2+\\sqrt{2}W_\\phi\\sigma+\\sigma^2+\\sqrt{2}\\phi_x \\sigma\\right)\\\\\nZ&=&\\int dx \\phi_x W_\\phi=W(\\phi)\\vert_{x=\\infty}-W(\\phi)\\vert_{x=-\\infty}\n\\end{eqnarray}\nSince\n\\begin{equation} \n\\lbrace Q_2,Q_2\\rbrace\\geq 0,\n\\end{equation}\nwe have from (\\ref{sec:central:alge}) that the energy of a state $\\vert \\phi \\rangle$ verifies the following inequality\n\\begin{equation}\nP_0 \\geq Z.\n\\end{equation}\nThis relation is clearly saturated for $Q_2\\vert \\phi \\rangle=0$. For static solutions, this condition is equivalent to the equation (\\ref{sec:central:charge})\n\\begin{equation}\n\\phi_x-F=0\\rightarrow \\phi_x=-W_\\phi-\\sqrt{2}\\sigma,\n\\end{equation}\nwhich is the BPS equation previously obtained. Note that if in the original model we replace $\\sigma\\rightarrow-\\sigma$ the SUSY algebra becomes\n\\begin{eqnarray}\n&&\\lbrace Q_2,Q_2\\rbrace=0,\\label{sec:central:alge1}\\\\\n&&\\lbrace Q_1,Q_1\\rbrace=2 P_0-2Z,\\\\\n&&\\lbrace Q_1,Q_2\\rbrace=\\lbrace Q_2,Q_1\\rbrace=0.\n\\end{eqnarray}\nAfter a convenient change $W_\\phi \\rightarrow -W_\\phi$ we get the saturating condition $\\phi_x= W_\\phi+\\sqrt{2}\\sigma$.\n\n\n\n\n\n\\subsection{The zero modes}\n\\subsubsection{The fermionic zero modes}\n\\label{ferm}\n\nLet us consider the following model\n \\begin{equation}\n \\mathcal{L}_{tot}\\vert=\\frac{1}{2}\\partial_\\mu \\phi\\partial^\\mu\\phi+\\frac{i}{2}\\psi^\\alpha\\partial_\\alpha^{\\,\\,\\beta}\\psi_\\beta+\\frac{1}{2}F^2+W_\\phi(\\phi)F-\\sqrt{2}\\phi_x \\sigma+\\sqrt{2}\\sigma F+\\frac{1}{2}W_{\\phi \\phi} (\\phi)\\psi^\\alpha\\psi_\\alpha.\\label{sec:ferm:full}\n \\end{equation}\nThe fermion zero mode equations are given by\n \\begin{eqnarray}\n\\partial_x \\psi_2-W_{\\phi \\phi}(\\phi)\\psi_2&=&0,\\label{sec:ferm:fer1}\\\\\n\\partial_x  \\psi_1+W_{\\phi \\phi}(\\phi)\\psi_1&=&0.\\label{sec:ferm:fer2}\n \\end{eqnarray}\nIn the standard case (without impurity), the fermionic zero mode is simply the derivative of the solitonic solution which obviously represents the translational zero mode of the soliton. Hence, the only normalizable solution for the soliton reads\n\\begin{equation}\n\\psi_\\alpha=\\left(\n\\begin{matrix}\n\\phi^s_x(x)\\\\\n0\n\\end{matrix}\\right),\n\\end{equation}\nwhile for the antisoliton \n\\begin{equation}\n\\psi_\\alpha=\\left(\n\\begin{matrix}\n0\\\\\n\\phi^a_x(x)\n\\end{matrix}\\right).\n\\end{equation}\nIn the presence of the impurity, the fermion zero mode equations are still given by (\\ref{sec:ferm:fer1}) and (\\ref{sec:ferm:fer2}) but there is only one (modified) BPS equation $\\phi_x=-W_{\\phi}-\\sqrt{2}\\sigma$. This leads to the following fermionic zero mode\n\\begin{equation}\n\\psi_\\alpha=\\left(\n\\begin{matrix}\n\\phi_x (x)\\exp\\left(\\sqrt{2} \\int_0^xdx\\frac{\\sigma_x(x)}{\\phi_x(x)}\\right)\\\\\n0\n\\end{matrix}\\right).\n\\end{equation}\n\n\\subsubsection{The bosonic zero modes}\n\\label{bos}\n\nThe linear fluctuation equation in the kink\/antikink background can be derived by inserting the decomposition $\\phi(t,x)=\\phi_c(x)+\\cos(\\omega t)\\eta(x)$, where $\\phi_c$ is a kink\/antikink solution. The resulting fluctuation equation is\n\\begin{equation}\n-\\eta_{xx}(x)+\\left(W_{\\phi }(\\phi_c)W_{\\phi \\phi \\phi}(\\phi_c)+W_{\\phi \\phi}(\\phi_c)^2+\\sqrt{2}W_{\\phi \\phi \\phi}(\\phi_c)\\sigma\\right)\\eta(x)=\\omega^2\\eta(x).\n\\end{equation}\nWe will show that the bosonic and fermionic zero modes coincide. Let us assume that $\\eta$ satisfies the fermionic zero mode equation (\\ref{sec:ferm:fer2})\n\\begin{equation}\n\\eta_x=-W_{\\phi \\phi} \\eta\n\\end{equation}\nwhere $\\phi$ is a BPS solution i.e., obeys\n\\begin{equation}\n\\phi_x=-W_\\phi-\\sqrt{2}\\sigma\n\\end{equation}\nThen, acting with $\\partial_x$ we get\n\\begin{equation}\n\\eta_{xx}=-W_{\\phi \\phi \\phi} \\phi_x \\eta - W_{\\phi \\phi} \\eta_x= W_{\\phi \\phi \\phi} (W_\\phi+\\sqrt{2}\\sigma) \\eta + W_{\\phi \\phi} W_{\\phi \\phi} \\eta\n\\end{equation}\nwhere we used the Bogomolnyi equation and the fermionic zero mode equation. But the last formula is exactly the bosonic mode equation. Hence, as one could expect, both modes exactly coincide. \n\n\\vspace*{0.2cm}\n\nHere we present an example of the BPS-impurity model which allows for the exact computation of the BPS soliton as well as the related zero mode. Let us take $W_\\phi=(1-\\phi^2)$ which corresponds to the $\\phi^4$ potential while the impurity is\n\\begin{equation}\n\\sigma (x)= \\frac{\\alpha-1}{\\sqrt{2}} \\frac{1}{\\cosh^2 \\alpha x}\n\\end{equation}\nwhere $\\alpha$ is a real parameter, $\\alpha \\neq 0$. Then, the Bogomolnyi equation $\\phi_x=-W_\\phi -\\sqrt{2}\\sigma$ has the following exact solution\n\\begin{equation}\n\\phi = -\\tanh \\alpha x\n\\end{equation}\nwhich is a kink (positive topological charge) for $\\alpha < 0$. For $\\alpha >0$ we get an antikink (negative topological charge). For $\\alpha =1$ we arrive at the usual $\\phi^4$ theory antikink.   \nThe zero mode can be also explicitly found and reads\n\\begin{equation}\n\\eta = - \\frac{\\alpha}{\\left(\\cosh \\alpha x \\right)^{2\/\\alpha}}.\n\\end{equation}\n\n\n\n\n\n\n\\subsection{Generalizing the BPS preserving impurity}\n\\label{gen}\n\n\nAs a matter of fact, the addition of the impurity preserving half of the BPS structure is not unique. As an example, let us assume the following, new impurity term \n\\begin{equation}\n\\mathcal{L}_{impurity}=-\\frac{1}{2}\\sigma \\phi_x^2 \\label{sec:exam:nont}\n\\end{equation}\nwhich is added to the standard bosonic, impurity free part of the model. As we will see, it is again possible to include some extra terms that will preserve a part of the BPS structure. The form of such new terms can be deduced using our supersymmetric approach. From the supersymmetric point of view, the impurity term in (\\ref{sec:exam:nont}) suggests a term of the form\n\\begin{equation}\n\\mathcal{L}_1=\\frac{1}{4}\\int d^2\\theta \\sigma D^\\alpha \\Phi D_\\alpha \\Phi.\n\\end{equation}\nBut because of the reasons discussed above, this term does not preserve supersymmetry, i.e., \n\\begin{equation}\n\\delta\\mathcal{L}_1\\vert_{\\epsilon^2=0}=\\sigma\\partial_x\\left(\\epsilon^1\\psi_1(\\phi_x+F)-\\epsilon^1\\psi_2 \\phi_t\\right)\\label{sec:gen:var},\n\\end{equation}\nwhich is not a total derivative. Note that the time component of (\\ref{sec:gen:var}) is a total derivative since $\\partial_t \\sigma(x)=0$. On the other hand, we introduce\n\\begin{equation}\n\\mathcal{L}_2=\\sigma_x\\phi(F+\\phi_x),\n\\end{equation}\nwith the following transformation properties\n\\begin{equation}\n\\delta\\mathcal{L}_2\\vert_{\\epsilon^2=0}=-\\sigma_x \\epsilon^1\\psi_1(F+\\phi_x)-\\sigma_x\\phi \\epsilon^1\\psi_{2,t}.\\label{sec:gen:var2}\n\\end{equation}\nAgain, if we combine (\\ref{sec:gen:var}) and (\\ref{sec:gen:var2}) we have the following term\n\\begin{equation}\n\\delta(\\mathcal{L}_1-\\mathcal{L}_2)\\vert_{\\epsilon^2=0}= \\epsilon^1\\partial_x\\left( \\sigma \\psi_1(F+\\phi_x)\\right)+\\epsilon^1\\partial_t (\\sigma_x\\phi\\psi_2).\n\\end{equation}\nAs a consequence, ($\\mathcal{L}_1-\\mathcal{L}_2$) preserves half of the supersymmetry. After including a linear $\\sigma$-model term and a superpotential, the full Lagrangian (for $\\phi_t=0$) reads\n\\begin{equation}\n\\mathcal{L}_{tot}\\vert=\\mathcal{L}_{0}+\\mathcal{L}_1-\\mathcal{L}_2\\vert=-\\frac{1}{2}\\phi_x^2\\left(1+\\sigma\\right)+\\frac{1}{2}F^2(1+\\sigma)+\\sigma_x\\phi(F+\\phi_x)+F W'(\\phi).\n\\end{equation}\nSolving for $F$\n\\begin{equation}\nF=\\frac{\\sigma_x \\phi-W'(\\phi)}{1+\\sigma},\\label{sec:exam:aux}\n\\end{equation}\nthe on-shell Lagrangian takes the form\n\\begin{equation}\n\\mathcal{L}_{tot, \\; on-shell}=-\\frac{1}{2}\\phi_x^2(1+\\sigma)-\\sigma_x\\phi_x\\phi-\\frac{1}{2}\\frac{\\left(W'(\\phi)-\\sigma_x\\phi\\right)^2}{1+\\sigma}.\\label{sec:exam:full}\n\\end{equation}\nFinally, taking into account (\\ref{sec:intro:tpsi}) and (\\ref{sec:exam:aux}), the BPS equation can be expressed as\n\\begin{equation}\n\\phi_x+\\frac{\\sigma_x \\phi-W'(\\phi)}{1+\\sigma}=0.\n\\end{equation}\n\nIt is clear from (\\ref{sec:exam:full}) that the addition of an impurity (in the last example simply, $\\sigma\\phi_x^2$) and the conservation of part of the BPS structure can lead to nontrivial  Lagrangians.  As we have shown here, the latter condition can be translated into a partial  explicit breaking of the SUSY generators of an underlying supersymmetric model.  As a consequence of this reasoning, one can construct a large family of (half) BPS preserving soliton-impurity models which in the limit $\\sigma \\rightarrow 0$ reproduce the pure (no impurity) model\n\\begin{equation}\n\\mathcal{L}_0=\\frac{1}{2}( \\partial_\\mu \\phi)^2 - U.\n\\end{equation}\nA general coupling to an impurity $\\sigma$ preserving half of the BPS-ness has the following form\n\\begin{equation}\n\\mathcal{L}_\\sigma=  \\frac{1}{2}H^2( \\partial_\\mu \\phi)^2 - \\frac{U}{H^2}  - G^2 - 2 \\frac{\\sqrt{U}G}{H} - \\sqrt{2}HG \\phi_x\n\\end{equation}\nwhere $H, G$ are functions of $\\phi$ and $\\sigma$ such that for the vanishing impurity i.e., when $\\sigma \\rightarrow 0$, $H(\\phi, \\sigma) \\rightarrow 1$ and $G(\\phi,\\sigma)\\rightarrow 0$ to recover the pure bosonic model. The static energy reads\n\\begin{eqnarray}\nE&=&\\int dx \\left[  \\frac{1}{2}H^2 \\phi_x^2 +\\frac{U}{H^2}  + G^2 + 2 \\frac{\\sqrt{U}G}{H} + \\sqrt{2}HG\\phi_x \\right] \\\\\n&=& \\int_{-\\infty}^\\infty dx \\left( \\frac{H}{\\sqrt{2}} \\phi_x + \\frac{\\sqrt{U}}{H} +G \\right)^2 - \\sqrt{2} \\int_{-\\infty}^\\infty dx \\phi_x \\sqrt{U} \\\\\n&\\geq&  - \\sqrt{2} \\int_{-\\infty}^\\infty dx \\phi_x \\sqrt{U} = - \\sqrt{2} \\int_{\\phi(-\\infty)}^{\\phi(\\infty)} d\\phi \\sqrt{U} =  - Q \\sqrt{2} \\int_{\\phi_+^v}^{\\phi_-^v} d\\phi \\sqrt{U}\n\\end{eqnarray}\nThe bound is saturated if the Bogomolnyi equation is obeyed\n\\begin{equation}\n\\frac{H}{\\sqrt{2}} \\phi_x + \\frac{\\sqrt{U}}{H} +G =0.\n\\end{equation}\nFor $H=1$ and $G=\\sigma$ we get the first BPS-impurity model, while $H=\\sqrt{1+\\sigma}$, $G=\\frac{\\sigma_x}{\\sqrt{1+\\sigma}}$ leads to the second model (up to a factor of $\\sqrt{2}$). Let us remark that such a generalisation reminds us of the construction of generalised BPS models as presented in \\cite{k-model-2}, \\cite{adam-santamaria}.\n\nNote, that $G$ is a completely arbitrary function of the field as well as the impurity (and can depend, for example, on its higher derivatives). As we will see in the next section, this result can be understood in a more general framework.\n\n\n\\section{ $\\mathcal{N}=1$ SUSY BPS impurities - a general construction } \\label{sect-3}\nThe standard $\\mathcal{N}=1$ SUSY algebra in $1+1$ dimensions has the following form\n\\begin{equation}\n[\\delta_\\epsilon,\\delta_\\eta]X=-2i \\epsilon^\\alpha\\eta^\\beta\\partial_{\\alpha\\beta}X.\n\\end{equation}\nAfter the restriction to the BPS subspace, $\\epsilon^2=0$ (or $\\epsilon^1=0$) it reduces to \n\\begin{equation}\n[\\delta_\\epsilon,\\delta_\\eta]X=-2i \\epsilon^1\\eta^1\\partial_t X.\n\\end{equation}\nSince the BPS subspace has only one SUSY parameter we have the following property\n\\begin{equation}\n[\\delta_\\epsilon,\\delta_\\eta]X=2\\delta_\\epsilon\\delta_\\eta X=-4 i\\epsilon^1\\eta^1\\partial_t X.\n\\end{equation}\nAs a consequence, for $Y=\\delta X$, i.e. $Y\\in\\text{Im}(\\delta)$, we have: if $Y\\in \\text{Im}(\\delta)\\Rightarrow \\delta_\\eta Y\\propto\\partial_t X$, or in words, the SUSY transformation of a term in the image of the SUSY transformation gives a time derivative. This has a nice consequence for the models with impurities, since the addition of any term of the form $\\sigma (x) Y$ with $Y\\in \\text{Im}(\\delta)$ will preserve the amount of SUSY preserved by the BPS soliton of the original model. Therefore, if $\\mathcal{L}$ is a BPS model with $\\mathcal{N}=1$ SUSY, then $\\forall X$, the following Lagrangian is also BPS\n\\begin{equation}\n\\tilde{\\mathcal{L}}=\\mathcal{L}+\\sigma(x)\\delta_\\epsilon X\\vert \n\\end{equation}\nwhere $\\vert$ means that the SUSY parameter has been removed. The following question arises naturally: is it possible to add a term $Y$, $Y\\not\\in \\text{Im}(\\delta)$ and such that 1\/2 of SUSY is preserved? Let us assume that the statement is true. In order to have $1\/2$ SUSY the following must hold\n\\begin{equation}\n\\delta\\tilde{\\mathcal{L}}=\\delta\\mathcal{L}+\\sigma(x)\\delta Y=\\epsilon^1 \\partial_\\mu j^\\mu_1+\\sigma(x)\\epsilon^1\\partial_t J.\n\\end{equation}\nBut if we choose $\\chi=\\int dt \\delta Y=\\epsilon^1 J$, we have $\\delta\\chi=\\int dt \\delta\\delta Y\\propto Y$, and therefore $Y\\in \\text{Im}(\\delta)$. On the other hand, if $\\delta Y=0$ then $Y\\not\\in \\text{Im}\\delta$. But if $\\delta Y=0$, Y does not depend on the fields and, therefore, Y must be a constant (or an impurity). Let us take $Y=\\mu(x)$, then if $\\sigma(x)\\mu(x)\\in L^1(\\mathbb{R})$, the model $\\tilde{\\mathcal{L}}$ is still BPS, but the topological bound is shifted by a amount $\\Delta=\\int dx \\sigma(x)\\mu(x)$ (the effect of the impurity is trivial: it does not affect the e.o.m.). \n\nNontrivial situation: $Y\\in \\text{Im}(\\delta)$. Since $Y$ contains bosonic terms (non-vanishing when $\\psi_\\alpha\\rightarrow 0$), it has to be the image of a term linear in fermions ($X$). We have two possibilities\n\\begin{equation}\nX=\\psi_1 H(F,\\phi,\\mu(x)), \\,\\, \\text{or} \\,\\, X=\\psi_2 H(F,\\phi,\\mu(x)),\n\\end{equation}\nwhere $\\mu(x)$ is another impurity. From the SUSY variation of fermions in the restricted subspace, we obtain\n\\begin{eqnarray}\nY&=&\\delta \\psi_1 H(F,\\phi,\\mu(x))+(\\text{fermions})\\propto \\partial_t\\phi H(F,\\phi,\\mu(x))+(\\text{fermions}) \\label{gen1}\\\\\n&&\\text{or}\\nonumber\\\\\nY&=&\\delta \\psi_2 H(F,\\phi,\\mu(x))+(\\text{fermions})\\propto (F+\\partial_x\\phi) H(F,\\phi,\\mu(x))+(\\text{fermions})\\label{gen2}.\n\\end{eqnarray}\n\nIn both cases the topological bound does not change: for (\\ref{gen1}), because the static energy functional does not change, while in the second, because the modification is proportional to the BPS equation. Note also that, depending on the form of $H$, it may be necessary to add fermionic terms in order to preserve $1\/2$ SUSY. We can gather these results in the following lemma: \n\n\\begin{lemma}\nLet $\\mathcal{L}$ be a $1+1$ dimensional scalar model with $\\mathcal{N}=1$ SUSY and a topological bound $T$. Let $\\tilde{\\mathcal{L}}$ be a BPS preserving impurity model based on $\\mathcal{L}$ of the form $\\tilde{\\mathcal{L}}=\\mathcal{L}-\\sigma(x)Y$ with topological bound $\\tilde{T}$. Then all possible impurity terms $Y$, preserving one supersymmetric generator (and one BPS sector) are of one of the following forms:\\\\\ni) $Y\\in\\text{Im}(\\delta)\\,(=\\delta X, \\forall X)$ and $\\tilde{T}=T$.\\\\\nii)$Y\\not\\in\\text{Im}(\\delta)\\,(=\\mu(x))$ and $\\tilde{T}=T+\\Delta$, with $\\Delta=\\int dx \\,\\sigma(x)\\mu(x)$.\n\\end{lemma}\n\n\n\\section{The impurity baby BPS Skyrme model}\nHaving systematically investigated the BPS preserving impurities in (1+1) dimensions, we want to apply the same supersymmetric approach to implement such impurities also in (2+1) dimensions. \n\\subsection{The model}\n\nWe begin our analysis of (2+1) dimensional theories with the baby BPS Skyrme model \\cite{bBPS}, \nwhich provides also a lower dimensional counterpart of the BPS Skyrme model \\cite{BPS Sk}. The static energy reads\n\\begin{equation}\nE_{baby BPS}=\\int d^2 x \\left[ \\lambda^2 B_0^2 + \\mu^2 \\mathcal{U} \\right]\n\\end{equation}\nwhere $\\mathcal{B}_0$ is a temporal component (topological charge density) of the pertinent topological current \n\\begin{equation}\n\\mathcal{B}^\\mu=\\frac{1}{8\\pi}  \\epsilon^{\\mu \\nu \\rho} \\vec{\\phi} \\cdot \\left( \\partial_\\nu \\vec{\\phi} \\times \\partial_\\rho \\vec{\\phi} \\right)\n\\end{equation}\nsuch that $B=\\int d^2 x \\mathcal{B}_0$ is the topological charge. \nFurthermore, $\\vec{\\phi} \\in \\mathbb{S}^2$ is a three component isovector with the unit length $\\vec{\\phi}^2=1$ and $\\mathcal{U}$ is a non derivative part, that is, a potential (with at least one isolated vacuum). It is known that the model possesses the following topological bound (where $d\\Omega_{\\mathbb{S}^2}$ is the \"volume\" (area) form on the (target space) unit sphere)\n\\begin{equation}\nE_{baby BPS} \\geq 2\\mu \\lambda |B|  \\left\\langle \\sqrt{\\mathcal{U}} \\right\\rangle, \\;\\;\\;  \\left\\langle \\sqrt{\\mathcal{U}} \\right\\rangle \\equiv \\frac{1}{\\mbox{vol}_{\\mathbb{S}^2}} \\int d\\Omega_{\\mathbb{S}^2} \\sqrt{\\mathcal{U}}.\n\\end{equation}\nThe bound is saturated if the following Bogomolnyi equation is obeyed,\n\\begin{equation}\n\\lambda \\mathcal{B}_0  \\pm \\mu \\sqrt{\\mathcal{U}} =0.\n\\end{equation}\nSolutions of this equation exist in any topological sector and can be found in an exact form once a potential is chosen. This model is a higher dimensional counterpart of the standard scalar model in (1+1) dimensions. Indeed, both Lagrangians are sums of a potential (a function of the fields) and the square of a topological current. \n\nAgain, it is possible to couple an impurity in such a way that one-half of the BPS property of the model remains preserved. Namely, the energy functional reads \n\\begin{equation}\nE=\\int d^2 x \\left[ \\lambda^2  \\mathcal{B}_0^2 + \\mu^2 \\mathcal{U} - 2 \\lambda \\sigma \\mathcal{B}_0 +2\\mu \\sigma \\sqrt{\\mathcal{U}} + \\sigma^2\\right]. \\label{sec:baby:ener}\n\\end{equation}\nThen we find the topological bound\n\\begin{eqnarray}\nE&=&\\int d^2 x \\left( \\lambda \\mathcal{B}_0 - \\mu \\sqrt{\\mathcal{U}}-\\sigma \\right)^2 + 2 \\lambda \\mu  \\int d^2 x  \\mathcal{B}_0 \\sqrt{\\mathcal{U}} \\\\\n&\\geq& 2 \\lambda \\mu  \\int d^2 x \\mathcal{B}_0 \\sqrt{\\mathcal{U}} =  2 \\lambda \\mu  B \\left\\langle \\sqrt{\\mathcal{U}} \\right\\rangle\n\\end{eqnarray}\nwhich is saturated if the following modified, impurity dependent, Bogomolnyi equation is obeyed,\n\\begin{equation}\n\\lambda  \\mathcal{B}_0 - \\mu \\sqrt{\\mathcal{U}} - \\sigma=0.\n\\end{equation}\nOne can easily prove that the Bogomolnyi equation implies the full Euler-Lagrange equation, see appendix \\ref{a1}. \n\nAs an example, we consider the potential $\\mathcal{U}=(1-\\phi^3)^2$. Then, we can find an exact solution of the Bogomolnyi equation for any radial impurity $\\sigma = \\sigma(r)$. In fact, introducing, via the usual stereographic projection, a complex field $u$\n\\begin{equation}\nu=\\frac{\\phi^1-i\\phi^2}{1+\\phi^3}\n\\end{equation}\nand assuming $u=\\sqrt{\\frac{g}{1-g}} e^{-in\\varphi}$, where $r,\\varphi$ are the polar coordinates and $n$ is a positive integer, the Bogomolnyi equation takes a linear form\n\\begin{equation}\n-\\lambda \\frac{n}{2\\pi} g_y=2\\mu g + \\sigma\n\\end{equation}\nHere, for convenience we use $y=r^2\/2$. The topologically nontrivial boundary conditions are $g(0)=1$, $g(\\infty)=0$. Then, for an exponentially localized impurity $\\sigma = \\alpha e^{-\\beta y}$ we find\n\\begin{equation}\ng(y)= e^{-\\frac{4\\pi \\mu }{n \\lambda} y} + \\frac{2\\pi}{n\\lambda}  \\frac{\\alpha}{\\beta -\\frac{4\\pi \\mu}{n \\lambda}} \\left( e^{-\\beta y} - e^{-\\frac{4\\pi \\mu }{n \\lambda} y}\\right)\n\\end{equation}\nObserve that  for $\\beta = \\frac{4\\pi \\mu }{n \\lambda}$, which might occur only for one value of the topological charge, the solution takes a different form, \n\\begin{equation}\ng(y)=\\left(1+\\frac{2\\pi\\alpha}{\\lambda n} y \\right) e^{-\\frac{4\\pi \\mu }{n \\lambda} y}.\n\\end{equation}\nThe corresponding topological charge is positive, $B=n>0$. Anti baby Skyrmions, i.e., solitons with negative topological charge, do not obey the Bogomolnyi equation and can only be found after solving the full Euler-Lagrange equations. This is, of course, a much more difficult task. \n\n\n\n\\subsection{SUSY in the impurity baby BPS Skyrme model}\n\nWe can use the same strategy as in Sec. \\ref{BPS} to build a SUSY BPS baby Skyrme model preserving the BPS structure. This model has a natural $\\mathcal{N}=2$ SUSY formulation \\cite{k-susy-8}, which can be used to introduce the impurity term. The BPS solitons in the original BPS baby Skyrme model satisfy the following BPS condition\n\\begin{equation}\n\\delta\\psi=0,\\,\\,\\, \\delta\\bar{\\psi}=0.\\label{sec:baby:fert}\n\\end{equation}\n\n\n\n\nThe component expansion of the SUSY BPS baby Skyrme model (after stereographic projection) has the form\n\\begin{equation}\n\\mathcal{L}_{\\text{baby}}\\vert=g(u,\\bar{u})\\left(\\partial_\\mu\\bar{u}\\partial^\\mu u+F\\bar{F}\\right)+h(u,\\bar{u})\\left((\\partial_\\mu u)^2(\\partial_\\nu\\bar{u})^2+2F\\bar{F}\\partial_\\mu \\bar{u}\\partial^\\mu u+(F\\bar{F})^2\\right),\n\\end{equation}\nwhere $h(u\\bar{u})=1\/(1+\\vert u\\vert^2)^4$. In order to reproduce (\\ref{sec:baby:ener}) we introduce\n\\begin{equation}\n\\mathcal{L}_{\\text{impurity}}=2\\sigma(x)\\sqrt{h(u,\\bar{u})}\\left(-\\partial_i u\\partial_i\\bar{u}+F\\bar{F}+i(\\partial_1\\bar{u}\\partial_2 u-\\partial_1 u\\partial_2\\bar{u})\\right).\\label{sec:baby:impu}\n\\end{equation}\nNote that the last term is the topological charge density. In terms of the (anti)holomorphic derivative, (\\ref{sec:baby:impu}) reads\n\\begin{equation}\n\\mathcal{L}_{\\text{impurity}}=2\\sigma(x)\\sqrt{h(u,\\bar{u})}\\left(F\\bar{F}-\\partial_z u \\partial_{\\bar{z}}\\bar{u}\\right).\\label{sec:baby:impuh}\n\\end{equation}\n\nThe addition of (\\ref{sec:baby:impu}) will preserve part of the SUSY if it belongs to the image of the restricted SUSY transformation (see Sec. \\ref{s:N2}). It is possible to verify that the following term is the preimage of (\\ref{sec:baby:impuh}) modulo fermionic terms\n\\begin{eqnarray}\nX&=& \\sqrt{h(u,\\bar{u})}\\left(\\psi_{\\bar{z}}\\left(\\partial_{\\bar{z}}\\bar{u}+\\bar{F}e^{i\\gamma}\\right)+\\bar{\\psi}_z\\left(\\partial_z u+Fe^{-i\\gamma}\\right)\\right),\\\\\n\\delta X\\vert&=&2\\sqrt{2}\\,\\bar{\\epsilon}_{\\dot{2}}\\,\\mathcal{L}_{\\text{impurity}}+(\\text{fermions}).\n\\end{eqnarray}\n\nWe have finally\n\\begin{eqnarray}\n\\mathcal{L}_{\\text{baby}}+\\mathcal{L}_{\\text{impurity}}\\vert&=&h(u,\\bar{u})\\left((\\partial_\\mu u)^2(\\partial_\\nu\\bar{u})^2 -(\\partial_\\mu\\bar{u}\\partial^\\mu u)^2\\right)-\\frac{g^2(u,\\bar{u})}{4 h(u,\\bar{u})}-\\sigma(x)\\frac{g(u,\\bar{u})}{\\sqrt{h(u,\\bar{u})}}+\\nonumber\\\\\n&+&2\\sigma(x)\\mathcal{B}_0\\sqrt{h(u,\\bar{u})}-\\sigma^2(x)\n\\end{eqnarray}\nwhich corresponds to (\\ref{sec:baby:ener}) after the identification $\\frac{g^2(u,\\bar{u})}{4 h(u,\\bar{u})}=\\mathcal{U}$ and $\\lambda=\\mu=1$.\n\n\\section{The impurity $CP^1$ model}\n\\label{sec-cp1}\n\\subsection{The model}\n\nNow we can turn to the simplest solitonic model in (2+1) dimensions (which is also integrable in its static version), that is, the $CP^1$ model\n\\begin{equation}\n\\mathcal{L}_{CP^1}=\\frac{1}{2} \\frac{ \\partial_\\mu u \\partial^\\mu \\bar{u}}{(1+|u|^2)^2}\n\\end{equation}\nwhose static energy reads\n\\begin{equation}\nE_{CP^1}= \\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left( u_z \\bar{u}_{\\bar{z}} + u_{\\bar{z}} \\bar{u}_z \\right)\n\\end{equation}\nwhere we use the complex plane coordinate $z=x+iy$. \nTopologically nontrivial solutions are just (anti)holomorphic maps of degree $n$, which obey\nthe Bogomolnyi equation (here the Cauchy-Riemann (CR) equations)\n\\begin{equation}\nu_z=0 \\;\\;\\; \\mbox{or} \\;\\;\\; u_{\\bar{z}}=0.\n\\end{equation}\nFor solutions of the CR equations, the energy equals the modulus of the topological charge (the degree of the map)\n\\begin{equation}\nQ=\\frac{1}{\\pi} \\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left( u_z \\bar{u}_{\\bar{z}} - u_{\\bar{z}} \\bar{u}_z \\right)\n\\end{equation}\n\nThe BPS preserving $CP^1$ model with an impurity is defined as follows (we restrict ourselves only to the static energy functional)\n\\begin{equation}\nE=E_{CP^1} + E_{impurity} \\label{cp-imp}\n\\end{equation}\nwhere\n\\begin{equation}\nE_{impurity}= \\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left( 2 \\sigma \\bar{\\sigma} +2 \\sigma u_z +2 \\bar{\\sigma} \\bar{u}_{\\bar{z}}  \\right) \n\\end{equation}\nThen, \n\\begin{eqnarray}\nE&=&\\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left[ 2 \\left( u_z+\\bar{\\sigma} \\right)\\left( \\bar{u}_{\\bar{z}} +\\sigma\\right)  - u_z \\bar{u}_{\\bar{z}} + u_{\\bar{z}} \\bar{u}_z\\right] \\\\\n&\\geq& \\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left[  - u_z \\bar{u}_{\\bar{z}} + u_{\\bar{z}} \\bar{u}_z\\right] = -\\pi Q .\\\\ \n\\end{eqnarray}\nThe topological bound is saturated if the following Bogomolnyi equation is satisfied,\n\\begin{equation}\nu_z+\\bar{\\sigma}=0.\n\\end{equation}\nThis is an inhomogeneous generalization of the anti-holomorphic CR equation. One can check that solutions of this equation obey the full EL equation, i.e., are the critical points of (\\ref{cp-imp}). As the Bogomolnyi equation is a non-homogeneous linear differential equation the BPS sector is still {\\it completely solvable}. This reminds us of the situation of the so-called integrable defect, that is, an impurity which does not affect the integrability of the underlying scalar field theory (like sine-Gordon) \\cite{integrability}. Here, a solitonic-impurity solution consists of the homogeneous part, which is still given by an arbitrary antiholomorphic map $f(\\bar z)$ and a pertinent unique solution of the non-homogenous part. Therefore, contrary to the pure $CP^1$ model, the total solution does not have to be purely antiholomorphic (or holomorphic). Of course, this strongly depends  on the impurity. Furthermore, the impurity may also contribute to the topological charge (degree) $Q$ of the full soliton-impurity solution.\n\nLet us consider an antiholomorphic impurity, for example, $\\sigma_m=B\\bar{z}^m$, where $m\\in \\mathbb{N}$ and $B$ is a complex constant. Hence, a BPS solution is \n\\begin{equation}\nu= A\\frac{(\\bar{z}-\\bar{z}_1)...(\\bar{z}-\\bar{z}_r)}{(\\bar{z}-\\bar{z}_1)...(\\bar{z}-\\bar{z}_s)} - \\frac{\\bar{B}}{m+1} z^{m+1}\n\\end{equation}\nwhere we assumed a rational map as a solution of the homogenous part of the Bogomolnyi equation. Here the polynomials in the numerator and denominator do not have common divisors. Furthermore, max$(r,s)=n$ and $A\\in \\mathbb{C}$. However, the computation of the degree of the solution is not a completely trivial task. Even in the case of the constant impurity $m=0$ and the linear antiholomorphic part the topological degree of the solution \n\\begin{equation}\nu(z,\\bar{z})=A\\bar{z} -\\bar{B}z\n\\end{equation}\ndepends on the constants $A, B$ \\cite{Sch-DM}. Concretely\n\\begin{equation}\n\\mbox{deg } u = \\left\\{\n\\begin{array}{rr}\n1 \\;\\;\\;\\; & |A| < |B| \\\\\n0 \\;\\; \\;\\; & \\mbox{if} \\;\\;\\;\\; |A|=|B| \\\\\n-1 \\;\\;\\;\\;& |A|>|B|\n\\end{array}\n\\right.\n\\end{equation}\n\nThis observation can have a nontrivial impact on the moduli space of soliton-impurity solutions, as different values of the parameters of the homogeneous part of a solution (with a fixed degree) can lead to different degrees of the total solution and, therefore, to energetically inequivalent configurations. We will address this problem (and its dependence on a particular choice of the impurity) in a separate paper. \n\nWe also remark that our constant impurity leads to a Bogomolnyi equation which is identical to the Bogomolnyi equation very recently found for the magnetic planar Skyrmions with the Dzyaloshinskii-Moriya interaction energy \\cite{Sch-DM}. This may suggest that there is a deeper relation between our impurity coupling and Dzyaloshinskii-Moriya like terms \\cite{DM}.  Of course, this is of high importance as far as a possible experimental realization of the BPS preserving impurity is concerned. Moreover, one can ask whether  there exist DM counterparts of our non-homogenous CR Bogomolnyi equations also in the case of more complicated (non-constant) BPS preserving impurities. \n\n\\vspace*{0.2cm}\n\nA related issue is the global $U(1)$ symmetry. Of course, for an impurity which trivially transforms under this group, the model (\\ref{cp-imp}) is no longer invariant under the global $U(1)$. However, this symmetry is effectively restored in the BPS sector. Indeed, the group can act nontrivially on the homogeneous part of BPS solutions leading to energetically equivalent configurations. \n\nA possibility to restore the global $U(1)$ symmetry completely is to assume that the impurity also transforms in the fundamental representation of it i.e., $\\sigma \\rightarrow e^{i\\varphi} \\sigma$, where $\\varphi$ is the transformation parameter. This may apply to impurities which originate as a spatially frozen lump, which is a mechanism proposed in \\cite{vortices} for the magnetic impurity in the Abelian Higgs model. \n\nFinally, one can introduce an impurity term which is $U(1)$ invariant, although the impurity transforms trivially. For example one can consider the following modification \n\\begin{equation}\n\\tilde{E}_{impurity}= \\int d^2 z \\frac{1}{(1+|u|^2)^2} \\left( 2 \\sigma \\bar{\\sigma} u\\bar{u}+2 \\sigma \\bar{u} u_z +2 \\bar{\\sigma} u \\bar{u}_{\\bar{z}}  \\right) \\label{cp-imp-inv}\n\\end{equation}\nThe resulting Bogomolnyi equation is also deformed and reads \n\\begin{equation}\nu_z+ u \\bar{\\sigma}=0.\n\\end{equation}\nHowever, one can still find exact solutions, where now the impurity term acts multiplicatively on the original antiholomorphic map\n\\begin{equation}\nu= A\\frac{(\\bar{z}-\\bar{z}_1)...(\\bar{z}-\\bar{z}_r)}{(\\bar{z}-\\bar{z}_1)...(\\bar{z}-\\bar{z}_s)}  e^{-\\int dz \\bar{\\sigma}}\n\\end{equation}\n\nA particular choice for the treatment of the global $U(1)$ transformation depends, of course, on the physical application one has in mind. Obviously, it will affect the possibility to promote this global symmetry to a local one, that is, the gauging of the impurity model. \n\n\n\n\n\n\\subsection{SUSY in the impurity $CP^1$ model}\n\\label{sub:CP}\n\nThe $\\mathcal{N}=2$ formulation of the $CP^1$ model is well-known. It can be constructed in terms of the following K\\\"ahler potential\n\\begin{equation}\n\\int d^2\\theta d^2\\bar{\\theta}\\,\\text{ln}(1+\\Phi^\\dagger\\Phi),\n\\end{equation}\nwhere $\\Phi(\\Phi^\\dagger)$ are chiral (antichiral) superfields. Inspired by our one-dimensional construction is not very difficult to guess the SUSY form of (\\ref{cp-imp}): \n\\begin{equation}\n\\mathcal{L}_{\\text{impurity}}=\\frac{1}{(1+\\vert u \\vert^2)^2}\\sigma(x)(F-\\partial_{\\bar{z}} u)+\\text{h.c.}+\\text{fermions}.\\label{sub:imp1}\n\\end{equation}\nIn addition, this term lies in the image of $\\delta$, as can be seen explicitly by taking the following transformation\n\\begin{equation}\n\\mathcal{L}_{\\text{impurity}}=\\sigma(x)\\,\\delta\\left(\\frac{1}{(1+\\vert u \\vert^2)^2}\\psi_1+\\text{h.c.}\\right)+\\text{h.c.}\\vert.\n\\end{equation}\nAs we will see (Sec. \\ref{s:N2}), this allows us to conclude that both the impurity (\\ref{sub:imp1}) and the BPS solutions of the model preserve $1\/4$ of SUSY.\n\n\\section{The impurity Abelian Higgs model at critical coupling}\n\\subsection{Magnetic impurity}\nIt was observed by Tong and Wong \\cite{vortices} that the standard Abelian Higgs model at the critical\ncoupling admits a half-BPS preserving impurity extension. The impurity couples to the magnetic field $B$ and therefore is referred to as the magnetic impurity $\\sigma_m$,\n\\begin{equation}\nE_{m}=\\frac{1}{2} \\int d^2 x \\left( B^2 +\\overline{D_i u} D_i u +\\frac{1}{4}(1+\\sigma_m-|u|^2)^2 -\\sigma_m B \\right)\n\\end{equation}\nHere, $F_{\\mu \\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$ is the field strength of the $U(1)$ gauge field $A_\\mu$ and $D_\\mu u=\\partial_\\mu u - iA_\\mu u$ is the covariant derivative of the complex Higgs field $u$. The pertinent Bogomolnyi equations read\n\\begin{eqnarray}\nD_x u +iD_y u&=&0 \\\\\nB -\\frac{1}{2}(1- |u|^2)&=& \\frac{1}{2}\\sigma_m,\n\\end{eqnarray}\nand its solutions saturate the following energy bound\n\\begin{equation}\nE_m \\geq \\pi N, \\;\\;\\; N=\\frac{1}{2\\pi} \\int d^2 x B.\n\\end{equation}\n\n\\subsection{Higgs impurity}\nAs we have shown in Sec. \\ref{sec-cp1}, the $CP^1$ model couples to an impurity in a half-BPS preserving manner, provided that the original CR equations get modified to an inhomogeneous version. The resulting model preserves the global U(1) symmetry (not only in the BPS sector) if the impurity is assumed to transform in the fundamental representation. This opens a new path to implement a partially BPS preserving impurity in the Abelian Higgs model. Now, contrary to the above case, the impurity multiplies the derivatives of the matter field and therefore we call it a Higgs impurity $\\sigma_h$. Specifically, the model is\n\\begin{equation}\nE_{h}=E_{AH}+E_{impurity}\n\\end{equation}\nwhere to the standard Abelian Higgs part\n\\begin{equation}\nE_{AH}=\\frac{1}{2} \\int d^2 x \\left( B^2 +\\overline{D_i u} D_i u +\\frac{1}{4}(1-|u|^2)^2 \\right)\n\\end{equation}\nwe add the impurity term\n\\begin{equation}\nE_{impurity}= \\frac{1}{2} \\int d^2 x \\left( \\sigma_h \\bar{\\sigma}_h + \\sigma_h \\overline{(D_x u + i D_y u)} + \\bar{\\sigma}_h (D_x u + i D_y u) \\right)\n\\end{equation}\nwhich, after using complex coordinates $z, \\bar{z}$, is a gauged version of the $CP^1$ impurity term\n\\begin{equation}\nE_{impurity}= \\frac{1}{2} \\int d^2 z \\left(\\frac{1}{2} \\sigma_h \\bar{\\sigma}_h + \\bar{\\sigma}_h D_{\\bar{z}} u + \\sigma_h \\overline{D_z u} \\right) \\label{higgs1}\n\\end{equation}\nThe lower energy bound reads\n\\begin{eqnarray}\nE_h &=& \\frac{1}{2} \\int d^2 x \\left( \\left( B -\\frac{1}{2}(1-u\\bar{u}) \\right)^2 +   \\overline{(D_x u + i D_y u + \\sigma_h)}(D_x u + i D_y u + \\sigma_h)  \\right) \\\\\n&+& \\frac{1}{2} \\int d^2 x B -\\frac{i}{2} \\int d^2 x  \\left( \\partial_x (\\bar{u} D_y u) - \\partial_y (\\bar{u} D_z u)\\right) \\\\\n&\\geq & \\pi N\n\\end{eqnarray}\nwhere the last term in the second line integrates to zero. The bound is saturated if the following Bogomolnyi equations are satisfied\n\\begin{eqnarray}\nD_x u +iD_y u+\\sigma_h&=&0 \\\\\nB -\\frac{1}{2}(1- |u|^2)&=&0.\n\\end{eqnarray}\nAs expected, this impurity enters only into the gauged CR equations, leaving the magnetic field equation unchanged. Of course, one should remember that this construction makes sense only if the local $U(1)$ transformation continues to be a symmetry also after the inclusion of the impurity. This implies that $\\sigma_h \\rightarrow e^{i\\varphi (x,y)} \\sigma_h$. As in the case of the $CP^1$ model, this can be relevant for an impurity originating from a spatially frozen vortex. \n\nIt is also possible to implement a partially BPS preserving impurity term which is invariant under the gauge group even for a trivially transforming impurity, $\\sigma_h \\rightarrow  \\sigma_h$. For example, one can use the globally $U(1)$ invariant $CP^1$ model with impurity as a hint. Hence,\n\\begin{equation}\n\\tilde{E}_{impurity}= \\frac{1}{2} \\int d^2 z \\left(\\frac{1}{2} \\sigma_h \\bar{\\sigma}_h u\\bar{u}+ \\bar{\\sigma}_h \\bar{u} D_{\\bar{z}} u + \\sigma_h u\\overline{D_z u} \\right) \\label{higgs2}\n\\end{equation}\nThis leads to the the following modification of the gauge CR equations,\n\\begin{equation}\nD_x u +iD_y u+ u \\sigma_h=0.\n\\end{equation}\nWe comment that, in contrast to the magnetic impurity, the Higgs impurity does not have any impact on the corresponding Bradlow law. \nNote that both impurities can be implemented in a completely independent way which results in the most general partially BPS preserving impurity Abelian Higgs model\n\\begin{equation}\nE_{m, \\; h} = E_{magnetic} + E_{impurity}\n\\end{equation}\nThis model possesses the same topological energy bound as the above impurity models, and the Bogomolnyi equation is of the following form,\n\\begin{eqnarray}\nD_x u +iD_y u+\\sigma_h&=&0 \\\\\nB -\\frac{1}{2}(1+\\sigma_m - |u|^2)&=&0.\n\\end{eqnarray}\nThe above remarks on the gauge invariance apply. \n\n\n\n\n\\subsection{SUSY in the impurity Abelian-Higgs model}\n\nThe magnetic impurity can be introduced in a $1\/2$ SUSY invariant way as long as it is compensated by the auxiliary field of the vector multiplet \\cite{vortices,mirror}. In its simplest form, it can be written as \n\\begin{equation}\n\\mathcal{L}^h_{\\text{impurity}}=\\sigma_m(x)(B-D).\\label{sub:AH:im}\n\\end{equation}\nIt is easy to see that (\\ref{sub:AH:im}) corresponds to the image of $\\lambda_\\alpha$ (the fermion in the vector multiplet) under the gauge invariant SUSY transformation. It should be noted that (\\ref{sub:AH:im}) is proportional to the gauged BPS equation and, therefore, the addition of the magnetic impurity cannot change the Bogomolnyi bound. The Higgs impurity, on the other hand, takes the form\n\\begin{equation}\n\\mathcal{L}^h_{\\text{impurity}}=\\frac{1}{(1+\\vert u \\vert^2)^2}\\sigma_h(x)(F-D_{\\bar{z}} u)\\bar{u}+\\text{h.c.}+\\text{fermions},\\label{sub:AH:h}\n\\end{equation}\nwhich is an obvious generalization of (\\ref{sub:imp1}), up to a factor. But, as in the impurity $CP^1$ model, the impurity (\\ref{sub:AH:h}) only preserves $1\/4$ of the SUSY. Since the subalgebra preserved by (\\ref{sub:AH:h}) is contained in the subalgebra preserved by (\\ref{sub:AH:im}), the presence of both impurities breaks the SUSY to $1\/4$. It is interesting to note that (\\ref{sub:AH:im}) and (\\ref{sub:AH:h}) are perhaps the simplest nontrivial impurities that one can introduce in $\\mathcal{N}=2$ theories, but they are not unique. In fact, in Sec. \\ref{s:N2}, we explicitly show how to generate an infinite number of them. \n\n\n\n\n\n\n\n\n\n\\section{$\\mathcal{N}=2$ SUSY BPS impurities - a general construction}\n\\label{s:N2}\n\nLet us consider a scalar theory in $2+1$ dimensions with a complex field $\\phi$. The general BPS equations in terms of the auxiliary fields can be written as follows\n\\begin{equation}\nF=\\partial_{z}\\phi e^{i\\gamma} \\,\\, \\text{or} \\,\\, F=\\partial_{\\bar{z}}\\phi e^{i\\gamma}\\label{sec:N2:BPS}\n\\end{equation}\nwhere $\\gamma$ is an arbitrary constant phase. Now we have two complex SUSY generators (four real supercharges) and the BPS structure is more complex. When $F=0$, for example in the $CP^1$ model, the BPS solutions take the familiar Cauchy-Riemann form and they preserve half of the supersymmetry. This can be seen easily from the SUSY transformation of the fermions\n\\begin{equation}\n\\delta\\psi_\\alpha =i\\sqrt{2}\\sigma^{\\mu\\,\\, \\dot{\\alpha}}_{\\,\\alpha} \\bar{\\xi}_{\\dot{\\alpha}}\\partial_\\mu\\phi+\\sqrt{2}\\xi_\\alpha F \\label{sec:N2:fer}\n\\end{equation}\n\nIt is easy to see that for \n\\begin{equation}\n\\bar{\\xi}_{\\dot{1}}=i \\bar{\\xi}_{\\dot{2}}\\,\\, \\text{and}\\,\\, \\xi_1=-i\\xi_2\\label{sec:12:BPS}\n\\end{equation}\nand $\\partial_z\\phi=0,\\,\\partial_t\\phi=0$ the condition $\\delta\\psi_\\alpha=0$ is satisfied. We say therefore that if $F=0$ the BPS solitons are $1\/2$-BPS. If $F\\neq 0$ we need an extra constraint, namely \n\\begin{equation}\n\\xi_2=i \\bar{\\xi}_{\\dot{2}}e^{-i\\gamma}.\\label{sec:14:BPS}\n\\end{equation}\n The expression (\\ref{sec:N2:fer}) can be rewritten as \n\\begin{eqnarray}\n\\delta\\psi_1&=&\\sqrt{2}\\bar{\\xi}_{\\dot{2}}\\left(\\partial_0\\phi+e^{-i\\gamma}F-\\partial_z\\phi\\right),\\\\\n\\delta\\psi_2&=&-i\\sqrt{2}\\bar{\\xi}_{\\dot{2}}\\left(\\partial_0\\phi-e^{-i\\gamma}F+\\partial_z\\phi\\right).\n\\end{eqnarray}\nThe condition $\\delta\\psi_\\alpha=0$ is again achieved for static solutions satisfying (\\ref{sec:N2:BPS}), but this time only preserving $1\/4$ of the supersymmetry. \n\n\\subsection{$1\/4$ and $1\/2$ BPS impurities: scalar sector}\n\nThe general commutator between two supersymmetric transformations has the form\n\\begin{equation}\n[\\delta_\\eta,\\delta_\\xi]=-2i \\left(\\eta \\sigma^\\mu \\bar{\\xi}-\\xi\\sigma^\\mu \\bar{\\eta}\\right)\\partial_\\mu\n\\end{equation} \nIf we restrict the SUSY algebra to the $1\/4$ subspace by imposing (\\ref{sec:12:BPS}) and (\\ref{sec:14:BPS}) we have\n\\begin{equation}\n[\\delta_\\eta,\\delta_\\xi]=8 e^{-i\\gamma}\\bar{\\eta}_{\\dot{2}}\\bar{\\xi}_{\\dot{2}}\\partial_0.\n\\end{equation}\nSince there is only one SUSY parameter we have\n\\begin{equation}\n\\delta_\\eta \\delta_\\xi=4 e^{-i\\gamma}\\bar{\\eta}_{\\dot{2}}\\bar{\\xi}_{\\dot{2}}\\partial_0.\\label{sec:14:algebra}\n\\end{equation}\nAs a consequence, the results found in Sec. 3 for $\\mathcal{N}=1$ SUSY apply here: $Y\\in \\text{Im}\\,\\delta\\Rightarrow \\delta_\\eta Y\\propto \\partial_0 X$. Therefore a impurity of the form $\\sigma(x)Y$ preserves $1\/4$ of SUSY. \nRegarding the $1\/2$ preserving impurities in the scalar sector, we do not have a general answer. But since the preservation of $1\/2$ SUSY requires $F=0$, the impurity cannot change the BPS equations of the underlying model. For example in the $CP^1$ model one can add the following term\n\\begin{equation}\n\\delta \\left( \\psi_1 \\left(F+\\partial_z u\\right)\\right)\\vert_{\\gamma=0}=\\left(F^2-(\\partial_zu)^2\\right)+X\\partial_0 u+\\text{fermions}. \\label{sec:12:example}\n\\end{equation}\nTaking into account that (\\ref{sec:12:example}) is the image of $\\delta$ it preserves at least $1\/4$ of SUSY, but since the solution $F=\\bar{F}=0$ is still available the BPS impurity solutions are $1\/2$ and the BPS equations are not modified.\n\n\n\n\n\n\n\\subsection{$1\/4$ and $1\/2$ BPS impurities: gauge sector}\n\nIn the gauge sector, the previous results for $1\/4$ SUSY still apply since the subalgebra (\\ref{sec:14:algebra}) holds. The possible impurities are again in the image of the SUSY transformation. As a consequence, if one considers an impurity of the form \n\\begin{equation}\n\\sigma(x)\\,\\delta \\left(\\lambda_\\alpha F(A_\\mu, D)\\right),\n\\end{equation}\nat least one supercharge will be preserved. Moreover, the SUSY transformation of the fermion in the vector multiplet has the form\n\\begin{equation}\n\\delta \\lambda= \\sigma^{\\mu\\nu} \\xi F_{\\mu\\nu}+i \\xi D. \n\\end{equation}\nThe reduction to the $1\/4$ BPS subspace leads to \n\\begin{equation}\n\\delta \\lambda_2\\vert_{\\text{static}}= i \\xi_2\\left(F_{12}-D \\right)\\label{sec:gauge}\n\\end{equation}\nwhich is proportional to the BPS equation. Therefore, as in the previous cases, the impurity does not change the BPS bound.  It is also interesting to note that if one considers an impurity of the form (\\ref{sec:gauge}) alone, not only $1\/4$ but $1\/2$ of SUSY is preserved. Unlike in the scalar sector, the gauge $1\/2$ BPS impurity also modifies the BPS equation because $D\\neq0$ allows for the existence of $1\/2$ BPS solitons.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Summary}\n\\label{sum}\n\nIn the present work, we have systematically studied the coupling of an impurity to BPS models in (1+1) and (2+1) dimensions, such that the resulting model preserves half of the BPS property. It means that half of the solitons existing in the original (no impurity) theory still obey the pertinent (impurity dependent) Bogomolnyi equations and saturate a topological energy bound. It turns out that an extremely useful tool for our analysis is provided by supersymmetry. In particular, in (1+1) dimensions, SUSY not only allows to understand both the existence of the impurity BPS models (by relating them to the SUSY transformations of the fermions) and the presence of a generalised translational symmetry in these models \\cite{imp-phi4} (relating it to the fermionic zero mode). It also permits to construct the general class of impurity BPS models, see Section \\ref{sect-3}. The construction of an impurity BPS model requires a BPS theory without impurity as a starting point, and in (2+1) dimensions not all field theories are BPS. In this case we demonstrated that, whenever a (gauged or ungauged) scalar field theory is BPS, it allows for an impurity BPS generalisation, and SUSY provides an easy way to construct this impurity BPS model. Further, this generalisation is not unique, and each BPS theory allows, in fact, for infinitely many BPS-preserving impurities (see Section \\ref{s:N2}). \n\nSo, one important result is that any BPS model in (1+1) and (2+1) dimensions can be extended to a half-BPS impurity model. Since the particular form of the impurity is rather arbitrary, this significantly enlarges the number of theories with the (half) BPS property. In particular, this applies to (2+1) dimensions where, in contrast to the lower dimensional case, the BPS-ness puts very strong restrictions on the form of the action. As a consequence, even in higher dimensions the BPS property is not as rare a feature as one might expect. Owing to the fact that impurities are easily realized in condensed matter, as for example dislocations or defects in a periodic lattice (crystal), we believe that some BPS-impurity models may find applications to realistic physical systems. \n\nThe importance of (near) BPS impurity models is not only related to the simplicity of their mathematical structure. The crucial observation is that they describe topological solitons (kinks, lumps, vortices and baby Skyrmions) which have zero (small) interaction energy with the impurities. This results in an extremely low energy cost for the manipulation of these objects, which might be very interesting, e.g., for the transport and storage of data by topological solitons (like, for instance, magnetic skyrmions).\n\nFurthermore, it is very intriguing that in the case of the planar models ($CP^1$ and Abelian Higgs models) the BPS preserving impurity has a form similar to the Dzyaloshinskii-Moriya energy which may additionally contribute to an experimental realization of this kind of impurities. \n\nThere are many directions in which one may continue our work. First of all, there are other physically important (2+1) dimensional models which enjoy the BPS property. Therefore, one can search for their half-BPS impurity versions. Here one may mention the non-abelian vortices at critical coupling \\cite{non-ab vortex}, other gauged planar soliton systems as the gauged BPS baby Skyrme model \\cite{g-bBPS}, \\cite{CS-bBPS} and the gauged $O(3)$ \\cite{O(3)} as well as the conformal magnetic Skyrmions \\cite{bazeia-magnetic}.\n\nSecondly, one should understand how the low energy dynamics, approximated by the geodesic motion on the corresponding moduli space, is affected by the inclusion of the half-BPS preserving impurity. This may find some applications also in the study of multi-soliton interactions in non-impurity models, especially in the cases when impurities would be connected to frozen solitons. Here, the simplest case can be given by the $CP^1$ model \\cite{leese}, where the soliton-impurity BPS solutions can be found in an exact form, which provides a very good starting point for (even) analytical investigations of the moduli space dynamics. A similar study has already been performed for the half-BPS impurity $\\phi^4$ model, where the scattering of the BPS antikink  on the impurity has been shown to be very well described by the motion on moduli space \\cite{imp-phi4}.  It would be even more interesting to analyze the existence of solitons and their dynamics in the Abelian Higgs model with half-BPS impurities of both magnetic \\cite{vot-imp-dyn} and Higgs types. \n\nThirdly, the construction presented here can also be carried over to theories with more than two spatial dimensions. The suspersymmetry algebra is obviously different, but the (2+1) dimensional examples can serve as a guidance for the construction of the bosonic sector. \n\nAnother interesting direction is related to the fact that our BPS\nimpurity models\npossess  (spatially) modulated vacua. They can even be of a periodic\nform if a periodic lattice of impurities\nis used. Modulated vacua have been recently investigated in Lorentz\ninvariant theories \\cite{bja2} (with higher order\nderivative terms). However, such vacua break the supersymmetry\ncompletely. In light of the results presented here,\nit is an interesting question whether also in Lorentz invariant theories a fractional\nsusy preserving modulated vacuum may be found.\n\nOf course, the most exiting direction is to find condensed matter systems which allow for such a specific, BPS preserving, coupling of the impurity. This requires further studies of the relation between the BPS preserving impurity and the Dzyaloshinskii-Moriya interaction observed here. We plan to investigate this issue in a forthcoming paper. \n\n\\hspace*{0.3cm}\n\n \n{\\bf Acknowledgements.}  C.A and A.W.  acknowledge financial support from the Ministry of Education, Culture, and Sports, Spain (Grant No. FPA2017-83814-P), the Xunta de Galicia (Grant No. INCITE09.296.035PR and Conselleria de Educacion), the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), Maria de Maeztu Unit of Excellence MDM-2016-0692, and FEDER. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nLet $\\mathbb{N}$ be the set of non-negative integers. A subset $M$ of $\\mathbb{N}$ is called a \\textbf{submonoid} of $(\\mathbb{N},+)$ if $M$ contains $0 \\in \\mathbb{N}$ and $M$ is closed under the sum in $\\mathbb{N}$. A \\textbf{numerical semigroup} is a submonoid $S$ of $(\\mathbb{N},+)$ such that $\\mathbb{N} \\setminus S$ is finite.\n\nIf $A$ is a non-empty subset of $\\mathbb{N}$, then we will write $\\langle A \\rangle$ for the submonoid of $(\\mathbb{N},+)$ generated by $A$, that is, \\[\\langle A \\rangle = \\Big\\{ \\sum_{i=1}^n u_i a_i\\ \\mid\\ n \\in \\mathbb{N}\\setminus\\{0\\},\\ \\{a_1, \\ldots, a_n\\} \\subseteq A\\ \\text{and}\\ \\{u_1, \\ldots, u_n\\} \\subset \\mathbb{N} \\Big\\}.\\] It is a well-known fact that $\\langle A \\rangle$ is a numerical semigroup if and only if $\\gcd(A) = 1$ (see, for instance, \\cite[Lemma 2.1]{libro}). \n\nIf $M$ is a submonoid of $(\\mathbb{N},+)$ such that $M = \\langle A \\rangle$ for some $A \\subseteq \\mathbb{N}$, then we will say that $A$ is \\textbf{system of generators} of $M$. Furthermore, if $M \\neq \\langle B \\rangle$ for every $B \\subsetneq A$, then we will say that $A$ is a \\textbf{minimal system of generators} of $M$. In \\cite[Corollary 2.8]{libro}, it is proved that every submonoid of $(\\mathbb{N},+)$ has a unique minimal system of generators, which in addition is finite. We will write $\\operatorname{msg}(M)$ for the minimal system of generators of $M$.\n\nIf $S$ is a numerical semigroup, the elements in $\\mathbb{N} \\setminus S$ are called \\textbf{gaps of $S$}. The cardinality of $\\mathbb{N} \\setminus S$ is the called \\textbf{genus of $S$} and is denoted $\\operatorname{g}(S)$. The smallest integer in $S \\setminus \\{0\\}$ is called the \\textbf{multiplicity of $S$} and is denoted $\\operatorname{m}(S)$. The greatest integer that does not belong to $S$ is called the \\textbf{Frobenius number of $S$} and is denoted $\\operatorname{F}(S)$. \n\nLet $S$ and $T$ be numerical semigroups. We say that $T$ is an \\textbf{extension} of $S$ if $S \\subseteq T$ and we say that $T$ is an \\textbf{arithmetic extension} of $S$ if there exist positive integer numbers $d_1, d_2, \\ldots, d_n$ such that \\[T = \\big\\{ x \\in \\mathbb{N}\\ \\mid\\ d_i x \\in S,\\ i = 1, \\ldots, n \\big\\}.\\] Not all the extensions of a numerical semigroup is arithmetic. In Theorem \\ref{Th 9}, we characterize the numerical semigroups whose extensions are all arithmetic.\n\nIf $S$ is a numerical semigroup and $d$ is a positive integer, then \\[\\frac{S}d := \\big\\{x \\in \\mathbb{N}\\ \\mid\\ d\\, x \\in S\\big\\}\\] is a numerical semigroup, too (see \\cite[Proposition 5.1]{libro}). The numerical semigroup $\\frac{S}d$ is called the \\textbf{quotient of $S$ by $d$}. The quotients of numerical semigroups by an element have been studied by different authors; see, for instance, \\cite{Comm-Alg, Swanson} and more recently \\cite{borzi}.\n\nNotice that $\\frac{S}d = \\mathbb{N}$ if and only if $d \\in S$, and observe that $\\frac{S}d$ is an arithmetic extension of $S$, and that an extension $T$ of $S$ is arithmetic if and only if there exists $\\{d_1, \\ldots, d_n\\} \\subseteq \\mathbb{N} \\setminus S$ such that \\[T = \\frac{S}{d_1} \\cap \\frac{S}{d_2} \\cap \\ldots \\cap \\frac{S}{d_n}.\\] Indeed, by definition, $x \\in T$ if and only if there exist positive integers $d_1, \\ldots, d_n$ such that $d_i x \\in S,$ for all $i \\in \\{1, \\ldots, n\\};$ equivalently, $x \\in \\frac{S}{d_i}$ for every $i \\in \\{1, \\ldots, n\\}.$  \n\nBesides a theoretical interest, a major motivation for the study of arithmetic extensions of numerical semigroups relies in the following two examples.\n\nIn \\cite{borzi} it is shown that quotients of Arf numerical semigroups (see \\cite{arf}) by a positive integer are Arf numerical  semigroups. Since, the intersection of finitely many Arf numerical semigroups is an Arf numerical semigroup (see, for instance, \\cite[Proposition 3.22]{libro}), we conclude that arithmetic extensions are stable for the Arf property. Arf numerical semigroups are a relevant family of numerical semigroups as they are semigroup of values of Arf rings (see \\cite{lipman}).\n\nIn \\cite{proportionally} the notion of proportionally modular numerical semigroup (PM-semigroup, for short) was introduced. In \\cite{semigroup} the authors characterize the numerical semigroups that can be written as an intersection of finitely many PM-semigroups, these semigroups are called SPM-semigroups. On the other hand, in \\cite{toms} Toms introduced a family of numerical semigroups, that we call T-semigrups, which are $K_0-$groups of certain $C^*-$algebras (see \\cite[Theorem 1.1]{toms}). In \\cite{canadian} it is shown that the T-semigruops are precisely the SPM-semigroups. As a consequence of \\cite[Corollary 4.1]{archiv} we have that a numerical semigroup is SPM-semigroup if and only if the exist positive integer numbers $a, d_1, \\ldots, d_n$ such that \\begin{equation}\\label{ecu0} S = \\frac{\\langle a, a+1 \\rangle}{d_1} \\cap \\ldots \\cap  \\frac{\\langle a, a+1 \\rangle}{d_n}.\\end{equation} Thus, we conclude that the T-semigroups are the arithmetic extensions of the numerical semigroups generated by two consecutive integers.\n\nOne of the main results in this paper is Theorem \\ref{Th 12} which determines the Ap\\'ery set of the quotient of a numerical semigroup by a positive integer. As a consequence, explicit formulas for the genus and the Frobenius number of PM-semigroups are obtained.\n\nTo conclude the introduction we outline how this note is organized: in Section \\ref{Sect1} we study the first properties of the set of arithmetic extensions of a numerical semigroup and we characterize the numerical semigroups whose extensions are all arithmetic. In Section \\ref{Sect2}, we see that if $S$ is a numerical semigroup and $n$ is an element in $S$ different from zero, then the Ap\\'ery set of $\\frac{S}d$ with respect $n$ can be easily computed in terms of the Ap\\'ery set of $S$ with respect $n$. This fact will provide us nice explicit formulas for the Frobenius number and the genus of the numerical semigroup $\\frac{\\langle a, a+1\\rangle}d$ (see Remark \\ref{Rem 13}) and consequently of any T-semigroup by Proposition \\ref{Prop 16}. Finally, in the last section, we take advantage of the results in the previous sections to formulate an algorithm for the computation of all the arithmetic extensions of a numerical semigroup. We finish the paper by giving a simple GAP \\cite{gap} implementation of our algorithm; this implementation requires the GAP package \\texttt{NumericalSgps} \\cite{numericalsgps}.\n\n\\section[The set of arithmetic extensions]{The set of arithmetic extensions of a numerical semigroup}\\label{Sect1}\n\nLet $\\Delta$ be a numerical semigroup and let $d_1, d_2, \\ldots, d_n$ be positive integer numbers. We write $\\Delta(\\{d_1, d_2, \\ldots, d_n\\}) = \\{x \\in \\mathbb{N}\\ \\mid\\ d_i x \\in \\Delta,\\ i = 1, \\ldots, n\\}$. Clearly, \\[\\Delta(\\{d_1, d_2, \\ldots, d_n\\}) = \\bigcap_{i=1}^n \\frac{\\Delta}{d_i} \\] is an arithmetic extension of $\\Delta$. By convention, we assume that $\\Delta(\\varnothing) = \\mathbb{N}$.\n\nIn the following, we will denote by $\\mathcal{F}(\\Delta)$ the set of arithmetic extensions of $\\Delta$. The proof of the following result is straightforward.\n\n\\begin{proposition}\\label{Prop1}\nIf $\\Delta$ is a numerical semigroup, then \\[\\mathcal{F}(\\Delta) = \\{\\Delta(X)\\ \\mid\\ X \\subseteq \\mathbb{N}\\setminus \\Delta\\}.\\] \n\\end{proposition}\n\n\\begin{theorem}\\label{Th1}\nIf $\\Delta$ is a numerical semigroup, then \\[\\mathcal{F}(\\Delta) = \\big\\{ \\Delta(X) \\mid X \\subseteq \\mathbb{N}\\setminus \\Delta\\ \\text{and}\\ X = \\operatorname{msg}(\\langle X \\rangle) \\big\\}.\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{A} =  \\big\\{ \\Delta(X) \\mid X \\subseteq \\mathbb{N}\\setminus \\Delta\\ \\text{and}\\ X = \\operatorname{msg}(\\langle X \\rangle) \\big\\}$. By Proposition \\ref{Prop1}, $\\mathcal{A} \\subseteq \\mathcal{F}(\\Delta)$. Let us prove the opposite inclusion. To this end, we consider $S \\in \\mathcal{F}(\\Delta)$. By definition, we have that there exists $X = \\{x_1, x_2, \\ldots, x_n\\} \\subseteq \\mathbb{N} \\setminus \\Delta$ such that $S = \\Delta(X) = \\frac{\\Delta}{x_1} \\cap \\frac{\\Delta}{x_2} \\cap \\ldots \\cap \\frac{\\Delta}{x_n}$. Let $Y = \\{y_1, y_2, \\ldots, y_m\\} = \\operatorname{msg}(X)$. Since $Y \\subseteq X$, we have that \\[\\Delta(X) \\subseteq \\Delta(Y).\\] On the other hand, if $y \\in \\Delta(Y)$, then $y y_j \\in \\Delta$, for every $j = 1, \\ldots, m$ and $y x_i \\in \\Delta$, because $x_i \\in \\langle y_1, y_2, \\ldots, y_m \\rangle$, for every $i = 1, \\ldots, n$. Thus $y \\in \\Delta(X)$. Therefore $\\Delta(X) = \\Delta(Y) \\in \\mathcal{A}$.\n\\end{proof}\n\nIn general, not all the extensions of numerical semigroup are arithmetic, as we will see later on. One of the aims of this section is to characterize the numerical semigroups $\\Delta$ such that $\\mathcal{F}(\\Delta)$ agrees with the set of extension of $\\Delta$.\n\nIf $S$ is a numerical semigroup. An element $x \\in \\mathbb{N} \\setminus S$ is said to be a \\textbf{fundamental gap} of $S$ if $\\{2x,3x\\} \\subset S$ (see Section 5 of Chapter 3 in \\cite{libro} or \\cite{fundamental}). The set of fundamental gaps of $S$ is denoted by $\\operatorname{FG}(S)$. Observe that $\\operatorname{F}(S) \\in \\operatorname{FG}(S)$, provided that $S \\neq \\mathbb{N}$.\n\nThe proof of the following result is trivial.\n\n\\begin{lemma}\\label{lemma2}\nLet $S$ be a numerical semigroup and let $x$ be a gap of $S$. Then $x \\in \\operatorname{FG}(S)$ if and only if $\\big\\{k\\,x \\mid k \\in \\mathbb{N} \\setminus \\{1\\} \\big\\} \\subseteq S$.\n\\end{lemma}\n\nThe proof of the next result is straightforward, too.\n\n\\begin{lemma}\\label{lemma3}\nIf $S \\neq \\mathbb{N}$ is a numerical semigroup, then\n\\begin{enumerate}[(a)]\n\\item $\\frac{S}d = \\mathbb{N}$ if and only if $d \\in S$.\n\\item $\\frac{S}{d} = \\langle 2, 3 \\rangle$ if and only if $d \\in \\operatorname{FG}(S)$. In particular, $\\frac{S}{\\operatorname{F}(S)} = \\langle 2, 3 \\rangle,$\n\\item $\\frac{S}2 \\cap \\frac{S}3 = S \\cup \\operatorname{FG}(S)$,\n\\item if $d \\in \\mathbb{N} \\setminus \\{0,1\\},$ then $S \\cup \\operatorname{FG}(S) \\subseteq \\frac{S}d$.\n\\end{enumerate}\n\\end{lemma}\n\nNotice that if $S$ is a numerical semigroup, then $\\mathbb{N}$, $\\frac{S}{\\operatorname{F}(S)}$ and $S \\cup \\operatorname{FG}(S)$ are (non-necessarily different) arithmetic extension of $S$. Furthermore, the first and the third ones are proper extensions of $S$ and the second one is proper if and only if $S \\neq \\langle 2, 3 \\rangle$. \n\n\\begin{example}\\label{Ex3}\\mbox{}\\par\n\\begin{enumerate}[1.]\n\\item The only proper extension of $\\langle 2,3\\rangle$ is $\\mathbb{N}$.\n\\item The proper extensions of $\\langle 3,4,5\\rangle$ are $\\mathbb{N}$ and $\\langle 2,3 \\rangle = \\frac{\\langle 3,4,5\\rangle}{2}$.\n\\item The proper extensions of $\\langle 2,5 \\rangle$ are $\\mathbb{N}$ and $\\langle 2,3 \\rangle = \\frac{\\langle 2,5 \\rangle}{3}$.\n\\item The proper extensions of $\\langle 3,5,7 \\rangle$ are $\\mathbb{N},$ $\\langle 2,3 \\rangle = \\frac{\\langle 3,5,7\\rangle}{4}$ and $\\langle 3,4,5\\rangle = \\frac{\\langle 3,5,7\\rangle}{2}$.\n\\item The only proper extensions of $\\langle 4,5,7 \\rangle$ are $\\mathbb{N}$, $\\langle 2,3 \\rangle = \\frac{\\langle 4,5,7\\rangle}{6},\\ \\langle 3,4,5 \\rangle = \\frac{\\langle 4,5,7\\rangle}{3},\\ \\langle 2,5 \\rangle = \\frac{\\langle 4,5,7\\rangle}{2}$ and $\\langle 4,5,6,7 \\rangle = \\frac{\\langle 4,5,7\\rangle}{2} \\cap \\frac{\\langle 4,5,7\\rangle}{3}$.\n\\end{enumerate}\nNotice that we have shown that all the extensions of the following numerical semigroups: $\\langle 2,3\\rangle,$ $\\langle 3,4,5\\rangle,$ $\\langle 2,5 \\rangle,$  $\\langle 3,5,7 \\rangle$ and $\\langle 4,5,7 \\rangle$ are arithmetic.\n\\end{example}\n\nNotice that the set $\\mathcal{F}(\\Delta)$ is partially ordered by inclusion. Moreover, we have the following:\n\n\\begin{proposition}\\label{Prop4}\nIf $\\Delta \\neq \\mathbb{N}$ is a numerical semigroup and $\\mathcal{F}(\\Delta)$ is the set of arithmetic extensions of $\\Delta$, then\n\\begin{enumerate}[(a)]\n\\item $\\max(\\mathcal{F}(\\Delta)) = \\mathbb{N}$, \n\\item $\\min(\\mathcal{F}(\\Delta)) = \\Delta$,\n\\item $\\max(\\mathcal{F}(\\Delta) \\setminus \\{\\mathbb{N}\\}) = \\langle 2, 3 \\rangle,$\n\\item $\\min(\\mathcal{F}(\\Delta) \\setminus \\{\\Delta\\}) = \\Delta \\cup \\operatorname{FG}(\\Delta)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n(a) Notice that that $\\frac{\\Delta}d = \\mathbb{N}$ for every $d \\in \\Delta$ and that all numerical semigroups are submonoids of $\\mathbb{N}$.\n\n(b) If $\\Delta \\neq \\mathbb{N}$, then $1 \\not\\in \\Delta$ and $\\frac{\\Delta}1 = \\Delta$. Now, it suffices to observe that $\\Delta \\subset \\frac{\\Delta}d$, for every $d \\in \\mathbb{N} \\setminus \\{0,1\\}$, to conclude that every arithmetic extension of $\\Delta$ contains $\\Delta$.\n\n(c) By part (b) of Lemma \\ref{lemma3}, we have that $\\langle 2,3\\rangle \\in \\mathcal{F}(\\Delta)$. Now, it suffices to observe that if $S \\neq \\mathbb{N}$ is a numerical semigroup, then $1 \\not\\in S$ and consequently $S \\subseteq \\langle 2,3 \\rangle$.\n\n(d) By part (c) of Lemma \\ref{lemma3} and the remark before Proposition \\ref{Prop4}, we have that $\\Delta \\cup \\operatorname{FG}(\\Delta) \\in \\mathcal{F}(\\Delta)\\setminus\\{\\Delta\\}$. If $S \\in \\mathcal{F}(\\Delta) \\setminus \\{\\Delta\\}$, then there exists $\\{d_1, \\ldots, d_n\\} \\subseteq (\\mathbb{N}\\setminus S) \\setminus \\{1\\}$ such that $S = \\frac{\\Delta}{d_1}\\cap \\ldots \\cap \\frac{\\Delta}{d_n}$. By part (d) of Lemma \\ref{lemma3}, we have that $\\Delta \\cup \\operatorname{FG}(\\Delta) \\subseteq \\frac{\\Delta}{d_i}$ for every $i = 1, \\ldots, n$. So, $\\Delta \\cup \\operatorname{FG}(\\Delta) \\subseteq S$.\n\\end{proof}\n\nThe next example shows (among other things) that there exist extensions of a numerical semigroup that are not arithmetic.\n\n\\begin{example}\\label{Ejem5}\nLet $S = \\langle 5,7,9 \\rangle$. By direct checking, one can see that $\\operatorname{FG}(S) = \\{6,8,11,13\\}$. Clearly, $S \\cup \\{13\\}$ is an extension $S$ and $S \\subsetneq S \\cup \\{13\\} \\subsetneq S \\cup \\operatorname{FG}(S)$. Now, by part (d) of Proposition \\ref{Prop4}, we have the $S \\cup \\{13\\}$ cannot be an arithmetic extension of $S$. \n\\end{example}\n\nNow, we can take advantage of Proposition \\ref{Prop4} to characterize the numerical semigroups whose extensions are all arithmetic.\n\n\\begin{theorem}\\label{Th 9}\nThe only numerical semigroups whose extensions are all arithmetic are $\\mathbb{N},\\ \\langle 2,3 \\rangle,\\ \\langle 3,4,5 \\rangle,\\ \\langle 2,5 \\rangle,\\ \\langle 3,5,7 \\rangle$ and $\\langle 4,5,7 \\rangle$.\n\\end{theorem}\n\nBefore proving the previous theorem, we need some lemmas.\n\n\\begin{lemma}\\label{lemma6}\nIf $S$ is a numerical semigroup and the cardinal of $\\operatorname{FG}(S)$ is greater than $1$, then $S$ has non-arithmetic extensions.\n\\end{lemma}\n\n\\begin{proof}\nSince $S \\cup \\{\\operatorname{F}(S)\\}$ is a proper extension of $S$ and $\\{\\operatorname{F}(S)\\} \\subsetneq \\operatorname{FG}(S)$, it follows that $S \\cup \\{\\operatorname{F}(S)\\} \\subsetneq S \\cup \\operatorname{FG}(S)$. Thus, by part (d) of Proposition \\ref{Prop4}, $S \\cup \\{\\operatorname{F}(S)\\}$ cannot be an arithmetic extension of $S$\n\\end{proof}\n\n\\begin{lemma}\\label{lemma7}\nIf $S$ is a numerical semigroup and $\\operatorname{FG}(S) = \\{\\operatorname{F}(S)\\}$, then the set gaps of $S$ is $\\big\\{d \\in \\mathbb{N} \\mid d\\ \\text{divides}\\ \\operatorname{F}(S) \\big\\}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $d$ be a gap of $S$, that is to say, $d \\in \\mathbb{N} \\setminus S$. Since $\\mathbb{N} \\setminus S$ is finite, there exists $n \\in \\mathbb{N} \\setminus \\{0\\}$ such that $n d \\in \\mathbb{N} \\setminus S$, and $\\big\\{2(nd), 3(nd)\\big\\} \\subset S$. So, $nd$ is a fundamental gap. Then $n d \\in \\operatorname{FG}(S) = \\{\\operatorname{F}(S)\\}$ by hypothesis, and we conclude that $d$ divides $\\operatorname{F}(S)$. Conversely, if $d$ divides $\\operatorname{F}(S)$, say $F(S) = n d$, then $d \\not\\in S$; otherwise, $F(S) = n d \\in S$ which is impossible. \n\\end{proof}\n\n\\begin{lemma}\\label{lemma8}\nLet $F$ be a positive integer number. Then \\[\\mathbb{N} \\setminus \\big\\{d \\in \\mathbb{N} \\mid d\\ \\text{divides}\\ F \\big\\}\\] is a numerical semigroup if and only if $F \\in \\{1,2,3,4,6\\}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\big\\{d \\in \\mathbb{N} \\mid d\\ \\text{divides}\\ F \\big\\}$ is a finite set, we have that the set $\\mathbb{N} \\setminus \\big\\{d \\in \\mathbb{N} \\mid d\\ \\text{divides}\\ F \\big\\}$ is a numerical semigroup if and only if given two positive integers $x$ and $y$ not dividing $F$, then their sum does not divides $F$ either. By \\cite[Lemma 2]{fundamental}, it is easy to prove this is only possible if, and only if, $F \\in \\{1,2,3,4,6\\}$.\n\\end{proof}\n\n\\noindent\\emph{Proof of Theorem \\ref{Th 9}.} Let $S \\neq \\mathbb{N}$ be a numerical semigroup whose extensions are all arithmetic. By Lemma \\ref{lemma6}, $\\operatorname{FG}(S) = \\{\\operatorname{F}(S)\\}$. Then, by Lemma \\ref{lemma7}, \\[S = \\mathbb{N} \\setminus \\big\\{d \\in \\mathbb{N} \\mid d\\ \\text{divides}\\ \\operatorname{F}(S) \\big\\}.\\] Now, since $S$ is a numerical semigroup, by Lemma \\ref{lemma8} we have that $\\operatorname{F}(S) \\in \\{1,2,3,4,6\\}$. Therefore $S$ is equal to one of the following numerical semigroups $\\mathbb{N} \\setminus \\{1\\} = \\langle 2,3 \\rangle,\\ \\mathbb{N} \\setminus \\{1,2\\} = \\langle 3,4,5 \\rangle,\\ \\mathbb{N} \\setminus \\{1,3\\} = \\langle 2, 5 \\rangle,\\ \\mathbb{N} \\setminus \\{1,2,4\\} = \\langle 3,5,7 \\rangle$ or $\\mathbb{N} \\setminus \\{1,2,3,6\\} = \\langle 4,5,7 \\rangle$.\n\nFinally, since all the extensions of the five above numerical semigroups are arithmetic, as it was shown in Example \\ref{Ex3}, we are done.\\qed\n\n\\section[The Ap\\'ery set of the quotient]{The Ap\\'ery set of the quotient of a numerical semigroup}\\label{Sect2}\n\nIf $S$ is a numerical semigroup, then the \\textbf{Ap\\'ery set of $S$ with respect to $n \\in S \\setminus \\{0\\}$} is $\\operatorname{Ap}(S,n) := \\{x \\in S \\mid x-n \\not\\in S\\}.$ It is known (see, for instance, \\cite[Lemma 2.4]{libro}) that \\begin{equation}\\label{ecu1}\\operatorname{Ap}(S,n) = \\{0=\\omega(0),\\omega(1), \\ldots, \\omega(w-1)\\}\\end{equation} where $\\omega(i)$ is the least element in $S$ such that $\\omega(i) \\equiv i\\ \\text{mod}\\  n$, for each $i = 0, \\ldots, n-1.$ In particular, we have that the cardinality of $\\operatorname{Ap}(S,n)$ is $n$.\n\n\\begin{remark}\\label{Rem10}\nNotice that given $x \\in \\mathbb{N}$ we have that $x \\in S$ if and only if there exists $(k,\\omega) \\in \\mathbb{N} \\times \\\n\\operatorname{Ap}(S,n)$ such that $x = k n + \\omega$. Therefore, $(\\operatorname{Ap}(S,n)\\setminus\\{0\\} ) \\cup \\{n\\}$ is system of generators of $S$.\n\\end{remark}\n\nThe Ap\\'ery sets of $S$ provides formulas for the computation of $\\operatorname{F}(S)$ and $\\operatorname{g}(S)$:\n\n\\begin{proposition}\\label{Prop11}\nIf $S$ is a numerical semigroup and $n \\in S \\setminus \\{0\\}$, then \n\\begin{enumerate}[(a)]\n\\item $\\operatorname{F}(S) = \\max \\operatorname{Ap}(S,n) - n$.\n\\item $\\operatorname{g}(S) = \\frac{1}n \\sum_{\\omega \\in \\operatorname{Ap}(S,n)} \\omega - \\frac{n-1}2.$  \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFor a proof, see for instance \\cite[Proposition 2.12]{libro}.\n\\end{proof}\n\nAs usual $\\lfloor r \\rfloor$ and $\\lceil r \\rceil$ stand for the integer part and the upper integer part of the rational number $r$, respectively. If $a$ and $b$ are integer numbers, then we write $(a\\ \\text{mod}\\  b)$ for the remainder of the Euclidean of $a$ by $b$, in symbols, $(a\\ \\text{mod}\\  b) = a - \\lfloor \\frac{a}b \\rfloor b$.\n\n\\begin{theorem}\\label{Th 12}\nLet $S$ be a numerical semigroup and $n \\in S \\setminus \\{0\\}$. If $\\operatorname{Ap}(S,n) = \\{0=\\omega(0),\\omega(1), \\ldots, \\omega(n-1),\\}$ and $a \\in \\mathbb{N} \\setminus S$, then \n\\[\\operatorname{Ap}\\left(\\frac{S}a,n\\right) = \\{0, n\\, \\kappa_1 + 1, n\\, \\kappa_2 + 2, \\ldots, n\\, \\kappa_{n-1} + n-1\\},\\] where $\\kappa_i = \\Big\\lceil \\frac{\\omega\\big(\nai\\ \\text{mod}\\  n\\big) - a\\,i}{a\\, n} \\Big\\rceil,\\ i = 1, \\ldots, n-1$.\n\\end{theorem}\n\n\\begin{proof}\nLet $i \\in \\{1, \\ldots, n-1\\}$ and $k \\in \\mathbb{N}.$ Then $k\\, n + i \\in \\frac{S}a$ if and only if $a(k\\, n + i) \\in S$. Thus, by Remark \\ref{Rem10}, we have that $a\\, k\\, n + a\\, i \\in S$ if and only if $a\\, k\\, n + a\\, i  \\geq \\omega\\big(ai\\ \\text{mod}\\  n\\big)$. Therefore, $k\\, n + i \\in \\frac{S}a$ if and only if \\[k \\geq \\frac{\\omega\\big(ai\\ \\text{mod}\\  n\\big) - a\\,i}{a\\, n}.\\] Hence, the smallest element in $\\frac{S}a$ that is congruent with $i$ modulo $n$ is $\\Big\\lceil \\frac{\\omega\\big(ai\\ \\text{mod}\\  n\\big) - a\\,i}{a\\, n} \\Big\\rceil n + i$. Now, by \\eqref{ecu1}, we are done. \n\\end{proof}\n\nGiven a numerical semigroup $S$, we can use the above result to compute the Ap\\'ery set of the quotient of $S$ by $b$, and thereafter use Proposition \\ref{Prop11}, to compute the Frobenius number and the genus of $\\frac{S}b$.  Let us illustrate these computations through an example.\n\n\\begin{example}\nLet $S = \\langle 7,8 \\rangle$. Then $\\mathrm{Ap}(S,7) = \\{\\omega(0)=0, \\omega(1)=8, \\omega(2)=16, \\omega(3)=24,  \\omega(4)=32, \\omega(5)=40, \\omega(6)=48\\}$. If $\\overline S = \\frac{S}3$, then, by Theorem \\ref{Th 12}, we have that \n\\[\\mathrm{Ap}\\left(\\overline S, 7\\right) = \\{0\\} \\cup \\{7\\, \\kappa_i + i \\mid i = 1, \\ldots, 6\\},\\] where $\\kappa_i = \\Big\\lceil \\frac{\\omega\\big(3i\\ \\text{mod}\\  7)\\big) - 3\\,i}{21} \\Big\\rceil,\\ i = 1, \\ldots, 6$. In this case, $\\kappa_1 = 1, \\kappa_2 = 2, \\kappa_3 = 1, \\kappa_4 = 2, \\kappa_5 = 0$ and $\\kappa_6 = 1$. Therefore, $\\mathrm{Ap}\\left(\\overline S, 7\\right) = \\{0,8,16,10,18,5,13\\}$ and, consequently, \\[\\operatorname{F}(\\overline S) = 18-7 = 11\\] and \\[\\operatorname{g}(\\overline S) = \\frac{1}7 (8+16+10+18+5+13)-\\frac{7-1}2 = 10 - 3 = 7\\] by Proposition \\ref{Prop11}.\n\\end{example}\n\nClearly, if $S$ and $T$ are numerical semigroups, then $S \\cap T$ is a numerical semigroup, too. Therefore, from expression \\eqref{ecu1} we deduce the following result.\n\n\\begin{proposition}\\label{Prop 16}\nLet $S$ and $T$ be numerical semigroups and $n \\in (S \\cap T) \\setminus \\{0\\}$. If $\\operatorname{Ap}(S,n) = \\{0=\\omega(0),\\omega(1), \\ldots, \\omega(n-1),\\}$ and $\\operatorname{Ap}(S',n) = \\{0=\\omega'(0),\\omega'(1), \\ldots, \\omega'(n-1),\\}$, then $\\operatorname{Ap}(S \\cap T, n) = \\{0=\\bar \\omega(0),\\bar \\omega(1), \\ldots, \\bar \\omega(n-1),\\}$ where $\\bar \\omega(i) = \\max\\big(\\omega(i), \\omega'(i)\\big),$ para todo $i = 1, \\ldots, n-1$.\n\\end{proposition}\n\nThus, combining Theorem \\ref{Th 12} and Proposition \\ref{Prop 16}, we can compute the Ap\\'ery set of all the arithmetic extension of $\\Delta$. \n\n\\begin{example}\nBy direct computation, one can see that the Ap\\'ery set of $\\langle 4,5,7 \\rangle$ with respect to $4$ is $\\{0,5,10,7\\}$. By Theorem \\ref{Th 12}, we have that the Ap\\'ery set of $\\frac{\\langle 4,5,7 \\rangle}2 = \\langle 2,5 \\rangle$ and $\\frac{\\langle 4,5,7 \\rangle}3 = \\langle 3,4,5 \\rangle$ with respect to $4$ are $\\{0,5,2,7\\}$ and $\\{0,5,6,3\\}$, respectively. Therefore, the Ap\\'ery set of $\\frac{\\langle 4,5,7 \\rangle}2 \\cap \\frac{\\langle 4,5,7 \\rangle}3 = \\langle 4,5,6,7 \\rangle$ is $\\{0,5,6,7\\}$.\n\\end{example}\n\nUsing the notation introduced in \\cite{proportionally}, a proportionally modular Diophantine inequality is an expression of the form \\begin{equation}\\label{ecu2} a\\, x\\ \\text{mod}\\  b \\leq c\\, x,\\end{equation} where $a,b$ and $c$ are positive integer numbers. Given $a,b$ and $c \\in \\mathbb{N} \\setminus \\{0\\},$ the set of positive integer solutions of \\eqref{ecu2} constitutes a numerical semigroup (\\cite[Theorem 13]{proportionally}). These semigroups are called proportionally modular numerical semigroups (PM-semigroups, for short). In \\cite[Corollary 3.5]{archiv} the following characterization of PM-semigroup was given.\n\n\\begin{proposition}\\label{Prop 13}\nIf $S$ is a PM-semigroup, there exist $a$ and $b \\in \\mathbb{N} \\setminus \\{0\\}$ such that $S = \\frac{\\langle a,a+1 \\rangle}b$.\n\\end{proposition}\n\n\\begin{remark}\\label{Rem 13}\nIt is a challenging problem to find formulas for the Frobenius number or the genus of $\\frac{\\langle a,a+1 \\rangle}b$ in terms of $a$ and $b$. However, since \\[\\operatorname{Ap}(\\langle a,a+1 \\rangle, a) = \\{\\omega(0)=0, \\omega(1)=a+1, \\ldots, \\omega(a-1) = (a-1)a + (a-1)\\},\\] we can use Proposition \\ref{Prop11} and Theorem \\ref{Th 12} to conclude that \n\\[\n\\operatorname{F}\\left(\\frac{\\langle a,a+1 \\rangle}b\\right) = \\max_{i=1, \\ldots, a-1} \\left(\\Big\\lceil \\frac{(bi\\ \\text{mod}\\ a) (a+1) - b\\,i}{a\\, b} \\Big\\rceil a + i\\right) - a\n\\]\nand \n\\begin{align*}\n\\operatorname{g}\\left(\\frac{\\langle a,a+1 \\rangle}b\\right) & = \\frac{1}a \\left(\\sum_{i=1}^{a-1}  \\Big\\lceil \\frac{(bi\\ \\text{mod}\\ a) (a+1) - b\\,i}{a\\, b} \\Big\\rceil a + i \\right) - \\frac{a-1}2 \\\\ & =\\sum_{i=1}^{a-1}  \\Big\\lceil \\frac{(bi\\ \\text{mod}\\ a) (a+1) - b\\,i}{a\\, b} \\Big\\rceil.\n\\end{align*}\nThus, by Proposition \\ref{Prop 13}, we have obtained formulas for the Frobenius number and the genus of any PM-semigroup. \n\nObserve that from Proposition \\ref{Prop 16} and Remark \\ref{Rem 13} we can deduce a formula for the Frobenius number of any T-semigroup (see expression \\eqref{ecu0}).\n\\end{remark}\n\n\\section[An algorithm to compute the arithmetic extensions]{An algorithm to compute the arithmetic extensions of a numerical semigroup}\\label{Sect3}\n\nLet $\\Delta$ be a numerical semigroup. Our main aim in this section is to give an algorithm for the computation of $\\mathcal{F}(\\Delta)$. \n\n\\begin{definition}\nLet $S$ be a numerical semigroup and let $n \\in S \\setminus \\{0\\}$. If $\\operatorname{Ap}(S,n) = \\{0 = \\omega(0), \\omega(1) = \\kappa_1 n + 1, \\ldots, \\omega(n-1) = \\kappa_{n-1} n + n-1\\}$, then the vector \\[\\varphi_n(S) := (\\kappa_1, \\ldots, \\kappa_{n-1})\\] is called the $n-$th \\textbf{Kunz-coordinates vector of $S$}.\n\\end{definition}\n\nThe Kunz-coordinates vector of a numerical semigroup where introduced by V.Blanco and J.Puerto in \\cite{BP}.  \n\nObserve that Theorem \\ref{Th 12} provides a formula for $\\varphi_n\\left(\\frac{\\Delta}a\\right)$ in terms of $a, n$ and $\\operatorname{Ap}(\\Delta,n)$.\n\n\\medskip\n\\textbf{Notation.}\nGiven $\\mathbf x = (x_1, x_2, \\ldots, x_{n-1})$ and $\\mathbf y = (y_1, y_2, \\ldots, y_{n-1}) \\in \\mathbb{N}^{n-1}$, we write $\\mathbf x \\vee \\mathbf y := \\big(\\max(x_1,y_1), \\max(x_2,y_2), \\ldots, \\max(x_{n-1},y_{n-1}) \\big).$\n\n\\medskip\nAs an immediate consequence of Proposition \\ref{Prop 16} we have the following result.\n\n\\begin{corollary}\\label{Cor17}\nIf $S$ and $T$ are numerical semigroups and $n \\in (S \\cap T) \\setminus \\{0\\}$, then $\\varphi_n(S \\cap T) = \\varphi_n(S) \\vee \\varphi_n(T)$.\n\\end{corollary}\n\nThe following result is one of the key point in our algorithm.\n\n\\begin{corollary}\nIf $\\Delta \\neq \\mathbb{N}$ is numerical semigroups and $n \\in S \\setminus \\{0\\}$, then set of $n-$th Kunz-coordinates vectors of the arithmetic extensions of $\\Delta$ different from $\\mathbb{N}$ is \\[K:= \\left\\{\\bigvee_{a \\in X} \\varphi_n\\left(\\frac{\\Delta}a\\right) \\mid X \\subseteq \\mathbb{N}\\setminus \\Delta,\\ X \\neq \\varnothing\\ \\text{and}\\ X = \\operatorname{msg}(\\langle X \\rangle) \\right\\}\\]\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem \\ref{Th1}, the set of arithmetic extensions, $\\mathcal{F}(\\Delta)$, of $\\Delta$ is equal to $\\Big\\{\\displaystyle{\\bigcap_{a \\in X} \\frac{\\Delta}a} \\mid X\\ \\text{is a non-empty subset of}\\ \\mathbb{N}\\setminus \\Delta\\ \\text{and}\\ X = \\operatorname{msg}(\\langle X \\rangle) \\Big\\} \\cup \\mathbb{N}$. Thus, by Corollary \\ref{Cor17}, our claim follows.\n\\end{proof}\n\nNow, we are in condition to give an algorithm for the computation of $\\mathcal{F}(\\Delta)$.\n\n\\begin{algorithm}\\label{14}\\mbox{}\\par\n\\textsc{Input:} A numerical semigroup $\\Delta$.\\par\n\\textsc{Output:} $\\mathcal{F}(\\Delta).$\n\\begin{itemize}\n\\item[1.] Set $\\mathcal{F}(\\Delta) = \\{\\mathbb{N},\\langle 2,3 \\rangle, \\Delta\\}$ and $S:=\\Delta \\cup \\operatorname{FG}(\\Delta)$.\n\\item[2.] Compute $m:=\\operatorname{m}(S)$ and $G:=\\mathbb{N} \\setminus S$.\n\\item[3.] For each $a \\in G$ compute $\\mathbf{v}_a := \\varphi_m\\left(\\frac{\\Delta}a\\right)$.\n\\item[4.] Compute the set $\\mathcal{X} = \\{X \\subseteq \\mathbb{N}\\setminus \\Delta,\\ X \\neq \\varnothing\\ \\text{and}\\ X = \\operatorname{msg}(\\langle X \\rangle)\\}.$ \n\\item[5.] Compute $K := \\Big\\{\\displaystyle{\\bigvee_{a \\in X} \\mathbf{v}_a} \\mid X \\in \\mathcal X\\Big\\}$.\n\\item[6.] For each $(\\kappa_1, \\ldots, \\kappa_{m-1}) \\in K$, append the numerical semigroup generated by $\\{ m, \\kappa_1 m + 1, \\ldots, \\kappa_{m-1} m + m-1\\}$ to $\\mathcal{F}(\\Delta)$.\n\\end{itemize}\n\\end{algorithm}\n\nThe following GAP \\cite{gap} function is a rude implementation of Algorithm \\ref{14}. This function requires the GAP package \\texttt{NumericalSgps} \\cite{numericalsgps}.\n\n{\\small\n\\begin{verbatim}\n ArithmeticExtensions:=function(D)\n  local FD,G,S,m,v,a,pow,C,x,sg;\n\n  FD:=[NumericalSemigroup([1]),NumericalSemigroup(2,3), D];\n  G:=Gaps(D);\n  S:=NumericalSemigroupByGaps(Difference(G,FundamentalGaps(D))); \n \n  m:=Multiplicity(S);\n  G:=GapsOfNumericalSemigroup(S);\n\n  v:=[];\n  for a in G do\n   Append(v,[KunzCoordinates(S\/a,m)]); \n  od;\n  v:=Set(v);\n\n  pow:=Combinations(v);\n  pow:=pow{[2..Length(pow)]};\n\n  C:=[];\n  for x in pow do\n   x:=TransposedMat(x); \n   Append(C,[List(x,i->Maximum(i))]);\n  od;\n  C:=Set(C);\n\n  for x in C do\n   sg:=Concatenation([m],List(x,i->m*i)+[1..Length(x)]);\n   Append(FD,[NumericalSemigroup(sg)]);\n  od;\n  return Set(FD);\n end;\n\\end{verbatim}\n}\n\n\\begin{example}\nWe can compute the arithmetical extension of $\\Delta = \\langle 4,6,7 \\rangle$ by using the following commands\n{\\small\n\\begin{verbatim}\n LoadPackage(\"NumericalSgps\");\n D:=NumericalSemigroup(4,6,7);\n ArithmeticExtensions(D);\n\\end{verbatim}}\nBy this way, we obtain that $\\Delta$ has four arithmetic extensions, say $\\mathbb{N},  \\langle 2,3 \\rangle, \\langle 2,5 \\rangle$ and $\\langle 4,6,7 \\rangle.$ \n\\end{example}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec-int}\nA massive star builds up onion-like layers of different chemical elements\nsynthesized by hydrostatic nuclear burning processes during its lifetime.\nAt the end of its evolution,\nthe innermost Fe core collapses into a neutron star,\nwhich triggers a core-collapse supernova (SN) explosion.\nThe detailed process of the explosion is complicated and poorly understood,\nbut a consensus from theoretical studies suggests that \nthe explosion should be asymmetric and turbulent, especially near the core\n\\citep[e.g.,][and references therein]{sum05,tak14,gil15}.\nMulti-dimensional numerical simulations have shown that \nthe explosion also leads to extensive mixing and inversion\namong the stratified layers by hydrodynamic instabilities\n\\citep[e.g.][]{kif03,kif06,ham10,mao15}.\nA distinct method to explore the explosion dynamics of SNe, therefore,  \nwould be to investigate the detailed chemical and kinematic properties \nof SN ejecta material in nearby young Galactic supernova remnants (SNRs)\nwhere the imprints of explosion remain.\n\nCassiopeia A (hereafter Cas A),\nat the age of $\\sim 340$ years \\citep{tho01,fes06b},\nis one of the best studied young Galactic SNRs.\nIts SN explosion was classified as Type IIb\nbased on the optical spectra of light echoes\n\\citep{kra08,res11},\nwhich implies that the progenitor was probably a star of $15$--$20~M_{\\sun}$\nthat had lost a significant portion, but not all,\nof its H-rich envelope before the explosion.\nOver the past several decades, Cas~A has been extensively studied in \nalmost all wavebands from radio to gamma-rays.\nThe complex spatial distribution of {\\em shocked} SN ejecta, such as\nthe X-ray-emitting Fe-rich ejecta plumes beyond the Si-rich material\nand O\/S-rich optical knots outlying the bright ejecta shell\n\\citep[e.g.,][]{hug00, ham08},\nindicates an explosion resulting in an inversion of the chemical layers.\nFurthermore, the inhomogeneous distribution of {\\em unshocked} SN ejecta\nradiating Si and Ti emission lines \\citep{ise10,gre14} implies that\nthe explosion was turbulent near the progenitor core.\n\nIn the visible waveband,\nmany bright optical knots are presented in and around Cas A.\nThey have been classified into two major groups based on\ntheir proper motions and line-of-sight velocities:\n(1) fast-moving knots (FMKs) and (2) quasi-stationary flocculi (QSFs).\nThe FMKs show large proper motions and high radial velocities,\ncorresponding to expansion velocities of up to $\\sim10^{4}~{\\rm km~s}^{-1}$\n\\citep[e.g.,][]{fes96,ham08}.\nThey have spectral features strongly enhanced in\nO and other heavy elements (e.g., S and Ar)\nthat are mostly synthesized from the nuclear burning process\nin a deep stellar layer,\nwhile showing no detectable H or He emission lines\n\\citep[e.g.,][]{fes96,hur96}.\nBased on their high expansion velocities and chemical composition,\nthey have been regarded as dense material ejected from\nthe disrupted layer of the progenitor after the SN explosion.\nThe dynamical and chemical properties of the QSFs are very different from\nthose of the FMKs.\nThe QSFs have considerably slower velocities ($v<500~{\\rm km~s}^{-1}$), and so\ntheir typical expansion age is $11{,}000\\pm2000$ years \\citep{van85}, \nwhich is much larger than the age of the remnant.\nThey are bright in\n[\\ion{N}{2}]\\ and H$\\alpha$\\ emission,\nwith a handful of other H and He emission lines,\nin optical spectra \\citep[e.g.,][]{hur96,ala14}.\nConsidering these properties,\nQSFs are believed to be dense circumstellar medium (CSM)\nblown out from the progenitor prior to the SN explosion.\nIn addition, some intermediate optical knots,\nthat is, the so called fast-moving flocculi (FMFs) or nitrogen knots (NKs),\nwere reported by \\citet{fes87}.\nWhile their spectra, which show strong\n[\\ion{N}{2}]\\ 6548, 6583 \\AA\\footnote{All wavelengths listed in this paper are ``air wavelengths.''}\naccompanied by weak H$\\alpha$\\ without any lines of O and S,\nare analogous to those of QSFs \\citep{fes87,fes06a,fes01a},\ntheir proper motions are larger than $0\\farcs3~{\\rm yr}^{-1}$,\ncorresponding to $\\sim 5000~{\\rm km~s}^{-1}$ \\citep{ham08}.\nMost of them have been found outside of the bright main ejecta shell.\nTherefore, these outlying N-rich knots have been interpreted as\nbeing fragments of the progenitor's photosphere\nexpelled by the SN blast wave at the time of explosion.\n\nThe dynamical and chemical properties of these different types of optical knots\nhave provided important clues to unveiling the SNR's origin and evolution.\nFor example, the expansion center and age were determined from\nthe proper motions of the FMKs\n\\citep[e.g.,][and references therein]{tho01,fes06b},\nand the three-dimensional (3D) structure of the ejecta knots\nreconstructed from spectral mapping observations\nhas shown that the SN ejecta is expanding spherically\nbut is systematically receding at a speed of $800~{\\rm km~s}^{-1}$\nat a distance of 3.4 kpc \\citep{ree95,del10,mil13,ala14}.\nThe dense, slow-moving QSFs indicate that \nthe progenitor had undergone significant and inhomogeneous mass loss\nduring the red supergiant phase \\citep{che03}.\n\nAlthough this velocity-based classification is an efficient way\nto classify the knots as SN ejecta and CSM,\nwhich have distinctive expansion velocities,\ni.e., a few $1000~{\\rm km~s}^{-1}$ vs. a few $100~{\\rm km~s}^{-1}$,\nit encounters limitations when characterizing the SN ejecta material from\nthe different nucleosynthetic layers.\nAccording to previous numerical simulations for core-collapse SN explosions,\nthe radial velocity profiles of heavy elements in SN ejecta\nare almost identical \\citep{kif06,jog09}.\nSeveral models in \\citet{jog09} predict\nthe velocity separation of up to $\\sim1000~{\\rm km~s}^{-1}$\nbetween different heavy elements.\nTheir velocity profiles, however, are very broad (a few $1000~{\\rm km~s}^{-1}$),\nso that their distributions largely overlap.\nThese numerical simulations may imply that\nSN material from different nucleosynthetic layers is barely distinguishable\nin velocity space.\nIn order to comprehend the explosion dynamics, \ntherefore, a more systematic classification of SN ejecta\nbased on their chemical composition is needed.\n\nIn this paper,\nwe report the results of broad near-infrared (NIR) spectroscopic observations\ntoward the main ejecta shell of Cas A,\nfocusing on classification of the emission knots\nbased on their {\\em spectrochemical} properties.\nThe NIR study of Cas A has been relatively limited in the literature,\nalthough there are many bright forbidden lines of various elements\nin the NIR waveband,\nsome of which may arise from the deep nucleosynthetic layers.\nAs far as we are aware,\nthe only NIR spectroscopic study covering the entire {\\it JHK} bandpass\nwas conducted by \\citet{ger01},\nwho obtained low-resolution ($R \\sim 700$) spectra\nof five FMKs and three QSFs that were previously known.\nThey showed that the spectra of the FMKs are dominated by\n[\\ion{S}{2}]\\ 1.029, 1.032, 1.034, 1.037 \\micron\\\n(hereafter [\\ion{S}{2}]\\ 1.03 \\micron\\ multiplets)\nand [\\ion{P}{2}]\\ 1.188 \\micron,\nas well as two high-ionization Si lines,\n[\\ion{Si}{6}]\\ 1.963 \\micron\\ and [\\ion{Si}{10}]\\ 1.430 \\micron,\nwhile those of the QSFs show strong He {\\small I}\\ 1.083 \\micron\\\naccompanied by H emission lines. \nA dozen bright [\\ion{Fe}{2}]\\ lines are also detected in both FMKs and QSFs.\nOur spectra confirm these features.\nHere we present the spectrochemical classification of knots \nand discuss their characteristics.\nThe organization of the paper is as follows. \nIn Section~\\ref{sec-obs},\nwe outline our spectroscopic observations and data reduction procedures.\nAn explanation of how we identified individual knots\nfrom two-dimensional (2D) dispersed images\nand derived their spectral properties\nis given in Section~\\ref{sec-ide}.\nIn Sections~\\ref{sec-pca} and \\ref{sec-dis}, \nwe carry out a classification of the knots using\nprincipal component analysis (PCA) and\ndiscuss the origin of knots in different classes.\nFinally, the paper is summarized in Section~\\ref{sec-sum}.\n\n\n\n\\section{Observations and Data Reduction} \\label{sec-obs}\n\\subsection{Near-infrared Spectroscopy} \\label{sec-obs-spt}\nWe carried out NIR spectroscopic observations of Cas A using TripleSpec\nmounted on the Palomar 5 m Hale telescope.\nTripleSpec is a cross-dispersed NIR spectrograph that\nprovides simultaneous wavelength coverage from 0.94 to 2.46 \\micron\\\nat a spectral resolving power of $R \\sim 2700$ \\citep{wil04,her08}.\nThe spectrograph uses two adjacent quadrants,\ni.e., $2048 \\times 1024$ pixels, of a Rockwell Scientific Hawaii-II array.\nThe slit width and length are 1\\arcsec\\ and 30\\arcsec, respectively.\nOn 2008 June 29 and August 8,\nwe obtained spectra at eight slit positions\nalong the main ejecta shell (Figure~\\ref{fig-slit}).\nThe orientations of Slits 1 and 4 were set perpendicular to the shell,\nwhile those of the others were largely parallel to the shell.\nThe nearby A0V star HD223386 was observed as a spectroscopic standard\nright before and\/or after the target observations at similar airmasses.\nIn addition to the target spectra,\nwe also obtained spectra of sky background by dithering along the slit\nor taking spectra of nearby sky,\ndepending on the complexity of the target fields.\nThe total on-source exposure time at each slit position\nranges from 300 to 1800 s.\nThe detailed parameters of the spectroscopic observations are provided in\nTable~\\ref{tbl-log}.\n\nWe developed an IDL-based data reduction pipeline\nto reduce the obtained TripleSpec data.\nThe pipeline first performed subtraction of a dark frame and flat fielding,\nfollowed by subtraction of sky background emission.\nFor the latter, we used sky background emission\nobtained in a dithered frame or in a frame from nearby empty sky,\ndepending on the complexity of the source emission in a given slit.\nBecause of the non-uniform dispersion by the three cross-dispersing prisms\nin TripleSpec \\citep{wil04,her08},\nsome orders of the TripleSpec spectra are severely curved.\nBy carefully tracing the spectra of standard stars and airglow emission lines,\nwe obtained a 2D wavelength solution for each order\nthat can correct this effect in a satisfactory manner.\nWe conducted fifth-order order polynomial fits\nto the wavelengths of the OH airglow emission lines \\citep{ost97,rou00}\nin the TripleSpec spectra\nto obtain the wavelength solutions at 0.5 \\AA\\ $1\\sigma$ uncertainty\nfor each order.\nHeliocentric velocity correction was also performed\nwhen calculating the velocity of the emission line\ndetected in the TripleSpec spectra.\nWe used an A0V-type standard star (HD223386)\nin the photometric calibration and confirmed that\nthe fluxes of the [\\ion{Fe}{2}]\\ 1.644 \\micron\\ emission line are consistent with\nthose from the narrow-band imaging observations of the same areas\n(see Section~\\ref{sec-obs-img}).\n\n\n\\subsection{Near-infrared Imaging} \\label{sec-obs-img}\nIn 2005 August 28 and 2008 August 11,\nwe performed NIR imaging observations for the remnant\nusing the Wide-field Infrared Camera \\citep[WIRC;][]{wil03}\nattached to the Palomar 5 m telescope.\nThe camera consists of\na single $2048 \\times 2048$ Rockwell Hawaii-II HgCdTe NIR detector\nwith a pixel scale of $0\\farcs2487$ pixel$^{-1}$,\nwhich provides a field of view of $8\\farcm7 \\times 8\\farcm7$.\nWe used an [\\ion{Fe}{2}]\\ narrow-band filter\nthat has a mean wavelength of 1.644 \\micron\\ and a bandwidth of 252 \\AA.\nWhile the single exposure time per frame in 2005 and 2008 is 60 s and 200 s,\nthe multiple dithering observations yield total integration times of\n1200 s and 5400 s, respectively.\nWe also obtained H-continuum narrow-band images\n(mean wavelength of 1.570 \\micron\\ and bandwidth of 236 \\AA)\nin order to subtract the bright stars in the [\\ion{Fe}{2}]\\ narrow-band images.\nThe average seeing throughout our observations was $\\sim 0\\farcs9$ FWHM.\nWe followed standard procedures for the reduction of NIR imaging data.\nFirst, the dark and sky background\nwere subtracted from each individual dithered frame.\nThen, all of the frames were divided by the normalized flat image.\nThe astrometric solution was derived on the basis of\nunsaturated stars around the remnant\nin the Two Micron All Sky Survey (2MASS) Point-Source Catalog\n\\citep[PSC;][]{skr06}.\nWe coadded all dithered frames which were astrometrically aligned.\nIn terms of photometric calibration,\nthe {\\it H}-band magnitude in the 2MASS system was used by assuming that\nthe [\\ion{Fe}{2}]\\ narrow-band magnitude is the same as the {\\it H}-band magnitude.\nThe uncertainty in the zero-point magnitude we derived\nis less than 0.1 mag, corresponding to the 10\\% of the flux.\n\n\n\n\\section{Identification of Knots and Line Parameters} \\label{sec-ide}\nFigure~\\ref{fig-2dspec} shows 2D dispersed images of\nstrong emission lines detected in our TripleSpec spectra:\nred for [\\ion{Fe}{2}]\\ 1.644 \\micron, green for He {\\small I}\\ 1.083 \\micron,\nand blue for [\\ion{P}{2}]\\ 1.188 \\micron\\ + [\\ion{S}{3}]\\ 0.953 \\micron.\nThe emission features are complex, often with multiple peaks,\nand so the identification of individual `knots' by visual inspection\nis not straightforward.\nWe used the IDL routine {\\sc Clumpfind} \\citep{wil94},\nwhich was developed for the identification of clumps in molecular clouds.\nThe routine identifies `clumps' by searching for local maxima\nabove some intensity threshold and\nfollowing them down them to low-intensity levels.\nFor a given 2D dispersed image,\na ``mask'' locating individual knots was generated from\n[\\ion{Fe}{2}]\\ 1.644 \\micron\\ emission features or, if they are weak,\n[\\ion{S}{3}]\\ 0.953 \\micron\\ or [\\ion{P}{2}]\\ 1.188 \\micron\\ emission features,\nand shifted along the wavelength\nto find other emission lines associated with the knots.\nThe details of this knot identification process are given in\nthe Supplement Material of \\citet{koo13}.\nIn total, we identified 63 knots of\ndistinctive kinematical and spectral properties in the 2D dispersed images.\nFigure~\\ref{fig-clumps} shows their locations\nand Table~\\ref{tbl-knot} lists their positions, sizes, and radial velocities.\n\nWe extracted one-dimensional (1D) spectra of individual knots\nusing their mask files (Figure~\\ref{fig-clumps}).\nWe identified 46 emission lines in total\nand derived their parameters by performing single Gaussian fits\nto the detected lines.\nTable~\\ref{tbl-flx} lists the derived line widths and fluxes.\n[\\ion{Fe}{2}]\\ 1.644 \\micron\\ line is detected in all 63 knots\nand their fluxes are also listed in Table~\\ref{tbl-knot}.\nAmong the 46 emission lines,\n43 lines are previously detected lines in SNRs\n\\citep{den81,rud94,hur96,ger01},\nwhereas three lines of [\\ion{Fe}{3}]\\ at 2.145, 2.218, and 2.242 \\micron\\\nare detected for the first time in SNRs.\nThe [\\ion{Fe}{3}]\\ lines originate from transitions\nbetween levels in $^3$G and $^3$H terms \nwith high excitation energies ($\\gtrsim 30{,}000$ K) and   \nhave been detected in a few objects, such as\nSgrA$^{\\ast}$\/IRS16 complex \\citep[][and references therein]{lut93},\nclassical novae \\citep{wag96},\nH {\\small II}\\ regions \\citep{oku01}, and planetary nebulae \\citep{lik06}.\nAmong the previously reported lines, on the other hand,\nthe O {\\small I} 1.1286, 1.1287 \\micron, [\\ion{Fe}{2}]\\ 1.1881 \\micron, and\n[\\ion{Si}{10}]\\ 1.430 \\micron\\ lines are not detected in our spectra.\n\\citet{ger01} reported detection of the\nO {\\small I} lines in the FMKs in Cas A,\nbut we were unable to confirm the detection with our spectra.\nThey also reported detection of\nthe highly ionized [\\ion{Si}{10}]\\ 1.430 \\micron\\ line,\nbut again we were unable to confirm the detection.\nThe [\\ion{Fe}{2}]\\ 1.1881 \\micron\\ line was included in the list of identified lines\nin three QSFs in Cas A and the Kepler SNR by \\cite{ger01},\nbut we consider that\nthis was a misidentification of the [\\ion{P}{2}]\\ 1.1883 \\micron\\ line.  \nThe expected flux of the [\\ion{Fe}{2}]\\ 1.1881 \\micron\\ line in typical conditions\n(e.g., $T \\lesssim 10^{4}$ K, $n_{\\rm e} \\lesssim 10^{5}$ cm$^{-3}$) in SNRs\nis almost negligible,\ni.e., its flux relative to the [\\ion{Fe}{2}]\\ 1.257 \\micron\\ line is\n$\\lesssim 10^{-5}$,\nwhereas the [\\ion{P}{2}]\\ 1.1883 \\micron\\ line could be as strong as\n$\\sim 1\/10$ of the [\\ion{Fe}{2}]\\ 1.257 \\micron\\ line for cosmic abundance,\nor even higher if Fe atoms are locked into dust grains \\citep{koo13}.\n\n\n\n\\section{Principal Component Analysis of Knots' Spectral Properties}\n\\label{sec-pca}\n\\subsection{Method} \\label{sec-pca-method}\nIn order to systematically characterize the spectral properties of the 63 knots,\nwhich have 46 emission lines in total, we applied the PCA method.\nPCA measures the variances among the original input variables\n(i.e., brightness of the emission lines in this study)\nand then sets new orthogonal axes called principal components (PCs)\nalong the largest variances.\nTherefore, the largest variance (or information) is contained in\nthe first PC (PC1), the second most in PC2, and so forth.\nIf there are significant correlations among the original input variables,\nthen the majority of the information is confined within the first few PCs,\nwhich makes it possible to categorize the objects into\na few groups based on the first few PCs.\n\nBefore performing the PCA, we apply an extinction correction \nusing the line ratio of the [\\ion{Fe}{2}]\\ 1.257 and 1.644 \\micron\\ emission.\nThese two lines originate from the same upper level ($a^{4}D_{7\/2}$)\nand therefore their intrinsic flux ratio\n($[F_{\\rm 1.257}\/F_{\\rm 1.644}]_{\\rm int}$)\ndepends on the Einstein A-coefficients ($A_{\\rm ki}$) and their wavelengths,\ni.e., $[F_{\\rm 1.257}\/F_{\\rm 1.644}]_{\\rm int}=(A_{ki,1.257}\/1.257)\/(A_{ki,1.644}\/1.644)$.\nThe A-coefficients, however, are considerably uncertain\nso the line ratio ranges from 0.98 to 1.49\n\\citep[][and references therein]{gia15,koo15} in literature.\nWe adopt a line ratio of 1.36,\nwhich is the value suggested by \\citet{nus88} and \\citet{deb10}.\n(We found that the uncertainty of the theoretical line ratio\ndoes not affect our classification of the knots\nbecause they are well grouped in PC spaces,\nas we will show in Section~\\ref{sec-pca-result-pc}.\nThe criteria of the groups, however, may change\ndepending on the intrinsic line ratio we adopt.\nIn Section~\\ref{sec-pca-result-pc}, we will describe this in more detail.)\nThen, by comparing the observed line flux ratio to the intrinsic ratio,\nwe obtain the extinction of the knots\nand deredden the observed fluxes of all of the lines.\nWe use the general interstellar extinction curve derived from\na carbonaceous-silicate grain model with a Milky Way size distribution\nfor $R_{V}=3.1$ \\citep{dra03}.\nThe derived extinctions are listed in Table~\\ref{tbl-knot}.\nIn our previous work \\citep{leeyh15},\nwe showed that the extinction toward the west is systematically larger than\nthat toward the east,\nwhich is consistent with the previous optical\/X-ray extinction estimates\n\\citep[see Figure~1 in][]{leeyh15}.\nWe further showed that the extinctions of red-shifted knots are\nsystematically higher than those of the blue-shifted knots,\nimplying the presence of a large amount of SN dust inside\nand around the main ejecta shell\n\\citep[see][for details]{leeyh15}.\nThe lines from the same upper level in the dereddened spectral data\ndo not provide independent information any more, \nso that the number of attributes in the PCA are now reduced from 46 to 23.\nIn order to prevent a few bright lines dominating the PCA,\nthe line intensities are standardized by\nsubtracting the mean and dividing by the standard deviation.\nSince we use the mean-subtracted data,\nthe zero PCs represent the location of mean brightness,\nnot the location of the zero fluxes of the lines (hereafter convergent point),\nwhich is important for our classification\n(see Section~\\ref{sec-pca-result-pc}).\nIn order to get the PC coefficients of a knot indicating the convergent point\nwhen all the emission lines of a knot get close to zero flux,\nwe add one artificial knot into the data set\nthat has emission lines with zero flux.\n\n\n\\subsection{Results} \\label{sec-pca-result}\n\\subsubsection{Principal Components and Classification of Knots}\n\\label{sec-pca-result-pc}\nTable~\\ref{tbl-pca} contains the relative and cumulative fraction of variances\ncontributed by the 10 most significant PCs.\nThe first three PCs account for the majority, i.e., $\\sim$85\\%,\nof the spectral information;\nthus, we use them in our classification of the knots.\nFigure~\\ref{fig-atr} shows the projection of the 23 attributes\non the combination plane of the three most significant PCs.\n(This type of plots is known as an {\\it h-plot};\nsee \\citet{ung97} and \\citet{neu07}.)\nWhile Figure~\\ref{fig-atr}(a) and \\ref{fig-atr}(b) show 2D projections\non the planes of (PC1 vs. PC2) and (PC1 vs. PC3),\nFigure~\\ref{fig-atr}(c) shows 3D projections for\nthe three PCs together.\nThe lengths of the vectors in the plots are proportional to\nthe fractional contributions by the spectral line to the given PC,\nand their quadratic sum is equal to unity.\nIn Figure~\\ref{fig-atr}(d),\nwe visualize the 3D projections using the coordinate of\n(Longitude vs. Latitude) representing the two projection angles\non the surface of a sphere.\nWe see in Figure~\\ref{fig-atr} that\nthe attributes can be largely divided into three groups,\neach of which is composed of several strongly correlated spectral lines. \n(Note that the cosine of the angle between the vectors on the plots\nmeasures the linear correlation between the emission lines;\nsee \\citet{neu07} for example.)\nThe first group (hereafter `He group') is composed of\nH {\\small I}, He {\\small I}, and [\\ion{N}{1}]\\ lines.\nThese lines are almost perfectly correlated with each other.\nThe lengths of their vectors are close to unity in the PC1-PC3 plane,\nwhich means that these spectral lines\nare properly accounted for by PC1 and PC3.\nThe second group (`S group') is composed of\n[\\ion{Si}{6}], [\\ion{P}{2}], and ionized S emission lines.\nThese lines are also strongly correlated with each other\nand mostly contributed by PC2 and PC3\nin the direction orthogonal to the He group lines.\nThe last group (`Fe group') is composed of ionized Fe emission lines,\ni.e., [\\ion{Fe}{2}]\\ and [\\ion{Fe}{3}]\\ lines.\nThese lines are rather loosely correlated\nbut are still generally well separated from the lines in the other two groups.\nThe [\\ion{Fe}{2}]\\ 2.046 and 2.224 \\micron\\ lines in particular appear\nsomewhat distinct from the other [\\ion{Fe}{2}]\\ lines. \nThis might be caused by the higher excitation energies of the two lines\nthan those of the other lines, i.e., $\\sim 30{,}000$ K vs. $\\sim 10{,}000$ K.\n\nWe plot the PC coefficients of the 63 knots on the PC planes\nin Figure~\\ref{fig-obj}. \nThe central positions of the planes, where (PC1, PC2, PC3) = (0, 0, 0),\nrepresent the spectrum made by averaging all of the spectra of all 63 knots.\nIn the lower panels of the figure,\nwhich are the enlarged plots of the central areas of the PC planes,\nwe draw dashed lines in order to group the knots (see below) originating from\n(PC1, PC2, PC3) = (-0.10, 0.23, 0.10).\nThis convergent point is the location of the virtual knot with zero flux\n(Section~\\ref{sec-pca-method}),\nand the radial distance from the convergent point is proportional to\nthe brightness of the emission line.\nThe distributions of the PC coefficients in Figure~\\ref{fig-obj} are\nvery similar to those in Figure~\\ref{fig-atr}.\nThere appear to be three groups of knots in Figure~\\ref{fig-obj}\nthat have PC coefficients similar to those of the three groups\nin Figure~\\ref{fig-atr}, i.e., the He, S, and Fe groups.\nWe therefore group the knots in Figure~\\ref{fig-obj} as\nHe-rich, S-rich, and Fe-rich knots using the dashed lines.\nAs a result,\nwe identify 7 He-rich knots, 45 S-rich knots, and 11 Fe-rich knots.\n\nFigure \\ref{fig-1dspec} shows sample 1D spectra of the three groups.\nAs expected, the He-rich knots have\nstrong lines of He {\\small I}\\ 1.083 \\micron\\ compared to [\\ion{Fe}{2}]\\,\nand some of them also show several emission lines of\nH {\\small I}, [\\ion{N}{1}], [\\ion{P}{2}], and [\\ion{Fe}{3}]\\ as well.\nThe H {\\small I}\\ and [\\ion{N}{1}]\\ lines are detected only in the He-rich knots.\nThe S-rich knots also have bright [\\ion{Fe}{2}]\\ lines,\nbut they have much stronger lines of\n[\\ion{S}{3}]\\ 0.953 \\micron, [\\ion{S}{2}]\\ 1.03 \\micron\\ multiplets,\n[\\ion{P}{2}]\\ 1.188 \\micron, and [\\ion{Si}{6}]\\ 1.963 \\micron.\nThe low-ionized emission lines of refractory elements, i.e., [\\ion{C}{1}]\\ and [\\ion{Si}{1}],\nare also detected in several S-rich knots.\nAlthough some S-rich knots show the He {\\small I}\\ 1.083 \\micron\\ line,\ntheir intensities are significantly less than those of the He-rich knots.\nThe Fe-rich knots have strong lines of [\\ion{Fe}{2}]\\ and\nsome of them have a weak He {\\small I}\\ 1.083~\\micron\\ line as well. \nThe brightest Fe-rich knot, Knot 10 in Slit 5, also emits \n[\\ion{Fe}{3}]\\ lines in the {\\it K} band.\nAs in the He-rich knots, \nno lines of C, Si, and S are detected in the Fe-rich knots.\n\nThe three knot groups are easily distinguishable\nwhen the flux ratios of [\\ion{S}{2}]\\ 1.03~\\micron\\ multiplets,\nHe {\\small I}\\ 1.083~\\micron, and [\\ion{P}{2}]\\ 1.188~\\micron\\ are compared\n(Figure~\\ref{fig-fluxratio}).\nThe He-rich knots are well separated from the other groups\nin F(He {\\small I}-1.083)\/F([\\ion{Fe}{2}]-1.644),\ni.e., the ratio is greater than two for the He-rich knots\nbut lower than two for the S-rich and Fe-rich knots.\nThe F([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644) and F([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644)\nof the He-rich knots are mostly smaller than\na few times 1.0 and 0.1, respectively.\nAlthough the S-rich and Fe-rich knots are not clearly distinguished in\nthe F([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644) and F([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644) comparisons,\nwe see that these ratios are higher in the S-rich knots\nthan the Fe-rich knots, i.e., F([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644) $\\gtrsim 5$ and\nF([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644) $\\gtrsim 0.3$ for the S-rich knots and vice versa.\nIt is worth noting that\nthe flux ratios of the knots in Figure~\\ref{fig-fluxratio}\nand the criteria mentioned above are based on the assumption that\nthe intrinsic flux ratio of [\\ion{Fe}{2}]\\ 1.257 and 1.644~\\micron\\ is 1.36\n(Section~\\ref{sec-pca-method}).\nIf we adopt different line ratios, e.g., 0.98 to 1.49\n\\citep[][and references therein]{gia15,koo15},\nthen the criteria of those three flux ratios will be\nF(He {\\small I}-1.083)\/F([\\ion{Fe}{2}]-1.644) = 1.0--2.4, \nF([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644) = 2.3--6.0, and\nF([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644) = 0.19--0.34.\n\n\n\\subsubsection{Physical Properties of Three Knot Groups}\n\\label{sec-pca-result-prop}\nFigure~\\ref{fig-hist} compares the distributions of the knot sizes,\nradial velocities, and line widths of the three knot groups.\nThe angular sizes of most of the knots are in the range 2\\arcsec--7\\arcsec\\\n(or 0.03--0.1 pc at a distance of 3.4 kpc)\nand there is no significant difference among the three groups in their sizes,\nalthough some of the S-rich and Fe-rich knots are as large as 10\\arcsec.\nOn the other hand, there are significant differences in\ntheir radial velocities and line widths.\nThe radial speeds of He-rich knots are $\\lesssim 200~{\\rm km~s}^{-1}$,\nwhile those of S-rich knots range from $-2000~{\\rm km~s}^{-1}$ to $+2000~{\\rm km~s}^{-1}$\nwith a median of $+630~{\\rm km~s}^{-1}$.\nThe radial velocities of Fe-rich knots range from $-500~{\\rm km~s}^{-1}$ to $+1500~{\\rm km~s}^{-1}$\nwith a median of $+330~{\\rm km~s}^{-1}$.\nIn line width,\nthe He-rich knots have widths of 5--10 \\AA,\nwhile the S- and Fe-rich knots have widths of 10--35 \\AA.\n(Note that our spectral resolution at 1.64 \\micron\\ is $\\sim 6$ \\AA.)\nFigure~\\ref{fig-hist} also compares the distribution of\nthe [\\ion{Fe}{2}]\\ 1.644~\\micron\\ line fluxes\namong the three knot groups.\nWhile the S-rich and Fe-rich knots have a similar distribution with\nan increased number of knots that have faint [\\ion{Fe}{2}]\\ emission,\nthe pattern is absent in the He-rich knots.\nThere is no apparent correlation among the\nfour physical parameters of the knots.\n\nOne of the physical parameters of the knots\nthat can be straightforwardly obtained is electron density\nusing [\\ion{Fe}{2}]\\ lines originating from levels with similar excitation energies,\nbecause their ratios are mainly determined by electron densities\n\\citep[e.g.,][]{koo16}.\n[\\ion{Fe}{2}]\\ 1.644~\\micron\\ and 1.677~\\micron\\ are such lines,\nand Figure~\\ref{fig-ne} compares their expected ratios as a function of density\nfor the assumed temperatures of 5000, 10{,}000, and 20{,}000 K (left panel) with\nobserved values as a function of the [\\ion{Fe}{2}]\\ 1.644~\\micron\\ flux (right panel).\nWe see that the ratio is quite insensitive to temperature and\ncan be used to estimate electron density in the range\n$\\sim 10^3$--$10^5~{\\rm cm^{-3}}$.\nThe electron density of the He-rich knots is a few $10^4~{\\rm cm^{-3}}$,\nwhile the S-rich knots show electron densities over a broad range of \n$10^3~{\\rm cm^{-3}}$ to $10^5~{\\rm cm^{-3}}$.\nThe Fe-rich knots have somewhat lower ($10^3$--$10^4~{\\rm cm^{-3}}$)\nelectron densities compared to the other two groups.\nThere appears to be no correlation between the electron densities and\n[\\ion{Fe}{2}]\\ 1.644~\\micron\\ line fluxes.\n\n\n\n\\section{Discussion} \\label{sec-dis}\nIn this section, we discuss the origin of the knots\nusing their spectral characteristics described in the previous section.\n\n\\subsection{He-rich and S-rich knots} \\label{sec-dis-hek}\nThe He-rich and S-rich knots have quite distinct spectral properties;\nthe He-rich knots have high F(He {\\small I}-1.083)\/F([\\ion{Fe}{2}]-1.644)\nand low F([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644), F([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644),\nwhile the S-rich knots have low F(He {\\small I}-1.083)\/F([\\ion{Fe}{2}]-1.644)\nand high F([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644), F([\\ion{P}{2}]-1.188)\/F([\\ion{Fe}{2}]-1.644).\nTheir kinematic properties are also quite different;\nthe He-rich knots have low ($\\lesssim 200~{\\rm km~s}^{-1}$) line-of-sight speeds,\nwhile the S-rich knots have high (up to $\\sim 2000~{\\rm km~s}^{-1}$) line-of-sight speeds.\nThese spectral and kinematical properties suggest that \nthe He-rich knots are dense, slow-moving CSM\nswept up by the SN blast wave,\nwhile the S-rich knots are fast-moving SN ejecta that have been shocked.\nThe same conclusion was reached by \\citet{koo13},\nwho performed an abundance analysis\nusing [\\ion{P}{2}]\\ 1.188~\\micron\\ and [\\ion{Fe}{2}]\\ 1.257~\\micron\\ lines.\nThey showed that the relative abundance of P\n(a major product of the stellar Ne-burning layer)\nto Fe (in number) for the He-rich knots is similar to the solar abundance,\nwhereas that of the S-rich knots is 10--100 times higher than\nthe solar abundance.\n\nThe characteristics of the two types of knots match \nwell with those of QSFs and FMKs known from previous optical studies\n(see Section~\\ref{sec-int} for a summary of their properties).\nSimilar to the optical QSFs,\nthe He-rich knots have bright He {\\small I}\\ lines\ntogether with H {\\small I}\\ and [\\ion{N}{1}]\\ lines;\nall the He-rich knots have a He {\\small I}\\ 1.083 \\micron\\ line,\n6 out of the 7 show H {\\small I}\\ Pa$\\beta$, Br$\\gamma$ lines,\nand the three brightest ones also have [\\ion{N}{1}]\\ 1.040, 1.041 \\micron\\ lines.\nLike the optical FMKs dominated by lines of ionized heavy elements O, S, Ar,\nthe S-rich knots show strong S lines plus [\\ion{P}{2}]\\ and [\\ion{Si}{6}]\\ lines;\nall the S-rich knots have [\\ion{S}{2}]\\ 1.03 \\micron\\ multiplets,\nand 39 out of 45 S-rich knots also have\n[\\ion{P}{2}]\\ 1.188 \\micron\\ and\/or [\\ion{Si}{6}]\\ 1.963 \\micron\\ lines.\nIndeed, the overall spectra of He-rich and S-rich knots are\nsimilar to the NIR spectra of QSFs and FMKs\nobtained by \\citet{ger01}, respectively.\nFurthermore, the radial velocities of the two NIR knot groups\nare well consistent with those of the optical groups.\nFor example, the radial velocity of the He-rich knots is $-50 \\pm 90~{\\rm km~s}^{-1}$,\nwhile the generally accepted radial velocity of QSFs is \n$-140 \\pm 300~{\\rm km~s}^{-1}$ \\citep{van85,ree95}.\nIn addition,\nthe median line-of-sight velocity of S-rich knots is $+620~{\\rm km~s}^{-1}$,\nwhile the systematic velocity of the FMKs\nis $+770\\pm40~{\\rm km~s}^{-1}$ \\citep{ree95}.\n\nRich emission lines of Si, P, and S\nin the S-rich knots and FMKs\nimply that they are the SN ejecta originated from the Ne- and O-burning layers.\nThe two bright emission lines, [\\ion{P}{2}]\\ 1.188~\\micron\\ and [\\ion{Fe}{2}]\\ 1.644~\\micron,\nhave comparable excitation energies and critical densities,\nand so their line ratios are strongly dependent on their abundance ratio,\n$X$(P\/Fe) \\citep{koo13}.\nAs seen in Figure~\\ref{fig-fluxratio}, \nthere is a large scatter in this line ratio for S-rich knots,\nwhich implies that the abundance ratio $X$(P\/Fe) varies\nalmost two orders of magnitude for these knots.\nWe also found that 13 out of 45 S-rich knots have clear\nbut relatively weak emission lines of He {\\small I}\\ and\/or [\\ion{C}{1}].\nThe detection of the He, C, and Fe lines in the S-rich ejecta,\nwhich are either lighter or heavier elements than the O-burning materials,\nmight infer microscopic mixing during the SN explosion.\nIn many S-rich knots,\na highly ionized Si line, [\\ion{Si}{6}]\\ 1.963 \\micron, is detected,\nwhile in a few S-rich knots, a [\\ion{Si}{1}]\\ 1.645 \\micron\\ line is also detected.\nThe detection of Si in very different ionization stages\nindicates the broad range of temperatures in the S-rich knots.\n\n\n\\subsection{Fe-rich knots} \\label{sec-dis-fek}\nIn Section~\\ref{sec-pca-result},\nwe found that the Fe-rich knots exhibit intermediate characteristics between\nHe-rich and S-rich knots; they emit strong [\\ion{Fe}{2}]\\ lines\nwithout any Si, P, and S lines,\nbut have high line-of-sight speeds of up to $\\sim 1500~{\\rm km~s}^{-1}$. \nA few knots also emit an He {\\small I}\\ 1.083~\\micron\\ line.\nThe high velocities, however, suggest that\nthey are not dense QSFs represented by the He-rich knots.\nTheir line widths are also considerably broader than those of He-rich knots,\ni.e., 10--35~\\AA\\ vs. 5--10~\\AA\\ (Figure~\\ref{fig-hist}).\nOn the other hand, the missing Si, P, and S lines indicate that\nthe abundances of these Ne- and O-burning elements are\nvery low in these Fe-rich knots.\nWe can consider two possible explanations\nregarding the origin of the Fe-rich knots:\n(1) swept-up CSM around contact discontinuity (CD) or\n(2) shocked SN ejecta enriched with Fe elements\nthat had been synthesized in explosive Si burning.\nThe ambient medium that the Cas A SN blast wave is propagating into \nis believed to be CSM with an $r^{-2}$\ndensity distribution \\citep[e.g.,][]{leejj14}.\nIn such a case,\n1D similarity solutions show that\nthe shocked CSM accumulates at the CD with infinite density asymptotically\n\\citep{che82}.\nWe thus expect ``dense'' CSM expanding at a speed comparable to\nthe shocked SN material.\nIn the real situation, however,\nthis interacting region between the shocked SN ejecta and the shocked CSM is\nhydrodynamically unstable and distorted,\nwith the density of the shocked CSM limited to $\\lesssim 10$ times the density\nat the ambient shock \\citep{che92,blo01,van09}.\nThe temperature of the shocked CSM near the CD, therefore,\nmay be lower than the typical temperature ($\\sim 2$ keV) of the shocked CSM\n\\citep{hwa12},\nbut probably no more than a factor of 10,\nand all of the Fe in the shocked CSM will be in high ionization stages.\nFurthermore, H {\\small I}\\ lines are not detected in all Fe-rich knots\nwith an upper limit of F(H {\\small I}-Pa$\\beta$)\/F([\\ion{Fe}{2}]-1.257) $\\lesssim 0.1$.\nNote that the observed ratio of these line intensities\nranges between 0.05 and 10 for SNRs\nwhile it is $\\sim 50$ for Orion,\nwhich should be the typical ratios for shocked and photoionized gases\nof cosmic abundance, respectively \\citep{koo15}.\nTherefore, H must be depleted in Fe-rich knots.\nThe non-detection of He and N lines might also indicate that\nthe abundances of these `circumstellar' elements are very low\nin Fe-rich knots,\nalthough this needs to be confirmed from other waveband observations.\n(See the next paragraph for an explanation of the faint He lines\ndetected in some Fe-rich knots.)\nWe may therefore conclude that the Fe-rich knots are not likely the shocked CSM.\n\nThis leaves the second possibility,\ni.e., the Fe-rich knots are Fe-enriched SN ejecta.\nThe high velocities and large velocity widths are consistent with \nSN ejecta being swept up by the reverse shock. \nThe low [\\ion{P}{2}]\\ and [\\ion{S}{2}]\\ fluxes compared to the [\\ion{Fe}{2}]\\ flux,  \nhowever, implies that the abundances of P and S, which are  \nNe- and O-burning materials, are very low,\nwhich is in sharp contrast to the S-rich knots.\nThese characteristics strongly suggest that \n{\\em the Fe-rich knots are most likely ``pure'' Fe ejecta\nsynthesized in the deepest stellar interior.}\nThe weak He {\\small I}\\ 1.083 \\micron\\ line detected in some Fe-rich knots\ncould be due to an $\\alpha$-rich freeze-out process during the\nexplosive Si burning;\njust after the explosion,\ncomplete Si burning with an $\\alpha$-rich freeze-out\noccurs under high temperatures and low density conditions\nin the stellar deep layer,\nand many alpha particles are ``frozen out''\nwithout participating in further nucleosynthetic processes\n\\citep{woo73}.\nSimilar dense, Fe-predominant ejecta,\nlikely from the $\\alpha$-rich freeze-out process,\nhave been detected in another young core-collapse SNR G11.2-0.3\n\\citep{moo09}.\n\nPure Fe ejecta have been detected in X-rays (see below)\nbut not in optical or NIR wavebands.\nThis is surprising considering the extensive optical\/NIR studies\ncarried out for Cas A since its discovery. \nFigure~\\ref{fig-knotpos} partly gives an answer.\nIn the right panel of Figure~\\ref{fig-knotpos},\nred is an [\\ion{Fe}{2}]\\ 1.644 \\micron\\ narrow-band image\nwhile green and blue are {\\it Hubble Space Telescope} ({\\it HST})\nACS\/WFC F850LP and F775W images \nwhich are dominated by ionized S and O lines, respectively\n\\citep{fes06a,ham08}.\nPrevious optical observations had been mostly toward \nthe northern ejecta shell bright in \nionized O, S, and Ar lines \\citep[e.g.,][]{che79,hur96,fes01b} or \ntoward FMKs outside the main shell \\citep[e.g.,][]{fes88,fes06a,fes01a}.\nFigure~\\ref{fig-knotpos}, however, shows that\nthe southwestern (SW) main ejecta shell, which is bright in the [\\ion{Fe}{2}]\\ line\nbut faint in the ionized O and S lines,\nis the region where Fe-rich ejecta can be found.\nIndeed, our result shown in the left panel of Figure~\\ref{fig-knotpos}\nconfirms this;\n9 out of 11 Fe-rich knots are located in Slits 4--6.\n(Note that the compact red knots in the interior and beyond the SW shell\nare mostly QSFs,\nand they are bright in [\\ion{N}{2}]\\ 6548, 6583 \\AA\\ and H$\\alpha$ images too\n\\citep{van85,ala14}.)\nThe slit positions are determined from an [\\ion{Fe}{2}]\\ 1.644 \\micron\\ image,\nso that some of them were placed\ntoward the red portions of the main ejecta shell and,\nby decomposing the emission into individual velocity components,\nwe were able to identify Fe-rich ejecta.\nIt is worth noting that\n\\citet{rho03} also noted the bright [\\ion{Fe}{2}]\\ emission in the SW shell\nin their [\\ion{Fe}{2}]\\ 1.644 \\micron\\ image of Cas A.\nMeanwhile, we can see that\nthe [\\ion{Fe}{2}]\\ 17.9 \\micron\\ emission is much brighter than the emission from\nO-burning elements such as Ar and S in the SW shell\nin the {\\it Spitzer} mid-infrared maps of ionic lines \\citep{enn06,smi09}.\nOur result suggests that this [\\ion{Fe}{2}]\\ emission-predominant area,\ni.e., the red area of the SW ejecta shell in Figure~\\ref{fig-knotpos},\nis probably mainly composed of Fe ejecta.\n\nIt is not easy to identify the counterpart of Fe-rich knots in optical\nimages because several velocity components are usually superposed\nalong the line of sight toward the main ejecta shell.\nThe brightest Fe-rich knot (Knot 10 in Slit 5; hereafter K10), however,\nis somewhat isolated and we can identify its counterpart.\nIn Figure \\ref{fig-k10},\nthe upper two images are [\\ion{Fe}{2}]\\ 1.644 \\micron\\ images at different epochs\nand they show that K10 is a clump of $\\sim 10\\arcsec \\times 3\\arcsec$\nelongated along the slit.\nThe two [\\ion{Fe}{2}]\\ images clearly show that\nthe clump is moving fast tangentially.\nThe proper motion is measured $0\\farcs28$ yr$^{-1}$,\nimplying a tangential velocity of $4500 \\pm 200~{\\rm km~s}^{-1}$.\nIn the lower F775W and F850LP images,\nwe see diffuse faint emission at the position of K10 (see the red contour).\nIts brightness distribution is different,\nwith two small ($\\lesssim 2\\arcsec$) bright spots (S1 and S2 in the figure)\nin the lower part of the clump.\nOne of these bright spots, S1, is coincident with an S-rich knot\n(Knot 9 in Slit 5),\nwhich is spatially coincident with K10\nbut has a line-of-sight velocity ($+1500~{\\rm km~s}^{-1}$)\nthat is very different from that of K10 ($-300~{\\rm km~s}^{-1}$).\nThe other bright spot, S2, must also be due to an S-rich knot\nnot detected in our spectroscopy.\nSo excluding these two bright knots,\nK10 appears faint in F775W and F850LP images.\nWe suspect that most of the diffuse emission in the F775W and F850LP images\nis due to optical [\\ion{Fe}{2}]\\ lines.\nThis can be confirmed by optical spectroscopy.\nRecently, there have been optical spectral mapping observations of Cas A \n\\citep{ree95,mil13,ala14} and, in principle,    \na similar analysis can be done to detect Fe-rich knots,\nalthough the optical [\\ion{Fe}{2}]\\ lines,\ne.g., [\\ion{Fe}{2}]\\ 7155 and 8617 \\AA\\ lines, will be much fainter than\nthe [\\ion{Fe}{2}]\\ 1.644 \\micron\\ line\nbecause of large interstellar extinction ($A_{\\rm V} = 5$--10 mag) toward Cas A\n\\citep[e.g.,][]{eri09,hwa12,leeyh15}.\n\nThe distribution and amount of Fe ejecta in Cas A\nhave been a subject of controversy.\nPrevious X-ray studies detected\nhot and diffuse ``pure'' Fe ejecta with mass $\\sim 0.1~M_{\\sun}$\nthat might have formed by $\\alpha$-rich freeze-out\nduring the complete Si burning \\citep{hwa03,hwa12}.\nThese X-ray-emitting shocked Fe ejecta are distributed mainly\nin the southeastern and northern regions of the remnant\n(Figure~\\ref{fig-fedist}).\nOn the other hand,\nthe hard X-ray emission from the radioactive decay of $^{44}$Ti\nhas been detected in the interior of the main ejecta shell\n\\citep[Figure~\\ref{fig-fedist};][]{gre14}.\nSince $^{44}$Ti is essentially synthesized in complete Si burning with\n$\\alpha$-rich freeze-out in the innermost region \\citep[e.g.,][]{mag10},\n$^{44}$Ti traces ``pure'' $^{56}$Ni or its stable nuclei $^{56}$Fe.\nThe majority of the observed $^{44}$Ti is inside the reverse shock and\ntherefore from unshocked Fe ejecta.\nThe inferred mass of the unshocked Fe ejecta is\n$\\sim 0.1~M_{\\sun}$ \\citep{gre14}.\nSuch unshocked Fe ejecta, however, have not yet been detected,\npresumably because they are cool\n\\citep[$T \\lesssim 40$ K; e.g.,][]{bar10,sib10,leeyh15}.\nInstead, the X-ray-emitting Fe ejecta\nlocated just outside the $^{44}$Ti emission was attributed to\nthe corresponding shocked ejecta \\citep[Figure~\\ref{fig-fedist};][]{gre14}.\nHowever, the missing X-ray-emitting Fe ejecta\ntoward the south and northeast directions from the explosion center\nhave been puzzling.\nWe note that the NIR [\\ion{Fe}{2}]-bright, red portion of the SW shell\nappears to be in contact with the interior $^{44}$Ti-emitting region\n(Figure~\\ref{fig-fedist}).\nThis might be the case\nfor the small red portion near Slit 2 in the northeastern shell, too.\nTherefore, if these [\\ion{Fe}{2}]-bright regions are composed of\nshocked, dense Fe ejecta, as implied from our spectroscopic result,\nit explains why we do not see X-ray-emitting diffuse Fe ejecta\ntoward these directions;\nthe unshocked ``pure'' Fe ejecta traced by\nthe radioactive $^{44}$Ti emission in the interior of the main shell\nis composed of both dense and diffuse ejecta,\nand when these are swept up by a reverse shock,\nwe observe either NIR [\\ion{Fe}{2}]\\ emission or X-ray emission\ndepending on their densities.\nWe do not see NIR emission associated with\nthe central bright $^{44}$Ti emission\nbut this could be because the shock is face-on, as suggested by\nthe large red-shifted central velocity (1100--3000 ${\\rm km~s}^{-1}$)\nof the $^{44}$Ti line \\citep{gre14}, or\nbecause the reverse shock has not yet reached the dense, unshocked Fe ejecta.\nRecent multi-dimensional simulations show that\nsuch global asymmetry in Fe (or $^{56}$Ni) density can arise from\nthe low-mode convection of the innermost region just after the core bounce\n\\citep[e.g.,][]{won13}.\nFuture NIR spectral mapping observations\nrevealing the 3D distribution of the dense Fe ejecta\nwill be helpful for understanding the SN explosion dynamics of\nthe innermost region.\n\n\n\n\\section{Summary} \\label{sec-sum}\nWe have carried out NIR spectroscopic observations toward\nthe main ejecta shell of the young SNR Cas A.\nIn total, 63 individual knots were identified from eight slit positions\nby using a clump-finding algorithm.\nEach of these knots has distinct kinematical and spectral properties.\nWithin the {\\it JHK} spectral range (0.94--2.46 \\micron),\nwe found 46 emission line features including a dozen bright [\\ion{Fe}{2}]\\ lines,\nforbidden lines of other metallic species, and H and He lines.\nWe employed the PCA method to classify the knots\nbased on their relative line fluxes into three distinctive groups:\nHe-rich knots of pre-supernova circumstellar wind material,\nplus S-rich and Fe-rich knots of SN ejecta material.\nThe He-rich and S-rich knots correspond to QSFs and FMKs\nstudied in the visible waveband,\nwhile Fe-rich knots, showing in general only [\\ion{Fe}{2}]\\ emission lines,\nare likely `pure' dense Fe ejecta from the innermost layer of the progenitor.\nWe summarize our main results as follows.\n\n\\noindent\n1. The PCA showed that the NIR spectral lines\ncan be grouped into three groups:\n(1) Group 1, composed of H {\\small I}\\ and He {\\small I}\\ lines\ntogether with [\\ion{N}{1}]\\ lines,\n(2) Group 2, composed of forbidden lines of Si, P, and S, and \n(3) Group 3, composed of forbidden Fe lines.\nThe lines in the first two PCs are strongly correlated with each other,\nwhile the correlation is rather weak among the forbidden Fe lines in Group 3.\nThese three spectral groups of the emission lines are almost independent in\n3D PC space (Figure~\\ref{fig-atr}).\n\n\\noindent\n2. The distribution of the knots in the PC planes matches well with \nthe above spectral groups,\nand we classified the knots into three groups:\n(1) He-rich, (2) S-rich, and (3) Fe-rich knots. \nThe knots belonging to these three groups are well separated\nfrom each other in\nF([\\ion{S}{2}]-1.03)\/F([\\ion{Fe}{2}]-1.644) vs. F(He {\\small I}-1.083)\/F([\\ion{Fe}{2}]-1.644) plane\n(Figure~\\ref{fig-fluxratio}),\nso that one may use these line ratios to classify the knots in Cas A.\nIt would be interesting to determine\nwhether this classification methodology applies for other core-collapse SNRs.\n\n\\noindent\n3. The He-rich knots show \nbright emission lines of He {\\small I}\\ 1.083~\\micron\\ and [\\ion{Fe}{2}]\\\ntogether with [\\ion{N}{1}]\\ and H {\\small I}\\ lines.\nTheir line-of-sight speeds are small ($\\lesssim 200~{\\rm km~s}^{-1}$).\nFrom these chemical and kinematical characteristics,\nwe conclude that the He-rich knots are dense CSM\nswept up by the SN blast wave.\nThese knots correspond to the previously known QSFs.\n\n\\noindent\n4. The S-rich knots show strong forbidden lines of S\ntogether with [\\ion{P}{2}]\\ and [\\ion{Si}{6}],\nand their line-of-sight speeds reach a few 1000 ${\\rm km~s}^{-1}$.\nThese chemical and kinematical properties indicate that\nthe S-rich knots are dense SN ejecta material\nmostly originating from the O-burning layers\nand swept up by a reverse shock.\nThese knots correspond to the FMKs\ndetected in previous optical studies.\n\n\\noindent\n5. The Fe-rich knots only show strong [\\ion{Fe}{2}]\\ and [\\ion{Fe}{3}]\\ lines,\nand no or weak He {\\small I}\\ 1.083~\\micron\\ lines.\nLike the S-rich knots,\nthey have large line-of-sight speeds (up to $\\sim 1500~{\\rm km~s}^{-1}$)\nand broad line widths (10--35 \\AA),\nbut they do not show the lines from Si, P, and S.\nSome Fe-rich knots show He {\\small I}\\ 1.083~\\micron\\,\nbut their fluxes compared to the [\\ion{Fe}{2}]\\ lines are much weaker than\nthose of the He-rich knots.\nThese spectroscopic properties suggest that\nthe Fe-rich knots are most likely ``pure'' dense Fe ejecta\nfrom the innermost layer of the SN.\nThe comparison of [\\ion{Fe}{2}]\\ 1.644 \\micron\\ images with\nthe {\\it HST} ACS\/WFC F850LP and F775W\nand {\\it NuSTAR} $^{44}$Ti images reveals that\nthese Fe ejecta are mainly distributed in the SW main ejecta shell,\njust outside the unshocked $^{44}$Ti in the interior.\nThis supports that\nthere could be a large amount of unshocked ``pure'' Fe ejecta\nassociated with $^{44}$Ti.\nTogether with the diffuse, X-ray-emitting ``pure'' Fe ejecta\ndetected by {\\it Chandra},\nour result implies that\nthe Fe ejecta synthesized in the innermost region develop\nlarge-scale non-uniformity during the SN explosion\nand are expelled asymmetrically.\nThis seems to be consistent with\nthe low-mode, convection-driven SN explosion model. \n\n\n\n\\acknowledgments\nWe wish to thank the anonymous referee\nfor the very useful comments and suggestions\nwhich helped us improve the quality of the paper.\nWe thank Brian Grefenstette and Fiona Harrison\nfor providing the {\\it NuSTAR} data.\nWe also want to thank John Raymond and Sung-Chul Yoon\nfor helpful discussions.\nThe interactive 3D figures were made by using {\\sc Asymptote}\nwhich is a descriptive vector graphics language\\footnote{\\url{http:\/\/asymptote.sourceforge.net}}.\nThis research was supported by Basic Science Research Program through\nthe National Research Foundation of Korea(NRF)\nfunded by the Ministry of Science, ICT and future Planning\n(2014R1A2A2A01002811).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\nHot alkali vapors are a workhorse medium for many atomic sensing tasks, including magnetometry \\cite{budker2007optical}, magnetic gradiometry \\cite{LimesPRAppl2020, LuciveroPRAppl2021}, comagnetometry \\cite{PhysRevLett.124.170401,RomalisVas-comag}, alkali-noble-gas gyroscopes \\cite{WalkerBook2016,FangSensors2012}  and quantum memories \\cite{Shaham2022}, as well as tabletop searches for new  physics \\cite{Afach2021,PhysRevLett.125.201802,pustelny2013global}. Recent works have also shown that these media support non-classical correlations \\cite{MitchellNatureCom,KuzmichSpinSqueezingQND,VasilakisPolzik2015,PhysRevA.102.012822,PRXQuantum.3.010305}, allowing quantum enhancement techniques, e.g., squeezing of light or of spins, to improve their performance \\cite{Troullinou,Julsgaard2001,Sherson2006,TrutleinHe3}.\nDetailed models for the behaviour of such devices include intra- and inter-atomic spin dynamics, response to probe and pumping light and external fields \\cite{seltzerthesis, SavukovRomalis}. In these models, spin-exchange (SE) collisions play an important role, and in practice often limit the spin-coherence time and consequently the performance of these devices. \n\nSE collisions \\cite{Grossette1,Grossette2}, together with the intra-atomic spin dynamics and coupling to external fields, lead to non-intuitive relationships among atomic number density, magnetic field, and coherence time \\cite{PhysRev.153.132}.  At low densities or high fields, SE collisions are a decoherence mechanism, causing spin coherence times to decrease with increasing density. In the so-called spin-exchange relaxation-free (SERF) regime of high density and low field, however, frequent SE collisions decouple the net spin from decoherence via the external field, and  coherence times increase with increasing density \\cite{PhysRevLett.31.273, PhysRevA.16.1877}.\n\n\nSpin noise spectroscopy (SNS)  provides a window into the non-intuitive spin dynamics of dense atomic vapors \\cite{lucivero2017sensitivity}. SNS measures the spin fluctuations of an unpolarized spin ensemble, driven only by random influences such as collisions and diffusion of spins into and out of the system. These spin dynamics show correlations that reflect the magnetic resonance spectral response \\cite{PhysRevA.104.063708,BruunPRL2009, MullerAPL2010}, and are acquired without excitation of the medium. In particular, the spin-noise spectrum of an unpolarized vapor reveals the vapor's transition from spin-exchange and SERF regimes, including the non-intuitive linewidth behavior \\cite{MitchellNatureCom,PhysRevA.104.063708,katsoprinakis}.\n\nDespite recent experimental and theoretical study of the noise properties of SERF-regime vapors, nearly all work has focused on the most prominent feature, which is a Lorentzian-like peak at the Larmor frequency (taking into account  nuclear slowing).  In this work we calculate from first principles the full spin-noise spectrum of a hot alkali vapor, including the effects of hyperfine, Zeeman, and spin-exchange effects on the vapor, and the effects of probe detuning on the coupling to the different ground hyperfine states $F_a$ and $F_b$. We find features not previously remarked, including a spectrally-broad spin-noise background that mimics optical shot noise, departure of the spin-noise spectrum from the Lorentzian shape, a probe-detuning-dependent ``interference'' of noise contributions from $F_a$ and $F_b$, and in some circumstances,  a significant reduction of spin-noise at the precession frequency. We also describe how, in typical experimental conditions, the integrated spin-noise power can deviate by as much as 20\\% from the predictions of simpler models. This can significantly affect the use of spin-noise of unpolarized vapors to benchmark quantum noise, e.g., in quantum sensing.  \n\n\n\n\nThe paper is organized as follows.  In \\autoref{sec:Model}, we present the statistical model for the spin dynamics, first in terms of a master equation describing the evolution of the single-atom density matrix $\\rho$, and then in terms of an equivalent stochastic Bloch equation that governs the collective spin observables  $\\mathbf{F}^{a}$ and  $\\mathbf{F}^{b}$. The spin readout, by Faraday rotation, is described by input-output relations for the probe beam's Stokes operators, including intrinsic polarization noise. In \\autoref{sec:Correlations}, we derive analytic results for correlation functions and spectra. In \\autoref{sec:PredictedPhenomena} we predict several observable SNS phenomena based on the analystic results. \n\n\n\n\n\n \n\n\n\\section{Model}\n\\label{sec:Model}\n\n\n\\subsection{Spin dynamics -- master equation}\n\\label{sec:MasterEquation}\n\nIn the presence of a constant magnetic field $\\mathbf{B}$, the angular momenta in the two hyperfine multiplets precess in opposite directions. Spin-exchange collisions transfer the atoms between the two hyperfine manifolds resulting in a spin-dephasing evolution captured by the single-atom density matrix equation \\cite{SavukovRomalis}\n\\begin{equation}\n \\frac{d \\rho}{d t} = -\\frac{i}{\\hbar} [A_{\\mathrm{hf}} \\mathbf{\\hat{I}} \\cdot \\mathbf{\\hat{s}} +g_s \\mu_{_B} \\mathbf{\\hat{s}} \\cdot \\mathbf{B}, \\rho] + R_{\\mathrm{se}} [\\hat{\\varphi} (\\mathbbm{1}+4 \\langle \\mathbf{\\hat{s}} \\rangle \\cdot \\mathbf{\\hat{s}})- \\rho], \\label{eq:density-matrix}\n \\end{equation} \nwhere $A_{\\mathrm{hf}}$ is the hyperfine coupling constant, $g_s \\approx 2$ is the electron g-factor, $\\mu_{_B} \\approx 9.27 \\times 10^{-24}$ \\si{\\joule\\per\\tesla} is the Bohr magneton, $R_{\\mathrm{se}}$ is the spin-exchange rate, $\\langle \\mathbf{\\hat{s}} \\rangle= \\mathrm{Tr}[\\rho \\mathbf{\\hat{s}}]$ is the expectation value of the electronic spin in the state $\\rho$ and  $\\hat{\\varphi} = \\sum_{i=0}^3 \\hat{s}_i \\rho \\hat{s}_i$ is the electron-spin-depolarized density operator. Here, $\\hat{s}_0 \\equiv \\mathbbm{1}\/2$ and $\\{\\hat{s}_1,\\hat{s}_2,\\hat{s}_3\\} = \\{\\hat{s}_x,\\hat{s}_y,\\hat{s}_z\\}$. The steady-state solution of Eq.\\eqref{eq:density-matrix} is the thermal spin state $\\rho_{\\mathrm{eq}} = e^{\\bm{\\beta} \\cdot \\mathbf{\\hat{f}}}\/\\mathrm{Tr}[e^{\\bm{\\beta} \\cdot \\mathbf{\\hat{f}}}]$ where $\\mathbf{\\hat{f}} \\equiv \\mathbf{\\hat{f}}^a + \\mathbf{\\hat{f}}^b$ is the total angular momentum in the ground state and $\\bm{\\beta}$ is the inverse spin-temperature related to the spin-polarization $\\mathbf{P} \\equiv \\langle \\mathbf{\\hat{s}} \\rangle\/s$ as $\\bm{\\beta}~||~\\mathbf{P}$, $\\beta_j= \\mathrm{ln}[ (1+P_j)\/(1-P_j)]$ with $j \\in \\{ x,y,z\\}$ \\cite{Appelt,HapperRev}. In Eq.\\eqref{eq:density-matrix} we have ignored the SE frequency shift,  which is negligible for a low-polarized vapor and the small interaction of the nuclear spin with the magnetic field. Finally, regarding the angular momentum degrees of freedom, the dimension of the Hilbert space in the ground state and subsequently of the operators entering Eq.\\eqref{eq:density-matrix} is $d\\equiv (2S+1)(2I+1)$.\n\n\nThe applied magnetic field exerts a torque both on the electron and the nuclear spin. Since the nuclear g-factor is significantly smaller than electron's $g_s$, the angular momenta in the two hyperfine levels precess according to $d\\langle \\mathbf{\\hat{f}^{\\alpha}}\\rangle\/dt\\approx \\langle \\mathbf{\\hat{s}^{\\alpha}}\\rangle \\times \\gamma_0 \\mathbf{B}$ with $\\alpha \\in\\{ a,b\\}$. The mean value of the electron spin operator, projected in the two hyperfine levels, is related to the corresponding total angular momentum operator through the relations: $\\langle \\mathbf{\\hat{s}^{a}}\\rangle =  \\langle \\mathbf{\\hat{f}^{a}}\\rangle\/(2I+1)$ and $\\langle \\mathbf{\\hat{s}^{b}}\\rangle =-  \\langle \\mathbf{\\hat{f}^{b}}\\rangle\/(2I+1)$, respectively \\cite{optical_mag}.\n\n\\subsection{Spin dynamics --  Bloch equations}\n\\label{sec:BlochEquations}\n\nHaving described the single-atom dynamics, the collective state of the ensemble is given by $\\tilde{\\rho}=\\rho^{\\otimesN_\\mathrm{at}}$, being a tensor product over all the single atom density operators. The expectation value of the collective spin  $\\mathbf{\\hat{F}}^{\\alpha}=\\sum_{i=1}^{N_\\mathrm{at} } \\mathbf{\\hat{f}}_{i}^{\\alpha}$, $\\alpha \\in \\{ a,b \\}$ is then $\\langle \\mathbf{F}^{\\alpha} \\rangle =\\mathrm{Tr}[\\tilde{\\rho} \\mathbf{F}^{\\alpha}]=\\mathrm{Tr}[\\rho^{\\otimesN_\\mathrm{at}} \\mathbf{F}^{\\alpha}]= N_\\mathrm{at} \\mathrm{Tr}[\\rho \\mathbf{f}^{\\alpha}]$. Similarly, correlations between collective operators scale as $\\langle \\hat{F}_i^{\\alpha} \\hat{F}_j^{\\beta}\\rangle = N_\\mathrm{at} \\langle \\hat{f}_i^{\\alpha} \\hat{f}_j^{\\beta} \\rangle$ with $i,j \\in \\{x,y,z \\}$ and $\\alpha , \\beta \\in \\{ a,b \\}$. In the following we will make use of collective spin operators.\nIn the low polarization limit, it has been shown that by expanding the density matrix in the spherical tensor operator basis one can arrive to a system of differential equations for $\\langle \\mathbf{\\hat{F}^a} \\rangle=\\mathrm{Tr}[\\tilde{\\rho}_a \\mathbf{\\hat{F}}]$ and $\\langle \\mathbf{\\hat{F}^b} \\rangle=\\mathrm{Tr}[\\tilde{\\rho}_b \\mathbf{\\hat{F}}]$, where $\\tilde{\\rho}=\\tilde{\\rho}_a +\\tilde{\\rho}_b$, describing the coupled spin precession and relaxation of the average hyperfine angular momenta in the two multiplets \\cite{PhysRevA.16.1877,PhysRevA.103.043116,katz2015coherent,PhysRevLett.110.263004,ofer}\n\\begin{eqnarray}\n\\frac{d \\langle \\mathbf{\\hat{F}^a} \\rangle}{dt} &=& \\langle \\mathbf{\\hat{F}^a} \\rangle \\times \\gamma_0 \\mathbf{B} -R_{\\mathrm{se}} \\kappa_{aa} \\langle \\mathbf{\\hat{F}^a} \\rangle +R_{\\mathrm{se}}  \\kappa_{ab} \\langle \\mathbf{\\hat{F}^b} \\rangle,\n\\label{eq:DiffEqA}\\\\\n\\frac{d \\langle \\mathbf{\\hat{F}^b} \\rangle}{dt}&=&- \\langle \\mathbf{\\hat{F}^b} \\rangle \\times \\gamma_0 \\mathbf{B} -R_{\\mathrm{se}}  \\kappa_{bb} \\langle \\mathbf{\\hat{F}^b} \\rangle +R_{\\mathrm{se}}  \\kappa_{ba} \\langle \\mathbf{\\hat{F}^a} \\rangle.\n\\hspace{6mm}\n\\label{eq:DiffEqB}\n\\end{eqnarray}\nHere $\\gamma_0=\\gamma_e\/(2I+1)$ is the atomic gyromagnetic ratio with $\\gamma_e = g_s \\mu_{_B}\/ \\hbar \\approx 2 \\pi \\times 2.8 \\times 10^4$ \\si{\\mega\\hertz\\per\\tesla} being the electron gyromagnetic ratio, when neglecting the small contribution of the nuclear Zeeman term, $\\omega_0=\\gamma_0 B$ the linear Larmor frequency resulting from the diagonalization of the Breit-Rabi Hamiltonian $\\hat{H}=A_{\\mathrm{hf}} \\mathbf{\\hat{I}} \\cdot \\mathbf{\\hat{s}} +g_s \\mu_{_B} \\mathbf{\\hat{s}} \\cdot \\mathbf{B}$ at low magnetic fields and $R_{\\mathrm{se}}= n \\sigma_{\\mathrm{se}} u$ is the spin-exchange rate with $\\sigma_{\\mathrm{se}} \\approx 2 \\times 10^{-14}$ cm$^2$ being the spin-exchange cross-section and $u$ the mean relative velocity of the pairwise collisions. The coefficients $\\kappa_{\\alpha\\beta}$ with $\\alpha,\\beta \\in \\{a,b\\}$ describe the relaxation and the coupling between the two hyperfine multiplets. Following Appendix \\ref{sec:spin-dyn}, for $a=I+1\/2$ and $b=I-1\/2$ it is useful to express the coefficients $\\kappa_{\\alpha \\beta}$ in terms of the nuclear spin\n\\begin{eqnarray}\n\\kappa_{ba}  &=&\\kappa_{aa}  = \\frac{2}{3}\\frac{I(2I-1)}{(2I+1)^2},\n\\\\\n\\kappa_{ab}&=& \\kappa_{bb}= \\frac{2}{3}\\frac{(2I+3)(I+1)}{(2I+1)^2}.\n\\end{eqnarray}\nFrom Eqs.~\\eqref{eq:DiffEqA} and \\eqref{eq:DiffEqB} we obtain that $d(\\langle \\mathbf{\\hat{F}^a} \\rangle+\\langle \\mathbf{\\hat{F}^b} \\rangle)\/dt$, does not depend on the SE dynamics and especially at $\\mathbf{B}=0$, where $\\langle \\mathbf{\\hat{F}^a} \\rangle+\\langle \\mathbf{\\hat{F}^b} \\rangle$ is a constant of motion, reflecting that SE collisions conserve the total angular momentum.\n \n\nThe coupled system of differential equations in Eqs.~\\eqref{eq:DiffEqA} and \\eqref{eq:DiffEqB} captures the peculiar features of the spin-exchange interaction in a DC magnetic field, specifically, the narrowing of the magnetic resonance spectrum in the SERF regime, as well as, the frequency shift of the resonance frequency $\\omega_0$ towards lower frequencies. Both these effects are a consequence of the rapid transfer of angular momentum between the two hyperfine multiplets and start to become significant when the spin-exchange rate is comparable or exceeds the spin-precession frequency $\\omega_0$.  \n\n\n\n\\subsection{Spin dynamics --  stochastic Bloch equations}\n\\label{sec:StochasticBlochEquations}\n\nEqs.~\\eqref{eq:DiffEqA} and \\eqref{eq:DiffEqB}, which describe the evolution of the statistical averages $\\langle \\mathbf{\\hat{F}}^\\alpha \\rangle$, predict a relaxation toward a zero mean polarization. This mean does not, however,  capture the full dynamics of $\\mathbf{\\hat{F}}^\\alpha$, which fluctuate about their averages in a correlated fashion. To account for spin-noise fluctuations in the thermal spin state we first suppose that the magnetic field is transverse to the probe beam, namely $\\mathbf{B}=(0,B,0)^T$ along the $\\mathbf{\\hat{y}}$-direction. Additionally, we define the angular momentum vector $\\mathbf{\\hat{X}}(t)\\equiv[\\hat{F}^{a}_x(t),\\hat{F}^{a}_y(t),\\hat{F}^{a}_z(t),\\hat{F}^{b}_x(t),\\hat{F}^{b}_y(t),\\hat{F}^{b}_z(t)]^T$ and the corresponding noise vector \n$d {\\cal{\\mathbf{\\hat{W}}}}(t)\\equiv[d\\mathcal{\\hat{W}}^{a}_x(t),\\ldots,d\\mathcal{\\hat{W}}^{b}_z(t)]^T$.\nHere, $d{\\mathcal{\\hat{W}}}^{\\alpha}_{j} (t)$ with $j \\in \\{x,y,z\\}$ and $\\alpha \\in \\{a,b\\}$ represents white Gaussian noise with zero mean and variance $dt$. In addition, we assume that the angular momentum vector $\\mathbf{\\hat{X}} (t)$ satisfies the stochastic differential equation $d \\mathbf{\\hat{X}} (t) = A \\mathbf{\\hat{X}}(t) dt+ Q  d{\\cal{\\mathbf{\\hat{W}}}}(t)$,   namely, we assume that the angular momentum operators are stochastic variables and we augment the deterministic differential equations with Gaussian source noise terms associated with the spontaneous fluctuations of the collective angular momenta in the two hyperfine multiplets \\cite{PhysRevLett.120.040503}. Here $Q$ is the noise-strength matrix resulting from the microscopic collisional dynamics and being associated with the spin-variance in the corresponding state.  As a consequence of the linearity of the stochastic differential equation, the mean of $\\mathbf{\\hat{X}} (t)$  follows the equation $d \\langle \\mathbf{\\hat{X}} (t) \\rangle =  A \\langle \\mathbf{\\hat{X}} (t) \\rangle dt$ with solution \n\\begin{equation}\n\\label{eq:AverageSpinDynamics}\n\\langle \\mathbf{\\hat{X}} (t) \\rangle = e^{A \\tau} \\langle \\mathbf{\\hat{X}} (0) \\rangle. \n\\end{equation}\nA simple formula relates the dynamics matrix $A$ with the noise-strength matrix $Q$ and the steady-state covariance $\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0)$ \\cite{gardiner2009stochastic}\n\\begin{equation}\n\\label{eq:FluctuationDissipation}\nA \\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0)+\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) A^{T} =QQ^{T}.\n\\end{equation}\n\nIn the following sections we define the steady-state covariance $\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0)$ at $\\tau=0$ while we explicitly calculate its $\\tau$-dependence and subsequently deduce the spin-noise spectrum. \n\n\\subsection{Intrinsic versus thermal spin-noise}\n\n\n\nIn quantum-limited probing of atomic spin systems, intrinsic spin-noise and the effect of measurement on the spin distribution can play an important role \\cite{vasilakis2011stroboscopic, VasilakisPolzik2015, Troullinou}.  The magnitude of these effects can be understood by considering the spin uncertainty relations \n\\begin{equation}\n\\label{eq:SpinUncertRel}\n\\delta \\hat{F}_i \\delta \\hat{F}_j \\ge \\frac{1}{2} |\\langle [\\hat{F}_i, \\hat{F}_j]\\rangle| = \\frac{1}{2} | \\langle i \\hat{F}_k\\rangle|   =\\frac{1}{2} |\\langle  \\hat{F}_k\\rangle|,\n\\end{equation}\nfor mutually-orthogonal directions $\\{i,j,k\\}$. For example, fully-polarized spin-$F$ states with $\\langle F_z \\rangle = N_\\mathrm{at} f$ have $\\delta \\hat{F}_x  \\delta \\hat{F}_y = N_\\mathrm{at} f\/2$, which saturates the uncertainty relation. For such polarized states, processes that reduce the uncertainty of one component, e.g. measurement or optical pumping, are required to introduce noise into other components, to satisfy the uncertainty relation. \n\nUnpolarized and weakly-polarized states also obey Eq.\\eqref{eq:SpinUncertRel}, but satisfy it in a different way. Unpolarized states have uncertainty products $\\delta \\hat{F}_i \\delta \\hat{F}_j  \\approx N_\\mathrm{at} f(f+1)\/3$, comparable to those of polarized states.  A much smaller fraction of their spin variance is imposed by the uncertainty relation, however: for unpolarized states, the r.h.s. of Eq.\\eqref{eq:SpinUncertRel} is $|\\langle \\hat{F}_k \\rangle| \\le \\langle F_k^2 \\rangle^{1\/2} \\sim N_\\mathrm{at} ^{1\/2} \\ll N_\\mathrm{at}$. The uncertainty relation thus imposes an additional variance that is negligible when compared to the thermal variance of the state. For this reason, the collective angular momentum can be accurately modeled as a classical stochastic variable. The quantum nature of the spin is nonetheless reflected in the structure of the dynamical equations, i.e., in ~Eqs. \\eqref{eq:density-matrix},  \\eqref{eq:DiffEqA} and \\eqref{eq:DiffEqB}. In the next paragraph we will introduce noise terms to complete the dynamics. We note that, because we are dealing with unpolarized states, it suffices to describe the thermal noise contribution. This contribution reflects the thermal state variance, and thus the spin structure of the atomic state, but does not derive from Eq.\\eqref{eq:SpinUncertRel}.   More extensive discussions of the  effects of measurement back-action on unpolarized states can be found in \\cite{toth2010singlet, BehboodPRL2013, BehboodPRL2014, MitchellNatureCom}.\n\n\n\\subsection{Faraday signal}\n\nThe dispersive interaction of a linearly polarized weak probe beam with a hot atomic vapor results in  paramagnetic Faraday rotation \\cite{PhysRev.163.12,optical_mag} \nby an angle $\\hat{\\phi}(t)\\approx (4 \\pi^2 n l\\nu\/c) \\mathrm{Re}[ \\hat{a}^{1}(\\nu,t)]$ where $n$ is the alkali number density in the vapor cell of length $l$, $\\nu$ the frequency of the monochromatic laser beam interacting with the atomic medium, $c$ the speed of light and $\\hat{a}^{1}(\\nu,t)$ is the  rank-1 irreducible vector component of the ground state polarizability tensor $\\hat{a}$. In writing the rotation angle, we have considered a detuning much larger than the hyperfine splitting in the excited electronic state such that the contribution of the rank-2 component of the polarizability tensor is negligible. Additionally, we assume that the pressure broadening due to the interaction of the alkali atoms with the atoms of a buffer gas is larger than the Doppler broadening of the optical transition. Under these assumptions the Faraday rotation takes the form \\cite{MitchellNatureCom}\n\\begin{equation}\n\\hat{\\phi} (t) \\approx g_b \\hat{F}_{z}^{b}(t)-g_{a}\\hat{F}_{z}^{a} (t),\n\\label{eq:Faraday Rotation}\n\\end{equation}\nwith $\\hat{F}_{z}^{\\alpha}=\\sum_{i=1}^{N_\\mathrm{at} } \\hat{f}_{i, z}^{\\alpha}$, $\\alpha \\in \\{ a,b \\}$ being the collective ground state angular momentum operator expressed as a uniformly weighted sum over all the single-atom angular momentum operators probed by the laser beam. The uniform weighting represents a homogeneous atom-light coupling \\cite{PhysRevA.89.033850}, while the probe beam is assumed to propagate along the $\\hat{\\mathbf{z}}$-direction. Here, $a=I+1\/2$ and $b=I-1\/2$ indicate the upper and lower hyperfine manifolds, respectively, with $I$ being the nuclear spin.  In Eq.\\eqref{eq:Faraday Rotation} we ignore the terms of the Faraday signal that oscillate at the hyperfine frequency since they significantly exceed the bandwidth of the usual photodetectors and are irrelevant to this work. The coupling constants $g_{\\alpha}$ with $\\alpha \\in \\{a,b \\}$ are given by\n\\begin{equation}\ng_{\\alpha} = \\frac{1}{2I+1} \\frac{c r_e f_{\\mathrm{osc}}}{A_\\mathrm{{eff}}} \\frac{\\nu-\\nu_\\alpha}{(\\nu-\\nu_\\alpha)^2+(\\frac{\\Delta \\nu}{2})^2},\n\\label{eq:gjDef}\n\\end{equation}\nwhere $r_e=2.83 \\times 10^{-13}$ cm is the classical electron radius, $f_{\\mathrm{osc}}$ is the oscillator strength associated with the particular optical transition, $2I+1$ is the nuclear spin multiplicity, $A_{\\mathrm{eff}}$ is the effective beam area, $\\Delta \\nu$ is the FWHM optical linewidth and $ \\nu-\\nu_{\\alpha}$ the optical detuning of the linearly polarized probe light from the electronic manifold of the excited state. Here, $\\nu_{\\alpha}$ with $\\alpha \\in \\{ a,b \\}$ are the optical resonance frequencies of the two ground state hyperfine levels. For the magnetic fields of interest in this paper, the Zeeman splittings are small compared to the detuning from the optical transition, hence, they can be safely neglected in the description of the atom-light coupling. \n\nIn Faraday probing, the optical rotation is usually measured with a balanced photodetector. The optical signal reaching the detector can be expressed in the Stokes parameter formalism using the input-output relations \\cite{RevModPhys.82.1041,Colangelo2017,colangelo2013quantum}\n\\begin{equation}\n\\mathcal{\\hat{S}}_y^{(\\mathrm{out})} (t) \\approx \\mathcal{\\hat{S}}_y^{(\\mathrm{in})} (t) + [g_b \\hat{F}_{z}^{b}(t)-g_{a}\\hat{F}_{z}^{a} (t)] \\mathcal{S}_{x}^{(\\mathrm{in})}(t), \\label{eq:polarimeter}\n\\end{equation}\nwhere $\\mathcal{\\hat{S}}_x \\equiv [\\hat{n}_{\\mathrm{ph}}(x)-\\hat{n}_{\\mathrm{ph}}(y) ]\/2  $ is the difference between the number of photons with linear polarization along the $\\mathbf{\\hat{x}}$- and $\\mathbf{\\hat{y}}$- directions and $\\mathcal{\\hat{S}}_y \\equiv [\\hat{n}_{\\mathrm{ph}}(+45^{\\circ})-\\hat{n}_{\\mathrm{ph}}(-45^{\\circ}) ]\/2  $ is the difference between the photon numbers  with linear polarization along the $\\pm 45^{\\circ}$, with respect to the x-axis. The input beam is linearly polarized along $\\mathbf{\\hat{x}}$, therefore $\\mathcal{\\hat{S}}_x^{(\\mathrm{in})}$ can be treated as a classical variable with $ \\mathcal{S}_{x}^{(\\mathrm{in})} \\equiv \\langle \\Phi \\rangle\/2$. Here $\\langle \\Phi \\rangle$ is the mean photon flux where we additionally assume that $\\mathcal{S}_x^{(\\mathrm{in})}(t) =\\mathcal{S}_x^{(\\mathrm{in})}= \\langle \\Phi \\rangle \/2$ is on average constant during the acquisition process \\cite{Julsgaard2001}. The applicability of Eq.\\eqref{eq:polarimeter} is limited to small Faraday rotation angles. For unpolarized or low-polarized vapors, the small-angle approximation is satisfied. Finally, due to the off-resonance probing considered here, the effects of optical absorption are negligible and therefore not accounted in Eq.\\eqref{eq:polarimeter}.\n\n\\section{Spin correlation functions}\n\\label{sec:Correlations}\n\nTo describe the statistical properties of spin-noise we will make extensive use of two-time correlation functions: for observables $\\hat{A}$ and $\\hat{B}$, the covariance is $\\mathcal{R}_{\\hat{A},\\hat{B}}(\\tau) \\equiv \\langle \\hat{A}(\\tau) \\hat{B}(0) + \\hat{B}(0)\\hat{A}(\\tau) \\rangle\/2= \\mathrm{Re}[\\langle \\hat{A}(\\tau) \\hat{B}(0) \\rangle]$ since $\\hat{A}$ and $\\hat{B}$ are Hermitian operators and $(\\hat{A} \\hat{B})^{\\dagger}=\\hat{B}^{\\dagger}\\hat{A}^{\\dagger}=\\hat{B}\\hat{A}$. Additionally, given a column vector of observables $\\mathbf{\\hat{V}}(t) = (\\hat{A},\\hat{B},\\hat{C},...)^T$ with stationary statistical properties, the covariance matrix is \n\\begin{equation}\n\\begin{split}\n\\mathcal{R}_{\\mathbf{\\hat{V}},\\mathbf{\\hat{V}}}(\\tau) &= \\frac{1}{2}\\Big \\{ \\langle \\mathbf{\\hat{V}}(\\tau)\\mathbf{\\hat{V}}^{T}(0)\\rangle + [\\langle \\mathbf{\\hat{V}}(0)\\mathbf{\\hat{V}}^{T}(\\tau)\\rangle]^{T} \\Big \\}\\\\\n&= \\mathrm{Re}[\\langle \\mathbf{\\hat{V}}(\\tau)\\mathbf{\\hat{V}}^{T}(0)\\rangle].\n\\label{eq:RDef}\n\\end{split}\n\\end{equation}  \nFor commuting or classical observables $\\mathbf{V}(t)$, this is simply \n$\\mathcal{R}_{\\mathbf{{V}},\\mathbf{{V}}}(\\tau) \\equiv \\langle \\mathbf{{V}}(\\tau)\\mathbf{{V}}^{T}(0)\\rangle$. Here, $\\langle \\cdot \\rangle $ denotes the average over different and independent realizations of the acquisition process. In fact, the main premise in spin-noise spectroscopy is that the spin dynamics have equilibrated; therefore, the recorded signal corresponds to a stochastic spin evolution probed for acquisition times $t \\gg T_2$, where the spin variance has transitioned from the time-dependent diffusive evolution to the stationary regime \\cite{gardiner2009stochastic}.\n\n\\subsection{Equal-time correlations}\n\\label{sec:EqualTimeCorrelations}\n\nThe thermal state $\\rho_\\mathrm{eq}^{\\otimesN_\\mathrm{at}}$, $\\rho_\\mathrm{eq} \\equiv \\mathbbm{1}\/d$ and  Eq.\\eqref{eq:RDef} imply the equal-time correlators\n\\begin{eqnarray}\n\\mathcal{R}_{\\hat{F}_i^{\\alpha},\\hat{F}_j^{\\beta}}(0) &=& \\frac{1}{2} \\left\\langle \\hat{F}_i^{\\alpha}(0)\\hat{F}_j^{\\beta}(0) + \n \\hat{F}_j^{\\beta}(0)\\hat{F}_i^{\\alpha}(0)\\right\\rangle\n\\nonumber \\\\ & = & \n\\delta_{ij} \\delta_{\\alpha \\beta} \\frac{f^{\\alpha}(f^{\\alpha}+1)(2f^{\\alpha}+1)}{6 (2I+1) }N_\\mathrm{at}.\n\\label{eq:variance}\n\\end{eqnarray}\nThis describes vanishing correlations among different components and different hyperfine levels at $\\tau=0$. It is nonetheless compatible with non-vanishing off-diagonal unequal-time correlations, to be computed in the following section. The spectral features of the resulting cross-correlations are similar to what has been shown for heterogeneous spin systems previously \\cite{roy2015cross,katz2015coherent}. For future reference we define $\\var{(F^{\\alpha})} \\equiv f^{\\alpha} (f^{\\alpha}+1) (2f^{\\alpha}+1) N_\\mathrm{at} \/ [6(2I+1)]$ being the variance of the angular momentum vector in the unpolarized state. \n\n\n\n\n\n\n\n\n\\subsection{Unequal-time correlations}\n\\label{sec:Unequal-time correlations}\n\n\n\nFor $\\tau>0$, the $\\tau$-derivative of $\\langle \\mathbf{\\hat{X}}(\\tau)\\mathbf{\\hat{X}}^{T}(0)\\rangle$ is given by\n\\begin{eqnarray}\n\\frac{d}{d\\tau} \\langle \\mathbf{\\hat{X}}(\\tau)\\mathbf{\\hat{X}}^{T}(0) \\rangle \n&=& \\langle \\frac{A \\mathbf{\\hat{X}} (\\tau)d\\tau+Q d {\\cal{\\mathbf{\\hat{W}}}}(\\tau)}{d\\tau}\\mathbf{\\hat{X}}^{T}(0)\\rangle \\nonumber \\\\\n&=&A  \\langle \\mathbf{\\hat{X}}(\\tau)\\mathbf{\\hat{X}}^{T}(0) \\rangle,\n\\end{eqnarray}\nwhere in the second step we used the properties of the Wiener increment assuming there is no correlation between $d {\\cal{\\mathbf{\\hat{W}}}}(\\tau)$ and the initial state vector $\\mathbf{\\hat{X}}^{T}(0)$. Consequently, the $\\tau$-dependence of the steady-state covariance matrix is given by\n\\begin{equation}\n  \\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(\\tau) =\n  \\begin{cases}\ne^{A\\tau}\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) & ,\\tau > 0\\\\\n\\\\\n\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) e^{-A^{T}\\tau} & ,\\tau < 0\n \\end{cases} \\label{eq:cov}\n\\end{equation}\nThis is an important result, indicating that the $\\tau$ evolution of the correlation $\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(\\tau)$  is governed by the same equation as the time evolution of the mean $ \\langle \\mathbf{\\hat{X}} (t) \\rangle$, c.f. Eq.\\eqref{eq:AverageSpinDynamics}. This result is known in the literature as the regression theorem \\cite{gardiner2009stochastic}.  Hereafter, we concentrate only on correlations for $\\tau>0$, whilst equivalent results can be obtained for $\\tau<0$. In Eq.\\eqref{eq:cov}, $A$ is the $6 \\times 6$ matrix\n\\begin{equation}\n\\label{eq:ADef}\n \\setlength{\\arraycolsep}{0.1pt}\n\\small{A=} \n\\begin{pmatrix} \n-R_{\\mathrm{se}} \\kappa_{aa}  & 0 & -\\omega_0 & R_{\\mathrm{se}} \\kappa_{bb} & 0 & 0 \\\\\n0& -R_{\\mathrm{se}} \\kappa_{aa} & 0& 0 &\nR_{\\mathrm{se}} \\kappa_{bb} & 0\n\\\\\n\\omega_0 & 0 & -R_{\\mathrm{se}} \\kappa_{aa} & 0 & 0 &  R_{\\mathrm{se}} \\kappa_{bb} \\\\ \nR_{\\mathrm{se}} \\kappa_{aa} & 0 & 0 &  -R_{\\mathrm{se}} \\kappa_{bb} & 0 & \\omega_0 \\\\\n0 & R_{\\mathrm{se}} \\kappa_{aa} & 0 & 0 & -R_{\\mathrm{se}} \\kappa_{bb} & 0 \\\\\n0 & 0 & R_{\\mathrm{se}} \\kappa_{aa} & -\\omega_0 & 0 & -R_{\\mathrm{se}} \\kappa_{bb} \n\\end{pmatrix} .\n\\end{equation}\n\n\nAnalytical expressions for the covariance components can be derived by eigen-decomposing the matrix exponential into $e^{A \\tau}= V e^{\\Lambda \\tau} V^{-1}$. The details of the eigenspectrum of $A$ are presented in Appendix \\ref{sec:Eigenspectrum}. In the presence of the magnetic field, the dynamics are separated into longitudinal and transverse, with vanishing correlations between the two. Simple analytical expressions are obtained for both components. The longitudinal covariances are given by\n\\begin{equation}\n\\mathcal{R}_{\\hat{F}_y^{a},\\hat{F}_y^{a}}(\\tau) = \\frac{\\var{(F^{a})}}{\\kappa_{aa}+\\kappa_{bb}}\\Big[\\kappa_{bb}+\\kappa_{aa}e^{-(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}\\tau} \\Big],\n\\label{eq:longitudinal1}\n\\end{equation}\n\\begin{equation}\n\\mathcal{R}_{\\hat{F}_y^{b},\\hat{F}_y^{b}}(\\tau) = \\frac{\\var{(F^{b})}}{\\kappa_{aa}+\\kappa_{bb}}\\Big[\\kappa_{aa}+\\kappa_{bb}e^{-(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}\\tau} \\Big],\n\\end{equation}\n\\begin{equation}\n\\mathcal{R}_{\\hat{F}_y^{a},\\hat{F}_y^{b}}(\\tau) = \\frac{\\var{(F^{b})}}{\\kappa_{aa}+\\kappa_{bb}}\\kappa_{bb}\\Big[1-e^{-(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}\\tau} \\Big], \n\\end{equation}\n\\begin{equation}\n\\mathcal{R}_{\\hat{F}_y^{b},\\hat{F}_y^{a}}(\\tau) = \\frac{\\var{(F^{a})}}{\\kappa_{aa}+\\kappa_{bb}}\\kappa_{aa}\\Big[1-e^{-(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}\\tau} \\Big]. \n\\label{eq:longitudinal4}\n\\end{equation}\nWe note that $\\kappa_{aa} \\var{(F^a)} = \\kappa_{bb} \\var{(F^b)}$ and thus, $\\mathcal{R}_{\\hat{F}_y^{a},\\hat{F}_y^{b}}(\\tau) = \\mathcal{R}_{\\hat{F}_y^{b},\\hat{F}_y^{a}}(\\tau) $. The transverse covariance components can be summarized in a simple formula\n\\begin{equation}\n\\mathcal{R}_{\\hat{F}_i^{\\alpha},\\hat{F}_j^{\\beta}}(\\tau) = 2 \\Big( \\mathrm{Re}[c_1 e^{-(\\Gamma_{-}+i \\Omega_{-})\\tau}]+\n \\mathrm{Re}[c_2 e^{-(\\Gamma_{+}+i \\Omega_{+})\\tau}] \\Big), \\label{eq:cov-tau}\n\\end{equation}\nwhere the coefficients $c_1 \\equiv c_1(F^{\\alpha}_{i},F^{\\beta}_{j})$ and $ c_2 \\equiv c_2(F^{\\alpha}_{i},F^{\\beta}_{j})$ are different for each covariance component and are discussed in Appendix \\ref{sec:time evolution}. For slow spin-exchange ($\\omega_0 \\gg R_{\\mathrm{se}}$), the relaxation rates $\\Gamma_{\\pm}$ entering Eq.\\eqref{eq:cov-tau} are given by $\\Gamma_{-} \\approx \\kappa_{aa}R_{\\mathrm{se}}$ and $\\Gamma_{+} \\approx \\kappa_{bb}R_{\\mathrm{se}}$ and are of comparable orders of magnitude. Additionally, the two modes $\\Omega_{\\pm}$ precess at the same frequency, in opposite directions, and with independent phases; therefore, they are related through $\\Omega_{+}= -\\Omega_{-}$. These approach $\\pm \\omega_0$ when $\\omega_0 \\gg R_{\\mathrm{se}}$, but can differ significantly from it in the rapid SE regime ($\\omega_0 \\ll R_{\\mathrm{se}}$) where the precession frequency is shifted down to\n\\begin{equation}\n\\Omega_+ = - \\Omega_{-} \\approx \\frac{3(2I+1)}{3+4I(I+1)}\\omega_0.\n\\end{equation} \nThe corresponding relaxation rates are approximately given by \\cite{PhysRevA.16.1877}\n\\begin{equation}\n\\Gamma_{+} \\approx(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}},\n\\end{equation}\n\\begin{equation}\n\\Gamma_{-} \\approx \\frac{\\omega_0^2}{R_{\\mathrm{se}}} \\frac{2 I(I+1)(2I-1)(2I+3)}{(4I^2 + 4I +3)}.\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{Real_parts.pdf} \n\\caption{Two times the real parts of the coefficients $c_1(F^{\\alpha}_{z},F^{\\beta}_{z})$ and $c_2(F^{\\alpha}_{z},F^{\\beta}_{z})$ entering the time-evolution of the transverse covariance components (a) $\\mathcal{R}_{F_z^{a},F_z^{a}}(\\tau)$, (b) $\\mathcal{R}_{F_z^{a},F_z^{b}}(\\tau) $ and $\\mathcal{R}_{F_z^{b},F_z^{a}}(\\tau)$, and (c) $ \\mathcal{R}_{F_z^{b},F_z^{b}}(\\tau)$, according to Eq.\\eqref{eq:cov-tau}. The single-atom variances in the two hyperfine multiplets are shown with blue dashed lines. The plot corresponds to a $^{87}$Rb vapor at a magnetic field of $B=10$ mG.}\n\\label{fig:cov_coefficients}\n\\end{figure} \n\nIn \\autoref{fig:cov_coefficients} the real parts of $c_1$ and $c_2$ are plotted against $R_{\\mathrm{se}}\/ \\omega_0$ while in Appendix \\ref{sec:time evolution} analytical expressions for the coefficients are presented for the covariance components along the $\\mathbf{\\hat{z}}$-direction. \n\n\\begin{figure*}[htp]\n\\centering\n\\includegraphics[width=18cm]{time-frequency-final.pdf} \n\\caption{$\\tau$-dependence  of the steady-state covariance components  probed by a linearly polarized electric field (top row) and the corresponding power spectra (bottom row) for a $^{87}$Rb vapor at four different values of the spin-exchange rate: $R_{\\mathrm{se}}\/ \\omega_0=0.1$ (purple), $1$ (blue), $10$ (yellow) and $100$ (red). The magnitude of the magnetic field is fixed to $B=10$ mG corresponding to a  Larmor frequency of $\\omega_0 \\approx 2 \\pi \\times 7.0$ \\si{\\kilo\\hertz} and a Larmor period of $\\tau_{_L}\\approx 0.1$ \\si{\\milli\\second}. In the time domain, an increasing coherence time for all covariance components as approaching the SERF regime is illustrated. On top of that, the cross-covariances in (b) reveal an increasing amplitude, indicating strong cross-correlations between the two hyperfine levels. The power spectra behave accordingly. In (a) and (c) the spectrum is given by a single Lorentzian centered at the corresponding precession frequency, while the spectrum in (b) is of dispersive form that tends towards a Lorentzian in the SERF regime. \nFor rapid SE, the Larmor frequency is slown-down to the value $|\\Omega_+|=|\\Omega_-| \\approx 3(2I+1)\\omega_0\/[3+4I(I+1)]\\approx 2 \\pi \\times 4.7$ \\si{\\kilo\\hertz}, also the linewidth is approximately given by $\\Gamma_{-} \\approx \\frac{\\omega_0^2}{R_{\\mathrm{se}}} [2 I(I+1)(2I-1)(2I+3)]\/(4I^2 + 4I +3)$, both constituting strong footprints of the SE interaction on the spin-noise spectrum. The spectral lines in the lower manifold are broader due to the large contribution of the fast relaxation rate $\\Gamma_{+}= (\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}$, affecting the wings of the resonance.}\n\\label{fig:time-cov}\n\\end{figure*} \n\nAt high magnetic fields the fast spin precession ``averages out'' correlations between the two oppositely precessing states. In contrast, the dynamics in the two hyperfine levels are interlocked by the rapid SE collisions as the spins precess slowly in the magnetic field \\cite{PhysRevA.16.1877}; thus, enabling unequal-time cross-correlations to build-up. These effects are imprinted on the behaviour of  $c_1(F^{\\alpha}_{i},F^{\\beta}_{j})$ and $ c_2(F^{\\alpha}_{i},F^{\\beta}_{j})$ as illustrated in \\autoref{fig:cov_coefficients}b). Both coefficients transition from a zero value at high magnetic fields to an amplitude roughly equal to that of $\\mathcal{R}_{\\hat{F}_z^{b},\\hat{F}_z^{b}}(\\tau)$, shown in \\autoref{fig:cov_coefficients}c). Further, since the absolute values of the two coefficients are equal, according to Eq.\\eqref{eq:cov-tau}, the relaxation rates $\\Gamma_+$ and $\\Gamma_-$ contribute equally in the $\\tau$-evolution. In contrast, as seen in \\autoref{fig:cov_coefficients}a) and c), the dynamics of  $\\mathcal{R}_{\\hat{F}_z^{a},\\hat{F}_z^{a}}(\\tau)$ and $\\mathcal{R}_{\\hat{F}_z^{b},\\hat{F}_z^{b}}(\\tau)$ are mostly dominated by a single eigenmode with relaxation rate $\\Gamma_{-}$ and $\\Gamma_{+}$, respectively. As a consequence, the two auto-correlations relax differently.\n\n\n\nApart from the correlated dynamics, \\autoref{fig:cov_coefficients} indicates also the way spin-noise power is distributed among the two transverse modes appearing in Eq.\\eqref{eq:cov-tau} and subsequently across the frequency spectrum. This is the case since $\\mathrm{Re}[c_1]$ and $\\mathrm{Re}[c_2]$ directly refer to the spin-noise power associated with the particular mode. In the SERF regime, it is apparent that in the upper hyperfine manifold, spin-noise power is mostly concentrated in a narrow spectral region, in the vicinity of the precession frequency $\\Omega_{-}$ since $\\lambda_1=\\Gamma_{-}+i \\Omega_{-}$ dominates the correlations. In contrast, the power in the lower manifold is distributed across a broad range of frequencies around $\\Omega_{+}$, reflecting the dominance of $\\lambda_2=\\Gamma_{+}+i\\Omega_{+}$.  Finally, the cross-correlation power is zero as expected according to Eq.\\eqref{eq:variance}. \n\nAn overall demonstration of the discussed effects is illustrated in \\autoref{fig:time-cov}. Both the time and the frequency domain of the covariance functions are plotted across the SE-SERF transition while the magnetic field is kept constant at $B=10$ mG. When the SE rate starts to exceed the Larmor frequency, $\\omega_0 \\approx 2 \\pi \\times 7$ \\si{\\kilo \\hertz}, an increasing coherence time and a frequency shift is demonstrated for all covariance components. In \\autoref{fig:time-cov}a) and c) the auto-covariance functions $\\mathcal{R}_{F_z^{a},F_z^{a}}(\\tau)$ and $\\mathcal{R}_{F_z^{b},F_z^{b}}(\\tau)$ in the upper and the lower hyperfine manifolds, respectively, are presented together with the corresponding power spectra. The spectral lines in the lower manifold are broad due to the large contribution of the fast relaxation\nrate $\\Gamma_{+}= (\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}$, affecting the wings of the resonance. In contrast, narrow resonant features with considerable amplitudes are obtained for the upper manifold consistent with \\autoref{fig:cov_coefficients}a). Finally, in \\autoref{fig:time-cov}b) the cross-correlation dynamics are illustrated. Details about the spectral features as well as implications of the peculiar lineshapes are discussed in the following sections.\n\n\n\n\\subsection{Spin-noise spectra}\n\nThe power spectral density is the Fourier transform of the auto-covariance function of the detected optical signal \\cite{PhysRevA.96.062702}\n\\begin{equation}\nS(\\nu) =  \\int_{-\\infty}^{+ \\infty} \\mathcal{R}_{\\mathcal{\\hat{S}}_y^{(\\mathrm{out})},\\mathcal{\\hat{S}}_y^{(\\mathrm{out})}}(\\tau) e^{-2 \\pi i  \\nu \\tau} d \\tau. \\label{eq:PSDmain}\n\\end{equation}\nPlugging Eq.\\eqref{eq:polarimeter} into Eq.\\eqref{eq:PSDmain} and given that the Stokes and the angular momentum operators are Hermitian, it is apparent that four distinct terms contribute to the covariance function of the Faraday signal: i) a white photon shot noise originating from $\\mathcal{\\hat{S}}_y^{(\\mathrm{in})}(t)$, ii) the atomic covariance indicating correlations of the atomic signal between different times and iii) the cross-terms corresponding to correlations between the atomic angular momentum and the fluctuations of the input light $\\mathcal{\\hat{S}}_y^{(\\mathrm{in})}(t)$. The photon shot-noise auto-covariance function for a coherent field is given by $\\mathcal{R}_{\\mathcal{\\hat{S}}_y^{(\\mathrm{in})},\\mathcal{\\hat{S}}_y^{(\\mathrm{in})}}(\\tau) = \\mathrm{Re}[\\langle \\mathcal{\\hat{S}}_y^{(\\mathrm{in})} (\\tau) \\mathcal{\\hat{S}}_y^{(\\mathrm{in})} (0) \\rangle] =  \\langle \\Phi \\rangle \\delta(\\tau)$ with $\\delta(\\tau)$ being the Dirac delta function. Hereafter, we assume that polarization fluctuations of the input light are not affecting the atomic variables and therefore they are not correlated with the spin-evolution i.e., the cross-terms are not contributing to the spin-noise spectrum. For an unpolarized vapor probed by an off-resonance beam, the backaction effect is negligible \\cite{optical_mag}. The resulting auto-covariance function of the detected optical signal can be therefore expressed as\n\n\\begin{widetext}\n\\begin{equation}\n\\mathcal{R}_{\\mathcal{\\hat{S}}_y^{(\\mathrm{out})},\\mathcal{\\hat{S}}_y^{(\\mathrm{out})}}(\\tau) \\approx  \\langle \\Phi \\rangle \\delta(\\tau)  + \\frac{\\langle\\Phi\\rangle^2}{4} \\left\\{\ng_a^2 \\mathcal{R}_{\\hat{F}_z^a,\\hat{F}_z^a}(\\tau)  + g_b^2  \\mathcal{R}_{\\hat{F}_z^b,\\hat{F}_z^b}(\\tau) - g_a g_b [\\mathcal{R}_{\\hat{F}_z^a,\\hat{F}_z^b}(\\tau) + \\mathcal{R}_{\\hat{F}_z^b,\\hat{F}_z^a}(\\tau) ] \n\\right\\}.\n\\label{Eq:auto-corr}\n\\end{equation}\n\\end{widetext}\n\n\n\nIn order to obtain the power spectrum given by Eq.\\eqref{eq:PSDmain} and Eq.\\eqref{Eq:auto-corr}, we calculate the Fourier transform of each of the covariance components as demonstrated in the previous section\n\\begin{equation}\n\\begin{split}\n S_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(\\omega)  &=\\int_{-\\infty}^{0}  \\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) e^{-A^{T}\\tau}  e^{i\\omega \\tau} d \\tau\\\\ \n&+ \\int_{0}^{+\\infty} e^{A\\tau}\\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) e^{-i\\omega \\tau} d \\tau , \\label{eq:power spec}\n\\end{split}\n\\end{equation}\nBy applying the spectral decomposition $e^{A\\tau}=V e^{\\Lambda \\tau} V^{-1}$ and by explicitly calculating the integral $\\int_{0}^{+\\infty} e^{[\\Lambda-i \\omega \\mathbb{1}]\\tau} d\\tau=(\\Lambda-i \\omega \\mathbb{1})^{-1}$ we obtain \n\\begin{equation}\n S_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(\\omega)  =  \n -V (\\Lambda-i\\omega \\mathbb{1})^{-1} V^{-1} \\mathcal{R}_{\\mathbf{\\hat{X}},\\mathbf{\\hat{X}}}(0) + \\mathrm{c.c}. \\label{eq:power spec1}\n\\end{equation}\nThe resulting power spectra are real for all covariance components with the longitudinal being centered at zero and the transverse at $\\pm \\Omega_{\\pm}$. Eq.\\eqref{eq:power spec1} shows the dependence of the power spectrum on the eigenspectrum $\\Lambda$ in the presence of SE collisions, as was first illustrated in \\cite{PhysRevA.16.1877}. Both $\\Gamma_{\\pm}$ are contributing to the broadening of the spectrum, each weighted according to \\autoref{fig:cov_coefficients}.\n\nIn \\autoref{fig:PSDvsB} we present the atomic power spectrum resulting from the Fourier transform of the atomic covariance components appearing in the expression for the dispersive Faraday rotation signal, Eq.\\eqref{Eq:auto-corr}.\nWe note that the spectrum is qualitatively in agreement with the spectral features reported in \\cite{MitchellNatureCom}. It is apparent that the lineshape is dominated by the dynamics in the upper hyperfine multiplet since the amplitude of $\\mathcal{R}_{\\hat{F}_z^a,\\hat{F}_z^a}(\\tau)$ greatly exceeds the amplitudes of the remaining components. Although the underlying generation mechanism of the spectrum is the sum of  eight complex Lorentzians appearing in Eq.\\eqref{eq:PowerSpec}, in practice, the experimentally relevant lineshape resembles a simple Lorentzian feature centered at $|\\Omega_{\\pm}|$.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.0cm]{total_cov_proc.pdf} \n\\caption{Atomic spin-noise power spectrum for a pure $^{87}$Rb vapor as a function of the magnetic field. The spectrum is calculated at a temperature of $T=200^{\\circ}$ C corresponding to an alkali number density of $n_{\\mathrm{Rb}}=9.2 \\times 10^{14}$ cm$^{-3}$ and consequently to a spin-exchange rate of $R_{\\mathrm{se}}\\approx 2 \\pi \\times 134$ \\si{\\kilo\\hertz}. The chosen detuning is $50$ \\si{\\giga\\hertz} blue detuned from the $D_1$ transition and the pressure broadening $\\Delta \\nu= 10$ \\si{\\giga\\hertz}.}\n\\label{fig:PSDvsB}\n\\end{figure} \n\nDue to the intriguing behaviour of the correlation functions in the SERF regime, in the next section we show that by appropriately choosing the optical detuning of the Faraday probe we can control the spectral distribution of spin-noise power and under certain circumstances obtain significant reduction as a consequence of the peculiar lineshape.\n\n\\section{Predicted phenomena in SNS }\n\\label{sec:PredictedPhenomena}\n\nWe now describe specific observable phenomena predicted by the theory described in the preceding sections. \n\n\\subsection{Probe detuning dependence}\n\\label{sec:ProbeDetuning}\n\nBesides $c_n(F^{\\alpha}_{i},F^{\\beta}_{j})$ in \\autoref{fig:cov_coefficients} that shape the spectrum and distribute the noise power for a given ratio of the alkali number density and the magnetic field, an equally significant weighting parameter is the optical detuning of the probe. In \\autoref{fig:detuning} we plot the coupling constants entering Eq.\\eqref{Eq:auto-corr} as a function of the frequency of the probe for a $^{87}$Rb vapor at low buffer gas pressure; corresponding to an optical broadening of $\\Delta \\nu =1 $ \\si{\\giga\\hertz}. Due to the unique behaviour of each covariance component, by suitably choosing the detuning, both the lineshape and the spectral distribution of the spin-noise power can be modified. \n\n\n\nIn SERF, as a consequence of the strong cross-correlations between the two hyperfine levels, untypical spin-noise spectra are obtained. For instance, by choosing a red detuning at the wing of the lower hyperfine resonance, we eliminate the contribution of the strong covariance component in the upper hyperfine manifold and consequently the lineshape is dominated only by cross-covariances and covariances in the lower manifold.\nAccording to \\autoref{fig:cov_coefficients}, both of the preceding components have comparable amplitudes, and hence, as seen in \\autoref{fig:detuning}, by appropriately choosing the detuning we can make their contribution in the Faraday signal equally significant. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8.0cm]{coupling_constant_vs_detuning_proc.pdf} \n\\caption{Coupling constants $g_a^2$, $g_b^2$ and $-g_a g_b$ appearing in Eq.\\eqref{Eq:auto-corr} as a function of the optical detuning from the $D_1$ transition of $^{87}$Rb. The spectrum corresponds to an optical linewidth of $\\Delta \\nu=\\SI{1}{\\giga\\hertz}$ and the lineshapes are normalized to the maximum value of $g_a^2$.}\n\\label{fig:detuning}\n\\end{figure} \n\n\\subsection{Partial cancellation of spin-noise at the precession frequency}\n\\label{sec:ProbeDetuning}\n\nIn addition, the opposite sign between the coupling constants $g_{b}^2$ and $-g_{a} g_{b}$ leads to unusual spectral features as illustrated in \\autoref{fig:spectrum_detuning}. Resonant dips is possible to be observed at the spin-precession frequency as indicated in \\autoref{fig:spectrum_detuning}b) and c). In this case, the spectrum resembles a band-rejection filter centered at the precession frequency with a spectral distribution such that spin-noise power is reduced on resonance. The spectrum is an effect of the coherent cross-correlations that are subtracted from the $\\lambda_2$-dominated spectrum in the lower manifold. Finally, we note that in the same regime, fine adjustment of the detuning across the red, near-wing of the lower hyperfine resonance, results in modified spectra like, for instance, in  \\autoref{fig:spectrum_detuning}a) \nwhich are significantly different from the Lorentzian lineshape usually encountered in SNS experiments.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.0cm]{spectrum_vs_detuning.pdf} \n\\caption{Spin-noise spectrum of $^{87}$Rb for various detunings. (\\textbf{a}) The probe laser frequency is tuned to $\\nu=-5.7$ \\si{\\giga\\hertz} from the $D_1$ electronic transition, (\\textbf{b}) and (\\textbf{c}) the laser frequency is $\\nu=-6.2$ \\si{\\giga\\hertz}. The optical linewidth is $\\Delta \\nu=1$ \\si{\\giga\\hertz}. The temperature is $T=200^{\\circ}$ C corresponding to a number density of $n_{\\mathrm{Rb}}=9.2 \\times 10^{-14}$ cm$^{-3} $ and to a spin-exchange rate of $R_{\\mathrm{se}} \\approx 2 \\pi \\times 134$ \\si{\\kilo\\hertz}. In (\\textbf{c}) we extend the plot of (\\textbf{b}) to higher frequencies. The dashed lines correspond to the precession frequencies $|\\Omega_{\\pm}|$ for each applied magnetic field.}\n\\label{fig:spectrum_detuning}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Variable integrated spin-noise power}\n\\label{sec:IntegratedPower}\n\nIn many experimental scenarios the spin-noise power of the thermal state is obtained either from the variance of the time-averaged signal in a sample time $\\Delta t$, $\n\\bar{\\hat{F}}_i^{\\alpha} (t) = \\frac{1}{\\Delta t} \\int_{t}^{t+\\Delta t} \\hat{F}_i^{\\alpha} (t') dt' $, after averaging many repetitions  \\cite{PhysRevLett.104.133601,VasilakisPolzik2015} or by fitting or numerically integrating the steady-state spectrum \\cite{katsoprinakis,Dellis,Squeezed-Spin-Noise,MitchellNatureCom}. In a time interval $[t,t+\\Delta t]$, each term in Eq.\\eqref{Eq:auto-corr} contributes to the total power of the detected Faraday signal. The spin-noise power in that time window is intimately related to the atomic covariance through\n\\begin{equation}\n\\begin{split}\n\\mathcal{P}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}} &= \\frac{1}{\\Delta t^2} \\int_{0}^{\\Delta t} \\tau d\\tau \\left[ -\\mathcal{R}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(-\\tau)-\\mathcal{R}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(\\tau) \\right]  \\\\\n&+ \\frac{1}{\\Delta t}  \\int_{-\\Delta t}^{\\Delta t} d\\tau \\mathcal{R}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(\\tau). \\label{eq:CorrectedVarTimeWindow}\n\\end{split}\n\\end{equation}\nThe derivation of the preceding is presented in Appendix \\ref{sec:MB}. In the frequency domain the power is obtained by realizing the fact that the correlation function is the Fourier transform of the power spectrum. From Eq.\\eqref{eq:CorrectedVarTimeWindow} we find \n\\begin{equation}\n\\mathcal{P}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}} =  \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} d\\omega S_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(\\omega) \\left[ \\mathrm{sinc}\\left(\\frac{\\omega \\Delta t}{2} \\right) \\right]^2, \\label{eq:NoisePowerSpecBandwidth}\n\\end{equation}\nwhere $\\mathrm{sinc}(x)=\\sin{(x)}\/x$. As $ \\Delta t \\rightarrow 0 $, from Eq.\\eqref{eq:NoisePowerSpecBandwidth} we obtain\n\\begin{equation}\n\\begin{split}\n\\mathcal{P}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}} &= \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} d\\omega S_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(\\omega)\\\\\n&=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} d\\omega \\mathcal{R}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(\\tau)e^{i \\omega \\tau}   d \\tau\\\\\n&= \\mathcal{R}_{\\hat{F}_z^{\\alpha},\\hat{F}_z^{\\beta}}(0),\n\\end{split}\n\\end{equation}\nwhere in the last step we used the property of the Dirac delta function $\\delta(\\tau-\\alpha)=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty} d\\omega e^{i \\omega (\\tau-\\alpha)}$. \nUsing the steady-state covariance at $\\tau=0$, the total power of the polarimeter output is expressed as\n\\begin{equation}\n\\mathcal{R}_{\\mathcal{\\hat{S}}_y^{(\\mathrm{out})},\\mathcal{\\hat{S}}_y^{(\\mathrm{out})}}(0)= \\mathrm{PSN} + \\frac{\\langle\\Phi\\rangle^2}{4} \n[g_a^2 \\var(F^a)  + g_b^2  \\var(F^b)],\n\\label{Eq:SN-power}\n\\end{equation}\nbeing in agreement with \\cite{vasilakis2011stroboscopic,shah-vasilakis}. Here $\\mathrm{PSN}$ is the total photon shot-noise power. As it is evident in Eq.\\eqref{Eq:SN-power},  although the cross-correlations are not contributing to the total spin-noise power, it is possible experimentally to measure a non-zero contribution due to the limited bandwidth of the acquisition process, defined as $\\mathrm{BW}=1\/(2 \\Delta t)$. As can be seen in the spectrum of \\autoref{fig:time-cov}(b), when in the SERF regime, the cross-correlation power, given by the integral under the spectral lineshape, is equally distributed between the resonant peak and the negative far wing that extends to the high-frequency part of the spectrum. Due to the limited bandwidth, frequencies higher than the Nyquist frequency are not entering the signal, thus resulting in a non-zero and positive cross-power. This means that according to Eq.\\eqref{eq:NoisePowerSpecBandwidth} and  Eq.\\eqref{Eq:auto-corr}, the amount of the remaining non-zero cross-correlation power will be subtracted from the total power of the Faraday signal, resulting in a reduced spin-noise power as opposed to the expected, given by Eq.\\eqref{Eq:SN-power}.\n\nApart from the non-zero cross-correlation power, we predict an additional and similar mechanism, realized in experimental implementations of SNS, that leads to further reduction of the noise power compared to the level expected from the thermal spin state. In the SERF regime, because of the fast decay $\\Gamma_{+}$ that enters both the auto-correlation functions, the corresponding spectra extend to high frequencies and are therefore constrained by the acquisition. Consequently, the integrated spectrum in Eq.\\eqref{eq:NoisePowerSpecBandwidth} contains only a portion of the presumed noise power of the thermal spin state. \n\nTo further illustrate these features, in \\autoref{fig:noise_det} we plot the spin-noise power as a function of the detuning for three different bandwidths. The power is obtained in two different ways. Either by integrating the spectrum up to the frequency defined by $\\mathrm{BW}$ or by directly applying Eq.\\eqref{eq:NoisePowerSpecBandwidth}. In both cases, the power has been normalized against the total power in the whole frequency range, given by Eq.\\ref{Eq:SN-power}. It is apparent that both methods produce similar results.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=8.5cm]{noise_detuning_proc.pdf} \n\\caption{Integrated spin-noise power as a function of the detuning. The spectrum is integrated up to three different bandwidths, $80$ \\si{\\kilo\\hertz} (green), $200$ \\si{\\kilo\\hertz} (purple) and $1$ \\si{\\mega\\hertz} (red). The bandwidth is related to the sampling time through $\\mathrm{BW}=1\/(2 \\Delta t)$. Solid lines correspond to direct integration of the power spectrum up to the bandwidth frequency, whereas the dashed lines correspond to the noise given by Eq.\\eqref{eq:NoisePowerSpecBandwidth}. The spectrum corresponds to a $^{87}$Rb vapor at a temperature of $T=200$ $^{\\circ}$C and a magnetic field of $B=10$ mG. The optical linewidth is $\\Delta \\nu=10$ \\si{\\giga\\hertz}. The response is normalized to the total atomic noise $g_a^2 \\var(F^a)  + g_b^2  \\var(F^b)$ obtained when integrating the spectrum in the whole frequency range.}\n\\label{fig:noise_det}\n\\end{figure} \nInterestingly, for a $^{87}$Rb vapor at realistic conditions, e.g., a cell temperature of $T=200$ $^{\\circ}$C, a magnetic field of $B=10$ mG, a pressure broadening of $\\Delta \\nu=10$ \\si{\\giga\\hertz} and a measurement bandwidth of $200$ \\si{\\kilo\\hertz} we obtain a spin-noise power at the far-wings of the optical transition of approximately $20$ \\% smaller than the noise of the thermal state. Apparently, for even smaller acquisition bandwidths, it can exceed $40$ \\%. In the limit where the bandwidth extends to infinity we obtain $g_a^2 \\var(F^a)  + g_b^2  \\var(F^b)$. The same results are also obtained when the noise-power is extracted from the time domain signal since there is direct relation between Eqs. \\eqref{eq:CorrectedVarTimeWindow} and \\eqref{eq:NoisePowerSpecBandwidth}.\n\n\\begin{figure*}[htp]\n\\centering\n\\includegraphics[width=15cm]{fit_spectrum.pdf} \n\\caption{$\\tau$-evolution of a) $\\mathcal{R}_{\\hat{F}_z^{a},\\hat{F}_z^{a}} (\\tau)$ and b) $\\mathcal{R}_{\\hat{F}_z^{b},\\hat{F}_z^{b}} (\\tau)$ with the corresponding power spectra presented in c) and d). The green areas represent a Lorentzian fit to the power spectrum of the form $\\gamma\/[(\\nu-\\nu_0)^2 +\\gamma^2]$. The spectrum corresponds to a $^{87}$Rb vapor at a magnetic field of $B=10$ mG and a spin-exchange rate 100 times larger than the Larmor frequency.}\n\\label{fig:fit spectrum}\n\\end{figure*} \n\n\nLastly, we emphasize that fitting the spectrum with a single Lorentzian, leads to a considerable underestimation of the spin-noise power, given by the integrated area under the Lorentzian curve. This is illustrated in \\autoref{fig:fit spectrum} where both the spectra in the two hyperfine states are fitted by simple Lorentzian functions, depicted by the green filled areas. We find that in the upper manifold the fit captures $83$ \\% of the total power whereas in the lower manifold only $16$ \\%. This is expected since the first is dominated by the slow and the latter by the fast relaxation as demonstrated in \\autoref{fig:fit spectrum}a) and b). In the SERF regime, the coupling between the two hyperfine multiplets is roughly characterized by $\\Gamma_{-}=(\\kappa_{aa}+\\kappa_{bb})R_{\\mathrm{se}}$ \\cite{PhysRevA.103.043116}, therefore it takes at least $1\/\\Gamma_{-}$ amount of time for the two manifolds to be correlated. This is reflected in the rapid jumps of the correlation functions close to $\\tau=0$. After that time interval, because of the strong interlock of the two states, the resulting correlations have long coherence times with amplitudes given in \\autoref{fig:cov_coefficients}.\n\n\n\\subsection{Observations}\n\\label{sec:Obervations}\n\nThe new features are the result of strong correlations between the two hyperfine manifolds in the SERF regime. These correlations result from the rapid transfer of angular momentum between the two hyperfine populations, and cause the two states to precess in a interlocked way.  \n\nThe effects are most visible when the probe frequency is tuned to the wings of the lower hyperfine level. Stronger effects are expected for low buffer gas pressures, for which the optical transitions from the ground state hyperfine levels are better resolved.\n\nFor simplicity, we have not included spin-destruction collisions, power broadening, or diffusion in the spin dynamics model. The first two of these can be included quite simply in the Bloch equations.  Diffusion can also be included, using a mode expansion \\cite{PhysRevA.102.012822}, at the cost of increasing the dimensionality of the model. \n\nAs just noted, the most dramatic new effects occur at low buffer gas pressure, and thus in conditions for which spin destruction collisions play a minor role.  Similarly, diffusion and power broadening can be made negligible by working with a large probe beam size \\cite{PhysRevA.96.062702}.  For these reasons, we believe the effects predicted here should be observable in appropriate conditions, and that other conditions can also be modeled through straightforward adaptation of the theory given here. \n\n\\section{Conclusions}\n\\label{sec:Conclusions}\n\nWe have derived analytic expressions for the spin-noise correlations and spin-noise spectra of hot alkali vapors, taking into account hyperfine, Zeeman, and spin-exchange interactions, and also accounting for the effects of probe detuning relative to the different atomic  hyperfine transitions. The results are applicable both in and out of the SERF regime, and show several interesting spectral behaviors not seen in phenomenological models. \n\nThe observation that spin-noise can be reduced at the Larmor frequency by proper choice of detuning is particularly intriguing for magnetic sensing and extensions such as magnetic gradiometry and comagnetometry. \n \n\n\n \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn recent years $p-$adic analysis has received a lot of attention due to its\napplications in mathematical physics, see e.g. \\cite{A-K2}, \\cite{A-B-K-O},\n\\cite{A-B-O}, \\cite{KH1}, \\cite{KH2}, \\cite{KO2}, \\cite{R-T}, \\cite{Va},\n\\cite{V-V-Z} and references therein. As a consequence new mathematical\nproblems have emerged, among them, the study of $p$-adic pseudo-differential\nequations, see e.g. \\cite{A-K-S}, \\cite{Chuong-Co}, \\cite{KO1}, \\cite{KH0},\n\\cite{KO2}, \\cite{KO3}, \\cite{KO4}, \\cite{KO5}, \\cite{Ro-Zu}, \\cite{V-V-Z},\n\\cite{Z-G1}, \\cite{Z-G2}, \\cite{Z-G3} and references therein. In this paper,\nwe study the solutions of $p$-adic elliptic pseudo-differential equations on\nSobolev spaces.\n\nA pseudo-differential operator $f(D,\\beta)$ is an operator of the form\n\\[\n\\left(  f(D,\\alpha)\\varphi\\right)  (x)=\\mathcal{F}_{\\xi\\rightarrow x}%\n^{-1}\\left(  |f(\\xi)|_{p}^{\\alpha}\\mathcal{F}_{x\\rightarrow\\xi}\\phi(x)\\right)\n,\\text{ \\ }\\phi\\in S,\n\\]\nwhere $\\mathcal{F}$ denotes the Fourier transform, $\\alpha$ is a positive real\nnumber, $S$ denotes the $\\mathbb{C}$-vector space of Bruhat-Schwartz functions\nover $\\mathbb{Q}_{p}^{n}$, and $f(\\xi)\\in\\mathbb{Q}_{p}[\\xi_{1},\\dotsc,\\xi\n_{n}]$. If $f(\\xi)$ is a homogeneous polynomial of degree $d$ satisfying\n\\[\nf(\\xi)=0\\text{ if and only if }\\xi=0,\n\\]\nthen the corresponding operator is called an elliptic pseudo-differential\noperator. At any case, the operator $f(D,\\beta)$ is continuous and has a\nself-adjoint extension with dense domain in $L^{2}(\\mathbb{Q}_{p}^{n})$. This\noperator is considered to be a $p$-adic analogue of a linear partial elliptic\ndifferential operator with constant coefficients. A $p$-adic\npseudo-differential equation is an equation of type\n\\[\nf(D,\\alpha)u=v.\n\\]\nIf $v\\in S$, then there exists a distribution $E_{\\alpha}$, a fundamental\nsolution, such that $u=E_{\\alpha}\\ast v$ is a solution. The existence of a\nfundamental solution for general pseudo-differential operators was established\nby the second author in \\cite{Z-G1} by adapting the proof given by Atiyah for\nthe Archimedean case \\cite{Atiyah}. However, it is unknown to which function\nspace $E_{\\alpha}\\ast v$ belongs. In this paper, we show that if $f(D,\\alpha)$\nis an elliptic operator, then $u=E_{\\alpha}\\ast v$ belongs to a certain\nSobolev space (see Theorem \\ref{mainresult}). Furthermore, we give conditions\nfor the continuity and uniqueness of $u$. By modifying the Sobolev norm, we\ncan establish that $f(D,\\alpha)$ gives an isomorphism between certain Sobolev\nspaces, (see Propositions \\ref{prop1}, \\ref{prop2} and Theorem \\ref{mainII}).\nOur approach is based on the explicit calculation of fundamental solutions of\npseudo-differential operators on certain function spaces and the fact that\nelliptic pseudo-differential operators behave like the Taibleson operator when\nacting on certain function spaces (see Theorems \\ref{fund solu Tai},\n\\ref{fund solu ellip}).\n\n\\textbf{Acknowledgement.} The authors wish to thank the referee for his\/her\ncareful reading of the original manuscript.\n\n\\section{Preliminary Results}\n\nWe summarize some basic facts about $p$-adic analysis that will be used in\nthis paper. For a complete exposition, we refer the reader to \\cite{TA},\n\\cite{V-V-Z}.\n\nLet $\\mathbb{Q}_{p}$ be the field of the $p$-adic numbers, and let\n$\\mathbb{Z}_{p}$ be the ring of $p$-adic integers. For $x\\in\\mathbb{Q}_{p}$,\nlet $v(x)\\in\\mathbb{Z}\\cup\\left\\{  \\infty\\right\\}  $ denote the valuation of\n$x$ normalized by the condition $v(p)=1$. By definition $v(x)=\\infty$ if and\nonly if $x=0$. Let $|x|_{p}=p^{-v(x)}$ be the normalized absolute value. Here,\nby definition $|x|_{p}=0$ if and only if $x=0$. We extend the $p$-adic\nabsolute value to $\\mathbb{Q}_{p}^{n}$ as follows:\n\\[\n||x||_{p}:=\\text{max}\\{|x_{1}|_{p},\\dotsc,|x_{n}|_{p}\\},\\text{ for }%\nx=(x_{1},\\dotsc,x_{n})\\in\\mathbb{Q}_{p}^{n}.\n\\]\n\n\nWe define the \\textit{exponent of local constancy} of $\\varphi(x)\\in\nS(\\mathbb{Q}_{p}^{n})$ as the smallest integer, $l\\geq0$, with the property\nthat, for any $x\\in\\mathbb{Q}_{p}^{n}$,\n\\[\n\\varphi(x+x^{\\prime})=\\varphi(x)\\text{ if }||x^{\\prime}||_{p}\\leq p^{-l}.\n\\]\n\n\nFor $x$, $y$ in $\\mathbb{Q}_{p}^{n}$, we put $x\\cdot y=\\sum_{i=1}^{n}\nx_{i}y_{i}$.\n\nLet $\\Psi$ denote an additive character of $\\mathbb{Q}_{p}$, trivial on\n$\\mathbb{Z}_{p}$, but not on $p^{-1}\\mathbb{Z}_{p}$. For $\\varphi\\in\nS(\\mathbb{Q}_{p}^{n})$, we define its Fourier transform as\n\\[\n(\\mathcal{F}\\varphi)(\\xi) = \\int_{\\mathbb{Q}_{p}^{n}} \\Psi(-x\\cdot\\xi\n)\\varphi(x)\\,dx,\n\\]\nwhere $dx$ denotes the Haar measure of $\\mathbb{Q}_{p}^{n}$ normalized in such\na way that $\\mathbb{Z}_{p}^{n}$ has measure one.\n\nWe denote by $\\chi_{r}$, $r\\in\\mathbb{Z}$, the characteristic function of the\npolydisc $B_{r}(0):=(p^{r}\\mathbb{Z}_{p})^{n}$. For any $\\varphi\\in S$, we\nset\n\\[\nr_{\\varphi}:=\\text{min}\\{r\\in\\mathbb{N}\\mid\\varphi|_{B_{r}\\left(  {0}\\right)\n}=\\varphi(0)\\}.\n\\]\n\n\n\\begin{Definition}\nWe set $\\mathcal{L}:=\\mathcal{L}(\\mathbb{Q}_{p}^{n})=\\{ \\varphi\\in S \\mid\n\\int_{\\mathbb{Q}_{p}^{n}} \\varphi(x)\\, dx =0\\}$, and $\\mathcal{W}%\n:=\\mathcal{W}(\\mathbb{Q}_{p}^{n})$ to be the $\\mathbb{C}$-vector space\ngenerated by the functions $\\chi_{r}$, $r \\in\\mathbb{Z}$.\n\\end{Definition}\n\nWe note that any $\\varphi\\in S$ can be written uniquely as $\\varphi\n_{\\mathcal{L}}+\\varphi_{\\mathcal{W}}$, where $\\varphi_{\\mathcal{W}%\n}=p^{r_{\\varphi}n}\\left(  \\int_{\\mathbb{Q}_{p}^{n}}\\phi(x)\\,dx\\right)\n\\chi_{r_{\\varphi}}\\in\\mathcal{W}$, and $\\varphi_{\\mathcal{L}}=\\varphi\n-\\varphi_{\\mathcal{W}}\\in\\mathcal{L}$. However, $S$ is not the direct sum of\n$\\mathcal{L}$ and $\\mathcal{W}$. The space $\\mathcal{W}$ was introduced in\n\\cite{Z-G2}, and $\\{\\mathcal{F}(\\varphi)\\mid\\varphi\\in\\mathcal{L}\\}$ is a\nLizorkin space of second class \\cite{A-K-S}.\n\n\\subsection{Elliptic Pseudo-differential Operators}\n\nLet $f(\\xi)\\in\\mathbb{Q}_{p}[\\xi_{1},\\dotsc,\\xi_{n}]$ be a nonconstant\npolynomial. A pseudo-differential operator $f(D,\\alpha)$, $\\alpha>0$, with\nsymbol $|f(\\xi)|_{p}^{\\alpha}$, is an operator of the form\n\\[\n\\left(  f(D,\\alpha)\\varphi\\right)  =\\mathcal{F}^{-1}\\left(  |f|_{p}^{\\alpha\n}\\mathcal{F}\\varphi\\right)  ,\n\\]\nwhere $\\varphi\\in S$.\n\n\\begin{Definition}\nLet $f(\\xi)\\in\\mathbb{Q}_{p}[\\xi_{1},\\dotsc,\\xi_{n}]$ be a nonconstant\npolynomial. We say that $f(\\xi)$ is an elliptic polynomial of degree $d$, if\nit satisfies: $(i)$ $f(\\xi)$ is a homogeneous polynomial of degree $d$, and\n$(ii)$ $f(\\xi)=0\\Leftrightarrow\\xi=0$.\n\\end{Definition}\n\n\\begin{Lemma}\n\\label{Lemma 1}\\cite[Lemma 1]{Z-G3} Let $f(\\xi)\\in\\mathbb{Q}_{p}[\\xi\n_{1},\\dotsc,\\xi_{n}]$ be an elliptic polynomial of degree $d$. There exist\npositive constants, $C_{0}(f)$ and $C_{1}(f)$, such that\n\\[\nC_{0}(f)||\\xi||_{p}^{d}\\leq|f(\\xi)|_{p}\\leq C_{1}(f)||\\xi||_{p}^{d},\\text{ for\nevery }\\xi\\in\\mathbb{Q}_{p}^{n}.\n\\]\n\n\\end{Lemma}\n\nWe note that if $f(\\xi)$ is elliptic, then $cf(\\xi)$ is elliptic for any\n$c\\in\\mathbb{Q}_{p}^{\\times}$. For this reason, we will assume from now on\nthat the elliptic polynomials have coefficients in $\\mathbb{Z}_{p}$.\n\n\\begin{Lemma}\n\\label{Lemma 2}\\cite[Lemma 3]{Z-G3} Let $f(\\xi)\\in\\mathbb{Q}_{p}[\\xi\n_{1},\\dotsc,\\xi_{n}]$ be an elliptic polynomial of degree $d$. Let\n$A\\subset\\mathbb{Q}_{p}^{n}$ be a compact subset such that $0\\notin A$. Then\nthere exists a positive integer $m=m(A,f)$ such that $|f(\\xi)|_{p}\\geq p^{-m}%\n$, for any $\\xi\\in A$. Furthermore, for any covering of $A$ of the form\n$\\cup_{i=1}^{L}B_{i}$, with $B_{i}=z_{i}+(p^{m}\\mathbb{Z}_{p})^{n}$, we have\n$|f(\\xi)|_{p}=|f(z_{i})|_{p}$ for any $\\xi\\in B_{i}$.\n\\end{Lemma}\n\n\\begin{Definition}\nLet $f(\\xi)\\in\\mathbb{Z}_{p}[\\xi_{1},\\dotsc,\\xi_{n}]$ be an elliptic\npolynomial of degree $d$. We will say that $|f|_{p}^{\\beta}$ is an elliptic\nsymbol, and that $f(D,\\beta)$ is an elliptic pseudo-differential operator of\norder $d$.\n\\end{Definition}\n\n\\subsection{Igusa's local zeta functions}\n\nLet $g(x)\\in\\mathbb{Q}_{p}[x]$, $x=(x_{1},\\dotsc,x_{n})$, be a non-constant\npolynomial. Igusa's local zeta function associated to $g(x)$ is the\ndistribution\n\\[\n\\langle|g|_{p}^{s},\\varphi\\rangle=\\int\\limits_{\\mathbb{Q}_{p}^{n}%\n\\smallsetminus g^{-1}(0)}|g(x)|_{p}^{s}\\varphi(x)\\,dx,\n\\]\nfor $s\\in\\mathbb{C}$, $\\text{Re}(s)>0$, where $\\varphi\\in S$, and $dx$ denotes\nthe normalized Haar measure of $\\mathbb{Q}_{p}^{n}$. The local zeta functions\nwere introduced by Weil and their basic properties for general $g(x)$ were\nfirst studied by Igusa. A central result in the theory of local zeta functions\nestablished that $|g|_{p}^{s}$ admits a meromorphic continuation to the\ncomplex plane such that $\\langle|g|_{p}^{s},\\varphi\\rangle$ is rational\nfunction of $p^{-s}$ for each $\\varphi\\in S$. Furthermore, there exists a\nfinite set $\\cup_{E\\in\\mathcal{E}}\\{(N_{E},n_{E})\\}$ of pairs of positive\nintegers such that\n\\[\n\\prod\\limits_{E\\in\\mathcal{E}}(1-p^{-n_{E}-N_{E}s})|g|_{p}^{s}%\n\\]\nis a holomorphic distribution on $S$. In particular, the real parts of the\npoles of $|g|_{p}^{s}$ are negative rational numbers see \\cite[Chap. 8]%\n{Igusa}. The existence of a meromorphic continuation for the distribution\n$|g|_{p}^{s}$ implies the existence of a fundamental solution for the\npseudo-differential operator with symbol $|g|_{p}^{\\alpha}$, \\cite{Z-G1}.\n\nFor a fixed $\\varphi\\in S$, we denote the integral $\\langle|g|_{p}^{s}%\n,\\varphi\\rangle$ by $Z_{\\varphi}(s,g)$. In particular, $Z(s,g)=Z_{\\chi_{0}%\n}(s,g)$.\n\n\\begin{Lemma}\nLet $f(x)\\in\\mathbb{Z}_{p}[x]$, $x=(x_{1},\\dotsc,x_{n})$, be an elliptic\npolynomial of degree $d$. Then\n\\[\nZ(s,f)=\\dfrac{L(p^{-s})}{1-p^{-ds-n}},\n\\]\nwhere $L(p^{-s})$ is a polynomial in $p^{-s}$ with rational coefficients.\nFurthermore, $s=-n\/d$ is a pole of $Z(s,f)$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $A=\\{x\\in\\mathbb{Z}_{p}^{n}\\mid\\text{ord}(x_{i})\\geq d,\\quad\ni=1,\\dotsc,n\\}$, and $A^{\\prime}=\\{x\\in\\mathbb{Z}_{p}^{n}\\mid\\text{ord}%\n(x_{i})<d,\\text{ for some }i\\}$. Then $\\mathbb{Z}_{p}^{n}$ is the disjoint\nunion of $A$ and $A^{\\prime}$ and\n\\begin{align*}\nZ(s,f)  &  =\\int_{A}|f(x)|_{p}^{s}\\,dx+\\int_{A^{\\prime}}|f(x)|_{p}^{s}\\,dx\\\\\n&  =p^{-ds-n}Z(s,f)+\\int_{A^{\\prime}}|f(x)|_{p}^{s}\\,dx,\n\\end{align*}\ni.e., $Z(s,f)=\\frac{1}{1-p^{-ds-n}}\\int_{A^{\\prime}}|f(x)|_{p}^{s}\\,dx$. Since\n$A^{\\prime}$ is compact, by applying Lemma \\ref{Lemma 2}, we find a covering\nof $A^{\\prime}=\\cup_{i=1}^{L}B_{i}$, where $|f|_{p}$ is constant on each\n$B_{i}$. Hence,\n\\[\n\\int_{A^{\\prime}}|f(x)|_{p}^{s}\\,dx=p^{-nm}\\sum_{i=1}^{L}|f(z_{i})|_{p}^{s},\n\\]\nand\n\\[\nZ(s,f)=\\dfrac{p^{-nm}\\sum_{i=1}^{L}|f(z_{i})|_{p}^{s}}{1-p^{-ds-n}}.\n\\]\n\n\\end{proof}\n\n\\subsection{The Riesz Kernel}\n\nWe collect some well-know results about the Riesz kernel that will be used in\nthe next sections, we refer the reader to \\cite{TA} or \\cite{V-V-Z} for\nfurther details.\n\nThe $p$-adic \\emph{Gamma function} $\\Gamma_{p}^{(n)}(s)$ is defined as\nfollows:\n\\[\n\\Gamma_{p}^{(n)}(s)=\\frac{1-p^{s-n}}{1-p^{-s}}\\text{, }s\\in\\mathbb{C}%\n,{\\ }s\\neq0.\n\\]\nThe Gamma function is meromorphic with simple zeros at $n+\\frac{2\\pi i}{\\ln\np}\\mathbb{Z}$ and unique simple pole at $s=0$. In addition, it satisfies\n\\[\n\\Gamma_{p}^{(n)}(s)\\Gamma_{p}^{(n)}(n-s)=1,\\text{ for }s\\notin\\{0\\}\\cup\n\\{n+\\frac{2\\pi i}{\\ln p}\\mathbb{Z}\\}.\n\\]\n\n\nThe \\emph{Riesz kernel} $\\mathit{R}_{s}$ is the distribution determined by the\nfunction\n\\[\n\\mathit{R}_{s }(x)=\\frac{||x||_{p}^{s -n}}{\\Gamma_{p}^{(n)}(s) }%\n,\\quad\\text{Re}(s)>0,{\\ }s \\notin n+\\frac{2\\pi i}{\\ln p}\\mathbb{Z} ,\\quad\nx\\in\\mathbb{Q}_{p}^{n}.\n\\]\n\n\nThe Riesz kernel has, as a distribution, a meromorphic continuation to\n$\\mathbb{C}$ given by\n\\begin{align*}\n\\left\\langle \\mathit{R}_{s}(x),\\varphi(x)\\right\\rangle  &  =\\frac{1-p^{-n}%\n}{1-p^{s-n}}\\varphi(0)+\\frac{1-p^{-s}}{1-p^{s-n}}\\int_{||x||_{p}>1}%\n||x||_{p}^{s-n}\\varphi(x)\\,dx\\\\\n&  +\\frac{1-p^{-s}}{1-p^{s-n}}\\int_{||x||_{p}\\leq1}||x||_{p}^{s-n}%\n(\\varphi(x)-\\varphi(0))\\,dx,\n\\end{align*}\nwith poles at $n+\\frac{2\\pi i}{\\ln p}\\mathbb{Z}$. In particular, for\n$\\text{Re}(s)>0$,\n\\[\n\\langle\\mathit{R}_{s}(x),\\varphi(x)\\rangle=\\frac{1-p^{-s}}{1-p^{s-n}}%\n\\int_{\\mathbb{Q}_{p}^{n}}\\varphi(x)||x||_{p}^{s-n}\\,dx,\\quad s\\notin\nn+\\frac{2\\pi i}{\\ln p}\\mathbb{Z},\n\\]%\n\\begin{equation}\n\\langle\\mathit{R}_{-s}(x),\\varphi(x)\\rangle=\\frac{1-p^{s}}{1-p^{-s-n}}%\n\\int_{\\mathbb{Q}_{p}^{n}}(\\varphi(x)-\\varphi(0))||x||_{p}^{-s-n}\\,dx.\n\\label{1}%\n\\end{equation}\n\n\nIn the case $s=0$, by passing to the limit, we obtain\n\\[\n\\langle\\mathit{R}_{0}(x),\\varphi(x)\\rangle:=\\lim_{s\\rightarrow0}\\left\\langle\n\\mathit{R}_{s}(x),\\varphi(x)\\right\\rangle =\\varphi(0),\n\\]\ni.e., \\ $\\mathit{R}_{0}(x)=\\delta\\left(  x\\right)  $, the Dirac delta\nfunction. Therefore, $\\mathit{R}_{s}\\in S^{\\prime}(\\mathbb{Q}_{p}^{n})$, for\n$s\\in\\mathbb{C\\setminus}\\left\\{  n+\\frac{2\\pi i}{\\ln p}\\mathbb{Z}\\right\\}  $.\n\n\\begin{Remark}\n\\label{continuation of |x|} The distribution $||x||_{p}^{s}$, $\\text{Re}%\n(s)>0$, admits the following meromorphic continuation,\n\\begin{align*}\n\\left\\langle ||x||_{p}^{s},\\varphi(x)\\right\\rangle  &  =\\frac{1-p^{-n}\n}{1-p^{-s -n}}\\varphi(0)+ \\int_{||x||_{p}>1}||x||_{p}^{s} \\varphi(x)\\,dx\\\\\n&  +\\int_{||x||_{p}\\leq1}||x||_{p}^{s} (\\varphi(x)-\\varphi(0))\\,dx,\n\\quad\\varphi\\in S.\n\\end{align*}\nIn particular, all the poles of $||x||_{p}^{s}$ have real part equal to $-n$.\n\\end{Remark}\n\n\\begin{Lemma}\n[{\\cite[Chap. III, Theorem 4.5]{TA}}]\\label{Fourier trans of Riesz} As element\nof $S^{\\prime}(\\mathbb{Q}_{p}^{n})$, $\\left(  \\mathcal{F}\\mathit{\\ R}%\n_{s}\\right)  (x)$ equals $||x||_{p}^{-s}$, for $s\\notin n+\\frac{2\\pi i}{\\ln\np}\\mathbb{Z}$.\n\\end{Lemma}\n\nThe following explicit formula will be used in the next sections.\n\n\\begin{Lemma}\n\\label{elliptic = Taibleson} Let $f(x)\\in\\mathbb{Q}_{p}[x]$, $x=(x_{1}%\n,\\dotsc,x_{n})$, be an elliptic polynomial of degree $d$. Then\n\\[\n|f|_{p}^{s}=\\frac{(1-p^{ds})L(p^{-s})}{(1-p^{-n})(1-p^{-ds-n})}\\mathit{R}%\n_{ds+n},\\quad s\\in\\mathbb{C}%\n\\]\nas distributions on $\\mathcal{W}$. Here $L(p^{-s})$ is the numerator of\n$Z(s,f)$ which is a polynomial in $p^{-s}$ with rational coefficients.\n\\end{Lemma}\n\n\\begin{proof}\nLet $\\varphi\\in\\mathcal{W}$, then\n\\[\n\\varphi(x)=\\sum_{i}c_{i}\\chi_{r_{i}}(x),\n\\]\nwhere $c_{i}\\in\\mathbb{C}$, $r_{i}\\in\\mathbb{Z}$ (recall that $\\mathcal{F}%\n(\\chi_{r})=p^{-nr}\\chi_{-r}$). The action of $|f|_{p}^{s}$ on $\\mathcal{F}%\n\\varphi$ can be explicitly described as follows:\n\\[\n\\langle|f|_{p}^{s},\\mathcal{F}\\varphi\\rangle=\\sum_{i}c_{i}\\langle|f|_{p}%\n^{s},p^{-nr_{i}}\\chi_{-r_{i}}\\rangle,\n\\]\nbut\n\\[\n\\langle|f|_{p}^{s},p^{-nr_{i}}\\chi_{-r_{i}}\\rangle=p^{-nr_{i}}\\int\n_{\\mathbb{Q}_{p}^{n}}|f(x)|_{p}^{s}\\chi_{-r_{i}}(x)\\,dx=p^{dr_{i}s}Z(s,f),\n\\]\nfor $\\text{Re}(s)>0$, thus\n\\[\n\\langle|f|_{p}^{s},\\mathcal{F}\\varphi\\rangle=Z(s,f)\\sum_{i}c_{i}p^{dr_{i}%\ns},\\quad\\text{Re}(s)>0.\n\\]\n\n\nOn the other hand,\n\\[\n\\langle\\dfrac{1-p^{ds}}{1-p^{-n}}\\mathit{R}_{ds+n}, p^{-nr_{i}}\\chi_{-r_{i}}\n\\rangle=\\langle\\dfrac{1-p^{-ds-n}}{1-p^{-n}}||x||_{p}^{ds}, p^{-nr_{i}}%\n\\chi_{-r_{i}} \\rangle= p^{dr_{i} s},\n\\]\nfor every $r_{i} \\in\\mathbb{Z}$ and $\\text{Re}(s)>0$. Then we have\n\\[\n\\langle|f|_{p}^{s},\\mathcal{F}\\varphi\\rangle= \\dfrac{1-p^{ds}}{1-p^{-n}}\nZ(s,f) \\langle\\mathit{R}_{ds+n},\\mathcal{F}\\varphi\\rangle,\n\\]\nfor $\\text{Re}(s)>0$. Now $Z(s,f)$ and $\\mathit{R}_{ds+n}$ have a meromorphic\ncontinuation to the complex plane, therefore this formula extends to\n$\\mathbb{C}$. Finally, since the Fourier transform establishes a $\\mathbb{C}%\n$-isomorphism on $\\mathcal{W}$, it is possible remove the Fourier transform symbol.\n\\end{proof}\n\n\\subsection{The Taibleson Operator}\n\n\\begin{Definition}\nThe Taibleson pseudo-differential operator $D_{T}^{\\alpha}$, $\\alpha>0$, is\ndefined as\n\\[\n(D_{T}^{\\alpha}\\varphi)(x)=\\mathcal{F}_{\\xi\\rightarrow x}^{-1}\\left(\n||\\xi||_{p}^{\\alpha}\\mathcal{F}_{x\\rightarrow\\xi}\\varphi\\right)  \\text{, for\n}\\varphi\\in S.\n\\]\n\n\\end{Definition}\n\nAs a consequence of the Lemma \\ref{Fourier trans of Riesz} and (\\ref{1}), one\ngets\n\\begin{align*}\n\\left(  D_{T}^{\\alpha}\\varphi\\right)  \\left(  x\\right)   &  =\\left(\n\\mathit{k}_{-\\alpha}\\ast\\varphi\\right)  \\left(  x\\right)  =\\\\\n&  \\frac{1-p^{\\alpha}}{1-p^{-\\alpha-n}}\\int_{\\mathbb{Q}_{p}^{n}}%\n||y||_{p}^{-\\alpha-n}(\\varphi(x-y)-\\varphi(x))\\,dy.\n\\end{align*}\n\n\nThe right-hand side of \\ previous formula makes sense for a wider class of\nfunctions than $S(\\mathbb{Q}_{p})$, for example, for the class $\\mathfrak{E}%\n_{\\alpha}(\\mathbb{Q}_{p}^{n})$ of locally constant functions $\\varphi(x)$\nsatisfying\n\\[\n\\int_{||x||_{p}\\geq1}||x||_{p}^{-\\alpha-n}|\\varphi(x)|\\,dx<\\infty.\n\\]\n\n\n\\begin{Remark}\nAs a consequence of the previous observations we may assume that the constant\nfunctions are contained in the domain of $D_{T}^{\\alpha}$, and that\n$D_{T}^{\\alpha}\\varphi=0$, for any constant function.\n\\end{Remark}\n\n\\section{Fundamental Solutions for the Taibleson Operator}\n\nWe now consider the following pseudo-differential equation:\n\\begin{equation}\n\\label{eq pseudo}D_{T}^{\\alpha}u=v, \\quad\\text{with $v\\in S$}, \\text{ and\n}\\alpha>0.\n\\end{equation}\n\n\nWe say that $E_{\\alpha}\\in S^{\\prime}$ is a \\textit{fundamental solution} of\n(\\ref{eq pseudo}) if $E_{\\alpha}\\ast v$ is a solution.\n\n\\begin{Lemma}\n\\label{funda + k} If $E_{\\alpha}$ is a fundamental solution of\n(\\ref{eq pseudo}), then for any constant $c$, $E_{\\alpha}+c$ is also a\nfundamental solution.\n\\end{Lemma}\n\n\\begin{proof}\nLet $E_{\\alpha}$ a fundamental solution for (\\ref{eq pseudo}), then\n\\begin{align*}\nD_{T}^{\\alpha}((E_{\\alpha}+c)\\ast v)  &  =D_{T}^{\\alpha}((E_{\\alpha}\\ast\nv)+(c\\ast v))\\\\\n&  =u+D_{T}^{\\alpha}(c\\ast v)=u,\n\\end{align*}\nbecause $u$ and the constant function, $c\\ast v$, are in the domain of\n$D_{T}^{\\alpha}$.\n\\end{proof}\n\n\\begin{Theorem}\n\\label{fund solu Tai} A fundamental solution of (\\ref{eq pseudo}) is\n\\[\nE_{\\alpha}(x)=%\n\\begin{cases}\n\\dfrac{1-p^{-\\alpha}}{1-p^{\\alpha-n}}||x||_{p}^{\\alpha-n} & \\text{ if\n$\\alpha\\neq n$}\\\\\n\\dfrac{1-p^{n}}{p^{n}\\ln p}\\ln(||x||_{p}) & \\text{ if $\\alpha=n$.}%\n\\end{cases}\n\\]\n\n\\end{Theorem}\n\n\\begin{proof}\nThe proof is based on the ideas introduced in \\cite{Z-G1}. The existence of a\nfundamental solution $E_{\\alpha}$ is equivalent to the existence of a\ndistribution $\\mathcal{F}E_{\\alpha}$ satisfying\n\\begin{equation}\n||x||_{p}^{\\alpha}\\mathcal{F}E_{\\alpha}=1, \\label{eq in distr}%\n\\end{equation}\nas distributions. Let $||x||_{p}^{s}=\\sum\\limits_{m\\in\\mathbb{Z}}%\nc_{m}(s+\\alpha)^{m}$ be the Laurent expansion at $-\\alpha$ with $c_{m}\\in\nS^{\\prime}$ for all $m$. The existence of this expansion is a consequence of\nthe completeness of $S^{\\prime}$ (see e.g. \\cite[pp. 65-66]{Igusa}). Since the\nreal parts of the poles of the meromorphic continuation of $||x||_{p}^{s}$ are\nnegative rational numbers (cf. Remark \\ref{continuation of |x|}),\n$||x||_{p}^{s+\\alpha}=||x||_{p}^{\\alpha}||x||_{p}^{s}$ is holomorphic at\n$s=-\\alpha$. Therefore, $||x||_{p}^{\\alpha}c_{m}=0$ for all $m<0$ and\n\\[\n||x||_{p}^{s+\\alpha}=||x||_{p}^{\\alpha}c_{0}+\\sum_{m=1}^{\\infty}%\n||x||_{p}^{\\alpha}c_{m}(s+\\alpha)^{m}.\n\\]\n\n\nBy using the Lebesgue dominated convergence theorem, one verifies that\n\\[\n\\underset{s\\rightarrow-\\alpha}{\\text{lim}}\\langle||x||_{p}^{s+\\alpha}%\n,\\phi\\rangle=\\int_{\\mathbb{Q}_{p}^{n}}\\phi(x)\\,dx=\\langle1,\\phi\\rangle,\n\\]\nand then we can take $\\mathcal{F}E_{\\alpha}=c_{0}$. Furthermore, if $-\\alpha$\nis not a pole of $||x||_{p}^{s}$,\n\\begin{equation}\n\\mathcal{F}E_{\\alpha}=\\underset{s\\rightarrow-\\alpha}{\\text{lim}}||x||_{p}^{s}.\n\\label{F E_a limit}%\n\\end{equation}\n\n\nTo calculate $c_{0}$, consider the following two cases.\n\n\\textbf{Case $\\mathbf{\\alpha\\neq n}$}.\n\nWe use (\\ref{F E_a limit}) and the Lemma \\ref{Fourier trans of Riesz}, i.e.,\n\\[\n\\int_{(\\mathbb{Q}_{p}^{\\times})^{n}} ||x||_{p}^{s} \\mathcal{F}(\\varphi)(x)\\,\ndx = \\dfrac{1-p^{s}}{1-p^{-s-n}}\\int_{(\\mathbb{Q}_{p}^{\\times})^{n}}\n||x||_{p}^{-s-n} \\varphi(x)\\, dx,\n\\]\nfor $s\\neq n+(2\\pi i\/\\ln p) \\mathbb{Z}$. If $\\alpha\\neq n$, by\n(\\ref{F E_a limit}),\n\\[\n\\langle E_{\\alpha},\\mathcal{F}(\\varphi) \\rangle=\\underset{s \\rightarrow\n-\\alpha}{\\text{lim}}\\int_{(\\mathbb{Q}_{p}^{\\times})^{n}} ||x||_{p}^{s}\n\\mathcal{F}(\\varphi)(x)\\, dx.\n\\]\n\n\nIf $\\alpha>n$, by the Lebesgue dominated convergence theorem, we can\ninterchange the limit and the integral. If $0<\\alpha<n$, by taking into\naccount that\n\\[\n\\int_{||x||_{p}\\leq1}||x||_{p}^{\\alpha-n}\\,dx<+\\infty,\\quad\\text{for }%\n0<\\alpha<n,\n\\]\nand by using Lebesgue dominated convergence theorem, we can exchange the limit\nand the integral. Therefore,\n\\begin{align*}\n\\langle E_{\\alpha},\\varphi\\rangle &  =\\frac{1-p^{-\\alpha}}{1-p^{\\alpha-n}}%\n\\int_{(\\mathbb{Q}_{p}^{\\times})^{n}}||x||_{p}^{\\alpha-n}\\varphi(x)\\,dx\\\\\n&  =\\frac{1-p^{-\\alpha}}{1-p^{\\alpha-n}}\\int_{\\mathbb{Q}_{p}^{n}}%\n||x||_{p}^{\\alpha-n}\\varphi(x)\\,dx.\n\\end{align*}\nSet $\\overset{\\sim}{\\varphi}(x)=\\varphi(-x)$, with $\\varphi\\in S$. The results\nfollow by replacing $\\varphi$ by $\\mathcal{F}(\\overset{\\sim}{\\varphi})$\nbecause $\\mathcal{F}\\bigl(\\mathcal{F}(\\overset{\\sim}{\\varphi})\\bigr)=\\varphi$.\n\n\\textbf{Case $\\mathbf{\\alpha= n}$}.\n\nWe compute the constant term, $c_{0}$, in the expansion\n\\[\n\\langle||x||_{p}^{s},\\mathcal{F}(\\varphi)\\rangle=\\sum_{m\\in\\mathbb{Z}}\\langle\nc_{m},\\mathcal{F}(\\varphi)\\rangle(s+n)^{m}.\n\\]\nSince\n\\begin{align*}\n\\langle||x||_{p}^{s},\\mathcal{F}(\\varphi)\\rangle &  =\\frac{1-p^{s}}%\n{1-p^{-s-n}}\\int_{\\mathbb{Q}_{p}^{n}}||x||_{p}^{-s-n}\\varphi(x)\\,dx\\\\\n&  =(1-p^{s})\\int_{\\mathbb{Q}_{p}^{n}}\\frac{p^{v(x)(s+n)}}{1-p^{-s-n}}%\n\\varphi(x)\\,dx,\n\\end{align*}\nwhere $x=(x_{1},\\dotsc,x_{n})$, $v(x):=\\underset{1\\leq i\\leq n}{\\min}v(x_{i}%\n)$, and $||x||_{p}=p^{-v(x)}$, by expanding\n\\begin{align*}\n\\dfrac{(1-p^{s})p^{v(x)(s+n)}}{1-p^{-s-n}}  &  =\\dfrac{1-p^{-n}}{\\ln\np}(s+n)^{-1}\\\\\n&  +\\dfrac{(1-p^{-1})v(x)\\ln p-\\frac{\\ln p}{p}+\\frac{(p-1)}{2p}\\ln p}{\\ln\np}+O(\\left(  s+n\\right)  ),\n\\end{align*}\none gets\n\\[\n\\langle E_{n},\\varphi\\rangle=\\langle c_{0},\\varphi\\rangle=\\int_{\\mathbb{Q}%\n_{p}^{n}}\\Bigl(\\frac{1-p^{n}}{p^{n}\\ln(p)}\\ln(||x||_{p})+\\frac{p^{n}-3}%\n{2p^{n}}\\Bigr)\\varphi(x)\\,dx.\n\\]\n\n\nThe announced results follow by replacing $\\varphi$ by $\\mathcal{F}%\n(\\overset{\\sim}{\\varphi})$, $\\varphi\\in S$, and using the fact that the\nfundamental solution is determined up to the addition of a constant (cf. Lemma\n\\ref{funda + k}).\n\\end{proof}\n\nIn the case $n=1$, the previous result is already known, see e.g.\n\\cite[Theorem 2.1]{KO4}.\n\n\\section{Fundamental Solutions for Elliptic Operators}\n\n\\begin{Theorem}\n\\label{fund solu ellip} Let $f(D,\\alpha)$ be an elliptic operator of order\n$d$. Then, a fundamental solution $E_{\\alpha}$ of $f(D,\\alpha)u=v$, $\\alpha\n>0$, and $v\\in\\mathcal{W}$, is given by\n\\[\nE_{\\alpha}(x)=%\n\\begin{cases}\n\\dfrac{L(p^{\\alpha})(1-p^{-d\\alpha})}{(1-p^{-n})(1-p^{d\\alpha-n})}%\n||x||_{p}^{d\\alpha-n} & \\text{ as a distribution on $\\mathcal{W}$, with\n$\\alpha\\neq n\/d$}\\\\\n\\, & \\\\\n\\dfrac{L(p^{n\/d})(1-p^{n})}{(1-p^{-n})(p^{n}\\ln p)}\\ln(||x||_{p}) & \\text{ as\na distribution on $\\mathcal{W}$, with $\\alpha=n\/d$},\n\\end{cases}\n\\]\nwhere $L(p^{-s})$ is the numerator of $Z(s,f)$.\n\\end{Theorem}\n\n\\begin{proof}\nAs we mention before, the problem of the existence of a fundamental solution,\n$E_{\\alpha}$, is equivalent to the existence of a distribution $\\mathcal{F}\nE_{\\alpha}$ satisfying\n\\[\n|f|_{p}^{\\alpha}\\mathcal{F}E_{\\alpha}=1 \\text{ in }S^{\\prime}.\n\\]\n\n\n\\noindent By Lemma \\ref{elliptic = Taibleson},\n\\[\n\\langle|f|_{p}^{\\alpha},\\varphi\\rangle=\\langle\\frac{(1-p^{d\\alpha\n})L(p^{-\\alpha})}{(1-p^{-n})(1-p^{-d\\alpha-n})}\\mathit{R}_{d\\alpha+n}%\n,\\varphi\\rangle\n\\]\n$\\varphi\\in\\mathcal{W}$, $s\\in\\mathbb{C}$. The result follows by reasoning as\nin the proof of Theorem \\ref{fund solu Tai}, and by the fact that the space\n$\\mathcal{W}$ is invariant under the Fourier transform.\n\\end{proof}\n\n\\begin{Corollary}\n\\label{cor1}With the hypotheses of the previous theorem, and assuming that\n$\\alpha\\neq n\/d$, we have\n\\[\n|\\mathcal{F}(E_{\\alpha}\\ast\\varphi)(x)|\\leq C(\\alpha)||x||_{p}^{-d\\alpha\n}|\\mathcal{F}(\\varphi)(x)|,\n\\]\nfor all $x\\in\\mathbb{Q}_{p}^{n}$, and $\\varphi\\in\\mathcal{W}$.\n\\end{Corollary}\n\n\\section{Solutions of Elliptic Pseudo-Differential Equations in Sobolev\nSpaces}\n\nGiven $\\phi\\in\\mathcal{S}$ and $l$ a non-negative number, we define\n\\[\n||\\phi||_{H^{l}}^{2}=\\int_{\\mathbb{Q}_{p}^{n}}[\\text{max}(1,||\\xi||_{p}%\n)]^{2l}|\\mathcal{F}(\\phi)(\\xi)|^{2}\\,d\\xi.\n\\]\n\n\nWe call the completion of $\\mathcal{S}$ with respect to $||\\cdot||_{H^{l}}$\nthe \\textit{$l$-Sobolev space } $H^{l}:=H^{l}(\\mathbb{Q}_{p}^{n})$.\n\nWe note that $H^{l}$ contains properly the space of test functions, $S$.\nIndeed, consider the function\n\\[\nf(x)=\n\\begin{cases}\n0 & \\text{ if }||x||_{p}\\leq1\\\\\n||x||_{p}^{-\\beta} & \\text{ if }||x||_{p}>1\n\\end{cases}\n\\]\nwith $\\beta>n$. A direct calculation shows that\n\\[\n||f||_{H^{l}}^{2}=\\int\\limits_{||\\xi||_{p}\\leq1}\\Bigl|\\frac{(1-p^{-n}%\n)(1-||\\xi||_{p}^{\\beta-n}p^{n-\\beta})}{(1-p^{n-\\beta})}-p^{-\\beta}||\\xi\n||_{p}^{\\beta-n}\\Bigr|^{2}\\,d\\xi.\n\\]\nThus, $||f||_{H^{l}}^{2}<\\infty$, but $f$ does not have compact support.\n\n\\begin{Lemma}\n\\label{Sobolev embedding Lemma} If $l>n\/2$, then there exists an embedding of\n$H^{l}$ into the space of uniformly continuous functions.\n\\end{Lemma}\n\n\\begin{proof}\nLet $\\phi\\in H^{l}$. Since the Fourier transform of a function in $L^{1}$ is\nuniformly continuous, it is \\ sufficient to show that $\\mathcal{F}(\\phi)\\in\nL^{1}$. By using the H\\\"{o}lder inequality and the fact that%\n\\[\n\\int_{\\mathbb{Q}_{p}^{n}}(\\text{max}(1,||\\xi||_{p}))^{-2l}\\,d\\xi\n<+\\infty\\text{, for }l>n\/2\\text{,}%\n\\]\nwe have%\n\\[\n\\int_{\\mathbb{Q}_{p}^{n}}|\\mathcal{F}(\\phi)(\\xi)|\\,d\\xi=\\int_{\\mathbb{Q}%\n_{p}^{n}}\\frac{(\\text{max}(1,||\\xi||_{p}))^{l}}{(\\text{max}(1,||\\xi\n||_{p}))^{l}}|\\mathcal{F}(\\phi)(\\xi)|\\,d\\xi\\leq C||\\phi||_{H^{l}}.\n\\]\n\n\\end{proof}\n\n\\begin{Lemma}\n\\label{Lemma 6} For any $\\alpha>0$ and $l\\geq0$, the mapping $f(D,\\alpha\n):H^{l+d\\alpha}\\rightarrow H^{l}$ is a well-defined continuous mapping between\nBanach spaces.\n\\end{Lemma}\n\n\\begin{proof}\nLet $\\phi\\in S$. Since $f(D,\\alpha)$ is an elliptic operator, by Lemma\n\\ref{Lemma 1}, we have that\n\\begin{align*}\n||f(D,\\alpha)\\phi||_{H^{l}}^{2}  &  =\\int_{\\mathbb{Q}_{p}^{n}}[\\text{max}%\n(1,||\\xi||_{p})]^{2l}|f(\\xi)|^{2\\alpha}|\\mathcal{F}(\\phi)(\\xi)|^{2}\\,d\\xi\\\\\n&  \\leq C_{1}\\int_{\\mathbb{Q}_{p}^{n}}[\\text{max}(1,||\\xi||_{p})]^{2(l+\\alpha\n)}|\\mathcal{F}(\\phi)(\\xi)|^{2}\\,d\\xi=C_{1}||\\phi||_{H^{l+\\alpha}}^{2}.\n\\end{align*}\nThe result follows from the fact that $S$ is dense in $H^{l+d\\alpha}$.\n\\end{proof}\n\n\\begin{Remark}\n\\label{remark3}Let $\\beta$ be a positive real number, and let\n\\[\nI(\\beta):=\\int\\limits_{||\\varepsilon||_{p}\\leq1}||\\varepsilon||_{p}^{\\beta\n}\\,d\\varepsilon.\n\\]\nThen\n\\[\nI(\\beta)=\\dfrac{1-p^{-n}}{1-p^{-n-n\\beta}},\\text{ for }\\beta>-n.\n\\]\nIndeed,\n\\begin{align*}\nI(\\beta) &  =\\int\\limits_{||\\varepsilon||_{p}<1}||\\varepsilon||_{p}^{\\beta\n}\\,d\\varepsilon+\\int\\limits_{||\\varepsilon||_{p}=1}\\,d\\varepsilon\\\\\n&  =\\int\\limits_{||\\varepsilon||_{p}<1}||\\varepsilon||_{p}^{\\beta\n}\\,d\\varepsilon+1-p^{-n}.\n\\end{align*}\nBy making the change of variables $\\varepsilon_{i}=px_{i}$, $i=1,\\dotsc,n$, we\nhave\n\\[\nI(\\beta)=p^{-n-n\\beta}I(\\beta)+1-p^{-n}.\n\\]\n\n\\end{Remark}\n\n\\begin{Theorem}\n\\label{mainresult}Let $f(D,\\alpha)$, $0<\\alpha<n\/2d$ be an elliptic\npseudo-differential operator of order $d$. Let $l$ be a positive real number\nsatisfying $l>n\/2$. Then, the equation\n\\[\nf(D,\\alpha)u=v\\quad\\left(  v\\in S \\right)  ,\n\\]\nhas a unique uniformly continuous solution $u\\in H^{l+d\\alpha}$.\n\\end{Theorem}\n\n\\begin{proof}\nLet $v\\in\\mathcal{S}$, then $v=v_{\\mathcal{W}}+v_{\\mathcal{L}}$, where\n$v_{\\mathcal{W}}\\in\\mathcal{W}$ and $v_{\\mathcal{L}}\\in\\mathcal{L}$. Thus, in\norder to prove the existence of a solution $u$, it is sufficient to show that\nthe two following equations have solutions:\n\\begin{equation}\nf(D,\\alpha)u_{\\mathcal{W}}=v_{\\mathcal{W}}, \\label{Eq v_W}%\n\\end{equation}%\n\\begin{equation}\nf(D,\\alpha)u_{\\mathcal{L}}=v_{\\mathcal{L}}. \\label{Eq v_L}%\n\\end{equation}\n\n\nWe first consider equation (\\ref{Eq v_W}). By Theorem \\ref{fund solu ellip},\n$u_{\\mathcal{W}}=E_{\\alpha}\\ast v_{\\mathcal{W}}$ is a solution of\n(\\ref{Eq v_W}), and by Corollary \\ref{cor1}, we have%\n\n\\begin{align*}\n||u_{\\mathcal{W}}||_{H^{l+d\\alpha}}^{2}  &  =\\int_{\\mathbb{Q}_{p}^{n}%\n}[\\text{max}(1,||\\xi||_{p})]^{2(l+d\\alpha)}|\\mathcal{F}(E_{\\alpha}\\ast\nv_{\\mathcal{W}})(\\xi)|^{2}\\,d\\xi\\\\\n&  =C(\\alpha,d,n)\\int_{\\mathbb{Q}_{p}^{n}}[\\text{max}(1,||\\xi||_{p}%\n)]^{2(l+d\\alpha)}|\\left\\Vert \\xi\\right\\Vert _{p}^{-2d\\alpha}|\\mathcal{F}%\n(v_{\\mathcal{W}})(\\xi)|^{2}\\,d\\xi\\\\\n&  =C(\\alpha,d,n)\\Bigl\\{\\int_{||\\xi||_{p}\\leq1}||\\xi||_{p}^{-2d\\alpha\n}|\\mathcal{F}(v_{\\mathcal{W}})(\\xi)|^{2}\\,d\\xi\\\\\n&  \\hspace{1cm}+\\int_{||\\xi||_{p}>1}||\\xi||_{p}^{2l}|\\mathcal{F}%\n(v_{\\mathcal{W}})(\\xi)|^{2}\\,d\\xi\\Bigr\\}.\n\\end{align*}\n\n\nWe now recall that $v_{\\mathcal{W}}(\\xi)=p^{rn}C\\chi_{r}(\\xi)$, with $r>0$.\nThen, $\\mathcal{F}(v_{\\mathcal{W}})(\\xi)=C\\chi_{-r}(\\xi)$ and\n\\begin{align*}\n||u_{\\mathcal{W}}||_{H^{l+d\\alpha}}^{2} &  \\leq C(\\alpha,d,n)\\Bigl\\{C^{2}%\np^{2rn}\\int\\limits_{||\\varepsilon||_{p}\\leq1}||\\varepsilon||_{p}^{-2d\\alpha\n}\\,d\\varepsilon\\\\\n&  \\hspace{2.5cm}+||v_{\\mathcal{W}}||_{H^{l}}^{2}\\Bigr\\}\\\\\n&  \\leq C(\\alpha,d,n)\\Bigl\\{C_{1}(\\alpha,d,n)+||v_{\\mathcal{W}}||_{H^{l}}%\n^{2}\\Bigr\\},\n\\end{align*}\nsince $-2d\\alpha>-n$, cf. Remark \\ref{remark3}. Therefore $u_{\\mathcal{W}}\\in\nH^{l+d\\alpha}$.\n\nWe now consider equation (\\ref{Eq v_L}). Since\n\\[\n\\mathcal{F}(u_{\\mathcal{L}})=\\mathcal{F}(v_{\\mathcal{L}})|f|_{p}^{-\\alpha},\n\\]\nand $f$ is elliptic,\n\\[\n|\\mathcal{F}(u_{\\mathcal{L}})(\\xi)|\\leq C||\\xi||^{-d\\alpha}|\\mathcal{F}\n(v_{\\mathcal{L}})(\\xi)|, \\text{(cf. Lemma \\ref{Lemma 1})}.\n\\]\nThen,\n\\begin{align*}\n||u_{\\mathcal{L}}||_{H^{l+d\\alpha}}^{2}  &  \\leq\\int_{||\\xi||_{p}\\leq1}%\n||\\xi||_{p}^{-2d\\alpha}|\\mathcal{F}(v_{\\mathcal{L}})(\\xi)|^{2}\\,d\\xi\\\\\n&  \\hspace{1cm}+\\int_{||\\xi||_{p}>1}||\\xi||_{p}^{2l}|\\mathcal{F}%\n(v_{\\mathcal{L}})(\\xi)|^{2}\\,d\\xi.\n\\end{align*}\n\n\nThe second integral is bounded by $||v_{\\mathcal{L}}||_{H^{l}}^{2}$. For the\nfirst integral, we observe that if $0<\\alpha<n\/2d$, then\n\\[\n\\int_{||\\xi||_{p}\\leq1}||\\xi||_{p}^{-2d\\alpha}|\\varphi(\\xi)|^{2}\\,d\\xi\\leq\nC||\\varphi||_{L^{2}},\n\\]\nfor any $\\varphi\\in S$. Therefore,\n\\[\n||u_{\\mathcal{L}}||_{H^{l+d\\alpha}}^{2}\\leq C||\\mathcal{F}(v_{\\mathcal{L}%\n})||_{L^{2}}+||v_{\\mathcal{L}}||_{H^{l}}^{2}.\n\\]\n\n\nIn this way, we established the existence of $u\\in H^{l+d\\alpha}$ which is\nuniformly continuous, by Lemma \\ref{Sobolev embedding Lemma}, such that\n$f(D,\\alpha)u=v$, for any $v\\in S$. Finally, we show that $u$ is unique.\nIndeed, if $f(D,\\alpha)u^{\\prime}=v$, then\n\\[\nf(D,\\alpha)(u-u^{\\prime})=0,\\qquad\\text{i.e., }|f|_{p}^{\\alpha}\\mathcal{F}%\n(u-u^{\\prime})=0,\n\\]\nand thus $\\mathcal{F}(u-u^{\\prime})(\\xi)=0$ if $\\xi\\neq0$, since $f$ is\nelliptic. Then $\\Psi(x\\cdot\\xi)(u-u^{\\prime})(\\xi)=0$ almost everywhere, and a\nfortiori $(u-u^{\\prime})(\\xi)=0$ almost everywhere, and by the continuity of\n$u-u^{\\prime}$, $u(\\xi)=u^{\\prime}(\\xi)$ for any $\\xi\\in\\mathbb{Q}_{p}^{n}$.\n\\end{proof}\n\n\\section{Solutions of Elliptic Pseudo-Differential Equations in Singular\nSobolev Spaces}\n\nIn this section, we modify the Sobolev norm to obtain spaces of functions on\nwhich $f(D,\\alpha)$ gives a surjective mapping.\n\n\\begin{Definition}\nGiven $\\varphi\\in S$ and $l$ a non-negative number, we set\n\\[\n||\\varphi||_{\\mathcal{H}^{l}}^{2}:=\\int_{\\mathbb{Q}_{p}^{n}}||\\xi||_{p}%\n^{2l}|\\mathcal{F}(\\varphi)(\\xi)|^{2}\\,d\\xi.\n\\]\nWe call the completion of $S$ with respect to $||\\cdot||_{\\mathcal{H}^{l}}$\nthe \\emph{$l$-singular Sobolev Space} $\\mathcal{H}^{l}:=\\mathcal{H}%\n^{l}(\\mathbb{Q}_{p}^{n})$. Note that $H^{l}\\subseteq\\mathcal{H}^{l}$, $l\\geq\n0$, since $||\\varphi||_{\\mathcal{H}^{l}}\\leq||\\varphi||_{H^{l}}$.\n\\end{Definition}\n\n\\begin{Lemma}\n\\label{lemma3}For any, $\\alpha>0$, $l\\geq0$, the mapping $f(D,\\alpha\n):\\mathcal{H}^{l+d\\alpha}\\rightarrow\\mathcal{H}^{l}$ is a well-defined\ncontinuous mapping between Banach Spaces.\n\\end{Lemma}\n\n\\begin{proof}\nSimilar to the proof of Lemma \\ref{Lemma 6}.\n\\end{proof}\n\nWe denote by $\\mathcal{L}^{l}$ and $\\mathcal{W}^{l}$, the respective\ncompletions of $\\mathcal{L}$ and $\\mathcal{W}$ with respect to $||\\cdot\n||_{\\mathcal{H}^{l}}$; furthermore, we set\n\\[\n\\mathcal{H}_{0}^{l}:=\\mathcal{L}^{l}+\\mathcal{W}^{l}\\subseteq\\mathcal{H}^{l}.\n\\]\n\n\n\\begin{Proposition}\n\\label{prop1}Let $f(D,\\alpha)$, $\\alpha>0$, be an elliptic pseudo-differential\noperator of order $d$, and let $l$ be a non-negative real number. Then\n$f(D,\\alpha):\\mathcal{H}^{l+d\\alpha}\\rightarrow\\mathcal{W}^{l}$, is a\nsurjective mapping between Banach spaces.\n\\end{Proposition}\n\n\\begin{proof}\nBy Lemma \\ref{lemma3}, the mapping is well-defined. Let $v\\in\\mathcal{W}^{l}$,\nand let $\\{v_{n}\\}$ a Cauchy sequence in $\\mathcal{W}$ converging to $v$. By\nTheorem \\ref{fund solu ellip}, there exits a sequence $\\{u_{n}\\}$ in\n$H^{l+d\\alpha}$ such that $f(D,\\alpha)u_{n}=v_{n}$. We now show that\n$\\{u_{n}\\}$ is a Cauchy sequence in $\\mathcal{H}^{l+d\\alpha}$ as follows:\n\\begin{align*}\n||u_{n}-u_{m}||_{\\mathcal{H}^{l+d\\alpha}}^{2}  &  \\leq C\\int_{\\mathbb{Q}%\n_{p}^{n}}||\\xi||_{p}^{2(l+d\\alpha)}||\\xi||_{p}^{-2d\\alpha}|\\mathcal{F}%\n(v_{n}-v_{m})(\\xi)|^{2}\\,d\\xi\\\\\n&  \\leq C||v_{n}-v_{m}||_{\\mathcal{H}^{l}}^{2}.\n\\end{align*}\nThus, there exists $u\\in\\mathcal{H}^{l+d\\alpha}$ such that $u_{n}\\rightarrow\nu$, and by the continuity of $f(D,\\alpha)$, $f(D,\\alpha)u=v$.\n\\end{proof}\n\n\\begin{Proposition}\n\\label{prop2}Let $f(D,\\alpha)$, $\\alpha>0$, be an elliptic pseudo-differential\noperator of order $d$, and let $l$ be a non-negative real number. Then,\n$f(D,\\alpha):\\mathcal{H}^{l+d\\alpha}\\rightarrow\\mathcal{L}^{l}$ is a\nsurjective mapping between Banach spaces.\n\\end{Proposition}\n\n\\begin{proof}\nBy Lemma \\ref{lemma3}, the mapping is well-defined. Let $v\\in\\mathcal{L}^{l}$,\nand let $\\{v_{n}\\}$ a Cauchy sequence in $\\mathcal{L}$ converging to $v$. By\nthe same reasoning given in proof Theorem \\ref{mainresult} for establishing\nthe existence of a solution for equation (\\ref{Eq v_L}), we obtain a sequence\n$\\{u_{n}\\}$ in $H^{l+d\\alpha}$ such that $f(D,\\alpha)u_{n}=v_{n}$. We now show\nthat $\\{u_{n}\\}$ is a Cauchy sequence in $\\mathcal{H}^{l+d\\alpha}$.\n\nBy using\n\\[\n|\\mathcal{F}(u_{n})(\\xi)|\\leq C||\\xi||^{-d\\alpha}|\\mathcal{F}(v_{n})(\\xi)|,\n\\]\none gets\n\\begin{align*}\n||u_{n}-u_{m}||_{\\mathcal{H}^{l+d\\alpha}}^{2}  &  \\leq C\\int_{\\mathbb{Q}%\n_{p}^{n}}||\\xi||_{p}^{2(l+d\\alpha)}||\\xi||_{p}^{-2d\\alpha}|\\mathcal{F}%\n(v_{n}-v_{m})(\\xi)|^{2}\\,d\\xi\\\\\n&  \\leq C||v_{n}-v_{m}||_{\\mathcal{H}^{l}}^{2}.\n\\end{align*}\nThus, there exists $u\\in\\mathcal{H}^{l+d\\alpha}$ such that $u_{n}\\rightarrow\nu$, and by the continuity of $f(D,\\alpha)$, $f(D,\\alpha)u=v$.\n\\end{proof}\n\nFrom the previous two lemmas we obtain the following result.\n\n\\begin{Theorem}\n\\label{mainII}Let $f(D,\\alpha)$ be an elliptic pseudo-differential operator of\norder $d$. Let $l$ be a positive \\ real number. Then the equation\n\\[\nf(D,\\alpha)u=v,\\quad v\\in\\mathcal{H}_{0}^{l}%\n\\]\nhas a unique solution $u\\in\\mathcal{H}^{l+d\\alpha}$.\n\\end{Theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\mlabel{sec:1}\n\n\nLet $\\mathcal{H}$ be a complex Hilbert space and \n$\\mathcal{M} \\subseteq B(\\mathcal{H})$ be a von Neumann algebra. \nA unit vector $\\Omega \\in \\mathcal{H}$ is called \n{\\it cyclic} for $\\mathcal{M}$ if $\\mathcal{M}\\Omega$ is dense in $\\mathcal{H}$, and \n{\\it separating} if the map $\\mathcal{M} \\to \\mathcal{H},M \\mapsto M\\Omega$ is injective. \nIf $\\Omega$ is both, cyclic  and separating for $\\mathcal{M}$, \nthen the Tomita--Takesaki Theorem (\\cite[Thm.~2.5.14]{BR87}) \nasserts in particular that the closed real subspace \n\\[ {\\tt V} := \\overline{\\{ M\\Omega \\: M = M^* \\in \\mathcal{M}\\}} \\] \nis {\\it standard}, i.e., \n\\begin{equation}\n  \\label{eq:stansub}\n  {\\tt V} \\cap i {\\tt V} = \\{0\\}\\ \\ ({\\tt V} \\ \\mbox{is separating}) \n\\quad \\mbox{ and } \\quad \\mathcal{H} = \\overline{{\\tt V} + i {\\tt V}}\n\\ \\ \\ \\ ({\\tt V} \\ \\mbox{is cyclic}) \n\\end{equation} \n(cf.\\ \\cite{Lo08} for the basic theory of standard subspaces). \nTo the standard subspace ${\\tt V}$, we associate a {\\it pair of modular \nobjects} $(\\Delta_{\\tt V}, J_{\\tt V})$: the {\\it modular operator} $\\Delta_{\\tt V}$ is a positive \nselfadjoint operator, $J_{\\tt V}$ is a {\\it conjugation} (an antiunitary \ninvolution), and these two operators satisfy the modular relation \n$J_{\\tt V} \\Delta_{\\tt V} J_{\\tt V} = \\Delta_{\\tt V}^{-1}$. The pair $(\\Delta_{\\tt V}, J_{\\tt V})$ is obtained\nby the polar decomposition $\\sigma_{\\tt V} = J_{\\tt V} \\Delta_{\\tt V}^{1\/2}$ of the closed operator \n\\[ \\sigma_{\\tt V} \\: {\\tt V} + i {\\tt V} \\to \\mathcal{H}, \\quad \nx + i y \\mapsto x- iy \\] \nwith ${\\tt V} = \\mathop{{\\rm Fix}}\\nolimits(\\sigma_{\\tt V})$. The main assertion of the \nTomita--Takesaki Theorem\nis  that \n\\[ J_{\\tt V} \\mathcal{M} J_{\\tt V} = \\mathcal{M}' \\quad \\mbox{ and } \\quad \n\\Delta_{\\tt V}^{it} \\mathcal{M} \\Delta_{\\tt V}^{-it}  = \\mathcal{M} \\quad \\mbox{ for }  \\quad t \\in {\\mathbb R}.\\] \nSo we obtain a one-parameter group of automorphisms of $\\mathcal{M}$ \n(the modular group) \nand a symmetry between $\\mathcal{M}$ and its commutant $\\mathcal{M}'$, implemented by $J_{\\tt V}$. \n\nIn Algebraic Quantum Field Theory (AQFT) \nin the sense of Haag--Kastler, one considers \n{\\it nets} of von Neumann algebras $\\mathcal{M}(\\mathcal{O}) \\subseteq B(\\mathcal{H})$, \nassociated to  regions $\\mathcal{O}$ in some space-time manifold~$M$ \n(\\cite{Ha96}). \nThe hermitian elements of the algebra $\\mathcal{M}(\\mathcal{O})$ are interpreted as observables  \nthat can be measured in the ``laboratory'' $\\mathcal{O}$. \nAccordingly, one requires {\\it isotony}, i.e., that $\\mathcal{O}_1 \\subseteq \\mathcal{O}_2$ implies \n$\\mathcal{M}(\\mathcal{O}_1) \\subseteq \\mathcal{M}(\\mathcal{O}_2)$. \nCausality enters by the {\\it locality} assumption that \n$\\mathcal{M}(\\mathcal{O}_1)$ and $\\mathcal{M}(\\mathcal{O}_2)$ commute if \n$\\mathcal{O}_1$ and $\\mathcal{O}_2$ are space-like separated, i.e., cannot correspond \nwith each other. One further assumes a unitary representation \n$U \\: G \\to \\mathop{\\rm U{}}\\nolimits(\\mathcal{H})$  of a Lie group~$G$, acting as a space-time symmetry \ngroup on $M$, such that $U(g) \\mathcal{M}(\\mathcal{O}) U(g)^* = \\mathcal{M}(g\\mathcal{O})$ for \n$g \\in G$. \nIn addition, one assumes a $U(G)$-fixed \nunit vector $\\Omega \\in \\mathcal{H}$, representing typically \na vacuum state \nof a quantum field. \nThe domains $\\mathcal{O} \\subseteq M$ for which \n$\\Omega$ is cyclic and separating for $\\mathcal{M}(\\mathcal{O})$ are of particular \nrelevance. For these domains $\\mathcal{O}$, the von Neumann algebra \n$\\mathcal{M}(\\mathcal{O})$ specifies a standard subspace ${\\tt V}(\\mathcal{O})\\subseteq \\mathcal{H}$ \nwhich determines a pair $(\\Delta_\\mathcal{O},J_\\mathcal{O})$ of modular objects and in particular \na modular automorphism group \n$\\alpha_t(M) = \\Delta_\\mathcal{O}^{-it\/2\\pi} M \\Delta_\\mathcal{O}^{it\/2\\pi}$. \nIt is now an interesting question if this modular group is ``geometric'' \nin the sense that it is implemented by a one-parameter subgroup of~$G$, hence \ncorresponds to a one-parameter group of symmetries of~$M$. For the modular \nconjugation $J_\\mathcal{O}$, we may likewise ask for the existence of an involutive \nautomorphism $\\tau_G$ of $G$ and an involution $\\tau_M$ on $M$ \nreversing the causal structure, such that \n\\[ J_\\mathcal{O} \\mathcal{M}(\\tilde\\mathcal{O}) J_\\mathcal{O} = \\mathcal{M}(\\tau_M(\\tilde \\mathcal{O})), \\qquad \nJ_\\mathcal{O} U(g) J_\\mathcal{O} = U(\\tau_G(g))\\quad \n \\mbox{ for } \\quad g \\in G, \\tilde \\mathcal{O} \\subseteq M,\\] \nand that $\\tau_M$ and $\\tau_G$ are compatible in the sense that \n\\[ \\tau_M \\circ g = \\tau_G(g) \\circ \\tau_M  \\quad \\mbox{ for } \\quad g \\in G. \\] \nWe are particularly interested in the geometric realizations of modular \ngroups and involutions in the sense explained above. \nThe present paper contributes to this project \nby exhibiting large classes of nets of standard subspaces \nwhere the action of the modular group is geometric. \n\nTo study this question systematically, we \nfirst observe that, passing from operator algebras \nto the corresponding standard subspaces \nis a tremendous reduction of information, but \nthe net of standard subspaces ${\\tt V}(\\mathcal{O})$ \nstill encodes the geometric features of the original theory \nand in particular it still reflects the action of the symmetry group \n$G$ and its extensions by involutions.  \n\nConversely, one can use the functorial process provided by Second Quantization \n(\\cite{Si74}) to associate to each standard subspace ${\\tt V} \\subseteq \\mathcal{H}$   \na pair $(\\mathcal{R}_\\pm({\\tt V}),\\Omega)$, where $\\mathcal{R}_\\pm({\\tt V})$ is a von Neumann algebra \non the bosonic\/fermionic Fock space $\\mathcal{F}_\\pm(\\mathcal{H})$, for which the \nvacuum vector is cyclic and separating. \nThis method has been developed by Araki and Woods  in the context of \nfree bosonic quantum fields (\\cite{Ar63, Ar64, AW63, AW68}); \nsome of the corresponding fermionic results are more recent \n(cf.\\ \\cite{EO73}, \\cite{BJL02}). Other statistics (anyons) \nare discussed in \\cite{Schr97} and more recent deformations of \nthis procedure are discussed in \\cite[\\S 3]{Le15}. \nThroughout this paper we only deal with nets of standard subspaces, \nbut it is important to keep in mind that there are functorial \nconstructions that associate to such nets various types of \nfree quantum fields on homogeneous spaces. \n\nThe current interest in standard subspaces arose \nin the 1990s from the work of\nBorchers \nand\nWiesbrock (\\cite{Bo92, Wi93}). \nThis in turn led to the concept of modular localization \nin Quantum Field Theory introduced by  \nBrunetti, Guido and Longo in \\cite{BGL02, BGL93}; see also \\cite{BDFS00} \nand \\cite{Le15, LL15} for important applications of this technique. \n\nWe start with  a unitary representation $(U,\\mathcal{H})$ of $G$ which extends  \nto an {\\it antiunitary representation} of the  extended group~$G_\\tau \n = G \\rtimes \\{\\mathbf{1},\\tau_G\\}$, containing $G$ as a subgroup of index~$2$. \nThen $J := U(\\tau_G)$ is a conjugation satisfying \n$J U(g) J = U(\\tau_G(g))$ for $g \\in G$. \nWe then obtain for each pair \n$(h,\\tau_G)$ for which $h$ is fixed by the Lie algebra \ninvolution $\\tau_{\\mathfrak g} := \\L(\\tau_G)$, a standard subspace \n${\\tt V} := {\\tt V}_{(h,\\tau_G,U)}$, specified by \n\\begin{equation}\n  \\label{eq:bgl}\nJ_{\\tt V} = U(\\tau_G) \\quad \\mbox{ and } \\quad \n\\Delta_{{\\tt V}}^{-it\/2\\pi} = U(\\exp th) \\quad \\mbox{ for } \\quad t \\in {\\mathbb R}.\n\\end{equation}\nThis assignment is called the {\\it Brunetti--Guido--Longo \n(BGL) construction} (see \\cite{BGL02}). \nAs a consequence, standard subspaces can be associated to antiunitary \nrepresentations in abundance, but only a few of them \ncarry interesting geometric \ninformation. In particular, we would like to understand \nwhen a standard subspace of the form ${\\tt V}_{(h,\\tau_G,U)}$ arises from a \nnatural family ${\\tt V}(\\mathcal{O})$ of real subspaces associated to open \nsubsets of a homogeneous space $M = G\/P$ and which domains $\\mathcal{O} \\subseteq M$ \n(so-called {\\it generalized wedge domains}) correspond to such standard subspaces.  \nThe geometric investigation of such domains in causal symmetric spaces \nwill be pursued further in \\cite{NO20}. \n\nIn the present paper we develop a method for the construction of  \nnets of standard subspaces for antiunitary representations of Lie groups \nin spaces of distributions which are boundary values of holomorphic functions. \nThis construction provides for a large class of triples\n $(h,\\tau_G, U)$ a realization of the corresponding standard subspace \n${\\tt V}_{(h,\\tau_G,U)}$ as some ${\\tt V}(\\mathcal{W})$, associated to an open subset \n$\\mathcal{W}$ of the group $G$ or of a homogeneous space. We obtain such a \nrealization on three levels: \n\\begin{itemize}\n\\item[(GL)] the group level, where $\\mathcal{W} \\subseteq G$ is an open subsemigroup. \n\\item[(SL)] the level of symmetric spaces, where $\\mathcal{W}$ is  an open \ndomain in the symmetric space $G\/G^{\\tau_G}$. \n\\item[(CL)] the conformal level, where $\\mathcal{W}$ is an open domain in a {\\it Jordan space-time}, \ni.e., the universal cover of the conformal completion \nof a euclidean Jordan algebra~$E$. These are the Jordan space-times in the sense of \nG\\\"unaydin \\cite{Gu93}, resp., the simple space-time manifolds in the sense of \nMack--de Riese (\\cite{MdR07}). \n\\end{itemize}\nIn Physics, these three levels arise for one-dimensional conformal \nfield theory, where $G$ is the $3$-dimensional M\\\"obius group \n${\\rm\\textsf{M\\\"ob}} \\cong \\mathop{{\\rm PSL}}\\nolimits_2({\\mathbb R})$, the corresponding Jordan space-time \nis the real line, considered as the simply connected covering of the circle, and \nthe corresponding causal symmetric space is the $2$-dimensional \nAnti-de Sitter space $\\mathop{{\\rm AdS}}\\nolimits^2$. For the universal cover \n$G = \\tilde\\mathop{{\\rm SO}}\\nolimits_{2,d}({\\mathbb R})_0$ of the conformal group of \n$d$-dimensional Minkowski space, \nthe corresponding Jordan space-time is the simply connected cover of the \nconformal completion of Minkowski space. \\\\\n\nThe following table contains some information on the \nsimple hermitian Lie algebras of tube type ${\\mathfrak g}$, the corresponding \nsimple euclidean Jordan algebra $E \\cong {\\mathfrak g}^1(h) = \\ker(\\mathop{{\\rm ad}}\\nolimits h-\\mathbf{1})$, the rank of \n$E$, which is the real rank of the simple \nreal Lie algebra ${\\mathfrak g}$, the subalgebra ${\\mathfrak h} = {\\mathfrak g}^0(h) = \n\\ker(\\mathop{{\\rm ad}}\\nolimits h)$, \nand the topology of the simple Jordan space-time~$M$, \nconsidered as a product of ${\\mathbb R}$ with a Riemannian symmetric space. Here \n$E_6^c$ stands for the simply connected compact Lie group of type $E_6$ and \n$F_4^c \\subseteq E_6^c$ for a connected subgroup of type $F_4$.\\\\\n\n\\hspace{-8mm}\n\\begin{tabular}{||l|l|l|l|l||}\\hline\n{} ${\\mathfrak g}$ \\mbox{(conf. Lie alg)}  &\\ $E$\\ \\mbox{(Jordan alg.)} & ${\\rm rk}(E)$ &\\ \\ ${\\mathfrak h} = {\\mathfrak g}^0(h)$ \\phantom{\\Big(}& $M$ \\\\ \n\\hline\\hline \n $\\so_{2,n}({\\mathbb R}), n > 2$ & \\phantom{\\Big(} ${\\mathbb R}^{1,n-1}$ & $2$ & ${\\mathbb R} \\oplus \\so_{1,n-1}({\\mathbb R})$  \n& ${\\mathbb R} \\times {\\mathbb S}^{n-1}$ \\\\ \n\\hline\n $\\sp_{2n}({\\mathbb R})$ &  $\\mathop{{\\rm Sym}}\\nolimits_n({\\mathbb R})$ & $n$ & ${\\mathbb R} \\oplus \\fsl_n({\\mathbb R}) \\cong \\gl_n({\\mathbb R})$  \n& ${\\mathbb R} \\times \\mathop{{\\rm SU}}\\nolimits_n({\\mathbb R})\/\\mathop{{\\rm SO}}\\nolimits_n({\\mathbb R})$ \\\\ \n\\hline\n $\\su_{n,n}({\\mathbb C})$ &  $\\mathop{{\\rm Herm}}\\nolimits_n({\\mathbb C})$ & $n$ & ${\\mathbb R} \\oplus \\fsl_n({\\mathbb C})$  \n& ${\\mathbb R} \\times \\mathop{{\\rm SU}}\\nolimits_n({\\mathbb C})$  \\\\ \n\\hline\n $\\so^*(4n)$ &  $\\mathop{{\\rm Herm}}\\nolimits_n(\\H)$ & $n$ & ${\\mathbb R} \\oplus \\fsl_n(\\H) \\cong \\gl_n(\\H)$ \n& ${\\mathbb R} \\times \\mathop{{\\rm SU}}\\nolimits_{2n}({\\mathbb C})\/\\mathop{\\rm U{}}\\nolimits_n(\\H)$  \\\\ \n\\hline\n ${\\mathfrak e}_{7(-25)}$ &  $\\mathop{{\\rm Herm}}\\nolimits_3(\\mathbf{O})$ & $3$ & ${\\mathbb R} \\oplus {\\mathfrak e}_{6(-26)}$  \n& ${\\mathbb R} \\times E_6^c\/F_4^c$  \\\\ \n\\hline\n\\end{tabular}\n\n\\vspace{2mm}\n\nOur construction of nets of standard subspaces is based on \ndistribution vectors of unitary representations. \nTo introduce this concept, we first observe that the subspace \n$\\mathcal{H}^\\infty \\subseteq \\mathcal{H}$ of vectors $v \\in \\mathcal{H}$ for which the orbit map \n$U^v \\: G \\to \\mathcal{H}, g \\mapsto U(g)v$, is smooth \n(the {\\it smooth vectors}) is dense \n(we assume throughout that $\\dim G < \\infty$). \nIt carries a natural Fr\\'echet topology for which the action of \n$G$ on this space is smooth (\\cite{Ne10}). \nThe space $\\mathcal{H}^{-\\infty}$ of continuous antilinear functionals $\\eta \\: \\mathcal{H}^\\infty \\to {\\mathbb C}$ \n(the {\\it distribution vectors}) \ncontains in particular Dirac's kets \n$\\langle \\cdot, v \\rangle$, $v \\in \\mathcal{H}$, so that \nwe obtain a {\\it rigged Hilbert space} \n\\[ \\mathcal{H}^\\infty \\hookrightarrow \\mathcal{H} \\hookrightarrow \\mathcal{H}^{-\\infty},\\] \nwhere $G$ acts on all three spaces \nby representations denoted $U^\\infty, U$ and $U^{-\\infty}$, respectively, \nand its Lie algebra ${\\mathfrak g}$ acts on $\\mathcal{H}^{\\infty}$ and $\\mathcal{H}^{-\\infty}$ \nby the derived representation \n(see Appendix~\\ref{app:a} for details). \n\nTo any real subspace ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ and every open subset \n$\\mathcal{O} \\subseteq G$, we associate the real subspace \n\\begin{equation}\n  \\label{eq:HE}\n {\\sf H}_{\\sf E}(\\mathcal{O}) := \\overline{{\\rm span}_{\\mathbb R} \\{ U^{-\\infty}(\\phi) \\eta \\: \n\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}), \\eta \\in {\\sf E}\\}},\n\\end{equation}\nwhere \n\\[ U^{-\\infty}(\\phi) = \\int_G \\phi(g)U^{-\\infty}(g)\\, dg, \\qquad \n\\phi \\in C^\\infty_c(G) \\] \ndenotes the integrated representation of the convolution algebra \n$C^\\infty_c(G)$ of test functions on $G$ on the space~$\\mathcal{H}^{-\\infty}$.  \nThis assignment has already \ntwo obvious properties: \n\\begin{itemize}\n\\item[\\rm(Iso)] Isotony: $\\mathcal{O}_1 \\subseteq \\mathcal{O}_2$ implies \n${\\sf H}_{\\sf E}(\\mathcal{O}_1) \\subseteq {\\sf H}_{\\sf E}(\\mathcal{O}_2)$. \n\\item[\\rm(Cov)] Covariance: $U(g){\\sf H}_{\\sf E}(\\mathcal{O}) = {\\sf H}_{\\sf E}(g\\mathcal{O})$. \n\\end{itemize}\n\nOne of our main results consists in \nspecifying sufficient conditions for \n\\eqref{eq:HE} to produce \nstandard subspaces including some ${\\tt V}_{(h,\\tau_G,U)}$ \nobtained from the Brunetti--Guido--Longo construction. \nImportant questions in this context are: \n\\begin{itemize}\n\\item[\\rm(RS)] Does ${\\sf H}_{\\sf E}(\\mathcal{O})$ generate $\\mathcal{H}$ \nfor each non-empty open subset $\\mathcal{O} \\subseteq G$? \n({\\it Reeh--Schlieder property}) \n\\item[\\rm(BW)] If ${\\sf H}_{\\sf E}(\\mathcal{O})$ is standard, when is its modular group implemented by a \none-parameter subgroup of $G$ as in \\eqref{eq:bgl}? \nIf this is the case for the one-parameter group \ngenerated by $h \\in {\\mathfrak g}$, we say that ${\\sf H}_{\\sf E}(\\mathcal{O})$ has the \n{\\it Bisognano--Wichmann property} with respect to~$h$.\n\\end{itemize}\n\nA structural property with strong impact in this context is the {\\it spectrum condition} \non the infinitesimal generators $\\partial U(x)$ of the one-parameter groups \n$(U(\\exp tx))_{t \\in {\\mathbb R}}$. This is the requirement that the closed convex cone \n\\[ C_U := \\{ x \\in {\\mathfrak g} \\: -i \\partial U(x) \\geq 0 \\} \\] \n(the {\\it positive cone of $U$}) \nis ``large'' in the sense that the ideal ${\\mathfrak g}_{C_U} := C_U - C_U$ \nsatisfies \\break ${\\mathfrak g} = {\\mathfrak g}_{C_U} + {\\mathbb R}  h$. \nWe shall see in Theorem~\\ref{thm:sw64-gen-abs} \nthat this already implies the Reeh--Schlieder property~(RS).\nUnder more specific assumptions \non ${\\sf E}$ and $\\mathcal{O}$, we show that ${\\sf H}_{\\sf E}(\\mathcal{O})$ is actually standard. \nWe refer to Section~\\ref{sec:3} for details. \n\n\\nin {\\bf Content of this paper:} \nSection~\\ref{sec:2} is devoted to properties of smooth and distribution \nvectors for representations for which $C_U$ is pointed and generating, i.e., \n${\\mathfrak g}_{C_U} = {\\mathfrak g}$. Then $U$ extends by \n\\[ U(g \\exp(ix)) = U(g) e^{i\\partial U(x)}\\]  \nto a representation of the complex Olshanski semigroup $S_{C_U} = G \\exp(iC_U)$ \nby contractions. \nIn Subsection~\\ref{subsec:2.1} we study the action of $S_{C_U}$ \non the space of distribution vectors and establish continuity \nand holomorphy of orbit maps in the weak-$*$-topology. \nThis is used in Subsection~\\ref{subsec:2.2} to verify the Reeh--Schlieder property \n(Theorem~\\ref{thm:sw64-gen-abs}). \n\nIn Section~\\ref{sec:3} we turn to a more specific situation, where \n${\\tt V} = {\\tt V}_{(h,\\tau_G,U)}$ is a standard subspace obtained from the BGL \nconstruction, and the semigroup \n\\begin{equation} \\label{eq:sv}\n  S_{\\tt V} = \\{ g \\in G \\: U(g){\\tt V} \\subseteq {\\tt V}\\} \n\\end{equation}\nof endomorphisms of ${\\tt V}$ is large in the sense that its Lie wedge \n\\[ \\L(S_{\\tt V}) = \\{ x \\in {\\mathfrak g} \\: \\exp({\\mathbb R}_+ x) \\subseteq S_{\\tt V} \\} \\] \n(the set of infinitesimal generators of one-parameter subsemigroups of $S_{\\tt V}$) \nspans the Lie algebra~${\\mathfrak g}$. Here we build on the previous work \n \\cite{Ne19, Ne19b} of the first author on these semigroups. \nThe main results of these two papers are easy to describe. \nIn \\cite{Ne19} the Lie wedge $\\L(S_{\\tt V})$ is calculated. \nAs a consequence of its explicit description, \nthe assumption that it spans ${\\mathfrak g}$ implies that \n$\\mathop{{\\rm ad}}\\nolimits h$ defines a $3$-grading in the sense that \n\\begin{equation}\n  \\label{eq:3grad}\n{\\mathfrak g} = {\\mathfrak g}^1 \\oplus {\\mathfrak g}^0 \\oplus {\\mathfrak g}^{-1} \\quad \\mbox{ for } \\quad \n{\\mathfrak g}^j := \\ker(\\mathop{{\\rm ad}}\\nolimits h - j\\mathop{{\\rm id}}\\nolimits_{\\mathfrak g}).\n\\end{equation}\nAssuming \\eqref{eq:3grad}, the semigroup $S_{\\tt V}$ has been completely \ndetermined in \\cite{Ne19b}. To describe the structure of $S_{\\tt V}$, \nlet \n\\[ C_\\pm := \\pm C_U \\cap {\\mathfrak g}_{\\pm 1},\\quad \\mbox{ and write } \\quad \nG_{\\tt V} = \\{ g \\in G \\: U(g) {\\tt V} = {\\tt V} \\}\\]\nfor the stabilizer group of ${\\tt V}$ in~$G$. Then  \n\\begin{equation}\n  \\label{eq:SE}\nS_{\\tt V} = \\exp(C_+) G_{\\tt V} \\exp(C_-) = G_{\\tt V} \\exp(C_+ + C_-).\n\\end{equation}\nIn the setting of \\eqref{eq:3grad}, we define in Subsection~\\ref{subsec:3.1} \nthe space ${\\tt V}^{-\\infty}$ of distribution vectors associated to the \nstandard subspace ${\\tt V}$. \nThis requires an extension of \nthe symplectic form on $\\mathcal{H}$ to $\\mathcal{H}^\\infty \\times \\mathcal{H}^{-\\infty}$ \nand the symplectic orthogonal space ${\\tt V}'$ of~${\\tt V}$, so that we can define \n${\\tt V}^{-\\infty}$ as the annihilator of ${\\tt V}' \\cap \\mathcal{H}^\\infty$. \nHere the difficulty  is to identify elements of this space. \nThis is achieved in Subsection~\\ref{subsec:3.2}, where we extend the following \ncharacterization of elements in a standard subspaces from \\cite[Prop.~2.1]{NOO20} \nto distribution vectors: An element $\\xi \\in \\mathcal{H}$ is contained in ${\\tt V}$ \nif and only if the orbit map $\\alpha^\\xi \\: {\\mathbb R} \\to \\mathcal{H}, \n\\alpha^\\xi(t) :=  \\Delta_{\\tt V}^{-it\/2\\pi}\\xi$ extends to a \ncontinuous map on the closure of the strip \n\\[ \\mathcal{S}_\\pi = \\{ z \\in {\\mathbb C} \\: 0 <\\Im z < \\pi\\}, \\] \nwhich is holomorphic on $\\mathcal{S}_\\pi$ and satisfies the boundary \nrelation $\\alpha^\\xi(\\pi i) = J_{\\tt V} \\xi$. \nA suitable extension of this requirement to distribution vectors specifies \na linear subspace $\\mathcal{H}^{-\\infty}_{{\\rm ext},J}\\subseteq \\mathcal{H}^{-\\infty}$ \nwhich is invariant under \n$U^{-\\infty}(S_{\\tt V}^0)$ and $U^{-\\infty}(C^\\infty(S_{\\tt V}^0))$. \nThis leads to the inclusion \n$\\mathcal{H}^{-\\infty}_{{\\rm ext},J}\\subseteq {\\tt V}^{-\\infty}$ \n(Lemma~\\ref{lem:key}) which is one of our key tools. \nWe conclude Section~\\ref{sec:3} with the observation that \n$\\mathcal{H}^{-\\infty}_{\\rm ext}$ is invariant under the Lie algebra and \ndiscuss the irreducible antiunitary representation of the $ax+b$-group \non $L^2({\\mathbb R}_+)$ in some detail. \n\nThe results in Section~\\ref{sec:3} are valid \nfor general Lie groups whose Lie algebra contains an invariant cone \n$C$ and an element $h$ defining a $3$-grading for which \nthe cones $C_\\pm = \\pm C \\cap {\\mathfrak g}^{\\pm 1}$ are generating. \nTo construct representations to which the theory developed in \nSection~\\ref{sec:3} applies, we specialize in Section~\\ref{sec:4} \nto semisimple Lie groups. Assuming, in addition, that ${\\mathfrak g}^0$ contains \nno non-zero ideals, ${\\mathfrak g}$ is a direct sum of simple hermitian ideals of tube type \nand $E := {\\mathfrak g}^1$ thus carries the structure of a unital euclidean Jordan algebra \nsuch that $C_+$ is the closure of the open positive cone $E_+$ of invertible \nsquares in~$E$ (see \\cite{FK94} and the table above). \n\nThis structure is used in Section~\\ref{sec:4} to show that every \nirreducible antiunitary representation $(U,\\mathcal{H})$ of $G_\\tau$, which is \n{\\it $C$-positive} in the sense that $C \\subseteq C_U$, can be realized on \na Hilbert space $\\mathcal{H}_\\rho$ of holomorphic vector-valued functions \non the tube domain $\\mathcal{T} := E + i E_+$ on which the group $G$ acts as a \ngroup of biholomorphic automorphisms,  and $\\tau_G$ corresponds to the \nantiholomorphic involution $\\tau_E(z) = -\\overline z$. The input for this construction \nis a real irreducible representation $(\\rho, \\mathcal{K}_{\\mathbb R})$ of the simply \nconnected Lie group $\\tilde H$ with Lie algebra ${\\mathfrak g}^0$ which satisfies \na certain positivity condition. Then $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$, \nthe space of holomorphic functions with values in the \ncomplexification $\\mathcal{K}$ of $\\mathcal{K}_{\\mathbb R}$. \nThe positivity condition for $(\\rho,\\mathcal{K}_{\\mathbb R})$ is hard to evaluate explicitly \nfor non-scalar representations, an explicit classification \ncan be derived by combining the classification \nof unitary highest modules of ${\\mathfrak g}$,  \ndue to Enright, Howe and Wallach (\\cite{EHW83}), \nwith the translation into the present context, carried out in \\cite{HN01}. \nThe main point of this realization is that evaluation in $0 \\in \\partial \\mathcal{T}$ \ndefines a continuous linear map $\\mathcal{H}_\\rho^\\infty \\to \\mathcal{K}$, which embeds \n$\\mathcal{K}$ into $\\mathcal{H}^{-\\infty}$. \nIn Section~\\ref{sec:5} we show that this leads to a \nreal subspace ${\\sf E} \\subseteq \\mathcal{K}$ contained in ${\\tt V}^{-\\infty}$, \nso that we can apply Section~\\ref{sec:3}. We then have the identity \n${\\tt V} = {\\sf H}_{\\sf E}(S_{\\tt V}^0)$ and obtain a net ${\\tt V}(\\mathcal{O}) := {\\sf H}_{\\sf E}(\\mathcal{O})$ \nof cyclic subspaces associated to non-empty open subsets $\\mathcal{O} \\subseteq G$. \nThe set of generalized wedge domains in $G$ in the sense of (GL) is the set \n\\[ \\mathcal{W} = \\{ g S \\: g \\in G \\}\\]\nof left translates of the connected component $S$ of the \nopen semigroup $S_{\\tt V}^0$ containing $e$ in its closure. If $\\mathcal{O}$ is contained \nin a left translate of the semigroup $S_{\\tt V}^0$, then \nthe cyclic subspace ${\\tt V}(\\mathcal{O})$ is standard. \n\nAs the real subspace ${\\sf E}$ is invariant under the subgroups \n$H := \\langle \\exp_G {\\mathfrak g}^0 \\rangle$ and $P^- := \\exp({\\mathfrak g}^{-1})H$ of $G$, \nit easily follows that ${\\tt V}(\\mathcal{O}) = {\\tt V}(\\mathcal{O} \\cdot H) = {\\tt V}(\\mathcal{O} \\cdot P^-)$, \nso that these nets of cyclic subspaces also define nets on the homogeneous spaces \n$G\/H$ and $G\/P^-$. Here $G\/H$ is a compactly causal symmetric spaces \nof Cayley type (\\cite{HO97}) und $G\/P^-$ is a  Jordan space-time in the sense of \nG\\\"unaydin \\cite{Gu93} and a simple space-time manifold in the sense of \nMack--de Riese (\\cite{MdR07}). \n\n\n\\nin {\\bf Perspectives:} In the present paper we construct nets \nof standard subspaces on homogeneous spaces which are associated \nto unitary representations $(U,\\mathcal{H})$ for which the positive cone $C_U$ \nis non-trivial. If this cone is trivial, one needs other \ntechniques to specify suitable subspaces ${\\sf E} \\subseteq {\\tt V}^{-\\infty}$. \nA prototypical example is  the action of the Lorentz \ngroup $G = \\mathop{{\\rm SO}}\\nolimits_{1,d}({\\mathbb R})_0$ on de Sitter space $\\mathop{{\\rm dS}}\\nolimits^d$, which is a \nnon-compactly causal symmetric space. \nIn the physics literature one finds several \nreplacements for the spectrum condition that are used to deal \nwith quantum fields on curved spaces such as de Sitter space. \nOf particular relevance is the \n``microlocal spectrum condition''; \nsee \\cite{TS16} and \\cite{PEGW19}.\n\nWe expect that the  reflection positivity condition and the \nKMS condition that characterize the modular objects \n(see the appendix of \\cite{NOO20} for details) \ncan be used to express a condition on the distribution vectors in ${\\sf E}$ \nto ensure that the subspace \n${\\sf H}_{\\sf E}(\\mathcal{W})$ associated to a ``wedge domain''  $\\mathcal{W} \\subseteq M$  is \nthe standard subspace corresponding to the pair \n$(h,\\tau)$. \n\nIn the forthcoming paper \\cite{NO20} we develop a theory of \ngeneralized wedge domains in causal symmetric spaces that \nwill shed additional light on the class of domains $\\mathcal{W} \\subseteq M$ \nfor which one may expect a relation like ${\\sf H}_{\\sf E}(\\mathcal{W}) = {\\tt V}$. \n\nIn all these constructions, a good understanding of the class of \ngeneralized wedge domains is of crucial importance. \nThis motivated the abstract approach to these domains \nin \\cite{MN20}, where the set \n\\[ \\mathcal{G}(G) := \\{ (x,\\sigma) \\in {\\mathfrak g} \\times (G \\times \\{\\tau_G\\}) \n\\subseteq {\\mathfrak g} \\times G_\\tau\\:\n\\mathop{{\\rm Ad}}\\nolimits(\\sigma) x = x, \\sigma^2 = e \\} \\] \nis studied as a candidate of an index set for nets of standard subspaces. \nIn this context the {\\it Euler couples}, i.e., those for which \n$\\mathop{{\\rm ad}}\\nolimits x$ is diagonalizable with eigenvalues $\\{-1,0,1\\}$ and \n$\\mathop{{\\rm Ad}}\\nolimits(\\sigma) = e^{\\pi i \\mathop{{\\rm ad}}\\nolimits x}$ are of particular relevance. \nFor instance, the assumptions in Section~\\ref{sec:3} imply \nthat $(h,\\tau)$ is an Euler couple. \n\n\\nin {\\bf Acknowledgment:} We thank Roberto Longo for asking us on several \noccasions to construct quantum field theories on Lie groups. The present paper \nis a first step in this direction. We are most \ngrateful to Daniel Oeh for several valuable \ncomments on an earlier version of this manuscript and for pointing \nout an error in Theorem~\\ref{thm:5.4}. \n\n\n\\subsection*{Notation} \n\n\\begin{itemize}\n\\item For a Lie group $G$ with neutral element $e$, the \nidentity component is denoted $G_0$. \nWe write ${\\mathfrak g}$ for its Lie algebra, \n$\\mathop{{\\rm Ad}}\\nolimits \\: G \\to \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g})$ for the adjoint action of $G$ on ${\\mathfrak g}$, induced by the \nconjugation action of $G$ on itself, and $\\mathop{{\\rm ad}}\\nolimits x(y) = [x,y]$ for the adjoint \naction of ${\\mathfrak g}$ on itself. \n\\item For a group action $G \\times M \\to M$ and $m \\in M$, we write \n$G^m := \\{ g \\in G \\: g.m = m\\}$ for the stabilizer subgroup of $m$. \n\\item For a connected Lie group $G$, we write $q_G \\: \\tilde G \\to G$ \nfor the universal covering morphism and \n$\\eta_G \\: G \\to G_{\\mathbb C}$ for the universal complexification. \n\\item If $M$ is a smooth manifold, we write $C^\\infty_c(M)$ for the space \nof complex-valued test functions on $M$, endowed with the natural LF topology, \nand $C^{-\\infty}_c(M)$ for the space of {\\bf antilinear} continuous \nlinear functionals  on this space, i.e., the space of \ndistributions on $M$. \n\\item We likewise consider tempered distributions \n$D \\in \\mathcal{S}'(E)$ on a real finite dimensional vector space $E$ \nas antilinear functionals on the Schwartz space $\\mathcal{S}(E)$. \nThe {\\it Fourier transform of an $L^1$-function $f$} on $E$ is defined by \n\\begin{equation}\n  \\label{eq:ftrafo}\n\\hat f(\\lambda) := \\int_E e^{-i\\lambda(x)} f(x) \\,  d\\mu_E(x), \\quad \n\\lambda \\in E^*, \n\\end{equation}\nwhere $\\mu_E$ is a Haar measure on $E$. \nFor tempered distributions $D \\in \\mathcal{S}'(E)$, we define the Fourier transform by \n\\begin{equation}\n  \\label{eq:2}\n\\hat D(\\phi) := D(\\tilde \\phi), \\quad \\mbox{ where} \n\\quad \n\\tilde\\phi(\\lambda) \n:= \\hat\\phi(-\\lambda) \n= \\int_E e^{i\\lambda(x)} \\phi(x) \\,  d\\mu_E(x).\n\\end{equation}\nThe distribution \n$D_f(\\phi) := \\int_E \\overline{\\phi(x)} f(x)\\, d\\mu_E(x)$, defined by \nan $L^1$-function, then satisfies  $\\hat{D_f} = D_{\\hat f}$. \n\\end{itemize}\n\n\\section{Distribution vectors of $C$-positive representations} \n\\mlabel{sec:2}\n\nLet $G$ be a Lie group with Lie algebra ${\\mathfrak g}$ \nand $C \\subseteq {\\mathfrak g}$ a closed convex pointed generating \n$\\mathop{{\\rm Ad}}\\nolimits(G)$-invariant cone (see \\cite{Ne99} for a structure theory of these \nconfigurations). \nIn this section we study distribution vectors of \nunitary representations $(U,\\mathcal{H})$ of $G$ which are \n$C$-positive in the sense that $C \\subseteq C_U$.\\begin{footnote}{In Section~\\ref{sec:3}, this \nsection will be applied to the \nideal ${\\mathfrak g}_C := C - C$ generated by the invariant cone~$C$.} \\end{footnote} \nThen the representation $U \\: G \\to \\mathop{\\rm U{}}\\nolimits(\\mathcal{H})$ extends analytically \nto a representation of an Olshanski semigroup $S_C= G \\exp(iC)$. \n\nIn Subsection~\\ref{subsec:2.1} we start with some results concerning \nthe action of the semigroup $S_C$ on the spaces of smooth and \ndistribution vectors. In particular, we show that \nthe closed semigroup $S_C$ acts on the space $\\mathcal{H}^{-\\infty}$ of distribution vectors  with weak-$*$-continuous orbit maps that are holomorphic on the \ninterior of~$S_C$. From this we derive \nthe interesting result that every distribution vector \n$\\eta \\in \\mathcal{H}^{-\\infty}$ defines a $G$-right equivariant map \n$\\mathcal{H}^{-\\infty} \\to \\mathop{{\\rm Hol}}\\nolimits(S_C^0)$ whose range consists of holomorphic \nfunctions with distributional boundary values \n(Theorem~\\ref{thm:2.5}). \n\nIn Subsection~\\ref{subsec:2.2} we turn to real subspaces ${\\sf H}_{\\sf E}(\\mathcal{O})$ \ngenerated by $U^{-\\infty}(C^\\infty_c(\\mathcal{O},{\\mathbb R})){\\sf E}$, where \n$\\mathcal{O} \\subseteq G$ is open and ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ \nis a real subspace. Here the main result is the \nReeh--Schlieder property (Theorem~\\ref{thm:sw64-gen-abs}), asserting \nthat ${\\sf H}_{\\sf E}(\\mathcal{O})$ is total in $\\mathcal{H}_{\\sf E}(G)$ for every \nnon-empty open subset $\\mathcal{O} \\subseteq G$ and every \nreal subspace ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$. \n\n\\subsection{The semigroup action on distribution vectors} \n\\mlabel{subsec:2.1} \n\nWe start by recalling the construction of the complex Olshanski semigroup \n$S_C$ associated to the invariant cone~$C \\subseteq {\\mathfrak g}$. \n\n\\begin{defn} \\mlabel{def:olshanski} (Olshanski semigroups) \nLet $G$ be a connected Lie group with Lie algebra ${\\mathfrak g}$ \n and $C \\subseteq {\\mathfrak g}$ be a pointed generating \n$\\mathop{{\\rm Ad}}\\nolimits(G)$-invariant closed convex cone. \nThen $C$ is {\\it elliptic} \nin the sense that, for every element $x \\in C^0$, \nthe derivation $\\mathop{{\\rm ad}}\\nolimits x$ is semisimple with purely imaginary spectrum \n(\\cite[Prop.~VII.3.4(b)]{Ne99}). \nThe corresponding {\\it Olshanski semigroup} \n$S_C$ is defined as follows.\n\nLet $q_G \\:  \\tilde G \\to G$ be the universal covering group \nof $G$ and $\\tilde G_{\\mathbb C}$ be the $1$-connected Lie group with Lie algebra ${\\mathfrak g}_{\\mathbb C}$, \nso that the universal complexification \n$\\eta_{\\tilde G} \\: \\tilde G \\to \\tilde G_{\\mathbb C}$ simply is the \ncanonical morphism of Lie groups \nfor which $\\L(\\eta_{\\tilde G}) \\: {\\mathfrak g} \\hookrightarrow {\\mathfrak g}_{\\mathbb C}$ is the inclusion. \n\\begin{footnote}{In general the map $\\eta_G$ is not injective, as the example $G = \\tilde\\mathop{{\\rm SL}}\\nolimits_2({\\mathbb R})$ with\n$G_{\\mathbb C} = \\mathop{{\\rm SL}}\\nolimits_2({\\mathbb C})$ shows.}   \n\\end{footnote}\nAs $\\tilde G_{\\mathbb C}$ is simply connected, the complex conjugation \non ${\\mathfrak g}_{\\mathbb C}$ integrates to an antiholomorphic \ninvolution $\\sigma \\: \\tilde G_{\\mathbb C} \\to \\tilde G_{\\mathbb C}$ with \n$\\sigma \\circ \\eta_{\\tilde G} = \\eta_{\\tilde G}$, and this implies that\n$\\eta_{\\tilde G}(\\tilde G)$ coincides with the subgroup $(\\tilde G_{\\mathbb C})^\\sigma$ \nof $\\sigma$-fixed points in $\\tilde G_{\\mathbb C}$.\n\\begin{footnote}{Since $\\tilde G_{\\mathbb C}$ is simply \nconnected, this subgroup is connected by \\cite[Thm.~IV.3.4]{Lo69}.}\n\\end{footnote}\nAs $C$ is elliptic, \nLawson's Theorem (\\cite[Thm.~XIII.5.6]{Ne99}) asserts that\n\\[S'_C := (\\tilde G_{\\mathbb C})^\\sigma \\exp(iC) \\subseteq \\tilde G_{\\mathbb C} \\]\nis a closed subsemigroup of $\\tilde G_{\\mathbb C}$ stable under the antiholomorphic\ninvolution $s^* := \\sigma(s)^{-1}$ and the polar map\n\\[ (\\tilde G_{\\mathbb C})^\\sigma \\times C \\to S_C, \\quad (g,x) \\mapsto g \\exp(ix) \\]\nis a homeomorphism. \nNext we observe that $\\ker \\eta_{\\tilde G}$ is a discrete subgroup of \n$\\tilde G$ and define \n$\\tilde S_C$ as the simply connected covering space of $S_C'$ and \n\\[ S_C := \\tilde S_C\/\\ker(q_G) \\] \n(\\cite[Def.~XI.1.11]{Ne99}). The quotient map is denoted \n$q_{S_C} \\: \\tilde S_C \\to S_C$. It is the universal covering of~$S_C$.\nWe write \n\\[ \\tilde\\exp \\: {\\mathfrak g} + i C \\to \\tilde S_C \\] \nfor the continuous lift of the exponential function \n$\\exp \\:  {\\mathfrak g} + i C = \\L(S_C')\\to S_C' \\subseteq \\tilde G_{\\mathbb C}$ \nand \n\\[ \\exp := q_{S_C} \\circ \\tilde\\exp \\: {\\mathfrak g} + i C \\to S_C.\\]\nThis function extends the exponential \nfunction of the group~$G$ and,  for every $x \\in {\\mathfrak g} + i C$, \nthe curve $\\gamma_x(t) := \\exp(tx)$ \nis a continuous one-parameter semigroup of $S_C$.  \nBasic covering theory implies that $S_C$ inherits an involution \ngiven by \n\\begin{equation}\n  \\label{eq:eq*}\n (g \\exp(ix))^* = \\exp(ix)g^{-1} = g^{-1} \\exp(\\mathop{{\\rm Ad}}\\nolimits(g)ix)\n\\end{equation}\n and a homeomorphic  polar map $G \\times C \\to S_C,  (g,x) \\mapsto g \\exp(ix)$. \n\nIf $C$ has interior points, then \nthe polar map maps $(\\tilde G_{\\mathbb C})^\\sigma \\times C^0$ \ndiffeomorphically onto the interior $S_{C^0}'$ of $S_C'$. \nHence $S_C^0 = S_{C^0}= G \\exp(i C^0)$ is a complex manifold \nwith a holomorphic multiplication, and the exponential function \n${\\mathfrak g} + i C^0 \\to S_{C^0}$ is holomorphic, \nwhereas the involution $*$ is antiholomorphic \n(\\cite[Thm.~XI.1.12]{Ne99}). \n\\end{defn}\n\nThe following theorem ensures the existence of extensions of a \nunitary representation~$U$ of $G$ to the semigroup $S_C$, provided its positive cone $C_U$ contains~$C$.\n\n\\begin{thm} {\\rm(Holomorphic Extension Theorem)} \\mlabel{thm:extens}\nIf $(U,\\mathcal{H})$ is a unitary representation of $G$ \nsatisfying $C \\subseteq C_U$, then it extends by \n\\[ U(g \\exp(ix)) = U(g) e^{i\\partial U(x)}, \\quad g \\in G, x \\in C, \\] \nto a $*$-representation \nof the closed complex Olshanski semigroup $S_C= G \\exp(iC)$ by contractions on $\\mathcal{H}$ \nwhich defines a continuous action of $S_C$ on $\\mathcal{H}$. \nIf $C$ has interior points, then $U$ defines a holomorphic map \n$S_{C^0} \\to B(\\mathcal{H})$. \n\\end{thm}\n\n\\begin{prf} The existence of the holomorphic extension to $S_C$ follows from \n\\cite[Thm.~XI.2.5]{Ne99} and its weak continuity from \n\\cite[Prop.~XI.3.7]{Ne99}. As it is a representation by contractions, \nstrong continuity follows from \\cite[Cor.~IV.1.18]{Ne99}. \nSince the representation of $S_C$ on $\\mathcal{H}$ is strongly continuous \nand by contractions, it defines a continuous action of $S_C$ on $\\mathcal{H}$ \nbecause \n\\[ \\|U(s')\\xi' - U(s)\\xi \\|\n\\leq  \\|U(s')(\\xi' - \\xi) \\| + \\|(U(s') - U(s))\\xi\\| \n\\leq \\|\\xi' - \\xi\\| +  \\|(U(s') - U(s))\\xi\\| \\] \nfor $\\xi,\\xi' \\in \\mathcal{H}$ and $s,s' \\in S_C$.  \n\\end{prf}\n\n\nThe main purpose of this subsection is to analyze the representation \nof the semigroup $S_C$ on the spaces of smooth and distribution \nvectors. The first crucial step is to show that $U(S_C)$ leaves \n$\\mathcal{H}^\\infty$ invariant (Proposition~\\ref{prop:2.2}).\n\n\\begin{lem}\n  \\mlabel{lem:2.1} \nThe operators $(U(s))_{s \\in S_C}$ preserve the dense subspace \n$\\mathcal{H}^\\infty$. For every smooth vector $\\xi \\in \\mathcal{H}^\\infty$, the orbit map \n\\[ U^\\xi \\: S_C \\to \\mathcal{H}^\\infty, \\quad \ns \\mapsto U(s)\\xi \\] \nis continuous on $S_C$ and holomorphic on the interior $S_C^0$.   \n\\end{lem}\n\n\\begin{prf} We consider $\\mathcal{H}$ as a subspace of $\\mathcal{H}^{-\\infty}$ \nas in \\eqref{eq:embindistr} in Appendix~\\ref{app:a}. \nAs the locally convex topology on $\\mathcal{H}^\\infty$ is defined by the \ntopological embedding \n\\begin{equation}\n  \\label{eq:topsmo}\n\\mathcal{H}^\\infty \\hookrightarrow \\prod_{D \\in \\mathcal{U}({\\mathfrak g})} \\mathcal{H}, \\quad \n\\xi \\mapsto (\\dd U(D)\\xi)_{D \\in \\mathcal{U}({\\mathfrak g})}\n\\end{equation}\n(see \\eqref{eq:tophinfty} in Appendix~\\ref{app:a}), \nwe have to show that: \n\\begin{itemize}\n\\item[\\rm(a)] For every $D \\in \\mathcal{U}({\\mathfrak g})$ and $\\xi \\in \\mathcal{H}^\\infty$, \nwe have $f(s) := \\dd U^{-\\infty}(D) U(s) \\xi \\in \\mathcal{H}$ \n(Lemma~\\ref{lem:charsmooth}). \n\\item[\\rm(b)] The function $f \\: S_C \\to \\mathcal{H}$ is continuous and holomorphic on~$S_C^0$. \n\\end{itemize}\n\nWe first observe that the adjoint action of $G$ on ${\\mathfrak g}$ \nextends to a locally finite holomorphic representation of the universal \ncomplexification $G_{\\mathbb C}$ (\\cite[Thm.~15.1.4]{HN12}). As the semigroup \n$S_C$ has a natural continuous homomorphism \n\\[ \\eta_S \\: S_C \\to G_{\\mathbb C}, \\quad \\eta_S(g \\exp(ix)) \n= \\eta_G(g) \\exp_{G_{\\mathbb C}}(ix) \\] \nwhich is holomorphic on the interior \n(Definition~\\ref{def:olshanski}),\nwe thus obtain in particular a continuous representation  \n\\[ \\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C} \\: S_C \\to \\mathop{{\\rm Aut}}\\nolimits(\\mathcal{U}({\\mathfrak g}))  \\] \nof $S_C$ on the complex enveloping algebra \n$\\mathcal{U}({\\mathfrak g})$ which is holomorphic on $S_C^0$. \nFor $g \\in G$ and $D \\in \\mathcal{U}({\\mathfrak g})$, we have the relation \n\\begin{equation}\n  \\label{eq:adrel1}\n\\dd U(D) U(g) = U(g) \\dd U(\\mathop{{\\rm Ad}}\\nolimits(g^{-1})D) \n\\: \\mathcal{H}^\\infty \\to \\mathcal{H}.\n\\end{equation}\nWe claim that \n\\begin{equation}\n  \\label{eq:s-id}\n \\dd U^{-\\infty}(D) U(s) = U(s) \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \n\\: \\mathcal{H}^\\infty \\to \\mathcal{H}^{-\\infty} \n\\quad \\mbox{ for }\\quad \ns \\in S_C.\n\\end{equation}\nNote that both sides define linear maps $\\mathcal{H}^\\infty \\to \\mathcal{H}^{-\\infty}$ \nand that the right hand side maps into~$\\mathcal{H}$. To verify \n\\eqref{eq:s-id}, we have to show that \n\\begin{equation}\n  \\label{eq:s-id2}\n\\langle \\xi, \\dd U^{-\\infty}(D) U(s)\\eta \\rangle \n= \\langle \\xi, U(s) \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\eta \\rangle\n\\quad \\mbox{ for } \\quad \n\\xi,\\eta \\in \\mathcal{H}^\\infty.\n\\end{equation}\nThe left hand side equals \n$\\langle \\dd U(D^*) \\xi,  U(s)\\eta \\rangle,$\nwhich is continuous on $S_C$ and holomorphic on the interior. \nFor the right hand side we obtain \n\\[ \\langle \\xi, U(s) \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\eta \\rangle \n= \\langle U(s^*)\\xi, \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\eta \\rangle.\\] \nHere $U(s^*)\\xi$ is continuous on $S_C$ and antiholomorphic \non $S_C^0$ and $\\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\eta$ is also \ncontinuous on $S_C$ and holomorphic on $S_C^0$. Therefore the \nright hand side of \\eqref{eq:s-id2} \nis continuous on $S_C$ and holomorphic on the \ninterior. As both sides of \\eqref{eq:s-id2} coincide on $G$ by \n\\eqref{eq:adrel1}, they are equal \n(\\cite[Lemma~A.III.6]{Ne99}). \nThis implies \\eqref{eq:s-id}, and hence that \n$\\dd U^{-\\infty}(D) U(s) \\mathcal{H}^\\infty \\subseteq \\mathcal{H}$. \nAs \n\\[ \\mathcal{H}^\\infty = \\{ \\xi \\in \\mathcal{H} \\subseteq \\mathcal{H}^{-\\infty} \\: (\\forall D \\in \\mathcal{U}({\\mathfrak g})) \n\\ \\dd U^{-\\infty}(D)\\xi \\in \\mathcal{H}\\} \\] \nby Lemma~\\ref{lem:charsmooth}, it follows that \n$U(S_C)$ preserves $\\mathcal{H}^\\infty$. \nSince the right hand side of \\eqref{eq:s-id} \ndefines a continuous linear map $\\mathcal{H}^\\infty \\to \\mathcal{H}$, \nthe definition of the topology on $\\mathcal{H}^\\infty$ by the embedding \n\\eqref{eq:topsmo} also shows that the restrictions \n\\begin{equation}\n  \\label{eq:dag}\n U^\\infty(s) := U(s)\\vert_{\\mathcal{H}^\\infty} \\: \\mathcal{H}^\\infty \\to \\mathcal{H}^\\infty, \\quad s \\in S_C,  \n\\end{equation}\nare continuous linear maps. \nIn particular, we have \n$f(s) \\in \\mathcal{H}^\\infty$ for $s \\in S_C$. To see that $f$ is continuous, \nwe write it with \\eqref{eq:s-id} as \n\\[ f(s) = U(s) \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\xi.\\] \nAs the representation of $S_C$ defines a continuous \naction on $\\mathcal{H}$ (Theorem~\\ref{thm:extens}),\nthe continuity of $f$ follows from the continuity of the map \n\\begin{equation}\n  \\label{eq:admap}\nS_C \\to \\mathcal{H}, \\quad s \\mapsto \\dd U(\\mathop{{\\rm Ad}}\\nolimits_{\\mathbb C}(s)^{-1}D) \\xi\n\\end{equation}\nwhich actually ``extends'' to a holomorphic map on all of~$G_{\\mathbb C}$. \n\nThat $f$ is holomorphic on $S_C^0$ follows likewise from the holomorphy \nof the action of $S_C^0$ on the complex manifold $\\mathcal{H}$ and the \nholomorphy of \\eqref{eq:admap} on $S_C^0$. \n\\end{prf}\n\nWith Lemma~\\ref{lem:2.1}, we immediately get: \n\n\\begin{prop} \\mlabel{prop:2.2}\nThe prescription \n\\[ U^{-\\infty}(s)\\alpha := \\alpha\\circ U^\\infty(s^*) \n \\quad \\mbox{ for } \\quad \n\\alpha \\in \\mathcal{H}^{-\\infty}, s \\in S_C, \\] \ndefines a representation of $S_C$ on $\\mathcal{H}^{-\\infty}$ \nwhose orbit maps are weak-$*$-continuous and weak-$*$-holomorphic on \n$S_C^0$. \n\\end{prop} \n\nAlthough the closed semigroup $S_C$ acts on smooth and distribution \nvectors, its interior $S_{C^0}$ has particular regularizing properties \nwhich are described in the following two lemmas. \n\n\\begin{lem} For $s \\in S_C^0$, we  have $U(s)\\mathcal{H}\\subseteq \\mathcal{H}^\\infty$, and the maps \n  \\begin{equation}\n    \\label{eq:smoothop}\n U(s) \\: \\mathcal{H} \\to \\mathcal{H}^\\infty, \\qquad s \\in S_C^0, \n  \\end{equation}\nare continuous.   \n\\end{lem}\n\n\\begin{prf} Let $\\xi \\in \\mathcal{H}$. \nThen the orbit map $U^\\xi \\:  S_C^0 \\to \\mathcal{H}, U^\\xi(s) := U(s)\\xi$ \nis  holomorphic, so that its range lies in particular in \nthe space of smooth vectors for $G$. \nFor $D = x_1 \\cdots x_n \\in \\mathcal{U}({\\mathfrak g})$, $x_j \\in {\\mathfrak g}$, we consider the \ndifferential operator defined by \n\\[ D^R \\: C^\\infty(S_C^0) \\to C^\\infty(S_C^0), \\quad \n(D^R\\phi)(s) \n= \\frac{\\partial^n}{\\partial t_1 \\cdots \\partial t_n}\\Big|_{t_j = 0} \n\\phi(\\exp(t_n x_n) \\cdots \\exp(t_1 x_1)s).\\] \nThat the maps \\eqref{eq:smoothop} \nare continuous follows with \\eqref{eq:tophinfty} in Appendix~\\ref{app:a} \nfrom the fact that, for every $D \\in \\mathcal{U}({\\mathfrak g})$,  the composition \n$\\dd U(D) \\circ U^\\xi$ is obtained by applying the right invariant \ndifferential operator $D^R$ \nto the smooth functions $g \\mapsto U^\\xi(gs)$, $s \\in S_C^0$. \n\\end{prf}\n\n\n\\begin{lem} \\mlabel{lem:9.14} \nFor $s \\in S_C^0$, we have $U^{-\\infty}(s)\\mathcal{H}^{-\\infty} \n\\subseteq \\mathcal{H}^\\infty$ and the orbit maps \n\\begin{equation}\n  \\label{eq:orbitmapint}\nU^{-\\infty,\\eta} \\: S_C^0 \\to \\mathcal{H}^\\infty, \\quad\ns \\mapsto U^{-\\infty}(s) \\eta, \\qquad \n\\eta \\in \\mathcal{H}^{-\\infty},  \n\\end{equation}\nare holomorphic. \n\\end{lem}\n\n\\begin{prf} For $D \\in \\mathcal{U}({\\mathfrak g})$,  let $D^L$ denote the corresponding \nleft invariant differential operator. \nFor $x_1, \\ldots, x_n \\in {\\mathfrak g}$ and  $D = x_1 \\cdots x_n$, it is defined by \n\\[ (D^L\\phi)(g) \n= \\frac{\\partial^n}{\\partial t_1 \\cdots \\partial t_n}\\Big|_{t_j = 0} \n \\phi(g \\exp(t_1x_1) \\cdots \\exp(t_n x_n)). \\] \n \nFor $\\xi \\in \\mathcal{H}$, the orbit map $U^\\xi \\: S^0_C \\to \\mathcal{H}$ is \nholomorphic  and $G$-equivariant with respect to left multiplications in the sense that \n\\begin{equation}\n  \\label{eq:leftcov}\nU^\\xi \\circ \\lambda_g = U(g) \\circ U^\\xi \\quad \\mbox{ for } \\quad g \\in G,\n\\lambda_g(h) = gh, \\end{equation}\nin particular $U^\\xi(S_C^0) \\subseteq \\mathcal{H}^\\infty$.\nIt follows that, for $D \\in \\mathcal{U}({\\mathfrak g})$, the function \n\\[ D^L U^\\xi \\: S_C^0 \\to \\mathcal{H}, \\] \nobtained by applying a left invariant differential operator, \nis also holomorphic and $G$-equivariant in the sense of \\eqref{eq:leftcov}, \nhence takes values in~$\\mathcal{H}^\\infty$. \n\nNext we observe that, for $\\eta \\in \\mathcal{H}^\\infty$ and \n$s \\in S_C^0$,  the relation \n\\[ \\langle U^\\xi(s), \\eta \\rangle = \\langle \\xi, U(s^*) \\eta \\rangle \\] \nleads for $D := x_1 \\cdots x_k$, $x_j \\in {\\mathfrak g}$, to \n\\begin{align*}\n\\langle (D^L U^\\xi)(s), \\eta \\rangle \n&= \\frac{\\partial^k}{\\partial t_1 \\cdots \\partial t_k}\\Big|_{t_1= \\cdots = t_k = 0} \n\\langle U^\\xi(s \\exp(t_1 x_1) \\cdots \\exp(t_k x_k)), \\eta \\rangle  \\\\\n&= \\frac{\\partial^k}{\\partial t_1 \\cdots \\partial t_k} \\Big|_{t_1= \\cdots = t_k = 0}  \n \\langle U(\\exp(t_1 x_1) \\cdots \\exp(t_k x_k))\\xi, U(s^*) \\eta \\rangle \\\\\n&=  \\langle \\dd U^{-\\infty}(D)\\xi, U(s^*) \\eta \\rangle \n=  \\langle U^{-\\infty}(s)\\dd U^{-\\infty}(D)\\xi, \\eta \\rangle.\n\\end{align*}\nHere we use that $U(s^*)\\eta \\in \\mathcal{H}^\\infty$ by Lemma~\\ref{lem:2.1} and that \n\\[ \\frac{\\partial^k}{\\partial t_1 \\cdots \\partial t_k} \n\\Big|_{t_1= \\cdots = t_k = 0}  \n  U(\\exp(t_1 x_1) \\cdots \\exp(t_k x_k))\\xi  \n= \\dd U^{-\\infty}(D) \\xi \\in \\mathcal{H}^{-\\infty} \\] \nholds in the weak-$*$-topology on $\\mathcal{H}^{-\\infty}$. \nFor $\\xi \\in \\mathcal{H}$, we thus obtain \n\\[ \\mathcal{H}^\\infty \\ni (D^L U^\\xi)(s) = U^{-\\infty}(s) \\dd U^{-\\infty}(D) \\xi.\\]\nAs $\\mathcal{H}^{-\\infty}$ is spanned by $\\dd U^{-\\infty}({\\mathfrak g})\\mathcal{H}$ \n(see Lemma~\\ref{lem:charsmooth}(b)), it follows that \n$U^{-\\infty}(s)\\mathcal{H}^{-\\infty} \\subseteq~\\mathcal{H}^\\infty$. \n\nTo see that the orbit maps \\eqref{eq:orbitmapint} \nare holomorphic, we have to show that the maps \n\\[  f\\: S_C^0 \\to \\mathcal{H}, \\quad\ns \\mapsto \\dd U(D) U^{-\\infty}(s) \\dd U^{-\\infty}(D')\\xi, \\qquad  \\xi \\in \\mathcal{H}, D, D' \\in \n\\mathcal{U}({\\mathfrak g}),\\] \nare holomorphic because $\\mathcal{H}^{-\\infty}$ is spanned by $\\dd U^{-\\infty}(\\mathcal{U}({\\mathfrak g}))\\mathcal{H}$ \n(Lemma~\\ref{lem:charsmooth}(b)). \nAs \\[ f = D^R (D')^L U^\\xi \\: S_C^0 \\to \\mathcal{H} \\] and $U^\\xi$ is holomorphic, \n$f$ is holomorphic as well. \n\\end{prf}\n\nThe following theorem shows that any distribution vector generates a \nsubrepresentation that can be realized in holomorphic functions on \n$S_C^0$ with distributional boundary values. \n\n\\begin{thm} {\\rm(Realization in holomorphic functions)} \n  \\mlabel{thm:2.5} Let $\\eta \\in \\mathcal{H}^{-\\infty}$. By \n\\[ j_\\eta^S \\: \\mathcal{H}^{-\\infty} \\to \\mathop{{\\rm Hol}}\\nolimits(S_C^0), \\quad \nj_\\eta^S(\\alpha)(s) := \\alpha(U^{-\\infty}(s^*)\\eta), \\] \nwe obtain a map which intertwines $U^{-\\infty}$ with the action of \n$S_C$ on $\\mathop{{\\rm Hol}}\\nolimits(S_C^0)$ by right translations. \nEvery function $j_\\eta^S(\\alpha) \\in \\mathop{{\\rm Hol}}\\nolimits(S_C^0)$ has the \ndistributional boundary value $j_\\eta^\\vee(\\alpha) \n:= j_\\eta(\\alpha)^\\vee\\in C^{-\\infty}(G)$ \nin the sense that \n\\begin{equation}\n  \\label{eq:ddag}\nj_\\eta^\\vee(\\alpha)(\\phi) \n= \\lim_{s \\to e} \\int_G \\overline{\\phi(g)} j_\\eta^S(\\alpha)(gs)\\, dg \n\\quad \\mbox{ for } \\quad \n\\phi \\in C^\\infty_c(G).  \n\\end{equation}\n\\end{thm}\n\n\\begin{prf} (a) \nAs $U^{-\\infty}(s)\\eta \\in \\mathcal{H}^\\infty$ for $s \\in S_C^0$ by \nLemma~\\ref{lem:9.14}, the function $j_\\eta^S(\\alpha) \\: S_C^0 \\to {\\mathbb C}$ \nis defined for each distribution vector $\\alpha$. \nThat it is holomorphic follows from the holomorphy of the orbit \nmaps \\eqref{eq:orbitmapint} in Lemma~\\ref{lem:9.14} \nand the antilinearity of $\\alpha$ on $\\mathcal{H}^\\infty$. \n\n\\nin (b) By definition (see \\eqref{eq:dist-inc2}), \nthe left hand side of \\eqref{eq:ddag} equals \n\\[ j_\\eta^\\vee(\\alpha)(\\phi)  = \\alpha(U^{-\\infty}(\\phi^\\vee)\\eta)\n\\quad \\mbox{ for } \\quad \n\\phi^\\vee(g) := \\Delta_G(g)^{-1} \\phi(g^{-1}).\\] \nUsing that \n\\[ \\int_G \\phi(g) U(g^{-1})\\, dg \n=   \\int_G \\phi^\\vee(g) U(g)\\, dg = U(\\phi^\\vee),\\] \nwe obtain for the integral on the right hand side of \\eqref{eq:ddag} \n\\[ \\int_G \\overline{\\phi(g)} \\alpha(U^{-\\infty}(s^*g^{-1})\\eta)\\, dg \n= \\alpha\\big(U^{-\\infty}(s^*) U^{-\\infty}(\\phi^\\vee)\\eta\\big) \n= (U^{-\\infty}(s)\\alpha)\\big(U^{-\\infty}(\\phi^\\vee)\\eta\\big).\\] \nEvaluating on the smooth vector $U^{-\\infty}(\\phi^\\vee)\\eta$, the assertion \nnow follows from the weak-$*$-continuity of the map \n$S_C \\to \\mathcal{H}^{-\\infty}, s \\mapsto U^{-\\infty}(s)\\alpha$ \n(Proposition~\\ref{prop:2.2}). \n\\end{prf}\n\n\\subsection{Real subspaces generated by distribution vectors} \n\\mlabel{subsec:2.2} \n\n\n\nIn this subsection we turn to real subspaces of $C$-positive \nunitary representations $(U,\\mathcal{H})$ of $G$. Our main result \nis the Reeh--Schlieder property (Theorem~\\ref{thm:sw64-gen-abs}). \nWe shall use the following terminology concerning real subspaces of \ncomplex Hilbert spaces, which is inspired by cyclic and separating \nvectors for Neumann algebras, as defined in  the introduction. \n\n\n\\begin{defn} Let ${\\sf H}$ be a closed real subspace of the complex Hilbert space~$\\mathcal{H}$.\n\n\\nin (a) We write \n\\begin{equation}\n  \\label{eq:sympdual}\n{\\sf H}' := \\{ \\xi \\in \\mathcal{H} \\: \n(\\forall \\eta \\in {\\sf H}) \\Im \\langle \\xi, \\eta \\rangle = 0\\} \n\\end{equation}\nfor its {\\it symplectic orthogonal space} and note that ${\\sf H} = {\\sf H}''$ follows \nfrom the closedness of ${\\sf H}$. \n\n\\nin (b) $\\mathcal{H}$ is said to be \n\\begin{itemize}\n\\item {\\it cyclic} if ${\\sf H} + i {\\sf H}$ is dense in $\\mathcal{H}$. \n\\item {\\it separating} if ${\\sf H} \\cap i {\\sf H} = \\{0\\}$. \nAs ${\\sf H} \\cap i {\\sf H} = ({\\sf H}' + i {\\sf H}')'$, this is equivalent to is ${\\sf H}'$ being cyclic. \n\\item {\\it standard} if it is cyclic and separating. \n\\end{itemize}\n\\end{defn}\n\n\\begin{defn} \\mlabel{def:HE}\nLet $(U,\\mathcal{H})$ be a unitary representation of $G$. \nIf ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ is a real linear subspace, then we consider \nfor an open subset $\\mathcal{O} \\subseteq G$ the subspaces \n\\begin{equation}\n  \\label{eq:he1}\n{\\sf H}_{\\sf E}(\\mathcal{O}) := \\overline{{\\rm span}_{\\mathbb R} \\big(U^{-\\infty}(C^\\infty_c(\\mathcal{O},{\\mathbb R})){\\sf E}\\big)} \n\\subseteq \\mathcal{H}_{\\sf E}(\\mathcal{O}) := \\overline{{\\rm span}_{\\mathbb C} \\big(U^{-\\infty}(C^\\infty_c(\\mathcal{O})){\\sf E}\\big)}\n\\subseteq \\mathcal{H}.\n\\end{equation}\nFor $\\eta \\in \\mathcal{H}^{-\\infty}$, we also put \n\\begin{equation}\n  \\label{eq:heta}\n {\\sf H}_\\eta(\\mathcal{O}) := {\\sf H}_{{\\mathbb R} \\eta}(\\mathcal{O}) \\quad \\mbox{ and } \\quad \n \\mathcal{H}_\\eta(\\mathcal{O}) := \\mathcal{H}_{{\\mathbb R} \\eta}(\\mathcal{O}).\n\\end{equation}\n By construction and \\eqref{eq:covl1}  in Appendix~\\ref{app:a}, we have\n \\begin{equation}\n   \\label{eq:hecov}\nU(g) {\\sf H}_{\\sf E}(\\mathcal{O}) = {\\sf H}_{\\sf E}(g\\mathcal{O}) \\quad \\mbox{ for } \\quad g \\in G, \n\\mathcal{O} \\subseteq G. \n \\end{equation}\n\\end{defn}\n\nIn this subsection we show that, if the cone $C_U$ has interior points, \nthen $\\mathcal{H}_\\eta(\\mathcal{O}) = \\mathcal{H}_\\eta(G)$ for every non-empty open subset \n$\\mathcal{O} \\subseteq G$ (Reeh--Schlieder property). \nIf $\\eta$ is cyclic, i.e., $\\mathcal{H}_\\eta(G) = \\mathcal{H}$, this means that \n${\\sf H}_\\eta(\\mathcal{O})$ is a cyclic real subspace. \n\n\n\\begin{lem} \\mlabel{lem:fragment} {\\rm(Fragmentation Lemma)} \nLet $\\emptyset\\not=\\mathcal{O} \\subseteq G$ be open.\nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] If $P \\subseteq G$ is a closed subgroup, then \n  \\begin{itemize}\n  \\item[\\rm(i)] every test function $\\phi \\in C^\\infty_c(\\mathcal{O} P,{\\mathbb R})$ is a finite sum of \ntest functions of the form \n\\[ \\psi \\circ \\rho_p \\: G \\to {\\mathbb C}, \\quad g \\mapsto \\psi(gp),\n\\quad \n\\psi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}), p \\in P.\\] \n  \\item[\\rm(ii)] every test function $\\phi \\in C^\\infty_c(P\\mathcal{O},{\\mathbb R})$ is a finite sum of \ntest functions of the form \n\\[ \\psi \\circ \\lambda_p \\: G \\to {\\mathbb C}, \\quad g \\mapsto \\psi(pg),\n\\quad \n\\psi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}), p \\in P.\\] \n  \\end{itemize}\n\\item[\\rm(b)] Every $\\phi \\in C^\\infty_c(G,{\\mathbb R})$ \nis a finite sum $\\sum_{j = 1}^n \\phi_j \\circ \\lambda_{g_j}$ \nwith $\\phi_j \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$ and $g_j \\in G$. \n\\end{itemize}\n\\end{lem}\n\n\\begin{prf} (a)(i) The family $(\\mathcal{O} p)_{p \\in P}$ is an open cover of the \ncompact subset $\\mathop{{\\rm supp}}\\nolimits(\\phi)$, so that there exist \n$p_1, \\ldots, p_n \\in P$ with \n\\[ \\mathop{{\\rm supp}}\\nolimits(\\phi) \\subseteq \\mathcal{O} p_1 \\cup \\cdots \\cup \\mathcal{O} p_n.\\] \nLet $\\chi_0, \\ldots, \\chi_n$ be a smooth partition of unity subordinated to the \nopen cover \n\\[ G \\setminus \\mathop{{\\rm supp}}\\nolimits(\\phi), \\quad \\mathcal{O} p_1, \\ldots, \\mathcal{O} p_n\\] \nof $G$. Then \n$\\phi = \\sum_{j = 1}^n \\phi_j$, where $\\phi_j := \\chi_j \\phi$ \nsatisfies $\\mathop{{\\rm supp}}\\nolimits(\\phi_j) \\subseteq \\mathcal{O} p_j$. \nThen $\\psi_j := \\phi_j \\circ \\rho_{p_j} \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$ and \n$\\phi = \\sum_{j= 1}^n \\psi_j \\circ \\rho_{p_j^{-1}}$. \n\n\\nin (a)(ii) and (b) are proved along the same lines. For (b), we use \nthe open cover $(g\\mathcal{O})_{g \\in G}$ of the group $G$. \n\\end{prf}\n\n\n\\begin{lem}  \\mlabel{lem:sw64-genb} \nLet $(U, \\mathcal{H})$ be a unitary representation of $G$, let \n${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ be a real linear subspace, \n$P \\subseteq G$ a closed subgroup \nand $\\emptyset\\not=\\mathcal{O} \\subseteq G$. \nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] ${\\sf H}_{\\sf E}(\\mathcal{O} P) = {\\sf H}_{\\sf E}(\\mathcal{O})$ if \n${\\sf E}$ is $P$-invariant. \n\\item[\\rm(b)] ${\\sf H}_{\\sf E}(P\\mathcal{O})$ is the closed real span of $U(P){\\sf H}_{\\sf E}(\\mathcal{O})$. \n\\item[\\rm(c)] The real subspace \nspanned by $U(G){\\sf H}_{\\sf E}(\\mathcal{O})$ is dense in ${\\sf H}_{\\sf E}(G)$. \n\\end{itemize}\n\\end{lem}\n\n\\begin{prf} (a) The inclusion \n${\\sf H}_{\\sf E}(\\mathcal{O}) \\subseteq {\\sf H}_{\\sf E}(\\mathcal{O} P)$ is trivial. \nConversely, \nfor $\\phi = \\psi \\circ \\rho_p$, $\\psi \\in C^\\infty_c(\\mathcal{O})$ and $p \\in P$, we \nobtain with \\eqref{eq:rightrel} in Appendix~\\ref{app:a}\n\\[ U^{-\\infty}(\\phi){\\sf E} \n= U^{-\\infty}(\\psi \\circ \\rho_p) {\\sf E}\n= \\Delta_G(p)^{-1} U^{-\\infty}(\\psi) U^{-\\infty}(p^{-1}){\\sf E}\n=  U^{-\\infty}(\\psi){\\sf E}\\subseteq {\\sf E}.\\] \nHence the assertion follows from Lemma~\\ref{lem:fragment}(a). \n\n\\nin (b)  From \\eqref{eq:hecov} we know  that \n$U(p) {\\sf H}_{\\sf E}(\\mathcal{O}) = {\\sf H}_{\\sf E}(p\\mathcal{O}) \\subseteq {\\sf H}_{\\sf E}(P\\mathcal{O})$ \nfor $p \\in P$. Now the assertion follows from Lemma~\\ref{lem:fragment}(b). \n\n\\nin (c) is an immediate consequence of (b), applied with $P = G$. \n\\end{prf}\n\n\nThe following lemma is of particular interest for semidirect products \nsuch as the Poincar\\'e group. \n\n\\begin{lem} \\mlabel{lem:semidir}\nLet $G= L \\ltimes_\\sigma N$ be a semidirect product group \nand $(U,\\mathcal{H})$ be a unitary representation of $G$. We write \n$U_N := U\\vert_N$ for the restriction to $N$. \nSuppose that ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}(U_N) \\subseteq \\mathcal{H}^{-\\infty}(U)$ \nis an $L$-invariant real subspace. Then we have for every non-empty open \n$\\sigma(L)$-invariant subset $\\mathcal{O} \\subseteq N$ the relation  \n\\[ {\\sf H}_{\\sf E}^N(\\mathcal{O}) \n:= \\overline{{\\rm span}_{\\mathbb R} U^{-\\infty}(C^\\infty_c(\\mathcal{O},{\\mathbb R})) {\\sf E}} \n= {\\sf H}_{\\sf E}(\\mathcal{O} L).\\] \n\\end{lem}\n\n\\begin{prf} First we show that \n${\\sf H}_{\\sf E}^N(\\mathcal{O}) \\subseteq  {\\sf H}_{\\sf E}(\\mathcal{O} L).$\nAs subsets of $G$, we have $\\mathcal{O} L = L \\mathcal{O}$, and the multiplication map \n$L \\times \\mathcal{O} \\to L\\mathcal{O}$ is a diffeomorphism onto an open subset of $G$. \nOn $L\\mathcal{O}$ we thus have test functions $\\phi \\in C^\\infty_c(L\\mathcal{O})$ of the \nform \n\\[ \\phi_n(l,x) = \\delta_n(l)\\psi(x), \\quad \\mbox{ where } \\quad \n\\psi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}), \\] \nand $(\\delta_n)_{n \\in {\\mathbb N}}$ is a \n$\\delta$-sequence in $C^\\infty_c(L,{\\mathbb R})$ \n(See Appendix~\\ref{subsec:app1}). \nA left invariant Haar measure of $G$ \nis given by \n\\[ d\\mu_G(l,n) = d\\mu_L(l) d\\mu_N(n), \\] \nwhere $\\mu_L$ and $\\mu_N$ are left invariant Haar measures on $L$ and \n$N$, respectively.  Then \n\\begin{equation}\n  \\label{eq:intprodform}\nU(\\phi_n) \n=  \\int_\\mathcal{O} \\int_L \\delta_n(l) U(l) \\psi(x) U(x)\\, d\\mu_L(l)\\, d\\mu_N(x) \n=   U(\\delta_n) U(\\psi) \n\\end{equation}\nleads for $\\eta \\in {\\sf E} \\subseteq \\mathcal{H}^{-\\infty}(U_N)$ to \n\\[ U^{-\\infty}(\\phi_n)\\eta = U(\\delta_n) U^{-\\infty}(\\psi) \\eta \\to \nU^{-\\infty}(\\psi) \\eta\\] \nfor $n \\to \\infty$. We conclude that \n${\\sf H}_{\\sf E}^N(\\mathcal{O}) \\subseteq {\\sf H}_{\\sf E}(\\mathcal{O} L)$. \n\nTo verify the converse inclusion, we recall \nthat the functions \n\\[ \\phi(l,x) = \\phi_L(l) \\phi_N(x) \\quad \\mbox{ where } \\quad \n\\phi_L \\in C^\\infty_c(L,{\\mathbb R}), \n\\phi_N \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}) \\] \nspan a dense subspace of $C^\\infty_c(L\\mathcal{O})$ \n(\\cite[Thm.~5.16]{Tr67}). In view of \\eqref{eq:intprodform}, \nit thus suffices to show that, \n\\[ U(\\phi_L) U^{-\\infty}(\\phi_N) {\\sf E} \\subseteq \nU(\\phi_L) {\\sf H}_{\\sf E}^N(\\mathcal{O}) \\subseteq {\\sf H}_{\\sf E}^N(\\mathcal{O}).\\] \nThe first inclusion is trivial, and the second one \nfollows from the $L$-invariance and the closedness of ${\\sf H}_{\\sf E}^N(\\mathcal{O})$, \nwhich in turn follows from the $\\sigma(L)$-invariance of $\\mathcal{O}$. \n\\end{prf}\n\n\\subsection{The Reeh--Schlieder property} \n\nThe following lemma is our key tool in the derivation of the Reeh--Schlieder \nproperty. \n\\begin{lem}  \\mlabel{lem:sw64-gen2} \nLet $(U, \\mathcal{H})$ be a unitary representation of the connected Lie \ngroup $G$ for which the positive cone $C_U$ \nhas interior points. If $\\xi, \\eta \\in \\mathcal{H}$ are such that the matrix coefficient \n\\[ U^{\\xi,\\eta} \\:  G\\to {\\mathbb C}, \\quad g \\mapsto \\langle \\xi, U(g) \\eta \\rangle \\] \nvanishes on an open subset of $G$, then $U^{\\xi,\\eta} =0$ on~$G$.\n\\end{lem} \n\n\n\\begin{prf} Passing to the quotient Lie group $G\/\\ker(U)$, we may \nw.l.o.g.\\ assume that $U$ is injective. \nThen the \nclosed convex cone $C_U\\subseteq {\\mathfrak g}$ is pointed, and by assumption \nit is also generating, so that it defines a closed complex Olshanski semigroup \n$S_{C_U} = G \\exp(iC_U)$ (Definition~\\ref{def:olshanski}), \nand the matrix coefficient \n$U^{\\xi,\\eta}$ extends to a continuous function on $S_{C_U}$ which is holomorphic on \nthe interior~$S_{C_U}^0 = G \\exp(i C_U^0)$, which is a complex manifold \n(Theorem~\\ref{thm:extens}). \n\nSuppose that $U^{\\xi,\\eta}$ vanishes on the non-empty open subset~$\\mathcal{O}$. \nReplacing $\\eta$ by $U(g_0)\\eta$ for some $g_0 \\in \\mathcal{O}$, we may assume that \n$e \\in \\mathcal{O}$. \nThe exponential function \n$\\exp \\: {\\mathfrak g} + i C_U \\to S_{C_U}$ \nis continuous, and holomorphic on ${\\mathfrak g} + i C_U^0$. \nTherefore the function $U^{\\xi,\\eta} \\circ \\exp \\: {\\mathfrak g} \\to {\\mathbb C}$ \nextends to a continuous function on the closed wedge \n${\\mathfrak g} + i C_U$, holomorphic on the interior and vanishing on \n$\\exp^{-1}(\\mathcal{O}) \\not=\\emptyset$. \nNow \\cite[Lemma~A.III.6]{Ne99} implies that \n$U^{\\xi,\\eta} \\circ \\exp = 0$. \nTherefore the regularity of the exponential \nfunction ${\\mathfrak g} + i C_U^0 \\to S_{C_U}^0$ near $0$ implies that \n$U^{\\xi,\\eta} = 0$ on $S_{C_U}$, and hence its restriction to $G$ vanishes. \n\\end{prf}\n\nThe following theorem is the main result of this section. \nIt is a key tool for geometric constructions of standard subspaces \nfor unitary representations satisfying a spectrum condition, \nsuch as $C \\subseteq C_U$. \n\n\n\\begin{thm}  \\mlabel{thm:sw64-gen-abs} \n{\\rm(Reeh--Schlieder property)} \nLet $\\emptyset\\not=\\mathcal{O}\\subseteq G$  be an open subset and \n$(U, \\mathcal{H})$ be a unitary representation of the connected Lie \ngroup $G$ for which the positive cone $C_U$ \nhas interior points. Then ${\\sf H}_{\\sf E}(\\mathcal{O})$ is total in $\\mathcal{H}_{\\sf E}(G)$ \nfor every real subspace ${\\sf E} \\subseteq  \\mathcal{H}^{-\\infty}$. \n\\end{thm}\n\n\\begin{prf}  Suppose that $\\xi \\in \\mathcal{H}_{\\sf E}(G)$ is orthogonal to \n${\\sf H}_{\\sf E}(\\mathcal{O})$ and let $\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$. \nThen $\\mathop{{\\rm supp}}\\nolimits(\\phi)$ is a compact subset of $\\mathcal{O}$, so that there exists an \nopen $e$-neighborhood $V \\subseteq G$ with $V \\mathop{{\\rm supp}}\\nolimits(\\phi) \\subseteq \\mathcal{O}$. \nFor $g \\in V$ and $\\eta \\in {\\sf E}$, we then have by \n\\eqref{eq:covl1} in Appendix~\\ref{app:a} \n\\[ \\xi \\bot U(g)U^{-\\infty}(\\phi) \\eta \n= U^{-\\infty}(\\lambda_g \\phi) \\eta \\quad \\mbox{ because } \\quad \n\\mathop{{\\rm supp}}\\nolimits(\\lambda_g\\phi) =g\\mathop{{\\rm supp}}\\nolimits(\\phi) \\subseteq \\mathcal{O}.\\] \nIf follows that, for $\\eta' := U^{-\\infty}(\\phi) \\eta\\in \\mathcal{H}$, \nthe continuous function \n\\[ U^{\\xi,\\eta'} \\: G \\to {\\mathbb C}, \\quad g \\mapsto \\langle \\xi, U(g) \\eta' \\rangle \\] \nvanishes on the identity neighborhood $V$, and by \nLemma~\\ref{lem:sw64-gen2} it vanishes on~$G$.\nWe conclude that $\\xi \\bot U(G) {\\sf H}_{\\sf E}(\\mathcal{O})$, so that \nLemma~\\ref{lem:sw64-genb} shows that $\\xi = 0$. \n\\end{prf}\n\n\\subsection{Support properties of corresponding distributions} \n\nAlthough we will not use them below, we record some immediate consequences \nof our discussion for supports of certain distributions. We expect these \nresults to become relevant in subsequent work. \nFor the abelian case, we refer to \\cite{NOO20} for more details. \n\nFrom Theorem~\\ref{thm:sw64-gen-abs} we immediately obtain \nthe following criterion for ${\\sf H}_{\\sf E}(\\mathcal{O})$ to be standard in \nterms of support properties of distributions. \n\n\\begin{cor} Let $\\mathcal{O}, \\mathcal{O}' \\subseteq G$ be two non-empty open subsets, let \n$(U,\\mathcal{H})$ be a unitary representation with $C_U^0\\not=\\emptyset$, \nlet $\\eta \\in  \\mathcal{H}^{-\\infty}$, and consider the distribution \n$D = j_\\eta(\\eta) \\in C^{-\\infty}(G)$ \n{\\rm(see \\eqref{eq:dist-inc} in Appendix~\\ref{app:a})}. \nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] ${\\sf H}_\\eta(\\mathcal{O}') \\subseteq {\\sf H}_\\eta(\\mathcal{O})'$ is equivalent to \n\\begin{equation}\n  \\label{eq:imdcond}\n\\Im D(\\psi^** \\phi) = 0 \\quad \\mbox{ for } \\quad \n\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R}), \\psi \\in C^\\infty_c(\\mathcal{O}',{\\mathbb R}). \n\\end{equation}\nIf this is the case, then ${\\sf H}_\\eta(\\mathcal{O})$ is a standard subspace of $\\mathcal{H}_\\eta(G)$. \n\n\\item[\\rm(b)] Suppose that $\\mathcal{O} \\subseteq G$ is an open subsemigroup with \n$e \\in \\overline{\\mathcal{O}}$. Then \n\\begin{equation}\n  \\label{eq:supprest}\n{\\sf H}_\\eta(\\mathcal{O}^{-1}) \\subseteq {\\sf H}_\\eta(\\mathcal{O})' \\quad \\Leftrightarrow \\quad \n \\mathop{{\\rm supp}}\\nolimits(\\Im D) \\subseteq G \\setminus \\mathcal{O}.\n\\end{equation}\n\\end{itemize}\n\\end{cor}\n\n\\begin{prf} (a) From Theorem~\\ref{thm:sw64-gen-abs}\nwe infer that ${\\sf H}_\\eta(\\mathcal{O})$ is total in $\\mathcal{H}_\\eta(G)$. \nWith the notation from \\eqref{eq:cHD}  and \\eqref{eq:scalproHd} \nin Appendix~\\ref{app:a}, we see that \n\\[ \\Im D(\\psi^* * \\phi) \n= \\Im \\langle \\phi * D, \\psi * D\\rangle_{\\mathcal{H}_D} \n= \\Im \\langle U^{-\\infty}(\\phi)\\eta, U^{-\\infty}(\\psi)\\eta \\rangle \\] \nvanishes  for all \n$\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$ and $\\psi \\in C^\\infty_c(\\mathcal{O}',{\\mathbb R})$ \nif and only if ${\\sf H}_\\eta(\\mathcal{O}') \\subseteq {\\sf H}_\\eta(\\mathcal{O})'.$ \nAs the real subspace ${\\sf H}_\\eta(\\mathcal{O}')$ is cyclic in $\\mathcal{H}_\\eta(G)$ \n(Theorem~\\ref{thm:sw64-gen-abs}), \nthe above condition implies that \nthe subspace ${\\sf H}_\\eta(\\mathcal{O})$ is also separating, hence standard. \n\n\\nin (b) For $\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$ and $\\psi \\in C^\\infty_c(\\mathcal{O}',{\\mathbb R})$, \nthe convolution product \n$\\psi^** \\phi$ is a test function on~$\\mathcal{O}$. \nUsing a $\\delta$-sequence in $C^\\infty_c(\\mathcal{O},{\\mathbb R})$, it follows that \n\\eqref{eq:imdcond} is equivalent to $\\Im(D)$ vanishing on \n$C^\\infty_c(\\mathcal{O},{\\mathbb R})$, which means that $\\mathop{{\\rm supp}}\\nolimits(\\Im D) \\subseteq G \\setminus \\mathcal{O}$.\n\\end{prf}\n\n\\begin{cor} Let $G$ be a connected Lie group, \n$D\\in C^{-\\infty}(G)$ be a positive definite distribution, \nand $(U_D,\\mathcal{H}_D)$ be the corresponding unitary representation of~$G$ \non the Hilbert subspace $\\mathcal{H}_D \\subseteq C^{-\\infty}(G)$. \nIf $C_U^0 \\not=\\emptyset$ then, for any non-empty open subset $\\mathcal{O} \\subseteq G$, we have: \n  \\begin{itemize}\n  \\item[\\rm(1)] The real subspace ${\\sf H}_D(\\mathcal{O}):= \n\\overline{C^\\infty_c(\\mathcal{O},{\\mathbb R}) * D}$ is cyclic in $\\mathcal{H}_D$. \n  \\item[\\rm(2)] If $\\mathcal{O}$ is an open subsemigroup with \n$e \\in \\overline{\\mathcal{O}}$ and $\\Im D$ vanishes on $G \\setminus \\mathcal{O}$, \nthen ${\\sf H}_D(\\mathcal{O})$ is a standard subspace of $\\mathcal{H}_D$ with \n${\\sf H}_D(\\mathcal{O}^{-1}) \\subseteq {\\sf H}_D(\\mathcal{O})'$. \n  \\end{itemize}\n\\end{cor}\n\n\\begin{lem} \\mlabel{lem:supp1} \nLet $\\eta \\in \\mathcal{H}^{-\\infty}$ and $\\mathcal{O} \\subseteq G$ be an open subset for which \n${\\sf H}_\\eta(\\mathcal{O})$ is total in $\\mathcal{H}$.\nIf $\\alpha \\in \\mathcal{H}^{-\\infty}$ is such that the distribution \n$j_\\eta(\\alpha)$ vanishes on the subset $V \\cdot \\mathcal{O}$, for an $e$-neighborhood $V$ \nin $G$, then $\\alpha =0$. \n\\end{lem}\n\n\\begin{prf} Let $(\\delta_n)_{n \\in {\\mathbb N}}$ be a $\\delta$-sequence in \n$C^\\infty_c(G,{\\mathbb R})$ and \n$\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$. \nThen $\\delta_n * \\phi \\in C^\\infty_c(V \\mathcal{O},{\\mathbb R})$ if \n$n$ is sufficiently large. We therefore have \n\\[ 0 \n= j_\\eta(\\alpha)(\\delta_n * \\phi) \n= \\alpha(U^{-\\infty}(\\delta_n * \\phi)\\eta) \n= \\alpha(U^{-\\infty}(\\delta_n) U^{-\\infty}(\\phi)\\eta) \n= \\langle U^{-\\infty}(\\phi)\\eta, U^{-\\infty}(\\delta_n^*)\\alpha \\rangle.\\] \nAs ${\\sf H}_\\eta(\\mathcal{O})$ is total, \n$U^{-\\infty}(\\delta_n)\\alpha = 0$ if $n$ is sufficiently large, \nand since this sequence converges to $\\alpha$ in $\\mathcal{H}^{-\\infty}$, \nwe obtain $\\alpha =0$. \n\\end{prf}\n\n\\begin{prop} \\mlabel{prop:distrubsupp} \n{\\rm(Full support property)} \nSuppose that \n$\\eta \\in \\mathcal{H}^{-\\infty}$ is such that \n${\\sf H}_\\eta(\\mathcal{O})$ is total in  $\\mathcal{H}_\\eta(G)$ for every \nnon-empty open subset $\\mathcal{O} \\subseteq G$.\nIf $\\alpha \\in \\mathcal{H}^{-\\infty}$ is such that $\\mathop{{\\rm supp}}\\nolimits(j_\\eta(\\alpha)) \\not= G$, \nthen $\\alpha = 0$. \n\\end{prop}\n\n\\begin{prf} If $\\mathop{{\\rm supp}}\\nolimits(j_\\eta(\\alpha)) \\not= G$, then \nthere exists a non-empty open subset $\\mathcal{O}_0 \\subseteq G$ on which \n$j_\\eta(\\alpha)$ vanishes. Let $\\mathcal{O} \\subseteq \\mathcal{O}_0$ be a non-empty \nopen relatively compact subset. Then there exists an \nidentity neighborhood $V \\subseteq G$ with $V \\mathcal{O} \\subseteq \\mathcal{O}_0$. \nAs ${\\sf H}_\\eta(\\mathcal{O})$ is total in $\\mathcal{H}$, Lemma~\\ref{lem:supp1} implies \nthat $\\alpha =0$.   \n\\end{prf}\n\nThe following corollary generalizes \nLemma~\\ref{lem:sw64-genb} to distributional matrix coefficients. \n\n\\begin{cor} For any unitary representation $(U,\\mathcal{H}_D)$ on a subspace \n$\\mathcal{H}_D \\subseteq C^{-\\infty}(G)$ for which $C_U^0 \\not=\\emptyset$, any \nnon-zero distribution $D' \\in \\mathcal{H}_D$ satisfies $\\mathop{{\\rm supp}}\\nolimits(D') = G$. \n\\end{cor}\n\n\\begin{prf} This follows by combining Theorem~\\ref{thm:sw64-gen-abs} \nwith Proposition~\\ref{prop:distrubsupp}. \n\\end{prf}\n\n\n\\section{$C$-positive representation \nof $3$-graded Lie groups} \n\\mlabel{sec:3}\n\nAfter the preparations in Section~\\ref{sec:2}, we now turn to standard \nsubspaces ${\\tt V} = {\\tt V}_{(h,\\tau_G,U)}$ specified by the BGL construction \nfor an antiunitary representation $(U,\\mathcal{H})$ of $G_\\tau \n= G \\rtimes \\{\\mathbf{1},\\tau_G\\}$ as in \\eqref{eq:bgl} by \n\\[ J_{\\tt V} = U(\\tau_G) \\quad \\mbox{ and } \\quad \n\\Delta_{{\\tt V}}^{-it\/2\\pi} = U(\\exp th) \\quad \\mbox{ for } \\quad t \\in {\\mathbb R}.\\] \nIn this section we assume that ${\\tt V}$ is {\\it regular} \nin the sense that the Lie wedge $\\L(S_{\\tt V})$ has interior \npoints. This  leads us to assumption (B4) below (\\cite[Thm.~4.4]{Ne19}).\n\nWe now consider the following setting which is a \nnon-abelian generalization of \\cite{NOO20}: \n\\begin{itemize}\n\\item[\\rm(B1)] $G$ is a connected Lie group with Lie algebra~${\\mathfrak g}$. \n\\item[\\rm(B2)] $\\tau_G \\in \\mathop{{\\rm Aut}}\\nolimits(G)$ is an involution and $h \\in {\\mathfrak g}$ \nis fixed by $\\tau = \\L(\\tau_G)\\in \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g})$. \n\\item[\\rm(B3)] $C \\subseteq {\\mathfrak g}$ is a pointed \nclosed convex cone invariant under $\\mathop{{\\rm Ad}}\\nolimits(G)$ and $-\\tau$, and \n${\\mathfrak g} = {\\mathfrak g}_C + {\\mathbb R} h$, where ${\\mathfrak g}_C = C - C$ is the ideal generated by~$C$. \nWe write $G_C \\trianglelefteq G$ for the connected normal \nsubgroup corresponding to ${\\mathfrak g}_C$ and $A := \\exp({\\mathbb R} h)$. \nIf $h \\in {\\mathfrak g}$, then $G = G_C$, and if $h \\not\\in{\\mathfrak g}$, then \n$G =  G_C A$. \n\\item[\\rm(B4)] $h$ is an {\\it Euler element}, i.e., \n$\\mathop{{\\rm ad}}\\nolimits h$ is diagonalizable with eigenvalues $\\{-1,0,1\\}$,  \nand $\\tau = e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h}\\vert_{{\\mathfrak g}}$. \n\\end{itemize}\n\nThen ${\\mathfrak g} = {\\mathfrak g}^{-1}\\oplus {\\mathfrak g}^0 \\oplus {\\mathfrak g}^1$, \nwhere $\\tau\\vert_{{\\mathfrak g}^j} = (-1)^j \\mathop{{\\rm id}}\\nolimits_{{\\mathfrak g}^j}$ and $h \\in {\\mathfrak g}^0 = {\\mathfrak g}^\\tau$ \ndefines the grading by ${\\mathfrak g}^j = \\ker(\\mathop{{\\rm ad}}\\nolimits h  - j \\mathbf{1})$. \nFurther, (B3\/4) imply that ${\\mathfrak q} := {\\mathfrak g}^{-\\tau}$ is generated by \nthe cone\n\\begin{equation}\n  \\label{eq:cq}\nC \\cap {\\mathfrak q} = C\\cap ({\\mathfrak g}_1 \\oplus {\\mathfrak g}_{-1}) = C_+ \\oplus - C_-\n\\end{equation}\n(cf.\\ \\cite[Lemma~3.1]{NOO20}). \n\nThe group $G_\\tau := G \\rtimes \\{\\mathbf{1},\\tau_G\\}$ has a natural grading \nand every  antiunitary representation $(U,\\mathcal{H})$ of $G_\\tau$ \ndefines a standard subspace ${\\tt V} = {\\tt V}_{(h,\\tau_G,U)}$ as in \\eqref{eq:bgl}  \n(\\cite{BGL02}, \\cite{NO17}). From \n\\cite[Thm.~3.4]{Ne19b} we know that, for $C = C_U$, \n\\begin{equation}\n  \\label{eq:sv2}\nS_{\\tt V} = \\exp(C_+) G_{{\\tt V}} \\exp(C_-) \n= G_{{\\tt V}} \\exp(C_+ + C_-), \n\\end{equation}\nwhere $\\L(G_{\\tt V}) = {\\mathfrak g}^0$ and $C_+ + C_-$ is a pointed generating cone in ${\\mathfrak q}$ \n(see (B3) and \\eqref{eq:cq}). \n\n\n\n\\begin{rem} If $C$ has interior points, then \n$\\Xi := G \\exp(i C^0) = S_C^0 \\subseteq S_C$ is a complex \nmanifold with the antiholomorphic involution \n\\[ \\tau_S(g \\exp(ix)) = \\tau_G(g) \\exp(-i \\tau(x)) \\] \nand the fixed point set \n\\[ S^c := G^{\\tau_G} \\exp(i (C^0 \\cap {\\mathfrak q})).\\] \nSince $C$ is invariant under $e^{{\\mathbb R} \\mathop{{\\rm ad}}\\nolimits h}$, we obtain for each $t \\in {\\mathbb R}$ \na unique automorphism $\\beta_t$ of $S_C$\nwhich induces the automorphism $e^{t\\mathop{{\\rm ad}}\\nolimits h}$ on the Lie wedge $\\L(S_C)$. \n\nNow $S_{\\tt V}^0$ plays the role of a \n``wedge  domain'' in the ``Shilov boundary'' $G$ of $S_C^0$, \non which we may ``realize'' the standard subspace~${\\tt V}$ \nas ${\\sf H}_{\\sf E}(S_{\\tt V}^0) \\subseteq \\mathcal{H}_{\\sf E}(G)$ for some \n${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$. \n\n\\nin (c) If $C$ is not generating in ${\\mathfrak g}$ and \n${\\mathfrak g} = {\\mathfrak g}_C + {\\mathbb R} h$, then $\\Xi := G_C \\exp(iC^0)$ \n(the open complex Olshanski semigroup associated to the pointed \ngenerating invariant cone $C \\subseteq {\\mathfrak g}_C$) \nis a complex manifold \nwith an ${\\mathbb R}^\\times$-action and we shall realize the representation \nin holomorphic functions on~$\\Xi$, with boundary value maps to~$G_C$. \nIn this case $S_{\\tt V} = (S_{\\tt V} \\cap G_C)A$ \nand we have the continuous inclusion \n\\[ \\mathcal{H}^\\infty(U) \\hookrightarrow  \\mathcal{H}^\\infty(U\\vert_{G_C}) \\] \nwhich dualizes by Lemma~\\ref{lem:charsmooth}(b) in the appendix \nto a linear injection \n\\begin{equation}\n  \\label{eq:disvecincl}\n \\mathcal{H}^{-\\infty}(U\\vert_{G_C})  = \\dd U^{-\\infty}(\\mathcal{U}({\\mathfrak g}_C)) \\mathcal{H} \\hookrightarrow \n\\dd U^{-\\infty}(\\mathcal{U}({\\mathfrak g})) \\mathcal{H} = \\mathcal{H}^{-\\infty}(U),  \n\\end{equation}\nand we have to work with elements \n$\\eta \\in \\mathcal{H}^{-\\infty}(U\\vert_{G_C})$. \n\\end{rem}\n\n\n\n\\subsection{Distribution vectors associated to standard subspaces} \n\\mlabel{subsec:3.1}\n\nIn this subsection we develop some results concerning the connection between \nstandard subspaces and distribution vectors. We shall assume (B1-4), that \n$(U,\\mathcal{H})$ is an antiunitary representation with \n$C = C_U$ pointed and that ${\\tt V} \\subseteq \\mathcal{H}$ is the standard subspace \nspecified by the triple $(h,\\tau_G,U)$. \nFor later reference, we note that \\eqref{eq:SE} implies \nthat, for the symplectic orthogonal space \n${\\tt V}'$ (see \\eqref{eq:sympdual}), \nwhich is the standard subspace specified by \nthe triple $(-h,\\tau_G, U)$, the endomorphism semigroup \n(cf.\\ \\eqref{eq:sv2}) is given by \n\\begin{equation}\n  \\label{eq:sv'}\n  S_{{\\tt V}'} = S_{{\\tt V}}^{-1}.\n\\end{equation}\n\n\\begin{defn} We write ${\\tt V}^\\infty := {\\tt V} \\cap \\mathcal{H}^\\infty$ for the subspace of smooth \nvectors contained in ${\\tt V}$ and \n\\[ {\\tt V}^{-\\infty} := \\{ \\alpha \\in \\mathcal{H}^{-\\infty} \\: \n\\Im\\alpha(({\\tt V}')^\\infty) = \\{0\\}\\},\\] \nwhere ${\\tt V}' = {\\tt V}^{\\bot_\\omega} = {\\tt V}_{(-h,\\tau_G, U)}$ \nis the symplectic orthogonal space of ${\\tt V}$ \n(see \\eqref{eq:sympdual}).\n\\end{defn}\n\n\\begin{lem}\n  \\mlabel{lem:vinfty} \nThe following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] ${\\tt V}^\\infty$ is dense in ${\\tt V}$ and \n$({\\tt V}')^\\infty$ is dense in ${\\tt V}'$. \n\\item[\\rm(b)] ${\\tt V}^{-\\infty}$ is the weak-$*$-closure of the real subspace \n${\\tt V}$ of $\\mathcal{H}^{-\\infty}$. \n\\item[\\rm(c)] ${\\tt V}^{-\\infty} = \\{ \\alpha \\in \\mathcal{H}^{-\\infty} \\: (\\forall \n\\phi \\in C^\\infty_c(S_{\\tt V}^0,{\\mathbb R})) \\, U^{-\\infty}(\\phi)\\alpha \\in {\\tt V} \\}$. \n\\item[\\rm(d)] ${\\tt V}^{-\\infty} \\cap i{\\tt V}^{-\\infty}= \\{0\\}$. \n\\item[\\rm(e)] ${\\tt V}^\\infty$ is $\\dd U({\\mathfrak g})$-invariant. \n\\item[\\rm(f)] ${\\tt V}^{-\\infty}$ is $\\dd U^{-\\infty}({\\mathfrak g})$-invariant. \n\\end{itemize}\n\\end{lem}\n\n\\begin{prf} (a) Let $(\\delta_n)_{n \\in {\\mathbb N}}$ be a $\\delta$-sequence in \n$C^\\infty_c(S_{\\tt V}^0,{\\mathbb R})$. \nFor $\\xi \\in {\\tt V}$, the sequence $U(\\delta_n)\\xi$ of smooth vectors is contained \nin ${\\tt V}$ and converges to~$\\xi$. We likewise conclude that \n$({\\tt V}')^\\infty$ is dense in ${\\tt V}'$. \n \n\\nin (b) An element $\\xi \\in \\mathcal{H}^\\infty$ annihilates the subspace ${\\tt V}$ of $\\mathcal{H}^{-\\infty}$ \nunder the real bilinear pairing \n\\[ \\mathcal{H}^{-\\infty} \\times \\mathcal{H}^\\infty \\to {\\mathbb R}, \\quad \n(\\alpha, \\xi) \\mapsto \\Re \\alpha(\\xi) \\] \nif and only if $\\Re \\langle \\xi, {\\tt V} \\rangle = \\{0\\}$, which is equivalent to \n$\\xi \\in (i {\\tt V}') \\cap \\mathcal{H}^\\infty = i ({\\tt V}')^\\infty$. \nTherefore the weak-$*$-closure of ${\\tt V}$ in $\\mathcal{H}^{-\\infty}$ is the annihilator \nof $i ({\\tt V}')^\\infty$, which consists of all elements $\\alpha$ satisfying \n$\\{0\\} = \\Re\\alpha(i ({\\tt V}')^\\infty)$, which is equivalent to \n$\\{0\\} = \\Im\\alpha(({\\tt V}')^\\infty)$, i.e., to $\\alpha \\in {\\tt V}^{-\\infty}$. \n\n\\nin (c) If $\\alpha \\in {\\tt V}^{-\\infty}$ and $\\phi \n \\in C^\\infty_c(S_{\\tt V}^0,{\\mathbb R})$, then any $\\xi \\in ({\\tt V}')^\\infty$ satisfies \n\\[ \\Im (U^{-\\infty}(\\phi)\\alpha)(\\xi) \n= \\int_G \\phi(g) \\Im (U^{-\\infty}(g)\\alpha)(\\xi)\\, dg  \n= \\int_G \\phi(g) \\Im \\alpha(U(g^{-1})\\xi)\\, dg = 0 \\] \nbecause, for $\\phi(g) \\not=0$, the relation $U(g^{-1})\\xi \\in ({\\tt V}')^\\infty$ \nfollows from $S_{{\\tt V}'} = S_{{\\tt V}}^{-1}$ (see \\eqref{eq:sv'}).\n\nSuppose, conversely, that $\\alpha \\in \\mathcal{H}^{-\\infty}$ is such that, for any \n$\\phi \\in C^\\infty_c(S_{{\\tt V}}^0,{\\mathbb R})$ the vector $U^{-\\infty}(\\phi)\\alpha$ is contained in ${\\tt V}$. \nAs $S_{\\tt V}$ has dense interior (\\cite[Thm.~4.4]{Ne19}),  \n$C^\\infty_c(S_{{\\tt V}}^0,{\\mathbb R})$ contains a $\\delta$-sequence $(\\delta_n)_{n \\in {\\mathbb N}}$. \nFor $\\xi \\in ({\\tt V}')^\\infty$, we then obtain \n\\[ \\Im \\alpha(\\xi) \n= \\lim_{n \\to \\infty} \\Im (U^{-\\infty}(\\delta_n) \\alpha)(\\xi) \n= \\lim_{n \\to \\infty} \\Im \\langle \\xi, U^{-\\infty}(\\delta_n) \\alpha \\rangle = 0.\\] \nThis means that $\\alpha \\in {\\tt V}^{-\\infty}$. \n\n\\nin (d) A distribution vector $\\alpha$ is contained in ${\\tt V}^{-\\infty}  \\cap i {\\tt V}^{-\\infty}$ \nif and only if $({\\tt V}')^\\infty = {\\tt V}' \\cap \\mathcal{H}^\\infty\\subseteq \\ker \\alpha$. \nLet $\\xi \\in {\\tt V}'$ and $(\\delta_n)_{n \\in {\\mathbb N}}$ be a $\\delta$-sequence of test functions \nsupported in the interior of $S_{{\\tt V}'} = \\tau_G(S_{\\tt V}) = S_{{\\tt V}}^{-1}$. \nThen $U(\\delta_n) \\xi \\in ({\\tt V}')^\\infty \\subseteq \\ker \\alpha$ implies that \n$U(\\delta_n)({\\tt V}' + i {\\tt V}') \\subseteq \\ker \\alpha$, and since \n$U(\\delta_n) \\: \\mathcal{H} \\to \\mathcal{H}^\\infty$ is continuous and ${\\tt V}'$ is standard, it follows that \n$U(\\delta_n) \\mathcal{H} \\subseteq \\ker \\alpha$. Now the assertion follows \nfrom $U(\\delta_n) v \\to v$ in $\\mathcal{H}^\\infty$ for every smooth vector~$v$, \na consequence of the continuity of the $G$-action on the Fr\\'echet space \n$\\mathcal{H}^\\infty$ (cf.\\ \\cite[Thm.~4.4]{Ne10}). \n\n\\nin (e) Let $\\xi \\in {\\tt V}^\\infty$ and $x \\in {\\mathfrak g}$. If $(\\delta_n)_{n \\in {\\mathbb N}}$ \nis a $\\delta$-sequence in $C^\\infty_c(S_{{\\tt V}}^0)$, then we have already observed \nunder (d) that $U(\\delta_n) \\xi \\to \\xi$ in the topology of $\\mathcal{H}^\\infty$. \nAs $\\dd U(x) U(\\delta_n) \\xi = U(x^R \\delta_n) \\xi \\in {\\tt V}$ \nby \\eqref{eq:derrep2} in Appendix~\\ref{app:a}, \npassing to the limit $n \\to \\infty$ \nimplies $\\dd U(x) \\xi \\in {\\tt V}^\\infty$. \n\n\\nin (f) For $\\alpha \\in {\\tt V}^{-\\infty}$, $\\xi \\in ({\\tt V}')^\\infty$ and $x \\in {\\mathfrak g}$, we have \n\\[ (\\dd U^{-\\infty}(x)\\alpha)(\\xi) \n=  -\\alpha(\\dd U(x)\\xi) \\in {\\mathbb R} \\] \nbecause (e), applied to the standard subspace ${\\tt V}'$, implies that \n$\\dd U(x) \\xi \\in({\\tt V}')^\\infty$. \n\\end{prf}\n\nSince we shall use it several times below, we recall the following \nimportant fact on standard subspaces \nfrom \\cite[Prop.~3.10]{Lo08}: \n\n\\begin{lem} \\mlabel{lem:lo-3.10} \nSuppose that  ${\\sf H}_1 \\subseteq {\\sf H}_2$ are standard subspaces of $\\mathcal{H}$. \nIf \n\\begin{itemize}\n\\item[\\rm(a)] $\\Delta_{{\\sf H}_2}^{it} {\\sf H}_1 = {\\sf H}_1$ for every $t \\in {\\mathbb R}$, or \n\\item[\\rm(b)] $\\Delta_{{\\sf H}_1}^{it} {\\sf H}_2 = {\\sf H}_2$ for every $t \\in {\\mathbb R}$, \n\\end{itemize}\nthen ${\\sf H}_1 = {\\sf H}_2$. \n\\end{lem}\n\n\\begin{prf} That (a) implies ${\\sf H}_1 = {\\sf H}_2$ follows from \n\\cite[Prop.~3.10]{Lo08}). From (b) we obtain by dualization \n${\\sf H}_2' \\subseteq {\\sf H}_1'$ with \n$\\Delta_{{\\sf H}_1'}^{it} {\\sf H}_2' = {\\sf H}_2'$ for $t \\in {\\mathbb R}$, so that \nwe obtain \n${\\sf H}_1' = {\\sf H}_2'$ with (a), hence ${\\sf H}_1 = {\\sf H}_2$ also holds in this case. \n\\end{prf}\n\n\nThe following theorem is an interesting tool \nto obtain nice descriptions of standard subspaces in concrete \nsituations. Here a subtle point is that we assume \n${\\sf E} \\subseteq {\\tt V}^{-\\infty}$, but we \nshall see in Lemma~\\ref{lem:key} below how this assumption can be verified \nin terms of the action of $A = \\exp({\\mathbb R} h)$ on $\\mathcal{H}^{-\\infty}$. \n\n\\begin{thm} \\mlabel{thm:3.6}\nFor a real subspace ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ invariant \nunder $A = \\exp({\\mathbb R} h)$ and satisfying \n$\\mathcal{H}_{\\sf E}(G) = \\mathcal{H}$, the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] If $\\emptyset\\not=\\mathcal{O}\\subseteq G$ is open, \nthen ${\\sf H}_{\\sf E}(\\mathcal{O})$ is cyclic {\\rm(Reeh--Schlieder property)}. \n\\item[\\rm(b)] If ${\\sf E} \\subseteq {\\tt V}^{-\\infty}$, then ${\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}.$ \n\\end{itemize}\n\\end{thm}\n\nIf $C= C_U$ has interior points, then $G = G_C$, and \nthe Reeh--Schlieder property \n(Theorem~\\ref{thm:sw64-gen-abs}) implies that ${\\sf H}_{\\sf E}(\\mathcal{O})$ is cyclic \nin $\\mathcal{H}_{\\sf E}(G)$ for every non-empty open subset $\\mathcal{O} \\subseteq G$. \nAssertion (a) above shows that this remains true under the weaker assumption~(B3).\n\n\\begin{prf}  (a) For $\\alpha \\in {\\sf E}$ and $\\phi \\in C^\\infty_c(\\mathcal{O},{\\mathbb R})$, \nthe support $\\mathop{{\\rm supp}}\\nolimits(\\phi)$ is a compact subset of $\\mathcal{O}$. Hence \nthere exists an $e$-neighborhood $V \\subseteq G_C$ with \n$V \\mathop{{\\rm supp}}\\nolimits(\\phi) \\subseteq \\mathcal{O}$. This shows that \n\\begin{equation}\n  \\label{eq:inc-abc}\n U(C^\\infty_c(V,{\\mathbb R})) U^{-\\infty}(\\phi)\\alpha \n\\subseteq U^{-\\infty}(C^\\infty_c(\\mathcal{O},{\\mathbb R}))\\alpha \\subseteq {\\sf H}_{\\sf E}(\\mathcal{O}).\n\\end{equation}\nApplying Theorem~\\ref{thm:sw64-gen-abs} to \n${\\sf E} = {\\mathbb R}\\eta$ for $\\eta := U^{-\\infty}(\\phi)\\alpha \\in \\mathcal{H} \\subseteq \\mathcal{H}^{-\\infty}$ \nand the open subset $V\\subseteq G_C$, we first see that \n${\\sf H}_{\\eta}(V)$ is total in $\\mathcal{H}_{\\eta}(G_C)$, \nand thus\n\\[ U(G_C)\\eta \\subseteq \\mathcal{H}_{\\eta}(G_C) =  \\mathcal{H}_\\eta(V) \n{\\buildrel{\\eqref{eq:inc-abc}} \\over \\subseteq} \\mathcal{H}_{\\sf E}(\\mathcal{O}).\\] \nThis implies that $\\mathcal{H}_{\\sf E}(\\mathcal{O})$ is $G_C$-invariant. \n\nLemma~\\ref{lem:fragment}(a)(ii) now implies that \n$\\mathcal{H}_{\\sf E}(\\mathcal{O}) = \\mathcal{H}_{\\sf E}(G_C\\mathcal{O}).$ \nAs $A$ is abelian and $G = G_CA$, the open subset $G_C\\mathcal{O} \\subseteq G$ \nis invariant under all inner automorphisms. \nTherefore the $A$-invariance of ${\\sf E}$ implies that \n$\\mathcal{H}_{\\sf E}(\\mathcal{O})$ is also $A$-invariant, hence $G$-invariant, \nand thus $\\mathcal{H}_{\\sf E}(\\mathcal{O}) = \\mathcal{H}_{\\sf E}(G)$ (Lemma~\\ref{lem:sw64-genb}). \n\n\\nin (b) First, Lemma~\\ref{lem:vinfty}(c) implies that \n${\\sf H}_{\\sf E}(S_{\\tt V}^0) \\subseteq {\\tt V}$, hence that ${\\sf H}_{\\sf E}(S_{\\tt V}^0)$ is separating. \nAs ${\\sf H}_{\\sf E}(S_{\\tt V}^0)$ is cyclic by (a), it is standard. \nThe invariance of ${\\sf E}$ under $U^{-\\infty}(A)$ and the invariance \nof $S_{\\tt V}^0$ under conjugation with $A$ entail that \n${\\sf H}_{\\sf E}(S_{\\tt V}^0)$ is invariant under $\\Delta_{\\tt V}^{i{\\mathbb R}} = U(A)$ \n(see \\eqref{eq:covl1} in the appendix). \nNow ${\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}$ follows from~Lemma~\\ref{lem:lo-3.10}.   \n\\end{prf}\n\nFor ${\\sf E} = {\\tt V}$, we have $\\mathcal{H}_{\\tt V}(G) = \\mathcal{H}$, and \nTheorem~\\ref{thm:3.6}(b) reduces to \nthe tautology ${\\sf H}_{\\tt V}(S_{\\tt V}^0) = {\\tt V}$, so that this theorem \nis most interesting if ${\\sf E}$ is small. Actually it is \nfinite-dimensional in many interesting situations (see Section~\\ref{sec:5}). \nIn particular, we would \nalso like to have that ${\\sf H}_{\\sf E}((S_{\\tt V}^0)^{-1}) = {\\sf H}_{{\\sf E}}(S_{{\\tt V}'}^0) \\subseteq {\\tt V}'$, \nbut this requires \n\\begin{equation}\n  \\label{eq:seinintersect}\n   {\\sf E} \\subseteq {\\tt V}^{-\\infty} \\cap ({\\tt V}')^{-\\infty}\n\\end{equation}\nby Lemma~\\ref{lem:vinfty}(c). \nThis is an interesting point because it may happen that the symplectic form \nis non-degenerate on ${\\tt V}$, i.e., ${\\tt V} \\cap {\\tt V}' = \\{0\\}$, \nbut that nevertheless \nthe subspaces ${\\tt V}^{-\\infty}$ and $({\\tt V}')^{-\\infty}$ have non-trivial \nintersection. \nAs we shall see in Subsection~\\ref{subsec:3.4}, \nthe irreducible antiunitary positive energy representation of \n$\\mathop{{\\rm Aff}}\\nolimits({\\mathbb R})$ provides an example where ${\\tt V} \\cap {\\tt V}' = \\{0\\}$ \nfollows from the fact that $\\Delta-\\mathbf{1}$ is injective, but in this \ncase ${\\tt V}^{-\\infty} \\cap ({\\tt V}')^{-\\infty}$ may contain $J_{\\tt V}$-fixed \n$\\Delta$-eigenvectors. Note that \n\\[ {\\tt V} \\cap {\\tt V}' = \\{ \\xi \\in \\mathcal{H} \\: \\Delta_{\\tt V} \\xi = \\xi = J_{\\tt V} \\xi \\} \n= \\ker(\\Delta_{\\tt V} - \\mathbf{1})^{J_{\\tt V}}\\] \nis contained in the $1$-eigenspace of $\\Delta_{\\tt V}$. \n\n\n\\subsection{Extending orbit maps of distribution vectors} \n\\mlabel{subsec:3.2}\n\nIn \\cite[Prop.~2.1]{NOO20} we have seen that an element \n$\\xi \\in \\mathcal{H}$ is contained in the standard subspace ${\\tt V}$ if and only \nif the orbit map \n\\[ \\alpha^\\xi \\:{\\mathbb R} \\to \\mathcal{H}, \\quad \n\\alpha^\\xi(t) :=  \\Delta_{\\tt V}^{-it\/2\\pi}\\xi \\] extends to a \ncontinuous map on the closed strip \n$\\overline{\\mathcal{S}_\\pi} \\to \\mathcal{H}$, holomorphic on $\\mathcal{S}_\\pi$,  \nand satisfying $\\alpha^\\xi(\\pi i) = J_{\\tt V} \\xi$. In this subsection \nwe consider a similar condition for distribution vectors. \nThis condition specifies a linear \nsubspace $\\mathcal{H}^{-\\infty}_{{\\rm ext},J}\\subseteq \\mathcal{H}^{-\\infty}$. We then show that this \nspace is invariant under \n$U^{-\\infty}(S_{\\tt V}^0)$ and $U^{-\\infty}(C^\\infty(S_{\\tt V}^0))$, which in turn \nleads to the important result that it is contained in $V^{-\\infty}$ \n(Lemma~\\ref{lem:key}). \n\n\n\\begin{defn}\nLet $\\eta \\in \\mathcal{H}^{-\\infty}$ be a distribution vector. We say that \n\\[ \\alpha^\\eta \\:  {\\mathbb R} \\to \\mathcal{H}^{-\\infty}, \\quad \n\\alpha^\\eta(t) := U^{-\\infty}(\\exp th) \\eta \\] \n{\\it extends} if there exists a weak-$*$-continuous extension \n$\\alpha^\\eta \\: \\overline{\\mathcal{S}_\\pi} \\to \\mathcal{H}^{-\\infty}$ which is weak-$*$-holomorphic \non $\\mathcal{S}_\\pi$. We write $\\mathcal{H}^{-\\infty}_{\\rm ext}$ for the linear subspace \nof distribution vectors with this property and \n$\\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ for the subspace \nof those $\\eta \\in \\mathcal{H}^{-\\infty}_{\\rm ext}$ \nfor which the extension to $\\overline{\\mathcal{S}_\\pi}$ satisfies \n$\\alpha^\\eta(\\pi i) = J_{\\tt V} \\eta$.   \n\\end{defn}\n\nNote that ${\\tt V} \\subseteq \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ follows from the continuity \nof the inclusion $\\mathcal{H} \\hookrightarrow \\mathcal{H}^{-\\infty}$ with respect to the weak-$*$-topology \n(\\cite[Prop.~2.1]{NOO20}). To address the invariance properties \nof $\\mathcal{H}^{-\\infty}_{\\rm ext}$, we start with an abstract lemma. \n\n\n\\begin{lem} \\mlabel{lem:cartes}\nLet $X$ be a locally compact space and \n$f \\: X \\to \\mathcal{H}^{-\\infty}$ be a weak-$*$-continuous map. \nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] $f^\\wedge \\: X \\times \\mathcal{H}^\\infty \\to {\\mathbb C}, f^\\wedge(x,\\xi) := f(x)(\\xi)$, \nis continuous. \n\\item[\\rm(b)] If, in addition, $X$ is a complex manifold and \n$f$ is antiholomorphic, then $\\overline{f^\\wedge}$ is holomorphic. \n\\end{itemize}\n\\end{lem}\n\n\\begin{prf} (a) Since the assertion is local in the first argument, we may w.l.o.g.\\ \nassume that $X$ is compact. Then $f(X)$ is pointwise bounded, \nhence equicontinuous because \n$\\mathcal{H}^\\infty$ is a Fr\\'echet space and therefore barrelled (\\cite[Thm.~33.2]{Tr67}). \nTo see that $f^\\wedge$ is continuous in $(x,\\xi)$, we note that \n\\[ f(x')(\\xi') - f(x)(\\xi) \n= f(x')(\\xi' - \\xi) + (f(x')-f(x))(\\xi). \\] \nAs $f(\\cdot)(\\xi)$ is continuous and $f(X)$ is equicontinuous, \nthe continuity of $f^\\wedge$ follows. \n\n\\nin (b) If, in addition, $X$ is a complex manifold \nand $f$ is antiholomorphic, \nthen $\\overline{f^\\wedge}$ is a continuous map on the product of two complex Fr\\'echet \nmanifolds which is holomorphic in each argument separately. \nHence the assertion follows from Hartogs' Theorem. Alternatively, \none may combine \\cite[Prop.~1.2.8]{GN} with \\cite[Thm.~2.1.12]{GN} \nto see that $\\overline{f^\\wedge}$ is holomorphic. \n\\end{prf}\n\nSince the semigroup $S_C$ is not a manifold because $C$ is a closed \ncone and $C$ may have empty interior, it is not at all clear what \nholomorphic  functions with values in $S_C$ should be. Here is a definition. \n\\begin{defn}  \\mlabel{def:holodisc} ($S_C$-valued holomorphic functions) \nLet \n\\[ q_S \\: \\tilde S_C \\to \\tilde G_{\\mathbb C}, \\quad \nq_S(g \\exp(ix)) = \\eta_{\\tilde G}(g) \\exp_{\\tilde G_{\\mathbb C}}(ix), \\quad  g\\in \\tilde G, \nx \\in C,\\] \nwhere $\\eta_{\\tilde G} \\: \\tilde G \\to \\tilde G_{\\mathbb C}$ \nis the universal homomorphism for which \n$\\L(\\eta_{\\tilde G}) \\: {\\mathfrak g} \\to {\\mathfrak g}_{\\mathbb C}$ is the inclusion, and $\\tilde G_{\\mathbb C}$ is a \n$1$-connected Lie group with Lie algebra~${\\mathfrak g}_{\\mathbb C}$ \n(cf.\\ Definition~\\ref{def:olshanski}).\nIf $M$ is a complex manifold, then we call a continuous map \n$f \\: M \\to \\tilde S_C$ {\\it holomorphic} if the composition \n\\break  $q_S \\circ f \\: M \\to \\tilde G_{\\mathbb C}$ is holomorphic. \n\nA map $f \\: M \\to S_C$ is called {\\it holomorphic}, if, for every \n$1$-connected open subset $U \\subseteq M$ a lift \n$\\tilde f_U \\: U \\to \\tilde S_C$ of $f\\vert_U \\: U \\to S_C$ \n(which exists if $U$ is $1$-connected) is holomorphic. \n\\end{defn} \n\n\n\\begin{lem} \\mlabel{lem:betaact} \nFor $s \\in S_C$, we write \n\\[ \\beta^s(t) = \\exp(th)s\\exp(-th) \\] \nfor the orbit map under the conjugation action of the \nmodular one-parameter group. \nFor every $s \\in S_{\\tt V}$, this orbit map extends to a continuous map \n$\\beta^s \\: \\overline{\\mathcal{S}_\\pi}\\to S_C$ on the closed strip \n$\\overline{\\mathcal{S}_\\pi}$ which is holomorphic on $\\mathcal{S}_\\pi$ \nin the sense of {\\rm Definition~\\ref{def:holodisc}}. The so obtained map \n\\begin{equation}\n  \\label{eq:contact}\n \\overline{\\mathcal{S}_\\pi} \\times S_{{\\tt V}} \\to S_C, \\quad \n(z, s) \\mapsto \\beta^s(z)\n\\end{equation}\nis continuous. \nIn addition, \n$\\beta^s(\\mathcal{S}_\\pi) \\subseteq S_C^\\circ := G \\exp(i C^\\circ)$ for $s \\in S_{\\tt V}^0$, \nwhere $C^\\circ$ denotes the interior of $C$ in its span ${\\mathfrak g}_C = C-C$.\n\\end{lem}\n\n\\begin{prf} For $s = g\\exp(x_1 + x_{-1})\\in S_{\\tt V}$ with $g \\in G_{\\tt V}$, \n$x_{\\pm 1} \\in C_\\pm$, we have \n\\begin{equation}\n  \\label{eq:betarel1}\n\\beta^s(z) = g\\exp(e^z x_1 + e^{-z} x_{-1}). \n\\end{equation}\nIf $s \\in S_{\\tt V}^0$, i.e., $x_{\\pm 1} \\in C_\\pm^0$, and \n$z \\in \\mathcal{S}_\\pi$, then \n$\\beta^s(z) \\in S_C^\\circ$ because, for $z = a + i b$, $0 < b <  \\pi$, we have  \n\\[ \\Im(e^z x_1 + e^{-z} x_{-1}) \n=\\sin(b)(e^a x_1 - e^{-a} x_{-1}) \\in C_{\\mathfrak q}^0 \\subseteq C^\\circ.\\]\nIn particular, we see that $\\beta^s(z) \\in S_C$, and \\eqref{eq:betarel1} shows that \nthe map \\eqref{eq:contact} is continuous. \nIt also shows that all maps $\\beta^s \\: \\mathcal{S}_\\pi \\to S_C$, \n$s \\in S_{\\tt V}$, are holomorphic in the sense of {\\rm Definition~\\ref{def:holodisc}}.\n\\end{prf}\n\n\n\\begin{lem} \\mlabel{lem:hinftyext}\nThe subspaces $\\mathcal{H}^{-\\infty}_{\\rm ext}$ and $\\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ \nare invariant under $U^{-\\infty}(S_{\\tt V}^0)$ and the algebra \n$U^{-\\infty}(C^\\infty_c(S_{\\tt V}^0,{\\mathbb R}))$. \n\\end{lem}\n\n\\begin{prf} {\\bf Step 1:} $U^{-\\infty}(S_{\\tt V}^0)\\mathcal{H}^{-\\infty}_{\\rm ext}\\subseteq \\mathcal{H}^{-\\infty}_{\\rm ext}$. \\\\\nLet $\\eta \\in \\mathcal{H}^{-\\infty}_{\\rm ext}$. \nFor $t \\in {\\mathbb R}$ and $g \\in S_{\\tt V}$, we have \n\\[ \\alpha^{U^{-\\infty}(g)\\eta}(t) \n= U^{-\\infty}(\\exp th) U^{-\\infty}(g) \\eta \n= U^{-\\infty}(\\beta^g(t)) U^{-\\infty}(\\exp th) \\eta \n= U^{-\\infty}(\\beta^g(t)) \\alpha^\\eta(t).\\] \nBy Lemma~\\ref{lem:betaact} and Proposition~\\ref{prop:2.2}, \nthis curve extends to the function \n\\[\\overline{\\mathcal{S}_\\pi} \\to \\mathcal{H}^{-\\infty}, \\quad \nz \\mapsto F(z,g) := U^{-\\infty}(\\beta^g(z)) \\alpha^\\eta(z).\\] \n\n\\nin {\\bf Claim 1:} \nThe map $F \\: \\overline{\\mathcal{S}_\\pi} \\times S_{\\tt V} \\to \\mathcal{H}^{-\\infty}, \nF(z,g) = U^{-\\infty}(\\beta^g(z)) \\alpha^\\eta(z)$ is \nweak-$*$-continuous. \\\\ \nAs $\\alpha^\\eta$ is weak-$*$-continuous on the \nlocally compact space $\\overline{\\mathcal{S}_\\pi}$, the weak-$*$-continuity of \n$F$ follows from \nLemma~\\ref{lem:cartes}(a) and the continuity of the maps \n\\begin{equation}\n  \\label{eq:16}\n\\overline{\\mathcal{S}_\\pi} \\times S_{\\tt V} \\to \\mathcal{H}^\\infty, \\quad \n(z,g) \\mapsto  U(\\beta^g(z)^*)\\xi,  \\qquad \\xi \\in \\mathcal{H}^\\infty,\n\\end{equation}\nwhich in turn follows from \nLemma~\\ref{lem:2.1} and Lemma~\\ref{lem:betaact}. \n\n\\nin {\\bf Claim 2:} For $g \\in S_{\\tt V}^0$, the map \n$F(\\cdot,g)$ is  weak-$*$-holomorphic on $\\mathcal{S}_\\pi$. \\\\\nAs $\\alpha^\\eta$ is weak-$*$-holomorphic on $\\mathcal{S}_\\pi$, \nClaim 2 follows from Lemma~\\ref{lem:cartes}(b) and the antiholomorphy of the maps \n\\[ \\mathcal{S}_\\pi \\to \\mathcal{H}^{\\infty}, \\quad z \\mapsto U(\\beta^g(z)^*)\\xi \n= U(\\beta^{g^{-1}}(\\overline z))\\xi,  \\qquad \\xi \\in \\mathcal{H}^\\infty,\\] \nwhich in turn follows from Lemma~\\ref{lem:2.1} \nand the holomorphy of $\\beta^g$ on $\\mathcal{S}_\\pi$. \\\\\n\\nin Claims 1 and 2 imply that, for $g \\in S_{\\tt V}^0$, we have \n $U^{-\\infty}(g)\\eta \\in \\mathcal{H}^{-\\infty}_{\\rm ext}$ with \n$\\alpha^{U^{-\\infty}(g)\\eta} = F(\\cdot,g)$. \n\n\\nin {\\bf Step 2:} $U^{-\\infty}(C^\\infty_c(S_{\\tt V}^0,{\\mathbb R}))\\mathcal{H}^{-\\infty}_{\\rm ext}\\subseteq \\mathcal{H}^{-\\infty}_{\\rm ext}$. \\\\\nFor $\\psi \\in C^\\infty_c(S_{\\tt V}^0,{\\mathbb R})$, we consider the map \n\\[ f\\:  \\overline{\\mathcal{S}_\\pi} \\to \\mathcal{H}^{-\\infty}, \n\\quad \nf(z) :=  \\int_{\\mathop{{\\rm supp}}\\nolimits(\\psi)} \\psi(g) F(z,g)\\, dg\n= \\int_{S_{\\tt V}} \\psi(g) U^{-\\infty}(\\beta^g(z))\\alpha^\\eta(z)\\, dg. \\] \nThat $f$ is weak-$*$-continuous follows from Claim 1 and \n\\cite[Lemma~1.1.11]{GN}. That \nit is weak-$*$-holomorphic on~$\\mathcal{S}_\\pi$ \nlikewise follows from Claim 2 with~\\cite[Lemma~1.3.15]{GN}. \nThis shows that $U^{-\\infty}(\\psi)\\eta \\in \\mathcal{H}^{-\\infty}_{\\rm ext}$ \nbecause $f = \\alpha^{U^{-\\infty}(\\psi)\\eta}$. \n\n\\nin {\\bf Step 3:} Invariance of $\\mathcal{H}^{-\\infty}_{{\\rm ext},J}$. \\\\\nWe now assume that $\\eta\\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$, so that \n$\\alpha^\\eta(\\pi i) = J_{\\tt V} \\eta$. Then \n\\begin{equation}\n  \\label{eq:17}\n\\alpha^{U^{-\\infty}(g)\\eta}(\\pi i) \n= F(\\pi i,g) \n= U^{-\\infty}(\\beta^g(\\pi i)) \\alpha^\\eta(\\pi i)\n= U^{-\\infty}(\\tau_G(g)) J_{\\tt V} \\eta \n= J_{\\tt V} U^{-\\infty}(g) \\eta \n\\end{equation}\nshows that  $U^{-\\infty}(g)\\eta \\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$. \nWe further obtain with \\eqref{eq:17} \n\\begin{align*}\n \\alpha^{U^{-\\infty}(\\psi)\\eta}(\\pi i) \n&= f(\\pi i) \n= \\int_{S_{\\tt V}} \\psi(g) U^{-\\infty}(\\beta^g(\\pi i))\\alpha^\\eta(\\pi i)\\, dg\\\\\n&= \\int_{S_{\\tt V}} \\psi(g) J_{\\tt V} U^{-\\infty}(g) \\eta\\, dg \n= J_{\\tt V} \\int_{S_{\\tt V}} \\psi(g)  U^{-\\infty}(g) \\eta\\, dg \n= J_{\\tt V} U^{-\\infty}(\\psi)\\eta.\n\\qedhere\\end{align*}\n\\end{prf}\n\nThe following technical lemma is of key importance in this section. It provides a \nsufficient condition for elements $\\eta \\in \\mathcal{H}$ to be contained \nin ${\\tt V}$ in terms of a rather weak holomorphic extension \nrequirement on the orbit map~$\\alpha^\\eta$. \nCombined with Lemma~\\ref{lem:hinftyext}, its helps us to construct \nelements $U^{-\\infty}(\\phi)\\eta \\in {\\tt V}$. \n\n\\begin{lem} \\mlabel{lem:key-abstract}\nLet $\\mathcal{D} \\subseteq \\mathcal{H}$ be a dense complex subspace, so that \n$v \\mapsto \\langle \\cdot, v \\rangle$ injects $\\mathcal{H}$ into the space \n$\\mathcal{D}^\\sharp$ of antilinear functionals $\\mathcal{D} \\to {\\mathbb C}$. \nLet ${\\tt V} \\subseteq \\mathcal{H}$ be a standard subspace for which $\\mathcal{D}$ is invariant \nunder $J_{\\tt V}$ and $\\Delta_{\\tt V}^{i{\\mathbb R}}$. \nSuppose that $\\eta \\in \\mathcal{H}$ is such that \nthe orbit map \n\\[ \\alpha^\\eta \\: {\\mathbb R}\\to \\mathcal{D}^\\sharp, \\quad t \\mapsto \n\\eta \\circ \\Delta_{\\tt V}^{it\/2\\pi} \\] extends to a map \n$\\alpha^\\eta \\: \\overline{\\mathcal{S}_\\pi} \\to \\mathcal{D}^\\sharp$ which is \npointwise continuous,  pointwise holomorphic on $\\mathcal{S}_\\pi$, and \nsatisfies $\\alpha^\\eta(\\pi i) = J_{\\tt V} \\eta$, \nfor $(J_{\\tt V} \\eta)(\\xi) := \\overline{\\eta(J_{\\tt V} \\xi)}$. \nThen $\\langle \\xi, \\eta \\rangle \\in {\\mathbb R}$ for every $\\xi \\in \\mathcal{D} \\cap {\\tt V}'$, \nand if $\\mathcal{D} \\cap {\\tt V}'$ is dense in ${\\tt V}'$, then $\\eta \\in {\\tt V}$. \n\\end{lem}\n\n\\begin{prf} Let $\\xi \\in \\mathcal{D} \\cap {\\tt V}'$. \nBy assumption, we have a continuous function \n\\[ f \\: \\overline{\\mathcal{S}_\\pi} \\to {\\mathbb C}, \\quad \nf(z) = \\alpha^\\eta(z)(\\xi) \\] \nwhich is holomorphic on $\\mathcal{S}_\\pi$. This function satisfies for $t \\in {\\mathbb R}$ \n\\begin{equation}\n  \\label{eq:s1}\nf(t + \\pi i) \n= \\alpha^\\eta(t + \\pi i)(\\xi) \n= \\alpha^\\eta(\\pi i)(\\Delta_{\\tt V}^{it\/2\\pi}\\xi) \n= (J_{\\tt V}\\eta)(\\Delta_{\\tt V}^{it\/2\\pi}\\xi) \n= \\overline{\\eta(\\Delta_{\\tt V}^{it\/2\\pi}J_{\\tt V}\\xi)}.\n\\end{equation}\nAs $\\xi \\in {\\tt V}'$, we also have for $t \\in {\\mathbb R}$ \n\\[ f(t) = \\alpha^\\eta(t)(\\xi) \n= \\langle \\Delta_{{\\tt V}}^{it\/2\\pi}\\xi, \\eta \\rangle,\\] \nand this function extends to a continuous function on \n$-\\overline{\\mathcal{S}_\\pi}$, holomorphic on $-\\mathcal{S}_\\pi$, given by \n\\[ f(z) =  \\langle \\Delta_{\\tt V}^{i \\overline  z\/2\\pi} \\xi, \\eta \\rangle.\\] \nWe thus obtain a continuous function \n\\[ f \\:  \\overline{\\mathcal{S}_{-\\pi,\\pi}} = \\{ z \\in {\\mathbb C} \\: -\\pi \\leq \\Im z \\leq \\pi \\} \\to {\\mathbb C}, \\] \nholomorphic on the complement of the real line in the interior \n$\\mathcal{S}_{-\\pi,\\pi}$, so that Morera's Theorem implies that it is \nholomorphic on~$\\mathcal{S}_{-\\pi,\\pi}$. On the lower boundary we have\n\\[   f(t - \\pi i) \n= \\langle \\Delta_{\\tt V}^{it\/2\\pi} \\Delta_{\\tt V}^{-1\/2}\\xi, \\eta \\rangle \n= \\langle \\Delta_{\\tt V}^{it\/2\\pi} J_{\\tt V} \\xi, \\eta \\rangle \n= \\eta(\\Delta_{\\tt V}^{it\/2\\pi} J_{\\tt V} \\xi) = \\overline{f(t + \\pi i)}.\\] \nTherefore the function \n\\[ F \\: \\{ z \\in {\\mathbb C} \\: -\\pi \\leq \\Im z \\leq \\pi \\} \\to {\\mathbb C}, \\quad \nF(z) := f(z) - \\overline{f(\\overline z)} \\] \nis continuous, holomorphic on the interior, and vanishes on the line \n${\\mathbb R} + \\pi i$. This implies that $F = 0$, and evaluating on the real line shows that \n$f({\\mathbb R}) \\subseteq {\\mathbb R}$. For $t = 0$, we obtain in particular \n$f(0) = \\langle \\xi, \\eta \\rangle \\in {\\mathbb R}$. \n\nIf $\\mathcal{D} \\cap {\\tt V}'$ is dense in ${\\tt V}'$, then this further leads to \n$\\eta \\in ({\\tt V}')' = {\\tt V}$. \n\\end{prf}\n\n\\begin{lem} \\mlabel{lem:key}\n$\\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\subseteq {\\tt V}^{-\\infty}$. \n\\end{lem}\n\n\\begin{prf} First we show that \n  \\begin{equation}\n    \\label{eq:step1}\n\\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\cap \\mathcal{H} \\subseteq {\\tt V}.\n  \\end{equation}\nWe apply Lemma~\\ref{lem:key-abstract} \nwith $\\mathcal{D} = \\mathcal{H}^\\infty$ and $\\eta \\in \\mathcal{H} \\cap \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$. \nAs ${\\tt V}' \\cap \\mathcal{H}^\\infty$ is dense in ${\\tt V}'$ by \nLemma~\\ref{lem:vinfty}(a), we obtain $\\eta \\in {\\tt V}$. \nThis proves \\eqref{eq:step1}. \n\nFor $\\eta \\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ and \n$\\psi \\in C^\\infty_c(S_{\\tt V}^0,{\\mathbb R})$, Lemma~\\ref{lem:hinftyext} \nimplies that \n$U^{-\\infty}(\\psi) \\eta \\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$, \nand since this is actually an element of $\\mathcal{H}$, \n\\eqref{eq:step1} shows that $U^{-\\infty}(\\psi) \\eta \\in {\\tt V}$. \nNow the assertion follows from Lemma~\\ref{lem:vinfty}(c). \n\\end{prf}\n\n\\begin{prop} Let $F \\subseteq \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ be a real linear subspace \nwhich is $G$-cyclic in the sense that $\\mathcal{H}_{\\sf F}(G) = \\mathcal{H}$, \nlet $A = \\exp({\\mathbb R} h)$, and \n\\[ {\\sf E} := {\\rm span}_{\\mathbb R} (U^{-\\infty}(A){\\sf F}) \\subseteq \\mathcal{H}^{-\\infty}.\\] \nThen \n\\[ {\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}.\\] \n\\end{prop}\n\n\\begin{prf} Since the subspace $\\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ \nis $A$-invariant, \nwe have ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\subseteq {\\tt V}^{-\\infty}$ \n(Lemma~\\ref{lem:key}). \nAs ${\\sf F}$ is $G$-cyclic, $\\mathcal{H}_{\\sf E}(G) \\supseteq \\mathcal{H}_{\\sf F}(G) = \\mathcal{H}$, \nso that Theorem~\\ref{thm:3.6}(b) implies that ${{\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}}$. \n\\end{prf}\n\n\\begin{rem} \\mlabel{rem:3.12} \n(a) If $\\eta \\in \\mathcal{H}^{-\\infty}$ is contained \nin a finite dimensional complex $A$-invariant subspace \n$\\mathcal{K} \\subseteq \\mathcal{H}^{-\\infty}$, \nthen the representation of $A$ on $\\mathcal{K}$ is continuous because \nit is continuous on its dual space $\\mathcal{H}^{\\infty}\/\\mathcal{K}^\\bot$. \nHence it extends to a holomorphic representation $\\rho$ of $A_{\\mathbb C} \\cong {\\mathbb C}$ \non the finite dimensional complex vector space $\\mathcal{K}$, and therefore \n$\\mathcal{K} \\subseteq \\mathcal{H}^{-\\infty}_{\\rm ext}$. \n\nIf, in addition,  $J\\mathcal{K} \\subseteq \\mathcal{K}$ and $J_\\mathcal{K} := J\\vert_{\\mathcal{K}}$, then \nthe real subspace \n\\[ {\\sf E} := \\mathop{{\\rm Fix}}\\nolimits(J_\\mathcal{K} \\rho(\\exp(\\pi i h))) \\subseteq \\mathcal{H}_{{\\rm ext},J} \\subseteq {\\tt V}^{-\\infty}\\]\nof $\\mathcal{K}$ satisfies \n\\[ {\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}.\\]\n\n\\nin (b) Below we shall need the following more general fact. \nWe assume that $(\\rho,\\mathcal{K})$ is a norm-continuous representation \nof $A\\cong {\\mathbb R}$ on the Hilbert space $\\mathcal{K}$ by symmetric operators \nand that we have \na continuous $A$-equivariant map $\\eta \\: \\mathcal{H}^\\infty \\to \\mathcal{K}$ \nwith dense range. \nThe adjoint map $\\eta^* \\: \\mathcal{K} \\to \\mathcal{H}^{-\\infty},\n\\eta^*(\\xi)(v) := \\langle \\eta(v), \\xi \\rangle$ defines a weak-$*$ continuous \nequivariant embedding. \nSince $\\eta^*$ is $A$-equivariant and $\\rho$ extends to a \nholomorphic representation of $A_{\\mathbb C}$ on $\\mathcal{K}$, we then have \n$\\eta^*(\\mathcal{K}) \\subseteq \\mathcal{H}^{-\\infty}_{\\rm ext}$. \n\\end{rem}\n\n\n\\begin{rem}\nWe assume that $\\ker(U)$ is discrete. \nClearly, regularity of ${\\tt V}$, i.e., that $\\L(S_{\\tt V})$ spans ${\\mathfrak g}$, \nimplies $e \\in \\overline{S_{\\tt V}^0}$, but the converse is not \nclear. In view of the Germ Theorem (\\cite[Thm.~4.1]{Ne19}), \n$e \\in \\overline{S_{\\tt V}^0}$ is equivalent to $e \\in \\overline{S_{C,{\\rm inv}}^0}$, \nwhere \n\\[ S_{C, {\\rm inv}} = \\{ g \\in G \\:  \n(\\forall z \\in \\overline{\\mathcal{S}_\\pi})\\ \\beta^g(z) \\in G \\exp(iC) \\}.\\] \nFrom \\cite[Lemma~5.4(ii)]{Ne19} it follows that $e \\in \\overline{S_{C,{\\rm inv}}^0}$ \nimplies that $\\tau = e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h}\\vert_{\\mathfrak g}$, hence \n$e^{2\\pi i \\mathop{{\\rm ad}}\\nolimits h} = \\mathop{{\\rm id}}\\nolimits_{{\\mathfrak g}_{\\mathbb C}}$, so that  $\\mathop{{\\rm ad}}\\nolimits h$ is diagonalizable \nwith integral eigenvalues. Presently we do not know how to derive from \n $e \\in \\overline{S_{C,{\\rm inv}}^0}$ that $\\L(S_{\\tt V})$ has interior points. \nThe main difficulty is to show that ${\\rm Spec}(\\mathop{{\\rm ad}}\\nolimits h) \\subseteq \\{-1,0,1\\}$. \n\\end{rem}\n\nAlthough we will not need it in the following, we record the following invariance \nproperty: \n\n\\begin{lem} The subspaces \n$\\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\subseteq \\mathcal{H}^{-\\infty}_{\\rm ext}$ are both invariant under \n$\\dd U^{-\\infty}({\\mathfrak g})$. \n\\end{lem}\n\n\\begin{prf} If suffices to prove invariance under $\\dd U^{-\\infty}({\\mathfrak g}^j)$ \nfor $j = 1,0,-1$. \nLet $\\eta \\in \\mathcal{H}^{-\\infty}_{\\rm ext}$ and \n$x \\in {\\mathfrak g}^j$, so that $[h,x] = j x$. \nFor $\\eta_x := \\dd U^{-\\infty}(x)\\eta$, we then we then have \n\\[ \\alpha^{\\eta_x}(t) \n= \\dd U^{-\\infty}(e^{t \\mathop{{\\rm ad}}\\nolimits h} x)\\alpha^\\eta(t) \n= e^{tj} \\dd U^{-\\infty}(x)\\alpha^\\eta(t) \\quad \\mbox{ for } \\quad t \\in {\\mathbb R}.\\] \nHence the map \n\\[ \\alpha^{\\eta_x} \\: \\overline{\\mathcal{S}_\\pi} \\to \\mathcal{H}^{-\\infty}, \\qquad \nz \\mapsto \n\\dd U^{-\\infty}(e^{z \\mathop{{\\rm ad}}\\nolimits h} x)\\alpha^\\eta(z) \n= e^{jz} \\dd U^{-\\infty}(x)\\alpha^\\eta(z) \\] \nis weak-$*$-continuous and weak-$*$-holomorphic on the interior. \nThis proves the invariance of $\\mathcal{H}^{-\\infty}_{\\rm ext}$. \nIf, in addition, $\\alpha^\\eta(\\pi i) = J \\eta$, then we further get \n\\[ \\alpha^{\\eta_x}(\\pi i) \n= \\dd U^{-\\infty}(e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h} x)\\alpha^\\eta(\\pi i) \n= \\dd U^{-\\infty}(\\tau(x)) J\\eta\n= J \\dd U^{-\\infty}(x) \\eta = J \\eta_x.\\qedhere\\] \n\\end{prf}\n\n\\subsection{An example: The $ax + b$-group} \n\\mlabel{subsec:3.4}\n\nWe consider the affine group $G = \\mathop{{\\rm Aff}}\\nolimits({\\mathbb R})_0$ of the real line with \n\\[ G_\\tau = \\mathop{{\\rm Aff}}\\nolimits({\\mathbb R}) \\cong {\\mathbb R} \\rtimes {\\mathbb R}^\\times \n\\quad \\mbox{ and } \\quad \n\\tau_G(b,a) = (-b,a).\\] \nThe Lie algebra data is given by \n\\[ C = [0,\\infty), \\quad h := (0,1), \\quad q := (1,0), \\quad \\mbox{ and } \\quad \n{\\mathfrak g}^0 = {\\mathbb R} h,\\quad {\\mathfrak g}^1 = {\\mathbb R} q.\\] \n\n\\begin{prop} For the irreducible antiunitary $C$-positive representation \n$(U,\\mathcal{H})$ of $\\mathop{{\\rm Aff}}\\nolimits({\\mathbb R})$, given by \n\\[ \\mathcal{H} = L^2\\Big({\\mathbb R}^\\times_+, \\frac{dp}{p}\\Big) \n\\quad \\mbox{ and } \\quad \n(U(b,e^t)f)(p) = e^{ibp} f(e^t p), \\quad \nU(0,-1)f = \\overline f,\\] \nthe following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] The power functions $(p^s)_{\\Re s > 0}$ define distribution vectors \n\\[ \\eta_s(f) := \\int_{{\\mathbb R}^\\times} \\overline{f(p)} p^s\\, \\frac{dp}{p}\\] \nfor the restriction $U_N$ to the translation group $N = {\\mathbb R} \\times \\{1\\}$. \n\\item[\\rm(b)] These distribution vectors transform under the action of the dilation \ngroup ${\\mathbb R}^\\times$ by \n\\begin{equation}\n  \\label{eq:trafos}\n U^{-\\infty}(0,a) \\eta_s = a^s \\eta_s \\quad \\mbox{ for } \\quad a > 0 \n\\quad \\mbox{ and } \\quad \nU^{-\\infty}(0,-1) \\eta_s = \\eta_{\\overline s}.\n\\end{equation}\n\\item[\\rm(c)]  The distribution $D_s := j_{\\eta_s}(\\eta_s) \\in \\mathcal{S}'({\\mathbb R})$ \n{\\rm(cf.\\ \\eqref{eq:dist-inc} in Appendix~\\ref{app:a})} \ncoincides with the Fourier transform of the measure \n$p^{2 \\Re s-1}\\, dp$ on ${\\mathbb R}_+$. \n\\end{itemize}\n\\end{prop}\n\n\\begin{prf} (a) As $\\mathcal{H}^{-\\infty}(U_N) = {\\rm span}_{\\mathbb C} \\big(\\dd U^{-\\infty}(\\mathcal{U}({\\mathbb R} q)) \\mathcal{H}\\big)$ \n(Lemma~\\ref{lem:charsmooth}), it \nsuffices to show that there exists a polynomial $F(p) \\in {\\mathbb C}[p]$ \nwith $F(p)^{-1} p^s \\in L^2({\\mathbb R}^\\times_+, \\frac{dp}{p})$. \nFor $F(p) := (1 + p^2)^n$, we have \n\\[ \\|F(p)^{-1}p_s\\|_2^2 \n= \\int_0^\\infty \\frac{p^{2 \\Re s}}{(1 + p^2)^{2n}}\\, \\frac{dp}{p},\\] \nand this integral is finite if and only if \n$\\Re s > 0$ and $2n > \\Re s$. \n\n\\nin (b) Next we note that \n\\[ \\big(U^{-\\infty}(0,e^t) \\eta_s\\big)(f) \n= \\eta_s(U(0,e^{-t})f)\n= \\int_{{\\mathbb R}^\\times} \\overline{f(e^{-t}p)} p^s\\, \\frac{dp}{p}\n= \\int_{{\\mathbb R}^\\times} \\overline{f(p)} (e^tp)^s\\, \\frac{dp}{p}\n= e^{ts} \\eta_s(f)\\] \nand \n\\[ \\big(U^{-\\infty}(0,-1) \\eta_s\\big)(f) \n= \\overline{\\eta_s(\\overline f)} \n= \\int_{{\\mathbb R}^\\times} \\overline{f(p)} p^{\\overline s}\\, \\frac{dp}{p}\n= \\eta_{\\overline s}(f),\\] \nso that $\\eta_s \\in \\mathcal{H}^{-\\infty}$ is an $\\exp({\\mathbb R} h)$-eigenvector satisfying \n\\eqref{eq:trafos}. \n\n\\nin (c) For $\\phi \\in C^\\infty_c({\\mathbb R},{\\mathbb R})$, we have \n\\[ (U^{-\\infty}(\\phi) \\eta_s)(p) \n= \\tilde \\phi(p) p^s, \\qquad \n\\tilde\\phi(p) = \\int_{\\mathbb R} e^{ipx} \\phi(x)\\, dx,   \\] \nbecause \n\\[ \\eta_s(U(\\phi^*)f) \n= \\int_0^\\infty p^s \\overline{\\tilde{\\phi^*}(p)} \\overline{f(p)}\\, \\frac{dp}{p}\n= \\int_0^\\infty p^s \\tilde \\phi(p) \\overline{f(p)}\\, \\frac{dp}{p}.\\]\nFor the distribution $D_s = j_{\\eta_s}(\\eta_s) \\in C^{-\\infty}({\\mathbb R})$, we thus \nobtain \n\\[ D_s(\\phi) = \\eta_s(U^{-\\infty}(\\phi)\\eta_s) \n= \\int_0^\\infty p^s \\overline{\\tilde\\phi(p) p^s}\\, \\frac{dp}{p}\n= \\int_0^\\infty p^{2 \\Re s} \\overline{\\tilde\\phi(p)}\\, \\frac{dp}{p}\n= \\int_0^\\infty p^{2 \\Re s} \\hat{\\overline\\phi}(p)\\, \\frac{dp}{p}.\\]\nThis is (c). \n\\end{prf}\n\nTo identify corresponding standard subspaces, we note that \n$\\alpha^{\\eta_s}(z) = e^{sz} \\eta_s$ implies that \n$\\alpha^{\\eta_s}(\\pi i) = e^{\\pi i s} \\eta_s$ and \n$J \\eta_s = \\eta_{\\overline s}$. For $s \\in {\\mathbb R}$, it follows that \n\\[ \\tilde \\eta_s := e^{-\\frac{s \\pi i}{2}} \\eta_s \n\\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\subseteq {\\tt V}^{-\\infty}.\\] \nAccordingly, ${\\tt V} \\subseteq \\mathcal{H}$ is generated by the functions \n\\[ \\tilde\\phi \\tilde \\eta_s = e^{-\\frac{s \\pi i}{2}} \\tilde \\phi \\eta_s \n\\quad \\mbox{ for  }\\quad \\phi \\in C^\\infty_c((0,\\infty),{\\mathbb R}).\\] \nFor $s \\not\\in {\\mathbb R}$, we put \n\\[ \\tilde \\eta_s := \n\\eta_s + e^{-\\pi i \\overline s} \\eta_{\\overline s}  \\] \nand obtain \n$J \\tilde \\eta_s  = e^{\\pi i s} \\eta_s + \\eta_{\\overline s} \n=  \\alpha^{\\tilde\\eta_s}(\\pi i),$ so that \n\\[ \\tilde\\eta_s \\in \\mathcal{H}^{-\\infty}_{{\\rm ext},J} \\subseteq {\\tt V}^{-\\infty}.\\] \n\n\n\\section{Realizing $C$-positive representations \non tubes} \n\\mlabel{sec:4} \n\nIn this section we show that, for semisimple Lie algebras \n${\\mathfrak g}$, which are direct sums of simple hermitian ideals of tube type, \nand any pointed generating invariant cone $C \\subseteq {\\mathfrak g}$, \nthe (irreducible) $C$-positive unitary representation can be realized \nin Hilbert spaces of holomorphic functions on a tube domain. This \nrealization will be used in Section~\\ref{sec:5} to exhibit a suitable \nreal subspace ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}_{{\\rm ext},J}$ to which the results \nfrom Section~\\ref{sec:3} apply.\n\n\nLet ${\\mathfrak g}$ be a semisimple Lie algebra \nand $h \\in {\\mathfrak g}$ be an Euler element for which \n${\\mathfrak h} := {\\mathfrak g}^0(h)$ contains no non-zero ideal of ${\\mathfrak g}$, i.e., the Lie algebra \n${\\mathfrak g}$ is generated by  ${\\mathfrak g}^{\\pm 1} = {\\mathfrak g}^{\\pm 1}(h)$. \nWe consider the Lie groups\n\\[ G := \\mathop{{\\rm Inn}}\\nolimits({\\mathfrak g}) = \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g})_0 \\quad \\mbox{ and } \\quad \nG_\\tau := G  \\rtimes \\{ \\mathbf{1}, \\tau \\} \n\\quad \\mbox{ for } \\quad \\tau = e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h}\\vert_{{\\mathfrak g}}. \\]\nThen $G_\\tau$ is a graded Lie group and we put $H := \\mathop{{\\rm Inn}}\\nolimits_{\\mathfrak g}({\\mathfrak h}) = (G^\\tau)_0$. \nWe also fix a Cartan involution $\\theta$ of ${\\mathfrak g}$ fixing the center \npointwise and satisfying $\\theta(h) = - h$. The corresponding \nCartan decomposition is written ${\\mathfrak g} = {\\mathfrak k} \\oplus {\\mathfrak p}$, \nand we write $K := \\langle \\exp {\\mathfrak k} \\rangle$ for the corresponding \nmaximal compact subgroup of~$G$.\n\nWe further assume that all simple ideals of ${\\mathfrak g}$ are \nhermitian, so that the existence of an Euler element \nimplies that they are tube type (cf.~\\cite[Thm.~3.12]{MN20}; \n\\cite[Thm.~5.6]{O91}) \nand that we may consider the eigenspace \n$E := {\\mathfrak g}^1(h)$ as a, not necessarily simple, \nunital euclidean Jordan algebra. We may further pick \nthe unit element $e\\in E$ in such a way that it is fixed by $H_K= H^\\theta$. Then \n\\[ H_K \\cong \\mathop{{\\rm Aut}}\\nolimits(E,e)_0 \\quad \\mbox{ and } \\quad \nH \\cong \\mathop{{\\rm Str}}\\nolimits(E)_0 \\subseteq H_{\\mathbb C} := \\mathop{{\\rm Str}}\\nolimits(E_{\\mathbb C})_0 \\] \nare the identity components of the real and complex structure groups. \nAccordingly, we identify them with linear groups acting on $E_{\\mathbb C}$. \nWe then have the two associated complex domains: \n\\begin{itemize}\n\\item the tube domain $\\mathcal{T} := \\mathcal{T}_{E_+} := E + i E_+ \\subseteq E_{\\mathbb C}$, and \n\\item the unit ball \n$\\mathcal{D} := \\{ z\\in E_{\\mathbb C} \\:  \\|z\\|_\\infty < 1 \\}$, \nwith respect to the spectral norm $\\|\\cdot\\|_\\infty$ of $E_{\\mathbb C}$, \nwhich is a bounded symmetric domain \n\\end{itemize}\nThe {\\it Cayley transform} \n\\[ p \\: \\mathcal{T} \\to \\mathcal{D}, \\quad \np(z) := (z - ie)(z+ie)^{-1} \\] \nmaps the tube $\\mathcal{T}$ biholomorphically onto $\\mathcal{D}$. Its inverse is given \nby \n\\[ c \\: \\mathcal{D} \\to \\mathcal{T}, \\quad \nc(z) := i(e+z)(e-z)^{-1}\\] \n(cf.~\\cite[Thm.~X.4.3]{FK94}).\nThe differential of the Cayley \ntransform is given by \n\\begin{equation}\n  \\label{eq:diffcayley}\n \\dd p(z) = 2i P(z + i e)^{-1} \\quad \\mbox{ for } \\quad z \\in \\mathcal{T},\n\\end{equation} \nwhere \n\\[ P \\: E_{\\mathbb C} \\to \\mathop{{\\rm End}}\\nolimits(E_{\\mathbb C}), \\qquad P(z) := 2 L(z)^2 - L(z^2), \\qquad \nL(z)w := zw, \\] \nis the {\\it quadratic representation} \n(\\cite[p.~192]{FK94}). \nWe have \n$P(E_{\\mathbb C}^\\times) \\subseteq \\mathop{{\\rm Str}}\\nolimits(E_{\\mathbb C})$ \n(\\cite[Prop.~VIII.2.4]{FK94}), hence in particular \n\\[ P(\\mathcal{T}) \\subseteq H_{\\mathbb C} \\quad \\mbox{ because} \\quad \nP(ie) = -P(e) = - \\mathbf{1} \\in {\\mathbb C}^\\times\\mathbf{1} = \\exp({\\mathbb C} h)\\subseteq H_{\\mathbb C}\\] \n(cf.\\ \\cite[p.~168]{FK94}). \n\nThe group \n$G\\cong \\mathop{\\rm Conf{}}\\nolimits(E)_0$ is generated by the subgroups \n$N^+ := \\exp({\\mathfrak g}^1(h))\\cong (E,+)$ (the translation group of $E$), \nthe subgroup $H$\nand an element $\\eta \\in Z(K)$ acting on $E_{\\mathbb C}^\\times$ by \nthe involution \n\\begin{equation}\n  \\label{eq:dereta}\n \\eta(z) = - z^{-1} \\quad \\mbox{ with differential } \\quad \n\\dd\\eta(z) = P(z)^{-1}\n\\end{equation}\n(\\cite[Prop.~II.3.3(i)]{FK94}). \nThis is the point symmetry in the point $ie$ of the Riemannian \nsymmetric space~$\\mathcal{T}$.\nThe element $\\eta \\in G$ implements the \nCartan involution $\\theta$ of $G$ by \n\\[ \\theta(g) = \\eta \\circ g \\circ \\eta^{-1}= \\eta \\circ g \\circ\n\\eta. \\] \n\nWe write $G_{\\mathbb C} := \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g}_{\\mathbb C})_0$ and observe that $H_{\\mathbb C} = \\langle \\exp {\\mathfrak h}_{\\mathbb C} \\rangle$. \nFor $g \\in H$, we put \n\\[ g^* := \\theta(g)^{-1},\\] \nand if $\\overline\\theta$ denotes the antiholomorphic involution of \n$H_{\\mathbb C}$ extending $\\theta$, we likewise put \n\\[ g^* := \\overline\\theta(g)^{-1} \\quad \\mbox{ for } \\quad g \\in H_{\\mathbb C}.\\] \nWe also use this notation for elements of the simply connected covering \ngroup $q_{H_{\\mathbb C}} \\: \\tilde H_{\\mathbb C} \\to H_{\\mathbb C}$ and the lift of $\\overline\\theta$ to \n$\\tilde H_{\\mathbb C}$. We note that the kernel \n of $q_{H_{\\mathbb C}}$ is a discrete central subgroup.\nAs $\\mathcal{T}$ is simply connected, there exists a holomorphic lift \n\\begin{equation}\n  \\label{eq:tildep}\n\\tilde P \\: \\mathcal{T} \\to \\tilde H_{\\mathbb C}. \n\\end{equation}\nAs $P(ie) = -\\mathbf{1}$, any such lift is determined by \n$\\tilde P(ie) \\in q_{H_{\\mathbb C}}^{-1}(-\\mathbf{1})$, which is a discrete subgroup \nof the center of $\\tilde H_{\\mathbb C}$; a coset of $\\ker(q_{H_{\\mathbb C}})$. \nWe also observe that \n$\\tilde P(ie)$ is contained in the connected \nsubgroup $(\\tilde H_{\\mathbb C})^{\\overline\\theta}$ of unitary elements \nwith respect to the involution $g^* = \\overline\\theta(g)^{-1}$. \n\nWe write $\\tau_{H_{\\mathbb C}}(g) = \\overline g = \\tau g \\tau$ \nfor the antiholomorphic automorphism of $H_{\\mathbb C}$ induced by \ncomplex conjugation on~$E_{\\mathbb C}$. Its lift to $\\tilde H_{\\mathbb C}$ \nis denoted $\\tau_{\\tilde H_{\\mathbb C}}$. We write \n$H^\\sharp = (\\tilde H_{\\mathbb C})^{\\tau_{\\tilde H_{\\mathbb C}}}$ for its \nconnected group of fixed points \n(\\cite[Thm.~IV.3.4]{Lo69}). \n\n\\subsection{Translating between tube and ball} \n\\mlabel{subsec:4.1} \n\nLet $G(\\mathcal{D}) = \\mathop{{\\rm Aut}}\\nolimits(\\mathcal{D})_0$ be the identity component of the \ngroup of holomorphic automorphisms of \n$\\mathcal{D}$ and $G(\\mathcal{T}) = \\mathop{{\\rm Aut}}\\nolimits(\\mathcal{T})_0 \\cong G$ be the corresponding group \nfor the tube domain $\\mathcal{T}$. Both are subgroups \nof the conformal group $\\mathop{\\rm Conf{}}\\nolimits(E_{\\mathbb C})$, and in this group \nthey are conjugate by the Cayley transform \n\\begin{equation}\n  \\label{eq:cayconj}\nG(\\mathcal{D})=c^{-1} G(\\mathcal{T})c \\subseteq G_{\\mathbb C}.\n\\end{equation}\nFor the simply connected covering groups \n$q_{G(\\mathcal{D})} \\: \\tilde G(\\mathcal{D}) \\to G(\\mathcal{D})$ and \n$q_{G(\\mathcal{T})} \\: \\tilde G(\\mathcal{T}) \\to G(\\mathcal{T})$ we define action \nand $\\mathcal{D}$ and $\\mathcal{T}$, respectively, by \n$g.z := \\alpha_{G(\\mathcal{D})}(g)(z)$ and $g.z := \\alpha_{G(\\mathcal{T})}(g)(z)$. \n\nLet \n\\begin{equation}\n  \\label{eq:liftalphac}\n\\tilde\\alpha_c \\: \\tilde G(\\mathcal{D}) \\to \\tilde G(\\mathcal{T}) \n\\end{equation}\nbe the isomorphism obtained by lifting the isomorphism \n\\[ \\alpha_c \\: G(\\mathcal{D}) \\to G(\\mathcal{T}),\\quad  \\alpha_c(g) := cgc^{-1}.\\] \nWe write \n\\[ K(\\mathcal{D}) := G(\\mathcal{D})^0 \\quad \\mbox{ and } \\quad \nK(\\tau) := G(\\mathcal{T})^{ie} \\] \nfor the stabilizer groups of the base points $0 \\in \\mathcal{D}$ and $ie \\in \\mathcal{T}$. \n\nOn the tube domain $\\mathcal{T} = E + i E_+$, we consider the \n{\\it universal kernel} \n\\[ Q \\: \\mathcal{T} \\times \\mathcal{T} \\to H_{\\mathbb C}, \\quad \nQ(z,w) := P\\Big(\\frac{z-\\overline w}{2i}\\Big), \\] \nwhich satisfies $Q(ie,ie) = P(e) = \\mathbf{1}$. It is sesquiholomorphic \nin the sense that it is holomorphic in the first and antiholomorphic \nin the second argument. As $\\mathcal{T}$ is simply \nconnected, this kernel lifts to a sesquiholomorphic kernel \n\\[ \\tilde Q \\: \\mathcal{T} \\times \\mathcal{T} \\to \\tilde H_{\\mathbb C} \n\\quad \\mbox{ with }\\quad \n\\tilde Q(ie,ie) = \\mathbf{1}.\\] \nThe action of $G(\\mathcal{T})$ on $\\mathcal{T}$ defines a cocycle \n\\[ J \\: G(\\mathcal{T}) \\times \\mathcal{T} \\to H_{\\mathbb C}, \\qquad \nJ(g,z) := J_g(z) := \\dd \\sigma_g(z) \\] \nsatisfying \n\\begin{equation}\n  \\label{eq:cocyc1}\nJ(g_1 g_2,z) = J(g_1,g_2.z) J(g_2,z) \n\\quad \\mbox{ for } \\quad \ng\\in G, z \\in \\mathcal{T}.  \n\\end{equation}\nSince $\\mathcal{T}$ is simply connected, this cocycle lifts to a \nsmooth cocycle \n\\[ \\tilde J \\: \\tilde G(\\mathcal{T}) \\times \\mathcal{T} \\to \\tilde H_{\\mathbb C} \n\\quad \\mbox{ determined  by } \\quad \\tilde J(e,ie) = \\mathbf{1}, \\] \nwhich is holomorphic in the second argument. The cocycle \nproperty of $\\tilde J$ follows by the uniqueness of lifts, \nwhich immediately implies $\\tilde J(e,z) = \\mathbf{1}$ for every $z \\in \\mathcal{T}$. \nWe have \n\\begin{itemize}\n\\item $\\tilde J_g = 1$ for translations $g(x) = x + v$, $ \\in E$, \n\\item On the connected subgroup $\\tilde H(\\mathcal{T}) = \\langle \\exp {\\mathfrak h} \\rangle \n\\subseteq \\tilde G(\\mathcal{T})$, the cocycle $\\tilde J$ \ndefines a morphism of Lie groups \n$\\tilde J\\vert_{\\tilde H(\\mathcal{T})} \\: \\tilde H(\\mathcal{T}) \\to H^\\sharp \\subseteq \\tilde H_{\\mathbb C}$ \nintegrating the inclusion ${\\mathfrak h} \\hookrightarrow {\\mathfrak h}_{\\mathbb C}$. \n\\end{itemize}\n\n\\begin{lem}\n  \\mlabel{lem:III.2.1} \n  \\begin{equation}\n    \\label{eq:kern-trafo-lift}\n  \\tilde Q(g.z, g.w) = \n\\tilde J(g,z) \\tilde Q(z,w) \\tilde J(g,w)^*\\quad \\mbox{  for } \\quad \nz \\in \\mathcal{T}, g \\in \\tilde G(\\mathcal{T}).\n \\end{equation}\n\\end{lem}\n\n\\begin{prf} (a) First we verify the corresponding relation \nfor the $H_{\\mathbb C}$-valued kernel~$Q$: \n\\[ Q(g.z, g.w) = J(g,z) Q(z,w) J(g,w)^* \n\\quad \\mbox{  for } \\quad z \\in \\mathcal{T}, g \\in \\tilde G(\\mathcal{T}).\\] \nSince $G = G(\\mathcal{T})$ is generated by $H$, $\\eta$ and translations \nand $J$ is a cocycle, we have to verify this relation only \nfor these three cases. \n\nFor $g \\in H$ we have \n\\[ Q(g.z, g.w) \n=   P({\\textstyle{g.z - \\overline {g.w} \\over 2i }})\n=   P(g.{\\textstyle{z - \\overline {w} \\over 2i }})\n \\ {\\buildrel {!} \\over =} \\  g   P({\\textstyle{z - \\overline {w} \\over 2i }}) g^* \n= J(g,z)  Q(z,w) J(g,w)^*,\\] \nwhere we use \\cite[p.~147]{FK94} for $!$. \nFor $u \\in E$ and the translation $g = \\exp u \\in N^+$, we obtain \n\\[ Q(z+u,w+u) \n=   P\\Big(\\frac{z + u - \\overline w - \\overline u}{2i}\\Big)\n=   P\\Big(\\frac{z - \\overline {w}}{2i}\\Big)\n=   Q(z,w) \n=   J(g,z)  Q(z,w)   J(g,w)^* \\] \nbecause $J(g,z) = \\mathbf{1}$. \nFinally we see with \\cite[Lemma~X.4.4(i)]{FK94} that \n\\begin{align*}\nQ(-z^{-1}, -w^{-1}) \n&=   P({\\textstyle{- z^{-1} + \\overline w^{-1} \\over 2i }}) \n= -{\\textstyle{1\\over 4}} P(- z^{-1} + {\\overline w}^{-1}) \\\\ \n&= -{\\textstyle{1\\over 4}} P(-z)^{-1} P(- z + \\overline w) P(\\overline w)^{-1}  \n= P(z)^{-1} P({\\textstyle{z - \\overline {w} \\over 2i }}) P(\\overline w)^{-1} \\\\ \n&= J(\\eta, z) Q(z,w) J(\\eta,w)^* \n\\end{align*}\nbecause \n$P(w)^* = \\overline{P(w)} = P(\\overline w)$  \nand $J(\\eta,z) = P(z)^{-1}$. \n\n\\nin (b) Now we turn to the lifted kernel $\\tilde Q$. \nFix $z,w \\in \\mathcal{T}$. Then both sides of \\eqref{eq:kern-trafo-lift} \ntake the same values in $g = e$. By (a), they are lifts of the \nsame maps to $H_{\\mathbb C}$. Hence the assertion follows from the uniqueness \nof continuous lifts. \n \\end{prf}\n\n\\begin{rem} The preceding lemma has some remarkable consequences. \n\n\\nin (a) For $g \\in H$, we have \n\\[ Q(g.ie, g.ie) = gg^*, \\] \nwhich is the quadratic representation of the Riemannian symmetric \nspace $E_+ \\cong i E_+ \\subseteq \\mathcal{T}$ with values in its isometry \ngroup~$(H,\\theta)$.\n\n\\nin (b) For elements of the stabilizer group $\\tilde K \n= \\tilde K(\\mathcal{T}) = \\tilde G(\\mathcal{T})^{ie}$, \nwe have \n\\[ \\tilde J(g,ie)  \\tilde J(g,ie)^* = \\mathbf{1} \\quad \\mbox{ for } \\quad \ng \\in \\tilde K, \\] \nso that $\\tilde J$ defines a homomorphism \n\\[ \\tilde J \\: \\tilde K \\to (\\tilde H_{\\mathbb C})^{\\overline\\theta} = \\tilde K(\\mathcal{D}).\\] \n\\end{rem}\nUsing the Cayley transform, we obtain a kernel on $\\mathcal{D}$ by \n\\begin{align} \\label{eq:2kerneltrafo}\n  Q^\\mathcal{D}(z, w) \n&:= 4\\cdot \\dd c(z)^{-1} Q(c(z), c(w)) (\\dd c(w)^{-1})^* \\notag \\\\ \n&= 4\\cdot \\dd p\\big(c(z)\\big) Q(c(z), c(w)) \\big(\\dd p(c(w))\\big)^*\\notag\\\\\n&{\\buildrel \\eqref{eq:diffcayley} \\over =}\\ \\  P\\Big(\\frac{c(z)+ie}{2}\\Big)^{-1}  Q(c(z), c(w)) \n\\Big(P\\Big(\\frac{c(z)+ie}{2}\\Big)^{-1}\\Big)^*. \n\\end{align}\nThe kernel $Q^\\mathcal{D}$ also takes values in $H_{\\mathbb C}$, \nand $c(0) = ie$ leads in particular to \n\\[ Q^\\mathcal{D}(0,0) \n= P(ie)^{-1}  Q(ie, ie) (P(ie)^{-1})^* = \\mathbf{1},\\] \nso that there exists a unique continuous lift\n\\begin{equation}\n  \\label{eq:qdlift}\n \\tilde Q^\\mathcal{D}\\colon \\mathcal{D} \\times \\mathcal{D} \\to \\tilde H_{\\mathbb C}\n\\quad \\mbox{ with } \\quad \\tilde Q^\\mathcal{D}(0,0) = \\mathbf{1}.\n\\end{equation}\nAs for the tube $\\mathcal{T}$, the cocycle \n\\[ J^\\mathcal{D}\\colon G(\\mathcal{D})\\times \\mathcal{D}\\to H_{\\mathbb C},\n\\qquad J^\\mathcal{D}(g,z):=\\dd g(z)\\] \nlifts to a cocycle \n\\[ \\tilde J^{\\cal D}\\colon \\tilde G(\\mathcal{D})\\times \\mathcal{D} \\to \\tilde H_{\\mathbb C}.\\] \nAs \n\\[ J^\\mathcal{D}(k,z)=k\\quad \\mbox{ for}\\quad  k\\in K(\\mathcal{D})= G(\\mathcal{D})^0, \\]\nthe restriction of $J^\\mathcal{D}$ provides an inclusion \n\\[  J^\\mathcal{D} \\: K(\\mathcal{D}) = G(\\mathcal{D})^0 \\hookrightarrow H_{\\mathbb C} \\] \n(cf.~\\cite[Thm.~X.5.3, Prop.~X.3.1]{FK94}). Its image is a maximal \ncompact subgroup, so that $H_{\\mathbb C} \\cong K(\\mathcal{D})_{\\mathbb C}$. \\\\\n\n\\subsection{Unitary representations} \n\\mlabel{subsec:4.2} \n\nNow we turn to unitary representations of the simply connected groups \n$\\tilde G(\\mathcal{D})$ and $\\tilde G(\\mathcal{T})$ on Hilbert spaces of holomorphic functions \non $\\mathcal{D}$ and $\\mathcal{T}$, respectively.  \nWe start with a holomorphic involutive representation \n$(\\rho, \\mathcal{K})$ of the complex involutive Lie group $\\tilde H_{\\mathbb C}$ \nwith $g^* = \\overline\\theta(g)^{-1}$ on the complex Hilbert space~$\\mathcal{K}$. \nThis representation does not have to be finite dimensional, but \nholomorphy implies that the operators of the derived representations \nare all bounded and that $\\rho$ is norm-continuous. Conversely, \nevery norm-continuous $*$-representation of $\\tilde H$ defines a \nholomorphic representation of its universal complexification $\\tilde H_{\\mathbb C}$. \n\n\\begin{defn} A holomorphic $*$-representation $(\\rho,\\mathcal{K})$ of \n$\\tilde H_{\\mathbb C}$ is said to be {\\it $\\mathcal{T}$-positive} if the kernel \n\\[ Q_\\rho := \\rho \\circ \\tilde Q, \\: \\mathcal{T} \\times \\mathcal{T} \\to B(\\mathcal{K}), \\quad  \nQ_\\rho(z,w) = P_\\rho\\Big(\\frac{z - \\overline w}{2i}\\Big) \\] is \npositive definite. \n\\end{defn}\n\nThen we obtain a representation of $\\tilde G(\\mathcal{D})$ \non the space $\\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ by \n\\begin{equation}\n  \\label{eq:piD}\n (U^\\mathcal{D}_\\rho(g)f)(z) \n:= J^\\mathcal{D}_\\rho(g, g^{-1}.z) f(g^{-1}.z) \n\\quad \\mbox{ for } \\quad g \\in \\tilde G(\\mathcal{D}), z \\in \\mathcal{D}, \nJ^\\mathcal{D}_\\rho := \\rho \\circ \\tilde J^\\mathcal{D},\n\\end{equation}\nand a representation of $\\tilde G(\\mathcal{T})$ \non the space $\\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$ by \n\\begin{equation}\n  \\label{eq:pitau}\n (U_\\rho(g)f)(z) := J_\\rho(g, g^{-1}.z) f(g^{-1}.z)  \n\\quad \\mbox{ for } \\quad g \\in \\tilde G(\\mathcal{T}), z \\in \\mathcal{T}, \nJ_\\rho := \\rho \\circ \\tilde J.\n\\end{equation}\nThe space $\\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ contains a unitary subrepresentation \nof $G(\\mathcal{D})$ if and only if \nthe kernel $Q_\\rho^\\mathcal{D} := \\rho \\circ \\tilde Q^\\mathcal{D}$ \nis positive definite (\\cite[Prop.~XII.2.1]{Ne99}; see \nalso \\cite[Thm.~XII.2.6]{Ne99} for the link to highest weight representations). \nTo see that the Cayley transform provides a natural intertwining operator, \nwe use a holomorphic lift \n$\\tilde P \\: \\mathcal{T} \\to \\tilde H_{\\mathbb C}$ of the quadratic representation~$P$ \n(cf.\\ \\eqref{eq:tildep}). \nAs \n\\begin{equation}\n  \\label{eq:jcrels}\n J(c,z) := J_c(z) := \\dd c(z) = \\dd p^{-1}(z) = \\dd p(c(z))^{-1} \n= \\frac{1}{2i} P(z + ie) \n= \\frac{2}{i} P\\Big(\\frac{z + ie}{2}\\Big),\n\\end{equation}\nup to a central factor $c_0 \\in Z(\\tilde H_{\\mathbb C})$, \n\\[ \\tilde J_c(z) := c_0 \\tilde P(z + ie) \\] \nis a lift of $J_c$. \n\nFor $g \\in \\tilde G(\\mathcal{D})$ and $\\tilde g := \\tilde\\alpha_c(g)\\in \\tilde G(\\mathcal{T})$, we have \n$\\tilde g.(c(z)) = c(g.z)$, and thus \n\\[ J_{\\tilde g}(c(z)) J_c(z) = J_c(g.z) J_g(z) \\quad \\mbox{ for } \\quad \nz \\in \\mathcal{D}.\\] \nFor any holomorphic lift $\\tilde J_c$ of $J_c$, we thus have the relation \n\\[ \\tilde J_{\\tilde g}(c(z)) \\tilde J_c(z) \\tilde J_g(z)^{-1} \n\\tilde J_c(g.z)^{-1} \\in \\ker(q_{H_{\\mathbb C}}).\\] \nFor $g = e$, this element is the identity of $\\tilde H_{\\mathbb C}$, and \nsince $\\ker(q_{H_{\\mathbb C}})$ is discrete,  \n\\[ \\tilde J_{\\tilde g}(c(z)) \\tilde J_c(z) \\tilde J_g(z)^{-1} \n\\tilde J_c(g.z)^{-1} = e \\quad \\mbox{ for } \\quad \ng \\in \\tilde G(\\mathcal{D}), z \\in \\mathcal{D}.\\] \nThis shows that \n\\begin{equation}\n  \\label{eq:lift-intertwice}\n\\tilde J_{\\tilde g}(c(z)) \\tilde J_c(z) = \n\\tilde J_c(g.z)\\tilde J_g(z) \\quad \\mbox{ for } \\quad \ng \\in \\tilde G(\\mathcal{D}), z \\in \\mathcal{D}.\n\\end{equation}\n\nFrom \\eqref{eq:2kerneltrafo} we first derive the relation \n\\begin{equation}\n  \\label{eq:2liftrel}\n  \\tilde Q^\\mathcal{D}(z, w) \n= \\tilde P\\Big(\\frac{c(z)+ie}{2}\\Big)^{-1}  \\tilde Q(c(z), c(w)) \n\\Big(\\tilde P\\Big(\\frac{c(w)+ie}{2}\\Big)^{-1}\\Big)^*. \n\\end{equation}\nComposing with the $*$-representation $\\rho$ \nand putting $P_\\rho := \\rho \\circ \\tilde P$, \nnow leads to\\begin{footnote}\n{There is a certain freedom in normalizing the kernel on the tube domain. \nWe have chosen a normalization for which $Q_\\rho(ie,ie) = \\mathop{{\\rm id}}\\nolimits_\\mathcal{K}$, and \nthis is why we use the $P_\\rho$-factors to transform the \nkernels and not the factors $J_\\rho(c,c^{-1}(z))$ obtained from \nthe differential of the Cayley transform $c \\: \\mathcal{D} \\to \\mathcal{T}$. The source \nof the trouble is the fact that $dc(0) = dp(ie)^{-1} = 2i \\mathbf{1}$ is not unitary.\n}\\end{footnote}\n\n\\begin{equation}\n  \\label{eq:posdefrel}\n  Q_\\rho^\\mathcal{D}(z, w) \n= P_\\rho\\Big(\\frac{c(z)+ie}{2}\\Big)^{-1}   Q_\\rho(c(z), c(w)) \nP_\\rho\\Big(\\frac{c(w)+ie}{2}\\Big)^{-*}\n\\end{equation}\nThis relation implies in particular that the kernel \n$\\mathcal{Q}_\\rho^\\mathcal{D}$ on $\\mathcal{D} \\times \\mathcal{D}$ is positive definite if and \nonly if the kernel $\\mathcal{Q}_\\rho$ on $\\mathcal{T} \\times \\mathcal{T}$ \nis positive definite, i.e., if the representation \n$\\rho$ is $\\mathcal{T}$-positive \n(cf.~\\cite[Thm.~I.1.4]{Ne99}). From now one we assume that this is the case. \nLet $\\mathcal{H}^\\mathcal{D}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ and \n$\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$ denote the reproducing kernel \nHilbert spaces with kernel $Q_\\rho^\\mathcal{D}$ and \n$Q_\\rho$, respectively. \n\n\n\\begin{lem} \\mlabel{lem:gamma-intertwine} \nThe map \n\\[ \\Gamma \\: \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K}) \\to \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K}), \\qquad \n\\Gamma(f)(z) := P_\\rho\\Big(\\frac{z + i e}{2}\\Big)^{-1} f(c(z)), \\]\nintertwines the representation of the groups \n$G(\\mathcal{T})$ and $G(\\mathcal{D})$ in the sense that \n\\[  \\Gamma \\circ U_\\rho(\\tilde\\alpha_c(g)) =   \nU_\\rho^\\mathcal{D}(g) \\circ \\Gamma \\quad \\mbox{ for } \\quad g \\in \\tilde G(\\mathcal{D}).\\] \nIf the kernel $Q_\\rho$ is positive definite, then it restricts to a unitary operator \n$\\Gamma \\: \\mathcal{H}_\\rho \\to \\mathcal{H}_\\rho^\\mathcal{D}$. \n\\end{lem}\n\n\\begin{prf} First we show that the slightly modified map \n$\\Gamma_0(f)(z) := \\tilde J_\\rho(c,z)^{-1} f(c(z))$ \nintertwines the two representation. \nFor $\\tilde g:= \\tilde\\alpha_c(g)  \\in \\tilde G(\\mathcal{T})$ and \n$f \\in \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ we have \n  \\begin{align*}\n\\Gamma_0(U_\\rho(\\tilde g)f)(z) \n&= \\tilde J_\\rho(c,z)^{-1} \\tilde J_\\rho(\\tilde g^{-1},c(z))^{-1} f(\\tilde g^{-1}.c(z)) \\\\ \n&= \\big(\\tilde J_\\rho(\\tilde g^{-1},c(z))\\tilde J_\\rho(c,z)\\big)^{-1} f(c(g^{-1}.z)) \\\\ \n& {\\buildrel {\\eqref{eq:lift-intertwice}} \\over =}\\ \\ \n \\big(\\tilde J_\\rho(c,g^{-1}.z)\\tilde J_\\rho(g^{-1},z)\\big)^{-1} f(c(g^{-1}.z)) \\\\ \n&= \\tilde J_\\rho(g^{-1},z)^{-1} \\tilde J_\\rho(c,g^{-1}.z)^{-1} f(c(g^{-1}.z))\\\\ \n&= \\big(U_\\rho(g) \\Gamma_0(f)\\big)(z).\n  \\end{align*}\nHere we have used the relation \n\\[   \\tilde J_\\rho(\\tilde g^{-1},c(z))\\tilde J_\\rho(c,z) \n= \\tilde J_\\rho(c,g^{-1}.z)\\tilde J_\\rho(g^{-1},z)\\]\nwhich follows from \\eqref{eq:lift-intertwice}. \nNext we observe that, by \\eqref{eq:jcrels}, \nthere exists a central element $z \\in \\tilde H_{\\mathbb C}$ such that \n$\\Gamma(f) = \\rho(z) \\Gamma_0(f)$. As composition with $\\rho(z)$ \ncommutes with the representation of $\\tilde G(\\mathcal{D})$, this \nimplies the intertwining property of~$\\Gamma$.\n\nTo see that $\\Gamma$ restrict to a \nunitary operator $\\mathcal{H}_\\rho \\to \\mathcal{H}^\\mathcal{D}_\\rho$, we show that \nit maps $\\mathcal{H}_\\rho$ isometrically onto the subspace with reproducing\nkernel $Q^\\mathcal{D}_\\rho$. Writing, for simplicity, $\\Gamma(f)(z) = \\beta(z) f(c(z))$ \nwith $\\beta(z) := P_\\rho\\big(\\frac{z + i e}{2}\\big)^{-1}$, \nthe injective map $\\Gamma$ maps $\\mathcal{H}_\\rho$ unitarily onto a reproducing \nkernel Hilbert space $\\mathop{{\\rm im}}\\nolimits(\\Gamma) \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ on which the point evaluations are given by \n\\[ \\mathop{{\\rm ev}}\\nolimits_z = \\beta(z) \\circ \\mathop{{\\rm ev}}\\nolimits_{c(z)} \\circ \\Gamma^{-1} \n= \\beta(z) Q_{\\rho, c(z)} \\Gamma^*.\\]\nTherefore the reproducing kernel of $\\mathop{{\\rm im}}\\nolimits(\\Gamma)$ is given by \n\\begin{align*}\n \\mathop{{\\rm ev}}\\nolimits_z \\mathop{{\\rm ev}}\\nolimits_w^* \n&=  \\beta(z) Q_{\\rho, c(z)} \\Gamma^* \\Gamma Q_{\\rho, c(w)}^* \\beta(w)^*\n=  \\beta(z) Q_{\\rho, c(z)} Q_{\\rho, c(w)}^* \\beta(w)^*\\\\\n&=  \\beta(z) Q_\\rho(c(z),c(w)) \\beta(w)^* \n\\ \\ {\\buildrel \\eqref{eq:posdefrel} \\over = } \\ \\  Q_\\rho^\\mathcal{D}(z,w).\n\\end{align*}\nThis implies that $\\Gamma(\\mathcal{H}_\\rho) = \\mathcal{H}_\\rho^\\mathcal{D}$ and that \n$\\Gamma\\vert_{\\mathcal{H}_\\rho}$ is unitary (\\cite[Lemma~I.1.5]{Ne99}). \n\\end{prf}\n\n\\subsection{Antiunitary extensions} \n\\mlabel{subsec:4.3} \n\nIn this subsection we discuss antiunitary extensions of the unitary \nrepresentations of $\\tilde G(\\mathcal{D})$ and $\\tilde G(\\mathcal{T})$ on \n$\\mathcal{H}_\\rho^\\mathcal{D}$ and $\\mathcal{H}_\\rho$ respectively. We refer to \\cite{NO17} for \nmore background on antiunitary representations. \nWe are mainly interested in antiunitary  extensions \nto the simply connected group $\\tilde G_\\tau \\cong \\tilde G(\\mathcal{T})_\\tau$. \nThe involution \n$\\tau$ acts on $\\mathcal{T}$ and $\\mathcal{D}$ by \n\\[ \\tau_\\mathcal{T}(z) = - \\overline z \\quad \\mbox{ and }\\quad \\tau_\\mathcal{D}(z) =  \\overline z.\\] \nFrom \n\\begin{equation}\n  \\label{eq:cayley-tau}\n\\tau_\\mathcal{T}(c(z)) = i(e +\\overline z)(e - \\overline z)^{-1}   \n=   c(\\overline z)=   c(\\tau_\\mathcal{D}(z))\n\\end{equation}\nwe derive that the Cayley transform intertwines \n$\\tau_\\mathcal{T}$ with $\\tau_\\mathcal{D}$. \nWe define \n\\[ \\tau_G(g) := \\tau_\\mathcal{D} \\circ g \\circ \\tau_\\mathcal{D} \n\\quad \\mbox{ for } \\quad g \\in G(\\mathcal{D}),\n \\quad \\mbox{ and } \\quad \n\\tau_G(g) := \\tau_\\mathcal{T} \\circ g \\circ \\tau_\\mathcal{T} \\quad \\mbox{ for } \\quad \ng \\in G(\\mathcal{T}).\\] \nThen \\eqref{eq:cayley-tau} implies that \n\\begin{equation}\n  \\label{eq:cayley-g-inter}\nc \\tau_G(g) c^{-1} = \\tau_G(cgc^{-1}) \\quad \\mbox{ for } \\quad  \ng \\in \\mathcal{G}(\\mathcal{D}).\n\\end{equation}\n \n\n\n\\begin{defn} \\mlabel{def:4.4}\nLet $\\tilde G(\\mathcal{D})$ denote the universal covering \nof the group $G(\\mathcal{D}) = \\mathop{{\\rm Aut}}\\nolimits(\\mathcal{D})_0$ and \n$\\tau_G$ denote the involution of $\\tilde G(\\mathcal{D})$ induced \nby conjugating with $\\tau_\\mathcal{D}$. \nAn antiunitary representation $(U, \\mathcal{H})$ of \n\\[ \\tilde G(\\mathcal{D})_{\\tau} := \\tilde G(\\mathcal{D}) \\rtimes \\{\\mathbf{1},\\tau_G\\} \\] \nis called a {\\it positive energy representation} if \n\\[ -i\\dd U(\\bd)  \\geq 0 \\quad \\mbox{ holds for } \\quad \\bd := -ih. \\] \n\\end{defn}\n\nPositive energy representations $(U,\\mathcal{H})$ of \n$\\tilde G(\\mathcal{D})_{\\tau}$  are direct sums \nof representations which are holomorphically induced from \nunitary representations of $\\tilde K(\\mathcal{D})$ \n(see \\cite{Ne13} for details and definitions), i.e., \nrealized in reproducing kernel spaces \n$\\mathcal{H}_\\rho^\\mathcal{D} \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ \nfor norm-continuous unitary representations $(\\rho, \\mathcal{K})$ of $\\tilde K(\\mathcal{D})$. \n\nFor any such representation, the unitary one-parameter group \n$U_t := U(\\exp t \\bd)$ acts on $f \\in \\mathcal{H}_\\rho^\\mathcal{D}$ by \n\\[ (U_t f)(z) \n= \\rho(\\exp t\\bd) f(\\exp(-t\\bd).z) \n= \\rho(\\exp t\\bd) f(e^{it}z),\\] \nso that \n\\[ - i \\partial U(\\bd) f \n= -i\\partial \\rho(\\bd) f + \\dd f(z)z,\\] \nwhere $(Ef)(z) := \\dd f(z)z$ is the Euler operator on holomorphic functions. \nTherefore \n\\begin{equation}\n  \\label{eq:specinf}\n{\\rm Spec}(-i\\partial U(\\bd)) \\subseteq \n{\\rm Spec}(-i\\partial \\rho(\\bd)) + {\\mathbb N}_0.\n\\end{equation}\nAs $\\rho$ is norm-continuous, the operator $\\partial \\rho(\\bd)$ is bounded, \nand thus $-i\\partial U(\\bd)$ is bounded from below. \nIf, in addition, $(\\rho,\\mathcal{K})$ is irreducible, then \n$-i \\dd \\rho(\\bd) = \\zeta \\mathbf{1}$ for some $\\zeta \\in {\\mathbb R}$ and \n\\[ - i \\partial U(\\bd) f \n= \\zeta f + \\dd f(z)z\\] \nshows that $\\mathcal{K} = \\ker(i\\partial U(\\bd) + \\zeta \\mathbf{1})$ is the minimal \neigenspace of $-i\\partial U(\\bd)$. \nHere we use that, if $f \\: \\mathcal{D} \\to \\mathcal{K}$ is holomorphic and \n$ f= \\sum_{n = 0}^\\infty f_n$ is its Taylor expansion, then \n\\[ \\dd f(z) z = \\sum_{n = 0}^\\infty n f_n(z).\\]  \n\nWe recall that the positive definiteness of the kernel \n$Q_\\rho^\\mathcal{D}$ implies that, for the \ncorresponding unitary representation $U_\\rho^\\mathcal{D}$, the operator \n$-i \\partial U_\\rho^\\mathcal{D}(\\bd)$ is bounded from below. As \n$\\bd$ belongs to a minimal invariant cone in ${\\mathfrak g}$, it even follows that \n$-i \\partial U_\\rho^\\mathcal{D}(\\bd) \\geq 0$ and hence that \n $-i\\partial \\rho(\\bd) \\geq 0$ (see \\cite[Thms.~X.4.1, XIV.1.3]{Ne99}). \nActually the positive cone $C_\\rho \\subseteq {\\mathfrak k}(\\mathcal{D})$ contains \nthe intersection $C_{U^\\mathcal{D}_\\rho} \\cap {\\mathfrak k}(\\mathcal{D})$, which  \nhas interior points. \n\n\n\\begin{prop} \\mlabel{prop:c.5}\nSuppose that the representation $(\\rho, \\mathcal{K})$ of $\\tilde H_{\\mathbb C}$ is irreducible.  \nFor the representations  \n$(U_\\rho^\\mathcal{D}, \\mathcal{H}^\\mathcal{D}_\\rho)$ of $\\tilde G(\\mathcal{D})$ and \n$(U_\\rho, \\mathcal{H}_\\rho)$ of $\\tilde G(\\mathcal{T})$,  the following are equivalent: \n\\begin{itemize}\n\\item[\\rm(a)] $(U_\\rho^\\mathcal{D}, \\mathcal{H}^\\mathcal{D}_\\rho)$ \nextends to an antiunitary representation of \n$\\tilde G(\\mathcal{D})_\\tau$. \n\\item[\\rm(b)] $(\\rho,\\mathcal{K})$ extends to an \nantilinear representation of $\\tilde H_{{\\mathbb C},\\tau}$, \ni.e., there exists a conjugation $J_\\mathcal{K}$ on $\\mathcal{K}$ satisfying \n$J_\\mathcal{K} \\rho(g) J_\\mathcal{K} = \\rho(\\tau_{\\tilde H_{\\mathbb C}}(g))$ for \n$g \\in \\tilde H_{\\mathbb C}$.\n\\item[\\rm(c)] $(U_\\rho, \\mathcal{H}_\\rho)$ \nextends to an antiunitary representation of \n$\\tilde G(\\mathcal{T})_\\tau$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{prf} (a) $\\Rightarrow$ (b): Suppose that $(U_\\rho^\\mathcal{D}, \\mathcal{H}^\\mathcal{D}_\\rho)$ \nextends to an antiunitary representation of \n$\\tilde G(\\mathcal{D})_\\tau$ by $J := U_\\rho^\\mathcal{D}(\\tau_G)$. Then \n\\[ J (-i\\partial U_\\rho^\\mathcal{D}(\\bd)) J \n= i\\partial U_\\rho^\\mathcal{D}( \\tau(\\bd)) \n= i\\partial U_\\rho^\\mathcal{D}(-\\bd) \n= -i\\partial U_\\rho^\\mathcal{D}(\\bd) \\] \nimplies that $J$ commutes with $i\\dd U_\\rho^\\mathcal{D}(\\bd)$, hence preserves its eigenspaces \nand in particular the minimal eigenspace $\\mathcal{K}$ of constant functions. \nTherefore  the restriction $J_\\mathcal{K} := J\\vert_{\\mathcal{K}}$ defines an \nantiunitary extension of the unitary representation \n$(\\rho,\\mathcal{K})$ of $\\tilde K(\\mathcal{D})$ to the group \n$\\tilde K(\\mathcal{D})_\\tau  = \\tilde K(\\mathcal{D}) \\rtimes \\{\\mathbf{1},\\tau\\}$. \nAs $\\tilde K(\\mathcal{D})$ and $\\tilde H_{\\mathbb C}$ are simply connected, \nand  the homomorphism $\\tilde J \\: \\tilde K(\\mathcal{D}) \\to \\tilde H_{\\mathbb C}$ induces  \non the Lie algebra level an inclusion ${\\mathfrak k}(\\mathcal{D}) \\to {\\mathfrak h}_{\\mathbb C}$ \nof a real form, $\\tilde J$ defines the universal complexification \nof $\\tilde K(\\mathcal{D})$. Hence \nthe finite dimensional representation $\\rho$ extends to a holomorphic representation \nof $\\tilde H_{{\\mathbb C}}$, so that we obtain a representation \nof $\\tilde H_{{\\mathbb C},\\tau}$. \n\n\\nin (b) $\\Rightarrow$ (c): Let $J_\\mathcal{K}$ be a conjugation on $\\mathcal{K}$ extending \nthe representation $\\rho$ to $\\tilde H_{{\\mathbb C},\\tau}$. We define \n\\[(Jf)(z) := J_\\mathcal{K} f(\\tau(z))= J_\\mathcal{K} f(-\\overline z).\\] \nIt is clear that $J$ is an antilinear involution on \n$\\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$. To see that it is antiunitary on $\\mathcal{H}_\\rho$, \nwe have to verify that \n\\[ Q_\\rho(-\\overline z, - \\overline w) \n= J_\\mathcal{K} Q_\\rho(w,z) J_\\mathcal{K} \\quad \\mbox{ for } \\quad z,w \\in \\mathcal{D}.\\] \nThe left hand side is given by \n\\[ Q_\\rho(-\\overline z, - \\overline w) \n= \\rho(\\tilde Q(-\\overline z, -\\overline w)),\\] \nand the right hand side by  \n\\[ J_\\mathcal{K} Q_\\rho(w,z) J_\\mathcal{K} \n= \\rho\\big(\\tau \\tilde Q(w,z) \\tau)\n= \\rho\\big(\\tau_{\\tilde H_{\\mathbb C}}(\\tilde Q(w,z))).\\] \nBy the uniqueness of lifts, it therefore suffices to verify that \n\\begin{equation}\n  \\label{eq:qrela}\n\\tau_{H_{\\mathbb C}}(Q(w,z)) \n= Q(\\tau(z), \\tau(w)) \\quad \\mbox{ for } \\quad z,w \\in \\mathcal{D}.\n\\end{equation}\n\nWe start with the observation that \n\\begin{equation}\n  \\label{eq:pconj1}\nP(-\\overline z) \n= P(\\overline z) = \\overline{P(z)} = \\tau_{H_{\\mathbb C}}(P(z)) \\quad \\mbox{ for }\\quad \nz \\in E_{\\mathbb C}^\\times.\n\\end{equation}\nWe further have \n\\begin{equation}\n  \\label{eq:pconj2}\nQ(\\tau(z), \\tau(w)) = Q(-\\overline z,-\\overline w) \n= P\\Big(\\frac{-\\overline z+ w}{2i}\\Big) = Q(w,z), \n\\end{equation}\nwhich leads to \n\\begin{align}\n  \\label{eq:pconj3}\nQ(\\tau(z), \\tau(w)) \n&= Q(w,z) \n= P\\Big(\\frac{w-\\overline z}{2i}\\Big) \n= \\tau_{H_{\\mathbb C}}\\Big(P\\Big(\\frac{-\\overline w+z}{-2i}\\Big)\\Big)\\notag \\\\\n&= \\tau_{H_{\\mathbb C}}\\Big(P\\Big(\\frac{z-\\overline w}{2i}\\Big)\\Big)\n= \\tau_{H_{\\mathbb C}}(Q(z,w)).\n\\end{align}\nThis proves \\eqref{eq:qrela}. \nLifting \\eqref{eq:pconj2} to $\\tilde H_{\\mathbb C}$ and composing with \nthe extension of $\\rho$ to $\\tilde H_{{\\mathbb C},\\tau}$ by \n$\\rho(\\tau) := J_\\mathcal{K}$, we get \n\\begin{equation}\n  \\label{eq:pconj4}\nQ_\\rho(\\tau(z), \\tau(w)) \n= J_\\mathcal{K} Q_\\rho(z,w) J_\\mathcal{K}.\n\\end{equation}\nThis implies that $(Jf)(z) = J_\\mathcal{K} f(-\\overline z)$ defines an \nantiunitary involution on $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$.  \n\nNext we show that $U_\\rho(\\tau) := J$ defines an antiunitary extension \nof $U_\\rho$ to $\\tilde G(\\mathcal{T})_\\tau$. As \n$\\tau(g) = \\tau g \\tau$ holds in $\\mathop{{\\rm Diff}}\\nolimits(\\mathcal{T})$ for any $g \\in G(\\mathcal{T})$, \nwe obtain for $g \\in \\tilde G(\\mathcal{T})$ the relation \n\\[ (J U_\\rho(g)f)(z) \n= J_\\mathcal{K} J_\\rho(g,g^{-1}.\\tau(z)) f(g^{-1}.\\tau(z)) \n\\ {\\buildrel{!}\\over =}\\ \nJ_\\rho(\\tau(g),\\tau(g)^{-1}.z) J_\\mathcal{K} f(\\tau(\\tau(g)^{-1}.z)) \\] \nfollows from \n\\begin{align*}\n \\tau_{H_{\\mathbb C}}(J(g,g^{-1}.\\tau(z))) \n&= \\tau_{H_{\\mathbb C}}(J(g^{-1},\\tau(z)))^{-1} \n= \\tau_{H_{\\mathbb C}}(\\dd(g^{-1})(-\\overline z))^{-1} \\\\\n&= \\dd(\\tau(g)^{-1})(z)^{-1} \n= J(\\tau(g), \\tau(g)^{-1}.z).  \n\\end{align*}\n\n\\nin (c) $\\Rightarrow$ (a): Since we have a \nunitary intertwining operator $\\Gamma \\: \\mathcal{H}_\\rho \\to \\mathcal{H}_\\rho^\\mathcal{D}$ \n(Lemma~\\ref{lem:gamma-intertwine}), \nit suffices to show that $\\Gamma$ is also compatible with the \ninvolutions $\\tau_G$ on the groups $\\tilde G(\\mathcal{D})$ and $\\tilde G(\\mathcal{T})$ in \nthe sense that \n\\[\\Gamma \\circ U_\\rho(\\tau_G(g)) = U^\\mathcal{D}_\\rho(\\tau_G(g)) \\circ \\Gamma \n\\quad \\mbox{ for } \\quad g \\in \\tilde G(\\mathcal{T}).\\] \nAs \n\\[  \\Gamma \\circ U_\\rho(\\tilde\\alpha_c(g)))\n =   U_\\rho^\\mathcal{D}(g) \\circ \\Gamma  \\] \nby Lemma~\\ref{lem:gamma-intertwine}, the assertion \nfollows from \\eqref{eq:cayley-g-inter}, which, in view of the uniqueness\nof lifts, implies that $\\tilde\\alpha_c(\\tau_G(g)) \n= \\tau_G(\\tilde\\alpha_c(g))$ for \n$g \\in \\tilde G(\\mathcal{D})$. \n\\end{prf}\n\n\n\\begin{lem} The homomorphism \n\\[ \\tilde J \\: \\tilde G^{\\tau_G} \\to \\tilde H_{\\mathbb C} \\] \nis injective and defines the universal complexification of $\\tilde G^{\\tau_G}$. \nIn particular, it defines an isomorphism $\\tilde G^{\\tau_G} \\to H^\\sharp \n= \\langle \\exp_{\\tilde H_{\\mathbb C}} {\\mathfrak h} \\rangle$. \n\\end{lem}\n\n\\begin{prf} Since the image of $\\tilde J$ is the connected Lie subgroup \nwith Lie algebra ${\\mathfrak h}$, it suffices to show that it is injective. \n\n\\nin (a) Recall that $\\tilde J^\\mathcal{D} \\: \\tilde K(\\mathcal{D}) \\to \\tilde H_{\\mathbb C}$ is the \nuniversal complexification of the simply connected group \n$\\tilde K(\\mathcal{D})$. We know from the Gelfand--Raikov Theorem \n(\\cite[Thm.~XI.5.2]{Ne99})  that the contractive holomorphic representations \nof $S_C$ separate the points of $S_C$. \nThis implies that the $\\mathcal{T}$-positive representations of $\\tilde H_{\\mathbb C}$ \nseparate the points of $\\tilde H_{\\mathbb C}$ because composition with \n$\\tilde J^\\mathcal{D}$ separates the points of $\\tilde K(\\mathcal{D})$. \n\n\\nin (b) Now let $g \\in \\tilde G^{\\tau_G}$ with $\\tilde J(g) = e$. \nAs the action of $\\tilde G^{\\tau_G}$ on $E_{\\mathbb C}$ factors through the action \nof $\\tilde H_{\\mathbb C}$, defined by the covering map into $H_{\\mathbb C}$, \n$g$ acts trivially on $\\mathcal{T}$. As $\\tilde J(g) = e$ also acts trivially \non $\\mathcal{K}$ for every $\\mathcal{T}$-positive representation $(\\rho,\\mathcal{K})$, \nit follows that $U_\\rho(g) = \\mathbf{1}$. From (a) we now derive that $g = e$. \n\\end{prf}\n\n\\begin{rem}  As any $\\mathcal{T}$-positive norm-continuous representation \n$(\\rho,\\mathcal{K})$ of $\\tilde H_\\tau$ defines a holomorphic representation \nof its universal complexification, we have $\\ker \\rho \\supseteq \\ker(\\eta_{\\tilde H})$, \ni.e., $\\rho$ factors through a representation of the subgroup \n$H^\\sharp \\subseteq \\tilde H_{\\mathbb C}$. In the preceding lemma we have seen that \nthis subgroup can be identified with $\\tilde G^{\\tau_G}$ via the isomorphism~$\\tilde J$.\n\\end{rem}\n\n\\begin{ex} For the Jordan algebra \n$E = \\mathop{{\\rm Sym}}\\nolimits_n({\\mathbb R})$, ${\\mathfrak g} =\\sp_{2n}({\\mathbb R})$ and \n${\\mathfrak h} = \\gl_n({\\mathbb R})$, we have \n\\[ \\tilde H_{\\mathbb C}\\cong \\tilde \\mathop{{\\rm GL}}\\nolimits_n({\\mathbb C}) \\cong {\\mathbb C} \\times \\mathop{{\\rm SL}}\\nolimits_n({\\mathbb C}).\\] \nThe real subgroup corresponding to ${\\mathfrak h}$ is the non-simply connected group\n$H^\\sharp \\cong {\\mathbb R} \\times \\mathop{{\\rm SL}}\\nolimits_n({\\mathbb R})$. The natural map \n$H^\\sharp \\to \\mathop{{\\rm Sp}}\\nolimits_{2n}({\\mathbb R})$ lifts to a homomorphism \n$H^\\sharp \\to \\tilde\\mathop{{\\rm Sp}}\\nolimits_{2n}({\\mathbb R})$ because the inclusion \n$\\mathop{{\\rm SO}}\\nolimits_n({\\mathbb R}) \\hookrightarrow \\mathop{\\rm U{}}\\nolimits_n({\\mathbb C})\\subseteq \\mathop{{\\rm Sp}}\\nolimits_{2n}({\\mathbb R})$ induces the trivial homomorphism \n$\\pi_1(\\mathop{{\\rm SO}}\\nolimits_n({\\mathbb R})) \\to \\pi_1(\\mathop{{\\rm Sp}}\\nolimits_{2n}({\\mathbb R}))$. \n\\end{ex}\n\n\n\\begin{rem} \\mlabel{rem:4.9}\n(a) In the proof of Proposition~\\ref{prop:c.5}, the implications \n(b) $\\Rightarrow$ (c) $\\Rightarrow$ (a) hold for any $\\mathcal{T}$-positive \nholomorphic $*$-representation \n$(\\rho,\\mathcal{K})$ of $\\tilde H_{\\mathbb C}$. Only the implication (a) $\\Rightarrow$ (b) requires \nadditional assumptions; such as irreducibility. \n\n\n\\nin (b) If the conditions of Proposition~\\ref{prop:c.5} \nare satisfied, then \nthe unitary intertwining operator $\\Gamma \\: \\mathcal{H}_\\rho \\to \\mathcal{H}_\\rho^\\mathcal{D}$ \nintertwines the conjugation $J$ on $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$ defined \nby $(Jf)(z) := J_\\mathcal{K} f(-\\overline z)$ with the conjugation $J^\\mathcal{D}$ \non $\\mathcal{H}_\\rho^\\mathcal{D}$ defined by \n$(J^\\mathcal{D} f)(z) := J_\\mathcal{K} f(-\\overline z)$. \nIn fact, we have \n\\begin{align*}\n(J^\\mathcal{D} \\Gamma(f))(z) \n&= J_\\mathcal{K} P_\\rho\\Big(\\frac{-\\overline z + ie}{2}\\Big) f(c(-\\overline z)) \n= P_\\rho\\Big(\\frac{z + ie}{2}\\Big) J_\\mathcal{K} f(- \\overline{c(z)}) \n= \\Gamma(J f)(z).\n\\end{align*}\n\n\\nin (c) For any unitary representation \n$(U,\\mathcal{H})$ of $\\tilde G$, the direct sum \n$U \\oplus (U^*\\circ \\tau_G)$  on $\\mathcal{H} \\oplus \\mathcal{H}^*$ \nextends to an antiunitary representation of $\\tilde G_\\tau$, \nso that the situations of Proposition~\\ref{prop:c.5} can always \nbe achieved by this doubling process \n(cf.\\ \\cite[Lemma~2.10]{NO17}). \nThe same argument applies to a holomorphic $*$-representation \n$(\\rho, \\mathcal{K})$ of the symmetric Lie group \n$(\\tilde H_{\\mathbb C}, \\overline\\theta)$, for which the representation \n$\\rho\\oplus (\\rho^*\\circ \\overline\\theta)$ on $\\mathcal{K} \\oplus \\mathcal{K}^*$ always \nextends to $\\tilde H_{{\\mathbb C},\\tau}$. \n\n\\nin (d) Uniqueness of extensions: From \\cite[Thm.~2.11]{NO17} we \nknow that any two extensions of a unitary representation \n$(U,\\mathcal{H})$ of $\\tilde G$ to an antiunitary representation of \n$\\tilde G_\\tau$ are unitarily equivalent, regardless of whether the \nrepresentation $U$ is irreducible or not. \n\n\\nin (e) Irreducible holomorphic representations \n$(\\rho, \\mathcal{K})$ of $(\\tilde H_{\\mathbb C}, \\overline\\theta)$ are finite dimensional \nand every finite dimensional irreducible  complex representation \n$(\\beta,\\mathcal{K})$ of the complex Lie algebra ${\\mathfrak h}_{\\mathbb C}$ on a complex vector space $\\mathcal{K}$ \nintegrates to a holomorphic representation \n$(\\rho,\\mathcal{K})$ of $\\tilde H_{\\mathbb C}$. Moreover, $\\mathcal{K}$ carries a Hilbert space \nstructure for which $\\rho$ is a $*$-representation if and only if, \nfor each $x \\in {\\mathfrak z}({\\mathfrak h})$, the operator \n$\\beta(x)$ is diagonalizable with real eigenvalues. \nHere we use that $\\theta(x) = x$ for $x \\in {\\mathfrak z}({\\mathfrak h})$.  \nThen the subgroup $\\tilde H_{\\mathbb C}^{\\overline\\theta}$ is represented \nby a compact group and the scalar product can be obtained by averaging any \ngiven scalar product on $\\mathcal{K}$. \n\n\\nin (f) A representation of $\\tilde H_{\\mathbb C}$ extends to \n$\\tilde H_{{\\mathbb C},\\tau}$ if and only if there exists a conjugation $J_\\mathcal{K}$ \non $\\mathcal{K}$ commuting with $\\beta({\\mathfrak h})$. This is equivalent to the representation \n$(\\beta, \\mathcal{K})$ of the Lie algebra ${\\mathfrak h}$ being {\\it real} \nin the sense that $\\mathcal{K}_{\\mathbb R} := \\mathop{{\\rm Fix}}\\nolimits(J_\\mathcal{K})$ is a real ${\\mathfrak h}$-invariant subspace \nand the representation on $\\mathcal{K}$ arising by complex linear extension. \nConversely, every real $*$-representation \n$(\\rho, \\mathcal{K}_{\\mathbb R})$ of ${\\mathfrak h}$, resp., the group $H^\\sharp \\subseteq \\tilde H_{\\mathbb C}$, \nextends to a complex $*$-representation \n$(\\rho, (\\mathcal{K}_{\\mathbb R})_{\\mathbb C})$ of $H^\\sharp_\\tau$ by $\\rho(\\tau)(x + iy) \n:= x - iy$ and this representation extends to a holomorphic representation \nof the universal complexification $\\tilde H_{\\mathbb C}$. \nTherefore the irreducible extendable representations of $\\tilde H_{{\\mathbb C},\\tau}$ \ncorrespond to the irreducible real $*$-representations of ${\\mathfrak h}$. \n\n\\nin (g) The classification of real irreducible representations can easily \nbe reduced to the classification of the complex ones by the following observations \n(\\cite[Ch.~8]{On20}, \\cite{Iw59}). If $(\\beta,V)$ is a real simple \n${\\mathfrak h}$-module, then we say it is of real, complex of quaternionic \ntype if $\\mathop{{\\rm End}}\\nolimits_{\\mathfrak h}(V) \\cong {\\mathbb R},{\\mathbb C}$ or $\\H$. \nIf $V$ is of real type, then $V_{\\mathbb C} \\cong \\overline{V_{\\mathbb C}}$ is simple. \nIf $V$ is of complex type, then $V$ \nitself is complex simple with $\\overline V \\not\\cong V$ \n(there is no antilinear automorphism), and if \n$V$ is of quaternionic type, then $V \\cong \\overline V$ is simple complex, \nand there exists an antilinear endomorphism $J$ with $J^2 = - \\mathbf{1}$. \nWe thus obtain for each simple real ${\\mathfrak h}$-module a simple complex \none, and the three types are reflected in the (non-)existence of \nantilinear endomorphisms $J$ with $J^2 = \\pm \\mathbf{1}$.\n\nThus every simple complex module $V$ with $\\overline V \\not\\cong V$ \ndefines by restriction of scalars a real simple module \n$V^{\\mathbb R}$ of complex type, and $\\overline V^{\\mathbb R} \\cong V^{\\mathbb R}$. \nIf $\\overline V \\cong V$, then  $V$ there exists an \nantilinear endomorphism $J$ with $J^2 = \\pm\\mathbf{1}$. \nIf $J^2 = \\mathbf{1}$, then $V^J = \\mathop{{\\rm Fix}}\\nolimits(J)$ is a real simple module of real type, \nand if $J^2 =- \\mathbf{1}$, then $V^{\\mathbb R}$ is a real simple module of \nquaternionic type. \n\nThe three types of complex simple modules can be identified in \nterms of their highest weights. \nLet ${\\mathfrak b} \\subseteq {\\mathfrak h}^{\\theta}$ be maximal abelian \nand ${\\mathfrak a} \\subseteq {\\mathfrak h}^{-\\theta} \\cap {\\mathfrak z}_{\\mathfrak h}({\\mathfrak b})$ be maximal abelian, \nso that ${\\mathfrak c} := {\\mathfrak a} \\oplus {\\mathfrak b}$ is a Cartan subalgebra \nof ${\\mathfrak h}$ which is ``maximally compact''. In particular ${\\mathfrak b}$ contains \nregular elements of ${\\mathfrak h}$. We parametrize irreducible complex representations \n$(\\beta_\\lambda, V_\\lambda)$ of ${\\mathfrak h}$ \nby their highest weights $\\lambda \\: {\\mathfrak c} \\to  {\\mathbb C}$ which are real on ${\\mathfrak a}$,  \npurely imaginary on ${\\mathfrak b}$ and dominant with respect to  a Weyl chamber \nintersecting $i{\\mathfrak b}$, so that the Weyl chamber can be chosen invariant \nunder $\\theta$. \nAs the weights of the complex conjugate representation are the complex \nconjugates of the weights and $-\\lambda \\circ \\theta = \\overline\\lambda$ on ${\\mathfrak c}$, \nthe highest weight of the complex conjugate representation is\n $\\overline{V_\\lambda}$ is $-w.(\\lambda \\circ \\theta) = w.\\overline\\lambda$, where \n$w \\in \\mathcal{W}$ is the longest element in the Weyl group. \nTherefore  $\\overline{V_\\lambda}\\cong V_\\lambda$ is equivalent to \n$\\lambda = w.\\overline\\lambda$. \nIf this is the case, then \nthere exists an antilinear endomorphism $J$ with $J^2 = (-1)^{\\varepsilon_\\lambda} \\mathbf{1}$. \nThe number $\\varepsilon_\\lambda$ can be determined from $\\lambda$ and the \nroot system; see \\cite[Thm.~8.3]{On20} and the subsequent discussion \nfor details. \n\\end{rem}\n\n\n\n\n\\subsection{$L^2$-realization} \n\\mlabel{subsec:4.4} \n\nThe representations $(\\rho, \\tilde H_{\\mathbb C})$ that are of interest for \nus are the $\\mathcal{T}$-positive ones. This condition \nis equivalent to the positive definiteness \nof the function \n$P_\\rho \\: \\mathcal{T} \\to B(\\mathcal{K})$ on the $*$-semigroup $\\mathcal{T}$ with involution \n$z^* := - \\overline z$, which in turn is equivalent to the existence \nof a unique $\\mathop{{\\rm Herm}}\\nolimits^+(\\mathcal{K})$-valued measure $\\mu$ on $E_+^\\star$ with \n\\[ P_\\rho = \\mathcal{L}(\\mu), \\qquad \n\\mathcal{L}(\\mu)(z) = \\int_{E^*} e^{-\\lambda(z)}\\, d\\mu(\\lambda)\\] \n(cf.\\ \\cite[Thm.~V.12]{HN01})\n\n\\begin{lem} The measure $\\mu$ has the following transformation properties:\n  \\begin{itemize}\n  \\item[\\rm(a)] $J_\\mathcal{K} \\mu J_\\mathcal{K} = \\mu$, i.e., $\\mu$ takes \nvalues in the subspace $\\mathop{{\\rm Sym}}\\nolimits^+(\\mathcal{K}^{J_\\mathcal{K}})$ of positive symmetric \noperators on real Hilbert space $\\mathcal{K}^{J_\\mathcal{K}}$. \n  \\item[\\rm(b)] $(g^{-1})_*\\mu = \\rho(g) \\mu \\rho(g)^*$ for $g \\in \n\\tilde H$; i.e., for every compactly supported continuous function we have \n\\[ \\int_{E^*} \\phi(g^{-1}.\\lambda)\\, d\\mu(\\lambda) \n = \\int_{E^*} \\phi(\\lambda)\\, \\rho(g)\\, d\\mu(\\lambda) \\, \\rho(g)^*.\\]\n  \\end{itemize}\n\\end{lem}\n\n\\begin{prf} (a)  The relation \n$\\tau_{H_{\\mathbb C}}(P(z)) = \\overline{P(z)} = P(\\overline z) = P(-\\overline z)$ \nfrom \\eqref{eq:pconj1} implies that \n\\[\\mathcal{L}(\\mu)(\\overline z) =  P_\\rho(\\overline z) \n= J_\\mathcal{K} P_\\rho(z) J_\\mathcal{K} = J_\\mathcal{K} \\mathcal{L}(\\mu)(z)  J_\\mathcal{K} \n= \\mathcal{L}(J_\\mathcal{K} \\mu J_\\mathcal{K})(\\overline z)\n \\quad \\mbox{ for } \\quad z \\in E_+ + i E = -i \\mathcal{T}.\\]  \nAs any tempered measure on $E_+^\\star$ is uniquely determined \nby its Laplace transform, the assertion follows.\n\n\\nin (b) Starting from the relation \n\\[ P_\\rho(g.z) = \\rho(g) P_\\rho(z) \\rho(g)^* \\quad \\mbox{ for } \\quad \ng \\in \\tilde H, z \\in \\mathcal{T},\\] \nwe obtain \n\\[ \\mathcal{L}((g^{-1})_*\\mu)(z) \n=  \\mathcal{L}(\\mu)(g.z) \n= \\rho(g) \\mathcal{L}(\\mu)(z) \\rho(g)^*\n= \\mathcal{L}(\\rho(g) \\mu \\rho(g)^*)(z).\\] \nAs in (a), this implies (b). \n\\end{prf}\n\nWith a slight modification of the normalizations \nin \\cite[\\S 3.2]{NOO20} (a factor $2$ in the exponent), it follows that \n\\begin{equation}\n  \\label{eq:Phi}\n\\Phi \\: L^2(E^*,\\mu,\\mathcal{K}) \\to \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T}, \\mathcal{K}), \\quad \n\\Phi(f)(z) = \\int_{E^*} e^{\\frac{i\\lambda(z)}{2}} d\\mu(\\lambda) f(\\lambda) \n\\end{equation}\nmaps $L^2(E^*,\\mu,\\mathcal{K})$ injectively \nonto the reproducing kernel Hilbert space $\\mathcal{H}_Q$ with $B(\\mathcal{K})$-valued kernel \n\\[ Q_\\rho(z,w) = \\mathcal{L}(\\mu)\\Big(\\frac{z- \\overline w}{2i}\\Big) \n= \\int_{E^*} e^{i\\lambda(z-\\overline w)\/2}\\, d\\mu(\\lambda).\\] \nNote that \n\\begin{equation}\n  \\label{eq:eval1}\n\\langle \\xi, \\Phi(f)(z) \\rangle  = \\langle e_{-i \\overline z\/2}\\, \\xi, \\mu \\cdot f \\rangle \n\\quad \\mbox{ for } \\quad v \\in \\mathcal{K}, z \\in \\mathcal{T}.\n\\end{equation}\n\nThe map $\\Phi$ is a unitary intertwining operator for the \nantiunitary representation of the group \n$E \\rtimes \\tilde H_\\tau$ on $L^2(E^*,\\mu,\\mathcal{K})$, defined by \n\\begin{equation}\n  \\label{eq:Urep}\n (U(x,g)f)(\\lambda) := e^{-\\frac{i\\lambda(x)}{2}} \\rho^\\theta(g) f(g^{-1}.\\lambda), \n\\qquad U(0,\\tau)f := J_\\mathcal{K} \\circ f, \n\\end{equation}\nwhere $\\rho^\\theta :=\\rho \\circ \\theta$ is the $\\theta$-twist of $\\rho$. \nHere we use that the representation of $E \\rtimes \\tilde H_\\tau$ on \n$\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$ has the form \n\\[  (U_\\rho(x,g)f)(z) \n:= \\rho(g) f(g^{-1}.(z - x)) \n\\quad \\mbox{ for } \\quad g \\in \\tilde H, x \\in E, z \\in \\mathcal{T},\\] \nand \n\\[ (U_\\rho(\\tau)f)(z) := J_\\mathcal{K} f(-\\overline z).\\] \n\n\\begin{ex} The case where $\\mathcal{K} = {\\mathbb C}$ is one-dimensional is particularly \ninteresting and easily described. Then the representation $\\rho \\: \\tilde H \n\\to \\mathop{{\\rm GL}}\\nolimits_1({\\mathbb R}) \\cong {\\mathbb R}^\\times$ is a real character. We now take a closer \nlook at these representations for the case where $E$ is simple. \n\nLet $E$ be a simple euclidean Jordan algebra or rank $r$ \nwhose Pierce subspaces are of dimension~$d$. For \n\\begin{equation}\n  \\label{eq:walachset}\ns \\in \\Big\\{0, \\cdots, (r-1)\\frac{d}{2}\\Big\\} \\cup \n\\Big((r-1)\\frac{d}{2}, \\infty\\Big), \n\\end{equation} \nwe consider the corresponding Riesz measure $\\mu_s$ on \n$E_+^\\star$ whose Fourier (Laplace) transform satisfies \n\\[ \\mathcal{L}(\\mu_s)(z)= \\tilde\\mu_s(iz) = \\Delta_E(z)^{-s} \\quad \\mbox{ for } \\quad \nz \\in E_+ + i E = -i \\mathcal{T},\\] where \n$\\Delta_E$ is the Jordan determinant (\\cite{FK94}). \nRecall the relations \n\\[ \\Delta_E(g.x) = {\\det}_E(g)^{r\/n} \\Delta_E(x)\\quad \\mbox{ and } \\quad \n\\Delta_E(P(y)x) =\\Delta_E(y)^2 \\Delta_E(x) \\] \n(\\cite[Prop.~III.4.2]{FK94}). \nFor the corresponding one-dimensional representation $(\\rho_s, {\\mathbb C})$ \nof $\\tilde H$, the relations \n\\[ \\Delta_E(z)^{-s} = P_{\\rho_s}(z) = \\rho_s(\\tilde P(z)) \n\\quad \\mbox{ and } \\quad \n\\tilde P(g.z) = g \\tilde P(z) g^* \\] \nnow lead to $\\rho_s(gg^*) = {\\det}_E(g)^{-rs\/n}$, which in turn shows that \n\\begin{equation}\n  \\label{eq:det-s-rel}\n \\rho(g)  = {\\det}_E(g)^{-rs\/2n}.\n\\end{equation}\nIt follows in particular that, as a representation of $\\tilde H$, \n$\\rho$ actually factors through the group $H$ itself. \n\\end{ex}\n\n\\subsection{Distribution vectors} \n\\mlabel{subsec:4.5} \n\nThe main purpose of this section is to realize $C$-positive \nunitary representations of $\\tilde G$ in such a way that \nwe get a nice picture of a sufficiently large space of distribution \nvectors. This is finally done in this subsection, where we formulate the \nprecise results on distribution vectors that we shall use in \nSection~\\ref{sec:5} to construct local nets of standard subspaces. \nThe distribution vectors are most easily identified in the $L^2$-realization. \n\n\\begin{prop} \\mlabel{prop:c.10} For $\\xi \\in \\mathcal{K}$, the corresponding constant \nfunction on $E^*$ defines a distribution vector $\\eta_\\xi$ of the \nunitary representation of $(E,+)$ on $L^2(E^*,\\mu;\\mathcal{K})$ by \n\\[ \\eta_\\xi(f) = \\int_{E^*} \\langle d\\mu(\\lambda) f, \\xi \\rangle.\\] \nThese distribution vectors have the following properties: \n\\begin{itemize}\n\\item[\\rm(a)] The subspace $\\{ \\eta_\\xi \\: \\xi \\in \\mathcal{K}\\}$ \nin $L^2(E^*,\\mu;\\mathcal{K})^{-\\infty}$ \nis cyclic for the group $(E,+)$. \n\\item[\\rm(b)] $U^{-\\infty}(0,g)\\eta_\\xi = \\eta_{\\rho^\\theta(g)\\xi}$ \nfor $g \\in \\tilde H$ and $\\xi \\in \\mathcal{K}$. \n\\item[\\rm(c)]  $U^{-\\infty}(0,\\tau)\\eta_\\xi = \\eta_{J_\\mathcal{K} \\xi}$. \n\\end{itemize}\n\\end{prop}\n\n\\begin{prf} Let $\\|\\cdot\\|$ denote the euclidean norm on $E^*$. \nSince the measure $\\mu$ is tempered \n(\\cite[Thm.~V.12]{HN01}), there \nexists an $m \\in {\\mathbb N}$ such that, for every $\\xi \\in \\mathcal{K}$, the integral \n\\[ \\int_{E^*} \\frac{1}{(1 + \\|\\lambda\\|^2)^{2m}}\\, \n\\langle \\xi, \\dd \\mu(\\lambda) \\xi \\rangle \\] \nis finite.  This implies that \n$\\xi \\in (1 + \\|\\lambda\\|^2)^m L^2(E^*,\\mu;\\mathcal{K}) \n\\subseteq L^2(E^*,\\mu;\\mathcal{K})^{-\\infty}$, because \nthe enveloping algebra $S(E)$ of the \nLie algebra of the abelian group $(E,+)$ acts on $L^2(E^*,\\mu;\\mathcal{K})$ \nby multiplication with polynomial functions on~$E^*$.\n\n\\nin (a) We first observe that, for \n$\\phi \\in C^\\infty_c(E)$, we have \n\\[ \\big(U(\\phi)f\\big)(\\lambda) \n= \\int_E \\phi(x) e^{-i\\lambda(x)\/2} f(\\lambda)\\, d\\mu_E(x) \n= \\hat\\phi\\Big(\\frac{\\lambda}{2}\\Big) f(\\lambda), \n\\quad \\mbox{ where } \\quad \n\\hat\\phi(\\lambda) :=  \\int_E \\phi(x) e^{-i\\lambda(x)}\\, d\\mu_E(x)\\] \nis the Fourier transform of $\\phi$. As \n$\\mathcal{S}(E^*) \\cdot \\mathcal{K} \\subseteq \\mathcal{S}(E^*,\\mathcal{K})$ \nis dense and $\\mathcal{S}(E^*,\\mathcal{K})$ maps to a dense subspace of $L^2(E^*,\\mu;\\mathcal{K})$, \nthe assertion follows. \n\n\\nin (b) For $g \\in \\tilde H$, we have \n\\begin{align*}\n\\eta_{\\rho(g)^*\\xi}(f) \n&= \\langle f, \\rho(g)^*\\xi \\rangle \n= \\int_{E^*} \\langle d\\mu(\\lambda) f(\\lambda), \\rho(g)^*\\xi \\rangle\n= \\int_{E^*} \\langle f(\\lambda), d\\mu(\\lambda) \\rho(g)^*\\xi \\rangle\\\\\n&= \\int_{E^*} \\langle \\rho(g^{-1})^* f(\\lambda), \n\\rho(g) d\\mu(\\lambda) \\rho(g)^*\\xi \\rangle\\\\\n&= \\int_{E^*} \\langle \\rho(g^{-1})^* f(g^{-1}.\\lambda), d\\mu(\\lambda) \\xi \\rangle\\\\\n&= \\int_{E^*} \\langle \\rho^\\theta(g) f(g^{-1}.\\lambda), d\\mu(\\lambda) \\xi \\rangle\n= \\int_{E^*} \\langle (U(0,g)f)(\\lambda), d\\mu(\\lambda) \\xi \\rangle \\\\\n&= \\eta_\\xi(U(0,g)f)= (U^{-\\infty}(0,g^{-1})\\eta_\\xi)(f).\n\\end{align*}\nThis implies~(b).\n\n\\nin (c) This assertion follows from \n\\begin{align*}\n\\eta_{J_\\mathcal{K}\\xi}(f) \n&= \\int_{E^*} \\langle d\\mu(\\lambda) f(\\lambda), J_\\mathcal{K}\\xi \\rangle\n= \\int_{E^*} \\langle f(\\lambda), d\\mu(\\lambda) J_\\mathcal{K} \\xi \\rangle\\\\\n&= \\int_{E^*} \\langle f(\\lambda), J_\\mathcal{K} d\\mu(\\lambda) \\xi \\rangle\n= \\int_{E^*} \\langle d\\mu(\\lambda) \\xi, J_\\mathcal{K} f(\\lambda)  \\rangle\\\\\n&= \\overline{\\eta_\\xi(U(0,\\tau)f)}\n= (U^{-\\infty}(0,\\tau)\\eta_\\xi)(f).\\qedhere \n\\end{align*}\n\\end{prf}\n\n\nFor later use, we collect some information on the boundary values of the \nholomorphic functions in $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$. \n\n\\begin{lem} \\mlabel{lem:C.11} \nFor $\\phi \\in C^\\infty_c(E)$ and $F \\in \\mathcal{H}_\\rho$, we consider the \nholomorphic function \n\\[F_\\phi = U(\\phi^*)F \\: \\mathcal{T} \\to \\mathcal{K}, \\quad \n F_\\phi(z) := \\int_E \\overline{\\phi(x)} F(z + x)\\, d\\mu_E(x), \\] \nwhere $dx = d\\mu_E(x)$ denotes a Haar measure on~$E$.\nThis function extends to a continuous function on \n$\\overline{\\mathcal{T}} = E + i \\overline{E_+}$ which is smooth on~$E$.\n\\end{lem}\n\n\\begin{prf} We write $F = \\Phi(f)$ for some \n$f \\in L^2(E^*,\\mu;\\mathcal{K})$ as in \\eqref{eq:Phi}. Then  the integrated \nrepresentation of \nthe convolution algebra $C^\\infty_c(E)$ on $L^2(E^*,\\mu;\\mathcal{K})$ is given \nby \n\\[ \\big(U(\\phi)f\\big)(\\lambda) \n= \\int_E \\phi(x) e^{-i\\lambda(x)\/2} f(\\lambda)\\, d\\mu_E(x) \n= \\hat\\phi(\\lambda\/2) f(\\lambda).\\] \nWe thus obtain \n\\begin{align*}\nF_\\phi(z) \n&= \\int_E \\phi^*(x) F(z-x)\\, d\\mu_E(x) \n = (U_\\rho(\\phi^*)F))(z) \n = (U_\\rho(\\phi^*)\\Phi(f))(z) \\\\\n& = (\\Phi(U(\\phi^*)f))(z) \n= \\int_{E^*} e^{i\\lambda(z)\/2} d\\mu(\\lambda)\\, \\overline{\\hat\\phi(\\lambda\/2)} f(\\lambda).\n\\end{align*}\nNext we observe that the function \n$\\overline{\\hat\\phi(\\lambda\/2)} f(\\lambda)$ on $E^*$ \nis $\\mu$-integrable because $\\hat\\phi$ \nis a Schwartz function, hence $L^2$. As \n\\[ |e^{i\\lambda(z)\/2}|\n= e^{-\\Im \\lambda(z)\/2}\n= e^{-\\lambda(\\Im z)\/2} \\leq 1 \\quad \\mbox{ for } \\quad \nz \\in \\overline\\mathcal{T},\\] \nit follows from the Dominated Convergence Theorem that \n$F_\\phi$ extends to a continuous function on~$\\overline\\mathcal{T}$. \nIts restriction to $E$ is the Fourier transform \nof the measure \n$d\\mu(\\lambda) \\overline{\\hat\\phi(\\lambda\/2)} f(\\lambda)$, \nwhich remains finite when multiplied by any polynomial. Therefore its \nFourier transform is smooth. \n\\end{prf}\n\n\\begin{rem} (a) The preceding lemma implies in particular that \n$\\bo(F)(\\phi) := F_\\phi(0)$ \nis defined for every $F \\in \\mathcal{H}_\\rho$ and $\\phi \\in C^\\infty_c(E)$.\nTo see that this defines a tempered distribution on~$E$, we note \nthat, for $F = \\Phi(f)$, $f \\in L^2(E^*,\\mu;\\mathcal{K})$, we have \n\\[ F_\\phi(0) \n= \\int_{E^*} \\overline{\\hat\\phi(\\lambda\/2)}\\, d\\mu(\\lambda)  f(\\lambda),\\] \nand that the measure $\\mu \\cdot f$ on $E^*$ is tempered. \nTherefore the continuity of the Fourier transform \n$\\mathcal{S}(E) \\to \\mathcal{S}(E^*)$ implies that \n$\\bo(F)$ defines a $\\mathcal{K}$-valued tempered distribution on $E$. \nThis leads to an injective linear map \n\\begin{equation}\n  \\label{eq:boundval}\n \\bo \\: \\mathcal{H}_\\rho \\to \\mathcal{S}'(E,\\mathcal{K}), \\quad \n\\bo(F)(\\phi) = F_\\phi(0),  \n\\end{equation}\nwhose image is a Hilbert space of tempered distributions \non which the scalar product is defined by the requirement that $\\bo$ is isometric. \n\n\\nin (b) To identify this Hilbert space \nas a reproducing kernel Hilbert space, \nwe write \n\\[ \\mathop{{\\rm ev}}\\nolimits_\\phi \\: \\bo(\\mathcal{H}_\\rho) \\to \\mathcal{K} \\] \nfor evaluation in $\\phi$, which is a linear map, but \nantilinear in $\\phi$. \nFor $F \\in \\mathcal{H}_\\rho$, we then have \n\\begin{align*}\n\\langle \\bo^* \\mathop{{\\rm ev}}\\nolimits_\\phi^*\\xi, F \\rangle \n&= \\langle \\xi, F_\\phi(0) \\rangle \n= \\int_{E^*} \\overline{\\hat\\phi(\\lambda\/2)}\\, \n\\langle \\xi, d\\mu(\\lambda) \\Phi^{-1}(F)(\\lambda) \\rangle \\\\ \n&= \\langle \\hat\\phi(\\cdot\/2)\\xi, \\Phi^{-1}(F) \\rangle_{L^2} \n= \\langle \\Phi({\\hat\\phi}(\\cdot\/2) \\xi), F \\rangle_{\\mathcal{H}_\\rho}.\n\\end{align*}\nWe conclude that \n\\[ \\bo^* \\mathop{{\\rm ev}}\\nolimits_\\phi^*\\xi \n= \\Phi({\\hat\\phi}(\\cdot\/2) \\xi) \\in \\mathcal{H}_\\rho.\\] \nThe corresponding reproducing kernel is therefore given by \n\\[  \\langle \\xi, \\mathop{{\\rm ev}}\\nolimits_\\phi \\mathop{{\\rm ev}}\\nolimits_\\psi^*\\eta \\rangle \n= \\langle \\bo^*\\mathop{{\\rm ev}}\\nolimits_\\phi^* \\xi, \\bo^*\\mathop{{\\rm ev}}\\nolimits_\\psi^*\\eta \\rangle \n= \\langle {\\hat\\phi}(\\cdot\/2) \\xi, {\\hat\\psi}(\\cdot\/2) \\eta \\rangle\n_{L^2} \n= \\int_{E^*} \\overline{\\hat\\phi(\\lambda\/2)} \n{\\hat\\psi}(\\lambda\/2) \n\\langle \\xi, d\\mu(\\lambda) \\eta \\rangle.\\] \n\nNext we note that the holomorphic function \n\\[ D \\: \\mathcal{T} \\to B(\\mathcal{K}), \\quad \nD(z) = \\int_{E^*} e^{i\\lambda(z)\/2}\\ d\\mu(\\lambda)  \\] \nhas distributional boundary values $D \\in \\mathcal{S}'(E,B(\\mathcal{K}))$, given by \n\\[ D(\\phi) \n= \\int_{E^*} \\tilde{\\overline\\phi}(\\lambda\/2)\\, d\\mu(\\lambda) \n= \\int_{E^*} \\overline{\\hat\\phi(\\lambda\/2)}\\, d\\mu(\\lambda)\\]  \n(\\cite[Lemma 3.22]{NOO20}). \nThis distribution satisfies \n\\[ D(\\psi^* * \\phi) \n= \\int_{E^*} \\overline{\\hat\\phi(\\lambda\/2)}\n{\\hat\\psi}(\\lambda\/2)\\, d\\mu(\\lambda) \n= \\mathop{{\\rm ev}}\\nolimits_\\phi \\mathop{{\\rm ev}}\\nolimits_\\psi^*,\\]  \nso that it represents the reproducing kernel \nof the subspace $\\bo(\\mathcal{H}_\\rho) \\subseteq \\mathcal{S}'(E,\\mathcal{K})$ \n(cf.\\ \\cite[Def.~7.1.5]{NO18}, which does not have the factor $\\frac{1}{2}$). \n\\end{rem}\n\n\n\n\\section{Nets of standard subspaces} \n\\mlabel{sec:5} \n\nLet ${\\mathfrak g}$ be a direct sum of simple hermitian Lie algebras of tube type \n(see the table in the introduction for a list of these \nLie algebras) \nand $(\\rho, \\mathcal{K})$ be a $\\mathcal{T}$-positive norm-continuous representation  \nof $\\tilde H_{{\\mathbb C},\\tau}$ (cf.~Remark~\\ref{rem:4.9}), \nso that we obtain an antiunitary representation \n$(U_\\rho, \\mathcal{H}_\\rho)$ of $\\tilde G_\\tau$ on the Hilbert subspace \n$\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$. \nIn this section we show that, for the standard subspace \n${\\tt V} \\subseteq \\mathcal{H}_\\rho$ specified by the pair $(h,\\tau_G)$, there exists a \nreal subspace ${\\sf E} \\subseteq \\mathcal{K}$ contained in ${\\tt V}^{-\\infty}$,  \nto which we can apply Section~\\ref{sec:3}. This leads to the identity \n${\\tt V} = {\\sf H}_{\\sf E}(S_{\\tt V}^0)$ and to a net ${\\tt V}(\\mathcal{O}) := {\\sf H}_{\\sf E}(\\mathcal{O})$ \nof cyclic subspaces associated to non-empty open subsets $\\mathcal{O} \\subseteq G$. \nAs a consequence of the construction in Section~\\ref{sec:4}, \nthis covers all irreducible \n$C$-positive antiunitary representations $(U,\\mathcal{H})$ of $\\tilde G_\\tau$ \n(Theorem~\\ref{thm:5.1}). \n\n\\subsection{Nets of standard subspaces on Lie groups} \n\n\nWe briefly recall our setting. In this section ${\\mathfrak g}$ denotes a semisimple Lie algebra. \n\\begin{itemize}\n\\item We assume that ${\\mathfrak g}$ contains a closed pointed generating invariant \nconvex cone $C \\subseteq {\\mathfrak g}$. \n\\item We fix an {\\it Euler element} $0 \\not=h \\in [{\\mathfrak g},{\\mathfrak g}]$, \ni.e., $\\mathop{{\\rm ad}}\\nolimits h$ is diagonalizable with ${\\rm Spec}(\\mathop{{\\rm ad}}\\nolimits h) \\subseteq  \\{-1,0,1\\}$. \nAll these elements are conjugate under inner automorphisms \n(cf.~\\cite[Ex.~6.1]{MN20}), so that it suffices to consider one \nsuch element $h$. Together with the preceding item, the \nexistence of such an element is equivalent \nto all simple non-compact ideals of ${\\mathfrak g}$ being hermitian of tube type \n(see the proof of~\\cite[Thm~3.12]{MN20} and \\cite[Thm.~5.6]{O91}). \n\\item We assume that ${\\mathfrak g}^0(h) = \\ker(\\mathop{{\\rm ad}}\\nolimits h)$ contains no proper ideal, \ni.e., that ${\\mathfrak g}$ is generated as a Lie algebra by the eigenspaces ${\\mathfrak g}^{\\pm 1}(h)$. \nThis excludes compact ideals in ${\\mathfrak g}$. \n\\item We consider on ${\\mathfrak g}$ the involution \n$\\tau_h := e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h}$ fixing~$h$. \n\\item Then the eigenspace $E := {\\mathfrak g}^1 = {\\mathfrak g}^1(h)$ \nis a unital euclidean Jordan algebra, where the unit element $e \\in E$ \ncan be chose such that the open cone $E_+$ of positive squares \nsatisfies \n\\[ \\overline{E_+} = E \\cap C = {\\mathfrak g}^1(h) \\cap C =: C_+.\\] \n\\end{itemize}\n\nAs in Section~\\ref{sec:4}, we consider the connected Lie group \n$G := \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g})_0$ and write \n$q_G \\: \\tilde G \\to G$ for its universal covering group. \nLet $\\tau_G \\in \\mathop{{\\rm Aut}}\\nolimits(\\tilde G)$ be the involutive automorphism \nintegrating $\\tau_h \\in \\mathop{{\\rm Aut}}\\nolimits({\\mathfrak g})$. Then \n\\[ \\tilde G_\\tau := \\tilde G \\rtimes \\{\\mathop{{\\rm id}}\\nolimits_G, \\tau\\} \\] \nis a graded Lie group with two connected components. \nWe write \n\\[ {\\mathfrak h} = {\\mathfrak g}^0 = \\ker(\\mathop{{\\rm ad}}\\nolimits h), \\qquad \nH := \\langle \\exp_G {\\mathfrak h} \\rangle \\subseteq G,  \\quad \\mbox{ and } \\quad \nH_\\tau := H \\times \\{\\mathop{{\\rm id}}\\nolimits, \\tau\\} \\subseteq G_\\tau.\\] \nWe also note that \n\\[ \\tilde G^{\\tau_G} = \\{ g\\in \\tilde G \\: \\tau_G(g)= g \\} \\] \nis connected because $\\tilde G$ is simply connected \n(\\cite[Thm.~IV.3.4]{Lo69}), so that \n$q_G$ restricts to a covering map \n\\[ q_H \\: \\tilde G^{\\tau_G} \\to H.\\] \n\n\n\\begin{thm} {\\rm(Realization Theorem)} \\mlabel{thm:5.1} \nFor each irreducible\nantiunitary representation $(U,\\mathcal{H})$ of $\\tilde G_\\tau$ \nfor which $C_U$ is pointed and generating, \nthere exists an irreducible involutive finite dimensional \nrepresentation $(\\rho, \\mathcal{K})$ of the product group \n\\[ \\tilde H_\\tau := \\tilde H \\times \\{\\mathbf{1},\\tau\\}\\] \nwith the following properties: \n\\begin{itemize}\n\\item[\\rm(a)] The holomorphic extension of $\\rho$ to \nthe universal complexification $\\tilde H_{\\mathbb C}$ is $\\mathcal{T}$-positive, i.e., \nthe $B(\\mathcal{K})$-valued kernel \n\\[ Q_\\rho(z,w) := P_{\\rho}\\Big(\\frac{z - \\overline w}{2i}\\Big) \\] \non the tube domain $\\mathcal{T} = E + i E_+$ is positive definite. \n\\item[\\rm(b)] $(U,\\mathcal{H})$ is equivalent \nto the representation of $\\tilde G_\\tau$ on the corresponding \nreproducing kernel Hilbert space \n$\\mathcal{H}_\\rho  \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K}),$  given by \n\\begin{equation}\n  \\label{eq:j1}\n(U_\\rho(g)f)(z) = \nJ_{\\rho}(g, g^{-1}.z) f(g^{-1}.z) \\quad \\mbox{ \nfor } \\quad g \\in \\tilde G, z \\in \\mathcal{T},\n\\end{equation} \nand \n\\begin{equation}\n  \\label{eq:j2}\n(U_\\rho(\\tau_G)f)(z) := J_\\mathcal{K} f(-\\overline z),\n\\end{equation}\nwhere  $J_\\mathcal{K} = \\rho(\\tau)$, \n$\\tilde J \\: \\tilde G \\times \\mathcal{T} \\to \\tilde H_{\\mathbb C}$ is the lift \nof the cocycle $J \\: G \\times \\mathcal{T} \\to H_{\\mathbb C}, J(g,z) := \\dd g(z)$, and \n$J_\\rho := \\rho \\circ \\tilde J$. \n\\end{itemize}\n\\end{thm}\n\n\\begin{prf} Recall the isomorphism $\\tilde\\alpha_c \\: \n\\tilde G(\\mathcal{D})_\\tau \\to \\tilde G_\\tau = \\tilde G(\\mathcal{T})_\\tau$ from \\eqref{eq:liftalphac}. \nComposing with $\\tilde\\alpha_c$, \nany irreducible antiunitary representation $(U,\\mathcal{H})$ of $\\tilde G_\\tau$ for which \n$C_U$ is pointed and generating defines a representation \n$(U^c,\\mathcal{H})$ of $\\tilde G(\\mathcal{D})_\\tau$ with the same property. \nFrom \\cite[Thm.~X.3.9]{Ne99} we know that this is a highest weight \nrepresentation for a suitably chosen positive system, hence \ncan be realized in the space $\\mathop{{\\rm Hol}}\\nolimits(\\mathcal{D},\\mathcal{K})$ of \nholomorphic functions on $\\mathcal{D}$, where \n$\\mathcal{K}$ carries an irreducible \nantiunitary representation $(\\rho^c,\\mathcal{K})$ of the simply \nconnected group $\\tilde K(\\mathcal{D})_\\tau$ \n(Proposition~\\ref{prop:c.5}, \\cite[Thm.~XII.2.6]{Ne99}). \nWe conclude that the kernel $Q_\\rho^\\mathcal{D} := \\rho \\circ \\tilde Q^\\mathcal{D}$ \nis positive definite (\\cite[Prop.~XII.2.1]{Ne99}), \nNow Lemma~\\ref{lem:gamma-intertwine} transports all this \nstructure from $\\mathcal{D}$ to $\\mathcal{T}$, so that \nwe obtain the required realization \nin a Hilbert space $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$, \nwhere $(\\rho, \\mathcal{K})$ is a finite-dimensional $*$-representation \nof $\\tilde H_\\tau$, and the representation is given by \n\\eqref{eq:j1} and \\eqref{eq:j2}. \n\\end{prf}\n\n\n\nWe now turn to the standard subspace ${\\tt V} = {\\tt V}_{(h,\\tau_G,U)}$ specified by the \nmultiplicative one-parameter group of $G_\\tau$, \ncorresponding to the pair $(h,\\tau_G)$ via \n\\begin{equation}\nJ_{\\tt V} = U(\\tau_G) \\quad \\mbox{ and } \\quad \n\\Delta_{{\\tt V}}^{-it\/2\\pi} = U(\\exp th) \\quad \\mbox{ for } \\quad t \\in {\\mathbb R}.\n\\end{equation}\nIn the following proposition, we shall see in particular \nthat the distribution vectors \n$\\Phi(\\eta_\\xi) \\in \\mathcal{H}_\\rho^{-\\infty}$ are invariant under the subgroup \n$N^- = \\exp({\\mathfrak g}^{-1}(h)) \\subseteq \\tilde G(\\mathcal{T})$. \nAs the action of this subgroup is not so easily  accessible \nin the $L^2$-picture, we study this problem in the holomorphic picture on $\\mathcal{H}_\\rho$. \n\n\\begin{prop} \\mlabel{prop:c.12} \nLet $(\\rho,\\mathcal{K})$ be a holomorphic $\\mathcal{T}$-positive $*$-representation \nof $\\tilde H_{\\mathbb C}$ and let $\\mathop{{\\rm ev}}\\nolimits_0 \\: \\mathcal{H}_\\rho^\\infty \\to \\mathcal{K}, F \\mapsto F(0)$ be the \nevaluation in $0 \\in \\overline{\\mathcal{T}}$. \nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] \nFor each $\\xi \\in \\mathcal{K}$, we obtain a distribution vector  \n$\\mathop{{\\rm ev}}\\nolimits_0^\\xi \\: \\mathcal{H}_\\rho^\\infty \\to {\\mathbb C}, \\mathop{{\\rm ev}}\\nolimits_0^\\xi(F) := \\langle F(0), \\xi \\rangle$. \n\\item[\\rm(b)] $U_\\rho(g) \\mathop{{\\rm ev}}\\nolimits_0^\\xi = \\mathop{{\\rm ev}}\\nolimits_0^{\\rho(\\overline\\theta(\\tilde J(g)))\\xi}$ \nfor $g \\in \\tilde G^{\\tau_G}\\subseteq \\tilde G$ and \n$\\tilde J(g) \\in \\tilde H_{\\mathbb C}$. \n\\item[\\rm(c)] $\\mathop{{\\rm ev}}\\nolimits_0^\\xi$ is invariant under the subgroup \n$N^- = \\exp({\\mathfrak g}^{-1}(h)) \\subseteq \\tilde G = \\tilde G(\\mathcal{T})$. \n\\item[\\rm(d)] \nThe real subspace \n\\begin{equation}\n  \\label{eq:defE}\n {\\sf E} := \\{ \\mathop{{\\rm ev}}\\nolimits_0^\\xi \\: e^{-\\pi i \\cdot \\dd \\rho(h)} \\xi = J_\\mathcal{K} \\xi\\} \n\\end{equation}\nis contained in $\\mathcal{H}^{-\\infty}_{\\rho, {\\rm ext}, J}$ \nand invariant under the subgroup $P^- := N^- \\tilde G^{\\tau_G}$. \n\\end{itemize}\n\\end{prop}\n\n\\begin{prf} (a) Let $F\\in \\mathcal{H}_\\rho$ be a smooth vector. \nThen it is in particular smooth for the representation \nof $(E,+) \\cong \\exp({\\mathfrak g}^1(h))$. \nBy the Dixmier--Malliavin Theorem (\\cite{DM78}), it is of the form \n$F = U_\\rho(\\phi)F_0$ for some \n$\\phi \\in C^\\infty_c(E)$ and $F_0 \\in \\mathcal{H}_\\rho$. \nThen \n\\[ F(z) = \\int_E \\phi(x) F_0(z-x)\\, d\\mu_E(x),\\] \nso that Lemma~\\ref{lem:C.11} implies that $F$ extends to a \ncontinuous function on $\\overline{\\mathcal{T}}$ which is smooth on~$E$. \nTherefore $\\mathop{{\\rm ev}}\\nolimits_0$ is defined.\n\nNext we note that, for $F = \\Phi(f)$ and $f \\in L^2(E^*,\\mu;\\mathcal{K})^\\infty$,  \n\\begin{align*}\n\\Phi(\\eta_\\xi)(F) \n&=  \\Phi(\\eta_\\xi)(\\Phi(f)) \n=  \\eta_\\xi(f) \n= \\int_{E^*} \\langle f(\\lambda), d\\mu(\\lambda) \\xi \\rangle\\\\\n&= \\int_{E^*} \\langle d\\mu(\\lambda) f(\\lambda),  \\xi \\rangle\n= \\langle \\Phi(f)(0), \\xi \\rangle \n= \\langle F(0), \\xi \\rangle =\\mathop{{\\rm ev}}\\nolimits_0^\\xi(F).\n\\end{align*}\nAs $\\eta_\\xi$ is a distribution vector for the representation \nof $(E,_+)$ on $L^2(E^*,\\mu;\\mathcal{K})$,  the same holds for \n$\\mathop{{\\rm ev}}\\nolimits_0^\\xi$ and the representation of the translation group $(E,+)$ \non $\\mathcal{H}_\\rho$. \n\n\\nin (b) follows from \n\\begin{align*}\n(U^{-\\infty}_\\rho(g) \\mathop{{\\rm ev}}\\nolimits_0^\\xi)(F)\n&=  \\mathop{{\\rm ev}}\\nolimits_0^\\xi(U^{\\infty}_\\rho(g^{-1}) F)\n= \\langle (U^{\\infty}_\\rho(g^{-1}) F)(0), \\xi \\rangle \\\\\n&= \\langle \\rho(g^{-1}) F(0), \\xi \\rangle \n= \\langle F(0), \\rho(g^{-1})^* \\xi \\rangle \n= \\langle F(0), \\rho(\\theta(g)) \\xi \\rangle.\n\\end{align*}\n\n\\nin (c) For $g \\in N^-$, we have $U_\\rho(g)F \\in \\mathcal{H}_\\rho^\\infty$ and \n\\[ (U_\\rho(g)F)(0) \n= \\lim_{z \\to 0} (U_\\rho(g)F)(z)\n= \\lim_{z \\to 0} J_\\rho(g,g^{-1}.z) F(g^{-1}.z) \n=  J_\\rho(g,g^{-1}.0) F(g^{-1}.0) = F(0) \\] \nbecause $\\dd g(0) = \\mathbf{1}$ for every $g \\in N^-$. \nThis implies that the distribution \nvectors $\\mathop{{\\rm ev}}\\nolimits_0^\\xi$ are $N^-$-invariant. \n\n\\nin (d) On $\\mathcal{T} \\subseteq E_{\\mathbb C}$, the one-parameter group $\\exp({\\mathbb R} h)$ acts by $\\exp(th)z = e^t z$. \nThen\n\\[ U^{-\\infty}(\\exp th) \\mathop{{\\rm ev}}\\nolimits_0^\\xi \n= \\mathop{{\\rm ev}}\\nolimits_0^{\\rho(\\exp -th)\\xi}\\quad \\mbox{ for every } \\quad \n\\xi \\in \\mathcal{K}\\] \nfollows from (b). \nThis shows that \n$\\mathop{{\\rm ev}}\\nolimits_0^\\xi \\in \\mathcal{H}^{-\\infty}_{\\rho, {\\rm ext},J}$ \nif and only if $e^{-\\pi i \\dd\\rho(h)} \\xi = J_\\mathcal{K} \\xi.$\nAs $J_\\mathcal{K}$ and $\\dd \\rho(h)$ commute with $\\rho(\\tilde H)$, the \nreal subspace ${\\sf E}$ is $U(\\tilde G^{\\tau_G})$-invariant. \nFrom (c) we have the invariance of each $\\mathop{{\\rm ev}}\\nolimits_0^\\xi$ under the subgroup \n$N^-$, so that ${\\sf E}$ is invariant under $P^- = N^- \\tilde G^{\\tau_G}$. \n\\end{prf}\n\nWe are now ready to prove the main theorem of this paper. \n\n\\begin{thm} \\mlabel{thm:5.4} \nFor a holomorphic $\\mathcal{T}$-positive $*$-representation $(\\rho, \\mathcal{K})$ of \n$\\tilde H_{\\mathbb C}$, we consider the irreducible antiunitary representation \n$(U_\\rho, \\mathcal{H}_\\rho)$ of $\\tilde G_\\tau$ from \\eqref{eq:j1} and \\eqref{eq:j2}\n and the real subspace \n${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}_\\rho$ \nfrom \\eqref{eq:defE}. \nThen the following assertions hold: \n\\begin{itemize}\n\\item[\\rm(a)] The standard subspace ${\\tt V}$ of $\\mathcal{H}_\\rho$, specified by the \nconjugation $J_{\\tt V} = U_\\rho(\\tau_G)$ and the modular group \n$\\Delta_{\\tt V}^{-it\/2\\pi} = U_\\rho(\\exp th)$, coincides with \n\\[ {\\sf H}_{\\sf E}(S) = \n\\overline{{\\rm span}_{\\mathbb R} \\big(U^{-\\infty}(C^\\infty_c(S,{\\mathbb R})){\\sf E}\\big)},\n\\quad \\mbox{ where } \\quad S := \\tilde G^{\\tau_G} \\exp(C_+^0 \\oplus C_-^0) \\] \nis an open connected subsemigroup of $G$. \n\\item[\\rm(b)] The open subsemigroup $S_{\\tt V}^0\\supseteq S$ also satisfies \n${\\sf H}_{\\sf E}(S_{\\tt V}^0) = {\\tt V}.$ \n\\item[\\rm(c)] The prescription ${\\tt V}(\\mathcal{O}) := {\\sf H}_{\\sf E}(\\mathcal{O})$ assigns to \nevery non-empty \nopen subset $\\mathcal{O} \\subseteq G$ a real cyclic subspace, and this net \nhas the following properties: \n\\begin{itemize}\n\\item[\\rm(I)] {\\rm Isotony:} $\\mathcal{O}_1 \\subseteq \\mathcal{O}_2$ implies ${\\tt V}(\\mathcal{O}_1) \\subseteq {\\tt V}(\\mathcal{O}_2)$. \n\\item[\\rm(Cov)] {\\rm Covariance:} ${\\tt V}(g\\mathcal{O}) = U(g){\\tt V}(\\mathcal{O})$ for $g \\in G$. \n\\item[\\rm(BW)] {\\rm Bisognano--Wichmann property:} For \n$g \\in G$ and $\\mathcal{W} := gS$, the subspace ${\\tt V}(\\mathcal{W})$ is standard with \n\\[ \\Delta_{{\\tt V}(\\mathcal{W})}^{-it\/2\\pi} = U(\\exp t \\mathop{{\\rm Ad}}\\nolimits(g)h)\\quad \\mbox{  for  } \\quad \nt \\in {\\mathbb R}, g \\in G\n\\quad \\mbox{ and } \\quad J_{{\\tt V}(\\mathcal{W})} = U(g \\tau_G(g)^{-1}) J_{\\tt V}.\\]\n\\item[\\rm(Inv)] ${\\tt V}(gS)' = {\\tt V}(gS^{-1}) = {\\tt V}(g\\tau_G(S))$ for $g \\in G$. \n\\end{itemize}\n\\item[\\rm(d)] If $\\emptyset \\not=\\mathcal{O} \\subseteq G$ is open and there exists a \n$g \\in G$ with $\\mathcal{O} \\subseteq gS$, then ${\\tt V}(\\mathcal{O})$ is standard. \n\\item[\\rm(e)] If $A\\mathcal{O} S \\subseteq \\mathcal{O} \\supseteq S$ and ${\\tt V}(\\mathcal{O})$ is standard, then \n$\\mathcal{O} \\subseteq S_{\\tt V}^0$. In particular\n\\begin{itemize}\n\\item[$\\bullet$] If $g \\in S^{-1}\\setminus S_{\\tt V}$, \nthen ${\\tt V}(AgS)$ is not standard. \n\\item[$\\bullet$] ${\\tt V}(G)$ is not standard. \n\\end{itemize}\n\\end{itemize}\n\\end{thm}\n\n\\begin{prf} (a) From Proposition~\\ref{prop:c.12}(d) and Lemma~\\ref{lem:key} \nit follows that ${\\sf E} \\subseteq \\mathcal{H}_{\\rho,{\\rm ext},J}^{-\\infty} \\subseteq {\\tt V}^{-\\infty}$. \n\nAs $S_{\\tt V}^0 = G_{\\tt V} \\exp(C_+^0 \\oplus C_-^0)$ by \n\\eqref{eq:SE} in the introduction, and $\\tilde G^{\\tau_G} = (G_{\\tt V})_0$, we have \n$S_{\\tt V}^0 = G_{\\tt V} S$ and in particular $S \\subseteq S_{\\tt V}$. For \n$\\phi \\in C^\\infty_c(S,{\\mathbb R})$, \nLemma~\\ref{lem:vinfty}(c) implies $U^{-\\infty}(\\phi){\\sf E} \\subseteq  {\\tt V}$, \nso that ${\\sf H}_E(S) \\subseteq {\\tt V}$. \nAs ${\\tt V}$ is standard, ${\\sf H}_{\\sf E}(S)$ is also separating, hence standard. \nThe invariance of ${\\sf E}$ under $U^{-\\infty}(A)$ and the invariance \nof $S$ under conjugation with $A$ show that \n${\\sf H}_{\\sf E}(S)$ is invariant under $\\Delta_{\\tt V}^{i{\\mathbb R}} = U(A)$. \nNow ${\\sf H}_{\\sf E}(S) = {\\tt V}$ follows from \nLemma~\\ref{lem:lo-3.10}. \n\n\\nin (b) From (a) we recall that $S_{\\tt V}^0 = G_{\\tt V} S$. \nBy (a), ${\\tt V} = {\\sf H}_{\\sf E}(S)$, and this subspace is invariant under $G_{\\tt V}$, \nso that Lemma~\\ref{lem:sw64-genb} shows that ${\\sf H}_{\\sf E}(S) \n= {\\sf H}_{\\sf E}(G_{\\tt V} S)= {\\sf H}(S_{\\tt V}^0)$. \n\n\\nin (c) Since ${\\sf E} \\subseteq \\mathcal{H}^{-\\infty}$ is $G$-cyclic in the sense that \n$\\mathcal{H}_{\\sf E}(G) = \\mathcal{H}$ \n(Proposition~\\ref{prop:c.10}(a)), Theorem~\\ref{thm:3.6}\nimplies that each subspace ${\\sf H}_{\\sf E}(\\mathcal{O})$, $\\mathcal{O} \\not=\\emptyset$, is cyclic. \nProperties (I) and (Cov) are trivial, as we have already observed in \nDefinition~\\ref{def:HE}. Property (BW) follows from (a) and the covariance \nof the BGL net. \n\nTo verify (Inv), we first note that, \nreplacing $h$ by $-h$ corresponds to replacing ${\\tt V}$ by \n${\\tt V}'$ (\\cite[Prop.~3.3]{Lo08}) and $S = \\tilde G^{\\tau_G} \\exp(C_+^0 + C_-^-)$ by \n$S^{-1} = \\tilde G^{\\tau_G}  \\exp(-C_+^0 - C_-^0)$. Therefore (a) implies \n${\\tt V}' = {\\sf H}_{\\sf E}(S^{-1}) = {\\tt V}(S^{-1})$. For $g \\in G$, we thus obtain \n\\[ {\\tt V}(gS)' = (U(g){\\tt V})' = U(g){\\tt V}' = U(g){\\tt V}(S^{-1}) = {\\tt V}(g S^{-1}).\\] \nFinally, we note that $\\L(\\tau_G)(C_\\pm) = - C_\\pm$ and $\\tau_G(\\tilde G^{\\tau_G}) = \\tilde G^{\\tau_G}$ \nimplies $\\tau_G(S) = S^{-1}$. \n\n\\nin (d) By (Cov) we have ${\\tt V}(\\mathcal{O}) \\subseteq U(g){\\tt V}(S)$, and the right hand side \nis standard. Hence the cyclic subspace ${\\tt V}(\\mathcal{O})$ is also separating and \ntherefore standard. \n\n\\nin (e) As $A\\mathcal{O} = \\mathcal{O}\\supseteq S$, \n(Cov) implies that the real subspace, ${\\tt V}(\\mathcal{O})$ is invariant \nunder the modular group $U(A)$ of ${\\tt V} = {\\tt V}(S) \\subseteq {\\tt V}(\\mathcal{O})$. \nIf ${\\tt V}(\\mathcal{O})$ is standard, then Lemma~\\ref{lem:lo-3.10} \nimplies that ${\\tt V}(\\mathcal{O}) = {\\tt V}$. This in turn shows that, \nfor $g \\in \\mathcal{O}$, we have  \n\\[ U(g){\\tt V} =U(g){\\tt V}(S) = {\\tt V}(g S) \n\\subseteq {\\tt V}(\\mathcal{O}) = {\\tt V}.\\]\nWe conclude that $\\mathcal{O} \\subseteq S_{\\tt V}$, and since $\\mathcal{O}$ is open, \nthat $\\mathcal{O} \\subseteq S_{\\tt V}^0$. \n\nAs $S_{\\tt V} \\not=G$, it follows in particular that ${\\sf H}_{\\sf E}(G)$ is not standard. \nFor $g \\in S^{-1}\\setminus S_{\\tt V}$, the relation\n $g^{-1} S \\subseteq S$ implies that \n$S \\subseteq gS \\subseteq A g S =: \\mathcal{O}$. Hence $\\mathcal{O}$ satisfies the assumptions from above. \nIf $\\mathcal{O} \\subseteq S_{\\tt V}^0$, then $e \\in \\overline{S}$ leads to the contradiction \n$g \\in S_{\\tt V}$. Therefore ${\\tt V}(\\mathcal{O})$ is not standard. \n\\end{prf} \n\nThe preceding theorem provides for every \n$\\mathcal{T}$-positive holomorphic representation \n$(\\rho,\\mathcal{K})$ of $\\tilde H_{{\\mathbb C},\\tau}$, or, equivalently, \nany norm-continuous real representation \n$(\\rho, \\mathcal{K}_{\\mathbb R})$ of $\\tilde G^{\\tau_G}$, a net of cyclic real subspaces, \nsome of which are standard. \nApplying second quantization functors, we thus \nobtain free quantum fields in the sense of Haag--Kastler on the \ngroup $\\tilde G$, where the left translates $\\mathcal{W} = gS$ of the semigroup \n$S$ play the role of wedge domains. \nHere the invariant cone $C \\subseteq {\\mathfrak g}$ defines a biinvariant causal structure \nin the sense of a cone field $(gC)_{g \\in \\tilde G}$ on the group $\\tilde G$. \nFor $\\mathcal{W} = S$, the modular involution corresponds to the involution \n$\\tau_G$ on $\\tilde G$ which exchanges $S$ and $S^{-1}$. The modular \none-parameter group acts on $S$ by the left translations \n$\\alpha_t(s) = (\\exp th)s$. By left translation, we obtain the geometric \nactions of the modular objects on the domains $gS$. \n\n\n\\begin{rem} The results of this section extend easily \nto quasihermitian Lie groups, i.e., where the Lie algebra ${\\mathfrak g}$ \nis of the form \n${\\mathfrak g} = {\\mathfrak g}_1 \\oplus {\\mathfrak g}_2$, where\n${\\mathfrak g}_1$ is a sum of simple hermitian ideals of tube type and ${\\mathfrak g}_2$ is a compact \nLie algebra. Then ${\\mathfrak h} = {\\mathfrak h}_1 \\oplus {\\mathfrak g}_2$, \nand irreducible representations \n$(\\rho, \\mathcal{K})$ of $\\tilde H \\cong \\tilde H_1 \\times \\tilde G_2$ \nare tensor products \n$\\rho \\cong \\rho_1 \\otimes \\rho_2$, where $\\rho$ is \n$\\mathcal{T}$-positive if and only if $\\rho_1$ has this property. \n\\end{rem}\n\n\\subsection{Jordan space-times and causal symmetric spaces} \n\n\nThe real subspace ${\\sf E}$ in Theorem~\\ref{thm:5.4} \nthat we used above is also invariant \nunder the closed connected subgroups $P_1 := \\tilde G^{\\tau_G}$ and $P_2 := N^- \n\\tilde G^{\\tau_G}$ (Proposition~\\ref{prop:c.12}). \nWe thus obtain on the simply connected \nhomogeneous spaces $M_j := \\tilde G\/P_j$ \ncovariant nets of cyclic\/standard subspaces as follows. \nLet  $q_j \\: \\tilde G \\to M_j$ denote the canonical projection and put \n\\[ {\\tt V}_{M_j}(\\mathcal{O}) := {\\tt V}(q_j^{-1}(\\mathcal{O})) \\quad \\mbox{ for } \\quad \n\\emptyset\\not= \\mathcal{O} \\subseteq M_j.\\] \nFor these nets isotony and $G$-covariance \n\\[ {\\tt V}_{M_j}(g\\mathcal{O}) = U(g){\\tt V}_{M_j}(\\mathcal{O}) \\] \nare clear. For $\\mathcal{W}_j := q_j(S)$ it follows from \nLemma~\\ref{lem:sw64-genb} that \n\\[ {\\tt V}_{M_j}(\\mathcal{W}_j) = {\\tt V}(q_j^{-1}(\\mathcal{W}_j)) = {\\tt V}(S P_j) = {\\tt V}(S).\\] \nHence the subspace ${\\tt V}_{M_j}(\\mathcal{W}_j)$ is standard and \nhas the Bisognano--Wichmann property. \nIts modular conjugation is implemented on $M_j$ by \n\\[ \\tau_{M_j}(gP_j) = \\tau_G(g) P_j,\\] \nand the modular group by \n\\[ \\alpha_t(g P_j) = \\exp(th) g P_j \n= \\exp(th) g \\exp(-th) P_j \\quad \\mbox{ for } \\quad g \\in G, t \\in {\\mathbb R}.\\] \n\n\nHere the manifold $M_1 = \\tilde G\/P_1$ is a symmetric space \nof Cayley type (cf.~\\cite{HO97})  \nand the manifolds $M_2 = \\tilde G\/P_2$ \ncoincide with the Jordan space-times \ndescribed by G\\\"unaydin in \\cite{Gu93} \nas a natural class of causal manifolds with conformal symmetries,\nrepresented by the group~$G$. \nThese manifolds also coincide with the \n{\\it simple space-time manifolds} \nclassified in  \\cite[Thm.~4]{MdR07} by Mack and de Riese. \n\n\n\n\\subsection{Affine groups} \n\nThe subgroup $P^+ := N^+ \\tilde G^{\\tau_G} \\subseteq \\tilde G$ acts \nby affine maps on the Jordan algebra $E \\cong {\\mathfrak g}^1(h)$, where \n$N^+ \\cong (E,+)$ acts by translations, and $G^{\\tau_G}$ by linear maps \nleaving the cone $E_+$ invariant. For the special case, where \n$E \\cong {\\mathbb R}^{1,n-1}$ is $n$-dimensional Minkowski \nspace,  $P^+ \\cong E \\rtimes {\\mathbb R}^\\times_+  {\\rm Spin}_{1,n-1}({\\mathbb R})$ \nis the Poincar\\'e group extended by dilations. Restricting everything from \n$\\tilde G$ to $P^+$, we immediately obtain \nnets of cyclic\/standard subspaces on $E$, which we may consider as an open \nsubset of the homogeneous space $G\/P^-$. In this case it is of some \ninterest to consider an even smaller group, such as the \nLorentz group $\\mathop{{\\rm SO}}\\nolimits_{1,n-1}({\\mathbb R})_0$ on $E = {\\mathbb R}^{1,n-1}$, which does not \nnecessarily contain the dilations. \n\nA natural setting for this is explored in \\cite{NOO20}, and \nwe now explain briefly how it connects to the present paper. \nLet $L \\subseteq H$ be a connected subgroup invariant under the \nCartan involution $\\theta$, and \n$q_L\\: \\tilde L \\to L$ be its universal covering group. \nWe consider a norm-continuous $*$-representation \n$(\\rho, \\mathcal{K})$ of $\\tilde L_\\tau = \\tilde L \\times \\{\\mathbf{1},\\tau\\}$ \nand a tempered $\\mathop{{\\rm Herm}}\\nolimits^+(\\mathcal{K})$-valued measure $\\mu$ on $E_+^\\star$ such that \n\\[ (g^{-1})_* \\mu = \\rho(g) \\mu \\rho(g)^* \\quad \\mbox{ for } \\quad \ng \\in \\tilde L.\\] \nThen the Laplace transform $\\mathcal{L}(\\mu)$ defines a holomorphic function on \n$\\mathcal{T}$, and \n\\[ Q_\\rho(z,w) := \\mathcal{L}(\\mu)\\Big(\\frac{z - \\overline w}{2i}\\Big) \\] \nis a positive definite $B(\\mathcal{K})$-valued kernel on $\\mathcal{T}$. \nWe thus obtain a \nunitary representation of $E \\rtimes \\tilde L_\\tau$ on the \ncorresponding Hilbert subspace $\\mathcal{H}_\\rho \\subseteq \\mathop{{\\rm Hol}}\\nolimits(\\mathcal{T},\\mathcal{K})$, given by \n\\begin{equation}\n  \\label{eq:pitau2}\n (U_\\rho(x,g)f)(z) := \\rho(g) f(g^{-1}.(z-x))\n\\quad \\mbox{ for } \\quad x \\in E, z \\in \\mathcal{T}, g \\in \\tilde L_\\tau.\n\\end{equation}\n\nNow we consider a element $h' \\in {\\mathfrak l}$, the Lie algebra of $L$, \nfor which $\\mathop{{\\rm ad}}\\nolimits h'$ defines a $3$-grading on ${\\mathfrak g}$, and the involution \n$\\tau'$ on $E \\rtimes \\tilde L$ integrating the Lie algebra involution \n$e^{\\pi i \\mathop{{\\rm ad}}\\nolimits h'}$ on $E \\rtimes {\\mathfrak l}$. For \n$J_\\mathcal{K} := \\rho(\\tau')$, the real subspace \n\\begin{equation}\n  \\label{eq:defE2}\n {\\sf E} := \\{ \\mathop{{\\rm ev}}\\nolimits_0^\\xi \\: e^{-\\pi i \\cdot \\dd \\rho(h')} \\xi = J_\\mathcal{K} \\xi\\} \n\\end{equation}\nis then contained in $\\mathcal{H}_\\rho^{-\\infty}(U_\\rho\\vert_E)$ and invariant under \n$U_\\rho(\\tilde L^0)$, \nwhere $\\tilde L^0 \\subseteq \\tilde L$ is the centralizer of~$h'$. \nNow the standard subspace \n${\\tt V} = {\\tt V}_{(h',\\tau',U_\\rho)}$ coincides with \n${\\sf H}_{\\sf E}^E(\\mathcal{W})$, where $\\mathcal{W}$ is the wedge domain \n\\[ \\mathcal{W} = (E_+ \\cap E^1(h')) \\oplus E^0(h') \\oplus (-E_+ \\cap E^{-1}(h')). \\] \nUsing Lemma~\\ref{lem:semidir}, we even get \n\\[ {\\sf H}_{\\sf E}^E(\\mathcal{W}) = {\\sf H}_{\\sf E}^{E \\rtimes \\tilde L^0}(\\mathcal{W} \\rtimes \\tilde L^0).\\] \nIn the special case where $\\mathop{{\\rm ad}}\\nolimits h\\vert_{E} = \\mathop{{\\rm id}}\\nolimits_E$, \nwe have $\\tilde L^0 = \\tilde L$. \nFor more details on this situation, we refer to \\cite{NOO20}. \n\n\n\\section{Perspectives} \n\\mlabel{sec:6}\n\n\\subsection{Extension to more general groups} \n\\mlabel{subsec:6.1} \n\nThe results of Sections~\\ref{sec:2} and \\ref{sec:3} work for \nnon-reductive Lie groups satisfying the conditions (B1)-(B4). There are various \ntypes of non-reductive groups with this property, such as \nthe extended Jacobi group, discussed in some detail in \\cite[Ex.~3.7]{Ne19b}. \nWe briefly recall some observations from \\cite{Ne19b}. \nIts Lie algebra is \n\\[ {\\mathfrak g} = \\hcsp({\\mathbb R}^{2n},\\omega) = \\heis({\\mathbb R}^{2n},\\omega) \\rtimes \\csp_{2n}({\\mathbb R}), \\] \nwhere $\\omega$ is the canonical symplectic form on ${\\mathbb R}^{2n}$, \n$\\heis({\\mathbb R}^{2n},\\omega) = {\\mathbb R} \\oplus {\\mathbb R}^{2n}$ is the corresponding Heisenberg algebra \nwith the bracket $[(z,v),(z',v')] = (\\omega(v,v'),0)$, and \n\\[ \\csp_{2n}({\\mathbb R}) := \\sp_{2n}({\\mathbb R}) \\oplus {\\mathbb R} \\mathbf{1} \\] \nis the {\\it conformal symplectic Lie algebra} of $({\\mathbb R}^{2n},\\omega)$. \nThe hyperplane ideal \n\\[ {\\mathfrak j} := \\heis({\\mathbb R}^{2n},\\omega) \\rtimes \\sp_{2n}({\\mathbb R}) \\] \n(the {\\it Jacobi algebra})  \ncontains a pointed generating invariant cone~$C$, \ncorresponding to the non-negative polynomials of degree $\\leq 2$ on ${\\mathbb R}^{2n}$. \nThe involution  $\\tau(q,p)= (-q,p)$ is antisymplectic, and \nthe operator $h := {\\textstyle{\\frac{1}{2}}}(\\mathbf{1} + \\tau)\\in \\csp_{2n}({\\mathbb R})$  defines \na $3$-grading of ${\\mathfrak g}$ for which the cones $C_\\pm$ generate ${\\mathfrak g}^{\\pm 1}(h)$. \nThe corresponding simply connected group $G$ has an \nirreducible unitary representation on \n\\[ \\mathcal{H} \n:= L^2\\Big({\\mathbb R}^\\times_+, \\frac{d\\lambda}{\\lambda}; L^2({\\mathbb R}^n)\\Big) \\cong \n L^2\\Big({\\mathbb R}^\\times_+ \\times {\\mathbb R}^n, \\frac{d\\lambda}{\\lambda} \\otimes dx\\Big),\\]\nwhere $\\lambda$ parametrizes a family of mutually \ninequivalent irreducible representations of the Jacobi group on $L^2({\\mathbb R}^n)$.\n It would be very interesting to identify nets of standard subspaces \nin this representation if it is extended to an antiunitary representation \nof $G_\\tau$ by \n\\[ (Jf)(\\lambda,x) := \\overline{f(\\lambda,-x)}.\\] \n\n\n\\subsection{Relation to causal structures} \n\\mlabel{subsec:6.2}\n\nIn Section~\\ref{sec:5} we  constructed nets of cyclic subspaces \non three levels: \n\\begin{itemize}\n\\item On the group level, where the invariant cone \n$C \\subseteq {\\mathfrak g}$ defines a biinvariant causal structure on the group manifold \n$G$ and the semigroup $S^0 = \\exp(C_+^0) (G^{\\tau_G})_0 \\exp(C_-^0)$ \nplays the role of a wedge domain. \n\\item On the symmetric space $M = G\/G^{\\tau_G}$, whose tangent space in the origin \nis ${\\mathfrak q} = {\\mathfrak g}^1 \\oplus {\\mathfrak g}^{-1}$, in which $C_{\\mathfrak q} := C \\cap {\\mathfrak q} = C_+ - C_-$ is a pointed \ngenerating $\\mathop{{\\rm Ad}}\\nolimits(G^{\\tau_G})$-invariant cone that defines an invariant causal \nstructure on $M$. Here the image of $S$ in $M$, which is \nthe set $\\mathop{{\\rm Exp}}\\nolimits(C_+ + C_-)$, where $\\mathop{{\\rm Exp}}\\nolimits \\: {\\mathfrak q} \\to M$ is the exponential function of \nthe symmetric space~$M$, plays the role of a wedge domain in $M$. Actually \nit also is the causal future of the base point for another \ncausal structure on $M$ corresponding to the cone $C_+ + C_-$. \nNote that the first causal structure turns $M$ into a compactly  causal  symmetric \nspace in the sense of \\cite{HO97} and the second one into a \nnon-compactly causal symmetric space. \n\\item For the Jordan space-time \n$M = \\tilde G\/\\tilde P^-$, the tangent space in the base point is \n${\\mathfrak g}^1$, which we considered also as a euclidean Jordan algebra $E$, \nand the causal structure is given by the cone $C_+ = \\overline{E_+}$. \nHere the image of $S$ coincides with the image of $\\exp(C_+)$ in $M$. \nIt plays the role of a wedge domain and the future of the base point for the \ncausal structure at the same time. \n\\end{itemize}\n\nThese observations show that, to proceed beyond the class of spaces \nconsidered here, we need a better theory of wedge domains in \nordered homogeneous spaces. Some first steps in this general program \nwill be  carried out in \\cite{NO20} for the classes of compactly causal and \nnon-compactly causal symmetric spaces, such as de Sitter spaces. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nMax-infinitely divisible processes (max-id) have gained much popularity in recent years for the flexible modeling of spatial extremes in phenomena  such as precipitation, wind speeds and temperatures \\citep{Padoan2013,Huser.etal:2020,Bopp.etal:2020, Huser.Wadsworth:2020, Zhong.Huser.Optiz:2020}. Such models encompass max-stable processes \\citep{Haan1984} as a subclass, which are currently the mainstream for modeling spatial extremes, as demonstrated by a large body of theoretical and applied literature \\citep[see, e.g.,][]{Davison.etal:2012,Dey2016,Davison.etal:2019}. Max-stability is a property that arises in asymptotic models for multivariate block maxima, but that is however often violated in finite samples \\citep{Huser.Wadsworth:2020}. Precisely, the dependence strength in max-stable processes does not depend on event magnitude, while most environmental and climatic processes suggest that it should become weaker at higher thresholds, with the most extreme events being typically more localized. To circumvent this limitation, more flexible max-id processes, which drop the restrictive max-stability assumption but retain natural properties of multivariate block maxima, have been considered. In particular, the max-id models proposed by \\citet{Huser.etal:2020}, \\citet{Bopp.etal:2020} and \\citet{Zhong.Huser.Optiz:2020}, extend certain popular classes of max-stable models (obtained as limiting cases on the boundary of the parameter space), in order to achieve a good compromise between the pragmatism of flexible max-id models and the strong theoretical foundations of the max-stable sub-class. \n\nThe theory behind max-id processes has been studied in depth \\citep{Gine1990, Resnick1987, Brown.Resnick:1977}. In particular, \\cite{Gine1990} showed that max-id processes possess a functional Poisson point process (PPP) representation.\nMore precisely, each max-id process $Z = \\{Z(\\bm s)\\}_{\\bm s \\in {\\mathcal{S}}}$ defined over the region ${\\mathcal{S}}\\subset\\mathbb{R}^d$, can be defined as the pointwise maximum of a potentially infinite number of random functions $\\{\\eta_i; \\ i=1,2,\\ldots\\}$ defined over ${\\mathcal{S}}$, which are sampled according to a PPP with mean measure $\\nu$, i.e.,\n\\begin{equation}\n\\label{eq:max-id}\nZ(\\bm s)=\\max_{i=1,2,\\ldots}\\eta_i(\\bm s),\\quad \\bm s\\in\\mathcal{S}.    \n\\end{equation}\nHowever, analytical forms of the multivariate distribution functions of such processes exist only in special cases, and by analogy with max-stable processes their numerical evaluation in high dimensions becomes impossible due to the combinatorial explosion of the number of terms to be calculated \\citep{Padoan.etal:2010,Castruccio.etal:2016,Huser.etal:2020}. Consequently, simulation of max-stable processes is often crucial to study their dependence characteristics, or for simulation-based inference \\citep{Erhardt2012,Hainy.etal:2016,Lee.etal:2018}, and the same is true for general classes of max-id processes. However, exact simulation of $Z(\\bm s)$ in \\eqref{eq:max-id} is not trivial, especially when the number of functions $\\eta_i$ is infinite and when these random functions are tricky to sample.\n\nFor the subclass of max-stable models, there are essentially two types of generic simulation algorithm that have been proposed based on variants of model representations stemming from \\eqref{eq:max-id} \\citep{Oesting.Strokorb:2019}.  These algorithms exploit the specific structure of max-stable processes---the so-called \\emph{spectral representation} \\citep{Haan1984}, whereby on the unit Fr\\'echet scale (i.e., ${\\rm Pr}\\{Z(\\bm s)>z\\}=1\/z$, $z>0$) the points of the Poisson process $\\{\\eta_i;i=1,2,\\dots\\}$ in \\eqref{eq:max-id} can be decomposed into two multiplicative and stochastically independent components as $\\eta_i(\\bm s)=R_iW_i(\\bm s)$. Precisely, $\\{R_i;\\ i=1,2,\\dots\\}$ here denotes a Poisson point process on $[0,\\infty)$ with measure measure $r^{-2}{\\rm d}r$, independent of the random functions $\\{W_i;\\ i=1,2,\\dots\\}$, which are independent copies of a random process $W(\\bm s)$ defined on ${\\mathcal{S}}$ such that ${\\rm E}[\\max\\{0,W(\\bm s)\\}]=1$. This decomposition always exists for max-stable processes with unit Fr\\'echet margins but is not unique. When the points $\\{R_i;\\ i=1,2,\\dots\\}$ have heavy-tailed intensity $r^{-(\\alpha+1)}{\\rm d}r$ for some $\\alpha>0$, the resulting process $Z$ remains max-stable, but with $\\alpha$-Fr\\'echet margins. Moreover, the points $\\{R_i^{-1};\\ i=1,2,\\dots\\}$ can be defined as the arrival times of a renewal process such that we can simulate them in decreasing order, i.e., $R_1>R_2>\\ldots$. When this is the case, the componentwise maximum of $\\{R_jW_j; \\ j=1,\\ldots,i-1\\}$ is less and less impacted by the $i$-th process $\\eta_i=R_iW_i$ as $i>1$ is iteratively incremented. If $W(\\bm s)$ is an almost surely bounded random process on ${\\mathcal{S}}$, the contribution of $\\eta_i=R_iW_i$ becomes even ultimately completely irrelevant. Based on this idea, \\cite{Schlather:2002} introduced the first exact simulation algorithm for max-stable processes by setting an appropriate stopping rule for the simulation of the Poisson points $R_iW_i$. \nIf $W(\\bm s)$ is unbounded, we can use certain alternative representations  to make it a bounded random process under a suitable change of measure \\citep{Dieker.Mikosch:2015,Oesting.Schlather.Zhou:2018,Dombry.Engelke.Oesting:2016}, but simulation from the bounded process is intricate and numerically challenging  even in moderate dimensions for many of the customary models \\citep{Kabluchko.etal:2009,Opitz:2013}. The second type of exact simulation algorithm for max-stable processes, introduced by \\cite{Dombry.Engelke.Oesting:2016}, relies on the concept of extremal functions \\citep{Dombry.Eyi-Minko:2013}. With this method, it is possible to directly simulate the random functions $\\{\\eta_i\\}$ on ${\\mathcal{S}}$ that contribute to the maximum at a finite number of locations. In practice, efficient conditional simulation of the random functions $\\eta_i(\\bm s)$ given its value at a single location $\\bm s_0$ is a prerequisite for efficient simulation with this approach.\n\nHowever, none of the approaches described above tackled exact simulation for general max-id processes, and we intend to fill this gap. In this work, we demonstrate that the idea of the exact simulation algorithm based on extremal functions, developed in \\cite{Dombry.Engelke.Oesting:2016} for max-stable processes carries over to the general max-id case after suitable adjustments, and we illustrate it with max-id models that have found interest in practical applications.  The paper is organized as follows. In \\S\\ref{sec:Max-id}, we recall some theoretical background and formally define the notions of max-id processes and extremal functions. Then, in \\S\\ref{sec:Exact-Simu}, we derive the specificities of the theory of max-id processes required for exact simulation based on extremal functions, and we present simulation algorithms focusing on two major classes of max-id processes. In particular, we start with some known max-id models of the form \\eqref{eq:max-id} where the random functions $\\eta_i(\\bm s)$ can be represented as Gaussian scale mixtures, and we then also explore new max-id models based on Gaussian location mixtures that possess appealing tail dependence characteristics. In \\S\\ref{sec:Simu-Study}, we conduct a simulation study to demonstrate the performance of our proposed simulation algorithm. We conclude with a discussion in \\S\\ref{sec:Discussion}.\n\n\\section{Max-id Processes and Extremal Functions}\n\\label{sec:Max-id}\nA process $\\{Z(\\bm s)\\}_{\\bm s \\in {\\mathcal{S}}}$ is called max-infinitely divisible (max-id) if the joint cumulative distribution function (cdf) $G$ of $\\bm Z(K_N)=\\{Z(\\bm s_1),\\ldots, Z(\\bm s_N)\\}^\\top$ at any finite collection of sites $K_N=\\{\\bm s_1,\\bm s_2,\\dots,\\bm s_N\\}\\subset {\\mathcal{S}}$ defines a valid cdf $G^t$ (with the notation $G^t(\\bm z)=\\{G(\\bm z)\\}^t$, $\\bm z=(z_1,\\ldots,z_N)^\\top\\in \\mathbb R^N$) for any  $t>0$. The distribution $G^t$ does not necessarily stay within the same location-scale family;\\ this property is only satisfied for max-stable distributions $\\tilde G$, for which $\\tilde G^m(\\bm a_m\\bm z+\\bm b_m)=\\tilde G(\\bm z)$, for any integer $m=1,2,\\dots$, and appropriately defined normalizing vectors $\\bm a_m\\in (0,\\infty)^{N}$ and $\\bm b_m\\in\\mathbb R^N$. As shown in Equation~\\eqref{eq:max-id}, any max-id process can be constructed by taking pointwise maxima over a Poisson point process defined on a suitable functions space. We assume that $\\{\\eta_i(\\bm s);\\  i=1,2,\\dots\\}_{\\bm s\\in{\\mathcal{S}}}$ in \\eqref{eq:max-id} is a Poisson point process with mean measure $\\Lambda$ on ${\\mathcal C}$, where ${\\mathcal C}$ denotes the space of functions with compact support ${\\mathcal{S}}$. Then, the extremal functions are defined as follows. \n\\begin{defn}\n\t Let $K\\subset {\\mathcal{S}}$ be a nonempty subset of the domain ${\\mathcal{S}}$, and $Z$ a max-id process. A function $\\phi\\in{\\mathcal C}$ is called $K$-extremal if there exists $\\bm s\\in K$ such that $Z(\\bm s)=\\phi(\\bm s)$, otherwise $\\phi$ is called $K$-subextremal. We denote ${\\mathcal C}^+_K$ the set of K-extremal functions and ${\\mathcal C}_K^-$ the set of $K$-subextremal functions.\n\\end{defn}\nIn other words, an extremal function is a Poisson point $\\eta_i$ in ${\\mathcal C}$ that contributes to the maximum process $Z$ at one or more locations. We further make an assumption ensuring the continuity of univariate marginal distributions of $Z$. Let $\\ell$ denote the lower bound of the marginal support of $Z$ such that $\\pr\\{Z(\\bm s) >\\ell\\}=1$ for all $\\bm s\\in{\\mathcal{S}}$. We assume that for any fixed $\\bm s_0\\in {\\mathcal{S}}$, \n\\begin{equation}\n\\label{eq:assumption}\n\t\\Lambda_{\\bm s_0}(z)=\\Lambda(\\{\\eta\\in{\\mathcal C}; \\eta(\\bm s_0)\\in [z,+\\infty) \\})\\ \\text{is continuous on}\\ (\\ell,+\\infty),\\ \\lim_{z\\downarrow \\ell}\\Lambda_{\\bm s_0}(z)=+\\infty.\n\\end{equation}\nUnder this assumption, \\cite{Dombry.Eyi-Minko:2013} showed that for singletons $K=\\{\\bm s_0\\}$ the set of extremal functions ${\\mathcal C}_{\\{\\bm s_0\\}}^+$ contains almost surely a single point, denoted by $\\phi_{\\{\\bm s_0\\}}^+$, i.e., the resulting max-id process $Z(\\bm s)$ is realized by one and only one Poisson point $\\eta_i$ at a given location $\\bm s_0$. This assumption implies that $Z$ has continuous univariate margins. \nThis restriction is not problematic since all strictly monotone marginal transformations of $Z$ yield again a max-id process. In particular, any max-id process $Z$ with continuous univariate distributions can be marginally transformed to satisfy assumption~\\eqref{eq:assumption}.\n\nLet $P_{\\bm s_0}(x,\\cdot)$ denote the conditional distribution of $\\eta$ given  $\\eta(\\bm s_0)=x$. It turns out that the extremal function $\\phi_{\\{\\bm s_0\\}}^+$ at location $\\bm s_0\\in {\\mathcal{S}}$, given $Z(\\bm s_0)=z$, follows the same distribution as $P_{\\bm s_0}(z,\\cdot)$ (see Proposition 4.1 in \\cite{Dombry.Eyi-Minko:2013}). In other words, the distribution of a Poisson point $\\eta_i$ conditional on the value at a particular location $\\eta_i(\\bm s_0) = z$ is the same if we pick an arbitrary point $\\eta_i$ or a point for which $z$ is the maximum at $\\bm s_0$ over all points $\\{\\eta_i\\}$ of the Poisson process. This property, which is related to the independent occurrence of points in Poisson processes, is key for the exact simulation algorithm proposed by \\cite{Dombry.Engelke.Oesting:2016}. Essentially, this algorithm requires simulating the extremal functions at the simulation sites $K_N=\\{\\bm s_1,\\ldots,\\bm s_N\\}$ sequentially, which therefore involves sampling from $P_{\\bm s_i}(z,\\cdot)$, $i=1,\\ldots,N$. If conditional simulation for an arbitrary Poisson point $\\eta_i$ is straightforward, then this advantage will thus also automatically carry over to the simulation of the corresponding max-id process constructed as $\\max_{i=1,2,\\ldots} \\eta_i(\\bm s)$. The algorithm for general max-id processes is described in more detail in \\S\\ref{sec:Exact-Simu}.\n\n\nIn particular, when the process $\\{Z(\\bm s)\\}_{\\bm s\\in{\\mathcal{S}}}$ is max-stable with unit Fr\\'echet margins, the Poisson point process $\\{\\eta_i;\\ i=1,2,\\dots\\}$ can be decomposed as $\\eta_i(\\bm s) = R_iW_i(\\bm s)$, $i=1,2,\\dots$, where $\\{R_i;\\ i=1,2,\\dots\\}$ is a Poisson point process on $(0,\\infty)$ with mean measure $\\kappa([r,\\infty])=r^{-1}, r>0$, and $\\{W_i(\\bm s)\\}_{\\bm s\\in {\\mathcal{S}}},\\  i=1,2,\\dots,$ are independent copies of a random process $\\{W(\\bm s)\\}_{\\bm s\\in {\\mathcal{S}}}$ with ${\\rm E}[\\max\\{W(\\bm s),0\\}]=1$, which are also independent of the points $\\{R_i;i=1,2,\\dots\\}$ \\citep{Haan1984,Schlather:2002}. A number of parametric max-stable models have been proposed for statistical applications, such as the Brown--Resnick model \\citep{Brown.Resnick:1977, Kabluchko.etal:2009}, and the extremal-$t$ model \\citep{Opitz:2013}, where the intensity function associated to the mean measure $\\Lambda$ has a closed form expression. In these special cases, the multivariate conditional distributions required for exact simulation are relatively easy to simulate from,\nand \\cite{Dombry.Engelke.Oesting:2016} provide the exact forms of the distributions $P_{\\bm s}(z,\\cdot)$ required for simulating these max-stable models.\n\nHowever, such simplifications are no longer available for general max-stable processes and for the classes of max-id (but not max-stable) process models proposed in the recent literature. Very often, the mean measure $\\Lambda(\\cdot)$ does not have a closed-form intensity function, and the corresponding conditional distributions are complicated; rather, calculations of relevant measures of the form $\\Lambda(B)$ and intensities often involve computing a one-dimensional integral \\citep{Huser.etal:2020, Zhong.Huser.Optiz:2020}. Therefore, simulating from $P_{\\bm s}(z,\\cdot)$ is more involved than in the max-stable case and requires further investigation into its form to propose efficient simulation techniques.\n\n\\section{Exact Simulation}\n\\label{sec:Exact-Simu}\n\n\\subsection{General simulation algorithm}\n\\label{sec:generalalgo}\n\nWe now extend the generic max-stable simulation algorithm based on extremal functions, as proposed  by \\cite{Dombry.Engelke.Oesting:2016}, to the general max-id case.   \nWe aim to simulate the max-id process $\\{Z(\\bm s)\\}_{s\\in{\\mathcal{S}}}$ at a the finite number $N$ of sites given as $K_N=\\{\\bm s_1,\\bm s_2,\\dots,\\bm s_N\\}\\subset {\\mathcal{S}}$. We define the subsets $K_n=\\{\\bm s_1,\\dots,\\bm s_n\\}$ of the first $n$ sites for  $n=1,\\dots,N$, and we denote by $Z_n = \\max_{\\eta_i\\in {\\mathcal C}^+_{K_n}}\\eta_i$ the process constructed from the extremal functions at the sites in $K_n$, such that  $\\bm Z(K_N)=\\bm Z_N(K_N)$. \nTo simulate from the joint distribution of the process $\\{Z(\\bm s)\\}$ at sites in $K_N$, we can iterate through the set of locations from $\\bm s_1$ to $\\bm s_N$ in order to simulate the extremal functions at each location. Therefore, a user-defined order of sites must be fixed before simulating, and below we will comment on how this order may be chosen in practice.\nRecall that we denote by $\\{\\phi_{\\bm s_n}^+\\}_{1\\leq n\\leq N}$\tthe sequence of extremal functions associated to the singletons $\\{\\bm s_n\\}$, $n=1,\\ldots,N$, where some of these functions may be identical to each other. The following theorem characterizes the conditional distributions of $\\phi_{\\bm s_{n}}$ given $\\{\\phi_{\\bm s_k}^+\\}_{1\\leq k\\leq n-1}$  for extremal functions at position $n=2,\\ldots,N-1$, which are required for iterative simulation. \n\n\\begin{theorem}[Conditional distributions of extremal functions]\\label{thm:2}\nConsider a max-id process  $\\{Z(\\bm s)\\}_{s\\in{\\mathcal{S}}}$  as defined in Section~\\ref{sec:Max-id}. Given sites $K_N=\\{\\bm s_1,\\bm s_2,\\dots,\\bm s_N\\}$,  the distribution of its extremal function $\\phi_{\\bm s_{1}}$ and of its conditional extremal functions $\\phi_{\\bm s_{n}}\\mid\\{\\phi_{\\bm s_k}^+\\}_{1\\leq k\\leq n-1}$ for $n>1$ can be characterized as follows:\n\\begin{enumerate}[(1)]\n    \\item Initial extremal function: given a realization $z$ of $Z(\\bm s_1)$, the extremal function $\\phi_{\\bm s_1}^+$ is distributed according to $P_{\\bm s_1}(z,\\cdot)$.\n    \\item Conditional extremal functions for $n\\geq 1$: given the extremal functions $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$ for sites in $K_{n}$, consider the Poisson measure  of functions  that  do not exceed the already simulated max-id values of sites in $K_{n}$ but that exceed the maximum of already simulated values at $\\bm s_{n+1}$:\n     \\begin{equation}\\label{eq:simu}\n      \\tilde\\Lambda(\\mathrm{d}f)=\\mathbb{I}\\left(f(\\bm s_i)<Z_{n}(\\bm s_i),1\\leq i\\leq n\\right)\\mathbb{I}\\left(f(\\bm s_{n+1})>Z_n(\\bm s_{n+1})\\right)\\,\\Lambda(\\mathrm{d}f),\n      \\end{equation}\n      where the indicator function $\\mathbb{I}(A)$ is $1$ if the condition $A$ is satisfied and $0$ otherwise. \n    Then, the extremal function $\\phi_{\\bm s_{n+1}}^+$ conditional on $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$ is given  as follows: \n    \\[\\tilde \\phi_{\\bm s_{n+1}}^+ = \\left\\{\\begin{array}{ll}\n    \\arg\\max_{\\phi\\in PPP(\\tilde\\Lambda)}\\phi(\\bm s_{n+1}), & \\tilde\\Lambda({\\mathcal C})\\neq0, \\\\\n     \\arg\\max_{\\phi\\in\\{\\phi_{\\bm s_1}^+,\\dots,\\phi_{\\bm s_{n}}^+\\}}\\phi(\\bm s_{n+1}),    &  \\tilde\\Lambda({\\mathcal C})=0.\n    \\end{array}\\right.\\]\n\n\n\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe proof follows closely the proof of Theorem 2 in \\cite{Dombry.Engelke.Oesting:2016}, which concerns max-stable processes, while we here treat max-id processes. \n\nTo simulate extremal functions, we proceed sequentially. The distribution of the first extremal function $\\phi_{\\bm s_1}^+$ given $\\phi_{\\bm s_1}^+(\\bm s_1) = z$ is exactly $P_{\\bm s_1}(z,\\cdot)$, as explained in \\S\\ref{sec:Max-id} above, and $\\phi_{\\bm s_1}^+(\\bm s_1)$ thus follows the marginal distribution of $Z(\\bm s_1)$, which has now a more general form than for max-stable processes. Notice also that given the continuity assumption in \\eqref{eq:assumption}, there is almost surely exactly one extremal function at each location.\n\nThen, to simulate the extremal function $\\phi_{\\bm s_{n+1}}^+$  at site $\\bm s_{n+1}$, for $n\\geq 1$, we take into account that it can be generated independently conditional on the already simulated extremal functions $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$, thanks to the Poisson process property.\nTherefore, $\\phi_{\\bm s_{n+1}}^+$ either corresponds to one of the functions in $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$, or it is a new extremal function, stochastically independent of the first $n$ extremal functions $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$. The mean  measure $\\tilde\\Lambda({\\mathcal C})$ in \\eqref{eq:simu}  defines the restricted Poisson process of candidate functions with positive Poisson intensity, and a new extremal function $\\phi_{\\bm s_{n+1}}^+$ can be generated according to this restricted point process with finite intensity measure. \n\tIf the measure $\\tilde\\Lambda({\\mathcal C})$ is zero, then there are  almost surely no points in the Poisson process PPP($\\tilde\\Lambda$) of points that exceed the extremal functions $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$ at the location $\\bm s_{n+1}$. Hence, in this case the value $Z(\\bm s_{n+1})$ of the max-id process corresponds to the maximum of the previous extremal functions $\\{\\phi_{\\bm s_i}^+\\}_{1\\leq i \\leq n}$, i.e., $Z(\\bm s_{n+1}) = Z_n(\\bm s_{n+1})$. If $\\tilde\\Lambda({\\mathcal C})>0$, then the realization of the restricted point process contains at least one point with positive probability. If there is at least one point, then the new extremal function at $\\bm s_{n+1}$ corresponds to the point realizing the maximum value at $\\bm s_{n+1}$. \n\\end{proof}\n\nAn exact simulation algorithm based on Theorem~\\ref{thm:2} is given in Algorithm~\\ref{algo}.\nAlthough the main structure of the algorithm resembles that proposed by \\citet{Dombry.Engelke.Oesting:2016} for max-stable processes, the key difficulty for general max-id processes resides in efficiently simulating from the distribution $P_{\\bm s}(z,\\cdot)$ given $Z(\\bm s)=z$. For suitable constructions of the Poisson point process $\\{\\eta_i;\\, i=1,2,\\dots\\}$, we can investigate the specific structure of this conditional distribution. In this paper, we consider two different and very general structures for $\\eta_i$, which are inspired from the construction of max-stable processes, yet they give rise to max-id models with much more flexibility in their joint tail decay rates. These two types of models, which encompass all of the commonly used max-stable models from the spatial extremes literature, are described in the following two subsections. \n\n\\begin{algorithm}[t]\n  \\caption{Exact simulation of a max-id process $Z$ at locations $K_N=\\{\\bm s_1,\\dots,\\bm s_N\\}.$\\label{algo}}\n  \\begin{algorithmic}[1]\n  \\Require  Stationary marginal distribution $G_0$ of $Z$. \n  \\Require  Dependence characterization of $Z$  allowing simulation of $P_{\\bm s_n}$, $n=1,\\ldots,N$. \n  \\State Simulate $E_0 {\\ {\\sim}\\ } \\rm{Exp}(1)$ and set $z=G_0^{-1}(\\exp(-E_0))$.\n  \\State Simulate $Y{\\ {\\sim}\\ } P_{\\bm s_1}(z,\\cdot)$. \n  \\State Set $Z(K_N) = Y(K_N)$.\n  \\ForAll {$n=2,\\dots,N$}\n  \\State Simulate $E_0 {\\ {\\sim}\\ } \\rm{Exp}(1)$ and set $z=G_0^{-1}(\\exp(-E_0))$.\n  \\While {$\\{z>Z(\\bm s_n)\\}$}\n  \\State Simulate $Y{\\ {\\sim}\\ } P_{\\bm s_n}(z,\\cdot)$.\n  \\If {$Y(\\bm s_i)<Z(\\bm s_i)$ for all $i=1,\\dots,n-1$}\n  \\State update $Z( K_N)$ by the componentwize $\\max(Z(K_N),Y(K_N))$.\n  \\EndIf\n  \\State Simulate $E_1 {\\ {\\sim}\\ } \\rm{Exp}(1)$ and update $E_0$ and $z$ by setting $E_0 = E_0 + E_1$ and $z=G_0^{-1}(\\exp(-E_0))$.\n  \\EndWhile\n  \\EndFor\n  \\Ensure Return $Z(K_N)$\n  \\end{algorithmic}\n \\end{algorithm}\n\n\\subsection{Representation of $\\eta_i$ as a Gaussian scale mixture}\n\\label{sec:gaussian-scale-mixture}\n\n\\subsubsection{Model description}\u00a0\\label{sec:gaussian-scale-mixture-description}\nOne possible structure of the general max-id process is achieved by representing $\\eta_i$ as a \\emph{Gaussian scale mixture}, i.e., $\\eta_i = R_iW_i$, where $\\{R_i\\}$ are the points of a Poisson point process on $\\mathbb R^+$ with $\\Lambda(0,\\infty)=\\infty$, and $\\{W_i\\}$ are independent Gaussian random fields on ${\\mathcal{S}}$, which are not necessarily identically distributed and may depend on the overall random ``magnitude'' $R_i$. This construction mimics  the max-stable spectral construction of extremal-$t$ processes \\citep{Opitz:2013} arising for a specific choice of $\\Lambda$ with $\\{R_i\\}$ and $\\{W_i\\}$ being independent. To extend max-stable processes, \\citet{Huser.etal:2020} proposed using a more flexible mean measure for $\\{R_i\\}$ and \\citet{Zhong.Huser.Optiz:2020} further relaxed the independence assumption between $\\{R_i\\}$ and $\\{W_i\\}$.\nSpecifically, they proposed a Weibull-tailed mean measure $\\kappa$ for $\\{R_i\\}$, where\n\\begin{equation}\\label{eq:measure_R}\n\\kappa(r)=\\kappa\\left([r,\\infty)\\right) = r^{-\\beta}\\exp\\{-\\alpha(r^{\\beta}-1)\/\\beta\\},\\quad r>0,\\quad (\\alpha,\\beta)^\\top\\in (0,\\infty)^2. \n\\end{equation}\nWith this specification, $\\kappa(r){\\longrightarrow} r^{-\\alpha}$ as $\\beta\\downarrow0$. Therefore, the model remains in the neighborhood of max-stable processes for which $\\kappa(r)=r^{-\\alpha}$. Given $\\{R_i\\}$, the processes $\\{W_i\\}$ may be chosen as standard Gaussian random fields with correlation function $\\rho(\\bm s_1,\\bm s_2;R_i)$, where $R_i$ modulates the dependence range of $W_i$ in such a way that large magnitudes $R_i$ create processes $W_i$ with weaker dependence. This gives extra flexibility for capturing the joint tail decay rate of the resulting max-id process $Z$ in the asymptotic independence setting, i.e., when $\\lim_{u\\to1}\\pr[G_0\\{Z(\\bm s_1)\\}>u\\mid G_0\\{Z(\\bm s_2)\\}>u]=0$ with $G_0$ being the marginal distribution of $Z$. For example, \\citet{Zhong.Huser.Optiz:2020} chose a non-stationary correlation $\\rho(\\bm s_1,\\bm s_2;R_i)$, which may be specified as\n\\begin{equation}\\label{eq:corr_function_W}\n\\rho(\\bm s_1,\\bm s_2;R_i) = (\\lambda_{\\bm s_1}\\lambda_{\\bm s_2})\\left(\\frac{\\lambda_{\\bm s_1}^2+\\lambda_{\\bm s_2}^2}{2}\\right)^{-1}\\exp\\left\\{-\\left(\\frac{\\lambda_{\\bm s_1}^2+\\lambda_{\\bm s_2}^2}{2}\\right)^{-1\/2}(1+R_i)^\\nu\\|\\bm s_1-\\bm s_2\\|\\right\\},\n\\end{equation}\nwhere $\\lambda_{\\bm s}>0$ is a spatially-varying range parameter surface that can be associated with spatial covariates, and $\\nu\\geq0$ is a parameter that controls how the scaling variable $R_i$ influences the dependence range of $W_i$. In the simulation examples below, we will illustrate some specific structures for $\\lambda_{\\bm s}$. When the value of $R_i$ is large and $\\nu>0$, the dependence within the Gaussian random field $W_i$ indeed decreases with increasing ``magnitude\" $R_i$. \nIn this model, the marginal exponent measure for the locations in $K\\equiv K_N$ may be written as\n$$\\Lambda_K([\\bm z_K,\\infty))=\\int_0^\\infty\\{1-\\Phi(\\bm z_K\/r;\\bm\\rho(K;r))\\}\\kappa(\\mathrm{d}r)$$\nwhere $\\Phi(\\cdot;\\bm\\rho(K;r))$ is the multivariate normal distribution function with zero mean and correlation matrix $\\bm\\rho(K;r)$ determined by the correlation function $\\rho(\\bm s_1,\\bm s_2;r)$ for $\\bm s_1,\\bm s_2\\in K$.\n\n\nTo illustrate the tail dependence properties of this max-id process, we use the level-dependent extremal coefficient \\citep{Padoan2013,Huser.etal:2020}. For a collection of sites $K_N \\subset {\\mathcal{S}}$ and $|K_N|=N$, the level-dependent extremal coefficient is defined as\n\\begin{equation}\\label{eq:ext_coef}\n\\theta_N(z_0) = {\\log\\{G(\\bm z_0)\\}\\over\\log\\{G_0(z_0)\\}\n\\end{equation}\nwhere $\\bm z_0 = (z_0,\\dots,z_0)^T\\in\\mathbb {R}^N$, $G_0$ is the marginal distribution of $Z$ and $G$ is the joint distribution of the random vector $\\bm Z(K_N)$ with observations at the locations in the set $K\\subset{\\mathcal{S}}$. This coefficient can be expressed in terms of the measure $\\Lambda$, and in the case where the margins are unit Fr\\'echet it is specifically equal to $\\theta_N(z_0) =z_0\\Lambda_{K_N}\\left\\{(-\\infty,\\infty)^N\\setminus (-\\infty,z_0]^N\\right\\}$. From \\eqref{eq:ext_coef} it is easy to see that $\\pr\\{\\bm Z(K_N)\\leq z_0\\} = G_0(z_0)^{\\theta_N(z_0)}$, such that this coefficient can be interpreted as the effective number of independent variables among the components of the vector $\\bm Z(K_N)$ at level $z_0$.  Higher values of $\\theta_N(z_0)$ imply weaker dependence at quantile level $z_0$. This dependence measure  is widely used in extreme-value studies (mainly with max-stable models), where modeling the tail dependence structure is the main interest. As an example, we consider the correlation function of $W_i$ conditional to the event magnitude $R_i$ chosen as $\\rho(\\bm s_1,\\bm s_2;R_i) =\\exp\\left\\{-2(1+R_i)^\\nu \\|\\bm s_1-\\bm s_2\\|\\right\\}$, which corresponds to the stationary and isotropic counterpart of \\eqref{eq:corr_function_W} with spatially-constant $\\lambda_{\\bm s}=1\/2$.  Figure~\\ref{fig:GS} shows the realizations of $\\{Z(s)\\}_{s\\in(0,1)}$ on a standard Gumbel scale generated using our proposed exact simulation algorithm, as well as the bivariate level-dependent extremal coefficients between two sites at distance $h=0.5$ for the parameters $\\alpha=1$, $\\beta=0,0.5,1,2$ and $\\nu=0,0.25,0.5,1$. When $\\beta=0$ (interpreted here as $\\beta\\downarrow0$), the extremal dependence persists with increasing level $z$. The curve of extremal dependence becomes a horizontal line when we $\\nu$ and $\\beta$ are both zero. In this case, the max-id process is the max-stable extremal-$t$ process. When $\\beta$ is different from zero, we get asymptotic independence (since $\\theta_2(z_0)\\to2$ when $z{\\longrightarrow}\\infty$), and the dependence level decreases as the values of $\\beta$ and $\\nu$ increase. The realizations were generated using the same random seed, and we can discern stronger fluctuations of the realizations as the value of  $\\beta$ increases, corresponding to a decreasing dependence strength.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{simulation_GS.pdf}\n\t\\includegraphics[width=1\\linewidth]{theta_GS.pdf}\n\t\\caption{Simulations (top row) and bivariate level-dependent extremal coefficients (bottom row) for models based on the Gaussian scale mixture construction. The parameters are chosen as $\\nu=0, 0.25,0.5,1$ (black, red, green, blue curves), where $\\nu$ controls the dependence between $R_i$ and $W_i$ by using correlation function $\\rho(\\bm s_1,\\bm s_2;R_i) =\\exp\\left\\{-2(1+R_i)^\\nu \\|\\bm s_1-\\bm s_2\\|\\right\\}$ conditional to $R_i$. The level $z$ is shown on a logarithmic scale.}\n\t\\label{fig:GS}\n\\end{figure}\n\n\n\\subsubsection{Simulation algorithm}\\label{sec:gaussian-scale-mixture-simulation}\n\nThe generic Algorithm~\\ref{algo} based on extremal functions can be applied, and we now specifically describe how to sample from $P_{\\bm s_0}(z,\\cdot)$ for some fixed site $\\bm s_0$ (see lines 2 and 7 of the algorithm) for the max-id model constructed from Gaussian scale mixtures $\\eta_i=R_iW_i$ as described in \\S\\ref{sec:gaussian-scale-mixture-description}. Simulation according to $P_{\\bm s_0}(z,\\cdot)$ at locations in $K$ implies that we have to simulate $\\eta_i=R_iW_i$ at $K$ conditional on $\\eta_i(\\bm s_0)=R_iW_i(\\bm s_0) = z$. Given the intensity function $f_{R}(r) =\\mathrm{d}\\Lambda(r)\/\\mathrm{d}r$ and the univariate standard Gaussian density $f_W$, it is straightforward to use variable transformations to derive the joint intensity of $R_i$ and $R_iW_i(\\bm s_0)$ as $f_{(R,RW)}(r,z) = f_R(r)\\times f_W(z\/r)\/r$. Then, using the definition of conditional intensities, the \n conditional density $f_{R\\mid RW}(r\\mid z)$ of ${R_i\\mid R_iW_i(\\bm s_0) = z}$ is $f_{(R,RW)}(r,z)\/f_{RW}(z)$, where  the  marginal intensity arising in the denominator is $f_{RW}(z) = \\int f_{(R,RW)} (r,z) \\, \\mathrm{d}r$. Since this involves an integral, which typically cannot be obtained in closed form and whose numerical calculation  in practice may be relatively slow and potentially unstable, we treat it as an unknown constant. We here explore the use of two simulation techniques where knowledge of the value of this constant is not required. Precisely, we can proceed according to the following three general steps:\n\\begin{enumerate}[(1)]\\label{simu:Gaussian-scale}\n    \\item Simulate a value $R$ from the distribution of $R_i\\mid R_iW_i(\\bm s_0)=z$, using either \n    \\begin{itemize}\n    \t\\item the Metropolis--Hastings algorithm \\citep[e.g.,][]{Robert2013}\n\n    \t\\item  the  adaptive rejection sampling method proposed in \\cite{Gilks.Wild:1992}.\n    \\end{itemize}\n    \\item Given $R_i=R$, simulate a vector $\\bm W$ from the distribution of $W_i\\mid RW_i(\\bm s_0)=z$ at the locations in $K_N$. \n    \\item Return $R\\bm W$. \n\\end{enumerate}\nNotice that the second step corresponds to simulating  from a conditional multivariate normal distribution and can easily be performed exactly \\citep[e.g.,][]{Varadhan2015} in dimension $N$ up to several thousands. \nTherefore, the key difficulty lies in the first step, where we have to simulate from the probability density $f_{R\\mid RW}(r\\mid z)$ to simulate the conditional event magnitudes. \n\n\\subsubsection{Options for simulating  conditional event magnitudes}\n\nWith the Metropolis--Hastings (MH) method in Step~2 of the above algorithm in \\S\\ref{sec:gaussian-scale-mixture-simulation}, we use the following symmetric random walk proposal: \n\\[\\log(R_i^{j}) \\mid R_i^{j-1}  {\\ {\\sim}\\ } \\mathcal{N}(\\log(R_i^{j-1}),\\sigma^2), \\quad j = 1,2,\\ldots,\\]\nwhere $\\sigma^2$ is the proposal variance and has to be tuned, and the initial value $R_i^{0}$ is either fixed by the user or drawn from some distribution that we can easily simulated from. \nMH is approximate (thus not exact) when the distribution of the initial value is different from the target distribution (which is always the case in practice), but its error becomes negligible when a sufficient number of iterations are performed. Here, the MH algorithm is performed on a single variable $R_i$ at a time, so each Markov chain Monte Carlo (MCMC) chain mixes well and converges very quickly to its stationary distribution, while iterations are of very small computational cost. Our experience shows that very few iterations are in fact needed in practice to get accurate results. We therefore regard Algorithm~\\ref{algo} combined with MH for simulating $R_i\\mid R_iW_i(\\bm s_0)=z$ as a general ``quasi-exact'' simulation method.\n\nAs an alternative to MH, we also consider adaptive rejection sampling (ARS), which is an exact simulation algorithm that is often used in combination with the Gibbs sampler. It can generate exact independent samples from the target distribution, here supposed to be the (unnormalized) density $f(r\\mid z)=f_{R\\mid RW}(r\\mid z)\\propto f_{R,RW}(r,z)$ with fixed $z$. The only requirement for using this method is that $h(r)=\\log f(r\\mid z)$ is concave in $r$ with a connected support, and that it is continuous and differentiable everywhere within the support. In particular, for the previously presented parametric family of Gaussian scale mixture models based on \\eqref{eq:measure_R}, it is straightforward to show that $h(r)$ is indeed a continuous, differentiable and concave function over its support $\\mathbb R$. The ARS algorithm adaptively constructs a piecewise upper hull function $u(r)$ and a piecewise lower bounding function $l(r)$ of $h(r)$, respectively, and then uses the distribution $s(r)=\\exp\\{u(r)\\}\/\\int \\exp\\{u(r')\\} \\mathrm{d}r'$ to generate proposals in the exact rejection sampling scheme. Figure \\ref{fig:ars} illustrates the adaptive rejection sampling scheme, where the upper hull function $u(r)$ is tangent to the target $h(r)$ at locations $\\bm r=\\{r_1,r_2,r_3, r_4\\}$. \n\\begin{figure}\n\t\\centering \n\t\\includegraphics[width=0.5\\textwidth]{ars.pdf}\n\t\\caption{An illustration of the adaptive rejection sampling scheme, where the upper hull function $u(r)$ is tangent to the logarithmic target density $h(r)$ at locations $\\bm r=\\{r_1,r_2,r_3, r_4\\}$. }\n\t\\label{fig:ars}\n\\end{figure}\nTherefore, $u(r)$ and $l(r)$ are uniquely determined by the locations in $\\bm r$ and the target function $f(r\\mid z)$. As we increase the number of points in $\\bm r$, the functions $u(r)$ and $l(r)$ will form an increasingly narrow envelope around the function $h(r)$, and the acceptance rate of the algorithm will increase. The steps of the ARS algorithm are as follows:\n\\begin{enumerate}[(1)]\n   \\item[(0)] Initialization: choose a set of anchor points $\\bm r$ and construct the upper and lower bounds $u(r)$ and $l(r)$, respectively, as illustrated in Figure~\\ref{fig:ars}.\n   \\item Sampling: sample $R^*{\\ {\\sim}\\ } s(r)$, and sample  $U {\\ {\\sim}\\ } \\rm{Unif}(0,1)$ independently of $R^*$. \n   \\item First acceptance test:  if $U \\leq \\exp\\{l(R^*)-u(R^*)\\}$ then accept $R^*$ and go to 5. \n  \n   \\item Second acceptance test: If $U \\leq \\exp\\{h(R^*)-u(R^*)\\}$ then accept $R^*$ and go to 5.\n   \\item If the first and second acceptance tests both fail, reject $R^*$; add $R^*$ to list of anchor points and update the bounds $u(r)$ and $l(r)$. Go to 1.\n   \\item Return $R^*$.\n\\end{enumerate}\nThe distribution of the simulated value given acceptance is thus \n\\begin{eqnarray*}\n\\pr(R^*\\in A\\mid\\text{acceptance of }R^*) &=&{\\int_A s(r) \\left\\{\\int_{0}^{\\exp\\{l(r)-u(r)\\}}{\\rm d}w + \\int_{\\exp\\{l(r)-u(r)\\}}^{\\exp\\{h(r)-u(r)\\}}{\\rm d}w\\right\\}{\\rm d}r\\over\\int_{\\mathbb R} s(r) \\left\\{\\int_{0}^{\\exp\\{l(r)-u(r)\\}}{\\rm d}w + \\int_{\\exp\\{l(r)-u(r)\\}}^{\\exp\\{h(r)-u(r)\\}}{\\rm d}w\\right\\}{\\rm d}r}\\\\\n&=&{\\int_A \\exp\\{h(r)\\}{\\rm d}r\\over\\int_{\\mathbb R} \\exp\\{h(r)\\}{\\rm d}r},\n\\end{eqnarray*}\nwhich corresponds  exactly to  the target distribution with density $f(r\\mid z)$. By exploiting the concavity property of $h$, ARS usually only needs a relatively small number of iterations before acceptance, and it may therefore be faster and more accurate than MH, as we will illustrate through the simulation experiments in \\S\\ref{sec:Simu-Study}. \n\n\\subsection{Representation of $\\eta_i$ as a Gaussian location mixture} \n\n\\subsubsection{Model description}\n\nAnother flexible and general max-id model family is the class of \\emph{Gaussian location mixtures}, where $\\eta_i$ has the form $\\eta_i=R_i+W_i$. Here, the location variables $\\{R_i\\}$ are the points of a Poisson point process over $\\mathbb R$ with mean measure $\\overline \\kappa$, and $\\{W_i\\}$ are independent Gaussian random fields. Again, the random variables $\\{R_i\\}$ can be interpreted as overall event magnitudes, while the random processes $\\{W_i\\}$ vary over $\\mathcal{S}$ and can be viewed as spatial ``profiles\".  For a general parametric form of the distribution of event magnitudes $\\{R_i\\}$,  we propose to define $\\overline \\kappa$ as\n\\begin{equation}\\label{eq:measure_R2}\n\\overline\\kappa(r) = \\exp\\left[-\\alpha\\left\\{\\mathbb{I}(r>0)|r|^{\\beta_1}+\\mathbb{I}(r<0)|r|^{\\beta_2}\\right\\}\\right], \\quad r\\in\\mathbb R,\\quad(\\alpha,\\beta_1,\\beta_2)^\\top\\in(0,\\infty)^2\\times(0,2),\n\\end{equation}\nwhere $\\overline\\kappa(r)$ is the short-hand notation for $\\overline\\kappa([r,\\infty))$. The parameter restrictions for $\\alpha$, $\\beta_1$ and $\\beta_2$ ensure that the mean measure $\\Lambda$ of the points $\\{\\eta_i\\}$ is indeed Radon (i.e., locally finite) as required to defined a valid Poisson point process. Max-id models based on Gaussian location mixtures have never been explored in the literature, and the new model~\\eqref{eq:measure_R2} is especially interesting in view of the flexible dependence structure that it yields and its close connection to max-stable Brown--Resnick processes \\citep{Kabluchko.etal:2009}. The parameter $\\beta_1$ controls the dependence strength in the upper tail, while $\\beta_2$ controls the dependence strength in the lower tail.  As in \\S\\ref{sec:gaussian-scale-mixture-description}, we may further allow for dependence between the point $R_i$ and the Gaussian field $W_i$ for each $i$. For a general and flexible correlation function of $W_i$, we may again use the correlation function in \\eqref{eq:corr_function_W} to drive the range of the Gaussian fields $W_i$ through the location variable $R_i$, though other options are of course possible. In this case, the mean measure of $\\{\\eta_i;i=1,2,\\dots\\}$ at a set of locations $K$ is given by \n\\[\\Lambda_K([\\bm z_K,\\infty))=\\int_{\\mathbb R}\\{1-\\Phi(\\bm z_K-r;\\bm \\rho(K;r))\\}\\overline\\kappa({\\rm d}r).\\]\n\nWhen $\\alpha=\\beta_1=\\beta_2=1$ and $\\{R_i\\}$ and $\\{W_i\\}$ are independent of each other, the proposed model reduces to the popular Brown--Resnick max-stable model \\citep{Kabluchko.etal:2009,Wadsworth2014}, although this model is often defined through intrinsically stationary Gaussian processes $\\{W_i\\}$ with some variogram function $\\gamma(h)$ rather than through a correlation function $\\rho(\\bm s_1,\\bm s_2)$. \n\n\\subsubsection{Simulation algorithm}\n\nBy analogy with the approach developed for Gaussian scale mixtures, we can here again use Algorithm~\\ref{algo} for the max-id location mixture process. Moreover, to sample from $P_{\\bm s_0}(z,\\cdot)$, we similarly propose a simulation algorithm, where the first step consists of sampling from the conditional (unnormalized) density $f(r\\mid z)$ of $R_i\\mid R_i+W_i(\\bm s_0)=z$. Again, change-of-variable formulas allow writing $f(r \\mid z)$ as a density with a normalizing constant defined through a univariate integral. However, here we observe that the conditional density is not logarithmically concave, and therefore we cannot use the adaptive resampling scheme to generate extremal functions from $P_{\\bm s_0}(z,\\cdot)$, but we can still use the quasi-exact Metropolis--Hastings algorithm, which provides high accuracy at low computational cost. Thus, our proposed procedure for simulating max-id processes of Gaussian location mixture type at a collection of sites $K_N$ relies on Algorithm~\\ref{algo} combined with the following three-step approach to sample from $P_{\\bm s_0}(z,\\cdot)$:\n\\begin{enumerate}[(1)]\\label{simu:Gaussian-location}\n    \\item Simulate a value  $R$ according to the distribution of $R_i\\mid R_i+W_i(\\bm s_0)=z$ by using the Metropolis--Hastings method with random walk proposal distribution $R_i^{j}\\mid R_i^{j-1} {\\ {\\sim}\\ } \\mathcal{N}(R_i^{j-1},\\sigma^2)$, $j=1,2,\\ldots$, and initial point $R_i^0$.\n    \\item Given $R_i=R$, simulate a vector $\\bm W$ according to the distribution of $W_i\\mid R+W_i(\\bm s_0)=z$ at the locations in $K_N$.\n    \\item Return $R+\\bm W$. \n\\end{enumerate}\nAs in the Gaussian scale mixture case, the second step consists in simulating from a conditional multivariate normal distribution, which is fast and straightforward. Figure~\\ref{fig:GL} shows examples of realizations of $\\{Z(s)\\}_{s\\in(0,1)}$ with standard Gumbel marginal distributions (generated using the proposed quasi-exact algorithm), as well as the bivariate level-dependent  extremal coefficients at distance $h=0.5$ for the parameters $\\beta_1 = 0.5,1,1.5$, $\\beta_2 = 0.5,1,1.5$ and $\\alpha=1$. The correlation function for $W_i$ is here chosen to be exponential as $\\rho(h) =\\exp\\left(-2h\\right)$. We see that when both $\\beta_1$ and $\\beta_2$ are equal $1$, the curve of the extremal coefficients is a horizontal line. In this case, the max-id process is the Brown--Resnick max-stable process. As the values for $\\beta_1$ and $\\beta_2$ increase, the dependence levels in the lower tail and the upper tail, which are controlled by $\\beta_2$ and $\\beta_1$, respectively, decrease. When $\\beta_1>1$, the extremal coefficient increases as the level $z{\\longrightarrow}\\infty$. By contrast, when $\\beta_1<1$, the extremal coefficient decreases as the level $z{\\longrightarrow}\\infty$. Similar results are observed for the dependence level in the lower tail with $\\beta_2$. Therefore, this model shows great flexibility through its ability to capture asymptotic dependence and asymptotic independence in the upper tail and different extremal dependence structures in the lower tail.\nIn these  realizations generated with the same random seed, we discern that the fluctuations are increasing as the dependence level decreases, as is expected.  \n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{simulation_GL.pdf}\n\t\\includegraphics[width=1\\linewidth]{theta_GL.pdf}\n\t\\caption{Simulations (top row) and bivariate level-dependent extremal coefficients (bottom row) for models based on the Gaussian location mixture construction. Parameters are chosen as $\\alpha=1$ and $\\beta_1=0.5,1,1.5$ (blue, black, red). The correlation function for $W_i$ is exponential and given by $\\rho(h) =\\exp\\left(-2h\\right)$. The level $z$ is shown in logarithmic scale.}\n\t\\label{fig:GL}\n\\end{figure}\n\t\n\\section{Simulation Study}\n\\label{sec:Simu-Study}\n\nIn this section, we use the Gaussian scale mixture representation described in \\S\\ref{sec:gaussian-scale-mixture} as an example to illustrate Algorithm~\\ref{algo} and to assess the performance of the two algorithmic choices detailed  in \\S\\ref{sec:gaussian-scale-mixture-simulation} for sampling from $P_{\\bm s_0}(z,\\cdot)$, i.e., of the Metropolis--Hastings scheme and the adaptive rejection sampling scheme. We also provide a comparison with respect to a naive simulation approach inspired from \\citet{Schlather:2002}, which consists in simulating the magnitudes $R_i$ in descending order and truncating the maximum in \\eqref{eq:max-id} as $Z(\\bm s)\\approx\\max_{i=1,\\ldots,n}\\eta_i(\\bm s)$ with $\\eta_i=R_iW_i(\\bm s)$ for a sufficiently large integer $n$. \n\n We perform the simulation on a $7\\times7$ regular grid in $(0,1)^2$, corresponding to $49$ locations in $K$. We fix $\\alpha=5,\\beta=2$ in \\eqref{eq:measure_R} and $\\nu=3,\\lambda_{\\bm s} = \\exp[2-0.5\\Phi\\{(s_x-0.5)\/0.25\\}]$ in \\eqref{eq:corr_function_W}, where $\\bm s=(s_x,s_y)^\\top$ and $\\Phi(\\cdot)$ is the standard normal distribution function. We implemented three algorithms using the \\texttt{R} software: the approximate naive sampling method (APPROX), Algorithm~\\ref{algo} combined with the quasi-exact random walk Metropolis--Hastings (MH), and Algorithm~\\ref{algo} combined with the exact adaptive rejection sampling (ARS). Our aim is to compare the simulation accuracy with respect to univariate distributions and to dependence properties. \n We tuned the settings of these three algorithms such that they require approximately the same amount of running time to generate $N=10^5$ independent samples of the max-id process on the grid. These settings are given as follows: \n\\begin{enumerate}[(1)]\n\t\\item Naive method: Simulate $\\eta_i=R_iW_i$,  $i=1,2,\\dots,n$ on the grid  $K$, where $n=100$.\n\t\\item Metropolis--Hastings scheme: In the MCMC, run $100$ iterations to generate one sample from $R_i\\mid R_iW_i(\\bm s_0)=z$.\n\t\\item Adaptive rejection sampling scheme: Use 6 points to initialize the upper hull function and lower hull function.\n\\end{enumerate}\nBased on  the generated samples, we compute the empirical Kullback--Leibler (KL) divergence $D_{\\bm s_0}(p\\|q)$ \\citep{Kullback.Leibler:1951} in order to compare the accuracy of simulated samples at each site $\\bm s_0\\in K\\subset{\\mathcal{S}}$. Precisely, we compute\n\\begin{equation}\n\tD_{\\bm s_0}(p\\|q) = \\sum_{i=1}^N p\\{Z_i(\\bm s_0)\\}\\log\\left[{q\\{Z_i(\\bm s_0)\\}\\over p\\{Z_i(\\bm s_0)\\}}\\right],\n\\end{equation}\nwhere $q$ is the empirical density of the samples while $p$ is the exact marginal density of $Z$. Therefore, we have $49$ empirical KL divergences in total, one for each point on the grid. Figure \\ref{fig:KL} shows the boxplot of the $49$ empirical KL divergences using the simulated datasets generated by the three algorithms. Both the adaptive rejection sampling scheme and the Metropolis--Hastings scheme outperform the naive method by a significant margin. Moreover, the median of the empirical KL divergence for the adaptive rejection sampling scheme is notably lower than  with  the Metropolis--Hastings scheme. Generating a single max-id replication on the grid took approximately 1.2 seconds using a single-core machine. \n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{KLD_test.pdf}\n\t\\label{fig:KL}\n\t\\caption{Boxplots of  empirical KL divergences for the $49$ grid locations based on the simulated data sets generated by the naive method (APPROX), the Metropolis--Hastings scheme (MH) and the adaptive rejection sampling scheme (ARS).}\n\\end{figure}\n\nTo further evaluate the performance of each sampling scheme with respect to the dependence structure, we calculated the bivariate (level-dependent) empirical extremal coefficients $\\hat\\theta_2$. For each pair of sites, we compared the empirical extremal coefficient computed from the $N=10^5$ simulated replicates with the theoretical counterpart computed by evaluating the mean measure $\\Lambda$ with a very accurate numerical unidimensional integral.  Figure~\\ref{fig:ext_coef} shows the results for the three simulation techniques. The adaptive rejection sampling scheme and the Metropolis--Hastings scheme appear to perform much better than the naive scheme, which is significantly biased at low quantile levels $z_0$, while the adaptive rejection sampling scheme seems to be perhaps slightly better than the Metropolis--Hastings scheme. Our results imply that if we draw statistical inferences based on simulated samples from the naive method, the strength of the dependence at relatively low quantile levels of extremes would be significantly overestimated in the current setting. Better results might be obtained for the naive method by increasing the number $n$ of simulated fields $\\{\\eta_i\\}$, but this comes with a computational cost. By contrast, our proposed algorithm always provides exact max-id simulations at a low computational cost, which may prove essential for simulation-based statistical inference.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[page=1,width=0.9\\textwidth]{scatter_plot_extremal.pdf}\n\n\t\\includegraphics[page=2,width=0.9\\textwidth]{scatter_plot_extremal.pdf}\n\n\t\\includegraphics[page=4,width=0.9\\textwidth]{scatter_plot_extremal.pdf}\n\n\t\\includegraphics[page=5,width=0.9\\textwidth]{scatter_plot_extremal.pdf}\n\t\\label{fig:ext_coef}\n\t\\caption{Bivariate scatterplots of theoretical against empirical bivariate level-dependent extremal coefficients at quantile levels $0.05,0.25,0.5,0.75,0.95$ (from top to bottom) for all pairs of locations, based on the three sampling schemes (left: Algorithm~\\ref{algo} with adaptive rejection sampling (ARS), middle: Algorithm~\\ref{algo} with Metropolis--Hastings (MH), right: naive approximate method).}\n\\end{figure} \n\nAs in \\citet{Dombry.Engelke.Oesting:2016}, it can be shown that the expected number of random fields $W_i$ to be simulated in the exact simulation algorithm equals to the number of simulation sites, i.e., the dimension $|K|$. To further illustrate the efficiency of our proposed algorithm, we simulated 10 replicas for different numbers of randomly sampled locations $K$ on the unit square $(0,1)^2$ using the ARS scheme and calculated the time and number of random fields $W_i$ that needed to be simulated. The results are shown in Figure~\\ref{fig:complexity}. The parameters are set to be $\\alpha=1,\\beta=1$ and the correlation function for $W_i$ is chosen to be $\\rho(h) = \\exp(-h)$. Since we use the Cholesky decomposition for simulating the process $W_i$ on $K$, whose complexity is ${\\mathcal{O}}(|K|^3)$, the expected overall complexity for simulating a max-id replicate is ${\\mathcal{O}}(|K|^4)$. The red curve on the left panel of Figure~\\ref{fig:complexity} illustrates this. Here, the computational time is about two minutes for one replicate in dimension $|K|=900$, which is already quite fast. If further speeds-up are required, approximate algorithms for simulating the Gaussian processes $W_i$ may be employed (e.g., based on Gaussian Markov random fields). As expected, the right panel of Figure~\\ref{fig:complexity} shows that the expected number of simulations of $W_i$ that are needed is indeed equal to $|K|$. We also investigated how the ordering of locations $\\bm s_1,\\ldots,\\bm s_n$ chosen in Algorithm~\\ref{algo} to iteratively sample from $P_{\\bm s_0}(z,\\cdot)$ might affect the algorithm's efficiency. Our results (not shown) suggest that there was very little difference among the four orderings considered (coordinate-wise, random, middle-out, or maximum-minimum orderings), in terms of the mean and variance of the number of processes $W_i$ to sample in each simulation. In our simulations, we used a coordinate-wise ordering for simplicity.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{complexity.pdf}\n\t\\caption{Left: Time in second needed to simulate one max-id replicate for different numbers of locations $|K|$. The red curve is the fit from a $4$-th order polynomial. Right: Number of random fields $W_i$ that were simulated for one max-id replicate for different numbers of locations $|K|$. The red line is the diagonal. The parameters are here set to be $\\alpha=1,\\beta=1$ and the correlation function for $W_i$ is chosen to be $\\rho(h) = \\exp(-h)$.}\n\t\\label{fig:complexity}\n\\end{figure}\n\\section{Discussion}\n\\label{sec:Discussion}\n\nIn this paper, we have generalized the exact simulation algorithm for max-stable processes based on extremal functions to the more general class of max-id processes. Whenever possible, we recommend using the adaptive rejection sampling scheme for the conditional simulation of magnitude variables, in order to achieve fast and exact simulations in cases where variables to be simulated do not correspond to some standard distribution from which samples are easily drawn. Although the specific implementation of the algorithm still depends on the chosen representation for $\\eta_i$, our approach is general and can be applied to very broad classes of max-id processes while preserving computational efficiency. Moreover, the specific examples showcased in this paper already have very flexible dependence structures. In a simulation study, we used three different sampling algorithms to compare their speed and accuracy. Our novel exact simulation algorithm using the adaptive rejection sampling scheme clearly outperforms the two alternatives. In cases where the target density in the adaptive rejection sampling scheme is not logarithmically concave, it is possible to use an appropriately defined quasi-exact Metropolis--Hastings scheme to achieve high accuracy.\n\nBesides standard uses of such fast and exact simulation techniques in Monte--Carlo-based statistical inferences involving quantities that cannot be calculated analytically, another interesting perspective concerns Approximate Bayesian Computation (ABC) for general max-id processes, which could be implemented by analogy with the approach developed for max-stable processes in \\citet{Erhardt2012}. As highlighted above, the computation of multivariate expression of densities and distributions functions is often numerically very expensive due to integrals that have to be evaluated numerically, such that likelihood-free inference techniques are promising alternatives. Fast simulation of large numbers of samples of the model is required in likelihood-free estimation methods such as ABC, and in other simulation-based inference methods such as maximum simulated likelihood \\citep[e.g.,][]{Hajivassiliou2000}.\n\n\\baselineskip=15pt\n\n\\bibliographystyle{CUP}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMuch of the progress in our understanding of the capacity limits of wireless networks over the past decade has come from the pursuit of progressively refined capacity approximations. Generalized degrees of freedom  (GDoF) characterizations represent a most significant step along this path because of their ability to capture arbitrary channel strength and channel uncertainty levels.  The GDoF framework may seem counter-intuitive at first because it allows exponential scaling of signal strengths with various exponents. An intuitive justification for the GDoF framework is as follows. It is important to remember that the goal behind GDoF is to seek capacity approximations for a given wireless network with its arbitrary \\emph{finite} signal strengths and channel uncertainty levels. Unlike the degrees of freedom (DoF) metric which linearly scales all signal strengths and loses the distinction of different channel strengths (every non-zero channel carries $1$ DoF), the GDoF formulation takes a more sophisticated approach.  The key to GDoF is the intuition that if the capacity of every link in a network is scaled by the same factor, then the capacity region of the network should scale by approximately the same factor as well. Normalizing the capacity of the network by the scaling factor then yields a capacity approximation for the original network. Following this intuition, one allows the scaling factor to approach infinity, while guaranteeing that the capacity is always normalized by the scaling factor. The asymptotic behavior of normalized capacity is potentially easier to characterize than a direct approximation of the capacity of the original network. Let $\\alpha_i$ represent the capacity of the $i^{th}$ link in the original network (in isolation from all other links), and let $\\log(P)$ be the scaling factor applied to every link capacity. Then we obtain channels whose capacity scales as $\\alpha_i\\log(P)$, i.e., channels whose strength scales as $P^{\\alpha_i}$, and, according to this intuitive reasoning, normalization of network capacity by $\\log(P)$ in the limit $P\\rightarrow\\infty$ presents the approximation of the capacity of the original network. This approximation is what is known as the GDoF characterization, and along with its abstractions into deterministic channel models, over the past decade it has been the key to finding capacity approximations for many networks whose exact capacity remains intractable. Thus, the linear scaling of capacity naturally corresponds to an exponential scaling of signal strengths in the GDoF model.\n\nGDoF studies started with settings where perfect CSIT is available \\cite{Etkin_Tse_Wang, Jafar_Vishwanath_GDOF, Karmakar_Varanasi}. The opposite extreme of no CSIT was also explored under strong assumptions of statistical equivalence between users \\cite{Huang_Jafar_Shamai_Vishwanath, Varanasi_noCSIT, Guo_noCSIT}. Lately, however, the focus has shifted to the broader assumption of finite precision CSIT  \\cite{Arash_Jafar_TC}, \\cite{Arash_Bofeng_Jafar_BC}. Some of the more sophisticated concepts such as interference alignment \\cite{Jafar_FnT} have turned out to be too fragile to be useful with finite precision CSIT, so that conventional achievable schemes are usually optimal. As such the  main challenge for GDoF studies under finite precision CSIT tends to be the proof of optimality, i.e., the converse, or the GDoF outer bound.  Finding tight GDoF outer bounds under finite precision CSIT is generally a hard problem, as exemplified by the conjecture of Lapidoth et al. \\cite{Lapidoth_Shamai_Wigger_BC} which remained unresolved for nearly a decade. The main idea for these outer bounds is the Aligned Image Sets (AIS) argument that was introduced in  \\cite{Arash_Jafar_PN} in order to settle the conjecture of Lapidoth et al. Generalizations of the AIS approach have also helped settle the GDoF in other settings such as the X channel and the $2$ user MISO BC under finite precision CSIT in \\cite{Arash_Jafar_TC}, and the 2 user MISO BC with arbitrary channel strengths and channel uncertainty levels in \\cite{Arash_Bofeng_Jafar_BC}. Of particular relevance to this work is  \\cite{Arash_Jafar_IC} where the sum GDoF of $K$-user symmetric interference channel (IC) is characterized under finite precision CSIT (see Figure \\ref{fig:Kusert}). This work is motivated by the goal of further broadening the scope of the AIS argument, so that the results of \\cite{Arash_Jafar_IC} may be generalized to MIMO settings.\n\n\\begin{figure*}[h]\n\\center\n\\begin{minipage}[c]{0.54\\textwidth}\n\t\\centerline{\\includegraphics[width=3.7in]{p2+.pdf}}\n\\end{minipage}~~~~~\n\\begin{minipage}[c]{0.33\\textwidth}\n\\begin{eqnarray*}\nd(\\alpha) = \\left\\{\n\\begin{array}{ll} \n1-\\alpha,&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\frac{K-2-(K-3)\\alpha}{K-1}, &\\frac{1}{2}<\\alpha\\leq\\frac{K}{K+1}\\vspace{0.03in}\\\\ \n1-\\left(\\frac{K-1}{K}\\right)\\alpha,&\\frac{K}{K+1}<\\alpha\\leq 1\\\\\n\\frac{\\alpha}{K},& 1<\\alpha\\leq K\\\\\n1, &K<\\alpha\n\\end{array}\n\\right.\n\\end{eqnarray*}\n\\end{minipage}\n\\caption{GDoF\/user of the Symmetric $K$ User Interference Channel with Finite Precision CSIT \\cite{Arash_Jafar_IC}.}\\label{fig:Kusert}\n\\end{figure*}\n\n\nIn this paper, we characterize the GDoF for the symmetric  $K$-user MIMO Interference Channel  under the assumption that the CSIT is limited to finite precision. In this symmetric setting, each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, each desired channel (i.e., a channel between a transmit antenna and a receive antenna belonging to the same user) has strength $\\sim P$, while each undesired channel has strength $\\sim P^\\alpha$, where $P$ is a nominal SNR parameter.  GDoF per user take the form of a $W$-curve with respect to $\\alpha$ for fixed values of $M$ and $N$. See Figure \\ref{Fig1}. As usual for finite precision CSIT, achievability is fairly straightforward. While ostensibly the main result of this work is the GDoF characterization for the $K$-user symmetric MIMO IC, the deeper significance of this paper resides in a key generalization of the AIS approach that allows comparisons in the GDoF sense of the entropies of  different numbers of linear combinations (finite precision versus perfectly known channels) of random variables under various power-level partitions. The generalization seems broadly useful for GDoF problems related to MIMO wireless networks. \n\n\n\n\n{\\it Notation:} The notation $|A|$ denotes the cardinality of the set $A$ and the notation $[n]$ is defined as $\\{1,2,\\cdots,n\\}$ for any $n\\in\\mathbb{N}$ where $\\mathbb{N}$ is the set of all positive integer numbers. The notations~ $X^{[T]}$ and $X_{i}^{[T]}$ also stand~ for $\\{X(1), X(2), \\cdots X(T)\\}$ and $\\{X_i(t): \\forall t\\in[T]\\}$, respectively. Moreover, we use the Landau $o(\\cdot)$ notation for the functions $f(x), g(x)$ from $\\mathbb{R}$ to $\\mathbb{R}$ as follows. $f(x)=o(g(x))$ denotes that $\\limsup_{x\\rightarrow\\infty}\\frac{|f(x)|}{|g(x)|}=0$. Finally, we define $\\lfloor x\\rfloor$  as the largest integer that is smaller than or equal to $x$  for any positive real number $x$ and  the smallest integer that is larger than or equal to $x$  for any negative real number $x$. $A^\\dagger$ is the transpose of matrix $A$. The support of a random variable $X$ is denoted as supp$(X)$.\n\n\n\n\n\n\n\\section{Definitions}\n\\begin{definition}\\label{bd}[Bounded Density Channel Coefficients \\cite{Arash_Jafar_PN}] Define a set of real valued random variables, $\\mathcal{G}$ such that the magnitude of each random variable $g\\in\\mathcal{G}$ is bounded away from  infinity, $ |g|\\leq\\Delta<\\infty$, for some positive constant $\\Delta$, and there exists a finite positive constant $f_{\\max}$, such that for all finite cardinality disjoint subsets $\\mathcal{G}_1, \\mathcal{G}_2$ of $\\mathcal{G}$, the joint probability density function of all random variables in $\\mathcal{G}_1$, conditioned on all random variables in $\\mathcal{G}_2$, exists and is bounded above by $f_{\\max}^{|\\mathcal{G}_1|}$. Without loss of generality we will assume that $f_{\\max}\\geq 1, \\Delta\\geq 1$.\n\\end{definition}\n\n\n\n\n\\begin{definition}[Power Levels] Consider integer valued random variables $X_i$ over alphabet $\\mathcal{X}_{\\lambda_i}$,\n\\begin{eqnarray}\n\\mathcal{X}_{\\lambda_i}&\\triangleq&\\{0,1,2,\\cdots,\\bar{P}^{\\lambda_i}-1\\}\n\\end{eqnarray}\nwhere $\\bar{P}^{\\lambda_i}$ is a compact notation for $\\left\\lfloor\\sqrt{P^{\\lambda_i}}\\right\\rfloor$ and the constant $\\lambda_i$ is a positive real number denoting the \\emph{power level} of $X_i$. \n\\end{definition}\n\n\n\n\n\n\\begin {definition}\\label{powerlevel} For $X\\in\\mathcal{X}_\\lambda$, and $0\\leq \\lambda_1\\leq\\lambda$, define the random variables $(X)_{\\lambda_1}^{\\lambda}$ as,\n \\begin{eqnarray}\n(X)_{\\lambda_1}^{\\lambda}&\\triangleq&\\left \\lfloor \\frac{X}{\\bar{P}^{\\lambda_1}} \\right \\rfloor\n\\end{eqnarray}\n\\end {definition}\nIn words, $(X)_{\\lambda_1}^{\\lambda}$ retrieves the top $\\lambda-\\lambda_1$ power levels of $X$. Similarly, for  the vector ${\\bf V}=\\begin{bmatrix}v_1&v_2&\\cdots&v_k\\end{bmatrix}^\\dagger$, we define $({\\bf V})^{\\lambda}_{\\lambda_1}$ as,\n \\begin{eqnarray}\n({\\bf V})^{\\lambda}_{\\lambda_1}&\\triangleq&\\begin{bmatrix}(v_1)^{\\lambda}_{\\lambda_1}&(v_2)^{\\lambda}_{\\lambda_1}&\\cdots&(v_k)^{\\lambda}_{\\lambda_1}\\end{bmatrix}^\\dagger\n\\end{eqnarray}\n\n\\begin {definition}\\label{deflc} For {real}  numbers $x_1,x_2,\\cdots,x_k{\\in\\mathcal{X}_{\\eta}}$ define the notation $L_j^b(x_i,1\\le i\\le k)$  to represent,\n\\begin {eqnarray}\nL^b_j(x_1,x_2,\\cdots,x_k )&=&\\sum_{1\\le i\\le k} \\lfloor g_{j_i}x_i\\rfloor\n\\end{eqnarray}\nfor  distinct random variables $g_{j_i}\\in\\mathcal{G}$. The subscript $j$ is used to distinguish among multiple linear combinations, and may be dropped if there is no potential for ambiguity.  For  the vector $V=\\begin{bmatrix}v_1&v_2&\\cdots&v_k\\end{bmatrix}^\\dagger$ define the notation $L^b_j(V)$  to represent,\n\\begin {eqnarray}\nL^b_j(V)=\\sum_{1\\le i\\le k} \\lfloor g_{j_i}v_i\\rfloor\n\\end{eqnarray}\nfor  distinct random variables $g_{j_i}\\in\\mathcal{G}$.\n\\end {definition}\n\n\\begin{definition}\\label{defvec} For the two vectors $V=\\begin{bmatrix}v_1&\\cdots&v_{k_1}\\end{bmatrix}^\\dagger$ and $W=\\begin{bmatrix}w_1&\\cdots&w_{k_2}\\end{bmatrix}^\\dagger$ define the vector $V\\bigtriangledown W$ as $\\begin{bmatrix}v_1&\\cdots&v_{k_1}&w_1&\\cdots&w_{k_2}\\end{bmatrix}^\\dagger$.\n\\end{definition}\n\n\n\n\n\\section{System Model} {\\label{sec-sys}}\nIn this work we consider only the setting where all variables take real values. Extensions to complex settings are cumbersome but conceptually straightforward  as in \\cite{Arash_Jafar_PN}.\n\\subsection{The Channel}\n\\noindent Define  random variables $\\mathbf{X}_{k}(t)$ and $\\mathbf{Y}_{k}(t)$, $\\forall k\\in[K]$ as,\n\\begin{align}\n\\mathbf{X}_{k}(t)=&\\begin{bmatrix}{X}_{k1}(t)&{X}_{k2}(t)&\\cdots&{X}_{kM}(t)\\end{bmatrix}^\\dagger\\\\\n\\mathbf{Y}_{k}(t)=&\\begin{bmatrix}{Y}_{k1}(t)&{Y}_{k2}(t)&\\cdots&{Y}_{kN}(t)\\end{bmatrix}^\\dagger\n\\end{align}\nwhere the channel uses are indexed by $t\\in[T]$. $X_{km}(t), k\\in[K],m\\in[M],t\\in[T]$ are the symbols sent from $m$-th transmit antenna of the $k$-th transmitter and are subject to unit power constraint, while $Y_{kn}(t), k\\in[K],n\\in[N],t\\in[T]$ are the symbols observed by the $n$-th antenna of the $k$-th receiver. Under the GDoF framework, the channel model for the $K$-user MIMO IC is defined by the following input-output equations\n\\begin{align}\n{\\bf Y}_{k}(t)=&\\sqrt{P}{\\bf G}_{kk}(t)\\mathbf{X}_k(t)+\\sqrt{{P}^{\\alpha}}\\sum_{\\hat{k}=1,\\hat{k}\\neq k}^K{\\bf G}_{k\\hat{k}}(t)\\mathbf{X}_{\\hat{k}}(t)+{\\bf \\Gamma}_{k}(t)\n\\end{align}\nfor all $k\\in[K]$ and $t\\in[T]$. The $N\\times M$ matrix ${\\bf G}_{k\\hat{k}}(t)$ is the channel fading coefficient matrix between the $k$-th receiver and the $\\hat{k}$-th transmitter for any $k,\\hat{k}\\in[K]$. The entry in the $n$-th row and $m$-th column of the matrix ${\\bf G}_{k\\hat{k}}(t)$ is ${G}_{k\\hat{k}nm}(t)$. $\\mathbf{\\Gamma}_{k}(t)$ are  $N\\times 1$  matrices whose components are zero mean unit variance additive white Gaussian noise (AWGN) experienced by $k$-th receiver.  Figure \\ref{Fig1rr} illustrates a $3$-user $3\\times 2$ MIMO IC.  $P$ is a nominal SNR parameter that approaches infinity for GDoF characterizations. CSIR is assumed to be perfect. However, CSIT is limited to finite precision. Under finite precision CSIT  we assume that $G_{k\\hat{k}nm}(t)\\in\\mathcal{G}$ for any $k,\\hat{k}\\in[K],n\\in[N],m\\in[M]$ and $t\\in[T]$, and since transmitters only know the probability density but not the realizations of channel coefficients, we assume that all ${\\bf X}_k(t), t\\in[T], k\\in[K]$ are independent of $\\mathcal{G}$.\n\\begin{figure}[tp]\n\\centering \n\\includegraphics[scale =0.28]{p8.pdf}\n\\caption{Three user $3 \\times 2$ MIMO IC.}\n\\label{Fig1rr}\n\\end{figure}\n\\subsection{GDoF}\nThe definitions of achievable rates $R_i(P)$ and capacity region $\\mathcal{C}(P)$ are standard. The GDoF region is defined as\n\\begin{eqnarray}\n\\mathcal{D}&=&\\{(d_1,d_2,\\cdots,d_K): \\exists (R_1(P),R_2(P), \\cdots,R_K(P))\\nonumber\\\\\n&&\\in\\mathcal{C}(P), \\mbox{ s.t. } d_k=\\lim_{P\\rightarrow\\infty}\\frac{R_k(P)}{\\frac{1}{2}\\log(P)}, \\forall k\\in[K]\\}\n\\end{eqnarray}\nThe maximum value of $d_1+d_2+\\cdots+d_K$ over $\\mathcal{D}$ is known as the sum GDoF value.\n\n\n\n\\section{Main Result}\n\n\\begin{theorem}\\label{theorem:GDoF1} The sum GDoF value for the $K$-user  symmetric MIMO IC for $M\\le \\frac{N}{K}$ is $KM$, and for $\\frac{N}{K}\\le M$ is\n\\begin{align}\n{\\sum_{k=1}^Kd_k}=&\\left\\{\n\\begin{array}{ll} \nK\\min(M,N)(1-\\alpha)+\\frac{K(N-M)^+\\alpha}{K-1},&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\min\\left(\\frac{K}{K-1}\\left((K-2)\\min(M,N)(1-\\alpha)+N(\\alpha)\\right)\\right., &\\vspace{0.03in}\\\\ \nN\\alpha+K\\min(M,N)\\left(1-\\alpha\\right)\\big),&\\frac{1}{2}<\\alpha\\leq 1\\\\\n\\min\\left(D(\\alpha),K\\min(M,N)\\right), &1<\\alpha\n\\end{array}\n\\right.\\label {fg<}\n\\end{align}\nwhere $N(\\alpha)$ and $D(\\alpha)$ are defined as,\n\\begin{eqnarray}\nN(\\alpha)&=&\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\\\\nD(\\alpha)&=&(N-(K-1)M)^++\\min(N,(K-1)M)\\alpha \n\\end{eqnarray}\n\\end{theorem}\n\\begin{figure}[tp]\n\\centering \n\\includegraphics[scale =0.4]{p2c.pdf}\n\\caption{Sum GDoF of the three user $3 \\times N$ MIMO IC.}\n\\label{Fig1}\n\\end{figure}\n\\begin{remark} The sum GDoF, i.e., \\eqref{fg<} for $N< M$ yields,\n\\begin{align}\n{\\sum_{k=1}^Kd_k}=&{KN}\\times  \\left\\{\n\\begin{array}{ll} \n(1-\\alpha),&0\\leq\\alpha\\leq \\frac{1}{2}\\vspace{0.03in}\\\\ \n\\frac{K-2-(K-3)\\alpha}{K-1}, &\\frac{1}{2}<\\alpha\\leq\\frac{K}{K+1}\\vspace{0.03in}\\\\ \n1-(\\frac{K-1}{K})\\alpha,&\\frac{K}{K+1}<\\alpha\\leq 1\\\\\n\\frac{\\alpha}{K},& 1<\\alpha\\leq K\\\\\n1, &K<\\alpha\n\\end{array}\n\\right.\\label {GDOFM>N}\n\\end{align}\n\\end{remark}\n\n\n\\section{Proof of Theorem \\ref{theorem:GDoF1}: Converse}\n The first step in the converse proof, identical to \\cite{Arash_Jafar_IC},\n is the transformation\ninto a deterministic setting such that a GDoF outer bound\non the deterministic setting is also a GDoF outer bound on\nthe original setting.  We start directly from the deterministic\nmodel.\n\n\\subsection{Deterministic Model}\\label{DM_1}\n\\vspace{-1em}\n\\begin{align}\n\\bar{\\mathbf{Y}}_{k}(t)&=[\\bar{Y}_{k1}(t)\\ \\bar{Y}_{k2}(t)\\ \\cdots\\ \\bar{Y}_{kN}(t)]^\\dagger\\\\\n\\bar{Y}_{kn}(t)&=L_{kn1}^b(t)\\left({(\\bar{\\mathbf{X}}_{k}(t))}^{\\max(1,\\alpha)}_{\\max(1,\\alpha)-1}\\right)+L_{kn2}^b(t)\\left({(\\bar{\\mathbf{X}}_{j}(t))}^{\\max(1,\\alpha)}_{\\max(1,\\alpha)-\\alpha},\\forall j\\in[K],j\\neq k\\right)\\label{dm1}\n\\end{align}\nfor all $k\\in[K],n\\in[N],t\\in[T]$. $\\bar{\\mathbf{X}}_{k}(t)$ are  defined as,\n\\begin{align}\n\\bar{\\mathbf{X}}_{k}(t)=&[\\bar{X}_{k1}(t)\\ \\bar{X}_{k2}(t)\\ \\cdots\\ \\bar{X}_{kM}(t)]^\\dagger\\label{ggf1}\n\\end{align}\nfor any $k\\in[K]$, $t\\in[T]$ where $\\bar{X}_{km}(t)\\in\\{0, 1, \\cdots, {\\bar{P}}^{\\max(1,\\alpha)}-1\\}$, $\\forall k\\in[K],m\\in[M],t\\in[T]$. \n\n\\subsection{ Key Lemma} \nThe following  lemma is the critical generalization of the AIS bound needed for Theorem \\ref{theorem:GDoF1}.\n\\begin{lemma}\\label{lemma} Define the two random variables $\\bar{\\bf U}_1$ and  $\\bar{\\bf U}_2$ as,\n\\begin{eqnarray}\n\\bar{\\bf U}_1&=&\\left({U}_{11}^{[T]},{U}_{12}^{[T]},\\cdots,{U}_{1N_1}^{[T]}\\right)\\label{lemmamimox1}\\\\\n\\bar{\\bf U}_2&=&\\left({U}_{21}^{[T]},{U}_{22}^{[T]},\\cdots,{U}_{2N_2}^{[T]}\\right)\\label{lemmamimox2}\n\\end{eqnarray}\nwhere for any $t\\in[T]$, $U_{1n}(t)$ and $U_{2n}(t)$ are defined as,\n\\begin{eqnarray}\nU_{1n}(t)&=&L_{1n}^b(t)\\left((\\bar{\\mathbf{V}}_1(t))^{\\eta}_{\\eta-\\lambda_{11}}\\bigtriangledown(\\bar{\\mathbf{V}}_2(t))^{\\eta}_{\\eta-\\lambda_{12}}\\bigtriangledown\\cdots\\bigtriangledown(\\bar{\\mathbf{V}}_l(t))^{\\eta}_{\\eta-\\lambda_{1l}}\\right), \\forall n\\in[N_1]\\label{lemmamimox3}\\\\\nU_{2n}(t)&=&L_{2n}^b(t)\\left((\\bar{\\mathbf{V}}_1(t))^{\\eta}_{\\eta-\\lambda_{21}}\\bigtriangledown(\\bar{\\mathbf{V}}_2(t))^{\\eta}_{\\eta-\\lambda_{22}}\\bigtriangledown\\cdots\\bigtriangledown(\\bar{\\mathbf{V}}_l(t))^{\\eta}_{\\eta-\\lambda_{2l}}\\right), \\forall n\\in[N_2]\\label{lemmamimox4}\n\\end{eqnarray}\nwhere $\\bar{\\mathbf{V}}_i(t)=\\begin{bmatrix}\\bar{V}_{i1}(t)&\\cdots&\\bar{V}_{iM_i}(t)\\end{bmatrix}^\\dagger$, $\\bar{V}_{im}(t)\\in\\mathcal{X}_{\\eta}$  are all independent of $\\mathcal{G}$, and $0\\le\\lambda_{1i},\\lambda_{2i}\\le\\eta$ for any $i\\in[l]$.  Without loss of generality,  $(\\lambda_{1i}-\\lambda_{2i})^+$ are sorted in descending order, i.e., $(\\lambda_{1i}-\\lambda_{2i})^+\\ge(\\lambda_{1j}-\\lambda_{2j})^+$ if $1\\le i< j\\le l$.  Then, for any acceptable\\footnote{Let $\\mathcal{G}(Z)\\subset\\mathcal{G}$ denote the set of all bounded density channel coefficients that appear in $\\bar{\\bf U}_1,\\bar{\\bf U}_2$.  $W$ is acceptable if  conditioned on any $\\mathcal{G}_o\\subset (\\mathcal{G}\/\\mathcal{G}(Z))\\cup \\{W\\}$, the channel coefficients $\\mathcal{G}(Z)$ satisfy the bounded density assumption. For instance, any random variable $W$ independent of $\\mathcal{G}$ can be utilized in  Lemma \\ref{lemma}.} random variable ${W}$, if $N_1\\le \\min(N_2, \\sum_{i=1}^lM_i)$ we have,\n\\begin{eqnarray}\n&&H({\\bar{\\bf U}}_1\\mid {W},\\mathcal{G})-H({\\bar{\\bf U}}_2\\mid {W},\\mathcal{G})\\nonumber\\\\\n&\\le&T\\big((N_1-\\sum_{i=1}^sM_i)(\\lambda_{1,s+1}-\\lambda_{2,s+1})^++\\sum_{i=1}^sM_i(\\lambda_{1i}-\\lambda_{2i})^+\\big)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{lemmamimox5}\n\\end{eqnarray}\nwhere $s$ must satisfy the condition  $\\sum_{i=1}^{s}M_i\\le N_1< \\sum_{i=1}^{s+1}M_i$.  \n\\end{lemma} \nProof of Lemma \\ref{lemma} is based on the AIS argument and is relegated to Appendix \\ref{lemmap}.\n\n\n\n\\subsection{Some Insights For the Three User $2\\times3$ MIMO IC}\n\\begin{figure}[!h]\n\\centerline{\\includegraphics[width=6in]{p8d.pdf}}\n\\caption{ Three user $2\\times3$ MIMO IC. The network is fully connected but only the channel strength parameters needed for the application of Lemma \\ref{lemma} are shown in this figure.}\\label{fig:intuit-}\n\\end{figure}\nTo gain some insights into the application of Lemma \\ref{lemma}, consider the three user $2\\times3$ MIMO IC illustrated in Figure \\ref{fig:intuit-} for $\\alpha\\le1$. To apply Lemma \\ref{lemma}, the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$, $\\bar{\\bf V}_2^{[T]}$ and $W$ are interpreted as $\\bar{\\bf Y}_2^{[T]}$, $\\bar{\\bf Y}_1^{[T]}$, $\\bar{\\bf X}_2^{[T]}$, $\\bar{\\bf X}_3^{[T]}$ and $\\bar{\\bf X}_1^{[T]}$, respectively. The first user receives the top $\\alpha$ power levels of $\\bar{\\bf X}_2^{[T]}$ and $\\bar{\\bf X}_3^{[T]}$ while second reciever sees  the top $1$ power levels of $\\bar{\\bf X}_2^{[T]}$ and  the top $\\alpha$ power levels of $\\bar{\\bf X}_3^{[T]}$.  So we have $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=\\alpha$. Therefore, $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=0$. From Lemma \\ref{lemma} we conclude,\n\\begin{align}\n&H(\\bar{\\bf Y}_2^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\mathcal{G})-H(\\bar{\\bf Y}_1^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\mathcal{G})\\nonumber\\\\\n\\le&T\\big(1\\times 0+2\\times(1-\\alpha)\\big)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{jl0}\n\\end{align}\n  Let us also explain how intuitively we expect \\eqref{jl0} to be true as well. Conditioned on $\\bar{\\bf X}_{1}^{[T]}$, $\\bar{\\bf Y}_{2}(t)$ is a linear combination of $\\bar{\\bf X}_{2}(t)$ and $(\\bar{\\bf X}_{3}(t))^{\\alpha}$ while $\\bar{\\bf Y}_{1}(t)$ is a linear combination of $(\\bar{\\bf X}_{2}(t))^{\\alpha}$ and $(\\bar{\\bf X}_{3}(t))^{\\alpha}$. Consider the channel illustrated in Figure \\ref{fig:intuit-}. First of all, observe that $\\bar{\\bf X}_{2}(t)$ appears in $\\bar{\\bf Y}_{2}(t)$ with the signal strength levels $1$ and appears in $\\bar{\\bf Y}_{1}(t)$ with the signal strength levels $\\alpha$. Thus, due to the bounded density assumption the maximum difference of $2(1-\\alpha)$ is possible in the GDoF sense between the two entropies. Note that, $\\bar{\\bf X}_{3}(t)$ appears in both the received signals  $\\bar{\\bf Y}_{1}(t)$ and  $\\bar{\\bf Y}_{2}(t)$ with the same signal strength levels of $\\alpha$. Therefore, it cannot contribute positive difference of entropies as in the finite precision CSIT no interference alignment is possible.\n  \nSimilarly, from Lemma \\ref{lemma} we have,  \n\\begin{align}\nH(\\bar{\\bf Y}_3^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\bar{\\bf X}_2^{[T]},\\mathcal{G})-H(\\bar{\\bf Y}_2^{[T]}\\mid \\bar{\\bf X}_1^{[T]},\\bar{\\bf X}_2^{[T]},\\mathcal{G})\\le&2T(1-\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{jl00}\n\\end{align} \nOn the other hand, writing Fano's inequality for all the three users (and suppressing $o(T)$ terms for simplicity) we obtain the following bounds,\n\\begin{eqnarray}\nTR_1&\\le& H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\mathcal{G})- H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\mathcal{G})\\label{jl1}\\\\\nTR_2&\\le& H(\\bar{\\bf Y}_{2}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\mathcal{G})- H(\\bar{\\bf Y}_{2}^{[T]}\\mid \\bar{\\bf X}_{1}^{[T]},\\bar{\\bf X}_{2}^{[T]},\\mathcal{G})\\label{jl2}\\\\\nTR_3&\\le& H(\\bar{\\bf Y}_{3}^{[T]}\\mid  \\bar{\\bf X}_{1}^{[T]},\\bar{\\bf X}_{2}^{[T]},\\mathcal{G})\\label{jl3}\n\\end{eqnarray}\nTherefore, for $\\alpha\\le1$, from \\eqref{jl0}-\\eqref{jl3} we have,\n\\begin{eqnarray}\nTR_1+TR_2+TR_3&\\le& H(\\bar{\\bf Y}_{1}^{[T]}\\mid \\mathcal{G})+4T(1-\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\\\\n&\\le&T(6-3\\alpha)\\log{\\bar{P}}+T~o~(\\log{\\bar{P}})\\label{oi}\n\\end{eqnarray}\n\\eqref{oi} is true as discrete entropy of any discrete random variable is bounded by logarithm of its cardinality.\n\n\\subsection{Equivalent Bounds}\nTheorem \\ref{theorem:GDoF1} is concluded from the following bounds,\n\\begin{enumerate}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\alpha\\le\\frac{1}{2}$, then \n\\begin{eqnarray}\n&&\\sum_{k=1}^Kd_k\\nonumber\\\\\n&\\le& \\frac{K\\left(\\min(M,N)(1-\\alpha)+(N-M)^+\\alpha\\right)+K(K-2)\\min(M,N)(1-\\alpha)}{K-1}\\label{b1}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\frac{1}{2}\\le\\alpha\\le1$, then \n\\begin{eqnarray}\n&&\\sum_{k=1}^Kd_k\\nonumber\\\\\n&\\le& \\frac{K\\left(\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\right)+K(K-2)\\min(M,N)(1-\\alpha)}{K-1}\\nonumber\\\\\n&&\\label{b1+}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,\\alpha\\le1$, then \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& N\\alpha+K\\min(M,N)(1-\\alpha)\\label{b2}\n\\end{eqnarray}\n\\item If $\\alpha\\in\\mathbb{R}^+,1\\le\\alpha$, then \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& (N-(K-1)M)^++\\min(N,(K-1)M)\\alpha\\label{b2+}\n\\end{eqnarray}\n\\item For any $\\alpha\\in \\mathbb{R}^+$, \n\\begin{eqnarray}\n\\sum_{k=1}^Kd_k&\\le& K\\min(M,N)\\label{b3}\n\\end{eqnarray}\n\\end{enumerate}\nThus, in order to prove Theorem \\ref{theorem:GDoF1}, the Bounds \\eqref{b1}-\\eqref{b3} should be proved.\n\\subsection{Proof of Bounds \\eqref{b1}-\\eqref{b3}}\\label{mt}\nThe last bound,  $\\sum_{k=1}^Kd_k\\le K\\min(M,N)$ is the trivial combination of single user bounds.  Let us prove the other four  bounds, i.e., \\eqref{b1}-\\eqref{b2+}. \n\\begin{enumerate}\n\\item {\\bf Proof of \\eqref{b1} and \\eqref{b1+}}\\\\\n\n Writing Fano's Inequality for the first $K-1$ receivers we have,\n\\begin{eqnarray}\nTR_1&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})\\label{c1}\\\\\nTR_k&\\le& I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G}), \\forall k\\in[K-1],k\\neq 1\\label{c2}\n\\end{eqnarray}\n{Summing}  \\eqref{c1} and \\eqref{c2}, we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K-1}R_k\\nonumber\\\\\n&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K-1} I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\\\\n&=& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{K}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&&+\\sum_{k=2}^{K-1} \\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})-H(\\bar{\\bf Y}^{[T]}_{k-1}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\right)\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{K}\\mid \\mathcal{G}\\right)+\\sum_{k=2}^{K-1} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c3}\n\\end{eqnarray}\nwhere the new random variable,  $\\bar{\\bf X'}_k(t)$ is defined as\\\\\n\\begin{eqnarray}\n\\bar{\\bf X'}_k(t)&=&\\begin{bmatrix}\\bar{X'}_{k1}(t)&\\bar{X'}_{k2}(t)&\\cdots&\\bar{X'}_{kN}(t)\\end{bmatrix}^\\dagger\\\\\n\\bar{X'}_{kn}(t)&=&{L}_{kn3}^b(t)\\left((\\bar{\\bf X}_{k}(t))^{\\alpha}\\right), \\forall n\\in[N]\n\\end{eqnarray}\nLet us explain how Lemma \\ref{lemma} {yields}  \\eqref{c3}. Substitute the random variables $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$, $\\bar{\\bf V}_2^{[T]}$ and $W$  in Lemma \\ref{lemma}  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf Y}^{[T]}_{k-1}$, $\\bar{\\bf X}_k^{[T]}$,  $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\notin[k]\\right)$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[k-1]\\right)$, respectively. Next,  we set $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=\\alpha,M_1=M,M_2=(K-k)M,N_1=N_2=N$. Thus, we have $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=0$. Therefore, from Lemma \\ref{lemma},  \\eqref{c3} is concluded. Similar to  \\eqref{c3}, by symmetry we have, \n\\begin{eqnarray}\n&&T\\sum_{k\\in[K],k\\neq j}R_k\\nonumber\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{j+1}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{j}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&&+(K-2) T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c4}\n\\end{eqnarray}\nfor all $j\\in[K]$. Summing \\eqref{c4} for all $j\\in[K]$ we have,\n\\begin{eqnarray}\n&&T(K-1)\\sum_{k=1}^KR_k\\nonumber\\\\\n&=&T\\sum_{j=1}^K\\sum_{k\\in[K],k\\neq j}R_k\\nonumber\\\\\n&\\le& \\sum_{k=1}^K\\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\right)+TK(K-2) \\min(M,N)(1-\\alpha)\\log{\\bar{P}}\\nonumber\\\\\n&&+T~o(\\log{\\bar{P}})\\label{xx}\n\\end{eqnarray}\nNow, let us consider the two cases of $\\alpha\\le\\frac{1}{2}$ and $\\frac{1}{2}\\le\\alpha\\le1$ separately.\n\\begin{enumerate}\n\\item{$\\alpha\\le\\frac{1}{2}$}\n\\begin{eqnarray}\n&&H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&\\le&  T\\left(\\min(M,N)(1-\\alpha)+(N-M)^+\\alpha\\right)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c5}\n\\end{eqnarray}\nLet us explain how \\eqref{c5} follows from Lemma \\ref{lemma}. Substitute the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf X'}^{[T]}_{k}$, $\\bar{\\bf X}_k^{[T]}$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq k\\right)$, respectively. Moreover,  setting $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=\\alpha, \\lambda_{22}=0$, we have $(\\lambda_{11}-\\lambda_{21})^+=1-\\alpha$ and $(\\lambda_{12}-\\lambda_{22})^+=\\alpha$. Therefore, from Lemma \\ref{lemma} we conclude \\eqref{c5}. From \\eqref{xx} and \\eqref{c5}, \\eqref{b1} is concluded.\n\\item{$\\frac{1}{2}\\le\\alpha\\le1$}\n\\begin{eqnarray}\n&&H(\\bar{\\bf Y}^{[T]}_{k}\\mid \\mathcal{G})-H\\left(\\bar{\\bf X'}^{[T]}_{k}\\mid \\mathcal{G}\\right)\\nonumber\\\\\n&\\le&  T\\left(\\min((K-1)M,N)\\alpha+(N-(K-1)M)^+(1-\\alpha)\\right)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c5g}\n\\end{eqnarray}\n \\eqref{c5g} follows from Lemma \\ref{lemma} similar to \\eqref{c5}. Substitute the random variables  $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with $\\bar{\\bf Y}^{[T]}_{k}$, $\\bar{\\bf X'}^{[T]}_{k}$, $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq k\\right)$ and $\\bar{\\bf X}_k^{[T]}$, respectively. The rest of the proof is concluded similar to \\eqref{c5}. From \\eqref{xx} and \\eqref{c5g}, \\eqref{b1+} is concluded.\n\\end{enumerate}\n\\item {\\bf Proof of \\eqref{b2}}\\\\\n\n Summing \\eqref{c1} and \\eqref{c2}, we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K}R_k\\nonumber\\\\\n&\\le& I(\\bar{\\bf Y}^{[T]}_{1};\\bar{\\bf X}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} I(\\bar{\\bf Y}^{[T]}_{k};\\bar{\\bf X}^{[T]}_{k}\\mid \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\label{c6+}\\\\\n&=& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})\\nonumber\\\\\n&&+\\sum_{k=2}^{K} \\left(H(\\bar{\\bf Y}^{[T]}_{k}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})-H(\\bar{\\bf Y}^{[T]}_{k-1}\\mid  \\bar{\\bf X}^{[T]}_{1},\\cdots,\\bar{\\bf X}^{[T]}_{k-1},\\mathcal{G})\\right)\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c6}\\\\\n&\\le& T\\left((N-M)^+\\alpha+\\min(M,N)\\right)\\log{\\bar{P}}+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)\\log{\\bar{P}}\\nonumber\\\\\n&&+T~o(\\log{\\bar{P}})\\label{cv6}\\\\\n&=& T(N\\alpha+K\\min(M,N)(1-\\alpha))\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c7}\n \\end{eqnarray}\n\\eqref{c6} follows similar to \\eqref{c3} and \\eqref{cv6}  is concluded as the entropy of a discrete random variable is   bounded by logarithm of the cardinality of its support, i.e., \\footnote{\\eqref{cv6} follows from Lemma \\ref{lemma}  by substituting $\\bar{\\bf U}_1$ and $\\bar{\\bf U}_2$ with $\\bar{\\bf Y}^{[T]}_{1}$ and ${ C}^{[T]}$ where ${\\bf C}^{[T]}$ is a $T$-letter constant variable.  Then, substituting $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$  with  $\\bar{\\bf X}_1^{[T]}$ and $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq 1\\right)$, \\eqref{cv6} is concluded. Here, we assume $\\eta=1, \\lambda_{11}=1, \\lambda_{21}=\\alpha, \\lambda_{12}=0, \\lambda_{22}=0,,M_1=M,M_2=(K-1)M,N_1=N_2=N$.}\n\\begin{eqnarray}\n H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})&\\le& T\\left((N-M)^+\\alpha+\\min(M,N)\\right)\\log{\\bar{P}}\n \\end{eqnarray}\nDividing \\eqref{c7} by $T\\log{\\bar{P}}$, \\eqref{b2}  is obtained.\n\\item {\\bf Proof of \\eqref{b2+}}\\\\\n\n Similarly, from (\\eqref{c6+}-\\eqref{c6}) we have,\n\\begin{eqnarray}\n&&T\\sum_{k=1}^{K}R_k\\nonumber\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+\\sum_{k=2}^{K} T\\min(M,N)(1-\\alpha)^+\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\\\\n&\\le& H(\\bar{\\bf Y}^{[T]}_{1}\\mid \\mathcal{G})+T~o(\\log{\\bar{P}})\\label{c6-}\\\\\n&\\le& (N-(K-1)M)^+T\\log{\\bar{P}}+\\min(N,(K-1)M)\\alpha T\\log{\\bar{P}}+T~o(\\log{\\bar{P}})\\label{c6--}\n \\end{eqnarray}\n\\eqref{c6-} is true as $1\\le\\alpha$ and \\eqref{c6--} follows similar\\footnote{\\eqref{c6--} follows similar to  \\eqref{cv6}  from Lemma \\ref{lemma}. Substitute $\\bar{\\bf U}_1$, $\\bar{\\bf U}_2$, $\\bar{\\bf V}_1^{[T]}$ and $\\bar{\\bf V}_2^{[T]}$ with $\\bar{\\bf Y}^{[T]}_{1}$, ${ C}^{[T]}$,  $\\left(\\bar{\\bf X}_{j}^{[T]},j\\in[K],j\\neq 1\\right)$ and $\\bar{\\bf X}_1^{[T]}$ and  assume $\\eta=\\alpha, \\lambda_{11}=\\alpha, \\lambda_{21}=1, \\lambda_{12}=0, \\lambda_{22}=0,M_1=(K-1)M,M_2=M,N_1=N_2=N$.} to  \\eqref{cv6}. Dividing \\eqref{c6--} by $T\\log{\\bar{P}}$, \\eqref{b2+}  is obtained.\n\n\n\n\\end{enumerate}\n\n\\section{Proof of Theorem \\ref{theorem:GDoF1}: Achievability}\n\\subsection{A Useful Lemma}\nConsider a $(M_1+M_2)$-user multiple access channel (MAC) where each transmitter is equipped with a single antenna, the receiver has $N$ antennas, $N< M_1+M_2$, and the $N\\times 1$ received signal vector ${\\bf Q}$ is represented as,\n\\begin{align}\n{\\bf Q}=&\\sqrt{P}\\sum_{k=1}^{M_1} {\\bf H}_k{ T}_k+\\sqrt{P^{\\alpha}}\\sum_{k=M_1+1}^{M_1+M_2} {\\bf H}_k{ T}_k+\\sum_{n=1}^{N} \\sqrt{P^{\\alpha_n}}{\\bf G}_nZ_n\\label{mac0}\n\\end{align}\nwhere  $T_1, T_2, \\cdots, T_{M_1+M_2}$ are the transmitted signals, and $Z_1,Z_2,\\cdots,Z_N$ are i.i.d. Gaussian zero mean unit variance noise terms. The ${\\bf H}_k, {\\bf G}_n$ are $N\\times 1$ generic vectors, i.e.,  generated from continuous distributions with bounded density, so that any $N$ of them are linearly independent almost surely. The transmit power constraint is  expressed as,\n\\begin {eqnarray}\n\\mbox{E}{|T_{k}|}^2&\\leq&P^{-\\eta_k},~\\forall k\\in[M_1+M_2]\\label{mac1}\n\\end{eqnarray}\nwhere for any $k\\in[M_1+M_2]$,  $\\eta_k$ is a non-negative integer. Further, define $\\gamma_k$ for $k\\in[M_1+M_2]$ as,\n\\begin {eqnarray}\n\\gamma_k&=&\\left\\{\n\\begin{array}{ll} \n{(1-\\eta_k)}^+,&k\\in[M_1]\\\\ \n{(\\alpha-\\eta_k)}^+, &\\text{Otherwise}\n\\end{array}\n\\right.\\label{gamma1}\n\\end{eqnarray}\nThus $\\gamma_k$ is the received power level of user $k$ in the GDoF sense. The GDoF region $\\mathcal{D}'$ is defined as\n\\begin{align}\n\\mathcal{D}'\\triangleq&\\{(d'_1,d'_2,\\cdots,d'_{M_1+M_2}): \\exists (R'_1(P),R'_2(P),\\cdots,R'_{M_1+M_2}(P))\\in\\mathcal{C}'(P),\\nonumber\\\\\n& \\mbox{ s.t. } d'_k=\\lim_{P\\rightarrow\\infty}\\frac{R'_k(P)}{\\frac{1}{2}\\log{(P)}}, \\forall k\\in[M_1+M_2]\\} \\label {region}\n\\end{align}\n where $\\mathcal{C}'(P)$ is the capacity region of MAC described in (\\ref{mac0}).\n \n \n\\begin{lemma}\\label{lemma:mac} The GDoF tuple $(d'_1, d'_2, \\cdots, d'_{M_1+M_2})$ is achievable in the multiple access channel described above if $\\forall k\\in[M_1+M_2]$, and $\\forall S\\subset[M_1+M_2]$ where $|S|=k$,\n\\begin{align}\n\\sum_{i\\in S} d'_i\n&\\le\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i -\\min_{S_1\\in[N],|S_1|=\\min(k,N)}\\sum_{i\\in S_1} \\alpha_i \\label{mac3}\n\\end{align}\n\\end{lemma} \nFor proof of Lemma \\ref{lemma:mac} see \\cite{Arash_Jafar_MIMOsym_ArXiv}. It is sufficient to derive the achievability for Theorem \\ref{theorem:GDoF1}, as Theorem \\ref{theorem:GDoF1} is automatically concluded from it.\n\n\\subsection{Proof of Achievability in Theorem \\ref{theorem:GDoF1}} \\label{app3}\nNow, let us achieve the bound (\\ref{fg<}). We will suppress the time-index $t$ in this section to simplify the notation. For any $k\\in[K]$ user $k$'s message $W_k$ is split into messages $(W_{kc},W_{kp})$, representing common message and private message, respectively.  Let us consider the three cases of $\\alpha\\le\\frac{1}{2}$, $\\frac{1}{2}\\le\\alpha\\le1$, and $1\\le\\alpha$ separately as,\n\\begin{enumerate}\n\\item {$\\alpha\\le\\frac{1}{2}$.} Our goal here is to achieve $\\min(M,N)(1-\\alpha)+\\frac{(N-M)^+\\alpha}{K-1}$ GDoF per user where results in $K\\min(M,N)(1-\\alpha)+\\frac{K(N-M)^+\\alpha}{K-1}$ GDoF totally. In order to achieve $\\min(M,N)(1-\\alpha)+\\frac{(N-M)^+\\alpha}{K-1}$ GDoF per user, for any $k\\in[K]$ the public message $W_{kc}$ is encoded into Gaussian codebooks $U_{k1c},U_{k2c},\\cdots,U_{kMc}$ with powers $\\mbox{E}{|U_k|}^2=1-P^{-\\alpha}$ each carrying $\\frac{(N-M)^+\\alpha}{(K-1)M}$ GDoF. These codewords are transmitted through $M$ antennas along  $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{M}}$.  The private message $W_{kp}$ is encoded into Gaussian codebooks $U_{k1p},U_{k2p},\\cdots,U_{k\\min(M,N)p}$ with powers $\\mbox{E}{|U_{kjp}|}^2=P^{-\\alpha}$ for any $j\\in[\\min(M,N)]$ so that the total power per transmitter is unity. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. Each of the  private messages is carrying $1-\\alpha$ GDoF. The transmitted and received signals are,\n\\begin{align}\n\\mathbf{X}_{k}=&\\sum_{j=1}^{M}\\mathbf{V}_{kj}{U}_{kjc}+\\sum_{j=1}^{\\min{M,N}}\\mathbf{V}_{kj}{U}_{kjp}\\label{TR21a-}\\\\\n\\mathbf{Y}_{k}=&\\sqrt{P}{\\bf G}_{kk}\\mathbf{X}_{k}+\\sum_{j=1,j\\neq k}^{K}\\sqrt{P^{\\alpha}}{\\bf {G}}_{kj}\\mathbf{X}_{j}+\\mathbf{\\Gamma}_{k}\\label{TR21a}\n\\end{align}\n Using Lemma \\ref{lemma:mac} we claim that each receiver, e.g., receiver $1$ can decode  all the signals $U_{kjc}$ and $U_{1jp}$ for all $k\\in[K]$ and $j\\in[M]$  treating all the other signals as noise. Set the variables $M_1=M+\\min(M,N)$, $M_2=(K-1)M$ and $\\alpha_n=0$ for all $n\\in[N]$. Moreover, define the codewords $T_{1},\\cdots,T_{KM+\\min(M,N)}$ as \n\\begin {align}\nT_j=&\\left\\{\n\\begin{array}{ll} \nU_{1jc},&1\\le j\\le {M}\\\\ \nU_{2(j-M)c},&M<j\\le 2M\\\\\n\\vdots &\\vdots\\\\\nU_{K(j-(K-1)M)c},&(K-1)M<j\\le KM\\\\\nU_{1(j-KM)p},&KM<j\\le KM+\\min(M,N)\n\\end{array}\n\\right.\n\\end{align}\nFrom \\eqref{gamma1}, $\\gamma_1,\\cdots,\\gamma_{KM+\\min(M,N)}$ are derived as,\n\\begin {align}\n\\gamma_j=&\\left\\{\n\\begin{array}{ll} \n1,&1\\le j\\le M\\\\ \n\\alpha,&M< j\\le KM\\\\ \n1-\\alpha,&KM< j\\le KM+\\min(M,N)\n\\end{array}\n\\right.\\label{ffd2}\n\\end{align}\nNote that $N\\le KM$ and $\\alpha\\le\\frac{1}{2}$. Thus, from the received signal in (\\ref{TR21a}), $T_1,\\cdots,T_{KM+\\min(M,N)}$ are decoded by  first receiver as (\\ref{mac3}) is satisfied for all $k\\in[KM+\\min(M,N)]$. For instance if we set $k=KM+\\min(M,N)$, the condition (\\ref{mac3}) is equivalent to,\n\n\\begin{align}\n\\sum_{i\\in S} d'_i=&\\frac{K(N-M)^+\\alpha}{K-1}+\\min(M,N)(1-\\alpha)\\nonumber\\\\\n\\le&\\min(M,N)+(N-M)^+\\alpha=\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i .\n\\end{align}\n\n\\item{$\\frac{1}{2}<\\alpha\\leq1$.} Let us achieve $d$ GDoF where $d$  is equal to,\n\\begin{align}\nd=\\min\\left(\\frac{K}{K-1}\\left((K-2)\\min(M,N)(1-\\alpha)+N(\\alpha)\\right),N\\alpha+K\\min(M,N)\\left(1-\\alpha\\right)\\right) \n\\end{align}\nSimilar to the case $\\alpha\\le\\frac{1}{2}$, the public message $W_{kc}$ is encoded into Gaussian codebooks $U_{k1c},U_{k2c},\\cdots,U_{kMc}$ with powers $\\mbox{E}{|U_k|}^2=1-P^{-\\alpha}$ each carrying $\\frac{d-K\\min(M,N)(1-\\alpha)}{KM}$ GDoF. These codewords are transmitted through $M$ antennas along  $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{M}}$.  The private ~message $W_{kp}$ ~is encoded~ into Gaussian~ codebooks ~$U_{k1p},U_{k2p},\\cdots,U_{k\\min(M,N)p}$ with powers $\\mbox{E}{|U_{kjp}|}^2=P^{-\\alpha}$ for any $j\\in[\\min(M,N)]$. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. Each of the  private messages is carrying $1-\\alpha$ GDoF. The transmitted and received signals follows the same as \\eqref{TR21a-} and \\eqref{TR21a}. From Lemma \\ref{lemma:mac} each receiver, e.g., receiver $1$ can decode all the codewords $U_{kjc}$ and $U_{1jn}$ for all $k\\in[K]$ and $j\\in[M]$  treating all the other signals as noise. The details how receiver $1$ can decode all these codewords follows the same as the case $\\alpha\\le\\frac{1}{2}$.\n\n\\item{$1\\le\\alpha$.} In this case, $\\min\\left(D(\\alpha),K\\min(M,N)\\right)$ is achieved as follows. Recall that $D(\\alpha)$ was defined as $(N-(K-1)M)^++\\min(N,(K-1)M)\\alpha$. All $\\min(M,N)$ messages of each transmitter are\npublic in this case and are encoded into Gaussian codebooks\n$U_{k1c},U_{k2c},\\cdots,U_{k\\min(M,N)c}$ with unit powers each carrying $\\min\\left(\\frac{D(\\alpha)}{K\\min(M,N)},1\\right)$ GDoF. These codewords are transmitted through $\\min(M,N)$ antennas along the $M\\times 1$ generic unit  vectors ${\\bf V}_{k1},{\\bf V}_{k2},\\cdots,{\\bf V}_{k{\\min(M,N)}}$. The transmitted and received signals are concluded similar to \\eqref{TR21a-} and \\eqref{TR21a} as,\n\\begin{align}\n\\mathbf{X}_{k}=&\\sum_{j=1}^{M}\\mathbf{V}_{kj}{U}_{kjc}\\label{TR21a-1}\\\\\n\\mathbf{Y}_{k}=&\\sqrt{P}{\\bf G}_{kk}\\mathbf{X}_{k}+\\sum_{j=1,j\\neq k}^{K}\\sqrt{P^{\\alpha}}{\\bf {G}}_{kj}\\mathbf{X}_{j}+\\mathbf{\\Gamma}_{k}\\label{TR21a1}\n\\end{align}\nEach receiver, e.g., receiver $1$ can decode  all the signals $U_{kjc}$ for all $k\\in[K]$ and $j\\in[\\min(M,N)]$  treating all the other signals as noise. Set the variables $M_1=\\min(M,N)$, $M_2=(K-1)\\min(M,N)$ and $\\alpha_n=0$ for all $n\\in[N]$ in Lemma \\ref{lemma:mac} . Moreover, define the codewords $T_{1},\\cdots,T_{K\\min(M,N)}$ as \n\\begin {align}\nT_j=&\\left\\{\n\\begin{array}{ll} \nU_{1jc},&1\\le j\\le {\\min(M,N)}\\\\ \nU_{2(j-\\min(M,N))c},&\\min(M,N)<j\\le 2\\min(M,N)\\\\\n\\vdots &\\vdots\\\\\nU_{K(j-(K-1)\\min(M,N))c},&(K-1)\\min(M,N)<j\\le K\\min(M,N)\n\\end{array}\n\\right.\n\\end{align}\nFrom \\eqref{gamma1}, $\\gamma_1,\\cdots,\\gamma_{K\\min(M,N)}$ are derived as,\n\\begin {align}\n\\gamma_j=&\\left\\{\n\\begin{array}{ll} \n1,&1\\le j\\le \\min(M,N)\\\\ \n\\alpha,&\\min(M,N)< j\\le K\\min(M,N)\n\\end{array}\n\\right.\\label{ffd2}\n\\end{align}\nSimilar to the case $\\alpha\\le \\frac{1}{2}$, $T_1,\\cdots,T_{K\\min(M,N)}$ are decoded by  first receiver as (\\ref{mac3}) is satisfied for all $k\\in[K\\min(M,N)]$. For instance if we set $k=K\\min(M,N)$, the condition (\\ref{mac3}) is equivalent to,\n\\begin{align}\n\\sum_{i\\in S} d'_i=&K\\min(M,N)\\times\\min\\left(\\frac{D(\\alpha)}{K\\min(M,N)},1\\right)\\nonumber\\\\\n\\le&\\min\\left((K-1)M,N\\right)\\alpha+\\left(N-(K-1)M\\right)^+=\\max_{S_2\\in S,|S_2|=\\min(k,N)}\\sum_{i\\in S_2} \\gamma_i \n\\end{align}\n\n\\end{enumerate}\n\n \n\\section{Conclusion}\nSymmetric $K$-user MIMO IC with $M$ antennas at each transmitter and $N$ antennas at each receiver is considered. Sum GDoF of this channel is derived. The Sum GDoF  is found to be a $W$ curve as a function of $\\alpha$ for fixed  $M$ and $N$ similar to the SISO case. Outer bound proof is obtained with the help of a key lemma that generalizes the AIS argument.  The achievability follows from the achievability of the GDoF region of a MAC, combined with the `treating interference as noise' scheme. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION \\label{sec:intro}}\n\nImages of spiral and irregular galaxies have been defined historically\nby regions of massive star formation, signposts demarcating important\nfeatures such as spiral arms, bars, and starbursts.  They remind us\nthat galaxies really are evolving, with continuous injection of energy\nand processed material into the galaxy.  Massive star-forming regions\n(MSFRs) present us with a microcosm of starburst astrophysics, where\nstellar winds from O and Wolf-Rayet (WR) stars compete with supernovae\n(SNe) to carve up the neutral medium from which they formed, igniting a\nnew generation of stars.  With X-ray observations, we see the sources\nthat shape the larger view of a galaxy, injecting energy and processed\nmaterial into the disk and halo ISM through winds, ionization fronts,\nSNe, superbubbles, and chimneys.  About 80\\% of all SNe occur in\nsuperbubbles, so cosmic rays generated by SNe also may be accelerated\nin superbubbles \\citep[e.g.][]{Higdon98, Parizot04}.  X-ray\nobservations probe different energetic components than traditional\nvisual and infrared (IR) studies, penetrating the obscuring material of\nthe natal cloud while minimizing confusion from foreground and\nbackground objects.\n\nX-ray studies also detect the presence of past SNe through the shocks\nin their extended remnants.  Some superbubbles are well-known bright\nX-ray sources, where multiple SNe from past OB stars produce soft\nX-rays filling bubbles 50--100~pc in size with $L_x \\sim\n10^{35-36}$~ergs~s$^{-1}$ \\citep{Chu90}, while others (not recently\nbrightened by such ``cavity'' SNe) are X-ray faint \\citep{Chu95}.\nMSFRs thus exhibit a complicated mixture of point and extended\nstructures that are easily confused by low-resolution X-ray\ntelescopes.  30~Doradus (30~Dor) is a classic example of this.  The\n{\\em Chandra X-ray Observatory} \\citep[Figure~\\ref{fig:csmoothimage}\nand][hereafter Paper~II]{Townsley05b} and {\\em XMM-Newton}\n\\citep{Dennerl01} share the highest-quality views yet achieved of this\nextraordinary and complicated X-ray field.\n\n30~Dor is a field of superlatives.  Located in the Large Magellanic\nCloud (LMC), it is the most luminous Giant Extragalactic H{\\sc II}\nRegion (GEHR) and ``starburst cluster'' in the Local Group.  It hosts\nthe massive compact star cluster R136 \\citep[][called ``RMC~136'' in\nSIMBAD]{Feast60}, a testbed for understanding recent and ongoing star\nformation in the 30~Dor complex \\citep{Walborn02}.  Although R136, with\na mass of $\\sim 6 \\times 10^4$~M$_{\\odot}$ \\citep{Brandl05}, does not\nquite qualify as a ``super star cluster'' \\citep{Meurer95}, it has more\nthan 50 times the ionizing radiation of the Orion Nebula.  Nearby, one\nfinds the large supernova remnant (SNR) N157B (30~Dor~B), embedded in\nan H{\\sc II} region generated by the OB association LH~99 \\citep{Chu92}\nand containing an X-ray pulsar \\citep{Marshall98}.  {\\em ROSAT} images\nshowed two fainter sources associated with WR stars near the central\nstar cluster, at least five plasma-filled superbubbles, and the N157B\nSNR \\citep{Wang95, Wang99}.  30~Dor's proximity and complexity provide\nus with a unique and important microscope into the starburst phenomenon\nin galaxies.\n\nFigure~\\ref{fig:csmoothimage} illustrates the spatial and spectral\ncomplexity of the X-ray emission from 30~Dor with a smoothed image of\nour {\\em Chandra} observation obtained with the Advanced CCD Imaging\nSpectrometer (ACIS).  The aimpoint of the $17\\arcmin \\times 17\\arcmin$\nACIS Imaging Array (ACIS-I) was the R136 cluster.  At $D=50$~kpc, the\ndistance we assume throughout this paper and Paper~II, $1^{\\prime} \\sim\n14.5$~pc, so ACIS-I covers $\\sim 240$~pc $\\times$ 240~pc.  Two off-axis\nCCDs from the ACIS Spectroscopy Array (ACIS-S) were operated for this\nobservation, adding a $\\sim 120$~pc $\\times$ 240~pc image of nearby\ninteresting structures (see Figure~\\ref{fig:introimage}), albeit with\npoor spatial resolution.  A more finely binned image of the ACIS-I\ndata, highlighting the X-ray point sources and showing a J2000\ncoordinate grid, can be found in Figure~1 of Paper~II.  \n\n30~Dor lies at the confluence of two LMC supergiant shells and just the\nmain nebula is $\\sim 250$~pc in diameter.  Since the LMC is viewed\nnearly face-on, confusion between 30~Dor and other disk structures is\nminimized.  SNe pervade the region but most go undetected due to age\nand environment \\citep{Chu90}.  Nearby are two pulsars (B0540$-$69.3 in\nthe SNR N158A and J0537$-$6910 in the SNR N157B) and SN1987A\n(Figure~\\ref{fig:introimage}).  Wide-field ground-based H$\\alpha$\nimages \\citep[e.g.][]{Meaburn84,Chu90,Wang99}, enhanced by recent {\\em\nHST} \\citep{Walborn02} and {\\em Spitzer}\ndata\\footnote{\\url{http:\/\/www.spitzer.caltech.edu\/Media\/releases\/ssc2004-01\/release.shtml}},\nshow that the combined actions of stellar winds and SNe from several\ngenerations of high-mass stars in 30~Dor have carved its ISM into an\namazing display of arcs, shells, pillars, voids, and bubbles, ranging\nover spatial scales of 1--100~pc \\citep{Brandl05}. \n\nThis field provides a unique bridge between Galactic giant H{\\sc II}\nregions (e.g.\\ W49A, W51A, NGC~3603) and GEHRs \\citep[e.g.\\ NGC~5471\nand other regions in M101,][]{Chen05}.  30~Dor is the result of\nmultiple epochs of high-mass star formation in a vast molecular cloud\ncomplex chemically enriched by many SNe; it has no Galactic counterpart\nin terms of mass or complexity of its star formation history and is not\neven equalled by other GEHRs in the Local Group.  It provides us with a\nunique view of the most fundamental building block of the starburst\nphenomenon in galaxies; currently we are not able to resolve comparable\nstar-forming complexes in starburst galaxies, only those many times\nlarger and more powerful \\citep[e.g.\\ NGC~253,\nNGC~7714\/7715;][]{Strickland02,Smith05}.\n\nIn \\S\\ref{sec:background} we provide a brief summary of X-ray\nobservations of the 30~Dor complex.  Our {\\em Chandra} observations and\ndata analysis are outlined in \\S\\ref{sec:observation}.  An overview of\nthe X-ray morphology and global spectral properties of the complex is\npresented in \\S\\ref{sec:global}.  The rest of the paper provides more\nin-depth studies of specific diffuse X-ray components of the 30~Dor\ncomplex:  the superbubbles centered on R136 (\\S\\ref{sec:superbubbles}),\nthe SNR N157B (\\S\\ref{sec:N157B}), and other diffuse structures, with a\ncomparison to the H$\\alpha$ kinematic study of 30~Dor by \\citet{Chu94}\n(\\S\\ref{sec:otherdiffuse}).  We conclude with a comparison of the X-ray\ndata to recent H$\\alpha$ and IR data and a summary of our results.  The\nX-ray point source population of 30~Dor revealed by this dataset is\ndescribed in Paper~II.\n\n\n\n\\section{PAST X-RAY OBSERVATIONS\n\\label{sec:background}}\n\n\\subsection{The Large Magellanic Cloud and 30~Doradus}\n\nX-rays were first detected from the LMC in 1968 using rocket-launched\nproportional counters \\citep{Mark69}.  An early survey of the LMC with\nthe {\\em Einstein Observatory} \\citep{Long81} found 75 discrete sources\nassociated with the galaxy, including 25 SNRs and about 25 other\nextended sources.  A re-analysis of the {\\em Einstein} LMC survey\nresulted in a point source catalog of 105 sources, ``roughly half''\nassociated with the LMC \\citep{Wang91c}; the other sources are\nforeground stars or background active galactic nuclei (AGN).  These\nauthors noted source confusion associated with the 30~Dor complex.\n\nDetailed studies of 30~Dor and other LMC superbubbles were performed\nwith {\\em Einstein} \\citep[e.g.][]{Chu90,Wang91a,Wang91b}; they\nconcluded that the bright, diffuse X-ray emission filling the 30~Dor\nsuperbubbles was probably created by off-center cavity SNe inside the\nsuperbubbles shocking the superbubble shells.  Extending the work of\n\\citet{Meaburn84}, \\citet{Chu94} performed a detailed echelle study of\nthe kinematics of 30~Dor and compared their findings to the {\\em\nEinstein} data, concluding that 30~Dor is dominated by a hierarchy of\nexpanding structures with spatial scales ranging over 1--100~pc and\nexpansion velocities of 20--200~km~s$^{-1}$.  The diffuse X-ray sources\nare coincident with these expanding shells, with the brightest X-ray\nemission coincident with high-velocity features.  \\citet{Chu94} argued\nthat both the high-velocity features and the X-ray emission are related\nto high-velocity shocks created by fast stellar winds and SNe.\n\nTwo catalogs of LMC point sources emerged from the {\\em ROSAT}\nObservatory; a few sources (often SNRs or SNR candidates) are\nassociated with the 30~Dor complex \\citep{Haberl99,Sasaki00}.  In\nmosaicked {\\em ROSAT} maps of the hot ISM in the LMC, 30~Dor is an\noutstanding feature---hotter, brighter, and at higher pressure than\nmost other parts of the galaxy \\citep{Snowden94,Points01,Sasaki02}.\n{\\em ROSAT} studies of individual LMC superbubbles showed that several\nare X-ray faint \\citep{Chu95}, consistent with earlier conclusions that\nsuperbubbles are only brightened in X-rays when SNe hit the shell\nwalls, while others exhibit ``breakout regions'' and may be venting hot\ngas into the LMC's ISM \\citep{Dunne01}.\n\nA {\\em ROSAT}\/PSPC study of 30~Dor reported 22 point sources within the\n30~Dor nebula, including sources associated with R136 and N157B\n\\citep{Norci95}.  Spectral fits to the diffuse emission yielded $N_H\n\\sim 8 \\times 10^{21}$~cm$^{-2}$ and $kT \\sim 0.4$~keV, with a\nluminosity (0.1--2.4~keV) of $L_X = $3--6$ \\times\n10^{37}$~ergs~s$^{-1}$ depending on choice of background.  Since the\nGalactic absorbing column toward 30~Dor is $7 \\times\n10^{20}$~cm$^{-2}$, most of the absorption found in their spectral fits\nis local to 30~Dor.  They confirm the {\\em Einstein} results\n\\citep{Wang91a} that the diffuse X-ray emission fills the voids in the\n$H\\alpha$ emission and further conclude that stars in 30~Dor contribute\n$\\leq 2$\\% of its total X-ray luminosity.  While Norci and {\\\" O}gelman\nnote that the central cluster R136 is too young to have produced X-ray\nbinaries that could contribute to the X-ray emission in the region, a\n{\\em ROSAT}\/HRI study suggested that the two point sources found by HRI\nwere associated with the WR stars R140a2 (a WN6 binary) and Melnick 34\n(WN4.5) and could be WR\/black hole binaries \\citep{Wang95}.  This\nconjecture was supported by {\\em ASCA} spectral results that indicated\nhard X-ray emission from the central regions of 30~Dor.  Further {\\em\nROSAT}\/HRI and {\\em ASCA} observations led \\citet{Wang99} to conclude\nthat the diffuse emission came from 2--9~MK thermal plasmas and that\nthe primary mechanism to explain the temperature and X-ray luminosity\nof this emission is mass-loading via mixing of the hot wind-generated\ngas with H{\\sc II} gas from the large number of ionization fronts in\nthe region \\citep{Scowen98}.\n\nThe first light image from the {\\em XMM-Newton Observatory} was\nobtained from a 106~ks pointing just west of the main 30~Dor nebula\n\\citep{Dennerl01}.  This observation yielded broad-band (0.1--10~keV)\nspectra for several important objects in the neighborhood of 30~Dor:\nN157B, the Honeycomb Nebula \\citep[a SNR,][]{Chu95b}, and faint diffuse\nemission northwest of 30~Dor \\citep{Dennerl01}, as well as several\nlikely AGN seen through the LMC disk \\citep{Haberl01}.  A 38~ks {\\em\nXMM} observation of N157B (Observation 0113020201, PI Aschenbach)\nimages even more of the main 30~Dor nebula; although it is beyond the\nscope of this effort to include that observation in this paper, future\nefforts to obtain spectra of various features in 30~Dor from this\ndataset would be worthwhile.\n\n\n\\subsection{N157B and PSR~J0537$-$6910}\n\\label{sec:n157bbkgd}\n\nThe N157B SNR has itself been the subject of many X-ray studies.  A\nknown radio source, it was first detected in X-rays by {\\em Einstein}\n\\citep{Long79}.  A later {\\em Einstein} study \\citep{Clark82}\ndemonstrated that its spectrum showed no strong emission lines and was\nnon-thermal, well-fit by a power law with slope $\\Gamma = 2.9$, $N_H =\n1.2 \\times 10^{21}$~cm$^{-2}$, and luminosity (0.5--4.5~keV) of $L_X =\n2.3 \\times 10^{36}$~ergs~s$^{-1}$; they suggested that a search for a\npulsar in N157B should commence.  \\citet{Chu92} determined the boundary\nof the SNR based on kinematic information and suggested that N157B is\nanother example of an off-center SNR expanding into a pre-existing\nsuperbubble and that the SNR may be interacting with a dark cloud on\nits southern edge; this dark cloud may shadow the soft X-ray emission\nfrom the SNR.\n\nA detailed {\\em ROSAT} and {\\em ASCA} study of N157B was presented by\n\\citet{Wang98a}.  Their spectrum is dominated by a non-thermal\ncomponent with power law slope $\\Gamma \\sim 2.5$, although there are\nhints of emission lines below 2~keV that they attributed to a thermal\nplasma from the SNR with $kT = $0.4--0.7~keV.  {\\em ROSAT}\/HRI revealed\na comet-shaped central nebula that dominates the X-ray emission,\nexplained as a synchrotron nebula from a pulsar moving through the ISM\nat a rate of $\\sim 1000$~km~s$^{-1}$, although no pulsed emission was\napparent in these data. \n\nThe long-awaited detection of PSR~J0537$-$6910 in N157B, with a period\nof 16~msec and a power law photon index of $\\Gamma = 2.6$, came\nserendipitously from an {\\em RXTE} observation of SN1987A\n\\citep{Marshall98} and was soon confirmed by {\\em BeppoSAX}\n\\citep{Cusumano98}.  A slightly later {\\em ROSAT}\/HRI detection\nprovided an arcsecond position for the pulsar \\citep{Wang98b}.  This is\nthe shortest period known for a pulsar associated with a SNR.  This\npulsar remains undetected in the radio \\citep{Crawford05} and no\nunambiguous optical counterpart has been found, even with deep {\\em\nHST}\/ACS imaging \\citep{Mignani05}.\n\nThe thermal emission component in N157B's spectrum is confirmed by {\\em\nXMM} \\citep{Dennerl01}, also confirming N157B as a composite SNR,\ncontaining a Crab-like synchrotron core with $\\Gamma = 2.83$\n\\citep{Dennerl01} surrounded by a thermal shell.  {\\em Chandra} has\nobserved N157B directly with both the HRC \\citep{Wang01} and ACIS-S\nusing subarray readout (ObsID 2783, PI Wang).  With the HRC data,\n\\citet{Wang01} were able to pinpoint the position of the pulsar to\n$<1\\arcsec$ and to resolve a compact pulsar wind nebula at the ``head''\nof the comet-shaped nebula seen by {\\em ROSAT} \\citep{Wang98a}.  This\n$\\sim 2\\arcsec \\times 7\\arcsec$ feature is centered on the pulsar and\noriented perpendicular to the larger cometary nebula.  \\citet{Wang01}\nexplain this feature as a toroidal pulsar wind nebula confined by the\nbow shock from the supersonic motion of the pulsar and depositing its\nenergy in the trailing cometary nebula.\n\t\n\n\\subsection{Past Work on This {\\em Chandra} Observation}\n\t\nOur short {\\em Chandra}\/ACIS-I observation of 30~Dor (described in\ndetail below) was one of the first observations made by {\\em Chandra}\nand was part of the ACIS Team's Guaranteed Time Observation program.  We\nhave reported preliminary results from this observation at several\nmeetings \\citep{Townsley99,Townsley00b,Townsley02} and have used it to\nestimate the position of PSR~J0537$-$6910 in the N157B SNR\n\\citep{Mignani05}.  With this study we finally are able to provide a\ndetailed analysis of this observation, after several years' worth of\ncode development and special analysis techniques were implemented.  Our\ncustom tools are also useful for analysis of other {\\em Chandra}\nfields, so we have made them publicly available (see\n\\S\\ref{sec:observation}).\n\nThis dataset has also been studied by other groups.  Recently,\n\\citet{Lazendic03} proposed two new SNR candidates in the 30~Dor nebula\nbased on radio continuum maps obtained with the {\\em Australia\nTelescope Compact Array}, although the {\\em Chandra} data showed no\nX-ray enhancements associated with these regions.  \\citet{Chu04}\nexamined this claim and concluded, based in part on the absence of\nassociated X-ray emission, that these small radio sources are not SNRs;\nrather one is likely a young star-forming region and the other a\nmolecular cloud.\n\n\\citet{PPL02} found 20 mostly pointlike sources in the inner $11\\arcmin\n\\times 11\\arcmin$ of the ACIS-I field of view and studied 11 of those\nin the central region, within $100\\arcsec$ from the core of R136.  They\nmatched these sources with known early-type (WR and O) stars, some\nknown to be binaries, and concluded that they are bright in X-rays\nbecause they are all probably colliding-wind binaries.\n\\citet{Stevens03} included a spectral analysis of unresolved X-ray\nemission around R136 in their study of the properties of super star\nclusters.  Some unspecified number of point sources was removed from\nthe inner $16\\arcsec$ of R136, then the remaining X-ray emission was\nfit with a thermal plasma model, resulting in fit parameters of $N_H =\n4.2 \\times 10^{21}$~cm$^{-2}$, $kT = 2.1$~keV, and absorption-corrected\nluminosity $L_X = 5.5 \\times 10^{34}$~ergs~s$^{-1}$.  In Paper II we\nshow that more X-ray point sources in R136 are discernable in this\ndataset than these two papers considered; their results would change\nsomewhat if these faint sources were included in their analyses.\n\nSince our observation included the two off-axis ACIS-S chips S3 and S4,\nwe were able to image an $8\\arcmin.5 \\times 17\\arcmin$ region to the\nsouthwest of 30~Dor, featuring SN1987A, the large superbubble 30~Dor~C,\nand the Honeycomb Nebula (see Figures~\\ref{fig:csmoothimage} and\n\\ref{fig:introimage}).  As mentioned above, the {\\em XMM-Newton}\nfirst-light image was centered on this field and gives much better\nspatial and spectral information on these structures than this short\nACIS exposure \\citep{Dennerl01}.  The X-ray brightening of SN1987A has\nbeen monitored by {\\em Chandra} since early in the mission; see\n\\citet{Park05} and references therein.  30~Dor~C, an unusual\nnon-thermal X-ray emitter, has been studied with {\\em Chandra}\n\\citep{Bamba04} and {\\em XMM} \\citep{Smith04}.  Given these better\ndatasets and existing studies, we will not include detailed analysis of\nthese sources in this paper.\n\nThe results we present below represent only one step in the\nhigh-resolution characterization of X-rays from the 30~Dor region.  A\n100~ks {\\em Chandra} ACIS-I observation of 30~Dor is scheduled for\nlater this year; it should give a $\\sim 5$-fold improvement in\nsensitivity and signal over the results reported here.\n\n\n\n\\section{{\\em CHANDRA} OBSERVATIONS AND ANALYSIS\n\\label{sec:observation}}\n\n\\subsection{Instrumental Set-up}\n\nWe observed 30~Dor and its environs for $\\sim 26$~ks with ACIS-I on\n1999 September 21--22.  R136 was placed at the aimpoint.  The focal\nplane temperature at this early point in the mission was $-110$C to\nfacilitate water outgassing; it was later reduced to $-120$C to improve\ndetector performance.  This higher focal plane temperature causes the\neffects of radiation damage to the frontside-illuminated CCDs in the\nACIS-I array to be particularly pronounced \\citep{Townsley02a}.  For\nexample, the spectral resolution at 1.5~keV varies across the device\nfrom $\\sim 75$~eV FWHM at low CCD row numbers to $\\sim 170$~eV at high\nrow numbers.  Unfortunately, the aimpoint of the ACIS-I array falls\nnear the top of the I3 CCD, so spectral resolution for the sources in\nR136 is particularly bad.  A summary of the observations is given in\nTable~\\ref{tbl:obslog}.  \n\nThe first $\\sim 1$~ks of the observation (Observation Identification or\n``ObsID'' 22) was made in the usual ``timed event, faint'' mode, with\n$3 \\times 3$ pixel events, 3.2~s CCD frames, and the Spectroscopy Array\n(ACIS-S) CCDs S2 and S3 operating in addition to the ACIS-I devices.\nThe remainder of the observation (ObsID 62520) was taken in ``timed\nevent, very faint'' mode ($5 \\times 5$ pixel events) using the\n``alternating'' readout mode with a 10:1 cycle; every ten CCD frames\nwere exposed for 3.3~s, then every eleventh frame had a short 0.3~s\nexposure.  The data processing pipeline divides these short and long\nframes from ObsID 62520 into separate event lists.  In order to image\ninteresting structures near the 30~Dor field, ACIS-S CCDs S3 and S4\nwere operational for this observation.\n\nWe chose to use the ACIS camera's alternating-exposure mode because the\n30~Dor field contains PSR~J0537$-$6910, a bright 16~ms pulsar\n\\citep{Marshall98}.  Although it is imaged $\\sim 7\\arcmin$ off-axis, we\nknew that it would still suffer from photon pile-up in regular-length\nCCD frames, so we included the short frames to try to mitigate pile-up\non this source.  In hindsight, this was not an ideal approach, as there\nare too few photons in the short frames to provide much information on\nPSR~J0537$-$6910 and the use of alternating mode this early in the\nmission caused problems for the data processing pipelines, resulting in\nabsent or incorrect Good Time Interval (GTI) tables, incorrect keyword\nvalues, and other complications.\n\nFor this analysis, we used Version 3 of the pipeline processing\n(created 26 July 2001), which has correct GTI tables.  Even this third\nversion of the data still contains some errors though; for example, the\nEXPOSURE and LIVETIME keywords in the 0.3~s event list of ObsID 62520\nare incorrect.  After recalculating these quantities for each CCD and\ncorrecting the header keywords in the event list, a total of $\\sim\n200$~s of data taken with 0.3~s CCD frames is obtained.\n\n\n\\subsection{Data Analysis}\n\nStarting with the Level 1 event lists, the data were reduced using our\nstandard methods, as described in \\citet{Townsley03} and\n\\citet{Getman05}, using {\\it CIAO} 3.0.2.  {\\em Chandra} source\npositions were determined from the source-finding routine {\\it\nwavdetect} \\citep{Freeman02} run on a $2 \\times 2$-pixel binned image\nof the hard band (2--7~keV) data from the ``long'' observations in\nObsID 62520; data below 2~keV were excluded to minimize the\ncontamination of source positions by the soft diffuse emission that\npervades the field.  ObsID 62520 was registered to the astrometric\nreference frame of 2MASS using 4 matches.  Offsets were $+0.95\\arcsec$\nin RA, $+1.0\\arcsec$ in Dec.  \n\nThe {\\em Clean55}\nalgorithm\\footnote{\\url{http:\/\/hea-www.harvard.edu\/\\-~alexey\/\\-vf\\_bg\/vfbg.html}}\ndeveloped by Alexey Vikhlinin was used to flag events in ObsID 62520;\nthese flagged events were removed to suppress background for\nsource-finding and image generation, but this flag can also erroneously\nremove events from the cores of point sources so it was not applied in\nspectral analysis.  Due to a strong background flare caused by\nparticles generated by solar activity, ObsID 62520 has some telemetry\nsaturation that only affects the S3 chip, from frame $\\sim 7248$ ($\\sim\n24300$~sec) to the end of the observation.  Background flares affected\nCCD S3 both early and late in the exposure and those times were\nfiltered from the S3 data; other CCDs were unaffected by these flares.\nThe {\\it CIAO} ``destreaking'' algorithm by John\nHouck\\footnote{\\url{http:\/\/asc.harvard.edu\/ciao\/threads\/destreak\/}} was\napplied to CCD8 to suppress noise on that device.\n  \nWe did not adjust the astrometry of ObsID 22 due to only two 2MASS\nmatches and inconsistent, small offsets with those matches.  Although\nwe would normally not remove the event position randomization added in\nthe standard data processing pipeline for such a short observation, it\nwas removed for ObsID 22 in order to combine these data with ObsID\n62520.  S3 was experiencing background flares for the duration of this\nshort observation.  Given that and the short amount of total\nintegration time for the area covered by CCD S2, only the ACIS-I array\ncomponent of this observation was merged with ObsID 62520.\n\nFor ObsID 62520, individual CCD columns were examined in chip\ncoordinates using our {\\it Event Browser} tool\\footnote{Code available\nat \\url{http:\/\/www.astro.psu.edu\/xray\/docs\/TARA\/}} \\citep{Broos00} to\nsearch for additional hot pixels and columns that were not part of the\npre-defined bad pixel file and to establish the energies of those noise\nevents.  At $-110$C, there are many more hot columns (exhibiting extra\nnoise below 1~keV) than at $-120$C (the focal plane temperature used\nfrom January 2000 to the present), especially on CCD S4.  Those events\nwere removed from both ObsID 62520 and ObsID 22 using CIAO 2.3 because\nthe necessary ``exclude'' syntax (to remove only certain energies from\nspecified spatial coordinates) was not working correctly in CIAO\n3.0.2.  Although the effects of these deleted events cannot be included\nin the exposure map (because only events below certain energies, not\nall events from a given column, were removed), we chose to suffer this\nslight calibration error rather than to retain the noisy events because\nthey become especially pronounced and distracting in smoothed soft-band\nimages.\n\nBoth datasets were corrected for charge transfer inefficiency (CTI)\nusing the Penn State CTI corrector, which was originally developed\nspecifically to address the CTI problem for this\ndataset\\footnote{Details, source code, and calibration products for CTI\ncorrection are available at\n\\url{http:\/\/www.astro.psu.edu\/users\/townsley\/cti}.} \\citep{Townsley00,\nTownsley02a}, due to our interest in spectral analysis of the many\npointlike and diffuse structures in this field.  It remains the only\nCTI corrector available for the backside-illuminated CCD S3 and for all\nACIS data taken at a focal plane temperature of $-110$C (where CTI is\nworse than at $-120$C for frontside-illuminated CCDs, as mentioned\nabove).  Custom response matrices (RMFs) and quantum efficiency\nuniformity (QEU) files were generated to facilitate spectral fitting\nfor $-110$C data \\citep{Townsley02b}.  We also refined the event\npositions in both datasets using the subpixel positioning algorithm\ndeveloped by Koji Mori\\footnote{Code available at\n\\url{http:\/\/www.astro.psu.edu\/users\/mori\/chandra\/subpixel\\_resolution.html}}\n\\citep{Mori01, Tsunemi01}.\n\nThe reduced event lists were then merged and the exposure maps added.\nA {\\em status=0} filter was applied to create an event list appropriate\nfor source searching and image generation.  The data were once again\nexamined in {\\it Event Browser}, showing that the best energy range to\nconsider for imaging studies is 350--7300~eV; the lower limit retains\nevents below 500 eV that are clearly from the diffuse superbubble\nstructures in the field while limiting low-energy noise, while the\nupper limit is set just below the instrumental Ni~K$\\alpha$ line, which\nmarks the beginning of enhanced high-energy background.\n\nNext we conducted a multistage process for identifying and extracting\nphotons of unresolved sources that is described in detail in Paper II.\nIt involves source identification using a wavelet transform and a\nmaximum likelihood image restoration algorithm.  Point source events\nfor the final list of 180 sources were then extracted using PSB's\nIDL-based code {\\it ACIS Extract} \\citep{Broos02} developed to treat\ncomplicated fields with multiple point-like and diffuse\ncomponents\\footnote{The {\\it ACIS Extract} code is available at\n\\url{http:\/\/www.astro.psu.edu\/xray\/docs\/TARA\/ae\\_users\\_guide.html}.}.\nFor analyzing the diffuse components in the 30~Dor field, the only\npoint source properties required are their locations and extraction\nregions; these are provided in Tables 1 and 2 of Paper II.  These are\nused to construct a point source mask and a source-free dataset\nsuitable for analysis of diffuse emission.\n\nWe used a mask produced by the {\\it ACIS Extract} tool {\\it\nae\\_optimal\\_masking} which identifies regions where the expected\nsurface brightness from the calibrated point sources is larger than one\nhalf the smoothed observed local background.  These point source masks\nare illustrated in Figure~\\ref{fig:mask_emap}, which shows the I-array\nexposure map for ObsID 62520.  Chip gaps and bad columns are seen as\nareas of reduced exposure, while the small white regions distributed\nacross the field and concentrated in the center show areas of zero\nexposure representing areas that were masked to exclude point sources.\nThis figure also illustrates that the {\\em Chandra} PSF is a strong\nfunction of off-axis angle; most of the X-ray point sources in the field\nare concentrated in R136 (field center), yet the sharp PSF on-axis leads\nto relatively little area removed for diffuse emission studies there.\n\nExtraction regions for a variety of diffuse structures were then\ndefined by hand, based on the apparent surface brightness in a\nsmoothed, sources-removed image.  Spectra, backgrounds, ARFs, and RMFs\nfor all point sources and diffuse structures were constructed by {\\it\nACIS Extract} and used for both automated and interactive spectral\nfitting using Keith Arnaud's {\\it XSPEC} fitting package\n\\citep{Arnaud96} and {\\it XSPEC} scripts provided by Konstantin Getman.\n\n\n\n\\section{GLOBAL PROPERTIES OF THE X-RAY EMISSION\n\\label{sec:global}}\n\nWe start by describing the large-scale features revealed by this ACIS\nobservation.  Figure~\\ref{fig:csmoothimage} shows the soft-band image\nof 30~Dor created with the adaptive-kernel smoothing tool {\\it csmooth}\nin {\\it CIAO} (a translation of the {\\it asmooth} code by Harald\nEbeling).  This smoothed image emphasizes the soft diffuse structures,\nmapping red to the spectral range 500--700~eV, green to 700--1120~eV,\nand blue to 1120--2320~eV.  The reduced, full-field, full-band binned\ndata are shown in Figure~\\ref{fig:introimage}.  The event data were\nbinned into $8\\arcsec \\times 8\\arcsec$ pixels to create this image.\nSome well-known objects are labeled, as are the ACIS CCDs used in this\nobservation.  Areas of reduced exposure between the CCDs are clearly\nvisible, as is the extensive diffuse X-ray emission present in this\nfield.  Several famous structures are imaged far off-axis on the S3 and\nS4 CCDs.\n\n\n\\subsection{X-ray Morphology\n\\label{sec:globalmorph}}\n\nIn these images we see a bright concentration of hard X-ray events\nassociated with the R136 star cluster at the center of the ACIS-I array\nand fainter emission from several other well-studied stellar objects,\nmostly WR stars (see Figure 1 in Paper II).  The bright SNR N157B is a\ndistinct, extended feature to the southwest of the main 30~Dor\ncomplex.  It is the hardest source in the field due to non-thermal\nemission from its central pulsar and synchrotron nebula, but the larger\nSNR softens with distance from the pulsar.  A number of\nwidely-distributed compact X-ray sources are scattered across the\nfield; some were seen in previous X-ray studies\n(\\S\\ref{sec:background}).  The hard sources are probably mostly\nbackground AGN (see Paper II).\n\nBy far the dominant features, however, are the large diffuse structures\nassociated with the superbubbles produced by existing and past OB\nassociations and their supernova events.  These structures were\nwell-known from {\\em Einstein} and {\\em ROSAT} studies, as outlined in\n\\S\\ref{sec:background} \\citep[see especially the HRI data\nin][]{Wang99}, but  higher on-axis spatial resolution of {\\em Chandra}\nand the ACIS camera's intrinsic spectral capabilities reveal a new\nlevel of complexity.  The center of the field is complicated by\ncrossing ACIS-I chip gaps but clearly contains some of the hardest\nemission in the field, due to the R136 cluster, R140, and R145.  Faint,\nsofter diffuse emission appears to pervade the central region.  Just\nnorthwest of R140 (due west of R139) is a bright, medium-soft clump\n$\\sim 1\\arcmin$ in size.\n\nShell 1 in Figure~\\ref{fig:introimage} contains several point sources\nbut is dominated by a bright, medium-energy clump of diffuse emission\nroughly $1\\arcmin.5 \\times 2\\arcmin$ in size.  An ACIS-I chip gap\nbifurcates this feature, so the two-lobed structure it shows in\nFigure~\\ref{fig:csmoothimage} is largely an artifact \\citep[see Figure\n1 in][]{Wang99}.  Shell 2 is shown as a larger structure in\nFigure~\\ref{fig:introimage} than originally defined\n\\citep{Meaburn84,Wang91a} because the kinematic study of \\citet{Chu94}\nshowed that the expanding shell is larger than the morphological Shell\n2 identified in an H$\\alpha$ image \\citep{Meaburn84}.  It is\nedge-brightened in X-rays, with a large central void and a sharp\nboundary along its southeast edge.  There are small clumps of emission\naround its periphery, many of which are consistent with point sources.\nIts faint emission is harder than the other shell structures in the\n30~Dor nebula.  At its upper center is an interesting comet-shaped\nfeature showing substantial spectral complexity, with a soft point\nsource at its head.\n\nShell 3 is also largely edge-brightened, quite faint and soft around\nits periphery but with a harder, nearly-complete, clumpy central ring\nabout $2\\arcmin.5$ in diameter (hereafter called the ``West Ring'').\nThis ring structure also shows spectral complexity, appearing harder on\nits western half.  A faint, small clump is nearly centered on this\nring, but it is too large and diffuse to be a point source.  The $\\sim\n20$~Myr old stellar cluster Hodge~301 \\citep{Grebel00} sits at the\nnortheast edge of this ring, but we do not resolve any X-ray point\nsources in it.  Shell 3 encompasses two dark voids, one on either side\nof its central ring.\n\nShells 4 and 5 are both distinctly soft and bright, and differ from the\nother shells in that they appear center-filled rather than\nedge-brightened.  Shell 5, unfortunately, has a chip gap running all\nthe way through its long dimension, so some of the structure we see may\nbe affected by that.  Both it and Shell 4 are spectrally uniform and\nsimilar to each other.  Shell 5 is quite clumpy, with a bright lower\nclump about $1\\arcmin \\times 2\\arcmin$ in size containing the\nWR star R144 and a string of smaller, fainter clumps populating\nits interior.\n\nThe off-axis ACIS-S CCDs appear to be pervaded by soft, faint diffuse\nemission.  (Small point-like features around the edges of this field\nseen in Figure~\\ref{fig:csmoothimage} are smoothing artifacts.)  The\nmost striking feature here is the superbubble 30~Dor~C, seen as a\nnearly complete ring of clumpy emission spanning S3 and S4; its western\nhalf is very hard in X-rays due to strong non-thermal emission (see\n\\S\\ref{sec:background}).  SN1987A is the bright point source on S4; its\nlight is spread across a region more than an arcminute in size due to\n{\\em Chandra's} broad PSF at this large off-axis angle ($>20\\arcmin$).\nThe bright, soft feature at the lower center of S4 is the Honeycomb\nSNR; superposed on it appears to be a large ring of soft, diffuse\nemission that arcs to the north and east of the Honeycomb.  Such faint,\nfar off-axis features are more clearly seen in the {\\em XMM}\nfirst-light image \\citep{Dennerl01}.\n\n  \n\\subsection{Other Views\n\\label{sec:diffsrcs}}\n\nGiven the great variety of diffuse X-ray structures apparent in this\nobservation, it is worthwhile to find new ways to highlight them.\nFigure~\\ref{fig:diffuseintro} shows a color composite of three\nadaptively-smoothed images \\citep[from PSB's adaptive kernel smoothing\ntool {\\em adaptive\\_density\\_2d},][]{Townsley03} where the 180 point\nsources in the field have been removed and the resulting ``holes''\nsmoothed over.  The images were made in three soft bands, 350--700~eV,\n700--1100~eV, and 1100--2200~eV, similar to the bands used to create\nFigure~\\ref{fig:csmoothimage}.  They were combined to show the spectral\nvariety that must be present in the diffuse structures, even on spatial\nscales too small for us to extract and fit spectra to demonstrate this\nvariety; the intensities were set to emphasize faint diffuse features.\n\nThis smoothing technique has the advantage (over that used in\nFigure~\\ref{fig:csmoothimage}) that it does not create artifacts around\nthe edges of the field, although the overall appearance is more\nblurry.  We think that the best impression of the field comes from\nactively comparing these two renderings.  For example, the West Ring in\nShell 3 is much more clearly portrayed by Figure~\\ref{fig:csmoothimage},\nbut there are two small, faint loops ($< 2\\arcmin$ in diameter) on its\nnorth and east edges that are only visible in\nFigure~\\ref{fig:diffuseintro}.  We do not have enough data to know\nwhether or not these features are real.\n\nWith the point sources gone from Figure~\\ref{fig:diffuseintro}, we are\nfree to concentrate on the diffuse structures.  Here the outer loop in\nShell 3 is more pronounced, as are other faint diffuse features,\nespecially on the S-array CCDs.  Another intriguing small loop not\napparent in Figure~\\ref{fig:csmoothimage} is seen in the eastern half\nof Shell 2, visible as a green arc $\\sim 1.5\\arcmin$ across just above\na bright green knot in the sharp lower edge of Shell 2.  Again, only\nmore data will allow us to understand this feature in detail.\n\n\n\\subsection{Spectra\n\\label{sec:globalspectra}}\n\nFor comparison with more distant GEHRs, Figure~\\ref{fig:globalspectra}\nshows spectra of large-scale regions of X-ray emission on the ACIS-I\narray and {\\it XSPEC} fits.  Fit results are given in\nTable~\\ref{tbl:globalspectable}.  Column 1 lists the various composite\nregions on which spectral fits were performed; rows 1--4 correspond to\nthe fits shown in Figure~\\ref{fig:globalspectra}.  Column 2 gives the\nnumber of counts used in the fit.  Columns 3--8 give the spectral fit\nparameters, while columns 9--12 show the elemental abundances relative\nto solar values from the $\\it vapec$ thermal plasma model\n\\citep{Smith01} for elements that required $Z > 0.3Z_{\\odot}$.  Columns\n13--17 give the X-ray luminosities based on the spectral fits for a\nvariety of wavebands, with and without correction for absorption.\n\nColumn 8 shows that many of these fits are not formally acceptable (the\nreduced $\\chi^2 > 1.1$).  This is due mainly to the large number of\ndegrees of freedom, which is in turn due to the large number of events\nin these spectra; while in principle more events should allow the\nreduced $\\chi^2$ to approach unity as signal-to-noise improves, in\npractice systematic errors in the ARF and RMF become more important and\nthe reduced $\\chi^2$ remains high.  The fits in\nFigure~\\ref{fig:globalspectra} show that the spectral models used in\nTable~\\ref{tbl:globalspectable} characterize the data quite well in\nspite of the formally unacceptable fit; a more complicated model\n(such as an additional thermal plasma component) does not improve the fit.  \n\nFigure~\\ref{fig:globalspectra}a illustrates the global spectral fit for\nall components of the main 30~Dor nebula as outlined in\nFigure~\\ref{fig:introimage}, including point sources, the SNR N157B,\nits pulsar, and its cometary nebula.  The spectrum is fit with a single\nabsorbing column via the {\\it wabs} model \\citep{Morrison83}, which\nassumes solar abundances for the absorbing material, and a spectral\nmodel consisting of two components:  a variable-abundance thermal\nplasma and a power law.  The average absorption is a factor of 2\nsmaller than the {\\em ROSAT} result \\citep{Norci95}, although the\nplasma temperature is similar.  \\citet{Norci95} do not state what\nabundances they assume for the absorbing material and \\citet{Mignani05}\nnote that abundances of $0.4Z_{\\odot}$ give $N_H$ values a factor of\ntwo higher than those for solar abundances when fitting the spectrum of\nPSR~J0537$-$6910, so perhaps different assumptions regarding the\nmetallicity of the intervening material account for this discrepancy in\nthe average absorption.  The ACIS soft-band intrinsic\n(absorption-corrected) luminosity for the main 30~Dor nebula is a\nfactor of 2.5--5 smaller than the {\\em ROSAT} 0.1--2.4~keV estimate\n\\citep{Norci95}. \n\nFigure~\\ref{fig:globalspectra}b covers the same region of X-ray\nemission as Figure~\\ref{fig:globalspectra}a minus the N157B SNR (a\ncomposite spectrum of the SNR is shown in \\S\\ref{sec:n157bspectra}).\nSince N157B is quite bright and contains substantial non-thermal\nemission that would not necessarily be present in a more distant GEHR\nunless it contained a recent supernova, removing it from the overall\nspectrum of 30~Dor might yield a more typical GEHR spectrum.  The {\\it\nwabs*(vapec + powerlaw)} fit is actually quite similar to that in\nFigure~\\ref{fig:globalspectra}a, with nearly the same $N_H$, $kT$, and\nplasma abundances as that fit, but a steeper power law slope.  This\nshows that, although the non-thermal N157B pulsar and cometary nebula\nare bright compared to other point sources in the field, the overall\nspectral shape of the GEHR is still dominated by a soft thermal plasma\nwith fairly simple spatially-averaged spectral properties---no\nprominent lines are present and there is no obvious need for multiple\nplasma components or absorbing columns to fit the composite spectrum.\nThe only indication of the presence of the pulsar is at high energies,\nwhere it flattens the power law component of the fit.\n\nThe same extraction region is used for Figure~\\ref{fig:globalspectra}c\nas was used in Figure~\\ref{fig:globalspectra}b, but now the 158 point\nsources that this region contains have been removed from the spectrum.\nThe composite spectrum of those point sources is shown in\nFigure~\\ref{fig:globalspectra}d.  With only the diffuse emission\ncomponent remaining in Figure~\\ref{fig:globalspectra}c, the power law\nis no longer needed in the spectral model; rather the emission can be\nrepresented by a single, slightly hotter thermal plasma with enhanced\nabundances.  The composite point source spectrum in\nFigure~\\ref{fig:globalspectra}d requires a hotter thermal plasma and a\nflatter power law; it constitutes less than 10\\% of the full-band\nemission in this region but dominates the hard-band (2--8~keV) emission\nby a factor of 3 over the diffuse component.\n\nThese fits suggest that unresolved GEHRs in more distant galaxies might\nappear as soft, moderately luminous ($L_X \\sim 10^{36}$~ergs~s$^{-1}$)\nX-ray sources, quite distinct from harder and often brighter ($L_X \\sim\n10^{37}$--$10^{40}$~ergs~s$^{-1}$) X-ray binaries.  Distinguishing\nGEHRs from individual SNRs, though, is probably only possible for young\nSNRs, which might present hotter thermal plasmas or occasionally\nnon-thermal emission and may have prominent line features.  Even though\nthe N157B SNR is quite young, it does not show prominent lines and,\nwhen averaged with the emission from the rest of 30~Dor, it is not\nbright enough to harden the composite spectrum.  If its non-thermal\ncentral components were absent, it would not be spectrally distinct\nfrom the other diffuse structures in 30~Dor.  This is illustrated in\nthe spectral fitting presented in \\S\\ref{sec:superbubbles} and\n\\S\\ref{sec:N157B} below.\n\nDespite its complex spatial morphology, 30~Dor's integrated thermal\nemission has simple spectral properties:  it is well-fit by a\nsingle-temperature plasma with $T \\simeq 4$~MK, roughly solar line\nstrengths, and a single absorbing column density.  We\nwill see in \\S\\ref{sec:superspectra} that a considerable variety of\nplasma temperatures and absorbing columns are present on small spatial\nscales.  In the ACIS soft band (0.5--2~keV), the X-ray luminosity comes\nprimarily from the diffuse structures, which are brighter by an order\nof magnitude than the point sources.  Above 2~keV, though, the diffuse\nemission disappears and the integrated emission is dominated by the\npoint sources.  Paper~II demonstrates that this point source emission\nis itself dominated by a few bright sources; one third of the counts\nthat compose the spectrum in Figure~\\ref{fig:globalspectra}d come from\na single source \\citep[probably a colliding-wind binary,][]{PPL02}.\n\n\n\n\\section{THE X-RAY SUPERBUBBLES\n\\label{sec:superbubbles}}\n\n\\subsection{Morphology\n\\label{sec:supermorph}}\n\nAs outlined in \\S\\ref{sec:globalmorph}, the five plasma-filled\nsuperbubbles first defined by \\citet{Wang91a} from {\\em Einstein}\nobservations of 30~Dor appear to be much more complicated structures in\nthese ACIS images, exhibiting knots, whisps, loops, and voids with a\nrange of surface brightnesses and X-ray colors.  In some cases the\nX-ray morphology may be affected by the clumpy molecular clouds in the\nregion \\citep{Johansson98}, the shreds of the material from which R136\nformed.  This neutral material may shadow soft X-rays; comparing CO\ncloud locations from \\citet{Johansson98} to our smoothed images, this\neffect may account for some of the X-ray voids close to R136, although\nprobably not the larger, more distant voids seen in Shells 2 and 3.\n\nSome X-ray regions show distinctive sharp edges, such as the long\nnorth-central bubble comprising Shell 5 and the long arc stretching\nacross the southeast edge of Shell 2.  Other regions such as the large\nloop cut off by the northwest edge of the ACIS-I array have softer\nedges that seem to blend into faint, larger-scale diffuse emission.\nThe two bubbles of Shells 4 and 5 running almost due north and south of\nR136 appear quite red in Figures~\\ref{fig:csmoothimage} and\n\\ref{fig:diffuseintro}, perhaps implying that they are less absorbed\nthan other parts of the field or are filled with cooler gas.  They are\nalso distinctly center-filled, while other superbubbles appear as loops\nwith central voids.  The ACIS-S structures 30~Dor~C and the faint loop\nthat arcs to the northeast of the Honeycomb SNR are good examples of\nsuch loops, but the main 30~Dor complex shows them as well.\n\nIn the hot thermal plasma models that we use for spectral fitting of\nthese data (see \\S\\ref{sec:globalspectra}), most of the emission\nbetween 0.5 and 2~keV comes from a large number of blended spectral\nlines, with only a small remainder caused by thermal bremsstrahlung.\nIn order to understand the morphology of 30~Dor's superbubbles and its\nenergy dependence, Figure~\\ref{fig:lineims}\\footnote{These images\n(point sources removed) were made with the {\\it adaptive\\_density\\_2d}\nprocedure in {\\it ACIS Extract} using the smoothing scales from a\nbroadband (350--2000~eV) image, so that all narrow-band images have the\nsame smoothing scales.  To account for the fact that each narrow-band\nimage was made from a different number of events, each image was\nnormalized by its median intensity, then all were scaled the same in\nthe display.} shows narrow-band smoothed images of the full ACIS field\ncentered on some of the most prominent spectral lines in the thermal\nplasma models, specifically those often seen in SNRs, following {\\it\nChandra} studies of Magellanic Cloud diffuse nebulae by \\citet{Behar01}\nand \\citet{Naze02}.  The continuum emission dominates the\nhighest-energy image ($\\sim$65\\%) but contributes only $\\sim$10--30\\%\nto the other narrow-band images.\n\nWhile small-scale changes between narrow-band images should be viewed\nwith caution due to limited photon statistics, it is clear that all of\nthese images hint at the presence of complex fine spatial structure and\nthat the morphology of large-scale structures changes substantially\nwith energy.  A good example of this is 30~Dor~C, which seems to emerge\nout of faint, unorganized swaths of soft emission to coalesce into the\nwell-defined hard loop that we see in broad-band images.  The large,\nsoft superbubble at the south-center of the ACIS-I array (Shell 4) and\nthe loop containing the Honeycomb SNR are prominent in the softer\nimages, but fade above 950~eV.  Conversely, the sharp southeast ridge\nseen on CCD I1 (edge of Shell 2) becomes more prominent at higher\nenergies and the void to its northwest appears to be filling in, with\nno void left in the 1780--1940~eV image.  Smaller-scale knots and\nwhisps near the center and west-center of 30~Dor also come and go with\nenergy, with some prominent in the softer panels and others bright at\nhigher energies.\n\nThese images show that substantial spectral variation exists in the\n30~Dor superbubbles but do not by themselves reveal whether the\nvariations are produced by spatial changes in absorption columns,\nplasma temperatures, abundances, or other phenomena.  To investigate\nthis question, we divide the diffuse emission on the ACIS-I array as\nshown in Figure~\\ref{fig:diffregions}.  These regions were chosen\nprimarily on the basis of apparent surface brightness of the diffuse\nemission and overall morphology of the 30~Dor complex, so they form a\nphenomenological rather than a physical parameterization of the\nX-ray emission.  We emphasize that they were not chosen by considering\nthe morphology of 30~Dor in any other waveband; this will become relevant\nlater in this paper.  \n\nThe three large green polygons labeled ``B'' are the regions used to\nobtain the background spectrum used in all spectral fitting of diffuse\nregions described in this paper.  From {\\em XMM} data of this region\n\\citep{Dennerl01} and {\\em ROSAT} maps of the LMC\n\\citep[e.g.][]{Points01,Sasaki02}, we know that these regions contain\nfaint diffuse emission, but this may be the most appropriate background\nto use, as the whole field may well be pervaded by such faint diffuse\nemission.\n\nThe regions in Figure~\\ref{fig:diffregions} are best understood by\nconsidering them hierarchically by size, starting with the outermost,\nlargest regions and working inward and to smaller sizes.  The large red\ncontour outlines the region used earlier for the global spectral\nproperties of 30~Dor.  The six white regions define large-scale diffuse\nstructures, including the N157B SNR.  Region 1 encompasses the\nbrightest emission from the whole nebula, including regions 3, 12, 16,\n24, n8, and the central polygon that these regions abut.  Region 2 is\nthe perimeter between region 1 and the outer red contour, also\nexcluding region 18.  The five purple contours define ``voids,'' or\nareas of low apparent surface brightness.  Two central green regions (9\nand 10) outline relatively bright central structures.  Contained within\nthese large-scale regions are smaller regions defining locally bright\nareas, in blue, yellow, cyan, and red.  A cyan contour in the far\nnortheastern corner (region 23) defines an area of faint diffuse\nemission known from {\\em ROSAT} observations \\citep{Points01}.  A\nseries of annular regions (n0--n7) is used to explore the spectral\nproperties of the N157B SNR; \\S\\ref{sec:N157B} gives more detail.\n\nAfter choosing these extraction regions based on the X-ray apparent\nsurface brightness in our {\\em Chandra} images, we compared them to the\n$\\sim 100$~ks {\\em XMM} first light image in \\citet{Dennerl01}, which\ncontains the western half of the main 30~Dor nebula.  This image\nclosely matches the {\\em Chandra} data, showing very similar diffuse\nX-ray structures, and justifies our choice of extraction regions even\nfor faint features such as the outer loop in Shell 3 that we delineate\nas region 18, soft features such as regions 3 and 4 that fill Shell 4,\nand the X-ray voids in regions 19, 20, and 21.\n\n\n\\subsection{Spectra\n\\label{sec:superspectra}}\n\nThe spectral variety in these diffuse regions is shown in\nTable~\\ref{tbl:diffspectable}.  Column 1 gives the identifier of the\nregions shown in Figure \\ref{fig:diffregions}.  Column 2 gives the net\nfull-band (0.5--8~keV) counts in each diffuse region.  Spectral fit\nparameters, allowing for both {\\em vapec} thermal plasma and power law\nfits, are given in Columns 3--7.  Column 8 gives the reduced $\\chi^2$\nvalue for the fit.  All elemental abundances were allowed to vary;\nColumns 9-12 show abundances relative to solar that exceeded the\nnominal value $0.3Z_{\\odot}$.  These abundances were not always\nwell-constrained in the fits; in cases where the fit was improved with\nhigher abundances but actual values were not well-determined, they are\nlisted as ``hi'' in the table.  Derived X-ray luminosities are given in\nColumns 13-16; since the diffuse emission is quite soft in most\nextended regions (apart from the center of N157B), we report the\nintrinsic (absorption-corrected) soft-band luminosity as well as the\nusual intrinsic full-band luminosity.  The area subtended by each\ndiffuse region is given in Column 17 and the table is completed by a\nlist of intrinsic soft-band surface brightnesses in Column 18.\n\nUsing methods similar to those developed by Jeremy Sanders for study of\nextragalactic plasmas \\citep[e.g.][]{Sanders05}, we have used thermal\nplasma fits and the regions defined in Figure~\\ref{fig:diffregions} to\nproduce maps of $N_H$, $kT$, and intrinsic X-ray surface brightness in\n30~Dor, as shown in Figure~\\ref{fig:nhktmaps}.  For uniformity, the\nspectral fits used to make these maps were single-temperature thermal\nplasmas, so the results are not necessarily the same as those presented\nin Table~\\ref{tbl:diffspectable}, where power law components were added\nfor some regions.  Characteristics of the largest regions are recorded\nfirst and are overlaid by those of smaller regions.  This approach has\nthe misleading effect of leaving apparent ``rings'' around void\nregions (e.g.\\ regions 15 and 19) where the underlying large-scale fit\nis not masked by an overlying smaller region.\n\nAlthough crude, these maps are useful for understanding the physical\nstate of the plasmas that make up the 30~Dor diffuse emission.  For\nexample, region 4 is not intrinsically more luminous than the\nsurrounding region 3, it is just less absorbed and, since it has the\nsame temperature as region 3, its apparent surface brightness is\nlarger.  The same explanation applies to region 8.  Region 11 is\nparticularly bright, even though it has the same plasma temperature as\nits surrounding region 10, because it is slightly less absorbed plus it\nhas intrinsically higher surface brightness.  Regions 25 and 26 suffer\nmore absorption than the surrounding region 24 and have cooler plasma\ntemperatures, but they stand out because they have higher intrinsic\nsurface brightness.  Regions 18 and 19 suffer similar high obscuration\nand have the same cool plasma; region 19 appears as a void because it\nis intrinsically fainter than its surrounding region 18, which is only\nvisible to us in regular images because it has relatively high\nintrinsic surface brightness.  Region 22 has similar obscuration and\nplasma temperature, but appears as a void because it lacks this\nenhanced intrinsic brightness.\n\nParticularly unusual is region 15, which has high obscuration and low\nintrinsic surface brightness so it appears to us as a void, but it has\nthe highest plasma temperature in the nebula.  Region 5 is also\ninteresting, exhibiting a low temperature and relatively high\nobscuration, but it is very bright in regular images due to its\nintrinsic brightness, one of the largest in the nebula.  Of even higher\nintrinsic brightness, though, is region 7, yet it is much less\nnoticeable because of its large obscuration.  The intrinsically faint\nand soft emission in region 23, probably associated with more eastern\nstructures rather than with 30~Dor itself, is visible only because of\nits minimal obscuration.\n\nIn order to get enough counts for reliable spectral fitting, the\nregions used here average over many tens of square parsecs.  From the\ncomplex energy variations seen in Figures~\\ref{fig:csmoothimage},\n\\ref{fig:diffuseintro}, and \\ref{fig:lineims}, it is quite possible\nthat each region averages over many distinct plasma components, with\ndifferent pressures, densities, temperatures, absorbing columns, and\npossibly even different abundances.  Study of such features and their\nenergetics will be possible when longer ACIS exposures of the region\nare made.\n\n\n\n\\section{THE SUPERNOVA REMNANT N157B AND PSR~J0537$-$6910\n\\label{sec:N157B}}\n\n\\subsection{Morphology\n\\label{sec:n157bmorph}}\n\nThe {\\em Chandra}\/HRC \\citep{Wang01} and ACIS images of the composite\nSNR N157B are incremental improvements to the {\\em ROSAT} data, which\nalso showed a comet-shaped nebula embedded in a diffuse remnant,\nsuggesting that the neutron star received a kick in the supernova blast\nand is plowing through the surrounding ISM \\citep{Wang98a}.  Smoothed,\nsoft-band and hard-band ACIS images of N157B are shown in\nFigure~\\ref{fig:N157B-regs}.  The soft-band image\n(Figure~\\ref{fig:N157B-regs}a) shows the point source extraction\nregions outlined in red; the hard-band image\n(Figure~\\ref{fig:N157B-regs}b) additionally shows the concentric\ndiffuse extraction regions n7--n0 as defined in\nFigure~\\ref{fig:diffregions} and  Table~\\ref{tbl:diffspectable}.  The\nextraction region for the pulsar is shown in white.\n\nThe ACIS soft-band smoothed image shows diffuse emission from N157B\nextending somewhat farther than that shown in the HRC data\n\\citep[][Figure 1]{Wang01}, giving dimensions for the X-ray SNR of\n$\\sim 3\\arcmin.3 \\times 3\\arcmin.5$, or roughly 50~pc in diameter.  The\nsoft X-ray emission might be partially shadowed along the southern edge\nof the SNR by the dark cloud described in \\citet{Chu92}.  The hard\nX-ray emission in Figure~\\ref{fig:N157B-regs}b, which would not be\nshadowed because it penetrates such material, does not appear to\nextend into this region, though.  Thus it does not appear that the\nSNR extends behind the southern dark cloud.   \n  \nFigure~\\ref{fig:imageN157B} shows ACIS images of the embedded cometary\nnebula and pulsar.  The SNR is imaged $6\\arcmin.9$ off-axis, where the\nPSF is $\\sim 5\\arcsec \\times 10\\arcsec$ in size (shown in white in\nFigure~\\ref{fig:imageN157B}a).  The central source is still quite\ncentrally peaked and sharp enough that photon pile-up affects its\nspectrum.  Since it is immersed in its pulsar wind nebula and may have\nfurther corruption from the surrounding cometary nebula and SNR, our\nestimate of the pulsar's power law slope ($\\Gamma = 2.0$) should be\nconsidered qualitative.  \\citet{Mignani05} estimate $\\Gamma = 1.8$ for\nthe pulsar using an ACIS subarray observation that minimizes photon\npile-up for the pulsar and images it on-axis, so the small PSF limits\nthe contamination from diffuse structures around the pulsar.\n\nFigure~\\ref{fig:imageN157B}b shows the maximum likelihood\nreconstruction of Figure~\\ref{fig:imageN157B}a, using the PSF of the\npulsar for the reconstruction of the whole field. We recover an image\nvery similar to the on-axis {\\em Chandra}\/HRC images in\n\\citet{Wang01}:  the pulsar is a circular point source centered on its\npulsar wind nebula and a bright, $\\sim 4\\arcsec \\times 7\\arcsec$ region\nof emission is oriented perpendicular to the larger, trailing cometary\nnebula.  Although this image is not useful for spectral analysis, it is\nreassuring that even a simple reconstruction algorithm can recover\ninformation far off-axis, for sufficiently bright sources.\n\n\n\\subsection{Spectra\n\\label{sec:n157bspectra}}\n\nFigure~\\ref{fig:n157bspectra}a shows the spectral fit for the entire\nN157B SNR (labeled in Figure~\\ref{fig:introimage} and seen as region n8\nin Figure~\\ref{fig:diffregions}), including its pulsar (suffering from\nmoderate photon pile-up with $\\sim 0.9$ counts per frame), its pulsar\nwind nebula, and the surrounding cometary nebula.  Fit parameters are\ngiven on the last line of Table~\\ref{tbl:globalspectable}.  Using the\nsame spectral model as for Figure~\\ref{fig:globalspectra}a and b, we\nobtain the same $N_H$ but a hotter thermal plasma.  Some elements\nrequire abundances $>0.3Z_{\\odot}$ but are not well-constrained.\n\nOur thermal plasma fit results are quite similar to the {\\em ROSAT}\nresults \\citep{Wang98a}, but the power law component is substantially\nflatter than the results given by other X-ray observations of N157B\n(see \\S\\ref{sec:n157bbkgd}).  Since N157B is a young SNR, we also\ncharacterized its spectrum with a variable-abundance non-equilibrium\nionization model, using the {\\it XSPEC} model {\\it vnei + powerlaw}.\nThe fit was essentially the same as the thermal plasma fit described\nabove, with $kT \\sim 0.9$~keV and $\\Gamma = 2.2$, but the fit\nparameters were not as well-constrained as those for the thermal plasma fit.\n\nRemoving all the point sources from this region leaves the diffuse\nemission associated with the SNR (Figure~\\ref{fig:n157bspectra}b).  Its\nspectral fit parameters are listed under region n8 in\nTable~\\ref{tbl:diffspectable}.  The cometary nebula associated with the\npulsar contributes the power law component of the spectrum.  The\nthermal emission averaged over the whole SNR is quite soft and possible\nemission lines of Ne and Mg are seen.  While our fit results are\nconsistent with the {\\em XMM} results for the thermal component and\noverabundances of Ne and Mg are consistent with {\\em XMM's} discovery\nof emission lines \\citep{Dennerl01}, our power law slope is slightly\nflatter than the $\\Gamma = 2.8$ result from {\\em XMM}\n\\citep{Dennerl01}, even though our fit excluded the pulsar.\n\\citet{Dennerl01} note that, if N157B is similar to the Crab Nebula, we\nwould expect a difference between the core power law slope and that for\nthe entire nebula of $\\sim 0.5$ due to synchrotron losses.  This is\nseen in our {\\em Chandra} data, confirming the analogy with the Crab.\n\nThe {\\em XMM} value for the absorption (assuming an abundance of\n$0.5Z_{\\odot}$) was $N_H = 1.9 \\times 10^{22}$~cm$^{-2}$, more than a\nfactor of 6 higher than our result; the highest absorption we see for\nany part of N157B is at its core, where our $N_H$ is still a factor of\n3 lower than the {\\em XMM} value.  As discussed above, our assumption\nof solar abundance for the absorption can account for a factor of $\\sim\n2$ discrepancy in the measured column, but the cause of the remaining\ndifference in $N_H$ estimates is unknown.  One possibility might be\ncalibration uncertainties in the early {\\em XMM} results or \ncross-calibration issues between the two observatories.  \n\nTo study the SNR on finer spatial scales, spectral fits to the annular\nregions n0--n7 defined in Figure~\\ref{fig:N157B-regs} were performed\nand are given in Table~\\ref{tbl:diffspectable}.  Our spectral fitting\nconfirms the results of \\citet{Dennerl01} and \\citet{Wang98a} that\nN157B is a composite nebula, with a thermal plasma showing hints of\nline emission in the outer regions giving over to synchrotron emission\nin the bright core.  The spectral model in these fits allowed for both\na thermal plasma and a power law component, reverting to a single\ncomponent when an adequate fit could be obtained with just the thermal\nplasma or the power law model.\n\nWe have included two point source extraction regions in this table and\nshow their spectra in Figure~\\ref{fig:n157bspectra}:  p1 contains the\npulsar (CXOU~J053747.41$-$691019.8), p2 (CXOU~J053745.61$-$691011.1) is the\npointlike component of the cometary nebula found by {\\em wavdetect}.\nThese sources are included because their large off-axis PSFs make it\ndifficult to isolate the point source emission from the surrounding\nbright diffuse emission, so these point sources include diffuse\nspectral components.  In fact it is not clear in the ACIS subarray\nobservation of N157B that p2 is a point source at all; it is most\nlikely another example of concentrated diffuse emission that is\nconsistent with the ACIS PSF at this large off-axis angle.  The\nspectral fits in Table~\\ref{tbl:diffspectable} are not identical to\nthose in Paper II because the former were performed by hand, while the\nlatter were performed automatically in {\\em ACIS Extract}.  The results\nare consistent to within errors. \n\nComparing the spectrum of p1 (core, Figure~\\ref{fig:n157bspectra}c) to\np2 (cometary nebula bright point, Figure~\\ref{fig:n157bspectra}d) and\nthe surrounding bright cometary nebula (n0, the smallest blue annulus\nin Figure~\\ref{fig:N157B-regs}b), we see that all three regions are\nadequately fit by a simple power law, but its slope steepens rapidly\naway from the pulsar.  As we progress through the larger concentric\nannular regions n1--n7, an additional thermal component is needed for\nan adequate fit and the power law component apparently becomes very\nsteep.  In the outermost annular region (n7), the power law component\nis no longer necessary.  The spectral fits of these annular regions are\nproblematic due to small number counts; there can be interplay between\nthe thermal plasma temperature, the power law slope, and the absorption\nthat can lead to incongruous results, as we see for region n6.  Here\nthe power law slope is quite flat, but its normalization is low.  The\ngeneral trend of these annular spectra, though, seems to indicate\nelectron cooling through the steepening power law slope, a transition\nfrom non-thermal to thermal spectra with distance from the pulsar, and\nhigher abundances in the SNR, possibly hinting at the presence of\nspectral lines.\n\n\n\n\\section{DIFFUSE STRUCTURES IN THE CONTEXT OF 30~DOR KINEMATICS\n\\label{sec:otherdiffuse}}\n\nFrom {\\em Einstein} and {\\em ROSAT} data, it has long been known that\nmany of the X-ray concentrations in the 30~Dor superbubbles are\nspatially associated with high-velocity optical emission line clouds\n\\citep[][hereafter CK94]{Chu94}.  These kinematic data are invaluable\nfor sorting out the many overlapping X-ray features seen in the {\\em\nChandra} data as well.  For example, the large arc that appears to be\nassociated with the Honeycomb SNR is actually not kinematically related\nto it, but kinematic data indicate that the Honeycomb itself is the\nresult of a cavity supernova explosion.  The part of it seen in X-rays\nis due to the collision of this cavity SNR with an intervening porous\ngas sheet that could be associated with its slow-moving giant shell\n\\citep{Chu95b,Redman99}.  \n  \nWe place our bright X-ray features in the context of CK94's echelle\nstudy in Table~\\ref{tbl:kinematics}.  Several prominent X-ray features\n(identified solely on the basis of their apparent X-ray surface\nbrightness in a smoothed ACIS image, see \\S\\ref{sec:supermorph}) show\nhigh velocities in the kinematic data:  our West Ring is adjacent to\nCK94's NW Loop; our region 5 is CK94's R139W; our region 6 is CK94's\nR136E.  This correspondence between high-velocity features and diffuse\nX-ray emission was noticed and explained by CK94 as X-rays produced in\nthe shocks between high-velocity material and the surrounding\nslow-moving shells.  The {\\em Chandra} data allow us to explore this\ngeneral idea in more detail, with spatial resolution  better matched to\nthat of CK94's echelle data.\n\n\n\\subsection{X-rays from High-velocity Features\n\\label{sec:highvel}}\n\nThe {\\em Chandra} data show a more complex spatial distribution of\nX-ray plasmas than that described by \\citet{Wang99}, who inferred from\n{\\em ASCA} data that the diffuse emission was hot in the core of the\nnebula and cooler in its outer regions.  This complex distribution more\nclosely matches the kinematic portrait of the nebula built by CK94,\nthough.  Many of the bright X-ray regions that match high-velocity\nfeatures listed in Table~\\ref{tbl:kinematics} show cooler plasma\ntemperatures than their surroundings (Figure~\\ref{fig:nhktmaps}b).  We\ninterpret this as evidence that these regions are denser due to the\nhigh-velocity shocks, hence they are able to cool faster than the \nsurrounding superbubbles.\n\nThe high velocities measured near this X-ray-emitting material are more\nconsistent with CK94's interpretation of them as supernova shocks\nrather than as regions of high mass-loading, as proposed by\n\\citet{Wang99}, although our data do not exclude mass loading as a\nsecondary mechanism for increasing X-ray luminosity in some regions.\nWe would expect this mechanism to be most important in regions where\nsubstantial neutral material is known to exist, thus mass loading may\ncontribute to the bright diffuse X-rays seen near the center of the\nnebula, which contains the remains of the GMC.\n\nOur bright, soft region 5 is coaligned with CK94's fast shell R139W,\nwhich shows blueshifted emission with velocities up to\n150~km~s$^{-1}$.  The region of bright X-ray emission is $\\sim 15$~pc\nin diameter and fills a prominent, well-known hole in the H$\\alpha$\nemission that also appears in the {\\em Spitzer} data (see\n\\S\\ref{sec:hotcold}).  Nearby is another bright, soft X-ray patch\n(region 6) also $\\sim 15$~pc in diameter that is well-matched to CK94's\nfast shell R136E.  This region exhibits both redshifted and blueshifted\nemission with speeds in excess of 130~km~s$^{-1}$.  The X-ray emission\nis centered on the WR star R145, but CK94 note that it is unlikely to\nbe caused solely by the winds from that one star because the energy\nrequirements to support the X-ray emission and expansion velocities\nseen would require at least 5 average WR stars.  Both X-ray regions 5\nand 6 are cooler and intrinsically brighter than their surrounding\nregion 9 (see Figure~\\ref{fig:nhktmaps}).  They may well be parts of\nthe bubble blown by R136 that have been brightened by off-center SNe\n\\citep{Chu90}.\n\nWe find that not every high-velocity feature shows bright X-ray\nemission.  A good example is region 15, which coincides with a 20~pc\nshell expanding at 110~km~s$^{-1}$ described by CK94.\nFigure~\\ref{fig:nhktmaps} shows that this region is intrinsically X-ray\nfaint and exhibits the hottest plasma in the 30~Dor nebula.  Perhaps\nthis is the site of a recent cavity supernova that occurred far enough\nfrom the edge of Shell 2 that it did not produce an X-ray-bright shock\nbut instead deposited its energy into heating the surrounding plasma\n\\citep{Chu90}.  A more detailed velocity study of this region is\nwarranted to elucidate the cause of this unusual X-ray feature.\n\n\n\\subsection{Is the West Ring a Cavity Supernova Remnant?\n\\label{sec:westring}}\n\nA prominent feature in all X-ray images featured in this paper is a\nnearly-complete, clumpy ring structure in the middle of Shell 3, as\ndescribed in \\S\\ref{sec:globalmorph}.  We called this structure the\n``West Ring'' and find that it sits $\\sim 1\\arcmin$ west of CK94's\nhigh-velocity ``NW Loop,'' which shows expansion velocities of\n200~km~s$^{-1}$.  CK94 note that such high velocities are never found\nin wind-blown bubbles.  In Figure~\\ref{fig:nhktmaps}, the West Ring is\ncontained in region 17, which is indistinguishable from its surrounding\nregion 16 in obscuration and temperature but has slightly higher\nintrinsic surface brightness.  \n\nThe West Ring could be a SNR:  \\citet{Meaburn88} performed an echelle\nstudy of this region and concluded that the large expansion velocities\nmake a supernova origin the most likely explanation.  It resembles the\n``shell'' SNRs catalogued in \\citet{Williams99}, both in size and\nstructure.  Such a ring structure is predicted by \\citet{Velazquez03}\nwhen a SNR hits its cluster's stellar winds.  Recent echelle\nspectroscopy concentrating on this region also finds knots and shells\nconsistent with a supernova interpretation \\citep{Redman03}.\n\nThis structure is also very similar to the central X-ray ring\n\\citep{Leahy85} in the over-sized Galactic SNR HB3 (G132.7+1.3),\nconsidered to be a cavity SNR \\citep{Routledge91}.  The West Ring is\nabout $2\\arcmin.5$ in diameter, or $\\sim 36$~pc.  The X-ray ring in HB3\nis about $35\\arcmin$ in diameter, which at $D = 2.3$~kpc corresponds to\n$\\sim 23$~pc.  HB3's X-ray ring is also very clumpy and not quite\ncomplete \\citep{Landecker87}; it is strikingly similar in appearance to\nthe West Ring.  From Table~\\ref{tbl:diffspectable}, diffuse region 17\nhas 0.5--8~keV intrinsic luminosity $L_{X,corr} = 4.7 \\times\n10^{35}$~ergs~s$^{-1}$ and $kT = 0.5$~keV.  The {\\em Einstein}\n(0.2--4~keV) X-ray luminosity of HB3 totals $1.6 \\times\n10^{35}$~ergs~s$^{-1}$ and shows a hot central region ($kT \\sim 1$~keV)\nevolving to a cooler limb ($kT \\sim 0.3$~keV) \\citep{Leahy85}.  Our\nspectral fit to region 17 shows solar abundances for O and Ne; a recent\nanalysis of a short {\\em XMM} observation of HB3 shows the possibility\nof enhanced abundances of O, Ne, and Mg in its X-ray ring\n\\citep{Lazendic05}.  Just as in the West Ring, HB3 lacks strong X-ray\nemission lines.\n\nExamining the visual and radio images from \\citet{Dickel94}, it is\nclear that there is no prominent visual or radio continuum source at\nthe location of the West Ring.  As noted by \\citet{Chu00} in their\nstudy of the large LMC X-ray ring RX~J050736-6847.8, though, SNRs in\nlow-density media are not expected to show prominent visual or radio\nfeatures.  Given the similarity of the West Ring to HB3 and other LMC\nSNRs, we suspect that it is in fact a cavity SNR, perhaps produced by a\nstar in the Hodge~301 cluster.\n\nAlthough more detailed comparisons of the data from CK94 and other\nkinematic studies to the diffuse X-ray structures revealed by {\\em\nChandra} are beyond the scope of this paper, CK94 noted that such\ncomparisons likely hold the key to understanding the complex ISM in\n30~Dor.  We will search for the parsec-scale high-velocity knots seen\nby CK94 and \\citet{Meaburn84} in the longer {\\em Chandra} observation.\nWe suspect that this dataset will reveal small-scale diffuse X-ray\nfeatures that will merit new kinematic studies as well.\n\n\n\n\\section{DISCUSSION\n\\label{sec:discussion}}\n\n\\subsection{The Relationship between Hot, Warm, and Cool Interstellar \nMaterial in 30~Dor\n\\label{sec:hotcold}}\n\nThe morphology of the hot X-ray emitting plasma is extremely\ncomplicated and bears little resemblance to theoretical calculations of\nindividual SNRs or superbubbles, which generally predict\nquasi-spherical structures.  Insight into the origins of these\nstructures emerges from comparison with the distribution of\ninterstellar material traced by H$\\alpha$ emission from ionized gas and\nIR emission from dust.  The comparison with H$\\alpha$ has been made\nsince the earliest X-ray images of 30~Dor were obtained by {\\em\nEinstein} \\citep[e.g.][]{Chu90,Walborn91,Wang91a}.\n\nFigure~\\ref{fig:Halpha} shows a high-resolution H$\\alpha$ image from\nthe MCELS project \\citep{Smith00} in red, with adaptively-smoothed\nsoft-band ACIS images (point sources removed) in green and blue.  This\nimage is reminiscent of earlier combinations of H$\\alpha$ and X-ray\ndata that showed that diffuse soft X-ray emission was anticorrelated\nwith H$\\alpha$ emission, often filling the cavities outlined by ionized\ngas \\citep[e.g.][]{Wang99}.  These high-resolution images allow us to\nrefine that picture somewhat, giving more detailed information on the\nrelative locations of harder and softer X-rays and H$\\alpha$ emission.\n\nComparing Figure~\\ref{fig:Halpha} to a similar image based on {\\em\nROSAT}\/HRI data in \\citet[][Figure 3]{Wang99}, both images show that\nthe northern Shell 5 and southern Shell 2 are very clearly outlined by\nlarge-scale H$\\alpha$ structures.  X-rays from the northern part of SNR\nN157B coincide with a dark cavity in the H$\\alpha$ emission caused by a\ndust lane \\citep{Chu92}.  Conversely, the outer loop of Shell 3,\nlabeled as X-ray emitting region 18 in Figure~\\ref{fig:diffregions},\ncoincides with a similar loop in H$\\alpha$; in Figure~\\ref{fig:Halpha}\nwe can see that the northern part of this loop is softer than the\nsouthern part.  Both images show that a prominent H$\\alpha$ dark region\nin Shell 2 (southeast of R136) contains faint X-ray emission, while the\nbright H$\\alpha$ filaments that outline it bifurcate our X-ray void\nlabeled region 15.  Figure~\\ref{fig:Halpha} shows that Shell 2 X-rays\nare relatively hard and Figure~\\ref{fig:nhktmaps} reveals that this is\ndue to a combination of intrinsic spectral hardness and heavy\nobscuration in this part of 30~Dor.  The West Ring is now more clearly\nseen to be a clumpy ring of X-ray emission, fainter in its center, with\nsome soft X-ray emission coincident with the Hodge 301 cluster on the\nnortheast edge of the ring.  Small-scale clumps (such as our regions 7\nand 8) are now distinct from the smoother underlying X-ray emission.\nWith {\\em Chandra's} high on-axis spatial resolution, we have been able\nto excise the point source X-ray emission from Figure~\\ref{fig:Halpha},\nso it is clearer that bright diffuse regions 5 and 6 are distinct from\nthe R136 cluster and from the nearby WR stars R139, R140, and R145.\n\nThe 8$\\mu$m {\\em Spitzer} data (Figure~\\ref{fig:spitzer}) highlighting\nwarm dust also add insight into the diffuse X-ray structures.  Hot\nX-ray plasma fills the interiors of superbubbles that are outlined by\nwarm dust and emission from PAHs (Brandl et al.\\ 2006, in\npreparation).  The X-ray morphology and possible confinement are more\nfully appreciated when anchored by these IR data.  Heated dust provides\nan envelope to the base of the cylindrical X-ray Shell 5.  X-ray\nemission could be suppressed in this region or it could be present but\nabsorbed.  An IR-bright V-shaped ridge of emission (reminiscent of the\nCarina Nebula) separating Shell 3 from Shell 5 is devoid of observable\nsoft X-ray plasma, while the bright X-ray spot dominating Shell 1 is\nnearly devoid of IR emission.  The thermal plasma that constitutes the\nouter regions of the N157B SNR fills a large cavity in the warm dust.\nOne of the brightest clumps of diffuse X-rays (our region 5) fills a\ndistinct hole in both the H$\\alpha$ and IR emission.\n\nTo give a more complete view of the wide range of emission from 30~Dor,\nFigure~\\ref{fig:3bands} combines the 8$\\mu$m data tracing PAH emission\nand warm dust (red) from Figure~\\ref{fig:spitzer}, the H$\\alpha$ data\ntracing ionized $10^4$~K gas (green) from Figure~\\ref{fig:Halpha}, and\nthe 1120--2320~eV X-ray data tracing $10^7$~K plasma (blue) from\nFigure~\\ref{fig:csmoothimage}.  Note that here the X-ray image comes\nfrom the ACIS data processed with {\\em csmooth}, with the point sources\nleft in place, not the adaptively-smoothed images used in\nFigures~\\ref{fig:Halpha} and \\ref{fig:spitzer} where the point sources\nwere removed. \n\nFigure~\\ref{fig:3bands} shows the full ACIS-I field of view and is scaled\nto show the full extent of faint diffuse X-ray emission across the\nfield.  The zoomed image in Figure~\\ref{fig:3bandszoom} is scaled to\nshow just the brighter patches of X-ray emission in the center of the\nfield and removes the color saturation in the core.  Although the {\\em\nChandra} data are not deep enough to match the spatial resolution seen\nin the H$\\alpha$ and {\\em Spitzer} data, it is clear that the 30~Dor\ncomplex cannot be understood by visual and IR studies alone, no matter\nhow high the quality of those datasets.  The high-energy emission is a\nnear-perfect complement to the longer-wavelength emission, filling\ncavities in the complex that are outlined by H{\\sc II} regions.  These\nare in turn outlined by warm dust, because ultraviolet radiation in the\nH{\\sc II} regions destroys PAHs.  \n\nX-rays from the northern parts of SNR N157B fill a prominent hole in\nthe 8$\\mu$m emission.  The eastern side of the Shell 5 X-ray\nemission, which appeared unconfined by warm dust in\nFigure~\\ref{fig:spitzer}, is clearly defined by a large region of\nH$\\alpha$ emission, while its western side shows a narrower H{\\sc II}\nregion bordered by dust.  The southwestern side of the 30~Dor complex\nshows more warm dust than the northern or eastern sides.\n\nThere is a notable absence of X-ray emission in the southeastern corner\nof the image, where the stellar cluster SL~639\n\\citep{Shapley63,Melnick87,Bica99} is clearly seen in the H$\\alpha$ and\n8$\\mu$m data.  Some ionized gas is present around the cluster along\nwith substantial amounts of heated dust.  This cluster may be too young\nto have produced a substantial wind-blown bubble or any supernovae, or\nits ionizing stars may not generate enough wind power to blow a\nsubstantial bubble.  Conversely, if it is older than $\\sim$10~Myr it\nmay have dispersed its hot gas.  It will be an interesting site to\nsearch for faint diffuse X-rays in longer observations.\n\nThe famous central arcs north and west of R136 are bright at 8$\\mu$m as\nwell as in H$\\alpha$, illustrating the transition layers from cold\nmolecular material to heated dust to ionized gas that characterize much\nof the 30~Dor complex.  Figure~\\ref{fig:3bandszoom} shows that the\nbright X-ray emission lies in the cavities interior to these ionization\nfronts, but the saturated center of Figure~\\ref{fig:3bands} suggests\nthat some diffuse X-rays are seen superposed on the H$\\alpha$ and IR\nemission.  This is true in other parts of the image as well and implies\nthat the soft X-rays come from regions that lie in front of the denser\nIR-emitting material since they avoided being absorbed.  This is a\nreminder that the complex ISM in 30~Dor may be absorbing similar soft\nX-rays along other lines of sight.  Thus the regions exhibiting hot\nplasma may be connected via tunnels or fissures that are not visible to\nus in these images.  The regions where we do not see soft X-ray\nemission are not necessarily lacking in hot plasma, especially if it is\nclear from the {\\em Spitzer} data that substantial absorbing material\nlies along the line of sight.\n\nThese data thus augment and support the ideas developed over the last\ntwenty years for the evolution of the 30~Dor complex.  Powerful stellar\nwinds from extremely massive stars are carving holes in an extensive\nGMC and filling those holes with hot plasma.  Additional hot plasma is\nadded as those stars explode inside the cavities they created.  In this\nprocess, cold gas is pushed aside and confines the hot plasma into\nshapes otherwise difficult to understand.  Since this plasma appears to\nemit only soft X-rays for most of its lifetime, some of it may be\nabsorbed by intervening molecular material.  Tunnels and fissures could\nexist in the GMC that allow hot plasma to flow between apparently\nunconnected regions, although none of the hot plasma seen by {\\em\nChandra} appears completely unconfined (there are no empty bubbles that\nappear to have vented their X-ray plasma into the surrounding ISM).\nThe H$\\alpha$ emission traces the ionization fronts, H{\\sc II} region\n``sheets'' that mark the transition between the million-degree plasma\nand the neutral material.  This transition layer is heated, evaporated,\nand is accelerating away from the cloud interface.  Evolved stars are\ndistributed across the entire 30~Dor nebula, providing the fuel for the\nSNRs that occasionally brighten the superbubbles in X-rays and the\nenhanced density that allows the plasma to cool.  Neutral clumps may\nalso enhance the X-ray emission, especially in regions close to the\nremains of the GMC.\n\n\n\\subsection{Summary of Findings}\n\nDiffuse emission dominates the morphology of 30~Dor in soft X-rays,\ncaused by hot plasma filling the large superbubbles created by\ngenerations of star formation and subsequent SNe in this region.  This\nis dramatically illustrated by combining the {\\em Chandra} data with\nH$\\alpha$ and {\\em Spitzer} images\n(Figures~\\ref{fig:Halpha}--\\ref{fig:3bandszoom}); the $10^6$--$10^7$~K\nX-ray plasmas are enveloped and probably confined in most cases by the\ncooler gas and warm dust that define the classic picture of 30~Dor.\nThis example illustrates the need for high-quality X-ray observations\nof star-forming regions; new facets of both stellar and diffuse\ncomponents are revealed by high-energy data.\n\nThe combined spatial and spectral resolution afforded by ACIS shows\nthat these superbubbles are not uniformly filled with a\nsingle-temperature gas; great variety is seen, on a range of spatial\nscales, in absorption, plasma temperature, and intrinsic surface\nbrightness (Figure~\\ref{fig:nhktmaps}).  Some bubbles are\ncenter-filled, perhaps revealing interactions with cold gas left in\nshell interiors \\citep{Arthur96,Wang99}, while others are\nedge-brightened with distinct central voids not caused by obscuring\nforeground material.  The faintest regions (voids and the periphery of\nthe main 30~Dor nebula) are a factor of 20 fainter in intrinsic surface\nbrightness than the brightest regions, which have surface brightnesses\n$>1 \\times 10^{33}$~ergs~s$^{-1}$~pc$^{-2}$.  Our diffuse regions range\nin size from $1\\arcmin$ ($\\sim 14.5$~pc) for small surface brightness\nenhancements to $>7\\arcmin$ for the large shells.  For comparison with\nother GEHRs, we have provided integrated spectra and fits for the main\n30~Dor nebula and for the N157B SNR (Table~\\ref{tbl:globalspectable}).\n\nSpectral fits to the diffuse emission show moderate absorption ($N_H =\n1$--$6 \\times 10^{21}$~cm$^{-2}$) and soft thermal plasmas with $kT =\n0.3$--0.8~keV (3--9~MK).  Many diffuse regions exhibit elevated\nabundances (above the nominal value of $0.3Z_{\\odot}$), perhaps\nindicative of line emission, but none show prominent emission lines.\nAlthough not a definitive demonstration of a supernova origin for the\nX-ray emission, these spectral fit results are consistent with that\ninterpretation.  SNR N157B, also possibly the result of an explosion\nwithin a pre-existing cavity, also lacks strong emission lines, as does\nthe Galactic example of a possible cavity SNR discussed here, HB3.\n\nThe brightest point source in the field is PSR~J0537$-$6910 in the SNR\nN157B, imaged almost $7\\arcmin$ off-axis.  Although the large PSF there\nprevents us from performing accurate spectral analysis of the pulsar\nand its pulsar wind nebula, these data clearly show spectral changes as\na function of distance from the pulsar; the pure non-thermal spectrum\nsteepens and combines with a thermal component, finally giving over to\na pure thermal spectrum in the outer regions of the SNR\n(\\S\\ref{sec:N157B}).\n\nSeveral interesting smaller-scale structures emerge with the\nhigh-resolution {\\em Chandra} observations.  One feature (the ``West\nRing'') may be the relic signature of a past cavity supernova that\nexploded inside a pre-existing wind-blown bubble generated by the\nHodge~301 cluster.  As first shown by CK94, bright patches of X-ray\nemission often coincide with high-velocity features known from\nH$\\alpha$ echelle studies, indicating that X-rays are produced in\nshocks in 30~Dor's ISM (\\S\\ref{sec:otherdiffuse}).\n\nThe MCELS H$\\alpha$ image, the {\\em Spitzer} image, and the recent\nmosaic of {\\em HST} data \\citep{Walborn02} are all dominated by\nhighly-structured arcs and shells on 1--10~pc scales; in the central\nregion, these reveal the interfaces between the central cavity created\nby R136 and the surrounding molecular clouds \\citep{Scowen98} and\ndemonstrate how R136 is shredding its natal environment\n(\\S\\ref{sec:hotcold}).  Here and throughout the 30~Dor nebula, an\nappreciation of the X-ray emission is necessary to further our\nunderstanding of the processes working to shape this complex.  As the\nimages in \\S\\ref{sec:hotcold} show, it is imperative that the diffuse\nX-ray emission become part of the ``classic'' picture of 30~Dor.\n\nNew stellar clusters are now forming in the dense knots that remain in\n30~Dor's GMC; their collapse was probably triggered by R136\n\\citep{Walborn02}.  These clusters are not yet resolved in X-rays,\ndemonstrating that there is much more work to be done to achieve a\ncomplete picture of the X-ray emission from 30~Dor.\n\n  \n   \n\\subsection{30~Dor and Other Massive Star-forming Regions\n\\label{sec:othermsfrs}}\n\n30~Dor's diffuse X-ray emission is, overall, substantially different\nthan that seen in Galactic MSFRs that are too young to have produced\nSNe.  M~17 shows strong diffuse soft X-rays probably due to wind-wind\nand\/or wind-cloud collisions \\citep{Townsley03}, but the primary\ncomponent of this emission is hotter (kT = 0.6~keV) than many 30~Dor\nregions and has low surface brightness\n($10^{31.9}$~ergs~s$^{-1}$~pc$^{-2}$).  These quantities are most\nsimilar to the regions we call ``voids'' in 30~Dor (regions 15, 19, 20,\n21, and 22 in Figure~\\ref{fig:diffregions}).  The diffuse X-ray\nemission reported for the massive Galactic clusters NGC~3603\n\\citep{Moffat02} and Arches \\citep{Yusef-Zadeh02} has higher surface\nbrightness more comparable to the brighter regions in 30~Dor\n\\citep[$10^{32.6}$ and $10^{33.1}$~ergs~s$^{-1}$~pc$^{-2}$\nrespectively, calculated from Table 4 of][]{Townsley03} but the plasmas\nare much hotter (kT = 3.1~keV and 5.7~keV respectively).  Either a very\ndifferent mechanism is generating hard diffuse emission in these\nclusters \\citep[e.g. wind collisions,][]{Canto00}, young SNe dominate\nthe diffuse emission, or substantial unresolved point source emission\nis corrupting the measurements.\n\n\\citet{Chu93} proposed that the soft diffuse X-ray emission seen in the\nCarina star-forming complex \\citep{Seward82} could be due to a cavity\nSNR inside a superbubble blown by Carina's many massive stellar\nclusters; thus the Carina complex should be a Galactic analog to one of\nthe superbubbles seen in 30~Dor.  The entire Carina complex shows\nintegrated X-ray emission with $kT \\sim 0.8$~keV and surface brightness\n$\\geq 10^{32.2}$~ergs~s$^{-1}$~pc$^{-2}$ \\citep{Seward82,Townsley03}.\n{\\em Chandra} and {\\em XMM} resolve this X-ray emission into thousands\nof harder but comparatively faint stellar sources\n\\citep[e.g.][]{Albacete03,Evans03} and bright diffuse emission\npervading the complex yet not centered on the O or WR stars or the\nstellar clusters.  No obvious SNR is present, but the complex is old\nenough to have produced SNe, as evidenced by the presence of evolved\nstars.  This bright, soft diffuse emission is quite comparable in\nsurface brightness and extent to the smaller regions sampled in\n30~Dor.  Using new {\\em Chandra} data centered on the Trumpler~14 OB\ncluster in Carina, we show that the brightest regions of diffuse\nemission are well-separated from the massive stars in the Carina\ncomplex and show filamentary morphology, both consistent with a cavity\nsupernova origin \\citep{TownsleyIAUS227}.  Thus we suspect that Carina\nserves as a good microscope for understanding the processes powering\nthe X-ray emission from 30~Dor.\n\nGiven the morphological complexity that 30~Dor displays at all\nwavelengths, it is surprisingly similar to other GEHRs, most notably\nNGC~604 in M33, the second-largest GEHR in the Local Group.  Although\nnot dominated by a massive central star cluster, the large-scale loops\nand voids of NGC~604 are also characteristic of superbubbles and are\nfilled with hot gas emitting soft X-rays \\citep[][Brandl et al.\\ 2006,\nin preparation]{Maiz04}.  The second-largest GEHR in the LMC, N11, also\nexhibits a central massive stellar cluster surrounded by a superbubble\nand containing bright, soft diffuse X-rays \\citep{MacLow98,Naze04}.\nAlthough more distant GEHRs are not resolvable with current technology,\nwe should expect all GEHRs to be made up of a complex mix of pointlike\nand diffuse X-ray components with the hard-band X-ray luminosity\ndominated by massive stars and the soft-band X-ray luminosity dominated\nby the effects of recent cavity SNe expoding near the edges of\nsuperbubbles, as we see in 30~Dor.\n\n\n\\subsection{Concluding Comments}\n\nWe have analyzed an early {\\em Chandra}\/ACIS observation of 30~Doradus,\nconcentrating here on the diffuse X-ray structures and in Paper II on\nthe point sources.  This study documents a wide variety of diffuse\nX-ray-emitting sources:  a complex hierarchy of diffuse structures,\nfrom small-scale knots and wisps to huge superbubbles; a composite SNR\nincluding its pulsar, pulsar wind nebula, and cometary tail; an X-ray\nring possibly due to a cavity SNR; bright X-ray patches associated with\nhigh-velocity H$\\alpha$ structures.  A coherent understanding of the\nstructures is beginning to emerge from a multiwavelength comparison of\nthe X-ray, H$\\alpha$, and mid-IR maps.  We demonstrate a variety of\ndata analysis tools for study of ACIS fields with mixtures of pointlike\nsources and diffuse structures.  All software used in this study is\npublicly available.\n\nFrom this work, we conclude the following:\n\\begin{itemize}\n\\item  30~Dor's integrated diffuse emission is well-fit by a\nsingle-temperature plasma with $T \\simeq 7$~MK, roughly solar line\nstrengths, and a single absorbing column density of $N_H \\simeq 3\n\\times 10^{21}$~cm$^{-2}$.  The soft-band luminosity is dominated by the\ndiffuse structures, but these disappear above 2~keV and the integrated\nemission is dominated by just a few bright point sources\n(\\S\\ref{sec:globalspectra} and Paper~II).\n\\item  ACIS spectra of 30~Dor's diffuse emission regions often suggest\nchemical enrichment, another argument in favor of a SNR origin for the\nX-ray emission (Table~\\ref{tbl:diffspectable}).\n\\item  Annular spectra around the SNR N157B, centered on\nPSR~J0537$-$6910, indicate electron cooling through a steepening power\nlaw slope, a transition from non-thermal to thermal spectra with\ndistance from the pulsar, and chemical enrichment in the SNR\n(\\S\\ref{sec:N157B}).\n\\item  A powerful method for understanding the complex spectral and\nspatial information contained in high-resolution X-ray studies of\nmassive star-forming complexes like 30~Dor is to study maps of column\ndensity, plasma temperature, and intrinsic surface brightness derived\nfrom X-ray spectral fitting (Figure~\\ref{fig:nhktmaps}).\n\\item  The main 30~Dor nebula exhibits several X-ray ``voids'' not\ncaused by obscuration, most notably the hot region 15\n(\\S\\ref{sec:superspectra}).  These resemble the wind collision plasma\nseen in nearby H{\\sc II} regions like M~17 (\\S\\ref{sec:othermsfrs}).\n\\item  Comparing the diffuse X-ray emission with kinematic studies of\nthe warm gas in H{\\sc II} region complexes is essential to disentangle\nthe confusing morphological information.  While the fast shocks found\nin 30~Dor by CK94 often correlate well with regions of high X-ray\nsurface brightness, the correlation is not complete; some fast shells\nlack bright X-rays while some X-ray features are not known to exhibit\nhigh velocities (\\S\\ref{sec:highvel}).\n\\item Some shells do not currently contain massive clusters to supply\nwind-generated X-rays.  The source of diffuse X-rays may be centered\nSNe that heated the interior of these shells without producing fast\nshocks against the shell walls \\citep{Chu90}.\n\\item  The X-ray structure that we call the West Ring (our region 17)\nis adjacent to CK94's NW Loop and has X-ray properties very similar to\nthe Galactic cavity SNR HB3.  We propose that it is a cavity SNR that\nexploded inside 30~Dor's Shell 3 and that its projenitor most likely\ncame from the massive cluster Hodge~301 (\\S\\ref{sec:westring}).\n\\item  Bright X-ray patches often have cooler plasma temperatures than\ntheir surrounding shells, possibly indicating that the increased\ndensities associated with fast shocks allow the associated X-ray plasma\nto cool more efficiently than the plasma associated with the large,\nlow-density shells.\n\\end{itemize}\n\nOur upcoming 100~ks ACIS-I observation of 30~Dor will provide the\nhigh-quality X-ray dataset that is needed to place the X-ray emission\nin context with state-of-the-art observations in other wavebands.  We\nexpect this observation to reveal more of the high-mass stellar\npopulation in 30~Dor and to give even more detailed information on the\ncomplex morphology of the wind-blown bubbles and superbubbles, both\nwith higher spatial resolution imaging and with spectroscopy on finer\nspatial scales.  True progress in understanding, though, will require\ncooperation.  By combining the power of today's Great Observatories and\nhigh-quality ground-based data, we are confident that unique insight\nawaits.\n\n\n\\acknowledgments\n\nSupport for this work was provided to Gordon Garmire, the ACIS\nPrincipal Investigator, by the National Aeronautics and Space\nAdministration (NASA) through NASA Contract NAS8-38252 and {\\em\nChandra} Contract SV4-74018 issued by the {\\em Chandra X-ray\nObservatory} Center, which is operated by the Smithsonian Astrophysical\nObservatory for and on behalf of NASA under contract NAS8-03060.  LKT\nappreciates technical contributions by Jeremy Sanders and Konstantin\nGetman and helpful discussions with Mike Eracleous and Bruce\nElmegreen.  We thank the Magellanic Cloud Emission Line Survey (MCELS)\nteam for the use of their H$\\alpha$ image of 30~Doradus.  We also\nthank our anonymous referee for investing the time to review this\npaper and for several helpful suggestions.\n\nThis work is based in part on observations made with the {\\em Spitzer\nSpace Telescope}, which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology under NASA Contract 1407.  This\nresearch made use of data products from the Two Micron All Sky Survey,\nwhich is a joint project of the University of Massachusetts and the\nInfrared Processing and Analysis Center\/California Institute of\nTechnology, funded by NASA and the National Science Foundation.  This\nresearch also made use of the SIMBAD database and VizieR catalogue\naccess tool, operated at CDS, Strasbourg, France.  We are grateful for\nthe invaluable tools of NASA's Astrophysics Data System.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Acknowledgements}\nThe authors thank the Amazon Mechanical Turk workers that participated in the user study. This work was supported by the DFG \u2013 EXC number 2064\/1 \u2013 project number 390727645, by the DFG: SFB 1233, Robust Vision: Inference Principles and Neural Mechanisms - project number: 276693517, by the ERC (853489 - DEXIM), and by the BMBF (FKZ: 01IS18039A). The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Leonard Salewski.\n\\section{Conclusion}\nWe introduced the novel CLEVR-X\\xspace{} dataset which contains natural language explanations for the VQA task on the CLEVR dataset.\nOur user study confirms that the explanations in the CLEVR-X\\xspace{} dataset are complete and match the questions and images.\nFurthermore, we have provided baseline performances using the PJ-X and FM frameworks on the CLEVR-X\\xspace{} dataset. The structured nature of our proposed dataset allowed the detailed evaluation of the explanation generation quality according to answer and question types. We observed that the generated explanations were of higher quality for easier answer and question categories.\nOne of our findings is, that explanations for counting problems are worse than for other answer types, suggesting that further research into this direction is needed.\nAdditionally, we find that the four NL\\rev{G} metrics used to evaluate the quality of the generated explanations exhibit different convergence patterns depending on the number of available ground-truth references. \n\nSince this work only considered two natural language generation methods for VQA as baselines, the natural next step will be the benchmarking and closer investigation of additional recent frameworks for textual explanations in the context of VQA on the CLEVR-X\\xspace{} dataset.\nWe hope that our proposed CLEVR-X\\xspace{} benchmark will facilitate further research to improve the generation of natural language explanations in the context of vision-language tasks.\n\\section{The CLEVR-X dataset}\\label{sec:dataclevrx}\n\n\nIn this section, we introduce the CLEVR-X dataset that consists of natural language explanations in the context of VQA\\@. The CLEVR-X dataset extends the CLEVR dataset with 3.6 million natural language explanations for 850k question-image pairs.\nIn Section~\\ref{sec:data-clevr}, we briefly describe the CLEVR dataset, which forms the base for our proposed dataset. Next, we present an overview of the CLEVR-X dataset by describing how the natural language explanations were obtained in Section~\\ref{sec:data-generation}, and by providing a comprehensive analysis of the CLEVR-X dataset in Section~\\ref{sec:data-analysis}.\nFinally, in Section~\\ref{sec:data-userstudy}, we present results for a user study on the CLEVR-X dataset.\n\n\n\n\\subsection{The  CLEVR dataset}\\label{sec:data-clevr}\nThe CLEVR dataset consists of images with corresponding full scene graph annotations which contain information about all objects in a given scene (as nodes in the graph) along with spatial relationships for all object pairs. The synthetic images in the CLEVR dataset contain three to ten (at least partially visible) objects in each scene, where each object has the four distinct properties \\texttt{size}, \\texttt{color}, \\texttt{material}, and \\texttt{shape}.\nThere are three shapes (\\texttt{box}, \\texttt{sphere}, \\texttt{cylinder}), eight colors (\\texttt{gray}, \\texttt{red}, \\texttt{blue}, \\texttt{green}, \\texttt{brown}, \\texttt{purple}, \\texttt{cyan}, \\texttt{yellow}), two sizes (\\texttt{large}, \\texttt{small}), and two materials (\\texttt{rubber}, \\texttt{metallic}). This allows for up to 96 different combinations of properties. \n\n\n\n\n\n\nThere are a total of 90 different question families in the dataset which are grouped into 9 different question types. Each type contains questions from between 5 and 28 question families.\nIn the following, we describe the 9 question types in more detail.\n\n\\myparagraph{\\emph{Hop} questions:}\nThe \\emph{zero hop}, \\emph{one hop}, \\emph{two hop}, and \\emph{three hop} question types contain up to three relational reasoning steps, e.g.\\ \\enquote{What color is the cube \\underline{to the left of} the ball?} is a \\emph{one hop} question.\n\n\\myparagraph{\\emph{Compare} and \\emph{relate} questions:}\nThe \\emph{compare integer}, \\emph{same relate}, and \\emph{comparison} question types require the understanding and comparison of multiple objects in a scene.\nQuestions of the \\emph{compare integer} type compare counts corresponding to two independent clauses (e.g.\\ \\enquote{Are there \\underline{more} cubes \\underline{than} red balls?}).\n\\emph{Same relate} questions reason about objects that have the same attribute as another previously specified object (e.g.\\ \\enquote{What is the color of the cube that has \\underline{the same size} as the ball?}).\nIn contrast, \\emph{comparison} question types compare the attributes of two objects (e.g.\\ \\enquote{\\underline{Is the color} of the cube \\underline{the same} as the ball?}).\n\n\\myparagraph{\\emph{Single and\/or} questions:}\n\\emph{Single or} questions identify objects that satisfy an exclusive disjunction condition (e.g.\\ \\enquote{How many objects are \\underline{either} red \\underline{or} blue?}).\nSimilarly, \\emph{single and} questions apply multiple relations and filters to find an object that satisfies all conditions (e.g.\\ \\enquote{How many objects are red \\underline{and} to the left of the cube.}).\\\\\n\n\\rev{Each CLEVR question can be represented by a corresponding functional program and its natural language realization. \nA functional program is composed of basic functions that resemble elementary visual reasoning operations, such as \\emph{filtering} \\rev{objects} by \\rev{one or more} properties, spatially \\emph{relating} objects to each other, or \\emph{querying} object properties.\nFurthermore, logical operations \\rev{like} \\emph{and} and \\emph{or}, as well as counting operations like \\emph{count}, \\emph{less}, \\emph{more}, and \\emph{equal} are used to build complex questions.\nExecuting the functional program associated with the question against the scene graph yields the correct answer to the question.}\nWe can distinguish between three different answer types: Binary answers (\\texttt{yes} or \\texttt{no}), counting answers (integers from \\texttt{0} to \\texttt{10}), and attribute answers (any of the possible values of \\texttt{shape}, \\texttt{color}, \\texttt{size}, or \\texttt{material}).\n\n\n\\subsection{Dataset generation}%\n\\label{sec:data-generation}\n\\begin{figure}[t]\n     \\centering\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 85mm 7mm 15mm,clip]{figures\/dataset\/PaperFigures_ask.pdf}\n        \\caption{%\n        CLEVR-X dataset generation: Generating a natural language explanation for a sample from the CLEVR dataset. Based on the question, the functional program for answering the question is executed on the scene graph and traced. A language template is used to cast the gathered information into a natural language explanation.\n        }%\n        \\label{fig:generation-and-examples}\n\\end{figure}\n\nHere, we describe the process for generating natural language explanations for the CLEVR-X dataset.\n\\rev{In contrast to image captions, the CLEVR-X explanations only describe image elements that are relevant to a specific input question.}\nThe explanation generation process for a given question-image pair is illustrated in Fig.~\\ref{fig:generation-and-examples}. It consists of three steps: Tracing the functional program, relevance filtering (not shown in the figure), and explanation generation. In the following, we will describe those steps in detail.\n\n\\myparagraph{Tracing the functional program.}\nGiven a question-image pair from the CLEVR dataset, we trace the execution of the functional program (that corresponds to the question) on the scene graph (which is associated with the image). The generation of the CLEVR dataset uses the same step to obtain a question-answer pair. When executing the basic functions that comprise the functional program, we record their outputs in order to collect all the information required for explaining a ground-truth answer.\n\n\n\\rev{In particular, we trace} the \\emph{filter}, \\emph{relate} and \\emph{same-property} functions \\rev{and record the returned objects and their properties, such as, for instance, \\texttt{shape} or \\texttt{size}.}\nAs a result, the tracing omits objects \\rev{in the scene} that are not relevant for the question.\n\\rev{As we are aiming for complete explanations for all question types, each explanation has to mention all the objects that were needed to answer the question, i.e.\\ all the evidence that was obtained during tracing.\nFor example, for \\emph{counting} questions, all objects that match the \\rev{\\emph{filter} function} preceding the \\emph{counting} step are recorded during tracing.\nFor \\emph{and} questions, we merge the tracing results of the preceding \\rev{functions} which results in short and readable explanations.}\n\\rev{In summary, the tracing} produces a \\emph{complete} \\rev{and \\emph{correct}} understanding of the objects and relevant properties which contributed to an answer.\n\n\n\n\\myparagraph{Relevance filtering.}\nTo keep the explanation at a reasonable length, we filter the object attributes that are mentioned in the explanation according to their relevance.\nFor example, the \\texttt{color} of an object is not relevant for a given question that asks about the \\texttt{material} of said object.\n\\rev{We deem all properties that were listed in the question to be relevant.}\nThis makes it easier to recognize the same referenced object in both the question and explanation.\nAs the \\texttt{shape} property also serves as a noun in CLEVR, our explanations always mention the \\texttt{shape} to avoid using generic shape descriptions like \\enquote{object} or \\enquote{thing}.\nWe distinguish between objects which are used to build the question (e.g.\\ {} \\enquote{[\\ldots] that is left of the \\textit{cube}?}) and those that are the subject of the posed question (e.g.\\ {} \\enquote{What color is the \\textit{sphere} that is left of the cube?}).\n\\rev{For the former, we do not mention any additional properties, and for the latter}, we mention the queried property (e.g.\\ \\texttt{color}) for question types yielding attribute answers.\n\n\\myparagraph{Explanation generation.} To obtain the final natural language explanations, each question type is equipped with one or more natural language templates with variations in terms of the wording used.\nEach template contains placeholders which are filled with the output of the previous steps, i.e.\\ the tracing of the functional program and subsequent filtering for relevance.\nAs mentioned above, our explanations use the same property descriptions that appeared in the question. \n\\rev{This is done to ensure that the wording of the explanation is consistent with the given question, e.g.\\ for the question \\enquote{Is there a \\underline{small} object?} we generate the explanation \\enquote{Yes there is a \\underline{small} cube.}%\n\\footnote{%\nThe explanation could have used the synonym \\enquote{box} instead of \\enquote{cube}. In contrast, \\enquote{tiny} and \\enquote{small} are also synonyms in CLEVR, but the explanation would not have been consistent with the question which used \\enquote{small}.}%\n}.\nWe randomly sample synonyms for describing the properties of objects that do not appear in the question.\nIf multiple objects are mentioned in the explanation, we randomize their order.\n\\rev{If the tracing step returned an empty set, e.g.\\ if no object exists that matches the given filtering function for an \\emph{existence} or \\emph{counting} question, we state that no relevant object is contained in the scene (e.g.\\ \\enquote{There is no red cube.}).}\n\nIn order to decrease the overall sentence length and to increase the readability, we aggregate repetitive descriptions (e.g.\\ \\enquote{There is a \\underline{red cube} and a \\underline{red cube}}) using numerals (e.g.\\ \\enquote{There are \\underline{two} red cubes.}).\nIn addition, if \\rev{a function of} the functional program merely restricts the output set of a preceding \\rev{function}, we only mention the outputs of the later \\rev{function}.\nFor instance, if a \\texttt{same-color} \\rev{function} yields a large and a small cube, and a \\rev{subsequent} \\texttt{filter-large} \\rev{function} restricts th\\rev{e output} to only the large cube, we do not mention the output of \\texttt{same-color}, as the output of the following \\texttt{filter-large} causes natural language redundancies%\n\\footnote{\\rev{E.g.\\ for the question: \\enquote{How many large objects have the same color as the cube?}, we do not generate the explanation \\enquote{There are a small and a large cube that have the same color as the red cylinder of which only the large cube is large.} but instead only write \\enquote{There is a large cube that has the same color as the red cylinder.}}}%\n.\n\n\n\\rev{The selection of different language templates, random sampling of synonyms and randomization of the object order (if possible) results in multiple different explanations.} \nWe uniformly sample up to 10 different explanations \\rev{per question }for our dataset.\n\n\\myparagraph{Dataset split.}\nWe provide explanations for the CLEVR training and validation sets, skipping only a negligible subset (less than $0.04\\permil$) of questions due to malformed question programs from the CLEVR dataset\\rev{, e.g.\\ due to disjoint parts of their abstract syntax trees. In total, this affected 25 CLEVR training and 4 validation questions.}\n\nAs the scene graphs and question functional programs are not publicly available for the \\rev{CLEVR} test set, we use the original CLEVR validation subset as the CLEVR-X test set. 20\\% of the CLEVR training set serve as the CLEVR-X validation set. We perform this split on the image-level to avoid any overlap between images in the CLEVR-X training and validation sets.\nFurthermore, we verified that the relative proportion of samples from each question and answer type in the CLEVR-X training and validation sets is similar, such that there are no biases towards specific question or answer types.\n\n\nCode for generating the CLEVR-X dataset and the dataset itself are publicly available at \\url{https:\/\/github.com\/ExplainableML\/CLEVR-X}.\n\n\n\n\n\\subsection{Dataset analysis}\\label{sec:data-analysis}\n\\input{sections\/body\/tables\/dataset_statistics}\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 3mm 3mm 4mm, clip]{figures\/dataset\/xlen_hist_train_v0.7.10-recut.pdf}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.495\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth,trim=3mm 3mm 3mm 4mm, clip]{figures\/dataset\/dataset_histogram_comparison_vqax_esnlive_clevr_train_v0.7.10-recut.pdf}\n     \\end{subfigure}\n     \\vspace{-2ex}\n    \\caption{%\n        Stacked histogram of the average explanation lengths measured in words for the 9 question types for the CLEVR-X training set (left).\n        Explanation length distribution for the CLEVR-X\\xspace{}, VQA-X\\xspace{}, and e-SNLI-VE\\xspace{} \\rev{training sets} (right).\n        The long tail of the e-SNLI-VE\\xspace{} distribution (125 words) was cropped out \\rev{for better} readability.\n    }%\n    \\label{fig:dataset-statisics}\n    \\vspace{-0.6em}\n\\end{figure}\n\n\n\nWe compare the CLEVR-X dataset to the related VQA-X and e-SNLI-VE datasets in Table~\\ref{tab:new_dataset_statistics}. Similar to CLEVR-X, VQA-X contains natural language explanations for the VQA task. However, different to the natural images and human explanations in VQA-X, CLEVR-X consists of synthetic images and explanations.\nThe e-SNLI-VE\\xspace{} dataset provides explanations for the visual entailment (VE) task. VE consists of classifying an input image-hypothesis pair into entailment \/ neutral \/ contradiction categories. \n\n\n\n\nThe CLEVR-X\\xspace{} dataset is significantly larger than the VQA-X\\xspace{} and e-SNLI-VE\\xspace{} datasets in terms of the number of images, questions, and explanations.\nIn contrast to the two other datasets, CLEVR-X\\xspace{} provides (on average) multiple explanations for each question-image pair in the train set.\nAdditionally, the average number of words per explanation is also higher. Since the explanations are built to explain each component mentioned in the question, long questions require longer explanations than short questions. Nevertheless, by design, there are no unnecessary redundancies. The explanation length in CLEVR-X\\xspace is very strongly correlated with the length of the corresponding question (Spearman's correlation coefficient between the number of words in the explanations and questions is $0.89$). \n\nFigure~\\ref{fig:dataset-statisics} (left) shows the explanation length distribution in the CLEVR-X dataset for the 9 question types.\nThe shortest explanation consists of 7 words, and the longest one has 53 words. On average, the explanations contain \\rev{21.53} words. In Fig.~\\ref{fig:dataset-statisics} (right) and Table~\\ref{tab:new_dataset_statistics}, we can observe that explanations in CLEVR-X tend to be longer than the explanations in the VQA-X dataset. Furthermore, VQA-X\\xspace has significantly fewer samples overall than the CLEVR-X dataset. The e-SNLI-VE dataset also contains longer explanations (that are up to 125 words long), but the CLEVR-X dataset is significantly larger than the e-SNLI-VE dataset.\nHowever, due to the synthetic nature and limited domain of CLEVR, the vocabulary of CLEVR-X is very small with only 96 different words.\nUnfortunately, VQA-X\\xspace{} and e-SNLI-VE\\xspace{} contain spelling errors, resulting in multiple versions of the same words.\nModels trained on CLEVR-X circumvent those aforementioned challenges and can purely focus on visual reasoning and explanations for the same.\nTherefore, Natural Language Generation (NLG) metrics applied to CLEVR-X indeed capture the factual correctness and completeness of an explanation.\n\n\n\n\n\n\\input{sections\/body\/user_study}\n\\section{Experiments}\nWe describe the experimental setup for establishing baselines on our proposed CLEVR-X dataset in Section~\\ref{sec:exp-setup}. \nIn Section~\\ref{sec:clevrx-results}, we present quantitative results on the CLEVR-X dataset. Additionally, we analyze the generated explanations for the CLEVR-X dataset in relation to the question and answer types in Section~\\ref{sec:type-results}.\nFurthermore, we study the behavior of the NL\\rev{G} metrics when using different numbers of ground-truth explanations for testing in Section~\\ref{sec:length-results}.\nFinally, we present qualitative explanation generation results on the CLEVR-X dataset in Section~\\ref{sec:quali-results}.\n\n\n\\subsection{Experimental setup}\\label{sec:exp-setup}\nIn this section, we provide details about the datasets and models used to establish baselines for our CLEVR-X dataset and about their training details. Furthermore, we explain the metrics for evaluating the explanation generation performance.\n\n\\myparagraph{Datasets.}\nIn the following, we summarize the datasets that were used for our experiments. In addition to providing baseline results on CLEVR-X, we also report experimental results on the VQA-X and e-SNLI-VE datasets.\nDetails about our proposed \\textbf{CLEVR-X} dataset can be found in Section~\\ref{sec:dataclevrx}.\nThe \\textbf{VQA-X} dataset~\\cite{hukpark2018MultimodalExplanationsJustifying} is a subset of the VQA~v2 dataset with a single human-generated textual explanation per question-image pair in the training set and $3$ explanations for each sample in the validation and test sets.\nThe \\textbf{e-SNLI-VE} dataset~\\cite{Do2020eSNLIVE20CV,kayser2021vil} is a large-scale dataset with natural language explanations for the visual entailment task.\n\n\\myparagraph{Methods.}\nWe used multiple frameworks to provide baselines on our proposed CLEVR-X dataset.\nFor the \\textbf{random words} baseline, we sample random word sequences of length $w$ for the answer and explanation words for each test sample. The full vocabulary corresponding to a given dataset is used as the sampling pool, and $w$ denotes the average number of words forming an answer and explanation in a given dataset.\nFor the \\textbf{random~explanations} baseline, we randomly sample an answer-explanation pair from the training set and use this as the prediction. \\rev{The explanations from this baseline are well-formed sentences. However, the answers and explanations most likely do not match the question or the image.}\nFor the random-words and random-explanations baselines, we report the NLG metrics for all samples in the test set (instead of only considering the correctly answered samples, since the random sampling of the answer does not influence the explanation).\nThe Pointing and Justification model \\textbf{PJ-X}~\\cite{hukpark2018MultimodalExplanationsJustifying} provides text-based post-hoc justifications for the VQA task. It combines a modified MCB~\\cite{fukui2016MultimodalCompactBilinear} framework, pre-trained on the VQA v2 dataset, with a visual pointing and textual justification module.\nThe Faithful Multimodal (\\textbf{FM}) model~\\cite{wu2019FaithfulMultimodalExplanation} aims at grounding parts of generated explanations in the input image to provide explanations that are \\textit{faithful} to the input image. It is based  on the Up-Down VQA model~\\cite{anderson2018bottom}. In addition, FM contains an explanation module which enforces consistency between the predicted answer, explanation and the attention of the VQA model.\nThe implementations for the PJ-X and FM models are based on those provided by the authors of~\\cite{kayser2021vil}.\n\n\\myparagraph{Implementation and training details.}\nWe extracted 14$\\times$14$\\times$1024 grid features for the images in the CLEVR-X dataset using a ResNet-101~\\cite{he2016deep}, pre-trained on ImageNet~\\cite{deng2009imagenet}. These grid features served as inputs to the FM~\\cite{wu2019FaithfulMultimodalExplanation} and PJ-X~\\cite{hukpark2018MultimodalExplanationsJustifying} frameworks. \nThe CLEVR-X explanations are lower case and punctuation is removed from the sentences.\nWe selected the best model on the CLEVR-X validation set\nbased on the highest \\rev{mean of the four NLG metrics, where explanations for incorrect answers were set to an empty string}. \\rev{This metric accounts for the answering performance as well as for the explanation quality.}\nThe final models were \nevaluated on the CLEVR-X test set.\nFor PJ-X, our best model was trained for \\rev{52} epochs, using the Adam optimizer~\\cite{kingma2014adam} with a learning rate of 0.0002 and a batch size of \\rev{256}.\nWe did not use gradient clipping for PJ-X.\nOur strongest FM model was trained for \\rev{30} epochs, using the Adam optimizer with a learning rate of 0.000\\rev{2}, a batch size of 128, and gradient clipping of 0.1. All other hyperparameters were taken from~\\cite{hukpark2018MultimodalExplanationsJustifying,wu2019FaithfulMultimodalExplanation}.\n\n\n\\myparagraph{Evaluation metrics.}\nTo evaluate the quality of the generated explanations, we use the standard natural language generation metrics BLEU~\\cite{papineni2001BLEUMethodAutomatic}, METEOR~\\cite{banerjee2005METEORAutomaticMetric}, ROUGE-L~\\cite{lin2004ROUGEPackageAutomatic} and CIDEr~\\cite{vedantam2015CIDErConsensusbased}.\nBy design, there is no correct explanation that can justify a wrong answer. \nWe follow~\\cite{kayser2021vil} and \\rev{report the quality of the generated explanations for the subset of correctly answered questions}.\n\n\n\n\\subsection{Evaluating explanations generated by state-of-the-art methods}\\label{sec:clevrx-results}\nIn this section, we present quantitative results for generating explanations for the CLEVR-X dataset (Table~\\ref{tab:clevr-results}).\nThe random words baseline exhibits weak explanation performance for all NLG metrics on CLEVR-X.\nAdditionally, the random answering accuracy is very low at 3.6\\%.\nThe results are similar on VQA-X\\xspace{} and e-SNLI-VE\\xspace{}.\nThe random explanations baseline achieves stronger explanation results on all three datasets, but is still significantly worse than the trained models.\nThis confirms that, even with a medium-sized answer space (28 options) and a small vocabulary (96 words), it is not possible to achieve good scores on our dataset using a trivial approach.\n\nWe observed that the PJ-X model yields a significantly stronger performance on CLEVR-X in terms of the NL\\rev{G} metrics for the generated explanations compared to the FM model, with METEOR scores of \\rev{58.9} and \\rev{52.5} for PJ-X and FM respectively. Across all explanation metrics, the scores on the VQA-X and e-SNLI-VE datasets are in a lower range than those on CLEVR-X. For PJ-X, we obtain a CIDEr score of \\rev{639.8} on CLEVR-X and \\rev{82.7} and \\rev{72.5} on VQA-X and e-SNLI-VE\\@. This can be attributed to the smaller vocabulary and longer sentences, which allow $n$-gram based metrics (e.g.\\ BLEU) to match parts of sentences more easily.\n\n\\rev{In contrast to the explanation generation performance, the FM model is better at answering questions than PJ-X on CLEVR-X with an answering accuracy of 80.3\\% for FM compared to 63.0\\% for PJ-X.}\n\\rev{Compared to recent models tuned to the CLEVR task, the answering performances of PJ-X and FM do not seem very strong. However, the PJ-X backbone MCB~\\cite{fukui2016MultimodalCompactBilinear} (which is crucial for the answering performance) preceded the publication of the CLEVR dataset.\nA version of the MCB backbone (CNN+LSTM+MCB in the CLEVR publication~\\cite{johnson2017CLEVRDiagnosticDataset}) achieved an answering accuracy of 51.4\\% on CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset}, whereas PJ-X is able to correctly answer 63\\% of the questions.\nThe strongest model discussed in the initial CLEVR publication (CNN+LSTM+SA in~\\cite{johnson2017CLEVRDiagnosticDataset}) achieved an answering accuracy of 68.5\\%.\n}\n\n\n\\input{sections\/body\/tables\/clevr_results}\n\n\\subsection{Analyzing results on CLEVR-X by question and answer types}\\label{sec:type-results}\n\nIn Fig.~\\ref{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt} (left and middle), we present the performance for PJ-X on CLEVR-X for the 9 question and 3 answer types. The explanation results for samples which require counting abilities (counting answers) are lower than those for attribute answers (57.3 vs.\\ 63.3).\nThis is in line with prior findings that VQA models struggle with counting problems~\\cite{Trott2018InterpretableCF}.\n\\rev{The explanation quality for binary questions is even lower with a METEOR score of only 55.6.}\nThe generated explanations are of higher quality for easier question types; \\textit{zero-hop} questions yield a METEOR score of 64.9 compared to 62.1 for \\textit{three-hop} questions. \nIt can also be seen that \\textit{single-or} questions are harder to explain than \\textit{single-and} questions.\nThese trends can be observed across all NL\\rev{G} explanation metrics.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.6775\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figures\/types\/question_answer_types.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.2825\\textwidth}\n        \\centering\n    \\includegraphics[width=\\linewidth,trim=4mm 4mm 4mm 4mm,clip]{figures\/average_gt_perf\/average_gt_perf.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\caption{%\n    Explanation generation results for PJ-X on the CLEVR-X test set according to question (left) and answer (middle) types \\rev{compared to the overall explanation quality}. Easier types yield higher METEOR scores.\n    NLG metrics using different numbers of ground-truth explanations on the CLEVR-X test set (right). CIDEr converges faster than the other NLG metrics.}%\n    \\label{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt}%\n\\end{figure}\n\n\n\\subsection{Influence of using different numbers of ground-truth explanations}\\label{sec:length-results}\nIn this section, we \nstudy the influence of using multiple ground-truth explanations for evaluation on the behavior of the NL\\rev{G} metrics. \nThis gives insights about whether the metrics can correctly rate a model's performance with a limited number of ground-truth explanations.\nWe set an upper bound $k$ on the number of explanations used and randomly sample $k$ explanations if a test sample has more than $k$ explanations for $k \\in\\{1,2,\\dots,10\\}$.\nFigure~\\ref{fig:clevrx-answer-question-type-full-results-and-metrics-with-different-gt} (right) shows the NL\\rev{G} metrics (normalized with the maximum value for each metric on the test set for all ground-truth explanations) for the PJ-X model depending on the average number of ground-truth references used on the test set.\n\nOut of the four metrics, BLEU-4 converges the slowest, requiring close to 3 ground-truth explanations to obtain a relative metric value of 95\\%. Hence, BLEU-4 might not be able to reliably predict the explanation quality on the e-SNLI-VE dataset which has only one explanation for each test sample.\nCIDEr converges faster than ROUGE and METEOR, and achieves 95.7\\% of its final value with only one ground-truth explanation.\n\\rev{This could be caused by the fact, that CIDEr utilizes a tf-idf weighting scheme for different words, which is built from all reference sentences in the subset that the metric is computed on. This allows CIDEr to be more sensitive to important words (e.g.\\ attributes and shapes) and to give less weight, for instance, to stopwords, such as \\enquote{the}.}\nThe VQA-X and e-SNLI-VE datasets contain much lower average numbers of explanations for each dataset sample (1.4 and 1.0). Since there could be many more possible explanations for samples in those datasets that describe different aspects than those mentioned in the ground truth, automated metric may not be able to correctly judge a prediction \\rev{even if it is correct and faithful w.r.t.\\ to the image and question.\n}\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\scriptsize\n        \\caption*{\\textbf{Question}: How many tiny red things are the same material as the big sphere?}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_014997.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        \\rev{1} $\\mid$ The tiny red metal block has the same material as a big sphere.\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        \\rev{1} $\\mid$ \\rev{There is the tiny red metal block which has the identical material as a big sphere.}\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\100.0 \/ 100.0 \/ 100.0 \/ 744.0}\n        \n        }%\n       \\label{fig:clevrx-qualitative-example-1}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{\\textbf{Question}: The cylinder has what size?\\\\}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_000079.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        Small $\\mid$ The cylinder is small.\\\\\\\\\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        Small $\\mid$ \\rev{The cylinder is tiny}.\\\\\\\\\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\100.0 \/ 100.0 \/ 100.0 \/ 462.4}\n        }%\n        \\label{ffig:clevrx-qualitative-example-2}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{\\textbf{Question}: Are there any small matte cubes?\\\\}\n        \\includegraphics[width=.75\\linewidth]{figures\/qualitative_examples\/CLEVR_val_000100.png}\n        \\caption*{%\n        \\textbf{GT Answer $\\mid$ Explanation}:\\\\\n        No $\\mid$ There are no small matte cubes.\\\\\\\\\n        \\textbf{Pred. Answer $\\mid$ Expl.}\\\\\n        Yes $\\mid$ There is a small matte cube.\\\\\\\\\n        \\rev{\\textbf{B4 \/ M \/ RL \/ C:}\\\\0.0 \/ 76.9 \/ 57.1 \/ 157.1}\n        }%\n        \\label{ffig:clevrx-qualitative-example-3}\n    \\end{subfigure}\n    \\caption{Examples for answers and explanations generated with the PJ-X framework on the CLEVR-X dataset, showing correct answer predictions (left, middle) and a failure case (right). \\rev{The NLG metrics obtained with the explanations for the correctly predicted answers are high compared to those for the explanation corresponding to the wrong answer prediction.}\n    }%\n    \\label{fig:clevrx-qualitative-examples}%\n\\end{figure}\n\n\n\\subsection{Qualitative explanation generation results}\\label{sec:quali-results}\nWe show examples for explanations generated with the PJ-X framework on CLEVR-X in Fig.~\\ref{fig:clevrx-qualitative-examples}. As can be seen across the three examples presented, PJ-X generates high-quality explanations which closely match the ground-truth explanations.\n\nIn the left-most example in Fig.~\\ref{fig:clevrx-qualitative-examples}, we can observe slight variations in grammar when comparing the generated explanation to the ground-truth explanation. However, the content of the generated explanation corresponds to the ground truth. \nFurthermore, some predicted explanations differ from the ground-truth explanation in the use of another synonym for a predicted attribute. For instance, in the middle example in Fig.~\\ref{fig:clevrx-qualitative-examples}, the ground-truth explanation describes the size of the cylinder as ``small'', whereas the predicted explanation uses the equivalent attribute ``tiny''.\nIn contrast to other datasets, the set of ground-truth explanations for each sample in CLEVR-X contains these variations. Therefore, the automated NLG metrics \\rev{do} not decrease when such variations are found in the predictions. \\rev{For the first and second example, PJ-X obtains the highest possible explanation score (100.0) in terms of the BLEU-4, METEOR, and ROUGE-L metrics.}\n\nWe show a failure case where PJ-X predicted the wrong answer in Fig.~\\ref{fig:clevrx-qualitative-examples}~(right). The generated answer-explanation pair shows that the predicted explanation is consistent with the wrong answer prediction and does not match the input question-image pair.\n\\rev{The NLG metrics for this case are significantly weaker with a BLEU-4 score of $0.0$, as there are no matching $4$-grams between the prediction and the ground truth.}\n\n\n\n\\section{Introduction}\n\nExplanations for automatic decisions form a crucial step towards increasing transparency and human trust in deep learning systems. In this work, we focus on natural language explanations in the context of vision-language tasks.\n\\blfootnote{A version of the contribution has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record will be available online at: \\url{https:\/\/doi.org\/10.1007\/978-3-031-04083-2_5}.}\n\nIn particular, we consider the\nvision-language task of Visual Question Answering (VQA) which consists of answering a question about an image. This requires multiple skills, such as visual perception, text understanding, and cross-modal reasoning in the visual and language domains. A natural language explanation for a given answer allows a better understanding of the reasoning process for answering the question and adds transparency. \nHowever, it is challenging to formulate what comprises a good textual explanation in the context of VQA involving natural images.\n\nExplanation datasets commonly used in the context of VQA, such as the VQA-X dataset~\\cite{hukpark2018MultimodalExplanationsJustifying} or the e-SNLI-VE dataset~\\cite{Do2020eSNLIVE20CV,kayser2021vil} for visual entailment,\ncontain explanations of widely varying quality since \nthey are generated by humans. The ground-truth explanations in VQA-X and e-SNLI-VE can range from statements that merely describe an image to explaining the reasoning about the question and image involving prior information, such as common knowledge. One example for a ground-truth explanation in VQA-X that requires prior knowledge about car designs from the 1950s can be seen in Fig.~\\ref{fig:teaser-dataset-comparison}. The e-SNLI-VE dataset contains numerous explanation samples which consist of repeated statements (``x because x'').\nSince existing explanation datasets for vision-language tasks contain immensely varied explanations, it is challenging to perform a structured analysis of strengths and weaknesses of existing explanation generation methods.\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\scriptsize\n        \\caption*{%\n        {\\bf VQA-X} \\\\ \\textbf{Question}: Does this scene look like it could be from the early 1950s?}\n        \\includegraphics[width=0.75\\linewidth,trim={0 3mm 0 1.5mm},clip]{figures\/teaser\/COCO_val2014_000000416220.jpg}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        Yes $\\mid$ The photo is in black and white and the cars are all classic designs from the 1950s\n}%\n       \\label{fig:vqax-example}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n       \\centering\n        \\caption*{{\\bf e-SNLI-VE} \\\\ \\textbf{Hypothesis}: A woman is holding a child.\\\\}\n        \\includegraphics[width=0.75\\linewidth,trim={0 6mm 0 6mm},clip]{figures\/teaser\/3047751696.jpg}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        Entailment $\\mid$ If a woman holds a child she is holding a child.}%\n        \\label{fig:esnlive-examples}\n    \\end{subfigure}\n     \\hfill\n    \\begin{subfigure}[t]{0.31\\textwidth}\n        \\centering\n        \\caption*{{\\bf CLEVR-X} \\\\\\textbf{Question}: There is a purple metallic ball; what number of cyan objects are right of it?}\n        \\includegraphics[width=0.75\\linewidth]{figures\/teaser\/CLEVR_val_005182.png}\n        \\caption*{\\textbf{Answer $\\mid$ Explanation:}\\\\\n        1 $\\mid$ There is a cyan cylinder which is on the right side of the purple metallic ball.}%\n       \\label{fig:clevrx-examples}\n    \\end{subfigure}\n  \n    \\caption{Comparing examples from the VQA-X\\xspace{} (left), e-SNLI-VE\\xspace{} (middle), and CLEVR-X\\xspace{} (right) datasets. The explanation in VQA-X\\xspace{} requires prior knowledge (about cars from the 1950s), e-SNLI-VE\\xspace{} argues with a tautology, and our CLEVR-X\\xspace{} only uses abstract visual reasoning.\n    \\label{fig:teaser-dataset-comparison}\n\\end{figure}\n\nIn order to fill this gap, we propose the novel, diagnostic CLEVR-X dataset for visual reasoning with natural language explanations. It extends the synthetic CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset} dataset through the addition of structured natural language explanations for each question-image pair. An example for our proposed CLEVR-X dataset is shown in Fig.~\\ref{fig:teaser-dataset-comparison}.\nThe synthetic nature of the CLEVR-X dataset results in several advantages over datasets that use human explanations.\nSince the explanations are synthetically constructed from the underlying scene graph, the explanations are \\textit{correct} and do not require auxiliary prior knowledge. The synthetic textual explanations do not suffer from errors that get introduced with human explanations. Nevertheless, the explanations in the CLEVR-X dataset are human parsable as demonstrated in the human user study that we conducted. Furthermore, the explanations contain all the information that is necessary to answer a given question about an image without seeing the image. This means that the explanations are \\textit{complete} with respect to the question about the image.\n\n\nThe CLEVR-X dataset allows for detailed diagnostics of natural language explanation generation methods in the context of VQA\\@. For instance, it contains a wider range of question types than other related datasets. We provide baseline performances on the CLEVR-X dataset using recent frameworks for natural language explanations in the context of VQA\\@. Those frameworks are jointly trained to answer the question and provide a textual explanation. Since the question family, question complexity (number of reasoning steps required), and the answer type (binary, counting, attributes) is known for each question and answer, the results can be analyzed and split according to these groups. In particular, the challenging counting problem~\\cite{Trott2018InterpretableCF}, which is not well-represented in the VQA-X dataset, can be studied in detail on CLEVR-X.\nFurthermore, our dataset contains multiple ground-truth explanations for each image-question pair. These capture a large portion of the space of correct explanations which allows for a thorough analysis of the influence of the number of ground-truth explanations used on the evaluation metrics.\nOur approach of constructing textual explanations from a scene graph yields a great resource which could be extended to other datasets that are based on scene graphs, such as the CLEVR-CoGenT dataset.\n\n\n\nTo summarize, we make the following four contributions:\n\\begin{enumerate*}[label=(\\arabic*)]\n    \\item We introduce the CLEVR-X dataset with natural language explanations for Visual Question Answering; \n    \\item We confirm that the CLEVR-X dataset consists of correct explanations that contain sufficient relevant information to answer a posed question by conducting a user study;\n    \\item We provide baseline performances with two state-of-the-art methods that were proposed for generating textual explanations in the context of VQA;\n    \\item We use the CLEVR-X dataset for a detailed analysis of the explanation generation performance for different subsets of the dataset and to better understand the metrics used for evaluation.\n\\end{enumerate*}\n\n\n\\section{Related work}\nIn this section, we discuss several themes in the literature that relate to our work, namely \\textit{Visual Question Answering}, \\textit{Natural language explanations (for vision-language tasks)}, and the \\textit{CLEVR dataset}. \n\n\\myparagraph{Visual Question Answering (VQA).} The VQA~\\cite{antol2015vqa} task has been addressed by several works that apply attention mechanisms to text and image features~\\cite{yang2016stacked,xu2016ask,zhu2016visual7w,shih2016look,fukui2016MultimodalCompactBilinear}.\nHowever, recent works observed that the question-answer bias in common VQA datasets can be exploited in order to answer questions without leveraging any visual information~\\cite{agrawal2016analyzing,agrawal2018don,johnson2017CLEVRDiagnosticDataset,zhang2016yin}.\nThis has been further investigated in more controlled dataset settings, such as the CLEVR~\\cite{johnson2017CLEVRDiagnosticDataset}, VQA-CP~\\cite{agrawal2018don}, and GQA~\\cite{hudson2019GQANewDataset} datasets. In addition to a controlled dataset setting, our proposed CLEVR-X dataset contains natural language explanations that enable a more detailed analysis of the reasoning in the context of VQA\\@.\n\n\n\n\\myparagraph{Natural language explanations.}\nDecisions made by neural networks can be visually explained with visual attribution that is determined by introspecting trained networks and their features~\\cite{simonyan2013deep,zeiler2014visualizing,selvaraju2019GradCAMVisual,bach2015pixel,zhang2018top}, by using input perturbations~\\cite{petsiuk2018rise,fong2019understanding,fong_iccv_2017}, or by training a probabilistic feature attribution model along with a task-specific CNN~\\cite{kim2021keep}. Complementary to visual explanations methods that tend to not help users distinguish between correct and incorrect predictions~\\cite{kim2021hive}, natural language explanations have been investigated for a variety of tasks, such as fine-grained visual object classification~\\cite{hendricks2018GroundingVisualExplanations,hendricks2018Generatingcounterfactualexplanations}, or self-driving car models~\\cite{kim2018textual}. The requirement to ground language explanations in the input image can prevent shortcuts, such as relying on dataset statistics or referring to instance attributes that are not present in the image.\nFor a comprehensive overview of research on explainability and interpretability, we refer to recent surveys~\\cite{Barredo_Arrieta_XAI_2020,brundage2020toward,gilpin2018explaining}.\n\n\n\\myparagraph{Natural language explanations for vision-language tasks.}\nMultiple datasets for natural language explanations in the context of vision-language tasks have been proposed, such as the VQA-X~\\cite{hukpark2018MultimodalExplanationsJustifying}, VQA-E~\\cite{li2018vqa}, and e-SNLI-VE datasets~\\cite{kayser2021vil}.   VQA-X~\\cite{hukpark2018MultimodalExplanationsJustifying} augments a small subset of the VQA~v2~\\cite{goyal2017MakingVVQA} dataset for the Visual Question Answering task with human explanations.\nSimilarly, the VQA-E dataset~\\cite{li2018vqa} extends the VQA~v2 dataset by sourcing explanations from image captions. However, the VQA-E explanations resemble image descriptions and do not provide satisfactory justifications whenever prior knowledge is required~\\cite{li2018vqa}. The e-SNLI-VE~\\cite{kayser2021vil,Do2020eSNLIVE20CV} dataset combines human explanations from e-SNLI~\\cite{Camburu2018eSNLINL} and the image-sentence pairs for the Visual Entailment task from SNLI-VE~\\cite{Xie2019VisualEA}.\nIn contrast to the VQA-E, VQA-X, and e-SNLI-VE datasets which consist of human explanations or image captions, our proposed dataset contains systematically constructed explanations derived from the associated scene graphs.\nRecently, several works have aimed at generating natural language explanations for vision-language tasks~\\cite{hukpark2018MultimodalExplanationsJustifying,wu2019FaithfulMultimodalExplanation,wu2020ImprovingVQAits,Marasovi2020NaturalLR,patro2020robust,kayser2021vil}.\nIn particular, we use the PJ-X~\\cite{hukpark2018MultimodalExplanationsJustifying} and FM~\\cite{wu2019FaithfulMultimodalExplanation} frameworks to obtain baseline results on our proposed CLEVR-X dataset.\n\n \n\n\\myparagraph{The CLEVR dataset.}\nThe CLEVR dataset~\\cite{johnson2017CLEVRDiagnosticDataset} was proposed as a diagnostic dataset to inspect the visual reasoning of VQA models.\nMultiple frameworks have been proposed to address the CLEVR task~\\cite{hudson2018CompositionalAttentionNetworks,perezFiLMVisualReasoning,hudson2019learning,johnson2017inferring,Suarez2018DDRprogAC,Shi2019ExplainableAE}. \nTo add explainability, the XNM model~\\cite{Shi2019ExplainableAE} adopts the scene graph as an inductive bias which enables the visualization of the reasoning based on the attention on the nodes of the graph.\nThere have been numerous dataset extensions for the CLEVR dataset, for instance to measure the generalization capabilities of models pre-trained on CLEVR (CLOSURE~\\cite{bahdanau2019closure}), to evaluate object detection and segmentation (CLEVR-Ref+~\\cite{liu2019clevr}), or to benchmark visual dialog models (CLEVR dialog~\\cite{kottur2019clevr}).\nThe Compositional Reasoning Under Uncertainty (CURI) benchmark uses the CLEVR renderer to construct a test bed for compositional and relational learning under uncertainty~\\cite{vedantam2021curi}. \\cite{Holzinger2021KANDINSKYPatternsA} provide an extensive survey of further experimental diagnostic benchmarks for analyzing explainable machine learning frameworks along with proposing the KandinskyPATTERNS benchmark that contains synthetic images with simple 2-dimensional objects. It can be used for testing the quality of explanations and concept learning.\nAdditionally,~\\cite{arras2020ground} proposed the CLEVR-XAI-simple and CLEVR-XAI-complex datasets which provide ground-truth segmentation information for heatmap-based visual explanations.\nOur CLEVR-X augments the existing CLEVR dataset with explanations, but in contrast to (heatmap-based) visual explanations, we focus on natural language explanations.\n\n\n\n\n\n\\section{First Section}\n\\subsection{A Subsection Sample}\nPlease note that the first paragraph of a section or subsection is\nnot indented. The first paragraph that follows a table, figure,\nequation etc. does not need an indent, either.\n\nSubsequent paragraphs, however, are indented.\n\n\\subsubsection{Sample Heading (Third Level)} Only two levels of\nheadings should be numbered. Lower level headings remain unnumbered;\nthey are formatted as run-in headings.\n\n\\paragraph{Sample Heading (Fourth Level)}\nThe contribution should contain no more than four levels of\nheadings. Table~\\ref{tab1} gives a summary of all heading levels.\n\n\\begin{table}\n\\caption{Table captions should be placed above the\ntables.}\\label{tab1}\n\\begin{tabular}{|l|l|l|}\n\\hline\nHeading level &  Example & Font size and style\\\\\n\\hline\nTitle (centered) &  {\\Large\\bfseries Lecture Notes} & 14 point, bold\\\\\n1st-level heading &  {\\large\\bfseries 1 Introduction} & 12 point, bold\\\\\n2nd-level heading & {\\bfseries 2.1 Printing Area} & 10 point, bold\\\\\n3rd-level heading & {\\bfseries Run-in Heading in Bold.} Text follows & 10 point, bold\\\\\n4th-level heading & {\\itshape Lowest Level Heading.} Text follows & 10 point, italic\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\noindent Displayed equations are centered and set on a separate\nline.\n\\begin{equation}\nx + y = z\n\\end{equation}\nPlease try to avoid rasterized images for line-art diagrams and\nschemas. Whenever possible, use vector graphics instead (see\nFig.~\\ref{fig1}).\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{fig1.eps}\n\\caption{A figure caption is always placed below the illustration.\nPlease note that short captions are centered, while long ones are\njustified by the macro package automatically.} \\label{fig1}\n\\end{figure}\n\n\\begin{theorem}\nThis is a sample theorem. The run-in heading is set in bold, while\nthe following text appears in italics. Definitions, lemmas,\npropositions, and corollaries are styled the same way.\n\\end{theorem}\n\\begin{proof}\nProofs, examples, and remarks have the initial word in italics,\nwhile the following text appears in normal font.\n\\end{proof}\nFor citations of references, we prefer the use of square brackets\nand consecutive numbers. Citations using labels or the author\/year\nconvention are also acceptable. The following bibliography provides\na sample reference list with entries for journal\narticles~\\cite{ref_article1}, an LNCS chapter~\\cite{ref_lncs1}, a\nbook~\\cite{ref_book1}, proceedings without editors~\\cite{ref_proc1},\nand a homepage~\\cite{ref_url1}. Multiple citations are grouped\n\\cite{ref_article1,ref_lncs1,ref_book1},\n\\cite{ref_article1,ref_book1,ref_proc1,ref_url1}.\n\\subsection{User study on explanation completeness and relevance}\\label{sec:data-userstudy}\n\n\nIn this section, we describe our user study for evaluating the completeness and relevance of the generated ground-truth explanations in the CLEVR-X\\xspace{} dataset.\n\\rev{%\nWe wanted to verify whether humans are successfully able to parse the synthetically generated textual explanations and to select complete and relevant explanations.\nWhile this is obvious for easier explanations like \\enquote{There is a blue sphere.}, it is less trivial for more complex explanations such as \\enquote{There are two red cylinders in front of the green cube that is to the right of the tiny ball.}\nThus, strong human performance in the user study indicates that the sentences are parsable by humans.\n}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=8.3cm]{figures\/user_study\/completeness.png}%\n        \\label{fig:user-study-example-completeness}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=8.3cm]{figures\/user_study\/relevance.png}%\n        \\label{fig:user-study-example-relevance}\n    \\end{subfigure}%\n    \\caption{Two examples from our user study to evaluate the completeness (left) and relevance (right) of natural language explanations in the CLEVR-X dataset.}\\label{fig:user-study-example}\n\\end{figure}\n\nWe performed our user study using Amazon Mechanical Turk (MTurk). It consisted of two types of Human Intelligence Tasks (HITs).\nEach HIT was made up of\n\\begin{enumerate*}[label=(\\arabic*)]\n\\item An explanation of the task;\n\\item A non-trivial example, where the correct answers are already selected;\n\\item A CAPTCHA~\\cite{Ahn2003CAPTCHAUH} to verify that the user is human;\n\\item The problem definition consisting of a question and an image;\n\\item A \\rev{user qualification step}, for which the user has to correctly answer a question about an image. \\rev{This ensures that the user is able to answer the question in the first place, a necessary condition to participate in our user study};\n\\item Two explanations from which the user needs to choose one.\n\\end{enumerate*}\nExample screenshots of the user interface for the user study are shown in Fig.~\\ref{fig:user-study-example}.\n\nFor the two different HIT types, we randomly sampled 100 explanations from each of the 9 question types, resulting in a total of 1800 samples for the completeness and relevance tasks. \nFor each task sample, we requested 3 different MTurk workers based in the US (with high acceptance rate of $>95\\%$ and over 5000 accepted HITs).\nA total of 78 workers participated in the completeness HITs. They took on average 144.83 seconds per HIT\\@.\nThe relevance task was carried out by 101 workers which took on average 120.46 seconds per HIT\\@.\nIn total, 134 people participated in our user study.\nIn the following, we describe our findings regarding the completeness and relevance of the CLEVR-X explanations in more detail.\n\n\n\n\n\\myparagraph{Explanation completeness.}\nIn the first part of the user study, we evaluated whether human users are able to determine if the ground-truth explanations in the CLEVR-X\\xspace{} dataset are complete (and also correct).\nWe presented the MTurk workers with an image, a question, and two explanations. \nAs can be seen in Fig.~\\ref{fig:user-study-example} (left), a user had to first select the correct answer (\\textit{yes}) before deciding which of the two given explanations was complete.\nBy design, one of the explanations presented to the user was the complete one from the CLEVR-X dataset and the other one was a modified version for which at least one necessary object had been removed. As simply deleting an object from a textual explanation could lead to grammar errors, we re-generated the explanations after removing objects from the tracing results. This resulted in incomplete, albeit grammatically correct, explanations.\n\n\nTo evaluate the ability to determine the completeness of explanations, we measured the accuracy of selecting the complete explanation.\nThe human participants obtained an average accuracy of \\rev{92.19}\\%, confirming that  complete explanations which mention all objects necessary to answer a given question were preferred over incomplete ones. \nThe performance was weaker for complex question types, such as \\textit{compare-integer} and \\textit{comparison} with accuracies of only 77.00\\% and 83.67\\% respectively, compared to the easier \\textit{zero-hop} and \\textit{one-hop} questions with accuracies of 100\\% and 98.00\\% respectively. \n\nAdditionally, there were huge variations in performance across different participants of the completeness study (Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(top left)), with the majority performing very well ($>$97\\% answering accuracy) for most question types. For the \\emph{compare-integer}, \\emph{comparison} and \\emph{single or} question types, some workers exhibited a much weaker performance with answering accuracies as low as $0\\%$.\nThe average turnaround time shown in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(bottom left) confirms that easier question types required less time to be solved than more complex question types, such as \\emph{three hop} and \\emph{compare integer} questions. Similar to the performance, the work time varied greatly between different users.\n\n\\input{sections\/body\/tables\/user_study_results}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=4mm 4mm 4mm 4mm,clip]{figures\/user_study\/boxplots\/joint_figure.pdf}\n    \\caption{%\n    \\rev{Average answering accuracies for each worker (top) and average work time (bottom) for the user study (left: completeness, right: relevance). The boxes indicate the mean as well as lower and upper quartiles, the lines extend 1.5 interquartile ranges of the lower and upper quartile. All other values are plotted as diamonds.\n   \n    }\n    }%\n    \\label{fig:user-study-worker-answer-accuracies-and-work-time}%\n\\end{figure}\n\n\n\\myparagraph{Explanation relevance.}\nIn the second part of our user study, we analyzed if humans are able to identify explanations which are relevant for a given image. For a given question-image pair, the users had to first select the correct answer. Furthermore, they were provided with a correct explanation and another randomly chosen explanation from the same question family (that did not match the image). The task consisted of selecting the correct explanation that matched the image and question content. Explanation~1 in the example user interface shown in Fig.~\\ref{fig:user-study-example} (right) was the relevant one, since Explanation 2 does not match the question and image. \n\nThe participants of our user study were able to determine which explanation matched the given question-image example with an average accuracy of 92.52\\%. Again, the performance for complex question types was weaker than for easier questions. The difficulty of the question influences the accuracy of detecting the relevant explanation, since this task first requires understanding the question. Furthermore, complex questions tend to be correlated with complex scenes that contain many objects which makes the user's task more challenging. The accuracy for \\textit{three-hop} questions was 89.00\\% compared to 99.67\\% for \\textit{zero-hop} questions. For \\textit{compare-integer} and \\textit{comparison} questions, the users obtained accuracies of 83.67\\% and 87.33\\% respectively, which is significantly lower than the overall average accuracy.\n\n\n\nWe analyzed the answering accuracy per worker in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(top right).\nThe performance varies greatly between workers, with the majority performing very well ($>$90\\% answering accuracy) for most question types.\nSome workers showed much weaker performance with answering accuracies as low as $0\\%$ (e.g.\\ for \\emph{compare-integer} and \\emph{single or} questions).\nFurthermore, the distribution of work time for the relevance task is shown in Fig.~\\ref{fig:user-study-worker-answer-accuracies-and-work-time}~(bottom right). The turnaround times for each worker exhibit greater variation on the completeness task (bottom left) compared to the relevance task (bottom right).\nThis might be due to the nature of the different tasks. For the completeness task, the users need to check if the explanation contains all the elements that are necessary to answer the given question. The relevance task, on the other hand, can be solved by detecting a single non-relevant object to discard the wrong explanation.\n\nOur user study confirmed that humans are able to parse the synthetically generated natural language explanations in the CLEVR-X dataset. Furthermore, the results have shown that users prefer complete and relevant explanations in our dataset over corrupted samples.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs is well known, the study of null geodesics at a given \nspacetime is not only \nrelevant\nfrom a theoretical point of view, but also from a practical perspective.\nTheir analysis, in addition to information about the causal structure of the \nspacetime, can also establish observable consequences of various \nastrophysical \nphenomena \\cite{stuchlik,cov,vsoc,fernando12,vo13,vv13,garcia13}.\n\nThe role of so-called circular null geodesics is not trivial. \nIn some situations, these geodesics can allow the existence of \nwhat is known as a photon surface, which, in the context of \nfour-dimensional spacetimes, is a \nthree-dimensional non-spacelike manifold $\\mathcal{P}$ , such \nthat every null geodesic whose\ntangent vector $l^a$ at a given point $p\\in \\mathcal{P}$ is contained in  \nthe tangent space $T_p\\mathcal{P}$ of $\\mathcal{P}$ at $p$, always remains \nin $\\mathcal{P}$. In particular, in \nthe case of spherically symmetric spacetimes,\na photon sphere can be defined as a $SO(3)\\times\\mathcal{R}$-invariant photon surface. \nFor rigorous \ndefinitions see\\cite{Ellis01}.\n\nThe study of photon spheres in the case of spherical \nsymmetry and circular null geodesics \nin axial symmetric spacetimes\n is important in astrophysics for several reasons: \n\\begin{enumerate}\n\\item[a)] They appear explicitly in the study of \nrelativistic images produced by black \nholes in the\nstrong lensing regime\\cite{Bozza02, Bozza10}.\n\\item[b)] They play an important role in the analysis of quasinormal \nmodes in black hole perturbations\\cite{Cardoso2009, Decanini10}.\n\\item[c)] They determine the shadow of black holes, or \n(which is the same), how they \nlook to outside observers\\cite{Atamurotov13}.\n\\item[d)] In the case of hairy black holes, they determine \na lower bound on the size of the hair\\cite{Hod11b}.\n\\item[e)] They allow a link between points a) and \nb), or more precisely between gravitational waves and \nlensing, as recently shown in\n\\cite{Stefanov10, Wei14}.\n\\end{enumerate}\nThey have also been analyzed in situations that do not contain black holes, as \nfor example, in Boson stars\\cite{Horvat}, where the authors showed that \nin certain configurations \nphoton spheres can occur, or in certain classes of regular metrics \nwith non-negative trace of the \nenergy-momentum tensor, where the existence of at least two \nphoton spheres was shown\\cite{Hod14}.\nRecently, a study of the thermodynamics of a quantum \nversion of photon spheres was presented\\cite{Baldiotti14}.\n\n\nAlthough in axially-symmetric spacetimes there is no notion yet of \nthe analog of photon spheres,\nthere are circular null geodesics on the equatorial plane, and they are \nalso useful for the\ndiscussion of the previous points. \n\nFor all these considerations, the characterization \nand localization of these particular surfaces or circular\nnull geodesics is relevant.\n\n\nRecently\\cite{Hod13}, Hod made some interesting observations \nabout this type of null \ngeodesics. In particular, in the framework of four-dimensional \ngeneral relativity (GR), he analyzed a general family of spherically \nsymmetric black holes,  which satisfy some natural asymptotic \nand energy conditions, finding \nan upper bound for the radius of photon spheres in terms of their ADM mass.\nIn the case of axial symmetry, he studied circular null geodesics on the \nequatorial plane of a \nKerr spacetime, finding that they are also the fastest way to circle a \nKerr black hole\\cite{Hod11} and conjecturing an expression for the \nminimum orbital time \nto circle any compact object in GR in terms of its mass. \nAs shown by Pradhan\\cite{Pradhan13}, the conclusion that \ncircular null geodesics are\nthe fastest way to circle black holes remains valid for the $n$-dimensional \nversion of the Kerr-Newman metric, namely, for charged Myers-Perry spacetimes. \n\n\nThe reason for and interest in the study of theories of\ngravity in higher dimensions is motivated by string\ntheory. As a possibility, the Einstein-Gauss-Bonnet (EGB) gravity\ntheory is selected by the low energy limit of the string\ntheory~\\cite{cuerdas1,cuerdas3}. In this theory,\ncorrective terms to Einstein gravity appear, which are quadratic in the\ncurvature of the spacetime. The effect of those Gauss-Bonnet terms is\nnontrivial for higher dimensions, so the theory of gravity, which\nincludes Gauss-Bonnet terms, is called Einstein-Gauss-Bonnet (EGB)\ngravity. \n\nEven without these Gauss-Bonnet corrections,\nthe $n$-dimensional version of GR is usually studied in \nastrophysical and \ntheoretical contexts, and in general it is also the case using \nother alternative gravitational\ntheories. For some works on these topics in EGB or pure GR see\n\\cite{Tsukamoto14, Cardoso2009, Cardoso2009b, Sadeghi14, Man14}.\n\nBecause of the evidence garnered of the importance of \ncircular null geodesics\nin the characterization of spherically and axially-symmetric \nblack holes, \nwe extend and generalize some of the Hod results to higher-dimensional \ngravitational theories. In particular we study photon spheres \nin $n$-dimensional GR and EGB theory and \ncircular null geodesics on the equatorial plane for generic \naxially-symmetric spacetimes.  \n\nThe article is organized as follows. In section II we review EGB theory, \nand prepare\nthe setting for the discussion of photon spheres.\nIn section III, we establish and prove two theorems which state \nupper bounds on the radii of\nphoton spheres for black holes in GR and EGB theories, in \nterms of their ADM mass. \nIn section IV, we analyze some of their implications, giving \na universal upper bound for the real \npart of quasinormal frequencies in the WKB limit and a lower \nbound for the first relativistic \nimage in the strong lensing regime.\nIn section V, we make some general comments on the relation between \ncircular null geodesics in \naxially-symmetric \nspacetimes and the fastest way to circle black holes. We also show, \nby using an explicit \ncounterexample, that a lower bound for the orbital period \nof circular null geodesics conjectured by Hod in the context of \nfour-dimensional GR\\cite{Hod11}, \ncannot be assumed to be valid also in alternative gravitational theories. \n\n\\section{Background and setting}\n\nThe action that describes Einstein-Gauss-Bonnet gravity coupled\nwith matter fields reads:\n\\begin{equation*}\nS = \\frac {1}{16\\pi } \\int d^nx \\sqrt {-g}  \\left[ R - 2\\Lambda +\n\\alpha (R_{a b c d } R^{a b c d } + \\right.\n\\end{equation*}\n\\begin{equation*}\n\\left. + R^2 - 4R_{a b } R^{a b} )\\right]+S_{\\text{matt}},\n\\end{equation*}\nwhere $S_{\\text{matt}}$ is the action associated with the matter\nfields, and $\\alpha $ is the Gauss-Bonnet coupling constant\nassociated in the string models, with the tension of these\nstrings. This constant introduces a length scale. In fact, the\ncorrections that this theory produces to GR, are noted at short\ndistances, given by the scale $l=\\sqrt{4 \\alpha}$.\n\nThe equations of motion resulting from $\\delta S=0$ are\n\\begin{eqnarray*}\n\\kappa T_{a b } &=& \\mathcal{G}_{a b}=G^{(0)}_{a b}+G^{(1)}_{a\nb}+G^{(2)}_{a b},\n\\end{eqnarray*}\nwhere $\\kappa=8\\pi G\/c^4$ is the gravitational constant, $T_{a b}$ \nis the energy-momentum tensor, \nrepresenting the matter-field distribution resulting from the variation \n$\\delta S_{\\text{matt}}\/\\delta g^{a b},$ and\n\\begin{eqnarray*}\nG^{(0)}_{a\nb}&=&\\Lambda g_{a b}  \\\\\n G^{(1)}_{a\nb}&=& R_{a b }-\\frac{1}{2}Rg_{a b}\\\\\nG^{(2)}_{a b}&=&-\\alpha \\left[\\frac {1}{2} g_{a b} (R_{ c j e\nk}R^{c\n j e k }-4R_{c j }R^{c j }+R^2) \\right. -   \\\\\n&-&\\left. 2RR_{a b}+4R_{a c}R^{c}_{b}+4R_{c j} R^{c j}_{ \\ \\ a\nb}-2R_{a c j e}R_{b}^{ \\ c j e} \\right ].\n\\end{eqnarray*}\n\nFrom now on, we will use unit where $c=1$ and we adopt $\\alpha$ \npositive since this\ncondition arises from the string theory. \nWe also assume asymptotically flat spacetimes, and therefore $\\Lambda=0$. \n\n\nLet us consider a spherically symmetric metric given by\n\\begin{equation}\\label{eq:metric}\nds^2=-e^{-2\\delta(r)}\\mu(r)dt^2+\\mu(r)^{-1}dr^2+r^2d\\Omega^2_{n-2},\n\\end{equation}\nwith $d\\Omega^2_{n-2}$\nbeing the metric of the $(n-2)$-sphere\n\\begin{equation}\nd\\Omega^2_{n-2} = d\\theta^2_1 + \\sum^{n-2}_{i=2}\\prod^{i-1}_{j=1}\n\\sin^{2}\\theta_j\\;d\\theta^2_i \\; ,\n\\end{equation}\nsolution to the Einstein-Gauss-Bonnet equations \nin higher dimensions. \nThe area of the unit $(n-2)$-sphere is given by \n\\begin{equation}\nS_{n-2}=\\frac{2\\pi^{\\frac{n-1}{2}}}{\\Gamma\\left(\\frac{n-1}{2}\\right)},\n\\end{equation} \nwith $\\Gamma(x)=\\int^\\infty_0 t^{x-1}e^{-t} dt$, the gamma function.\n\nThe Einstein-Gauss-Bonnet equations in terms of \nthe components of the \nenergy momentum tensor, \n$T^t_t=-\\rho$, $T^r_r=p_r$ and\n$T^{\\theta_i}_{\\theta_i}=p_\\bot$ reads\n\\begin{equation}\n\\begin{split}\\label{eq:rhogb}\n\\kappa\\rho=&-\\frac{1}{2r^2}(n-2)[r\\mu'+(n-3)(\\mu-1)]\\\\\n&+\\frac{\\widehat{\\alpha}}{2r^4}(n-2)(\\mu-1)[2r\\mu'+(n-5)(\\mu-1)],\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\\label{eq:prgb}\n\\kappa p_r=&-(n-2)\\left\\{\\frac{1}{2r^2}[2r\\mu\\delta'-r\\mu'-(n-3)(\\mu-1)]\\right.\\\\\n&\\left.-\\frac{\\widehat{\\alpha}}{2r^4}(\\mu-1)[-2r\\mu'+4r\\mu\\delta'-(\\mu-1)(n-5)]\\right\\},\n\\end{split}\n\\end{equation}\nwhere $\\widehat{\\alpha}=\\alpha (n-3)(n-4)$.\nWe have not written the angular-angular equations \nbecause we do not need them. Instead, \nwe will use the only nontrivial component\nof the energy-momentum conservation equation $\\nabla_aT^a_r=0$, which reads\n\\begin{equation}\\label{eq:conser}\np'_r=-\\frac{(e^{-2\\delta}\\mu)'}{2e^{-2\\delta}\\mu}(\\rho+p_r)+\\frac{n-2}{r}(p_\\bot-p_r).\n\\end{equation}\nFrom eqs.(\\ref{eq:rhogb}) and (\\ref{eq:prgb}) it follows that\n\\begin{equation}\\label{eq:delta}\n\\mu[2\\alpha(\\mu-1)-r^2]\\delta'=\\kappa\\frac{r^3}{(n-2)}(\\rho+p_r).\n\\end{equation}\nWe are interested in regular black holes.\nIn particular, we assume that at the horizon $r_H$, $\\mu(r)$ and $\\delta(r)$ satisfy\n\\begin{eqnarray}\\label{crh}\n\\mu(r_H)=0,\\;\\;\\;\\;\\;  \\mu'(r_H)\\geq 0,\\\\\n\\delta(r_H)\\leq\\infty,\\;\\;\\;\\;\\;  \\delta'(r_H)\\leq\\infty.\n\\end{eqnarray}\nThese conditions, together with (\\ref{eq:delta}) imply\n\\begin{equation}\n\\rho(r_H)+p_r(r_H)=0.\n\\end{equation}\n\n\n\nBecause of the asymptotic flatness requirement, we also assume\n\\begin{eqnarray}\n\\mu(r\\rightarrow \\infty)=1,\\label{eq:mu}\\\\\n\\delta(r\\rightarrow \\infty)\\rightarrow 0.\n\\end{eqnarray}\nNote that, by requiring a GR limit as $\\widehat{\\alpha}\\rightarrow 0$,\nthe general solution of (\\ref{eq:rhogb}) can be written as\n\\begin{equation}\n\\mu=1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} M(r)}\n{(n-2)S_{n-2}r^{n-1}}}\\right),\\label{eq:muM}\n\\end{equation}\nwith $M(r)$ given by\n\\begin{equation}\nM(r)=M_H+S_{n-2}\\int^r_{r_H} \\rho r^{n-2}dr;\n\\end{equation}\nwhich can be shown to be the generalized \nMisner-Sharp mass\\cite{Maeda08}. In particular,\n$M_H$ is the horizon mass, and when $r$ goes \nto infinity,  $M(r)$ goes to\nthe ADM mass $\\mathcal{M}$. In order to have a \nfinite ADM mass, $\\rho$ must satisfy\n\\begin{equation}\n\\lim\\limits_{r\\rightarrow\\infty} r^{n-1}\\rho=0.\\label{eq:rholim}\n\\end{equation} \nIn the vacuum case ($T^a_b=0$), we have  \n\\begin{equation}\n\\mu=1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha}\\mathcal{M}}\n{(n-2)S_{n-2}r^{n-1}}}\\right);\n\\end{equation}\nand the associated vacuum metric is known as the \nBoulware-Deser-Wheeler (BDW) black hole.\n\nIf $\\hat{\\alpha}\\rightarrow 0$, Eq.(\\ref{eq:muM}) reduces to its GR \nlimit,\n\\begin{equation}\n\\mu=1-\\frac{2\\kappa M(r)}\n{(n-2)S_{n-2}r^{n-3}}.\\label{eq:muMe}\n\\end{equation}\n \n\n\n\nFinally, \n\\begin{equation}\nT=-\\rho+p_r+(n-2)p_\\bot;\n\\end{equation} \ndenotes the trace of the energy momentum tensor and we assume \nthat matter satisfies the dominant energy condition (DEC) \n$\\rho\\geq 0$, $\\rho\\geq |p_r|,|p_\\bot|$. From\nthe DEC, we see then that $M_H\\leq M(r)\\leq \\mathcal{M}.$\n\n\n\\section{Upper bound on the photon sphere radii}\nWe are now prepared to establish the following theorems:\\\\\n\n\\textbf{Theorem 1. Hod's theorem ($n$-dimensional GR version):} \\textit{Let $(\\tilde{M},g_{ab})$ \nbe a spherically symmetric spacetime, such that: i) it has a regular \nevent horizon, ii) it is asymptotically flat, iii) the dominant \nenergy condition \nis satisfied and the \nenergy-momentum trace is nonpositive, iv) it satisfies the $n$-dimensional \nEinstein equations. \nThen this metric admits at least one photon sphere, which in the coordinates \ngiven by (\\ref{eq:metric}) is characterized by\na radius $r_\\gamma$, which is bounded by the\nfollowing expression in terms of its total ADM mass $\\mathcal{M}$:\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}} \\mathcal{M}\\right)^{\\frac{1}{n-3}}.\\label{boundrgr}\n\\end{equation}\n}\n\nThis theorem can be extended to Einstein-Gauss-Bonnet theory \nin $n$-dimensions. In particular, \nin five dimensions we can obtain an upper bound for the photon sphere radius in terms of \nthe ADM mass and the constant $\\hat\\alpha$.\nFor $n>5$, although we cannot give a general expression for the \nphoton sphere radius bound in terms of the ADM mass \n(with the exception of $n=9$), \nwe can state a more general result. These properties \nare summarized in the following theorem.\\\\\n\n\\textbf{Theorem 2. $n$-dimensional EGB version:} \\textit{Let $(\\tilde{M},g_{ab})$ be a \nspherically symmetric spacetime such that: a) it satisfies conditions i), ii) \nand iii) of Theorem $1$, b) it satisfies the $n$-dimensional \nEGB equations. Then this metric admits at least one photon sphere, \nwhose radius $r_\\gamma$ in the coordinates given by (\\ref{eq:metric}) is always \nbounded\nby the photon sphere radius $r^*_\\gamma$ of a vacuum spherically \nsymmetric BDW black hole \nwith the same total ADM mass $\\mathcal{M}$, i.e. $r_\\gamma\\leq r^*_\\gamma$.\nIn particular in five dimensions the photon sphere radius is bounded by the\nfollowing expression in terms of its total ADM mass $\\mathcal{M}$:\n\\begin{equation}\nr_\\gamma\\leq\\frac{\\sqrt{6}[\\kappa\\mathcal{M}\n(\\kappa\\mathcal{M}-3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\\label{rg5}\n\\end{equation}\n}\n \nNote that the inequality (\\ref{rg5}) is saturated for a five-dimensional BDW \nblack hole with mass\n$\\mathcal{M}$.\\\\\n\n\n\\textbf{Proof:} \\textit{Common part.} Since the Einstein equations \ncan be recovered from (\\ref{eq:rhogb}) and (\\ref{eq:prgb}) with \n$\\hat{\\alpha}=0$, in the proof \nthere is a common part for both theorems. \nBasically, we follow the same steps as Hod's proof\\cite{Hod13}, although some \nchanges are \nneeded to incorporate the $n$-dimensional case. \nBecause of the spherical symmetry, null geodesics may without \nloss of generality be taken to be \non the equatorial plane. \nFrom the metric (\\ref{eq:metric}) it is easy to show that the equation\nfor circular null geodesics $\\dot r_\\gamma=(\\dot r_\\gamma)'=0$ \n(where a dot means derivative with\nrespect to an affine parameter and $'$ means derivative with respect to $r$)\nreads\\cite{Chandra83,Cardoso2009}\n\\begin{equation}\n\\tilde{N}(r_\\gamma)=0;\n\\end{equation}\nwith\n\\begin{equation}\n\\tilde{N}(r)=2e^{-2\\delta(r)}\\mu(r)-r[e^{-2\\delta(r)}\\mu(r)]'.\n\\end{equation}\nBy defining the function\n\\begin{equation}\nN(r)=e^{2\\delta(r)}\\tilde{N}(r),\\label{eq:No}\n\\end{equation}\nwe similarly obtain\n\\begin{equation}\\label{N1}\nN(r_\\gamma)=2[1+r_\\gamma\\delta'(r_\\gamma)]\\mu(r_\\gamma)-r_\\gamma\\mu'(r_\\gamma)=0.\n\\end{equation}\n\nFrom the trace of the energy-momentum tensor and (\\ref{eq:No}),\nwe can write (\\ref{eq:conser}) as\n\\begin{equation}\np'_r=\\frac{1}{2\\mu r}\\left(N(\\rho+p_r)+2\\mu T-2n\\mu p_r\\right).\\label{eq:prp}\n\\end{equation}\nNow if we define:\n\\begin{equation}\nP=r^np_r,\n\\end{equation}\nwe arrive at\n\\begin{equation}\\label{consef}\nP'(r)=\\frac{r^{n-1}}{2\\mu}\\left(N(\\rho+p_r)+2\\mu T\\right). \n\\end{equation}\nIn terms of the energy-momentum components, $N$ reads\n\\begin{equation}\\label{N2}\n\\begin{split}\nN=&\\frac{1}{2\\widehat{\\alpha}(\\mu-1)-r^2}\\left\\{(n-1)\\widehat{\\alpha}\\mu^2\\right.\\\\\n&\\left.-[(n-1)r^2+2(n-3)\\widehat{\\alpha}]\\mu+(n-5)\\widehat{\\alpha}+(n-3)r^2\\right.\\\\\n&\\left.+\\frac{2}{n-2}\\kappa r^4p_r\\right\\};\n\\end{split}\n\\end{equation}\nwhich in the $n$-dimensional GR\ncase reduces to\n\\begin{equation}\nN_{GR}=(n-1)\\mu-(n-3)-\\frac{2\\kappa}{(n-2)}r^2p_r.\\label{NGR}\n\\end{equation}\nLet us observe that (\\ref{N1}) admits at least one solution as follows \nfrom the fact that \nthe conditions (\\ref{crh}) and (\\ref{eq:No}) imply \n$N(r_H)\\leq 0$, and taking into account that \n$\\lim\\limits_{r\\rightarrow\\infty} r^2p_r=0$ (which follows \nfrom the dominant energy condition and (\\ref{eq:rholim})), we see that\n$N(r\\rightarrow\\infty)\\rightarrow 2$; therefore, there \nis a $r_\\gamma$ where $N(r_\\gamma)=0$.\nWhat is more, because we are interested in the innermost \nnull circular orbit, $N(r)$\n must satisfy \n\\begin{equation}\\label{N3}\nN(r_H\\leq r<r_\\gamma)<0.\n\\end{equation}\nThen, by using the dominant energy condition \nwe obtain\n\\begin{equation}\np_r(r_H)=-\\rho(r_H)\\leq 0, \\;\\; \\Rightarrow p_r(r_H)\\leq 0 \\Leftrightarrow P(r_H)\\leq 0.\\label{prh1}\n\\end{equation}\nSimilarly from (\\ref{consef}), (\\ref{N3}) and the condition of nonpositive trace $T\\leq 0$ , \nwe see that\n $P'(r_H\\leq r<r_\\gamma)\\leq 0.$ \nFrom this last condition and (\\ref{prh1}) we have\n\\begin{equation}\np_r(r_\\gamma)\\leq 0.\\label{eq:prf}\n\\end{equation}\nLet us analyze the implications of this relation in the Einstein \nand Einstein-Gauss-Bonnet cases\nseparately.\\\\\n\n{\\textsl{ Completion of the proof of Theorem 1.}}\nIn the Einstein case, from (\\ref{eq:prf}) and (\\ref{NGR}) we conclude that\n\\begin{equation}\n\\mu(r_\\gamma)\\leq\\frac{n-3}{n-1},\n\\end{equation}\nwhich  by using (\\ref{eq:muMe}) implies\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}}{M(r_\\gamma)}\\right)^{\\frac{1}{n-3}},\n\\end{equation}\nor in terms of the total ADM mass $\\mathcal{M}$,\n\\begin{equation}\nr_\\gamma\\leq\\left(\\frac{\\kappa(n-1)}{(n-2)S_{n-2}} \\mathcal{M}\\right)^{\\frac{1}{n-3}}.\n\\end{equation}\n\n\n\\textsl{Completion of the proof of Theorem 2.} Let us start by \nnoting that in (\\ref{N2}) \nthe following expression appears in \nthe denominator\n\\begin{equation}\n2\\widehat{\\alpha}(\\mu-1)-r^2=-\\frac{r^2}{2\\widehat{\\alpha}}\\sqrt{1\n+\\frac{8\\kappa \\widehat{\\alpha} \nM(r)}{(n-2)S_{n-2}r^{n-1}}}<0.\n\\end{equation} \nTherefore from (\\ref{N2}) and (\\ref{eq:prf}) we deduce that at $r_\\gamma$,\n\n\\begin{equation}\\label{Ngb}\n\\begin{split}\n&N^*\\equiv\\left\\{(n-1)\\widehat{\\alpha}\\mu^2-[(n-1)r^2+2(n-3)\\widehat{\\alpha}]\\mu\n+(n-5)\\widehat{\\alpha}\n\\right.\\\\\n&\\left.+(n-3)r^2\\right\\}=-\\frac{2}{n-2}\\kappa r^4p_r\\geq0.\n\\end{split}\n\\end{equation}\nThe implications of this equation for the allowed $r_\\gamma$ \nwill be discussed \nfirst in five dimensions because this case is exactly solvable, \nand after that, we will\nextend the analysis to higher dimensions\n\nBy writing $N^*$  \nas\n\\begin{equation}\nN^*=(n-1)\\widehat{\\alpha}(\\mu-B_{(n)-})(\\mu-B_{(n)+}),\\label{eq:NbmbM}\n\\end{equation}\nwe see that in five dimensions, (\\ref{Ngb}) implies that at $r_\\gamma$, some\nof these conditions hold:\n \\begin{equation}\\label{eq:mup}\n\\mu\\leq\\frac{1}{2\\widehat{\\alpha}}(\\widehat{\\alpha}+r^2\n-\\sqrt{\\widehat{\\alpha}^2+r^4})\\equiv B_{(5)-},\n\\end{equation}\n\nor\n\\begin{equation}\\label{eq:munp}\n\\mu\\geq\\frac{1}{2\\widehat{\\alpha}}(\\widehat{\\alpha}+r^2+\\sqrt{\\widehat{\\alpha}^2+r^4})\\equiv B_{(5)+};\n\\end{equation}\nwhere in order to write these inequalities we use the fact \nthat $B_{(5)-}$ and $B_{(5)+}$ are both \npositive real \nfunctions and that $B_{(5)-}<B_{(5)+}$.\n\nWe also note [using $M_H\\leq M(r)\\leq \\mathcal{M}$] that independently of \nthe dimension,\n$\\mu$ is bounded from below by\n\\begin{equation}\\label{eq:muMadm}\n\\mu\\geq 1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} \\mathcal{M}}\n{(n-2)S_{n-2}r^{n-1}}}\\right)\\equiv \\mu_{(n)\\mathcal{M}};\n\\end{equation}\nand by \n\\begin{equation}\\label{eq:muMH}\n\\mu\\leq 1+\\frac{r^2}{2\\widehat{\\alpha}}\\left(1-\\sqrt{1+\\frac{8\\kappa \\widehat{\\alpha} {M_H}}\n{(n-2)S_{n-2}r^{n-1}}}\\right)\\equiv \\mu_{(n){M_H}};\n\\end{equation}\nfrom above.\nConsequently, the validity of the conditions (\\ref{eq:mup}) or (\\ref{eq:munp}) implies\n\\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r_\\gamma)\\leq B_{(5)-}(r_\\gamma);\\label{in1}\n\\end{equation}\nor \n\\begin{equation}\n\\mu_{(5)M_H}(r_\\gamma)\\geq B_{(5)+}(r_\\gamma);\\label{in2}\n\\end{equation}\nrespectively.\nMoreover, the four functions $\\mu_{\\mathcal{M}}(r)$, $\\mu_{(5)M_H}$, $B_{(5)+}(r)$ and \n$B_{(5)-}(r)$ are \nall monotonically increasing functions.\nFrom this fact, and observing that $B_{(5)+}(r)\\geq 1$, and $\\mu_{(5)M_H}(r\\rightarrow\\infty)=1$, \nit follows that \n\\begin{equation}\n\\mu(r)\\leq\\mu_{(5)M_H}(r) < B_{(5)+}(r)\\; \\forall r,\n\\end{equation} \nthus,  \nit is not possible for any real $r_\\gamma$ to satisfy the \ninequality (\\ref{eq:munp}). \n\nLet us study now the inequality (\\ref{in1}). \nObserving that for $r\\geq r_H$,\n\\begin{equation}\n0<B_{(5)-}(r)<B_{(5)-}(r\\rightarrow\\infty)=\\frac{1}{2},\n\\end{equation}\nand \n\\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r=r_H)\\leq 0,\\;\\; \\mu_{(5)\\mathcal{M}}(r\\rightarrow\\infty)=1,\n\\end{equation} \nwe see that (\\ref{in1}) will be valid for all $r$ such that \n$r_H\\leq\\, r\\leq r^*_\\gamma$, with\n$r^*_\\gamma$ solution of \n \\begin{equation}\n\\mu_{(5)\\mathcal{M}}(r^*_\\gamma)=B_{(5)-}(r^*_\\gamma).\n\\end{equation}\nThe only positive real solution of this equation is \n\\begin{equation}\nr^*_\\gamma=\\frac{\\sqrt{6}[\\kappa\\mathcal{M}(\\kappa\\mathcal{M}-3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\n\\end{equation}\nOn the other hand, as $\\mu(r_H)=0$, $B_{(5)-}(r_H)>0$, and $\\mu(r)\\geq \\mu_{(5)\\mathcal{M}}(r)$, \nwe conclude that  $\\mu(r)> B_{(5)-}(r)$ for all $r> r^*_\\gamma$.\nConsequently, the radius $r_\\gamma$ which satisfies (\\ref{eq:mup}) \nmust necessarily be\nbounded from above by $r^*_\\gamma$, that is\n\\begin{equation}\nr_\\gamma\\leq\\frac{\\sqrt{6}[\\kappa\\mathcal{M}(\\kappa\\mathcal{M}-\n3\\widehat{\\alpha}\\pi^2)]^{1\/4}}{3\\pi}.\\label{eqrg}\n\\end{equation}\n\n\nIn the $n$-dimensional case,  as was mentioned above, \nwe cannot find an explicit upper bound in terms of the ADM mass;\nhowever, we can show that even in these situations, the \nphoton sphere radius will always \nbe bounded by the photon sphere radius of a BDW black \nhole with the same mass.\nIt can be shown as follows. \n\nFirst, in the general case, following a similar analysis \nas in Eqs.(\\ref{eq:NbmbM}), (\\ref{eq:mup}) \nand (\\ref{eq:munp}), we see that (\\ref{Ngb}) is satisfied at $r_\\gamma$ if\n\\begin{equation}\\label{eq:mupN}\n\\mu\\leq\\frac{1}{2\\widehat{\\alpha}}\\left(\\frac{2(n-3)}{n-1}\\widehat{\\alpha}+r^2-\n\\sqrt{\\frac{16}{(n-1)^2}\\widehat{\\alpha}^2\n+r^4}\\right)\\equiv B_{(n)-},\n\\end{equation}\nor if \n\\begin{equation}\\label{eq:mupN2}\n\\mu\\geq\\frac{1}{2\\widehat{\\alpha}}\\left(\\frac{2(n-3)}{n-1}\\widehat{\\alpha}+r^2+\n\\sqrt{\\frac{16}{(n-1)^2}\\widehat{\\alpha}^2\n+r^4}\\right)\\equiv B_{(n)+},\n\\end{equation}\nwhich using (\\ref{eq:muMadm}) and (\\ref{eq:muMH}) implies\n\\begin{equation}\n\\mu_{(n)\\mathcal{M}}(r_\\gamma)\\leq B_{(n)-}(r_\\gamma);\\label{inn1}\n\\end{equation}\nor \n\\begin{equation}\n\\mu_{(n)M_H}(r_\\gamma)\\geq B_{(n)+}(r_\\gamma).\\label{inn2}\n\\end{equation}\nFor the same reasons as in the five-dimensional case [$B_{(n)+}(r)\\geq 1$, \n$\\mu_{(n){M_H}}(r\\rightarrow\\infty)=1$, and both being \nmonotonically increasing functions], \nwe conclude that there is no real $r_\\gamma$ so that (\\ref{inn2}) [and hence \n(\\ref{eq:mupN2})] \ncan be satisfied.\nOn the other hand, from the fact that $B_{(n)-}$ and $\\mu_{(n)\\mathcal{M}}$ are\nmonotonically increasing functions,\nand taking into account that\n\\begin{equation}\n0\\leq B_{(n)-}(r)< B_{(n)-}(r\\rightarrow\\infty)=\\frac{n-3}{n-1}<1\\;\\forall r\\geq r_H;\n\\end{equation} and $\\mu_{(n)\\mathcal{M}}(r_H)\\leq 0$, \n$\\mu_{(n)\\mathcal{M}}(r\\rightarrow\\infty)=1$, we see that (\\ref{inn1}) holds\nfor all $r_H\\leq r<r^*_\\gamma$ with $r^*_\\gamma$ satisfying\n\\begin{equation}\n\\mu_{(n)\\mathcal{M}}(r^*_\\gamma)=B_{(n)-}(r^*_\\gamma).\\label{eq:genAB}\n\\end{equation}\nAs \n$\\mu(r_H)=0, B_{(n)-}(r_H)>0$ and $\\mu(r)\\geq \\mu_{(n)\\mathcal{M}}(r)$; and following the\nsame arguments as in the five-dimensional situation, we conclude that  \nthe radius $r_\\gamma$ where (\\ref{eq:mup}) holds, will also\nsatisfy $r_\\gamma\\leq r^*_\\gamma$.\nFurthermore, from (\\ref{N2}), (\\ref{N3}) and (\\ref{eq:NbmbM}), and observing that\n\\begin{equation}\nN^*=[2\\widehat{\\alpha}(\\mu-1)-r^2]N-\\frac{2}{n-2}\\kappa r^4p_r,\n\\end{equation} \nwe see that (\\ref{eq:genAB}) is equivalent \nto requiring $N(r^*_\\gamma)=0$ for a \nBDW black hole with ADM-mass $\\mathcal{M}$ ($\\rho=p_r=p_\\bot=0$).\nIn other words, $r^*_\\gamma$ can be interpreted as the radius \nof the photon sphere for a \nBDW  black hole with mass $\\mathcal{M}$. As a result, \nthe bound is \nsaturated in these cases. $\\blacksquare$\\\\\n\n\nLet us make some remarks before continuing. In order to see how \nthe bound (\\ref{boundrgr}) varies \nin terms of the dimension, we plot the quotient $r_\\gamma\/(G\\mathcal{M})^{\\frac{1}{n-3}}$ \nin Fig.(1), assuming a \ncontinuous $n$. The physical values \nmust be taken\nonly for natural $n$ in the graphic. We see that in eight dimensions \nthis quotient reaches its \nminimum value. However, as for $n>4$, the dependence in the mass is not linear, \nthe dimension which \nminimizes the value of the upper bound\nfor the radius $r_\\gamma$ at a given fixed $\\mathcal{M}$ will\ndepend on the value of this mass.\n\\begin{figure}[!h]\n\\centering \n\\includegraphics[clip,width=80mm]{rgammaGMfixed5b.eps}\n\\caption{Upper bound for the quotient $r_\\gamma\/(G\\mathcal{M})^{\\frac{1}{n-3}}$ \nin terms of the \ndimension $n$ of the spacetime.\n}\n\\label{fig:distrib}\n\\end{figure}\n\nLet us also observe that if we take $\\alpha\\rightarrow 0$ in (\\ref{eqrg}), \nwe recover the upper \nbound that we found in the GR \ncase.\n\nAs a final comment, we mention that (\\ref{eq:genAB}) can also be \nsolved in nine dimensions, obtaining \nthe result\n\\begin{equation}\nr_\\gamma\\leq 98(14)^{1\/3}\\left[\\frac{h^{2\/3}-686(14)^{2\/3}\\pi^2\\kappa\\hat\\alpha\\mathcal{M}\n}{\\pi^3h^{1\/3}}\\right]^{1\/4},\n\\end{equation}\nwith\n\\begin{equation}\nh=\\pi\\kappa\\mathcal{M}\\left[18\\kappa\\mathcal{M}+\\sqrt{2\\kappa\\mathcal{M}\n(162\\kappa\\mathcal{M}+7\\pi^4\\hat\\alpha^3)}\\right].\n\\end{equation}\n\n\n\\section{Applications: Quasinormal modes and strong lensing}\n\nRecently(\\cite{Cardoso2009}), Cardoso {\\it et al.}, showed a \ncorrespondence between\nthe quasinormal modes associated to a black hole,\nin the eikonal limit, and some properties \nof photon spheres.\nIn particular, in this limit, the quasinormal\nfrequencies $\\omega_{QNM}$ can be computed from \nanalytical WKB \napproximation methods\\cite{Iyer87},\nobtaining\n\\begin{equation}\n\\omega_{QNM}=l\\Omega_\\infty-i|\\lambda|(m+1\/2),\\label{eq:wqnm}\n\\end{equation}\nwhere \n\\begin{equation}\n\\Omega_\\infty=\\frac{[e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]^{1\/2}}{r_\\gamma},\\label{omegainfty}\n\\end{equation} \nis the angular velocity\nof circular null geodesics at $r_\\gamma$ as\nmeasured by asymptotic observers, $\\lambda$ is the \nLyapunov exponent associated with\nthis kind of geodesics, \n$l$ is the angular momentum of the perturbation \n(and it is assumed that $l\\gg 1$) \nand $m$ is the overtone number. \nBased on this fact, Hod in\\cite{Hod13} presented a simple \nand universal bound\nfor the real part of the quasinormal frequencies\nin the WKB limit, expressed in terms of the event horizon \nradius of the associated black hole.\nNow, we generalize this expression to $n$-dimensional \nGR and EGB theory, and we also\ndiscuss other simple bounds for other observables. \n\nFrom (\\ref{eq:wqnm}), we see that the real part of these frequencies is given by\n\\begin{equation}\n\\omega_l\\equiv\\Re{[\\omega_{QNM}]}=l\\Omega_\\infty.\\label{omegalq}\n\\end{equation}\nBy using (\\ref{eq:mupN}), which can be re-written as\n\\begin{equation}\n\\mu(r_\\gamma)\\leq 2\\frac{\\frac{n-5}{n-1}\\hat\\alpha+\\frac{n-3}{n-1}r^2_\\gamma}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}};\n\\end{equation}\nand using the fact that $e^{-2\\delta}\\leq 1$, [which follows from (\\ref{eq:delta}) and \nthe asymptotic \ncondition for $\\delta(r)$], we see that (\\ref{omegainfty}) implies \n\\begin{equation}\n\\Omega_\\infty\\leq 2\\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}.\n\\end{equation}\nHence, from relation (\\ref{omegalq}) we obtain\n\\begin{equation}\\label{eq:omegalp}\n{\\omega_l}\\leq 2 l \\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}.\n\\end{equation}\nIn order to compare this with the Hod bound, we will give a \nweaker inequality, but one \nwhich is simpler and more universal than (\\ref{eq:omegalp})\nin the sense that it does not depend on the \nparticular properties of the\nblack hole.\nLet us note that, in our study cases, $\\hat\\alpha\\geq 0$ and $r_\\gamma\\geq r_H$; \ntherefore,\n\n\\begin{equation}\n\\begin{split}\n{\\Omega}_\\infty & \\leq 2  \\left[\\frac{\\frac{n-5}{(n-1)r^2_\\gamma}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\hat\\alpha+r^2_\\gamma+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_\\gamma}}\\right]^{1\/2}\\\\\n& \\leq 2  \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{\\frac{n-3}{n-1}\\hat\\alpha+r^2_H+\\sqrt{\\frac{16\\hat\\alpha^2}{(n-1)^2}+r^4_H}}\\right]^{1\/2}\\\\\n& \\leq   \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{r^2_H}\\right]^{1\/2}.\\label{omegacot}\n\\end{split}\n\\end{equation}\n\nCombining this inequality with (\\ref{omegalq}), we finally obtain\n \\begin{equation}\n\\begin{split}\n{\\omega}_l \n\\leq l  \\left[\\frac{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n{r^2_H}\\right]^{1\/2};\n\\end{split}\n\\end{equation}\nwhich implies\n \\begin{equation}\n\\begin{split}\n{\\omega}_l r_H \n& \\leq l  \\left[\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}\\right]^{1\/2}.\\label{eq:omeg}\n\\end{split}\n\\end{equation}\nThis expression generalizes to $n$-dimensional EGB theory, \nthe universal bound found\nby Hod in the framework of four-dimensional GR\\cite{Hod13}.\nIn particular, in the GR case it reduces to \n\\begin{equation}\n{\\omega}_l r_H  \\leq l  \\left[\\frac{n-3}{n-1}\\right]^{1\/2},\n\\end{equation}\nwhich, when $n=4$ agrees with the expression given in \\cite{Hod13}.\nNote also that in five dimensions the first factor of \n(\\ref{eq:omeg}) does not survive,  \nand we get the simple bound\n\\begin{equation}\n{\\omega}_l r_H \\leq \\frac{1}{\\sqrt{2}} l.\n\\end{equation}\n\nAt this point, we can ask if there is a similar \nuniversal bound for the \nLyapunov exponent; \nhowever, the answer is negative.\nThe Lyapunov exponent for circular null geodesics \nis defined by\\cite{Cardoso2009}\n\\begin{equation}\n\\lambda=\\sqrt{\\frac{r^2_\\gamma [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]}\n{2L^2}V''(r_\\gamma)},\\label{lambda}\n\\end{equation}\nwith \n\\begin{equation}\nV''(r_\\gamma)=\\frac{L^2}{r^4_\\gamma e^{-2\\delta{(r_\\gamma)}}}\\left\\{e^{-2\\delta{(r_\\gamma)}}\n\\mu(r_\\gamma)-r^2_\\gamma [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]''\\right\\},\n\\end{equation}\nand $L$ the orbital angular momentum of the circular null geodesics.\nAfter some simple computations it can be shown that in $n$-dimensional GR,\n\\begin{equation}\nV''(r_\\gamma)=\\frac{L^2 N'(r_\\gamma)}{r^3_\\gamma}.\n\\end{equation}\nThis makes it possible to write (\\ref{lambda}) as\n\\begin{equation}\n\\lambda=\\sqrt{ [e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)]\n\\frac{ N'(r_\\gamma)}{2r_\\gamma}}.\\label{lambda2}\n\\end{equation}\nFrom (\\ref{NGR}) we have \n\\begin{equation}\n\\begin{split}\nN'(r_\\gamma)&=(n-1)\\mu'(r_\\gamma)-\\frac{2\\kappa}{(n-2)}\\left[2r_\\gamma p_r(r_\\gamma)+\nr^2_\\gamma p'_r(r_\\gamma)\\right]\\\\\n&\\geq (n-1)\\mu'(r_\\gamma)+2{\\kappa} r_\\gamma p_r(r_\\gamma),\\label{Npf}\n\\end{split}\n\\end{equation}\nwhere in order to establish the last inequality, we have taken into account that \nat $r_\\gamma$, $p_r$ and $P'$\nare both nonpositive numbers.\nHowever, from the first factor in (\\ref{lambda2}), which is the factor \n$e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)$, \nwe  can only ensure that $e^{-2\\delta(r_\\gamma)}\\mu(r_\\gamma)\\leq\\mu(r_\\gamma)$, \nand therefore\nthis inequality and (\\ref{Npf}) impose bounds in opposite \ndirections, thereby preventing \na universal bound (independent of the matter content) \nfrom being written \nfor the Lyapunov exponent.\n\nComing back to the inequality (\\ref{omegacot}), it can also be used to \nimpose a universal bound over the location of the first\nrelativistic image in the strong lensing regime. In general, if we assume \nthat the observer\nis at a distance $D_{ol}$ from the lens, then the first relativistic \nimage subtends an \nangle $\\theta_\\infty$ \\cite{Bozza02,Stefanov10}, which\ncan be expressed in terms of the circular orbital frequency $\\Omega_\\infty$ as\n\\begin{equation}\n\\theta_\\infty=\\frac{1}{D_{ol}\\Omega_\\infty}.\n\\end{equation}\nFrom this relation and using (\\ref{omegacot}),\nwe obtain the universal and simple lower bound\n\\begin{equation}\n\\theta_\\infty\\geq \\frac{1}{D_{ol}}\\left[\\frac{r^2_H}{\\frac{n-5}{(n-1)r^2_H}\\hat\\alpha+\\frac{n-3}{n-1}}\n\\right]^{1\/2}.\n\\end{equation}\nIn the GR limit it reduces to\n\\begin{equation}\n\\theta_\\infty\\geq \\frac{\\, r_H}{D_{ol}}\\left[{\\frac{n-1}{n-3}}\n\\right]^{1\/2},\n\\end{equation}\nwhich, for the 4-dimensional case gives\n\\begin{equation}\n\\theta_\\infty\\geq \\sqrt{3}\\frac{\\, r_H}{D_{ol}}.\n\\end{equation}\n\\section{General comment on the fastest way to circle \naxially-symmetric spacetimes}\nIt would be very interesting to find a generalization of some \nof the previous results\nfor the case of axial symmetry. Even in this case, the \ncircular null geodesics on the\nequatorial plane share some of the properties found \nin the spherical symmetric case. \nFor example, \nWei and Liu (\\cite{Wei14}) found universal relations between \nthe first relativistic image and the quasinormal frequencies by \nanalyzing equatorial circular null \ngeodesics in arbitrary axially-symmetric black holes, thus \nextending some of the \nresults of spherical symmetry.\n\n Recently, Hod\\cite{Hod11} also showed that the fastest way to circle a \nblack hole in general \nrelativity is through circular null geodesics. In particular, \nhe demonstrated \nthat for any spherically \nsymmetric spacetime in four dimensions, the minimum orbital \ntime as measured by an \nasymptotic observer is realized by circular null geodesics, \nindependently of the underlying gravitational theory. Moreover, \nhe showed that a similar \nresult can be obtained in GR for the case of \na Kerr black hole by noting that the equatorial circular \norbit with minimum traveling\ntime coincides with the solution obtained by solving the equation governing \ncircular null geodesics.\nMore recently, in \\cite{Pradhan13}, Pradhan, in his study of charged \nMyers-Perry black holes \nin higher dimensions, made the observation that Hod's \nconclusion regarding the fastest way \nto circle a black hole \nremains valid in this more general family of metrics.\n\n\nAs mentioned, the fact that circular null geodesics minimize \nthe orbital period in \nthe case of spherically symmetric spacetimes is always valid. \nHowever, in\\cite{Hod11}, when Hod analyzed the Kerr metric,  \nhe obtained two different equations (one for the \nfastest circular orbit, and another for the \ncircular\nnull geodesics, see Eqs.(21) and (31) in his paper), and\nhe showed that they admit the same kind of solutions; therefore, \nit was not clear from his \ndiscussion how general these results were. In other words, \none can ask whether the agreement \nbetween equatorial \ncircular null geodesics and the fastest way to circle\nblack holes is a property unique to GR or it is a \ngeometrical property \ngeneral to any axially-symmetric \nmetric, independently of any gravity theory. \n\n\nAdditionally, Hod made the interesting conjecture that\nthere should be a lower bound for the traveling time, given \nin terms of the ADM mass \nof the black hole\\cite{Hod11}. \nMore precisely, he conjectured a lower bound on the orbital periods \n$T_\\infty$ of circular null geodesics around compact objects \nwith mass $\\mathcal{M}$ \n(as seen from far away \nobservers). In mathematical terms he states that (in units with $G=c=1$)\n\\begin{equation}\nT_\\infty\\geq 4\\pi \\mathcal{M}\\label{Tinf}.\n\\end{equation}\nIn particular, the equality is satisfied by an extreme Kerr black hole.\nThis conjecture is physically motivated by taking into \naccount the rotational dragging \nwhich is maximal in the case of an extremal black hole.\nIf this conjecture were correct, it could also be used to \nestablish another universal bound for\nthe possible location of the first relativistic image of an \nobject lensed by a black hole.\nIn fact, if (\\ref{Tinf}) were valid we should obtain \n$\\Omega_\\infty=2\\pi\/T_\\infty\\leq\\frac{1}{2\\mathcal{M}},$ which would imply\n\\begin{equation}\n\\theta_\\infty\\geq\\frac{2\\mathcal{M}}{D_{ol}}.\\label{thetab}\n\\end{equation}\nIn other words, the expression (\\ref{thetab}) would also be valid\nfor a generic rotating black hole, at least in the GR regime.\n\nEven given the reasonableness of conjecture (\\ref{Tinf}), \nit has not yet been proven. \nHowever, it leads one to wonder whether this conjecture \ncan be kept as potentially valid in alternative \ngravitational theories. \nIt is our purpose in this section to answer these questions. \nIn particular, we \nmake the observation \nthat the orbital period for circular orbits {\\it always} \ncoincides with the  circular null geodesics for \nany axially-symmetric spacetime, \nindependently of the gravitational theory.\nWe also answer the second question negatively \non the validity of (\\ref{Tinf}) for alternative\nblack hole candidates \nby giving an explicit counterexample where the bound \nassumed by Hod is not satisfied. \nWhat is more, we will show that the conjecture can be violated \neven in the case of a nonrotating \nblack hole.\nIn order to do that, we will discuss a special family of \nKaluza-Klein black holes.\n\nLet us start with the first question.\nWe begin, with the more general conformally stationary and\naxially-symmetric metric in $n$-dimensions, without making \nreference to any gravitational theory.\n\nThese kinds of metrics are given by\n\\begin{equation}\n\\begin{split}\nds^2&=e^{2\\Phi(r,\\theta,t)}\\left(g_{tt}dt^2+2g_{t\\phi}dtd\\phi+g_{rr}dr^2\\right.\n\\\\&+\\left. g_{\\theta\\theta}d\\theta^2+g_{\\phi\\phi}d\\phi^2+r^2\\cos(\\theta)^2d\\Omega^2_{n-3}\\right),\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\nd\\Omega^2_{n-3}=d\\phi^2_2+\\sin^2\\phi_2\\left[d\\phi^2_3+\\sin^2\\phi_3(\\cdots d\\phi^2_{n-4})\\right],\n\\end{equation}\nand with the components of the metric not depending on $\\phi$.\nAs this metric is axially-symmetric, there are equatorial orbits, \nand in particular we\ncan compute from this metric the orbital time for circular null curves, \ni.e., curves \nsuch that $ds^2=0, \\theta=\\pi\/2$ and $r=r_\\gamma$. \nThe orbital period for these curves is:\n\\begin{equation}\\label{eq:T}\nT_\\infty=2\\pi\\frac{-g_{t\\phi}\\pm\\sqrt{\\Delta}}{g_{tt}},\n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta=g_{t\\phi}^2-g_{tt}g_{\\phi\\phi},\n\\end{equation}\nand the $+\/-$ signs correspond to counter-rotating\/co-rotating orbits, \nrespectively.\nIf we compute their extremes we obtain\n\\begin{equation}\n\\frac{dT}{dr}=\\frac{\\pi}{\\sqrt{\\Delta}}G=0,\n\\end{equation}\nwith \n\\begin{equation}\\label{eq:G}\n\\begin{split}\nG_\\pm&=\\frac{1}{g_{tt}^2}\\left[(\\mp 2g_{t\\phi}\\sqrt{\\Delta}\n+\\Delta+g_{t\\phi}^2)\\frac{dg_{tt}}{dr}\\right.\\\\\n&\\left.+(-2g_{tt}g_{t\\phi}\\pm 2g_{tt}\n\\sqrt{\\Delta})\\frac{dg_{t\\phi}}{dr}+g_{tt}^2\\frac{dg_{\\phi\\phi}}{dr}\\right].\n\\end{split}\n\\end{equation}\nSo, if we assume that $\\Delta$ is a regular function and we \nonly consider the region exterior\nto all horizons (where $\\Delta=0$), the minimum orbital \nperiod is obtained for \nthose curves which satisfy $G_\\pm=0$.\n\nOn the other hand, we can study the condition for the \nexistence of circular null geodesics on the\nequatorial plane.\nWe start from the conservation equations\n\\begin{eqnarray}\nE&=&-\\left(g_{tt}\\dot{t}+g_{t\\phi}\\dot{\\phi}\\right),\\\\\nL&=&g_{t\\phi}\\dot{t}+g_{\\phi\\phi}\\dot{\\phi},\\\\\np_r&=&g_{rr}\\dot{r},\n\\end{eqnarray}\nwhich represent the energy of the null particle as measured by an \nasymptotic observer,\nthe orbital angular momentum and the radial component of the linear \nmomentum, respectively.\nThese equations can be solved for $\\dot{t}$, and $\\dot\\phi$ and replaced in\nthe Hamiltonian $H$,\n\\begin{equation}\nH=-E\\dot{t}+L\\dot\\phi+p_r\\dot{r}=0;\n\\end{equation} \nobtaining the following equation for $\\dot{r}$:\n\\begin{equation}\n\\dot{r}^2=V_r;\n\\end{equation}\nwith \n\\begin{equation}\nV_r=-\\frac{E^2}{g_{rr}\\Delta}\\left(g_{\\phi\\phi}+2g_{t\\phi}b\n+g_{tt}b^2\\right),\n\\end{equation}\nand $b=L\/E$, the impact parameter. \nCircular null geodesics must satisfy $\\dot{r}_\\gamma=\\dot{r}'_\\gamma=0$, or equivalently \n$V_r=V'_r=0$.\nFrom  $V_r=0$, we obtain\n\\begin{equation}\nb=\\frac{\\pm\\sqrt{\\Delta}-g_{t\\phi}}{g_{tt}},\\label{eq:b}\n\\end{equation}\nand from $V'_r=0$ it follows that\n\\begin{equation}\n\\frac{d}{dr}g_{\\phi\\phi}+2b\\frac{d}{dr}g_{t\\phi}+b^2\\frac{d}{dr}g_{tt}=0,\n\\end{equation}\nFinally, by replacing the expression for $b$ \ngiven by (\\ref{eq:b}) in the last equation,\nwe conclude that the circular null geodesics, if they exist, \nmust satisfy\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{g_{tt}^2}\\left[(\\mp 2g_{t\\phi}\\sqrt{\\Delta}\n+\\Delta+g_{t\\phi}^2)\\frac{dg_{tt}}{dr}\\right.\n\\\\&\\left.+(-2g_{tt}g_{t\\phi}\\pm 2g_{tt}\\sqrt{\\Delta})\n\\frac{dg_{t\\phi}}{dr}+g_{tt}^2\\frac{dg_{\\phi\\phi}}{dr}\\right]=0,\n\\end{split}\n\\end{equation}\nwhich agrees with the expression for $G_\\pm$ given previously.\nAs a result, its critical points always coincide with \nthe values of the circular geodesics, generalizing in this way \nthe results given by Hod for\nspherically symmetric spacetimes, and the particular cases of Kerr\\cite{Hod11} \nand Myers Perry\\cite{Pradhan13}. Let us remark that \nthere is not inconsistency \nbetween this result\nand the fact that in\\cite{Hod11} Hod obtained\ntwo different algebraic equations [Eqs.(21) and (31) in \\cite{Hod11} for \nthe minimum orbital period  \nand geodesic motion, respectively]. In fact, \nthe circular null geodesics are obtained by solving $\\dot{r}_\\gamma=0$ \nand  $\\dot{r}'_\\gamma=0$, \nboth being functions of $b$. If we first solve $\\dot{r}'_\\gamma=0$ for $b$ \nand replace its value in \n$\\dot{r}_\\gamma=0$, (as the path followed by Hod) we obtain Eq.(31) \nof\\cite{Hod11}. Alternatively, \nif we first solve for $b$ \nfrom $\\dot{r}_\\gamma=0$ and replace it in $\\dot{r}'_\\gamma=0$, \nwe arrive to his Eq.(21).   \n\nAs mentioned above, in \\cite{Hod11}, by comparing the total \norbital period for rotating Kerr\nblack holes with those of Schwarzschild ones, \na lower bound for the orbital \nperiods \n$T_\\infty$ (as measured from far away \nobservers) of circular null geodesics around compact objects with \nmass $\\mathcal{M}$, \nEq.(\\ref{Tinf}) was conjectured. \nNow, by giving an explicit counterexample, we will show that, \nin contrast to the geometrical \nstatus\nof fastest way to circle black holes, this conjecture cannot be \ngenerally valid in other black holes candidates coming from \nalternative gravitational theories.\n\nThe counterexample is based on the Kaluza-Klein \nblack hole\\cite{Aliev13}. Shadows of these kinds of \nblack holes were recently analyzed in\\cite{Eiroa13}.\nLet us consider the following metric, which satisfies \nthe vacuum Einstein equations in five dimensions. \n\\begin{equation}\nds^2=-(1-\\frac{2m}{r})dt^2+\\frac{dr^2}{1-\\frac{2m}{r}}\n+r^2(d\\theta^2+\\sin^2(\\theta)d\\phi^2)+dy^2.\n\\end{equation}\nThen, by doing a compactification of the extra dimension \nand a boost transformation with \nvelocity $v$ in the $y$-direction, \nand by projecting in the $4$-manifold, a new metric is obtained \nrepresenting a charged \nspherically symmetric black hole together with a \ndilaton field\\cite{Aliev13}. \nThe four-dimensional metric reads:\n\\begin{equation}\n\\begin{split}\nds^2&=-\\frac{1-\\frac{2m}{r}}{B}dt^2+\\frac{B}{1-\\frac{2m}{r}}dr^2+\\\\\n& Br^2\\left(d\\theta^2+\\sin(\\theta)^2d\\phi^2\\right),\n\\end{split}\n\\end{equation}\nwith\n\\begin{equation}\nB=\\sqrt{1+\\frac{2mv^2}{r(1-v^2)}}.\n\\end{equation}\nThis metric together with the\nscalar field \n\\begin{equation}\n\\Phi=-\\frac{\\sqrt{3}}{2}\\ln{B},\n\\end{equation}\nand the electromagnetic potential $\\mathcal{A}$\n\\begin{equation}\n\\mathcal{A}=\\frac{v}{2(1-v^2)}\\frac{1-\\frac{2m}{r}}{B^2}dt,\n\\end{equation}\nsatisfies the set of Einstein-Maxwell-dilaton field equations \nthat follow from the action:\n\\begin{equation}\nS=\\int d^4x\\sqrt{-g}\\left[R+2(\\nabla\\Phi)^2\n-e^{2\\sqrt{3}\\Phi}F^2\\right].\n\\end{equation}\nThis metric represents a black hole with ADM mass $\\mathcal{M}$ and\ncharge $Q$ given in terms of the parameter $m$ and the velocity $v$ \nby:\n\\begin{eqnarray}\n\\mathcal{M}&=&m\\left[1+\\frac{v^2}{2(1-v^2)}\\right],\\\\\nQ&=&\\frac{mv}{1-v^2}.\n\\end{eqnarray}\n\nOne of the characteristics of this type of metric is that the ADM mass,\nthe dilation field and the charge depend on the boost parameter.\nNote that the event horizon is located at $r_H=2m$, which \nfor a fixed mass $\\mathcal{M}$, shrinks \nto zero when $v$ goes to $1$.\nLet us compute now the minimum orbital time that a circular null geodesic \ntakes to orbit this black hole.\nIn order to do this, we must solve for the co-rotating orbits, that is $G_{-}=0$, \nwhich, using the expression (\\ref{eq:G}) in this case gives\n\\begin{equation}\n\\begin{split}\n&(v^2-2)^2r^2-2\\mathcal{M}(v^2-2)(4v^2-3)r\\\\\n&-16\\mathcal{M}^2v^2(1-v^2)=0.\n\\end{split}\n\\end{equation}\nThe physical solution of this equation is\n\\begin{equation}\nr_\\gamma=\\frac{3-4v^2+\\sqrt{9-8v^2}}{2-v^2}\\mathcal{M}.\n\\end{equation}\nBy replacing this radius in (\\ref{eq:T}), \nwe obtain\n\\begin{equation}\nT_\\infty=2\\pi\\frac{3-4{v}^{2}+\\sqrt{-8{v}^{2}+9}}{2-{v}^{2}}\\sqrt{\\frac{\n3+\\sqrt{-8{v}^{2}+9}}{-1+\\sqrt{-8{v}^{2}+9}}}\\mathcal{M}.\n\\end{equation}\nA plot of this function is shown in Fig. 2. At $v=0$ (Schwarzschild case)\nit takes the value $T_\\infty=6\\sqrt{3}\\pi\\mathcal{M}$, \nand it goes to zero when $v\\rightarrow 1$.\n\n\\begin{figure}[]\n\\centering\n\\includegraphics[clip,width=80mm]{Tinfinity.eps}\n\\caption{Plot of the quotient $T_\\infty\/\\mathcal{M}$ as \na function of $v$ We assume $G=1$.\n}\n\\label{fig:Tinfi}\n\\end{figure}\nConsequently, the orbital period around a black hole with mass $\\mathcal{M}$\ncan be made arbitrarily small.\nTherefore, there is a value of $v$ from which the Hod conjecture \ncannot be satisfied for this type of black holes. \nNumerically we found that for $v>0.96591$ all the orbital periods $T_\\gamma$ \nare smaller than $4\\pi\\mathcal{M}$.  \n\n\\section{Summary}\nIn this work, we have found upper bounds for the radius of \nphoton spheres in $n$-dimensional\nGR and EGB theories, \nthereby generalizing the previous work of Hod in\\cite{Hod13}.\nIn the general situation of a black hole dressed with matter \nfields, we have seen that the photon\nsphere radius is always smaller than that corresponding \nto a vacuum black hole with the same mass. \nIt would be interesting to know how generic these results \nare, in the sense of\nhow much they depend on gravitational \nfield equations. In particular, it would be natural to study whether these results \nhold for the more general \nLovelock gravitational theory or if they can be extended to other\ntheories like $f(R)$ ones.\n\nWith respect to the study of circular null geodesics in axial \nsymmetry, we have observed that equatorial circular null geodesics \nalways minimize \nthe orbital time around a black hole. It is a geometrical result. Given \nthe astrophysical importance of this type of geodesics, it would \nalso be interesting to attempt a proof of the Hod conjecture (\\ref{Tinf}). \nWe wish to deal with these and other associated problems in the near future.\\\\\n\n\n\n\\section*{Acknowledgments}\nE. Gallo thanks the Instituto de F\\'isica y\nAstronom\\'ia, Universidad de Valpara\\'iso for their kind hospitality, \nwhile working on this paper. \nE. Gallo acknowledge financial support from CONICET and SeCyT-UNC.\nJ. R. Villanueva is supported by Comisi\\'on Nacional de Investigaci\\'on Cient\\'ifica \ny Tecnol\\'ogica through FONDECYT grants No  11130695.\nIn this post-publisher version two typos were corrected: the signature of the metric in eq.(1) and a factor in eq.(102). We thanks Gary Gibbons and Chris Pope for bringing to our attention these typos. \n\n\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nHierarchical structure formation is one of the most profound predictions of the standard model of cosmology --- the Lambda--Cold Dark Matter ($\\Lambda$CDM) cosmology \\citep[e.g.,][]{10.1093\/mnras\/183.3.341,10.1086\/183911,10.1038\/311517a0,2010gfe..book.....M}. In a $\\Lambda$CDM universe, dark matter is attracted to local density peaks of initial fluctuations, laid down during inflation, forming dark matter haloes. These dark matter haloes grow and merge with one another, and are believed to be the nests within which visible galaxies develop. The spatial distribution of dark matter haloes is hence a powerful prediction of the $\\Lambda$CDM model, and can be compared with observations of galaxy distribution to test our understanding of cosmology and galaxy formation physics. \n\nDespite the nonlinearity in the formation and merging of dark matter haloes, we have a good general picture of the distribution of dark matter haloes, thanks to extensive studies using both mathematical approximations and numerical simulations. In particular, the excursion set formalism provides an analytic description of the halo mass function, halo mass assembly histories, and halo clustering properties  (\\citealt{Press1974,1991ApJ...379..440B}; for a review see \\citealt{10.1142\/S0218271807010511} and Chapter 7 of \\citealt{2010gfe..book.....M}). In this context, the clustering properties of haloes are a function of halo mass alone. This is commonly referred to as ``halo bias'' and can be approximated analytically \\citep[e.g.,][]{Kaiser84,Cole1989,Mo1996,Sheth2001} or calibrated against numerical simulations \\citep[e.g.,][]{Tinker2010}. These analytic descriptions of halo bias are powerful tools within galaxy models that rely on halo mass as the link between haloes and the galaxies that they host, such as the Halo Occupation Distribution \\citep[e.g.,][]{Peacock2000,Seljak2000,Berlind2002,Zheng2005,Zehavi2005} and the Conditional Luminosity Function \\citep[e.g.,][]{Yang2003,vandenBosch2013}, to predict galaxy clustering efficiently. \n\nOn the other hand, it has also been shown, within cosmological $N$-body simulations, that halo clustering also depends on halo properties other than mass, amongst which the most notable is halo assembly history  \\citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. This is commonly referred to as ``halo assembly bias.'' Although halo assembly bias is a smaller effect than mass-dependent halo bias, it has drawn increasing attention from researchers who model the galaxy--halo connection \\citep[e.g.,][]{1507.01948,Hearin2016,Zentner2016,1611.09787,Lehmann2017,2017MNRAS.469.1809R}, because it is very plausible that the galaxy assembly history is to some extent connected to the assembly history of its host halo. If this is the case, one would need to understand halo assembly bias to predict accurately galaxy clustering and to mitigate potential bias in any inference from clustering measurements \\citep{Reddick2013,Zentner2014}. \n\nIn the excursion set formalism, a characteristic mass $M^*$, below which most haloes have already collapsed and formed, is defined to satisfy $\\sigma(M^*)=\\delta_c D(a)$, where $\\delta_c \\simeq 1.686$ is the critical overdensity, $D(a)$ is the linear growth rate, and $\\sigma(M)$ is the squared root of the mass variance with a top-hat filter of mass $M$. \nStudies have found that, below the characteristic mass $M^*$, haloes that form earlier are more strongly clustered \\citep{2001PhDT.........7W,Gao2005,Wechsler2006,2008MNRAS.389.1419L}. However, above the characteristic mass $M^*$, the signal of halo assembly bias is less clean. \\citet{Wechsler2006} found that, above the collapse mass, haloes with lower \\emph{concentration} are more clustered, but detected no clear bias as a function of halo formation time, despite the correlation between concentration and formation time \\citep{Wechsler2002}. Similarly, \\citet{2008MNRAS.389.1419L} inspected several different definitions of halo formation time, but found little assembly bias for high-mass haloes upon splitting haloes by their formation times. Attempts have been made to explain the physical origin of halo assembly bias \\citep{Sandvik2007,10.1142\/S0218271807010511,Desjacques2007,Dalal2008,2009MNRAS.396.2249W}, and while they provided some heuristic insights, they do not explain why different proxies of halo assembly history (e.g., concentration and formation time) exhibit assembly biases of very different magnitudes for massive haloes.\n\nOn the observational front, recent efforts have been made to detect halo assembly bias with galaxy clusters \\citep{Yang2006,Tinker2012,Miyatake2015,More2016,Dvornik2017}. However, results remain inconclusive as various systematic effects can masquerade as observed halo assembly bias \\citep{Lin2016,2016arXiv161100366Z,1702.01682}. One challenge facing observations is that, because halo properties are not directly observable, one has to use some observational proxy to approximate halo assembly history. In the studies of \\citet{Miyatake2015,More2016,Dvornik2017}, they use the average distance of member galaxies in the cluster as a proxy of halo formation time. The motivation for such a proxy is the well-documented correlation between halo concentration and halo formation time \\citep{Wechsler2002} and the assumed correlation between halo concentration and member galaxy distances. Nevertheless, even if a bias signal due to the average distance of member galaxies had been robustly detected observationally, it would not be clear whether or not this bias is of the same physical origin as the halo assembly bias identified in cosmological $N$-body simulations. \n\nIn addition to halo assembly bias, the clustering of haloes also depends on other halo properties, such as spin, shape, and substructure abundance \\citep[e.g.,][]{2001PhDT.........7W,Wechsler2006,Bett2007,Gao2007,2007MNRAS.375..489H,Faltenbacher2010}. {More recently, there are also studies that explore the relationships amongst different kinds of halo assembly bias \\citep[e.g.,][]{1612.04360,1708.08451}.}  In much of the literature, these other \\emph{secondary halo biases} (i.e., dependence of halo clustering on halo properties other than mass) are also commonly referred to as ``assembly bias,'' regardless of whether or not the secondary halo properties have a direct connection to halo assembly history. This nomenclature could result in some confusion as to whether all the different kinds of secondary halo bias have the same physical origin. While halo assembly history is arguably the most important property other than halo mass that characterizes individual haloes, when it comes to the clustering properties, it is still unclear if or how the different secondary halo biases connect with one another. \n\n\n\nIn this study, we join the exploration of halo secondary bias with two specific aims. First, we examine the dependence of halo clustering on a set of secondary halo properties for cluster-size haloes using a modern, large-volume cosmological simulation to validate previous results, some of which may have suffered from limited volume or resolution. Secondly, we inspect the correlations between different halo properties and how they connect to the secondary bias, to better understand the relation between assembly bias and all different kinds of secondary biases. To this end, we present a novel way to characterize secondary bias, which helps us to gain insight into these questions. \n\nWe focus our current study on cluster-size haloes for three reasons. First, in the cluster-mass regime, the behaviour of assembly bias due to different proxies, such as halo concentration and formation time, is poorly understood. Secondly, at the high-mass end of the halo mass spectrum, halo assembly bias is thought not to be due to nonlinear interactions between neighbouring haloes, but reflective of the initial conditions for structure formation \\citep{10.1142\/S0218271807010511,Dalal2008}. Therefore, it seems plausible that assembly bias is simpler at high mass.\nThird, there have been, and likely will soon be more, observational attempts to directly measure halo assembly bias with galaxy clusters. Hence, this is a timely and crucial study to facilitate the interpretation of current measurements and preparation for forthcoming observations.  \n\nThis paper is structured as follows. In \\autoref{sec:methods}, we introduce the simulation used in this study, define the halo properties, and explain how we remove mass dependence in the bias. We present our main results in \\autoref{sec:results}, where we show and compare the secondary halo biases due to different halo properties, with \\autoref{fig:bias} summarizing this result, and also present a novel way to characterize secondary bias. In \\autoref{sec:discussion}, we turn to explore the interplay between correlation and bias, demonstrate that the correlation between different halo properties does not determine the secondary bias they exhibit, and also discuss the implication for galaxy assembly bias. \\autoref{fig:history} highlights the absence of assembly bias (in its strict definition) at this mass scale, despite the existence of other secondary halo biases. We conclude in \\autoref{sec:conclusion}. We also include a summary of the correlations amongst all secondary halo properties used in this study and their dependence on the environment in Appendix~\\ref{sec:appendix}.\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{marks.pdf}\n\\caption{\\label{fig:marks}%\nEach scatter plot shows the relation between halo mass and one of the secondary halo properties: concentration parameter \\ensuremath{c}, spin parameter \\ensuremath{\\lambda}, half-mass scale \\ensuremath{a_{1\/2}}, accretion rate before peak mass \\ensuremath{\\Gamma_\\text{peak}}, number of subhaloes \\ensuremath{N_\\text{sub}}, and averaged subhalo distance \\ensuremath{R_\\text{mem}}. All distinct haloes in the sample are plotted and coloured by their mark values, which are always between 0 and 1 and are assigned according to the rank of the secondary property within each halo mass bin as explained in \\autoref{sec:marks}.\n}\n\\end{figure*}\n\n\\section{Methods}\n\\label{sec:methods}\n\n\\subsection{Simulations}\n\\label{sec:sims}\n\nIn this study we use the MultiDark Planck 2 (MDPL2) simulation \\citep{Klypin2016}. MDPL2 is a cosmological gravity-only $N$-body simulation, run with the \\textsc{L-Gadget2} code. It has a periodic volume of 1\\,\\ensuremath{h^{-1}}\\,Gpc$^3$, with 3840$^3$ particles. The mass resolution of each particle is $1.51\\times10^9$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}, and the physical force resolution ranges from 5 to 13\\,\\ensuremath{h^{-1}}\\,kpc (smaller at lower redshift). \nMDPL2 adopts the Planck 2013 $\\Lambda$CDM cosmology \\citep{Planck2013cosmology}, and the actual values used in MDPL2 are: total matter density $\\Omega_\\text{m} = 0.307115$, dark energy density $\\Omega_\\Lambda = 1 - \\Omega_\\text{m}$, baryon density $\\Omega_\\text{b} = 0.048206$, Hubble parameter $h = 0.6777$, scalar spectral index $n_s = 0.96$, and the amplitude of mass density fluctuation $\\sigma_8 = 0.8228$.\n\nThe MDPL2 simulation has been analysed by the \\textsc{Rockstar} halo finder and the \\textsc{Consistent Trees} merger tree builder \\citep{2013ApJ...762..109B,2013ApJ...763...18B}. \nThe halo mass definition used here is the virial mass; in this cosmology the virial overdensity corresponds to approximately 100 times the critical density \\citep{Bryan1998}.\nA halo is called a ``subhalo'' if its centre is within the virial radius of any larger halo. Any halo that is not a subhalo is called a ``distinct halo.''\n\nIn this study, we use haloes from the present-day ($z=0$) halo catalogue. We select all distinct haloes with a present-day virial mass $\\ensuremath{M_\\text{vir}} \\geq 10^{14}$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}{} as our full halo sample. There are 27,029 distinct haloes in this sample, corresponding to a number density of $2.7\\times 10^{-5}\\,h^3\\,\\text{Mpc}^{-3}$.\n\nAlthough the MDPL2 halo catalogues contain information about halo mass assembly histories, we do not have direct access to the full assembly history of each individual halo. Hence we also use the DarkSky-Gpc simulation when the full assembly history is required (\\autoref{fig:history}). DarkSky-Gpc ({\\tt ds14\\_b}) is part of the Dark Sky Simulations \\citep{Skillman2014}, run with the \\textsc{2HOT} code \\citep{Warren2013}. It also has a periodic volume of 1\\,\\ensuremath{h^{-1}}\\,Gpc$^3$, and was run with 10240$^3$ particles and a mass resolution of $7.63\\times10^7$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}. \nDarkSky-Gpc has also been analysed by the \\textsc{Rockstar} halo finder; however, the full \\textsc{Rockstar--Consistent Trees} halo catalogues and merger trees have only been constructed on a downsampled version, which has only $10240^3\/32 \\simeq 3225^3$ particles and an effective mass resolution of $2.44\\times10^9$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}%\n\\footnote{See also \\citet{2016PhDT........76M,Lehmann2017}. We note that these two studies mistakenly reported the DarkSky-Gpc particle mass to be twice its actual value.}, similar to the mass resolution of MDPL2. In this study we only use the downsampled version of DarkSky-Gpc. DarkSky-Gpc adopts a flat cosmology close to the Planck 2013 $\\Lambda$CDM cosmology, with $h = 0.688$, $\\Omega_\\text{m} = 0.295$, $n_s = 0.968$, $\\sigma_8 = 0.834$. We also use the virial overdensity as the halo mass definition for DarkSky-Gpc.\n\nWhile we present most of our result (all except for \\autoref{fig:history}) using the MDPL2 simulation because it is publicly available and has slightly better mass resolution, we have verified that the DarkSky-Gpc simulation produces very similar result, with no qualitative difference and little quantitative difference. Our result holds in both simulations.\n\n\n\\subsection{Secondary halo properties}\n\\label{sec:properties}\n\nIn this study we select six different halo properties other than halo mass and investigate the secondary halo bias due to these properties. The properties we choose are as follows:\n\n\\begin{enumerate}\n\\itemsep0.5\\baselineskip\n\n\\item Concentration parameter (\\ensuremath{c}), obtained by fitting the dark matter density profile to a Navarro--Frenk--White (NFW) profile \\citep{Navarro1996,Navarro1997}.\n\n\\item Spin parameter (\\ensuremath{\\lambda}), as defined in \\citet{Peebles1969}.\n\n\\item Half-mass scale (\\ensuremath{a_{1\/2}}), defined as the scale factor at which a distinct halo reaches more than or equal to half of its present-day ($z=0$) mass on its main branch.\n\n\\item Accretion rate {of} peak mass (\\ensuremath{\\Gamma_\\text{peak}}), the average halo mass accretion rate (in \\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}\\,yr$^{-1}$) {of peak mass, i.e., \\[\\left[\\ensuremath{M_\\text{peak}}(z=0) - \\ensuremath{M_\\text{peak}}(z=0.5)\\right]\/\\left[t(z=0) - t(z=0.5)\\right].\\]} Since our halo sample contains the most massive haloes in the simulation, the majority (68.7\\%) have $\\ensuremath{z_\\text{peak}} = 0$ {(the redshift when peak mass takes place)}. The median \\ensuremath{z_\\text{peak}}{} for haloes whose present-day mass is not peak mass (i.e., $\\ensuremath{z_\\text{peak}} > 0$) is 0.093. Hence, for our halo sample, \\ensuremath{\\Gamma_\\text{peak}} is basically the average halo mass accretion rate for $0 < z < 0.5$ .\n\n\\item Number of subhaloes (\\ensuremath{N_\\text{sub}}). We count the number of subhaloes (identified by the `upid' value in the \\textsc{Rockstar--Consistent Trees} catalogue) that have a peak maximal circular velocity (\\ensuremath{V_\\text{peak}}) above 135\\,\\text{km\\,s$^{-1}$}. The peak maximal circular velocity is defined as the largest value of the maximal circular velocity on the main branch of the subhalo in consideration. \n\n\\item Average subhalo distance (\\ensuremath{R_\\text{mem}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo. The subhalo definition is the same as the definition when we calculate the number of subhaloes for each distinct halo.\n\n\\end{enumerate}\n\nFor \\ensuremath{c}, \\ensuremath{\\lambda}, \\ensuremath{a_{1\/2}}, and \\ensuremath{\\Gamma_\\text{peak}}, we use the values in the \\textsc{Rockstar--Consistent Trees} catalogue directly.\n\n\n\\subsection{Definition of mass-normalized marks}\n\\label{sec:marks}\n\nFormally, we say a secondary halo bias exists if two groups of haloes that have the same halo mass but different values of a secondary propriety cluster differently. However, practically, because it is difficult to measure spatial clustering within an infinitesimally thin mass bin due to the finite number of haloes even within our large-volume simulations, we need to remove any bias that may be induced by halo mass when we measure the secondary bias. \n\nTo do so, for each secondary halo property in consideration, we assign a ``mass-normalized'' mark value to each halo to represent the secondary halo property. We first bin haloes by halo mass, and then in each mass bin, we assign the mark value based on the rank within that mass bin. Hence, within each mass bin and also overall, the mark values always span $[0,1]$ uniformly. The mark values are always dimensionless.\n\nFor secondary halo properties that are discrete, such as the number of subhaloes and the half-mass scale (which is discrete because of the time interval between the simulation snapshots that are saved), if a group of haloes in a particular mass bin share the same exact value, they are randomly assigned different mark values in a way that still preserve the overall ranks. For example, when assigning the mark for the number of subhaloes, we add a random number drawn from a continuous uniform distribution on $[0, 1)$ to the number of subhaloes before the ranking process. In this fashion, haloes with one subhalo would have different mark values, but their mark values will always be smaller than the mark values of haloes with two subhaloes. This procedure hence ensures that the mark is uniformly distributed and that it is not clustered at a certain value, yet it does not introduce noise in the overall ranks.\n\nThe sample of distinct haloes used in this study spans the mass range of $14 \\leq \\log\\,\\left[M\/(\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$})\\right] < 15.55$. We split this mass range into 30 bins. The bin widths increase quadratically in log mass, such that the lowest mass bins do not contain the vast majority of haloes in the sample. \\autoref{fig:marks} shows the relations between the halo mass and each of the secondary halo properties we considered here. The mark value we assigned to each halo is represented by the colour of each point. We can observe the binning effect on the mark values for properties that change more rapidly with halo mass (\\ensuremath{\\Gamma_\\text{peak}}, \\ensuremath{N_\\text{sub}}, and \\ensuremath{R_\\text{mem}}). Nevertheless, we have tested and find our results are insensitive to the binning schemes. When we use different binning schemes, including uniformly spacing in log mass and also different number of bins, the result still holds.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Secondary halo bias manifested in the bias function}\n\\label{sec:bias}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{bias.pdf}\n\\caption{\\label{fig:bias}%\nThe ratio between the pair counting function ($n_\\text{pairs}$) of only haloes with high marks and that of only haloes with low marks for each of the six secondary halo properties: (solid lines from top to bottom) \\ensuremath{\\lambda}, \\ensuremath{R_\\text{mem}}, \\ensuremath{N_\\text{sub}}, \\ensuremath{\\Gamma_\\text{peak}}, \\ensuremath{a_{1\/2}}, and \\ensuremath{c}. A high (or low) mark value means a mark value above (or below) 0.5. The pair counting function is calculated by counting all pairs of haloes that have three-dimensional distances between 2 to 40\\, \\ensuremath{h^{-1}}\\,Mpc, and binned in distance ($r$), normalized by the square of number of haloes. For the concentration parameter (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties. For comparison, a thin horizontal black line shows a ratio of unity, and the grey band shows the 3$\\sigma$ deviation for a randomly-assigned mark values. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property.\n}\n\\end{figure}\n\nThe most straightforward way to evaluate halo bias is to calculate the ratio between the pair counts of two samples of haloes. This is sometimes called the bias function. If the two samples have different numbers of total haloes, one needs to normalize the pair count first, usually by dividing out the expected number of pairs of a set of uniformly distributed random points. As for secondary halo bias, one can then calculate the bias function between two halo samples that differ in a secondary halo property, but not in halo mass. \n\nHere we present in \\autoref{fig:bias}, for each of the six secondary halo properties listed in \\autoref{sec:properties}, the bias function (or more precisely, the ratio between the pair counting functions) of two samples of haloes, split by the mark values. For each case, one sample has all haloes with mark values above 0.5, and the other has all haloes with mark values below 0.5. That is, we compare haloes in the upper half of the mark distribution with those in the lower half of the mark distribution at \\emph{fixed} mass (cf., \\autoref{fig:marks}). Note that because the mark values are assigned in mass bins, the split by mark already excludes any clustering dependence upon mass, and the bias we observe is the secondary bias.\n\nThe pair counting function is evaluated in bins of pair distance ($r$). Here we use 11 equally-spaced bins in logarithmic scale between 2 to 40\\,\\ensuremath{h^{-1}}\\,Mpc, and we have verified that our result is insensitive to the binning scheme.\nWe also estimate how much of the bias signal in the ratio of pair counting functions can come from noise. We uniformly at random assign mark values of $1\/N, 2\/N, \\ldots, 1$ to all haloes in the sample, calculate the ratio of pair counting functions for high and low randomly assigned marks, and repeat this procedure until we obtain convergence of the 99.7\\% (3$\\sigma$) distribution of the bias signal; this is  shown by the grey band in \\autoref{fig:bias}. Hence, deviation beyond this grey band indicates a secondary halo bias signal $>3\\sigma$.\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{markdist.pdf}\n\\caption{\\label{fig:markdist}%\nThe probability distributions of mark values of each of the six secondary halo properties, when only counting paired haloes. A ``paired halo'' is any distinct halo that is neighboured with at least one other distinct halo in the full halo sample within 10\\,\\ensuremath{h^{-1}}\\,Mpc. If a halo has in multiple neighbour haloes, its mark value is counted only once in the distribution shown by an orange solid line, and counted multiple times (as many as the number of its neighbours) in the distribution shown by a red dashed line. \nFor comparison, a thin horizontal black line shows a uniform distribution, which, by construction, is the distribution of the mark values when including all, paired or not, haloes. The deviation of the coloured lines from the horizontal black line shows a bias due to the secondary halo property. The grey band shows a typical 3$\\sigma$ deviation for a uniform random distribution of the same sample size.\n}\n\\end{figure*}\n\n\\autoref{fig:bias} shows that, amongst the six secondary halo properties, the two that directly quantify some aspect of the mass accumulation histories of haloes, namely the half-mass scale $\\ensuremath{a_{1\/2}}$, and the accretion rate before peak mass $\\ensuremath{\\Gamma_\\text{peak}}$, do not exhibit secondary bias; in both cases, the ratio between the the pair counts is consistent with random marks on all scales.\n\nAll other halo properties exhibit clear secondary bias. Haloes with higher concentrations are less clustered, while haloes with higher spins, more subhaloes, or a larger average subhalo distances are more clustered. In terms of the magnitude of the secondary bias, the spin parameter exhibits the strongest amongst these properties. The concentration parameter, number of subhaloes, and the average subhalo distance all produce a secondary bias of the roughly same magnitude. \\autoref{fig:bias} also shows that, the secondary bias due to the properties listed decreases with scale. Nevertheless, between scales of 2 to 40\\,\\ensuremath{h^{-1}}\\,Mpc, this secondary bias is always statistically significant. Moreover, on all of the scales that we have considered, the secondary bias induced by any individual property never crosses the black horizontal line at unity, meaning that the secondary bias is always of the same sense. \n\nThis result is broadly consistent with previous studies. In this mass regime, the inverted concentration bias has been shown in \\citet{Wechsler2006}, the strong spin bias has been shown in \\citet{Gao2007}, the lack of assembly bias when split by formation time has been shown in \\citet{Gao2005,2008MNRAS.389.1419L}, and the bias due to subhalo distance has been shown in \\citet{More2016}. Our study confirms that the secondary bias signals in previous studies can be reproduced with a larger-volume and higher-resolution simulation. {However, we caution the reader that we present the pair counting ratio in \\autoref{fig:bias}, but this quantity \ndoes not directly translate into the ratio of linear biases of the subsamples. We will not quote linear biases in this work because the \nMDPL2 particle data are not publicly available.}\n\nIt is interesting to note that the term ``assembly bias'' has been widely use to refer to any secondary halo bias, despite the fact that {when splitting cluster-size haloes by their half-mass scales or recent accretion rates, the two sample have indistinguishable halo clustering properties}%\n\\footnote{We note that for low-mass haloes, which we do not address in this work, {there exists assembly bias even with this strict definition}.}.\nThe reason for this nomenclature is very likely due to the common understanding that halo assembly history is correlated with many halo properties, including concentration, spin, and so on. Nevertheless, the fact is that {some} direct measures of halo assembly history (e.g., $\\ensuremath{a_{1\/2}}$ and $\\ensuremath{\\Gamma_\\text{peak}}$) result in no significant secondary bias at this mass regime, and this causes the ``assembly bias'' nomenclature to be potentially confusing and misleading. We will discuss this seemingly counter-intuitive result further in \\autoref{sec:discussion}. \n\n\\subsection{Secondary halo bias through the demography of paired haloes}\n\\label{sec:markdist}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.9\\textwidth]{scatter.pdf}\n\\caption{\\label{fig:scatter}%\nA matrix of two-dimensional distributions of the mark values for each pair of the six secondary halo properties. In each cell, the $x$- and $y$-axes both go from 0 to 1 and show the mark value of the corresponding labels. The lower triangular cells of the matrix (with blue points) depict the distributions when only including haloes that are \\emph{not} in pairs, while the upper triangular cells (with orange points) exhibit the distributions including only paired haloes. Paired haloes are those with at least one other distinct halo within 10\\,\\ensuremath{h^{-1}}\\,Mpc. The number in each cell is the value of the Spearman correlation coefficient. In the absence of correlations, the absolute value of Spearman correlation coefficient would be smaller than $0.03$ at the $99\\%$ level  ($3\\sigma$) for a sample the size of our halo sample. Diagonal cells are left blank as only a trivial perfect correlation would appear in those cells. For each cell in the upper triangular portion of the matrix, the marginal distribution of the points would match the orange solid lines in \\autoref{fig:markdist}.\n}\n\\end{figure*}\n\nA different, yet arguably more intuitive, way to demonstrate the halo bias is to first split the halo sample by whether or not the halo contributes to the correlation (pair counting) function. Here we define a ``paired'' halo as any distinct host halo that is within 10\\,\\ensuremath{h^{-1}}\\,Mpc of another cluster-size distinct halo, and an ``unpaired'' halo is an isolated halo that has no other cluster-size distinct haloes within 10\\,\\ensuremath{h^{-1}}\\,Mpc.\nThe choice of 10\\,\\ensuremath{h^{-1}}\\,Mpc is, to some extent, arbitrary, but we have verified that our results are insensitive to the choice of this radius in the range of 5--20\\,\\ensuremath{h^{-1}}\\,Mpc. \nOnce we divide the full halo sample into paired and unpaired subsets, we can inspect the difference between the demography of these two samples, and the difference is a manifestation of halo secondary biases.\n\nWe start with \\autoref{fig:markdist}, showing the mark distribution of the six secondary halo properties for paired haloes only. If paired haloes form an unbiased subset of all haloes, the mark distribution of only paired haloes should be identical to the distribution of the full sample, which \\emph{by construction} is uniformly distributed between $[0,1]$. \nConsequently, the deviations from the uniform distributions in \\autoref{fig:markdist} highlight the secondary halo bias. Here we find that paired haloes are strongly biased towards lower concentration and higher spin, somewhat biased towards higher number of subhaloes and larger average subhalo distance, and nearly unbiased in half-mass scale and accretion rate before peak mass. All signals are consistent with the secondary bias signals in \\autoref{fig:bias}. To verify that these signals are robust, we also show the mark distribution for all paired haloes with repeated counting, to account for the fact that some haloes contribute to multiple pairs and those haloes contribute more to the bias function. Nevertheless, in both counting schemes, the results are generally the same.\n\nNote that this ``demography of paired haloes'' approach, demonstrated with \\autoref{fig:markdist}, contains similar, but not identical, information to the bias function approach shown in \\autoref{fig:bias}. For example, the cross-correlation between high- and low-mark haloes does not contribute to \\autoref{fig:bias}, while any halo in pairs contributes to \\autoref{fig:markdist}. Furthermore, \\autoref{fig:markdist} helps us to understand the demography of paired haloes by allowing us to inspect the actual mark distribution for paired haloes. We can see that, for instance, the distributions of marks that exhibit secondary bias, are generally monotonic with respect to the mark values. This feature indicates that the secondary bias is not due to intermediate mark values but mostly due to high- or low-end tails in the mark distribution. In the next section we will further see the usefulness of this ``demography of paired haloes'' approach. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Correlations between halo properties}\n\\label{sec:scatter}\n\nThe term ``halo assembly bias'' has been widely used to refer to any kind of secondary halo bias. While this is an issue of nomenclature, the extensive use of the term ``halo assembly bias'' has a significant drawback, as it implicitly suggests that all secondary halo bias may have originated from differences amongst the assembly histories of haloes. However, for haloes with the characteristic masses of galaxy clusters, we do not find significant secondary bias for \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}, the two halo properties that we study which are directly related to halo assembly histories. This finding is in broad agreement with the detailed findings in the previous literature, though not always with the physical interpretation of them. In particular, despite the lack of assembly bias in its strict sense, there is still clear secondary halo bias for other properties that are correlated with halo assembly history, such as halo concentration. How can we understand better these counter-intuitive and seemingly contradictory results? \n\nWe start by inspecting the correlations amongst the secondary halo properties. \\autoref{fig:scatter} shows the scatter plot and the Spearman correlation coefficient between the mark values for each pair amongst the six secondary properties. \\autoref{fig:scatter} shows these distributions for two subsets of haloes, namely paired haloes only (upper triangular cells in orange) and unpaired haloes only (lower triangular cells in blue). In order to interpret \\autoref{fig:scatter}, it is useful to consider that in the absence of correlations, the Spearman correlation coefficient for a sample of the size of our halo subsets would be limited to an absolute value less than 0.03 at the 99.7\\% ($3\\sigma$) level. Therefore, for example, the correlation coefficient of $-0.06$ describing the correlation between $\\lambda$ and $\\ensuremath{R_\\text{mem}}$ is a weak, but likely real, correlation. Correlation coefficients with a magnitude larger than this are very highly significant.\n\nAt first glance, most correlations are as expected based upon the previous literature. The half-mass scale \\ensuremath{a_{1\/2}}{} and the accretion rate before peak mass \\ensuremath{\\Gamma_\\text{peak}}{} are strongly correlated, because the assembly histories of haloes are rather universal and in most cases can be well described by a one-parameter function \\citep{Wechsler2002,Wu2013}. \nHalo concentration \\ensuremath{c}{} is well correlated with both \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}, consistent with the finding of \\citet{Wechsler2002}. Halo concentration \\ensuremath{c}{} is also fairly correlated with the number of subhaloes \\ensuremath{N_\\text{sub}}{}, consistent with the finding of \\citet{Zentner2005,Mao2015}. The halo spin parameter \\ensuremath{\\lambda}{} is correlated with \\ensuremath{c}{}, \\ensuremath{a_{1\/2}}{}, and \\ensuremath{\\Gamma_\\text{peak}}{}, also consistent with the finding of \\cite{2007MNRAS.378...55M}. \nThe average subhalo distance does not exhibit strong correlations with the other five halo properties, but correlates somewhat weakly with \\ensuremath{a_{1\/2}}{} and \\ensuremath{\\Gamma_\\text{peak}}{}; this is consistent with the findings of \\citep{More2016}.\n\nA more unexpected feature of \\autoref{fig:scatter} is that the correlations amongst these halo properties \\emph{seem} to be nearly the same for paired and unpaired haloes, in terms of both the features in the scatter plots and the values of the Spearman correlation coefficients. This may be counter-intuitive as one might naively expect that if two properties are well correlated (such as concentration and half-mass scale), then both of them should result in secondary biases of roughly the same magnitude. However, this naive expectation is mathematically unfounded. As we demonstrate next, the distribution of the orange points (paired haloes) differs slightly from the distribution of the blue points (unpaired haloes). Because of this small difference, the correlation between two secondary properties, say concentration and half-mass scale, is far from a guarantee of that these two properties would result in similar secondary biases. \n\n\n\\subsection{Correlation does not imply secondary bias}\n\\label{sec:demo}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{demo.pdf}\n\\caption{\\label{fig:demo}%\nAn illustration of two highly correlated variables that result in different biases when a subset is selected. The main panel shows the two-dimensional distribution (corresponding to \\autoref{fig:scatter}) of two fictitious variables $X$ and $Y$, both uniformly distributed between $[0,1]$ for all points. The top and right-hand panels show the marginal probability distributions (corresponding to \\autoref{fig:markdist}) of $X$ and $Y$, respectively. A subset of points (shown in orange, approximately 40\\% of the total points) exhibit bias in $Y$ but not in $X$.  The numbers in parentheses are the Spearman correlation coefficient.\n}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.8\\textwidth]{c-ahalf.pdf}\n\\caption{\\label{fig:c-ahalf}%\nLeft panel shows the two-dimensional distribution of the marks on the concentration parameter (\\ensuremath{c}{}) and the half-mass scale (\\ensuremath{a_{1\/2}}{}), for unpaired (blue points) and paired (orange points) haloes. The distributions are the same as those in \\autoref{fig:scatter}, but here we overlay them for a direct comparison. Right panel shows the mean value of the \\ensuremath{c}{}-mark, conditioned on the \\ensuremath{a_{1\/2}}{}-mark; the black dashed, blue solid, and orange solid lines show the sample of all, paired, and unpaired haloes, respectively. \nThe standard error of the mean is shown by error bars, but too small to be seen.\n}\n\\end{figure*}\n\nTo demonstrate this important statement that two highly correlated halo properties can result in distinctly different secondary biases, consider a fictitious sample of 30,000 points that are described by two highly correlated variables $X$ and $Y$. The variables $X$ and $Y$ yield a Spearman correlation coefficient of $-0.90$, far more significant than the correlations amongst any of our halo secondary properties aside from the correlation of $\\ensuremath{a_{1\/2}}$ with $\\ensuremath{\\Gamma_\\text{peak}}$. From amongst these 30,000 points, we select approximately 40\\% of the total points and put them in a subset $S$. \\autoref{fig:demo} shows the two-dimensional and marginal distributions of $X$ and $Y$, for both points in and out of the subset $S$. We can see that the subset $S$ exhibits bias in $Y$ but not in $X$, despite the very strong correlation between $X$ and $Y$. Of course, the selection of the points in $S$ is not random. The subset $S$ is constructed by preferably selecting points with lower $Y$ in bins of $X$, and hence it is by construction that this subset results in a different bias in $X$ and $Y$. Mathematically speaking, for the full sample we have $P(X)=P(Y)=1$ and $P(X,Y)=P(X|Y)=P(Y|X)$. However, for the subset $S$, we alter the conditional distribution $P_{S}(Y|X)$ so that it differs from $P_{S}(X|Y)$, and hence the marginal distributions $P_S(X)$ and $P_S(Y)$ differ.\n\nThe two-dimensional distribution in \\autoref{fig:demo} is an analogy for \\autoref{fig:scatter}, and the marginal distributions in \\autoref{fig:demo} are analogies for \\autoref{fig:markdist}. The subset of points $S$ is analogous to the subset of paired haloes. In fact, the marginal distributions for each cell in the upper triangular part of \\autoref{fig:scatter} exactly corresponds to the mark distributions of paired haloes, shown by the orange solid lines in \\autoref{fig:markdist}. Hence, even though by eye the two-dimensional distributions of two marks for paired and unpaired haloes look very similar, the small difference between them can result in markedly different secondary clustering biases.\n\nTo take an even closer look at the particular case of halo concentration and half-mass scale, we overlay the two-dimensional distributions of the mark values of halo concentration and half-mass scale for paired and unpaired haloes, and show them in the left-hand panel of \\autoref{fig:c-ahalf}. Clearly, $\\ensuremath{a_{1\/2}}$ and $\\ensuremath{c}$ are similarly correlated for both subsamples. Indeed, they have Spearman correlation coefficients that are consistent with each other given the sample size. However, we can already see the small difference between the two-dimensional distributions for paired and unpaired haloes by eye. To quantify this difference, in the right-hand panel of \\autoref{fig:c-ahalf}, we show the mean value of the concentration mark, conditioned on half-mass scale mark, for the sample of all, paired, and unpaired haloes.\nWe find that, although concentration and half-mass scale are similarly correlated for paired and unpaired haloes, paired haloes have a slightly lower concentration at a given half-mass scale than unpaired haloes, and this small difference gives rise to their different behaviours in secondary bias.\n\nAt this point, it may be useful to summarize the phenomenology of \\ensuremath{c}{}- and \\ensuremath{a_{1\/2}}{}-dependent halo clustering. Haloes exhibit secondary bias based on \\ensuremath{c}{} because paired haloes constitute a subset of haloes with preferentially lower concentrations (in the mass range we consider). However, this subset of paired haloes also has the \\emph{same} distribution of \\ensuremath{a_{1\/2}}{} as unpaired haloes (as shown in \\autoref{fig:markdist}). Consequently, high-mass haloes exhibit \\ensuremath{c}{}-dependent secondary bias, but do not exhibit $\\ensuremath{a_{1\/2}}$-dependent (or $\\ensuremath{\\Gamma_\\text{peak}}$-dependent) secondary bias despite \\ensuremath{a_{1\/2}}{} and \\ensuremath{c}{} being strongly correlated. Selecting haloes based upon whether or not they have a neighbour alters the conditional distributions $P(\\ensuremath{c}|\\ensuremath{a_{1\/2}})$ and $P(\\ensuremath{a_{1\/2}}|\\ensuremath{c})$, in a highly non-trivial manner. The underlying physical reasons for this shift in halo properties induced by proximity to neighbour haloes are not immediately apparent, but our work suggests that halo properties are related to environment in a manner that is considerably more complex than is commonly assumed.\n\nInterestingly, the story of $\\ensuremath{R_\\text{mem}}$ is nearly the opposite of that for \\ensuremath{a_{1\/2}}{}. While \\ensuremath{R_\\text{mem}}{} exhibits a secondary bias similar to \\ensuremath{c}, \\ensuremath{\\lambda}, and \\ensuremath{N_\\text{sub}}{} (see \\autoref{fig:markdist}), it is only weakly correlated with those three properties (see \\autoref{fig:scatter}), for both paired and unpaired haloes. The lack of correlation of two variables by no means implies they cannot have similar or even the same marginal distribution, which is indeed what happens here. For example, the mark distributions of \\ensuremath{R_\\text{mem}}{} and \\ensuremath{N_\\text{sub}}{} for paired haloes are very similar, but these two properties are also the least correlated amongst the properties studied here. \n\nAs a consequence, one should be very cautious when studying the secondary bias due to one halo property through the use of a different halo property as a proxy. Even when the two properties, namely the halo property of interest and the proxy, have been shown to be strongly correlated, it is not true that the two properties must exhibit similar secondary biases. Likewise, when two properties both exhibit similar secondary biases, they may still have little correlation with each other.\n\n\n\\subsection{Implication for galaxy assembly bias}\n\\label{sec:galaxy-bias}\n\nThe term ``assembly bias'' is also frequently used to refer to ``galaxy assembly bias,'' which does not have a single clear definition. In most contexts galaxy assembly bias means that the clustering properties of galaxies depend on some galaxy properties at a fixed host halo mass. With what we have learned here, we shall take a closer look at the idea of galaxy assembly bias. Consider a galaxy property $G$, which depends on a halo property $H$. If $G$ and $H$ are highly correlated and their conditional distributions $P(G|H)$ and $P(H|G)$ are not altered by the presence of nearby haloes (e.g., in the context of our examples, $P(G|H)$ and $P(H|G)$ stay the same for both paired and unpaired haloes), then the secondary halo bias due to $H$ induces a ``galaxy assembly bias'' due to $G$. However, if the conditional distributions $P(G|H)$ and $P(H|G)$ are altered for haloes in pairs, such as $(X,Y)$ and $(\\ensuremath{a_{1\/2}},\\ensuremath{c})$ in our examples above, then the secondary halo bias due to $H$ does \\emph{not} guarantee any bias signal due to $G$. Similarly, any bias signal due to $G$ cannot be used to infer the existence of an underlying secondary halo bias due to $H$. Likewise, seeing both biases due to $G$ and $H$ does not guarantee a correlation between the galaxy property $G$ and the halo property $H$.\n\nIn short, the correlation between two variables and the dependence of clustering properties on these two variables do not have a firm connection. This statement is particularly evident for the secondary halo biases for cluster-size haloes, but it is a general, mathematical statement. Hence, using ``assembly bias'' to refer to different kinds of secondary halo bias and even bias in galaxy clustering is potentially misleading. \n\n\\subsection{Physical robustness of the secondary halo bias signals}\n\\label{sec:robust}\n\nDespite some of the counter-intuitive and seemingly contradictory results that we have presented, the secondary halo biases we show in this work are, in fact, physically robust. One may suppose that either \\ensuremath{a_{1\/2}}{} or \\ensuremath{\\Gamma_\\text{peak}}{} or both do not exhibit significant secondary halo clustering bias because they are measured with considerably more noise than the other halo properties that we explore or because these measures do not probe the particularly important epochs of halo formation. As we will show, this is not the case. Moreover, the existence of other secondary halo biases (due to concentration, spin, and subhalo properties) is also physically robust. \n\nTo demonstrate the robustness of the secondary halo biases that we present above we proceed as follows. For each of the six secondary properties, we identify another, similar property that has approximately the same physical meaning but defined differently. We then investigate whether or not this similar halo property exhibits a similar secondary halo bias. This new set of six properties are as follows.\n\n\\begin{enumerate}\n\\itemsep0.5\\baselineskip\n\n\\item Maximal circular velocity (\\ensuremath{V_\\text{max}}). For a halo that follows a NFW profile perfectly, the maximal circular velocity is a simple function of halo mass and concentration. At fixed mass, higher \\ensuremath{V_\\text{max}}{} implies higher halo concentration.\n\n\\item Spin parameter (\\ensuremath{\\lambda_\\text{Bullock}}), as defined in \\citet{Bullock2001}, which has a different normalization compared to the Peebles spin parameter.  \n\n\\item Scale of last major merger(\\ensuremath{a_\\text{LMM}}), defined as the scale factor at which the halo experiences its last major merger on its main branch. A major merger is defined as a merging event of two haloes with a mass ratio that is larger than one-third. \n\n\\item Present-day instantaneous accretion rate (\\ensuremath{\\Gamma_\\text{inst}}), it is defined as the mass change rate between two adjacent snapshot outputs on the main branch. For MDPL2, this rate is calculated between $z=0$ and 0.0224, which corresponds to approximately 300\\,Myr.\n\n\\item Peak maximal circular velocity of the largest subhalo (\\ensuremath{V_\\text{peak}^\\text{(1st sub)}}). Here largest subhalo means the subhalo that has the largest \\ensuremath{V_\\text{peak}}{} values of all subhaloes in that distinct halo. At a fixed distinct halo mass, this value is correlated with the concentration of the host halo and with the number of subhaloes \\citep{Mao2015}. \n\n\\item Average subhalo distance weighted by subhalo mass (\\ensuremath{R_\\text{mem}^\\text{(weighted)}}), defined as the average three-dimensional distance between all subhaloes and the centre of the main halo, with the contribution of each subhalo weighted by the subhalo mass. The subhalo definition is the same as the definition used to calculate the number of subhaloes for each distinct halo.\n\n\\end{enumerate}\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{bias_more.pdf}\n\\caption{\\label{fig:bias-more}%\nSame as \\autoref{fig:bias} but for six different (but related) secondary halo properties: (solid lines from top to bottom) \\ensuremath{\\lambda_\\text{Bullock}}, \\ensuremath{R_\\text{mem}^\\text{(weighted)}}, \\ensuremath{V_\\text{peak}^\\text{(1st sub)}}, \\ensuremath{a_\\text{LMM}}, \\ensuremath{\\Gamma_\\text{inst}}, and \\ensuremath{V_\\text{max}}.\nFor \\ensuremath{V_\\text{max}} (blue solid line), we also show the inverse ratio as a blue dashed line to guide an easy comparison with bias due to other secondary halo properties.\n}\n\\end{figure}\n\n\n\\autoref{fig:bias-more} shows the secondary bias (as ratio of pair counting functions) for these six secondary properties, and the behaviours of the secondary biases are very similar to the six properties used in \\autoref{fig:bias}. The spin parameter with \\citet{Bullock2001} definition still exhibits the largest bias. The maximal circular velocity exhibits the second most significant secondary bias, similar to, but slightly stronger than, the secondary bias exhibited by the concentration parameter. The two assembly history-related properties (\\ensuremath{a_\\text{LMM}}{} and \\ensuremath{\\Gamma_\\text{inst}}{}) again exhibit clustering that is consistent with random sampling and do not indicate secondary halo bias based upon halo mass assembly history. Lastly, the two subhalo-related properties (\\ensuremath{V_\\text{peak}^\\text{(1st sub)}}{} and \\ensuremath{R_\\text{mem}^\\text{(weighted)}}{}) show moderate secondary bias. \n\nAs our discussion in \\autoref{sec:demo} points out, correlation does not imply similar clustering bias. Hence the similar secondary bias signals we observe in \\autoref{fig:bias} and \\ref{fig:bias-more} can be interpreted to indicate that both properties in each pair affect the clustering properties. It also implies that the two conditional distributions of each pair of properties stay roughly the same for paired and unpaired haloes. In other words, the small change in definitions, in these particular cases, does not alter the joint distribution in a way that would result in different marginal distributions for paired or unpaired haloes (see \\autoref{fig:conditional-mean} for a full comparison of the conditional distributions amongst all properties). \n\n\n\\begin{figure*}\n\\centering\\includegraphics[width=\\textwidth]{history.pdf}\n\\caption{\\label{fig:history}%\nStacked main-branch assembly histories for halo mass $M(a)\/M(a=1)$ (upper row) and for maximal circular velocity $\\ensuremath{V_\\text{max}}(a)\/\\ensuremath{V_\\text{max}}(a=1)$ (lower row).\nIn each panel, the full sample of cluster-size haloes is split to show the difference between the stacked assembly histories of the two subsamples. \nIn the three columns, the sample is split by concentration mark (left column; orange dashed line for the 50\\% low-concentration haloes and blue solid line for the 50\\% high-concentration),\nby whether or not the haloes reside in pairs (middle column; orange dashed line for paired haloes and blue solid line for unpaired), \nand by large-scale density mark (left column; orange dashed line for the 50\\% haloes in high-density regions and blue solid line for the 50\\% haloes in low-density regions).\nHere paired haloes are haloes that have neighbour haloes within 10\\,\\ensuremath{h^{-1}}\\,Mpc), and the large-scale density is calculated by summing the total mass$^\\text{\\ref{footnote:halo-proxy}}$ within a 20\\,\\ensuremath{h^{-1}}\\,Mpc-radius sphere.\nFor each subsample under consideration, the line shows the median mass assembly history, and the corresponding band shows the 16$^\\text{th}$ and 84$^\\text{th}$ percentiles.\nThis plot is made with the DarkSky-Gpc simulation. \n}\n\\end{figure*}\n\n\\subsection{{Full halo assembly histories and bias}}\n\\label{sec:history}\n\nSo far, we have inspected four different summary statistics of halo mass assembly history: \\ensuremath{a_{1\/2}}, \\ensuremath{a_\\text{LMM}}, \\ensuremath{\\Gamma_\\text{peak}}, and \\ensuremath{\\Gamma_\\text{inst}}, and \\emph{none of them} exhibits any statistically significant secondary clustering bias. Given the similarity of halo mass assembly histories, which can be described well by one or two parameters, {one might \\emph{speculate} that, for high-mass haloes, the entire halo mass assembly history is independent of the environment}. With the ``demography for paired haloes'' approach that we introduced in \\autoref{sec:markdist}, we can directly inspect the mass assembly history for haloes {in different environments to further investigate the relationship between bias and assembly history.}\n\nThe upper row of \\autoref{fig:history} shows the stacked (median) mass assembly histories for main-branch progenitors as a function of scale factor, $M(a)\/M(a=1)$, using three different ways to split the full cluster-size halo sample into two subsamples. To produce \\autoref{fig:history}, we used the DarkSky-Gpc simulation to obtain the full mass assembly histories for all distinct haloes that have a present-day mass $\\ensuremath{M_\\text{vir}} \\geq 10^{14}\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}$. On the left of \\autoref{fig:history}, we split haloes by their concentration mark. In the middle, we split haloes by whether or not they are in pairs; here a ``paired halo'' is again defined as any distinct halo that has at least one other distinct halo closer than 10\\,\\ensuremath{h^{-1}}\\,Mpc. On the right, we split haloes by their {mass-normalized mark values of} large-scale matter density; we calculate the matter density by summing up the masses of all resolved haloes%\n\\footnote{We calculate the mass by summing up halo masses within 20\\,\\ensuremath{h^{-1}}\\,Mpc spheres because we do not have direct access to the full particle snapshot of DarkSky-Gpc. To make this approximation as close as the actual matter distribution, we use all distinct haloes identified in the DarkSky-Gpc halo catalogue, down to a minimal halo mass of $4.88 \\times 10^{10}$\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}{} (20 particles). These haloes, in total, contain 42.5\\% of the total matter mass in the simulation box. For our purpose of ranking the large-scale matter densities, this method provides good approximation, as we have verified using independent simulations. \\label{footnote:halo-proxy}}\nwithin a 20\\,\\ensuremath{h^{-1}}\\,Mpc sphere around each distinct halo in our sample, and then calculate the mass-normalized mark values using the procedure outlined in \\autoref{sec:marks}.\n\nWe can immediately see that the stacked mass assembly histories for paired and unpaired haloes are \\emph{essentially identical}. The 16$^\\text{th}$ and 84$^\\text{th}$ percentiles also match between the two samples, indicating that the variety of possible assembly histories is quite similar for both paired and unpaired haloes. \nFurthermore, when the halo sample is split by the large-scale matter density around the haloes, haloes in denser regions have nearly identical stacked assembly history as those haloes in less dense regions, as shown in the upper right-hand panel of \\autoref{fig:history}.\nThe lack of difference between the assembly histories of paired and unpaired haloes (or of haloes in high- and low-density regions) is \\emph{not} caused by the stacking procedure. As the upper left-hand panel of \\autoref{fig:history} shows, when the haloes are split by the concentration mark, there is a clear difference in the stacked assembly histories. The trend is consistent with our expectation, that high-concentration haloes form early. {One should also note that these two groups of haloes \\emph{do} have different clustering biases, as we know the concentration bias does exist at this mass scale.}\n\nWe have also verified that the lack of difference in the assembly histories for haloes in different environments is insensitive to the radii used in the definitions of paired haloes and large-scale density (the result holds when using {5, 10, 20, 30, and 40}\\,\\ensuremath{h^{-1}}\\,Mpc), and is also insensitive to how we split the density mark (the result holds when selecting the 25\\% or 10\\% most extreme mark values).\nIn addition, we further inspect the history of the maximal circular velocity (\\ensuremath{V_\\text{max}}) for main-branch progenitors, which represents the mass assembly history for the core of a halo. As the lower row of \\autoref{fig:history} shows, we again find that the stacked \\ensuremath{V_\\text{max}}{} histories of the paired and unpaired haloes, or of haloes in high- and low-density regions, are nearly identical. The concentration-split histories show a difference that is consistent with the difference in mass assembly history. We also find that the probability distributions of the number of major mergers (mass ratio between $1\/3$ and 1) that happened along the main branch are also essentially the same for paired and unpaired haloes.\n\nThese findings are in good agreement with our main results and also with \\citet{Gao2005} and \\citet{2008MNRAS.389.1419L}. {At this mass scale ($\\gtrsim 10^{14}$\\,\\ensuremath{h^{-1}}\\text{M$_\\odot$}), haloes that are in different environments do not have significantly different assembly histories.\nHowever, this does not guarantee that a correlation of any form between large-scale halo clustering and halo assembly history does not exist. Our results demonstrate that any such correlation is not evident either from the summary statistics that have been explored here or in the stacked assembly histories. The reasons for this phenomenon are twofold. First, at this mass scale, the strength of the secondary bias is small, and hence even if the $c-\\ensuremath{a_{1\/2}}$ relation were not biased for haloes in denser environments, the difference in the stacked assembly histories of haloes in different environments would still be modest. Secondly, as we discussed in \\autoref{sec:demo}, the $c-\\ensuremath{a_{1\/2}}$ relation is slighted biased for haloes in denser environments, and it cancels out any remaining difference in the assembly histories for haloes in different environments.}\n\n{It is possible to estimate the relative sizes of these two effects. In \\autoref{fig:history_diff}, we plot the relative differences between the stacked mass assembly histories of subgroups of haloes split by various halo properties. In each case, we split the samples into two equal-sized subgroups about a mark value of 0.5. For example, the green line in \\autoref{fig:history_diff} shows the normalized difference between the blue and orange lines in the upper left-hand panel of \\autoref{fig:history}. Splitting haloes by their \n\\ensuremath{c}{}, \\ensuremath{a_{1\/2}}{}, or \\ensuremath{\\Gamma_\\text{peak}}{} (green solid, orange dash--dotted, and red dotted lines respectively), yields significant differences in mass assembly histories, as one would expect. On the other hand, splitting haloes by large-scale density (blue solid line) yields assembly histories that are extraordinarily similar. This is another representation of our previous results. \n}\n\n{\nWe can now explore what would be expected of the mass assembly difference between haloes selected by density due only to the fact that density is correlated with concentration. To compute this, we place haloes into narrow bins of concentration mark and then randomly shuffle the density marks. We then split haloes based upon the shuffled density mark. In this manner, the selection upon density becomes meaningless, but the two sub-populations have concentration mark distributions that are identical to the subgroups split on actual density. The mass accretion history difference constructed in this way is shown by the purple dashed line in \\autoref{fig:history_diff}. This line shows the slightly different mass accretion histories of haloes due to the fact that selecting upon density also introduces small differences in the concentration distributions of the two halo subgroups. While this difference is small relative to selecting upon concentration or formation time directly, it is interesting that this difference is significantly larger than the difference in mass accretion histories induced by selecting on {\\em actual} density. Despite the fact that selecting upon density also selects populations with slightly different concentration distributions, selecting upon {\\em actual} density yields a nearly undetectable difference in mass accretion histories. The relation between concentration and mass accretion history is density dependent in such a way as to render the different mass accretion histories nearly identical. The difference in mass accretion histories exhibited by the purple dashed line in \\autoref{fig:history_diff} is also what we would have observed if the secondary bias due to \\ensuremath{a_{1\/2}}{} or \\ensuremath{\\Gamma_\\text{peak}}{} were as significant as the concentration bias.\n}\n\n{A common assumption has been that the secondary bias due to halo concentration is merely a consequence of the simple halo assembly bias. In other words, it has been commonly believed that secondary bias due to concentration was induced by the combination of assembly bias and the formation time--concentration relation. Given what we have learned here, the concentration bias seems more intriguing at this mass scale. In fact, one extremely interesting feature in \\autoref{fig:history_diff} is that when the haloes are split by the concentration mark, the difference in the mass assembly history {is similar to} the difference when the haloes are split by half-mass scale or accretion rate {only at early time ($a < 0.35$)}. This suggests that, at this mass scale, halo concentration, as a summary statistic, {captures only early-time assembly history and is more correlated with large-scale bias.} We leave this intriguing phenomenon for future study. }\n\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\columnwidth]{history_diff.pdf}\n\\caption{\\label{fig:history_diff}%\n{Similar to \\autoref{fig:history} but showing the relative difference in the stacked (median) main-branch mass assembly histories for two groups of haloes, for several different ways to split the haloes: by mark values of large-scale (20 \\ensuremath{h^{-1}}\\,Mpc) density (blue solid, bottom-most), concentration (green solid, top-most), half-mass scale (orange dash--dotted), or accretion rate after peak mass (red dotted lines), and shuffled density (purple dashed line); all splits are made at the mark value of 0.5. The case when the haloes are split by shuffled density, each of the two groups has the same concentration distribution as in the case of split by large-scale density. \nThis plot is made with the DarkSky-Gpc simulation. }\n}\n\\end{figure}\n\n\n\\subsection{Secondary biases due to subhalo properties}\n\nIn \\autoref{fig:bias} and \\ref{fig:bias-more} we see that the number of subhaloes and the average subhalo distance (both weighted and unweighted) give significant secondary bias, and that the peak maximal circular velocity of the largest subhalo, which at a fixed distinct halo mass represents the gap between the first subhalo and the distinct halo, also exhibits a modest secondary bias signal.\nSimilar to halo concentration, both the number of subhaloes and the average subhalo distance are correlated with the assembly history of the parent distinct halo, and both of them exhibit secondary bias despite the lack of assembly bias. This result is in good agreement with the findings in \\citet{More2016}. \n\nThe secondary halo biases due to these subhalo properties are particularly interesting because they are more likely to be directly observable; at this mass scale, most large subhaloes would host galaxies.\nFor example, if the way galaxies \\emph{populate} subhaloes is not influenced by large-scale environment (although there is no clear evidence for such a dependence, it could exist), then the secondary bias due to subhalo occupation would directly translate to observable galaxy occupation bias (i.e., the dependence of central galaxy clustering on the member occupation or richness). Similarly, the average galaxy member distance will exhibit a bias signal if the galaxy--subhalo connection is not influenced by large-scale environment.\n\nThere are, however, complications to these potential observables. First, as we have discussed in \\autoref{sec:galaxy-bias}, the galaxy occupation bias or the average galaxy distance bias does not directly translate to the halo concentration bias or the halo assembly bias (in fact, we have already shown the latter does not exist for cluster-size haloes).\nSecondly, observationally, projection effects and redshift-space distortions can contaminate member assignment, and potentially produce artificial bias signal \\citep{2016arXiv161100366Z,1702.01682}. \nHence, one should be cautious when interpreting the galaxy occupation bias or the average galaxy distance bias.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this study, we have revisited the complex phenomenon of secondary halo bias, which is commonly referred to as ``halo assembly bias,'' for cluster-size haloes ($\\ensuremath{M_\\text{vir}} \\geq 10^{14}\\,\\ensuremath{h^{-1}}\\,\\text{M$_\\odot$}$) using large-volume simulations. As part of this investigation, we presented a novel approach to highlight secondary halo bias. Our approach was to study the demographics of paired and unpaired haloes, enabling us to determine whether or not the distributions of halo properties are different for haloes in pairs compared to the full halo sample. \n\nUsing both halo two-point functions and paired halo demographics, we found that halo concentration, halo spin, number of subhaloes, and average subhalo distance all exhibit significant secondary halo biases at the cluster mass scale. Amongst these properties, halo spin exhibits the strongest secondary halo bias, in the sense that high-spin haloes cluster more strongly. Low-concentration haloes cluster more weakly than high-concentration haloes, a dependence that is the converse of concentration-dependent secondary halo bias at lower masses. Cluster-size haloes cluster more strongly as a function of both subhalo number and the average distance between subhaloes and the halo centre.\n\nWe have identified no statistically significant secondary halo bias at this mass scale for any of four halo properties that directly measure the mass assembly history: half-mass scale, accretion rate before peak mass, instantaneous accretion rate, and time of last major merger. We have found that the entire main-branch mass assembly histories of paired and unpaired haloes (or haloes in different large-scale densities) are statistically identical. {This suggests that the assembly histories of massive, cluster-size haloes do not correlate with their environments in a simple fashion. This is not equivalent to the statement that there exists no features of halo assembly histories that do correlate with environment and\/or clustering strength. Indeed, some particular features of the halo assembly histories, such as those captured by halo concentration, may still correlate with the large-scale environment.}\n\nWith our halo demographic approach, we further investigated the seemingly contradictory result of the lack of secondary halo bias due to assembly history-related properties, given the clear correlation between halo concentration and halo assembly history. We have demonstrated that the correlation between two variables is, in general, not relevant to the question of whether or not the two variables will result in similar secondary halo clustering biases. If the conditional distributions of the two variables are altered, even only slightly, for paired haloes, then the two variables can easily result in very different secondary clustering biases. For instance, halo concentration and half-mass scale are similarly correlated for paired and unpaired haloes, yet paired haloes have a slightly lower concentration at a given half-mass scale, which results in their different secondary bias signals. Likewise, the fact that two variables (e.g., halo concentration and average subhalo distance) yield similar secondary clustering biases by no means implies  a correlation between the two variables. These statements have the following important consequence: Even when the presence of a galaxy in a halo is a function of a halo property that exhibits secondary halo bias, this does \\emph{not} necessarily imply galaxy ``assembly bias.''\n\nOur study has provided a comprehensive view of secondary halo biases for cluster-size haloes, and leads to a different perspective on secondary halo biases. In particular, we caution against use of the term ``assembly bias,'' particularly for cluster-size haloes.  The term ``assembly bias,'' when used to refer to secondary biases other than those due to the assembly history (e.g., concentration-dependent halo clustering), implies that all such biases have a common origin rooted in some aspect of the mass assembly histories of haloes. \n{While these different secondary biases may still all have some connections with the halo assembly history, those connections are more complex than simple correlations amongst different halo properties. \nIn particular, the halo secondary bias due to concentration is not a direct consequence of the difference in bias between early- and late-forming haloes.}\n\nOur results regarding halo assembly histories have an important consequence for the interpretation of halo clustering. At face value, our result is in qualitative agreement with early analytic studies of halo abundance and clustering using excursion set theory, which predicted no assembly bias (specifically, \\citealt{Kaiser84,Cole1989,1991ApJ...379..440B,Mo1996,Sheth2001}; see \\citealt{10.1142\/S0218271807010511} for a review and subsequent developments). Yet, these predictions stem from ad hoc assumptions adopted for computational ease, rather than for any well-established physical reasons. Several authors have proposed more physically-motivated analytic interpretations of secondary halo bias based upon the assembly histories of haloes \\citep{10.1142\/S0218271807010511,Desjacques2007,Dalal2008}. {However, given our findings, these analytic interpretations of halo assembly bias do not manifest in all different kinds of secondary halo biases in a straightforward manner. Indeed, these interpretations contradict our finding that high-mass haloes cluster independently of halo formation time and other simple metrics of halo age. The explicit relations amongst large-scale environment, halo assembly history, and sundry internal halo properties, such as concentration and spin, remain unclear.}\n\nWhile this study has demonstrated that a very complex phenomenology of secondary halo bias is mathematically possible, it has yet to provide a solid physical explanation for the existence of the concentration bias, the spin bias, the subhalo abundance bias, and the average subhalo distance bias. Previous attempts to explain these secondary biases that rely upon the existence of halo assembly bias in its restricted definition are not valid in this mass regime. It is also important to understand why the correlations between particular halo properties (e.g., between subhalo abundance and assembly history) depend on large-scale environment. We have not yet identified a plausible working theory that is able to explain all of the correlations. We hence leave this interesting problem for future study, with the hope that what we have laid out in this study will provide useful insights. \n\n\n\n\\section*{Acknowledgements}\n\nThe authors thank Andreas Berlind, Philipp Busch, Neal Dalal, \\mbox{Andrew} Hearin, Ari Maller, Surhud More, Rita Tojeiro, \\mbox{Li-Cheng} Tsai, Frank \\mbox{van den Bosch}, Antonio \\mbox{Villarreal}, and Kuan Wang for useful discussions.\nThis research made use of the MDPL2 simulation; the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.\\ (\\http{www.gauss-centre.eu}) and the Partnership for Advanced Supercomputing in Europe (PRACE, \\http{www.prace-ri.eu}) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, \\http{www.lrz.de}), and also thank Peter Behroozi for the direct access to the MDPL2 halo catalogues on SLAC servers.\nThis research made use of the DarkSky-Gpc (\\texttt{ds14\\_b}) simulation, which was part of the Dark Sky Simulations (\\http{darksky.slac.stanford.edu}) produced using an INCITE 2014 allocation on the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, a U.S.\\ Department of Energy Office; the authors thank Sam Skillman, Mike Warren, Matt Turk, and other Dark Sky collaborators for their efforts in creating these simulations and for providing access to them.\nThis research made use of computational resources at SLAC National Accelerator Laboratory, a U.S.\\ Department of Energy Office; YYM and RHW thank the support of the SLAC computational team.\nYYM is supported by the Samuel P.\\ Langley PITT PACC Postdoctoral Fellowship.\nARZ is supported, in part, by grants AST 1516266 and AST 1517563 from the U.S.\\ National Science Foundation as well as by the Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PITT PACC) at the University of Pittsburgh.\nRHW received partial support from the U.S.\\ Department of Energy contract to SLAC No.\\ DE-AC02-76SF00515.\nThis work was completed at the Kavli Institute for Theoretical Physics at the program ``The Galaxy--Halo Connection Across Cosmic Time,'' and supported in part by the National Science Foundation under Grant No.\\ NSF PHY-1125915.\n\nThis research made use of Python, along with many community-developed or maintained software packages, including\nIPython \\citep{ipython},\nJupyter (\\http{jupyter.org}),\nMatplotlib \\citep{matplotlib},\nNumPy \\citep{numpy},\nPandas \\citep{pandas},\nand SciPy \\citep{scipy}.\nThis research made use of NASA's Astrophysics Data System for bibliographic information.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{secIntro}\n\nWith deductive database technology,\nthe range of tasks a database can do is increasing:\nIt can not only fetch and compose data,\nas classical SQL queries,\nbut a larger part of the application can be specified declaratively\nin a pure logic programming language.\nIt is common that database systems offer stored procedures\nand triggers today,\nbut this program code is written in an imperative language,\nnot declaratively as queries in SQL.\nIn addition, still other (non-declarative) languages are used\nfor the application development itself.\nIt is the claim of deductive databases like LogicBlox~\\cite{Aref15}\nto unify all this.\n\nMaking deductive databases a successful programming platform\nfor future applications has many aspects,\nincluding the design of language features\nfor declarative, rule-based output\n(see, e.g.,~\\cite{Bra12}).\nBut improving the performance of query evaluation \/ program execution\nis still an important task.\nSome older deductive database prototypes\nhave been not very good from the performance viewpoint.\nTypically, people who are not yet convinced of the declarative approach\t\nask about performance as one of their first questions.\nFurthermore, the amount of data to be processed\nis constantly increasing.\nNew processor architectures offer new opportunities,\nwhich may not be utilized in old code.\n\nBenchmarks can be a useful tool to evaluate and compare the performance\nof systems.\nFurthermore,\nthey can be motivating for new system developers.\nIn the area of deductive databases,\nOpenRuleBench~\\cite{LFWK09} is a well-known benchmark suite.\nIt is a quite large collection of problems,\nbasically 12~logic programs,\nbut some with different queries and different data files.\nThe original paper~\\cite{LFWK09} contains 18~tables with benchmark results.\nThe logic programs used for the tests\n(with different syntax variants and settings for different systems),\nthe data files and supporting shell scripts can be downloaded from\n\\begin{center}\n\\url{http:\/\/www3.cs.stonybrook.edu\/~pfodor\/openrulebench\/download.html}\n\\end{center}\nThe tests have been re-run in 2010 and 2011,\nthe results are available at:\n\\begin{center}\n\\url{http:\/\/rulebench.semwebcentral.org\/}\n\\end{center}\nIn this paper,\nwe look at only five of the benchmark problems,\nbut in significantly more detail\nthan the original OpenRuleBench paper.\nWe are currently developing an implementation\nof our ``Push''-method for bottom-up evaluation~\\cite{Bra10CPP,BS15WLP},\nand would like to document and to share our insights\nfrom many preformance tests with different variants.\nWe compare our method with XSB~\\cite{SSW94},\nYAP~\\cite{CRD12},\nand dlv~\\cite{LPFEGPS06}.\nIt is also of some value that we did these measurements\nagain five years later with new versions of these systems\non a different hardware.\nWe also put more emphasis on the loading time and overall runtime\nthan the authors of the original OpenRuleBench paper~\\cite{LFWK09}.\nIn addition,\nwe did some measurements on main memory consumption.\n\nAs we understand the rules of OpenRuleBench,\nthe options for each benchmark and system\nwere manually selected in cooperation with the system developers.\nThis means that a good automatic optimization\ndoes not ``pay off'' in the benchmark results,\nalthough it obviously is of importance for the usability of the system.\nHowever,\nthis also helps to separate two different aspects:\nThe basic performance of program execution\nand having an intelligent optimizer,\nwhich finds a good query execution plan\nand index structures.\nFor systems that allow to influence query evaluation\nwith many options,\nthe optimizer becomes less important for the OpenRuleBench results.\n\nSince our query execution code\nis in part manually created\nand we still try different data structures,\nthis can be seen as an extreme case of the manual tuning.\nIt is our goal to test and demonstrate the potential\nof our approach,\nbut we are still far from a complete system.\nNaturally,\nbefore investing maybe years to develop a real deductive database system,\nit is good to see what can be reached,\nand whether it is worth to follow this path.\n\nIn contrast,\neach of the three systems we use as a comparison\nhas been developed over more than a decade.\nThey are mature systems which offer a lot of programming features\nwhich are important for usage in practical projects.\nAlthough our performance numbers are often significantly better,\nour prototype could certainly not be considered\nas a competitor for real applications.\n\nOur approach is to translate Datalog (pure Prolog without function symbols)\nto \\CPP,\nwhich is then compiled to native code\nby a standard compiler.\nWe have an implementation of this translation (written in Prolog),\nand a library of data structures used in the translated code\nwritten in \\CPP:\n\\begin{center}\n\\url{http:\/\/www.informatik.uni-halle.de\/~brass\/push\/}\n\\end{center}\nAt the moment,\nthe result of the translation must still be manually copied\nto a \\CPP\\ class frame,\nand compiler and linker are manually executed,\nbut it should be an easy task to automize this.\nThe tests reported in the current paper discuss\ndifferent code structures,\ntherefore it seemed better to first find the optimal structure\nbefore programming the complete translation.\nHowever,\nthe current implementation of the Datalog-to-\\CPP\\ translation\nalready generates different variants of the main query execution code.\n\nXSB compiles into machine code of an abstract machine\n(XWAM) and interprets that code.\nThe same is true for YAP~\\cite{CRD12}\n(the abstract machine is called YAAM).\nDLV probably uses a fully interpreted approach.\nSince we compile to native code,\nthis obviously gives us an advantage.\nIf one compares the numbers given in~\\cite{Costa99}\nfor Sicstus Prolog native code vs.~emulated bytecode,\nthe speedup for native code is between 1.5 and~5.6,\nwith median value~2.6.\nOf course,\nthe speedup also depends on the granularity of the operations\nthat are executed.\nFor instance,\nif many operations are index lookups or other large database operations,\nthese are in native code in the virtual machine emulator anyway,\nso that the overhead of interpretation is not important in this case.\nBut this means that the speedup that must be attributed\nto using native code might even be less than what is mentioned above.\n\n\n\nOne of the strengths of the ``Push'' method\ninvestigated in this paper is that it tries to avoid copying\ndata values (as far as possible).\nIn the standard implementation of bottom-up evaluation,\nthe result of applying a rule\nis stored in a derived relation.\nIn contrast,\nProlog implementations do not copy variable values\nor materialize derived literals\n(except with tabling).\nIt might be that the ``Push'' method\nmakes bottom-up evaluation competitive because it gives it something\nwhich Prolog systems already had.\nHowever,\nas we will see,\ngood performance depends on many factors\n(including, e.g., efficient data structures).\nThe message of this paper is certainly not\nthat the ``Push'' method alone could make us win benchmarks.\n\n\n\n\\section{Query Language, General Setting}\n\n\\label{secQueryLanguage}\n\nIn this paper,\nwe consider the very basic case of Datalog,\ni.e.~pure Prolog without negation\nand without function symbols\n(i.e.~terms can only be variables or constants).\nA logic program is a set of rules,\nfor example\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{}}\nanswer(X) :- grandparent(sam, X).\\\\\ngrandparent(X, Z) :- parent(X, Y), parent(Y, Z).\n\\end{tabular}\n\\end{tt}\n\\end{center}\nThis computes the grandparents of~\\code{sam},\ngiven a database relation~\\code{parent}.\nFor our bottom-up method,\nwe assume that there is a ``main'' predicate \\code{answer},\nfor which we want to compute all derivable instances.\nFor the other systems,\nthe body of the \\code{answer}-rule is directly posed as query\/goal.\n\nWe require range-restriction (allowedness),\ni.e.~all variables in the head of the rule\nmust also appear in a body literal.\nTherefore,\nwhen rules are applied from right (body) to left (head),\nonly facts are derived,\ni.e.~variable-free atomic formulas\nlike \\code{grandparent(sam,~john)}.\n\nAs usual in deductive databases,\nthe predicates are classified into\n\\begin{itemize}\n\\item\nEDB-predicates (``extensional database''),\nwhich are defined only by facts\n(usually a large set of facts\nstored in a database,\nor maybe specially formatted files).\nIn the above example,\n\\code{parent} is an EDB predicate.\n\\item\nIDB-predicates (``intensional database''),\n\\kern-0.8pt\nwhich are defined by rules.\n\\kern-0.8pt\nIn the\nexample,\n\\code{grandparent} and \\code{answer}\nare IDB predicates.\n\\end{itemize}\n\nThe execution of a logic program\/query has three distinct phases:\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\thicklines\n\\begin{picture}(156,50)(0,0)\n\\put(  0, 0){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tLogic Program\\\\\n\t\t\t\t(Rules for IDB Predicates)\n\t\t\t\t\\end{tabular}}}\n\\put( 22,20){\\vector(0,1){10}}\n\\put(  0,30){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tCompilation\n\t\t\t\t\\end{tabular}}}\n\\put( 44,40){\\vector(1,0){12}}\n\\put( 56, 0){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tData File\\\\\n\t\t\t\t(Facts for EDB Predicates)\n\t\t\t\t\\end{tabular}}}\n\\put( 78,20){\\vector(0,1){10}}\n\\put( 56,30){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tData Load\n\t\t\t\t\\end{tabular}}}\n\\put(100,40){\\vector(1,0){12}}\n\\put(112, 0){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tOutput\\\\\n\t\t\t\t(Query Result)\n\t\t\t\t\\end{tabular}}}\n\\put(134,30){\\vector(0,-1){10}}\n\\put(112,30){\\framebox(44,20){\\begin{tabular}{@{}c@{}}\n\t\t\t\tExecution\n\t\t\t\t\\end{tabular}}}\n\\end{picture}\n\\end{center}\nThe input to our transformation are the rules for the IDB-predicates,\ni.e.~these are known at ``compile time''.\nThe concrete data for the EDB-predicates\nare only known at ``run time''.\nTherefore,\nthe \\CPP\\ program resulting from our translation\ncan be executed for different database states\n(sets of facts).\nIn this way,\noptimizations or precomputations done at compile time\ncan be amortized over many program executions.\nActually,\nsince the database is usually large\nand the program relatively small,\neven a single execution might suffice\nso that work invested during the transformation ``pays off''.\n\n\nIn the OpenRuleBench tests,\nthe runtime is again split into the time required\nfor loading the data file\nand the real execution time\nwhen the data is already in main memory\n(and useful data structures have been built).\nIn the OpenRuleBench paper~\\cite{LFWK09},\nthe authors are interested only in the time for the inference\n(i.e.~the execution phase),\nand only this time is shown in the paper.\nWe show the times for both,\nthe data load and the execution,\nand the overall runtime measured externally.\nOnly in case of DLV\nit was not possible to separate this\n(we did measure the time for only loading the data,\nbut it is not clear how much of the data structure building\nis done if the data is not used).\n\nOf course,\nhandling the data load separately\nis interesting if many queries are executed later\non the data in main memory,\ni.e.~if one had a main memory DBMS server.\n\nFor the most part,\nwe did not consider the compilation time\nwhich might be considered as giving us an unfair advantage.\nSystems which have no separate compilation phase\ncannot profit from this.\n\n\\section{Data Structures}\n\n\\label{secDataStructures}\n\nWhen the facts from a data file are loaded,\nthey are stored in main-memory relations.\nA large part of the \\CPP\\ library\nwhich is used in the translation result\nimplements various data structures for main-memory relations.\nMost relations are \\CPP\\ templates\nwhich are parameterized with the tuple type.\nHowever,\nthere are a few exceptions\n(experimental\/old implementations, or data structures for very special cases).\n\nActually,\nall arguments are currently mapped to integers.\nWe have implemented a string table\nusing a radix tree similar to~\\cite{LKN13}\nin order to assign unique, sequential numbers to strings.\nIf we know that different arguments have different domains\n(which are never joined)\nwe use different such ``string tables''\nto keep the numbers dense\n(which is good for array based data structures).\n\nThere are basically three types of relation data structures:\n\\begin{itemize}\n\\item\nLists,\nwhich support access to the tuples\nwith binding pattern ``\\code{ff\\ldots f}'',\ni.e.~all arguments are free (output arguments).\nOne can open a cursor over the list and iterate over all its elements (tuples).\nLists are implemented as a tree of 4~KB pages,\neach containing an array of tuples\n(i.e.~the tuples are stored in consecutive memory locations\nwhich helps to improve CPU cache utilization).\nThe next level of the tree contains pages with pointers\nto the data pages\n(similar to the management of blocks in the original Unix file system).\n\\item\nSets,\nwhich support access to the tuples\nwith binding pattern ``\\code{bb\\ldots b}'',\ni.e.~all arguments are bound (input arguments).\nOne can insert a tuple and gets the information\nwhether the insertion was successful (new tuple)\nor not (the tuple was contained in the set already).\nWe are experimenting with different set implementations,\ncurrently:\n(a)~a simple hash table of fixed size,\n(b)~a dynamic hash table which doubles its size if it gets too full,\nand (c)~an array of bitmaps\n(some of the benchmarks have only a small domain of integers from~0 to~1000).\n\\item\nMaps (or really multimaps) for all other cases,\ne.g.~a binding pattern like~\\code{bf}:\nGiven values for the input arguments\n(in this case, the first one),\none can open a cursor over the corresponding (tuples of) values\nfor the output arguments.\nThe current state is that we have only a flexible array implementation\nfor a single integer input argument,\nbut a general map will be implemented soon.\n\\end{itemize}\n\nTemporary relations might be needed during execution\nof a logic program:\nIf a rule has more than one IDB body literal,\npreviously derived facts matching these literals must be stored\n(except possibly one of the literals,\nif we know that its facts arrive last\nand we can immediately process them).\nIn this case it is important\nthat when a cursor is opened,\nthe set of tuples it loops over is not affected by future insertions.\n\nTemporary sets are also needed for duplicate elimination\nduring query evaluation.\nFor some benchmarks,\nthis has a big influence on speed.\n\n\n\\section{The Push Method for Bottom-Up Evaluation}\n\n\\label{secPushMethod}\n\nThe push method has been introduced in~\\cite{Bra10CPP},\nand a version with partial evaluation\nhas been defined in~\\cite{BS15WLP}.\nThe following is just a quick reminder\nin order to be able to understand different variants of the code.\n\nThe basic method only works with rules with at most one IDB body literal\n(as generated, e.g., by the SLDMagic method~\\cite{Bra00}).\nRules with multiple IDB body literals\nare normalized by creating intermediate storage\nfor previously derived tuples\n(similar to seminaive evaluation with a single tuple as ``delta'' relation,\nsee~\\cite{BS15WLP}).\n\nIn the push method,\na fact is represented\nby a position in the program code\nand values of \\CPP\\ variables.\nA fact type is a positive IDB literal\nwith constants and \\CPP\\ variables as arguments,\nfor instance \\code{p(a,~v{\\U}1,~v{\\U}2)}.\nIf execution reaches a code position corresponding to this fact type,\nthe corresponding fact with the current values\nof the variables \\code{v{\\U}1} and \\code{v{\\U}2}\nhas been derived.\nNow this derived fact is immediately ``pushed''\nto rules with a matching body literal.\nA rule application consists of\n\\begin{itemize}\n\\item\na rule of the given program,\n\\item\na fact type that is unifiable with the IDB body literal of this rule,\nor a special marker ``\\code{init}'' if the rule has only EDB body literals,\n\\item\na fact type that is general enough to represent\nall facts which can be derived with the rule\napplied to the input fact type.\n\\end{itemize}\nA rule application graph consists of the fact types as nodes\n(plus an ``\\code{init}'' node),\nand the rule applications as edges.\n\nOne can use different sets of fact types.\nIn~\\cite{Bra10CPP} the fact types had the form\n\\code{p(p{\\U}1,~p{\\U}2,~\\ldots)},\ni.e.~there was one \\CPP\\ variable\nfor each argument position of each IDB predicate.\nThis requires a lot of copying in a rule application.\nThe approach of~\\cite{BS15WLP} did avoid this by computing\na set of fact types as a fixpoint of an abstract~\\m{\\TP}-operator\nworking with \\CPP\\ variables instead of real constants.\nThese variables were introduced if a data value was needed\nfrom an EDB body literal (the real values are not known at compile time).\nThis avoids copying,\nbut the set of generated fact types can ``explode''\n(as, e.g., in the wine ontology benchmark).\nTherefore,\nour current implementation supports both approaches,\nand we work on mixed variants which combine the advantages of both.\n\nThere is one code piece for each rule application.\nAt the end of this code piece,\na fact has been derived and the variables of the resulting fact type\nhave been set.\n\nOf course,\na derived fact might match body literals of several rules.\nIn that case control jumps to one such rule application,\nand the other possibilities are later visited\nvia backtracking.\nIf variables are changed when there might still be backtrack points\non the stack which need the old value,\nthese are saved on a stack, too.\nIn both of the above variants of the push method,\nthis happens only for recursive rule applications.\n\nIt is also possible that a rule application can produce several facts.\nOnly one is computed at a time and immediately used,\nother facts are computed later via backtracking.\nTherefore,\nthe code piece for a rule application has two entry points:\nOne for computing the first fact (``\\code{START}''),\nand one for computing an additional fact (``\\code{CONT}'').\nBefore control jumps to a place where the derived fact is used,\nthe ``\\code{CONT}''-label is put on the backtrack stack.\nIn case a rule application is unsuccessful\nor there is no further fact that can be derived,\nthe main loop is continued, which takes the next rule application\nfrom the backtrack stack and executes its code piece.\n\n\n\n\\section{About the Time and Memory Measurements}\n\n\\label{secBenchmark}\n\nWe compared our method with the following systems:\n\\begin{itemize}\n\\item\nXSB Prolog version 3.6 (Gazpatcho)\n\\item\nYAP Prolog version 6.2.2\n\\item\nDLV version x86-64-linux-elf-static\n\\end{itemize}\n\nBoth Prolog systems support tabling.\nThe DLV system uses a variant of magic sets transformation\nand then does bottom-up evaluation.\n\nWe executed the benchmarks on\na machine with\ntwo 6272 AMD Opteron\n16~Core~CPUs.\nHowever,\nthe current version of our program does not use multiple threads\n(we will consider this in future research).\nThe machine has 64~GB of RAM.\nThis is quite a lot,\nbut we also measure how much memory is really used by the systems.\nThe operating system is Debian \\code{x86{\\U}64} GNU\/Linux~3.2.63.\n\nThe overall execution time (``elapsed wall clock time'')\nand the memory (``maximum resident set size'')\nfor each test\nwas measured with the Linux \\code{time} program.\nThe time for loading the data and for executing the query\nare measured by functions of each system.\n\nEvery test was run ten times and the time average values were calculated. \nAdditionally, for the XSB and YAP systems the times for loading the\ndata and program files and for executing the queries were separately \nmeasured. \n\nFor the DLV system the use of the \\code{-stats} option is officially\ndiscouraged for benchmark purposes, so for estimating the loading time\nthe loading of the data files was measured once using the \n\\code{time}\nprogram; the value appears in parentheses in the tables.\n\nXSB Prolog compiles the input files to a \\code{xwam} file which can\nbe loaded quite fast, but for better comparison with the other systems dynamic\nloading was used.\n\n\n\\section{The DBLP Benchmark}\n\n\\label{secDBLP}\n\nFor the DBLP benchmark~\\cite{LFWK09},\ndata from the DBLP bibliography\n(\\code{dblp.uni-trier.de})\nis stored as a large set of facts\nfor an EAV-relation\n\\begin{center}\n\\code{att(DocID, Attribute, Value)}.\n\\end{center}\nThe file contains slightly more than 2.4~million facts of this type,\nfor instance:\n\\begin{center}\n\\code{att('node12eo5mnsvx56023',title,'On the Power of Magic.').}\n\\end{center}\nIt is about 122~MB large.\nThe test query is\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{}}\nanswer(Id, T, A, Y, M) :-\\\\\n~~~~att(Id, title,~ T),\\\\\n~~~~att(Id, year,~~ Y),\\\\\n~~~~att(Id, author, A),\\\\\n~~~~att(Id, month,~ M).\n\\end{tabular}\n\\end{tt}\n\\end{center}\nIn this case,\n\\code{att} is an EDB-predicate,\nand \\code{answer} is the only IDB-predicate.\nThe file contains data of about 215.000~documents\n(i.e.~11.3~facts per document).%\n\\footnote{%\nThe DBLP data are available as XML file from\n\\url{http:\/\/dblp.uni-trier.de\/xml\/}.\nThe DBLP has now data of 3.4~million documents,\nthe XML-file is 1.8~GB large.\nWe plan to transform this larger data file\nto the same type of Datalog facts\nas the official OpenRuleBench DBLP test file.}\n\nIt seems that the key to success for this benchmark problem\nare the data structures to represent the facts.\nWhen the data is loaded,\nwe must store it in relation data structures\nfor later evaluation of the EDB body literals.\nNote that the program\/query is known\nwhen the data is loaded,\nso we can try to create an optimal index structure for each EDB body literal.\nIn particular,\nselections for constants known at compile time\ncan already be done when the data is loaded.\nIn the example,\nall body literals contain different constants,\ntherefore we create a distinct relation\nfor each of them.\n\nE.g.~there is a relation~\\code{att{\\U}title},\nwhich represents \\code{att}-facts\nthat match\nthe first body literal\n\\code{att(Id,~title,~T)}.\nOf course, the constant is not explicitly stored,\ntherefore the relation has only two columns.\nWe use a nested loop\/index join.\nThe first occurrence of each variable binds that variable,\nand at all later occurrences,\nits value is known.\nSince this is the first body literal,\nboth variables are still free,\nand the relation \\code{att{\\U}title}\nis accessed with binding pattern~\\code{ff},\ni.e.~it is a list.\n\nFor the second body literal \\code{att(Id,~year,~Y)},\nthere is a relation~\\code{att{\\U}year}\nwhich represents only \\code{att}-facts with \\code{year} as second argument.\nAgain,\nthis argument is not explicitly represented in the stored tuples.\nSince the value of the variable~\\code{Id} is known\nwhen this body literal is evaluated,\nthe relation must support the binding pattern~\\code{bf}.\nTherefore it is a map\n(or really multimap, since we do not know that there is only one year\nfor each document).\n\nIn the same way,\nthe other two body literals are turned into exactly matching relations\nof type~\\code{bf}.\nThis means that although the loader has to look at all facts in the data file,\nit stores only those facts which are relevant for the query\n(which are\n35\\% of the facts in the data file).\n\nThe strings that occur in the data are mapped to unique, sequential integers.\nSince the three arguments of the predicate \\code{att} are never compared,\nwe use three distinct ``string tables'':\nThis makes it simpler to implement the maps based on a kind of flexible array\n--- for other map data structures, this would probably have no advantage.\nHowever,\nalso bitmap implementations of sets\nprofit from a small, dense domain.\nIn summary,\nthe main data structures are:\n\\vspace*{1mm}\n\\begin{center}\n\\begin{tabular}{@{}|l|r|r|@{}}\n\\hline\nData Structure&Rows\/Strings&Memory (KB)\\\\\n\\hline\n\\hline\nString table for Arg.~1&214\\kern1pt 905&91\\kern1pt 612\\\\\nString table for Arg.~2&19&4\\\\\nString table for Arg.~3&923\\kern1pt 881&284\\kern1pt 052\\\\\n\\hline\n\\hline\nList for \\code{att(Id,~title,~T)}&209\\kern1pt 943&1\\kern1pt 648\\\\\nMap for \\code{att(Id,~year,~Y)}&209\\kern1pt 944&844\\\\\nMap for \\code{att(Id,~author,~A)}&440\\kern1pt 934&3\\kern1pt 688\\\\\nMap for \\code{att(Id,~month,~A)}&2\\kern1pt 474&844\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{1mm}\n\n\n\nOne might expect that creating these indexes slows down\nthe loading of data\n(in effect, some time is moved from the execution phase to the load phase).\nHowever,\nour loader is quick.\nHere are the benchmark results:\n\\vspace*{1mm}\n\\begin{center}\n\\begin{tabular}{|l|r|r|r|r|r|r|}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)&Mem.~Diff.\\\\\n\\hline\n\\hline\nPush&2\\kern1pt 565\\phantom{)}&22&2\\kern1pt 610&1.0\n\t\t&385\\kern1pt 172\n\t\t&383\\kern1pt 990\\\\\nXSB&89\\kern1pt 045\\phantom{)}&2\\kern1pt 690&92\\kern1pt 390&35.4\n\t\t&415\\kern1pt 500\n\t\t&404\\kern1pt 535\\\\\nXSB (trie\n\t\n&90\\kern1pt 710\\phantom{)}&269&91\\kern1pt 275&35.0\n\t\t&380\\kern1pt 259\n\t\t&369\\kern1pt 294\\\\\n\tYAP&24\\kern1pt 878\\phantom{)}&7\\kern1pt 370&32\\kern1pt 438&12.4\n\t\t&813\\kern1pt 760\n\t\t&808\\kern1pt 911\\\\\n\tDLV&(20\\kern1pt 110)&---&25\\kern1pt 898&9.9\n\t\t&926\\kern1pt 864\n\t\t&926\\kern1pt 167\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{1mm}\nThe total time is dominated by the loading time,\nhowever the pure execution time is also interesting.\nBy the way,\njust reading the file character by character with the standard library\ntakes about the same time as our loader\n(which reads the file in larger chunks).\nIn the OpenRuleBench programs,\nXSB did not use the trie index,\t\t\t\t\t\t\t\t\t\nbut this dramatically improves execution time\t\t\t\t\t\n(the influence on the total time is not big).\n\nThe main memory consumption of our ``Push'' implementation is acceptable\n(383~MB, which is about the same as XSB,\nand less than half of YAP and XSB).\nHowever,\nsince only one third of the facts are actually stored,\nand only with a specific binding pattern,\nmemory could become an issue for other application scenarios.\n\nIn order to check how much memory was used by the data for the benchmark\nand how much is program code in the system\n(including possibly large libraries)\nwe determined the memory used by just starting and stopping\neach system (without loading data or compiling rules).\nThe result is:\n\\begin{center}\n\\begin{tabular}{@{}|l|r|@{}}\n\\hline\nSystem&Base Memory (KB)\\\\%&Mem.~Diff.~(KB)\\\\\n\\hline\\hline\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nPush&1\\kern1pt 182\\\\%&383\\kern1pt 990\\\\\nXSB&10\\kern1pt 965\\\\%&404\\kern1pt 535\\\\\nYAP&4\\kern1pt 849\\\\%&808\\kern1pt 911\\\\\nDLV&697\\\\%&926\\kern1pt 167\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nThe column ``Mem.~Diff.'' in the benchmark result table\ncontains the difference of the memory used in the benchmark\nminus this baseline.\n\n\n\n\nSince this query is a standard database task,\nwe also tried HSQLDB~2.3.4.\nIt loads all data into main memory when the system starts,\nthis took 32\\kern1pt 471~ms.\nExecuting the following SQL query took~3\\kern1pt 863~ms:\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{ }l@{}}\nSELECT&COUNT(*)\\\\\nFROM  &ATT ATT{\\U}TITLE,\n      ATT ATT{\\U}YEAR,\n      ATT ATT{\\U}AUTHOR,\n      ATT ATT{\\U}MONTH\\\\\nWHERE &ATT{\\U}TITLE.ATTRIBUTE = 'title'\\\\\nAND   &ATT{\\U}YEAR.ATTRIBUTE  = 'year'\\\\\nAND   &ATT{\\U}AUTHOR.ATTRIBUTE  = 'author'\\\\\nAND   &ATT{\\U}MONTH.ATTRIBUTE  = 'month'\\\\\nAND   &ATT{\\U}TITLE.DOC{\\U}ID = ATT{\\U}YEAR.DOC{\\U}ID\\\\\nAND   &ATT{\\U}TITLE.DOC{\\U}ID = ATT{\\U}AUTHOR.DOC{\\U}ID\\\\\nAND   &ATT{\\U}TITLE.DOC{\\U}ID = ATT{\\U}MONTH.DOC{\\U}ID\n\\end{tabular}\n\\end{tt}\n\\end{center}\n\nThe total time for starting the database, executing the query\nand shutting down the database was 40\\kern1pt 777~ms\n(the shutdown took 4\\kern1pt 330~ms).\nHSQLDB did use three threads (300\\%~CPU)\nand 3~GB of main memory\n(i.e.~three times the system resources of the deductive systems above).\nHSQLDB is written in Java\n(and was executed in the OpenJDK 64-bit Server VM: IcedTea~2.6.6, Java~1.7.0).\nWe used an index over \\code{(ATTRIBUTE, DOC{\\U}ID)}.\nWe tried also an index with the two attributes inversed,\nor all three attributes,\nbut all this did not change much.\nAlthough there might be further options for tuning HSQLDB,\nit is at least not easy\nto get much better runtimes with a relational database.\n\n\\section{The Join~1 Benchmark}\n\n\\label{secJoinOne}\n\nThe Join1 example from~\\cite{LFWK09}\ncontains the following rules:\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{}}\n\ta(X, Y)~ :- b1(X, Z), b2(Z, Y).\\\\\n\tb1(X, Y) :- c1(X, Z), c2(Z, Y).\\\\\n\tb2(X, Y) :- c3(X, Z), c4(Z, Y).\\\\\n\tc1(X, Y) :- d1(X, Z), d2(Z, Y).\n\\end{tabular}\n\\end{tt}\n\\end{center}\nThe EDB predicates are \\code{c2}, \\code{c3}, \\code{c4}, \\code{d1}, \\code{d2}.\nThere are three data files:\nOne with 10\\kern0.2em 000~facts each\n(i.e.~50\\kern0.2em 000~facts in total),\none with 50\\kern0.2em 000~facts each,\nand one with 250\\kern0.2em 000~facts each.\nThe data values are randomly generated integers between~1 and~1000.\nDifferent queries are considered in~\\cite{LFWK09},\nnamely \\code{a(X,~Y)}, \\code{b1(X,~Y)}, \\code{b2(X,~Y)},\nand the same with bound first or second argument,\ne.g.~the query~\\code{a(1,~Y)}.\nIn our test,\nwe only tried the query~\\code{a(X,~Y)}.\n\n\n\n\n\n\n\n\n\nThe benchmark results for the small data file\n(10\\kern0.2em 000~facts per EDB-predicate, 624~KB) are:\n\\vspace{-1mm}\n\\begin{center}\n\\begin{tabular}[t]{|l|r|r|r|r|r|}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)\\\\\n\\hline\n\\hline\nPush (Bitmap)&14\\phantom{)}&1\\kern0.2em 772&1\\kern0.2em 787&1.0\n\t\t&7\\kern0.2em 311\\\\\nPush (Dyn.Hashtab)&11\\phantom{)}&10\\kern0.2em 372&10\\kern0.2em 383&5.8\n\t\t&45\\kern0.2em 688\\\\\nxsb-btc&141\\phantom{)}&22\\kern0.2em 432&22\\kern0.2em 827&12.8\n\t\t&82\\kern0.2em 667\\\\\nXSB\t\t&148\\phantom{)}&24\\kern0.2em 551&25\\kern0.2em 002&14.0\n\t\t&82\\kern0.2em 526\\\\\nYAP&421\\phantom{)}&17\\kern0.2em 557&18\\kern0.2em 067&10.1\n\t\t&10\\kern0.2em 933\\\\\nDLV&&&147\\kern0.2em 172&82.4\n\t\t&461\\kern0.2em 646\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{2mm}\nThe benchmark scripts (from the authors of~\\cite{LFWK09})\nused a version of XSB with ``batched scheduling''\nfor this test.\nHowever,\nit gives only a relatively small improvement over the standard version\n(with ``local scheduling'').\n\nThe domain of values are in all cases integers from~1 to~1000,\nso many duplicates will be generated.\nActually,\nthis test is dominated by the time for duplicate elimination.\nWith an ``array of bitmaps'' implementation,\nwe are much faster than with a dynamic hash table\n(it doubles its size if the chains get too long).\nHere are the relations used in our Push implementation:\n\\vspace{2mm}\n\\begin{center}\n\\begin{tabular}{@{}|l|l|r|r|@{}}\n\\hline\nTable&Comment&Rows&Memory (KB)\\\\\n\\hline\n\\hline\n\\code{c2{\\U}bf}&EDB-Relation (Map)&10\\kern0.2em 000&88\\\\\n\\code{c3{\\U}ff}&EDB-Relation (List)&10\\kern0.2em 000&84\\\\\n\\code{c4{\\U}bf}&EDB-Relation (Map)&10\\kern0.2em 000&88\\\\\n\\code{d1{\\U}ff}&EDB-Relation (List)&10\\kern0.2em 000&84\\\\\n\\code{d2{\\U}bf}&EDB-Relation (Map)&10\\kern0.2em 000&88\\\\\n\\hline\n\\code{b1{\\U}fb}&Temporary IDB-Relation (Map)\n\t&634\\kern0.2em 088\n\t&4\\kern0.2em 976\\\\\n\\hline\n\\code{b1{\\U}bb}&Duplicate Check for IDB-Pred.~(Set)\n\t&634\\kern0.2em 088\n\t&14\\kern0.2em 004\\\\\n\\code{b2{\\U}bb}&Duplicate Check for IDB-Pred.~(Set)\n\t&95\\kern0.2em 954\n\t&2\\kern0.2em 524\\\\\n\\code{c1{\\U}bb}&Duplicate Check for IDB-Pred.~(Set)\n\t&95\\kern0.2em 570\n\t&2\\kern0.2em 520\\\\\n\\code{a{\\U}bb}&Duplicate Check for IDB-Pred.~(Set)\n\t&1\\kern0.2em 000\\kern0.2em 000\n\t&19\\kern0.2em 724\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{2mm}\nIn the version with bitmap duplicate check,\nthe last four sets need only 128~KB each.\nThe temporary IDB-relation is needed because\nthe first rule has two IDB-body literals:\n\\begin{center}\na(X, Y) :- b1(X, Z), b2(Z, Y).\\\\\n\\end{center}\nTherefore,\nwe compute first the \\code{b1}-facts and store them in~\\code{b1{\\U}fb}.\nLater,\nwhen we derive a \\code{b2}-fact,\nwe use it immediately to derive \\code{a}-facts with this rule.\nSince in this case \\code{Z} is bound from the \\code{b2}-fact,\nwe need the binding pattern~\\code{fb}\nfor the intermediate storage of~\\code{b1}-facts.\n\nIf one does not do any duplicate elimination,\none gets 99.9~million result tuples,\ni.e.~on average every answer is computed 100~times.\nOur implementation of this needed 2\\kern0.2em 501~ms execution time,\ni.e.~was quite fast.\nThis shows again that this benchmark depends a lot\non the efficiency of duplicate elimination.\n\nOur results with the hash table are not impressive and could be fully explained\nwith using native code instead of emulating an abstract machine.\nHowever,\nit scales better than XSB as the other data sets show.\nXSB timeouts already on the middle data set with 50\\kern0.2em 000~rows\nper EDB predicate\n(timeout is set at 30~minutes).\nOur Push implementation (with hash table) needs 106~s.\nThis means that it is better by at least the factor~17.\n\nHowever, we found that tabling all IDB predicates for XSB and YAP and introducing\nindexes for the EDB relations in XSB dramatically improved the \nruntime performance of\nthese systems (in the original program files, tabling was only done \nfor the predicate \\code{a\/2} in XSB, and no indices were used). This led to\nthe following results:\n\\vspace{-1mm}\n\\begin{center}\n\\begin{tabular}[t]{|l|r|r|r|r|r|}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Memory (KB)\\\\\n\\hline\n\\hline\nXSB&414\\phantom{)}&11\\kern0.2em 302&11\\kern0.2em 927\n\t\t&123\\kern0.2em 885\\\\\nYAP&437\\phantom{)}&6\\kern0.2em 719&7\\kern0.2em 256\n\t\t&133\\kern0.2em 622\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{2mm}\nDue to creating indexes, the loading time of XSB has increased, and both \nsystems need more memory for the tables. \n\n\n\n\n\n\n\n\n\n\\section{The Transitive Closure Benchmark with Both Arguments Free}\n\n\\label{secTCFF}\n\nAn important test for a deductive database system is the transitive closure\nprogram:\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{}}\n\ttc(X, Y) :- par(X, Y).\\\\\n\ttc(X, Y) :- par(X, Z), tc(Z, Y).\n\\end{tabular}\n\\end{tt}\n\\end{center}\nOf course,\nthis is one of the OpenRuleBench benchmark problems for recursion.\nIt uses 10~data files of different size (see below),\nwe first concentrated on the file\n\\code{tc{\\U}d1000{\\U}parsize50000{\\U}xsb{\\U}cyc.P}\nwhere\nthe arguments of the \\code{par}-relation are taken\nfrom a domain of 1000~integers (1~to~1000)\n(i.\\,e.\\ the graph has 1000~nodes);\nthere are 50\\kern0.2em 000~randomly generated \\code{par}-facts\n(i.\\,e.\\ the graph has 50\\kern0.2em 000~edges);\nthe graph is cyclic;\nand the file size is 673~KB.\nOpenRuleBench used three test queries:\n\\code{tc(X,\\kern0.2em Y)},\n\\kern0.3em\n\\code{tc(1,\\kern0.2em Y)},\n\\kern0.3em\nand \\code{tc(X,\\kern0.2em 1)}%\n\\footnote{Actually, anonymous variables were used in the test,\nbecause the answers are not further processed.}.\nHere are the benchmark results\nfor the first query asking for the entire \\code{tc} relation\n(i.\\,e.\\ with binding pattern~\\code{ff}):\t\t\t\t\n\\vspace{1mm}\n\\begin{center}\n\\begin{tabular}{@{}|l|r|r|r|r|r|@{}}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)\\\\\n\\hline\n\\hline\nPush&11\\phantom{)}&2\\kern0.2em 171&2\\kern0.2em 190&1.0\n\t\t&21\\kern0.2em 832\\\\\nSeminaive&10\\phantom{)}&4\\kern0.2em 598&4\\kern0.2em 613&2.1\n\t\t&29\\kern0.2em 587\\\\\nXSB&1\\kern0.2em 095\\phantom{)}&11\\kern0.2em 357&12\\kern0.2em 641&5.8\n\t\t&134\\kern0.2em 024\\\\\nYAP&424\\phantom{)}&19\\kern0.2em 407&19\\kern0.2em 867&9.1\n\t\t&145\\kern0.2em 566\\\\\nDLV&(260)&---&109\\kern0.2em 673&50.1\n\t\t&404\\kern0.2em 737\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{1mm}\nObviously,\none must detect duplicates in order to ensure termination.\nBut even with non-cyclic data,\nthere are many duplicates:\nEach node is linked to~5\\% of the other nodes.\nSo the probability is quite high\nthat even after short paths the same node is reached multiple times.\nXSB uses tabling\nto solve this problem~\\cite{SSW94,CW96SLG,Swi99}.\nThe relations we used for this problem are:\n\\begin{center}\n\\begin{tabular}{@{}|l|l|r|r|@{}}\n\\hline\nTable&Comment&Rows&Memory (KB)\\\\\n\\hline\n\\hline\n\\code{par{\\U}ff}&EDB-relation (as list), used in first rule&50000&396\\\\\n\\code{par{\\U}fb}&EDB-relation (as map), used in second rule&50000&400\\\\\n\\code{tc{\\U}bb}&Duplicate check for result (set)&1000000&19\\kern0.2em 536\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nThe first rule produces initial \\code{tc}-facts\nand simply loops over the entire \\code{par}-relation,\t\t\t\ntherefore it needs a version implemented as list.\nThe second (recursive) rule is activated in the ``Push'' method\nwhen a new \\code{tc}-fact is derived.\nTherefore,\na value for variable~\\code{Z} is known\nand the EDB-relation is accessed with binding pattern~\\code{fb}.\n\nWe also implemented a standard semina\\\"ive bottom-up evaluation\nfor comparison.\nIt used the same data structures as our Push implementation,\nit only needed an additional list \\code{tc{\\U}ff} for the result.\nIt is slower than the ``Push'' method.\nProbably, this is due to the different memory accesses:\nIn each iteration, freshly derived tuples are first stored in the result list,\nand they are accessed again in the next iteration.\n\nThe following table\nlists the benchmark results for the full set of 10~data files\nof various size and characteristics.\nWe compared only the execution times with XSB.\nWe used here the original settings for XSB from the OpenRuleBench scripts.\nIn the result shown above,\na \\code{trie}-index was added,\nwhich improved the performance (12.6\\kern0.2em s instead of 15.5\\kern0.2em s).\n\\begin{center}\n\\begin{tabular}{@{}|r|r|c|r|r|r|@{}}\n\\hline\n\\multicolumn{1}{@{}|c}{\\vrule height 1.05em depth 0.5em width 0pt Rows}&\n\t\\multicolumn{1}{|c}{Dom}&\n\t\\multicolumn{1}{|c}{cyc}&\n\t\\multicolumn{1}{|c}{XSB}&\n\t\\multicolumn{1}{|c}{Push}&\n\t\\multicolumn{1}{|c|@{}}{Factor}\\\\\n\\hline\n\\hline\n50\\kern0.2em 000&1000&N&\n\n\n\n\n\n\n\n\t5.980\\kern0.1em s\n\t&0.732\\kern0.1em s\n\t&8.2\n\n\n\t\\\\\n50\\kern0.2em 000&1000&Y&\n\n\n\n\n\n\n\n\t15.513\\kern0.1em s\n\t&2.160\\kern0.1em s\n\t&7.2\n\n\n\t\\\\\n250\\kern0.2em 000&1000&N&\n\n\n\n\n\n\n\n\t33.274\\kern0.1em s\n\t&4.350\\kern0.1em s\n\t&7.6\n\n\n\t\\\\\n250\\kern0.2em 000&1000&Y&\n\n\n\n\n\n\n\n\t82.497\\kern0.1em s\n\t&12.440\\kern0.1em s\n\t&6.6\n\n\n\t\\\\\n500\\kern0.2em 000&1000&N&\n\n\n\n\n\n\n\n\t76.673\\kern0.1em s\n\t&10.230\\kern0.1em s\n\t&7.5\n\n\n\t\\\\\n500\\kern0.2em 000&1000&Y&\n\n\n\n\n\n\n\n\t187.200\\kern0.1em s\n\t&30.620\\kern0.1em s\n\t&6.1\n\n\n\t\\\\\n500\\kern0.2em 000&2000&N&\n\n\n\n\n\n\n\n\t139.881\\kern0.1em s\n\t&23.870\\kern0.1em s\n\t&5.9\n\n\n\t\\\\\n500\\kern0.2em 000&2000&Y&\n\n\n\n\n\n\n\n\t329.873\\kern0.1em s\n\t&61.460\\kern0.1em s\n\t&5.4\n\n\n\t%\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\t\\\\\n\\kern-3pt 1\\kern0.2em 000\\kern0.2em 000&2000&N&\n\n\n\n\n\n\n\n\t297.870\\kern0.1em s\n\t&56.623\\kern0.1em s\n\t&5.3\n\n\n\t\\\\\n\\kern-3pt 1\\kern0.2em 000\\kern0.2em 000&2000&Y&\n\n\n\n\n\n\n\n\t714.721\\kern0.1em s\n\t&150.270\\kern0.1em s\n\t&4.8\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{2mm}\nThe first column gives the number of rows in the data\\-base,\ni.\\,e.\\ the size of the \\code{par}-relation.\nThe second column lists the size of the domain of values\nwhich occur in the \\code{par} columns.%\n\\footnote{Some of the graphs which claim to be non-cyclic\nactually do contain cycles.\nThe largest non-cyclic graph\nwith 1000~nodes is \\m{\\setCond{(i,j)}{1\\le i\\lt j\\le 1000}}.\nThis has only \\m{999*1000\/2=499\\:500}~edges.}\nThe last column shows the speed improvement factor\nof our Push implementation over XSB.\nFor loading the last relation (1~million rows, 14.2~MB),\nXSB needs 90.2\\kern0.1em s,\nour Push implementation needs 0.2\\kern0.1em s.\n\n\n\n\\section{The Transitive Closure Benchmark with the First Argument Bound}\n\n\\label{secTCBF}\nIn this benchmark,\none is not interested in all connected pairs in the transitive closure,\nbut only in nodes reachable from node~\\code{1}.\nSo the binding pattern for accessing the predicate is~\\code{bf}.\n\nDifferent systems use different methods to pass bindings\nto called predicates.\nE.g.~for XSB and YAP,\nthis is a feature of SLD-resolution.\nFor a system based on bottom-up evaluation,\nthe magic set method is the standard solution~\\cite{BMSU86,BR87}.\nProbably DLV does this.\nHowever, since magic sets are known\nto have problems with tail recursions~\\cite{Ros91},\nwe use SLDMagic~\\cite{Bra00} for our ``Push'' method instead\n(the ``Push'' method alone is a pure bottom-up method\nwould not pass query bindings).\nThe output of the transformation is:\n\\begin{center}\n\\begin{tt}\n\\begin{tabular}{@{}l@{}}\np1(A) :- p1(B), par(B,A).\\\\\np1(A) :- par(1,A).\\\\\np0(A) :- p1(B), par(B,A).\\\\\np0(A) :- par(1,A).\\\\\ntc(1,A) :- p0(A).\n\\end{tabular}\n\\end{tt}\n\\end{center}\nThe predicate \\code{p0} is not really needed,\nbut since the SLDMagic prototype produces this program,\nit would be unfair to improve it manually.\nNevertheless,\nit can be evaluated extremely fast,\nsince it reduced the arity of the recursive predicate.\nHere are the benchmark results:\n\\vspace*{1mm}\n\\begin{center}\n\\begin{tabular}{@{}|l|r|r|r|r|r|@{}}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)\\\\\n\\hline\n\\hline\nPush&10\\phantom{)}&14&30&1.0\n\t&9\\kern0.2em 716\\\\\nXSB&1\\kern0.2em 098\\phantom{)}&7\\kern0.2em 142&8\\kern0.2em 418&280.6\n\t&86\\kern0.2em 936\\\\\nYAP&440\\phantom{)}&8\\kern0.2em 743&9\\kern0.2em 246&308.2\n\t&91\\kern0.2em 325\\\\\nDLV&(260)&---&110\\kern0.2em 779&3\\kern0.2em 692.6\n\t&404\\kern0.2em 736\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{1mm}\nAt least the standard version of tabling\nhas the same problem with tail recursions\nas magic sets:\nOne cannot materialize the derived \\code{tc}-facts in this case,\nsince they are not only for the type \\code{tc(1,\\kern0.2em Y)},\nbut all facts \\code{tc(X,\\kern0.2em Y)}\nfor every~\\code{X} reachable from~\\code{1}.\nDepending on the graph,\nthis can make the difference between a linear number of facts\nand a quadratic number of facts.\nXSB and YAP might be affected by this problem.\n\nIt could \nbe argued that the \ngood results for the Push Method are mainly due to the SLDMagic program\ntransformation.\nTherefore,\nin a further test,\nthe SLDMagic-transformed program was\nalso given as input to the other systems.\nIndeed, they did profit\nfrom the transformation\n(in runtime as well as in memory usage), \nbut the Push Method is still \nin the first place:\n\\vspace*{1mm}\n\\begin{center}\n\\begin{tabular}{@{}|l|r|r|r|r|r|@{}}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)\\\\\n\\hline\n\\hline\nPush&10\\phantom{)}&14&30&1.0\n\t&9\\kern0.2em 716\\\\\nXSB&1\\kern0.2em 093\\phantom{)}&10&1\\kern0.2em 292&43.1\n\t&14\\kern0.2em 298\\\\\nYAP&449\\phantom{)}&36&562&18.7\n\t&16\\kern0.2em 696\\\\\nDLV&(260)&---&389&13.0\n\t&11\\kern0.2em 997\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{1mm}\nFor the other data files,\nwe compared the execution times only with XSB\n(using SLDMagic for the Push method, but not for XSB):\n\\begin{center}\n\\begin{tabular}{@{}|r|r|c|r|r|r|@{}}\n\\hline\n\\multicolumn{1}{@{}|c}{\\vrule height 1.05em depth 0.5em width 0pt Rows}&\n\t\\multicolumn{1}{|c}{Dom}&\n\t\\multicolumn{1}{|c}{cyc}&\n\t\\multicolumn{1}{|c}{XSB}&\n\t\\multicolumn{1}{|c}{Push}&\n\t\\multicolumn{1}{|c|@{}}{Factor}\\\\\n\\hline\n\\hline\n50\\kern0.2em 000&1000&N\n\t&1.296\\kern0.1em s\n\t&0.010\\kern0.1em s\n\t&130\n\n\n\n\n\t\\\\\n50\\kern0.2em 000&1000&Y\n\t&6.912\\kern0.1em s\n\t&0.010\\kern0.1em s\n\t&691\n\n\n\n\n\t\\\\\n250\\kern0.2em 000&1000&N\n\t&9.309\\kern0.1em s\n\t&0.030\\kern0.1em s\n\t&310\n\n\n\n\n\t\\\\\n250\\kern0.2em 000&1000&Y\n\t&35.098\\kern0.1em s\n\t&0.030\\kern0.1em s\n\t&1170\n\n\n\n\n\t\\\\\n500\\kern0.2em 000&1000&N\n\t&19.989\\kern0.1em s\n\t&0.050\\kern0.1em s\n\t&400\n\n\n\n\n\t\\\\\n500\\kern0.2em 000&1000&Y\n\t&69.929\\kern0.1em s\n\t&0.060\\kern0.1em s\n\t&1165\n\n\n\n\n\t\\\\\n500\\kern0.2em 000&2000&N\n\t&36.067\\kern0.1em s\n\t&0.060\\kern0.1em s\n\t&601\n\n\n\n\n\t\\\\\n500\\kern0.2em 000&2000&Y\n\t&154.110\\kern0.1em s\n\t&0.070\\kern0.1em s\n\t&2202\n\n\n\n\n\t\\\\\n\\kern-3pt 1\\kern0.2em 000\\kern0.2em 000&2000&N\n\t&80.117\\kern0.1em s\n\t&0.130\\kern0.1em s\n\t&616\n\n\n\n\n\t\\\\\n\\kern-3pt 1\\kern0.2em 000\\kern0.2em 000&2000&Y\n\t&300.719\\kern0.1em s\n\t&0.150\\kern0.1em s\n\t&2005\n\n\n\n\n\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nThe case with the second argument bound\nstill has to be investigated.\nThe SLDMagic method is no advantage in this case.\nSo our Push method would need the same time\nas in the \\code{tc(X,\\kern0.2em Y)} case,\nwhich means 2.171\\kern0.2em s for execution.\nXSB needs only 0.020\\kern0.2em s for execution,\ni.e.~is better by a factor of~109!\nSince our data loader is quicker,\nthe factor for total time is less than~2,\nbut XSB is still better.\nOf course,\nwe are working on improving the SLDMagic method.\nBut that is a different topic.\n\n\\section{The Wine Ontology Benchmark}\n\n\\label{secWineOntology}\n\nThe wine ontology benchmark\nconsists of 961~rules with 225~IDB-predicates,\nof which all but one are recursive,\nand 113~EDB-predicates.\nThe program is basically one big recursive component.%\n\\footnote\nActually,\nthe wine ontology program does not do what it is supposed to do:\nThe test query is for californian wines,\nbut the result contains\nother objects, too.\nWe checked that XSB produces the same output.\nNevertheless,\nthe program is a challenging test for a deductive system,\nno matter whether it\nis meaningful.\nThe wine ontology was developed for\nthe OWL guide \\url{http:\/\/www.w3.org\/TR\/owl-guide\/}.\nIt links to a site no longer available,\nbut the ontology is probably the one available here:\n\\url{https:\/\/www.movesinstitute.org\/exi\/data\/DAML\/wines.daml}.\nThanks to Boris Motik,\nwho referred us to the paper~\\cite{HKRT08}:\nThis introduces an approximate translation from OWL DL TBoxes to Datalog.\nAlthough it is not completely clear yet that this translation was used,\nthe approximation would explain that the result might contain wrong answers.}\n\\begin{center}\n\\begin{tabular}{@{}|l|r|r|r|r|r|@{}}\n\\hline\nSystem&Load (ms)&Execution (ms)&Total time (ms)&Factor&Memory (KB)\\\\\n\\hline\n\\hline\nPush&1\\phantom{)}&2\\kern0.2em 255&2\\kern0.2em 260&1.0&9\\kern0.2em 236\\\\\nXSB&106\\phantom{)}&8\\kern0.2em 548&8\\kern0.2em 851&3.9&322\\kern0.2em 894\\\\\nYAP&52\\phantom{)}&10\\kern0.2em 793&10\\kern0.2em 899&4.8&334\\kern0.2em 761\\\\\nDLV&(60)&---&31\\kern0.2em 743&14.0&42\\kern0.2em 452\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nThis is an example\nwhere the Push method with partial evaluation\nas introduced in~\\cite{BS15WLP}\n``explodes'':\nIt produces a lot of different specializations of the rules,\nof which there are already many in the input program.\nTherefore,\nwe used the version of the push method without partial evaluation\nfrom~\\cite{Bra10CPP} here.\nEven with that,\nthe resulting \\CPP\\ program is large (34\\kern0.2em 294~lines).\nThe detection of duplicates is essential for termination.\nWe used the hash table here,\nprobably a bitmap would further improve performance.\n\nStandard implementations of bottom-up evaluation\nwould iterate all rules in a recursive clique\nuntil one such iteration did not produce a new fact.\nIn this example,\nthis would be very inefficient\nbecause basically all rules form a single recursive clique,\nbut in each iteration only a few rules\nactually ``fire''.\nIn contrast,\nthe ``Push'' method only looks at rules\nwhich are activated by a new fact for an IDB body literal.\n\n\n\\section{The Mondail Benchmark}\n\n\n\n\n\n\n\\section{Related Work}\n\n\\label{secRelatedWork}\n\nThe push method has been studied in~\\cite{Bra10CPP},\nand compared with ``Pull' and ``Materialization'' methods\nof bottom-up evaluation,\nbut only with artificial examples,\nno real data loaded from files,\nand no index structures.\nThe paper~\\cite{BS15WLP}\ndefines a different version of the push method,\nwhich does not create variables for every argument of each IDB predicate,\nbut for variables occurring in EDB literals in rule bodies.\nThis helps to reduce or nearly eliminate the copying of values,\nat the price of creating several specializations of the same rule\n(this reduces runtime, but increases code size, sometimes significantly).\nIt is basically this version which was used in our benchmarks,\nwith certain improvements.\nFurthermore,\nour transformation program can also fall back\nto the version of~\\cite{Bra10CPP},\nif the generated program would otherwise become unacceptably large.\nActually,\nboth versions can be combined for a single input program.\n\nThe idea of immediately using derived facts\nto derive more facts is not new in those papers.\nFor instance,\nvariants of semi-naive evaluation have been studied\nwhich work in this way~\\cite{Sch93,SU99}.\nIt also seems to be related to the propagation of updates\nto materialized views.\nHowever,\nthe representation of tuples at runtime\nand the code structure is different from~\\cite{Sch93}\n(and this is essential for the reduction of copying values).\nThe paper~\\cite{SU99} \ntranslates from a temporal Datalog extension to Prolog,\nwhich makes any efficiency comparison dependend\non implementation details of the used Prolog compiler.\n\nIn relational databases, the benefits of not materializing intermediate\nresults have been recognized for a long time; this is known as \n\\emph{pipelining}. Pushing data up a pipeline has been studied in \n\\cite{Neu11}; \nhowever, they work with relational algebra expressions rather than\nDatalog rules.\n\n\n\n\\section{Conclusion}\n\n\\label{secConclusion}\n\nThis is a paper about some experiences gained\nwhile implementing a deductive database system\nand testing its performance.\nOf course, the system is not yet finished,\nand more problems and insights are waiting along the road.\nNevertheless,\nthe understanding reached so far seems interesting and useful:\n\\begin{itemize}\n\\item\nA system based on a bottom-up method can compete with\nsystems based on SLD-resolution with tabling.\nOlder deductive database systems like Coral~\\cite{RSS94,SFH96}\nhad no chance against XSB~\\cite{SSW94}.\nAvoiding the copying of data values\nand the materialization of derived tuples\nseems to be the main reason for the success of the Push method.\n\\item\nFast data structure implementations,\nespecially for duplicate elimination,\nare very important.\nFor duplicate tests,\ntuples were materialized in ``set'' data structures.\nWe plan to use nested tables in future\t\t\t\t\t\t\t\t\t\t\nin order to keep the advantage of reducing the copying of values.\n\\item\nThe SLDMagic method proved very useful\nfor the transitive closure with binding pattern~\\code{bf}.\nFor the binding pattern~\\code{fb},\nwork is still needed.\n\\item\nSome variants in the code\nthat intuitively seemed important\nhad no effect on performance.\nProbably,\nthe \\code{g++} compiler is clever enough\nto optimize the different versions in the same way.\nFor instance,\nwe first tried a static procedure which only counted the derived facts,\nbut did not return them.\nWhen we later used an object with a cursor interface\nwhere one can fetch each result\nthere was no difference\n(although there were many more procedure calls,\nand what had been local or static variables before\nhad to become attributes).\nFurthermore,\nreplacing a \\code{switch} and \\code{goto}\n(our standard translation)\nwith \\code{while}-loops (where possible)\nhad no important influence on performance\n(but improves the readability of the generated code, of course).\n\\end{itemize}\nWe plan to define an abstract machine\t\t\t\t\t\t\t\t\t\t\nand translate alternatively into this machine.\nThen it would not always be necessary to use a \\CPP\\ compiler.\t\t\t\t\t\n\nWe also work on a parallelized version of our method.\nAt times\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nwhere even small PCs have at least four cores,\none should obviously make use of this resource.\nThe easier parallelization is also one motivation\t\t\t\t\t\t\t\t\nfor declarative programming.\n\nOf course,\nalso adding function symbols (term constructors),\nnegation and aggregation functions are on the agenda,\t\t\t\t\t\t\nplus more exotic things like ordered predicates and declarative output.\t\t\n\n\n\\section{Translation Method}\n\n\n\n\n\n\n\\bibliographystyle{eptcs}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{References}}\n\\begin{document}\n\n\n    \\long\\def \\cblu#1{\\color{blue}#1}\n    \\long\\def \\cred#1{\\color{red}#1}\n    \\long\\def \\cgre#1{\\color{green}#1}\n    \\long\\def \\cpur#1{\\color{purple}#1}\n\n\\newcommand{\\eric}[1]{{\\color{blue}#1}}\n\\newcommand{\\guido}[1]{{\\color{violet}#1}}\n\\newcommand{\\matthias}[1]{{\\color{blue}#1}}\n\\newcommand{\\fabian}[1]{{\\color{blue}#1}}\n\\newcommand{\\di}[1]{{\\color{blue}#1}}\n\\newcommand{\\ericC}[1]{{\\color{red}\\textit{\\textbf{Eric:} #1}}}\n\\newcommand{\\guidoC}[1]{{\\color{red}\\textit{\\textbf{Guido:} #1}}}\n\\newcommand{\\matthiasC}[1]{{\\color{red}\\textit{\\textbf{Matthias:} #1}}}\n\\newcommand{\\fabianC}[1]{{\\color{red}\\textit{\\textbf{Fabian:} #1}}}\n\\newcommand{\\diC}[1]{{\\color{red}\\textit{\\textbf{Di:} #1}}}\n\n\n\\def\\FileRef{\n\\input FName\n{\n\\newcount\\hours\n\\newcount\\minutes\n\\newcount\\min\n\\hours=\\time\n\\divide\\hours by 60\n\\min=\\hours\n\\multiply\\min by 60\n\\minutes=\\time\n\\\n\\advance\\minutes by -\\min\n{\\small\\rm\\em\\the\\month\/\\the\\day\/\\the\\year\\ \\the\\hours:\\the\\minutes\n\\hskip0.125in{\\tt\\FName}\n}\n}}\n\n\\mathchardef\\muchg=\"321D\n\\let\\na=\\nabla\n\\let\\pa=\\partial\n\n\\let\\muchg=\\gg\n\n\\let\\t=\\tilde\n\\let\\ga=\\alpha\n\\let\\gb=\\beta\n\\let\\gc=\\chi\n\\let\\gd=\\delta\n\\let\\gD=\\Delta\n\\let\\ge=\\epsilon\n\\let\\gf=\\varphi\n\\let\\gg=\\gamma\n\\let\\gh=\\eta\n\\let\\gj=\\phi\n\\let\\gF=\\Phi\n\\let\\gk=\\kappa\n\\let\\gl=\\lambda\n\\let\\gL=\\Lambda\n\\let\\gm=\\mu\n\\let\\gn=\\nu\n\\let\\gp=\\pi\n\\let\\gq=\\theta\n\\let\\gr=\\rho\n\\let\\gs=\\sigma\n\\let\\gt=\\tau\n\\let\\gw=\\omega\n\\let\\gW=\\Omega\n\\let\\gx=\\xi\n\\let\\gy=\\psi\n\\let\\gY=\\Psi\n\\let\\gz=\\zeta\n\n\\let\\lbq=\\label\n\\let\\rfq=\\ref\n\\let\\na=\\nabla\n\\def{\\dot{I}}{{\\dot{I}}}\n\\def{\\dot{q}}{{\\dot{q}}}\n\\def{\\dot{\\phi}}{{\\dot{\\phi}}}\n\n\\def\\bar{\\sigma}{\\bar{\\sigma}}\n\\def\\bar{\\eta}{\\bar{\\eta}}\n\\def\\bar{\\gamma}{\\bar{\\gamma}}\n\\def\\bar{\\Phi}{\\bar{\\Phi}}\n\\def\\bar{\\Psi}{\\bar{\\Psi}}\n\n\\def\\bar{F}{\\bar{F}}\n\\def\\bar{j}{\\bar{j}}\n\\def\\bar{J}{\\bar{J}}\n\\def\\bar{p}{\\bar{p}}\n\\def\\bar{P}{\\bar{P}}\n\\def\\bar{x}{\\bar{x}}\n\n\\def\\hat{\\phi}{\\hat{\\phi}}\n\\def\\hat{\\theta}{\\hat{\\theta}}\n\n\\def\\hat{T}{\\hat{T}}\n\\def\\hat{R}{\\hat{R}}\n\\def\\hat{b}{\\hat{b}}\n\\def\\hat{h}{\\hat{h}}\n\\def\\hat{z}{\\hat{z}}\n\n\\let\\gG=\\Gamma\n\\def{\\tilde{A}}{{\\tilde{A}}}\n\\def{\\tilde{B}}{{\\tilde{B}}}\n\\def{\\tilde{G}}{{\\tilde{G}}}\n\\def{\\tilde{p}}{{\\tilde{p}}}\n\\def{\\tilde{v}}{{\\tilde{v}}}\n\\def{\\tilde{\\Phi}}{{\\tilde{\\Phi}}}\n\n\\def{\\bm{\\xi}}{{\\bm{\\xi}}}\n\\def{\\bm{\\zeta}}{{\\bm{\\zeta}}}\n\n\\def{\\bf e}{{\\bf e}}\n\\def{\\bf k}{{\\bf k}}\n\\def{\\bf i}{{\\bf i}}\n\\def{\\bf j}{{\\bf j}}\n\\def{\\bf l}{{\\bf l}}\n\\def{\\bf n}{{\\bf n}}\n\\def{\\bf r}{{\\bf r}}\n\\def{\\bf u}{{\\bf u}}\n\\def{\\bf v}{{\\bf v}}\n\\def{\\bf x}{{\\bf x}}\n\n\\def\\vec{B}{\\vec{B}}\n\\def\\vec{e}{\\vec{e}}\n\\def\\vec{h}{\\vec{h}}\n\\def\\vec{l}{\\vec{l}}\n\\def\\vec{v}{\\vec{v}}\n\\def\\vec{\\nu}{\\vec{\\nu}}\n\\def\\vec{\\kappa}{\\vec{\\kappa}}\n\\def\\vec{\\gt}{\\vec{\\gt}}\n\\def\\vec{\\xi}{\\vec{\\xi}}\n\\def\\vec{\\zeta}{\\vec{\\zeta}}\n\n\\def{\\bf A}{{\\bf A}}\n\\def{\\bf B}{{\\bf B}}\n\\def{\\bf D}{{\\bf D}}\n\\def{\\bf E}{{\\bf E}}\n\\def{\\bf J}{{\\bf J}}\n\\def{\\bf V}{{\\bf V}}\n\\def{\\bf X}{{\\bf X}}\n\n\\def\\R#1#2{\\frac{#1}{#2}}\n\\def\\begin{tabular}{\\begin{tabular}}\n\\def\\end{tabular}{\\end{tabular}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\ebox#1{\n  \\begin{eqnarray}\n    #1\n\\end{eqnarray}}\n\n\n\\def \\cred#1{{\\color{red}(#1)}}\n\\def \\cblu#1{{\\color{blue}#1}}\n\n\n\n\\title[Particle-based JOREK impurity model]{Collisional-radiative non-equilibrium impurity treatment for JOREK simulations}\n\\author{D. Hu$^{1}$, G.T.A. Huijsmans$^{2,3}$, E. Nardon$^2$, M. Hoelzl$^4$, M. Lehnen$^5$, D. Bonfiglio$^6$ \\& JOREK Team\\footnote{See the author list of Ref \\cite{Hoelzl2021NF} for a list of current team members}}\n \\ead{hudi2@buaa.edu.cn}\n\\address{\n$^1$Beihang University, No. 37 Xueyuan Road, Haidian District, 100191 Beijing, China.\n}\n\\address{\n$^2$CEA, IRFM, F-13108 Saint-Paul-Lez-Durance, France\n}\n\\address{\n$^3$Eindhoven University of Technology, De Rondom 70 5612 AP Eindhoven, the Netherlands.\n}\n\\address{\n$^4$Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching b. M., Germany\n}\n\\address{\n$^5$ITER Organization, Route de Vinon sur Verdon, CS 90 046,13067 Saint Paul-lez-Durance, Cedex, France.\n}\n\\address{\n$^6$Consorzio RFX-CNR, ENEA, INFN, Universit\\`{a} di Padova, Acciaierie Venete SpA. I-35127 Padova, Italy.\n}\n\\ead{hudi2@buaa.edu.cn}\n\n\\vspace{10pt}\n\\begin{indented}\n\\item[]\\today\n\\end{indented}\n\n\n\\begin{abstract}\nA collisional-radiative non-equilibrium impurity treatment for JOREK 3D nonlinear magneto-hydrodynamic (MHD) simulations has been developed. The impurities are represented by super-particles flowing along the fluid velocity field lines, while ionizing and recombining independently according to ADAS data and local fluid density and temperature. The non-equilibrium impurity contributions are then projected back to the fluid field for self-consistent time evolution. A 2D test case is used to compare the new non-equilibrium impurity model against previous Coronal Equilibrium (CE) impurity treatment, as well as to compare the non-equilibrium impurity behavior between the single and the two temperature model. Further, we conduct benchmark with previously published coronal non-equilibrium results by other 3D nonlinear MHD codes such as M3D-C1 and NIMROD. The new non-equilibrium treatment is shown to successfully capture the early phase cooling by weakly ionized impurities which the CE model missed. The benchmarks with M3D-C1 and NIMROD show general agreement in both the integrated quantities and the 2D profile evolution, despite the difference in the atomic model used. The above comparison and benchmark cases demonstrate the capability of the non-equilibrium impurity model for JOREK, paving the way for more sophisticated 3D non-linear Massive Material Injection (MMI) simulations which have important applications in disruption mitigation studies.\n\\end{abstract}\n\n\\maketitle\n\n\n\\section{Introduction}\n\\label{s:Intro}\n\nUnmitigated disruptions could represent a severe threat to future high performance tokamaks such as ITER \\cite{Lehnen2015JNM}, thus effective and efficient Disruption Mitigation Systems (DMSs) are essential for the sustained operation of such devices. The injection of massive amounts of impurity atoms or hydrogen isotopes are the most promising techniques currently in development \\cite{Hollmann2015POP,Baylor2015FST}, and there have been extensive experimental explorations concerning Massive Gas Injection (MGI) \\cite{Lehnen2011NF,Reux2015JNM,Pautasso2011NF,\nHollmann2011JNM,Duan2017NF,Chen2018NF,Ding2018PST}, Shattered Pellet Injection (SPI) \\cite{Shiraki2018NF,Raman2020NF,Sweeney2020NF,\nXu2018FST,Li2018RSI,Baylor2019NF,Park2020FED} as well as the newly emerged Dispersive Shell Pellet Injection (DSPI) techniques \\cite{Hollmann2019PRL}. For the better understanding of the aforementioned injection dynamics, numerous MGI simulations have been carried out \\cite{Izzo2008POP,Izzo2013POP,Izzo2015NF,Fil2015POP,Nardon2017PPCF,Zafar2021PST}, while explorations into SPI \\cite{Di2018NF,Kim2019POP,Hoelzl2020POP,Nardon2020NF,Di2021NF} and DSPI \\cite{Izzo2020NF} are also being conducted by 3D nonlinear magneto-hydrodynamic (MHD) codes such as JOREK, M3D-C1 and NIMROD which provide important insight on the interplay between the MHD modes and the injected material. Understanding such interplay is instrumental for obtaining deeper knowledge of the MHD destabilization and the particle transport after the injections, and ultimately important to the endeavor of finding the most efficient DMS configuration.\n\nThe JOREK code is a 3D nonlinear code with both reduced MHD and full MHD capability \\cite{Hoelzl2021NF}. It has been used to investigate impurity Massive Material Injection (MMI) dynamics \\cite{Di2021NF,Nardon2021PPCF} assuming a Coronal Equilibrium (CE) distribution of the charge states \\cite{Mosher1974PRA}. Such assumption, however, is not always accurate in a massive injection scenario, especially in the early phase of the injection and the cooling.\nIndeed, the timescale of the injection and the transport during this phase could be comparable with the timescale needed for the impurity charge state distribution to reach the equilibrium \\cite{Smith2010APJ}, so that the radiation power density predicted by the CE assumption could suffer inaccuracy compared with that of a more detailed non-equilibrium model as is shown in Fig.\\,2 of Ref.\\,\\cite{Di2021NF}. This is especially important if the temperature drop is not fast enough during the early injection period or during the Thermal Quench (TQ) when sudden outgoing heat flux reheats the outer plasma region, since the CE assumption tends to underestimate the line radiation contribution from the weakly ionized states.\n\nTo address the aforementioned potential issue, we hereby develop a particle-based collisional-radiative non-equilibrium model for a better treatment of the impurity charge state distribution. The model relies on so-called ``marker-particles'' which flow along the fluid velocity field line while ionizing and recombining independently according to local plasma temperature and their own ionization-recombination probability. The resulting charge state distribution is then projected to the finite element grid and used in the fluid equations being solved. To demonstrate the capability of this new treatment, we will present a set of comparison and benchmark concerning this non-equilibrium impurity treatment.\n\nThe rest of the report is arranged as follows. First, we introduce the implementation of the particle-based impurity treatment in Section \\ref{s:Implementation} and describe the axisymmetric equilibrium and simulation setup in Section \\ref{s:Eq}. We then show the comparison of the particle-based model against the previous CE impurity model in Section \\ref{s:ParModvsCEmod}. The comparison between the single temperature model and the two temperature model both utilizing the non-equilibrium impurity treatment are shown in Section \\ref{s:TeTiComparison}. Further, we benchmark the particle-based model with previous M3D-C1 and NIMROD axisymmetric simulations of a central massive gas injection in a DIII-D-like plasma with both argon and neon injections in Section \\ref{s:Benchmark}. Last, the conclusion is given in Section \\ref{s:Conclusion}. We further demonstrate that very good agreement could be reached with the M3D-C1 and NIMROD results when using an older version of the ADAS data in \\ref{ap:Atomic}.\n\n\\section{The implementation of the particle-based impurity treatment}\n\\label{s:Implementation}\n\nJOREK has implemented the infrastructure for both test particle and coupled particle-fluid simulations in recent years \\cite{VanVugt2018EPS,VanVugt2019POP,VanVugtThesis}. Our non-equilibrium impurity treatment is based on the existing infrastructure, but with some modification to the particle pusher and ionization-recombination process.\n\nIn this section, we introduce the implementation of the particle-based impurity model. We first show how the particles are evolved which includes the particle pusher and the evolution of its charge states. We then introduce how the particles contribute to the fluid evolution by their moment projection.\n\n\n\\subsection{Marker-particle pusher and particle generation}\n\\label{ss:MarkerPartPusher}\n\nThe ``marker-particles'' considered in out treatment do not evolve according to their own respective Newton's law, but are simply advected with the fluid velocity field, hence the name. This simplification is in accordance with our previous assumption that the inter-species friction and the charge exchange is strong enough so that all species and their different charge states share the same velocity \\cite{Di2021NF}. Currently, the additional diffusive transport of the fluid density is not yet reflected in the particle advection due to its sub-dominant role.\n\nIn JOREK, the fluid velocity field can be directly calculated by the flow function and the parallel velocity field:\n\\begin{eqnarray}\n\\lbq{eq:MagField}\n{\\bf B}\n=\nF_0\\na\\gj\n+\\na\\gy\\times\\na\\gj\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:VelField}\n{\\bf v}\n=\nv_\\|{\\bf B}\n-R^2\\na u\\times\\na \\gj\n.\\end{eqnarray}\nHere $F_0\/R$ is the toroidal magnetic field while $F_0$ is assumed to be constant in space and time. The poloidal magnetic flux is $\\gy$, while $v_\\|{\\bf B}$ is the parallel velocity, $R$ is the major radius, $\\gj$ is the toroidal angle and $u$ is the poloidal flow potential.\nUsing the above expressions, we can write down the velocity as\n\\begin{eqnarray}\n\\lbq{eq:VelFieldExplicit}\n{\\bf v}\n&\n=\n&\n\\left(-R\\pa_Zu+\\R{v_\\|}{R}\\pa_Z\\gy\\right){\\bf e}_{R}\n+\\left(R\\pa_Ru-\\R{v_\\|}{R}\\pa_R\\gy\\right){\\bf e}_{Z}\n+\\R{F_0}{R} v_\\|{\\bf e}_\\gj\n.\\end{eqnarray}\nThe marker-particle pusher advects the marker-particles with the fluid velocity, {\\it i.e.} the particle velocity is the same as the local fluid velocity.\n\nIn JOREK, the particle time step is set to be a fraction of the fluid time step, so that the fluid fields are evolved one time step after a certain number of particle steps. The pusher hence pushes the marker-particles for several particle time steps under a given fluid field before the fluid field is updated. From past simulations, setting the particle time step to be smaller than $10^{-8}s$ is enough to assure accuracy in the particle evolution \\cite{VanVugt2019POP}.\n\nAt the end of each fluid time step, a number of marker particles are added to the simulation according to the impurity density source term in the fluid evolution, which in turn is determined by the gas injection source or the pellet ablation. The amplitude and shape of such density source is detailed in Ref.\\,\\cite{Di2021NF}. The generated super-particles are equal in their weight (the amount of real particles represented by a given super-particle), and their generation location is obtained by the reject-sampling method according to the aforementioned source terms. We assume all particles are generated as neutrals.\n\n\\subsection{The ionization, recombination and radiation}\n\\label{ss:IonRecon}\n\nIn a MMI scenario, the total injected atom number could easily be as high as $10^{22}$ impurity atoms for large devices such as ITER. This either demands using a tremendous amount of super-particles, or requires each super-particle to represent a large population of real particles. The former is undesirable in terms of simulation cost, while the latter is detrimental to the smoothness of particle moment projection onto the fluid field. The large super-particle weight could be especially problematic when we are concerned with the dynamic balance of the ionization and the recombination where we have the cancellation of two large terms.\n\nOn the other hand, due to the use of the marker-particle pusher, we are free to allocate a charge state density array to each super-particle instead of a single charge number, reflecting the charge state distribution of the real particles represented by that super-particle. This has the benefit of providing a smoother ionization-recombination representation compared with the traditional single charge number treatment, enabling us to simulate a MMI scenario with moderate number of super-particles. Typically, we use $10^7$ super-particles to represent $10^{21}$ to $10^{22}$ impurity atoms, thus the typical weight of marker-particles is $10^{14}$ to $10^{15}$. Empirically, depending on the number of ``active'' super-particles, the additional computational time due to the super-particle evolution ranges from negligible in the early simulations to tens of seconds per fluid time-step late into the simulation. As a reference, the fluid time-step typically takes thirty seconds for an axisymmetric run on four nodes, each with 32 FeiTeng FT-2000+ processors.\n\nThe charge state distribution for each super-particle is evolved as the following \\cite{Hughes1985ApJ}:\n\\begin{eqnarray}\n\\lbq{eq:Ion-Rec}\n\\R{d f_i}{dt}\n=\nn_e\\left[\nS_{i-1}(T_e)f_{i-1}\n-\\left(S_{i}(T_e)f_{i}+\\ga_{i}(T_e)\\right)f_i\n+\\ga_{i+1}f_{i+1}\n\\right]\n.\\end{eqnarray}\nHere, $S_i$ and $\\ga_i$ are the ionization rate and recombination rate of charge state $i$ respectively, while $f_i$ is the number density of that charge state, and $n_e$ is the electron density. Both the ionization and recombination data are obtained from the open ADAS atomic database \\cite{OPENADAS}. The open ADAS data is calculated by using the collisional-radiative model, as opposed to the coronal model ADPAK data used in Ref. \\cite{Lyons2019PPCF}, which may account for some deviation that we will observe in the following sections. To be specific, we are using the 1989 version of the argon ADAS data and the 1996 version of the neon ADAS data. Indeed, better agreement is found with the M3D-C1 and NIMROD result when older versions of ADAS data are used as shown in \\ref{ap:Atomic}. However, due to the availability issue of some of the old atomic data, in the main article we will continue to use the newer versions as specified above. We will also show the comparison of the line radiation, the ionization and the recombination coefficient of neon between the ADAS data and the ADPAK data in \\ref{ap:Atomic}.\n\n\n\\subsection{The projection of particle contribution}\n\\label{ss:PartProj}\n\nThe last piece in the JOREK particle-fluid coupling is the projection of particle moment onto the fluid fields. The exact projection method is detailed in Ref.\\,\\cite{VanVugtThesis}. In our treatment, we mainly project the following five quantities onto the fluid fields.\n\nThe first is the ionization power density, which is representing the energy taken from the electron thermal energy to compensate the impurity's ionization energy. At each particle time step as the charge state distribution is evolved, we collect the change in the ionization energy for each super-particle:\n\\begin{eqnarray}\n\\lbq{eq:Eion}\n\\gD E_{ion}\n=\nn_e\\gD t\\sum_{i=0}^{Z-1}\\left(\nN_{imp}^{i+}S_{i}(T_e)-N_{imp}^{i+1+}\\ga_{i}(T_e)\n\\right)E_{ion}^{i+}\n.\\end{eqnarray}\nHere, $N_{imp}^{i+}$ is the number of real particles at charge state $i+$, $E_{ion}^{i+}$ is the ionization energy when ionizing from $i+$ charge state to $i+1+$ charge state, $\\gD t$ is the particle time step. This energy change will act on the electron fluid thermal energy as a source-sink term by moment projection. It should be noted that, physically, the ionization energy would be lost by recombination radiation upon recombination. Here in the modelling, we added the ionization energy back to the electron thermal energy to cancel part of the ADAS recombination radiation, to avoid double counting the part of recombination energy that is taken from the ionization potential energy.\n\nThe second is the radiation power density. With a given charge state distribution for a given super-particle, the total radiation power $P_{rad}$ is then a function of the electron density and temperature \\cite{Summers1979JPB}.\n\\begin{eqnarray}\n\\lbq{eq:Prad}\nP_{rad}\n&\n=\n&\nn_e n_{imp} L_{rad}(n_e,T_e)\n\\nonumber\n\\\\\n&\n=\n&\nn_e\\sum_{i=0}^{Z}n_{imp}^{i+}\\left(L_L^{i+}(n_e,T_e)+L_R^{i+}(n_e,T_e)+L_B^{i+}(n_e,T_e)\\right)\n.\\end{eqnarray}\nHere, $n_{imp}$ is the total density of the impurity species, while $n_{imp}^{i+}$ is that of each charge state. The radiation power function $L_L^{i+}$, $L_R^{i+}$ and $L_B^{i+}$ are the line radiation, the recombination radiation and the bremsstrahlung radiation of each charge state respectively.\nThe radiation power functions are also obtained from the open ADAS database \\cite{OPENADAS}.\n\nThe third is the effective charge of the impurity species which is defined as\n\\begin{eqnarray}\n\\lbq{eq:EffectiveCharge}\nZ_\\textrm{eff}\n\\equiv\n\\R{\\sum_i{n_iZ_i^2}}{\\sum_i{n_iZ_i}}\n\\nonumber\n.\\end{eqnarray}\nNote that the summation here includes all species and their charge states. The effective charge affects the calculation of the Spitzer-like resistivity \\cite{Hirshman1978POF}:\n\\begin{eqnarray}\n\\lbq{eq:Resistivity}\n\\gh\n=\nZ_\\textrm{eff}\\R{\\gh_0}{\\max{\\left(T_e,T_{thres}\\right)}^{3\/2}}\\times\n\\R{1+1.198\\,Z_\\textrm{eff}+0.222\\,Z_\\textrm{eff}^2}{1+2.966\\,Z_\\textrm{eff}+0.753\\,Z_\\textrm{eff}^2}\n.\\end{eqnarray}\nHere we take the cut-off temperature $T_{thres}=1eV$, below which the temperature dependency is not accounted for.\n\nThe fourth quantity is the mean charge of the impurity species, which is used to calculate the electron density assuming quasi-neutrality. The last projected quantity is the impurity number density.\n\n\\subsection{The governing equations with particle moment contributions}\n\\label{ss:GovEq}\n\nThe formulation of our single and two temperature models under the CE assumption has been detailed in Ref. \\cite{Di2021NF}. For the completeness of the paper, we again write down the governing equations with the particle moment contributions in this subsection, but we will only focus on the particle moment contributions instead of going over all the details.\n\nFirst, we have the induction equation:\n\\begin{eqnarray}\n\\lbq{eq:InductionEq}\n\\R{\\pa\\gy}{\\pa t}\n&\n=\n&\n\\gh\\left(T_e\\right)\\gD^*\\gy\n-R\\left\\{u,\\gy\\right\\}\n-F_0\\R{\\pa u}{\\pa\\gj}\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:AmpereEq}\nj\n&\n=\n&\n\\gD^*\\gy\n,\\quad\nj_\\gj\n=\n-j\/R\n,\\end{eqnarray}\nwith Poisson bracket $\\left\\{f,g\\right\\}\\equiv R\\left(\\na f\\times\\na g\\right)\\cdot\\na\\gj$. As mentioned in Section \\ref{ss:PartProj}, the particle moment contributes to the resistivity through the effective charge $Z_\\textrm{eff}$.\n\nSecond, the continuity equation for both the total plasma mass density and that for the impurity species is not affected:\n\\begin{eqnarray}\n\\lbq{eq:ContinuityEq}\n\\R{\\pa \\gr}{\\pa t}\n&\n=\n&\n-\\na\\cdot\\left(\\gr{\\bf v}\\right)\n+\\na\\cdot\\left(D\\na\\gr\\right)\n+S_{bg}\n+S_{imp}\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:ImpEq}\n\\R{\\pa \\gr_{imp}}{\\pa t}\n=\n-\\na\\cdot\\left(\\gr_{imp}{\\bf v}\\right)\n+\\na\\cdot\\left(D\\na\\gr_{imp}\\right)\n+S_{imp}\n.\\end{eqnarray}\nHere, the impurity density source $S_{imp}$ determines the particle generation by reject-sampling as mentioned in Section \\ref{ss:MarkerPartPusher}. We will show later in Section \\ref{s:ParModvsCEmod} that the fluid impurity density evolution described here are mostly consistent with that from the particle moment projection before the plasma termination.\n\nThird, we have the perpendicular and the parallel momentum equations:\n\\begin{eqnarray}\n\\lbq{eq:VorticityEq}\nR\\na\\cdot\\left[R^2\\R{\\pa}{\\pa t}\\left(\\gr\\na_{pol}u\\right)\\right]\n&\n=\n&\n\\R{1}{2}\\left\\{R^2\\left|\\na_{pol}u\\right|^2,R^2\\gr\\right\\}\n+\\left\\{R^4\\gr\\gw,u\\right\\}\n\\nonumber\n\\\\\n&&\n-R\\na\\cdot\\left[R^2\\na_{pol}u\\na\\cdot\\left(\\gr{\\bf v}\\right)\\right]\n+\\left\\{\\gy,j\\right\\}\n\\nonumber\n\\\\\n&\n&\n-\\R{F_0}{R}\\R{\\pa j}{\\pa\\gj}\n+\\left\\{P,R^2\\right\\}\n+R\\gm_\\bot\\left(T_e\\right)\\na_{pol}^2\\gw\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:VorticityDef}\n\\gw\n&\n=\n&\n\\R{1}{R}\\R{\\pa}{\\pa R}\\left(R\\R{\\pa u}{\\pa R}\\right)\n+\\R{\\pa^2 u}{\\pa Z^2}\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:ParallelEq}\nB^2\\R{\\pa}{\\pa t}\\left(\\gr v_\\|\\right)\n&\n=\n&\n-\\R{1}{2}\\gr\\R{F_0}{R^2}\\R{\\pa}{\\pa \\gj}\\left(v_\\|B\\right)^2\n-\\R{\\gr}{2R}\\left\\{B^2v_\\|^2,\\gy\\right\\}\n-\\R{F_0}{R^2}\\R{\\pa P}{\\pa \\gj}\n\\nonumber\n\\\\\n&\n&\n+\\R{1}{R}\\left\\{\\gy,P\\right\\}\n-B^2\\na\\cdot\\left(\\gr{\\bf v}\\right)v_\\|\n+B^2\\gm_\\|\\na_{pol}^2v_\\|\n.\\end{eqnarray}\nThe vorticity equation Eq.\\,(\\rfq{eq:VorticityEq}) is obtained by applying $\\na\\gj\\cdot\\na\\times\\left(R^2\\cdots\\right)$ on both sides of the momentum equation. The total pressure consists of the electron and the ion contribution $P=P_e+P_i$, with $P_i=\\left(n_{bg}+n_{imp}\\right)T_i$ and $P_e=n_e T_e=\\left(n_{bg}+Z_{imp}n_{imp}\\right)T_e$. Here, $n_{bg}$ is the number density of the background hydrogen isotopes, $n_{imp}$ is the impurity number density, $Z_{imp}$ is the projected mean charge number. The electron and the ion temperature are represented by $T_e$ and $T_i$ respectively. For the single temperature model, we have $T_e=T_i$.\n\nLast, we write down the pressure equation for the single and the two temperature model separately. For the single temperature model,\n\\begin{eqnarray}\n\\lbq{eq:PressureEq}\n\\R{\\pa P}{\\pa t}\n&\n=\n&\n-{\\bf v}\\cdot\\na P\n-\\gg P\\na\\cdot{\\bf v}\n+\\R{\\gg-1}{R^2}\\gh\\left(T_e\\right)j^2\n+\\na\\cdot\\left(\\gk_\\bot\\na_\\bot T\n+\\gk_\\|\\na_\\| T\\right)\n\\nonumber\n\\\\\n&\n&\n+\\left(\\gg-1\\right)\\gm_\\|\\left[\\na_{pol}\\left(v_\\|B\\right)\\right]^2\n-\\left(\\gg-1\\right)\\left(P_{rad}+P_{ion}\\right)\n\\nonumber\n\\\\\n&&\n+\\R{\\gg-1}{2}{\\bf v}\\cdot{\\bf v}\\left(S_{bg}+S_{imp}\\right)\n.\\end{eqnarray}\nHere, $P_{rad}$ is the projected radiative power. The ionization power is defined by $P_{ion}\\equiv \\gD E_{ion}\/\\gD t$, where $\\gD E_{ion}$ is the projected ionization energy loss during the fluid time step $\\gD t$. For the two temperature model, the equations are:\n\\begin{eqnarray}\n\\lbq{eq:i_TemperatureEq502}\n\\R{\\pa}{\\pa t}P_i\n&\n=\n&\n-{\\bf v}\\cdot \\na P_i\n-\\gg P_i\\na\\cdot{\\bf v}\n+\\na\\cdot\\left(\\gk_\\bot\\na_\\bot T_i\n+\\gk_{i,\\|}\\na_\\| T_i\\right)\n\\nonumber\n\\\\\n&&\n+\\R{\\gg-1}{2}{\\bf v}\\cdot{\\bf v}\\left(S_{bg}+S_{imp}\\right)\n+\\left(\\gg-1\\right)\\gm_\\|\\left[\\na_{pol}\\left(v_\\|B\\right)\\right]^2\n\\nonumber\n\\\\\n&&\n+\\left(n_{bg}+n_{imp}\\right)\\left(\\pa_t T_i\\right)_{c,e}\n,\\end{eqnarray}\n\\begin{eqnarray}\n\\lbq{eq:e_TemperatureEq502}\n\\R{\\pa}{\\pa t}P_e\n&\n=\n&\n-{\\bf v}\\cdot \\na P_e\n-\\gg P_e\\na\\cdot{\\bf v}\n+\\na\\cdot\\left(\\gk_\\bot\\na_\\bot T_e\n+\\gk_{e,\\|}\\na_\\| T_e\\right)\n\\nonumber\n\\\\\n&&\n+\\R{\\gg-1}{R^2}\\gh\\left(T_e\\right)j^2\n-\\left(\\gg-1\\right)\\left(P_{rad}+P_{ion}\\right)\n+n_e\\left(\\pa_t T_e\\right)_{c,i}\n.\\end{eqnarray}\nThe additional thermalization terms $\\left(n_{bg}+n_{imp}\\right)\\left(\\pa_t T_i\\right)_{c,e}$ and $n_e\\left(\\pa_t T_e\\right)_{c,i}$ are also dependent on the projected effective charge since the collisional ion-electron thermalization rate is proportional to the charge number squared of the ion species.\n\nTogether, Eq.\\,\\rfq{eq:InductionEq} to Eq.\\,\\rfq{eq:e_TemperatureEq502} form our governing equations for the non-equilibrium impurity treatment.\n\n\\section{The axisymmetric equilibrium and the simulation setup}\n\\label{s:Eq}\n\nWe provide several comparison and benchmark cases using a DIII-D equilibrium which has been used by the published M3D-C1 \\& NIMROD benchmark study \\cite{Lyons2019PPCF}. The equilibrium corresponds to the DIII-D shot 137611 at 1950ms. The initial plasma consists of pure deuterium with spatially uniform ion and electron density, $n_e=n_i=10^{20}m^{-3}$. In Ref. \\cite{Lyons2019PPCF}, both NIMROD and M3D-C1 were running with the single temperature model. As in Ref. \\cite{Lyons2019PPCF}, all runs are axisymmetric and nonlinear. Constant thermal conductivity and particle diffusivity are used here for direct comparison with Ref \\cite{Lyons2019PPCF}. To be explicit, we use the isotropic density diffusivity $D=10m^2\/s$, and the perpendicular thermal diffusivity is set to $100m^2\/s$. The parallel thermal diffusivity is set to $10^8m^2\/s$. For the two temperature case, the same heat conduction is used for both electrons and ions. The viscosity corresponds to a momentum diffusivity of $100m^2\/s$ initially, but with the same temperature dependence of the resistivity so that the magnetic Prandtl number is constant, as opposed to the cases in Ref. \\cite{Lyons2019PPCF} where the momentum diffusivity is constant. This deviation does not significantly impact our result, however, due to the lack of strong MHD activities in an axisymmetric modelling.\n\nWe use two different kinds of resistivity models as in Ref. \\cite{Lyons2019PPCF}, namely the constant resistivity and the Spitzer-like resistivity. For the constant resistivity, the resistivity is fixed to $10^{-5}\\gW m$, while for the Spitzer-like resistivity the expression is given in Eq.\\,(\\rfq{eq:Resistivity}), with $\\gh_0$ set to $1.83339\\times10^{-8}\\gW m$. Note that, as in Ref. \\cite{Lyons2019PPCF}, this resistivity corresponds to an enhanced resistivity $2.444$ times higher than the physical one. In our simulations, the temperature dependency of the resistivity is cut off below $T_{thres} = 1eV$ to avoid numerical instability.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{Initial_Eq.pdf}\n}\n\\end{tabular}\n\\caption{The initial equilibrium. $\\gY_N$ is the normalized poloidal flux.}\n\\label{fig:01}\n\\end{figure*}\n\nWe use a Gaussian shape distribution for our impurity density source with the following shape:\n\\begin{eqnarray}\nS_n\n\\propto\n\\exp{\\left(-\\R{\\left(R-R_{f}\\right)^2+\\left(Z-Z_{f}\\right)^2}{\\gD r^{2}_{NG}}\\right)}\n\\times\\exp{\\left(-\\left(\\R{\\gj-\\gj_{f}}{\\gD\\gj_{NG}}\\right)^2\\right)}\n.\\end{eqnarray}\nIn this axisymmetric simulation, we simply set $\\gD\\gj_{NG}$ to be a very large number, and we set $\\gD r^{2}_{NG}=0.356m$, $R_0=1.77037m$ and $Z_0=0.01447m$ so that we have generally the same deposition shape with Ref. \\cite{Lyons2019PPCF}. It should be noted that this shape still differs from that is used in Ref. \\cite{Lyons2019PPCF} by a $R_0\/R$ factor, which would not have any qualitative impact for the plasma dynamics given the geometry concerned. The initial equilibrium profile can be seen in Fig.\\,\\ref{fig:01}, where the initial electron temperature, electron density, toroidal current density and safety factor profiles are shown as functions of the normalized flux. As in Ref. \\cite{Lyons2019PPCF}, the volume integrated impurity atom injection rate is approximately $4.37\\times10^{23}$ per second.\n\n\n\\section{Comparison of the particle-based model against the CE fluid model}\n\\label{s:ParModvsCEmod}\n\nBefore we venture to benchmark with the M3D-C1 and the NIMROD results, we first present the comparison between the new particle-based model and the CE model used in Ref. \\cite{Di2021NF}. We will be looking at argon injections described by both the single temperature model and the two temperature model, with the Spitzer-like resistivity in both cases. We mainly compare the evolution of the integrated quantities in this section, for a typical 2D profile evolution of the non-equilibrium impurity case please see Fig.\\ref{fig:09} and Fig.\\ref{fig:10} later on in Section \\ref{s:Benchmark}.\n\nFor the single temperature model, the comparison between the particle-based treatment and the CE treatment is shown in Fig.\\,\\ref{fig:02}. The radiation power from both cases are shown in red, the ionization power in blue and the Ohmic heating power in green. The black lines represent the combined radiation and ionization power. The particle-based results are shown in solid lines while the CE ones are shown in dash-dot lines. The noisiness in the CE ionization power curve is due to the numerical noise when we reconstruct the ionization power from the total ionization energy by finite temporal difference. It can be seen that although the CE model initially start with larger ionization cooling due to the impurities jumping to the equilibrium charge distribution, overall the combined cooling power of the CE model exhibit slower increase compared with the non-equilibrium one. This causes the CE model to underestimate the initial cooling, and the combined cooling power evolution exhibits some delay relative to the non-equilibrium case as can be seen by comparing the black solid and dash-dot lines in Fig.\\,\\ref{fig:02}, although the peak cooling power is comparable between the two models. A notable deviation is that the non-equilibrium case quickly reached the balance between the combined cooling power and the Ohmic heating at $t\\approx 0.7ms$, while the CE case only does so very late into the simulation.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig01-S4.pdf}\n}\n\\end{tabular}\n\\caption{The radiation power (red), ionization power (blue) and Ohmic power (green) for the particle model (solid lines) and the CE model (dash-dot lines) with single temperature treatment. The black solid and dash-dot lines are the combined radiation and ionization power for the particle model and the CE model respectively.}\n\\label{fig:02}\n\\end{figure*}\n\nFor the two temperature model, a similar deviation between the particle-based model and the CE model is observed as shown in Fig.\\,\\ref{fig:03}, with the same color scheme as Fig.\\,\\ref{fig:02}. Again, the solid lines correspond to that of the particle-based model, and the dash-dot lines correspond to that of the CE one. Similar to the single temperature case, the CE model exhibits a stronger ionization cooling due to the immediate jump of the impurity ions to the equilibrium charge state distribution, but the overall cooling power in the early injection phase is underestimated compared with that of the non-equilibrium model, only reaching similar intensity when the plasma is cooled down significantly. A similar delay in the rise of the Ohmic power between the CE model and the non-equilibrium model is also observed. Different from the single temperature model shown in Fig.\\,\\ref{fig:02}, it can be seen that after a while the radiation and ionization power are balanced by the Ohmic power for both cases, although this happens slightly later for the CE model. The better CE model behavior in the two temperature case probably is due to the fact that the electron temperature is cooled down faster compared with that of the single temperature case. Both the peak cooling power and the peak Ohmic power are comparable for the CE and non-equilibrium models.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig02-S4.pdf}\n}\n\\end{tabular}\n\\caption{The radiation power (red), ionization power (blue) and Ohmic power (green) for the particle model (solid lines) and the CE model (dash-dot lines) with two temperature treatment. The black solid and dash-dot lines are the combined radiation and ionization power for the particle model and the CE model respectively.}\n\\label{fig:03}\n\\end{figure*}\n\nTo demonstrate the deviation of the impurity charge state distribution from the CE, we now show the mean impurity charge profile at three different times during the non-equilibrium two temperature argon injection with Spitzer-like resistivity in Fig.\\,\\ref{fig:Z_imp_compare}(a)-(c). For comparison, we also present the CE mean impurity charge profile for the same run at the same times in Fig.\\,\\ref{fig:Z_imp_compare}(d)-(f). It is important to note that, Fig.\\,\\ref{fig:Z_imp_compare}(d)-(f) show the CE mean impurity charge value calculated from the temperature and the density profile of the non-equilibrium impurity simulation, instead of that from a CE simulation. In this way we can directly show how much deviation we have between the non-equilibrium charge number to its equilibrium value. At time $t=0.26ms$, Fig.\\,\\ref{fig:Z_imp_compare}(a) and (d) correspond to the early phase of the injection, and it is clear that the mean impurity charge is low compared with its equilibrium value due to the existence of large weakly ionized population. At time $t=0.45ms$ which coincides with the first ``turning point'' in Fig.\\,\\ref{fig:03}, the non-equilibrium result mostly agrees with the CE result, although the CE result already begins to drop in the plasma core due to the central cooling as is shown in Fig.\\,\\ref{fig:Z_imp_compare}(e) . This is not yet reflected in Fig.\\,\\ref{fig:Z_imp_compare}(b) due to the fastness of the thermal collapse. Finally, at time $t=0.78ms$, we see that the inside-out cooling results in a halo of strongly ionized impurities under the CE assumption, as is shown in Fig.\\,\\ref{fig:Z_imp_compare}(f). Such hollow behavior also exists in the non-equilibrium result shown in Fig.\\,\\ref{fig:Z_imp_compare}(c), but to a lesser extent. The non-equilibrium result also shows higher ionization level since it takes some time to recombine and reach the dropping CE ionization level.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{ccc}\n\\parbox{2.0in}{\n    \\includegraphics[scale=0.2]{Z_imp_00300.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{Z_imp_00450.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{Z_imp_00700.png}\n}\n\\\\\n(a)&(b)&(c)\n\\\\\n\\parbox{2.0in}{\n    \\includegraphics[scale=0.2]{Z_imp_00300_CE.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{Z_imp_00450_CE.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{Z_imp_00700_CE.png}\n}\n\\\\\n(d)&(e)&(f)\n\\end{tabular}\n\\caption{The mean impurity charge for the non-equilibrium treatment at time (a) $t=0.26ms$, (b) $t=0.45ms$ and (c) $t=0.78ms$. For comparison, the corresponding CE mean impurity charge at the same time with (a)-(c) are shown in (d)-(f) respectively. Note the different color scale used in some figures.}\n\\label{fig:Z_imp_compare}\n\\end{figure*}\n\nSince we now have two representations of the impurity density profile, one from the fluid description and the other from the projection of particle moments, it is of interest to compare the consistency between the two. The contour comparison of the two temperature argon injection with Spitzer-like resistivity is shown in Fig.\\,\\ref{fig:N_imp_compare}. In the figure, the red contours corresponds to that from the particle moment projection, while the white ones are from the fluid representation. The contours from each color are on exactly the same level, so that the matching of the contours between the two colors indicate quantitative agreement between the two representations. It is seen that, throughout the majority of the time evolution, good agreement could be found between the two representations as is shown in Fig.\\,\\ref{fig:N_imp_compare}(a) and (b). Later on at the time of the plasma termination at $t=1.10ms$, some local deviation could be seen, especially in the region with elongated density profile, likely due to the lack of the diffusion contribution in our current particle pusher. We are working on a better pusher to include this effect in the near future. Meanwhile, we do not expect such effect to cause significant impact in future 3D simulations since the stochastic convection along the field line would dominate over the normal diffusion process.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{ccc}\n\\parbox{2.0in}{\n    \\includegraphics[scale=0.2]{N_imp_comparison_00450.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{N_imp_comparison_00700.png}\n}\n&\n\\parbox{2.0in}{\n\t\\includegraphics[scale=0.2]{N_imp_comparison_00950.png}\n}\n\\\\\n(a)&(b)&(c)\n\\end{tabular}\n\\caption{The impurity density contour comparison between the fluid description (white contours) and the particle moment projection (red contours) at (a) $t=0.45ms$, (b) $t=0.78ms$ and (c) $t=1.10ms$. Note that the contours from each color are on exactly the same scale, so that matching contour lines indicate quantitative match of the two representations.}\n\\label{fig:N_imp_compare}\n\\end{figure*}\n\nAnother interesting comparison could be made between the particle treatment and the CE fluid treatment if we enforce the CE charge state distribution upon the super-particles, instead of letting them evolve independently. This could serve to demonstrate the difference between particle representation and the fluid representation directly. A comparison of the cooling power and the ohmic power for the so-called ``CE particle'' against the normal CE fluid representation can be seen in Fig.\\,\\ref{fig:CE_particle}. Apart from some delay in time close to the plasma termination, it can be seen that very good agreement is found between the CE particle (solid lines) and the CE fluid (dash-dot lines) representation, both in terms of the time of signature events and in terms of the peak value of the cooling power and the ohmic heating. The late stage deviation could be due to the slight deviation in the particle and fluid impurity density representation as previously discussed in Fig.\\ref{fig:N_imp_compare}.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{CE_particle.pdf}\n}\n\\end{tabular}\n\\caption{The radiation power (red), ionization power (blue) and Ohmic power (green) for the CE particle model (solid lines) and the CE fluid model (dash-dot lines) with two temperature treatment for argon injection. The black solid and dash-dot lines are the combined radiation and ionization power for the particle model and the CE model respectively.}\n\\label{fig:CE_particle}\n\\end{figure*}\n\nAll in all, the non-equilibrium model shows a stronger early phase cooling than the CE model. This discrepancy is due to the deviation of impurity charge state distribution from the CE early in the injection. The peak cooling power and Ohmic heating power are comparable between the two models, however the CE model exhibits a delay in reaching those peaks due to the aforementioned inaccuracy in the early cooling phase. In both cases, the total cooling power becomes balanced with the Ohmic power after some time, although the non-equilibrium model reaches this balance faster due to previously explained reasons. The non-equilibrium treatment also shows delayed charge state evolution compared with CE as would be expected. Good agreement is found between the fluid and the particle representation of the impurity density for the majority of the plasma evolution.\n\n\\section{Comparison between the single and the two temperature cases}\n\\label{s:TeTiComparison}\n\nIn this section, we use the axisymmetric DIII-D non-equilibrium impurity simulation to demonstrate the difference in behaviours between the single and the two temperature cases.\n\nWe compare the time evolution of integrated quantities during an axisymmetric DIII-D argon injection with non-equilibrium impurity treatment in Fig.\\,\\ref{fig:12}. The solid lines indicates the quantities of the two temperature model while the dash-dot lines indicate that of the single temperature model. The red lines are the radiation power, the blue lines are the ionization power and the green lines are the Ohmic power. For the two temperature model, the magenta solid line indicates the thermal energy transfer rate between the ions and the electrons, where a positive value indicates that the ions are transferring energy to the electrons. The black lines indicate the combination of the radiation power and the ionization power.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig04a.pdf}\n}\n\\end{tabular}\n\\caption{The comparison of JOREK non-equilibrium treatment under the single temperature (dash-dot lines) and the two temperature (solid lines) case. The red lines are the radiation power, the blue ones the ionization power, the black ones represent the above two combined. The Ohmic power is shown in green and the ion-electron energy transfer for the two temperature model is shown in magenta. Apart from the magenta line, this figure is essentially the overlay of the non-equilibrium curves from Fig.\\ref{fig:02} and Fig.\\ref{fig:03}.}\n\\label{fig:12}\n\\end{figure*}\n\n\nThe first peak in the cooling power indicates the plasma is cooled down below $100eV$, thus this ``turning point'' coincides with the rise of the Ohmic power. This is also when the ion-electron temperature difference is largest, as suggested by the peak in the ion-electron energy transfer. Then the cooling power and the Ohmic power begin to balance each other, resulting in the plateau region. At this time the inter-species temperature difference is greatly diminished due to thermalization, as indicated by the magenta solid line going to zero. We also see later in Fig.\\,\\ref{fig:14} that by this time, most of the thermal energy is already depleted and it is the Ohmic conversion of the magnetic energy that is balancing the radiative loss, in agreement with Fig.\\,\\ref{fig:12} here. Finally there is another peak in the Ohmic power and the cooling power before the plasma terminates.\nIt can be seen that the two models generally exhibit the same trend of evolution. Both models show the rise of the combined cooling power before the plasma is significantly cooled. Both models show the rise of the Ohmic heating coincides with the turning point of the combined cooling power. Both models exhibit the balance between the Ohmic heating and the radiative cooling, and both models show the spike in the Ohmic heating and the radiative cooling just before the plasma terminates towards the end of the simulations. The two temperature model shows a somewhat reduced radiation peak, as well as an earlier rise of the Ohmic power. This is due to the deviation between the ion and electron temperature in the two temperature case, as opposed to the single temperature case where the two species thermalize immediately.\n\nThe comparison for the neon injection case is similarly shown in Fig.\\,\\ref{fig:13}. The solid lines still represent the two temperature model and the dash-dot lines represent the single temperature model. The cooling power shows a plateau before rising into the first peak, which corresponds to the time of plasma cooling down to $100eV$ range. After the ``turning point'' the Ohmic power balances the cooling power, until a second peak marks the termination of the plasma. Once again, the two models agree which each other on the general trend of the time evolution, as well as the peak value of the cooling power and the Ohmic heating power. However, the neon single temperature model exhibits a more pronounced delay than the argon case in all the key characteristic events compared with the two temperature model. This is again due to the fact that the electrons in the two temperature model are cooled down faster in the early phase of the injection, thus reaching the ``turning point'' in the radiation curve faster than their single temperature counterparts.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig04b.pdf}\n}\n\\end{tabular}\n\\caption{The comparison of JOREK non-equilibrium treatment under the single temperature (dash-dot lines) and the two temperature (solid-lines) case. The red lines are the radiation power, the blue ones the ionization power, the black ones represent the above two combined. The Ohmic power is shown in green, and the inter-specie energy transfer is shown in magenta.}\n\\label{fig:13}\n\\end{figure*}\n\nThe ion and electron thermal energy evolution for the above argon and neon injection cases is shown in Fig.\\,\\ref{fig:14}. The ion thermal energies are shown in blue lines while the electron ones are shown in red lines. The argon injection case is shown with solid lines and the neon one is shown in dash-dot lines. Since the radiative cooling is acting mostly on the electron, one would naively expect to see the electron thermal energy decreasing faster than the ion one. Somewhat anti-intuitively, approximately after $t=0.6ms$ in both cases, the electron thermal energy is higher than the ion one in both cases, this is simply due to the larger particle number of electrons as the electron temperature in general is lower than the ion temperature. Another noteworthy feature is that by the time we have the balance between the cooling power and the Ohmic power in Fig.\\,\\ref{fig:12} and Fig.\\,\\ref{fig:13}, the thermal energies from both species are already mostly depleted, and only the Ohmic conversion of the magnetic energy is balancing the radiative cooling.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig_Eth.pdf}\n}\n\\end{tabular}\n\\caption{The time evolution of the ion and the electron thermal energy in simulations with the two-temperature model.}\n\\label{fig:14}\n\\end{figure*}\n\nOverall, the two temperature model shows a faster cooling electron population compared with the single temperature case, although in both cases the peak value of the cooling power and the Ohmic heating power are comparable. Depending on species, the single temperature model could see some time delay in the occurrence of the key characteristic events compared with the two temperature model. This might have some impact on the disruption mitigation behaviors especially considering the interplay between the MHD modes and the fragment locations.\n\n\\section{Benchmark with NIMROD \\& M3D-C1}\n\\label{s:Benchmark}\n\nIn this section, we provide the benchmark against the published M3D-C1 and NIMROD axisymmetric simulations from Ref. \\cite{Lyons2019PPCF}. We look into argon and neon injections with both constant resistivity and Spitzer-like resistivity as explained in Section \\ref{s:Eq}. We only use the single temperature model of JOREK in this section.\n\n\\subsection{Argon and neon cases with constant resistivity}\n\\label{ss:ArConstEta}\n\nWe begin with argon and neon injections using constant resistivity. We directly extracted the combined radiation and ionization power as a function of time from Ref. \\cite{Lyons2019PPCF}. We have chosen to only show the NIMROD data here since the M3D-C1 and the NIMROD data follow each other rather well. We also extracted the total electron number increase due to the injected impurities, as an indication of the ionization state evolution.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig00a-S5.pdf}\n}\n\\end{tabular}\n\\caption{The comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with constant resistivity. The red lines are the argon injection, the black ones are the neon injection.}\n\\label{fig:04}\n\\end{figure*}\n\nWe compare the total electron number increase between the JOREK and NIMROD results in Fig.\\,\\ref{fig:04}. The solid lines are the JOREK results while the dash-dot lines are the NIMROD results. The red lines indicates the argon injection results, while the black lines indicate the neon injection results. Very good agreement is obtained in the early phase. However, discrepancies begin to occur near the ``turning points'' of the curves, which correspond to the cooling of the plasma from $100eV$ to $10eV$. We will see later that for argon this is mostly due to a time difference of the plasmas reaching such a temperature region. For the neon case, we will show in \\ref{ap:Atomic} that it may be caused by the difference in the recombination probability due to the different atomic models used.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig01-S5.pdf}\n}\n\\end{tabular}\n\\caption{The argon injection comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with constant resistivity. The red solid line is the radiation power, the blue one the ionization power, the black one represents the above two combined. The Ohmic power is not shown due to being constant as the current density change is small.}\n\\label{fig:05}\n\\end{figure*}\n\nThe cooling power comparison between the JOREK argon injection and the corresponding NIMROD data for the constant resistivity case is shown in Fig.\\ref{fig:05}. Comparing the black solid line and the dash-dot line, we can see that the combined radiation and ionization power as a function of time share the same general shape, as well as similar peak level. The JOREK result seems to have its peak delayed by $0.15ms$ compared with the NIMROD result, this is due to the difference in the atomic data used. Indeed, we demonstrate in \\ref{ap:Atomic} that much better agreement is achieved if we switch to older versions of the ADAS data in JOREK. Nonetheless, the JOREK case and the NIMROD case deplete the thermal energy by the same time, as is shown by the black solid and the dash-dot lines reaching the constant Ohmic heating level at the same time, showing the robustness of the plasma dynamics. In the later half of the simulation, the red solid line differs slightly from the black solid line. This means that recombination contributes significantly to the radiated power. Despite the deviation in cooling details caused by the difference in the atomic data, the two codes generally agree with each other in both the time evolution of the cooling power curve and the peak cooling power.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig03-S5.pdf}\n}\n\\end{tabular}\n\\caption{The neon injection comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with constant resistivity. The red solid line is the radiation power, the blue one the ionization power, the black one represents the above two combined. The Ohmic power is not shown due to being constant as the current density change is small.}\n\\label{fig:06}\n\\end{figure*}\n\nA similar comparison for neon injections is shown in Fig.\\,\\ref{fig:06}. Again, the solid lines are the JOREK non-equilibrium treatment results, while the dash-dot lines are the NIMROD results. Very good agreement exists before $t=1.2ms$, but the JOREK result shows a higher radiation peak and subsequent rapid drop of the radiation power, while the NIMROD result plateaus at a lower level for some time. The quick drop of the JOREK radiation power is the result of the rapid recombination shown in Fig.\\,\\ref{fig:04}, since the radiation power is dependent on the electron density and the charge state distribution as is shown in Eq.\\,\\rfq{eq:Prad}.\n\n\\subsection{Argon and neon case with the Spitzer-like resistivity}\n\\label{ss:ArSpEta}\n\nWe now compare the argon and neon injection simulated by JOREK and NIMROD with Spitzer-like resistivity. The total electron number from the injected impurities is shown in Fig.\\,\\ref{fig:07}. Compared with the constant resistivity case, the agreement between the JOREK result and the NIMROD one is better, although the JOREK neon case still shows significant recombination when the plasma core is cooled down below $100eV$ at $t=1.2ms$, corresponding to the turning point in the black solid line. Again, this difference in behaviour is most likely caused by the difference in the atomic data used.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig00b-S5.pdf}\n}\n\\end{tabular}\n\\caption{The comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with Spitzer-like resistivity. The red lines are the argon injection, the black ones are the neon injection.}\n\\label{fig:07}\n\\end{figure*}\n\nThe comparison of the cooling power between the JOREK particle-based treatment and the NIMROD data for the Spitzer-like resistivity case is shown in Fig.\\,\\ref{fig:08}. Similar to the constant resistivity case, although the cooling peak occurs a bit later for the JOREK case, the two codes generally agree on the peak cooling power and the time evolution. Further, the Ohmic power evolution shows general agreement, and the two codes exhibit the same ``plateau'' behavior on the same power level when the Ohmic power comes to balance the cooling power as the plasma cools down. The two codes also show the same ``bursting'' feature in the Ohmic power and the cooling power at the same time towards the end of the simulation, although in the NIMROD case the Ohmic heating seems to be stronger. It should be noted that such deviation towards the end of the simulation is also present between the NIMROD and M3D-C1 comparison in Ref. \\cite{Lyons2019PPCF}, probably due to the difference in the details of the current filament dynamics.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig02-S5.pdf}\n}\n\\end{tabular}\n\\caption{The comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with Spitzer-like resistivity. The red solid line is the radiation power, the blue one the ionization power, the black one represent the above two combined. The Ohmic power for both cases is shown in green solid and dash-dot lines respectively. The JOREK time traces here is the same with the time trace of the non-equilibrium treatment shown in Fig.\\ref{fig:02}.}\n\\label{fig:08}\n\\end{figure*}\n\nThe agreement can also be seen from the contour evolution of both the temperature and the current density. The electron temperature evolution for the JOREK case with argon injection and Spitzer-like resistivity is shown in Fig.\\,\\ref{fig:09}.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{cc}\n\\parbox{2.8in}{\n    \\includegraphics[scale=0.28]{T_e_00600.png}\n}\n&\n\\parbox{2.8in}{\n\t\\includegraphics[scale=0.28]{T_e_01300.png}\n}\n\\\\\n(a)&(b)\n\\\\\n\\parbox{2.8in}{\n  \t\\includegraphics[scale=0.28]{T_e_02050.png}\n}\n&\n\\parbox{2.8in}{\n\t\\includegraphics[scale=0.28]{T_e_03350.png}\n}\n\\\\\n(c)&(d)\n\\end{tabular}\n\\caption{The electron temperature evolution during the argon-injection simulated by the JOREK particle-based impurity model with Spitzer-like resistivity at time (a) $t=0.408ms$, (b) $t=0.585ms$, (c) $t=0.890ms$, $t=1.106ms$. Note the different color scale used in some figures.}\n\\label{fig:09}\n\\end{figure*}\n\nCompared with Fig.\\,7 of Ref. \\cite{Lyons2019PPCF}, it can be seen that the JOREK electron temperature evolution follows the same characteristic trend compared with the M3D-C1 and NIMROD results, although the exact timing may slightly differ. From Fig.\\,\\ref{fig:09}(a) to (b), we see the inside-out core temperature collapse due to the on-axis injection of argon, and a hollow electron temperature profile. This trend of outward cooling continues until there is only a thin electron temperature ``halo'' remaining as shown in Fig.\\,\\ref{fig:09}(c). After this ``halo'' finally collapsed, we begin to see the re-heating of a very localized portion of the plasma, which is caused by a re-concentration of the toroidal current as will be shown below.\n\nThe accompanying toroidal current density profile at the same time slices is shown in Fig.\\,\\ref{fig:10}. Once again, the characteristic trend of the current density profile evolution agrees with the other two codes presented in Ref. \\cite{Lyons2019PPCF}, as can be seen by comparing directly with Fig.\\,8 of Ref. \\cite{Lyons2019PPCF}. From Fig.\\,\\ref{fig:10}(a) to (b), an inside-out current halo develops as the result of the hollow temperature profile. This current halo in the JOREK case is not as steep as in the other two codes, probably due to the different atomic data used causing the electron temperature profile to differ subtly, thus impacting the details of axisymmetric current redistribution. Nonetheless, as the plasma cools down further, we see the thin current halo forming as shown in Fig.\\,\\ref{fig:10}(c), exhibiting a similar behaviour as is shown in Ref. \\cite{Lyons2019PPCF}. This thin current layer corresponds to the thin electron temperature shell shown in Fig.\\,\\ref{fig:09}(c), as it is the Ohmic heating that maintain the electron temperature. Last, towards the end of the simulation, we see the re-concentration of the toroidal current near the new magnetic axis as is shown in Fig.\\,\\ref{fig:10}(d), responsible for the heat spot in Fig.\\,\\ref{fig:09}(d). This thin current filament can exist only due to the axisymmetric nature of our simulation, otherwise it would terminates by a kink motion due to such a concentration of current density.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{cc}\n\\parbox{2.8in}{\n    \\includegraphics[scale=0.28]{J_phi_00600.png}\n}\n&\n\\parbox{2.8in}{\n\t\\includegraphics[scale=0.28]{J_phi_01300.png}\n}\n\\\\\n(a)&(b)\n\\\\\n\\parbox{2.8in}{\n  \t\\includegraphics[scale=0.28]{J_phi_02050.png}\n}\n&\n\\parbox{2.8in}{\n\t\\includegraphics[scale=0.28]{J_phi_03350.png}\n}\n\\\\\n(c)&(d)\n\\end{tabular}\n\\caption{The toroidal current density evolution during the argon-injection simulated by the JOREK particle-based impurity model with Spitzer-like resistivity at time (a) $t=0.408ms$, (b) $t=0.585ms$, (c) $t=0.890ms$, $t=1.106ms$. Note the different color scale used in some figures.}\n\\label{fig:10}\n\\end{figure*}\n\nLike-wise, the cooling power comparison of the neon injection is shown in Fig.\\,\\ref{fig:11}. The solid lines indicate the JOREK result while the dash-dot lines indicate the NIMROD results. Good agreement is once again found in the early to middle phase of the simulation, but the JOREK result still shows somewhat higher radiation peak when the plasma is cooled down below $100eV$ at $t=1.3ms$. Nevertheless, the plasma termination occurs approximately at the same time for the two codes, as can be seen by comparing the peak time of the two green curves near $t=1.7ms$.\n\n\\begin{figure*}\n\\centering\n\\noindent\n\\begin{tabular}{c}\n\\parbox{6.5in}{\n    \\includegraphics[scale=0.45]{fig04-S5.pdf}\n}\n\\end{tabular}\n\\caption{The comparison between JOREK non-equilibrium treatment (solid lines) and the NIMROD data (dash-dot lines) with Spitzer-like resistivity. The red solid line is the radiation power, the blue one the ionization power, the black one represents the above two combined. The Ohmic power for both cases is shown in green solid and dash-dot lines respectively.}\n\\label{fig:11}\n\\end{figure*}\n\nOverall, despite the difference in the atomic models used, the JOREK particle-based impurity model shows reasonable agreement with M3D-C1 and NIMROD simulations both in terms of the time evolution of integrated quantities and in terms of the characteristic 2D profile evolution of the current density and the temperature. Apart from demonstrating the capability of the JOREK non-equilibrium impurity treatment, such agreement also highlights the robustness of the radiative cooling dynamics.\n\n\n\n\\section{Conclusion and discussion}\n\\label{s:Conclusion}\n\nIn this paper, we have introduced the new particle-based collisional-radiative non-equilibrium impurity treatment in JOREK. This new treatment is based on the coupled evolution of marker-particles and the fluid fields. The marker-particles are generated according to the impurity density source term, and are pushed along the fluid velocity field lines while ionizing and recombining independently according to local plasma parameters. We then project the particle contribution of the ionization power, the radiative power, the effective and mean charge and the impurity density back to the fluid evolution.\n\nTo illustrate the performance of this new non-equilibrium model, we presented a series of comparison and benchmark cases both against the CE result and against M3D-C1 and NIMROD non-equilibrium impurity injection simulations. As for the comparison against the JOREK CE treatment using Spitzer-like resistivity, we find the particle-based model successfully catches the early cooling dynamics which were missed by the previous CE treatment, although the peak radiation value is comparable between the non-equilibrium and the CE treatment.\n\nUnder the single temperature model, the CE treatment missed the quick rise of the Ohmic heating power in the middle of the simulation, as it underestimates the early phase cooling and did not bring the electron temperature down fast enough. Only towards the end of the simulation, we see the balance between the Ohmic heating and the radiative cooling. Under the two temperature treatment, the CE model behaves better as the peak radiative power, the peak Ohmic power and the power balance are comparable with the non-equilibrium model, although the CE result still show some delay in time.\n\nAs a demonstration, we also show the evolution of the non-equilibrium mean charge number profile and the comparison with its CE value using the non-equilibrium, two temperature argon injection case. The mean charge number generally show some delay in the time evolution due to the finite time requirement to reach the CE state, highlighting the importance of the non-equilibrium description when describing the impurity dynamics in a rapidly changing plasma background.\n\nComparing the non-equilibrium impurity behavior between the single and the two temperature treatment, it is found that the electrons in the two temperature model generally are cooled down faster compared with their single temperature counterparts. This results in earlier ``turning point'' in the radiation power curve, although the peak cooling power is comparable between the single and the two temperature models. This time difference is especially pronounced in the neon injection case. Such comparison has shown that the two temperature model is capable of capturing the temperature deviation between species at the time of the radiative collapse. Such deviation could be important for the MHD and the ablation dynamics during a DMS scenario.\n\nMoreover, compared with the coronal non-equilibrium simulation of M3D-C1 and NIMROD, the JOREK non-equilibrium result shows good agreement with the other two codes in terms of the peak cooling power, the Ohmic power, the power balance and the time evolution of those quantities. For the argon case, the time by which the first radiation peak is reached differs slightly between the JOREK result and that of the other two codes, due to the difference in the atomic data used. For the neon case, the JOREK results show stronger radiative power and recombination when the plasma is cooled down below $100eV$. Despite these deviations, the later plasma evolution and the plasma termination process show the same characteristic behaviours among the three codes, highlighting the robustness of the plasma dynamics.\n\nSince we are only pushing the marker particles along the velocity field lines thus only taking into consideration the convective transport of the impurities, our current method may suffer some inconsistency in describing the impurity density evolution should there be a strong impurity diffusion. However, since our current scenarios of interest are mostly dominated by convective transport, this is not immediately foreseen to be problematic. We are working on an improved particle pusher to take into account the diffusive transport of the marker particles so that we can later on expand our study to cases with strong impurity diffusion.\n\nOverall, we have shown that the JOREK non-equilibrium impurity model is now ready for production. It is capable of accurately capturing the early injection phase cooling and is compatible with both the single and the two temperature model, paving the way for future simulation of important DMS scenarios.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven the ubiquity of viscous phenomena around us, it is only natural to wonder how cosmological observations can constrain the presence, or not, of viscous effects at different stages in the large scale evolution of the universe. In fact, dissipative processes in the early universe have been studied for quite some time\n\\cite{WeinbergViscosityCosmology,Belinsky:1979,Gron:1990ew}.  In isotropic and\nhomogeneous spacetimes dissipation can only appear from scalar sources,\nwhich motivated early on the study of bulk viscosity in the expansion of\nthe universe \\cite{Murphy:1973zz,Belinsky:1977}. Since this effect is expected to contribute\nnegatively to the pressure of an expanding universe, bulk viscosity-driven\ninflation has also been examined\n\\cite{Padmanabhan:1987dg,Hiscock:1991sp,Pavon:1990qf,Zakari:1993yhk,Maartens:1995wt,Zimdahl:1996ka}.\nMore recently, after the discovery of the\ncurrent accelerated expansion of the universe \\cite{Peebles:2002gy},\nthe possibility of unifying dark matter and dark energy as a single\nviscous fluid generated a lot of\ninterest \\cite{Fabris:2005ts,Colistete:2007xi,LiBarrow,Avelino,Hipolito1,Hipolito2,Gagnon,Piattella_et_al,Velten:2012uv,VeltenSchwarz,BG,Disconzi_Kephart_Scherrer_2015,DisconziKephartScherrerNew,\nCruz:2018psw,Cruz:2019uya} (for a recent review, see \\cite{Brevik:2017msy}). \n\nBefore statements concerning the suitability of viscous dark matter models to match cosmological observations can be reliably made, it is important to keep in mind that there are still fundamental theoretical questions concerning the description of viscous effects in general-relativistic fluids that are very relevant to this problem. In fact, in standard approaches viscous processes fundamentally alter the equations of motion of relativistic fluids through the addition of new terms containing space-like derivatives\\footnote{Hence, only spatial derivatives appear in the energy-momentum tensor in the fluid's local rest frame.} of the hydrodynamic variables in the fluid's energy-momentum tensor. This occurs, for instance, in the famous theories pioneered by Eckart \\cite{EckartViscous} and Landau and Lifshitz \\cite{LandauLifshitzFluids}. \nHowever, these modifications seem to be incompatible with general relativity. Indeed, the theories proposed by Eckart and Landau and Lifshitz are acausal \\cite{PichonViscous,Hiscock_Lindblom_instability_1985}, which makes them unsuitable for the investigation of real-time  viscous processes in relativity. Furthermore, it is known that such theories are unstable against perturbations around the thermodynamical equilibrium state  \\cite{Hiscock_Lindblom_instability_1985}. This is a consequence of the more general statement that acausal dissipative theories cannot be stable in relativity \\cite{Gavassino:2021owo}.\n\nOne could at first sight think that the Eckart and Landau and Lifshitz theories\nare still suitable for cosmological investigations despite their acausality and\ninstability because the symmetry assumptions made in cosmological models imply\nthat the dynamic evolution is described by a system of ordinary differential\nequations, whereas causality and stability are concepts applicable only to\npartial differential equations.\\footnote{One can of course talk about stability of ordinary differential equations. But the type of stability discussed in the context of Eckart and Landau and Lifshitz theories requires at least two independent variables.} Nevertheless, such a description given in terms of ordinary differential equations implicitly assumes that the underlying system of partial differential equations (from which the ordinary differential equations\narise upon imposing symmetry conditions) is well posed. More precisely, since the symmetry assumptions of cosmology are only approximately satisfied (e.g., the universe is not perfectly homogeneous nor isotropic in large scales), the actual system's description is given in terms of the full Einstein-matter equations as partial differential equations. Only if the ordinary differential equations provide a good approximation to the underlying partial differential equations can one take their solutions as a good approximation to the actual, not perfectly symmetric, system. This requires the system of partial differential equations to be well-posed \\cite{Rezzolla_Zanotti_book}, a property that fails for the Eckart and Landau and Lifshitz theories \\cite{PichonViscous}. \n\nIn sum, the above facts hamper the application of Eckart's and Landau and Lifshitz's theories in  questions concerning the cosmological evolution of the universe. The same can be said about the fate of cosmological fluctuations in such viscous fluid models. Therefore, conclusions obtained from such models must be taken with a grain of salt (at best).\n\n\nIsrael and Stewart (IS) put forward an approach \\cite{MIS-5} where linearized disturbances around global equilibrium can be causal and stable \\cite{Hiscock_Lindblom_stability_1983}, if certain conditions for the fluid's equation of state and transport coefficients (e.g., bulk and shear viscosities) are fulfilled. However, despite recent progress \\cite{Bemfica:2019cop,Bemfica:2020xym}, very little is known about the properties and the constraints that must be fulfilled in these theories in the nonlinear regime, which can be important in simulations already in flat spacetime \\cite{Plumberg:2021bme} and, also, when embedding such fluid models in dynamical spacetimes. In fact, in the context of viscous dark fluid modeling, it is not known how the recently found nonlinear constraints \\cite{Bemfica:2019cop,Bemfica:2020xym} coming from causality affect previous conclusions drawn from such Israel-Stewart-like models (e.g., \\cite{Maartens:1995wt,Cruz:2018psw,Cruz:2019uya}). The constraints become especially relevant in the far-from-equilibrium regime where viscous effects are large, which is probed in viscous dark fluid models that attempt to unify dark matter and dark energy in an accelerating universe.  \nIn addition, well-posedness of the Israel-Stewart equations remains an open question, except in some very particular cases \\cite{Bemfica:2019cop}. This is a potential drawback given the importance\nof well-posedness for an accurate description of the evolution, as explained above.\n\n\nIn this work, we investigate how viscous effects can affect the evolution of\nfluids in isotropic, homogeneous, and spatially flat spacetimes using the new\ngeneral effective-theory formalism originally proposed in\n\\cite{BemficaDisconziNoronha} and further developed in\n\\cite{Kovtun:2019hdm,Bemfica:2019knx,Hoult:2020eho,Bemfica:2020zjp}. In this\napproach, known as BDNK in the fluid dynamical literature after the initials of\nthe authors of \\cite{BemficaDisconziNoronha,Kovtun:2019hdm}, the viscous\ncontribution to the energy-momentum tensor is expanded according to a\nwell-defined power-counting scheme in terms of all the possible time-like and\nspace-like derivatives of the hydrodynamic variables (e.g. density, flow\nvelocity) compatible with the symmetries, in contrast to standard formulations \n\\cite{EckartViscous,LandauLifshitzFluids} where only terms defined using\nspace-like derivatives are included. The full system of equations of motion\ndescribing the evolving viscous fluid coupled to Einstein's equations has been\nproven  to be causal and strongly hyperbolic \\cite{Bemfica:2020zjp}, hence\nwell-posed, even in the full nonlinear regime.\\footnote{Hydrodynamic stability around equilibrium has also been established, see \\cite{BemficaDisconziNoronha,Kovtun:2019hdm,Bemfica:2019knx,Hoult:2020eho,Bemfica:2020zjp}.} Therefore, this framework is uniquely suited to investigate real-time dynamical problems, and also mathematical questions, concerning the  coupling of Einstein's equations to viscous fluids in general relativity. Numerical solutions of this theory can already be found in \\cite{Pandya:2021ief,Pandya:2022pif,Bantilan:2022ech,Pandya:2022sff} while systematic derivations of BDNK theory from kinetic theory and holography were presented in \\cite{Rocha:2022ind,Hoult:2021gnb} (see also \\cite{Biswas:2022cla}). Some recent applications can be found  in Refs.\\ \\cite{Danielsson:2021ykm,Armas:2022wvb}.\n\nWe focus in this work on the simple dynamics displayed by a viscous fluid in  Friedmann-Lema\\^itre-Robertson-Walker spacetime \\cite{Weinberg_GR_book}, in the absence of a cosmological constant ($\\Lambda=0$). We show through a variety of examples that an initially dust-like matter component (cold dark matter) can drive\nan accelerated expansion at late times when viscous effects are included. This implies that the previous intuition concerning the effects of viscosity acquired from inconsistent, or less well understood, fluid models was well motivated. However, the consistent treatment of causality at all levels in our approach predicts a curious new effect that is only possible in general relativity: the  viscous fluid does not asymptotically achieve\na constant equilibrium density in accelerating cosmologies; instead, this density decays away as a power of the scale factor. This occurs even though the cosmological constant is set to zero. In other words, the equilibrium contribution of the viscous fluid disappears at late times leaving only the acceleration of the universe behind even though there is no cosmological constant. Surprisingly enough, this phenomenon is fully consistent with Einstein's equations. \n\nThis ``Cheshire Cat\"-like behavior \\cite{carroll:1865} during accelerated\nexpansion is a consequence of causality in this approach and it cannot be reproduced by any previous model without a cosmological constant where the equilibrium density must remain nonzero when the universe is accelerating. Although different types of behavior for the evolution of the cosmological scale factor are possible, for completeness we show that \nan appropriate choice of model parameters can produce evolution that almost exactly mimics $\\Lambda$CDM. While we are not advocating here that viscous fluids provide an alternative way to fully describe cosmological observations by unifying dark energy and dark matter, it is amusing to see that the more sophisticated and theoretically consistent framework employed in this work does not seem to be incompatible with this idea (at least when it comes to the average properties of the universe). \n\nThis paper is organized as follows. In the next section we lay out the general properties of first-order theories of relativistic fluid dynamics and discuss their dynamics in Friedmann-Lema\\^itre-Robertson-Walker spacetime. We discuss the case of radiation and also present some new features of such theories in the case of accelerated expansion. In section \\ref{fixedpointsection} we analyze the stability property of the equations of motion and their fixed points. In section \\ref{viscouscosmo} we consider whether the theory presented here can lead to cosmological evolution consistent\nwith observations. Our final remarks can be found in section \\ref{conclusions}.\n\n\\emph{Notation}: We use natural units $\\hbar=c=k_B=1$, a 4-dimensional spacetime metric $g_{\\mu\\nu}$ with a mostly plus signature, and Greek indices\nrun from 0 to 3 while Latin indices run from 1 to 3. \n\n\\section{Cosmology with first-order general-relativistic viscous fluid dynamics}\\label{section2}\n\nLet us briefly review the effective theory approach to relativistic viscous fluid dynamics introduced in Ref.\\ \\cite{BemficaDisconziNoronha} and further developed in \\cite{Kovtun:2019hdm,Bemfica:2019knx,Hoult:2020eho,Bemfica:2020zjp}. As usual \\cite{Rezzolla_Zanotti_book}, one starts by decomposing the energy-momentum tensor of a fluid in a general out of equilibrium state in terms of irreducible structures constructed using a time-like future-directed  4-velocity vector $u^\\mu$ (where $u_\\mu u^\\mu=-1$):\n\\begin{equation}\nT^{\\mu\\nu}=\\mathcal{E}u^\\mu u^\\nu+ \\mathcal{P}\\Delta^{\\mu\\nu}+\\pi^{\\mu\\nu}+u^\\mu\\mathcal{Q}^\\nu+u^\\nu\\mathcal{Q}^\\mu,\n\\label{general_tensor}\n\\end{equation}\nwhere $\\mathcal{E} = u_\\mu u_\\nu T^{\\mu\\nu}$ is the total energy density seen by an observer comoving with the fluid, $\\mathcal{P} = \\Delta_{\\mu\\nu}T^{\\mu\\nu}\/3$ is the total fluid pressure defined using the space-like projector $\\Delta_{\\mu\\nu}=g_{\\mu\\nu}+u_\\mu u_\\nu$ orthogonal to $u^\\mu$, $\\mathcal{Q}^\\mu = - \\Delta^\\mu_\\lambda u_\\nu T^{\\nu\\lambda}$ describes energy diffusion, and $\\pi^{\\mu\\nu} = \\Delta^{\\mu\\nu\\alpha\\beta}T_{\\alpha\\beta}$ is the shear-stress tensor defined using the symmetric, rank-4 traceless projector $\\Delta^{\\mu\\nu\\alpha\\beta} = \\frac{1}{2}\\left(\\Delta^{\\mu\\beta}\\Delta^{\\nu\\alpha} + \\Delta^{\\mu\\alpha}\\Delta^{\\nu\\beta}\\right)-\\frac{1}{3}\\Delta^{\\mu\\nu}\\Delta^{\\alpha\\beta}$ \\cite{Rezzolla_Zanotti_book}.\n\nIn an equilibrium state, $\\pi^{\\mu\\nu}$ and $\\mathcal{Q}^\\mu$ vanish, $\\mathcal{E}$ becomes the equilibrium energy density $\\rho$, and $\\mathcal{P}$ the corresponding thermodynamic pressure $P(\\rho)$ of the system determined by its equation of state. The corresponding equilibrium energy-momentum tensor is then $T_0^{\\mu\\nu} = \\rho u^\\mu u^\\nu + P\\Delta^{\\mu\\nu}$. In a general out-of-equilibrium state, one may write $\\mathcal{E} = \\rho+\\mathcal{A}$ and $\\mathcal{P} = P + \\Pi$, where $\\mathcal{A}$ and the bulk scalar $\\Pi$ represent the out of equilibrium corrections to the energy density and pressure, respectively, as long as they vanish in equilibrium. In this case, the most general energy-momentum tensor that can describe an out-of-equilibrium state can be written as \n\\begin{equation}\nT^{\\mu\\nu}=(\\rho+\\mathcal{A})u^\\mu u^\\nu+\\left (P(\\rho)+\\Pi\\right )\\Delta^{\\mu\\nu}+\\pi^{\\mu\\nu}+u^\\mu\\mathcal{Q}^\\nu+u^\\nu\\mathcal{Q}^\\mu.\n\\label{novotensorBDNfodaotche}\n\\end{equation}\nConstraints on the out-of-equilibrium contributions can be readily obtained by the dominant energy condition \\cite{WaldBookGR1984}, which imposes that $\\rho +\\mathcal{A}\\geq 0$  and $\\mathcal{Q}_\\mu\\mathcal{Q}^\\mu \\leq (\\rho+\\mathcal{A})^2$. This naturally places a bound on the size of some of the out-of-equilibrium corrections. However, we note that the conservation of energy and momentum, $\\nabla_\\mu T^{\\mu\\nu}=0$, is not enough to fully determine the evolution described by the 14 dynamical variables $\\{\\rho,u^\\mu, \\mathcal{A},\\Pi,\\mathcal{Q}^\\mu, \\pi^{\\mu\\nu}\\}$. Therefore, some  procedure must be implemented to fully specify the system's dynamics. Instead of treating the non-equilibrium corrections as new degrees of freedom (and consequently postulating new additional equations of motion for them) as in Israel-Stewart-based approaches and extended irreversible thermodynamics \\cite{JouetallBook}, here we consider the case where the effective theory describing the macroscopic motion of the system is defined solely in terms of the standard hydrodynamic variables already present in equilibrium, which in our case are $\\{\\rho,u^\\mu\\}$. In this approach,\nthe dissipative contributions must be given in terms of the hydrodynamic fields $\\{\\rho,u^\\mu\\}$ and their derivatives, which may be organized through a relativistic derivative expansion. Assuming that deviations from equilibrium are small, the most general theory compatible with the symmetries that can be written in terms of first-order derivatives is defined by\n\\begin{equation}\n\\mathcal{A}=\\chi_1\\frac{u^\\alpha\\nabla_\\alpha \\rho}{\\rho+P}+\\chi_2 \\nabla_\\alpha u^\\alpha,\\,\\, \\Pi=\\chi_3\\frac{u^\\alpha\\nabla_\\alpha \\rho}{\\rho+P}+\\chi_4 \\nabla_\\alpha u^\\alpha,\\,\\, \n\\mathcal{Q}_\\mu=\\lambda\\left (\\frac{\\Delta^{\\nu}_\\mu\\nabla_\\nu P}{\\rho+P}+u^\\alpha\\nabla_\\alpha u_\\mu\\right), \\,\\, \\pi_{\\mu\\nu} = -2\\eta \\sigma_{\\mu\\nu},\n\\label{BDNdef}\n\\end{equation}\nwhere $\\sigma_{\\mu\\nu}=\\Delta_{\\mu\\nu}^{\\alpha\\beta}\\nabla_\\alpha u_\\beta$ is the shear tensor.\nAbove, the shear viscosity $\\eta$, and the coefficients $\\lambda$ and $\\chi_1$, $\\chi_2$, $\\chi_3$, and $\\chi_4$ are in principle known functions of $\\rho$, which are determined from the underlying microscopic theory. The bulk viscosity coefficient is given by the combination \\cite{Kovtun:2019hdm,Bemfica:2019knx} \n\\begin{equation}\n\\zeta = \\chi_3-\\chi_4 + c_s^2 (\\chi_2-\\chi_1),\n\\end{equation}\nwhere $c_s^2 = dP\/d\\rho$ is the equilibrium speed of sound squared. The transport coefficients $\\eta$ and $\\zeta$ determine how the long wavelength limit of hydrodynamic modes (i.e., sound and shear disturbances) are damped and the amount of entropy produced \\cite{Kovtun:2019hdm}, while three out of the four $\\chi$ coefficients determine the scales associated with non-hydrodynamic\\footnote{Those describe linearized disturbances around equilibrium that carry energy even in the homogeneous limit.} modes.     \n\nIt is important to stress a few properties of the expressions above. First, given that in equilibrium $u_\\mu\/T$ (with $T$ being the temperature) is a Killing vector \\cite{Hiscock_Lindblom_stability_1983} and $\\rho = \\rho(T)$, every single term in $\\mathcal{A}$ and $\\Pi$ separately vanishes in equilibrium, while  $ \\frac{\\Delta^{\\nu}_\\mu\\nabla_\\nu T}{T}+u^\\alpha\\nabla_\\alpha u_\\mu=0 \\Longrightarrow \\mathcal{Q}_\\mu=0$. We also note that in this approach time-like derivatives of the density, $u^\\alpha \\nabla_\\alpha \\rho$, appear in the constitutive relations. This fact is crucial for ensuring that the evolution is causal \\cite{BemficaDisconziNoronha} and linearly stable around equilibrium. Indeed, precise conditions for the coefficients $\\{\\eta,\\lambda,\\chi_1,\\chi_2,\\chi_3,\\chi_4\\}$ can be found that guarantee causality, stability, strong hyperbolicity and, thus, local well-posedness of solutions of Einstein's equations coupled to the viscous fluid equations \\cite{Bemfica:2019knx,Bemfica:2020zjp}. These conditions are violated in the Landau-Lifshitz theory \\cite{LandauLifshitzFluids}, which corresponds to setting $\\chi_1=\\chi_2=\\chi_3=\\lambda=0$ and $\\chi_4 = -\\zeta$. \n\n\n\n\nIn this work we initiate the investigation of the possible cosmological consequences of this approach. We consider the viscous fluid theory defined by  \\eqref{novotensorBDNfodaotche} and \\eqref{BDNdef} coupled to Einstein's equations (without a cosmological constant) in spatially flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) spacetime described by the line element \\cite{WeinbergCosmology}\n\\begin{equation}\n\\label{1FRW}\nds^2=-dt^2+a^2(t)\\delta_{ij}dx^i dx^j,\n\\end{equation}\nwhere $a(t)$ is the cosmological scale factor.\nHomogeneity and isotropy impose that the shear-stress tensor and the energy diffusion exactly vanish, and our tensor becomes simply \n\\begin{equation}\nT^{\\mu\\nu}=\\left(\\rho+\\chi_1\\frac{u^\\alpha\\nabla_\\alpha \\rho}{\\rho+P}+\\chi_2 \\nabla_\\alpha u^\\alpha\\right)u^\\mu u^\\nu+\\left (P+\\chi_3\\frac{u^\\alpha\\nabla_\\alpha \\rho}{\\rho+P}+\\chi_4 \\nabla_\\alpha u^\\alpha\\right )\\Delta^{\\mu\\nu}.\n\\end{equation}\nWe use $u^\\alpha=(1,0,0,0)$ and $\\nabla_\\mu u^\\mu = 3H(t)$, where $H(t)=\\dot{a}(t)\/a(t)$ is the Hubble expansion rate (with notation $\\dot{a}=da\/dt$). Using \\eqref{1FRW}, Einstein's equations dictate that the spatial derivative $\\partial_i\\rho=0$, and one finds the following set of equations of motion for $\\rho$ and $H$:\n\\begin{subequations}\n\\label{11FRW}\n\\begin{eqnarray}\n\\dot{H}+H^2&=&-\\frac{1}{6}\\left [\\rho+3P+\n\\frac{3\\chi_3+\\chi_1}{\\rho+P}\\dot{\\rho}\n+3(3\\chi_4+\\chi_2)H\\right ],\\label{11aFRW}\\\\\nH^2&=&\\frac{1}{3}\\left (\\rho+\\frac{\\chi_1}{\\rho+P}\\dot{\\rho}\n+3\\chi_2 H\\right ),\\label{11bFRW}\n\\end{eqnarray}\n\\end{subequations} \nwhere we have appropriately normalized the fields above to take into account the $8 \\pi G$ constant factor present in Einstein's equations. \nAs in the ideal fluid case, the equations of motion of the fluid $\\nabla_\\mu T^{\\mu\\nu}=0$ follow directly from those above. Since causality requires $\\chi_1>0$ \\cite{Bemfica:2019knx}, it is convenient to rewrite \\eqref{11aFRW} as\n\\begin{equation}\n\\label{13FRW}\n\\dot{H}+H^2=-\\frac{1}{2}\\left [P-\\frac{\\chi_3}{\\chi_1}\\rho+\\frac{\\chi_1+3\\chi_3}{\\chi_1}H^2-3\\chi_2\\left (\\frac{\\chi_3}{\\chi_1}-\\frac{ \\chi_4}{\\chi_2}\\right )H\\right ].\n\\end{equation}\nFinally, it is useful to define the variable \n\\begin{equation}\nw=\\frac{P}{\\rho}\n\\end{equation}\nto investigate the out-of-equilibrium properties of the fluid for different types of equation of state.  \n\n\\subsection{Radiation}\n\nFor radiation $w=1\/3$ and the imposition of  conformal invariance implies that $\\chi=\\chi_1=\\chi_2 = 3\\chi_3=3\\chi_4\\sim \\rho^{3\/4}$ (hence, $\\zeta=0$) \\cite{BemficaDisconziNoronha} and we obtain from \\eqref{11bFRW} and \\eqref{13FRW}\n\\begin{equation}\n\\dot\\rho + 3 H \\rho \\left(1+w_{eff}^{(r)}\\right)=0 \\qquad \\textrm{and} \\qquad  \\dot{H}+2H^2 = 0,\n\\label{raditioneq}\n\\end{equation}\nwhere $w_{eff}^{(r)} = \\frac{1}{3} + \\frac{4}{9H\\chi}\\left(\\rho-{3H^2}\\right)$.\nOne recognizes that the Hubble parameter decouples from the energy density as it obeys the well-known\nequation found for radiation in equilibrium in FLRW \\cite{WeinbergCosmology},\nwith general solution $H(t) = H_0\/(1+2H_0 t)$. Furthermore, we note that $\\rho = 3H^2$ is a solution of the equation of motion for the energy density. Though at first one may think that the equation of motion for $\\rho$ in \\eqref{raditioneq}\nshould have a complicated solution, the uniqueness property to the solutions of the equations of motion (i.e., well-posedness)\ndirectly implies that the general solution for the energy density equation is indeed simply $\\rho = 3H^2$, just as in the ideal fluid case. Therefore, as expected, out-of-equilibrium corrections vanish exactly for pure (conformal) radiation in FLRW where $\\zeta=0$.\nThis is also true in the Landau-Lifshitz theory. \n\nBesides indicating theoretical consistency, this result is also important from the standpoint of observational cosmology as the behavior of the universe when it is dominated by\nradiation is tightly constrained both by Big Bang nucleosynthesis (BBN) \\cite{Cyburt:2015mya} and by observations of fluctuations in\nthe cosmic microwave background (CMB) \\cite{Planck2018}.  Thus, any model for viscosity that\nstrongly alters the expansion history of the universe\nduring the radiation-dominated era can be ruled out.  As our\ntheory produces no change in the radiation equation of state, it automatically\nsatisfies this constraint.\n\n\\subsection{Zero entropy production limit away from conformal invariance}\n\nAbove, we saw that for conformal radiation  viscous effects drop out of the Friedmann equation entirely and no entropy is produced, as expected.  We\nhave no similar requirement on the values of the $\\chi$'s for other equations of state,\nbut we can derive the requirements on these parameters such that out-of-equilibrium corrections\nvanish. One can see that if $\\chi_3=\\chi_4$ in \\eqref{13FRW} and $\\chi_1=\\chi_2$ in \\eqref{11bFRW}, then again $3\\rho=H^2$ is a solution of the equation of the equations of motion, which reduce to those of an ideal fluid. Again, well-posedness implies that the solutions are unique and, thus, as long as these two conditions for the $\\chi$ parameters are satisfied, the out-of-equilibrium corrections\nto the Friedmann equations vanish.  Note that this conclusion is independent of\nthe $\\rho$-dependence of any of the $\\chi$ parameters. Also, we remark that when $\\chi_1=\\chi_2$ and $\\chi_3=\\chi_4$ one finds that $\\zeta=0$, so indeed no entropy \\cite{Kovtun:2019hdm} is produced in this case.\n\n\\subsection{Landau-Lishitz theory}\n\nIn the absence of a cosmological constant, in Landau-Lifshitz theory where $\\chi_1=\\chi_2=\\chi_3=0$ and $\\chi_4=-\\zeta$  the\nviscous dark matter evolves, at late times, to a fluid with a constant density thus mimicking the evolution of dark energy. This can be easily understood because in this model there are no\nout-of-equilibrium corrections to the energy density ($\\mathcal{A}=0$) and Einstein's equations\nbecome (taking $w=0$ for simplicity)\n\\begin{equation}\nH^2 = \\frac{\\rho}{3} \\qquad \\mathrm{and}\\qquad  \\dot{H}+H^2 = -\\frac{1}{6}\\left(\\rho - 9 \\zeta\nH\\right).\n\\end{equation} \nOne can see that here a constant energy density implies a constant Hubble expansion rate,\nwith nonzero solution $H = \\zeta$ that is positive for $\\zeta>0$. Therefore, in Landau-Lifshitz theory\nit is possible for viscous matter to behave at late times as dark energy at the background\nlevel.\\footnote{The same result holds for standard IS theories such as those considered in \\cite{BemficaDisconziNoronha_IS_bulk}.} This result motivated the creation of many unified viscous dark matter scenarios \\cite{Brevik:2017msy} and, in fact, it is known that this model can provide a good description of the background expansion of the universe \\cite{LiBarrow}. However, density perturbations in this scenario based on Landau-Lifshitz theory are rapidly damped out, which leads to severe constraints when attempting to\nreconcile it with precision cosmology data \\cite{LiBarrow,Velten:2011bg,Velten:2012uv}. \n\n\nWe emphasize that such constraints relied on theories known to be acausal\nor for which nonlinear causality remains open. We contend that decisive conclusions about the \nviability of the viscous dark matter idea should be based exclusively \non viscous theories that satisfy causality and well-posedness, in the nonlinear\nregime and also when  coupling to Einstein's equations, and for which\nlinear stability in flat spacetime also holds. In Ref.\\ \\cite{BemficaDisconziNoronha_IS_bulk} it was proven that IS theories, in the absence of shear viscosity and heat flow, fulfill these requirements (we note that cosmological perturbations in bulk viscous IS-like theories were studied in \\cite{Piattella_et_al}). However, we point out that since effects from shear and heat flow do contribute when the spacetime is not homogeneous and isotropic, their influence on the evolution of cosmological perturbations must also be investigated \\cite{Barbosa:2017ojt}. To the best of our knowledge, the general first-order theory studied here is now the only framework that fulfills the consistency conditions mentioned above and can, thus, be used to reliably study the effects of bulk, shear, and heat flow even in viscous inhomogeneous cosmological applications in a model independent manner. \n \n\\subsection{The Cheshire Cat mechanism} \n\nWe show below that our approach differs from earlier attempts to model dark energy as a viscous phenomenon in an interesting way.  \nIn the model investigated here, the equilibrium component of the density of the viscous dark matter is driven asymptotically to zero.\nBecause of the way that the Friedmann equation is altered by viscous effects in this model,\nthe universe continues to accelerate even after the driver of this acceleration effectively disappears. Therefore,\nwe refer to this as the ``Cheshire Cat\" mechanism for generating accelerated expansion\\footnote{Note that the term ``Cheshire Cat\" has been used\nin an entirely different way in the context of quantum measurement theory \\cite{Aharonov}.}.  \n\nThis effect can be most easily illustrated by the following analytical example. For simplicity, let us assume a constant non-negative $w \\ll 1$ so the viscous matter has very small equilibrium pressure, which works as\nthe small parameter in the perturbative argument that follows.\nAssume that the matter is such that $\\eta\\geq 0$ and $\\zeta=2\\eta w (\\frac{1}{3}-w)$. The conditions for causality, well-posedness,\nand stability are satisfied if, for instance, $\\chi_1=\\lambda=4\\eta w$, $\\chi_2 = 2\\eta(1-w)$, $\\chi_3 = w \\chi_1$, and $\\chi_4=4\\eta w\/3$ \\cite{Bemfica:2019knx}.\nWe note that the viscosity coefficients are, thus, very small and when $w=0$ we recover an ideal fluid\nwith a matter-like, pressureless equation of state\\footnote{Note that\n$\\eta$ and $\\lambda$ need not to be zero even though shear and heat flux vanish. The latter vanish because of symmetries and not because the\ncoefficients are zero.}.\nWe take $\\eta$ (and, thus, $\\zeta$) to be constant. Under these conditions, assuming that $H$ is constant and non-negative, the general solution of the equations of motion \\eqref{11bFRW} and \\eqref{13FRW} can be found analytically \n\\begin{equation}\nH = H_0 =\\frac{\\zeta}{1+w} \\qquad \\textrm{and} \\qquad \\rho(t) = \\frac{\\rho_0\\,\\beta}{(\\rho_0+\\beta)e^{t\\beta\/\\alpha}-\\rho_0},\n\\end{equation}\nwhere $\\alpha = 4\\eta w\/(1+w)$ and $\\beta = \\alpha H_0(3-w)\/2w$. Therefore, even though $H$ is constant,\nwe note that the energy density still varies with time, decreasing exponentially\ntowards zero even though there is no cosmological constant\\footnote{For a comparison, consider the evolution of an ideal fluid with equation of state given by $P=w\\rho$, in the presence\nof a positive cosmological constant $\\Lambda$ (constant dark energy).\nAsymptotically, $3H^2 \\to \\Lambda$ and the ideal fluid energy density vanishes.\nThe difference here in the viscous case is that the energy density evolves toward zero\nwhen $H$ is constant and in the absence of a cosmological constant.}. We remark that for this type of matter the Hubble constant is very small, as small as $w$, and that the energy density still varies in time (i.e, it is not a cosmological constant). \n\nIn an accelerated expansion driven by dark energy,\nthe universe expands at a constant rate and the dark\nenergy density remains constant.\nIn the example above, the cosmological scale factor $a(t)$ increases exponentially\nbut causality generally imposes that the energy density must lag behind, varying on time scales of the order of $\\chi_1\/(\\rho(1+w))$. According to the theory presented here, this is\na general consequence of causality and energy-momentum conservation in out of equilibrium systems described by first-order theories of relativistic viscous fluid dynamics. We shall return to the Cheshire Cat mechanism in Section \\ref{viscouscosmo}. \\\\\n\n\\section{Fixed point analysis}\\label{fixedpointsection}\n\nHere we set $w=0$, i.e., $P=0$, and study the fixed points and the stability properties of the equations of motion. We also assume that $\\chi_1 >0$. We start by rewriting the relevant equations of motion as\n\\begin{equation}\n\\chi_1\\dot\\rho + 3 H \\rho \\chi_2  + \\rho\\left(\\rho-{3H^2}\\right)=0\n\\end{equation}\nand\n\\begin{equation}\n2 \\chi_1 {\\dot{H}} = \\chi_3 \\rho - {3H^2}\\left(\\chi_1+\\chi_3\\right)+3H\\left(\\chi_2\\chi_3-\\chi_1\\chi_4\\right).\n\\end{equation}\nWe assume as before that $\\chi_i = \\chi_i(\\rho)$. Let $\\rho_0$ and $H_0$ be the fixed points of the equations above so $\\dot{H_0}=\\dot{\\rho_0}=0$. Consider now fluctuations around the fixed points $\\rho(t) \\to \\rho_0 + \\delta \\rho(t)$ and $H(t)\\to H_0 + \\delta H(t)$. Note that fluctuations also act on the transport coefficients, i.e., $\\chi_i(\\rho) \\to \\chi_i(\\rho_0) +  \\chi_i'(\\rho_0)\\delta \\rho(t)$, where $\\chi_i' = d\\chi_i\/d\\rho$. To zeroth order in the fluctuations we obtain \n\\begin{equation}\n \\rho_0\\left(\\rho_0+3 H_0  \\chi_2(\\rho_0)  - {3H_0^2}\\right)=0\n \\label{0thorder_rho}\n\\end{equation}\nand\n\\begin{equation}\n3H_0 \\chi_1(\\rho_0)\\left({H_0}+\\chi_4(\\rho_0)\\right) =\n\\chi_3(\\rho_0)\\left(\\rho_0 + 3H_0 \\chi_2(\\rho_0)- {3H_0^2}\\right).\n \\label{0thorder_H}\n\\end{equation}\nWe see that Eq.\\ \\eqref{0thorder_rho} implies that $\\rho_0=0$ or $\\rho_0+3 H_0 \n\\chi_2(\\rho_0)= {3H_0^2}$. Clearly, the latter can be complicated since one must know how $\\chi_2$ depends on $\\rho$ to solve it. \n\n\\subsection{$\\rho_0=0$ and $H_0=0$}\n\nIt is easy to see that $\\rho_0=H_0=0$ is a fixed point. In fact, this fixed point is very general as it does not depend on the properties of the $\\chi_i$'s. Let us now study the linear stability properties of this fixed point. The linearized equations for the fluctuations are\n\\begin{eqnarray}\n\\delta \\dot\\rho &=& 0, \\\\\n2 \\delta \\dot{H} &=& \\frac{\\chi_3(0)}{\\chi_1(0)}\\delta \\rho + 3\\left(\\chi_2(0)\\chi_3(0)-\\chi_1(0)\\chi_4(0)\\right)\\delta H.\n\\end{eqnarray}\nWe can write this in matrix form, which reveals that the\neigenvalues of the matrix\nare $0$ and $(3\/2)\\left(\\chi_2(0)\\chi_3(0)-\\chi_1(0)\\chi_4(0)\\right)$.\nSince one of the eigenvalues vanishes\n(in fact, the determinant of the matrix vanishes),\nthis is a marginal case where one does not have an isolated fixed point. In\nthis case, a linear stability analysis is not guaranteed to give\nthe correct information about the stability properties of the\nsystem \\cite{strogatz:2000}.  In any event, this is not a physically interesting\ncase.\n\n\\subsection{$\\rho_0=0$ and $H_0\\neq 0$}\n\nIn this case, assuming that $\\chi_1(0)+\\chi_3(0)\\neq 0$, one finds\n\\begin{equation}\nH_0 = \\frac{\\left(\\chi_2(0)\\chi_3(0)-\\chi_1(0)\\chi_4(0)\\right)}{\\chi_1(0)+\\chi_3(0)}.\n\\end{equation}\nWe will now find the conditions that ensure that $H_0>0$ and $\\rho_0=0$ is an attractor (i.e., a stable isolated fixed point). In this case, the equations for the linearized fluctuations $\\vec{v} = {\\delta \\rho \\choose \\delta H}$ become \n\\begin{equation}\n\\frac{d\\vec{v}}{dt} = A\\, \\vec{v},\n\\end{equation}\nwhere $A$ is a 2 x 2 matrix that depends on $\\chi_i(0)$ and $\\chi_i'(0)$. The eigenvalues of $A$ are\n\\begin{equation}\na_1 = {3H_0}\\left(H_0-  \\chi_2(0)\\right),\\qquad a_2 = -\\frac{3}{2 \\chi_1(0)}\\left[\\chi_2(0) \\chi_3(0)-\\chi_1(0)\\chi_4(0)\\right].\n\\end{equation}\nThus, a stable fixed point occurs when $a_1 < 0$ and $a_2 <0$ (note that both eigenvalues are real). We see that $a_2 < 0$ implies that  \n\\begin{equation}\n\\chi_2(0) \\chi_3(0)-\\chi_1(0)\\chi_4(0)>0.\n\\end{equation}\nSince that quantity appears in $H_0$, which we consider to be positive, we see that this occurs then if $\\chi_1(0)+\\chi_3(0)>0$. The condition that $a_1<0$ then occurs when $\\chi_4 > -\\chi_2$. When those conditions are met, the fixed point is \\emph{hyperbolic} (see Chapter 6 of Ref.\\ \\cite{strogatz:2000}), which means that the qualitative behavior of the system's phase portrait near the attractor is not changed even when small nonlinear terms are included. As a matter of fact, the Hartman-Grobman theorem \\cite{strogatz:2000} states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait obtained via linearization and, thus, the conclusions regarding the stability of the fixed point are the same as in the linearized system (in other words, the phase portrait near this attractor is structurally stable). \n\nSummarizing, we conclude that $\\rho=0$ with constant $H_0>0$ is an attractor when the following conditions are met:\n\\begin{eqnarray}\n&&\\chi_2(0) \\chi_3(0)-\\chi_1(0)\\chi_4(0)>0\\\\\n&& \\chi_1(0)+\\chi_3(0)>0 \\\\\n&& \\chi_4 > -\\chi_2,\n\\end{eqnarray}\nwhich are compatible with the conditions found for causality and also linear stability around equilibrium presented in Ref.\\ \\cite{Bemfica:2019knx}.\n\n\n\n\\section{Toward a Realistic Cosmology}\\label{viscouscosmo}\n\n\n\nIn this section we investigate whether the viscous fluid  presented here can\nproduce cosmological evolution consistent with observations, within a variety of different scenarios.  In particular, we will\nbe interested in the types of models discussed in Refs.\\\n\\cite{Avelino,Hipolito1} in which the universe contains a pressureless\ndark matter component whose viscosity drives the accelerated expansion.\n\nFrom an observational point of view, it is useful to redefine our\nevolution equations in terms of the scale factor $a$ instead of the time.\nMeasurements of the dark energy density are effectively determinations\nof $H(z)$, where the redshift $z$ is related to the scale factor as\n$a = 1\/(1+z)$.  Using $d\/dt = Ha\\,d\/da$, we can rewrite Eqs.\\\n(\\ref{11aFRW})-(\\ref{11bFRW}) with the scale factor as the independent variable:\n\\begin{eqnarray}\n\\label{Fried1}\nH^\\prime a + \\frac{3}{2}H &=& -\\frac{P}{2H} -\\frac{1}{2} \\chi_3 \\frac{\\rho^\\prime}{\\rho +P} a - \\frac{3}{2} \\chi_4,\\\\\n\\label{Fried2}\nH^2 &=& \\frac{1}{3} \\left( \\rho + \\chi_1 \\frac {\\rho^\\prime}{\\rho + P}aH\n+ 3 \\chi_2 H \\right),\n\\end{eqnarray}\nwhere we denote\n${}^\\prime = \\frac{d}{da}$. When $P=0$, Eqs.\\ (\\ref{Fried1}) and (\\ref{Fried2}) become\n\\begin{eqnarray}\n\\label{Fried1cdm}\nH^\\prime a + \\frac{3}{2}H &=& -\\frac{1}{2} \\chi_3 \\frac{\\rho^\\prime}{\\rho} a - \\frac{3}{2} \\chi_4,\\\\\n\\label{Fried2cdm}\nH^2 &=& \\frac{1}{3} \\left( \\rho + \\chi_1 \\frac {\\rho^\\prime}{\\rho}aH\n+ 3 \\chi_2 H \\right).\n\\end{eqnarray}\nThe $\\chi$ parameters must satisfy the conditions for nonlinear causality and\nlinear stability, which were obtained in Ref.\\ \\cite{Bemfica:2019knx}.  When $P=0$ these conditions are\n\\begin{eqnarray}\n\\label{condition1}\n\\lambda \\chi_1 \\ge \\lambda \\chi_3 + \\chi_2  \\chi_3 + \\chi_1\\left(\\frac{4 \\eta}{3} - \\chi_4 \\right) &\\ge& 0,\\\\\n\\label{condition2}\n\\chi_1^2\\left(\\frac{4 \\eta}{3} - \\chi_4 \\right) + \\lambda \\chi_3(\\lambda + \\chi_2)\n+\\chi_1 \\chi_2 \\chi_3 &\\ge& 0,\\\\\n\\label{condition3}\n\\lambda + \\chi_1 \\ge \\chi_3 + \\frac{4 \\eta}{3} - \\chi_4 &\\ge& 0,\n\\end{eqnarray}\nsubject to the constraints $\\lambda,\\chi_1 > 0$, $\\eta \\ge 0$, and $\\lambda \\ge \\eta$.  Furthermore, when\n$P=0$, the bulk viscosity becomes $\\zeta = \\chi_3 - \\chi_4$, which must be nonnegative in accordance with the second law of thermodynamics.  Hence,\nwe have the further requirement here that\n\\begin{equation}\n\\label{condition4}\n\\chi_3 \\ge \\chi_4.\n\\end{equation}\n\n\\subsection{Constant $\\chi_1$, $\\chi_2$, $\\chi_3$, and $\\chi_4$}\n\nLet us first consider the case where \nall of $\\chi$ coefficients are constant.\nAs noted in the previous section, there is generically\nan attractor solution with $H \\rightarrow H_0 = \\mathrm{constant}$, with\n\\begin{equation}\nH_0 = \\frac{\\chi_2 \\chi_3 - \\chi_1 \\chi_4}{\\chi_1 + \\chi_3},\n\\end{equation}\nas long as both the numerator and denominator on the right-hand side are positive.\nSubstituting this attractor into Eqs.\\ (\\ref{Fried1})-(\\ref{Fried2}),\nwe find that $a \\rho^\\prime\/\\rho$\nalso evolves to a constant, given by\n\\begin{equation}\na \\frac{\\rho^\\prime}{\\rho} = -3 \\left(\\frac{\\chi_2 + \\chi_4}{\\chi_1 + \\chi_3} \\right).\n\\label{rhoprimeeq}\n\\end{equation}\nFurthermore, we define an effective $w_\\mathrm{eff}$ given by\n\\begin{equation}\n\\label{wdef}\n1 + w_\\mathrm{eff} = -\\frac{1}{3} a \\frac{\\rho^\\prime}{\\rho}\n\\end{equation}\nand we note that, in particular, the special case $\\chi_2 + \\chi_4 = \\chi_1 + \\chi_3$ gives $w_\\mathrm{eff} = 0$. Because the general solution of \\eqref{rhoprimeeq} is \n\\begin{equation}\n\\rho \\propto a^{-3(1+w_\\mathrm{eff})},\n\\end{equation}\none can again see the \nCheshire Cat behavior noted earlier.  When $H \\rightarrow \\mathrm{constant}$, the scale factor\nevolves as $a \\sim e^{H_0 t}$, so the equilibrium energy density of a  fluid with constant $w_\\mathrm{eff}$ will decay as a power-law\nin $a$, but exponentially in $t$.\n\nThe evolution is particularly simple for the special case $\\chi_3 = 0$.  In this case,\nwe can solve Eq.\\ (\\ref{Fried1cdm}) exactly to yield\n\\begin{equation}\n\\label{L-L}\nH = C a^{-3\/2} - \\chi_4,\n\\end{equation}\nwith $C$ a constant.\nAs long as $\\chi_4 < 0$, the Hubble parameter will evolve from initial dark matter dominated evolution\n($H \\sim a^{-3\/2}$) to evolution resembling a cosmological constant dominated universe ($H \\sim\n\\mathrm{constant}$).  Note, however, that $\\rho$ in this case does not evolve asymptotically to a constant.\nIf we take $H$ to be equal to its asymptotic constant value ($H = -\\chi_4$) in Eq.\\\n(\\ref{Fried2cdm}), we obtain\n\\begin{equation}\n\\rho = \\frac{3 \\chi_4 (\\chi_2 + \\chi_4)}{D a^{3(\\chi_2 + \\chi_4)\/\\chi_1} + 1},\n\\end{equation}\nwith $D$ a constant.\nThe causality and stability conditions require $\\chi_1 > 0$, and we need $\\chi_2 + \\chi_4 > 0$ to ensure a positive density.\n  Although Eq.\\ (\\ref{Fried1cdm}) in this case\nbecomes exactly what one would obtain in Landau-Lifshitz theory\nwith constant $\\zeta$ (and the evolution of $H(a)$\nis therefore identical), the value of $\\rho$ in the corresponding Landau-Lifshitz case evolves\nto a constant.  In our approach, we once again see $\\rho$ evolving to zero while viscous effects\nmimic a constant-density evolution for $H$.\n\n\\subsection{Power-law density dependence of the $\\chi$ parameters}\n\nNow consider the more general case where the $\\chi$ parameters are not constant but evolve as functions of $\\rho$.  For simplicity, we will assume a power law behavior and take all of the\n$\\chi$ parameters to evolve as the same power of $\\rho$, namely, $\\chi_i \\to \\chi_i \\rho^m$, $i = 1,2,3,4$,\nwhere the $\\chi_i$'s are constant. This is the analog of the case considered by \\cite{LiBarrow} in Landau-Lishitz theory where it was assumed that $\\zeta\\sim \\rho^m$.\n\nFor this case, Eqs.\\ (\\ref{Fried1})-(\\ref{Fried2}) become\n\\begin{eqnarray}\n\\label{Fried1m}\nH^\\prime a + \\frac{3}{2}H &=& -\\frac{1}{2} \\chi_3 \\rho^m \\frac{\\rho^\\prime}{\\rho}\na - \\frac{3}{2} \\chi_4 \\rho^m,\\\\\n\\label{Fried2m}\nH^2 &=& \\frac{1}{3} \\left( \\rho + \\chi_1 \\rho^m \\frac {\\rho^\\prime}{\\rho}aH\n+ 3 \\chi_2 \\rho^m H \\right).\n\\end{eqnarray}\nAt early times, the density of the dark energy is observed to be\nnegligible, and the universe is dominated\nby dark matter with a density scaling as $a^{-3}$.\nThus, if we require that the ``dark energy\"\ncorresponds to viscous corrections to the dark matter evolution, then these corrections must vanish in the limit\nwhere $a \\rightarrow 0$.  In this limit, we require $\\rho$ to scale as $a^{-3}$ and $H$ to scale as $a^{-3\/2}$.\nThen in order for the viscous corrections to be subdominant in Eqs.\\ (\\ref{Fried1m})-(\\ref{Fried2m})\nas $a \\rightarrow 0$, we need either $\\chi_1 =\n\\chi_2$ and $\\chi_3 = \\chi_4$ (so that there are no\nviscous corrections at all), or $m \\le 1\/2$.\n\nFor $0 < m \\le 1\/2$, we find attractor solutions with $w \\rightarrow \\mathrm{constant}$,\nbut $H \\rightarrow 0$.  Hence, these do not correspond to the observed universe\nif\nwe want viscous effects to drive the present-day accelerated expansion.\nWhen $m < 0$, we find an attractor solution for which $H \\rightarrow \\infty$.\nThis corresponds to phantom-like behavior \\cite{Caldwell:1999ew}, and can\nbe consistent with observations depending on the exact parameters of\nthe expansion.\n\n\\subsection{Inclusion of baryons}\n\nFinally, in order to derive results that could in principle be compared with observations, we must include\nboth a viscous dark matter component and the nonviscous baryons (see, e.g., similar\ntreatments in Refs. \\cite{Colistete:2007xi,LiBarrow}).  If $\\rho_B$ is the density of baryons\nand $\\rho_D$ is the density of viscous pressureless dark matter, then we can rewrite\nEqs.\\ (\\ref{Fried1cdm})-(\\ref{Fried2cdm}) as\n\\begin{eqnarray}\n\\label{Fried1semifinal}\nH^\\prime a + \\frac{3}{2}H &=& -\\frac{1}{2} \\chi_3 \\frac{\\rho_D^\\prime}{\\rho_D} a - \\frac{3}{2} \\chi_4,\\\\\n\\label{Fried2semifinal}\nH^2 &=& \\frac{1}{3} \\left( \\rho_D + \\rho_B + \\chi_1 \\frac {\\rho_D^\\prime}{\\rho_D}aH\n+ 3 \\chi_2 H \\right).\n\\end{eqnarray}\nThese equations can be re-expressed in the form\n\\begin{eqnarray}\n\\label{Fried2final}\nH^\\prime a &=& - \\frac{3}{2}\\left(1+ \\frac{\\chi_3}{\\chi_1}\\right)H + \\frac{3}{2}\\left(\\frac{\\chi_2 \\chi_3}{\\chi_1}-\\chi_4\\right)\n+ \\frac{1}{2} \\left(\\frac{\\chi_3}{\\chi_1}\\right) \\frac{\\rho_D + \\rho_B}{H},\\\\\n\\label{Fried1final}\n\\rho_D^\\prime a &=& \\frac{\\rho_D}{\\chi_1}\\left(3H - \\frac{\\rho_D + \\rho_B}{H} - 3 \\chi_2\\right).\n\\end{eqnarray}\nThe baryon density $\\rho_B$ scales exactly as $a^{-3}$.  In standard nonviscous models of cold dark matter we also\nhave $\\rho_D \\propto a^{-3}$ but here the evolution of $\\rho_D$ is determined instead by\nEq.\\ (\\ref{Fried1final}).  However, observational limits on the present-day dark matter density combined with\nhigh-redshift estimates of $\\rho_D$ from the cosmic microwave background indicate that\n$\\rho_D$ must evolve approximately as $a^{-3}$ up to the present.\n\nBy an appropriate choice of the $\\chi$ coefficients it is possible to derive a model satisfying this constraint on $\\rho_D$ that also\nclosely approximates $\\Lambda$CDM.  We first take\n\\begin{eqnarray}\n\\label{chi3choice}\n\\chi_3 = - \\chi_1,\\\\\n\\label{chi4choice}\n\\chi_4 = - \\chi_2,\n\\end{eqnarray}\nwhere we choose $\\chi_1, \\chi_2 > 0$, so that $\\chi_3, \\chi_4 < 0$.\nThis choice is consistent with the causality constraints. Indeed, if we substitute\nEqs.\\ (\\ref{chi3choice}) and (\\ref{chi4choice}) into the\ncausality constraint equations (\\ref{condition1})-(\\ref{condition3}), we find that\nthere exist values for $\\lambda$ and $\\eta$ for which all of the constraint equations are satisfied as long as\n$4\/3 \\ge \\chi_2\/\\chi_1 \\ge 1$.\nSubstituting these values for $\\chi_3$ and $\\chi_4$ into our evolution equations above, Eq.\\ (\\ref{Fried1final}) is unchanged, while\nEq.\\ (\\ref{Fried2final}) becomes\n\\begin{equation}\n\\label{Fried3final}\nH^\\prime a = - \\frac{1}{2}\\frac{\\rho_D + \\rho_B}{H}.\n\\end{equation}\nIf we neglect $\\rho_B$, these equations can be solved exactly.  Dividing Eq.\\ (\\ref{Fried1final}) by Eq.\\ (\\ref{Fried3final})\nyields\n\\begin{equation}\n\\frac{d\\rho_D}{dH} - \\frac{2}{\\chi_1} \\rho_D = - \\frac{6}{\\chi_1} H^2 + 6 \\frac{\\chi_2}{\\chi_1}H,\n\\end{equation}\nwith solution\n\\begin{equation}\n\\rho_D = C e^{2H\/\\chi_1} + 3 H^2 - 3(\\chi_2 - \\chi_1)H - \\frac{3}{2} \\chi_1(\\chi_2 - \\chi_1),\n\\end{equation}\nwhere $C$ is again a constant of integration.  At early times ($H \\rightarrow \\infty$) we must have $\\rho = 3H^2$, so $C=0$\nand our solution is\n\\begin{equation}\n\\label{Friedsolution}\n\\rho_D = 3 H^2 - 3(\\chi_2 - \\chi_1)H - \\frac{3}{2} \\chi_1(\\chi_2 - \\chi_1).\n\\end{equation}\nNote that when $\\chi_1 = \\chi_2$, we once again obtain the evolution appropriate for nonviscous matter, namely\n$\\rho_D = 3 H^2$ (which is consistent with the fact that in this case $\\zeta=0$).  Substituting Eq.\\ (\\ref{Friedsolution}) back into Eq.\\ (\\ref{Fried1final}),\nwe see that $\\rho_D$ evolves as\n\\begin{equation}\n\\label{rhoDsolution}\n\\frac{\\rho_D^\\prime}{\\rho_D}a = -3 + \\frac{3}{2}(\\chi_2 - \\chi_1)\/H.\n\\end{equation}\n\nIn order for Eqs.\\ (\\ref{Friedsolution}) and (\\ref{rhoDsolution})\nto mimic $\\Lambda$CDM, we make one further requirement:  we take $\\chi_2\/\\chi_1 = 1+\\epsilon$,\nwith $0< \\epsilon \\ll 1 $.  (Note that this assumption means that the causality constraint $4\/3 \\ge \\chi_2\/\\chi_1 \\ge 1$ will\nautomatically be satisfied.)  With this assumption, the second term on the right-hand side of  (\\ref{Friedsolution})\nis always subdominant, and we have\n\\begin{equation}\n\\label{HLCDM}\nH^2 = \\frac{1}{3}\\rho_D + \\frac{1}{2} \\chi_1 (\\chi_2 - \\chi_1). \n\\end{equation}\nThis has the form of standard $\\Lambda$CDM, where we identify $\\rho_\\Lambda = (3\/2) \\chi_1 (\\chi_2 - \\chi_1).$\nFurthermore, Eq.\\ (\\ref{HLCDM}) implies that $H^2 > (1\/2) \\chi_1 (\\chi_2 -\n\\chi_1)$, so that $\\frac{3}{2}(\\chi_2-\\chi_1)\/H \\ll 1$.  Thus,\n$\\rho_D$ scales almost exactly as $a^{-3}$, as required.\nNote\nthat this behavior for $H$, while identical\nto $\\Lambda$CDM, is once again an example of Cheshire Cat evolution:\nthe acceleration is driven by viscous effects from the\ndark matter, whose equilibrium density is driven to zero by the expansion.\nWhile we neglected $\\rho_B$ in this derivation, it is easy to see\nthat the evolution will be unchanged when $\\rho_B$ is included since\nit scales in exactly the same way as $\\rho_D$ ($\\sim a^{-3}$), a result we have verified\nwith numerical integration of Eqs.\\ (\\ref{Fried2final})\nand (\\ref{Fried1final}).\n\nFinally, we do not mean to imply that our choices for the $\\chi$ coefficients in this\ncase are the single set of ``correct\" values of these parameters for pressureless\ndark matter.  Instead, we simply wish to demonstrate that causal and stable first-order viscous fluid theories\ncan reproduce $\\Lambda$CDM for at least one choice of these parameters.  It\nis quite possible that other choices for these parameters can similarly\nreproduce the current observations.\n\n\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\nThis work represents the first examination of cosmology with a causal, stable, first-order theory of relativistic viscous fluid dynamics (the BDNK theory).\nWe have shown that this effective theory approach to relativistic viscous fluids has two very attractive properties from the\nstandpoint of cosmology.  First, viscosity has no effect on the behavior of radiation, i.e.,\nfluids with $w=1\/3$.  Thus, the standard cosmology during the radiation-dominated era\nsimply carries over in this case without modification, including all of the successes of\nBBN and the CMB.  Second, under very general conditions on the viscosity parameters,\na matter-like fluid (i.e., one with $w=0$) can generate an accelerated expansion, just\nas in the case of Landau-Lifshitz theory. \n\nThe major difference between this approach for relativistic viscous fluids and other previously-investigated models such\nas the Landau-Lifshitz model\nlies in the modification to Eq.\\ (\\ref{11bFRW}).  Previous models have modified Eq.\\ (\\ref{11aFRW}),\naltering the effective pressure, but leaving the relationship between $H$ and $\\rho$ as in\nthe standard cosmology (i.e., $H^2  = \\rho\/3$).  By altering this relationship, we decouple\nthe behavior of $H$ from that of the equilibrium energy density $\\rho$.\nThus, while Landau-Lifshitz viscosity can cause a zero-pressure\ndark matter fluid to behave as an effective\ndark energy component with constant density at late times,\nthe BDNK theory can produce\nan accelerated expansion even as the equilibrium density of the dark\nmatter fluid goes to zero, an effect we have dubbed the\nCheshire Cat mechanism. It is important to remark that it is $H(a)$, and not the evolution\nof the dark energy density, that is actually\nthe observable quantity in cosmology, so the model presented here can be made\nconsistent with current observations of dark energy.  Indeed,\nby a suitable choice of model parameters, this model can nearly\nexactly mimic $\\Lambda$CDM at the background level.\n\nAs we have already noted, previous attempts to use Landau-Lifshitz viscosity with\npressureless dark matter to account for the accelerated expansion\nof the universe have produced acceptable results\nat background level but have\nfoundered on the issue of perturbation growth\n\\cite{LiBarrow,Velten:2011bg,Velten:2012uv}.  Thus,\nthe results presented here are necessary but not sufficient\nevidence that a viable cosmology can be constructed with\ndark matter and BDNK viscous effects alone.\nThe next step will be to examine perturbation growth in this theory to see if it survives\nthis further level of scrutiny. Only then one would be able to conclusively answer whether or not viscous effects are compatible with cosmological observations. \n\n\nFinally, it is important to remark that an accurate description of this vanishing equilibrium energy density behavior is, formally, beyond the regime of applicability of the hydrodynamics expansion. This occurs because in this case the out of equilibrium correction, $\\mathcal{A}$, becomes larger than the equilibrium piece $\\rho$. This issue is also present when considering higher-order theories, such as the generalized Israel-Stewart theories constructed using a general hydrodynamic frame in Ref.\\ \\cite{Noronha:2021syv}. Those should also display the properties found here, given that BDNK can be seen as the first-order truncation of such generalized 2nd order theories. Therefore, further investigation of this type of solution is needed. We defer a systematic investigation of that to future work.\n\n\n\\section*{Acknowledgements} \n\nWe thank L.~Gavassino for comments on this work. Part of this work was done while F.S.B. was visiting Vanderbilt University. M.M.D. is partially supported by a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, NSF grant DMS-2107701,\nand a Vanderbilt's Seeding Success Grant. J.N. is partially supported by the U.S.~Department of Energy, Office of Science, Office for Nuclear Physics under Award No. DE-SC0021301. \nR.J.S. was supported in part by the Department of Energy (DE-SC0019207).\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter*{Abstract}\nX-rays are a valuable diagnostic tool for the study of high energy accelerated electrons. Bremsstrahlung X-rays produced by, and directly related to, high energy electrons accelerated during a flare, provide a powerful diagnostic tool for determining both the properties of the accelerated electron distribution, and of the flaring coronal and chromospheric plasmas. This thesis is specifically concerned with the study of {\\it spatial, spectral and polarization} properties of solar flare X-ray sources via both modelling and X-ray observations using the Ramaty High Energy Solar Spectroscopic Imager ({\\it RHESSI}). Firstly, a new \nmodel is presented, accounting for finite temperature, pitch angle scattering and initial pitch angle injection. This is developed to accurately infer the properties of the acceleration region from the observations of dense coronal X-ray sources. Moreover, examining how the spatial properties of dense coronal X-ray sources change in time, interesting trends in length, width, position, number density and thermal pressure are found and the possible causes for such changes are discussed. Further analysis of data in combination with the modelling of X-ray transport in the photosphere, allows changes in X-ray source positions and sizes due to the X-ray albedo effect to be deduced. Finally, it is shown, for the first time, how the presence of a photospheric X-ray albedo component produces a spatially resolvable polarization pattern across a hard X-ray (HXR) source. It is demonstrated how changes in the degree and direction of polarization across a single HXR source can be used to determine the anisotropy of the radiating electron distribution.\n\n \n\n\\chapter*{Acknowledgements}\n\n\\addcontentsline{toc}{chapter}{Acknowledgements}\n\n\\thispagestyle{empty}\n\nIt must be mentioned that each chapter (Chapters \\ref{ref:Chapter2}, \\ref{ref:Chapter3}, \\ref{ref:Chapter5} and \\ref{ref:Chapter6}) is published and hence I wish to thank my publication co-authors: Drs. Eduard P. Kontar, Nicolas H. Bian and A. Gordon Emslie, and highlight their contribution to the work residing within this thesis. I would also like to thank Dr. Brian Dennis and the {\\it RHESSI} team at NASA GSFC for their help with {\\it RHESSI\\,} imaging and spectroscopy during my short stay at Goddard.\n\\\\\\\\\nHowever, I wish to solely acknowledge and express my sincerest gratitude to Dr. Eduard Kontar, for his invaluable help and insightful guidance during my postgraduate study and undergraduate summer projects. \n\n\n\\chapter{Calculating the photon stepsize}\\label{ref:App2}\n\nIn Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6}, a Monte Carlo simulation is used to model the movement and interaction of photons through the solar photosphere. Before an interaction, each photon must move a certain distance called the step size $ss$, which is a certain fraction of its mean free path $l$. The mean free path of a photon is given by,\n\\begin{equation}\\label{ap2_1}\nl=\\frac{1}{n\\sigma_{T}}=\\frac{1}{n_{H}\\left(\\sigma_{a}+\\sigma_{c}\\right)}\n\\end{equation}\nwhere $n$ is the number density of the photosphere and $\\sigma_{T}$ is the total cross section, that is, the sum of all attenuation processes, which for this case is the sum of the Compton scattering $\\sigma_{c}$ and absorption $\\sigma_{a}$ cross sections, as discussed in Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6}.\n\nTo calculate the photon step size through the solar photosphere, accounting for the processes of absorption and scattering, the {\\it Beer-Lambert law} \\citep[cf.][]{2002phat.book.....H} is used. This is given by\n\\begin{equation}\\label{ap2_2}\nI(z)=I_{0}(z)\\exp(-z\/l)=I_{0}(z)\\exp(-n_{H}(\\sigma_{c}+\\sigma{a})z)\n\\end{equation}\nwhere $I$ is the intensity of light after travelling a distance $z$ through a material (in this case) that absorbs and scatters a proportion of the light, and $I_{0}$ is the starting intensity.\n\nThe probability density function (PDF) used to describe this can then be given by,\n\\begin{equation}\\label{ap2_3}\np(ss)=\\frac{1}{l}\\exp(-ss\/l)\n\\end{equation}\nwhere $z$ has been replaced by the step size $ss$. The {\\it Inverse transform Method} is then employed; this maps each value of the PDF to a randomly generated number $\\zeta_{step}$ between $0$ and $1$ using the cumulative distribution of the PDF \\citep[e.g.,][]{Salvatetal2008}, given by,\n\\begin{equation}\\label{ap2_4}\n\\zeta_{step}=\\int_{0}^{ss}p(ss^{'})dss^{'}\n=\\int_{0}^{ss}\\frac{1}{l}\\exp(-ss^{'}\/l)dss^{'}=1-\\exp(-ss\/l)\n\\end{equation}\nRe-arranging Equation (\\ref{ap2_4}) then gives,\n\\begin{equation}\\label{ap2_5}\nss=-l\\ln(1-\\zeta_{step})\\equiv-l\\ln(\\zeta_{step}),\n\\end{equation}\nwhich is the step size $ss$ used in the Monte Carlo simulation in Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6}.\n\n\n\\chapter{Introduction}\n\\label{ref:Chapter1}\n\n\\newcommand{\\hilight}[1]{\\colorbox{green}{#1}}\n\\newcommand{\\hili}[1]{\\colorbox{red}{#1}}\n\n\\section{The Sun, its atmosphere and solar flares}\\label{intro_intro}\nOur star, the Sun is a G2 main sequence star. It has a mass, radius, luminosity and effective surface temperature of M$_{\\bigodot}=1.99\\times10^{33}$ g, R$_{\\bigodot}=6.96\\times10^{10}$ cm, $\\mathcal{L}_{\\bigodot}=3.84\\times10^{33}$ erg s$^{-1}$ and T$_{\\bigodot}=5778$ K respectively \\citep[e.g.,][]{2004suin.book.....S}, with an estimated age of 4.6 Gyr \\citep{2011MNRAS.418.1217H}. The solar atmosphere, which extends into the solar wind, is the largest continuous structure in the solar system, permeating the entire heliosphere. The solar magnetic field governs the evolution of the solar corona and hence it is widely believed to be responsible for transient phenomenon such as solar flares. Solar flares are uninterestingly defined as a ``rapid, sudden brightening in the solar atmosphere\", yet they are responsible for the largest release of energy in our solar system, which can be greater than $10^{32}$ erg. Most solar flares occur within active regions on the Sun; regions where the solar magnetic field is particularly strong. The physics associated with the production of, and processes throughout, a solar flare is immense; in order to fully understand the entire flare mechanism, large scale processes describing the evolution of the magnetic field within an entire active region must be coupled with the small scale processes describing the interactions of high energy particles accelerated during the flare. This thesis is concerned with the latter.\n\nThe solar atmosphere is a continuous structure with many layers of varying temperature and number density. A semi-empirical model of the solar atmosphere is shown in Figure \\ref{temp_num}. It is usual to split the solar atmosphere into three layers defined as the: photosphere, chromosphere, and the corona, which eventually extends into, and is renamed, the solar wind at roughly $3R_{\\bigodot}$, filling the entire heliosphere. The photosphere is the optical `surface' of the Sun; the point at which the solar atmosphere becomes opaque to optical wavelengths. The temperature $T$ and number density $n$ of the photosphere fall with increasing height, with $T$ falling from $\\sim6000$ K to $\\sim4000$ K at the highest point of the photosphere, known as the \\textit{temperature minimum} region. Hydrogen number densities within the photosphere are of the order $10^{17}$ cm$^{-3}$, falling to around $10^{15}$ cm$^{-3}$ at the temperature minimum region \\citep{2008ApJS..175..229A, Vernazzaetal1981}. Within hydrogen number densities of the order $10^{17}$ cm$^{-3}$, high energy X-rays can interact with free or bound electrons, and a significant proportion of this thesis is dedicated to studying these interactions (Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6}). After the temperature minimum region, there is a $\\sim2000$ km layer known as the chromosphere, where the temperature of the solar atmosphere begins to rise, reaching $\\sim2\\times10^{4}$ K at the top. At the top of chromosphere, hydrogen number densities have fallen to around $10^{11}$ cm$^{-3}$ (Figure \\ref{temp_num}). The higher hydrogen number densities deeper within the chromosphere collisionally stop high energy electrons transported to the chromosphere during a solar flare, producing bremsstrahlung X-rays. At the top of the chromosphere lies the {\\it transition region}. Here, there is sudden two magnitude increase in temperature and decrease in number density over a very small height of around $100$ km. After the transition region, there is the final and largest layer of the solar atmosphere; the corona. The lower corona is a low $\\beta$ plasma where the thermal pressure is much less than that the magnetic pressure, of the order $\\sim10^{-2}$. However $\\beta$ can vary dramatically with coronal height and solar activity \\cite[e.g., models by][]{2001SoPh..203...71G}. However, in general the corona is magnetically dominated and highly conductive. At quiet Sun times, the corona has a high temperature of $\\sim1-2$ MK and hence can be observed at X-ray energies. The high temperature of the corona is indicated by the presence of lines from highly ionised elements such as iron (Fe) and calcium (Ca) in the coronal emission spectrum. The method of heating the corona to such high temperatures is still not properly understood and is an outstanding problem in astrophysics \\cite[e.g.][]{2012RSPTA.370.3217P}. The energy release process that causes the onset of a solar flare is believed to occur within the corona, where the temperature of the plasma in the vicinity of the region of energy release can be tens of mega Kelvin. The number density of the quiet corona is low; $\\sim10^{8}-10^{9}$ cm$^{-3}$ or less. During a solar flare, regions of the corona can have a number density as high as $10^{11}$ cm$^{-3}$, possibly from heated material moving into the corona from the denser chromosphere below; this is known as chromospheric evaporation \\citep[cf.,][]{1980ApJ...239..725D,1983SoPh...86...67A}. As in the chromosphere, high coronal densities are important for the interaction of particles, mainly electrons, via Coulomb collisions with the background plasma, and the emission of X-rays. This is particularly important in Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3} of this thesis.\n\\begin{figure}\n\\centering\n\\includegraphics[width=14cm]{numT_2.png}\n\\caption[The changing number density and temperature structure of the solar atmosphere.]{Original figure taken from \\cite{2004psci.book.....A} and then adapted. The figure shows how electron number density $n_{e}$, hydrogen number density $n_{H_{0}}$ and electron temperature $T_{e}$ change with height above the solar photosphere. The photosphere, chromosphere, corona, temperature minimum region and transition region are noted on the figure.}\n\\label{temp_num}\n\\end{figure}\n\nIt is widely believed that the onset of a solar flare is caused by the release of stored magnetic energy in the corona, due to reconnecting magnetic fields \\cite[cf.,][]{2000mare.book.....P}. During a flare, coronal plasma in the vicinity of the energy release region is heated to temperatures greater than $10$ MK. Particles, primarily electrons, but also protons and heavier ions, are accelerated to high energies greater than $\\sim20$ keV and often up to MeV and even GeV energies, out of the background thermal plasma. The acceleration of a large number of particles during a solar flare requires an efficient acceleration mechanism. This is a topic of ongoing debate within the solar physics community. Popular candidates are: DC electric field acceleration, stochastic acceleration (second order Fermi acceleration) and shock acceleration (first order Fermi acceleration) \\cite[see][as a recent review of such mechanisms]{2011SSRv..159..107H}. The energy released during a solar flare propagates into the lower layers of the corona, transition region and chromosphere, either in the form of precipitating high energy electrons, protons and heavier ions, or by thermal conduction, due to the even steeper temperature gradient created between the corona and chromosphere during a flare. The chromosphere and transition region react to this heating; dense, heated chromospheric material bound by the magnetic field has to expand up into the corona, causing the chromospheric evaporation mentioned in the previous paragraph.\n\\begin{figure}\n  \\begin{minipage}[c]{0.55\\textwidth}\n   \n   \\includegraphics[width=\\textwidth]{yohkoh_invert.png}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.40\\textwidth}\n\\caption[{\\it Yohkoh} soft and hard X-ray images of a flare from the 13th January 1992.]{X-ray image of a flare (13th January 1992) using Soft and Hard X-ray Telescopes (SXR and HXR) on-board {\\it Yohkoh}. HXR contours are overlaid onto the SXR loop. The positions of X-ray sources are discussed in Section \\ref{intro_imaging}. This image is taken and adapted from \\url{http:\/\/hesperia.gsfc.nasa.gov\/hessi\/images\/fd-close.gif}.}\n\\label{sfm_with_xray}\n  \\end{minipage}\n\\end{figure}\n\nDuring a solar flare, radiation is emitted across the entire electromagnetic spectrum from radio to X-rays and even gamma rays for the largest flares; from the corona to the photosphere. Hard X-rays (HXRs) with energies greater than $\\sim10$ keV are produced collisionally by the electrostatic interactions of electrons with background particles in both the corona and chromosphere, mainly as free-free bremsstrahlung emission. Soft X-rays (SXRs) in the range of $\\sim0.1-10$ keV are also produced as bremsstrahlung but mainly from particles interacting within a high temperature plasma.  Gamma-rays, if present, above around $300$ keV can also be produced by the interaction of protons, heavier ions and flare produced neutrons. For example gamma-rays can be emitted from the photosphere by the interactions of neutrons combining with neutral hydrogen to form deuterium \\citep[e.g.,][]{2009RAA.....9...11C}.\n\nSolar flare sizes are classified by their soft X-ray flux; specifically by the 1-8~\\AA flux measured by the Geostationary Orbiting Environmental Satellites ({\\it GOES\\,}) at 1 AU. The flare classifications are A, B, C, M and X with an X-class flare being the largest. The flux of each class increases by an order of magnitude. The flux of an X-class flare is equal to or greater than $10^{-4}$ W m$^{-2}$, while the flux of a smaller M-class flare is of the order $10^{-5}$ W m$^{-2}$. For classes A to M, the numbers 1 to 10 also denote the strength of the flare, that is, a M10 flare has a higher flux than a M5 flare. There is no limit on the numbers for an X-class flare \\cite[e.g.,][]{2011SSRv..159...19F}.\n\nX-rays, and even more so, gamma-rays if present, only represent a small proportion of the total flare radiative output \\citep{2004GeoRL..3110802W,2006JGRA..11110S14W,2011A&A...530A..84K}, with the majority of the emission actually coming from larger wavelength emissions of extreme ultraviolet, ultraviolet and visible light. However, the chromosphere and corona are optically thin at high X-ray and gamma-ray energies, and studying their temporal, energetic, spatial and polarization properties can provide a direct link not only to the accelerated electrons, protons and ions responsible for their production, but also the conditions in the corona or chromosphere during a flare; the main topics of study within this thesis. Therefore, the rest of this chapter will discuss the observation and analysis of solar flare X-rays, starting with a brief review of the particle interactions and emission mechanisms required for the production of solar flare X-rays in the solar atmosphere.\n\n\\section{Electron and ion interactions the solar atmosphere}\n\\subsection{Coulomb collisions}\nIn a fully or partially ionised plasma such as the solar corona or chromosphere, electrons and ions will interact by the Coulomb electrostatic force, via `Coulomb collisions'. When an electron passes close to an ion or another electron, it is deflected by some angle $\\theta_{D}$ due to the Coulomb electric field of the ion. This is shown in Figure \\ref{cc}.  In the simplest model describing Coulomb collisions, an electron moves through a background plasma of heavy, stationary ions. This is known as a \\textit{Lorentz model}. The background electrons required for neutrality in the plasma are neglected, since the Lorentz model assumes that the ion atomic number $Z$ is large, meaning that the electron-ion collisions (e-i) have a dominant effect over the electron-electron (e-e) collisions. The cross section $\\sigma_{R}$ for the small angle scatter of a moving electron due to the Coulomb field of a heavy, stationary ion can be given by the Rutherford formula \\cite[cf.,][]{1981phki.book.....L}:\n\\begin{equation}\\label{ruth_cs}\n\\sigma_{R}=\\frac{4\\pi Ze^{2}}{m_{e}^{2}v_{e}^{4}}\\int_{b_{min}}^{b^{max}}\\frac{db}{b}\n\\end{equation}\nwhere $e$ [esu] is the charge of an electron, $m_{e}$ [g] is the mass of the electron and $v_{e}$ [cm s$^{-1}$] is the total electron speed. The encounter is characterised by $b$ [cm], the impact parameter; the expected closest distance of approach between the electron and ion, had the electron not been deflected during their encounter (as shown in Figure \\ref{cc}). The integral\n\\begin{equation}\\label{clog}\n\\int_{b_{min}}^{b^{max}}\\frac{db}{b}=\\ln\\frac{\\lambda_{D}}{b_{min}}=\\ln\\Lambda\n\\end{equation}\nis defined as the Coulomb logarithm $\\ln\\Lambda$. The role of the Coulomb logarithm is to take into account the total effect of all deflections with different impact parameters $b$ ranging from a minimum $b_{min}$ to $b_{max}=\\lambda_{D}$, where $\\lambda_{D}$ is the Debye length. Within a flaring solar corona, where the plasma is fully ionised, values of $\\ln\\Lambda\\sim20$ are often used. An electron undergoing many deflections within a field of ions will experience a \\textit{collisional drag force}, causing an electron to lose momentum in its direction of travel, and this is transferred to the components of momentum perpendicular to the direction of travel. The time $\\tau_{0}$ (or equivalently the frequency $\\nu_{0}$) it takes for all particle momentum to be lost in the direction of travel, is given by,\n\\begin{equation}\\label{lorentz_nu_cgs}\n\\centering\n\\tau_{0}^{-1}=\\nu_{0}=n_{i}v_{e}\\sigma=\\frac{4\\pi n_{e}Ze^{4}\\ln\\Lambda}{m_{e}^{2}v_{e}^{3}}=\\frac{\\Gamma}{v^{3}},\\;\\;\\;\\;\\;\\;\\;\\;\\;{\\rm where}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\Gamma=\\frac{4\\pi n_{e}Ze^{4}\\ln\\Lambda}{m_{e}^{2}}\n\\end{equation}\nand $n_{e},n_{i}$ are the electron and ion number densities [cm$^{-3}$]. This is known as the \\textit{Lorentz collisional time or frequency}. In a Lorentz model, where the background particles are stationary, there is no {\\it exchange of energy} between the electrons and the ions.\n\\begin{figure*}\n\\includegraphics[width=15cm]{collision3.png}\n\\caption[Diagram of a Coulomb collision between an electron and an ion.]{{\\it Left:} Electron deflected by a heavy ion in a Lorentz collisional model. {\\it Right:} In general, both particles are deflected during a Coulomb collision, and momentum and energy are transferred.}\n\\label{cc}\n\\end{figure*}\n\nIn solar flare conditions, collisions are not fully described by the Lorentz model, where there will be e-i, e-e and i-i collisions. Electrons and ions will be in motion and {\\it exchange energy} during an interaction.  In this case, there are two main timescales to consider:\n\\begin{enumerate}\n\\item{the momentum loss time $\\tau^{p}$, and}\n\\item{the energy exchange time $\\tau^{E}$.}\n\\end{enumerate}\nAssuming, during a solar flare, the background distribution of particles are Maxwellian in form, and in thermal equilibrium, then 1. describes the time it takes for a particle distribution to isotropize in angle with the thermal background, and 2. describes the time required for a particle distribution to form an energy equilibrium with the thermal background. Each timescale is slightly different depending on the particle species involved in the collision.\nThe timescales for {\\it energy exchange} can be related by,\n\\begin{equation}\\label{timescales}\n\\tau_{ee}^{E}:\\tau_{ii}^{E}:\\tau_{ei}^{E}\\sim1:\\left(\\frac{m_i}{m_e}\\right)^{1\/2}:\\left(\\frac{m_{i}}{m_{e}}\\right)\n\\end{equation}\nshowing that the quickest equilibrium time for, and hence the largest change in energy occurs during, an $e-e$ interaction \\citep{1981phki.book.....L} . For $e-e$ collisions, the energy loss is given by,\n\\begin{equation}\\label{high_energy_loss}\n\\frac{dE}{dt}=-\\frac{E}{\\tau_{ee}^{E}}=-\\frac{2E}{\\tau_{0}}=-\\frac{2E\\Gamma}{v_{e}^{3}}=-\\frac{Kn_{e}}{E}v_{e}\n\\end{equation}\nwhere $K=\\Gamma m_{e}^{2}\/2n_{e}$. Equation \\ref{high_energy_loss} is often used to describe collisions in solar flare physics vis a {\\it collisional thick target model} \\cite[e.g.,][]{1971SoPh...18..489B,1972SvA....16..273S}. The loss in electron energy over a distance along $z$ from an initial energy $E_{0}$ is then found to be,\n\\begin{equation}\\label{energy_loss_2}\nE^{2}=E_{0}^{2}-2K\\int_{0}^{z}n(z^{'})dz^{'}\\;\\;\\;\\; {\\rm where\\;the\\;column\\;density\\;\\;}\\;N(z)=\\int_{0}^{z}n(z^{'})dz^{'}\n\\end{equation}\nAssuming the density of the target is constant such that $N(z)=n_{0}z$, then a typical chromospheric density of $n_{0}=10^{13}$ cm$^{-3}$ would collisionally stop a 30 keV electron over a distance $\\Delta z\\sim0.5\\mbox{\\ensuremath{^{\\prime\\prime}}}$. Within a target density of $n_{0}=10^{11}$ cm$^{-3}$, which is the number density of a dense corona (Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3}), then a 30 keV electron would lose all energy over a distance $\\Delta z\\sim47\\mbox{\\ensuremath{^{\\prime\\prime}}}$. If the target is not fully ionised \\citep[e.g.,][]{1978ApJ...224..241E} then the value of the Coulomb logarithm is decreased. Often values of $\\ln\\Lambda\\sim7$ are used in the chromosphere to account for the presence of atoms in cooler chromospheric regions. Equations \\ref{high_energy_loss} and \\ref{energy_loss_2} only describe the energy loss of an electron in the high energy limit, that is when $E>>E_{th}$, where $E_{th}$ is the average thermal energy of the background plasma. The energy variation of electrons close to the average thermal energy of a background plasma is discussed in Chapter \\ref{ref:Chapter2}.\n\n\\section{Solar flare X-rays: bremsstrahlung}\nDuring a Coulomb collision, on average only a very small fraction of the energy lost by an accelerated electron is radiated as a photon. The radiation that is emitted is termed bremsstrahlung and means ``braking radiation\". Although other emission mechanisms may contribute in the corona, such as free-bound emission \\citep[e.g.,][]{1970MNRAS.151..141C,2010A&A...515C...1B} from the recombination of an ion and electron for example, overall bremsstrahlung is the most important emission mechanism for the production of X-rays during a solar flare \\citep{1967SvA....11..258K} and is produced by both electron-ion and electron-electron Coulomb collisions \\citep{1975SoPh...45..453H,2007ApJ...670..857K}, in the solar corona and chromosphere. Below $\\sim300$ keV, the bulk of solar flare bremsstrahlung emission comes from electron-ion interactions.\n\n\\subsection{Bremsstrahlung produced by a single accelerated electron}\nIn the simplest situation, where a single electron is moving at a non-relativistic velocity, the total power $P_{rad}$ [erg s$^{-1}$] radiated by the accelerated electron is given by Larmor's formula\n\\begin{equation}\\label{Larmor_cgs}\nP_{rad}=\\left(\\frac{dE}{dt}\\right)_{rad}=\\frac{2e^{2}|{\\dot{\\bf v}}|^{2}}{3c^{3}}\n\\end{equation}\nwhere $e$ [esu] is the electron charge, $\\dot{\\bf v}$ is the electron acceleration [cm s$^{-2}$] and $c$ [cm s$^{-1}$] is the speed of light. Larmor's formula gives the radiation loss rate in the frame of the electron. The total energy per unit frequency $\\frac{dE}{d\\omega}$ emitted the entire time a single charge is accelerated can be found via Parseval's theorem \\citep[cf.,][]{1981cup..book.....L},\n\\begin{equation}\\label{longair}\n\\int_{-\\infty}^{\\infty}\\frac{dE}{dt}dt\n=\\int_{-\\infty}^{\\infty}\\frac{2e^{2}}{3c^{3}}|{\\bf \\dot{v}(\\omega)}|^{2}d\\omega\n=2\\int_{0}^{\\infty}\\frac{2e^{2}}{3c^{3}}|{\\bf \\dot{v}(\\omega)}|^{2}d\\omega\n\\end{equation}\ngiving,\n\\begin{equation}\\label{I_2}\nI(\\omega)=\\frac{4e^{2}}{3c^{3}}|{\\bf \\dot{v}(\\omega)}|^{2},\n\\end{equation}\nthe frequency spectrum of a single accelerated charge.\n\n\n\\subsection{Bremsstrahlung X-rays from a solar flare}\nIn solar flare physics, the general form of the total angle-averaged X-ray distribution $I$ [photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$] produced by an electron flux density [electrons cm$^{-2}$ s$^{-1}$ keV$^{-1}$] undergoing Coulomb collisions in the corona or chromosphere is given by,\n\\begin{equation}\\label{I_1}\nI(\\epsilon)=\\frac{1}{4\\pi R^{2}}\\int_{\\epsilon}^{\\infty}\\int_{V}n({\\rm\\textbf{r}})F(E,{\\rm\\textbf{r}})\\sigma(\\epsilon,E)dEd^{3}{\\rm\\textbf{r}},\n\\end{equation}\nwhere $R=1$ AU is the Sun-Earth distance, $\\epsilon$ [keV] is the photon energy, $V$ [cm$^{3}$] is the emitting volume, $n$ [cm$^{-3}$] is the number density of the emitting region, ${\\rm\\textbf{r}}$ is the position on the Sun and $\\sigma$ [cm$^{2}$] is the angle-averaged bremsstrahlung cross section.\nOften for clarity or simplicity the electron-ion (e-i) bremsstrahlung cross section is approximated by the Kramers formula $\\sigma=Q_{K}$ or can be better estimated by the non relativistic Bethe-Heitler cross section $\\sigma=Q_{BH}$. Both are given by\n\\begin{equation}\\label{K_cross}\nQ_{K}(\\epsilon,E)=\\frac{Z^{2}8\\alpha}{m_{e}c^{2}r_{0}^{2}\\epsilon E} ,\\,\\,\\,\\,\\,\\,\\,\\,\\, Q_{BH}(\\epsilon,E)=Q_{K}\\ln\\left(\\frac{1+\\sqrt{1-\\epsilon\/E}}{1-\\sqrt{1-\\epsilon\/E}}\\right),\n\\end{equation}\nwhere $\\alpha\\sim1\/137$ is the fine structure constant and $r_{0}=2.82\\times10^{-13}$ cm is the classical electron radius \\citep[cf.][]{2011SSRv..159..301K}. Equation \\ref{I_1} is an inverse problem; from observation the ultimate goal is to deduce the electron flux distribution $F$ from a measured photon flux $I$. The form of the inferred electron distribution is dependent upon the form of the bremsstrahlung cross section. Both the Kramers and non-relativistic Bethe-Heitler forms are often used as they allow the analytical deduction of $F(E,{\\rm\\textbf{r}})$ \\cite[e.g.,][]{1971SoPh...18..489B,1977ApJS...35..419R,2002SoPh..210..373B}. However, \\cite{1997A&A...326..417H} noted that relativistic changes to the e-i bremsstrahlung cross section should be taken into account above even $\\sim30$ keV, and found an analytical form up to semi-relativistic energies. The full form of the angle-averaged e-i bremsstrahlung cross section is shown in \\cite{1959RvMP...31..920K}, formula 3BN, with a more useable form for numerical simulation given by \\cite{1997A&A...326..417H}.\nMeasuring the X-ray photon spectrum alone without any spatial information, implies that Equation \\ref{I_1} can be spatially integrated to give,\n\\begin{equation}\\label{I_2a}\nI(\\epsilon)=\\frac{1}{4\\pi R^{2}}\\int_{\\epsilon}^{\\infty}\\left[\\bar{n}V\\bar{F(E)}\\right]\\sigma(\\epsilon,E)dE\n\\end{equation}\nwhere $\\left[\\bar{n}V\\bar{F(E)}\\right]$ is known as the mean electron flux spectrum \\citep{2003ApJ...595L.115B}.\nAs well as being dependent upon the X-ray energy $\\epsilon$, initial electron energy $E$ and the atomic number of the target $Z$, the bremsstrahlung cross section $\\sigma$ is also angular and polarization dependent. The angular dependent\n\\begin{figure}\n\n    \\includegraphics[width=7cm]{massone_1.png}\n    \\hspace{2 cm}\n    \\includegraphics[width=5cm]{massone_angles.png}\n \n \n\\caption[A polar diagram of the angular dependent e-i bremsstrahlung cross section.]{{\\it Left:} Figure taken from \\cite{Massoneetal2004}. Angle dependent e-i bremsstrahlung cross section for a 100 keV electron and the emission of a 30 keV (solid), 50 keV (dotted) and 80 keV (dashed) photon. The radial distance gives the size of the cross section while the angle from the x-axis is the angle between the photon emission and the incoming electron. {\\it Right:} Diagram showing the X-ray emission angle $\\theta$, the electron angle to the guiding field $\\beta$, the electron azimuthal angle $\\phi$ and the angle between $\\theta$ and $\\beta$, $\\Theta$.}\n\\label{massone_figures}\n\\end{figure}\ne-i cross section summed over all polarization states is given by \\cite{GlucksternHull1953}.  A polar diagram showing the form of the polarization integrated angular dependent e-i bremsstrahlung cross section is shown in Figure \\ref{massone_figures} (left) and is taken from \\cite{Massoneetal2004}. It is plotted for a $100$ keV electron emitting either a $30$, $50$ or $80$ keV photon. This figure shows the e-i cross section is larger for lower energy photons. It is more likely a low energy photon will be emitted during an interaction and the direction of emission is more likely to peak away from the direction of the incoming electron as the emitted photon energy increases. Using an angle dependent e-i bremsstrahlung cross section $\\sigma$, the angular and energy dependent photon flux distribution $I(\\epsilon,\\theta)$ can be given by,\n\\begin{equation}\\label{I_3}\nI(\\epsilon,\\theta)\\propto\\int_{E=\\epsilon}^{\\infty}\\int_{\\phi=0}^{2\\pi}\\int_{\\beta=0}^{\\pi}F(E,\\beta)\\sigma(E,\\epsilon,\\beta,\\theta,\\phi)\\sin\\beta d\\beta d\\phi dE,\n\\end{equation}\nwhere $\\theta$ is the photon emission angle, $\\beta$ is the electron angle to the guiding field and $\\phi$ is the electron azimuthal angle in the plane perpendicular to the guiding field. Each angle is related by,\n\\begin{equation}\n\\cos\\Theta=\\cos\\theta\\cos\\beta+\\sin\\theta\\sin\\beta\\cos\\Phi.\n\\label{cosTheta_intro}\n\\end{equation}\nThis is further described in Chapter \\ref{ref:Chapter6}, and each angle can be seen pictorially in Figure \\ref{massone_figures} (right).\nDepending upon the disk location (viewing angle) of the X-ray source and the electron anisotropy, \\cite{Massoneetal2004} found that using the angle-averaged, instead of the angle-dependent e-i bremsstrahlung cross section can cause significant changes to the inferred electron flux distribution, particularly above 50 keV, leading to suspect inferred mean electron spectra and total injected energies.\n\\cite{GlucksternHull1953} gives the polarization dependent parallel and perpendicular components of the angular dependent e-i bremsstrahlung cross section. These are used in Chapter \\ref{ref:Chapter6} of this thesis and hence are further discussed there. The total polarization and angular dependent e-i bremsstrahlung cross section $\\sigma$ is then the sum of the components of the cross section parallel $\\sigma_{||}$ and perpendicular $\\sigma_{\\perp}$ to the plane of X-ray emission\n\\begin{equation}\\label{Q_par_perp}\n\\sigma=\\sigma_{||}+\\sigma_{\\perp}.\n\\end{equation}\nImportantly, the angular distribution of the X-ray and hence electron distribution is positively correlated with the X-ray polarization. This will be discussed further in Section \\ref{intro_pol} and Chapter \\ref{ref:Chapter6}.\n\n\\subsection{Electron-ion versus electron-electron bremsstrahlung}\\label{eeb}\nAs mentioned, bremsstrahlung X-rays can be produced by both electron-ion and electron-electron Coulomb collisions. In the previous section, only e-i bremsstrahlung was considered, as most solar flare problems only need to account for the electron-ion collisions. Below $\\sim300$ keV the e-e bremsstrahlung cross section decreases rapidly and the emission is negligible compared to that of e-i bremsstrahlung \\citep{1975SoPh...45..453H,2007ApJ...670..857K}. However, \\cite{2007ApJ...670..857K} found that the presence of e-e bremsstrahlung should not be ignored above $\\sim300$ keV. For a given X-ray distribution, the presence of an e-e bremsstrahlung component requires a steeper electron spectrum at higher energies.\nUnlike e-i bremsstrahlung, e-e bremsstrahlung cannot produce X-rays of all energies up to the energy of the emitting electron. The maximum e-e bremsstrahlung energy is bounded by the angle between the direction of the incoming electron and the emitted X-ray. In general the bremsstrahlung cross section should be a combination of both e-i and e-e interactions \\citep{1975SoPh...45..453H,1998SoPh..178..341H,2007ApJ...670..857K}, given by\n\\begin{equation}\\label{total_brem_cs}\n\\sigma(\\epsilon,E)=Z^{2}\\sigma_{ei}(\\epsilon,E)+Z\\sigma_{ee}(\\epsilon,E),\n\\end{equation}\nwhere $Z$ is the effective atomic number of a plasma or quasi-neutral target. In Chapter \\ref{ref:Chapter6}, only the angular and polarization dependent e-i bremsstrahlung cross section is used, since the majority of the work in Chapter \\ref{ref:Chapter6} studies X-rays in the range of 20-50 keV, where the emission due to e-e interactions is negligible.\n\n\\subsection{Thermal bremsstrahlung}\\label{tb}\nThermal bremsstrahlung is the term given to the production of bremsstrahlung X-rays by a distribution of electrons in thermal equilibrium. Often the spectrum of lower energy X-rays (below $\\sim30$ keV) during a solar flare has an exponential form, representative of the emission from a thermal distribution of particles, \\cite[e.g., first suggested via observation by][]{1966JGR....71.3611C}. Although realistically, the flaring region will have a temperature distribution, it is often useful to fit this part of the X-ray spectrum with a single isothermal function (see Figure \\ref{x_gam_ray_spect}), in order to obtain an average temperature $T$ [K] and emission measure $EM=n^{2}V$ [cm$^{-3}$], where $V({\\rm\\textbf{r}})$ [cm$^{3}$] is the volume of the emitting flare region. The electron flux density $F(E,{\\rm\\textbf{r}})$ of such a distribution and the resulting photon flux distribution $I(\\epsilon,{\\rm\\textbf{r}})$ can be given by,\n\n\\begin{equation}\\label{F_th}\nF(E,{\\rm\\textbf{r}})=\\frac{2^{3\/2}}{(\\pi m_{e}^{1\/2})}\\frac{n({\\rm\\textbf{r}})E}{(k_{B}T^{3\/2}({\\rm\\textbf{r}}))}\\exp\\left(\\frac{-E}{k_{B}T({\\rm\\textbf{r}})}\\right)\\;\\;\\rightarrow\\;\\;I(\\epsilon,{\\rm\\textbf{r}})\\propto\\frac{n^{2}({\\rm\\textbf{r}})V({\\rm\\textbf{r}})}{\\epsilon T^{1\/2}({\\rm\\textbf{r}})}\\exp\\left(\\frac{-\\epsilon}{k_{B}T({\\rm\\textbf{r}})}\\right)\n\\end{equation}\nwhere \\textbf{r} is the position on the Sun.\n\n\\subsection{Non-thermal bremsstrahlung}\\label{ntb}\nOften in X-ray solar flare physics, the higher energies of the X-ray distribution have a form that can be approximated by either a single or double power law (see Figure \\ref{x_gam_ray_spect}) \\cite[e.g.,][]{1968JGR....73..434C,1981ApJ...251L.109L,1985SoPh..100..465D}. Hence, this means that the parent electron energy distribution can also be approximated by a power law,\n\\begin{equation}\\label{ntb1}\nF(E)\\propto E^{-\\delta}\\;\\;\\;\\;\\;\\rightarrow\\;\\;\\;\\;\\;I(E)\\propto \\epsilon^{-\\gamma}.\n\\end{equation}\nIn a collisional thick target model, the electrons lose all of their kinetic energy in the target region. In general, the spectral index of the target electron spectrum differs from the injected electron spectral index by $\\delta_{T}\\sim\\delta-2$ and the spectral index of the resulting X-ray distribution is given by,  $\\gamma_{thick}=\\delta+1$ \\cite[e.g.,][]{1971SoPh...18..489B}. The chromosphere and also the corona, depending on its density, can act as a thick target during a solar flare. In a thin target model, electrons do not lose all of their energy as they move through a thin target region and the resulting spectral index of the photon distribution is given by $\\gamma_{thin}=\\delta-1$. A low density corona may act as a thin target. The above approximations are for non-relativistic e-i interactions. The relationship between the spectral index of the electron distribution $\\delta$ and the spectral index of the X-ray distribution $\\gamma$ flattens if the X-ray emission is due to relativistic e-i interactions or e-e collisions. For example, e-e interactions in a thick target model produce an X-ray spectrum of the form $\\delta\\sim\\gamma$ \\citep{1989A&A...218..330H,2007ApJ...670..857K}.\n\n\\section{Solar flare X-rays: photon interaction processes}\nIn a dense plasma or neutral atmosphere X-ray photons can interact with free or bound electrons by Compton scattering.\nThe corona and chromosphere are mainly optically thin to X-ray and gamma-ray energies. However, this is not true in the high densities of the photosphere ($\\sim10^{17}$ cm$^{-3}$), as noted in Section \\ref{intro_intro}. Compton scattering in the photosphere is the cause of photospheric albedo which is discussed in Section \\ref{intro_albedo} of this chapter and in Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6} of this thesis. In the absence of energy exchange in the low energy limit below $\\sim1$ keV, Compton scattering can be described by, and is equivalent to Thomson scattering.\n\n\\subsection{Thomson scattering}\nThomson scattering describes the interaction of an incident plane wave with an electron, and is a purely classical process \\cite[cf.,][]{1962clel.book.....J,1981cup..book.....L}. The radiation field of the incident wave causes the electron to oscillate, with the direction of oscillation dependent upon the polarization of the incident wave. The oscillating electron is an accelerated charge and subsequently radiates its own radiation in a different direction; the scattered wave. For low energy interactions described by Thomson scattering, the wavelength of the incident wave is equal to wavelength of the scattered wave. The energy per unit time, $dP$ scattered into a solid angle $d\\Omega$ or equivalently radiated by the electron is given by Equation \\ref{dpdomega} (left) and the time-averaged energy flux of the incident electromagnetic wave (time-averaged Poynting flux) $\\langle U \\rangle$, [erg s$^{-1}$ cm$^{-2}$], is given by Equation \\ref{dpdomega} (right),\n\\begin{equation}\\label{dpdomega}\n\\frac{dP}{d\\Omega}=\\frac{e^{4}\\mathcal{E}^{2}}{8\\pi m_{e}^{2}c^{3}}\\sin^{2}\\Theta,\\;\\;\\;\\;\\;\\;\\;\\langle U\\rangle=\\frac{c}{8\\pi}\\mathcal{E}^{2}\n\\end{equation}\nwhere $\\mathcal{E}$ is the oscillating electric field of the scattered electromagnetic wave and $\\Theta$ is the {\\it angle between the direction of electron acceleration and the propagation direction of the outgoing radiation, not the scattering angle}.\nThe differential Thomson scattering cross section is the ratio of these two quantities, giving,\n\\begin{equation}\\label{thom_dcs_pol}\n\\frac{d\\sigma_{thom}}{d\\Omega}=\\left(\\frac{e^{2}}{m_{e}c^{2}}\\right)^{2}\\sin^{2}\\Theta.\n\\end{equation}\nEquation \\ref{thom_dcs_pol} describes completely polarized radiation. The angle $\\Theta$ is related to the polar scattering angle $\\theta$ and the direction of the incoming polarization $\\Psi$ by $\\cos\\Theta=\\sin\\theta\\cos\\Psi$. Each of these angles is shown in Figure \\ref{thom_scatter} (left).\nRearranging gives $\\sin^{2}\\Theta=1-\\sin^{2}\\theta\\cos^{2}\\Psi$. For unpolarized radiation, the average polarization angle is required and hence this gives $\\sin^{2}\\Theta=1-\\sin^{2}\\theta\\langle\\cos^{2}\\Psi\\rangle=1-\\sin^{2}\\theta\/2=\\left(1+\\cos\\theta\\right)\/2$. The Thomson scattering differential cross section for unpolarized incident radiation is then given by,\n\\begin{equation}\\label{thom_dcs_unpol}\n\\frac{d\\sigma_{thom}}{d\\Omega}=\\left(\\frac{e^{2}}{m_{e}c^{2}}\\right)^{2}\\frac{1}{2}\\left(1+\\cos^{2}\\theta\\right)\n\\end{equation}\nwhere $\\theta$ is the scattering angle. Integrating either Equation \\ref{thom_dcs_pol} or \\ref{thom_dcs_unpol} over solid angle $d\\Omega=2\\pi\\sin\\theta d\\theta$ gives,\n\\begin{equation}\\label{thom_cs}\n\\sigma_{thom}=\\left(\\frac{e^{2}}{m_{e}c^{2}}\\right)^{2}\\frac{2\\pi}{2}\\int_{0}^{\\pi}\\left(1+\\cos^{2}\\theta\\right)\\sin\\theta d\\theta\n=\\frac{8\\pi}{3}\\left(\\frac{e^{2}}{m_{e}c^{2}}\\right)^{2},\n\\end{equation}\nwhich is the total Thomson scattering cross section $\\sigma_{thom}$. There is no energy exchange during a Thomson scattering since the electron does not recoil. Figure \\ref{thom_scatter} (left) depicts the Thomson scattering of completely polarized radiation. Figure \\ref{cs_graphs} plots the total Thomson cross section against energy and the unpolarized differential Thomson cross section against scattering angle (Equation \\ref{thom_dcs_unpol}).\n\\begin{figure}\n\\centering\n\\hspace{-25pt}\n\\includegraphics[width=7cm]{thom_diagram.png}\n\\includegraphics[width=8cm]{cs_diagram_2.png}\n\\caption[Diagrams of Thomson and Compton scattering]{{\\it Left:} Cartoon showing a polarized incident plane wave interacting with an electron, causing it to oscillate and re-radiate. The original figure was taken from \\url{http:\/\/www.exul.ru\/education\/1\/Note3b.pdf} and then adapted for this thesis. Each of the angles $\\theta$ (angle between the incident and scattered radiation), $\\Theta$ (angle between the direction of electron acceleration and the propagation direction of the outgoing radiation) and $\\Psi$ (the direction of the incoming polarization measured from the x-axis) are shown. {\\it Right:} diagram of the Compton interaction between a photon and an electron.}\n\\label{thom_scatter}\n\\end{figure}\n\n\\subsection{Compton scattering}\nIn general, when a photon scatters from an electron, there is energy exchange and the energy of the outgoing photon is decreased. Arthur Compton's original result \\citep{1923PhRv...21..483C} was derived from experiment and the formula was given in terms of a shift in photon wavelength, $\\Delta\\lambda$. This can be found easily by studying the kinematics of the collision, assuming that the incident radiation acts as a particle, that is a photon. The resulting wavelength $\\lambda$ or energy $\\epsilon$ of the outgoing photon can be found from,\n\\begin{equation}\\label{cs_1}\n\\Delta\\lambda=\\lambda-\\lambda_{0}=\\frac{h}{m_{e}c}\\left(1-\\cos\\theta\\right),\\;\\;\\;\\;\\;\\;\n\\frac{\\epsilon}{\\epsilon_{0}}=\\frac{1}{1+\\frac{\\epsilon}{m_{e}c^{2}}\\left(1-\\cos\\theta\\right)}.\n\\end{equation}\n$\\lambda_{0}$ and $\\epsilon_{0}$ are the incoming photon wavelength and energy respectively, $h$ is the Planck constant and $\\theta$ is the scattering angle of the outgoing photon relative to the direction of incoming photon. Equation \\ref{cs_1} (left hand side) is easily converted to photon energy (right hand side) by $\\epsilon=\\frac{hc}{\\lambda}$.\n\\begin{figure}\n  \\begin{minipage}[c]{0.55\\textwidth}\n    \\includegraphics[width=\\textwidth]{comp_cs.pdf}\n     \\includegraphics[width=\\textwidth]{make_cs_theta_with_thom.pdf}\n\\includegraphics[width=\\textwidth]{make_cs_theta_2.pdf}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.35\\textwidth}\n\\caption[The full and differential Thomson and Compton scattering cross sections plotted against X-ray energy and scattering angle.]{{\\it Top:} A comparison of the KN $\\sigma_{c}$ (red solid) and Thomson $\\sigma_{thom}$ (grey dashed) scattering cross sections in units of $\\sigma_{thom}$. Both cross sections only match at low energies less than $\\sim1$ keV and hence the KN scattering cross section should be used to describe the Compton interaction in X-ray solar flare studies. {\\it Middle:} The differential KN Compton scattering cross section $d\\sigma_{c}\/d\\Omega$ versus scattering angle $\\theta_{S}$. {\\it Bottom} the azimuthal angle-averaged differential KN scattering cross section $d\\sigma_{c}\/d\\theta_{S}$ versus scattering angle $\\theta_{S}$. The case for Thomson scattering is shown by a grey dashed line in each case.}\n\\label{cs_graphs}\n\\end{minipage}\n\\end{figure}\nThe original formulation of Thomson scattering only described light as a plane wave, it did not take into account its quantum particle properties as a photon, and hence it can not account for the energy exchange between the incoming photon and the electron, which recoils during the interaction, gaining energy at the loss of the photon. A simple diagram of the Compton interaction is shown in Figure \\ref{thom_scatter} (right). In order to find the Compton scattering cross section $\\sigma_{c}$, modifications have to be made to the Thomson scattering cross section $\\sigma_{thom}$ in order to account for the change in energy. From an approximate quantum mechanical derivation, the Compton scattering differential cross section is found to be,\\\\\n\\begin{equation}\n\\frac{d\\sigma_{c}}{d\\Omega}=\\frac{d\\sigma_{thom}}{d\\Omega}\\left(\\frac{\\epsilon}{\\epsilon_{0}}\\right)^{2}.\n\\end{equation}\\\\\nA proper derivation of the differential Compton scattering cross section, fully taking into account both quantum and relativistic effects is performed in quantum electrodynamics. This gives the Klein-Nishina (KN) Compton differential scattering cross section \\citep{KleinNishina1929},\\\\\n\\begin{equation}\\label{comp_cs_intro}\n\\frac{d\\sigma_{c}}{d\\Omega}=\\frac{1}{2}r_{0}^{2}\\left(\\frac{\\epsilon}{\\epsilon_{0}}\\right)^{2}{\\bigg\\lgroup}\\frac{\\epsilon}{\\epsilon_{0}}+\n\\frac{\\epsilon_{0}}{\\epsilon}-\\sin^{2}\\theta_{S}{\\bigg\\lgroup}1-Q\\cos2\\phi_{S}\n -U\\sin2\\phi_{S}{\\bigg\\rgroup}{\\bigg\\rgroup},\n\\end{equation}\\\\\nwhere $\\phi_{S}$ is the azimuthal scattering angle and $Q$ and $U$ are linear Stokes parameters used to describe linear polarization. This is discussed in detail in Chapter \\ref{ref:Chapter6}. The total KN cross section against energy and the differential KN cross section against scattering angle for the completely unpolarized case (that is setting Q and U to zero in Equation \\ref{comp_cs_intro}) are plotted in Figures \\ref{cs_graphs}. From Figure \\ref{cs_graphs}, it can be seen that Compton cross section deviates greatly from the constant Thomson cross section at high energies, decreasing due to the fact that a high energy photon is less influenced by an electron. In Figure \\ref{cs_graphs}, both $d\\sigma_{c}\/d\\Omega$ and $d\\sigma_{c}\/d\\theta_{S}=(d\\sigma_{c}\/d\\Omega) 2\\pi\\sin\\theta_{S}$ are plotted. At low energies, the unpolarized $d\\sigma_{c}\/d\\Omega$ matches that of the Thomson case and is symmetrical, with the smallest value occurring at a scattering angle of $90^{\\circ}$, where the scattering angle is measured from the direction of the incident photon. As the incident photon energy increases, the scattered radiation becomes more and more forward beamed, it is scattered at a smaller angle and there is a smaller change in photon energy. Removing the azimuthal dependency and plotting $d\\sigma_{c}\/d\\theta_{S}$ shows that the photons are more likely to be scattered between $50^{\\circ}$ and $130^{\\circ}$ at low energies, and the importance of this is discussed in Chapter \\ref{ref:Chapter5}. At higher energies, the maximum scattering angle falls to a lower $\\theta_{S}$ due to the forward beaming.\n\n\\section{Solar flare X-rays: observations}\nIn this section, the main X-ray observables during a solar flare: the X-ray temporal evolution, the X-ray spectrum, the X-ray source location and spatial properties, and finally the X-ray polarization will be discussed. The space-borne satellite, {\\it The Ramaty High Energy Solar Spectroscopic Imager (RHESSI\\,)} \\citep{2002SoPh..210....3L} is currently used for high resolution imaging spectroscopy of solar flare X-rays from $3$ keV. {\\it RHESSI} is discussed in Section \\ref{intro_rhessi}.\n\n\\subsection{X-ray temporal evolution of a solar flare}\\label{intro_temp}\nThe duration of a solar flare can usually be separated into three stages: the rise or precursor stage ({\\it stage 1}), the impulsive stage ({\\it stage 2}) and lastly, the decay stage ({\\it stage 3}).  {\\it RHESSI} and {\\it Geostationary Operational Earth Satellites (GOES)} light curves showing the typical temporal evolution of soft X-rays (SXRs) $\\leq10$ keV and hard X-rays (HXRs) $\\geq10$ keV are shown in Figure \\ref{intro_lc}. Each stage is labelled on the figure. During {\\it stage 1}, there is usually a slow, gradual increase in SXRs and lower energy HXRs ($\\sim 1-20$ keV for the flare shown in Figure \\ref{intro_lc}), where the coronal plasma is being heated to tens of mega-Kelvin.  During {\\it stage 2}, there is usually a sudden, fast increase in HXRs above 20 keV, lasting for only $\\sim$ 1 or 2 minutes, where a large number of electrons are accelerated to high non-thermal energies. The SXR and lower energy HXR emission usually peaks after the impulsive HXR emission, and then starts to gradually decrease. This denotes {\\it stage 3}. The overall time and the length of each stage is individual for each flare; for example, the SXR emission may take hours to decrease during {\\it stage 3}, while for other flares it decays over a much quicker period.\n\\begin{figure}\\centering\n\\includegraphics[width=8cm]{intro_lc_2.png}\n\\caption[{\\it GOES} and {\\it RHESSI} lightcurves for a flare that occurred on the 20th September 2002.]{Figure taken from \\cite{2011ApJ...733...37F} showing both {\\it GOES} and {\\it RHESSI} lightcurves during a solar flare occurring on the 20th September 2002. {\\it Stages 1, 2 and 3} are labelled on the figure and described in \\ref{intro_temp}.}\n\\label{intro_lc}\n\\end{figure}\n\\subsection{The X-ray and gamma-ray solar flare energy spectrum}\\label{intro_spect}\n\\begin{figure}\\centering\n\\includegraphics[width=15cm]{x_gam_ray_spectrum_2.pdf}\n\\caption[An example X-ray and gamma ray solar flare spectrum.]{This figure was taken and then adapted from \\cite{2002SoPh..210....3L}. A typical composite X-ray and gamma-ray spectrum from a solar flare. The X-ray spectrum is dominated by both thermal (1a. red) and non-thermal (1b. blue) bremsstrahlung emission. Prominent line emissions at 6.7 keV and 8.1 keV can often be observed (2. purple) due to highly ionised Fe and Ni in the corona. Above $\\sim500$ keV, the gamma-ray spectrum is more complex, with continuum emission from bremsstrahlung, line emissions from nuclear interactions (3. green) and emission from pion decay (4. violet). Each process in described in \\ref{intro_spect}.}\n\\label{x_gam_ray_spect}\n\\end{figure}\nA general example of an expected solar flare X-ray and gamma-ray spectrum is shown in Figure \\ref{x_gam_ray_spect}. The continuum emission in the spectrum from 1 keV onwards to 100 MeV, is predominantly bremsstrahlung emission produced by mostly e-i Coulomb collisions below $\\sim400$ keV (see Section \\ref{eeb}) and both e-i and e-e Coulomb collisions at higher energies. The spectrum is usually exponential in form at lower energies below $\\sim30$ keV, suggesting the emission comes from collisions within a hot, thermal plasma. Spectral fits often suggest temperatures of $18-30$ MK and the use of imaging spectroscopy with instruments such as {\\it RHESSI} shows that the majority of the thermal emission originates from the corona, possibly close to point of energy release during the flare. In this range, instruments like {\\it RHESSI} can often see two line emissions: one at 6.7 keV due to highly ionised iron (Fe) and one at 8.1 keV due to highly ionised Fe and nickel (Ni) in the solar corona. Both the peaks and widths of these lines are highly dependent on the temperature and iron abundance of the corona and act as a useful diagnostic tool \\citep{2006ApJ...647.1480P,2012ApJ...748...52P}. At higher energies $\\geq 25$ keV, a non-thermal bremsstrahlung spectrum can be fitted by either one or two power laws, suggesting the emission comes from high energy particles accelerated out of the background thermal distribution. In general, but not always (see the following sections), the bulk of the HXR emission comes from the chromosphere. Line emissions in the gamma-ray range above $\\sim500$ keV are mostly produced by nuclear interactions of accelerated protons and heavier ions. At 511 keV and 2.223 MeV, two clear emission lines can be seen; 511 keV is the electron-positron annihilation line and 2.223 MeV is the neutron capture line. This is the capture of neutrons by hydrogen in the photosphere. In an extremely rare case, if the spectrum can be seen up to 100 MeV, gamma-ray emission may be produced from pion decay \\citep[e.g.,][]{1979ApJS...40..487R,2009RAA.....9...11C,2011SSRv..159..167V}.\n\n\\newpage\n\\subsection{X-ray imaging of a solar flare}\\label{intro_imaging}\n\\subsubsection{The locations of X-ray sources}\nTypically, there are X-ray sources located in both the chromosphere and corona during a solar flare. In a standard flare model, it is expected that the bulk of the HXR emission will be produced in the chromosphere, which is dense enough to stop high energy electrons accelerated in the corona. For the majority of flares, chromospheric HXR sources or {\\it HXR footpoints}, as they are known, are observed, usually at X-ray energies greater than $\\sim25$ keV. HXR footpoints generally sit at the bottom of the legs of a loop, formed by the reconnecting magnetic field (see Figure \\ref{sfm_with_xray}), and hence for many flares they come in pairs at a given time and energy; one footpoint at the end of each loop leg. In the context of other observations, the HXR footpoints usually straddle the `magnetic inversion line', an imaginary line in the photosphere that separates regions of opposite vertical magnetic polarity in active regions. The majority of flare X-ray source morphologies are {\\it footpoint dominated} \\citep[e.g.,][]{1981ApJ...246L.155H, 1982SoPh...78..107A, 1982SoPh...81..137D, 1994PhDT.......335S,1996AdSpR..17...67S}. Most footpoint dominated flares also produce coronal SXR and HXR emission which can be a mixture of thermal, thin target and thick target emissions depending on the properties of the corona.  Usually this emission is observed up until $\\sim30$ keV, with the majority emanating from the loop-top region (again see Figure \\ref{sfm_with_xray}), possibly close to the energy release site. Many flares of this type were observed with instruments such as {\\it Yohkoh} and now routinely with {\\it RHESSI}. An excellent example of such a flare morphology is shown in Figure \\ref{xray_sources} (top left).\n\n\\begin{figure*}[t]\n\\centering\n\\hspace{-30pt}\n\\includegraphics[width=8cm]{may_image1_2}\n\\hspace{-50pt}\n\\includegraphics[width=7 cm]{sui_holman_2.png}\n\\hspace{-30pt}\n\\end{figure*}\n\\begin{figure*}[h]\n\\centering\n\\hspace{-40pt}\n\\includegraphics[width=8cm]{aug_image1}\n\\hspace{-10pt}\n\\includegraphics[width=5.5 cm]{ed_fleish_2.png}\n\\hspace{-40pt}\n\\caption[Four different examples of solar flare X-ray source morphologies.]{Four different flare X-ray source morphologies. {\\it Top left:} `standard' flare with strong HXR footpoint emission and lower energy coronal emission. {\\it Top right:} as top left with the observation of an additional above the loop-top X-ray source (figure taken from \\cite{2003ApJ...596L.251S}). {\\it Bottom left:} a coronal thick target source with very little HXR footpoint emission. {\\it Bottom right:} a rare event with only the observation of HXR footpoints (figure taken from \\cite{2011ApJ...731L..19F}). The energy bands of each X-ray source are displayed on the figure. Legends have been added to the top right and bottom right figures.}\n\\label{xray_sources}\n\\end{figure*}\nFor some flares, a HXR source above the lower energy X-ray loop-top can also be observed \\citep[e.g.,][]{1994Natur.371..495M}. Although it is speculated that this HXR emission comes from the site of energy release itself, and may offer evidence for the formation of a current sheet in the corona during magnetic reconnection, it is only observed on rare occasions. One interesting case in particular, investigated by \\cite{2003ApJ...596L.251S} with follow-on studies by \\cite{2004ApJ...612..546S,2008ApJ...676..704L} is shown in Figure \\ref{xray_sources} (top right). For this event both a loop top and an above the loop top source can be observed. While the height of the loop top source at a given energy decreased before the peak in X-ray emission, the above the loop-top source showed the opposite trend and moved away from the Sun at a high inferred velocity of $\\sim300$ km s$^{-1}$. Sometimes, the above the loop-top source can be observed at very high energies, greater than that of the X-ray source within the loop \\citep[e.g.,][]{1994Natur.371..495M,2011ApJ...737...48I,2014ApJ...780..107K}. However, the observation of an above the loop-top source is rare; this may be due to the fact they do not exist for every flare morphology or simply because instruments such as {\\it RHESSI} have a limited dynamic range ($\\sim10:1$) and are not sensitive to the low intensity emission from a low density corona.\n\nAnother type of flare morphology that is particularly important for this thesis, are flares where the majority of the X-ray emission comes from not the chromosphere, but the corona. Often this type of coronal X-ray source is interpreted as a {\\it thick target coronal source} \\citep[e.g.,][]{1995SoPh..158..283W, 2004ApJ...603L.117V}, where a high coronal density of the order $1\\times10^{11}$ [cm$^{-3}$] allows the corona to stop electrons up to $30$ keV or so. Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3} are dedicated to studying this type of event, and their properties will be further explained in these chapters, and in the following section. An example of this type of flare morphology is shown in Figure \\ref{xray_sources} (bottom left).\n\nIn a very rare case \\cite{2011ApJ...731L..19F} observed and analysed an interesting event with {\\it only} non-thermal HXR footpoint emission. It was found that electrons were accelerated up to $100$ keV or so, but the measured temperature during the flare did not exceed 6.1 MK. An example of this event is shown in Figure \\ref{xray_sources} (bottom right).\n\\subsubsection{Observations of X-ray source spatial properties}\\label{fp_loop}\nOften HXR footpoints in the chromosphere exhibit a circular or elliptical shape.  \\citet*{2010ApJ...717..250K}, performed a study examining the changing spatial properties of HXR sources with X-ray energy. This study used a forward fitting method known as Visibility Forward Fitting (Vis FwdFit) that `fits' simple shapes to the X-ray visibilities ({\\it RHESSI} is a non-direct imager and creates an image from Fourier components in $uv$ space; this is discussed further, along with the imaging algorithm of Vis FwdFit in Section \\ref{rhessi_imaging}), allowing the source spatial properties to be studied.  This study built upon previous spatial studies with {\\it RHESSI}, specifically \\cite{2008A&A...489L..57K} where the Vis FwdFit method was employed and simple circular Gaussians were fitted to the HXR footpoint sources visibilities. \\citet*{2010ApJ...717..250K} advanced the study by using an elliptical source that allowed the radial height and the shape of the HXR footpoint to be analysed; giving both the semi-major and semi-minor axes of the ellipse and hence the HXR footpoint. For this study, a flare located at the solar limb was chosen, where the changes in radial distance could be interpreted as height changes and the semi-major and semi-minor axes give the horizontal and vertical extents of the HXR source respectively. It was found that all three measured spatial properties decreased with increasing X-ray energy, and properties of the chromosphere could be deduced from the {\\it RHESSI} observations. An image of the HXR footpoints for this event and a Vis FwdFit ellipse fitted to the bright southern footpoint is shown in Figure \\ref{footpointsandloop}. The graphs for radial height, horizontal size and vertical size are also shown in Figure \\ref{footpointsandloop}. By analysing the changes in both radial height and vertical extent of the HXR source together, it was found that the number density structure of the target chromosphere could not be explained by a simple uniform density structure but was much better fitted by a multi-threaded number density structure (see \\citet*{2010ApJ...717..250K} for details); this would not have been deduced from the radial height measurements alone. The observation of a decreasing HXR source horizontal size with X-ray energy, and therefore with height in the chromosphere, suggests that the magnetic field guiding the electrons through the chromosphere is converging. From the conservation of magnetic flux $B_{1}A_{1}=B_{2}A_{2}$ where $B$ and $A$ are the magnetic field strength and area perpendicular to the field respectively, the magnetic field strength $B$ must increase as the cross sectional area $A$ decreases at lower heights in the chromosphere as described by \\cite{2008A&A...489L..57K}.\n\\begin{figure*}\n\\centering\n\\vspace{20pt}\n\\includegraphics[width=15cm]{fp_spat_1.pdf}\n\\includegraphics[width=15cm]{fp_spat_2.pdf}\n\\caption[Changes in X-ray spatial parameters with energy, for both a chromospheric HXR footpoint (top) and coronal X-ray source (bottom).]{Changes in X-ray spatial parameters with energy. {\\it Top:} Changes in radial height and semi-major and semi-minor axes for a chromospheric HXR footpoint. {\\it Bottom:} Changes in loop length and loop width for a coronal X-ray source. The bottom figure is taken from \\cite{2011ApJ...730L..22K}. See Section \\ref{fp_loop} for details.}\n\\label{footpointsandloop}\n\\end{figure*}\n\\\\\nThe observation, simulation and analysis of thick target coronal X-ray source spatial properties, of the type shown in Figure \\ref{xray_sources} (bottom left), are discussed and then further examined in Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3}. As is discussed in Chapter \\ref{ref:Chapter3}, the changing lengths and widths of these loops with energy have also been previously analysed with the Vis FwdFit imaging algorithm. Many studies have found \\citep{2008ApJ...673..576X, 2011ApJ...730L..22K, 2012ApJ...755...32G,2013ApJ...766...28G} that the source length parallel to the guiding field grows with energy, indicative of a collisional thick target model in a high density corona of the order $10^{11}$ cm$^{-3}$. The width changes of these loops perpendicular to the guiding field are particularly interesting. \\cite{2011ApJ...730L..22K} performed a detailed study of one flare examining how the loop width increased proportionally with X-ray energy. However, unlike increases in coronal loop length, changes in width are more difficult to explain since the electrons are bound to the guiding field and classical cross field transport should be negligible. \\cite{2011ApJ...730L..22K} and \\cite{2011A&A...535A..18B} inferred that the width increase could be due to the presence of magnetic turbulence within the loop. An example of one such coronal X-ray source displaying these trends in length and width, found from the Vis FwdFit algorithm, is shown in Figure \\ref{footpointsandloop}. Again, these types of study indicate the usefulness of observing changes in the spatial properties of X-ray sources, particularly in determining the properties of the chromosphere and corona during a flare.\n\n\\subsection{Solar flare X-ray and gamma ray polarization}\\label{intro_pol}\nThe pitch angle distribution of solar flare electrons in both the corona and chromosphere should be determinable through X-ray and gamma ray linear polarization measurements. The bremsstrahlung X-rays emitted from a highly beamed distribution of electrons for instance, should be highly polarized. In general the level of polarization detected is dependent on the photon energy, the level of beaming, the location of the X-ray source on the solar disk and whether Coulomb collisions have isotropised the electrons as they are transported along the guiding field. This has been extensively modelled \\citep[e.g.][]{ElwertHaug1970,Brown1972, Haug1972, BaiRamaty1978, LeachPetrosian1983} for both thin and thick target scenarios. \\cite{1980ApJ...237.1015E} demonstrated that the emission from a purely thermal source should have some level of polarization. In a simple model, the degree of polarization should increase with viewing angle. This means, that for the same distribution of electrons producing X-rays during a flare, the degree of polarization will be highest for sources viewed at the solar limb, at angles perpendicular to the guiding field. Figure \\ref{bai_ramaty_3} taken from \\cite{BaiRamaty1978} plots, for a given simulation model using an electron distribution with pitch angles uniformly distributed in a cone with a half opening angle of $30^{\\circ}$, the resulting degree of polarization versus solar heliocentric angle. For this model, the highest degree of polarization for low energies (10-20 keV) is at a viewing angle of $90^{\\circ}$, equivalent to viewing the flare at the solar limb. The plot also shows how the sign of the degree of polarization changes from negative to positive at high energies of $\\sim300$ keV. This indicates that the preferred direction of polarization, the polarization angle, changes from being aligned to the local radial direction (the direction along the line connecting the centre of the X-ray source with the solar centre) to the perpendicular to radial direction. This is discussed further in Chapter \\ref{ref:Chapter6}.\n\\begin{figure}\\centering\n\\includegraphics[width=12cm]{bairamaty3.png}\n\\caption[The degree of polarization plotted against X-ray emission angle for different energies.]{Figure taken from \\cite{BaiRamaty1978} showing the degree of polarization against X-ray emission angle for different energies. The sign determines the polarization angle, see Section \\ref{intro_pol}.}\n\\label{bai_ramaty_3}\n\\end{figure}\nMeasuring the spatially integrated polarization angle provides information about the overall geometry of the flaring loop and hence the orientation of the magnetic field \\citep{Emslieetal2008}. If the magnetic field of the loop is tilted away form the local solar vertical direction, then the polarization angle will not lie along the radial direction. The work of \\cite{Emslieetal2008} grew from the non-radial polarization angle observations from \\cite{McConnelletal2003} and the suggestions of \\cite{2003ApJ...595L..81S}. Figure \\ref{emslie_pol} taken from \\cite{Emslieetal2008} plots the azimuthal X-ray emission angle on the solar disk $\\Phi$ against viewing angle $i$ for two different loop tilts $\\tau=0^{\\circ},30^{\\circ}$ for an X-ray energy of $\\epsilon=40$ keV. The direction of each arrow gives the angle of polarization. For a loop tilt of $0^{\\circ}$, the polarization angle is always equal to $0^{\\circ}$ and the arrows lie along the horizontal radial direction. However for the case of a loop tilt of $30^{\\circ}$, the polarization angle can have values of $\\Psi\\ne0^{\\circ}$ depending on the X-ray source position on the solar disk. It will also be shown in Chapter \\ref{ref:Chapter6} that the direction of the polarization angle parallel or perpendicular to the radial direction is dependent upon the highest energies in the electron distribution and the degree of beaming.\n\\begin{figure}\n\\includegraphics[width=15cm]{emslie_pol.png}\n\\caption[The azimuthal X-ray emission angle plotted against the polar X-ray emission angle, showing the changing polarization angle with loop tilt.]{Figure taken from \\cite{Emslieetal2008}. The azimuthal X-ray emission angle $\\Phi$ against polar X-ray emission angle (heliocentric angle) $i$ for an X-ray energy of 40 keV, for two different loop tilt angles $\\tau$ of $0^{\\circ}$ and $30^{\\circ}$. The arrow direction from the horizontal line gives the polarization angle $\\Psi$.}\n\\label{emslie_pol}\n\\end{figure}\nThe {\\it RHESSI} satellite, which is discussed in detail in Section \\ref{intro_rhessi} has limited polarization capabilities. Rare measurements for seven flares \\citep{2006SoPh..239..149S} showed a range of polarization degrees from $0-60\\%$ using an X-ray energy range of $100-350$ keV. \\cite{Boggsetal2006} used {\\it RHESSI} gamma-ray observations to measure the polarization of two flares. They found that the degree of polarization at these higher energies was consistent with a beamed distribution of electrons. However, a number of other observations with {\\it RHESSI} over the last decade suggest that the emitted X-ray distribution, and hence the radiating electron distribution, is close to being isotropic, which should produce a very low level of polarization. One such method that has been used to determine this isotropy is that of X-ray albedo \\citep[e.g.,][]{KontarBrown2006,Kasparovaetal2007,2013SoPh..284..405D}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.6cm]{sg_pol.pdf}\n\\includegraphics[width=7.6cm]{sg_pol_2.pdf}\n\\caption[Solar flare polarization measurements from {\\it RHESSI}.]{Figure taken from \\cite{2006SoPh..239..149S}. {\\it Left:} Solar flare polarization measurements from {\\it RHESSI} for one event. {\\it Right:} the measured degree of polarization for all seven events.}\n\\end{figure}\n\\subsection{X-rays from the photosphere and albedo emission}\\label{intro_albedo1}\nThe corona and chromosphere are optically thin at X-ray and gamma-ray energies. Therefore X-rays emitted at these wavelengths from chromosphere and corona can travel down into the denser layers of the solar atmosphere until they reach the much higher densities of the photosphere. For X-ray photons, the photosphere acts as either an absorber or `mirror', scattering a proportion of the X-rays back into the chromosphere and hence out into interplanetary space. The scattered X-ray flux depends on many factors: the energy of the X-rays, the heliocentric angle of the X-ray emitter in the chromosphere, the height of the X-ray emitter above the photosphere and the proportion of the X-rays emitted towards the photosphere. X-rays scattered from the photosphere are known as the albedo X-rays and the scattering mechanism is Compton scattering. The study of albedo is a major part of this thesis. The first studies of the X-ray photospheric albedo component were examined analytically by \\cite{Tomblin1972} and through simulation by \\cite{Santangeloetal1973}. A more comprehensive study was provided by \\cite{BaiRamaty1978}. The peak of the X-ray albedo flux appears at around 20-50 keV due to: (1) lower energy X-rays below 10 keV are more likely to be photoelectrically absorbed than scattered and (2) higher energy electrons, particularly above 100 keV, are lost within the depths of the photosphere. This produces the well-known albedo reflectivity curve, the ratio of reflected flux to directly emitted primary flux which is shown in the bottom plot of Figure \\ref{intro_albedo}. The scattered albedo flux also varies with X-ray emission angle or equivalently heliocentric angle on the solar disk; X-ray sources at the solar centre have the greatest proportion of albedo photons, while X-ray sources located at the solar limb have the smallest proportion, since the fraction of reflected photons seen by an observer is smaller at the solar limb ($90^{\\circ}$) than at the solar centre ($0^{\\circ}$). The properties inferred from an observed X-ray source will be tainted if it is assumed that the X-ray source consists of only directly emitted X-rays from the chromosphere. The albedo component will change the spectral, spatial and polarization properties of the observed X-ray emission. Some of the spatial and polarization changes are analysed for the first time in Chapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6} of this thesis and are published in \\cite{KontarJeffrey2010} and \\cite{2013ApJ...766...75J}. The known spectral and polarization changes due to photospheric albedo are now briefly reviewed.\n\\begin{figure}\n\\centering\n\\includegraphics[width=15cm]{albedo_pic_4.png}\n\\caption[A cartoon of solar flare X-ray interactions in the photosphere, after being emitted from the chromosphere.]{Cartoon showing the X-rays emitted directly from electrons stopped collisionally in the dense chromospheric thick target. Many of the X-rays are directly emitted away from the solar surface. These are known as the primary X-rays, while some travel down into the photosphere. Here they interact by photoelectric absorption or Compton scattering. Those X-rays Compton backscattered out of the photosphere are known as albedo X-rays from the albedo patch in the photosphere.}\n\\end{figure}\n\\subsubsection{Spectral changes and the determination of the electron distribution}\nAn albedo component produces a `bump' in the photon spectrum which peaks at around 30-50 keV, due to the peak albedo flux at these energies. Although the presence of an albedo component from an X-ray spectrum should always be accounted for, it should be noted that other effects can produce similar flattening at lower energies in the spectrum (10-50 keV) such as return current, count pile-up in the {\\it RHESSI} detectors \\citep[cf.][]{2011SSRv..159..107H} and even wave-particle interactions \\cite{2012A&A...539A..43K}. The size of the `bump' is dependent on the factors mentioned in the previous section and changes in the photon spectrum have been extensively studied through the use of simulations with a known primary X-ray spectrum \\cite[e.g.,][]{BaiRamaty1978}. However, this approach is not so useful for actual observations with instruments such as {\\it RHESSI}, where the properties of the initial primary X-ray distribution are unknown and have to be disentangled from the albedo spectrum. In \\cite{Kontaretal2006}, a Green's function approach was used to try and remove the albedo component from the unknown primary X-ray spectrum by adapting \\cite{MagdziarzZdziarski1995}. The primary distribution $I_{P}$ can be found using,\n\\begin{equation}\\label{I_greens}\nI(\\epsilon)=I_{P}(\\epsilon)+I_{S}(\\epsilon)=I_{P}(\\epsilon)+\\int_{\\epsilon}^{\\epsilon_{max}}I_{P}G(\\mu,\\epsilon,\\epsilon_{0})d\\epsilon_{0}\n\\end{equation}\nwhere $I$ is the total distribution and $I_{S}$ is the scattered albedo distribution, which is the integral over energy of the primary distribution $I_{P}$ and a Green's function $G(\\mu,\\epsilon,\\epsilon_{0})$, where $\\mu,\\epsilon$ and $\\epsilon_{0}$ are the cosine of heliocentric angle, scattered X-ray energy and incoming X-ray energy respectively. Writing the problem discretely, gives,\n\\begin{equation}\\label{I_greens_2}\nI(\\epsilon_{i})=\\left(1_{ij}+\\alpha G_{ij}\\right)I_{P}(\\epsilon_{j}).\n\\end{equation}\nwhere $G_{ij}$ is the Green's matrix, $1_{ij}$ is a diagonal matrix with values of 1 and $\\alpha$ is the anisotropy of the X-ray distribution \\citep{Kontaretal2006}. The best solution of $I_{P}$ can be found numerically and is incorporated into the {\\it RHESSI} OSPEX software via the detector response matrix (as discussed in Section \\ref{intro_rhessi}). Figure \\ref{intro_albedo} shows a RHESSI photon spectrum before and after the correction. The presence of an albedo component should produce the `bump' discussed above in the observed spectrum between $\\sim10-100$ keV, which can be seen in the black solid line in Figure \\ref{intro_albedo}. The albedo corrected spectrum is shown by the black dashed line. Determining the mean electron spectrum from the observed X-ray distribution often produces a strange minimum at $\\sim40$ keV. However, properly accounting for an X-ray albedo component can remove this strange feature \\citep{Kontaretal2006} and clearly indicates the importance of accounting for the albedo component when interpreting information from an observed X-ray spectrum.\n\\begin{figure}\n\\centering\n\\includegraphics[width=15cm]{kontar2006.png}\n\\caption[X-ray albedo reflectivity and an X-ray spectrum with and without an albedo contribution, calculated using a Green's function. The figure also shows a real flare X-ray spectrum observed with {\\it RHESSI} before and after albedo correction.]{Figures taken from \\cite{Kontaretal2006}. {\\it Left:} a photon spectrum calculated from Green's functions and an initial primary photon distribution of $I_{P}(\\epsilon)\\sim\\epsilon^{-3}$, where the black solid line represents the primary spectrum, the black dotted line the albedo spectrum and the black dashed line the total spectrum. The bottom figure plots the reflectivity (flux down\/flux up). The black diamonds plot the reflectivity from \\cite{BaiRamaty1978}. {\\it Right:} The spectrum of a flare that occurred on the 17th September 2002. The observed spectrum is corrected by accounting for the Compton scattered albedo component with the solid and dashed lines showing the spectrum before and after the correction respectively.}\n\\label{intro_albedo}\n\\end{figure}\n\\subsubsection{Changes in total polarization}\nThe albedo component will change the measured polarization of an X-ray source. For the case of a completely isotropic non-thermal X-ray distribution, the total degree of polarization will be zero for an X-ray source viewed at all solar heliocentric angles. However the degree of polarization for the scattered albedo component will {\\it increase} with emission angle, as the observer views more and more X-rays scattered at angles closer to $90^{\\circ}$. The polarization of the scattered contribution is hence $0\\%$ for an X-ray source at the solar centre and becomes $\\sim20\\%$ for an isotropic distribution at the solar limb. This is shown in Figure \\ref{br_pol} taken from \\cite{BaiRamaty1978}. However, this will only produce a degree of polarization of $\\sim4\\%$ at the solar limb due to the much higher flux of the primary X-ray component at large heliocentric angles. Larger changes in polarization are possible for more anisotropic X-ray distributions. In contrast, the backscattered albedo X-rays generally act to reduce the polarization of an anisotropic source \\citep{Henoux1975, LangerPetrosian1977,BaiRamaty1978}. This is discussed in greater detail in Chapter \\ref{ref:Chapter6}, where the changes in spatially resolvable polarization due to albedo are discussed for the first time in solar physics.\n\\begin{figure}\n  \\begin{minipage}[c]{0.55\\textwidth}\n    \\includegraphics[width=\\textwidth]{bairamaty.png}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.40\\textwidth}\n\\caption[The varying magnitude of polarization for a completely isotropic source viewed at different locations on the solar disk due to the presence of a photospheric backscattered albedo component.]{Figure taken from \\cite{BaiRamaty1978} showing how the degree of polarization changes for a completely isotropic source viewed at different locations on the solar disk due to the presence of a photospheric backscattered albedo component.}\n\\label{br_pol}\n\\end{minipage}\n\\end{figure}\n\\subsubsection{Determining the electron anisotropy using albedo}\n\\cite{KontarBrown2006} used a technique to effectively separate the `upward' and `downward albedo' components of X-ray flux that contribute to the measured X-ray spectrum and hence electron spectrum. Determining the X-ray albedo flux allows the anisotropy of a single flare to be measured, using a single instrument such as {\\it RHESSI}. The results of this study determined that both flares observed had extremely isotropic distributions, a result that is not expected if the distribution of electrons is beamed; an assumption often made in a standard flare collisional thick target model. Follow on studies by \\cite{2013SoPh..284..405D} performed a larger examination of eight events and again found a lack of anisotropy for each event below $\\sim150$ keV. An example of one of the flares examined in \\cite{2013SoPh..284..405D} is shown in Figure \\ref{ed_figures}. A centre-to-limb statistical survey of 398 flare spectral indices in three different energy bands was performed by \\cite{Kasparovaetal2007}. They found that there was a clear change in spectral index in the low energy $15-20$ keV band, with the spectral index increasing towards the limb. This is consistent with the presence of an albedo component flattening the X-ray spectrum at low energies. Further, \\cite{Kasparovaetal2007} determined the directivity of the X-ray emission for a number of their flares, using the albedo Green's function method of \\cite{Kontaretal2006}. From this the ratio of downward to upward flux was found to lie anywhere between $0.2$ and $5$ in the $15-20$ keV band (see Figure \\ref{ed_figures} (right)). Hence this study gave no clear conclusion regarding the X-ray anisotropy, that is, the results are consistent with the predictions of a beamed downward distribution of electrons {\\it and} with an isotropic distribution.\n\\begin{figure}\n \n    \\includegraphics[width=7 cm]{ed_pic_1.pdf}\n    \\includegraphics[width=8 cm]{kkb_intro.pdf}\n \n \n\\caption[Measured X-ray anisotropy from {\\it RHESSI} observations using two different methods that take advantage of the X-ray albedo component.]{{\\it Left:} Figure taken from \\cite{2013SoPh..284..405D} showing the anisotropy of the electron spectrum (Flux down \/Flux up) for one flare where dark grey represents a 1-sigma and light grey a 3-sigma, confidence interval. {\\it Right:} Figure taken from \\cite{Kasparovaetal2007} showing determined directivity $\\alpha=$Flux down\/Flux up for different heliocentric angles $\\mu$ for an energy range of $15-20$ keV.}\n\\label{ed_figures}\n\\end{figure}\n\\section{Current X-ray telescopes and X-ray imaging methods}\\label{intro_rhessi}\nThe X-ray observations shown in Chapter \\ref{ref:Chapter3} of this thesis and some of the simulation work shown in Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter5} are concerned with the current solar X-ray and gamma ray imaging spectroscopy performed by the {\\it Ramaty High Energy Solar Spectroscopic Imager, RHESSI} \\citep{2002SoPh..210....3L} . Hence, here in Chapter \\ref{ref:Chapter1}, a brief review of the instrument is given.\n\\subsection{RHESSI: instrument overview}\n{\\it RHESSI} is a NASA-led mission, launched in February 2002. For more than a decade, it has provided unparalleled hard X-ray observations of the Sun and solar flares in particular. {\\it RHESSI} observes the full disk of the Sun from a low Earth orbit over the energies of $3$ keV to $17$ MeV. {\\it RHESSI} performs imaging spectroscopy of X-rays and gamma-rays and it was designed specifically to study particle acceleration and energy release in solar flares. The {\\it RHESSI} instrument consists of a spectrometer; nine cooled Germanium detectors placed at the rear of the spacecraft, with additional imaging apparatus consisting of nine pairs of widely spaced grids at a distance of $1.5$ m called rotating modulation collimators (RMCs) placed in front of each detector. The instrument setup is shown in Figure \\ref{fig:rhessi_1}.\n\\begin{figure}\n\\includegraphics[width=15cm]{rhessi_1.png}\n\\caption[Diagram of the {\\it RHESSI} grids and detectors.]{Diagram of {\\it RHESSI} grids and detectors. This image was taken from \\cite{2002SoPh..210...61H}.}\n\\label{fig:rhessi_1}\n\\end{figure}\n\\subsection{RHESSI imaging}\\label{rhessi_imaging}\nDue to their high energies, X-rays above $1$ keV are difficult to image directly, although astrophysical missions such as {\\it The Nuclear Spectroscopic Telescope Array} NuSTAR \\citep{2013ApJ...770..103H} and recent balloons missions: {\\it The Focusing Optics X-ray Solar Imager} FOXSI \\citep[e.g.][]{2011SPIE.8147E...4K} and {\\it The High Energy Replicated Optics to Explore the Sun} HEROES \\citep[e.g.][]{2013SPD....44...76C} are pioneering the technology and techniques required for future direct X-ray imaging in solar physics. {\\it RHESSI} on the other hand, creates X-ray images via a non-direct Fourier technique using its RMCs. {\\it RHESSI} rotates continuously around its axis pointing towards the Sun. As the spacecraft rotates, the incoming X-ray signal passes through the slits and slats of its nine RMCs, with the slits and slats either impeding or allowing the X-rays path to the detectors. This produces a time modulated signal, that is dependent upon the position and size of the X-ray source \\cite[cf.,][]{2002SoPh..210...61H}. Figure \\ref{fig:rhessi_2} (left) shows a diagram of the X-rays passing through the front and rear grids of one RMC and onto a detector while Figure \\ref{fig:rhessi_2} (right) shows examples of different modulation patterns, and how the pattern varies with X-ray source position and size. The time modulated signal can be stacked per roll bin (fraction of a spacecraft rotation) over so many spacecraft rotations, creating {\\it X-ray visibilities}. The X-ray visibilities $V$ are the two-dimensional Fourier components of the X-ray source in $uv$ space, given by,\n\\begin{equation}\\label{rhessi_vis}\nV(u,v;\\epsilon)=\\int_{x}\\int_{y}I(x,y;\\epsilon)e^{2\\pi i(xu+yv)}dxdy.\n\\end{equation}\n\\begin{figure}\n\\hspace{-15pt}\n\\includegraphics[width=16.2cm]{rhessi_2a.png}\n\\caption[An example of a photon entering a {\\it RHESSI} RMC and {\\it RHESSI} time modulation curves.]{{\\it Left:} Diagram showing how photons entering a {\\it RHESSI} RMC at a given time, during a spacecraft rotation, are either blocked by the slats of a grid or travel towards the detector, depending on their incident position. {\\it Right:} An example of {\\it RHESSI} time modulation curves for different simulated sources sitting on the solar disk. Both figures were taken from \\cite{2002SoPh..210...61H}.}\n\\label{fig:rhessi_2}\n\\end{figure}\nThe inverse Fourier transform of Equation \\ref{rhessi_vis} gives the X-ray image $I$ at a given energy $\\epsilon$ in the real $xy$ plane. The X-ray visibilities are represented in $uv$ space by circles of constant radius, with each circle representing the angular resolution of RMC 1-9. This is shown for RMCs 3-9 in Figure \\ref{fig:rhessi_uv_plane}. The grids of RMC 1 have the finest spatial resolution of 2.26 arc second and the grids of RMC 9 have the coarsest spatial resolution of 183.2 arc second, where the spatial resolution of each RMC increases by factor $\\sqrt{3}$, with increasing number from 1 to 9. The creation of an image from Equation \\ref{rhessi_vis} or directly from the time modulated signal is an inverse problem and performing a two-dimensional Fourier transform produces a Back projection image \\cite{1986JOSAA...3.2167M}. However this method produces a poor image quality with side lobes and hence a number of different imaging algorithms have been created, or adapted from radio astronomy in order to solve the problem and improve the image, such as CLEAN \\citep{1974A&AS...15..417H, 2002SoPh..210...61H}, Pixon \\citep{1993PASP..105..630P,1996ApJ...466..585M} and forward fitting algorithms such as Visibility Forward Fitting (Vis FwdFit) \\citep{2002SoPh..210...61H,2007SoPh..240..241S}. The CLEAN, Pixon and Vis FwdFit imaging algorithms are used in this thesis and hence are briefly discussed:\n\\begin{enumerate}\n\\item{CLEAN - this algorithm assumes that the X-ray image consists of a superposition of many X-ray point sources. The process of CLEAN-ing is an iterative process that continuously searches for the highest intensity pixel in the image. At each iteration, once the highest pixel is found, a chosen proportion of the highest intensity, convolved with the Point Spread Function (PSF) of the instrument, is centred at the highest pixel and subtracted from the image. This process usually repeats for a chosen number of iterations or until the peak flux in the image is negative. The resulting final image is a CLEANed map consisting of the positions and amplitudes of each chosen pixel at each iteration, convolved with the PSF.}\n\\item{Pixon - this algorithm wants to construct the simplest model for the image that is consistent with the data \\citep{2002SoPh..210...61H}. Pixon uses different size pixels or `pixons' together to try and reproduce the X-ray modulation patterns and aims to use the least number of pixons to achieve this and reconstruct the image.}\n\\item{Vis FwdFit - imaging methods such as CLEAN or Pixon try to find a solution to Equation \\ref{rhessi_vis} and often errors in the reconstructed images occur due to problems such as finite coverage in Fourier space. This can make estimating X-ray source spatial parameters difficult using CLEAN and Pixon \\citep*{2010ApJ...717..250K}. Therefore, in certain situations, such as estimating X-ray source spatial parameters, it may be more helpful to use forward fitting methods. One such method is Vis FwdFit. This algorithm works by taking one of three simple shapes: a circular Gaussian, an elliptical Gaussian or a curved elliptical Gaussian, and matches the chosen shape with the X-ray visibilities given by Equation \\ref{rhessi_vis}. Unlike other imaging algorithms, if a good comparison between the chosen model and the actual X-ray visibilities is achieved then the spatial properties of the X-ray source can be determined through the moments of the chosen Gaussian model: the position from the first moment and the spatial extent from the second moment. Another major advantage of this type of algorithm is the estimation of errors for each X-ray source spatial parameter, which can be found by propagating the error associated with the difference between the model and actual X-ray visibilities. The major drawback of this type of forward fitting method is the lack of source shapes, and hence only a limited number of X-ray sources that match well with one of the three simple Gaussian shapes, should ever by analysed using this method. However, the use of Vis FwdFit has, in the last couple of years, allowed the spatial properties of both chromospheric and coronal sources to be estimated, which is much harder with algorithms such as CLEAN and Pixon.}\n\\end{enumerate}\nVis FwdFit is discussed further in Chapters \\ref{ref:Chapter2}, \\ref{ref:Chapter3} and \\ref{ref:Chapter5} of this thesis. An example of two CLEANed images is shown in Figure \\ref{footpointsandloop}: one for HXR chromospheric footpoints at the limb and another of an X-ray coronal source, also at the limb. Over-plotted are Vis FwdFIt contours; an elliptical Gaussian was fitted to a HXR footpoint and a curved elliptical Gaussian was fitted to the coronal source. The parameters found from each fit are also plotted in the figure.\n\\begin{figure}\n  \\begin{minipage}[c]{0.60\\textwidth}\n    \\includegraphics[width=\\textwidth]{rhessi_uv_plane.png}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[c]{0.30\\textwidth}\n\\caption[Diagram of the {\\it RHESSI} $uv$ plane.]{Diagram of the {\\it RHESSI} $uv$ plane. Detector 9 with the largest angular resolution produces the smallest circle. Only detectors 3-9 are shown in this diagram. Image taken and adapted from \\cite{2009ApJ...703.2004M}.}\n\\label{fig:rhessi_uv_plane}\n\\end{minipage}\n\\end{figure}\n\\subsection{RHESSI spectroscopy and polarimetry}\nThe nine cooled ($<75$ K) Germanium detectors make up the {\\it RHESSI} spectrometer. The spectrometer has an energy resolution of $\\le$ $1$ keV at $3$ keV and this increases to $\\sim5$ keV at $5$ MeV \\citep{2002SoPh..210....3L}. Each Ge detector consists of two segments: front and rear. The front segments can absorb photons up to around $\\sim250$ keV and the rear segments up to $\\sim17$ MeV. The photons that reach the detectors and cause a small current are registered as counts. Therefore, {\\it RHESSI} produces a {\\it count rate spectrum} and this is related to the {\\it X-ray photon spectrum} via,\n\\begin{equation}\\label{crs}\n\\textit{\\textbf{C=B+SRM\\;I}}\n\\end{equation}\nwhere $\\textit{\\textbf{C}}$ is the count rate spectrum, $\\textit{\\textbf{B}}$ is the background, $\\textit{\\textbf{SRM}}$ is called the spectrometer or detector response matrix and $\\textit{\\textbf{I}}$ is the photon rate spectrum. {\\it RHESSI} is an unshielded spacecraft and hence any background counts due to high energy cosmic rays and trapped high energy electrons and protons in the Earth's Van Allen belt, for example must be removed before a detailed spectroscopy of a flare is performed. The detector response matrix $\\textit{\\textbf{SRM}}$ accounts for the many effects that could modify the input count spectrum such as photons being Compton scattered in multiple detectors or detector radiation damage. If an effect only changes the efficiency of the instrument to detect a photon at its proper energy, then it contributes to the diagonal elements of the detector response matrix. If an effect changes the value of photon energy from its true value, then the effect contributes to the off-diagonal elements of the matrix \\citep[cf.][]{2002SoPh..210...33S}. Each detector has two shutters or `attenuators'. The job of the attenuators is to stop the detectors saturating at high X-ray count rates, say, during a large solar flare. There are two different shutters: a thin and a thick shutter, and there are three attenuation states or combinations: A0 - no shutters, A1- thin shutter only and A3 - thin and thick shutter. The spectral analysis of a solar flare is performed in software called OSPEX \\citep{2002SoPh..210..165S}, where different functions can be fitted to the data. {\\it RHESSI} spectroscopy is performed in Chapter \\ref{ref:Chapter3} and the functions fitted to the data are discussed there, for the flares analysed.\n\n{\\it RHESSI} is also capable of measuring the linear polarization of solar flare HXRs \\citep{McConnelletal2002}. For this purpose from $20-100$ keV, {\\it RHESSI} has a beryllium (Be) scatterer. {\\it RHESSI} polarimetry is achievable since, as described above, the Ge detectors consist of two parts: a front and a rear segment. Photons in the range of $20-100$ keV should not be able to reach the rear segments and hence any detection of these low energy photons in the Ge rear segments must be due to firstly Compton scattering in the Be scatterer. The magnitude and direction of polarization is then found by measuring the scattered photon count rates in each rear segment of the detector. As mentioned in Section \\ref{intro_pol}, {\\it RHESSI} is also capable of measuring the polarization of higher energies above $100$ keV. For this, the same method is used, as described, except that the possible scattering between different Ge detectors is utilised at higher energies, instead of scattering from the Be scatterer. The simulated polarization measurements described in Chapter \\ref{ref:Chapter6} are not currently possible with {\\it RHESSI} or other dedicated astrophysical polarimeters, and hence Chapter \\ref{ref:Chapter6} looks towards future instruments with imaging polarimetry capabilities.\n\n\\chapter{The variation of solar flare coronal X-ray source sizes with energy}\n\\label{ref:Chapter2}\n\n\\normalsize{\\it This work can be found in the publication \\cite{2014ApJ...787...86J}}\n\n\\section{Introduction to the chapter}\\label{intro}\nChapter \\ref{ref:Chapter1} discussed how, during a solar flare, the surrounding plasma is heated to tens of mega-Kelvin and electrons are accelerated to deka-keV energies and beyond. In a simple model, electrons travel through a tenuous corona and deposit energy into a dense chromospheric `thick target' via Coulomb collisions, with only a small fraction ($\\sim 10^{-5}$) of the energy emitted as bremsstrahlung hard X-rays, mostly at the dense chromospheric footpoints.  Hard X-rays emitted from the corona are usually interpreted as predominantly thermal bremsstrahlung from a hot coronal plasma or as a combination of thermal and thin-target emissions.\n\nOver the last decade, the Ramaty High Energy Solar Spectroscopic Imager \\citep[{\\em RHESSI};][]{2002SoPh..210....3L}\nhas provided unprecedented imaging spectroscopy observations of both chromospheric and coronal X-ray sources\n\\citep[for recent reviews of this topic see][]{2011SSRv..159..107H,2011SSRv..159..301K}.\nChapter \\ref{ref:Chapter1} discussed how the design of the {\\em RHESSI} instrument is such that spatial information is fundamentally encoded as two-dimensional\nFourier transforms, or {\\it visibilities}.  The subsequent development of sophisticated and reliable visibility-based image reconstruction algorithms, such as visibility forward\nfitting \\citep{2002SoPh..210...61H,2007SoPh..240..241S}\nand \\textit{uv\\_smooth} \\citep{2009ApJ...703.2004M}, coupled with the use of {\\it electron visibilities}, which are spectral inversions of the count visibility data provided by {\\em RHESSI } \\citep{2007ApJ...665..846P}, have\nallowed the quantitative analysis of solar hard X-ray sources in both photon and electron space.\n\n{\\em RHESSI} observations have revealed the morphology details of flares with high plasma density \\citep[e.g.,][]{1980ApJ...238L..43M,1981A&A....97..210C,1994ApJ...424..444F}, in which the bulk of the hard X-rays come from the corona, with only very weak or no footpoint emission from the chromosphere \\citep[e.g.,][]{2004ApJ...603L.117V,2004ApJ...612..546S,2007ApJ...666.1256B,2008ApJ...673..576X,2013ApJ...769L..11L}.\nThe behaviour of the source extent with energy is not consistent with a thermal source characterised by a temperature distribution with a peak at the loop-apex, since for such a source the source size should decrease with energy.  Rather, the X-ray source extent {\\it grows} with energy \\citep{2008ApJ...673..576X}, indicative of a non-thermal model in which the propagation distance increases with energy. Apparently, the density within the coronal region in such sources is high enough to stop electrons prior to reaching the chromosphere; the source is a coronal ``thick target''.\n\nStudying these events is particularly valuable since: (1) the coronal X-ray component and hence acceleration region can be studied without contamination from an intense chromospheric source; and (2) such sources exhibit trends in source extent with energy \\citep{2008ApJ...673..576X, 2011ApJ...730L..22K, 2012ApJ...755...32G,2013ApJ...766...28G}\nand time \\citep[and Chapter \\ref{ref:Chapter3}]{2013ApJ...766...75J}, which can be used to study particle acceleration and transport processes \\citep[e.g.,][]{2012SoPh..277..299G,2013SoPh..284..489G}.  Further, unlike footpoint-dominated solar flares \\citep[e.g.,][]{1982SoPh...78..107A,1982SoPh...81..137D,1995PASJ...47..355T,1996AdSpR..17...67S,1999ApJ...527..945P,2003ApJ...595L.107E,2004A&A...415..377M,2007A&A...461..315T,2011A&A...533L...2B,2011ApJ...731L..19F,2013ApJ...777...33C},\nthe HXR spectra of such ``coronal thick target sources'' tends to be softer than, and the sources higher than, chromospheric sources, which generally reduces the albedo contribution to X-ray images \\citep[and see Chapter \\ref{ref:Chapter5}]{KontarJeffrey2010}, making the interpretation of the spectro-spatial structure of such sources more straightforward.\n\nObservations of compact coronal non-thermal hard X-ray sources typically show that the extent of the source parallel to the guiding magnetic field increases approximately quadratically with photon energy. Since the collisional stopping distance of an electron in a plasma also increases quadratically with particle energy, \\cite{2008ApJ...673..576X} explained this behaviour in terms of an extended acceleration region, from which accelerated electrons emerge and subsequently undergo collisional transport in a background medium of uniform density.  As shown by \\citet{2008AIPC.1039....3E}, application of such a model allows parameters such as the number density $n$ of the region and the specific electron acceleration rate $\\eta$ (electrons~s$^{-1}$ per ambient electron) to be estimated.\n\nHowever, the simple one-dimensional cold target approximation used by these authors is not completely realistic, for two main reasons. Firstly, it assumes that the injected electron trajectories are completely aligned with the guiding magnetic field, and it does not take into account pitch angle scattering (collisional or otherwise) of the accelerated electrons in the target. Secondly, it neglects effects associated with the finite temperature of the ambient medium; electrons with energies comparable to the thermal energy of the plasma $\\sim k_BT$, where $k_{B}$ is the Boltzmann constant and $T$ is the temperature, are just as likely to gain\nas lose energy during a collision, unlike the monotonic energy loss experienced by suprathermal electrons interacting with a cold plasma \\citep[e.g.,][]{1978ApJ...224..241E}.  Even for electrons that do lose energy, they do so at a rate that is not the same as in a cold target, so that a quadratic behaviour of source extent with energy is not necessarily expected.\n\n\\cite{2003ApJ...595L.119E} and \\cite{2005A&A...438.1107G} investigated analytically the effects of a finite target temperature, and both found that the associated velocity diffusion cannot be neglected when interpreting the results of flare hard X-ray spectra. \\cite{2003ApJ...595L.119E} found that, because of the reduced energy losses suffered by accelerated electrons in  warm target, the inferred energy content of the injected electron distribution was significantly reduced. Indeed, he showed that allowance for this effect obviated the need to introduce a low-energy cutoff in the electron distribution. \\cite{2005A&A...438.1107G} found that changes occurring close to the thermal energy of the plasma meant that many flare X-ray spectra may not be well fitted by a simple isothermal-plus-power-law model as discussed in Chapter \\ref{ref:Chapter1}.\n\nThe motivation of Chapter \\ref{ref:Chapter2} is to incorporate the effects of pitch angle scattering and finite target temperature in models of the variation of X-ray source size with electron energy.  How the inclusion of each of these processes changes the behaviour of the variation of source extent with electron energy and the estimation of parameters such as number density $n$ and acceleration region length $L_{0}$, is investigated. The conclusion (Section \\ref{ref:chap2_con}) also briefly discusses how inferred parameters such as the filling factor $f$ and specific electron acceleration rate $\\eta$ will change.\n\n\\section{Electron collisional transport in a cold plasma}\\label{cold_theory}\n\nFirstly electron transport within a uniform cold target is briefly reviewed. Here the electron energy $E>>k_BT$ where $k_B$ is Boltzmann's constant and $T$ is the target temperature.\nThe variation of energy $E$ [erg] with position $z$ [cm] in such a model is given by \\citep[cf.][]{Brown1972,1978ApJ...224..241E}\n\n\\begin{equation}\\label{eq:et}\nE(E_{0},z)=\\sqrt{E_{0}^{2}-2KN(z)}\n=\\sqrt{E_{0}^{2}-2Kn \\, |z-z_{0}|} \\,\\,\\, ,\n\\end{equation}\nwhere $z_{0}$ is the (single) point of injection, $K=2\\pi e^{4} \\ln\\Lambda$ (where $e$ [esu] is the electron charge and $\\ln\\Lambda$ the Coulomb logarithm), and $N$ and $n$ are the column density [cm$^{-2}$] along the trajectory and ambient number density [cm$^{-3}$], respectively.\n\nThis expression allows the stopping position $L_{S}$ of an electron of initial energy $E_{0}$ within a plasma of density $n$ to be found \\citep[cf.][]{2002SoPh..210..373B}, viz.\n\n\\begin{equation}\\label{eq: sd2}\nL_{S}=z_{0}+\\frac{E_{0}^{2}}{2Kn} \\,\\,\\, .\n\\end{equation}\n\nUsing Equation~(\\ref{eq:et}) and the one-dimensional continuity equation, the form of the electron spectrum $F(E,z)$ [electrons s$^{-1}$ cm$^{-2}$ erg$^{-1}$] as a function of position $z$ in the target can also be obtained:\n\n\\begin{equation}\\label{eq: fex}\nF(E,z)=F_{0}(E_0) \\, \\frac{dE_{0}}{dE} = F_{0}(E_0) \\, \\frac{E}{E_0}\n= F_{0}(E_0[E,z]) \\, E(E^{2}+2Kn|z-z_{0}|)^{-\\frac{1}{2}} \\,\\,\\, .\n\\end{equation}\n\nSetting $z_{0}=0$ for simplicity and assuming a power-law injection spectrum $F_{0}(E_0) \\propto E_{0}^{-\\delta}$ gives,\n\\begin{equation}\\label{eq:fex1}\nF(E,z)=E(E^{2}+2Kn|z|)^{-\\frac{(\\delta+1)}{2}} \\,\\,\\,,\n\\end{equation}\nand an expression for the source extent as the square root of the variance $var(E)$ can be derived\n\n\\begin{equation}\\label{eq: varfex1}\nvar(E)=\\frac{\\int_{0}^{\\infty} z^2 \\, F(E,z) \\, dz}\n{\\int_{0}^{\\infty} F(E,z) \\, dz}=\n\\frac{\\int_{0}^{\\infty} z^2 \\, (E^{2}+2Kn|z|)^{-(\\delta+1)\/2} \\, dz}\n{\\int_{0}^{\\infty} (E^{2}+2Kn|z|)^{-(\\delta+1)\/2} \\, dz} \\,\\,\\, ,\n\\end{equation}\nwhere the symmetry about $z=0$ has been used.  Evaluating the integrals gives\n\n\\begin{equation}\\label{eq:stdfex}\nstd(E)=\\sqrt{var(E)}\n=\\frac{1}{2Kn} \\, \\sqrt{\\frac{8}{(\\delta-3)(\\delta-5)}} \\, E^{2} \\,\\,\\,\n\\end{equation}\nwhere $std(E)$ is the standard deviation. The spatial extent at a given energy $E$ depends on the spectral index $\\delta$; for $\\delta=7$ the form of the stopping distance is obtained, $std(E)=L_{s}$ given by Equation~(\\ref{eq: sd2}). It should be noted that Equation (\\ref{eq: varfex1}), and hence the spatial extent defined by Equation~(\\ref{eq:stdfex}), is applicable only for $\\delta >5$; for $\\delta \\leq 5$, the integral on the numerator diverges at the upper limit. This is related to the fact that the collisional stopping length is an increasing function of energy $\\propto E^2$, so that large energies give the largest contribution to the integral for $\\delta \\leq 5$.  This issue can be formally avoided by imposing an upper energy cut-off $E_{max}$ to $F_0(E_0)$, so that the upper limit in the integral (\\ref{eq: varfex1}) is finite, given by $E_{max}^2\/2Kn$.\n\nEquations \\ref{eq: varfex1} and \\ref{eq:stdfex} assumed the initial electron distribution was injected as a point source at $z=z_{0}=0$. However, if the initial electron distribution is injected over a finite region, with the injected flux profile having the form of (say) a Gaussian distribution with standard deviation $d$, then the equation  for $F(E,z)$ becomes \\citep[see, e.g.,][]{2014ApJ...780..176K}\n\\begin{figure*}\n\\label{fig: fex_num_g_2}\n\\centering\n\\vspace{-20pt}\n\\includegraphics[width=15.5cm]{fig1a.pdf}\n\\includegraphics[width=15.5cm]{fig1b.pdf}\n\\caption[The standard deviation and FWHM plotted against electron energy for a point source and a source of Gaussian standard deviation $d=10\\mbox{\\ensuremath{^{\\prime\\prime}}}$.]{{\\it Top panels}: the standard deviation $\\sigma$ calculated for a point source (left) and a source of Gaussian standard deviation $d=10\\mbox{\\ensuremath{^{\\prime\\prime}}}$ (right), using the moment-based Equation~(\\ref{eq: varfex1}) and the distribution of electron flux with energy and position given by Equation~(\\ref{eq:fexgauss}), for a target density $n=1\\times10^{11}$~cm$^{-3}$. For the point source, the curves calculated using Equation~(\\ref{eq:stdfex}) for $\\delta=6-9$ and a maximum injected energy of $30$~keV are over-plotted as dashed lines of the same colour. {\\it Bottom panels}: Gaussian FWHM calculated by fitting Gaussian curves to $F(E,z)$ for a point source (left) and a $10''$ source (right). Equation~(\\ref{L}) is fitted to each curve:  the corresponding values of $L_0$ and $\\alpha$ are shown on each panel. The curve FWHM $=2 \\, \\sqrt{2 \\ln 2} \\, d + E^2\/2Kn$ \\citep[black dashed curve; used by previous authors - e.g., ][]{2011ApJ...730L..22K} is overplotted.}\n\\label{fex_good}\n\\end{figure*}\n\\begin{equation}F(E,z)\\label{eq:fexgauss}\n\\sim \\frac{1}{d\\sqrt{2\\pi}} \\, \\int_{-\\infty}^{\\infty} E \\, \\left(E^{2}+2Kn \\, \\vert z-z^{'}\\vert \\right)^{-(\\delta+1)\/2}\n\\exp{\\left(-\\frac{{z^{'}}^2}{2d^{2}}\\right)} \\, dz^{'} \\,\\,\\, .\n\\end{equation}\\\\\nFor this case, the evaluation of $F(E,z)$ and the corresponding standard deviation $std(E)$ cannot be evaluated analytically, and hence is calculated numerically. Figure~\\ref{fex_good} (top) shows the numerical results for $std(E)$ for $\\delta=4,5,6,7,8$ and $9$ using the initial source sizes of $d=0\\mbox{\\ensuremath{^{\\prime\\prime}}}\\;\\mbox{and}\\;10\\mbox{\\ensuremath{^{\\prime\\prime}}}$ and a number density $n=1\\times10^{11}$~cm$^{-3}$. For the $d=0\\mbox{\\ensuremath{^{\\prime\\prime}}}$ case, and for cases with $\\delta > 5$ (cf. Equation~(\\ref{eq:stdfex})), the $std(E)$ results calculated from the point-injection case~(Equation~(\\ref{eq:stdfex})) are over-plotted for comparison and match well with the numerically calculated curves as expected.\n\nThe form of the spatially resolved spectrum $F(z)$ at a given energy $E$ at distances further away from the peak, where $F(E,z) \\sim 0.15\\max(F(E))$ (cf. Equation~(\\ref{eq:fexgauss})) is not well determined by the {\\em RHESSI} observations. Since {\\em RHESSI} data is created as two-dimensional spatial Fourier transforms or X-ray visibilities (Chapter \\ref{ref:Chapter1}, Section \\ref{intro_rhessi}), the source sizes in practice are determined by fitting a Gaussian-like shape to the observed visibilities. Due to this indirect imaging approach and the finite dynamic range of the instrument, the brightest part of the image is the most reliable. Thus, $std(E)$ is calculated not through a moment-based approach, but rather through a shape-based analysis that focuses  on the high-intensity ``core'' of the spatial distribution of flux at a given electron energy $E$.\n\nTherefore, Gaussian curves are fitted to $F(E,z)$ in order to determine the Gaussian standard deviation $std_{G}(E)$ at each energy. This allows the calculation of the Gaussian Full Width at Half Maximum FWHM$=2 \\, \\sqrt{2 \\ln 2 \\,var_{G}(E)}$.\nThe curves for $\\delta= $4, 7, and 9 are plotted in the bottom panels of Figure~\\ref{fex_good}. In general, and as expected, the $std_{G}(E)$ values deduced from the shape of the core of the $F(E,z)$ profile are smaller than the $std_{G}(E)$ values deduced from the moment-based analysis.\n\nEach curve in Figure~\\ref{fex_good} (bottom) was then fitted with an equation of the form\n\n\\begin{equation}\n\\mbox{FWHM}(E)=L_{0}+\\alpha E^{2}\n\\label{L}\n\\end{equation}\nand the values of $L_{0}$ and $\\alpha$ are shown on each plot. In the bottom panels of Figure~\\ref{fex_good},  FWHM$=2 \\, \\sqrt{2 \\ln 2} \\, d + E^2\/2Kn$ is also over-plotted for comparison, since this simple approximation (basically the stopping distance approximation) has been\nused \\citep[e.g.,][]{2011ApJ...730L..22K,2012A&A...543A..53G} to infer information from observations; it is given by the black dashed curve.\n\nFrom Figure~\\ref{fex_good}, two main points are noted:\n\n\\begin{enumerate}\n\\item{For a given energy $E$, $std(E)$ decreases with increasing spectral index $\\delta$.  This is because as $\\delta$ increases there are a lower proportion of higher energy electrons in the overall electron distribution.  The lower energy electrons that are representative of steeper spectra travel a smaller distance through the plasma.}\n\\item{For a given spectral index $\\delta$, the value of the quadratic coefficient $\\alpha$ decreases somewhat with source size $d$. This is because of the increased contribution of the acceleration region to the overall source extent; the ``propagation'' region is to a large extent contained within the acceleration region itself.}\n\\end{enumerate}\nObservationally, $L_{0}$ has been used to infer the size of the acceleration region, while $\\alpha\\propto 1\/n$ has allowed the\nnumber density of the propagation region to be inferred. This is assumed to be the same as the density of the acceleration region. Using the simplest one-dimensional cold plasma approximation ($\\alpha=1\/2Kn$), $n$ can be inferred easily. However, from Figure~\\ref{fex_good}, it can be seen that, in general $\\alpha=\\mathcal{B}\/2Kn$, where the value of the dimensionless number $\\mathcal{B}$ and hence the number density $n$, depends upon the properties of both the acceleration region and the electron distribution.\n\nFurther, Equations~(\\ref{eq:stdfex}) and~(\\ref{eq:fexgauss}) do not account for three important processes expected to occur within a real flaring coronal plasma:\n\\begin{enumerate}\n\\item {a finite range of pitch angles in the injected pitch angle distribution,}\n\\item{any form of pitch angle scattering (collisional or non-collisional) within the target, and}\n\\item{the finite temperature of the plasma through which the electrons travel.}\n\\end{enumerate}\nAll of these physically important effects impact the form of $E(E_0,z)$, the variation of electron energy with position in the source, and incorporating them will thus change the resulting forms of $std(E)$ and FWHM$(E)$, in a manner which will now be investigated in the following sections of this chapter.\n\n\\section{Electron transport in a hot plasma with collisional pitch angle scattering}\n\n\\subsection{The Fokker-Planck Equation and coefficients}\n\nIn order to describe the transport of electrons through a coronal plasma of finite temperature $T$, accounting for collisional pitch angle scattering, a Fokker-Planck type equation can be used, as was briefly mentioned in Chapter \\ref{ref:Chapter1}. For the purposes required in this investigation, a three-dimensional form from e.g., \\cite{1981phki.book.....L,1986CoPhR...4..183K} in spherical coordinates is used. Assuming azimuthal symmetry and adding a source term for electrons $S$, this is given by\n\n\\begin{equation}\\label{eq:fp}\n\\frac{df(v,z,\\beta,t)}{dt}=\\frac{\\partial f}{\\partial t}+v\\cos\\beta \\frac{\\partial f}{\\partial z}=-\\frac{1}{v^{2}} \\, \\frac{\\partial}{\\partial v}\n\\left(v^{2} \\, J_{v}\\right) - \\frac{1}{v\\sin\\beta}\\frac{\\partial}{\\partial \\beta}\n\\left(\\sin\\beta \\, J_{\\beta}\\right) +S(v,z,\\beta,t),\n\\end{equation}\nwhere $f(v,z,\\beta,t)$ is the phase-space distribution function [electrons~cm$^{-3}$~{[cm~s$^{-1}$]}$^{-3}$], $v$ [cm~s$^{-1}$] is the total particle speed, $\\beta$ is the particle pitch angle to the guiding magnetic-field (along the direction $z$ [cm]), $t$ is time [s] and $J_{v}$ and $J_{\\beta}$ are given by\n\n\\begin{equation}\\label{eq:SS}\nJ_{v}=-D_{vv} \\, \\frac{\\partial f}{\\partial v} + F_{v} \\, f\n \\,\\,\\, , \\qquad J_{\\beta}=-D_{\\beta\\beta} \\, \\frac{1}{v} \\, \\frac{\\partial f}{\\partial \\beta} \\,\\,\\, .\n\\end{equation}\nHere $D_{vv}$ and $D_{\\beta\\beta}$ are the velocity and pitch angle diffusion terms while $F_{v}$ is the velocity collisional friction term. These three terms are respectively given by\n\n\\begin{eqnarray}\\label{eq:dcvv}\nD_{vv}=\\frac{\\Gamma}{2v} \\, \\left(\\frac{{\\rm erf}(u)}{u^{2}\n}-\\frac{{\\rm erf}^{'}(u)}{u}\\right) &\\equiv & \\frac{\\Gamma}{v} \\, G(u)\n\\cr\nD_{\\beta\\beta}=\\frac{\\Gamma}{4v}\\left( \\left[2-\\frac{1}{u^{2}}\\right]{\\rm erf}(u)\n+ \\frac{{\\rm erf}^{'}(u)}{u} \\right) &\\equiv & \\frac{\\Gamma}{2v}\\biggl ( {\\rm erf}(u)-G(u) \\biggr )\n\\cr\nF_{v}=-\\frac{\\Gamma}{v^{2}}\\left({\\rm erf}(u)-u \\, {\\rm erf}^{'}(u)\\right)\n&\\equiv & -\\frac{2 \\, \\Gamma}{v^{2}} \\, u^{2} \\, G(u) \\,\\,\\, ,\n\\end{eqnarray}\nwhere the dimensionless velocity $u=v\/(\\sqrt{2} \\, v_{th})$, $v_{th}=\\sqrt{k_{B}T\/m_{e}}$ where $k_{B}$ is the Boltzmann constant [erg K$^{-1}$], $T$ is the temperature of the background plasma [K] and $m_{e}$ is the electron rest mass [g], $\\Gamma=4\\pi e^{4} \\ln\\Lambda \\, n \/m_{e}^{2}$ where $e$ [esu] is the electron charge, $n$ is the number density [cm$^{-3}$] and $\\ln\\Lambda$ is the Coulomb logarithm. ${\\rm erf}(u)$ is the error function and $G(u)$ is the Chandrasekhar function,\n\n\\begin{equation}\\label{eq:gcha}\n{\\rm erf}(u)\\equiv (2\/\\sqrt{\\pi})\\int\\limits_{0}^{u}\\exp(-t^2) \\, dt\n\\,\\,\\,\\,\\,\\,\\,{\\rm and}\\,\\,\\,\\,\\,\\,\\,\nG(u)=\\frac{{\\rm erf}(u)-u \\, {\\rm erf}^{'}(u)}{2u^{2}} \\,\\,\\, .\n\\end{equation}\nSubstituting into the Fokker-Planck equation~(\\ref{eq:fp}) gives\n\n\\begin{eqnarray}\\label{eq:fpv}\n\\frac{d f(v,z,\\beta,t)}{d t}\n&= & \\frac{\\Gamma}{2v^{2}} \\left \\{ \\frac{\\partial}{\\partial v}\\left(2 \\, v \\, G(u) \\, \\frac{\\partial f(v,z,\\beta,t)}{\\partial v}\n+4 \\, G(u) \\, u^{2} \\, f(v,z,\\beta,t)\\right) + \\right . \\cr\n&+ & \\left . \\frac{1}{v\\sin\\beta} \\, \\frac{\\partial}{\\partial \\beta}\\left(\\sin\\beta \\, \\biggl [ {\\rm erf}(u) -G(u) \\biggr ] \\, \\frac{\\partial f(v,z,\\beta,t)}{\\partial \\beta} \\right) \\right \\}+S(v,z,\\beta,t). \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n\\end{eqnarray}\n\\\\\nCurrent imaging spectroscopy X-ray observations with instruments such as {\\em RHESSI} have a time resolution the order of several seconds; it takes a full spacecraft rotation period $\\sim$4 s to yield a reliable image, which is much longer than the timescale for transport of deka-keV electrons ($v \\sim 10^{10}$~cm~s$^{-1}$) along the typical length of a coronal loop ($\\sim 10^9$~cm).  Therefore, it is appropriate to consider the time-independent case. It is also convenient to convert from the variable $\\beta$ to the variable $\\mu=\\cos{\\beta}$, giving\n\n\\begin{eqnarray}\\label{eq:fpvxmu}\n\\mu \\, v \\, \\frac{\\partial f(v,z,\\mu)}{\\partial z}\n&= & \\frac{\\Gamma}{2v^{2}} \\left \\{ \\frac{\\partial}{\\partial v}\\left(2 \\, v \\, G(u) \\, \\frac{\\partial f(v,z,\\mu)}{\\partial v}\n+4 \\, u^{2} \\, G(u) \\, f(v,z,\\mu)\\right) + \\right . \\cr\n&+ & \\left . \\frac{1}{v} \\, \\frac{\\partial}{\\partial \\mu}\\left( (1-\\mu^2) \\, \\biggl [ {\\rm erf}(u) -G(u) \\biggr ] \\, \\frac{\\partial f(v,z,\\mu)}{\\partial \\mu} \\right) \\right \\}+S(v,z,\\mu). \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n\\end{eqnarray}\n\\\\\nIt is assumed that the source term $S(v,z,\\mu)$ is separable in $v,\\;\\mu$ and $z$, with the spatial variation assumed to have a Gaussian form:\n\n\\begin{equation}\\label{eq:source_v}\nS(v,z,\\mu)=f_{0}(v) \\, \\frac{1}{\\sqrt{2\\pi d^{2}}}\\exp{\\left(-\\frac{z^{2}}{2d^{2}}\\right)} \\, H(\\mu) \\,\\,\\, ,\n\\end{equation}\nwhere $f_{0}(v)$ and $H(\\mu)$ are the initial velocity and pitch angle distribution functions. Equation~(\\ref{eq:fpvxmu}) describes the evolution of an injected electron distribution through a non-evolving finite temperature background Maxwellian distribution.\n\n\\subsection{Steady-state solution}\nFor a background plasma with a finite temperature $T$, the input velocity distribution will evolve to a thermal distribution of the form\n\n\\begin{equation}\\label{eq:fv_the}\nf(v)\\sim\\exp{\\left(-\\frac{m_{e}v^{2}}{2k_{B}T}\\right)} \\,\\,\\, ,\n\\end{equation}\nleading to an average kinetic energy of\n\\begin{equation}\\label{eq:fv_kT}\n\\left\\langle\\frac{m_{e}v^{2}}{2}\\right\\rangle\n=\\frac{\\int_{0}^{\\infty}\\frac{m_{e}v^{2}}{2}f(v)d^{3}v}{\\int_{0}^{\\infty}f(v)d^{3}v}\n=\\frac{3}{2} \\, k_{B}T \\,\\,\\, .\n\\end{equation}\n\n\\subsection{High velocity limit}\nIn the high electron velocity limit $u\\gg1$, one finds  ${\\rm erf} (u)\\rightarrow 1$ and $G(u)\\rightarrow 1\/2u^2=(v_{th}\/v)^{2}$. In this limit Equation~(\\ref{eq:fpvxmu}) becomes\n\n\\begin{eqnarray}\\label{eq:fpvxmu_he}\n\\mu \\, v \\, \\frac{\\partial f(v,z,\\mu)}{\\partial z}\n&= & \\frac{\\Gamma}{v^2} \\left \\{ \\frac{\\partial}{\\partial v} \\left( \\frac{v_{th}^2}{v} \\frac{\\partial f(v,z,\\mu)}{\\partial v} \\,\n+ \\, f(v,\\mu,z)\\right) + \\right . \\cr\n&+ & \\left . \\frac{1}{2v}\n\\frac{\\partial}{\\partial \\mu} \\left( (1-\\mu^2) \\, \\frac{\\partial f(v,z,\\mu)}{\\partial \\mu} \\right) \\right \\}  + S(v,z,\\mu)\\,\\,\\, .\n\\end{eqnarray}\n\n\\subsection{Cold plasma limit}\nIf the temperature of the plasma is also small compared to the typical particle energies, then it can be formally taken that $T=0$ (that is, $v_{th}=0$).  Equation (\\ref{eq:fpvxmu_he}) then becomes\n\n\\begin{equation}\\label{eq:fpvxmu_noT}\n\\mu \\, \\frac{\\partial f(v,z,\\mu)}{\\partial z}\n= \\frac{\\Gamma}{v^3} \\, \\left \\{ \\frac{\\partial f(v,z,\\mu)}{\\partial v} + \\frac{1}{2 v} \\, \\frac{\\partial}{\\partial \\mu} \\left( (1-\\mu^2) \\, \\frac{\\partial f(v,z,\\mu)}{\\partial \\mu} \\right) \\right \\}  + S(v,z,\\mu)\\,\\,\\, ,\n\\end{equation}\nwhich is the transport equation for a cold plasma with azimuthal symmetry, a more familiar form often used in solar physics\n\\citep[e.g.,][]{1981SvA....25..215K}.\n\n\\subsection{Conversion to the electron flux distribution}\nThe electron flux spectrum $F(E,z,\\mu)$ [electrons~cm$^{-2}$~s$^{-1}$~keV$^{-1}$] as a function of field-aligned coordinate $z$ [cm], energy $E$ [keV] and pitch angle cosine $\\mu$ is related to the three-dimensional phase-space distribution function $f(v,z,\\mu)$ by\n\n\\begin{equation}\\label{con_fF1}\nv \\, f(v,z,\\mu) \\, d^{3}v = v \\, f(v,z,\\mu) \\, v^2 \\, dv = F(E,z,\\mu) \\, dE \\,\\,\\, ,\n\\end{equation}\nso that\n\\begin{equation}\\label{con_fF}\nf(v,z,\\mu)=\\frac{dE}{dv}\\frac{1}{v^{3}}\\,F(E,z,\\mu)= \\frac{m_{e}}{v^{2}} \\, F(E,z,\\mu) = \\frac{m_{e}^{2}}{2E} \\, F(E,z,\\mu) \\,\\,\\, .\n\\end{equation}\n\\\\\nUsing this relation, the Fokker-Planck equation~(\\ref{eq:fpvxmu}) can be re-written in terms of electron energy $E$\nand the electron flux distribution $F(E,z,\\mu)$, which is a more useful form for comparison with observations.  The result is\n\n\\begin{eqnarray}\\label{eq: fp_e}\n\\mu \\, \\frac{\\partial F}{\\partial z} &= & \\Gamma m_{e}^2 \\left \\{ \\frac{\\partial}{\\partial E} \\left[ G (u[E] ) \\, \\frac{\\partial F}{\\partial E} + \\frac{G (u[E] )}{E} \\, \\left ( \\frac{E}{k_B T}-1 \\right ) \\, F \\right] + \\right . \\cr\n&+ & \\left . \\frac{1}{8E^2} \\, \\frac{\\partial}{\\partial \\mu} \\left [ (1-\\mu^{2}) \\biggl ( {\\rm erf} (u[E] ) - G (u[E] ) \\biggr ) \\, \\frac{\\partial F}{\\partial \\mu} \\right ] \\right \\} + S_{F}(E,z,\\mu),\\,\\,\\,\\,\n\\end{eqnarray}\n\\\\\nwhere $u(E)=\\sqrt{E\/k_B T}$ is used.\nThe solar corona also contains elements other than hydrogen, and for an element with atomic number $Z$, the Coulomb energy loss scales as $Z^2$ \\citep[e.g.,][]{1978ApJ...224..241E}, and these additional elements are accounted for by adopting an effective atomic number $Z_{eff}=\\sum_{i} n_{i} Z_{i}^{2}\/\\sum_i n_i$. Defining $\\Gamma_{eff}=\\Gamma Z_{eff} m_{e}^{2}$ and $G(u)=G\\left(\\sqrt{\\frac{E}{k_{B}T}}\\right)$, Equation \\ref{eq: fp_e} becomes\n\n\\begin{eqnarray}\\label{eq: fp_e_zefftext}\n\\mu \\, \\frac{\\partial F}{\\partial z} &= &\n\\Gamma_{eff} \\left \\{ \\frac{\\partial}{\\partial E} \\left [ G \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\, \\frac{\\partial F}{\\partial E} +\\frac{1}{E} \\, \\left ( \\frac{E}{k_B T} - 1 \\right ) \\, G \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\, F \\right ] \\right .\n+ \\cr\n&+ & \\left . \\frac{1}{8E^2} \\, \\frac{\\partial}{\\partial \\mu} \\left [ (1-\\mu^{2}) \\, \\left ( {\\rm erf} \\left ( \\sqrt{\\frac{E}{k_B T}} \\, \\right ) -G\\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\right ) \\frac{\\partial F}{\\partial \\mu} \\right ]\\right \\} \n+ \\cr\n&+ &\n+ S_{F}(E,z,\\mu)),\n\\end{eqnarray}\nThe source term again consists of three separable functions for $E,\\,\\mu$ and $z$:\n\\begin{equation}\\label{eq:source}\nS_{F}(E,z,\\mu)) = F_{0}(E) \\; \\frac{1}{\\sqrt{2\\pi d^{2}}} \\, \\exp{\\left(-\\frac{z^{2}}{2d^{2}}\\right)} \\, H(\\mu) \\,\\,\\, ,\n\\end{equation}\nwhere $F_0(E)\\propto E_{0}^{-\\delta}$ and $H(\\mu)$ describe the forms of the initial energy spectrum and pitch angle distribution, respectively.\n\n\\subsection{Derivation of the stochastic differential equations}\nFor use in the simulations, Equation \\ref{eq: fp_e_zefftext} must be converted to a set of stochastic differential equations (SDE) for the field-aligned coordinate $z$ [cm], energy $E$ [keV] and pitch angle cosine $\\mu$. If the source term is ignored, focusing on electron transport, Equation~(\\ref{eq: fp_e_zefftext}) can be rewritten in the form\n\n\\begin{equation}\\label{eq: fp_eABin}\n\\mu \\, \\frac{\\partial F}{\\partial z}\n=\\frac{\\partial}{\\partial E}\\biggl ( A_E(E) \\, F \\biggr )\n+ \\frac{\\partial^{2}}{\\partial E^{2}} \\biggl ( D_{EE}(E) \\, F \\biggr )\n+\\frac{\\partial}{\\partial \\mu} \\biggl ( A_\\mu(E,\\mu) \\, F \\biggr )\n+\\frac{\\partial^2}{\\partial \\mu^2} \\biggl ( D_{\\mu\\mu}(E,\\mu) \\, F \\biggr )  \\,\\,\\, ,\n\\end{equation}\nwhere the coefficients are given by\n\\begin{align}\\label{eq:A-D}\nA_E(E) &=&  \\frac{\\Gamma_{eff}}{2E} \\left [ {\\rm erf} \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) -2 \\sqrt{\\frac{E}{k_B T}} \\, {\\rm erf}^{'} \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\right ] \\,\\,\\, && \\cr\n&\\equiv& \\frac{\\Gamma_{eff}}{2E} \\, g_{th} \\left (\\sqrt{\\frac{E}{k_B T}} \\right )\\,\\,\\, &;& \\cr\nD_{EE}(E)\\equiv\\frac{1}{2} \\, B_E^2(E) &=&   \\, \\Gamma_{eff} \\, G \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right )  \\,\\,\\, &;&\n\\cr\nA_\\mu(E,\\mu) &=& \\frac{\\mu \\, \\Gamma_{eff}}{4E^2} \\left [ {\\rm erf} \\left ( \\sqrt{\\frac{E}{k_B T}} \\, \\right ) -G \\left ( \\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\right ] \\,\\,\\, &;&\n\\cr\nD_{\\mu\\mu}(E,\\mu)\\equiv\\frac{1}{2} \\, B_\\mu^2(E,\\mu) &=&\n \\frac{(1-\\mu^{2}) \\, \\Gamma_{eff}}{8 E^2} \\, \\left [ {\\rm erf}\\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) - G \\left (\\sqrt{\\frac{E}{k_B T}} \\, \\right ) \\right ] \\,\\,\\, &.&\n\\end{align}\n\nThis general form of the Fokker-Planck equation is equivalent to the following stochastic differential equations (SDE) for $E$ and $\\mu$ in the It\\^o form \\cite[cf.][]{1991A&A...251..693M,1994hsmp.book.....G}\n\n\\begin{equation}\\label{eq:sto_Emu}\n{dE}= - \\, A_E \\, ds + B_E \\, \\, dW_{E}\n \\,\\,\\, ; \\qquad {d\\mu}= - \\, A_\\mu \\,ds  + B_\\mu \\, \\, dW_{\\mu} \\,\\,\\, ,\n\\end{equation}\nwhere the independent Wiener processes $W_{\\mu}$ and $W_{E}$ are stochastic processes with independent increments. These two equations suggest the numerical stepping algorithm\n\n\\begin{equation}\\label{eq:sto_x}\nz_{j+1}=z_{j}+\\mu_{j} \\, \\Delta s \\,\\,\\, ;\n\\end{equation}\n\\begin{equation}\\label{eq:sto_E}\nE_{j+1}=E_{j}-\\frac{\\Gamma_{eff}}{2E_{j}} \\, g_{th}(u_j) \\, \\Delta s\n+\\sqrt{2 \\, \\Gamma_{eff} \\, G(u_{j}) \\, \\Delta s} \\,\\, W_{E} \\,\\,\\, ; \\,\\,\\,\n\\end{equation}\n\\begin{equation}\\label{eq:sto_mu}\n\\mu_{j+1}=\\mu_{j}-\\frac{\\Gamma_{eff} \\biggl ( {\\rm erf}(u_{j})-G(u_{j}) \\biggr ) } {4 E_{j}^{2}} \\,\\, \\mu_{j} \\, \\Delta s\n+\\sqrt{\\frac{ (1-\\mu_{j}^{2}) \\, \\Gamma_{eff} \\, \\biggl ( {\\rm erf}(u_{j})-G(u_{j}) \\biggr ) } {4 E_{j}^{2}} \\, \\Delta s} \\, \\, W_{\\mu} \\,\\,\\, ,\n\\end{equation}\nwhere $u_j=\\sqrt{E_j\/k_B T}$ and $W_E$ and $W_{\\mu}$ are drawn at random from the Gaussian distribution $N(0,1)$ such that $\\langle W_{\\mu}\\rangle=\\langle W_{E}\\rangle  =0$, $\\langle W^2_{\\mu}\\rangle=\\langle W^2_{E}\\rangle= 1$. Equations~(\\ref{eq:sto_x}) through~(\\ref{eq:sto_mu}) are\nthe form of the SDEs used in the numerical simulations, which must be amended for low energies. This will be discussed in Section \\ref{low-energy-limit}.\n\nIt should be noted that a root mean square (rms) atomic number of $Z_{eff}=1$ is taken for simplicity (that is, a pure hydrogen target), but the equation for a general $Z_{eff}$ is provided as it may prove useful in other studies.\n\nThe coefficients $A_E$, $A_\\mu$, $B_E (=\\sqrt{2D_{EE}})$ and $B_\\mu (=\\sqrt{2D_{\\mu\\mu}})$ are plotted against energy $E$ in Figure~\\ref{fig:check_ABCD}, for a number of different plasma temperatures $T$ ranging from $T = 0$ (cold plasma) to $T=100$~MK. For ease of presentation, the $A_\\mu$ and $B_\\mu$ terms are shown as a function of $E$ for a fixed value of $\\mu$ ($\\mu=1$ for $A_\\mu$ and $\\mu=0$ for $B_\\mu$). Below an energy $E_c\\simeq k_BT$ the coefficient $A_E$ becomes negative; that is, electrons on average {\\it gain} energy; the value of $E_c$ for which $A_E=0$ increases linearly with the ambient\n\\begin{sidewaysfigure*}[ht]\n\\centering\n\\includegraphics[width=23cm]{fig2.pdf}\n\\caption[Plots of the energy $A_E$, $B_{E}$ and pitch angle $A_\\mu(E,\\mu=1)$, $B_{\\mu}(E,\\mu=0)$ coefficients against\nelectron energy $E$ for different plasma temperatures from $T=0 - 100$~MK.]{Plots of the energy $A_E$, $B_{E}$ and pitch angle $A_\\mu(E,\\mu=1)$, $B_{\\mu}(E,\\mu=0)$ coefficients against\nelectron energy $E$ for different plasma temperatures from $T=0 - 100$~MK. The colours corresponding to each plasma temperature are shown on the bottom right plot. The $A_E$ coefficient is plotted three times (top row) so that all the features in different energy ranges can be seen clearly.}\n\\label{fig:check_ABCD}\n\\end{sidewaysfigure*}\n\\clearpage\ntemperature. In order that these features can be seen clearly, the coefficient $A_E$ is plotted (top row of Figure~\\ref{fig:check_ABCD}) over three different energy ranges: two (below 1~keV, and below 30~keV) plotted on linear $y$-axes and 1-50~keV plotted on a logarithmic $y$-axis.  Further, the stochastic term $B_{E}$ peaks at $\\simeq k_B T$. Therefore, in a warm plasma, electrons with $E \\sim k_B T$ are more likely to gain energy, both secularly and through diffusion, rather than to lose it.\n\n\n\nTo get reliable results from the simulations, an appropriate value of the length step $\\Delta s$ (Equations~(\\ref{eq:sto_x}) through~(\\ref{eq:sto_mu})) must be chosen. This was chosen by calculating the thermal collision length (mean free path) $\\lambda_c(E)$ and ensuring that $\\Delta s$ was much smaller than $\\lambda_c$ for all $E$ of interest. The thermal collisional length is given by $\\lambda_{c}=v\\tau_{c}$, where $\\tau_{c}$ is the thermal collisional time as discussed in Chapter \\ref{ref:Chapter1}.  The mean-free path $\\lambda_c$ for a 1~keV electron in a cold target of density $n=1 \\times 10^{11}$~cm$^{-3}$ is approximately $10^6$~cm; the mean-free paths in warm targets are even greater.  For all simulations, a length step $\\Delta s=1\\times10^{5}$~cm is used, much smaller than the mean free path in all cases and this is shown in Figure \\ref{plot_tau}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.8]{plot_tau.pdf}\n\\caption[Electron collisional length versus electron energy in a cold plasma (black) with the thermal collisional lengths over-plotted for $T=1$, $10$, $20$, $30$ and $100$ MK.]{Electron collisional length versus electron energy in a cold plasma (black) with the thermal collisional lengths over-plotted for $T=1$, $10$, $20$, $30$ and $100$ MK. In the simulations, sensible flaring plasma temperatures of $T=10$, $20$ and $30$ MK are used and hence a stochastic distance step is chosen to be $\\Delta s=1\\times10^{5}$ cm, which is at least one order of magnitude smaller than the thermal collisional length for $T=10$ MK.}\n\\label{plot_tau}\n\\end{figure*}\n\\subsection{The low-energy limit}\\label{low-energy-limit}\n\nAs the plots in Figure~\\ref{fig:check_ABCD} show, $A_E$, $A_\\mu$ and $B_{\\mu}$ diverge as $E\\rightarrow 0$. Therefore, following \\citet{2009JCoPh.228.1391L} and \\citet{2010ITPS...38.2394C}, for low energies $E$ the finite difference Equation~(\\ref{eq:sto_E}) is replaced with an analytic expression for the energy evolution. To obtain this expression, the functions ${\\rm erf}(u)$ and ${\\rm erf'}(u)$ for small $u$ are expanded in a MacLaurin series, so that the coefficients $A_E$ and $B_E$ become\n\n\\begin{equation}\\label{low_e_A}\nA_E = \\frac{\\Gamma_{eff}}{2E} \\left ( {\\rm erf}(u)-2u \\, {\\rm erf}^{'}(u) \\right )\n\\simeq - \\, \\frac{\\Gamma_{eff}}{\\sqrt{\\pi}E} \\, u \\,\\,\\, ,\n\\end{equation}\n\n\\begin{equation}\\label{low_e_B}\nB_E = \\sqrt{2 \\, \\Gamma_{eff} \\, G (u)}\n\\simeq \\sqrt{\\frac{4 \\, \\Gamma_{eff}}{3 \\, \\sqrt{\\pi}} \\, u} \\,\\,\\, ,\n\\end{equation}\nand in the low-energy limit, $E\\rightarrow 0$, the energy equation becomes\n\n\\begin{equation}\\label{eq:low_de_sde}\n\\frac{dE}{ds} \\simeq \\frac{\\Gamma_{eff}}{E} \\, \\sqrt{\\frac{E}{\\pi k_{B} T}} + \\left ( \\frac{4 \\, \\Gamma_{eff}}{3} \\, \\sqrt{\\frac{E}{\\pi k_B T}} \\, \\right )^{1\/2} W^{E} \\,\\,\\, .\n\\end{equation}\nFor low values of $E$, the second (stochastic) term can be neglected in comparison with the first (secular) term to give\n\n\\begin{equation}\\label{eq:low_de_sde_A}\n\\frac{dE}{ds}\\simeq \\frac{\\Gamma_{eff}}{\\sqrt{\\pi k_{B} T}} \\, \\frac{1}{\\sqrt{E}} \\,\\,\\, ,\n\\end{equation}\nwhich can be integrated analytically, giving\n\n\\begin{equation}\\label{eq:low_e_sde_A_2}\nE=\\left[E_{0}^{3\/2}+\\frac{3 \\, \\Gamma_{eff}}{2\\sqrt{\\pi k_{B} T}} \\, (s-s_0)\\right]^{2\/3} \\,\\,\\, .\n\\end{equation}\nEquation (\\ref{eq:low_e_sde_A_2}) was used for energies below\n\n\\begin{equation}\nE_{low} = \\left[\\frac{3\\Gamma_{eff}}{2\\sqrt{\\pi k_{B} T}} \\, \\Delta s  \\right]^{2\/3}\\,\\,\\, ,\n\\end{equation}\nthus guaranteeing that $E \\ge 0$ everywhere. To avoid divergence, the pitch angle cosine $\\mu$ for energies $E\\leq E_{low}$ was sampled from a uniform distribution between $-1$ and $1$. Since electrons with $E_{low}<<\\bar{E}$, where $\\bar{E}$ is the average thermal energy of the plasma, are likely to be part of the background thermal distribution, then it is sensible to draw their pitch angle from an isotropic distribution.\n\\\\\\\\\nIn the cold plasma limit $T\\rightarrow0$, the stochastic equation for $E$ becomes\n\n\\begin{equation}\\label{eq:cold_E}\nE_{j+1}=E_{j}-\\frac{\\Gamma_{eff}}{2 E_{j}} \\, \\Delta s \\,\\,\\, ,\n\\end{equation}\nwhich can be solved to give the usual cold target result\n\n\\begin{equation}\nE_{j+1}=\\sqrt{E^2_{j} - 2 K n \\, \\Delta s} \\,\\,\\, ,\n\\end{equation}\nwhere $K=2\\Gamma_{eff}\/n$. In this limit, the pitch angle behaviour is given by\n\n\\begin{equation}\\label{eq:cold_mu}\n\\mu_{j+1}=\\mu_{j}-\\frac{\\Gamma_{eff}}{4 E_{j}^{2}} \\, \\mu_{j} \\, \\Delta s + \\sqrt{\\frac{\\Gamma_{eff}}{4 E_{j}^2} \\, (1-\\mu^{2}) \\, \\Delta s} \\,\\, W_{\\mu}\\,\\,\\, .\n\\end{equation}\n\n\\section{Simulations}\n\nThe aim of the simulations is to determine how collisional pitch angle scattering and the finite temperature of the target plasma affect the transport of electrons through the plasma compared to the one-dimensional cold target result, and hence to determine how the observed length of a hard X-ray source varies with electron energy in a more realistic physical scenario.  The simulations use the stochastic equations for $z$, $E$, and $\\mu$ given by Equations~(\\ref{eq:sto_x}) through~(\\ref{eq:sto_mu}) with initial conditions for each injected electron provided by sampling the source term $S(E,z,\\mu)$ -- see Equation (\\ref{eq:source}). The simulations model the evolution of an injected distribution of electrons, moving either within a cold plasma or a plasma of finite temperature, they do not account for the evolution of the background plasma; the properties of the background plasma remain constant throughout a simulation.\n\n\\subsection{Simulation input, boundary and end conditions}\\label{sec:input}\n\nAll simulations use a common set of certain input parameters.  The electron number density is set to $n = 1 \\times 10^{11}$~cm$^{-3}$, a relatively high value for the coronal density, but one which is necessarily high in order for the deka-keV electrons to be stopped in the corona and which is chosen to correspond to recent analyses of thick target coronal sources \\citep[e.g.,][]{2008ApJ...673..576X,2011ApJ...730L..22K,2013ApJ...766...75J}.\nFor the Coulomb logarithm a typical coronal value of $\\ln{\\Lambda}=20$ is used. The plasma temperature is assumed uniform\nalong the $z$ direction, at a value of either $0$~MK, $10$~MK, $20$~MK or $30$~MK. The initial spatial distribution of injected\nflux (``acceleration region size'') is assumed to be a Gaussian centred at $z=0$ (which is the position of the coronal loop apex) with an input standard deviation of $d=10\\mbox{\\ensuremath{^{\\prime\\prime}}}$, corresponding to a FWHM$=2\\sqrt{2\\ln 2} \\, d=23\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$.  The initial pitch angle distribution is taken to be either {\\it completely beamed} (that is, half the distribution has $\\mu=1$ and the other half $\\mu=-1$) or {\\it isotropic}. The injected electron energy flux distribution $F_0(E)$ has the form of a power law with spectral index $\\delta=4$ or $\\delta=7$, up to a maximum energy of $50$ keV, above which the energy-integrated electron flux is negligibly small.\n\nThe upper boundary of the $z$ domain is set at a value sufficiently large that no electrons ever {\\it spatially} leave the region of computation. For the runs that use the cold target energy loss formula, electrons lose energy monotonically.  Hence an electron is removed from the simulation once its  energy is below $1$~keV. However, for the warm target simulations, electrons of very low energy can still gain energy through Coulomb collisions with more energetic neighbours, as the ensemble evolves to a thermal (Maxwellian) distribution.  Thus electrons in the warm target runs are {\\it never} removed; for such runs the particle number is conserved and the electron distribution asymptotically approaches the Maxwellian distribution $F(E)\\sim E \\exp (-E\/k_B T)$.  For this distribution, the flux-averaged energy is\n\n\\begin{equation}\\label{average-energy}\n{\\overline E} = \\frac {\\int^{\\infty}_{0} E \\, F(E) \\, dE}  {\\int^{\\infty}_{0} F(E) \\, dE} = 2 \\, k_{B} T \\,\\,\\, .\n\\end{equation}\nTherefore a simulation is terminated when the average energy of the distribution is $2 k_B T$ and the pitch angle distribution becomes approximately isotropic, conditions that approximate the essential features of a Maxwellian. Note that $\\overline E$ is {\\it not} the average kinetic energy of the three-dimensional phase space distribution $f(v,\\mu,z)$ (which is $\\langle{mv^{2}}\/{2}\\rangle=\\frac{3}{2}k_{B}T$). After each distance step $\\Delta s$, the values of the electron distribution function $F(E,\\mu,z)$ are saved into an array. These arrays represent the distribution functions resulting from the continuous injection of electrons with the source function given by Equation (\\ref{eq:source}).\n\n\\subsection{Gaussian fitting and the determination of the source length FWHM}\n\nThe arrays generated by each simulation are energy-binned to give $F(z,\\mu)$ in increasing energy bins from $1$~keV to $30$~keV.\nThe longitudinal extent of the source could be identified as the standard deviation $std(E)=\\sqrt{var(E)}$ of the $F(z,\\mu)$ spatial distribution in each energy bin, calculated from the second spatial moment\nof $F(z,\\mu)$.  However, in part because the injected flux\ndistribution is {\\it assumed} to be Gaussian, the forms of $F(z,\\mu)$ generally also closely resemble Gaussian forms,\nexcluding relatively low-intensity components at high $|z|$.  Therefore, just as in Section~\\ref{cold_theory}, a Gaussian distribution is instead fitted to each $F(z,\\mu)$ distribution and thus used to determine the associated standard\ndeviations $std_{G}(E)$ and corresponding FWHM$=2\\sqrt{2 \\ln 2} \\, std_{G}(E)$ in each energy bin. In this way, the extent of the source is characterised through the {\\it shape} of its core spatial form, rather than through a {\\it moment} of the entire distribution. Again, as in Section~\\ref{cold_theory}, FWHM$(E)=L_{0}+\\alpha E^{2}$ (Equation~(\\ref{L})) was fitted to the FWHM versus electron energy results, and values of $\\alpha$ and $L_0$ found.\n\nFor a cold plasma with an initially beamed pitch angle distribution and no collisional pitch angle scattering, it is expected that $L_{0}=L_{init}=2\\sqrt{2 \\ln 2} \\, d$, the Gaussian FWHM of the input distribution, has a value of $\\alpha$ equal\nto that found numerically from the fits to $\\delta=4$ and $\\delta=7$ curves in Figure~\\ref{fex_good}. However,\nthe presence of a finite plasma temperature $T$, an initially broad pitch angle distribution, and\/or collisional pitch angle scattering will all change the values of $L_{0}$ and $\\alpha$ obtained. The inferred values of the acceleration region density $n$ depend on the value of $\\alpha$ ($\\alpha \\propto 1\/n$).  The values of other parameters inferred from $n$ and the acceleration region length $L_0$ -- see Section~\\ref{ref:chap2_con} -- are thus dependent upon both the assumed electron distribution and the properties of the target plasma. The results will be used to find, for instance, if the inappropriate use of a one-dimensional cold target assumption changes the inferred number density by a factor larger than the observational uncertainty, and thus determine if a correction should be applied to the results.\n\\begin{figure}[h]\n\\vspace{-20pt}\n\\centering\n\\includegraphics[width=75mm]{fig3a.pdf}\n\\includegraphics[width=75mm]{fig3b.pdf}\n\\includegraphics[width=75mm]{fig3c.pdf}\n\\includegraphics[width=75mm]{fig3d.pdf}\n\\caption[The energy $E$ of a single electron and $<E>$ of the entire distribution for $T=0,10,20,30$ MK simulations plotted as a function of the overall distance $\\sum \\Delta s$ travelled.]{{\\it Top panels}: $E$ of a single electron for $T=0,10,20,30$ MK simulations as a function of the overall distance $\\sum \\Delta s$ travelled. For the chosen $\\Delta s = 10^5$~cm $\\simeq 10^{-3}$ arc seconds, the change in energy over a single step is small. The randomness in the $T$=10, 20 and 30~MK cases is due to thermal fluctuations that increase with $T$; the error associated is difficult to estimate for a single particle. {\\it Bottom panels}: $\\langle E \\rangle$ of the entire distribution versus $\\sum \\Delta s$ travelled, for the parameters given in Section \\ref{sec:input}.  In contrast to the results for a single particle, these show smooth curves, with only small fluctuations for the $T$=10, 20 and 30~MK cases. Black dashed dot curve (0 MK): analytical cold target solution and orange dashed dot curves (10, 20 and 30 MK): final average energy of the $F(E)$ distribution.}\n\\label{fig:check_errors}\n\\end{figure}\n\n\\subsubsection{Simulation accuracy and limiting cases}\n\nIn general, consideration of the errors associated with stochastic simulations are a complex problem and beyond the scope of this thesis. However,  convergence of the simulation results against limiting analytical solutions can be checked. In the various plots shown in Figure~\\ref{fig:check_errors} the energy of a single electron versus the overall step distance travelled (top) and the average energy of the entire distribution against the distance travelled (bottom) are plotted.  This was done for $\\delta=7$, and for $T$=0, 10~MK, 20~MK and 30MK. For the cold ($T=0$) case, the error in the energy of a single electron is very small; the stochastic terms in the difference equations~(\\ref{eq:sto_x}) through~(\\ref{eq:sto_mu}) are negligible and individual electron energies (and hence the average energy of the entire distribution) follow the analytical results very well. However, for a finite temperature target, the stochastic part of the difference equations plays a significant role, the dominance of which increases with $T$. Hence the energy of a single electron fluctuates significantly, especially at low energies.  However, due to ensemble averaging, even for finite target temperatures the average energy of the {\\it distribution} exhibits a relatively smooth transition from the starting average energy of the distribution to the final average value of the distribution $F(E)$.\n\\begin{sidewaysfigure*}\n\\centering\n\\includegraphics[width=21.5cm]{fig4.pdf}\n\\caption[Gaussian FWHM versus electron energy $E$ for all cold plasma simulation runs.]{Gaussian FWHM versus electron energy $E$ for all cold plasma simulation runs with $n=1\\times10^{11}$ cm$^{-3}$ and $\\delta=4$ (left plot) and $\\delta=7$ (right plot). The cases shown are for: (A) beamed, no pitch angle scattering (orange, blue), (B) beamed, with pitch angle scattering (red, green), (C) isotropic, no pitch angle scattering (pink, navy) and (D) isotropic, with pitch angle scattering (purple, grey). An equation of the form $L_{0}+\\alpha E^{2}$ was fitted to each curve and the values thus found for $L_{0}$ and $\\alpha$ are shown on each plot. The fits use energies in the range $\\sim 8-25$ keV, matching the energy range fitted to {\\em RHESSI} observations. The dashed-dot lines represent the Gaussian FWHM curves fitted to the results of Equation~(\\ref{eq:fexgauss}), as in the bottom panels of Figure \\ref{fex_good}. As expected, these match well with scenario (A).}\n\\label{fig:cold_runs}\n\\end{sidewaysfigure*}\n\\clearpage\n\n\\subsection{Numerical results}\n\n\\subsubsection{Cold plasma with collisional pitch angle scattering}\n\nFirstly, the case of a cold target is considered, with different pitch angle injection and scattering scenarios. Eight simulations were performed, corresponding to two spectral indices ($\\delta=4$ and $\\delta=7$) and:\n\n\\begin{itemize}\n\\item {(A) an injected bi-directional beamed distribution of electrons ($\\mu=-1,1$) without collisional pitch angle scattering,}\n\\item {(B) an injected bi-directional beamed distribution of electrons ($\\mu=-1,1$) undergoing collisional pitch angle scattering,}\n\\item {(C) an initially isotropic pitch angle distribution of electrons without collisional pitch angle scattering, and}\n\\item {(D) an initially isotropic pitch angle distribution of electrons undergoing collisional pitch angle scattering.}\n\\end{itemize}\n\n\nFigure~\\ref{fig:cold_runs} shows the Gaussian spatial FWHM plotted against electron energy $E$ for cases (A), (B), (C) and (D), together with fits using Equation (\\ref{L}) between $\\sim8-25$ keV. This energy range is chosen to match with the energy ranges often used for such observations by {\\em RHESSI}. The corresponding values of $\\alpha$ and $L_{0}$ for each scenario are shown on Figure~\\ref{fig:cold_runs}, and there are two general statements that can be made regarding the results. Firstly, the broader the initial pitch angle distribution, the smaller the source length at a given energy and secondly, the presence of collisional pitch angle scattering acts to slightly decrease the source length at a given electron energy.  Both effects occur because electrons with $|\\mu| < 1$ move a correspondingly smaller distance along the magnetic field. The latter effect is greater at higher electron energies but overall the change is rather small (Figure~\\ref{fig:cold_runs}).\n\nThe case of an initially isotropic distribution, with or without pitch angle scattering, produces the flattest (lowest value of $\\alpha$) results for each $\\delta$. For example, compared with the initially beamed, scatter-free cases for $\\delta=4,7$, the isotropic, scatter-free $\\alpha$'s are lower by factors of $\\sim2.6$ and $\\sim3.5$, respectively.\n\nSince the coefficient $\\alpha$ (Equation~(\\ref{eq: sd2})) in a one-dimensional cold target formulation is inversely proportional to the ambient density $n$, {\\it the reduced penetration distance associated with the presence of an initially broad pitch angle distribution and\/or collisional scattering will lead to an overestimate of $n$ if the results are interpreted using the one-dimensional cold target result},  with the exact reduction factor dependent upon the properties of the initial electron distribution and background plasma.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{fig5.pdf}\n\\caption[For each cold target simulation scenario -- (A), (B), (C) and (D) -- the value of the coefficient $\\alpha$ calculated by fitting each curve in Figure \\ref{fig:cold_runs} is used to infer a number density $n$ using two different one-dimensional cold target approaches.]{For each cold target simulation scenario -- (A), (B), (C) and (D) -- the value of the coefficient $\\alpha$ calculated by fitting each curve in Figure \\ref{fig:cold_runs} is used to infer a number density $n$ using two different one-dimensional cold target approaches: (1) point injection $\\alpha=1\/2Kn$ (red) and (2) an extended Gaussian input that is initially beamed with no pitch angle scattering (blue). The actual number density of $1\\times10^{11}$ cm$^{-3}$ is given by the grey dashed line and the inferred value of $n$ is $\\sim$ equal or greater than the actual value.}\n\\label{fig:find_num_6}\n\\end{figure*}\n\nValues of $n$ were inferred for each of the cases (A), (B), (C) and (D), using two different interpretive approaches:\n\n\\begin{enumerate}\n\\item {$\\alpha_{1}=1\/2Kn$, i.e., simple one-dimensional propagation within a cold target, giving $\\alpha_{1}=0.026$ arcsecond keV$^{-2}$ for $n=1\\times10^{11}$ cm$^{-3}$}\n\\item {$\\alpha_{2}$, found using an extended Gaussian input for an initially beamed distribution with no pitch angle scattering, i.e., Equation~(\\ref{eq:fexgauss}) and scenario (A).  From the lower right panel of Figure~\\ref{fex_good}, for $n=1\\times10^{11}$ cm$^{-3}$, $\\alpha_{2}=0.026$ arcsecond~keV$^{-2}$ for $\\delta=4$ and $\\alpha_{2}=0.012$ arcsecond~keV$^{-2}$ for $\\delta=7$.}\n\\end{enumerate}\n\nIn Figure \\ref{fig:find_num_6}, the actual number density of the region $n=1\\times10^{11}$ cm$^{-3}$ is shown by the dashed grey line and the values of $n$ inferred from approaches (1) and (2) are shown by the red and blue points, respectively. The inferred number density can be up to six times too large, with the largest effect being for steep spectra (the $\\delta=7$ case) and isotropic injection (cases (C) and (D)).\n\n\\subsubsection{Hot plasma and collisional pitch angle scattering}\n\nIn this section it is studied how the effect of a finite-temperature target (in the presence of collisional pitch angle scattering) changes the electron transport through the plasma and hence the extent of the source with energy. Six further simulations were considered corresponding to three target temperatures (10~MK, 20~MK and 30~MK), and pitch angle scenario (B), an injected beamed electron distribution including pitch angle scattering, for both $\\delta=4$ and $\\delta=7$.\n\nFigure~\\ref{fig:spectra_x} shows both the spatially-integrated spectra and the spectrally-integrated spatial distributions for five different simulations: one-dimensional (beamed) cold target (black), cold target with isotropic injection (grey), and beamed injection in three warm target cases: $T$=10~MK (orange), 20~MK (green) and 30~MK (blue). Figure~\\ref{fig:spectra_x} shows only the spatially and spectrally integrated evolutions of the injected electron distribution and does not include the background cold or Maxwellian distribution. The total spatially-integrated spectra are plotted in the top row of panels, for $\\delta=4$ (left) and $\\delta=7$ (right); the spatial distribution of the spectrally-integrated flux is plotted in the bottom row of panels, again for $\\delta=4$ (left) and $\\delta=7$ (right).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=15.5cm]{fig6a.pdf}\n\\includegraphics[width=15.5cm]{fig6b.pdf}\n\\caption[Plots of the spatially-integrated spectra and energy-integrated spatial distributions for both cold and hot plasma simulation runs.]{{\\it Top panels}: spatially-integrated spectra; {\\it bottom panels}: energy-integrated spatial distributions for the following scenarios: (1) cold plasma, initially beamed distribution with pitch angle scattering (black); (2) cold plasma, initially isotropic distribution with pitch angle scattering (grey); and warm target cases with (3) $T$=10~MK (orange), (4) $T$=20~MK (green) and (5) $T$=30~MK (blue), with pitch angle scattering.  Results are shown for both $\\delta=4$ (left) and $\\delta=7$ (right). The red dashed lines in the bottom panels indicate the the source function $S(E,\\mu=1,z)$.}\n\\label{fig:spectra_x}\n\\end{figure*}\n\nNot surprisingly, higher temperature targets tend to make the overall electron spectrum more thermal in form. The lower the temperature of the background Maxwellian plasma, the greater the distinction between the thermal part of the distribution at lower energies and the nonthermal power-law component at higher energies. Also, the inclusion of thermal effects tends to broaden the spatial distribution of the electron distribution, with the effect being more pronounced at higher temperatures.  The spatial spread for a given input distribution is larger for a smaller spectral index because of the larger fraction of higher energy electrons in such flat distributions.  It was also found (not shown) that, not surprisingly, the initially beamed distribution (case (B)) shows greater spreads in $z$ than for the same six runs performed for the isotropic injection case (case (D)), see Figure~\\ref{fig:spectra_x}.\n\nFigure~\\ref{hot_errors} shows the results of the Gaussian fits to the computed spatial distributions for all six warm target scenarios, together with the corresponding results for the cold target case.  Compared to the cold target case, the addition of thermal effects results in changes that affect the inferred values of both $n\\propto 1\/\\alpha$ {\\it and} $L_{0}$. Firstly, it is obvious from all panels in Figure~\\ref{hot_errors} that the value of the $y$-axis intercept $L_0$ (the inferred acceleration region length) increases with temperature; it was found that the magnitude of this increase depends somewhat on the number density $n$ and is relatively independent of the power-law index $\\delta$. This effect is purely due to the thermal diffusive nature of the electron transport, both energetically and spatially, at low energies. This result suggests that the temperature of the background plasma must be accounted for, when estimating $L_0$ from such observations. The determination of the actual acceleration region length from the inferred length is discussed further in Section \\ref{landn}.\n\nJust as before, curves of the form of Equation (\\ref{L}) are fitted between $\\sim8-25$~keV, for a better comparison with observations when using imaging algorithms such as VisFwdFit and {\\it uv$\\_$smooth}.  These are shown by the purple dashed lines and the values of $L_0$ and $\\alpha$ from the purple fits are shown on each panel of Figure~\\ref{hot_errors}.  However, the presence of a finite background temperature causes the lower energies of the distribution, in particular, to be dominated by thermal diffusion and hence analysis of the curves in Figure~\\ref{hot_errors} shows that overall, the FWHM over the entire plotted energy range is not so well-fitted by a single curve of the form FWHM$(E)=L_0+\\alpha E^{2}$. This can be clearly seen for the 20 MK, $\\delta=4$ curve. Therefore, two other FWHM$(E)=L_0+\\alpha E^{2}$ curves are fitted to the results; one component representing the lower energy values that are controlled mainly by thermal diffusion (grey curve) and another component representing higher energies mainly controlled by collisional friction, since the FWHM values should return to match those of a cold target case when $E>>k_{B}T$. The $L_0$ and $\\alpha$ values found from the grey and black curves are also shown on each panel of Figure~\\ref{hot_errors}.\n\nTo illustrate, for the $T$=10~MK case, the FWHM values match those of the cold case (red or green dashed line) after $\\sim10$~keV, for both the $\\delta=4$ and $\\delta=7$ cases. This is because the temperature diffusion is limited to energies below $\\sim8$~keV (grey curve); Figure~\\ref{hot_errors} clearly shows this transition. Therefore for the 10 MK case, the $8-25$~keV fits (purple) match that of the higher energy black fits and cold cases reasonably well for both $\\delta=4$ and $\\delta=7$. By $T$=20~MK, the energy range between $8-25$ keV is not so well fitted by a curve of the form of Equation (\\ref{L}) and occurs because the trend of the FWHM moves from being dominated by the effects of thermal diffusion to being dominated by the effects of collisional friction at approximately $15$ keV, right in the middle of the range used for the fit. This is clear for the $\\delta=4$ case but harder to see for $\\delta=7$ case due to the smaller values of $\\alpha$. The $\\alpha$ values of the friction-dominated fits (black curves) are only approximately the same as for the cold plasma case after $\\sim17$ keV. Also the diffusion at 20~MK noticeably influences the length values at all energies plotted, with the FWHM values above $\\sim$17~keV lying above those for the cold case. By $T$=30~MK, the entire plotted energy range and the fitted energy range between $8-25$ keV is mainly controlled by thermal diffusion and the $\\alpha$ values for both the $\\delta=4$ and $\\delta =7$ cases are similar. All plotted FWHM values are much larger than that of equivalent cold cases, over $10\\mbox{\\ensuremath{^{\\prime\\prime}}}$ at $1$ keV. For the $8-25$ keV fits, the $\\delta=4$ value is smaller than that of the equivalent cold case, and the $\\delta=7$ value is slightly larger.\n\n\\begin{sidewaysfigure*}\n\\centering\n\\includegraphics[width=18.5cm]{fig7a.pdf}\n\\includegraphics[width=18.5cm]{fig7b.pdf}\n\\caption[Plots of FWHM versus electron energy for the finite temperature plasma simulation runs.]{Plots of FWHM versus electron energy for $\\delta=4$ (top) and $\\delta=7$ (bottom), and $T=$ 10 MK, 20 MK and 30 MK (left to right) and $n=1\\times10^{11}$ cm$^{-3}$. Fits of the form FWHM=$L_0+\\alpha E^{2}$ are shown on each plot. The red and green dashed curves show the corresponding results for the  beamed, cold plasma case with pitch angle scattering (scenario (B)). The purple dashed lines show the best fit in the energy range 8-25~keV, the range used in the fit to {\\em RHESSI} observations. Also shown, are the two-component fits, one component representing the thermal diffusion at lower energies (grey dashed curve) and another component representing collisional friction that dominates at higher energies (black dashed curve).}\n\\label{hot_errors}\n\\end{sidewaysfigure*}\n\\clearpage\n\n\\subsubsection{Inferring the acceleration region length $L_0$ and density $n$}\\label{landn}\n\nThe thermal diffusion-component (grey dashed) curves in Figure~\\ref{hot_errors} use Equation~(\\ref{L}) to fit the FWHM values at lower energies, and hence give $L_0$, the inferred length of the acceleration region. For a given temperature, the values of $L_0$ found for both $\\delta=4$ and $\\delta=7$ are approximately the same, with an average value of $25\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=10~MK, $29\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=20~MK and $34\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=30~MK.\n\nFor the reasons discussed above, it is more likely that the $L_0$ values obtained from the $\\sim8-25$~keV fits will be most reliable from observation. Averaged over the two values of $\\delta$, these give values of $24\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=10~MK, $28\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=20~MK, and $33\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for $T$=30~MK.  These values are only slightly smaller than the values found from the grey dashed fits at lower energies. However, if viable, as low an energy as possible should be used to find the inferred value of $L_0$. Figure \\ref{hot_runs_2} (left) plots the values of $L_0$ found for the thermal diffusion-dominated (grey curve) and $8-25$ keV fits against $T$. Each is fitted with a curve of the form,\n\\begin{equation}\nL_0(T,n)=L_{0}(T=0)+\\xi(n) T^{2}=23\\mbox{\\ensuremath{^{\\prime\\prime}}}.5+\\xi(n) T^{2} \\,\\,\\, .\n\\label{eq:LT}\n\\end{equation}\nBy fitting Equation (\\ref{eq:LT}) to each, $\\xi$ is found for both ``global'' and thermal diffusion-dominated fits, and an average value of $\\langle\\xi(n=1\\times10^{11})\\rangle=0.011$ arcsecond MK$^{-2}$ is calculated empirically from the four fits.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=16.5cm]{fig8.pdf}\n\\caption[Inferred acceleration region length $L_0$ and quadratic fit parameter $\\alpha$ versus plasma temperature.]{{\\it Left panel}: $L_0$ versus $T$. The blue band represents the area containing the $L_{0}$ values for both the $\\delta=4$ and $\\delta=7$ low-energy and 8-25 keV fits from Figure \\ref{hot_errors} (grey and purple respectively), plotted against temperature $T$. For each of these curves, a function of the form $L_0=L_{0}(T=0)+\\xi T^{2}$ is fitted, and average values of $\\langle L_{0}(T=0)\\rangle=23\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$ and $\\langle\\xi(n=1 \\times 10^{11}$~cm$^{-3})\\rangle=0.011$ [arcsec MK$^{-2}$] are found, with $L_0=\\langle L_{0}(T=0)\\rangle+\\langle \\xi\\rangle T^{2}$ represented by the orange dashed line.  {\\it Right panel:} $\\alpha$ from the 8-25 keV fits (Figure~\\ref{hot_errors}) versus $T$, for $\\delta=4$ (red) and $\\delta=7$ (green).}\n\\label{hot_runs_2}\n\\end{figure*}\nTo summarize, if the size $L_0(T=0)$ and number density $n$ of the region have been inferred from a cold target analysis, and $n$ is close to $n=1\\times10^{11}$ cm$^{-3}$ (as it must be for a viable thick target coronal source to appear), then {\\it the actual extent of the acceleration region is less than would be inferred using a cold target formula.}  Quantitatively, the actual size of the acceleration region $L_0$ can be approximated by the expression\n\\begin{equation}\\label{T-correction}\nL_0=L_0(T=0)-0.011 \\, T^{2} \\,\\,\\, ,\n\\end{equation}\nwhere $L_0(T=0)$ is the value deduced from a fit using the cold target formula to an observation.\n\nThe right panels in Figure~\\ref{hot_runs_2} also show how $\\alpha$ from the $8-25$ keV fits changes with $T$ for both $\\delta=4$ and $\\delta=7$. For $\\delta=4$, $\\alpha$ decreases between $T$=10~MK and $T$=30~MK.  This is expected, since for higher temperatures, particle diffusion is controlling the shape of the curve and the $\\delta=4$ cold target case has a relatively high $\\alpha$ value. However this is not the case for $\\delta=7$, where between 10-30 MK $\\alpha$ increases with $T$.\n\\\\\\\\\nFrom the plots in Figure \\ref{hot_runs_2}, the values of $\\alpha$ for the fits between 8-25 keV can be used to infer a number\ndensity from observations. Two cold target approaches are used: (1) $\\alpha={1}\/2Kn$, and (2) an extended source Gaussian input as found from Equation (\\ref{eq:fexgauss}). Also, using the results from the cold-plasma cases (2) can be expanded to account for the initial beaming of the distribution so that a range of $n$ can be found.  Finally, (3) can be used, which is the same as (2) but accounts for pitch angle scattering.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{fig9.pdf}\n\\caption[Cold plasma fits are applied to the different hot plasma simulation curves to determine an inferred density that can be compared with the actual density of the region.]{Cold plasma fits are applied to the different hot plasma simulation curves to determine an inferred density that can be compared with the actual density of the region: (1) (red lines): $\\alpha_{1}={1}\/2Kn$. (2) (blue hashed areas): an extended Gaussian injection model with no pitch angle scattering that is initially either beamed (Equation~(\\ref{eq:fexgauss})) or isotropic (found from the cold plasma simulation -- see Figure~\\ref{fig:cold_runs}). (3) (orange regions): as for (2), but with collisional pitch angle scattering included.  For both (2) and (3), the spread in $n$ occurs due to different electron pitch angle distributions from completely beamed to isotropic. The highest inferred values for $n$ are for a completely beamed distribution.}\n\\label{new_simn}\n\\end{figure*}\n\nThe inferred values of $n$ for $T=10, 20, 30$~MK and for $\\delta=4,7$ are shown in Figure~\\ref{new_simn}. For $\\delta=4$ the largest inferred value  is $\\sim1.7$ times larger than the actual density and the smallest is around three times smaller; for $\\delta=7$, the largest value is $\\sim3.3$ times larger and the smallest value is again about 3 times smaller. In general, (1) (cold target, point injection, red lines) produces the largest differences, which is not surprising since the input was an extended Gaussian, rather than point-injection, source. However, even this simple analytical case, that accounts very poorly for the true physical properties of the electron distribution, only increases the number density by a factor of about $3$ (for a beamed finite temperature case). (2) and (3) (extended injection models, without and with collisional scattering in the target, respectively) that do not account for the finite temperature of the plasma provide an inferred value for $n$ that is quite close to the true value of $n$, with the biggest uncertainty due to the unknown degree of beaming of the injected distribution.\n\n\\section{Discussion and conclusions}\\label{ref:chap2_con}\nThe aim of Chapter \\ref{ref:Chapter2} was to understand how the presence of different injected pitch angle distributions, plus the effects of collisional pitch angle scattering and of a finite target temperature change electron transport through a plasma and hence the spatial properties of compact hard X-ray sources in solar flares.\n\nThe simulations show three main results:\n\n\\begin{enumerate}\n\n\\item{Collisional pitch angle scattering alone does not dramatically change the behaviour of source length with electron energy.}\n\n\\item{Beaming of the initial electron pitch angle distribution does produce a significant change in the variation of the length of the X-ray source with energy; distributions that are initially beamed produce a larger variation of length with energy, a consequence of the fact that the collisional stopping distance is now projected onto the direction defined by the guiding magnetic field.  The difference in the coefficient $\\alpha$ can be up to a factor of $6$ if a beamed approximation is used for a distribution that is in fact completely isotropic. The uncertainty in the initial angular distribution of the injected electrons produces the largest uncertainty in the inferred number density $n$.}\n\n\\item{The finite temperature of the target atmosphere leads to thermal diffusion, both in energy and space, and an increase of the inferred acceleration region length. The FWHM versus energy consists of two competing components, one due to thermal diffusion that is dominant at lower energies, and another due to advection that is dominant at higher energies.  Which component predominates depends on a complicated way on the temperature of the region, on the density $n$, and even on the spectral index $\\delta$. Therefore the use of a cold target approximation with a single fitted curve to infer properties of the acceleration region should always be used with caution. The results show that applying a cold model to a warm plasma changes the inferred acceleration length $L_0$ by several arc seconds (see Equation~(\\ref{T-correction})) and the inferred number density by up to a factor of $3$ (in either direction), depending mainly on the initial beaming of the electron distribution (see Figure~\\ref{new_simn}).}\n\\end{enumerate}\n\nThe influence of the effects studied in this chapter also influence the determination of other quantities, such as the acceleration region filling factor $f$ (the fraction of the apparent source volume in which acceleration occurs) and the specific acceleration rate (the fraction of the ambient electron population that is accelerated per unit time). The filling factor $f$ is defined by\n\n\\begin{equation}\\label{filling-factor}\nf = \\frac{EM}{n^2 V} \\,\\,\\, ,\n\\end{equation}\nwhere $V = (\\pi W^2\/4) L_0$ is the volume of the acceleration region, determined from the inferred value of $L_0$ and the observed lateral extent $W$ of the (cylindrical) acceleration volume, and the emission measure $EM$ is determined from, for example, fits to the spatially-integrated soft X-ray spectrum of the flare.  The effects studied in this chapter show that in general, application of a one-dimensional cold target formula leads to erroneously high inferred values for both the acceleration region length $L_0$ (see Figure~\\ref{hot_errors}) and density $n$ (see Figure~\\ref{new_simn}).  Use of such erroneously high values of $L_0$ and $n$  leads to an overestimate of the denominator in Equation~(\\ref{filling-factor}) and so an underestimate of the filling factor $f$.\n\nIn a study of 24 coronal thick target events, using the one-dimensional cold target result~(\\ref{cold_theory}) to estimate $L_0$ and $n$, \\citet{2013ApJ...766...28G} found filling factors $f$ that were generally somewhat less than unity. The results of this chapter therefore lend support to a value of $f$ being even closer to unity than previously thought. Indeed, given that $f$ cannot exceed unity, this may place constraints on the allowable values of $n$ and $L_0$.  And, since the inferred values of $n$ depend significantly on the pitch angle distribution of the injected electrons, this could conceivably be used to constrain the form of the injected pitch angle distribution.  In particular, broad injected distributions lead to relatively small values of the coefficient $\\alpha$ (see Figure~\\ref{fig:cold_runs}) and hence to inferred densities that are higher than the actual target density (Figure~\\ref{new_simn}).  Correcting for such an effect in the interpretation of a particular event could imply an actual target density that was too small to be compatible with the observationally-inferred emission measure, thus ruling out the hypothesis of a broad injected distribution of accelerated electrons.\nThe inference of the acceleration region length $L_0$, lateral extent $W$, and density $n$ also gives the number of electrons available for acceleration:\n\n\\begin{equation}\\label{number_of_particles}\n{\\cal N} = n \\, V = n \\, \\left ( \\frac{\\pi W^2}{4} \\right ) \\, L_0 \\,\\,\\, .\n\\end{equation}\nThis, combined with the inference of $d{\\cal N}(E_0)\/dt$, the rate of electron acceleration beyond energy $E_0$ (obtained rather straightforwardly from spatially-integrated hard X-ray data) provides the value of the {\\it specific acceleration rate} (electrons~s$^{-1}$ per ambient electron)\n\n\\begin{equation}\\label{acceleration-rate}\n\\eta(E_0) = \\frac{1}{\\cal{N}} \\, \\frac{d {\\cal N}(E_0)}{dt} \\,\\,\\, .\n\\end{equation}\nOverestimating the value of the acceleration region volume and density through the use of an over-simplistic one-dimensional\ncold target model thus causes an overestimate of ${\\cal N}$ and, since $d{\\cal N}(E_0)\/dt$ is fixed, this causes an underestimate\nof $\\eta(E_0)$.  In their multi-event study, \\citet{2013ApJ...766...28G} found typical values for $\\eta(E_0=20~{\\rm keV})$ were of the order $10^{-2}$~s$^{-1}$ and they compared these values with those predicted from different acceleration models: large scale electric field acceleration (super-Dreicer) \\cite[e.g.][]{1993SoPh..146..127L,2008AIPC.1039....3E} and stochastic acceleration \\cite[e.g.][]{1996ApJ...461..445M,2012ApJ...754..103B}, both of which could be made to account for such values. The application of the physically realistic source models considered herein will increase $\\eta$ even further, and place more profound constraints on the electron acceleration mechanism.\n\n\n\\chapter{The temporal and spatial evolution of solar flare coronal X-ray sources}\n\\label{ref:Chapter3}\n\n\\normalsize{\\it This is work can be found in the publication \\cite{2013ApJ...766...75J}}\n\n\\section{Introduction to the chapter}\\label{ref:intro}\nUsing simulations, Chapter \\ref{ref:Chapter2} studied the variation of coronal X-ray source lengths with electron energy. This study was motivated by recent {\\it RHESSI} observations of dense coronal X-ray sources and the information that could be inferred from the length increases with X-ray energy. In their study \\cite{2008ApJ...673..576X} found that the coronal X-ray loop width, the direction perpendicular to the guiding field, of each event also increased with X-ray energy. \\cite{2011ApJ...730L..22K} examined the width changes of one coronal X-ray source and found that the loop width increased proportionally with X-ray energy. However, unlike increases in coronal loop length, changes in width are more difficult to explain since the electrons are bound to the guiding field and collisional cross field transport should be negligible. \\cite{2011ApJ...730L..22K} and \\cite{2011A&A...535A..18B} inferred that the width increase could be due to the magnetic diffusion of field lines perpendicular to the direction of the field, which is caused by the presence of magnetic turbulence within the loop. The interesting spatial trends shown by these observations and the information inferred indicate the usefulness of observing coronal X-ray loop spatial properties. Chapter \\ref{ref:Chapter3} now goes on to further explore this, by examining how the spatial properties: length, width and position of such coronal X-ray sources change in time before, during and after the impulsive phase of the flare.\n\n\\subsection{Past studies of coronal loop spatial properties}\nPast observations concentrated on studying changes in coronal loop positions, due to the difficultly in trying to quantitatively infer source size changes with {\\it RHESSI} and other instruments, as was discussed in Chapter \\ref{ref:Chapter1}.\n\\cite{1996ApJ...459..330F} used the Soft X-ray Telescope (SXT) on-board {\\it Yohkoh} to observe the changing locations of post flare loops, which was interpreted as the decrease in height that open field lines undergo after they have reconnected to form closed loops. This study looked at two long duration events near the solar limb and found the presence of loop shrinkage that matched the shrinkage predicted by a simple model of the reconnecting field, but overall the entire flare loop system grew with time.\n\\cite{2003ApJ...596L.251S}, \\cite{2004ApJ...612..546S}, \\cite{2006A&A...446..675V} and \\cite{2009ApJ...706.1438J} all noted a decrease in the altitude of coronal loop top sources\nduring the impulsive phase of the flare, until the peak X-ray emission and an increase in altitude after the impulsive phase.\n\\cite{2003ApJ...596L.251S} also found evidence for an above the loop top source and interpreted the situation as the formation\nof a reconnection current sheet between the loop top source and the higher coronal source.\n\\cite{2006A&A...446..675V} interpreted the decrease as a collapsing magnetic trap \\citep{1997ApJ...485..859S,2004A&A...419.1159K}.\nThe contraction and expansion of the loop source has also been observed in other wavelengths of EUV \\citep{2009ApJ...696..121L,2009ApJ...706.1438J}\nand radio \\citep{2005ApJ...629L.137L,2010ApJ...724..171R}.  \\cite{2010ApJ...724..171R} found both changes in the radio loop span and height with time.\nMore recently, \\cite{2012ApJ...749...85G} looked for evidence of collapsing fields using Solar Dynamics Observatory ({\\it SDO\/AIA} and {\\it HMI}) observations.\nThe loops rose slowly and then moved into a collapse phase during the impulsive phase of the flare, where the loop tops contracted. Lower loops contracted earlier than higher loops and the loop contraction was interpreted as a reduction of magnetic energy as the system relaxed to a state of lower energy, that is, relaxation theory \\citep{1974PhRvL..33.1139T}.\n\nIn this chapter, three flares with dense coronal X-ray loops are studied, in order to find how the emission lengths, widths and positions change with time at three energy ranges\nbetween 10-25 keV. Using spectroscopy, it is also found how plasma parameters such as emission measure and plasma temperature vary with time, for each flare.\nUsing a combination of imaging and spectroscopy parameters, the X-ray loop corpulence, volume, plasma number density, thermal pressure\nand thermal energy density are inferred during the time evolution of the flare. This chapter will also propose some explanations describing the trends and the processes occurring within the coronal loops.\n\n\\section{Chosen events with coronal X-ray emission}\n\\begin{table}\\scriptsize\n\\caption{Table showing the main parameters of Flares 1, 2 and 3.}\\label{table1}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n & GOES Class & Date & Obs. time & Peak time (10 keV) & Footpoints (30-40 keV) \\\\\n\\hline\nFlare 1 &  M3.0 & 23-August-2005 & 14:22:00-14:40:00 & 14:30:00 & 14:36:00 onwards\\\\\n\\hline\nFlare 2 &  M4.1 & 14\/15-April-2002 &23:58:00-00:20:00 & 00:12:00 & 00:05:00 onwards\\\\\n\\hline\nFlare 3 & M2.6 & 21-May-2004 & 23:42:00-23:58:00 & 23:50:00 & 23:42:00 onwards\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nThe three events studied are: 23rd August 2005 from 14:22:00 (Flare 1),\n14th\/15th April 2002 from 23:58:00 (Flare 2) and 21st May 2004 from 23:40:00 (Flare 3).\nAll three flares share similar characteristics: {\\it GOES\\,} M-class flares with similar lightcurves, strong coronal X-ray loop top emission\nand only relatively weak footpoint emission. Since the aim of the study is to examine how the properties of coronal X-ray sources\nvary with time, these events were chosen as they show a clear coronal X-ray source throughout the rise, peak and decay stages\nof X-ray emission and their spatial properties have been previously studied in energy by \\cite{2008ApJ...673..576X} and \\cite{2011ApJ...730L..22K}.\nThe coronal X-ray emission during each flare appears clearly, for study, up to $\\sim 30$ keV and appears as a simple loop-like shape\nconnecting weak 30-40 keV HXR footpoints, during certain time intervals. The main parameters of each flare: {\\it GOES} class, date,\nobservation time, peak time at 10 keV and the time of footpoint appearance are given\nin Table~\\ref{table1}. As discussed in Chapter \\ref{ref:Chapter2}, length variations of each of these coronal loops with X-ray energy were studied by \\cite{2008ApJ...673..576X}.\nLength and width changes with X-ray energy for Flare 2 were also studied by \\cite{2011ApJ...730L..22K}. It should be noted that Flare 1\nand Flare 3 show similar results as Flare 2 in \\cite{2011ApJ...730L..22K}, where both the length and width increase with\nenergy as $\\sim\\epsilon^{2}$ and $\\sim\\epsilon$ respectively but this chapter will concentrate on the size, position and spectral parameter\nchanges with time.\n\\subsection{Lightcurves for each event}\nThe lightcurves for each event are shown in Figure \\ref{fig:is_paras} (top row). Flares 1 and 3 have similar lightcurves;\na simple shape with one peak. The lightcurves of four energy bands between 10-40 keV for Flare 1 are shown in Figure \\ref{fig:is_paras} (top row, left plot). The study of this event begins\nat 14:22:00. At this time, X-ray emission from the 10-20 keV energy bands are slowly rising and reach a peak at $\\sim$14:30:00.\nAfter this point, there is a gradual decrease in X-ray emission until $\\sim$ 14:40:00, and continues to decrease until 14:50:00, where {\\it RHESSI} enters into a night phase and can no longer view the flare. In the 20-40 keV band there are a series of peaks between the observation range of 14:22:00 and 14:40:00.\nThe lightcurve for Flare 3 is shown\nin Figure \\ref{fig:is_paras} (top, right plot). From the start of the study at 23:40:00, the X-ray emission from four energy bands between 14-40 keV rises and peaks around 23:50:00. After this peak, the X-ray emission from each energy band decreases. The lightcurves for Flare 2 are more complex and are shown\nin Figure \\ref{fig:is_paras} (top row, middle plot) for four energy bands from 10-40 keV. During the observational time there are two main peaks in X-ray emission at $\\sim$00:03:00 and $\\sim$00:12:00, possibly more peaks, which are most prominent in the 20-40 keV energy range.\n\n\\begin{figure*}\n\\vspace{-10pt}\n\\textbf{Flare 1}\\\\\n\\includegraphics[width=15.8cm]{aug232005_visplot_levels_newc.pdf}\n\\vspace{-10pt}\n\\textbf{Flare 2}\\\\\n\\includegraphics[width=15.8cm]{apr142002_visplot_levels_newc.pdf}\n\\vspace{-10pt}\n\\textbf{Flare 3}\\\\\n\\includegraphics[width=15.8cm]{may212004_visplot_levels_newc.pdf}\n\n\\caption[CLEAN images and Vis FwdFit contours for Flares 1,2 and 3, at different observational energies and times.]{CLEAN background image (green) at one time and over-plotted Vis FwdFit contours (50\\% maximum intensity)\nat four times and energies of 10-12 keV (left), 12-15 keV (middle) and 15-20 keV (right) for Flare 1 (top) and Flare 2 (middle) and 14-16 keV (left), 16-20 keV (middle) and 20-25 keV (right) for Flare 3 (bottom). The asterisks denote the loop position for each time. The cyan Vis FwdFit contour matches the time of the CLEAN image.}\n\\label{images_flare}\n\\end{figure*}\n\n\\subsection{Imaging of each event}\nEach event was studied in X-rays with {\\it RHESSI} using the imaging algorithms of CLEAN \\citep{1974A&AS...15..417H, 2002SoPh..210...61H},\nPixon \\citep{1993PASP..105..630P,1996ApJ...466..585M} and Vis FwdFit \\citep{2002SoPh..210...61H,2007SoPh..240..241S} (see Chapter \\ref{ref:Chapter1}, Section \\ref{intro_rhessi}).\nFirstly, each event was studied using only CLEAN and Pixon. These imaging algorithms were used to confirm the loop shape of each coronal source and find the energy ranges over which an X-ray  coronal source was present in each flare. Confident that the\nchosen events only had a simple and similar loop shape given by CLEAN and Pixon, each event was then studied using Vis FwdFit by fitting a curved elliptical Gaussian to each loop. It is important that the coronal source has a simple, singular loop-like shape so that Vis FwdFit can effectively fit a curved elliptical Gaussian to the X-ray visibilities and give reliable and realistic estimates with errors for the source parameters. Amongst other parameters, Vis FwdFit provides the following spatial parameters: loop length FWHM (full width half maximum), loop width FWHM and the $(x,y)$ centroid position of the loop, which are the mean coordinates of the loop shape. For this study of radial position, the $(x,y)$ position of the loop top is required, not the mean position of the loop shape itself. However, Vis FwdFit creates the elliptical Gaussian shape by placing a set of circular Gaussian sources along the length of the loop. Therefore, the coordinates for the coronal loop top position were simply obtained by extracting the coordinates of the central circular Gaussian.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=7.15cm]{aug232005_visfits}\n\\hspace{50pt}\n\\includegraphics[width=7.15cm]{apr142002_visfits}\n\\includegraphics[width=7.15cm]{may212004_visfits}\n\\caption[The observed {\\it RHESSI\\,} visibility amplitudes plus the error bars at one chosen time bin for Flare 1, Flare 2 and Flare 3.]{The observed {\\it RHESSI\\,} visibility amplitudes (black stars) versus RMC (the minor x-axis gives the position angle of the RMC\/180$^{\\circ}$) plus error bars (blue) at one chosen time bin for Flare 1 (top), Flare 2 (bottom left) and Flare 3 (bottom right). The fitted elliptical Gaussian model is shown by the red line. The difference between the observed and fitted amplitudes is shown in the bottom plots by the green triangles.}\n\\label{visfits}\n\\end{figure*}\nThis is important as a loop that is very curved and approaching the shape of a ring will pull the shape centroid towards the ends of the loop; this could mask small changes in loop position with time and\/or energy. This was especially significant for the large looped shape of the 23rd August 2005 event (Flare 1). In order to study changes in time at a specific energy range, the coronal sources of Flares 1, 2 and 3 were imaged over two minute intervals where possible and four minute intervals where the count rates were lower, during the rise and decay phases of the X-ray emission.\nThe exact time bins used for each flare are shown in Figures \\ref{fig:is_paras} and \\ref{fig:com_paras}.\nThe energy ranges of 10-12 keV, 12-15 keV and 15-20 keV were chosen for Flare 1 and Flare 2  and the energy\nranges of 14-16 keV, 16-20 keV and 20-25 keV were chosen for Flare 3, allowing the spatial parameters of the loops to be studied in both energy and time. The energy ranges from 10-20 keV or 14-25 keV were chosen since the X-ray loop appears clearly over these energy ranges allowing Vis FwdFit to be used reliably. An example of how well the loop model fits the observed {\\it RHESSI\\,} visibility amplitudes of each rotating modulation collimator (RMC) is shown in Figure \\ref{visfits}. Images for Flares 1, 2 and 3 are shown\nin Figure \\ref{images_flare} respectively. Figure \\ref{images_flare} shows that Flare 1\nis a limb event while both Flares 2 and 3 are disk events. The main parameters of each flare are given in Table~\\ref{table1}. Figure \\ref{images_flare} plots background CLEAN images of the coronal source for each flare at one chosen time interval\nand over-plots Vis FwdFit contours for selected time intervals, one corresponding to the same time interval as the CLEAN image.\nComparing the shape and size of the Vis FwdFit contour with the CLEAN background image at the selected time interval\nfor each flare shows good agreement between both algorithms. Qualitatively, Vis FwdFit also agrees well with the CLEAN and Pixon images at other times not shown in Figure \\ref{images_flare}. For each algorithm, Flare 1 used detectors 3 to 8 due to its relative large size while Flares 2 and 3 used detectors 3 to 7. It should be noted that sources sizes observed with the CLEAN algorithm can be manually changed using a parameter known as the \\textit{clean$\\_$beam$\\_$width$\\_$factor} parameter. This parameter can be thought of as representing the instrument point spread function (PSF) which is convolved with the image reconstruction from CLEAN (see Chapter \\ref{ref:Chapter1}, Section \\ref{rhessi_imaging} and \\cite{DennisPernak2009}). However since CLEAN and Pixon were only used to confirm the qualitative shape of the looped sources, the fact that the source sizes observed with CLEAN can be manually changed was of no initial concern and a \\textit{clean$\\_$beam$\\_$width$\\_$factor}$=$1.8 was chosen for each flare, since this was suggested by the observations of recent HXR chromospheric sources \\citep*{2010ApJ...717..250K}. However, an additional study of source width was performed by plotting the intensity profiles of the CLEAN, Pixon and Vis FwdFit images, along a line through the centre of the loop top, midway and perpendicular to the line connecting the weak footpoints. This is shown in Figure \\ref{cuts} (top left). A measure of loop width was then found by: (1) the standard deviation ($std$) of the profile distribution and (2) the $std$ of a Gaussian fit to the profile distribution, providing the loop width FWHM via FWHM$=2\\sqrt{2\\ln2}\\;std$. Figure \\ref{cuts} shows good agreement between the widths at each time and those found from Vis FwdFit. The standard deviation of the profile in particular shows for Flare 1, a \\textit{clean$\\_$beam$\\_$width$\\_$factor}$\\sim$3.0 matches better with the results of Vis FwdFit than a \\textit{clean$\\_$beam$\\_$width$\\_$factor}$\\sim$1.8. From Figure \\ref{images_flare}, it should be noted that the results for Flare 3\nare probably less reliable than those of Flare 1 and Flare 2. From the position of the footpoints in the CLEAN image,\nit appears as through the southern `loop leg' is tucked underneath the observer's line of sight.\nThis means that it is harder for Vis FwdFit with a loop to fit it with a correctly shaped loop and usually fits it with a loop\nthat is slightly too large or departs from a loop-shape. Hence, events analysed for study with Vis FwdFit must be chosen with caution.\n\\begin{figure*}\n\\vspace{-20pt}\n\\hspace{-35pt}\n\\centering\n\\includegraphics[width=16.5cm]{aug232005_ic_clean}\n\\caption[For Flare 1, a comparison of the standard deviation of a chosen intensity profile along a line through the loop top perpendicular to line midpoint joining the footpoints, found from the second moment of the distribution, for both CLEAN and Vis FwdFit algorithms.]{{\\it Top left:} Flare 1 CLEAN image at one time and energy range of 10-20 keV. The positions of 30-40 keV footpoints are shown by green dots. Loop top intensity profiles are found along the grey band, perpendicular to line midpoint joining the footpoints. {\\it Top right:} total intensity profile (grey) and a Gaussian fit to the profile (light blue). {\\it Bottom left:} comparison of the standard deviation of the intensity profile, found from the second moment of the distribution, for CLEAN intensity profiles, with either a \\textit{clean$\\_$sigma$\\_$beam$\\_$width}$=1.8$ (dark blue) or $3.0$ (light blue) and the Vis FwdFit intensity profile (green). The values given by the Vis FwdFit algorithm are also plotted (pink). {\\it Bottom right:} as bottom left, but for the Gaussian FWHM, as found from the fits to the image profiles or from the Vis Fwd Fit algorithm.}\n\\label{cuts}\n\\end{figure*}\nAs was discussed in Chapter \\ref{ref:Chapter2}, flares of this type only show relatively weak HXR emission from the chromosphere. It was found that Flare 1 has one weak but clear southern footpoint in the 30-40 keV range. This appears at $\\sim$14:36:00, after the peak emission time from 10-20 keV and 14 minutes after the start of the observation start time, corresponding to a bump in the 20-40 keV band shown in the lightcurve for this event. Flare 3 has two very weak HXR footpoints in the 30-40 keV band during the entire observational period. The lightcurve for this event shows that the 25-40 keV band follows the trend of the lower energy bands, all roughly peaking at 23:50:00. Flare 2 is more complex than Flares 1 and 3 due to its multiple lightcurve peaks but it also has two weak footpoints in the 30-40 keV energy range. These appear at $\\sim$00:05:00. Figure \\ref{fig:is_paras} shows the imaging parameters for Flare 1 (column 1), 2 (column 2) and 3 (column 3):\nloop width FWHM (row 2), loop length FWHM (row 3) and loop-top radial position (row 4) for each imaging\nenergy band. Above these plots, the lightcurve is plotted for each of the imaging energy bands to allow\ncomparison with changes in the spatial parameters. The dashed lines drawn in each plot represent\nthe time interval over which the peak X-ray emission occurs for the three energy bands.\nFor each flare, in general, peaks in the lightcurve represent changes of width, length and source position\nparameters with time and this will be discussed in Section \\ref{ref:results}\n\\subsection{Spectroscopy of each event}\nFor each of the imaging time intervals for Flares 1, 2 and 3, energy spectra\nwere created. The spectra of each flare at each time interval were fitted with a thermal component (continuum only), a non-thermal\ncomponent corresponding to thick target bremsstrahlung and two Gaussian line functions for the Fe and Ni lines at 6.7 keV and 8.1 keV. The total fit function used is v$\\_$th(continuum)$+$thick2$+$line$+$line. The thermal fits provide information about coronal loop plasma: the emission measure, $EM$, and\nthe plasma temperature, $T$, that can be used with the imaging parameters to infer the thermal plasma number density, pressure and energy density.\nSpectra for Flares 1, 2 and 3 are shown in Figure \\ref{fig:spectra} at three selected time bins corresponding to a rise, peak\nand decay stage of X-ray emission for each flare. The last two rows of Figure \\ref{fig:is_paras} plot how the thermal fit parameters $EM$ and $T$ vary with time for each flare.\n\n\\section{Spatial and spectral changes with time}\\label{ref:results}\n\\begin{figure*}\n\\vspace{-20pt}\n\\includegraphics[scale=.31]{aug232005_spect_new_135pos.pdf}\n\\includegraphics[scale=.31]{apr142002_spect_new_169pos.pdf}\n\\includegraphics[scale=.31]{may212004_spect_new_135pos.pdf}\n\\caption[Spectra for Flares 1, 2 and 3, at three chosen imaging time bins (during the X-ray rise, peak and decay stages).]{Spectra for Flares 1 ({\\it top}), 2 ({\\it middle}) and 3 ({\\it bottom}), at three chosen imaging time bins (during the X-ray rise, peak and decay stages). Black = data - background, Olive = background, red = thermal fit, green = thick target bremsstrahlung fit. The residuals are plotted below each plot.}\n\\label{fig:spectra}\n\\end{figure*}\n\\subsection{Emission measure and plasma temperature}\nFor all three events, the emission measure $EM$ rises throughout the observation times,\neither slowing or decreasing slightly during the last few minutes for each event.\nFor each event, the plasma temperature $T$ peaks before the peak in X-ray emission\nand then slowly decreases after this point, as noted by \\cite{1978ApJ...220.1137A}. The plasma temperature decreases much\nslower than if it were decreasing by thermal conduction only, suggesting prolonged energy release at the later decay stages of the flare.\n\n\\subsection{Loop width}\nFor all three flares, there is a general pattern for emitting loop width $W$ changes with time.\nBefore a peak in X-ray emission, the source width $W$ for each flare tends to decrease\nand after a peak in X-ray emission, $W$ for each flare increases.\nIt is also notable, that the rates of both width expansion and contraction (width change in time $dW\/dt$) are\napproximately the same, as can be seen in Figure \\ref{fig:dvwl}.\n\n{\\it Flare 1-} For all energies plotted, $W$ decreases until the peak X-ray emission at $\\sim$14:30:00-14:32:00 UT\nand then increases after this point. The largest change occurs for the highest energy of 15-20 keV.\nThis falls from $14.5''$ to $11.5''$ at 14:28:00-14:30:00 and then increases until 14:40:00\nwhere the width peaks at $\\sim 20''$, producing a final larger loop width than seen at the beginning of the observational time.\n\n{\\it Flare 2-} For the energies of 10-12 keV and 12-15 keV, $W$ decreases before the first peak in X-ray emission\nat $\\sim$00:03:00. After this point $W$ increases before dropping at 00:12:00-00:13:00 where there is another X-ray peak\nin the lightcurve. After this peak, $W$ continues to grow again. Before 00:03:00, the width of the 10-12 keV source\nfalls from around $\\sim$10$''$ to $\\sim5''$ and after 00:13:00 rises from $\\sim$6$''$ to $\\sim$13$''$. The 15-20 keV\nsource also shows this pattern except there is a larger peak at 00:05:00 and then a more pronounced\ndecrease in $W$ until 00:12:00-00:13:00 UT.\n\n{\\it Flare 3-} Again, the change in loop width $W$ with time follows a similar pattern as Flares 1 and 2. For all three energies\nranges considered, $W$ decreases from 23:42:00 to the peak in X-ray emission at 23:50:00.\n$W$ then increases after this time. The 14-16 keV and 16-20 keV sources $\\sim$ fall\nfrom 7-8$''$ at 23:42:00 to $\\sim$5$''$ at 23:50:00 and then increase up to 8-9$''$ at 23:58:00.\nThe 20-25 keV source falls from $11''$ to $5''$. Note the missing data at the fourth and sixth time bins for the 20-25 keV energy range, as Vis FwdFit was unable to fit successfully at these times, for this energy range.\n\n\\subsection{Loop length}\n\nAs with loop width, each flare shows general pattern for emitting loop length $L$ changes with time. Before a peak in X-ray emission,\nthe source length for each flare tends to decrease and after a peak in X-ray emission, the source length either\nincreases slightly or remains approximately constant within the errors.\n\n{\\it Flare 1 - } For all energies plotted, there is a rapid decrease in $L$ until the peak X-ray emission\nat $\\sim$14:30:00-14:32:00. After the peak, $L$ remains approximately constant (within the error).\nThe smallest decrease in loop length before the X-ray peak emission occurs for the 10-12 keV source which falls\nfrom $54''$ to $38''$. The decrease in $L$ grows with energy and the highest drop in $L$ occurs\nfor the highest energies plotted at 15-20 keV, which fall from $74''$ to $38''$ before the peak in X-ray emission.\n\n{\\it Flare 2 - } The pattern for length changes with time are similar to that of the width changes. For the 10-12 keV source,\n$L$ before the 00:03:00 peak in the lightcurve, falls from $\\sim$26$''$ to $\\sim$20$''$, rises\nto $\\sim$24$''$ at 00:11:00, falls to $\\sim$20$''$ at 00:15:00 and then increases to $21''$ at 00:17:00. Due to the multiple peaks, $L$ tends to increase after a peak in the lightcurve.\n\n{\\it Flare 3 - } For all energies, $L$ drops rapidly between 23:42:00 and 23:50:00 from the range of $25''$ to 30$''$\nto $\\sim$10$''$ for all energies plotted. Again as with Flare 1, after the X-ray peak the length of the loop remains\napproximately steady until the final plotted time of 23:58:00 for all energies.\n\n\\subsection{Loop radial position}\n\nFor all three flares, peaks in each X-ray lightcurve tend to denote times where the trends in loop radial position $R$ change.\n\n{\\it Flare 1- }Since this is a limb event,\nthe radial position $R$ can show whether the source is moving away or towards the limb. Before the X-ray emission peak\nat 14:30:00-14:32:00, the source moves towards the limb, falling a distance of $\\sim2''$ at 10-12 keV, 12-15 keV and 15-20 keV.\nThen after the peak, the source moves away from the limb. Plotting the actual source positions, shows that the\nentire loop structure moves in a U-shape during the time interval of 14:22:00-14:40:00. This was also seen for a number of flares in \\cite{2008ApJ...686L..37S}. From the\nresults of Flare 1, the changes in $R$ with time are comparable to the width changes and smaller than the length changes. The largest change in position is only $2-3''$ while the width decreases by $3''$ and increases by $6''$.\nThe length shows the largest change with a decrease of $\\sim17''$ or more before the peak in X-ray emission.\n\n{\\it Flare 2 - }Overall, the radial positions $R$ of the 10-12 keV, 12-15 keV and 15-20 keV sources increase with time.\nAt points of peak X-ray emission, that is, at $\\sim$00:03:00 and $\\sim$00:12:00, there are changes in the gradient.\nThe slope steepens between $\\sim$00:03:00 and 00:12:00. Since this is a disk event, it is difficult to say whether there is a change\nin source altitude at these peaks (as for Flare 1).  At 10-12 keV, between 23:58:00\nand 00:20:00 the source radial direction changes by 9.5$''$. Therefore the overall change in position is larger than the individual changes in loop width and loop length for Flare 2.\n\n{\\it Flare 3 - }For all energies, the radial distance $R$ falls with increasing time. There does not seem to be any significant\ndifference in the radial distance trend after the X-ray emission peak at 23:50:00, apart from the steadier decrease in radial distance\nafter this time. As with Flare 2, Flare 3 is a disk source and hence it is difficult to determine if there is any altitude change. For all three energies, the radial distance decreases by $\\sim$8$''$ between 23:40:00 and 23:58:00.\n\n\\begin{figure*}\n\\vspace{-35pt}\n\\includegraphics[width=5cm]{aug232005_new_plots_3.pdf}\n\\includegraphics[width=5cm]{apr142002_new_plots_3.pdf}\n\\includegraphics[width=5cm]{may212004_new_plots_3.pdf}\n\\caption{{\\it Left:} 23-Aug-2005,\n{\\it middle:} 14-Apr-2002 and {\\it right:} 21-May-2004. row 1: lightcurves, row 2: width, row 3: length, row 4: radial position, row 5: emission measure and row 6: plasma temperature, vs. time.\nDashed lines: peak X-ray emission.}\n\\label{fig:is_paras}\n\\end{figure*}\n\n\\section{Corpulence, volume and other inferred parameters}\n\n\\subsection{Loop corpulence}\nIn the following sections of this chapter, the word `corpulence' will be used to define the shape of a loop. Here corpulence $\\mathcal{C}$ will be defined as the ratio of the loop width $W$ to loop length $L$\n\\begin{equation}\\label{corpulence}\n\\mathcal{C}=\\frac{W}{L}.\n\\end{equation}\nThis means that for a given loop length FWHM $L$, the loop corpulence $\\mathcal{C}$ will increase with increasing loop width FWHM $W$. Therefore a loop has a very high corpulence if $\\mathcal{C}>1$ and a low corpulence if $\\mathcal{C}<<1$. For a corpulence of 1, the length and width of the loop are equal. Loop corpulence for each event is plotted in the second row of Figure \\ref{fig:com_paras} for Flares 1, 2 and 3. In general, the loop corpulence increases with time. This is particularly noticeable for Flare 1, where the loop corpulence increase throughout the observation time for each of the three energy bands. For example, in the 12-15 keV band $\\mathcal{C}$ increases from $\\sim0.2$ to $\\sim0.45$. For Flare 2, the changing corpulence with time is more complex, as expected, and it follows the same trend of both the length and width parameters that are very similar for Flare 2. However, overall $\\mathcal{C}$ is larger at the final observational time than at the start time. The corpulence for Flare 3 is similar to that of Flare 1. Overall, it increases throughout the observational time and is larger at the end time than at the start time, with only a small dip at the time of peak X-ray emission. For the 12-15 keV band, $\\mathcal{C}$ increases from $\\sim0.3$ to $\\sim0.8$.\n\n\\subsection{Volume, number density, thermal pressure and energy density}\nFrom the loop width FWHM $W$ and loop length FWHM $L$, the general changes in the emitting loop volume, $V$, can be inferred\nfor each flare over time, at each energy band. It is assumed the volume of the loop is given by\n$V=\\pi W^{2}L\/4$ that is, assuming a cylindrical loop. The changes in emitting loop volume with time for Flares 1, 2 and 3 are plotted\nin Figure \\ref{fig:com_paras} (second row). In general, for all three events, the source volume decreases before a peak\nin X-ray emission and increases after a peak in X-ray emission. The changes in plasma number density, thermal pressure and thermal energy density can all be calculated using combinations of loop volume, emission measure\nand plasma temperature. The plasma number density, $n$, can be obtained via\n$n=\\sqrt{EM\/V}$, the pressure, $P$, from $P=nk_{B}T$, where $k_{B}$ is the Boltzmann constant\nand finally the energy density, $U=3nk_{B}T$. The variation of these quantities with time for Flares 1, 2 and 3\nare shown in Figure \\ref{fig:com_paras}  (third, fourth and fifth rows respectively). It should be noted that it is assumed that the entire loop volume is emitting, that is the filling factor $f=1$. This means that the calculated values for number density, thermal pressure and energy density are a lower limit, and will increase if $f\\le1$.\n\n{\\it Flare 1 - }As expected from the width and length results,\nthe loop volume falls between 14:22:00 and the peak in the X-ray lightcurve at $\\sim$14:30:00 and then rises\nafter this time for all three energies. For all three energy bands, the decrease and increase in loop volume occurs at roughly the same rate. The largest decrease is for the highest 15-20 keV band, falling from $\\sim4.7\\times10^{27}$ cm$^{3}$ to $\\sim1.4\\times10^{27}$ cm$^{3}$ at 14:29:00 and then rising after this time to $\\sim4.7\\times10^{27}$ cm$^{3}$ at the last observational time. The 10-12 keV band falls from $\\sim2.4\\times10^{27}$ cm$^{3}$ to $\\sim1.3\\times10^{27}$ cm$^{3}$ at 14:31:00 and then rises back to $\\sim2.5\\times10^{27}$ cm$^{3}$ at the final observational time. The number density, thermal pressure and energy density for all three energy bands tend to follow\nthe same pattern, rising to a peak at some time after the peak X-ray emission and then slowly decreasing. For the 10-12 keV band, the number density rises from $1.5\\times10^{10}$ cm$^{-3}$ at 14:22:00\nto $6.5\\times10^{10}$ cm$^{-3}$ at 14:35:00. It then falls to $\\sim5.6\\times10^{10}$ cm$^{-3}$ at 14:38:00. The 12-15 keV and 15-20 keV bands follow similar patterns, peaking at $\\sim$14:33:00.\nThe pressure rises from $\\sim40$ g\/[cm s$^{2}$] at 14:22:00\nand reaches $170$ g\/[cm s$^{2}$]  at 14:31:00, the time where the X-ray emission peaks. After this time the pressure\nremains approximately constant at $\\sim160-170$ g\/[cm s$^{2}$] until the last observation time where it falls to $\\sim140$ g\/[cm s$^{2}$]. The 12-15 keV and 15-20 keV energy bands peak at $\\sim$14:33:00.\nThe thermal energy density of the plasma is just the thermal plasma pressure multiplied by 3 and hence it follows the same pattern as plasma pressure throughout the observed duration of the flare.\nThe energy density peaks at $500$ ergs cm$^{-3}$ in the 12-15 keV band at $\\sim$14:33:00, after the peak X-ray emission.\n\n{\\it Flare 2 - }For the 10-12 keV and 12-15 keV sources, the loop volume falls until the first peak at 00:03:00\nand then increases until it reaches 00:12:00, drops at 00:13:00 and then increases again after this time.\nAs for Flare 1, the number density, thermal pressure and energy\ndensity all follow the same pattern for each energy band, rising to a time at or just after the peak in X-ray emission and then decreasing after this point. The highest number density, thermal pressure and energy density for each energy band occurs at 00:12:00 to 00:14:00, just after the peak in X-rays at 00:11:00-00:13:00. The number density peaks at around $18\\times10^{10}$ cm$^{-3}$ in the 12-15 keV band, while the thermal pressure and energy density peak at $\\sim500$ g\/[cm s$^{2}$] and $1500$ ergs cm$^{-3}$ respectively at this time and energy.\n\n{\\it Flare 3 - }The loop volume again falls before the peak in X-ray emission at 23:50:00\nand then rises again after this time, for all three energies.\nFor this flare again, the number density, thermal pressure and energy density all peak just after the peak in X-ray emission. At 23:50:00-23:52:00 the number density peaks at $23\\times10^{10}$ cm$^{-3}$, the thermal pressure peaks at $750$ g\/[cm s$^{2}$] and the energy density at $\\sim2300$ ergs cm$^{-3}$, in the 16-20 keV band.\n\n\\begin{figure*}\n\\vspace{-30pt}\n\\includegraphics[width=5cm]{aug232005_spect_compare_pc_4.pdf}\n\\includegraphics[width=5cm]{apr142002_spect_compare_pc_4.pdf}\n\\includegraphics[width=5cm]{may212004_spect_compare_pc_4.pdf}\n\\caption{{\\it Left:} 23-Aug-2005,\n{\\it middle:} 14-Apr-2002 and {\\it right:} 21-May-2004. row 1: lightcurves, row 2: corpulence, row 3: volume,\nrow 4: number density, row 5: thermal pressure and row 6: thermal energy density, versus time.}\n\\label{fig:com_paras}\n\\end{figure*}\n\n\\section{Summary and discussion}\n\nUsing visibility forward fitting, a dedicated study was performed for the first time of changing spatial and spectral properties\nof three coronal X-ray loops with time during the flare. All X-ray loops exhibited similar changes in both their spatial and spectral properties and hence the results indicate that a common process is\noccurring for all three events; the emitting flaring loop volume is decreasing before the peak in X-rays and increases after the peak in X-rays is reached.\nBefore the peak X-ray emission, the emitting lengths and widths of each coronal loop decreased with time, indicating that the X-ray emitting region of the loop volume\nwas contracting; there was a reduction in loop width and length, as the X-ray emission from the region grew. After the X-ray peak, the loop width increased at approximately the same rate as during the contraction stage. For Flares 1 and 3, with one peak in their lightcurves, once the minimum X-ray loop length was reached during the X-ray peak it remained approximately constant, at least within the errors of the results. It was found that a property defined as the loop corpulence (equal to the loop width divided by loop length), in general, increases with time during the observation time.\n\nSimilar to previous studies \\citep{1978ApJ...220.1137A,1980ApJ...242.1243S,1984ApJ...285..835G,1999ApJ...514..472M},\nspectroscopy for each event showed that the plasma temperature initially grew but began to decrease before the peaks in X-ray emission and emission measure. The emission measure for each flare generally grew with dips observed during the final observational times of Flares 2 and 3. At the same time, the number density, thermal pressure and energy density of the plasma\nalso increased as the X-ray emission grew.  The plasma temperature decreased much slower than by thermal conduction only, even during the X-ray decay phase, suggesting at later stages additional energy\nrelease is required \\citep[e.g.][]{2011A&A...531A..57K} to explain the longer lasting X-ray loop emission.\nFor the limb event (Flare 1), there is a decrease in loop altitude before the peak in X-ray emission of roughly $2''\\simeq1.4$ Mm, which is comparable in magnitude to the decrease in loop width but overall, the largest changes occurred for the X-ray loop volume, not the X-ray loop position. Decreases in loop altitude before the peak in X-ray emission have been well noted before and are often referred to as coronal implosion or loop contraction. Some of these observations were briefly discussed in the introduction of this chapter (Section \\ref{ref:intro}).\n\n\\cite{2004ApJ...612..546S} also studied the loop position changes of Flare 2. From their observations, they did conclude\nthat the loop was indeed decreasing in altitude before the first peak in the lightcurve at 00:04:00.\nThey concluded that the loop decreased by $\\sim2''$ over a 4 minute period, which is consistent with the radial distance results for Flare 2. However, no altitude decrease before the larger peak in X-ray emission at 00:12:00 was noted, where again there is a decrease in loop length and loop width.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm]{aug232005_dvwl_plots_1020keV.pdf}\n\\caption[For Flare 1: lightcurve (row 1), $dW\/dt$ (row 2), $dL\/dt$ (row 3) and $dr\/dt=v$ (row 4).]{For Flare 1 (23-Aug-2005), lightcurve (row 1), $dW\/dt$ (row 2), $dL\/dt$ (row 3) and $dR\/dt=v$ (row 4). Both $dW\/dt$ and $v$ can be\nfitted using a straight line indicating constant accelerations $a_{W}$ and $a_{v}$ and hence constant forces.}\n\\label{fig:dvwl}\n\\end{figure}\n\nIt has been suggested that a decrease in loop altitude may be an indication of collapsing magnetic trap acceleration\/heating \\citep{1966PASJ...18...57T,1997ApJ...485..859S,2002A&ARv..10..313P,2004A&A...419.1159K,2006A&A...446..675V,2012A&A...546A..85G}.\nAlthough this may be the case for other events, it is not believed that the collapsing magnetic trap is the prime solution\nto these observations, firstly due to overall larger loop volume changes over relatively small position changes.\n\\cite{2006A&A...446..675V} observed the coronal loop of a {\\it GOES} X-class flare and found downward velocities as large\nas $\\sim14$ km\/s in 10-15 keV band and $\\sim29$ km\/s in the 15-20 keV band prior to the peak in X-ray emission.\nThis is not what is observed for these events. For Flare 1, there is an average downward velocity in the 10-20 keV\nband of $\\sim4$ km\/s, which is generally comparable to the decrease in loop width during this period.\nAverage changes in loop radial position for Flares 2 and 3 are also $\\sim4$ km\/s. More convincingly, in a collapsing magnetic\ntrap model, simple compressive heating during the contraction stage would imply that\n$NT\\propto1\/A$ \\citep{1978ApJ...223.1058M,1981ApJ...244..653E}, where $N$ is the number of particles in the region,\n$T$ is the plasma temperature and $A$ is the cross-sectional area of the region, given by $\\pi W^{2}$, where $W$ is the loop width.\nFigure \\ref{fig:NTA} plots $\\log{NT}$ against $\\log{A}$ for each flare and shows this not to be the case.\nStraight lines fits to both the contraction and expansion phases show\nthat the gradients are either greater or less than $-1$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=15.5cm]{aug232005_eit_rs.pdf}\n\\caption[{\\it SOHO} EIT $195\\AA$ images for Flare 1 at the times of 14:21:12 and 14:34:51, corresponding\nto the times of rise and peak in X-ray emission.]{{\\it SOHO} EIT $195$~{\\AA} images for Flare 1 at the times of 14:21:12 and 14:34:51, corresponding\nto the times of rise and peak in X-ray emission. {\\it RHESSI} 10-12 keV X-ray contours are over plotted in pink.\nNote the lack of bright EUV emission at the rise stage of this flare.}\n\\label{fig:fig_eit}\n\\end{figure*}\n\nObservations by \\cite{2011ApJ...730L..22K} showed how the loop width of the 14\/15th April 2002 event (Flare 2) increased with\nenergy during two times corresponding to the first rise and first decay stages of the observations and suggested that the presence of magnetic\nturbulence (the diffusion of field lines) was the cause of the energetic width increases. Hence, the suggestion that magnetic\nturbulence in the region can account for the energetic changes in loop width, may also be able to account for the extra\nenergy in the loop, since both \\cite{2011ApJ...730L..22K} and \\cite{2011A&A...535A..18B} found the energy density of\nmagnetic fluctuations to be significant and could be comparable to that of the flaring plasma and higher than the energy density of non-thermal particles. Flares 1 and 3 also show similar length and width increases with energy at a single time range, also suggesting the presence of magnetic turbulence.\n\nMany observations \\citep{2005ApJ...629L.137L,2009ApJ...696..121L,2009ApJ...706.1438J,2010ApJ...724..171R,2012ApJ...749...85G} of loop height and length changes have been explained in terms of a reduction in magnetic\npressure. Usually the reduction in magnetic pressure is referred to as Taylor relaxation but this only refers to a special case where the resulting field is linear force free \\citep{1974PhRvL..33.1139T}. The reduction in magnetic pressure could also account for the reduction of loop width\nor cross-sectional area, as shown in simulations by \\cite{2007A&A...472..957J}, and hence the observed trends of number density and pressure.\n\\cite{2009ApJ...696..121L} studied coronal implosion of one coronal source. They explained the reduction in height of a coronal source\nbefore the peak in X-ray emission in terms of Taylor relaxation. They also suggested that this type of event can only occur\nif the coronal loop is already filled with hot, dense plasma before the onset of a new event, that is from a previous event in that region.\nThe fact there is no EUV emission during the rise phase of Flare 1 seems to correlate with the observations\nand suggestions of \\cite{2009ApJ...696..121L}. Figure \\ref{fig:fig_eit} shows {\\it SOHO} (Solar and Heliospheric Observatory)\nEIT (Extreme Ultraviolet Imaging Telescope) 195~{\\AA} images at the times of 14:21:12 and 14:34:51 for Flare 1.\nX-ray emission contours at 10-12 keV for 14:22:00-14:26:00 and 14:34:00-14:36:00 are also over plotted.\nFrom Figure \\ref{fig:fig_eit}, during the rise phase there is no bright 195~{\\AA} EUV emission emanating from the loop,\nonly 10-20 keV X-ray emission. After the peak in X-ray emission, EUV emission can be observed from the loop.\nThere is an overall increase in the number of particles, $N$, within the loop region throughout the\nduration of all three events. Due to low coronal densities, chromospheric evaporation probably accounts for this increasing $N$, initially driven by thermal conduction and possibly at later times\nby electrons reaching the chromosphere, where there is weak footpoint emission and EUV emission from the loop.\n\nIt should be noted that a study of the 23rd July 2002 flare by \\cite{2010ApJ...725L.161C} calculated temporal volume changes using the CLEAN algorithm and assuming an elliptical geometry. Overall this flare shows a general trend consistent with the results; an overall decrease in volume before the peak in X-ray emission and an overall increase in volume after the peak in X-ray emission. This flare also shows a peak in plasma temperature before the peak in X-ray emission and a high number density (and hence thermal pressure) after the X-ray emission first peaks.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=7.5cm]{aug232005_NTA.pdf}\n\\hspace{50pt}\n\\includegraphics[width=7.5cm]{apr142002_NTA.pdf}\n\\includegraphics[width=7.5cm]{may212004_NTA.pdf}\n\\caption[Plots of $\\log{NT}$ against $\\log{1\/A}$ for Flares 1, 2 and 3.]{Plots of $\\log{NT}$ against $\\log{1\/A}$ for Flares 1 (top), 2 (bottom left) and 3 (bottom right).  The star represents the start time\nof each event while the triangle represents the peaks in X-ray emission for each event. The red and blue\ndashed lines represent straight line fits during the compressive (red) and expansive (blue) phases.\nFor Flare 2, only the first compressive and final expansion phase have been fitted. For simple compressive\nheating, it is expected the gradient of each line would be $\\gamma=-1$.}\n\\label{fig:NTA}\n\\end{figure*}\n\n\\subsection{Three temporal phases and suggested explanations for the observations}\n\n\\begin{figure*}\n\\includegraphics[width=5.0cm]{aug232005_3phases.pdf}\n\\includegraphics[width=5.0cm]{apr142002_3phases.pdf}\n\\includegraphics[width=5.0cm]{may212004_3phases.pdf}\n\\caption[Observations of plasma temperature, X-ray emission, loop width and thermal pressure are replotted together for Flares 1,\n2 and 3 at one energy band of 10-20 keV (14-25 keV for Flare 3). ]{Observations of plasma temperature, X-ray emission, loop width and thermal pressure are replotted together for Flares 1 (left),\n2 (middle) and 3 (right) at one energy band of 10-20 keV (14-25 keV for Flare 3). The orange bars represent three phases: 1. a peak in plasma temperature, 2. a peak in X-ray emission, generally coinciding with the smallest loop width and 3. a peak in thermal pressure.}\n\\label{fig:modphases}\n\\end{figure*}\n\nFor each flare, Figure \\ref{fig:modphases} replots the plasma temperatures, X-ray emissions, emitting loop widths and thermal pressures, but now at a single energy band of 10-20 keV (14-25 keV for Flare 3). From Figures \\ref{fig:is_paras} and \\ref{fig:com_paras} and now more clearly in Figure \\ref{fig:modphases}, it can be observed collectively, that the observations of each flare display three distinct phases and each of these phases will form the basis of the suggested explanation. Each phase is represented by a shaded orange bar. During Phase 1, there is a peak in plasma temperature and during Phase 2, a peak in X-ray emission. At Phase 2 the smallest emitting loop width, length and hence volume occurs. Finally, during Phase 3, the thermal pressure of the region peaks. For Flare 1 (left column), each phase is well separated and can be clearly seen. The pattern can also be seen for Flares 2 (middle column) and 3 (right column) but each phase is not as clearly defined as for Flare 1. During Flare 3, each phase occurs over much shorter time intervals and therefore each of the phases overlap slightly. During Flare 2, there are multiple peaks, which along with the shorter timescales for each process, makes each individual phase harder to see. However, the overall pattern is observed for all three flares.\nFigure \\ref{fig:modphases} shows that each phase can only be easily seen for slower events. The quicker the event, the harder it is to distinguish between each of the three phases as each phase overlaps in time. Many flares may exhibit a similar pattern but each phase may occur at timescales too fast to be currently observed.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=15.5cm]{chapt3_loop_pic_2.png}\n\\caption[Simple cartoon showing the suggested coronal loop evolution with time. ]{Simple cartoon showing the suggested coronal loop evolution with time. At time 1, the coronal region (pink) has a number density $n$ (red dots) and a temperature $T$. At time 2, number density of the region increases due to the cross-sectional area of the loop decreasing and the expansion of material from lower atmospheric layers. At time 3, the loop expands from thermal pressure due to chromospheric evaporation.}\n\\label{fig:chapt3_dcfig}\n\\end{figure*}\n\nIn order to understand the observations, it is crucial to understand the decreasing X-ray widths before the X-ray peak. Length variations along the field can always be explained in terms of changing number density, but in order to explain the width variations, where electrons are tied to the magnetic field, the most plausible explanation is the movement or the diffusion of magnetic field lines. The energetic loop width increases at a given time for each flare has already been noted, suggesting the presence of turbulence in the region, but overall in time the width at a given energy band shrinks until the X-ray peak is reached, implying the presence of both turbulence and the shrinking of the cross-sectional area of the field at this stage of the flare. Therefore, it is suggested the plasma within the emitting X-ray loop region is tied to the magnetic field lines and hence the contraction of the emitting loop cross-section or width during the rise phase is ultimately due to the contraction of the cross-sectional area of field lines that thread the region or possibly the expansion of field lines above the region. It is also sensible to assume that the loop region in X-rays actually consists of multiple coronal loops that cannot be resolved in X-rays using {\\it RHESSI}. Although the region of the loop emitting X-rays can only be observed, it is assumed the entire loop region is contracting since the plasma is tied to the field lines. Hence from these observations, the reason for this cross-sectional width shrinking is unknown, but it can be speculated for further study that it may be due to a reduction in magnetic pressure within the X-ray loop region, as often suggested to describe height decreases (coronal implosion or loop contraction) discussed earlier \\citep{2005ApJ...629L.137L,2009ApJ...696..121L,2009ApJ...706.1438J,2010ApJ...724..171R,2012ApJ...749...85G}. The magnetic pressure decreases as the magnetic field relaxes from a non-potential state. Although it is reasonable to assume that during a solar eruptive event such as a flare, the field will reside in a non-potential state, X-ray observations of these events possibly provide evidence for non-potentiality, again through the inference of magnetic turbulence and hence a non-parallel field component within the loop region. The non-potential state and decreasing magnetic pressure could be due to reconnection above the region or maybe even along the loop itself \\citep{2004ApJ...608..540V,2011ApJ...729..101G,2012SoPh..277..299G,2013SoPh..284..489G}. One such model is caused by a kink instability as shown in recent simulations by \\cite{2011ApJ...729..101G,2012SoPh..277..299G,2013SoPh..284..489G}. In this model, the energy of twisted loops is released by reconnection inside the loop and transferred to plasma heating and particle accelerations.\n\n{\\it Phase 1:} During phase 1, the process causing the contraction of loop width\/cross-section is also probably responsible for the comparable decrease in loop altitude for Flare 1, and at least the change in loop position observed for Flares 2 and 3 sitting on the solar disk. Plots of $dW\/dt$ (width contraction\/expansion) and $dR\/dt=v$ (centroid velocity) for Flare 1 plotted in Figure \\ref{fig:dvwl} show that each parameter can be fitted using a straight line during both the contraction and expansion phases. $dL\/dt$ (length contraction\/expansion) is also plotted in Figure \\ref{fig:dvwl} but its trend cannot be described by a straight line fit, unlike $dW\/dt$ and $dR\/dt$.\nThe increasing temperature of the region means that energy will be thermally conducted towards the lower levels of the solar atmosphere causing gentle chromospheric evaporation of the denser coronal and chromospheric layers below. The observations show the plasma temperature peaking relatively early before the peak in X-ray emission and then slowly decreasing; slower than by thermal conduction, implying that energy is still being supplied to the loop plasma. This is most likely via the conversion of magnetic energy. Chromospheric evaporation drives plasma into the region producing the increasing number density and hence thermal pressure, along with the shrinking width at this phase. The increasing number density is responsible for the rapid non-linear decreasing X-ray loop length, since electrons accelerated within the region will travel shorter distances before interacting.\n\n{\\it Phase 2:} During Phase 2, after the peak in plasma temperature, the loop width stops shrinking. The number density and thermal pressure within the loop are still increasing due to chromospheric evaporation and the length of the emitting region also reaches its lowest point. The loop width may stop shrinking because the process causing the shrinking ceases or it may be due to the balancing of forces within the region. For example, if the reduction of loop width is due to the reduction of magnetic pressure in the loop then at this phase, the growing thermal pressure may finally be high enough to balance the reduction in $B$ pressure.\n\n{\\it Phase 3:} During the final phase, the thermal pressure continues to rise within the loop due to the increasing number density from chromospheric evaporation. It is believed the growing thermal pressure in the region is now responsible for the expanding loop width. This expansion, in turn, eventually halts the increasing number density and thermal pressure at a time after the peak in X-ray emission. After Phase 3, the loop width continues to increase and slow decreases in both number density and thermal pressure are observed. The X-ray emission continues to decrease during and after Phase 3 and the emitting loop length during this period remains approximately constant, equal to the minimum loop length in Phase 2, even with a decreasing number density. It is sensible to assume that the acceleration mechanism in the loop is slowing during this time. However, Flare 2 is an exception to this trend with multiple events\/X-ray peaks in the lightcurve.\n\nThis observational study shows the usefulness of measuring changes in the spatial properties of coronal X-ray sources and combining such observations with the parameters deduced from spectral analysis, in order to deduce how the properties of the flaring corona change during the flare. It is hoped that future observations of such flares will have complementary EUV data from other solar missions such as SDO\/AIA. \n\\chapter{Solar flare X-ray albedo and the positions and sizes of hard X-ray (HXR) footpoints}\n\\label{ref:Chapter5}\n\n\\normalsize{\\it This work can be found in the publication \\cite{KontarJeffrey2010} and also \\cite{2011A&A...536A..93J}.}\n\\\\\\\\\n\\section{Introduction}\nChapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6} examine solar flare X-ray albedo; an effect that can change the observed properties of HXR sources and hence the interpretation of the observations from instruments such as {\\it RHESSI}. A brief discussion of this was given in Chapter \\ref{ref:Chapter1}, Section \\ref{intro_albedo1}. Here in Chapter \\ref{ref:Chapter5}, it is discussed quantitatively for the first time how an albedo component will change the positions and sizes of observed HXR sources and hence the interpretation of the results. The solar atmosphere above HXR sources is optically thin and the X-rays emitted as bremsstrahlung, for example, are directly related to the emitting target electrons. However, depending on the anisotropy of the HXR source and hence the anisotropy of the target electron distribution in the chromosphere, a certain proportion of the X-rays are emitted downwards, towards the denser layers of the solar atmosphere, namely the photosphere. Here they can interact with\nfree or bound electrons and can be back-scattered towards the observer. This phenomenon was first discussed by \\cite{Tomblin1972} and \\cite{Santangeloetal1973} and the X-rays back-scattered and emerging from the dense photosphere, ``the albedo patch\", are known as the albedo X-rays. The albedo X-rays are viewed alongside those X-rays directly emitted from the HXR source, which are often called the primary X-rays; together as a single observed HXR footpoint.\nAn isotropic HXR source produces the minimum albedo and even its\nflux can account for up to 40\\% of the detected flux in the peak albedo energy range between 20 and 50 keV \\citep{BaiRamaty1978,ZhangHuang2004,Kontaretal2006,Kasparovaetal2007}. Therefore, all X-ray sources at the solar disk should be viewed as a combination of both the primary and backscattered albedo fluxes. As discussed in Chapter \\ref{ref:Chapter1}, Section \\ref{intro_albedo1}, accounting for the albedo effect is important for all X-ray solar observations, which can only view disk sources as a combination of the direct X-ray flux and the backscattered X-ray flux. The backscattered component often taints the primary HXR source properties, changing the observed angular, energy, spatial and polarization distributions. At the same time, the usefulness of albedo as diagnostic of electron directivity will be discussed in Chapter \\ref{ref:Chapter6}.\nPast studies concentrated on the observations of the era; looking at changes in X-ray spectra or the total integrated polarization of the HXR source. Albedo changes the shape of the spatially integrated X-ray spectrum, which is flattened at lower energies up to around 20-30 keV and at higher energies above around 70 keV, the spectrum is steeper than expected from a primary X-ray spectrum alone. Albedo can even produce artificial spectral features in observed spectra \\citep{2008SoPh..252..139K}. \\cite{Kontaretal2006} developed and implemented an albedo correction for spectral X-ray {\\it RHESSI} analysis using a Green's function approximation by \\cite{MagdziarzZdziarski1995}. This was discussed in Chapter \\ref{ref:Chapter1}. For a more in-depth introduction on the topic of X-ray albedo, an excellent review can be found in \\cite{2011SSRv..159..301K}.\nDue to the height of a HXR source in the chromosphere ($\\sim2\\mbox{\\ensuremath{^{\\prime\\prime}}}$), the reflected albedo X-rays come from a rather large area in the photosphere, the albedo patch. The surface brightness of the albedo patch at the solar surface is therefore rather low \\citep{BaiRamaty1978} since the flux is spread over this large area. This fact explains the difficulty in imaging the albedo patch \\citep{SchmahlHurford2002}. However, imaging algorithms such as VisFwdFit, discussed in Chapter \\ref{ref:Chapter1}, have allowed the positions and sizes of HXR footpoints to be found (\\cite{2008A&A...489L..57K}, \\citet*{2010ApJ...717..250K}), using the spatially integrated moments of the X-ray distribution. In \\citet*{2010ApJ...717..250K}, changes in radial height and vertical and horizontal spatial extents with energy, of a HXR footpoint situated at the solar limb, were found. This work is discussed in Chapter \\ref{ref:Chapter1}, Section \\ref{fp_loop}, demonstrating how the careful use of forward fitting algorithms and the spatially integrated moments of the X-ray distribution can be used to infer the positions, sizes and even shapes of HXR footpoints. It must be ensured that the albedo component, which should be present as part of every HXR footpoint source, is properly accounted for, before the conditions within the chromosphere or the properties of the radiating electron distribution are deduced from these spatial changes.  Even though the albedo patch is difficult to image directly, the use of imaging algorithms such as Vis FwdFit may actually help to `see' the presence of an albedo component. Since the moments of the distribution are integrated over the full area of the source, algorithms such as Vis FwdFit can better account for the low intensity and diffusiveness of the albedo component, than other imaging methods currently available.\n\n\n\\section{The modelling of X-ray transport in the photosphere}\nIn order to study the backscattered X-ray flux and albedo effect, Monte Carlo (MC) simulations are used to model photon transport in the photosphere, starting with a hundred million photons per run from a chosen HXR source created in the chromosphere. Each of the main steps involved in the MC photon transport code are described in the following sections and can be seen graphically as a flow chart in Figure \\ref{fig:mcs_flow_chart}.\n\\subsection{The modelling of a hard X-ray footpoint source}\nAn unpolarized HXR source is modelled in space as\na two-dimensional circular Gaussian distribution in the plane\nparallel to the solar surface\n\\begin{equation}\nI(x,y)\\propto\\exp\\left(-\\frac{x^{2}}{2 var_{x}}-\\frac{y^{2}}{2 var_{y}}\\right),\n\\label{eq:I_xy}\n\\end{equation}\nwith width (standard deviation) $=\\sqrt{var_{x}}=\\sqrt{var_{y}}$, at a chosen height $h=1$ Mm ($\\sim1\\mbox{\\ensuremath{^{\\prime\\prime}}}.4$) in the chromosphere, above a layer defined as the photosphere (see Section \\ref{ref:photo}). It is assumed that the HXR source has zero extent in $z$, the direction perpendicular to the solar surface. Although, the results of \\citet*{2010ApJ...717..250K} showed that HXR sources at lower energies sit at a higher point in the chromosphere and have a greater extent in $z$, modelling $z$ with a finite extent would unnecessarily complicate the initial results and hence is neglected for clarity. For the HXR energy range, a source at $1$ Mm above the photosphere is chosen to match with recent X-ray observations\n\\citep{Aschwandenetal2002,2008A&A...489L..57K,Pratoetal2009,SaintHilaireetal2010,MrozekKowalczuk2010,2010ApJ...717..250K,BattagliaKontar2011}.\nThe X-ray energy spectrum in the chromosphere is simply input as a power law,\n\\begin{equation}\nI(\\epsilon)\\sim\\epsilon^{-\\gamma}\n\\end{equation}\nwith a spectral\nindex of $\\gamma$, for X-ray energies $\\epsilon$ between 3 keV and 300 keV, an energy range typical of HXR footpoints observed by {\\it RHESSI}. Typical HXR footpoint spectral index values are also used: $\\gamma=2,3,4$ \\citep[e.g.,][]{1991ApJ...379..381M, Kasparovaetal2007}\n\n\\subsection{X-ray transport and interaction in the photosphere}\\label{ref:photo}\n\\begin{figure}\n\\includegraphics[width=15cm]{mcs_flow_chart_4.png}\n\\centering\n\\caption[A flow chart showing the main steps involved in the Monte Carlo photon transport simulations in the photosphere.]{A flow chart showing the main steps involved in the Monte Carlo photon transport simulations in the photosphere and the creation of the HXR distribution in the chromosphere.}\n\\label{fig:mcs_flow_chart}\n\\end{figure}\nIn this model, it is assumed that X-rays move freely without interaction until they reach the photosphere and hence their positions on the solar surface are simply calculated from their initial $(x,y)$ positions and emission angles $(\\theta,\\phi)$.\nThe photosphere is defined as a layer with a hydrogen number density of $1.16\\times10^{17}$ cm$^{-3}$ \\citep{Vernazzaetal1981}.  An\nX-ray interaction with the photospheric medium can either be by Compton scattering or by\nphotoelectric absorption. For X-rays with energies less than $\\sim10$ keV, photoelectric\nabsorption is the more probable process while Compton scattering dominates above $\\sim10$ keV. This can be seen in Figure~\\ref{fig:cross sections}. Within the photosphere, each X-ray photon moves a\nstep-size $ss$ before an interaction and this is calculated using $ss=-l\\ln \\zeta_{step}$ (see Appendix \\ref{ref:App2} for more information), where $l$ is the\nphoton mean free path and $\\zeta_{step}\\in[0,1]$ is drawn from a uniform random distribution. The photon mean free path is calculated by\n$l=1\/n_{H}\\sigma_{total}$, where $\\sigma_{total}=\\sigma_{c}+\\sigma_{a}$, the addition of\nthe Compton scattering cross section $\\sigma_{c}$ and\nphotoelectric absorption cross section $\\sigma_{a}$. For a number density of $1.16\\times10^{17}$ cm$^{-3}$, the photon mean free path, $l$, is of the order 100 km. Hence, the earlier assumption that photons travel freely until they reach the photosphere is valid since even a high chromospheric density of $\\sim1\\times10^{15}$ cm$^{-3}$ would give a mean free path $l\\sim1000$ km $=1$ Mm, the same size as the chosen source height. When a photon is absorbed, it is simply removed from the simulations.  For each\nphoton, one of the two processes is chosen by calculating the ratio of $\\sigma_{c}\/\\sigma_{total}$.\nAnother random number $\\zeta_{pick}$ is then sampled from a uniform distribution between 0 and 1.\nIf the ratio is greater than $\\zeta_{pick}$ then the photon is Compton scattered and if the ratio\nis less than $\\zeta_{pick}$, then the photon is absorbed.\nThese simulations also differ from previous work as the curvature of the Sun is included and a photon exits the photosphere when it satisfies the condition\n$z>z_{\\bigodot}=\\sqrt{R_{\\bigodot}^{2}-x^{2}-y^{2}}-R_{\\bigodot}$, where $R_{\\bigodot}$ is the\nradius of the Sun which is taken to be $6.96\\times10^{10}$ cm $\\sim960''$. The extent of the albedo patch is limited by properly modelling the curvature of the Sun. The photons are allowed to scatter\nmultiple times until they exit the photosphere or are removed by\nabsorption. Photons are also removed when their energy falls below 3 keV since X-rays below this energy cannot be observed by {\\it RHESSI} and a photon cannot gain energy from an interaction with an electron in photospheric conditions. Most photons will leave the photosphere during their first scatter, with subsequent scatterings\nproducing less and less photons. This is shown in terms of Green's functions in \\cite{Kontaretal2006}.\nPhotons that exit the photosphere with $\\cos\\theta > 0$ are collected into selected angular\nor energy bins corresponding to HXR sources sitting at any chosen heliocentric\nangle on\/above the solar disk.\n\\begin{figure}\n\\includegraphics[width=15cm]{albedo_cartoon_2}\n\\centering\n\\caption[Cartoon showing how X-rays emitted in the chromosphere via the Coulomb interaction can travel to the photosphere, Compton scatter, head out into interplanetary space and then be detected alongside X-rays directly emitted from the chromosphere.]{Cartoon showing how X-rays emitted in the chromosphere via the Coulomb interaction can travel to the photosphere, Compton scatter, head out into interplanetary space and then be detected alongside X-rays directly emitted from the chromosphere. The polar coordinates of the emitted X-rays $\\theta_{\\epsilon}$ and $\\phi_{\\epsilon}$ are also shown.}\n\\label{fig:ac2}\n\\end{figure}\n\\subsection{Photoelectric absorption}\nFor photons with energies below $\\sim $10 keV, photoelectric absorption is the most probable photon\ninteraction in the photosphere. The process of absorption is heavily dependent upon the abundance of chemical\nelements within the photosphere. Absorption was therefore\nmodelled using the latest known solar photospheric abundances taken from \\cite{Asplundetal2009}.\nAbsorption cross section codes for the most important elements of H, He, C, N, O, Ne, Na, Mg, Al,\nSi, S, Cl, Ar, Ca, Cr, Fe and Ni were adapted from \\cite{Balucinska-ChurchMcCammon1992}. For energies\nhigher than $10$ keV, the absorption\ncross section was approximated by $\\sigma_a(\\epsilon_{0})\\propto\\epsilon_{0}^{-3}$.  A comparison of the Compton scattering and absorption\ncross sections is shown in Figure \\ref{fig:cross sections} (with $\\sigma_{a}$ and $\\sigma_{c}$ multiplied by $10^{24}\n\\epsilon^{3}$ ($\\epsilon$ in keV) for comparison with \\cite{MorrisonMcCammon1983}). Any differences between\nFigure \\ref{fig:cross sections} and \\cite{MorrisonMcCammon1983} are\ndue to the newer element abundances \\citep{Asplundetal2009} and updated absorption cross section codes\n(in particular Helium) being used in these simulations.\n\\begin{figure}\n\\centering\n\\includegraphics[width=12cm]{crossfinal.pdf}\n\\caption[Absorption $\\sigma_{a}$ and Compton\n$\\sigma_{c}$ cross sections plotted at low energies below 10 keV.]{Absorption  $\\sigma_{a}$ (black solid) and Compton\n$\\sigma_{c}$ (red dashed) cross sections plotted at low energies below 10 keV and\nmultiplied by $10^{24}\\epsilon^{3}$ ($\\epsilon$ in keV) for clarity. The absorption cross section\nis calculated using photospheric element abundances by \\cite{Asplundetal2009}.}\n\\label{fig:cross sections}\n\\end{figure}\n\\subsection{Compton scattering}\\label{ref:cs}\nSimilar to previous MC simulations \\citep{BaiRamaty1978, MagdziarzZdziarski1995}, Compton scattering is modelled using the Klein-Nishina \\citep{KleinNishina1929} differential scattering cross section for unpolarized X-ray radiation. This is valid\nat all energies of interest and is given by,\n\\begin{equation}\n\\frac{d\\sigma_{c}(\\theta_{S},\\epsilon)}{d\\Omega}=\\frac{1}{2}r_{0}^{2}\\left(\\left(\\frac{\\epsilon}{\\epsilon_{0}}\\right)^{3}+\n\\frac{\\epsilon_{0}}{\\epsilon}-\\left(\\frac{\\epsilon}{\\epsilon_{0}}\\right)^{2}\\sin^{2}\\theta_{S}\\right).\n\\label{eq:sigma_c}\n\\end{equation}\nwhere $\\epsilon_{0}$ is the initial photon energy, $\\epsilon$ is the new photon energy, $\\theta_{S}$ is the angle between the initial and after scattering photon directions and $r_{0} = 2.82\\times10^{-13}$ cm is the classical electron radius. In MC simulations, when a Compton scattering occurs, the properties of the outgoing photon: energy $\\epsilon$ and polar\nscattering angle $\\theta_{S}$, need to be updated.\nSince MC simulations operate by drawing numbers randomly from a given distribution, this means\nthat $\\theta_{S}$ can be easily found by matching each value of\n$\\theta_{S}\\in[0^{\\circ},180^{\\circ}]$ with a random number $\\zeta_{\\theta}\\in[0,1]$ drawn from a uniform distribution, for every value of\n$\\epsilon$ using,\n\\begin{equation}\n\\zeta_{\\theta}=\\frac{2\\pi}{\\sigma_{c}}\\int_{0}^{\\theta_{S}=\\pi}\\frac{d\\sigma_{c}(\\theta_{S},\\epsilon)}{d\\Omega}\\sin\\theta_{S}\\;\\;d\\theta_{S}.\n\\end{equation}\nNew values of $\\theta_{S}$ are simply\ndrawn at each scattering using the photon energy before a scattering and a random, uniform number $\\zeta_{\\theta}$ between\n$0$ and $1$. Once the new scattering angles $\\theta_{S}$ are obtained then the new photon energy $\\epsilon$ can\nbe easily found using,\n\\begin{equation}\n\\epsilon=\\frac{\\epsilon_{0}}{1+\\frac{\\epsilon_{0}}{mc^{2}}(1-\\cos\\theta_{S})}.\n\\end{equation}\n\nIn the simulations, the Klein-Nishina cross section is\nmultiplied by $Z_{photo}=1.18$ to take account of elements higher than hydrogen that are present\nwithin the photosphere. $Z_{photo}$ indicates the average atomic number and the\nnumber of electrons per hydrogen atom in the photosphere, and is given by\n\\begin{equation}\nZ_{photo}=\\frac{\\sum\\limits _{Z}Z10^{A_{Z}}}{10^{A_{H}}}\n\\end{equation}\nwhere $A_{Z}$ is the $\\log_{10}$ abundance of an element with atomic number Z relative to\nhydrogen while $A_{H}=12$ is the $\\log_{10}$ abundance of hydrogen \\citep{Asplundetal2009}.\nIn this case, where the X-ray distribution is completely isotropic and unpolarized, the azimuthal scattering angle in the plane perpendicular to the incoming direction of the photon during a scattering $\\phi_{S}$ can just be drawn randomly from a uniform distribution between 0 and 2$\\pi$.  This is not the case for a polarized X-ray distribution, which will be described in Chapter \\ref{ref:Chapter6}. In the photosphere, it is very unlikely that the opposite case where the photon gains energy during an interaction with an electron, named inverse Compton scattering, will occur. For this to occur, a photon must interact with an electron with a kinetic energy larger than that of the photon energy, and this is unlikely in photospheric conditions.\n\n\\section{The position and sizes of backscattered and observed hard X-ray sources}\n\nThe escaping photons are accumulated to create the brightness distribution $I(x,y)$ over a given energy and solid angle. The total primary or reflected flux is then just an integral over the corresponding area $\\int I(x,y)dxdy$ which is the zeroth moment of the brightness distribution. In these simulations, the source positions and sizes of each component\n(primary and albedo) and the total source (primary plus albedo) can be found using the first and second moments of the distribution; the first and second normalised moments which are the mean and variance of the distribution respectively.\n\n\\subsection{The moments of the hard X-ray distribution}\nUsing solar disk centred coordinates, the centroid position $(\\bar{x},\\bar{y})$ of both the albedo source alone and total observed source can be found by the mean,\n\\begin{equation}\n\\bar{x}=\\frac{\\int_{0}^{\\infty}xI(x,y)dxdy}{\\int_{0}^{\\infty}I(x,y)dxdy}\\;\\;\\;,\\;\\;\\;\n\\bar{y}=\\frac{\\int_{0}^{\\infty}yI(x,y)dxdy}{\\int_{0}^{\\infty}I(x,y)dxdy},\n\\end{equation}\nand the spatial extent in each direction $(x,y)$ by the variance $(var_{x},var_{y})$\n\\begin{equation}\nvar_{x}=\\frac{\\int_{0}^{\\infty}(x-\\bar{x})^{2}I(x,y)dxdy}{\\int_{0}^{\\infty}I(x,y)dxdy}\\;\\;\\;,\\;\\;\\;\nvar_{y}=\\frac{\\int_{0}^{\\infty}(y-\\bar{y})^{2}I(x,y)dxdy}{\\int_{0}^{\\infty}I(x,y)dxdy}.\n\\end{equation}\nAlthough the primary distribution is initially\nGaussian, the albedo (and hence total observed) distribution will have a complex shape that is no longer Gaussian. To quantify the sizes we use `Gaussian' Full Width Half Maximum (FWHM) defined as FWHM$_{x,y}=2\\sqrt{2\\ln2var_{x,y}}$, using the square of the variance and not the actual FWHM of the complex distribution. This allows a simple comparison with {\\it RHESSI} measurements \\citep{2008A&A...489L..57K, DennisPernak2009, Pratoetal2009}. All FWHM values in this chapter and also Chapter \\ref{ref:Chapter6} are measured in this way.\n\n\n\\subsection{Resulting brightness distributions}\nFigure \\ref{fig:paper_plots} shows the primary and escaping photon brightness distributions for a completely isotropic HXR footpoint located at the disk centre. Similar to previous results \\citep{BaiRamaty1978}, for a chosen compact primary source of size $d=1.5h$ (FWHM$_{x,y}\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}} .9$), the back-scattered albedo photons are reflected from an area much larger than the primary source. The reflected photons change the spatial distribution of the observed photons and produce a halo around the primary source.\n\\clearpage\n\\begin{sidewaysfigure*}[ht]\n\\centering\n\\includegraphics[width=22cm]{paper_plots.pdf}\n\\caption[The X-ray scatter distributions of the primary photons and the Compton back-scattered photons.]{The X-ray scatter distributions of the primary photons (red dots) and the Compton back-scattered photons (blue dots) for a primary source at $h=1$ Mm with width $d=\\sqrt{var_{x,y}}=1.5h$ (giving a FWHM$_{x,y}\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}} .9$) between 20 and 50 keV for four viewing angles given by $\\mu$. The yellow and green ellipses show the FWHM sizes for the primary and combined sources respectively.}\n\\label{fig:paper_plots}\n\\end{sidewaysfigure*}\n\\clearpage\nThe scattered X-ray flux depends on the cosine of the heliocentric angle of the source ($\\mu=\\cos(\\theta)$) or equivalently on the position of the source at the solar disk, $\\mu=\\sqrt{1-(x^{2}+y^{2})\/R_{\\bigodot}^{2}}$. A circular X-ray source located above the centre of the disk will produce a circular albedo patch, as can be seen in the first plot of Figure \\ref{fig:paper_plots}. Naturally, the location of the HXR source and albedo patch will coincide at the disk centre, so albedo will not change the source position. However, the albedo will make the source larger than the input primary size. The albedo contribution becomes asymmetric if the source is located away from the disk centre at a given heliocentric angle $\\theta$ (Figure \\ref{fig:paper_plots} b-d). A diagram depicting this scenario for different locations on the solar disk is shown in Figure \\ref{fig:albedo_pic_fb}.\n\nDue to the spherical symmetry of the Sun, there are two distinct directions: radial along the line connecting the centre of the Sun and the X-ray source $r$, and perpendicular to the radial $r_{\\perp}$. This is shown in Figure \\ref{fig:albedo_pic_fb}. There is no change in centroid position in the $r_{\\perp}$ direction for a spherically symmetric primary source. Qualitatively, in the $r$ direction, the albedo component causes a centroid shift towards the disk centre that rises from $0$ at $\\mu=1$ (solar centre), peaks at a position less than $\\mu=1$ (this will be discussed further in the following sections) and reduces to $0$ at $\\mu=0$ (solar limb). This pattern emerges since the centroid position of the albedo component is located at a position $h\\sin\\theta$ disk-ward of the primary centroid position, where $\\theta$ is the heliocentric angle of the source (see Figure \\ref{fig:albedo_pic_fb}). However, the intensity of the albedo component falls as we move closer to the limb and hence the position of the total source peaks at a heliocentric angle where the combined contribution of $h\\sin\\theta$ and the albedo intensity is greatest.\n\\begin{figure}\n\\centering\n\\includegraphics[width=16.5cm]{albedo_pic_fb.png}\n\\caption[Diagram showing a HXR primary source at three different heliocentric angles $\\theta$ above the solar disk and the corresponding albedo patch at a shifted location of $h\\sin\\theta$.]{Diagram showing a HXR primary source (orange) at three different heliocentric angles $\\theta$ above the solar disk and the corresponding albedo patch (blue) at a shifted location of $h\\sin\\theta$, where $\\theta=0^{\\circ}$ ($\\mu=1$) is located at the solar centre and $\\theta=90^{\\circ}$ ($\\mu=0$) is located at the edge of the Sun, the solar limb. The radial $r$ and perpendicular to radial $r_{\\perp}$ directions are also shown.}\n\\label{fig:albedo_pic_fb}\n\\end{figure}\nFigure \\ref{fig:paper_plots} also shows how the source size varies in the $r_{\\perp}$ direction, with the FWHM of the total source generally decreasing at lower $\\mu$, since the albedo intensity falls as $\\mu\\rightarrow0$. In the radial direction, the FWHM of the total and primary sources decrease close to linear due to a simple projection effect. To study the albedo component only, source sizes are only examined in the $r_{\\perp}$ direction and the source positions are examined in the radial direction $r$. Similar to the spatially integrated albedo \\citep{Kontaretal2006}, the shift in centroid position and the growth of the source size are also energy and $\\mu$ dependent. In the following section, the position and source size changes are studied for various: (a) spectral index of the HXR primary source, (b) HXR primary source size, and (c) X-ray directivity (the ratio of downward to upward emitted photons), separately. All the results are shown in Figure \\ref{fig:albedo_2}.\n\n\\subsection{Changes due to hard X-ray spectral index}\nIn order to study changes due to spectral index only, a completely isotropic HXR primary source is input into the simulation. An isotropic source should produce the minimum albedo contribution and hence the smallest changes in source size and position. Using an isotropic HXR primary source with a FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ and three spectral indices of $\\gamma=2,3,4$, Figure \\ref{fig:albedo_2} a-d shows that the albedo contribution from a smaller spectral index $\\gamma$ produces the largest shift in position and a larger total source size. The lowest modelled spectral index of $\\gamma=2$ produces the greatest shift of $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$ at $\\mu=0.5-0.6$ and $\\sim30$ keV. This spectral index also produces the largest source size, compared with the other spectral indices of $\\gamma=3,4$ modelled, and has a resulting FWHM$\\sim9\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$ at $\\mu=1$.\n\n\\subsection{Changes due to hard X-ray primary source size}\nFor the same reasons in the previous section, an isotropic source is input into the simulations to study changes due to an input primary HXR source size. For an isotropic HXR primary source with fixed spectral index of $\\gamma=3$, three primary source sizes are tested: FWHM$\\sim0\\mbox{\\ensuremath{^{\\prime\\prime}}}$ (point), FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ and FWHM$\\sim14\\mbox{\\ensuremath{^{\\prime\\prime}}}.6$. As expected, it is found all primary source sizes produce the same shift in centroid position. The maximum shift in position occurs at $\\mu=0.5-0.6$ and $\\sim30$ keV for all sources modelled. These results can be seen in Figure \\ref{fig:albedo_2} e-h.  Although the FWHM of the total source grows with increasing primary size, it is observed that the relative size of the total to the primary source is smaller for a larger primary source. This indicates that a larger primary source should have a smaller relative size increase due to albedo since the brightness distribution of a large primary source is less influenced by the reflected photons but nevertheless the source will look larger than the true primary size of a source between $10-100$ keV at all heliocentric angles. More importantly, even a primary point source will be seen as a source of finite size and an initial point source produces a total source with a FWHM peaking around $7\\mbox{\\ensuremath{^{\\prime\\prime}}}$ (Figure \\ref{fig:albedo_2} f, h). There is even an increase of $\\sim5\\mbox{\\ensuremath{^{\\prime\\prime}}}$ at the solar limb between $20-50$ keV where the albedo contribution is smallest.\n\n\\subsection{Changes due to hard X-ray anisotropy}\nUsing a HXR primary source with a FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ and $\\gamma=3$, three photon anisotropies are modelled using the ratio of downward to upward flux of: (1) $I_{down}\/I_{up}=1$ (isotropic), (2) $I_{down}\/I_{up}=2$ and (3) $I_{down}\/I_{up}=5$. The shift in centroid position is larger for a higher initial downward anisotropy for all $\\mu$ and energies, shown in Figure \\ref{fig:albedo_2} i, k. All shifts follow the general trend and tend towards zero at the centre ($\\mu=1$) and the limb ($\\mu=0$). A directivity of $5$ produces a peak difference of $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ and even an isotropic source produces a peak difference of $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.4$. The shift in source position peaks near $\\mu=0.4-0.6$ and $\\sim$30 keV for a downward anisotropy of $2$ and an isotropic source, but the shift peaks at a lower $\\mu=0.4-0.5$ for a downward directivity of $5$. The stronger downward beaming of the primary source also leads to larger apparent source sizes for all $\\mu$ and energies (Figure \\ref{fig:albedo_2} j, l). It should be observed that the total FWHM produced for a directivity of $5$ peaks at $\\mu=0.15$ (Figure \\ref{fig:albedo_2} 2p) giving an apparent FWHM$\\sim13\\mbox{\\ensuremath{^{\\prime\\prime}}}$. Since the fraction of reflected photons reduces with $\\mu$ the FWHM in perpendicular direction can be expected to slowly decrease from disk centre to limb, but the FWHM actually increases, peaks at $\\mu\\sim0.15$ and only then starts to decrease. This effect is due to the angular dependence of the Compton cross section. This is because the azimuthal-independent (that is, assuming that the scattering is isotropic in azimuthal angle) Compton cross section is anisotropic and peaks at angles less than $90^{\\circ}$ (see Figure \\ref{cs_graphs} in Chapter \\ref{ref:Chapter1}), which allows a larger number of photons to scatter into an observer direction for flares close to the limb. It is this anisotropy in the scattering of the photons that causes the FWHM to peak at an angle smaller than $\\mu=1.0$. The observation of this effect is particularly clear in the case of high downward directivity (Figure \\ref{fig:albedo_2} l). It should be noted that if the anisotropy of the photon distribution is created from a given radiating electron distribution, then as the directivity of the electron distribution increases,\n\\clearpage\n\\begin{sidewaysfigure*}[ht]\n\\centering\n\\includegraphics[width=16.5cm]{report_spect.pdf}\n\\includegraphics[width=16.5cm]{report_size.pdf}\n\\includegraphics[width=16.5cm]{report_direct.pdf}\n\n\n\\caption[Plots of the source position shift in the radial direction and source size FWHM in the perpendicular to radial direction due to albedo, against X-ray energy and heliocentric angle.]{Spectral index dependency (panels a-d)): the source position shift is in the radial direction due to albedo (source $\\mu= 0.55$) and source size FWHM is in the perpendicular to radial direction for various spectral indices $\\gamma$ for an isotropic source with FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$: green-$\\gamma= 4$; red-$\\gamma= 3$; purple-$\\gamma=2$. Primary source size dependency (panels e-h)): isotropic primary source with $\\gamma= 3$: orange-point source, red-FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$, blue-FWHM$\\sim14\\mbox{\\ensuremath{^{\\prime\\prime}}}.6$. Anisotropy dependency (ratio of downward to upward directed fluxes) (panels i-l)): simulations are for the primary source with FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ and spectral index $\\gamma= 3$: red-anisotropy $= 1$ (isotropic), blue-anisotropy$=2$, green-anisotropy$= 5$. Graphs as a function of energy are for $\\mu = 0.55$ and the graphs as a function of $\\mu$ are for energies between 20 and 50 keV.}\n\\label{fig:albedo_2}\n\\end{sidewaysfigure*}\n\\clearpage\na higher proportion of the X-ray distribution will be concentrated over a narrower solid angle of emission, instead of a higher proportion of X-rays emitted downward at all angles, as was assumed in this simple simulation. The photon directivity is properly modelled from a chosen electron distribution in Chapter \\ref{ref:Chapter6}.\n\n\\section{Discussion and conclusions}\n\nThe results of the simulations show that albedo can substantially affect the precise position and source size measurements of X-ray sources. Therefore, the effect of albedo should always be considered when the sizes or positions of X-ray sources are analysed. The only exception is occulted flares or possibly limb flares. However, this assumption should be used with caution particularly if the anisotropy of the source is high. The albedo displacement of source position is radially directed towards the disk centre and depends on the anisotropy of X-ray radiation, the X-ray source size and the spectral index of the primary source. Similar to total reflected flux, the displacement of HXR source position is energy dependent. The largest displacement can be observed in the range between $30-50$ keV at $\\mu\\sim0.5$ (heliocentric angle $\\sim60^{\\circ}$). The shift in centroid position in this energy range is $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.1-0\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$ for an isotropic (minimum albedo) source $1\\mbox{\\ensuremath{^{\\prime\\prime}}}.4$ above the photosphere and this can be up to $\\sim0\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ for a downward beaming with factor of $5$. Because of the albedo, X-ray source sizes will be energy dependent, larger in the perpendicular to radial direction, and elliptical even for a spherically symmetric primary source. In the perpendicular to radial direction, the largest growth in source size occurs for sources close to the solar disk centre, in the energy range between $30-50$ keV, where albedo is the strongest. Thus, an isotropic primary source with FWHM$\\sim4\\mbox{\\ensuremath{^{\\prime\\prime}}}.9$ at $1\\mbox{\\ensuremath{^{\\prime\\prime}}}.4$ above the photosphere will have an apparent FWHM size of $\\sim9\\mbox{\\ensuremath{^{\\prime\\prime}}}$ in the energy range $20-50$ keV for sources in a wide range of heliocentric angles from $0^{\\circ}$ to $80^{\\circ}$.\n\nThe simulations demonstrate that X-ray sources will have a minimum size. An isotropic point source at $1$ Mm above the photosphere will be measured by {\\it RHESSI} as a source with a FWHM size of $\\sim7\\mbox{\\ensuremath{^{\\prime\\prime}}}$ across. This result can explain larger X-ray footpoint sizes than EUV or optical ones e.g. \\cite{Kasparovaetal2005}. \\cite{DennisPernak2009} reported that the average semi-minor axis of 18 double source flares is about $4\\mbox{\\ensuremath{^{\\prime\\prime}}}$, while a few of the X-ray source sizes were found to be consistent with line sources along the flare ribbons. \n\nThe energy dependent character of albedo predicts that the source size as measured by {\\it RHESSI} should grow with energy from $10$ keV up to $\\sim30$ keV. Considering a large primary source of $14\\mbox{\\ensuremath{^{\\prime\\prime}}}.6$ across, for example a flaring loop, it is found that the source will grow up to $\\sim18\\mbox{\\ensuremath{^{\\prime\\prime}}}$ at $\\sim30$ keV. The X-ray anisotropy results show that spatial changes due to albedo have a great diagnostic potential for finding the anisotropy of the radiating electron distribution in the chromosphere. The source size changes due to albedo were not applied to the coronal X-ray sources studied in Chapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3} because coronal X-ray sources sit at much greater heights ($\\sim\\ge15\\mbox{\\ensuremath{^{\\prime\\prime}}}$) than the chromospheric X-ray sources studied here ($1$ Mm) and hence, changes in X-ray source size and position will be smaller at coronal heights due to albedo. However, future work will fully investigate the albedo contribution when studying coronal X-ray source sizes and positions. \n\\chapter{Solar flare X-ray albedo and spatially resolved polarization of hard X-ray (HXR) footpoints}\n\\label{ref:Chapter6}\n\n\\normalsize{\\it This work can be found in the publications \\cite{2011A&A...536A..93J} and \\cite{KontarJeffrey2010}.}\n\n\n\\section{Introduction}\n\nAs discussed in Chapter \\ref{ref:Chapter1}, a major insight regarding the angular properties of HXR footpoints comes directly from the X-ray polarization. The anisotropy and polarization of an X-ray distribution produced by bremsstrahlung will increase with the\nanisotropy of the electron distribution \\citep[e.g.][]{1959RvMP...31..920K, GlucksternHull1953, ElwertHaug1970,Brown1972, Haug1972, LeachPetrosian1983},\nand hence in theory HXR polarization allows the anisotropy of the\nemitting electron distribution in the chromosphere to be inferred. However, compared with the other HXR observables: energy, spatial\nlocation, source size and time of arrival for example, polarization measurements through the years\n\\citep[see][as a review]{2011SSRv..159..301K} have been fraught with difficulties and the measurements\noften met with skepticism. Nonetheless many missions have reported measurements\nof HXR polarization from solar flares \\citep{Tindoetal1970, Tindoetal1972, Nakadaetal1974,\nTindoetal1976, Lemenetal1982, McConnelletal2004, Boggsetal2006, 2006SoPh..239..149S, McConnelletal2007}.\n\nCompton scattering is polarization dependent; the total integrated polarization of an HXR source will be\naltered by its albedo component. However, the albedo\ncomponent will not only change the total polarization of the observed HXR footpoint; there will be spatial variations in polarization across the extent of the albedo source and hence the observed HXR footpoint.\nChapter \\ref{ref:Chapter5} discussed the importance of compensating for the Compton backscattered albedo component when interpreting the positions and sizes of HXR sources. In Chapter \\ref{ref:Chapter6}, the usefulness of albedo polarization as a valuable diagnostic tool will be discussed.\nSpatially resolved polarization measurements across a HXR source caused by albedo, have the advantage over spatially integrated measurements since both the magnitude and the direction of polarization will change with X-ray directivity; allowing maps of the albedo and primary components to be created. Understanding how these two parameters change with the photon anisotropy is essential and provides a new method of investigating the entire photon anisotropy from a single HXR source.\nHowever, although there are a\nnumber of simulations for the spatially integrated polarization signal in flares\n\\citep{ElwertHaug1970,Haug1972,Zharkovaetal1995,Emslieetal2008,Zharkovaetal2010} and how an albedo component changes the spatially integrated polarization \\citep[e.g.][]{Henoux1975,LangerPetrosian1977,BaiRamaty1978}, until now spatially resolved polarization had not been investigated. The work shown in this chapter and published in \\cite{2011A&A...536A..93J} is the only known prediction of the spatially resolved hard X-ray\npolarization due to albedo.\n\nFor the first time in solar physics, spatially resolved polarization across HXR sources at various locations\non the solar disk, taking into account the influence of albedo, is computed, for various emitting electron\npopulations. The simulations \\citep{2011A&A...536A..93J} will also predict the angular resolution and preferred energy range required for such future polarization observations. Finally, the usefulness of future observations such as these will be discussed. Chapter \\ref{ref:Chapter6} will also briefly examine how changing the maximum electron energy available during bremsstrahlung can alter spatially integrated polarization measurements with photon energy, possibly providing a new method for finding the maximum electron cutoff energy.\n\n\\section{Defining the polarization of an X-ray distribution}\nThe polarization state of incoherent radiation can be completely described using four Stokes\nparameters \\citep{Stokes1852,Chandrasekhar1960}. The Stokes pseudovector consists of these four\nparameters and takes the form of\n\\begin{equation}\nS= \\left[\n\\begin{matrix}\nI \\\\\nQ\\\\\nU\\\\\nV\n\\end{matrix}\n\\right]=[I\\:Q\\:U\\:V]^{T}.\n\\end{equation}\nA pseudovector is a vector-like object that is invariant under an inversion of its coordinate axes and is often called an axial vector. The first Stokes parameter $I$ is the normalised\ntotal intensity of the photon beam and hence always equal to 1, while $Q\/I$, $U\/I$ and $V\/I$ will have values between $-1$ and\n$1$. The second and third normalised Stokes parameters are used to define linear polarization with $1$ or $-1$\nindicating that the beam or photon packet is completely polarized with the sign providing the\ndirection of polarization. The fourth parameter is used to describe circular polarization. However,\nbremsstrahlung emission in the solar corona or chromosphere only produces radiation that is\nlinearly polarized. In order to produce circularly polarised radiation via bremsstrahlung, the\nspins of the radiating electrons need to be aligned and the magnetic field in the\ncorona or chromosphere is not strong enough for this alignment. Also, Compton scattering cannot produce circularly polarized emission, that is linear photons can not become circularly polarized during a Compton scattering. This means that only the\nfirst three Stokes parameters are required and the fourth can be set to zero throughout the\nsimulations. Generally in X-ray and gamma ray astronomy the polarization of radiation is measured\nusing the degree of polarization ($DOP$) and the polarization angle $\\Psi$, which is the preferred\ndirection of the electric field. These are defined using the Stokes parameters as,\n\\begin{equation}\nDOP=\\frac{\\sqrt{Q^{2}+U^2}}{I},\n\\label{eq:DOP}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:Psi}\n\\Psi=\\frac{1}{2}\\arctan\\left(\\frac{-U}{-Q}\\right),\n\\end{equation}\nwhere the angle $\\Psi$ is chosen to lie within the quadrant between $[-180^{\\circ},180^{\\circ}]$,\nso that $\\arctan\\left(\\frac{+0}{+0}\\right)=+0^{\\circ},\\; \\arctan\\left(\\frac{+0}{-0}\\right)=+180^{\\circ},\\; \\arctan\\left(\\frac{-0}{+0}\\right)=-0^{\\circ}$ and\n$\\arctan\\left(\\frac{-0}{-0}\\right)=-180^{\\circ}$. The negatives introduced into Equation (\\ref{eq:Psi}) ensure that a\nnegative $Q$ gives $0$ and a positive $Q$ gives $90^{\\circ}$. Hence with this definition, when\n$\\Psi=0^{\\circ}$, the observed radiation is polarized parallel to the radial direction at the solar\ndisk and when $\\Psi=90^{\\circ}$, the radiation is polarized perpendicular to the radial direction.\nThe opposite definition of radiation polarized parallel to the radial direction having $\\Psi=90^{\\circ}$ is equally valid as long as the definition is consistent.\n\\begin{figure}\n\\includegraphics[width=15.5cm]{stokes.png}\n\\caption[Diagram showing the preferred direction of the electric field for a photon travelling out of the page,  for each of the possible values of the linear Stokes parameters $Q$ and $U$.]{Diagram showing the preferred direction of the electric field for a photon travelling out of the page,  for each of the possible values of the linear Stokes parameters $Q$ and $U$. A completely unpolarised source has $Q=0$ and $U=0$ and no preferred direction of the electric field.}\n\\label{fig:stokes}\n\\end{figure}\nThe Stokes parameters are also frame dependent and hence have to be updated by the\nuse of rotation matrices \\citep[e.g.,][]{Hovenier1983} when moving between different coordinate frames. In the simulations, the Stokes pseudovector will be initially defined in the source frame and must be rotated to the scattering frame during each Compton scattering. The pseudovector must then be rotated back to the source frame before the results are examined. These rotations will be properly described in Section \\ref{ref:upps}. The $DOP$ remains unchanged by a rotation but the polarization angle $\\Psi$ is measured with respect to the new frame.\n\nFor simplicity, it is assumed that the flare loop and the dominant direction of the electrons lie parallel with the local solar vertical. This means that both the spatially integrated and spatially resolved bremsstrahlung polarization direction $\\Psi$ only ever equal $0^{\\circ}$ or $90^{\\circ}$, since $U$ is always close to zero \\citep{BaiRamaty1978} (see Equation \\ref{eq:Psi}). Here, it should be noted that the HXR source is always assumed to be much smaller than the solar disk. However, Compton scattering in the photosphere can\nproduce non-zero values of $U$. This means that the spatially resolved Compton scattered $\\Psi$ can have values other than $0^{\\circ}$ or $90^{\\circ}$ in the solar disk frame, which is the frame of the HXR source and can therefore provide us with additional information regarding the anisotropy of the electron distribution. The spatially integrated albedo Stokes parameters again sum to produce $\\Psi$ values of either $0^{\\circ}$ or $90^{\\circ}$. Therefore this means the spatially integrated polarization angle $\\Psi$ never provides us with any information regarding the anisotropy of the electron distribution. However, \\cite{Emslieetal2008} found that the spatially integrated direction of the\npolarization angle is related to the dominant direction of electrons or equivalently, the tilt of the guiding field or loop with respect to the local solar vertical; with values of spatially integrated $\\Psi$ other than $0^{\\circ}$ or $90^{\\circ}$ revealing that the loop does not lie parallel to the local solar vertical.\n\n\\section{HXR footpoint bremsstrahlung polarization}\n\n\\subsection{The radiating electron distribution}\nIn order to create X-ray distributions in the chromosphere with varying anisotropy and polarization, for input into the simulations, an X-ray emitting electron distribution in the chromosphere is chosen to have the following form,\n\\begin{equation}\nF(E,\\beta)\\propto E^{-\\delta_{T}}\\exp\\left({-\\frac{(1+\\cos\\beta)^{2}}{\\Delta\\nu^{2}}}\\right).\n\\label{eq:F_E_beta}\n\\end{equation}\n$F(E,\\beta)$ is the electron flux distribution [electrons cm$^{-2}$ s$^{-1}$ keV$^{-1}$] in the chromosphere, $E$ is the electron energy and $\\beta\\in[0^{\\circ},180^{\\circ}]$ is the pitch-angle of the emitting electrons\nvelocity to the local magnetic field with $\\beta=0^{\\circ}$ directed away from the Sun along the local solar vertical. The energy dependence follows a power law as shown by observations and is produced by an injected electron distribution of $\\delta=\\delta_{T}+2$ \\citep[see][as recent\nreviews and Chapter \\ref{ref:Chapter1}]{2011SSRv..159..107H,2011SSRv..159..301K}. The electron angular distribution is modelled as a Gaussian distribution; allowing the angular anisotropy of the electron distribution to be easily controlled by a single parameter $\\Delta\\nu$. The smaller the value of $\\Delta\\nu$, the greater the proportion of the electron distribution, and hence the resulting bremsstrahlung X-ray emission directed towards the photosphere.\n\n\\subsection{The emitted primary X-ray photon distribution}\n\nThe intensity of an X-ray photon distribution $I(\\epsilon,\\theta)$ [photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$] produced by bremsstrahlung for a\nchosen electron distribution $F(E,\\beta)$ is given by\n\\begin{eqnarray}\nI(\\epsilon,\\theta)\\propto\\int^{\\infty}_{E=\\epsilon}\\int^{2\\pi}_{\\Phi=0}\\int^{\\pi}_{\\beta=0}F(E,\\beta)\n\\sigma(E,\\epsilon,\\Theta) \\sin\\beta d\\beta d\\Phi dE\n\\label{eq:I}\n\\end{eqnarray}\nwhere $\\epsilon$ is the X-ray energy and $\\sigma(E,\\epsilon,\\Theta)$ is the total (averaged over all polarization\nstates) angle-dependent bremsstrahlung cross-section\n\\citep{ElwertHaug1970,BaiRamaty1978,Massoneetal2004}. $\\theta\\in[0^{\\circ},180^{\\circ}]$ is the\nphoton polar emission angle measured from the local solar vertical with $\\theta=0^{\\circ}$ directed\naway from the Sun, as for $\\beta$.\n$\\Phi\\in[0^{\\circ},360^{\\circ}]$ is the corresponding electron azimuthal angle measured in the plane perpendicular to the local solar vertical and\n$\\Theta(\\beta,\\Phi,\\theta)$ is the angle between the plane of emission (at angle $\\beta$)\nand the plane of observation (at angle $\\theta$). Figure \\ref{fig:epfig3} depicts this scenario pictorially showing the angles $\\beta$, $\\Phi$, $\\theta$ and $\\Theta$.\nThe X-ray emission angle\nis described by $\\mu=\\cos\\theta$, where $\\mu$ from $0$ to $1$ corresponds to emission away from the\nSun, and $\\mu$ from $-1$ to $0$ corresponds to emission towards the solar surface. The X-ray emission\nangle $\\mu=\\cos\\theta$ is related to the electron pitch-angle $\\beta$ by:\n\\begin{equation}\n\\cos\\Theta=\\cos\\theta\\cos\\beta+\\sin\\theta\\sin\\beta\\cos\\Phi.\n\\label{eq:cosTheta}\n\\end{equation}\nViewing the outward emission from $\\mu=0$ to $\\mu=1$ corresponds to observing the HXR source at a selected\nheliocentric angle on the solar disk, that is $\\mu=0$ corresponds to $90^{\\circ}$ and is equivalent to viewing a HXR footpoint source sitting at the solar limb.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=14cm]{epfig4.png}\n\\caption[A cartoon of a typical solar flare scenario where an electron in the chromosphere, transported along the guiding field from the corona interacts by Coulomb collisions producing a HXR photon. ]{{\\it Left:} A cartoon of a typical solar flare scenario where an electron in the chromosphere, transported along the guiding field from the corona interacts by Coulomb collisions producing an HXR photon. {\\it Right:} Diagram indicating all the angles defined in Equation \\ref{eq:cosTheta}.}\n\\label{fig:epfig3}\n\\end{figure*}\n\nJust as with the X-ray intensity, $I$, the linear Stokes parameters $Q$ and $U$ can be calculated in a similar manner:\n\\begin{equation}\nQ(\\epsilon,\\theta)\\propto\\int^{\\infty}_{E=\\epsilon}\\int^{2\\pi}_{\\Phi=0}\\int^{\\pi}_{\\beta=0}F(E,\\beta)\\sigma_{Q}(E,\\epsilon,\\Theta)\n \\sin\\beta d\\beta d\\Phi dE,\n\\label{eq:Q}\n\\end{equation}\n\\begin{equation}\nU(\\epsilon,\\theta)\\propto\\int^{\\infty}_{E=\\epsilon}\\int^{2\\pi}_{\\Phi=0}\\int^{\\pi}_{\\beta=0}F(E,\\beta)\\sigma_{U}(E,\\epsilon,\\Theta)\n\\sin\\beta d\\beta d\\Phi dE ,\n\\label{eq:U}\n\\end{equation}\n\\\\\nwith the only difference being the use of either $\\sigma(E,\\epsilon,\\Theta)$,\n$\\sigma_{Q}(E,\\epsilon,\\Theta)$ or $\\sigma_{U}(E,\\epsilon,\\Theta)$.\n$\\sigma(E,\\epsilon,\\Theta)$, $\\sigma_{Q}(E,\\epsilon,\\Theta)$ and $\\sigma_{U}(E,\\epsilon,\\Theta)$ are the polarization\ndependent cross-sections for bremsstrahlung taken from \\cite{GlucksternHull1953} and also following\nthe form used in \\cite{Haug1972} and \\cite{Emslieetal2008}. They are given by\n\\begin{equation}\n\\sigma(E,\\epsilon,\\Theta)=\\sigma_{\\perp}(E,\\epsilon,\\Theta)+\\sigma_{\\parallel}(E,\\epsilon,\\Theta),\n\\end{equation}\n\\begin{equation}\n\\sigma_{Q}(E,\\epsilon,\\Theta)=(\\sigma_{\\perp}(E,\\epsilon,\\Theta)-\\sigma_{\\parallel}(E,\\epsilon,\\Theta))\\cos2\\Theta ,\n\\end{equation}\nand\n\\begin{equation}\n\\sigma_{U}(E,\\epsilon,\\Theta)=(\\sigma_{\\perp}(E,\\epsilon,\\Theta)-\\sigma_{\\parallel}(E,\\epsilon,\\Theta))\\sin2\\Theta,\n\\end{equation}\nwhere $\\sigma_{\\perp}(E,\\epsilon,\\Theta)$ and $\\sigma_{\\parallel}(E,\\epsilon,\\Theta)$ are the perpendicular and parallel components of the\nbremsstrahlung cross-section respectively. $Q(\\epsilon,\\theta)$ and $U(\\epsilon,\\theta)$ are normalised between [-1,1] by dividing\nthrough by $I(\\epsilon,\\theta)$.\n\n\\section{Photon transport in the photosphere and changes in hard X-ray polarization}\n\n\\subsection{Monte Carlo simulation inputs}\nIn order to study changes in HXR polarization due to a given target-averaged electron angular distribution, the energy, angular and polarization properties of the input\nphoton distributions for the MC code are determined via the chosen input electron distributions\ngiven by Equation \\ref{eq:F_E_beta}. In all simulation runs, a mean electron spectrum of spectral index $\\delta_{T}=2$ is used. This means that the injected electron distribution has a typical solar flare value of $\\delta=4$. In order to test how both the total integrated and spatially resolved HXR polarization change with electron directivity, three values of $\\Delta\\nu$ are used to describe various pitch-angle distributions of electrons in the chromosphere: $\\Delta\\nu=4.0$ , $\\Delta\\nu=0.5$ or $\\Delta\\nu=0.1$. A $\\Delta\\nu=4.0$ electron distribution produces an approximately isotropic, unpolarised\nphoton distribution, while the $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$ electron distributions produce photon distributions with progressively greater beaming towards the photosphere.\nIn the MC simulations, distributions of energy $\\epsilon$, angle $\\theta$ and polarization $Q$ and $U$\nare numerically created from $I(\\epsilon,\\theta)$, $Q(\\epsilon,\\theta)$ and $U(\\epsilon,\\theta)$ (Equations \\ref{eq:I}, \\ref{eq:Q} and \\ref{eq:U}). Each photon input azimuthal angle $\\phi$ is simply drawn from a uniform, random distribution between $0$ and $2\\pi$. The spatial properties are determined via Equation \\ref{eq:I_xy}, as described in Chapter \\ref{ref:Chapter5}, again at a height $h=1$ Mm. To achieve the best statistics feasible, $10^8$ photons are used in every MC code run.\nOnce the HXR source is created from a given target electron distribution, the MC code runs as described in Chapter \\ref{ref:Chapter5} with the only differences due to the inclusion of polarization, which will be described in the following sections. An updated flow chart showing the main steps of the MC photon transport code, including polarization, is shown in Figure \\ref{fig:mcs_flow_chart_pol}.\n\n\\begin{figure}\n\\includegraphics[width=15cm]{mcs_flow_chart_pol_4.png}\n\\centering\n\\caption{An updated version of the steps in the MC simulations including polarization and the creation of a HXR distribution via a chosen electron distribution in the chromosphere.}\n\\label{fig:mcs_flow_chart_pol}\n\\end{figure}\n\n\\subsection{Photoelectric absorption and hard X-ray polarization}\nChapter \\ref{ref:Chapter5} describes how photoelectric absorption is the dominant interaction process in the photosphere below $\\sim $10 keV. The probability of photoelectric absorption\nis assumed to be independent of polarization \\citep{Poutanenetal1996}.\nOnly the angular distribution of the ejected electron is dependent upon the photon polarization,\nwhich is not modelled in these simulations.\n\n\\subsection{Compton scattering and hard X-ray polarization}\\label{ref:upps}\nAs described in Chapter \\ref{ref:Chapter5}, above $\\sim$10 keV, Compton scattering is the dominant interaction process in the photosphere. The polarization\ndependent differential Compton scattering cross-section is given by \\cite{KleinNishina1929}, but\nthe form used in these simulations is from \\cite{McMaster1961} and \\cite{BaiRamaty1978}\n\\begin{equation}\n\\frac{d\\sigma_{c}}{d\\Omega}=\\frac{1}{2}r_{0}^{2}\\left(\\frac{\\epsilon}{\\epsilon_{0}}\\right)^{2}{\\bigg\\lgroup}\\frac{\\epsilon}{\\epsilon_{0}}+\n\\frac{\\epsilon_{0}}{\\epsilon}-\\sin^{2}\\theta_{S}{\\bigg\\lgroup}1-Q\\cos2\\phi_{S}\n -U\\sin2\\phi_{S}{\\bigg\\rgroup}{\\bigg\\rgroup},\n\\end{equation}\nwhere $r_{0}=2.82\\times10^{-13}$cm is the classical electron radius, $\\epsilon_{0}$ is the energy\nof the incoming photon, $\\epsilon$ is the energy of the outgoing photon, $\\theta_{S}$ is the polar\nscattering angle, $\\phi_{S}$ is the azimuthal scattering angle and $Q$ and $U$ are the linear\nStokes parameters respectively \\citep{McMaster1961, BaiRamaty1978}. The maximum change in $DOP$\noccurs when $\\theta_{S}=90^{\\circ}$ and no change in $DOP$ occurs for a backscattering at\n$180^{\\circ}$. The azimuthal scattering angle $\\phi_{S}$ also has a non uniform dependency on the incoming polarization state. If the HXR photon distribution is completely isotropic and hence unpolarized, then the Klein-Nishina cross section returns to the unpolarized form used in Chapter \\ref{ref:Chapter5} Equation \\ref{eq:sigma_c}, since the linear Stokes parameters $Q$ and $U$ are zero.\n\nIn the MC simulations, when a Compton scattering occurs, the properties of the outgoing photon: energy $\\epsilon$, polar\nscattering angle $\\theta_{S}$, azimuthal angle $\\phi_{S}$ and Stokes parameters $Q$ and $U$, need to be updated. New polar scattering angles $\\theta_{S}$\nfor each photon can be found by integrating the polarization dependent Klein-Nishina differential cross section over $\\phi_{S}$ to produce a differential cross-section that is only dependent on $\\epsilon$ and $\\theta_{S}$, that is the unpolarized form given by Equation \\ref{eq:sigma_c}. The new energy $\\epsilon$ and scattering angle $\\theta_{S}$ can then be found by the same method shown in Chapter \\ref{ref:Chapter5} Section \\ref{ref:cs} while the new $\\phi_{S}$ and Stokes parameters $Q$ and $U$ are found by the method described below.\n\n\\subsection{Updating photon polarization states}\\label{ref:uppps}\nIf the photon distribution is completely isotropic and unpolarised then the azimuthal scattering\nangle $\\phi_{S}$ can just be sampled from a uniform distribution between $0$ and $2\\pi$, but this\nis not true for the more general polarization dependent case. The probability density function of\nobtaining a value of $\\phi_{S}$ between $\\phi_{S}$ and $\\phi_{S}+d\\phi_{S}$ can be described by:\n\\begin{equation*}\nP(\\phi_{S})=\\frac{1}{2\\pi}\\frac{d\\sigma_{c}(\\epsilon,\\theta_{S},\\phi_{S})\/d\\Omega}{d\\sigma_{c}(\\epsilon,\\theta_{S})\/d\\Omega}=\n\\end{equation*}\n\\begin{equation}\n=\\frac{1}{2\\pi}\\frac{\\frac{\\epsilon_{0}}{\\epsilon}+\\frac{\\epsilon}{\\epsilon_{0}}-\n\\sin^{2}\\theta_{S}(1-Q\\cos2\\phi_{S}-U\\sin2\\phi_{S})}{\\frac{\\epsilon_{0}}{\\epsilon}+\\frac{\\epsilon}{\\epsilon_{0}}-\\sin^{2}\\theta_{S}}\n\\end{equation}\nwith the maximum value of this function given by:\n\\begin{equation}\nP_{max}(\\phi_S)=\\frac{1}{2\\pi}\\frac{\\frac{\\epsilon_{0}}{\\epsilon}+\\frac{\\epsilon}{\\epsilon_{0}}-\n\\sin^{2}\\theta_{S}\\left(1-\\sqrt{Q^{2}+U^{2}}\\right)}{(\\frac{\\epsilon_{0}}{\\epsilon}+\n\\frac{\\epsilon}{\\epsilon_{0}}-\\sin^{2}\\theta_{S})}.\n\\end{equation}\nFirstly, a value of $\\phi_{S}$ is sampled between $0$ and $2\\pi$. The condition that\n$P(\\phi_S)<P_{max}(\\phi_S)$ is then used to accept a value of $\\phi_{S}$ and provides a method for\nsampling values of $\\phi_{S}$ for each photon, using the new values $\\theta_S$ and $\\epsilon$\nalready calculated for each photon. This method is repeated until the condition is\nsatisfied for each photon and each photon is provided with an azimuthal scattering angle\n\\citep{Salvatetal2008}.\n\nDue to Compton scattering, the Stokes parameters have to be updated using the scattering matrix T\n\\citep{McMaster1961, BaiRamaty1978}\t\t\t\t\\begin{equation} \t\tT(\\epsilon,\\theta_{S})= \t\t\n\\left( \t\t\\begin{array}{ccc}\n  \t\t\\frac{\\epsilon}{\\epsilon_{0}}+\\frac{\\epsilon_{0}}{\\epsilon}+\\sin^{2}\\theta_{s}&\n  \\sin^{2}\\theta_{s}  &0   \\\\\n  \t\t\\sin^{2}\\theta_{s}&\\cos^{2}\\theta_{s}+1   & 0  \\\\\n  \t\t0& 0  &2\\cos\\theta_{s}\n\t\t\\end{array} \t\t\\right).\n\\label{eq:T_matrix}\n\t\t\\end{equation} Before a scattering, the Stokes parameters have to be rotated by $\\phi_S$ so\nthat they are defined relative to the plane of scattering, using the rotation matrix $M(\\phi_{S})$ given by:\n\t\t\n\t\t\\begin{equation} \t\tM(\\phi_{S})= \t\t\\left( \t\t\\begin{array}{ccc} \t\t1&0&0 \\\\\n  \t\t0 &\\cos2\\phi_{S}  &\\sin2\\phi_{S} \\\\\n  \t\t0 &-\\sin2\\phi_{S}  &\\cos2\\phi_{S} \\\\\n\t\t\\end{array} \t\t\\right). \t\t\\end{equation}\n\t\t\n After a scattering, the Stokes\nparameters have to be rotated again so that they are defined relative to the starting position of\nthe source, and are rotated by the rotation matrix\n$M(-\\Xi)$.\n\t\t\n\t\t\\begin{equation} \t\tM(-\\Xi)= \t\t\\left( \t\t\\begin{array}{ccc} \t\t1&0&0 \\\\\n  \t\t0 &\\cos-2\\Xi  &\\sin-2\\Xi \\\\\n  \t\t0 &-\\sin-2\\Xi  &\\cos-2\\Xi \\\\\n\t\t\\end{array} \t\t\\right), \t\t\\end{equation}\n\\begin{figure}\n\\centering\n\\includegraphics[width=90mm]{hovenier.png}\n\\caption[The position of the photon before scattering and after scattering\nand the angle $\\Xi$ that determines the final rotation of the Stokes parameters back into the\nframe of the source from the scattering frame.]{The position of the photon before scattering (blue) and after scattering (green)\nand the angle $\\Xi$ that determines the final rotation of the Stokes parameters back into the\nframe of the source from the scattering frame.}\n\\label{fig:rotation}\n\\end{figure}\nwhere\n\\begin{equation}\n\\cos\\Xi=\\pm\\frac{w'-\\cos\\theta_{s}w}{\\sqrt{1-\\cos^{2}\\theta_{s}}\\sqrt{1-w^2}}.\n\\label{eq:cosGamma}\n\\end{equation}\n$w$ and $w'$ are the current and previous $z$ direction cosines respectively \\citep{Hovenier1983}.\nThe $\\pm$ in equation (\\ref{eq:cosGamma}) is present due to the negative sign being used when\n$\\pi\\leq\\phi_{s}\\leq2\\pi$ and the positive sign for $0\\leq\\phi_{S}<\\pi$. The angle $\\Xi$ is\nshown in Figure \\ref{fig:rotation}. $\\Xi$ is the angle between the scattering plane and the\nnormal plane in the frame of the source. Therefore during a Compton scattering the order of the\nrotations on the Stokes pseudovector $[I Q U]^{T}$ is $M(\\phi_{S})T(\\epsilon,\\theta_{S})M(-\\Xi)$.\n\n\\section{Integrated distribution of hard X-ray polarization}\n\n\\subsection{Hard X-ray polarization and electron directivity}\n\nFigure \\ref{fig:Fig1} shows the resulting total integrated flux and degree of polarization for each simulation run using the $\\Delta\\nu=4.0$, $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$ electron pitch-angle distributions.\nEach HXR component: primary towards the observer only (orange), backscattered albedo (blue) and also the total observed HXR emission (green) are plotted against emission angle or equivalently heliocentric angle $\\mu=\\cos\\theta$. Each are shown between\n20-50 keV, where the HXR albedo emission peaks. As expected, even though the\n$\\Delta\\nu=0.1$ distribution has a greater downward beaming and hence a smaller proportion of its emission is directed towards the observer than the $\\Delta\\nu=0.5$ distribution, within the 20-50 keV range the difference in anisotropy between the two primary distributions cannot be clearly seen. However, there is a clear difference in the $DOP$ of both distributions with disk location over 20-50 keV (second row Figure \\ref{fig:Fig1}). The total integrated polarization angle $\\Psi$ is not shown due to the fact that it remains constant with emission angle between the energy range of 20-50 keV at $\\Psi=0^{\\circ}$ and hence provides no information concerning the directivity of the electron distribution in the chromosphere. In Figure \\ref{fig:Fig1} $\\Psi=0^{\\circ}$ is indicated by the negative value of spatially integrated $DOP$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{dmu40501mu.pdf}\n\\caption[Plots of the photon flux and spatially integrated $DOP$ against heliocentric angle for the upward primary,\nalbedo and total components, for each MC simulation input.]{Photon flux (top row) and spatially integrated $DOP$ (bottom row) for the upward primary (orange),\nalbedo (blue) and total (green) components for both the $\\Delta\\nu= 4.0$\n(1st column), the $\\Delta\\nu= 0.5$ (2nd column) and the $\\Delta\\nu=0.1$ (last column) created\nphoton distributions respectively, at the peak albedo energies of 20-50 keV, for different\nlocations on the solar disk from the centre ($\\mu=1.0$) to the limb ($\\mu=0.0$). Note here that a\nnegative $DOP$ denotes that the direction of polarization $\\Psi$ across the source is parallel to the\nradial direction. The $DOP$ for $\\Delta\\nu=4.0$ photon distribution (near isotropic\ndistribution) shows nearly the same result as \\cite{BaiRamaty1978}, as expected.}\n\\label{fig:Fig1}\n\\end{figure*}\n\n\\subsection{Hard X-ray polarization and the high energy cutoff in the electron distribution}\nSpatially integrated polarization is dependent on the highest energy in the electron distribution, called the high cutoff energy\n \\citep{Heristchi1987}. When calculating spatially integrated polarization,\nequation (\\ref{eq:DOP}) reduces to $DOP=Q$ and Equation (\\ref{eq:Psi}) reduces to $\\Psi=\\frac{1}{2}\\arctan\\left(\\frac{-0}{-Q}\\right)$\nas $U$ sums to zero for a single measurement across the entire source. A negative $DOP$ indicates that the polarization angle\nis parallel to the radial direction ($\\Psi=0^{\\circ}$), while a positive $DOP$ indicates that the polarization angle is\nperpendicular to the radial direction ($\\Psi=90^{\\circ}$).\n\nFor the three electron distributions of $\\Delta\\nu=4.0$, $\\Delta\\nu=0.5$ and\n$\\Delta\\nu=0.1$, simulations were run with two high cutoff electron energies of $E_{cutoff}=500$\nkeV and $E_{cutoff}=2$ MeV. Figure \\ref{fig:energy_dnu_all} plots the flux and spatially integrated polarization across the total source (green)\nand the primary source only (orange) against photon energy $\\epsilon$ at four disk locations\n$\\mu\\in[0.20-0.25],[0.60-0.65],[0.80-0.85],[0.95-1.00]$ for $\\Delta\\nu=4.0$, $\\Delta\\nu=0.5$ and\n$\\Delta\\nu=0.1$ (top to bottom) respectively. The important property to observe here is not the\nmagnitude but the sign of the $DOP$ or whether $\\Psi=0^{\\circ}$ or $\\Psi=90^{\\circ}$.\n\nUsing an electron distribution with $\\Delta\\nu=4.0$ (Figure \\ref{fig:energy_dnu_all} rows 1 and 2) and a cutoff energy of $E_{cutoff}=500$ keV\nproduces a photon distribution with a negative $DOP$ at all photon energies and disk locations, while using the same distribution\nwith a cutoff energy of $E_{cutoff}=2$ MeV creates a photon distribution where the $DOP$ changes from negative to positive at\n$\\sim100-200$ keV at all disk locations.  During bremsstahlung, in order to conserve energy, electrons with higher energies will scatter through larger angles.\nWhen a photon is scattered through a large angle, its polarization is more likely to be directed perpendicular to the plane of emission\n($\\Psi=90^{\\circ}$) rather than parallel to the plane of emission ($\\Psi=0^{\\circ}$). Therefore, a change in the direction of polarization\n(from $\\Psi=0^{\\circ}$ to $\\Psi=90^{\\circ}$) indicates the presence of higher energies in the electron distribution, greater than $\\sim1$ MeV.\n\nAs the beaming of the electron distribution increases ($\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$ distributions), the above statement\ndoes not hold and it becomes more likely that both electron distributions with cutoff energies of either $E_{cutoff}=500$ keV and $E_{cutoff}=2$ MeV\nwill produce photons with $\\Psi=90^{\\circ}$. For the $\\Delta\\nu=0.5$ distribution (Figure\n\\ref{fig:energy_dnu_all} rows 3 and 4), as the source moves towards the solar centre, the\nphoton distribution created by the $E_{cutoff}=500$ keV also produces photons at higher energies (again $\\sim100-200$ keV)\nwith $\\Psi=90^{\\circ}$. For the very beamed $\\Delta\\nu=0.1$ distribution (Figure \\ref{fig:energy_dnu_all} rows 5 and 6), both\nthe $E_{cutoff}=500$ keV and $E_{cutoff}=2$ MeV distributions produce high ($\\sim100-200$ keV) energy photons with $\\Psi=90^{\\circ}$\nat all disk locations.\n\\begin{figure}[h]\n\\vspace{-20pt}\n\\centering\n\\includegraphics[width=14.55cm]{dmu4energy.pdf}\n\\includegraphics[width=14.55cm]{dmu05energy.pdf}\n\\includegraphics[width=14.55cm]{dmu01energy.pdf}\n\\caption[Plots of photon flux and spatially integrated $DOP$ for the upward primary,\nalbedo and total components against X-ray energy.]{Flux and spatially integrated $DOP$ for total source (green), primary (orange) and albedo (blue) vs. energy $\\epsilon$ using $E_{cutoff} = 500$ keV (solid) and $E_{cutoff} = 2$ MeV (dashed) for $\\Delta\\nu=4$ (top), $\\Delta\\nu=0.5$ (middle) and $\\Delta\\nu=0.1$ (bottom).}\n\\label{fig:energy_dnu_all}\n\\end{figure}\nTherefore using the direction of the spatially integrated polarization angle $\\Psi$ as an indicator for high energies in the electron\ndistribution becomes less and less useful as the beaming of the photon\/electron distribution\nincreases. The increased beaming causes lower and lower energy photons to scatter at larger angles, especially at locations closer to the solar centre,\nhence producing photons with $\\Psi=90^{\\circ}$ in the $E_{cutoff}=500$ keV distributions.\nHowever this method may be useful when the anisotropy of the HXR distribution in the chromosphere is close to isotropic as has been suggested by recent observations \\citep{KontarBrown2006,2013SoPh..284..405D}.\n\n\\section{Spatial distribution of hard X-ray polarization}\n\n\\subsection{Single Compton scatter for an isotropic unpolarised source}\n\nIn order to demonstrate the spatial variation in polarization due to Compton scattering, the\neasiest example to consider is the albedo patch created by an initially isotropic, unpolarised\npoint source at a height $h$ above the photosphere (see Chapter \\ref{ref:Chapter5} and \\cite{KontarJeffrey2010}). For this example, the variation in\npolarization across the source can be described analytically by \\citep{McMaster1961} if no energy losses are assumed,\n\\begin{equation}\nDOP=\\frac{1-\\cos^{2}\\theta_{S}}{1+\\cos^{2}\\theta_{S}},\n\\label{eq:DOP_1scat}\n\\end{equation}\nwhere\n\\begin{equation}\n\\cos\\theta_{S}=\\cos\\theta\\cos(\\pi-\\theta_{i})+\\sin\\theta\\sin(\\pi-\\theta_{i})\\cos\\phi.\n\\label{eq:cosTheta_S}\n\\end{equation}\n$\\theta_{S}$ is the scattering angle, $\\theta_{i}$ is the emission angle measured from the local\nsolar vertical, $\\theta$ is the heliocentric angle on the solar disk and $\\phi$ is the azimuthal angle\nmeasured in the solar disk plane. In Equation (\\ref{eq:DOP_1scat}), the scattering angle\n$\\theta_{S}$ determines the $DOP$. $\\theta_{S}$ is related to distance $r$ from the centre of the\nalbedo patch by Equation (\\ref{eq:cosTheta_S}) though $r=h\\tan(\\pi-\\theta_{i})$, and hence the $DOP$ at any point across the albedo patch located at any heliocentric angle on the solar disk can be\neasily calculated. Note that for this simple example, Equation (\\ref{eq:DOP_1scat}) assumes that\nthe energy difference between the incoming and scattered photon is negligible,\nthough this is only the case for low HXR energies of $\\sim10$ keV. The $DOP$ for higher energies must be calculated using the $T$ scattering matrix, which is used in the simulations (Equation (\\ref{eq:T_matrix}) and discussed in Section \\ref{ref:uppps}) \\citep{McMaster1961} .\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=14cm]{newpolfig.pdf}\n\\caption[Diagram of a single Compton scattering in the photosphere\nfor three heliocentric angles of $0^{\\circ}$, $45^{\\circ}$ and $90^{\\circ}$.]{Diagram of a single Compton scattering in the photosphere\nfor three heliocentric angles of $0^{\\circ}$, $45^{\\circ}$ and $90^{\\circ}$. For a single scatter,\nthe DOP tends to $100\\%$ as the scattering angle approaches $90^{\\circ}$, producing a variation in\npolarization across the extent of the source.}\n\\label{fig:cartoon1s}\n\\end{figure}\n\nFigure \\ref{fig:cartoon1s} shows a cartoon of this single scattering from an isotropic point source\nfor three different heliocentric angles of $0^{\\circ}$, $45^{\\circ}$ and $90^{\\circ}$.  For a\nsource located exactly above the solar centre ($0^{\\circ}$) at a height $h$, the resulting variation across the\nphotospheric albedo patch is radially symmetric. As the radial distance $r$ from the source centre\nincreases, the scattering angle $\\theta_{S}$ of any observed radiation will decrease from $180^{\\circ}$ towards $90^{\\circ}$,\ncausing the $DOP$ to grow from $0\\%$ to $\\sim 100\\%$.\nRadiation scattered in the photosphere at a location directly below the HXR source, which is a $180^{\\circ}$ backscatter,  and emitted\ntowards the observer will experience no change in its $DOP$. This\nstatement is true for HXR sources at any heliocentric angle $\\theta_{i}$ but the\nprojection effects at angles $\\theta_{i}>0^{\\circ}$ will create an asymmetry in the polarization pattern along the radial direction,\nwhereas the polarization pattern in the perpendicular to radial direction always remains\nsymmetrical. The described pattern can be seen in Figure \\ref{fig:mapsss} which shows the\npolarization maps for a single Compton scattering at four disk locations\nranging from the solar centre to the limb, along $Y=0''$ at $X=214'' , 543'' , 750'', 936''$.\nThese locations are equivalent to $\\mu=0.97,0.82,0.62,0.22$ ($\\theta_{i}=14^{\\circ},35^{\\circ},52^{\\circ},77^{\\circ}$) and denote the approximate positions of the total observed HXR source,\nwhich are shifted from the primary HXR position due to the albedo component \\citep{KontarJeffrey2010}, which was discussed in Chapter \\ref{ref:Chapter5}.\nThe polarization across the HXR source at any location on the solar disk can always be measured with respect to the radial line connecting the solar\ndisk centre and the centre of the source. Therefore due to the symmetry of the problem, source\nlocations at $Y=0''$ considered in this thesis can straightforwardly be applied to any solar disk\nlocation.  In Figure \\ref{fig:mapsss}, the dotted blue ellipse denotes the FWHM of the diffuse albedo\ncomponent, the solid green ellipse denotes the FWHM of the total observed source and the orange,\nblue and green asterisks indicate the centroid positions of primary, albedo and total observed\nsources respectively. The polarization angle $\\Psi$ follows a steady pattern across all\nsources. $\\Psi$ is always at an angle tangential to the line connecting the desired position and\nthe location of the source centroid position. Hence at disk centre locations the $\\Psi$ pattern is also symmetrical\nacross the source.\n\nFigure \\ref{fig:mapsms} shows the same albedo\npolarization maps as in Figure \\ref{fig:mapsss} but for multiple Compton scatterings.\nThe overall pattern for the $DOP$ and $\\Psi$ are preserved but due to multiple scatterings the overall $DOP$ across\nall locations over the albedo patch has decreased. A single scattering $DOP$ of $\\sim100\\%$ near the\nedge of the source has been reduced to $\\sim50\\%$ by multiple scatterings. All other simulations\nshown below use multiple Compton scattering in the photosphere.\n\nThe albedo pattern from a primary source at a greater\nheight than 1 Mm (say from a coronal source) should produce the same albedo polarization\npattern but over a much greater area in the photosphere. The albedo patch for such a source should therefore be very large with a very low intensity and hence should not alter the properties of coronal sources to the same extent as chromospheric sources. The polarization pattern is always plotted at the peak albedo energies 20-50 keV since this is where the greatest change in source polarization should occur.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17cm]{mapsss.pdf}\n\\caption[Albedo polarization maps for an isotropic, unpolarised point source sitting above the\nphotosphere at four different locations after a single Compton scatter in the photosphere.]\n{Albedo polarization maps for an isotropic, unpolarised point source sitting above the\nphotosphere\nat four different radial locations of $X=214'', 543'', 750'', 936''$ (left to right) at $Y=0''$ (corresponding\n$\\mu=0.97,0.82,0.62,0.22$) after a single Compton scatter in the photosphere. All results are shown\nat the peak albedo energies of 20-50 keV. The length of each red arrow indicates the $DOP$ and the\ndirection of each arrow depicts the polarization angle $\\Psi$ within the chosen plotting bin. The\nsolar radial direction (or $X$ axis for this case) is defined as the $\\Psi = 0^{\\circ}$ position. An\narrow length of $2''$ corresponds to a maximum $DOP$ of $100 \\%$. The green and blue ellipses give\nthe FWHM of the total and albedo sources respectively, while the green, blue and orange asterisks\ngive the centroid position of the total, albedo and primary sources.}\n\\label{fig:mapsss}\n\\centering\n\\includegraphics[width=17cm]{mapsms.pdf}\n\\caption[Albedo polarization maps as in Figure \\ref{fig:mapsss}, but for the case of multiple\nCompton scatterings\nin the photosphere.]{Albedo polarization maps as in Figure \\ref{fig:mapsss}, but for the case of multiple\nCompton scatterings\nin the photosphere. Multiple scatterings have acted to decrease the $DOP$ at all points across each\nsource but the polarization angle at each point on the map remains the same.}\n\\label{fig:mapsms}\n\\end{figure*}\n\n\\subsection{Anisotropic source at a height of h=1~Mm (1''.4) and size of 5''}\n\nFor a chosen chromospheric HXR source height of $h =1$ Mm $(1\\mbox{\\ensuremath{^{\\prime\\prime}}}.4)$ and a primary source size of\nFWHM$\\sim5''$, simulations were performed for all three photon distributions created by the\n$\\Delta\\nu= 4.0$, $\\Delta\\nu= 0.5$ and $\\Delta\\nu= 0.1$ electron distributions. This is the same height that was used in Chapter \\ref{ref:Chapter5} and again was chosen to match chromospheric HXR source height measurements\n\\citep{2008A&A...489L..57K,Pratoetal2009,SaintHilaireetal2010,MrozekKowalczuk2010,2010ApJ...717..250K,BattagliaKontar2011}.\nAgain, all the results shown here are for the energy range of 20-50 keV, where albedo emission peaks,\nproducing the largest distortion to the primary component but the best opportunity for the\ndetection of the albedo component.\n\nFigures \\ref{fig:maps_dnu4}, \\ref{fig:maps_dnu05} and \\ref{fig:maps_dnu01} each plot the resulting polarisation maps for four HXR sources\n(resulting from the primary and albedo components) created by the\n$\\Delta\\nu= 4.0$, $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$ electron distributions respectively. As with Figures \\ref{fig:mapsss} and \\ref{fig:mapsms},\neach figure plots four HXR sources positioned at $\\mu\\sim0.97,0.82,0.62,0.22$.\nIn Figures \\ref{fig:maps_dnu4}, \\ref{fig:maps_dnu05} and \\ref{fig:maps_dnu01}, the dotted ellipses denote the FWHM of the total source (green),\nthe primary source (orange) and the albedo source (blue) and\nthe correspondingly coloured asterisks denote the $(x,y)$ centroid position of the total source and\nthe primary and albedo components respectively. Figures \\ref{fig:slicesX_dnu4}, \\ref{fig:slicesX_dnu05} and \\ref{fig:slicesX_dnu01} plot intensity,\n$I$ (top row), $DOP$ (middle row) and $\\Psi$ (bottom row) along the radial\ndirection $X$ centred at $Y= 0''$ (across a of bin width$= 2''$) for each of the maps in Figures \\ref{fig:maps_dnu4}, \\ref{fig:maps_dnu05} and \\ref{fig:maps_dnu01}. Figures\n\\ref{fig:slicesY_dnu4},  \\ref{fig:slicesY_dnu05} and \\ref{fig:slicesY_dnu01} plot the $DOP$ (top row) and $\\Psi$ (bottom row) along the perpendicular to radial direction $Y$\ncentred at X$=213.6'' ,543.9'' ,750.1'' ,936.6''$ (again across a bin width$=2''$ ) for each of the maps in Figures \\ref{fig:maps_dnu4}, \\ref{fig:maps_dnu05} and \\ref{fig:maps_dnu01}.\n\n\\subsubsection{Quasi-isotropic distribution of electrons with $\\Delta\\nu=4.0$}\n\nThe HXR photon distribution produced by the $\\Delta\\nu=4.0$ electron distribution is approximately\nunpolarised and isotropic. Therefore, both the spatially integrated and spatially resolved polarization measurements across the primary source at all locations on the solar disk produce\n$DOP\\sim 0\\%$ and $\\Psi=0^{\\circ}$ (radial) at $20-50$ keV.\nThe albedo component produces asymmetrical $DOP$ and $\\Psi$ variations along the source radial direction (Figure \\ref{fig:slicesX_dnu4})\nwhile along the source perpendicular to radial direction (Figure \\ref{fig:slicesY_dnu4}),\nvariations in albedo $DOP$ and $\\Psi$ are approximately symmetrical, since the centroid positions of the primary and albedo components always coincide in the perpendicular to radial direction (see Chapter \\ref{ref:Chapter5}).\nThe simulated HXR sources plotted in Figure \\ref{fig:maps_dnu4} have a finite source size of $\\sim5''$.\nCompared with a point source (Figure \\ref{fig:mapsms}), this produces two main differences:\ni) photons leave the source from different positions above the photosphere and\nii) for certain distributions and disk locations, the primary polarization will dominate over the extent of the primary source.\nThe first property reduces the $DOP$ at all points across the source, compared with the albedo patch created by a point source. Due to the second property, the polarization variation caused by albedo may be slightly masked by the primary component within the source FWHM, especially for cases where the primary component is dominant, that is, isotropic or near isotropic distributions.\n\n\\begin{figure}[h]\n\\vspace{-30pt}\n\\includegraphics[width=17cm]{maps4.pdf}\n\\caption[Total X-ray brightness and polarization maps for $\\Delta\\nu=4.0$ electron distribution.]{Total X-ray brightness and polarization maps for $\\Delta\\nu=4.0$. The total sources sit at 4 disk locations of $X=213.6'', 542.9'', 750.1'', 936.6''$ at $Y=0''$(corresponding to $\\mu\\sim0.97,0.82,0.62,0.22$). Green, blue and orange ellipses\/dots give the FWHM and centroid positions of the total, albedo and primary sources.}\n\\label{fig:maps_dnu4}\n\\includegraphics[width=17cm]{slicex4.pdf}\n\\caption[$I$, $DOP$ and $\\Psi$ radial slices along $X$ at $Y=0''$ for the sources in Figure \\ref{fig:maps_dnu4} for the\n$\\Delta\\nu=4.0$ electron distribution.]{$I$ and $DOP$ radial slices along $X$ at $Y=0''$ for the sources in Figure \\ref{fig:maps_dnu4}\n($\\Delta\\nu=4.0$). Colours as in Figure~\\ref{fig:maps_dnu4} and dash-dot lines denote the centroid positions and the FWHMs of the total observed source, primary and albedo sources.}\n\\label{fig:slicesX_dnu4}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{slicey4labels.pdf}\n\\caption[Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu4} for the $\\Delta\\nu=4.0$ electron distribution.]{Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu4} for the $\\Delta\\nu=4.0$ distribution.\nEach of the perpendicular to radial slices are taken along $Y$  at $X=213.6'', 542.9'', 750.1'', 936.6''$ for the $DOP$ and polarization angle $\\Psi$. The lines and\ncolours are as in Figures \\ref{fig:maps_dnu4}, \\ref{fig:slicesX_dnu4}. The $DOP$ and the magnitude\nof the polarization angle $\\Psi$ remain symmetrical along the source perpendicular to radial direction.}\n\\label{fig:slicesY_dnu4}\n\\end{figure*}\nFor the quasi-isotropic $\\Delta\\nu=4.0$ distribution, the primary component is the dominant component at\nall four disk locations, with the albedo contribution falling as the source location moves towards\nthe limb (Figure \\ref{fig:slicesX_dnu4}- first row). Hence, the primary component dominates within the\nFWHM of the total source, while the albedo component dominates after this boundary.\nThe second and third rows of Figure \\ref{fig:slicesX_dnu4} demonstrate common radial trends in $DOP$\nand $\\Psi$, not only across the extent of each individual source but also between sources at different disk locations.\nFor a quasi-isotropic distribution, at a particular disk location (other than the disk centre), the highest $DOP$ along the radial direction is observed at the disk-centre-side of the source (where the albedo dominates). This falls to approximately zero within the FWHM of the total source (where the primary dominates) and then increases again towards the limb-side of the source (where the albedo again dominates), but always remains lower than the $DOP$ at the disk-centre-side. Comparing the four disk locations, the $DOP$\nat all points along the radial direction of a single source decreases as the source location nears the limb.\nIn the radial direction, spatially resolved $DOP$ can achieve values as high as $\\sim30\\%$ at disk centre locations.\nThe albedo component produces the distinctive $\\Psi$ variation shown in Figure \\ref{fig:maps_dnu4}.\nAlong the radial direction, $\\Psi=90^{\\circ}$ at the disk-centre-side of the source, falls to zero within the FWHM extent and then rises again at the limb-side of the source.\nAlong the perpendicular to radial direction of a single source (Figure \\ref{fig:slicesY_dnu4}), $DOP$ (first row) and the magnitude of $\\Psi$ (second row) are symmetrical due to no projection effects and the centres of the primary and albedo components always coinciding. As with the radial direction, in the perpendicular to radial direction, spatially resolved $DOP$ can achieve values as high as $\\sim30\\%$.\nWhile the magnitude of $\\Psi$ across the\nsource is symmetrical at each disk location in the perpendicular to radial direction,\n$\\Psi$ itself behaves as an odd function along Y, with a $180^{\\circ}$ rotational symmetry about the source centre, increasing from the radial at the upper source edge to $\\left|{\\Psi>0^{\\circ}}\\right|$ and then back to the radial direction at the lower source edge.\n\n\\subsubsection{Beamed electron distributions $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$}\n\nAll plots for the photon distribution created by the $\\Delta\\nu=0.5$ electron distribution are shown in Figures (\\ref{fig:maps_dnu05}-\\ref{fig:slicesY_dnu05}), while Figures (\\ref{fig:maps_dnu01}-\\ref{fig:slicesY_dnu01}) show all plots for the photon distribution created by the $\\Delta\\nu=0.1$ electron distribution. Comparison of Figures \\ref{fig:maps_dnu4}, \\ref{fig:maps_dnu05} and \\ref{fig:maps_dnu01} demonstrate that increased beaming towards the photosphere produces smaller, more concentrated and intense albedo patches\n\nFor the $\\Delta\\nu=0.5$ distribution, Figure \\ref{fig:slicesX_dnu05} plots the radial intensity, $I$ (top) (along $X$ at $Y=0''$). The first two disk locations are albedo dominated, the third disk location has approximately equal contributions from the primary and albedo components and only the disk location closest to the limb is primary dominated. The primary $DOP$ can rise as high as $\\sim20\\%$ at the limb and the primary $\\Psi$ is radial at all locations.\nFigure \\ref{fig:slicesX_dnu01} (top) plots radial intensity slices (along $X$ at $Y=0''$) for the $\\Delta\\nu=0.1$ distribution. As expected, the first three disk locations are albedo dominated and the primary $DOP$ can reach $\\sim40\\%$ at the limb. Again, the primary $\\Psi$ is radial at all disk locations.\n\n\\begin{figure}[h]\n\\vspace{-30pt}\n\\includegraphics[width=17cm]{maps05.pdf}\n\\caption[Total X-ray brightness and polarization maps for the photon distribution created by the $\\Delta\\nu=0.5$ electron distribution.]{Total X-ray brightness and polarization maps for the photon distribution created by the $\\Delta\\nu=0.5$ electron distribution for a $5''$ primary source. Colours and symbols as in Figure~\\ref{fig:maps_dnu4}.}\n\\label{fig:maps_dnu05}\n\n\\includegraphics[width=17cm]{slicex05.pdf}\n\\caption[Radial slices (along $X$) through $Y=0''$ for the intensity, $I$, the $DOP$ and $\\Psi$\nfor each of the sources in Figure \\ref{fig:maps_dnu05} for the $\\Delta\\nu=0.5$ electron distribution.]{Radial slices (along $X$) through $Y=0''$ for the intensity, $I$, the $DOP$ and $\\Psi$\nfor each of the sources in Figure \\ref{fig:maps_dnu05} for the $\\Delta\\nu=0.5$ distribution. Again, the source locations, colours and symbols are the same as in Figures \\ref{fig:maps_dnu4} and \\ref{fig:slicesX_dnu4} for $\\Delta\\nu=4.0$.\n\\label{fig:slicesX_dnu05}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{slicey05labels.pdf}\n\\caption[Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu05} for the $\\Delta\\nu=0.5$ electron distribution.]{Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu05} for the $\\Delta\\nu=0.5$ distribution.\nEach of the perpendicular to radial slices are taken along $Y$  at $X=213.1'', 542.6'', 750.0'', 936.5''$ for the $DOP$ and polarization angle $\\Psi$. The lines and\ncolours are as in Figures \\ref{fig:maps_dnu05}, \\ref{fig:slicesX_dnu05}.}\n\\label{fig:slicesY_dnu05}\n\\end{figure*}\n\nAs with the quasi-isotropic $\\Delta\\nu=4.0$ distribution, common trends can be observed across individual sources at particular disk locations and between disk locations for both the $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$ distributions. More importantly for observations and anisotropy deduction purposes, trends between each of the three simulated distributions ($\\Delta\\nu=4.0$, $\\Delta\\nu=0.5$ and $\\Delta\\nu=0.1$) can be observed, along the radial and perpendicular to radial directions, at any chosen disk location. The most notable trends are observed in the radial (X) direction, and it is these trends that may help to deduce the anisotropy of the photon distribution for a HXR source sitting at a given disk location. Trends can be observed at all disk locations, but in this example the patterns are most noticeable in the second and third disk locations plotted. In both of these locations, the disk-centre-side $DOP$ falls with increased beaming while the limb-side $DOP$ rises with increased beaming.\n\n\\begin{figure}[h]\n\\vspace{-30pt}\n\\includegraphics[width=17cm]{maps01.pdf}\n\\caption[Total X-ray brightness and polarization maps for the photon distribution created by the $\\Delta\\nu=0.1$ electron distribution.]{Total X-ray brightness and polarization maps for the photon distribution created by the $\\Delta\\nu=0.1$ electron distribution for a $5''$ primary source. The source locations, colours and symbols as in Figure~\\ref{fig:maps_dnu4}.}\n\\label{fig:maps_dnu01}\n\\includegraphics[width=17cm]{slicex01.pdf}\n\\caption[Radial slices (along $X$) through $Y=0''$ for the intensity, $I$, the $DOP$ and $\\Psi$\nfor each of the sources in Figure \\ref{fig:maps_dnu01} for the $\\Delta\\nu=0.1$ electron distribution.]{Radial slices (along $X$) through $Y=0''$ for the intensity, $I$, the $DOP$ and $\\Psi$\nfor each of the sources in Figure \\ref{fig:maps_dnu01} for the $\\Delta\\nu=0.1$ distribution. Again, the locations, colours and symbols are the same as in Figures \\ref{fig:maps_dnu4} and \\ref{fig:slicesX_dnu4} for $\\Delta\\nu=4.0$.}\n\\label{fig:slicesX_dnu01}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{slicey01label.pdf}\n\\caption[Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu01} for the $\\Delta\\nu=0.1$ electron distribution.]{Perpendicular to radial slices through each of the sources shown in Figure\n\\ref{fig:maps_dnu01} for the $\\Delta\\nu=0.1$ distribution.\nEach of the perpendicular to radial slices are taken along $Y$  at $X=213.0'', 542.5'', 749.9'', 936.4''$ for the $DOP$ and polarization angle $\\Psi$. The lines and\ncolours are as in Figures \\ref{fig:maps_dnu01}, \\ref{fig:slicesX_dnu01}.}\n\\label{fig:slicesY_dnu01}\n\\end{figure*}\n\nComparing the third disk location (for example) in Figures \\ref{fig:slicesX_dnu4}, \\ref{fig:slicesX_dnu05} and \\ref{fig:slicesX_dnu01} shows how the radial $DOP$ at the limb-side of the source rises with increased beaming from $\\sim2\\%$ for the $\\Delta\\nu=4.0$ distribution to $\\sim18\\%$ for the $\\Delta\\nu=0.5$ distribution to $\\sim30\\%$ for the $\\Delta\\nu=0.1$ distribution. The radial $DOP$ at the disk-centre-side of the source falls with increased beaming from $\\sim18\\%$ ($\\Delta\\nu=4.0$) to $\\sim14\\%$ ($\\Delta\\nu=0.5$)  to $\\sim4\\%$ ($\\Delta\\nu=0.1$). The polarization angle $\\Psi$ also produces similar patterns with changing anisotropy. A clear example of this can be observed by comparing the second disk location plotted in Figures \\ref{fig:slicesX_dnu4}, \\ref{fig:slicesX_dnu05} and \\ref{fig:slicesX_dnu01}. Along the radial direction, disk-centre-side $\\Psi$ generally stays at $\\Psi=90^{\\circ}$ for all photon anisotropies, while the outer limb-side $\\Psi$ falls significantly with increased beaming, from $\\Psi=60^{\\circ}$ ($\\Delta\\nu=4.0$) to $\\Psi=20^{\\circ}$ ($\\Delta\\nu=0.5$) to $\\Psi=0^{\\circ}$ ($\\Delta\\nu=0.1$).\n\nTherefore, the $DOP$ and $\\Psi$ patterns prescribe how spatially resolved polarization measurements could be used to determine the beaming of the photon distribution. It should be noted that a (near) disk-centre source produces a slightly different trend in radial DOP with increasing photon anisotropy. The $DOP$ at the disk-centre-side remains approximately the same for all photon anisotropies while the $DOP$ at the limb-side falls with increased beaming (this is the opposite trend to other disk locations).\n\nComparing each disk location along the perpendicular to radial direction (Y) in Figures \\ref{fig:slicesY_dnu4}, \\ref{fig:slicesY_dnu05} and \\ref{fig:slicesY_dnu01} shows that greater beaming increases the $DOP$ over the whole extent of the source at any given location (except at the disk centre where the spatially resolved polarization along Y is approximately the same for all three distributions). This spatial increase is most noticeable at limb locations where from the source\ncentre to the source edge, $DOP$ increases from $\\sim0\\%$ to $\\sim20\\%$ ($\\Delta\\nu=4.0$), from\n$\\sim20\\%$ to $\\sim40\\%$ ($\\Delta\\nu=0.5$) and from $\\sim40\\%$ to $\\sim55\\%$ ($\\Delta\\nu=0.1$).\n\n\\section{Discussion and conclusions}\nThe simulation results show that Compton backscattering produces a clear and distinct albedo polarization pattern across a HXR source at the peak albedo energies of 20-50 keV. Trends can be observed for both of the measured polarization parameters, $DOP$ and $\\Psi$, and are clear indications of the existence of an albedo component in comparison with the constant, radial polarization of the primary emission. This means that spatially resolved polarization can be used to probe structure within HXR footpoint sources, helping to distinguish between the bremsstrahlung source and the albedo source. More importantly, at a single disk location, spatially resolved $DOP$ and $\\Psi$ exhibit clear variations with changing photon directivity, along both the radial and perpendicular to radial directions and can be used to determine the anisotropy of the electron distribution.\nTherefore, to take advantage of these properties requires reliable polarization measurements with an angular resolution of $\\sim5''-10''$ over the peak albedo energies of $\\sim20-50$ keV.\n\nThe simulations also suggest that for approximately isotropic HXR sources, spatially integrated polarization angle measurements, $\\Psi$, from low to high energies,\nwith consideration of the changes due to albedo, could help indirectly infer the highest energy in\nthe electron distribution. For near isotropic sources implied by {\\it RHESSI} X-ray observations\n\\citep{Kasparovaetal2007,KontarBrown2006}, changes in $\\Psi$ from the radial ($\\Psi=0^{\\circ}$) to the perpendicular to radial direction ($\\Psi=90^{\\circ}$) may help indicate the presence of high energy electrons ($\\geq1$ MeV) present in the electron distribution. Changes in spatially integrated $DOP$ measurements, from low to high energies, will also help determine the anisotropy of the photon distribution.\n\nCurrently, when observing solar flares, the instrumentation required to produce spatially resolved polarization measurements is not available. It is doubtful whether proposed {\\it near} future missions such as the Gamma-Ray Imager\/Polarimeter for Solar Flares ({\\it GRIPS}) \\citep{Shihetal2009} may have the capability to measure polarization over albedo energies, even though it should be able to measure polarization across $12\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$. The best polarization measurements are likely to be over the range of 150-650 keV with a $\\sim$4$\\%$ minimum detectable polarization \\citep{Shihetal2009}. At energies greater than $\\sim$100 keV, the albedo flux drops off steeply, thus it is unlikely that any albedo component could be detected at these energies and the polarization across the observed HXR source would only be from the bremsstrahlung emission. Therefore, good polarization measurements at 150-650 keV from flares with high fluxes will give a direct measurement of the primary component and $DOP\/\\Psi$ measurements at these energies may be used to infer the high energy cutoff in the electron distribution, for relatively isotropic distributions. However, it should be noted that most flares have a relatively low flux after $\\sim100$ keV since the X-ray flux generally decreases as a power law. Therefore, only the highest X-class flares will probably give enough counts for a reliable polarization measurement. Hence, the importance of understanding the albedo component and how it can be used as a beneficial diagnostic tool is stressed since the most reliable future polarization measurements from flares will probably be over an energy range of 10-50 keV, where the photon flux is high. \n\\chapter{Conclusions and final remarks}\n\\label{ref:Chapter7}\n\nThe aim of this thesis was to study changes in the spatial, spectral and polarization properties of solar flare X-ray sources, in order to determine how these properties are related to the energetic and angular properties of an emitting electron distribution, the properties of a flaring coronal or chromospheric plasma, and X-ray interactions in the photosphere such as Compton scattering. The work in this thesis was performed by modelling and using observations with {\\it RHESSI}.\n\nIn Chapter \\ref{ref:Chapter2}, stochastic simulations were used to model electron transport in a dense corona. Different injected electron pitch angle distributions, undergoing collisional pitch angle scattering within a finite temperature plasma were simulated, and this work can also be found in \\cite{2014ApJ...787...86J}. This study was partly in response to recent {\\it RHESSI} observations of coronal thick target X-ray sources. The thick target model is applicable to the corona for these events because of their high number densities of the order $10^{11}$ cm$^{-3}$. The lengths of such X-ray sources increase quadratically with energy and often a simple one-dimensional cold target collisional transport model is used to estimate the number density and the length of an acceleration region, using a quadratic fit function of the form $FWHM=L_0+\\alpha\\;E^{2}$ where $\\alpha\\propto1\/n$. The simulations showed how the presence of collisional pitch angle scattering alone does not dramatically change the behaviour of source length with electron energy. However, it was shown that the beaming of the injected electron pitch angle distribution did produce a significant change in the variation of X-ray source length with energy. It was found that injecting a beamed electron distribution produced a larger variation of length with energy, since electrons with velocity completely aligned to the guiding magnetic field, move the greatest distance through a plasma. The uncertainty in the initial angular distribution of the injected electrons produces the largest uncertainty in the inferred number density $n$, which can be up to a factor of $\\sim6$ larger. The finite temperature of the target atmosphere leads to thermal diffusion, both in energy and space, and causes an increase of the inferred acceleration region length $L_0$. The results show that the application of a one-dimensional cold target approach to a warm target changes the inferred acceleration length $L_0$ by up to $\\sim10\\mbox{\\ensuremath{^{\\prime\\prime}}}$ for a $30$ MK plasma, and the equation $L_0=L_0(T=0)-0.011 \\, T^{2}$ was found empirically in order to estimate the true length of the acceleration region from observation, for number density values close to $n\\sim1\\times10^{11}$ [cm$^{-3}$]. It was also found that the FWHM versus energy curve consisted of two competing components, one due to thermal diffusion that is dominant at lower energies, and another due to collisional friction that is dominant at higher energies where $E>>k_{B}T$. The dominant component depends on the temperature of the region, on the density, and the spectral index of the electron distribution, and it was shown that the application of a one-dimensional cold target approach to a warm target produced an inferred number density different by a factor of $\\sim3$ (in either direction), again, depending mainly on the initial beaming of the electron distribution.\n\nIt is often assumed that the initial accelerated electron distribution and even the target electron distribution in the corona or chromosphere is completely beamed. However, many recent {\\it RHESSI observations} are consistent with a lack of anisotropy, including \\cite{KontarBrown2006}, \\cite{Kasparovaetal2007} and \\cite{2013SoPh..284..405D}; with all studies using the presence of a photospheric albedo component to determine the electron distribution isotropy via a single spacecraft stereoscopic method or a centre-to-limb statistical method. Hence simulating possibly more realistic isotropic electron distributions, in general produces a more gradual variation of source length with energy, that is, smaller values of the quadratic fit coefficient $\\alpha$. Therefore, depending on the electron distribution spectral index, observed steep behaviours (high values of $\\alpha$) may be indicative of other processes at work within the coronal region. For instance, throughout the simulations it was assumed that the length of the acceleration region length $L_0$ was independent of electron energy $E$. However, depending on the acceleration process, this may not be the case. For example, if $L_0$ grows with energy, this may produce a larger value of $\\alpha$ than expected and hence the analysis of this effect may indicate the properties of the acceleration mechanism itself. As was discussed in Chapter \\ref{ref:Chapter2}, the stochastic simulations are not self-consistent, that is, they do not account for changes in the properties of the background plasma, specifically the temperature increase of the background plasma due to the energy loss of the injected electron distribution. Further improvements could also be made to include spatial variations in temperature and\/or density along the loop, as would be expected in a real flaring region.  Also, it should be noted that a recent study by \\citet{2014ApJ...780..176K} shows how the presence of {\\it non-collisional} pitch angle scattering (for example, involving magnetic field inhomogeneities) results in a different non-quadratic predicted behaviour for the variation of source length with energy. Moreover, the code developed for this work could be rather straightforwardly extended to the study of magnetic diffusion of particles across the guiding field in a warm target \\citep[e.g.,][]{2011A&A...535A..18B} and hence to study the variation of source length with energy in this alternative scenario.\n\nIn Chapter \\ref{ref:Chapter3}, temporal changes of the spatial properties of dense coronal X-ray sources were analysed for the first time, using observations from {\\it RHESSI}. The work can also be found in \\cite{2013ApJ...766...75J}. Similar to the studies of both coronal and chromospheric X-ray source spatial parameters with energy, this study was made possible by the use of the forward-fitting imaging algorithm {\\it Visibility Forward Fitting}. This algorithm can only be used for specific X-ray source shapes where one or two X-ray sources have a simple form that can be well-fitted by either a circular, elliptical or curved elliptical Gaussian shape. Since many X-ray sources have a more complex shape, only three dense coronal X-ray flares were analysed reliably using this method. However, the study found that all three X-ray loops exhibited similar and interesting temporal trends in both their observed spatial properties and inferred physical properties such as temperature, number density and thermal pressure, estimated from a combined spectral and spatial analysis of each X-ray source. Peaks in X-ray emission denoted periods where there were changes in the loop spatial dynamics and most interestingly of all, a study of temperature, volume, number density and thermal pressure showed how the X-ray loops went through three phases denoted by three clear peaks: the first in temperature, the second in X-ray emission and the third in thermal pressure. Before the peak in X-ray emission, the loop length, loop width and radial position decreased. After the X-ray peak the loop width and radial position rose again while the loop length seemed to remain approximately constant - at least for the two flares that only have one clear X-ray peak in their lightcurve. It was found that at the start of the observation time, the X-ray loops are relatively long and thin and then they become smaller, before growing in width at the later decay stages of X-ray emission. Hence in order to describe this changing morphology, a new parameter named {\\it corpulence, $\\mathcal{C}$} was defined; the ratio of loop width $W$ to loop length $L$, and it was found, in general, that corpulence increased with time. Overall, the volume of the loop decreased before the peak in X-ray emission, but the relationship between temperature and volume did not support simple compressive heating, as in a collapsing magnetic trap model. The most difficult observation to explain is that of the decreasing loop widths before the peak in X-ray emission, since electrons should be unable to move across the guiding field lines threading the corona. This leads to the suggestion that the field lines themselves are being squeezed together during this time. In the discussion of Chapter \\ref{ref:Chapter3}, it was tentatively suggested that this could be due to Taylor relaxation, where the release of magnetic energy during a flare may cause the surrounding field lines to shrink or contract in some manner. However it should be stressed that the cause is unknown, and further future investigation of similar coronal X-ray loops is required to help explain this observation. Another possible cause of this contraction may be the sausage-pinch effect, whereby the emitting plasma is pushed together by a generated Lorentz force due to the current flowing in the parallel direction producing an azimuthal magnetic field that compresses the plasma. However, we would expect the parallel field along which electrons are flowing to cancel any sausage-pinch effects. However, the lack of flares with this type of X-ray source morphology, and especially those with {\\it RHESSI} observations that can be analysed with algorithms such as Vis FwdFit, since the launch of {\\it SDO}\/AIA has proved rather frustrating. The observation of the spatial and temporal properties of separate loops within the `blob' region viewed in X-rays may help to better understand the trends in loop width and position. Further, it was suggested that thermal conduction causes chromospheric evaporation, leading to the increasing number density and thermal pressure in the loop. This could cause the loop lengths to decrease, as electrons interact at shorter and shorter distances. Eventually, the increasing thermal pressure in the region could balance the process causing the loop to shrink, and cause the increase in loop width after the peak X-ray emission.\n\nIn Chapter \\ref{ref:Chapter5}, Compton scattering and photoelectric absorption of X-rays in the photosphere were studied via a Monte Carlo simulation of photon transport, in order to study how X-ray albedo photons, backscattered from the photosphere, can alter the spatial properties of a HXR source (the primary source), namely its size and position. The results of simulations can also be found in the publication \\cite{KontarJeffrey2010}. These changes were inspected by simulating HXR chromospheric footpoint sources at different heliocentric angles above the solar disk, at a typical chromospheric footpoint height of $1\\mbox{\\ensuremath{^{\\prime\\prime}}}.4$ and using X-ray energies between 3 and 300 keV, matching recent {\\it RHESSI} observations, as discussed in Chapter \\ref{ref:Chapter5}. The results from all the simulations showed a general trend for both the measured source sizes and the shifts from the true primary source position. The greatest alteration in source size and position occurred between the energies of 20-50 keV, the energy range over which the albedo emission peaks. Generally the largest source increases due to albedo are observed nearer the solar centre since the greatest fraction of albedo X-rays are emitted towards an observer at this heliocentric angle, hence with the smallest source size increases occurring at solar limb. The largest shifts in source position occur at mid-heliocentric angles with the measured position nearing its true primary position both at the solar centre, where the positions of both the primary and albedo sources coincide, and near the solar limb, where the albedo flux is minimal. The resulting albedo component, and hence total observed source from primary HXR sources, with different spectral indices, source sizes and anisotropy were simulated. It was found that the lower the spectral index of the photon distribution, that is, the harder the spectrum, the greater the source size increase and position shift over all disk positions and photon energies, compared with similar HXR sources with larger spectral indices. It is observed that the smaller the true HXR source size, the greater the relative increase due to the albedo X-rays. Therefore, the results show that interestingly even a HXR point source that is isotropic, and hence has the minimum backscattered albedo flux, may be observed as a HXR source as large as $7\\mbox{\\ensuremath{^{\\prime\\prime}}}$ across, when located near the solar centre and viewed at a peak albedo energy range of 20-50 keV. The larger the initial HXR source, the smaller the relative increase due to the albedo X-rays, over all energies and heliocentric angles. It was found that the position shifts seem to be generally independent of primary HXR source size if all other factors such as spectral index and beaming remain the same. The contribution from an albedo component depends greatly on the anisotropy of the HXR source. If the HXR source initially throws a greater proportion of photons into the photosphere than towards e.g.,{\\it RHESSI}, then the albedo component can be very large compared to the true direct component. Large downward directivities from anisotropic sources can produce shifts as large as $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.8$.\n\nThe results of Chapter \\ref{ref:Chapter5} and \\cite{KontarJeffrey2010}, and as discussed in the conclusion of Chapter \\ref{ref:Chapter5}, suggest inevitably that all HXR sources in the chromosphere, and {\\it even} those with an isotropic pitch angle distribution, should have an albedo component capable of altering their spatial properties, and hence the information deduced from such observations. Hence, the results were also suggestive of the initial HXR emission coming from a volume in the chromosphere smaller than suggested by {\\it RHESSI} observations. For example, \\citet*{2011ApJ...739...96K} examined both high resolution optical and hard X-ray observations of a flare that occurred on the 6th December 2006. The G-band observations resolved the width of the flare ribbon to be somewhere between $0\\mbox{\\ensuremath{^{\\prime\\prime}}}.5$ and $1\\mbox{\\ensuremath{^{\\prime\\prime}}}.8$. However, the {\\it RHESSI} hard X-ray observations were unresolved and suggested that the HXR source width was even smaller than that of the G-band observations. Even though this flare is located close to the solar limb, the results shown here suggest that its spatial properties should be tainted by an albedo component that would increase the size of the observed HXR source. The simulations shown in Chapter \\ref{ref:Chapter5} and \\cite{KontarJeffrey2010}, were used to estimate the size of the primary HXR source which should be smaller than that found from observations. However, the size of the primary HXR source could not be easily estimated since the HXR source was unresolved and hence the observed HXR source width of $\\sim1\\mbox{\\ensuremath{^{\\prime\\prime}}}.1$ {\\it with} the presence of an albedo component was an upper limit, and suggestive that the HXR sources were extremely point-like in this flare.\n\nIn Chapter \\ref{ref:Chapter6}, a new study of solar flare X-ray polarization due to the presence of a photospheric albedo component was presented. For the first time in solar physics, this chapter simulated how an albedo component can produce spatial changes in polarization across a single HXR source. The results of Chapter \\ref{ref:Chapter6} are also published in \\cite{2011A&A...536A..93J}. The Monte Carlo simulations used in Chapter \\ref{ref:Chapter5} were adapted to study spatially resolved polarization due to albedo. In order to achieve this, HXR sources created from three different electron directivities were input into the simulations; from the near isotropic to a highly beamed distribution. The first purpose of the study was to examine how the degree {\\it and} direction of polarization change in space, across a single observed HXR footpoint, due to the inevitable presence of a Compton scattered albedo component. The polarization changes in space across an X-ray source were presented as polarization maps and cuts were taken along the radial (line connecting the Sun centre and the centre of the HXR source) and perpendicular to radial lines. Clear spatial changes in both the degree and direction of polarization were found for each case, which importantly are dependent on the beaming of the electron distribution. In general, for most HXR source locations, not located close to the disk centre, it was found for a given heliocentric angle, that the radial $DOP$ at the limb-side of the HXR source rises with increased beaming, while the $DOP$ at the disk-centre-side of the HXR source decreases with increased beaming. Similarly, for a radial measurement of $\\Psi$, the angle of polarization along the source, the limb-side $\\Psi$ decreases with increased beaming while the disk-side $\\Psi$ generally stays at $90^{\\circ}$. For example, for the HXR source simulated at a disk location of $\\mu\\sim0.60$, the radial $DOP$ at the limb-side of the source rises with increased beaming from $\\sim2\\%$ for the isotropic $\\Delta\\nu=4.0$ distribution to $\\sim18\\%$ for the (mildly anisotopic) $\\Delta\\nu=0.5$ distribution to $\\sim30\\%$ for the (very anisotropic) $\\Delta\\nu=0.1$ distribution. The radial $DOP$ at the disk-centre-side of the source falls with increased beaming from $\\sim18\\%$ ($\\Delta\\nu=4.0$) to $\\sim14\\%$ ($\\Delta\\nu=0.5$) to $\\sim4\\%$ ($\\Delta\\nu=0.1$). The polarization angle $\\Psi$ for a HXR source at a disk location of $\\mu\\sim0.8$ shows a similar pattern. Along the radial direction, disk-centre-side $\\Psi$ generally stays at $\\Psi =90^{\\circ}$ for all photon anisotropies, while the outer limb-side $\\Psi$ falls significantly with increased beaming, from $\\Psi=60^{\\circ}$ ($\\Delta\\nu=4.0$) to $\\Psi=20^{\\circ}$ ($\\Delta\\nu=0.5$) to $\\Psi=0^{\\circ}$ ($\\Delta\\nu=0.1$).\n\nThe second purpose of the simulations was to assess the usefulness of measuring spatially resolvable polarization, across a single HXR source, as a possible future diagnostic tool. The simulations found that not only are these results useful, they also require no manipulation, that is, there is no need to separate the primary and albedo components as in Chapter \\ref{ref:Chapter5}; these are now used together as one single measurement to determine the photon, and hence electron anisotropy in the chromosphere. As well as other methods involving the use of albedo, X-ray polarization and specifically as suggested here, {\\it spatially resolvable albedo polarization} measurements could provide another method of reliably determining the anisotropy of the electron distribution from an {\\it individual flare}. Spatially resolvable polarization measurements are not only useful for finding changes across a single HXR source but, more realistically near-future polarization missions might be able to determine the total source polarization of each HXR footpoint in the chromosphere and that of an X-ray coronal source individually. This could provide a method of mapping the electron pitch angle distribution during the flare from the corona to the chromosphere, with the contribution of photospheric albedo providing additional information about individual changes across a single HXR source. Even a polarization mission with a lower angular resolution than that required to see changes across individual HXR sources from $10-100$ keV as discussed in Chapter \\ref{ref:Chapter6} could provide essential electron pitch angle information from the corona and chromosphere separately.\n\nMuch of the work within this thesis was based on the excellent imaging spectroscopy methods currently available with {\\it RHESSI}. The work in this thesis also looks towards future X-ray missions; instruments with higher angular resolution, possibly with direct X-ray imaging capabilities that can more reliably examine the changing sub-arc second lengths, widths and positions of X-ray sources. It is hoped that the work in this thesis can also encourage the development of future X-ray polarisation instruments that can eventually measure spatially resolvable polarization across a single HXR source. \n\n\n\n\\chapter*{Preface}\n\n\\addcontentsline{toc}{chapter}{Preface}\n\nChapter \\ref{ref:Chapter1} provides a brief introduction to the topics and theory required for the following chapters: the interactions of electrons and ions in a plasma, the emission mechanisms required to create solar flare X-rays, the interactions of solar flare X-rays in the photosphere (the albedo effect) and our current understanding of solar flare X-ray observations, using instruments such as Ramaty High Energy Solar Spectroscopic Imager ({\\it RHESSI\\,}).\n\nChapters \\ref{ref:Chapter2} and \\ref{ref:Chapter3} examine an interesting flare type with strong coronal X-ray emission from a dense loop, with little or no emission from the chromosphere. Observations of these events with instruments such as {\\it RHESSI\\,} have enabled the detailed study of their structure, revealing that amongst other interesting trends, the spatial parameter parallel to the guiding field increases with X-ray energy. This variation has been discussed in the context of a beam of non-thermal electrons in a one-dimensional cold target model, and the results used to constrain both the physical extent of, and density within, an electron acceleration region believed to be situated within the coronal loop itself. In Chapter \\ref{ref:Chapter2}, the investigation is extended to a physically realistic model of electron transport that takes into account the finite temperature of the ambient plasma, the initial pitch angle distribution of the accelerated electrons, and the effects of collisional pitch angle scattering. The implications of the results when determining parameters such as number density and acceleration region length from observation are discussed. In Chapter \\ref{ref:Chapter3}, the observational analysis of such flare types is further advanced, and the spatial and spectral properties of three dense coronal X-ray loops are studied temporally before, during, and after the peak X-ray emission. Using observations from {\\it RHESSI\\,}, the temporal changes in emitting X-ray length, width, volume, position, number density and thermal pressure are deduced. Collectively, the observations also show for the first time three temporal phases given by peaks in temperature, X-ray emission, and thermal pressure, with the minimum volume coinciding with the X-ray peak. The possible explanations for the observed trends are discussed.\n\nChapters \\ref{ref:Chapter5} and \\ref{ref:Chapter6} examine solar flare X-ray albedo, an effect produced by the Compton backscattering of solar flare produced X-rays in the photosphere. This is studied via Monte Carlo simulations of X-rays in the photosphere. Chapter \\ref{ref:Chapter5} investigates quantitatively for the first time the resulting positions and sizes of solar flare hard X-ray chromospheric sources due to the presence of an albedo component, for various chromospheric X-ray source sizes, spectral indices and directivities. It is shown how the albedo effect can alter the true source positions and substantially increase the measured source sizes; this is greater for flatter primary X-ray spectra, stronger downward anisotropy, and for sources closer to the solar disk centre, between the peak albedo energies of 20 and 50 keV. Chapter \\ref{ref:Chapter5} demonstrates how the albedo component should be taken into account when X-ray footpoint positions, footpoint motions and source sizes are observed and analysed by instruments such as {\\it RHESSI\\,}. In Chapter \\ref{ref:Chapter6}, this study is extended to investigate the polarization of solar flare chromospheric X-ray sources, by investigating how the presence of an X-ray albedo component produces a variation in the spatial distribution of polarization across a single X-ray source. From this, polarization maps for each of the modelled electron distributions are calculated at various heliocentric angles from the solar centre to the solar limb. The investigation shows how Compton scattering produces a distinct polarization variation across the albedo patch at peak albedo energies of 20-50 keV. It discusses how spatially resolved hard X-ray polarization measurements from future X-ray polarimeters could provide important information about the directivity and energetics of the radiating electron distribution, using both the degree and direction of polarization.\n\nChapter \\ref{ref:Chapter7} provides conclusions, discussion and some final remarks regarding the thesis as a whole, in the context of current solar flare understanding and possible future missions. Unless indicated, $CGS$ units are used throughout the thesis. \n\n\\thispagestyle{empty}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:1motiv}\n\nProviding IT services in an inter-organizational manner is a complex and often error-prone task. Managing IT services is often characterized by applying the classic FCAPS partitioning: fault, configuration, accounting, performance, and security management. In this article, we focus on the technical functionality as well as the organizational aspects of fault management in the context of inter-organizationally operated IT services. \nOur work is primarily motivated by the interaction of the following three challenges:\n\n\\paragraph*{\\textit{The \\textbf{Outsourcing} problem}}: \nOne characteristic of the last decade is that many organizations have outsourced their IT services to external parties, either entirely (e.g. email, file storage, and web servers) or just partially. Consequently, many processes and workflows have been transferred to and restructured by these external service providers. Outsourcing is performed in order to reduce the organization's IT costs, but also to facilitate good technical support. Related to these goals, ITIL v3 (see \\cite{itil07d}) describes the migration from the \\textit{value-chain-model} -- also known as hierarchical service delivery model -- to the \\textit{value-network-model}, which contains horizontal (non-hierarchical) relationships between the involved providers. Within this scope, different sourcing strategies are defined.\n\n\\paragraph*{\\textit{The problem of \\textbf{heterogeneity and autonomy} in multi-domain environments}}:\nFrom an organizational point of view, IT service providers collaborate with each other in very diverse ways. This makes it difficult to specify a single, universal methodology for effective and efficient inter-organizational fault management (\\emph{ioFM})\nThe common denominator of the organizational models found in practice is the \\textit{heterogeneity} and \\textit{autonomy}, especially concerning the deployed IT systems and management tools. We therefore have to face the challenge of specifying fault management concepts that are to be deployed in cross-organizational or multi-domain environments and deal with these characteristics.\n\n\\paragraph*{\\textit{The \\textbf{service delivery diversity}}}:\nRegarding the service delivery process as a productive process, a big difference between the real organizations concerning  process control, communication, and many other IT service management (ITSM) aspects can be observed. Therefore very useful reference processes exist.\nReference processes for fault management have been described, for example, in \\cite{itil07d} and \\cite{etom} for hierarchical service delivery, and in \\cite{Hamm09} for heterarchical (i.e. non-hierarchical) service delivery. Based on these reference processes, other related work, and real-world scenarios, we have extracted the requirements for an ioFM architecture as presented in this paper. \n\n\n\\begin{figure}[] \n\t\\centerline{\\includegraphics[width=0.75\\columnwidth]{motivComplex.jpg}}\n\t\\caption{Propagation of faults in inter-organizational environments}\n\t\\label{fig:complex} \n\\end{figure}\n\nThe above stated problems are those characteristics of inter-organizational IT environments that are most relevant for ioFM; a simple example is given in Figure \\ref{fig:complex}: A provider delivers its services to customers A, B, and C in different ways: It outsourced three of the services (labeled Service 1, 2, and 3 respectively) to other service providers. Up to this point we deal with a vertical service chain, which represents the classical type of \\textit{hierarchical service delivery}. Both services 1 and 2 are delivered by only one service provider (providers 1 and 2 are the subcontractors of the service provider). Opposed to these two services, Service 3 is provided by multiple cooperating service providers (Provider 3, 4, and 5). Each of these three providers is required to deliver its part of the service, but none of them has a superior role; instead, they are on a par with each other: These service providers coexist on the same ,,service layer'' regarding the service functionality. They deliver ,,service parts'' (as discussed in \\cite{hmy08}) (Service Part 1, 2, and 3 respectively) which together lead to the delivery of a single horizontal service. These service parts are concatenated within the same service layer, so the horizontal service chain represents a \\textit{heterarchical service delivery}. \n\n\nIt is usual that each real world organization aligns itself on its own requirements, workflows, and processes. It also uses different IT infrastructures, systems, and tools. As a consequence, each organization we deal with needs to be analyzed first, and typically there is a lack of tool interoperability whenever multiple service providers are about to be coupled in order to jointly provide an IT service. In this context, management tool support is of utmost importance, because the complexity of the IT infrastructure as well as of each service increases with the number of involved providers.\n\nTaking into account the above stated challenges, the scenario described here is clearly a heterogeneous one. Following issue is important here: \nA fault, e.g., within the Provider 4's domain, will -- independent of its root cause -- make the whole Service 3 fail because of this issue within Service Part 2. This fault will be propagated to the Service Provider, and thus the customers will face a quality-degraded or unavailable service. This fault can have more or less follow-ups depending on the service customization for each individual customer. Nevertheless, in such inter-organizational scenarios it is very difficult to precisely locate such a fault, to correlate it with other unsolved faults, and to track and steer the progress of the handling and correction.\n\nFor a single IT service provider's infrastructure already several approaches and best-practices concerning fault management exist. But regarding ioFM there is a lack of both research and best practices.\nOur work faces the additional practical challenge that IT service providers from different countries are involved, which in turn increases both the technical complexity as well as the organizational and legal constraints, resulting in even more complex delivery processes. \n\nRegarding outsourcing as well as multi-domain IT service delivery from a process-oriented point of view, a well defined and proper ioFM is needed on the \\emph{system layer}. In order to meet this demand, our work focuses on an ioFM Architecture (ioFMA). This article presents our methodology and the results of our ioFMA requirements analysis. It is structured as follows: In Section~\\ref{sec:2relwork} we sum up the related work that has influenced our methodology and ioFMA design. In Section~\\ref{sec:3design}, we present details about our design rationale and the MDA-based approach that has been taken. Section~\\ref{sec:4scen} outlines the inter-organizational scenarios we have analyzed. Section~\\ref{sec:4req} specifies the roles and actors relevant to ioFM on which the organizational model bases and on this basis we then present the identified use cases and the derived requirements. In Section~\\ref{sec:6funcModel} we are giving an overview on the functional model of the ioFMA. A summary and an outlook to our future work concludes this paper in Section~\\ref{sec:5further}. \n\n\n\n\\section{Related Work}\\label{sec:2relwork}\n\n\\subsection{Management architectures and their submodels}\n\nIn Hegering et al. \\cite{han99} the building blocks of \\textit{management architectures (MA)} are described. The primary goal of each management architecture is to establish an \\emph{integrated} management approach by providing a valid system management framework instead of using several management tools independently of each other. The MA is composed of four complementary submodels: the \\textit{information model (IM)}, the \\textit{organizational model (OM)}, the \\textit{communication model (CM)}, and the \\textit{functional model (FM)}. The \\textit{IM} represents the description and modeling of the managed objects (management-relevant information to be exchanged). The \\textit{OM} describes the roles as well as the responsibilities and specifies the communication patterns within the \\textit{MA}. The \\textit{CM} specifies the communication procedures for the exchange of management information. The FM splits the management task into several components and provides dedicated management functionalities: \\textit{fault management, configuration management, accounting management, performance management, and security management (also known as FCAPS)}.\n\nThe MA concept along with its submodels is very valuable for this work, because it the base for holistic integrated network management. Thus our work will be aligned to the four submodels of such a MA. They have to be extended to take inter-organizational conditions into account, which have not been considered by previous MA variations yet. Also the functional area of fault management (FM) will be taken into account and refined to additional ioFM functionalities that are tailored for inter-organizational environments.  \n\n\\subsection{IT Service Management}\n\nITSM frameworks, such as ITIL v3 \\cite{itil07d}, ISO\/IEC 20000 \\cite{iso20000}, and eTOM \\cite{etom} have been established to design management processes that follow the continual improvement strategy of Deming's plan-do-check-act life cycle. These ITSM frameworks have been used primarily for process definition in hierarchical service delivery scenarios. For non-hierarchical service delivery, a new concept has been developed in \\cite{Hamm09}. \n\nThese approaches give guidelines for the inter-organizational service delivery \\emph{processes} as a whole. Nevertheless, on the (technical) \\emph{system layer} there is no underlying concept for inter-organizational service delivery defined yet. Our work focuses on refining the given reference processes and designing an integrated system-level MA.\n\n\\subsection{Service Composition}\n\nAs we take into account services delivered in an inter-organizational environment, the concept of service composition is a key enabler for our research. In \\cite{Dreo02}, Dreo distinguishes between two types of supply chains: vertical and horizontal. By vertical the well known hierarchical service delivery is meant. The horizontal supply chain addresses the issue of peering. \nDespite the partially overlapping scope between these results and our work, the non-hierarchical service delivery taken in account by our research does not only cover peering. The underlying necessity has also been postulated by Hedlund \\cite{Hed05}, whose work uses the term \\emph{heterarchy} for the non-hierarchical organizational forms, which we also address.\n\nIn their work \\cite{vpmb07} on service composition applied to network management, Vianna et al. show that service composition can indeed be realized by using traditional management technologies. The application of technologies created to support service composition will bring important advantages to the network management discipline. However, they consider only services based on a hierarchical chain of compositions. \n\nKlie et. al analyze the automatic web service composition as a possibility to further automate network management in \\cite{kgf07}. They compare several web service composition technologies in order to describe an approach using a composition engine for network management. This automatic web service composition can be used to simplify complex network management tasks. It also enables the automatic composition for covering large parts of several network management tasks; this approach is valuable as a guideline for the implementation of the ioFMA.\n\n\\subsection{Fault Management related Tasks}\n\nIn \\cite{mdkp09} a framework for problem determination is proposed.  It is based on the monitoring of event streams that are generated by the different components of an IT service. A generic representation of a problem through spatial-temporal patterns is given. Additionally, efficient algorithms are described in order to sustain building blocks for a hierarchical heuristic for detecting generic patterns. Even though some of these concepts are distantly related to our approach, their work is merely based on hierarchical service structures. Also in \\cite{GPM08a} the automation of the incident management is proposed.\n\nIn our former work \\cite{MSGL09} we specified a methodology for handling faults in non-hierarchical service delivery environments, which we called Service Provider Coalitions. This approach's goal was the correlation of fault reports generated by different incident ticketing systems in multi-enterprise environments. We now propose to realize the fault management on a higher level of abstraction.\n\n\n\n\n\\section{Design rationale}\\label{sec:3design}\n\nThis section describes the methodology used in designing the architecture, several of the taken design decisions, and the consequences for the ioFMA. \n\n\\subsection{Model Driven Architecture}\\label{subsec:mda}\nOur design of the management architecture follows the \\textit{Model Driven Architecture (MDA)} \\cite{OMG03} approach. Its iteratively refining character is outlined in Figure \\ref{fig:mdaallg}.\nMDA contains three models: \n\n\\begin{enumerate}\n\n\\item The \\textit{computation independent model (CIM)} provides a general view on the system, as well as on the environment in which this system will be deployed.\n\n\\item The \\textit{platform independent model (PIM)} provides a view on the system independently of the platform that it will be deployed on. Consequently, this model is still generic and can be applied to several platforms of similar type.\n\n\\item The \\textit{platform specific model (PSM)} takes the specification from the PIM and describes its application to a specific platform.\n\n\\end{enumerate}\n\nAs a result, the three models build upon each other and descend from a higher level of abstraction (CIM) to a lower one (PSM).\nThe design of our ioFMA is done in analogy to MDA. In our design process, the requirements elicitation and its model design correspond to MDA's CIM view.  \n\nThe scenarios' description (one hierarchical and one heterarchical scenario) and their generalization are part of the requirement analysis. From the resulting general scenario we derive use cases and several implicit requirements on the ioFMA.\n\nA three-tier procedure for the model design is used:\n\n\\begin{enumerate} \n\n\\item The \\textit{process view} corresponds to CIM and contains reference processes regarding Incident Management (for hierarchy we used \\cite{itil07d} and for heterarchy we used \\cite{Hamm09}).\n\n\\item The \\textit{architecture view} corresponds to PIM and contains the ioFMA as well as its sub models, which correspond to the described processes in the upper layer. \n\n\\item The \\textit{system view}, which is representing the PSM in our approach, contains the implementation of the overlaid architecture on any specific platform on the system layer.\n\n\\end{enumerate}\n\n \\begin{figure}\n\t\\centering\n  \t\\includegraphics[width=0.75\\columnwidth]{mdaallg}\n\t\\caption{Our methodology, which is based on the MDA approach}\n\t\\label{fig:mdaallg} \n \\end{figure}\n\nFurthermore, the design methodology of our ioFMA is split into two parts: the requirements analysis and the model design.\n\n\n\n\\subsection{Methodology of requirements analysis}\n\nIn order to elicit ioFMA requirements, we have analyzed two real world scenarios: The IntegraTUM scenario as an representative example of a hierarchical inter-organizational service delivery, and the G\\'EANT scenario representing the heterarchical service delivery in inter-organizational environments. Based on these practical scenarios, we derived a more abstract generic scenario and its use cases. The textual description of the use cases has been performed with a focus on management architectures (cf. section \\ref{sec:2relwork}) and their sub models. Functional and non-functional requirements have then been derived from these use cases.\n\n\\subsection{Methodology of model design}\n\nBased on the requirements and on the reference process for incident management (cf. section \\ref{sec:2relwork}), the sub models of our ioFMA are specified in the following order: \n\n\\begin{enumerate}\n\n\\item The \\textit{functional model}, which has to underline the most important functionalities concerning fault management, comes first.\n\n\\item The \\textit{organizational model} follows and reveals the roles and responsibilities in inter-organizational environments that are required in order to conduct efficient ioFM.\n\n\\item The \\textit{communication model} then delivers the required information communication exchange measures and procedures.\n\n\\item The \\textit{information model} finally specifies the data format for the ioFM information exchange and processing.\n\n\\end{enumerate}\n\nIn the next step, the \\textit{ioFMA} will be transformed to a PIM; then it will be instantiated for hierarchical, heterarchical, and mixed forms of service delivery. All of them will be mapped onto PSMs. \nIn the next section, we present details about the first step in this methodology, i.e. the requirements analysis.\n\n\\section{Scenarios for inter-organizational fault management}\\label{sec:4scen}\n\nIn order to design and implement an ioFMA, we have chosen the following two scenarios, one for each inter-organizational service delivery model: hierarchy (IntegraTUM) and heterarchy (G\\'EANT).\n\n\\subsection{IntegraTUM}\nIn the IntegraTUM project \\cite{IntegraTUM}, which has been funded by the German Research Foundation (DFG) and initiated by the Technische Universit\\\"at M\\\"unchen (TUM), several university IT services, which were previously operated by the various TUM institutions (e.g. library, administration, and faculties) themselves, have been reorganized and recentralized at the Leibniz Supercomputing Center (LRZ).\n\nTUM's staff and students are automatically granted access to all relevant services, such as the university web portal, learning management system, and computer labs based on an identity management process that is coupled with the student enrolment process and the human resources (HR) management software. Thus, TUM is LRZ's customer and the scenario fulfills the criteria of the hierarchical inter-organizational service delivery model as outlined above. \nA fault management process has been established between the both organizations in this hierarchy and is described in detail in \\cite{hokn10}.\n\n\\subsection{G\\'EANT}\nThe End-to-End (E2E) Link service in the G\\'EANT2 multi-national network \\cite{Geant10} is an example of services delivered by a heterarchical service provider organization.\n \nCo-funded by the European Commission as well as Europe's national research and education networks (NRENs), and managed by DANTE, the G\\'EANT network connects 34 countries via 30 NRENs. On the technical layer, multiple 10Gbps wavelengths are used to set up dedicated E2E links. One representative customer is the Large Hadron Collider (LHC) project at CERN in Switzerland. It is expected that its recently started experiments will produce 15 petabytes of scientific data each year. In order to meet the bandwidth and quality of service requirements of large-scale research projects, dedicated optical E2E Links must be set up. These links span multiple countries and allow the unrestricted utilization of the physically possible bandwidth. \n\nE2E Links connect organizations located in different countries and cross the networks of different providers. When providing the E2E Link services, each provider (member of the service provider coalition) has to collaborate w.r.t. setup, maintenance, and management with the other providers. Major challenges in the realization of these services are the heterogeneity concerning the technical implementations, the used software tools, various people related issues, and many more.\nIn \\cite{Hamm09}, Hamm introduced a reference incident management process for E2E Links.\n\n\n\\section{Use cases and requirements elicitation}\\label{sec:4req}\n\nBoth of the scenarios outlined above provide plenty of use cases for the elicitation of ioFMA requirements, although fault management obviously is only one of a lot of aspects that need to be addressed in such complex service provider constellations. One of the characteristics common to both scenarios is that the service providers, which are involved in the delivery process, are communicating and cooperating with each other in a kind of \\textit{,,provider network\"}. To better address such specifics, we first define the roles for ioFM in the next section. They have been generalized based on the roles and responsibilities we found in the real world scenarios.\n\n\\subsection{Defining roles for inter-organizational fault management}\\label{subsec:roles}\n\nOne of the most important roles in ioFM is the \\textit{user}. This is the role that typically initiates the fault management process by means of fault notifications that are stored in trouble ticket systems (TTS). In inter-organizational environments this role can be assigned to a service provider that is using a certain service as a \\textit{user}, e.g., due to outsourcing. \n\n\\textit{Service Provider (SP)} is the role that is responsible for the delivery of a service and for the fulfillment of the Service Level Agreements (SLAs) agreed with its \\textit{users}. These \\textit{SP}s are also essential to the ioFM as they constitute the \\textit{provider network} and deliver IT services in a cooperative manner.\n\nWithin the different service provider domains, there is always a role that is responsible for the local fault management. We called this role the \\textit{Domain Fault Manager (DFM)}. The \\textit{DFM} does not only communicate within its domain, but also with the \\textit{DFM}s of other domains.\n\nOn the local level also a \\textit{Domain Fault Operator (DFO)} is required in order to isolate, correct, and log a fault within her own domain. Even though these both are intra-organizational roles, the \\textit{DFO} has a purely operational role, whereas the \\textit{DFM} primarily has coordinating responsibilities.\n\nIn ioFM, the so-called \\textit{Global Fault Coordination Manager (GFCM)} has the overall coordination role: It addresses all the domains that are involved in the service delivery process. The GFCM's main tasks include: monitoring of confirmed and potential faults, forwarding of fault-related information between the different domains, and facilitating inter-domain communication. In the hierarchical case the role of the \\textit{GFCM} is identical to \\textit{DFM} for obvious reasons. However, in a heterarchy, the role of \\textit{GFCM} will be assigned temporarily to each of the domains in an on-demand manner.\n\nLast but not least the \\textit{Domain Monitoring System (DMS)} is responsible within a domain for system and component monitoring. This role announces fault notifications or alarms about malfunctions of the system. Using these roles the use cases are described in the following section.\n\nThe important roles defined here are the base for the \\textit{organizational model} of the ioFMA. \n\n\n\n\n\\subsection{Identifying use cases}\\label{subsec:usecase}\n\nAbove we describe and analyze the two real-world scenarios in order to elicit use cases needed for the requirements analysis.\nTherefore we have identified the following different classes of use cases: \\textit{fault localization}, \\textit{fault resolution progress management}, \\textit{monitoring}, \\textit{reporting}, and \\textit{handling false-positives}. These also represent the core functionalities that an ioFMA should offer.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Use cases for fault localization]\n {\\includegraphics[width=0.49\\textwidth]{ucfaultloc.jpg}\\label{fig:ucfaultloc}}\n\\subfigure[Use cases for fault monitoring]\n {\\includegraphics[width=0.49\\textwidth]{ucmonitor.jpg}\\label{fig:ucmonitor}}\n\\caption{Use cases for fault localization and monitoring}\n\\end{figure}\n\n\\subsubsection{Fault Localization}\nThe main functionality of the ioFMA has to be the precise localization of faults. Depending on the place where the fault will be localized, there can be multiple variations as shown in Figure \\ref{fig:ucfaultloc}: \nThe \\textbf{\\textit{fault localization within one's own domain (L01)}} is initiated by the \\textit{user}, or by the \\textit{DMS} respectively, and will be localized by the \\textit{DFM} if a known fault occurs; otherwise, i.e. if it is an unknown fault, it will be the \\textit{DFM}'s task with the support of the \\textit{DFO}. \n\nIf the fault cannot be isolated within this domain, the issue will be forwarded to another domain. The \\textbf{\\textit{fault localization in an undefined domain (L02)}} will therefore be initiated. The \\textit{DFM} is reporting the fault to the \\textit{GFCM}, which will forward it to all \\textit{DFM}s involved in the service delivery. In collaboration with the \\textit{DFO}s, the fault will -- in the best case -- be found in one of the domains and back reported to the \\textit{GFCM}. However, in the case that the fault cannot be isolated in this way, an escalation procedure has to be initiated. \nA derivate of this use case is \\textbf{\\textit{fault localization within a specific domain (L03)}}; here, the \\textit{GFCM} has to forward the fault only to a certain (known) domain and not to all involved partners.\n\n\\subsubsection{Fault Resolution Progress Management}\nA status display informs about the \\textit{progress of the fault resolution} or the \\textit{progress of the maintenance work}. The \\textbf{\\textit{progress of the fault resolution (P01)}} is initiated by the \\textit{DFM} that wants to know the progress of the fault resolution within his own or any other involved domain. It can also be initiated by the \\textit{GFCM} in order to get an overview of the whole inter-organizational network with respect to the fault resolution process instances. Consequently, the \\textit{DFM} and\/or \\textit{GFCM} query the \\textit{DFM}s regarding the progress of the fault resolution in their respective domains. The \\textit{DFM}s will retrieve this information from their \\textit{DFO}s and give feedback to the \\textit{DFM} or \\textit{GFCM} from which the query originates. For the \\textbf{\\textit{progress of the maintenance work P02}} the same steps will be run through, but with a different scope. The case when a user wishes to be informed about the status of the fault resolution and\/or maintenance is a secondary scenario within this use case, which results in a query forwarded by the \\textit{DFM} or \\textit{GFCM}.\n\n\n\\subsubsection{Monitoring}\nIn both the hierarchical and the heterarchical case, monitoring is a very important feature that the ioFMA should have. By means of continuous monitoring, faster fault localization is enabled. We distinguish between \\textit{domain monitoring}, \\textit{overall monitoring}, and \\textit{service monitoring} (see figure \\ref{fig:ucmonitor}). The \\textbf{\\textit{domain monitoring (M01)}} is responsible for the fault monitoring within a domain. It can be initiated by the \\textit{user}, \\textit{DFM}, or \\textit{GFCM}. They will be querying the DFM of a certain domain about the general status of the faults within this domain. The result will be retrieved from the \\textit{DMS}, which is always updated concerning the alarms and fault notifications. One exception that needs to be dealt with is when the \\textit{user} or \\textit{DFM} does not have the necessary access rights to fetch monitoring information about another domain. \\textbf{\\textit{Overall monitoring (M02)}} is responsible for the monitoring of the whole provider network. It can be initiated by the \\textit{GFCM} or by any other \\textit{DFM} that has sufficient access rights. This results in querying the entire domain \\textit{DFM}s about their monitoring status. If all of the domains are replying with a valid status, then the overall monitoring is enabled; otherwise only a partial monitoring of the provider network can be established. As many providers (but not all of those within the provider network) are involved in the delivery of a certain service, the \\textbf{\\textit{service monitoring (M03)}} is denoting that only these involved domains will be monitored. This is a special case of the former one, as it monitors only a well-defined subset of the provider network.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Use cases for fault resolution progress management]\n {\\includegraphics[width=0.49\\textwidth]{ucresproc}\\label{fig:ucresproc}}\n\\subfigure[Use cases for false positives]\n {\\includegraphics[width=0.49\\textwidth]{ucfalsepos}\\label{fig:ucfalsepos}}\n\\caption{Use cases for fault resolution progress management and false positives}\n\\end{figure}\n\n\\subsubsection{Reporting}\n\nReports are supporting different processes, such as fault management. They give an \\textit{overview of actual measurements, metrics, accounting data}, \\textit{Quality of Service (QoS) parameters}, but also information based on historical data, e.g. in order to facilitate a \\textit{trend analysis}. First the realization of\\textbf{ \\textit{statistical plots and accounting data reports (R01)}} will be specified. This is usually initiated by the \\textit{GFCM}, which is about to retrieve all this data from all \\textit{DFM}s in the provider network. In the best case all the domains send the requested information so that a report and statistical plots from all the involved domains can be conducted. In the case that some domains do not respond to the information request, incomplete statistical plots or\/and accounting data will be shown. The \\textbf{\\textit{QoS parameter (R02)}} will be retrieved in order to check the fulfillment of the agreed SLAs and to evaluate the follow-up of different faults that have occurred in the past. Based on historical information, \\textbf{\\textit{trend analysis (R03)}} can be done by predicting the liability of the system to some specific faults with various follow-ups according to various statistical models. Potential future faults could therefore be resolved or by-passed before they really occur.\n\n\\subsubsection{False-positives}\nIn order to be assured that information concerning faults is valid, false positives (i.e. wrongly announced faults) have to be identified and removed. This use case is very important as in many cases the search for non-existing faults impedes the normal functionality of an IT service. The \\textbf{\\textit{localization of false positives (F01)}} is initiated by the \\textit{GFCM} or by one of the \\textit{DFM}s. In the case that a potential false fault notification is given that cannot be mapped onto the behavior of the system, the \\textit{GFCM} or \\textit{DFM} is querying the responsible \\textit{DFM} about this issue. The \\textit{DFM} has to consult the \\textit{DFO} and figure out whether this fault really is a false positive. The result will be reported back to the GFCM. The \\textbf{\\textit{removal of false positives (F02)}} requires that it has reliably been identified as such first. Thus, the \\textit{DFO} identifies the non-existing fault and removes the false positive (manually or tool-supported) from the monitoring system. This action is then reported to the \\textit{DFM}.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|} \\hlin\n\n& \\multicolumn{3}{c|}{{\\textsf{Scenario}}} \\\\ \\cline{2-4\n\n\\textsf{Use case} & \\textsf{IntegraTUM}&\\textsf{G\\'EANT} &\\textsf{Generalized}\\\\\n & \\textsf{(hierarchy)}&\\textsf{heterarchy} &\\textsf{(mixed)}\\\\\n\\hline\n\\hline\n\n\\textsf{L01:}& \\ding{51} & \\ding{51}&\\ding{51}\\\\ \n\\textsf{L02:}&\t\t\t & \\ding{51}& \\ding{51}\\\\ \n\\textsf{L03:}&\t\\ding{51}& &\\ding{51} \\\\ \n\\textsf{P01:}& \\ding{51}& \\ding{51}& \\ding{51}\\\\ \n\\textsf{P02:}&\\ding{51} & \\ding{51}&\\ding{51}\\\\ \n\\textsf{M01:}&\\ding{51} & \\ding{51}&\\ding{51}\\\\  \n\\textsf{M02:}&  \t\t& \\ding{51}&\\ding{51}\\\\ \n\\textsf{M03:}&\\ding{51} &\\ding{51} & \\ding{51}\\\\ \n\\textsf{R01:}&\\ding{51} & \\ding{51}&\\ding{51} \\\\ \n\\textsf{R02:}&\\ding{51}\t& \\ding{51}&\\ding{51} \\\\ \n\\textsf{R03:}& \\ding{51}& \\ding{51}& \\ding{51}\\\\ \n\\textsf{F01:}& \t\t\t&\\ding{51} & \\ding{51}\\\\ \n\\textsf{F02:}& \t\t\t&\\ding{51} & \\ding{51} \\\\ \n\n\\hline \n\\end{tabular}\n\\caption{Coverage of the scenarios by the use cases}\n\\label{tab:usecase2szen}\n\\end{center}\n\\end{table}\n\n\\subsection{Deriving requirements}\n\nTable \\ref{tab:usecase2szen} summarizes the different use case occurrence as requirements for the functional model of the \\textit{ioFMA}. Additional to these, the following two additional requirements have to be considered:\n\n\\begin{itemize}\n\\item \\textbf{FM-01}: In order to increase the legibility of the fault information, a \\textbf{\\textit{visual presentation}} is necessary.\n\\item \\textbf{FM-02}: Especially regarding the use cases for fault resolution progress management and in the removal of false positives the possibility to \\textbf{\\textit{change or remove fault data}} has to be given.\n\\end{itemize}\n\nIn order to support the realization of the use cases described above some requirements on the sub-models of the \\textit{ioFMA} have to be fulfilled.\n\nWe identified the following requirements regarding the \\emph{information model} of the \\textit{ioFMA}:\n\n\\begin{itemize}\n\\item \\textbf{IM-01}: A \\textbf{\\textit{common data format}} for fault information is needed in order to facilitate the inter-domain data exchange and the communication. This should consist of a set of common attributes or properties.\n\\item \\textbf{IM-02}: Another additional or coexisting requirement to the first one is the existence of \\textbf{\\textit{conversion methods}} between the data format in the different domains.\n\\item \\textbf{IM-03}: \\textbf{\\textit{Interface definition}} across different domains have to be defined.\n\\item \\textbf{IM-04}: The ioFMA has to support all the \\textit{\\textbf{life cycle phases}} of a fault resolution process (detection, isolation, repairing\/recovery, and forecast\/prevention).\n\\item \\textbf{IM-05}: Also the use of \\textit{\\textbf{standard metrics}} has a supporting role in the monitoring, and respectively in the reporting. An example of such a set of standard metrics is the IP Performance Metrics (IPPM) \\cite{IPPM} (e.g., One Way Delay (OWD \\cite{RFC2679}), IP Delay Variation (\\cite{RFC3393}), Packet Loss (\\cite{RFC2680}), and others).\n\\item \\textbf{IM-06}: As the correlation\/interrelation between the metrics of different domains has to be provided, a suitable \\textbf{\\textit{aggregation function}} has to be defined.\n\\end{itemize}\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|p{0.12\\textwidth}|c|c|c|c|p{0.12\\textwidth}|} \\hline\n\n\n\\multicolumn{2}{|c|}{\\multirow{2}{*}{\\textsf{Requirements}}} & \\multicolumn{4}{c|}{{\\textsf{Phases of the fault life cycle}}} \\\\ \\cline{3-6}\n\\multicolumn{2}{|c|}{}&\\textsf{Detection}&\\textsf{Isolation} &\\textsf{Repairing}& \\textsf{Forecast\/\\-Prevention}\\\\\n\\hline\n\\hline\n\\multirow{ 7}{*}{\\textsf{for the IM}}& \n\n\\textsf{IM-01}&&\\ding{51}&&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{IM-02}&\\ding{51}&&&\\ding{51}\\\\ \\cline{2-6}\n& \\textsf{IM-03}&\\ding{51}&\\ding{51}&\\ding{51}&\\\\ \\cline{2-6}\n&\\textsf{IM-04}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{IM-05}&\\ding{51}&&&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{IM-06}&\\ding{51}&&&\\ding{51}\\\\ \\hline\n\n\\multirow{ 2}{*}{\\textsf{for the OM}}&\n\\textsf{OM-01}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{OM-02}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\hline\n\n\n\\multirow{ 15}{*}{\\textsf{for the FM}}& \n\n\\textsf{FM-L01}&\\ding{51}&\\ding{51}&&\\\\ \\cline{2-6}\n&\\textsf{FM-L02}&\\ding{51}&&&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-L03}&\\ding{51}&\\ding{51}&&\\\\\\cline{2-6}\n\n&\\textsf{FM-P01}&&&\\ding{51}&\\\\\\cline{2-6}\n&\\textsf{FM-P02}&&&\\ding{51}&\\\\\\cline{2-6}\n\n&\\textsf{FM-M01}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-M02}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-M03}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\\\cline{2-6}\n\n&\\textsf{FM-R01}&\\ding{51}&&&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-R02}&\\ding{51}&&&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-R03}&&&&\\ding{51}\\\\\\cline{2-6}\n\n&\\textsf{FM-F01}&&\\ding{51}&&\\ding{51}\\\\\\cline{2-6}\n&\\textsf{FM-F02}&&&\\ding{51}&\\ding{51}\\\\\\cline{2-6}\n\n&\\textsf{FM-01}&\\ding{51}&\\ding{51}&&\\ding{51}\\\\  \\cline{2-6}\n&\\textsf{FM-02}&&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\hline\n\n\n\\multirow{ 3}{*}{\\textsf{for the KM}}&\n\\textsf{KM-01}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\\\cline{2-6} \n&\\textsf{KM-02}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{KM-03}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n\\hline\n\\multirow{ 8}{*}{\\textsf{NFA}}& \n\n\\textsf{NF-01}&&\\ding{51}&\\ding{51}&\\\\ \\cline{2-6}\n&\\textsf{NF-02}&&\\ding{51}&\\ding{51}&\\\\ \\cline{2-6}\n&\\textsf{NF-03}&\\ding{51}&&&\\ding{51}\\\\ \\cline{2-6}\n& \\textsf{NF-04}&\\ding{51}&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{NF-05}&&\\ding{51}&\\ding{51}&\\\\ \\cline{2-6}\n&\\textsf{NF-06}&\\ding{51}&\\ding{51}&&\\\\ \\cline{2-6}\n&\\textsf{NF-07}&&\\ding{51}&\\ding{51}&\\ding{51}\\\\ \\cline{2-6}\n&\\textsf{NF-08}&&\\ding{51}&\\ding{51}&\\\\ \n\\hline\n\\end{tabular}\n\\caption{Coverage of the phases of the fault life cycle by the requirements}\n\\label{tab:lifecycle2req}\n\\end{center}\n\\end{table}\n\nFurthermore, requirements regarding the \\emph{organizational model} of the \\textit{ioFMA} must be considered:\n\n\\begin{itemize}\n\\item \\textbf{OM-01}: The \\textbf{\\textit{inter-organizational service delivery models}} have to be supported.\n\\item \\textbf{OM-02}: \\textbf{\\textit{Definition of roles and responsibilities}} according to the use cases described above.\n\\end{itemize}\n\nHowever, also the following requirements regarding the \\emph{communication model} of the \\textit{ioFMA} must be kept in mind:\n\n\\begin{itemize}\n\\item \\textbf{KM-01}: \\textbf{\\textit{Communication mechanism}}, such as pull or push models have to be supported by the ioFMA.\n\\item \\textbf{KM-02}: \\textbf{\\textit{Inter-domain communication}} is a very important requirement as the ioFMA will be deployed in an inter-organizational environment. Different networks with heterogeneous technologies exchange different data with each other. The inter-domain communication is also important, because in the absence of a central unit for coordination and communication between different networks at least a minimal set of information has to be exchanged.\n\\item \\textbf{KM-03:} In order to support the data exchange within different networks a \\textbf{\\textit{communication protocol}} has to be defined. The complexity of the inter-organizational environment with their different provider, networks, and protocols is the challenge we are facing here.\n\\end{itemize}\n\nFinally, we argue that the functional requirements regarding the sub-models of the \\emph{ioFMA} must be complemented by the following series of non-functional requirements:\n\n\\begin{itemize}\n\\item \\textbf{NF-01}: An \\textbf{\\textit{access control}} mechanism has to be part of the \\emph{ioFMA}.\n\\item \\textbf{NF-02}: Protection against data loss and deliberate data altering especially in the fault localization, reporting, and false-positive \\textbf{\\textit{data integrity}} has to be provided all the time.\n\\item \\textbf{NF-03}: The \\textbf{\\textit{up-to-dateness}} of the data in the \\emph{ioFMA} has to be guaranteed.\n\\item \\textbf{NF-04}: Especially fault localization, monitoring, and false-positives management require a well-designed \\textbf{\\textit{scalability}} of the tools in order to provide the discussed functionality.\n\\item \\textbf{NF-05}: Adequate \\textbf{\\textit{performance}} in the realization of the above named functionalities has to be achieved.\n\\item \\textbf{NF-06}: The \\textbf{\\textit{automation}} of as many possible functionalities as possible has to be realized in order to speed up the fault resolution process.\n\\item \\textbf{NF-07}: A \\textbf{\\textit{common data base}} for all the providers involved in the inter-organizational fault management process.\n\\item \\textbf{NF-08}: Last but not least all processes and functionalities have to be properly \\textbf{\\textit{documented}}.\n\\end{itemize}\n\nAs we take the whole fault resolution process into account, the requirements have to be related to all relevant life cycle phases. Table \\ref{tab:lifecycle2req} shows which requirements have to be fulfilled in the different phases of the fault life cycle (detection, isolation, repairing\/recovery, and forecast\/prevention).\n\n\\section{Conclusions and future work}\\label{sec:5further}\n\nIn this article we presented a full requirement analysis in order to design an inter-organizational fault management architecture. We also discussed the core aspects of the functional and organizational models based on the elicited use cases and requirements. The next steps in our research are to complete the architecture with a communication and an information model. After that we will deliver a full model of ioFMA on the PIM layer as well as its transformation to the system layer. Our implementation will be customized for the LHC optical private network (LHCOPN), which is operated by the European G\\'EANT network.\n\n\n\n\\section{Core aspects of the functional model}\\label{sec:6funcModel}\n\nThis section addresses the functional model of the ioFMA. As a base for its design the use cases described in section \\ref{subsec:usecase} are applied. As stated in \\cite{han99}, the functional model contains the functional areas which integrate all the required functionalities of a management architecture. For the ioFMA, we elicited three functional areas related to the organizational domain in which it is deployed: \n\n\\begin{itemize}\n \\item Provider management -- this the part of the ioFMA concerned with local ,,arrangements'' and integrating them with intra-organizational fault management\n \\item Inter-organizational Management -- this is the core part of the functional model of the ioFMA as it contains all inter-organizational aspects \n \\item Customer management -- is placed on a more abstract level above the both former functional areas as it is connected to both of them and is the enabler of the provider and inter-organizational management, respectively. \n\\end{itemize}\n\n\\subsection{Provider Management}\n\nWithin the service provider domain, different management functions in order to support the inter-organizational fault management have to be implemented. These management functions rely on the described use cases.\n\nFault localization within one's own domain is the first management function which has to be realized in a domain as a part of an ioFMA. The progress management for the fault resolution as well as the progress management for the maintenance work have to be performed within the service provider domain and connect to the inter-organizational management. Finding and removing false positives as well as performing data changes (under the strict control of the inter-organizational management) have also to be implemented within the domain.\n\n\\subsection{Inter-organizational Management}\n\nAs the core of the functional model, the inter-organizational management has to coordinate, integrate, put together information and functions from the different involved service provider domains. \nThe management functions, which the inter-organizational management comprises, are: fault localization in an unspecified domain and within a specific domain, progress management for the fault resolution and for the maintenance work, overall monitoring and service monitoring, creation of statistical plots and accounting data reports, representation of QoS parameter and realization of trend analysis as well as detecting respective removing false positive fault reports. It can be observed that these are mainly the use cases defined previously. In addition to this a very important management function -- \\emph{data change} -- has to be added. This has to be allowed but only under control of the inter-organizational management.\n\n\\subsection{Customer Management}\n\nThe customer management is the key enabler for both the provider management and the inter-organizational management. It actually contains all the management functions listed above, but has additional functionality. For example, from the customer's perspective the opening and updating of fault reports has to be supported. It serves as both a trigger and a feedback channel and is an essential core component of IT service management architectures.  \n\n\n\\begin{figure}[] \n\t\\centerline{\\includegraphics[width=0.95\\columnwidth]{functionalModel.jpg}}\n\t\\caption{Overview on the functional model of the ioFMA}\n\t\\label{fig:functionalModel} \n\\end{figure}\n\n\\subsection*{Autors}\n\\label{sec:shortBio}\n\\normalsize\n\\label{sec:PatriciaMarcu}\n\\parpic[r]{\\includegraphics[width=0.2\\textwidth]{.\/BildMarcu.jpg}}\n\\textbf{Patricia Marcu} received her diploma in Computer Science in 2006 at the LMU Munich. In 2007 she joined the MNM-Team at Leibniz Supercomputing Centre as a research assistant and pursues her Ph.D. degree in Computer Science. She is currently working on the further development of the Customer Network Managemnt (CNM) tool and on the visualization of the LHCOPN within the European Geant project. Her research focuses on inter-organizational fault management and IT Service Management.\n\\\\\n\n\\label{sec:WolfgangHommel}\n\\parpic[r]{\\includegraphics[width=0.2\\textwidth]{.\/BildHommel.jpg}}\n\n\\textbf{Wolfgang Hommel} has a Ph.D. in computer science from LMU Munich, and heads the network services planning group at the Leibniz Supercomputing Centre. His current research focuses on IT security and privacy management in large distributed systems, including identity federations and Grids. Emphasis is put on a holistic perspective, i.e., the problems and solutions are analyzed from the design phase through software engineering, deployment in heterogeneous infrastructures, and during the\noperation and change phases according to IT service management process frameworks, such as ISO\/IEC 20000-1. \n\n\n\\section*{Acknowledgments}\nThe authors would like to thank their colleagues at the Leibniz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities (see \\url{http:\/\/www.lrz.de\/}) for helpful discussions and valuable comments about this paper.\n\nThe authors wish to thank the members of the Munich Network Management Team (MNM-Team) for helpful discussions and\nvaluable comments on previous versions of this paper. The MNM Team directed by Prof. Dr. Dieter Kranzlm\\\"uller and Prof.\nDr. Heinz-Gerd Hegering is a group of researchers at Ludwig-Maximilians-Universit\\\"at M\\\"unchen, Technische Universit\\\"at\nM\\\"unchen, the University of the Federal Armed Forces and the Leibniz Supercomputing Centre of the Bavarian Academy of\nSciences and Humanities. See \\url{http:\/\/www.mnm-team.org\/}.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nApproximate Bayesian computation (ABC) algorithms are a popular method of inference for a wide class of otherwise intractable probability models in applications such as population genetics, ecology, epidemiology and systems biology \\citep{Beaumont:2010, Marin:2012}.  They select parameter vectors $\\theta$ for which datasets $y$ simulated from the model of interest are sufficiently close to the observations.  A bottleneck is the computational cost of producing the large quantity of model simulations needed, which becomes increasingly severe for more detailed models.  However, it is often clear during a simulation that it is unpromising.  For example it is likely to produce a poor match or to require excessive computation time.  This paper presents \\emph{lazy ABC}, an importance sampling method which abandons some such simulations, a step referred to as \\emph{early stopping}, exploiting information from incomplete simulations to save time.  The result is an ABC algorithm which is more scalable to applications where simulation is computationally demanding. \n\nIn more detail, standard ABC is based on a random likelihood estimator $\\hat{L}_{\\text{ABC}}$, which is 1 for a close match of simulated and observed data, and zero otherwise.  The algorithm can be shown to target a distribution corresponding to the approximate likelihood $L_{\\text{ABC}}(\\theta) = \\E[\\hat{L}_{\\text{ABC}} | \\theta]$.  Lazy ABC is based on an alternative estimator $\\hat{L}_{\\text{lazy}}$.  This equals zero with probability $1-\\alpha$ -- if early stopping is performed -- and otherwise equals the $\\hat{L}_{\\text{ABC}}$ estimator multiplied by a weight.  Letting the weight equal $1\/\\alpha$ makes $\\hat{L}_{\\text{lazy}}$ an unbiased estimator of $L_{\\text{ABC}}(\\theta)$.  Results on random likelihood estimates show that importance sampling\nalgorithms based on $\\hat{L}_{\\text{lazy}}(\\theta)$ therefore target the same distribution as standard ABC.  No further approximation has been introduced.\n\nThe lazy ABC estimator trades off an increase in variance for a reduction in computation time.  It is shown that for this to be most advantageous $\\alpha$ should be (1) larger when there is a high probability of the simulated dataset being a close match to the observations (2) smaller when the expected time to complete the simulation is large.  To achieve this, $\\alpha$ is based on $X$, a random variable encapsulating some preliminary information about the simulated dataset $Y$, also a random variable.  The final likelihood estimator is based on $X$ and $Y$.  However when early stopping occurs a realised value of zero is obtained without drawing a value of $Y$. This is an example of the computer science concept of \\emph{lazy evaluation} \\citep[``functional programming'' entry]{Ralston:2003}, which is the basis for the method's name.\n\nThe paper presents theoretical results on the optimal tuning of $\\alpha$ in lazy ABC, making precise the two properties just outlined.  This choice is asymptotically optimal in terms of maximising efficiency, which is defined as effective sample size (ESS) per unit CPU time.  Based on this, a framework for tuning in practice is also presented.  The main requirements are the estimation of the probability of close matches and of expected remaining computation times.  Both of these are conditional on $\\theta$ and $x$ (realised values of $X$).  As $x$ is typically high dimensional this estimation is not feasible, so instead it is recommended to base the choice of $\\alpha$ on a vector of low dimensional \\emph{decision statistics} $\\phi(\\theta, x)$.  A computationally demanding example is presented based on the spatial extremes application of \\cite{Erhardt:2012} where lazy ABC increases the efficiency by roughly 3 times for modestly sized data and 8 times for a larger example.\n\nIt should be emphasised that this approach to tuning is \\emph{optional}.\nIn many applications a simple ad-hoc choice of $\\alpha$ may produce significant efficiency gains,\nespecially where application knowledge can be used to create heuristic criteria which quickly identify many unpromising simulations.\n\nThe focus of this paper is on importance sampling, which is widely used by ABC practitioners and particularly amenable to parallelisation.  However the lazy ABC approach is also applicable to other algorithms, such as Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC), as discussed in the final section.\n\nSeveral recent papers have proposed speeding up ABC by fitting a model to $(\\theta, y)$ pairs, simulated either in a preliminary stage or in earlier ABC iterations \\citep{Buzbas:2013, Meeds:2014, Moores:2014, Wilkinson:2014}.  This model is then sometimes or always used in place of the original model of interest in the inference algorithm.  A potential application of lazy ABC is to make use of such approximate models (their predictions given $\\theta$ forming $X$ in the notation above) to gain speed benefits without incurring additional approximation errors.  More generally, there has been much interest over the past decade in Bayesian inference algorithms with random weights \\citep[e.g.][]{Beaumont:2003, Andrieu:2009, Fearnhead:2010, Tran:2014}.  A novelty of lazy ABC is that it introduces a random factor to the weights to reduce computation time, rather than to deal with intractability.  \n\n\nFinally, a byproduct of the paper's theory, Corollary \\ref{cor:ABC-IS}, is of independent interest.\nThis gives importance densities which optimise asymptotic ESS per unit CPU time for:\n(a) ABC importance sampling (b) importance sampling with random weights.\n\nThe remainder of the paper is structured as follows.  Section \\ref{sec:background} contains background material on ABC and importance sampling. \nSection \\ref{sec:ESalgorithm} gives the lazy ABC algorithm and proves it targets the correct distribution.  Section \\ref{sec:tuning} presents theory and practical methods for tuning the algorithm, as well as the corollary mentioned above.\nSection \\ref{sec:epidemics} illustrates the method for a simple epidemic model.\nR code to implement this is available at\n\\texttt{https:\/\/github.com\/dennisprangle\/lazyABCexample}.\nSection \\ref{sec:extremes} contains a more challenging spatial extremes application and\nSection \\ref{sec:discussion} is a discussion.\nAppendices contain proofs and discuss some extensions:\nlazy ABC with multiple stopping decisions\nand application of similar methods outside ABC.\n\n\\section{Importance sampling} \\label{sec:background}\n\nConsider analysing data $y_{\\text{obs}}$ under a probability model with density $\\pi(y|\\theta)$ and parameters $\\theta$.  The likelihood is defined as $L(\\theta) = \\pi(y_{\\text{obs}}|\\theta)$.  Bayesian inference introduces a prior distribution with density $\\pi(\\theta)$ and aims to find the posterior distribution $\\pi(\\theta | y_{\\text{obs}}) = \\pi(\\theta) L(\\theta) \/ \\pi(y_{\\text{obs}})$, where $\\pi(y_{\\text{obs}}) = \\int \\pi(\\theta) L(\\theta) d\\theta$, or at least to estimate the posterior expectation $\\E[h(\\theta) | y_{\\text{obs}}]$ of a generic function $h(\\theta)$.  Importance sampling is a simple method to do this.  Parameter values $\\theta_{1:N}$ are simulated independently from an importance density $g(\\theta)$ and given weights $w_i = L(\\theta_i) \\pi(\\theta_i) \/ g(\\theta_i)$ (n.b.~$\\theta_{1:N}$ represents the sequence $(\\theta_i)_{1 \\leq i \\leq N}$.  Similar notation is used later.)  It is assumed throughout that $g(\\theta)>0$ whenever $\\pi(\\theta)>0$.  Each of the $(\\theta_i, w_i)$ pairs can be computed in parallel, allowing for efficient implementation.\n\nA Monte Carlo estimate of $\\E[h(\\theta) | y_{\\text{obs}}]$ is $\\mu_h = \\frac{\\sum_{i=1}^N h(\\theta_i) w_i}{\\sum_{i=1}^N w_i}$.\nTwo properties of importance sampling estimates are\n\\begin{align}\n&\\mu_h \\to \\E[h(\\theta) | y_{\\text{obs}}] \\quad \\text{almost surely as } N \\to \\infty, \\label{eq:IS1} \\\\\n&E[N^{-1} \\sum_{i=1}^N w_i] = \\pi(y_{\\text{obs}}). \\label{eq:IS2}\n\\end{align}\nSee \\cite{Geweke:1989} for proof that \\eqref{eq:IS1} holds under weak conditions.  To prove \\eqref{eq:IS2} note that each $w_i$ is an unbiased estimator of $\\pi(y_{\\text{obs}})$.  Estimating this is of interest for model comparison.  \n\n\\subsection{Notation}\n\nThe remainder of the paper is largely concerned with the distribution of random variables produced in an iteration of various importance sampling algorithms.  Henceforth, expectations and probabilities that involve quantities produced by importance sampling should be read as being with respect to this distribution.  In particular this means that below the marginal density of $\\theta$ is taken to be $g(\\theta)$.  The preceding material in this section is the only time that a marginal density of $\\pi(\\theta)$ is used instead.\n\n\\subsection{Random weights} \\label{sec:randomweightIS}\n\nAlgorithm \\ref{alg:RW-IS} describes random weight importance sampling (RW-IS), an importance sampling algorithm in which likelihood evaluations are replaced with random estimates of the likelihood.  Under the condition that these estimates are non-negative and unbiased, the algorithm produces valid output, in the sense that \\eqref{eq:IS1} and \\eqref{eq:IS2} continue to hold.  This can be seen by noting that Algorithm \\ref{alg:RW-IS} is equivalent to a deterministic weight importance sampling algorithm with augmented parameters $(\\theta, \\ell)$, prior density $\\pi(\\theta) \\pi(\\ell| \\theta)$, importance density $g(\\theta) \\pi(\\ell| \\theta)$ and likelihood $\\ell$.  Here $\\ell$ is the realisation of the likelihood estimator and $\\pi(\\ell| \\theta)$ is the conditional density of this estimator.  This algorithm gives the correct marginal posterior for $\\theta$.\nSee  \\cite{Tran:2014} for a more detailed proof and \\cite{Fearnhead:2010} for further background\n(including the observation that the non-negativity condition above can be removed.)\n\n\\begin{algorithm}[htp]\n\n\\rule{\\textwidth}{0.2mm} \\\\\n{\\bf Input:}\n\\begin{itemize}\n\\item Prior density $\\pi(\\theta)$ and importance density $g(\\theta)$.\n\\item Number of iterations to perform $N$.\n\\item Likelihood estimator $\\hat{L}$.\n\\end{itemize}\nRepeat the following steps $N$ times.\n\\begin{enumerate}\n\\item[1] Simulate $\\theta^*$ from $g(\\theta)$.\n\\item[2] Simulate $\\ell^*$ from $\\hat{L} | \\theta$.\n\\item[3] Set $w^* = \\ell^* \\pi(\\theta^*) \/ g(\\theta^*)$.\n\\end{enumerate}\n{\\bf Output:}\\\\ A set of $N$ pairs of $(\\theta^*, w^*)$ values.\\\\\n\\rule{\\textwidth}{0.2mm}\n\\caption{Random weight importance sampling (RW-IS) \\label{alg:RW-IS}}\n\\end{algorithm}\n\n\\begin{algorithm}[htp]\n\n\\rule{\\textwidth}{0.2mm} \\\\\n{\\bf Input:}\n\\begin{itemize}\n\\item Prior density $\\pi(\\theta)$ and importance density $g(\\theta)$.\n\\item Number of iterations to perform $N$.\n\\item Observed data $y_{\\text{obs}}$.\n\\item Summary statistics $s(\\cdot)$, distance function $d(\\cdot,\\cdot)$ and threshold $\\epsilon \\geq 0$.\n\\end{itemize}\n{\\bf Algorithm:}\\\\\nRepeat the following steps $N$ times.\n\\begin{enumerate}\n\\item[1] Simulate $\\theta^*$ from $g(\\theta)$.\n\\item[2] Simulate $y^*$ from $Y | \\theta^*$\n\\item[3] Set $\\ell^* = \\mathbbm{1}[d(s(y^*), s(y_{\\text{obs}})) \\leq \\epsilon]$.\n\\item[4] Set $w^* = \\ell^* \\pi(\\theta^*) \/ g(\\theta^*)$.\n\\end{enumerate}\n{\\bf Output:}\\\\ A set of $N$ pairs of $(\\theta^*, w^*)$ values.\\\\\n\\rule{\\textwidth}{0.2mm}\n\\caption{ABC importance sampling (ABC-IS) \\label{alg:ABC-IS}}\n\\end{algorithm}\n\n\\subsection{Approximate Bayesian computation}\n\nMany interesting models are sufficiently complicated that it is not feasible to calculate exact likelihoods or useful (i.e.~reasonably low variance) unbiased estimators.  ABC algorithms instead base inference on simulation from the model.\nAlgorithm \\ref{alg:ABC-IS} (ABC-IS) is a standard importance sampling implementation of this idea.\n\nFor later theoretical work, it is helpful to interpret ABC-IS in the framework of RW-IS.\nIn particular, Algorithm \\ref{alg:ABC-IS} can be obtained from Algorithm \\ref{alg:RW-IS} by replacing the likelihood estimator $\\hat{L}$ with a Bernoulli estimator \n\\begin{equation} \\label{eq:LhatABC}\n\\hat{L}_{\\text{ABC}} = \\mathbbm{1}[d(s(Y), s(y_{\\text{obs}})) \\leq \\epsilon].\n\\end{equation}\nThis equals 1 if the distance between summary statistics of the simulated and observed datasets is less than or equal to a threshold $\\epsilon$.\nIt is typically a biased estimate of the likelihood and ABC-IS targets an approximate likelihood $L_{\\text{ABC}}(\\theta) = E[\\hat{L}_{\\text{ABC}} | \\theta]$.\n\nA special case of ABC-IS is when $g(\\theta)=\\pi(\\theta)$.  The weights in this case equal zero or one, and it is often referred to as ABC rejection sampling.  A generalisation of ABC-IS, considered in Section \\ref{sec:discussion}, is to use as a likelihood estimator $K(d(s(Y), s(y_{\\text{obs}}))\/\\epsilon)$, where $K$ is a density function known as the ABC kernel.  Algorithm \\ref{alg:ABC-IS} uses a uniform kernel.\n\nAs $\\epsilon \\to 0$, the target distribution of ABC-IS converges to $\\pi(\\theta | s(y_{\\text{obs}}))$.  However, $\\epsilon>0$ is typically required to achieve a reasonable number of non-zero weights, so a trade-off must be made.  The observed summary statistics $s(y_{\\text{obs}})$ should ideally preserve most of the information on $\\theta$ available from $y_{\\text{obs}}$.  However analysis of ABC algorithms shows that the quality of the approximation deteriorates with the dimension of $s(y)$.  Therefore the choice of $s(\\cdot)$ involves a trade-off between low dimension and informativeness.  For further background details on all aspects of ABC see the review articles of \\cite{Beaumont:2010} and \\cite{Marin:2012}.\n\n\\section{Lazy ABC} \\label{sec:ESalgorithm}\n\nLazy ABC is Algorithm \\ref{alg:lazyABC}. Given proposed parameters $\\theta^*$, step 2 performs an initial part of data simulation, with results $x^*$. A continuation probability $\\alpha(\\theta^*, x^*)$ is calculated. With this probability the simulation is completed, resulting in $y^*$, and a weight is calculated by steps 5 and 6. Otherwise the iteration is given weight zero. The desired behaviour is that simulating $x^*$ is computationally cheap but can be used to quickly reject many unpromising importance sampling iterations.\n\n\\begin{algorithm}[htp]\n\n\\rule{\\textwidth}{0.2mm} \\\\\n{\\bf Input:}\n\\begin{itemize}\n\\item Prior density $\\pi(\\theta)$ and importance density $g(\\theta)$.\n\\item Number of iterations to perform $N$.\n\\item Observed data $y_{\\text{obs}}$.\n\\item Summary statistics $s(\\cdot)$, distance function $d(\\cdot,\\cdot)$ and threshold $\\epsilon \\geq 0$.\n\\item Continuation probability function $\\alpha(\\theta,x)$ taking values in $[0,1]$.\n\\item Random variable $X$ representing an initial stage of simulation.\n\\end{itemize}\n{\\bf Algorithm:}\\\\\nRepeat the following steps $N$ times.\n\\begin{enumerate}\n\\item[1] Simulate $\\theta^*$ from $g(\\theta)$.\n\\item[2] Simulate $x^*$ from $X | \\theta^*$ and let $a^* = \\alpha(\\theta^*, x^*)$.\n\\item[3] With probability $a^*$ continue to step 4. Otherwise perform \\emph{early stopping}: let $\\ell^*=0$ and go to step 6.\n\\item[4] Simulate $y^*$ from $Y | \\theta^*, x^*$.\n\\item[5] Set $\\ell^*_{\\text{ABC}} = \\mathbbm{1}[d(s(y^*), s(y_{\\text{obs}})) \\leq \\epsilon]$ and $\\ell^* = \\ell^*_{\\text{ABC}} \/ a^*$.\n\\item[6] Set $w^* = \\ell^* \\pi(\\theta^*) \/ g(\\theta^*)$.\n\\end{enumerate}\n{\\bf Output:}\\\\ A set of $N$ pairs of $(\\theta^*, w^*)$ values.\\\\\n\\rule{\\textwidth}{0.2mm}\n\\caption{Lazy ABC \\label{alg:lazyABC}}\n\\end{algorithm}\n\nThe following conditions are required for Algorithm \\ref{alg:lazyABC} to be well defined:\n\\begin{itemize}\n\\item[C1] $\\alpha(\\theta,x) > 0$ whenever $\\Pr(\\hat{L}_{\\text{ABC}}>0 | \\theta, x) > 0$\n\\item[] (recall $\\hat{L}_{\\text{ABC}}$ is defined by \\eqref{eq:LhatABC})\n\\item[C2] The random variable $X$ is such that both $X | \\theta$ and $Y | \\theta, x$ can be simulated from.\n\\end{itemize}\n\nLazy ABC and ABC-IS both perform importance sampling inference for the same approximate likelihood $L_{\\text{ABC}}(\\theta)$.\nTo see this first observe that Algorithm \\ref{alg:lazyABC} can be interpreted as a RW-IS algorithm using a likelihood estimator of the form\n\\begin{equation} \\label{eq:es}\n\\hat{L}_{\\text{lazy}} = \\begin{cases} \n  \\hat{L}_{\\text{ABC}} \/ \\alpha(\\theta, X) & \\text{with probability } \\alpha(\\theta, X) \\\\\n  0 & \\text{otherwise}\n\\end{cases}\n\\end{equation}\nBy the arguments of Section \\ref{sec:background}, to show that lazy ABC and ABC-IS target the same likelihood it is sufficient to prove the following result.\n\n\\begin{theorem} \\label{thm:es}\nConditional on $\\theta$, $\\hat{L}_{\\text{lazy}}$ is a non-negative unbiased estimator of $L_{\\text{ABC}}(\\theta)$.\n\\end{theorem}\n\n\\begin{proof}\nNon-negativity is immediate.  For unbiasedness first observe that $\\E(\\hat{L}_{\\text{lazy}} | \\theta, x)$ equals zero when $\\alpha(\\theta,x)=0$ and $E(\\hat{L}_{\\text{ABC}} | \\theta, x)$ otherwise.\nBy C1 if $\\alpha(\\theta,x)=0$ then $\\Pr(\\hat{L}_{\\text{ABC}}>0 | \\theta,x)=0$ and so $\\E(\\hat{L}_{\\text{ABC}} | \\theta, x) = 0$.  Hence $\\E(\\hat{L}_{\\text{lazy}} | \\theta, X) = \\E(\\hat{L}_{\\text{ABC}} | \\theta, X)$.  Taking expectations over $X$ gives the required result.\n\\end{proof}\n\n\\subsection{Examples}\n\nThe following are examples of situations in which lazy ABC can be useful. The first three form the main focus of the paper.  Each assumes that $Y$ is a deterministic function of a latent vector $X_{1:p}$ such that it is possible to simulate from $X_1 | \\theta$ and $X_i | \\theta, x_{1:i-1}$ for all $1<i\\leq p$.  Further examples are given in Appendix \\ref{sec:multistop} which use the same framework to consider multiple stopping decisions.\n\\begin{itemize}[leftmargin=*]\n\\item[] \\emph{Example 1: Partial simulation}  Let $X=X_{1:t}$ for some $t<p$.\n\\item[] \\emph{Example 2: Partial calculation of $s$} Assume that computing $s(Y)$ involves calculating variables $X'_{1:q}$ which are deterministic transformations of $Y$, and that this is the most expensive part of simulating $\\hat{L}_{\\text{ABC}}$.  Let $X=(X_{1:p}, X'_{1:t})$ for some $t<q$.  (This is applied in Section \\ref{sec:simmethods}.)\n\\item[] \\emph{Example 3: Random stopping times} As for either previous example but with $t$ replaced by a random stopping time variable $T$.  This allows a stopping decision once a particular event has occurred.\n\\item[] \\emph{Example 4: Deterministic stopping}\nSuppose $s(Y)=(s_1(X_{1:t}), s_2(X_{t+1:p}))$.\nLet $X=X_{1:t}$ and $\\alpha(\\theta,x) = \\mathbbm{1}[\\Pr(\\hat{L}_{\\text{ABC}}>0 | \\theta, x) > 0]$. That is, early stopping occurs if and only if $s_1$ is too extreme for the ABC acceptance criterion to be met.\n\\end{itemize}\n\n\\subsection{Terminology and notation} \\label{sec:termnot}\n\nSimulation from $X | \\theta$ is referred to as the \\emph{initial simulation stage} and from $s(Y) | \\theta, x$ as the \\emph{continuation simulation stage}.  It is often useful later to have $\\alpha(\\theta, x) = \\alpha(\\phi(\\theta, x))$ where $\\phi(\\theta, x)$ is a vector of summaries referred to as the \\emph{decision statistics}.\n\nNotation is now introduced for expected CPU times: $\\bar{T}_1(\\theta)$ is for steps 2 and 3 in Algorithm \\ref{alg:lazyABC} conditional on $\\theta$, $\\bar{T}_2(\\theta, \\phi)$ is for steps 4--6 conditional on $(\\theta, \\phi)$ and $\\bar{T}(\\theta)$ is steps 2--4 of Algorithm \\ref{alg:ABC-IS} conditional on $\\theta$.  The first two are roughly the times of the initial simulation and continuation stages, but also cover the other steps of Algorithm \\ref{alg:lazyABC} given $\\theta$.\nThe following definitions are also used later: $\\bar{T}_1 = \\E[\\bar{T}_1(\\theta)]$ and $\\bar{T}_2(\\phi) = \\E[\\bar{T}_2(\\theta, \\phi) | \\phi]$.\nNote that $\\bar{T}_1$ is a constant.\nThe other terms above are deterministic functions of one or both of $\\theta$ and $\\phi$ and\nmay therefore be random variables when their arguments are.\n\n\n\n\\section{Tuning} \\label{sec:tuning}\n\nThere is considerable freedom to tune lazy ABC through the choice of $X$ (when to consider stopping) and $\\alpha$ (the function assigning continuation probabilities).  Section \\ref{sec:tuning theory} provides theory on the most efficient choice of $\\alpha$.  This is used\nin Section \\ref{sec:tuning practice} to motivate practical tuning methods.\n\n\n\\subsection{Theory} \\label{sec:tuning theory}\n\nA commonly used tool for the analysis of importance sampling algorithms is the effective sample size (ESS).  \\cite{Liu:1996} argued that typically the variance of the importance sampling estimator is roughly equal to that of $N_{\\text{eff}}$ independent samples where\n\\[\nN_{\\text{eff}} = N E(W)^2 \/ E(W^2),\n\\]\n$N$ is the number of importance sampling iterations\nand the random variable $W$ is the weight generated in an iteration of importance sampling (with zero representing rejection).\nThe argument of \\citeauthor{Liu:1996} generalises immediately to RW-IS algorithms through the interpretation of them as importance sampling algorithms on an augmented parameter space given in Section \\ref{sec:randomweightIS}.\n\nDefine efficiency as $N_{\\text{eff}} \/ T$ where $T$ is the CPU time of the algorithm (i.e.~ignoring any execution time savings due to parallelisation.)  \nVarious results on asymptotic efficiency for large $N$ now follow.\nThese are proved in Appendix \\ref{sec:alpha} as corollaries of Theorem \\ref{thm:tuning} which is stated there.\n\nThe choice of $\\alpha(\\theta,x)$ which maximises asymptotic efficiency is as follows for some $\\lambda \\geq 0$,\n\\begin{equation} \\label{eq:tuning1}\n\\alpha(\\theta,x) = \\min\\left\\{1, \\lambda \\frac{\\pi(\\theta)}{g(\\theta)} \\left[\\frac{\\gamma(\\theta,x)}{\\bar{T}_2(\\theta,x)}\\right]^{1\/2} \n\\right\\},\n\\end{equation}\nwhere $\\gamma(\\theta, x) = \\Pr \\left(d(s(Y), s(y_{\\text{obs}})) \\leq \\epsilon | \\theta, x \\right)$ and $\\bar{T}_2(\\theta, x)$ is as defined in Section \\ref{sec:termnot}. Roughly, $\\gamma(\\theta, x)$ is the probability that continuing the simulation will meet the ABC acceptance criterion and $\\bar{T}_2(\\theta, x)$ is the expected time it will take. The interpretation of $\\lambda$ is not clear, nor does a simple closed form expression for its optimal value seem possible.\n\n\nThis optimal choice of $\\alpha$ is of limited practical use, as typically $X$ is high dimensional and thus estimation of $\\gamma(\\theta, x)$ and $\\bar{T}_2(\\theta, x)$ is not feasible. Estimation is more feasible if $(\\theta, x)$ is replaced by some low dimensional vector of summaries, $\\phi(\\theta,x)$, which will be referred to as the \\emph{decision statistics}. Consider decision statistics which satisfy the condition\n\\begin{itemize}\n\\item[C3] There is a function $u(\\phi)$ such that $u(\\phi(\\theta,x)) = \\tfrac{\\pi(\\theta)}{g(\\theta)}$.\n\\end{itemize}\nThen the optimal $\\alpha$ amongst those of the form $\\alpha(\\theta,x)=\\alpha(\\phi(\\theta,x))$ is as follows for some $\\lambda \\geq 0$,\n\\begin{equation} \\label{eq:tuning2}\n\\alpha(\\phi) = \\min\\left\\{1, \\lambda u(\\phi) \\left[\\frac{\\gamma(\\phi)}{\\bar{T}_2(\\phi)}\\right]^{1\/2} \\right\\},\n\\end{equation}\nwhere $\\gamma(\\phi) = \\Pr \\left(d(s(Y), s(y_{\\text{obs}})) \\leq \\epsilon | \\phi \\right)$ and $\\bar{T}_2(\\phi)$ is as defined in Section \\ref{sec:termnot}.\nSee Figures \\ref{fig:SIR}D and \\ref{fig:SErep}D for example $\\alpha(\\phi)$ functions based on estimating \\eqref{eq:tuning2}.\n\nSome cases in which condition C3 holds are:\n$\\pi(\\theta)=g(\\theta)$ (ABC rejection sampling); $\\phi(\\theta,x)=(\\theta,\\ldots)$ (the decision statistics include $\\theta$); $\\phi(\\theta,x)=(\\tfrac{\\pi(\\theta)}{g(\\theta)}, \\ldots)$.\nThe optimal $\\alpha$ when this condition does not hold is described by Theorem 2 in Appendix C.\n\n\\subsubsection{Tuning $g$}\n\nIt is not clear what the optimal choice of $g(\\theta)$ is for lazy ABC. The examples later use typical choices from the ABC literature, but a better choice may improve performance further.\nThis paper's theory gives the following corollary on the optimal choice of $g(\\theta)$ for ABC-IS and RW-IS, which is of some general interest.\n\n\\begin{corollary} \\label{cor:ABC-IS}\nThe asymptotic efficiency of ABC-IS is maximised by $g(\\theta) \\propto \\pi(\\theta) \\left[\\dfrac{\\gamma(\\theta)}{\\bar{T}(\\theta)}\\right]^{1\/2}$,\nwhere $\\gamma(\\theta) = \\E(\\hat{L}_{\\text{ABC}} | \\theta)$.\nThis also holds for RW-IS with $\\gamma(\\theta) = \\E(\\hat{L}^2 | \\theta)$.\n\\end{corollary}\n\n\\begin{proof} See Appendix \\ref{sec:g}. \\end{proof}\n\n\n\\subsection{Methods} \\label{sec:tuning practice}\n\nThe theory above motivates choosing $\\alpha$ by estimating \\eqref{eq:tuning2}.  This section details a method to implement this approach.  Its effectiveness is discussed in Section \\ref{sec:discussion}.\n\nThe method is outlined here and more detail is given in Sections \\ref{sec:tbarest} to \\ref{sec:movingepsilon}.\nTuning begins with a pilot run of $N'$ iterations of ABC-IS,\nrecording intermediate simulation states and timings of interest.\nThis is used to estimate $\\gamma(\\phi)$ and $\\bar{T}_2(\\phi)$ for various choices of $X$ and $\\phi$, considering only $\\phi$ such that condition C3 holds.  Under each of these choices, $\\lambda$ is found by numerically maximising an estimate of efficiency.  The optimal choice of $X$ and $\\phi$ is then made.\nFollowing tuning, $N$ iterations of lazy ABC are performed (unless the estimated efficiency gains are judged inadequate, in which case ABC-IS can be used).\n\nA simpler variation on this method is possible if the set of possible $\\phi(\\theta,x)$ values is finite and small. A pilot run is performed as before. Then $\\alpha(\\phi)$ values are chosen by direct numerical optimisation of an estimate of the algorithm's efficiency.\n\n\n\n\\subsubsection{Estimation of $\\bar{T}_2(\\phi)$} \\label{sec:tbarest}\n\nIt may often suffice to treat $\\bar{T}_2(\\phi)$ as constant and estimate it as the mean CPU time of the continuation stage in the pilot run.  This is the case if knowledge of the simulation process shows the number of computational operations required is unaffected by $\\phi$, or if the pilot run shows $\\bar{T}_2(\\phi)$ varies little relative to $\\gamma(\\phi)$.  Alternatively, statistical methods such as regression can be used for estimation, which is straightforward when $\\phi$ is low dimensional.\n\n\\subsubsection{Estimation of $\\gamma(\\phi)$} \\label{sec:gammaest}\n\nEstimation of $\\gamma(\\phi)$ is more difficult.  Two approaches are suggested: the ``standard'' approach, producing $\\hat{\\gamma}^{(1)}$, attempts accurate estimation but typically involves strong assumptions; the ``conservative'' approach, producing $\\hat{\\gamma}^{(2)}$, sacrifices accuracy to improve robustness.  They are based on two equivalent expressions for $\\gamma(\\phi)$: $\\Pr(d(s(Y),s(y_{\\text{obs}})) \\leq \\epsilon | \\phi)$ and $\\E[\\hat{L}_{\\text{ABC}} | \\phi]$.  Examples of successful implementations of both approaches are given in Sections \\ref{sec:epidemics} and \\ref{sec:extremes}.\n\nThe standard approach is to model the relationship between $\\phi$ and $d(s(Y), s(y_{\\text{obs}}))$ and use this to estimate $\\Pr(d(s(Y),s(y_{\\text{obs}})) \\leq \\epsilon | \\phi)$.\nHowever a difficulty is that for most $\\phi$ values this involves extrapolating into the tails of the distribution of $d(s(Y),s(y_{\\text{obs}})) | \\phi$.  See Figure \\ref{fig:SErep}A for example.  This creates a danger of underestimating the optimal $\\alpha$ values and potentially producing very large importance sampling weights.\n(This can be avoided for simple summary statistics by first modelling $s(Y)|\\phi$; see Section \\ref{sec:epidemics} for example.)\n\n\nThe conservative approach is to select $\\epsilon_1$ following the pilot run such that a sufficiently large number of its simulations $y_{1:N'}$ satisfy $d(s(y_i),s(y_{\\text{obs}})) \\leq \\epsilon_1$.  Let $z_i$ be indicator variables denoting meeting this condition and model the relationship between $z_i$ and the simulated $\\phi_i$ values.  One method, used in the application later, is non-parametric logistic regression following \\cite{Wood:2011}.  This approach is effectively tuning the method based on an $\\epsilon$ value larger than that of interest.  This is an inefficient way to sample from the target of interest.  However, if tuning can be done well for the larger $\\epsilon$ value then this method is safe from producing any dangerously large importance weights, as discussed further in Section \\ref{sec:movingepsilon}.  Nonetheless, for $\\phi$ regions where there are no $z_i=1$ values the conservative estimate is still based on extrapolation and unlikely to be accurate.  Consequences of this are discussed in Section \\ref{sec:discussion}.\n\n\\subsubsection{Estimating efficiency} \\label{sec:effest}\n\nThe tuning method outlined above requires the use of $N'$ pilot run iterations to estimate the efficiency of lazy ABC under various choices of tuning details (in particular $X$, $\\phi$ and $\\alpha$).  It is sufficient to estimate $[\\E(W^2) \\E(T)]^{-1}$, as this equals efficiency up to a constant of proportionality.  This can be used to estimate efficiency relative to ABC-IS, which is a particularly interpretable form of the results as it shows the efficiency improvement of using lazy ABC.\n\nAssume that for a particular choice of tuning details the following are available for $1 \\leq i \\leq N'$: $t^{(1)}_i$ - initial simulation stage time; $t^{(2)}_i$ - continuation simulation stage time; $\\alpha_i$ - continuation probability; $\\hat{\\gamma}_i$ - estimate of $\\E(\\hat{L}_{\\text{ABC}} | \\phi_i)$; $u_i$ - ratio $\\pi(\\theta)\/g(\\theta)$.  An estimate up to proportionality of efficiency is then $[\\widehat{W^2} \\hat{T}]^{-1}$ where \n$\\widehat{W^2} = N'^{-1} \\sum_{i=1}^{N'} u_i^2 \\hat{\\gamma}_i\/\\alpha_i$ and\n$\\hat{T} = \\sum_{i=1}^{N'} t^{(1)}_i + \\sum_{i=1}^{N'} \\alpha_i t^{(2)}_i$.\nAn estimate of efficiency of ABC-IS is formed by taking $\\alpha \\equiv 1$.  Note that this often overestimates $\\E(T)$ as interrupting the simulation to record intermediate states and timings reduces algorithm efficiency.\nA more precise estimate would be possible using further pilot simulations without this interruption.\n\n\\subsubsection{Combining pilot and main run output} \\label{sec:combining}\n\nTo make efficient use of the pilot run, it can be used in the final output as well as for tuning.  This is done by appending the pilot sequence of $(\\theta,w)$ pairs to that from the main algorithm.  Loosely speaking, since each individual sequence targets the same distribution, so does the combined sequence.  More technically, it is straightforward to see that ABC versions of relations \\eqref{eq:IS1} and \\eqref{eq:IS2} are roughly true for the combined sequence when $N$ and $N'$ are large, and are exactly true as $N \\to \\infty$ regardless of $N'$. Also note that on appending the sequences, gains in efficiency are possible by multiplying the weights of one sequence by a constant, but this is not implemented here as little improvement was observed in the application later.\n\n\\subsubsection{Choice of $\\epsilon$} \\label{sec:movingepsilon}\n\nIn ABC-IS, an appropriate value $\\epsilon$ is often unknown a priori and is instead chosen based on the simulated $d(s(Y),s(y_{\\text{obs}}))$ values.  For lazy ABC in this situation one can use the pilot run to select a preliminary conservative choice of $\\epsilon_1$ as in Section \\ref{sec:gammaest}\nand perform lazy ABC with $\\epsilon=\\epsilon_1$.  Alternative values of $\\epsilon$ can then be investigated by updating the realised $\\hat{L}_{\\text{ABC}}$ values in the weight calculations.  For $\\epsilon < \\epsilon_1$ this simply reduces the number of non-zero weights.  However $\\epsilon \\gg \\epsilon_1$ is not recommended as this may introduce large weights and destabilise the importance sampling approximation.\n\n\\section{Example: infectious diseases} \\label{sec:epidemics}\n\nThis section illustrates a basic implementation of lazy ABC and the tuning approach of Section \\ref{sec:tuning practice} on a simplified version of the standard Markovian SIR infectious disease model.\nWhile ABC is not necessary for inference under this model \\citep[see][]{Andersson:2000}, it is used for more complex variants \\citep{McKinley:2009,Brooks:2014}.\n\nSuppose the random variables $S(t), I(t), R(t)$ follow a discrete time Markov chain.\nThese represent the population size which is susceptible, infectious or recovered after $t$ transitions.\nTwo transitions are possible.\nLet $M=S(0)+I(0)+R(0)$ be the total population size.\nWith probability $k \\tfrac{R_0}{M} S(t) I(t)$, an infection occurs giving $S(t+1)=S(t) - 1$, $I(t+1)=I(t)+1$ and $R(t+1)=R(t)$.\nWith probability $k I(t)$, a recovery occurs giving $S(t+1)=S(t)$, $I(t+1)=I(t)-1$ and $R(t+1)=R(t)+1$.\nThe constant $k$ is chosen so that the probabilities sum to 1.\nWhen $I(t)=0$ the chain terminates\nand a simple random sample of size $100$ is taken from the population.\nThe observed data $y_{\\text{obs}}$ is the number of these which are recovered.\nThe parameter of interest is the \\emph{basic reproduction number} $R_0$,\nwhich is given a prior of $\\text{Gamma}(3,1)$.\nIt is assumed that $M=10^5$, $I(0)=10^3$ and $R(0)=0$, giving $S(0)=9.9 \\times 10^4$.\n\nStandard ABC can be implemented here simply by taking $s(y)=y$ and $d(s_1,s_2)=|s_1-s_2|$.\nStandard ABC rejection sampling was performed on a simulated dataset with $y_{obs}=73$ using $N=10^4$ iterations and $\\epsilon=1$.\nThese settings are also used for the three lazy ABC analyses that follow.\n\nFirst a simple version of lazy ABC with ad-hoc tuning choices is presented.\nThis considers stopping at $t=1000$, so that $X=(S(1000),I(1000),R(1000))$.\nThe decision statistic $\\phi(\\theta,X)$ is $I(1000)$.\nIt seems reasonable to stop if there is evidence that the number of infectious is not growing.\nSo let $\\alpha(\\phi) = 0.1$ if $\\phi \\leq 1000$, and $\\alpha(\\phi)=1$ otherwise.\n\nThe second lazy ABC analysis uses the standard tuning method of Section \\ref{sec:tuning practice}, with the same choice of $X$ and $\\phi$ as above.\nA pilot run of 1000 standard ABC simulations was performed.\nFigures \\ref{fig:SIR}A and B plot pilot run realisations of $\\phi$ against $y$, the observation, and the continuation stage simulation time.\nFrom this $\\bar{T}_2(\\phi)$, expected remaining simulation time given $\\phi$, was estimated by fitting a non-parametric regression, using a log link to enforce positivity.\nAlso $\\hat{p}(\\phi)$, the expectation given $\\phi$ of the proportion of the population who are recovered when the Markov chain terminates, was estimated by fitting a non-parametric binomial GLM.\nThis GLM model fits the pilot data well as illustrated by Figure \\ref{fig:SIR}A,\nwhich shows the resulting expected number recovered in the final subsample and a 95\\% confidence interval.\nAll non-parametric regressions were fitted using the approach of \\cite{Wood:2011}.\nAn estimate of the probability of ABC acceptance, $\\hat{\\gamma}(\\phi)$ can be directly calculated from $\\hat{p}(\\phi)$ and is shown in Figure \\ref{fig:SIR}C.\nIt is now possible to estimate $\\alpha(\\phi)$ for a given value of $\\lambda$ from \\eqref{eq:tuning2}.\nFor each $\\lambda$ the efficiency estimate of Section \\ref{sec:effest} can be calculated.\nNumerical optimisation shows this is maximised for $\\lambda=3.45$.\nFigure \\ref{fig:SIR}D shows the estimated optimal $\\alpha$ function.\n\nFinally lazy ABC using the conservative tuning approach is considered.\nThis proceeds as for the standard tuning approach just described, except for the estimation of $\\gamma(\\phi)$.\nA variable $z$ is introduced to indicate whether the pilot run data has ABC distance less than or equal to $\\epsilon_1=3$.\nThis variable equals 1 for 50 cases and 0 for the remainder.\nThen $\\gamma(\\phi)$ is estimated by fitting a non-parametric logistic regression of $z$ on $\\phi$, with the results are shown in Figure \\ref{fig:SIR}C.\nThis estimate is more conservative than that above in that it always assigns a higher probability of ABC acceptance.\nNumerical optimisation chooses $\\lambda=2.54$ and the resulting $\\alpha(\\phi)$ function is shown in Figure \\ref{fig:SIR}D.\n\nTable \\ref{tab:SIR} shows the results of the analyses described above.\nNote that each analysis used the same sequence of random seeds so that their $(\\theta, X, Y)$ simulations were the same.\nEach analysis accepted the same 194 parameter values, except for lazy ABC with conservative tuning which accepted a subset of 177 of these.\nUnder lazy ABC with ad-hoc tuning all weights were 1.\nWith standard tuning 189 weights equalled 1 with the others below 3.\nWith conservative tuning 131 weights equalled 1, with the others below 4.\nEstimates of posterior quantities from all methods were very similar.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics{SIRout.pdf}\n\\end{center}\n\\caption{\\label{fig:SIR}\nTuning details of a simulation study applying lazy ABC to an SIR epidemic model.\n\\emph{Panel A} Pilot run realisation of $I(1000)$ (the decision statistic: number of infectious at $t=1000$) and $y$ (number recovered in subsample).\nThe solid line is the prediction under a non-parametric logistic regression and the dashed lines are corresponding 0.025 and 0.975 quantiles.\nThe dotted line shows the observed data.\n\\emph{Panel B} Pilot run realisations of $I(1000)$ and $T_2$ (continuation stage simulation time, in seconds).\nThe solid line is a fitted non-parametric regression.\n\\emph{Panel C} Estimated $\\gamma(\\phi)$ functions -- probability of eventual ABC acceptance -- under standard (solid) and conservative (dotted) tuning. Standard tuning derives this curve from panel A. Conservative tuning fits a logistic regression to the indicator variable represented by vertical marks. This indicates which pilot runs result in an ABC distance of 3 or less.\n\\emph{Panel D} $\\alpha$ functions (continuation probabilities) under ad-hoc (dashed), standard (solid) and conservative (dotted) tuning. The latter two cases are derived from the pilot run as described in Section \\ref{sec:tuning practice}.\nThe marks on the horizontal axis indicate the ABC simulations which were accepted.\n}\n\\end{figure*}\n\n\\begin{table*}[htp]\n\\centering\n\\fbox{\n\\begin{tabular}{ccccc|cc}\n\\multirow{2}{*}{ABC method} & \\multirow{2}{*}{Tuning} & \\multirow{2}{*}{Time (s)} & \\multirow{2}{*}{ESS} & Relative & \\multicolumn{2}{c}{Estimated posterior} \\\\\n           &              &          &       & efficiency & Mean & Standard deviation \\\\\n\\hline\nStandard   & -            & 18124    & 194   & 1    & 1.803 & 0.1267 \\\\\nLazy       & Ad-hoc       & 17848    & 194   & 1.02 & 1.803 & 0.1267 \\\\\nLazy       & Standard     & 5113     & 192   & 3.51 & 1.804 & 0.1276 \\\\\nLazy       & Conservative & 3318     & 167   & 4.70 & 1.796 & 0.1212\n\\end{tabular}}\n\\caption{\\label{tab:SIR}Simulation study on a SIR epidemic model. Four ABC analyses were performed on the same observations, sharing the same random seeds for their simulations. All used the same choice of $N=10^4$ and $\\epsilon=1$. Iterations were run in parallel and computation times are summed over all cores used. Efficiency (ESS\/CPU time) relative to standard ABC is shown. The pilot run required for tuning the last two rows tuning took a further 2016 seconds, with the remaining tuning calculations taking less than 2 seconds. For simplicity results are for lazy ABC output only, pilot run simulations have not been appended as in Section \\ref{sec:combining}.}\n\\end{table*}\n\nThis example shows that it is straightforward to implement lazy ABC using ad-hoc tuning.\nThe time savings here are small as the $\\alpha$ function is quite conservative, only allowing early stopping when the epidemic is already dying out at time $t=1000$.\nA more effective $\\alpha$ could be sought using results on the typical behaviour of this model \\citep[see][]{Andersson:2000}.\nThe above instead considers the tuning method of Section \\ref{sec:tuning practice} and shows it is simple to implement here and produces significant time savings.\nConservative tuning does better, but this may reflect variability of results for a relatively small number of iterations.\n\n\\section{Example: spatial extremes} \\label{sec:extremes}\n\nThis section uses lazy ABC in a computationally demanding application of ABC to spatial extremes introduced by \\cite{Erhardt:2012}. As well as illustrating the method for a complicated application, simulation studies are performed to investigate its efficiency.\n\n\\subsection{Background}\n\nThe observation $y_{t,d}$ represents the maximum measurement (e.g.~of rainfall) during year $t$ at location $x_d \\in \\mathbb{R}^2$.  There are $D$ locations and $M$ years.  The data are treated as $M$ independent replications of a spatial distribution.  Several models based on extreme value theory have been proposed, and \\citeauthor{Erhardt:2012} concentrate on the \\emph{Schlather process} \\citep{Schlather:2002}.\n\nThe details of the Schlather process follow, together with the method \\citeauthor{Erhardt:2012} used to calculate ABC summary statistics. The full mathematical details are not essential to understanding the implementation of lazy ABC. Instead, what is most important is the order of operations required to simulate a dataset and summary statistics. This is summarised as Algorithm \\ref{alg:extremes sim} for later reference.\n\nThe Schlather process is based on independent identically distributed mean zero stationary Gaussian processes $U_i(x)$ where $i=1,2,\\ldots$ (indexing an infinite number of Gaussian processes) and $x \\in \\mathbb{R}^2$ (representing location). \n\nThe correlation between locations $x$ and $x'$ is given by the correlation function $\\rho(h)$ where $h=||x-x'||_2$.  Let $s_i$ be draws from a Poisson process on $s \\in (0,\\infty)$ with intensity $\\mu^{-1} s^{-2}$, where $\\mu=\\E[\\max(0,U(x))]$.  Then the Schlather process is\n\\begin{equation} \\label{eq:Schlather}\nY(x) = \\max_i s_i \\max(0, U_i(x)).\n\\end{equation}\n\\citeauthor{Erhardt:2012} focus on the Whittle-Mat\\'ern correlation function with zero nugget\n\\begin{equation} \\label{eq:Whittle-Matern}\n\\rho(h; c, \\nu) = \\frac{2^{1-v}}{\\Gamma(\\nu)} \\left(\\frac{h}{c}\\right)^\\nu K_\\nu \\left(\\frac{h}{c}\\right),\n\\end{equation}\nwhere $\\Gamma$ is the gamma function and $K_\\nu$ is the modified Bessel function of the third kind with order $\\nu$.  This has two parameters: range $c>0$ and smoothness $\\nu > 0$.\n\nA density function for the Schlather process is not available for $D>2$, making inference difficult.  \\cite{Schlather:2002} provides a near-exact algorithm to simulate from the process based on only a finite number of copies of $U_i$,  motivating the use of ABC by \\citeauthor{Erhardt:2012}.\nThey applied ABC rejection and importance sampling with a uniform prior on $[0,10]^2$ and investigated several choices of summary statistics.  The analysis here focuses on the choice they find most successful, based on \\emph{tripletwise extremal coefficient estimators}.  Given a triple of 3 locations, $i,j,k$, this estimator is\n\\begin{equation} \\label{eq:extremalcoeff}\n\\hat{\\theta}_{ijk} = \\frac{M}{\\sum_{t=1}^{M}} 1\/\\max(y_{t,i}, y_{t,j}, y_{t,k}).\n\\end{equation}\nThere are $O(D^3)$ such summaries, so \\citeauthor{Erhardt:2012} calculate a vector $m$ of mean values within $100$ clusters of triples, and use these as summary statistics.  Their clustering process\nfinds triples of similar shapes, ignoring differences of location and rotation.  The ABC distance function between two vectors $m_1$ and $m_2$ of cluster means is\n\\begin{equation} \\label{eq:sedist}\nd(m_1, m_2) = \\sum_{i=1}^{100} |m_{1i} - m_{2i}|.\n\\end{equation}\nAlthough applying dimension reduction techniques to such high dimensional summaries has been shown to often improve ABC results \\citep{Fearnhead:2012}, this is not investigated here as the aim is to investigate the efficiency improvements of lazy ABC.\n\n\nImplementing the analysis below used the R packages ``SpatialExtremes'' \\citep{Ribatet:2013} to simulate from the Schlather process and ``ABCExtremes'' to implement some details of the approach of \\citeauthor{Erhardt:2012}.\n\n\\begin{algorithm}[htp]\n\n\\rule{\\textwidth}{0.2mm} \\\\\n{\\bf Input:}\n\\begin{itemize}\n\\item Parameters: range $c>0$ and smoothness $\\nu>0$.\n\\item Locations $x_1,\\ldots,x_D$.\n\\item Number of years $M$.\n\\end{itemize}\n{\\bf Data simulation:}\n\\begin{enumerate}\n\\item[1] Calculate $D \\times D$ covariance matrix $\\Sigma(c,\\nu)$ using \\eqref{eq:Whittle-Matern}.\n\\item[2] Repeat the following steps for $t=1,\\ldots,M$.\n\\begin{enumerate}\n\\item[a] Sample values $s_1, s_2, \\ldots, s_q$ from a Poisson process (see text for intensity).\n\\item[b] For each $1\\leq i \\leq q$, sample $u_i(x_1), u_i(x_2), \\ldots, u_i(x_D)$ from $N(0,\\Sigma(c,\\nu))$.\n\\item[c] Calculate Schlather process realisations for year $t$ by \\eqref{eq:Schlather}.\n\\end{enumerate}\n\\end{enumerate}\n{\\bf Summary statistic calculation:}\n\\begin{enumerate}\n\\item[1] Calculate extremal coefficient estimate $\\hat{\\theta}_{ijk}$ for all triples of locations $ijk$ using \\eqref{eq:extremalcoeff}.\n\\item[2] Calculate mean $\\hat{\\theta}_{ijk}$ values for each cluster of triples.\n\\end{enumerate}\n{\\bf Output:}\n\\begin{itemize}\n\\item Summary statistic vector comprising the cluster means.\n\\end{itemize}\n\\rule{\\textwidth}{0.2mm}\n\\caption{Overview of simulation of Schlather process data and summary statistics. Full details of the steps are given in the text. \\label{alg:extremes sim}}\n\\end{algorithm}\n\n\\subsection{Methods} \\label{sec:simmethods}\n\nExploratory investigation of ABC code with $D=20$ and $M=100$ showed that the majority of time was spent simulating the data (7.1ms\/iteration) and calculating extremal coefficient estimates (17.9ms\/iteration), with the remaining steps being brief (3.1ms\/iteration).  The time costs of the first two of these scaled with $D$ as roughly proportional to $D$ and $D^3$ respectively, so the latter is expected to dominate for large $D$.  Furthermore, interrupting and then resuming operations during the calculation of extremal coefficients is much simpler to implement than during simulation of data.  Therefore the initial simulation stage of the lazy ABC analysis was chosen to be simulating the data at all locations, and extremal coefficient estimates at a subset of locations $L$.  The continuation simulation stage was to calculate the remaining extremal coefficient estimates.\n\nThe decision statistic $\\hat{d}$ was constructed as follows.  Let $m_{1i}$ be the $i$th cluster mean for the observed data.  Let $\\hat{m}_{2i}$ be the $i$th cluster mean for the simulated data using only extremal coefficient estimates available at the initial simulation stage, and $B$ be the set of clusters for which any such estimates are available.  Then define $\\hat{d} = \\sum_{i \\in B} |m_{1i} - \\hat{m}_{2i}|$.  This is an estimate of the ABC distance $d$ \\eqref{eq:sedist}.  It could be improved by estimating typical $\\hat{m}_{2i}$ values for $i \\not \\in B$ but including such constant terms has no effect on the analysis below.\n\nIt was assumed that $\\bar{T}_2(\\hat{d})$ is constant as, given $D$, $L$ and $M$, the continuation stage always involves the same number of calculations.  The value was estimated by the mean CPU time for this stage in the pilot run.  \nAnalyses were performed using both the standard and conservative $\\gamma$ estimators.\nTo calculate $\\hat{\\gamma}_1(\\hat{d})$ from pilot run output, the relationship between $\\hat{d}$ and $d$ was modelled statistically.  Exploratory analysis showed that there was a roughly linear relationship, but for some choices of $L$ this was heteroskedastic (see Figure \\ref{fig:SErep}A).  Furthermore, for small $\\hat{d}$ the distribution of $d | \\hat{d}$ was skewed.  So $\\Pr(d \\leq \\epsilon | \\hat{d})$ was estimated based on a linear regression of $d$ on $\\hat{d}$ with a Box-Cox transformation, using only simulations with nearby values of $\\hat{d}$.  This was done for several $\\hat{d}$ values and interpolated estimates elsewhere formed  $\\hat{\\gamma}_1(\\hat{d})$.\nFor the importance sampling case, $\\log u$ was also included in each regression giving a number of functions mapping $u$ to estimates of $\\Pr(d \\leq \\epsilon | \\hat{d}, u)$ for various $\\hat{d}$ values, which were used for interpolation.\nCalculation of $\\hat{\\gamma}_2$ was as described in Section \\ref{sec:gammaest}, taking $\\epsilon_1$ to give 100 acceptances in the pilot run.\nGiven estimates of $\\bar{T}_2$ and $\\gamma$, tuning was performed as described in Section \\ref{sec:tuning practice}, with optimisation over possible choices of $L$ by backwards selection.\n\nThree simulation studies were performed.\nThe first replicated the rejection sampling analysis of \\citeauthor{Erhardt:2012} on several simulated datasets.  These used $D=20, M=100$ and true parameter values shown in Table \\ref{tab:SErep}.  Each dataset used a different set of observation locations with integer coordinates sampled from $[0,10]^2$.  The first analysis was a replication of the standard ABC analysis, using $\\epsilon$ values corresponding to 200 acceptances.  Then lazy ABC was performed on the same datasets under each method of estimating $\\gamma$.  To compare the methods fairly, lazy ABC used the same $\\epsilon$ value as standard ABC and reused its random seeds so that the sequence of $(\\theta, X, Y)$ realisations is also the same.\n\nThe second simulation study investigated rejection sampling for a single larger simulated dataset with $D=35$, $c=0.5$ and $\\nu=1$.  Locations were chosen as before.  As in a real application $\\epsilon$ was not assumed to be known in advance and the approach of Section \\ref{sec:movingepsilon} was used to select this post-hoc.  A complication for this dataset was that the simulation of Gaussian processes was difficult when both parameters were large: the default ``direct method'', based on Choleski decomposition, sometimes produced numerical errors.  Simulation was possible via the turning bands method (TBM) but much slower (roughly 150 times the CPU time).  A two stage simulation method was implemented.  First the direct method was attempted and if this failed TBM was used.  To save time lazy ABC was implemented with multiple stopping decisions, the first taking place after attempting the direct method.  This has a binary decision statistic indicating success or failure.  The second stopping decision is as described earlier.  Tuning was performed as described in Appendix \\ref{sec:onecont}, using $\\hat{\\gamma}_1$ fitted as described above by either the standard or conservative tuning method.  The standard method used $\\epsilon_1$ to give 30 acceptances in the pilot run.  As before all analyses reused the same random seeds.\n\nFinally an importance sampling analysis was performed on the larger dataset.  A sample of $10^4$ log parameter values was taken from simulations of the preceding standard ABC analysis with distances below the 0.3 quantile.  A Gaussian mixture distribution was constructed with locations given by this sample and variances equal to twice the empirical variance of the sample.  After truncation to the prior support, this was used to give $g(\\theta)$, where $\\theta$ now represents the log parameters.  This choice follows the suggestions of \\cite{Beaumont:2009},\nnoting that using the log scale produced a better fit to the sample and that the subsample was used to avoid slow density calculations.  The preceding $D=35$ analysis was then repeated.\nAs discussed in Section \\ref{sec:tuning practice}, $u=\\pi(\\theta)\/g(\\theta)$ was included as a decision statistic.  Estimation of $\\gamma$ and $\\bar{T}_2$ was performed as before with $u$ included in the $\\gamma$ estimate as described earlier.\n\nAll ABC analyses performed $10^6$ total iterations.  For lazy ABC $10^4$ of these comprised the pilot run.\n\n\\subsection{Results}\n\nFigure \\ref{fig:SErep} illustrates some details of tuning for one case of the $D=20$ study.  The results are shown in Table \\ref{tab:SErep}.  For all datasets lazy ABC is roughly 4 times more efficient under the standard tuning method and 3 times under the conservative method.  \nEfficiency gains for conservative tuning are slightly less than estimated.  This is because the estimate is made for a choice of $\\epsilon_1$ larger than the final $\\epsilon$.\nThe mean weights were also investigated, as these are useful in model selection as an estimate of $\\pi(y_{\\text{obs}})$.  All lazy ABC estimates differed from the standard ABC estimate by no more than 4\\%.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics{ESrepout.pdf}\n\\end{center}\n\\caption{\\label{fig:SErep} Details of tuning lazy ABC in a simulation study on spatial extremes corresponding to the first row of Table \\ref{tab:SErep}.  Panels A-C concentrate on the standard tuning approach.  \\emph{Panel A} Pilot run values of $\\hat{d}$ (estimated distance) and $d$ (actual distance).  The dashed line shows the value of $\\epsilon$.  \\emph{Panel B} Estimates of efficiency (ESS\/time) for different values of $\\lambda$ (parameter of \\eqref{eq:tuning2} which must be determined numerically).  The dashed line shows the realised efficiency (ESS\/time of the final lazy ABC algorithm).  \\emph{Panel C} Estimated efficiency for the best choices of $L$ (subset of locations from which early stopping is considered) of various lengths output by backwards selection.  The dashed line shows the realised efficiency.  \\emph{Panel D} Values of $\\hat{d}$ and $\\alpha$ (continuation probability) from non-pilot simulations under standard (solid line) and conservative (dashed line) tuning.  The marks on the horizontal axis indicate the simulations which resulted in positive weights.  (For this panel conservative tuning was performed using $L$ as selected by standard tuning.)}\n\\end{figure*}\n\n\\begin{table*}[pht]\n\\centering\n\\fbox{ \\footnotesize\n\\begin{tabular}{cc|c|ccc|cc}\n      &        & Standard       & \\multicolumn{3}{|c|}{Lazy} & \\multicolumn{2}{|c}{Relative efficiency}                \\\\\nRange & Smooth & Time ($10^3$s) & Time ($10^3$s)             & Sample size & ESS           & Estimated   & Actual      \\\\ \n  \\hline                                        \n 0.5  & 1      & 32.0           & 8.0  (11.6)                & 196 (199)   & 196.0 (198.7) & 4.08 (3.28) & 4.00 (2.79) \\\\ \n 1    & 1      & 31.3           & 7.3  (9.8)                 & 200 (200)   & 199.9 (200.0) & 4.34 (4.31) & 4.35 (3.25) \\\\ \n 1    & 3      & 31.3           & 8.2  (11.2)                & 194 (198)   & 182.5 (196.5) & 3.77 (3.43) & 3.51 (2.79) \\\\ \n 3    & 1      & 31.2           & 7.7  (11.1)                & 194 (200)   & 189.9 (200.0) & 4.18 (3.56) & 3.89 (2.86) \\\\ \n 3    & 3      & 31.2           & 7.4  (11.0)                & 192 (199)   & 175.8 (199.0) & 4.43 (3.65) & 3.79 (2.87) \\\\ \n 5    & 3      & 31.3           & 8.3  (11.1)                & 200 (200)   & 200.0 (200.0) & 3.73 (3.49) & 3.85 (2.87)\n\\end{tabular}}\n\\caption{\\label{tab:SErep}Simulation study on spatial extremes replicating \\cite{Erhardt:2012}.  Each row represents the analysis of a simulated dataset under the given values of range $c$ and smoothness $\\nu$ parameters (used in \\eqref{eq:Whittle-Matern}). In each analysis a choice of $\\epsilon$ was made under standard ABC so that the accepted sample size (and therefore ESS) was 200, and the same value was used for lazy ABC.  Lazy ABC figures are shown for both the standard $\\hat{\\gamma}$ estimate and, in brackets, the conservative estimate.  The lazy ABC output includes the pilot run as described in Section \\ref{sec:combining}, and also includes the tuning time (roughly 120 seconds for the standard approach and 210 for the conservative).  Iterations were run in parallel and computation times are summed over all cores used.  For all datasets efficiency (ESS\/time) to 1 significant figure was 0.006 for standard ABC and 0.02 or 0.03 under either approach to lazy ABC.}\n\\end{table*}\n\nTable \\ref{tab:SEbig} shows results for the $D=35$ dataset.  In the initial rejection sampling analysis lazy ABC improved efficiency by roughly 8 times.  For importance sampling the improvement factor is 2, showing that lazy ABC still improves efficiency, although this is harder when $g$ concentrates on plausible choices of $\\theta$.  For example, standard ABC now spends negligible time on TBM simulations.  As before lazy ABC estimates of $\\pi(y_{\\text{obs}})$ differed from those of standard ABC by no more than 4\\%.  In both cases a post-hoc selection of $\\epsilon$ has been used successfully.\n\nTable \\ref{tab:SEbig} shows that under rejection sampling the standard tuning ESS is considerably smaller than the sample size.  This is due to two simulations which are given importance weights of 10.  These have $\\hat{\\gamma}$ values of roughly $10^{-4}$ which appears to be an underestimate.  Conservative tuning avoids large importance weights to give an ESS of roughly 200 and improves the relative efficiency for the same final choice of $\\epsilon$.  For importance sampling conservative tuning also performs better.  The reason here is not obvious but may be a better final selection of $L$.\n\n\\begin{table*}[pht]\n\\centering\n\\fbox{ \\footnotesize\n\\begin{tabular}{l|c|ccc|ccc|c}\n                      &            & \\multicolumn{3}{|c|}{Standard} & \\multicolumn{3}{|c|}{Lazy} & Relative                                                        \\\\\n                      & $\\epsilon$ & Time ($10^3$s)                 & Sample size                & ESS  & Time ($10^3$s)        & Sample size & ESS   & efficiency \\\\\n\\hline                                        \nRS standard           & 2.61       & 241.4                          & 210                        & 210  & 22.8                  & 200         & 139.6 & 7.0        \\\\\nRS conservative       & 2.61       & 241.4                          & 207                        & 207  & 25.5                  & 200         & 197.7 & 9.0        \\\\\n\\hline\nIS standard           & 2.33       & 136.4                          & 209                        & 168  & 61.9                  & 200         & 165   & 2.2        \\\\\nIS conservative       & 2.33       & 136.4                          & 209                        & 168  & 51.0                  & 200         & 162   & 2.6        \\\\ \n\n\\end{tabular}}\n\\caption{\\label{tab:SEbig}Simulation study on a spatial extremes dataset with $D=35$.  Results are shown for rejection and importance sampling with standard and conservative tuning.  The rejection sampling output was used to create the importance density.  The final choice of $\\epsilon$ is shown.  For IS the two $\\epsilon$ values are equal but there is a small difference for RS.   The lazy ABC output includes the pilot run and the tuning time.}\n\\end{table*}\n\n\\section{Discussion} \\label{sec:discussion}\n\nThis paper has introduced lazy ABC, a method to speed up inference by ABC importance sampling without introducing further approximations.  The approach is to abandon some unpromising simulations before they are complete.  By using a probabilistic stopping rule and weighting the accepted simulations accordingly, the algorithm targets exactly the same distribution as standard ABC, in the sense that Monte Carlo estimates of functions $h(\\theta)$ and of the model evidence converge to unchanged values.\n\nResults have been provided on the optimal tuning of the lazy ABC stopping rule and used to motivate a practical tuning method.  This has been demonstrated for a simple epidemiological example and a computationally challenging application where it has produced improvements in efficiency (ESS\/CPU time) over standard ABC of up to 8 times.\n\nThe tuning method is based on estimating the optimal choice of $\\alpha(\\phi)$, \\eqref{eq:tuning2}.  The most difficult part was estimating $\\gamma(\\phi) = \\Pr \\left(d(s(Y), s(y_{\\text{obs}})) \\leq \\epsilon | \\phi \\right)$ from pilot run data.  Two approaches to this were described, a standard approach of direct estimation and a conservative approach of estimation using a larger $\\epsilon$ value than is of interest for ABC.  The latter approach improves robustness and make estimation simpler at the cost of some inefficiency.  Both approaches performed well in the simulation studies but some improvements are desirable.  Firstly, estimation of $\\gamma(\\phi)$ often involves extrapolation which may produce inaccurate results.  Secondly, several choices by the user are required, especially for the standard approach.  A more automated approach would be useful for lazy versions of ABC SMC algorithms, where a new choice of $\\alpha$ would be needed for each $\\epsilon$ value,  or alternatively for lazy ABC algorithms which adapt $\\alpha$ as more simulations become available.  It would be of interest to find suboptimal but robust choices of $\\alpha$ addressing these issues.\n\nA related point is that lazy ABC can be generalised to allow a non-uniform ABC kernel.\nThis may allow alternative approaches to tuning.\n\nIn some situations likelihood-based inference is possible, but calculating the likelihood or an unbiased estimate is expensive.\nGeneralising lazy ABC ideas to this situation is possible.\nAppendix \\ref{sec:LIS} describes this \\emph{lazy importance sampling} algorithm and shows that the theoretical results of the paper carry over to it.\nIt also discusses why practical application seems more challenging than for lazy ABC.\nNonetheless this is an interesting topic for future research.\n\nLazy ABC with multiple stopping decisions is another extension to the framework of the main paper and is described in Appendix \\ref{sec:multistop},\nwith an example of implementation in Section \\ref{sec:extremes}.\nA tuning method is given when the decision statistics for all stopping decisions are discrete, and also some cases where one decision statistic is continuous.  For more complex cases tuning results are not available.  For now it is recommended to discretise most decision statistics to avoid this difficulty.\n\n\nThis paper has concentrated on importance sampling, which is widely used by ABC practitioners, but the lazy ABC approach can be extended to ABC versions of MCMC and SMC, which are more efficient algorithms.  The tuning results are applicable to SMC algorithms, but further practical methods are needed, as mentioned above.  Further theory on optimal tuning is necessary for MCMC, although good performance may be possible with ad-hoc tuning.\nIt would also be of interest\nto design algorithms in which the initial simulation stages can be resampled and continued many times.\n\n\\paragraph{Acknowledgements}\nThanks to Chris Sherlock, Richard Everitt, Scott Sisson, Christian Robert and two anonymous referees for helpful suggestions, and Robert Erhardt for advice on the spatial extremes example.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nQuantum cosmology \\cite{Halliwell:1990uy,Bojowald:2015iga} aims at\nunderstanding how gravitational fields describing cosmological setups,\nusually treated as purely classical backgrounds \\cite{PeterUzan2009},\nmay be affected by quantization. In~particular, the most important issue\nnot addressed by classical general relativity, i.e., the singularity from\nwhich our Universe ensues, could be tackled by imposing physically\nrelevant boundary conditions on the wave function.\n\nThe Universe being by definition unique, and quantum measurements being\nunderstood by means of ensemble averages, i.e., repeated experiments, the\nmeaning of the wave function of the Universe seems rather unclear. There\nexists however a formulation of quantum mechanics, originally developed\nby de Broglie \\cite{deBroglie:1925cca} and Bohm\n\\cite{Bohm:1951xw,Bohm:1951xx} and based on trajectories\n\\cite{Holland:1993ee} that, as it happens, is easily applicable to\ncosmology \\cite{Pinto-Neto:2013toa}. It is in this framework that one\ncan assign actual values at each instant of time to the scale factor\n(the quantum trajectory) \\cite{AcaciodeBarros:1997gy,Peter:2016kan} and\neven address the question of time \\cite{AcaciodeBarros:1998nb}.\n\nIn the following, I briefly recap how gravitation may be quantized {\\sl\n\\`a la} Wheeler De Witt and how does the restriction to\nFriedmann-Lema\\^itre-Roberston-Walker (FLRW) minisuperspace\nprovides a time-dependent Schr\\\"odinger-like equation when a perfect\nfluid is considered to be the source of Einstein equations. The trajectory\napproach then permits to derive a fully quantum time-dependent scale\nfactor whose properties are examined in detail.\n\n\\section{General Setup}\n\\vspace{-6pt}\n\n\n\n\\subsection{Classical Hamiltonian General Relativity}\n\nSince the purpose is to quantize general relativity (GR)\n\\cite{Kiefer:2007ria}, one starts from the usual Einstein-Hilbert action\non a compact space $\\mathcal{M}$ with boundary $\\partial \\mathcal{M}$,\nincluding a possible cosmological constant $\\Lambda$,\n\\begin{equation}\n\\mathcal{S} = \\frac{1}{16\\piG_{_\\mathrm{N}}} \\left[ \\int_\\mathcal{M} \\!\\!\\! \\sqrt{-g}\n\\ \\left( R -2\\Lambda \\right)\\mathrm{d}^4 x + 2 \\int_{\\partial \\mathcal{M}} \\!\\!\n\\!\\sqrt{h} K^i_{\\ i} \\mathrm{d}^3 x \\right] + \\mathcal{S}_\\mathrm{matter} \\left[\n\\Phi\\left(x\\right)\\right],\n\\label{EH}\n\\end{equation}\nwhere the Ricci scalar $R$ is coupled to matter fields symbolically\nnamed $\\Phi$. Figure \\ref{ADM} shows the usual 3+1 split of spacetime\nwhen the metric takes the form\n\\begin{equation}\n\\mathrm{d} s^2 = g_{\\mu\\nu} \\mathrm{d} x^\\mu \\mathrm{d} x^\\nu = - N^2 \\mathrm{d} t^2\n+ h_{ij} \\left( \\mathrm{d} x^i + N^i \\mathrm{d} t\\right) \\left( \\mathrm{d} x^j + N^j \\mathrm{d} t\\right).\n\\label{Metric}\n\\end{equation}\n\nIn \\eqref{EH}, the extrinsic curvature of each leaf $\\Sigma_t$ is given\nby\n\\begin{equation}\nK_{ij} = - \\nabla^{(h)}_j n_i = \\frac{1}{2 N} \\left( \\nabla^{(h)}_j N_i +\n\\nabla^{(h)}_i N_j - \\frac{\\partial h_{ij}}{\\partial t}\\right),\n\\label{Kij}\n\\end{equation}\nwhere $\\nabla^{(h)}$ is the covariant derivative associated with\nthe intrinsic metric $h_{ij}$. From \n\\begin{equation}\n\\mathcal{S} = \\int L\\mathrm{d} t = \\frac{1}{16\\piG_{_\\mathrm{N}}} \\int \\mathrm{d} t \\,\n\\left[\\int \\mathrm{d}^3 x\\, N\\sqrt{h} \\left(\nK_{ij} K^{ij} -K^2 +^3\\!\\! R-2\\Lambda \\right) + L_\\mathrm{matter} \\right],\n\\label{SLdt}\n\\end{equation}\none derives the canonical momenta\n\\begin{equation}\n\\pi^{ij} \\equiv \\frac{\\delta L}{\\delta \\dot{h}_{ij}} = -\\frac{\\sqrt{h}}{16\\piG_{_\\mathrm{N}}} \\left( \nK^{ij} - h^{ij} K\\right), \\ \\ \\ \\ \\ \\pi^{0} \\equiv \\frac{\\delta L}{\\delta \\dot{N}} = 0, \n\\ \\ \\ \\ \\ \n\\pi^{i} \\equiv \\frac{\\delta L}{\\delta \\dot{N^i}} = 0,\n\\label{Momenta}\n\\end{equation}\nthe last two providing primary constraints, as well as the momentum associated\nwith the matter component, $\\pi^\\Phi$ say. The Hamiltonian is therefore\n\\begin{equation}\nH \\equiv \\int \\mathrm{d}^3 x \\left( \\pi^0 \\dot{N} + \\pi^i \\dot{N}_i +\\pi^{ij}\n\\dot{h}_{ij} + \\pi^\\Phi \\dot\\Phi \\right) - L =  \\int \\mathrm{d}^3 x \\left(\n\\pi^0 \\dot{N} + \\pi^i \\dot{N}_i + N H + N_i \\mathcal{H}^i \\right).\n\\label{Hamil}\n\\end{equation}\n\nVariations of \\eqref{Hamil} yields the Hamiltonian description of GR.\n\n\\begin{figure}[H]\n\\centering \\includegraphics[width=7 cm]{3and1} \\includegraphics[width=7cm]{Evol_05}\n\\caption{ (\\textbf{a}) Spacetime foliation in terms of hypersurfaces\n\t$\\Sigma_t$, each labelled by a time-like parameter $t$. The diagram\n\tdefines the normal $n^\\mu$ and makes explicit the decomposition of a\n\ttangent to the worldline $x^i=\\mathrm{const}$ through\n\tthe lapse function $N$ and shift vector $N^i$.  (\\textbf{b}) A\n\ttime-evolved example with a few typical trajectories showing a quantum\n\tnon-singular bouncing universe. }\\label{ADM}\n\\end{figure}   \n\n\\subsection{Quantization}\n\nQuantum mechanics proceeds by first defining a Hilbert space of\naccessible states. In the GR case, it is the space of all the 3-metrics\n$h_{ij}$ and matter fields compatible with diffeomorphism invariance; it~is called superspace. The wave functional is then $\\Psi\\left[\nh_{ij}\\left(x\\right),\\Phi\\left(x\\right)\\right]$ and depends on the\ncoordinates $\\{x^\\mu\\}$, now understood as mere parameters.\n\nUpon adopting the Dirac canonical quantization procedure whereby\ncanonical momenta are replaced by $-i$ times the functional derivative\nwith respect to the variable they are the momenta of,~i.e.,\n\\begin{equation}\n\\pi^{ij} \\to - i \\frac{\\delta}{\\delta h_{ij}}, \\ \\ \\ \\ \\ \n\\pi^{0} \\to - i \\frac{\\delta}{\\delta N}, \\ \\ \\ \\ \\ \n\\pi^{i} \\to - i \\frac{\\delta}{\\delta N_i}, \\ \\ \\ \\ \\\n\\pi^{\\Phi} \\to - i \\frac{\\delta}{\\delta \\Phi},\n\\label{Dirac}\n\\end{equation}\none finds that the primary constraints translate into the fact\nthat the wave function depends neither on the lapse function\nnor on the shift vector, that it is unchanged under diffeomorphisms,\nand finally that the Wheeler De Witt equation\n\\begin{equation}\nH\\Psi = \\left[ -16\\piG_{_\\mathrm{N}} \\mathcal{G}_{ijkl}\n\\frac{\\delta^2}{\\delta h_{ij} \\delta h_{kl}} + \\frac{\\sqrt{h}}{16\\piG_{_\\mathrm{N}}}\\left(\n-^3R+2\\Lambda + 16\\piG_{_\\mathrm{N}} \\hat{T}^{00}\\right) \\right]\\Psi = 0\n\\label{WDW}\n\\end{equation}\nholds, with the De Witt metric defined as\n\\begin{equation}\n\\mathcal{G}_{ijkl} = \\frac{1}{2\\sqrt{h}}\\left( h_{ik} h_{jl} + h_{il} h_{jk}\n-h_{ij} h_{kl} \\right),\n\\label{DWm}\n\\end{equation}\nand $^3R$ is the curvature associated with the metric $h_{ij}$. In\n\\eqref{WDW}, $\\hat{\\bm{T}}$ is the operator version on superspace of the\n\tstress energy tensor relevant for the matter fields.\n\n\\subsection{Minisuperspace}\n\nAs it is essentially out of question to solve \\eqref{WDW} in general, one\nrestricts attention to the special FLRW case for which one replaces the\ngeneral 3D metric $h_{ij}$ by\n\\begin{equation}\nh_{ij} \\mathrm{d} x^i \\mathrm{d} x^j \\ \\ \\mapsto \\ \\ \\ a^2(t) \\left[  \\frac{\\mathrm{d} r^2}{1-\\mathcal{K} r^2} +\nr^2 \\left( \\mathrm{d}\\theta^2 + \\sin^2\\theta \\mathrm{d}\\phi^2 \\right)\\right],\n\\label{mini}\n\\end{equation}\nleading to a numerical parameter, the spatial curvature $\\mathcal{K}$, in\npractice set to zero in agreement with observational data, and a\ndynamical function, the scale factor $a(t)$. Under the assumption that\nthe 3D metric takes the form \\eqref{mini}, the Wheeler De Witt equation\nbecomes a Schr\\\"odinger-like equation for, say, the 2 degrees of freedom\nwave function $\\Psi\\left[ a(t),\\phi(t) \\right]$. There are many points,\nboth mathematical and physical, that can be raised about the\nminisuperspace approach, but they shall not concern us in the framework\nof this paper, and we refer the reader to, for instance,\nRefs.~\\cite{Kiefer:2008deh,Pinto-Neto:2013toa} and the references therein\nfor that matter.\n\nWe want instead here focus on the simplest possibility, namely that of\nvanishing spatial curvature $\\mathcal{K}\\to 0$ and consider as the matter\ncomponent a perfect fluid which we treat using Schutz formalism~\\mbox{\\cite{Schutz1970,Schutz1971}}. In this formalism, the full Hamiltonian\nreads\n\\begin{equation}\n\\mathrm{d} s^2 = N^2(\\tau) \\mathrm{d}\\tau - a^2(\\tau) \\gamma_{ij} \\mathrm{d} x^i \\mathrm{d} x^j \\ \\ \\ \n\\hbox{and} \\ \\ \\ p=w\\rho \\ \\ \\ \\Longrightarrow \\ \\ \\ \nH = N \\left( - \\frac{\\pi_a^2}{4 a} + \\frac{\\pi_t}{a^{3w}}\\right),\n\\label{Hfluid}\n\\end{equation}\nwhere the variable $t$ is associated to the velocity potentials and we\nkeep the lapse function $N(\\tau)$ unfixed for later convenience. With\nthe choice $N\\to a^{3w}$ and for a radiation fluid having $w=\\frac13$,\nthe~replacement $\\pi_t\\to -i\\partial_t$ transforms the Wheeler De Witt\nequation $H\\Psi = 0$ into $i\\partial_t\\Psi = \\frac14 \\partial_a^2 \\Psi$,\nwhich~is, up to a sign, a time-dependent Schr\\\"odinger equation for a\nfree particle \\cite{AcaciodeBarros:1997gy}. Not surprisingly, the~fluid\nhas permitted to define a global time variable.\n\n\\section{Quantum Trajectories}\n\nAlthough it is not the purpose of this paper to review the trajectory method in\nquantum mechanics, let me summarize it shortly.\n\nSince we have seen that the matter content merely serves in our case to\ndefine a time variable in the time-dependent Schr\\\"odinger equation, I\nnow consider a canonical transformation for the scale factor, namely\n$\\left(a ,\\pi_a \\right) \\mapsto \\left(A ,\\Pi_A \\right)$, leading to\n$H\\left(a,\\pi_a\\right)\\mapsto \\mathcal{H} \\left(A,\\Pi_A\\right)$ by means of a\ngenerating function $F(a,A;t)$ satisfying\n\\begin{equation}\n\\dot a \\pi_a - H = \\dot A \\Pi_A - \\mathcal{H} +\\frac{\\mathrm{d} F}{\\mathrm{d} t} \\ \\ \\ \\ \n\\Longrightarrow \\ \\ \\ \\ \\mathrm{d} F = \\pi_a \\mathrm{d} a - \\Pi_A\\mathrm{d} A +\n\\left( \\mathcal{H}-H\\right)\\mathrm{d} t.\n\\label{dFdt}\n\\end{equation}\n\nFrom \\eqref{dFdt}, one infers that \n\\begin{equation}\n\\mathrm{d} \\left( F+A\\Pi_A \\right) = \\pi_a \\mathrm{d} a + A\\mathrm{d} \\Pi_A + \\left( \\mathcal{H}-H\\right)\\mathrm{d} t,\n\\label{dS}\n\\end{equation}\nshowing that the function $\\left( F+A\\Pi_A \\right)$ depends on $a$,\n$\\Pi_A$ and $t$.\n\nWe now choose the canonical transformation such that the new Hamiltonian $\\mathcal{H}$\nidentically vanishes on shell. Hamilton equations then imply \n$$\n\\dot A = \\frac{\\partial \\mathcal{H}}{\\partial \\Pi_A} = 0 \\ \\ \\ \\ \\hbox{and} \\ \\ \\ \\\n\\dot \\Pi_A = - \\frac{\\partial \\mathcal{H}}{\\partial A} = 0 \\ \\ \\ \\ \\Longrightarrow\n\\ \\ \\ \\ \\frac{\\mathrm{d} \\left(F+A\\Pi_A\\right)}{\\mathrm{d} t} = \\dot a \\pi_a - H = L,\n$$\n$L$ being the original Lagrangian [see Equation~\\eqref{SLdt}]. Therefore, one\nmay identify the function $\\left( F+A\\Pi_A \\right)$ with the action\n$\\mathcal{S} = \\int L\\mathrm{d} t$. Equation~\\eqref{dS}, taken on shell, now reads\n\\begin{equation}\n\\mathrm{d} \\mathcal{S} = \\pi_a \\mathrm{d} a - H \\mathrm{d} t \\ \\ \\ \\ \\Longrightarrow \\ \\ \\ \\\n\\frac{\\partial \\mathcal{S}}{\\partial t} = - H \\ \\ \\ \\ \\hbox{and} \\ \\ \\\n\\ \\frac{\\partial \\mathcal{S}}{\\partial a} = \\pi_a, \\label{PW}\n\\end{equation}\nwhich can be recast in the more usual Hamilton-Jacobi form\n\\begin{equation}\nH\\left( a,\\frac{\\partial \\mathcal{S}}{\\partial a} \\right) +\n\\frac{\\partial S}{\\partial t} = 0, \\label{HJ}\n\\end{equation}\nand the last equation of \\eqref{PW} relates the actual value of the\nmomentum to the gradient of the action; as~we shall see below, this will\nbe equivalent to the pilot-wave equation when the action is identified\nwith the phase of the wave function.\n\nIn a quantum framework, the modified Hamilton-Jacobi equation is\nobtained as the real part of the Schr\\\"odinger equation when the wave\nfunction is written explicitly as an amplitude $R\\in \\mathbb{R}$ and a\nphase $S\\in \\mathbb{R}$, namely when setting $\\Psi(a,t) = R \\mathrm{e}^{i S}$.\nThis is one way to identify the phase of the wave function with the\naction. As a result, it is natural to assume that an actual\ntrajectory can be obtained as the solution of the canonical\ntransformation eikonal relation $\\pi_a|_\\mathrm{actual} \\propto \\dot\na = \\partial S\/\\partial a$, the dot denoting a time derivative.\n\nIt is instructive to note as well that setting $\\rho = R^2$ and\nreplacing the time derivative $\\dot a$ by an actual velocity $v$, one\nfinds that the imaginary part of Schr\\\"odinger equation may be written\nas\n$$\n\\frac{\\partial \\rho}{\\partial t} + \\mathrm{div} \\left( \\rho v\\right) =0,\n$$\nwhere the divergence is with respect to the variable $a$. In other\nwords, this formulation of quantum physics is very close to ordinary\nfluid mechanics, and the same methods (Eulerian or Lagrangian) actually\napply.\n\nSolving the trajectory equation is all but trivial, and many methods,\nmostly numerical, have~been devised \\cite{Wyatt:2005uc}. The one\nwe use in the examples presented below is based on a simple radial\nbasis interpolation, but it can be extended to include a moving mesh\nmethod. In fact, if the initial distribution of points $\\{a_n|_{t=t_0}\\}$\nat which the trajectories are calculated is $|\\Psi|^2$, then it remains\ndistributed along the square of the wave function at all subsequent\ntimes, so that whatever the behavior of $\\Psi$, one is sure to\ncover a domain that always remains where the wave function is\nlarge. For our illustrative purpose however, this refinement is\nnot necessary.\n\n   \n\nIn most cases of cosmological relevance, even if one wants to compare\nthe canonical quantization procedure with less usual ones\n\\cite{Bergeron:2017ddo}, or even when a larger minisuperspace is\nconsidered, e.g., to~account for a possible anisotropy (Bianchi Universe)\n\\cite{Peter:2016kan}, one ends up essentially solving a Schr\\\"odinger\nequation in a potential. Figure \\ref{Trajs} exemplifies a particular\ncase, showing the real and imaginary parts of the wave function, its\namplitude, and the derivative of its phase with respect to the scale\nfactor. It~illustrates that not only is the singularity resolved by\nquantum mechanics effects in this approach, but that the resulting\nactual trajectory depends crucially on the initial condition of the\nscale factor.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm]{Evol_01}\n\\includegraphics[width=6cm]{Evol_02}\n\\includegraphics[width=6cm]{Evol_03}\n\\includegraphics[width=6cm]{Evol_04}\n\\caption{Time evolution of a wave function and typical trajectories:\n(\\textbf{a}) initial condition, the phase gradient has both\npositive and negative values, so some trajectories are expanding while\nother are contracting, depending on their initial value. (\\textbf{b}) An\ninstant later, the phase gradient has essentially reversed, so all\ntrajectories have bounced in one way or another. (\\textbf{c}) After some\ntime, the~phase gradient tends to increase to become almost everywhere\npositive. (\\textbf{d}) Finally, the Universe starts expanding forever.\nAlso shown is the average value of the scale factor, which is seen to be\npotentially very different from the typical trajectories (and it is not even\nthe same as the trajectory having the same initial value of $a$, i.e., that\nfor which $a(t_0) = \\langle a \\rangle$).}\n\\label{Trajs}\n\\end{figure}\n\n\\section{Conclusions}\n\nThe trajectory method, also known as de Broglie Bohm pilot wave, permits\na clearer understanding of how quantum effects may affect cosmology near\nthe singularity, resolving the latter. However, it also shows that\ndefining the state itself may not be sufficient, as the initial\ncondition fixes the subsequent evolution of the scale factor: it may\nbounce once or many times depending on its initial value! If one\ncalculates perturbations in a self-consistent way \\cite{Peter:2008qz} on\ntop of such a trajectory, they will depend explicitly on which\ntrajectory has been chosen. This could actually provide a means of\nmeasuring the time evolution of the very early scale factor.\n\\vspace{6pt}\n\n\n\n\\acknowledgments{I gratefully acknowledge enlightening conversations with\nJ.-P. Gazeau, N. Pinto-Neto and particularly S. Vitenti, who also provided\nfigures. I would like to thank the Labex Institut Lagrange de Paris\n(reference ANR-10-LABX-63) part of the Idex SUPER, within which this work\nhas been partly done. I am hosted at Churchill College, Cambridge, partially\nsupported by a fellowship funded by the Higher Education, Research and\nInnovation Dpt of the French Embassy to the United-Kingdom.}\n\n\\conflictsofinterest{The author declares no conflict of interest.} \n\n\\reftitle{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nTwin-width is a measure of graph\\footnote{We consider simple, finite, undirected graphs $G$ with vertex-set $V(G)$ and edge-set $E(G)$. A \\defn{graph class} is a set of graphs closed under isomorphism. A graph class is \\defn{hereditary} if it is closed under taking subgraphs. A graph class is \\defn{monotone} if it is closed under taking induced subgraphs. A graph $H$ is a \\defn{minor} of a graph $G$ if $H$ is isomorphic to a graph obtained from a subgraph of $G$ by contracting edges. A graph class $\\mathcal{G}$ is \\defn{proper minor-closed} if $\\mathcal{G}$ is closed under taking minors, and some graph is not in $\\mathcal{G}$.} complexity recently introduced by \\citet{TW-I} (inspired by the work of \\citet{MT04} and \\citet{GM14}). The topic has attracted widespread interest~\\citep{PS,Przyb22,PSZ21,TW-I,TW-II,TW-III,TW-IV,TW-VI,BNOST,BH21,BKRTW,GPT21,AHKO,SS21,BBD21,DGJOR22}, often motivated by connections to model theory, logic, graph sparsity, fixed parameter tractability, enumerative combinatorics, and permutations.\n\nWe start with an informal description. Given a graph $G$, choose two vertices $u$ and $v$ in $G$, identify $u$ and $v$ into a single vertex, and colour each edge incident to exactly one of $u$ and $v$ red. Repeat this step until the graph has only one vertex. At each stage, the introduced red edges indicate an `error' in the reduction sequence. The goal is to find a sequence of identifications with small error. Twin-width measures the error by the maximum degree of the red graph (minimised over all reduction sequences). \n\nTo formalise this idea we need the following definitions. A \\defn{trigraph} is a triple $G = ( V, E, R )$ where $V$ is a finite set, and $E$ and $R$ are disjoint subsets of $\\binom{V}{2}$. Elements of $V$ are \\defn{vertices}. Elements of $E\\cup R$ are \\defn{edges}, edges in $E$ are \\defn{black}, and edges in $R$ are \\defn{red}. Let $V(G):=V$ and $E(G):=E$ and $R(G):=R$. Let $\\widetilde{G}$ be the spanning subgraph of $G$ consisting of the red edges. For distinct vertices  $u,v\\in V(G)$, let \\defn{$G\/u,v$} be the trigraph $(V',E',R')$ with:\n\\begin{itemize}\n\\item $V' = ( V \\setminus \\{u, v\\} ) \\cup \\{w\\}$ where $w\\not\\in V$, \n\\item $G-\\{u,v\\} = ( G\/u,v ) - w$, and\n\\item for all $x\\in V\\setminus\\{u,v\\}$:\n\\begin{itemize}\n\t\\item $wx \\in E'$ if and only if $ux \\in E$ and $vx \\in E$,\n\t\\item $wx \\not\\in E' \\cup R'$ if and only if $ux \\not\\in E \\cup R$ and $vx \\not\\in E \\cup R$, and\n\t\\item $wx \\in R'$ otherwise.\n\\end{itemize}\n\\end{itemize}\nThe \\defn{underlying graph} (or \\defn{total graph}) of a trigraph $G$ is the graph $H$ with $V(H)=V(G)$ and $E(H)=E(G)\\cup R(G)$.\n\nA sequence of trigraphs $G_n,G_{n-1},\\dots,G_1$ is a \\defn{reduction sequence} of $G_n$ (also called \\defn{contraction sequence}) if for each $i\\in\\{2,\\dots,n\\}$ we have $G_{i-1}=G_i\/u,v$ for some $u,v\\in V(G_i)$,  and $G_1$ is a trigraph with one vertex. In this case, each `prefix' $G_n, G_{n-1}, \\ldots, G_i$ is called a \\defn{partial reduction sequence to} $G_i$. A (\\defn{partial}) \\defn{reduction sequence} of a graph $G$ is a (partial) reduction sequence of the trigraph $(V(G),E(G),\\emptyset)$ (with no red edges).\n\nGiven a graph $G$, it is natural to ask for a reduction sequence $G_n,\\dots,G_1$ such that the red graphs $\\widetilde{G}_n,\\dots,\\widetilde{G}_1$ have desirable properties. For example, a graph has a reduction sequence with no red edges if and only if it is a cograph \\citep{TW-I} (and cographs are considered to be particularly well-behaved). \\citet{TW-I} cared about the maximum degree of the red graphs. They defined the \\defn{twin-width} of a graph $G$, denoted by \\defn{$\\tww(G)$}, to be the minimum $k\\in\\mathbb{N}_0$ such that there is a reduction sequence $G_n,G_{n-1},\\dots,G_1$ of $G$ where $\\widetilde{G}_i$ has maximum degree at most $k$ for each $i\\in\\{1,\\dots,n\\}$. \n\nThis paper studies reduction sequences where the red graph has other properties in addition to bounded maximum degree. For any graph parameter\\footnote{A \\defn{graph parameter} is a function $f$ such that $f(G)\\in\\mathbb{N}_0$ for every graph $G$, and $f(G_1)=f(G_2)$ for all isomorphic graphs $G_1$ and $G_2$. Examples of relevance to this paper include \n maximum degree $\\Delta(G)$, \n %\n bandwidth $\\bw(G)$, \n pathwidth $\\pw(G)$, \n treewidth $\\tw(G)$, \n clique-width $\\cw(G)$, \nand boolean-width $\\boolw(G)$.\n %\n A graph parameter $f$ is \\defn{monotone} if for each $k\\in\\mathbb{N}_0$ the graph class $\\{ G : f(G)\\leq k\\}$ is monotone. A graph parameter $f$ is \\defn{hereditary} if for each $k\\in\\mathbb{N}_0$ the graph class $\\{ G : f(G)\\leq k\\}$ is hereditary. A graph parameter $f$ is \\defn{union-closed} if $f(G\\cup H) \\leq \\max(f(G), f(H))$ for all disjoint graphs $G$ and $H$.} $f$, let \\defn{reduced~$f$} be the graph parameter~$\\reduced{f}$, where for any graph $G$,  \\defn{$\\reduced{f}(G)$} is the minimum $k\\in\\mathbb{N}$ such that there is a reduction sequence $G_n,G_{n-1},\\dots,G_1$ of $G$ where $f(\\widetilde{G}_i)\\leq k$ for each $i\\in\\{1,\\dots,n\\}$. So  reduced maximum degree~$\\reduced{\\Delta}$ is the same as  twin-width.\n\nEvery graph has a reduction sequence in which every red graph is a star (just repeatedly identify the centre of the star with any other vertex). So it makes sense to consider graph properties that are unbounded on the class of all stars (such as maximum degree).\n\nThis line of research was initiated by \\citet{TW-VI}, who considered the following  parameter\\footnote{\\citet{TW-VI} also considered the total number of edges in the red graph, but with a slightly different notion of reduction sequence in which red loops appear on identified vertices. The resulting parameter is called \\defn{total twin-width}.}. Let $\\star(G)$ be the maximum number of vertices in a connected component of a graph $G$. Then $\\reduced{\\star}$ is called the \\defn{component-twinwidth} \\citep{TW-VI}; here the goal is to find a reduction sequence such that every red graph has small components. \\citet{TW-I} proved that every graph $G$ satisfies $\\reduced{\\star}(G) \\leq 2^{\\boolw(G)+1}$, which implies $\\reduced{\\star}(G) \\leq 2^{\\tw(G)+2}$ and $\\reduced{\\star}(G) \\leq 2^{\\cw(G)+1}$. Note that $\\Delta(G) \\leq \\star(G)-1$ and thus $\\reduced{\\Delta}(G) \\leq \\reduced{\\star}(G)-1$. \n\n\n\\citet{TW-I} proved that every proper minor-closed graph class has bounded twin-width (amongst other results). The primary contribution of this paper is a qualitative strengthening of this result, where reduced maximum degree is replaced by reduced bandwidth. The \\defn{bandwidth} of a graph $G$ is the minimum $k\\in\\mathbb{N}_0$ such that there is an ordering $v_1,\\dots,v_n$ of $V(G)$ satisfying $|i-j|\\leq k$ for every edge $v_iv_j\\in E(G)$. We prove that every proper minor-closed class has bounded reduced bandwidth (\\cref{MinorClosedPower}). \n\nNote that $\\Delta(G)\\leq2\\,\\bw(G)$, implying $\\reduced{\\Delta}(G)\\leq 2\\,\\reduced{\\bw}(G)$. Thus, our upper bound on the reduced bandwidth of proper minor-closed classes implies the above-mentioned analogous result for twin-width, and indeed is qualitatively stronger since there are graph classes with bounded maximum degree and unbounded bandwidth. Complete binary trees are a simple example~\\citep{CS89}. There are even trees with maximum degree 3, pathwidth 2, and unbounded bandwidth\\footnote{\\label{QnHere} Let $Q_n$ be the tree consisting of disjoint paths $P_1,\\dots,P_n$, each with $n$ vertices, plus an edge joining the first vertex in $P_i$ and the first vertex in $P_{i+1}$ for each $i\\in\\{1,\\dots,n-1\\}$. Observe that $Q_n$ has maximum degree 3, pathwidth 2, $n^2$ vertices, diameter less than $3n$, and bandwidth at least $\\frac{n}{3}$ (since $|V(G)| \\leq \\bw(G)\\diam(G)+1$ for every graph $G$; see \\citep{CS89}).}. Generally speaking, graphs with bounded bandwidth are considered to be particularly well-behaved. Indeed, every graph with bandwidth $k$ is a subgraph of the $k$-th power of a path\\footnote{For a graph $G$ and $r\\in \\mathbb{N}$, the \\defn{$r$-th power} of $G$, denoted by \\defn{$G^r$}, is the graph with vertex-set $V(G)$ where two vertices $u$ and $v$ are adjacent in $G^r$ if and only if the distance between $u$ and $v$ in $G$ is at most $r$. The 2-nd power of $G$ is called the \\defn{square} of $G$.}. \n\nIn many cases, our results are also quantitatively stronger than previous bounds. The improvements for planar graphs are most significant. The previous proofs that planar graphs have bounded twin-width gave no explicit bounds, but it can be seen that all the previous bounds \\citep{TW-VI,TW-I} were at least $2^{1000}$. We show that every planar graph has reduced bandwidth at most $466$ and twin-width at most $583$. The proof method generalises for graphs embeddable on any surface\\footnote{The \\defn{Euler genus} of a surface with $h$ handles and $c$ crosscaps is $2h+c$. The \\defn{Euler genus} of a graph $G$ is the minimum Euler genus of a surface in which $G$ embeds without edge crossings. For $\\gamma\\in\\mathbb{N}_0$, the class of graphs of Euler genus at most $\\gamma$ is a proper minor-closed class.}. In particular, we show that every graph with Euler genus $\\gamma$ has reduced bandwidth at most $164\\gamma+466$ and twin-width at most $205\\gamma+583$ (\\cref{ReducedBandwidthGenus,TwinwidthGenus}).\n\nA key tool in our proofs are recent product structure theorems, which say that every graph of bounded Euler genus is a subgraph of the strong product of a graph with bounded treewidth and a path; see \\cref{ProductStructure} for details. Our results hold for any graph class that has such a product structure, which includes several non-minor-closed graph classes. For example, a graph is \\defn{$(\\gamma, k)$-planar} if it has a drawing in a surface of Euler genus $\\gamma$ such that every edge is involved in at most $k$ crossings (assuming no three edges cross at a single point)~\\citep{DEW17}. We prove that every $(\\gamma, k)$-planar graph has reduced bandwidth $2^{O(\\gamma k)}$ (\\cref{gkPlanar}). \n\nWe also strengthen the above-mentioned results by showing that fixed powers of graphs in any proper minor-closed class have bounded reduced bandwidth (\\cref{MinorClosedPower}).  Since FO-transductions preserve bounded twin-width~\\cite[Section 8]{TW-I}, it was previously known that these graphs have bounded twin-width. Note that powers of sparse graphs can be dense; for instance, $2$-powers of stars are complete graphs. As an example of our results for graphs powers, we consider map graphs, which are well-studied generalisations of graphs embedded in surfaces. We prove that map graphs of Euler genus $\\gamma$ have  reduced bandwidth $O(\\gamma^4)$. We emphasise there is no dependence on degree, and that these graphs might be dense. \n\n\\cref{Expanders} considers limitations of reduced bandwidth. We show that any infinite class of expander graphs excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth (\\cref{ExpanderClass}). This result separates reduced bandwidth from twin-width, since \\citet{TW-II} showed there are bounded-degree expanders (thus excluding a fixed complete bipartite subgraph) with twin-width at most 6. The theme of tied and separated parameters is continued in \\cref{Tied} where we show that the reduced versions of several natural parameters are separated. We conclude in \\cref{OpenProblems} by presenting a number of open problems.\n\n\\section{Preliminaries}\n\\label{Definitions}\n\nLet $\\mathbb{N}:=\\{1,2,\\dots\\}$ and $\\mathbb{N}_0:=\\{0,1,\\dots\\}$. \n\nFor a graph $G$ and $S\\subseteq V(G)$, let \\defn{$G[S]$} be the \\defn{induced} subgraph of $G$ with vertex-set $S$ and edge-set $\\{vw\\in E(G):v,w\\in S\\}$. For $F\\subseteq E(G)$, let \\defn{$G-F$} be the graph obtained from $G$ by removing edges in $F$.\nFor two graphs $G$ and $H$, let \\defn{$G\\cup H$} be the graph with vertex-set $V(G)\\cup V(H)$ and edge-set $E(G)\\cup E(H)$.\nA \\defn{clique} in $G$ is a (possibly empty) set of pairwise adjacent vertices. For disjoint sets $S, T\\subseteq V(G)$, we say that $S$ is \\defn{complete} to $T$ in $G$ if every vertex in $S$ is adjacent to every vertex in $T$.\n\nFor vertices $v,w\\in V(G)$, a \\defn{$(v,w)$-path} in $G$ is a path with end-vertices $v$ and $w$. For vertices $v,w\\in V(G)$, let \\defn{$\\dist_G(v,w)$} be the length of a shortest $(v,w)$-path in $G$. For a vertex $v$ in $G$, let \\defn{$N_G(v)$} $:=\\{w\\in V(G):vw\\in E(G)\\}$ and \\defn{$N_G[v]$} $:=\\{v\\}\\cup N_G(v)$. Let \\defn{$\\deg_G(v)$} $:=|N_G(v)|$, called the \\defn{degree} of $v$ in $G$. For each $r\\in\\mathbb{N}$,  let \\defn{$N^r_G[v]$} $:=\\{w\\in V(G):\\dist_G(v,w)\\leq r\\}$.\nFor $S\\subseteq V(G)$, let \\defn{$N_G(S)$} $:=\\bigcup_{v\\in S}N_G(v)\\setminus S$ and  \\defn{$N_G[S]$} $:=\\bigcup_{v\\in S}N_G[v]$. \n\nFor a vertex $v$ in a trigraph $G$, let \\defn{$N_G(v)$} $:=\\{w\\in V(G):vw\\in E(G)\\cup R(G)\\}$. \n\n\\subsection{Tree-decompositions}\n\nA \\defn{tree-decomposition} of a graph $G$ is a pair $(T, \\mathcal{B})$ consisting of a tree $T$ and a collection $\\mathcal{B}=(B_x\\subseteq V(G):x\\in V(T))$ of subsets of $V(G)$ (called \\defn{bags}) indexed by the nodes of $T$, such that:\n\\begin{enumerate}[label=(\\alph*)]\n\\item for every edge $uv\\in E(G)$, some bag $B_x$ contains both $u$ and $v$, and\n\\item for every vertex $v\\in V(G)$, the set $\\{x\\in V(T):v\\in B_x\\}$ induces a non-empty subtree of~$T$.\n\\end{enumerate}\nThe \\defn{width} of a tree-decomposition is the size of the largest bag minus 1. The \\defn{treewidth} \\defn{$\\tw(G)$} of a graph $G$ is the minimum width of a tree-decomposition of $G$. These definitions are due to \\citet{RS-II}. Treewidth is recognised as the most important measure of how similar a given graph is to a tree. Note that a connected graph with at least two vertices has treewidth 1 if and only if it is a tree. \nA \\defn{path-decomposition} is a tree-decomposition in which the underlying tree is a path. The \\defn{pathwidth} \\defn{$\\pw(G)$} of a graph $G$ is the minimum width of a path-decomposition of $G$. It is well-known and easily proved that for every graph $G$, \n\\[ \\tw(G) \\leq \\pw(G) \\leq \\bw(G).\\]\n\nLet $(T, \\mathcal{B}=(B_x:x\\in V(T)))$ be a tree-decomposition of a graph $G$. \nThe \\defn{torso} of a bag $B_u$ is the subgraph obtained from $G[B_u]$ by adding, for each edge $uv\\in E(T)$, all edges $xy$ where $x$ and $y$ are distinct vertices in $B_u\\cap B_v$. \n\nA \\defn{separation} of a graph $G$ is a pair $(A, B)$ of subsets of $V(G)$ such that $A\\cup B=V(G)$ and there is no edge of $G$ between $A\\setminus B$ and $B\\setminus A$.\n\nWe sometimes consider a tree-decomposition $(T, \\mathcal{B}=(B_x:x\\in V(T)))$ to be \\defn{rooted} at a specific \\defn{root bag} $B_r$ for some $r\\in V(T)$. In this case, for every node $t\\neq r$, let $t'$ be the node of $T$ adjacent to $t$ in $T$ and on the $(r,t)$-path in $T$. We say that $B_{t'}$ is the \\defn{parent} of $B_t$ and $B_t$ is a \\defn{child} of $B_{t'}$. A bag $B_x$ is a \\defn{descendant} of a bag $B_y$ if $y$ lies on the $(r,x)$-path in $T$. \n\nLet $t$ be a node of $T$, and let $\\{t_1, \\ldots, t_d\\} $ be a set of children of $t$. Let $C$ be the union of $B_t$ and every bag that is a descendant of $B_{t_i}$ for some $i\\in \\{1, 2, \\ldots, d\\}$. Let $D:=(V(G)\\setminus C)\\cup B_t$. As illustrated in \\cref{fig:rootedsep}, $(C, D)$ is a separation of $G$ with $C\\cap D=B_t$, said to be a \\defn{rooted separation at $B_t$} and a \\defn{rooted separation from $(T, \\mathcal{B})$}.\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[scale=0.35]{rootedsep} }\n\\caption{A rooted separation $(C,D)$ in a rooted tree-decomposition with root bag $B_r$.}\n\\label{fig:rootedsep}\n\\end{figure}\n\nFor a non-root degree-1 node $t$ of $T$, we say that $B_t$ is a \\defn{leaf bag};  every other bag is said to be \\defn{internal}. Note that a root bag is always internal.\n\nFor $k,q\\in\\mathbb{N}$ with $q\\ge k+1$, a rooted tree-decomposition $(T, \\mathcal{B})$ is \\defn{$(k,q)$-rooted} if:\n\\begin{itemize}\n \\item the root bag is empty,\n \\item every internal bag has at most $k+1$ vertices, and\n \\item for every leaf bag $B$ with parent $B'$, $\\abs{B\\setminus B'}\\le q$.\n\\end{itemize}\nA rooted tree-decomposition $(T, \\mathcal{B})$ is \\defn{$(k, \\infty)$-rooted} if the root bag is empty, and  every internal bag has at most $k+1$ vertices (so  leaf bags can be arbitrarily large). \n\n\\subsection{Sparsity}\n\nFor $d\\in\\mathbb{N}_0$, a graph $G$ is \\defn{$d$-degenerate} if every subgraph of $G$ has minimum degree at most $d$. The minimum such $d$ is the \\defn{degeneracy} of $G$. \n\n\\citet{KY03} introduced the following definition. For a graph $G$, total order $\\preceq$ of $V(G)$, vertex $v\\in V(G)$, and  $s\\in\\mathbb{N}$, let \\defn{$\\reach_s(G,\\preceq,v)$} be the set of vertices $w\\in V(G)$ for which there is a path $v=w_0,w_1,\\dots,w_{s'}=w$ of length $s'\\in \\{0,\\dots,s\\}$ such that $w\\preceq v$ and $v\\prec w_i$ for all $i\\in \\{0,\\dots, s'-1\\}$. For a graph $G$ and  $s\\in\\mathbb{N}$, the \\defn{$s$-strong colouring number} $\\col_s(G)$ is the minimum integer $k$ for which there is a total order~$\\preceq$ of $V(G)$ with $|\\reach_s(G,\\preceq,v)|\\leq k$ for every vertex $v$ of $G$. Strong colouring numbers interpolate between degeneracy and treewidth~\\citep{KPRS16}. Indeed, $\\col_1(G)$ equals the degeneracy of $G$ plus 1. At the other extreme, \\citet{GKRSS18} showed that $\\col_s(G)\\leq \\tw(G)+1$ for all $s\\in\\mathbb{N}$, and indeed $$\\lim_{s\\to\\infty}\\col_s(G)=\\tw(G)+1.$$\n\nObserve that a graph $H$ is a minor of a graph $G$ if and only if there are pairwise vertex-disjoint subtrees $(T_v)_{v\\in V(H)}$ in $G$ such that for each edge $vw\\in E(H)$ there is an edge between $T_v$ and $T_w$ in $G$. For $r\\in\\mathbb{N}$, if each such tree $T_v$ has radius at most $r$, then $H$ is an \\defn{$r$-shallow} minor of $G$. Let $$\\nabla_r(G):=\\max_H\\frac{|E(H)|}{|V(H)|},$$ \ntaken over all $r$-shallow (non-empty) minors $H$ of $G$. A graph class $\\mathcal{G}$ has \\defn{bounded expansion} if there is a function $f:\\mathbb{N}_0\\to\\mathbb{R}$ such that $\\nabla_r(G)\\leq f(r)$ for every graph $G\\in\\mathcal{G}$ and $r\\in\\mathbb{N}_0$. A graph class $\\mathcal{G}$ has \\defn{linear} or \\defn{polynomial expansion} respectively if there is a linear or polynomial expansion function.\n\n\\subsection{Product Structure Theorems}\n\\label{ProductStructure}\n\nFor graphs $G$ and $H$, the \\defn{strong product} $G\\boxtimes H$ is the graph with vertex-set $V(G)\\times V(H)$, where vertices $(v,w)$ and $(x,y)$ are adjacent if:\n\\begin{itemize}\n    \\item  $v=x$ and $wy\\in E(H)$, or \n    \\item $w=y$ and $vx \\in E(G)$, or\n    \\item $vx \\in E(G)$ and $wy\\in E(H)$.\n\\end{itemize}\n\n\n\nThe proofs of our main theorems depend on the following recent product structure results. \\citet{BDJMW} defined the \\defn{row-treewidth} of a graph $G$ to be the minimum $k\\in\\mathbb{N}_0$ such that $G$ is isomorphic to a subgraph of $H\\boxtimes P$ for some graph $H$ with treewidth $k$ and path $P$. The motivation for this definition is the following `Planar Graph Product Structure Theorem' of \\citet{DJMMUW20} (improved by \\citet{UWY}). \n\n\\begin{thm}[\\citep{DJMMUW20,UWY}]\n\\label{PlanarProductStructure}\nEvery planar graph has row-treewidth at most 6.\n\\end{thm}\n\n\\cref{PlanarProductStructure} was generalised for graphs of given Euler genus.\n\n\\begin{thm}[\\citep{DJMMUW20,UWY}]\n\\label{GenusProductStructure}\nEvery graph of Euler genus $\\gamma$ has row-treewidth at most $2\\gamma+6$.\n\\end{thm}\n\nMore generally, \\citet{DJMMUW20} proved that a minor-closed class has bounded row-treewidth if and only if it excludes some apex graph\\footnote{A graph $X$ is \\defn{apex} if $X-v$ is planar for some vertex $v$.}. For an arbitrary  proper minor-closed class, \\citet{DJMMUW20} obtained by the following `Graph Minor Product Structure Theorem', where $A+B$ is the \\defn{complete join} of graphs $A$ and $B$ (obtained from disjoint copies of $A$ and $B$ by adding every edge with one end-vertex in $A$ and one end-vertex in $B$).\n\n\n\n\\begin{thm}[\\citep{DJMMUW20}] \n\\label{MinorFree}\nFor every graph $X$, there exist $k,a\\in\\mathbb{N}$ such that every $X$-minor-free graph has a tree-decomposition in which every torso is a subgraph of $(H\\boxtimes P)+K_a$ for some graph $H$ of treewidth at most $k$ and some path $P$. \n\\end{thm}\n\nProduct structure theorems for several non-minor-closed classes are known \\citep{DMW,HW21b}. Here is one example. \n\n\n\\begin{thm}[\\citep{DMW}] \n\\label{gkPlanarProduct}\nEvery $(\\gamma,k)$-planar graph has row-treewidth $O(\\gamma k^6)$.\n\\end{thm}\n\n\n\n\nWith these tools in hand we now give the intuition behind our proofs. \nFirst note that \\citet{TW-II} showed that for all graphs $G$ and $H$, \n$$\\tww(G\\boxtimes H) \\leq \\max\\{\\tww(G)(\\Delta(H)+1) + 2\\Delta(H), \\tww(H) + \\Delta(H) \\} .$$\nHowever, this is not enough to conclude results about subgraphs of $G\\boxtimes H$ since twin-width is not monotone. Consider a subgraph $G$ of $H\\boxtimes P$ where $H$ has bounded tree-width and $P$ is a path. As mentioned in \\cref{Introduction}, $H$ has bounded component twin-width. This says there is a reduction sequence for $H$ such that each red component $X$ has bounded size. Observe that $X\\boxtimes P$ has bounded bandwidth. Our strategy is to construct a  reduction sequence for $G$ so that each red component is of the form $X\\boxtimes P$ where $X$ is a bounded-size subgraph of $H$, implying each red subgraph has bounded bandwidth. A key to the proof is to find vertex identifications so that the resulting graph stays a subgraph of some $H\\boxtimes P$. The above intuitive description has some inaccuracies. Implementing this strategy rigorously needs several further ideas, especially in the setting of powers (\\cref{BoundedRowTreewidth}). Finally, to apply \\cref{MinorFree} for $K_t$-minor-free graphs we need more ideas to cater for apex vertices and tree-decompositions (\\cref{ProperMinorClosedClasses}). \n\n\\section{Neighbourhood Diversity}\n\\label{NeighbourhoodComplexity}\n\nThe following concept will help to optimise our bounds on reduced bandwidth and twin-width, and is of independent interest because of connections to VC-dimension~\\citep{PP20,GHOORRVS17,RVS19} and sparsity theory~\\citep{EGKKPRS17,RVS19}. \n\nFix a graph $G$ and $r\\in\\mathbb{N}$. \nFor vertices $v,w\\in V(G)$, let\n\\[ \\dist^r_G(v,w) :=\n\\begin{cases}\n \\dist_G(v,w) & \\text{if $\\dist_G(v,w)\\le r$,}\\\\\n \\infty & \\text{otherwise.}\n  \\end{cases}\n\\]\nFor $A\\subseteq V(G)$ and $v\\in V(G)\\setminus A$, the \\defn{distance-$r$ profile of $v$ on $A$} is\n\\[\\pi^r_G(v,A):= \\{ (w,\\dist^r_G(v,w) ) : w\\in A \\},\\]\nand the \\defn{distance-$r$ diversity on $A$} is\n\\[\\pi^r_G(A) := | \\{ \\pi^r_G(v,A) : v \\in V(G)\\setminus A \\} |. \\]\nThis definition is different to similar definitions in \\citep{EGKKPRS17,RVS19} in that we only consider $v\\in V(G)\\setminus A$. We use different notation to avoid confusion. Clearly, $\\pi^r_G(A)\\le (r+1)^{|A|}$. \n\n\n Our upper bounds on reduced bandwidth and twin-width are expressed in terms of $\\pi^r_G(A)$ for sets $A$ of a given size (see \\cref{RowTreewidthNeighbourhoodPowers}). Motivated by this connection, \\cref{FirstNeigh} presents various upper bounds on $\\pi^1_G(A)$ that are tight  for graphs of given Euler genus (\\cref{SurfaceNeighbours}) and $K_t$-minor-free graphs (\\cref{MinorfreeNeighbour}). \\cref{SecondNei} gives an upper bound on $\\pi^2_G(A)$ where $G$ is a graph of given Euler genus. \n\n\\subsection{First Neighbourhoods} \n\\label{FirstNeigh}\n\nThis subsection presents bounds on $\\pi^1_G(A)$ for various graphs $G$. Note that \n \\[\\pi^1_G(A)=|\\{N_G(u)\\cap A:u\\in V(G)\\setminus A\\}|.\\] \nSo in a monotone class, we may assume that $G$ is bipartite with bipartition $\\{A,V(G)\\setminus A\\}$, where $N_G(u)\\cap X\\neq N_G(v)\\cap X$ for distinct $u,v\\in V(G)\\setminus A$. \n\nThe following lemma is a more precise version of a result by \\citet{GHOORRVS17} (see \\cref{NeighNabla}). For a graph $H$ and $k\\in\\mathbb{N}_0$, let \\defn{$C(H,k)$} be the number of cliques of order $k$ in $H$, where $\\emptyset$ is considered to be the only clique of order $0$ in $G$. So $C(H,0)=1$, $C(H,1)=|V(H)|$, and $C(H,2)=|E(H)|$. Let \\defn{$C(H,\\leq k)$} be the number of cliques of order at most $k$ in $H$, and let \\defn{$C(H)$} be the total number of cliques in $H$. \n\n\\begin{lem}\n\\label{Neighbourlyness}\nLet $G$ be a bipartite graph with bipartition $\\{X,Y\\}$, where $K_t$ is not a 1-shallow minor of $G$. Then there is a 1-shallow minor $H$ of $G$ on $|X|$ vertices, such that \n$$|\\{N_G(u):u\\in Y\\}| \\leq \n\\begin{cases}\nC(H,\\leq t-2) & \\text{if }t\\geq 4\\\\\nC(H,\\leq 2) & \\text{if }t=3.\\\\\n\\end{cases}$$\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $N_G(u)\\neq N_G(v)$ for all distinct $u,v\\in Y$. \nFor $i\\in\\mathbb{N}_0$, let $Y_i:=\\{v\\in Y: \\deg_G(v)=i\\}$. \nFor each $v\\in Y_2$, let $X_v:=N_G(v)$. \nBy assumption, $X_v\\neq X_w$ for distinct $v,w\\in Y_2$. \nLet $A$ be a maximal set such that:\n\\begin{itemize}\n    \\item $Y_2\\subseteq A\\subseteq Y\\setminus(Y_0\\cup Y_1)$, and \n    \\item for each $v\\in A$ there exists $X_v\\subseteq N_G(v)$ with $|X_v|=2$ where $X_v\\neq X_w$ for all distinct $v,w\\in A$.\n\\end{itemize} \n\nLet $H$ be the graph obtained from $G[X\\cup A]$, where for each $v\\in A$, we pick one $w\\in X_v$ and contract the edge $vw$. So $H$ is a 1-shallow minor of $G$ (with each branch set centred at a~vertex in $X$), where $|X|=|V(H)|$ and $|A|=|E(H)|=C(H,2)$. \n\nLet $B:=Y\\setminus(Y_0\\cup Y_1\\cup A)$. Consider a vertex $w\\in B$ with $d:= \\deg_G(w)$. So $d\\geq 3$ since $Y_2\\subseteq A$. By the maximality of $A$, we have $N_G(w)$ is a clique in $H$, implying $K_{d+1}$ is a 1-shallow minor of $G$. Thus $d\\in\\{3,4,\\dots,t-2\\}$. \nFor distinct $v,w\\in B$, we have $N_G(v)\\neq N_G(w)$. So $|B|\\leq \\sum_{i=3}^{t-2} C(H,i)$, and $B=\\emptyset$ if $t\\leq 4$. \n\nAlso $|Y_0|\\leq 1=C(H,0)$ and $|Y_1|\\leq|X|=C(H,1)$. \nTherefore $|Y|=|Y_0|+|Y_1|+|A|+|B|\\leq C(H,0)+C(H,1)+C(H,2)+|B|$. If $t\\leq 4$ then $B=\\emptyset$ and \n$|Y|\\leq C(H,\\leq 2)$; otherwise $|Y|\\leq C(H,0)+C(H,1)+C(H,2)+\n\\sum_{i=3}^{t-2} C(H,i) = C(H,\\leq t-2)$.\n\\end{proof}\n\nIt is well known (see \\citep{Wood-GC07,Wood16}) that every $d$-degenerate graph $G$ with $n\\geq d$ vertices satisfies:\n\\begin{itemize}\n    \\item $C(G,k)\\leq \\binom{d}{k-1}(n-\\frac{(k-1)(d+1)}{k})$ for all $k\\in\\{0,1,\\dots,d+1\\}$,\n    \\item $C(G)\\leq 2^d(n-d+1)$.\n\\end{itemize}\nSo \\cref{Neighbourlyness} implies:\n\n\\begin{cor}\n\\label{NeighDegen}\nFor every graph $G$ and set $A\\subseteq V(G)$, if every 1-shallow minor of $G$ on $|A|$ vertices is $d$-degenerate, then \n$\\pi^1_G(A) \\leq \\min\\{ 2^d(|A|-d+1), 2^{|A|} \\} $.\n\\end{cor}\n\n\\cref{NeighDegen} is applicable with $d=\\floor{2\\nabla_1(G) }$ since every 1-shallow minor of a graph $G$ is $\\floor{2\\nabla_1(G)}$-degenerate. We obtain the following lemma, which is a slight strengthening  of a result of  \\citet[Lemma~4.3]{GHOORRVS17}. \n\n\\begin{lem}\\label{NeighNabla}\nFor every graph $G$ with $\\nabla_1(G) \\leq k$ and for every set $A\\subseteq V(G)$, \n$$\\pi^1_G(A) \\leq \n\\min\\{ 2^{\\floor{2k}}(|A|-\\floor{2k}+1), 2^{|A|}\\} .$$\n\\end{lem}\n\n\\begin{lem}\n\\label{NeighCol}\nFor every graph $G$ with $\\col_5(G)\\leq c$ and for every set $A\\subseteq V(G)$, \n\\[ \\pi^1_G(A) \\leq \\min\\{ 2^{c-1}( |A|-c+2), 2^{|A|} \\}.\\]\n\\end{lem}\n\n\\begin{proof}\n\\citet[Lemma~20]{HW21b} proved that for every graph $G$ and every $r$-shallow minor $G'$ of $G$, we have \n$\\col_s(G') \\leq \\col_{2rs+2r+s}(G)$. \nEvery graph $G$ is $(\\col_1(G)-1)$-degenerate.\nThus every 1-shallow minor of a graph $G$ is $(\\col_5(G)-1)$-degenerate. The result follows from \\cref{NeighDegen}.\n\\end{proof}\n\n\\begin{lem}\\label{NeighGammak}\nFor every $(\\gamma,k)$-planar graph $G$ and for every set $A\\subseteq V(G)$, \n\\[ \\pi^1_G(A) \\leq \\min\\{ 2^{22(2\\gamma+3)(k+1)-1}(|A|-22(2\\gamma+3)(k+1)+2), 2^{|A|} \\}.\\]\n\\end{lem}\n\n\\begin{proof}\nVan den Heuvel and Wood~\\citep{vdHW18} proved that $\\col_s(G)\\leq 2(2\\gamma+3)(k+1)(2s+1)$.\nThe result follows from \\cref{NeighCol}. \n\\end{proof}\n\n\\begin{lem}\n\\label{NeighRowTreewidth}\nFor every graph $G$ with row-treewidth at most $k$ and for every set $A\\subseteq V(G)$, \n$$\\pi^1_G(A) \\leq \\min\\{ 2^{11k+10}(|A|-11k-9), 2^{|A|} \\}.$$\n\\end{lem}\n\n\\begin{proof}\nBy assumption, $G$ is isomorphic to a subgraph of $H\\boxtimes P$, where $\\tw(H)\\leq k$ and $P$ is a path. \\citet[Lemma~22]{HW21b} showed that $$\\col_s(H \\boxtimes P) \\leq (2s+1) \\col_s(H) \\leq (2s+1) (\\tw(H)+1 ).$$ \nIn particular, $\\col_5(G) \\leq \\col_5(H \\boxtimes P) \\leq 11(k+1)$. The result follows from \\cref{NeighCol}.\n\\end{proof}\n\nWe get improved bounds for minor-closed classes. For every $n$-vertex $K_t$-minor-free graph $G$, \\citet{Kostochka84} and \\citet{Thomason84} independently showed that $G$ has  $O(t\\sqrt{\\log t}\\, |V(G)|)$ edges, and \\citet{FW17} showed that $G$ has at most $3^{2t\/3+o(t)}|V(G)|$ cliques. \\cref{Neighbourlyness} implies:\n\n\\begin{cor}\\label{MinorfreeNeighbour}\nFor every $K_t$-minor-free graph $G$ and for every set $A\\subseteq V(G)$,\n$$\\pi^1_G(A) \\leq 3^{2t\/3+o(t)}|A|+1.$$\n\\end{cor}\n\nWe now show that this bound is tight up to the $o(t)$ term. We may assume $2t=3p+2$ for some even $p\\in\\mathbb{N}$. Let $H$ be the complete $p$-partite graph $K_{2,\\dots,2}$, which can be obtained from $K_{2p}$ by deleting a perfect matching $v_1w_1,\\dots,v_pw_p$. Obviously, the largest complete subgraph in $H$ is $K_p$. \\citet{Wood-GC07} showed that the largest complete graph minor in $H$ is $K_{3p\/2}$ (with branch sets $\\{v_1\\},\\dots,\\{v_p\\},\\{w_1,w_2\\},\\dots,\\{w_{p-1},w_p\\}$). So  $H$ is $K_t$-minor-free (since $t>\\frac32 p$). It is well known that $H$ has exactly $3^p$ cliques (since if $C_i$ is any element of $\\{\\emptyset,\\{v_i\\},\\{w_i\\}\\}$, then $C_1\\cup\\dots\\cup C_p$ is a clique, giving $3^p$ cliques in total, and every clique in $H$ is obtained this way). Let $G$ be the bipartite graph with bipartition $\\{X,Y\\}$, where $X=V(H)$ and for each clique $C$ in $H$, there is a one vertex $y_C$ in $Y$ with $N_G(y_C)=C$. So $G$ can be obtained from $H$ by clique-sums with complete graphs of order at most $p+1$ (and edge-deletions). So $G$ is also $K_t$-minor-free, and every vertex in $Y$ has a unique neighbourhood. Thus  $\\pi^1_G(X)=|Y|=3^p=3^{(2t-2)\/3}=3^{2t\/3-O(\\log t)}|X|$. \n\n\\begin{cor}\n\\label{NeighbourlynessTreewidth}\nFor every graph $G$ with treewidth $k\\in\\mathbb{N}$ and for every set $A\\subseteq V(G)$, \n\\[ \\pi^1_G(X) \\leq \n\\begin{cases}\n2|A| & \\text{if }k=1\\\\\n2^{|A|} & \\text{if }k\\geq 2\\text{ and }|A|\\leq k\\\\\n(2^k-1)(|A|-k)+2^k & \\text{if }k\\geq 2\\text{ and }|A|\\geq k\\\\\n\\end{cases}\\]\n\\end{cor}\n\n\\begin{proof}\nLet $Y:=V(G)\\setminus A$. First suppose that $k=1$. So $G$ and every minor of $G$ is a forest. So \\cref{Neighbourlyness} is applicable with $t=3$. Thus, there is a minor $H$ of $G$ on $|A|$ vertices, such that \n$\\pi^1_G(A)=|\\{N_G(u)\\cap A : u\\in Y\\}| \\leq C(H)\\leq 2|A|$ since $H$ is a forest.  \n\nNow assume that $k\\geq 2$. The class of treewidth-$k$ graphs is proper minor-closed and has no $K_{k+2}$ minor. So  \\cref{Neighbourlyness} is applicable with $t=k+2\\geq 4$. Thus there is a minor $H$ of $G$ on $|A|$ vertices, such that $|\\{N_G(u):u\\in Y\\}| \\leq C(H,\\leq k)$. This is at most $2^{|A|}$ if $|A|\\leq k$. Now assume that $|A|\\geq k+1$. Every graph with treewidth at most $k$ is $k$-degenerate. So  $H$ is $k$-degenerate, implying \n\\[\\pi^1_G(A)= |\\{N_G(u):u\\in Y\\}| \\leq \\sum_{i=0}^{k} \\tbinom{k}{i-1} \\big(|A|-\\tfrac{(i-1)(k+1)}{i}\\big) = 2^k(|A|-k+1) -|A|+k.\\qedhere\\]\n\\end{proof}\n\nThe bound in \\cref{NeighbourlynessTreewidth} is tight: Let $H$ be a $k$-tree on $n\\geq k$ vertices, which has $2^k(n-k+1)$ cliques and $n-k$ cliques of size $k+1$~\\citep{Wood16}. Let $G$ be the bipartite graph with bipartition $\\{X,Y\\}$, where $X=V(H)$ and for each clique $C$ with $|C|\\leq k$ in $H$, there is a one vertex $y_C$ in~$Y$ with $N_G(y_C)=C$. So $|Y|=C(H,\\leq k)=2^{k}(n-k+1)-n+k$. Given a tree-decomposition of $H$ with width $k$, for each clique $C$ in $H$ with $|C|\\leq k$, which must be in some bag $B$, add one new bag $B'=B\\cup\\{y_C\\}$ adjacent to $B$ to obtain a tree-decomposition of $G$ with width $k$.\n\nWe have the following bound for graphs of given Euler genus. It follows from Euler's formula that $|E(G)|\\leq 3(|V(G)|+\\gamma-2)$ for every graph $G$ with Euler genus $\\gamma$; moreover, $|E(G)|\\leq 2(|V(G)|+\\gamma-2)$ if $G$ is bipartite. \n\n\\begin{lem}\n\\label{SurfaceNeighbours}\nFor every graph $G$ with Euler genus $\\gamma$ and for every set $X\\subseteq V(G)$ with $|X|\\geq 2$,\n\\[\\pi^1_G(X) \\leq  6|X|+5\\gamma -9.\\]\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $G$ is bipartite with bipartition $\\{X,Y\\}$, and that $N_G(u)\\neq N_G(v)$ for distinct $u,v\\in Y$ (otherwise delete one of the vertices). \nFor $i\\in\\mathbb{N}_0$, let $Y_i:= \\{u\\in Y: \\deg_G(u)=i\\}$. \nLet $G':=G-Y_0-Y_1$. \nSince $G'$ is bipartite, \n\\begin{equation*}\n|E(G)|-|Y_1| \n= |E(G')|\\leq 2(|V(G')|+\\gamma-2) =\n2( |V(G)|-|Y_0|-|Y_1|+\\gamma-2).\n\\end{equation*}\nThus\n\\begin{equation*}\n\\sum_{i\\geq 0} i |Y_i| \n= |E(G)| \\leq 2 |V(G)|- 2|Y_0| - |Y_1| +2\\gamma-4 \n= 2\\Big( |X|+\\sum_{i\\geq 0} |Y_i| \\Big) - 2|Y_0| - |Y_1| +2\\gamma-4.\n\\end{equation*}\nHence\n\\begin{equation*}\n|Y| - |Y_0| - |Y_1| - |Y_2| = \n\\sum_{i\\geq 3} |Y_i| \\leq \n\\sum_{i\\geq 3} (i-2) |Y_i| \\leq \n2 |X| +2\\gamma - 4,\n\\end{equation*}\nimplying $|Y| \\leq  2 |X| + |Y_0| + |Y_1| +|Y_2| +2\\gamma- 4$. Since $N_G(u)\\neq N_G(v)$ for distinct $u,v\\in Y$, we have $|Y_0|\\leq 1$ and $|Y_1|\\leq|X|$. If $H$ is obtained from $G[X\\cup Y_2]$ by contracting one edge incident to each vertex in $Y_2$, then $H$ has no parallel edges and $|X|$ vertices, implying $|Y_2|=|E(H)|\\leq 3(|X|+\\gamma-2)$. Hence \\begin{equation*}\n    |Y|  \\leq 2|X| + 1 + |X| +  3(|X|+\\gamma-2) +2\\gamma - 4 = 6|X|+5\\gamma-9.\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\\cref{SurfaceNeighbours} is also tight: Let $G_0$ be a triangulation of a surface with Euler genus $\\gamma$ with at least four vertices. Let $G$ be obtained from $G_0$ as follows: add one vertex adjacent to the three vertices of each face of $G_0$, subdivide each edge of $G_0$, add one vertex adjacent to each vertex of $G_0$, and add one isolated vertex. So $G$ is bipartite with bipartition $\\{X,Y\\}$ where $X:=V(G_0)$ and $Y:=V(G)\\setminus X$. No two vertices in $Y$ have the same neighbourhood, and $|Y|=|Y_0|+|Y_1|+|Y_2|+|Y_3|=1+|X|+3(|X|+\\gamma-2)+2(|X|+\\gamma-2)=6|X|+5\\gamma-9$.\n\n\n\\subsection{Second Neighbourhoods}\n\\label{SecondNei}\n\nThis section gives bounds on the second neighbourhood diversity in graphs of given Euler genus. These results are useful for bounding the reduced bandwidth of squares and map graphs. \n\n\\begin{lem}\\label{SurfaceSecondNeighbourhood} \nLet $G$ be a graph of Euler genus $\\gamma$, and let $X\\subseteq V(G)$ with $|X|\\geq 2$. Let $Y:=N_G(X)$ and $Z:=V(G)\\setminus (X\\cup Y)$. Then $$| \\{ N^2_G(v)\\cap X : v\\in Z \\}| \\leq  \n(60 \\gamma^2 + 125 \\gamma + 68)|X|-120 \\gamma^2 -250 \\gamma -132.$$\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $N^2_G(v)\\cap X\\neq N^2_G(w)\\cap X$ for all distinct $v,w\\in Z$. Let $Z^0:=\\{v\\in Z: |N^2_G(v)\\cap X|=0\\}$ and $Z^1:=\\{v\\in Z: |N^2_G(v)\\cap X|=1\\}$ and $Z^2:=\\{v\\in Z: |N^2_G(v)\\cap X|\\geq 2\\}$.  \nThus $|Z^0|\\leq 1$ and $|Z^1|\\leq|X|$. \nBy \\cref{SurfaceSecondNeighbourhoodInduction} below, \n$|Z^2|\\leq (60 \\gamma^2 + 125 \\gamma + 67)(|X|-2)+1$. \nIn total, $|Z|\\leq \n(60 \\gamma^2 + 125 \\gamma + 67)(|X|-2)+|X|+2\n= \n(60 \\gamma^2 + 125 \\gamma + 68)|X|-120 \\gamma^2 -250 \\gamma -132$, as desired.\n\\end{proof}\n\nThe proof of \\cref{SurfaceSecondNeighbourhoodInduction} uses the next lemma, which follows from  \\citep[Proposition~4.2.7]{MoharThom} and the discussion after it.\n\n\\begin{lem}\n\\label{FindContractiblecycle}\nIf $K_{2,2\\gamma+2}$ is embedded in a surface of Euler genus $\\gamma$, then some 4-cycle in $K_{2,2\\gamma+2}$ is contractible.\n\\end{lem}\n\n\\begin{lem}\n\\label{SurfaceSecondNeighbourhoodInduction}\nLet $G$ be a graph of Euler genus $\\gamma$, and let $X\\subseteq V(G)$ with $|X|\\geq 2$. Let $Y:=N_G(X)$ and $Z:=V(G)\\setminus(X\\cup Y)$. \nAssume that $|N^2_G(v)\\cap X| \\geq 2$ for every vertex $v\\in Z$, and that $N^2_G(v)\\cap X\\neq N^2_G(w)\\cap X$ for all distinct $v,w\\in Z$. \nThen $|Z|\\leq (60 \\gamma^2 + 125 \\gamma + 67)(|X|-2)+1$. \n\\end{lem}\n\n\\begin{proof}\nWe prove that $|Z|\\leq c(|X|-2)+1$ by induction on $|Y|$, where $c:=60 \\gamma^2 + 125 \\gamma + 67$. (This choice of $c$ will become clear at the end of the proof.)\\ In the base case, if $|X|=2$ then $|Z|\\leq 1=c(|X|-2)+1$. Now assume that $|X|\\geq 3$. We may assume that each of $X$, $Y$ and $Z$ are independent sets. \n\nFirst suppose that there is a set $Y_0\\subseteq Y$ and distinct vertices $x_1,x_2\\in X$ such that $|Y_0|\\geq 2\\gamma+2$ and $N_G(y)\\cap X=\\{x_1,x_2\\}$ for each $y\\in Y$. Thus $G[\\{x_1,x_2\\}\\cup Y_0]$ contains a $K_{2,2\\gamma+2}$ subgraph. By \\cref{FindContractiblecycle}, there is a \ncontractible 4-cycle $C=(x_1,y_1,x_2,y_2)$ in $G$, for some $y_1,y_2\\in Y_0$. \n\nThus $G=G_1\\cup G_2$ for some induced subgraphs $G_1$ and $G_2$ of $G$ with $G_1\\cap G_2=C$, where $C$ is the boundary of a face of both $G_1$ and $G_2$. Let $G'$ be the graph obtained from $G_1$ by identifying $y_1$ and $y_2$ into a vertex $y'$. Let $G''$ be the graph obtained from $G_2$ by identifying $y_1$ and $y_2$ into a vertex $y''$. Since $y_1$ and $y_2$ are on a common face before their identification, $G'$ and $G''$ have Euler genus at most $g$. \n\nLet $X':=V(G')\\cap X$ and $X'':=V(G'')\\cap X$. \nNote that $X'\\cap X''=\\{x_1,x_2\\}$.\nLet $Y':=(V(G')\\cap Y)\\cup\\{y'\\}$ and \n$Y'':=(V(G'')\\cap Y)\\cup\\{y''\\}$. \nNote that $|Y'|<|Y|$ and $|Y''|<|Y|$. \nLet $Z':=V(G')\\cap Z$ and $Z'':=V(G'')\\cap Z$. \nNote that $Z'$ and $Z''$ partition $Z$. \n\nWe claim that $|N^2_{G'}(v)\\cap X'|\\geq 2$ for each $v\\in Z'$. Suppose that $|N^2_{G'}(v)\\cap X'|\\leq 1$. Then there is a vertex from $N^2_G(v)\\cap X$ in $G''-V(G')$. Since $C$ is separating, $v$ is adjacent to $y_1$ or $y_2$ in~$G$, implying $x_1,x_2\\in N^2_{G'}(v)$ and $|N^2_{G'}(v)\\cap X'|\\geq 2$, as desired. Similarly, $|N^2_{G''}(v)\\cap X''|\\geq 2$ for each $v\\in Z''$. \n\nSince $N_{G'}(y')\\cap X'=N_G(y_1)\\cap X=N_G(y_2)\\cap X=\\{x_1,x_2\\}$, we have $N^2_{G'}(v)\\cap X' = N^2_G(v)\\cap X$ for every $v\\in Z'$. Hence $N^2_{G'}(v)\\cap X'\\neq N^2_{G'}(w)\\cap X'$ for distinct vertices $v,w\\in Z'$. Similarly, $N^2_{G''}(v)\\cap X''\\neq N^2_{G''}(w)\\cap X''$ for distinct vertices $v,w\\in Z''$. \n\nBy assumption, there is at most one vertex $v\\in Z$ with  $N^2_G(v)\\cap X=N_G(y_1)\\cap X$. Without loss of generality, if such a vertex $v$ exists, then $v$ is not in $Z''$. Add a new vertex $z$ to $Z''$ and to $G''$ only adjacent to $y''$ in $G''$. Then $x_1,x_2\\in N^2_{G''}(z)\\cap X''$ and $|N^2_{G''}(z)\\cap X''|\\geq 2$ as required. If $N^2_{G''}(z)\\cap X'' = N^2_{G''}(v)\\cap X''$ for some vertex $v\\in Z''\\setminus\\{z\\}$, then $N^2_{G''}(v)\\cap X'' = N^2_{G''}(z)\\cap X''=N_G(y_1)$, which contradicts the above property of $Z''$. Hence $N^2_{G''}(z)\\cap X'' \\neq N^2_{G''}(v)\\cap X''$ for every vertex $v\\in Z''\\setminus\\{z\\}$.\n\nNow $|Z_1|+|Z_2|=|Z|+1$. We have shown that $X'$, $Y'$ and $Z'$ satisfy the assumptions of the inductive hypothesis within $G'$. Since $|Y'|<|Y|$, by induction, $|Z'|\\leq c(|X'|-2)+1$. Similarly, $|Z''|\\leq c(|X''|-2)+1$. Hence\n\\begin{align*}\n|Z|\\,=\\,|Z'|+|Z''|-1  & \\leq c(|X'|-2)+1 +  c(|X''|-2)+1-1  \\\\\n&= c( |X'|+|X''|-4)+1 \\\\\n& = c(|X|-2)+1,\n\\end{align*}\nas desired. Now assume that there are no such vertices $x_1,x_2\\in X$ and set $Y_0$. \n\nAs illustrated in \\cref{Hgraph}, let $Y^1$ be the set of vertices in $Y$ with exactly one neighbour in $X$. Let $Y^1_1,\\dots,Y^1_p$ be the partition of $Y^1$, where for all $v\\in Y^1_i$ and $w\\in Y^1_j$ we have $N_G(v)\\cap X=N_G(w)\\cap X$ if and only if $i=j$. Let $y^1_i$ be a vertex in $Y^1_i$ and let $x_i$ be the neighbour of $y^1_i$ in $X$. \n\nLet $Y^2$ be the set of vertices in $Y$ with exactly two neighbours in $X$. \nLet $Y^2_1,\\dots,Y^2_q$ be the partition of $Y^2$, where for all $v\\in Y^2_i$ and $w\\in Y^2_j$ we have $N_G(v)\\cap X=N_G(w)\\cap X$ if and only if $i=j$. As shown above, $|Y^2_i|\\leq 2\\gamma+1$ for each $i\\in\\{1,\\dots,q\\}$. Let $y^2_i$ be a vertex in $Y^2_i$. \n\nLet $Y^3$ be the set of vertices in $Y$ with at least three neighbours in $X$. Let $Y^3_1,\\dots,Y^3_r$ be the partition of $Y^3$, where for all $v\\in Y^3_i$ and $w\\in Y^3_j$ we have $N_G(v)\\cap X=N_G(w)\\cap X$ if and only if $i=j$. Since $K_{3,2\\gamma+3}$ has Euler genus greater than $\\gamma$, $|Y^3_i|\\leq 2\\gamma +2$ and $|Y^3|\\leq (2\\gamma+2)r$. Let $y^3_i$ be a vertex in $Y^3_i$. \n\nBy construction, the vertices $y^1_1,\\dots,y^1_p,y^2_1,\\dots,y^2_q,y^3_1,\\dots,y^3_r$ have pairwise distinct non-empty neighbourhoods in $X$. By \\cref{SurfaceNeighbours} applied to the bipartite graph between $X$ and $\\{y^1_1,\\dots,y^1_p,y^2_1,\\dots,y^2_q,y^3_1,\\dots,y^3_r\\}$, we have $p+q+r\\leq (6|X|+5\\gamma-9)-1=6|X|+5\\gamma-10$. In fact, $p\\leq |X|$ and $q+r\\leq 5|X|+5\\gamma-10$.\n\n\\begin{figure}\n\\centering\n\\includegraphics{Hgraph}\n\\caption{(a) The sets $X,Y^1_1,\\dots,Y^1_p,Y^2_1,\\dots,Y^2_q,Y^3_1,\\dots,Y^3_r,Z$ in $G$. (b) The graph $H$.}\n\\label{Hgraph}\n\\end{figure}\n\nLet $H$ be the graph obtained from $G$ as follows: \ndelete $X\\setminus\\{x_1,\\dots,x_p\\}$, \ndelete any edges between $\\{x_1,\\dots,x_p\\}$ and $Y^2\\cup Y^3$, and \nfor each $i\\in\\{1,\\dots,p\\}$ contract $\\{x_i\\}\\cup Y^1_i$ (which induces a star) into a new vertex $h_i$. Note that $H$ is bipartite and planar, with one colour class $Z$ and the other colour class $\\{h_1,\\dots,h_p\\}\\cup Y^2\\cup Y^3$, which has size at most \n$$p+|Y^2|+|Y^3| \n\\leq p + (2\\gamma+1)q + (2\\gamma+2)r \n\\leq  p + (2\\gamma+2)(q+r)\n\\leq  |X| + (2\\gamma+2)(5|X|+5\\gamma-10).$$ \nFor each $v\\in Z$, \n$$N^2_G(v)\\cap X = \n\\{ x_i : h_i\\in N_H(v) \\} \\cup \n\\Big( \\bigcup_{u\\in N_H(v)\\setminus\\{h_1,\\dots,h_p\\}} \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! N_G(u)\\cap X \\Big).$$\nThus $N^2_G(v)\\cap X$ is determined by $N_H(v)$. \nFor all distinct $v,w\\in Z$, since\n$N^2_G(v)\\cap X\\neq N^2_G(w)\\cap X$, we have $N_H(v)\\neq N_H(w)$.\nBy \\cref{SurfaceNeighbours} applied to $H$, \n$$|Z|\n\\leq 6 (p+|Y^2|+|Y^3|) +5\\gamma - 10\n\\leq 6\\big( |X| + (2\\gamma+2)(5|X|+5\\gamma-10) \\big) +5\\gamma - 10\n \\leq c(|X|-2)+1,$$ \nsince $|X|\\geq 3$. Indeed, $c$ is defined so that this final inequality holds. \n\\end{proof}\n\n\n\n\\begin{lem}\n\\label{Nu2}\nLet $f:\\mathbb{N}\\to\\mathbb{N}$ be a function and let $\\mathcal{G}$ be a monotone class, such that for every $G\\in\\mathcal{G}$ and $X\\subseteq V(G)$ and $Z\\subseteq V(G)\\setminus N_G[X]$,\n\\[| \\{ N^2_G(v)\\cap X : v\\in Z \\}| \\leq f(|X|).\\]\nThen $\\pi^2_G(X)\\leq \\pi^1_G(X)\\, f(|X|)$. \n\\end{lem}\n\n\\begin{proof}\nLet $s:= \\pi^1_G(X)$ and $t:= f(|X|)$. \nLet $Y:=V(G)\\setminus X$. \nLet $Y_1,\\dots,Y_s$ be a partition of $Y$ where $v,w\\in Y_i$ if and only if $N_G(v)\\cap X=N_G(w)\\cap X$.\nFor each $i\\in \\{1, \\ldots, s\\}$, let $G_i$ be the graph obtained from $G$ by deleting the edges between $Y_i$ and $X$. \nSince $\\mathcal{G}$ is monotone, $G_i\\in\\mathcal{G}$, implying\n$|\\{ N^2_{G_i}(v) \\cap X : v \\in Y_i \\}| \\leq t$.\nLet $Y_{i,1},\\dots,Y_{i,t}$ be a partition of $Y_i$ where\n$v,w\\in Y_{i,j}$ if and only if $N^2_{G_i}(v)\\cap X = N^2_{G_i}(w)\\cap X$. It follows that for each $i\\in \\{1, \\ldots, s\\}$ and $j\\in\\{1, \\ldots, t\\}$, \nwe have $\\pi_2^G(v,X) = \\pi_2^G(w,X)$ for all $v,w\\in Y_{i,j}$. Thus $\\pi^2_G(X)\\leq st$, as desired. \n\\end{proof}\n\n\\cref{SurfaceNeighbours,SurfaceSecondNeighbourhood,Nu2} imply the following bound on $\\pi^2_G$ for graphs of given Euler genus. \n\n\\begin{cor}\\label{SurfaceNu2}  \nFor every graph $G$ of Euler genus $\\gamma$ and for every set $X\\subseteq V(G)$ with $|X|\\geq 2$, \n\\[ \\pi^2_G(X) \\leq (6 |X| + 5\\gamma -9)\n( (60 \\gamma^2 + 125 \\gamma + 68)|X|-120 \\gamma^2 -250 \\gamma -132). \\]\n\\end{cor}\n\n\\section{Bounded Row Treewidth Classes}\n\\label{BoundedRowTreewidth}\n\\label{PlanarGraphs}\n\nThis section presents upper bounds on reduced bandwidth and twin-width for powers of graphs with bounded row-treewidth, which includes planar graphs, graphs with Euler genus $\\gamma$, $(\\gamma,k)$-planar graphs, and map graphs. The heart of the proof is \\cref{ProductPath} below, which depends on the following definition. As illustrated in \\cref{fig:sxq}, for $x,q\\in\\mathbb{N}$ with $q\\geq 2$, let \\defn{$S^*_{x,q}$} be the graph where:\n\\begin{itemize}\n    \\item $V(S^*_{x,q})$ is the disjoint union of sets $Q,A_1,\\dots,A_x,B_1,\\dots,B_x,C_1,\\dots,C_x$ with $|Q|=2q-1$ and $|A_i|=|B_i|=|C_i|=q$ for all $i\\in\\{1,\\dots,x\\}$, and\n    \\item $Q\\cup A_1$ is a clique, \n$Q\\cup B_1$ is a clique, \n$Q\\cup C_1$ is a clique, and for all $\\in\\{1,\\dots,x-1\\}$, \n$A_i\\cup A_{i+1}$ is a clique, \n$B_i\\cup B_{i+1}$ is a clique, and\n$C_i\\cup C_{i+1}$ is a clique.\n\\end{itemize}  \nFor $x,q,r\\in\\mathbb{N}$ with $q\\geq 2$, let $S_{x,q,r}$ be the graph obtained from $(S^*_{x,q})^r$ by removing all edges between $B_1\\cup\\dots\\cup B_x$ and $C_1\\cup\\dots\\cup C_x$. We call $Q$ the \\defn{center} of $S_{x,q,r}$. Note that the vertices in $Q$ have maximum degree in $S_{x,q,r}$. Thus \n\\begin{equation}\n\\label{DeltaS}\n\\Delta(S_{x,q,r})\\le (3r+2)q-2.\n\\end{equation}\nConsidering the vertex-ordering $A_x,A_{x-1},\\dots,A_1,Q,B_1,C_1,B_2,C_2,\\dots,B_x,C_x$ we see that \n\\begin{equation}\n\\label{bwS}\n\\bw(S_{x,q,r})\\leq |Q|+\\sum_{i=1}^r(|B_i|+|C_i|)-1=(2q-1)+2qr-1=(2r+2)q-2.\n\\end{equation}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[scale=0.3]{sxq} }\n\\caption{An illustration of $S_{x,q}^*$.}\n\\label{fig:sxq}\n\\end{figure}\n\n\n\\begin{lem}\\label{ProductPath}\nLet $k, q, r\\in\\mathbb{N}$ with $q\\ge k+1$. Let $f$ be a monotone and union-closed graph parameter and let $g:\\mathbb{N}\\times \\mathbb{N}\\to\\mathbb{R}$ be a function such that $f(S_{x,q,r})\\le g(q,r)$ for all $x\\in \\mathbb{N}$.\nLet $P$ be a path and let $H$ be a graph admitting a $(k,q)$-rooted tree-decomposition $(T, \\mathcal{B})$. Let $F$ be a trigraph with  $V(F)\\subseteq V(H\\boxtimes P)$ such that: \n \\begin{itemize}\n    \\item \\textup{(\\defn{red edge condition})} for every red edge $vw$ of $F$, there is a leaf bag $B$ with parent $B'$ in $(T, \\mathcal{B})$ such that $v,w\\in (B\\setminus B')\\times V(P)$;\n     \\item \\textup{(\\defn{separation condition})} for every rooted separation $(C, D)$ of $H$ from $(T, \\mathcal{B})$ and every $z\\in V(P)$, we have\n $\\left| \\left\\{N_{F}(v)\\cap (D\\times V(P)):v\\in ((C\\setminus D)\\times \\{z\\})\\cap V(F)\\right\\}  \\right| \\le q$; and\n   \\item \\textup{(\\defn{neighbourhood condition})} for every $z\\in V(P)$ and $v\\in (V(H)\\times \\{z\\}) \\cap V(F)$, we have $N_{F}[v]\\subseteq V(H)\\times N^r_P[z]$.\n \\end{itemize}\nThen $\\reduced{f}(F)\\le g(q, r)$.\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $V(F)=V(H\\boxtimes P)$ because adding isolated vertices preserves the above three conditions and $\\reduced{f}(F)$. Say $P=(w_1,w_2,\\dots,w_{\\ell})$. Let $\\mathcal{T}:=(T, \\mathcal{B})$ and let $R$ be the root bag of $\\mathcal{T}$.\nWe proceed by induction on the number of bags of $\\mathcal{T}$. Since the root bag is empty, $\\mathcal{T}$ has at least two bags.\n\nFirst suppose that $\\mathcal{T}$ consists of exactly two bags. \nSince $(T,\\mathcal{B})$ is $(k,q)$-rooted, $|V(H)|\\leq q$. \nBy the neighbourhood condition, the underlying graph of $F$  is isomorphic to a subgraph of $S_{\\ell,q,r}$. By assumption, $f(S_{\\ell,q,r})\\le g(q,r)$, and since $f$ is monotone, for every subgraph $Y$ of $S_{\\ell,q,r}$, $f(Y)\\le g(q, r)$. We obtain a reduction sequence of $F$ as follows. For $i=1,\\dots,\\ell-1$, arbitrarily identify $V(H)\\times \\{w_i\\}$ into a vertex,  and then identify the resulting vertex with a vertex in $V(H)\\times \\{w_{i+1}\\}$. Lastly, we identify $V(H)\\times \\{w_\\ell\\}$ into a vertex. The underlying graph of every trigraph in this reduction sequence is isomorphic to a subgraph of $S_{\\ell, q,r}$, which shows that $\\reduced{f}(F)\\le g(q,r)$.\n\n\nNow assume that $\\mathcal{T}$ has at least three bags. Let $B$ be an internal bag at maximum distance in $\\mathcal{T}$ from $R$. So all the children of $B$ are leaf bags. If $B=R$, then let $Y:=\\emptyset$; otherwise, let $Y:=B\\cap B'$ where $B'$ is the parent of $B$. \n\nFirst suppose that $B$ has at least two child bags $Q$ and $Q'$. \nIf $\\abs{(Q\\cup Q')\\setminus B}\\le q$, then \nwe obtain a tree-decomposition from $\\mathcal{T}$ by removing $Q$ and $Q'$ and attaching a leaf bag $Q\\cup Q'$ to $B$. This results in a new tree-decomposition that satisfies the given conditions and has one fewer bag. So we are done by  induction. Now assume that \n$\\abs{(Q\\cup Q')\\setminus B}> q$.\n\nSince  $(T, \\mathcal{B})$ is $(k,q)$-rooted, $\\abs{(Q\\cup Q')\\setminus B}\\le  2q$.\nFurthermore, since $V(F)=V(H\\boxtimes P)$, for each $w\\in V(P)$, we have $\\abs{((Q\\cup Q')\\setminus B)\\times \\{w\\}}=\\abs{(Q\\cup Q')\\setminus B}> q$.\nLet \n$C:=Q\\cup Q'\\cup B$ and \n$D:=(V(H)\\setminus (Q\\cup Q')) \\cup B$.\nNote that $(C, D)$  is a rooted separation of $H$ at $B$. \nBy the separation condition, for each $w\\in V(P)$, \n \\[\\left| \\left\\{N_{F}(v)\\cap (D\\times V(P)):v\\in ((C\\setminus D)\\times \\{w\\})\\cap V(F)\\right\\}  \\right| \\le q.\\] \nSo, for each $w\\in V(P)$, there are distinct vertices $y$ and $z$ in $((Q\\cup Q')\\setminus B)\\times \\{w\\}$ having the same neighbourhood on $D\\times V(P)$. \n\nFor $i=1,2,\\dots,\\ell$, reduce $((Q\\cup Q')\\setminus B)\\times \\{w_i\\}$ into a set of $q$ vertices, by repeatedly choosing two vertices having the same neighbourhood on $D\\times V(P)$. Note that we create no red edge incident with $D\\times V(P)$. \n\n\nSuppose that $U$ is the red graph constructed immediately  after identifying some vertices of $((Q\\cup Q')\\setminus B)\\times \\{w_i\\}$ for some $i$. We claim that  $f(U)\\le g(q,r)$.\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[scale=0.3]{sxq2-index} }\n\\caption{Identifications on $\\bigcup_{j\\in \\{1, 2, \\ldots, \\ell\\}} ((Q\\cup Q')\\setminus B)\\times \\{w_j\\}$, when $r=1$.}\n\\label{fig:sxq2}\n\\end{figure}\n\nFor each $j<i$, $((Q\\cup Q')\\setminus B)\\times \\{w_j\\}$ has been identified to a set of $q$ vertices, say $W_j$. For each $j>i$, $(Q\\setminus B)\\times \\{w_j\\}$ and $(Q'\\setminus B)\\times \\{w_j\\}$ are not yet identified, and so there are only black edges between $\\bigcup_{j>i} (Q\\setminus B)\\times \\{w_j\\}$ and $\\bigcup_{j>i} (Q'\\setminus B)\\times \\{w_j\\}$ by the red edge condition.  Also, $((Q\\cup Q')\\setminus B)\\times \\{w_i\\}$ has been identified to at most $2q-1$ vertices, because at least one pair of vertices is identified. Call it $W_i$.  See \\cref{fig:sxq2} for an illustration. \n\nLet \n\\[A_1:=\\bigcup_{j\\le i}W_i\\cup \\left(\\bigcup_{j>i} (Q\\setminus B)\\times \\{w_j\\}\\right) \\cup \\left(\\bigcup_{j>i} (Q'\\setminus B)\\times \\{w_j\\}\\right), \\]\nand let $A_2:=V(U)\\setminus A_1$. Observe that $A_2=D\\times V(P)$.\nSince we create no red edge incident with $D\\times V(P)$ during the identifications, there is no edge between $A_1$ and $A_2$ in $U$. Furthermore, by the red edge condition and the neighbourhood condition, for each component of $U[A_2]$, its underlying graph is a subgraph of $S_{\\ell,q,r}$.\nThus, $f(U[A_2])\\le g(q,r)$. \nAlso, the underlying graph of $U[A_1]$ is a subgraph of $S_{\\ell, q, r}$ where $W_i$ is a subset of the center of $S_{\\ell, q,r}$. So, $f(U[A_1])\\le g(q,r)$. Since $f$ is union-closed, $f(U)\\le g(q,r)$.\n\n\nNow we explain how to apply the induction hypothesis to the resulting trigraph.\nLet $F'$ be the resulting trigraph. Let $H'$ be the graph obtained from $H$ by removing $(Q\\cup Q')\\setminus B$ and adding a clique $Z$ of size $q$ that is complete to $B$. We obtain a tree-decomposition $(T', \\mathcal{B}')$ of $H'$ from $(T, \\mathcal{B})$ by removing bags $Q$ and $Q'$, and adding a new bag $Z\\cup B$ incident with $B$ (and $|(Z\\cup B)\\setminus B|\\leq q$ as desired). Observe that $F'$ is a trigraph with $V(F')\\subseteq V(H'\\boxtimes P)$ satisfying the red edge and neighbourhood conditions. \n\nWe verify that \n$F'$, $H'$, and $(T', \\mathcal{B}')$ satisfy the separation condition. Let $(C', D')$ be a rooted separation of $F'$ of $H'$ from $(T', \\mathcal{B}')$, and let $z\\in V(P)$. \n\n\\textbf{Case 1.} $Z\\subseteq C'$: \n\nLet $(C_{pre}, D')$ be the rooted separation of $H$ obtained from $(C', D')$ by removing $Z$ from $C'$ and adding $(Q\\cup Q')\\setminus B$. Since we identified two vertices in $(C_{pre}\\setminus D')\\times \\{z\\}$ that have the same neighbourhood on $D' \\times V(P)$, \n\\begin{align*}\n    &\\left| \\left\\{N_{F'}(v)\\cap (D'\\times V(P)):v\\in ((C'\\setminus D')\\times \\{z\\})\\cap V(F')\\right\\}  \\right| \\\\\n    \\le &\\left| \\left\\{N_{F}(v)\\cap (D'\\times V(P)):v\\in ((C_{pre}\\setminus D')\\times \\{z\\})\\cap V(F)\\right\\}  \\right| \\le q.\n\\end{align*}\n\n\\textbf{Case 2.} $Z\\subseteq D'$:\n\nLet $(C', D_{pre})$ be the rooted separation of $H$ obtained from $(C', D')$ by removing $Z$ from $D'$ and adding $(Q\\cup Q')\\setminus B$. Again, since we identified two vertices in $(D_{pre}\\setminus C')\\times \\{z\\}$ that have the same neighbourhood on $C'\\times V(P)$, \n\\begin{align*}\n    &\\left| \\left\\{N_{F'}(v)\\cap (D'\\times V(P)):v\\in ((C'\\setminus D')\\times \\{z\\})\\cap V(F')\\right\\}  \\right| \\\\\n    \\le &\\left| \\left\\{N_{F}(v)\\cap (D_{pre}\\times V(P)):v\\in ((C'\\setminus D_{pre})\\times \\{z\\})\\cap V(F)\\right\\}  \\right| \\le q.\n\\end{align*}\n\nThus, $F'$, $H'$, and $(T', \\mathcal{B}')$ satisfy the separation condition.\n\nSince $(T', \\mathcal{B}')$ has one fewer bag than $(T, \\mathcal{B})$, by induction, there is a reduction sequence $L'$ of~$F'$, where for every trigraph $G$ in $L'$, $f(\\widetilde{G})\\le g(q,r)$. Together with the partial reduction sequence producing $F'$ from $F$, this gives the desired reduction sequence for $F$.\n\n\nTo finish the proof, it remains to consider the case in which $B$ has exactly one child  bag $Q$. Since $\\mathcal{T}$ has at least three bags, $B$ has its parent $B'$ and $Y=B\\cap B'$. Now, $(B\\cup Q)\\setminus Y$ has at most $q+k+1\\le 2q$ vertices, and we can do the same procedure in Cases 1 or 2 to reduce $((B\\cup Q)\\setminus Y)\\times \\{w\\}$ (for each $w\\in V(P)$) to a set of at most $q$ vertices by identifying two vertices having the same neighbourhood on $\\bigcup_{j\\in \\{1, 2, \\ldots, \\ell\\}} Y\\times \\{w_j\\}$. This will correspond to vertices of $H$ forming one bag with $Y$.\nNote that at the beginning, all edges between $\\bigcup_{j\\in \\{1, 2, \\ldots, \\ell\\}} ((B\\cup Q)\\setminus Y)\\times \\{w_j\\}$ and $\\bigcup_{j\\in \\{1, 2, \\ldots, \\ell\\}} Y\\times \\{w_j\\}$ are black.\nSo, the underlying graphs of red graphs constructed from $\\bigcup_{j\\in \\{1, 2, \\ldots, \\ell\\}} ((B\\cup Q)\\setminus Y)\\times \\{w_j\\}$ will be subgraphs of $S_{\\ell,q,r}$.\n\nLet $F'$ be the resulting trigraph. Let $H'$ be the graph obtained from $H$ by removing $(B\\cup Q)\\setminus Y$ and adding a clique $Z$ of size $q$ that is complete to $Y$.\nWe obtain a tree-decomposition $(T', \\mathcal{B}')$ of $H'$ from $(T, \\mathcal{B})$ by removing bags $Q$ and $B$, and adding a new bag $Z\\cup Y$ incident with $B'$ (and $|Z\\setminus B'|\\leq q$ as desired). \nIt is not difficult to see that $F', H', (\\mathcal{T}', \\mathcal{B}')$ satisfy the red edge, separation, and neighbourhood conditions. \nSo, we can apply induction, which completes the proof of the lemma.\n \\end{proof}\n\n\nWe now rewrite \\cref{ProductPath} in a more useful form. \n\n\\begin{lem}\n\\label{RowTreewidthNeighbourhoodPowers} \nLet $k,r,\\pi^*\\in\\mathbb{N}$. Let $G$ be a graph with row-treewidth $k$, such that $\\pi^r_G(X) \\leq \\pi^* $ for every set $X\\subseteq V(G)$ with $|X|\\leq (2r+1)(k+1)$. Then \n\\[\\reduced{\\bw}(G^r) \\leq (2r+2)\\pi^*-2\n\\quad\\text{and}\\quad\n\\tww(G^r) \\leq (3r+2)\\pi^*-2.\\]\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $G\\subseteq H\\boxtimes P$, where $\\tw(H)\\leq k$ and $P$ is a path. We now show that \\cref{ProductPath} is applicable, where $F$ is the trigraph obtained from $G^r$ with no red edges. The red edge condition holds trivially. \n\nConsider a  rooted tree-decomposition $(T, \\mathcal{B})$ of $H$ with width at most $k$. Let $(C, D)$ be a rooted separation of $(T, \\mathcal{B})$. Consider $z\\in V(P)$ and $v\\in ((C\\setminus D)\\times \\{z\\})\\cap V(F)$. Every path in $G$ from $v$ to $D\\times V(P)$ with length at most $r$ must intersect $(C\\cap D)\\times N^r_P[z]$. Thus $N_{F}(v)\\cap (D\\times V(P))$ is determined by the distance-$r$ profile of $v$ on $(C\\cap D) \\times N^r_{P}[z]$, which has at most $(2r+1)(k+1)$ vertices. Thus\n\\begin{align*}\n  \\left| \\left\\{N_{F}(v)\\cap (D\\times V(P)):v\\in ((C\\setminus D)\\times \\{z\\})\\cap V(F)\\right\\}  \\right| \n\\,\\le\\, \n\\pi^r_G((C\\cap D) \\times N^r_{P}[z]) \n\\,\\leq\\, \\pi^*.\n\\end{align*}\nHence the separation condition in \\cref{ProductPath} holds with $q=\\pi^*$. The neighbourhood condition holds since $N_F[v]=N^r_G[v]$ and $G\\subseteq H\\boxtimes P$.\n\nFor the upper bound on $\\reduced{\\bw}$, we may apply \\cref{ProductPath} with $f=\\bw$ and $g(q,r)=(2r+2)q-2$ by \\cref{bwS}. Thus $\\reduced{\\bw}(G)\\leq g(q,r) = (2r+2)\\pi^*-2$. \n\nFor the upper bound on $\\tww$, we may apply \\cref{ProductPath} with $f=\\Delta$ and $g(q,r)=(3r+2)q-2$ by \\cref{DeltaS}. Thus $\\tww(G)\\leq g(q,r) = (3r+2)\\pi^*-2$.\n\\end{proof}\n\n\nSince $\\pi^r_G(A)\\leq(r+1)^{|A|}$, \\cref{RowTreewidthNeighbourhoodPowers} is applicable with \n$\\pi^*=(r+1)^{(2r+1)(k+1)}$. Thus:\n\n\\begin{cor}\n\\label{RowTreewidPowers} \nFor every graph $G$ with row-treewidth $k$ and for $r\\in\\mathbb{N}$,\n$$\\reduced{\\bw}(G^r) \\leq 2 (r+1)^{(2r+1)(k+1)+1} -2\n\\quad\\text{and}\\quad\n\\tww(G^r) \\leq (3r+2)(r+1)^{(2r+1)(k+1)}-2.$$\n\\end{cor}\n\n\\cref{RowTreewidPowers,GenusProductStructure} imply:\n\n\\begin{cor}\n\\label{GenusPowers} \nFor every graph $G$ with Euler genus $\\gamma$ and for $r\\in\\mathbb{N}$,\n$$\\reduced{\\bw}(G^r) \\leq 2 (r+1)^{(2r+1)(2\\gamma+7)+1} -2\n\\quad\\text{and}\\quad\n\\tww(G^r) \\leq (3r+2)(r+1)^{(2r+1)(2\\gamma+7)}-2.$$\nIn particular, for every planar graph $G$ and $r\\in\\mathbb{N}$,\n$$\\reduced{\\bw}(G^r) \\leq 2 (r+1)^{14r+8} -2\n\\quad\\text{and}\\quad\n\\tww(G^r) \\leq (3r+2)(r+1)^{14r+7}-2.$$\n\\end{cor}\n\nApplying the neighbourhood complexity bounds from \\cref{FirstNeigh} we obtain the following improved bounds in the $r=1$ case. \n\n\n\\begin{thm}\n\\label{ReducedBandwidthGenus}\n\\label{TwinwidthGenus}\n\\label{PlanarReducedBandwidth}\n\\label{PlanarTwinwidth}\nFor every graph $G$ with Euler genus $\\gamma$,\n$$\\reduced{\\bw}(G)\\leq 164\\gamma+466\\quad\\text{and}\\quad\n\\tww(G)\\leq 205\\gamma+583.$$\nIn particular, for every planar graph $G$,\n\\[\\reduced{\\bw}(G) \\leq 466\\quad\\text{and}\\quad\\tww(G) \\leq 583.\\]\n\\end{thm}\n\n\\begin{proof}\nBy \\cref{GenusProductStructure}, $G$ has row-treewidth at most $k:=2\\gamma+6$. By \\cref{SurfaceNeighbours}, for every $X\\subseteq V(G)$ we have $\\pi^1_G(X)\\le \\max\\{4,6|X|+5\\gamma-9\\}$, which is at most $41\\gamma+117$ when $|X| \\le 3(k+1)=6\\gamma+21$. The result thus follows from \\cref{RowTreewidthNeighbourhoodPowers} with $r=1$ and $\\pi^*=41\\gamma+117$.\n\\end{proof}\n\n\n\\begin{thm}\n\\label{ReducedBandwidthTwinwidthCol}\nFor every graph $G$ with row-treewidth $k$ and $\\col_5(G)\\leq c$,\n\\[\n\\reduced{\\bw}(G^r) \\leq 5 \\min\\{ 2^{c-1}( 3k-c+1), 2^{3(k+1)} \\}\n\\quad\\text{and}\\quad\n\\tww(G^r) \\leq 8 \\min\\{ 2^{c-1}( 3k-c+1), 2^{3(k+1)} \\}\n\\]\n\\end{thm}\n\n\n\n\\begin{proof}\nBy \\cref{NeighCol}, for every $X\\subseteq V(G)$ we have \n$ \\pi^1_G(X) \\leq \\min\\{ 2^{c-1}( |X|-c+2), 2^{|X|} \\}$, \nwhich is at most \n$\\min\\{ 2^{c-1}( 3k-c+1), 2^{3(k+1)} \\}$ \nwhen $|X| \\le 3(k+1)$. The result thus follows from \\cref{RowTreewidthNeighbourhoodPowers} with $r=1$ and \n$\\pi^*=\\min\\{ 2^{c-1}( 3k-c+1), 2^{3(k+1)} \\}$.\n\\end{proof}\n\n\n\\begin{thm}\n\\label{gkPlanar}\nEvery $(\\gamma,k)$-planar graph $G$ has reduced bandwidth, \n$$\\reduced{\\bw}(G)\\leq 2^{O(\\gamma k)}.$$\n\\end{thm}\n\n\\begin{proof}\nBy \\cref{gkPlanarProduct}, $G$ has row-treewidth $O(\\gamma k^6)$. By \\cref{NeighGammak}, for every $X\\subseteq V(G)$ we have\n$\\pi^1_G(X) \\leq 2^{22(2\\gamma+3)(k+1)-1}|X|$, which is \nin $2^{O(\\gamma k)}$ when $|X| \\in O(\\gamma k^6)$. The result thus follows from \\cref{RowTreewidthNeighbourhoodPowers} with $r=1$ and  $\\pi^*=2^{O(\\gamma k)}$.\n\\end{proof}\n\n\nWe have the following result for squares of graphs of given Euler genus. \n\n\\begin{thm}\n\\label{GenusSquare}\nFor every graph $G$ of Euler genus $\\gamma$,\n\\begin{align*}\n    \\reduced{\\bw}(G^2) & \\leq\n234000 \\gamma^4 + 1983300 \\gamma^3  + 5769330 \\gamma^2   + 6671550 \\gamma + 2711084\n\\quad\\text{and}\\\\\n\\tww(G^2) & \\leq \n312000 \\gamma ^4\n+ 2644400 \\gamma ^3\n + 7692440 \\gamma ^2\n + 8895400 \\gamma \n + 3614782.\n\\end{align*}\nIn particular, for every planar graph $G$,\n\\begin{align*}\n    \\reduced{\\bw}(G^2)  \\leq 2711084\n\\quad\\text{and}\\quad\n\\tww(G^2) & \\leq  3614782.\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nBy \\cref{GenusProductStructure}, $G$ has row-treewidth at most $k:=2\\gamma+6$. By \\cref{SurfaceNu2}, for every set $X\\subseteq V(G)$,\n\\[ \\pi^2_G(X) \\leq \\max\\{ 2, (6 |X| + 5\\gamma -9) ( (60 \\gamma^2 + 125 \\gamma + 68)|X|-120 \\gamma^2 -250 \\gamma -132) \\}. \\]\nThe result thus follows from \\cref{RowTreewidthNeighbourhoodPowers} with $r=2$ and \n\\begin{align*}\n\\pi^* \n& = \n( 6 ( 5(k+1) ) + 5\\gamma -9) ( (60 \\gamma^2 + 125 \\gamma + 68) 5(k+1) -120 \\gamma^2 -250 \\gamma -132)\\\\\n& = \n39000 \\gamma^4 + 330550 \\gamma^3+ 961555 \\gamma^2 + 1111925 \\gamma + 451848 .\\qedhere\n\\end{align*}\n\\end{proof}\n\nWe now give an example of the application of \\cref{GenusSquare}. \nMap graphs are defined as follows. Start with a graph $G_0$ embedded in a surface of Euler genus $\\gamma$, with each face labelled a `nation' (or a `lake'). Let $G$ be the graph whose vertices are the nations of $G_0$, where two vertices are adjacent in $G$ if the corresponding faces in $G_0$ share a vertex. Then $G$ is called a \\defn{map graph} of Euler genus $\\gamma$. A map graph of Euler genus 0 is called a (plane) \\defn{map graph}; see \\citep{FLS-SODA12,CGP02} for example. Map graphs generalise graphs embedded in surfaces, since a graph $G$ has Euler genus at most $\\gamma$ if and only if  $G$ has a representation as a map graph of Euler genus at most $\\gamma$ with the extra property that each vertex of $G_0$ is incident with at most three nations \\citep{DEW17}. Our results for map graphs are independent of the number of nations incident to each vertex of $G_0$.  It follows from the definition that for each map graph $G$ of Euler genus $\\gamma$, there is a bipartite graph $H$ with bipartition $\\{A,B\\}$, such that $H$ has Euler genus at most $\\gamma$ and $V(G) = A$ and $vw \\in E(G)$ whenever $v,w \\in N_H(b)$ for some $b \\in B$ (see \\citep{DEW17}); $G$ is called the \\defn{half-square} of $H$. Note that $G$ is an induced subgraph of $H^2$. Since reduced bandwidth is hereditary\\footnote{It is an easy exercise to show that $\\reduced{f}$ is hereditary for every monotone graph parameter $f$; see \\citep[Section~4.1]{TW-I} for a proof sketch in the case of twin-width.}, the results in \\cref{GenusSquare} also hold for map graphs of Euler genus $\\gamma$.  We emphasise there is no dependence on the maximum degree (which is customary when considering map graphs). \n\nThe proof of \\cref{ProductPath} is constructive and the desired reduction sequence can be found in time $O(|V(F)|^2)$ when $H$ and $P$ are given. For all the above examples, the graphs $H$ and $P$ can be found in $O(|V(G)|^2)$ time; see \\citep{DJMMUW20,Morin21,DMW,BMO} for details and speed-ups. So the desired reduction sequence can be found in  $O(|V(G)|^2)$ time.\n\n\n\\section{Proper Minor-Closed Classes}\n\\label{ProperMinorClosedClasses}\n\nThis section shows that fixed powers of graphs in any proper minor-closed class have bounded reduced bandwidth. To use the product structure theorem for $K_t$-minor-free graphs (\\cref{MinorFree}), we first prove an upper bound on the reduced bandwidth of the $r$-th power of a subgraph of $(H\\boxtimes P)+K_a$  where $H$ has bounded treewidth, $P$ is a path, and $a\\in \\mathbb{N}$. To do so, in \\cref{lem:productpathpowerrpower} below we prove an extension of \\cref{ProductPath} that allows for apex vertices. \nConsider the $r$-th power of a subgraph of $(H\\boxtimes P)+K_a$. There may exist an edge $vw$ with $v,w\\in V(H)\\times V(P)$ where the corresponding vertices in $P$ are far apart in $P$, because there may exist a path of length at most $r$ through some vertices in $K_a$. So the neighbourhood condition of \\cref{ProductPath} is no longer relevant. Instead, red edges are controlled by a new `red edge condition', and black edges are controlled by two separation conditions. The second separation condition is necessary to deal with the base case when the tree-decomposition of $H$ has one non-empty bag. \n\n\n\n\n\\begin{lem}\\label{lem:productpathpowerrpower}\nLet $a, k, q, r\\in\\mathbb{N}$ with $q\\ge k+1$. \nLet $f$ be a monotone and union-closed graph parameter and let $g:\\mathbb{N}\\times \\mathbb{N}\\to\\mathbb{R}$ be a function such that $f(S_{x,q,r})\\le g(q,r)$ for all $x\\in \\mathbb{N}$.\nLet $P=(w_1,w_2,\\dots,w_{\\ell})$ be a path and let $H$ be a graph admitting a $(k,q)$-rooted tree-decomposition $(T, \\mathcal{B})$.\n Let $F$ be a trigraph with  $V(F)\\subseteq V((H\\boxtimes P)+K_a)$ such that: \n   \\begin{itemize}\n   \\item \\textup{(\\defn{red edge condition})} for every red edge $vw$ of $F$, there is a leaf bag $B$ with parent $B'$ in $(T, \\mathcal{B})$ and there are vertices $x,y$ in $P$ with $\\dist_P(x,y)\\le r$, such that $v,w\\in (B\\setminus B')\\times \\{x,y\\}$, \n  \\item \\textup{(\\defn{first separation condition})} for every rooted separation $(C, D)$ of $H$ from $(T, \\mathcal{B})$ and every $z\\in V(P)$, \n \\[\\left| \\left\\{N_{F}(v)\\cap M_1:v\\in ((C\\setminus D)\\times\\{z\\})\\cap V(F)\\right\\}  \\right| \\le q,\\] where $M_1=\\Big(D\\times V(P)\\Big) \\cup \\Big(C\\times (V(P)\\setminus N^r_P[z])\\Big)\\cup V(K_a)$,\n \\item \\textup{(\\defn{second separation condition})} for every $t\\in \\{1,\\ldots, \\ell\\}$, \n \\[|\\{N_F(v)\\cap M_2: v\\in (V(H)\\times \\{w_1,\\dots,w_t\\})\\cap V(F)\\}|\\le q,\\] where $M_2=\\big(V(H)\\times \\{w_{t+r+1},\\dots,w_\\ell\\}\\big)\\cup V(K_a)$.\n \\end{itemize}\n Then:\n \\begin{enumerate}[(1)]\n     \\item there is a partial reduction sequence identifying $(V(H)\\times V(P))\\cap V(F)$ into at most $q$ vertices such that for every trigraph $G$ in the sequence, $G$ has no red edge incident with $V(K_a)$ and $f(\\widetilde{G})\\le g(q,r)$, and\n     \\item  $\\reduced{f}(F)\\le  g(a+q,r)$.\n \\end{enumerate}\n  \\end{lem}\n\\begin{proof}\n     We identify the $V(H)\\times V(P)$ part using a partial reduction sequence similar to the one in the proof of~\\cref{ProductPath}. The main difference in the induction step is that when we consider a rooted separation $(C,D)$ from $(T, \\mathcal{B})$ and $z\\in V(P)$, instead of choosing two vertices having the same neighbourhood on $D\\times V(P)$, here we choose two vertices having the same neighbourhood on \\[\\Big(D\\times V(P)\\Big) \\cup \\Big(C\\times (V(P)\\setminus N^r_P[z])\\Big)\\cup V(K_a).\\] This clearly preserves the new red edge and separation conditions. So, as in the proof of~\\cref{ProductPath}, choose an internal bag $B$ at maximum distance in $(T, \\mathcal{B})$ from the root bag. Then for $i=1,2,\\dots,\\ell$, merge $(Q\\setminus B)\\times \\{w_i\\}$ and $(Q'\\setminus B)\\times \\{w_i\\}$ for each $w_i\\in V(P)$ if $B$ has two child bags $Q$ and $Q'$, and otherwise, merge $(B\\setminus B')\\times \\{w_i\\}$ and $(Q\\setminus B')\\times \\{w_i\\}$ for the unique child $Q$ of $B$ and the parent $B'$ of $B$.\n     Because of the choice of identified vertices, for each red component of an intermediate trigraph, its underlying graph is isomorphic to a subgraph of $S_{\\ell, q, r}$, and has $f$ at most $g(q,r)$.\n    \n    Now, we consider the base case where $(T, \\mathcal{B})$ has two bags $R$ and $B$, where $R$ is the empty root bag.\n     We need a new argument for this case, because if we arbitrarily identify two vertices, then we may create some red edges between two vertices contained in $V(H)\\times\\{x\\}$ and $V(H)\\times \\{y\\}$ where $x$ and $y$ are far from each other in $P$, and we cannot guarantee that the red graphs have bounded $f$-values.\n     \n     We prove by induction on $\\ell$ that there is a partial reduction sequence identifying $(V(H)\\times V(P))\\cap V(F)$ into at most $q$ vertices, such that for every trigraph $G$ in the sequence, \\begin{itemize}\n        \\item $G$ has no red edge incident with $V(K_a)$, and\n    $f(\\widetilde{G})\\le g(q,r)$. \n     \\end{itemize}\n    We may assume that $H$ is a complete graph on $q$ vertices. If $\\ell=1$, then $V(H)\\times V(P)$ has at most $q$ vertices, and there is nothing to show. \n    \n    We assume that $\\ell>1$.\n    If $|(B\\times \\{w_1\\})\\cup (B\\times \\{w_2\\})|\\le q$, then remove $B\\times \\{w_1\\}$ and replace $B\\times \\{w_2\\}$ with $(B\\times \\{w_1\\})\\cup (B\\times \\{w_2\\})$. Thus we can reduce the length of $P$, and are done by  induction.\n    Now assume that $|(B\\times \\{w_1\\})\\cup (B\\times \\{w_2\\})|> q$.\n    By the second separation condition, \n    there are two vertices in $(B\\times \\{w_1\\})\\cup (B\\times \\{w_2\\})$\n    that have the same neighbourhood on \\[M_2=\\big(V(H)\\times \\{w_{2+r+1},\\dots,w_\\ell\\}\\big)\\cup V(K_a)\\]\n    By repeatedly identifying two vertices having the same neighbourhoods on $M_2$, we identify $(B\\times \\{w_1\\})\\cup (B\\times \\{w_2\\})$ into at most $q$ vertices. In this way, we can reduce the length of $P$, and we are done by induction. This proves (1).\n    \n    For (2), observe that the statement (1) holds when we replace $g(q,r)$ with $g(a+q, r)$, because every subgraph of $S_{\\ell, q, r}$ is also a subgraph of $S_{\\ell, a+q, r}$. We obtain the desired reduction sequence by arbitrarily identifying the resulting trigraph on at most $a+q$ vertices into a 1-vertex graph.\n \\end{proof}\n\n\n We extend \\cref{lem:productpathpowerrpower} to deal with the internal node case of the tree-decomposition of $X$-minor free graphs. We now allow bags of size more than $q$, and we start by identifying vertices corresponding to those bags. The first three conditions in \\cref{lem:productpathpowerrpower2} are the same as the conditions in \\cref{lem:productpathpowerrpower}. We use \\cref{lem:productpathpowerrpower} as a base case.\n\n\\begin{lem}\\label{lem:productpathpowerrpower2}\nLet $a, k, q, r, t\\in\\mathbb{N}$ with $q\\ge k+1$. Let $f$ be a monotone and union-closed graph parameter. Let $g:\\mathbb{N}\\times \\mathbb{N}\\to\\mathbb{R}$ be a function such that $f(S_{x,q,r})\\le g(q,r)$  for all $x\\in \\mathbb{N}$. Let $P=(w_1,w_2,\\dots,w_{\\ell})$ be a path and let $H$ be a graph admitting a $(k,\\infty)$-rooted tree-decomposition $(T, \\mathcal{B})$. Let $F$ be a trigraph with  $V(F)\\subseteq V((H\\boxtimes P)+K_a)$ such that the red edge, first separation and second separation conditions from \\cref{lem:productpathpowerrpower} are satisfied, and in addition: \n \\begin{itemize}\n \\item \\textup{(\\defn{large leaf bag condition})} \n for every leaf bag $B$ with parent bag $B'$ and $|B\\setminus B'|>q$, \n \\begin{itemize}\n     \\item $B\\times V(P)\\subseteq B\\times \\{x,y\\}$ for some adjacent vertices $x,y$ of $P$,\n     \\item there is a partial reduction sequence identifying $((B\\setminus B')\\times V(P))\\cap V(F)$ into at most $q$ vertices such that for every trigraph $G$ in the reduction sequence, $G$ has no red edge incident with $((V(H)\\setminus (B\\setminus B'))\\times V(P))\\cup V(K_a)$ and $f(\\widetilde{G})\\le t$.\n \\end{itemize} \n \\end{itemize}\n  Then:\n \\begin{enumerate}[(1)]\n     \\item There is a reduction sequence identifying $(V(H)\\times V(P))\\cap V(F)$ into at most $q$ vertices such that for every trigraph $G$ in the reduction sequence, $G$ has no red edge incident with $V(K_a)$ and $f(\\widetilde{G})\\le \\max (g(q,r), t)$.\n     \\item  $\\reduced{f}(F)\\le \\max (g(a+q,r), t)$.\n \\end{enumerate}\n \\end{lem}\n \n \\begin{proof}\n We say that a leaf bag $B$ of $(T, \\mathcal{B})$ with parent bag $B'$ is \\defn{large} if $|B\\setminus B'|> q$.\n We prove (1) by induction on the number of large leaf bags.\n If there are no large leaf bags, then $(T, \\mathcal{B})$ is $(k,q)$-rooted, and the result follows from \\cref{lem:productpathpowerrpower}.\n Now assume that $(T, \\mathcal{B})$ contains a large leaf bag.\n \n Let $B$ be a large leaf bag with parent bag $B'$.\n Let $U_1:=(B\\setminus B')\\times V(P)$ and $U_2:=((V(H)\\setminus (B\\setminus B'))\\times V(P))\\cup V(K_a)$.\n \n Let $L_B$ be a partial reduction sequence identifying $U_1\\cap V(F)$, as described in the large leaf bag condition. By the red edge condition, there is no red edge between $U_1$ and $U_2$. Apply $L_B$ to $U_1\\cap V(F)$. By assumption, we always identify two vertices having the same neighbourhoods on $U_2$, and therefore, it creates no red edge incident with $U_2$. Since $B\\times V(P)\\subseteq B\\times \\{x,y\\}$ for some adjacent vertices $x,y$ of $P$, the resulting trigraph satisfies the red edge condition. Also, each red graph of an intermediate trigraph constructed from $U_1$ has $f$ at most $t$, by assumption. \n  \nLet $F'$ be the resulting trigraph. Let $H'$ be the graph obtained from $H$ by removing $B\\setminus B'$ and adding a clique $Z$ of size $q$ that is complete to $B'$. We obtain a tree-decomposition $(T', \\mathcal{B}')$ of $H'$ from $(T, \\mathcal{B})$ by removing the bag $B$, and adding a new bag $Z\\cup B'$ incident with $B'$ (and $|(Z\\cup B')\\setminus B'|\\leq q$ as desired). Observe that $F'$ is a trigraph with $V(F')\\subseteq V(H'\\boxtimes P)\\cup V(K_a)$ satisfying the four conditions. \n\nSince $(T', \\mathcal{B}')$ has one fewer large leaf bag than $(T, \\mathcal{B})$, by induction, there is a reduction sequence identifying $(V(H')\\times V(P))\\cap V(F')$ into at most $q$ vertices such that for every trigraph $G$ in the reduction sequence, $G$ has no red edge incident with $V(K_a)$ and $f(\\widetilde{G})\\le \\max (g(q,r), t)$. Together with the reduction sequence producing $F'$ from $F$, we obtain a desired reduction sequence for $F$.\n\nSimilar to \\cref{lem:productpathpowerrpower},\n    the statement (1) holds when we replace $g(q,r)$ with $g(a+q, r)$, because every subgraph of $S_{\\ell, q, r}$ is also a subgraph of $S_{\\ell, a+q, r}$. We obtain the desired reduction sequence by arbitrarily identifying the resulting trigraph on at most $a+q$ vertices into a 1-vertex graph.\n    \\end{proof}\n \n \n\\begin{thm}\\label{thm:Hminorfreepower}\n Let $t, r\\in\\mathbb{N}$. Let $f$ be a monotone and union-closed graph parameter  and let $g:\\mathbb{N}\\times \\mathbb{N}\\to\\mathbb{R}$ be a function such that $f(S_{x,q,r})\\le g(q,r)$ for all $x\\in \\mathbb{N}$.\nThen there exists $c\\in\\mathbb{N}$ such that $\\reduced{f}(G^r)\\leq c$ for every $K_t$-minor-free graph $G$. \n\\end{thm}\n\n\\begin{proof}\nBy \\cref{MinorFree}, there exist $k,a\\in\\mathbb{N}$ such that $G$ has  a tree-decomposition $\\mathcal{T}=(T, \\mathcal{B})$ in which every torso of a bag is a subgraph of $(H\\boxtimes P)+K_a$ for some graph $H$ of treewidth at most $k$  and some path $P$. Consider $\\mathcal{T}$ to be rooted at an arbitrary bag $R$ of $\\mathcal{T}$. Let $\\beta:= 2(k+1)+a$. Note that the size of a maximum clique of $(H\\boxtimes P)+K_a$ is at most $\\beta$.\nLet $h$ be the longest path from an internal bag to $R$ in $\\mathcal{T}$. \nDefine\n\\[\\theta:=a+\\tbinom{\\beta}{2}(r+1)+\\beta r,\n\\quad\nq:=(r+1)^{(k+1)(2r+1)+\\theta}\n\\quad \\text{and} \\quad\nc:=g(q,r).\\]\n\nFor each bag $B$ of $\\mathcal{T}$,  if $B=R$ then let $Y_B:=\\emptyset$; otherwise, let $Y_B:=B\\cap B'$ where $B'$ is the parent of $B$.\nAlso, let $U_B$ be the union of $B$ and all the descendants of $B$. Note that $0\\le h-\\dist_T(B,R)\\le h$.\n\nWe prove by induction on $h-\\dist_T(B, R)$ that \n\\begin{itemize}\n    \\item[($\\ast$)] there is a partial reduction sequence identifying $U_B\\setminus Y_B$ into at most $q$ vertices in $G^r$ such that:\n    \\begin{itemize}\n        \\item we always identify two vertices $u$ and $v$ where the corresponding vertices in $G$ have the same distance-$r$ profiles on $Y_B$ in $G[U_B]$, and\n        \\item for every trigraph $G^*$ in the sequence,\n    $f(\\widetilde{G^*})\\le c$.\n    \\end{itemize}\n\\end{itemize}\n\nLet $B$ be a bag. Let $G_1$ be the graph obtained from $G[U_B]$ by adding, for each pair of vertices $y_1, y_2$ of $Y_B$ with $\\dist_{G-E(G[U_B])}(y_1, y_2)\\le r$, a new path $X_{y_1, y_2}$ of length $\\dist_{G-E(G[U_B])}(y_1, y_2)$ whose end-vertices are $y_1$ and $y_2$, and then adding, for every vertex $y$ of $Y_B$, a path $W_y$ of length $r-1$ whose end-vertex is $y$.\n\nWe claim that for any two vertices $v,w\\in U_B$,\n\\begin{itemize}\n    \\item $\\dist_G(v,w)\\le r$ if and only if $\\dist_{G_1}(v,w)\\le r$.\n\\end{itemize} \n\nAssume that there is a path $Z$ of length at most $r$ between $v$ and $w$ in $G$. We construct a path $Z^*$ of $G_1$ from $Z$ as follows.\n\\begin{itemize}\n    \\item We repeatedly choose a maximal subpath $Q$ of $Z$ whose end-vertices are in $U_B$ and all internal vertices are not in $U_B$. Assume its end-vertices are $y_1$ and $y_2$. Then we replace $Q$ with $X_{y_1,y_2}$.\n\\end{itemize}\nBy construction, $|E(X_{y_1, y_2})|=\\dist_{G-E(G[U_B])}(y_1, y_2)\\le |E(Q)|$. So, the resulting path $Z^*$ has length at most $r$, which shows that $\\dist_{G_1}(v, w)\\le r$. \n\nFor the other direction, assume there is a path $Z$ of length at most $r$ between $v$ and $w$ in $G_1$. Similarly, we construct a walk of length at most $r$ in $G$ from $Z$ as follows.\n\\begin{itemize}\n    \\item We repeatedly choose a maximal subpath $Q$ of $Z$ of length at least $2$ whose end-vertices are in $U_B$ and all internal vertices are not in $U_B$.\n    Note that $Q$ is $X_{y_1, y_2}$ for some $y_1, y_2\\in Y$. By the construction, there is a path $Q_{y_1, y_2}$ in $G-E(G[U])$  of length $|E(X_{y_1,y_2})|$. Then we replace $Q$ with $Q_{y_1, y_2}$.\n\\end{itemize}\nThe resulting walk has length at most $r$ between $v$ and $w$ in $G$. Thus, $\\dist_G(v,w)\\le r$.\n\nBy the claim, $G^r[U_B]=(G_1)^r[U_B]$.\n\nNow focus on $G_1$. First observe that $G_1[B]=G[B]$ and its torso is a subgraph of $(H\\boxtimes P)+K_a$ for some graph $H$ of treewidth at most $k$ and some path $P$. Let $(T_B, \\mathcal{B}_B)$ be a rooted tree-decomposition of $H$ of width at most $k$. Let $P=(w_1,w_2,\\dots,w_{\\ell})$.\n\nObserve that $Y_B\\cup (\\bigcup_{\\{y_1, y_2\\}\\in \\binom{Y_B}{2}}V(X_{y_1, y_2}))\\cup (\\bigcup_{y\\in Y_B}V(W_y))$ contains at most  $\\beta+\\binom{\\beta}{2}(r-1)+\\beta(r-1)$ vertices. Since $G_1[B]$ is a subgraph of $(H\\boxtimes P)+K_a$, the graph \\[G_1[B]\\cup \\bigcup_{\\{y_1, y_2\\}\\in \\binom{Y_B}{2}} X_{y_1,y_2}\\cup \\bigcup_{y\\in Y_B}W_y.\\] \ncan be regarded as a subgraph of $(H\\boxtimes P)+K_{\\theta}$, where \n$Y_B\\cup (\\bigcup_{\\{y_1, y_2\\}\\in \\binom{Y_B}{2}}V(X_{y_1, y_2}))\\cup (\\bigcup_{y\\in Y_B}V(W_y))$ \nis placed in the $K_{\\theta}$ part.  Let \n\\[G_2:=G_1[B]\\cup \\bigcup_{\\{y_1, y_2\\}\\in \\binom{Y_B}{2} } X_{y_1,y_2}\\cup \\bigcup_{y\\in Y_B}W_y.\\]\n\nNext, we extend the product structure for $G_2$ into one for $G_1$. We modify $H$ to $H^*$ as follows. Let $B'$ be a child of the bag $B$. Observe that $B\\cap B'$ is a clique in the torso of $B$. Therefore, the vertices of $B\\cap B'$ lie on $(N_B\\times \\{y_1,y_2\\})\\cup V(K_{\\theta})$ for some consecutive two vertices $y_1$ and $y_2$ in $P$ and some bag $N_B$ of $(T_B, \\mathcal{B}_B)$.  \nTo $H$, we add a clique $Q_{B'}$ of size $|U_{B'}|$ that is complete to $N_B$. \nWe add a leaf bag $Q_{B'}\n\\cup N_B$ adjacent to $N_B$, and embed the vertices of $U_{B'}\\setminus B$ into $Q_{B'}\\times \\{y_1\\}$. We do this process for every child of $B$. \n\nLet $H^*$ be the resulting graph obtained from $H$, and let $(T_B^*, \\mathcal{B}_B^*)$ be the resulting rooted tree-decomposition of $H^*$.\nSince $(T_B, \\mathcal{B}_B)$ has width at most $k$, $(T_B^*, \\mathcal{B}_B^*)$ is $(k, \\infty)$-rooted. Also \n$G_1$ is a subgraph of $(H^*\\boxtimes P)+K_{\\theta}$. \n\nWe now apply \\cref{lem:productpathpowerrpower2} to $(G_1)^r$ and $(T_B^*, \\mathcal{B}_B^*)$. To do so, we verify the four conditions. Since $(G_1)^r$ has no red edges, the red edge condition holds. \n\n(First separation condition) Let $(C,D)$ be a rooted separation of $H^*$ from $(T_B^*, \\mathcal{B}_B^*)$ and let $z\\in V(P)$. Let \\[M_1:=\\Big(D\\times V(P)\\Big) \\cup \\Big(C\\times (V(P)\\setminus N^r_P[z])\\Big)\\cup V(K_\\theta).\\]\nObserve that if there is a path of length at most $r$ in $G_1$ between $((C\\setminus D)\\times \\{z\\})\\cap V(G_1)$ and $M_1$, \nthis path intersects $((C\\cap D)\\times N^r_P[z])\\cup V(K_\\theta)$. Thus two vertices $v, w\\in ((C\\setminus D)\\times \\{z\\})\\cap V(G_1)$ having the same distance-$r$ profiles on \n$((C\\cap D)\\times N^r_P[z])\\cup V(K_\\theta)$ in $G_1$ have the same neighbourhood on $M_1$ in $(G_1)^r$. Therefore, \n\\begin{align*}\n    \\left| \\left\\{N_{(G_1)^r}(v)\\cap M:v\\in ((C\\setminus D)\\times \\{z\\})\\cap V(G_1)\\right\\}  \\right|  \n    \\le\\; & \\pi^r_{G_1}\\Big(((C\\cap D)\\times N^r_P[z])\\cup V(K_\\theta)\\Big) \\\\\n    \\le\\; & (r+1)^{(k+1)(2r+1)+\\theta} \\,=\\, q.\n\\end{align*}\n\n(Second separation condition) Let $t\\in \\{1, 2, \\ldots, \\ell\\}$ and let \\[M_2:=\\big(V(H^*)\\times \\{w_{t+r+1},\\dots,w_\\ell\\}\\big)\\cup V(K_\\theta).\\]\nObserve that if there is a path of length at most $r$ in $G_1$ between $(V(H^*)\\times \\{w_1,\\dots,w_t\\})\\cap V(G_1)$ and $M_2$, then this path intersects $V(K_{\\theta})$. So, any two vertices in $(V(H^*)\\times \\{w_1,\\dots,w_t\\})\\cap V(G_1)$ having the same distance-$r$ profiles on $V(K_\\theta)$  have the same neighbourhood on $M_2$ in $(G_1)^r$.\nThus,\n\\begin{align*}\n    |\\{N_{(G_1)^r}(v)\\cap M_2: v\\in (V(H)\\times \\{w_1,\\dots,w_t\\})\\cap V(G_1)\\}| \n    \\le\\; \\pi^r_{G_1}(V(K_\\theta)) \\,\\le\\, (r+1)^{\\theta} \\,\\le\\, q.\n\\end{align*} \n\n(Large leaf bag condition)\n   Let $A$ be a leaf bag of $(T_B^*, \\mathcal{B}_B^*)$ with parent bag $A'$ with $|A\\setminus A'|>q$.\n   By the construction of $H^*$ and $(T_B^*, \\mathcal{B}_B^*)$, we know that $A\\times V(P)\\subseteq A\\times \\{y_1, y_2\\}$ for some adjacent vertices $y_1$ and $y_2$ in $P$. \n   Also, $((A\\setminus A')\\times \\{y_1, y_2\\})\\cap V(G_1)=U_{B'}\\setminus B$ for some child $B'$ of $B$. \n    By induction, \n    there is a partial reduction sequence identifying $U_{B'}\\setminus Y_{B'}$ into at most $q$ vertices in $G^r$ such that: \n    \\begin{itemize}\n        \\item we always identify two vertices $u$ and $v$ where the corresponding vertices in $G$ have the same distance-$r$ profiles on $Y_{B'}$ in $G[U_{B'}]$, and\n        \\item for every trigraph $G^*$ in the sequence,\n    $f(\\widetilde{G^*})\\le c$.\n    \\end{itemize}\n    Since $G^r[U_B]=(G_1)^r[U_B]$, \n    this sequence is also a partial reduction sequence for $(G_1)^r[U_{B'}\\setminus Y_{B'}]$.\n    Note that if two vertices have the same distance-$r$ profiles on $Y_{B'}$ in $G[U_{B'}]$, then they have the same distance-$r$ profiles on $Y_B$ in $G[U_B]$.\n    Thus, by the first bullet, \n    this sequence does not create any red edge incident with  $V(K_\\theta)$.\n\n   We deduce that $(G_1)^r$ and $(T_B^*, \\mathcal{B}_B^*)$ satisfy the large leaf bag condition of \\cref{lem:productpathpowerrpower2}.\n\n    Therefore, by \\cref{lem:productpathpowerrpower2}, \n    there is a partial reduction sequence identifying $(V(H^*)\\times V(P))\\cap V(G_1)$ into at most $q$ vertices in $(G_1)^r$ such that for every trigraph $G^*$ in the sequence, $G^*$ has no red edge incident with $V(K_\\theta)$, and $f(\\widetilde{G^*})\\le \\max (g(q,r), c)$. \n    \n    Now apply the same reduction sequence to $G^r$. We claim that we always identify two vertices having the same distance-$r$ profiles on $Y_B$ in $G[U_B]$. Suppose for contradiction that we identified two vertices $u$ and $v$ having distinct distance-$r$ profiles on $Y_B$ in $G[U_B]$.\n    We may assume that for some $x\\in Y_B$ and $\\ell_1 \\in \\{1, 2, \\ldots, r\\}$ and $\\ell_2\\in \\{\\ell_1+1, \\ldots r\\}\\cup \\{\\infty\\}$, we have $\\dist_{G[U_B]}(u,x)=\\ell_1$ and $\\dist_{G[U_B]}(v,x)=\\ell_2$. But then on the path $W_x$, there is a vertex adjacent to $u$ but not to $y$ in $(G_1)^r$. So, when we identify these vertices in $(G_1)^r$, we create a red edge incident with $V(K_{\\theta})$, a contradiction. Therefore, the claim holds. \n    \n    We conclude that the statement $(\\ast)$ holds by induction.  \n    When $B=R$, we know that $U_B=V(G)$ and $Y_B=\\emptyset$. Therefore, \n    there is a partial reduction sequence identifying $V(G)$ into at most $q$ vertices in $G^r$ such that\n    for every trigraph $G^*$ in the sequence,\n    $f(\\widetilde{G^*})\\le c$. We obtain the desired reduction sequence by arbitrarily identifying the resulting trigraph on at most $q$ vertices into a 1-vertex graph. For every trigraph $G^{**}$ in this sequence, $f(\\widetilde{G^{**}})\\le g(q,r)=c$, as every graph on at most $q$ vertices is a subgraph of $S_{1,q,r}$.\n    This completes the proof of the theorem.\n\\end{proof}\n\n\\cref{thm:Hminorfreepower} with $f=\\bw$ and $g(q,r)=(2r+2)q-2$ implies the following result, which is the main contribution of the paper. \n \n\\begin{thm}\n\\label{MinorClosedPower}\nFor all $t,r\\in\\mathbb{N}$ there exists $c\\in\\mathbb{N}$ such that for every $K_t$-minor-free graph $G$, \\[ \\reduced{\\bw}(G^r)\\leq c.\\]\n\\end{thm}\n\n\n\\section{Expanders}\n\\label{Expanders}\n\n\nThis section shows that bounded degree expanders have unbounded reduced bandwidth. In fact, we show a stronger result for expanders excluding a fixed complete bipartite subgraph. For $s\\in\\mathbb{N}$ and $\\beta\\in\\mathbb{R}^+$, a graph $G$ is an \\defn{$(s,\\beta)$-expander} if $G$ contains no $K_{s,s}$ subgraph, and for every $S \\subseteq V(G)$ with $|S| \\leq \\frac{|V(G)|}{2}$ we have $\\abs{N_G(S)} \\ge \\beta \\abs{S}$. \nExpanders can be constructed probabilistically or constructively; see the survey \\citep{HLW06} for example. In the following result it is necessary to exclude some fixed complete bipartite subgraph, as otherwise complete bipartite graphs would be counterexamples. \n\n\\begin{thm}\n\\label{ExpanderClass}\nLet $\\mathcal{G}$ be an infinite class of $(s,\\beta)$-expanders for some $s\\in\\mathbb{N}$ and $\\beta\\in\\mathbb{R}^+$. \nThen $\\mathcal{G}$ has unbounded $\\reduced{\\bw}$.\n\\end{thm}\n\n\\begin{proof}\nAssume for contradiction that there exists $k\\in\\mathbb{N}$ such that $\\reduced{\\bw}(G) \\le k$ for every $G \\in \\mathcal{G}$. Consider an $n$-vertex graph $G\\in\\mathcal{G}$ where $n\\gg k,s,\\beta^{-1}$ (as detailed below). \n\nLet $G=G_n, G_{n-1}, \\dots, G_1$ be a reduction sequence for $G$ such that every red graph has bandwidth at most $k$. It is convenient to consider the corresponding partition sequence \n  \\[\\{\\{v\\} : v \\in V(G)\\}=\\mathcal{P}_n, \\mathcal{P}_{n-1}, \\dots, \\mathcal{P}_1=\\{V(G)\\},\\]\n  and the associated trigraphs $G_{\\mathcal{P}_n}, G_{\\mathcal{P}_{n-1}}, \\dots, G_{\\mathcal{P}_1}$ isomorphic to $G_n, G_{n-1}, \\dots, G_1$ respectively. If $G_{i-1}=G_i\/u,v$ and $u\\in X$ and $v\\in Y$, then $X$ and $Y$ are distinct parts of $\\mathcal{P}_i$, and $\\mathcal{P}_{i-1}$ is obtained from $\\mathcal{P}_i$ by replacing $X$ and $Y$ by $X\\cup Y$. We consider the parts of $\\mathcal{P}_i$ to be the vertices of $G_{\\mathcal{P}_i}$. Say a part $X\\in \\mathcal{P}_i$ is \\defn{big} if $|X|\\geq s$ and \\defn{small} otherwise. Observe that for each black edge $XY$ in $G_{\\mathcal{P}_i}$, there is a complete bipartite subgraph in $G$ between $X$ and $Y$, which implies that $X$ or $Y$ is small. Moreover, if $X$ is big and $Y_1,\\dots,Y_r$ are the neighbours of $X$ for which $XY_i$ is black, then there is a complete bipartite subgraph in $G$ between $X$ and $Y_1\\cup\\dots\\cup Y_r$, implying $|Y_1\\cup\\dots\\cup Y_r|\\leq s-1$. \n\nConsider the first trigraph $G_{\\mathcal{P}_m}$ of the sequence (that is, the largest $m$) such that $\\mathcal{P}_m$ contains a part $X_0$ of size at least $\\sqrt n$. Thus $X_0$ is big (for large $n$). Let $x:=\\abs{X_0}$ and $t := \\lceil \\sqrt x \\rceil$. By the choice of $m$, we have $\\sqrt n \\le x \\le 2 \\sqrt n$, and every part in $\\mathcal{P}_m$, except $X_0$, has size less than~$\\sqrt{n}$. \n  \n  We now show there are distinct parts $X_0,X_1,\\ldots,X_t \\in \\mathcal{P}_m$ such that $\\abs{X_i} \\ge \\frac{\\beta x}{4ki}$ for each $i\\in\\{1,\\dots,t\\}$, and $\\{X_0,X_1,\\dots,X_t\\}$ induce a connected red subgraph in $G_{\\mathcal{P}_m}$. Assume that $X_0,X_1,\\dots,X_i$ satisfy these properties for some $i\\in\\{0,\\dots,t-1\\}$. We now show how to find $X_{i+1}$. For $j\\in\\{1,\\dots,i\\}$,\n\\[|X_j|\n\\geq\\frac{\\beta x}{4kj}\n\\geq\\frac{\\beta x}{4k(t-1)} \n\\geq\\frac{\\beta x}{4k\\sqrt{x}} \n\\geq\\frac{\\beta \\sqrt{x}}{4k} \n\\geq\\frac{\\beta n^{1\/4}}{4k} \n\\geq s \\text{ (for large $n$)}.\\] \nSo $X_0,X_1,\\dots,X_i$ are all big. \nLet $Y_1,\\dots,Y_p$ be the parts of $\\mathcal{P}_m \\setminus\\{ X_0,\\dots,X_i\\}$ joined to $X_0\\cup\\dots\\cup X_i$ by black edges. \nAs argued above, $|Y_1\\cup\\dots\\cup Y_p| \\leq (i+1)(s-1)$. \nLet $Z_1,\\dots,Z_q$ be the parts of $\\mathcal{P}_m \\setminus\\{ X_0,\\dots,X_i\\}$ joined to $X_0\\cup\\dots\\cup X_i$ by red edges. \nEach of $X_0,\\dots,X_i$ are incident to at most $2k$ red edges in $G_{\\mathcal{P}_m}$ (since graphs of bandwidth $k$ have maximum degree at most~$2k$). So $q\\leq 2k(i+1)$. \nOn the other hand, \n$\\abs{X_0\\cup\\dots\\cup X_i} \\leq (i+1)x \\leq tx \\le \\frac{n}{2}$, implying\n$|Y_1\\cup\\dots\\cup Y_p|+|Z_1\\cup\\dots\\cup Z_q| \\geq |N_G(X_0\\cup\\dots\\cup X_i)|\\geq \\beta x$ and\n$|Z_1\\cup\\dots\\cup Z_q|\\geq \\beta x-(i+1)(s-1) \\geq\\frac{\\beta x}{2}$ (for large $n$).  \nLet $X_{i+1}$ be the largest of $Z_1,\\dots,Z_q$. \nSo $|X_{i+1}|\\geq \\frac{\\beta x}{2q} \\geq \\frac{\\beta x}{4k(i+1)}$, \nand $\\{X_0,X_1,\\dots,X_{i+1}\\}$ induces a connected red graph in $G_{\\mathcal{P}_m}$, as desired. Hence there are distinct parts $X_0,X_1,\\ldots,X_t \\in \\mathcal{P}_m$ such that $\\abs{X_i} \\ge \\frac{\\beta x}{4ki}$ for each $i\\in\\{1,\\dots,t\\}$, and $\\{X_0,X_1,\\dots,X_t\\}$ induces a connected red subgraph in $G_{\\mathcal{P}_m}$. As shown above, $X_0,\\dots,X_t$  are all big. \n\nLet $H$ be the connected component of $G_{\\mathcal{P}_m}$ containing $X_0,\\dots,X_t$. Consider a vertex-ordering of~$H$ with bandwidth at most~$k$. Since $\\{X_0, X_1, \\ldots, X_t\\}$ induces a connected red subgraph of~$H$, there is a set $I$ of $kt+1$ consecutive vertices in this ordering of $H$ containing $X_0,\\dots,X_t$ and with at most $k-1$ parts strictly between `consecutive' $X_i$s. \n\nLet $Z$ be the union of all the big parts in~$I$. \nSo $\\abs{Z} \\le (kt+1)x \\le \\frac{n}{2}$ (for large $n$), implying \n\\[\\abs{N_G(Z)}\\geq \\beta \\abs{Z}\n\\ge \\beta \\abs{X_1\\cup\\dots\\cup X_t} \n\\ge \\frac{\\beta^2 x}{4k} \\sum_{i=1}^t\\frac{1}{i} \n\\ge \\frac{\\beta^2 x \\ln t}{4k} \n.\\]\nWe now upper bound $|N_G(Z)|$. Each vertex in $N_G(Z)$ is either in: \n\\begin{itemize}\n    \\item a small part of $\\mathcal{P}_m$ in $I$, \n    \\item a part of $\\mathcal{P}_m$ adjacent to $I$ in $H$, or \n    \\item a part of $\\mathcal{P}_m$ outside $H$.\n\\end{itemize}\nThere are at most $(kt+1)(s-1)$ vertices in small parts of $\\mathcal{P}_m$ in $I$. There are at most $2k$ parts of $\\mathcal{P}_m$ adjacent to $I$ in $H$ ($k$ to the left of $I$, and $k$ to the right), and each such part has at most $x$ vertices, thus contributing at most $2kx$ vertices to $N_G(Z)$. Each big part in~$I$ is incident to at most $s-1$ vertices in parts outside $H$ (since the corresponding edges are black), so there are at most  \n$(kt+1)(s-1)$ vertices in $N_G(Z)$ of the third type. \nHence\n\\[ \\frac{\\beta^2 x\\ln t}{8k} \n\\leq |N_G(Z)| \n\\leq (kt+1)(s-1)  + 2kx + (kt+1)(s-1). \\]\nThis is the desired contradiction, since  the left-hand side is $\\Theta(\\sqrt{n} \\log n)$ and right-hand side is $\\Theta(\\sqrt{n})$ for fixed $k,s,\\beta$.\n\\end{proof}\n\n\\section{Tied and Separated Parameters}\n\\label{Tied}\n\nLet $\\preceq$ be the preorder, where for graph parameters $f_1$ and $f_2$, define \\defn{$f_1\\preceq f_2$} if there exists a~function~$g$ such that $f_1(G)\\leq g( f_2(G))$ for every graph $G$. Graph parameters $f_1$ and $f_2$ are \\defn{tied} (also called \\defn{functionally equivalent}) if $f_1\\preceq f_2$ and $f_2\\preceq f_1$. For example, \\citet{TW-VI} showed that total twin-width is tied to linear rank-width, while component twin-width,  clique-width and boolean-width are tied. \n\nDefine \\defn{$f_1 \\precneq f_2$} to mean $f_1\\preceq f_2$ and  $f_2\\not\\preceq f_1$. For example, $\\tww \\precneq \\reduced{\\bw}$ (since for every graph~$G$, we have $\\Delta(G)\\leq 2 \\bw(G)$ implying $\\tww(G)\\leq 2 \\,\\reduced{\\bw}(G)$, but there are bounded degree expanders with twinwidth 6 \\citep{TW-II} and unbounded reduced bandwidth by \\cref{ExpanderClass}). \nGraph parameters $f_1$ and $f_2$ are \\defn{separated} if \n$f_1\\precneq f_2$ or $f_2\\precneq f_1$ (that is, they are not tied). \n\nRecall that in the context of reduction sequences, it only makes sense to consider graph parameters that are unbounded on stars. This motivates the following definition. For a graph parameter $f$ let \\defn{$f+\\Delta$} be the graph parameter defined by $(f+\\Delta)(G):=f(G)+\\Delta(G)$. This section shows that $\\reduced{\\Delta}$, $\\reduced{(\\Delta+\\tw)}$, $\\reduced{(\\Delta+\\pw)}$ and $\\reduced{\\bw}$ are separated (even within graphs).\n\n\\begin{thm}\\label{par-separation}\n$\\reduced{\\Delta} \\precneq \\reduced{(\\Delta+\\tw)} \\precneq \\reduced{(\\Delta+\\pw)} \\precneq \\reduced{\\bw}$.\n\\end{thm}\n\nThe proof of \\cref{par-separation} uses a lemma of \\citet{BBD21}. \n\nFor every graph $G$, let \\defn{$\\operatorname{red}(G)$} be the trigraph $G'=(V(G')=V(G),E(G')=\\emptyset,R(G')=E(G))$ obtained by making all its edges red. An \\defn{induced subtrigraph} of~$G$ is another trigraph~$H$ obtained by removing some vertices from $G$, and $G$ is then called a \\defn{supertrigraph} of~$H$. Note that $\\reduced{f}(G)$ is well-defined for any graph parameter $f$ and any trigraph $G$ (since reduction sequences are  defined for trigraphs $G$).\n\nThe \\defn{2-blowup} of a graph $G$, denoted by \\defn{$G \\,\\rotatebox{90}{$\\bowtie$} K_2$}, is the graph obtained by replacing each vertex $u \\in V(G)$ by two non-adjacent vertices $u_1, u_2$, and replacing each edge $uv \\in E(G)$ by four edges $u_1v_1, u_1v_2, u_2v_1, u_2v_2$. \n\n\n\n\\begin{lem}[essentially Lemma~10 in \\citep{BBD21}]\\label{lem:trigraph-to-graph}\nFor every connected graph~$H$ of maximum degree~$d$, there is a graph $G$ admitting a partial reduction sequence $G_n, \\ldots, G_i$ such that:\n\\begin{itemize}\n    \\item  $\\Delta(\\widetilde{G_k}) \\leq 2d$ for every $k \\in \\{i,i+1,\\ldots,n\\}$, \n    \\item each connected component of $\\widetilde{G_k}$ is a subgraph of $G \\,\\rotatebox{90}{$\\bowtie$} K_2$,\n    \\item $G_i$ is isomorphic to $\\operatorname{red}(H)$, \n    \\item and every (full) reduction sequence of $G$ goes through a supertrigraph of $\\operatorname{red}(H)$ or a~trigraph with red maximum degree at least $|V(H)|$.\n\\end{itemize}\n\\end{lem}\n\\begin{proof}[Proof Sketch]\nIn Lemma 10 of~\\cite{BBD21}, the second item does not appear, and the second option of the fourth item consists of going through a trigraph of red maximum degree at least $2d+1$.\nThe lemma is actually shown in greater generality (see Lemma 11 of~\\cite{BBD21}) and accounts for trigraphs possibly having black edges and a disconnected red graph. Since we do not need this general form here, we can simplify the construction.\n\nThe graph $G$ is built as follows, where \n$t := \\abs{V(H)}^{\\abs{V(H)}}+\\abs{V(H)}$. \nFor every vertex $v \\in V(H)$, add to $G$ a clique $v_1, \\ldots, v_t$.\nWe informally refer to  $\\{\\{v_1,\\ldots,v_t\\} : v \\in V(H)\\}$ as the \\defn{cliques of~$G$}.\nFor each edge $vw \\in E(H)$, add to $E(G)$ the matching $v_1w_1, v_2w_2, \\ldots, v_tw_t$. \n\nWe now describe the partial reduction sequence of $G$ satisfying the first three items.\nFor every $v \\in V(H)$, identify $v_1$ and $v_2$, and call the resulting vertex $v'_2$.\nThen for every $v \\in V(H)$, identify $v'_2$ and $v_3$, and call the resulting vertex $v'_3$; and so on, until every clique of every $v \\in V(H)$ has been identified to a single vertex. It is straightforward to check that this partial reduction sequence satisfies the first three items.\n\nThe proof of the fourth item follows the proof of Statement~5 in Lemma~11 in \\citep{BBD21}. We sketch the argument here, since the situation is simpler. If two parts of $G$ intersecting different cliques of $G$ are identified, the red degree of the resulting part is at least the number $s$ of other parts intersecting these two cliques, possibly minus 2.\nThus the second condition of the item is satisfied unless $s < \\abs{V(H)}+2$.\nWe may now assume that immediately  before the first such identification, there is a part $P_v$ of size at least $\\abs{V(H)}^{\\abs{V(H)}-1}$ entirely contained within the clique of $v \\in V(H)$.\nLet $w$ be a neighbour of $v$ in $H$. \nFor the red degree of $P_v$ to be less than $\\abs{V(H)}$, the $\\abs{P_v}$ matched vertices in the clique of $w$, have to be in less than $\\abs{V(H)}$ parts.\nThis means that the largest of those parts, $P_w$, has size at least $\\abs{V(H)}^{\\abs{V(H)}-2}$.\nSince $H$ is connected, we continue this reasoning along a spanning tree of $H$, and exhibit a collection $\\{P_u : u \\in V(H)\\}$ of parts inducing the trigraph $\\operatorname{red}(H)$.\n\\end{proof}\n\nThe next lemma says that, for graph parameters $f$ and $g$ satisfying various properties, $f \\precneq g$ implies $\\reduced{f} \\precneq \\reduced{g}$. A~parameter $f$ is \\defn{2-blowup-closed} if there is a function $g: \\mathbb N_0 \\to \\mathbb N_0$ such that $f(G \\,\\rotatebox{90}{$\\bowtie$} K_2) \\leq g(f(G))$ for every graph~$G$. \n\n\n\\begin{lem}\\label{lem:par-to-reduced-par}\nLet $f$ and $g$ be monotone graphs parameters such that:\n\\begin{itemize}\n\\item $f \\precneq g$,\n\\item $f$ is union-closed and \\mbox{2-blowup-closed}, \n\\item there is a non-decreasing function $q': \\mathbb N_0 \\to \\mathbb N_0$ satisfying $\\lim_{n\\to\\infty} q'(n) = \\infty$ and $g(G) \\geq q'(\\Delta(G))$ for every graph $G$, \nand\n\\item \nthere is a constant $c \\in \\mathbb N$ such that for every $n \\in \\mathbb N$ there is a connected graph $H_n$ on at least $n$ vertices with $\\lim_{n\\to\\infty} g(H_n) =\\infty$ and $\\reduced{f}(\\operatorname{red}(H)) \\leq c$.\n\\end{itemize}    \nThen $\\reduced{f} \\precneq \\reduced{g}$.\n\\end{lem}\n\\begin{proof}\nLet $t(H)$ be the graph $G$ from \\cref{lem:trigraph-to-graph} given graph~$H$. Define $\\mathcal F := \\{t(H_n) : n \\in \\mathbb{N}\\}$. \nSince $f \\preceq g$, it is immediate that $\\reduced{f} \\preceq \\reduced{g}$. To prove that~$\\reduced{g} \\not\\preceq \\reduced{f}$, we show that $\\reduced{f}$ is bounded on $\\mathcal F$, but  $\\reduced{g}$ is unbounded. \n\nWe first show that $\\reduced{f}$ is bounded on $\\mathcal F$.\nLet $G =t(H) \\in \\mathcal F$. By~\\cref{lem:trigraph-to-graph}, there is a partial reduction sequence $G_{\\abs{V(G)}}, \\ldots, G_i$ of $G$ such that $G_i$ is isomorphic to $\\operatorname{red}(H)$, and every red graph $\\widetilde{G_k}$ (for $k \\in \\{i, \\ldots, \\abs{V(G)}\\}$) is a disjoint union of subgraphs of the 2-blowup of $H$.\nSince $\\reduced{f}(\\operatorname{red}(H)) \\leq c$, in particular, $f(H) \\leq c$.\nSince $f$ is monotone, union-closed and 2-blowup-closed, there is a function $q: \\mathbb N_0 \\to \\mathbb N_0$ such that $f(\\widetilde{G_k}) \\leq f(H \\,\\rotatebox{90}{$\\bowtie$} K_2) \\leq q(f(H)) \\leq q(c)$.\nBy assumption $\\reduced{f}(G_i) \\leq c$, thus $G$ has a (full) reduction sequence witnessing that $\\reduced{f}(G) \\leq \\max(q(c),c)$.\n\nWe now show that $\\reduced{g}$ is unbounded on $\\mathcal F$.\nFor the sake of contradiction, suppose there exists $c' \\in \\mathbb N$ such that $\\reduced{g}(G') \\leq c'$ for every $G' \\in \\mathcal F$.\nLet $n \\in \\mathbb N$ be such that $g(H_n)>c'$ and $q'(n) > c'$. \nBy assumption such an integer always exists.\nConsider $G :=t(H_n) \\in \\mathcal F$. \nBy~\\cref{lem:trigraph-to-graph}, every reduction sequence of $G$ either goes through a~trigraph $Y$ with red maximum degree \\mbox{$\\abs{V(H_n)} \\geq n$} or through a~supertrigraph $Z$ of $\\operatorname{red}(H_n)$.\nIn the former case, $g(\\widetilde{Y}) \\geq q'(\\Delta(\\widetilde{Y})) \\geqslant q'(n) > c'$.\nIn the latter case, since $g$ is monotone, $\\reduced{g}(Z) \\geq \\reduced{g}(\\operatorname{red}(H_n)) \\geq g(H_n) > c'$.\nBoth cases imply that $\\reduced{g}(G) >  c'$, which contradicts the boundedness of $\\reduced{g}$ on~$\\mathcal F$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{par-separation}]\nFirst observe that $\\Delta$, $\\Delta+\\tw$, $\\Delta+\\pw$ and $\\bw$ are monotone, union-closed and \\mbox{2-blowup-closed} with function $n \\mapsto 2n+1$. If $g$ is any of these parameters, then \n$g(G) \\geq \\lfloor\\frac{\\Delta(G)}{2}\\rfloor$, which provides the $q'$ function in \\cref{lem:par-to-reduced-par}.\n\nThe first claim, $\\reduced{\\Delta} \\precneq \\reduced{(\\Delta+\\tw)}$, follows from  \\cref{lem:par-to-reduced-par} where $H_n$ is the \n$n\\times n$ planar grid graph, which has $(\\Delta+\\tw)(H_n) \\geq n$ (see \\citep{HW17}) and $\\reduced{\\Delta}(\\operatorname{red}(H_n))=4$ (see \\citep{TW-I}).\n\nThe second claim, $\\reduced{(\\Delta+\\tw)} \\precneq \\reduced{(\\Delta+\\pw)}$, follows from  \\cref{lem:par-to-reduced-par} where $H_n$ is the complete binary tree of height $n$, which has $(\\Delta+\\pw)(H_n) \\geq \\frac{n}{2} $ (see \\citep{Scheffler89}) and $\\reduced{(\\Delta+\\tw)}(\\operatorname{red}(H_n))\\leq 4$ (see~\\citep{TW-I}).\n\nFinally, $\\reduced{(\\Delta+\\pw)} \\precneq \\reduced{\\bw}$ follows from  \\cref{lem:par-to-reduced-par} where $H_n$ is the tree defined in \\cref{QnHere} on page~\\pageref{QnHere}, which has $\\bw(H_n)\\geq \\frac{n}{3}$ and $\\reduced{(\\Delta+\\pw)}(\\operatorname{red}(H_n))\\leq 5$. The latter can be seen by iteratively identifying any leaf with its parent (in any order), which yields only subtrigraphs of $H_n$ and can be done until the trigraph has a single vertex. Throughout, the red graphs have maximum degree at most~3 and pathwidth at most~2.\n\\end{proof}\n\n\n\\section{Open Problems}\n\\label{OpenProblems}\n\nOur results lead to several interesting open problems. \n\n\\paragraph{Parameter tied to reduced bandwidth?} \n\nRecall that component twin-width and clique-width are tied~\\citep{TW-VI}. Is there a `natural' graph parameter tied to reduced bandwidth? This is related to the question of which dense graph classes have bounded reduced bandwidth. Our results for powers give such examples. Complements provide other examples: if $\\overline{G}$ is the complement of a graph $G$, then $\\reduced{\\bw}(\\overline{G})=\\reduced{\\bw}(G)$, since any reduction sequence for $G$ defines a reduction sequence for $\\overline{G}$ with equal red graphs. In general, $\\reduced{f}(G)=\\reduced{f}(\\overline{G})$ for every $G$ and~$f$. \n\\citet{TW-III} proved that unit interval graphs have  twin-width~2; in fact, they have a reduction sequence in which every red graph is a disjoint union of paths. Thus unit interval graphs have reduced bandwidth 1.\n\n\n\\paragraph{Best possible parameters:}\nIs bandwidth the largest possible parameter $f$ such that planar graphs have bounded $\\reduced{f}$? We formalise this question as follows. Say a graph parameter $f$ is \\defn{continuous} if there exists a function $g$ such that\n$f(G) \\leq g( f(G-e) )$ for every graph $G$ and edge $e\\in E(G)$, and\n$f(G) \\leq g( f(G-v) )$ for every graph $G$ and isolated vertex $v$ of $G$. \nAll the graph parameters studied in this paper are continuous. \nIs there a continuous\\footnote{We need to assume $f$ is continuous because of the following example. For a graph $G$, define $f(G):=\\bw(G)$ if $\\bw(G)\\leq 466$ and $f(G):=|V(G)|$ otherwise. So $\\bw\\precneq f$ and planar graphs have bounded $\\reduced{f}$, but $f$ is not continuous.} graph  parameter $f$ such that $\\bw \\precneq f$ and planar graphs have bounded $\\reduced{f}$?  Maximum component size is not such a parameter, since planar grid graphs have unbounded clique-width~\\citep{GR00} and therefore have unbounded $\\reduced{\\star}$. In general, for a graph class $\\mathcal{G}$, what is a continuous graph parameter $f$ such that $\\reduced{f}$ is bounded on $\\mathcal{G}$, and $\\reduced{g}$ is unbounded on $\\mathcal{G}$ for every parameter $g$ with $f \\precneq g$. Does such a graph parameter always exist?\n\n\n\\paragraph{Lower Bounds:}\n\nA non-trivial lower bound on reduced bandwidth is \\cref{ExpanderClass} for expanders. Stronger lower bounds are plausible. For example, does every monotone graph class excluding a fixed complete bipartite subgraph and with bounded reduced bandwidth have polynomial expansion? It is even plausible that the degree of the polynomial is an absolute constant. Does every monotone graph class excluding a fixed complete bipartite subgraph and with bounded reduced bandwidth have linear expansion? Graphs with bounded row-treewidth have linear expansion. A good example to consider is the class of 3-dimensional grids, which we guess have unbounded reduced bandwidth.\n\n\n\n\\paragraph{Subdivisions:}\n\n\\citet{TW-II} showed that $t$-subdivisions of $K_n$ have bounded twin-width if and only if $t \\in\\Omega(\\log n)$. No explicit bound on the twin-width was given. Recently, \\citet{BBD21} proved that every $(\\geq 2 \\log n)$-subdivision of any $n$-vertex graph has twin-width at most 4. The proof  constructs a reduction sequence in which every red graph is a forest. So this class of graphs has exponential expansion, is  $K_{2,2}$-free,  and has  $\\reduced{(\\tw+\\Delta)}\\leq 5$. The proof in \n\\citep{BBD21} can be adapted\\footnote{In the proof of Theorem~7 in~\\citep{BBD21}, replace the virtual binary red tree by an $n$-vertex red path, and observe that running the same sequence produces a red caterpillar with maximum degree 4 and at most one vertex of degree 4; refer to Figures~3 and 4 in~\\cite{BBD21}. Such caterpillars have bandwidth at most 2.} to show that every $(\\geq n)$-subdivision of any $n$-vertex graph has reduced bandwidth at most 2. Is this result best possible, in the sense that if $K_n^{(p)}$ is the $p$-subdivision of $K_n$ and $\\reduced{\\bw}(K_n^{(p)})\\leq O(1)$, must $p\\in\\Omega(n)$? Since $\\Omega(n)$-subdivisions of $n$-vertex graphs have linear expansion, a positive answer to this question would be good evidence for the suggestion above that graph classes with bounded reduced bandwidth and excluding a fixed complete bipartite subgraph have linear expansion.\n\n\n\\subsection*{Acknowledgements}\nThis research was initiated at the Dagstuhl workshop \\emph{Sparsity in Algorithms, Combinatorics and Logic} (September 2021). Many thanks to the organisers and other participants. \n\nAfter the initial announcement of our results, \\citet{JP22} proved that planar graphs have twin-width at most 183. Their proof is based on the proof of \\cref{GenusProductStructure}. \n\n\\def\\soft#1{\\leavevmode\\setbox0=\\hbox{h}\\dimen7=\\ht0\\advance \\dimen7\n  by-1ex\\relax\\if t#1\\relax\\rlap{\\raise.6\\dimen7\n  \\hbox{\\kern.3ex\\char'47}}#1\\relax\\else\\if T#1\\relax\n  \\rlap{\\raise.5\\dimen7\\hbox{\\kern1.3ex\\char'47}}#1\\relax \\else\\if\n  d#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.9ex \\char'47}}#1\\relax\\else\\if\n  D#1\\relax\\rlap{\\raise.5\\dimen7 \\hbox{\\kern1.4ex\\char'47}}#1\\relax\\else\\if\n  l#1\\relax \\rlap{\\raise.5\\dimen7\\hbox{\\kern.4ex\\char'47}}#1\\relax \\else\\if\n  L#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.7ex\n  \\char'47}}#1\\relax\\else\\message{accent \\string\\soft \\space #1 not\n  defined!}#1\\relax\\fi\\fi\\fi\\fi\\fi\\fi}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\\subsection{}\n\\subsubsection{}\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nMany current and future applications require ultra-high repetition frequency light pulse\nsources \\cite{RAF07}. Among these applications most also require highly regular pulse arrival times.\nMode-locked (ML) solid state lasers can fulfill these requirements.\nHowever, such devices are too expensive for large scale use. Due to this limitation extensive\nresearch has gone into semiconductor ML lasers. The most attractive mode-locking technique, due to\nits simplicity of production and handling, is passive mode-locking, which does not require any\nexternal RF modulation source. However, due to the absence of an external reference\nclock passively ML lasers exhibit relatively large fluctuations in the temporal positions\nof pulses compared with a perfectly periodic pulse train \\cite{LIN10c}. This phenomenon is referred\nto as pulse timing jitter. Recently, it was proposed to use optical feedback to significantly reduce\nthe  timing jitter of passively ML lasers \\cite{SOL93,LIN10e,OTT12a,OTT14b}. Other methods of pulse\nstream stabilisation which have been investigated include hybrid mode-locking \\cite{FIO10,ARK13} and optical\ninjection \\cite{REB10,REB11}.\nTo characterize the performance of such devices, with respect to the timing regularity, the timing\njitter is calculated. Experimentally this is done using the von Linde method, which involves\nintegrating over the sidebands of the power spectrum of the laser output. However, for the\nnumerical investigation of ML lasers the von Linde method can be impractical as it is computationally\nvery expensive. In this paper we therefore propose a semi-analytical method of calculating the pulse\ntiming jitter for a set of delay differential equations (DDEs) proposed earlier to describe passive\nmode-locking in semiconductor lasers \\cite{VLA04,VLA04a,VLA05}. The method is of general nature and can be used to estimate the\nvariance of timing fluctuations in a wide range of time periodic dynamical systems described by\nautonomous systems of DDEs subject to weak additive noise.\n\nTheoretical analysis of the influence of noise on ML pulses propagating in a laser cavity was first\nperformed by H. Haus using a master equation \\cite{HAU93a}.  Later this technique was extended by\ntaking into account the finite carrier density relaxation rate in semiconductor lasers \\cite{JIA01}. The\nmaster equation has secant-shaped ML pulses as a solution, and a small perturbation of\nthis state can be studied using the linearized equation of motion. The perturbed pulse is described\nby four parameters: the perturbations of the pulse amplitude, phase, frequency, and timing. Using\nthe orthogonality of the solutions of the linearized equation to the solutions of the adjoint\nhomogeneous linear system, coupled first order differential equations of motion, driven by noise,\ncan be written out.  However, due to multiple simplifying assumptions underlying the Haus master\nequation, this approach is not directly applicable to the analysis of semiconductor laser devices.\nThis is why the theoretical estimation\nof timing jitter in ML semiconductor lasers has been previously performed using the direct numerical\nsimulations of travelling wave \\cite{ZHU97,MUL06} and delay-differential equation (DDE)\n\\cite{OTT12a,OTT14b,JAU15a,SIM14} models. As purely computational approaches are time-consuming, the influence of\nnoise on the dynamics of ML pulses has been studied only in limited parameter regions. In a recent\npaper \\cite{PIM14b} a new semi-analytical method to estimate timing jitter in the DDE-model\n\\cite{VLA04,VLA04a,VLA05} of a passively ML semiconductor laser  was proposed. This method was used\nto study the effect of nonlinear\nphenomena such as bifurcations and bistability on timing jitter, and the numerical results were found to be in\ngood qualitative agreement with experimental data. In this paper we consider a generalisation of\nthe semi-analytical method to study passively ML lasers\nwith multiple delayed feedback.  We then use this semi-analytical method to\nderive a formula for the timing jitter for resonant feedback delay lengths.\n\nIn Section~II \\ref{sec:method} we introduce an autonomous DDE model of a laser\noperating in a passive ML regime and describe the parameters used in our calculations.\nIn Sec.~III, by linearizing the model equations near the ML periodic solution and projecting the\nperturbation term on the neutral eigenfunctions corresponding to the time and phase shift\nsymmetries of the unperturbed equations, we derive a semi-analytical expression for the variance of the\npulse timing fluctuations \\cite{HAL66,HAL77}.\nSection~IV is devoted to the comparison of the results obtained using this expression with those\nof direct numerical calculations of pulse timing jitter, and a derivation of the dependence of the\ntiming jitter on the feedback delay time in the particular case of resonant feedback. Finally, in\nSec.~V we conclude with a brief discussion of our results.\n\\cite{FLU07}\n\n\\section{DDE Model}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ringcavitydualfb2.pdf}\n\\caption{Schematic diagram of a two section ring cavity laser subject to optical feedback from two external cavities (EC). The yellow region represents the gain section, the blue region corresponds to the saturable absorber (SA) section and the green bar indicates the spectral filtering element.\n}\\label{model}\n\\end{center}\n\\end{figure}\n\nWe use a DDE model for a passively ML ring cavity laser subject to optical feedback from $M$\nexternal cavities,\nbased on the model introduced in \\cite{OTT12a}, a schematic diagram of the model is shown in Fig.~\\ref{model} for the case of\ntwo feedback cavities.\nThis model is an extension of the DDE model proposed in \\cite{VLA04,VLA05}. A detailed description\nand derivation of the\nfeedback terms for a laser with a single feedback cavity can be found in \\cite{OTT12a}. The final\nset of three coupled delay differential equations is\n\\begin{eqnarray}\n& \\qquad \\dot{\\mathcal{E}}\\left(t\\right)=-\\left( \\gamma+i\\omega\\right)\\mathcal{E}\\left(t\\right)\n+\\gamma R\\left(t-T\\right) e^{-i \\left(\\Delta \\Omega+\\omega\\right) T}\n\\mathcal{E}\\left(t-T\\right)\\nonumber\\\\\n &+\\gamma\\sum_{m=1}^M\\sum^{\\infty}_{l=1} K_{m,l} e^{-i l C_m} R\\left(t-T-l\\tau_m\\right) e^{-i\\left( \\Delta \\Omega+\\omega \\right) \\left(T+l\\tau_m\\right)}\n \\mathcal{E}\\left(t-T-l\\tau_m\\right)+D\\xi\\left(t\\right), \\label{subeq:E}\\\\\n &\\dot{G}\\left(t\\right)= J_g-\\gamma_g G\\left(t\\right)-e^{-Q\\left(t\\right)}\\left(e^{G\\left(t\\right)}-1\\right)|\\mathcal{E}\\left(t\\right)|^2,\\label{subeq:G}\\\\\n &\\dot{Q}\\left(t\\right)=J_q-\\gamma_q Q\\left(t\\right)-r_s e^{-Q\\left(t\\right)}\\left(e^{Q\\left(t\\right)}-1\\right)|\\mathcal{E}\\left(t\\right)|^2,\\label{subeq:Q}\n\\end{eqnarray}\nwith\n\\begin{equation}\n R\\left(t\\right)\\equiv\\sqrt{\\kappa} e^{\\frac{1}{2}\\left(\\left(1-i \\alpha_g\\right)G\\left(t\\right)-\\left(1-i \\alpha_q\\right)Q\\left(t\\right)\\right)}.\\label{defR}\n\\end{equation}\nThe dynamical variables are the slowly varying electric field amplitude $\\mathcal{E}$, the saturable\ngain $G$ and the saturable loss $Q$. The saturable gain $G$ and saturable loss $Q$ are related to\nthe carrier inversion in the gain and absorber sections, respectively.  In Eq.~\\eqref{subeq:G} $J_g$\nis related to the current pumped into the gain section and $J_q$ in Eq.~\\eqref{subeq:Q} describes\nthe unsaturated absorption. The carrier lifetimes in the gain and absorber sections are given by\n$1\/\\gamma_g$ and $1\/\\gamma_q$, respectively.  The factor $r_s$ is the ratio of the saturation\nintensities in the gain and absorber sections. The $M+1$ delay times in this system are the cold\ncavity round-trip time $T$ and the external cavity round-trip times (delay times) $\\tau_m$ of the\n$M$ feedback cavites. The cold cavity round trip time is defined as $T$\\,$\\equiv$\\,$v\/L$, where $L$\nis the length of the ring cavity. The bandwidth of the laser is limited by the finite width of the\ngain spectrum, which is taken into account by a Lorentzian-shaped filter function of width\n$\\gamma$.  $\\omega$ describes the shift between the reference frequency and the central\nfrequency of the spectral filter. The possibility of detuning between this latter frequency and the\nfrequency of the nearest cavity mode is allowed for by the inclusion of $\\Delta \\Omega$. The optical\nfeedback is described by the sum in Eq.~\\eqref{subeq:E}. Here $l$ is the number of round-trips in\nthe external cavity, $K_{m,l}$ is the round-trip dependent feedback strength of the $m$th feedback\ncavity and $C_m$ is the phase shift that accumulates over one round-trip in the external\ncavity. Below we consider feedback contributions only from light that has made one round-trip in the\nexternal cavities ($K_{m,1}=K_m$). The last term in Eq.~\\eqref{subeq:E} models spontaneous emission\nnoise using a complex Gaussian white noise term $\\xi(t) = \\xi_1(t) + i \\xi_2(t)$ with strength $D$,\n$$ \\left\\langle \\xi_i(t)\\right\\rangle = 0\\qquad \\mbox{and}\n\\qquad\\left\\langle\\xi_i(t)\\xi_j(t')\\right\\rangle=\\delta_{i,j}\\delta(t-t').$$ Equation~\\eqref{defR}\ndescribes the amplification and losses of the electric field during one round-trip in the laser\ncavity.  Internal and out-coupling losses are taken into account in the attenuation factor $\\kappa$\nand the linewidth enhancement factors ($\\alpha$-factor) in the gain and absorber sections are denoted\n$\\alpha_g$ and $\\alpha_q$, respectively.\n\n\\begin{table}[htbp]\n  \\centering\n  \\begin{tabular}[c]{l l l l l}\n    \\hline\n    symbol & value && symbol & value\\\\\n    \\toprule\\hline\n     $T$ & $25$ ps &&$\\gamma$ &  $2.66$ ps$^{-1}$\\\\\n     $\\gamma_g$ & $1$ ns$^{-1}$ &&$\\gamma_q$ & $75$ ns$^{-1}$\\\\\n    $J_g$ & $0.12$ ps$^{-1}$&&$J_q$ & $0.3$ ps$^{-1}$\\\\\n    $r_{s}$ & $25.0$  && $C_m$ & $0$ \\\\\n    $\\kappa$ & $0.1$ &&$\\Delta\\Omega$& 0 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Parameter values used in numerical simulations, unless stated otherwise.}\n  \\label{tab:Simparams}\n\\end{table}\n\n\\section{Perturbation analysis}\n\\label{sec:method}\n\nVarious methods of calculating the timing jitter are discussed in\n\\cite{MUL06,OTT14b,KEF08,PIM14b,LIN86}. In this section, we consider an extension of the\nsemi-analytical method of timing jitter estimation proposed in \\cite{PIM14b} for the DDE model of\npassively\nML laser to the system \\eqref{subeq:E}-\\eqref{subeq:Q} with external feedback and, hence, multiple\ndelay times. Details of the derivation of the semi-analytical expression for the\nestimation of pulse timing jitter are presented in the Appendix. As we do not use a specific form of equations \\eqref{subeq:E}-\\eqref{subeq:Q},\nthe same approach can be applied to the analysis of the effect of small additive noise on stable periodic\nsolutions in other physical systems described by autonomous DDEs with multiple delays.\n The advantage of the proposed method, compared with the von Linde technique or the so called long term jitter\ncalculation \\cite{OTT14b}, is that it is based on the numerical solution of deterministic equations and\ntherefore requires much shorter computation times. Furthermore, when the spontaneous emission noise\nis modeled by a Gaussian white noise term,\nthe fluctuations of the pulse arrival times behave like a random walk \\cite{OTT14b}, making the\ntiming jitter calculated from the semi-analytical method proportional to the rms timing jitter given\nby the von Linde method. This is useful for comparison with experiments.\n\nWe consider a periodic ML solution, $\\psi_0 = (\\Re \\mathcal{E}_0, \\Im \\mathcal{E}_0, G_0, Q_0)^{\\mathrm{T}}$\nof the system \\eqref{subeq:E}-\\eqref{subeq:Q} for $D = 0$, with period $T_0$. One should note that due to the rotational symmetry,\nthere is a family of such solutions $\\Gamma_\\varphi\\cdot \\psi_0 = (\\Re (e^{i\\varphi}\\mathcal{E}_0), \\Im (e^{i\\varphi}\\mathcal{E}_0), G_0, Q_0)^{\\mathrm{T}}$,\nwhere $\\Gamma_\\varphi$ denotes the corresponding matrix of rotation of the $\\mathcal{E}_0$ plane.\nThe noise perturbation is assumed to be reasonably small, $D\\ll 1$,\nand we restrict our analysis to the situation when solutions remain at a distance of order $D$\nfrom the torus of stable periodic solutions $\\Gamma_\\varphi\\cdot \\psi_0(t+\\theta)$  at all times (that is, the probability of a large fluctuation of the solution\nis assumed to be negligible during the typical time interval of system observation).\nUnder this assumption, the noise results in a slow diffusion of the phase $\\theta$ of the solution, as well as a slow diffusion of the angular variable $\\varphi$.\nFurthermore, one expects that the variance of the phase $\\theta$ and of the variable $\\varphi$ increases linearly with\ntime, that is $\\langle \\theta -\\bar \\theta\\rangle^2 \\propto t$, which expresses a simple\ndiffusion process \\cite{Daffertshofer}. We use the coefficient of proportionality\nin this relationship as a measure of the timing jitter.\n\nThe phase of a solution can be defined in several ways \\cite{rice}, which, in practice, lead\nto equivalent or close results when applied for the evaluation of the phase diffusion rate.\nIn particular, the definition of the asymptotic phase is based on the fact that every solution $\\psi(t)$\nof the unperturbed system \\eqref{subeq:E}-\\eqref{subeq:Q} with $D = 0$ converges to\na periodic solution $\\Gamma_\\varphi \\cdot \\psi_0(t+\\theta)$ in the limit $t\\to\\infty$\nwhere the constant $\\theta$, called the asymptotic phase, and the angle $\\varphi$ are specific\nto the initial state of the solution $\\psi(t)$.\nRecall that states of system \\eqref{subeq:E}-\\eqref{subeq:Q} are functions defined on the interval $[-\\tau_M',0]$.\nThe asymptotic phase $\\theta$ and the angle $\\varphi$ remain constant along the trajectories of the unperturbed system.\nHowever, in the perturbed system, the asymptotic phase $\\theta$ and the angular variable $\\varphi$ evolve as functions of the evolving state $\\psi(t+r)$ ($r\\in [-\\tau_M',0]$).\n\nAs dynamics are restricted to a small neighborhood of the limit cycle $\\psi_0$ (and its rotations $\\Gamma_\\varphi \\cdot \\psi_0$), the evolution of the phase can be deduced\nfrom the linearization \\eqref{linsys} of system  Eqs.\\,\\eqref{subeq:E}-\\eqref{subeq:Q}\naround this cycle. Details on the analysis of the dynamics of the solutions of the linear system \\eqref{linsys} as well as its effect on the evolution of the phase can be found in Appendix. \nNoise results in a slow diffusion of the variables $\\theta$ and $\\varphi$ along\nthe neutral periodic eigenmodes of the linearized unperturbed system \\eqref{eq:linearh} with the variance proportional to time.\nThere are two such neutral modes,\n\\begin{equation}\\label{eq:neutral}\n \\delta\n\\psi_{1}(t)\n(\\Re\\dot{\\mathcal{E}}_0(t), \\Im \\dot {\\mathcal{E}}_0(t),\\dot G_0(t),\\dot Q_0(t))^{\\mathrm{T}},\\qquad\n\\delta \\psi_{2}(t)= (-\\Im \\mathcal{E}_0(t), \\Re \\mathcal{E}_0(t),0,0)^{\\mathrm{T}},\n\\end{equation}\nwhich correspond to the time-shift and rotational symmetries of the unperturbed ($D=0$) nonlinear system\n\\eqref{subeq:E}-\\eqref{subeq:Q}, respectively; all the other Floquet modes are exponentially decaying.\nTwo properly normalized \\eqref{eq:orth} neutral modes $\\delta \\psi_{1}^\\dagger(t)$ and $\\delta \\psi_{2}^\\dagger(t)$  of the adjoint linear system \\eqref{eq:linearadj} can be used for\ncalculating the projections of noise onto the eigendirections $\\delta \\psi_{1}$ and $\\delta \\psi_{2}$.\nUsing the perturbation expansion with respect to the small parameter $D$, and adapting the asymptotic analysis from \\cite{REB11}, we obtain the following equations\nfor the noise-driven slow evolution of the phase $\\theta$ and the angular variable $\\varphi$ of solutions to Eqs.\\,\\eqref{subeq:E}-\\eqref{subeq:Q}:\n\\begin{equation}\\label{rate}\n\\dot \\theta = D\\,\\delta \\psi_{1}^\\dagger(t+\\theta) \\Gamma_{-\\varphi} w(t),\\qquad \\dot \\varphi =D\\,\\delta \\psi_{2}^\\dagger(t+\\theta) \\Gamma_{-\\varphi} w(t)\n\\end{equation}\nwith the Langevin term $\\Gamma_{-\\varphi}w(t)=(\\xi_1(t)\\cos\\varphi+\\xi_2(t)\\sin\\varphi,-\\xi_1(t)\\sin\\varphi+\\xi_2(t)\\cos\\varphi,0,0)^{\\rm T}$\nand the $T_0$-periodic coefficients $\\delta\\psi_{1}^\\dagger$ and $\\delta \\psi_{2}^\\dagger$.\n\nThe coefficients of the Fokker-Planck equation for the joint probability density\n$p(t,\\theta,\\varphi)$ of the stochastic process \\eqref{rate} are also periodic with respect to time.\nSince, for $D\\ll 1$, the probability density function $p(t,\\theta,\\varphi)$ changes slowly, Eqs.~\\eqref{rate} and the corresponding\nFokker-Planck equation can be averaged over the period $T_0$ of the functions $\\delta \\psi_{i}^\\dagger(t+\\theta)$,\nresulting in the diffusion equation with constant coefficients \\cite{averaging}. The diffusion coefficient \n\\begin{equation}\\label{dbar}\n\\bar d_{11}= \\frac{D^2}{T_0} \\int_0^{T_0}  \\bigl(\\delta \\psi_{1, 1}^\\dagger (s) \\bigr)^2+ \\bigl(\\delta \\psi_{1, 2}^\\dagger(s)\\bigr)^2\\, ds\n\\end{equation}\nof the time averaged Fokker-Planck equation approximates the rate of diffusion of the phase $\\theta$ (see Appendix). \nFinally, since the pulse timing jitter is usually calculated over a long time interval $n \\tilde T_0$ with $n \\gg 1$ and the average period $\\tilde T_0 \\approx T_0$,\nand is normalized by the number of round-trips $n$, we make the estimate of timing jitter as the product of the diffusion rate by the period\n\\begin{equation}\\label{eq:jitter}\n\\sigma_{\\mbox{var}}^2 = \\bar d_{11} T_0 = D^2 \\int_0^{T_0} \\bigl(\\delta \\psi_{1, 1}^\\dagger (s) \\bigr)^2+ \\bigl(\\delta \\psi_{1, 2}^\\dagger(s)\\bigr)^2\\, ds.\n\\end{equation}\nThis value is approximately equal to the variance of $\\theta(n \\tilde T_0)$ divided by $n \\gg 1$. We note that for the number of roundtrips $n \\geq 1$ that is not sufficiently large,\nthe numerically calculated timing jitter is not\napproximated by \\eqref{eq:jitter} since the numerically calculated value is affected by amplitude noise, or, in other words, stable eigendirections play role as well (see Fig. \\ref{fig1_1} (a)).\n\nFor the case of resonant optical feedback, expression \\eqref{eq:jitter} for the timing jitter can be further simplified, to ascertain the dependence on the feedback delay\nlength. This will be shown in the next section where we compare the analytic result with a numerical estimate of the timing jitter.\n\n\\section{Results}\n\\subsection{Comparison of semi-analytical and numerical methods of timing jitter calculation.}\n\nIn this section we compare the timing jitter calculated using Eq.~\\eqref{eq:jitter} with that\nobtained from the variance of the pulse timing fluctuations (long-term timing jitter) through numerical integration of the\nstochastic system (Eqs.~\\eqref{subeq:E}-\\eqref{subeq:Q} with $D\\neq 0$).\nThe latter (numerical) method is described in detail in \\cite{OTT14b}. We will focus mainly on the\ncase of one feedback cavity, $M=1$, and compare the two approaches to the timing jitter calculation\nat different feedback delay times ($\\tau_1\\equiv\\tau$) and the feedback strengths ($K_1\\equiv K$).\n\nFirst, we apply the semi-analytical method of the timing jitter calculation to the case of a\npassively ML semiconductor laser without feedback, i.e. $K_m \\equiv 0$ in\nEqs.\\,\\eqref{subeq:E}-\\eqref{subeq:Q}.\nIn \\cite{OTT14b} it was shown that after a sufficiently large number of roundtrips $n$ within the laser\ncavity the variance of the pulse timing fluctuations grows linearly with the round-trip number.\nIn the numerical method the timing fluctuations are therefore calculated over many thousands of cavity\nroundtrips. In Fig.~\\ref{fig1_1} (a) the timing jitter is plotted as a function of the round\ntrip number $n$. The initial decrease of the numerically calculated timing fluctuation variance (green line) with\n$n$ (for small $n$) can be attributed to the impact of the eigenfunctions with $\\Re \\lambda <0$ (see Appendix).\nUsing DDE-BIFTOOL \\cite{ENG01}, for $\\gamma T \\gg 1$ (or $\\gamma \\tau_m\\gg1$), one can typically observe that many characteristic exponents\n$\\lambda$ of the ML solution have real parts close to $0$, and, therefore, the equation of motion\n\\eqref{eq:motion} suggests that such exponents will have a non-negligible impact on the numerically\ncalculated timing  jitter even after many cavity round-trips. Since the eigenfunctions with $\\Re\n\\lambda <0$ are neglected in the semi-analytical approach, the value of the timing jitter estimated using this approach does not depend on\n$n$ (dashed red line in Fig.~\\ref{fig1_1} (a)). In the limit of large $n$ this value is in agreement with the data obtained by direct numerical\nintegration of Eqs.\\,\\eqref{subeq:E}-\\eqref{subeq:Q}, as shown in Fig.~\\ref{fig1_1} (a).\nFigure \\ref{fig1_1} (b) shows the timing jitter, obtained using both methods, in dependence of the noise strength $D$. It is seen\nthat good quantitative agreement is obtained for small to moderate levels of noise.\n\n\\begin{figure}[t!]\n\\centering{}%\n\\includegraphics[clip,width=0.45\\textwidth]{ltJitter_vs_rtn_k0_tau0.pdf}\n\\label{fig1_1a}\n\\includegraphics[clip,width=0.45\\textwidth]{ltJitter_vs_noise_k0_tau0.pdf}\n\\label{fig1_1b}\n\n\\caption{(a) Comparison of the results of numerical calculation of pulse timing jitter (green solid line),\nobtained for different numbers of round-trips $n$, with the timing jitter value from formula\n\\eqref{eq:jitter} (red dashed line).\n(b) Estimation of timing jitter, calculated using formula \\eqref{eq:jitter} (red solid line) and the\nnumerical method (green dots), vs noise strength $D$.\nParameters: $K=0$, $\\tau=0$, $T= 25\\mbox{ ps}$, $\\kappa=0.3$, $\\gamma^{-1}=125\\mbox{ fs}$, $\\gamma_g^{-1}=500\\mbox{\nps}$,\n$\\gamma_q^{-1}=5\\mbox{ ps}$, $s=10$, $q_0^{-1}=10\\mbox{ ps}$, $g_0^{-1}=250\\mbox{ ps}$,\n$\\alpha_g=2$, $\\alpha_q=1$.}\n\\label{fig1_1}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{ltJitter_vs_noise_ag0_aq0_k0.1_tau70.98_real300.pdf}\n\\includegraphics[width=0.45\\textwidth]{lttj_semiana_vs_num.pdf}\n\\caption{(a) Timing jitter in dependence of the noise strength, calculated using the semi-analytic method\n(red line) and the numerical method described in \\cite{OTT14b} (green dots) for $\\tau=70T_{ISI,0}$.\n(b) Timing jitter in dependence of the feedback delay time,\ncalculated using the semi-analytic method (red dashed line) and the numerical method (green line) for $D=0.2$.\nParameters: $\\alpha_g=0$, $\\alpha_q=0$, $K=0.1$. Other parameters are as in Table \\ref{tab:Simparams}.\n}\\label{pl2}\n\\end{center}\n\\end{figure}\nNext, let us consider a system with feedback from one external cavity. Figure \\ref{pl2} (a)\nshows a comparison of the timing jitter calculated from the two methods in dependence of the noise\nstrength. For the numerical timing jitter calculation method (green dots) the timing fluctuations that\narise over $40000$ round-trips in the laser cavity are calculated, and the variance of these timing\nfluctuations is then calculated for $300$ noise realisations. For the semi-analytical method (red\nline) the solutions to the adjoint linearized homogeneous system~\\eqref{eq:linearadj} are\nnumerically calculated. In both cases we simulate for a sufficiently long time (approximately 5000 roundtrips) before\nstarting the calculation of the timing jitter to avoid transient effects. We find very good\nagreement between the results obtained using the two methods. For the simulations presented in\nFig.~\\ref{pl2} (a) the feedback delay time was chosen to be resonant with the  ML pulse repetition\nperiod (inter-spike interval time) $T_{ISI,0}$ of a solitary laser (ML laser without feedback),\nmeaning that the condition $\\tau=qT_{ISI,0}$ is fulfilled, where $q$ is an integer.\nResonant feedback applied in the fundamental ML regime does not significantly affect the dynamical behaviour of the system,\nhence the laser output remains periodic and the semi-analytic method is applicable.\nWhen the feedback delay time is tuned from one resonance to the next, bifurcations\ncan occur and the dynamical behaviour can change. This is described in detail in \\cite{OTT14} and\n\\cite{OTT12a}. In Fig.~\\ref{pl2} (b) the numerically calculated dependence of the timing jitter on\nthe delay time $\\tau$ is compared to that estimated semi-analytically, spanning from the $67$th to\nthe $68$th resonance ($q=67$ and $q=68$, respectively). Within the frequency-pulling\nregions of the main resonances there is very good agreement between the results obtained using the\ntwo methods. The frequency pulling regions are the $\\tau$ ranges about the main resonances within which there is\none pulse in the laser cavity and the repetition rate tunes with $\\tau$ \\cite{OTT12a}. In Fig.~\\ref{pl2} (b) these\nregions can be identified by the low timing jitter about the main resonances.\nAt the edges of the frequency-pulling regions there is a sharp increase in the timing jitter. This very\nlarge timing jitter coincides with saddle-node bifurcation points of the deterministic system\n(Eqs.~\\eqref{subeq:E}-\\eqref{subeq:Q} with $D=0$) \\cite{OTT14b}. At the edge of the $67$th resonance\nthere is a large discrepancy between the semi-analytical and numerical methods. This is because in the stochastic system\nnoise induced switching between bistable solutions,\nwhich arise due to the saddle-node bifurcations, occurs. Away from the bifurcation points there is good agreement between the two methods, also\nbetween the main resonances, because although the dynamical behaviour changes between the main resonances, i.e. multiple feedback induced\npulses, the solutions remain periodic and therefore the semi-analytical method is applicable.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{ltJitter_100tisi_ag2_aq1.5.pdf}  \\includegraphics[width=0.48\\textwidth]{ltJitter_adjoint_100tisi_ag2_aq1.5_it2.3E8.pdf}\n\\caption{Timing jitter in dependence of the feedback cavity delay time and feedback strength,\ncalculated numerically (a) and using the semi-analytical method (b). The timing jitter is indicated by the colour code and $\\sigma_{lt,0}$ is the\ntiming jitter of the solitary laser. Regions in white indicate a\ntiming jitter greater then $20$fs. In subplot (b) black marks the regions where the deterministic system has a non-periodic solution and\nthe semi-analytical method cannot be applied.\nParameters: $D=0.2$, $\\alpha_g=2$, $\\alpha_q=1.5$, others are as in table \\ref{tab:Simparams}.\n}\\label{pl3}\n\\end{center}\n\\end{figure}\n\nFor the parameters used in Fig.~\\ref{pl2} (b) the system is well behaved and the solutions are periodic, however\nfor other parameters, particularly for larger feedback strengths and non-zero amplitude-phase coupling,\nthis is not the case; quasi-periodic or chaotic dynamics can be observed. In such regions the semi-analytic approach\nis invalid, however the timing jitter calculated by numerical methods is not meaningful in these non-periodic region either.\nIn Fig.~\\ref{pl3} the timing jitter, calculated from the numerical (a) and semi-analytical (b) methods, is plotted in\ndependence of $K$ and $\\tau$ for $\\alpha_g=2$ and $\\alpha_q=1.5$. The timing jitter is given by the colour code, where blue regions\nindicate a reduction in the timing jitter with respect to the solitary laser, red tones indicate an increase and white regions indicate a\ntiming jitter greater than $20$fs, indicative of a non-periodic pulse stream. In the black regions in Fig.~\\ref{pl3} (b)\nthe solutions of the DDE system are non-periodic and the semi-analytic method is not applied. Good\nagreement is observed between these two methods over\nmost of the parameter range depicted. The non-periodic regions indicated in subplot (b) coincide\nwith the very high timing jitter estimations obtained using the\nnumerical method.\n\nA key difference between the two methods is that the semi-analytic method is based on the\nnumerical simulation of deterministic equations, while the purely numerical method requires\nintegration of a system of stochastic DDEs. Using the latter method one can run into problems that\narise due to the multiplicity of stable solutions found in this system. Since timing jitter estimation\nrequires averaging over many noise realisations, depending on the particular realisation, due to transient effects, the system\ncan land on different solutions. As different ML solutions can have slightly different inter-spike\ninterval times, the fully numerical estimation of the timing jitter can lead to erroneously large\nvalues in such case \\cite{SIM14}. This makes it difficult to perform\ntiming jitter calculations over a large parameter domain, as it is not easy to distinguish between\nthe above mentioned effect and a destabilisation of the pulse stream due to the feedback conditions.\nNote that this is a different effect to switching between solutions within one time series. Such\ndifficulties are eliminated when using the\nsemi-analytic method, as in this case the estimation of the variance is based on the integration of\ndeterministic equations. Therefore, there are two main advantages to using the semi-analytic method\nto calculate the timing jitter, compared with brute force methods involving numerical integration of\nstochastic differential equations. Firstly, the aforementioned difficulties can be avoided, and\nsecondly, the computation times can be greatly reduced (by over a factor of 100) as averaging over many noise realisation is\nnot needed. This means that it can become feasible to calculate the timing jitter for longer\nfeedback delay times, which is of interest\ndue to the improved timing jitter reduction predicted for increased delay times \\cite{OTT14} and for\nbetter comparison with experiments, where typically very long feedback cavities are used\n\\cite{ARS13,LIN10e}.\n\n\\subsection{Delay length dependence of timing jitter}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ltJitter_ag0_aq0_tau0-209.999_K0.1_1D.pdf}  \\includegraphics[width=0.49\\textwidth]{ltJitter_ag0_aq0_tau0-209.999_K0.1_.png}\n\\caption{(a) and (b) Timing jitter $\\sigma_{lt}$ in dependence of the feedback cavity delay time.\nThe colour code indicates the\ntiming jitter according to the colour bar given in subplot (c). The black dashed line indicates the\ntiming jitter of\nthe solitary laser. (c) Timing jitter $\\sigma_{lt}$ in dependence of the feedback cavity delay time,\nwhere $\\tau=\\tau_0+\\tau_1$ for any given point. The horizontal axis spans one $T_{ISI,\\tau=0}$ and\nis centered on an exact main resonance. The vertical axis indicates the number of the main\nresonance. The timing jitter is indicated by the colour code and $\\sigma_{lt,0}$ is the\ntiming jitter of the solitary laser.\nParameters: $K=0.1$, $D=0.2$, $\\alpha_g=0$, $\\alpha_q=0$, others as in table \\ref{tab:Simparams}.\n}\\label{pl4}\n\\end{center}\n\\end{figure}\n\nWe now use the semi-analytic method to investigate how the timing jitter decreases with increased\nresonant feedback delay times and how the width of the\nfrequency-pulling regions is affected by this\nincrease. In Fig.~\\ref{pl4} the timing jitter is plotted as a function of $\\tau$ in subplots (a)\nand (b) for a short and a long $\\tau$ range, respectively. The black dashed line indicates the\ntiming jitter of the solitary laser. The delay times are plotted in\nunits of $T_{ISI,\\tau=0}$, the inter-spike interval time for zero delay\nfeedback (instantaneous feedback, $\\tau=0$ and $K\\neq0$), meaning that the resonant feedback\noccurs at the integer delay values. ($T_{ISI,\\tau=0}$ and $T_{ISI,0}$ only differ slightly. Here we choose $T_{ISI,\\tau=0}$\nas our reference because the period is the same for all $\\tau=qT_{ISI,\\tau=0}$, $q\\in\\mathbb{N}$, and we will\nuse this property in subsequent calculations.) In both (a) and (b) a timing jitter reduction is\nobserved for resonant feedback. For the longer delay times depicted in subplot (b) the timing jitter\nreduction is greater and the frequency-pulling region\nabout the main resonances is wider. Changes in the frequency-pulling regions are not discernible over small $\\tau$\nranges. To show the change\nin dependence of $\\tau$ more clearly a map of the timing jitter is shown in a $\\tau-\\tau$ plot in\nsubplot (c). In this plot both axes are related\nto the delay time, the $\\tau_1$ axis shows changes over one\n$T_{ISI,\\tau=0}$-interval , whereas the $\\tau_0$ axis shows changes from one resonance to\nthe next. For\neach point on this map the feedback delay time is given by $\\tau=\\tau_0+\\tau_1$. The $\\tau_1$ axis\nis centered on the exact main resonances $\\tau=qT_{ISI,\\tau=0}$\nand the $\\tau_0$ axis gives the number $q$ of the main resonances. The timing jitter is\ngiven by the colour code. Regions in blue and green indicate\na reduction in the timing jitter with respect to the solitary laser ($K=0$) and regions in red indicate an\nincrease in the timing jitter. In the green\nregions the timing jitter is reduced by a factor of 10 or greater. For all $q$ values a reduction in\nthe timing jitter is achieved at the exact main resonances and for increasing $q$ the decrease in\nthe timing jitter can clearly be seen. It is seen from Fig.~\\ref{pl4}(c) that for short\ndelays the width of the frequency-pulling regions, with reduced timing jitter, increases\napproximately linearly with the number $q$. The edges of the frequency pulling region are marked by the\ndashed black lines. At about $q=50$ the frequency-pulling region is\nintersected by the solutions that correspond to higher order resonances ($p\\tau=qT_{ISI,\\tau=0}$, where $p=2,3,4,...$). This is due to a bistability\nbetween the main and higher order resonant solutions \\cite{OTT12a}. For the results\npresented in subplot (c) of Fig.~\\ref{pl4}, the same\ninitial conditions were used in the numerical simulations for all delay values. By performing a\nsweep in $\\tau$ (using the previous $\\tau$ solution as the initial conditions for the next $\\tau$) one can stay on the main resonant solution\nin the bistable regions.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{ltJitter_ag0_aq0_tau0-209.999_K0.1_restau_tstep0.001.pdf}\n\\caption{Timing jitter $\\sigma_{lt}$ at the exact main resonances (red solid line) and the minimum timing jitter in each\nresonance region (blue solid line) as a function of the number $q$ of\nthe main resonance, calculated using the semi-analytic\nmethod. The dashed line shows the timing jitter at the exact main resonances given by the analytic expression Eq.~\\eqref{sigmares}. The dot-dashed\nline shows the fit of Eq.~\\eqref{sigmin} to the minimum timing jitter in each\nresonance regions.\nParameters: $K=0.1$, $D=0.2$, $\\alpha_g=0$, $\\alpha_q=0$, others as in table \\ref{tab:Simparams}.\n}\\label{pl5}\n\\end{center}\n\\end{figure}\n\nIn order to quantify the decrease in the timing jitter with increasing number $q$, we\nhave plotted the timing jitter at the main resonances in Fig.~\\ref{pl5}. The red line shows the\nresults of the semi-analytic method for the exact main resonances $\\tau=qT_{ISI,\\tau=0}$ ($\\tau$ values corresponding to the white dashed line in Fig.~\\ref{pl4} (b)) and the\nblue line shows the results of the semi-analytic method for the minimum timing jitter in each main\nresonance frequency-pulling region ($\\tau$ values corresponding to the white dot-dashed line in Fig.~\\ref{pl4} (b)). The expression for the timing jitter at the main resonances, $\\tau=qT_{ISI,\\tau=0}$, can be\nderived analytically using Eq.~\\eqref{eq:jitter} and the bilinear form \\eqref{eq:form}. At the\nexact main resonances the solutions to Eqs.~\\eqref{subeq:E}-\\eqref{subeq:Q} are identical for all\n$q$, and\nthe periodicity is the same as that of the laser with zero delay (instantaneous) feedback $T_0$ ($T_{ISI,\\tau=0}$).\nTherefore, for $\\tau=qT_{ISI,\\tau=0}$, Eq.~\\eqref{eq:form} can be expressed as\n\\begin{gather}\n\\left[ \\delta \\psi^\\dagger, \\delta \\psi \\right](t) = \\delta \\psi^\\dagger(t) \\delta \\psi(t)+\\int_{-T}^{0}\n \\delta\\psi^\\dagger\\left(t+r+T\\right)B_0\\left(t+r\\right)\\delta\\psi\\left(t+r\\right)dr\\nonumber\\\\\n  +K\\int_{-T}^{0}\n \\delta\\psi^\\dagger\\left(t+r+T\\right)B_1\\left(t+r\\right)\\delta\\psi\\left(t+r\\right)dr \\nonumber \\\\\n +K\\int_{-T-qT_{ISI,\\tau=0}}^{-T}\n \\delta\\psi^\\dagger\\left(t+r+T\\right)B_1\\left(t+r\\right)\\delta\\psi\\left(t+r\\right)dr. \\label{bilin2}\n\\end{gather}\nThe last term on the right-hand side can be further simplified due to the time shift invariance and periodicity of the integrand, giving\n\\begin{equation}\n\\left[ \\delta \\psi^\\dagger, \\delta \\psi \\right]=\\left[ \\delta \\psi^\\dagger, \\delta \\psi \\right]^{\\tau=0} +Kq\\int_{-T_{ISI,\\tau=0}}^{0}\n \\left(\\delta\\psi^\\dagger\\left(t+r+T\\right)\\right)^{\\mathrm{T}}B_1\\left(t+r\\right)\\delta\\psi\\left(t+r\\right)dr, \\label{bilin3}\n\\end{equation}\nwhere the first three terms on the rhs of Eq.~\\eqref{bilin2} are now expressed as $\\left[ \\delta \\psi^\\dagger, \\delta \\psi \\right]^{\\tau=0}$, which is the bilinear form for $\\tau=0$ ($q=0$).\nEquation~\\eqref{eq:jitter} can thus be expressed as\n\\begin{equation}\n \\sigma_{var}=\\sqrt{D^2\\int_{0}^{T_0}\\left(\\frac{\\delta\\psi_{0t,1}^{\\dagger *}\\left(t\\right)}{\\left[\\delta\\psi_{1}^{\\dagger *},\\delta\\psi_{1}^*\\right]^{\\tau=0} +Kq \\mathcal{F}'\\left( K\\right)}\\right)^2+\\left(\\frac{\\delta\\psi_{0t,2}^{\\dagger *}\\left(t\\right)}{\\left[\\delta\\psi_{1}^{\\dagger *},\\delta\\psi_{1}^*\\right]^{\\tau=0} +Kq \\mathcal{F}'\\left( K\\right)}\\right)^2 dt}, \\label{svar2}\n\\end{equation}\nwhere $\\frac{\\delta\\psi_{1}^{\\left( \\dagger \\right) *}}{\\left[ \\delta\\psi_{1}^{\\dagger *},\\delta\\psi_{1}^*\\right]}=\\delta\\psi_{1}^{\\left( \\dagger \\right) }$ and\n\\begin{equation}\n \\mathcal{F}'\\left( K\\right) =\\int_{-T_{ISI,\\tau=0}}^{0}\n \\delta\\psi_{0t}^{\\dagger *}\\left(t+r+T\\right)B_1\\left(t+r\\right)\\delta\\psi_{1}^*\\left(t+r\\right)dr, \\nonumber\n\\end{equation}\nwhich is a function of $K$ but not of $\\tau$. Finally, Eq.~\\eqref{svar2} can be simplified to\n\\begin{equation}\n \\sigma_{var}=\\frac{1}{1+Kq\\mathcal{F}' \\left( K\\right)}\\sqrt{D^2\\int_{0}^{T_0}\\left(\\delta\\psi_{0t,1}^{\\dagger \\tau=0}\\left(t\\right)\\right)^2+\\left(\\delta\\psi_{0t,2}^{\\dagger \\tau=0}\\left(t\\right)\\right)^2 dt}, \\label{svar3}\n\\end{equation}\nwhere $\\delta\\psi_{0t}^{\\dagger \\tau=0}=\\left( \\delta\\psi_{0t,1}^{\\dagger \\tau=0},\\delta\\psi_{0t,2}^{\\dagger \\tau=0},\\delta\\psi_{0t,3}^{\\dagger \\tau=0},\\delta\\psi_{0t,4}^{\\dagger \\tau=0}\\right)^\\mathrm{T}$\nis the solution fulfilling the biorthogonality condition for $\\tau=0$ and $\\mathcal{F}\\left( K \\right) =\\frac{\\mathcal{F}'\\left( K\\right)}{\\left[\\delta\\psi_{1}^{\\dagger *},\\delta\\psi_{1}^*\\right]^{\\tau=0}}$. The timing jitter for\nresonant feedback, $\\tau=qT_{ISI,\\tau=0}$, is therefore given by\n\\begin{equation}\n \\sigma_{lt}^{\\tau=qT_{ISI,\\tau=0}}= \\frac{\\sigma_{lt}^{\\tau=0}\\left( K\\right)}{1+Kq\\mathcal{F} \\left( K\\right)}, \\label{sigmares}\n\\end{equation}\nwhere $\\sigma_{lt}^{\\tau=0}\\left( K\\right)$ is the timing jitter for $\\tau=0$. The curve obtained using this analytic expression is shown by dashed black line in Fig.~\\ref{pl5}. A formula for the minimum jitter can not be derived in the same way as the inter-spike interval time changes with $q$. However,\nfitting the minimum jitter curve for various feedback strengths we find that the relation\n\\begin{equation}\n\\sigma_{lt}^{\\mathrm{min}}\\approx\\frac{\\sigma_{lt}^{\\tau=0}\\left( K\\right)}{1+Kq}, \\label{sigmin}\n\\end{equation}\nholds well for low feedback strengths. The fit is plotted in the black dot-dashed line in\nFig.~\\ref{pl5}.\n\nIn the derivation of Eq.~\\eqref{sigmares} contributions to the timing jitter from\neigenfunctions with negative eigenvalues, $\\lambda<0$, are neglected. However, for\nincreased feedback delay lengths, the number of weakly stable Floquet multipliers close to\none increases. This leads to long transients in numerical simulations of the deterministic\nsystem (Eqs.~\\eqref{subeq:E}-\\eqref{subeq:Q} $D=0$). These transient effects are accompanied\nby fluctuations in the pulse heights, which have the periodicity of the feedback delay time.\nIncluding noise in the system excites these transient amplitude fluctuations, which results in an\nincreased timing jitter, as, via the interaction with the gain and absorber media, changes in the\npulse height also lead to slight changes in the pulse positions.\nEquation~\\eqref{sigmares} is therefore only valid in the limit in which such effects can be neglected. For\nthe parameter values used in our simulations Eq.~\\eqref{sigmares} holds for up\nto $q\\approx300$. These noise induced transient effects were observed\nexperimentally as side peaks in the phase noise spectra \\cite{HAJ12,ARS13,DRZ13a}.\n\n\\section{Conclusions}\n\nWe have investigated the influence of optical feedback on the timing jitter of a passively ML semiconductor laser.\nFor resonant feedback we have derived an expression, Eq.~\\eqref{sigmares}, for the analytical dependence of the timing jitter on the\nfeedback delay length, showing that the timing jitter drops off as approximately $1\/\\tau$ for $\\tau\\gg T$, as long as\namplitude jitter effects can be neglected. About the main resonant feedback delay lengths, frequency-pulling regions form,\nin which the timing jitter is reduced with respect to the solitary laser. For small\nfeedback strengths $K$ the widths of these frequency-pulling regions\nincrease linearly with the number $q$ of the main resonance.\nThese results were obtained using a semi-analytical method, presented in this paper,\nof calculating timing fluctuations in a DDE system describing the dynamics of a\npassively ML semiconductor laser subject to optical feedback from an arbitrary number of feedback\ncavities. The semi-analytical method shows good agreement\nwith methods based on direct numerical integration of the stochastic model and has the advantage of greatly reduced computation times.\n\n\\begin{acknowledgments}\nThe authors are thankful to D. Turaev for a very useful discission of the asymptotic analysis method.\nL. Jaurigue thanks B. Lingnau for fruitful discussions and acknowledges support from the GRK\n1558 funded by the DFG. A. Pimenov and A. G. Vladimirov acknowledge the support of SFB 787 of the\nDFG, project B5. A. G. Vladimirov also acknowledges the support of 14-41-00044 of RSF at the Lobachevsky\nUniversity of Nizhny Novgorod. D. Rachinskii acknowledges the support of NSF through grant\nDMS-1413223.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusions}\\label{sec:conclusion}\nIn this paper, we proposed an automatic face recognition system for unconstrained video-based face recognition tasks. The proposed system consists of modules for face detection and alignment, face association and tracking, face representation, subspace learning and subspace-to-subspace similarity-based matching. We evaluated our system on four video datasets. The experimental results demonstrate the superior performance of the proposed system.\n\n\\section{Experiments}\\label{sec:exp}\nIn this section, we report video-based face recognition results for the proposed system on two challenging unconstrained multimedia face recognition dataset, IARPA Janus Benchmark B (IJB-B) \\cite{ijbb} and IARPA Janus Surveillance Video Benchmark  (IJB-S), and compare with other baseline methods. We also provide results on Multiple Biometric Grand Challenge (MBGC) Version 1 \\cite{MBGC}, and Face and Ocular Challenge Series (FOCS) \\cite{FOCS} datasets, to demonstrate the effectiveness of the proposed system. We discuss below the details of datasets, protocols and our training and testing procedures.\n\n\\subsection{Datasets}\n\\textbf{IARPA Janus Benchmark B (IJB-B):} IJB-B \\cite{ijbb} dataset is an unconstrained face recognition dataset. It contains 1845 subjects with 11,754 images, 55,025 frames and 7,011 multiple-shot videos. IJB-B is a template-based dataset where a template consists of a varying number of still images or video frames from different sources. A template can be either image-only, or video-frame-only, or mixed media template.  Sample frames from this dataset are shown in Figure~\\ref{fig:ijbb_example}.\n\nIn this paper, we only focus on the 1:N video protocol of IJB-B. It is an open set 1:N identification protocol where each given probe is collected from a video and is searched among all gallery faces. Gallery candidates are ranked according to their similarity scores to the probes. Top-K rank accuracy and True Positive Identification Rate (TPIR) over False Positive Identification Rate(FPIR) are used to evaluate the performance. The gallery templates are separated into two splits, $G_1$ and $G_2$,  all consisting of still images. For each video, we are given the frame index with face bounding box of the first occurrence of the target subject, as shown in Figure~\\ref{fig:ijbb_example}. Based on this anchor, all the faces in that video with the same identity should be collected to construct the probes. The identity of the first occurrence bounding box will be considered as the template identity for evaluation.\n\n\\textbf{IARPA Janus Surveillance Video Benchmark (IJB-S):} Similar to IJB-B, the IJB-S dataset is also a template-based, unconstrained video face recognition dataset. It contains faces in two separate domains: high-resolution still images for galleries and low quality, remotely captured surveillance videos for probes. It consists of 202 subjects from 1421 images and 398 single-shot surveillance videos. The number of subjects is small compared to IJB-B, but it is even more challenging due to the low quality nature of surveillance videos.\n\nBased on the choices of galleries and probes, we are interested in three different surveillance video-based face recognition protocols: surveillance-to-single protocol, surveillance-to-booking protocol and surveillance-to-surveillance protocol. These are all open set 1:N protocols where each probe is searched among the given galleries. Like IJB-B, the probe templates are collected from videos, but no annotations are provided. Thus raw face detections should be grouped to construct templates with the same identities.\n\nGalleries consist of only single frontal high resolution image for surveillance-to-single protocol. Galleries are constructed by both frontal and multiple-pose high resolution images for surveillance-to-booking protocol. For the most challenging surveillance-to-surveillance protocol, galleries are collected from surveillance videos as well, with given bounding boxes. In all three protocols, gallery templates are split into two splits, $G_1$ and $G_2$. During evaluation, the detected faces in videos are first matched to the ground truth bounding boxes to find their corresponding identity information. The majority of identities appears in each template will be considered as the identity of the template, and will be used for further identification evaluation. Example frames are shown in Figure~\\ref{fig:cs6_example}. Notice the remote faces are acquired with very low quality.\n\n\\textbf{Multiple Biometric Grand Challenge (MBGC):} The MBGC Version 1 dataset contains 399 walking (frontal face) and 371 activity (profile face) video sequences from 146 people. Figure~\\ref{fig:mbgc_focs_example} shows some sample frames from different walking and activity videos. In the testing protocol, verification is specified by two sets: target and query. The protocol requires the algorithm to match each target sequence with all query sequences. Three verification experiments are defined: walking-vs-walking (WW), activity-vs-activity (AA) and activity-vs-walking (AW). \n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[b]{0.23\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/MBGC_W1.jpg}\n\\caption{MBGC Walking}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.23\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/MBGC_A2.jpg}\n\\caption{MBGC Activity}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}[b]{0.23\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/FOCS_W1.jpg}\n\\caption{FOCS Walking}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.23\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/FOCS_A1.jpg}\n\\caption{FOCS Activity}\n\\end{subfigure}\n\\caption{Examples of MBGC and FOCS datasets}\n\\label{fig:mbgc_focs_example}\n\\end{figure}\n\n\\textbf{Face and Ocular Challenge Series (FOCS):}\nThe video challenge of FOCS \\cite{FOCS} is designed for frontal and non-frontal video sequence matching. The FOCS UT Dallas dataset contains 510 walking (frontal face) and 506 activity (non-frontal face) video sequences of 295 subjects with frame size of 720$\\times$480 pixels. Like MBGC, FOCS specifies three verification protocols: walking-vs-walking, activity-vs-walking, and activity-vs-activity. In these experiments, 481 walking videos and 477 activity videos are chosen as query videos. The size of target sets ranges from 109 to 135 video sequences. Sample video frames from this dataset are shown in Figure~\\ref{fig:mbgc_focs_example}.\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/MBGC_WW_newer.png}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/MBGC_AW_newer.png}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/MBGC_AA_newer.png}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/FOCS_WW_newer.png}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/FOCS_AW_newer.png}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.32\\textwidth}\n\\includegraphics[width=\\textwidth]{figs\/FOCS_AA_newer.png}\n\\end{subfigure}\n\\caption{Verification results on MBGC and FOCS datasets}\n\\label{fig:mbgc_focs_v}\n\\end{figure*}\n\n\\subsection{Implementation Details}\\label{sec:imp}\nIn this part, we discuss the implementation details for each dataset respectively.\n\\subsubsection{IJB-B}\nFor the IJB-B dataset, we employ the SSD face detector \\cite{icipssd} to extract the face bounding boxes in all images and video frames. We employ the facial landmark branch of All-in-One Face \\cite{allinone} for fiducial detection on every detected bounding boxes and apply facial alignment based on these fiducials using the seven-point similarity transform.\n\nThe aligned faces are further represented using three networks proposed in \\cite{magazine}. We denote them as Network A, Network B and Network C. Network A modifies the ResNet-101 \\cite{He2015residual} architecture. It has an input size of dimensions $224\\times224$ and adds an extra fully connected layer after the last convolutional layer to reduce the feature dimensionality to 512. Also it replaces the original softmax loss with the crystal loss \\cite{crystal} for more stable training. Network B uses the Inception-ResNet-v2 \\cite{inception} model as the base network. Similar to Network A, an additional fully-connected layer is added for dimensionality reduction. Naive softmax followed by cross entropy loss is used for this network. Network C is based on the face recognition branch in the All-in-One Face architecture \\cite{allinone}. The branch consists of seven convolutional layers followed by three fully connected layers.\n\nNetwork A and Network C are trained on the MSCeleb-1M dataset \\cite{msceleb} which contains 3.7 million images from 57,440 subjects. Network B is trained on the union of three datasets called the Universe dataset: 3.7 million still images from the MSCeleb-1M dataset, 300,000 still images from the UMDFaces dataset \\cite{UMDFaces}, and about 1.8 million video frames from the UMDFaces Video dataset. For each network, we further reduce its dimensionality into 128 by triplet probabilistic embedding (TPE) \\cite{SwamiBTAS} trained on the UMDFaces dataset.\n\nFor face association, we use the kernelized correlation filter (KCF) tracker \\cite{ofstracker} to pre-associate the initial face bounding boxes for target subjects. The tracking algorithm is applied on the face detection bounding boxes for the first k = 50 frames after the annotated frame. If a pre-associated face detection bounding box has an IoU with the tracking bounding box less than 0.3, we discard the bounding box to prevent undesirable pre-associations resulting from the drifting of the tracker. For the parameters of face association, two bounding boxes with IoU ratio less than $\\gamma = 0.1$ are enforced by a cannot-link constraint.\n\nA dataset of 160,498 face images from 1,710 subjects is collected to model the negative background subjects.\nWe use the weighted LIBLINEAR implementation \\cite{liblinear} with L2-regularized L2-loss support vector classification setting to learn the weight vector, and its cost parameter C is set to 10.\n\nThen, features from associated bounding boxes are used to construct the probe templates. We use quality-aware pooling for both gallery and probe templates to calculate their exemplars (\\textbf{QCos}) where $t = 7$ and $q=0.3$ are used for detection score normalization. Subspaces are built by applying quality-aware subspace learning method (\\textbf{QSub}) on each template and taking the top three eigenvector with the largest corresponding eigenvalues. When fusing the cosine similarity and variance-aware projection similarity metric (\\textbf{VPM}), we use $\\lambda = 1$ so two similarity scores are fused equally. We compute the subspace-to-subspace similarity score for each network independently, and combine the similarity scores from three networks by score-level fusion. We also implement baseline methods using combinations of exemplars from vanilla average pooling (\\textbf{Cos}), subspaces learned by regular PCA (\\textbf{Sub}) and projection similarity metric (\\textbf{PM}).\n\n\\begin{table*}[t]\n\\centering\\resizebox{0.8\\textwidth}{!}\n{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n Methods &  Rank=1 & Rank=2 & Rank=5 & Rank=10 & Rank=20 & Rank=50 & FPIR=0.1 & FPIR=0.01 \\\\\n \\hline\n\\cite{TFA} with Iteration 0 & 55.94\\% & - & 68.40\\% & 72.89\\% & - &  83.71\\% & 44.60\\% & 28.73\\%\\\\\n \\hline\n\\cite{TFA} with Iteration 3 & 61.01\\% & - & 73.39\\% & 77.90\\% & - &  87.62\\% & 49.73\\% & 34.11\\%\\\\\n \\hline\n \\cite{TFA} with Iteration 5 & 61.00\\% & - & 73.46\\% & 77.94\\% & - &  87.69\\% & 49.78\\% & 33.93\\%\\\\\n \\hline\n \\hline\nCos & 78.37\\% & 81.35\\% & 84.39\\% & 86.29\\% & 88.30\\% & 90.82\\% & 73.15\\% & 52.19\\%\\\\\n\\hline\nQCos & 78.43\\% & 81.41\\% & 84.40\\% & 86.33\\% & 88.34\\% & 90.88\\% & \\textbf{73.19\\%} & \\textbf{52.47\\%}\\\\\n\\hline\nCos+Sub-PM & 77.99\\% & 81.45\\% & 84.68\\% & 86.75\\% & 88.96\\% & 91.91\\% & 72.31\\% & 38.44\\%\\\\\n\\hline\nQCos+Sub-PM & 78.02\\% & 81.46\\% & 84.76\\% & 86.72\\% & 88.97\\% & 91.91\\% & 72.38\\% & 38.88\\%\\\\\n\\hline\nQCos+QSub-PM & 78.04\\% & 81.47\\% & 84.73\\% & 86.72\\% & 88.97\\% & 91.93\\% & 72.39\\% & 38.91\\%\\\\\n\\hline\nQCos+QSub-VPM & \\textbf{78.93\\%} & \\textbf{81.99\\%} & \\textbf{84.96\\%} & \\textbf{87.03\\%} & \\textbf{89.24\\%} & \\textbf{92.02\\%} & 71.26\\% & 47.35\\%\\\\\n\\hline\n\\end{tabular}}\n\\caption{1:N Search Top-K Average Accuracy and TPIR\/FPIR of IJB-B video search protocol}\n\\label{tab:ijbb}\n\\end{table*}\n\n\\subsubsection{IJB-S}\nFor the IJB-S dataset, we employ the multi-scale face detector DPSSD to detect faces in surveillance videos. We only keep face bounding boxes with detection scores greater than 0.4771, to reduce the number of false detections. We use the facial landmark branch of All-in-One Face \\cite{allinone} as the fiducial detector. Face alignment is performed using the seven-point similarity transform.\n\nDifferent from IJB-B, since IJB-S does not specify the subject of interest, we are required to localize and associate all the faces for different subjects to yield the probe sets. Since IJB-S videos are single-shot, we use SORT \\cite{SORT} to track every face appearing in the videos. Faces in the same tracklet are grouped to create a probe template. Since some faces in surveillance videos are of extreme pose, blur and low-resolution, to improve precision, tracklets consisting of such faces should be rejected during the recognition stage. By observation, we find that most of the short tracklets are of low quality and not reliable. The average of the detection score provided by DPSSD is also used as an indicator of the quality of the tracklet. On the other hand, we also want to take the performance of face detection into consideration to strike a balance between recall and precision. Thus in our experiments, we use two configurations for tracklets filtering: 1) We keep those tracklets with length greater than or equal to 25 and average detection score greater than or equal to 0.9 to reject low-quality tracklets and focusing on precision. It is referred to as \\textbf{with Filtering}. 2) Following the settings in \\cite{ijbs}, we produce results without any tracklets filtering and focusing on both precision and recall. It is referred to as \\textbf{without Filtering}.\n\nBecause of the remote acquisition scenario and blurred probes in the IJB-S dataset, we retrain Network A with the same crystal loss but on the Universe dataset used by Network B. We denote it as Network D. We also retrain Network B with the crystal loss \\cite{crystal} on the same training data. We denote it as Network E. As a combination of high capacity network and large scale training data, Network D and E are more powerful than Networks A, B, and C. As before, we reduce feature dimensionality into 128 using the TPE trained on the UMDFaces dataset.\n\nIn IJB-S, feature aggregation and matching parts are the same as IJB-B except that we combine the similarity score  by score-level fusion from Network D and E. One thing that needs to be mentioned is that for the surveillance-to-surveillance protocol, we only use single Network D for representation as Network E is ineffective for low-quality gallery faces in this protocol.\n\n\\subsubsection{MBGC and FOCS}\nFor MBGC and FOCS datasets, we use All-in-One Face for both face detection and facial landmark detection. The MBGC and FOCS datasets contain only one person in a video in general. Hence, for each frame, we directly use the face bounding box with the highest detection score as the target face. Similar to IJB-S, bounding boxes are filtered based on detection scores. From the detected faces, deep features are extracted using Network D. Since MBGC and FOCS datasets do not provide training data, we also use the TPE trained on UMDFaces dataset to reduce feature dimensionality into 128. For MBGC and FOCS, feature aggregation and matching parts are the same as IJB-B and IJB-S.\n\n\\subsection{Evaluation Results}\nIn the following section, we first show some face association results on IJB-B and IJB-S datasets. Then we compare the performance of the proposed face recognition system with several baseline methods. For each dataset, all the baseline methods listed below use deep features extracted from the same network and with the same face detector.\n\\begin{itemize}\n\\item \\textbf{Cos:} We compute the cosine similarity scores directly from the average pooled exemplars of the deep representation.\n\\item \\textbf{QCos:} We compute the cosine similarity scores from the quality-aware average pooled exemplars of the deep representation.\n\\item \\textbf{Cos+Sub-PM:} Subspace-to-subspace similarity is computed by fusing the plain cosine similarity and plain projection metric, and subspaces are learned by plain PCA.\n\\item \\textbf{QCos+Sub-PM:} Subspace-to-subspace similarity is computed by fusing the quality-aware cosine similarity and plain projection metric, and subspaces are learned by plain PCA.\n\\item \\textbf{QCos+QSub-PM:} Subspace-to-subspace similarity is computed by fusing the quality-aware cosine similarity and plain projection metric, and subspaces are learned by quality-aware subspace learning.\n\\item \\textbf{QCos+QSub-VPM:} Subspace-to-subspace similarity is computed by fusing the quality-aware cosine similarity and variance-aware projection metric, and subspaces are learned by quality-aware subspace learning.\n\\end{itemize}\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/ijbb_result_1A.png}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/ijbb_result_2A.png}\n\\end{subfigure}\n\\caption{Examples of face association results by the proposed TFA method on IJB-B. The target annotation is in red box, and the associated faces of the target subject are in magenta boxes.}\n\\label{fig:ijbb_result}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/ijbb_result_1.png}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/ijbb_result_2.png}\n\\end{subfigure}\n\\caption{Associated faces by the proposed TFA method corresponding to examples in Figure~\\ref{fig:ijbb_result}. Face images are in the order of the confidence of face association.}\n\\label{fig:ijbb_result_box}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_result_1.jpg}\n\\end{subfigure}\n\\\\\n\\vspace{2pt}\n\\begin{subfigure}[b]{0.95\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_result_2.jpg}\n\\end{subfigure}\n\\caption{Associated faces using SORT in IJB-S. Face images are in their temporal order. Notice the low-quality faces at the boundaries of tracklets since the tracker cannot reliably track anymore.}\n\\label{fig:cs6_result_box}\n\\end{figure}\n\n\\noindent \\textbf{IJB-B:} Figures~\\ref{fig:ijbb_result} and \\ref{fig:ijbb_result_box} show some examples of our face association results using the proposed TFA in IJB-B dataset. Table~\\ref{tab:ijbb} shows the Top-K Accuracy results for IJB-B video protocol. In this dataset, besides the baselines, our method is compared with original results in \\cite{TFA} corresponding to different iteration numbers. Results shown are the average of two galleries. Notice that our proposed system and \\cite{TFA} use the same face association method, but we have different networks and feature representation techniques.\n\n\\begin{table*}[t]\n\\centering\\resizebox{\\textwidth}{!}\n{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n  \\multirow{2}{*}{Methods} & \\multicolumn{6}{c|}{Top-K Average Accuracy \\textbf{with Filtering}} & \\multicolumn{6}{c|}{EERR metric \\textbf{without Filtering}}\\\\ \n \\cline{2-13}\n & R=1 & R=2 & R=5 & R=10 & R=20 & R=50 & R=1 & R=2 & R=5 & R=10 & R=20 & R=50\\\\\n \\hline\nArc-Cos \\cite{arcface} & 52.03\\% & 56.83\\% & 63.16\\% & 69.05\\% & 76.13\\% & 88.95\\% & 24.45\\% & 26.54\\% & 29.35\\% & 32.33\\% & 36.38\\% & 44.81\\%\\\\\n\\hline\nArc-QCos+QSub-PM & 60.92\\% & 65.06\\% & 70.45\\% & 75.19\\% & 80.69\\% & 90.29\\% & 28.73\\% & 30.44\\% & 32.98\\% & 35.40\\% & 38.70\\% & 45.46\\%\\\\\n \\hline\n \\hline\nCos & 64.86\\% & 70.87\\% & 77.09\\% & 81.53\\% & 86.11\\% & 93.24\\% & 29.62\\% & 32.34\\% & 35.60\\% & 38.36\\% & 41.53\\% & 46.78\\%\\\\\n\\hline\nQCos & 65.42\\% & 71.34\\% & 77.37\\% & 81.78\\% & 86.25\\% & 93.29\\% & 29.94\\% & 32.60\\% & 35.85\\% & 38.52\\% & 41.70\\% & 46.78\\%\\\\\n\\hline\nCos+Sub-PM & 69.52\\% & 75.15\\% & 80.41\\% & 84.14\\% & 87.83\\% & 94.27\\% & 32.22\\% & 34.70\\% & 37.66\\% & 39.91\\% & 42.65\\% & 47.54\\%\\\\\n\\hline\nQCos+Sub-PM & 69.65\\% & 75.26\\% & 80.43\\% & 84.22\\% & 87.81\\% & 94.25\\% & 32.27\\% & 34.73\\% & 37.66\\% & 39.91\\% & 42.67\\% & 47.54\\%\\\\\n\\hline\nQCos+QSub-PM & \\textbf{69.82\\%} & \\textbf{75.38\\%} & \\textbf{80.54\\%} & \\textbf{84.36}\\% & \\textbf{87.91\\%} & \\textbf{94.34\\%} & \\textbf{32.43\\%} & \\textbf{34.89\\%} & \\textbf{37.74\\%} & \\textbf{40.01\\%} & \\textbf{42.77\\%} & \\textbf{47.60\\%}\\\\\n\\hline\nQCos+QSub-VPM & 69.43\\% & 75.24\\% & 80.34\\% & 84.14\\% & 87.86\\% & 94.28\\% & 32.19\\% & 34.75\\% & 37.68\\% & 39.88\\% & 42.56\\% & 47.50\\%\\\\\n\\hline\n\\end{tabular}}\n\\caption{1:N Search results of IJB-S surveillance-to-single protocol. Using both Network D and E for representation.\n\\label{tab:cs6_sg2s}\n\\end{table*}\n\n\\begin{table*}[t]\n\\centering\\resizebox{\\textwidth}{!}\n{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n  \\multirow{2}{*}{Methods} & \\multicolumn{6}{c|}{Top-K Average Accuracy \\textbf{with Filtering}} & \\multicolumn{6}{c|}{EERR metric \\textbf{without Filtering}}\\\\ \n \\cline{2-13}\n & R=1 & R=2 & R=5 & R=10 & R=20 & R=50 & R=1 & R=2 & R=5 & R=10 & R=20 & R=50\\\\\n \\hline\nArc-Cos \\cite{arcface} & 54.59\\% & 59.12\\% & 65.43\\% & 71.05\\% & 77.84\\% & 89.16\\% & 25.38\\% & 27.58\\% & 30.59\\% & 33.42\\% & 37.60\\% & 45.05\\%\\\\\n\\hline\nArc-QCos+QSub-VPM & 60.86\\% & 65.36\\% & 71.30\\% & 76.15\\% & 81.63\\% & 90.70\\% & 28.66\\% & 30.64\\% & 33.43\\% & 36.11\\% & 39.57\\% & 45.70\\%\\\\\n \\hline\n \\hline\nCos & 66.48\\% & 71.98\\% & 77.80\\% & 82.25\\% & 86.56\\% & 93.41\\% & 30.38\\% & 32.91\\% & 36.15\\% & 38.77\\% & 41.86\\% & 46.79\\%\\\\\n\\hline\nQCos & 66.94\\% & 72.41\\% & 78.04\\% & 82.37\\% & 86.63\\% & 93.43\\% & 30.66\\% & 33.17\\% & 36.28\\% & 38.84\\% & 41.88\\% & 46.84\\%\\\\\n\\hline\nCos+Sub-PM & 69.39\\% & 74.55\\% & 80.06\\% & 83.91\\% & 87.87\\% & \\textbf{94.34\\%} & 32.02\\% & 34.42\\% & 37.59\\% & 39.97\\% & 42.64\\% & \\textbf{47.58\\%}\\\\\n\\hline\nQCos+Sub-PM & 69.57\\% & 74.78\\% & 80.06\\% & 83.89\\% & 87.94\\% & 94.33\\% & 32.16\\% & 34.61\\% & 37.62\\% & 39.99\\% & 42.71\\% & 47.57\\%\\\\\n\\hline\nQCos+QSub-PM & 69.67\\% & 74.85\\% & 80.25\\% & 84.10\\% & 88.04\\% & 94.22\\% & 32.28\\% & 34.77\\% & 37.76\\% & 40.11\\% & \\textbf{42.76\\%} & 47.57\\%\\\\\n\\hline\nQCos+QSub-VPM & \\textbf{69.86\\%} & \\textbf{75.07\\%} & \\textbf{80.36\\%} & \\textbf{84.32\\%} & \\textbf{88.07\\%} & 94.33\\% & \\textbf{32.44\\% }& \\textbf{34.93\\%} & \\textbf{37.80\\%} & \\textbf{40.14\\%} & 42.72\\% & \\textbf{47.58\\%}\\\\\n\\hline\n\\end{tabular}}\n\\caption{1:N Search results of IJB-S surveillance-to-booking protocol. Using both Network D and E for representation.}\n\\label{tab:cs6_b2s}\n\\end{table*}\n\n\\begin{table*}[t]\n\\centering\\resizebox{\\textwidth}{!}\n{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n  \\multirow{2}{*}{Methods} & \\multicolumn{6}{c|}{Top-K Average Accuracy \\textbf{with Filtering}} & \\multicolumn{6}{c|}{EERR metric \\textbf{without Filtering}}\\\\ \n \\cline{2-13}\n & R=1 & R=2 & R=5 & R=10 & R=20 & R=50 & R=1 & R=2 & R=5 & R=10 & R=20 & R=50\\\\\n \\hline\nArc-Cos \\cite{arcface} & 8.68\\% & 12.58\\% & 18.79\\% & 26.66\\% & 39.22\\%\t&68.19\\% & 4.98\\% & 7.17\\% & 10.86\\% & 15.42\\% & 22.34\\% & 37.68\\%\\\\\n\\hline\nArc-QCos+QSub-PM & 8.64\\% & 12.57\\% & 18.84\\% & 26.86\\% & 39.78\\% & \\textbf{68.21\\%} & \\textbf{5.26\\%} & \\textbf{7.44\\%} & \\textbf{11.31\\%} & 15.90\\% & \\textbf{22.68\\%} & \\textbf{37.83\\%}\\\\\n \\hline\n \\hline\nCos(D+E) & 9.24\\% & 12.51\\% & 19.36\\% & 25.99\\% & 32.95\\% & 52.95\\% & 4.74\\% & 6.62\\% & 10.70\\% & 14.88\\% & 19.29\\% & 30.64\\%\\\\\n \\hline\nQCos+QSub-VPM(D+E) & \\textbf{9.56\\%} & \\textbf{13.03\\%} & 19.65\\% & 27.15\\% & 35.39\\% & 56.02\\% & 4.77\\% & 6.78\\% & 10.88\\% & 15.52\\% & 20.51\\% & 32.16\\%\\\\\n \\hline\n \\hline\nCos(D) & 8.54\\% & 11.99\\% & 19.60\\% & 28.00\\% & 37.71\\% & 59.44\\% & 4.42\\% & 6.15\\% & 10.84\\% & 15.73\\% & 21.14\\% & 33.21\\%\\\\\n\\hline\nQCos(D) & 8.62\\% & 12.11\\% & 19.62\\% & 28.14\\% & 37.78\\% & 59.21\\% & 4.46\\% & 6.20\\% & 10.80\\% & 15.81\\% & 21.06\\% & 33.17\\%\\\\\n\\hline\nCos+Sub-PM(D) & 8.19\\% & 11.79\\% & 19.56\\% & 28.62\\% & 39.77\\% & 63.15\\% & 4.26\\% & 6.25\\% & 10.79\\% & 16.18\\% & 22.48\\% & 34.82\\%\\\\\n\\hline\nQCos+Sub-PM(D) & 8.24\\% & 11.82\\% & 19.68\\% & 28.68\\% & 39.68\\% & 62.96\\% & 4.27\\% & 6.25\\% & 10.92\\% & 16.18\\% & 22.39\\% & 34.69\\%\\\\\n\\hline\nQCos+QSub-PM(D) & 8.33\\% & 11.88\\% & 19.82\\% & 28.65\\% & 39.78\\% & 62.79\\% & 4.33\\% & 6.21\\% & 10.96\\% & 16.19\\% & 22.48\\% & 34.69\\%\\\\\n\\hline\nQCos+QSub-VPM(D) & 8.66\\% & 12.27\\% & \\textbf{19.91\\%} & \\textbf{29.03\\%} & \\textbf{40.20\\%} & 63.20\\% & 4.30\\% & 6.30\\% & 10.99\\% & \\textbf{16.23\\%} & 22.50\\% & 34.76\\%\\\\\n\\hline\n\\end{tabular}}\n\\caption{1:N Search results of IJB-S surveillance-to-surveillance protocol. D stands for only using Network D for representation. D+E stands for using both Network D and E for representation.\n\\label{tab:cs6_s2s}\n\\end{table*}\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.8\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/subspace_example_23767.png}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.8\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/subspace_example_100476.png}\n\\end{subfigure}\n\\caption{Visualization of example templates in IJB-S. Each sample is a dot in the plot with their first two principal components as the coordinates. Samples with $d_i \\geq 0.7$ are in \\textbf{blue} dots and the rest samples are in \\textbf{red} dots. \\textbf{Grey} line and \\textbf{black} line are the projection of the first subspace basis learned by \\textbf{Sub} and \\textbf{QSub} respectively.}\n\\label{fig:cs6_subspace}\n\\end{figure}\n\n\\noindent \\textbf{IJB-S:} Figure~\\ref{fig:cs6_result_box} shows some examples of our face association results using SORT in IJB-S dataset. Table~\\ref{tab:cs6_sg2s}, Table~\\ref{tab:cs6_b2s} and Table~\\ref{tab:cs6_s2s} show the results for IJB-S surveillance-to-single protocol, surveillance-to-booking protocol and surveillance-to-surveillance protocol respectively. Notice that under the \\textbf{with Filtering} configuration, we use the regular top-K average accuracy for evaluation. Under the \\textbf{without Filtering} configuration, we use the End-to-End Retrieval Rate (EERR) metric proposed in \\cite{ijbs} for evaluation. For surveillance-to-surveillance protocol, we show results for two different network configurations as well. We also implement state-of-the-art network ArcFace \\cite{arcface} on IJB-S and compare our method with it. Results from ArcFace are shown with the prefix \\textbf{Arc-}.\n\nTwo recent works \\cite{cfan} and \\cite{rean} have reported results on the IJB-S dataset. These works mainly focused on face recognition and not detection so that they built video templates by matching their detections with ground truth bounding boxes provided by the protocols and evaluated their methods using identification accuracy and not EERR metric. Our system focuses on detection, association and recognition. Therefore after detection, we associate faces across the video frames to build templates without utilizing any ground truth information and evaluate our system using both identification accuracy and EERR metric. Since these two template building procedures are so different, a directly comparison is not meaningful.\n\n\n\\begin{table*}[t]\n\\centering \\resizebox{\\textwidth}{!}\n{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n \\multirow{3}{*}{Methods}  & \\multicolumn{6}{c|}{MBGC} & \\multicolumn{6}{c|}{FOCS}\\\\\n \\cline{2-13}\n & \\multicolumn{2}{c|}{WW} & \\multicolumn{2}{c|}{AW} & \\multicolumn{2}{c|}{AA} & \\multicolumn{2}{c|}{WW} & \\multicolumn{2}{c|}{AW} & \\multicolumn{2}{c|}{AA}\\\\ \n \\cline{2-13}\n & FAR=0.01 & FAR=0.1 & FAR=0.01 & FAR=0.1 & FAR=0.01 & FAR=0.1 & FAR=0.01 & FAR=0.1 & FAR=0.01 & FAR=0.1 & FAR=0.01 & FAR=0.1\\\\\n \\hline\n Arc-Cos \\cite{arcface} & 84.40\\% & 92.20\\% & 53.88\\% & 75.00\\% & 32.47\\% & 66.49\\% & 98.18\\% & \\textbf{99.09\\%} & 48.61\\% & 69.44\\% & 48.36\\% & 78.87\\%\\\\\n \\hline\n Arc-QCos+QSub-PM & \\textbf{85.32\\%} & 92.20\\% & \\textbf{55.58\\%} & 75.00\\% & 32.99\\% & 64.43\\% & \\textbf{98.64\\%} & \\textbf{99.09\\%} & 52.31\\% & 74.07\\% & 50.23\\% & 79.81\\%\\\\\n\\hline\n \\hline\n DFRV$_{deep}$ \\cite{DFRV_2012} & 78.90\\% & \\textbf{95.87\\%} & 43.69\\% & 71.36\\% & 33.51\\% & 64.95\\% & 87.73\\% & 96.36\\% & 42.13\\% & 78.70\\% & 56.81\\% & 84.51\\%\\\\\n \\hline\n Hybrid \\cite{hybrid} & 77.06\\% & 94.04\\% & 48.06\\% & \\textbf{79.37\\%} & 42.53\\% & 71.39\\%& 95.00\\% & 97.73\\% & 47.69\\% & 79.63\\% & 50.23\\% & 80.75\\%\\\\\n \\hline\n \\hline\n Cos & 77.52\\% & 92.66\\% & 45.87\\% & 76.94\\% & \\textbf{43.30\\%} & 71.65\\% & 94.09\\% & 96.36\\% & 50.46\\% & 81.48\\% & 57.75\\% & 83.57\\%\\\\\n \\hline\n QCos & 77.52\\% & 92.66\\% & 47.57\\% & 76.94\\% & \\textbf{43.30\\%} & 71.13\\% & 95.91\\% & \\textbf{99.09\\%} & \\textbf{53.70\\%} & 80.09\\% & \\textbf{58.22\\%} & 83.57\\%\\\\\n \\hline\n Cos+Sub-PM & 77.98\\% & 94.95\\% & 47.57\\% & 79.13\\% & 41.24\\% & 72.68\\% & 91.82\\% & 97.27\\% & 49.07\\% & \\textbf{83.33\\%} & 54.93\\% & 85.45\\%\\\\\n\\hline\nQCos+Sub-PM & 77.98\\% & 94.95\\% & 48.30\\% & 78.64\\% & 41.75\\% & \\textbf{73.71\\%} & 95.91\\% & 98.64\\% & 52.78\\% & 82.87\\% & 55.40\\% & \\textbf{85.92\\%}\\\\\n\\hline\nQCos+QSub-PM & 77.52\\% & 94.95\\% & 48.54\\% & 78.64\\% & 41.75\\% & 73.20\\% & 95.91\\% & \\textbf{99.09\\%} & 52.31\\% & 81.02\\% & 55.87\\% & \\textbf{85.92\\%}\\\\\n\\hline\nQCos+QSub-VPM & 77.06\\% & 94.95\\% & 48.06\\% & 78.16\\% & 41.24\\% & 72.68\\% & 95.91\\% & \\textbf{99.09\\%} & \\textbf{53.70\\%} & 81.94\\% & 56.34\\% & \\textbf{85.92\\%}\\\\\n\\hline\n\\end{tabular}}\n\\caption{Verification results on MBGC and FOCS datasets}\n\\label{tab:mbgc_focs_v}\n\\vspace{-15pt}\n\\end{table*}\n\n\\noindent\\textbf{MBGC:} The verification results for the MBGC dataset are shown in Table~\\ref{tab:mbgc_focs_v} and Figure~\\ref{fig:mbgc_focs_v}. We compare our method with the baseline algorithms, \\textbf{Hybrid} \\cite{hybrid} and \\cite{DFRV_2012} using either raw pixels as \\textbf{DFRV$_{px}$} (reported in their paper) or deep features as \\textbf{DFRV$_{deep}$} (our implementation). We also report the results of the proposed method applied on the ArcFace features with the prefix \\textbf{Arc-}. Figure~\\ref{fig:mbgc_focs_v} does not include all the baselines, for a clearer view. The result of \\cite{DFRV_2012} is not in the table because the authors did not provide exact numbers in their paper.\n\n\\noindent \\textbf{FOCS:}  The verification results of FOCS dataset are shown in Table~\\ref{tab:mbgc_focs_v} and Figure~\\ref{fig:mbgc_focs_v}. O'Toole et al. \\cite{OToole2011} evaluated the human performance on this dataset.\nIn the figures, \\textbf{Human} refers to human performance with all bodies of target subjects seen and \\textbf{Human\\_Face} refers to performance that only faces of the target subjects are seen. Here besides baseline algorithms and \\textbf{Hybrid} \\cite{hybrid}, we also compare our method with \\cite{DFRV_2012} in either raw pixels as \\textbf{DFRV$_{px}$} (reported in their paper) or deep features as \\textbf{DFRV$_{deep}$} (our implementation). We also report the results using ArcFace features.\nSimilarly, the results of \\cite{DFRV_2012} and human performance are not in the table since they did not provide exact numbers.\n\n\\subsection{Observation and Discussion}\n\nFor the IJB-B dataset, we can see that using the TFA approach, our system produces high quality face association results. We do see some incorrect association results in the examples. But since they are of relatively low quality, our deep subspace representation can filter out these samples, and the proposed system performs consistently better than all the results in \\cite{TFA} and the baseline \\textbf{Cos} on identification accuracy. For open-set metric TPIR\/FPIR, the proposed quality-aware cosine similarity achieves better results, but the proposed subspace similarity metric still performs better than \\cite{TFA} with a large margin. For IJB-S results, we have similar observations: our system produces reliable tracking results. The low-quality faces on the tracking boundaries are handled by our robust subspace representation as the proposed system with subspace-to-subspace similarity measure performs better than \\textbf{Cos} on surveillance-to-single and surveillance-to-booking protocols, by relatively large margin. It also achieves better accuracy than \\textbf{Cos} on the surveillance-to-surveillance protocol. We notice that the fusion of Network D and E does not work well on surveillance-to-surveillance protocol, especially at higher rank accuracy. Such observations are consistent under both tracklets filtering configurations and their corresponding metrics: \\textbf{with Filtering} with Top-K average accuracy and \\textbf{without Filtering} with the EERR metric. The proposed system also outperforms ArcFace with larger margin in surveillance-to-single and surveillance-to-booking protocols of IJB-S. For MBGC and FOCS datasets, from the tables and plots we can see that in general, the proposed approach performs better than \\textbf{Cos} baseline, \\textbf{DFRV}$_{deep}$, \\textbf{DFRV}$_{px}$ and \\textbf{Hybrid}.\n\nFigure~\\ref{fig:cs6_subspace} shows the visualization of two templates in IJB-S dataset in PCA-subspace, which illustrates the advantage of the proposed subspace learning method. In the plot, each dot corresponds to a sample in the template, where x- and y-axes correspond to the first two principal components of the samples, learned from each template respectively. Relatively high-quality detections with detection score greater than or equal to 0.7 are represented by blue dots. Relatively low-quality detections with detection score less than 0.7 are represented by red dots. The projections of the first subspace bases learned by \\textbf{Sub} and the proposed \\textbf{QSub} onto the PCA-subspace are grey and black straight lines in the plot, respectively. From the plot we can see that, with quality-aware subspace learning, the subspaces learned by the proposed method put more weights on the high-quality sample. It fits the high-quality samples better than the low-quality ones. But the plain PCA takes each sample into account equally, which is harmful for the representation of the template.\n\nWe also compare our system with other baseline methods as part of an ablation study, from baseline cosine distance \\textbf{Cos} to the proposed quality-aware subspace-to-subspace similarity \\textbf{QCos+QSub-VPM}. As we gradually modify the method by including quality-aware cosine distance \\textbf{QCos}, quality-aware subspace learning \\textbf{QSub} and variance-aware projection metric \\textbf{VPM}, we can see the performance also gradually improves, especially for IJB-B and IJB-S datasets.\n\nFrom the results above, we observe the following:\n\\begin{itemize}\n\\item The proposed system performs the best in general, which shows the effectiveness of 1) learning subspace as template representation, 2) matching video pairs using the subspace-to-subspace similarity measure and 3) utilizing quality and variance information to compute exemplars, learn subspaces and measure similarity.\n\n\\item \\textbf{QCos} generally performs better than \\textbf{Cos}, which shows that quality-aware exemplars weigh the samples according to their quality and better represent the image sets than plain average exemplars.\n\n\\item In most of the cases, \\textbf{Cos+Sub-PM} achieve higher performance than \\textbf{Cos}. It implies that subspace can utilize the correlation information between samples and is a good complementary representation of exemplars as global information.\n\n\\item \\textbf{QCos+QSub-PM} performs better than \\textbf{QCos+Sub-PM} in general. It shows that similar to \\textbf{QCos}, we can learn more representative subspaces based on the quality of samples.\n\n\\item \\textbf{QCos+QSub-VPM} works better than \\textbf{QCos+QSub-PM} in most of the experiments. It implies that by considering the variances of bases in the subspaces, \\textbf{VPM} similarity is more robust to variations in the image sets.\n\n\\item The improvement of the proposed system over the compared algorithms is consistent under both \\textbf{with filtering} and \\textbf{without filering} configurations on the IJB-S dataset. It shows that our method is effective for both high-quality and low-quality tracklets in surveillance videos.\n\n\\item For IJB-S, the performance on surveillance-to-surveillance protocol is in general lower than the performance on other protocols. This is because the gallery templates of this protocol are from low-quality surveillance videos, while the rest two protocols have galleries from high-resolution still images.\n\n\\item The fusion of Network D and E does not perform as well as single Network D on surveillance-to-surveillance protocol, especially at higher rank accuracy. It is probably because of the low-quality galleries in this protocol which Network E cannot represent well. \n\n\\item On IJB-S, the proposed method performs better than state-of-the-art network ArcFace \\cite{arcface} in general, especially on surveillance-to-single and surveillance-to-booking protocols, which shows the discriminative power of the features from the proposed networks. ArcFace still performs better on surveillance-to-surveillance protocol. But the results also show that using the quality-aware subspace-to-subspace similarity improves the performance for ArcFace features as well.\n\n\\item On MBGC and FOCS, ArcFace performs better in the walking-vs-walking protocol but Network D outperforms ArcFace on more challenging protocols like activity-vs-activity. Also, by applying the proposed subspace-to-subspace similarity on both features, the performance consistently improves, which shows its effectiveness on different datasets and using different features.\n\n\\item For the FOCS dataset, the performance of our system surpasses the human performance, which again demonstrates the effectiveness of the proposed system. \n\\end{itemize}\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\nFace recognition is one of the most actively studied problems in computer vision and biometrics. Nowadays, video-based face recognition is an active research topic because of a wide range of applications including visual surveillance, access control, video content analysis, etc. Compared to still image-based face recognition, video-based face recognition is more challenging due to a much larger amount of data to be processed and significant intra\/inter-class variations caused by motion blur, low video quality, occlusion, frequent scene changes, and unconstrained acquisition conditions.\n\nTo develop the next generation of unconstrained video-based face recognition systems, two datasets have been recently introduced, IARPA Benchmark B (IJB-B) \\cite{ijbb} and IARPA Janus Surveillance Video Benchmark (IJB-S) \\cite{ijbs}, acquired under more challenging scenarios, compared to the Multiple Biometric Grand Challenge (MBGC) dataset \\cite{MBGC} and the Face and Ocular Challenge Series (FOCS) dataset \\cite{FOCS} which are collected in relatively controlled conditions. IJB-B and IJB-S datasets are captured in unconstrained settings and contain faces with much more intra\/inter class variations on pose, illumination, occlusion, video quality, scale and etc.\n\nThe IJB-B dataset is a template-based dataset that contains 1845 subjects with 11,754 images, 55,025 frames and 7,011 videos where a template consists of a varying number of still images and video frames from different sources. These images and videos are collected from the Internet and are totally unconstrained, with large variations in pose, illumination, image quality etc. Samples from this dataset are shown in Figure~\\ref{fig:ijbb_example}. In addition, the dataset comes with protocols for 1-to-1 template-based face verification, 1-to-N template-based open-set face identification, and 1-to-N open-set video face identification. For the video face identification protocol, the gallery is a set of still-image templates. The probe is a set of videos (e.g. news videos), each of which contains multiple shots with multiple people and one bounding box annotation to specify the subject of interest. Probes of videos are searched among galleries of still images. Since the videos are composed of multiple shots, it is challenging to detect and associate the faces for the subject of interest across shots due to large appearance changes. In addition, how to efficiently leverage information from multiple frames is another challenge, especially when the frames are noisy.\n\nSimilar to the IJB-B dataset, the IJB-S dataset is also an unconstrained video dataset focusing on real world visual surveillance scenarios. It consists of 202 subjects from 1421 images and 398 surveillance videos, with 15,881,408 bounding box annotations. Samples of frames from IJB-S are shown in Figure~\\ref{fig:cs6_example}. Three open-set identification protocols accompany this dataset for surveillance video-based face recognition where each video in these protocols is captured from a static surveillance camera and contains single or multiple subjects: (1) in surveillance-to-single protocol, probes collected from surveillance videos are searched in galleries consisting of one single high-resolution still image; (2) in surveillance-to-booking protocol, same probes are searched among galleries consisting of seven high-resolution still face images covering frontal and profile poses. Probe templates in (1) and (2) should be detected and constructed by the recognition system itself; (3) in the most challenging surveillance-to-surveillance protocol, both gallery and probe templates are from videos, which implies that probe templates need to be compared with relatively low quality gallery templates.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.95\\linewidth]{figs\/ijbb_example.png}\n\\caption{Example frames of a multiple-shot probe video in the IJB-B dataset. The target annotation is in red box and face detection results from face detector are in green boxes.}\n\\label{fig:ijbb_example}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[b]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_example1.jpg}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_example2.jpg}\n\\end{subfigure}\n\\\\\n\\vspace{1em}\n\\begin{subfigure}[b]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_example3.jpg}\n\\end{subfigure}\n~\n\\begin{subfigure}[b]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{figs\/cs6_example4.jpg}\n\\end{subfigure}\n\\caption{Example frames of two single-shot probe videos in the IJB-S dataset.}\n\\label{fig:cs6_example}\n\\end{figure}\n\nFrom these datasets, we summarize the four common challenges in video-based face recognition as follows:\n\\begin{enumerate}\n\\item For video-based face recognition, test data are from videos where each video contains tens of thousands of frames and each frame may have several faces. This makes the scalability of video-based face recognition a challenging problem. In order to make the face recognition system to be operationally effective, each component of the system should be fast, especially face detection, which is often the bottleneck in recognition.\n\n\\item Since faces are mostly from unconstrained videos, they have significant variations in pose, expression, illumination, blur, occlusion and video quality. Thus, any face representations we design must be robust to these variations and to errors in face detection and association steps.\n\n\\item Faces with same identity across different video frames need to be grouped by a reliable face association method. Face recognition performance will degrade if faces with different identities are grouped together. Videos in the IJB-B dataset are acquired from multiple shots involving scene and view changes, while most videos in IJB-S are low-quality remote surveillance videos. These conditions increase the difficulty of face association. \n\n\\item Since each video contains different number of faces for each identity, the next challenge is how to efficiently aggregate a varying-length set of features from the same identity into a fixed-size or unified representation. Exploiting the correlation information in a set of faces generally results in better performance than using only a single face.\n\\end{enumerate}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figs\/diagram_v3.pdf}\n\\caption{Overview of the proposed system.}\n\\label{fig:overview}\n\\end{figure*}\n\nRecently, with the availability of powerful GPUs and large amounts of labeled training data, deep convolutional neural networks (DCNNs) have demonstrated impressive performances for many computer vision tasks such as object recognition \\cite{He2015residual}, object detection \\cite{fasterrcnn} and semantic segmentation \\cite{deeplab}. DCNNs have also produced state-of-the-art results for face detection and still face-based recognition tasks as reported in \\cite{allinone}, \\cite{ssh}, \\cite{Taigman2014}, \\cite{Parkhi15}, \\cite{Schroff2015}, \\cite{JC15}. Thus, by utilizing the power of deep networks, we could detect faces more accurately from videos and produce robust deep representations for these faces than traditional approaches. However, the speed of face detectors is still a critical bottleneck in the face recognition pipeline. Single Shot MultiBox Detector (SSD) \\cite{ssd} and YOLO \\cite{yolo} provide a solution for fast-speed face detector. These one-step face detectors achieve comparable results to state-of-the-art two-step detectors \\cite{fasterrcnn} \\cite{maskrcnn}, but are several times faster. Also, instead of using the traditional cross-entropy loss, Ranjan \\emph{et al.} \\cite{crystal} recently introduced the crystal loss, which functions by constraining the deep features to lie on a hypersphere, and achieves impressive result on the challenging face recognition dataset, IARPA JANUS Benchmark C (IJB-C) \\cite{ijbc}. \n\nMany tracking techniques \\cite{trackparticle}, \\cite{robustmft}, \\cite{OnlineMD}, \\cite{SORT} have been proposed to associate face images of a subject in videos containing a single shot, by utilizing spatial, temporal, and appearance affinity. Simple Online and Realtime Tracking (SORT) \\cite{SORT} is a very fast and efficient tracker for tracking multi-target bounding boxes. However, SORT is not effective for multiple-shot videos where frequent scene changes are present across shots. To address the large variations across shots, we use a face association method, Target Face Association (TFA) proposed in \\cite{TFA}, which adaptively updates face representations by one-shot SVM kernel to retrieve a set of representative face images in a multiple-shot video for video-based face recognition. More details are described in Section~\\ref{faceas}.\n\nFor feature aggregation in video-based face recognition, temporal deep learning model such as Recurrent Neural Network (RNN) can be applied to yield a fixed-size encoded face representation. However, large-scale labeled training data is needed to learn robust representations, which is very expensive to collect in the context of video-based recognition problem. This is also true for the adaptive pooling method \\cite{qan}, \\cite{nan} for image set-based face recognition problem. For IJB-B and IJB-S datasets, lack of large-scale training data makes it impossible to train an RNN-based method. On the contrary, representative and discriminative models based on manifolds and subspaces have also received attention for image set-based face recognition \\cite{Wang2012} \\cite{wang2009}. These methods model sets of image samples as manifolds or subspaces and use appropriate similarity metric for set-based identification and verification. One of the main advantages of subspace-based methods is that different from sample mean, the subspace representation encodes the correlation information between samples. In low-quality videos, faces have significant variations due to blur, extreme poses and low resolution. Exploiting correlation between samples by subspacess will help learn a more robust representation to capture these variations. Also, a fixed-size representation can be learned from an arbitrary number of images or video frames.\n\nTo summarize, we propose an automatic system by integrating deep learning components to overcome the challenges in unconstrained video-based face recognition. The proposed system first detects faces and facial landmarks using state-of-the-art DCNN face detectors. Single Shot Detector (SSD) for faces \\cite{icipssd} and Deep Pyramid Single Shot Face Detector (DPSSD) \\cite{systbiom} are used for different scenarios. The former is an efficient face detector which is effective for videos collected from the Internet where faces are relatively larger than the ones in surveillance videos. The latter is a multi-scale face detector that can produce reliable and accurate face detections at different scales, thus is capable of detecting tiny and blurred faces which are common in surveillance videos. Both detectors are fast, which can tackle the first challenge of processing speed. Next, we extract deep features from the detected faces using state-of-the-art DCNNs for face recognition. We use crystal loss to train the DCNN network, to address the second challenge of generating a robust representation. Besides SORT for single-shot videos, TFA is used to cluster target faces from multiple-shot videos. By incorporating SORT and TFA into our system, together with robust face detectors, face SSD and DPSSD, our system overcomes the third challenge of video-based face detection and association. Finally, in the proposed face recognition system, we learn a subspace representation from each video template and match pairs of templates using principal angles-based subspace-to-subspace similarity measure on the learned subspace representations. This helps us to handle the fourth challenge of representation aggregation. An overview of the proposed system is shown in Figure~\\ref{fig:overview}.\n\nWe evaluate our face recognition system on the challenging IJB-B and IJB-S datasets, as well as MBGC and FOCS datasets, and the results demonstrate that the proposed system achieves improved performance over other deep learning-based baselines and state-of-the-art approaches.\n\nThe main contributions of the proposed system are summarized as\nfollows:\n\n\\begin{itemize}\n\\item We propose an automatic video-based face recognition system with components including face\/fiducial detection, face association, and face recognition.\n\\item We propose a quality-aware subspace learning approach for face feature aggregation.\n\\item We compute the video template-to-template similarity using a subspace-to-subspace similarity metric for video-based face recognition. A quality-aware subspace-to-subspace similarity metric is also proposed.\n\\end{itemize}\n\nThe rest of this paper is organized as follows: in Section~\\ref{sec:related}, we briefly review some related works. In Section~\\ref{sec:method}, we introduce the proposed system in detail. In Section~\\ref{sec:exp}, we discuss the implementation details and present the experimental results on four datasets. Finally, conclusions are presented in Section~\\ref{sec:conclusion}.\n\n\\section*{Acknowledgment}\n\nThis research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R\\&D Contract No. 2019-022600002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.\n\n}\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Proposed Method}\\label{sec:method}\n\n For each video, we first detect faces from video frames and align them using the detected fiducial points. Deep features are then extracted for each detected face using our DCNN models for face recognition. Based on different scenarios, we use face association or face tracking to construct face templates with unique identities. For videos with multiple shots, we use the proposed face association technique TFA \\cite{TFA} to collect faces from the same identities across shots. For single-shot videos, we use the face tracking algorithm SORT introduced in \\cite{SORT} to produce tracklets of faces.\nAfter templates are constructed, in order to aggregate the representation of videos, subspaces are learned using quality-aware principal component analysis. Subspaces along with quality-aware exemplars of templates are used to produce the similarity scores between video pairs by a quality-aware principal angle-based subspace-to-subspace similarity measure.\nIn the following sections, we discuss the proposed video-based face recognition system in details.\n\n\\subsection{Face\/Fiducial Detection}\nThe first step in our face recognition pipeline is to detect faces in images (usually for galleries) and videos. The challenges mostly come from the low quality of surveillance videos. Our detector should balance the precision and recall of bounding boxes since we do not want too many non-faces to be input to the rest of the pipeline. Also we do not want our detector to miss too many faces since they can provide useful information for recognition. Thus we use two DCNN-based detectors in our pipeline based on different distributions of input.\n\n\\subsubsection{SSD Face Detector}\nFor regular images and video frames, faces are relatively bigger and with higher resolution. We use SSD trained with the WIDER face dataset as our face detector \\cite{icipssd}. \n\n\\subsubsection{DPSSD Face Detector}\nFor small and remote faces in surveillance videos, because of the domain difference, a traditional face detector cannot perform well on detecting these tiny faces. Thus for surveillance videos, we use a novel DCNN-based face detector, called Deep Pyramid Single Shot Face Detector (DPSSD) \\cite{systbiom} for face detection. It is fast and capable of detecting tiny faces, which is very suitable for face detection in videos\n\nAfter raw face detection bounding boxes are generated using either SSD or DPSSD detectors, we use All-in-One Face \\cite{allinone} for fiducial localization. It is followed by a seven-point face alignment step based on the similarity transform on all the detected faces.\n\n\\subsection{Deep Feature Representation}\nAfter faces are detected and aligned, we use the DCNN models to represent each detected face. The models are state-of-the-art networks with different architectures for face recognition. Different architectures provide different error patterns during testing. After fusing the results from different models, we achieve performance better than a single model. The overview of these networks along with their training details are described in Section~\\ref{sec:imp}.\n\n\\subsection{Face Association}\\label{faceas}\nIn previous steps, we obtain raw face detection bounding boxes using our detectors. Features for the detected bounding boxes are extracted using face recognition networks. Since video frames are of relatively low quality, the detector outputs will consist of both faces and non-face objects. Even for correct detection, faces from different identities will appear in the same video containing multiple subjects. Many popular video-based face recognition datasets have testing protocols in which faces are matched in templates (aka image sets) instead of single images. If the constructed templates are not clean, outliers in the template will have negative influence on the discriminative power of the face representation and adversely affect the face recognition performance. Thus, the next important step in our face recognition pipeline is to combine the detected bounding boxes from the same identity to construct templates for good face recognition result.\n\n\\subsubsection{Face Association for Single-Shot Videos}\n\nFor single-shot videos, which means the bounding boxes of a certain identity will probably be contiguous, we rely on SORT \\cite{SORT} to build tracklets for each identity.\nSORT is a real-time online tracking algorithm which approximates the dynamics with linear Gaussian state space models and associates detection bounding boxes in every frame using Kalman Filters.\n\n\\subsubsection{Face Association for Multi-Shot Videos}\\label{sec:TFA}\nFor multi-shot videos, it is challenging to continue tracking across different scenes. Thus, face association based on the appearance (deep representation in our case) of the face bounding boxes instead of their position is more robust to scene changes in these videos. In the proposed system, we use \\cite{TFA} to adaptively update the face representation through one-shot SVM. The details are described below.\n\n\\textbf{a) Face Pre-Association by Tracking}\nStarting from the annotated face, an off-the-shelf tracking technique \\cite{ofstracker} is first applied to track the target face, which is called pre-association, to collect subsequent faces of high quality. Faces in the same tracklet are utilized as the initial positive training set.\n\nA tracklet is built by associating the face detection bounding box that has the highest IoU ratio with the tracking bounding box in each frame. We only track faces in the first k frames, or stop at scene boundaries, since tracking will be unreliable if occlusion, motion and scene changes occur.\n\n\\textbf{b) One-Shot SVM Learning}\nIn a video, suppose the annotated target face is indicated by a bounding box $b_0$ in frame $f_0$. There are a total of $m$ bounding boxes discovered by the detector. These face bounding boxes are denoted as $b_1, b_2,\\dots, b_m$, which are present in frames $f_1, f_2, \\dots, f_m$, respectively. The feature corresponding to the face bounding box $b_i$ is denoted as $\\mathbf{x}_i$. Given the initial positive training set, a subject-specific linear SVM is trained to establish the intra\/inter-shot face association of the target face.\n\nThe index set of initial positive instances is represented as $S_p = \\{0\\}\\cup T$, where $T$ are indices of pre-associated bounding boxes. The negative training instances can be discovered by the cannot-link relation. The cannot-link relation between the $i$th and $j$th bounding boxes is defined as\n\\vspace{-5pt}\n\\begin{align}\ng_{i,j} = \\begin{cases}\n1 & \\text{if } r_{i,j} \\leq \\gamma,\\;f_i = f_j,\\;i, j \\in \\{0, 1, \\dots , m\\}\\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\end{align}\n\\vspace{-5pt}\n\nwhere $r_{i,j}$ is the IoU ratio between bounding box $b_i$ and $b_j$, $\\gamma$ is the corresponding IoU threshold. Thus, $g_{i,j} = 1$ indicates that the $i$th and $j$th bounding boxes appear in the same frame, cannot belong to the same face, and should not be identified as the same subject. Therefore the index set of within-video negative instances is represented as $S_n = \\cup_{j\\in S_p}\\{i |g_{i,j} = 1\\}$.\n\nIf there is no within-video training instance, a background negative set $\\{\\mathbf{x}_i\\}_{i=m+1}^{m+l}$ with size $l$ collected from an external face dataset will be used with corresponding index set $S_b = \\{m + 1,m + 2, \\dots, m + l\\}$.\n\nThen, the linear SVM is trained by the combined training data $\\{(\\mathbf{x}_i, y_i)|i \\in (S_p \\cup S_n \\cup S_b)\\}$, where the data label is defined as\n\\begin{equation}\ny_i = \\begin{cases}\n1 & \\text{if } i \\in S_p,\\\\\n-1 & \\text{otherwise.}\n\\end{cases}\n\\end{equation}\n\nThe weight vector $\\mathbf{w}$ of the linear SVM is solved using the max-margin framework\n\\begin{align}\\label{svm}\n\\operatorname*{minimize}_{\\mathbf{w}}&\\frac{1}{2}\\mathbf{w}^T\\mathbf{w} + C_p\\sum_{i\\in S_p}\\max[0, 1-y_i\\mathbf{w}^T\\bar{\\mathbf{x}}_i]^2\\nonumber\\\\\n&+ \\mathbbm{1}[S_n\\neq\\varnothing]C_n\\sum_{i\\in S_n}\\max[0, 1-y_i\\mathbf{w}^T\\bar{\\mathbf{x}}_i]^2\\nonumber\\\\\n&+ \\mathbbm{1}[S_b\\neq\\varnothing]C_b\\sum_{i\\in S_b}\\max[0, 1-y_i\\mathbf{w}^T\\bar{\\mathbf{x}}_i]^2\n\\end{align}\n\nwhere $C_p$, $C_n$ and $C_b$ are weights related to the number of training samples, $\\bar{\\mathbf{x}}_i=\\begin{bmatrix}\\mathbf{x}_i^T\/\\|\\mathbf{x}_i\\|, 1\\end{bmatrix}^T$ is the normalized feature, $\\mathbbm{1}[\\cdot]$ is the indicator function. An illustration of the training instances is shown in Figure~\\ref{fig:TFA}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{figs\/tfa_example.png}\n\\caption{Example of training instances for the SVM.}\n\\label{fig:TFA}\n\\end{figure}\n\nAfter the SVM is learned, faces that are classified as positive will be regarded as the associated face set to the target subject as\n\n\\begin{equation}\nA = \\{0\\}\\cup\\{i|\\mathbf{w}^T\\mathbf{x}_i > 0, i=1,2,\\dots,m\\}\n\\end{equation}\n}\nSince faces in set $A$ may come from the same frame, $A$ is further refined by iteratively remove the least likely\ninstance among those instances that violate the cannot-link constraints.\n\nSubsequently, face assignment and refinement procedure is applied on every annotated bounding box, and thus we obtain the associated face set for each annotation. \n\n\\subsection{Model Learning: Deep Subspace Representation}\\label{Subspace}\n\nOnce deep features are extracted for each face template, since each template contains a varying number of faces, these features need to be further aggregated into a fixed-size or a unified representation for efficient face recognition.\n\nThe simplest representation of a set of samples is the sample mean. However, this video template contains faces with different quality and large variations in illumination, blur and pose. Since the average treats all the samples equally, the outliers may deteriorate the discriminative power of the representation. As compared to other feature aggregation approaches that require a large amount of extra training data which are not available for datasets like IJB-B and IJB-S, we propose to represent face templates by subspaces.\n\n\\subsubsection{Subspace Learning from Deep Features}\nA $d$-dimensional subspace $S$ can be uniquely defined by a set of orthonormal bases $\\mathbf{P}\\in\\mathbb{R}^{D\\times d}$, where $D$ is the dimensionality of features. Given face features from a video sequence $\\mathbf{Y}\\in\\mathbb{R}^{D\\times N}$, where $N$ is the sequence length, $\\mathbf{P}$ can be found by optimizing:\n\\begin{align}\\label{PCA}\n\\operatorname*{minimize}_{\\mathbf{P}, \\mathbf{X}}\\;\\|\\mathbf{Y} - \\mathbf{P}\\mathbf{X}\\|_F^2\\quad s.t.\\;\\mathbf{P}^T\\mathbf{P} = \\mathbf{I}\n\\end{align}\nwhich is the reconstruction error of features $\\mathbf{Y}$ in the subspace $S$. It is exactly the principal component analysis (PCA) problem and can be easily solved by eigenvalue decomposition. Let $\\mathbf{Y}\\mathbf{Y}^T = \\mathbf{U}\\boldsymbol{\\Lambda}\\mathbf{U}^T$ be the eigenvalue decomposition, where $\\mathbf{U} = \\begin{bmatrix}\\mathbf{u}_1, \\mathbf{u}_2,\\cdots,\\mathbf{u}_D\\end{bmatrix}$ are eigenvectors and $\\boldsymbol{\\Lambda} = diag\\{\\lambda_1, \\lambda_2, \\dots, \\lambda_D\\}$ with $\\lambda_1\\geq\\lambda_2\\geq\\dots\\geq\\lambda_D$ are the corresponding eigenvalues, we have $\\mathbf{P} = \\begin{bmatrix}\\mathbf{u}_1, \\mathbf{u}_2,\\cdots,\\mathbf{u}_d\\end{bmatrix}$ which consists of the first $d$ basis in $\\mathbf{U}$. We use \\textbf{Sub} to denote this basic subspace learning algorithm.\n\n\\subsubsection{Quality-Aware Subspace Learning from Deep Features}\nIn a face template from videos, faces contain large variations in pose, illumination, occlusion, etc. Even in a tracklet, faces have different poses because of the head movement, or being occluded at some frames because of the interaction with the environment. When learning the subspace, treating the frames equally is not an optimal solution. In our system, the detection score for each face bounding box given by the face detector can be used as a good indicator of the quality of faces, as shown in \\cite{crystal}. Hence, following the quality pooling proposed in \\cite{crystal}, we propose quality-aware subspace learning based on detection scores. The learning problem modifies \\eqref{PCA} as\n\\begin{align}\n\\operatorname*{minimize}_{\\mathbf{P}, \\mathbf{X}}\\;\\sum_{i=1}^{N}\\tilde{d}_i\\|\\mathbf{y}_i - \\mathbf{P}\\mathbf{x}_i\\|_2^2\\quad s.t.\\;\\mathbf{P}^T\\mathbf{P} = \\mathbf{I}\n\\end{align}\nwhere $\\tilde{d}_i=softmax(q l_i)$ is the normalized detection score of face $i$, $q$ is the temperature parameter and\n\\begin{equation}\\label{li}\nl_i = \\min(\\frac{1}{2}\\log\\frac{d_i}{1-d_i}, t)\n\\end{equation}\nwhich is upper bounded by threshold $t$ to avoid extreme values when the detection score is close to 1.\n\nLet $\\tilde{\\mathbf{Y}} = \\begin{bmatrix}\\sqrt{d_1}\\mathbf{y}_1,\\cdots,\\sqrt{d_N}\\mathbf{y}_N\\end{bmatrix}$ be the normalized feature set, and the corresponding eigenvalue decomposition to be $\\tilde{\\mathbf{Y}}\\tilde{\\mathbf{Y}}^T = \\tilde{\\mathbf{U}}\\tilde{\\boldsymbol{\\Lambda}}\\tilde{\\mathbf{U}}^T$.\nWe have\n\\begin{equation}\\label{PD}\n\\mathbf{P}_D = \\begin{bmatrix}\\tilde{\\mathbf{u}}_1, \\tilde{\\mathbf{u}}_2,\\cdots,\\tilde{\\mathbf{u}}_d\\end{bmatrix}\n\\end{equation}\nwhich consists of the first $d$ bases in $\\tilde{\\mathbf{U}}$. The new subspace is therefore learned by weighting samples differently according to their quality. This quality-aware learning algorithm is denoted as \\textbf{QSub}.\n\\subsection{Matching: Subspace-to-Subspace Similarity for Videos}\n\nAfter deep subspace face representations are learned for video templates, inspired by manifold-to-manifold distance \\cite{Wang2012}, we measure the similarity between two video templates of faces using a subspace-to-subspace similarity measure. In this part, we first introduce the widely used measure based on principal angles. Then we propose several weighted subspace-to-subspace measures which take the importance of bases into consideration.\n\n\\subsubsection{Principal Angles and Projection Metric}\nOne of the mostly used subspace-to-subspace similarity is based on principal angles. The principal angles $0\\leq \\theta_1\\leq\\theta_2\\leq\\dots\\leq\\theta_r\\leq \\frac{\\pi}{2}$ between two linear subspaces $S_1$ and $S_2$ can be computed by Singular Value Decomposition (SVD).\n\nLet $\\mathbf{P}_1\\in\\mathbb{R}^{D\\times d_1}$, $\\mathbf{P}_2\\in\\mathbb{R}^{D\\times d_2}$, denoting the orthonormal basis of $S_1$ and $S_2$, respectively. The SVD of $\\mathbf{P}_1^T\\mathbf{P}_2$ is $\\mathbf{P}_1^T\\mathbf{P}_2 = \\mathbf{Q}_{12}\\boldsymbol{\\Lambda}\\mathbf{Q}_{21}^T$, where $\\boldsymbol{\\Lambda} = diag\\{\\sigma_1,\\sigma_2,\\dots,\\sigma_r\\}$. $\\mathbf{Q}_{12}$ and $\\mathbf{Q}_{21}$ are orthonormal matrices. The singular\nvalues $\\sigma_1,\\sigma_2,\\dots,\\sigma_r$ are exactly the cosine of the principal angles as $\\cos\\theta_k = \\sigma_k,\\; k=1, 2, \\dots, r$.\n\nProjection metric \\cite{PM} is a popular similarity measure based on principal angles:\n\\begin{equation}\ns_{PM}(S_1,S_2)=\\sqrt{\\frac{1}{r}\\sum_{k=1}^r\\cos^2\\theta_k}\n\\end{equation}\n\nSince $\\|\\mathbf{P}_1^T\\mathbf{P}_2\\|_F^2 = \\|\\mathbf{Q}_{12}\\boldsymbol{\\Lambda}\\mathbf{Q}_{21}^T\\|_F^2=\\|\\boldsymbol{\\Lambda}\\|_F^2=\\sum_{k=1}^r\\sigma_k^2=\\sum_{k=1}^r\\cos^2\\theta_k$, we have\n\\begin{equation}\\label{PM}\ns_{PM}(S_1,S_2) = s_{PM}(\\mathbf{P}_1,\\mathbf{P}_2)=\\sqrt{\\frac{1}{r}\\|\\mathbf{P}_1^T\\mathbf{P}_2\\|_F^2}\n\\end{equation}\nand there is no need to compute the SVD explicitly. We use \\textbf{PM} to denote this similarity measure.\n\n\\subsubsection{Exemplars and Basic Subspace-to-Subspace Similarity}\nExisting face recognition systems usually use cosine distance between exemplars to measure the similarity between templates. Exemplar of a template is defined as its sample mean, as $\\mathbf{e}=\\frac{1}{L}\\sum_{i=1}^{L}\\mathbf{y}_{i}$,\nwhere $\\mathbf{y}_i$ are samples in the template. Exemplars mainly capture the average and global representation of the template. On the other hand, the projection metric we introduced above measures the similarity between two subspaces, which models the correlation between samples. Hence, in the proposed system, we make use of both of them by fusing their similarity scores as the subspace-to-subspace similarity between two video sequences.\n\nSuppose subspaces $\\mathbf{P}_1\\in\\mathbb{R}^{D\\times\nd_1}$ and $\\mathbf{P}_2\\in\\mathbb{R}^{D\\times d_2}$ are learned from a pair of video templates $\\mathbf{Y}_1\\in\\mathbb{R}^{D\\times L_1}$ and\n$\\mathbf{Y}_2\\in\\mathbb{R}^{D\\times L_2}$ in deep features respectively, by either \\textbf{Sub} or \\textbf{QSub} methods introduced in Section~\\ref{Subspace}. Their exemplars are\n$\\mathbf{e}_1=\\frac{1}{L_1}\\sum_{i=1}^{L_1}\\mathbf{y}_{1i}$ and\n$\\mathbf{e}_2=\\frac{1}{L_2}\\sum_{i=1}^{L_2}\\mathbf{y}_{2i}$ respectively. Combining the orthonormal bases and exemplars, the subspace-to-subspace similarity can be computed as:\n\\begin{align}\ns(\\mathbf{Y}_1,\\mathbf{Y}_2)&=s_{Cos}(\\mathbf{Y}_1,\\mathbf{Y}_2)+\\lambda s_{PM}(\\mathbf{P}_1, \\mathbf{P}_2)\\nonumber\\\\\n&=\\frac{\\mathbf{e}_1^T\\mathbf{e}_2}{\\|\\mathbf{e}_1\\|_2\\|\\mathbf{e}_2\\|_2} + \\lambda\\sqrt{\\frac{1}{r}\\|\\mathbf{P}_1^T\\mathbf{P}_2\\|_F^2}\n\\end{align}\nwhere $s_{Cos}(\\mathbf{Y}_1,\\mathbf{Y}_2)$\nis the cosine distance between exemplars, denoted as \\textbf{Cos}, and $s_{PM}(\\mathbf{P}_1, \\mathbf{P}_2)$ is computed by \\eqref{PM}. Since the DCNN features are more robust if we keep their signs, instead of using $s_{Cos}^2(\\mathbf{Y}_1,\\mathbf{Y}_2)$ as in \\cite{Wang2012} where the sign information is lost, we use $s_{Cos}(\\mathbf{Y}_1,\\mathbf{Y}_2)$ in our formulation. Accordingly, we also take the square root of the principal angle term to keep the scale consistent. $\\lambda$ here is a hyperparameter that balances the cosine similarity and principal angle similarity. If $\\mathbf{P}_i$'s are learned by \\textbf{Sub}, we denote the whole similarity measure (including exemplars computing and subspace learning) as \\textbf{Cos+Sub-PM}. If $\\mathbf{P}_i$'s are learned by the proposed \\textbf{QSub}, we denote the similarity as \\textbf{Cos+QSub-PM}.\n\n\\subsubsection{Quality-Aware Exemplars}\nIn either \\textbf{Cos+Sub-PM} or \\textbf{Cos+QSub-PM} we are still using simple average pooling to compute the exemplars. But as discussed in Section~\\ref{Subspace}, templates consist of faces of different quality. Treating them equally for pooling will let some low-quality faces degrade the global representation of the template. Therefore, we propose to use the same normalized detection score as in Section~\\ref{Subspace} to compute the quality-aware exemplars by $\\mathbf{e}_D=\\frac{1}{L}\\sum_{i=1}^L\\tilde{d}_i\\mathbf{y}_{i}$,\nwhere $\\tilde{d}_i=softmax(ql_i)$ and $l_i$ are computed by \\eqref{li}.\nThen, the cosine distance between the quality-aware exemplars is\n\\begin{equation}\\label{cosd}\ns_{QCos}(\\mathbf{Y}_1,\\mathbf{Y}_2)=\\frac{\\mathbf{e}_{D1}^T\\mathbf{e}_{D2}}{\\|\\mathbf{e}_{D1}\\|_2\\|\\mathbf{e}_{D2}\\|_2}\n\\end{equation}\nand we denote it as \\textbf{QCos}. Using the new cosine distance, the similarity becomes\n\\begin{align}\ns(\\mathbf{Y}_1,\\mathbf{Y}_2)&=s_{QCos}(\\mathbf{Y}_1,\\mathbf{Y}_2)+\\lambda s_{PM}(\\mathbf{P}_1, \\mathbf{P}_2)\n\\end{align}\nIf $P_i$'s are learned by \\textbf{QSub}, the similarity is further denoted by \\textbf{QCos+QSub-PM}.\n\\subsubsection{Variance-Aware Projection Metric}\nAs previously discussed, the projection metric $S_{PM}(S_1, S_2)$ is the square root of the mean square of principle angles between two subspaces and it treats each basis in each subspace equally. But these bases are actually eigenvectors of an eigenvalue decomposition problem. Different basis corresponds to different eigenvalue, which represents the variance of data in each basis direction. Obviously, those bases with larger variances contain more information than those with smaller variances. Therefore, based on the variance of each basis, we propose a variance-aware projection metric as:\n\\begin{equation}\\label{VPM}\ns_{VPM}(\\mathbf{P}_1,\\mathbf{P}_2)=\\sqrt{\\frac{1}{r}\\|\\tilde{\\mathbf{P}}_1^T\\tilde{\\mathbf{P}}_2\\|_F^2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{\\mathbf{P}}_i = \\frac{1}{tr(\\log(\\boldsymbol{\\Lambda}_i))}\\mathbf{P}_i\\log(\\boldsymbol{\\Lambda}_i)\n\\end{equation}\n$\\boldsymbol{\\Lambda}_i$ is a diagonal matrix whose diagonals are eigenvalues corresponding to eigenvectors in $\\mathbf{P}_i$. $\\frac{1}{tr(\\log(\\boldsymbol{\\Lambda}_i))}$ is the normalization factor. We use the logarithm of variance to weight different bases in a subspace. This similarity measure is inspired by the Log-Euclidean distance used for image-set classification in \\cite{cov}. Empirically, we use $\\max(0, \\log(\\boldsymbol{\\Lambda}_i))$ instead of $\\log(\\boldsymbol{\\Lambda}_i)$ to avoid negative weights. We use \\textbf{VPM} to denote this similarity measure.\n\n\\subsubsection{Quality-Aware Subspace-to-Subspace Similarity}\nBy combining the quality-aware subspace learning, quality-aware exemplars and variance-aware projection metric, we propose the quality-aware subspace-to-subspace similarity between two video templates as:\n\\begin{align}\ns(\\mathbf{Y}_1,\\mathbf{Y}_2)&=s_{QCos}(\\mathbf{Y}_1,\\mathbf{Y}_2) + \\lambda s_{VPM}(\\mathbf{P}_{D1},\\mathbf{P}_{D2})\n\\end{align}\nwhere $s_{QCos}$ is defined in \\eqref{cosd}, $\\mathbf{P}_{Di}$'s are learned by \\eqref{PD} and $s_{VPM}$ is defined in \\eqref{VPM}. This similarity measure is denoted as \\textbf{QCos+QSub-VPM}.\nComparisons of the proposed similarity measures and other baselines on several challenging datasets will be discussed in Section~\\ref{sec:exp}. \n\\section{Related Work}\\label{sec:related}\n\n\\textbf{1. Deep Learning for Face Recognition:}\nTaigman \\emph{et al.} \\cite{Taigman2014} learned a DCNN model on the frontalized faces generated from 3D shape models built from face dataset. Sun \\emph{et al.} \\cite{Sun2014} \\cite{Sun2014Sparse} achieved results surpassing human performance for face verification on the LFW dataset \\cite{LFW}.\nSchroff \\emph{et al.} \\cite{Schroff2015} adopted the GoogLeNet trained for object recognition to face recognition and trained on a large-scale unaligned face dataset. Parkhi \\emph{et al.} \\cite{Parkhi15} achieved impressive results using a very deep convolutional network based on VGGNet for face verification. Ding \\emph{et al.} \\cite{Ding2016} proposed a trunk-branch ensemble CNN model for video-based face recognition. Chen \\emph{et al.} \\cite{JC15} trained a 10-layer CNN on CASIAWebFace dataset \\cite{casia} followed by a joint Bayesian metric and achieved state-of-the-art performance on the IJB-A \\cite{IJBA} dataset. Chen \\emph{et al.} \\cite{jc_ijcv} further extended \\cite{JC15} and designed an end-to-end system for unconstrained face recognition and reported very good performance on IJB-A, JANUS CS2, LFW and YouTubeFaces \\cite{YTF} datasets. In order to tackle the training bottleneck for face recognition network, Ranjan \\emph{et al.} \\cite{crystal} proposed the crystal loss to train the network on very large scale training data. It achieved state-of-the-art result on the challenging IJB-C dataset for unconstrained face recognition. \n\n\\textbf{2. Face Detection:}\nNajibi \\emph{et al.} \\cite{ssh} proposed a single-stage fully convolutional network for face detection. It is fast and achieves state-of-the-art results on WIDER \\cite{WIDER}, FDDB \\cite{FDDB} and Pascal-Faces \\cite{PascalFace} datasets. HyperFace \\cite{hyperface} is the first multi-task network that can simultaneously detect faces, extract fiducials, estimate pose and recognize gender. In \\cite{allinone}, Ranjan \\emph{et al.} built on \\cite{hyperface} and used multi-task learning to simultaneously obtain face-related information like face bounding boxes, fiducials, facial pose, gender and identity. Chen \\emph{et al.} \\cite{icipssd} proposed a multi-task face detector based on the single stage SSD detector with extra branches. It achieved competitive performance on FDDB, AFW and PASCAL-Faces datasets with the similar speed as SSD.\n\n\\textbf{3. Face Retrieval and Tracking:}\nIn the literature on face retrieval, many methods have been proposed \\cite{SwamiBTAS}, \\cite{Ortiz}, \\cite{Cinbis}. Sivic et al. \\cite{Sivic} proposed a method that retrieves the target subject using a set of images that contains extensive variations of exemplars. Arandjelovic \\emph{et al.} \\cite{Arandjelovic} proposed an end-to-end video face retrieval system with several processing steps.\n\nThere are also many works on face tracking and association. Zhou \\emph{et al.} \\cite{trackparticle} incorporated appearance-based models in a particle filter to realize robust visual tracking. \\cite{robustmft} proposed a multi-pose face tracking approach in two stages using multiple cues. Comaschi \\emph{et al.} \\cite{OnlineMD} also proposed an online multi-face tracker using detector confidence and a structured SVM. An efficient tracker, SORT \\cite{SORT}, achieves comparable results to other state-of-the-art methods with 20 times faster speed. Du \\emph{et al.} \\cite{du_association} proposed a conditional random field (CRF) framework to associate faces by utilizing the similarity of facial appearance, location, motion, and body appearance.\n\n\n\\textbf{4. Image Set\/Video-based Recognition:}\nFor image set-based recognition, Wang \\emph{et al.} \\cite{Wang2012} proposed a Manifold-to-Manifold Distance (MMD) for face recognition based on image set. In \\cite{cov}, the proposed approach models the image set with its second-order statistic for image set classification. Chen \\emph{et al.} \\cite{DFRV_2012} and \\cite{DFRV_2015} proposed a video-based face recognition algorithm using sparse representation and dictionary learning. Zheng \\emph{et al.} \\cite{hybrid} proposed a hybrid dictionary learning and matching approach for video-based face recognition.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOver the last few decades ~\\cite{Cormack1964Representation, Hounsfield1974Computerized}, X-ray Computed Tomography (CT) has demonstrated its prominent practical value and wide range of applications including clinical diagnosis, safety inspection and industrial detection ~\\cite{Wang2008An}. Especially in the past year, due to the global spread of the Corona Virus Disease 2019 (COVID-19), the term CT has become well-known to the public as an essential auxiliary technology. However, the radiation dose brought by CT has a nonnegligible side effect on the human body. Since it has a latent risk of inducing cancers, radiation dose reduction is becoming more and more crucial under the principle of ALARA (as low as reasonably achievable) ~\\cite{Krishnamoorthi2011Effectiveness, Slovis2002CT, Mccollough2009Strategies, Mccollough2006CT}.\n\nGenerally speaking, there are two approaches to reduce radiation dose. The approach of tube current (or voltage) reduction ~\\cite{Poletti2007Low-DoseVersus, Tack2003DoseReduction} lowers the x-ray exposure in each view but suffers from the increased noise in projections. Although the approach of projection number reduction ~\\cite{Bian2010Evaluation, Bian2012Optimization} (also known as sparse-view CT) can avoid the former problem and realize the additional benefit of accelerated scan and calculation, it leads to severe image quality degradation of increased streaking artifacts brought by its missing projections. In this paper, we focus on effectively repairing and reconstructing sparse-view CT so as to acquire high-quality CT images.\n\nSparse-view CT reconstruction has always been a classic inverse problem which has attracted wide attention \\cite{Jin2016Deep}. In the past few decades, iterative reconstruction methods have become the dominant approach to solve inverse problems \\cite{Andersen1984Simultaneous, Wu2017Iterative, Hu2017An, Zhang2019JSR}. With the advent of compressed sensing \\cite{Candes2006Robust} and its related regularizers, the quality of reconstructed images has been improved to a certain extent. One of the most typical regularizers is the total variation (TV) method, algorithms based on which include TV-POCS \\cite{Sidky2008Image}, TGV method \\cite{Niu2014Sparse}, SART \\cite{Andersen1984Simultaneous} and SART-TV \\cite{Sidky2009Accurate} etc. In addition, dictionary learning is also commonly used as a regularizer. For example, \\cite{Xu2012Low} constructs a global dictionary and an iterative adaptive dictionary to solve the problem of low-dose CT reconstruction.\n\nIn recent years, with the improvement of computing power, there comes a rapid growth in deep learning \\cite{LeCun2015DeepLearning}. Subsequently, neural networks have been widely applied in image analysis tasks, such as image classification \\cite{Russakovsky2015ImageNet}, image segmentation \\cite{Long2015Fully, Ronneberger2015U, Soltaninejad2020Three}, especially inverse problems in image reconstruction, such as artifacts reduction \\cite{Dong2015Compression, Guo2016Building}, denoising \\cite{Xie2012Image} and inpainting \\cite{Kulkarni2016ReconNet}. Since GAN (Generative Adversarial Networks) was designed elaborately by Goodfellow in 2014 \\cite{goodfellow2014generative}, it has been adopted in many image processing tasks due to its prominent performance in realistically predicting image details. Therefore, GANs are also naturally applied to improving the quality of low-dose CT images \\cite{Bai2018Limited, Xie2019Artifact, Zhao2018Sparse}. In addition, Ye et al. explored the relationship between deep learning and classical signal processing methods in \\cite{Ye2017Deep}, explained the reason why deep learning can be employed in imaging inverse problems, and provided a theoretical basis for the application of deep learning in low-dose CT reconstruction.\n\nSome researchers adopt deep learning-based architectures to complement and restore the limit-view Radon data \\cite{Dong2019A, fu2020a, Bai2018Limited, Anirudh2018Lose, Dai2019Limited, ghani2018deep}. Dong et al. \\cite{Dong2019A} used U-Net \\cite{Ronneberger2015U} to predict the missing Radon data, then reconstruct it to the image through FBP \\cite{katsevich2002theoretically}. Jian Fu et al. \\cite{fu2020a} built a network that involves the tight coupling of the deep learning neural network and DPC-CT (Differential phase-contrast CT) reconstruction algorithm in the domain of DPC projection sinograms. The estimated result is a complete phase-contrast projection sinogram. Rushil Anirudh et al. established CTNet \\cite{Anirudh2018Lose}, a system of 1D and 2D convolutional neural networks, which operates on the limited-view sinogram to predict the full-view sinogram, and then fed it to the standard analytical and iterative reconstruction algorithms to obtain the final result.\n\nOther researchers carried out post-processing on reconstructed images with deep learning models, so as to remove the artifacts and noises for upgrading the quality of these images\\cite{Han2018Framing, Zhang2018A, zhang2016image, Xie2019Artifact, Wang2020Deep, kuanar2019low, guan2020fully}. In 2016, a deep convolutional neural network \\cite{zhang2016image} was proposed to learn an end-to-end mapping between the FBP and artifact-free images. In 2018, Yoseob Han and Jong Chul Ye designed a dual frame and tight frame U-Net \\cite{Han2018Framing} which satisfies the frame condition and performs better for recovery of high frequency edges in sparse-view CT. In 2019, Xie et al. \\cite{Xie2019Artifact} built an end-to-end cGAN model with joint function used for removing artifacts from limited-angle CT reconstruction images. In 2020, Wang et al. \\cite{Wang2020Deep} developed a limited-angle TCT image reconstruction algorithm based on U-Net, which could suppress the artifacts and preserve the structures. Experiments have shown that U-Net-like structures are efficacious for image artifacts removal and texture restoration \\cite{Ye2017Deep,Dong2019A,Han2018Framing,Wang2020Deep,guan2020fully} .\n\nSince neural networks are capable of predicting unknown data in the Radon and image domains, a natural idea is to combine these two domains \\cite{Lee2018High,liang2018comparison,Zhao2018Sparse,Zhu2020Low,hammernik2017a,Zhang2020Artifact} to acquire better restoration results. Specifically, it first complements the Radon data, and then remove the residual artifacts and noises on images converted from the full-view Radon data. In 2018, Zhao et al. proposed SVGAN \\cite{Zhao2018Sparse}, an artifacts reduction method for low-dose and sparse-view CT via a single model trained by GAN. In 2019, Liang et al. \\cite{liang2018comparison} proposed a comprehensive network combining projection and image domains. The projection estimation network is based on Res-CNN structure, and the image domain network takes the advantage of U-Net. In 2020, Zhu et al. designed ADAPTIVE-NET \\cite{Zhu2020Low} to conduct joint denoising on the acquired sinogram and the reconstructed CT image, while reconstructing CT image in an end-to-end manner. In the past three years, experiments have proved that this sort of two-stage algorithm is quite conducive to image quality improvement.\n\nAll the current mainstream methods mentioned above make us notice that they solely process on each single CT image, while neglecting the solid fact the scanned object is always highly continuous. Consequently, there is abundant spatial information lies in these obtained consecutive CT images, which is largely left to be exploited. This enlightens us to propose a novel cascade model called LS-AAE (Lightweight Spatial Adversarial Autoencoder) that mainly focus on availably utilizing the spatial information between greatly correlated images. It has been proved in our experiments that this sort of structure design manages to efficaciously remove streaking artifacts in sparse-view CT images, and outruns other prevailing methods with its remarkable performance.\n\nIt is the social trend now to make healthcare mobile and portable. In lots of deep learning-based methods, however, scholars improve accuracy at the expense of sacrificing computing resources. Such computational complexity usually exceeds the capabilities of many mobile and embedded applications. This paper adopts the inverted residual with linear bottleneck \\cite{Sandler2018MobileNetV2} in the module design to propose a mobile structure that reduce model parameters to one-eighth of its original without sacrificing its performance.\n\nAlthough enhancing the sparsity of sparse-view CT can bring benefits of accelerated scanning and related calculations, it will cause additional imaging damage. Balancing image quality and X-ray dose level has become a well-known trade-off problem. Thus, in order to explore the limit of sparsity in sparse-view CT reconstruction, we conduct sparse sampling at intervals of 4\u00b0, 8\u00b0 and most importantly, 16\u00b0. Even under such sampling sparsity, our model can still exhibit its remarkable robustness and the state-of-the-art performance.\n\nWe introduce our proposed method exhaustively in Section II, the experimental results and corresponding discussion are described in section III, and conclusion is stated in section IV.\n\n\\section{Methods}\n\\subsection{Preliminaries}\n\\subsubsection{Utilize Spatial Information for Artifact Removal}\n\n\n\nAs is known to all, consecutive CT images usually contain high spatial coherency since the scanned object is usually spatially continuous. On account of that, we can imagine these CT images as adjacent frames in a video which contains much more information than a still image. This high correlation within the sequence of images can improve the performance of artifact removal from two aspects. Firstly, the extension of search regions from two-dimensional image neighborhoods to three-dimensional spatial neighborhoods provide extra information which can be used to denoise the reference image. Secondly, using spatial neighbors helps to reduce streaking artifacts as the residual error in each image is correlated.\n\nAlso, we cannot help but notice that the task of artifact removal between consecutive images is similar to video denoising. Therefore, after investigating lots of research work on video denoising \\cite{pascanu2013on, caballero2017real, maggioni2012video, arias2018video, vogels2018denoising, ehret2019model, claus2019videnn, davy2018non, tassano2019dvdnet, chen2016deep}, we find out that current state-of-the-art methods lay lots of emphasis on motion estimation due to the strong redundancy along motion trajectories. To conclude, in order to more effectively remove streaking artifacts from sparse-view CT images, we need to design a structure that can not only look into the three-dimensional spatial neighborhood, but also capture the motion between consecutive images.\n\n\\subsubsection{Enhance Mobility of Neural Networks}\nIn recent years, lots of research has been invested into tuning deep neural networks to achieve an optimal balance between efficiency and performance. Among them, depthwise separable convolutions \\cite{howard2017mobilenets} exhibits its extraordinary capability and has gradually become an essential building block for numerous lightweight neural networks \\cite{howard2017mobilenets, chollet2017xception, zhang2018shufflenet}. It aims to decompose the standard convolutional layer into two separate layers, namely the depthwise convolutional layer and the pointwise convolutional layer. The former layer is designed to perform lightweight filtering through employing a single convolutional filter per input channel, the latter one conducts $1\\times1$ convolution to construct new features by computing linear combinations of input channels.\n\nFor the standard convolutional layer with input tensor size $(c_{in}, h, w)$, kernel size $(c_{out}, c_{in}, k, k)$ and output tensor size $(c_{out}, h, w)$, its computational cost equals to $c_{in} \\cdot h \\cdot w\\cdot (k^2 \\cdot  c_{out})$. However, in depthwise separable convolutions, the depthwise convolutional layer has a computational cost of $c_{in} \\cdot  h \\cdot w \\cdot k^2$ since it merely operates on a single input channel, and the pointwise convolutional layer has a computational cost of $c_{in} \\cdot  h \\cdot  w \\cdot  c_{out}$. Therefore, we only need a computational cost of $h \\cdot  w \\cdot  c_{in} \\cdot  (k^2 + c_{out}) $ for depthwise separable convolutions, which is almost the one-ninth ($k$ equals to 3 in our case) of the standard convolution. Most importantly, depthwise separable convolutions manage to lower the computational complexity to a large extent without sacrificing its accuracy, which would make it perfect to be inserted into our module design.\n\n\\subsection{Overall Structure}\n\\subsubsection{Structure Overview}\n\n\\begin{figure*}[!htb]\n\\setlength{\\abovecaptionskip}{-5pt}\n\\centerline\n{\\includegraphics[width=\\linewidth]{fig1.eps}}\n\\caption{Structure overview. The sparse-view Radon data $\\bold{X}$ is first sent to the neural network $F$ for completion, then the restored full-view Radon data $\\bold{X}^\\prime$ is converted to the image $\\bold{Y}$, which is feed into the neural network $G$ for artifacts removal and we can finally obtain the ideal high-quality image $\\bold{Y}^\\prime$.}\n\\label{fig1}\n\\end{figure*}\n\nWe can learn from the universal approximation theorem \\cite{hornik1989multilayer} that multilayer feedforward networks are capable of approximating various continuous functions. This inspires us to think that neural networks can be used to learn complex mappings that are difficult to solve through mathematical analysis. Thus, in this paper, we utilize a deep learning-based structure that combines the Radon domain and the image domain (Figure \\ref{fig1}) to solve the task of sparse-view CT reconstruction and inpainting.\n\nFirstly, we want to make full use of the prior information in the Radon domain by converting the sparse-view Radon data $\\bold{X}$ to the full-view Radon data $\\bold{X}^{\\prime}$ so as to complement the missing data in some scanning angles. This process can be represented by the mapping: $\\bold{X} \\xrightarrow{f} \\bold{X}^{\\prime}$ according to the universal approximation theorem, where function $f$ can be approximated through our proposed neural network $F$. After we obtain the full-view Radon data $\\bold{X}^{\\prime}$, we transform it to the image $\\bold{Y}$ through FBP. Although the first stage manages to alleviate the severe imaging damage from the original sparse-view CT image, there are still lots of streaking artifacts existing in Y that need to be removed to acquire the high-quality restored result $\\bold{Y}^{\\prime}$. We represent the restoration process into the mapping: $\\bold{Y} \\xrightarrow{g} \\bold{Y}^{\\prime}$, where function $g$ can be approximated through our proposed neural network $G$. Through the above two-stage structure that combines the Radon domain with the image domain, we can finally get the ideal restored results.\n\n\\subsubsection{Stage One: Data Completion in the Radon Domain}\n\\begin{figure}[!tb]\n\\vspace{-7pt}\n\\setlength{\\abovecaptionskip}{-5pt}\n\\centerline{\\includegraphics[scale=0.7]{fig2_3.eps}}\n\\caption{The diagram of our proposed L-AAE, which is composed of a L-AE and a discriminator that help restore the image texture.}\n\\label{fig2}\n\\vspace{-5pt}\n\\end{figure}\n\nWe first adopt linear interpolation to convert the original sparse-view Radon data to full-view Radon data so as to satisfy the structural characteristics of our proposed neural network, which requires the input and output images to have the same resolution. Then we build a lightweight adversarial autoencoder (L-AAE) in Figure \\ref{fig2} to restore the Radon data, the structure of its autoencoder (L-AE) can be seen from Figure \\ref{fig3} and Table \\ref{table1}, which is composed of the encoder and the decoder that are highly symmetrical.\n\n\\begin{figure}[!tb]\n\\setlength{\\abovecaptionskip}{-10pt}\n\\centerline{\\includegraphics[width=\\linewidth]{fig3_2.eps}}\n\\caption{The detailed structure of the L-AE. Input images are first feed into the encoder for feature extraction and then sent into the decoder for texture restoration, where skip connections are added to merge low-level features.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{table}[!htb]\n  \\centering\n  \\caption{Parametric structure of the L-AE}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{m{5.5em}<{\\centering} m{5.5em}<{\\centering}cc m{4em}<{\\centering} m{4em}<{\\centering} } \\hline \\hline\n    \\multicolumn{1}{m{5.5em}<{\\centering}}{Layer} & \\multicolumn{1}{m{5.5em}<{\\centering} }{$IC$} & \\multicolumn{1}{m{1.6em}<{\\centering} }{$OC$} & \\multicolumn{1}{m{2em} <{\\centering}}{Stride} & Input Size & Output Size\\\\\\hline\n    Conv1 & 1     & 32    & 2     & 192$\\times$512 & 96$\\times$256 \\\\\n    Block1 & 32    & 16    & 1     & 96$\\times$256 & 96$\\times$256 \\\\\n    Block2\\_1 & 16    & 32    & 2     & 96$\\times$256 & 48$\\times$128 \\\\\n    Block2\\_2 & 32    & 32    & 1     & 48$\\times$128 & 48$\\times$128 \\\\\n    Block3\\_1 & 32    & 64    & 2     & 48$\\times$128 & 24$\\times$64 \\\\\n    Block3\\_2 & 64    & 64    & 1     & 24$\\times$64 & 24$\\times$64 \\\\\n    Block3\\_3 & 64    & 64    & 1     & 24$\\times$64 & 24$\\times$64 \\\\\n    Block4\\_1 & 64    & 128   & 2     & 24$\\times$64 & 12$\\times$32 \\\\\n    Block4\\_2 & 128   & 128   & 1     & 12$\\times$32 & 12$\\times$32 \\\\\n    Block4\\_3 & 128   & 128   & 1     & 12$\\times$32 & 12$\\times$32 \\\\\n    Block4\\_4 & 128   & 128   & 1     & 12$\\times$32 & 12$\\times$32 \\\\\n    Trans Conv1 & 128   & 64    & 2     & 12$\\times$32 & 24$\\times$64 \\\\\n    Block5\\_1 & \\multicolumn{1}{m{6em}}{64+64(Concat)} & 64    & 2     & 24$\\times$64 & 24$\\times$64 \\\\\n    Block5\\_2 & 64    & 64    & 1     & 24$\\times$64 & 24$\\times$64 \\\\\n    Block5\\_3 & 64    & 64    & 1     & 24$\\times$64 & 24$\\times$64 \\\\\n    Trans Conv2 & 64    & 32    & 2     & 24$\\times$64 & 48$\\times$128 \\\\\n    Block6\\_1 & \\multicolumn{1}{m{6em}}{32+32(Concat)} & 32    & 1     & 48$\\times$128 & 48$\\times$128 \\\\\n    Block6\\_2 & 32    & 32    & 1     & 48$\\times$128 & 48$\\times$128 \\\\\n    Trans Conv3 & 32    & 16    & 2     & 48$\\times$128 & 96$\\times$256 \\\\\n    Block7 & \\multicolumn{1}{m{6em}}{16+16(Concat)} & 32    & 1     & 96$\\times$256 & 96$\\times$256 \\\\\n    Block8 & \\multicolumn{1}{m{6em}}{32+32(Concat)} & 32    & 1     & 96$\\times$256 & 96$\\times$256 \\\\\n    Trans Conv4 & 32    & 16    & 2     & 96$\\times$256 & 192$\\times$512 \\\\\n    Conv9 & 16    & 1     & 1     & 192$\\times$512 & 192$\\times$512 \\\\ \\hline \\hline\n    \\end{tabular}%\n  \\label{table1}%\n  \\vspace{-10pt}\n\\end{table}%\n\nWe perform four down sampling in the encoder to obtain high-level semantic features of the input image, which is initially downsampled through conv1 layer with a stride of 2, and the subsequent downsampling is separately accomplished by the first building block of each unit. Each downsampling will halve the height and width of the activation map and double the number of channels.\n\nAs for the decoder, we correspondingly conduct four upsampling to restore the texture of the input image. Deconvolution is adopted here for upsampling with its kernel size and stride both equal to 2, so that each upsampling will double the height and width of the activation map and halve the number of channels. In addition, we add skip connections \\cite{he2016identity} between the encoder and decoder feature maps of the same resolution. Since the final feature map of the encoder has a relatively low resolution due to multiple downsampling, it will have an undesirable effect on the restoration of the image texture in the decoder. While the skip connection incorporates low-level features from the encoder which have a high resolution and contain abundant detailed information that will help to accurately restore the image texture. This sort of multi-scale, U-Net-like architectures have been proved to be effective in processing medical images.\n\nThe detailed structure of our building block can be seen from Figure \\ref{fig4}, it adopts the inverted residual with linear bottleneck referring to \\cite{Sandler2018MobileNetV2}, each block is composed of three convolutional layers. The first layer expands (characterized by the expansion factor $exp$) a low-dimensional compressed representation to high dimension with a kernel size of $1 \\times 1$. The intermediate expansion layer adopts lightweight depthwise convolutions mentioned above so as to significantly decreases the number of operations (ops) and memory needed while sustaining the same performance. The last layer projects the feature back to a low-dimensional representation with a linear convolution like the first layer. All these layers are followed by batch normalization \\cite{ioffe2015batch} and ReLU \\cite{glorot2011deep} as the non-linear activation except for the last layer that only followed by a batch normalization layer.\n\n\\begin{figure*}[!htb]\n\\setlength{\\abovecaptionskip}{3pt}\n\\centerline{\\includegraphics[width=14cm]{fig4.eps}}\n\\caption{The detailed diagram of the building blocks in L-AAE, which adopt the inverted residual with linear bottleneck.}\n\\label{fig4}\n\\end{figure*}\n\nIn Figure \\ref{fig4}, $IC$ and $OC$ stand for the input and output channel of building blocks respectively. All convolutional layers in all building blocks have a stride of 1 except for Block2\\_1, Block3\\_1 and Block4\\_1 that have a stride of 2 to conduct downsampling.Expansion factor $exp$ is 1 for Block1, Block7 and Block8 to avoid large ops and memory cost, we set up $exp$ to be 3 for Block5\\_1 and Block6\\_1, and every block expect these mentioned above have an $exp$ of 6. Besides, shortcut connections are implemented in blocks that have the same resolution between its input and output feature maps to enhance information flow and also improve the ability of a gradient to propagate across multiplier layers. We adopt $1\\times1$ convolution in shortcuts when the number of channel in the input and output feature maps is different.\n\nThe discriminator in our L-AAE aims to strengthen model's ability to restore the detailed texture of images, its structure is almost the same as the encoder above, except that its Block4\\_3 and Block4\\_4 have an $OC$ of 64 and 1 respectively. The output of Block4\\_4 is flattened and sent to sigmoid function for probability prediction, which we average to get the final output that represents the input image's probability to be a real image. This novel lightweight AAE enables us to acquire the well restored Radon data that are complete in every scanning angle, and the computational cost is about 8 times smaller than that of standard convolutions without sacrificing its accuracy.\n\n\n\n\\subsubsection{Stage Two: LS-AAE -- Image Inpainting through spatial information}\nAfter stage one, we transform the acquired full-view Radon data to images and find out that, we successfully enrich the information in the Radon domain and alleviate streaking artifacts from the original sparse-view CT imaging. Now in stage two, we will mainly focus on removing artifacts, restoring image to an ideal level. As mentioned above, we need a neural network that not only look into the three-dimensional spatial neighborhood, but also capture the motion between consecutive images, so as to efficaciously utilize the abundant spatial information between consecutive images to remove artifacts from the input image.\n\nGenerally speaking, motion estimation always brings an additional degree of complexity that is adverse to model's implementation in reality. It means that we need a structure that can manage to deploy motion estimation without much resource cost, we refer to \\cite{tassano2020fastdvdnet} and its general structure appears to be a cascaded two-step architecture that inherently embed the motion of objects. Inspired by this, we propose a model named Lightweight Spatial Adversarial Autoencoder (LS-AAE) which can be seen from Figure \\ref{fig5}. It slightly modifies the L-AE from Figure \\ref{fig3} as its inpainting block, details are shown in Table \\ref{table1}. The replacement from 2D convolution to 3D convolution enables our model to look into the three-dimensional spatial neighborhood for extra information.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{From 2D convolution to 3D convolution}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{m{4em}<{\\centering}m{3.5em}<{\\centering}m{1.4em}<{\\centering}m{1.4em}<{\\centering}m{2.5em}<{\\centering}m{2.5em}<{\\centering}m{2.8em}<{\\centering}}\\hline\\hline\n          & Layer & \\multicolumn{1}{m{1.4em}<{\\centering}}{$IC$} & \\multicolumn{1}{m{1.4em}<{\\centering}}{$OC$} & Kernel Size & Stride & Padding \\\\ \\hline\n    \\multicolumn{1}{c}{2D Convolution} & Conv1 & 1     & 16    & (3,3) & (2,2) & (1,1) \\\\ \\hline\n    \\multicolumn{1}{c}{\\multirow{2}[4]{*}{3D Convolution}} & Conv1\\_1 & 1     & 16    & (3,3,3) & (1,2,2) & (1,0,0) \\\\\n     & Conv1\\_2 & 16    & 32    & (3,3,3) & (2,1,1) & (0,1,1) \\\\ \\hline\\hline\n    \\end{tabular}%\n  \\label{table2}%\n \n\\end{table}%\n\nAs shown in Figure \\ref{fig5}, five consecutive images $\\left\\{\\bold{I}_{i-2}, \\bold{I}_{i-1}, \\bold{I}_i, \\bold{I}_{i+1}, \\bold{I}_{i+2}\\right\\}$ are sent into the LS-AAE to restore the middle one. We firstly treat these inputs as triplets of consecutive images $\\left\\{\\bold{I}_{i-2}, \\bold{I}_{i-1}, \\bold{I}_i\\right\\}$, $\\left\\{\\bold{I}_{i-1}, \\bold{I}_i, \\bold{I}_{i+1}\\right\\}$ and $\\left\\{\\bold{I}_i, \\bold{I}_{i+1}, \\bold{I}_{i+2}\\right\\}$, then enter them into the Inpainting Blocks 1. Subsequently, we obtain the outputs of these blocks and combine them into triplet $\\left\\{\\bold{I}^{\\prime}_{i-1}, \\bold{I}^{\\prime}_i, \\bold{I}^{\\prime}_{i+1}\\right\\}$ which will be sent into Inpainting Block 2 to acquire the ultimate estimation $\\bold{I}^{\\prime\\prime}_i$ corresponding to the central image $\\bold{I}_i$. The LS-AAE digs deep into the three-dimensional space and implicitly handles motion without any explicit motion compensation stage on account of the traits of its architecture. Besides, the three Inpainting Blocks in step one share the same weights so as to avoid memory cost. We also add a discriminator in stage two to better restore the image texture, the predicted image $\\bold{I}^{\\prime\\prime}_i$ and its corresponding ground truth (the full-view CT imaging) $\\bold{I}^{GT}_i$ are both send into this discriminator, its structure is exactly the same as it is in stage one..\n\n\\begin{figure*}[!htb]\n\\setlength{\\abovecaptionskip}{-10pt}\n\\centerline{\\includegraphics[width=\\linewidth,scale=0.8]{fig5_2.eps}}\n\\caption{The diagram of our proposed LS-AAE. It combines an autoencoder that fully utilizes the spatial correlation between consecutive CT images and a discriminator that help refine image details.}\n\\label{fig5}\n\\end{figure*}\n\n\\subsection{Network Training}\nStage one and stage two are trained separately. For the autoencoders in these two models, we employ the multi-loss function below, which is consists of three parts $l_{MSE}$, $l_{Adv}$ and $l_{Reg}$ with their respective hyperparameters $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$.\n\n\\begin{equation}\n\\label{eq1}\n{{l}_{AE}}={{\\alpha}_{1}}{{l}_{MSE}}+{{\\alpha}_{2}}{{l}_{Adv}}+{{\\alpha}_{3}}{{l}_{Reg}}\n\\end{equation}\n\n$l_{MSE}$ calculates the mean square error between the restored image and its corresponding ground truth, it is widely used in various image inpainting tasks since it provides an intuitive evaluation for the model's prediction. The expression of  $l_{MSE}$ can be seen from Equation (\\ref{eq2}).\n\n\\begin{equation}\\label{eq2}\n{{l}_{MSE}}=\\frac{1}{W\\times H}\\sum\\limits_{x=1}^{W}{\\sum\\limits_{y=1}^{H}{{{(\\bold{I}_{x,y}^{GT}-{{G}_{AE}}{{({{\\bold{I}}^{Input}})}_{x,y}})}^{2}}}}\n\\end{equation}\n\n\\noindent Where function $G_{AE}$ stands for the autoencoder, $\\bold{I}^{Input}$ and $\\bold{I}^{GT}$ are the input image and its corresponding ground truth, $W$ and $H$ are the width and height of the input image respectively.\n\n$l_{Adv}$ refers to the adversarial loss. The autoencoder manages to fool the discriminator by making its prediction as close to the ground truth as possible, so as to achieve the ideal image restoration outcome. Its expression can be seen from Equation (\\ref{eq3}).\n\n\\begin{equation}\\label{eq3}\n{{l}_{adv}}=1-D({{G}_{AE}}({\\bold{I}^{Input}}))\n\\end{equation}\n\n\\noindent Where function $D$ and $G_{AE}$ stands for the discriminator and the autoencoder respectively, $\\bold{I}^{Input}$ is the model's input image.\n\n$l_{Reg}$ is the regularization term of our multi-loss function. Since noises will have a side effect on our restoration result, we add a regularization term to maintain the smoothness of the image and also prevent overfitting. TV Loss is widely used in image analysis tasks, it reduces the variation between adjacent pixels to a certain extent. Its expression can be seen from Equation (\\ref{eq4}).\n\n\\begin{equation}\\label{eq4}\n{{l}_{Reg}}=\\frac{1}{W\\times H}\\sum\\limits_{x=1}^{W}{\\sum\\limits_{y=1}^{H}{\\left\\| \\left. \\nabla {{G}_{AE}}{{({\\bold{I}^{Input}})}_{x,y}} \\right\\| \\right.}}\n\\end{equation}\n\n\\noindent Where function $G_{AE}$ represents the autoencoder, $I^{Input}$ is the model's input image, $W$ and $H$ are the width and height of the input image respectively.  $\\nabla$ calculates the gradient, $\\left\\| \\cdot \\right\\|$ obtains the norm.\n\nTo optimize the discriminator of these two stages, their loss function should enable them to better distinguish between real and fake inputs. The loss function $l_{Dis}$ is shown in Equation (\\ref{eq5}).\n\n\\begin{equation}\\label{eq5}\n{{l}_{D\\text{is}}}=1-D\\left( {\\bold{I}^{GT}} \\right)+D\\left( {{G}_{AE}}\\left( {\\bold{I}^{Input}} \\right) \\right)\n\\end{equation}\n\n\\noindent Where function $D$ and $G$ stands for the discriminator and the autoencoder respectively, $\\bold{I}^{Input}$ and $\\bold{I}^{GT}$ are the input image and its corresponding ground truth. The discriminator outputs a scalar between 0 to 1 which represents the probability that the input image is real. Therefore, minimizing $1-D(\\bold{I}^{GT})$\/maximizing $D(\\bold{I}^{GT})$ enables the discriminator to recognize real images, while minimizing $D(G_{AE}(\\bold{I}^{Input}))$ enables the discriminator to distinguish fake images that generated from the autoencoder from all input images.\n\nDuring the training process, we adopt the Adam algorithm \\cite{kingma2015adam} for optimization. the learning rate is set to 1e-4 initially. For the multi-loss function, $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$ are set to 1, 1e-3, and 2e-8 respectively. We implement our whole structure using PyTorch \\cite{paszke2017automatic} on two GeForce RTX 2080 Ti.\n\n\\section{Experiments}\nWe adopt the LIDC-IDRI \\cite{armato2011the} as our dataset, which includes 1018 cases and approximately 240,000 DCM files of corresponding CT images. Cases 1 to 100 are divided into test set, cases 101 to 200 are divided into validation set, and the rest are divided into train set. Such a large amount of data allows us to train our models from scratch without overfitting. We utilize NumPy to read from these DCM files and conduct sparse sampling at intervals of 4, 8 and 16 (the corresponding full-view Radon data has 180 projections). Subsequently, we first analyze our overall structure through a series of ablation studies, and then compare our experimental results with other current methods to prove its superiority and robustness.\n\n\\subsection{Ablation Study}\nWith all these innovations we make in our overall structure design, it would be appropriate for us to conduct corresponding ablation studies to prove their necessity. In this part, all the experimental results are acquired from sparse-view CT data with an interval of 4 if there is no specific mention.\n\n\\subsubsection{The L-AE's Trade-off between Mobility and Performance}\nAs is known to all, U-Net has extraordinary performance in numerous medical image processing tasks, \\cite{Han2018Framing} implemented it for sparse-view CT imaging restoration and obtained outstanding restoration results. To testify that our proposed autoencoder can achieve a good balance between performance and mobility, we replace it with U-Net in the first stage and compare the restoration results and model parameters of this stage with ours, as shown in Table \\ref{table3}. The images mentioned in Table \\ref{table3} are reconstructed from the Radon data restored through stage one.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{U-Net VS. L-AE}\n   \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{m{3em}<{\\centering}ccccm{4.945em}<{\\centering}}\\hline\\hline\n    \\multicolumn{1}{c}{} & \\multicolumn{1}{m{3em}<{\\centering}}{Radon PSNR} & \\multicolumn{1}{m{3em}<{\\centering}}{Radon SSIM} & \\multicolumn{1}{m{3em}<{\\centering}}{Image PSNR} & \\multicolumn{1}{m{3em}<{\\centering}}{Image SSIM} & Parameters \\\\\\hline\n    U-Net & 57.582 & 0.998 & 29.598 & 0.874 & 10.401M \\\\\n    L-AE  & 57.66 & 0.998 & 29.609 & 0.874 & 1.675M \\\\\\hline\\hline\n    \\end{tabular}%\n  \\label{table3}%\n \n\\end{table}%\n\nAs we can see from Table \\ref{table3}, whether in the Radon domain or in the image domain, L-AE has competitive performance compared with U-Net. Moreover, it significantly reduces model parameters, making it suitable for situations where computational resources are extremely limited. This exhibits our model's ability in efficiently restoring CT images, thus adapting to the social trend of deploying portable medical devices.\n\n\\subsubsection{The Discriminator}\nWe establish discriminators in both two stages, hoping to further improve our model's performance in restoring sparse-view CT data through the adversarial learning between the autoencoders and the discriminators. In order to verify this point of view, we send the test set into stage one where there is merely an autoencoder and compare its restoration results with ours, which can be seen from Table \\ref{table4}. The images mentioned in Table \\ref{table4} are reconstructed from the Radon data restored through stage one.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{The Role of the Discriminator}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{m{5.2em}<{\\centering}cccc}\\hline\\hline\n    \\multicolumn{1}{c}{} & \\multicolumn{1}{m{3em}<{\\centering}}{Radon PSNR} & \\multicolumn{1}{m{3em}<{\\centering}}{Radon SSIM} & \\multicolumn{1}{m{3em}<{\\centering}}{Image PSNR} & \\multicolumn{1}{m{3em}<{\\centering}}{Image SSIM} \\\\\\hline\n    L-AE Only & 48.904 & 0.985 & 28.448 & 0.871 \\\\\n     \\textbf{L-AAE} &  \\textbf{57.660} &  \\textbf{0.998} &  \\textbf{29.609} &  \\textbf{0.874} \\\\ \\hline\\hline\n    \\end{tabular}%\n  \\label{table4}%\n \n\\end{table}%\n\nFrom the above table, we can realize the significance of our proposed discriminator, it indeed assists our model to achieve a better level of restoration under the evaluation of PSNR and SSIM. Its precise structure (refers to Sec II) also ensures a high degree of mobility, which enables our overall structure to be portable and accurate at the same time.\n\n\n\n\\subsubsection{The Two-step Architecture -- LS-AAE}\nAs we state in Sec II, this sort of cascaded two-step structure inherently embeds the motion of objects which can largely help to remove image artifacts due to the strong redundancy between these consecutive images. Consequently, we design an experiment with reference to \\cite{tassano2020fastdvdnet} to prove this view. In the second stage, instead of sending five consecutive images into this two-step LS-AAE, we directly input them into a single Inpainting Block (SIB) that is slightly modified in the three-dimensional convolution part to handle five images, that means we adopt a stride of 2 in the Conv1\\_1 layer (refers to Table \\ref{table1}). The experimental results can be seen from Table \\ref{table5} below.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Restoration Results of SIB and LS-AAE}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{p{4.5em}<{\\centering}cc}\\hline\\hline\n    \\multicolumn{1}{c}{} & \\multicolumn{1}{p{6em}<{\\centering}}{Image PSNR} & \\multicolumn{1}{p{6em}<{\\centering}}{Image SSIM} \\\\\\hline\n    SIB   & 38.972 & 0.941 \\\\\n     \\textbf{LS-AAE} &  \\textbf{40.305} &  \\textbf{0.948} \\\\\\hline\\hline\n    \\end{tabular}%\n  \\label{table5}%\n \n\\end{table}%\n\nNow the SIB no longer owns this built-in cascade structure to implicitly conduct motion estimation, it suffers from a obvious drop in PSNR and SSIM. Therefore, we can arrive at the conclusion that, LS-AAE manages to effectively improve model's capability of restoring CT images with its cascaded two-step architecture that inherently capture the motion between consecutive images.\n\n\\subsubsection{The 3D convolution in LS-AAE}\nWe mention in Sec II that, the extension of search regions from two-dimensional image neighborhoods to three-dimensional spatial neighborhoods provide extra information for image restoration. Also, extracting spatial features is conducive to remove streaking artifacts as the residual error in each image is correlated. In order to realize this extension of search regions, three-dimensional convolution is employed in every Inpainting Block of LS-AAE. To verify the cruciality of these three-dimensional convolutions, we conduct an experiment in which 3D convolution are replaced back to 2D convolution, where the number of input images is regarded as the number of input channel (refers to Table \\ref{table2}). The inpainting results of these two models are shown in Table \\ref{table6}.\n\n\\begin{table}[htbp]\n\n  \\centering\n  \\caption{Restoration Results of 2D and 3D LS-AAE}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{p{6em}<{\\centering}cc}\\hline\\hline\n    \\multicolumn{1}{c}{} & \\multicolumn{1}{p{6em}<{\\centering}}{Image PSNR} & \\multicolumn{1}{p{6em}<{\\centering}}{Image SSIM} \\\\\\hline\n    2D LS-AAE & 39.472 & 0.944 \\\\\n     \\textbf{3D LS-AAE} &  \\textbf{40.305} &  \\textbf{0.948} \\\\\\hline\\hline\n    \\end{tabular}%\n  \\label{table6}%\n  \n\\end{table}%\n\nWe can see that the inpainting outcome suffers from a drop about 0.9dB in PSNR, proving that three-dimensional convolutions assist model in restoring CT images to a certain extent without significantly consuming computational resources.\n\n\n\n\\subsubsection{The Image Interval of LS-AAE's Input}\nIn all the experiments above, we set the image interval between input consecutive CT images of LS-AAE to the default value of 1. However, we cannot help but wonder that whether increasing the interval value can help the model obtain more spatial information, thereby enhancing its ability in removing image artifacts. In the following experiment, we set this image interval $T$ to 1, 2, 3, 4 and 5 respectively, their corresponding results are shown in Table \\ref{table7}.\n\n\\begin{table}[htbp]\n\n  \\centering\n  \\caption{The Image Interval's Effect on Restoration Results}\n   \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{p{4em}<{\\centering}cc}\n    \\hline\\hline\n    \\multicolumn{1}{c}{} & \\multicolumn{1}{p{6em}}{Image PSNR} & \\multicolumn{1}{p{6em}}{Image SSIM} \\\\ \\hline\n    $\\bm{T=1}$   & \\textbf{40.305} & \\textbf{0.948} \\\\\n    $T=2$   & 39.961 & 0.948 \\\\\n    $T=3$   & 40.032 & 0.948 \\\\\n    $T=4$   & 40.147 & 0.948 \\\\\n    $T=5$   & 40.195 & 0.950 \\\\\\hline\\hline\n    \\end{tabular}%\n  \\label{table7}%\n  \n\\end{table}%\n\nIt can be learnt from Table \\ref{table7} that this hyperparameter T does not have much impact on the final restoration result. Spatial correlation seems to be well utilized when the image interval is set to 1, which would be a decent default choice.\n\n\\subsubsection{The Radon Domain VS. the Image Domain}\nIn this paper, we adopt a two-stage structure that combines the Radon domain and the image domain to obtain high-quality sparse-view CT images. Since each stage of the overall structure conducts restoration in their separate domains and both remarkably upgrade the restoration results, this leads us to think, what role do these two domains play? Subsequently, we feed our test set into these three structures: L-AAE in stage one that concentrates on the Radon domain, LS-AAE in stage two that focus on the image domain and of course, our overall structure that contains these two stages. The quantitative inpainting results of the above three structures can be referred from Table \\ref{table8}, the intuitive outcome can also be seen in Figure \\ref{fig6}.\n\n\\begin{figure}[!htb]\n\\centerline{\\includegraphics[width=\\linewidth,scale=0.8]{fig6.eps}}\n\\caption{The intuitive restoration results obtained by different domains.}\n\\label{fig6}\n\\vspace{5pt}\n\\end{figure}\n\n\\begin{table}[htbp]\n\\vspace{5pt}\n  \\centering\n  \\caption{Restoration Results obtained by different domains}\n   \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{p{9em}<{\\centering}cc} \\hline\\hline\n    \\multicolumn{1}{c<{\\centering}}{} & \\multicolumn{1}{p{6em}<{\\centering}}{Image PSNR} & \\multicolumn{1}{p{6em}<{\\centering}}{Image SSIM} \\\\ \\hline\n    The Radon Domain & 30.310 & 0.905 \\\\\n    The Image Domain & 34.135 & 0.888 \\\\\n    \\textbf{Dual Domains} &  \\textbf{40.305} &  \\textbf{0.948} \\\\\\hline\\hline\n    \\end{tabular}%\n  \\label{table8}%\n \n\\end{table}%\n\nIt can be seen from above that, restoration in each domain has its pros and cons. For the Radon domain, it demonstrates its superiority in enhancing the structural similarity of images so as to perform well under the evaluation of SSIM. While as for the image domain, it exhibits great ability in alleviating distortion, thus has a relatively good performance under the evaluation of PSNR. Naturally, we acquire extraordinary restoration results when combining these two domains to merge their respective advantages. Besides, we solely utilize the spatial correlation in the Image domain due to our discovery that, the spatial information between continuous Radon slices has little impact on the final inpainting outcome. We suppose this is because the texture in Radon slices does not have much similarity with CT images, thus cannot be restored in this way.\n\n\\subsection{Methods Comparison}\nAfter verifying the rationality of our overall structural design, we want to testify its robustness through applying it to sparse-view CT data with a higher level of sparsity, which means, conducting sparse sampling at intervals of 4, 8 and even 16 (the corresponding full-view Radon data has 180 projections). In addition, we compare our method with other current ones to prove its prominent capability of restoring sparse-view CT images and removing streaking artifacts. The experimental results are shown in Table \\ref{table9}, and the intuitive outcome can be seen from Figure \\ref{fig7}.\n\n\\begin{figure}[!htb]\n\\centerline{\\includegraphics[width=\\linewidth,scale=0.8]{fig7.eps}}\n\\caption{The intuitive restoration results of various methods at different sampling intervals.}\n\\label{fig7}\n\\end{figure}\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Methods Comparison}\n  \\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{p{4.1em}<{\\centering}p{2.7em}<{\\centering}p{2.7em}<{\\centering}p{2.7em}<{\\centering}p{2.7em}<{\\centering}p{2.7em}<{\\centering}p{2.7em}<{\\centering}} \\hline\\hline\n    \\multicolumn{1}{c}{\\multirow{2}[4]{*}{}} & \\multicolumn{2}{c}{Interval=4} & \\multicolumn{2}{c}{Interval=8} & \\multicolumn{2}{c}{Interval=16} \\\\\n\\cline{2-7}    \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{PSNR} & \\multicolumn{1}{c}{SSIM} & \\multicolumn{1}{c}{PSNR} & \\multicolumn{1}{c}{SSIM} & \\multicolumn{1}{c}{PSNR} & \\multicolumn{1}{c}{SSIM} \\\\\n    \\hline\n    FBP   & 12.080 & 0.498 & 12.065 & 0.485 & 12.032 & 0.471 \\\\\n    SART-TV & 19.179 & 0.665 & 19.061 & 0.632 & 18.777 & 0.602\\\\\n    U-Net & 34.018 & 0.885 & 31.944 & 0.843 & 28.767 & 0.798 \\\\\n    \\textbf{Ours} & \\textbf{40.305} & \\textbf{0.948} & \\textbf{37.633} & \\textbf{0.937} & \\textbf{34.052} & \\textbf{0.910} \\\\ \\hline\\hline\n    \\end{tabular}%\n  \\label{table9}%\n \n\\end{table}%\n\n\nAs we can see, our method exhibits extraordinary capability of restoring sparse-view CT imaging, effectively removes streaking artifacts and outruns other methods by a large margin. Also, it can be applied to extreme sparsity while still obtaining prominent inpainting outcome. Particularly, our method still exceeds others when the sampling rate is one-fourth of them, thereby demonstrating its remarkable robustness and superiority.\n\n\\section{Conclusion}\nIn this paper, we propose a lightweight structure that efficaciously restores sparse-view CT with its two-stage architecture combining the Radon domain and the image domain. Most importantly, we groundbreakingly exploit the abundant spatial information existing between consecutive CT images, so as to achieve a remarkable restoration outcome even if our method encounters extreme sparsity.\n\nIn the first stage, a mobile model named L-AAE is proposed to complement the original sparse-view CT in the Radon domain, it adopts the inverted residual with linear bottleneck in order to significantly reduce computational resource requirements while maintaining outstanding performance. In the second stage, after reconstructing the restored full-view Radon data into images through FBP, we establish a lightweight model called LS-AAE. It is designed to implicitly conduct motion estimation and dig into the three-dimensional spatial neighborhood with a relatively low memory cost. Therefore, it manages to concentrates on fully utilizing the strong spatial correlation between continuous CT images, so as to productively remove streaking artifacts and finally acquire high-quality restoration results.\n\nEventually, for the sparse-view CT with a sampling interval of 4, we achieve a PSNR of 40.305 and a SSIM of 0.948, realizing a remarkable restoration result that can effectively eliminate image artifacts. In addition, our method also performs well when it comes to extreme sparsity (the sampling interval is 8 or even 16), exhibiting its prominent robustness.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{section:introduction}\n\n\n\\IEEEPARstart{I}{n} this paper and in a sequel \\cite{OIS} we consider the problem of interpolating all values of a discrete, periodic  signal $f\\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{C}$, $N \\ge 2$, when $d<N$ values of $f$ are known.  One solution is a discrete form of the classical Nyquist-Shannon theorem, where the spectrum of the signal is assumed to vanish outside a contiguous band of frequencies; see \\cite{Will:thesis}, for example.  At the other extreme is the new and important area of compressed sensing, where no assumptions on the spectrum are made. For this, of the many papers we mention only \\cite{candes:robust},  \\cite{Venk-Bresler} and  \\cite{Mishali-Eldar}, since we will refer to this work later. \n\nOur approach to the problem is in between, though we begin by formulating a very general definition. \n\\begin{defn} \\label{def:interpolating-system}\nLet $\\mathbb{Y}$ be a $d$-dimensional subspace of $\\mathbb{C}^N$, let $\\mathcal{I} \\subset [0: N-1]$ be an index set of size $d$, and let $\\mathcal{U}_\\mathcal{I}=\\{u_i \\colon i \\in \\mathcal{I}\\}$ be a set of $d$ vectors in $\\mathbb{Y}$. We say that $(\\mathcal{I},\\mathcal{U}_\\mathcal{I})$ is an \\emph{interpolating system} if each $f\\in \\mathbb{Y}$ can be written as\n\\begin{equation} \\label{eq:interp-formula}\nf=\\sum_{i\\in\\mathcal{I}} f(i)u_i.\n\\end{equation}\nWe call $\\mathcal{I}$ a \\emph{sampling set} and $\\mathcal{U}_\\mathcal{I}$ an \\emph{interpolating basis}. When we refer simply to a sampling set we always mean that it is associated with an interpolating basis. If the vectors $u_i$ are orthogonal we say that $(\\mathcal{I},\\mathcal{U}_\\mathcal{I})$ is an \\emph{orthogonal interpolating system} and that $\\mathcal{U}_\\mathcal{I}$ is an \\emph{orthogonal interpolating basis}. \n\\end{defn}\n\nThe point of the definition is that the interpolation of all values of $f$ uses the sampled values $f(i)$, $i\\in\\mathcal{I}$, which might be thought of as measurements of $f$ with respect to the fixed, natural basis of the ambient space $\\mathbb{C}^N$, while the basis $\\mathcal{U}_\\mathcal{I}$ is tailored to $\\mathbb{Y}$ and $\\mathcal{I}$.\\footnote{We could make the definition even more general and allow $\\mathbb{Y}$ to be a subspace of any finite-dimensional vector space $\\mathbb{X}$, and sample $f\\in \\mathbb{Y}$ with respect to any fixed basis of $\\mathbb{X}$, but the present definition suffices.}  Note that $\\mathcal{I}$ need not consist of uniformly spaced indices, so the sampling may be irregular. Indeed, the results described here and in \\cite{OIS} were originally motivated by questions from colleagues in medical imaging who had observed that irregular sampling patterns could often give excellent results with less computation. \n\nFor us, to solve the interpolation problem for $\\mathbb{Y}$ is to find an interpolating system. It is a linear theory in all aspects. Every subspace has an interpolating system, though it may not be unique, but {not} every subspace has an orthogonal interpolating system. For a given subspace it is also {not} true that any index set is a sampling set for some interpolating basis, so the intervals between samples are not arbitrary. The only subspaces that have \\emph{orthonormal} interpolating systems are the coordinate subspaces. All of this is discussed in Section \\ref{section:general-properties}.  Orthogonal interpolating systems are the subject of \\cite{OIS}, and we find interesting connections with difference sets, perfect graphs, tiling, and we answer affirmatively a discrete version of a conjecture of Fuglede. \n\nIn Section \\ref{section:general-properties} we provide some basic results on interpolating systems in general. We quickly move, in Section \\ref{section:bandlimited}, to study bandlimited spaces, $\\mathbb{B}^\\mathcal{J}$, defined as signals whose discrete Fourier transforms are supported on $\\mathcal{J}$. We do not require that $\\mathcal{J}$ be a set of contiguous indices, so this is more general than the situation in the discrete Nyquist-Shannon theorem (though we continue to use the term ``bandlimited\" for short). \n\n In Section \\ref{section:universal} we begin to concentrate on universal sampling sets, namely index sets $\\mathcal{I}$ that are sampling sets for \\emph{any} bandlimited space $\\mathbb{B}^\\mathcal{J}$ with $|\\mathcal{J}|=|\\mathcal{I}|$. That is, $\\mathcal{I}$ is universal if the sampling pattern specified by $\\mathcal{I}$ can be used for interpolation of signals from any $\\mathbb{B}^\\mathcal{J}$. Universal sampling sets were used in \\cite{Venk-Bresler} for multicoset sampling and in \\cite{Mishali-Eldar} in connection with compressed sensing. Here our central result  gives several necessary and sufficient conditions for an index set to be universal when $N$ is a prime power.   A mathematical consequence of our result is a generalization of Chebotarev's theorem on the invertibility of submatrices of the Fourier matrix. \n \n \n \nIn Section \\ref{section:maximal} we show that a universal sampling set has an interesting structure as a disjoint union of what we call elementary universal sets, and through this analysis we are  able to count the number of  universal sampling sets of a given size. We also introduce maximal (and minimal) universal sampling sets which in turn enter naturally into the uncertainty principles that we discuss in Section \\ref{section:uncertainty}.  As an application of uncertainty and universality we prove a ``random'' uncertainty principle, and deduce a generalization of the Cauchy-Davenport theorem from  additive number theory. Our debt to the work in \\cite{tao:uncertainty} and \\cite{candes:robust}  is clear. Many of our results assume that $N$ is a prime power, and naturally we wonder whether this can be generalized.\n\nThe definitions we introduce and the methods we use are based primarily on properties of index sets when the elements are reduced modulo powers of a prime. With a few exceptions (e.g., minimal and cyclotomic polynomials) these can be considered elementary, and it is surprising (to us) how far they lead. The methods here also seem rather different from those of compressed sensing. In compressed sensing, which is nonlinear in theory and practice, the recovery of a signal from samples  does not require knowledge of the frequency spectrum, whereas linear theories like ours cannot do without knowledge of the spectrum.  Nevertheless, with universality the sampling patterns in our approach do not depend on the frequencies, the reconstruction of a signal from its samples is by linear operations, and the samples are ``samples'' in the classical sense instead of random projections of the signal onto a measurement basis as is done in compressed sensing.  Both approaches start with discrete signals, but one needs to sample an analog signal in the first place and this analog sampling generally needs some knowledge of the frequency spectrum.  Works such as \\cite{Venk-Bresler} and  \\cite{Mishali-Eldar} confront this issue through ``spectrum blind\" sampling, and  they end up needing the idea of universality in the process. It is also interesting that the linear theory here can be used to prove a  random uncertainty principle without the necessity of nonlinear techniques, though our result is not as strong as the result in \\cite{candes:robust}. We hope to pursue the connections and differences further. We refer to \\cite{Will:thesis} and \\cite{Aditya:thesis} for additional results, discussion, and examples. See also Appendix \\ref{section:additional-references} for references to papers on universality for continuous-time signals. \n\n\n\\section{General Properties, Existence of Interpolating Systems} \\label{section:general-properties}\n\nThis section is a summary of elementary properties of interpolating systems, including existence theorems in both an algebraic and geometric formulation.   The ideas are simple enough, but they fit together nicely and are an essential foundation for the less simple work to follow.\n\nWe fix some notation. Without further comment we will identify a vector in $\\mathbb{C}^N$ with its $N$-periodic extension and vice versa, and we typically index vectors from $0$ to $N-1$. (We assume periodicity because the discrete Fourier transform will soon enter the picture.) For $i\\in[0 : N-1]$ we let $\\delta_i\\colon \\mathbb{Z}_N\\longrightarrow \\mathbb{C}$ be the (periodized) discrete $\\delta$-function shifted to $i$, so that $\\{\\delta_0,\\delta_1,\\dots, \\delta_{N-1}\\}$ is the natural basis of $\\mathbb{C}^N$. The components of a vector in $\\mathbb{C}^N$ will always be in terms of the natural basis, but any fixed basis of $\\mathbb{C}^N$ would do for the following development. If $\\mathcal{I} \\subset [0:N-1]$ we let\n\\[\n\\mathbb{C}^\\mathcal{I} = \\rm{span}\\{\\delta_i\\colon i \\in \\mathcal{I}\\}.\n\\]\n\n\n\nOur first goal is to establish\n\\begin{theorem} \\label{theorem:existence}\nAny subspace $\\mathbb{Y}$ of $\\mathbb{C}^N$ has an interpolating system. \n\\end{theorem}\nWe will give two proofs, one geometric and one algebraic, and both are straightforward. \n\nIn the following, $\\mathbb{Y}$ is always a subspace of dimension $d$ and $\\mathcal{I}$ is always an index set of size $d$. Let $\\mathcal{I}'=[0 :N-1]\\setminus \\mathcal{I}$.  We record several facts.  \n\n\nAn interpolating basis for a subspace $\\mathbb{Y}$ is trying to be the natural basis in the slots specified by the index set. In fact this is a characterization of interpolating bases. \n\\begin{proposition} \\label{proposition:characterization}\n(i) A basis $\\mathcal{U}= \\{u_i\\colon i\\in \\mathcal{I}\\}$ for $\\mathbb{Y}$ is an interpolating basis if and only if \n\\[\nu_j(i) =\\delta_j(i) \\quad i,j\\in\\mathcal{I}.\n\\]\n\n(ii) Any natural basis vector $\\delta_k$ lying in $\\mathbb{Y}$ is an element of any interpolating basis of $\\mathbb{Y}$. \n\n(iii) An interpolating basis is determined by its index set, more precisely, if $\\{u_i\\colon i\\in \\mathcal{I}\\}$ and $\\{v_i\\colon i\\in \\mathcal{I}\\}$ are interpolating bases for $\\mathbb{Y}$ then $u_i=v_i$. for all $i \\in \\mathcal{I}$\n\\end{proposition}\n\nExpanding on the first point in Proposition  \\ref{proposition:characterization}, the elements of an interpolating basis are perturbations of the natural basis vectors by vectors outside $\\mathbb{Y}$:\n\\begin{proposition} \\label{proposition:perturbation}\n(i) Any interpolating basis $\\{u_i \\colon i \\in \\mathcal{I}\\}$ of $\\mathbb{Y}$ is of the form\n\\[\nu_i = \\delta_i + v_i, \n\\]\nwhere $v_i \\in \\mathbb{C}^{\\mathcal{I}'}$. If $v_i \\in \\mathbb{Y}$ then $v_i=0$. \n\n(ii) The subspaces of $\\mathbb{C}^N$ having an orthogonal interpolating system are of the form $\\mathbb{Y}= {\\rm span}\\{\\delta_i+v_i\\colon i \\in\\mathcal{I}\\}$ where  the nonzero $v_i$ are orthogonal vectors in $\\mathbb{C}^{\\mathcal{I}'}$.\n\\end{proposition}\n\n\nWe omit the proofs of Propositions \\ref{proposition:characterization} and \\ref{proposition:perturbation}. Part (ii) of Proposition \\ref{proposition:perturbation} can be applied in the negative to find examples of subspaces that do not have an orthogonal interpolating basis -- this is a much larger topic -- and it also follows from part (ii) that the only subspaces having an orthonormal interpolating basis are the coordinate subspaces. Both of these points were raised in the introduction.\n\n\\begin{IEEEproof}[Geometric Proof of Theorem \\ref{theorem:existence}] \nIt is easy to see that there is an index set $\\mathcal{J}$ of size $N-d$ such that $\\mathbb{C}^N = \\mathbb{Y} \\oplus \\mathbb{C}^\\mathcal{J}$. Let $P\\colon \\mathbb{Y}\\oplus \\mathbb{C}^\\mathcal{J} \\rightarrow \\mathbb{Y}$ be the projection of $\\mathbb{C}^N$ onto $\\mathbb{Y}$ along $\\mathbb{C}^\\mathcal{J}$. If $f\\in \\mathbb{Y}$ then, on the one hand,\n\\[\nf = \\sum_{i=1}^N f(i) \\delta_i.\n\\]\nOn the other hand, since $\\mathbb{C}^\\mathcal{J} = {\\rm ker} P$ and $Pf = f$ we have\n\\[\nf = Pf = \\sum_{i=1}^N f(i)P\\delta_i =\\sum_{i\\not\\in\\mathcal{J}} f(i) P\\delta_i.\n\\]\nThus the $u_i=P\\delta_i$ form an interpolating basis of $\\mathbb{Y}$ indexed by $\\mathcal{I} = [0 : N-1]\\setminus \\mathcal{J}$.\n\\end{IEEEproof}\n\nWe see from this why an interpolating basis need not be unique. The ambiguity in choosing an interpolating basis arises from the ambiguity in choosing a complement; if there is not a unique choice of the complement $\\mathbb{C}^\\mathcal{J}$ of $\\mathbb{Y}$, and generally there is not, then there is not a unique interpolating basis for $\\mathbb{Y}$. However,  the existence of an interpolating basis  \\emph{produces} a complement to $\\mathbb{Y}$:\n\\begin{proposition}\nLet $\\mathcal{U}=\\{u_i\\colon i \\in \\mathcal{I}\\}$ be an interpolating basis of $\\mathbb{Y}$. Then\n$\n\\mathbb{C}^N = \\mathbb{Y} \\oplus \\mathbb{C}^{\\mathcal{I}'}$.\n\\end{proposition}\n\\begin{IEEEproof} If we show that $\\mathbb{Y}\\cap \\mathbb{C}^{\\mathcal{I}'}=\\{{0}\\}$ then $\\mathcal{U} \\cup \\{\\delta_j\\colon j \\in \\mathcal{I}'\\}$ forms a basis for $\\mathbb{C}^N$. For this, let $f \\in \\mathbb{Y}\\cap \\mathbb{C}^{\\mathcal{I}'}$. Then\n\\begin{equation} \\label{eq:f-in-Y}\nf = \\sum_{i\\in \\mathcal{I}} f(i)u_i\n\\end{equation}\nbecause $\\mathcal{U}$ is an interpolating basis for $\\mathbb{Y}$, and also\n\\[\nf = \\sum_{j\\in\\mathcal{I}'} f(j)\\delta_j.\n\\]\nThus\n\\[\n \\sum_{i\\in \\mathcal{I}} f(i)u_i = \\sum_{j\\in\\mathcal{I}'} f(j)\\delta_j.\n\\]\nLet $k\\in \\mathcal{I}$ and evaluate both sides at $k$:\n\\[\n\\begin{aligned}\n \\sum_{i\\in \\mathcal{I}} f(i)u_i(k) &= \\sum_{j\\in\\mathcal{I}'} f(j)\\delta_j(k),\\\\\nf(k) &= 0.\n\\end{aligned}\n\\]\nBy \\eqref{eq:f-in-Y}, $f = {0}$ and we are done.\n\\end{IEEEproof}\n\nThe algebraic proof of Theorem \\ref{theorem:existence} is in terms of matrices.   \nAssociate with an index set $\\mathcal{I}=\\{i_1,i_2,\\dots, i_d\\}$ the $N \\times d$ matrix $E_\\mathcal{I}$ whose $d$ columns are the basis vectors $\\delta_{i_1}$, $\\delta_{i_2}$, \\dots, $\\delta_{i_d}$. If $R$ is an a $N\\times M$ matrix then $E_\\mathcal{I}^\\textsf{T}R$ is $d \\times M$ submatrix of $R$ obtained by choosing the rows indexed by $\\mathcal{I}$. In particular, operating by $E_\\mathcal{I}^\\textsf{T}$ on an $N$-vector $f$ produces the $d$-vector with components $f(i_1)$, $f(i_2)$,\\dots, $f(i_d)$. If $R$ is an $M \\times N$ matrix then  $RE_\\mathcal{I}$ is the $M\\times d$ submatrix of $R$ obtained by choosing the columns indexed by $\\mathcal{I}$. \n\nWe note three general facts. First,\n\\(\nE_\\mathcal{I}^\\textsf{T} E_\\mathcal{I} = I_d,\n\\)\nwhere $I_d$ is the $d\\times d$ identity matrix. \nSecond, if $S$ is a $d\\times d$ matrix then \n\\(\n E_\\mathcal{I}^\\textsf{T}(RS)=(E_\\mathcal{I}^\\textsf{T} R)S\\,.\n\\)\nFinally, if $\\mathcal{U}=\\{u_{i_1},u_{i_2},\\dots, u_{i_d}\\}$ is a basis for $\\mathbb{Y}$ and $U$ is the $N \\times d$ matrix whose columns are the $u_i$ then the condition \\eqref{eq:interp-formula} that $\\mathcal{U}$ be an interpolating basis can be written in matrix form as\n\\begin{equation} \\label{eq:interpolation-formula-1}\nf = UE_\\mathcal{I}^\\textsf{T}f\n\\end{equation}\nfor all $f \\in \\mathbb{Y}$. Here $UE_\\mathcal{I}^\\textsf{T}$ is an $N \\times N$ matrix and we see that \n$\\mathcal{U}$ is an interpolating basis for $\\mathbb{Y}$ with sampling set $\\mathcal{I}$ if and only if $\\mathbb{Y} = {\\rm ker}(I_N-UE_\\mathcal{I}^\\textsf{T})$.\n\nNow we have\n\n\\begin{IEEEproof}[Algebraic Proof of Theorem \\ref{theorem:existence}] \nTake any basis $\\mathcal{V}=\\{v_1,v_2,\\dots,v_d\\}$ of $\\mathbb{Y}$ and let $R$ be the $N \\times d$ matrix whose columns are the basis vectors $v_k$; thus $R_{jk}=v_k(j)$. Since $R$ has rank $d$ it has a $d\\times d$ invertible submatrix, and possibly many such submatrices. Let $\\mathcal{I}$ be the index set corresponding to the $d$ rows chosen from $R$ to form the invertible submatrix $E_\\mathcal{I}^\\textsf{T} R$. The columns of the $N\\times d$ matrix $R(E_\\mathcal{I}^\\textsf{T} R)^{-1}$ are again a basis of $\\mathbb{Y}$. We write them as $u_{i_1}$, $u_{i_2}$, \\dots, $u_{i_d}$, indexed by $\\mathcal{I}$. Since\n\\[\nE_\\mathcal{I}^\\textsf{T}(R(E_\\mathcal{I}^\\textsf{T} R)^{-1}) = (E_\\mathcal{I}^\\textsf{T} R)(E_\\mathcal{I}^\\textsf{T} R)^{-1} = I_d\\,,\n\\]\n the $u_{i_j}$ are as in Proposition \\ref{proposition:characterization}, and hence comprise an interpolating basis of $\\mathbb{Y}$. \n\\end{IEEEproof}\n\nThis proof shows how to produce an interpolating basis provided one can find a $d\\times d$ invertible submatrix $E_\\mathcal{I}^\\textsf{T} R$, indexed by $\\mathcal{I}$. The more such submatrices the more interpolating bases for $\\mathbb{Y}$. On the opposite side, in general not every index set $\\mathcal{I}$ is sampling set for an interpolating basis since, in general,  not every choice of a $d \\times d$ submatrix is invertible. \n\nA slightly different way of arranging the algebraic proof also gives an interpolation formula, making \\eqref{eq:interpolation-formula-1} more explicit.  As above, let $\\mathcal{V}=\\{v_1,v_2,\\dots,v_d\\}$ be a basis of $\\mathbb{Y}$ and let $R$ be the corresponding $N \\times d$ matrix. If $f \\in \\mathbb{Y}$ then\n\\[\nf = \\sum_{k=1}^N f(k)\\delta_k \\quad \\text{and also}\\quad f = \\sum_{k=1}^d \\alpha_k v_k,\n\\]\nfor some constants $\\alpha_k$. We want to solve for the $\\alpha_k$ in terms of $d$ of the values $f(k)$. \nWrite the second equation for $f$ as\n\\[\nf = R{\\alpha}, \\quad \\alpha=(\\alpha_1,\\alpha_2, \\dots, \\alpha_d)^\\textsf{T}.\n\\]\nNow $R$ has an invertible $d \\times d$ submatrix, say $E_\\mathcal{I}^\\textsf{T} R$ for an index set $\\mathcal{I}$, and\nso\n\\[\nE_\\mathcal{I}^\\textsf{T} f = E_\\mathcal{I}^\\textsf{T}(R {\\alpha}) = (E_\\mathcal{I}^\\textsf{T} R){\\alpha}.\n\\]\nWe can then solve for ${\\alpha}$ via\n\\[\n{\\alpha} = (E_\\mathcal{I}^\\textsf{T} R)^{-1}(E_\\mathcal{I}^\\textsf{T} f),\n\\]\nresulting in\n\\begin{equation} \\label{eq:interpolation-formula-2}\nf = R(E_\\mathcal{I}^\\textsf{T} R)^{-1}(E_\\mathcal{I}^\\textsf{T}f).\n\\end{equation}\nThis equation writes $f$ in terms of the components $f(i)$, $i \\in \\mathcal{I}$. \n\n\nCarrying the algebraic line of reasoning a little further, we also see how two interpolating bases for $\\mathbb{Y}$ are related to each other.\n\\begin{theorem} \\label{theorem:matrix-formulation}\nFix an interpolating basis of $\\mathbb{Y}$, indexed by $\\mathcal{J}$, and let $R$ be the corresponding $N \\times d$ matrix. If $S$ is the matrix of another interpolating basis of $\\mathbb{Y}$, indexed by $\\mathcal{I}$, then $E_\\mathcal{I}^\\textsf{T} R$ is invertible and\n\\[\nS=R(E_\\mathcal{I}^\\textsf{T} R)^{-1}\\,.\n\\]\n\\end{theorem}\n\n\\begin{IEEEproof}\nLet $\\{v_i\\colon i\\in\\mathcal{I}\\}$ be the interpolating basis of $\\mathbb{Y}$ that are the columns of $S$ and let $\\{u_j\\colon j\\in\\mathcal{J}\\}$ be the columns of $R$.  Since the $u_j$ are an interpolating basis we can write, for each $i\\in \\mathcal{I}$,\n\\[\nv_i= \\sum_{j\\in\\mathcal{J}} v_i(j) u_j.\n\\]\nIn matrix form this is\n\\[\nS = R(E_\\mathcal{J}^\\textsf{T} S).\n\\]\nNow multiply on the left by $E_\\mathcal{I}^\\textsf{T}$, resulting in\n\\[\nE_\\mathcal{I}^\\textsf{T} S= E_\\mathcal{I}^\\textsf{T}(R(E_\\mathcal{J}^\\textsf{T} S)) = (E_\\mathcal{I}^\\textsf{T} R)(E_\\mathcal{J}^\\textsf{T} S).\n\\]\nBut $E_\\mathcal{I}^\\textsf{T} S$ is the $d\\times d$ identity matrix, so this shows that $E_\\mathcal{I}^\\textsf{T}R$ is invertible,  that\n$(E_\\mathcal{I}^\\textsf{T} R)^{-1} = E_\\mathcal{J}^\\textsf{T} S$, and then that\n\\[\nS=R(E_\\mathcal{I}^\\textsf{T} R)^{-1}.\n\\]\n\\end{IEEEproof} \n\nFinally, we look a little more closely at the interpolating basis provided by \n$R(E_\\mathcal{I}^\\textsf{T} R)^{-1}$ in relation to the geometric construction.\n  From the $d\\times  d$ matrix $(E_\\mathcal{I}^\\textsf{T} R)^{-1}$ form a $d \\times N$ matrix by adding $N-d$ columns of zeros in the slots $\\mathcal{I}'$. Call this matrix $T$. Then $RT$ is an $N \\times N$ matrix and one sees that \n\\[\nRT\\delta_i=\n\\begin{cases}\nu_i,&\\quad i \\in \\mathcal{I}\\\\\n{0},& \\quad i \\in \\mathcal{I}'\n\\end{cases}\n\\] \nThus $RT$ is the projection of $\\mathbb{C}^N$ onto $\\mathbb{Y}$ along $\\mathbb{C}^{\\mathcal{I}'}$ and  we are back to the idea of the geometric argument. Observe that whereas the geometric argument started with a complement $\\mathbb{C}^{\\mathcal{I}'}$ to $\\mathbb{Y}$ and produced the interpolating basis via projection, here we started with an interpolating basis for $\\mathbb{Y}$ and produced the projection and the complement.\n\n\n\n\\section{Discrete Bandlimited Spaces} \\label{section:bandlimited}\n\nBandlimited signals are defined by the vanishing of the discrete Fourier transform  outside a set of specified indices. They form a particularly interesting class of subspaces. \n\nFor notation,  let \n\\[\n\\zeta_n= e^{-2\\pi i \/n},\n\\]\nsimplified to just $\\zeta$ when $n=N$, and let\n$\\omega\\colon\\mathbb{Z}_N \\longrightarrow \\mathbb{C}$ be the discrete complex exponential,\n\\[\n\\omega(m) = \\zeta^m. \n\\]\nThe discrete Fourier transform is then\n\\[\n{{\\mathcal{F}}} f = \\sum_{n=0}^{N-1} f(n) \\omega^{n}.\n\\]\nAs usual, we also regard ${{\\mathcal{F}}}$ as an $N \\times N$ matrix whose $mn$-entry is ${{\\mathcal{F}}}_{mn}=\\omega^{n}(m) = \\zeta^{mn}$.  We recall that ${{\\mathcal{F}}}^{-1} = (1\/N){{\\mathcal{F}}}^*$ (the adjoint of ${{\\mathcal{F}}}$).\n\n\n\n\\begin{defn} \\label{def:bandlimited}\nLet $\\mathcal{J}\\subseteq [0:N-1]$. The $|\\mathcal{J}|$-dimensional space of bandlimited signals with frequency support $\\mathcal{J}$ is\n\\[\n\\mathbb{B}^\\mathcal{J} = {{\\mathcal{F}}}^{-1}(\\mathbb{C}^\\mathcal{J}). \n\\]\n\\end{defn}\n\nIn words, $f \\in \\mathbb{B}^\\mathcal{J}$ if ${{\\mathcal{F}}} f$ has zeros in the slots $\\mathcal{J}'=[1:N]\\setminus \\mathcal{J}$. There might be more zeros of ${{\\mathcal{F}}} f$ for a given $f$ but there are at least these zeros.  We do not assume that the indices in $\\mathcal{J}$ are contiguous, so ${{\\mathcal{F}}} f$ is not necessarily supported on a ``band'' of frequencies, but we maintain the use of the term ``bandlimited'' in all cases. \n\nSince ${{\\mathcal{F}}}^* \\delta_n(m) =\\zeta^{-mn}$, we get a basis for $\\mathbb{B}^\\mathcal{J}$ by  pulling out of ${{\\mathcal{F}}}^*$ the columns indexed by $\\mathcal{J}$. Thus we get an interpolating basis with sampling set $\\mathcal{I}$ if and only if $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}}^*E_\\mathcal{J}$ is invertible, or equivalently if and only if $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is invertible. We prefer to use the latter, with ${{\\mathcal{F}}}$ instead of ${{\\mathcal{F}}}^*$.\n\nFor the remainder of this paper, interpolating systems for bandlimited spaces will be our main concern. Spaces of bandlimited functions having orthogonal interpolating bases are the subject of \\cite{OIS}, but we do have one general observation here: such spaces cannot be too big.\n \\begin{proposition} \\label{prop:N\/2>2}\n If $\\mathbb{B}^\\mathcal{J}$ has an orthogonal interpolating basis then $|\\mathcal{J}| \\le N\/2$.\n \\end{proposition}\n \\begin{IEEEproof}\n Suppose $\\mathbb{B}^\\mathcal{J}$ has an orthogonal interpolating basis indexed by $\\mathcal{I}$. Then $|\\mathcal{I}|=|\\mathcal{J}|$. Let $\\mathcal{I}'=[0\\ : N-1]\\setminus \\mathcal{I}$. By Proposition \\ref{proposition:perturbation} we can write \n \\[\n \\mathbb{B}^{\\mathcal{J}}= {\\rm span}\\{\\delta_i+v_i\\colon i \\in\\mathcal{I}\\} ,\n \\]\nwhere the $v_i$ are orthogonal vectors in $\\mathbb{C}^{\\mathcal{I}'}$,  or some possibly $0$. But none of the $v_i$ can be zero, for ${{\\mathcal{F}}} \\delta_k = \\omega^{-k}$ which never vanishes. There are $|\\mathcal{I}|$ of the $v$'s, and if $|\\mathcal{J}|=|\\mathcal{I}| > N\/2$ then $|\\mathcal{I}'| <N\/2$ and we would have more than $N\/2$ orthogonal vectors in a space of dimension less than $N\/2$.\n\\end{IEEEproof}\n\n\\subsection{Necklaces and Bracelets} \\label{subsection:necklaces-bracelets}\n\n Sampling sets for bandlimited spaces have more algebraic structure than it might appear. Namely, the property of being a sampling set for a particular $\\mathbb{B}^\\mathcal{J}$ is preserved under the action of the dihedral group.  To explain, on $\\mathbb{Z}_N$ we denote the operations of {translation} (by $1$) and {reflection} by $\\tau$ and $\\rho$, respectively:\n\\[\n\\begin{aligned}\n&\\tau \\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{Z}_N, \\quad \\tau(n) = n-1 \\mod N,\\\\\n& \\rho \\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{Z}_N, \\quad \\rho(n) = -n \\mod N .\n\\end{aligned}\n\\]\nThen \n\\[\n\\tau^N = \\rm{id}.\\quad \\rho^2 = \\rm{id} \\quad \\text{and} \\quad \\rho\\tau\\rho = \\tau^{-1} \\quad \\text{or} \\quad (\\rho\\tau)^2 = \\rm{id},\n\\]\nso $\\tau$ and $\\rho$ generate the dihedral group  ${\\rm Dih}_N$. Clearly ${\\rm Dih}_N$ can act on an index set $\\mathcal{I}$ via\n\\[\n\\tau\\mathcal{I} = \\{\\tau(i) \\colon i \\in \\mathcal{I}\\}, \\quad \\rho\\mathcal{I} = \\{\\rho(i) \\colon i \\in \\mathcal{I}\\}.\n\\]\nWe define the \\emph{bracelet} of $\\mathcal{I}$ to be the orbit of $\\mathcal{I}$ under the action of ${\\rm Dih}_N$. The \\emph{necklace} of $\\mathcal{I}$ is  the orbit of $\\mathcal{I}$ under the action of the cyclic subgroup $\\langle \\tau \\rangle$ of ${\\rm Dih}_N$. Think of $\\mathcal{I}\\subset [0:N-1]$ as specifying a pattern of $N$ beads on a loop,  with black beads in the locations in $\\mathcal{I}$ separated by white beads in the locations in the complement $\\mathcal{I}'$, as in Figure \\ref{fig:bracelets}.  A necklace is worn around the neck, and if the cyclic group acts then the spacing of the black and white beads is the same however the necklace is rotated. But a bracelet can be worn on either wrist, introducing a reflection, and the symmetry group is ${\\rm Dih}_N$. See Appendix \\ref{appendix:bracelets} for a formula that counts distinct bracelets, and for references. \n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[scale=0.6]{Fig-1a}\n\t\t\\includegraphics[scale=0.6]{Fig-1b}\t\n        \\caption{Two different bracelets with $N=12$ and $|\\mathcal{I}|=4$. On top the index set is $\\mathcal{I}=\\{0,2,5,7\\}$, on the bottom the index set is $\\mathcal{I} = \\{0,3,5,6\\}$}\n        \\label{fig:bracelets}\n\\end{figure}\n\n\nWith these definitions we now have\n\\begin{proposition} \\label{prop:bracelet}\nIf $\\mathcal{I}$ is a sampling set for $\\mathbb{B}^\\mathcal{J}$ then any index set in the bracelet of $\\mathcal{I}$ is a sampling set for $\\mathbb{B}^\\mathcal{J}$.\n\\end{proposition}\n\n\\begin{IEEEproof}\nLet $\\mathcal{I} = \\{m_1, m_2, \\ldots, m_d\\}$, $\\mathcal{J} = \\{n_1, n_2, \\ldots, n_d\\}$ and let \n$\\mathcal{K}=\\tau\\mathcal{I}$. Then the new\nsubmatrix $E_\\mathcal{K}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is given by\n\n\\begin{align*}\n\\label{eq:brace1}\nE_\\mathcal{K}^T{{\\mathcal{F}}} E_\\mathcal{J}&=\n\\begin{bmatrix}\n\\zeta^{ (m_1-1)n_1}  & \\zeta^{(m_1-1)n_2}   & \\cdots & \\zeta^{(m_1-1)n_d}   \\\\\n\\zeta^{(m_2-1)n_1} & \\zeta^{(m_2-1)n_2} &  \\cdots & \\zeta^{(m_2-1)n_d}\\\\\n \\vdots & \\vdots &  \\ddots & \\vdots \\\\\n\\zeta^{(m_d-1)n_1} & \\zeta^{(m_d-1)n_2}\n & \\cdots & \\zeta^{(m_d-1)n_d}\n\\end{bmatrix}\\\\\n\\vspace{0.1in}\n& = \\begin{bmatrix}\n\\zeta^{m_1n_1} & \\zeta^{m_1n_2} & \\cdots & \\zeta^{m_1n_d} \\\\\n\\zeta^{m_2n_1} & \\zeta^{m_2n_2} &  \\cdots & \\zeta^{m_2n_d}  \\\\\n \\vdots & \\vdots &   \\ddots & \\vdots \\\\\n\\zeta^{m_dn_1}  & \\zeta^{m_dn_2}&   \\cdots & \\zeta^{m_dn_d}\n\\end{bmatrix}  \\times \\\\\n& \\hspace{.35in} \\begin{bmatrix}\n\\zeta^{-n_1}& 0 & 0 & \\cdots & 0 \\\\\n0 & \\zeta^{-n_2} & 0 & \\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & \\zeta^{-n_d}\n\\end{bmatrix}\\\\\n& = (E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}) \\times\\text{an invertible diagonal matrix}.\n\\end{align*}\nHence $E_\\mathcal{K}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is invertible whenever $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}}\nE_\\mathcal{J}$ is, and the same is true for any translation of $\\mathcal{I}$.\n\nNext suppose $\\mathcal{K}$ is obtained by reversing $\\mathcal{I}$, namely $\\mathcal{K} = \\{N-m_1,\nN-m_2, \\ldots, N-m_d\\}$. Then $E_\\mathcal{K}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is just the\nconjugate of $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$, so again, $E_\\mathcal{K}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is\ninvertible whenever $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is.\n\\end{IEEEproof}\n\n\\section{Universal Sampling Sets} \\label{section:universal}\n\nThere is a kind of interchange duality for bandlimited spaces between sampling sets and frequency support sets. On the one hand, the sampling problem is to start with $\\mathbb{B}^\\mathcal{J}$ and ask which index sets $\\mathcal{I}$ are sampling sets. On the other hand, one could also start with an index set $\\mathcal{I}$ and ask which $\\mathbb{B}^\\mathcal{J}$ result from this sampling pattern. These two questions are equivalent.\n\n\\begin{proposition} \\label{prop:interchange}\n$\\mathbb{B}^\\mathcal{J}$ has $\\mathcal{I}$ as a sampling set if and only if $\\mathbb{B}^\\mathcal{I}$ has $\\mathcal{J}$ as a sampling set.\n\\end{proposition}\n\n\\begin{IEEEproof} The subspace $\\mathbb{B}^\\mathcal{J}$ has $\\mathcal{I}$ as a sampling set if and only if $E_\\mathcal{I}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$ is invertible, and this is true if and only if its transpose $E_\\mathcal{J}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{I}$ is invertible.\n\\end{IEEEproof}\n\nThough the sampling problem may seem the more natural one, we will concentrate on the second, equivalent question and ask which frequency patterns, that is which $\\mathbb{B}^\\mathcal{J}$, can arise from a given sampling set $\\mathcal{I}$. It may be that the space $\\mathbb{B}^\\mathcal{J}$ is not known exactly, or that we may have some erroneous estimate $\\tilde{\\mathcal{J}}$ of $\\mathcal{J}$. The  question is whether we can pick sampling locations $\\mathcal{I}$ that are robust for these estimation errors. We will find some interesting phenomena, and the results can easily be translated to apply to the sampling problem.  The extreme case is captured by the following definition.\n\n\\begin{defn} An index set $\\mathcal{I} \\subset [0:N-1]$ is a \\emph{universal sampling set}  if $\\mathcal{I}$ is a sampling set for each $\\mathbb{B}^{\\mathcal{J}}$ with $|\\mathcal{J}| = |\\mathcal{I}|$. \n\\end{defn}\nSee also \\cite{Venk-Bresler} and \\cite{Mishali-Eldar}.\n\nIf $\\mathcal{I}$ is a universal sampling set, then while an interpolating basis of a space $\\mathbb{B}^\\mathcal{J}$ still depends on $\\mathcal{J}$, \\emph{where} the samples are taken does \\emph{not} depend on $\\mathcal{J}$. In Section \\ref{section:maximal} we will show that there are universal sampling sets of any given size; in fact, we will count them.\n\nVery concretely, to ask if $\\mathcal{I}$ is a universal sampling set is to ask if there are rows of ${{\\mathcal{F}}}$ indexed by $\\mathcal{I}$, $|\\mathcal{I}|=d$, such that \\emph{any} $d \\times d$ submatrix of ${{\\mathcal{F}}}$ formed with these rows is invertible. Phrased this way, standard properties of Vandermonde determinants applied to ${{\\mathcal{F}}}$ allow us to conclude:\n\\begin{proposition} \\label{prop:universal-1}\n(i) If $\\mathcal{I}$ is a set of $d$ consecutive indices, reduced mod $N$, \n\\[\n\\mathcal{I}=\\{i_0,i_0+1,\\dots, i_0+(d-1)\\} \\mod N,\n\\]\n then $\\mathcal{I}$ is a universal sampling set.\n\n(ii) If $\\mathcal{I}$ is a set of $d$ indices in arithmetic progression, reduced mod $N$,\n\\[\n\\mathcal{I}=\\{i_0, i_0+s, i_0+2s,\\dots, i_0+(d-1)s\\} \\mod N, \n\\]\nwhere $s$ is coprime to $N$, then $\\mathcal{I}$ is a universal sampling set.\n\\end{proposition} \\hspace*{\\fill}~{\\rule{2mm}{2mm}}\\par\\endtrivlist\\unskip\n\nMuch deeper is the following theorem of Chebotarev.\n\n\\begin{theorem}\n(Chebotarev) If $N$ is prime, then every square submatrix of  ${{\\mathcal{F}}}$ is invertible.\n\\end{theorem} \n\nAnd so, if $N$ is prime then any index set $\\mathcal{I}$ is a universal sampling set.  Chebotarev's theorem dates to 1948 (the original paper is in Russian) and there are now several published (and unpublished) proofs, see, e.g., \\cite{prasolov:linear-algebra}, \\cite{Frenkel:Chebotarev}, but this is by no means a trivial result.  \n\n\n\nWe will generalize Chebotarev's theorem when $N$ is a prime power, and we will offer several characteristic properties of universal sampling sets. We are indebted to the works of Tao \\cite{tao:uncertainty} and Delvaux and Van Barel \\cite{delvaux:rank-defficient}.\n\nThe key is a quantitative, almost statistical comparison of $\\mathcal{I}$ to the simplest universal sampling set,\n\\[\n\\mathcal{I}^* = [0 : d-1], \n\\]\nwhen the  elements of both $\\mathcal{I}$ and $\\mathcal{I}^*$ are reduced modulo prime powers.  We need several additional definitions to state our main results. \n\n\\subsection{Multisets and the Size of Congruence Classes} \\label{subsection:multisets}\n\nWe have found it conceptually helpful to use \\emph{multisets} in the description of one of the central ideas, and we briefly review this concept. Informally, a multiset is a finite, unordered list $\\widetilde{A}$ whose elements are drawn from a finite set $A$, and where, to distinguish a multiset from simply a set,  elements of the list may be repeated. More formally, a multiset  is a pair $(A, \\widetilde{\\chi}_A)$ where $\\widetilde{\\chi}_A$ is the multiplicity function (generalizing the characteristic function):\n\\[\n\\begin{aligned}\n&\\widetilde{\\chi}_A \\colon A \\longrightarrow \\mathbb{N}, \\\\\n\\widetilde{\\chi}_A(a) &= \\text{the number of times $a\\in A$ is listed in $\\widetilde{A}$}.\n\\end{aligned}\n\\]\nTwo multisets $\\widetilde{A}$ and $\\widetilde{B}$ are equal if $\\widetilde{\\chi}_A = \\widetilde{\\chi}_B$, so the individual elements are the same and so are their multiplicities. The cardinality of $\\widetilde{A}$ is\n\\[\n|\\widetilde{A}| = \\sum_{a\\in A} \\widetilde{\\chi}_A(a).\n\\]\n\n\n It is common practice to use the standard set notation in writing a multiset. Thus, for example, drawing from $\\{a,b,c,d\\}$ we write a multiset as $\\{ a,a,c,d,d\\}$. The tilde notation $\\widetilde{A}$ for a multiset drawn from $A$ is helpful in discussing general principles but, like all general notations, it has its limitations in particular cases. It is a notation often used for covering spaces, as we comment on below.\n\nAssociated with a multiset $\\widetilde{A}$ is another multiset \n\\[\n\\mathcal{M}(\\widetilde{A}) = \\{\\widetilde{\\chi}_A(a) \\colon a \\in A\\},\n\\]\nwhich we call the \\emph{multiplicity multiset} of $\\widetilde{A}$. Thus $\\mathcal{M}(\\widetilde{A})$ records as a multiset the counts of the elements of $\\widetilde{A}$ and also includes a zero for each element of $A$ that does not appear in $\\widetilde{A}$.  One can think of $\\mathcal{M}(\\widetilde{A})$ as providing some statistics of $\\widetilde{A}$, a kind of histogram of $\\widetilde{A}$  with bins from $A$, except that the bins are not ordered. \n\n\nNext, let $p$ be a prime, $k \\ge 0$ an integer, and for $x \\in \\mathbb{N}$ let $[x]_k$  be the \nresidue of $x$ reduced mod $p^k$.   \nFor an index set $\\mathcal{I}$ let \n\\[\n\\mathcal{I}\/{p^k}=\\{[i]_k \\colon i \\in \\mathcal{I}\\}\n\\]\n be the set of  \n residues mod $p^k$ of the elements of $\\mathcal{I}$, and let   \n $(\\mathcal{I}\/p^k)^\\sim$ be the corresponding multiset, meaning that each residue is listed according to its multiplicity, i.e, the size of its congruence class. We regard the elements of  $(\\mathcal{I}\/p^k)^\\sim$  \n to be drawn from $[0:p^k-1]$, all possible residues, and we write $\\widetilde{\\chi}_k \\colon [0:p^k-1] \\longrightarrow \\mathbb{N}$ for the multiplicity function for the multiplicity multiset $\\mathcal{M}( (\\mathcal{I}\/p^k)^\\sim)$. To be explicit, for $a \\in[0:p^k-1]$\n\\begin{equation} \\label{eq:chi_k}\n\\begin{aligned}\n\\widetilde{\\chi}_k(a) = &\\,\\text{the number of elements of $\\mathcal{I}$}\\\\ \n&\\,\\text{that leave a remainder of $a$ on dividing by $p^k$.}\n\\end{aligned}\n\\end{equation}\n\nIn particular,  $\\widetilde{\\chi}_k(a)=0$ means that no element of $\\mathcal{I}$ leaves a remainder of $a$ on dividing by $p^k$. In this case we speak of an empty congruence class in $\\mathcal{I}\/p^k$. For $ a \\in [0:p^k-1]$ it will also be helpful to use the notation\n\\[\n\\mathcal{I}_{ka} = \\{i\\in\\mathcal{I} \\colon i \\equiv a \\text{ mod  $p^k$}\\}    \n\\]\nfor the elements of the congruence class of $a$ mod $p^k$.  Then $\\widetilde{\\chi}_k(a) = |\\mathcal{I}_{ka}|$. \n\nWhen we need to emphasize the index set, especially in Section \\ref{section:maximal}, we will write $\\widetilde{\\chi}_k(a\\,;\\mathcal{I})$. We note the obvious properties:\n\\begin{itemize}\n\\item If $\\mathcal{I}$ and $\\mathcal{J}$ are disjoint then $\\widetilde{\\chi}_k(a\\,;\\mathcal{I} \\cup \\mathcal{J}) = \\widetilde{\\chi}_k(a\\,;\\mathcal{I}) + \\widetilde{\\chi}_k(a\\,;\\mathcal{J})$.\n\\item  $\\mathcal{I} \\subseteq \\mathcal{J} \\implies \\widetilde{\\chi}_k(a\\,;\\mathcal{I}) \\le \\widetilde{\\chi}_k(a\\,;\\mathcal{J})$.\n\\end{itemize}\n\n\n\nObserve for $k=0$ that $(\\mathcal{I}\/1)^\\sim$ just consists of $|\\mathcal{I}|$ zeros and $\\widetilde{\\chi}_0(0)=|\\mathcal{I}|$. More generally,\n\\begin{equation} \\label{eq:cardinality-chi}\n| \\mathcal{I} | = \\sum_{a=0}^{p^k-1} \\widetilde{\\chi}_k(a).\n\\end{equation}\n We also note that the multiplicity multiset $\\mathcal{M}((\\mathcal{I}\/p^k)^\\sim)$ depends only on the bracelet of $\\mathcal{I}$. While the multisets $(\\mathcal{I}\/p^k)^\\sim$ will generally change if $\\mathcal{I}$ is shifted or reversed, the counts of the residues on dividing by $p^k$ will be the same:\n\\begin{equation} \\label{eq:multiplicity-bracelet}\n\\begin{aligned}\n\\mathcal{M}((\\tau\\mathcal{I}\/p^k)^\\sim) & = \\mathcal{M}((\\mathcal{I}\/p^k)^\\sim) \\quad \\text{and}\\\\\n\\mathcal{M}((\\rho\\mathcal{I}\/p^k)^\\sim) & = \\mathcal{M}((\\mathcal{I}\/p^k)^\\sim).\n\\end{aligned}\n\\end{equation}\n\n\\begin{remark} \\label{remark:covering-spaces} Introducing the multiset $(\\mathcal{I}\/p^k)^\\sim$ is reminiscent of introducing covering spaces (for Riemann surfaces) to resolve the problem of multivalued functions. Here we have the remainder map $r\\colon\\mathcal{I} \\longrightarrow \\mathcal{I}\/p^k$, $r(i)=[i]$, which is generally not injective and so has a multivalued inverse. Think of the residues (with multiplicity) in $(\\mathcal{I}\/p^k)^\\sim$ as tagged by the number they come from, say as a pair $([i],i)$, which serves to distinguish them much as we think of tagging points on different sheets of a covering space of a Riemann surface. Then we have the commutative diagram\n\\[\n\\xymatrix{\n\t\t& (\\mathcal{I}\/p^k)^\\sim \\ar[d]^{\\text{pr}}\\\\\n\t\t\\mathcal{I} \\ar[ur]^{\\widetilde{r}} \\ar[r]_r &\\mathcal{I}\/p^k  }\n\\]\t\t\nwhere $\\text{pr}$ is the projection map, $([i],i) \\mapsto [i]$ and the  lift $\\widetilde{r}(i) =([i],i)$, of $r$  is bijective.  \nThe value of the multiplicity function $\\widetilde{\\chi}_k(i)$ is then the number of elements in the preimage $\\text{pr}^{-1}([i])$, \nanalogous to the number of sheets over $[i]$. It will generally vary with $[i]$.\n\n\\end{remark}\n\nReturning to our primary considerations, we write $\\widetilde{\\chi}_k^*$ to distinguish the special case when $\\mathcal{I}=\\mathcal{I}^*$.   We will need the following property of $\\widetilde{\\chi}_k^*$: \n\\begin{equation} \\label{eq:chi-I*}\n|\\widetilde{\\chi}_k^{*}(a) - \\widetilde{\\chi}_k^*(b)| \\leq 1,\n\\end{equation}\nfor all $a,b \\in [0:p^k-1]$ and all $k$. In words, when reducing the elements of $\\mathcal{I}^*=[0:d-1]$ modulo $p^k$ for any $k$, the conjugacy classes are all of about the same size.  Or, pursuing the analogy above, the preimages $\\text{pr}^{-1}([i])$ of the individual residues all have approximately the same number of elements and one might say that $\\mathcal{I}^*\/p^k$ is \\emph{uniformly covered} for each $k$. \n\nThe inequality in  \\eqref{eq:chi-I*} is easy to see. For some background calculations we have found it helpful to have a formula  for $\\widetilde{\\chi}_k^*$ (from which \\eqref{eq:chi-I*} also follows).  If $\\ell \\in \\mathcal{I}^*$ with $[\\ell]_k= a \\in [0:p^k-1]$ then  $\\ell = a + \\alpha p^k$ for an integer $\\alpha \\ge 0$, and since $\\ell \\le d-1$ we must have $0 \\le \\alpha \\le (d-1-a)\/p^k$. The number of integers $\\alpha$ for which this inequality holds is the number of $\\ell$ whose residue is $a$. Thus\n\\begin{equation} \\label{eq:chi^*-explicit}\n\\widetilde{\\chi}_k^*(a) = \\left\\lfloor \\frac{d-1-a}{p^k} +1\\right\\rfloor.\n\\end{equation}\n\n\n\\subsection{A Characterization of Universal Sampling Sets} \\label{subsection:universal-characterization}\n\nOur main result is:\n\n\\begin{theorem} \\label{theorem:universal}\nLet $\\mathcal{I}$ be an index set in $[0:p^M-1]$. The following are equivalent:\n\\begin{enumerate}\n\\item[(i)]  $\\widetilde{\\chi}_k = \\widetilde{\\chi}_k^*$ \n for all $0 \\le  k \\le M$.\n\\item[(ii)] $|\\widetilde{\\chi}_k(a)-\\widetilde{\\chi}_k(b)| \\le 1$ for all $a,b \\in [0:p^k-1]$ and $0 \\le k \\le M$.\n\\item[(iii)] $\\mathcal{I}$ is a universal sampling set.\n\\end{enumerate}\n\\end{theorem}\n\n\n\nAccording to Proposition \\ref{prop:bracelet} and the relations \\eqref{eq:multiplicity-bracelet}, any index set in the bracelet of $\\mathcal{I}$ is also a universal sampling set. Likewise, any index set in the bracelet of $\\mathcal{I}^*$ can serve as a model universal sampling set. Only condition (i) directly compares $\\mathcal{I}$ to $\\mathcal{I}^*$, and in terms of multisets it could be stated equivalently as\n\\[\n\\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim) = \\mathcal{M}(({\\mathcal{I}^*\/p^k})^\\sim).\n\\]\n Condition (i) for $k=0$ guarantees that $\\mathcal{I}$ and $\\mathcal{I}^*$ have the same size, from \\eqref{eq:cardinality-chi}. Computing $ \\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim)$ and  $\\mathcal{M}(({\\mathcal{I}^*\/p^k})^\\sim)$ for $k \\ge M$ is redundant; since all elements in $\\mathcal{I}$ and $\\mathcal{I}^*$ are in $[0:p^M-1]$, $\\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim)$ for $k \\ge M$ is just indicative of the cardinality of $\\mathcal{I}$ and $\\mathcal{I}^*$. Namely, for $k \\ge M$,  each of $\\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim)$ and $\\mathcal{M}(({\\mathcal{I}^*\/p^k})^\\sim)$ contains $|\\mathcal{I}|$ ones and $p^k-|\\mathcal{I}|$ zeros.  Condition (ii), a property only of $\\mathcal{I}$, indirectly compares $\\mathcal{I}$ to $\\mathcal{I}^*$ via \\eqref{eq:chi-I*}. It says that  $\\mathcal{I}\/p^k$, like $\\mathcal{I}^*\/p^k$, is uniformly covered for each $k$.\n\n\n\nBefore we embark on the proof of the theorem, here is an example. Let $N=2^3$, and $\\mathcal{I} = \\{ 0, 1, 3, 4, 6 \\}$. The following are the multisets for $k=1,2,3$:\n\\[\n\\begin{aligned}\n&({\\mathcal{I}\/2})^\\sim = \\{0, 1, 1, 0, 0\\},  \\quad \\mathcal{M}(({\\mathcal{I}\/2})^\\sim)= \\{3, 2\\}; \\nonumber\\\\\n& ({\\mathcal{I}\/2^2})^\\sim= \\{0, 1, 3, 0, 2\\},  \\quad \\mathcal{M}(({\\mathcal{I}\/2^2})^\\sim) = \\{2, 1, 1, 1\\};\\nonumber\\\\\n&({\\mathcal{I}\/2^3})^\\sim = \\{0, 1, 3, 4, 6\\}, \\\\\n& \\hspace{.3in} \\mathcal{M}(({\\mathcal{I}\/2^3})^\\sim) = \\{1, 1, 0, 1, 1, 0, 1, 0\\} \\nonumber.\n\\end{aligned}\n\\]\n\nThe computations for $\\mathcal{I}^*=\\{0,1,2,3,4\\}$ yield\n\\[\n\\begin{aligned}\n&({\\mathcal{I}^*\/2})^\\sim = \\{0, 1, 0, 1, 0\\}, \\quad \\mathcal{M}(({\\mathcal{I}^*\/2})^\\sim) = \\{3, 2\\};\\nonumber\\\\\n&({\\mathcal{I}^*\/2^2})^\\sim = \\{0, 1, 2, 3, 0\\}, \\quad  \\mathcal{M}(({\\mathcal{I}^*\/2^2})^\\sim) = \\{2, 1, 1, 1\\}; \\nonumber\\\\\n&({\\mathcal{I}^*\/2^3})^\\sim = \\{0, 1, 2, 3, 4\\}, \\\\\n& \\hspace{.3in}  \\mathcal{M}(({\\mathcal{I}^*\/2^3})^\\sim) = \\{1, 1, 1, 1,1 , 0, 0, 0\\}. \\nonumber\n\\end{aligned}\n\\]\nWe see that $ \\mathcal{M}(({\\mathcal{I}\/2^k})^\\sim) =  \\mathcal{M}(({\\mathcal{I}^*\/2^k})^\\sim)$ for $k=1,2,3$, and hence $\\mathcal{I}$ is a universal sampling set. So in case the reader has ever wondered, for the $8 \\times 8$ Fourier matrix any $5\\times 5$ submatrix built from the rows indexed by $\\mathcal{I}$, or from the rows of an index set in the bracelet of $\\mathcal{I}$, is invertible.\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{theorem:universal}, (i) $\\Longleftrightarrow$ (ii)] Note: This equivalence does not require that $N$ be a prime power.\nThe implication (i) $\\implies$ (ii) is immediate from \\eqref{eq:chi-I*}. Assume (ii) holds and \nlet \n\\[\n\\chi = \\min_a\\widetilde{\\chi}_k(a).\n\\]\nFrom (ii) it follows that any $\\widetilde{\\chi}_k(a)$ is either $\\chi$ or $\\chi+1$. Suppose $r$ of the $p^k$ numbers\n$\\widetilde{\\chi}_k(a) $ are equal to $\\chi + 1$ and the rest\nare equal to $\\chi$. The cardinality equation, \\eqref{eq:cardinality-chi},\n\\begin{equation}\n\\label{eq:univ-alt-1}\n\\sum_{a=0}^{p^k-1}  \\widetilde{\\chi}_k(a) = |\\mathcal{I}|=d,\n\\end{equation}\n then gives\n\\[\np^k \\chi + r = d, \\quad \\text{ with } \\quad 0\\leq r < p^k.\n\\]\nThis means that $\\chi$ is the quotient on dividing $d$ by $p^k$ and\n$r$ is the remainder. In other words, (ii) and\n\\eqref{eq:univ-alt-1} together uniquely determine the multiset $\\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim) = \\{\n\\widetilde{\\chi}_k(a)\\colon a \\in [0,p^k-1] \\}$. Since $\\mathcal{I}$ and $\\mathcal{I}^*$ both satisfy (ii) and\n\\eqref{eq:univ-alt-1}, we must have $\\mathcal{M}(({\\mathcal{I}\/p^k})^\\sim) = \\mathcal{M}(({\\mathcal{I}^*\/p^k})^\\sim)$, or $\\widetilde{\\chi}_k = \\widetilde{\\chi}_k^*$.\n\\end{IEEEproof}\n\n\n\n \n \n\nWe need two lemmas to prove that condition (i) implies that $\\mathcal{I}$ is a universal sampling set. The first is a very old theorem on Vandermonde determinants, \\cite{Mitchell:determinants},  as updated in \\cite{delvaux:rank-defficient}: \n\n\\begin{lemma}[Delvaux and Van Barel]\n\\label{thm:genvan}\nLet \n\\begin{equation}\n\\label{eq:genvan1}\nV=\n\\begin{bmatrix}\n x_1^{m_1} & x_2^{m_1} & x_3^{m_1} & \\cdots & x_d^{m_1} \\\\\n x_1^{m_2} & x_2^{m_2} & x_3^{m_2} & \\cdots & x_d^{m_2} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n x_1^{m_d} & x_2^{m_d} & x_3^{m_d} & \\cdots & x_d^{m_d}\n\\end{bmatrix}\n\\end{equation}\nbe a $d \\times d$ generalized Vandermonde matrix. Then the determinant of $V$ is given by\n\\begin{equation}\n\\label{eq:genvan}\n\\det V = \\left(\\prod_{i<j}(x_j - x_i)\\right)S(x_1, x_2, \\ldots, x_d), \n\\end{equation}\nwhere $S(x_1, x_2, \\ldots, x_d)$ is a symmetric polynomial in $x_1, x_2, \\ldots x_d$ with integer coefficients such that \n\\[\nS(1,1,\\ldots,1) =  \\frac{\\prod_{0\\leq i <j \\leq d-1}(m_j-m_i)}{\\prod_{0\\leq i <j \\leq d-1}(j-i)}.\n\\]\n\\end{lemma}\n\nThe polynomial $S$ is called a \\emph{Schur polynomial}, see, for example, \\cite{stanley:enumerative-combinatorics}. Based on this lemma we deduce a second result that is itself already a sufficient condition for an index set to be a universal sampling set.\n\\begin{lemma}\n\\label{lem:univ-suff}\nLet $\\mathcal{I} = \\{m_0, m_1, m_2, \\ldots, m_{d-1}\\}$. If \n\\begin{equation}\n\\label{eq:magic-frac}\n\\mu= \\frac{\\prod_{0\\leq i <j \\leq d-1}(m_j-m_i)}{\\prod_{0\\leq i <j \\leq d-1}(j-i)}\n\\end{equation}\nis coprime to $p$, then $\\mathcal{I}$ is a universal sampling set.\n\\end{lemma}\n\nNote that without Lemma \\ref{thm:genvan}, it would not even be clear that $\\mu$ is an integer. An intuitive idea for why this should be so is given below. The proof of Lemma \\ref{lem:univ-suff} is along the  lines of the proof of Chebotarev's theorem in \\cite{evans:generalized-vandermonde}, and also in \\cite{tao:uncertainty}.\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:univ-suff}]  We make use of Lemma \\ref{thm:genvan} in the case when $V = E_{\\mathcal{I}}^T \\mathcal{F} E_{\\mathcal{J}} $. Each $x_\\ell$ in \\eqref{eq:genvan1} is then a power of $\\zeta=e^{-2\\pi i \/N}$, $x_\\ell = \\zeta^{j_\\ell}$, where $\\mathcal{J} = \\{j_1,j_2,\\ldots,j_d\\}$. \n\nSuppose $\\det V = 0$. From \\eqref{eq:genvan}, this means that $S(x_1, x_2, \\ldots, x_d) = 0$. Substituting $x_\\ell = \\zeta^{j_\\ell}$ in $S(x_1, x_2, \\ldots, x_d) = 0$, we obtain an equation of the form $s(\\zeta) = 0$, where $s(x)$ is a polynomial in one variable with integer coefficients. This means that $\\zeta$ is a root of $s(x)$ and since $s(x)$ has only integer coefficients, $s(x)$ must contain the minimal polynomial of $\\zeta$ over $\\mathbb{Z}$ as a factor. \n\nFor $N=p^M$, the minimal polynomial of $\\zeta$ over $\\mathbb{Z}$ is  $\\phi_N(x) = 1 + x^{p^{M-1}} + x^{2p^{M-1}} + x^{3p^{M-1}} + \n\\cdots +x^{(p-1)p^{M-1}}$ (the $N$'th cyclotomic polynomial). So we have $\\phi_N(x) \\mid s(x)$, where\n\\[\n\\phi_N(x) = 1 + x^{p^{M-1}} + x^{2p^{M-1}} + x^{3p^{M-1}} + \\cdots +x^{(p-1)p^{M-1}} . \n\\]\nNow $\\phi_N$ and $s$ are both polynomials with integer coefficients, hence  $\\phi_N(1) \\mid s(1)$. However, $\\phi_N(1) = p$, and $s(1) = S(1,1,\\ldots,,1) = \\mu$. Thus \n\\[\np \\mid  \\mu \\, \\text{ if $\\det V = 0$}.\n\\]\nThis proves the lemma.\n\\end{IEEEproof}\n\n\n\nChebotarev's theorem follows from this result. If $N$ is a prime $p$ then $\\mu$ is coprime to $p$ because every factor in the numerator and denominator of $\\mu$ is an integer strictly between $-p$ and  $p$. \n\nWe can now complete the proof of one direction of the implications in Theorem \\ref{theorem:universal}. \n\\begin{IEEEproof}[Proof of Theorem \\ref{theorem:universal}: (i) $\\implies$ (iii)] \nLet $\\mathcal{I} = \\{m_1,m_2,m_3,\\ldots,m_d\\}$ and consider the product of differences\n\\[\nA=\\prod_{1\\leq i < j \\leq d}(m_j - m_i). \n\\]\n \nThere are $\\widetilde{\\chi}_k(\\ell)$ elements of $\\mathcal{I}$ that leave a remainder of $\\ell$ when divided by $p^k$. Moreover, $m_i \\equiv m_j \\mod p^k$ if and only if  $p^k \\mid (m_j - m_i)$. The number of differences that have a factor of $p^k$ (or higher: $p^r$ for $r>k$) is \n\\[\n\\sum_{l=0}^{p^k-1} \\binom{\\widetilde{\\chi}_k(l)}{2},\n\\]\nand hence the number of differences that have a factor of exactly $p^k$ is given by \n\\[\n\\sum_{l=0}^{p^k-1} \\binom{\\widetilde{\\chi}_k(l)}{2} - \\sum_{l=0}^{p^{k+1}-1} \\binom{\\widetilde{\\chi}_{k+1}(l)}{2}.\n\\]\nThe largest power of $p$ that divides $A$ is then $p$ raised to\n\\begin{equation} \\label{eq:largest-power-of-p}\n{\\sum_k k\\left( \\sum_{l=0}^{p^k-1} \\binom{\\widetilde{\\chi}_k(l)}{2} - \\sum_{l=0}^{p^{k+1}-1} \\binom{\\widetilde{\\chi}_{k+1}(l)}{2}\\right)} .\n\\end{equation}  \nThe expression \\eqref{eq:largest-power-of-p} depends only on the values of $\\widetilde{\\chi}_k$, but the hypothesis is that   $\\widetilde{\\chi}_k = \\widetilde{\\chi}_k^*$ for $0 \\le k \\le N$, and therefore the products  $A=\\prod(m_j - m_i)$ and $B = \\prod(j-i)$ have the same powers of $p$ as factors. Hence  $\\mu = A\/B$ is coprime to $p$ and from  Lemma \\ref{lem:univ-suff} we conclude that $\\mathcal{I}$ is a universal sampling set.\n\\end{IEEEproof}\n\n\\begin{remark} \nThe argument above  also gives an insight, if not a proof, as to why $\\mu=A\/B$ in \\eqref{eq:magic-frac} is an integer. Suppose  $\\mathcal{M}((\\mathcal{I}\/p^k)^\\sim)= \\{r_1, r_2, r_3, \\ldots, r_d\\}$. The power of $p^k$ in $A = \\prod (m_i - m_j)$ is given by \n\\[\n\\sum_{i=1}^d {r_i \\choose 2} = \\frac{1}{2}\\left(\\sum_{i=1}^d r_i^2 - \\sum_{i=1}^d r_i\\right).\n\\]\nNow, $\\sum_{i=1}^d r_i$ is the cardinality of $\\mathcal{I}$ so\n\\[\n\\sum_{i=1}^d r_i = d.\n\\]\nHence for a set $\\mathcal{I}$ which has the minimum power of $p^k$ in $A$ it must be that $\\mathcal{M}((\\mathcal{I}\/p^k)^\\sim) = \\{r_1, r_2,\\ldots r_d\\}$ is a solution to  \n\\begin{align*}\n&\\text{minimize }r_1^2 + r_2^2 + \\cdots +r_d^2 \\\\\n&\\text{subject to }r_1 + r_2 + \\cdots r_d = d.\n\\end{align*}\nOn the reals the optimal solution satisfies $r_1 = r_2 = \\cdots = r_d$. This suggests that the set $\\mathcal{I}$ with the smallest power of $p^k$ in $A$ must have roughly an equal number of elements in each congruence class. $\\mathcal{I}^* = \\{0,1,2,\\ldots, d-1\\}$ is one such set. Thus  the power of $p^k$ is smaller in $B = \\prod(i-j)$ than in $A = \\prod (m_i - m_j)$ for each $p$ and $k$, and, if the reasoning is to trusted, $\\mu = A\/B$ is an integer.\n\n\\end{remark}\n\n\nTo finish the proof of Theorem \\ref{theorem:universal} we will derive the following bounds on $\\widetilde{\\chi}_k$. \n\n\\begin{lemma} \\label{lemma:chi-upper-and-lower}\nIf $\\mathcal{I}\\subseteq [0:p^M-1]$ is a universal sampling set of size $d$ then\n\\begin{equation} \\label{eq:chi-upper-and-lower}\n\\left\\lfloor \\frac{d}{p^k}\\right\\rfloor \\le \\widetilde{\\chi}_k(s) \\le \\left\\lceil \\frac{d}{p^k}\\right\\rceil, \\quad s\\in[0:p^k-1]\\,, 0 \\le k \\le M.\n\\end{equation}\n\\end{lemma}\n\nIt follows immediately from \\eqref{eq:chi-upper-and-lower} that if $\\mathcal{I}$ is a universal sampling set then\n\\[\n|\\widetilde{\\chi}_k(a) - \\widetilde{\\chi}_k(b)| \\le 1, \\quad a, b \\in [0:p^k-1].\n\\]\nThis is condition (ii), and with this result the proof of Theorem \\ref{theorem:universal} will be complete.  Incidentally, for the case  $\\mathcal{I}=\\mathcal{I}^*$, \\eqref{eq:chi-upper-and-lower} is a simple consequence of \\eqref{eq:chi^*-explicit} and \\eqref{eq:chi-I*}.\n\n\n\n\nThe argument for Lemma \\ref{lemma:chi-upper-and-lower} is through constructing submatrices of the Fourier matrix of known rank to obtain upper and lower bounds for $\\widetilde{\\chi}_k$. The first step is to build a particular model submatrix, and this requires some  bookkeeping. \n\nLet $\\mathcal{I} \\subseteq [0:p^M-1]$,  at this point not assumed to be a universal sampling set. Fix $k\\le M$ and $s\\in[0:p^k-1]$, and recall that we let\n\\[\n\\mathcal{I}_{ks} = \\{ i \\in \\mathcal{I} \\colon i \\equiv s \\text{ mod $p^k$}\\}.\n\\]\nThe set $\\mathcal{I}_{ks}$ has $\\widetilde{\\chi}_k(s)$ elements. List them, in numerical order, as $i_0, i_1,i_2,\\dots, i_c$, where we put $c=\\chi_k(s)-1$ to simplify notation. Let $r$ be a positive integer and define the column vector of length $c$ by \n\\[\n \\mathfrak{z}^r = \\begin{bmatrix}\n\\zeta_{N}^{i_0r} & \\zeta_{N}^{i_1r} & \\zeta_{N}^{i_2r} & \\cdots & \\zeta_N^{i_cr}\n\\end{bmatrix}^\\textsf{T}.\n\\]\n\nNow let $\\mathfrak{Z}^r$ be the $c \\times p^k$ matrix obtained by repeating $p^k$ copies of the column $\\mathfrak{z}^r$:\n\\[\n\\mathfrak{Z}^r =\\underbrace{\\begin{bmatrix} \\mathfrak{z}^r  & \\mathfrak{z}^r & \\mathfrak{z}^r & \\cdots \\mathfrak{z}^r \\end{bmatrix}}_\\text{$p^k$ times},\n\\]\nand let $\\mathfrak{D}^s$ be the $p^k \\times p^k$ diagonal matrix\n\\[\n\\mathfrak{D}^s =\n\\begin{bmatrix}\n1 & 0 & 0 & \\dots & 0\\\\\n0 & \\zeta_{p^k}^{s} & 0  \\dots  & 0\\\\\n0 & 0 & \\zeta_{p^k}^{2s} & \\dots & 0\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & \\zeta_{p^k}^{(p^k-1)s}\n\\end{bmatrix}  .\n\\]\n\nFinally,  let $k'=M-k$, and set \n\\begin{equation} \\label{eq:JJ_k'r}\n\\mathcal{J}_{k'r} = \\{0\\cdot p^{k'}+r, 1\\cdot p^{k'} +r, 2\\cdot p^{k'}+r, \\dots, (p^k-1)p^{k'}+r\\} .\n\\end{equation}\nFrom the Fourier matrix ${{\\mathcal{F}}}$ we choose $c$ rows indexed by $\\mathcal{I}_{ks}$ and $p^k$ columns indexed by $\\mathcal{J}_{k'r}$. The result of these choices, we claim, results in\n\\begin{equation} \\label{eq:F_rs-1}\nE_{\\mathcal{I}_{ks}}^\\textsf{T} {{\\mathcal{F}}} E_{\\mathcal{J}_{k'r}} = \\mathfrak{Z}^r\\mathfrak{D}^s.\n\\end{equation}\n\n\nAfter the preparations, the derivation of \\eqref{eq:F_rs-1} is straightforward. The $(a,b)$-entry of $E_{\\mathcal{I}_{ks}}^\\textsf{T} {{\\mathcal{F}}} E_{\\mathcal{J}_{k'r}}$ is\n\\[\n\\begin{aligned}\n\\zeta_{N}^{i_a(bp^{k'}+r)}&=\\exp\\left(-\\frac{(2\\pi i)i_a(bp^{M-k}+r)}{p^M}\\right)\\\\\n&= \\exp\\left(-\\frac{(2\\pi i ) i_ab}{p^k}\\right)\\exp\\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right) .\n\\end{aligned}\n\\]\nBut now recall that, by definition, when $i_a\\in \\mathcal{I}_{ks}$ is divided by $p^k$ it leaves a remainder of $s$, and thus\n\\[\n\\begin{aligned}\n&\\exp\\left(-\\frac{(2\\pi i ) i_ab}{p^k}\\right)\\exp\\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right)\\\\\n &= \\exp\\left(-\\frac{(2\\pi i ) sb}{p^k}\\right)\\exp\\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right) \\\\\n&=\\zeta_{p^k}^{sb}\\,\\zeta_N^{i_a r}.\n\\end{aligned}\n\\]\n\n\nThis construction is the basis for the proof of Lemma \\ref{lemma:chi-upper-and-lower}, but applied in block form.\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lemma:chi-upper-and-lower}]  To deduce the upper bound $\\widetilde{\\chi}(s) \\le \\lceil d\/p^k\\rceil$ we begin by letting\n\\[\n\\mathcal{J}= \\mathcal{J}_{k'0}\\cup \\mathcal{J}_{k'1} \\cup \\mathcal{J}_{k'2} \\cup \\cdots \\cup \\mathcal{J}_{k'd'}, \\quad d'=\\left\\lceil \\frac{d}{p^k}\\right\\rceil -1,\n\\]\nwhere $\\mathcal{J}_{k'r}$ is defined as in \\eqref{eq:JJ_k'r}. Note that $\\mathcal{J}$ is a union of $\\lceil d\/p^k\\rceil$ disjoint sets. Each $\\mathcal{J}_{k'r}'$, $0\\le r \\le d'=\\lceil d\/p^k\\rceil-1$ indexes the choice of $p^k$ columns from ${{\\mathcal{F}}}$ and applying \\eqref{eq:F_rs-1} we have\n\\[\n\\begin{aligned}\n&E_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}= \\\\\n&\\begin{bmatrix}\nE_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_{\\mathcal{J}_{k'0}} & E_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_{\\mathcal{J}_{k'1}} & \\cdots & E_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_{\\mathcal{J}_{k'd'}}\n\\end{bmatrix}\\\\\n&=\n\\begin{bmatrix}\n\\mathfrak{Z}^0\\mathfrak{D}^s & \\mathfrak{Z}^1\\mathfrak{D}^s &\\cdots & \\mathfrak{Z}^{d'}\\mathfrak{D}^s\n\\end{bmatrix}\\\\\n&=\n\\begin{bmatrix}\n\\mathfrak{Z}^0 & \\mathfrak{Z}^1 &  \\cdots & \\mathfrak{Z}^{d'}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathfrak{D}^s & 0 & \\cdots & 0 \\\\\n0 & \\mathfrak{D}^s & \\cdots &0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\mathfrak{D}^s \n\\end{bmatrix} .\n\\end{aligned}\n\\]\nThe diagonal matrix in this product is invertible, hence\n\\begin{equation} \\label{eq:rank-bound-1}\n\\begin{aligned}\n&\\text{Rank of $E_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$} = \\text{Rank of $\\begin{bmatrix}\n\\mathfrak{Z}^0 & \\mathfrak{Z}^1 & \\mathfrak{Z}^2 & \\cdots & \\mathfrak{Z}^{d'}\n\\end{bmatrix}$}  \\\\\n&\\hspace{.5in} \\le \\text{Number of distinct columns} = \\left\\lceil\\frac{d}{p^k}\\right\\rceil.\n\\end{aligned}\n\\end{equation}\nNow, the number of columns of $E_{\\mathcal{I}_{ks}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is equal to \n\\begin{align*}\n|\\mathcal{J}| &= |\\mathcal{J}_{k'0} \\cup \\mathcal{J}_{k'1} \\cup \\mathcal{J}_{k'2} \\cup \\ldots \\cup \\mathcal{J}_{k'd'}| \\\\\n& = \\sum_{r=0}^{ \\lceil d\/p^k \\rceil -1} |\\mathcal{J}_{k'r}| = p^k\\lceil d\/p^k \\rceil \\geq d,\n\\end{align*}\nso there are at least $d$ columns. Hence if $\\mathcal{I}$ is a universal sampling set of size $d$ then  $E_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ must be of full row rank. In particular, since $\\mathcal{I}_{ks} \\subseteq \\mathcal{I}$, it must be that \n$E_{\\mathcal{I}_{ks}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is also of full row rank, for each $s$. Next, the number of rows in $E_{\\mathcal{I}_{ks}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is equal to $|\\mathcal{I}_{ks}| = \\widetilde{\\chi}_k(s)$ by definition. From \\eqref{eq:rank-bound-1} we know that the rank of $E_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is at most $\\lceil d \/p^k \\rceil$, and so we have\n\\[\n\\text{(Number of rows) $\\widetilde{\\chi}_k(s)$} \\leq \\left\\lceil \\frac{d}{p^k} \\right\\rceil.\n\\]\n\nThe proof of the lower bound $\\widetilde{\\chi}_k(s) \\ge \\lfloor d\/p^k \\rfloor$ is very similar. This time we construct a set $\\mathcal{J}$ with $|\\mathcal{J}| \\leq d$, and observe that if $\\mathcal{I}$ is a universal sampling set of size $d$, then $E_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is of full column rank. \n\nLet\n\\[\n\\mathcal{J} = \\mathcal{J}_{k'0} \\cup \\mathcal{J}_{k'1} \\cup \\mathcal{J}_{k'2} \\cup \\ldots \\cup \\mathcal{J}_{k' d''}, \\quad d''=\\lfloor d\/p^k \\rfloor -1.\n\\] \nThen just as above,\n\\[\n\\begin{aligned}\n\\text{Rank of $E_{\\mathcal{I}_{ks}}^\\textsf{T}{{\\mathcal{F}}} E_\\mathcal{J}$} &=  \\text{Rank of $\\begin{bmatrix}\n\\mathfrak{Z}^0 & \\mathfrak{Z}^1 & \\mathfrak{Z}^2 & \\cdots & \\mathfrak{Z}^{d''}\n\\end{bmatrix}$}  \\\\\n&\\le \\text{Number of distinct columns} = \\left\\lfloor\\frac{d}{p^k}\\right\\rfloor.\n\\end{aligned}\n\\]\n\nThe number of rows of $E_{\\mathcal{I}_{ks}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}$ is $|\\mathcal{I}_{ks}| = \\widetilde{\\chi}_k(s)$, and so we must have\n\\begin{equation}\n\\text{Rank of }E_{\\mathcal{I}_{ks}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\leq \\min \\{\\lfloor d\/p^k \\rfloor ,\\widetilde{\\chi}_k(s)\\}. \\label{eq:univ-nec-31}\n\\end{equation} \nFurthermore, \n\\[\nE_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\nonumber\\\\\n= \\begin{bmatrix}E_{\\mathcal{I}_{k0}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\\\ E_{\\mathcal{I}_{k1}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\\\\n\\vdots\\\\\nE_{\\mathcal{I}_{k(p^k-1)}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}}\\\\\n \\end{bmatrix} , \\nonumber\n\n \\]\nwhence\n\\begin{align}\n&\\text{Row rank of }E_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\nonumber \\\\\n&\\quad \\leq \\text{Rank of }E_{\\mathcal{I}_{k0}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} + \\text{Rank of }E_{\\mathcal{I}_{k1}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\nonumber\\\\\n& \\hspace{.5in}+ \\ldots + \\text{Rank of }E_{\\mathcal{I}_{k(p^k-1)}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} \\nonumber\\\\\n&\\quad \\leq \\sum_{s=0}^{p^k-1} \\min \\{\\lfloor d\/p^k \\rfloor ,\\widetilde{\\chi}_k(s)\\} .  \\label{eq:univ-nec-32}\n\\end{align}\n\nNow, the number of columns indexed by $\\mathcal{J}$ is $p^k \\lfloor d\/p^k \\rfloor \\leq d$. Hence if $\\mathcal{I}$ is a universal sampling set of size $d$, we need $E_{\\mathcal{I}}^\\textsf{T}\\mathcal{F} E_{\\mathcal{J}} $ to be of full column rank. From \\eqref{eq:univ-nec-32}, this means we must have \n\\[\n\\text{(Number of columns) } p^k \\lfloor d\/p^k \\rfloor \\leq \\sum_{s=0}^{p^k-1} \\min \\{\\lfloor d\/p^k \\rfloor ,\\widetilde{\\chi}_k(s)\\}. \n\\] \nThis inequality will not be satisfied unless $\\lfloor d\/p^k \\rfloor \\leq \\widetilde{\\chi}_k(s)$ for all $s$. This completes the proof. \n\\end{IEEEproof}\n\n\\begin{remark}\nFor many values of $d$, it is enough to prove one side of the inequality \\eqref{eq:chi-upper-and-lower}. If we know that $\\widetilde{\\chi}_k(s) \\leq \\lceil d \/p^k \\rceil$, then from $\\sum_s\\widetilde{\\chi}_k(s) =d$ and a recurrence relation \\eqref{eq:chi-recurrence}, below,  it is possible to prove that $\\lfloor d \/p^k \\rfloor \\leq \\widetilde{\\chi}_k(s)$. Such cases include\n\\begin{enumerate}\n\\item $N=p^M$, $d = c_0 p^k + c_1 p^{k-1}$ for $c_0, c_1 \\in \\{0,1,2,\\ldots,p-1\\}$.\n\\item $N=2^M$, $d = c_0 2^k + c_1 2^{k-1} + c_22^{k-2}$ for $c_0, c_1, c_2 \\in \\{0,1\\}$\n\\item $N=2^M$, $d = 2^k + 2^{k-1} + 2^{k-2} + \\ldots + 2^{k-r+1}$ for some $r$,\n\\end{enumerate}\n\\end{remark}\n\n\n\n\n\\subsection{Digit Reversal and Universal Sampling Sets} \\label{subsection:digit-reversal}\n\nThere is another interesting characterization of universal sampling sets in terms of \\emph{digit reversal}. Expanding in base $p$, any integer $a \\in [0:p^m-1]$, $m \\ge 1$, can be written uniquely as\n\\[\na=\\alpha_0+\\alpha_1p+\\alpha_2p^2+\\cdots \\alpha_{m-1}p^{m-1},\n\\] \nwhere the $\\alpha$'s are in $[0:p-1]$. We define a permutation $\\pi_m\\colon [0:p^m-1] \\longrightarrow [0:p^m-1]$ by\n\\[\n\\begin{aligned}\n&\\pi_m(\\alpha_0+\\alpha_1p+\\alpha_2p^2+\\cdots \\alpha_{m-1}p^{m-1}) = \\\\\n&\\hspace{.5in} \\alpha_{m-1}+\\alpha_{m-2}p+\\alpha_{m-3}p^2+\\cdots +\\alpha_0p^{m-1}.\n\\end{aligned}\n\\]\nThe $\\alpha$'s are the digits in the base $p$ expansion of $a \\in [0:p^m-1]$ and applying $\\pi_m$ to $a$ produces the number in $[0:p^m-1]$ with the digits reversed.  For example (an example we will use again  in Section \\ref{section:maximal}), take $[0:7]$. Then $\\pi_3([0:7])=\\{0,4,2,6,1,5,3,7\\}$ in that order.  Such digit reversing permutations were used in \\cite{delvaux:rank-defficient} to find rank-one submatrices of the Fourier matrix. \n\n\n\n\n\n\n\nThe issue for universal sampling sets is how the numbers $\\pi_M(\\mathcal{I})$ are dispersed within the interval $[0:p^M-1]$, where, as before, $N=p^M$. To make this precise, take $k\\ge 1$ and partition $[0:p^M-1]$ into $p^k$ equal parts:\n\\[\n[0:p^M-1] = \\bigcup_{a=0}^{p^k-1}[ap^{k'}:(a+1)p^{k'}-1], \\quad k'=M-k.\n\\] \nFor any $\\mathcal{J} \\subseteq [0:p^M-1]$ and $a \\in [0:p^k-1]$, let\n\\[\n\\phi_k(a\\,;\\mathcal{J}) = |\\mathcal{J} \\cap [ap^{k'},(a+1)p^{k'}-1]|.\n\\]\nWe say that $\\mathcal{J}$ is \\emph{uniformly dispersed} in $[0,p^M-1]$ if\n\\begin{equation} \\label{eq;uniformly-dispersed}\n|\\phi_k(a\\,;\\mathcal{J})  - \\phi_k(b\\,;\\mathcal{J})| \\le 1\n\\end{equation}\nfor all $a, b\\in[0:p^k-1]$, and $1\\le k \\le M$.  Thus $\\mathcal{J}$ is uniformly dispersed if roughly equal numbers\nof its elements are in each of the intervals $[ap^{k'}: (a+1)p^{k'}-1]$ for all $1 \\le k \\le M$, $k'=M-k$. \n\nWe will show \n\\begin{equation} \\label{eq:phi-chi}\n\\phi_k(\\pi_k(a)\\,; \\pi_M(\\mathcal{I})) = \\widetilde{\\chi}_k(a), \\quad a \\in[0:p^k-1].\n\\end{equation}\nThus, to the three equivalent conditions in Theorem \\ref{theorem:universal} we can add a fourth:\n\\begin{quote}\n\\begin{enumerate}\n\\item[(iv)] $\\pi_M(\\mathcal{I})$ is uniformly dispersed. \n\\end{enumerate}\n\\end{quote}\n\n\nThe derivation of \\eqref{eq:phi-chi} uses the following lemma.\n\\begin{lemma}\n\\label{lemma:univ-alt-2}\nIf $j \\in [0:p^M-1]$ is given by $j = b + ap^{k'}$, $0\\leq b \\leq\np^{k'}-1 $, then $\\pi_M(j) = \\pi_k(a) + p^k\\pi_{k'}(b)$. \n\\end{lemma}\n\nThe proof is straightforward, and the argument for \\eqref{eq:phi-chi} then goes very quickly. As defined, for any index set $\\mathcal{J}$, $\\phi_k(a\\,; \\mathcal{J})$ is the number of elements in $\\mathcal{J}$ that lie in $[ap^{k'} : (a+1)p^{k'}-1]$, and these are precisely the $j \\in \\mathcal{J}$ of the form $ap^{k'}+b$ with $0\\le b \\le p^{k'}-1$.  Thus for $i \\in [0:p^k-1]$,\n\\[\n\\begin{aligned}\n&{\\phi}_k(\\pi_k(i)\\,;{\\pi_M(\\mathcal{I})}) = \\\\\n&\\hspace{.25in}\\text{the number of }j \\in\n\\pi_M(\\mathcal{I}) \\\\\n&\\hspace{.35in}\\text{ of the form }\\pi_k(i) p^{k'} + b, \\ 0\\leq b \\leq\np^{k'}-1 \\nonumber \\\\\n&= \\text{ number of }j \\in\n\\mathcal{I} \\text{ of the form } \\\\\n&\\hspace{.35in}p^{k}\\pi_{k'}(b) + i, \\ 0\\leq b \\leq\np^{k'}-1 \\text{ (from Lemma \\ref{lemma:univ-alt-2})} \\\\\n& = \\text{ number of }j \\in\n\\mathcal{I} \\text{ that leave a remainder of  } \\\\\n&\\hspace{.35in} i \\text{ on dividing by }p^k\n\\nonumber \\\\\n& =  \\widetilde{\\chi}_k(i). \\nonumber\n\\end{aligned}\n\\]\n \n\n\n\\section{Structure and Enumeration of Universal Sampling Sets} \\label{section:maximal}\n\n In this section we analyze in detail the structure of universal sampling sets. Specifically we show that when $N=p^M$ is a prime power  such a set $\\mathcal{I}$ is the disjoint union of smaller, \\emph{elementary universal sets} that depend on the base $p$ expansion of $|\\mathcal{I}|$. The method is algorithmic, allowing us to construct universal sets of a given size, and to find a formula that counts the number of universal sets as a function of $p^M$ and $|\\mathcal{I}|$. In particular the formula  answers the question:  How likely is it that a randomly chosen index set is universal? Not very likely, but there are several subtle aspects to the answer. For example, we exhibit plots of the counting function showing some striking phenomena depending on the prime $p$.    Our approach is via \\emph{maximal universal sampling sets} which, in turn, enter naturally in studying the relationship between universal sampling sets and uncertainty principles. We take up the latter topic in the next section.\n\n\n\n \n\n\\subsection{A Recurrence Relation and Tree for $\\widetilde{\\chi}$}\n\nWhen $N=p^M$ the condition that an index set be a universal sampling set depends on the values of $\\widetilde{\\chi}_k$ for different $k$. To study this we use a recurrence relation in $k$ for $\\widetilde{\\chi}_k(a)$. The formula holds even when $N$ is not a prime power.  \n\n\\begin{lemma} \\label{lemma:recursion}\nLet $\\mathcal{I} \\subseteq [0:N-1]$. Then\n \\begin{equation} \\label{eq:chi-recurrence}\n\\widetilde{\\chi}_{k-1}(a) = \\sum_{j=0}^{p-1}\\widetilde{\\chi}_k(a + jp^{k-1}),  \\quad \n \\end{equation}\n{for all $a\\in[0:p^{k-1}-1]$.}\n \\end{lemma}\n\\begin{IEEEproof}\nAn integer $x \\in \\mathcal{I}$ that leaves a remainder of $a$ when divided by $p^{k-1}$ is of the form $x=\\alpha p^{k-1} + a$. Let $\\alpha = \\beta p + \\gamma$ for $\\gamma \\in [0:p-1]$. Then $x = \\beta p^k + \\gamma p^{k-1} + a$, that is, $x$ leaves a remainder of either $0\\cdot p^{k-1}+a, 1\\cdot p^{k-1}+a, 2\\cdot p^{k-1}+ a,\\dots$  or $(p-1)\\cdot p^{k-1}+a$ on dividing by $p^k$. The result follows.  \n\\end{IEEEproof}\n\nWhen $N=p^M$ the recurrence formula and the relation it expresses between conjugacy classes has an appealing interpretation in terms of a $p$-ary tree. Several arguments in this section will be based on this configuration. \n\nLet $\\mathcal{I}\\subseteq [0: p^M-1]$. We construct a tree with $M+1$ levels and $p^k$ nodes in level $k$, $0 \\le k \\le M$. The nodes in level $k$ are identified by a pair $(k,a)$, with $a \\in [0:p^{k-1}]$.  Call the\nnodes at the level $M$ the leaves. At the node $(k,a)$ we imagine placing the congruence class $\\mathcal{I}_{ka} =\\{i\\in \\mathcal{I} \\colon i \\equiv a \\mod p^k\\}$. The root is $\\mathcal{I}_{00}=\\mathcal{I}$ and the nodes at the leaves host the sets $\\mathcal{I}_{Ma}$, $a \\in [0:p^M-1]$, each of which is either a singleton or empty. We assign a weight of $\\widetilde{\\chi}_k(a) = |\\mathcal{I}_{ka}|$ to the node $(k,a)$. Further, at each level we arrange the nodes according to the digit reversing permutation, i.e., nodes at level $k$ are arranged as $\\pi_k([0:p^k-1])$, where $\\pi_k$ is the digit reversing permutation from Section \\ref{subsection:digit-reversal}. (This is similar to the starting step of the FFT algorithm, where the indices are sorted according to the reversed digits.)  Figure \\ref{fig:tree1} shows the case  $N=2^3$, a binary tree with four levels, $k=0, 1, 2, 3$. In  the third level of the tree the nodes are ordered $0, 4, 2, 6, 1, 5, 3, 7$, which is $\\pi_3([0:7])$. Then:\n\n\\begin{enumerate}\n\\item The set $\\mathcal{I}_{ka}$ at level $k$ is the disjoint union of the sets at its children nodes at level $k+1$.\n\\item  The value of $\\widetilde{\\chi}_k(a)$ at the node $(k,a)$ is \n  the sum of the values of $\\widetilde{\\chi}_{k+1}$ at its children\n  nodes at level $k+1$. In other words, the weight of a parent is the sum of the weights of its children; this is the recurrence relation. Consequently, the value of $\\widetilde{\\chi}_k$ at any\n  node is the sum of the values of $\\widetilde{\\chi}_M$ at the\n  leaves at level $M$ descended from the node. \n  \\end{enumerate}\n  For example, in Figure \\ref{fig:tree1} we have\n\\[\n\\begin{aligned}\n  \\widetilde{\\chi}_0(0) &= \\sum_{a=0}^7\\widetilde{\\chi}_3(a), \\\\\n \\widetilde{\\chi}_1(0) &=\\sum_{a=0}^3\\widetilde{\\chi}_3(2a),\\\\\n \\widetilde{\\chi}_1(1) &= \\sum_{a=0}^3\\widetilde{\\chi}_3(2a+1), \n \\end{aligned}\n \\]\n and so on.  \n\nIn fact, a more general conclusion is the following: Fix a level $k$. Then the value of $\\widetilde{\\chi}_r$ at any\n  node $(r,a)$, for $r \\leq k$ is the sum of the values of $\\widetilde{\\chi}_k$ at the level-$k$ nodes descending from the tree node $(r,a)$.  \n\n\nWhen the root is $[0:p^M-1]$, the extreme case, the leaves are all singletons and the nodes at level $k$ are each of weight $p^{M-k}$.\n\n\\begin{figure*}\n\\centering\n\\begin{tikzpicture}[scale=1.0]\n\\node (0) at (0,8) [circle, minimum size = 1.0cm, fill=gray!10, draw] {$\\mathcal{I}_{00}$};\n\\node (1) at (-4,6) [circle, minimum size = 1.0cm, fill=gray!10,\ndraw] {$\\mathcal{I}_{10}$};\n\\node (2) at (4,6) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{11}$};\n\\node (3) at (-6,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{20}$};\n\\node (4) at (-2,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{22}$};\n\\node (5) at (2,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{21}$};\n\\node (6) at (6,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{23}$};\n\\node (7) at (-7,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{30}$};\n\\node (8) at (-5,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{34}$};\n\\node (9) at (-3,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{32}$};\n\\node (10) at (-1,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{36}$};\n\\node (11) at (1,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{31}$};\n\\node (12) at (3,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{35}$};\n\\node (13) at (5,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{33}$};\n\\node (14) at (7,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{37}$};\n\\draw (node cs:name=0) -- (node cs:name =1);\n\\draw (node cs:name=0) -- (node cs:name =2);\n\\draw (node cs:name=1) -- (node cs:name =3);\n\\draw (node cs:name=1) -- (node cs:name =4);\n\\draw (node cs:name=2) -- (node cs:name =5);\n\\draw (node cs:name=2) -- (node cs:name =6);\n\\draw (node cs:name=3) -- (node cs:name =7);\n\\draw (node cs:name=3) -- (node cs:name =8);\n\\draw (node cs:name=4) -- (node cs:name =9);\n\\draw (node cs:name=4) -- (node cs:name =10);\n\\draw (node cs:name=5) -- (node cs:name =11);\n\\draw (node cs:name=5) -- (node cs:name =12);\n\\draw (node cs:name=6) -- (node cs:name =13);\n\\draw (node cs:name=6) -- (node cs:name =14);\n\\end{tikzpicture}\n\\caption{A tree representing the relations between the congruence classes, and the recurrence relation satisfied by\n          $\\widetilde{\\chi}_k(a)$. The value of\n          $\\widetilde{\\chi}_k(a)$ at any node is the sum of\n          the values of $\\widetilde{\\chi}_k(a)$ at its children nodes in level $k+1$.}\n          \\label{fig:tree1}\n\\end{figure*}\n\n\n\\subsection{Elementary and Maximal Sets} \\label{subsection:elementary-maximal}\n\nTo study the structure of universal sampling sets we need a series of definitions. When $N$ is a prime power the building blocks are the elementary sets:\n\\begin{defn} \\label{definition:elementary}\nA set $\\mathcal{E} \\subseteq [0:p^M-1]$ is a $k$-\\emph{elementary set} if\n\\[\n\\widetilde{\\chi}_k(a) = 1, \\quad \\text{~for all $a\\in [0:p^k-1]$}.\n\\]\n\\end{defn}\nNote that $|\\mathcal{E}|=p^k$.\n\nAs a first application of the  formula \\eqref{eq:chi-recurrence} we can add the adjective ``universal'' to the description of elementary sets.\n\\begin{lemma} \\label{lemma:elementary-universal}\nA $k$-elementary set $\\mathcal{E}$ is a universal sampling set.\n\\end{lemma}\n\n\\begin{IEEEproof}\nFrom $\\widetilde{\\chi}_k(a) = 1$ and  \\eqref{eq:chi-recurrence} it follows that $\\mathcal{E}$ has an equal number of elements in each\ncongruence class modulo $p^s$, $s \\leq k$. More precisely,\n\\begin{equation} \\label{eq:chi-constant}\n\\widetilde{\\chi}_s(a) = p^{k-s},\n\\end{equation}\nfor all $s \\le k$. Also from \\eqref{eq:chi-recurrence}, for $s > k$ all the congruence classes are of size $0$ or $1$, i.e.\n\\begin{equation} \\label{eq:chi-in-{0,1}}\n\\widetilde{\\chi}_s(a) \\in\\{0,1\\}.\n\\end{equation}\nTherefore\n\\[\n|\\widetilde{\\chi}_s(a)-\\widetilde{\\chi}_s(b)| \\le1,\n\\]\nfor all $a,b \\in [0: p^k-1]$ and all $s$, and we conclude that $\\mathcal{E}$ is a universal sampling set. \n\\end{IEEEproof}\n\n\nNext, a fruitful approach to understanding the structure of universal sampling sets is to ask how well an arbitrary index set is approximated from within by universal sets. \n\n \n\n\\begin{defn} \\label{definition:maximal}\nLet $\\mathcal{I} \\subseteq [0:N-1]$. A \\emph{maximal universal sampling set} for $\\mathcal{I}$ is a universal sampling set of largest cardinality that is contained in $\\mathcal{I}$. \n\\end{defn}\n\nNote that the definition does not require $N$ to be a prime power, though this will most often be the case. There is an allied notion of a minimal universal set. We define this in Subsection \\ref{subsection:maximal-minimal} below, and show how they are related to maximal sets. Maximal and minimal sets enter naturally and together in connection with uncertainty principles, discussed in Section \\ref{section:uncertainty}.\n\n\nFinding  a maximal universal sampling set for a given $\\mathcal{I}$ is a finitary process, so existence is not an issue. However, maximal universal sampling sets need not be unique. For example, take $N=3^2$ and $\\mathcal{I}=\\{0, 1, 2, 3, 6 \\}$. The set $\\mathcal{I}$ is not itself a universal sampling set, and both $\\{0, 1, 2, 3\\}$ and $\\{0, 1,2,6 \\}$ are maximal universal sampling sets contained in $\\mathcal{I}$. \n\nDespite the lack of uniqueness it will be convenient to have a notation, and we let $\\Omega(\\mathcal{I})$ denote a generic maximal universal sampling set in $\\mathcal{I}$.  The cardinality $|\\Omega(\\mathcal{I})|$ is well-defined; by definition $|\\mathcal{J}| \\le |\\Omega(\\mathcal{I})|$ for any universal sampling set $\\mathcal{J} \\subseteq \\mathcal{I}$.  \n\n\nElementary sets and maximal sets are related through an important construction of an elementary set. \n\n\\begin{defn}\nLet $\\mathcal{I}\\subseteq[0:p^M-1]$ and let $\\bar{k}$ be the largest integer such that no congruence class in $\\mathcal{I}\/p^{\\bar{k}}$ is empty.  (It might be that $\\bar{k}=0$.) Let $\\mathcal{I}_{\\bar{k}}^\\dagger$ denote an elementary set obtained by choosing one element from each congruence class in $\\mathcal{I}\/p^{\\bar{k}}$. \n\\end{defn}\n\nBy Lemma \\ref{lemma:elementary-universal}, $\\mathcal{I}_{\\bar{k}}^\\dagger$ is a universal sampling set, and is of order $p^{\\bar{k}}$. \nWe now have\n\n\\begin{theorem}\n\\label{thm:maximal-bounds}\nLet  $\\mathcal{\\mathcal{I}} \\subseteq [0:p^M-1]$, and $\\mathcal{I}_{\\bar{k}}^\\dagger$ as above. \nThen\n\\begin{enumerate}\n\\item[(i)] $p^{\\bar{k}}\\leq|\\Omega(\\mathcal{I}) | < p^{\\bar{k}+1}$.\n\\item[(ii)] There exists a maximal universal sampling set contained in $\\mathcal{I}$ and containing $\\mathcal{I}_{\\bar{k}}^\\dagger$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{IEEEproof} \nThe lower bound in (i) follows from the definition of a maximal set and the comments above,\n\\[\n p^{\\bar{k}} = |\\mathcal{I}_{\\bar{k}}^\\dagger| \\le |\\Omega(\\mathcal{I})|.\n \\]\n   To prove the upper bound, suppose  $\\mathcal{J} \\subseteq \\mathcal{I}$ has $|\\mathcal{J}| \\geq p^{\\bar{k}+1}$. By the\ndefinition of $\\bar{k}$ at least one congruence class  in $\\mathcal{J}\/p^{\\bar{k}+1}$ is empty, so  $\\widetilde{\\chi}_{\\bar{k}+1}(a\\,;\\mathcal{J}) = 0$ for some $a\\in[0:p^{\\bar{k}+1}-1]$. From the cardinality equation \\eqref{eq:cardinality-chi},\n\\[\n\\sum_{\\ell=0}^{p^{\\bar{k}+1}-1}\\widetilde{\\chi}_{\\bar{k}+1}(\\ell \\,; \\mathcal{J}) = |\\mathcal{J}| \\geq p^{\\bar{k}+1}.\\]\nThis implies  that at least one congruence class in $\\mathcal{J}\/p^{\\bar{k}+1}$ has at least two elements, or $\\widetilde{\\chi}_{\\bar{k}+1}(b\\,;\\mathcal{J}) \\geq 2$ for some $b$. We then have\n\\[\n|\\widetilde{\\chi}_{\\bar{k}+1}(b\\,;\\mathcal{J}) - \\widetilde{\\chi}_{\\bar{k}+1}(a\\,;\\mathcal{J})| = 2 > 1,\n\\]\nand $\\mathcal{J}$ cannot be a universal sampling set. \n\nFor part {(ii)}, we first show that any maximal universal sampling set $\\Omega(\\mathcal{I})$ set must contain at least one element from each congruence class in $\\mathcal{I}\/p^{\\bar{k}}$. By way of contradiction, suppose that $\\widetilde{\\chi}_{\\bar{k}}(a\\,;\\Omega(\\mathcal{I}))=0$ for some $a$. Since $\\Omega(\\mathcal{I})$ is universal we must then have $\\widetilde{\\chi}_{\\bar{k}}(b\\,;\\Omega(\\mathcal{I}))\\le 1$ for all $b$. By \\eqref{eq:cardinality-chi},\n\\[\n|\\Omega(\\mathcal{I})|= \\sum_{b=0}^{p^{\\bar{k}-1}} \\widetilde{\\chi}_{\\bar{k}}(b\\,;\\Omega(\\mathcal{I})) <p^{\\bar{k}},\n\\]\ncontradicting the lower bound in (i).\n\nLet $\\mathcal{K} \\subseteq \\Omega(\\mathcal{I})$ be an elementary set, of size $p^{\\bar{k}}$, that contains one element from each congruence class in $\\mathcal{I}\/p^{\\bar{k}}$, guaranteed to exist from what we just showed. Assuming $\\mathcal{K} \\ne \\mathcal{I}_{\\bar{k}}^\\dagger$, since otherwise we are done, we will use $\\mathcal{K}$ and $\\Omega(\\mathcal{I})$ to construct a (new) maximal universal set that contains $\\mathcal{I}_{\\bar{k}}^\\dagger$. \n\nSet up a $p$-ary tree, as above, with root $\\Omega_{00}=\\Omega(\\mathcal{I})$ and  $(\\ell,a)$-node the congruence class\n\\[\n\\Omega_{\\ell a}= \\{  i \\in \\Omega(\\mathcal{I}) \\colon  i \\equiv a \\text{ mod $p^\\ell$}\\},  \\quad |\\Omega_{\\ell a}|=\\widetilde{\\chi}_\\ell(a),  \n\\]\nfor $a \\in [0:p^\\ell-1]$. Recall that $\\Omega_{\\ell a}$, at level $\\ell$, is the disjoint union of the sets at its children nodes at level $\\ell+1$. \n\nFigure \\ref{fig:tree2} is an example for $p=3$ and $M\\ge 3$, showing only three levels for reasons of space. The shading has to do with the rest of the proof, as we now explain.\n\n\\begin{figure*}\n\\centering\n\\begin{tikzpicture}[scale=1.2]\n\\node (0) at (0,6) [circle, minimum size = 1.0cm, fill=gray!10, draw] {$\\Omega_{00}$};\n\\node (1) at (-3,3) [circle, minimum size = 1.0cm, fill=gray!10,\ndraw] {$\\Omega_{10}$};\n\\node (2) at (0,3) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\Omega_{11}$};\n\\node (3) at (3,3) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\Omega_{12}$};\n\\node (4) at (-4.5,0) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{{\\text{blue}}}{\\Omega_{20}}$};\n\\node (5) at (-3.3,0) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\Omega_{23}}$};\n\\node (6) at (-2.2,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\Omega_{26}$};\n\\node (7) at (-1.15,0) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\Omega_{21}}$};\n\\node (8) at (0,0) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\Omega_{24}}$};\n\\node (9) at (1.1,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\Omega_{27}$};\n\\node (10) at (2.1,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\Omega_{22}$};\n\\node (11) at (3.2,0) [circle, minimum size = 1cm, fill=red!50!blue!50, draw]\n{$\\underset{\\text{purple}}{\\Omega_{25}}$};\n\\node (12) at (4.3,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\Omega_{28}$};\n\\draw (node cs:name=0) -- (node cs:name =1);\n\\draw (node cs:name=0) -- (node cs:name =2);\n\\draw (node cs:name=0) -- (node cs:name =3);\n\\draw (node cs:name=1) -- (node cs:name =4);\n\\draw (node cs:name=1) -- (node cs:name =5);\n\\draw (node cs:name=1) -- (node cs:name =6);\n\\draw (node cs:name=2) -- (node cs:name =7);\n\\draw (node cs:name=2) -- (node cs:name =8);\n\\draw (node cs:name=2) -- (node cs:name =9);\n\\draw (node cs:name=3) -- (node cs:name =10);\n\\draw (node cs:name=3) -- (node cs:name =11);\n\\draw (node cs:name=3) -- (node cs:name =12);\n\\end{tikzpicture}\n\\caption{The congruence class tree for $\\Omega(\\mathcal{I})$. The node ${\\Omega}_{\\ell a}$ is the congruence class of $a$ modulo $p^\\ell$ in $\\Omega(\\mathcal{I})$, so that $\\Omega_{00}=\\Omega(\\mathcal{I})$ and $|{\\Omega}_{\\ell a} | = \\widetilde{\\chi}_\\ell(a)$. }\n\\label{fig:tree2}\n\\end{figure*}\n\n\nBoth $\\mathcal{I}_{\\bar{k}}^\\dagger$ and $\\mathcal{K}$ are assembled by choosing single elements from sets at the nodes in the $\\bar{k}$-level   (call these the assembly nodes) for a total of $p^{\\bar{k}}$ elements for $\\mathcal{I}_{\\bar{k}}^\\dagger$ and $\\mathcal{K}$ each. Observe that the sets at the nodes in the $\\bar{k}+1$ level are either empty or singletons. This is so because by definition of $\\bar{k}$ there must be some $a\\in [0:p^{\\bar{k}+1}-1]$ for which $\\widetilde{\\chi}_{\\bar{k}+1}(a) = 0$, and hence by universality $\\widetilde{\\chi}_{\\bar{k}+1}(b) \\le 1$ for all $b\\in [0:p^{\\bar{k}+1}-1]$. And then, according to how the tree is structured, the sets at all nodes farther down in the tree must as well be either empty or singletons.\n\n\n Let $\\mathcal{L} \\supseteq \\mathcal{I}_{\\bar{k}}^\\dagger$ be the set of elements in $\\mathcal{I}$ that leave the same remainders as do the elements in $\\mathcal{I}_{\\bar{k}}^\\dagger$ when divided by $p^{\\bar{k}+1}$, more precisely,\n\\[\n\\mathcal{L}= \\{j \\in \\mathcal{I} \\colon j \\equiv  i \\text{ mod $p^{\\bar{k}+1}$ for some $i\\in \\mathcal{I}_{\\bar{k}}^\\dagger$}\\}.\n\\]\nLikewise let  $\\mathcal{L}' \\supseteq \\mathcal{K}$ be\n\\[\n\\mathcal{L}'= \\{j \\in \\mathcal{I} \\colon j \\equiv  i \\text{ mod $p^{\\bar{k}+1}$ for some $i\\in \\mathcal{K}$}\\}.\n\\]\n $\\mathcal{L}$ is the union of the assembly nodes for $\\mathcal{I}_{\\bar{k}}^\\dagger$ and $\\mathcal{L}'$ is the union of the assembly nodes for $\\mathcal{K}$. The collections  may overlap.\n \n We color a node red if it contributes to $\\mathcal{L}$ and blue if it contributes to $\\mathcal{L}'$, and both red and blue (otherwise known as purple) if it contributes to both $\\mathcal{L}$ and $\\mathcal{L}'$.  In the figure we take $\\bar{k}=1$, so $\\mathcal{I}_{\\bar{k}}^\\dagger$ and $\\mathcal{K}$ live at the middle level in the tree, as shown.\n\nFocus on each red node in turn. The red node contains an element in $\\mathcal{I}^\\dagger_{\\bar{k}}$, say $i$.\n\\begin{enumerate}\n\\item If $\\Omega(\\mathcal{I})$ contains an element from this red node, say $j$ (which may or may not be equal to $i$), we replace $j\\in \\Omega(\\mathcal{I})$ with $i$. This neither changes the size of $\\Omega(\\mathcal{I})$ nor the universality.\n\n\\item Now suppose $\\Omega(\\mathcal{I})$ does not contain an element from this red node. We know that the sibling blue node (i.e. the blue node that shares the parent with this red node) contains an element of $\\mathcal{K}$ (and hence of $\\Omega(\\mathcal{I})$), say $j$. Replace $j \\in \\Omega(\\mathcal{I})$ with $i$. This neither changes the size, nor the universality; we are just exchanging one element from a node with its sibling, so the value of $\\widetilde{\\chi}$ at the parent node does not change.  \n\\end{enumerate}\n\nThese operations preserve size and universality, and repeating them for each red node ensures that the resultant set contains $\\mathcal{I}^\\dagger_k$.\n \n\\end{IEEEproof}\n\n\nA stronger version of the upper bound in (i)  is the following.\n\\begin{corollary}\n\\label{cor:uncer-2bnd}\nLet $\\mathcal{I} \\subseteq[0:p^M-1]$ and let $\\underline{k}$ be the smallest integer such that $\\widetilde{\\chi}_{\\underline{k}}(a\\,;\\mathcal{I})=0$ for some $a$. Then,\n\\[\n|\\Omega(\\mathcal{I})| \\leq |\\{a: \\widetilde{\\chi}_{\\underline{k}}(a; \\mathcal{I})\\neq 0 \\}|.\n\\] \n\\end{corollary}\n\\begin{IEEEproof} From the definition of $\\underline{k}$, we have $\\widetilde{\\chi}_{\\underline{k}}(a_0\\,; \\mathcal{I}) = 0 $ for some $a_0$. Hence by universality, $\\Omega(\\mathcal{I})$ must satisfy $|\\widetilde{\\chi}_{\\underline{k}}(b\\,; \\Omega(\\mathcal{I}))| \\leq 1$ for all $b$, an observation we used above and will use again. From the cardinality equation \\eqref{eq:cardinality-chi}\n\\[\n|\\Omega(\\mathcal{I})|  = \\sum_b \\widetilde{\\chi}_{\\underline{k}}(b\\,; \\Omega(\\mathcal{I})) \\leq  |\\{a: \\widetilde{\\chi}_{\\underline{k}}(a\\,; \\mathcal{I})\\neq 0 \\}|.\n\\]\n\\end{IEEEproof}\n\n\n\nUltimately we will show that when $N=p^M$ any maximal universal sampling set, and in particular any universal sampling set, is a disjoint union of elementary sets. In general, however, the union of two disjoint, elementary sets need not be universal. For example, take $N=2^3$, $\\mathcal{E}=\\{0, 1\\}$, $\\mathcal{E}'=\\{4,5\\}$. Then $\\mathcal{E}$ and $\\mathcal{E}'$ are elementary but their union $\\mathcal{E} \\cup \\mathcal{E}' =\\{0,1,4,5\\}$ is not universal. What is needed is a kind of independence condition on a collection of elementary sets. The following lemma, whose converse we will also show, makes this latter point precise and introduces the main features of the structure of universal sets. \n\n\\begin{lemma} \\label{lem:univ-structure-forward}\nLet $N=p^M$. Suppose there exists a finite sequence of nonincreasing integers $k_1 \\ge k_2 \\ge \\cdots \\ge 0$ and sets $\\mathcal{E}_{r}\\subseteq[0:N-1]$, $r =1, 2, \\dots$,  such that\n\\begin{enumerate}[(i)]\n\\item $\\mathcal{E}_{r}$ is $k_r$-elementary.\n\\item For each $r\\ge 1$\n\\[\n\\mathcal{E}_{r} \\cap \\left(\\bigcup_{j=1}^{r-1} \\mathcal{L}_j\\right) = \\emptyset,\n\\]\nwhere\n\\end{enumerate}\n\\[\n\\mathcal{L}_j = \n\\{x  \\in [0:N-1] \\colon x \\equiv e \\text{ mod $p^{k_j+1}$ for some $ e \\in\n\\mathcal{E}_{j}$}\\} .\n\\]\n\nLet\n\\[\n\\mathcal{I}= \\bigcup_r \\mathcal{E}_{r}.\n\\]\nThen  $\\mathcal{I}$ is a universal sampling set.\n\\end{lemma}\n\n Obviously it is condition (ii) that requires further comment. The set $\\mathcal{L}_r$ is defined much as in the proof of Theorem \\ref{thm:maximal-bounds}, and we will illustrate the point of (ii) again by means of a tree. Observe first that the $\\mathcal{E}_{r}$ are disjoint. This follows from (ii), since $\\mathcal{L}_j \\supseteq \\mathcal{E}_{j}$. \n\n \n \nWe build a congruence tree with root the full interval $[0:N-1]$. Write this as $\\mathcal{N}_{00}$ and write $\\mathcal{N}_{ka}$ for the congruence class of $a$ modulo $p^k$ in $[0:N-1]$, so that $|\\mathcal{N}_{ka} | = \\widetilde{\\chi}_k(a\\,;[0:N-1])$.  All the nodes represent non-singletons, except the bottom-most level, $M$. As before, Figure \\ref{fig:count-tree} has $p=3$, $M\\ge 3$ and shows the tree only up to the third level.    \n \n  \n  Suppose $k_1 =1$, so $\\mathcal{E}_{1}$, as an elementary set, contains one element from each node at the middle  level in the figure. In turn, suppose $\\mathcal{E}_1$ comes from picking one element from each of the red nodes. The set $\\mathcal{L}_1$ is the union of the red nodes. Now, the set $\\mathcal{E}_2$ comes from choosing one element from each node at the $k_2$-level, and the sequence $k_r$ is nonincreasing so $\\mathcal{E}_2$ is drawn from nodes in a level at or higher up in the tree than $\\mathcal{E}_1$  (in this example $k_2$ is either $1$ or $0$). Condition (ii) requires that $\\mathcal{E}_2$ be disjoint from the red nodes, not just from $\\mathcal{E}_1$ which is a (small) subset of the red nodes. \n  \n  In the general case, think of $k_1$ as large (eventually it will be chosen as in Theorem \\ref{thm:maximal-bounds}), so $\\mathcal{E}_1$ comes from a level far down the tree from the root, and then $\\mathcal{E}_2, \\mathcal{E}_3, \\dots$ are, at least, no further down since $k_1 \\ge k_2 \\ge \\cdots$. Condition (ii) requires that $\\mathcal{E}_r$ be assembled from nodes that were not used in assembling any of the $\\mathcal{E}_s$ for $s<r$. It is this property that we exploit to show that $\\bigcup_r\\mathcal{E}_r$ is universal.\n  \n  \n\n\n\\begin{figure*}\n\\centering\n\\begin{tikzpicture}[scale=1.2]\n\\node (0) at (0,6) [circle, minimum size = 1.0cm, fill=gray!10, draw] {$\\mathcal{N}_{00}$};\n\\node (1) at (-3,3) [circle, minimum size = 1.0cm, fill=gray!10,\ndraw] {$\\mathcal{N}_{10}$};\n\\node (2) at (0,3) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\mathcal{N}_{11}$};\n\\node (3) at (3,3) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\mathcal{N}_{12}$};\n\\node (4) at (-4.4,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{20}$};\n\\node (5) at (-3.35,0) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{N}_{23}}$};\n\\node (6) at (-2.3,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{26}$};\n\\node (7) at (-1.1,0) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{N}_{21}}$};\n\\node (8) at (0,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{24}$};\n\\node (9) at (1,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{27}$};\n\\node (10) at (2.2,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{22}$};\n\\node (11) at (3.3,0) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{N}_{25}}$};\n\\node (12) at (4.4,0) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{N}_{28}$};\n\\draw (node cs:name=0) -- (node cs:name =1);\n\\draw (node cs:name=0) -- (node cs:name =2);\n\\draw (node cs:name=0) -- (node cs:name =3);\n\\draw (node cs:name=1) -- (node cs:name =4);\n\\draw (node cs:name=1) -- (node cs:name =5);\n\\draw (node cs:name=1) -- (node cs:name =6);\n\\draw (node cs:name=2) -- (node cs:name =7);\n\\draw (node cs:name=2) -- (node cs:name =8);\n\\draw (node cs:name=2) -- (node cs:name =9);\n\\draw (node cs:name=3) -- (node cs:name =10);\n\\draw (node cs:name=3) -- (node cs:name =11);\n\\draw (node cs:name=3) -- (node cs:name =12);\n\\end{tikzpicture}\n\\caption{Similar to Figure \\ref{fig:tree2} but with root $\\mathcal{N}_{00}=[0:N-1]$,\n this tree shows the relationship between $\\mathcal{N}_{ka}$, for\n  $p=3$, $M\\geq 3$.}\n  \\label{fig:count-tree}\n\\end{figure*}\n\n\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:univ-structure-forward}]\nFix $r$ and $s$ with $k_r< s$, and note that \n\\begin{equation} \\label{eq:chi-in-{0,1}-again}\n\\widetilde{\\chi}_s(a\\,;\\mathcal{E}_{r}) \\in\\{0,1\\}, \\quad  a\\in [0:p^{s}-1],\n\\end{equation}\n from \\eqref{eq:chi-in-{0,1}}. Now suppose \n$\\widetilde{\\chi}_s(a\\,;\\mathcal{E}_{r}) =1$, so one element in $\\mathcal{E}_{r}$ leaves a remainder of $a$ on dividing by $p^s$. Then \n\\begin{equation} \\label{eq:chi=0-t>r}\n\\widetilde{\\chi}_s(a\\,;\\mathcal{E}_{t})=0 \\quad \\text{for all $t>r$},\n\\end{equation}\n i.e., none of the $\\mathcal{E}_{t}$ for $t>r$ will have an element from the congruence class of $a$  modulo $p^s$. This follows (just as described for the tree) from $\\mathcal{E}_{t} \\cap \\mathcal{L}_r=\\emptyset$, and also\n\\begin{align}\n\\mathcal{L}_r &= \\{x  \\in [0:N-1] \\colon x \\equiv e \\text{ mod $p^{k_r+1}$ for some $e \\in\n\\mathcal{E}_{r}$}\\}    \\nonumber \\\\\n&\\supseteq \\{x \\in [0:N-1]: x \\equiv e \\text{ mod $p^s$ for some }e \\in\n\\mathcal{E}_{r} \\} \\nonumber \\label{eq:univ-structure-1}.\n\\end{align}\n\nFrom \\eqref{eq:chi-in-{0,1}-again} and \\eqref{eq:chi=0-t>r} we conclude that\n\\begin{equation} \\label{eq:sum-chi-in-{0,1}}\n\\sum_{r} \\widetilde{\\chi}_s(a\\,;\\mathcal{E}_r) \\in\\{0,1\\}\n\\end{equation}\nfor all $a$, where the sum is over all $r$ with $k_r <s$. \n\nWith this we can show that $\\mathcal{I} = \\bigcup_r \\mathcal{E}_r$ is universal. For any $s$, and for any $a,b \\in [0:p^s-1]$,\n\\begin{align}\n&\\widetilde{\\chi}_s(a\\,;\\mathcal{I}) - \\widetilde{\\chi}_s(b\\,;\\mathcal{I})= \\sum_r \\widetilde{\\chi}_s(a\\,;\\mathcal{E}_r)  - \\sum_r \\widetilde{\\chi}_s(b\\,;\\mathcal{E}_r) \\nonumber \\\\\n& = \\left(\\sum_{r \\,(k_r\\geq s)} \\widetilde{\\chi}_s(a\\,;\\mathcal{E}_r) - \\sum_{r\\,(\n  k_r\\geq s)} \\widetilde{\\chi}_s(b\\,;\\mathcal{E}_r)\\right) +\\\\\n & \\hspace{.5in} \\left(\\sum_{r\\,(\nk_r<s)} \\widetilde{\\chi}_s(a\\,;\\mathcal{E}_r) - \\sum_{r\\,( k_r<s)}\n\\widetilde{\\chi}_s(b\\,;\\mathcal{E}_r) \\right)\\nonumber \\\\\n& = \\sum_{r\\,(k_r<s)} \\widetilde{\\chi}_s(a\\,;\\mathcal{E}_r) - \\sum_{r\\, (k_r<s)}\n\\widetilde{\\chi}_s(b\\,;\\mathcal{E}_r)\\nonumber \\\\  \n&\\hspace{.5in}\\text{ (the first two sums cancel, by \\eqref{eq:chi-constant})}.\\nonumber \n\\end{align}\nFrom \\eqref{eq:sum-chi-in-{0,1}} we have that\n$|\\widetilde{\\chi}_s(a\\,;\\mathcal{I}) - \\widetilde{\\chi}_s(b\\,;\\mathcal{I})| \\leq 1$, \nso $\\mathcal{I}$ is universal. \n\\end{IEEEproof}\n\n\n\\subsection{An Algorithm to Construct Maximal Universal Sets} \\label{subsection:algorithm-maximal}\n\nConsider now the problem of finding\na maximal universal sampling set contained in a  given $\\mathcal{I} \\subseteq [0:p^M-1]$.  Build the congruence class tree with root $\\mathcal{I}$, as in Figure \\ref{fig:tree1}, up to level $M$. The leaves having weight $1$ are singletons in $\\mathcal{I}$, and $\\widetilde{\\chi}_k(a\\,;\\mathcal{I})$, $a \\in [0:p^k-1]$, is the {total weight} at node $(k,a)$.  The problem of constructing $\\Omega(\\mathcal{I})$ is to pick a subset of the leaves so that the tree with root $\\Omega(\\mathcal{I})$ is well balanced at each level. By `well balanced' we mean that at any given level, all the subtrees have roughly equal weight, corresponding to the condition $|\\widetilde{\\chi}_k(a\\,;\\Omega(\\mathcal{I})) -\\widetilde{\\chi}_k(b\\,;\\Omega(\\mathcal{I}))| \\le ~1$. The following algorithm realizes this and provides the value of $|\\Omega(\\mathcal{I})|$.  It marries the construction of elementary sets in Theorem \\ref{thm:maximal-bounds} with an iterative version of the method used in the proof of Lemma \\ref{lem:univ-structure-forward}. \n\nLet $\\mathcal{I} \\subseteq [0:p^M-1]$. Initialize with $\\mathcal{I}_1 = \\mathcal{I}$, and $r=1$. \n\\begin{enumerate}\n\\item Let ${k}_r$ be the largest integer such that no congruence class in $\\mathcal{I}_r\/p^{{k_r}}$ is empty. \n\n\\item Construct an elementary set $\\mathcal{I}_{r}^\\dagger \\subseteq \\mathcal{I}_r$ by choosing one element of $\\mathcal{I}_r$ from each congruence class modulo $p^{{k}_r}$. \n(There may not be a unique choice, and this is the reason why there may be many universal sets contained in $\\mathcal{I}$.)\n\\item Define $\\mathcal{L}_r \\supseteq \\mathcal{I}_r^\\dagger$ by \\[\n\\mathcal{L}_r= \\{j \\in \\mathcal{I} \\colon j \\equiv  i \\text{ mod $p^{{k}_r+1}$ for some $i\\in \\mathcal{I}_r^\\dagger$}\\}.\n\\]\n\n\\item Let $\\mathcal{I}_{r+1} = \\mathcal{I}_r \\setminus \\mathcal{L}_r$.  Stop if $\\mathcal{I}_{r+1} = \\emptyset$. Else increment $r$ to ${r+1}$ and go to (1). \n\\end{enumerate}\n\nNote the following:\n\\begin{enumerate}[(i)]\n\\item At each step of the algorithm the size of $\\mathcal{I}_r$ is reduced by\n  $|\\mathcal{L}_r|\\geq |\\mathcal{I}_r^\\dagger|=p^{k_r} \\geq 1$:\n  \\[\n  |\\mathcal{I}_{r+1}| \\leq |\\mathcal{I}_r| -\n  p^{k_r}.\n  \\]\n   Since $\\mathcal{I}= \\mathcal{I}_1$ is a\n  finite set, the algorithm terminates at some point. \n\\item The $k_r$ are nonincreasing: \n\\[\n k_1 \\geq k_2 \\geq k_3 \\geq \\ldots.\n \\]\n \\end{enumerate}\n \n We can now state\n \\begin{theorem}\n\\label{theorem:maximal-precise}\nWith $k_r$, $r \\geq 1$, defined as above, we have \n\\begin{equation} \\label{eq:|Omega(I)|}\n|\\Omega(\\mathcal{I})| = \\sum_r p^{k_r}.\n\\end{equation}\n One possible maximal universal sampling set is \n\\begin{equation} \\label{eq:maximal-union}\n  \\Omega(\\mathcal{I}) = \\bigcup_r \\mathcal{I}_r^\\dagger.\n\\end{equation}\n  By construction this is a disjoint union.\n\\end{theorem}\n\nHere is an example of the algorithm in action. Let $N=2^5$ and \n\\[\n\\mathcal{I} = \\{0,1,2,3,4,6,7,8,9,10,12,14, 15\\}= \\mathcal{I}_1.\n\\] \n\\begin{enumerate}\n\\item $(r=1)$ Note that $\\widetilde{\\chi}_3(5\\,;\\mathcal{I}_1) = 0$, and that no values $\\widetilde{\\chi}_2(i\\,;\\mathcal{I}_1)$ are zero. Hence $k_1 = 2$. Form $\\mathcal{I}_1^\\dagger$ by taking one element from each congruence class in $\\mathcal{I}_1$ modulo $2^{k_1} = 4$, e.g. $\\mathcal{I}_1^\\dagger= \\{0,1,2,3\\}$.\nThen $\\mathcal{L}_1=\\{0,1,2,3,8,9,10 \\}$ is the set of all elements of $\\mathcal{I}_1$ that leave a remainder of $0,1,2$ or $3$ on dividing by $2^{k_1+1} = 8$. Removing such numbers from $\\mathcal{I}_1$, we have $\\mathcal{I}_2 = \\mathcal{I}_1 \\setminus \\mathcal{L}_1 = \\{4, 6, 7, 12, 14, 15 \\}$. \n\\item $(r=2)$ Now $\\widetilde{\\chi}_2(1\\,;\\mathcal{I}_2) = 0$ while $\\widetilde{\\chi}_1(0\\,;\\mathcal{I}_2)$, $\\widetilde{\\chi}_1(1\\,;\\mathcal{I}_2) \\neq 0$ so $k_2 = 1$. Let $\\mathcal{I}_2^\\dagger = \\{4, 7\\}$. Then $\\mathcal{L}_2=\\{4,7,12,15\\}$ is the set of all elements in $\\mathcal{I}_2$ that leave a remainder of $4 \\text{ mod }4 = 0 $ or $7 \\text{ mod }4 = 3$ on dividing by $2^{k_2+1} = 4$. Removing such numbers from $\\mathcal{I}_2$, we have $\\mathcal{I}_3 = \\mathcal{I}_2 \\setminus \\mathcal{L}_2 = \\{6, 14 \\}$. \n\\item $(r=3)$ Now clearly $k_3 = 0$. Let $\\mathcal{I}_3^\\dagger = \\{6\\}$. Then $\\mathcal{L}_4 = \\{6,14\\}$, $\\mathcal{I}_4 = \\emptyset$ and the algorithm terminates.\n\\end{enumerate}\nAccording to the theorem, we have $|\\Omega(\\mathcal{I})| = 2^{k_1} + 2^{k_2} + 2^{k_3} = 7 $, and an example $\\Omega(\\mathcal{I})$ is given by $\\mathcal{I}_1^\\dagger \\cup \\mathcal{I}_2^\\dagger\\cup \\mathcal{I}_3^\\dagger = \\{0,1,2,3,4, 6, 7\\}$. \n\nWe have several additional comments. First, we can say more about the formula for $|\\Omega(\\mathcal{I})|$. Since the $k_r$'s are nonincreasing,  a typical sequence is, say,\n\\[\n\\underbrace{l_1, l_1,\\ldots}_{\\alpha_1 \\text{ times }} \\quad \\underbrace{l_2, l_2, \\ldots}_{ \\alpha_2 \\text{ times }} \\quad \\underbrace{l_3, l_3, \\ldots}_{ \\alpha_3 \\text{ times }}, \\quad \\dots\n\\]\nwith $l_1 > l_2 > l_3$. Given this, equation \\eqref{eq:|Omega(I)|} appears as \n\\begin{equation} \\label{eq:Omega(I)-2}\n|\\Omega(\\mathcal{I})| = \\alpha_1p^{l_1} + \\alpha_2p^{l_2} + \\alpha_3p^{l_3} + \\ldots .\n\\end{equation}\nIn fact, effectively, Theorem \\ref{theorem:maximal-precise} constructs a\nbase $p$ expansion of $|\\Omega(\\mathcal{I})|$ because each power of $p$ appears at most $p-1$ times.\n\n\\begin{corollary} \\label{corollary:base-p-expansion}\nLet $\\mathcal{I} \\subseteq [0:p^M-1]$. The formula \\eqref{eq:Omega(I)-2} is of the form,\n\\[\n|\\Omega(\\mathcal{I})| = \\sum_r p^{k_r} = \\sum_s \\alpha_s p^{l_s},\n\\]\nwith $l_1 > l_2 > l_3 > \\ldots$ and $\\alpha_s \\in [0:p-1]$ for all $s$.\n\\end{corollary}\n\n\\begin{IEEEproof}  Begin with $\\Omega(\\mathcal{I}) = \\bigcup_r\\mathcal{I}_r^\\dagger$.\nSince the $\\mathcal{I}_r^\\dagger$ are disjoint, we have\n\\begin{equation}\n\\label{eq:uncen-phi-k-0}\n\\begin{aligned}\n\\sum_{r=1}^{\\alpha_1} \\widetilde{\\chi}_{l_1+1}(a\\,;\\mathcal{I}_r^\\dagger) &= \\widetilde{\\chi}_{l_1+1}(a\\,;\\bigcup_{r=1}^{\\alpha_1}\\mathcal{I}_r^\\dagger) \\nonumber\\\\\n& \\leq \\widetilde{\\chi}_{l_1+1}(a\\,; \\Omega(\\mathcal{I})), \\quad a\\in[0:p^{l_1+1}-1].\n\\end{aligned}\n\\end{equation}\nSumming this over all $a \\in [0:p^{l_1 +1}-1]$ we have \n\\begin{equation}\n\\label{eq:uncen-phi-k}\n\\begin{aligned}\n\\alpha_1p^{l_1} &= \\sum_{r=1}^{\\alpha_1} |\\mathcal{I}_r^\\dagger| = \\sum_{r=1}^{\\alpha_1} \\sum_a\n\\widetilde{\\chi}_{l_1+1}(a\\,;\\mathcal{I}_r^\\dagger)\\\\\n& \\leq \\sum_i\\widetilde{\\chi}_{l_1+1}(a\\,; \\Omega(\\mathcal{I})) = |\\Omega(\\mathcal{I})| < p^{l_1+1}, \n\\end{aligned}\n\\end{equation}\nso $\\alpha_1 < p$. For the last inequality in\n\\eqref{eq:uncen-phi-k} we have used the upper bound from part (ii) in Theorem\n\\ref{thm:maximal-bounds}. We have also used that $|\\mathcal{I}_r| = p^{k_r}$.\nThe proof for other $\\alpha_s$ is similar. For example, to prove that\n$\\alpha_2<p$ we start with \n\\[\n\\sum_{r=\\alpha_1+1}^{\\alpha_2} \\widetilde{\\chi}_{l_2+1}(a\\,;\\mathcal{I}_r^\\dagger) \\leq\n\\widetilde{\\chi}_{l_2+1}\\left(a\\,; \\Omega(\\mathcal{I}_{a_1+1})\\right)\n\\]\ninstead of \\eqref{eq:uncen-phi-k-0}. \n \\end{IEEEproof}\n\n\n\nIf the algorithm above were initialized with a universal set $\\mathcal{I}$,\nthen from Theorem \\ref{theorem:maximal-precise} we would obtain\n$\\mathcal{I}= \\Omega(\\mathcal{I})  =\n\\bigcup _ r \\mathcal{I}_r^\\dagger$. \nThis allows us to conclude that any universal set $\\mathcal{I}$ is a union of\nelementary universal sets.\nMoreover, the sets $\\mathcal{I}^\\dagger_r$ defined by the algorithm satisfy\nconditions in Lemma \\ref{lem:univ-structure-forward}. For condition (ii), note that \nin the algorithm the set $\\mathcal{I}_r$ is recursively defined as $ \\mathcal{I}_r = \\mathcal{I}_{r-1}\\setminus\n\\mathcal{L}_{r-1}$, so that \n\\[\n\\mathcal{I}_r = \\left(\\left(\\left(\\mathcal{I} \\setminus\n    \\mathcal{L}_1\\right)\\setminus \\mathcal{L}_2\\right) \\ldots\n\\setminus \\mathcal{L}_{r-1}\\right)= \\mathcal{I} \\setminus \\left(\n\\bigcup_{j=1}^{r-1}\\mathcal{L}_j \\right).\n\\]\nHence $\\mathcal{I}_r \\cap\n\\left(\\bigcup_{j=1}^{r-1} \\mathcal{L}_j\\right) = \\emptyset$. Then the sets $\\mathcal{I}^\\dagger_r$, obtained by the algorithm, satisfy $\\mathcal{I}_r^\\dagger \\cap\n\\left(\\bigcup_{j=1}^{r-1} \\mathcal{L}_j\\right) = \\emptyset$, since\n$\\mathcal{I}^\\dagger_r \\subseteq \\mathcal{I}_r$. Putting all these comments together  we have the converse of\n  Lemma \\ref{lem:univ-structure-forward}, and then adding Theorem \\ref{theorem:maximal-precise} we can state\n\n\\begin{corollary}\n\\label{cor:univ-structure-final}\n $\\mathcal{I} \\subseteq [0:p^M-1]$ is universal if and only if there\nexist \n\\begin{enumerate}[(i)]\n\\item A nonincreasing finite sequence $k_1 \\ge k_2 \\ge \\cdots \\ge 0$, with each value of $k_r$ repeating at most $p-1$ times;\n\\item Sets $\\mathcal{I}^\\dagger_r\\subseteq \\mathcal{I}$ with  $\\mathcal{I} = \\bigcup_r \\mathcal{I}^\\dagger_r$;\n\\end{enumerate}\nsuch that \n\\begin{enumerate}\n\\item[(iii)] $\\mathcal{I}_r^\\dagger$ is a $k_r$-elementary universal set;\n\\item[(iv)]  $\\mathcal{I}_r^\\dagger \\cap \\left(\\bigcup_{j=1}^{r-1} \\mathcal{L}_j\\right)\n  = \\emptyset$, where\n\\[\n\\mathcal{L}_j=\\{x \\in [0:N-1]: x \\equiv i \\text{ mod $p^{k_j}+1$ for some $i \\in\n\\mathcal{I}_j^\\dagger$}\\}. \\nonumber\n\\]\n\\end{enumerate}\n\\end{corollary}\n\nNote that from (i), (ii) and (iii) we can also\nconclude that \\[\n|\\mathcal{I}| = \\sum_r |\\mathcal{I}_r^\\dagger| = \\sum_r p^{k_r},\n\\]\nso the $k_r$ are the powers of $p$ appearing in the base-$p$\nexpansion of $|\\mathcal{I}|$, taken with repetitions.\nFor example with $N=9$, $|\\mathcal{I}| = 7 = 2\\cdot 3^1 + 1 \\cdot 3^0$, we expect the\nuniversal set $\\mathcal{I} = \\mathcal{I}_1^\\dagger \\cup \\mathcal{I}_2^\\dagger \\cup\n\\mathcal{I}_3^\\dagger$ with $\\mathcal{I}_1^\\dagger$ and $\\mathcal{I}^\\dagger_2$ being\n$1$-elementary, and $\\mathcal{I}_3^\\dagger$ being $0$-elementary.\nCorollary \\ref{cor:univ-structure-final} implies that the $k_r$ read\noff from the base-$p$ expansion of $|\\mathcal{I}|$ must be the same as the\n$k_r$ generated by the algorithm if $\\mathcal{I}$ is universal.\n\n\\begin{remark}[Universal sets of prescribed order]\n\\label{rem:algo-mod}  As~it~stands, the  \nalgorithm finds a universal set of the largest size contained in $\\mathcal{I}$. With Corollary \\ref{cor:univ-structure-final} we can now modify the algorithm to solve the following problem:\n\\begin{quote}\n\\textit{Given  a set $\\mathcal{I} \\subseteq [0:p^M-1]$, and  \n$d \\le |\\Omega(\\mathcal{I})|$, find a universal set $\\mathcal{J} \\subseteq \\mathcal{I}$ with $|\\mathcal{J}| = d$.}\n\\end{quote}\nWe follow the algorithm as in steps 1-4, but we change the definition of $k_r$ in Step 1. Write the base-p expansion of $d$ with repetitions, $d = \\sum _r p^{k_r}$, read off the $k_r$ as the powers of $p$ that appear in the expansion, and arrange the $k_r$ in nonincreasing order. This ensures that condition (i) in Corollary \\ref{cor:univ-structure-final} is satisfied. The construction of the $\\mathcal{I}_r^\\dagger$ in Steps 2-4 of the algorithm will ensure that (iii) and (iv) are satisfied. We conclude that with the $\\mathcal{I}_r^\\dagger$ so obtained by the algorithm the set $\\mathcal{J} = \\bigcup_r \\mathcal{I}_r^\\dagger \\subset \\mathcal{I}$ is universal, and it is of the right size by definition of the $k_r$.\n\\end{remark}\n\n\nFinally, we have\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{theorem:maximal-precise}] As observed above, the $\\mathcal{I}_r^\\dagger$ generated by the algorithm satisfy\nthe hypotheses of Lemma \\ref{lem:univ-structure-forward},  so the set\n$\\bigcup_r \\mathcal{I}^\\dagger_r$ is universal. If we show \n\\[\n|\\Omega(I)| \\leq \\sum_rp^{k_r},\n\\]\nthen Theorem \\ref{theorem:maximal-precise} follows.\n\nFor this we prove\n\\begin{equation} \\label{eq:maximal-<=}\n|\\Omega(\\mathcal{I}_r)| \\le p^{k_r}+ |\\Omega(\\mathcal{I}_{r+1})|.\n\\end{equation}\nWe appeal to Theorem \\ref{thm:maximal-bounds} to find a maximal universal sampling set $\\mathcal{A}$ with $\\mathcal{I}_r^{\\dagger} \\subseteq \\mathcal{A} \\subseteq \\mathcal{I}_r$, and we will show\n\\begin{equation}\n\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger \\quad \\text{is universal},\\label{eq:last-condition-1} \n\\end{equation}\n\\begin{equation}\n\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger \\subseteq \\mathcal{I}_{r+1}. \\label{eq:last-condition-2}\n\\end{equation}\nSince $|\\mathcal{A}| = |\\Omega(\\mathcal{I}_r)|$ These imply\n\\[\n|\\Omega(\\mathcal{I}_r)| - p^{k_r} = |\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger| \\le |\\Omega(\\mathcal{I}_{r+1})|,\n\\]\nwhich is \\eqref{eq:maximal-<=}.\n\n\nFirst \\eqref{eq:last-condition-1}. Now,\n\\[\n\\widetilde{\\chi}_s(a \\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger) = \\widetilde{\\chi}_s(a \\,;\\mathcal{A}) - \\widetilde{\\chi}_s(a \\,;\\mathcal{I}_r^\\dagger), \\quad a \\in[0:p^s-1],\n\\]\nand for $s \\le k_r$ the second term is constant,\n\\[\n\\widetilde{\\chi}_s(a\\,;\\mathcal{I}_r^\\dagger) = p^{k_r-s},\n\\]\nfrom \\eqref{eq:chi-constant}. Since $\\mathcal{A}$ is universal, $|\\widetilde{\\chi}_{s}(a\\,;\\mathcal{A}) - \\widetilde{\\chi}_{s}(b\\,;\\mathcal{A})| \\le 1$ for all $a, b \\in [0:p^s-1]$ and for all $s$, so we at least have\n\\[\n|\\widetilde{\\chi}_s(a\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger) - \\widetilde{\\chi}_s(b\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger)| \\le 1,\n\\]\nfor all $s \\le k_r$. We need to check that this inequality continues to hold for $s \\ge k_r+1$.\n\nAs we have argued before,  by the definition of $k_r$ at least one congruence class in $\\mathcal{I}_r\/p^{s}$ is empty when $s \\ge k_r+1$, so $\\widetilde{\\chi}_s(a_0\\,;\\mathcal{I}_r)=0$ for some $a_0$, and because $\\mathcal{A} \\subseteq \\mathcal{I}_r$ is universal we have $\\widetilde{\\chi}_s(a\\,;\\mathcal{A}) \\le 1$ for all $a$. Furthermore,  $\\mathcal{I}_r^\\dagger \\subseteq \\mathcal{A}$ implies\n\\[\n0 \\le  \\widetilde{\\chi}_s(a\\,;\\mathcal{A}) - \\widetilde{\\chi}_s(a\\,;\\mathcal{I}_r^\\dagger) = \\widetilde{\\chi}_s(a\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger).\n\\]\nHence the values of\n$\\widetilde{\\chi}_s(a\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger)$ are in $\\{0,1\\}$ and consequently\n\\[\n|\\widetilde{\\chi}_s(a\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger)-\\widetilde{\\chi}_s(b\\,;\\mathcal{A}\\setminus\\mathcal{I}_r^\\dagger)| \\le 1,\n\\]\nfor all $s \\ge k_r+1$. This establishes that $\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger$ is universal.\n\nWe prove \\eqref{eq:last-condition-2} by contradiction.  If it were not true that $\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger \\subseteq \\mathcal{I}_{r+1}$ then there would exist an $x \\in (\\mathcal{A}\\setminus \\mathcal{I}_r^\\dagger) \\cap \\mathcal{L}_r$. Then $\\widetilde{\\chi}_{k_r+1}([x]_{k_r+1}\\,;\\mathcal{A}) = 2$, for on dividing by $p^{k_r+1}$,  $x$ leaves a remainder of $[x]_{k_r+1}$ (by definition) and so does one other element in $\\mathcal{I}_r^\\dagger$. But this contradicts  $\\widetilde{\\chi}_s(a\\,;\\mathcal{A}) \\le 1$ for $s \\ge k_r+1$ from the preceding paragraph. \n\nThis completes the proof of Theorem  \\ref{theorem:maximal-precise}. \n\\end{IEEEproof}\n\n\\begin{remark}\nWe can give an upper bound for the computational complexity of the algorithm for constructing a universal sampling set of size $d$ (including constructing a maximal universal sampling set). Within an iteration, in the worst case the algorithm makes a complete\npass over all the nodes of the tree once, and the the number of nodes is $O(N)$. Further, the number of iterations is $\\alpha_1+\\alpha_2+\\cdots +\\alpha_M$ where\n\\[\nd = \\alpha_1p^{M-1} + \\alpha_{2}p^{M-2} + \\ldots + \\alpha_{M-1}p + \\alpha_M.\n\\]\nHence the largest number of iterations is $(p-1)M$, and the complexity\nof the algorithm is at most $O(N \\log N)$.\n\\end{remark}\n\n\n\\subsection{Counting Universal Sets} \\label{subsection:counting-universal}\n\nThe preceding structure theorems\n  allow us to find the number of universal sampling sets $\\mathcal{I} \\subseteq [0:p^M-1]$ of size $d$. The formula uses the digits from the   base-$p$ expansion of $d$, and as above we let\n\\begin{equation}\n\\nonumber\nd = \\alpha_1p^{M-1} + \\alpha_{2}p^{M-2} + \\ldots + \\alpha_{M-1}p + \\alpha_M,\n\\end{equation}\nwhere $0 \\leq \\alpha_i < p$. For $i=0,1,\\dots, M$ define\n\\[\nd_i= \\sum_{j=i+1}^M\\alpha_jp^{M-j}.\n\\]\nHence $d_0=d$ and $d_M=0$.\n\n\n\\begin{theorem} \\label{thm:count}\nThe number of universal sampling sets in $[0:p^M-1]$ of size $d$ is \n\\[\n\\mathcal{C}(d,p^M) = \\prod_{i=1}^M \\binom{p}{\\alpha_i+1}^{d_i}\\binom{p}{\\alpha_i}^{p^{M-i}-d_i} .\n\\]\n\\end{theorem}\n\n\\begin{IEEEproof} The proof goes by establishing a recurrence relation for $\\mathcal{C}$ in the $d_i$.\\footnote{We are grateful to a reviewer for suggesting a way to make greater use of the recursive aspect of our original argument, resulting in a much shorter and cleaner proof.} Let $\\mathcal{I}$ be a universal sampling set of size $d$ and construct the congruence tree as in Figure \\ref{fig:tree1} with root $\\mathcal{I}_{00}=\\mathcal{I}$. We first note that $d_1$ of the nodes at level $M-1$ have weight $\\alpha_1+1$ and the remaining $p^{M-1}-d_1$ nodes have weight $\\alpha_1$, where\n\\[\nd_1 = \\sum_{i=2}^M \\alpha_i p^{M-i}.\n\\]\n The proof for this is along the same lines as the argument in the proof of Theorem \\ref{theorem:universal}, $(i) \\Longleftrightarrow (ii)$. \nFigure \\ref{fig:tree10} illustrates this. The singleton blue nodes at the bottom level are the elements of $\\mathcal{I}$, and the other nodes (which would be the singletons $\\{6\\}$ and $\\{7\\}$) are empty. The red nodes at the penultimate\nlevel represent the nodes that have weight $\\alpha_1 +1$ (and there\nare $d_1$ of them). \n\n\\begin{figure*}\n\\centering\n\\begin{tikzpicture}[scale=1.2]\n\\node (0) at (0,8) [circle, minimum size = 1.0cm, fill=gray!10, draw] {$\\mathcal{I}_{00}$};\n\\node (1) at (-2,6) [circle, minimum size = 1.0cm, fill=gray!10,\ndraw] {$\\mathcal{I}_{10}$};\n\\node (2) at (2,6) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\mathcal{I}_{11}$};\n\\node (3) at (-2.75,4) [circle, minimum size = 1.0cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{I}_{20}}$};\n\\node (4) at (-1.25,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{22}$};\n\\node (5) at (1.25,4) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{I}_{21}}$};\n\\node (6) at (2.75,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{23}$};\n\\node (7) at (-4,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{0\\}}$};\n\\node (8) at (-2.8,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{4\\}}$};\n\\node (9) at (-1.6,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{2\\}}$};\n\\node (10) at (-0.55,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\{\\,\\}$};\n\\node (11) at (0.5,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{1\\}}$};\n\\node (12) at (1.7,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{5\\}}$};\n\\node (13) at (2.9,2) [circle, minimum size = 1cm, fill=blue!50, draw]\n{$\\underset{\\text{blue}}{\\{3\\}}$};\n\\node (14) at (4,2) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\{\\,\\}$};\n\\draw (node cs:name=0) -- (node cs:name =1);\n\\draw (node cs:name=0) -- (node cs:name =2);\n\\draw (node cs:name=1) -- (node cs:name =3);\n\\draw (node cs:name=1) -- (node cs:name =4);\n\\draw (node cs:name=2) -- (node cs:name =5);\n\\draw (node cs:name=2) -- (node cs:name =6);\n\\draw (node cs:name=3) -- (node cs:name =7);\n\\draw (node cs:name=3) -- (node cs:name =8);\n\\draw (node cs:name=4) -- (node cs:name =9);\n\\draw (node cs:name=4) -- (node cs:name =10);\n\\draw (node cs:name=5) -- (node cs:name =11);\n\\draw (node cs:name=5) -- (node cs:name =12);\n\\draw (node cs:name=6) -- (node cs:name =13);\n\\draw (node cs:name=6) -- (node cs:name =14);\n\\end{tikzpicture}\n\\caption{The congruence class tree for $N=8$. The universal sampling set $\\{0,1,2,3,4,5\\}$ of size\n  $d=6$ is represented by the blue nodes at the bottom level. The\n  red nodes at the penultimate level represent the nodes that\nhave weight 2, the rest of the nodes at the penultimate level have\nweight 1.}\n\\label{fig:tree10}\n\\end{figure*}\n\nNow remove the bottom level of the tree, effectively making\n$N=p^{M-1}$, and resulting in Figure \\ref{fig:tree11}. If the starting set (the blue nodes in\nFigure \\ref{fig:tree10}) is universal, then so must be the set formed\nby the red nodes in Figure \\ref{fig:tree11}. Hence the number of ways of\nchoosing the red nodes is the same as the number of universal sampling sets of\nsize $d_1$ in $[0:p^{M-1}-1]$, that is $\\mathcal{C}(d_1,p^{M-1})$..\n\nOnce the red nodes are chosen, we need to choose the blue nodes by taking\n$\\alpha_1+1$ elements from the red nodes and $\\alpha_1$ elements from\nthe remaining (non-red) nodes, which can be done in\n\\[\n\\binom{p}{\\alpha_1+1}^{d_1}\\binom{p}{\\alpha_1}^{p^{M-1}-d_1}\n\\]\nways. Hence\n\\begin{equation} \\label{eq:count-recurrence}\n\\mathcal{C}(d, p^M) = \\binom{p}{\\alpha_1+1}^{d_1}\\binom{p}{\\alpha_1}^{p^{M-1}-d_1}\\mathcal{C}(d_1, p^{M-1}).\n\\end{equation}\nThis full formula follows.\n\\end{IEEEproof}\n\n\\begin{figure\n\\centering\n\\begin{tikzpicture}[scale=1.2]\n\\node (0) at (0,8) [circle, minimum size = 1.0cm, fill=gray!10, draw] {$\\mathcal{I}_0$};\n\\node (1) at (-2,6) [circle, minimum size = 1.0cm, fill=gray!10,\ndraw] {$\\mathcal{I}_{10}$};\n\\node (2) at (2,6) [circle, minimum size = 1.0cm, fill=gray!10, draw]\n{$\\mathcal{I}_{11}$};\n\\node (3) at (-2.75,4) [circle, minimum size = 1.0cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{I}_{20}}$};\n\\node (4) at (-1.25,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{22}$};\n\\node (5) at (1.25,4) [circle, minimum size = 1cm, fill=red!50, draw]\n{$\\underset{\\text{red}}{\\mathcal{I}_{21}}$};\n\\node (6) at (2.75,4) [circle, minimum size = 1cm, fill=gray!10, draw]\n{$\\mathcal{I}_{23}$};\n\\draw (node cs:name=0) -- (node cs:name =1);\n\\draw (node cs:name=0) -- (node cs:name =2);\n\\draw (node cs:name=1) -- (node cs:name =3);\n\\draw (node cs:name=1) -- (node cs:name =4);\n\\draw (node cs:name=2) -- (node cs:name =5);\n\\draw (node cs:name=2) -- (node cs:name =6);\n\n\\end{tikzpicture}\n\\caption{Remove the bottom level of the tree in Figure \\ref{fig:tree10}. The resulting red nodes\n  are a universal sampling set in $[0:3]$}\n  \\label{fig:tree11}\n\\end{figure}\n\n\nOne special case of the counting formula is easy to evaluate.\n\n\\begin{corollary}\nLet $d = p^k$ where $k <M$. Then the number of universal sets of size $d$ in $[0:p^M-1]$ is $(p^M\/d)^d$.  \n\\end{corollary}\n\n\nIn particular when $N=2^M$, and $d = 2^{M-1}=N\/2$, the number of universal sets is $2^{N\/2}$.  \nOn the other hand, the total number of sets of size $2^{M-1}$ in $[0:2^N-1]$ is \n\\[\n\\binom{N}{N\/2} \\approx 2^N\/\\sqrt{\\pi N}\n\\] \nby Stirling's approximation. Hence the fraction of sets that are universal is approximately $\\sqrt{\\pi N}\/2^{N\/2}$, which decreases exponentially with $N$. \n\n\n\nThe function $\\mathcal{C}(d,p^M)$ is certainly complicated, but \nit has some remarkable properties.  Though not clear from the formula, we have \n\\[\n\\mathcal{C}(d,p^M) = \\mathcal{C}(p^M-d,p^M).\n\\]\n This follows from the following lemma, which is itself a simple but interesting property of universal sampling sets.\n\n\\begin{lemma} \\label{lemma:complement-universal-is-universal}\n If $\\mathcal{A} \\subseteq[0:p^M-1]$ is a universal sampling set then so is $\\mathcal{A}'=[0:p^M-1] \\setminus \\mathcal{A}$.\n\\end{lemma}\n\nThis extends the bracelet property of universal sampling sets, though for bracelets we need not assume that $N$ is a prime power. \n\n\\begin{IEEEproof}\nFor any $0 \\le k \\le M$ and $a \\in [0:p^k-1]$,\n\\[\n\\begin{aligned}\n\\widetilde{\\chi}_k(a\\,;\\mathcal{A}') &= \\widetilde{\\chi}_k(a\\,;[0:p^M-1]) - \\widetilde{\\chi}_k(a\\,;\\mathcal{A})\\\\\n&= p^{M-k} - \\widetilde{\\chi}_k(a\\,;\\mathcal{A}).\n\\end{aligned}\n\\]\n \nNext, since \n\\[\n|\\widetilde{\\chi}_k(a\\,;\\mathcal{A}) - \\widetilde{\\chi}_k(b\\,;\\mathcal{A})| \\leq 1,\n\\]\nfor all $a,b \\in [0:p^k-1]$, it follows that \n\\[\n|\\widetilde{\\chi}_k(a\\,;\\mathcal{A}') - \\widetilde{\\chi}_k(b\\,;\\mathcal{A}')| \\leq 1 .\n\\]\n\\end{IEEEproof}\n\n\nFigure \\ref{fig:count} displays $\\log \\mathcal{C}(d,5^M)$ as a function of $d$ as $M$ takes\nincreasing values. The plots show the symmetry, $\\mathcal{C}(d,p^M) = \\mathcal{C}(p^M-d,p^M)$, but they show much more. We can observe the following:\n\\begin{enumerate}[(i)]\n\\item  There are a series of bumps on several (visible) scales. One\n  cannot fail to notice that at each scale the number of bumps in the\n  graph is $5$, which is the prime $p$ here. Experiments with other\n  primes have similar plots and in each case indicate   that the number of bumps is equal to the prime. \n  \\end{enumerate}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width =\\textwidth]{Fig-7}\n\\caption{Plots of $\\log\\mathcal{C}(d,p^M)$ \\textit{vs} $d$ for powers of $p=5$. Note the $5$ bumps on different scales.}\n\\label{fig:count}\n\\end{figure*}\n\n\n\\begin{enumerate}\n\\item[(ii)] With increasing $M$ the plots of the\n  count are somehow converging in shape -- they all start to look\n  similar. \n \\end{enumerate}\n  The second point can indeed be quantified. One can show that for each \n  $\\alpha \\in [0,1]$, \n\\[\n\\lim_{M \\rightarrow \\infty}\\frac{\\log \\mathcal{C}(\\lfloor\\alpha p^M\\rfloor,p^M)}{p^M}\n\\]\nexists. See \\cite{Aditya:thesis}.  This compares nicely with the fact that a similar\nfunction with $\\mathcal{C}(d,N)$ replaced by $\\binom{N}{d}$  also converges, and to the entropy function:\n\\[\n\\begin{aligned}\n\\lim_{M \\rightarrow \\infty}\\left(\\frac{1}{p^M}\\log \\binom{p^M}{\\lfloor\\alpha  p^M\\rfloor}\\right) &= \\alpha \\log \\frac{1}{\\alpha} + (1-\\alpha)\\log\\frac{1}{1-\\alpha}\\\\\n& =H(\\alpha).\n\\end{aligned}\n\\]\nThis is the limiting case of counting \\emph{all} index sets. \n\n\nPlots of\n\\begin{equation} \\label{eq:count-limit}\n\\mathcal{H}_p(\\alpha) = \\lim_{M \\rightarrow \\infty}\\frac{\\log \\mathcal{C}(\\lfloor\\alpha p^M\\rfloor,p^M)}{p^M}, \\quad 0\\le \\alpha \\le 1,\n\\end{equation}\nare shown in Figures \\ref{fig:count-entropy} and\n\\ref{fig:count-converging} for several values of $p$, along with a plot of $H(\\alpha)$.  \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = .9\\textwidth]{Fig-8}\n\\caption{Plots of the limit of the counting functions for $p=2,5$ compared to the Entropy function. Note the self-similarity as it depends on the prime.}\n\\label{fig:count-entropy}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width = .9\\textwidth]{Fig-9}\n\\label{fig:count-converging-1}\n\\caption{Similar to Figure \\ref{fig:count-entropy} with $p=3,7$}\n\\label{fig:count-converging}\n\\end{figure*}\n\n\n\n\nThe plots of $H_p(\\alpha)$ seem to satisfy observation (i), that the curves have $p$ bumps at each scale.  Here is an explanation. In the notation of Theorem \\ref{thm:count}, suppose $\\alpha_1=0$ (i.e., $d <p^{M-1}$). Then $d_1=d$ and we have, as in \\eqref{eq:count-recurrence},\n\\[\n\\begin{aligned}\n\\mathcal{C}(d,p^M) &= \\binom{p}{1}^{d_1}\\binom{p}{0}^{p^{M-1}-d_1}\\mathcal{C}(d_1,p^{M-1})\\\\\n& = p^d\\mathcal{C}(d,p^{M-1}).\n\\end{aligned}\n\\]\nLet $M \\rightarrow \\infty$, so $d\/p^M \\rightarrow \\alpha$ (with $\\alpha < p$). Then with reference to \\eqref{eq:count-limit}, \n\\[\n\\mathcal{H}_p(\\alpha) = \\lim_{M\\rightarrow \\infty}\\left(\\frac{d}{p^M}\\log p + \\mathcal{H}_p(p\\alpha)\\right) = \\alpha \\log p +\\mathcal{H}_p(p\\alpha),\n\\]\nleading to the self-similar plots we observe. \n\n\n\\subsection{Maximal and Minimal Universal Sampling Sets} \\label{subsection:maximal-minimal}\n\nAlong with maximal universal sets is the allied notion of  minimal universal sets.\n\n\\begin{defn} \\label{definition:minimal}\nLet $\\mathcal{I} \\subseteq [0:N-1]$. A \\emph{minimal universal sampling set} for $\\mathcal{I}$ is a universal sampling set of smallest cardinality that contains $\\mathcal{I}$.\n\\end{defn}\n\nAgain we need a notation and  we let $\\Phi(\\mathcal{I})$ denote a generic minimal universal sampling set containing $\\mathcal{I}$. Thus $|\\Phi(\\mathcal{I})| \\le |\\mathcal{J}|$ for any universal sampling set $\\mathcal{J} \\supseteq \\mathcal{I}$.\n\nLet us show one way that maximal and minimal universal sampling sets are related. The proof relies on Lemma \\ref{lemma:complement-universal-is-universal} from the previous subsection.\n\n\\begin{theorem} \\label{theorem:maximal-minimal}\nLet $\\mathcal{I} \\subset [0:p^M-1]$, $\\mathcal{I}'=[0:p^M-1]\\setminus \\mathcal{I}$. Then\n\\[\n|\\Phi(\\mathcal{I})| = p^M-|\\Omega(\\mathcal{I}')|.\n\\]\n\\end{theorem}\n\n\\begin{IEEEproof}\nLet $\\mathcal{A}'=[0:p^M-1]\\setminus \\Phi(\\mathcal{I})$. Then $\\mathcal{A}'$ is universal by Lemma \\ref{lemma:complement-universal-is-universal}. Since $\\Phi(\\mathcal{I}) \\supseteq \\mathcal{I}$ we have $\\mathcal{A}' \\subset [0:N-1] \\setminus \\mathcal{I} =\\mathcal{I}'$ and hence\n\\[\np^M-|\\Phi(\\mathcal{I})| = |\\mathcal{A}'| \\le |\\Omega(\\mathcal{I}')|.\n\\]\n\nSimilarly, let $\\mathcal{B}'=[0:p^M-1]\\setminus \\Omega(\\mathcal{I}')$. Then $\\mathcal{B}'$ is universal, it contains $[0:p^M-1]\\setminus \\mathcal{I}'=\\mathcal{I}$ and so\n\\[\np^M-|\\Omega(\\mathcal{I}')| = |\\mathcal{B}'| \\ge |\\Phi(\\mathcal{I})|.\n\\]\nTaken together the two inequalities prove the theorem.\n\\end{IEEEproof}\n\n\n\n\n\n\\section{An Uncertainty Principle, Random Signals, and Sumsets} \\label{section:uncertainty}\n\nGenerally speaking, an ``uncertainty principle'' is an inequality relating the supports of a nonzero function and its Fourier transform, in the present setting  $f\\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{C}$,  and ${{\\mathcal{F}}} f\\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{C}$.  The notions of  maximal and minimal universal sampling sets lead immediately to an additive uncertainty principle. Without the language of universality, Tao \\cite{tao:uncertainty} made this connection in the case when $N$ is a prime using Chebotarev's theorem, see Corollary \\ref{corollary:uncertainty-prime}, though, as he states, it was probably already known as a folk theorem.\n\n\nLet \n\\[\n\\mathcal{Z}(f) = \\{i \\colon f(i) = 0\\} \n\\]\nbe the zero set of $f$. The support is the complement of the zero set, and we denote it by ${\\rm supp}(f)$. Our result is\n\\begin{theorem} \\label{theorem:uncertainty}\nIf $f$ is not the zero function then\n\\begin{equation} \\label{eq:uncertainty-1}\n\\begin{aligned}\n|{\\rm supp}({{\\mathcal{F}}} f)| &\\ge 1+|\\Omega(\\mathcal{Z}(f))|, \\; \\\\\n|{\\rm supp} (f)| &\\ge 1+|\\Omega(\\mathcal{Z}({{\\mathcal{F}}} f))|;\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation} \\label{eq:uncertainty-2}\n\\begin{aligned}\n|\\mathcal{Z}({{\\mathcal{F}}} f)| +1 &\\le |\\Phi({\\rm supp}(f))|,\\; \\\\\n |\\mathcal{Z}( f)| +1 &\\le |\\Phi({\\rm supp}({{\\mathcal{F}}} f))|.\n \\end{aligned}\n\\end{equation}\n\\end{theorem}\n\nWe are not assuming that $N$ is a prime power here. However, we immediately deduce\n\\begin{corollary}[Tao]  \\label{corollary:uncertainty-prime}\nIf $N$ is prime and $f$ is not the zero function then \n\\[\n|{\\rm supp}({{\\mathcal{F}}}{f})|  + |{\\rm supp} (f)| \\geq N +1.\n\\]\n\\end{corollary}\n\\begin{IEEEproof} If $N$ is prime, then by Chebotarev's theorem every index set\nis universal. In particular the set $\\mathcal{Z}(f)$ is universal. Hence\n$\\Omega(\\mathcal{Z}(f)) = \\mathcal{Z}(f)$. From Theorem \\ref{theorem:uncertainty},\n \\[\n \\begin{aligned}\n|{\\rm supp}({{\\mathcal{F}}} {f})| &\\geq 1 + |\\Omega(\\mathcal{Z}(f))| \\\\\n&= 1 + |\\mathcal{Z}(f)| = 1 + N - |{\\rm supp}(f)|.\n\\end{aligned}\n\\]\n\\end{IEEEproof}\n\nWe also have\n\n\\begin{corollary} \\label{corollary:uncertainty-consecutive}\nSuppose $f$ vanishes on a set of consecutive integers $\\mathcal{I}$. Then\n$|{\\rm supp}({{\\mathcal{F}}}{f})| \\geq |\\mathcal{I}|+1$. If $\\mathcal{J}$ is a set of\nintegers such that ${{\\mathcal{F}}} f(\\mathcal{J}) = 0$, then $|\\mathcal{I}| + |\\mathcal{J}| \\leq N - 1$.\n\\end{corollary}\n\n\\begin{IEEEproof}We observed previously that any set of consecutive integers, $\\mathcal{I}$ in this case, is\nuniversal. Since $\\mathcal{I} \\subseteq \\mathcal{Z}(f)$, we have $|\\Omega(\\mathcal{Z}(f))| \\geq |\\mathcal{I}|$. From Theorem\n\\ref{theorem:uncertainty}, this implies $|{\\rm supp}({{\\mathcal{F}}}{f})| \\geq\n|\\mathcal{I}| + 1$. Further, if ${{\\mathcal{F}}}{f}(\\mathcal{J}) = 0$ then $N - |\\mathcal{J}| \\geq |{\\rm supp} ({{\\mathcal{F}}} f)|$ and\nso  $N - |\\mathcal{J}| \\geq  |\\mathcal{I}| + 1$.\n\\end{IEEEproof}\n\nThe proof of Theorem \\ref{theorem:uncertainty} itself is very brief. \n\\begin{IEEEproof}[Proof of Theorem \\ref{theorem:uncertainty}] Suppose\n$|{\\rm supp} ({{\\mathcal{F}}}{f})| \\leq |\\Omega(\\mathcal{Z}(f))|$. From\n$\\Omega(\\mathcal{Z}(f)) \\subseteq \\mathcal{Z}(f)$ it follows that $f$ vanishes on $\\Omega(\\mathcal{Z}(f))$. Since $\\Omega(Z)$ is a universal sampling set this implies that ${{\\mathcal{F}}}{f}\n\\equiv 0$,  contradicting the assumption that $f$ is not the zero function. This proves the first statement in \\eqref{eq:uncertainty-1}.  A similar argument establishes the second statement.\n\nFor the proof of \\eqref{eq:uncertainty-2}, write $\\mathcal{Z} = \\mathcal{Z}(\\mathcal{F}f)$ and $\\mathcal{A}=\\Phi(\\text{supp}(f))$. Then\n\\[\n\\mathcal{F}f (\\mathcal{Z}) = 0 \\text{ and so } E_{\\mathcal{Z}}^\\textsf{T}\\mathcal{F}f  = 0.\n\\]\nHowever $f$ is supported within $\\mathcal{A}$, and so we may\nwrite $f = E_{\\mathcal{A}}g$, where $ g = f(\\mathcal{A}) \\neq 0$. This means we must have\n\\begin{equation}\n\\label{eq:alt-proof}\n E_{\\mathcal{Z}}^\\textsf{T}\\mathcal{F}E_{\\mathcal{A}} g = 0, \\text{ for  some } g \\neq 0, \n\\end{equation}\ni.e. the columns of  $E_{\\mathcal{Z}}^\\textsf{T}\\mathcal{F}E_{\\mathcal{A}}$ are\ndependent. This is expected if $|\\mathcal{Z}|< |\\mathcal{A}|$. However, if $|\\mathcal{Z}| \\geq |\\mathcal{A}|$,\nthis contradicts the universality of $\\mathcal{A}$. Hence we\nmust have $|\\mathcal{Z}| \\leq |\\mathcal{A}|-1$, which is the first inequality in \\eqref{eq:uncertainty-2}. A similar argument establishes the second statement.\n\\end{IEEEproof}\n\nIt is interesting that when $N$ \\emph{is} a prime power the two statements \\eqref{eq:uncertainty-1} and \\eqref{eq:uncertainty-2} are equivalent. To see this we first derive \\eqref{eq:uncertainty-2} from \\eqref{eq:uncertainty-1} when $N=p^M$. This appeals to Theorem \\ref{theorem:maximal-minimal} on the relation between maximal and minimal sets, with ${\\rm supp}(f) = [0:N-1]\\setminus \\mathcal{Z}(f)$. Thus, from \\eqref{eq:uncertainty-1}, $|{\\rm supp}({{\\mathcal{F}}} f)| \\ge 1+|\\Omega(\\mathcal{Z}(f))|$, and substituting from Theorem \\ref{theorem:maximal-minimal},\n\\[\n|{\\rm supp}({{\\mathcal{F}}} f)| \\ge 1 + N - |\\Phi({\\rm supp}(f))|.\n\\]\nBut $|{\\rm supp}({{\\mathcal{F}}} f)| = N-|\\mathcal{Z}({{\\mathcal{F}}} f)|$, so\n\\[\nN-|\\mathcal{Z}({{\\mathcal{F}}} f)| \\ge 1+N-|\\Phi({\\rm supp}(f))|,\n\\]\nwhich is the same as the first statement in \\eqref{eq:uncertainty-2}. Again, the second statement in \\eqref{eq:uncertainty-2} follows in a similar manner. We could have started instead with \\eqref{eq:uncertainty-2} and from this derived \\eqref{eq:uncertainty-1}. \n\n\nIn cases where $\\mathcal{Z}(f)$ itself is a universal sampling set, the uncertainty\nprinciple in Theorem \\ref{theorem:uncertainty} can be as strong as the uncertainty\nprinciple for the prime $N$ case.\n\n\\begin{remark}\nReaders familiar with the seminal paper of Donoho and Stark \\cite{donoho-stark:uncertainty} will wonder if the additive uncertainty principle in Theorem \\ref{theorem:uncertainty} can be applied to the problem of reconstruction of a signal corrupted\nby sparse noise. (See also \\cite{studer-signal-recovery} for more recent work.)  The answer is yes, and we refer to \\cite{osw:reconstruction}.\n\n\\end{remark}\n\n\n\\subsection{Random Index Sets and Random Signals}\n\nWe will give several applications of these ideas. First we combine Theorem \\ref{theorem:uncertainty} with a probabilistic estimate on the size of a maximal universal sampling set for randomly chosen index sets. We must revert to the assumption that $N$ is a prime power.\n\n\\begin{theorem}\n\\label{theorem:rand-samp}\nLet $N=p^M$. Let $\\mathcal{R}_s$ be an index set of $s$ numbers chosen at random from $[0:N-1]$. Let $\\lambda = (N-s)\/N$. If $d, \\delta>0$ satisfy\n\\begin{equation}\n\\label{eq:ran-uncen-1}\nN\\log(1\/\\lambda) \\geq (1+ \\delta) d \\log d, \n\\end{equation}\nthen $|\\Omega(\\mathcal{R}_s)| \\geq d$ with probability at least $1- d^{-\\delta}$.\n\\end{theorem}\n\nThis means that if we can choose a large $d$ satisfying \\eqref{eq:ran-uncen-1}, which is possible, for example, if $N$ is large and $\\lambda$ is small, then $|\\Omega(\\mathcal{R}_s)| \\geq d$ with high probability. Thus while it is unlikely that a randomly chosen index set will be universal, it is quite likely that such an index set will contain a large universal set as a subset.\n\n\n\nWe will apply Theorem \\ref{theorem:rand-samp} to the case when $\\mathcal{R}_s$ is the zero set of $f\\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{C}$. Then $\\lambda = |\\text{supp}(f)|\/N$, i.e., $\\lambda$ is the fraction of nonzero entries in $f$.\n\n\\begin{IEEEproof} The proof uses the bound in part (ii) of Theorem \\ref{thm:maximal-bounds}. Let $k$ be the largest integer such that no congruence classes in $\\mathcal{R}_s\/p^k$ are empty. Note that $k$ is random since $\\mathcal{R}_s$ is random. Then $|\\Omega(\\mathcal{R}_s)| \\le d-1$ implies \n\\[\np^k \\le |\\Omega(\\mathcal{R}_s)| \\le d-1,\n\\]\nby Theorem \\ref{thm:maximal-bounds}. Therefore\n\\begin{align}\n&\\text{Prob}\\left(|\\Omega(\\mathcal{R}_s)| \\leq d-1 \\right) \\leq \\text{Prob}(p^k \\leq d-1) \\nonumber \\\\\n&\\quad = \\text{Prob}(k \\leq \\lfloor \\log_p(d-1) \\rfloor) \\nonumber \\\\\n&\\quad = \\text{Prob}(\\text{at least one congruence class in } \\nonumber\\\\\n& \\hspace{.5in} \\mathcal{R}_s\/p^{\\lfloor \\log_p(d-1) \\rfloor +1} \\text{ is empty}).\n\\end{align}\nWe will compute the last probability.\n\n\nLet $b = {\\lfloor \\log_p(d-1) \\rfloor +1}$, and let $\\mathcal{N}_{ba}$ be the set of elements in $[0:N-1]$ that leave a remainder of $a\\in[0:p^b-1]$ when divided by $p^b$.  Since $N=p^M$ all of the $\\mathcal{N}_{ba}$ have size $t = N\/p^b  = p^M \/ p^{\\lfloor \\log_p(d-1) \\rfloor +1}$.  \n\nFix a particular residue $a$. The probability that $\\mathcal{N}_{ba}\\cap \\mathcal{R}_s $ is empty (in words, the probability that a particular congruence class goes missing in $\\mathcal{R}_s$) is  $\\binom{N-t}{s}\/\\binom{N}{s}$. This is because the number of ways of picking $\\mathcal{R}_s$ is $\\binom{N}{s}$ while the number of ways of picking $\\mathcal{R}_s $ so that $\\mathcal{N}_{ba} \\cap \\mathcal{R}_s = \\emptyset$ is the number of ways of picking $s$ elements from \n\\[ |[0:N-1] \\setminus \\mathcal{N}_{ba}| = N- t\n\\]\n elements. Then\n\\begin{align}\n&\\text{Prob}\\left(\\mathcal{N}_{ba}\\cap \\mathcal{R}_s = \\emptyset\\right) \\nonumber \\\\ \n&= \\binom{N-t}{s}\\Big{\/} \\binom{N}{s} \\nonumber \\\\\n&= \\frac{(N-(t-1)\n  -s)(N-(t-2)-s)\\ldots(N-s)}{(N-t+1)(N-t+2)\\ldots N} \\nonumber \\\\\n&\\quad = \\left(1- \\frac{s}{N-t+1} \\right) \\left(1- \\frac{s}{N-t+2}\n\\right)\\ldots \\left(1- \\frac{s}{N} \\right) \\nonumber \\\\\n&\\quad \\leq \\left(1- \\frac{s}{N} \\right) \\left(1- \\frac{s}{N}\n\\right)\\ldots \\left(1- \\frac{s}{N} \\right) =\n\\left(1-\\frac{s}{N}\\right)^t. \\label{eq:rnd-1} \n\\end{align}\nFrom this,\n\\begin{align}\n &\\text{Prob}(\\text{at least one congruence class in\n   } \\nonumber \\\\\n & \\hspace{.7in} \\mathcal{R}_s\/p^{\\lfloor \\log_p(d-1) \\rfloor +1} \\text{ is empty})\n \\nonumber \\\\\n &\\quad = \\text{Prob}\\left(\\bigcup_i \\left(\\mathcal{N}_{ba}\\cap \\mathcal{R}_s = \\emptyset \\right)\\right)\n \\nonumber \\\\\n&\\quad \\leq \\sum_i  \\text{Prob}\\left(\\mathcal{N}_{ba}\\cap \\mathcal{R}_s = \\emptyset\\right) \\nonumber \\\\\n&\\quad \\leq  \\frac{N}{t}\\left(1-\\frac{s}{N}\\right)^t = N\\lambda^t\/t.\\label{eq:end-2}\n\\end{align}\nHence we have from \\eqref{eq:end-2},\n\\begin{equation}\n\\label{eq:ran-bnd-1}\n\\text{Prob}\\left(|\\Omega(\\mathcal{R}_s)| \n\\leq d-1 \\right) \\leq N\\lambda^t\/t. \n\\end{equation}\nNow,  $ t =  N \/ p^{\\lfloor \\log_p(d-1) \\rfloor +1} \\geq N\/d$, since $\\lfloor x \\rfloor \\leq x$. Using this in \\eqref{eq:ran-bnd-1},\n\\begin{align}\n&\\text{Prob}\\left(|\\Omega(\\mathcal{R}_s)| \\leq d-1 \\right) \\nonumber\\\\ \n& \\leq  N\\lambda^t\/t \\leq d\\lambda^{N\/d} \\nonumber \\\\\n& =\\exp\\left(\\log d - \\frac{N\\log(1\/\\lambda)}{d} \\right) \\nonumber \\\\\n& = \\exp\\left(\\log d\\left(1- \\frac{N\\log(1\/\\lambda)}{d\\log d} \\right)\\right) \\nonumber \\\\\n& \\leq \\exp\\left(-\\delta \\log d\\right) \\text{ (from the hypothesis of the theorem) } \\nonumber \\\\\n& = d^{-\\delta}.\n\\end{align}\nWe conclude that $\\text{Prob}\\left(|\\Omega(\\mathcal{R}_s)| \\geq d \\right) \\geq 1- d^{-\\delta}$.\n\\end{IEEEproof}\n\n We can now state a probabilistic uncertainty principle. Afterward we will comment on how this compares to the result of Candes, Romberg and Tao \\cite{candes:robust}. \n \n \\begin{theorem}\n\\label{thm:rnd-uncen} \nLet $N=p^M$. Let $\\mathcal{G}_{N,r}$ be the set of all signals $g\\colon \\mathbb{Z}_N \\longrightarrow \\mathbb{C}$ with support of size $r$. Let $g \\in \\mathcal{G}_{N,r}$ be a signal whose support is drawn at random from the set of all index sets of size $r$. Let the values of $g$ on the support set be drawn according to some arbitrary distribution. For $\\delta>0$ let \n\\[\na_{N, \\delta} = \\frac{N}{(1+\\delta)\\log N}\\left(1+ \\log (1+ \\delta) + \\log \\log N\\right).\n\\]\nThen \n\\begin{equation}\n\\label{eq:rnd-uncen}\n|\\text{supp}(g)| + |\\text{supp}(\\mathcal{F}g)| \\geq 1 + a_{N,\\delta} \n\\end{equation}\nwith probability at least $1- (a_{N,\\delta}-r)^{-\\delta}$.\n\\end{theorem}\n\nIf $r$ is small compared to $a_{N,\\delta}$, Theorem \\ref{thm:rnd-uncen} states\nthat almost all signals $g$ in $\\mathcal{G}_{N,r}$ satisfy the \nuncertainty principle above; roughly speaking\n\\[\n|\\text{supp}(g)| + |\\text{supp}(\\mathcal{F}g)| \\geq N(1+ \\log \\log\nN)\/\\log N\n\\]\nfor most $g$.\n\\begin{IEEEproof}\nPicking the support of $g$  at random among sets of size $r$ is equivalent to picking the zero set of $g$  at random among all index sets of size $N-r$. The proof now makes use of Theorem \\ref{theorem:rand-samp} to get a lower bound on $|\\Omega(\\mathcal{Z}(g))|$. For this we need to choose $d, \\delta$ so that\n\\begin{equation}\n\\label{eq:ran-uncen-repeat}\nN \\log (1\/\\lambda) = N \\log N\/r > (1+\\delta)d \\log d.\n\\end{equation}\nFix any $\\delta>0$ and let $d = N \\log (N\/r)\/(1+\\delta)\\log N$. We check that $d, \\delta$ satisfy $\\eqref{eq:ran-uncen-repeat}$:\n\\begin{align*}\n(1+\\delta)d\\log d &= \\frac{N \\log (N\/r)}{\\log N} \\log \\left(\\frac{N \\log (N\/r)}{(1+\\delta)\\log N} \\right)\\\\\n&<\\frac{N \\log (N\/r)}{\\log N} \\log N = N \\log N\/r,  \n\\end{align*}\nThen  from Theorem  \\ref{theorem:rand-samp}, \n\\[\n|\\Omega(\\mathcal{Z}(g))| \\geq N\\log (N\/r) \/ (1+\\delta)\\log N \n\\] \nwith probability $1-d^{-\\delta}$. From the uncertainty principle Theorem \\ref{theorem:uncertainty}, we now have\n\\[\n\\begin{aligned}\n|\\text{supp}(\\mathcal{F}g)| &\\geq 1 + |\\Omega(\\mathcal{Z}(g))|\\\\\n& \\geq 1 + N\\log (N\/r) \/ (1+\\delta)\\log N \n\\end{aligned}\n\\]\nwith probability  $1-d^{-\\delta}$.\n\nThe final step in the proof uses a lower bound on $d = N \\log (N\/r)\/(1+\\delta)\\log N$. We have set apart this technical result as Lemma \\ref{lem: rnd-ucen-cvx}, below. This gives\n\\[\n|\\text{supp}(\\mathcal{F}g)| \\geq 1 + a_{N,\\delta} - r \\quad\n\\]\nwith probability $1-d^{-\\delta}$.\nSince $1-d^{-\\delta} \\geq 1- (a_{N,\\delta} - r)^{-\\delta}$, we can say\n\\[\n|\\text{supp}(\\mathcal{F}g)| \\geq 1 + a_{N,\\delta} - r \n\\]\nwith probability  $1-(a_{N,\\delta} - r)^{-\\delta}$.\nThe result follows since $ r = |\\text{supp}(g)|$.\n\\end{IEEEproof}\n\n\\begin{lemma}\n\\label{lem: rnd-ucen-cvx}\nLet \n\\[\nd = \\frac{N\\log(N\/r)}{(1+\\delta)\\log N}\n\\]\nand \n\\[\na_{N, \\delta} = \\frac{N}{(1+\\delta)\\log N}\\left(1+ \\log (1+ \\delta) + \\log \\log N\\right),\n\\]\nas  in Theorem \\ref{thm:rnd-uncen}. Then $d \\geq a_{N, \\delta} - r$.\n\\end{lemma}\n\n\\begin{IEEEproof}\nThe convex function $\\log (N\/r)$ is bounded below by its tangent at any point $r_0>0$. Thus\n\\[\n\\log (N\/r) \\geq \\log (N\/r_0) + \\left(-\\frac{1}{r_0}(r - r_0)\\right).\n\\]\nFor \n\\[\nr_0 = \\frac{N}{(1+\\delta)\\log N}\\,,\n\\]\n this reads\n\\[\n\\begin{aligned}\n\\log (N\/r) &\\geq \\log \\left((1+\\delta)\\log N\\right)\\\\\n& \\hspace{.25in} + \\left(-\\frac{(1+\\delta)\\log N}{N}\\left(r - \\frac{N}{(1+\\delta)\\log N}\\right)\\right).\n\\end{aligned}\n\\]\nMultiplying by $N\/(1+\\delta)\\log N$, we have\n\\begin{align*}\nd &= \\frac{N\\log(N\/r)}{(1+\\delta)\\log N} \\\\\n&\\geq \\frac{N\\log \\left((1+\\delta)\\log N\\right)}{(1+\\delta)\\log N} - \\left(r - \\frac{N}{(1+\\delta)\\log N}\\right) \\\\\n&= \\frac{N}{(1+\\delta)\\log N}\\left(\\log(1+\\delta) + 1 + \\log \\log N \\right) - r \\\\\n&= a_{N, \\delta} - r.\n\\end{align*}\n\\end{IEEEproof}\n\n\\begin{remark}\nThe robust uncertainty principle of Candes, Romberg and Tao in \\cite{candes:robust} is as follows: for $M >0$ there exists a constant $C_M$ such that\n\\[\n|\\text{supp}(g)| + |\\text{supp}(\\mathcal{F}g)| \\geq C_M N(\\log N)^{-1\/2},\n\\]\nwith probability $1- O(N^{-M})$.\nThis inequality is stronger than that of Theorem \\ref{thm:rnd-uncen} by about $(\\log N)^{-1\/2}$. Also, Theorem \\ref{thm:rnd-uncen} holds for $N=p^M$, whereas the inequality above holds for all $N$.\n\nIn our proof of Theorem \\ref{theorem:rand-samp} we have only used the bound $|\\Omega(\\mathcal{Z}(g))| \\geq p^k$ from Theorem \\ref{thm:maximal-bounds}. By using the exact formula for $|\\Omega(\\mathcal{Z}(g))|$ in Theorem \\ref{theorem:maximal-precise} (or by a better lower bound) it might be possible to tighten the uncertainty principle of Theorem \\ref{thm:rnd-uncen} and remove the factor  $(\\log N)^{-1\/2}$.  \n\\end{remark}\n\n\\subsection{Sumsets and the Cauchy-Davenport Theorem}\n\nOur final application is a  generalization of the Cauchy-Davenport theorem \\cite{davenport:residues}, from additive number theory, on the size of sumsets. Again the inspiration comes from Tao's approach, \\cite{tao:uncertainty},  to the original Cauchy-Davenport theorem via Chebotarev's theorem. \n\n\\begin{theorem}\n\\label{thm:sumset-univ}\nLet $\\mathcal{X}, \\mathcal{Y} \\subseteq [0:N-1]$. If either $\\mathcal{X}$ or $\\mathcal{Y}$ is a universal sampling set, then \n\\begin{equation}\n\\label{eq:sumset-univ}\n|\\mathcal{X}+\\mathcal{Y}| \\geq |\\mathcal{X}| + |\\mathcal{Y}| -1,\n\\end{equation}\nwhen $|\\mathcal{X}| + |\\mathcal{Y}| -1 \\leq N$. \n\nHere $\\mathcal{X}+\\mathcal{Y}$ is the sumset defined as\n\\[\n\\mathcal{X} + \\mathcal{Y} = \\{x + y: x \\in \\mathcal{X}, y \\in \\mathcal{Y} \\},\n\\]\nwhere the addition is modulo $N$. \n\\end{theorem}\n\nWe are not assuming that $N$ is a prime power, while the classical theorem has $N=p$ and there are no assumptions on $\\mathcal{X}$ or $\\mathcal{Y}$. \nThat form of the result follows from Theorem \\ref{thm:sumset-univ}, since all index sets in $[0:N-1]$ are universal when $N$ is prime. \n\nAs a corollary we get a statement on the size of $|\\mathcal{X}+\\mathcal{Y}|$ without making an assumption on $\\mathcal{X}$ or $\\mathcal{Y}$.\n\\begin{corollary}\n\\label{cor:sumset-1}\nLet $\\mathcal{X}, \\mathcal{Y} \\subseteq [0:N-1]$ be index sets. Then, \n\\begin{equation}\n\\label{eq:sumset-gen}\n|\\mathcal{X}+\\mathcal{Y}| \\geq \\max\\{ |\\Omega (\\mathcal{X})| + |\\mathcal{Y}| -1, |\\mathcal{X}| + |\\Omega (\\mathcal{Y})| -1 \\}.\n\\end{equation}\n\\end{corollary}\n\\begin{IEEEproof} Since $\\Omega(\\mathcal{X}) \\subseteq \\mathcal{X}$, it follows that $\\Omega(\\mathcal{X}) + \\mathcal{Y} \\subseteq \\mathcal{X} + \\mathcal{Y}$. Now,\n\\begin{align*}\n|\\mathcal{X}+ \\mathcal{Y}| \\geq |\\Omega(\\mathcal{X}) + \\mathcal{Y}| \\geq |\\Omega(\\mathcal{X})| + |\\mathcal{Y}| -1 \\text{ from Theorem \\ref{thm:sumset-univ}}.\n\\end{align*}\nThe inequality $|\\mathcal{X} + \\mathcal{Y}| \\geq |\\mathcal{X}| + |\\Omega(\\mathcal{Y})| -1 $ follows similarly.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:sumset-univ}]\nFirst note that \\eqref{eq:sumset-univ} follows trivially when either $X$ or $Y$ is a singleton. (More precisely, if, say,  $\\mathcal{X}$ is a singleton, then $\\mathcal{X}+\\mathcal{Y}$ is just a translate of $\\mathcal{Y}$, and so \\eqref{eq:sumset-univ} holds with equality). For the rest of the proof, we assume that $|\\mathcal{X}|, |\\mathcal{Y}| \\geq 2$. Let $|\\mathcal{X}|=r$, $|\\mathcal{Y}| =s$.\n\n\nAssume without loss of generality that $\\mathcal{X}$ is universal. Let\n\\[\nf_1 \\in \\mathbb{B}^{\\mathcal{X}} \\text{ be such that }f_1(\\left[1:r\\right]) = (\\underbrace{0, 0, \\ldots, 0}_{r-1 \\text{ times }}, 1). \n\\]\nSuch an $f_1$ exists because the set $[1:r]$, as an index set of $r$ consecutive integers, is a universal sampling set, so is in particular a sampling set for $\\mathbb{B}^\\mathcal{X}$. Similarly let\n\\[\nf_2 \\in \\mathbb{B}^\\mathcal{Y} \\text{ be such that }f_2(\\left[r:r+s-1\\right]) = (\\underbrace{0, 0, \\ldots, 0}_{s-1 \\text{ times }}, 1),\n\\]\n again possible because $[r:r+s-1]$ is a set of $s$ consecutive integers, and hence a sampling set for $\\mathbb{B}^\\mathcal{Y}$. Note that $f_1f_2 \\in \\mathbb{B}^{\\mathcal{X}+\\mathcal{Y}}$ and so $|\\mathcal{X}+\\mathcal{Y}| \\geq  {\\rm supp}(\\mathcal{F}(f_1f_2))$. Note also that the zero set  $\\mathcal{Z}(f_1f_2)$ of $f_1f_2$ contains $[1:r+s-2]$, and hence, since the latter is a universal sampling set, $ |\\Omega \\left(\\mathcal{Z}(f_1f_2)\\right)| \\geq r+s-2 = |\\mathcal{X}|+|\\mathcal{Y}|-2$.\n \n Now we apply the uncertainty principle of Theorem \\ref{theorem:uncertainty} to $f_1f_2$. We have, so long as $f_1f_2 \\neq 0$,\n\\begin{align} \\label{eq:sumset-0}\n|\\mathcal{X}+\\mathcal{Y}| &\\geq {\\rm supp}\\left(\\mathcal{F}(f_1f_2)\\right) \\nonumber \\\\\n&\\geq 1 + |\\Omega\\left(\\mathcal{Z}(f_1f_2)\\right)| \\nonumber \\\\\n& \\geq 1 + |\\mathcal{X}|+|\\mathcal{Y}|-2 = |\\mathcal{X}| + |\\mathcal{Y}| -1, \n\\end{align}\nSo we have proved that $|\\mathcal{X}+\\mathcal{Y}| \\geq |\\mathcal{X}| + |\\mathcal{Y}| -1 $ if we know that $ f_1 f_2 \\neq 0$. \n\nFor this, again from Theorem \\ref{theorem:uncertainty} we have \n\\[\n|\\mathcal{Z}(f_1)| \\leq |\\Phi({\\rm supp}(\\mathcal{F}f_1))| - 1 \\leq |\\Phi(\\mathcal{X})| - 1,\n\\]\nsince $f_1 \\in \\mathbb{B}^\\mathcal{X}$. But $\\mathcal{X}$ is universal, so $\\Phi(\\mathcal{X})=\\mathcal{X}$ and\n\\begin{equation}\n\\label{eq:sumset-1}\n|\\mathcal{Z}(f_1)| \\leq |\\mathcal{X}| -1. \n\\end{equation}\nBy definition of $f_1$, the set $[1:r-1]=[1:|\\mathcal{X}|-1]$ is already in $\\mathcal{Z}(f_1)$. Together with \\eqref{eq:sumset-1}, this implies that $f_1$ cannot have any more zeros. In particular, $f_1(r+s -1) \\neq 0$. Since $f_2(r+s -1) = 1$, $f_1f_2$ cannot be identically zero and \\eqref{eq:sumset-0} applies.   \\end{IEEEproof}\n\nAn important generalization of the Cauchy-Davenport theorem to any finite abelian group, not necessarily of prime order, is due to Kneser, \\cite{kneser:sumset}. \n\\begin{theorem}[Kneser]\n Let $G$ be a finite abelian group. Let $A, B \\subseteq G$ be non empty subsets of $G$. Let $H$ be the set of periods, defined by $H = \\{h \\in G : h + (A+B) = A+B \\}$. (Thus $A+B$ is periodic if $H \\ne \\{0\\}$.) \n Then \n \\[\n |A+B| \\geq |A| + |B| - |H|.\n \\]\nHence unless $A + B$ is periodic, $|A+B| \\geq |A| + |B| - 1$. \n\\end{theorem}  \n\nThough the form is similar, this result neither implies nor is implied by Theorem \\ref{thm:sumset-univ}. We give two examples. Let $N=8$, $\\mathcal{X}= \\{0,1\\}$, $\\mathcal{Y} = \\{0,4\\}$. Then $\\mathcal{X}$ is universal and $\\mathcal{X} + \\mathcal{Y} = \\{0, 1, 4, 5\\}$ is periodic with period $4$. So Theorem \\ref{thm:sumset-univ} applies, but Kneser's theorem does not. Next let $N=16$, $\\mathcal{X} = \\{0,2\\}, \\mathcal{Y} = \\{0,2,4\\}$. Then $\\mathcal{X}+\\mathcal{Y} = \\{0,2,4,6,8,10\\}$, which is not periodic, and neither $\\mathcal{X}$ nor $\\mathcal{Y}$ is universal. So Kneser's theorem applies, but Theorem \\ref{thm:sumset-univ} does not. We hope to understand this more thoroughly.\n\n\n \n\\appendices{\\section{Condition Number Associated with the Universal Sampling Set $\\mathcal{I}^*$} \\label{appendix:condition-number}}\n\nAn index set of consecutive integers is the simplest universal sampling set, but there is a catch in using it.\nLet $\\mathcal{I}$ be a universal sampling set of size $d$, $f\\in \\mathbb{C}^N$, and $f_\\mathcal{I}$ the $d$-vector obtained from $f$ by sampling at locations in $\\mathcal{I}$. If $f$ is in some bandlimited space $\\mathbb{B}^\\mathcal{J}$, $|\\mathcal{J}|=d$, then the interpolation formula \\eqref{eq:interpolation-formula-2} reads\n\\[\nf = \\mathcal{F} E_\\mathcal{J} (E_\\mathcal{I}^T \\mathcal{F} E_\\mathcal{J})^{-1} f_\\mathcal{I}.\n\\]\nThe practical difficulty is the computation of the inverse of $E_\\mathcal{I}^T \\mathcal{F} E_\\mathcal{J}$. Suppose we use $\\mathcal{I} = \\mathcal{I}^* =[0:d-1]$ as a universal sampling set. We give a lower bound on the condition number of $E_\\mathcal{I}^T \\mathcal{F} E_\\mathcal{J}$ that can be quite large for some $\\mathcal{J}$, even though the matrix $E_\\mathcal{I}^T \\mathcal{F} E_\\mathcal{J}$ is invertible for all $\\mathcal{J}$. \n\nFor $\\mathcal{I} = [0:d-1]$, note that\n\\begin{align*}\n|\\det\\left(E_{\\mathcal{I}}^T \\mathcal{F} E_{\\mathcal{J}} \\right)| &= |\\det(\\zeta_N^{ij})_{i \\in \\mathcal{I}, j \\in \\mathcal{J}} |\\\\\n&= \\prod_{j_1, j_2 \\in \\mathcal{J}}|\\zeta_N^{j_1} - \\zeta_N^{j_2}| \\\\\n&= \\prod_{j_1, j_2 \\in \\mathcal{J}}\\left|2\\sin\\frac{2\\pi(j_1-j_2)}{N}\\right|.\n\\end{align*}\n\nIf $\\{\\sigma_i\\}$ are the singular values of $A = E_{\\mathcal{I}}^T \\mathcal{F} E_{\\mathcal{J}}$, then \n\\begin{equation}\n\\label{eq:minsgn}\n\\det(A) = \\sigma_1\\sigma_2\\sigma_3\\ldots \\sigma_d \\geq \\sigma_{\\min}^d.\n\\end{equation}\nAlso if $a_{rk} = \\exp(-2\\pi i rj_k\/N)$ are the entries of $A$, then \n\\begin{equation}\n\\label{eq:maxsgn}\nd^2 = \\sum_{r,k = 0}^{d-1}|a_{rk}|^2 = \\text{tr}(A^*A) = \\sum_{r=0}^{d-1} \\sigma_r^2 \\leq d\\sigma_{\\text{max}}^2,\n\\end{equation}\nand so $\\sigma_{\\text{max}}^2 \\geq d$. \n\nFrom \\eqref{eq:minsgn} and \\eqref{eq:maxsgn}, the condition number satisfies\n\\[\n\\frac{\\sigma_{\\max}}{\\sigma_{\\min}} \\geq \\sqrt{d}\\left(\\frac{1}{\\prod_{j_1, j_2 \\in \\mathcal{J}}|2\\sin\\frac{2\\pi(j_1-j_2)}{N}|}\\right)^{1\/2d}.\n\\]\nA possible scenario may be when $d$ is very small and $N$ is very large. In this case, the condition number can be very large if the frequency slots $\\mathcal{J}$ are clustered.\n\n\n\\section{Counting Bracelets} \\label{appendix:bracelets}\n\nSeveral of our results, Theorem \\ref{theorem:universal} for example, depend only on the bracelet of an index set rather than on the index set itself. Thus it is useful to know how many bracelets there are and how to enumerate them. Counting  bracelets -- actually, multicolored bracelets -- is a standard application in combinatorics of the orbit stabilizer theorem, and the problem is treated in many places. Our situation is slightly different because we want a count that specifies the number of black beads  in a black-and-white bracelet, corresponding to the size of the index set that determines the locations of the black beads. Nevertheless, the orbit stabilizer theorem can still be applied, and we have the following results.\n\n\\begin{theorem} Let $\\phi$ denote Euler's totient function.\nWhen $N$ is odd, the number of black-and-white bracelets of length $N$ with exactly $d$ black beads is\n\\[\n\\begin{array}{ll}\n\\frac{1}{2}  { { {(N-1)}\/{2}  } \\choose {  d\/2 } }       + \\frac{1}{2N}    \\sum_{k|N , k|d} \\frac{\\phi(k)}{N} { {N\/k} \\choose {d\/k}} \n\t\t& \\quad \\textrm{for even } d, \\\\ \\\\\n\\frac{1}{2}   { { {(N-1)}\/{2}  } \\choose {  ({d-1})\/{2} } }      +  \\frac{1}{2N}    \\sum_{k|N , k|d} \\frac{\\phi(k)}{N} { {N\/k} \\choose {d\/k}} \n\t\t& \\quad \\textrm{for odd } d.\n\t\t\\end{array} \n\\]\n\nWhen $N$ is even, the number of black-and-white bracelets of length $N$ with exactly $d$ black beads is\n\\[\n\\begin{array}{ll}\n\\frac{1}{2}  {{ N\/2  } \\choose {  d\/2 } }   +  \\frac{1}{2N}    \\sum_{k|N , k|d} \\frac{\\phi(k)}{N} { {N\/k} \\choose {d\/k}} \n\t\t& \\quad \\textrm{for even } d, \\\\ \\\\\n\\frac{1}{2}  {{(N\/2) - 1} \\choose {{(d-1)}\/{2} }}    +  \\frac{1}{2N}    \\sum_{k|N , k|d} \\frac{\\phi(k)}{N} { {N\/k} \\choose {d\/k}} \n\t\t& \\quad \\textrm{for odd } d.\n\t\t\\end{array} \n\\]\t\n\\end{theorem}\n\nWe omit the proof; see \\cite{Will:thesis}. An efficient algorithm for enumerating bracelets has been devised only recently by Sawada \\cite{sawada:bracelets}. An algorithm for determining when two index sets are in the same necklace is due to J.P. Duval \\cite{duval:necklace}. It can also be used for bracelets. See \\cite{Will:thesis} for examples of both of these.\n\n\n\\section{Additional References} \\label{section:additional-references}\n\nThough our work has concerned discrete-time signals exclusively, there is also a notion of universal sampling sets for continuous-time signals. We will not give the definition; it is interesting and not clear what the relations between the two may be. Here we cite only a few sources, starting with the paper of Landau \\cite{landau:density} that featured the renowned necessary density condition on sampling sets. More recently, many interesting results have been obtained by Olevskii and Ulanovskii \\cite{olevskii-ulanovskii:universal}, \\cite{olevskii-ulanovskii:universal-2} on universal sampling and stable reconstruction, by Matei and Meyer \\cite{matei-meyer:variant-compressed}, who work with lattices and make contact with compressed sensing, and by Bass and Gr\\\"ochenig  \\cite{Bass04randomsampling}, who consider random sampling. Of course, anyone writing on so fundamental a topic as sampling and interpolation will encounter an enormous literature, and most probably miss an equal or greater amount.  We apologize to the authors of works we have missed.\n\n\\section*{Acknowledgments}\n\nThere are many people to thank for their interest, insight, and encouragement over quite some time, in particular S. Boyd, M. Chudnovsky,  A. El Gamal, J.T. Gill, S. Gunturk, B. Hassibi, J. Sawada,  J. Smith, and M. Tygert.   We also thank the reviewers for their thorough and thoughtful comments. \n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{preprocessing network by weitghing the edges}\n\n\n\n\\section{Introduction}\n\nIn many sciences---for example Sociology~\\citep{scott2017social}, Biology~\\citep{chung2015bridging}, and Computer Science~\\citep{hendrickson2000graph}---units under study often belong to communities, and thus, may behave similarly.\nHence, understanding the community structure of units is critical in these sciences, and much work has been devoted to the development of methods to detect these communities~\\citep{fortunato2010community}.  \n\nCommunity detection methods often use the framework of mathematical networks: units are nodes (vertices) in the graph, and edges are drawn between two nodes if the corresponding units interact with each other.\nUnder this framework, the problem of community detection becomes a graph partitioning problem where units within each block of the partition are ``optimally'' connected under some objective.\nCommonly, communities are identified through edge prevalence; edges are more frequently observed between nodes in the same community than between nodes in different communities.\n\nOptimal detection of communities is often $\\mathcal{NP}$-hard because the number of possible partitions grow exponentially  as the number of units $n$ grows.\nMany (mostly heuristic) methods have been developed for the detection of communities under large $n$ settings.\nTwo such methods include the Louvain algorithm~\\citep{blondel2008fast} and the CNM~\\citep*{clauset2004finding} algorithm.\nMoreover, the ability of these algorithms to accurately identify communities may be improved through incorporating additional graph structure, which are typically integrated into these algorithms through the use of edge weights~\\citep{de2013enhancing,sun2014weighting, khadivi2011network}.\n\n\n\nOf note, previous work has shown that measures quantifying the cyclic structure of the network may be useful in detecting communities~\\citep{newman2003structure, radicchi2004defining, kim2005cyclic, vragovic2006network, zhang2008clustering, shakeri2017network}.\nThe intuition is that, if edges are more prevalent within a community than across communities, cycles (especially small cycles) may be more prevalent within communities as well. \nHence, edge weights that incorporate information about the density of cycles within a graph may be useful in detecting communities within that graph.\nHowever, some of these methods may be computationally prohibitive to implement as network sizes grow very large.\n\n\n\nOn the other hand, random walks have been proposed as a computationally efficient tool for uncovering the structure of networks. \nWhen the density of edges is higher within communities than across communities, random walks will spend a majority of their time traveling within communities~\\citep{hughes1995random, lai2010enhanced}. \nHence, network structure may be uncovered by following the paths of these random walks.\nAn important variant is the non-backtracking random walk (NBRW), in which the random walk is prohibited from returning back to a node in exactly two steps~\\citep{alon2007non, fitzner2013non}.\nWhile work on NBRW currently focuses on its fast convergence rate, we exploit another useful property of NBRW---its ability to identify cyclic structure.  \n\nIn this paper, we consider the process in which a NBRW is performed until it forms a cycle, at which point the walk terminates and restarts---we call this process the renewal non-backtracking random walk (RNBRW).\nWe are particularly interested in the \\textit{retracing probability} of an edge---the probability that the edge completes the cycle in one iteration of RNBRW. \nIntuitively, edges with higher retracing probabilities are more critical to the formation of a cycle.\nHence, the cyclic structure of a network may be incorporated into community detection algorithms through weighting edges by their retracing probability.\nAlthough analytically obtaining exact values for retracing probabilities is difficult, repeated iterations of RNBRW can yield accurate estimates of these probabilities.  \nAdditionally, these iterations can be performed in parallel, allowing these probabilities to be estimated extremely fast, even for networks containing millions of nodes.  \n\nWe show through simulation that weighting edges by their estimated retracing probabilities through RNBRW improves substantially the ability for Louvain and CNM methods to detect communities.\nThe improvement is especially noticeable in the case of sparse graphs.\nAdditionally, we show that using Louvain and CNM with these weights may offer comparable performance to that of state-of-the-art community detection methods with considerably less computation.\n\n\n\n\\section{Notation and preliminaries}\\label{sec:prelim}\nLet $G=(V,E)$ denote a graph with node set $V$ and edge set $E$. \nWe do not currently make any restrictions on whether $G$ is directed or undirected. \nLet $n = |V|$ and $m = |E|$. \nIf $G$ is undirected, we will denote the edge between vertices $i$ and $j$ by $ij$, and if $G$ is directed, we denote the edge traveling from $i$ to $j$ by $\\vec{ij}$.\nFor convenience, we may refer to edges in an undirected graph $G$ using the latter notation; \nin this case, $\\vec{ij}$ and $\\vec{ji}$ refer to the same edge $ij$.\nEdges $\\vec{ij} \\in E$ may be weighted; let $w_{\\vec{ij}}$ denote the value of this weight.\nFor ease of notation, we assume an unweighted graph has default edge weights $w_{\\vec{ij}} = 1$.\nAn \\textit{adjacency matrix} $A$ for a graph $G$ is an $n \\times n$ matrix \nwhere element $A_{ij} =1$ if edge $\\vec{ij} \\in E$ and $A_{ij} = 0$ otherwise. \n\nThe (possibly weighted) \\textit{degree} of node $i$, denoted $d_i$, is the sum of the edge weights that begin from node $i$. \n\\begin{equation}\n    \\label{eq:definedi}\n   d_i \\equiv \\sum_{j = 1}^n w_{\\vec{ij}}\n\\end{equation}\nWe assume edge weights are scaled so that\n\\begin{equation}\n    \\sum_{i=1}^n d_i = 2m.\n\\end{equation}\nNote that, in the unweighted case, $d_i$ is simply the number of edges beginning from node $i$. \n\nWe deviate from conventional literature by defining a walk by the edges it traverses; a \\textit{walk} $(\\overrightarrow{v_0v_1}, \\overrightarrow{v_1v_2}, \\ldots, \\overrightarrow{v_{k-1}v_k})$ of length $k$ is a vector of edges  $\\overrightarrow{v_{\\ell-1}v_{\\ell}} \\in E$, $\\ell = 1,2,\\ldots, k$ connecting (possibly non-distinct) nodes $v_0,v_1, v_2, \\ldots,v_k \\in V$.\nA \\textit{random walk} $(e_1, e_2, \\ldots, e_k)$ is a walk such that:\n\\begin{align}\n    &P\\left(e_{\\ell+1} = \\overrightarrow{v_{\\ell}v_{\\ell+1}}  | e_{\\ell} = \\overrightarrow{v_{\\ell-1}v_{\\ell}}, \\ldots, e_1 = \\overrightarrow{v_{0}v_{1}}\\right)\\nonumber \\\\\n     =\\;&  \n     P\\left(e_{\\ell+1} = \\overrightarrow{v_{\\ell}v_{\\ell+1}} | e_{\\ell} = \\overrightarrow{v_{\\ell-1}v_{\\ell}}\\right) = \\frac{w_{\\overrightarrow{v_{\\ell}v_{\\ell+1}}}}{d_\\ell}\n\\end{align} \nA walk $c$ is a \\textit{simple cycle} \nif $v_{0} = v_{k}$ and $v_0,v_1, v_2, \\ldots,v_{k-1}$ are distinct.\n\n\nThe nodes $V$ are partitioned into $q$ communities, numbered 1 through $q$.\nLet $g_v \\in \\{1, \\ldots, q\\}$ denote the community to which $v$ belongs and $\\mathbf g = (g_1, g_2, \\ldots, g_n)$.\nIn problems of community detection, we wish to uncover the true label $g_v$ for each node $v \\in V$.\n\n\\subsection{Community detection through maximizing modularity}\nOne popular approach for the problem of community detection is to choose communities that corrrespond to the optimal solution of an integer programming problem.  \nA common objective function for community detection is \nthe graph modularity~\\citep{newman2003structure}: \n\\begin{equation}\\label{eq_Modularity}\n    M(\\mathbf g; A) = \\frac{1}{2m}\\sum_{i,j}\\left(A_{ij}-\\frac{d_i d_j}{2m} \\right) \\delta_{g_i g_j}.\n\\end{equation}\nHere, $\\delta_{g_i g_j}$ is the Dirac delta function equal to 1 if\n$g_i = g_j$---that is, if $i$ and $j$ belong to the same community.\nIntuitively, for a given community label $\\mathbf g$, the modularity function measures how much within-group and across-group edge formation deviates from independence.\nBetter choices of communities $\\mathbf g$ correspond to larger values of $M(\\mathbf g)$.  \nChoosing communities $\\mathbf g$ that maximize the modularity---or almost any other graph partitioning objective---is an $\\mathcal{NP}$-hard problem; hence, this approach often focuses on finding heuristic or approximately optimal solutions~\\citep{brandes2008modularity}.\nWe now detail two such approaches, the CNM method~\\citep{clauset2004finding} and the Louvain method~\\citep{blondel2008fast}.\n\n\n\n\n\n\\subsubsection{CNM method}\nThe CNM method is a greedy algorithm in which communities are iteratively merged so that each merger maximizes the change in modularity.\nLet $\\Delta Q_{g_ig_j}$ denote the change in modularity obtained when communities $g_i$ and $g_j$ are merged together.\nFor CNM, each node initially begins in its own singleton community.\nIn this case, the change of modularity obtained from merging nodes $i$ and $j$ into a single community is\n\\begin{equation}\\label{eq_CNMstep}\n    \\Delta Q_{ij} = \\begin{cases} \n    1\/{m}-2d_id_j\/(2m)^2 & \\text{if }A_{ij}\\neq 0 \\\\\n    0 & \\text{otherwise}\n\\end{cases}\n\\end{equation}\nOnce the initializations are complete, the algorithm repeatedly selects the merger that maximizes modularity, and then updates the $\\Delta Q$ values, until no further mergers are possible.\nCNM then selects communities according to the largest found modularity.\nThe computational complexity of the CNM algorithm is\n$O(n^2\\log n)$.\n\\subsubsection{Louvain method}\nThe Louvain method is divided into two phases.  \nThe first phase initializes each node into its own community \nand considers increasing modularity locally by changing a node's community label to that of a neighboring community. \nFor example, the modularity change obtained by moving singular node $i$ from its community to a neighboring community $g$ is calculated by:\n\\begin{equation}\\label{eq_LouvStep}\n\\begin{split}\n\\Delta Q_{-i,g}={\\bigg [}{\\frac {\\Sigma _{in}+d_{i,in}}{2m}}-{\\bigg (}{\\frac {\\Sigma _{tot}+d_{i}}{2m}}{\\bigg )}^{2}{\\bigg ]}\n-{\\bigg [}{\\frac {\\Sigma _{in}}{2m}}-{\\bigg (}{\\frac {\\Sigma _{tot}}{2m}}{\\bigg )}^{2}-{\\bigg (}{\\frac {d_{i}}{2m}}{\\bigg )}^{2}{\\bigg ]}\n\\end{split}\n\\end{equation}\nwhere $\\Sigma_{in}$  is sum of internal edge weights within $g$, $ \\Sigma _{tot}$ is the sum of all edge weights incident to nodes in $g$, $d_{i,in} $ is the sum of the weights of edges between node $i$ and nodes in $g$, \nand $m$ is the sum of the edge weights of the entire network. \nFor each node $i$, this change in modularity is computed for each community neighboring $i$, and $i$ is then assigned the community label that corresponds with the largest change.  \nIf no positive change in modularity is possible, then $i$ does not change its community label.  \n\nIn the next phase, the algorithm uses the final network from the first phase to generate a new network.  \nEach community from phase one is condensed into a single node, and edges between two (possibly non-distinct) communities are condensed into a single edge with weight equal to the sum of the corresponding between-community edge weights.\nIn particular, self-loops in the new network may be formed if the corresponding community in the previous network has more than one node.\nPhase one is then applied to this new network.\n\n\nThese two phases are repeated iteratively until no local improvement in the modularity is possible.\nThe Louvain algorithm examines fewer mergers (only adjacent ones) compared to CNM, and hence, the time complexity is reduced---it appears to run in linearithmic time for sparse graphs \\citep{lancichinetti2009community}.\n\n\n\n\\subsection{Using edge weights to improve community detection}\n The quality of community detection may be improved by adding a pre-processing step to weight edges of the graph.  \n We briefly go over some proposed weighting schemes in the literature.\n\n\\subsubsection{Weights based on random walks}\nInspired by message propagation, \\citet{de2013enhancing} introduces a method called weighted edge random walk-K path (WERW-Kpath) that runs a $k$-hop random walk and weights edges on the walk based on their ability in passing a message. \nWERW-Kpath initially assigns to each edge $e\\in E$ a weight $w_e = 1$.\nAt each iteration, WERW-Kpath chooses a node at random and and runs a biased random walk of at most $k$ steps; the probability of traveling from node $i$ to $j$ is fraction of times that $j$ has been visited from node $i$ over the total number of times $i$ has been visited. \nFinal edge weights are obtained after many iterations of WERW-Kpath.\n\\citet{lai2010enhanced} suggests a similar approach for using random walks to explore local structures of communities, and ultimately obtain an edge weight for edge $ij$ as the cosine distance between $i$ and $j$.\n\n\n\n\\subsubsection{Preprocessing the graph by analyzing cyclic topologies}\\label{sec:LitRevWeighting}\nThe idea of using cyclic structure to identify communities relies on the fact that, if edges are more prevalent within communities than across communities, then flows will also tend to stay within communities. \nFor example,  \\citet{klymko2014using} focused on using triangles (cycles of length 3) to improve community detection in directed networks by weighting  the edges based on \\textit{3-cycle cut ratio}.\n\\citet{radicchi2004defining} incorporate the importance of triangles using the edge clustering coefficient.\nThis method is similar to \\citet{newman2004finding} in which at each iteration the edge with the smallest clustering coefficient is removed.\nThe complexity of the algorithm in~\\citet{radicchi2004defining} is $O(m^4\/n^2)$---roughly $O(n^2)$ for sparse graphs. \nThis algorithm may perform poorly in graphs with few short cycles.\n\n\n\\citet{castellano2004self} modified the edge clustering coefficient for the weighted networks, in which number of cycles is multiplied by the edge weight. \n\\citet{zhang2008clustering} extended this to bipartite graphs with \ncycles of even length.\n\\citet{vragovic2006network} introduced the node loop centrality measure such that communities are built around nodes with high centrality in graph cycles---this algorithm has time complexity of $O(nm)$.\nCommunities with rich cyclic structures--many short loops--are known to have higher quality; flows tend to stay longer and information disseminates faster in their nodes.\n\n\\citet{shakeri2017network} suggest weighting edges using a method called \\textit{loop modulus} (LM). \nLM finds a probability distribution over all cycles such that two random cycles from this distribution have \"minimum edge overlap.\"\nThe weight of an edge is the probability that this edge is overlapped.\n\n\n\n\n\n\n \n\n\n\n\\subsubsection{Other weighting method based on local and global network structure}\\label{sec:k_path}\nThe use of global measures for determining edge weights requires the knowledge of entire networks. \nWhile these edge weights may be helpful in community detection, their use requires substantial computation, and sometimes the weights lack the necessary specificity to the community structures.\nLocal measures are often much faster to compute and may estimate local network subtlety that global methods cannot, but are unable to obtain thorough knowledge of global network structure.  \nWe now summarize the use of global and local measures in the literature.\n\n\n\\citet{newman2004finding} propose to use of edge-betweeness centrality (EBC) as a way to evaluate the community structure of networks. \nEBC was introduced by \\citet{freeman1977set} and, for each edge $e$, measures the fraction of geodesic paths between all pairs of nodes that passes through $e$. \n\\citet{khadivi2011network} proposed to use common neighbor ratio in addition to EBC to balance the local and global measurements and weigh the graph edges. \nThis will allow for smaller weights of edges formed between clusters and also facilitates greedy algorithms in identifying the hierarchical structure of the clusters. \nThe runtime complexity of this weighting strategy is dominated by computing the edge-betweenness centrality which is in $\\mathcal{O}(nm)$ \\citep{brandes2001faster} \nThis runtime is prohibitive for large networks.\nFurthermore, tuning the hyperparameters requires the design of efficient heuristics to improve the performance of the weighting algorithm.\n\\citet{zhang2015novel} focused on the problem of local and global weighting of the edges to improve the separation of communities. They proposed to measure the similarity of each pair of nodes based on their similarity to other nodes through an iterative function called SimRank.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Renewal non-backtracking random walk (RNBRW)} \\label{sec_3}\n\n\n\n\nWe now introduce our method for weighting edges called the renewal non-backtracking random walk (RNBRW).\nThis method is effective in capturing network cyclic structure while maintaining the computational advantages given by random walk methods. \nWe begin with some definitions.\n\nRecall the definition of a random walk in Section~\\ref{sec:prelim}.\nA non-backtracking random walk (NBRW) is a random walk that cannot return to a node $i$ in exactly two steps. \nThat is, for all nodes $j$,\n\\begin{eqnarray}\n    P(e_{\\ell+1} = \\vec{ji} | e_{\\ell} = \\vec{ij}) = 0.\n\\end{eqnarray}\nNote that, while NBRW is not Markovian on the nodes of the graph, we can make NBRW Markovian on the edges by replacing each undirected edge $uv$ in $E$ with two directed edges $\\vec{uv}$ and $\\vec{vu}$. \n\nFor an unbiased NBRW, the transition probability is \\citep{kempton2016non}\n\\begin{equation}\\label{eq_NBRW_transition}\n\\Pr(\\vec{jk},\\vec{ij})=\n \\begin{cases} \n1\/(d_j-1),& \\text{ if } \\vec{ij}, \\vec{jk} \\in E \\text{ and } k \\neq i, \\\\\n0, & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere $d_j$ is the out-degree of node $j$.\nA primary benefit of NBRW is that it explores the graph more efficiently than random walks; using NBRW, the proportion of time spent on each edge converges faster to the stationary distribution.\n\nIn previous works, such as \\citet{de2013enhancing}, the random walk has been used to quantify the idea of information passing and to identify nodes or edges that are more capable of passing information.\nBoth random walk and NBRW behave similarly in the long run and their stationary distribution depends only on the degrees of nodes \\citep{kempton2015high}. \nTo reduce this dependability on node-degree, we keep the walk length small by stopping the NBRW.\nHowever, instead of stopping NBRW after a given number of iterations, we instead make a stopping rule that may help in uncovering cyclic structure in the network, thereby obtaining the renewal non-backtracking random walk. \n\n\\begin{definition}[Renewal non-backtracking random walk (RNBRW)]\nA renewal non-backtracking random walk (RNBRW) is a NBRW that terminates and restarts once the walk completes a cycle.\nEach NBRW begins at a directed edge selected completely at random.\nEach new run of a NBRW is an iteration of RNBRW.\n\\end{definition}\n\\begin{definition}[Retraced edge]\n\tThe retraced edge for a run of RNBRW is the edge that forms a cycle. \n\tMore precisely, supposing that $j$ is the first node revisited by a NBRW, and supposing that the NBRW travels to $j$ from $i$, then $\\vec{ij}$ is the retraced edge.  \n\\end{definition}\n\\begin{definition}[Retracing probability]\n\tThe retracing probability $\\pi_e$ of an edge $e$ is the probability that $e$ is retraced in one iteration of RNBRW given that the NBRW terminates by forming a cycle. \n\tThe retracing probability $\\pi_{ij}$ of an undirected edge $ij$ is obtained by summing the retracing probabilities for both $\\vec{ij}$ and $\\vec{ji}$.\n\\end{definition}\n\nFigure~\\ref{fig:RetracedNBRW} gives an example of one RNBRW iteration on a graph.\nHeuristically, edges with large retracing probabilities are important to the formation of cycles.\nAs discussed in Section~\\ref{sec:LitRevWeighting}, the presence of cycles is a helpful indicator in discovering communities.\nThus, it seems reasonable that quantifying the cyclic structure by the retracing probabilities may lead to improved performance of algorithms designed for detecting communities.\nFigure~\\ref{fig:HouseNBRW} gives undirected retracing probabilities for a small graph.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[clip, angle = 90, width=.4\\columnwidth]{RetracedNBRW.pdf}%\n\t\\caption{One iteration of RNBRW on a graph. Gray lines denote edges not traversed by the NBRW.  The walk begins with the edge in black.  The walk completes after forming the cycle in blue. The dotted edge represents the retraced edge.}\n\t\\label{fig:RetracedNBRW}\n\\end{figure}\n\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[clip,width=.5\\columnwidth]{House3.pdf}%\n\t\t\\caption{Undirected retracing probabilities for a house graph with 5 nodes and 6 edges. \n\t\n\t\n\t\tEdges with more overlapping loops have higher retracing probabilities.  \n\t\tFor this graph, edge $dc$ appears to be the most important edge for the formation of cycles. \n\t\t}\\label{fig:HouseNBRW}\n\t\\end{figure}\n\nThe retracing probabilities $\\pi_e$ for each edge $e$ may be difficult to compute analytically, even for small graphs.  \nInstead, as we show in Section~\\ref{subsec:asymptoticprops} these probabilities can be estimated precisely by running many iterations of RNBRW.\nPrecisely, the estimated retracing probability for edge $e$, denoted $\\hat \\pi_e$, is the fraction of RNBRW runs for which $e$ is the retracing edge.\nNote that $\\sum_{e} \\pi_e = \\sum_{e} \\hat \\pi_e = 1$.\n\n\n\nA particularly useful property of RNBRW is that iterations are mutually independent.\nIn particular, unlike other random walk methods (\\textit{e.g.} \\citet{de2013enhancing}), each run is not affected by weights obtained from previous runs.\nHence, multiple RNBRW instances can be run in parallel, and thus, retracing probabilities can be estimated quickly, even for large graphs.\nFor example, in \\ref{sec:results}, we apply RNBRW on graphs containing one-million nodes.\n\nFinally, note that a RNBRW iteration may terminate in one of two ways: either the run forms a cycle or the NBRW reaches a node with degree 1.\nIn the latter case, the iteration is discarded---no retracing edge is recorded---and a new iteration of RNBRW is started.\nFortunately, the probability that this case occurs may become vanishingly small as the number of nodes $n$ gets large.\nFor example, for Erd\\H{o}s-R\\'enyi graphs, the probability that a run terminates by visiting a node with degree 1 decreases exponentially with $n$~\\citep{tishby2017distribution}.\n\n\\subsection{RNBRW for community detection}\n\nRNBRW can be implemented to improve community detection algorithms as follows.\nFirst, the estimated undirected retracing probability $\\hat \\pi_e$  is obtained for each edge $e$ through performing many iterations of RNBRW.\nThen, each edge $e$ in the graph given the weight $w_e = 2m\\pi_e$, where $m$ is the number of edges.\nNext, the weighted degree $d_i$ in~\\eqref{eq:definedi} is computed for each node $i$.\nNote that multiplying $\\hat \\pi_e$ by $2m$ ensures that the sum of degrees $\\sum_{i=1}^n d_i$ is the same when moving from unweighted edges to weighted edges. \nThese weighted degrees are then plugged into the modularity function~\\eqref{eq_Modularity}.\nFinally, community detection algorithms such as CNM or Louvain are then performed using the weighted modularity function.\n\nFigure~\\ref{fig:weightLFR} demonstrates the usefulness of RNBRW weighting in community detection.\nThis figure shows an LFR benchmark graph~\\citep{lancichinetti2008benchmark} with four communities before and after weighting RNBRW.\nWeights of edges formed within communities are much larger on average than weights of edges across communities.\n\n\\begin{figure}\n\t\\centering\n\t\\subfloat[]{\t\t\n\t\t\\includegraphics[clip,width=.35\\columnwidth]{LFR_unweighted.pdf}%\n\t}~~~~~~~~~\n\t\\subfloat[]{%\n\t\t\\includegraphics[clip,width=.35\\columnwidth]{LFR_weighted.pdf}%\n\t}\\\\\n\n\n\t\\caption{(a) An LFR benchmark graph. (b) The same graph weighted by RNBRW. Observe that within-community edges have larger weights than across-community edges.} \n\n\t\\label{fig:weightLFR}\n\\end{figure}\n\nFigure~\\ref{fig:iteration} gives a simple illustration of how many iterations of RNBRW is necessary in order to obtain satisfactory detection of communities after weighting.\nAs the figure indicates, \nthere is a large improvement in the detection of communities when the number of walkers is on the order of the number of edges in the graph. \nRunning additional walkers will still improve this precision;\nhowever, the corresponding improvement in community detection is fairly small.\nA theoretical justification for this number of iterations\ncan be found in Section~\\ref{subsec:asymptoticprops}.\n\n\n\n\t\\begin{figure}\n\n\t\\centering\n\n\n\n\t\\includegraphics[clip,width=.5\\columnwidth]{NumberWalker.pdf}\n\n\t\\caption{Performance of community detection with RNBRW weighting increasing the number of walkers in both CNM and Louvain using RNBRW weighting for LFR networks with $n = 10,000$ and average degrees $9$ and $27$.}\n\t\\label{fig:iteration}\n\\end{figure}\n\n\n\n\n\n\n\n \n\n\n\n\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\\n\n\n\n\n\n\n\n\n\n\\subsection{Algorithm for RNBRW}\n\nFor completeness, we now describe in detail the RNBRW algorithm. \nEach run of the algorithm runs a NBRW and returns the retracing edge (if any) as the sample of retracing edge. \nThe number of times each edge has been retraced by a NBRW is approximately proportional to its retracing probability.\nThe algorithm only requires knowledge the graph and each run is independent of the other. \nTherefore, we can collect samples in parallel leading to fast convergence. \nEach run of the algorithm is comprised of the following steps:\n \\begin{itemize}\n\t\\item[1] Choose a random edge $\\overrightarrow{v_0v_1}$ in $E$.\n\t\\item[2] Form the walk $w = \\left(\\overrightarrow {v_0v_1}\\right)$. \n\t\\item[3] (For  $k=\\{1,\\cdots\\}$) The walker continues her walk from ${v_k}$ to a neighboring node $v_{k+1}\\neq v_{k-1}$.\n\t\\item[4] If $v_{k+1}$ has degree 1, return immediately to Step~1.  \n\tIf $v_{k+1}$ is already in $w$, return $\\overrightarrow{v_kv_{k+1}}$ as the retracing edge and return to Step~1.   \n\tOtherwise add $v_{k+1}$ to $w$ and go to Step 3 incrementing $k = k+1$.\n\\end{itemize}\nIn Algorithm~1, we present the pseudo-code for RNBRW.\nBy employing a swarm of walkers and recording their retracing edges, we are able to get an accurate estimate of the retracing probability for each edge.\nNote that these walkers can be initialized independently from each other and retraced edges can be collated at the very end of the process. \nTherefore, one can execute this process as an array of jobs on a cluster of computers efficiently. \n\n\\begin{figure}\n\t\\begin{minipage}{\\linewidth}\\label{algNBRW}\n\t\t\\begin{algorithm}[H]\n\t\t\t\\caption{Algorithm for  RNBRW from a random directed edge $\\vec{uv}$.}\n\t\t\t\\begin{algorithmic}[1]\n\t\t\t\t\\State \\textit{walk} $\\leftarrow$empty set\n\t\t\t\t\\State \\textbf{choose} edge $\\vec{uv}$ at random\n\t\t\t\t\\State \\textbf{add} $\\vec{uv}$ to \\textit{walk}\n\t\t\t\t\\While{True}\n\t\t\t\t\\State \\textit{nexts}$\\leftarrow$neighbors of $v$\n\t\t\t\t\\State \\textbf{remove} $u$ from \\textit{nexts}\n\t\t\t\t\\State \\textbf{if} nexts is empty \\textbf{break}\n\t\t\t\t\\State \\textit{next} $\\leftarrow$\\textbf{choose} a node from \\textit{nexts} randomly\n\t\t\t\t\\If {\\textit{next }$\\in$\\textit{walk}}\n\t\t\t\t\\State \\textbf{return} ($v$,\\textit{next}) as the retracing edge\n\t\t\t\t\\State \\textbf{add} \\textit{next} to \\textit{walk}\n\t\t\t\t\\State $u\\leftarrow v$\n\t\t\t\t\\State $v \\leftarrow $\\textit{next}\n\t\t\t\t\\EndIf\n\t\t\t\t\\EndWhile\n\t\t\t\\end{algorithmic}\n\t\t\\end{algorithm}\n\t\\end{minipage}\n\\end{figure}\n\n\n\n\n\\section{Asymptotic properties of RNBRW \\label{subsec:asymptoticprops}}\n\nWe now give proof that the RNBRW estimates for the retracing probabilities converge almost surely to the true retracing probabilities.\nThe proof also gives insight as far as how many iterations are necessary in order to obtain a good estimate of these probabilities.\nWe begin with a few definitions.\n\nSuppose there are $\\rho$ iterations of RNBRW that conclude by returning a retracing edge.\nFor a given directed edge $\\vec e$, define $Y_{\\vec e,k}$ as a random variable indicating that the $k$th iteration returns edge ${\\vec e}$ as the retracing edge.\nThat is,\n\\begin{equation}\n\tY_{\\vec e,k}=\n\t\t\\begin{cases}\n\t\t1,&  \\text{if the $k$th RNBRW returns $e$ as the retracing edge}, \\\\\n\t\t0, &  otherwise.\n\t\t\\end{cases}\n\\end{equation}\nNote that $Y_{\\vec e,k}$ is a \\textit{Bernoulli}$(\\pi_{\\vec e})$ random variable and that\n\\begin{equation}\n    \\overline Y_{\\vec e} = \\frac{1}{\\rho}\\sum_{k = 1}^\\rho Y_{\\vec e,k} = \\hat \\pi_{\\vec e}.\n\\end{equation}\n\nWe first make a statement about how close any arbitrary estimated retracing probability comes to the true probability.\n\\begin{lemma}\\label{lemma:justhoeff}\n    \\begin{equation}\n    \\label{eq:justhoeff}\n        P(|\\overline{Y_{\\vec e}}-\\pi_{\\vec e}| \\geq \\epsilon) \\leq 2 \\exp(-2\\rho \\epsilon^2).\n    \\end{equation}\n\\end{lemma}\n\\begin{proof}\n    This result follows from direct application of Hoeffding's inequality~\\citep{hoeffding1963probability}.\n\\end{proof}\n\nWe now obtain a result on how estimates of undirected retracing probabilities uniformly converge to the true probabilities.\nSuppose that, in the original NBRW setup, undirected edge $e$ is replaced with directed edges $\\vec e_1$ and $\\vec e_2$. \nHence, $\\pi_{e} = \\pi_{\\vec e_1} + \\pi_{\\vec e_2}$, and\nthis undirected probability is estimated by $\\overline Y_e = \\overline Y_{\\vec e_1} + \\overline Y_{\\vec e_2} = \\hat \\pi_{\\vec e_1} + \\hat \\pi_{\\vec e_2}$.\n\\begin{lemma}\\label{lemma:maxsumhoeff}\n    \\begin{equation}\n    \\label{eq:maxsumhoeff}\n        P(\\max_{e} |\\overline{Y_{e}}-\\pi_{e}| \\geq \\epsilon) \\leq 4 m \\exp(-\\rho \\epsilon^2\/2).\n    \\end{equation}\n\\end{lemma}\n\\begin{proof}\nTo see this, note that\n\\begin{equation}\n    \\label{eq:halfthehoeff}\n\tP(|\\overline{Y_{e}}-\\pi_{e} |\\geq \\epsilon)=\n\t\tP(|\\overline{Y}_{\\vec e_1}-\\pi_{\\vec e_1} + \\overline{Y}_{\\vec e_2}-\\pi_{\\vec e_2} |\\geq \\epsilon) \\leq\n\t\tP\\left(\\bigcup_{\\ell = 1}^2\\left\\{|\\overline{Y}_{\\vec e_\\ell}-\\pi_{\\vec e_\\ell} |\\geq \\epsilon\/2\\right\\}\\right).\n\\end{equation}\nThus, by~\\eqref{eq:halfthehoeff} and Lemma~\\ref{lemma:justhoeff}, we have\n    \\begin{eqnarray}\n\t\tP(\\max_{e}|\\overline{Y_{e}}-\\pi_{e} |\\geq \\epsilon) &=& \n\t\t\tP\\left(\\bigcup_{e}\\left\\{\\left|\\overline{Y_{ e}}-\\pi_{e} \\right|\\geq \\epsilon\\right\\}\\right)  \\nonumber \\\\ &\\leq&\n\t\n\t\t\tP\\left(\\bigcup_{e}\\bigcup_{\\ell = 1}^2\\left\\{\\left|\\overline{Y}_{ \\vec e_{\\ell}}-\\pi_{\\vec e_\\ell} \\right|\\geq \\epsilon\/2\\right\\}\\right)\t \\nonumber \\\\\n\t\t&\\leq& \\sum_{{e}}\\sum_{{\\ell = 1}}^2 P\\left(\\left|\\overline{Y}_{\\vec e_\\ell}-\\pi_{\\vec e_\\ell} \\right|\\geq \\epsilon\/2\\right) \\nonumber \\\\ &\\leq&\n\t\t 4m \\exp(-\\rho \\epsilon^2\/2).\n    \\end{eqnarray}\n\\end{proof}\n\\noindent\nThat is, we have shown that the estimated retracing probabilities converge exponentially fast to the true retracing probabilities as number of iterations of RNBRW (that terminate by forming a cycle) increases.  \n\nLemma~\\ref{lemma:maxsumhoeff} also gives insight as far how many iterations of RNBRW are necessary to obtain good estimates of the retracing probability.  \nWe now give a result that demonstrates how accurate these estimates are if the number of RNBRW runs is approximately equal to the number of edges in the graph.\n\\begin{corollary}\n    Given $\\rho = m$ iterations of RNBRW, the estimated retracing probabilities are guaranteed to be within $\\sqrt{\\frac{4\\log(2m)}{m}}$ of the true probabilities\n    with probability no smaller than $(m-1)\/m$.\n    That is,\n    \\begin{equation}\n        \tP\\left(\\max_{e}|\\overline{Y_{e}}-\\pi_{e} |\\geq \\sqrt{\\frac{4\\log(2m)}{m}}\\right) \\leq \\frac{1}{m}.\n    \\end{equation}\n\\end{corollary}\n\\begin{proof}\n    To see this result, set $\\rho = m$ and $\\epsilon = \\sqrt{\\frac{4\\log(2m)}{m}}$ in~\\eqref{eq:maxsumhoeff}.\n\\end{proof}\n\nFinally, we show that the estimated retracing probabilities converge almost surely and uniformly to the true retracing probabilities.\n\\begin{theorem}[Strong and uniform consistency]\n    As the number of iterations of RNBRW that terminate in a cycle $\\rho \\to \\infty$, then $\\hat \\pi_e \\overset{a.s.}{\\to} \\pi_e$ for each undirected edge $e$.  Moreover this convergence is uniform across all edges $e$.\n\\end{theorem}\n\t\\begin{proof}\n\tUsing~\\eqref{eq:maxsumhoeff}, we observe that\n\t\\begin{eqnarray}\n\t\t\\lim_{\\rho \\to \\infty}\\sum_{n = 1}^\\rho P(\\max_{e} |\\overline{Y_{e}}-\\pi_{e}| \\geq \\epsilon) \\leq\n\t\t\t\\lim_{\\rho \\to \\infty}\\sum_{n = 1}^\\rho 4 m \\exp(-\\rho \\epsilon^2\/2) < \\infty.\n\t\\end{eqnarray}\t \n\tThe theorem then follows directly from the Borel-Cantelli Lemma \\citep{rosenthal2006first}.\n\t\\end{proof}\n\t\n\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n\n\n\n\n\n\\section{Simulation results}\\label{sec:results}\nWe investigate the performance of two community detection methods, Louvain and CNM, with and without the proposed weighting methods.\nIn addition to RNBRW, we consider these preprocessing weighting methods: WERW-Kpath, SimRank, Loop modulus, and the weighting method considered in~\\citet{khadivi2011network}.\nAdditionially, we compare the performance from preprocessing with RNBRW to other scalable community detection algorithms such as\nInfo map \\citep{rosvall2007maps}, Label propagation \\citep{raghavan2007near}, Edge betweenness \\citep{girvan2002community}, Spin glass \\citep{reichardt2006statistical}, and Walk trap \\citep{pons2005computing}. \nTo perform our comparisons, we apply the community detection algorithms on LFR benchmarks \\citep{lancichinetti2008benchmark} and measure their similarity to the ground truth data using normalized mutual information (NMI)  \\citep{danon2005comparing}. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison of weighting methods}\n\n\nFigure~\\ref{fig:InfoMixModulusNBRW} compares the performance in detecting communities for various weighting methods coupled with the Louvain algorithm as the mixing parameter $\\mu$ increases.\nFigure~\\ref{fig:Khadivi_L} compares these methods for sparse graphs (average degree is $\\approx \\log(n)$) given a fixed mixing parameter but as the number of nodes increases.\nIn all cases, preprocessing the graph with RNBRW substantially improves the detection of communities when compared to the unweighted algorithm.\nAdditionally, the improvement provided by RNBRW is substantially larger than that from the WERW-KPath, SimRank, and the Khadavi \\textit{et.~al.} methods.\nThe performance of RNBRW seems comparable to that of loop modulus.  \nHowever, a good comparison between RNBRW and loop modulus is difficult as current methods for computing loop modulus require prohibitive computational cost for graphs containing more than a couple thousand nodes. \nHence, loop modulus is omitted from \nFigure~\\ref{fig:Khadivi_L}.\nAppendix \\ref{app_CNM} gives these results for the CNM algorithm.\n\n\n\\begin{figure}\n\n\t\\centering\n\t\\includegraphics[width=.60\\textwidth]{InfoMix_L.pdf}\n\t\\caption{\n\tPerformance analysis of community detection when coupling weighting methods with the Louvain algorithm. \n\tWe use LFR benchmark networks with  $n = 500$, average degree $7$, and community sizes ranging from $30$ to $70$.\n\t\tThe mixing rate $\\mu$ adjusts the ratio of within-communities links over all links.}\\label{fig:InfoMixModulusNBRW}\n\t\n\t\n\t\n\t\n\t\t\t\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure}\n\n\t\\centering\t\t\n\t   \n\t\t\\includegraphics[width=.65\\textwidth]{Info_degree_retracing.pdf}%\n\t\n\t\\caption{Performance analysis of weighting methods on LFR benchmark networks with $n=500$ and varying average degree. We compare the performance of weighting RNBRW with the raw algorithm (no weighting) and weighting WERW-Kpath with the raw algorithm (no weighting).}\\label{fig:InfoMixW}\n\\end{figure}\n\\end{comment}\n\n\n\n\t\\begin{figure}\n\t\n\t\t\\centering\n\t\n\n\t\t\t\\includegraphics[clip, width=.60\\columnwidth]{sparse_khadivi_L.pdf}%\n\t\t\\caption{Performance improvement of Louvain algorithm for different weighting algorithms  for sparse LFR benchmark graphs ($\\mu = 0.4$, $\\hat{d} = \\log(n)$). Loop modulus is omitted in this example due to expensive computational costs.}\\label{fig:Khadivi_L}\n\t\\end{figure}\n\t\n\n\n\n\n\n\n\n\\subsection{Performance of RNBRW to other scalable algorithms}\n\nWe now compare the performance of Louvain and CNM equipped with RNBRW to five popular scalable algorithms: Infomap, Label propagation, Edge betweenness, Spinglass, and Walktrap.\nWe first begin our comparison on a small LFR benchmark graph with identical parameters to that of Figure~\\ref{fig:InfoMixModulusNBRW}.\nFigure~\\ref{fig:InfoMixLFR} suggests that---on an albeit small graph of $n=500$ nodes---Louvain with RNBRW is as good or better at discovering communities as the aforementioned algorithms across all mixing parameters considered.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[clip,width=.60\\columnwidth]{InfoMixLFR500_single.pdf}%\n\t\\end{center}\n\t\\caption{Comparing the performance of Louvain equipped with RNBRW to Infomap, Label propagation, Edge betweenness, Spinglass, and Walktrap algorithms. \n\tThe LFR benchmark networks have  $500$ nodes with average degree $7$, and community sizes ranging from $30$ to $70$.\n\n\t\t}\\label{fig:InfoMixLFR}\n\\end{figure}\n\nWe now focus our analysis on larger graphs with varying degrees of sparseness.\nSparse graphs tend to be a particularly challenging case for community detection algorithms.\nWhile \\citet{zhao2012consistency} showed consistent detection of communities if the average degree grows at least logarithmically with network size, in practice, some algorithms may still struggle when average degree increases sub-linearly with the number of nodes.\nAnother problem that arises with modularity maximization methods is their resolution limit in detecting small communities.\nThis is a critical shortcoming since small size communities are common in large social networks and their size  are  not necessarily growing with graph size. \n\\citet{fortunato2007resolution} identify that communities with number of internal edges less than $\\sqrt{2|E|}$ are most likely misdetected. \nThis leads to a resolution limit that holds back heuristics for  modularity maximization.\n\nWe test these community detection algorithms and see how they withstand the challenges of low average degree and existence of small communities for LFR benchmarks.\nWe consider average degrees that grow logarithmically with the number of nodes.\nThe community sizes vary between an upper and a lower bound that falls below the resolution limit and the average degree is changing for different size graphs. \n\nFor these larger benchmarks\\footnote{We use Beocat for running simulations on the large networks. Beocat is a computer cluster located in Kansas State University \\url{https:\/\/support.beocat.ksu.edu\/BeocatDocs\/index.php\/Main_Page}.} we use a constant mixing rate $\\mu =0.3$ and consider graphs with $10,000$, $100,000$ and $1,000,000$ nodes.\nEdge betweenness, Spinglass, and the previously considered weighting algorithms are not included due to prohibitive computational cost with larger network sizes.\n\nWe first analyze how the use of RNBRW can improve the performance of Louvain and CNM for sparse graphs.\nFigure~\\ref{fig:100k} plots the mutual information for graphs of $n = 10,000$ and $n = 100,000$ nodes for CNM and Louvain with and without RNBRW.  \nWe demonstrate that, even when the average degree is on the order of $\\log(n)$, these algorithms equipped with RNBRW achieve a mutual information very close to 1.  \nIn particular, for $n = 10,000$ and average degree $\\log(n)$, equipping CNM with RNBRW increases the mutual information from $0.263$ to $0.974$.\n\n\n\n\n\\begin{figure}\n\n\t\\centering\n\t\\includegraphics[clip,width=1\\columnwidth]{Info_degree_retracing10k_single.pdf}%\n\t\\caption{Plots of NMI for sparse graphs.  \n\tWe use LFR networks with $n = 10,000$ (left) and $n = 100,000$ nodes (right) with mixing parameter $\\mu = 0.3$.  \n\tAverage degree varies from $\\log(n)$ to $3\\log(n)$.\n\tWeighting by RNBRW leads to substantial performance increases for both the CNM and Louvain algorithms. }\\label{fig:100k}\n\\end{figure}\n\n\t\\begin{table}\n\t\t\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{ |l|c|c|c|c|c|c|c| c| }\n\t\\hline\n\tnetwork size & average degree & Infomap & LP & WT &CNM & Louvain &    RNBRW+CNM&  RNBRW+Louvain\\\\\n\t\\hline\n\t\\multirow{3}{*}{$10,000$} & $\\log n$  & $0.912$ &   $\\textbf{0.992}$ & $0.763$ & $0.263$ & $0.740$   &  $0.974$ &   $0.970$\\\\\n\t& $2\\log n$ &  $\\textbf{1}$ &   $0.995$ & $\\textbf{1}$  &$0.276$ & $0.757$   &   $\\textbf{1}$ &   $\\textbf{1}$\\\\\n\t& $3\\log n$ &   $\\textbf{1}$ &   $\\textbf{1}$ & $\\textbf{1}$ &$0.232$ & $0.882$   &   $\\textbf{1}$ &   $\\textbf{1}$\\\\\n\t\\hline\t\n\t\\multirow{3}{*}{$100,000$}& $\\log n$ & $0.73$ &   $\\textbf{0.988}$ & $0.644$ & $0.154$ & $0.524$   &    $0.975$ &   $0.960$\\\\\t\n\t& $2\\log n$ &  $\\textbf{1}$ &   $\\textbf{1}$ & $\\textbf{1}$ & $0.159$ & $0.649$   &   $\\textbf{1}$ &   $\\textbf{1}$\\\\\t\n\t& $3\\log n$ & $\\textbf{1}$ &   $\\textbf{1}$ & $\\textbf{1}$ &$0.118$ & $0.713$   &     $\\textbf{1}$ &   $\\textbf{1}$\\\\\t\n\t\\hline\t\t\t  \n\t$1,000,000$ & $\\log n$ & $0.994$ &   $\\textbf{0.998}$ & $-$ & $0.050$ & $0.192$   &   $0.989$ &   $0.969$\\\\\t\n\t\\hline\n\\end{tabular}}\n\\caption{Performance measured by NMI for scalable community detection algorithms for sparse LFR networks with $\\mu =0.3$ and average degree $\\approx \\log n$.  \nWe compare CNM and Louvain with and without RNBRW weighting with Infomap, Label propagation (LP), and Walktrap (WT).  }\n\\label{tab_results}\n\\end{table}\n\nWe now compare CNM and Louvain equipped with RNBRW with other efficient community detection algorithms.\nSummaries of these simulations are be found in Table~\\ref{tab_results}. \nWe first note that, again, equipping CNM and Louvain with RNBRW improves community detection dramatically; for example, in the 1,000,000 node example, equipping these community detection algorithms with RNBRW leads to an increase NMI in CNM from 0.050 to 0.989 and in Louvain from 0.192 to 0.969. \nSimilar increases occur across all other network sizes and average degrees.\nMoreover, the detection of communities for CNM and Louvain with RNBRW is comparable to other scalable community detection algorithms.\n\nAlthough not included in the table, the runtime of RNBRW---especially coupled with Louvain---is much lower than the other considered methods.  \nFor example, because RNBRW is embarrassingly parallelizable, because Louvain has a small asymptotic runtime, and because RNBRW weights and the Louvain algorithm are performed sequentially, the total runtime of Louvain weighted by RNBRW is smaller than\nInfomap $\\mathcal{O}(n^2)$, Label propagation $\\mathcal{O}(n^2)$, Spinglass $\\mathcal{O}(n^{3.2})$ , and \nWalktrap $\\mathcal{O}(n^2\\log n)$. \nTherefore, adding a RNBRW preprocessing to Louvain seems to be a very reasonable method for obtaining efficient community detection with performance comparable to that of the best scalable community detection algorithms.\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nCommunities can be identified by the ``richness'' of cycles; more short cycles occur within a community than across communities.  \nHence, existing community detection algorithms may be enhanced by incorporating information about the participation of edges in rhe cyclic structure of the graph. \nWe develop renewal non-backtracking random walks (RNBRW) as a way of quantifying the cyclic stucture of a graph.\nRNBRW quantifies edge importance as the likelihood of an edge completing a cycle in a non-backtracking random walk, providing a scalable alternative to analyse real-world networks.\nWe describe how RNBRW weights can be coupled with other popular scalable community detection algorithms, such as CNM and Louvain, to improve their performance.\n\n\nWe show that weighting the graph with RNBRW can substantially improve the detection of ground-truth communities.\nThis improvement is most notable when the network is sparse. \nFurthermore, the RNBRW preprocessing step can overcome the problem of detecting small communities known as resolution limit in modularity maximization methods.\nWe show that performance appears to be equal or superior to that of other weighting methods, and of other scalable (non-weighted) commnity detection algorithms.\nIn the case of large, sparse, networks, RNBRW may be quite effective; RNBRW can improve the efficacy of available community detection algorithms without sacrificing their computational efficiency.\n\n\n\n\n\\bibliographystyle{chicago}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLarge-scale magnetic fields with fascinating \nquasi-regular spatio-temporal behavior \nare ubiquitous in solar and  stellar settings.  \nUnderstanding the mechanisms for the generation of such fields and \ntheir spatio-temporal variations is still a major challenge for \ndynamo theory. \nThe solar magnetic field has three particularly\nimportant features: quasi-regular oscillations, reversal of polarity\nand equatorward migration. \nDirect numerical simulations of solar-like convective dynamos\nhave been able to generate large-scale magnetic fields \n\\citep{bro+bro+bru+mie+nel+tom07,brown10}\nwhich in some cases show oscillatory behavior,\nbut the fields exhibit either rather weak equatorward migration at high\nlatitudes~\\citep{charb} or \nanti-solar (i.e.\\ poleward) migration~\\citep{gil83,kapyla_etal2010}. \n\nA useful tool for studying these dynamical phenomena is mean-field (MF)\nelectrodynamics \\citep[e.g.][]{Kra+Rad80,bra+sub05}, where the effects\nof turbulence are characterized by turbulent magnetic diffusion\nand an $\\alpha$ effect.\nAccording to MF theory, equatorward migration\nis expected if there is negative radial shear accompanied by a\npositive (negative) $\\alpha$ effect in the northern (southern) \nhemisphere \\citep{Kra+Rad80}.\nDirect numerical simulations (DNS) of helical turbulence with shear \nhave confirmed the presence of migratory dynamo waves \n\\citep{bra+big+sub02,kap+bra09}. It is, however, \nunclear whether this is really what is going on in the Sun,\nsince there the layer with negative radial shear is rather thin and\nonly concentrated near the surface (see, e.g., \\cite{bra05} and references therein).\nThe other alternative is that meridional circulation might change the\ndirection of migration \\citep{cho+sch+dik95}, but evidence for\nthis has not yet been seen in DNS.\n\nIn this Letter we present a completely different mechanism for\npolarity reversal and equatorward migration of dynamo activity.\nIn the context of MF models this mechanism is connected with the \nantisymmetry of the profiles\nof $\\alpha$ across the equator \\citep{Rud+Hol04,bra+can+cha09}.\nWe demonstrate the operation\nof this mechanism in DNS of the equations of magnetohydrodynamics (MHD).\nOur model consists of a spherical wedge-shaped\nshell in which the turbulence in the fluid is maintained by a random\nhelical forcing.\nMotivated by the Sun, we choose our forcing to have opposite\nsigns of helicity in the two hemispheres\n(negative in the north and positive in the south).\nWe emphasize that, even though our model\ndoes not explicitly include convection, stratification and \nrotation,\nthe helical forcing used here does partially\nmodel these features implicitly.\n\nOur model shows large-scale magnetic fields\nin excess of the equipartition value. More importantly,\nwe find oscillations of the magnetic field which show opposite\nsigns in different hemispheres with periodic reversals of polarity.\nFurthermore, the magnetic\nfield develops at higher latitudes and migrates equatorward where the two\ndifferent polarities of magnetic field annihilate and the cycle repeats itself\nas shown in the top panel of Fig.~\\ref{fig:butterfly}. \nTo our knowledge, such\ndynamical features of the large-scale magnetic field have not been observed\nearlier in DNS of MHD turbulence.\nBelow we introduce our model, discuss its oscillatory solutions,\nand briefly compare our DNS results with those obtained\nfrom corresponding MF models.\n\n\\begin{figure}[h]\n\\includegraphics[width=\\linewidth]{fig\/butterfly_three_eqopen.ps}\n\\caption{Space-time diagrams of the azimuthal component\nof the large-scale magnetic field for\nDNS over both the hemispheres (top panel),\nDNS over only the northern hemisphere with an antisymmetric\ncondition at the equator (middle panel), and\nMF simulation over only the northern hemisphere with \nan antisymmetry condition at the equator (bottom panel).}\n\\label{fig:butterfly}\n\\end{figure}\n\\begin{figure}[h]\n\\includegraphics[width=\\linewidth]{fig\/constr_large_domain.ps}\n\\caption{Orthographic projection of the toroidal magnetic field $B_\\phi$\nat $r=0.85$ for Run~{\\tt S5}.\nThe projection is tilted by $15$ degrees towards the viewer. \n} \n\\label{fig:contour}\n\\end{figure}\n\\section{The model}\nIn our simulations, we solve the equations for compressible MHD\nin terms of the velocity $\\u$, the logarithmic density $\\ln\\rho$,\nand the magnetic vector potential $\\A$,\n\\begin{eqnarray}\n\\label{mhd1}\n\\Dt\\u &=& -c_{\\rm s}^2\\grad\\ln\\rho + \\frac{1}{\\rho}\\j\\times\\B \n + \\bm{F}_{\\rm visc} + \\f, \\\\\n\\Dt\\ln\\rho &=& -\\grad\\cdot\\u, \\\\\n\\delt\\A &= & \\u\\times\\B + \\eta\\lap\\A ,\n\\end{eqnarray}\nwhere $\\bm{F}_{\\rm visc}=(\\mu\/\\rho)(\\lap\\u + \\frac{1}{3}\\grad\\dive\\u)$\nis the viscous force, $\\mu$ is the dynamic viscosity, \n$\\B = \\curl \\A$ is the magnetic field,\n$\\j = \\curl \\B\/\\mu_0$ is the current density,\n$\\mu_0$ is the vacuum permeability,\n$c_{\\rm s}$ is the (constant) speed of sound in the medium,\n$\\eta$ is the magnetic diffusivity,\nand $D_t \\equiv \\delt + \\u\\cdot\\grad$ is the advective derivative.\nOur computational domain is a spherical wedge with \nradius $r \\in [r_1,r_2]$ symmetric about the equator with \ncolatitude $\\theta \\in [\\Theta,\\pi-\\Theta]$\nand azimuth $\\phi \\in [0,\\Phi]$. \nThe radial, meridional and azimuthal extents of our domain are respectively,\n$\\lr \\equiv r_2-r_1$, $\\lt \\equiv r_2(\\pi-2\\Theta)$ and\n$\\lp \\equiv r_2 \\Phi$. All lengths are measured\nin the units of $r_2$.\nOur main results do not depend on our choices of $\\Theta$ and $\\Phi$.\n\nIn Equation~\\ref{mhd1} $\\fxt$ is an external white-in-time random helical forcing \nconstructed using\nthe Chandrasekhar-Kendall functions \\citep{cha+ken57} as described below.\nIn spherical coordinates a helical vector function can be expressed\nin terms of a scalar potential function:\n\\begin{equation}\n\\psi\\big(\\beta(t),\\ell(t),m(t)\\big)=\n      \\mbox{Re}\\, z_\\ell(\\beta r)Y^m_\\ell(\\theta,\\phi)\\exp[{\\rm i}\\xi_m(t)],\n\\label{solhelmoholtz}\n\\end{equation}\nwith \n$z_\\ell(\\beta r)=a_\\ell j_\\ell(\\beta r) + b_\\ell n_\\ell(\\beta r) $.\nHere $j_\\ell$ and $n_\\ell$ are spherical Bessel functions of the first and\nsecond kind respectively, $a_\\ell$ and $b_\\ell$ are constants determined by the \nboundary conditions and\n$\\xi_m$ is a random angle uniformly distributed between $0$ and $2\\pi$.\nThe helical forcing $\\f$ is then given by the equation $\\curl \\f = \\beta \\f $,\nwhere $\\f = {\\bm T} + {\\bm S}$,\n${\\bm T} = \\curl({\\bm e}\\psi)$,\nand ${\\bm S} = \\beta^{-1}\\curl{\\bm T}$,\nwhere ${\\bm e}(t)$ is a unit vector chosen randomly on the unit sphere. \nAs to the choice of boundary conditions, we demand that $\\f$ is zero at the \ntwo radial boundaries $r=r_1$ and $r=r_2$ which yields the following\ntranscendental equation relating $a_\\ell$, $b_\\ell$ and $\\beta$:\n\\begin{equation}\na_\\ell j_\\ell(\\beta r_1) + b_\\ell n_\\ell(\\beta r_1)\n= a_\\ell j_\\ell(\\beta r_2) + b_\\ell n_\\ell(\\beta r_2) = 0.\n\\label{alphaeqn}\n\\end{equation}\nWe first construct a table of values of $m$, $\\ell$ and $\\beta$ in the\nfollowing way. \nAs we use periodic boundary conditions along the azimuthal direction,\n$m_{\\rm min}= 2\\pi\/\\Phi$. We choose $m = p m_{\\rm min}$, and for a fixed\n$m$ we choose $\\ell$ to be odd, $\\ell = 2(m+q)+1$, because we want the forcing\nto go to zero at the equator. \nHere $p$ and $q$ are integers which range between $3$ to $5$.\nFor a fixed $\\ell$ and $m$ we solve Eq.~(\\ref{alphaeqn}) by Newton-Raphson\nmethod and list the solutions which have $3$ to $5$ zeros in the domain. \nTo randomize the resulting forcing we randomly choose a triplet \nof $m$, $\\ell$ and $\\beta$\nfrom the table. We also randomize the unit vector ${\\bm e}$ on the unit sphere.\nTwo different signs of helicity are imposed by choosing negative \n(positive) $\\beta$\nin the northern (southern) hemisphere. The choice of parameters\nimplies that we have a scale separation between $3$ to $5$ in our simulations.\nOur results are fairly robust under the change of different parameters\nof forcing. We need scale separation of 3 or more to excite a \nlarge-scale dynamo \\citep{HBD04},\nwhich invariably shows oscillations and equatorward migration. \nOur simulations are performed using the \n\\textsc{Pencil Code}\\footnote{{\\tt http:\/\/pencil-code.googlecode.com}};\nsee \\cite{mit+tav+bra+mos08} for details\nregarding the implementation of spherical polar coordinates.\n\\begin{deluxetable*}{c|c|c|c|c|c|c|c|c|c|c|c|c|c}\nRun & Grid & $\\lt$  & $\\lp$  & $\\Bbar_{\\rm rms}\/B_{\\rm eq}$  & $\\Rey$ & $\\Rm$ &\n $\\kf\/\\kone$ & $\\nu\\times10^{5}$ & $\\eta\\times10^{5}$ & $\\etat\\times10^{5}$&\n$\\omc\\times10^{3}$ &$T\\times10^{-3}$ & $t_{\\rm max}$ \\\\\n\\hline\n{\\tt S1} &$32\\times 64\\times 32$ &$\\pi\/5$ & $\\pi\/10$  & $0.88$  & $5$ & $12$ &\n$3$ & $5$ & $2$ & $5.3$ & $2.5$ & $0.18$ & $\\sim10T$ \\\\\n{\\tt S2} &$64\\times 128\\times 64$ &$\\pi\/5$ &$\\pi\/10$  & $0.79$  & $8$ & $21$ & \n$4$ & $3$ & $1.2$ & $6.2$ & $2.5$ & $0.16$ &  $\\sim5T$\\\\\n{\\tt S3} &$32\\times 64\\times 64$ &$\\pi\/5$ & $\\pi\/5$ &  $1.16$ & $2$ & $4$ &\n$7$ & $5$ & $2$ & $3.6$ & $2$ & $0.27$ & $\\sim10T$\\\\\n{\\tt S4} &$64\\times 128\\times 128$ & $\\pi\/5$ & $\\pi\/5$ & $1.04$  & $4$ & $10$ & \n$7$ & $2$ & $1$ & $4.9$ & $2.6$ & $0.2$ & $\\sim5T$\\\\\n{\\tt S5} &$64\\times 128\\times 128$ & $9\\pi\/10$ & $\\pi\/2$ & $2.03$  & $7$ & $13$ & \n$7$ & $2$ & $1$ & $4.4$ & $-$ & $-$ & $\\sim T$. \\\\\n\\caption{Summary of our parameters including \ngrid size, the meridional and azimuthal extents of our domain, \nrms value of the azimuthally averaged field,\nReynolds number $\\Rey$ and  \nmagnetic Reynolds number $\\Rm$.\nThe forcing amplitude, $f_{\\rm amp}=0.2$, is chosen such that the Mach number is \nof the order of 0.1, making the flow essentially incompressible. $t_{\\rm max}$\nthe duration of each run. The run {\\tt S5}\nhas not been ran long enough to accurately measure $\\omc$.} \n\\label{paratable}\n\\end{deluxetable*}\n\nWe use periodic boundary conditions along the azimuthal direction\nand set the normal component of the magnetic field to zero on all\nother boundaries (perfect-conductor boundary condition). \nThis is implemented by setting the two tangential components of $\\A$\nto zero.\nAs an estimate of the characteristic Fourier mode \nof the forcing we define $\\kf=w_{\\rm rms}\/\\urms$,\n(column 8 of Table~\\ref{paratable})\nwhere $w_{\\rm rms}$ and $\\urms$ are the rms values of small-scale\nvorticity and velocity, respectively.\nWe introduce the fluid and magnetic Reynolds numbers \nas $\\Rey=\\urms\/\\nu\\kf$ and $\\Rm=\\urms\/\\eta\\kf$, respectively. \nHere, $\\nu = \\mu\/\\rho_0$ is the mean kinematic\nviscosity, where $\\rho_0$ is the\ninitial and the mean density in the volume.\nA representative list of parameters is given in Table~\\ref{paratable}.\n\\section{Results}\nWe start our simulations with a random seed magnetic field\nof no particular parity about the equator.\nAfter a transient time, of about one turbulent diffusion time, \nwe find\nthat a large-scale magnetic field is generated with energy of the order \nof or exceeding the equipartition strength in all runs.\nThe  magnetic field encompasses the whole azimuthal extent of \nthe domain. \nA contour plot of the toroidal component of the magnetic field \non a surface with constant radius from Run~{\\tt S5} is shown in \nFig.~\\ref{fig:contour}.\nWe define the large-scale magnetic field via averages over the azimuthal \nand radial directions, i.e.,\n$\\Bbar = \\bra{{\\bm B}}_{r\\phi}$,\nsuch that the resultant magnetic field is solely a function \nof latitude and time.\nWe normalize the magnetic field with the equipartition field strength,\n$B_{\\rm eq}=\\bra{\\mu_0\\rho {\\bm u}^2}^{1\/2}$,\nwhere ${\\bm u} = \\u-\\Ubar$ is the small-scale velocity. \nThe field first develops at higher \nlatitudes and then with time migrates equatorward.\nIn each hemisphere the field shows oscillations\nand reversals of polarity. These features can be seen in the space-time diagram\nshown in the top panel of Fig.~\\ref{fig:butterfly} where we plot\n$\\overline{B}_\\phi$ as a function of latitude and time.\nThe principal frequency \nof oscillations, $\\omc$ (column 12 of Table~\\ref{paratable} and the\ninset of Fig.~\\ref{fig:osc+helm})\nis obtained by Fourier transforming the \ntime series of ${\\overline B}_{\\phi}(\\theta,t)$ in time at a \ngiven $\\theta$ ($\\theta=\\pi\/20$ say)\nand determining the frequency corresponding to the dominant mode.\nNormalized energy in the large-scale magnetic field also shows  \noscillations as a function of time, but with frequency\n$ 2 \\omc$. \nA characteristic dynamical time scale\nis the turbulent diffusion time corresponding to the length scale $\\lt$ defined by\n$T\\equiv (k_{\\theta}^2 \\etat)^{-1}$ where $k_{\\theta} = 2\\pi\/\\lt$ and for $\\etat$\nwe take the expression from the first-order-smoothing approximation,\n$\\etat=\\urms\/3\\kf$. In all our runs we find the product $\\omc T$ to be \nof order unity (see the bottom panel of Fig.~\\ref{fig:osc+helm}).\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig\/freq_vs_Rm_inset.ps}\n\\caption{\nThe frequency of the oscillations multiplied\nby the turbulent diffusion time $T=(\\eta_tk^2_{\\theta})^{-1}$ \nas a function of $\\Rm$ \nfor $\\lt = \\pi\/5 $.\nData from DNS (closed circles) and MF models (open circles) are shown.\nFor the MF runs $\\etat=1$, \nfor the DNS runs $\\etat$ is given in \nTable~\\ref{paratable}. \nThe inset shows $\\omc \\times10^{3}$ (column 12 Table~\\ref{paratable})\nas a function of $\\Rm$ for the DNS data.    \n}\\label{fig:osc+helm}\n\\end{figure}\n\nWe note that the radial and azimuthal components of the large-scale\nmagnetic field are almost antisymmetric about the equator.\nThis allows a further simplification of our model by simulating only one\nhalf of the domain (e.g., the northern hemisphere), while keeping exactly the\nsame forcing function (e.g., a forcing that is random, and negatively helical \nin the northern hemisphere going smoothly to zero at the equator), but choosing\nthe boundary condition $B_r=B_\\phi=0$ at the equator. Such simulations \nproduce exactly the same oscillations (as can be seen by comparing the top and\nthe middle panels of Fig.~\\ref{fig:butterfly}) as those obtained in the\nDNS with both hemispheres. \nThis implies that these oscillations can be studied with \nhalf the number of grid points and appropriately chosen boundary \nconditions at the equator. \n\nGiven the large values of the magnetic Reynolds number in solar\/stellar \nsettings, an important question is how the frequency $\\omc$ scales with \n$\\Rm$?\nThis question cannot be answered from DNS because the  \nmagnetic Reynolds numbers reached are far from \nthe asymptotic limit of large $\\Rm$~(see Column 7 \nof Table~\\ref{paratable})\nA way forward is to use analogous MF models.\nThe appropriate setting would be that of an $\\alpha^2$ dynamo\nwith dynamical $\\alpha$ quenching~\\citep{BB02}, which incorporates \nconservation of magnetic helicity, given by the equations\n\\begin{eqnarray}\n\\partial_t \\Bbar &=& \\curl (\\alpha \\Bbar) + (\\eta+\\etat)\\lap \\Bbar, \\\\\n\\partial_t \\alpha &=& -2\\eta\\kf^2 \\left( \n                                    \\frac{\\alpha \\bra{\\Bbar^2} -\\etat \\bra{\\Bbar\\cdot \\Jbar}}{\\Beq^2} +\n                                    \\frac{\\alpha-\\alpha_{\\rm K}}{\\Rm\/3}\n                                   \\right) ,\n\\end{eqnarray}\nwhere $\\Rm\/3=\\etat\/\\eta$ and $\\Beq$ is the equipartition field strength.\nWe use\n$\\kf\/\\kone=6$ in our MF simulations. \nIn view of the discussion above, we solve the MF equations in only the northern\nhemisphere with appropriate boundary condition at the equator.   \nIn the MF approach the helical nature of turbulence is modelled by the $\\alpha$\ncoefficient ($\\alpha_{\\rm K}$).\nWe choose $\\alpha_{\\rm K} = g(\\theta)\\alpha_0$ and $\\etat = 1$. \nThe profile function $g$ takes positive (negative) values in the northern \n(southern) hemisphere, going smoothly to zero at the equator.\nThis reflects the fact that, according to MF theory, the kinetic $\\alpha$\neffect usually has the opposite sign to the mean kinetic helicity.\nWe have used three different functional forms for $g$, namely\n$g = \\theta-\\pi\/2$, $g = \\sin(\\theta-\\pi\/2) $ and\n$ g = \\tanh(\\theta-\\pi\/2)$, without any significant change in our results.\nWe need $\\alpha_0 \\geq 16$ to excite a dynamo but once excited the oscillatory\nand migratory properties of the dynamo do not depend on $\\alpha_0$.  \nWe use perfect-conductor boundary conditions along the radial direction\nand our magnetic Reynolds number (changed by varying $\\eta$)\nranges between $ 10 \\lesssim \\Rm \\lesssim 10^{8}$. \nWe have also used domains with\nlarger latitudinal extents than those used in our DNS.\n\nHere we briefly mention a few important outcomes of\nour MF results relevant to our discussion above:\n(a) Our DNS results -- namely oscillatory behavior as well as\nmigration towards the equator -- are qualitatively reproduced by the MF\nsolutions in the range of parameters reported here. \nAn example of the space-time\ndiagram produced by our MF runs is shown in the bottom\npanel of Fig.~\\ref{fig:butterfly}.\n(b) The frequency of oscillations remains almost constant as a \nfunction of $\\Rm$, see Fig.~\\ref{fig:osc+helm}. \nThe dependence of the oscillation period on $\\Rm$\nseen in DNS may be related to $\\Rm$-dependence of\nthe turbulent magnetic diffusivity. \nA similar behavior has been observed earlier in Cartesian DNS\n\\citep{kap+bra09}.\n(c) To show the robustness of our results with respect to the\nsize of the domain, we also studied domain sizes extended\nin the meridional direction from $\\lt = \\pi\/5$ to $(178\/180)\\pi$ \nwhich corresponds respectively to  $\\Theta= 72 $ degrees and $1$ degree.\nWe find that the oscillations and the migratory behavior do not change.\n(d) For the MF model considered here the mean value of \nthe large-scale magnetic field decreases as $\\Rm^{-1}$, i.e., the field is catastrophically \nquenched for large values of $\\Rm$.  \nIn the DNS, however, such quenching could be alleviated by magnetic \nhelicity fluxes across the equator \\citep{bra+can+cha09,mitra_etal2010}.\n\nTo test the robustness of our simulations with respect \nto the choice of boundary condition in the radial direction \nwe repeated our simulations with the vertical field boundary \ncondition, which makes the two tangential components of the magnetic \nfield vanish at the radially outward boundary. \nThese simulations also show oscillations and\nequatorward migration of magnetic activity, but in this case the\noscillations are less regular and the frequency is marginally higher. \n\\section{Conclusions}\nWe have found large-scale fields, oscillations on\ndynamical time scales and polarity reversals with equatorward\nmigration of magnetic activity \nin direct numerical simulations of helically\nforced MHD equations in spherical wedge domains. \nDespite its simplicity, it is quite\nstriking how our model can reproduce these important features of\nthe solar dynamo. As far as we are aware, these features \nhave not been observed earlier in DNS.\nWe have elucidated our DNS results by considering\nanalogous $\\alpha^2$ MF models which support\nour conclusions. \nWe have further used these MF models\nto explore magnetic Reynolds numbers that are \nat present inaccessible to DNS. \nThis has enabled us to show that the frequency \nof the oscillations is almost\nindependent of $\\Rm$\nfor large $\\Rm$.\nSuch MF models have been known to have oscillatory solutions\nif $\\alpha$ changes sign in the computational domain \n\\citep[see, e.g.][]{bar+shu87,ste+ger03,Rud+Hol04,bra+can+cha09},\nbut their migratory property had not been studied before.\nAntisymmetry of $\\alpha$ with depth also produces oscillatory \nsolutions \\citep{bar+shu87,ste+ger03}, but not the equatorward\nmigration. \n\nThe helical forcing used in our DNS, with its different\nsigns of helicity in different hemispheres, implicitly \nmodels only the helical aspect of the effects of rotation \nand stratification present in the Sun. \nPhysically, a more complete picture should emerge from DNS of convective\nturbulent dynamo as done, for example, by \n\\cite{gil83,bro+bro+bru+mie+nel+tom07,brown10,kapyla_etal2010,charb}.\nSuch simulations also generate differential rotation and lead\nto another dynamo mode of operation -- the $\\alpha\\Omega$ dynamo\n-- which also produces oscillatory behavior.\nHowever, in order to get equatorward\nmigration radial shear must be negative. \nHelioseismology has shown that negative radial shear exists only\nnear the surface of the convection zone.\nThis feature has so far not been reproduced by global DNS.\nWhether or not an $\\alpha\\Omega$ dynamo is the dominant mechanism operating\nin the Sun remains unclear.\nIt is therefore important to keep in mind \nthat there exists alternative mechanisms \nfor producing oscillatory behavior with \nequatorward migration, such as the one discussed here.\n\\acknowledgments\nThis work was supported in part by\nthe European Research Council under the AstroDyn Research Project No.\\ 227952\nand the Swedish Research Council Grant No.\\ 621-2007-4064.\nDM is supported by the Leverhulme Trust.\nPJK is supported by the Academy of Finland grant No.\\ 121431.\nComputational resources were granted by\nUKMHD, QMUL HPC facilities purchased under the SRIF initiative,\nthe National Supercomputer Centre in Link\\\"oping in Sweden, and \nCSC--IT Center for Science in Espoo, Finland.\n\n\\newcommand{\\ypre}[3]{ #1, {Phys.\\ Rev.\\ E,} {#2}, #3}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe classification of compact, complex surfaces $S$ of general type with $\\chi(\\mathcal{O}_S)=1$, i.e. $p_g(S)=q(S)$, is currently an active area of research. For all these surfaces we have $p_g \\leq 4$, and the cases $p_g=q=4$ and $p_g=q=3$ are nowadays completely described. Regarding the case $p_g=q=2$, a complete classification has been recently obtained when $K_S^2=4$: in fact, these are surfaces on the Severi line $K_S^2=4 \\chi (\\mathcal{O}_S)$. By contrast, the classification in the case $p_g=q=2$, $K_S^2 \\geq 5$ is still missing, albeit some interesting examples were recently discovered. We refer the reader to the paper \\cite{1} and the references contained therein for a historical account on the subject and more details. \n\nThe purpose of this note is to show how monodromy representations of braid groups can be concretely applied to the fine classification of surfaces with $p_g=q=2$ and maximal Albanese dimension, allowing one to rediscover old examples and to find new ones.\n\nThe idea is to consider degree $n$, generic covers of $\\SS$, the symmetric square of a smooth curve of genus $2$, simply branched over the diagonal $\\delta$. In fact, if such a cover exists, then it is a smooth surface $S$ with\n\\begin{equation*}\n\\chi(\\mathcal{O}_S)=1, \\quad K_S^2=10-n.\n\\end{equation*} \n\nOn the other hand, by the Grauert-Remmert extension theorem and the GAGA principle, isomorphism classes of degree $n$, connected covers \n\\begin{equation*}\nf \\colon S \\to \\SS, \n\\end{equation*}\nbranched at most over $\\delta$, correspond to group homomorphisms \n\\begin{equation*}\n\\varphi \\colon \\pi_1(\\SS - \\delta) \\to \\mathfrak{S}_n\n\\end{equation*}\nwith transitive image, up to conjugacy in $\\mathfrak{S}_n$. The group $\\pi_1(\\SS - \\delta)$ is isomorphic to $\\mathsf{B}_2(C_2)$, the braid group on two strings on $C_2$; furthermore, our condition that the branching is simple can be translated by requiring that $\\varphi(\\sigma)$ is a transposition, where $\\sigma$ denotes the homotopy class in $\\SS - \\delta$ of a topological loop in $\\SS$ that ``winds once around $\\delta$\". \n\nA group homomorphism $\\mathsf{B}_2(C_2) \\to \\mathfrak{S}_n$ satisfying the requirements above will be called a \\emph{generic monodromy representation} of $\\mathsf{B}_2(C_2)$. By using the Computer Algebra System \\verb|GAP4|  we computed the number of generic monodromy representations  for $2 \\leq n \\leq 9$. In particular, such a number is zero for $n \\in \\{5, \\, 7, \\, 9 \\}$, so there exist no generic covers in these cases.\n\nAs an application of the general theory, we end the paper with a detailed discussion of the cases $n=2$ and $n=3$.\n\n\\section{Braid groups on closed surfaces}\n\nIn this section we collect some preliminary results on surface braid groups that are needed in the sequel of the work.\n\n\\begin{definition-conf}\nLet $X$ be a topological space. The $k$th \\emph{ordered configuration space} of $X$ is defined as\n\\begin{equation} \\label{eq:n-conf}\n\\mathrm{Conf}_k (X):=\\{(x_1,\\ldots, x_k) \\in X^k \\, | \\, \\ x_i \\neq x_j \\;\\; \\textrm{for all} \\; \\;i \\neq j\\}, \n\\end{equation}\nnamely $\\mathrm{Conf}_k (X) = X^k - \\Delta$, where $\\Delta$ is the big diagonal. \n \nThe quotient of $\\mathrm{Conf}_k (X)$ by the natural free action of the symmetric group $\\mathfrak{S}_k$ is called the $k$th \\emph{unordered configuration space} of $X$, and it is denoted by $\\mathrm{UConf}_k (X)$.  \n\nThen $\\mathrm{UConf}_k (X) = \\mathrm{Sym}^k(X) - \\delta$, where $\\delta$ denotes the image of $\\Delta$ in the symmetric product. \n\\end{definition-conf}\n\n\\begin{remark}\nIf $X$ is a smooth, compact, $n$-dimensional manifold, then both the configuration spaces $\\mathrm{Conf}_k (X)$ and $\\mathrm{UConf}_k (X)$ are smooth, open, $kn$-dimensional manifolds.  \n\\end{remark}\n\n\\begin{definition-braid}\nLet $\\Sigma_g$ be a closed topological surface of genus $g$, and let  $\\mathscr{P} = \\{p_1, \\ldots, p_k\\} \\subset \\Sigma_g$ be a set of $k$ distinct points. A \\emph{geometric braid} on $\\Sigma_g$ based at $\\mathscr{P}$ (also called a \\emph{braid on} $k$ \\emph{strings}) is a $k$-ple $(\\alpha_1, \\ldots, \\alpha_k)$ of paths $\\alpha_i \\colon [0, \\, 1] \\to \\Sigma_g$ such that \n\\begin{itemize}\n\\item $\\alpha_i(0) = p_i, \\quad i=1, \\ldots, k$;\n\\item $\\alpha_i(1) \\in \\mathscr{P}, \\quad i=1, \\ldots, k$;\n\\item the points $\\alpha_1(t), \\ldots, \\alpha_k(t) \\in \\Sigma_g$ are pairwise distinct for all $t \\in [0, \\, 1]$.\n\\end{itemize}     \nA geometric braid such that $\\alpha_i(0)=\\alpha_i(1)$ for all $i \\in \\{1, \\ldots, k\\}$ is called a \\emph{pure geometric braid}.\n\\end{definition-braid}\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}\n\\braid[number of strands=3, style strands={1}{red}, style strands={2}{blue}, style strands={3}{olive}, line width=1.5pt]\n(braid) a_1 a_2 a_1^{-1};\n\\end{tikzpicture}\n\\medskip\n\\caption{A non-pure braid on $3$ strings} \\label{fig:braids}\n\\end{figure}\nThe \\emph{braid group} on $k$ strings on $\\Sigma_g$ is the group $\\mathsf{B}_{k}(\\Sigma_g)$ whose elements are the braids  based at $\\mathscr{P}$ and whose operation is the usual product of paths, up to homotopies among braids. The \\emph{pure braid group} is the subgroup $\\mathsf{P}_{k}(\\Sigma_g)$ of $\\mathsf{B}_{k}(\\Sigma_g)$ given by the homotopy classes of pure braids. It can be shown that $\\mathsf{B}_{k}(\\Sigma_g)$ and $\\mathsf{P}_{k}(\\Sigma_g)$ do not depend on the choice of the set $\\mathscr{P}$, and that there is a short exact sequence of groups\n\\begin{equation} \\label{eq:pure-nonpure}\n1 \\to \\mathsf{P}_{k}(\\Sigma_g)\\to \\mathsf{B}_{k}(\\Sigma_g)\\to \\mathfrak{S}_k \\to 1.\n\\end{equation}\nMoreover, there are isomorphisms \n\\begin{equation} \\label{eq:iso-braids}\n\\mathsf{P}_{k}(\\Sigma_g) \\simeq \\pi_1(\\mathrm{Conf}_k (\\Sigma_g)), \\quad \\mathsf{B}_{k}(\\Sigma_g) \\simeq \\pi_1(\\mathrm{UConf}_k (\\Sigma_g)),\n\\end{equation}\nso that we can interpret \\eqref{eq:pure-nonpure} as the short exact sequence of fundamental groups induced by the $\\mathfrak{S}_k$-covering \n\\begin{equation}\n\\mathrm{Conf}_k (\\Sigma_g) \\to \\mathrm{UConf}_k (\\Sigma_g). \n\\end{equation}\nBraid groups are an important and flexible tool used in several areas of science, such as \n\\begin{itemize}\n\\item[-] Knot Theory (Alexander's theorem)\n\\item[-] Mathematical Physics (Yang-Baxter's equation)\n\\item[-] Mechanical Engineering (robot motion planning)\n\\item[-] Algebraic Geometry (monodromy invariants).\n\\end{itemize}\nWe will focus on the last topic, explaining\nhow the representation monodromy of surface braid groups onto the symmetric group can be used to produce interesting examples of projective surfaces defined over the field of complex numbers. \\\\\n\nWe are primarily interested in the case $g=k=2$. In that case, a simple presentation for the braid group is provided by the following result.\n\\begin{bellingeri} \nThe braid group $\\mathsf {B}_2(\\Sigma_2)$ can be generated by five elements $a_1, \\, a_2, \\, b_1, \\, b_2, \\, \\sigma,$\nsubject to the eleven relations below$:$\n\\begin{equation} \\label{eq:relations}\n\\begin{split}\n(R2) \\quad &  \\sigma^{-1} a_1 \\sigma^{-1} a_1= a_1 \\sigma^{-1} a_1 \\sigma^{-1} \\\\ &  \\sigma^{-1} a_2 \\sigma^{-1} a_2= a_2 \\sigma^{-1} a_2 \\sigma^{-1} \\\\ &\n\\sigma^{-1} b_1 \\sigma^{-1} b_1 = b_1 \\sigma^{-1} b_1 \\sigma^{-1} \\\\ & \\sigma^{-1} b_2 \\sigma^{-1} b_2 = b_2 \\sigma^{-1} b_2 \\sigma^{-1}\\\\ \n&  \\\\\n(R3) \\quad & \\sigma^{-1} a_1 \\sigma a_2 = a_2 \\sigma^{-1} a_1 \\sigma \\\\ &  \\sigma^{-1} b_1 \\sigma b_2 = b_2 \\sigma^{-1} b_1 \\sigma \\\\\n& \\sigma^{-1} a_1 \\sigma b_2 = b_2 \\sigma^{-1} a_1 \\sigma \\\\\n& \\sigma^{-1} b_1 \\sigma a_2 = a_2 \\sigma^{-1} b_1 \\sigma \\\\\n & \\\\\n(R4) \\quad & \\sigma^{-1} a_1 \\sigma^{-1} b_1 = b_1 \\sigma^{-1} a_1 \\sigma \\\\\n & \\sigma^{-1} a_2 \\sigma^{-1} b_2 = b_2 \\sigma^{-1} a_2 \\sigma \\\\\n & \\\\\n (TR) \\quad &  [a_1, \\, b_1^{-1}] [a_2, \\, b_2^{-1}]= \\sigma^2.  \n\\end{split}\n\\end{equation}\n\\end{bellingeri}\nHere the $a_i$ and the $b_i$ are pure braids coming from the representation of the topological surface $\\Sigma_2$ as a polygon of $8$ sides with the standard identification of the edges, whereas $\\sigma$ is a non-pure braid exchanging the two points $p_1$, $p_2 \\in \\Sigma_2$. These braids are depicted in Figure \\ref{fig:2}; note that, both in the cases of $a_i$ and $b_j$, the only non-trivial string is the first one, which goes through the wall $\\alpha_i$, respectively the wall $\\beta_j$. \n\\begin{figure}[H]\n \\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]\n\\clip(-6.17,-1.27) rectangle (15.43,5.43);\n\\fill[line width=0.4pt,color=rvwvcq,fill=rvwvcq,fill opacity=0.08] (-2.5,4.61) -- (-4.26,4.63) -- (-5.518650070512054,3.3996342007354072) -- (-5.538650070512054,1.6396342007354078) -- (-4.3082842712474605,0.38098413022335364) -- (-2.548284271247461,0.3609841302233536) -- (-1.289634200735407,1.5913499294879463) -- (-1.269634200735407,3.351349929487946) -- cycle;\n\\fill[line width=0.8pt,color=rvwvcq,fill=rvwvcq,fill opacity=0.07] (3.21,4.55) -- (1.45,4.57) -- (0.1913499294879455,3.3396342007354085) -- (0.17134992948794458,1.579634200735409) -- (1.4017157287525365,0.32098413022335403) -- (3.161715728752536,0.30098413022335313) -- (4.420365799264591,1.531349929487945) -- (4.440365799264592,3.2913499294879447) -- cycle;\n\\fill[line width=0.4pt,color=rvwvcq,fill=rvwvcq,fill opacity=0.08] (8.97,4.53) -- (7.21,4.55) -- (5.951349929487945,3.3196342007354067) -- (5.931349929487945,1.559634200735406) -- (7.161715728752538,0.3009841302233509) -- (8.921715728752538,0.2809841302233509) -- (10.180365799264594,1.511349929487944) -- (10.200365799264593,3.2713499294879446) -- cycle;\n\\draw [line width=0.4pt,color=rvwvcq] (-5.538650070512054,1.6396342007354078)-- (-4.3082842712474605,0.38098413022335364);\n\\draw [line width=0.4pt,color=rvwvcq] (-4.3082842712474605,0.38098413022335364)-- (-2.548284271247461,0.3609841302233536);\n\\draw [line width=0.4pt,color=rvwvcq] (-2.548284271247461,0.3609841302233536)-- (-1.289634200735407,1.5913499294879463);\n\\draw [line width=0.4pt,color=rvwvcq] (-1.289634200735407,1.5913499294879463)-- (-1.269634200735407,3.351349929487946);\n\\draw [line width=0.8pt,color=rvwvcq] (3.21,4.55)-- (1.45,4.57);\n\\draw [line width=0.8pt,color=rvwvcq] (1.45,4.57)-- (0.1913499294879455,3.3396342007354085);\n\\draw [line width=0.8pt,color=rvwvcq] (4.420365799264591,1.531349929487945)-- (4.440365799264592,3.2913499294879447);\n\\draw [line width=0.8pt,color=rvwvcq] (4.440365799264592,3.2913499294879447)-- (3.21,4.55);\n\\draw [line width=0.4pt,color=rvwvcq] (8.97,4.53)-- (7.21,4.55);\n\\draw [line width=0.4pt,color=rvwvcq] (7.21,4.55)-- (5.951349929487945,3.3196342007354067);\n\\draw [line width=0.4pt,color=rvwvcq] (5.951349929487945,3.3196342007354067)-- (5.931349929487945,1.559634200735406);\n\\draw [line width=0.4pt,color=rvwvcq] (5.931349929487945,1.559634200735406)-- (7.161715728752538,0.3009841302233509);\n\\draw [line width=0.4pt,color=rvwvcq] (7.161715728752538,0.3009841302233509)-- (8.921715728752538,0.2809841302233509);\n\\draw [line width=0.4pt,color=rvwvcq] (8.921715728752538,0.2809841302233509)-- (10.180365799264594,1.511349929487944);\n\\draw [line width=0.4pt,color=rvwvcq] (10.180365799264594,1.511349929487944)-- (10.200365799264593,3.2713499294879446);\n\\draw [line width=0.4pt,color=rvwvcq] (10.200365799264593,3.2713499294879446)-- (8.97,4.53);\n\\draw [line width=0.8pt,color=ffqqqq] (-3.98,2.49)-- (-1.8820135571336331,3.9778069722401543);\n\\draw [->,line width=0.8pt,color=ffqqqq] (-4.912072950290508,3.992580374435121) -- (-3.98,2.49);\n\\draw [line width=0.8pt,color=ffqqqq] (1.73,2.43)-- (0.7983789804885912,0.9381906887922181);\n\\draw [->,line width=0.8pt,color=ffqqqq] (3.787196012305263,0.9124086770670292) -- (1.73,2.43);\n\\draw [->,line width=1pt,color=qqqqcc] (-5.5386500705120545,1.6396342007354092) -- (-5.518650070512055,3.399634200735406);\n\\draw [->,line width=1pt,color=qqqqcc] (-5.518650070512055,3.399634200735406) -- (-4.26,4.63);\n\\draw [->,line width=1pt,color=qqqqcc] (-1.269634200735407,3.3513499294879456) -- (-2.5,4.61);\n\\draw [->,line width=1pt,color=qqqqcc] (-2.5,4.61) -- (-4.26,4.63);\n\\draw [->,line width=1pt,color=qqqqcc] (0.1913499294879455,3.3396342007354085) -- (0.17,1.51);\n\\draw [->,line width=1pt,color=qqqqcc] (0.17139736171360578,1.6297503150282006) -- (1.41,0.33);\n\\draw [->,line width=1pt,color=qqqqcc] (3.17,0.33) -- (1.41,0.33);\n\\draw [->,line width=1pt,color=qqqqcc] (4.421030852899942,1.5898746493988644) -- (3.17,0.33);\n\\draw [shift={(8.11,2.4)},line width=0.8pt,color=ffqqqq]  plot[domain=-0.016127633843636246:3.125465019746157,variable=\\t]({1*0.6200806399171007*cos(\\t r)+0*0.6200806399171007*sin(\\t r)},{0*0.6200806399171007*cos(\\t r)+1*0.6200806399171007*sin(\\t r)});\n\\draw [->,line width=2pt,color=ffqqqq] (8.639103545440543,2.723341055546999) -- (8.73,2.39);\n\\draw [shift={(8.11,2.4)},line width=0.8pt,color=ffqqqq]  plot[domain=3.125465019746157:6.26705767333595,variable=\\t]({1*0.6200806399170989*cos(\\t r)+0*0.6200806399170989*sin(\\t r)},{0*0.6200806399170989*cos(\\t r)+1*0.6200806399170989*sin(\\t r)});\n\\draw [->,line width=2pt,color=ffqqqq] (7.563712410516943,2.106623331573913) -- (7.49,2.41);\n\\draw [->,line width=2pt,color=ffqqqq] (7.563712410516943,2.106623331573913) -- (7.49,2.41);\n\\draw (-3.63,-0.35) node[anchor=north west] {$a_i$};\n\\draw (2.37,-0.35) node[anchor=north west] {$b_j$};\n\\draw (7.69,-0.35) node[anchor=north west] {$\\sigma$};\n\\begin{scriptsize}\n\\draw [fill=black] (-3.98,2.49) circle (1.5pt); \\draw[color=black]\n(-3.89,2.08) node {$p_1$}; \\draw [fill=black] (-2.74,2.47)\ncircle(1.5pt); \\draw[color=black] (-2.65,2.04) node {$p_2$}; \\draw\n[fill=black] (1.73,2.43) circle (1.5pt); \\draw[color=black]\n(1.83,2.02) node {$p_1$}; \\draw [fill=black] (2.97,2.41) circle\n(1.5pt); \\draw[color=black] (3.05,1.98) node {$p_2$}; \\draw\n[fill=black] (7.49,2.41) circle (1.5pt); \\draw[color=black] (7.14,2.41) node {$p_1$};  \\draw [fill=black]\n(8.73,2.39) circle (1.5pt); \\draw[color=black] (9.10,2.39) node {$p_2$}; \n\\draw[color=qqqqcc] (-5.80,2.46) node\n{$\\beta_i$}; \n\\draw[color=qqqqcc] (-5.15,4.1) node {$\\alpha_i$};\n\\draw[color=qqqqcc] (-1.6,4) node {$\\alpha_i$};\n\\draw[color=qqqqcc] (-3.20,4.88) node {$\\beta_i$};\n\\draw[color=qqqqcc] (-0.11,2.46) node {$\\alpha_j$};\n\\draw[color=qqqqcc] (0.57,0.82) node {$\\beta_j$};\n\\draw[color=qqqqcc] (2.50,0) node {$\\alpha_j$}; \\draw[color=qqqqcc]\n(4.07,0.82) node {$\\beta_j$};\n\\end{scriptsize}\n\\end{tikzpicture}\n\\medskip\n\\caption{Generators of $\\mathsf{B}_2(\\Sigma_2)$}\\label{fig:2}\n\\end{figure}\nRegarding the generator $\\sigma$, in terms of the isomorphism \n\\begin{equation}\n\\mathsf{B}_{2}(\\Sigma_2) \\simeq \\pi_1(\\mathrm{UConf}_2 (\\Sigma_2)) = \\pi_1(\\mathrm{Sym}^2(\\Sigma_2)-\\delta), \n\\end{equation}\nit corresponds to the homotopy class in $\\mathrm{UConf}_2(\\Sigma_2)$ of a topological loop in $\\mathrm{Sym^2}(\\Sigma_2)$ that ``winds once around the diagonal $\\delta$\".\n\n\\section{Finite coverings and monodromy representations}\n\nLet us recall now the classification of branched coverings \n$f \\colon X \\to Y$ of complex, projective varieties via the classification of monodromy representations of the fundamental group $\\pi_1(Y-B)$, where $B \\subset Y$ is the branch locus of $f$. The main technical tools needed are the \\emph{Grauert-Remmert extension theorem} and the GAGA \\emph{principle}, that we recall below. \n\n\\begin{GR}\nLet $Y$ be a normal analytic space over $\\mathbb{C}$ and $Z \\subset Y$ a closed analytic subspace such that $U=Y - Z$ is dense in $Y$. Then any finite, analytic, unramified covering $f^{\\circ} \\colon V \\to U$ can be extended to a normal, analytic, finite covering $f \\colon X \\to Y$, branched at most over $Z$. Furthermore, such an extension is unique up to analytic isomorphisms.   \n\\end{GR}\n\n\\begin{GAGA}\nLet $X$, $Y$ be projective varieties over $\\mathbb{C}$, and $X^{\\rm an}$, $Y^{\\rm an}$ the underlying complex analytic spaces. Then\n\\begin{itemize}\n\\item[$\\boldsymbol{(i)}$] every analytic map $X^{\\rm an} \\to Y^{\\rm an}$ is algebraic$;$\n\\item[$\\boldsymbol{(ii)}$] every coherent analytic sheaf on $X^{\\rm an}$ is algebraic, and its algebraic cohomology coincides with its analytic one.\n\\end{itemize} \n\\end{GAGA}\nFrom this, we deduce the following important consequences.\n\\begin{proj-ext}  \nLet $Y$ be a smooth, projective variety over $\\mathbb{C}$ and $Z \\subset Y$ be a smooth, irreducible divisor. Set $U=Y - Z$. Then any finite, unramified analytic covering $f^{\\circ} \\colon V \\to U$ can be extended in a unique way to a finite covering $f \\colon X \\to Y,$ branched at most over $Z$. \\vskip 3mm\n\nMoreover, there exists on $X$ a unique structure of smooth projective variety that makes $f$ an algebraic finite covering. \n\\end{proj-ext}\n\n\n\n\\begin{monodromy} \nLet $Y$ be a smooth projective variety over $\\mathbb{C}$ and $Z \\subset Y$ be a smooth, irreducible divisor. Then isomorphism classes of connected coverings of degree $n$\n\\begin{equation*}\nf \\colon X \\to Y,\n\\end{equation*}\nbranched at most over $Z$, are in bijection to group homomorphisms with transitive image\n\\begin{equation} \n\\varphi \\colon \\pi_1(Y - Z) \\to \\mathfrak{S}_n,\n\\end{equation}\nup to conjugacy in $\\mathfrak{S}_n$. Furthermore, $f$ is a Galois covering if and only if the subgroup $\\mathrm{im}\\, \\varphi$ of $\\mathfrak{S}_n$ has order $n$, and in this case $\\mathrm{im}\\, \\varphi$ is isomorphic to the Galois group of $f$.\n\\end{monodromy}\nThe group homomorphism \n\\begin{equation} \n\\varphi \\colon \\pi_1(Y - Z) \\to \\mathfrak{S}_n,\n\\end{equation}\nis called the \\emph{monodromy representation} of the covering $f \\colon X \\to Y$, and its image $\\mathrm{im}\\, \\varphi \\subseteq \\mathfrak{S}_n$ is called the \\emph{monodromy group} of $f$. The last corollary implies that, if $f$ is a Galois covering, then the monodromy group of $f$ is isomorphic to its Galois group (coinciding with the group $D(X\/Y)$ of deck transformations of the covering).\n\n\\section{Generic coverings of $\\mathrm{UConf}_2(\\Sigma_2)$}\n\nWe now apply the previous theory in the special case\n\\begin{equation}\nY = \\mathrm{UConf(\\Sigma_2)}=\\mathrm{Sym}^2(\\Sigma_2)-\\delta, \\quad  Z  = \\delta.\n\\end{equation}\nWe consider $\\Sigma_2$ as a \\emph{compact Riemann surface}, namely we fix a complex structure on it. Then the Abel-Jacobi map\n\\begin{equation}\n\\pi \\colon \\mathrm{Sym}^2(\\Sigma_2) \\to J(\\Sigma_2)\n\\end{equation}\nis a birational morphism onto the the Abelian surface $J(\\Sigma_2)$, more precisely it is the blow-down of the unique rational curve $E \\subset \\mathrm{Sym}^2(\\Sigma_2)$, namely the $(-1)$-curve given by the graph of the hyperelliptic involution on $\\Sigma_2$. \n\nWe have $\\delta E=6$, because the curve $E$ intersects the diagonal $\\delta$ transversally at the six points corresponding to the six Weierstrass points of $\\Sigma_2$. Writing $\\Theta$ for the numerical class of a Theta divisor in $J(\\Sigma_2)$, it follows that the image $D:=\\pi_*\\delta \\subset J(\\Sigma_2)$ is an irreducible curve with an ordinary sextuple point and no other singularities, whose numerical class is $4 \\Theta$.  \n\n\\begin{definition-gen-cov}\nLet $f \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2)$ be a connected covering of degree $n$\nbranched over the diagonal $\\delta$, with ramification divisor $R \\subset S$. Then $f$ is called \\emph{generic} if  \n\\begin{equation*}\nf^* \\delta= 2R + R_0,\n\\end{equation*}\nwhere the restriction $\\left.f\\right|_{R} \\colon R \\to \\delta$ is an isomorphism and $R_0$ is an effective divisor over which $f$ is not ramified. \n\\end{definition-gen-cov}\nGeneric coverings are never Galois, unless $n=2$, in which case $R_0$ is empty and  $f^* \\delta = 2R$. \nSince $\\delta$ is smooth, the genericity condition in the previous definition  is equivalent to requiring that the fibre of $f$ over every point of $\\delta$ has cardinality $n-1$.  \n\nSetting $\\alpha:=\\pi \\circ f$, the case where the curve  $Z=f^*(E)$ is irreducible is illustrated in Figure \\ref{fig:3}.\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[xscale=-1,yscale=-0.25,inner sep=0.7mm,place\/.style={circle,draw=black!100,fill=black!100,thick}] \n\\draw (-0.7,-0.5) rectangle (0.6,5);\n\n\\draw[red,rotate=92,x=6.28ex,y=1ex] (0.9,-0.85) cos (1,0) sin (1.25,1) cos (1.5,0) sin (1.75,-1) cos (2,0) sin (2.25,1) cos (2.5,0) sin (2.75,-1) cos (3,0) sin (3.25,1) cos (3.5,0) sin (3.6,-0.85);\n\\draw[rotate=92] (0.5,-0.025) .. controls (1.75,0.035) and (2.75,0.035) .. (4,-0.025);\n\n\\draw (-6.7,-0.5) rectangle (-5.4,5);\n\n\\draw[red,rotate=92,x=6.28ex,y=1ex,xshift=6,yshift=170] (0.9,-0.85) cos (1,0) sin (1.25,1) cos (1.5,0) sin (1.75,-1) cos (2,0) sin (2.25,1) cos (2.5,0) sin (2.75,-1) cos (3,0) sin (3.25,1) cos (3.5,0) sin (3.6,-0.85);\n\\draw[rotate=92,xshift=6,yshift=170] (0.5,-0.025) .. controls (1.75,0.035) and (2.75,0.035) .. (4,-0.025);\n\n\\draw (-6.7,15.5) rectangle (-5.4,21);\n\n\\draw[red,xscale=-1,yscale=-4] (6.05,-4.55) node(A0) [place,scale=0.2]{} to [in=5,out=55,looseness=8mm,loop] () to [in=65,out=115,looseness=8mm,loop] () to [in=125,out=175,looseness=8mm,loop] () to [in=185,out=235,looseness=8mm,loop] () to [in=245,out=295,looseness=8mm,loop] () to [in=305,out=355,looseness=8mm,loop] ();\n\n\\draw[xscale=-1,yscale=-4] (0.2,-0.1) node(B0) []{\\textcolor{red}{\\footnotesize{$R$}}};\n\\draw[xscale=-1,yscale=-4] (0.25,-1.05) node(0B) []{\\footnotesize{$Z$}};\n\n\\draw[xscale=-1,yscale=-4] (6.25,-0.1) node(B1) []{\\textcolor{red}{\\footnotesize{$\\delta$}}};\n\\draw[xscale=-1,yscale=-4] (6.25,-1.05) node(1B) []{\\footnotesize{$E$}};\n\n\\draw[xscale=-1,yscale=-4] (6.4,-5) node(B2) []{\\textcolor{red}{\\footnotesize{$D$}}};\n\n\\draw[xscale=-1,yscale=-4] (-0.9,-0.125) node(C0) []{$S$};\n\\draw[xscale=-1,yscale=-4] (7,-0.125) node(C1) []{$\\quad \\quad \\quad \\;  \\mathrm{Sym}^2(\\Sigma_2)$};\n\\draw[xscale=-1,yscale=-4] (7,-5.00) node(C2) []{$\\quad \\quad J(\\Sigma_2)$};\n\n\\draw[->] (-0.9,2.25) -- (-5.2,2.25) node[midway,above] {$f$};\n\n\\draw[->] (-6.05,5.8) -- (-6.05,14.7) node[midway,right] {$\\pi$};\n\n\\draw[->] (-0.9,4.8) -- (-5.2,15.5) node[midway,below=3pt] {$\\alpha$}; \n\\end{tikzpicture}\n\\medskip\n\\caption{A generic covering of $\\mathrm{Sym}^2(\\Sigma_2)$ branched over $\\delta$ }\\label{fig:3}\n\\end{figure}\n\nFrom now on, by \\emph{surface} we mean a smooth, complex, projective variety $S$ with $\\dim_{\\mathbb{C}}(S)=2$. For such a surface\n \\begin{itemize}\n\\item $K_S=\\wedge^2 \\Omega_S^1$ denotes the \\emph{canonical line bundle} \n\\item $p_g(S)=h^0(S, \\, K_S)$ is the \\emph{geometric genus}\n\\item $q(S)=h^1(S, \\, K_S)$ is the \\emph{irregularity}\n\\item $\\chi(\\mathcal{O}_S)=1-q(S)+p_g(S)$ is the \\emph{holomorphic Euler-Poincar\\'e characteristic}.\n\\end{itemize}\nThe zero locus of a meromorphic section of $K_S$ defines a class in $H_2(S, \\, \\mathbb{Z})$, whose Poincar\\'e \ndual $[K_S] \\in H^2(S, \\, \\mathbb{Z})$ is called the \\emph{canonical class} of $S$. Its self-intersection is the integer number defined by the cup product\n\\begin{equation}\nK_S^2=[K_S] \\cup [K_S] \\in H^4(S, \\, \\mathbb{Z}) \\simeq \\mathbb{Z}.\n\\end{equation}\nGiven a surface $S$, we can define some important meromorphic maps on it: the \\emph{pluricanonical maps} and the \\emph{Albanese map}. \n\\begin{definition-pluri}\nSet $N(r):=\\dim H^0(S, \\, K_S^{\\otimes r})-1$ and let $\\{ \\sigma_0, \\ldots, \\sigma_{N(r)} \\}$ be a basis for $H^0(S, \\, K_S^{\\otimes r})$. Then the $r$-\\emph{th} \\emph{pluricanonical map} of $S$ is  the rational map\n\\begin{equation}\n\\psi_r \\colon S \\to \\mathbb{P}^{N(r)}(\\mathbb{C}), \\quad x \\mapsto [\\sigma_0(x): \\ldots : \\sigma_{N(r)}(x)].\n\\end{equation}\nWe say that $S$ is \\emph{of general type} if the image of $\\psi_r$ is a surface for $r$ large enough (i.e., if $\\psi_r$ is generically finite onto its image for $r$ large enough).\n\\end{definition-pluri}\n\\begin{definition-alb}\nThe \\emph{Albanese map} of $S$ is the rational morphism\n\\begin{equation}\na_S \\colon S \\to \\mathrm{Alb}(S):=H^0(S, \\, \\Omega^1_S)^*\/H_1(S, \\, \\mathbb{Z}), \n\\end{equation}\ndefined by the integration of global, holomorphic $1$-forms on $S$ (this is a generalization of the Abel-Jacobi map $C \\to J(C)$, sending a smooth complex curve into its Jacobian). Note that $\\mathrm{Alb}(S)$ is a complex torus (actually, an Abelian variety) of dimension $q(S)$.\n\nWe say that $S$ is \\emph{of maximal Albanese dimension} if the image of its Albanese map is a surface \n(i.e., if $a_S$ is generically finite onto its image); note that this condition implies $q(S) \\geq 2$.\n\\end{definition-alb}\nSurfaces of general type satisfy $\\chi(\\mathcal{O}_S) \\geq 1$, and those with $\\chi(\\mathcal{O}_S)=1$ are usually difficult to construct. This explains the relevance of the following result.\n\\begin{theoremP1}\nLet $f \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2)$ be a generic covering of degree $n$ and whose branch locus is the diagonal $\\delta$. Then $S$ is a surface of maximal Albanese dimension with\n\\begin{equation*}\n\\chi(\\mathcal{O}_S)=1, \\quad K_S^2= 10-n. \n\\end{equation*}\nMoreover, if $ 2 \\leq n \\leq 9$ then $S$ is of general type.\n\\end{theoremP1}\nOur aim is now to use the theory developed before in order to construct generic coverings $f \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2)$.\n\n\\begin{definition-gen-mod} \nA \\emph{generic monodromy representation} of the braid group $\\mathsf{B}_2(\\Sigma_2)$ is a group homomorphism \n\\begin{equation}\n\\varphi \\colon \\mathsf{B}(\\Sigma_2) \\to \\mathfrak{S}_n\n\\end{equation}\nwith transitive image and such that $\\varphi(\\sigma)$ is a \\emph{transposition}.\n\\end{definition-gen-mod}\nBy the previous discussion, since $\\mathsf{B}_2(\\Sigma_2) \\simeq \\pi_1(\\mathrm{Sym}^2(\\Sigma_2) - \\delta)$, generic coverings and generic monodromy representations are related by the following\n\\begin{theoremP2} \nIsomorphism classes of generic coverings of degree $n$ \n\\begin{equation}\nf \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2),\n\\end{equation}\nwith branched locus $\\delta$, are in bijective correspondence to generic monodromy representations\n\\begin{equation}\n\\varphi \\colon \\mathsf{B}_2(\\Sigma_2) \\to \\mathfrak{S}_n,\n\\end{equation}    \nup to conjugacy in $\\mathfrak{S}_n$. \nFor $2 \\leq n \\leq 9$, the number of such representations is given in the table below$:$  \n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c}\n$ n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ \\\\\n \\hline\n$\\mathrm{Number \\; of}$ $\\varphi$ & $16$ & $3 \\cdot 80$ & $6 \\cdot 480$ & $0$ & $15 \\cdot 2880$  & $0$ & $28 \\cdot 172800$ & $0$\\\\\n\\end{tabular}\n\\end{center}\n\\end{table} In particular, for $n \\in \\{5, \\, 7, \\, 9\\}$ there exist no generic coverings.\n\\end{theoremP2}\n\n\\bigskip \\bigskip\n\nAs an application of the general theory, let us finish this note by discussing in detail the cases $n=2$ and $n=3$.\n\\begin{example-2}\nIn this case we are looking for generic monodromy representations\n\\begin{equation}\n\\varphi \\colon \\mathsf {B}_2(\\Sigma_2) \\to \\mathfrak{S}_2 = \\{(1), \\, (1 \\, 2)\\}.\n\\end{equation} \nSince $\\mathsf{B}_2(\\Sigma_2)$ is generated by five elements $a_1$, $a_2$, $b_1$, $b_2$, $\\sigma$, and necessarily $\\varphi(\\sigma)=(1 \\, 2)$, we see that there are $2^4=16$ possibilities for $\\varphi$. \n\nThe group $\\mathfrak{S}_2$ is abelian, so there is no conjugacy relation to consider and we get sixteen isomorphism classes of double coverings $f \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2)$, branched over $\\delta$ and with \n\\begin{equation}\n\\chi(\\mathcal{O}_S)=1, \\quad K_S^2=8.\n\\end{equation}\nThese coverings correspond to the sixteen square roots of $\\delta$ in  the Picard group of $\\mathrm{Sym}^2(\\Sigma_2)$. One covering coincides with the natural projection $f \\colon C_2 \\times C_2 \\to \\mathrm{Sym}^2(\\Sigma_2)$, in fact \n\\begin{equation*}\np_g(C_2 \\times C_2) = q(C_2 \\times C_2) =4, \\quad K_{C_2 \\times C_2}=8.\n\\end{equation*}\nThe remaining fifteen coverings are surfaces of general type  with\n\\begin{equation}\np_g(S)=q(S)=2, \\quad K_S^2=8.\n\\end{equation}\n\\end{example-2}\n\\begin{example-3}\nIn this case we are looking for generic monodromy representations\n\\begin{equation*}\n\\varphi \\colon \\mathsf{B}_2(\\Sigma_2) \\to \\mathfrak{S}_3, \n\\end{equation*} \nup to conjugacy in $\\mathfrak{S}_3$. By hands, or by using a Computer Algebra software like $\\mathsf{GAP4}$ (see \\cite{5}), we find that the total  number of monodromy representations is $240$. For every such a representation we have $\\mathrm{im}\\, \\varphi = \\mathfrak{S}_3$. Moreover, each orbit for the conjugacy action of $\\mathfrak{S}_3$ on the set of monodromy representations consists of six elements, and consequently the orbit set has cardinality $240\/6 = 40$. \nBy our last theorem, this implies that there are $40$ isomorphism classes of generic coverings $f \\colon S \\to \\mathrm{Sym}^2(\\Sigma_2)$ of degree $3$ and branched over $\\delta$. For all of them, $S$ is a surface of general type with\n\\begin{equation*}\n  p_g(S)=q(S)=2, \\quad K_S^2=7 \n\\end{equation*}\nand its Albanese map $\\alpha \\colon S \\to J(\\Sigma_2)$ is a generically finite covering of degree $3$.\n\nThese surfaces were previously studied by R. Pignatelli and the Author in \\cite{6}, by using completely different methods. In fact, they showed that they all lie in the same deformation class, and that their moduli space is a connected, quasi-finite cover of degree $40$ of $\\mathcal{M}_2$, the coarse moduli space of curves of genus $2$.\n\\end{example-3}\n\n\\section*{Acknowledgments}\nThe Author wishes to thank the organizers of Group 32 - The 32nd \\emph{International Colloquium on Group Theoretical Methods in Physics}, held on Czech Technical University (Prague) on July 9-13, 2018, for the invitation and the hospitality.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe work described in this paper arises from a problem posed to me by Ted Newman during my first visit to the University of Pittsburgh in the mid-1970s. It turns out that this problem can be solved \nby Newman-style methods and that it leads on to making interesting connections with other areas of Ted's work.\n\nThe problem is as follows: given a cut $\\Sigma$ of the future-null infinity ${\\cal{I}}^+$ of Minkowski space $\\mathbb{M}$, how do you reconstruct a space-like 2-surface $S$ inside \n$\\mathbb{M}$ such that $\\Sigma$ is the intersection with ${\\cal{I}}^+$ of $\\dot{J}(S)$, the boundary of the future of $S$? This is related to a version of the ``fuzzy point'' idea which \nwas current at that time: if $\\Sigma$ is a cut of the ${\\cal{I}}^+$ of a non-flat but asymptotically-flat space-time ${\\mathcal{M}}$ arising from a point $p$ in ${\\mathcal{M}}$ (known then as a {\\it{light-cone cut}}) then, when \ntransferred to the ${\\cal{I}}^+$ of $\\mathbb{M}$, $\\Sigma$ will determine a null hypersurface ${\\mathcal{N}}$ which does not converge to a point; however ${\\mathcal{N}}$ may nearly converge to a point \nand may determine small 2-surfaces $S$ which are nearly points, or are fuzzy points. If so, then by taking all possible $\\Sigma$ for all possible $p$ in ${\\mathcal{M}}$, one might obtain a representation of \nthe curved space-time ${\\mathcal{M}}$ as fuzzy points in the flat space-time $\\mathbb{M}$.\n\n\\medskip\n\nThe plan of this paper is as follows:\n\nIn section 2 I discuss convex bodies in $\\mathbb{R}^3$. The theory of convex bodies centres on the {\\it{St\\\"utzfunktion}} or support-function, which I'll anglicise as stutzfunction, and I review some of this theory.\n\nIn section 3, I turn to Minkowski space and identify the relation between the stutzfunction of a convex body and the cut-function which the boundary of the future of the body defines at ${\\cal{I}}^+$. This effectively \nsolves the problem posed above, and it also illuminates some of the theory in section 2.\n\nIn section 4, I sketch some further developments of the theory of convexity for 2-surfaces in Minkowski space. These include an approach to an inequality found by Gibbons and Penrose \\cite{rp1}, \\cite{g1}, as a prediction of the cosmic censorship hypothesis.\n\n\\medskip\n\nIt gives me great pleasure to dedicate this paper to Ted Newman in his 60th year\\footnote{Note the previous footnote.}, to acknowledge his long-standing and beneficial influence and to record my debt and gratitude to him.\n\n\\section{Convex bodies in $\\mathbb{R}^3$}\nIn this section, I develop some standard theory of convex bodies in $\\mathbb{R}^3$ following \\cite{bla} and \\cite{egg}, but with the kind of formalism that I learned from Ted Newman.\n\nWe may define a {\\it{convex body}} $B$ in $\\mathbb{R}^3$ to be a closed body such that, if $p,q$ are two points of $B$ then the line segment $tp+(1-t)q$ for $0\\leq t\\leq 1$ lies entirely in $B$. Then a {\\it{convex surface}} $S$\nis the surface of a convex body.\n\nWe define the Gauss map in the familiar way: choose an orthonormal triad and parametrise a unit vector $\\ell$ by spherical polars as\n\\begin{equation}\\label{1}\n\\ell=\\ell(\\theta,\\phi)=(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta),\\end{equation}\nin the triad. Thus $\\ell$ corresponds to a point on the unit sphere $S^2$. Given a choice of $\\ell$, that is a choice of $(\\theta,\\phi)$, take the plane with normal $\\ell$ that is tangent to the convex surface $S$, with $\\ell$ \nthe outward normal. If this happens at \n$p\\in S$ then the Gauss map from $S$ to $S^2$ takes $p$ to the point labelled $(\\theta,\\phi)$ on the unit sphere. For a smooth, strictly convex body, the Gauss map is one-one as we shall see. In that case we have introduced coordinates $(\\theta,\\phi)$ \non $S$. The tangent plane to $S$ at $p$ has the equation\n\\begin{equation}\\label{2}\n{\\bf{x\\cdot\\ell}(\\theta,\\phi)}=H(\\theta,\\phi),\\end{equation}\nwhere $H(\\theta,\\phi)$ is the perpendicular distance from the origin (which we'll assume to be inside $S$) to the tangent plane. A knowledge of $H(\\theta,\\phi)$ \ndetermines $S$ as an envelope of tangent planes and $H$ is the {\\it{St\\\"utzfunktion}} \n\\cite{bla} or support function, which we'll call the stutzfunction.\n\nTo obtain a parametric expression for the surface $S$ we can solve (\\ref{2}) and its derivatives for $(x,y,z)$. To this end, we introduce the Newman-Penrose operator `eth' \\cite{np2} defined on a spin-weight $s$ function $\\eta$ by\n\\[\n\\eth\\eta:=\\frac{1}{\\sqrt{2}}(\\sin\\theta)^s\\left(\\frac{\\partial}{\\partial\\theta}+\\frac{i}{\\sin\\theta}\\frac{\\partial}{\\partial\\phi}\\right)(\\sin\\theta)^{-s}\\eta,\n\\]\nand define\n\\[{\\bf m}=\\eth\\ell=\\frac{1}{\\sqrt{2}}(\\cos\\theta\\cos\\phi-i\\sin\\phi,\\cos\\theta\\sin\\phi+i\\cos\\phi,-\\sin\\theta)\\]\n\\[\\overline{\\bf m}=\\overline{\\eth}\\ell=\\frac{1}{\\sqrt{2}}(\\cos\\theta\\cos\\phi+i\\sin\\phi,\\cos\\theta\\sin\\phi-i\\cos\\phi,-\\sin\\theta).\\]\nThese have $s=1,-1$ respectively and, by differentiating again,\\\n\\[\n\\eth {\\bf m}=0=\\overline{\\eth}\\overline{\\bf m},\\;\\;\\;\\overline{\\eth} {\\bf m}=\\eth\\overline{\\bf m}=-\\ell.\\]\nThe positive-definite metric of $\\mathbb{R}^3$ can be written\n\\begin{equation}\\label{5}\n\\delta=\\ell\\ell+{\\bf m\\overline{m}}+{\\bf \\overline{m}m}.\n\\end{equation}\nNote also, as usual, that\n\\begin{equation}\\label{6}\n(\\eth\\overline{\\eth}-\\overline{\\eth}\\eth)\\eta=-s\\eta,\n\\end{equation}\nwhen $\\eta$ has spin-weight $s$.\n\nTo obtain the convex surface parametrically we must solve (\\ref{2}) simultaneously with \n\\[{\\bf{x}}\\cdot {\\bf m}=\\eth H,\\;\\;\\;{\\bf{x}}\\cdot \\overline{\\bf m}=\\overline{\\eth} H,\\]\nwhich with the aid of (\\ref{5}) can be solved to give \n\\begin{equation}\\label{7}\n{\\bf x}={\\bf x}(\\theta,\\phi)=H\\ell+\\overline{\\eth}H{\\bf m}+\\eth H\\overline{\\bf m}.\n\\end{equation}\nThis gives an explicit parametrisation of $S$. Using the standard theory of surfaces in $\\mathbb{R}^3$ (see e.g. \\cite{doc}) we find the area element of $S$ to be\n\\begin{equation}\\label{9}\ndA=\\left((H+\\eth\\overline{\\eth}H)^2-\\eth^2H\\overline{\\eth}^2H\\right)\\sin\\theta d\\theta d\\phi.\\end{equation}\nBy general theory, the Jacobian of the Gauss map is the Gauss curvature $k$ so that\n\\begin{equation}\\label{10}\nk^{-1}=R_1R_2=(H+\\eth\\overline{\\eth}H)^2-\\eth^2H\\overline{\\eth}^2H\n\\end{equation}\nin terms of principal radii of curvature $R_1,R_2$. A similar calculation gives the mean curvature $h$:\n\\[h=\\frac12(R_1^{-1}+R_2^{-1})=(H+\\eth\\overline{\\eth}H)k,\\]\nso that\n\\[hdA=(H+\\eth\\overline{\\eth}H)\\sin\\theta d\\theta d\\phi.\\]\nFor use below note that then\n\\begin{equation}\\label{h}\n\\int_ShdA=\\int_{S^2}H\\sin\\theta d\\theta d\\phi,\\end{equation}\nsince $\\eth\\overline{\\eth}$ is a constant multiple of the 2-sphere Laplacian, so the second term integrates to zero.\n\n\nFor strict convexity we require $R_1,R_2>0$ which is equivalent to $h,k>0$. Since necessarily $h^2\\geq k$ it is sufficient to require $k>0$ ($h$ will be positive somewhere on $S$ since we are using the outward normal). \nFinally, since this is the Jacobian of the Gauss map, the Gauss map is one-one and onto precisely for smooth, strictly convex surfaces. This imposes restrictions on $H$ which we shall discuss. First we note how \nsimple the Gauss-Bonnet theorem is in this context:\n\\begin{prop}{\\bf The Gauss-Bonnet Theorem}\n\nFor a strictly convex surface $S$\n\\[\\int_S kdA=4\\pi.\\]\n\\end{prop}\n\n{\\bf Proof}\n\nFrom (\\ref{9}) and (\\ref{10}), $kdA=\\sin\\theta d\\theta d\\phi$.\n\n\\medskip\n\n\\noindent Now what conditions do we require $H$ to satisfy for it to be the stutzfunction of a smooth, strictly convex surface? We need $H$ positive and in (\\ref{10}) we want the right-hand-side to be positive. If it fails to be positive then the surface enveloped by (\\ref{2}) will have cusps. However if $k$ from (\\ref{10}) \nis not positive then it can be made positive by adding a positive constant to $H$.\n\nThe process of adding a positive constant to $H$ is an interesting transformation that changes a convex surface $S$ into another, $S'$, which is {\\it parallel} to it in the sense of \\cite{ste}. In that \nreference, the idea is motivated by imagining rolling a sphere $K$ of constant radius $s$ over \nthe surface $S$. The locus of the centre of $K$ defines $S'$. Equivalently one moves the centre of $K$ over $S$ and takes $S'$ to be envelope swept out by $K$. In this second form, one sees a connection with the \nidea of {\\it Huyghens' secondary wavelets} which will reappear in section 3.\n\nGiven a convex surface $S$, one can consider a sequence of surfaces parallel to $S$ with larger and larger separations $s$. In this way one arrives at the following string of theorems.\n\\begin{prop}{\\bf Steiner's Theorem}\n\nAlong such a sequence, the area is given by\n\\begin{equation}\\label{11}\nA(s)=A+2sM+4\\pi s^2,\n\\end{equation}\nwhile the volume contained is\n\\begin{equation}\\label{12}\nV(s)=V+sA+s^2M+\\frac{4}{3}\\pi s^3.\\end{equation}\nHere $A$ and $V$ are the area and volume of $S$ and $M$ is the integral of mean curvature:\n\\begin{equation}\\label{13}\nM=\\int_ShdA.\\end{equation}\n\\end{prop}\n\n{\\bf Proof}\n\nClearly the surface parallel to $S$ at distance $s$ has stutzfunction $H+s$. Substitute into (\\ref{9}) and expand in powers of $s$ to obtain (\\ref{11}) (using (\\ref{h}) along the way). Integrate (\\ref{11}) to obtain (\\ref{12}).\n\n\\medskip\n\n\\noindent Along a sequence of parallel surfaces, the surfaces should become ``rounder''. This intuitive feeling is made precise in the following (which I won't prove):\n\\begin{prop}{\\bf The Brunn-Minkowski Theorem}\n\n Define $R(s)=(V(s))^{1\/3}$ then $R$ is convex in $s$, in that $\\frac{d^2R}{ds^2}\\leq 0$.\n\\end{prop}\n\\noindent Take this to be true and calculate the derivative, then for positive $s$:\n\\begin{equation}\\label{14}\n(6MV-2A^2)+2s(12\\pi V-AM)+2s^2(4\\pi A-M^2)\\leq 0.\\end{equation}\nFrom this we may deduce {\\it Minkowski's inequality}\n\\begin{equation}\\label{15}\nM^2\\geq 4\\pi A,\\end{equation}\nas well as the {\\it {isoperimetric inequality}}\n\\begin{equation}\\label{16}\n36\\pi V^2\\leq A^3.\\end{equation}\nAlthough (\\ref{15}) follows from (\\ref{14}) there is a straightforward direct proof due to Blaschke and using the stutzfunction (see \\cite{bla}):\n\n\\medskip\n\n\\noindent{\\bf Blaschke's proof of Minkowski's Inequality}\n\n\\medskip\n\nFrom (\\ref{9}),(\\ref{6}) and integration by parts\n\\[A=\\int(H^2-\\eth H\\bar{\\eth}H)\\sin\\theta d\\theta d\\phi\\]\nwhile from (\\ref{h})\n\\[M=\\int H\\sin\\theta d\\theta d\\phi.\\]\n Set $H=H_0+H_1$ where $\\int H_1\\sin\\theta d\\theta d\\phi=0$ and $H_0$ is constant then\n \\[M=4\\pi H_0,\\;\\;A=4\\pi H_0^2+\\int(H_1^2-\\eth H_1\\bar{\\eth}H_1)\\sin\\theta d\\theta d\\phi.\\]\n It follows by expanding $H_1$ in spherical harmonics that the integral contribution to $A$ is strictly negative unless $H_1$ is a combination of $\\ell=1$ spherical harmonics. This case corresponds to a sphere with a translated \n origin, so that (\\ref{15}) is proved, with equality only for a round sphere.\n \n \\medskip\n \n To conclude this section, I shall record another way of obtaining the surface $S$ from the stutzfunction $H$. Define\n \\[\\hat{H}(r,\\theta,\\phi)=rH(\\theta,\\phi)=F(\\hat{x},\\hat{y},\\hat{z}),\\]\n where $\\hat{x},\\hat{y},\\hat{z}$ are expressed in terms of spherical polar coordinates $r,\\theta,\\phi$ in the usual way. Then (\\ref{7}) is equivalent to the parametrisation given by\n \\[x=\\frac{\\partial F}{\\partial \\hat{x}},\\;\\;y=\\frac{\\partial F}{\\partial \\hat{y}},\\;\\;z=\\frac{\\partial F}{\\partial \\hat{z}},\\]\n where, after the differentiation, $\\hat{x},\\hat{y},\\hat{z}$ are again eliminated.\n \\section{Stutzfunction and cut function}\n In Minkowski space $\\mathbb{M}$ we introduce the null tetrad\n \\[\n \\ell^a=(1,\\ell),\\;\\;m^a=\\eth\\ell^a=(0,{\\bf m}),\\;\\;\\overline{m}^a=\\overline{\\eth}\\ell^a=(0,\\overline{\\bf m}),\\;\\;n^a=\\frac12(1,-\\ell),\\]\n with $\\ell,{\\bf m},\\eth$ as in section 2.\nThe Minkowski metric can then be written\n\\[\\eta^{ab}=2\\ell^{(a}n^{b)}-2m^{(a}\\overline{m}^{b)}.\\]\n Note that\n \\[\n \\overline{\\eth} m^a=(0,-\\ell)=-\\frac12\\ell^a+n^a.\\]\n Define the unit time-like vector\n \\begin{equation}\\label{24}\n t^a=(1,{\\bf 0})\\end{equation}\n so that also $\\eta_{ab}t^a\\ell^b=1$, and introduce advanced null polar coordinates $(u,r,\\theta,\\phi)$ by\n \\[\n x^a=ut^a+r\\ell^a(\\theta,\\phi)\\]\n (see e.g. \\cite{knt}). Then $(u,\\theta,\\phi)$ are coordinates on ${\\cal{I}}^+$ which is located at $r=\\infty$.\n \n A cut of ${\\cal{I}}^+$ is defined by a function\n \\[u=V(\\theta,\\phi)\\]\n where $V$ can conveniently be called \\emph{the cut function} for the cut. If we choose an arbitrary point $p$ with coordinates $x_0^a$ and a null-vector $\\ell^a(\\theta_0,\\phi_0)$ at $p$ then \n the null geodesic from $p$ in the direction of $\\ell^a(\\theta_0,\\phi_0)$ meets ${\\cal{I}}^+$ at \n \\begin{equation}\\label{27}u=x_0^a\\ell_a(\\theta_0,\\phi_0),\\;\\;\\;\\theta=\\theta_0,\\;\\;\\;\\phi=\\phi_0.\\end{equation}\nNow suppose we are given a convex surface $S$ in the form\n\\[x^a=(0,{\\bf x}(\\theta,\\phi))\\]\nwith ${\\bf x}(\\theta,\\phi)$ determined by a stutzfunction $H$ according to (\\ref{7}). The boundary of the future of $S$, $\\dot{J}(S)$, is ruled by null geodesics that meet $S$ orthogonally. From the definitions so \nfar made, the outward null normal at the point $p$ of $S$ labelled by $(\\theta_0,\\phi_0)$ is $\\ell^a(\\theta_0,\\phi_0)$ and the null geodesic from $p$ in this direction meets ${\\cal{I}}^+$ at $u$ given by \n(\\ref{27}). As $p$ runs over $S$, the cut $\\Sigma=\\dot{J}(S)\\cap{\\cal{I}}^+$ is generated with the cut function\n\\begin{equation}\\label{29} u=x^a\\ell_a(\\theta,\\phi)=-{\\bf x\\cdot}\\ell(\\theta,\\phi).\\end{equation}\nComparing (\\ref{29}) with (\\ref{2}) we conclude that \\emph{the cut function is minus the stutzfunction}.\n\n\\medskip\n\nConversely, if we are given the cut $\\Sigma$ and its cut function $V$ then (\\ref{29}) determines a null hypersurface $\\mathcal{N}$\n\\[x^a\\ell_a(\\theta,\\phi)=V(\\theta,\\phi),\\]\nwhich, with its angular derivatives, we can solve for $\\mathcal{N}$ parametrically as\n\\[x^a=x^a(\\lambda,\\theta,\\phi)=-Hn^a+\\overline{\\eth} Hm^a+\\eth H\\overline{m}^a+\\lambda \\ell^a\\]\nfor arbitrary real $\\lambda$. Now intersecting $\\mathcal{N}$ with hypersurfaces of constant $t$, i.e. hypersurfaces orthogonal to $t^a$ in (\\ref{24}), gives a sequence of parallel surfaces in the sense of section 2. \nThis shows how the parallel-surface idea is related to Huyghen's secondary wavelets: if a convex surface is momentarily lit up then the resulting expanding (out-going) wave-front traces out a sequence of surfaces parallel to the first.\n\nOf course this converse is incomplete in the following sense: if what we are given is just the cut function then we can define the null surface $\\mathcal{N}$ but we cannot fix a unique value of $\\lambda$ to represent \nthe 2-surface $S$ without some extra input. If we are trying to make precise the fuzzy-point idea then we might want to pick out an instant of {\\it minimum volume} or of {\\it best focus}. This could also involve boosting the cut function or \nconsidering a different set of constant-time hypersurfaces.\n\\section{Further developments}\nIn this last section I describe some attempts to carry over other parts of the theory of convex 2-surfaces into Minkowski space. I shall work with the GHP formalism \\cite{ghp} and omit proofs.\n\nA space-like 2-surface $S$ in ${\\mathbb{M}}$ defines a pair of future-pointing null normals $\\ell^a,n^a$ (where we shall take $\\ell^a$ to be the outward normal and $n^a$ the inward normal) or equivalently a normalised spinor \ndyad $(o^A,\\iota^A)$. The second fundamental form of $S$ is coded by the GHP formalism into weighted scalars $(\\rho,\\rho',\\sigma,\\sigma')$ (see e.g. \\cite{t1} for an account of this). In terms of these, I shall \nsay that $S$ is\n\\[\\mbox{future convex iff }\\rho<0,\\;\\;\\;\\rho^2-\\sigma\\overline{\\sigma}>0\\]\n\\[\\mbox{past convex iff }\\rho'>0,\\;\\;\\;\\rho'^2-\\sigma'\\overline{\\sigma}'>0.\\]\nWe recall that the Gauss curvature of $S$ is twice the real part of the \\emph{complex curvature} $Q=-\\rho\\rho'+\\sigma\\sigma'$ \\cite{pr}.\n\\begin{prop}\n\n If $S$ is future and past convex then the Gauss curvature of $S$ is everywhere positive.\n \n\\end{prop}\n\n{\\bf Proof:} This is elementary since\n\\[k=-2\\rho\\rho'+\\sigma\\sigma'+\\overline{\\sigma}\\overline{\\sigma}'\\geq-2\\rho\\rho'-2|\\sigma\\sigma'|\\]\nand\n\\[(-\\rho\\rho'-|\\sigma\\sigma'|)^2=(\\rho^2-\\sigma\\overline{\\sigma})(\\rho'^2-\\sigma'\\overline{\\sigma}')+(\\rho|\\sigma'|+\\rho'|\\sigma|)^2.\\]\n\n\\noindent The quantities occuring in the above definitions of convexity arise in the various Gauss maps that can be defined for $S$. If we choose and fix a constant normalised spinor dyad $(\\alpha^A,\\beta^A)$ then we \ncan define a \\emph{future Gauss map} by\n\\[f:S\\rightarrow {\\mathbb{CP}}^1;\\;\\;\\;\\;p\\mapsto \\zeta=\\frac{o_A\\alpha^A}{o_A\\beta^A}\\]\nand a \\emph{past Gauss map} by\n\\[f:S\\rightarrow {\\mathbb{CP}}^1;\\;\\;\\;\\;p\\mapsto \\eta=\\frac{\\iota_A\\alpha^A}{\\iota_A\\beta^A}.\\]\nEquivalently, these express $o^A$ and $\\iota^A$ in terms of $(\\alpha^A,\\beta^A)$  as\n\\[o^A=\\lambda\\frac{(\\alpha^A+\\zeta\\beta^A)}{(1+\\zeta\\overline{\\zeta})^{1\/2}},\\;\\;\\;\\iota^A=\\mu\\frac{(\\alpha^A+\\eta\\beta^A)}{(1+\\eta\\overline{\\eta})^{1\/2}},\\]\nwhere $\\lambda,\\mu$ are not fixed by the Gauss maps, but note that\n\\[t^a\\ell_a=\\frac{1}{\\sqrt{2}}\\lambda\\overline{\\lambda},\\;\\;\\;t^an_a=\\frac{1}{\\sqrt{2}}\\mu\\overline{\\mu},\\]\nwhere $t^{AA'}= \\frac{1}{\\sqrt{2}}(\\alpha^A\\overline{\\alpha}^{A'}+\\beta^A\\overline{\\beta}^{A'})$, so that $t^a$ is a unit time-like vector determined by the chosen tetrad. (These Gauss maps are similar to but \ndifferent from those defined by Kossowski \\cite{kos}.)\n\nThere is a third Gauss map, conveniently called the \\emph{complex Gauss map} which can be defined by\n\\[2o^{(A}\\iota^{B)}=\\zeta\\alpha^A\\alpha^B+2\\eta \\alpha^{(A}\\beta^{B)}+\\xi\\beta^A\\beta^B\\]\nwhere this $\\zeta,\\eta$ are to be distinguished from the previous. This maps $S$ to the complex quadric $\\mathbb{Q}$ defined by\n\\[\\zeta\\xi-\\eta^2=1\\]\nin $\\mathbb{C}^3$ (this Gauss map has also been considered by Roger Penrose).\n\nThe images of the future and past Gauss maps carry volume forms $4d\\zeta d\\overline{\\zeta}(1+\\zeta\\overline{\\zeta})^{-2}$ and $4d\\eta d\\overline{\\eta}(1+\\eta\\overline{\\eta})^{-2}$ while $\\mathbb{Q}$ admits \nthe holomorphic 2-form\n\\[\\frac{d\\zeta\\wedge d\\xi}{\\eta}= 2\\frac{d\\zeta\\wedge d\\eta}{\\zeta}=2\\frac{d\\eta\\wedge d\\xi}{\\xi},\\]\n so that we can calculate the Jacobians for the Gauss maps as in section 2.\n \\begin{prop}\n \n  For the future Gauss map we find\n  \\[\n  \\frac{4d\\zeta d\\overline{\\zeta}}{(1+\\zeta\\overline{\\zeta})^{2}}=\\frac{1}{(t^a\\ell_a)^2}(\\rho^2-\\sigma\\overline{\\sigma})dA,\\]\n  for the past Gauss map\n  \\[\n  \\frac{4d\\eta d\\overline{\\eta}}{(1+\\eta\\overline{\\eta})^{2}}=\\frac{1}{(t^an_a)^2}(\\rho'^2-\\sigma'\\overline{\\sigma}')dA,\\]\n  and for the complex Gauss map\n  \\[\n  \\frac{d\\zeta\\wedge d\\xi}{\\eta}=(-\\rho\\rho'+\\sigma\\sigma')dA.\\]\n    \n \\end{prop}\n\\noindent As in section 2, we integrate these expressions over $S$:\n\\begin{prop}{\\bf Generalised Gauss-Bonnet Theorem}\n\n For the three cases treated above integration gives:\n\\begin{equation}\\label{41} \\int_S\\frac{1}{(t^a\\ell_a)^2}(\\rho^2-\\sigma\\overline{\\sigma})dA=\\int_S\\frac{1}{(t^an_a)^2}(\\rho'^2-\\sigma'\\overline{\\sigma}')dA=\\frac{4\\pi}{t^at_a}\\end{equation}\n\\[ \\int_S(-\\rho\\rho'+\\sigma\\sigma')dA=2\\pi.\\]\n\\end{prop}\n\\noindent In (\\ref{41}) I have included the term $t^at_a$ explicitly both to give a slightly more general formula (valid when $t^a$ is any constant time-like vector) and to point up the resemblance \nto Newman's expression for the ${\\mathcal{H}}$-space metric \\cite{new1}.\n\n\\medskip\n\nNext we turn to consideration of possible generalisations of the notion of parallel bodies and Propositions 2.2-2.4. For this we need to write down and solve the \\emph{Sachs equations} which are the NP spin-coefficient \nequations for the evolution of $\\rho,\\sigma$ and $\\rho', \\sigma'$ \\cite{np1} Given $S$ we consider the null hypersurface $\\mathcal{N}$ generated by the outgoing null normals to $S$. We scale $\\ell^a$ to be affinely \nparametrised:\n\\[D\\ell^a\\equiv \\ell^b\\nabla_b\\ell^a=0,\\]\nand choose an affine parameter $s$ with\n\\[Ds=1,\\;\\;\\;s=0\\mbox{  at  }S.\\]\nThen the Sachs equations are\n\\[D\\rho=\\rho^2+\\sigma\\overline{\\sigma},\\;\\;\\;D\\sigma=2\\rho\\sigma,\\]\nwhile the area element is carried along $\\ell^a$ according to\n\\[D(dA)=-2\\rho dA.\\]\nThen the Sachs equations can be solved explicitly as\n\\begin{equation}\\label{47}\n\\rho(s)=\\Delta^{-1}(\\rho_0-s(\\rho_0^2-\\sigma_0\\overline{\\sigma}_0)),\\;\\;\\;\\sigma(s)=\\Delta^{-1}\\sigma_0\\end{equation}\nwith\n\\[\\Delta=1-2s\\rho_0+s^2(\\rho^2_0-\\sigma_0\\overline{\\sigma}_0)\\]\nand $\\rho_0,\\sigma_0$ are the values at $S$. For the area element we similarly find\n\\[dA(s)=\\Delta dA_0.\\]\nWe deduce at once the following proposition:\n\\begin{prop}\n For a future convex surface, the outgoing null hypersurface encounters no caustics to the future.\n\\end{prop}\n{\\bf Proof: } Caustics to the future are signalled by singularities in $\\rho(s)$, or equivalently zeroes in $\\Delta$, for positive $s$ but from the definition of future convex $\\Delta$ is positive definite in this range.\n\n\\medskip \n\nThere is a corresponding statement for past convex.\n\n\\medskip\n\nFor the analogue of Proposition 2.2, Steiner's Theorem, we need to integrate $dA(s)$. However there is a problem of \\emph{weights} in the GHP sense: at this point we have the freedom to rescale $\\ell^a$ at $S$ by\n\\begin{equation}\\label{49}\\ell^a\\rightarrow \\Omega(\\theta,\\phi)\\ell^a,\\end{equation}\nand under this transformation\n\\[s\\rightarrow \\Omega^{-1}s,\\;\\;\\;\\rho_0\\rightarrow \\Omega \\rho_0,\\;\\;\\;\\sigma_0\\rightarrow \\Omega \\sigma_0,\\]\nso that we would not get a formula like (\\ref{11}) by simply integrating $dA(s)$. The simplest way to resolve this difficulty is to choose a constant unit time-like vector $t^a$ and define\n\\[\\hat{s}=st^a\\ell_a,\\]\nas $\\hat{s}$ is unchanged by (\\ref{49}) and now we can integrate $dA(s)$:\n\\begin{prop}Generalised Steiner Theorem\n\\[ A(\\hat{s})=A_0+2\\hat{s}\\hat{M}+4\\pi\\hat{s}^2,\\]\nwhere\n\\begin{equation}\\label{53}\\hat{M}=-\\int_S\\frac{\\rho_0}{t^a\\ell_a}dA\\end{equation}\nand we have used Proposition 4.3.\n\\end{prop}\n\\noindent Now we might hope to prove a counterpart of the Brunn-Minkowski Theorem, Proposition 2.3, or of Minkowski's Inequality, Proposition 2.4. However this is impossible since with $\\hat{M}$ \nas in (\\ref{53}) examples can be found to show that it is not the case that $\\hat{M}^2\\geq 4\\pi A$. In fact the Isoperimetric Inequality (\\ref{16}) can also be violated in Minkowski space in the following sense: \ngiven a space-like 2-surface $S$ one may be \nable to find space-like 3-surfaces spanning $S$ on which the volume $V$ enclosed by $S$ and the area $A$ of $S$ have\n\\[36\\pi V^2>A^3.\\] \nThe correct inequality to generalise (\\ref{15}) would seem to be one proposed by Gibbons and Penrose in an investigation of Cosmic Censorship, \\cite{rp1,g1}. This may be phrased as follows: consider \nthe vector\n\\begin{equation}\\label{55}\nP^a=\\frac12\\int_S(\\rho'\\ell^a-\\rho n^a)dA=\\int_Sp^adA,\\end{equation}\nwith $p^a=\\frac12(\\rho'\\ell^a-\\rho n^a)$. If $S$ is future \nconvex with $\\rho'>0$ as well, or past convex with $\\rho<0$ as well, then one conjectures the inequality:\n\\begin{equation}\\label{56}\nP_aP^a\\geq 2\\pi A,\\end{equation}\nwhere $A$ is the area of $S$. (This is not the form in which the inequality is stated by Gibbons and Penrose but I believe it to be equivalent.) \n\nNote that the \\emph{mean curvature vector} $H_a$ of $S$, equivalently the trace of the second fundamental form, is\n\\begin{equation}\\label{57}\nH_a=\\frac12(\\rho'\\ell_a+\\rho n_a).\\end{equation}\nThe significance of $H_a$ is that, given a vector field $X^a$ on $S$, the rate of change of the area of $S$ under displacement along $X^a$ is\n\\[\\dot{A}=\\int_sH_aX^adA.\\]\nWe see that the vector $p^a$ in (\\ref{55}) lies in the normal 2-plane to $S$ and is orthogonal to $H_a$ so it defines the direction in which $dA$ does not change (to first order). Also, by taking $X^a$ to be a constant \ntranslation, under which the area will not change, it is clear that\n\\[\\int_SH_adA=0\\mbox{  and so  }\\int_S\\rho'\\ell^adA=-\\int_S\\rho n^adA.\\]\nThus we can write $P^a$ as\n\\[P^a=\\int_S(-\\rho)n^adA=\\int_S\\rho'\\ell^adA.\\]\nIn this form it is clear that $P^a$ is time-like and future pointing for past or future convex $S$.\n\n\\medskip\n\nAs partial confirmation of (\\ref{56}) we note that if $S$ lies in a flat space-like 3-surface with unit time-like normal $t^a$ then\n\\[P^a=\\frac{1}{\\sqrt{2}}Mt^a,\\]\nwith $M$ as in (\\ref{13}). Thus in this case (\\ref{56}) is Minkowski's inequality. Further, if $S$ lies in an in- or out-going null cone then (\\ref{56}) can be established directly, \nas it reduces to an inequality for functions on the unit sphere, \\cite{rp1,g1}, which can be proved \\cite{t1,t2}. Finally one can verify (\\ref{56}) for surfaces infinitesimally close to a round sphere in a flat hyperplane. \nWhat is still lacking is a proof of (\\ref{56}) in full generality, subject only to the conditions of convexity\\footnote{The claimed proof in \\cite{g2} is defective: see e.g. \\cite{mm}.}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n     Low energy cross sections from the collision of nucleons with light mass \nnuclei show sharp as well as broad resonances upon a smooth, energy dependent \nbackground. Those resonances may correlate to states in the discrete spectrum \nof the target.    To interpret such scattering data requires use of a complex \ncoupled-channel reaction theory. We have developed such a theory~\\cite{Am03}; \none that has very important improvements over those used heretofore.     This\ntheory, a multi-channel algebraic scattering (MCAS) theory, finds solution of \nthe \ncoupled Lippmann-Schwinger equations for the scattering of quantal systems in\nmomentum space, and is a quite general one. To date, however, we have limited \nstudy to nucleon scattering from targets of zero ground state spin.\n\nThe prime information sought are   scattering ($S$) matrices which are easily\nextracted from the $T$-matrices generated by MCAS.      The approach involves \nusing matrix algebra on matrices built using Sturmian-state expansions of the \nrelevant nucleon-nucleus potential matrix.   With this method, all resonances \nin any energy range, can be identified       and their centroids, widths, and \nspin-parities determined. Similarly the energies and spin-parities of   bound \nstates of the compound system sub-threshold can be determined.\n\n               It has long been known that collective model prescriptions  of \nnucleon-nucleus scattering  violate the Pauli principle.   It seems that this   \nhas been the case in all such  calculations made prior to those reported   in \nRef.~\\cite{Am03}.              However, the MCAS procedure enables use of  an \northogonalizing pseudo-potential (OPP) approximation   by   which such  Pauli \nprinciple  violation can be alleviated.    Doing so is crucial to finding the \nparameter values specifying the interaction that  simultaneously   gives  the\nsub-threshold  compound nucleus spectrum and the low energy scattering  cross \nsections.\n\n \nThe MCAS approach that is based upon using Sturmian functions as a basis set, \nis outlined next along with the process by  which           resonances can be \nidentified and located. Results of calculations made using a collective model \nprescription for the interaction potential matrix are then discussed. In that \ncollective model, the interaction field is allowed  to  be deformed      from \nsphericity.  That deformation has been taken to second order in   the case of \nneutron scattering from $^{12}$C allowing coupling to the     ground $0^+_1$, \n$2^+_1$ (4.4389 MeV), and $0^+_2$ (7.64 MeV) states. Results have been  found \nfor nucleon energies to 6 MeV. Some new results for the ${}^{6}$He+$p$ system \nare reported as well.\n\n   The theory also allows formation of the optical potential by appropriately\nsubsuming  the  coupled  channels  equations  into  effective elastic channel \nscattering equations.     The optical potentials thus constructed, as well as\nallowing for the Pauli principle, are very non-local and energy dependent. We \npresent typical results for the $n+{}^{12}$C case.\n\n\\section{The MCAS theory (in brief)}\n\\label{multiT}\n\n    For a system of $\\Gamma$ channels for each allowed scattering spin-parity\n$J^\\pi$ let the index $c\\ (=1,\\Gamma)$ denote the set of quantum numbers that \nidentify each channel uniquely.    Let $c = 1$ designate the elastic channel.\nThe integral equation approach in momentum space   for   potential   matrices \n$V_{cc'}^{J^\\pi}(p,q)$, requires solution of coupled  Lippmann-Schwinger (LS)\nequations giving a multichannel $T$-matrix of the form\n\\begin{eqnarray}\nT_{cc'}(p,q;E) &=& V_{cc'}(p,q)\\, \n+\\, \\frac{2\\mu}{\\hbar^2} \n\\left[ \\sum_{c'' = 1}^{\\rm open} \\int_0^\\infty V_{cc''}(p,x) \n\\frac{x^2}{k^2_{c''} - x^2 + i\\epsilon} T_{c''c'}(x,q;E)\n\\ dx \\right.\n\\nonumber\\\\\n&&\\hspace*{2.8cm}\n\\left.- \\sum_{c'' = 1}^{\\rm closed} \\int_0^\\infty \nV_{cc''}(p,x) \\frac{x^2}{h^2_{c''} + x^2} \nT_{c''c'}(x,q;E) \\ dx \\right],\\;\n\\label{multiTeq}\n\\end{eqnarray}\nwhere the conserved values of $J^\\pi$ is to be understood.       The open and\nclosed channels contributions have channel wave numbers \n$k_c = \\sqrt{\\frac{2\\mu}{\\hbar^2}(E - \\epsilon_c)}$ and \n$h_c = \\sqrt{\\frac{2\\mu}{\\hbar^2}(\\epsilon_c - E)}$,\nfor $E > \\epsilon_c$ and $E < \\epsilon_c$ respectively.       $\\epsilon_c$ is\nthe threshold energy of channel $c$ and  $\\mu$ is the reduced mass. Solutions\nof Eq.~(\\ref{multiTeq}) are sought using (finite rank) separable   expansions\nof the potential matrix elements, \n$V_{cc'}(p,q) \\sim  \\sum^N_{n = 1} \\chi_{cn}(p)\\ \\eta^{-1}_n\\ \\chi_{c'n}(q)$.\n\n     In the MCAS method, the Sturmians that lead to specification of the form \nfactors $\\chi_{cn}(p)$, are solutions of Schr\\\"odinger equations with     the\nchosen input matrix of potentials.   In coordinate space, if those potentials\nare defined from a collective model prescription of the nucleon-target system \nwith local forms $V_{cc'}(r)$,  the Pauli principle can be satisfied by using\nan OPP  method in the determination of the Sturmians~\\cite{Am03,Ca05}.\n\nThe link between the multichannel $T$- and $S$-matrices is\n\\begin{equation}\nS_{cc'} = \\delta_{cc'} - i^{l_{c'} - l_c +1} \\pi \\frac{2\\mu}{\\hbar^2} \n\\sum_{n,n' = 1}^N \\sqrt{k_c} \\chi_{cn}(k_c) \\left([\\mbox{\\boldmath $\\eta$} \n- {\\bf G}_0]^{-1} \\right)_{nn'} \\ \\chi_{c'n'}(k_{c'})\\sqrt{k_{c'}}\\ ,\n\\label{multiS}\n\\end{equation}\nwhere now $c,c'$ refer to open channels only.         In this representation, \n\\mbox{\\boldmath $\\eta$} has matrix elements      $\\left( \\eta \\right)_{nn'} = \n\\eta_n\\ \\delta_{nn'}$      while      those     of   \\textbf{${\\bf G}_0$} are \n$\\left( {G}_0 \\right)_{nn'} = (G)$,\n\\begin{equation}\n(G) = \\frac{2\\mu}{\\hbar^2}\\left[ \\sum_{c = 1}^{\\rm open} \n\\int_0^\\infty \\chi_{cn}(x) \\frac{x^2}{k_c^2 - x^2 + i\\epsilon} \n\\chi_{cn'}(x)\\ dx - \\sum_{c = 1}^{\\rm closed} \\int_0^\\infty \n\\chi_{cn}(x) \\frac{x^2}{h_c^2 + x^2} \\chi_{cn'}(x)\\ dx \\right],\n\\label{xiGels}\n\\end{equation}\nBound states of the compound system are defined by the zeros of the    matrix\ndeterminant when the energy is $E < 0$. They link to zeros of \n$\\{ \\left| \\mbox{\\boldmath $\\eta$}-{\\bf G}_0\\right| \\}$\nwhen all channels in Eq.~(\\ref{xiGels}) are closed.        For details of the\nSturmian expansions and their use, see Ref.~\\cite{Am03}.\n\nAn essential feature of the MCAS approach, of importance in studies when very\nnarrow    resonances    are    to    be    observed, is the resonance finding \nscheme~\\cite{Am03}.           Essentially that requires recasting the elastic \nscattering $S$-matrix (for each $J^\\pi$) as\n\\begin{equation}\nS_{11} = 1 - i \\pi \\frac{2\\mu}{\\hbar^2} \\sum_{nn'=1}^M k \\ \\chi_{1n}(k) \n\\frac{1}{\\sqrt{\\eta_n}} \\left[\\left({\\bf 1} - \n\\mbox{\\boldmath $\\eta$}^{-\\frac{1}{2}}\n{\\bf G}_0\\mbox{\\boldmath $\\eta$}^{-\\frac{1}{2}}    \n\\right)^{-1}\\right]_{nn'} \\frac{1}{\\sqrt{\\eta_{n'}}} \\chi_{1n'}(k)\\, .\n\\end{equation}\nHere, the elements of the diagonal (complex) matrix \n\\mbox{\\boldmath $\\eta$}$^{-\\frac{1}{2}}$ are\n$\\frac{1}{\\sqrt{\\eta_n}}\\delta_{nn'}$.    Then the  complex-symmetric matrix, \n\\mbox{\\boldmath $\\eta$}$^{-\\frac{1}{2}}{\\bf G}_0$ \n\\mbox{\\boldmath $\\eta$}$^{-\\frac{1}{2}}$   can    be diagonalized to find the \nevolution of its complex eigenvalues, $\\zeta_r$,      with respect to energy. \nResonant behavior occurs when one of the complex $\\zeta_r$ eigenvalues passes \nclose to the point (1,0) in the Gauss plane. The complex part of that   limit\neigenvalue relates to the width of the resonance.         The elastic channel\n$S$-matrix has a  pole  structure  at  the  corresponding energy where one of \nthese eigenvalues approach unity.\n\n\\section{Results using MCAS for $n+A$ systems}\n\n    Results for the scattering of neutrons to $\\sim$6 MeV from ${}^{12}$C are\ndisplayed in Fig.~1. Therein the elastic scattering cross section is compared\nwith data with respect to the ground state of ${}^{12}$C as zero energy.   On\nthe same scale, in the right hand parts of \n\\begin{minipage}[t]{7.0cm}\n\\vspace*{2mm}\n\\scalebox{0.3}{\\includegraphics*{AmosK-fig1.eps}}\n\\vspace*{2mm}\n\\end{minipage} \\hfill\n\\begin{minipage}[t]{7.0cm}\nthe figure,     the experimental and theoretical sub-threshold  and resonance \nstates in ${}^{13}$C are compared.                  We have noted in a recent \npublication~\\cite{Ca05} how crucial it is to account  for the Pauli principle \nvia the OPP scheme to achieve these results.\n\\vspace*{5mm}\n\n{\\footnotesize {\\bf FIGURE 1.} Spectra of ${}^{12,13}$C   and   the   elastic \ncross section (barn) for the $n+{}^{12}$C system.  The labels with the states \ndesignate values of $2J$ and parity.}\n\\vspace*{2mm}\n\\end{minipage} \\hfill\n\nIn Fig.~2., the elastic cross section for neutron scattering  is shown in the\ntop panel \n\\begin{minipage}[t]{7.0cm}\n\\vspace*{2mm}\n\\scalebox{0.38}{\\includegraphics*{AmosK-fig2.eps}}\n\\vspace*{2mm}\n\\end{minipage} \\hfill\n\\begin{minipage}[t]{7.0cm}\nwhile polarizations at two scattering angles are shown underneath.        The \nresonance structures most evident in the cross section are well reproduced by\nMCAS and those details are confirmed in the polarizations.                The \nspin-parities of the resonances also coincide with those of known states   in\nthe spectrum of ${}^{13}$N.  Notably the\n\n\\vspace*{5mm}\n{\\footnotesize {\\bf FIGURE 2.}\nThe elastic cross section for n-${}^{12}$C scattering (top) and polarizations\nat 50$^\\circ$ (middle) and at 135$^\\circ$ (bottom) compared to MCAS results.}\n\\vspace*{2mm}\n\\end{minipage}\\hfill\nprominent $\\frac{5}{2}^+$ resonance at 2 MeV,     and the two $\\frac{3}{2}^+$ \nresonances spanning 2.8 - 4.0 MeV, are found with very good widths,    narrow\nand broad respectively.\n\\begin{minipage}[t]{7.0cm}\n\\vspace*{2mm}\n\\scalebox{0.55}{\\includegraphics*{AmosK-fig3.eps}}\n\\vspace*{2mm}\n\\end{minipage} \\hfill\n\\begin{minipage}[t]{7.0cm}\nIn Fig.~3.,    we show the spectra that one obtains for the $p+{}^6$He system \nwhen the target   (${}^6$He)   structure  is taken as three states; the $0^+$ \n(ground), and two $2^+$ states at 1.78 and 5.68 MeV respectively.           A \nquadrupole deformation was considered and first and second order deformations\nallowed for each excitation.\n\n\\vspace*{5mm}\n{\\footnotesize {\\bf FIGURE 3.}\nThe spectra of ${}^7$Li as determined using MCAS for the $p+{}^6$He system.}\n\\vspace*{2mm}\n\\end{minipage}\\hfill\nThe base interaction potential required a large diffuseness consistent   with\nthe target having an extended neutron matter distribution. The results on the\nfar left came from evaluations   with   deformation  set to zero and show the \nsub-structural origins of each state in the actual      spectrum~\\cite{Pi05}. \nAgain use of the OPP to account for Pauli blocking of     occupied states was \ncrucial in finding these results.\n\n\\section{MCAS and the optical potential}\n\n\n       Assuming a local form for the elastic channel element of the potential \nmatrix, the optical potential for elastic scattering is defined by\n\\begin{equation}\nV^{opt}(r,r';E) \n= V_{1 1}(r) + \n\\sum^\\Gamma_{c,c'=2} V_{1 c}(r)\\ \nG^{(Q)}_{cc'}(r,r';E)\\\nV_{c' 1}(r')\\ ,\n\\label{opteq}\n\\end{equation}\nwhere    the    second    term   is  the dynamic polarization potential (DPP)\n$\\Delta U(r,r';E)$.   This makes the formulated optical potential complex (if\nthe energy allows more than one open channel), nonlocal, and energy dependent\nas $G^{(Q)}_{cc'}$ are the full Green functions referring to the       $Q\\ (= \n\\Gamma -1)$ excluded channels. \n\n    In the MCAS approach~\\cite{Am03}, with ${\\mbox{\\boldmath $\\Lambda$}}(E) = \n\\left[ {\\mbox{\\boldmath $\\eta$}}    -          {\\bf G}_0^{(Q)}(E)\\right]^{-1} \n- {\\mbox{\\boldmath $\\eta$}}^{-1}$, the DPP is \n\\begin{equation}\n\\Delta U(r,r';E) = \\sum^N_{n, n' = 1} \\chi_{1 n}(r) \n\\left[{\\mbox{\\boldmath $\\Lambda$}}(E) \\right]_{nn'}(E) \\chi_{n' 1}(r')\\ .\n\\label{DPP}\n\\end{equation}\nwhere $\\chi_{n' 1}(r)$ are Bessel transforms of the from factors \n$\\chi_{n' 1}(k)$.\n\n\\begin{minipage}[t]{7.0cm}\n\\vspace*{2mm}\n\\scalebox{0.5}{\\includegraphics*{AmosK-fig4.eps}}\n\\vspace*{2mm}\n\\end{minipage} \\hfill\n\\begin{minipage}[t]{7.0cm}\n\\vspace*{2mm}\nThe  energy-volumes of the DPP evaluated for 2.73 MeV neutrons on  ${}^{12}$C \nare plotted in Fig.~4. for three partial waves, $\\ell = 0, 1, 2$. The central\ndepths of   each   are  indicated on the right as are the energy steps of the \ncontour lines. As the energy is below the first inelastic threshold,  the DPP\nis purely real.       But the coupled channel effects make that aspect of the\noptical potential extremely non-local.  The DPP is also very angular momentum\ndependent.\n\n\\vspace*{5mm}\n{\\footnotesize {\\bf FIGURE 4.}\nThe  DPP for three partial waves for the scattering of 2.73 MeV neutrons from \n${}^{12}$C.}\n\\vspace*{2mm}\n\\end{minipage}\\hfill\n\n\n\\section{Conclusions}\n\nThe MCAS approach to analyze (low-energy) nucleon-nucleus scattering is built\nfrom a model structure of the interaction potentials between a nucleon    and\neach of the target states taken into consideration.    All resonances (narrow\nand broad) in the cross section within the selected  (positive)  energy range\ncan be found on a background.         With negative energies, the MCAS method\nspecifies the sub-threshold bound states of the compound nucleus. Inherent is\na resonance finding procedure by which all resonances      (and sub-threshold\nbound) states will be found within a chosen energy range.  Also, when the OPP\nis used in definition of  the Sturmians,        those functions are assured to be \northogonal to any single nucleon bound state that is Pauli blocked.      That\nallowance for  the  influence  of  the Pauli principle was crucial in finding \nresults that concur well with observation.\n\n\\bibliographystyle{aipproc}  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nDetecting Out-of-Distribution (OOD) data \\cite{Yang2021GeneralizedOD}, i.e. input data that does not originate from the same distribution a deep neural network (DNN) is trained on, is a task that is attracting an increasing amount of attention in the domain of deep learning for computer vision \\cite{Liang2018EnhancingTR,Liu2020EnergybasedOD,Du2022VOSLW,Hendrycks2017ABF,Hendrycks2019BenchmarkingNN,Fort2021ExploringTL,Hsu2020GeneralizedOD, Techapanurak_2020_ACCV,Sun2021ReActOD,Wang2022ViMOW,Huang2021MOSTS,Lee2018ASU,Pearce2021UnderstandingSC,Yang2021GeneralizedOD,Zhang2021OnTO,Nalisnick2019DoDG}. This is important for applications where passing predictions on OOD data downstream may incur a large cost, \\eg in medicine or autonomous driving. \n\nDeep Ensembles \\cite{Lakshminarayanan2017SimpleAS} have been shown to reliably be superior for OOD detection compared to single models \\cite{Malinin2020EnsembleDD, Ryabinin2021ScalingED, Kim2021AUB}. An existing intuition for this is that the diversity of a Deep Ensemble is an intrinsic indicator of input distributional shift, as the members will disagree more on data that are further away from training data \\cite{Malinin2020EnsembleDD, Mukhoti2021DeterministicNN}. In this work we: \n\\begin{inparaenum}\n    \\item Show that this intuition is not valid for OOD detection for convolutional neural networks (CNNs) at ImageNet-scale.\n    \\item Present an alternative explanation for Deep Ensembles' superior performance -- they are ensembles of binary classifiers that are diverse in their OOD detection errors.\n    \\item Show that even better OOD detection performance can be achieved by averaging a task specific score such as Energy \\cite{Liu2020EnergybasedOD} over a Deep Ensemble.\n\\end{inparaenum}\n\\section{Preliminaries}\n\\subsection{OOD Detection}\nWe use the problem setting defined in \\cite{Yang2021GeneralizedOD} for OOD detection. For a $K$-class classification problem, we wish to predict labels $y\\in\\mathcal Y=\\{\\omega_k\\}_{k=1}^K$ given inputs $\\b x\\in \\mathcal X = \\mathbb{R}^D$. We train a DNN with parameters $\\b \\theta$ on finite dataset $\\mathcal D_\\text{train} = \\{y^{(n)},\\b x^{(n)}\\}_{n=1}^{N}$ sampled independently from the true joint data distribution $p_\\text{train}(y,\\b x)$. The network typically models the conditional categorical distribution $P(y|\\b x;\\theta)$ using a softmax output layer, and is trained using cross entropy loss. Predictions $\\hat y$ given test inputs $\\b x^*$ can then be made as $\\hat y = \\argmax_\\omega P(\\omega|\\b x^*;\\theta)$.\n\nDuring deployment, we expect $\\b x^*$ to be sampled either from the training input distribution $p_\\text{train}(\\b x)$, or $p_\\text{OOD}(\\b x)$, where the \\emph{label} space of the latter distribution has no intersection with $\\mathcal Y$. Predictions on the OOD samples will thus be invalid. In order to detect these samples the objective is then to design a binary classification function,\n\\begin{equation}\\label{eq:bin}\n    g(\\b x;t) = \\begin{cases}\n    \\text{ID}, &\\text{if }U(\\b x) < t\\\\\n    \\text{OOD}, &\\text{if }U(\\b x) \\geq t~,\n    \\end{cases}\n\\end{equation}\nwhere $U$ is a predictive uncertainty score (or $-U$ confidence), commonly extracted from the network. The performance can be evaluated using standard metrics for binary classification such as Area Under the ROC curve (AUROC$\\uparrow$) and False Positive Rate at 95\\% True Positive Rate (FPR@95$\\downarrow$), where arrows ($\\uparrow\\downarrow$) indicate whether higher or lower is better. The problem is challenging, as the OOD data distribution $p_\\text{OOD}(\\b x)$ is not known to the practitioner before deployment.\n\\subsection{Deep Ensembles}\nDeep Ensembles \\cite{Lakshminarayanan2017SimpleAS} are ensembles of DNNs trained on the same dataset with different random parameter initialisations and data shuffling, $\\Theta = \\{\\b \\theta^{(m)}\\}_{m=1}^M$. The ensemble members can be interpreted as samples from different modes of the Bayesian parameter posterior $p(\\b \\theta|\\mathcal D_\\text{train})$ \\cite{izmailov2021bayesian}, which allows their predictions to be more diverse compared to unimodal approximate Bayesian approaches \\cite{Fort2019DeepEA}. They have been shown to be a reliable and scalable approach for improving both the predictive performance and quality of uncertainty estimates of DNNs \\cite{Lakshminarayanan2017SimpleAS,ovadia2019can,Malinin2020EnsembleDD,Ryabinin2021ScalingED}.\n\nFollowing the Bayesian interpretation of Deep Ensembles, we obtain an approximate posterior predictive distribution by averaging over the softmax outputs of the individual ensemble members,\n\\begin{equation}\n      P(y|\\b{x},\\mathcal{D}_{\\text{train}}) \\approx \\frac{1}{M}\\sum_{m=1}^M P(y|\\b{x}, \\b{\\theta}^{(m)}) = P(y|\\b{x}, \\Theta),\n\\end{equation}\nas well as an uncertainty decomposition \\cite{Depeweg2018DecompositionOU} in the form of\n\n\\begin{multline}\\label{eq:approx_decomp}\n   \\underbrace{\\mathcal{I}[y, \\b{\\theta}|\\b{x}, \\mathcal{D}_{\\text{train}}]}_{\\text{mutual information}}  \\approx  \\underbrace{\\mathcal{H}\\left[\\frac{1}{M}\\sum_{m=1}^M P(y|\\b{x}, \\b{\\theta}^{(m)})\\right]}_{\\text{ensemble entropy}}\\\\  -\\underbrace{\\frac{1}{M}\\sum_{m=1}^M\\mathcal{H}\\left[P(y|\\b{x}, \\b{\\theta}^{(m)})\\right]}_{\\text{average entropy}},\n\\end{multline}\nwhere $\\mathcal H[\\cdot]$ is the entropy of a probability distribution and $\\mathcal{I}[\\cdot,\\cdot|\\cdots]$ the (conditional) mutual information (MI) between two random variables. We will refer to the other terms in the above equation as ensemble entropy (Ens.$\\mathcal H$) and average entropy (Av.$\\mathcal H$). MI, when approximated using \\cref{eq:approx_decomp}, measures the \\emph{diversity} of the ensemble, as it is the mean KL-divergence between the predictive distributions of the ensemble members and the ensemble overall,\n\\begin{multline}\\label{eq:kl}\n    \\mathcal{I}[y, \\b{\\theta}|\\b{x}, \\mathcal{D}_{\\text{train}}] \\approx \\\\\n      \\frac{1}{M}\\sum_{m=1}^M\\left[\\text{KL}\\left[P(y|\\b{x}, \\b{\\theta}^{(m)})||P(y|\\b{x},\\Theta)\\right]\\right].\n\\end{multline}\nAv.$\\mathcal H$ is a measure of \\textit{average uncertainty} over the ensemble, and we can say that Ens.$\\mathcal H$ measures \\textit{total uncertainty} by combining \\textit{diversity} and \\textit{average uncertainty}. Note in the following text, when we refer to MI, it is referring to the right hand side of \\cref{eq:kl}, i.e. the empirical approximation.\n\nNote that we will avoid the discussion of epistemic and aleatoric uncertainty \\cite{Hllermeier2021AleatoricAE, Mukhoti2021DeterministicNN, Pearce2021UnderstandingSC, Malinin2020EnsembleDD} in this work and instead directly focus the discussion on the task of OOD detection.\n\\subsection{Experimental Setup}\nIn order to contruct our Deep Ensemble, we train five ResNet-50 \\cite{He2016DeepRL} models independently using random seeds $\\{1,\\dots,5\\}$ and standard ImageNet settings.\\footnote{We use the default settings found here \\url{https:\/\/github.com\/pytorch\/examples\/blob\/main\/imagenet\/main.py}. For full details see \\cref{app:training}} \n\nFor our ID dataset we use ImageNet-200 \\cite{Kim2021AUB}, which uses a subset of 200 ImageNet-1k \\cite{Russakovsky2015ImageNetLS} classes. We use a large selection of ImageNet-scale OOD datasets that exhibit a wide range of semantics and difficulty in being identified. Near-ImageNet-200 \\cite{Kim2021AUB} is composed of ImageNet-1k classes semantically similar but disjunct to ImageNet-200. Caltech-45 \\cite{Kim2021AUB} is a subset of the Caltech-256 \\cite{Griffin2007Caltech256OC} dataset with disjoint classes to ImageNet-200. Openimage-O \\cite{Wang2022ViMOW} is a subset of the Open Images V3 \\cite{openimages} dataset chosen to be OOD to ImageNet-1k. \niNaturalist \\cite{Huang2021MOSTS} and Textures \\cite{Wang2022ViMOW} are the same for their respective original datasets \\cite{inat, cimpoi14describing}. Colorectal \\cite{Kather2016MulticlassTA} is composed of histological photographs of human colorectal cancer, whilst Colonoscopy is constructed from individual frames extracted from colonoscopic video of gastrointestinal lesions \\cite{Mesejo2016ComputerAidedCO}. Noise is a collection of square images where the pixel intensities, image resolution and image contrast are randomly generated (for details see \\cref{app:images}). Finally, ImageNet-O \\cite{Hendrycks2021NaturalAE} is adversarially constructed by selecting OOD images that a trained ResNet makes confident predictions on. Example images from each dataset, as well as their sizes can be found in \\cref{app:images}.\n\n\\section{Existing Intuition Doesn't Hold Up}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.7\\linewidth]{MI_illust.pdf}\n    \\caption{\\%FPR@95$\\downarrow$ for different uncertainty metrics. MI (measuring diversity) performs poorly across all OOD datasets. Ens.$\\mathcal H$ and Av.$\\mathcal H$ have similar performance and are able to improve over the single model baseline, although Av.$\\mathcal H$ is better in 2\/3 cases. This suggests MI is not reliably useful by itself, or as part of Ens.$\\mathcal H$. The ID dataset is ImageNet-200, the model ResNet-50.}\n    \\label{fig:MI_illust}\n\\end{figure}\n\n\\begin{table*}\n    \\centering\n \\resizebox{.8\\textwidth}{!}{   \n\\begin{tabular}{lllaallllllll}\n\\toprule\n & \\textbf{ID:} & \\textbf{ImageNet-200} & \\multicolumn{2}{c}{\\textbf{OOD mean}} & \\multicolumn{2}{c}{\\textbf{Near-ImageNet-200}} & \\multicolumn{2}{c}{\\textbf{Caltech-45}} & \\multicolumn{2}{c}{\\textbf{Openimage-O}} & \\multicolumn{2}{c}{\\textbf{iNaturalist}} \\\\\n\\textbf{Model} &  & \\textbf{Method} & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ \\\\\n\\midrule\n\\multirow[c]{8}{*}{\\begin{sideways}\\shortstack[l]{\\textbf{ResNet-50} \\\\ens. \\%Err: 16.28 \\\\ av. \\%Err: 19.01}\\end{sideways}} & \\multirow[c]{3}{*}{\\begin{sideways}\\textbf{Single}\\end{sideways}} & MSP & 84.47 & 66.55 & 77.56 \\scriptsize \u00b11.2 & 82.21 \\scriptsize \u00b11.1 & 84.54 \\scriptsize \u00b11.2 & 67.83 \\scriptsize \u00b14.0 & 85.09 \\scriptsize \u00b11.0 & 71.43 \\scriptsize \u00b11.8 & 88.40 \\scriptsize \u00b10.9 & 60.83 \\scriptsize \u00b12.9 \\\\\n &  & $\\mathcal{H}$ & 85.98 & 62.12 & 78.18 \\scriptsize \u00b11.2 & 82.33 \\scriptsize \u00b11.7 & 86.13 \\scriptsize \u00b11.0 & 63.96 \\scriptsize \u00b13.6 & 86.21 \\scriptsize \u00b11.0 & 70.76 \\scriptsize \u00b12.0 & 89.64 \\scriptsize \u00b10.8 & 59.22 \\scriptsize \u00b13.5 \\\\\n &  & Energy & \\underline{88.56} & \\underline{47.71} & 75.96 \\scriptsize \u00b11.2 & 82.57 \\scriptsize \u00b12.1 & \\underline{88.23 \\scriptsize \u00b10.8} & \\underline{54.32 \\scriptsize \u00b13.7} & 86.43 \\scriptsize \u00b10.6 & \\underline{62.24 \\scriptsize \u00b11.6} & 90.78 \\scriptsize \u00b10.4 & \\underline{46.18 \\scriptsize \u00b11.9} \\\\\n \\cmidrule{2-13}\n & \\multirow[c]{5}{*}{\\begin{sideways}\\textbf{5--Ens.}\\end{sideways}} & MSP & 86.67 & 63.53 & \\underline{79.28} & 81.27 & 86.71 & 64.48 & 87.24 & 68.39 & 89.95 & 59.15 \\\\\n &  &Ens. $\\mathcal{H}$ & \\underline{88.21} & 57.97 & \\underline{79.70} & \\underline{80.85} & 88.15 & 60.04 & \\underline{88.14} & 67.75 & \\underline{91.05} & 57.19 \\\\\n &  & Av. $\\mathcal{H}$ & 88.07 & \\underline{57.48} & \\textbf{79.75} & \\textbf{80.40} & \\underline{88.22} & \\underline{59.31} & \\underline{88.29} & \\underline{66.50} & \\underline{91.67} & \\underline{52.31} \\\\\n &  & MI & 83.46 & 78.84 & 77.94 & 84.18 & 83.75 & 79.36 & 83.46 & 78.40 & 81.97 & 90.05 \\\\\n &  & Av. Energy & \\textbf{90.16} & \\textbf{43.38} & 77.60 & \\underline{80.47} & \\textbf{89.99} & \\textbf{48.23} & \\textbf{88.30} & \\textbf{57.44} & \\textbf{92.43} & \\textbf{40.14} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\resizebox{.8\\textwidth}{!}{\n\\begin{tabular}{lllllllllllll}\n\\toprule\n& \\textbf{ID:} & \\textbf{ImageNet-200} & \\multicolumn{2}{c}{\\textbf{Textures}} & \\multicolumn{2}{c}{\\textbf{Colonoscopy}} & \\multicolumn{2}{c}{\\textbf{Colorectal}} & \\multicolumn{2}{c}{\\textbf{Noise}} & \\multicolumn{2}{c}{\\textbf{ImageNet-O}} \\\\\n\\textbf{Model} &  & \\textbf{Method} & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ & AUROC$\\uparrow$ & FPR@95$\\downarrow$ \\\\\n\\midrule\n\\multirow[c]{8}{*}{\\begin{sideways}\\shortstack[l]{\\textbf{ResNet-50} \\\\ens. \\%Err: 16.28 \\\\ av. \\%Err: 19.01}\\end{sideways}} & \\multirow[c]{3}{*}{\\begin{sideways}\\textbf{Single}\\end{sideways}} & MSP & 86.95 \\scriptsize \u00b11.4 & 60.49 \\scriptsize \u00b14.0 & 90.67 \\scriptsize \u00b11.8 & 53.16 \\scriptsize \u00b19.0 & 89.82 \\scriptsize \u00b15.2 & 50.88 \\scriptsize \u00b111.9 & 81.70 \\scriptsize \u00b16.5 & 71.13 \\scriptsize \u00b115.6 & 75.50 \\scriptsize \u00b10.6 & 80.96 \\scriptsize \u00b10.7 \\\\\n &  & $\\mathcal{H}$ & 88.69 \\scriptsize \u00b11.3 & 53.84 \\scriptsize \u00b13.8 & 92.94 \\scriptsize \u00b11.6 & 42.88 \\scriptsize \u00b19.1 & 91.95 \\scriptsize \u00b14.9 & 41.10 \\scriptsize \u00b115.7 & 83.61 \\scriptsize \u00b15.3 & \\underline{64.78 \\scriptsize \u00b122.8} & 76.51 \\scriptsize \u00b10.7 & 80.18 \\scriptsize \u00b10.3 \\\\\n &  & Energy & \\underline{92.93 \\scriptsize \u00b11.0} & \\underline{32.65 \\scriptsize \u00b12.8} & \\underline{95.40 \\scriptsize \u00b11.5} & \\underline{28.66 \\scriptsize \u00b112.1} & \\underline{98.25 \\scriptsize \u00b11.7} & \\underline{9.87 \\scriptsize \u00b19.7} & \\underline{91.99 \\scriptsize \u00b17.6} & \\textbf{38.43 \\scriptsize \u00b123.3} & 77.10 \\scriptsize \u00b10.9 & \\underline{74.47 \\scriptsize \u00b11.3} \\\\\n \n  \\cmidrule{2-13}\n \n & \\multirow[c]{5}{*}{\\begin{sideways}\\textbf{5--Ens.}\\end{sideways}} & MSP & 90.19 & 53.49 & 92.93 & 46.49 & 90.98 & 49.22 & 84.43 & 71.29 & 78.30 & 77.95 \\\\\n &  &Ens. $\\mathcal{H}$ & \\underline{91.86} & \\underline{44.92} & 95.22 & \\underline{31.60} & 93.63 & 36.18 & \\underline{86.95} & 65.79 & \\underline{79.17} & 77.45 \\\\\n &  & Av. $\\mathcal{H}$ & 91.23 & 48.00 & \\underline{95.26} & 31.65 & \\underline{93.97} & \\underline{34.46} & 85.67 & 65.75 & 78.56 & 78.95 \\\\\n &  & MI & 88.43 & 61.90 & 89.11 & 76.07 & 82.14 & 88.66 & 84.92 & 77.39 & \\textbf{79.41} & \\underline{73.55} \\\\\n &  & Av. Energy & \\textbf{94.55} & \\textbf{27.07} & \\textbf{96.87} & \\textbf{19.31} & \\textbf{99.08} & \\textbf{4.48} & \\textbf{94.00} & \\underline{41.13} & \\underline{78.64} & \\textbf{72.15} \\\\\n\\bottomrule\n\\end{tabular}\n}\n    \\caption{\\%AUROC$\\uparrow$ and \\%FPR@95$\\downarrow$ for different uncertainty scores over all OOD datasets. We show mean$\\pm2$std. for single model results. \\textbf{Bold} is best performance, \\underline{underline} 2nd or 3rd best. MI performs poorly compared to other uncertainties and even single models. Ens.$\\mathcal H$ and Av.$\\mathcal H$ consistently improve over single model $\\mathcal H$. Av. Energy is able to improve over single model Energy and is best overall.}\n    \\label{tab:results}\n\\end{table*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.7\\textwidth]{resnet50_imagenet200_DU_KUOOD_tog.pdf}\n    \\caption{Top: empirical conditional histograms of MI$|$Av.$\\mathcal H$. The behaviour of MI is quite similar between ID and OOD data, with the exception of Textures being slightly higher. Bottom: joint kernel density estimate plots of MI and Av.$\\mathcal H$. Contours represent increasing (lighter) values of Ens.$\\mathcal H$. MI is low for high Av.$\\mathcal H$ OOD data. MI does not contribute significantly to Ens.$\\mathcal H$.}\n    \\label{fig:conditional}\n\\end{figure*}\nIn the literature there is an existing intuition, that the \\emph{diversity} of a Deep Ensemble intrinsically indicates distributional shift of the input $\\b x$. Thus, uncertainty scores $U$ directly derived from diversity should be useful for OOD detection \\cite{Malinin2020EnsembleDD,Ryabinin2021ScalingED,Mukhoti2021DeterministicNN}. Moreover, consequently the superiority of measures of total uncertainty compared to single models (\\eg Ens.$\\mathcal H$) should be due to their diversity component (MI in \\cref{eq:approx_decomp}). The intuition relies on the idea that as the input data moves further away from the training distribution, the ensemble members will be less constrained by the training data, and so will behave differently to each other. This intuition tends to be demonstrated on toy, low-dimensional data \\cite{Malinin2020EnsembleDD, Mukhoti2021DeterministicNN}. However, we will show that it doesn't hold for ImageNet-scale OOD detection with CNNs.\n\n\n\\cref{fig:MI_illust} shows \\%FPR@95$\\downarrow$ for different uncertainty measures $U$ over three OOD datasets (full results can be found in \\cref{tab:results}). We see that although Ens.$\\mathcal H$ and Av.$\\mathcal H$ are both superior to the average performance of single-model $\\mathcal H$, MI is unable to outperform the single model baseline. In fact, on iNaturalist, it is $\\sim$30\\% worse. This shows that directly using the diversity of the ensemble to detect OOD data is actually a poor choice in this scenario. We note that inferior MI performance, at ImageNet-scale, has been shown before in \\cite{Ryabinin2021ScalingED}. Additionally, the performance is very close between Av.$\\mathcal H$ and Ens.$\\mathcal H$, but Av.$\\mathcal H$ beats Ens.$\\mathcal H$ in 2\/3 datasets, suggesting MI is a hindrance there. Similar behaviour over most of the other datasets can be observed in \\cref{tab:results}.\n\n\\subsection{Mutual Information Performs Poorly}\nIn order to understand the results in \\cref{fig:MI_illust} and \\cref{tab:results}, we can consider empirical plots of the \\emph{conditional} distribution of MI \\textit{given} Av.$\\mathcal H$, as well as the joint distribution. These are shown for ImageNet-200 (ID) and various OOD datasets in \\cref{fig:conditional}. The graded contours represent different values of Ens.$\\mathcal H$ (lighter means higher), as it is the sum of Av.$\\mathcal H$ and MI (\\cref{eq:approx_decomp}). Av.$\\mathcal H$ is much lower on OOD data -- see \\cite{Pearce2021UnderstandingSC} for a discussion of why. Intuitively, if the average uncertainty is high, then it is difficult for an ensemble to be diverse, as for high diversity the members need to disagree confidently (\\cref{eq:approx_decomp}). Thus, the MI of high Av.$\\mathcal H$ OOD data is actually quite low, leading to MI's poor performance. In other words, MI is failing because Av.$\\mathcal H$ is performing well. \n\nThe notable exception is ImageNet-O, where MI is able to perform the best. We hypothesise that this is due to the adversarial nature of the ImageNet-O data causing the members to be individually more confident (and thus diverse) on the OOD data. A similar effect is observed in \\cite{Malinin2021UncertaintyEI} for neural machine translation where diversity measures are able to successfully detect pathological overconfidence when models copy OOD inputs rather than translating them.\n\n\\subsection{Mutual Information Does Not Reliably Contribute to Ensemble Entropy}\n\n We see that in \\cref{fig:conditional} (top) for a \\emph{given} value of Av.$\\mathcal H$, the behaviour of MI is not that different between the ID and OOD datasets -- the diversity does not seem to act as a strong indicator of distributional shift \\textit{in addition} to the average uncertainty. It can also be observed that the actual variation in MI is much lower in comparison to Av.$\\mathcal H$. Thus, Ens.$\\mathcal H$ is in fact dominated by Av.$\\mathcal H$ and explains why their performance is so close. On Textures Ens.$\\mathcal H$ performs better compared to Av.$\\mathcal H$, and this suggests that MI is useful in addition to Av.$\\mathcal H$. \\cref{fig:conditional} shows that MI given Av.$\\mathcal H$ is slightly higher on Textures. However, on the other two datasets the opposite is the case -- including diversity in addition to average uncertainty harms performance, as the conditional distributions are very similar between ID and OOD. \n\nWe note similar results to \\cref{fig:conditional} are shown in \\cite{Abe2022DeepEW}, however the problem framing and conclusions reached in that work are different to ours.\n\\section{An Alternative Explanation}\nUncertainty scores derived directly from ensemble diversity don't seem to reliably contribute to the OOD detection performance of Deep Ensembles at ImageNet-scale. However, the question still remains: \\textit{why do Deep Ensembles perform better than single models for OOD detection?}\n\\subsection{Deep Ensembles as Binary Classifiers}\nThe fact that Av.$\\mathcal H$ of the ensemble is able to consistently improve OOD detection performance compared to single model $\\mathcal H$ (\\cref{tab:results}) points to a straightforward explanation. If we consider again the binary classification model presented in \\cref{eq:bin}, we can simply view Deep Ensembles for OOD detection through the same lens we view them for ID prediction. By being diverse in the individual errors they make for OOD detection, their overall OOD detection performance after combination will be better, and this diversity arises from the multimodal nature of the ensemble in parameter space \\cite{Fort2019DeepEA}. For a more formal explanation we can consider Jensen's inequality for the binary negative log-likelihood loss for OOD detection,\n\\begin{equation}\n    -\\log \\left(\\frac{1}{M}\\sum_{m=1}^M P(s|\\b{x}, \\b{\\theta}^{(m)})\\right) \\leq - \\frac{1}{M}\\sum_{m=1}^M \\log P(s|\\b{x}, \\b{\\theta}^{(m)})\n\\end{equation}\nwhere $s\\in\\{\\text{ID},\\text{OOD}\\}$ and \n\\begin{equation}\n    P(s=\\text{OOD}|\\b x, \\b \\theta) = aU(\\b x;\\b \\theta) + b,\n\\end{equation}\nwhere $a, b$ are arbitrary parameters such that $U$ is mapped to a valid probability. We can thus see how averaging $U = \\mathcal H$ over the ensemble leads to better OOD detection. \n\nThe \\emph{key intuition} is that by ensembling binary classifiers that are diverse such that they make OOD detection errors on different data samples, the ensemble members are able to compensate for each other's miss-predictions and raise the ensemble performance above their individual average.\n\\subsection{Using a Better OOD Detection Score}\nExisting work benchmarking Deep Ensembles focuses exclusively on uncertainty scores $U$ extracted from the softmax outputs of the members. Given the explanation in the previous section, it is natural to ask whether we can improve performance by ensembling a better $U$. Using Energy \\cite{Liu2020EnergybasedOD} ($U = -\\log\\sum_k\\exp v_k$, where $\\b v$ are the logits), and averaging over the ensemble, it can be seen in \\cref{tab:results} that average ensemble Energy is able to improve over single model Energy, as well as produce the best OOD detection results in the whole comparison. We remark that although we have chosen Energy here, we expect any $U$ with better performance than single model $\\mathcal H$ to receive a similar uplift upon ensembling.\n\\section{Concluding Remarks}\nIn this work, we show that the existing intuition, that the diversity of a Deep Ensemble is a useful indicator for whether data is OOD, does not hold for ImageNet-scale CNNs. We instead suggest an alternative explanation for Deep Ensembles' superior OOD detection performance compared to single models: OOD detection is binary classification, and individual Deep Ensemble members produce diverse errors on this binary classification task, so combining them together yields better performance.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro.sct}\n\nWhen dealing with the behaviour of Petri nets~\\cite{reisig-2013,murata-89} there are two opposite approaches.\nWe can {\\it analyse} a Petri net, building a variety of descriptions of its behaviour from sets of\nfiring sequences~\\cite{hack-lang} to event structures~\\cite{npw81}. One of the most common forms for describing the\nsequential behaviour is the reachability graph, containing the reachable markings \nas nodes together with edges denoting transitions that fire to reach \none marking from another.\nIn the reverse direction, i.e.\\ {\\it synthesis}~\\cite{bbd15}, we can try to find a Petri \nnet\\footnote{we generally use Petri nets with arc weights} that \nbehaves like a given specification, e.g. a labelled transition system (\\lts).\nSince using labelled Petri nets would always allow a trivial solution (isomorphic to the \\lts),\nwe restrict ourselves to unlabelled Petri nets. As the reachability problem for\nPetri nets is \\EX-hard~\\cite{lipton}, unlabelled nets can have a very complex behaviour, but on the\nother hand there are even simple words, i.e.\\ linearly ordered \\lts, that are\nnot behaviours of such Petri nets~\\cite{besw16}. This can make synthesis quite difficult;\neven for the rather small \\lts in the left of Fig.~\\ref{f.gen1}, the resulting synthesised net on the right\nis not immediately obvious.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[LTS]\n\\node[state,label=above:$s_0$] (s0) {};\n\\path[edge] (s0) ++(-0.5,0) edge (s0);\n\\node[state,below=of s0,label=left:$s_1$] (s1) {};\n\\node[state,right=of s1,label=below:$s_2$] (s2) {};\n\\node[state,right=of s0,label=above:$s_3$] (s3) {};\n\\node[state,below=of s1,label=left:$s_4$] (s4) {};\n\\node[state,right=of s2,label=below:$s_5$] (s5) {};\n\\node[state,right=of s4,label=above:$s_{10}$] (s10) {};\n\\node[state,below=of s10,label=left:$s_6$] (s6) {};\n\\node[state,right=of s5,label=above:$s_{12}$] (s12) {};\n\\node[state,below=of s12,label=right:$s_7$] (s7) {};\n\\node[state,above=of s5,label=above:$s_8$] (s8) {};\n\\node[state,right=of s6,label=right:$s_9$] (s9) {};\n\\node[state,right=of s10,label=below:$s_{11}$] (s11) {};\n\\node[state,right=of s12,label=below:$s_{13}$] (s13) {};\n\\node[state,above=of s13,label=above:$s_{14}$] (s14) {};\n\\draw[edge] (s0) edge node[auto,swap]{$a$} (s1);\n\\draw[edge] (s0) edge[bend left=10] node[auto]{$b$} (s3);\n\\draw[edge] (s1) edge node[auto,swap]{$d$} (s2);\n\\draw[edge] (s1) edge[bend left=10] node[auto]{$c$} (s4);\n\\draw[edge,densely dashed] (s2) edge node[auto,xshift=1mm,yshift=1mm]{$f$} (s0);\n\\draw[edge] (s2) edge node[auto]{$e$} (s5);\n\\draw[edge,densely dashed] (s3) edge[bend left=10] node[auto,xshift=1mm,yshift=0.5mm]{$c$} (s0);\n\\draw[edge,densely dashed] (s4) edge[bend left=10] node[auto]{$b$} (s1);\n\\draw[edge] (s4) edge node[auto,swap]{$a$} (s6);\n\\draw[edge] (s5) edge node[auto,xshift=-0.8mm,yshift=-0.8mm]{$a$} (s7);\n\\draw[edge] (s5) edge[bend left=10] node[auto]{$b$} (s8);\n\\draw[edge] (s6) edge[bend left=10] node[auto]{$c$} (s9);\n\\draw[edge] (s6) edge node[auto,swap]{$d$} (s10);\n\\draw[edge,densely dashed] (s7) edge[bend left=10] node[auto]{$c$} (s11);\n\\draw[edge] (s7) edge node[auto,swap]{$d$} (s12);\n\\draw[edge,densely dashed] (s8) edge[bend left=10] node[auto]{$c$} (s5);\n\\draw[edge,densely dashed] (s9) edge[bend left=10] node[auto]{$b$} (s6);\n\\draw[edge,densely dashed] (s10) edge node[auto,swap]{$f$} (s4);\n\\draw[edge] (s10) edge node[auto]{$e$} (s11);\n\\draw[edge,densely dashed] (s11) edge[bend left=10] node[auto,xshift=-1mm,yshift=-0.5mm]{$b$} (s7);\n\\draw[edge,densely dashed] (s12) edge node[auto,swap]{$f$} (s5);\n\\draw[edge] (s12) edge node[auto]{$e$} (s13);\n\\draw[edge] (s13) edge[bend left=10] node[auto]{$b$} (s14);\n\\draw[edge,densely dashed] (s14) edge[bend left=10] node[auto]{$c$} (s13);\n\\end{tikzpicture}\\hspace*{1.5cm}\n\\begin{tikzpicture}\n\\draw(0.3,3) node[place,tokens=2,label=left:$p_1$] (p1) {};\n\\draw(1.7,3) node[place,tokens=1,label=left:$p_2$] (p2) {};\n\\draw(0,1.5) node[transition] (a) {$a$};\n\\draw(1,1.5) node[transition] (b) {$b$};\n\\draw(2,1.5) node[transition] (c) {$c$};\n\\draw(0.3,0) node[place,label=left:$p_3$] (p3) {};\n\\draw(1.7,0) node[place,label=left:$p_4$] (p4) {};\n\\draw(3.5,0) node[transition] (d) {$d$};\n\\draw(3.5,1.5) node[place,label=right:$p_5$] (p5) {};\n\\draw(3,3) node[transition] (e) {$e$};\n\\draw(4,3) node[transition] (f) {$f$};\n\\draw(p1) edge[-latex,thick] (a);\n\\draw(p2) edge[-latex,thick] (a);\n\\draw(a) edge[-latex,thick] (p3);\n\\draw(a) edge[-latex,thick] (p4);\n\\draw(p2) edge[-latex,thick] (b);\n\\draw(b) edge[-latex,thick] (p4);\n\\draw(p4) edge[-latex,thick] (c);\n\\draw(c) edge[-latex,thick] (p2);\n\\draw(p3) edge[-latex,thick,bend right=30] (d);\n\\draw(p4) edge[-latex,thick] (d);\n\\draw(d) edge[-latex,thick] (p5);\n\\draw(p5) edge[-latex,thick] (e);\n\\draw(e) edge[-latex,thick] (p2);\n\\draw(p5) edge[-latex,thick] (f);\n\\draw(f) edge[-latex,thick,bend right=35] (p1);\n\\draw(f) edge[-latex,thick,bend right=25] (p2);\n\\end{tikzpicture}\n\\caption{An \\lts (left) that can be synthesised to a reduced asymmetric choice Petri net (right).\nSolid edges in this \\lts form a spanning tree $E$, other edges (`chords') are dashed.\nStates have Parikh vectors according to their spanning tree walks, e.g.\\ $\\parikh_E(s_9)=\\parikh(acac)=2a+2c$ since $s_0\\step{acac}_Es_9$.\nThe Parikh vector of the chord $s_7\\step{c}s_{11}$ (with the spanning tree walks $s_0\\step{adea}_Es_7$ and\n$s_0\\step{acade}_Es_{11}$) is $\\parikh_E(s_7\\step{c}s_{11})=\\parikh_E(s_{7})+1c-\\parikh_E(s_{11})=(2a+1d+1e)+1c-(2a+1c+1d+1e)=\\zero$.\nAll other chords have Parikh vectors $1b+1c$ or $1a+1d+1f$}\n\\label{f.gen1}\n\\end{figure}\n\nRegion theory~\\cite{er89,bd99} connects states of an \\lts with markings of a Petri net\nand determines two kinds of problems that need to be solved\nfor a successful synthesis. {\\it State separation problems} demand that distinguished\nstates in the \\lts must correspond to different markings in a Petri net. {\\it Event-state\nseparation problems} enforce the non-firability of a Petri net transition in a marking\nif the \\lts has no outgoing edge with the matching label in the corresponding state.\nBoth kinds of problems can be formulated as linear inequality systems~\\cite{bds16,sw18} and tackled\nvia e.g.\\ SMT-solvers. Seeking solutions in the integer domain (as generally required for Petri nets)\nis \\NP-complete, but often rational solutions are sufficient, \nand for this case Karmarkar~\\cite{karmarkar} provided an algorithm with a polynomial worst case complexity.\nGenerally, if all constant terms in the linear inequality systems are zero, we can use rational solutions\nand multiply them by appropiate factors to lift them to integers.\nE.g.,\\ the inequality system $x+1\\le y\\le x+y\\le 2\\le 4x$ with the contant term $2$ has the rational solution\n$x=0.5$, $y=1.5$, but no integer solutions.\n\nOften, we would like to find a Petri net with some additional properties, i.e.\\ we target for\nsome subclass of Petri nets (see also Fig.~\\ref{f.classes}). The most common case is the class of bounded Petri nets, which are\nexactly the nets with finite \\lts as behaviours, allowing us to work directly with the \\lts as\ninput. Very limited classes like marked graphs (\\MG)~\\cite{bd17b} and choice-free nets (\\CF)~\\cite{bds17} \nhave been investigated to determine whether structural analysis of an \\lts allows to reduce\nthe size of the linear inequality systems to be solved for synthesis, or even forego them altogether.\nSome overview of properties of Petri nets has been done~\\cite{s16}, with the result that some subclasses can\neasily be targetted (with canonical approaches that are also combinable), while other simple properties\ncannot be tackled at all. For the subclass \\EC of bounded equal-conflict Petri nets (where the postsets of places \nare either identical up to arc weights or completely disjoint), a structural analysis \nof the \\lts is necessary first~\\cite{sp17}. This analysis either\ndetermines that the synthesis must fail or it provides the modifications required for the\nlinear inequality systems (in which case the synthesis may still fail). These modifications enforce\nthe result to be an equal-conflict net, if successful. The well-known (extended) free-choice nets (\\EFC)~\\cite{bes87} are the subclass\nof equal-conflict nets where arc weights are limited to one (i.e.\\ plain equal-conflict nets).\nEqual-conflict nets can be synthesised in polynomial worst case complexity. For \nfree-choice synthesis no such result is known due to the limit on arc weights,\nbut at least an algorithm in \\NP{} exists.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,0) circle [radius=4mm];\n\\node(0,0) {\\MG};\n\\draw[thick](0,-1.0) ellipse [x radius=8mm,y radius=1.7cm];\n\\draw(0.0,-2.5) node {\\CF};\n\\draw(0,0) circle [radius=1.2cm];\n\\draw(0.0,1.0) node {\\EFC};\n\\draw[thick](0,-0.75) circle [radius=2cm];\n\\draw(1.0,-2.2) node {\\EC};\n\\draw(0,0) circle [radius=1.6cm];\n\\draw(0.0,1.4) node {\\RAC};\n\\draw(0,0) circle [radius=2cm];\n\\draw(0.0,1.8) node {\\BRAC};\n\\draw(-0.5,0) ellipse [x radius=3.2cm,y radius=2.2cm];\n\\draw(-3.4,0.0) node {\\ACC};\n\\draw(-0.6,-0.25) ellipse [x radius=3.7cm,y radius=2.65cm];\n\\draw(-3.3,-1.6) node {\\WAC};\n\\draw[thick](0.6,-0.25) ellipse [x radius=3.7cm,y radius=2.65cm];\n\\draw(3.8,-0.25) node {\\WPI};\n\\draw(0.0,-0.1) ellipse [x radius=4.7cm,y radius=3.0cm];\n\\draw(0,2.7) node {bounded nets};\n\\end{tikzpicture}\n\\caption{Relations between classes of Petri nets relevant for our work: marked graph (\\MG), choice-free (\\CF), \nextended free-choice (\\EFC), reduced asymmetric choice (\\RAC), block-reduced asymmetric choice (\\BRAC), \nasymmetric choice (\\ACC), equal-conflict (\\EC), asymmetric choice with arc weights (\\WAC), and weighted comparable preset (\\WPI).\nThe last three classes and \\CF allow arc weights greater than one \n\\label{f.classes}}\n\\end{figure}\n\nFor free-choice nets, structural and behavioural properties are strongly connected.\nCommoner's Theorem and the Rank Theorem~\\cite{de95} are two well-known examples linking\nliveness and well-formedness of a net to siphons\/marked traps and the rank of the net's\nincidence matrix, respectively.\nIn modern research areas like business processes and web services, these properties play\nan important role, but the modelled systems are seldom free-choice.\nAsymmetric choice nets (\\ACC)~\\cite{commoner,bes87} are an extension of free-choice nets\nwith a broader applicability where the important theorems still hold at least \npartially~\\cite{de95,jcl04} and their properties have also been investigated with\nrespect to complexity issues~\\cite{lj15}. \\ACC nets allow confusion to happen, \nan asymmetric combination of choice and concurrency. The postsets of any two places \nmay be disjoint or identical (like for free-choice) or one postset may\nbe contained in the other. Extensions have been defined~\\cite{akd98}, but in this paper we will\nbe more interested in restrictions, i.e.\\ classes between free-choice and\nasymmetric choice, in the hope of finding some class where synthesis\nis possible.\n\nCompared to \\ACC,\nreduced asymmetric choice (\\RAC)~\\cite{commoner} limits the number of transitions in the postsets of two places\nthat are properly contained in one another: one postset must be a singleton, the other contains exactly two transitions.\nFig.~\\ref{f.gen1} shows an example where the postset of $p_1$ is contained in that of $p_2$\n(written as $W(p_1,\\cdot)=1a\\le 1a+1b=W(p_2,\\cdot)$ or $p_1{}^\\bullet=\\{a\\}\\subseteq\\{a,b\\}=p_2{}^\\bullet$). A similar relation\nholds for $p_3$ and $p_4$ while $p_5$ is independent and presents a free choice between $e$ and $f$.\nWe will also look at classes with arc weights. Figure~\\ref{f.rac} shows some example nets from\nthese classes. In this paper, we will synthesise \\lts targetting at some superclasses of reduced asymmetric\nchoice nets, but not the full class of asymmetric choice nets. As shown in~\\cite{bes87},\nthere are strong relations between the classes of asymmetric choice and reduced asymmetric choice\nnet via a marking simulation, but in this paper we rather look at \\lts and thus labelled firing sequences.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,0) node[transition] (t1) {$t_1$};\n\\draw(1,0) node[transition] (t2) {$t_2$};\n\\draw(2,0) node[transition] (t3) {$t_3$};\n\\draw(0,2) node[place,label=above:$p_1$] (p1) {};\n\\draw(1,2) node[place,label=above:$p_2$] (p2) {};\n\\draw(2,2) node[place,label=above:$p_3$] (p3) {};\n\\draw(p1) edge[-latex,thick] node[auto,swap]{$2$} (t1);\n\\draw(p2) edge[-latex,thick] node[auto,swap]{$2$} (t2);\n\\draw(p2) edge[-latex,thick] node[auto,swap,near end,xshift=-1mm,yshift=1mm]{$2$} (t3);\n\\draw(p3) edge[-latex,thick] node[auto,swap,near start,xshift=-1mm]{$3$} (t2);\n\\draw(p3) edge[-latex,thick] node[auto]{$3$} (t3);\n\\end{tikzpicture}\\hspace*{1cm}\n\\begin{tikzpicture}\n\\draw(0,0) node[transition] (t1) {$t_1$};\n\\draw(1,0) node[transition] (t2) {$t_2$};\n\\draw(2,0) node[transition] (t3) {$t_3$};\n\\draw(0,2) node[place,label=above:$p_1$] (p1) {};\n\\draw(1,2) node[place,label=above:$p_2$] (p2) {};\n\\draw(2,2) node[place,label=above:$p_3$] (p3) {};\n\\draw(p1) edge[-latex,thick] node[auto,swap]{$2$} (t1);\n\\draw(p2) edge[-latex,thick] node[auto,swap,xshift=1mm]{$2$} (t1);\n\\draw(p2) edge[-latex,thick] (t2);\n\\draw(p3) edge[-latex,thick] node[auto,near end,xshift=-2mm]{$2$} (t1);\n\\draw(p3) edge[-latex,thick] node[auto,xshift=-1mm]{$2$} (t2);\n\\draw(p3) edge[-latex,thick] node[auto]{$3$} (t3);\n\\end{tikzpicture}\\hspace*{1cm}\n\\begin{tikzpicture}\n\\draw(0,0) node[transition] (t1) {$t_1$};\n\\draw(1,0) node[transition] (t2) {$t_2$};\n\\draw(2,0) node[transition] (t3) {$t_3$};\n\\draw(0,2) node[place,label=above:$p_1$] (p1) {};\n\\draw(1,2) node[place,label=above:$p_2$] (p2) {};\n\\draw(2,2) node[place,label=above:$p_3$] (p3) {};\n\\draw(p1) edge[-latex,thick] node[auto,swap]{$2$} (t1);\n\\draw(p2) edge[-latex,thick] node[auto,swap]{$4$} (t2);\n\\draw(p2) edge[-latex,thick] node[auto,xshift=-1mm]{$3$} (t3);\n\\draw(p3) edge[-latex,thick] node[auto]{$2$} (t3);\n\\end{tikzpicture}\n\\caption{Left: an equal-conflict net, $t_2$ and $t_3$ are in conflict and \nboth are concurrent to $t_1$. Middle: a weighted asymmetric choice net where the three places have increasing\npostsets with $W(p_1,\\cdot)\\lneqq W(p_2,\\cdot)\\lneqq W(p_3,\\cdot)$. Right: a weighted reduced asymmetric choice net, only structures forming an `N' are allowed\nfor asymmetric choices, $W(p_3,\\cdot)=2t_3\\lneqq 4t_2+3t_3=W(p_2,\\cdot)$\n\\label{f.rac}}\n\\end{figure}\n\nIn the next section, we will introduce the basic concepts around labelled transition systems and\nPetri nets as well as a short description of synthesis and how separation problems are defined.\nSection~\\ref{sect.3} draws conclusions from an \\lts for a synthesised net when we target one of two specific\nclasses. The first class, {\\em block-reduced asymmetric choice} (\\BRAC), lies between asymmetric choice\nand reduced asymmetric choice. The other class (\\WPI) is a superclass of \\BRAC which lies askew to\nasymmetric choice nets. In \\WPI (weighted comparable presets) the presets of transitions \nare either disjoint or comparable and we allow arbitrary arc weights.\nOne important conclusion is that self-loops (edges with the same source and target node) in the \\lts make synthesis intrinsically\ndifficult. In Section~\\ref{sect.4} we investigate the problems that stem from self-loops for the class \\WPI,\nand in Section~\\ref{sect.5} we solve these problems for the class \\BRAC.\nFinally, we give a summary and an outlook in Section~\\ref{sect.O}.\n\n\\section{Basic concepts}\n\\label{basic-not.sct}\n\n\\DEF{d.lts}{LTS}\nA {\\it labelled transition system} (\\lts) with initial state is a tuple $TS=(S,\\Sigma,\\to,s_0)$\nwith nodes $S$ (a countable set of states),\nedge labels $\\Sigma$ (a finite set of letters),\nedges $\\mathord{\\to}\\subseteq(S\\times \\Sigma\\times S)$,\nand an initial state $s_0\\in S$. An edge $(s,t,s')\\in\\mathord{\\to}$ may be written as\n$s\\step{t}s'$.  \nA {\\it walk} $\\sigma\\in \\Sigma^*$ from $s$ to $s'$, written as\n$s\\step{\\sigma}s'$,\nis given inductively by $s=s'$ for the empty word $\\sigma=\\varepsilon$ and by\n$\\exists s''\\in S$: $s\\step{w}s''\\step{t}s'$ for $\\sigma=wt$\nwith $w\\in \\Sigma^*$ and $t\\in \\Sigma$. A walk $s\\step{\\sigma}s'$ is a {\\it cycle}\nif and only if $s=s'$, we call it a {\\it self-loop} if $\\sigma\\in\\Sigma$.\nThe set $\\step{s}$ for $s\\in S$ is the set of all states reachable\nfrom $s$, $\\step{s} = \\{s'\\,|\\,\\exists \\sigma\\in \\Sigma^*:\\;s\\step{\\sigma}s'\\}$.\nThe {\\it Parikh vector} $\\parikh(\\sigma)\\colon\\Sigma\\to\\Z$ of a word $\\sigma\\in \\Sigma^*$ maps each letter $t\\in \\Sigma$\nto its number of occurrences in $\\sigma$, it will often be written as an element of the\ngroup spanned by $\\Sigma$. The neutral element is written as \\zero, comparisons are done componentwise\nwith $\\mathord{\\lneqq}$ meaning ``less or equal in all components, but not entirely equal''.\nWe map to $\\Z$ here instead of $\\N$ to be able to extend the notion of a Parikh vector later and\nto handle differences of Parikh vectors more easily.\n\nA {\\it spanning tree} $E$ of $TS$ is a set of edges $E\\subseteq\\mathord{\\to}$ \nsuch that for every $s\\in S$ there is a unique walk from $s_0$ to $s$ using edges in $E$ only.\nThis implies that $E$ is cycle-free.\nA {\\it walk in $E$} is a walk that uses edges in $E$ only (and not any of $\\mathord{\\to}\\backslash E$).\nEdges in $\\mathord{\\to}\\backslash E$ are called {\\it chords}.\nThe {\\it Parikh vector of a state} $s$ in a spanning tree $E$ is $\\parikh_E(s) = \\parikh(\\sigma)$\nwhere $s_0\\step{\\sigma}s$ is the unique walk in $E$. The {\\it Parikh vector of an edge}\n$s\\step{t}s'$ in $TS$ is $\\parikh_E(s\\step{t}s') = \\parikh_E(s)+1t-\\parikh_E(s')$, see\nFig.~\\ref{f.gen1} for an example. \nNote that Parikh vectors of edges in $E$ always evaluate to zero; for chords the Parikh vector may even \ncontain negative values. For a chord $s\\step{t}s'$, $s$ and $s'$ have a latest common predecessor $r$ \nin $E$, $t',t''\\in T$ with $t'\\not=t''$, $\\sigma,\\sigma'\\in T^*$ with two walks\n$r\\step{t'\\sigma}s\\step{t}s'$ and $r\\step{t''\\sigma'}s'$ in $E$. These two walks form a {\\it cycle in the\nLTS' underlying undirected graph}. If we follow the cycle in the direction of the chord\nand sum up the edges, we obtain the chord's Parikh vector $\\parikh_E(s)+1t-\\parikh_E(s')$. \nThe {\\it Parikh vector of a walk} $s_1\\step{t_1}s_2\\ldots s_n\\step{t_n}s_{n+1}$ is defined as \n$\\parikh_E(s_1\\step{t_1\\ldots t_n}s_{n+1}) = \\sum_{i=1}^n\\parikh_E(s_i\\step{t_i}s_{i+1})$.\nObviously, $\\parikh_E(s_1\\step{t_1\\ldots t_n}s_{n+1}) = \\parikh_E(s_1)+\\parikh(t_1\\ldots t_n)-\\parikh_E(s_{n+1})$.\nIf the walk is a cycle (with $s_1=s_{n+1}$), we thus find \n$\\parikh(t_1\\ldots t_n)=\\sum_{i=1}^n\\parikh_E(s_i\\step{t_i}s_{i+1})$ where all non-zero\nParikh vectors in the sum stem from chords. \nThe set $\\{\\parikh_E(s\\step{t}s')\\,|\\,(s,t,s') \\in \\mathord{\\to}\\backslash E\\}$ is then a generator\nfor all Parikh vectors of cycles (the latter being linear combinations of its elements).\nBy simple linear algebra, \nwe can compute a basis from this generator. This {\\it cycle base} $\\Gamma$ contains\nat most $|\\Gamma|\\le|\\Sigma|$ different Parikh vectors.\nThere will be no need to distinguish between cycles in the \\lts and in its underlying undirected\ngraph at all.\n\nAn \\lts $TS=(S,\\Sigma,\\to,s_0)$ is {\\it finite} if $S$ is finite and it is {\\it deterministic} \nif $s\\step{t}s'$ and $s\\step{t}s''$ implies $s'=s''$ for all $s\\in S$ and $t\\in \\Sigma$. We call\n$TS$ {\\it reachable} if for every state $s\\in S$ exists some $\\sigma\\in \\Sigma^*$ with $s_0\\step{\\sigma}s$.\nReachability implies the existence of a spanning tree.\nTwo labelled transition systems $TS_1=(S_1,\\Sigma_1,\\to_1,s_{01})$ and $TS_2=(S_2,\\Sigma_2$, $\\to_2,s_{02})$ are\n{\\it isomorphic} if $\\Sigma_1=\\Sigma_2$ and there is a bijection $\\zeta\\colon S_1\\to S_2$ with $\\zeta(s_{01})=s_{02}$ and\n$(s,t,s')\\in\\;\\to_1\\,\\Leftrightarrow(\\zeta(s),t,\\zeta(s'))\\in\\;\\to_2$, for all $s,s'\\in S_1$.\n\\ENDDEF{d.lts}\n\nAn example for a spanning tree $E$ and some Parikh vectors of states and chords is shown in Fig.\\ref{f.gen1}.\n\n\\DEF{d.pn}{Petri nets} \nAn {\\em (initially marked) Petri net} is denoted as $N=(P,T,W,M_0)$ where $P$ is a finite set of places,\n$T$ is a finite set of transitions,\n$W$ is the weight function $W\\colon((P\\times T)\\cup(T\\times P))\\to\\N$ \nspecifying the arc weights,\nand $M_0$ is the initial marking\n(where a marking is a mapping $M\\colon P\\to\\N$, indicating the number of tokens in each place).\nThe {\\em preset} of a place or transition $x\\in P\\cup T$ is defined as ${}^\\bullet x = \\{y\\mid W(y,x)>0\\}$\nand its {\\em postset} is $x^\\bullet = \\{y\\mid W(x,y)>0\\}$. We canonically extend this notion to\n${}^\\bullet X$ and $X^\\bullet$ for sets $X\\subseteq P\\cup T$.\nA transition $t\\in T$ is {\\it enabled} at a marking $M$,\ndenoted by $M\\step{t}$, if $\\forall p\\in P\\colon M(p)\\geq W(p,t)$.\nThe {\\it firing} of $t$ leads from $M$ to $M'$, denoted by $M\\step{t}M'$,\nif $M\\step{t}$ and $M'(p)=M(p)-W(p,t)+W(t,p)$.\nThis can be extended, by induction as usual, to $M\\step{\\sigma}M'$ for words $\\sigma\\in T^*$,\nand $\\step{M} = \\{M'\\,|\\,\\exists\\sigma\\in T^*\\colon M\\step{\\sigma}M'\\}$\ndenotes the set of markings reachable from $M$.\nThe {\\it reachability graph} $RG(N)$ of a Petri net $N$\nis the labelled transition system with the set of nodes $\\step{M_0}$,\ninitial state $M_0$, label set $T$,\nand set of edges $\\{(M,t,M')\\mid M,M'\\in\\step{M_0}\\land M\\step{t}M'\\}$.\nIf a labelled transition system $TS$ is isomorphic to the reachability graph $RG(N)$ of \na Petri net $N$ we say that $N$ {\\em PN-solves} (or simply {\\em solves}) $TS$,\nand that $TS$ is {\\em synthesisable} to $N$. \n\nA Petri net $N$ is {\\it bounded} if there is some $k\\in\\N$ such that $\\forall p\\in P$ $\\forall M\\in\\step{M_0}$: $M(p)\\le k$,\nwhich is equivalent to its reachability graph being finite.\nWe define some subclasses of bounded Petri nets by certain properties.\nA Petri net $(P,T,W,M_0)$ is\n\\begin{itemize}\n\\item {\\em plain} if $W\\colon (P\\times T)\\cup(T\\times P)\\to \\{0,1\\}$.\n\\item {\\em choice-free} (\\CF) if $\\forall p\\in P, t,t'\\in T\\colon W(p,t)>0 \\wedge t\\neq t' \\then W(p,t')=0$.\n\\item {\\em equal-conflict} (\\EC) if $\\forall t,t'\\in T\\colon {}^\\bullet t\\cap {}^\\bullet t'\\neq\\emptyset \\then W(\\cdot,t)=W(\\cdot,t')$.\\footnote{$W(\\cdot,t)\\colon P\\to\\N$ is the submapping of $W$ with fixed second parameter $t$} \n\\item {\\em (extended) free-choice} (\\EFC) if it is \\EC and plain.\n\\item {\\em weighted comparable preset} (\\WPI) if $\\forall t,t'\\in T\\colon {}^\\bullet t\\cap {}^\\bullet t'\\neq\\emptyset \\then W(\\cdot,t)\\geq W(\\cdot,t') \\vee W(\\cdot,t)\\leq W(\\cdot,t')$.\\footnote{Comparisons of mappings are done componentwise, cf.\\ Parikh vectors}\n\\item {\\em weighted asymmetric choice} (\\WAC) if $\\forall p,p'\\in P\\colon p^\\bullet\\cap p'{}^\\bullet\\neq\\emptyset \\then W(p,\\cdot)\\geq W(p',\\cdot) \\vee W(p,\\cdot)\\leq W(p',\\cdot)$.\n\\item {\\em asymmetric choice} (\\ACC) if it is \\WAC and plain.\n\\item {\\em reduced asymmetric choice} (\\RAC) if it is plain and $\\forall p,p'\\in P\\colon p^\\bullet\\cap p'{}^\\bullet\\neq\\emptyset \\then (|p^\\bullet|=1\\wedge|p'{}^\\bullet|\\le 2\\wedge {}^\\bullet(p'{}^\\bullet)=\\{p,p'\\})\\vee(|p'{}^\\bullet|=1\\wedge|p{}^\\bullet|\\le 2\\wedge {}^\\bullet(p{}^\\bullet)=\\{p,p'\\})$.\n\\item {\\em block-reduced asymmetric choice} (\\BRAC) if it is plain and $\\forall p,p'\\in P\\colon p^\\bullet\\cap p'{}^\\bullet\\neq\\emptyset \\then (p^\\bullet=p'{}^\\bullet \\vee \\exists T_1,T_2\\subseteq T\\colon (p^\\bullet=T_1\\wedge p'{}^\\bullet=T_1\\cup T_2\\wedge {}^\\bullet T_1=\\{p,p'\\}\\wedge {}^\\bullet T_2=\\{p'\\}) \\vee (p'{}^\\bullet=T_1\\wedge p^\\bullet=T_1\\cup T_2\\wedge {}^\\bullet T_1=\\{p,p'\\}\\wedge {}^\\bullet T_2=\\{p\\}))$.\n\\end{itemize}\nAll property names are also used as class names, e.g.\\ AC is the class of all bounded asymmetric choice Petri nets.\n\\ENDDEF{d.pn}\n\nNote that the free-choice property \\EFC can be alternatively defined via\n$\\forall p,p'\\in P\\colon p^\\bullet \\cap p'{}^\\bullet \\neq\\emptyset \\then W(p,\\cdot)=W(p',\\cdot)$\n(plus plainness), but for \\EC a place-based definition is not possible.\nIn \\EC, all transitions consuming tokens from a place must take the same amount,\nwhile the place-based alternative definition would demand that a transition must\ntake the same number of tokens from each place in its preset.\nAs a consequence, the classes \\WPI and \\WAC are not identical and have \\EFC in \ntheir intersection, but not \\EC. The right net in Fig.~\\ref{f.rac} is in \\WAC\nbut not in \\WPI as $W(p_2,t_3)\\lneqq W(p_2,t_2)$. If we swap arc weights, setting\n$W(p_2,t_2)=2$ and $W(p_3,t_3)=4$, the net is in \\WPI but not in \\WAC.\n\nThe definition of \\RAC stems from~\\cite{commoner} and can also be found in~\\cite{bes87}. A transition that does not share its preset with another\ntransition may have an arbitrary preset (with $|p^\\bullet|=1$ and $|p'{}^\\bullet|=1$).\nOther transitions (in the postset of $p$\/$p'$ with $|p^\\bullet|=2$ or $|p'{}^\\bullet|=2$) may share\ntheir preset with at most one other transition. One of $p$\/$p'$ will be in the preset of only one\nof these transitions, the other in both, forming the known `N'-structure in the Petri net. For transitions\nwith identical presets, this preset may only contain a single place.\n\n\\BRAC extends this (see Fig.~\\ref{f.brac}) by allowing {\\em blocks of transitions} with the same preset taking the place of\nsingle transitions in \\RAC. An {\\em asymmetric choice block} consists of two separate blocks of transitions, here \n$T_1=\\{t_6,t_7,t_8\\}$ and $T_2=\\{t_4,t_5\\}$, \nwhere all transitions in $T_2$ have the same (single) place ($p_4$) in their presets. \nThe presets in the other block ($T_1$) contain the same additional place ($p_5$).\nIf we have two places (here $p_4$, $p_5$) with non-identical postsets but sharing a transition,\nthe condition ${}^\\bullet T_1=\\{p,p'\\}$ demands that only those two places can be involved. Once\na third place comes into play (as with $\\{p_1,p_2,p_3\\})$ all places must have identical postsets,\nforming a {\\em free-choice block}.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,0) node[transition] (t1) {$t_1$};\n\\draw(1,0) node[transition] (t2) {$t_2$};\n\\draw(2,0) node[transition] (t3) {$t_3$};\n\\draw(4,0) node[transition] (t4) {$t_4$};\n\\draw(5,0) node[transition] (t5) {$t_5$};\n\\draw(6,0) node[transition] (t6) {$t_6$};\n\\draw(7,0) node[transition] (t7) {$t_7$};\n\\draw(8,0) node[transition] (t8) {$t_8$};\n\\draw(0,2) node[place,label=above:$p_1$] (p1) {};\n\\draw(1,2) node[place,label=above:$p_2$] (p2) {};\n\\draw(2,2) node[place,label=above:$p_3$] (p3) {};\n\\draw(5,2) node[place,label=above:$p_4$] (p4) {};\n\\draw(7,2) node[place,label=above:$p_5$] (p5) {};\n\\draw(p1) edge[-latex,thick] (t1);\n\\draw(p1) edge[-latex,thick] (t2);\n\\draw(p1) edge[-latex,thick] (t3);\n\\draw(p2) edge[-latex,thick] (t1);\n\\draw(p2) edge[-latex,thick] (t2);\n\\draw(p2) edge[-latex,thick] (t3);\n\\draw(p3) edge[-latex,thick] (t1);\n\\draw(p3) edge[-latex,thick] (t2);\n\\draw(p3) edge[-latex,thick] (t3);\n\\draw(p4) edge[-latex,thick] (t4);\n\\draw(p4) edge[-latex,thick] (t5);\n\\draw(p4) edge[-latex,thick] (t6);\n\\draw(p4) edge[-latex,thick] (t7);\n\\draw(p4) edge[-latex,thick] (t8);\n\\draw(p5) edge[-latex,thick] (t6);\n\\draw(p5) edge[-latex,thick] (t7);\n\\draw(p5) edge[-latex,thick] (t8);\n\\end{tikzpicture}\n\\caption{The two kinds of preset structures allowed in block-reduced asymmetric choice nets:\neither extended free-choice with arbitrarily large presets (left) or two blocks of transitions\nforming asymmetric choices where all transitions in one block have the same preset of either one\nor two places (right)\n\\label{f.brac}}\n\\end{figure}\n\n\\KOR{c.pn}{Class Inclusions}\nWe get some simple inclusions, also shown in Fig.~\\ref{f.classes}:\\pagebreak\n\n\\begin{itemize}\n\\item $\\EFC \\subseteq \\RAC \\subseteq \\BRAC \\subseteq \\ACC \\subseteq \\WAC$,\n\\item $\\CF \\subseteq \\EC \\subseteq \\WAC$,\n\\item $\\EFC \\subseteq \\EC$, and\n\\item $\\BRAC \\subseteq \\WPI$.\n\\end{itemize}\n\\ENDKOR{c.pn}\n\nMost of the inclusions are quite obvious, for $\\RAC\\subseteq\\BRAC$ the condition\n$|p'{}^\\bullet|\\le 2$ is split such that the case $|p'{}^\\bullet|=1$ implies $p^\\bullet=p'{}^\\bullet$ \nwhile $|p'{}^\\bullet|=2$ translates to $p'{}^\\bullet=T_1\\cup T_2$ in the \\BRAC definition,\nwith $T_1$ and $T_2$ being singleton sets.\nFor $\\BRAC\\subseteq\\WPI$, a place $p\\in {}^\\bullet t\\cap {}^\\bullet t'$ is either the only place in these\npresets or the \\BRAC-condition holds. In both cases, the presets are contained in one another.\n\nWe will mainly be interested in \\WPI and \\BRAC, but the results we show also hold for classes $C$\nwith $\\BRAC\\subseteq C\\subseteq \\WPI\\cap\\WAC$, which includes classes with arc weights. Essentially,\nwe can have arbitrary arc weights in the postsets of transitions and for presets of transitions\nthat fall under the \\EC-condition. Whenever a transition takes part in an asymmetric choice,\nthe arc weights in its presets are mostly limited to one.\n\n\\DEF{d.synth}{Synthesis~\\cite{bbd15}}\nA {\\it region} $r=(R,B,F)$ of an LTS $(S,\\Sigma,\\to,s_0)$ consists of three functions\n$R$: $S\\to\\N$, $B$: $\\Sigma\\to\\N$, and $F$: $\\Sigma\\to\\N$ such that for all edges $s\\step{t}s'$\nin the LTS we have $R(s)\\ge B(t)$ and $R(s')=R(s)-B(t)+F(t)$. This mimics the firing\nrule of Petri nets and makes regions essentially equivalent to places, i.e.\\ a place\n$p$ can be defined from $r$ via $M_0(p)=R(s_0)$, $W(p,t)=B(t)$, and $W(t,p)=F(t)$ for\nall $t\\in \\Sigma$. When a Petri net is constructed from a set of regions\nof a reachable LTS,\nthis implies\na {\\it uniquely defined marking} $\\M(s)$ for each state $s$ with \n$\\M(s)(p)=M_0(p)+\\sum_{t\\in \\Sigma}\\parikh(\\sigma)(t)\\cdot(W(t,p)-W(p,t))$\nfor an arbitrary walk $s_0\\step{\\sigma}s$.\n\nThe construction of a Petri net $N=(P,T,W,M_0)$ with one place in $P$\nfor each region of the LTS guarantees $s\\step{t} \\then \\M(s)\\step{t}$,\nbut has three issues: (1) $P$ might become infinite, (2) $\\M$ may not be injective,\nand (3) $\\M(s)\\step{t} \\then s\\step{t}$ need not hold. Failing (2) means\nthat there are states $s,s'\\in S$ with $s\\not=s'$ that are identified in $RG(N)$ (leading to non-isomorphism).\nA {\\it state separation problem} (SSP) is a pair $(s,s')\\in S\\times S$ with $s\\not=s'$.\nA region $r$ solves an SSP $(s,s')$ if $R(s)\\not=R(s')$ (and thus $\\M(s)\\not=\\M(s')$).\nAn example of an SSP in Fig.~\\ref{f.genx} is $(s_3,s_7)$, for which no solving region exists\ndue to the fact that $R(s_3)=R(s_4)-B(b)+F(b)-B(a)+F(a)=R(s_4)-B(b)+F(b)-B(a)+F(a)=R(s_7)$.\nFailing (3) results in an edge $\\M(s)\\step{t}$ in $RG(N)$ but not in the LTS,\n$\\neg s\\step{t}$.\nAn {\\it event\/state separation problem} (ESSP) is a pair $(s,t)\\in S\\times \\Sigma$\nwith $\\neg s\\step{t}$. A region $r$ solves an ESSP $(s,t)$ if $R(s)<B(t)$\n(and thus $\\neg\\M(s)\\step{t}$). \nAn example is the ESSP $(s_2,b)$ in Fig.~\\ref{f.genx}, which is unsolvable due to\nthe fact that $R(s_0)\\ge B(b)$ (as $s_0\\step{b}$), $R(s_2)=R(s_0)-B(a)+F(a)<B(b)$ (as $\\neg s_2\\step{b}$),\nand $R(s_4)=R(s_2)-B(a)+F(a)\\ge B(b)$ (as $s_4\\step{b}$). Then, $R(s_0)\\le R(s_2)\\le R(s_4)$ or\n$R(s_0)\\ge R(s_2)\\ge R(s_4)$, contradicting one of the comparisons to $B(b)$.\nThe set of all {\\it separation problems}, $\\{(s,s')\\in S\\times S\\,|\\,s\\not=s'\\}\\cup\\{(s,t)\\in S\\times \\Sigma \\mid \\neg s\\step{t}\\}$, \nis finite for finite LTS, and finding\na solution for every separation problem solves all three issues, making $RG(N)$\nisomorphic to the finite LTS, thus making the latter synthesisable.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[LTS]\n\\node[state,label=below:$s_0$] (s0) {};\n\\path[edge] (s0) ++(-0.5,0) edge (s0);\n\\node[state,above=of s0,label=above:$s_1$] (s1) {};\n\\foreach \\x\/\\y in {0\/2,2\/4,4\/6} {\n \\node[state,right=of s\\x,label=below:$s_{\\y}$] (s\\y) {}; }\n\\foreach \\x\/\\y in {1\/3,3\/5,5\/7} {\n \\node[state,right=of s\\x,label=above:$s_{\\y}$] (s\\y) {}; }\n\\draw[edge] (s0) edge node[auto,swap]{$b$} (s1);\n\\draw[edge] (s0) edge node[auto]{$a$} (s2);\n\\draw[edge] (s2) edge node[auto]{$a$} (s4);\n\\draw[edge] (s4) edge node[auto]{$a$} (s6);\n\\draw[edge] (s4) edge node[auto]{$b$} (s5);\n\\draw[edge] (s5) edge node[auto]{$a$} (s3);\n\\draw[edge] (s3) edge node[auto]{$b$} (s2);\n\\draw[edge] (s6) edge node[auto]{$b$} (s7);\n\\end{tikzpicture}\n\\caption{An LTS with unsolvable SSP $(s_3,s_7)$ and ESSP $(s_2,b)$}\n\\label{f.genx}\n\\end{figure}\n\nAn SSP $(s,s')$ can be written as a linear inequality system~\\cite{bds16}\nwith $\\forall\\gamma\\in\\Gamma$: $(F-B)\\tp\\cdot\\gamma = 0$ (all cycles must have\neffect zero in a region), $(F-B)\\tp\\cdot(\\parikh_E(s)-\\parikh_E(s')) \\neq 0$\n($\\M(s)\\neq\\M(s')$, compared via $R(s_0)$), \nand $\\forall s''\\step{t}s'''$: $R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s'')\\ge B(t)$\n(definition of a region, but without checking cycles).\nHere $F\\ge\\zero$, $B\\ge\\zero$ and $R(s_0)\\ge 0$ are the variables, \na solution forming a region solving the SSP. Note that if the\nfirst two formulas can be solved, $R(s_0)$ can always be selected \nhigh enough such that the third one is also fulfilled. \n\nAn ESSP $(s,t)$ can be written as a linear inequality system~\\cite{sw18} with\n$\\forall\\gamma\\in\\Gamma$: $(F-B)\\tp\\cdot\\gamma = 0$,\n$\\forall s'\\step{t'}s''$: $R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s')\\ge B(t')$,\nand $R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s)<B(t)$ ($\\neg \\M(s)\\step{t}$).\nWith variables $F\\ge\\zero$, $B\\ge\\zero$, and $R(s_0)\\ge 0$ a solution\nforms a region solving the ESSP. \n\\ENDDEF{d.synth}\n\nIf the LTS is finite, the (also finite)\nlinear inequality systems can be solved by standard means, e.g.\\ employing \nan ILP- or SMT-solver~\\cite{smtinterpol}. If an LTS is not reachable or not deterministic, it cannot be structurally isomorphic\nto a reachability graph, i.e.\\ no Petri net solving it can be found.\nFor these reasons, we will assume all LTS to\nbe finite, reachable, and deterministic in the remainder of this paper.\n\nFig.~\\ref{f.gen1} shows an \\lts (left side) and a \\RAC Petri net (right side) with a reachability graph\nisomorphic to the \\lts. All separation problems are solvable. The SSP $(s_4,s_5)$ e.g.\\\nis solved by the region $r=(R,B,F)$ with $R(s_0)=0$, $F(a)=1=B(d)$, and all other entries being zero.\nWith $s_0\\step{ac}s_4$ and $s_0\\step{ade}s_4$ we find \n$R(s_4)=R(s_0)-B(a)+F(a)-B(c)+F(c)=1\\neq 0=R(s_0)-B(a)+F(a)-B(d)+F(d)-B(e)+F(e)=R(s_5)$.\nThe region $r$ also fulfills the cycle conditions $F(b)-B(b)+F(c)-B(c)=0$ and\n$F(a)-B(a)+F(d)-B(d)+F(f)-B(f)=0$ since $bc$ and $adf$ are cycles in the \\lts.\nIt corresponds to the place $p_3$.\n\nFor the ESSP $(s_{13},a)$ with $s_0\\step{adeade}s_{13}$ and $\\neg s_{13}\\step{a}$ the region\n$r=(R,B,F)$ with $R(s_0)=2$, $B(a)=1=F(f)$, and all other entries zero is a solution.\nWe obtain inequalities\n$R(s_{13})=R(s_0)-2B(a)+2F(a)-2B(d)+2F(d)-2B(e)+2F(e)=2-2=0<1=B(a)$ forbidding $a$ at $s_{13}$\nand allowing it at $s_0$, $s_4$, and $s_5$ by $R(s_0)=2\\ge 1=B(a)$, $R(s_4)=R(s_0)-B(a)+F(a)-B(c)+F(c)=2-1\\ge 1=B(a)$,\nand $R(s_5)=R(s_0)-B(a)+F(a)-B(d)+F(d)-B(e)+F(e)=2-1\\ge 1=B(a)$.\nThe cycle conditions $F(a)-B(a)+F(d)-B(d)+F(f)-B(f)=0$ and $F(b)-B(b)+F(c)-B(c)=0$ are also fulfilled.\nThis region $r$ corresponds to the place $p_1$, which prohibits $a$ at $s_{13}$ but not at the three\nstates where it must occur.\n\n\\section{Synthesis by implication and deactivation properties}\\label{sect.3}\n\\newcommand\\IP[3]{\\ensuremath{\\mathrm{I#1}_{#2,#3}}\\xspace}\n\\newcommand\\DP[2]{\\ensuremath{\\mathrm{D}_{#1,#2}}\\xspace}\n\\newcommand\\IPS[1]{\\ensuremath{\\mathrm{I#1}}\\xspace}\n\\newcommand\\DPS{\\ensuremath{\\mathrm{D}}\\xspace}\n\nIn this section we will show that membership of a net in \\WPI or \\BRAC\nenforces some structural properties in its reachability graph. For pairs of\nlabels, these properties determine whether the corresponding transitions\nin the net must have identical, properly included, or disjoint presets.\n\n\\DeclareRobustCommand\\rhddisj{\\mathrel{\\scriptstyle\\|\\hspace*{0.4mm}}\\joinrel\\mathrel\\rhd}\n\\DEF{d.impl}{Implication Properties}\nFor any finite, reachable, deterministic \\lts $(S,\\Sigma,\\to,s_0)$ define the following relations\nover $\\Sigma$. For two labels $a,b\\in\\Sigma$, let:\n\\begin{itemize}\n\\item $a\\equiv b$ $\\iff$ $\\forall s\\in S\\colon s\\step{a} \\iff s\\step{b}$,\n\\item $a\\rhd b$ $\\iff$ $(\\forall s\\in S\\colon s\\step{a} \\then s\\step{b} \\wedge \\exists s\\in S\\colon s\\step{b} \\wedge \\neg s\\step{a})$,\n\\item $a\\bowtie b$ $\\iff$ ($a\\rhd b \\vee b\\rhd a$), and\n\\item $a\\interleave b$ $\\iff$ ($\\exists s\\in S\\colon s\\step{a}\\wedge\\neg s\\step{b} \\wedge \\exists s\\in S\\colon s\\step{b}\\wedge\\neg s\\step{a}$).\n\\end{itemize}\n\\ENDDEF{d.impl}\n\nTake a look at the \\lts on the left side of Fig.~\\ref{f.gen1}. The labels $e$ and $f$ both occur at\n$s_2$, $s_{10}$, and $s_{12}$ but nowhere else, thus we get $e\\equiv f$. At $s_0$, $s_4$, and $s_5$\nboth $a$ and $b$ are allowed, but $b$ is also present at $s_9$, $s_{11}$, and $s_{13}$. We conclude\n$a\\rhd b$ and thus $a\\bowtie b$. Similarly, $c$ and $d$ are possible at $s_1$, $s_6$, and $s_7$, and further $c$'s occur\nat $s_3$, $s_8$, and $s_{14}$, so $d\\rhd c$ and $c\\bowtie d$ hold. All other pairs of labels are related via $\\interleave$.\nLooking at the synthesised Petri net in the figure, we find that $e\\equiv f$ suggests identical\npresets for $e$ and $f$, while for $a\\rhd b$ and $d\\rhd c$ we find proper inclusion of presets.\nPairs of labels in the $\\interleave$-relation have disjoint presets in the Petri net. \n\nThe three relations $\\equiv$, $\\bowtie$, and $\\interleave$ obviously partition the set of pairs of labels\nfor any \\lts.\n\n\\KOR{c.impl}{Implication Properties are mutually exclusive}\nFor any finite, reachable, deterministic LTS $(S,\\Sigma,\\to,s_0)$ and two\nlabels $a,b\\in\\Sigma$, exactly one of the three properties $a\\equiv b$,\n$a\\bowtie b$, and $a\\interleave b$ holds.\nAlso, $a\\rhd b$ and $b\\rhd a$ are mutually exclusive, i.e.\\ exactly one of them is true if $a\\bowtie b$.\n\\ENDKOR{c.impl}\n\nTo detect whether two transitions must have a common place in their preset, the following\nproperty is useful.\n\n\\DEF{d.dp}{Deactivation Property}\nFor any finite, reachable, deterministic LTS $(S,\\Sigma,\\to,s_0)$ and two\nlabels $a,b\\in\\Sigma$, we define the relation $\\merge$ by\n$a\\merge b$ $\\iff$ $\\exists s,s'\\in S\\colon (s\\step{a}\\wedge s\\step{b}s' \\wedge\\neg s'\\step{a})\n \\vee (s\\step{b}\\wedge s\\step{a}s' \\wedge\\neg s'\\step{b})$.\n\\ENDDEF{d.dp}\n\nAssume now a given \\lts $TS=(S,\\Sigma,\\to,s_0)$ which is finite, reachable, and deterministic\nas well as two labels $a,b\\in\\Sigma$.\nWe check each combination of one of $a\\equiv b$, $a\\bowtie b$, and $a\\interleave b$\nwith either $a\\merge b$ or $\\neg a\\merge b$ to find out how far this determines the structure\nof a possible synthesized net in the class \\WPI or \\BRAC. Overall, we have six cases.\n\n{\\bf Case 1:} $a\\interleave b \\wedge a\\merge b$.\\\\\nFrom $a\\merge b$ we conclude that the presets of transitions $a$ and $b$ in a \\WPI or \\BRAC net $N=(P,T,W,M_0)$\nthat solves $TS$ are not disjoint, as one can deactivate the other. This contradicts\n$a\\interleave b$, telling us that neither $W(\\cdot,a)\\le W(\\cdot,b)$ nor $W(\\cdot,b)\\le W(\\cdot,a)$\nhold, i.e.\\ the presets of $a$ and $b$ must be disjoint.\nSynthesis must fail in this case, there exists no \\WPI or \\BRAC net solving $TS$.\nNote that this case may occur in reachability graphs of \\ACC nets.\nThen, the presets of $a$ and $b$ would necessarily overlap, but not be contained in one another.\n\n{\\bf Case 2:} $a\\interleave b \\wedge \\neg a\\merge b$.\\\\\nLike argued in the previous case, $a\\interleave b$ dictates that transitions $a$ and $b$ must have disjoint presets\nin any \\WPI net solving $TS$.\n\n{\\bf Case 3:} $a\\equiv b \\wedge a\\merge b$.\\\\\nAs one of the labels can deactivate the other in the LTS, the transitions $a$ and $b$ in a\nPetri net $N=(P,T,W,M_0)$ solving $TS$ must have a common place $p$ with $W(p,a)>0$ and $W(p,b)>0$. So, the\npresets of $a$ and $b$ are either identical or there is a proper inclusion. Assume the preset\nof $b$ is properly included in that of $a$, i.e.\\ we have an asymmetric choice.\nWe construct a new Petri net $N'=(P,T,W',M_0)$.\nFor every place $q$ with $W(q,a)>W(q,b)$ we set $W'(q,b)=W(q,a)$ and $W'(b,q)=W(b,q)+W(q,a)-W(q,b)$.\nFor all other combinations of $x,y\\in P\\cup T$ let $W'(x,y)=W(x,y)$. \nIf $N$ is \\WPI, so is $N'$ since the transition $a$'s preset compares correctly to all other transitions\nand $b$ gets this same preset. If $N$ is \\BRAC, so is $N'$, as the asymmetric choice between $a$ and $b$\nis removed in $N'$ compared to $N$. In the definition of \\BRAC, this means $b$ moves from the block $T_2$\nof transitions to $T_1$, but the condition remains true.\nFiring any transition changes the marking\nin the same way in both nets, $N$ and $N'$, as $W(x,y)-W(y,x)=W'(x,y)-W'(y,x)$ for all $x,y\\in P\\cup T$.\nAll transitions except $b$ are either enabled both or disabled both in $N$ and $N'$ under\nthe same marking. Under any marking, the transition $b$ is enabled in $N$ if it is enabled in $N'$.\nNow assume $b$ is enabled in $N$ and for some state $s\\in S$ with $s\\step{b}$ \nand the marking $M(s)$ \ncorresponding to $s$ in $N$ we have $M(s)\\ge W(\\cdot,b)$.\nThen, due to $a\\equiv b$, we get $s\\step{a}$ and $M(s)\\ge W(\\cdot,a)=W'(\\cdot,b)$.\nThus, $b$ is enabled under the same markings in $N$ and $N'$ and the reachability graphs of $N$\nand $N'$ must be isomorphic. So, if $N$ solves $TS$, also $N'$ solves $TS$ with $W'(q,a)=W'(q,b)$.\nBy symmetry, the same argument holds for reversed roles of $a$ and $b$, so synthesis is always possible\nin such a way that $a$ and $b$ have an identical preset (considering arc weights).\n\n{\\bf Case 4:} $a\\equiv b \\wedge \\neg a\\merge b$.\\\\\nIf no state $s\\in S$ with $s\\step{a}$ and $s\\step{b}$ exists, $a$ and $b$ do not occur at all\nin $TS$ and can be eliminated from the alphabet $\\Sigma$. Now let $s\\in S$ be a state with\n$s\\step{a}s'$ and $s\\step{b}$. As $a$ does not deactivate $b$, $s'\\step{b}$ and thus also $s'\\step{a}$ holds.\nIf $a$ is activated at $s$, it is activated at its successor, and by induction at all states\nreachable from $s$ via $a$'s only. \nAs the \\lts is finite, there must be some $a$-cycle with\n$s\\step{a^m}s'\\step{a^n}s'$ for some $m,n\\in\\N$ with $n\\ge 1$. We conclude\n$M(s')=M(s')+n\\cdot(W(a,\\cdot)-W(\\cdot,a))$ by the firing rule. We obtain $W(\\cdot,a)=W(a,\\cdot)$\nand $a$ leaves the marking unchanged, i.e.\\ $M(s')\\step{a}M(s')$ as well as $M(s)\\step{a}M(s)$. \nAs $s$ is the unique state with marking $M(s)$, we find $s\\step{a}s$ in the LTS. \nTherefore, $a$ (and by symmetry also $b$) forms self-loops wherever in the \\lts it occurs.\n\nAssume now a \\WPI or \\BRAC net $N=(P,T,W,M_0)$ solving $TS$ such that the presets of $a$ and $b$\nare properly contained in one another or disjoint. We construct a new net\n$N'=(P,T,W',M_0)$ with $W'(\\cdot,a)=W'(a,\\cdot)=W(\\cdot,b)$ and for $t\\in T\\backslash\\{a\\}$ with\n$W'(\\cdot,t)=W(\\cdot,t)$ and $W'(t,\\cdot)=W(t,\\cdot)$. As $a$ now has the same preset as $b$,\n$N'$ is in \\WPI. If $N$ is in \\BRAC, $b$ may be involved in an asymmetric choice (i.e.\\ for some places, $b$\nappears in one of the blocks $T_1$ or $T_2$ of transitions as per definition of \\BRAC). Since $a$\ngets the same preset as $b$ in $N'$, it becomes a member in the same block as $b$, and the\ndefining conditions for \\BRAC also hold for $N'$.\\footnote{This does not work for synthesis \nof \\RAC-nets. If ${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$ in $N$, $b$ could already be involved\nin an asymmetric choice via some $c$ with ${}^\\bullet c\\subseteq{}^\\bullet b$. In $N'$, we would\nget ${}^\\bullet c\\subseteq{}^\\bullet a={}^\\bullet b$, violating the \\RAC condition.} \nIf $b$ is part of a free-choice block of\ntransitions (as in the left of Fig.~\\ref{f.brac}), $a$ becomes a member in this block, too,\nagain as it has now the same preset as $b$. In both cases, $N'$ is in \\BRAC.\nSince $a$ and $b$ occur at the\nsame reachable states $s$ and markings $M(s)$ of $N$ and $N'$, $a$ is still activated in $N'$\nexactly when $b$ is. So, if the \\lts $TS$ is synthesisable, we find a solution where $a$ and $b$\nhave identical presets.\n\n{\\bf Case 5:} $a\\bowtie b \\wedge a\\merge b$.\\\\\nAs one of $a$ and $b$ can deactivate the other due to $a\\merge b$, their presets in a \\WPI net\n$N=(P,T,W,M_0)$ solving $TS$ cannot be disjoint. As $a\\bowtie b$ proves the existence of a state\n$s\\in S$ with $s\\step{a}$ and $\\neg s\\step{b}$ (or the other way around), the presets of\n$a$ and $b$ cannot be identical. Therefore, $W(\\cdot,a)\\ge W(\\cdot,b)$ or $W(\\cdot,a)\\le W(\\cdot,b)$,\ndepending on which label occurs at $s$ (i.e.\\ whether $a\\rhd b$ or $b\\rhd a$ holds).\nIn both cases, we have proper containment between the presets and know in which direction.\n\n{\\bf Case 6:} $a\\bowtie b \\wedge \\neg a\\merge b$.\\\\\nW.l.o.g. assume $b\\rhd a$ ($a\\rhd b$ is handled symmetrically).\nIn a \\WPI\/\\BRAC net $N=(P,T,W,M_0)$ solving $TS$,\n$W(\\cdot,a)\\neq W(\\cdot,b)$. So, either the presets are disjoint or\n$W(\\cdot,a)\\lneqq W(\\cdot,b)$. Let $s\\in S$ be a state with\n$s\\step{a}$ and $s\\step{b}$ (otherwise $b$ would not occur in $TS$ at all).\nAs we have already seen in case~4, if $a$'s form a cycle in the \\lts, all $a$'s are automatically self-loops.\nIf $a$ is not a self-loop at $s$, there is a maximal $k\\in\\N$ with\n$s\\step{a^k}s'$ (due to finiteness of $TS$).\nIf $W(\\cdot,a)\\le W(\\cdot,b)$ held, we would find that with\n$\\neg s'\\step{a}$ also $\\neg s'\\step{b}$ was true, so somewhere\nin the walk $s\\step{a^k}s'$ the label $b$ got deactivated by $a$,\ncontradicting $\\neg a\\merge b$. Thus, when $a$ is not a self-loop,\nwe find that $a$ and $b$ have disjoint presets in $N$.\n\nIf $a$ forms a self-loop at $s$, we conclude $M(s)\\step{a}M(s)$\nin $N$ for the marking $M(s)$ corresponding to $s$. Therefore,\n$W(\\cdot,a)=W(a,\\cdot)$. The presets of $a$ and $b$ may be disjoint or $W(\\cdot,a)\\lneqq W(\\cdot,b)$.\nWith the help of Fig.~\\ref{f.case6}, we can show that both situations may occur and that we\nmay not be able to choose whether we want disjointness or proper containment.\n\n\\tikzset{loop\/.style={to path={.. controls +(315:1) and +(45:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[LTS]\n\\node[state,label=below:$s_0$] (s0) {};\n\\path[edge] (s0) ++(-0.5,0) edge (s0);\n\\path(s0) +(30:1cm) node[state,label=above:$s_1$] (s1) {};\n\\path(s0) +(-30:1cm) node[state,label=below:$s_2$] (s2) {};\n\\draw[edge] (s0) edge node[auto]{$a$} (s1);\n\\draw[edge] (s1) edge node[auto]{$b$} (s2);\n\\draw[edge] (s2) edge node[auto,xshift=1mm]{$d$} (s0);\n\\draw[edge] (s1) edge[loop] node[auto,swap]{$c$} (s1);\n\\draw[edge] (s2) edge[loop] node[auto,swap]{$c$} (s2);\n\\path(s0) to (0,-1.6);\n\\end{tikzpicture}\\hspace*{1.5cm}\n\\begin{tikzpicture}\n\\draw(0,0) node[transition] (b) {$b$};\n\\draw(0,0) +(30:1.5) node[place,label=above:$p_2$] (p2) {};\n\\draw(0,0) +(-30:1.5cm) node[place,label=below:$p_1$] (p1) {};\n\\draw(0,0) +(-30:3cm) node[transition] (a) {$a$};\n\\draw(0,0) +(30:3cm) node[transition] (d) {$d$};\n\\draw(0,1.5) +(-30:3cm) node[place,tokens=1,label=left:$p_0$] (p0) {};\n\\draw(1,1.5) +(-30:3cm) node[place,label=above right:$p_3$] (p3) {};\n\\draw(2.5,1.5) +(-30:3cm) node[transition] (c) {$c$};\n\\draw(p1) edge[-latex,thick] (b);\n\\draw(b) edge[-latex,thick] (p2);\n\\draw(p2) edge[-latex,thick] (d);\n\\draw(d) edge[-latex,thick] (p0);\n\\draw(p0) edge[-latex,thick] (a);\n\\draw(a) edge[-latex,thick] (p1);\n\\draw(a) edge[-latex,thick,bend right=20] (p3);\n\\draw(p3) edge[-latex,thick,bend right=20] (d);\n\\draw(c) edge[-latex,thick,bend right=20] (p3);\n\\draw(p3) edge[-latex,thick,bend right=20] (c);\n\\end{tikzpicture}\\\\[5mm]\n\\begin{tikzpicture}[LTS]\n\\node[state,label=below:$s_0$] (s0) {};\n\\path[edge] (s0) ++(-0.5,0) edge (s0);\n\\path(s0) +(30:1cm) node[state,label=above:$s_1$] (s1) {};\n\\path(s0) +(-30:1cm) node[state,label=below:$s_2$] (s2) {};\n\\draw[edge] (s0) edge node[auto]{$a$} (s1);\n\\draw[edge] (s1) edge node[auto]{$a$} (s2);\n\\draw[edge] (s2) edge[bend left=30] node[auto]{$b$} (s1);\n\\draw[edge] (s1) edge[loop] node[auto,swap]{$c$} (s1);\n\\draw[edge] (s2) edge[loop] node[auto,swap]{$c$} (s2);\n\\path(s0) to (0,-1.6);\n\\end{tikzpicture}\\hspace*{1.5cm}\n\\begin{tikzpicture}\n\\path(-0.25,0);\n\\draw(0,0) +(-30:3cm) node[transition] (a) {$a$};\n\\draw(0,0) +(30:3cm) node[transition] (b) {$b$};\n\\draw(0,1.5) ++(-30:3cm) ++(-1,0) node[place,tokens=2,label=left:$p_0$] (p0) {};\n\\draw(1,1.5) +(-30:3cm) node[place,label=above right:$p_1$] (p1) {};\n\\draw(2.5,1.5) +(-30:3cm) node[transition] (c) {$c$};\n\\draw(p1) edge[-latex,thick,bend right=20] node[auto,swap]{$2$} (b);\n\\draw(b) edge[-latex,thick,bend right=10] (p1);\n\\draw(b) edge[-latex,thick,bend right=20] (p0);\n\\draw(p0) edge[-latex,thick,bend right=20] (a);\n\\draw(a) edge[-latex,thick,bend right=20] (p1);\n\\draw(c) edge[-latex,thick,bend right=20] (p1);\n\\draw(p1) edge[-latex,thick,bend right=20] (c);\n\\end{tikzpicture}\n\\caption{Left: Two \\lts (left) with $b\\rhd c$ and $\\neg b\\merge c$. Right: \\WPI Petri net solutions \nfor the \\lts on the left. The upper net is also in (B)\\RAC}\n\\label{f.case6}\n\\end{figure}\n\nIn the upper part of Fig.~\\ref{f.case6} we see an \\lts on the left with $b\\rhd c$ and $\\neg b\\merge c$.\nThe \\lts is synthesisable to a \\WPI net shown on the right. In this net, the presets of $b$ and $c$ \nare disjoint. There is no solution where the preset of $c$ is properly contained in that of $b$.\nNote that $b\\interleave d$, $\\neg b\\merge d$, $d\\rhd c$, and $d \\merge c$ hold.\nBy case 2, $b$ and $d$ have disjoint presets, and by case 5, the preset of $c$ is properly contained\nin that of $d$, so $b$ and $c$ must have disjoint presets.\n\nIn the lower part of Fig.~\\ref{f.case6} we have again an \\lts with $b\\rhd c$ and $\\neg b\\merge c$,\nwith a \\WPI net solving it on the right side. Here, the preset of $c$ is properly contained in that\nof $b$. We see that $a\\interleave c$ and $\\neg a\\merge c$ imply disjoint presets by case~2. If $b$ and\n$c$ also had disjoint presets, only $c$ could remove tokens from its own preset. But as $c$ is\na self-loop, this does not happen either. As $a$ activates $c$ by its first occurrence, it must\nput a token in $c$'s preset and then $s_1\\step{ab}s_1$ would increase this number of\ntokens, contradicting the fact that it is a cycle and the Petri net to be constructed must always be at the\nsame marking when reaching state $s_1$. Thus, no solution with disjoint presets for $b$ and $c$\nexists.\n\nThat we cannot simply decide for disjointness or proper containment when self-loops are involved\nin case~6 poses a problem for synthesis, which we bypass for the\nremainder of this section by disallowing self-loops in the \\lts.\nWe might argue that such self-loops are not too interesting as they typically denote idle actions anyway, \nnot changing the system's state. Note that for a solvable \\lts all edges with \nthe same label are synthesised to a single transition, so either all or none of them\nmust be a self-loop.\nFor \\lts without self-loops we find:\n\n\\SATZ{l.cases1}{Label relations in the self-loop free case}\nLet $TS=(S,\\Sigma,\\to,s_0))$ be an \\lts without self-loops. For every pair of labels\n$a,b\\in\\Sigma$ we can determine whether the transitions $a$ and $b$ in a \\WPI or \\BRAC\nnet solving $TS$ have disjoint, identical, or properly contained presets, i.e. which\nof the four relations ${}^\\bullet a\\cap {}^\\bullet b=\\emptyset$, ${}^\\bullet a ={}^\\bullet b$,\n${}^\\bullet a \\subsetneqq {}^\\bullet b$, or ${}^\\bullet a \\supsetneqq {}^\\bullet b$ holds.\n\\ENDSATZ\n\\BEW\nWe have seen above that any pair $(a,b)$ falls under one of six cases.\nCase~1 immediately renders the \\lts non-synthesisable and case~4 can only occur\nif self-loops are present. \nCase~2 and~6 (without self-loops) imply disjoint presets and case~3 determines them to be identical.\nIn the remaining case~5, one of the presets is contained in the other and\n$a\\rhd b$ implies ${}^\\bullet a \\supsetneqq {}^\\bullet b$ while\n$b\\rhd a$ implies ${}^\\bullet a \\subsetneqq {}^\\bullet b$.\n\\ENDBEW{l.cases1}\n\n\\subsection{Solving separation problems for \\WPI synthesis}\n\nWe can now formulate inequality systems for separation problems that incorporate the above relations\nbetween labels. Since there is no limit on the number of places in the preset of a transition for\n\\WPI nets, we can tackle each separation problem with its own inequality system and find a region\nsolving it.\nFor an ESSP $(s,a)$ in an \\lts $TS=(S,\\Sigma,\\mathord{\\to},s_0)$ we then obtain\n\n\\begin{eqnarray*}\n                              && R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s) < B(a) \\\\\n\\forall s'\\step{t}s''\\in\\mathord{\\to}\\colon   && R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s')\\ge B(t) \\\\\n\\forall \\gamma\\in\\Gamma\\colon && (F-B)\\tp\\cdot\\gamma = 0 \\\\\n\\forall b\\in T\\colon a\\equiv b\\then && B(b)=B(a) \\\\\n\\forall b\\in T\\colon a\\rhd b\\wedge a\\merge b\\then && B(b)\\le B(a) \\\\\n\\forall b\\in T\\colon b\\rhd a\\wedge a\\merge b\\then && B(a)\\le B(b) \\\\\n\\forall b\\in T\\colon a\\bowtie b \\wedge \\neg a\\merge b\\then && B(b)=0 \\\\\n\\forall b\\in T\\colon a\\interleave b \\wedge \\neg a\\merge b\\then && B(b)=0 \n\\end{eqnarray*}\nThe last five lines stand for the cases~3, 5, 5, 6, and~2, in this order.\nFor an SSP $(s,s')$ the first line changes to\n\\begin{eqnarray*}\n(F-B)\\tp\\cdot(\\parikh_E(s)-\\parikh_E(s')) & \\neq & 0. \\\\\n\\end{eqnarray*}\nSince the label $a$ is not part of the problem instance for the SSP $(s,s')$,\nwe must construct inequality systems for each label $a\\in T$. If at least one of them\nis solvable, the SSP is also solved, and we can construct a place from one of the solutions.\n\n\\subsection{Solving separation problems for \\BRAC synthesis}\n\nWhen targetting a \\BRAC net, the situation is more complicated. First, we must forbid\narc weights by adding\n$\\forall t\\in T\\colon B(t)\\le 1 \\wedge F(t)\\le 1$ to every inequality system we create.\nThe constant terms $1$ in these inequalities make solving these inequality systems \\NP-complete.\nThen, we must ensure that transitions involved in an asymmetric choice have only one\nor two places in their preset. So, if case~5 occurs with $a\\bowtie b \\wedge a\\merge b$\nfor some labels $a$ and $b$, we may construct only \ntwo inequality systems for all separation problems concerning $a$ and $b$ together.\nWe start by solving ESSPs and take a look at SSPs afterwards.\n\n\nAssume, targetting at \\BRAC, we have an asymmetric choice for two labels $a,b\\in T$ with $a\\rhd b$ and $a\\merge b$\n(since case~5 is the only case with properly included presets).\nAll ESSPs $(s,b)$ with $\\neg s\\step{b}$ must be solved by the same region, i.e.\\ a single place.\nIf $b$ can occur exactly $k$ times consecutively at a state $s$,\n$s\\step{b^k}$ and $\\neg s\\step{b^{k+1}}$, we have $R(s)=k$, $B(b)=1$, and $F(b)=0$.\\footnote{Otherwise\n$b$ would be a self-loop or the solving net would be unbounded.}\nFurthermore, $B(a)=1$ (by case~5) and all $t\\in T$ with either $a\\equiv t$ or $b\\equiv t$\n($t$'s with presets identical to that of either $a$ or $b$ by case~3 or~4) also yield $B(t)=1$.\nFor all other transitions $t'\\in T$ we get $B(t')=0$ (by the remaining cases~2 or~6).\nBecause $a$ will have two places in its preset, $F(a)=1$ does not mean that $a$ is a self-loop,\ni.e.\\ the value of $F(a)$ is not predetermined. The same holds for $t\\in T$ with $a\\equiv t$.\nWe solve the inequality system from above, modified by the known, fixed values and one copy of\nthe first line, $R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s) < B(b)$, for each ESSP $(s,b)$ with $\\neg s\\step{b}$,\nto obtain the region\/place in the preset of $b$. This also solves all ESSPs for $t\\in T$ with $b\\equiv t$,\nas these labels share $B(t)=1$ and $F(t)=0$ as well as the number of consecutive activations at any state $s$, \ni.e. the value $R(s)$, with $b$.\n\nA second region must solve all ESSPs $(s,a)$ with $\\neg s\\step{a}$ but $s\\step{b}$ at the same time\n\\footnote{ESSPs $(s,a)$ with $\\neg s\\step{b}$ are already solved by the first region since ${}^\\bullet b\\subsetneqq {}^\\bullet a$.}\nWe add the first line of our inequality system,\n$R(s_0)+(F-B)\\tp\\cdot\\parikh_E(s) < B(a)$, for each state $s$ with such an ESSP $(s,a)$.\nLabels $t$ with $a\\equiv t$ share this line with $a$ as $B(a)=B(t)$, so all ESSPs $(s,t)$ are tackled\nat the same time automatically.\\footnote{The values $F(a)$ and $F(t)$ for $a\\equiv t$, which may actually\ndiffer, are mostly determined via $F\\cdot\\parikh_E(s')$ for different $s'$ in the first and second line of our generic \ninequality system, but also restricted by the cycles of $\\Gamma$ in the third line.}\nA solution, if it exists, forms the second place in the preset of $a$\n\\footnote{If any constructed inequality system is unsolvable, so is the corresponding ESSP, \nand the \\lts is not synthesisable.}\n\nAll labels not involved in asymmetric choices form blocks with presets disjoint to each other (and from the asymmetric\nchoices), while all labels inside one block have identical presets. ESSPs for labels in these blocks\ncan be handled using the generic inequality system from above. Each ESSP may necessitate its own region and\nthus its own place.\n\nSSPs $(s,s')$ are best solved apart from asymmetric choice blocks first. We simply replace\nthe two lines for properly included presets in our above generic inequality system for SSPs\/\\BRAC by\n\\begin{eqnarray*}\n\\forall b\\in T\\colon a\\rhd b\\wedge a\\merge b\\then && B(a)=0 \\wedge B(b)=0 \\\\\n\\forall b\\in T\\colon b\\rhd a\\wedge a\\merge b\\then && B(a)=0 \\wedge B(b)=0 \n\\end{eqnarray*}\nIf such a system has a solution, we can convert the found region to a place in one\nof the free-choice blocks in our \\BRAC net. Otherwise, the SSP $(s,s')$ must be\nsolved by an asymmetric choice block. If no region constructed so far solves this SSP, one of the regions we have\ngenerated for an asymmetric choice block above is not sufficient. Since there may be more than one unsolved SSP,\nwe must solve those inequality systems again, and for each unsolved SSP $(s,s')$\nwe must either add the line $(F-B)\\tp\\cdot(\\parikh_E(s)-\\parikh_E(s')) \\neq 0$ or\nleave it out (if the SSP will be solved via the other of the two regions or even\nin a different asymmetric choice block), in all combinations. Clearly, this becomes exponential. Still,\nguessing which unsolved SSP will be solved by which asymmetric choice block and which of two regions\nis in \\NP, so the whole problem could be \\NP-complete in the worst case. \n\n\\KOR{k.sfsynth}{Synthesis to self-loop free \\WPI or \\BRAC nets}\nA \\WPI or \\BRAC net can be synthesised from an \\lts by solving all modified separation problems (as shown above)\nand converting solving regions for the inequality systems into places canonically.\nWhile solving for \\WPI nets has a polynomial time complexity, the extra condition for \\BRAC nets\ninvolving constant terms in the inequality systems only allows us to deduce membership in \\NP.\n\\ENDKOR{k.sfsynth}\n\nSo, we have two problems that may prevent a polynomial runtime for \\BRAC synthesis: \nthe limit on arc weights and the combinatorial problem of SSPs and asymmetric choice blocks.\n\nWe can attempt to solve each one of these problems (but not both at the same time) by extending the class \\BRAC.\nTo allow solving the SSPs in polynomial time, we can extend the class such that an asymmetric\nchoice block $T'$ of transitions can have more than two places in its preset. Each of the two \nsubblocks of transitions, $T'=T_1\\cup T_2$ with $T_1\\cap T_2=\\emptyset$, must have a common preset, i.e. \n$\\forall i\\in\\{1,2\\}\\forall t,t'\\in T_i\\colon {}^\\bullet t={}^\\bullet t'$.\nBetween blocks, proper containment of presets holds, i.e.\n$\\forall t_1\\in T_1, t_2\\in T_2\\colon {}^\\bullet t_2\\subsetneqq{}^\\bullet t_1$.\nUnder the condition that all arc weights (in the presets) are equal to one, we can formulate\ntwo inequality systems for each (unsolved) SSP, one for the sought region\/place in the preset of $T_1$,\nthe other for $T_2$ only. If one is solved, we can add a place accordingly to the constructed net.\nIf no solution exists, the SSPs must be handled in some other asymmetric choice block.\n\nWe can also try to remove the condition $\\forall t\\in T\\colon F(t)\\le 1\\wedge B(t)\\le 1$ to make our\ninequality systems solvable in polynomial time. But this will require us to define weighted\nasymmetric choices that also respect the \\WPI condition. In an asymmetric choice block\n$T'=T_1\\cup T_2$ with two places $p$, $p'$ with $p^\\bullet=T_1$ and $p'{}^\\bullet=T'$ we must\ndemand $W(p,t_1)\\le W(p',t_1)\\ge W(p',t_2)$ for all $t_1\\in T_1$ and $t_2\\in T_2$.\nThe first condition can only be formulated if we merge the two inequality systems for an\nasymmetric choice block into one (with still two separate sets of variables). Then we can add\nthe above comparisons. \n\nNow consider what happens if we make both extensions to \\BRAC at the same time.\nWe might have two (or more) different, so far unsolved SSPs that can be addressed\nin the same asymmetric choice block. With arc weights limited to one, we just had to check\nif an SSP could be solved with one of the two predetermined presets, but possibly different \npostsets for the transitions.\nWith arbitrary arc weights, the solutions might now require different arc weights\nin the preset of the asymmetric choice block.\nSince places can still only have one of two different postsets in an asymmetric choice,\nwe must decide which SSPs need to be solved by {\\it this} asymmetric choice block. \nBetween all asymmetric choices and all SSPs to solve, we arrive again at a combinatorial problem.\n\nWe make the following conjecture.\n\n\\CON{c.2}{Extensions of \\BRAC}\\\\\nFor every extension of the class \\BRAC still allowing only weighted asymmetric choices,\nthe synthesis is \\NP-complete.\n\\ENDCON{c.2}\n\n\n\n\\section{Handling self-loops in the \\lts}\\label{sect.4}\n\n\\tikzstyle{myRightArrowBaseline}=[baseline=-0.6ex]\n\\tikzstyle{myFullArrow}=[-stealth]\n\\tikzstyle{myBackArrow}=[stealth-]\n\\tikzstyle{myLine}=[-]\n\\tikzstyle{myDashedArrow}=[myFullArrow, dash pattern=on 1.5pt off 1.5pt]\n\\tikzstyle{myBackDashedArrow}=[myBackArrow, dash pattern=on 1.5pt off 1.5pt]\n\\tikzstyle{myNone}=[]\n\\newcommand{\\myArrow}[1]{\\mathbin{\\tikz [myRightArrowBaseline]{\n        \\draw [#1] (0,0) -- (1.2em,0);\n}}}\n\\newcommand{\\myNoArrow}{\\mathbin{\\tikz [myRightArrowBaseline]{\n        \\path (0,0) -- (1.2em,0);\n}}}\n\\newcommand{\\disjorincl}{\\myArrow{myDashedArrow}}\n\\newcommand{\\included}{\\myArrow{-stealth}}\n\\newcommand{\\backdisjorincl}{\\myArrow{myBackDashedArrow}}\n\\newcommand{\\backincluded}{\\myArrow{myBackArrow}}\n\\newcommand{\\equivalent}{\\myArrow{myLine}}\n\\newcommand{\\disjoint}{\\myNoArrow}\n\n\\DEF{d.grl}{Graphical preset relations}\nFor a finite, reachable, deterministic \\lts $TS=(S,\\Sigma,\\to,s_0)$ with two\nlabels $a,b\\in\\Sigma$, the six cases in the previous section enforce six\npossible preset relations for $a$ and $b$ in a theoretical, synthesised\n\\WPI net $N=(P,T,W,M_0)$. We denote by\n\\begin{itemize}\n\\item $a\\equivalent b$ that $W(\\cdot,a)=W(\\cdot,b)$ via case~3 or~4 (equivalence),\n\\item $a\\disjoint b$ that ${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$ via case~2 or~6 (disjointness),\n\\item $a\\included b$ that $W(\\cdot,a)\\lneqq W(\\cdot,b)$ via case~5 (proper inclusion),\n\\item $a\\backincluded b$ that $W(\\cdot,a)\\gneqq W(\\cdot,b)$ via case~5 (proper inclusion),\n\\item $a\\disjorincl b$ that $a$ is a self-loop in $TS$ with either ${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$ or $W(\\cdot,a)\\lneqq W(\\cdot,b)$ via case~6 (disjoint or properly included), or\n\\item $a\\backdisjorincl b$ that $b$ is a self-loop in $TS$ with either ${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$ or $W(\\cdot,a)\\gneqq W(\\cdot,b)$ via case~6 (disjoint or properly included).\n\\end{itemize}\nEach pair $(a,b)$ is in exactly one of these relations unless case~1 disallows\nthe synthesis of the \\lts at all.\n\\ENDDEF{d.grl}\n\nWe will now study the preset relations between more than two labels. Consider e.g.\\\ntwo labels $a$ and $b$ with $a\\equivalent b$ and a third label $c$. Intuitively,\nwe would assume that the relations between $a$ and $c$ and between $b$ and $c$\nare the same, as $a$ and $b$ are somehow equivalent. Let us check that.\n\n\\LEM{l.equiv}{Adapting equivalence}\nLet $TS=(S,\\Sigma,\\to,s_0)$ be a finite, reachable, deterministic \\lts.\nFor any three labels $a,b,c\\in\\Sigma$ with $a\\equivalent b$, we can\nmake the following conclusions:\n\\begin{itemize}\n\\item $a\\equivalent c \\then b\\equivalent c$,\n\\item $a\\disjoint c \\then (b\\disjoint c \\vee b\\disjorincl c \\vee b\\backdisjorincl c)$,\n\\item $a\\included c \\then (b\\included c \\vee b\\disjorincl c)$,\n\\item $a\\backincluded c \\then (b\\backincluded c \\vee b\\backdisjorincl c)$,\n\\item $a\\disjorincl c \\then (b\\disjorincl c \\vee b\\included c \\vee b\\disjoint c)$,\n\\item $a\\backdisjorincl c \\then (b\\backdisjorincl c \\vee b\\backincluded c \\vee b\\disjoint c)$,\n\\end{itemize}\nWe can strengthen the relations by deducing for any edge $b\\disjorincl c$ either that\n$b\\included c$ holds or that $b\\disjoint c$ holds, similarly for reverse edges.\nThen, for any of the six above relations $R$ holds $aRc \\then bRc$ and\n$\\equivalent$ becomes an equivalence respecting the other five relations.\n\\ENDLEM\n\\BEW\nAll points follow directly from $W(\\cdot,a)=W(\\cdot,b)$ and the preset relation\nbetween $a$ and $c$. All relations that allow this preset comparison are then\npossible between $b$ and $c$.\n\nIn the cases $a\\disjoint c$, $a\\included c$, and $a\\backincluded c$, we know\nthe exact relation, ${}^\\bullet a\\cap{}^\\bullet c=\\emptyset$ or\n$W(\\cdot,a)\\lneqq W(\\cdot,c)$ or $W(\\cdot,a)\\gneqq W(\\cdot,c)$, respectively.\nWe may then conclude that $b$ must be in that same relation to $c$, so\nwe may strengthen $b\\disjorincl c$ to either $b\\included c$ or $b\\disjoint c$,\nand $b\\backdisjorincl c$ to either $b\\backincluded c$ or $b\\disjoint c$.\nThis removes all dashed edges from the first four points.\n\nWe can also remove the relations $b\\included c$, $b\\backincluded c$, and\n$b\\disjoint c$ from the last two points by symmetry (swapping the roles of $a$ and $b$).\nAs a total effect, $a$ and $b$ then are in the same relation to $c$.\n\\ENDBEW{l.equiv}\n\nThis means after checking for inconsistencies (case~1) and strengthening\n$\\disjorincl$ and $\\backdisjorincl$ wherever possible, we can deal with\none representative from each equivalence class of $\\equivalent$ and forget\nabout the latter relation at all.\n\nNote that edges $a\\disjorincl b$ that were strengthend to $a\\included b$ do not share\nthe deactivation property with original edges $a\\included b$. For the former we have\n$\\neg a\\merge b$ while for the latter $a\\merge b$ holds. When strengthening the\nequivalence $\\equivalent$ in this way as shown in Lemma~\\ref{l.equiv}, at least\none member in an equivalence class of labels has an original edge $\\included$,\nand we should choose this label as representative for the class. The deactivation\nproperty will allow us to draw additional conclusions.\n\nIgnoring equivalence, two labels can be related in five ways. If we consider\nour problem case $a\\disjorincl b$, the two labels can be related to a third\nlabel $c$ in $5^2=25$ ways. Figure~\\ref{f.25} shows all possibilities.\nSome of these lead directly to a contradiction, shown by the symbol $\\times$\ninside the triangle. Contradictions happen either because the arrows form\na cycle\\footnote{Impossible as $x\\rhd y$ in case~5 and~6 means that the \\lts contains more\nedges with label $y$ than with label $x$} as in 8), 10), 18), and 20),\nor because self-loops $a$ and $c$ need to deactivate each other, as in 22) and 23).\nIn 12) proper inclusion follows from transitivity while in 13) disjointness for $a$ and $b$\ncontradicts the \\WPI property. For 3) and 6) proper inclusion for $a$ and $b$ contradicts \ntransitivity and in 2) again the \\WPI property comes into play.\nFinally, in 17) $a$ cannot deactivate $c$ as it is a self-loop, thus $c$ can deactivate $a$.\nIf $a\\disjorincl b$ means proper inclusion, the preset of $a$ will be contained in both\nof $b$ and $c$, i.e. the edge $b\\disjorincl c$ also means proper inclusion. As $c$ can\ndeactivate $a$, it can also deactivate $b$ then, contradicting $b\\disjorincl c$ and case~6.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,8) node(a11) {$a$};\n\\draw(a11) +(90:1cm) node(b11) {$b$};\n\\draw(a11) +(30:1cm) node(c11) {$c$};\n\\draw[-latex,dashed] (a11) -- (b11);\n\\draw(a11) +(60:5.5mm) node {?};\n\\draw(a11) +(135:6.5mm) node {$1)$};\n\n\\draw(2.4,8) node(a21) {$a$};\n\\draw(a21) +(90:1cm) node(b21) {$b$};\n\\draw(a21) +(30:1cm) node(c21) {$c$};\n\\draw[-latex,dashed] (a21) -- (b21);\n\\draw[-latex] (a21) -- (c21);\n\\draw(a21) +(60:5.5mm) node {$\\interleave$};\n\\draw(a21) +(135:6.5mm) node {$2)$};\n\n\\draw(4.8,8) node(a31) {$a$};\n\\draw(a31) +(90:1cm) node(b31) {$b$};\n\\draw(a31) +(30:1cm) node(c31) {$c$};\n\\draw[-latex,dashed] (a31) -- (b31);\n\\draw[-latex] (c31) -- (a31);\n\\draw(a31) +(60:5.5mm) node {$\\interleave$};\n\\draw(a31) +(135:6.5mm) node {$3)$};\n\n\\draw(7.2,8) node(a41) {$a$};\n\\draw(a41) +(90:1cm) node(b41) {$b$};\n\\draw(a41) +(30:1cm) node(c41) {$c$};\n\\draw[-latex,dashed] (a41) -- (b41);\n\\draw[-latex,dashed] (a41) -- (c41);\n\\draw(a41) +(60:5.5mm) node {?};\n\\draw(a41) +(135:6.5mm) node {$4)$};\n\n\\draw(9.6,8) node(a51) {$a$};\n\\draw(a51) +(90:1cm) node(b51) {$b$};\n\\draw(a51) +(30:1cm) node(c51) {$c$};\n\\draw[-latex,dashed] (a51) -- (b51);\n\\draw[-latex,dashed] (c51) -- (a51);\n\\draw(a51) +(60:5.5mm) node {?};\n\\draw(a51) +(135:6.5mm) node {$5)$};\n\n\\draw(0,6) node(a12) {$a$};\n\\draw(a12) +(90:1cm) node(b12) {$b$};\n\\draw(a12) +(30:1cm) node(c12) {$c$};\n\\draw[-latex,dashed] (a12) -- (b12);\n\\draw[-latex] (b12) -- (c12);\n\\draw(a12) +(60:5.5mm) node {$\\interleave$};\n\\draw(a12) +(135:6.5mm) node {$6)$};\n\n\\draw(2.4,6) node(a22) {$a$};\n\\draw(a22) +(90:1cm) node(b22) {$b$};\n\\draw(a22) +(30:1cm) node(c22) {$c$};\n\\draw[-latex,dashed] (a22) -- (b22);\n\\draw[-latex] (a22) -- (c22);\n\\draw[-latex] (b22) -- (c22);\n\\draw(a22) +(60:5.5mm) node {?};\n\\draw(a22) +(135:6.5mm) node {$7)$};\n\n\\draw(4.8,6) node(a32) {$a$};\n\\draw(a32) +(90:1cm) node(b32) {$b$};\n\\draw(a32) +(30:1cm) node(c32) {$c$};\n\\draw[-latex,dashed] (a32) -- (b32);\n\\draw[-latex] (c32) -- (a32);\n\\draw[-latex] (b32) -- (c32);\n\\draw(a32) +(60:5.5mm) node {$\\times$};\n\\draw(a32) +(135:6.5mm) node {$8)$};\n\n\\draw(7.2,6) node(a42) {$a$};\n\\draw(a42) +(90:1cm) node(b42) {$b$};\n\\draw(a42) +(30:1cm) node(c42) {$c$};\n\\draw[-latex,dashed] (a42) -- (b42);\n\\draw[-latex,dashed] (a42) -- (c42);\n\\draw[-latex] (b42) -- (c42);\n\\draw(a42) +(60:5.5mm) node {?};\n\\draw(a42) +(135:6.5mm) node {$9)$};\n\n\\draw(9.6,6) node(a52) {$a$};\n\\draw(a52) +(90:1cm) node(b52) {$b$};\n\\draw(a52) +(30:1cm) node(c52) {$c$};\n\\draw[-latex,dashed] (a52) -- (b52);\n\\draw[-latex,dashed] (c52) -- (a52);\n\\draw[-latex] (b52) -- (c52);\n\\draw(a52) +(60:5.5mm) node {$\\times$};\n\\draw(a52) +(135:6.5mm) node {$10)$};\n\n\\draw(0,4) node(a13) {$a$};\n\\draw(a13) +(90:1cm) node(b13) {$b$};\n\\draw(a13) +(30:1cm) node(c13) {$c$};\n\\draw[-latex,dashed] (a13) -- (b13);\n\\draw[-latex] (c13) -- (b13);\n\\draw(a13) +(60:5.5mm) node {?};\n\\draw(a13) +(135:6.5mm) node {$11)$};\n\n\\draw(2.4,4) node(a23) {$a$};\n\\draw(a23) +(90:1cm) node(b23) {$b$};\n\\draw(a23) +(30:1cm) node(c23) {$c$};\n\\draw[-latex,dashed] (a23) -- (b23);\n\\draw[-latex] (a23) -- (c23);\n\\draw[-latex] (c23) -- (b23);\n\\draw(a23) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a23) +(135:6.5mm) node {$12)$};\n\n\\draw(4.8,4) node(a33) {$a$};\n\\draw(a33) +(90:1cm) node(b33) {$b$};\n\\draw(a33) +(30:1cm) node(c33) {$c$};\n\\draw[-latex,dashed] (a33) -- (b33);\n\\draw[-latex] (c33) -- (a33);\n\\draw[-latex] (c33) -- (b33);\n\\draw(a33) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a33) +(135:6.5mm) node {$13)$};\n\n\\draw(7.2,4) node(a43) {$a$};\n\\draw(a43) +(90:1cm) node(b43) {$b$};\n\\draw(a43) +(30:1cm) node(c43) {$c$};\n\\draw[-latex,dashed] (a43) -- (b43);\n\\draw[-latex,dashed] (a43) -- (c43);\n\\draw[-latex] (c43) -- (b43);\n\\draw(a43) +(60:5.5mm) node {?};\n\\draw(a43) +(135:6.5mm) node {$14)$};\n\n\\draw(9.6,4) node(a53) {$a$};\n\\draw(a53) +(90:1cm) node(b53) {$b$};\n\\draw(a53) +(30:1cm) node(c53) {$c$};\n\\draw[-latex,dashed] (a53) -- (b53);\n\\draw[-latex,dashed] (c53) -- (a53);\n\\draw[-latex] (c53) -- (b53);\n\\draw(a53) +(60:5.5mm) node {?};\n\\draw(a53) +(135:6.5mm) node {$15)$};\n\n\\draw(0,2) node(a14) {$a$};\n\\draw(a14) +(90:1cm) node(b14) {$b$};\n\\draw(a14) +(30:1cm) node(c14) {$c$};\n\\draw[-latex,dashed] (a14) -- (b14);\n\\draw[-latex,dashed] (b14) -- (c14);\n\\draw(a14) +(60:5.5mm) node {?};\n\\draw(a14) +(135:6.5mm) node {$16)$};\n\n\\draw(2.4,2) node(a24) {$a$};\n\\draw(a24) +(90:1cm) node(b24) {$b$};\n\\draw(a24) +(30:1cm) node(c24) {$c$};\n\\draw[-latex,dashed] (a24) -- (b24);\n\\draw[-latex] (a24) -- (c24);\n\\draw[-latex,dashed] (b24) -- (c24);\n\\draw(a24) +(60:5.5mm) node {$\\interleave$};\n\\draw(a24) +(135:6.5mm) node {$17)$};\n\n\\draw(4.8,2) node(a34) {$a$};\n\\draw(a34) +(90:1cm) node(b34) {$b$};\n\\draw(a34) +(30:1cm) node(c34) {$c$};\n\\draw[-latex,dashed] (a34) -- (b34);\n\\draw[-latex] (c34) -- (a34);\n\\draw[-latex,dashed] (b34) -- (c34);\n\\draw(a34) +(60:5.5mm) node {$\\times$};\n\\draw(a34) +(135:6.5mm) node {$18)$};\n\n\\draw(7.2,2) node(a44) {$a$};\n\\draw(a44) +(90:1cm) node(b44) {$b$};\n\\draw(a44) +(30:1cm) node(c44) {$c$};\n\\draw[-latex,dashed] (a44) -- (b44);\n\\draw[-latex,dashed] (a44) -- (c44);\n\\draw[-latex,dashed] (b44) -- (c44);\n\\draw(a44) +(60:5.5mm) node {?};\n\\draw(a44) +(135:6.5mm) node {$19)$};\n\n\\draw(9.6,2) node(a54) {$a$};\n\\draw(a54) +(90:1cm) node(b54) {$b$};\n\\draw(a54) +(30:1cm) node(c54) {$c$};\n\\draw[-latex,dashed] (a54) -- (b54);\n\\draw[-latex,dashed] (c54) -- (a54);\n\\draw[-latex,dashed] (b54) -- (c54);\n\\draw(a54) +(60:5.5mm) node {$\\times$};\n\\draw(a54) +(135:6.5mm) node {$20)$};\n\n\\draw(0,0) node(a15) {$a$};\n\\draw(a15) +(90:1cm) node(b15) {$b$};\n\\draw(a15) +(30:1cm) node(c15) {$c$};\n\\draw[-latex,dashed] (a15) -- (b15);\n\\draw[-latex,dashed] (c15) -- (b15);\n\\draw(a15) +(60:5.5mm) node {?};\n\\draw(a15) +(135:6.5mm) node {$21)$};\n\n\\draw(2.4,0) node(a25) {$a$};\n\\draw(a25) +(90:1cm) node(b25) {$b$};\n\\draw(a25) +(30:1cm) node(c25) {$c$};\n\\draw[-latex,dashed] (a25) -- (b25);\n\\draw[-latex] (a25) -- (c25);\n\\draw[-latex,dashed] (c25) -- (b25);\n\\draw(a25) +(60:5.5mm) node {$\\times$};\n\\draw(a25) +(135:6.5mm) node {$22)$};\n\n\\draw(4.8,0) node(a35) {$a$};\n\\draw(a35) +(90:1cm) node(b35) {$b$};\n\\draw(a35) +(30:1cm) node(c35) {$c$};\n\\draw[-latex,dashed] (a35) -- (b35);\n\\draw[-latex] (c35) -- (a35);\n\\draw[-latex,dashed] (c35) -- (b35);\n\\draw(a35) +(60:5.5mm) node {$\\times$};\n\\draw(a35) +(135:6.5mm) node {$23)$};\n\n\\draw(7.2,0) node(a45) {$a$};\n\\draw(a45) +(90:1cm) node(b45) {$b$};\n\\draw(a45) +(30:1cm) node(c45) {$c$};\n\\draw[-latex,dashed] (a45) -- (b45);\n\\draw[-latex,dashed] (a45) -- (c45);\n\\draw[-latex,dashed] (c45) -- (b45);\n\\draw(a45) +(60:5.5mm) node {?};\n\\draw(a45) +(135:6.5mm) node {$24)$};\n\n\\draw(9.6,0) node(a55) {$a$};\n\\draw(a55) +(90:1cm) node(b55) {$b$};\n\\draw(a55) +(30:1cm) node(c55) {$c$};\n\\draw[-latex,dashed] (a55) -- (b55);\n\\draw[-latex,dashed] (c55) -- (a55);\n\\draw[-latex,dashed] (c55) -- (b55);\n\\draw(a55) +(60:5.5mm) node {?};\n\\draw(a55) +(135:6.5mm) node {$25)$};\n\\end{tikzpicture}\n\n\\caption{Relations between labels $a$, $b$, $c$ with $a\\protect\\disjorincl b$ for \\WPI synthesis.\nSome cases are contradictory (shown by a $\\times$ sign), while for others\nthe relation between $a$ and $b$ can be strengthened immediately (to\ndisjointness $\\interleave$ or proper inclusion $\\subsetneqq$). If the\nedge $a\\protect\\disjorincl b$ cannot be resolved, there is a question mark\n\\label{f.25}}\n\\end{figure}\n\n\\drop{\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,4) node(a1) {$a$};\n\\draw(a1) +(90:1cm) node(b1) {$b$};\n\\draw(a1) +(30:1cm) node(c1) {$c$};\n\\draw[-latex,dashed] (a1) -- (b1);\n\\draw[-latex,dashed] (c1) -- (a1);\n\\draw[-latex,dashed] (b1) -- (c1);\n\n\\draw(0,2) node(a2) {$a$};\n\\draw(a2) +(90:1cm) node(b2) {$b$};\n\\draw(a2) +(30:1cm) node(c2) {$c$};\n\\draw[-latex,dashed] (a2) -- (b2);\n\\draw[-latex] (c2) -- (a2);\n\\draw[-latex,dashed] (b2) -- (c2);\n\n\\draw(0,0) node(a5) {$a$};\n\\draw(a5) +(90:1cm) node(b5) {$b$};\n\\draw(a5) +(30:1cm) node(c5) {$c$};\n\\draw[-latex,dashed] (a5) -- (b5);\n\\draw[-latex] (c5) -- (a5);\n\\draw[-latex,dashed] (c5) -- (b5);\n\n\\draw(2,4) node(a3) {$a$};\n\\draw(a3) +(90:1cm) node(b3) {$b$};\n\\draw(a3) +(30:1cm) node(c3) {$c$};\n\\draw[-latex,dashed] (a3) -- (b3);\n\\draw[-latex,dashed] (c3) -- (a3);\n\\draw[-latex] (b3) -- (c3);\n\n\\draw(2,2) node(a4) {$a$};\n\\draw(a4) +(90:1cm) node(b4) {$b$};\n\\draw(a4) +(30:1cm) node(c4) {$c$};\n\\draw[-latex,dashed] (a4) -- (b4);\n\\draw[-latex] (c4) -- (a4);\n\\draw[-latex] (b4) -- (c4);\n\n\\draw(2,0) node(a6) {$a$};\n\\draw(a6) +(90:1cm) node(b6) {$b$};\n\\draw(a6) +(30:1cm) node(c6) {$c$};\n\\draw[-latex,dashed] (a6) -- (b6);\n\\draw[-latex] (a6) -- (c6);\n\\draw[-latex,dashed] (c6) -- (b6);\n\n\\draw[loosely dotted] (3.5,-0.5) -- (3.5,5.5);\n\n\\draw(4,0) node(a7) {$a$};\n\\draw(a7) +(90:1cm) node(b7) {$b$};\n\\draw(a7) +(30:1cm) node[label=right:$\\then$](c7) {$c$};\n\\draw[-latex,dashed] (a7) -- (b7);\n\\draw[-latex] (a7) -- (c7);\n\\draw[-latex] (c7) -- (b7);\n\\draw(5.7,0) node(a8) {$a$};\n\\draw(a8) +(90:1cm) node(b8) {$b$};\n\\draw(a8) +(30:1cm) node(c8) {$c$};\n\\draw[-latex] (a8) -- (b8);\n\\draw[-latex] (a8) -- (c8);\n\\draw[-latex] (c8) -- (b8);\n\n\\draw(7.7,0) node(a9) {$a$};\n\\draw(a9) +(90:1cm) node(b9) {$b$};\n\\draw(a9) +(30:1cm) node[label=right:$\\then$](c9) {$c$};\n\\draw[-latex,dashed] (a9) -- (b9);\n\\draw[-latex] (c9) -- (a9);\n\\draw[-latex] (c9) -- (b9);\n\\draw(9.4,0) node(a10) {$a$};\n\\draw(a10) +(90:1cm) node(b10) {$b$};\n\\draw(a10) +(30:1cm) node(c10) {$c$};\n\\draw[-latex] (a10) -- (b10);\n\\draw[-latex] (c10) -- (a10);\n\\draw[-latex] (c10) -- (b10);\n\n\\draw[loosely dotted] (3.5,1.5) -- (10.9,1.5);\n\n\\draw(4,2) node(a11) {$a$};\n\\draw(a11) +(90:1cm) node(b11) {$b$};\n\\draw(a11) +(30:1cm) node[label=right:$\\then$](c11) {$c$};\n\\draw[-latex,dashed] (a11) -- (b11);\n\\draw[-latex] (c11) -- (a11);\n\\draw(5.7,2) node(a12) {$a$};\n\\draw(a12) +(90:1cm) node(b12) {$b$};\n\\draw(a12) +(30:1cm) node(c12) {$c$};\n\\draw[-latex] (c12) -- (a12);\n\n\\draw(7.7,2) node(a13) {$a$};\n\\draw(a13) +(90:1cm) node(b13) {$b$};\n\\draw(a13) +(30:1cm) node[label=right:$\\then$](c13) {$c$};\n\\draw[-latex,dashed] (a13) -- (b13);\n\\draw[-latex] (b13) -- (c13);\n\\draw(9.4,2) node(a14) {$a$};\n\\draw(a14) +(90:1cm) node(b14) {$b$};\n\\draw(a14) +(30:1cm) node(c14) {$c$};\n\\draw[-latex] (b14) -- (c14);\n\n\\draw(4,4) node(a15) {$a$};\n\\draw(a15) +(90:1cm) node(b15) {$b$};\n\\draw(a15) +(30:1cm) node[label=right:$\\then$](c15) {$c$};\n\\draw[-latex,dashed] (a15) -- (b15);\n\\draw[-latex] (a15) -- (c15);\n\\draw(5.7,4) node(a16) {$a$};\n\\draw(a16) +(90:1cm) node(b16) {$b$};\n\\draw(a16) +(30:1cm) node(c16) {$c$};\n\\draw[-latex] (a16) -- (c16);\n\n\\draw(7.7,4) node(a17) {$a$};\n\\draw(a17) +(90:1cm) node(b17) {$b$};\n\\draw(a17) +(30:1cm) node[label=right:$\\then$](c17) {$c$};\n\\draw[-latex,dashed] (a17) -- (b17);\n\\draw[-latex] (a17) -- (c17);\n\\draw[-latex,dashed] (b17) -- (c17);\n\\draw(9.4,4) node(a18) {$a$};\n\\draw(a18) +(90:1cm) node(b18) {$b$};\n\\draw(a18) +(30:1cm) node(c18) {$c$};\n\\draw[-latex] (a18) -- (c18);\n\\draw[-latex,dashed] (b18) -- (c18);\n\n\\end{tikzpicture}\n\\caption{Relations between labels $a$, $b$, $c$ with pairwise non-identical presets, \nwhere the preset of $a$ is disjoint to or contained in that of $b$,\nshown by a dashed edge. Solid edges mean containment, non-existing ones disjointness.\nLabel $c$ can be related in 25 ways to $a$ and $b$, six of which (on the left)\nare contradictory, either because \\IPS{2} demands acyclicity or because $\\neg\\DPS$\ndoes not fit with the self-loops for $a$ and $c$ (the lower two pictures). In six more cases,\nthe exact relation of $a$ and $b$ can be concluded, either disjointness (upper right)\nor containment (lower right)\n\\label{f.rel}}\n\\end{figure}\n}\n\nFigure~\\ref{f.25} also provides strengthenings of $\\disjorincl$ to $\\included$ via 12) and 13). \nThe strengthened edge can be part of another triangle, as shown in Fig.~\\ref{f.indirect}.\nIn all six situations, the edge $a\\disjorincl b$ is strengthened to $a\\included b$ via 12) or\n13). The triangle consisting of $a$, $b$, and $d$ is now either that of 22) or 23), but we\ncannot draw a conclusion directly. For 22) and 23) it is essential that $a$ and $b$ (named\n$a$ and $c$ in Fig.~\\ref{f.25}) can deactivate each other, but here that is not the case.\nWe can still draw the same conclusion in an indirect way. Take the middle picture in the\nupper row of Fig.~\\ref{f.indirect} as an example: We have triangles $acd$ and $bcd$ to\nwhich 22) or 23) can be applied, showing the inconsistency we originally wanted to derive\nfrom the triangle $abd$. In this way, if an edge $\\included$ stems from a strengthening\ndisallowing the application of 22) or 23), we fall back to the triangle responsible for\nthe strengthened edge. This can even be done over several steps. Note that there may be\ndifferent conclusions though: In the pictures on the left and right hand sides of Fig.~\\ref{f.indirect}\nwe get disjointness and proper inclusion as a result, respectively.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,2) node(a1) {$a$};\n\\draw(a1) +(90:1cm) node(b1) {$b$};\n\\draw(a1) +(30:1cm) node(c1) {$c$};\n\\draw(a1) +(150:1cm) node(d1) {$d$};\n\\draw[-latex,dashed] (a1) -- (b1);\n\\draw[-latex] (a1) -- (c1);\n\\draw[-latex] (c1) -- (b1);\n\\draw[-latex,dashed] (b1) -- (d1);\n\\draw[-latex,dashed] (a1) -- (d1);\n\\draw(a1) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a1) +(120:5.5mm) node {$\\interleave$};\n\\draw(d1) +(-0.5,0.5) node {$12)+2)+3)$};\n\n\\draw(4,2) node(a3) {$a$};\n\\draw(a3) +(90:1cm) node(b3) {$b$};\n\\draw(a3) +(30:1cm) node(c3) {$c$};\n\\draw(a3) +(150:1cm) node(d3) {$d$};\n\\draw[-latex,dashed] (a3) -- (b3);\n\\draw[-latex] (a3) -- (c3);\n\\draw[-latex] (c3) -- (b3);\n\\draw[-latex,dashed] (b3) -- (d3);\n\\draw[-latex,dashed] (a3) -- (d3);\n\\draw[-latex,dashed] (c3) edge[bend left=90] (d3);\n\\draw(a3) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a3) +(120:5.5mm) node {$\\times$};\n\\draw(d3) +(-0.5,0.5) node {$12)+22)+23)$};\n\n\\draw(8,2) node(a5) {$a$};\n\\draw(a5) +(90:1cm) node(b5) {$b$};\n\\draw(a5) +(30:1cm) node(c5) {$c$};\n\\draw(a5) +(150:1cm) node(d5) {$d$};\n\\draw[-latex,dashed] (a5) -- (b5);\n\\draw[-latex] (a5) -- (c5);\n\\draw[-latex] (c5) -- (b5);\n\\draw[-latex,dashed] (b5) -- (d5);\n\\draw[-latex,dashed] (a5) -- (d5);\n\\draw[-latex] (c5) edge[bend left=90] (d5);\n\\draw(a5) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a5) +(120:5.5mm) node {$\\subsetneqq$};\n\\draw(d5) +(-0.5,0.5) node {$12)+12)+13)$};\n\n\\draw(0,0) node(a2) {$a$};\n\\draw(a2) +(90:1cm) node(b2) {$b$};\n\\draw(a2) +(30:1cm) node(c2) {$c$};\n\\draw(a2) +(150:1cm) node(d2) {$d$};\n\\draw[-latex,dashed] (a2) -- (b2);\n\\draw[-latex] (c2) -- (a2);\n\\draw[-latex] (c2) -- (b2);\n\\draw[-latex,dashed] (b2) -- (d2);\n\\draw[-latex,dashed] (a2) -- (d2);\n\\draw(a2) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a2) +(120:5.5mm) node {$\\interleave$};\n\\draw(d2) +(-0.5,0.5) node {$13)+3)+3)$};\n\n\\draw(4,0) node(a4) {$a$};\n\\draw(a4) +(90:1cm) node(b4) {$b$};\n\\draw(a4) +(30:1cm) node(c4) {$c$};\n\\draw(a4) +(150:1cm) node(d4) {$d$};\n\\draw[-latex,dashed] (a4) -- (b4);\n\\draw[-latex] (c4) -- (a4);\n\\draw[-latex] (c4) -- (b4);\n\\draw[-latex,dashed] (b4) -- (d4);\n\\draw[-latex,dashed] (a4) -- (d4);\n\\draw[-latex,dashed] (c4) edge[bend left=90] (d4);\n\\draw(a4) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a4) +(120:5.5mm) node {$\\times$};\n\\draw(d4) +(-0.5,0.5) node {$13)+23)+23)$};\n\n\\draw(8,0) node(a6) {$a$};\n\\draw(a6) +(90:1cm) node(b6) {$b$};\n\\draw(a6) +(30:1cm) node(c6) {$c$};\n\\draw(a6) +(150:1cm) node(d6) {$d$};\n\\draw[-latex,dashed] (a6) -- (b6);\n\\draw[-latex] (c6) -- (a6);\n\\draw[-latex] (c6) -- (b6);\n\\draw[-latex,dashed] (b6) -- (d6);\n\\draw[-latex,dashed] (a6) -- (d6);\n\\draw[-latex] (c6) edge[bend left=90] (d6);\n\\draw(a6) +(60:5.5mm) node {$\\subsetneqq$};\n\\draw(a6) +(120:5.5mm) node {$\\subsetneqq$};\n\\draw(d6) +(-0.5,0.5) node {$13)+13)+13)$};\n\\end{tikzpicture}\n\\caption{How to resolve 22)\/23) of Fig.~\\ref{f.25} when the edge $a\\protect\\included b$\nin the triangle $a$\/$b$\/$d$ is created from strengthening $a\\protect\\disjorincl b$ via $c$ and 12) or 13).\nAll possible relations between $c$ and $d$ are shown.\nThe $\\subsetneqq$ sign in the right triangle denotes the strengthening of the $a\\protect\\disjorincl b$ edge,\nthe sign in the left triangle belongs to both edges leading to $d$, $a\\protect\\disjorincl d$\nand $b\\protect\\disjorincl d$. The numbers at each picture show which parts of Fig.~\\ref{f.25} are applied\nto the three triangles containing $c$\n\\label{f.indirect}}\n\\end{figure}\n\n\nNote how the middle column of Fig.~\\ref{f.25} is now fully resolved: If there is no contradiction\nprohibiting synthesis, we know exactly whether the edge $a\\disjorincl b$ stands for disjointness\nor for proper inclusion. An unresolved situation $c\\included a\\disjorincl b$ does not occur anymore.\nAs a consequence, a self-loop $a$ with $a\\disjorincl b$ can only be deactivated in a synthesised\n\\WPI net by a label\/transition whose preset encompasses that of $a$.\nSince 7) is the only unresolved situation in the second column,\na proper inclusion edge starting at $a$ implies a proper inclusion edge at $b$ to the\nsame target label.\nUnfortunately, we find no more simple conclusions like this one as every other row and column of the figure\nstill contains more than one unresolved situation.\n\nThe above conclusions reduce the number of $\\disjorincl$ edges in our relations, but\nsome of them may remain. For all such edges, the possibilities, disjointness and\nproper inclusion, have to be tested all in conjunction when constructing inequality systems \nfor synthesis.\nSince the number of these edges is not limited in general, synthesis to \\WPI nets \nmay need exponential time with respect to them.\nWe might argue that we expect relatively few labels compared to states ($O(|\\Sigma|)\\subseteq O(\\log|S|)$)\nso that the synthesis keeps its polynomial time complexity, or that we have only\none `idle' label forming a self-loop in the \\lts, giving us just a constant factor\nas overhead. We can reduce the problem by the shown strenghtening of edges, which\ncan be computed in $O(|S|\\cdot|\\Sigma|^2+|\\Sigma|^3)$ compared to roughly $O(|S|^6)$ for solving a\nsingle inequality system in the rational numbers~\\cite{karmarkar}. But in the end,\nwe still have a potentially exponential problem. We may guess for each edge\n$\\disjorincl$ if it means disjointness or proper inclusion, bringing us into\nthe class \\NP.\n\n\\CON{c.1}{Self-loops might make \\WPI synthesis exponential}\\\\\nIf an \\lts has no self-loops or just one, synthesis to the target class \\WPI\ncan be done in polynomial time. With an arbitrary number of labels forming self-loops,\nwe rather expect the problem to be \\NP-complete,\nbut we also expect `realistic' \\lts to fit into the first category.\n\\ENDCON{c.1}\n\n\\section{Self-loops and the class \\BRAC}\\label{sect.5}\n\nWhen targetting our second class of nets, \\BRAC, tackling self-loops is considerably\neasier. Any asymmetric choice block of transitions consists of two subblocks, where\none has a single, common place as preset, and the other has this place and one other\nas a common preset. Two asymmetric choice blocks do not have common transitions.\nTherefore, the situations shown in Fig.~\\ref{f.ntbrac} are impossible when targetting\nat \\BRAC nets\n\\footnote{\nFor \\ACC nets only the third situation in Fig.~\\ref{f.ntbrac} is impossible as all places\nin ${}^\\bullet a\\cup{}^\\bullet b$ have the common postset transition $c$ and must thus have\ncomparable postsets. The fifth situation can easily occur when $a$, $b$, and $c$ share a\ncommon preset place but $b$ and $c$ also have unshared preset places.\n}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}\n\\draw(0,0) node(a1) {$a$};\n\\draw(a1) +(90:1cm) node(b1) {$b$};\n\\draw(a1) +(30:1cm) node(c1) {$c$};\n\\draw[-latex] (a1) -- (b1);\n\\draw[-latex] (a1) -- (c1);\n\\draw[-latex] (b1) -- (c1);\n\\draw(a1) +(60:5.5mm) node {$\\times$};\n\n\\draw(2,0) node(a2) {$a$};\n\\draw(a2) +(90:1cm) node(b2) {$b$};\n\\draw(a2) +(30:1cm) node(c2) {$c$};\n\\draw[-latex,dashed] (a2) -- (b2);\n\\draw[-latex] (a2) -- (c2);\n\\draw[-latex] (b2) -- (c2);\n\\draw(a2) +(60:5.5mm) node {$\\times$};\n\n\\draw(4,0) node(a3) {$a$};\n\\draw(a3) +(90:1cm) node(b3) {$b$};\n\\draw(a3) +(30:1cm) node(c3) {$c$};\n\\draw[-latex] (a3) -- (c3);\n\\draw[-latex] (b3) -- (c3);\n\\draw(a3) +(60:5.5mm) node {$\\times$};\n\n\\draw(6,0) node(a4) {$a$};\n\\draw(a4) +(90:1cm) node(b4) {$b$};\n\\draw(a4) +(30:1cm) node(c4) {$c$};\n\\draw[-latex] (a4) -- (b4);\n\\draw[-latex] (a4) -- (c4);\n\\draw[-latex,dashed] (b4) -- (c4);\n\\draw(a4) +(60:5.5mm) node {$\\times$};\n\n\\draw(8,0) node(a5) {$a$};\n\\draw(a5) +(90:1cm) node(b5) {$b$};\n\\draw(a5) +(30:1cm) node(c5) {$c$};\n\\draw[-latex] (a5) -- (b5);\n\\draw[-latex] (a5) -- (c5);\n\\draw(a5) +(60:5.5mm) node {$\\times$};\n\\end{tikzpicture}\n\\caption{For \\BRAC, proper inclusions with more than two labels involved are possible only if two labels are equivalent.\nIt does not matter whether the proper inclusion arrows are consecutive or begin or end at the same label\n\\label{f.ntbrac}}\n\\end{figure}\n\nWhen looking at Fig.~\\ref{f.25}, this means that in 7), 9), 11), 14), and 15) the edge $a\\disjorincl b$ needs\nto represent disjointness and can be strengthened accordingly. In the remaining unresolved cases\n1), 4), 5), 16), 19), 21), 24), and 25) all labels only have edges $\\disjorincl$ or show disjointness,\nproper inclusion edges do not occur anymore. Thus, the transitions in a synthesised \\BRAC net corresponding\nto these labels form a block with presets completely disjoint to all other transitions\n(aside from equivalent labels, which we have ignored).\n\nFor two edges $a\\disjorincl b\\disjorincl c$ we do not see immediately which of the two edges enforces\nproper inclusion of presets (if any), but clearly not both of them can have this meaning.\nA closer look shows:\n\n\\LEM{l.firstdoiisdisj}{The front edge $\\disjorincl$ means disjointness}\nLet $TS=(S,\\Sigma,\\to,s_0)$ be a finite, reachable, deterministic \\lts with label relations\nstrengthened according to Section~\\ref{sect.4}.\nLet there be three labels $a,b,c\\in\\Sigma$ with $a\\disjorincl b\\disjorincl c$. \nThen, if $TS$ is synthesisable to \\BRAC, we can always find a \\BRAC net $N=(P,T,W,M_0)$ solving $TS$ with \n${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$.\n\\ENDLEM\n\\BEW\nNote first, that by case~6 $a$ and $b$ are both self-loops in $TS$. There may be labels\nequivalent to $a$ or $b$, but by Lemma~\\ref{l.equiv} these also all are self-loops in $TS$.\nAssume now that $a\\disjorincl b$ stands for proper inclusion, otherwise we are done.\nSince $a$ and $b$ and their equivalents are then related to all other non-equivalent labels by disjointness,\nno other transition $t$ in a \\BRAC net solving $TS$ can take tokens from a place $p$\nin the preset of $a$ ($W(p,a)>0\\then a\\equivalent t\\vee b\\equivalent t\\vee W(p,t)=0$). Assume a region $r=(R,B,F)$\nsolving an ESSP $(s,a)$ or an SSP $(s,s')$ with $s,s'\\in S$. Then $B(b)=F(b)$, and with\n$B'(b')=F'(b')=0$ for all $b'\\in\\Sigma$ with $b\\equivalent b'$ and $B'(t)=B(t)\\wedge F'(t)=F(t)$ for all\nother labels $t$, the region\n$r'=(R,B',F')$ is also a solution for the inequality system for the same ESSP or SSP.\nThus, by making a disjoint copy of the preset of $a$ and disconnecting $a$ and $b$ (and their equivalent labels) from\neach other's new preset, we obtain a solution for $TS$ with ${}^\\bullet a\\cap{}^\\bullet b=\\emptyset$.\nClearly, the \\BRAC conditions still hold, as there are no other (non-equivalent) transitions involved\nthat take tokens from the preset of $a$ (old or new). \n\\ENDBEW{l.firstdoiisdisj}\n\nWe can now, in every chain $a\\disjorincl b\\disjorincl c$, replace the first edge by disjointness\nsimultaneously. What remains are two sets $A$ and $B$ of labels, such that inside the sets\n$\\forall a,a'\\in A\\colon a\\disjoint a'\\vee a\\equivalent a'$ and\n$\\forall b,b'\\in B\\colon b\\disjoint b'\\vee b\\equivalent b'$ hold\nwhile between the sets we have\n$\\forall a\\in A, b\\in B\\colon a\\disjoint b\\vee a\\disjorincl b$.\n\n\\tikzset{rloop\/.style={to path={.. controls +(325:1) and +(35:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{lloop\/.style={to path={.. controls +(145:1) and +(215:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{dloop\/.style={to path={.. controls +(235:1) and +(305:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{drloop\/.style={to path={.. controls +(280:1) and +(350:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{tloop\/.style={to path={.. controls +(55:1) and +(125:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{trloop\/.style={to path={.. controls +(0:1) and +(70:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\tikzset{tlloop\/.style={to path={.. controls +(160:1) and +(90:1) .. (\\tikztotarget) \\tikztonodes}}}\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[LTS]\n\\node[state,label=above:$s_0$] (s0) {};\n\\path[edge] (s0) ++(-0.5,0) edge (s0);\n\\node[state,right of=s0,label=right:$s_1$] (s1) {};\n\\draw[edge] (s0) edge node[auto,xshift=-1mm]{$a$} (s1);\n\\draw[edge] (s1) edge[tlloop] node[auto,swap]{$c$} (s1);\n\\node[state,above right of=s1,label=below:$s_2$] (s2) {};\n\\draw[edge] (s1) edge node[auto,xshift=1.5mm]{$a$} (s2);\n\\node[state,below right of=s1,label=above:$s_3$] (s3) {};\n\\draw[edge] (s1) edge node[auto,swap,xshift=0.5mm,yshift=0.5mm]{$b$} (s3);\n\\node[state,below right of=s2,label=left:$s_4$] (s4) {};\n\\draw[edge] (s2) edge node[auto,xshift=-1.5mm]{$b$} (s4);\n\\draw[edge] (s2) edge[tloop] node[auto]{$c$} (s2);\n\\draw[edge] (s3) edge node[auto,swap,xshift=-0.5mm]{$a$} (s4);\n\\draw[edge] (s3) edge[drloop] node[auto]{$c$} (s3);\n\\node[state,below right of=s4,label=below:$s_5$] (s5) {};\n\\draw[edge] (s4) edge node[auto,xshift=-0.5mm,yshift=-1mm]{$b$} (s5);\n\\draw[edge] (s4) edge[trloop] node[auto]{$c$} (s4);\n\\draw[edge] (s5) edge[rloop] node[auto]{$c$} (s5);\n\\node[state,below left of=s1,label=above:$s_6$] (s6) {};\n\\draw[edge] (s4) .. controls +(0.5,-2.3) and +(0,-2.3) .. node[auto]{$d$} (s6);\n\\draw[edge] (s6) edge node[auto,swap,yshift=0.5mm]{$e$} (s1);\n\\node[state,below right of=s6,label=left:$s_7$] (s7) {};\n\\draw[edge] (s6) edge node[auto,yshift=-0.5mm]{$b$} (s7);\n\\draw[edge] (s7) edge node[auto,yshift=-0.5mm]{$e$} (s3);\n\\draw[edge] (s6) edge[lloop] node[auto]{$c$} (s6);\n\\draw[edge] (s7) edge[drloop] node[auto]{$c$} (s7);\n\\end{tikzpicture}\\hspace*{1cm}\n\\begin{tikzpicture}\n\\draw(0,3) node[place,label=left:$p_1$,tokens=2] (p1) {};\n\\draw(0,2) node[place,label=left:$p_2$] (p2) {};\n\\draw(0,0) node[place,label=left:$p_3$] (p3) {};\n\\draw(0,1) node[transition] (b) {$b$};\n\\draw(1.5,2.25) node[transition] (a) {$a$};\n\\draw(1.5,0.75) node[transition] (d) {$d$};\n\\draw(3,2.25) node[place,label=below:$p_4$] (p4) {};\n\\draw(3,0.75) node[place,label=above:$p_5$] (p5) {};\n\\draw(4.5,2.25) node[transition] (c) {$c$};\n\\draw(5.5,1.5) node[transition] (e) {$e$};\n\\draw(p1) edge[-latex,thick] (a);\n\\draw(a) edge[-latex,thick] (p2);\n\\draw(p2) edge[-latex,thick] (b);\n\\draw(b) edge[-latex,thick] (p3);\n\\draw(p2) edge[-latex,thick,bend left=20] (d);\n\\draw(d) edge[-latex,thick,bend left=20] (p2);\n\\draw(p3) edge[-latex,thick] (d);\n\\draw(d) edge[-latex,thick] (p5);\n\\draw(a) edge[-latex,thick] (p4);\n\\draw(p4) edge[-latex,thick,bend left=20] (c);\n\\draw(c) edge[-latex,thick] (p4);\n\\draw(p4) edge[-latex,thick] (e);\n\\draw(p5) edge[-latex,thick] (e);\n\\draw(e) edge[-latex,thick,out=90,in=0] (p1);\n\\path(p3) +(0,-1);\n\\end{tikzpicture}\\vspace*{-5mm}\n\\caption{An \\lts synthesisable to the shown (B)\\RAC net. Note that $s_0\\step{a}s_1$ activates $c$ and $a$ must\nput a token in $c$'s preset. The cycle $s_1\\step{abde}s_1$ is token-neutral, so one of its transitions\nmust remove a token from $c$'s preset}\n\\label{f.bracnodisj}\n\\end{figure}\n\nUnfortunately, we cannot choose all the remaining $a\\disjorincl b$ edges to mean disjointness.\nFigure~\\ref{f.bracnodisj} shows an \\lts together with a \\BRAC (and also \\RAC) net solving the \\lts.\nWe find $c\\disjorincl b$, $c\\disjorincl d$, $c\\disjorincl e$, and $b\\included d$,\nall other pairs are not connected. Now, $b\\included d$ forces $c$ to be disjoint from $b$ and $d$,\notherwise we obtain one of the situations of Fig.~\\ref{f.ntbrac}. If $c$ and $e$ also have\ndisjoint presets in a synthesised \\BRAC net, we will conclude that the cycle $s_1\\step{abde}s_1$\ncannot take tokens from $c$'s preset, so it cannot put tokens there either. But $a$ must put a\ntoken there as $s_0\\step{a}s_1$ activates $c$. So, ${}^\\bullet c\\subsetneqq{}^\\bullet e$ holds in\nevery \\BRAC net solving this \\lts. As we cannot enforce disjointness, we must compute whether\nany edge $\\disjorincl$ means proper inclusion or if disjointness is possible.\n\n\\SATZ{l.esspbrac}{Linear number of inequality systems for an ESSP} \nFor a finite, reachable, deterministic \\lts $TS=(S,\\Sigma,\\to,s_0)$ assume label relations\nto be constructed according to Def.~\\ref{d.grl} and strengthened via Lemma~\\ref{l.equiv}, Fig.~\\ref{f.25},\nFig.~\\ref{f.ntbrac}, and Lemma~\\ref{l.firstdoiisdisj}.\nWhen targetting at \\BRAC,\nfor each ESSP $(s,a)$ with $s\\in S$ and $a\\in\\Sigma$ we need to solve at most $|\\Sigma|$ inequality systems.\n\\ENDSATZ\n\\BEW\nIf there is no edge $\\disjorincl$ or $\\included$ adjacent to $a$ in the label relations, \n$a$ is not involved in any asymmetric choice. Each ESSP $(s,a)$ with $s\\in S$ can then be solved\nseparately as $a$ can have as many places in its preset as necessary. If any of these\ninequality systems is unsolvable, the \\lts is not synthesisable to a \\BRAC net.\n\nIf an edge $\\included$ starts or ends at $a$, $a$ is in an asymmetric choice for certain.\nAll ESSPs $(s,a)$ for $s\\in S$ must be solved by the same inequality system, as shown in\nSection~\\ref{sect.3}. If the inequality system is not solvable, synthesis fails.\n\nIf an edge $b\\disjorincl a$ exists with $b\\in\\Sigma$, $b$ is a self-loop and firing $b$\nin a synthesised \\BRAC net (if it exists) will not change the net's marking, especially\nin the preset of $a$. \nWe check whether we are able to construct a preset for $a$ if all edges\n$t\\disjorincl a$ with $t\\in\\Sigma$ are interpreted as disjointness, trying to solve one\ninequality system per ESSP. If this is not possible,\n$TS$ cannot be synthesised at all. Checking whether any edge $\\disjorincl$ ending at $a$ must mean proper\ninclusion is postponed to the following case.\n\nAssume now $a\\disjorincl b$ for some $b\\in\\Sigma$. We try to interpret all edges $\\disjorincl$\nstarting at $a$ as disjointness and solve each ESSP $(s,a)$ with $s\\in S$ separately.\nSuppose this is possible, i.e.\\ all these ESSPs are solvable.\nSince $a$ cannot change any token distribution in a synthesised net, this will not help $b$ in any way.\nAs $b$ is arbitrary, sharing preset places with $a$ never helps in solving other ESSPs. We may therefore\nchoose to make a free-choice block of $a$ and its equivalents in a \\BRAC net.\n\nIf not all ESSPs $(s,a)$ are solvable when interpreting all edges $a\\disjorincl$ as disjointness,\none of the edges, $a\\disjorincl b$ for some $b\\in\\Sigma$ must mean proper inclusion,\n${}^\\bullet a\\subsetneqq{}^\\bullet b$, but we do not know which edge.\nFor each possibility, we construct one inequality system for all ESSPs $(s,a)$ and try to solve it.\nIf the inequality system for $a\\disjorincl b$ meaning proper inclusion is not solvable, the edge\nmust be interpreted as disjointness. We collect all pairs $(a,b)$ where proper inclusion for\nthe edge $a\\disjorincl b$ leads to a solvable inequality system in a set $\\Lambda$.\n\nAfter dealing with all the edges $\\disjorincl$ in this way, $\\Lambda$ contains a number of pairs $(t,t')$. If, for a given $t$, for two pairs\n$(t,t'),(t,t'')\\in \\Lambda$ holds $t'\\equivalent t''$, \nwe choose $t'$ as a representative for the equivalence and remove $(t,t'')$ from $\\Lambda$.\nIf for some $t$ there remain pairs $(t,t'),(t,t'')\\in\\Lambda$ where $t'$ and $t''$ are not\nequivalent, we must find out if the proper inclusion should hold towards $t'$ or $t''$\n(or possibly another label). This is a bipartite matching problem: Given the labels and\nedges occuring in the bipartite graph formed by $\\Lambda$, we must select\n$|\\{a\\mid (a,b)\\in\\Lambda\\}|$ edges such that no label (i.e.\\ node of the graph) is\nadjacent to more than one of them. Due to the graph structure, $|\\{a\\mid (a,b)\\in\\Lambda\\}|$\nis also the maximal number of edges that can ever be selected with this property.\nThis maximum bipartite matching problem~\\cite{mv80} can be solved in polynomial time, in\nour case in $O(|\\Sigma|^{2.5})$. If the maximum matching has exactly $|\\{a\\mid (a,b)\\in\\Lambda\\}|$\nedges, all our problems are solved, we know which edges $\\disjorincl$ mean proper inclusion, \nand we refer to the previously computed solutions for our inequality\nsystem to construct the places for the synthesised \\BRAC net. \n\nThe maximal number of inequality systems to solve for a single ESSP $(s,a)$ is then $|\\Sigma|$, \none to test for disjointness and (at most) one for each possible edge $a\\disjorincl b$ \nor $b\\disjorincl a$ with $b\\in\\Sigma\\backslash\\{a\\}$.\n\\ENDBEW{l.esspbrac}\n\nNote that there are still two sources for a possibly non-polynomial runtime, one of which\nwe cannot avoid:\nIntegrating SSPs into the inequality systems for ESSPs (as explained in Section~\\ref{sect.3}) \nmay still lead to an exponential number of inequality systems\nand\/or solving inequality systems remains \\NP-complete for the class \\BRAC.\n\n\n\\section{Conclusion}\\label{sect.O}\n\nWe have shown that synthesis of an \\lts to obtain a Petri net from the class \\WPI can be done\nin polynomial time if the \\lts contains only a fixed number of labels forming self-loop edges, \ni.e. cycles of length one. Otherwise, we only have an exponential-time algorithm in \\NP. \nThe true goal of this paper was the synthesis to nets in some class of asymmetric choice,\nthough. Due to the nature of the asymmetric choice condition, synthesis to the general class\nof asymmetric choice nets (\\ACC) cannot even be formulated in the context of separation problems.\nFor one subclass of \\ACC, \\BRAC (block-reduced asymmetric choice nets), we\nhave shown that the synthesis can be done at all, but we have also identified several sources\nof a possible non-polynomial runtime. If we allow arc weights greater than one (reducing\nproblems of solving inequality systems to rational solutions with a polynomial runtime),\nwe run into combinatorial problems with state separation (SSPs). Self-loops, which posed\nthe biggest problem for \\WPI synthesis, can essentially be dealt with in polynomial time\nfor the class \\BRAC.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nDue to the advancement of network science, it has become possible to incorporate the salient features of opinion formation and spreading into large-scale network models \\cite{sznajd2000,deffuant2000,hegselmann2002,krapivsky2003,schwaemmle2007,jao2009,ramos15}. The study of political mobilization is one example of such opinion dynamics, which allowed remarkable observations such as the identification of universal features in elections \\cite{fortunato07}, the importance of easily persuadable individuals for opinion cascades \\cite{watts07}, the impact of online social influence \\cite{bond2012} and the necessity of social reinforcement in order to convince people \\cite{chwe99,granovetter78,rogers2010diffusion}.\n\nIn recent years, political mobilization and the associated campaigns became more and more sophisticated due to online social media, the availability of data on personal preferences and the huge financial campaign support\\footnote{Insidegov.com, Compare Presidential Candidates 2016, \\url{http:\/\/presidential-candidates.insidegov.com\/}, retrieved August 31, 2017.} \\cite{ramswell2017,boettcher_petitions}. Still, the influence of campaign characteristics such as the number of political activists, political clout or the size of the campaign budget on actual outcomes is not well understood. A more profound understanding of political campaigns is necessary to appropriately interpret the corresponding outcomes and to possibly redesign certain aspects of campaign policies \\cite{Gersbach2017}. Thus, we  propose a novel mathematical framework to describe political campaign dynamics on networks. More specifically, we account for the fact that mobile political activists can convince persuadable individuals under certain efficiency and cost restrictions. We demonstrate the existence of one unique stationary solution and rigorously describe the interplay between activists, political clout, budgets and campaign costs on this state, thereby understanding how candidates can win elections. We find that a given budget might allow different choices of the number of activists, such that campaign groups find themselves in an activist-choice game. In these strategic situations, a campaign group chooses activists as a best response to the choice of the other campaign group, which may be crucial for winning the election. Interestingly, some activist combinations lead to strong competition and thus to a large campaign budget. Furthermore, an advantage in terms of political clout can overcome a substantial budget disadvantage or a lower number of activists. This is illustrated for the US presidential election of 2016, in which the winner had a huge budget disadvantage and a lower number of activists to support him.\n\nElection campaigns are an essential component of democratic politics. They have been studied extensively by scholars across social sciences and political sciences, in particular, as surveyed recently in Ref.~\\cite{jacobson2015_2}. Three characteristics from this rich body of literature are particularly important for the construction of our model and its investigation:\n\\begin{enumerate}\n\\item Some voters are amenable to persuasion and opinion switching, i.e., they might be induced to vote for either candidate if approached by a corresponding campaign activist. In the US presidential election of 2016, the proportion of undecided voters was estimated to be around 20\\% to 25\\% \\cite{kenski2010}. The share of persuadable voters is also large for elections below presidential level,\n\\item The most effective tactics in campaigns are personal interactions between activists and persuadable voters, such as door-to-door interaction or phone calls \\cite{jacobson2015},\n\\item \\lb{Money matters in campaigns. The benefit of campaign expenditures is positive, but typically low and subject to rapid decay \\cite{jacobson2015}.}\n\\end{enumerate}\n\\section*{Materials and methods}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Fig1-eps-converted-to.pdf}\n  \\caption{\\textbf{Illustration of the campaign model.} An illustration of the campaign model where nodes and their corresponding edges are represented by colored circles and black lines respectively. In this example, an $A^+$ activist node is going to occupy an empty node $O$ to convince  $B^0$ nodes. The utility condition $\\rho_A \\#B^0-c_A > 0$ in this case is, for example, fulfilled for $\\rho_A=1$ and $c_A=2$.}\n \\label{fig:model}\n\\end{figure}\nWe consider a network with $N$ nodes. Each node is either occupied by an individual having an opinion $A$ or $B$ or it is in an unoccupied state $O$. The total number of nodes is thus $N=N_A+N_B+N_O$. Both campaign groups consist of political activists $A^+$ and $B^+$ and persuadable individuals $A^0$ and $B^0$, such that $N_A=N_{A^+}+N_{A^0}$ and $N_B=N_{B^+}+N_{B^0}$. We illustrate the campaign model in Fig.~\\ref{fig:model}. Activists have a fixed opinion and they are able to move from their current location to unoccupied nodes to convince persuadable individuals who do not change their location. Each campaign group is given a finite budget $\\mathcal{B}_A$ and $\\mathcal{B}_B$ respectively, and a certain cost $c_A$ and $c_B$, respectively, has to be paid for every attempt at convincing a group of persuadable individuals. More specifically, an activist $A^+$ has the possibility to occupy an empty node $O$ from any place in order to change with probability $\\rho_A \\in (0,1]$ the opinion of neighboring $B^0$ nodes. The activist $A^+$ only moves and convinces other $B^0$ nodes if the utility condition $\\rho_A \\# B^0 -c_A >0$ is fulfilled and if enough budget $\\mathcal{B}_A>0$ is available. Here $\\# B^0$ represents the number of neighboring nodes of the empty node which are in state $B^0$. If the $A^+$ activist does move onto the empty node, independent of the number of persuaded $B^0$ nodes, the cost $c_A$ is deducted from the budget $\\mathcal{B}_A$ \\cite{boettcher14}. The same applies to activists $B^+$, however, with probability $\\rho_B$, cost $c_B$ and budget $\\mathcal{B}_B$.\n\n\\lb{From now on, we consider the fractions $a=A\/N$, $b=B\/N$ and $o=O\/N$ of the states $A$, $B$ and $O$ respectively.} The initial values are given by $a^0(0)$, $b^0(0)$, $a^+(0)$, $b^+(0)$ and $o(0)$.\nBy definition of our model, we know that only $a^0$ and $b^0$ change over time. Thus, $\\dot{a}^+=0$, $\\dot{b}^+=0$, $\\dot{o}=0$ and thus $a^+(t)=a^+(0)$, $b^+(t)=b^+(0)$ and $o(t)=o(0)$. It is now possible to derive mean-field rate equations of $\\dot{a}^0(t)$ and $\\dot{b}^0(t)$ by assuming a perfectly mixed population in the thermodynamic limit. Mean-field approximations have been proven useful to gain important insights about a given spreading dynamics \\cite{keeling-rohani2008,marro05}, and many real-world social networks are well described by mean-field approximations \\cite{gleeson12}. Our analytical results are supported by stochastic kinetic Monte Carlo simulations \\cite{gillespie76,gillespie77}. For analytical tractability, we assume a \\emph{regular network with fixed degree} $k$, i.e.~a fixed number of neighbors. A mean-field approach for general degree distributions $f_k$ is presented in Refs.~\\cite{boettcher162, boettcher171} and is based on an additional weighted sum accounting for different degrees in the network. We only focus on the derivation of $\\dot{a}^0(t)$, since $b^0=1-a^+-b^+-o-a^0$. Our first observation is that the utility condition $ \\rho_A \\# B^0 -c_A >0$ corresponds to a threshold \\cite{granovetter78,chwe99,watts02,macy07,easley2010,gleeson2013,boettcher162,boettcher171} in the sense that $A^+$ activists only move to empty nodes whose neighborhoods contain at least $\\ceil*{\\frac{c_A}{\\rho_A}}$ persuadable nodes in state $B^0$. Here $\\ceil*{\\cdot}$ denotes the ceil function since the quantity $\\frac{c_A}{\\rho_A}$ might not have an integer value. We define the transition functions $f_{A^0\\rightarrow B^0}(t)$ and $f_{B^0\\rightarrow A^0}(t)$ describing transitions $A^0\\rightarrow B^0$ and $B^0\\rightarrow A^0$, respectively, and find for our dynamics\n\\begin{equation}\n\\begin{split}\n\\dot{a}^0(t)= & f_{B^0\\rightarrow A^0}(t)-f_{A^0\\rightarrow B^0}(t)  \\\\\n= & o \\rho_A \\Theta(\\mathcal{B}_A) \\sum_{j=\\ceil*{\\frac{c_A}{ \\rho_A}}}^{k} \\sum_{l,m.n,r \\in S_{k- j}} j \\binom{k}{j,l,m,n,r} {b^0}^j {a^+}^{l+1} {b^+}^m {a^0}^n o^r\\\\ \n& -o \\rho_B \\Theta(\\mathcal{B}_B) \\sum_{j=\\ceil*{\\frac{c_B}{ \\rho_B}}}^{k} \\sum_{l,m.n,r \\in S_{k -j}} j \\binom{k}{j,l,m,n,r}{a^0}^j {a^+}^l {b^+}^{m+1} {b^0}^n o^r,\n\\end{split}\n\\label{eq:rate}\n\\end{equation}\nwhere $\\Theta(\\cdot)$ denotes the Heaviside step function accounting for the fact that opinion changes only occur when enough budget is available and $\\binom{k}{j,l,m,n,r}$ is the multinomial coefficient, such that $j+l+m+n+r=k$ and $S_{k- j}=\\left\\{ l,m,n,r \\geq 0, l+m+n+r=k-j \\right\\}$. The prefactor $o$ in Eq.~\\eqref{eq:rate} accounts for the necessity for a node to be empty before an activist can occupy it, and $\\rho_A$, $\\rho_B$ denote the probabilities to persuade others, henceforth \\emph{convincing probabilities}. The first double sum describes transitions $B^0\\rightarrow A^0$ where at least $\\ceil*{\\frac{c_A}{\\rho_A}}$ nodes of type $B^0$ need to be available according to the utility condition. The second term describes reverse transitions $A^0\\rightarrow B^0$. We notice that \\emph{no change} occurs if $\\ceil*{\\frac{c_A}{\\rho_A}}>k$ and $\\ceil*{\\frac{c_B}{\\rho_B}}>k$. The cost deductions from the respective budgets are described by \n\\begin{align}\n& \\dot{\\mathcal{B}}_A(t)=-\\frac{c_A}{\\rho_A} f_{B^0\\rightarrow A^0}(t), \\label{eq:budget_A} \\\\\n& \\dot{\\mathcal{B}}_B(t)=-\\frac{c_B}{\\rho_B} f_{A^0\\rightarrow B^0}(t),\n\\label{eq:budget_B}\n\\end{align}\nwith the initial conditions $\\mathcal{B}_A(0)=\\mathcal{B}_A^0$, $\\mathcal{B}_B(0)=\\mathcal{B}_B^0$ where $\\mathcal{B}_A^0$ and $\\mathcal{B}_B^0$ are the campaign budgets of the two campaign groups, respectively. \\lb{In general, our model could also be extended to more than two campaign groups, i.e.~more than two candidates. It is then necessary to account for the additional corresponding costs, convincing probabilities and transition functions.}\n\nIn the \\emph{Appendix}, we demonstrate that the dynamics captured by Eq.~\\eqref{eq:rate} exhibits a unique stationary solution for a given set of initial conditions. Furthermore, we show that there is a complex interplay between the fractions of activists, convincing probabilities and costs. In the case of $\\rho_A=\\rho_B$ and $c_A=c_B$, the campaign group with the larger fraction of activists obtains the majority. Otherwise, different convincing probabilities and different costs lead to a shift of the stationary states. In particular, a large convincing probability and small cost is advantageous for the respective campaign group. Convincing probabilities cannot be interchanged with the fractions of activists. \\lb{Small convincing probabilities and large costs lead to longer transients and to stationary states that are closer together, i.e., small gaps between the final campaign results. Unlike the activist fractions, convincing probabilities and costs also enter in the threshold condition.} We also discuss one example in the \\emph{Appendix}, which illustrates how the influence of $\\rho$ and $c$ might, surprisingly, lead to a change in the majority structure.\n\\section*{Results}\n\\label{sec:results}\n\\subsection*{Budget limitations}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{Fig2-eps-converted-to.pdf}\n  \\caption{\\textbf{Influence of budget restrictions.} The time evolution of the fractions of persuadable nodes $a^0(t)$, $b^0(t)$, activists $a^+(t)$, $b^+(t)$, empty nodes $o(t)$ as well as the budgets $\\mathcal{B}_A(t)$ and $\\mathcal{B}_B(t)$. The shown curves are the results of a stochastic simulation on a random regular network with $N=10^5$ nodes, $k=10$ and the following initial conditions: $a^0(0)=0.54$, $b^0(0)=0.01$, $a^+(0)=0.1$, $b^+(0)=0.15$ and $o(0)=0.2$. The convincing probabilities are fixed to $\\rho_A=\\rho_B=1$ and the costs to $c_A=c_B=2$. The thick grey curves are the numerical mean-field solutions of Eq.~\\eqref{fig:stochastic_1}. In the left panel, the budgets are exhausted after some time due to the small initial values $\\mathcal{B}_A(0)=2$ and $\\mathcal{B}_B(0)=2$, and thus the evolution changes. On the other hand, in the right panel, the initial budgets $\\mathcal{B}_A(0)=8.4$ and $\\mathcal{B}_B(0)=9.1$ are chosen to be large enough not to influence the evolution during the given time horizon.}\n \\label{fig:stochastic_1}\n\\end{figure}\nA political campaign typically lasts a fixed time $T$. We assume that political campaigns have information about the convincing probabilities $\\rho_A$ and $\\rho_B$ as well as the costs $c_A$ and $c_B$. A typical goal of each campaign is to convince as many people as possible by spending the available budget in such a way that there is no money left at time $T$ when campaigns start at time 0. Intuitively, the amount of budget spent does not only depend on the costs and the convincing probabilities but also on the number of activists in each campaign group. In particular, the budget $\\mathcal{B}_B$ will be exhausted faster than budget $\\mathcal{B}_A$ if the initial budgets, the costs and the convincing probabilities are equal but the fraction of activists of campaign group $B$ is larger than the one of group $A$. This might be problematic if the campaign lasts longer than the time until budget exhaustion, since campaign group $B$ will lose its majority sooner or later. We illustrate this effect in Fig.~\\ref{fig:stochastic_1} (left) where numerical simulations are presented together with the corresponding mean-field approximation. The finite size and the finite degree of the underlying random regular network lead to deviations from the mean-field perfect mixing assumption. For large degrees one finds better agreement between simulations and mean-field results \\cite{boettcher162}. Still, the simulated dynamics is qualitatively captured by Eq.~\\eqref{eq:rate}. To better understand the influence of budget restrictions, we now focus on two relevant points related to the latter example: (i) we want to determine the minimum necessary budget for given fractions of activists $a^+$, $b^+$ and a given campaign duration, and (ii) we want to discuss strategies \\emph{when} to send \\emph{how} many activists to effectively use a given budget so that it lasts until the campaign ends.\n\nWe begin with the discussion of point (i) and find that the minimum necessary budgets are \n\\begin{align}\n& \\mathcal{B}_A^{\\mathrm{min}}(T)=\\int_0^T \\frac{c_A}{\\rho_A}f_{B^0\\rightarrow A^0}(t) \\, \\mathrm{d}t, \\label{eq:min_budget_A} \\\\\n& \\mathcal{B}_B^{\\mathrm{min}}(T)=\\int_0^T \\frac{c_B}{\\rho_B}f_{A^0\\rightarrow B^0}(t) \\, \\mathrm{d}t,\n\\label{eq:min_budget_B}\n\\end{align}\naccording to Eqs.~\\eqref{eq:budget_A} and \\eqref{eq:budget_B}.\nTo solve the integrals defined by Eqs.~\\eqref{eq:min_budget_A} and \\eqref{eq:min_budget_B}, one first has to solve Eq.~\\eqref{eq:rate}, assuming sufficient resources, i.e.~$\\Theta(\\mathcal{B}_A)=1$ and $\\Theta(\\mathcal{B}_B)=1$ for all times. This solution has then to be used to compute the integrals, which yields unique solutions. We again look at Fig.~\\ref{fig:stochastic_1} (left), where the initial budget $\\mathcal{B}_B(0)=2$ turns out to be insufficient for a duration $T=15$. What is the necessary minimum in this case? We solve Eqs.~\\eqref{eq:min_budget_A} and \\eqref{eq:min_budget_B} and find that the mean-field initial budgets are $\\mathcal{B}_A^{\\mathrm{min}}(T)=8.4$ and $\\mathcal{B}_B^{\\mathrm{min}}(T)=9.1$. We clearly see in Fig.~\\ref{fig:stochastic_1} (right) that these budgets are sufficient. Due to the perfect mixing assumption, the mean-field dynamics tends to be slightly faster than the one of the simulations.\n\n\\subsection*{Strategic choices of activists}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{Fig3-eps-converted-to.pdf}\n  \\caption{\\textbf{Minimal budgets for different activist fractions.} We set $k=10$ and used the following initial conditions: $a^0(0)=b^0(0)=(1-a^+-b^+-o)\/2$ with $o(0)=0.2$. The convincing probabilities are fixed to $\\rho_A=\\rho_B=1$ and the costs to $c_A=c_B=2$. In the left panel, the minimal budgets $\\mathcal{B}_A^{\\mathrm{min}}(T)$ and $\\mathcal{B}_B^{\\mathrm{min}}(T)$ according to Eqs.~\\eqref{eq:min_budget_A} and \\eqref{eq:min_budget_B} are displayed as a function of the fraction $a^+$ for different values of $b^+$. The black solid line indicates a given budget $\\mathcal{B}=4.5$ for campaign of duration $T=15$ and the orange areas represent the corresponding regimes of possible choices for $a^+$ for which the given budget $\\mathcal{B}$ suffices. The right panel shows a three-dimensional illustration of the budget $\\mathcal{B}_A$ as a function of $a^+$ and $b^+$. The plane again represents a given budget $\\mathcal{B}=4.5$. All values of $a^+$ and $b^+$ leading to budget values below or on the plane are possible choices to have a sufficient budget.}\n \\label{fig:budget_1}\n\\end{figure}\nFor the discussion of point (ii), it is important to realize that while Eqs.~\\eqref{eq:min_budget_A} and \\eqref{eq:min_budget_B} yield unique minimal budgets for given $a^+$, $b^+$ and $T$, finding the two fractions of activists, for given campaign budgets and a duration $T$, is not a unique operation.\n\nHere we solely focus on choosing constant fractions of activists $a^+,b^+$ within the interval $ (0,0.3]$, since $a^+$ and $b^+$ are typically not too large. Fig.~\\ref{fig:budget_1} (left) helps to gain some intuition for the existence of multiple possible choices of $a^+$ and $b^+$ for given campaign budgets. We assumed equal convincing probabilities and costs. In this case, the campaign group with the larger fraction of activists wins the campaign in the limit of long times, as shown in the \\emph{Appendix}. Fig.~\\ref{fig:budget_1} (left) shows the solutions of Eqs.~\\eqref{eq:min_budget_A} and \\eqref{eq:min_budget_B} for $a^+\\in(0,0.3]$ and three values of $b^+\\in\\{0.04, 0.2, 0.3\\}$. All solutions are parabolas with a characteristic maximum originating from the competition between the two campaign groups. If one or the other campaign dominates, a relatively small budget suffices due to the small average number of opinion changes per unit of time, i.e.~small transition functions in Eqs.~\\eqref{eq:budget_A} and \\eqref{eq:budget_B}. However, if the two activist fractions are of similar size, the resulting competition leads to a larger necessary budget due to the larger transition functions in Eqs.~\\eqref{eq:budget_A} and \\eqref{eq:budget_B}. Considering a limited small budget, it is thus unfavorable for campaign groups to have activist numbers facilitating strong competition. Consequently, choosing the right number of activists is crucial for winning the election. In the more general case, one has to take the other parameters $\\rho_A$, $\\rho_B$, $c_A$ and $c_B$ into account which can lead to shifted maxima.\n\nWe now want to graphically determine a tuple $(a^+,b^+)$ such that a budget $\\mathcal{B}$ given to each campaign group suffices throughout the campaign duration. We assume a budget $\\mathcal{B}=4.5$ for campaign of duration $T=15$ and observe that, for example, in the case of $b^+=0.04$ it would allow for two solution branches, indicated by the orange areas in Fig.~\\ref{fig:budget_1} (left), where $a^+>0.27$ or $a^+<0.05$. In general, a given small budget restricts the possible choices of $a^+$ and $b^+$ to values on or below a plane, as shown in Fig.~\\ref{fig:budget_1} (right). If $\\mathcal{B}$ exceeds the maximum of the parabolas, all constellations of $a^+$ and $b^+$ are possible for the budgets to be sufficient, i.e.~$a^+, b^+ \\in (0,1)$. Interestingly, in the case of a small budget leading to two solution branches as in Fig.~\\ref{fig:budget_1} (left), one can interpret the situation as follows. Campaign group $B$, for example, takes a value of $b^+=0.3$. Then group $A$ can either send a very small fraction or one close or equal to $b^+$. In this case, group $B$ would never lose, but depending on the choice of group $A$, the difference between the final numbers of individuals in each campaign group would be smaller for larger $a^+$. However, if group $B$ takes a smaller value such as $b^+=0.04$, campaign group $A$ has the possibility to send a larger fraction to win the campaign. Therefore, the decision for a certain activist fraction made by one campaign group affects the possible choices of the other group. The right initial choice is important for winning the election.\n\nNext we look at the Nash equilibria of the strategic choices in the ensuing activist game as illustrated in Fig.~\\ref{fig:budget_1} (left). Formally, we investigate the best responses of the game defined by \n\\begin{equation}\n\\max_{\\mathcal{B}_A^{\\mathrm{min}}(T)\\leq 4.5,~b^+~\\mathrm{given}}\\{a^+\\},~\\max_{\\mathcal{B}_B^{\\mathrm{min}}(T)\\leq 4.5,~a^+~\\mathrm{given}}\\{b^+\\}.\n\\end{equation}\nA Nash equilibrium is a constellation $(a^+,b^+)$ that solves both problems. In our example a larger fraction of activists increases the chance of winning and we obtain three best responses for a fixed $b^+\\in \\{0.04,0.2,0.3\\}$\n\\begin{equation*}\n\\begin{split}\n& (a^+, b^+)_{b^+~\\mathrm{given}}=(0.3,0.04), \\\\\n& (a^+, b^+)_{b^+~\\mathrm{given}}=(0.03,0.2), \\\\\n& (a^+, b^+)_{b^+~\\mathrm{given}}=(0.3,0.3).\n\\end{split}\n\\end{equation*}\nAs a consequence of the parameters and initial conditions used in Fig.~\\ref{fig:budget_1}, the tuples obey the following symmetry: $(x, y)_{b^+~\\mathrm{given}}=(y,x)_{a^+~\\mathrm{given}}$. The game admits multiple equilibria. Besides $(0.3,0.3)$, for instance, there also exists another symmetric equilibrium around $(0.044, 0.044)$. Moreover, two campaigns with identical parameters, clout and budget are compatible with a constellation in which $A$ wins with a majority. The opposite constellation also exists, however. In practice, when the activists are not chosen simultaneously, we can look at how a campaign group can generate a first-mover advantage by choosing a large but still affordable number of activists that forces the other group to choose a lower number of activists. The preceding observation suggests that in a race of two campaigns having identical parameters and budgets, the campaign wins that first has assembled the right number of activists.\n\\subsection*{Can clout compensate a lower budget and fewer activists?}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\textwidth]{Fig4-eps-converted-to.pdf}\n  \\caption{\\textbf{Clout versus budget and activists.} The time evolution of the fractions of persuadable voters $a^0(t)$, $b^0(t)$, activists $a^+(t)$, $b^+(t)$, empty nodes $o(t)$ as well as the budgets $\\mathcal{B}_A(t)$ and $\\mathcal{B}_B(t)$. The shown curves are the results of a stochastic simulation on a random regular network with $N=10^5$ nodes, $k=10$ and the following initial conditions: $a^0(0)=0.32$, $b^0(0)=0.205$, $a^+(0)=0.175$, $b^+(0)=0.1$ and $o(0)=0.2$. The cost is set to $c_A=c_B=0.3$. The grey curves are the numerical mean-field solutions of Eq.~\\eqref{fig:stochastic_1}. The left panel shows the situation where both campaign groups have activists with $\\rho_A=\\rho_B=0.1$. In the right panel, the convincing probability in campaign group $B$ has been increased to a value of $\\rho_B=0.18$, leading to a change of the majority structure.}\n\\label{fig:stochastic_2}\n\\end{figure}\nThe model outlined above can be used to study political campaigns for specific parameter environments and to address campaign regulation. As an example we illustrate an application of the model and illustrate a phenomenon observed in the recent presidential election in the US battleground states. Three characteristics are widely discussed. First, Donald Trump had a significantly lower budget than Hillary Clinton \\footnote{Insidegov.com, Compare Presidential Candidates 2016, \\url{http:\/\/presidential-candidates.insidegov.com\/}, retrieved August 31, 2017.}. More precisely, he had half of Clinton's budget (404 versus 807 million US dollars). Second, Clinton had a higher number of volunteers than Trump (7\\% versus 4\\%)\\footnote{Ariel Edwards-Levy, Volunteering For A Campaign Or Going To Rallies? You're In The Minority, Huffpost, 11.01.2016, \\url{http:\/\/www.huffingtonpost.com\/entry\/campaign-volunteering-rallies-poll_us_581906b8e4b0f96eba968ca7}, retrieved August 31, 2017.}. Third, Trump was able to flip millions of white Obama supporters to his side. We thus investigate whether an advantage in terms of clout could compensate both a significantly lower budget and a lower share of activists. The clout is quantified through the convincing probabilities $\\rho_A$ and $\\rho_B$. In Fig.~\\ref{fig:stochastic_2} we illustrate a campaign situation roughly comparable to the one of the US presidential election 2016. We interpret campaign group $A$ as the Clinton group, whereas group $B$ represents Trump's campaign. We set the initial values of persuadable individuals to $a^0(0)=0.32$ and $b^0(0)=0.205$, taking into account the initial majority structure of the actual US presidential election 2016 poll data\\footnote{Poll Chart 2016 General Election: Trump vs.~Clinton, Huffpost Pollster, \\url{http:\/\/elections.huffingtonpost.com\/pollster\/2016-general-election-trump-vs-clinton}, retrieved August 31, 2017.}. Considering the aforementioned presidential election characteristics, we set $a^+=0.175$, $b^+=0.1$, $\\mathcal{B}_A^0=4$, $\\mathcal{B}_B^0=2$ to account for the different budgets and numbers of activists ($\\mathcal{B}_A^0\/\\mathcal{B}_B^0 \\approx 807\/404$ and $a^+\/b^+=7\/4$). Furthermore, we assume equal costs $c_A=c_B=0.3$ and first study the case of equal convincing probabilities $\\rho_A=\\rho_B=0.1$, as illustrated in Fig.~\\ref{fig:stochastic_2} (left). \\lb{As demonstrated in the \\emph{Appendix}, only the relative fraction $\\lambda=a^+\/b^+$ matters in determining the winning campaign if the convincing probabilities and costs are equal---campaign group $A$ wins if $\\lambda > 1$. This is the reason for our choice of $a^+\/b^+ = 0.175\/0.1=7\/4$.} Clearly, the Clinton campaign group $A$ dominates the dynamics due to the larger fraction of activists. However, in Fig.~\\ref{fig:stochastic_2} (right), we show that additional clout can compensate a significantly lower number of activists and a lower budget. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig5-eps-converted-to.pdf}\n \\caption{\\textbf{Poll and simulation results of the US presidential election 2016.} \\lb{Blue and red curves represent the poll results for Hillary Clinton and Donald Trump, respectively. We compare the poll results with our simulation (grey line) using a random regular network with $N=10^5$ nodes, $k=10$ and the following initial conditions: $a^0(0)=0.32$, $b^0(0)=0.205$, $a^+(0)=0.175$, $b^+(0)=0.1$ and $o(0)=0.2$. The cost is set to $c_A=c_B=0.3$ and the convincing probabilities to $\\rho_A=\\rho_B=0.1$. The results of the simulations have been rescaled to match the initial values of the polls and the corresponding duration. The data has been taken from \\url{http:\/\/elections.huffingtonpost.com\/pollster\/2016-general-election-trump-vs-clinton}.}}\n\\label{fig:trump_vs_clinton}\n\\end{figure}\n\\lb{Of course, one cannot say that Donald J.\\ Trump won the US presidential elections only because of more political clout. Other factors such as targeted campaigns or the sophisticated use of social media have also been important for his success \\cite{trump_media,trump_cambridge}. The example discussed here just illustrates the strong influence of political clout on the campaign outcome.}\n\nThis observation has broader implications for the resilience of democracy, as it is related to recent phenomena associated with populism. Candidates who promise to protect citizens from unemployment or income shocks caused by disruptions due to globalization or automation may be able to generate sufficient clout \\cite{rodrik17}. Such candidates may win election campaigns even if they will not be able to keep their promises or even enact detrimental policies.\n\n\\lb{Finally, we compare the simulation results presented in Fig.~\\ref{fig:stochastic_2} (left) with the US presidential election polls 2016, bearing in mind that the poll data did not reproduce the final result, as illustrated in Fig.~\\ref{fig:trump_vs_clinton}. For this comparison we rescaled the simulation results to match the initial values of the polls and the corresponding poll duration. As outlined above, our simulations take into account the empirically determined fractions of activists and assume values of $\\rho_A=\\rho_B=0.1$ and $c_A=c_B=0.3$. It has not been possible to determine these values empirically. Since convincing probabilities and costs are equal in this example, Clinton's campaign group wins due to the larger number of activists. We find that the simulated poll trajectories qualitatively correspond to the actual poll data. Assuming other values of convincing probabilities and costs would imply a different characteristic convergence time scale and a different gap between the final campaign results as shown in the \\emph{Appendix}. One would, however, still observe a qualitatively similar convergence behavior.}\n\\section*{Discussion}\nWe have developed a novel framework to study political campaign dynamics on social networks based on three essential features: moving activists, political clout and campaign budgets. We illustrated how the complex interplay between these factors determines the success of a campaign group. Our results imply that the right initial choice of the number of activists might lead to an important advantage in winning a campaign. In addition, we also showed that typical campaign characteristics can be taken into account by our model allowing to apply it to the US presidential election of 2016. What is more, our model opens up many further applications. For instance, we could introduce caps on campaign budgets to study how such regulations impact the outcome of elections. Furthermore, our framework allows to integrate additional channels such as media or targeting \\cite{boettcher_targeted} which strongly influence campaigns. Finally, by introducing preferences of citizens about policy-making once a candidate is in office, we can identify conditions under which a politician may win and enacts policies that will harm a majority of citizens.\n\\section*{Acknowledgments}\nWe acknowledge financial support from the ETH Risk Center and ERC Advanced grant number FP7-319968 FlowCCS of the European Research Council. We thank Moritz Hoferer for his input regarding the strategic choices of activists and for proofreading the manuscript. L.B.~wants to thank Sergio Sol\\'{o}rzano for interesting discussions.\n\n\\section*{Author Contributions}\n\\label{sec:contr}\nConceived and designed the experiments: LB HJH HG. Performed the experiments: LB. Analyzed the data: LB. Wrote the paper: LB HJH HG.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nGiven oracle access to\na\nmetric\nspace $(\\{1,2,\\ldots,n\\},d)$,\nthe {\\sc metric $1$-median}\nproblem asks for a point with the minimum average distance to all points.\nIndyk~\\cite{Ind99, Ind00} shows that {\\sc metric $1$-median} has a\nMonte-Carlo $O(n\/\\epsilon^2)$-time $(1+\\epsilon)$-approximation algorithm\nwith an $\\Omega(1)$ probability of success.\nThe more general {\\sc metric $k$-median} problem asks for\n$x_1$,\n$x_2$, $\\ldots$, $x_k\\in\\{1,2,\\ldots,n\\}$ minimizing\n$\\sum_{x\\in\\{1,2,\\ldots,n\\}}\\,\\min_{i=1}^k\\,d(x_i,x)$.\nRandomized\nas well as\nevasive\nalgorithms are well-studied for\n{\\sc metric $k$-median} and the related $k$-means problem~\\cite{GMMMO03,\nMP04, AGKMMP04, Che09, KSS10, JKS12},\nwhere $k\\ge 1$ is part of the input rather than a constant.\n\n\nThis paper focuses on {\\em deterministic sublinear-query} algorithms for\n{\\sc metric $1$-median}.\nGuha et al.~\\cite[Sec.~3.1--3.2]{GMMMO03} prove\nthat {\\sc metric $k$-median} has\na deterministic\n$O(n^{1+\\epsilon})$-time\n$O(n^\\epsilon)$-space\n$2^{O(1\/\\epsilon)}$-approximation algorithm\nthat reads distances in a single pass, where $\\epsilon>0$.\nChang~\\cite{Cha13} presents a deterministic\nnonadaptive\n$O(n^{1.5})$-time $4$-approximation algorithm for {\\sc metric $1$-median}.\nWu~\\cite{Wu14} generalizes Chang's result by showing an\n$O(n^{1+1\/h})$-time\n$2h$-approximation\nalgorithm for\nany integer $h\\ge 2$.\nOn the negative side,\nChang~\\cite{Cha12}\nshows that {\\sc metric $1$-median}\nhas no\ndeterministic\n$o(n^2)$-query\n$(3-\\epsilon)$-approximation algorithms for any constant\n$\\epsilon>0$~\\cite{Cha12}.\nThis\npaper\nimproves upon his result by showing\nthat {\\sc metric $1$-median} has no deterministic\n$o(n^2)$-query $(4-\\epsilon)$-approximation algorithms for any\nconstant $\\epsilon>0$.\n\nIn social network analysis,\nthe importance of an actor in a network\nmay\nbe\nquantified by\nseveral\ncentrality measures, among which the closeness centrality of an actor is\ndefined to be\nits average distance to other actors~\\cite{WF94}.\nSo\n{\\sc metric $1$-median}\ncan\nbe interpreted as the problem of finding the\nmost important point in a metric space.\nGoldreich and Ron~\\cite{GR08} and Eppstein and Wang~\\cite{EW04} present\nrandomized algorithms for approximating the closeness centralities of\nvertices in undirected graphs.\n\n\n\n\\section{Definitions}\\label{definitionssection}\n\nFor $n\\in\\mathbb{N}$,\ndenote\n$[n]\\equiv \\{1,2,\\ldots,n\\}$.\nTrivially, $[0]=\\emptyset$.\nAn $n$-point metric space $([n],d)$ is the set $[n]$, called the groundset,\nendowed with a function\n$d\\colon [n]\\times[n]\\to\\mathbb{R}$ satisfying\n\\begin{enumerate}[(1)]\n\\item\\label{nonnegative}\n$d(x,y)\\ge 0$ (non-negativeness),\n\\item\n$d(x,y)=0$ if and only if $x=y$ (identity of indiscernibles),\n\\item\\label{symmetry}\n$d(x,y)=d(y,x)$ (symmetry), and\n\\item\n$d(x,y)+d(x,z)\\ge d(y,z)$ (triangle inequality)\n\\end{enumerate}\nfor all $x$, $y$, $z\\in [n]$.\nAn\nequivalent definition\nrequires the triangle inequality only for distinct\n$x$, $y$, $z\\in [n]$, axioms~(\\ref{nonnegative})--(\\ref{symmetry}) remaining.\n\n\n\n\nAn algorithm with oracle access to a metric space $([n],d)$\nis given $n$ and may query\n$d$ on\nany\n$(x,y)\\in[n]\\times[n]$ to obtain $d(x,y)$.\nWithout loss of generality,\nwe\nforbid\nqueries for $d(x,x)$, which trivially return $0$,\nas well as repeated queries, where\na query\nfor $d(x,y)$\nis considered to repeat that for\n$d(y,x)$.\nFor convenience,\ndenote\nan algorithm ALG with oracle access to $([n],d)$\nby $\\text{ALG}^d$.\n\nGiven oracle access to a finite metric space $([n],d)$, the\n{\\sc metric $1$-median} problem asks for a point\nin\n$[n]$\nwith the minimum average distance to\nall\npoints.\nAn algorithm for\nthis problem is\n$\\alpha$-approximate if\nit outputs\na point $x\\in[n]$\nsatisfying\n$$\\sum_{y\\in[n]}\\,d\\left(x,y\\right)\n\\le\\alpha\\,\\min_{x^\\prime\\in[n]}\\,\\sum_{y\\in[n]}\\,d\\left(x^\\prime,y\\right),$$\nwhere\n$\\alpha\\ge 1$.\n\nThe following theorem is due to Chang~\\cite{Cha13} and generalized by\nWu~\\cite{Wu14}.\n\n\\begin{theorem}[{\\cite{Cha13, Wu14}}]\\label{nonadaptiveupperbound}\n{\\sc Metric $1$-median} has a deterministic nonadaptive $O(n^{1.5})$-time\n$4$-approximation algorithm.\n\\end{theorem}\n\n\\section{Lower bound}\n\nFix arbitrarily a\ndeterministic\n$o(n^2)$-query algorithm\n$A$\nfor {\\sc metric $1$-median}\nand a constant\n$\\delta\\in(0,0.1)$.\nBy padding queries, we may assume the existence of a function\n$q\\colon\\mathbb{Z}^+\\to\\mathbb{Z}^+$ such that $A$ makes exactly\n$q(n)=o(n^2)$ queries given oracle access to any metric space with groundset\n$[n]$.\n\nWe introduce some notations concerning a function\n$d\\colon[n]\\times[n]\\to\\mathbb{R}$\nto be determined later.\nFor $i\\in[q(n)]$, denote the $i$th query of $A^d$ by\n$(x_i,y_i)\\in[n]\\times[n]$; in other\nwords, the $i$th query of $A^d$ asks for $d(x_i,y_i)$.\nNote that $(x_i,y_i)$ depends only on $d(x_1,y_1)$, $d(x_2,y_2)$,\n$\\ldots$, $d(x_{i-1},y_{i-1})$ because $A$\nis deterministic and\nhas been fixed.\nFor $x\\in[n]$ and $i\\in\\{0,1,\\ldots,q(n)\\}$,\n\\begin{eqnarray}\nN_i(x)\n&\\stackrel{\\text{def.}}{=}&\n\\left\\{\ny\\in[n]\\mid\n\\left\\{\n\\left(x,y\\right), \\left(y,x\\right)\n\\right\\}\n\\cap \\left\\{\\left(x_j,y_j\\right)\\mid\nj\\in\\left[i\\right]\\right\\}\n\\neq\\emptyset\n\\right\\},\n\\label{neighborhoodinsubgraph}\\\\\n\\alpha_i(x)\n&\\stackrel{\\text{def.}}{=}&\n\\left|\\,\nN_i(x)\n\\,\\right|,\n\\label{numberoffrozenincidentdistances}\n\\end{eqnarray}\nfollowing Chang~\\cite{Cha12} with a slight change in notation.\nEquivalently, $\\alpha_i(x)$ is the degree of $x$ in the undirected graph\nwith vertex set $[n]$ and edge set $\\{(x_j,y_j)\\mid j\\in[i]\\}$.\nAs $[0]=\\emptyset$,\n$\\alpha_0(x)=0$ for $x\\in[n]$.\nNote that\n$\\alpha_i(\\cdot)$ depends only on\n$(x_1,y_1)$, $(x_2,y_2)$, $\\ldots$, $(x_i,y_i)$.\nDenote the\noutput of $A^d$\nby $p$.\nBy adding at most $n-1=o(n^2)$ dummy queries,\nwe may\nassume without loss of generality that\n\\begin{eqnarray}\n\\left(p,y\\right)\\in\\left\\{\\left(x_i,y_i\\right)\\mid i\\in\\left[q(n)\\right]\\right\\}\n\\label{algorithmoutputheavilyqueried}\n\\end{eqnarray}\nfor all $y\\in[n]\\setminus\\{p\\}$.\nConsequently,\n\\begin{eqnarray}\n\\alpha_{q(n)}(p)=n-1.\\label{outputallasked}\n\\end{eqnarray}\nFix any set $S\\subseteq [n]$ of size $\\lceil\\delta n\\rceil$,\ne.g., $S=[\\lceil\\delta n\\rceil]$.\n\nWe proceed to\nconstruct $d$ by gradually freezing distances.\nFor brevity, freezing\nthe value of\n$d(x,y)$\nimplicitly freezes $d(y,x)$ to the same value, where $x$, $y\\in[n]$.\nInductively, having\nanswered the first $i-1$ queries of $A^d$ by freezing\n$d(x_1,y_1)$, $d(x_2,y_2)$, $\\ldots$,\n$d(x_{i-1},y_{i-1})$, where $i\\in[q(n)]$,\nanswer the $i$th query by\n\\begin{eqnarray}\nd\\left(x_i,y_i\\right)\n&=&\\left\\{\n\\begin{array}{ll}\n3, &\\text{if $x_i$, $y_i\\in S$;}\\\\\n3, &\\text{if $x_i\\in S$, $y_i\\notin S$ and $\\alpha_{i-1}(x_i)\\le\\delta n$;}\\\\\n3, &\\text{if $y_i\\in S$, $x_i\\notin S$ and $\\alpha_{i-1}(y_i)\\le\\delta n$;}\\\\\n4, &\\text{if $x_i\\in S$, $y_i\\notin S$ and $\\alpha_{i-1}(x_i)>\\delta n$;}\\\\\n4, &\\text{if $y_i\\in S$, $x_i\\notin S$ and $\\alpha_{i-1}(y_i)>\\delta n$;}\\\\\n2, &\\text{if $x_i$, $y_i\\notin S$ and $\\max\\{\\alpha_{i-1}(x_i),\\alpha_{i-1}(y_i)\\}\\le\\delta n$;}\\\\\n4, &\\text{if $x_i$, $y_i\\notin S$ and $\\max\\{\\alpha_{i-1}(x_i),\\alpha_{i-1}(y_i)\\}>\\delta n$.}\n\\end{array}\n\\right.\n\\label{distanceassignment}\n\\end{eqnarray}\nIt is not hard to verify that\nthe seven cases\nin equation~(\\ref{distanceassignment})\nare\nexhaustive and\nmutually\nexclusive.\nWe have now\nfrozen\n$d(x_i,y_i)$ for all $i\\in[q(n)]$ and none of the other distances.\nAs repeated queries are forbidden, equation~(\\ref{distanceassignment})\ndoes not freeze one distance twice,\npreventing inconsistency.\n\nSet\n\\begin{eqnarray}\nB\n&\\stackrel{\\text{def.}}{=}&\n\\left\\{\nx\\in[n]\\mid \\alpha_{q(n)}(x)>\\delta n\n\\right\\},\\label{badpoints}\\\\\n\\hat{p}\n&\\stackrel{\\text{def.}}{=}&\n\\mathop{\\rm argmin}_{x\\in S}\\, \\alpha_{q(n)}(x),\n\\label{trueoptimal}\n\\end{eqnarray}\nbreaking ties arbitrarily.\nFor all distinct\n$x$, $y\\in[n]$\nwith $(x,y)$, $(y,x)\\notin\\{(x_i,y_i)\\mid i\\in[q(n)]\\}$,\nlet\n\\begin{eqnarray}\nd\\left(x,y\\right)\n=\n\\left\\{\n\\begin{array}{ll}\n1, &\\text{if $x=\\hat{p}$, $y\\notin S\\cup B$;}\\\\\n1, &\\text{if $y=\\hat{p}$, $x\\notin S\\cup B$;}\\\\\n3, &\\text{if $x$, $y\\in S\\cup B$;}\\\\\n4, &\\text{if $x\\in (S\\cup B)\\setminus \\{\\hat{p}\\}$ and $y\\notin (S\\cup B\\cup\\{\\hat{p}\\})$;}\\\\\n4, &\\text{if $y\\in (S\\cup B)\\setminus \\{\\hat{p}\\}$ and $x\\notin (S\\cup B\\cup\\{\\hat{p}\\})$;}\\\\\n2, &\\text{otherwise.}\n\\end{array}\n\\right.\n\\label{completingthemetric}\n\\end{eqnarray}\nClearly, the six cases in equation~(\\ref{completingthemetric}) are\nexhaustive and\nmutually\nexclusive.\nFurthermore, equation~(\\ref{completingthemetric}) assigns the same value\nto $d(x,y)$ and $d(y,x)$.\nFinally, for all $x\\in[n]$,\n\\begin{eqnarray}\nd\\left(x,x\\right)=0.\\label{trivialdistance}\n\\end{eqnarray}\nEquations~(\\ref{distanceassignment}),~(\\ref{completingthemetric})~and~(\\ref{trivialdistance})\ncomplete the construction of $d$ by freezing all distances.\n\n\nThe following lemma is straightforward.\n\n\\begin{lemma}\\label{distancesarezeroto4}\nFor all distinct $x$, $y\\in[n]$,\n$d(x,y)\\in\\{1,2,3,4\\}$.\n\\end{lemma}\n\nBelow is an immediate consequence of equation~(\\ref{trueoptimal}).\n\n\\begin{lemma}\\label{optimalisinpreservedregion}\n$\\hat{p}\\in S$.\n\\end{lemma}\n\nThe following lemma is a consequence of\nequations~(\\ref{neighborhoodinsubgraph})--(\\ref{numberoffrozenincidentdistances}) and\nour\nforbidding\nrepeated queries.\n\n\\begin{lemma}\\label{monotonicity}\nFor all $x\\in[n]$ and $i\\in[q(n)]$,\n\\begin{eqnarray*}\n\\alpha_i(x)-\\alpha_{i-1}(x)\n=\\left\\{\n\\begin{array}{ll}\n0, &\\text{if $x\\notin \\{x_i,y_i\\}$;}\\\\\n1, &\\text{otherwise.}\n\\end{array}\n\\right.\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nThe case of $x\\notin \\{x_i,y_i\\}$ is immediate from\nequations~(\\ref{neighborhoodinsubgraph})--(\\ref{numberoffrozenincidentdistances}).\nSuppose that $x\\in \\{x_i,y_i\\}$.\nBy symmetry,\nwe may assume\n$x=x_i$.\nSo by equation~(\\ref{neighborhoodinsubgraph}),\n\\begin{eqnarray}\nN_i(x)=N_{i-1}(x)\\cup\\left\\{y_i\\right\\}.\\label{newneighbor}\n\\end{eqnarray}\nAs\n$(x,y_i)=(x_i,y_i)$ is the $i$th query\nand we\nforbid\nrepeated\nqueries,\n\\begin{eqnarray}\ny_i\\notin N_{i-1}(x)\\label{reallynewneighbor}\n\\end{eqnarray}\nby equation~(\\ref{neighborhoodinsubgraph}).\\footnote{In detail,\nif $y_i\\in N_{i-1}(x)$, then\n$(x_j,y_j)\\in\\{(x,y_i),(y_i,x)\\}$\nfor some $j\\in[i-1]$ by\nequation~(\\ref{neighborhoodinsubgraph}); hence the $i$th query $(x_i,y_i)\n=(x,y_i)$\nrepeats the $j$th query, a contradiction.}\nEquations~(\\ref{numberoffrozenincidentdistances})~and~(\\ref{newneighbor})--(\\ref{reallynewneighbor})\ncomplete the\nproof.\n\\end{proof}\n\nIn short, Lemma~\\ref{monotonicity} says that adding the edge $(x_i,y_i)$\nto an undirected graph without that edge increases the degree of $x$\nby $1$ if and only if $x\\in\\{x_i,y_i\\}$.\n\n\n\n\\begin{lemma}\\label{monotonicitysame}\nFor all $x\\in[n]$ and $i\\in[q(n)+1]$,\nif $\\alpha_{i-1}(x)>\\delta n$, then\n$x\\in B$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{monotonicity}, $\\alpha_{q(n)}(x)\\ge \\alpha_{i-1}(x)$.\nInvoking equation~(\\ref{badpoints}) then completes the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{sumofdegrees}\n$$\\sum_{x\\in[n]}\\, \\alpha_{q(n)}(x)= 2\\, q(n).$$\n\\end{lemma}\n\\begin{proof}\nRecall that the left-hand side\nis the sum of degrees in the undirected graph with vertex set $[n]$\nand edge set $\\{(x_i,y_i)\\mid i\\in[q(n)]\\}$.\nAs we\nforbid\nrepeated queries, $\\left|\\,\\{(x_i,y_i)\\mid\ni\\in[q(n)]\\}\\,\\right|=q(n)$\nFinally, it is a basic fact in graph\ntheory that\nthe sum of degrees in an undirected graph equals\ntwice the number of edges.\n\\end{proof}\n\n\\begin{lemma}[{Implicit in~\\cite[Lemma~13]{Cha12}}]\\label{fewbadpoints}\n$|B|=o(n)$.\n\\end{lemma}\n\\begin{proof}\nWe have\n$$\n|B|\\,\\delta n\n\\stackrel{\\text{equation~(\\ref{badpoints})}}{\\le} \\sum_{x\\in B} \\alpha_{q(n)}(x)\n\\le \\sum_{x\\in [n]} \\alpha_{q(n)}(x)\n\\stackrel{\\text{Lemma~\\ref{sumofdegrees}}}{=}\n2\\,q(n).\n$$\nThis gives $|B|=o(n)$ as $\\delta\\in(0,0.1)$ is a constant and $q(n)=o(n^2)$.\n\\end{proof}\n\n\\begin{lemma}\\label{sparselyaskedpoint}\nFor all sufficiently large $n$ and all $i\\in[q(n)+1]$,\n\\begin{eqnarray}\n\\alpha_{i-1}\\left(\\hat{p}\\right)&\\le& \\delta n.\n\\label{sparselyaskedpointequation}\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{fewbadpoints}, $|S|=\\lceil\\delta n\\rceil$ and $\\delta\\in(0,0.1)$\nbeing a constant,\n$S\\setminus B\\neq\\emptyset$ for all sufficiently large $n$.\nBy equation~(\\ref{badpoints}), $S\\setminus B\\neq\\emptyset$\n$\\alpha_{q(n)}(x)\\le \\delta n$ for some $x\\in S$,\nwhich together with equation~(\\ref{trueoptimal})\ngives $\\alpha_{q(n)}(\\hat{p})\\le\\delta n$.\nFinally, Lemma~\\ref{monotonicity} and $\\alpha_{q(n)}(\\hat{p})\\le\\delta n$\nimply inequality~(\\ref{sparselyaskedpointequation}) for all $i\\in[q(n)+1]$.\n\\comment\nFurthermore,\n$$\n|S|\\cdot\\alpha_{q(n)}\\left(\\hat{p}\\right)\n\\stackrel{\\text{equation~(\\ref{trueoptimal})}}{\\le}\n\\sum_{x\\in S}\\, \\alpha_{q(n)}(x)\n\\le\n\\sum_{x\\in[n]}\\, \\alpha_{q(n)}(x)\n\\stackrel{\\text{Lemma~\\ref{sumofdegrees}}}{=}\n2\\cdot q(n).\n$$\nSo\n$\\alpha_{q(n)}(\\hat{p})\\le 2\\cdot q(n)\/|S|=o(n)$,\nwhich together with\nLemma~\\ref{monotonicity}\nshows\n$\\alpha_{i-1}(\\hat{p})\n\\le\\alpha_{q(n)}(\\hat{p})\n\\le\\delta n$\nfor all sufficiently large $n$\nand all $i\\in[q(n)+1]$.\n\n\\end{proof}\n\nHenceforth, assume $n$ to be sufficiently large to satisfy\ninequality~(\\ref{sparselyaskedpointequation}) for all $i\\in[q(n)+1]$.\n\n\\begin{lemma}\\label{onlysourceofdistance1}\nFor all $x$, $y\\in[n]$, if $d(x,y)=1$, then\none of the following conditions is true:\n\\begin{itemize}\n\\item $x=\\hat{p}$ and $y\\notin S\\cup B$;\n\\item $y=\\hat{p}$ and $x\\notin S\\cup B$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nInspect equation~(\\ref{completingthemetric}), which\nis the only equation that may set distances to $1$.\n\\end{proof}\n\n\\comment\nBelow is a consequence of Lemma~\\ref{onlysourceofdistance1}.\n\n\\begin{lemma}\\label{sourceofshortdistances}\nFor all distinct\n$x$, $y$, $z\\in[n]$, if $d(x,y)=d(x,z)=1$, then $x=\\hat{p}$ and $y$,\n$z\\notin S\\cup B$.\n\\end{lemma}\n\n\n\n\n\n\n\\begin{lemma}\\label{ordinarydistancesare2}\nFor all distinct $x$, $y\\in[n]\\setminus (S\\cup B)$,\n$d\\left(x,y\\right)=2$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{monotonicitysame},\n$\\max\\{\\alpha_{i-1}(x_i),\\alpha_{i-1}(y_i)\\}>\\delta n$\nmeans $\\{x_i,y_i\\}\\cap B\\neq\\emptyset$, where $i\\in[q(n)+1]$.\nSo\nonly the second-to-last case\nin equation~(\\ref{distanceassignment}), which sets $d(x_i,y_i)=2$,\nmay be consistent\nwith $x_i$, $y_i\\notin S\\cup B$.\n\nBy\nLemma~\\ref{optimalisinpreservedregion},\n$\\hat{p}\\in S$.\nSo\nonly the last case\nin equation~(\\ref{completingthemetric}),\nwhich sets $d(x,y)=2$,\nmay be consistent\nwith $x$, $y\\notin S\\cup B$.\n\\end{proof}\n\n\n\n\\comment\n\\begin{lemma}\\label{askeddistancesincidentonoptimalpoint}\nFor all $i\\in[q(n)]$,\n$\\alpha_{i-1}(\\hat{p})\\le\\delta n$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{sparselyaskedpoint},\n$\\hat{p}\\in S\\setminus B$.\n$\\hat{p}\\notin B$, which\ntogether with equation~(\\ref{badpoints}) and\nLemma~\\ref{monotonicity}\ncompletes the proof.\n\\end{proof}\n\n\n\\begin{lemma}\\label{optimalpointdistances1or3}\nFor all $x\\in[n]\\setminus\\{\\hat{p}\\}$,\n$d(\\hat{p},x)\\in\\{1,3\\}$.\n\\end{lemma}\n\\begin{proof}\nBy\nLemma~\\ref{optimalisinpreservedregion} and\ninequality~(\\ref{sparselyaskedpointequation}),\nonly\nthe first three\ncases in\nequation~(\\ref{distanceassignment}),\nwhich set $d(x_i,y_i)=3$,\nmay be consistent with\n$x_i=\\hat{p}$ or $y_i=\\hat{p}$.\n\nAgain by Lemma~\\ref{optimalisinpreservedregion},\nonly the first three cases in equation~(\\ref{completingthemetric}),\nwhich set $d(x,y)\\in\\{1,3\\}$,\nmay be consistent with $x=\\hat{p}$ or $y=\\hat{p}$.\n\\end{proof}\n\n\\begin{lemma}\\label{illegaldistances1}\nThere do not exist distinct $x$, $y$, $z\\in[n]$\nwith $d(x,y)=1$ and $\\{d(x,z), d(y,z)\\}=\\{2, 4\\}$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{onlysourceofdistance1},\n$d(x,y)=1$ implies $\\hat{p}\\in\\{x,y\\}$.\nBy symmetry, assume $x=\\hat{p}$.\nThen $d(x,z)\\in\\{1,3\\}$ by\nLemma~\\ref{optimalpointdistances1or3}.\n\\end{proof}\n\n\\begin{lemma}\\label{illegaldistances2}\nThere do not exist distinct $x$, $y$, $z\\in[n]$\nwith $d(x,y)=d(x,z)=1$ and $d(y,z)\\in\\{3, 4\\}$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{onlysourceofdistance1},\n$d(x,y)=d(x,z)=1$ implies $x=\\hat{p}$ and $y$, $z\\notin S\\cup B$.\nThen $d(y,z)=2$ by\nLemma~\\ref{ordinarydistancesare2}.\n\\end{proof}\n\n\nLemmas~\\ref{illegaldistances1}--\\ref{illegaldistances2}\nforbid all possible violations of the triangle inequality,\nyielding the following lemma.\n\n\n\\begin{lemma}\\label{itismetric}\n$([n],d)$ is a metric space.\n\\end{lemma}\n\\begin{proof}\nLemmas~\\ref{distancesarezeroto4}~and~\\ref{illegaldistances1}--\\ref{illegaldistances2}\nestablish the triangle inequality for $d$.\nFurthermore, $d$ is symmetric because (1)~freezing $d(x,y)$ automatically\nfreezes $d(y,x)$ to the same value,\n(2)~forbidding\nrepeated queries\nprevents equation~(\\ref{distanceassignment})\nfrom assigning inconsistent values to one distance and\n(3)~equation~(\\ref{completingthemetric}) is symmetric.\nAll the other axioms for metrics\nare easy to verify.\n\\end{proof}\n\nRecall that $p$ denotes the output of $A^d$.\nWe proceed to\ncompare $\\sum_{x\\in[n]}\\,d(p,x)$ with $\\sum_{x\\in[n]}\\,d(\\hat{p},x)$.\n\n\\begin{lemma}\\label{identifyingjumps}\nThere exist\n$k(1)$, $k(2)$, $\\ldots$, $k(n-1)\\in [q(n)]$\nand distinct $z_{k(1)}$, $z_{k(2)}$, $\\ldots$, $z_{k(n-1)}\\in[n]$\nsuch that\n\\begin{eqnarray}\n\\alpha_{k(t)-1}(p)&=&t-1,\\label{beforejumping}\\\\\\\n\\alpha_{k(t)}(p)&=&t,\\label{afterjumping}\\\\\n\\left(p,z_{k(t)}\\right)\n&\\in&\\left\\{\\left(x_{k(t)}, y_{k(t)}\\right), \\left(y_{k(t)}, x_{k(t)}\\right)\n\\right\\}\n\\label{algorithmoutputparticipate}\n\\end{eqnarray}\nfor all $t\\in[n-1]$.\n\\end{lemma}\n\\begin{proof}\nBy\nLemma~\\ref{monotonicity}, equation~(\\ref{outputallasked}) and the easy fact\nthat $\\alpha_0(p)=0$,\nthere exist distinct $k(1)$, $k(2)$, $\\ldots$, $k(n-1)\\in [q(n)]$ satisfying\nequations~(\\ref{beforejumping})--(\\ref{afterjumping})\nfor all $t\\in[n-1]$.\\footnote{Observe that\n$\\alpha_i(p)$ must go through all of $0$, $1$, $\\ldots$, $n-1$ as $i$\nincreases from $0$ to $q(n)$.}\nLemma~\\ref{monotonicity} and\nequations~(\\ref{beforejumping})--(\\ref{afterjumping}) imply\n$p\\in\\{x_{k(t)}, y_{k(t)}\\}$, establishing the existence of\n$z_{k(t)}$ satisfying\nequation~(\\ref{algorithmoutputparticipate}).\nIf $z_{k(1)}$, $z_{k(2)}$, $\\ldots$, $z_{k(n-1)}$ are not distinct, then\nthere are repeated queries by\nequation~(\\ref{algorithmoutputparticipate}), a contradiction.\n\\end{proof}\n\nFrom now on, let\n$k(1)$, $k(2)$, $\\ldots$, $k(n-1)\\in [q(n)]$\nand distinct $z_{k(1)}$, $z_{k(2)}$, $\\ldots$, $z_{k(n-1)}\\in[n]$\nsatisfy\nequations~(\\ref{beforejumping})--(\\ref{algorithmoutputparticipate})\nfor all $t\\in[n-1]$.\n\n\\begin{lemma}\\label{algorithmoutputtypicaldistances}\nFor each $t\\in[n-1]$, if $t\\ge \\lceil\\delta n\\rceil+2$ and $z_{k(t)}\\notin S$,\nthen\n$d(p,z_{k(t)})=4$.\n\\end{lemma}\n\\begin{proof}\nAssume in equation~(\\ref{algorithmoutputparticipate}) that\n$p=x_{k(t)}$ and $z_{k(t)}=y_{k(t)}$; the other case will be symmetric.\nBy equation~(\\ref{beforejumping}),\n\\begin{eqnarray}\n\\alpha_{k(t)-1}\\left(x_{k(t)}\\right)=t-1> \\delta n.\n\\label{algorithmoutputverymuchasked}\n\\end{eqnarray}\n\\begin{enumerate}[{Case }1:]\n\\item $x_{k(t)}\\in S$.\nBy equation~(\\ref{distanceassignment}),\n$x_{k(t)}\\in S$\nand $y_{k(t)}=z_{k(t)}\\notin S$,\n\\begin{eqnarray}\nd\\left(x_{k(t)},y_{k(t)}\\right)\n=\n\\left\\{\n\\begin{array}{ll}\n3, &\\text{if $\\alpha_{k(t)-1}(x_{k(t)})\\le \\delta n$;}\\\\\n4, &\\text{if $\\alpha_{k(t)-1}(x_{k(t)})> \\delta n$.}\n\\end{array}\n\\right.\n\\label{distancesfromalgorithmoutput1}\n\\end{eqnarray}\n\\item $x_{k(t)}\\notin S$.\nBy equation~(\\ref{distanceassignment}),\n$x_{k(t)}\\notin S$\nand $y_{k(t)}=z_{k(t)}\\notin S$,\n\\begin{eqnarray}\nd\\left(x_{k(t)},y_{k(t)}\\right)\n=\n\\left\\{\n\\begin{array}{ll}\n2, &\\text{if $\\max\\{\\alpha_{k(t)-1}(x_{k(t)}),\\alpha_{k(t)-1}(y_{k(t)})\\}\\le \\delta n$;}\\\\\n4, &\\text{if $\\max\\{\\alpha_{k(t)-1}(x_{k(t)}),\\alpha_{k(t)-1}(y_{k(t)})\\}> \\delta n$.}\n\\end{array}\n\\right.\n\\label{distancesfromalgorithmoutput2}\n\\end{eqnarray}\n\\end{enumerate}\nEquation~(\\ref{algorithmoutputverymuchasked})\ntogether with any one of\nequations~(\\ref{distancesfromalgorithmoutput1})--(\\ref{distancesfromalgorithmoutput2})\nimplies $d(x_{k(t)},y_{k(t)})=4$.\nHence $d(p,z_{k(t)})=d(x_{k(t)},y_{k(t)})=4$.\n\\end{proof}\n\nWe are now able to analyze the quality of $p$ as a solution to\n{\\sc metric $1$-median}.\n\n\\begin{lemma}\\label{analyzingalgorithmoutputassuboptimal}\n$$\\sum_{x\\in[n]}\\,d\\left(p,x\\right)\\ge\n4\\left(n-2\\left\\lceil\\delta n\\right\\rceil-2\\right).$$\n\\end{lemma}\n\\begin{proof}\nBy the distinctness of\n$z_{k(1)}$, $z_{k(2)}$, $\\ldots$, $z_{k(n-1)}$ in\nLemma~\\ref{identifyingjumps},\n\\begin{eqnarray}\n\\sum_{x\\in[n]}\\,d\\left(p,x\\right)\n\\ge\\sum_{t\\in[n-1]}\\,d\\left(p,z_{k(t)}\\right).\n\\label{initialinequalityinanalyzingalgorithmoutput}\n\\end{eqnarray}\nWrite\n$A=\\{t\\in[n-1]\\mid z_{k(t)}\\in S\\}$.\nAs $z_{k(1)}$, $z_{k(2)}$, $\\ldots$, $z_{k(n-1)}$ are distinct,\n\\begin{eqnarray}\n|A|\\le |S|.\n\\end{eqnarray}\nFurthermore,\n\\begin{eqnarray}\n&&\\sum_{t\\in[n-1]}\\,d\\left(p,z_{k(t)}\\right)\\nonumber\\\\\n&\\ge&\\sum_{t\\in[n-1],\\,t\\ge \\lceil\\delta n\\rceil+2,\\,t\\notin A}\\,\nd\\left(p,z_{k(t)}\\right)\\nonumber\\\\\n&\\stackrel{\\text{Lemma~\\ref{algorithmoutputtypicaldistances}}}{=}&\n\\sum_{t\\in[n-1],\\,t\\ge \\lceil\\delta n\\rceil+2,\\,t\\notin A}\\,\n4\\nonumber\\\\\n&\\ge& 4\\left(n-\\left\\lceil\\delta n\\right\\rceil-2-|A|\\right).\n\\label{lastinequalityinanalyzingalgorithmoutput}\n\\end{eqnarray}\nEquations~(\\ref{initialinequalityinanalyzingalgorithmoutput})--(\\ref{lastinequalityinanalyzingalgorithmoutput})\nand $|S|=\\lceil\\delta n\\rceil$ complete the proof.\n\\end{proof}\n\nWe now analyze the quality of $\\hat{p}$ as a solution to\n{\\sc metric $1$-median}.\nThe following lemma is immediate from\nequation~(\\ref{completingthemetric}).\n\n\\begin{lemma}\\label{optimalpointhasmanydistancesbeing1}\nFor all $y\\in [n]\\setminus(S\\cup B)$,\nif\n$y\\neq \\hat{p}$ and\n$(\\hat{p},y)$, $(y,\\hat{p})\\notin \\{(x_j,y_j)\\mid j\\in[q(n)]\\}$,\nthen $d(\\hat{p},y)=1$.\n\\end{lemma}\n\n\\begin{lemma}\\label{analyzingoptimalpoint}\n$$\\sum_{y\\in[n]}\\,d\\left(\\hat{p},y\\right)\n\\le\nn+3\\cdot\\left(\\left\\lceil\\delta n\\right\\rceil+o(n)+\\delta n\\right).\n$$\n\\end{lemma}\n\\begin{proof}\nBy equation~(\\ref{neighborhoodinsubgraph}),\n$$N_{q(n)}\\left(\\hat{p}\\right)\n=\\left\\{\ny\\in[n]\\mid\n\\left\\{\n\\left(\\hat{p},y\\right), \\left(y,\\hat{p}\\right)\n\\right\\}\n\\cap \\left\\{\\left(x_j,y_j\\right)\\mid\nj\\in\\left[q(n)\\right]\\right\\}\n\\neq\\emptyset\n\\right\\}.$$\nThis and\nLemma~\\ref{optimalpointhasmanydistancesbeing1} imply\n$d(\\hat{p},y)=1$ for all $y\\in [n]\\setminus(S\\cup B)$ with\n$y\\neq \\hat{p}$ and\n$y\\notin N_{q(n)}(\\hat{p})$.\nTherefore,\n\\begin{eqnarray}\n\\sum_{y\\in [n]\\setminus(S\\cup B\\cup\nN_{q(n)}(\\hat{p}))}\\,d\\left(\\hat{p},y\\right)\n\\le\nn-\\left|\\,S\\cup B\\cup\nN_{q(n)}\\left(\\hat{p}\\right)\\,\\right|.\n\\label{distance1parts}\n\\end{eqnarray}\nClearly,\n\\begin{eqnarray}\n\\sum_{y\\in S\\cup B\\cup N_{q(n)}(\\hat{p})}\\,d\\left(\\hat{p},y\\right)\n\\stackrel{\\text{Lemma~\\ref{distancesarezeroto4}}}{\\le}\n\\sum_{y\\in S\\cup B\\cup N_{q(n)}(\\hat{p})}\\,4\n= 4\\cdot\\left|\\,S\\cup B\\cup\nN_{q(n)}\\left(\\hat{p}\\right)\\,\\right|\n\\label{largedistancesparts}\n\\end{eqnarray}\nFurthermore,\n\\begin{eqnarray}\n\\left|\\,N_{q(n)}\\left(\\hat{p}\\right)\\,\\right|\n\\stackrel{\\text{equation~(\\ref{numberoffrozenincidentdistances})}}{=}\n\\alpha_{q(n)}\\left(\\hat{p}\\right)\n\\stackrel{\\text{inequality~(\\ref{sparselyaskedpointequation})}}{\\le}\n\\delta n.\\nonumber\n\\end{eqnarray}\nThis and Lemma~\\ref{fewbadpoints}\nimply\n\\begin{eqnarray}\n\\left|\\,S\\cup B\\cup N_{q(n)}\\left(\\hat{p}\\right)\\,\\right|\n\\le \\left\\lceil\\delta n\\right\\rceil+o(n)+\\delta n\n\\label{fewneighborsforoptimalpoint}\n\\end{eqnarray}\nas $|S|=\\lceil\\delta n\\rceil$.\nTo complete the proof, sum\nup\ninequalities~(\\ref{distance1parts})--(\\ref{largedistancesparts})\nand then use\ninequality~(\\ref{fewneighborsforoptimalpoint})\nin the trivial way.\n\\end{proof}\n\n\n\nCombining\nLemmas~\\ref{itismetric},~\\ref{analyzingalgorithmoutputassuboptimal}~and~\\ref{analyzingoptimalpoint}\nyields our main theorem, stated below.\n\n\\begin{theorem}\\label{maintheorem}\n{\\sc Metric $1$-median} has no deterministic $o(n^2)$-query\n$(4-\\epsilon)$-approximation algorithm for any constant $\\epsilon>0$.\n\\end{theorem}\n\\begin{proof}\nLemma~\\ref{itismetric} asserts that $([n],d)$ is a metric space.\nBy\nLemmas~\\ref{analyzingalgorithmoutputassuboptimal}~and~\\ref{analyzingoptimalpoint},\n$$\n\\sum_{x\\in[n]}\\,d\\left(p,x\\right)\n\\ge 4\\left(1-8\\delta-o(1)\\right)\\sum_{x\\in[n]}\\, d\\left(\\hat{p},x\\right).\n$$\nThis proves the theorem because\nthe deterministic $o(n^2)$-query algorithm $A$\nand\nthe constant $\\delta\\in(0,0.1)$\nare picked arbitrarily (note that $p$ denotes the output of $A^d$).\n\\end{proof}\n\nTheorem~\\ref{maintheorem}\ncomplements\nTheorem~\\ref{nonadaptiveupperbound}.\n\nIt is possible to\nsimplify equation~(\\ref{completingthemetric})\nat the expensive of an additional assumption.\nWithout loss of generality, we may assume that $\\alpha_{q(n)}(x)=n-1$\nfor all $x\\in B$; this increases the query complexity by a multiplicative\nfactor of $O(1)$ by equation~(\\ref{badpoints}).\nTherefore, if $x\\in B$ or $y\\in B$, then\n$d(x,y)$ will be frozen by equation~(\\ref{distanceassignment}).\nSo the third to fifth\ncases\nin equation~(\\ref{completingthemetric}),\nwhich satisfies $x\\in B$ or $y\\in B$,\ncan be omitted.\n\n\\comment\nDefine\n$$Q\\equiv \\left\\{\n\\text{unordered pair }\n\\left(x,y\\right)\\in[n]^2\\mid A^d\n\\text{ ever queries for }\nd\\left(x,y\\right)\n\\right\\}$$\nto be the set of\nqueries of $A^d$\ntreated as unordered\npairs.\nWithout loss of generality, assume $(x,x)\\notin Q$ for all $x\\in[n]$.\nLet $G=([n],Q)$ be the simple undirected graph with vertex set $[n]$ and\nedge set $Q$.\nDenote the degree of $x\\in[n]$ in $G$ by\n$\\text{deg}_G(x)=|\\{y\\in[n]\\mid (x,y)\\in Q\\}|$,\nand\n\\begin{eqnarray}\nB\\equiv \\left\\{x\\in[n]\\mid \\text{deg}_G(x)\\ge\\epsilon n\\right\\}.\n\\label{setofbadvertices}\n\\end{eqnarray}\n\nIn the sequel, we will specify $d$ incrementally in several steps.\nNote that $Q$ and\n$B$ are independent of\n$d$\nbecause of the nonadaptivity of $A$;\nhence\nthey will remain intact\nduring our\nspecification of $d$.\n\nBelow is an easy lemma.\n\n\\begin{lemma}[{Implicit in~\\cite{Cha12}}]\\label{numberofbadvertices}\n$|\\,B\\,|=o(n)$.\n\\end{lemma}\n\\begin{proof}\nWe have\n$$\n\\epsilon n\\,\n|\\,B\\,|\n= \\sum_{x\\in B}\\,\\epsilon n\n\\stackrel{\\text{Eq.~(\\ref{setofbadvertices})}}{\\le}\n\\sum_{x\\in B}\\,\\text{deg}_G(x)\n\\le \\sum_{x\\in[n]}\\,\\text{deg}_G(x)\n=2\\,|\\,Q\\,|,$$\nwhere the last equality follows from the well-known fact that the sum of\ndegrees in an undirected graph is\ntwice the\nnumber of edges.\nThis\ncompletes the proof\nbecause\n$|\\,Q\\,|=o(n^2)$ is $A$'s query complexity and $\\epsilon$ is a constant.\n\\end{proof}\n\nHenceforth we will assume $n\\in\\mathbb{Z}^+$ to be sufficiently large so\nthat\n\\begin{eqnarray}\nn-|\\,B\\,|-1-\\epsilon n>0\n\\label{numberofremainingpointstobemadegood}\n\\end{eqnarray}\nby Lemma~\\ref{numberofbadvertices}.\nFor all $x\\in[n]$,\n\\begin{eqnarray}\nd\\left(x,x\\right)\\equiv 0.\n\\label{zerodistancestoself}\n\\end{eqnarray}\nFor all $(x,y)\\in[n]^2\\setminus\\{(x,x)\\mid x\\in[n]\\}$ with $x\\in B$, $y\\in B$ or $(x,y)\\in Q$,\n\\begin{eqnarray}\nd(x,y)\\equiv\n\\left\\{\n\\begin{array}{ll}\n4, &\\text{if }x\\in B \\text{ or } y\\in B;\\\\\n2, &\\text{otherwise}.\n\\end{array}\n\\right.\n\\label{distancesonquerysetandbadvertices}\n\\end{eqnarray}\nClearly, this does not assign different values to $d(x,y)$ and $d(y,x)$.\n\nAs\nEq.~(\\ref{distancesonquerysetandbadvertices})\nspecifies\n$d$ on a superset of $Q$ (which is the set of $A$'s queries)\nand $A$ is deterministic,\nthe output of $A^d$\nhas\nnow\nbeen\nfixed\neven though\n$d$\nis not fully specified yet.\nLet\n$p\\in[n]\\setminus B$ and $p^\\prime\\in B$\nbe\nsuch that $\\{p,p^\\prime\\}$ contains\nthe output of $A^d$.\n\n\\begin{lemma}\\label{notbadandnotconnectedwithalgorithmoutput}\n$$\\left|\\,\\left([n]\\setminus \\left(B\\cup\\{p\\}\\right)\\right)\n\\cap \\left\\{x\\in[n]\\mid (p,x)\\notin Q\\right\\}\\,\\right|\n\\ge n-\n\\left|\\,B\\cup\\{p\\}\\,\\right|-\\epsilon n.$$\n\\end{lemma}\n\\begin{proof}\nEq.~(\\ref{setofbadvertices}) and $p\\notin B$ imply $\\text{deg}_G(p)<\\epsilon\nn$, i.e., $|\\,\\{x\\in[n]\\mid(p,x)\\in Q\\}\\,|<\\epsilon n$.\n\\end{proof}\n\n\\comment\n\\begin{eqnarray}\n\\hat{z}\\equiv\\left\\{\n\\begin{array}[ll]\n\\mathop{argmin}_{y\\in([n]\\setminus B)\\cap\\{x\\in[n]\\mid (x,z)\\notin Q\\}}\\,\n\\text{deg}_G(y),& \\text{if }z\\notin B;\\\\\n\\mathop{argmin}_{y\\in[n]\\setminus B}\\, \\text{deg}_G(y),& \\text{otherwise}.\n\\end{array}\n\\right.\n\\label{nearoptimalpoint}\n\\end{eqnarray}\n\n\nTake\n\\begin{eqnarray}\n\\hat{p}\\in \\left([n]\\setminus \\left(B\\cup\\{p\\}\\right)\\right)\n\\cap\\left\\{x\\in[n]\\mid\n\\left(x,p\\right)\\notin Q\\right\\}\n\\label{pickingourpoint}\n\\end{eqnarray}\narbitrarily,\nas\ncan be done\nby\nLemma~\\ref{notbadandnotconnectedwithalgorithmoutput}\nand Eq.~(\\ref{numberofremainingpointstobemadegood}).\nTrivially, $\\hat{p}\\notin B$.\n\nWe now complete the specification of $d$.\nFor all $(x,y)\\in [n]^2\\setminus (Q\\cup \\{(x,x)\\mid x\\in[n]\\})$ with $x\\notin B$ and $y\\notin\nB$,\\footnote{These are precisely the pairs whose $d$-distances are not\nspecified by\nEqs.~(\\ref{zerodistancestoself})--(\\ref{distancesonquerysetandbadvertices}).}\n\\begin{eqnarray}\nd(x,y)\\equiv\\left\\{\n\\begin{array}{ll}\n3,& \\text{if }\n((x=\\hat{p})\\land (y= p))\\text{ or }((y=\\hat{p})\\land (x=p));\\\\\n1, & \\text{if }((x=\\hat{p})\\land (y\\neq p))\\text{ or }((y=\\hat{p})\\land (x\\neq\np));\\\\\n4, &\\text{if }((x=p)\\land(y\\neq \\hat{p}))\\text{ or }((y=p)\\land(x\\neq\\hat{p}));\\\\\n2, &\\text{otherwise}.\n\\end{array}\n\\right.\n\\label{makingourpointgoodandalgorithmpointbad}\n\\end{eqnarray}\nThe four cases\nin Eq.~(\\ref{makingourpointgoodandalgorithmpointbad})\nare mutually exclusive because $p\\neq \\hat{p}$ by\nEq.~(\\ref{pickingourpoint}).\nClearly, Eq.~(\\ref{makingourpointgoodandalgorithmpointbad}) does not assign\ndifferent values to $d(x,y)$ and $d(y,x)$.\n\nThe following lemma is straightforward from\nEqs.~(\\ref{zerodistancestoself})--(\\ref{distancesonquerysetandbadvertices})~and~(\\ref{makingourpointgoodandalgorithmpointbad}).\n\n\\begin{lemma}\\label{rangeofdistances}\nFor all $x$, $y\\in[n]$, $d(x,y)\\in\\{0,1,2,3,4\\}$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{speciallydesignedpointisgood}\n$$\\sum_{y\\in[n]}\\,d\\left(\\hat{p},y\\right)\\le \\left(1+4\\epsilon\\right)n+o(n).$$\n\\end{lemma}\n\\begin{proof}\nBy Lemmas~\\ref{numberofbadvertices}~and~\\ref{rangeofdistances},\n\\begin{eqnarray}\n\\sum_{y\\in B}\\,d\\left(\\hat{p},y\\right)=o(n).\n\\label{distancesfromourpointtobad}\n\\end{eqnarray}\nFurthermore,\n\\begin{eqnarray}\n\\sum_{y\\in[n]\\text{ s.t.\\ }(\\hat{p},y)\\in Q}\\,d\\left(\\hat{p},y\\right)\n\\stackrel{\\text{Lemma~\\ref{rangeofdistances}}}{\\le} \\sum_{y\\in[n]\\text{ s.t.\\ }(\\hat{p},y)\\in Q}\\,4\n= 4\\,\\text{deg}_G\\left(\\hat{p}\\right)<4\\epsilon n,\n\\label{distancesfromoutpointtoqueried}\n\\end{eqnarray}\nwhere the last inequality follows from Eq.~(\\ref{setofbadvertices})\nand $\\hat{p}\\notin B$.\nWe have\n\\begin{eqnarray}\n\\sum_{y\\in[n]\\setminus(B\\cup\\{p,\\hat{p}\\})\\text{ s.t.\\ }\n(\\hat{p},y)\\notin Q}\\,d\\left(\\hat{p},y\\right)\n\\le n\n\\label{distancesfromourpointtononbadnonqueried}\n\\end{eqnarray}\nbecause all summands are $1$ by\nEq.~(\\ref{makingourpointgoodandalgorithmpointbad}) and $\\hat{p}\\notin B$.\nBy Lemma~\\ref{rangeofdistances},\n\\begin{eqnarray}\n\\sum_{y\\in\\{p,\\hat{p}\\}}\\,d\\left(\\hat{p},y\\right)=O(1).\n\\label{thetrivialdistance}\n\\end{eqnarray}\nSumming up\nEqs.~(\\ref{distancesfromourpointtobad})--(\\ref{thetrivialdistance})\ncompletes the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{outputpointisterribleifnotbad}\n$$\\sum_{y\\in[n]}\\,d\\left(p,y\\right)\\ge 4\\left(n-o(n)-\\epsilon n\\right).$$\n\\end{lemma}\n\\begin{proof}\nRecall that $p\\notin B$.\nWe have\n\\begin{eqnarray*}\n&&\\sum_{y\\in[n]}\\,d\\left(p,y\\right)\\\\\n&\\ge& \\sum_{y\\in[n]\\setminus (B\\cup\\{p,\\hat{p}\\})\n\\text{ s.t.\\ }(p,y)\\notin Q}\\,d\\left(p,y\\right)\\\\\n&\\stackrel{\\text{Eq.~(\\ref{makingourpointgoodandalgorithmpointbad})}}{=}&\n\\sum_{y\\in[n]\\setminus (B\\cup\\{p,\\hat{p}\\})\n\\text{ s.t.\\ }(p,y)\\notin Q}\\,4\\\\\n&\\ge&\n4\\left(\n\\left|\\,\n\\left\\{y\\in[n]\\setminus \\left(B\\cup\\{p\\}\\right)\\mid \\left(p,y\\right)\n\\notin Q\\right\\}\n\\,\\right|-1\\right)\\\\\n&\\stackrel{\\text{Lemma~\\ref{notbadandnotconnectedwithalgorithmoutput}}}{\\ge}&\n4\\left(n-|\\,B\\,|-\\epsilon n-2\\right)\\\\\n&\\stackrel{\\text{Lemma~\\ref{numberofbadvertices}}}{=}&\n4\\left(n-o(n)-\\epsilon n\\right).\n\\end{eqnarray*}\n\\end{proof}\n\nThe next lemma is immediate from Eq.~(\\ref{distancesonquerysetandbadvertices})\nand $p^\\prime\\in B$.\n\n\\begin{lemma}\\label{outputpointisterribleifbad}\n$$\\sum_{y\\in[n]\\setminus\\{p^\\prime\\}}\\,\nd\\left(p^\\prime,y\\right)= 4\\left(n-1\\right).$$\n\\end{lemma}\n\nWe proceed to prove that $([n],d)$ is a metric space through a few lemmas.\n\n\n\n\n\n\n\n\nThe following\nlemma is\nimmediate from\nEqs.~(\\ref{distancesonquerysetandbadvertices})~and~(\\ref{makingourpointgoodandalgorithmpointbad}).\n\n\\begin{lemma}\\label{onlybadandalgorithmoutputcanhavedistance4}\nFor all $x,$ $y\\in[n]$, if $d(x,y)=4$, then $\\{x,y\\}\\cap (B\\cup\n\\{p\\})\\neq\\emptyset$.\n\\end{lemma}\n\n\nBelow is a consequence of $p\\notin B$ and\nEqs.~(\\ref{pickingourpoint})--(\\ref{makingourpointgoodandalgorithmpointbad}).\n\n\\begin{lemma}\\label{distancebetweennonbadoutputandourdesignedpoint}\n$d(\\hat{p},p)=3$.\n\\end{lemma}\n\n\\begin{lemma}\\label{constructeddistanceismetric}\n$([n],d)$ is a metric space.\n\\end{lemma}\n\\begin{proof}\nWe only need to prove the triangle inequality for $d$ because all the other\naxioms\nare easy to verify.\nConsider\nthe following cases for all distinct $x$, $y$, $z\\in[n]$:\n\\begin{itemize}\n\\item $d(x,y)=1$, $d(x,z)=1$ and $d(y,z)=4$.\nBy Lemma~\\ref{sourceofshortdistances},\n$x=\\hat{p}$.\nHence if $y=p$ (resp., $z=p$), then $d(x,y)=3$ (resp., $d(x,z)=3$)\nby\nLemma~\\ref{distancebetweennonbadoutputandourdesignedpoint},\na\ncontradiction.\nTherefore, $p\\notin\\{y,z\\}$, which together with\nLemma~\\ref{onlybadandalgorithmoutputcanhavedistance4} forces $\\{y,z\\}\\cap\nB\\neq\\emptyset$.\nBut\nif\n$y\\in B$ (resp., $z\\in B$), then\n$d(x,y)=4$ (resp., $d(x,z)=4$)\nby Eq.~(\\ref{distancesonquerysetandbadvertices}), a contradiction.\n\\item $d(x,y)=1$, $d(x,z)=1$ and $d(y,z)=3$.\nBy Lemma~\\ref{sourceofshortdistances},\n$x=\\hat{p}$.\nOn the other hand, $d(y,z)=3$ means $(y,z)\\in\\{(\\hat{p},p),(p,\\hat{p})\\}$ by\nEq.~(\\ref{makingourpointgoodandalgorithmpointbad}) (which is the only\nequation that may set distances to $3$),\ncontradicting $x=\\hat{p}$.\n\\item $d(x,y)=1$, $d(x,z)=2$ and $d(y,z)=4$.\nBy Lemma~\\ref{onlybadandalgorithmoutputcanhavedistance4},\n$\\{y,z\\}\\cap(B\\cup\\{p\\})\\neq\\emptyset$.\nBut if $y\\in B$ (resp., $z\\in B$), then $d(x,y)=4$ (resp., $d(x,z)=4$) by\nEq.~(\\ref{distancesonquerysetandbadvertices}), a contradiction.\nTherefore, $p\\in\\{y,z\\}$.\nFurthermore, $\\hat{p}\\in\\{x,y\\}$ by Lemma~\\ref{onlysourceofdistance1}.\nConsequently, $(p,\\hat{p})\\in\\{(x,y),(x,z),(y,z)\\}$\n(note that $p\\neq \\hat{p}$ by Eq.~(\\ref{pickingourpoint})),\nimplying $3\\in\\{d(x,y),d(x,z),d(y,z)\\}$\nby\nLemma~\\ref{distancebetweennonbadoutputandourdesignedpoint},\na contradiction.\n\\end{itemize}\nWe have excluded all possibilities of $d(x,y)+d(x,z)<d(y,z)$, where\n$x$, $y$, $z\\in[n]$.\n\\end{proof}\n\n\nCombining\nLemmas~\\ref{speciallydesignedpointisgood}--\\ref{outputpointisterribleifbad},~\\ref{constructeddistanceismetric}\nand that\n$\\{p,p^\\prime\\}$ contains the output of $A^d$\nyields our main theorem.\n\n\\begin{theorem}\\label{maintheorem}\n{\\sc Metric $1$-median} has no deterministic nonadaptive $o(n^2)$-query\n$(4-\\epsilon)$-approximation\nalgorithms\nfor any constant $\\epsilon>0$.\n\\end{theorem}\n\n\n\nTheorem~\\ref{maintheorem}\nshows that\nthe approximation ratio of $4$ in\nTheorem~\\ref{nonadaptiveupperbound} cannot be improved to any constant\n$c<4$.\n\n\\comment\nFor all $y\\in[n]\\setminus B$ with $(\\hat{z},y)\\notin Q$,\n\\begin{eqnarray}\nd\\left(\\hat{z},y\\right)\\equiv\\left\\{\n\\begin{array}[ll]\n3,& \\text{if }y=z\\text{ and } z\\notin B;\\\\\n1, &\\text{otherwise}.\n\\end{array}\n\\right.\n\\label{distancesofnearoptimalpoint}\n\\end{eqnarray}\nClearly, this\nand Eq.~(\\ref{distancesonquerysetandbadvertices}) uniquely determines\n$d(\\hat{z},y)$ for all $y\\in [n]$.\n\nTo complete specifying $d$,\nfor all $(x,y)\\in [n]^2\\setminus Q$ with $x\\notin B$, $y\\notin B$\n\\begin{eqnarray}\nd(x,y)\\equiv\\left\\{\n\\begin{array}[ll]\n4,& \\text{if }x=z \\text{ or }y=z;\\\\\n2,& \\text{otherwise}.\n\\end{array}\n\\right.\n\\end{eqnarray}\n\n\n\n\nFor a metric space $([n],d)$ and a set $Q\\subseteq [n]^2$\nof unordered pairs,\nlet $G_Q=([n],Q)$ be the\nundirected graph with\nvertex set $[n]$ and\nedge set $Q$.\nAssign to each edge\n$(x,y)$ of $G_Q$\nthe\nlength $d(x,y)$.\nFor $x$, $y\\in[n]$, denote by $d_Q(x,y)$ the\nshortest-path distance between $x$ and $y$ in $G_Q$.\nClearly, $d_Q(x,y)\\ge d(x,y)$ for all $x$, $y\\in[n]$.\nFor a finite set $D$ and a function $f\\colon D\\to \\mathbb{R}$,\nthe $\\ell_1$ norm of\n$f$\nis $\\lVert f\\rVert_1=\\sum_{x\\in D}\\,|\\,f(x)\\,|$.\nThe following corollary investigates, with respect to the normalized\n$\\ell_1$ norm, the\ninapproximability of\nmetrics\nby small sets of distances.\n\n\\begin{corollary}\nThere do not exist sets $Q\\subseteq[n]^2$ of unordered pairs\nsatisfying\n\\begin{eqnarray}\n|\\,Q\\,|&=&o\\left(n^2\\right),\\label{numberofqueries}\\\\\n\\frac{\\lVert d_Q-d\\rVert_1}{\\lVert d\\rVert_1}&\\le&\n1-\\Omega(1)\\label{recoveryerror}\n\\end{eqnarray}\nfor all metric spaces $([n],d)$.\n\\end{corollary}\n\\begin{proof}\nSuppose for contradiction that $Q\\subseteq[n]^2$ satisfies\nEqs.~(\\ref{numberofqueries})--(\\ref{recoveryerror}).\nLet\n\\begin{eqnarray}\n\\tilde{z}&=&\\mathop{\\rm argmin}_{z\\in[n]}\\,\n\\sum_{x\\in[n]}\\,d_Q\\left(z,x\\right),\n\\label{optimalsolutiononpseudodistances}\\\\\nz^*&=&\\mathop{\\rm argmin}_{z\\in[n]}\\,\\sum_{x\\in[n]}\\,d\\left(z,x\\right).\n\\nonumber\n\\end{eqnarray}\nSo $z^*$ is\nthe optimal solution to {\\sc Metric $1$-median} with respect to $([n],d)$.\nNow,\n\\begin{eqnarray}\n&&\\sum_{x\\in[n]}\\,d\\left(\\tilde{z},x\\right)\\label{qualityofpseudo1median}\\\\\n&\\le& \\sum_{x\\in[n]}\\,d_Q\\left(\\tilde{z},x\\right)\\nonumber\\\\\n&\\stackrel{\\text{Eq.~(\\ref{optimalsolutiononpseudodistances})}}{\\le}&\n\\frac{1}{n}\\sum_{z\\in[n]}\\,\\sum_{x\\in[n]}\\,d_Q\\left(z,x\\right)\\nonumber\\\\\n&=&\\frac{1}{n}\\left\\lVert d_Q\\right\\rVert_1\\nonumber\\\\\n&\\stackrel{\\text{Eq.~(\\ref{recoveryerror})}}{\\le}&\n\\frac{2-\\Omega(1)}{n}\n\\left\\lVert d\\right\\rVert_1\\nonumber\\\\\n&=&\\frac{2-\\Omega(1)}{n}\\sum_{x,y\\in[n]}\\,d\\left(x,y\\right)\\nonumber\\\\\n&\\le&\\frac{2-\\Omega(1)}{n}\\sum_{x,y\\in[n]}\\,\\left(d\\left(z^*,x\\right)+d\\left(z^*,y\\right)\\right)\n\\nonumber\\\\\n&=&\\left(2-\\Omega(1)\\right)\\cdot\n2\\sum_{x\\in[n]}\\,d\\left(z^*,x\\right).\\label{optimalvalue4times}\n\\end{eqnarray}\nBy Eq.~(\\ref{optimalsolutiononpseudodistances}),\nwe may find $\\tilde{z}$ with $|\\,Q\\,|=o(n^2)$ queries,\nwhich together with\nEqs.~(\\ref{qualityofpseudo1median})--(\\ref{optimalvalue4times})\ncontradict Theorem~\\ref{maintheorem}.\n\\end{proof}\n\n\\comment\n\\section{Additional section --- to be modified}\n\nThis section modifies XXX slightly to XXX.\n\n\\begin{figure}\n\\begin{algorithmic}[1]\n\\FOR{each $(q,r)\\in S$}\n\t\\FOR{each $(q^\\prime,r^\\prime)\\in S$}\n\t\t\\IF{$q$, $q^\\prime \\le \\lfloor(n-1)\/m\\rfloor-1$}\n\t\t\t\\STATE Query for $d(q m+r,q^\\prime m+r)$;\n\t\t\t\\STATE Query for $d(q^\\prime m+r,q^\\prime m+r^\\prime)$;\n\t\t\t\\STATE $\\tilde{d}(q m+r,q^\\prime m+r^\\prime)\\leftarrow d(q m+r,q^\\prime m+r)+d(q^\\prime m+r,q^\\prime m+r^\\prime)$;\n\t\t\\ELSE\n\t\t\t\\STATE Query for $d(q m+r,q^\\prime m+r^\\prime)$;\n\t\t\t\\STATE $\\tilde{d}(q m+r,q^\\prime m+r^\\prime)\\leftarrow d(q m+r,q^\\prime m+r^\\prime)$;\n\t\t\\ENDIF\n\t\\ENDFOR\n\\ENDFOR\n\\STATE $(\\hat{q},\\hat{r})\\leftarrow\\mathop{\\rm argmin}_{(q,r)\\in S}\n\\sum_{(q^\\prime,r^\\prime)\\in S}\\, {\\tilde{d}}^2(q m+r,q^\\prime m+r^\\prime)$,\nbreaking ties arbitrarily;\n\\STATE Output $\\hat{q} m+\\hat{r}$;\n\\end{algorithmic}\n\\caption{Algorithm {\\sf Approx-Centroid}.}\n\\label{deterministic16approximationalgorithm}\n\\end{figure}\n\n\nFor all $(q,r)$, $(q^\\prime,r^\\prime)\\in S$ and $x\\in\\{0,1,\\ldots,n-1\\}$,\ndefine\n\\begin{eqnarray}\nf\\left(q,r,q^\\prime,x\\right)\n\\equiv\n\\left\\{\n\\begin{array}{ll}\nd(x,q^\\prime m+r), & \\text{if $q$, $q^\\prime\\le\\lfloor(n-1)\/m\\rfloor-1$;}\\\\\n0, & \\text{otherwise.}\n\\end{array}\n\\right.\n.\\label{additionalterm}\n\\end{eqnarray}\nThe same definition is made by Chang~\\cite{Cha13}.\n\n\\begin{fact}[{\\cite[Lemma~2]{Cha13}}]\\label{pseudodistanceupper}\nFor all $(q,r)$, $(q^\\prime,r^\\prime)\\in S$ and $x\\in\\{0,1,\\ldots,n-1\\}$,\n$$\n\\tilde{d}\\left(qm+r,q^\\prime m+r^\\prime\\right)\n\\le d\\left(x,qm+r\\right)+d\\left(x,q^\\prime\nm+r^\\prime\\right)+2f\\left(q,r,q^\\prime,x\\right)\n$$\nafter finishing the loop in lines~1--12 of {\\sf Approx-Centroid}.\n\\end{fact}\n\n\\begin{fact}[{\\cite[Lemma~4]{Cha13}}]\nFor all $(q,r)$, $(q^\\prime,r^\\prime)\\in S$,\n$$\nd\\left(qm+r,q^\\prime m+r^\\prime\\right)\n\\le\n\\tilde{d}\\left(qm+r,q^\\prime m+r^\\prime\\right)\n$$\nafter finishing the loop in lines~1--12 of {\\sf Approx-Centroid}.\n\\end{fact}\n\nIn the following two lemmas,\n$(q,r)$ and $(q^\\prime,r^\\prime)$ are\nindependent and uniformly random elements in $S$.\n\n\\begin{lemma}\\label{distancessquareexpected}\nFor all $x\\in\\{0,1,\\ldots,n-1\\}$,\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(qm+r,q^\\prime m+r^\\prime\\right)\\,\\right]\n\\le 8\\cdot\\mathop{\\rm E}\\left[\\,d^2\\left(x,qm+r\\right)\\,\\right]\n+8\\cdot\\mathop{\\rm E}\\left[\\,f^2\\left(q,r,q^\\prime,x\\right)\\,\\right].\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nBy Fact~\\ref{pseudodistanceupper},\n\\begin{eqnarray*}\n&&\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(qm+r,q^\\prime\nm+r^\\prime\\right)\\,\\right]\\\\\n&\\le&\n\\mathop{\\rm E}\\left[\\,\\left(\nd\\left(x,qm+r\\right)+d\\left(x,q^\\prime m+r^\\prime\\right)\n+f\\left(q,r,q^\\prime,x\\right)+f\\left(q,r,q^\\prime,x\\right)\n\\right)^2\\,\\right]\\\\\n&\\le&\n4\\cdot \\mathop{\\rm E}\\left[\\,\nd^2\\left(x,qm+r\\right)+d^2\\left(x,q^\\prime m+r^\\prime\\right)\n+f^2\\left(q,r,q^\\prime,x\\right)+f^2\\left(q,r,q^\\prime,x\\right)\n\\,\\right]\\\\\n&=&8\\cdot\\mathop{\\rm E}\\left[\\,d^2\\left(x,qm+r\\right)\\,\\right]\n+8\\cdot\\mathop{\\rm E}\\left[\\,f^2\\left(q,r,q^\\prime,x\\right)\\,\\right],\n\\end{eqnarray*}\nwhere the second inequality follows from Cauchy's inequality.\n\\end{proof}\n\n\\begin{lemma}\\label{fsquarelemma}\nFor all $x\\in\\{0,1,\\ldots,n-1\\}$,\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,f^2\\left(q,r,q^\\prime,x\\right)\\,\\right]\n\\le \\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r^\\prime\\right)\\,\\right].\n\\end{eqnarray*}\n\\end{lemma}\n\\begin{proof}\nBy Eq.~(\\ref{additionalterm}),\n{\\small\n\\begin{eqnarray}\n\\mathop{\\rm E}\\left[\\,f^2\\left(q,r,q^\\prime,x\\right)\\,\\right]\n=\n\\Pr\\left[\\,q,q^\\prime\\le \\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right]\n\\cdot\n\\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r\\right)\\mid q,q^\\prime\\le\n\\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right].\n\\label{fsquare}\n\\end{eqnarray}\n\nObserve that,\nconditional on any realization\nof $q$ and $q^\\prime$ with $q$, $q^\\prime\\in \\lfloor(n-1)\/m\\rfloor-1$,\nboth $r$ and $r^\\prime$ are uniformly distributed over $\\{0,1,\\ldots,m-1\\}$.\nTherefore,\n{\\smal\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r\\right)\\mid q,q^\\prime\\le\n\\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right]\n=\n\\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r^\\prime\\right)\\mid q,q^\\prime\\le\n\\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right].\n\\end{eqnarray*}\n\nThis and inequality~(\\ref{fsquare})\nimply\n\\begin{eqnarray*}\n&&\\mathop{\\rm E}\\left[\\,f^2\\left(q,r,q^\\prime,x\\right)\\,\\right]\\nonumber\\\\\n&=&\n\\Pr\\left[\\,q,q^\\prime\\le \\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right]\n\\cdot\n\\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r^\\prime\\right)\\mid q,q^\\prime\\le\n\\left\\lfloor\\frac{n-1}{m}\\right\\rfloor-1\\,\\right]\\\\\n&\\le& \\mathop{\\rm E}\\left[\\,d^2\\left(x,q^\\prime m+r^\\prime\\right)\\,\\right].\n\\end{eqnarray*}\n\\end{proof}\n\n\nBelow is a consequence of\nLemmas~\\ref{distancessquareexpected}--\\ref{fsquarelemma}.\n\n\\begin{lemma}\\label{therecomestheratioof16}\nFor all $x\\in\\{0,1,\\ldots,n-1\\}$,\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(qm+r,q^\\prime m+r^\\prime\\right)\\,\\right]\n\\le 16\\cdot \\mathop{\\rm E}\\left[\\,d^2\\left(x,qm+r\\right)\\,\\right].\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{theorem}\n{\\sc Metric $1$-median} has a deterministic $O(n^{3\/2})$-query\n$16$-approximation algorithm.\n\\end{theorem}\n\\begin{proof}\nBy line~13 of {\\sf Approx-Centroid},\n\\begin{eqnarray*}\n\\sum_{(q^\\prime,r^\\prime)\\in\nS}\\,{\\tilde{d}}^2\\left(\\hat{q}m+\\hat{r},q^\\prime m+r^\\prime\\right)\n\\le \\frac{1}{n}\\cdot \\sum_{(q,r)\\in S}\\,\\sum_{(q^\\prime,r^\\prime)\\in S}\\,\n{\\tilde{d}}^2\\left(qm+r,q^\\prime m+r^\\prime\\right).\n\\end{eqnarray*}\n\\comment\nEquivalently,\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(\\hat{q}m+\\hat{r},q^\\prime\nm+r^\\prime\\right)\\,\\right]\n\\le\n\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(qm+r,q^\\prime\nm+r^\\prime\\right)\\,\\right].\n\\end{eqnarray*}\n\nBy Lemma~\\ref{therecomestheratioof16},\n\\begin{eqnarray*}\n\\mathop{\\rm E}\\left[\\,{\\tilde{d}}^2\\left(qm+r,q^\\prime m+r^\\prime\\right)\\,\\right]\n\\le \\min_{x=0}^{n-1}\\,\n\\end{eqnarray*}\n\\end{proof}\n\n\\section{Something new}\n\nFor\na metric space\n$([n],d)$,\nuniformly random points\n$\\bs{u}$, $\\bs{v}$\nin $[n]$ and\nthe output $z$ of\nIndyk's algorithm given oracle access to $([n],d)$,\n\\begin{eqnarray*}\n&&\\mathop{\\rm\nE}\\left[\\,\\left|\\,\\frac{1}{2}\\left(d\\left(\\bs{u},z\\right)+d\\left(\\bs{v},z\\right)\\right)-d\\left(\\bs{u},\\bs{v}\\right)\\,\\right|\\,\\right]\\\\\n&\\le&\n\\mathop{\\rm\nE}\\left[\\,\\left|\\,\\frac{1}{2}d\\left(\\bs{u},z\\right)-\\frac{1}{2}d\\left(\\bs{u},\\bs{v}\\right)\\,\\right|\\,\\right]\n+\\mathop{\\rm\nE}\\left[\\,\\left|\\,\\frac{1}{2}d\\left(\\bs{v},z\\right)-\\frac{1}{2}d\\left(\\bs{u},\\bs{v}\\right)\\,\\right|\\,\\right]\\\\\n&=&\n\\mathop{\\rm\nE}\\left[\\,\\left|\\,d\\left(\\bs{u},z\\right)-d\\left(\\bs{u},\\bs{v}\\right)\\,\\right|\\,\\right]\\\\\n&\\le&\\mathop{\\rm E}\\left[\\,d\\left(z,\\bs{v}\\right)\\,\\right].\n\\end{eqnarray*}\nThat is,\nwriting\n$\\tilde{d}(x,y)\\equiv (d(x,z)+d(y,z))\/2$ for all $x$, $y\\in[n]$,\n$$\\left\\|\\,\\tilde{d}-d\\,\\right\\|_1\n\\le \\mathop{\\rm E}\\left[\\,d\\left(z,\\bs{v}\\right)\\,\\right].$$\nThis and the easily verifiable fact that $\\tilde{d}$ is a metric on $[n]$\nshow how to recover $d$ in $O(n)$ time with a bounded $\\ell_1$ error.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIf $w=w(x_1,\\ldots,x_n)$ is a group word, then for every group $G$ we have a\nverbal mapping\n\\[\n\\begin{matrix}\nw & : & G^{(n)} & \\longrightarrow & G\n\\\\\n& & (g_1,\\ldots,g_n) & \\longmapsto & w(g_1,\\ldots,g_n)\n\\end{matrix}\n\\]\nby substitutions of the $n$-tuples.\\\\\n\nWe use the notation $N(G,w)$ for the number of solutions to the equation $w=1$ in\nthe group $G$, that is, for the cardinality of $w^{-1}(1)$.\nMore generally, if $S$ is a subset of $G^{(n)}$, we write $N(S,w)$ for the number of\nsolutions to $w=1$ lying in $S$.\nThus $N(G,w)$ can be seen as a shortcut for $N(G^{(n)},w)$. \\\\\n\nIf $G$ is abelian, then the verbal mapping $w:G^{(n)}\\rightarrow G$ is a\nhomomorphism and so each non-empty fibre $w^{-1}(g)$ is a coset of the kernel\n$w^{-1}(1)$. Besides, since $G$ is also finite (a) $\\left\\vert w^{-1}%\n(1)\\right\\vert \\geq\\left\\vert G^{(n)}\\right\\vert \/\\left\\vert G\\right\\vert $\nand (b) all non-empty fibres have the same size. If $G$ is finite and nilpotent, but not abelian, then (b) is no longer true in\ngeneral. Whether (a) still holds is an open problem (that it does is\nconjectured by Amit [A]). The aim of this paper is to show that the conjecture of Amit is true for finite groups of nilpotency class at most 2.\\\\\n\n \\begin{thm}\n Let $G$ be a finite group of nilpotency class at most $2$, and let $w=w(x_1,\\ldots,x_n)$\nbe a group word. Then $N(G,w)\\ge |G|^{n-1}$.\n \\end{thm} \n If the theorem holds for two groups $A$ and $B$of nilpotency class at most 2, then it holds for their direct product $G = A\\times B$ since the word values can be solved componentwise.\n \n  Let $\\mathfrak{g} = \\mathfrak{a}\\cdot \\mathfrak{b}=(a_1, ..., a_n)\\cdot (b_1, ..., b_n)\\in A^{(n)}\\times B^{(n)}=G^{(n)}$, then  \n$w(\\mathfrak{g}) = w(\\mathfrak{a}).w(\\mathfrak{b})$ which implies $N(G,w)\\geq N(A,w).N(B,w)$\nand the result follows from $|G| = |A|.|B|$. \n Any finite nilpotent group is a direct product of its Sylow subgroups. Hence it will be enough to show the theorem for $p$-groups:\n \n \\begin{thm}\n Let $G$ be a finite $p$-group of nilpotency class at most $2$, and let $w=w(x_1,\\ldots,x_n)$\nbe a group word. Then $N(G,w)\\ge |G|^{n-1}$.\n \\end{thm} \n \nThis result was also independently obtained bt Matthew Levy [L].\n\\section{Useful lemmas}\n\nIf $w_1,\\ldots,w_k$ is a finite family of words in the same\nvariables $x_1,\\ldots,x_n$, then we can consider the mapping\n\\[\n\\begin{matrix}\nf_{w_1,\\ldots,w_k} & : & G^{(n)} & \\longrightarrow & G^{(k)}\n\\\\\n& & (g_1,\\ldots,g_n) & \\longmapsto & (w_1(g_1,\\ldots,g_n),\\ldots,w_k(g_1,\\ldots,g_n)).\n\\end{matrix}\n\\]\nOn the other hand, if $w=w(x_1,\\ldots,x_n)$ and $u_1,\\ldots,u_n$ are words in the\nsame variables $y_1,\\ldots,y_k$, then we can define the \\emph{composition\\\/}\n$w(u_1,\\ldots,u_n)$, which results from substituting each word $u_i$ for the indeterminate\n$x_i$ in the expression for $w$.\nIn other words, if $F$ is the free group on $x_1,\\ldots,x_n$ and $F^*$ is the free group\non $y_1,\\ldots,y_k$, then $w(u_1,\\ldots,u_n)$ is the image of $w$ under the homomorphism\n$F\\rightarrow F^*$ sending each $x_i$ to $u_i$.\n\n\\begin{lem}\n\\label{reduction}\nLet $w=w(x_1,\\ldots,x_n)$ and $u_1,\\ldots,u_n$ be group words, and put\n$w'=w(u_1,\\ldots,u_n)$.\nIf $G$ is a group for which the map $f_{u_1,\\ldots,u_n}$ is bijective,\nthen we have $N(S,w')=N(S,w)$ for every subset $S$ of $G^{(n)}$.\n\\end{lem}\n\n\n\n\\begin{proof}\nThis is clear, since $f_{w'}$ is the composition of $f_{u_1,\\ldots,u_n}$ and\n$f_w$.\n\\end{proof}\n\n\n\nThe previous lemma will be used in conjunction with the following one.\nRecall that, if $G$ is a finite $p$-group and $m$ is an integer which is not\ndivisible by $p$, then the map $g\\mapsto g^m$ is bijective.\nThis is an immediate consequence of B\\'ezout's identity.\n\n\n\n\\begin{lem}\n\\label{equivalent}\nLet $u_1,\\ldots,u_n$ be words of the form $u_i=x_i^{m_i} v_i$, where $v_i$ is\na word which only depends on $x_{i+1},\\ldots,x_n$.\nIf $G$ is a finite $p$-group and $(m_i,p)=1$ for every $i=1,\\ldots,n$, then\nthe mapping $f_{u_1,\\ldots,u_n}$ is bijective on $G^{(n)}$.\n\\end{lem}\n\n\n\n\\begin{proof}\nSince $f_{u_1,\\ldots,u_n}$ goes from $G^{(n)}$ to $G^{(n)}$, it suffices to prove\nthat it is an injective map.\nIf $(g_1,\\ldots,g_n)$ and $(h_1,\\ldots,h_n)$ are two tuples with the same image\nunder $f_{u_1,\\ldots,u_n}$, we prove that $g_i=h_i$ for all $i=1,\\ldots,n$ by reverse\ninduction on $i$.\nFor $i=n$ this is an immediate consequence of the remark preceding this lemma,\nsince the word $u_n$ is of the form $x_n^{m_n}$ with $(m_n,p)=1$.\nAssume now that $g_{i+1}=h_{i+1},\\ldots,g_n=h_n$, and let us see that $g_i=h_i$.\nSince $u_i(g_i,\\ldots,g_n)=u_i(h_i,\\ldots,h_n)$ and $u_i=x_i^{m_i} v_i$ for a word\n$v_i$ involving only $x_{i+1},\\ldots,x_n$, it follows that $g_i^{m_i}=h_i^{m_i}$,\nand we conclude that $g_i=h_i$ as before.\n\\end{proof}\n\n\n\nOf course, the previous lemma admits variations which follow by reordering the\nvariables.\nFor example, if $n=3$ and $u_1=x_1$, $u_2=x_2x_1$, $u_3=x_3x_2$, then the map\n$f_{u_1,u_2,u_3}$ is bijective on $G^{(3)}$.\n\n\n\n\\vspace{5pt}\n\n\n\nThe following result will also be used on several occasions.\nThe proof is obvious.\n\n\n\n\\begin{lem}\n\\label{simpler word}\nLet $w$ be a group word in the free group $F$, and let $G$ be a finite $p$-group\nof order $p^n$ and class $c$.\nIf $w'$ is a word such that $w\\equiv w' \\pmod{\\gamma_{c+1}(F)F^{p^n}}$, then\n$f_w=f_{w'}$ on $G^{(n)}$.\n\\end{lem}\n\n\n\nAs a consequence, if $G$ is a finite $p$-group of class $2$ and we want to study the\nnumber $N(G,w)$ for a group word $w$, we may assume without loss of generality that\n$w$ is of the form\n\\[\nx_1^{k_1}\\ldots x_n^{k_n} \\prod_{1\\le i<j\\le n} \\, [x_i,x_j]^{k_{ij}}.\n\\]\nAlso, if $w=u^k v$ for some words $u$ and $v$ and for some exponent $k$, then we\nmay replace $k$ with any other integer $k'$ such that $k\\equiv k'\\pmod{|G|}$.\n\n\n\n\\vspace{5pt}\n\n\n\nIn the next theorem we consider a particular kind of words, which will be crucial\nfor the proof of our main result.\nWe need the following very simple lemma.\n\n\n\n\\begin{lem}\n\\label{[g,G]}\nLet $G$ be a nilpotent group of class at most $2$, and let $g$ be an element of $G$.\nThen $[g,G]=\\{[g,h]\\mid h\\in G\\}$, and every element of $[g,G]$ can be represented\nin $|C_G(g)|$ different ways in the form $[g,h]$ with $h\\in G$.\nHence $|[g,G]|=|G:C_G(g)|$.\n\\end{lem}\n\n\n\n\\begin{proof}\nThe equality $[g,G]=\\{[g,h]\\mid h\\in G\\}$ follows from the fact that the commutator\nis multilinear in a nilpotent group of class at most $2$.\nOn the other hand, since\n\\[\n[g,h_1]=[g,h_2] \\Longleftrightarrow g^{h_1}=g^{h_2} \\Longleftrightarrow\nh_1h_2^{-1}\\in C_G(g) \\Longleftrightarrow C_G(g)h_1=C_G(g)h_2,\n\\]\nthe elements which give the same commutator with $g$ form a right coset of $C_G(g)$.\nThus the number of different representations of a commutator with $g$ coincides\nwith the order of $C_G(g)$.\n\\end{proof}\n\n\\section{Proof of the theorem}\nLet's first show the proof for a particular word which is the main step in the main proof:\n\\begin{thm}\n\\label{particular case}\nLet $G$ be a finite $p$-group of class at most $2$, and let $w$ be a group word\nof the form\n\\begin{equation}\n\\label{particular word}\nw = x_1^m \\prod_{2\\le i<j\\le n} \\, [x_i,x_j]^{k_{ij}}.\n\\end{equation}\nThen $N(G'\\times G^{(n-1)},w)\\ge |G|^{n-1}$.\n\\end{thm}\n\n\n\n\\begin{proof}\nWe argue by induction on $n$, the case $n=1$ being trivial.\nDue to the form of the word $w$, it is impossible to have $n=2$, so we suppose\nnow that $n\\ge 3$.\nIn particular, we may assume that not all the exponents $k_{ij}$ are $0$.\nChoose the pair $(r,s)$ such that the $p$-part of the exponent $k_{rs}$ is\nas small as possible.\nBy renaming the variables, we may assume that $r=n-1$ and $s=n$.\nFor simplicity, let us write $k$ for $k_{n-1,n}$.\nFor every pair $(i,j)$ with $2\\le i<j\\le n$, let $k'_{ij}$ be a solution\nto the congruence\n\\[\nk'_{ij}k\\equiv k_{ij} \\pmod{|G|}.\n\\]\n(Such a solution exists by the choice of $(r,s)$, and because $|G|$ is a power\nof $p$.)\nBy the remark preceding Lemma 5, we may assume that\n\\[\nw = x_1^m \\prod_{2\\le i<j\\le n} \\, [x_i,x_j]^{k'_{ij}k}.\n\\]\nIn the previous expression, we collect all commutators which contain either\n$x_{n-1}$ or $x_n$, and get\n\\[\nw = x_1^m v [x_{n-1},x_n]^k [x_{n-1},v_{n-1}]^k [v_n,x_n]^k,\n\\]\nfor some words $v$, $v_{n-1}$, and $v_n$ involving only the variables $x_i$\nwith $2\\le i\\le n-2$.\nThus\n\\[\nw = x_1^m v' [x_{n-1}v_n,x_nv_{n-1}]^k,\n\\]\nwhere $v'=[v_{n-1},v_n]^k v$ again only involves $x_2,\\ldots,x_{n-2}$.\nNow, by using Lemmas 3 and 4 with $u_{n-1}=x_{n-1}v_n$,\n$u_n=x_nv_{n-1}$ and $u_i=x_i$ for $i=1,\\ldots,n-2$ (see the remark after Lemma\n4), we get\n$N(G'\\times G^{(n-1)},w)=N(G'\\times G^{(n-1)},w')$ for the word\n\\[\nw' = x_1^m v' [x_{n-1},x_n]^k = x_1^m v' [x_{n-1}^k,x_n].\n\\]\n\nSince $w''=x_1^m v'$ is of the form in the statement of the theorem and involves $n-2$\nvariables, the result is true for $w''$.\nOn the other hand,\n\\begin{align*}\nN(G'\\times G^{(n-1)}\n&,w)\n\\\\\n&=\n\\big| \\{ (g_1,\\ldots,g_n)\\in G'\\times G^{(n-1)}\n\\mid w''(g_1,\\ldots,g_{n-2})=[g_{n-1}^k,g_n]^{-1} \\} \\big|.\n\\end{align*}\nBy Lemma 6, each element of the commutator subgroup $[g_{n-1}^k,G]$ is\nof the form $[g_{n-1}^k,g_n]$, and there are $|C_G(g_{n-1}^k)|$ choices of $g_n$\ngiving rise to the same element.\nConsequently,\n\\begin{align*}\n&N(G'\\times G^{(n-1)},w) =\n\\\\\n& \\sum_{g_{n-1}\\in G} \\, |C_G(g_{n-1}^k)| \\,\n\\big| \\{ (g_1,\\ldots,g_{n-2})\\in G'\\times G^{(n-3)}\n\\mid w''(g_1,\\ldots,g_{n-2})\\in [g_{n-1}^k,G] \\} \\big|.\n\\end{align*}\nNow, if we put $\\overline G=G\/[g_{n-1}^k,G]$, we have\n\\begin{multline*}\n\\big| \\{ (g_1,\\ldots,g_{n-2})\\in G'\\times G^{(n-3)}\n\\mid w''(g_1,\\ldots,g_{n-2})\\in [g_{n-1}^k,G] \\} \\big|\n\\\\\n=\n|[g_{n-1}^k,G]|^{n-2} N(\\overline G'\\times \\overline G^{(n-3)},w'').\n\\end{multline*}\nSince the result is true for $w''$, we get\n\\begin{align*}\nN(G'\\times G^{(n-1)},w)\n&\\ge\n\\sum_{g_{n-1}\\in G} \\, |C_G(g_{n-1}^k)| \\, |[g_{n-1}^k,G]|^{n-2} \\, |\\overline G|^{n-3}\n\\\\\n&=\n\\sum_{g_{n-1}\\in G} \\, |C_G(g_{n-1}^k)| \\, |[g_{n-1}^k,G]| \\, |G|^{n-3}\n\\\\\n&=\n\\sum_{g_{n-1}\\in G} \\, |G|^{n-2}\n=\n|G|^{n-1},\n\\end{align*}\nwhere we have used again Lemma 6.\n\\end{proof}\n\n\n\n\n\\begin{thm}\n\\label{general case}\nLet $G$ be a finite $p$-group of class at most $2$, and let $w=w(x_1,\\ldots,x_n)$\nbe a group word.\nThen $N(G,w)\\ge |G|^{n-1}$.\n\\end{thm}\n\n\n\n\\begin{proof}\nBy working in the abelian group $F\/F'$, we can write $w=(x_1^{k_1}\\ldots x_n^{k_n})^m v$,\nwhere $v\\in F'$, $m$ is a power of $p$, and at least one $k_i$ is not divisible by $p$.\nAfter renaming the variables if necessary, we may assume that $i=1$.\nIf we put\n\\[\nu_1=x_1^{k_1}\\ldots x_n^{k_n},\\ u_2=x_2,\\ \\ldots \\ ,\\ u_n=x_n,\n\\]\nthen we can apply Lemmas 3 and 4 to obtain $N(G,w)=N(G,w')$,\nwhere now $w'$ is of the form $w' = x_1^m v'$, for some word $v'\\in F'$.\nSince $G$ has class at most $2$, we may assume that $v'$ is of the form\n\\[\n\\prod_{1\\le i<j\\le n} \\, [x_i,x_j]^{k_{ij}}.\n\\]\nWe can write $v'=[x_1,v_1']v_2'$, where $v_1'$ and $v_2'$ depend only on $x_2,\\ldots,x_n$,\nand where $v_2'\\in F'$.\nIf we put $w''=x_1^m v_2'$, then $N(G,w)\\ge N(G'\\times G^{(n-1)},w'')$.\nNow the result follows by applying Theorem 7.\n\\end{proof}\n\\textbf{References}\n\\begin{flushleft}\n[A] A. Amit, On equations in nilpotent groups, unpublished\\medskip\n\n\n[L] M. Levy, On the probability of satisfying a word in nilpotent groups of class 2, \\emph{http:\/\/arxiv.org\/abs\/1101.4286}\\medskip\n\n\n\\end{flushleft}\n\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Sec:Introduction}\n\nSymplectic homology \nis a version of Hamiltonian Floer homology for a class of open symplectic \nmanifolds with contact boundary, including Liouville manifolds \\cites{SymplecticHomology,ViterboSH}. \nSome of the applications of symplectic homology include special cases of \nthe Weinstein conjecture \\cite{ViterboSH}, and a proof that there are \ninfinitely many distinct symplectic structures on $\\mathbb{R}^{2n}$, for $n\\geq 4$ \\cite{McLean1}.\n\nOne of the interests of symplectic homology is the\nfact that it relates to other important tools of symplectic\ntopology, including contact homology \\cite{BOExactSequence},\nRabinowitz Floer homology \\cite{CieliebakFrauenfelderOancea} and the Fukaya category\n\\cites{AbouzaidGeneration,GanatraThesis}, as well as to string\ntopology \\cite{AbbondandoloSchwarzRing}.  In this paper, we explore a\nrelation between symplectic homology and quantum cohomology. For other\nrelations between these two invariants, see Seidel \\cite{SeidelLFIII} \nas well as Borman and Sheridan \\cite{BormanSheridan}.\n\nSymplectic homology has a wealth of algebraic structures, including a product\nand a Batallin--Vilkovisky operator, as observed by Seidel\n\\cite{SeidelBiasedView}. This paper will not address the computation of these\nalgebraic structures, but we intend to explore that direction in future work\nusing the methods developed here.   \n\nDespite its great relevance, symplectic homology is often very hard to\ncompute. Among the few known results is the fact that it vanishes for\nsubcritical Weinstein manifolds (including $(\\mathbb{R}^{2n},\\omega_{std})$)\n\\cite{CieliebakHandleAttaching} and \nfor flexible Weinstein domains \\cite{BEEsurgery}.\nSymplectic homology of the cotangent\nbundle of a closed, orientable spin manifold $M$ is isomorphic\nas a ring to the Chas--Sullivan string topology ring of $M$\n\\cites{AbbondandoloSchwarzRing,AbouzaidViterbo}.  There is also a\nsurgery formula for the symplectic homology of a Weinstein manifold\n\\cites{BEEsurgery,BEEProduct}, which has been used to compute new examples\n\\cites{EkholmNg,EtguLekiliKoszul}.\n\nIn this paper, we explore the problem of computing the symplectic homology\nof a special class of symplectic manifolds, namely complements of certain\nsymplectic hypersurfaces $\\Sigma$ in closed symplectic manifolds $X$. An\nimportant source of examples comes from smooth ample divisors in projective\nvarieties.  Allowing for $\\Sigma$ to be a normal crossings divisor would\nbe a significant improvement in generality, but is currently beyond\nour reach. Important progress has been made in that direction, including\n\\cites{Pascaleff,NguyenThesis,GanatraPomerleano1,GanatraPomerleano2,McLeanMBSequence}.\n\n\nUnder these assumptions on $X$ and $\\Sigma$, the complement\n$X\\setminus\\Sigma$ is the interior of a Liouville domain, hence its\nsymplectic homology can be defined.  The goal of this paper is to\ncompute the symplectic homology groups of such divisor complements, in terms of counts of rigid\npseudoholomorphic spheres in $\\Sigma$ and $X$. These counts can sometimes be expressed as\nabsolute Gromov--Witten invariants of $(\\Sigma,\\omega_\\Sigma)$ and relative Gromov--Witten\ninvariants of $(X,\\Sigma,\\omega)$. \n\nThe following is the simplest formulation of our main result. We call a Morse function {\\em perfect} \nif the corresponding Morse differential vanishes.  \n\n\\begin{theorem}\nIf $\\Sigma$ and $X \\setminus \\Sigma$ admit perfect Morse functions, then\nthere is a chain complex computing the symplectic homology of $X\\setminus\n\\Sigma$, whose differential is expressed in terms of Gromov--Witten invariants of\n$(\\Sigma,\\omega_\\Sigma)$, relative Gromov--Witten invariants of $(X,\\Sigma,\\omega)$ and\nthe Morse differentials in the contact boundary $Y$ and in $X\\setminus \\Sigma$.  \n\\end{theorem}\n\nFor more general and precise statements, see Theorem \\ref{T:differential}\nand Lemmas \\ref{lacunary GW} and \\ref{augmentation rel GW}. \nThe strategy of the proof is inspired by\n\\cite{BOExactSequence}, which relates the symplectic homology of a symplectic manifold with contact type boundary with the \ncontact homology of the boundary. The idea consists of stretching the neck along the\nboundary $Y$ of a tubular neighborhood of $\\Sigma$ in $X$, and keeping track of \nthe degenerations of Floer cascades that contribute to the differential.\nThe limit configurations will include pieces in the symplectization of\nthe contact manifold $Y$, and pieces in (the completion of) $X\\setminus\n\\Sigma$. The former project to pseudohomolorphic spheres\nin $\\Sigma$. \nThese are related to Gromov--Witten invariants of $(\\Sigma,\\omega_\\Sigma)$.\nThe curves in $X \\setminus \\Sigma$ \nare related to pseudoholomorphic spheres in $X$ with tangency\nconstraints in $\\Sigma$. Under suitable hypotheses, they are counted by relative Gromov--Witten invariants of $(X,\\Sigma,\\omega)$. \n\n\\begin{remark}\n\nThe theorem also holds if $X$ and $\\Sigma$ are semipositive\n\\cite{McDuffSalamon}*{Definition 6.4.1} with {\\em negative} monotonicity\nconstants.\nSuch symplectic manifolds have no non-constant \n$J$-holomorphic spheres for regular $J$ \\cite{McDuffSalamonOld}*{proof of Lemma 5.1.3},\nand hence the symplectic homology differential is obtained from purely Morse\ntheoretic terms.\n(This was pointed out to us by Fran\\c cois Charest). \n\n\n\n\\end{remark}\n\nAn important step in our argument is the proof of a correspondence between moduli spaces of solutions to Floer's equation in $\\mathbb{R}\\times Y$ and moduli spaces of punctured pseudoholomorphic spheres in $\\mathbb{R}\\times Y$. This is done by showing that the {\\em difference} $\\bf e$ (in a sense we make precise) between a Floer solution and a corresponding pseudoholomorphic curve solves a certain PDE, and then proving that solutions to this PDE exist and are unique (see Sections \\ref{S:ansatz_Floer_to_holo} and \\ref{S:ansatz_holo_to_Floer}). \n\n\\begin{remark}\n As mentioned earlier, this paper drew much inspiration from\n \\cite{BOExactSequence} and its use of neck-stretching. We should\n point out some of the main differences between the two papers. In\n \\cite{BOExactSequence}, the authors consider a more general\n situation, in which the symplectic manifold is not necessarily obtained\n from the complement of a smooth symplectic divisor.  For this reason, the\n contact boundaries $Y$ they consider are not necessarily prequantization\n bundles over $\\Sigma$, as in our case.  The fact that our $Y$ are so\n special has a number of advantages which we use,\n notably in Part \\ref{Ansatz}. \n This enables\n us to obtain a bijection between moduli spaces of Floer trajectories\n and of pseudoholomorphic curves (rather than just a continuation map relating symplectic homology and a non-equivariant version of contact homology),\n which is then used to compute the symplectic homology differential\n explicitly. \n As studied in our paper \\cite{DiogoLisiSplit}, this also allows us to achieve\n transversality by geometric arguments involving monotonicity and automatic\n transversality.\n For more on the relation between this work and \\cite{BOExactSequence}, see Remark \\ref{Cone}.\n\\end{remark}\n\n\\begin{remark}\nThe contact boundaries $Y$ of the Liouville domains $X\\setminus\n\\Sigma$ are such that the Reeb flow is a free $S^1$-action. As explained in\n\\cite{SeidelBiasedView}*{(3.2)}, there is a spectral sequence converging\nto symplectic homology, whose first page consists of homology groups of\n$Y$ and $X\\setminus \\Sigma$. One can think of this paper as computing\nthe differential on that page of the spectral sequence (and there happen to\nbe no higher order differentials). This is also related to Remark \\ref{Cone}.   \n\\end{remark}\n \nThis paper is complemented by \\cite{DiogoLisiSplit} and is organized as follows. \nPart 1 provides more details about\nour geometric setup and the almost complex structures that we use. \nPart 2 describes the chain complexes that compute symplectic\nhomology, before and after stretching the neck. We use a Morse--Bott\napproach, so the differentials count {\\em Floer cylinders with cascades}. The full discussion of the stretched  \ncase (which we refer to as {\\em split}) is given in \\cite{DiogoLisiSplit}, including the relevant transversality results\nand an explanation of how our monotonicity assumptions imply that the Floer cylinders with cascades \nthat can contribute to the split differential can only be of four simple types. \nPart 3 is the most technical of this paper, where we explain why counting punctured \nFloer cylinders in $\\mathbb{R}\\times Y$ is equivalent to counting punctured \npseudohomolorphic cylinders in $\\mathbb{R}\\times Y$, and how the latter can be\nexpressed in terms of counts of pseudoholomorphic spheres in $\\Sigma$\n(which are in turn related to Gromov--Witten invariants).  Finally, in Part 4, \nwe illustrate this computation scheme by calculating the symplectic homology of $T^*S^2$ \n(which, as we mentioned earlier, is isomorphic to the well-known homology of the free loop \nspace of $S^2$, see for instance \\cite{CohenJonesYan}). \n\n\nBesides the references mentioned throughout the paper, this article shares \nsome important ideas with several works in the literature. A few of these are \n\\cites{BourgeoisThesis,GHK,EliashbergPolterovichQuasi,RitterNegativeBundles,\nFabertII,Uebele1,Uebele2,Fauck,Kerman,AlbersGuttHein,KimKwonLee}.\n\n\\subsection*{Acknowledgements}\nOur work has benefitted greatly from help and guidance we have received from\nmany people. We thank especially Yasha Eliashberg. We are also grateful for\nhelpful discussions with and feedback from\nMohammed Abouzaid, \nPaul Biran,\nStrom Borman,\nFr\\'ed\\'eric Bourgeois,\nKai Cieliebak,\nOliver Fabert,\nJoel Fish,\nEleny Ionel,\nJanko Latschev,\nMelissa Liu,\nDusa McDuff,\nMark McLean,\nLeonid Polterovich,\nDaniel Pomerleano,\nDietmar Salamon,\nMohammad Tehrani,\nJean-Yves Welschinger, and \nAleksey Zinger.\nThanks are also due to the anonymous referee for a careful reading of the text and helpful suggestions. \n\nS.L.~was partially supported by the ERC Starting Grant of Fr\\'ed\\'eric Bourgeois\nStG-239781-ContactMath and also by Vincent Colin's ERC Grant geodycon.\n\nL.D.~thanks Stanford University, ETH Z\\\"urich, Columbia University and Uppsala University \nfor excellent working conditions. L.D.~was partially supported by the Knut and Alice Wallenberg Foundation.  \n\n \n\\part{Setup, almost complex structures and neck-stretching}\n\n\\section{Almost complex structures and neck-stretching}\n\nIn this section, we consider a local model of the divisor, and identify a class\nof almost complex structures that are well-behaved in this local model.\n\n\n\\subsection{Symplectic hyperplane sections}\n\nWe adopt some of our terminology from \\cite{BiranKhanevsky}*{Section 2.2}, \nwith minor modifications.\nA variant on this construction is also explicitly described in\n\\cite{Albers_Frauenfelder_Negative_Line_Bundles}*{Section 3.1}.\n\nLet  $(X,\\omega)$ be a closed symplectic manifold such that\n$[K\\omega]$ admits a lift to $H^2(X;\\mathbb{Z})$ for some $K>0$.\nWe refer to a closed codimension 2 symplectic submanifold\n$\\Sigma^{2n-2}\\stackrel{\\iota}{\\subset} X$ as a \\defin{symplectic\ndivisor}.  Write $\\omega_\\Sigma= \\iota^*\\omega$.\n\n\n\n\nLet $\\pi \\colon E\\to \\Sigma$ be a complex line bundle over $\\Sigma$\nsuch that $c_1(E)= [K\\omega_\\Sigma]$ in $H^2(\\Sigma;\\mathbb{R})$, and  \nchoose a Hermitian metric on $E$. \nThen, there exists a connection 1-form $\\Theta\\in\n\\Omega^1(E\\setminus \\Sigma)$ such that $d\\Theta = - K \\pi^*\\omega_\\Sigma$. \n\\footnote{In \\cite{BiranKhanevsky}, $\\Theta$ is referred to as\nthe global angular 1-form $\\alpha^{\\nabla}$.} \n If the coordinate $\\rho$ stands for the norm in $E$ measured with\nrespect to the Hermitian metric, and $f \\colon [0, +\\infty) \\to (0, 1]$\nis a smooth function with $f(0) = 1$, $f' < 0$, then \n\\begin{equation} \\label{E:connectionForm}\n\\omega_E:= -d \\left( \\frac{f( \\rho^2)}{K} \\Theta \\right) = -\\frac{2 \\rho f'( \\rho^2)}{K} d\\rho \\wedge \\Theta + f(\\rho^2) \\pi^* \\omega_\\Sigma \n\\end{equation}\nis a symplectic form on $E\\setminus \\Sigma$ that extends smoothly to all of $E$. \nIts restriction to the $0$-section is $\\omega_\\Sigma$. Following  \n\\cite{BiranKhanevsky}, we will explicitly use $f(\\rho^2) = \\operatorname{e}^{-\\rho^2}$. \nBy construction, this symplectic form is exact on $E \\setminus 0$.\nThe Liouville vector field dual to the primitive $-\\frac{f( \\rho^2)}{K} \\Theta$ is \n\\[\n    \\frac{f(\\rho^2)}{2\\rho f'(\\rho^2)} \\frac{\\partial}{\\partial \\rho},\n\\]\nwhich points inwards along the fibres. \nThis then induces a contact form on the $\\rho_0$--sphere bundle, and we denote this space \nby $Y$. This contact form is \n\\[\n    \\alpha_{\\rho_0} = -\\frac{f(\\rho_0^2)}{K} \\Theta|_Y.\n\\]\nObserve that by following the flow of the Liouville vector field, we obtain that \n$E\\setminus \\Sigma$ \nequipped with the symplectic form $\\omega_E$ is then symplectomorphic \nto a piece of symplectization $\\left ( (-\\infty, -\\ln f(\\rho_0^2)) \\times Y,\nd(\\operatorname{e}^r \\alpha_{\\rho_0}) \\right )$. \n\nWith a choice of contact form coming from this construction, \n$(Y,\\alpha)$ is a {\\em prequantization bundle} associated to $(\\Sigma, K\\omega_\\Sigma)$. \nDifferent choices of $\\rho_0$ give diffeomorphic sphere bundles (by scaling in the\nfibre direction), but change the scaling of the contact form. It will be convenient to\ntake $\\alpha = - \\frac{1}{K} \\Theta|_Y$. This corresponds to a real blow-up\nof the $0$-section (i.e.~a limit as $\\rho_0 \\to 0$). \n\n\nSay that a {\\em symplectic tubular neighborhood} of the symplectic divisor $\\Sigma \\stackrel{\\iota}{\\subset} X$ is an extension of $\\iota$ to a\nsymplectomorphism $\\varphi \\colon \\mathcal{U} \\to X$, where $\\mathcal{U}$ is a disk neighborhood of the zero section on a complex line bundle $E$ over $\\Sigma$, equipped with a Hermitian metric and a symplectic form $\\omega_E$ as above.  \nIt is well known that every symplectic divisor has a symplectic tubular\nneighborhood (see \\cite{McDuffSalamonIntro}), where $E$ is the symplectic normal bundle to $\\Sigma$ in $X$.\n\n\\begin{definition}\nThe symplectic divisor $\\Sigma$ is a \\defin{symplectic hyperplane section}\nof $X$ if $[\\Sigma]$ is Poincar\\'e dual (over $\\mathbb{Q}$) to $[K\\omega]$ for some $K >0$.\n\n\\label{hyperplane}\n\\end{definition}\n\nThe condition of $\\Sigma$ being a symplectic hyperplane section\nsometimes includes the assumption that the complement of a neighborhood\nof $\\Sigma$ is a Weinstein domain. We will not need this assumption for\nmost of the paper, and will point out the one point where it is used,\nin the statement of Theorem \\ref{T:differential}.\n\n\nSources of examples of symplectic hyperplane sections \ninclude smooth ample divisors in smooth complex projective\nvarietes \\cite{BiranBarriers} and Donaldson divisors in symplectic\nmanifolds with rational $[\\omega]$ \\citelist{\\cite{GirouxOpenBook}\n\\cite{Giroux_remarks_Donaldson}}. Note that\nthis latter class of examples typically does not satisfy the\nadditional monotonicity requirements we impose on the divisor.\n\nWe now observe that the complement of the neighbourhood of the \nhyperplane section is a Liouville domain. In different language, this\nis \\cite{Opshtein_Polarizations}*{Lemma 2.1}. See also\n\\cite{Giroux_remarks_Donaldson}*{Proposition 5}.\n\n\n\\begin{lemma} \\label{L:WisLiouville}\n\n    Let $\\Sigma \\subset X$ be a symplectic hyperplane section, as in\n    Definition \\ref{hyperplane}, and let $E \\to \\Sigma$ be the normal bundle\n    to $\\Sigma$, endowed with a Hermitian metric and symplectic form\n    $\\omega_E$ as in Equation \\eqref{E:connectionForm}.\n\n    Then, \n    \\begin{enumerate}[(i)]\n        \\item the symplectic form $\\omega|_{X \\setminus \\Sigma}$ is exact on\n            $X \\setminus \\Sigma$; \n        \\item there exist a primitive $\\lambda$ on $X \\setminus \\Sigma$,\n            an open $\\rho_0$-disk bundle $\\mathcal{U} \\subset E$ and a\n            symplectic embedding $\\varphi \\colon \\mathcal{U} \\to X$ such that $\\varphi$\n            restricted to the $0$-section is the inclusion and \n            \\[\n                \\varphi^*\\lambda = - \\frac{f(\\rho^2)}{K} \\Theta\n            \\]\n            on $\\mathcal{U} \\setminus \\Sigma$. \n    \\end{enumerate}\n\n    In particular then, $X\\setminus \\Sigma$ is symplectomorphic to\n    $\\overline{W} \\setminus \\partial \\overline{W}$ where $\\overline{W}$ is a\n    Liouville domain with boundary $Y$, where $Y$ is diffeomorphic to the \n    $\\rho_0$-circle bundle in $E$. \n    The primitive $\\lambda$ extends to the\n    boundary and induces the contact form \n    \\[\n        \\alpha = - \\frac{1}{K} \\Theta|_Y\n    \\]\n\\end{lemma}\n\\begin{proof}\n\nFirst, the assumption that $\\Sigma$ is Poincar\\'e dual to a multiple of\n$\\omega$ implies \n    that $[\\omega|_{X \\setminus \\Sigma}] = 0 \\in H^2(X \\setminus\n    \\Sigma; \\mathbb{R})$. Thus, \n    $\\omega|_{X \\setminus \\Sigma} = d\\lambda_0$ for some 1-form\n    $\\lambda_0$ on $X \\setminus \\Sigma$. \n\nNow, by the symplectic neighbourhood theorem (see, for instance,\n\\cite{McDuffSalamonIntro}*{Theorem 3.30}), there exists a neighbourhood of the\n$0$-section $\\mathcal{U}$ (which we may take to be a $\\rho_0$-disk bundle) and a symplectic\nembedding \n\\[\n    \\varphi_0 \\colon \\mathcal{U} \\to X,\n\\]\ninducing the inclusion on the $0$--section.\n\nWriting $\\lambda_E = -\\frac{f(\\rho^2)}{K} \\Theta$, we have $\\omega_E = d\\lambda_E$.\n\nWe now have \n\\[\n    d \\left ( \\varphi_0^*\\lambda_0 - \\lambda_E \\right ) = 0\n\\]\non $\\mathcal{U} \\setminus \\Sigma$.\n\nThe deleted neighbourhood $\\mathcal{U} \\setminus \\Sigma$ retracts on to a $\\rho$-circle\nbundle $Y$ (for any $0 < \\rho < \\rho_0$). \nThe Gysin sequence applied to the circle bundle $Y \\to \\Sigma$ gives\n\\[\n        0 \\to H^1(\\Sigma; \\mathbb{R}) \\xrightarrow{\\pi^*} H^1(Y; \\mathbb{R}) \\xrightarrow{\\pi_*}\n        H^0(\\Sigma; \\mathbb{R}) \\xrightarrow{\\cup [K\\omega_\\Sigma]} H^2(\\Sigma; \\mathbb{R}).\n    \\]\n    The last map is an injection from $H^0(\\Sigma; \\mathbb{R})$ \n    to $H^2(\\Sigma; \\mathbb{R})$ and so $\\pi^*$ is an isomorphism. \n\nIt follows then that \n\\[\n    \\varphi_0^*\\lambda_0 - \\lambda_E = \\pi^*\\mu + dg\n\\]\nfor some closed 1-form $\\mu$ on $\\Sigma$ and some function $g \\colon \\mathcal{U} \\to \\mathbb{R}$. \n\nExtend $g\\circ \\varphi_0^{-1}$, defined on $\\varphi_0(\\mathcal{U})$, to a function on $X$\nthat we also denote by $g$. \nThe desired primitive is then $\\lambda = \\lambda_0 - dg$.\nIn particular then, \\[\n    \\varphi_0^*\\lambda = -\\frac{f(\\rho^2)}{K} \\Theta + \\pi^*\\mu.\n\\]\n\nWe will now construct a symplectomorphism $\\phi$ of $E$, sending the\n$0$-section to the $0$-section, so that $\\phi^*\\varphi^*\\lambda = \\lambda_E$.\n\n\n    Let $V$ be a vector field on $\\Sigma$ defined by $\\omega_\\Sigma(V, \\cdot)\n    =  \\mu$. The flow of $V$ induces a symplectomorphism since $\\mu$ is\n    closed. Now, construct a horizontal lift $\\tilde V$ to $E$, with the\n    properties that $\\rho^2 \\Theta(\\tilde V) = 0$ and $\\rho d\\rho(\\tilde V) = 0$. \n    In particular, $\\tilde V$ is defined on the total space $E$ and its\n    restriction to the $0$-section is $V$.\n\nLet $\\phi^t$ be the\n    flow of the vector field $-\\frac{1}{f(\\rho^2)} \\tilde V$. \n    Notice that this flow preserves the\n    levels $\\rho = \\text{constant}$.\n    \n    Also notice that $\\pi^*\\mu(\\tilde V) =\n    \\mu(V) =  \\omega_\\Sigma(V, V) = 0$ and $\\omega_E(\\tilde V, \\cdot) =\n    f(\\rho^2) \\pi^*\\omega_\\Sigma(V, \\cdot) = f(\\rho^2)\\pi^*\\mu$.\n    \n    Finally, we calculate:\n    \\begin{align*}\n        \\frac{d}{dt} (\\phi^t)^*\\left( \\lambda_E - t\\pi^*\\mu \\right ) \n        &= (\\phi^t)^* \\left( \\frac{1}{f(\\rho^2)}L_{\\tilde V} \\lambda_E - \\pi^*\\mu \\right ) \\\\\n        &= (\\phi^t)^* \\left( \\operatorname{e}^{\\rho^2} \\omega_E(\\tilde V, \\cdot) - \\pi^*\\mu \\right )\\\\\n        &= (\\phi^t)^* \\left( \\pi^*\\omega_\\Sigma( \\tilde V, \\cdot) - \\pi^*\\mu \\right ) \\\\\n        &= 0.  \n    \\end{align*}\n\n    Thus, by precomposing with the time-1 map $\\phi^1$, we then have \n    $(\\phi^1)^*\\varphi^*\\lambda|_Y = \\lambda_E|_Y = \\alpha_{\\rho_0}$, as required.\n\n\n    Notice now that \n    \\begin{align*}\n        [0, +\\infty) &\\to (-\\infty, 0]\\\\\n        \\rho &\\mapsto \\log f(\\rho^2)\n    \\end{align*} \n    is a diffeomorphism, using the fact that $f(0) = 1$ and $f' < 0$. \n    Let $\\rho(r)$ denote the inverse function.\n    In particular then, $\\operatorname{e}^r = f( \\rho(r)^2 )$.\n    \n    Then, if $Y$ is the $\\rho_0$-circle bundle in $E$, and $\\alpha = -\n    \\frac{1}{K}\\Theta|_Y$, we have a diffeomorphism\n    \\begin{align*}\n        (\\log f(\\rho_0^2), 0) \\times Y \\to (\\mathcal{U} \\setminus \\Sigma)\\\\\n        (r, y) \\mapsto \\frac{\\rho(r)}{\\rho_0}y\n    \\end{align*}\n    Notice that $| \\frac{\\rho(r)}{\\rho_0}y| = \\rho(r)$, so this pulls-back the\n    1-form $\\lambda_E$ to $-\\frac{e^r}{K}\\Theta|_Y = \\operatorname{e}^r \\alpha$.\n    \n    This therefore shows that $\\varphi(\\mathcal{U}) \\setminus \\Sigma$ is\n    symplectomorphic to $(\\log( f(\\rho_0^2) ), 0) \\times Y$ with symplectic\n    form given by $d(\\operatorname{e}^r \\alpha)$. We may now compactify by gluing in\n    $\\partial \\overline{W} = \\{0 \\} \\times Y$. \n\n\tNote that this implies that the vector field dual to $\\lambda$\n\tpoints outwards at the boundary, and is thus a Liouville vector field\n\tand $\\lambda$ is a Liouville form for $\\omega$.  \n        \\footnote{This fact was also pointed out to us by McLean, using a different\n        argument.}\n\n\\end{proof}\n\n\n\nFix now the data of the complex line bundle $E$ with Hermitian metric and\nconnection form, together with a neighbourhood of the $0$-section $\\mathcal{U}$ and\nsymplectic neighbourhood $\\varphi \\colon \\mathcal{U} \\to X$. \nWe will also henceforward assume that $f(\\rho^2) = \\operatorname{e}^{-\\rho^2}$ for\nconcreteness.\nAssume furthermore that $\\mathcal{U}$ is chosen so that $\\varphi$ extends to an\nembedding $\\varphi \\colon \\overline{\\mathcal{U}} \\to X$ of the closure of $\\mathcal{U}$. \nWe will allow our almost complex structures on $X$ to be perturbed in $\\mathcal{V} = X \\setminus \\varphi(\\overline \\mathcal{U})$, which is non-empty.\n\n\n\n\\subsection{Almost complex structures}\n\\label{sec:almost complex structures}\n\nSince the Hermitian connection on the complex line bundle $E$ above defines a horizontal distribution,\nany almost complex structure $J_\\Sigma$ on \n$(\\Sigma, \\omega_\\Sigma)$ may be lifted to an almost complex structure $J_E$ \nby requiring that $J_E$ preserve the horizontal and vertical subspaces,\nis $i$ on the vertical subspaces and is the lift of $J_\\Sigma$ to the horizontal subspaces. In particular, if $\\pi \\colon E \\to \\Sigma$ is the bundle projection map, we have $d\\pi \\cdot J_E = J_\\Sigma \\cdot d\\pi$.\n\n\\begin{definition} \\label{admissible J}\nAn almost complex structure $J_X$ on $(X, \\omega)$, \ncompatible with $\\omega$, is called\n\\defin{admissible} if, in the fixed neighbourhood $\\mathcal{U}$ of $\\Sigma$,  \n$J_X = \\varphi_*J_E$, where $J_E$ is the lift of an almost complex structure $J_\\Sigma$ on $\\Sigma$.\nIn particular, $\\Sigma$ has $J_X$ invariant tangent spaces \n(i.e.~is $J_X$-holomorphic).  \n\\end{definition}\n\n\nIn order to compute the symplectic homology of $X \\setminus \\Sigma$, we will need\nto study Floer trajectories in the completion $W$ of $X \\setminus \\Sigma$ (which we construct below).\nThis will involve SFT-style splitting (neck-stretching) along a contact-type hypersurface\nin $W$. We will construct a suitable family of almost complex structures on $W$ to implement this process.\n\n\n\\begin{remark}\nThe idea of this construction is very simply illustrated by an example where this \nbecomes trivial: let $X = S^2$ with area 1, and let $\\Sigma$ be a point.\nIt follows that $X \\setminus \\Sigma$ is symplectomorphic to the open unit disk. \nIts completion involves attaching $[0, +\\infty) \\times S^1$ to this open disk\nto give $\\mathbb{C}$ with its standard symplectic form.\n\nHowever, $X$ can be identified with $\\mathbb{CP}^1$\nand thus $X \\setminus \\Sigma$ is, holomorphically, the complex plane, which, as we argued, is the completion of the disk.\n\nMore generally, this same phenomenon occurs. First, $X \\setminus \\Sigma$ must be completed\nby attaching a cylindrical end. This completion is then biholomorphic (but not symplectomorphic) to $X \\setminus \\Sigma$.\nOur construction is to start with an admissible almost complex structure $J_X$ on $X$, complete $X \\setminus \\Sigma$ and equip it with a family \nof almost complex structures that allow us to simultaneously define\nsymplectic homology and stretch the neck. In the trivial example of $X = \\mathbb{CP}^1$, we \nstretch along a copy of $S^1$. The resulting split\nalmost complex manifold (as in \\cite{SFTcompactness}) will consist of two levels: \na lower level that is $\\mathbb{C}$ and an upper level that is $\\mathbb{R} \\times S^1$. \nThe lower level will then be biholomorphic to $X\\setminus \\Sigma$ with $J_X$, and the upper level will be biholomorphic to the normal \nbundle of $\\Sigma$ with the 0-section removed.\n\\end{remark}\n\n\nLet $Y \\subset E$ be the $\\rho_0$ circle bundle with respect to the\nHermitian metric on $E$. Let \n$\\alpha = -\\frac{1}{K}\\Theta$. Recall that the induced contact form on $Y$,\nseen as the $\\rho_0$-circle bundle, is $\\alpha_{\\rho_0} = \\operatorname{e}^{-\\rho_0^2} \\alpha$.\n\nAs observed earlier, there is an exact symplectomorphism \n\\begin{align*}\n    \\psi_1 \\colon (E \\setminus \\Sigma, \\omega_E) &\\to \\left ( (-\\infty, 0) \\times Y, d(\\operatorname{e}^{r} \\alpha) \\right ) \\\\\n    v &\\mapsto \\big (- |v|^2 , \\frac{\\rho_0 v}{|v|} \\big )\n\\end{align*}\nIn particular then, we have $X \\setminus \\Sigma$ (with primitive $\\lambda$) \nhas a neighbourhood of $\\Sigma$ exact symplectomorphic \n(by composing $\\varphi^{-1}$ with $\\psi_1$) to $(-\\rho_0^2, 0) \\times Y$.\n\n\nSince $\\Sigma$ has a neighborhood in $X$ modelled after a neighborhood\nof the zero section in $E$, we use $\\psi_1$ to glue a copy of\n$\\big([0,\\infty)\\times Y,d(\\operatorname{e}^r \\alpha)\\big)$ to \n$(X\\setminus \\Sigma, \\omega)$. We then obtain a symplectic manifold we denote\nby $W$, and call it the \\defin{completion} of $X \\setminus \\Sigma$.\nBy Lemma \\ref{L:WisLiouville}, there is a global primitive $\\lambda$ of the\nsymplectic form that agrees with $\\operatorname{e}^r \\alpha$ on the cylindrical end $[0,\n\\infty) \\times Y$.\n\n\n\\begin{remark}\nObserve that $J_1\\coloneq (\\psi_1)_*J_E$ becomes singular as $r \\to 0$. Indeed, if $R$ is\nthe Reeb vector field on $Y$ for the contact form $\\alpha = - \\frac{1}{K}\\Theta$, we obtain:\n\\[\nJ_1 \\partial_r = -\\frac{1}{4r \\pi K}R.\n\\]\nConsider now the diffeomorphism  (that is not symplectic!)\n\\begin{align*}\n\\psi_2 \\colon E\\setminus \\Sigma &\\to \\mathbb{R}\\times Y \\\\\nv &\\mapsto (- \\frac{1}{2\\pi K} \\ln|v|, \\frac{\\rho_0 v}{|v|})  \n\\end{align*}\nLet $J_Y \\coloneqq (\\psi_2)_*J_E$ be the pushforward complex structure. \nObserve that \\[\n(\\psi_2)_*( -2\\pi K \\rho \\partial_\\rho )= \\partial_r.\\]\nWe also have $\\Theta( J_E \\rho \\partial_\\rho ) = \\frac{1}{2\\pi}$. \nHence, $\\Theta( J_Y \\partial_r ) = -K$. \nIt follows then that $J_Y \\partial_r = R$. \nFurthermore, $\\psi_2$ commutes with the projection maps $\\pi \\colon E\n\\to \\Sigma$ and $\\pi_\\Sigma \\colon Y \\to \\Sigma$. In particular then, this\ngives the translation invariance of $J_Y|_{\\ker \\alpha}$ and that $J_Y$\npreserves the contact structure. \nThis verifies that $J_Y$ is a cylindrical almost complex structure on $\\mathbb{R}\n\\times Y$, \nadapted to the contact form $\\alpha$.\nNotice that \n$J_Y$ is both $\\mathbb{R}$--translation\ninvariant and Reeb-flow invariant since these two actions correspond to \nthe $\\mathbb{C}^*$ action on \nthe Hermitian line bundle $E$. Even though $\\psi_2$ is not a symplectomorphism,\n$J_Y$ is compatible with the symplectic form $d(\\operatorname{e}^r \\alpha)$.\n\\end{remark}\n\n\nWe now want to define an almost complex structure $J_W$ on $W$ that agrees\nwith $J_X$ on a compact subset and is cylindrical in the complement of a\nslightly larger compact subset. To achieve this, we will use an interpolation\nbetween the complex structures $J_1$ and $J_Y$.\n\nFix a small number $0 < \\epsilon \\le \\rho_0^2$ such that $(-\\epsilon,\\epsilon)\\times Y\\subset W$. \nThe interpolating region will be contained in the piece of the symplectization given\nby \n$(-\\epsilon\/2,-\\epsilon\/4)\\times Y$.\nLet $g\\colon (-\\infty,0) \\to \\mathbb{R}$ be a diffeomorphism (whose graph is given in Figure \\ref{F:g}), such that \n\\begin{equation*}\n\\begin{cases}\ng(r) = r $ in $ (-\\infty,-\\epsilon\/2) \\\\\ng(-\\epsilon\/4) = -\\epsilon\/4 \\\\\ng' > 0 \\\\\ng'(r) = -\\frac{1}{4\\pi K r} $ for $r \\geq -\\epsilon\/4.\n\\end{cases}\n\\end{equation*}\n\n\\begin{figure} \n\\begin{tikzpicture}[domain=0:4]\n %\n    \\draw[->] (-4.2,0) -- (1,0) node[right] {$x$};\n    \\draw[->] (0,-4.2) -- (0,2.2) node[above] {$g(x)$};\n    \\draw[line width=1.5pt, color=black] (-4.2,-4.2) -- (-2.8,-2.8);\n    \\draw[line width=1.5pt,domain=-1.7:-.001,smooth,variable=\\x,black] plot ({\\x},{-0.5*ln(-\\x)+0.5*ln(1.5) - 1.5});\n    \\draw[line width=1.5pt, color=black] (-2.8,-2.8)  ..  controls(-2.2,-2.2) and (-2.2,-1.733333) .. (-1.7,-1.5626);\n    \\draw [dashed, color=black] (-3,0.1) -- (-3,-3);\n    \\draw [dashed, color=black] (-3,-3) -- (0.1,-3);\n    \\draw [dashed, color=black] (-1.5,0.1) -- (-1.5,-1.5);\n    \\draw [dashed, color=black] (-1.5,-1.5) -- (0.1,-1.5);\n    \\node at (-3,.3) {$-\\epsilon\/2$};\n    \\node at (.5,-3) {$-\\epsilon\/2$};\n    \\node at (-1.5,.3) {$-\\epsilon\/4$};\n    \\node at (.5,-1.5) {$-\\epsilon\/4$};\n     %\n  %\n %\n  %\n  %\n\\end{tikzpicture}\n\\caption{The function $g$}\n\\label{F:g}\n\\end{figure}\n\nLet $G \\colon (-\\infty,0)\\times Y\\to \\mathbb{R}\\times Y$ be the diffeomorphism $G(r,x)= (g(r),x)$. \nDefine the almost complex structure $J_W$ on $W$ as follows:\n\\begin{equation*}\nJ_W := \n\\begin{cases}\nJ_X & \\text{ on } W\\setminus \\big([-\\epsilon\/2,\\infty)\\times Y\\big) \\\\\nG_*J_1 & \\text{ on } [-\\epsilon\/2,-\\epsilon\/4)\\times Y \\\\\nJ_Y & \\text{ on } [-\\epsilon\/4,\\infty)\\times Y\n\\end{cases}\n\\end{equation*}\n\nNotice that we may use $G$ to construct a diffeomorphism $\\hat G \\colon X\n\\setminus \\Sigma \\to W$ as follows\n\\begin{align*}\n    &\\hat G(x) = x \\quad \\text{ for } x \\in X \\setminus \\mathcal{U} \\\\\n    &\\hat G(\\varphi \\circ \\psi^{-1}(r, y)) = G(r, y) \\quad \\text{ for } (r, y)\n    \\in (-\\rho_0^2, 0) \\times Y.\n\\end{align*}\nThen, it follows immediately that $\\hat G_*J_X = J_W$.\n\nThe following result compares $J_W$-holomorphic planes with\n$J_X$-holomorphic spheres, and will be useful to relate the symplectic\nhomology differential with relative Gromov--Witten invariants. \n\n\\begin{lemma} \\label{planes = spheres}\nThere is a diffeomorphism $\\psi \\colon W \\to X \\setminus \\Sigma$ such that $\\psi_*J_W = J_X$. This map defines a bijection between $J_W$-holomorphic planes in $W$ asymptotic to Reeb orbits of multiplicity $k$ in $Y$ and $J_X$-holomorphic spheres in $X$ intersecting $\\Sigma$ at precisely one point, with order of contact $k$.\n\\end{lemma}\n\\begin{proof}\nDefine $\\psi$ to be the inverse of the diffeomorphism $\\hat G$ defined above. \nIt follows from our definitions that $\\psi_*J_W = J_X$. \n\nThe $J_W$-plane then corresponds to a punctured sphere in $X$ with finite\nenergy. Applying Gromov's removal of singularities theorem and observing that\nthe multiplicity of the Reeb orbit corresponds to windings around $\\Sigma$ of\na loop near the puncture. This then gives the claimed order of contact.\n\\end{proof}\n\n\n\nHaving defined the class of admissible almost complex structures on $W$,\ncylindrical and Reeb--invariant at infinity, we now also describe how we stretch the\nneck of $W$ along $\\{ 0 \\} \\times Y$ \n(in the sense of SFT compactness \\cite{SFTcompactness}).\n\n\n\\begin{lemma} \\label{L:stretch}\n\tSuppose that $\\Sigma$ is a symplectic hyperplane section of $(X, \\omega)$\n\tand let $J_X$ be an admissible almost complex structure on \n\t$X$. \n\n\tThen, there exists a 1-parametric family of almost complex structures\n\t$J_\\kappa$ on the completion $W$ of $X \\setminus \\Sigma$, which are\n\tcylindrical outside a compact set, so that $(W, J_\\kappa)$ converge\n\tin the SFT sense as $\\kappa\\to \\infty$ to a split almost complex manifold whose \n\tbottom level is $(W, J_W)$ and whose upper level is $(\\mathbb{R} \\times Y,J_Y)$.\n\\end{lemma}\n\n\\begin{proof}\n\nWe explicitly define a family $J_\\kappa$, for $\\kappa>>0$, of almost complex\nstructures on $W$. Let $J_W$ be the almost complex structure corresponding to\n$J_X$ by the construction above, for fixed diffeomorphism $g \\colon (-\\infty, 0)\n\\to \\mathbb{R}$. \n\nFor each $\\kappa \\geq \\epsilon\/4$ take a diffeomorphism \n$$f_\\kappa: [-\\epsilon\/4,\\epsilon\/4] \\to [-\\kappa,\\kappa]$$\nsuch that $f_\\kappa' \\equiv 1$ in a neighborhood of $\\pm \\epsilon\/4$ and $f_\\kappa'> 0$.\nDefine the diffeomorphisms \n\\begin{align*}\nF_\\kappa: [-\\epsilon\/4,\\epsilon\/4]\\times Y &\\to [-\\kappa,\\kappa]\\times Y \\\\\n(r,x) &\\mapsto (f_\\kappa(r),x)\n\\end{align*}\nLet \n\\begin{equation}\nJ_\\kappa := \n\\begin{cases}\nJ_W & \\text{ on } W\\setminus \\big([-\\epsilon\/4,\\infty)\\times Y\\big) \\\\\n(F_\\kappa)^*J_Y & \\text{ on } [-\\epsilon\/4,\\epsilon\/4]\\times Y \\\\\nJ_Y & \\text{ on } (\\epsilon\/4,\\infty)\\times Y\n\\end{cases}\n\\label{J_kappa}\n\\end{equation}\nSee Figure \\ref{Jkappa_fig}. Note that if $\\kappa=\\epsilon\/4$ and $f_\\kappa(r) = r$, then $J_{\\epsilon\/4} = J_W$.\n\n\n\\begin{figure}%\n  \\begin{center}\n    \\def0.8\\textwidth{0.5\\textwidth}\n    %\n\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  }\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}\n    \\renewcommand\\transparent[1]{}%\n  }\n  \\providecommand\\rotatebox[2]{#2}\n  \\ifx0.8\\textwidth\\undefined\n    \\setlength{\\unitlength}{294.97670279pt}\n  \\else\n    \\setlength{\\unitlength}{0.8\\textwidth}\n  \\fi\n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,1.02731484)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{JR.pdf}}%\n    \\put(0.62996209,0.47386319){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$-\\epsilon\/2$}}}%\n    \\put(0.63921907,0.56346689){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$-\\epsilon\/4$}}}%\n    \\put(0.66193603,0.64250162){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$0$}}}%\n    \\put(0.67381388,0.75380551){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\epsilon\/4$}}}%\n    \\put(0.67735345,0.83118533){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\epsilon\/2$}}}%\n    \\put(0.90781135,0.87608029){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$J_Y$}}}%\n    \\put(0.95759067,0.27313536){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$J_W$}}}%\n    \\put(0.81830753,0.22993992){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$J_X$}}}%\n    \\put(0.82329991,0.63681824){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$(F_\\kappa)^*J_Y$}}}%\n  \\end{picture}%\n\\endgroup\n\n   \\end{center}\n  \\caption[The almost complex structure $J_\\kappa$]{The almost complex structure $J_\\kappa$ in\n      \\protect{\\eqref{J_kappa}}}\n  \\label{Jkappa_fig}\n\\end{figure}\n\n\n\nBy SFT compactness \\cite{SFTcompactness}, a sequence of finite energy $J_{\\kappa_i}$-holomorphic curves $u_{\\kappa_i}$ in $W$, with $\\kappa_i \\to \\infty$, has a subsequence converging to a pseudoholomorphic building with levels mapping to either $(\\mathbb{R} \\times Y, J_Y)$ or $(W,J_W)$. \n\\end{proof}\n\n\nRecall that the symplectic homology of $W$ is independent of choice of almost complex structure. \nIn our context, it will be useful to use $J_\\kappa$, for $\\kappa$ large enough.  \n\n\n \n\\part{Moduli spaces and symplectic chain complex}\n\n\n\\section{The chain complex} \\label{S:chain complex}\n\nWe will describe two chain complexes associated to $W$ whose generators are (essentially) the same, but for which the differentials are \\textit{a priori} different.\nWe first define the group underlying these chain complexes.\n\n\n\\begin{figure} \n\\begin{tikzpicture}[domain=0:6]\n %\n    \\draw[->] (-0.2,0) -- (3.1,0) node[right] {$\\operatorname{e}^r = \\rho$};\n    \\draw[->] (0,-1.7) -- (0,2.7) node[above] {$h(\\operatorname{e}^r)$};%\n    \\draw[line width=1.5pt, color=black] (0,0) -- (1,0);\n    \\draw[line width=1.5pt,domain=1:2.7,smooth,variable=\\x,black] plot ({\\x},{(\\x-1)*(\\x-1)});\n    \\draw[line width=0.5pt, color=black] (0,-1.25) -- (3,1.75);\n    \\draw [dashed, color=black] (1.5,-0.1) -- (1.5,0.25);\n    \\node at (1.65,-.3) {$b_k$};\n    \\node at (-.65,-1.25) {$-\\mathcal{A}(\\gamma_k)$};\n     %\n  %\n %\n  %\n  %\n\\end{tikzpicture}\n\\caption{An admissible Hamiltonian and the graphical procedure for computing the action of a periodic orbit}\n\\label{F:Jshaped}\n\\end{figure}\n\n\n\n\\begin{definition} \\label{def:J_shaped}\nConsider a function $h \\colon (0, +\\infty) \\to \\mathbb{R}$ with the following properties:\n\\begin{enumerate}[(i)]\n\\item $h(\\rho) = 0$ for $\\rho \\le 2$;\n\\item $h'(\\rho) > 0$ for $\\rho > 2$;\n\\item $h'(\\rho) \\to +\\infty$ as $\\rho \\to \\infty$;\n\\item $h''(\\rho) > 0$ for $\\rho > 2$; \n\\end{enumerate}\n\n\nFrom this, we obtain a Hamiltonian function $H$ on either $\\mathbb{R} \\times\nY$ or on $W$ by setting $H(r, y) = h(\\operatorname{e}^r)$ on $\\mathbb{R}^+ \\times Y$ and\nextending it by $0$ elsewhere. We will refer to such  Hamiltonians as\n\\defin{admissible}. See Figure \\ref{F:Jshaped}.\n\n\\end{definition}\n\n\n\nSince $\\omega = d(\\operatorname{e}^r \\alpha)$ on $\\mathbb{R}_+\\times Y$, the Hamiltonian vector field associated to $H$ is $h'(\\operatorname{e}^r)R$, where $R$ is the Reeb vector\nfield associated to $\\alpha$. Recall that $(Y,\\alpha)$ is a prequantization bundle over $(\\Sigma,\\omega_\\Sigma)$, and that the corresponding periodic Reeb orbits correspond to covers of the $S^1$-fibres of $Y\\to \\Sigma$. The periods of these orbits are positive integer multiples of $\\frac{1}{K}$, giving the multiplicities of the covers.  The 1-periodic orbits of $H$ are thus of two types:\n\\begin{enumerate}\n \\item constant orbits: one for each point in \n \\[\n     W_0 \\coloneqq \\{w\\in W \\,|\\, (dH)_w=0\\} = \\overline{W \\setminus \\supp (dH)};\n \\]\n \\item non-constant orbits: for each $k\\in \\mathbb{Z}_+$, there is a $Y$-family of 1-periodic $X_H$-orbits, contained in the level set $Y_k\n \\coloneqq \\{b_k\\}\\times Y$, for the unique $b_k>\\log 2$ such that $h'(\\operatorname{e}^{b_k}) = kK$. Each point in $Y_k$ is the starting point of one such orbit.\n\\end{enumerate}\n\n\\begin{remark}\nIt is important to observe that these Hamiltonians are Morse--Bott\nnon-degenerate \\textit{except} at the boundary $\\partial W_0 = \\partial \\supp (dH)$. In Section \\ref{S:compactness}, we prove that the moduli spaces of curves we consider will not interact with these degenerate, constant orbits.\n\nRecall that a family of periodic Hamiltonian orbits for a time-dependent Hamiltonian vector field\nis said to be Morse--Bott non-degenerate if the parametrized 1-periodic orbits form a manifold, \nand the tangent space of the family of orbits at a point is given by the eigenspace of $1$ for the corresponding Poincar\\'e return map.\n(Morse non-degeneracy requires the return map not to have 1 as an eigenvalue\nand hence these periodic orbits must be isolated.)\n\\end{remark}\n\n\nWe also fix some auxiliary data, consisting of Morse functions and\nvector fields.  Fix throughout a Morse function $f_\\Sigma \\colon\n\\Sigma \\to \\mathbb{R}$ and a gradient-like vector field $Z_\\Sigma\\in\\mathfrak X(\\Sigma)$, which means that \n$\\frac{1}{c} |df_\\Sigma|^2 \\le df_\\Sigma(Z_\\Sigma)\\le c |df_\\Sigma|^2$ for some constant $c>0$.\nDenote the time-$t$ flow\nof $Z_\\Sigma$ by $\\varphi^t_{Z_\\Sigma}$. \nGiven $p\\in \\Crit (f_\\Sigma)$, its stable and unstable manifolds (or ascending and descending manifolds, respectively) are\n\\begin{equation}\\label{(un)stable}\nW^s_{\\Sigma}(p) := \\left\\{ q\\in \\Sigma | \\lim_{t\\to \\infty} \\varphi_{Z_\\Sigma}^{-t}(q) = p \\right\\}\n \\text{,} \\,\nW^u_{\\Sigma}(p) := \\left\\{ q\\in \\Sigma | \\lim_{t\\to -\\infty} \\varphi_{Z_\\Sigma}^{-t}(q) = p \\right\\}.\n\\end{equation}\nNotice the sign of time in the flow. \nWe further require that $(f_\\Sigma,Z_\\Sigma)$ be a Morse--Smale pair, i.e.~that all stable and unstable\nmanifolds of $Z_\\Sigma$ intersect transversally.\n\n\nThe contact distribution $\\xi$ defines an Ehresmann connection on the\ncircle bundle $S^1 \\hookrightarrow Y\\stackrel{\\pi_\\Sigma}{\\to} \\Sigma$ (by an abuse of notation, we will sometimes \nalso denote the projection $\\mathbb{R}\\times Y\\to \\Sigma$ by $\\pi_\\Sigma$). Denote the horizontal\nlift of $Z_\\Sigma$ by $\\pi_\\Sigma^*Z_\\Sigma\\in \\mathfrak X(Y)$.  We fix\na Morse function $f_Y \\colon Y \\to \\mathbb{R}$ %\nand a gradient-like vector field $Z_Y\\in \\mathfrak\nX(Y)$ such that $(f_Y,Z_Y)$ is a Morse--Smale pair and the vector field\n$Z_Y-\\pi_\\Sigma^*Z_\\Sigma$ is vertical (tangent to the $S^1$-fibres).\nUnder these assumptions, flow lines of $Z_Y$ project under $\\pi_\\Sigma$\nto flow lines of $Z_\\Sigma$.\n\nObserve that critical points of $f_Y$ must lie in the fibres above the critical points \nof $f_\\Sigma$ (and these are zeros of $Z_Y$ and $Z_\\Sigma$ respectively). \nFor notational simplicity, we suppose that $f_Y$ has two critical points in each\nfibre. \nIn the following, given a critical point for $f_\\Sigma$, $p \\in \\Sigma$, we denote the two critical points in the fibre above $p$ by $\\widehat p$ and $\\widecheck p$, the fibrewise maximum and fibrewise minimum of $f_Y$, respectively.\n\nWe will denote by $M(p)$ the Morse index of a critical point $p \\in \\Sigma$ of $f_\\Sigma$, and by $\\tilde M (\\tilde p) = M(p) + i(\\tilde p)$ the Morse index of the critical point $\\tilde p = \\widehat p$ or $\\tilde p = \\widecheck p$ of $f_Y$. The fibrewise index has \n$i(\\widehat p) = 1$ and $i(\\widecheck p) = 0$.\n\n\nSince $H$ is admissible, we can identify $W \\setminus \\overline W$ with\n$(\\log 2,\\infty) \\times Y$. Fix also a Morse function $f_W$ and a gradient-like vector field $Z_W$ on $W$, such\nthat $(f_W,Z_W)$ is a Morse--Smale pair and $Z_W$ restricted to $[-\\epsilon\/4,\\infty)\\times Y$ is the constant\nvector field $\\partial_r$, where $r$ is the coordinate function on the\nfirst factor and $\\epsilon>0$ is as in Section \\ref{sec:almost complex structures}. \nWe denote also by $(f_W,Z_W)$ the Morse--Smale pair that is induced on $X\\setminus\\Sigma$ by the diffeomorphism \nin Lemma \\ref{planes = spheres}.\nDenote by $M(x)$ the Morse index of $x\\in \\Crit(f_W)$ with respect to $f_W$.\n\n\nWe now define the Morse--Bott symplectic chain complex of $W$ and\n$H$. Recall that for every $k>0$, each point in $Y_k \\coloneqq \\{ b_k \\} \\times Y \\subset \\mathbb{R}^+ \\times Y$ is the starting point of a \n1-periodic orbit of $X_H$, which covers $k$ times the simple Reeb orbit through the corresponding point in $Y$. \nFor each critical\npoint $\\tilde p = \\widehat p$ or $\\tilde p = \\widecheck p$ of $f_Y$,\nthere is a generator corresponding to the pair $(k, \\tilde p)$. We will\ndenote this generator by $\\tilde p_k$.  The complex is then given by:\n\\begin{equation} \\label{eqn:chain complex}\nSC_*(W,H) = \\left(\\bigoplus_{k>0} \\,\n        \\bigoplus_{p\\in \\text{Crit}(f_\\Sigma)}\\mathbb{Z}\n            \\langle \\widecheck p_k, \\widehat p_k \\rangle\\right) \n        \\oplus \\left(\\bigoplus_{x\\in \\text{Crit}(f_W)}\\mathbb{Z}\\langle x \\rangle\\right) \n\\end{equation}\n\nRecall that $\\lambda$ is a Liouville form on $W$. \n\n\\begin{definition} \\label{def:symplectic action}\nThe \\defin{Hamiltonian action} of a loop $\\gamma \\colon S^1 \\to W$ is \n$$\\mathcal{A}( \\gamma ) = \\int \\gamma^*( \\lambda - H dt).$$ \n\\end{definition}\n\nIn particular, for any constant orbit $\\gamma \\in \\overline W$, $\\mathcal{A}(\\gamma ) = 0$ and for any orbit $\\gamma_k \\in Y_k$, we have \n\\begin{equation} \\label{E:HamiltonianAction}\n\\mathcal{A}( \\gamma_k) = \\operatorname{e}^{b_k} h'(\\operatorname{e}^{b_k}) - h( \\operatorname{e}^{b_k})>0,\n\\end{equation}\nwhere $b_k$ is as above. %\nThe action of $\\gamma_k$ is the negative of the $y$-intersept of the tangent line to the graph of $h$ at $\\operatorname{e}^{b_k}$. See Figure \\ref{F:Jshaped}. Note that the convexity of $h$ implies that $\\mathcal{A}(\\gamma_k)$ is monotone increasing in $k$.\n\nWe will now define the gradings of the generators. For more details on these formulas, see \\cite{DiogoLisiSplit}*{Section 3.1}. \nFor a critical point $\\widetilde p$ of $f_Y$, \nand a multiplicity $k$, we define\n\\begin{equation} \\label{eqn:grading Y}\n| \\widetilde p_k | = \\tilde M( \\widetilde p) + 1-n + 2\\frac{\\tau_X - K}{K} k \\in \\mathbb{R}, \n\\end{equation}\nwhere we recall that $\\tau_X$ is the monotonicity constant of $X$ and $c_1(N\\Sigma) = [K \\omega_\\Sigma]$.\n\n\n\nFor a critical point $x$ of $f_W$, we define\n\\begin{equation} \\label{eqn:grading W}\n|x| = n - M(x).\n\\end{equation}\n\n\n\\section{Floer moduli spaces before and after splitting}\n\nIn this section, we discuss the differential in the Morse--Bott symplectic homology of $W$, before and after stretching the neck. \n\n\\subsection{Morse--Bott symplectic homology}\n\nConsider $H \\colon W \\to \\mathbb{R}$ as defined in Section \\ref{S:chain complex}, together with the auxiliary data of $(f_\\Sigma, Z_\\Sigma)$, $(f_Y, Z_Y)$ and $(f_W, Z_W)$.\nFix an almost complex structure $J_\\kappa$ as in \\eqref{J_kappa} for a large $\\kappa>0$.\n\n\\begin{definition} \\label{D:Floer cylinder}\nA map $\\tilde v \\colon \\mathbb{R} \\times S^1 \\to W$ is a \\defin{Floer cylinder} if\n\\begin{equation}\n\\partial_s \\tilde v + J_\\kappa(\\partial_t \\tilde v - X_H(\\tilde v)) = 0.\n\\label{Floer eq W}\n\\end{equation}\n\nLet $\\mathcal M(x_-, x_+)$ denote the space of all such Floer cylinders such that, for fixed 1-periodic orbits $x_\\pm$ of $X_H$, $\\lim_{s \\to \\pm \\infty} \\tilde v(s,.) = x_{\\pm}$. \n\nLet $S_-$ and $S_+$ denote connected spaces of 1-periodic $X_H$-orbits (either $\\overline W$ or $Y_k$). Then we define\n\\[\n\\mathcal M(S_-, S_+) = \\bigcup_{x_- \\in S_-, x_+ \\in S_+} \\mathcal M(x_-, x_+).\n\\]\n\nGiven a Floer cylinder $\\tilde v$, we denote its asymptotic limits by $\\tilde\nv(+\\infty) = x_+$ and $\\tilde v(-\\infty) = x_-$.\n\\end{definition}\n\nNote that $\\mathcal{A}(\\tilde v(s_0,.)) \\leq \\mathcal{A}(\\tilde v(s_1,.))$ if $s_0 \\leq s_1$. \n\n\\begin{definition} \\label{D:presplit energy}\nThe {\\em energy} of a Floer cylinder $\\tilde v \\colon \\mathbb{R} \\times S^1 \\to W$ is given by:\n\\[\nE(\\tilde v) = \\int_{\\mathbb{R} \\times S^1} \\tilde v^*d\\lambda - \\tilde v^*dH \\wedge dt \n    = \\int_{\\mathbb{R}\\times S^1} || \\tilde v_s ||^2_{J_\\kappa} \\,ds \\wedge dt,\n\\]\nwhere $|| \\tilde v_s ||^2_{J_\\kappa} = d\\lambda(\\tilde v_s, J_\\kappa \\tilde v_s)$.\n\\end{definition}\n\n\nSince the symplectic form is exact, a Floer cylinder $\\tilde v$ with $\\tilde v(\\pm\\infty) = x_\\pm$ has\n\\[\nE(\\tilde v) = \\mathcal{A}(x_+) - \\mathcal{A}(x_-) =  \\int x_+^*( \\lambda - h(\\operatorname{e}^r) dt) - \\int x_-^*(\\lambda - h(\\operatorname{e}^r) dt).\n\\]\nA Floer cylinder is non-trivial if $E(\\tilde v) > 0$, or equivalently, if\n$\\tilde v$ is not of the form $(s, t) \\mapsto \\gamma(t)$ for some 1-periodic $X_H$-orbit $\\gamma$. \n\nLet us recall some convergence properties of finite energy Floer trajectories. \nThis is a combination of statements of several theorems from the literature,\nwith slightly different hypotheses.\nIn the Morse--Bott case, we refer to\n\\citelist{\\cite{BOSymplecticHomology} \\cite{BourgeoisThesis} \\cite{HWZ4}}.\nIn the non-degenerate case, the relevant ideas have appeared in \n\\citelist{\\cite{RobbinSalamonAsymptotics} \\cite{HWZ1}}, and of course the\noriginal work was done by Floer \\cite{Floer88}.\n\\begin{lemma} \\label{L:convergence to orbit}\nSuppose that $\\tilde v\\colon \\mathbb{R} \\times S^1 \\to W$ is a finite energy Floer cylinder contained in a compact subset of $W$.\nThen, for any sequence $s_k \\to +\\infty$ [resp.,  $s_k \\to -\\infty$] there is a subsequence we also denote $(s_k)_{k=1}^\\infty$ and a 1-periodic orbit $\\gamma(t)$ of the Hamiltonian vector field so that $\\tilde v(s_k, t) \\to \\gamma(t)$ in $C^\\infty$.\n\\begin{enumerate}\n\\item If $\\gamma$ is a Morse--Bott non-degenerate orbit, then the limit $\\gamma$ does not depend on the initial choice of sequence, and furthermore, we have the much stronger result that $\\tilde v(s,t)$ converges to $\\gamma$ exponentially fast in $s$;\n\\item If $\\gamma$ is a non-isolated degenerate orbit, the limit may depend on the initial sequence $(s_k)_{k=1}^\\infty$. Any two limits are connected by a family of periodic orbits of the same action.\n\t\\label{2}\n\\end{enumerate}\n\\end{lemma}\n\n\nFurthermore, if the curve converges exponentially fast, \nat a negative [resp., positive] puncture the rate of \nconvergence is governed by the smallest positive [largest negative] eigenvalue of the\ncorresponding asymptotic operator \\cite{SiefringAsymptotics}.\n\n\n\\begin{remark} \\label{R:not MB not converge}\nThere are several constructions of examples where the limit is not\nunique in the degenerate case. In particular, Siefring has carefully\nconstructed such examples in the case of pseudoholomorphic curves in\nsymplectizations \\cite{SiefringMultipleLimits}.  Similarly, a gradient\ntrajectory of a smooth function that is not Morse--Bott can fail to\nconverge to a single critical point, and may contain sequences converging\nto different critical points.\n\nEven if the limit were unique, these examples illustrate also that convergence\nwill not typically be at an exponential rate, and thus will not provide a good\nFredholm problem.\n\\end{remark}\n\n\n\\begin{remark}\n\nIn the case of a Morse--Bott limit, a cylinder contained in a compact subset of \n$W$ has finite energy if and only if it converges to 1-periodic\nHamiltonian orbits at its punctures, see\n\\citelist{\n    \\cite{SalamonFloerHomology}*{Proposition 1.21}\n    \\cite{SFTcompactness}*{Proposition 5.13}\n}.\n\n Yoel Groman pointed out to us that the assumption that $\\tilde v$\n has finite energy already implies that its image is contained in\n a compact subset of $W$, so the assumptions of Lemmas \\ref{L:convergence\n to orbit} and \\ref{no + limits in W} can be simplified.  This is\n because the action functional for our Hamiltonians satisfies the\n Palais--Smale condition \\cite{GromanOpen}.\n\n\\end{remark}\n\n\nThe next result shows that there are no non-trivial Floer cylinders\nwhose positive asymptote $x_+$ is in $W_0$.  Note that if\nsuch a cylinder has asymptotic limits at $+\\infty$ and $-\\infty$,\nthen it is contained in a compact subset of $W$, by the maximum\nprinciple.\n\n\n\\begin{lemma} \\label{no + limits in W}\nLet $\\tilde v\\colon \\mathbb{R} \\times S^1 \\to W$ be a solution of \\eqref{Floer eq W}\nthat has finite energy and is contained in a compact subset of $W$. If there is\na sequence $s_k^+\\to \\infty$ such that the $C^0$-limit $\\lim_{k\\to \\infty}\\tilde\nv(s_k^+,.) = x_+ \\in W_0$, then $\\tilde v$ is constant.  \n\\end{lemma}\n\\begin{proof}\nTake any sequence $s_k^- \\to -\\infty$. Lemma \\ref{L:convergence to orbit} \nimplies that there are subsequences (denoted also by $s_k^\\pm$) such that\n$\\tilde v(s_k^\\pm,.)$ converge in $C^1$ (actually $C^\\infty$) to 1-periodic\n$X_H$-orbits $x_\\pm$, with $x_+\\in W_0$. Then,\n\\begin{align*}\n0 \\leq E(\\tilde v) &= \\lim_{k\\to \\infty} E(\\tilde v|_{[s_k^-,s_k^+]\\times S^1}) = \\lim_{k\\to \\infty} \\big(\\mathcal{A}(\\tilde v(s_k^+,.)) - \\mathcal{A}(\\tilde v(s_k^-,.)) \\big) = \\\\\n&= \\mathcal{A}(x_+) - \\mathcal{A}(x_-) .\n\\end{align*}\nSince $\\mathcal{A}(x_+) = 0$ and $\\mathcal{A}(\\gamma)\\geq 0$ for every 1-periodic $X_H$-orbit, we conclude that $E(\\tilde v) = 0$. This implies the result. \n\\end{proof}\n\n\n\n\nThe following is the main definition of this section. \n\n\\begin{definition} \\label{def:FloerCylinderWithCascades}\nFix $N \\ge 0$. Let $S_0, S_1, \\dots, S_{N}$ be a collection of connected spaces\nof orbits, which can either be one of the $Y_k$ or $W_0$. \nLet $(f_i, Z_i)$, $i=0, \\dots, N+1$ be the pair of Morse function and \ngradient-like vector field of $f_i = f_Y$, $Z_i = Z_Y$ if $S_i = Y_k$ for some\n$k$, and $f_i = -f_W$, $Z_i=-Z_W$ if $S_i = W_0$.\n\nLet $x$ be a critical point of $f_0$ and $y$ a critical point of $f_N$ (so $x$ and $y$ are generators of the chain complex (\\ref{eqn:chain complex})). \n\nA {\\em Floer cylinder with $0$ cascades} ($N = 0$), with positive end at $y$ and negative end at $x$, consists of a \npositive gradient trajectory $\\nu \\colon \\mathbb{R} \\to S_0$, such that $\\nu(-\\infty) = x$, $\\nu(+\\infty) = y$ and $\\dot \\nu =  Z_0(\\nu)$.\n\nA {\\em Floer cylinder with $N$ cascades}, $N \\ge 1$, with positive end at $y$ and negative end at $x$, consists of the following data:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item $N-1$ length parameters $l_i > 0, i=1, \\dots, N-1$;\n\\item Two half-infinite gradient trajectories, $\\nu_0 \\colon (-\\infty, 0] \\to S_0$ and $\\nu_{N} \\colon [0, +\\infty) \\to S_{N}$\nwith $\\nu_0(-\\infty) = x$, $\\nu_N(+\\infty) = y$ and $\\dot \\nu_i = Z_i(\\nu_i)$ for $i=0$ or $N$;\n\\item $N-1$ gradient trajectories $\\nu_i$ defined on intervals of length $l_i$, $\\nu_i \\colon [0, l_i] \\to S_i$ for $i=1, \\dots, N-1$ \nsuch that $\\dot \\nu_i = Z_i(\\nu_i)$;\n\\item $N$ non-trivial Floer cylinders $\\tilde v_i \\colon \\mathbb{R} \\times S^1 \\to W$, \n    $i = 1, \\dots, N$, satisfying equation (\\ref{Floer eq W})\n\twhere (defining $l_0 = 0$)\n\t\\[\n            \\tilde v_i(+\\infty, \\cdot) = \\nu_i(0) \\in S_i, \n                \\quad \\tilde v_i(-\\infty, \\cdot) = \\nu_{i-1}(l_{i-1}) \\in S_{i-1}.\n\t\\]\n\\end{enumerate}\t\n\\label{Floer cascade}\nIn the case of a Floer cylinder with $N \\ge 1$ cascades, we refer to the \nnon-trivial Floer cylinders $\\tilde v_i$ as {\\em sublevels}. \nSee Figure \\ref{fig:Floer cylinder with cascades} for a schematic illustration.\n\\end{definition}\n\n\n\\begin{figure}\n  \\begin{center}\n    \\def0.8\\textwidth{0.5\\textwidth}\n    %\n\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  }\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}\n    \\renewcommand\\transparent[1]{}%\n  }\n  \\providecommand\\rotatebox[2]{#2}\n  \\ifx0.8\\textwidth\\undefined\n    \\setlength{\\unitlength}{323.78738945pt}\n  \\else\n    \\setlength{\\unitlength}{0.8\\textwidth}\n  \\fi\n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,0.79842498)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{cascade_presplit.pdf}}%\n    \\put(-0.00164074,0.01853505){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$x$}}}%\n    \\put(0.97698142,0.72810202){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$y$}}}%\n    \\put(0.23839863,0.13196028){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v_1$}}}%\n    \\put(0.47843802,0.36408628){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v_2$}}}%\n    \\put(0.75013097,0.60940121){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v_3$}}}%\n    \\put(0.09068209,0.06865309){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\nu_0$}}}%\n    \\put(0.35446167,0.29814131){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\nu_1$}}}%\n    \\put(0.61296557,0.55400752){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\nu_2$}}}%\n    \\put(0.87938298,0.76503116){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\nu_3$}}}%\n  \\end{picture}%\n\\endgroup\n   \\end{center}\n  \\caption{A Floer cylinder with 3 cascades, as in Definition \\protect{\\ref{def:FloerCylinderWithCascades}}, with positive end at $p$ and negative end at $q$.} \n   \\label{fig:Floer cylinder with cascades}\n\\end{figure}\n\n\nThe energy of a Floer cylinder with $N$ cascades is the sum of the energies of each of its $N$ sublevels. \n\n\\begin{lemma} \nFloer cylinders with cascades do not contain sublevels $\\tilde v_i$ such \nthat $\\lim_{s\\to+\\infty} \\tilde v_i(s,.) \\in W_0$. If $\\lim_{s\\to-\\infty} \\tilde\nv_i(s,.) = x_-\\in W_0$, then $i=1$ and $x_- \\notin \\partial W_0$. \n\\label{non-MB}\n\\end{lemma}\n\\begin{proof}\nThe case $s\\to +\\infty$ follows from Lemma \\ref{no + limits in W}.  \nFor the case $s\\to -\\infty$, observe that $\\nu_{i-1}$ is a negative flow line of \n$Z_W$, which agrees with $\\partial_r$ on $[-\\epsilon\/4,\\infty) \\times Y$ (in\nparticular on $\\partial W_0$). \nIf $i>0$, then the sublevel $\\tilde v_{i-1}$ must be such that\n$\\lim_{s\\to+\\infty} \\tilde v_{i-1}(s,.) \\in W_0$, which as we saw is impossible. \nIf $i=1$, then $\\nu_0(-\\infty) \\in \\Crit(f_W)$, and all critical points of $f_W$\nare ``below $\\partial W_0$''. Therefore, $x_-\\notin \\partial W_0$. \n\\end{proof}\n\n\n\\begin{remark} \\label{R:not MB}\nThe points in $\\partial W_0 = \\partial \\supp dH$, which were ruled out as asymptotic orbits of sublevels \nby the previous Lemma, correspond to constant orbits of $X_H$ that are not of Morse--Bott type. \nConvergence to orbits that are not Morse--Bott non-degenerate would introduce\nsevere analytical difficulties\n(as discussed in Remark \\ref{R:not MB not converge}).\n\\end{remark}\n\n\n\\begin{lemma}\nLet $S_0, S_1, \\dots, S_N$ be the spaces of orbits of a Floer cylinder with $N$\ncascades. Let $k(S_i) = k$ if $S_i = Y_k$ and let $k(S_i) = 0$ if $S_i = W_0$. \nThen, $k(S_i)$ is monotone increasing in $i$.  \n\\label{monotone multiplicities}\n\\end{lemma}\n\\begin{proof}\nThis follows immediately from the fact that $0 < E(\\tilde v_i) = \\mathcal{A}( \\tilde v_i(+\\infty)) - \\mathcal{A}( \\tilde v_i(-\\infty))$ and from the monotonicity of $\\mathcal{A}$ in the multiplicity $k$. \n\\end{proof}\n\n\n\\begin{remark}\n\tThese lemmas have a number of important consequences. In particular, \na Floer cylinder with cascades between two critical points of $f_W$ \nmust have 0 cascades, which is to say that it consists of a flow line of $-Z_W$. \n\nFurthermore, a Floer cylinder with cascades whose positive end is on a $Y_k$, $k\n\\ge 1$, and whose negative end is on $W_0$ must have the first sublevel\nconnecting an orbit in a $Y_l$ to a (constant) orbit in the interior of $W_0$. \nThis is the unique sub-level in the cascade that converges to a point in $W_0$.\n\\label{0 cascades W}\n\\end{remark}\n\nWe are now able to define a differential on the chain complex (\\ref{eqn:chain complex}).\nGiven generators $x, y$, denote by\n$$\n\\mathcal{M}_{H,N}(x,y;J_\\kappa)\n$$\nthe space of Floer cylinders with $N$ cascades from $x$ to\n$y$ (i.e. with negative end at $x$ and positive end at $y$).\n\n\nWe then define\n\\begin{equation} \\label{eqn:differential}\n    \\partial_{\\text{pre}} \\, y = \\sum_{|x|=|y| - 1} \\# \\left( \\mathcal{M}_{H, 0}(x,y; J_\\kappa) \/ \\mathbb{R} \\right ) x\n            + \\sum_{|x| = |y| - 1} \\sum_{N = 1}^\\infty \\# \\left ( \\mathcal{M}_{H, N}(x,y; J_\\kappa) \/ \\mathbb{R}^N\n            \\right ) x.\n\\end{equation}\n\nObserve that if $N = 0$, the Floer cylinder with $0$ cascades is a flow line, and the $\\mathbb{R}$ action is given by its standard reparametrization. If, instead, $N \\ge 1$, the $\\mathbb{R}^{N}$ action is given by domain\ntranslation on the $N$ sublevels.\n\nWe call $\\partial_{\\text{pre}}$ the {\\em presplit} Floer differential, to distinguish it from the {\\em split} differential that will be discussed next. \n\n\\begin{remark}\nTo give a complete definition of $\\partial_{\\text{pre}}$, we should\ndiscuss the orientations of the moduli spaces involved and the associated\nsigns. Instead, we will relate the presplit Floer differential with the\nsplit Floer differential, and refer to the discussion of orientations in\n\\cite{DiogoLisiSplit}*{Section 7}. Orientations for the split differential\nwill be further studied in Section \\ref{compute signs} below.\n\\end{remark}\n\n\n\\subsection{Morse--Bott split symplectic homology}\n\n\\label{cascades after splitting}\n\nRecall the definition of the almost complex structures $J_\\kappa$, $J_Y$\nand $J_W$ in Section \\ref{sec:almost complex structures}. Given a sequence\n$\\tilde v_{\\kappa_n}$ of finite energy $J_{\\kappa_n}$-Floer cylinders in\n$W$, with $\\kappa_n\\to \\infty$, SFT compactness \\cite{SFTcompactness}\nimplies that there is a subsequence converging to an SFT building.\nObserve that the Hamiltonian $H$ is supported in $(\\log 2, \\infty)\\times\nY$, which is above the hypersurface $\\{0\\}\\times Y$ along which we stretch\nthe neck. This makes it possible to apply the usual SFT compactness\nargument.  A gluing argument gives that a transverse SFT-type building\ncan be glued to obtain a ``presplit'' Floer cylinder, as in the previous\nsection.\n\n\nThis Hamiltonian induces a function on $\\mathbb{R}\\times Y$, vanishing on\n$\\mathbb{R}_-\\times Y$, which we also denote by $H$.  The non-constant orbits\nagain form manifolds $Y_k$, where $k\\geq 1$. The critical points of the\nMorse function $f_W \\colon W\\to \\mathbb{R}$ are below the neck-stretching region.\n\nThis then gives a different description of the differential on the\nchain complex (\\ref{eqn:chain complex}) (defined above in terms of Floer\ncylinders with cascades) by counting {\\em split} Floer cylinders with\ncascades. In principle, these could become quite complicated. Thanks\nto the monotonicity assumptions we impose on $X$ and on $\\Sigma$,\nthe cascades that have the correct Fredholm index to appear in the\ndifferential turn out to be relatively simple. This is analyzed in detail\nin \\cite{DiogoLisiSplit}, see in particular \n\\cite{DiogoLisiSplit}*{Section 6.1}.  \nWe now provide a brief summary of the cascades that\ncontribute to the differential in split symplectic homology.\n\nFirst, we describe the component pieces. The upper level of a cascade in the\ndifferential will consist of a (possibly punctured) Floer cylinder\n$\\tilde v \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma \\to \\mathbb{R} \\times Y$,\nwhere $\\Gamma$ is either empty or contains at most one point $P \\in \\mathbb{R} \\times\nS^1$. We refer to such a $P$ as an \\textit{augmentation puncture}. \nThis solves the Floer equation\n\\begin{equation} \\label{Floer eq Y}\n\\partial_s \\tilde v + J_Y (\\partial_t \\tilde v - X_H(\\tilde v)) = 0,\n\\end{equation}\nand has finite \\textit{hybrid energy}, given\nby\n\\begin{equation} \\label{E:hybridEnergy}\n    E( \\tilde v ) = \\sup \\left \\{ \n\\int_{\\mathbb{R} \\times S^1} \\tilde v {}^{*} \\left ( d(  \\eta \\alpha ) - dH \\wedge dt\n\\right) \\, | \\, \\eta \\colon \\mathbb{R} \\to [0, \\infty), \\eta' \\ge 0, \\eta(r) = \\operatorname{e}^r\n\\text{ for } r \\ge 0\n\\right \\}.\n\\end{equation}\nSuch a cylinder converges to a non-constant closed Hamiltonian orbit at $+\\infty$, to a\nReeb orbit in $\\{ -\\infty \\} \\times Y$ at the puncture $P$ and, at $-\\infty$,\nconverges either to a non-constant closed Hamiltonian orbit or to a Reeb orbit \nin $\\{ -\\infty \\} \\times Y$. \nA cascade may also have a lower level consisting of a $J_W$-holomorphic plane\nconverging to a Reeb orbit in $\\{+\\infty \\} \\times Y$ at its positive puncture.\n\nNotice that by the Reeb invariance of both the Hamiltonian $H$ and of the almost\ncomplex structure $J_Y$, such a (punctured) Floer cylinder projects to a\n$J_\\Sigma$ holomorphic sphere (with removable singularities at the punctures\n$\\pm \\infty$ and $P$). \n\nThe following result is a restatement of \\cite{DiogoLisiSplit}*{Propositions 6.2 and 6.3}. \nIt states \nthat under our monotonicity assumptions, the cascades contributing to the split Floer differential can be of only four simple types.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\def0.8\\textwidth{0.3\\textwidth}\n    %\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  } \n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded} \n    \\renewcommand\\transparent[1]{}%\n  } \n  \\providecommand\\rotatebox[2]{#2} \n  \\ifx0.8\\textwidth\\undefined \n    \\setlength{\\unitlength}{153.65167524pt} \n  \\else \n    \\setlength{\\unitlength}{0.8\\textwidth} \n  \\fi \n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,0.52879344)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{case_1.pdf}}%\n    \\put(-0.0634575,0.02310476){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\widehat q_{k_-}$}}}%\n    \\put(0.95149337,0.47648333){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\widecheck p_{k_+}$}}}%\n    \\put(0.52131919,0.24979405){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v$}}}%\n    \\put(0.902131919,0.164979405){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\mathbb{R}\\times Y$}}}%\n  \\end{picture}%\n\\endgroup\n\n   \\end{center}\n  \\caption[Case 1]{Case 1 in Proposition \\protect{\\ref{P:split SH cases}}}\n  \\label{case_1_fig}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\def0.8\\textwidth{0.25\\textwidth}\n    %\n\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  }\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}\n    \\renewcommand\\transparent[1]{}%\n  }\n  \\providecommand\\rotatebox[2]{#2}\n  \\ifx0.8\\textwidth\\undefined\n    \\setlength{\\unitlength}{155.47384928pt}\n  \\else\n    \\setlength{\\unitlength}{0.8\\textwidth}\n  \\fi\n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,1.34900323)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{case_2.pdf}}%\n    \\put(-0.08341697,0.77649313){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\widehat p_{k_-}$}}}%\n    \\put(0.95206187,1.29539538){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\widecheck p_{k_+}$}}}%\n    \\put(0.54043248,1.11827007){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v$}}}%\n    \\put(0.6052953,0.09792872){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$U$}}}%\n    \\put(0.95206187,0.77649313){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\mathbb{R}\\times Y$}}}%\n    \\put(1.04206187,0.097649313){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$W$}}}%\n  \\end{picture}%\n\\endgroup\n\n   \\end{center}\n  \\caption[Case 2]{Case 2 in Proposition \\protect{\\ref{P:split SH cases}}}\n  \\label{case_2_fig}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\def0.8\\textwidth{0.2\\textwidth}\n    %\n\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  }\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}\n    \\renewcommand\\transparent[1]{}%\n  }\n  \\providecommand\\rotatebox[2]{#2}\n  \\ifx0.8\\textwidth\\undefined\n    \\setlength{\\unitlength}{146.34692076pt}\n  \\else\n    \\setlength{\\unitlength}{0.8\\textwidth}\n  \\fi\n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,1.81416906)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{case_3.pdf}}%\n    \\put(-0.00144135,0.41693098){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$x$}}}%\n    \\put(0.94907221,1.75491512){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\widecheck p_{k_+}$}}}%\n    \\put(0.5192222,1.27663133){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v_1$}}}%\n    \\put(0.5192222,0.3684972){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\tilde v_0$}}}%\n    \\put(0.94907221,1.12663133){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\mathbb{R}\\times Y$}}}%\n    \\put(1.04907221,0.3284972){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$W$}}}%\n  \\end{picture}%\n\\endgroup\n\n   \\end{center}\n  \\caption[Case 3]{Case 3 in Proposition \\protect{\\ref{P:split SH cases}}} \n  \\label{case_3_fig}\n\\end{figure}\n\n\\begin{proposition} \\label{P:split SH cases}\nIn what follows, $q,p\\in \\Crit(f_\\Sigma)$ and $x\\in \\Crit(f_W)$. There are the following contributions to the \nsplit symplectic homology differential:\n\\begin{enumerate}\n\\item[(0)] An index $1$ gradient trajectory in $Y_k$, connecting $\\tilde q_k$ to $\\tilde p_k$ or \nan index $1$ gradient trajectory in $W$, connecting two critical points of $f_W$.\n\\item A half-infinite gradient trajectory $\\gamma_- \\colon (-\\infty, 0] \\to\n    Y_{k_-}$, with $\\gamma_-(-\\infty) = \\widehat q_{k_-}$, followed by a Floer\n    cylinder $\\tilde v \\colon \\mathbb{R} \\times S^1 \\to \\mathbb{R} \\times Y$ \n   %\n    followed by a\n    half-infinite gradient trajectory $\\gamma_+ \\colon [0, +\\infty) \\to\n    Y_{k_+}$, with $\\gamma_+(+\\infty) = \\widecheck p_{k_+}$. \n    Writing $A \\in H_2(\\Sigma; \\mathbb{Z})$ for the (spherical) homology class\n    represented by the projection of $\\tilde v$ to $\\Sigma$, these satisfy the\n    conditions\n    \\begin{align*}\n        &k_+ - k_- = K \\omega(A) > 0 \\\\\n        &\\gamma_-(0) = \\lim_{s \\to -\\infty} \\tilde v(s, 0) \\\\\n        &\\lim_{s \\to +\\infty} \\tilde v(s, 0) = \\gamma_+(0).\n    \\end{align*}\n    %\n    %\n    %\n    %\n    %\n        See Figure \\ref{case_1_fig}. \n    \\item A half-infinite gradient trajectory $\\gamma_- \\colon (-\\infty, 0] \\to\n        Y_{k_-}$ with $\\gamma_-(-\\infty) = \\widehat p_{k_-}$, followed by a\n        Floer cylinder $\\tilde v \\colon \\mathbb{R} \\times S^1 \\setminus \\{ P \\} \\to \\mathbb{R}\n        \\times Y$ and one augmentation puncture, with trivial projection to $\\Sigma$, and a\n$J_W$--holomorphic plane $U \\colon \\mathbb{C} \\to W$, followed by a\nhalf-infinite gradient trajectory $\\gamma_+ \\colon [0, +\\infty) \\to Y_{k_+}$\nwith $\\gamma_+(+\\infty) = \\widecheck p_{k_+}$.  \nWriting $B \\in H_2(X; \\mathbb{Z})$ for the homology class represented by the plane $U$,\nthese satisfy\n\\begin{align*}\n    &k_+-k_-= K \\omega(B) > 0 \\\\\n    &\\gamma_-(0) = \\lim_{s \\to -\\infty} \\tilde v(s, 0) \\\\\n    &\\lim_{s \\to +\\infty} \\tilde v(s, 0) = \\gamma_+(0),\n\\end{align*}\nand the Reeb orbit to which $\\tilde v$ converges at $P$ is the same Reeb orbit\nto which $U$ converges at $+\\infty$.\nNote that $\\widecheck p, \\widehat p \\in \\Crit(f_Y)$ both project to the same\ncritical point $p\\in \\Crit(f_\\Sigma)$. \nSee Figure \\ref{case_2_fig}.\n\\item A half-infinite gradient trajectory $\\gamma_- \\colon [0, +\\infty) \\to W$, with\n    $\\gamma_-(+\\infty) = x$, \n    followed by a pseudoholomorphic plane $\\tilde v_0$ in $W$, followed by a\n    Floer cylinder $\\tilde v_1$ in $\\mathbb{R} \\times Y$, and a half-infinite gradient\n    trajectory $\\gamma_+ \\colon [0, +\\infty) \\to Y_{k_+}$ with\n    $\\gamma_+(+\\infty) = \\widecheck p_{k_+}$. The Floer cylinder\n    $\\tilde v_1$ projects to a constant in $\\Sigma$.  \n    These satisfy\n    asymptotic matching conditions as follows: $\\gamma(0) = \\tilde v_0(0)$, the\n    plane $\\tilde v_0$ converges at its positive puncture to the same Reeb orbit\n    as $\\tilde v_1$ converges to at its negative puncture, which has\n    multiplicity $k_+$, and $\\gamma_+(0) = \\lim_{s \\to \\infty} \\tilde v_1(s,\n    0)$.\nSee Figure \\ref{case_3_fig}.\n\\end{enumerate}\n\\end{proposition}\n\nIn Cases 1, 2 and 3, call the (possibly punctured) Floer cylinder in $\\mathbb{R}\\times Y$ the {\\em upper level} of the split Floer cylinder. \nIn Case 2, the augmentation plane $U$ is considered modulo its domain\nautomorphisms. In Case 3, notice that the gradient trajectory is descending from\n$x$ to $\\tilde v_0(0)$. \n\nWe say that a Floer cylinder is simple if \n$\\tilde v \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$ projects to \na somewhere injective curve in $\\Sigma$, or if it projects to a constant.\nUsing the notation from \\cite{DiogoLisiSplit}, \nwe write \n$$\n\\mathcal{M}^*_{H,l,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y)\n$$\nfor the space of simple Floer cylinders $\\tilde v \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$, \nwhere $\\Gamma$ is a set of $l$ augmentation punctures, $\\lim_{s\\to \\pm\\infty} \\tilde v(s,.)$ are Hamiltonian orbits of \nmultiplicity $k_\\pm$ and $\\pi_\\Sigma \\circ \\tilde v$ represents the homology\nclass $A\\in H_2(\\Sigma;\\mathbb{Z})$. \n\nThe spaces of Floer cylinders with cascades in Case 1 are unions of fibre products \n\\begin{equation}\n W^s_{Y}(\\widehat q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\widecheck p).\n\\label{fib prod1}\n\\end{equation}\nThe fibre products are defined with respect to the inclusion maps\n\\begin{align*}\nW^s_{Y}(\\widehat q), W_{Y}^u(\\widecheck p) &\\to Y\n\\end{align*}\nand the evaluation maps\n\\begin{align*}\n\\widetilde{\\operatorname{ev}}_Y \\colon \\mathcal{M}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) &\\to Y\\times Y. %\n\\end{align*}\n\nTo give a similar description of Cases 2 and 3, we denote by\n$$\n\\mathcal{M}_H^*(B;J_W)\n$$\nthe space of simple $J_W$-holomorphic cylinders in $W$ with a removable singularity at $-\\infty$, realizing the class \n$B\\in H_2(X;\\mathbb{Z})$. Denote the quotient of $\\mathcal{M}_H^*(B;J_W)$ by the $\\mathbb{C}^*$-action as $\\mathcal{M}_X^*(B;J_W)$. We also define \n$\\mathcal{M}^*_{H,k_+}(0;J_Y)$ as the space of Floer cylinders $\\tilde v \\colon \\mathbb{R}\\times S^1 \\to \\mathbb{R} \\times Y$ such that \n$\\lim_{s\\to\\infty}\\tilde v(s,.)$ is a Hamiltonian orbit of multiplicity $k_+$, $\\lim_{s\\to-\\infty}\\tilde v(s,.)$ \nis the corresponding Reeb orbit and $\\pi_\\Sigma \\circ \\tilde v$ is constant in $\\Sigma$. \n \nThe spaces of Floer cylinders with cascades in Case 2 can now be written as unions of fibre products \n\\begin{equation}\n W^s_{Y}(\\wh p) \\times_{\\operatorname{ev}} \\big(\\mathcal{M}_X^*(B;J_W) \\times_{\\tilde \\operatorname{ev}} \\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\big) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p).\n\\label{fib prod2}\n\\end{equation}\nand in Case 3 we have unions of \n\\begin{equation}\n W^u_{W}(x) \\times_{\\operatorname{ev}^1_-} \\left(\\mathcal{M}^*_{H}(B;J_W)  \\prescript{}{\\operatorname{ev}^1_+\\,}\\times\\prescript{}{\\operatorname{ev}^2_-}{}  \\mathcal{M}^*_{H,k_+}(0;J_Y) \\right) \\times_{\\operatorname{ev}^2_+} W_{Y}^u(\\wc p).\n\\label{fib prod3}\n\\end{equation}\n\nThe relevant transversality results for the spaces \\eqref{fib prod1}, \\eqref{fib prod2} and \\eqref{fib prod3} are established in \n\\cite{DiogoLisiSplit}*{Section 5}. These spaces are oriented using the fibre product orientation convention.\nFor details, see \\cite{DiogoLisiSplit}*{Section 7} and Section \\ref{compute signs} below.\n\n\\ \n\nGiven our assumptions on $H$ and $J_Y$, there are geometric actions on the spaces of solutions to Floer's \nequation in $\\mathbb{R}\\times Y$. \nLet $\\tilde v = (b,v) \\colon \\mathbb{R}\\times S^1 \\setminus \\Gamma \\to \\mathbb{R} \\times Y$ be a solution to Floer's equation (\\ref{Floer eq Y}). \nThere is an action of $S^1 \\times S^1$ on the space of such solutions, with each circle factor acting by rotation on the domain and on the target, respectively:\n$$\n (\\theta_1,\\theta_2) . \\tilde v = \\tilde v_{(\\theta_1,\\theta_2)}\n$$\nwhere\n$$\n\\tilde v_{(\\theta_1,\\theta_2)} (s,t) = (b(s,t+\\theta_1), \\phi^{\\theta_2}_R \\circ v(s,t+\\theta_1)).\n$$\nHere, $\\phi^\\theta_R$ denotes the Reeb flow for time $\\theta$.\nIf we identify the images of the simple Reeb orbits underlying $x_\\pm(t) = \\lim_{s\\to \\pm\\infty} v(s,t)$ with $S^1$, then the following map keeps track of the asymptotic effect of the action: \n\\begin{align*}\n \\rot_{\\tilde v}\\colon S^1 \\times S^1 &\\to S^1 \\times S^1 \\\\\n (\\theta_1,\\theta_2) &\\mapsto \\left(\\lim_{s\\to -\\infty} v(s,\\theta_1) + \\theta_2, \\lim_{s\\to +\\infty} v(s,\\theta_1) + \\theta_2 \\right).\n\\end{align*}\nIf $k_\\pm$ are the multiplicities of the periodic orbits $x_\\pm (t)$, then, $\\rot_{\\tilde v}$ is the linear map represented in matrix form as \n\\begin{equation}\\label{rot v}\n\\rot_{\\tilde v} = \\begin{pmatrix}\nk_- & 1 \\\\\nk_+ & 1\n\\end{pmatrix}.\n\\end{equation}\n \n\nSuppose now that $\\tilde v_0 \\colon \\mathbb{R} \\times S^1 \\to W$ is a non-constant\n$J_W$-holomorphic cylinder with removable singularity at $-\\infty$ (as\nin Case 3 in Proposition \\ref{P:split SH cases}). There is an $S^1$-action on such\ncurves, by rotation on the domain: $$ \\theta . \\tilde v_0 (s,t) = \\tilde v_0^\\theta (s,t) \n= \\tilde v_0(s,t+\\theta).  $$ Identifying the image of the simple Reeb\norbit underlying $\\gamma(t) = \\lim_{s\\to \\infty} \\pi_Y(\\tilde v_0(s,t))$ with\n$S^1$, we have \n\\begin{align} \\label{rot v0}\n \\rot_{\\tilde v_0}\\colon S^1 &\\to S^1 \\\\ \n \\theta &\\mapsto \\lim_{s\\to +\\infty}\n \\tilde v_0(s,\\theta). \\nonumber\n\\end{align} \nIf the multiplicity of the periodic Reeb orbit $\\gamma(t)$\nis $k$, then $\\rot_{\\tilde v_0}$ is multiplication by $k$.\n\n\n\\section{Sketch of isomorphism between the two Floer complexes}\\label{S:compactness}\n\n\\begin{proposition}\nFor a generic choice of almost complex structures of the cylindrical type $J_\\kappa$, $J_W$ and $J_Y$ described in Section \\ref{sec:almost complex structures}, and for $R>0$ large enough,\nthe presplit and split chain complexes are well-defined and are chain isomorphic.\nFurthermore, these complexes compute the symplectic homology of $W$.\n\\end{proposition}\n\nWe will provide a sketch of the proof of this proposition. \nThe main steps of such a proof are as follows:\n\\begin{itemize}\n\t\\item Show that these complexes are well-defined. For this, we need \n\t\tto obtain transversality for generic almost complex structures\n\t\tin the (very restrictive) class we consider, which are both $\\mathbb{R}$ and\n\t\t$S^1$ (Reeb) invariant in the cylindrical end. This problem is\n\t\tsimilar in both the presplit and split complexes, and is addressed\n                in detail in \\cite{DiogoLisiSplit}*{Section 5}  \n\t\tin the more difficult setting of the split complex. \n\t\t(The additional difficulty in the split complex comes from \n\t\tFloer cylinders with cascades that have punctures capped\n\t\twith planes in $W$. The planes can be made transverse\n\t\tby a perturbation of $J$ supported in $\\mathcal{V} \\subset \\overline W$. \n                Such a perturbation\n\t\tdoes not make the upper level(s) transverse. \n\t\tIn the presplit case,\n\t\tsuch a building corresponds to a single cylinder with ``toes''\n\t\tthat dip in to $\\mathcal{V} \\subset \\overline W$, and thus \n\t\ta perturbation in $\\mathcal{V}$ suffices to obtain \n\t\ttransversality. See Figure\n\t\t\\ref{fig:transversality split presplit}.)\n\t\t\n\t\t\n\t\\item For these complexes to be well-defined, we also need to establish\n\t\tthat there are no curves counted either in $\\partial$ or in the proof of $\\partial^2=0$ that\n\t\tare asymptotic to the degenerate constant orbits at \n\t\t$\\partial \\supp dH$. As we pointed out in Remark \\ref{R:not MB}, these orbits fail to even be Morse--Bott,\n\t\tand thus represent a break-down of the standard analytical theory.\n\t\tWe saw in Lemma \\ref{non-MB} why these orbits don't appear in $\\partial$. A similar argument (applying \n\t\tLemma \\ref{no + limits in W}) implies that they also don't need to be considered when proving $\\partial^2 = 0$.\n\t\\item Compactness and gluing arguments allow us to identify the presplit and split moduli spaces\n\t\tfor sufficiently large stretching parameter. Notice that by formula \\eqref{J_kappa}, \n\t\teach of the almost complex structures $J_\\kappa$ used in \n\t\tstretching is biholomorphic to $J_W$, though the support of $H$ is moved towards $+\\infty$\n\t\tby the biholomorphism. Thus, a slight modification \n\t\tof the standard SFT compactness theorem works in our setting.\n\t\tIt is important to point out that after stretching the neck, the component of a Floer cylinder that is contained in $\\mathbb{R}\\times Y$ is connected, which is why the split Floer cascades we described above contain information about all the presplit Floer cascades. \n\t\tThis follows from an argument in \\cite{BOExactSequence}*{Step 1 in proof of Proposition 5}.  \n\t\tBy a gluing argument, for each subcomplex of\n\t\tthe split complex given by a bound in action, there exists a sufficiently large stretching \n\t\tparameter for which we get an identification with the corresponding\n\t\tsubcomplex of the presplit complex.\n\t\\item Show that the presplit complex is quasi-isomorphic to a symplectic homology complex\n\t\tobtained from a non-degenerate Hamiltonian. This involves\n\t\tconstructing a continuation map $\\Phi$ connecting the two chain complexes. \n\t\t%\n\t\tThe delicate part of the\n\t\tproof that this is a chain map\n\t\tstems again from the failure of $H$ to be \n\t\tMorse--Bott non-degenerate along the boundary $\\partial \\supp dH$.\n\t\tWe address this difficulty by means of the Abouzaid--Seidel\n\t\tLemma \\ref{non-MB} below.\n\t\t%\n\t\tA usual action filtration argument\n\t\tgives that this is a chain isomorphism. \n\\end{itemize}\n\n\n\n\\begin{figure}\n  \\begin{center}\n    \\def0.8\\textwidth{0.8\\textwidth}\n    %\n\n\\begingroup\n  \\makeatletter\n  \\providecommand\\color[2][]{%\n    \\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}\n    \\renewcommand\\color[2][]{}%\n  }\n  \\providecommand\\transparent[1]{%\n    \\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}\n    \\renewcommand\\transparent[1]{}%\n  }\n  \\providecommand\\rotatebox[2]{#2}\n  \\ifx0.8\\textwidth\\undefined\n    \\setlength{\\unitlength}{505.90228431pt}\n  \\else\n    \\setlength{\\unitlength}{0.8\\textwidth}\n  \\fi\n  \\global\\let0.8\\textwidth\\undefined\n  \\makeatother\n  \\begin{picture}(1,0.41457556)%\n    \\put(0,0){\\includegraphics[width=\\unitlength]{presplit_split.pdf}}%\n    \\put(0.10819533,0.24029146){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$q$}}}%\n    \\put(0.15491812,0.35196881){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$W$}}}%\n    \\put(0.94931704,0.26739761){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$\\mathbb{R}\\times Y$}}}%\n    \\put(0.96558072,0.02018953){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$W$}}}%\n    \\put(0.64779644,0.24029146){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$q$}}}%\n    \\put(0.40744728,0.39208592){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$p$}}}%\n    \\put(0.94704832,0.39208592){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{$p$}}}%\n    \\put(0.47599853,0.29229921){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{stretch}}}%\n    \\put(0.46526508,0.23096499){\\color[rgb]{0,0,0}\\makebox(0,0)[lb]{\\smash{the neck}}}%\n  \\end{picture}%\n\\endgroup\n   \\end{center}\n  \\caption{ By perturbing $J$ in $\\mathcal{V}$\n(i.e.~away from the cylindrical end of $W$), the curve on the left can\nbe made transverse. Such a perturbation only makes the plane in $W$\ntransverse for the building on the right.} \n   \\label{fig:transversality split presplit}\n\\end{figure}\n\n\\ \n\nWe now address the difficulty in the proof of the fact that a continuation map between presplit symplectic homology and a Morse perturbed version is a chain map.\n\nFirst, we will explain what the construction would entail if all\nof the 1-periodic orbits of the admissible Hamiltonian $H$ (as in Definition \\ref{def:J_shaped}) \nwere Morse--Bott non-degenerate.\n\nRecall that we defined a differential $\\partial_{pre}$ on $SC_*(W,H)$, as in\n\\eqref{eqn:chain complex}, which we called the presplit differential. \nWe then construct a non-degenerate Hamiltonian $\\tilde H\\colon S^1 \\times W \\to \\mathbb{R}$ such that $\\tilde H(t,w) = H(w)$ outside a small neighbourhood of the periodic orbits of $X_H$. \nWe require that $\\tilde H$ be $C^2$-small and Morse on $W \\setminus \\big((-\\varepsilon,\\infty)\\times Y\\big)$ \nfor some $\\varepsilon >0$. In addition, \n$\\tilde H$ is a small time-dependent perturbation of $H$ near the non-constant periodic orbits of $X_H$, using auxiliary Morse functions on the manifolds of orbits in a manner similar to \\cite{BOSymplecticHomology}*{page 73}.  \nPicking a generic almost complex structure on $W$, we get a chain complex $SC_*(W,\\tilde H)$ that computes the symplectic homology of $W$. Denote its differential by $\\tilde \\partial$.  \n\nWe construct a chain map \n$$\n\\Phi \\colon SC_*(W,H) \\to SC_*(W,\\tilde H)\n$$\nas follows. Let $\\overline H\\colon \\mathbb{R}\\times S^1 \\times W \\to \\mathbb{R}$ be such that\n\\begin{itemize}\n \\item $\\overline H(s,t,w) = H(w)$ if $s> 1$;\n \\item $\\overline H(s,t,w) = \\tilde H(t,w)$ if $s<-1$;  \n \\item $\\partial_s \\overline H \\leq 0$.\n\\end{itemize}\nWe will impose two more technical conditions on $\\overline H$, in Lemma \\ref{lem:noConvergeToBadOrbits}. \n\nPicking a domain-dependent almost complex structure $\\overline{J}$ interpolating\nbetween the ones used in the two chain complexes, we define\n$\\Phi$ via counts of index 0 Floer cylinders with cascades for\nthe data of the pair ($\\overline H$, $\\overline J$) \n(these are cascades that satisfy \\eqref{eq:0,1 continuation} below with $\\beta = dt$).\nWe will refer to such cylinders with cascades as \\textit{continuation map cascades}.\n\nIf all the periodic orbits of $H$ were Morse--Bott non-degenerate, \nthe argument is standard: we consider index $1$ families of cascades solving\nthe continuation map equation. These families are compact, and their\nboundaries are unions of terms that correspond to configurations either in\n$\\Phi \\circ \\partial_{pre}$ or in $\\tilde \\partial \\circ \\Phi$. Furthermore,\nany such configuration in $\\Phi \\circ \\partial_{pre}$ or in $\\tilde \\partial\n\\circ \\Phi$ can be glued, showing that all such configurations arise as\nboundary components of these 1-dimensional moduli spaces. \n\nThe presence of degenerate orbits of $X_H$ along $\\partial \\supp dH$\nintroduces two main difficulties. The first is in the consideration of index\n$0$ and index $1$ cascades solving the continuation map equation, specifically\nin the case of $\\Phi(x)$ where $x$ is a generator of the complex coming from a \ncritical point of $f_W$ (or the index $1$ analogue). In this case, the\nasymptotic boundary condition for the cascade requires that the positive end\nconverge to a (constant) orbit in $W$ in the \\textit{ascending} manifold of\n$x$. This ascending manifold is non-compact, and thus the family of solutions is not necessarily\ncompact. We will show that in actual fact, for energy reasons and for suitably\nchosen $\\tilde H$, no non-constant cascade of this form exists. \n\nThe second potential problem comes from the compactification of the moduli\nspace of index $1$ continuation map cascades. A portion of such cascades can\nconverge in $C^\\infty_{loc}$ to a Floer cylinder with an end that converges\n(badly) to the degenerate orbits in $\\partial \\supp dH$. (We may arrange for\nsuch convergence to happen at the $+\\infty$ puncture by changing our\nsubsequence and shifts.) We will again show that this kind of behaviour does\nnot occur for energy reasons.\n\n\nThe rest of this section will now prove these two claims. \nIn order to rule both of these problem configurations out, we will \ncombine \nthe convergence of sequences \nin Lemma \\ref{L:convergence to orbit} \nand a careful application of the following result \noriginally due to Abouzaid and Seidel (see \\cite{AbouzaidSeidel}*{Lemma 7.2} \nand also \\cite{Ritter}*{Lemma 19.3}). \nWe will state the result in slightly greater generality than required for our\npurposes here.\nIn the following lemma, for instance, the annulus $S = (-\\epsilon, 0] \\times\nS^1$ should be thought of as a subset of the punctured cylinder, and $\\beta$\nwill then be the restriction of the form $dt$ to this annulus. \n\n\\begin{lemma} \\label{L:ASestimate}\nLet $Y$ be a co-oriented contact manifold with contact form $\\alpha$ and denote the corresponding \nReeb vector field by $R$. \nLet $S$ be the annulus $S = (-\\epsilon, 0] \\times S^1 \\subset \\mathbb{R} \\times S^1$,\nwith coordinates $\\sigma \\in (-\\epsilon, 0]$ and $\\tau \\in S^1 = \\mathbb{R} \/ \\mathbb{Z}$.\nLet $i$ denote the complex structure on $S$ with $i \\partial_\\sigma =\n\\partial_\\tau$.\n\nLet $J$ be an $S$-dependent family of almost complex structures on $[r_0, r_0 + \\delta] \\times Y$ for small $\\delta>0$\nand let $H \\colon S \\times [r_0, r_0 + \\delta] \\times Y \\to \\mathbb{R}$ be an $S$-dependent family of Hamiltonians\nwith the following locally radial behaviour:\n\\begin{enumerate}\n\\item $J(z, r, y) \\partial_r = R$, $J(z, r, y) \\ker \\alpha = \\ker \\alpha$,\n\\item $H(z, r, y) = h(z, \\operatorname{e}^r)$, for some function $h(z,\\rho)$.\n\\end{enumerate} \nLet $\\beta$ be a closed $1$-form on $S$. \n\nSuppose that $\\tilde v \\colon S \\to [r_0, r_0 + \\delta) \\times Y$ satisfies the following equation:\n\\begin{equation}\n\\label{eq:0,1 continuation}\n0 = 2 (d\\tilde v - X_H \\otimes \\beta)^{0,1} = d\\tilde v + J(z, \\tilde v) d\\tilde v \\circ i - X_H \\otimes \\beta - J(z, \\tilde v) X_H \\otimes \\beta \\circ i,\n\\end{equation}\nand $\\tilde v( \\{ 0 \\} \\times S^1 ) \\subset \\{ r_0 \\} \\times Y$. \n\n\nThen, taking $\\lambda = \\operatorname{e}^r \\alpha$,\n\\begin{align*}\n\\int_{ \\{ 0 \\} \\times S^1 } \\tilde v^*\\lambda - H(z, \\tilde v(z)) \\beta &\\le \\int_{ \\{ 0 \\} \\times S^1 } ( \\lambda(X_H) - H ) \\beta = \\\\\n&=\\int_{ \\{ 0 \\} \\times S^1 } \\left ( \\operatorname{e}^{r_0} \\frac{\\partial h}{\\partial \\rho}(z, \\operatorname{e}^{r_0}) - h(z, \\operatorname{e}^{r_0}) \\right ) \\beta\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n\nApplying $\\lambda$ to \\eqref{eq:0,1 continuation} and using $\\lambda \\circ J = \\operatorname{e}^r dr$ and $\\lambda( JX_H) = 0$, we obtain\n\\[\n\\tilde v^*\\lambda + \\operatorname{e}^r dr( d\\tilde v \\circ i) = \\lambda(X_H) \\beta.\n\\]\nBy hypothesis, we have that $r( \\tilde v(0,\\tau) ) \\equiv r_0$. \nFurthermore, since the image of the annulus $S$ is not below $r=r_0$, \nwe have $dr( d\\tilde v \\circ i)(\\frac{\\partial}{\\partial \\tau}) \\ge 0$ along $\\{0\\} \\times S^1$, hence\n\\[\n\\tilde v^*\\lambda(\\frac{\\partial}{\\partial \\tau}) \\le \\lambda(X_H) \\beta(\\frac{\\partial}{\\partial \\tau}).\n\\]\nThe result now follows.\n\\end{proof}\n\n\n\n\nFollowing \\cites{AbouzaidSeidel,Ritter}, we introduce the following terminology:\n\n\n\\begin{definition}\nLet $(S, j)$ be a Riemann surface and let $\\beta$ be a 1-form on $S$ so $d\\beta \\le 0$. \nLet $(W, \\omega)$ be a symplectic manifold, and let $J$ be an $S$-dependent family of almost complex structures on $W$, compatible with $\\omega$.\nLet $H \\colon S \\times W \\to \\mathbb{R}$ be as $S$-dependent Hamiltonian. We write $dH$ to indicate the (total) differential of $H$ and $d_W H$ to denote the $S$-dependent family of 1-forms on $W$ given by $d_W H|_{z \\in S} = d (H(z, \\cdot))$.\n\nFor a map $\\tilde v \\colon S \\to W$, we define its\n\\begin{itemize}\n\\item {\\em Topological energy}: $E_\\mathrm{topo}(\\tilde v) = \\int_{S} \\tilde v^*( \\omega - d(H \\beta) )$;\n\\item {\\em Geometric energy}: $E_\\mathrm{geom}(\\tilde v) = \\int_{S} \\tilde v^*\\omega - (\\tilde v^*d_W H) \\wedge \\beta$.\n\\end{itemize}\n\\end{definition}\n\nNote that for any solution to Floer's equation \\eqref{eq:0,1 continuation}, $E_\\mathrm{geom}(\\tilde v) = \\int_S \\| \\partial_s \\tilde v\\|^2 \\geq 0$ is (pointwise) non-negative. \nWe say that the triple $(\\beta, H,J)$ is \\defin{monotone} if the topological energy dominates\nthe geometric energy pointwise (in the sense that the integrand in the geometric energy is equal to the integrand in the topological energy times a function $f\\leq 1$). Observe that they are equal if $H$ is $z$-independent and $\\beta = dt$.\n\n\n\n\\begin{lemma} \\label{lem:noConvergeToBadOrbits}\nLet $(W, d\\lambda)$ be an exact symplectic manifold with cylindrical end $((r_0, +\\infty) \\times Y, d( \\operatorname{e}^r \\alpha ))$, $r_0 < \\log 2$.\n\nLet $(\\beta,H, J)$ be a monotone triple on the cylinder $(\\mathbb{R} \\times S^1, i)$ with 1-form $\\beta = dt$ and \nwith $H_{(+\\infty, t)}$ an admissible Hamiltonian as in Definition \\ref{def:J_shaped}.\n\nLet $\\gamma_+$ be a (constant) 1-periodic orbit of $H_{(+\\infty, t)}$ contained in the level $\\{ r_1 \\} \\times Y$ for some $r_0< r_1 \\leq \\log 2$\nand let $\\gamma_-$ be a 1-periodic orbit\nof $H_{(-\\infty, t)}$, contained entirely below the the level $ \\{r_1 \\} \\times Y$. \n\n\nSuppose furthermore that $H \\colon \\mathbb{R} \\times S^1 \\times W \\to \\mathbb{R}$ and $J$ are locally radial (in the sense of Lemma \\ref{L:ASestimate}) \non $(r_1-\\delta, r_1 + \\delta ) \\times Y$ for small $\\delta>0$, with the additional condition that \n$H(s,t, r, y) = h(s, \\operatorname{e}^r)$ for some function $h(s,\\rho)$, with \n\\begin{align*}\n\\operatorname{e}^r \\frac{\\partial h}{\\partial \\rho}(s, \\operatorname{e}^r) - h(s, \\operatorname{e}^r) &\\ge 0 \\\\\n\\partial_s \\Big( \\operatorname{e}^r \\frac{\\partial h}{\\partial \\rho}(s, \\operatorname{e}^r) - h(s, \\operatorname{e}^r) \\Big) &\\le 0.\n\\end{align*}\n\nThen, there does not exist any $\\tilde v \\colon \\mathbb{R} \\times S^1 \\to W$ solving Floer's equation \\eqref{eq:0,1 continuation}\nfor which there exists a sequence $s_k \\to +\\infty$ along which $\\tilde v(s_k, t) \\to \\gamma_+(t) \\subset \\{ r_1 \\} \\times Y$  \nand for which $\\tilde v(s,t) \\to \\gamma_-(t), s\\to -\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose there is a $\\tilde v$ as in the statement. Consider a smooth function $\\tilde r \\colon W \\to \\mathbb{R}$ that is equal to the first coordinate $r$ on $(r_1-\\delta\/2, r_1 + \\delta\/2) \\times Y$ and constant outside of $(r_1-\\delta, r_1 + \\delta) \\times Y$.\nWe can assume that $\\delta>0$ is small enough that $\\tilde r \\circ \\gamma_-(t) < r_1-\\delta\/2$. \nTake a regular value $r_1-c$ of $\\tilde r \\circ \\tilde v \\colon \\mathbb{R} \\times S^1 \\to \\mathbb{R}$ in the interval $(r_1-\\delta\/2, r_1)$. \n\nBy hypothesis, there exists a $k_0$ sufficiently large so that for each $k \\ge k_0$, $\\min_{t \\in S^1} \\tilde r( \\tilde v(s_{k}, t) ) > r_1-c\/2$. Fix such a $k$.\n\nDefine the subdomain $S_k$ of $\\mathbb{R} \\times S^1$ by \n\\[\n S_k = \\{ (s,t) \\in (-\\infty, s_{k}) \\times S^1 \\, | \\, \\tilde r( \\tilde v(s,t)) > r_1-c \\}.\n\\]\nNote that while this subdomain may be disconnected, its closure in $\\mathbb{R} \\times S^1$ is compact, due the fact that $\\tilde v(s, t) \\to \\gamma_-(t)$ as $s \\to -\\infty$ and that $\\tilde r \\circ \\gamma_-(t) < r_1-\\delta\/2$.  Since $k \\ge k_0$, this set is non-empty.\n\nBy the fact that $r_1-c$ is a regular value of \n$\\tilde r \\circ \\tilde v$, we have that $\\partial S_k$ is a collection of smooth, embedded closed curves.\nLet $\\Gamma$ be the collection of embedded closed curves in $\\mathbb{R} \\times S^1$ so that the boundary of $S_k$ is $\\partial S_k = \\Gamma \\cup \\{ s_{k} \\} \\times S^1$. Let $\\Gamma$ be oriented with the orientation induced as the boundary of $S_k$. \nAs an aside, note that the projection of $\\Gamma \\subset \\mathbb{R} \\times S^1$ to $S^1$ has total degree $-1$, since it separates the cylinder.\n\nBy construction, $\\tilde v(\\Gamma) \\subset \\{ r_1- c \\} \\times Y$, and by definition $\\tilde r (\\tilde v(S_k))\\subset (r_1-c,r_1+\\delta]$. \nCombining Stokes's Theorem with Lemma \\ref{L:ASestimate} (and implicitly using biholomorphisms between neighborhoods of the connected components of $\\Gamma$ in $S_k$ and annuli of the form $(-\\epsilon,0]\\times S^1$), we obtain\n\\begin{align*}\nE_{\\mathrm{topo}} ( \\tilde v|_{S^k} ) &=  \\int_{ \\{ s_k \\} \\times S^1 } \\tilde v^*\\lambda - H(s,t,\\tilde v(s,t) ) dt  + \\int_{\\Gamma} \\tilde v^*\\lambda -H(s,t,\\tilde v(s,t) ) dt \\\\\n\t&\\le \\int_{ s=s_k } \\tilde v^*\\lambda - H(s_k,t,\\tilde v(s_k,t) ) dt + \\int_{\\Gamma} (\\operatorname{e}^{r_1-c} \\frac{\\partial h}{\\partial \\rho}(s, \\operatorname{e}^{r_1-c}) - h(\\operatorname{e}^{r_1-c}) ) dt \\\\\n\t&= \\int_{ s=s_k }  \\tilde v^*\\lambda  - H(s_k,t,\\tilde v(s_k,t) ) dt - \\\\\n\t\t\t\t\t\t& \\hphantom{mmmmmmm}  -  \\int_{ s=s_k} ( \\operatorname{e}^{r_1-c} \\frac{\\partial h}{\\partial \\rho}(s, \\operatorname{e}^{r_1-c}) - h(\\operatorname{e}^{r_1-c}) ) dt + \\\\\n\t\t\t\t\t\t& \\hphantom{mmmmmmm}  + \\int_{S_k} \\partial_s \\big( \\operatorname{e}^{r_1-c} \\frac{\\partial h}{\\partial \\rho}(s, \\operatorname{e}^{r_1-c}) - h(\\operatorname{e}^{r_1-c}) \\big) \\, ds \\wedge dt \\\\\n\t&\\le \\int_{s=s_k} \\tilde v^*\\lambda  - H(s_k,t,\\tilde v(s_k,t) ) dt.  \n\\end{align*}\n\nAs the parameter $k \\to +\\infty$, the right hand side converges to the action of $\\gamma_+$, which is $0$. Furthermore, since the Floer data is assumed to be monotone, $E_{\\mathrm{geom}}( \\tilde v|_{S_k}) \\le E_{\\mathrm{topo}}( u|_{S_k})$ and since the $S_k$ are nested, \n$E_{\\mathrm{geom}}(\\tilde v|_{S_k})$ is monotone non-decreasing in $k$. It follows that $E_{\\mathrm{geom}}( \\tilde v|_{S_k}) = 0$ for all $k$. As $S_k$ has non-empty interior, we conclude that $\\partial_s\\tilde v = 0$, which contradicts the fact that $\\tilde v(s,t) \\to \\gamma_-$ as $s \\to -\\infty$.\n\\end{proof}\n\nWe now explain how the previous result completes the argument that none of the problematic configurations described earlier occur, and thus $\\Phi$ is a chain map. \n\nWe take $\\overline H$ satisfying the radial\nconditions of Lemma \\ref{lem:noConvergeToBadOrbits}. \nThe lemma implies that the moduli\nspace of index $0$ continuation cascades that are counted in $\\Phi$ are\ncompact. Furthermore, none has positive end with an asymptotic limit in the\ndegenerate orbits at $\\partial \\supp dH$.  \n\nNow, consider index $1$ continuation cascades. First of all, by this lemma,\nnone of the curves has a positive puncture at a degenerate orbit. A sequence of such\ncascades, with a fixed number of cascades and with a uniform bound on the\naction of the positive puncture, will then either converge to a broken\ncylinder with cascades (corresponding to configurations in \n$\\tilde \\partial \\circ \\Phi$ or in $\\Phi \\circ \\partial_{pre}$) or will admit\na sequence of domain reparametrizations along which it converges in\n$C^\\infty_{loc}$ to a cylinder of the kind ruled out by Lemma\n\\ref{lem:noConvergeToBadOrbits}.\n\nFinally, we observe that Lemma \\ref{L:ASestimate} gives us control over the configurations that \ncan occur as possible limits after splitting. Specifically, this lemma, understood as a maximum \nprinciple, is behind the requirement that a split Floer\ncylinder with negative end asymptotic to a non-constant Hamiltonian orbit in\n$\\mathbb{R} \\times Y$ must have its positive end asymptotic to a non-constant\nHamiltonian orbit. (This is by an argument similar to \n\\cite{BOExactSequence}*{Proof of Proposition 5, Step 1}.)\n\n\n\n \n\\part{Ansatz}\n\n\\label{Ansatz}\n\n\\section{A correspondence between Floer and pseudoholomorphic cylinders}\n\nThe content of this section is technical, so we begin\nwith an overview and explanation of its relevance to the study of the split symplectic\nhomology differential. The main results are Propositions \\ref{Ansatz1}\nand \\ref{Ansatz2}, which we summarize in the following result. For precise\nstatements, notably of the uniqueness, see the full statements of the\npropositions in the later sections.\n\n\n\\begin{proposition} \\label{P:summary ansatz}\n There is a bijection \n $$\n \\{\\tilde v\\colon \\mathbb{R} \\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y\\} \\longleftrightarrow \\{(\\tilde u,{\\bf e})\\} \/ \\mathbb{R}\n $$\n where \n \\begin{itemize}\n     \\item $\\tilde v$ is an augmented Floer cylinder of finite hybrid energy (as\n         in Equation \\eqref{E:hybridEnergy}),\n  \\item $\\tilde u\\colon \\mathbb{R} \\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$ is a finite energy pseudoholomorpic curve with negative ends at $\\Gamma \\cup \\{-\\infty\\}$ and positive end at $+\\infty$,\n  \\item ${\\bf e}\\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times S^1$ is an essentially unique solution \n      to a partial differential equation that depends on $\\tilde u$ (Equation \\eqref{E:cylinder1}),\n  \\item if we write $\\tilde u = (a,u)$ and ${\\bf e} = (e_1,e_2)$, the quotient\n      is by the $\\mathbb{R}$-action given, for $c \\in \\mathbb{R}$, by\n  $$\n  c \\cdot ( (a,u),(e_1,e_2)) = ((a+c,u),(e_1-c,e_2) ), \\qquad c\\in \\mathbb{R},\n  $$\n  \\item if $\\tilde v$ converges to a Hamiltonian orbit of the form $(b,\\gamma)$ in $\\mathbb{R}\\times Y$ at one of its ends (where $b$ is a \n  constant and $\\gamma$ is a Reeb orbit), then $\\tilde u$ converges to the Reeb orbit $\\gamma$ at the corresponding end; if $\\tilde v$ \n  converges to a Reeb orbit in $Y$ at one of its ends, then $\\tilde u$ converges to the same orbit at the corresponding end.\n \\end{itemize}\n\\end{proposition}\n\n\nRecall that by Proposition \\ref{P:split SH cases}, there are four types of contributions to the differential, referred to as Cases 0, 1, 2 and 3.  \n\nCase 0 configurations are gradient flow lines in $Y$, so they can be described by the Morse differential in $Y$. \n\n\\\n\nCase 1 configurations are elements of 1-dimensional fibre products \\eqref{fib prod1}\n$$\nW^s_{Y}(\\wh q) \\times_{\\operatorname{ev}} \\widetilde{ \\M }^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\n$$\nwhere $q\\neq p\\in \\Crit(f_\\Sigma)$ and $A\\in H_2(\\Sigma;\\mathbb{Z})$. Recall that $\\widetilde{ \\M }^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y)$ is the \nspace of parametrized unpunctured Floer cylinders $\\tilde v$, going from orbits of multiplicity $k_-$ (as $s\\to -\\infty$) to orbits \nof multiplicity $k_+$ (as $s\\to +\\infty$), where $k_+>k_-$.\nThe fibre product above has a free action of $\\mathbb{R}$ by translations of $\\tilde v$ in the $s$-direction, so it is a finite disjoint union of copies of $\\mathbb{R}$.\n\n\nBy Proposition \\ref{P:summary ansatz}, we get a bijection between $\\widetilde{ \\M }^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y)$ \nand the space of $J_Y$-holomorphic cylinders in $\\mathbb{R}\\times Y$, going from Reeb orbits of multiplicity $k_-$ to Reeb orbits of multiplicity $k_+$. \nIn Section \\ref{S:meromorphic}, we will relate such pseudoholomorphic cylinders to \npseudoholomorphic spheres in $\\Sigma$ and meromorphic sections of line bundles over $\\mathbb{CP}^1$.  \n\n\\ \n\nCase 2 configurations are elements of 1-dimensional fibre products \\eqref{fib prod2} \n$$\nW^s_{Y}(\\wh p) \\times_{\\operatorname{ev}} \\left(\\mathcal{M}^*_X(B;J_W) \\times_{\\tilde\\operatorname{ev}} \\widetilde{ \\M }^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\right) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\n$$\nwhere $p\\in \\Crit(f_\\Sigma)$, $B\\in H_2(X;\\mathbb{Z})$. Recall that $\\mathcal{M}^*_X(B;J_W)$ is a space of unparameterized $J_W$-holomorphic planes $U$ in $W$ \n(which can be identified with a space of $J_X$-holomorphic spheres in $X$, by Lemma \\ref{planes = spheres}) and \n$\\widetilde{ \\M }^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y)$ is a space of Floer cylinders $\\tilde v\\colon \\mathbb{R}\\times S^1\\setminus \\{P\\} \\to \\mathbb{R} \\times Y$ \nprojecting to a point in $\\Sigma$, asymptotic at $-\\infty$ (resp. $+\\infty$) to an orbit of multiplicity $k_-$ (resp. $k_+$) and with \na negative puncture at $P\\in \\mathbb{R}\\times S^1$ that is asymptotic to a Reeb orbit of multiplicity $k_+-k_-$. The stable and unstable manifolds \nin the fibre product force $\\tilde v$ to project to $p\\in \\Sigma$. Hence, $U$ is asymptotic to a Reeb orbit of multiplicity \n$k_+-k_- = B\\bullet \\Sigma$ over $p$. Proposition \\ref{P:summary ansatz} now implies that $\\widetilde{ \\M }^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y)$ \ncan be identified with the space of punctured $J_Y$-holomorphic cylinders $\\mathbb{R}\\times S^1 \\setminus \\{P\\} \\to \\mathbb{R}\\times Y$ that are branched covers of a trivial cylinder, with \nnegative punctures at $-\\infty$ and $P$ asymptotic to orbits of multiplicity $k_-$ and $k_+-k_-$, respectively, and with a positive \npuncture at $+\\infty$ asymptotic to an orbit of multiplicity $k_+$. \n\n\\ \n\nFinally, Case 3 configurations are elements of fibre products \\eqref{fib prod3}\n$$\n W^u_{W}(x) \\times_{\\operatorname{ev}^1_-} \\left(\\mathcal{M}^*_{H}(B;J_W)  \\prescript{}{\\operatorname{ev}^1_+\\,}\\times\\prescript{}{\\operatorname{ev}^2_-}{}  \\mathcal{M}^*_{H,k_+}(0;J_Y) \\right) \\times_{\\operatorname{ev}^2_+} W_{Y}^u(\\wc p)\n$$\nwhere $p\\in \\Crit(f_\\Sigma)$, $x\\in \\Crit(f_W)$ and $B\\in H_2(X;\\mathbb{Z})$. Here, $\\mathcal{M}^*_{H}(B;J_W)$ is a space of \n$J_W$-holomorphic planes $\\tilde v_0$ in $W$, asymptotic to a Reeb orbit of multiplicity $k_+=B\\bullet\\Sigma$ \n(recall that $H=0$ in $W$ after neck-stretching, so we can think of $J_W$-holomorphic planes as Floer cylinders that asymptote to \nconstants at $-\\infty$). An element in $\\mathcal{M}^*_{H,k_+}(0;J_Y)$ is a Floer cylinder $\\tilde v_1$ in $\\mathbb{R}\\times Y$ that projects to a point in $\\Sigma$, is asymptotic at $+\\infty$ \nto a Hamiltonian orbit of multiplicity $k_+$, and asymptotic at $-\\infty$ to a Reeb orbit of multiplicity $k_+$. The unstable manifold \nat the right of the fibre product forces $\\tilde v_1$ to project to $p$ in $\\Sigma$. The fibre product forces the positive asymptotic orbit of $\\tilde v_0$ to be the same as the negative asymptote of $\\tilde v_1$ (which projects to $p$ in $\\Sigma$). \nProposition \\ref{P:summary ansatz} implies that $\\mathcal{M}^*_{H,k_+}(0;J_Y)$ can be identified with the \nspace of trivial $J_Y$-holomorphic spheres in $\\mathbb{R}\\times Y$, over Reeb orbits of multiplicity $k_+$. \n\n\n\\subsection{Sobolev spaces and Morse--Bott Riemann--Roch}\n\nBefore we state and prove the main results in this section, we need to briefly set-up \nthe appropriate Fredholm theory. We refer to \n\\cite{DiogoLisiSplit}*{Section 5.2} for more details. \n\nWe will need to consider exponentially weighted Sobolov spaces of sections of the trivial complex \nline bundle over a Riemann surface $\\dot S= \\mathbb{R}\\times S^1\\setminus \\Gamma$, where $\\Gamma$ is a finite set.  \nWe think of this as a punctured sphere, with punctures at $\\Gamma \\cup \\{ \\pm\n    \\infty \\}$. \nThese are partitioned into a single positive puncture $\\Gamma_+ = \\{ + \\infty\n\\}$ and negative punctures $\\Gamma_- = \\Gamma \\cup \\{ -\\infty \\}$. Near each\npuncture, we fix cylindrical coordinates with the sign corresponding to the\nsign of the puncture. \nFor $\\delta > 0$, we \ndenote by $W^{1,p,\\delta}(\\dot S, \\mathbb{C})$\nthe space of sections that decay exponentially like $\\operatorname{e}^{-\\delta |s|}$ near the\npunctures (in the cylindrical coordinates already chosen near the punctures).\nIn certain circumstances, we will find it useful to consider sections with \nexponential growth, and denote these by taking $\\delta < 0$.\n\nLet \n\\begin{align*}\n D &\\colon W^{1,p,\\delta}(\\dot S,\\mathbb{C}) \\to L^{p,\\delta}(\\dot S,\\Lambda^{0,1}T^*\\dot S)\n\\end{align*}\nbe a Cauchy--Riemann operator. Denote by $\\mathbf{A}_z\\colon W^{1,p}(S^1,\\mathbb{C}) \\to L^p(S^1,\\mathbb{C})$ \nthe asymptotic operator at the puncture $z\\in \\Gamma \\cup \\{ \\pm \\infty \\}$. We do not assume \nthat it is non-degenerate, since we want to study Morse--Bott moduli spaces. \nLet $\\mathbf V$ be a collection of vector spaces, associating to each\npuncture $z$ a vector subspace $V_z$ of $\\ker (\\mathbf{A}_z)$.\nFor $\\delta >0$, we then consider the space $W^{1,p,\\delta}_{\\mathbf V}(\\dot S, \\mathbb{C})$ \nof sections that converge exponentially at each puncture $z$ to a vector\nin the corresponding vector space $V_z$. \n\nDenote the spectrum of an asymptotic operator $\\mathbf{A}_z$ as\nabove by $\\sigma(\\mathbf{A}_z)$. The operator $\\mathbf{A}_z$ is of\nthe form $-i\\frac{d}{dt} - A_z(t)$, for some path $A_z(t)$ of\nsymmetric matrices. If $\\delta\\in \\mathbb{R}$ is such that $\\mathbf A_z +\n\\delta$ is a non-degenerate operator, then it has an associated\npath of symplectic matrices, given by the fundamental matrix of the\nODE $\\frac{d}{dt}x = i (A_z(t)-\\delta) x$.  This path has a\nConley--Zehnder index, that we denote by $CZ(\\mathbf{A}_z + \\delta)$\n(see also \\cite{DiogoLisiSplit}*{Section 5.2.1}).\n\n\n\n\n\\begin{theorem} \\label{T:RiemannRochMB}\nLet $\\delta > 0$ be sufficiently small that \nfor each puncture $z \\in \\Gamma \\cup \\{ \\pm \\infty \\}$,\n$[-\\delta, \\delta]\\cap\\sigma(\\mathbf{A}_z) \\subset \\{ 0 \\}$.\nLet $\\mathbf{V}$ be a collection of vector spaces, associating to each \n$z \\in \\Gamma \\cup \\{ \\pm \\infty \\}$ the vector subspace $V_z \\subset \\ker \\mathbf{A}_z$. \n\nThen, \n\\[\nD \\colon W^{1,p, {\\delta}}_{\\mathbf V}(\\dot S, \\mathbb{C}) \\to L^{p, \\mathbf{\\delta}}(\\dot S, \\Lambda^{0,1} T^*\\dot S).\n\\]\nis Fredholm, and its Fredholm index is given by\n\\[\n\\operatorname{Ind}(D) = \\chi_{\\dot S} + \\sum_{z \\in \\Gamma_+} \\big( \\text{CZ}( \\mathbf{A}_z + \\delta) + \\dim(V_z)\\big)\n- \\sum_{z \\in \\Gamma_-} \\big(\\text{CZ}( \\mathbf{A}_z + \\delta ) + \\operatorname{codim}(V_z)\\big).\n\\]\n\\end{theorem}\n\n\n\nThe following will be useful in determining the relevant Conley--Zehnder indices in our setting. \n\n\\begin{lemma}\\label{L:spectrum}\nGiven a constant $C\\geq 0$, the spectrum $\\sigma(A_C)$ of the operator\n$$\nA_C:= - i \\frac{d}{dt} - %\n\\begin{pmatrix} C & 0 \\\\\n0 & 0  \n\\end{pmatrix} %\n\\colon W^{1,p}(S^1,\\mathbb{C}) \\to L^p(S^1,\\mathbb{C})\n$$\nis the set\n $$\\left \\{\\frac{1}{2}\\left(- C - \\sqrt{ C^2 + 16 \\pi^2 k^2}\\right) \\, | \\, k\\in \\mathbb{Z} \\right\\} \\cup \\left \\{\\frac{1}{2}\\left(- C + \\sqrt{ C^2 + 16 \\pi^2 k^2}\\right) \\, | \\, k\\in \\mathbb{Z} \\right\\}.$$\nIf $\\lambda$ is an eigenvalue associated to $k\\in \\mathbb{Z}$, then the winding number of the corresponding eigenvector is $|k|$ if $\\lambda\\geq 0$ and $-|k|$ if $\\lambda\\leq 0$. If $C=0$, then all eigenvalues have multiplicity 2. \nIf $C>0$, then the same is true except for the eigenvalues $-C$ and $0$, corresponding to $k=0$ above, both of which have multiplicity 1.  \n \nIn particular, the $\\sigma(A_0) = 2\\pi\\mathbb{Z}$ and the winding number of $2\\pi k$ is $k$. \n\\end{lemma}\n\n\\begin{corollary} \\label{C:CZ computation}\nTake $C\\geq 0$ and $\\delta>0$ such that $[-\\delta,\\delta] \\cap \\sigma(A_C) = \\{0\\}$. Then\n$$\n\\text{CZ}(A_C + \\delta) = \\begin{cases}\n                     0 & \\text{ if } C > 0 \\\\\n                     -1 & \\text{ if } C = 0 \n                    \\end{cases}\n\\qquad \\text{and} \\qquad\n\\text{CZ}(A_C - \\delta) = 1. \n$$\n\n\\end{corollary}\n\n\n\\subsection{From Floer cylinders to pseudoholomorphic curves} \\label{S:ansatz_Floer_to_holo}\n\n\\label{Floer_to_holo}\n\n\nWe show that every Floer cylinder as in \\eqref{Floer eq Y} \ncan be obtained from a $J_Y$-holomorphic curve and a solution\nto an auxiliary equation.\n\nIt is useful to first recall that for every integer $T \\geq 1$,\nthere is a $Y$-family of Reeb orbits of period $T$, and a bijective\ncorrespondence to the $Y$-family of 1-periodic Hamiltonian orbits\ncontained in $\\{b\\}\\times Y \\subset \\mathbb{R} \\times Y$, where $h'(\\operatorname{e}^b)\n= T$.  \nA Reeb orbit $\\gamma$ \\defin{corresponds} to the Hamiltonian\norbit $t \\mapsto (b,\\gamma(Tt))$.  We say that two Reeb orbits of\nthe same period are the {\\em same as unparametrized orbits} if they\ndiffer by a constant translation on the domain.\n\n\n\\begin{proposition}\n\\label{Ansatz1}\nLet $\\Gamma\\subset \\mathbb{R}\\times S^1$ be either the empty set or the singleton $\\{P\\}$. Suppose $\\tilde v = (b, v)\\colon \\mathbb{R}\\times S^1 \\setminus \\Gamma \\rightarrow \\mathbb{R} \\times Y$\nis the upper level of a split Floer cylinder as in Cases (1) and (2) of Proposition \\ref{P:split SH cases}.\nThen, there exists a pair $(\\tilde u, {\\bf e})$ consisting of a map $\\tilde u = (a, u)\\colon \\mathbb{R}\\times S^1 \\setminus \\Gamma \\rightarrow\n\\mathbb{R} \\times Y$ and of a map ${\\bf e} = (e_1, e_2)\\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\rightarrow \\mathbb{R} \\times S^1$, with the\nfollowing properties:\n\\begin{enumerate}\n\\item The map $\\tilde u$ is a finite Hofer energy $J_Y$-holomorphic curve.\n\\item If $\\tilde v$ converges to a Hamiltonian orbit at $\\pm \\infty$, then $u$ converges to the \n\tcorresponding parametrized Reeb orbit $\\gamma_\\pm$ at $\\pm\\infty$, i.e.~$v(\\pm \\infty, t) = u(\\pm \\infty, t)$.\n\\item If $\\Gamma = \\{P\\}$, then $u$ is asymptotic to the same unparameterized Reeb orbit that $v$ converges to at $P$, i.e.~the asymptotic limit is the same orbit up to a phase shift.\n\\item The map ${\\bf e} \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R} \\times S^1$ satisfies the equation   \n\t\\begin{equation} \\label{E:cylinder0}\n\tde_1 -de_2 \\circ i +  h'(\\operatorname{e}^{b}) ds = 0 .\n\t\\end{equation}\n\\item The original Floer solution is given by \n\\[\n(b, v) = (a + e_1, \\phi_R^{e_2} \\circ u )\n\\] \n\twhere $\\phi_R^t \\colon Y\\to Y$ denotes Reeb flow for time $t\\in S^1$. \n\\end{enumerate}\n\n\\noindent \nThe pair $(\\tilde u,{\\bf e})$ is unique, up to replacing $\\tilde u = (a,u)$ with $(a+c_1,u)$ and replacing ${\\bf e}= (e_1,e_2)$ with $(e_1 - c_1, e_2)$, for some constant $c_1\\in \\mathbb{R}$. \n\n\n\nSuppose now that $\\Gamma = \\emptyset$ and that $\\tilde v$ is the upper level of\na split Floer cylinder as in Case (3) of Proposition \\ref{P:split SH cases}.\nThen, there exists a pair $(\\tilde u, \\bf e)$ as above, with the following difference:\n\\begin{enumerate}\n \\item[(2')] If $\\tilde v$ converges to a Hamiltonian orbit at $+\\infty$ and to a Reeb orbit at $-\\infty$, then $u$ converges to the corresponding Reeb orbit $\\gamma_+$ at $+\\infty$ and to the same Reeb orbit $\\gamma_-$ at $-\\infty$. \n\\end{enumerate}\n\n\\noindent\nThe pair $(\\tilde u,{\\bf e})$ is again unique up to an $\\mathbb{R}$-action.\n\n\\end{proposition}\n\n\nIn the following, we will construct the function $\\mathbf{e}$ in\nProposition \\ref{Ansatz1} by studying the differential equation\n\\eqref{E:cylinder0}.\n\nTo this end, we need to construct spaces of maps $\\mathbf{e} \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma \\to \\mathbb{R} \\times S^1$.\nThese will be subspaces of  $W^{1,p}_{\\text{loc}}(\\mathbb{R}\\times S^1\\setminus \\Gamma, \\mathbb{R}\\times S^1)$ satisfying different types of asymptotic conditions.\nThe maps $\\mathbf{e}$ we consider will induce trivial maps of fundamental groups.\nIt will be more convenient therefore to consider the lift to the universal cover\n$\\tilde \\mathbf{e} \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma \\to \\mathbb{C}$, and study Equation \\eqref{E:cylinder0} on this space. This lift is well-defined up to an additive constant.\n\n\n\n\nAs we will see below, all the punctures have the same asymptotic operator, namely\n\\begin{equation} \\label{asymptotic operator A}\n A := - i \\frac{d}{dt} \\colon W^{1,2}(S^1,\\mathbb{C}) \\to L^2(S^1,\\mathbb{C}).\n\\end{equation}\nNote that $A$ is self-adjoint as a partially defined operator $L^2(S^1,\\mathbb{C}) \\to L^2(S^1,\\mathbb{C})$.\nLemma \\ref{L:spectrum} describes the spectrum of $A$. For the remainder of this section, we will fix $0 < \\delta < 2\\pi$, which is smaller than the absolute value of every non-zero eigenvalue of $A$. \nWe will use this $\\delta$ below when defining weighted Sobolev spaces.\n\n\nWe introduce some notation first.  For a given $z \\in \\mathbb{R} \\times S^1$, or $z =\n\\pm \\infty$, we fix $\\mu_z$ to be a bump function supported near $z$ and\nidentically 1 near $z$.  \nRecall that the space of functions $W^{1,p,\\delta}_{(0,\\mathbb{C})}( \\mathbb{R} \\times S^1, \\mathbb{C})$ consists of complex-valued functions\nexponentially decaying to $0$ at $-\\infty$ and exponentially decaying\nto an unspecified constant at $+\\infty$.  In the case of a punctured\ncylinder, $W^{1,p,\\delta}_{(0, \\mathbb{C}; \\mathbb{C})}(\\mathbb{R} \\times S^1 \\setminus \\{ P \\}, \\mathbb{C})$ \ndenotes the space of functions exponentially decaying to $0$\nat $-\\infty$, decaying to a free constant at $P$ and decaying to a free\nconstant at $+\\infty$.\n\nIn the case with $\\Gamma= \\{P\\}$, we fix a parametrization \n\\begin{align*}\n\\varphi_P \\colon (-\\infty, -1]\\times S^1 &\\to \\mathbb{R}\\times S^1 \\setminus \\Gamma \\\\\n(\\rho,\\eta) &\\mapsto P + \\operatorname{e}^{2\\pi(\\rho + i \\eta)}\n\\end{align*}\nof a neighborhood of $P$, under the identification of $\\mathbb{R}\\times S^1$ with $\\mathbb{C}\/(z\\sim z + i)$. \nRecall that the measure on $\\mathbb{R} \\times S^1 \\setminus \\Gamma$ and the norm on derivatives \nused for defining $W^{1,p}$ and $L^p$\ncomes from these cylindrical coordinates near $P$.\nIn particular, we make the following observation that will be used multiple times,\n\\begin{equation} \\label{E:ds_is_exponential}\n\\varphi_P^* ds = 2\\pi \\operatorname{e}^{2\\pi \\rho} \\cos(2\\pi \\theta) d\\rho - 2\\pi \\operatorname{e}^{2\\pi \\rho} \\sin(2\\pi\\theta) d\\theta\n\\end{equation}\nIn particular then, $| \\varphi_P^*ds | = C \\operatorname{e}^{2\\pi \\rho}$.\n\n\nWe will consider solutions to equation \\eqref{E:cylinder0} in three different spaces, corresponding to \nCases 1, 2 and 3 appearing in Proposition \\ref{P:split SH cases} (recall Figures \\ref{case_1_fig}, \\ref{case_2_fig} and \\ref{case_3_fig}). \n\n    If $\\tilde v = (b, v)$ is the upper level of a split Floer cylinder as in Cases\n    1,2 or 3 of Proposition \\ref{P:split SH cases}, it converges at\n    $+\\infty$ to a Hamiltonian orbit of the form $(b_+, \\gamma(T_+ t))$, where\n    $\\gamma$ is a parametrization of a simple (embedded) Reeb orbit in $Y$\n    (and thus satisfying $h'(\\operatorname{e}^{b_+}) = T_+$). Similarly, at $-\\infty$,\n    $\\tilde v$ converges either to a Hamiltonian trajectory of the form $(b_-,\n    \\gamma(T_- t))$ or $v$ converges to $\\gamma(T_- t)$ at $-\\infty$. In the\n    following, these periods $T_\\pm$ will be used to define the relevant\n    function spaces.\n\n\n\n\\begin{itemize}\n\\item[Case 1:] $\\Gamma = \\emptyset$ and both asymptotic limits of $\\tilde v$ are Hamiltonian orbits. \n\nWe say $\\tilde \\mathbf{e} \\in \\tilde \\mathcal{X}^1$ if\n\\[\n\\tilde \\mathbf{e} +  (T_+ s + i 0) \\, \\mu_{+\\infty} +  (T_- s + i 0) \\, \\mu_{-\\infty} \\in W^{1,p, \\delta}_{(\\mathbb{C}; \\mathbb{C})} (\\mathbb{R} \\times S^1, \\mathbb{C}).\n\\]\n\n\n\\item[Case 2:] $\\Gamma = \\{ P\\}$ and both asymptotic limits of $\\tilde v$ are Hamiltonian orbits. \n    %\nWe say $\\tilde \\mathbf{e} \\in \\tilde \\mathcal{X}^2$ if\n\\[\n\\tilde \\mathbf{e} +  (T_+ s + i 0) \\, \\mu_{+\\infty} +  (T_- s + i 0) \\, \\mu_{-\\infty} \\in W^{1,p,\\delta}_{(\\mathbb{C}, \\mathbb{C}; \\mathbb{C})}( \\mathbb{R} \\times S^1 \\setminus \\{ P \\}, \\mathbb{C} )\n\\]\n\n\\item[Case 3:] $\\Gamma = \\emptyset$ and the $+\\infty$ asymptotic limit of $\\tilde v$ is a Hamiltonian orbit,\nwhereas at $-\\infty$, it converges to an orbit cylinder over a Reeb orbit in $\\{ -\\infty \\} \\times Y$ (this is (2') in Proposition \\ref{Ansatz1}). \nWe say $\\tilde \\mathbf{e} \\in \\tilde \\mathcal{X}^3$ if \n\\[\n\\tilde \\mathbf{e} +  (T_+ s + i 0) \\, \\mu_{+\\infty} \\in W^{1,p\\delta}_{(\\mathbb{C}; \\mathbb{C})}( \\mathbb{R} \\times S^1, \\mathbb{C}).\n\\]\n\n\\end{itemize}\nWe then define $\\mathcal{X}^1, \\mathcal{X}^2, \\mathcal{X}^3$ to be the set of functions obtained as compositions of functions in $\\tilde \\mathcal{X}^1, \\tilde \\mathcal{X}^2, \\tilde \\mathcal{X}^3$ with the projection map $\\mathbb{C} \\to \\mathbb{R} \\times S^1$.\n\n    Notice that $\\tilde \\mathcal{X}^1, \\tilde \\mathcal{X}^2, \\tilde \\mathcal{X}^3$ are described as \naffine spaces modelled on $W^{1,p,\\delta} \\oplus \\mathbb{C}^k$, where $k=2$ in Cases 1 and 3, and $k=3$ in Case 2, so $\\mathcal{X}^i$ are Banach manifolds.\n\n\n\\begin{remark}\nThe summands $(T_\\pm s + i0) \\, \\mu_{\\pm\\infty}$ keep track of the fact that the Floer trajectory $\\tilde v=(b,v)$ and the pseudoholomorphic curve $\\tilde u=(a,u)$ in Proposition \\ref{Ansatz1} have different asymptotic behavior at $\\pm\\infty$, and that $e_1 = b-a$. \n\nObserve that in all three cases, we impose $\\delta$-decay to unspecified constants at $\\pm\\infty$ (and possibly $P$). \n\n\\end{remark}\n\n\nDenote the spaces of solutions to \\eqref{E:cylinder0} in $\\tilde \\mathcal{X}^j$\nby $\\tilde {\\mathcal{C}}^{j}(\\tilde v)$, where $j\\in \\{1,2,3\\}$, and let\n$\\mathcal C^j(\\tilde v)$ be the corresponding space of maps to $\\mathbb{R} \\times S^1$. We can\nwrite $\\tilde {\\mathcal C}^j(\\tilde v)$ as the preimage of zero under\nthe operator\n\\begin{align*}\n \\tilde \\mathcal{X}^j &\\to L^{p,\\delta}(T^*( \\mathbb{R}\\times S^1 \\setminus \\Gamma)) \\\\\n\\tilde \\mathbf{e} = (e_1,e_2) &\\mapsto d e_1 - de_2 \\circ i + h'(\\operatorname{e}^b) ds.\n\\end{align*}\nObserve that this map descends to a map $\\mathcal{X}^j \\to L^{p,\\delta}(T^*(\\mathbb{R} \\times S^1\n\\setminus \\Gamma))$. We then have that $\\mathcal C^j(\\tilde v)$ is the preimage\nof zero under this quotient operator.\n\n\nIt will be useful to consider the corresponding linearized operators. \nLet $\\mathbf{e} \\in \\mathcal C^j(\\tilde v)$. The corresponding linearized operator is \n\\begin{equation} \\label{E:cylLinear0}\n\\begin{aligned} \nD^j \\colon T_{\\bf e} \\mathcal{X}^j &\\to L^{p,\\delta}(T^*( \\mathbb{R} \\times S^1 \\setminus \\Gamma) ) \\\\\nE = E_1 + i E_2 &\\mapsto dE_1 - d E_2 \\circ i\n\\end{aligned}\n \\end{equation}\nwhere \nin Cases 1 and 3, we have $T_{\\bf e} \\mathcal{X}^j = W^{1,p,\\delta}_{(\\mathbb{C}; \\mathbb{C})}(\\mathbb{R}\\times S^1, \\mathbb{C})$ and \nand in Case 2, we have $T_{\\bf e} \\mathcal{X}^2 = W^{1,p,\\delta}_{(\\mathbb{C}, \\mathbb{C}; \\mathbb{C})}(\\mathbb{R}\\times S^1 \\setminus \\Gamma, \\mathbb{C})$.\n\nEvaluating \\eqref{E:cylLinear0} at $\\partial_s$ and $\\partial_t$ and rearranging terms, we get operators with values in $L^{p,\\delta}(\\mathbb{R}\\times S^1 \\setminus \\Gamma, \\mathbb{C})$ given by \n\\begin{equation} \\label{E:cylLinear0 s}\n\\frac{\\partial E}{\\partial s} + i \\frac{\\partial E}{\\partial t}.\n\\end{equation} \nBy an abuse of notation, denote these operators also by $D^j$. Observe that the\nasymptotic operators to $D^j$ at all punctures are given by \n$-i\\frac{d}{dt}$.\n\n\n\n\\begin{lemma} \\label{L:cylTransversality1}\nFix $\\operatorname{e} \\in \\mathcal C^{j}(\\tilde v)$, for $j\\in \\{1,2,3 \\}$. Then, the linearized operator $D^{j}$ has index 2, is surjective and its kernel includes the span of the generator of the $S^1$-action on the target $\\mathbb{R}\\times S^1$.\nAs a consequence, the space $\\mathcal C^j(\\tilde v)$ has virtual dimension 2. \n\\end{lemma}\n\n\\begin{proof}\nSince the operator $D^j$ in \\eqref{E:cylLinear0} is just the standard Cauchy--Riemann operator, a more elementary proof of this result is possible, but we illustrate here the methods also used in the more technical setting of the next section. \n\nWe start with the Cases $j=1$ or $3$. Fix ${\\bf e}$ as in the statement. Recall that the asymptotic operator is $A = - i\\frac{d}{dt}$ at all punctures. \nWe need to compute the Conley--Zehnder index of the perturbed asymptotic operator $A+\\delta$, which is $-1$ by Corollary \\ref{C:CZ computation}. \nTherefore using the fact that all vector spaces in $\\mathbf{V}$ are the kernels of the corresponding asymptotic operators, which we identified with $\\mathbb{C}$, Theorem \\ref{T:RiemannRochMB} implies that \n$$\n\\operatorname{Ind} D^{j} = (-1+2)-(-1) = 2.\n$$\nFollowing the notation of Wendl \\cite{WendlAutomatic}*{Equation (2.5)}, we have \n$$\nc_1(E,l,\\textbf{A}_\\Gamma) = \\frac{1}{2}(2-2) = 0 < 2 = \\operatorname{Ind} \\widehat D^{j}\n$$\nand surjectivity and the statement about the kernel follow from \\cite{WendlAutomatic}*{Proposition 2.2}\ntogether with \\cite{DiogoLisiSplit}*{Lemma 5.20}. \n\n\nCase 2 is slightly different, since there is an additional puncture at $P$. Its asymptotic operator is again $A$. The Euler characteristic of the domain is now $-1$. \nTherefore, \n$$\n\\operatorname{Ind} D^{2} = -1 + \\big((-1+2)-(-1)-(-1)\\big) = 2,\n$$\n$$\nc_1(E,l,\\textbf{A}_\\Gamma) = \\frac{1}{2}(2-2) = 0 < 2 = \\operatorname{Ind} \\widehat D^{2}\n$$\nand we can once more apply \\cite{WendlAutomatic}*{Proposition 2.2} together with\n\\cite{DiogoLisiSplit}*{Lemma 5.20}. \n\\end{proof}\n\n\n\\begin{lemma} \\label{smooth_solutions_Ansatz0}\n    Let $\\mathbf{e} \\in \\mathcal C^j$, $j=1, 2, 3$. Then, $\\mathbf{e}$ admits a smooth extension \n    $\\mathbb{R} \\times S^1 \\to \\mathbb{R} \\times S^1$ that satisfies \n    Equation \\eqref{E:cylinder0}.\n\\end{lemma}\n\\begin{proof}\n    Let $\\mathbf{e} \\in \\mathcal C^j$ be a solution. Then, in Cases 1 and 3, the result\n    follows immediately from elliptic regularity and the fact that $h'(\\operatorname{e}^b)$\n    is a smooth function on the cylinder.\n\n    For Case 2, observe that $h'(\\operatorname{e}^r)$ vanishes for $r << 0$. \n    Therefore, $h'(\\operatorname{e}^b)$ vanishes in a neighbourhood of the puncture $P$,\n    and thus extends as a smooth function on the cylinder. Furthermore, \n    $\\mathbf{e}$ is holomorphic in this same punctured neighbourhood, and the\n    requirement it be in $W^{1,p, \\delta}_\\mathbb{C}( \\mathbb{R}^- \\times S^1)$ forces it to\n    have a continuous and thus smooth extension across the puncture.\n\n    It follows therefore that $\\mathbf{e}$ is smooth in Case 2 as well.\n\\end{proof}\n\n\\begin{lemma} \\label{L:aprioriMarker1}\nLet $\\tilde \\mathbf{e} = (e_1, \\tilde e_2) \\colon \\mathbb{R}\\times S^1 \\to \\mathbb{C}$ \nbe a lift of a solution $\\operatorname{e} =(e_1, e_2) \\in\n\\mathcal C^j$. \nThen, for every $t\\in S^1$\n\\[\n\\lim_{s\\to -\\infty } \\tilde e_2(s,t) = \\lim_{s\\to \\infty } \\tilde e_2(s,t).\n\\]\n\\end{lemma}\n\\begin{proof}\n    By Lemma \\ref{smooth_solutions_Ansatz0}, it follows that $\\tilde \\mathbf{e}$ is\n    smooth and satisfies the equation\n    \\[\n        de_1 - d \\tilde{e_2} \\circ i + h'(\\operatorname{e}^b) ds =0.\n    \\]\n\nFor each $c >> 1$, let $S_c := [-c,c] \\times S^1 \\subset \\mathbb{R} \\times S^1$. \nThen, \n\\begin{align*}\n\\int_{\\partial S_c} \\tilde e_2 \\, dt \n&= \\int_{S_c} d\\tilde e_2 \\wedge dt = \\int_{S_c} de_2 \\wedge dt \\\\\n&= \\int_{S_c} de_2 \\circ i \\wedge dt\\circ i \\\\\n&= \\int_{S_c} ( de_1 + h'(\\operatorname{e}^{b})ds ) \\wedge ds \\\\\n&= \\int_{S_c} de_1 \\wedge ds \\\\\n&= \\int_{\\partial S_c} e_1 \\, ds =0\n\\end{align*}\nsince $ds|_{\\partial S_c} = 0$. Hence, we obtain \n\\[\n\\int_{\\{c\\}\\times S^1} \\tilde e_2(c, t) dt \n\t = \\int_{\\{-c\\}\\times S^1} \\tilde e_2(-c, t) dt.\n\\]\nThe result now follows from taking a limit as $c \\to +\\infty$.\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{L:inhomoCRexistence}\nEquation \\eqref{E:cylinder0} has a solution in $\\mathcal{X}^j$ for $j\\in \\{1,2,3\\}$, which is unique up to adding a constant in the target $\\mathbb{R}\\times S^1$.\n\\end{lemma}\n\n\\begin{proof}\nWe will instead prove the result for the lifts $\\tilde \\mathbf{e} \\in \\tilde \\mathcal{X}^j$, solving the same differential equation.\nNotice that any solution $\\mathbf{e}$ admits a $\\mathbb{Z}$ family of lifts. It suffices then to show that the solutions\n$\\tilde \\mathbf{e} = (e_1, \\tilde e_2)$ are unique up to adding a constant in $\\mathbb{C}$.\n\n\nWrite \\eqref{E:cylinder0} as an inhomogeneous Cauchy--Riemann equation on $\\mathbb{R}\\times S^1 \\setminus \\Gamma$:\n\\begin{equation}\\label{E:inhomoCR}\nd{e}_1 - d\\tilde{e}_2 \\circ i = -h'(\\operatorname{e}^b)ds.\n\\end{equation}\n\n\nWe will consider Cases 1, 2 and 3. Let $\\nu_j: \\mathbb{R} \\to \\mathbb{R}$ be smooth functions such that \n\\begin{equation*}\n\\begin{cases}\n \\nu_j(s) = T_- s \\text{ for } s<<0 \\text{ and } \\nu_j(s) = T_+ s \\text{ for } s>>0 & j = 1, 2 \\\\\n \\nu_j(s) = 0 \\text{ for } s<<0 \\text{ and } \\nu_j(s) = T_+ s \\text{ for } s>>0 & j = 3 \\\\\n\\end{cases}\n\\end{equation*}\n  \nInstead of solving equation \\eqref{E:inhomoCR}, we will consider the equivalent problem of finding ${\\bf g} = (g_1, g_2 ) = (e_1 + \\nu_j, \\tilde{e}_2)$ such that%\n\\[\n{\\bf g} \\in \n\\begin{cases}\n W^{1,p,\\delta}_{(\\mathbb{C}; \\mathbb{C})}(\\mathbb{R}\\times S^1, \\mathbb{C}), \\quad &\\text{in cases }  j = 1, 3 \\\\\n W^{1,p,\\delta}_{(\\mathbb{C}, \\mathbb{C}; \\mathbb{C})}(\\mathbb{R}\\times S^1 \\setminus \\Gamma, \\mathbb{C}),  &\\text{in case } j = 2\n\\end{cases}\n\\]\nsolves the equation\n\\begin{equation} \\label{E:inhomoCRg}\ndg_1 - dg_2 \\circ i =-h'(\\operatorname{e}^b) ds + d\\nu_j. \n\\end{equation}\nNotice that $\\mathbf{g}$ is taken in the function space $T_{\\mathbf{e}}\\mathcal{X}^j$.\n\n\n\nLet $K = -h'(\\operatorname{e}^b) ds + d\\nu_j $. \nFirst, recall that $b$ converges exponentially fast to $b_\\pm$. \nIn our setting of Morse--Bott degeneracy, exponential convergence follows from \n\\cite{BourgeoisThesis} and \\cite{SFTcompactness}*{Appendix A}. Then, by\n\\cite{SiefringAsymptotics}, the exponential rate of convergence is given by\nthe spectral gap, which is larger than $\\delta$ by construction.\n\nIn Cases 1 and 3, the fact that $K \\in L^{p,\\delta}(T^*(\\mathbb{R}\\times S^1))$ follows immediately from the exponential rate of convergence of $b$ at $\\pm \\infty$ to $b_\\pm$, and $T_\\pm = h'(\\operatorname{e}^{b_\\pm})$. \n\nIn Case 2, the fact $K \\in L^{p, \\delta}(T^*(\\mathbb{R} \\times S^1 \\setminus \\{ P \\})$ follows from this and also from Equation \\eqref{E:ds_is_exponential} so that $\\nu'_2(s)ds$ has exponential decay near $P$.\n\nAccording to Lemma \\ref{L:cylTransversality1}, the linearized operators \\eqref{E:cylLinear0 s} have index 2 and are surjective.\nTheir kernels have dimension 2 and consist of constant functions. \nLet now ${\\bf g} \\in T_{\\bf e} \\mathcal{X}^j$, with $T_{\\bf e} \\mathcal{X}^j$ defined as in \\eqref{E:cylLinear0}, be such that \n$$\ndg_1 - dg_2 \\circ i = K.\n$$\n\nAny two solutions differ then by an element of the kernel of this linear operator, \nand thus differ by a constant.\nFrom this, we obtain the same statement for $\\tilde \\mathbf{e}$ and hence for $\\mathbf{e}$.\n \\end{proof}\n \n\n\\begin{proof}[Proof of Proposition \\ref{Ansatz1}]\n Using the fact that $X_H = h'(\\operatorname{e}^r) R$, the Floer equation \\eqref{Floer eq Y} satisfied by $\\tilde v = (b,v)$ is equivalent to:\n\\begin{equation}\n\\begin{cases}\ndb - v {}^{*} \\alpha \\circ i + h'(\\operatorname{e}^b) ds = 0 \\\\\nd\\pi_Y \\circ dv + J \\circ d\\pi_Y \\circ dv \\circ i = 0\n\\end{cases}\n\\end{equation}\nBy Lemma \\ref{L:inhomoCRexistence}, we have a solution \n${\\bf e}$ to \\eqref{E:cylinder0}. Now, $a := b - e_1$ and $u := \\phi_R^{-e_2} \\circ v$ are such that\n\\begin{align*}\nda - u {}^{*} \\alpha \\circ i \n\t&= db - de_1 - v {}^{*} \\alpha \\circ i - de_2 \\circ i\\\\\n\t&=db - v {}^{*} \\alpha \\circ i + h'(\\operatorname{e}^b) ds\\\\\n\t& = 0,\\\\\nd\\pi_Y \\circ du + J \\circ d\\pi_Y \\circ du \\circ i \n\t&= d\\pi_Y \\circ dv + J \\circ d\\pi_Y \\circ dv \\circ  i \\\\\n\t& = 0.\n\\end{align*}\nSo, $\\tilde u:=(a,u)$ is $J_Y$-holomorphic. The asymptotic constraints on the spaces $\\mathcal{X}^i$ are such that $\\tilde u$ converges to Reeb orbits at $\\pm\\infty$ (and at $P$, if $\\Gamma = \\{P \\}$).\nThe uniqueness statement in Lemma \\ref{L:inhomoCRexistence}, together with Lemma \\ref{L:aprioriMarker1}, imply that there is an $\\mathbb{R}$-family of solutions ${\\bf e} = (e_1,e_2)\\in\\mathcal{X}^j$, such that \n$$\n\\lim_{s\\to \\pm\\infty} e_2(s,t) = 0\\in S^1,\n$$\nwhere distinct solutions differ by adding a constant to $e_1$.\nThis implies the existence statements in Proposition \\ref{Ansatz1}, and uniqueness up to an $\\mathbb{R}$-action.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{From pseudoholomorphic curves to Floer cylinders} \\label{S:ansatz_holo_to_Floer}\n\n\\label{holo_to_Floer}\n\nProposition \\ref{Ansatz1} implies that every split Floer cylinder $\\tilde v \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$ has an underlying $J_Y$-holomorphic curve $\\tilde u \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$. \nWe will show that every $J_Y$-holomorphic curve $\\tilde u \\colon \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$ underlies a split Floer cylinder.\n\n\n\\begin{proposition}\\label{Ansatz2}\n Let $\\Gamma\\subset \\mathbb{R}\\times S^1$ be either the empty set or the singleton $\\{P\\}$. Suppose $\\tilde u = (a, u)\\colon \\mathbb{R}\\times S^1 \\setminus \\Gamma \\rightarrow \\mathbb{R} \\times Y$\nis a non-constant, finite Hofer energy $J_Y$-holomorphic curve, with a negative\npuncture at $\\Gamma$ if $\\Gamma \\ne \\emptyset$. Let $\\gamma_\\pm$ denote the\nReeb orbits to which $u$ is asymptotic at $\\pm \\infty$.\nThen, there exists a pair $(\\tilde v, {\\bf e})$ consisting of a map $\\tilde v = (b, v)\\colon \\mathbb{R}\\times S^1 \\setminus \\Gamma \\rightarrow\n\\mathbb{R} \\times Y$ and of a map ${\\bf e} = (e_1, e_2)\\colon \\mathbb{R}\\times S^1 \\rightarrow \\mathbb{R} \\times S^1$, with the\nfollowing properties:\n\\begin{enumerate}\n\\item The map $\\tilde v$ solves the Floer equation \\eqref{Floer eq Y}.\n\n\n\\item At each of $\\pm \\infty$, the map $\\tilde v$ converges to the Hamiltonian orbit to which \n    $\\gamma_\\pm$ corresponds.\n\\item If $\\Gamma = \\{P\\}$, then $v$ is asymptotic to the same unparameterized Reeb orbit that $u$ converges to at $P$.\n\\item The map ${\\bf e} \\colon \\mathbb{R}\\times S^1\\setminus \\{P\\} \\to \\mathbb{R} \\times S^1$ satisfies the equation\n\t\\begin{equation} \\label{E:cylinder1}\n\tde_1 -de_2 \\circ i +  h'(\\operatorname{e}^{a+e_1}) ds = 0.\n\t\\end{equation}\n\\item The original $J_Y$-holomorphic curve is given by \n\\[\n(a, u) = (b - e_1, \\phi_R^{-e_2} \\circ v).\n\\] \n\\end{enumerate}\n\n\\noindent \nThe pair $(\\tilde v,{\\bf e})$ with these properties is unique.\n\nIn the case that $\\Gamma = \\emptyset$ and $\\tilde u(s,t)=(Ts + c,\\gamma(Tt))$ is a trivial cylinder, for some Reeb orbit $\\gamma$ of period $T$, there also exists a pair $(\\tilde v, \\bf e)$ as above, except replacing property (2) by\n\\begin{enumerate}\n \\item[(2')] $\\tilde v$ converges to a Hamiltonian orbit whose underlying Reeb orbit is $\\gamma$ at $+\\infty$, and $v$ converges to $\\gamma$ at $-\\infty$. \n\\end{enumerate}\nThe pair $(\\tilde v,{\\bf e})$ is now unique only up to replacing $b(s,t)$ with\n$\\overline b(s,t)=b(s+s_0,t)$, for some constant $s_0\\in \\mathbb{R}$, and replacing $e_1$\nwith $\\overline{e}_1 = \\overline{b}-a$. \n\n\n\\end{proposition}\n\n\n\nEquation \\eqref{E:cylinder1} has a solution ${\\bf e}= (e_1,e_2)$ iff there is a solution ${\\bf f} = (f_1, f_2) = (e_1 + a,e_2)$ to the equation \n\\begin{equation} \\label{E:cylinder}\n\\begin{aligned}\n\t{\\bf f}=&(f_1,f_2) \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma  \\rightarrow \\mathbb{R} \\times S^1 \\\\\n\t&df_1 - df_2 \\circ i + h'(\\operatorname{e}^{f_1}) ds - da = 0,\n\\end{aligned}\n\\end{equation}\nso we will study this equation instead. As we will see below equation \\eqref{E:cylLinear2 s}, the linearized operator associated to \\eqref{E:cylinder} has asymptotic operators of the form \n$$\nA_C = -i\\frac{d}{dt} - \\begin{pmatrix}\n                       C & 0 \\\\\n                       0 & 0\n                      \\end{pmatrix} \\colon W^{1,p}(S^1,\\mathbb{C}) \\to L^{p}(S^1,\\mathbb{C})\n$$\nfor some $C\\geq 0$. Lemma \\ref{L:spectrum} implies that the kernel of $A_C$ has real dimension 2 if $C = 0$ (given by constant functions $c_1 + i c_2 \\in \\mathbb{C}$) and 1 if $C\\neq 0$ (given by constant functions $i c_2 \\in i\\mathbb{R}$). \n\nRecall that we write \n$W^{1,p,\\delta}_{(0, \\mathbb{C}; i\\mathbb{R})}(\\mathbb{R} \\times S^1 \\setminus \\{ P \\}, \\mathbb{C})$ to denote\nmaps converging exponentially fast to $0$ at $-\\infty$, to an arbitrary complex constant at $P$ and to an imaginary constant at $+\\infty$, and write $W^{1,p, \\delta}_{(0, i\\mathbb{R})}(\\mathbb{R} \\times S^1, \\mathbb{C})$ to denote maps converging exponentially fast to $0$ at $-\\infty$ and to an arbitrary imaginary constant at $+\\infty$.\nRecall that we have chosen cylindrical coordinates near $P$, $\\varphi \\colon\n(-\\infty, -1] \\times S^1 \\to \\mathbb{R} \\times S^1$ by $\\varphi(\\rho, \\theta) = P +\n\\operatorname{e}^{2\\pi(\\rho + i \\theta)}$, and also a bump function $\\mu_P$ that is \nidentically $1$ near $P$ and supported in the image of $\\varphi$.\n\n\nSimilarly to the previous section, we will study solutions to (\\ref{E:cylinder}) in $W^{1,p}_{\\text{loc}}(\\mathbb{R}\\times S^1\\setminus \\Gamma, \\mathbb{R}\\times S^1)$ satisfying three types of asymptotic conditions.%\n\nNotice that if $\\tilde u$ is a $J_Y$-holomorphic [punctured] cylinder as in\nProposition \\ref{Ansatz2}, it converges at $\\pm \\infty$ \nto closed Reeb orbits in $\\mathbb{R} \\times Y$, of period $T_\\pm$. \nLet $b_\\pm \\in \\mathbb{R}$ be the unique solutions to $T_\\pm =\nh'(\\operatorname{e}^{b_\\pm})$. (Recall that such solutions exist and are unique by the\nadmissibility conditions imposed on the Hamiltonian in Definition \\ref{def:J_shaped}. Furthermore, we then have $b_\\pm > \\log 2$.)\n\n\\begin{itemize}\n\\item[Case 1:] $\\Gamma = \\emptyset$. Define the space $\\mathcal{Y}^1$ by the condition $\\mathbf{f} \\in \\mathcal{Y}^1$ if it admits a lift $\\tilde \\mathbf{f} \\in W^{1,p}_\\text{loc}(\\mathbb{R} \\times S^1, \\mathbb{C})$ with \n\\[\n\\tilde{\\mathbf{f}} - (b_+ + i 0) \\, \\mu_{+\\infty} -  (b_- + i 0) \\, \\mu_{-\\infty} \\in W^{1,p,\\delta}_{(0, i\\mathbb{R})}(\\mathbb{R} \\times S^1, \\mathbb{C})\n\\]\n\n\n\\item[Case 2:]\n $\\Gamma = \\{ P\\}$. Define the space $\\mathcal{Y}^2$ by the condition that $\\mathbf{f} \\in \\mathcal{Y}^2$ if it admits a lift $\\tilde \\mathbf{f} \\in W^{1,p}_\\text{loc}(\\mathbb{R} \\times S^1, \\mathbb{C})$ with \n \\[\n \\tilde \\mathbf{f} -  (b_+ + i 0) \\, \\mu_{+\\infty} -  (b_- + i 0) \\, \\mu_{-\\infty} - \\mu_P \\, a \\in W^{1, p, \\delta}_{(0, \\mathbb{C}; i\\mathbb{R})}( \\mathbb{R} \\times S^1 \\setminus \\{ P \\}, \\mathbb{C}).\n \\]\n\n\n\\item[Case 3:] $\\Gamma = \\emptyset$. Define the space $\\mathcal{Y}^3$ by the condition that $\\mathbf{f}$ admits a lift $\\tilde \\mathbf{f}$ with\n\\[\n\\tilde\\mathbf{f} -  (b_+ + i 0) \\, \\mu_{+\\infty} \\in W^{1,p, \\delta}_{(\\mathbb{C}; i\\mathbb{R})}( \\mathbb{R} \\times S^1, \\mathbb{C})\n\\]\n(this corresponds to (2') in Proposition \\ref{Ansatz2}).\n\n\n\\end{itemize}\nAs before, we notice that the spaces $\\mathcal{Y}^i$ are Banach manifolds modelled\n    on $W^{1,p,\\delta} \\oplus \\mathbb{R}^k$, where $k$ depends on which of the 3 cases\nwe are considering.\n\n\\begin{remark}\n As in the previous section, the first part of Proposition \\ref{Ansatz2}\n is related to Cases 1 and 2 (see Figures \\ref{case_1_fig} and\n \\ref{case_2_fig}, respectively).  The last part of the Proposition,\n where $\\tilde u$ is assumed to be a trivial orbit cylinder, is\n related to Case 3 (see Figure \\ref{case_3_fig}).\n\n It is important to observe that the asymptotic conditions imposed in this section are different from the ones that were imposed in the previous section. This will have an effect on the indices of the relevant Fredholm problems. \nJust to give an example, in Case 1 we are considering functions that converge at\n$+\\infty$ to an unspecified constant in $\\{b_+\\}\\times S^1 \\subset \\mathbb{R}\\times S^1$, \nand that converge at $-\\infty$ to $(b_-,0)\\in \\mathbb{R}\\times S^1$. \n \\end{remark}\n\nWe will consider auxiliary parametric versions of Equation\n\\eqref{E:cylinder} in order to deform the solutions to \\eqref{E:cylinder}\nto solutions of a PDE for which we can explicitly describe the\nsolution space.  The deformation will give an identification of the\nsolution spaces.\n\nFor Case 1, we consider the map\n\\begin{equation} \\label{E:parametric operator1}\n\\begin{aligned}\n\\mathcal F \\colon \\mathcal Y^1 \\times [0,1] &\\to L^{p,\\delta}(\\mathbb{R}\\times S^1,(\\mathbb{R}^2)^*) \\\\\n({\\bf f},\\tau) = ((f_1,f_2),\\tau) &\\mapsto df_1 - df_2 \\circ i + h'(\\operatorname{e}^{f_1}) ds - \\tau da - (1-\\tau) d\\nu\n\\end{aligned}\n\\end{equation}\nwhere $\\nu \\colon \\mathbb{R}\\to \\mathbb{R}$ is a smooth function such that $\\nu(s) = T_- s$ for $s <<0$ and $\\nu(s) = T_+ s$ for $s >>0$ (just like $\\nu_1$ in the proof of Lemma  \\ref{L:inhomoCRexistence}). We will study solutions to equation\n\\begin{equation} \\label{E:parametric cylinder1}\n\\begin{aligned}\n\\mathcal F({\\bf f},\\tau) = 0,\n\\end{aligned}\n\\end{equation}\nwhich interpolates between (\\ref{E:cylinder}) and the equation\n\\begin{equation} \\label{E:deformed cylinder1}\ndf_1 - df_2 \\circ i + h'(\\operatorname{e}^{f_1}) ds - d\\nu= 0.\n\\end{equation}\nThe latter has the advantage that $d\\nu$ is independent of $t$. Denote the space of solutions to (\\ref{E:parametric cylinder1}) in $\\mathcal Y^1$ for fixed $\\tau$ by $\\mathcal C^{1,\\tau}(\\tilde u)$.  \n\n\nIn Case 2, it will be more convenient to study functions $\\gg=(g_1,g_2)=(f_1 - \\mu_P \\, a, f_2)$.\nWe think of ${\\bf g}$ as an element of the subspace $\\hat{ \\mathcal{Y}^2}\n\\subset W^{1,p}_\\text{loc}(\\mathbb{R}\\times S^1\\setminus \\Gamma,\\mathbb{R}\\times S^1)$, with\n$\\gg \\in \\hat \\mathcal{Y}^2$ if it has a lift $\\tilde \\gg$ with\n\\[\n\\tilde \\gg -  (b_+ + i 0) \\, \\mu_{+\\infty} -  (b_- + i 0) \\, \\mu_{-\\infty} \\in W^{1,p,\\delta}_{(0, \\mathbb{C}; i\\mathbb{R})}( \\mathbb{R} \\times S^1 \\setminus \\{ P \\}, \\mathbb{C}).\n\\]\nWe will consider the map \n\\begin{equation}  \\label{E:parametric operator2}\n\\begin{aligned}\n\\mathcal G \\colon \\hat{\\mathcal{Y}^2} \\times [0,1] &\\to L^{p,\\delta}(\\mathbb{R}\\times S^1 \\setminus \\Gamma,(\\mathbb{R}^2)^*) \\\\\n({\\bf g},\\tau) = ((g_1,g_2),\\tau) &\\mapsto dg_1 - dg_2 \\circ i + h'(\\operatorname{e}^{g_1+\\tau\n\\mu_P a}) ds - \\tau d((1-\\mu_P)a) - (1-\\tau) d\\nu\n\\end{aligned}\n\\end{equation}\nwhere $\\nu \\colon \\mathbb{R}\\to \\mathbb{R}$ is as above. \n\nWe will study solutions to \n\\begin{equation} \\label{E:parametric cylinder2}\n\\begin{aligned}\n\\mathcal G({\\bf g},\\tau) = 0.\n\\end{aligned}\n\\end{equation}\nThis equation interpolates between the equivalent of equation \\eqref{E:cylinder} for $\\bf g$:\n$$\ndg_1 - dg_2 \\circ i + h'(\\operatorname{e}^{g_1+ \\mu_P a}) ds - d((1-\\mu_P)a) = 0\n$$\nand equation \\eqref{E:deformed cylinder1} applied to $\\bf g$:\n$$\ndg_1 - dg_2 \\circ i + h'(\\operatorname{e}^{g_1}) ds - d\\nu= 0,\n$$\nThe latter has again the advantage that $d\\nu$ is independent of $t$. Denote the space of solutions to (\\ref{E:parametric cylinder2}) in $\\hat{\\mathcal Y^2}$ for fixed $\\tau$ by $\\mathcal C^{2,\\tau}(\\tilde u)$. \n\nAlso, denote the space of solutions to (\\ref{E:cylinder}) in $\\mathcal Y^3$ by $\\mathcal C^{3}(\\tilde u)$.  In Case 3, we will study solutions directly and do not require any deformation argument.\n\nWe need the following smoothness in order to apply the implicit function\ntheorem, which we will need in the proof of Proposition \\ref{Ansatz2}.\n\\begin{lemma} \\label{L:smooth parameter}\n The nonlinear operators $\\mathcal F$ and $\\mathcal G$ are $C^1$ in \n $\\mathcal Y^1 \\times [0,1] \\to L^{p,\\delta}(\\mathbb{R}\\times S^1,(\\mathbb{R}^2)^*)$ \n and \n $\\hat{\\mathcal Y^2} \\times [0,1] \n \\to  L^{p,\\delta}(\\mathbb{R}\\times S^1\\setminus \\{ P \\},(\\mathbb{R}^2)^*)\n $, respectively. \n\\end{lemma}\n\\begin{proof}\n    Notice that $\\mathcal F( \\mathbf{f}, \\tau)$ is a Floer-type differential\n    operator with an inhomogeneous term depending on $\\tau$. \n    The claimed result holds for $\\mathcal F$ since $-da + \\nu'(s) \\, ds$ is in\n    $L^{p, \\delta}$, which follows from the fact that $a(s,t)$ converges \n    exponentially fast, together with its derivatives, to $T_\\pm s + c_\\pm$ at $\\pm \\infty$. \n\n    In order to show the result for $\\mathcal G( \\mathbf{g}, \\tau)$, it suffices\n    to show that the following map is $C^1$, since the remaining terms in $\\mathcal G$ are $C^1$ by standard arguments.\n    \\begin{equation} \\label{E:smoothness}\n    \\begin{aligned}\n       \\mathcal H \\colon W^{1,p, \\delta}_{(0, \\mathbb{C}; i\\mathbb{R})}( \\mathbb{R} \\times S^1 \\setminus \\Gamma, \\mathbb{C}) \n                    &\\to L^{p, \\delta}( T^*(\\mathbb{R} \\times S^1 \\setminus \\Gamma))  \\\\\n        (g_1, g_2, \\tau) &\\mapsto h'( \\operatorname{e}^{g_1 + \\tau \\mu_P a}) \\, ds.\n    \\end{aligned}\n    \\end{equation}\n    First, we restrict our attention to a $W^{1,p,\\delta}_{(0, \\mathbb{C};\n    i\\mathbb{R})}$-ball of radius $R$. We will show the map is $C^1$ smooth\n    on each such ball by showing its second derivative is bounded\n    by a constant that depends on $R$. Exhausting the total space\n    by such balls will prove the result.\n\n    Notice that the $W^{1,p,\\delta}$ bound implies a uniform $C^0$\n    bound on $g_1$. Also notice that $\\mu_P a$ is bounded above.\n    Thus, $g_1 + \\tau \\mu_Pa$ is bounded above (where the bound\n    depends on $R$). \n    Hence, there exists a $C$ (depending on $R$)\n    so that \n    \\begin{align*}\n        &0 \\le h'(\\operatorname{e}^{g_1+ \\tau \\mu_P a}) \\le C,\\\\\n        &0 \\le h''(\\operatorname{e}^{g_1+ \\tau \\mu_P a})\\operatorname{e}^{g_1+ \\tau \\mu_P a} \\le C\\\\\n        &| h'''(\\operatorname{e}^{g_1+ \\tau \\mu_P a})\\operatorname{e}^{2(g_1+ \\tau \\mu_P a)}|\\le C.  \n    \\end{align*}\n\n    The weak derivative of $\\mathcal H$ is given by \n    \\[\n    d{\\mathcal H}(\\mathbf{g}, \\tau)(\\mathbf G, \\tau') \n    \t= G_1 h''(\\operatorname{e}^{g_1+ \\tau \\mu_P a})\\operatorname{e}^{g_1+ \\tau \\mu_P a} \\, ds \n\t\t\t+ \\tau' h''(\\operatorname{e}^{g_1+ \\tau \\mu_P a})\\operatorname{e}^{g_1+ \\tau \\mu_P a} \\mu_P a \\, ds\n    \\]\n    where $\\mathbf{G} = (G_1, G_2)$ is a variation of $\\mathbf{g} = (g_1,\n    g_2)$ and $\\tau'$ is a variation of $\\tau$.\n    %\nNotice that this expression is in $W^{1,p,\\delta}$ away from\n    $P$. Near $P$, in the cylindrical coordinates $(\\rho, \\theta)$,\n    we note that this is bounded by \n    \\[\n    C (|G_1| + |\\tau'| (T_P|\\rho| + D)) \\operatorname{e}^{2\\pi \\rho} (|d\\rho| + |d\\theta|) \\]\n    for suitable constants $C, D$ and where $T_P$ is the period of the Reeb orbit to \n    which $(a, u)$ converges at $P$. \n    From this, we obtain that $d{\\mathcal H}$ is uniformly bounded on each ball of radius $R$, \n    showing $\\mathcal H$ is Lipschitz on each ball of radius $R$, and hence $\\mathcal H$ is continuous.\n    \n    By a similar argument, we estimate the second derivative to show $C^1$ smoothness of $\\mathcal H$. \n    The $C^1$-smoothness of $\\mathcal G$ then follows.    \n\\end{proof}\n\nWe now discuss the linearized operators associated to $\\mathcal F$ and $\\mathcal G$, for fixed values of $\\tau$. In Case 1, writing a solution to \\eqref{E:parametric cylinder1} as ${\\bf f} = f_1 + i f_2$, the linearization of the operator $\\mathcal F^\\tau$ is\n\\begin{equation} \\label{E:cylLinear1}\n\\begin{aligned}\nD^{1,\\tau} \\colon T_{\\bf f}\\mathcal Y^1 &\\to L^{p,\\delta}(\\mathbb{R}\\times S^1, (\\mathbb{R}^2)^*) \\\\\nF = F_1 + iF_2 &\\mapsto dF_1 - dF_2 \\circ i + h''(\\operatorname{e}^{f_1}) \\operatorname{e}^{f_1} F_1 ds.\n\\end{aligned}\n\\end{equation}\nwhere $T_{\\bf f} \\mathcal Y^1 = W^{1,p,\\delta}_{\\mathbf{V}}(\\mathbb{R} \\times S^1, \\mathbb{C})$ with $ V_- = 0$ and $V_+ = i\\mathbb{R}$. \n\nIn Case 2, writing ${\\bf g} = g_1 + i g_2$ and again fixing $\\tau$, the linearized operator is \n\\begin{equation} \\label{E:cylLinear2} \n\\begin{aligned}\nD^{2,\\tau} \\colon T_{\\bf g} \\hat{\\mathcal Y^2} &\\to L^{p,\\delta}(\\mathbb{R}\\times S^1 \\setminus \\Gamma, (\\mathbb{R}^2)^*) \\\\\nG = G_1 + iG_2 &\\mapsto dG_1 - dG_2 \\circ i + h''(\\operatorname{e}^{g_1+ \\tau \\mu a}) \\operatorname{e}^{g_1+ \\tau \\mu a} G_1 ds.\n\\end{aligned}\n\\end{equation}\nwhere $T_{\\bf g}\\hat{\\mathcal Y^2} = W^{1,p,\\delta}_{\\mathbf{V}}(\\mathbb{R} \\times S^1 \\setminus \\Gamma, \\mathbb{C})$ with $V_- = 0$, $V_+ = i\\mathbb{R}$ and $V_P = \\mathbb{C}$.\n\nIn Case 3, the linearized operator\n$$\nD^3\\colon T_{\\bf f} \\mathcal Y^3 \\to L^{p,\\delta}(\\mathbb{R}\\times S^1, (\\mathbb{R}^2)^*), \n$$\nwhere $T_{\\bf f} \\mathcal Y^3 = W^{1,p,\\delta}_{\\mathbf{V}}(\\mathbb{R} \\times S^1, \\mathbb{C})$ with $V_- = \\mathbb{C}$ and $V_+ = i\\mathbb{R}$, is also given by \\eqref{E:cylLinear1} \n\nEvaluating \\eqref{E:cylLinear1} and \\eqref{E:cylLinear2} at $\\partial_s$ and $\\partial_t$ and rearranging, we get operators with values in $L^{p,\\delta}(\\mathbb{R}\\times S^1, \\mathbb{C})$, given by \n\\begin{equation} \\label{E:cylLinear1 s}\n\\frac{\\partial F}{\\partial s} + i \\frac{\\partial F}{\\partial t} + h''(\\operatorname{e}^{f_1}) \\operatorname{e}^{f_1} F_1\n\\end{equation} \nand \n\\begin{equation} \\label{E:cylLinear2 s}\n\\frac{\\partial G}{\\partial s} + i \\frac{\\partial G}{\\partial t} + h''(\\operatorname{e}^{g_1 + \\tau \\mu a}) \\operatorname{e}^{g_1 + \\tau \\mu a} G_1,\n\\end{equation} \nrespectively. \n\n\n\n\n\nThe asymptotic operators at $\\pm\\infty$ in Cases 1 and 2, and at $+\\infty$ in Case 3, are \n\\[\nA_\\pm := -i \\frac{d}{dt} \n- \n\\begin{pmatrix} h''(\\operatorname{e}^{b_\\pm}) \\operatorname{e}^{b_\\pm} \n& 0 \\\\\n0 &\n0  \n\\end{pmatrix} : W^{1,p}(S^1,\\mathbb{C})\\to L^{p}(S^1,\\mathbb{C})\n\\]\n\nIn Case 2, there is also an asymptotic operator at the augmentation puncture:\n\\[\nA_P := -i \\frac{d}{dt} : W^{1,p}(S^1,\\mathbb{C})\\to L^{p}(S^1,\\mathbb{C})\n\\]\n\nIn Case 3, we have the asymptotic operator at $-\\infty$ given by $A_-=-i\\frac{d}{dt}$.\n\nAll these asymptotic operators are self-adjoint, as partially defined operators $L^{2}(S^1,\\mathbb{C})\\to L^{2}(S^1,\\mathbb{C})$. \nTheir eigenvalues can be computed using Lemma \\ref{L:spectrum}. Note that they all have eigenvalue 0. For the \nremainder of this section, we will {fix $0 < \\delta < 2\\pi$} such that $\\delta < C_\\pm = h''(\\operatorname{e}^{b_\\pm}) \\operatorname{e}^{b_\\pm} > 0$ \nand $\\delta < \\frac{1}{2}\\left(- C_\\pm + \\sqrt{ C_\\pm^2 + 16 \\pi^2}\\right)$.\nSo, $\\delta$ is smaller than the absolute value of every non-zero eigenvalue of $A_\\pm$ and $A_P$. This $\\delta$ will \nbe used in the definitions of the relevant weighted Sobolev spaces. \n\n\n\n\n\n\n\\begin{lemma} \\label{L:cylTransversality}\nFix $\\tau\\in [0,1]$ and ${\\bf f} \\in \\mathcal C^{1,\\tau}(\\tilde u)$. Then, \nthe linearized operator $D^{1,\\tau}$ is Fredholm of index 0 and is an isomorphism. \n\nFix $\\tau\\in [0,1]$ and ${\\bf g} \\in \\mathcal C^{2,\\tau}(\\tilde\nu)$. Then, the linearized operator $D^{2,\\tau}$ is Fredholm of index 0\nand is an isomorphism.\n\nFor fixed ${\\bf f} \\in \\mathcal C^{3}(\\tilde u)$, $D^3$ is Fredholm of index\n2, is surjective and its kernel includes the span of the generator\nof the $S^1$-action on the target $\\mathbb{R}\\times S^1$.  \n\\end{lemma}\n\n\n\\begin{proof}\nThe proof has the same structure as that of Lemma \\ref{L:cylTransversality1}. \nWe start with Case 1. Fix $\\tau$ and ${\\bf f}$ as in the statement. \nCorollary \\ref{C:CZ computation} implies that the Conley--Zehnder indices of the perturbed asymptotic operators $A_\\pm +\\delta$ are $0$. \nRecall that the kernels of the asymptotic operators $A$ are identified with $i\\mathbb{R}$ and that the vector spaces in $\\mathbf V$ are $V_- =0$ and $V_+=i\\mathbb{R}$. \nTheorem \\ref{T:RiemannRochMB} now implies that\n$$\n\\operatorname{Ind} \\widehat D^{1,\\tau} = (0 + 1) - (0 + 1) = 0.\n$$\nFollowing the notation of Wendl \\cite{WendlAutomatic}*{Equation (2.5)}, we have \n$$\nc_1(E,l,\\textbf{A}_\\Gamma) = \\frac{1}{2}(0-2) = -1 < 0 = \\operatorname{Ind} \\widehat D^{1,\\tau}\n$$\nso $\\widehat D^{1,\\tau}$ is an isomorphism by \\cite{WendlAutomatic}*{Proposition 2.2}. \n\nIn Case 3, recall that the asymptotic operator at $-\\infty$ is now $A_-=-i\\frac{d}{dt}$. By Corollary \\ref{C:CZ computation}, the Conley--Zehnder index of the perturbed asymptotic operator $A_- +\\delta$ is $-1$. \nThe kernel of the asymptotic operator $A_-$ can be identified with $\\mathbb{C}$, and the vector spaces in $\\mathbf V$ are now $V_- =\\mathbb{C}$ and $V_+=i\\mathbb{R}$. \nBy Theorem \\ref{T:RiemannRochMB},\n$$\n\\operatorname{Ind} \\widehat D^3 = (0+1)-(-1+0) = 2,\n$$\n$$\nc_1(E,l,\\textbf{A}_\\Gamma) = \\frac{1}{2}(2-2) = 0 < 2 = \\operatorname{Ind} \\widehat D^3\n$$\nand $\\widehat D^{3}$ is surjective by \\cite{WendlAutomatic}*{Proposition 2.2}. \n\nIn Case 2, we have the additional puncture $P$, with asymptotic operator $A_P=-i\\frac{d}{dt}$, and the Euler charateristic of the domain is $-1$. \nWe now have $V_-=0$, $V_+=i\\mathbb{R}$ and $V_P =\\mathbb{C}$, so Theorem \\ref{T:RiemannRochMB} implies\n$$\n\\operatorname{Ind} \\widehat D^{2,\\tau} = -1 + (0+1) - (0+1)- (-1 + 0) = 0,\n$$\n$$\nc_1(E,l,\\textbf{A}_\\Gamma) = \\frac{1}{2}(0-2) = -1 < 0 = \\operatorname{Ind} \\widehat D^{2,\\tau}\n$$\nand we can apply once more \\cite{WendlAutomatic}*{Proposition 2.2}, to obtain \nthat $D^{2, \\tau}$ is an isomorphism. \n\\end{proof}\n\n\n\n\n\nWe now prove a result analogous to Lemma \\ref{L:aprioriMarker}.\n\\begin{lemma} \\label{L:aprioriMarker}\n\nLet $\\tau \\in [0,1]$.\n\nSuppose that \n$\\mathbf{f} \\in \\mathcal{Y}^1$, $\\gg \\in \\hat \\mathcal{Y}^2$ or $\\mathbf{f} \\in \\mathcal{Y}^3$ satisfy\n\\[ \\mathcal{F}(\\mathbf{f}, \\tau) = 0, \\quad \\mathcal {G}(\\gg, \\tau) = 0, \\text{ or } df_1 - df_2 \\circ i +h'(\\operatorname{e}^{f_1}) -da = 0, \\]\nrespectively.\n\nThen, the asymptotic constants satisfy\n$f_2(+\\infty, \\cdot) = f_2(-\\infty, \\cdot)$, $g_2(+\\infty, \\cdot) = g_2(-\\infty, \\cdot)$ \nand $f_2(+\\infty, \\cdot) = f_2(-\\infty, \\cdot)$.\n\nIn particular, in Cases 1 and 2, these asymptotic constants are $0$.\n\n\\end{lemma}\n\n\\begin{proof}\n\tThe proof is very similar to the proof of Lemma \\ref{L:aprioriMarker1}. \n\n\tNotice that if we have such a solution $\\mathbf{f}$, $\\gg$, the resulting function \n\tis smooth on the punctured cylinder, by standard elliptic regularity arguments.\n\n\tWe will prove the result for functions that solve the equation\n\t\\[\n\t\tdf_1 - df_2 \\circ i + h'(\\operatorname{e}^{f_1+x})\\, ds - \\tau \\, dy - (1-\\tau)\\, d\\nu = 0,\n\t\\]\n\ton $\\mathbb{R} \\times S^1 \\setminus \\Gamma$, \n\twhere $x, y \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma \\to \\mathbb{R}$ are \n        functions satisfying the following:\n\t\\begin{itemize}\n\t\t\\item[Case 1:] $x=0, y=a$, $\\Gamma = \\emptyset$;\n\t\t\\item[Case 2:] $x= \\tau \\mu_P a$, $y = (1-\\mu_P) a$, $\\Gamma = \\{ P \\}$;\n\t\t\\item[Case 3:] $\\tau =1$, $x=0$, $y = a$, $\\Gamma = \\emptyset$.\n\t\\end{itemize}\n\tWe assume furthermore that $f_2$ converges exponentially fast to a \n\tconstant at $\\pm \\infty$, and that $f_1$ converges exponentially fast to \n\ta constant at $P$ in Case 2.\n\n\tNotice in particular that if the cylinder has a puncture, \n\t$y$ vanishes in a neighbourhood of the puncture.\n\n\tFor each $c >> 1$, let $S_c := [-c,c] \\times S^1 \\setminus D_P(1\/c) \n\t\\subset \\mathbb{R} \\times S^1 \\setminus \\Gamma$, \n\twhere $D_P(\\epsilon)$ denotes a disk of radius $\\epsilon$ centered at $P$. \n\n\tWe then obtain (noticing that $d\\nu = \\nu'(s) \\, ds$)\n \\begin{align*}\n \\int_{\\partial S_c} \\tilde f_2 dt \n &= \\int_{S_c} d\\tilde f_2 \\wedge dt = \\int_{S_c} df_2 \\wedge dt \\\\\n &= \\int_{S_c} df_2 \\circ i \\wedge dt\\circ i \\\\\n &= \\int_{S_c} ( df_1 + h'(\\operatorname{e}^{f_1+x}) ds -\\tau dy - (1-\\tau) d\\nu) \\wedge ds \\\\\n &= \\int_{\\partial S_c} ( f_1 -\\tau y ) \\wedge ds\n \\intertext{\n This vanishes in Cases 1 and 3, and we are done. \n In Case 2, $\\partial S_c$ also contains a \n small loop around $P$. In that case, for $c$ sufficiently large,  we obtain:}\n &= \\int_{\\partial D_P(1\/c)} (f_1- \\tau y) \\, ds \\\\\n &= \\int_{\\partial D_P(1\/c)} f_1 \\, ds. \n \\end{align*}\n Notice that $\\int_{\\partial D_P(1\/c)} f_1 \\, ds$ decays like $\\frac{1}{c^{2\\pi\\delta+1}}$.\n\n The result now follows by taking the limit as $c \\to +\\infty$.\n\\end{proof}\n\n\n\n\n\nWe are now ready for the key techincal result of this section, which guarantees existence and uniqueness of solutions to our auxiliary equations. Existence of solutions will be obtained from reducing the problem to an ODE. \nUniqueness will follow from positivity of intersections of pseudoholomorphic curves in symplectic 4-manifolds.\n\n\\begin{lemma} \\label{L:countingSolutions1}\nThere is a unique solution to \\eqref{E:deformed cylinder1} in $\\mathcal Y^1$ and in $\\hat{\\mathcal Y^2}$.\n\n\nIf $a = Ts + c$, then there is an 2-parameter family of solutions to \\eqref{E:cylinder} in $\\mathcal Y^3$, parametrized by $\\mathbb{R}\\times S^1$. \n\n\n\n\n\\end{lemma}\n\\begin{proof}\nWe begin with Case 1. Equation \\eqref{E:deformed cylinder1} can be rewritten in coordinates as\n\\begin{equation}\\label{E:system}\n\\begin{cases}\n\\partial_s f_1 - \\partial_t f_2 + h'(\\operatorname{e}^{f_1}) - \\nu' = 0 \\\\\n\\partial_t f_1 + \\partial_s f_2 = 0\n\\end{cases}\n\\end{equation}\n\nLet us start by considering solutions where $f_2\\equiv k$ is constant, and $f_1(s,t)=f_1(s)$ is independent of $t$. Then, we get the $s$-dependent ODE\n$$\n\\frac{d f_1}{d s} + h'(\\operatorname{e}^{f_1})- \\nu' = 0. \n$$\nWrite it as\n$$\n\\frac{d f_1}{d s} = F(f_1,s)\n$$\nwhere $F(x,s) = - h'(\\operatorname{e}^x) + \\nu'(s)$. \n\\begin{claim}\nFor every $s_0\\in \\mathbb{R}$ and every $c_1\\in \\mathbb{R}$, there is a unique solution $f_1:\\mathbb{R}\\to \\mathbb{R}$ such that $f_1(s_0) = c_1$. \n\\end{claim}\nTo prove the claim, recall the assumptions that $h'(\\rho)\\geq 0$ and $h'(\\rho) = 0$ for $\\rho\\leq 2$. These and the asymptotic behavior of $\\nu$ imply that there are constants $C$ and $\\tilde C$ such that:\n\\begin{itemize}\n \\item $F(x,s)\\leq C$ for all $(x,s)$;\n \\item $F(x,s) \\geq \\tilde C$ for $x\\leq \\ln 2$, uniformly in $s$. \n\\end{itemize}\nThe claim now follows from standard existence and uniqueness theory for solutions to ODE's.\n\nThe facts that for $s<<0$ we have that $b_-$ is the only zero of the function $x \\mapsto F(x,s)$ and that $\\partial_x F(x,s)\\leq 0$ (a consequence of $h''(x) \\geq 0$) imply that there is a unique solution $\\tilde f_1$ of the ODE such that $\\lim_{s\\to -\\infty} \\tilde f_1(s) = b_-$, see Figure \\ref{F:tilde f_1}.\n\n\\begin{figure} \n\\begin{tikzpicture}[domain=0:6]\n %\n    \\draw[->] (-4,0) -- (4.1,0) node[right] {$s$};\n    \\draw[->] (0,-1) -- (0,4) node[above] {$\\tilde f_1(s)$};\n    \\draw [dashed, color=black] (0,3) -- (4,3);\n    \\draw [dashed, color=black] (-4,1) -- (0,1);\n    \\draw [dashed, color=black] (-2,-1) -- (-2,4.3);\n    \\draw [dashed, color=black] (2,-1) -- (2,4.3);\n    \\draw[line width=1.5pt, color=black] (-4,1) -- (-2,1);\n    \\draw[line width=1.5pt, color=black] (-2,1) ..  controls(-1,1) and (1,2.2) .. (2,2.594);\n    \\draw[line width=1.5pt,domain=2:4,smooth,variable=\\x,black] plot ({\\x},{3-3*exp(-\\x)});\n    \\node at (-.3,3) {$b_+$};\n    \\node at (.3,1) {$b_-$};\n    \\node at (-3.2,0.5) {$F\\geq 0$};\n    \\node at (-3.2,2) {$F\\leq 0$};\n    \\node at (-3.2,-0.5) {$\\nu = T_- s$};\n    \\node at (3.2,1.5) {$F\\geq 0$};\n    \\node at (3.2,3.5) {$F\\leq 0$};\n    \\node at (3.2,-0.5) {$\\nu = T_+ s$};\n     %\n  %\n %\n  %\n  %\n\\end{tikzpicture}\n\\caption{The function $\\tilde f_1$}\n\\label{F:tilde f_1}\n\\end{figure}\n\n\nNow, ${\\bf f}^0(s,t)=(\\tilde f_1(s),0)$ is a solution to \\eqref{E:system} in $\\mathcal Y^1$. The fact that it has the correct exponential convergence properties at $\\pm \\infty$ follow from $\\tilde f_1$ solving the ODE above. We want to show that there are no other solutions to \\eqref{E:system} in $\\mathcal Y^1$. The proof will be an application of positivity of intersections of pseudoholomorphic curves in dimension 4. \n\n\nSuppose that there is a solution ${\\bf f}^1=(f_1^1,f_2^1)$ of \\eqref{E:system} in $\\mathcal Y^1$ such that $f_2^1$ is not constant. By the definition of $\\mathcal Y^1$ and by Lemma \\ref{L:aprioriMarker}, $\\lim_{s\\to \\pm\\infty} f_2^1(s,t) = 0$. Since $f_2^1$ is not constant, there is $(s_0,t_0)\\in \\mathbb{R}\\times S^1$ such that $f_2^1(s_0,t_0)\\neq 0$. Let $f_1^2:\\mathbb{R}\\to\\mathbb{R}$ be the unique solution to the ODE above, for which $f_1^2(s_0) = f_1^1(s_0,t_0)$. Then, ${\\bf f}^2(s,t)=(f_1^2(s),f_2^1(s_0,t_0))$ is another solution to \\eqref{E:system} (although $(f^2_1,0)$ does not have the correct asymptotic behavior to belong to $\\mathcal Y^1$). \n\nNow, observe that \\eqref{E:system} is the Floer equation on the K\\\"ahler manifold $(\\mathbb{R}\\times S^1,dx\\wedge dy,i)$, for the time-dependent Hamiltonian \n\\begin{align*}\nH\\colon \\mathbb{R} \\times (\\mathbb{R}\\times S^1) &\\to \\mathbb{R} \\\\\n(s,x,y) &\\mapsto \\int_0^x h'(\\operatorname{e}^z) dz - \\nu'(s)x\n\\end{align*}\nBy Gromov's trick \\cite{Gromov85}, solutions to the Floer equation can be thought of as pseudoholomorphic curves $\\mathbb{R} \\times S^1 \\to M:= (\\mathbb{R}\\times S^1)\\times (\\mathbb{R}\\times S^1)$, for some twisted almost complex structure on $M$. The projection of these curves to the first cylinder factor is the identity. \n\nSince $\\lim_{s\\to -\\infty} f_2^1 = \\lim_{s\\to -\\infty} f_2^0$, Lemma \\ref{L:aprioriMarker} implies that they have lifts $\\tilde f_2^1$ and $\\tilde f_2^0$ to $\\mathbb{R}$, with the same limits as $s\\to \\pm \\infty$. \nBy assumption, ${\\bf f}^1$ and ${\\bf f}^0$ are both in $\\mathcal Y^1$, which implies that $\\lim_{s\\to \\pm \\infty} f_1^1(s,t)=\\lim_{s\\to \\pm \\infty} f_1^0(s,t)=b_\\pm$. Therefore, there is a homotopy from ${\\bf f}^1$ to ${\\bf f}^0$ that is $C^0$-small on neighborhoods of $\\pm\\infty\\times S^1$ in $\\mathbb{R} \\times S^1$. This gives a homotopy from the graph of ${\\bf f}^1$ to the graph of ${\\bf f}^0$ in $M$, and the intersections with the graph of ${\\bf f}^2$ during the homotopy will remain in a compact region of $M$. So, we have an equality of signed intersection numbers\n$$\n\\#(Graph({\\bf f}^1)\\cap Graph({\\bf f}^2))=\\#(Graph({\\bf f}^0)\\cap Graph({\\bf f}^2))=0,\n$$\nsince the second components of ${\\bf f}^0$ and ${\\bf f}^2$ are different constants. But then positivity of intersections of pseudoholomorphic curves in dimension 4 \\cite{McDuffSalamon}*{Exercise 2.6.1} implies that the graphs of ${\\bf f}^1$ and ${\\bf f}^2$ do not intersect, which, by the construction of ${\\bf f}^2$, is a contradiction. \nTherefore, there is no such ${\\bf f}^1$ to start with, which proves the first statement in the Lemma in Case 1.\n\nCase 2 can be dealt with by the same argument, applied to ${\\bf g} \\in \\hat{\\mathcal Y^2}$. \nWe again apply Lemma \\ref{L:aprioriMarker}.\nThe argument for Case 1 above produces a solution ${\\bf g} = (g_1(s), 0) \\in\n\\mathcal{Y}^1$. We need to argue that such a solution is indeed in $\\hat \\mathcal{Y}^2$. \nThis means that $({\\bf g} \\circ \\varphi_P) \\in W^{1,p,\\delta}_{\\mathbb{C}_-}((-\\infty,-1]\\times S^1,\\mathbb{R}\\times S^1)$. This will be a consequence of the fact that $g_1$ is $C^1$, and of its asymptotics. Write $P = (P_s,P_t)$. Then,\n$$\n({\\bf g} \\circ \\varphi_P)(\\rho,\\theta) = (g_1(\\operatorname{e}^{2\\pi \\rho} \\cos(2\\pi\\theta) + P_s),0)\n$$\nand for $\\rho << -1$ the Mean Value Theorem gives\n$$\n|g_1(\\operatorname{e}^{2\\pi \\rho} \\cos(2\\pi\\theta) + P_s) - g_1(P_s)| \\leq ||g_1'||_{L^\\infty}\\,\\operatorname{e}^{2\\pi \\rho} \n$$\n where the sup norm is finite by the exponential convergence of $g_1$ at $\\pm \\infty$. Since $\\delta < 2\\pi$, we get the desired exponential convergence at $P$. \n\n\nLet us now consider the case when $\\tilde u = (a,u)$ is such that $a = Ts + c$, and look in $\\mathcal{Y}^3$ for solutions to \\eqref{E:cylinder}. \nThis is equivalent to the system of equations   \n\n\\begin{equation}%\n\\begin{cases}\n\\partial_s f_1 - \\partial_t f_2 + h'(\\operatorname{e}^{f_1}) - T = 0 \\\\\n\\partial_t f_1 + \\partial_s f_2 = 0\n\\end{cases}\n\\end{equation}\nWe again have solutions of the form ${\\bf f}(s,t)=(f_1(s),f_2)$ with $f_2$ constant. There is an $\\mathbb{R}\\times S^1$ family of such solutions, parametrized by $\\lim_{s\\to -\\infty} (f_1(s)-Ts, f_2)\\in \\mathbb{R}\\times S^1$. The union of the graphs of these solutions foliates the region $(\\mathbb{R}\\times S^1)\\times ((-\\infty,b_+)\\times S^1) \\subset (\\mathbb{R}\\times S^1)\\times (\\mathbb{R}\\times S^1)$. \nWe can apply Lemma \\ref{L:aprioriMarker} to equation \\eqref{E:cylinder}, since it is a special case of \\eqref{E:parametric cylinder1} when $\\tau = 1$. The argument above using positivity of intersections can then be adapted to prove that there are no other solutions with the same asymptotics. \nTo get different solutions, we can replace $(f_1(s),f_2)$ with $(\\overline f_1(s),\\overline f_2)=(f_1(s+c_1),f_2+c_2)$, for a constant $(c_1,c_2)\\in \\mathbb{R}\\times S^1$. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{Ansatz2}]\n Most of the claims made in the statement follow from the existence of a solution $\\bf e$ to \\eqref{E:cylinder1} with the appropriate asymptotics. Indeed, given such ${\\bf e} = (e_1,e_2)$, one can construct $\\tilde v = (b,v)$ by taking \n $$\n (b,v) = (a + e_1, \\phi_R^{e_2} \\circ u).\n $$\n As we pointed out, the existence of such $\\bf e$ is equivalent to the existence of a solution $\\bf f$ to \\eqref{E:cylinder}. Let us consider different cases separately. \n \n In Case 1, note that Lemmas \\ref{L:cylTransversality} and \\ref{L:smooth parameter} and the Implicit Function Theorem imply that solutions to \\eqref{E:parametric cylinder1} vary smoothly in $\\tau$. Lemma \\ref{L:countingSolutions1} guarantees the existence and uniqueness of solutions to \\eqref{E:deformed cylinder1} in $\\mathcal{Y}^1$ (which corresponds to $\\tau = 0$). \n Therefore, we conclude that \\eqref{E:cylinder} (which corresponds to $\\tau = 1$) also has a unique solution in $\\mathcal{Y}^1$, which finishes the proof. \n %\n %\n  \n Case 2 works the same way, with solution $\\bf g$ to \\eqref{E:parametric cylinder2} instead of solutions $\\bf f$ to \\eqref{E:parametric cylinder1}.\n \n Case 3 corresponds to the final part of Proposition \\ref{Ansatz2}, when $\\tilde u$ is assumed to be a trivial cylinder. Lemma \\ref{L:countingSolutions1} implies that there is an $\\mathbb{R}\\times S^1$-family of solutions to \\eqref{E:cylinder} in $\\mathcal{Y}^3$. By Lemma \\ref{L:aprioriMarker}, there is an $\\mathbb{R}$-family of such solutions with $\\lim_{s\\to \\pm \\infty} f_2(s,t) = 0 \\in S^1$. \n This finishes the proof.\n \\end{proof}\n\n\n \\begin{remark}\n     Proposition \\ref{Ansatz2} relates Floer cylinders in $\\mathbb{R} \\times Y$ \n     with pseudoholomorphic cylinders and cylinders satisfying \n     equation \\eqref{E:cylinder1}. \n     We will now discuss the relationship between the Fredholm indices of these problems.\n     Indeed, suppose that $\\tilde u$ is a holomorphic curve, and $\\mathbf{e}$\n     is a solution to Equation \\eqref{E:cylinder1}. This then describes a\n     Floer solution $\\tilde v$.\n     We will now see that the Fredholm index describing deformations of\n     $\\tilde v$ is one less than the sum of the Fredholm index for $\\tilde u$\n     and of the index of the problem given by \\eqref{E:cylinder1} (the latter \n     of which is give in Lemma \\ref{L:cylTransversality}). This\n     difference is explained by the need to take a quotient by an $\\mathbb{R}$-action,\n     though this action is to be understood differently in the 3 cases we\n     consider.\n     \n     First, in Case 1 and in Case 2, we notice that \n     the index of the Floer\n     solution is one less than the index of the corresponding holomorphic\n     [punctured] cylinder in $\\mathbb{R} \\times Y$. This index difference comes from\n     the fact that the Floer cylinder corresponds to an equivalence class of\n     pairs $(\\tilde u, \\mathbf{e})$, where the equivalence comes from\n     translation of $\\tilde u$ in the symplectization and a corresponding\n     translation of $\\mathbf{e}$.\n\n     In Case 3, the Floer solution is contained in a fibre of $\\mathbb{R} \\times Y \\to\n     \\Sigma$. In this case, for any point $p \\in \\Sigma$, there is a $2$\n     parameter family of Floer solutions in the fibre over $p$, with positive\n     end at a closed Hamiltonian orbit and negative end at a Reeb orbit in\n     $\\{-\\infty \\} \\times Y$. \n     These 2-parameters are given by domain automorphisms. Notice that the\n     domain automorphisms are generated by rotations and translations. The\n     domain rotation and the Reeb flow are conjugate in this case. \n     Similarly, the trivial\n     orbit cylinders in $\\mathbb{R} \\times Y$ come in a 2-parameter family, which can\n     also be described by domain automorphism (or by the $\\mathbb{C}^*$-action on $\\mathbb{R}\n     \\times Y$, which is the same thing for these curves). \n     In particular then, we again obtain the claimed equality. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\section{Pseudoholomorphic spheres in \\texorpdfstring{$\\Sigma$}{Sigma} and meromorphic sections}\n\n\\label{S:meromorphic}\n\nThe previous sections explain that split Floer cylinders are in\nsome sense equivalent to $J_Y$-holomorphic curves. \nWe will now describe $J_Y$-holomorphic curves in a manner that is more suitable\nfor computing the Floer differential.\n\nLet $\\tilde u = (a,u): \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R} \\times Y$ be a $J_Y$-holomorphic curve, where $\\Gamma$ is either empty or the singleton $\\{P\\}$. Since $J_Y$ is $\\mathbb{R}\\times S^1$-equivariant, it is a lift of an almost complex structure $J_\\Sigma$ in $\\Sigma$, and one can project $\\tilde u$ to obtain a $J_\\Sigma$-holomorphic map $w: \\mathbb{R}\\times S^1\\setminus \\Gamma \\to\\Sigma$. \nSince punctures of finite energy pseudoholomorphic curves in $\\mathbb{R}\\times Y$ are asymptotic to Reeb orbits in $Y$, and these are \nmultiple covers of the fibres of $Y\\to \\Sigma$, $w$ extends to a $J_\\Sigma$-holomorphic map $\\mathbb{CP}^1 \\to \\Sigma$ \n\\cite{McDuffSalamon}*{Theorem 4.1.2}. We use the standard identification of $\\mathbb{R}\\times S^1$ with $\\mathbb{C}^* \\subset \\mathbb{CP}^1$. \n\nThe symplectization $\\mathbb{R} \\times Y$ can be identified with the complement of the zero section in a complex line bundle $E \\to \\Sigma$ (the dual to the normal bundle to $\\Sigma$ in $X$). \nWe have the following commutative diagram\n\n\n\n\\tikzset{node distance=2cm, auto}\n\n\\begin{center}\n\\begin{tikzpicture}\n\\node (P) {$w {}^{*} (E\\setminus 0)$};\n  \\node (B) [right of=P] {$\\mathbb{R}\\times Y$}; \n  \\node (B') [right of=B] {\\hspace{-2cm} $= E \\setminus 0$};\n  \\node (A) [below of=P] {$\\mathbb{CP}^1$};\n  \\node (C) [below of=B] {$\\Sigma$};\n  \\node (P1) [node distance=2cm, left of=A] {$\\mathbb{R}\\times S^1\\setminus \\Gamma$};\n  \\draw[->] (P) to node {} (B);\n  \\draw[->] (P) to node {} (A);\n  \\draw[->] (A) to node {$w$} (C);\n  \\draw[->] (B) to node {} (C);\n  \\draw[right hook->] (P1) -- (A) [anchor=west]{};\n  \\draw[->, bend left=60] (P1) to node {$(a,u)$} (B);\n  \\draw[->, dashed, bend left] (A) to node {$s$} (P);\n\\end{tikzpicture}\n\\end{center}\nwhere $s$ is a section of the pullback bundle $w^*E \\to \\mathbb{CP}^1$ with a zero at $0$ (and at $P$, if $\\Gamma = \\{P\\}$) and a pole at $\\infty$. \nThe line bundle $w^*E$ has an induced complex linear Cauchy--Riemann operator, in the sense of \\cite{McDuffSalamon}*{Section C.1}. Complex linearity is a consequence of the Reeb-invariance of $J_Y$. Such a Cauchy--Riemann operator corresponds to a unique holomorphic structure on $w^*E$ \\cite{Kobayashi}*{Proposition 1.3.7}. The section $s$ is in the kernel of this operator and is therefore meromorphic. This proves the following result.\n\n\n\n\n\\begin{lemma} \\label{L:cylinders to spheres}\nEvery $J_Y$-holomorphic curve $\\tilde u = (a,u): \\mathbb{R}\\times S^1\\setminus \\Gamma \\to \\mathbb{R} \\times Y$ defines a $J_\\Sigma$-holomorphic map $w: \\mathbb{CP}^1 \\to \\Sigma$ and a meromorphic section of $w^*E \\to \\mathbb{CP}^1$, with zero at  $0$ (and at $P$, if $\\Gamma = \\{P\\}$) and pole at $\\infty$.  \n\\end{lemma}\n\n\nThe converse to the Lemma (the statement that a meromorphic section induces a $J_Y$-holomorphic curve) is immediate. Therefore, \nwe can reduce the question of counting punctured $J_Y$-holomorphic maps $\\tilde u$ to that of counting $J_\\Sigma$-holomorphic curves \n$w: \\mathbb{CP}^1 \\to \\Sigma$, together with meromorphic sections $s$ of $w^*E$, with prescribed zeros and poles. The count of maps $w$ is \nrelated with the computation of genus zero Gromov--Witten numbers of $\\Sigma$ \\cite{McDuffSalamon}. More about this will be said below. \nWe can also give a complete description of the relevant meromorphic sections $s$. Given a divisor of points $D$ in $\\mathbb{CP}^1$ and a holomorphic line bundle $L$ over $\\mathbb{CP}^1$, where both $D$ and $L$ have degree $d$, there is a $\\mathbb{C}^*$-family of meromorphic sections of $L$, such that the divisor associated with each section is $D$. \nOne can justify this fact by reducing it to the simplest case of trivial $L$: use a trivialization of $L$ over $\\mathbb{C} \\subset \\mathbb{CP}^1$ to identify meromorphic sections of $L$ with meromorphic functions on $\\mathbb{CP}^1$ \n\\cite{Miranda}*{pp 342--345}.\n\n\nFix a $J_\\Sigma$-holomorphic map $w: \\mathbb{CP}^1 \\to \\Sigma$ and let $\\tilde u = (a,u) \\colon \\mathbb{R} \\times S^1\\setminus \\Gamma \\to \\mathbb{R}\\times Y$ be a $J_Y$-holomorphic lift. Given $(\\theta_1,\\theta_2) \\in S^1\\times S^1$, we can produce a new $J_Y$-holomorphic curve \n$$\n(\\theta_1,\\theta_2) . \\tilde u = \\tilde u_{(\\theta_1, \\theta_2)}\n$$\nby rotating in the domain and in the target: \n$$\n \\tilde u_{(\\theta_1, \\theta_2)}(s,t) = (a(s,t+\\theta_1), \\phi_R^{\\theta_2}\\circ u(s,t+\\theta_1)).\n$$\n\n\\begin{lemma} \\label{number of lifts}\nThe number of such lifts that have prescribed asymptotic markers at $\\pm\\infty$ is \n$k_+ - k_- = - \\langle c_1(E \\to \\Sigma), w_*[\\mathbb{CP}^1] \\rangle + \\sum_{P\\in \\Gamma} k_P > 0$. \n\\end{lemma}\n\\begin{proof}\n    This result is similar to \\cite{DiogoLisiSplit}*{Lemma 5.43}. \n    We identify the simple Reeb orbits to which $u$ converges at $\\pm \\infty$\n    with $S^1 \\times S^1$. Then, the action of the torus of rotation in the\n    domain and in the fibre can be\n    seen as a linear map ${\\,T}^2 \\to {\\,T}^2$ \n    represented by the matrix $\\begin{pmatrix} k_- &\n    1\\\\ k_+ & 1 \\end{pmatrix}. $ See Equation \\eqref{rot v} for the analogous discussion \n    in the case of lifts to solutions of Floer's equation in $\\mathbb{R} \\times Y$, instead of pseudoholomorphic curves. \n\n    \nThe number of lifts we want is the absolute \nvalue of the degree of this map $T^2 \\to T^2$, \nwhich is the absolute value of the determinant \nof the matrix, i.e.~$k_+ - k_-$.\n\nNotice now that $-k_+ + k_- + \\sum_{P \\in \\Gamma} k_P\n= \\langle c_1(E \\to \\Sigma), w_*[\\mathbb{CP}^1] \\rangle$ is the degree of the\nbundle $E$, which is negative by construction.\n\\end{proof}\n\n\n\\begin{remark}\nWe have related a Floer cylinder $\\tilde v$ in $\\mathbb{R}\\times Y$ with a pseudoholomorphic sphere $w$ in $\\Sigma$, by relating $\\tilde v$ with a pseudoholomorphic curve $\\tilde u$ in $\\mathbb{R}\\times Y$ and then projecting $\\tilde u$ to $w$. The difference between $\\tilde v$ and $\\tilde u$ is given by ${\\bf e} \\colon \\mathbb{R} \\times S^1 \\setminus \\Gamma \\to \\mathbb{R}\\times S^1$ solving \\eqref{E:cylinder0}, and the information lost when we project $\\tilde u$ to $w$ is a meromorphic section of a line bundle $L$ over $\\mathbb{CP}^1$. Alternatively, we could have projected $\\tilde v$ to $w$ directly. The information lost in this process would now be a section of $L$ solving an equation analogous to \\eqref{E:cylinder0}. We could have written analogues of Sections \\ref{S:ansatz_Floer_to_holo} and \\ref{S:ansatz_holo_to_Floer} to analyse such Hamiltonian-perturbed sections of $L$, but decided to separate the analytical and topological aspects of the problem by factoring through the pseudoholomorphic map $\\tilde u$.   \n\\end{remark}\n\n\n\n\n\n\n\n\n\\section{Orientations}\n\n\\label{compute signs}\n\nWe want to determine the signs of the contributions to the split symplectic homology differential. The orientation conventions \nfor Cases 0 through 3 are explained in \\cite{DiogoLisiSplit}*{Section 7}. As we saw before, Cases 1 through 3 can be related to counts \nof pseudoholomorphic spheres in $\\Sigma$ and $X$. \nWe wish to compare the signs that appear in the differential to the signs involved in counts of pseudoholomorphic spheres (as in Gromov--Witten theory).\n\nRecall that Floer cascades are oriented using the following fibre sum convention\n(see also \\cite{DiogoLisiSplit}*{Definition 7.4}). \n\n\\begin{definition} \\label{fibre sum orient} \nGiven linear maps between oriented vector spaces $f_i\\colon V_i \\to W$, $i = 1,2$, such that $f_1 - f_2 \\colon V_1\\oplus V_2 \\to W$ is surjective, the {\\em fibre sum orientation} on $V_1 \\oplus_{f_i} V_2 = \\ker (f_1 - f_2)$ is such that \n\\begin{enumerate}\n \\item $f_1 - f_2$ induces an isomorphism $(V_1 \\oplus V_2) \/ \\ker (f_1 - f_2) \\to W$ which changes orientations by $(-1)^{\\dim V_2 . \\dim W}$,\n \\item where a quotient $U\/V$ of oriented vector spaces is oriented in such a way that the isomorphism $V \\oplus (U\/V) \\to U$ (associated to a section of the quotient short exact sequence) preserves orientations. \n\\end{enumerate}\n\\end{definition}\n\nIt is useful to begin with some general properties of the fibre sum orientation. %\nThe first result is Proposition 7.5.(a) in \\cite{JoyceCorners} (which uses the same fibre sum orientation convention as Definition \n\\ref{fibre sum orient}). We include a proof here because the structure of its argument will serve as a model for other proofs in this section. \n\n\\begin{lemma} \\label{fibre sum commute} \n Let $f_i\\colon V_1 \\to W$ be as in Definition \\ref{fibre sum orient}. Consider the map $r\\colon V_1 \\oplus V_2 \\to V_2 \\oplus V_1$, such that $r(v_1,v_2) = (v_2,v_1)$. Then, $r$ induces an identification \n $$\n \\ker\\big((f_1 - f_ 2)\\colon V_1 \\oplus V_2 \\to W \\big) \\to  \\ker\\big((f_2 - f_ 1)\\colon V_2 \\oplus V_1 \\to W \\big),\n $$\n which changes the fibre sum orientations by $(-1)^{(\\dim V_1 +\\dim W)(\\dim V_2 + \\dim W)}$. \n\\end{lemma}\n\\begin{proof}\n Call $f\\coloneqq(f_1-f_2)\\colon V_1 \\oplus V_2 \\to W$ and $g\\coloneqq(f_2-f_1)\\colon V_2 \\oplus V_1 \\to W$. Denoting also by $f$ and $g$ the appropriate isomorphisms in Part (1) of Definition \\ref{fibre sum orient}, we have an isomorphism \n $$\n g \\circ r \\circ f^{-1} \\colon W \\to W,\n $$\nwhich is $-\\operatorname{Id}$, hence it changes orientation by $(-1)^{\\dim W}$. \n\nDenote also by $r$ the induced map $(V_1 \\oplus V_2)\/\\ker f \\to (V_2 \\oplus V_1)\/\\ker g$. A simple calculation yields the following.\n\n\\begin{claim}\n Let $s_{12} \\colon (V_1 \\oplus V_2)\/\\ker f \\to V_1 \\oplus V_2$ be a right-inverse for the projection $\\pi_{12} \\colon V_1 \\oplus V_2 \\to (V_1 \\oplus V_2)\/\\ker f$. Then, \n $s_{21}\\coloneqq r \\circ s_{12} \\circ r^{-1} \\colon (V_2 \\oplus V_1)\/\\ker g \\to V_2 \\oplus V_1$ is a right-inverse for $\\pi_{21} \\colon V_2 \\oplus V_1 \\to (V_2 \\oplus V_1)\/\\ker g$. \n\\end{claim}\nWe can now justify the sign in the statement of the Lemma. By the Claim, the following diagram commutes: \n\\begin{diagram}\n\\ker f \\bigoplus \\, \\big((V_1\\oplus V_2)\/\\ker f\\big) & \\rTo^{i_{12}+s_{12}\\,} & V_1\\oplus V_2 \\\\\n\\dTo_{r\\oplus r}                                & & \\dTo_{r}       \\\\\n\\ker g \\bigoplus \\, \\big((V_2\\oplus V_1)\/\\ker g\\big) & \\rTo^{i_{21}+s_{21}\\,} & V_2\\oplus V_1\n\\end{diagram}\nwhere $i_{12}$ and $i_{21}$ are inclusion maps. \n\nBy Part (1) in Definition \\ref{fibre sum orient} and the argument above the Claim, $r\\colon (V_1\\oplus V_2)\/\\ker f \\to (V_2\\oplus V_1)\/\\ker g$ changes orientations by $(-1)^{(\\dim V_1 + \\dim V_2 + 1) \\dim W}$. By Part (2) in Definition \\ref{fibre sum orient} and the fact that $r\\colon V_1\\oplus V_2 \\to V_2\\oplus V_1$ changes orientations by $(-1)^{\\dim V_1 \\dim V_2}$, we get the remaining contribution to the sign in the Lemma. \n\\end{proof}\n\n\\begin{definition} \\label{preimage orient} \nLet $f_1,f_2$ be as in Definition \\ref{fibre sum orient}. Denote by $\\Delta \\subset W\\oplus W$ the diagonal, and orient it as the image of the map $w\\mapsto (w,w)$. The {\\em preimage orientation} on the fibre sum $V_1 \\prescript{}{f_1\\,}\\oplus\\prescript{}{f_2}{} V_2  = (f_1,f_2)^{-1}(\\Delta) \\subset V_1 \\oplus V_2$ is so that, if $H\\subset V_1\\oplus V_2$ is a complementary subspace to $(f_1,f_2)^{-1}(\\Delta)$, the following isomorphisms preserve orientations\n\\begin{enumerate}\n\\item $H \\oplus  (f_1,f_2)^{-1}(\\Delta) \\to (-1)^{\\dim V_2(\\dim V_1 + \\dim W)}V_1\\oplus V_2$;\n\\item $(f_1,f_2)(H) \\oplus \\Delta \\to W\\oplus W$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} \\label{GuilleminPollack} \nIf $g_1\\colon X \\to Y$ and $g_2\\colon Z \\to Y$ are transverse maps of smooth oriented manifolds, then the previous definition can be applied to maps of tangent spaces to orient the fibre product $X \\prescript{}{g_1\\,}\\times\\prescript{}{g_2}{} Z$. In the special case when $g_2$ is the inclusion of a submanifold, this agrees with the orientation on the preimage, as defined in \\cite{GuilleminPollack}.\nThis is the reason for the name used for this orientation convention, and for the sign in Part (1) above.\n\\end{remark}\n\n\\begin{lemma} \\label{fibre sum vs preimage}\nThe identification of $\\ker f$ with the fibre sum orientation (as in Definition \\ref{fibre sum orient}) and $(f_1,f_2)^{-1}(\\Delta)$ with preimage orientation (as in Definition \\ref{preimage orient}) changes orientations by $(-1)^{(\\dim V_1 + \\dim W)(\\dim V_2 +\\dim W)}$. \n\\end{lemma}\n\\begin{proof}\nFor convenience, denote Parts (1) and (2) in Definition \\ref{preimage orient} by (P1) and (P2), respectively, and similarly denote by (F1) and (F2) the Parts in Definition \\ref{fibre sum orient}. Note that to write (F2) in this case, we need a section $s\\colon \\frac{V_1\\oplus V_2}{\\ker f} \\to V_1 \\oplus V_2$. We will take $H = s(\\frac{V_1\\oplus V_2}{\\ker f})$ in (P1). \n\nThe result follows from a sequence of oriented isomorphisms:\n\\begin{align*}\nW \\oplus W &\\oplus\\ker f \\cong  (-1)^{(\\dim V_1 + \\dim V_2 - \\dim W)\\dim W} W \\oplus \\ker f \\oplus W \\cong \\\\\n&\\stackrel{(F1)}{\\cong} (-1)^{\\dim V_1 \\dim W + \\dim W} W \\oplus \\ker f \\oplus \\frac{V_1 \\oplus V_2}{\\ker f } \\cong\\\\ \n& \\stackrel{(F2)}{\\cong} (-1)^{\\dim V_1 \\dim W + \\dim W} W\\oplus V_1 \\oplus V_2 \\cong \\\\\n&\\stackrel{(P1)}{\\cong} (-1)^{\\dim V_1 \\dim W + \\dim V_1 \\dim V_2 + \\dim V_2 \\dim W + \\dim W}. \\\\\n&\\hspace{5cm} .W \\oplus H \\oplus (f_1,f_2)^{-1}(\\Delta) \\cong \\\\\n&\\cong (-1)^{\\dim V_1 \\dim W + \\dim V_1 \\dim V_2 + \\dim V_2 \\dim W + \\dim W}. \\\\\n&\\hspace{5cm} .\\Delta \\oplus (f_1,f_2)(H) \\oplus (f_1,f_2)^{-1}(\\Delta) \\cong \\\\\n&\\cong (-1)^{\\dim V_1 \\dim W + \\dim V_1 \\dim V_2 + \\dim V_2 \\dim W}. \\\\\n&\\hspace{5cm} .(f_1,f_2)(H) \\oplus \\Delta \\oplus  (f_1,f_2)^{-1}(\\Delta) \\cong \\\\\n&\\stackrel{(P2)}{\\cong} (-1)^{\\dim V_1 \\dim W + \\dim V_1 \\dim V_2 + \\dim V_2 \\dim W} W \\oplus W \\oplus (f_1,f_2)^{-1}(\\Delta) \\cong\\\\\n&\\stackrel{\\varphi}{\\cong} (-1)^{\\dim V_1 \\dim W + \\dim V_1 \\dim V_2 + \\dim V_2 \\dim W + \\dim W} W \\oplus W \\oplus (f_1,f_2)^{-1}(\\Delta) \n\\end{align*}\nwhere $\\varphi = \\begin{pmatrix} 1 & 1 \\\\\n                                                1 & -1 \n                          \\end{pmatrix} \\oplus \\operatorname{id} \\colon W \\oplus W \\oplus (f_1,f_2)^{-1}(\\Delta) \\to W \\oplus W \\oplus (f_1,f_2)^{-1}(\\Delta)$. \nNote that $\\varphi$ changes orientations by $(-1)^{\\dim W}$. The composition of the chain of isomorphisms above is of the form \n$$\n\\begin{pmatrix} 2 & * \\\\\n                        0 & 1 \n                          \\end{pmatrix} \\oplus \\operatorname{id} \\colon W \\oplus W \\oplus \\ker f \\to W \\oplus W \\oplus (f_1,f_2)^{-1}(\\Delta)\n$$\nso it preserves orientations iff the identity $\\ker f \\to (f_1,f_2)^{-1}(\\Delta)$ does. The result now follows. \n\\end{proof}\n\n\\begin{remark}\nSince the signs appearing in Lemmas \\ref{fibre sum commute} and \\ref{fibre sum vs preimage} are the same, the order-reversing \nmap $r$ in Lemma \\ref{fibre sum commute} induces an orientation-preserving isomorphism between $\\ker(f_1-f_2)$ with the fibre \nsum orientation and $(f_2,f_1)^{-1}(\\Delta)$ with the preimage orientation (note the orders of the $f_i$).\n\\end{remark}\n\n\nLet us now study the signs that appear in the split symplectic homology differential. We start with Case 1 configurations. \nLet $p,q \\in \\Crit(f_\\Sigma)$ and consider a Floer cylinder with one cascade giving a contribution of $\\wh{q}_{k_-}$ to the \ndifferential of $\\wc{p}_{k_+}$.\nAs we saw in \\eqref{fib prod1}, these cascades form spaces \n\\begin{equation}\n \\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)\\coloneqq W^s_{Y}(\\wh q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\n \\label{fib prod 1}\n\\end{equation}\nfor suitable $A\\in H_2(\\Sigma;\\mathbb{Z})$. In \\cite{DiogoLisiSplit}*{Section 7}, these spaces are oriented using the fibre sum rule. \nthe spaces of cylinders are given a coherent orientation and the critical manifolds are oriented as follows. \n\nFor any oriented manifold $S$ with a Morse--Smale pair $(f_S,Z_S)$, fixing an orientation on a critical manifold of a critical point $p$ \nalso fixes an orientation on the other critical manifold, using the convention\n\\begin{equation}\n T_p W^s_S(p) \\oplus T_p W^u_S(p) \\cong T_p S.\n \\label{orient stable and unstable}\n\\end{equation}\n\nPick orientations on all unstable manifolds of $\\Sigma$ and $W$. Given $p\\in \\Crit(f_\\Sigma)$, assume that \nthe restrictions of $\\pi_\\Sigma\\colon Y \\to \\Sigma$ to\n\\begin{equation}\n W^u_Y(\\widecheck p) \\to W^u_\\Sigma(p) \\qquad \\text{and} \\qquad W^s_Y(\\widehat p) \\to W^s_\\Sigma(p)\n \\label{orient point lift}\n\\end{equation}\nare orientation-preserving diffeomorphisms. \n\nIt will also be useful to assume that, with respect to the connection induced by the contact form of $Y$ on the $S^1$-bundles \n$\\pi_\\Sigma^{-1}\\left( W^{u\/s}_\\Sigma(p) \\right) \\to W^{u\/s}_\\Sigma(p)$, the splittings \n\\begin{equation}\n T_{\\tilde x} \\, \\pi_\\Sigma^{-1}\\left( W^u_\\Sigma(p) \\right) \\cong T_x W^u_\\Sigma(p) \\oplus \\mathbb{R} \\qquad \\text{and} \\qquad \n T_{\\tilde x} \\, \\pi_\\Sigma^{-1}\\left( W^s_\\Sigma(p) \\right) \\cong T_x W^s_\\Sigma(p) \\oplus \\mathbb{R}\n \\label{orient full lift}\n\\end{equation}\nare orientation-preserving (where $\\pi_\\Sigma(\\tilde x) = x$). In particular, when $p$ is a maximum or minimum, we get the correct relation between the orientations of $Y$ \nand $\\Sigma$. \n\n\\begin{remark}\nNote that we have codimension zero inclusions of oriented manifolds \n$W^u_Y(\\widehat p) \\subset \\pi_\\Sigma^{-1}\\left( W^u_\\Sigma(p) \\right)$ and \n$W^s_Y(\\widecheck p) \\subset \\pi_\\Sigma^{-1}\\left( W^s_\\Sigma(p) \\right)$. Under our conventions, these maps \nare always orientation-preserving for $W^u_Y(\\widehat p)$, and preserve orientations for $W^s_Y(\\widecheck p)$ iff $M(p)$ is even.\n\\end{remark}\n\n\n\n\nEach cascade in \\eqref{fib prod 1} projects to a chain of pearls in $\\Sigma$ from $q$ to $p$, with exactly one sphere, contained in\n\\begin{equation}\n \\mathcal{M}(q,p;A)\\coloneqq W^s_{\\Sigma}(q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) \\times_{\\operatorname{ev}} W_{\\Sigma}^u(p),\n \\label{fib prod 2}\n\\end{equation}\nwhere $\\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)$ is the space of simple $J_\\Sigma$-holomorphic spheres in class $A$, with no marked points. \nNote that $\\mathcal{M}(q,p;A)$ can be identified with $\\operatorname{ev}_\\Sigma^{-1}\\left( W^s_{\\Sigma}(q) \\times W_{\\Sigma}^u(p)\\right)$, for\n\\begin{align*}\n\\operatorname{ev}_\\Sigma \\colon \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) &\\to \\Sigma \\times \\Sigma \\\\\n w &\\mapsto (w(0),w(\\infty))\n\\end{align*}\n\n\\begin{definition} \\label{GW orient case 1} \n The {\\em Gromov--Witten orientation} on $\\mathcal{M}(q,p;A)$ is the one induced by the identification with \n $\\operatorname{ev}_\\Sigma^{-1}\\left( W^s_{\\Sigma}(q) \\times W_{\\Sigma}^u(p)\\right)$, where the latter is equipped \n with the preimage orientation. The {\\em Gromov--Witten sign} of an element in $\\mathcal{M}(q,p;A)$ is given \n by the corresponding orientation on the zero-dimensional $\\mathcal{M}(q,p;A)\/\\mathbb{C}^*$. \n\\end{definition}\n\n\\begin{remark} \\label{GW signs} \nTo understand the reason for this terminology, suppose that these stable and unstable manifolds defined cycles in $H_*(\\Sigma)$. An element in $\\mathcal{M}(q,p;A)$ would then contribute to a two-point Gromov--Witten \ninvariant of $\\Sigma$, with insertions given by the stable and unstable manifolds. The sign of this contribution would be the the one prescribed by the Gromov--Witten orientation (see \\cite{McDuffSalamon}*{Exercise 7.1.2}).\n\\end{remark}\n\nOur goal is to compare \n\\begin{itemize}\n \\item the sign of the contribution of a Floer cylinder with cascades in \\eqref{fib prod 1} to the symplectic homology differential\n \\item to the sign induced on the projected chain of pearls in \\eqref{fib prod 2} by the Gromov--Witten orientation.\n\\end{itemize}\nThis will be the content of Proposition \\ref{case 1 and GW}. To compare the two signs, it will be useful to also consider the fibre product\n\\begin{equation}\n \\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)}\\coloneqq \\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q) \\right) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\times_{\\operatorname{ev}} \\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right).\n\\label{fib prod 3}\n\\end{equation}\nThe notation is justified by the fact that this space contains \n\\begin{equation*}\n \\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)\\coloneqq W^s_{Y}(\\wc q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\wh p)\n\\end{equation*}\nas a dense subspace. \n\nRecall that $\\pi_\\Sigma^{-1}\\left(W^{s\/u}_{\\Sigma}(q) \\right)$\nis an $S^1$-bundle over $W^{s\/u}_{\\Sigma}(q)$, and that these were given orientations in \\eqref{orient full lift}.\nThe submanifolds $W^s_{Y}(\\wh q)$ and $W^u_{Y}(\\wc q)$ specify sections of these bundles, which we use to trivialize \n$\\pi_\\Sigma^{-1}\\left(W^{s\/u}_{\\Sigma}(q) \\right)\\cong W^{s\/u}_{\\Sigma}(q) \\times S^1$ (identifying $W^s_{Y}(\\wh q)$ \nwith $W^s_{\\Sigma}(q) \\times \\{1\\}$ and $W^u_{Y}(\\wc q)$ with $W^u_{\\Sigma}(q) \\times \\{1\\}$). \nWe can now write \n\\begin{equation}\nW^s_{Y}(\\wh q)= \\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q) \\right) \\prescript{}{\\wc s\\,}\\times\\prescript{}{i} \\{1\\} \\text{ and }\nW^u_{Y}(\\wc p)= \\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right) \\prescript{}{\\wh u\\,}\\times\\prescript{}{i} \\{1\\},\n\\label{fib prod crit}\n\\end{equation}\nwhere the maps in the fibre products are $\\wc s\\colon \\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q)\\right) \\to S^1$, the analogous \n$\\wh u \\colon \\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right) \\to S^1$ and $i\\colon \\{1\\} \\hookrightarrow S^1$. \nRecall that the left sides in \\eqref{fib prod crit} were given orientations in \\eqref{orient point lift}, which agree with \nthe fibre sum orientations on the right sides.\n\n\n\n\n\nNote that $\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)$ is a codimension 2 submanifold of $\\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)}$. \nGiven an element $(x,u,y) \\in \\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)$, there is a splitting\n\\begin{equation}\nT_{(x,u,y)}\\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)} \\cong \\left( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\right) \\oplus \\mathbb{R}_{\\domain} \\oplus \\mathbb{R}_{\\target}\n\\label{orient splitting}\n\\end{equation}\nwhere $\\mathbb{R}_{\\domain}$ is the direction spanned by an infinitesimal rotation of the Floer cylinder $u$ on the domain (in the $t$-direction) and $\\mathbb{R}_{\\target}$ is spanned by the infinitesimal rotation of $u$ on the target (in the Reeb direction). \n\n\n\n\n\n\\begin{lemma} \\label{orient lifts free vs fixed} \nIf we take the fibre sum orientation on \\eqref{fib prod 1} and \\eqref{fib prod 3}, then the splitting \\eqref{orient splitting} preserves orientations iff $M(q)$ is even. \n\\end{lemma}\n\n \n\\begin{proof}\nUsing the associativity of the fibre sum orientation and Lemma \\ref{fibre sum commute} on the third identity, we have the oriented diffeomorphisms (dropping 0, $\\mathbb{R}\\times Y$ and $J_Y$ from the notation)\n\\begin{align*}\n \\mathcal{M}(&\\wh q_{k_-},\\wc p_{k_+};A) = W^s_{Y}(\\wh q) \\times_{\\operatorname{ev}}  \\mathcal{M}^*_{H;k_-,k_+}(A) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p) = \\\\\n &= \\left(\\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q) \\right) \\prescript{}{\\wc s\\,}\\times\\prescript{}{i} \\{1\\}\\right) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H;k_-,k_+}(A) \\times_{\\operatorname{ev}} \\left( \\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right) \\prescript{}{\\wh u\\,}\\times\\prescript{}{i} \\{1\\}\\right) = \\\\\n &= (-1)^{\\dim \\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q)\\right) +1}. \\\\\n &\\qquad .\\{1\\} \\prescript{}{i\\,}\\times\\prescript{}{\\wc s}{} \\big( \\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q) \\right) \\times_{\\operatorname{ev}}  \\mathcal{M}^*_{H;k_-,k_+}(A) \\times_{\\operatorname{ev}} \\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right) \\big) \\prescript{}{\\wh u\\,}\\times\\prescript{}{i} \\{1\\} = \\\\\n &= (-1)^{M(q)} \\{1\\} \\prescript{}{i\\,}\\times\\prescript{}{\\wc s}{} \\, \\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+})} \\, \\prescript{}{\\wh u\\,}\\times\\prescript{}{i} \\{1\\} \n\\end{align*}\n\n\nWe get an oriented isomorphism of vector spaces\n$$\n T_{(x,u,y)} \\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\cong (-1)^{M(q)} 0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} \\big(T_{(x,u,y)} \\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)} \\big) \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0\n$$\n(where we wrote $0$ instead of $T_{\\{1\\}}{\\{1\\}}$). \n\n\n\n\nOn the other hand, abbreviating $\\mathbb{R}^2\\coloneqq \\mathbb{R}_{\\domain} \\oplus \\mathbb{R}_{\\target}$ and thinking of the direct sum as a fibre sum (of maps to the zero vector space), we get\n\\begin{align*}\n 0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} &\\big( \\left( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\right) \\oplus \\mathbb{R}^2 \\big) \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0  = \\\\\n &= 0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} \\big( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\oplus (\\mathbb{R}^2 \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0 )\\big) = \\\\\n &= (-1)^{\\dim \\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)} 0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} \\big( (\\mathbb{R}^2 \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0 ) \\oplus T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)\\big) = \\\\\n &= - \\big( 0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} \\, \\mathbb{R}^2 \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0 \\big) \\oplus T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) = \\\\\n &= T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) .\n\\end{align*}\nThe first identity above uses the associativity of the fibre sum orientation, the second uses Lemma \\ref{fibre sum commute}, \nthe third uses the fact that $\\dim \\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) = 1$ (since we are studying contributions to the \ndifferential) and again the associativity property. The last identity follows from the fact that the fibre sum \norientation on the zero dimensional vector space \n$0 \\prescript{}{di\\,}\\oplus\\prescript{}{d\\wc s}{} \\, \\mathbb{R}^2 \\prescript{}{d\\wh u\\,}\\oplus\\prescript{}{di} \\,0$ \nis negative, which is a simple calculation (the reason ultimately being the fact that the matrix in \\eqref{rot v} has \nnegative determinant). \n\nThe Lemma follows from combining the previous two computations. \n\\end{proof}\n\n\nIn what follows, it is useful to work from a slightly more abstract point of view. \n Let $\\pi_A\\colon \\tilde A \\to A$ and $\\pi_B\\colon \\tilde B \\to B$ be $S^1$-bundles. Suppose that there are $S^1$-equivariant maps $\\tilde f_1\\colon \\tilde A \\to Y$ and $\\tilde f_2\\colon \\tilde B \\to Y$, and\n denote their projections by $f_1\\colon A \\to \\Sigma$ and $f_2\\colon B \\to \\Sigma$. Then, the fibre product $\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$ is an $S^1$-bundle over $A \\prescript{}{f_1\\,}\\times\\prescript{}{f_2}B$ (with `diagonal' $S^1$-action on $\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$). \n \nRecall that the contact form on $Y$ induces an Ehresmann connection on $\\pi_\\Sigma\\colon Y \\to \\Sigma$. The $S^1$-equivariance of $\\tilde f_1$ implies that there is a unique Ehresmann connection on $\\tilde A$ inducing the top arrow in the commutative diagram\n\\begin{equation}\n\\begin{diagram} \nT_a A\\oplus \\mathbb{R} & \\rTo^{\\,} & T_{\\tilde a}\\tilde A \\\\\n\\dTo^{d f_1 \\oplus \\operatorname{id}}   & & \\dTo_{d\\tilde f_1}       \\\\\nT_{f_1(a)}\\Sigma \\oplus \\mathbb{R} & \\rTo^{\\,} & T_{\\tilde f_1(\\tilde a)}Y\n\\end{diagram} \n\\label{connection A}\n\\end{equation}\nwhere $\\tilde a\\in \\tilde A$ and $a = \\pi_A(\\tilde a)$. Do the same for $\\tilde B$. \n\n\n\n\n These connections induce a connection on the $S^1$-bundle $\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$ as follows. \n Given $(\\tilde a, \\tilde b) \\in \\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$, we can take the horizontal subspace $H_{\\tilde a} \\prescript{}{d(\\pi_\\Sigma\\circ \\tilde f_1)\\,}\\oplus\\prescript{}{d(\\pi_\\Sigma\\circ \\tilde f_2)}{} H_{\\tilde b}$. \nThis can be identified with $T_{(a,b)} (A \\prescript{}{f_1\\,}\\times\\prescript{}{f_2}B)$, via the restriction of $(\\pi_A,\\pi_B)$ to the fibre sum. \nThis gives an identification \n\\begin{equation} \nT_{(\\tilde a, \\tilde b)} (\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B) \\cong \\left( T_{(a,b)} (A \\prescript{}{f_1\\,}\\times\\prescript{}{f_2}B) \\right) \\oplus \\mathbb{R},\n\\label{split lift}\n\\end{equation}\nwhere $\\mathbb{R}$ stands for the vector space generated by the infinitesimal generator of the $S^1$-action on $\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$. \nWe rewrite \\eqref{split lift} as an isomorphism\n$$\n\\chi\\colon \\ker (d f_1 - d f_2) _{(a,b)}\\oplus \\mathbb{R} \\to \\ker (d\\tilde f_1 - d\\tilde f_2)_{(\\tilde a,\\tilde b)}.\n$$\n\n\nPick an orientation on $A$ and orient $\\tilde A$ so that \nthe splitting $T_{\\tilde a}\\tilde A \\cong T_a A \\oplus \\mathbb{R}$ induced by the connection on $\\tilde A$ preserves orientations. \nPick an orientation on $B$ and orient $\\tilde B$ in an analogous manner. \n\n \\begin{lemma} \\label{fib prod lift}\nThe identification \\eqref{split lift} preserves orientations, if we use the fibre sum orientation on both fibre products, and \norient $\\mathbb{R}$ in the direction of the infinitesimal $S^1$-action on $\\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$.\n \\end{lemma}\n\\begin{proof}\nThe structure of the argument is parallel to that of Lemma \\ref{fibre sum commute}.  \n\nFix $(\\tilde a,\\tilde b)\\in \\tilde A \\prescript{}{\\tilde f_1\\,}\\times\\prescript{}{\\tilde f_2}{} \\tilde B$, so that \n$\\pi_A(\\tilde a)= a$ and $\\pi_B(\\tilde b)=b$. Write $d \\tilde f\\coloneqq(d \\tilde f_1- d\\tilde f_2)\\colon T\\tilde A \\oplus T\\tilde B \\to T Y$ \nand $d f\\coloneqq(d f_1- d f_2)\\colon T A \\oplus T B \\to T \\Sigma$, where we omited subscripts $a,b,\\tilde a,\\tilde b$ to make \nthe notation lighter. We will do this throughout the proof. Denote by $\\xi\\colon T\\Sigma \\oplus \\mathbb{R} \\to TY$ the isomorphism induced \nby the contact structure on $Y$. The commutative diagram\n\\begin{diagram}\n\\frac{T A\\oplus T B}{\\ker (d f)} \\oplus \\mathbb{R} & \\rTo^{[df]\\oplus \\operatorname{id} \\,} & T \\Sigma \\oplus \\mathbb{R} \\\\\n\\dTo_{\\varphi}                                & & \\dTo_{\\xi}       \\\\\n\\frac{T\\tilde A\\oplus T\\tilde B}{\\ker (d \\tilde f)} & \\rTo^{[d\\tilde f] \\,} & T Y\n\\end{diagram} \ndefines the isomorphism $\\varphi$. \n\nDenote by $\\psi$ the composition $T A \\oplus T B \\oplus \\mathbb{R} \\oplus \\mathbb{R} \\to T A \\oplus \\mathbb{R} \\oplus T B \\oplus \\mathbb{R} \\to T \\tilde A \\oplus T\\tilde B$, where the first isomorphism permutes the second and third factors and the second isomorphism is induced by the connections on $\\tilde A$ and $\\tilde B$. %\n\n\\begin{claim}\n Let $s \\colon \\frac{T A\\oplus T B}{\\ker (d f)} \\to TA \\oplus TB$ be a right-inverse for the projection $\\pi \\colon TA \\oplus TB \\to \\frac{T A\\oplus T B}{\\ker (d f)}$. Then, \n $\\tilde s\\coloneqq \\psi\\circ (s\\oplus (1,0)) \\circ \\varphi^{-1} \\colon \\frac{T\\tilde A\\oplus T\\tilde B}{\\ker (d \\tilde f)} \\to T\\tilde A \\oplus T\\tilde B$ is a right-inverse for $\\tilde \\pi \\colon T\\tilde A \\oplus T\\tilde B \\to \\frac{T\\tilde A\\oplus T\\tilde B}{\\ker (d \\tilde f)}$. \n\\end{claim}\n\nThe claim follows from the fact that the following diagram commutes, which is a simple calculation (using \\eqref{connection A} and its analogue for $\\tilde B$).\n\\begin{diagram}\nT A \\oplus T B \\oplus \\mathbb{R} \\oplus \\mathbb{R} & \\rTo^{\\pi\\oplus \\begin{pmatrix} 1 & -1 \\end{pmatrix} \\,} & \\frac{T A\\oplus T B}{\\ker (d f)} \\oplus \\mathbb{R} \\\\\n\\dTo_{\\psi}                                & & \\dTo_{\\varphi}       \\\\\nT\\tilde A\\oplus T\\tilde B & \\rTo^{\\tilde \\pi \\,} & \\frac{T\\tilde A\\oplus T\\tilde B}{\\ker (d \\tilde f)}\n\\end{diagram}\n\nWe can now justify the sign in the statement of the Lemma. The Claim can be used to show that the following diagram commutes: \n\\begin{diagram}\n\\big(\\ker (d f)\\oplus \\mathbb{R}\\big) \\bigoplus \\, \\left(\\frac{T A\\oplus T B}{\\ker (d f)} \\oplus \\mathbb{R}\\right) & \\rTo^{i+\\alpha+s+\\beta\\,} & T  A \\oplus T B \\oplus \\mathbb{R} \\oplus \\mathbb{R} \\\\\n\\dTo_{\\chi \\oplus \\varphi}                                & & \\dTo_{\\psi}       \\\\\n\\ker (d \\tilde f) \\bigoplus \\, \\frac{T\\tilde A\\oplus T\\tilde B}{\\ker (d \\tilde f)} & \\rTo^{\\tilde i + \\tilde s\\,} & T \\tilde A \\oplus T\\tilde B \n\\end{diagram}\nwhere $\\alpha \\oplus \\beta \\colon \\mathbb{R}\\oplus \\mathbb{R} \\to \\mathbb{R}^2$ is the linear map represented by \n$\\begin{pmatrix}\n1 & 1 \\\\\n1 & 0\n\\end{pmatrix}$.\n\nBy Part (1) in Definition \\ref{fibre sum orient}, $\\varphi$ changes orientations by $(-1)^{\\dim B \\dim \\Sigma + \\dim \\tilde B \\dim Y} = (-1)^{\\dim B + 1}$. The map $\\psi$ changes orientations by $(-1)^{\\dim B}$ (the sign comes from the permutation, since the orientations on $\\tilde A$ and $\\tilde B$ behave well with the connections). \nNote that to get the map $\\alpha \\oplus \\beta$ (which has negative determinant), we need to permute $\\mathbb{R}$ with $\\frac{T A\\oplus T B}{\\ker (d f)}$ at the top of the diagram. The permutation picks up no sign, since $\\dim \\frac{T A\\oplus T B}{\\ker (d f)} = \\dim \\Sigma$ is even. \nBy Part (2) in Definition \\ref{fibre sum orient} and the commutativity of the diagram, we conclude that the isomorphism $\\chi$ changes orientations by $(-1)^{\\dim B + 1 + \\dim B + 1} = 1$, as wanted. \n\\end{proof}\n\nThe space \\eqref{fib prod 3} is an $S^1$-bundle over \\eqref{fib prod 2}. With respect to the connections discussed above (see \\eqref{connection A}), we have a decomposition\n\\begin{equation}\nT_{(x,u,y)}\\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)} \\cong \\left( T_{(\\pi_\\Sigma(x),\\pi_\\Sigma\\circ u,\\pi_\\Sigma(y))}\\mathcal{M}(q,p;A) \\right) \\oplus \\mathbb{R}\n\\label{orient splitting 2}\n\\end{equation}\n\n\\begin{lemma} \\label{orient spheres and lifts} \nThe isomorphism \\eqref{orient splitting 2} preserves orientations, if the fibre products on the left and right are given their fibre sum orientations.\n\\end{lemma}\n\n\\begin{proof}\n This follows from applying Lemma \\ref{fib prod lift} twice, to the $S^1$-bundles \n $\\pi_\\Sigma^{-1}\\left(W^s_{\\Sigma}(q) \\right) \\to W^s_{\\Sigma}(q)$, \n ${\\mathcal{M}}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y) \\to \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)$ and \n $\\pi_\\Sigma^{-1}\\left(W^u_{\\Sigma}(p) \\right) \\to W^u_{\\Sigma}(p)$. \n\n Note that to apply Lemma \\ref{fib prod lift}, we need the identifications $T_{\\tilde a}\\tilde A\\cong T_a A\\oplus \\mathbb{R}$ and \n $T_{\\tilde b}\\tilde B\\cong T_b B\\oplus \\mathbb{R}$ (introduced above the statement) to be orientation-preserving.\n This holds when $A$ or $B$ are critical manifolds of $(f_\\Sigma,Z_\\Sigma)$ and when $\\tilde A$ and $\\tilde B$ are the \ncorresponding preimages under $\\pi_\\Sigma$, by \\eqref{orient full lift}. \nOn the other hand, from \\cite{DiogoLisiSplit}*{Lemma 5.22 and discussion after Lemma 7.3} we get this for the coherent orientation on \n$\\tilde B = {\\mathcal{M}}^*_{H,0,\\mathbb{R}\\times Y;k_-,k_+}(A;J_Y)$ and the usual complex orientation on $B=\\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)$.\n %\n\\end{proof}\n\n\n\\begin{lemma} \\label{fib prod vs preim in Sigma}\nThe fibre sum orientation on \\eqref{fib prod 2} differs from the preimage orientation on \\eqref{fib prod 2} by $(-1)^{\\dim W^s_{\\Sigma}(q) \\dim W_{\\Sigma}^u(p)}$. \n\\end{lemma}\n\\begin{proof}\n This follows from applying Lemma \\ref{fibre sum vs preimage} twice. Writing \\eqref{fib prod 2} as \n $$\n(W^s_{\\Sigma}(q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)) \\times_{\\operatorname{ev}} W_{\\Sigma}^u(p)\n $$\nwe apply Lemma \\ref{fibre sum vs preimage} to $W^s_{\\Sigma}(q) \\hookrightarrow \\Sigma$ and $\\operatorname{ev}_\\Sigma^1\\colon \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)\\to \\Sigma$ to get a sign \n$$(-1)^{(\\dim W^s_{\\Sigma}(q) + \\dim \\Sigma)(\\dim \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) + \\dim \\Sigma)} = 1$$\nsince $\\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma)$ and $\\Sigma$ are even dimensional. \nWe then apply Lemma \\ref{fibre sum vs preimage} to $\\operatorname{ev}^2\\colon W^s_{\\Sigma}(q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) \\to \\Sigma$ and $W_{\\Sigma}^u(p) \\hookrightarrow \\Sigma$ to get the sign \n\\begin{align*}\n(-1&)^{\\dim (W^s_{\\Sigma}(q) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) + \\dim \\Sigma) (\\dim W_{\\Sigma}^u(p) + \\dim \\Sigma)} \\\\\n&= (-1)^{\\dim W^s_{\\Sigma}(q) \\dim W_{\\Sigma}^u(p)}.\n\\end{align*}\n\\end{proof}\n\nCombining these results, we get the following.\n\\begin{proposition} \\label{case 1 and GW}\nThe sign with which an element of \\eqref{fib prod 1} contributes to the symplectic homology differential agrees with \nthe Gromov--Witten sign of the projection of this element to \\eqref{fib prod 2}  \niff $M(q)$ is even.\n\\end{proposition}\n\\begin{proof}\nCombining Lemmas \\ref{orient lifts free vs fixed} and \\ref{orient spheres and lifts}, we can relate the fibre sum orientations on \\eqref{fib prod 1} and \\eqref{fib prod 2}:\n\\begin{align*}\n\\left( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\right) \\oplus &\\mathbb{R}_{\\domain} \\oplus \\mathbb{R}_{\\target} 0\\cong (-1)^{M(q)} T_{(x,u,y)}\\overline{\\mathcal{M}(\\wc q_{k_-},\\wh p_{k_+};A)} \\cong \\\\\n&\\cong (-1)^{M(q)} \\left( T_{(\\pi_\\Sigma(x),\\pi_\\Sigma\\circ u,\\pi_\\Sigma(y))}\\mathcal{M}(q,p;A) \\right) \\oplus \\mathbb{R}\n\\end{align*}\nThe factors $\\mathbb{R}_{target}$ on the left and $\\mathbb{R}$ on the right can be identified, yielding an orientation-preserving isomorphism \n\\begin{align*}\n\\left( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\right) \\oplus \\mathbb{R}_{\\domain} \\cong (-1)^{M(q)} T_{(\\pi_\\Sigma(x),\\pi_\\Sigma\\circ u,\\pi_\\Sigma(y))}\\mathcal{M}(q,p;A).\n\\end{align*}\n\nObserve that $\\mathcal{M}(q,p;A)$ in \\eqref{fib prod 2} can be identified with \n \\begin{equation}\n \\mathcal{M}^*_{0,\\Sigma}(A;J_\\Sigma) \\times_{\\operatorname{ev}_\\Sigma} \\big(W^s_{\\Sigma}(q) \\times W_{\\Sigma}^u(p)\\big).\n \\label{fib prod 22}\n \\end{equation}\n \nThe sign with which an element $(x,u,y)\\in\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)$ contributes to the symplectic \nhomology differential is the sign of the zero-dimensional vector space obtained by taking the quotient \nof $T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A)$ by translation in the $s$-variable in the domain of the \nFloer cylinder $u$. This oriented zero-dimensional vector space coincides with the quotient of \n$\\left( T_{(x,u,y)}\\mathcal{M}(\\wh q_{k_-},\\wc p_{k_+};A) \\right) \\oplus \\mathbb{R}_{\\domain}$ by the infinitesimal \n$\\mathbb{C}^*$-action on the domain of $u$. \nOn the other hand, by Definition \\ref{GW orient case 1}, \nthe Gromov--witten sign of $(\\pi_\\Sigma(x),\\pi_\\Sigma\\circ u,\\pi_\\Sigma(y))\\in\\mathcal{M}(q,p;A)$ \nis the sign of the zero-dimensional vector space obtained as the quotient of \n$T_{(\\pi_\\Sigma(x),\\pi_\\Sigma\\circ u,\\pi_\\Sigma(y))}\\mathcal{M}(q,p;A)$ by the infinitesimal $\\mathbb{C}^*$-action,\nwhere $\\mathcal{M}(q,p;A)$ is identified with \\eqref{fib prod 22} with the preimage orientation. \nThe result now follows from \n\\begin{claim}\n If we equip \\eqref{fib prod 2} with the fibre sum orientation and \\eqref{fib prod 22} with the preimage orientation, then the natural identification of the two spaces is orientation-preserving.\n\\end{claim}\n\nBy Lemma \\ref{fib prod vs preim in Sigma}, the fibre sum and preimage orientations on \\eqref{fib prod 2} differ by $(-1)^{\\dim W^s_{\\Sigma}(q) \\dim W_{\\Sigma}^u(p)}$. \nBut this is also the difference between the preimage orientations on \\eqref{fib prod 2} and on \\eqref{fib prod 22}. The Claim now follows.  \n\\end{proof}\n\n\n\\begin{remark}\n The Claim in the previous proof is also a consequence of the following argument. If we equip \\eqref{fib prod 2} and \n \\eqref{fib prod 22} with their fibre sum orientations, then their natural identification preserves orientations, by \n \\cite{JoyceCorners}*{Proposition 7.5.(c)}. \n On the other hand, the fibre sum and preimage orientations on \\eqref{fib prod 22} are the same, by Lemma \\ref{fibre sum vs preimage}.  \n\\end{remark}\n\n\\\n\n\nThere is an analogous argument for Case 2 contributions to the symplectic homology differential. As we saw in \\eqref{fib prod2},\nthey come from elements of fibre products\n\\begin{equation}\n W^s_{Y}(\\wh p) \\times_{\\operatorname{ev}} \\big( \\mathcal{M}_X^*(B;J_W) \\times_{\\tilde \\operatorname{ev}} \\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\big) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p),\n\\label{fib prod case 2}\n\\end{equation}\nwhich are given the fibre sum orientation. An element in this fibre product has an underlying unparameterized plane in $\\mathcal{M}_X^*(B;J_W)$ \n(that can also be thought of as a space of spheres in $X$), which is asymptotic to a Reeb orbit over $p\\in \\Sigma$. Denote by \n$\\operatorname{ev}\\colon \\mathcal{M}_X^*(B;J_W) \\to \\Sigma$ the evaluation map. \n\n\n\\begin{definition} \\label{GW orient case 2}\n Given a generic point $pt \\in \\Sigma$, the preimage orientation on $\\operatorname{ev}^{-1}(pt)$ is called its \n {\\em Gromov--Witten orientation}. The corresponding sign associated to an element in the zero-dimensional\n $\\operatorname{ev}^{-1}(pt)$ is its {\\em Gromov--Witten sign}.\n\\end{definition}\n\nObserve that this is oriented diffeomeorphic to $\\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\{pt\\}$,\nby Lemma \\ref{fibre sum vs preimage}. Analogously to what was pointed out in Remark \\ref{GW signs}, \nthis orientation determines the sign with which an element in $\\operatorname{ev}^{-1}(pt)$ contributes to a \nrelative Gromov--Witten invariant of the pair $(X,\\Sigma,\\omega)$, with a single insertion of the point \n$pt$ (chosen to be generic) in $\\Sigma$, with order of contact $B\\bullet \\Sigma$.  \n\n\n\n\\begin{proposition} \\label{case 2 and gw}\nThe sign with which an element of \\eqref{fib prod case 2} contributes to the symplectic homology \ndifferential agrees with the Gromov--Witten sign of the underlying $J_W$-holomorphic plane in $W$ \n(thought of as a sphere in $X$ with one point constraint in $\\Sigma$ of order of contact \n$k_+-k_-=B\\bullet \\Sigma$ at a generic $p\\in \\Sigma$).\n\\end{proposition}\n\n\n\\begin{proof}\nThe proof is analogous to that of Proposition \\ref{case 1 and GW}. We have oriented isomorphisms\n\\begin{align*}\n \\big[&W^s_{Y}(\\wh p) \\times_{\\operatorname{ev}} \\big( \\mathcal{M}_X^*(B;J_W) \\times_{\\tilde \\operatorname{ev}} \\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\big) \\big]\\times_{\\operatorname{ev}} W_{Y}^u(\\wc p) \\\\\n & \\cong \\big[\\big(\\mathcal{M}_X^*(B;J_W) \\times_{\\tilde \\operatorname{ev}} \\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\big)\\times_{\\operatorname{ev}} W^s_{Y}(\\wh p) \\big] \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p) \\\\ \n & \\cong \\big[\\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\big(\\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\times_{\\operatorname{ev}} W^s_{Y}(\\wh p) \\big)\\big] \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)  \\\\\n & \\cong \\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\big[\\big(\\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\times_{\\operatorname{ev}} W^s_{Y}(\\wh p) \\big) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\\big]  \\\\\n & \\cong (-1)^{1 + M(p)} \\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\big[\\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\times_{\\operatorname{ev}} \\big(W^s_{Y}(\\wh p)  \\times W_{Y}^u(\\wc p)\\big)\\big]  \\\\\n & \\cong (-1)^{1 + M(p)}  \\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\big[\\text{sign}(S^1_d\\times S^1_t \\to S^1_- \\times S^1_+) \\\\\n & \\hspace{4.5cm}  (\\mathcal{M}^*_{1,\\Sigma}(0;J_\\Sigma)\/\\mathbb{C}^*) \\times_{\\operatorname{ev}} \\big(W^s_{\\Sigma}(p) \\times W_{\\Sigma}^u(p)\\big) \\mathbb{R}_p \\big]  \\\\\n & \\cong (-1)^{M(p)}\\big(\\Delta_\\Sigma \\cap (W^s_{\\Sigma}(p)\\times W^u_{\\Sigma}(p))\\big) \\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\mathbb{R}_p \\\\\n & \\cong \\big(\\mathcal{M}_X^*(B;J_W) \\times_{\\operatorname{ev}} \\{p\\}\\big) \\mathbb{R}_p %\n\\end{align*}\nwhere we identified the space of Floer cylinders from $\\wh p$ to $\\wc p$ with the real line $\\mathbb{R}_p$ in the last three lines, with the orientation induced by $s$-translation on the domain (any two such cylinders differ by translation by a constant $s_0\\in \\mathbb{R}$). \nThe first isomorphism preserves orientations, by Lemma \\ref{fibre sum commute} ($\\dim \\big( \\mathcal{M}_X^*(B;J_W) \\times_{\\tilde \\operatorname{ev}} \\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\big) + \\dim Y$ is even). \nThe second and third isomorphisms are orientation-preserving, by the associativity of the fibre sum orientation. The sign in the fourth isomorphism comes from \\cite{JoyceCorners}*{Proposition 7.5.(c)}. \nNote that by Lemma \\ref{fibre sum vs preimage}, the fibre sum and preimage orientations on $\\big[\\mathcal{M}^*_{H,1,\\mathbb{R}\\times Y;k_-,k_+}(0;J_Y) \\times_{ev} \\big(W^s_{Y}(\\wh p)  \\times W_{Y}^u(\\wc p)\\big)\\big]$ agree. \nIn the fifth isomorphism, $\\text{sign}(S^1_d\\times S^1_t \\to S^1_- \\times S^1_+)=-1$ means that the evaluation map\ntaking domain and target rotations to the $S^1$-fibres at the orbits at $-\\infty$ and $+\\infty$ reverses orientations \n(again by the fact that the matrix in \\eqref{rot v} has negative determinant). \nIn the penultimate line, $\\big(\\Delta_\\Sigma \\cap (W^s_{\\Sigma}(p)\\times W^u_{\\Sigma}(p))\\big)$ stands for \nthe sign of the intersection, which is the same as the sign of $(-1)^{M(p)} (W^s_{\\Sigma}(p)\\cap W^u_{\\Sigma}(p))$, \nby Remark \\ref{GuilleminPollack}. The sign of $W^s_{\\Sigma}(p)\\cap W^u_{\\Sigma}(p)$ is 1, by \\eqref{orient stable and unstable}.\n\\end{proof}\n\n\n\n\nFinally, Case 3 contributions to the symplectic homology differential come from\n\\begin{equation}\n W^u_{W}(x) \\times_{\\operatorname{ev}^1_-} \\left(\\mathcal{M}^*_{H}(B;J_W)  \\prescript{}{\\operatorname{ev}^1_+\\,}\\times\\prescript{}{\\operatorname{ev}^2_-}{}  \\mathcal{M}^*_{H,k_+}(0;J_Y) \\right) \\times_{\\operatorname{ev}^2_+} W_{Y}^u(\\wc p)\n \\label{fib prod case 3}\n\\end{equation}\nwith the fibre sum orientation, as in \\eqref{fib prod3} (fix orientations on unstable manifolds of $(-f_W,-Z_W)$, which induces \norientations on the stable manifolds via \\eqref{orient stable and unstable}). \nDespite the subscript (which reminds us that we haven't quotiented by domain automorphisms), the space \n$\\mathcal{M}^*_{H}(B;J_W)$ consists of pseudoholomorphic cylinders in $W$ (recall that $H=0$ in $W$ after stretching the neck),\nwhich by Lemma \\ref{planes = spheres} can be identified with a space of pseudoholomorphic spheres in $X$.\nRecall from Lemma \\ref{planes = spheres} that the map $\\psi \\colon W \\to X$ is a diffeomorphism onto its image \n$X\\setminus \\Sigma$. Denote by $\\operatorname{ev}\\colon \\mathcal{M}^*_{H}(B;J_W)\/\\mathbb{C}^* \\to X\\times \\Sigma$ the evaluation map at $(0,\\infty)$\n(where we quotiented by the space $\\mathbb{C}^*$ of domain automorphisms). Write $W^u_{X}(x)$ for $\\psi\\big(W^u_{W}(x)\\big)$.\n\n\n\n\\begin{definition}\n The preimage orientation on $\\operatorname{ev}^{-1}\\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right)$ is called \n its {\\em Gromov--Witten orientation}. The corresponding sign associated to an element in this \n zero-dimensional manifold is its {\\em Gromov--Witten sign}.\n\\label{GW orient case 3}\n\\end{definition}\n\nIf $W^u_{X}(x) \\subset X$ and $W_{\\Sigma}^u(p)\\subset \\Sigma$ represent homology classes, then the Gromov--Witten orientation determines the signs with which elements in $\\operatorname{ev}^{-1}\\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right)$ contribute to a relative Gromov--Witten invariant. \n\n\\begin{proposition}\nThe sign with which an element of \\eqref{fib prod case 3} contributes to the symplectic homology differential agrees with \nthe Gromov--Witten sign of the underlying $J_W$-holomorphic plane in $W$ (thought of as a sphere in $X$ \nwith one point constraint at $W^u_{X}(x)\\subset X$ and one point constraint in $\\Sigma$ of order of contact \n$k=B\\bullet \\Sigma$ at $W_{\\Sigma}^u(p)\\subset \\Sigma$).\n\\label{case 3 and gw}\n\\end{proposition}\n\\begin{proof}\nDenoting all evaluation maps by $\\operatorname{ev}$, we have the oriented diffeomorphisms\n\\begin{align*}\nW^u_{W}&(x) \\times_{\\operatorname{ev}} \\left(\\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,k_+}(0;J_Y) \\right) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p) \\\\\n&\\cong W^u_{W}(x) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} (\\mathcal{M}^*_{H,k_+}(0;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p))) \\\\\n&\\cong (-1)^{\\dim W_{Y}^u(\\wc p)} \\big(W^u_{W}(x) \\times_{\\operatorname{ev}}  \\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\\big) \\times \\mathbb{R}_2\\\\\n&\\cong (-1)^{\\dim W_{Y}^u(\\wc p) + \\dim W^u_{W}(x)} \\big(\\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} \\left(W^u_{W}(x) \\times W_{Y}^u(\\wc p)\\right)\\big)\\times \\mathbb{R}_2\n\\end{align*}\nwhere we denote by $\\mathbb{R}_2$ the domain of the $s$-coordinate of a Floer cylinder in $\\mathcal{M}^*_{H,k_+}(0;J_Y)$. We used \nthe associativity of the fibre sum orientation, the oriented diffeomorphism $\\mathcal{M}^*_{H,k_+}(0;J_Y)\\cong \\mathbb{R}_2 \\times Y$ \n(see \\cite{DiogoLisiSplit}*{Lemma 5.22}), the fact that $W$ and $\\mathcal{M}^*_{H}(B;J_W)$ are \neven dimensional and \\cite{JoyceCorners}*{Proposition 7.5.(a)\\&(c)}. \nNow, if we think of $\\mathcal{M}^*_{H}(B;J_W)$ \nas a space of pseudoholomorphic spheres in $X$, then using the fibre sum orientation convention, we also have the oriented diffeomorphisms\n\\begin{align*}\n&\\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} \\left(W^u_{W}(x) \\times W_{Y}^u(\\wc p)\\right) \\\\\n &\\quad \\cong (-1)^{\\dim W^u_{W}(x) + \\dim W_{\\Sigma}^u(p)} \\left((\\mathcal{M}^*_{H}(B;J_W)\/\\mathbb{C}^*) \\times_{\\operatorname{ev}} \\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right)\\right)\\times \\mathbb{R}_1\n\\end{align*}\nwhere $\\mathbb{R}_1$ is the domain of the $s$-coordinate of a Floer cylinder in $\\mathcal{M}^*_{H}(B;J_W)$. By Lemma \n\\ref{fibre sum vs preimage}, the fibre sum and preimage orientations agree on \n$(\\mathcal{M}^*_{H}(B;J_W)\/\\mathbb{C}^*) \\times_{\\operatorname{ev}} \\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right)$. Combining the two \ncalculations, we get an oriented diffeomorphism between \n$$\n W^u_{W}(x) \\times_{\\operatorname{ev}}  \\mathcal{M}^*_{H}(B;J_W) \\times_{\\operatorname{ev}} \\mathcal{M}^*_{H,k_+}(0;J_Y) \\times_{\\operatorname{ev}} W_{Y}^u(\\wc p)\n$$\nwith the fibre sum orientation (which gives rigid contributions to the symplectic homology differential when we quotient by $\\mathbb{R}_1\\times \\mathbb{R}_2$) and \n$$\n\\left((\\mathcal{M}^*_{H}(B;J_W)\/\\mathbb{C}^*) \\times_{\\operatorname{ev}} \\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right)\\right )\\times \\mathbb{R}_1 \\times \\mathbb{R}_2\n$$\nwith the preimage orientation \n(which gives the Gromov--Witten sign when we quotient by $\\mathbb{R}_1\\times \\mathbb{R}_2$, according to Definition \\ref{GW orient case 3}). \nThe result now follows. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\nWe now define some curve counts that will be useful when we write\na formula for the symplectic homology differential.  Each of the\nCases 1 through 3 motivates one of the following definitions.\n\n\\begin{definition} \\label{D:n_A}\n Let $q,p\\in \\Crit(f_\\Sigma)$ and $A \\in H_2(\\Sigma;\\mathbb{Z})$. Define\n $$\n n_A(q,p) := \\# \\mathcal{M}_A(q,p),\n $$\n where (using the notation of \\eqref{fib prod 2})\n $$\n \\mathcal{M}_A(q,p) := \\mathcal{M}(q,p;A) \/ \\mathbb{C}^*.\n $$\nHere, $\\mathcal{M}(q,p;A)$ is given the Gromov--Witten orientation (see Definition \\ref{GW orient case 1}). \n\\end{definition} \n \nThis makes sense when $ \\dim \\mathcal{M}_A(q,p) = 2\\, \\langle c_1(T\\Sigma),A\\rangle + M(p) - M(q)-2 = 0$. \n\n\n\n\n\\begin{definition} \\label{D:n_B}\nLet $B\\in H_2(X;\\mathbb{Z})$ and define\n$$\nn_B := \\# \\mathcal{M}_B,\n$$\nwhere\n \\begin{align*}\n \\mathcal{M}_B :=\\operatorname{ev}^{-1}(pt).\n \\end{align*}\nHere, $\\operatorname{ev}^{-1}(pt)$ is as in Definition \\ref{GW orient case 2}, and it is given its Gromov--Witten orientation. \n\\footnote{We thank Viktor Fromm for observing that the correct constraint is a generic point in $\\Sigma$, rather than a submanifold of possibly higher dimension.}\n\\end{definition}\n\nThis makes sense when $ \\dim \\mathcal{M}_B = 2\\, \\langle c_1(TX),B\\rangle - 2 (B\\bullet\\Sigma) - 2 = 0$. \n\n\n\n\\begin{definition} \\label{D:n_B(x,p)}\n Let $p\\in \\Crit(f_\\Sigma)$, $x\\in \\Crit(f_W)$ and $B \\in H_2(X;\\mathbb{Z})$. Define\n $$\n n_B(x,p) := \\# \\mathcal{M}_B(x,p)\n $$\n where, using the notation from Definition \\ref{GW orient case 3},\n \\begin{align*}\n \\mathcal{M}_B(x,p) := \\operatorname{ev}^{-1}\\left(W^u_{X}(x) \\times W_{\\Sigma}^u(p)\\right).\n \\end{align*}\n\\end{definition}\n\nThe formula makes sense when $ \\dim \\mathcal{M}_B(x,p) = 2\\, \\langle c_1(TX),B\\rangle - 2(B\\bullet\\Sigma) + M(p) + M(x) -2n = 0$ (see the index formula\nfor relative Gromov--Witten invariants, for instance in \\cite{IonelParkerRelative}*{Lemma 4.2}).\n\n\n\n\n\\begin{remark} \\label{index bounds}\nThe coefficients $n_A(q,p)$ above can only be non-trivial if \n$$\n\\langle c_1(T\\Sigma),A\\rangle = (\\tau_X - K) \\, \\omega(A) \\leq n.\n$$\n\nby the dimension formula for $\\mathcal{M}_A(q,p)$. Similarly, the coefficients $n_B$ can \nonly be non-trivial if \n$$\n\\, \\langle c_1(TX),B\\rangle - B\\bullet\\Sigma = (1 - K\/\\tau_X) \\, \\langle c_1(TX),B\\rangle = (\\tau_X - K) \\, \\omega(B) = 1.\n$$\nThe coefficients $n_B(x,p)$ can only be non-zero if\n$$\n\\, \\langle c_1(TX),B\\rangle - B\\bullet\\Sigma = (1 - K\/\\tau_X) \\, \\langle c_1(TX),B\\rangle = (\\tau_X - K) \\, \\omega(B) \\leq n.\n$$\nThese inequalities are useful when computing the symplectic homology differential, as we will see in an example below. \n\\end{remark}\n\n \n\n\n\n\\section{A formula for the symplectic homology differential}\n\n\nRecall that the symplectic chain complex \\eqref{eqn:chain complex} is \n$$\nSC_*(W,H) = \\left(\\bigoplus_{k>0} \\bigoplus_{p_k\\in \\text{Crit}(f_\\Sigma)}\\mathbb{Z}\\langle \\widecheck p_k, \\widehat p_k \\rangle\\right) \\oplus \\left(\\bigoplus_{x\\in \\text{Crit}(f_W)}\\mathbb{Z}\\langle x \\rangle\\right)\n$$\nas an abelian group, where $f_\\Sigma: \\Sigma \\to \\mathbb{R}$ and $f_W : W \\to \\mathbb{R}$ are Morse functions, with associated \nMorse--Smale gradient-like vector fields $Z_\\Sigma$ and $Z_W$, respectively. \nWe have assigned orientations to all critical manifolds in $\\Sigma$, $Y$ and $W$, in Section  \\ref{compute signs}.\nThe orientation conventions for the Morse differential were specified in \\cite{DiogoLisiSplit}*{Section 7}. \n\n\n\n\n\n\nWe are now ready to write the differential on $SC_*(W,H)$.\n\\begin{theorem} \\label{T:differential}\n The differential on \\eqref{eqn:chain complex}, denoted $\\partial$, is as follows. Given $p\\neq q \\in \\Crit(f_\\Sigma)$, the coefficient of $\\widehat q_{k_-}$ in $\\partial \\widecheck p_{k_+}$ is \n \\begin{align*}\n \\langle \\partial (\\widecheck p_{k_+}) , \\widehat q_{k_-} \\rangle = (-1)^{M(q)} (k_+-k_-) &\\sum_{A \\in H_2(\\Sigma;\\mathbb{Z})} \\delta_{k_+-k_-,\\langle c_1(N\\Sigma),A \\rangle} \\,\\, n_A(q,p) + \\\\\n & + \\delta_{k_+,k_-} \\, \\langle  \\partial_{f_Y}(\\widecheck p_{k_+}),  \\widehat q_{k_-} \\rangle\n \\end{align*}\n where $2\\langle c_1(T\\Sigma),A \\rangle + M(p) - M(q) -2 = 0$, $\\partial_{f_Y}$ is the Morse differential in $Y$ and the $\\delta_{*,*}$ are Kronecker deltas. \n \n If $p\\in \\Crit (f_\\Sigma)$, then \n $$\n \\langle \\partial (\\widecheck p_{k_+}) , \\widehat p_{k_-} \\rangle =  (k_+-k_-) \\, \\sum_{B \\in H_2(X;\\mathbb{Z})} \\delta_{k_+-k_-,(B\\bullet\\Sigma)} \\, n_B\n $$\n where $(1-K\/\\tau_X) \\, \\langle c_1(T X),B \\rangle - 1 = 0$. If $W$ is Weinstein, this term is trivial if $n\\geq 3$ and the minimal Chern number of spheres in $\\Sigma$ is at least 2.\n \n \n Given $p\\in \\Crit(f_\\Sigma)$ and $x\\in \\Crit(f_W)$, \n $$\n \\langle \\partial (\\widecheck p_{k_+}) , x \\rangle = k_+ \\sum_{B \\in H_2(X;\\mathbb{Z})} \\delta_{k_+,(B\\bullet\\Sigma)} \\, n_B(x,p)\n $$\n where $2(1-K\/\\tau_X) \\, \\langle c_1(TX),B\\rangle + M(p) + M(x) -2n = 0$. \n \n Given $x\\in \\Crit(f_W)$,\n $$\n \\partial(x) = \\partial_{-f_W}(x),\n $$\n where $\\partial_{-f_W}$ is the Morse differential in $W$ with respect to the function $-f_W$.\n\\end{theorem}\n\n\\begin{proof}\nThe split Floer differential on \\eqref{eqn:chain complex} was introduced at the end of Section \\ref{cascades after splitting}. Proposition \\ref{P:split SH cases} describe those split Floer cylinders with cascades that can contribute to the differential.  \n\nThe two types of contributions in $\\langle \\partial (\\widecheck p_{k_+}) , \\widehat q_{k_-} \\rangle$ are (1) and (0), respectively, in Proposition \\ref{P:split SH cases}. The former are given by Floer cylinders in $\\mathbb{R} \\times Y$ and correspond to Case 1 in Sections \\ref{Floer_to_holo} and \\ref{holo_to_Floer}. According to Propositions \\ref{Ansatz1} and \\ref{Ansatz2}, counts of Floer cylinders in $\\mathbb{R}\\times Y$ are equivalent to counts of pseudoholomorphic cylinders in $\\mathbb{R} \\times Y$. By Lemma \\ref{L:cylinders to spheres} and the discussion that follows, the latter are equivalent to counts of rigid pseudoholomorphic spheres in $\\Sigma$ (given by the $n_A(q,p)$), with additional factors $k_+-k_-$ in accordance with Lemma \\ref{number of lifts}. \nThe sign comes from Proposition \\ref{case 1 and GW}. \n\nThe contributions in $\\langle \\partial (\\widecheck p_{k_+}) , \\widehat p_{k_-} \\rangle$ correspond to (2) in Proposition \\ref{P:split SH cases}, namely split Floer cylinders with one augmentation. They correspond to Case 2 in Sections \\ref{Floer_to_holo} and \\ref{holo_to_Floer}. We can appeal again to Propositions \\ref{Ansatz1} and \\ref{Ansatz2}, to relate punctured Floer cylinders in $\\mathbb{R}\\times Y$ to counts of punctured pseudoholomorphic cylinders in $\\mathbb{R} \\times Y$. The latter are branched covers of trivial cylinders in this case, and project to constant spheres in $\\Sigma$. \nThe number of lifts of these constant spheres is $k_+-k_-$, again by Lemma \\ref{number of lifts} (in the notation of that Lemma, $k^- + k^-_P = k^+$ in this case, since the spheres in $\\Sigma$ are constant). \nThe counts of augmentation planes in $W$, capping the punctures in these cylinders, are given by the numbers $n_B$, by Lemma \\ref{planes = spheres}. \nThe comment about $W$ Weinstein follows from Lemma \\cite{DiogoLisiSplit}*{Lemma 6.6} %\n(and, by Lemma \\ref{augmentation rel GW} below, it can be interpreted as saying that a certain relative Gromov--Witten invariant of the triple $(X,\\Sigma,\\omega)$ vanishes).  \n\nThe contributions in $\\langle \\partial (\\widecheck p_{k_+}) , x \\rangle$ are the ones in Proposition \\ref{P:split SH cases}, \nand consist of a split Floer cylinder in $\\mathbb{R}\\times Y$ with one end asymptotic to a Reeb orbit in $Y$, matching the asymptotics \nof a pseudohomorphic plane in $W$, which is in turn connected to a critical point of $f_W$ by a flow line of $Z_W$. They correspond \nto Case 3 in Sections \\ref{Floer_to_holo} and \\ref{holo_to_Floer}. We can appeal to Propositions \\ref{Ansatz1} and \\ref{Ansatz2}, \nto relate the Floer cylinders in $\\mathbb{R}\\times Y$ to trivial pseudoholomorphic cylinders in $\\mathbb{R} \\times Y$. \nThe component in $W$ is a pseudoholomorphic cylinder $\\tilde v_0 \\colon \\mathbb{R}\\times S^1 \\to W$ with a removable singularity at \n$-\\infty$. At $+\\infty$, the cylinder converges to a Reeb orbit of multiplicity $k$. Therefore, precomposing $\\tilde v_0$ with \nconstant rotations on the domain (as in the discussion around \\eqref{rot v0}), we get $k_+$ many maps \nwith the correct asymptotic marker condition at $+\\infty$. By Lemma \\ref{planes = spheres}, such $\\tilde v_0$ correspond to \npseudoholomorphic spheres in $X$ with appropriate tangency condition to $\\Sigma$. The contribution of these configurations to \nthe differential will then be $k_+ \\, n_B(x,p)$, as wanted.\n\n\nThe only contributions to the differential of $x\\in \\Crit(f_W)$ are from the Morse differential, because $W$ is exact and hence \nhas no non-constant pseudoholomorphic spheres. They correspond to the second part of Case 0 in Proposition \\ref{P:split SH cases}. \n\\end{proof}\n\n\\begin{remark}\nFor every $p\\in \\Crit(f_\\Sigma)$ and $k_+>0$, we have $\\partial_{f_Y}(\\widehat{p}_{k_+}) = 0$. There are two flow lines of $-Z_Y$ from $\\widehat{p}_{k_+}$, which cancel one another.  \n\\end{remark}\n\n\\begin{remark}\nAt least if there were no contributions $\\langle \\partial (\\widecheck p_{k_+}) , x \\rangle$, it would be immediate from Theorem \\ref{T:differential} that $\\partial^2 = 0$.\n\\end{remark}\n\n\n\\subsection{Relation to Gromov--Witten invariants}\n\nAll the coefficients contributing to the differential in Theorem \\ref{T:differential} are either combinatorial or topological, except for the  $n_A(q,p)$, the $n_B$ and the $n_B(x,p)$. These are harder to determine, but can sometimes be related to Gromov--Witten invariants, absolute or relative. \n\nWe denote by $\\gw^\\Sigma_{0,k,A}(C_1,\\ldots,C_k)$ the genus 0 Gromov--Witten invariant of $(\\Sigma,\\omega_\\Sigma)$, counting pseudohomolorphic spheres in $\\Sigma$ in class $A\\in H_2(\\Sigma;\\mathbb{Z})$ with $k$ marked points constrained to go through representatives of the classes $C_i\\in H_*(\\Sigma;\\mathbb{Z})$. To see a definition of these invariants when $(\\Sigma,\\omega_\\Sigma)$ is monotone, see \\cite{McDuffSalamon}.  \nWe also denote denote by $\\gw^{(X,\\Sigma)}_{0,k,(s_1,\\ldots,s_l),B}(C_1,\\ldots,C_k;D_1,\\ldots,D_l)$ the genus 0 relative Gromov--Witten invariant of $(X,\\Sigma,\\omega)$, counting pseudohomolorphic spheres in $X$ class $B\\in H_2(X;\\mathbb{Z})$ with $k$ marked points constrained to go through representatives of the classes $C_i\\in H_*(X;\\mathbb{Z})$, and $l$ additional marked points constrained to be tangent to $\\Sigma$ with order of contact $s_j$ and go through representatives of the classes $D_j\\in H_*(\\Sigma;\\mathbb{Z})$. For details on how to define these invariants, see \\cites{IonelParkerRelative,LiRuan,TehraniZingerSSF} (for an algebro-geometric approach, see \\cite{LiRelative}). \nNote that our assumptions that $(X,\\omega)$ is monotone with monotonicity constant $\\tau_X$, that $\\Sigma$ is Poincar\\'e-dual to $K\\omega$ and that $\\tau_X - K > 0$ imply that the tuple $(X,\\Sigma,\\omega)$ is {\\em strongly semi-positive}, as in \\cite{TehraniZingerSSF}*{Definition 4.7}. \n\n\nRecall that a Morse function is {\\em perfect} if the corresponding Morse differential vanishes. Furthermore, it is {\\em lacunary} if it does not have critical points of consecutive Morse index.\n\\begin{lemma}\\label{lacunary GW}\n If the Morse function $f_\\Sigma$ is perfect, then $W^s_\\Sigma(q)$ and $W^u_\\Sigma(p)$ represent classes in $H_*(\\Sigma;\\mathbb{Z})$ and \n $$\n n_A(q,p) = \\gw^\\Sigma_{0,2,A}\\big([W^s_\\Sigma(q)],[W^u_\\Sigma(p)]\\big).\n $$\n  \n If the Morse function $f_W$ is also perfect, then $W^u_{X}(x)\\coloneq\\psi\\big(W^u_W(x)\\big)$ represents a class in the image of the map $H_*(X\\setminus \\Sigma;\\mathbb{Z}) \\to H_*(X;\\mathbb{Z})$ and \n $$\n n_B(x,p) = \\gw^{X,\\Sigma}_{0,1,(B\\bullet\\Sigma),B}\\big([W^u_{X}(x)];[W^u_\\Sigma(p)]\\big).\n $$\n  \n If $f_\\Sigma$ is lacunary, then \n $$\n \\langle \\partial_{f_Y}(\\widecheck p_{k_+}), \\widehat q_{k_+}\\rangle = \\, \\left\\langle c_1(N\\Sigma),\\big(W^s_{\\Sigma}(q) \\cap W^u_{\\Sigma}(p)\\big)\\right\\rangle \n $$\n where $N\\Sigma$ is the normal bundle to $\\Sigma$ in $X$. \n \n\\end{lemma}\n\n\\begin{proof}\nFor the first statement, the fact that the stable and unstable submanifolds of $f_\\Sigma$ define homology classes in $\\Sigma$ implies that the $n_A(q,p)$ are the Gromov--Witten invariants in the statement, and not merely numbers defined at the (Morse) chain level. \nThe orientation conventions in Definition \\ref{D:n_A} are precisely the ones used in Gromov--Witten theory (recall Remark \\ref{GW signs}). \nThe second statement is proven along similar lines. Note that the $W^u_W(x)$ define homology classes in $W$, but that the $W^s_W(x)$ (which we do not consider) would define relative classes in $H_*(W,Y;\\mathbb{Z})$ instead. \nOn a technical note, an admissible $J_X$ as in Definition \\ref{admissible J} satisfies the vanishing normal Nijenhuis tensor condition \\cite{TehraniZingerSSF}*{(4.6)}, and is hence suitable for defining relative Gromov--Witten invariants. \n\nWe now discuss the last statement. For index reasons, the coefficients $\\langle \\partial_{f_Y}(\\widecheck p_{k_+}),\\widehat q_{k_+}\\rangle$ \ncan only be non-zero if $\\ind_{f_\\Sigma} (p) - \\ind_{f_\\Sigma}(q) = 2$. Since $f_\\Sigma$ is assumed to be lacunary, \n$W^s_{\\Sigma}(q) \\cap W^u_{\\Sigma}(p)$, together with $p$ and $q$, form a 2-sphere $S$. The restriction of $Y \\to \\Sigma$ to $S$ \nis an $S^1$-bundle $E$ over $S$, and $(f_Y,Z_Y)$ restricts to a Morse--Smale pair $(f_E,Z_E)$ on $E$. By the description of the \nGysin sequence in Morse homology \\cite{OanceaThesis}*{(3.26)}, $\\langle \\partial_{f_E}(\\widecheck p), \\widehat q\\rangle$ is the \nEuler class of $E\\to S$ integrated over $S$, which is precisely what we wanted to show. \n\\end{proof}\n\nThe following result does not need any hypotheses on the auxiliary Morse functions. \n\\begin{lemma} \\label{augmentation rel GW}\nThe count of augmentation planes\n $$\n n_B = \\gw^{X,\\Sigma}_{0,0,(B\\bullet\\Sigma),B}(\\emptyset;[pt])\n $$ \nis the genus 0 relative Gromov--Witten invariant of $(X,\\Sigma,\\omega)$, in class $B\\in H_2(X,\\mathbb{Z})$, with no point constraints in $X$ and order of contact $B\\bullet\\Sigma$ to a fixed (generic) point in $\\Sigma$.\n\\end{lemma}\n\nThe proof is as in the second case of Lemma \\ref{lacunary GW}.\n\n\n\\begin{remark}\nWhen $f_\\Sigma$ is not a perfect Morse function, the stable and unstable manifolds of its critical points may not represent classes in $H_*(\\Sigma;\\mathbb{Z})$. In such cases, the $n_A(q,p)$ can still be thought of as {\\em chain level Gromov--Witten numbers} of $\\Sigma$. \nThese are not invariants, and may thus be harder to compute. Similarly, if $f_W$ is not perfect, the $n_B(x,p)$ are only chain level relative Gromov--Witten numbers. \n\nWe will see below an example in which we compute Gromov--Witten invariants of $(\\mathbb{CP}^1\\times \\mathbb{CP}^1,\\Delta,\\omega_{FS}\\oplus \\omega_{FS})$ using the integral $J$, even though it is not `cylindrical near $\\Sigma$', as we require of admissible $J_X$ in Definition \\ref{admissible J}. That is not a problem, though, since the numbers we are computing are invariants. \n\n\\end{remark}\n\n\n\n\n\\begin{remark} \\label{R:computations}\n\nComputing Gromov--Witten invariants can be a very hard problem, both in\nthe absolute and relative case, but it has been done for a large class\nof important examples, especially using tools from algebraic geometry.\nWe implicitly assume then that the definitions of Gromov--Witten\ninvariants in algebraic geometry and in symplectic geometry agree in\nsettings where they are both defined.  For more details about the relation\nbetween symplectic and algebraic absolute Gromov--Witten invariants,\nsee \\cites{SiebertGW,LiTian}.\n\nFor computations of absolute invariants, see for instance\n\\cites{Beauville,ZingerCompleteIntersections}.  Gathmann wrote\na computer program called \\texttt{GROWI} to compute absolute and\nrelative Gromov--Witten invariants of hypersurfaces in projective spaces\n\\cite{growi}.\n\nRelative invariants are often harder to compute than absolute\nones. Nevertheless, under certain assumptions, a relative\ninvariant where all intersections with $\\Sigma$ are transverse\n(i.e.~have order of tangency 1) can be related to an absolute\ninvariant where  one forgets $\\Sigma$ (in the spirit of\nthe divisor axiom for absolute Gromov--Witten invariants)\n\\cites{McDuffAbsoluteRelative,HuRuan,TehraniZingerAbsoluteRelative}.\n\nIt is also worth pointing out that the relative invariants\nof a tuple $(X,\\Sigma,\\omega)$ can in principle be obtained\nfrom the absolute invariants of $X$ and of $\\Sigma$ (including\nmore complicated insertions like $\\psi$-classes, in general)\n\\cite{MaulikPandharipande}. Nevertheless, it can be computationally\nquite challenging to use this fact to compute relative invariants of\n$(X,\\Sigma,\\omega)$.\n\nDepending on what part of symplectic homology one wants to compute, it may also\nbe possible to ignore the contributions coming from relative Gromov--Witten\ninvariants. \nBy Theorem \\ref{T:differential}, if \n$X$ is at least 6 dimensional, the minimal Chern number of $\\Sigma$ is at least 2 \nand $W$ is Weinstein, then the symplectic homology differential does not have\nany $n_A$ contributions (i.e.~corresponding to augmentation planes).\nIn this case then, the \n{\\em positive} (or {\\em large-action} or {\\em high energy}) \nsymplectic homology differential of $W$ \\cite{BOExactSequence} then ignores\nterms of the form $n_B(x,p)$ and thus \nonly has\ncontributions from the (absolute) Gromov--Witten invariants.\n\\end{remark}\n\n\n\\begin{remark} \\label{Cone}\nThe following was pointed out to us by D. Pomerleano. We also thank R. Leclercq and M. Sandon for helpful conversations about their related work in progress.\n Theorem \\ref{T:differential} may seem unsatisfactory since it relates the differential $\\partial$, which is a chain level object depending on the model we use to compute symplectic homology, to counts of pseudoholomorphic spheres which, at least under the assumption of Lemma \\ref{lacunary GW}, are given by Gromov--Witten invariants, which are homological. \n The chain complex $SC_*$ in \\eqref{eqn:chain complex} has a quotient complex, referred to in \\cite{BOExactSequence} as $SC_*^+$, obtained by modding out generators coming from critical points of $f_W$. \n We can write $SC_*^+$ as a complex of the form \n \\begin{equation}\n C_*(\\Sigma)[t]t \\oplus \\big(C_*(\\Sigma)[t]t \\big) [1]\n\\label{SC+}\n \\end{equation}\n where $C_*(\\Sigma)$ denotes the Morse complex of $f_\\Sigma$ and $t^k$ encodes generators associated to Reeb orbits of multiplicity $k$. The variable $t$ has degree $2\\frac{\\tau_X - K}{K}$, by \\eqref{eqn:grading Y} (which should also force us to take a global shift of $C_*(\\Sigma)$ by $1-n$; we don't do this to avoid making the notation even heavier). The first summand contains the orbits decorated with $\\widecheck{} \\,$ and the second summand, with the degree shift up by 1, contains the orbits decorated with $\\widehat{} \\,$. \n Theorem \\ref{T:differential} implies that the differential in \\eqref{SC+} is given by a matrix \n $$\n \\partial^+ = \\begin{pmatrix}\n  d_\\Sigma & \\Delta \\\\\n  0 & d_\\Sigma\n \\end{pmatrix} \n $$\n where $d_\\Sigma$ is the Morse differential in $\\Sigma$.\n The off-diagonal term $\\Delta\\colon C_*(\\Sigma)[t]t \\to C_*(\\Sigma)[t]t$ is obtained by counting flow lines in $\\Sigma$ connecting critical points of index difference 2, pseudohomolorphic spheres in $\\Sigma$ (the terms involving $n_A(p,q)$), and \n pseudoholomorphic spheres in $X$ (the terms involving $n_B$). \nThis means that $SC_*^+$ is a cone complex on $\\Delta$. There is an induced long exact sequence\n$$\n\\ldots \\to H_*(\\Sigma)[t]t[1] \\to SH_{*}^+(W,H) \\to H_{*}(\\Sigma)[t]t \\stackrel{[\\Delta]}{\\to} H_{*}(\\Sigma)[t]t[1] \\to \\ldots\n$$\nwith connecting homomorphism $[\\Delta]$ (we are not being careful keeping track of the degrees $*$ in the sequence). \nThis can be thought of as the long exact sequence in \\cite{BOExactSequence}, in the special case when the Liouville manifold is the completion of $X\\setminus \\Sigma$.  \nIt is also related to the spectral sequence in \\cite{SeidelBiasedView}*{(3.2)}.\n\nWe can think of this exact sequence as a more invariant consequence of Theorem \\ref{T:differential}. Note that if we were interested in computing $SH_*^+(W)$ with field coefficients, then the exact sequence suggests that it should be enough to compute Gromov--Witten invariants, instead of chain level counts of pseudoholomorphic spheres, even if one does not start with a perfect Morse function in $\\Sigma$. \nObserve that $SC_*(W)$ is also a cone complex on a chain map $SC_*^+(W) \\to C_*(W)$, counting pseudoholomorphic spheres in $X$ with tangency conditions in $\\Sigma$ (the terms involving $n_B(p,x)$). The same argument as above should enable us to compute $SH_*(W)$ with the additional information of relative Gromov--Witten invariants of the triple $(X,\\Sigma,\\omega)$, instead of the non-invariant counts $n_B(p,x)$, at least over field coefficients. \n\n\n\\end{remark}\n \n\\part{Example: \\texorpdfstring{$T^*S^2$}{T*S2}}\n\n\n\n\n\n\n\nWe now illustrate the results in this paper with the computation of the symplectic homology of the completion $W$ of $X\\setminus \\Sigma$, where $(X,\\Sigma, \\omega) = (\\mathbb{CP}^1 \\times \\mathbb{CP}^1,\\Delta,\\omega_{FS}\\oplus \\omega_{FS})$, where $\\Delta$ is the diagonal and $\\omega$ restricts to the Fubini-Study form of area 1 in each factor. The manifold $W$ is symplectomorphic to $T^*S^2$ (see Exercise 6.20 in \\cite{McDuffSalamonIntro}), so its symplectic homology is isomorphic to the homology of the based loop space of $S^2$. We will see that our computation recovers this result.  \n$X$ and $\\Sigma$ are both monotone, with $\\tau_{X} = 2$ and $\\tau_\\Sigma = 1$. Also, $[\\Sigma]$ is Poincar\\'e-dual to $[\\omega]$, so $K=1$. The $S^1$-bundle $Y\\to \\Sigma$ is $\\mathbb{R} P^3$.\n\n\\section{The coefficients in the differential}\nThe manifold $\\Sigma = \\mathbb{CP}^1$ admits a lacunary Morse function $f_\\Sigma: \\Sigma\n\\to \\mathbb{R}$, with one critical point of Morse index 0, denoted by $m$ (for {\\em minimum}),\nand one critical point of index 2, denoted by $M$ (for {\\em maximum}).\nLemma \\ref{lacunary GW} implies that all the numbers $n_A(q,p)$ (giving the Case 1 contributions to the differential) are absolute Gromov--Witten invariants of $\\Sigma = \\mathbb{CP}^1$. Definition \\ref{D:n_A} and Remark \\ref{index bounds} indicate that the only potentially non-trivial invariant is \n$$\nn_L(M,m) = \\gw_{L,2}^{\\mathbb{CP}^1}\\left([pt],[pt] \\right) = 1\n$$\nwhere $L\\in H_2(\\mathbb{CP}^1;\\mathbb{Z})$ is the class of a line. This invariant counts the\nnumber of lines in $\\mathbb{CP}^1$ through two generic points, which is 1 (the integral\ncomplex structure on $\\mathbb{CP}^1$ is regular). \n\nThe manifold $W = T^*S^2$ also admits a lacunary Morse function $f_W: W \\to \\mathbb{R}$\ngrowing at infinity, with one critical point of index 0, denoted by $e$, and one\ncritical point of index 2, denoted by $c$. \nThe manifold $W^u(c)$ represents the zero section in $T^*S^2$. We orient it as\n$L_1 - L_2$, where $L_1,L_2 \\in H_2(\\mathbb{CP}^1 \\times \\mathbb{CP}^1;\\mathbb{Z})$ are homology classes\nrepresenting the factors $\\mathbb{CP}^1 \\times \\{ pt \\}$ and $\\{ pt \\} \\times \\mathbb{CP}^1$,\nrespectively. We will refer to a sphere representing the classes $L_1$ and $L_2$\nas \\textit{horizontal} and \\textit{vertical}, respectively.\n\n\nDefinition \\ref{D:n_A} and Remark \\ref{index bounds} again indicate that the only potentially non-trivial coefficients in Cases 2 and 3 are \n$$\nn_{L_i}, \\quad n_{L_i}(e,M), \\quad n_{L_i}(c,m), \\quad n_{2L_i}(e,m) \\quad \\text{and} \\quad n_{L_1+L_2}(e,m).\n$$\n\nLemmas \\ref{lacunary GW} and \\ref{augmentation rel GW} imply that these numbers are relative Gromov--Witten invariants of $(\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta,\\omega)$. We will now compute these invariants.\n\n\n\n\n\\begin{proposition} \n \\begin{enumerate}\n  \\item $n_{L_i} = \\gw_{L_i,0,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1,\\Delta}\\left(\\emptyset; pt \\right) = 1$,\n  \n\\noindent \nfor $i=1,2$ (the point constraint is in $\\Delta$, not in $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$);\n  \n  \\item $n_{L_i}(e,M) = \\gw_{L_i,1,(1)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta}\\left([pt]; [\\Delta] \\right) = 1$;\n  \n  \\item $n_{L_i}(c,m) = \\gw_{L_i,1,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1, \\Delta}([S^2];[pt]) = (-1)^i$\n \n \\noindent \n  where $S^2 \\subset T^* S^2$ is the zero section, oriented so as to represent the homology class $L_1 - L_2$; \n  \n  \\item $n_{2L_i}(e,m) = \\gw_{2L_i,1,(2)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta}\\left([pt]; [pt] \\right) = 0$;\n  \n  \\item $n_{L_1+L_2}(e,m) = \\gw_{L_1+L_2,1,(2)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta}\\left([pt]; [pt] \\right) = 1$.\n   \\end{enumerate}\n   \\label{P:rel GW computations}\n\\end{proposition}\n\nWe need an auxiliary result. \n\n\\begin{lemma} \\label{L:integrable J}\n The product complex structure on $X = \\mathbb{CP}^1 \\times \\mathbb{CP}^1$ is regular for all\n immersed holomorphic spheres $U \\colon \\mathbb{CP}^1 \\to \\mathbb{CP}^1 \\times \\mathbb{CP}^1$.\nIn particular, this is true when $U_*[\\mathbb{CP}^1] \\in \\{L_1,L_2,L_1 + L_2\\}$. \n\\end{lemma}\n\\begin{proof}\n Observe that the symplectic form is integral and so for any holomorphic sphere\n $U$, $\\int_{\\mathbb{CP}^1} U^*\\omega \\geq 1$. \n By \\cite{McDuffSalamon}*{Lemma 3.3.3}, we just need to make sure that $\\langle c_1(TX),U_*[\\mathbb{CP}^1]\\rangle \\geq 1$, but $\\langle c_1(TX),U_*[\\mathbb{CP}^1]\\rangle = 2 \\int_{\\mathbb{CP}^1} U^*\\omega \\geq 2$.\n\\end{proof}\n\nFor the proof of cases (1) through (3) of Proposition \\ref{P:rel GW\ncomputations}, one could appeal to the relation between relative and\nabsolute Gromov--Witten invariants when all the tangencies to $\\Sigma$\nare of order 1, as mentioned in Remark \\ref{R:computations}.  This is\nnot necessary, though, since Lemma \\ref{L:integrable J} allows us to\ndo the computations explicitly using the product complex structure on\n$\\mathbb{CP}^1 \\times \\mathbb{CP}^1$. This satisfies the vanishing normal Nijenhuis\ntensor condition \\cite{TehraniZingerSSF}*{(4.6)}, and can thus be used\nfor computing relative Gromov--Witten invariants.\n\n\\begin{proof}[Proof of Proposition \\ref{P:rel GW computations}]\n\n  \\begin{enumerate}\n  \\item Given $i=1,2$,\n$$\n\\gw_{L_i,0,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1,\\Delta}\\left(\\emptyset; [pt] \\right) = 1\n$$\nis the fact that if one fixes any point $p\\in\\Delta$, then there is exactly one holomorphic sphere in class $L_i$ that goes through $p$.\n  \\item Given $i=1,2$, \n$$\n\\gw_{L_i,1,(1)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta}\\left([pt]; [\\Delta] \\right) = 1\n$$\nexpresses the fact that, if we fix any point $p \\in \\mathbb{CP}^1 \\times \\mathbb{CP}^1 \\setminus \\Delta$, there is a unique holomorphic sphere in class $L_i$ that goes through $p$ and intersects $\\Delta$. \n  \\item Given $i,j=1,2$,\n$$\n\\gw_{L_i,1,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1, \\Delta}(L_j;[pt]) = 1-\\delta_{i,j}.\n$$\nThis means, for instance, that there is a unique vertical sphere in $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ intersecting a horizontal sphere and a generic point $p \\in\\mathbb{CP}^1\\times \\mathbb{CP}^1$, but there is no horizontal sphere intersecting another horizontal sphere and a generic $p$. \nThis is because two different horizontal spheres do not intersect. Since $S^2 = L_1 - L_2$,\n$$\n\\gw_{L_i,1,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1, \\Delta}([S^2];[pt]) = \\gw_{L_i,1,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1, \\Delta}(L_1;[pt]) - \\gw_{L_i,1,(1)}^{\\mathbb{CP}^1\\times \\mathbb{CP}^1, \\Delta}(L_2;[pt]) = (-1)^i\n$$\n \\item Given $i = 1,2$,  \n $$\n \\gw_{2L_i,1,(2)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1,\\Delta}([pt];[pt]) = 0\n $$\nencodes the fact that holomorphic curves in classes $2 L_i \\in H_2(\\mathbb{CP}^1 \\times \\mathbb{CP}^1; \\mathbb{Z})$ are covers of either vertical or horizontal spheres, and therefore cannot go through one generic point in $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ and another generic point in $\\Delta$. \n\n\n \\item To prove that \n $$\n \\gw_{L_1+L_2,1,(2)}^{\\mathbb{CP}^1 \\times \\mathbb{CP}^1, \\Delta}\\left(pt; pt \\right) = 1\n $$\nwe show that, up to domain automorphism, there is a unique holomorphic map $U: \\mathbb{CP}^1 \\to \\mathbb{CP}^1 \\times \\mathbb{CP}^1$, such that $U_*[\\mathbb{CP}^1] = L_1+L_2 \\in H_2(\\mathbb{CP}^1 \\times \\mathbb{CP}^1;\\mathbb{Z})$, $U(\\infty) = (\\infty, 1)$, $U(0) = (0,0)$, and such that $U$ intersects the diagonal $\\Delta \\subset \\mathbb{CP}^1 \\times \\mathbb{CP}^1$ in a non-transverse way. Since \n$$\nU(\\infty) = U \\big([1;0]\\big) = (\\infty, 1) = \\big([1;0],[1;1]\\big),\n$$\nwe can write in homogeneous coordinates \n$$\nU\\big([z;1]\\big) = \\big([a z + b; 1], [z + c; z + d]\\big)\n$$ \nwhich we will abbreviate as \n$$\nU(z) = \\left(a z + b, \\frac{z + c}{z + d}\\right).\n$$\nSince $U(0) = (0,0)$, we get $b=c=0$, and so $U(z) = \\big(az, z\/(z+d)\\big)$. Now, for the tangency condition, note that\n$$\nU'(z) = \\left(a, \\frac{d}{(z + d)^2}\\right)\n$$\nand so $U'(0) = \\big(a, 1\/d\\big)$. So $U$ is tangent to the diagonal at $(0,0)$ precisely when $a = 1\/d$. Therefore, the space of maps $U$ can be identified with the space of $a\\in \\mathbb{C} ^*$. Taking a quotient by the group $\\mathbb{C}^*$ of automorphisms of the domain $(\\mathbb{CP}^1, \\{0,\\infty\\})$, we get the uniqueness of $U$.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{The group \\texorpdfstring{$SH_*(T^*S^2)$}{SH*(T*S2)}}\n\nWe can now compute the symplectic homology chain complex for $T^*S^2$. Using the functions $f_\\Sigma$ and $f_W$ introduced above, we have\n$$\nSC_*(T^*S^2) = \\mathbb{Z} \\left\\langle e,c\\right\\rangle \\oplus \\bigoplus_{k>0} \\mathbb{Z}\\left\\langle \\wc m_k, \\wh m_k, \\wc M_k, \\wh M_k \\right\\rangle\n$$\nWe can use Theorem \\ref{T:differential}, Lemma \\ref{lacunary GW} and Proposition \\ref{P:rel GW computations} to compute the differential. Given $k\\geq 2$,\n\\begin{align*}\n\\partial \\wc m_{k+1} &=  2 \\, n_L(M,m) \\, \\wh M_{k-1} \\,+ \\,\\big( n_{L_1} \\, + \\, n_{L_2} \\big)\\, \\wh m_k = \\\\\n&= 2 \\times 1 \\, \\wh M_{k-1} \\, + \\, \\big(1 + 1 \\big)\\, \\wh m_k = 2 \\, \\wh M_{k-1} + 2 \\, \\wh m_k \n\\end{align*}\nand\n\\begin{align*}\n\\partial \\wc M_{k} &= \\big( n_{L_1} \\, + \\, n_{L_2} \\big)\\, \\wh M_{k-1} \\,+\\langle c_1(\\mathbb{R} P^3 \\to \\mathbb{CP}^1),\\mathbb{CP}^1\\rangle \\, \\wh m_{k} = 2 \\, \\wh M_{k-1} + 2 \\, \\wh m_k  \n\\end{align*}\n\nThe remaining differentials are\n\\begin{align*}\n\\partial \\wc m_{2} &= \\big(n_{L_1} \\, + \\, n_{L_2}\\big)\\, \\wh m_1 + 2 \\, n_{L_1 + L_2}(e,m) \\, e = 2 \\, \\wh m_1 + 2 \\, e ,\n\\end{align*}\n\\begin{align*}\n\\partial \\wc M_{1} &= \\big(n_{L_1}(e,M) + n_{L_2}(e,M) \\big) \\, e \\,+\\, \\langle c_1(\\mathbb{R} P^3 \\to \\mathbb{CP}^1),\\mathbb{CP}^1\\rangle \\, \\wh m_1 = \\\\\n&= 2 \\, e + 2 \\, \\wh m_1 \n\\end{align*}\nand\n\\begin{align*}\n\\partial \\wc m_{1} &= \\,\\big(n_{L_1}(c,m) + n_{L_2}(c,m) \\big) \\, c = (-1+1)\\, c = 0. \n\\end{align*}\n\n\n\nSumming up, the non-trivial contributions to the differential are\n\\[\\left\\{\n\\begin{array}{l}\n\\partial \\wc{m}_{k+1} = 2 \\, \\wh{m}_{k} + 2 \\, \\wh{M}_{k-1} \\\\\n\\partial \\wc{M}_{k} = 2 \\wh{m}_{k} + 2 \\, \\wh{M}_{k-1} \\\\\n\\partial \\wc{m}_2 = 2 \\, \\wh{m}_1 + 2 \\, e \\\\\n\\partial \\wc{M}_1 = 2 \\, \\wh{m}_1 + 2 \\, e\n\\end{array}\n\\right.\n\\]\nfor $k\\geq 2$. We can conclude the following.\n\\begin{proposition}\n\\[\nSH_*(T^*S^2 ; \\mathbb{Z}) = \\mathbb{Z} \\left \\langle c, \\wc{m}_{1}, e, \\wc{M}_{k} - \\wc{m}_{k+1}, \\wh{M}_{k} \\right \\rangle \\oplus \\mathbb{Z} \/ 2 \\left \\langle e + \\wh{m}_{1}, \\wh{M}_{k} + \\wh{m}_{k+1} \\right \\rangle\n\\]\nwhere we take all $k \\geq 1$. \n\\end{proposition}\n\n\n\nWe can compare these results with earlier computations of the symplectic homology of $T^*S^2$. By Viterbo's Theorem, it is isomorphic as a ring to the homology of the free loop space of $S^2$ \\cite{AbbondandoloSchwarzRing}. As a graded abelian group, this was computed by McCleary \\cite{McClearyGeodesics}. \nAs a ring, it was computed in \\cite{CohenJonesYan} to be \n\\begin{equation} \\label{CJY}\n H_*(LS^2;\\mathbb{Z}) \\cong ( \\Lambda(b) \\otimes \\mathbb{Z}[a,v] ) \/ (a^2 , ab, 2av)\n\\end{equation}\nfor some $a\\in H_0(LS^2;\\mathbb{Z})$, $b\\in H_1(LS^2;\\mathbb{Z})$ and $v\\in H_4(LS^2;\\mathbb{Z})$. The following tables show how our computations match these, at least as graded abelian groups. For the free part, we get\n\\[\n\\begin{array}{c|c|c|c|c|c}\nSH_d(T^*S^2) & c & -\\wc{m}_{1} & e & \\wc{M}_{k} - \\wc{m}_{k+1} & \\wh{M}_{k} \\\\% & \\wc{M}_{2} - \\wc{m}_{3} & - \\wh{m}_{3} & \\wc{M}_{3} - \\wc{m}_{4} & - \\wh{m}_{4} & \\ldots \\\\\n\\hline\nH_d(L S^2) & a & b & 1 & bv^k & v^k \\\\\n\\hline\nd & 0 & 1 & 2 & 2k+1 & 2k+2\n\\end{array}\n\\]\nfor $k\\geq 1$. \nFor the $\\mathbb{Z} \/ 2$-torsion part:\n\\[\n\\begin{array}{c|c|c}\nSH_d(T^*S^2) & e + \\wh{m}_{1} & \\wh{M}_{k} + \\wh{m}_{k+1} \\\\% & \\wh{M}_{2} + \\wh{m}_{3} & \\wh{M}_{3} + \\wh{m}_{4} & \\ldots \\\\\n\\hline\nH_d(L S^2) & av & av^{k+1} \\\\\n\\hline\nd & 2 & 2k+2\n\\end{array}\n\\]\nfor $k\\geq 1$. One can also adapt the arguments presented above to describe the ring structure on symplectic homology and show that it matches \\eqref{CJY}, but that is beyond the scope of this paper, and will be addressed in future work. \n\n\n \n\n\n\\def$'${$'$}\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{AbouzaidGeneration}{article}{\n      author={Abouzaid, Mohammed},\n       title={A geometric criterion for generating the {F}ukaya category},\n        date={2010},\n        ISSN={0073-8301},\n     journal={Publ. Math. Inst. Hautes \\'Etudes Sci.},\n      number={112},\n       pages={191\\ndash 240},\n         url={http:\/\/dx.doi.org.umiss.idm.oclc.org\/10.1007\/s10240-010-0028-5},\n      review={\\MR{2737980}},\n}\n\n\\bib{AbouzaidViterbo}{incollection}{\n      author={Abouzaid, Mohammed},\n       title={Symplectic cohomology and {V}iterbo's theorem},\n        date={2015},\n   booktitle={Free loop spaces in geometry and topology},\n      series={IRMA Lect. Math. Theor. 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Topol.},\n      volume={21},\n      number={6},\n       pages={3313\\ndash 3389},\n         url={https:\/\/doi.org\/10.2140\/gt.2017.21.3313},\n      review={\\MR{3692968}},\n}\n\n\\bib{EkholmNg}{article}{\n      author={Ekholm, Tobias},\n      author={Ng, Lenhard},\n       title={Legendrian contact homology in the boundary of a subcritical\n  {W}einstein 4-manifold},\n        date={2015},\n        ISSN={0022-040X},\n     journal={J. Differential Geom.},\n      volume={101},\n      number={1},\n       pages={67\\ndash 157},\n         url={http:\/\/projecteuclid.org\/euclid.jdg\/1433975484},\n      review={\\MR{3356070}},\n}\n\n\\bib{EliashbergPolterovichQuasi}{unpublished}{\n      author={{Eliashberg}, Yakov},\n      author={{Polterovich}, Leonid},\n       title={Symplectic quasi-states on the quadric surface and {L}agrangian\n  submanifolds},\n        date={2010},\n        note={arXiv:1006.2501},\n}\n\n\\bib{FabertII}{unpublished}{\n      author={Fabert, Oliver},\n       title={Higher algebraic structures in {H}amiltonian {F}loer theory\n  {II}},\n        date={2014},\n        note={arXiv:1412.2682},\n}\n\n\\bib{Fauck}{article}{\n      author={Fauck, Alexander},\n       title={Rabinowitz--{F}loer homology on {B}rieskorn spheres},\n        date={2015},\n        ISSN={1073-7928},\n     journal={Int. Math. Res. Not. IMRN},\n      number={14},\n       pages={5874\\ndash 5906},\n         url={http:\/\/dx.doi.org\/10.1093\/imrn\/rnu109},\n      review={\\MR{3384461}},\n}\n\n\\bib{SymplecticHomology}{article}{\n      author={Floer, Andreas},\n      author={Hofer, Helmut},\n       title={Symplectic homology. {I}. {O}pen sets in {${\\bf C}^n$}},\n        date={1994},\n        ISSN={0025-5874},\n     journal={Math. Z.},\n      volume={215},\n      number={1},\n       pages={37\\ndash 88},\n         url={https:\/\/doi.org\/10.1007\/BF02571699},\n      review={\\MR{1254813}},\n}\n\n\\bib{Floer88}{article}{\n      author={Floer, Andreas},\n       title={The unregularized gradient flow of the symplectic action},\n        date={1988},\n        ISSN={0010-3640},\n     journal={Comm. Pure Appl. 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Math. Inst. Hautes \\'Etudes Sci.},\n      volume={122},\n       pages={65\\ndash 168},\n         url={http:\/\/dx.doi.org\/10.1007\/s10240-015-0073-1},\n      review={\\MR{3415066}},\n}\n\n\\bib{GirouxOpenBook}{inproceedings}{\n      author={Giroux, Emmanuel},\n       title={G\\'{e}om\\'{e}trie de contact: de la dimension trois vers les\n  dimensions sup\\'{e}rieures},\n        date={2002},\n   booktitle={Proceedings of the {I}nternational {C}ongress of\n  {M}athematicians, {V}ol. {II} ({B}eijing, 2002)},\n   publisher={Higher Ed. Press, Beijing},\n       pages={405\\ndash 414},\n      review={\\MR{1957051}},\n}\n\n\\bib{Giroux_remarks_Donaldson}{article}{\n      author={Giroux, Emmanuel},\n       title={Remarks on {D}onaldson's symplectic submanifolds},\n        date={2017},\n        ISSN={1558-8599},\n     journal={Pure Appl. Math. Q.},\n      volume={13},\n      number={3},\n       pages={369\\ndash 388},\n      review={\\MR{3882202}},\n}\n\n\\bib{GanatraPomerleano1}{unpublished}{\n      author={Ganatra, Sheel},\n      author={Pomerleano, Daniel},\n       title={A {L}og {PSS} morphism with applications to {L}agrangian\n  embeddings},\n        date={2016},\n        note={arXiv:1611.06849},\n}\n\n\\bib{GanatraPomerleano2}{unpublished}{\n      author={Ganatra, Sheel},\n      author={Pomerleano, Daniel},\n       title={Symplectic cohomology rings of affine varieties in the\n  topological limit},\n        date={2018},\n        note={arXiv:1811.03609},\n}\n\n\\bib{GuilleminPollack}{book}{\n      author={Guillemin, Victor},\n      author={Pollack, Alan},\n       title={Differential topology},\n   publisher={Prentice-Hall, Inc., Englewood Cliffs, N.J.},\n        date={1974},\n      review={\\MR{0348781}},\n}\n\n\\bib{GromanOpen}{unpublished}{\n      author={Groman, Yoel},\n       title={Floer theory on open manifolds},\n        date={2015},\n        note={arXiv:1510.04265},\n}\n\n\\bib{Gromov85}{article}{\n      author={Gromov, Mikhail},\n       title={Pseudo holomorphic curves in symplectic manifolds},\n        date={1985},\n        ISSN={0020-9910},\n     journal={Invent. Math.},\n      volume={82},\n      number={2},\n       pages={307\\ndash 347},\n         url={https:\/\/doi.org\/10.1007\/BF01388806},\n      review={\\MR{809718}},\n}\n\n\\bib{HuRuan}{article}{\n      author={Hu, JianXun},\n      author={Ruan, YongBin},\n       title={Positive divisors in symplectic geometry},\n        date={2013},\n        ISSN={1674-7283},\n     journal={Sci. China Math.},\n      volume={56},\n      number={6},\n       pages={1129\\ndash 1144},\n         url={http:\/\/dx.doi.org\/10.1007\/s11425-012-4557-x},\n      review={\\MR{3063960}},\n}\n\n\\bib{HWZ4}{incollection}{\n      author={Hofer, Helmut},\n      author={Wysocki, Kris},\n      author={Zehnder, Eduard},\n       title={Properties of pseudoholomorphic curves in symplectisation. {IV}.\n  {A}symptotics with degeneracies},\n        date={1996},\n   booktitle={Contact and symplectic geometry ({C}ambridge, 1994)},\n      series={Publ. Newton Inst.},\n      volume={8},\n   publisher={Cambridge Univ. 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(2)},\n      volume={157},\n      number={1},\n       pages={45\\ndash 96},\n         url={http:\/\/dx.doi.org.umiss.idm.oclc.org\/10.4007\/annals.2003.157.45},\n      review={\\MR{1954264}},\n}\n\n\\bib{JoyceCorners}{incollection}{\n      author={Joyce, Dominic},\n       title={On manifolds with corners},\n        date={2012},\n   booktitle={Advances in geometric analysis},\n      series={Adv. Lect. Math. (ALM)},\n      volume={21},\n   publisher={Int. Press, Somerville, MA},\n       pages={225\\ndash 258},\n      review={\\MR{3077259}},\n}\n\n\\bib{Kerman}{article}{\n      author={Kerman, Ely},\n       title={Rigid constellations of closed {R}eeb orbits},\n        date={2017},\n        ISSN={0010-437X},\n     journal={Compos. 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Soc., Providence, RI},\n       pages={143\\ndash 170},\n      review={\\MR{1492536}},\n}\n\n\\bib{McClearyGeodesics}{incollection}{\n      author={McCleary, John},\n       title={Homotopy theory and closed geodesics},\n        date={1990},\n   booktitle={Homotopy theory and related topics ({K}inosaki, 1988)},\n      series={Lecture Notes in Math.},\n      volume={1418},\n   publisher={Springer, Berlin},\n       pages={86\\ndash 94},\n         url={http:\/\/dx.doi.org\/10.1007\/BFb0083695},\n      review={\\MR{1048178}},\n}\n\n\\bib{McDuffAbsoluteRelative}{unpublished}{\n      author={McDuff, Dusa},\n       title={Comparing absolute and relative {G}romov--{W}itten invariants},\n        date={2008},\n        note={arXiv:math.SG\/0809.3534},\n}\n\n\\bib{McLean1}{article}{\n      author={McLean, Mark},\n       title={Lefschetz fibrations and symplectic homology},\n        date={2009},\n        ISSN={1465-3060},\n     journal={Geom. Topol.},\n      volume={13},\n      number={4},\n       pages={1877\\ndash 1944},\n         url={http:\/\/dx.doi.org.umiss.idm.oclc.org\/10.2140\/gt.2009.13.1877},\n      review={\\MR{2497314}},\n}\n\n\\bib{McLeanMBSequence}{misc}{\n      author={McLean, Mark},\n       title={Computing symplectic homology of affine varieties (using spectral\n  sequences)},\n        date={2016},\n  url={http:\/\/www.math.stonybrook.edu\/~markmclean\/talks\/spectralsequencealltogether.pdf},\n        note={Slides from minicourse given at SFT VIII},\n}\n\n\\bib{Miranda}{book}{\n      author={Miranda, Rick},\n       title={Algebraic curves and {R}iemann surfaces},\n      series={Graduate Studies in Mathematics},\n   publisher={American Mathematical Society, Providence, RI},\n        date={1995},\n      volume={5},\n        ISBN={0-8218-0268-2},\n         url={https:\/\/doi.org\/10.1090\/gsm\/005},\n      review={\\MR{1326604}},\n}\n\n\\bib{MaulikPandharipande}{article}{\n      author={Maulik, Davesh},\n      author={Pandharipande, Rahul},\n       title={A topological view of {G}romov--{W}itten theory},\n        date={2006},\n     journal={Topology},\n      volume={45},\n      number={5},\n       pages={887\\ndash 918},\n      review={\\MR{MR2248516 (2007e:14092)}},\n}\n\n\\bib{McDuffSalamon}{book}{\n      author={McDuff, Dusa},\n      author={Salamon, Dietmar},\n       title={{$J$}-holomorphic curves and symplectic topology},\n      series={American Mathematical Society Colloquium Publications},\n   publisher={American Mathematical Society, Providence, RI},\n        date={2004},\n      volume={52},\n        ISBN={0-8218-3485-1},\n         url={https:\/\/doi.org\/10.1090\/coll\/052},\n      review={\\MR{MR2045629}},\n}\n\n\\bib{McDuffSalamonOld}{book}{\n      author={McDuff, Dusa},\n      author={Salamon, Dietmar},\n       title={{$J$}-holomorphic curves and quantum cohomology},\n      series={University Lecture Series},\n   publisher={American Mathematical Society, Providence, RI},\n        date={1994},\n      volume={6},\n        ISBN={0-8218-0332-8},\n         url={https:\/\/doi.org\/10.1090\/ulect\/006},\n      review={\\MR{1286255}},\n}\n\n\\bib{McDuffSalamonIntro}{book}{\n      author={McDuff, Dusa},\n      author={Salamon, Dietmar},\n       title={Introduction to symplectic topology},\n     edition={Second},\n      series={Oxford Mathematical Monographs},\n   publisher={The Clarendon Press, Oxford University Press, New York},\n        date={1998},\n        ISBN={0-19-850451-9},\n      review={\\MR{1698616}},\n}\n\n\\bib{NguyenThesis}{thesis}{\n      author={Nguyen, Khoa},\n       title={On symplectic homology of the complement of a positive normal\n  crossing divisor in a projective variety},\n        type={Ph.D. Thesis},\n        date={2014},\n}\n\n\\bib{OanceaThesis}{thesis}{\n      author={Oancea, Alexandru},\n       title={La suite spectrale de {L}eray--{S}erre en la cohomologie de\n  {F}loer pour vari\\'et\\'es symplectiques compactes \\`a bord de type contact},\n        type={Ph.D. Thesis},\n        date={2003},\n}\n\n\\bib{Opshtein_Polarizations}{article}{\n      author={Opshtein, Emmanuel},\n       title={Polarizations and symplectic isotopies},\n        date={2013},\n        ISSN={1527-5256},\n     journal={J. Symplectic Geom.},\n      volume={11},\n      number={1},\n       pages={109\\ndash 133},\n  url={http:\/\/projecteuclid.org.umiss.idm.oclc.org\/euclid.jsg\/1362146735},\n      review={\\MR{3022923}},\n}\n\n\\bib{Pascaleff}{unpublished}{\n      author={Pascaleff, James},\n       title={On the symplectic cohomology of log {C}alabi--{Y}au surfaces},\n        date={2013},\n        note={arXiv:1304.5298},\n}\n\n\\bib{Ritter}{article}{\n      author={Ritter, Alexander~F.},\n       title={Topological quantum field theory structure on symplectic\n  cohomology},\n        date={2013},\n        ISSN={1753-8416},\n     journal={J. Topol.},\n      volume={6},\n      number={2},\n       pages={391\\ndash 489},\n         url={http:\/\/dx.doi.org.umiss.idm.oclc.org\/10.1112\/jtopol\/jts038},\n      review={\\MR{3065181}},\n}\n\n\\bib{RitterNegativeBundles}{article}{\n      author={Ritter, Alexander~F.},\n       title={Floer theory for negative line bundles via {G}romov--{W}itten\n  invariants},\n        date={2014},\n        ISSN={0001-8708},\n     journal={Adv. Math.},\n      volume={262},\n       pages={1035\\ndash 1106},\n         url={http:\/\/dx.doi.org\/10.1016\/j.aim.2014.06.009},\n      review={\\MR{3228449}},\n}\n\n\\bib{RobbinSalamonAsymptotics}{article}{\n      author={Robbin, Joel~W.},\n      author={Salamon, Dietmar~A.},\n       title={Asymptotic behaviour of holomorphic strips},\n        date={2001},\n        ISSN={0294-1449},\n     journal={Ann. Inst. H. Poincar\\'{e} Anal. Non Lin\\'{e}aire},\n      volume={18},\n      number={5},\n       pages={573\\ndash 612},\n         url={https:\/\/doi.org\/10.1016\/S0294-1449(00)00066-4},\n      review={\\MR{MR1849689}},\n}\n\n\\bib{SalamonFloerHomology}{incollection}{\n      author={Salamon, Dietmar},\n       title={Lectures on {F}loer homology},\n        date={1999},\n   booktitle={Symplectic geometry and topology ({P}ark {C}ity, {UT}, 1997)},\n      series={IAS\/Park City Math. Ser.},\n      volume={7},\n   publisher={Amer. Math. Soc., Providence, RI},\n       pages={143\\ndash 229},\n         url={http:\/\/dx.doi.org\/10.1016\/S0165-2427(99)00127-0},\n      review={\\MR{1702944}},\n}\n\n\\bib{SeidelBiasedView}{incollection}{\n      author={Seidel, Paul},\n       title={A biased view of symplectic cohomology},\n        date={2008},\n   booktitle={Current developments in mathematics, 2006},\n   publisher={Int. Press, Somerville, MA},\n       pages={211\\ndash 253},\n      review={\\MR{2459307 (2010k:53153)}},\n}\n\n\\bib{SeidelLFIII}{unpublished}{\n      author={Seidel, Paul},\n       title={{F}ukaya {$A_\\infty$}-structures associated to {L}efschetz\n  fibrations. {III}},\n        date={2016},\n        note={arXiv:1608.04012},\n}\n\n\\bib{SiefringAsymptotics}{article}{\n      author={Siefring, Richard},\n       title={Relative asymptotic behavior of pseudoholomorphic\n  half-cylinders},\n        date={2008},\n        ISSN={0010-3640},\n     journal={Comm. Pure Appl. Math.},\n      volume={61},\n      number={12},\n       pages={1631\\ndash 1684},\n         url={http:\/\/dx.doi.org\/10.1002\/cpa.20224},\n      review={\\MR{2456182 (2009k:32029)}},\n}\n\n\\bib{SiefringMultipleLimits}{article}{\n      author={Siefring, Richard},\n       title={Finite-energy pseudoholomorphic planes with multiple asymptotic\n  limits},\n        date={2017},\n        ISSN={0025-5831},\n     journal={Math. 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+{"text":"\\section{Introduction}\n\\label{sec_intro}\n\nOwing to its high accuracy-to-cost ratio, Kohn-Sham density functional theory (KS DFT) is presently the primary\nbuilding block of successes of quantum chemistry in disciplines that stretch from biochemistry to materials\nscience. \\cite{Bur-JCP-12,PGB15,MH17,GHBE17,VT20,Sim2022} \nDFT calculations consume a significant fraction of the world's supercomputing power \\cite{Sherrill2020}\nand tens of thousands of scientific papers report DFT calculations with the number ever growing. \\cite{PGB15} \nKS DFT is in principle exact, but in practice it requires approximations to one piece of the total energy, the so-called exchange-correlation (XC) functional, which encodes the quantum,  fermionic and Coulombic nature of electrons. \n\n\nThe construction of modern XC approximations draws from different approaches. \nSome of them are\nbased on \nforms fulfilling\n some known exact constraints, \\cite{Bur-JCP-12,SunRuzPer-PRL-15} some have been fitted to large\ndatabases,\\cite{MH17,VT20} and the most recent XC approximations are machine learned. \\cite{Kalita2021,fn_dm21,Nagai2020}\nRegardless of these differences in\ntheir design, nearly all current DFT approximations are constructed\nfrom the same ingredients (or features)\nthat form the ``Jacob's ladder''.\\cite{PerSch-AIP-01,Ham-SCI-17} \n\nDespite the progress,\\cite{fn_dm21} state-of-the-art XC approximations have been greatly successful mainly in describing only weak and moderate electronic correlations. \\cite{MH17,GHBE17} The inability of state-of-the-art DFT to capture strong correlations hampers its reliability and predictive power. \\cite{Bur-JCP-12,CohMorYan-CR-12,PGB15,Sim2022}\nOver the last two decades,  the strongly interacting limit of DFT (SIL)  \\cite{Sei-PRA-99,SeiGorSav-PRA-07,GorVigSei-JCTC-09,ButDepGor-PRA-12,FriGerGor-arxiv-22} has been explored and \na rigorous theory has been established. \nThis theory reveals mathematical objects that are very different from the ingredients that are used for building standard XC approximations (semilocal quantities and KS orbitals forming the Jacob's ladder). By offering building blocks for XC functionals\ntailored to describe strong correlations, the SIL has a potential to solve the long-standing problem of DFT simulations of strong electronic correlations. \n\nHere we give a summary of the development of the SIL in different contexts:\nthe development of the theory itself, its practical realization, and the development of approximations drawing from it.\nWe discuss paths for using this limit in different ways to solve the problem of strong correlations within DFT and discuss how it has enabled the construction of a range of quantities that can guide the further development of DFT. We also give an overview of how the SIL has motivated the development of methods that go beyond DFT, such as wavefunction methods delivering highly accurate noncovalent interactions.\\cite{DaaFabDelGorVuc-JPCL-21}\n\n\n\n\\section{Theory}\n\\label{sec_theory}\n\n\n\\subsection{Exchange-correlation functional in DFT}\n\\label{sec_xc}\n\nUsing the Levy-Lieb (LL) constrained-search formalism\\cite{Lev-PNAS-79,Lie-IJQC-83} the ground state energy and density of a many-electron system with an external potential $v$ can be obtained as\n\\begin{equation}\\label{eq:gs}\nE_{\\rm GS}[v] = \\min_\\rho \\left\\lbrace F[\\rho]+\\int v({\\bf r}) \\rho ({\\bf r}) \\mathrm{d}{\\bf r} \\right\\rbrace,\n\\end{equation}\nwhere $F[\\rho]$ is the $\\lambda=1$ (physical) value of the generalized universal LL functional for arbitrary coupling constant $\\lambda$,\n\\begin{equation}\\label{eq.LLalpha}\nF^{\\lambda}[\\rho]= \\min_{\\Psi\\mapsto\\rho} \\langle \\Psi|\\hat{T} +\\lambda\\,\\hat{V}_{ee} | \\Psi \\rangle.\n\\end{equation}\nIn the Kohn-Sham formalism, the functional $F[\\rho]=F^{\\lambda=1}[\\rho]$ is partitioned as\n\\begin{equation}\\label{eq:funi_part}\nF[\\rho]= T_s[\\rho]+U[\\rho]+E_{xc}[\\rho],\n\\end{equation}\nwhere $T_s[\\rho]=F^{\\lambda=0}[\\rho]$ is the KS non--interacting kinetic energy, $U[\\rho]$ is the Hartree (mean-field) energy, and $E_{xc}[\\rho]$ is the exchange--correlation energy.\nThe adiabatic connection (AC) formula for the XC functional reads\\cite{LanPer-SSC-75,GunLun-PRB-76}\n\\begin{equation}\nE_{xc}[\\rho]= \\int_0^1 W_\\lambda[\\rho]\\mathrm{d}\\lambda,\n\t\t\\label{eq:ac_xc}\n\\end{equation}\nwhere $W_\\lambda[\\rho]$ is the global AC integrand:\n\\begin{equation}\nW_\\lambda[\\rho]= \\langle \\Psi_{\\lambda}[\\rho]  | \\hat{V}_{ee}| \\Psi_{\\lambda}[\\rho]\\rangle - U[\\rho],\n\t\t\\label{eq:w1}\n\\end{equation}\nand $\\Psi_{\\lambda}[\\rho]$ is the minimizing wavefunction in Eq.~\\eqref{eq.LLalpha}.\nWe can also write $W_\\lambda[\\rho]$ as\n\\begin{equation} \\label{eq:wl}\nW_\\lambda[\\rho]=\\int w_\\lambda({\\bf r}) \\rho({\\bf r}) \\mathrm{d}{\\bf r},\n\\end{equation}\nwhere $ w_\\lambda({\\bf r}) $  is a $\\lambda$-dependent XC energy density that is not uniquely defined. \nIn the present work, we adhere to the  definition \nin terms of the electrostatic potential of the XC hole ({\\it conventional DFT gauge}),\\cite{BurCruLam-JCP-98,VucIroSavTeaGor-JCTC-16,VucGor-JPCL-17,VucLevGor-JCP-17}\n\\begin{equation} \\label{eq:wxc_def}\n    w_\\lambda({\\bf r}) = \\frac{1}{2} \\int_0^\\infty \\frac{h_{\\rm xc}^\\lambda({\\bf r},u)}{u}4 \\pi u^2 \\mathrm{d}u, \n\\end{equation}\nwhere $h_{\\rm xc}^\\lambda({\\bf r},u)$ is the spherical average (over directions of ${\\bf u}={\\bf r}'-{\\bf r}$) of the XC hole around a given position ${\\bf r}$. The XC hole, in turn, is determined by the pair density associated to the wavefunction $\\Psi_\\lambda[\\rho]$.\n\n\\section{Strongly interacting limit of DFT}\n\\label{sec_sil}\nIn this section, we briefly review the physical ideas behind the strongly interacting limit of DFT. \nFor a mathematically more rigorous and comprehensive overview, we recommend ref.~\\onlinecite{FriGerGor-arxiv-22}.\n\nThe strongly interacting limit of DFT corresponds to the situation in which the electron-electron repulsion dominates in $F^{\\lambda}[\\rho]$ of Eq.~\\eqref{eq.LLalpha}, yielding\\cite{Sei-PRA-99, SeiGorSav-PRA-07}\n\\begin{equation} \\label{eq.asy2}\n   \\lim_{\\lambda\\to\\infty} \\tfrac{1}{\\lambda}\\, F^\\lambda[\\rho] = V_{ee}^{\\rm SCE}[\\rho],\n\\end{equation} \nwhere $V_{ee}^{\\rm SCE}[\\rho]$\nis the {\\it strictly correlated electrons (SCE) functional} defined by the minimization of the electronic repulsion over wavefunctions $\\Psi$ with density $\\rho({\\bf r})$:\n\\begin{equation} \\label{eq.VSCE}\n   V_{ee}^{\\rm SCE}[\\rho] = \\min_{\\Psi\\mapsto\\rho}\n   \\langle\\Psi| \\hat{V}_{ee}|\\Psi\\rangle.\n\\end{equation}\nThe limit in eq. \\eqref{eq.asy2} has been established rigorously \\cite{CotFriKlu-CPAM-13,CotFriKlu-ARMA-18, Lew-CRM-18} with convergence of the `energies', i.e., the value of the functional $F^\\lambda[\\rho]$ divided by $\\lambda$ tends to $V_{ee}^{\\rm SCE}[\\rho]$, and qualitative convergence of the wave-functions squared, i.e., for any $(\\Psi_{\\lambda})_{\\lambda>0}$  minimizing \\eqref{eq.LLalpha} the $N$-body position density $P_\\lambda^N=\\sum_{\\rm spins}|\\Psi_\\lambda|^2$ converges to a limiting $N$-body  distribution $P^N_\\infty[\\rho]$. The latter minimizes the following alternative definition of the SCE functional \n\\begin{equation}\n\\label{eq:infmin}\n   V_{ee}^{\\rm SCE}[\\rho]\n   = \\operatorname*{min}_{P^N \\mapsto \\rho} \\int \\sum_{i,j\\, j>i}^N\\frac{1}{r_{ij}} \\,P^N({\\bf r}_1,...,{\\bf r}_N)\\mathrm{d} {\\bf r}_1...\\mathrm{d} {\\bf r}_N.\n\\end{equation}\nHere the minimum is over all symmetric $N$-body probability distributions with one-body density $\\rho$. Interestingly, unlike its absolute value squared, the wavefunction $\\Psi_\\lambda$ itself does not converge to any meaningful limit.\nSince $P^N_{\\infty}[\\rho]$ minimizes only the electronic repulsion, one can think of it as a natural analogue of the Kohn-Sham noninteracting wavefunction $\\Psi_0[\\rho]$, which minimizes the kinetic energy functional only. \n\n\\vspace{2mm}\n\\noindent\n\\textbf{Links to the XC functional:}\nThe functional \n$V_{ee}^{\\rm SCE}[\\rho]$ also corresponds to a well defined limit of the XC functional. \nIn fact, the $\\lambda \\to \\infty$ limit of the AC integrand of Eq.~\\eqref{eq:w1} is equal to\n\\begin{equation}\nW_\\infty[\\rho]=  V_{ee}^{\\rm SCE}[\\rho] - U[\\rho].\n\t\t\\label{eq:winfty}\n\\end{equation}\nMoreover, there is a well-known relation\\cite{LevPer-PRA-85,LevPer-PRB-93} between scaling the coupling strength $\\lambda$ and performing uniform coordinate scaling on the density, $\\rho_\\gamma ({\\bf r}) = \\gamma ^3 \\rho (\\gamma\\, {\\bf r}) $ (with $\\gamma >0$), which implies that the exact XC functional tends to $W_\\infty[\\rho]$ in the low-density ($\\gamma\\to 0$) limit. The SCE limit is thus complementary to exchange, which yields the high-density limit ($\\gamma\\to\\infty$) of $E_{xc}[\\rho]$,\n\\begin{equation}\n\\label{eq:ld-hd}\n\\lim_{\\gamma\\to \\infty}\\frac{E_{xc}[\\rho_\\gamma]}{\\gamma} = E_x[\\rho],\\qquad  \\lim_{\\gamma\\to 0}\\frac{E_{xc}[\\rho_\\gamma]}{\\gamma} = W_\\infty[\\rho].\n\\end{equation}\n\n\n\n\n\\vspace{2mm}\n\\noindent\n\\textbf{The SCE state:} As a candidate for the\n wave-function squared $P^N_\\infty[\\rho]$, Seidl and co-workers\\cite{Sei-PRA-99,SeiGorSav-PRA-07} proposed to restrict the minimization in \\eqref{eq:infmin} over singular distributions having the form: \n\\begin{equation}\\label{eq.psiSCE} \n\\begin{split}\n  &P^N_{\\rm SCE} = \\dfrac{1}{N!}\\sum_{\\mathcal{P}}\\int d\\mathbf{s}\\dfrac{\\rho(\\mathbf{s})}{N}\\delta({\\bf r}_1-{\\bf f}_{\\mathcal{P}(1)}(\\mathbf{s}))\\times  \\\\ \n&\\times\\delta({\\bf r}_2-{\\bf f}_{\\mathcal{P}(2)}(\\mathbf{s}))\\times\\dots\\times\\delta({\\bf r}_N-{\\bf f}_{\\mathcal{P}(N)}(\\mathbf{s})), \n\\end{split}\n\\end{equation}\nwhere ${\\bf f}_1,..,{\\bf f}_N$ are the so-called {\\it co-motion functions}, with ${\\bf f}_1({\\bf r})={\\bf r}$, $\\mathcal{P}$ is permutation of $\\{1,...N\\}$, and $\\delta\\big({\\bf r}-{\\bf f}_i({\\bf r})\\big)$ denotes the delta function of ${\\bf r}$ (alias Dirac measure) centered at  ${\\bf f}_i({\\bf r})={\\bf f}_i([\\rho];{\\bf r})$. The singular densities \\eqref{eq.psiSCE} are concentrated on the d-dimensional set  $\\Omega_0\\subset \\mathbb{R}^{dN}$,\n\\begin{equation} \\label{eq:Omega0}\n     \\Omega_0 = \\{ {\\bf r}_2 = f_2({\\bf r}),...,{\\bf r}_N=f_{N}({\\bf r})\\},\n\\end{equation}\nand its permutations.\nIntuitively speaking, such a $N$-body density describes a state in which the position of one of the electrons, say ${\\bf r}\\in\\mathbb{R}^d$, can be freely chosen according to the density $\\rho$, but this then uniquely fixes the position of all the other electrons through the  {\\it co-motion maps} ${\\bf f}_i({\\bf r})$, that is, ${\\bf r}_2=f_2({\\bf r}), \\dots,{\\bf r}_N=f_N({\\bf r})$. Thus states of form \\eqref{eq.psiSCE} are  called {\\it strictly correlated states}, or SCE states for short.\nIn other words, if a reference electron is at ${\\bf r}$, the other electrons in the SCE state can be found nowhere else, but at the ${\\bf f}_i({\\bf r})$ positions. Besides yielding minimal electronic repulsion, the co-motion functions need to satisfy group properties,\\cite{Sei-PRA-99,SeiGorSav-PRA-07,ColDepDim-CJM-15} accounting for the indistinguishability of electrons, and the pushforward conditions for the density constraint.\\cite{SeiGorSav-PRA-07,ColDepDim-CJM-15}  \n\nConstructing the co-motion functions is not simple, except in some special cases such as one-dimensional and spherically-symmetric systems.\\cite{Sei-PRA-99,SeiGorSav-PRA-07} In those cases, the co-motion maps are obtained from {\\it constrained integrals of the density}.\nThis is illustrated in Fig.~\\ref{fig_cmf}, which shows a simple one-dimensional example of the optimal solution for \\eqref{eq.VSCE}, which has the form \\eqref{eq.psiSCE}, with strictly correlated positions separated by ``chunks'' of density that integrate to integers.  We should stress that this solution has been rigorously proven to be exact for one-dimensional systems,\\cite{ColDepDim-CJM-15} which means that in this case the exact XC functional in the low-density limit is entirely determined by these constrained integrals rather than by any of the traditional Jacob's ladder ingredients.\n\nFor finite systems the SCE potential is defined as the functional derivative of the SCE functional $V_{ee}^{\\rm SCE}[\\rho]$ with respect to the density,\n$v_{\\rm SCE}({\\bf r})=\\delta V_{ee}^{\\rm SCE}[\\rho]\/\\delta \\rho({\\bf r})$, with the convention that $v_{\\rm SCE}({\\bf r})$ tends to zero as $|{\\bf r}|\\to\\infty$.  \nGiven an SCE state of Eq.~\\eqref{eq.psiSCE}, the SCE functional and potential\n can be simply written in terms of the co-motion functions:\\cite{MirSeiGor-JCTC-12,MalGor-PRL-12,MalMirCreReiGor-PRB-13}\n\\begin{align}\n\tV_{ee}^{\\rm SCE}[\\rho] & = \\int \\frac{\\rho({\\bf r})}{2}\\sum_{i=2}^{N}\\frac{1}{\\left | {\\bf r}-{\\bf f}_i({\\bf r}) \\right|}\\mathrm{d}{\\bf r} \n\\label{eq:vee_comotion} \\\\\n\\nabla v_{\\rm SCE}({\\bf r}) & =-\\sum_{i=2}^N \\frac{{\\bf r}-{\\bf f}_i({\\bf r})}{|{\\bf r}-{\\bf f}_i({\\bf r})|^3}. \\label{eq:forcev}\n\t\\end{align}\nEquation~\\eqref{eq:forcev} has a simple physical interpretation:  $v_{\\rm SCE}({\\bf r})$ is the one-body potential that corresponds to the net force exerted on an electron at position ${\\bf r}$ by the other $N-1$ electrons.\n\nIs actually the SCE state of  Eq.~\\eqref{eq.psiSCE} always the \\textit{true} minimizer of Eq.~\\eqref{eq:infmin}? It has been proven that this is true when $N=2$ in any dimension $d\\geq 1$\\cite{CotFriKlu-CPAM-13,ButDepGor-PRA-12} and when $d=1$ for any number of electrons.\\cite{ColDepDim-CJM-15} In general, the SCE state of  Eq.~\\eqref{eq.psiSCE} is not guaranteed to yield the absolute minimum for the electronic repulsion for a given arbitrary density $\\rho({\\bf r})$.\\cite{ColStr-M3AS-15,SeiDiMGerNenGieGor-arxiv-17}\nNevertheless,\nnumerical evidence suggests that the energetic difference between\n$V_{\\rm ee}^{\\rm SCE} [\\rho]$ from the SCE state (Eq.~\\eqref{eq:vee_comotion}) \nand the true minimum of Eq.~\\eqref{eq:infmin} is very small.\\cite{SeiDiMGerNenGieGor-arxiv-17}\n\n\\vspace{2mm}\n\\noindent\n\\textbf{Other $V_{\\rm ee}^{\\rm SCE} [\\rho]$ formulations.} In addition to  the co-motion functions formulation (Eq.~\\ref{eq:vee_comotion}), there are other equivalent formulations for $V_{ee}^{\\rm SCE}[\\rho]$ arising from mass transporation theory. The link between the SCE functional and mass transportation (or optimal transport) theory was found, independently, by Buttazzo {\\it et al.}\\cite{ButDepGor-PRA-12} and by Cotar {\\it et al.}\\cite{CotFriKlu-CPAM-13}.\nFrom the optimal transport viewpoint, the SCE functional defines a multimarginal problem, in which all the marginals are the same, so that the SCE mass-transportation problem corresponds to a reorganization of the ``mass pieces'' within the same density.  From optimal transport theory, the dual Kantorovich formulation\nfor $V_{ee}^{\\rm SCE}[\\rho]$ can be also deduced,\\cite{ButDepGor-PRA-12} defining the Kantorovich potential $u({\\bf r})$,\n\\begin{align}\n    &\tV_{ee}^{\\rm SCE}[\\rho]=\t\\label{eq:vee_kant} \\\\ \\nonumber\n    &\t\t \\max_{u}\\left \\{  \\int u({\\bf r})\\rho ({\\bf r})\\mathrm{d} {\\bf r}:\\sum_{i=1}^{N}u({\\bf r}_i)\\leqslant \\sum_{i=1}^{N}\\sum_{j>i}^{N}\\frac{1}{ |  {\\bf r}_{i}-{\\bf r}_{j} |}\\right \\},\n\\end{align}\n and can be reformulated \\cite{MenLin-PRB-13} by a nested optimization akin to the Legendre-Fenchel transform of Eq.~\\eqref{eq:infmin}: \n\t\\begin{align}\n\t\\label{eq:maxmin}\n\tV_{ee}^{\\rm SCE}[\\rho ]=\\max_{u}\\left \\{  \\int u({\\bf r})\\rho({\\bf r})\\mathrm{d}{\\bf r}+g[u]  \\right \\},\n\t\\end{align}\nwith $g[u]$ the minimum\nof the classical potential energy,\n\\begin{equation}\\label{eq:epotmin}\n    E^{\\rm pot}_{{\\rm SCE}}({\\bf r}_1,\\dots,{\\bf r}_N)=\\sum_{i<j}\\frac{1}{ |  {\\bf r}_{i}-{\\bf r}_{j} |}-\\sum_{i=1}^N u({\\bf r}_i),\n\\end{equation}\nover ${\\bf r}_1,...,{\\bf r}_N$.\n\t\nOne can show that the potentials  $u({\\bf r})$ and $v_{\\rm SCE}({\\bf r})$ differ only by a well-defined constant whose physical meaning has been explored in Refs.~\\onlinecite{VucWagMirGor-JCTC-15} and~\\onlinecite{VucLevGor-JCP-17}.\n\t\n\n\n\n\\vspace{2mm}\n\n\\noindent\n\\textbf{Next leading term:} More information about the exact LL functional at low density can be gained by studying the next leading term in Eq.~\\eqref{eq.asy2}. Under the assumption that the minimizer in \\eqref{eq.VSCE} is of the SCE type \\eqref{eq.psiSCE}, the classical potential energy \\eqref{eq:epotmin} is minimum on the manifold $\\Omega_0$ parametrised by the co-motion functions. \nThe conjecture is then that the next leading term is given by zero-point oscillations in the directions perpendicular to the SCE manifold\\cite{GorVigSei-JCTC-09}\n\\begin{align} \\label{eq:nexttermtot}\nF^\\lambda[\\rho] & \\;\\underset{\\lambda \\to \\infty}{\\sim}  \\; \\lambda V_{ee}^{\\rm SCE}[\\rho]+\\sqrt{\\lambda}\\,F^{\\rm ZPE}[\\rho],\n\\end{align}\\vspace{-6mm}\n\\begin{equation}\n\\text{where   }\\hspace{2mm}     F^{\\rm ZPE}[\\rho]=\\frac{1}{2}\\int_{\\mathbb{R}^d} \\frac{\\rho({\\bf r})}{N}{\\rm Tr}\\left(\\sqrt{\\mathbb{H}({\\bf r})}\\right)\\mathrm{d}{\\bf r},  \\label{eq:ZPOfunc}\n\\end{equation}\nand $\\mathbb{H}({\\bf r})$ is the hessian matrix evaluated on $\\Omega_0$ for fixed ${\\bf r}$. The intuition that this next term should be given by zero-point oscillations around the manifold parametrized by the co-motion functions appeared for the first time in Seidl's seminal work, \\cite{Sei-PRA-99} and calculations for small atoms (He to Ne) are reported in Ref.~\\onlinecite{GorVigSei-JCTC-09}. A rigorous proof in the one-dimensional case for any $N$ has been provided recently.\\cite{ColDMaStra-arxiv-21}\n\n\\vspace{2mm}\n\\noindent\n\\textbf{The spin state:} Besides the expansion of eq~\\eqref{eq:nexttermtot} in terms of powers of $\\lambda$, which is semiclassical in nature, it is conjectured\\cite{GorVigSei-JCTC-09,GorSeiVig-PRL-09} that the effect of the spin state will enter at large-$\\lambda$ through orders $e^{-\\sqrt{\\lambda}}$, which corresponds to the overlap of gaussians centered in different co-motion functions. This conjecture has been confirmed numerically for $N=2$ electrons in 1D.\\cite{GroKooGieSeiCohMorGor-JCTC-17}\n\\vspace{2mm}\n\n\\noindent\n\\textbf{Numerical realization of the SCE functional:} The SCE functional cannot at the moment be accurately and efficiently computed for general 3D densities and large $N$. But accurate numerical methods are available for small $N$ or special situations,  \nand novel methods aimed at large $N$ are under development. In particular, the very recent genetic column generation method  \\cite{FriSchVoe-21} appears in test examples to scale favourably with system size.\n\nIn Table~\\ref{tab_sce} we give an overview of the proposed algorithms for computing the SCE functional and potential\nand refer to the book chapter\\cite{FriGerGor-arxiv-22} for a more detailed review. From Table~\\ref{tab_sce}, we can see that for general 3D densities, numerical solutions were reported only for up to 10 electrons (the first method being limited to radial densities, whereas for the last method 3D tests are not yet available). This  indicates again the level of complexity and ultra non-locality of the SCE functional. \n\nIn fact, in the worst case scenario, the computational complexity of simple algorithms scales exponentially with the number $N$ of electrons\\cite{linHoCutJor-arXiv-19} and computing the SCE functional may be NP-hard. \\cite{AltBoi-arXiv-20,AltBoi-Dis-21, FriSchVoe-21} In the discrete setting, where the single particle density $\\rho$ is supported on $\\ell$ points, eq. \\eqref{eq:infmin} is equivalent to a linear programming problem with $\\ell N$ constraints and $\\ell^N$ variables. \n\nDespite these limitations in solving the SCE problem exactly, rather accurate approximations, retaining some of the SCE non-locality, have been recently proposed and they will be detailed in the next section. \n\n\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig\/cmf.pdf}\n\\caption{A pedagogical example: a sample of strictly correlated positions for a 1D density\nintegrating to 6 electrons.  \nA reference electron is placed at $x=-2$ (black point) \nand the SCE position of the other electrons is determined by \n$f_i(x)$\nand are represented by other colors. \nNotice that the shaded area \nbetween two adjacent SCE positions \nintegrates to $1$ and this is what defines $f_i(x)$ for 1D densities. \nInset is showing the co-motion functions for the given density.}\n\\label{fig_cmf}\n\\end{figure}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{An overview of proposed SCE algorithms. The third column shows $N_{\\rm max}$, which indicates up to how many electrons a given algorithm was applied to. This is not meant as a direct comparison of methods as the reported results differ in the fineness of discretization and the accuracy achieved. \nAlso, \nonly the second, fourth and last algorithms are free of additional approximations beyond discretization. \n} \\label{tab_sce}\n    \\begin{tabular}{|l|l|l|}\n        \\hline\n        Algorithm                                                    & References                                                                                & $N_{\\rm max}$ \\\\ \\hline\n         SGS approach based on co-motion functions (radial densities only) &  \\onlinecite{Vuc-Thesis-17}, see also \\onlinecite{SeiGorSav-PRA-07,SeiVucGor-MP-16} & 100           \\\\ \n         Linear programming applied to the N-body formulation \\eqref{eq:infmin}                     & \\onlinecite{CheFriMen-JCTC-14}              & 2             \\\\\n         Multi-marginal Sinkhorn algorithm                            & \\onlinecite{BenCarCutNenPey-SIAM-15,BenCarNen-SMCISE-16,DMaGerNen-TOOAS-17,DMaGer-JSC-20,GerGroGor-JCTC-20}                 & 5             \\\\ \n        Algorithms based on the Kantorovich  formulation \\eqref{eq:maxmin}           & \\onlinecite{MenLin-PRB-13,VucWagMirGor-JCTC-15}                                           & 6             \\\\ \n        Algorithm based on representability constraints for the pair density                       & \\onlinecite{KhoYin-SIAMJSC-19}                & 10             \\\\\n        Langevin dynamics with moment constraints & \n        \\onlinecite{AlfCoyEhrLom-21, AlfCoyEhr-21} & (to be assessed) \\\\ \n        Genetic column generation (3D tests not yet avaliable)                  & \\onlinecite{FriSchVoe-21}                 & 30           \\\\\n        \\hline\n    \\end{tabular}\n\\end{table*}\n\n\n\n\\section{Approximations to the SCE functional}\n\\label{sec_approx}\n\n\n\n\\begin{table*}\n\\caption{Approximate $w_\\infty({\\bf r})$ energy densities yielding $W_\\infty[\\rho]$ from: $W_\\infty[\\rho]= \\int \\rho({\\bf r}) w_\\infty({\\bf r}) \\mathrm{d}{\\bf r}$.\n   PC stands for point-charge plus continuum (PC) model, \\cite{SeiPerKur-PRA-00} LDA stands for the local density approximation, GEA for the gradient expansion approximation,\n   GGA for the generalized gradient approximation, and hPC \\cite{SDGF22} stands for\n   the harmonium PC based on a GGA form,\\cite{PerBurErn-PRL-96} whose parameters \n   $a$ and $b$ are\n   trained on the SCE energetics for the harmonium atom (for their numerical values see  Ref.~\\onlinecite{SDGF22}).\n   The reduced density gradient, $s$, is given by $s({\\bf r})=\n\\left| \\nabla \\rho ({\\bf r})\n\\right| \/ \\bigl( 2 (3\\pi^2)^{1\/3} \\rho({\\bf r})^{4\/3} \\bigr)$.\nThe nonlocal radius functional (NLR) \\cite{WagGor-PRA-14} approximates the strong coupling limit of the XC hole, whose depth is given by Eq.~\\ref{eq:u1} and is calculated from the integrals over the spherically averaged density (Eq.~\\eqref{eq:sadens}).\nThe shell model \\cite{BahZhoErn-JCP-16} adds a positive shell to the NLR hole, and the radii of the negative and positive shell, $u_s({\\bf r})$ and $u_c({\\bf r})$, respectively, are obtained at each ${\\bf r}$ from the uniform electron gas constraint and the normalization constraint on the underlying XC hole.\nThe approximate $w_\\infty({\\bf r})$ from PC-LDA, NLR and shell model are in the gauge of Eq.~\\eqref{eq:wxc_def} and thereby directly approximate $w_\\infty({\\bf r})$ of Eq.~\\eqref{eq:winf}. \n}\\label{tab_ap}\n    \\begin{tabular} {|c|c|c|}\n    \\hline \n    Approximation & $w_\\infty({\\bf r})$ form & Refs. \\\\ \\hline    \n    PC-LDA & $\\displaystyle -\\frac{9}{10}\\left(\\frac{4\\pi}{3}\\right)^{1\/3}\\rho({\\bf r})^{1\/3}$ & \\onlinecite{SeiPerKur-PRA-00}    \\\\ \\hline\n    PC-GEA & $\\displaystyle w_{\\infty}^{\\rm PC-LDA}({\\bf r})+9\\frac{ 2^{1\/3} \\pi}{175}  \\rho({\\bf r})^{1\/3} s({\\bf r})^2$  & \\onlinecite{SeiPerKur-PRA-00}   \\\\ \\hline\n    GGA (hPC) & $\\displaystyle w_{\\infty}^{\\rm PC-LDA}({\\bf r})\\, \\frac{1+ a\\, s({\\bf r})^2}{1+b\\, s({\\bf r})^2}$ & \\onlinecite{SDGF22} \\\\ \\hline\n    NLR & $\\displaystyle -2\\pi\\int_0^{u_1({\\bf r})} \\tilde{\\rho}({\\bf r},u)u~ \\mathrm{d} u$  & \\onlinecite{WagGor-PRA-14} \\\\ \\hline\n   shell model & $\\displaystyle -2\\pi\\int_0^{u_s({\\bf r})} \\tilde{\\rho}({\\bf r},u)u~ \\mathrm{d} u+2\\pi\\int_{u_s({\\bf r})}^{u_c({\\bf r})} \\tilde{\\rho}({\\bf r},u)u~ \\mathrm{d} u$ & \\onlinecite{BahZhoErn-JCP-16} \\\\ \\hline\n   \\end{tabular}\n   \n\\end{table*}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig\/be1.pdf}\\\\[2ex]\n\\includegraphics[width=\\linewidth]{fig\/be2.pdf}\n\\caption{Upper panel:  Difference between the exact (SCE) and approximate strong coupling limit energy density for the beryllium atom, $\\delta w_\\infty({\\bf r})=w_\\infty({\\bf r})-w_\\infty^{\\rm model}({\\bf r})$, as a function of the distance from the nucleus, r \/ a.u. The inset in the upper panel is focusing on the error of the shell and NLR model for larger $r$. Lower panel: The quantity from the upper panel multiplied by the density and the spherical volume element.}\n\\label{fig_be}\n\\end{figure}\n\n\nExisting approximations to the SCE functional are summarized in Table~\\ref{tab_ap}.  \nThey include the \npoint-charge plus continuum (PC) model of Seidl and coworkers,\\cite{SeiPerKur-PRA-00} which is a gradient expansion (GEA), and the recent harmonium PC (hPC) model, which is a generalized gradient approximation (GGA).\\cite{SDGF22}\nThe nonlocal radius functional (NLR) and the shell model retain some of the SCE non-locality \\cite{WagGor-PRA-14,BahZhoErn-JCP-16} through the integrals of the spherically averaged density, which is defined as:\n\\begin{equation}\n\\tilde{\\rho}({\\bf r},u) = \\frac{1}{4\\pi} \\int \\rho(\\mathbf{r} + \\mathbf{u}) \\mathrm{d}\\mathbf{\\Omega_{\\mathbf{u}} }.\n\\label{eq:sadens}\n\\end{equation}\nNLR approximates the XC hole in the strong coupling limit whose depth (``nonlocal radius''), $u_1({\\bf r})$, is implicitly defined through the following integral, inspired by the exact SCE functional for 1D systems,\n\\begin{equation}\n\\label{eq:u1}\n4 \\pi \\int_0^{u_1({\\bf r})}  u^2 \\tilde{\\rho}({\\bf r},u) \\mathrm{d}u =1.\n\\end{equation}\nOnce $u_1({\\bf r})$ is computed, the energy density from the electrostatic potential of the NLR XC hole is computed, which in turn, defines $W_\\infty[\\rho]$ within NLR. \nThe shell model is built upon NLR and makes it exact\nfor the SCE limit of the uniform electron gas.\\cite{BahZhoErn-JCP-16} \n\nIn Figure~\\ref{fig_be}, we explore the accuracy of different SCE approximations for energy densities (see Eq.~\\ref{eq:winf} below). From this figure, we can see that the shell model \nis the most accurate approximation locally. The PC-GEA model is the best performer globally here, and generally it gives a  rather accurate $W_\\infty[\\rho]$. \nHowever, the functional derivative of the PC GEA diverges in the exponentially-decaying density tails,\n \\cite{FabSmiGiaDaaDelGraGor-JCTC-19,SmiCon-JCTC-20,SDGF22} making self-consistent KS calculations impossible. \nThis problem is solved by turning to GGA's. \\cite{SmiCon-JCTC-20,SDGF22} In particular,\nthe very recently proposed hPC functional\\cite{SDGF22} preserves the accuracy of $W_\\infty[\\rho]$ from PC-GEA, while making self-consistent KS calculations possible.\n\n\n\n\\section{From SCE to practical methods}\n\n\nThe {\\it bare} SCE functional is not directly applicable in chemistry as it over-correlates electrons.\nIf we take the dissociation curve of H$_2$ as an example \\cite{CheFriMen-JCTC-14,VucWagMirGor-JCTC-15},\nwe can see that the SCE, unlike nearly all available XC approximations, dissociates the H$_2$ correctly without artificially breaking any symmetries, but predicts far too low energies around equilibrium and too short bond lengths.\nFor this reason,  \nSCE is not directly applicable in chemistry. \nInstead one should devise smarter strategies for incorporating the SCE in an approximate XC functional.\nThe challenge is then to use the SCE information to equip new functionals with the ability to capture strong electronic correlations,  while maintaining the accuracy of the standard DFT for weakly and moderately correlated systems. These strategies and challenges that come along the way are discussed in the next sections. \n\n\\subsection{Functionals via {\\it global} interpolations between weak and strong coupling limit of DFT}\n\\label{sec_global}\nXC approximations of different classes have been constructed from \nmodels to the global AC integrand (Eqs.~\\ref{eq:ac_xc}).\\cite{Ern-CPL-96,BurErnPer-CPL-97,Bec-JCP-93,MorCohYan-JCPa-06,song2021density} \nA possible way to avoid bias towards the weakly correlated regime present in nearly all XC approximations is to also include\nthe information from the strongly interacting limit of $W_\\lambda[\\rho]$.\nSuch an approach, called the interaction strength interpolation (ISI), where \n$W_\\lambda[\\rho]$ is obtained from an interpolation between its  weakly- and strongly- interacting limits, has been proposed by Seidl and coworkers.\\cite{SeiPerLev-PRA-99} \nSince the ISI approach has been proposed, different interpolation forms with different input ingredients have been tested.\\cite{SeiPerKur-PRL-00,SeiPerKur-PRA-00,GorVigSei-JCTC-09,LiuBur-PRA-09,ZhoBahErn-JCP-15,BahZhoErn-JCP-16,VucIroSavTeaGor-JCTC-16,VucIroWagTeaGor-PCCP-17} \nThese approaches typically use the exact information from the weakly interacting limit (exact exchange and the correlation energy from the second-order G\u00f6rling\u2013Levy perturbation theory (GL2)\\cite{GorLev-PRB-93}). \nExcept for some proof-of-principle calculations,\\cite{MalMirGieWagGor-PCCP-14,VucIroSavTeaGor-JCTC-16,KooGor-TCA-18}\nthe ISI scheme uses the approximate ingredients from \nthe large-$\\lambda$ limit ($W_\\infty[\\rho]$ and the next leading term described by $F^{\\rm ZPE} [\\rho]$) and these are\ntypically modeled at a semilocal level. In some cases,\\cite{ZhoBahErn-JCP-15,BahZhoErn-JCP-16} the ISI forms have been tested in tandem with the $W_\\infty[\\rho]$ approximations that retain some of the SCE nonlocality (see Sec.~\\ref{sec_approx}).\n\nA potential problem of the ISI functionals is the lack of size-consistency, which, however, can be easily corrected for interaction energies when there are no degeneracies. \\cite{VucGorDelFab-JPCL-18} \nThe ISI functionals have been tested on several chemical data sets and systems and they perform reasonably well for \ninteraction energies (energy differences). \\cite{FabGorSeiDel-JCTC-16,GiaGorDelFab-JCP-18,VucGorDelFab-JPCL-18} \nWhen applied in the post-SCF fashion, the ISI approach seems more promising when used in tandem with Hartree-Fock (HF) than with semilocal Kohn-Sham orbitals. \nThis finding has initiated the study of the strongly interacting limit in the Hartree-Fock theory\\cite{SeiGiaVucFabGor-JCP-18,DaaGroVucMusKooSeiGieGor-JCP-20} and the successes of approaches based on it will be briefly described in Sec.~\\ref{sec_hf}.\nRecently, the correlation potential from the ISI approach, which is needed for self-consistent ISI calculations to obtain the density and KS orbitals, has been computed. \\cite{FabSmiGiaDaaDelGraGor-JCTC-19} It has been shown that it is rather accurate for a set of small\natoms \nand diatomic molecules (see Fig.~\\ref{fig:ne}, where we show that the ISI correlation potential provides a substantial improvement over that from GL2 for the neon atom). \\cite{FabSmiGiaDaaDelGraGor-JCTC-19}\nThe computed ISI correlation potentials have enabled fully self-consistent ISI calculations that have been recently reported in Ref.~\\cite{SDGF22}. \n\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{fig\/ne.pdf}\n\\caption{Correlation potentials $v_c$ as a function of the distance from the nucleus ($r$) for the neon atom. The accurate correlation potential has been obtained from Quantum Monte Carlo (QMC). All the data have been taken from Ref.~\\onlinecite{FabSmiGiaDaaDelGraGor-JCTC-19}. }\n\\label{fig:ne}\n\\end{figure}\n\n\n\\subsection{Functionals via {\\it local} interpolations between weak and strong coupling limit of DFT}\n\\label{sec_local}\n\nIn addition to building models for the XC energy via global interpolations between the strongly- and weakly- interacting limits of $W_\\lambda[\\rho]$, one can also perform \nthe interpolation locally (i.e. in each point of space). \\cite{VucIroSavTeaGor-JCTC-16,VucIroWagTeaGor-PCCP-17}\nThis can be done by interpolating between the weakly and strongly interacting limits of $w_\\lambda([\\rho];{\\bf r})$, which is the $\\lambda$-dependent XC energy density of Eq.~\\eqref{eq:wxc_def}. \nThe main advantage of local interpolations\\cite{VucIroSavTeaGor-JCTC-16,VucIroWagTeaGor-PCCP-17,KooGor-TCA-18} over their global counterparts is that the former are size-consistent by construction if the interpolation ingredients are size-consistent.\\cite{GorSav-JPCS-08,Sav-CP-09}  Thereby, local interpolations, unlike their global counterparts, do not require the size-consistency correction .\\cite{VucLevGor-JCP-17} \n\nAs mentioned earlier, there is no unique definition for $w_\\lambda([\\rho];{\\bf r})$. \nRef.~\\onlinecite{VucLevGor-JCP-17} explored the suitability of different definitions of the $\\lambda$-dependent energy densities and it has been found that the energy densities definition of Eq.~\\eqref{eq:wxc_def} ('electrostatic potential of the XC hole') is the best choice so far in this context. \nWithin this definition, $w_\\lambda([\\rho];{\\bf r})$ reduces to the exact exchange energy density when $\\lambda = 0$, whereas in the $\\lambda \\to \\infty$ (within the SCE formulation), it is defined in terms of the co-motion functions \\cite{MirSeiGor-JCTC-12}: \n\\begin{align}\n\tw_\\infty([\\rho];{\\bf r})= \\sum_{i=2}^{N}\\frac{1}{\\left | {\\bf r}-{\\bf f}_i({\\bf r}) \\right|}  -\\frac{1}{2}v_H({\\bf r}),\n\\label{eq:winf}\n\t\\end{align}\nwhere $v_H({\\bf r})$ is the Hartree potential. In addition to these two, a closed form expression for the local initial slope for $w_\\lambda([\\rho];{\\bf r})$ has been derived in Ref.~\\onlinecite{VucIroSavTeaGor-JCTC-16} from second order perturbation theory. \n\nThe accuracy of different local interpolation forms have been tested with both exact \\cite{VucIroSavTeaGor-JCTC-16,KooGor-TCA-18} and approximate~\\cite{VucIroSavTeaGor-JCTC-16,ZhoBahErn-JCP-15,BahZhoErn-JCP-16} ingredients. Relative to the global interpolations, local interpolations typically give improved results for tested small chemical systems, \\cite{VucIroSavTeaGor-JCTC-16} but usually do not fix the failures of global interpolations.\\cite{VucIroWagTeaGor-PCCP-17} Nevertheless, the accuracy of XC functionals based on the local interpolation is still underexplored. This local interpolation framework can also be used to improve the latest XC approximations, such as the deep learned {\\it local hybrids},\\cite{fn_dm21} especially when it comes to the treatment of strong electronic correlations. \n\n\\subsection{Fully nonlocal multiple radii functional - inspired by the exact SCE form}\n\nThe mathematical form of the SCE functional has inspired new fully nonlocal approximations, called the multiple radii functional (MRF). \\cite{VucGor-JPCL-17,VJCTC19,VG19} MRF approximates the XC energy denities of Eq.~\\eqref{eq:wxc_def} at arbitrary $\\lambda$ in the following way:\n\\begin{equation}\nw_\\lambda^{\\rm MRF}({\\bf r})= \\frac{1}{2} \\sum_{i=2}^{N} \\frac{1}{R_i^{\\lambda}([\\rho];{\\bf r})}-\\frac{1}{2}v_H({\\bf r}). \n\\label{eq:mrf1}\n\\end{equation}\nEquation~\\ref{eq:mrf1} can be thought as the generalization of Eq.~\\eqref{eq:winf}, where starting from a reference electron at ${\\bf r}$, the remaining electrons are assigned effective radii or distances from ${\\bf r}$. The radii are then constructed from the integrals over the spherically averaged density and are implicitly defined by\n\\begin{equation}\n4 \\pi \\int_0^{R_i^{\\lambda}({\\bf r}) }  u^2 \\tilde{\\rho}({\\bf r},u) \\mathrm{d}u = i - 1 + \\sigma_i^{\\lambda}({\\bf r}),\n\\end{equation}\nwhere $\\sigma_i^{\\lambda}({\\bf r})$ is the so-called {\\it fluctuation function}. \nThe construction of the XC functional within MRF essentially reduces to building $\\sigma_i^{\\lambda}({\\bf r})$ , \nand  already very simple forms for $\\sigma_i^{\\lambda=1}({\\bf r})$ yield very accurate \natomic $w_\\lambda^{\\rm MRF}({\\bf r})$ at the physical regime for atoms, while also accurately capturing the physics of stretched bonds. This shows that the forms inspired by the SCE can work for the physical regime if properly re-scaled. \nFurthermore, despite its full non-locality, the cost of MRF is  $O(N^3)$ within seminumerical schemes.\\cite{BahmannKaupp2015} \n\nBy construction, MRF has very appealing properties: (1) it gives XC energies in the gauge of Eq.~\\eqref{eq:wxc_def} making it highly suitable to be used in the local interpolations described in Sec.~\\ref{sec_local}; (2) these energy densities have the correct asymptotic behavior; (3) MRF captures the physics of bond breaking; (4) it is fully nonlocal so it can better describe the physics of strong electronic correlations that the usual semilocal DFT functionals; (5) its form is universal and does not change as dimensionality\/interactions between particle changes as demonstrated in Ref.~\\onlinecite{VG19}.\nAll these features of MRF and its flexibility make it very promising for building the next-generation of DFT approximations. There are ongoing efforts to \ntransform these appealing features into robust XC functionals by developing improved MRF forms and efficiently implementing the MRF package into standard quantum-chemical codes. \n\n\n\\subsection{Other applications of SCE: Lower bounds to XC energies and correlation indicators}\n\nBesides being used to build XC approximations, the SCE approach has also  proven very useful in understanding general features of the exact XC functional and the nature of electronic correlations.\nFor example, the SCE limit is directly connected to the Lieb-Oxford (LO) inequality, \\cite{Lie-PLA-79,LieOxf-IJQC-81} a key exact property used in the construction of XC approximations.\\cite{Per-INC-91,SunRuzPer-PRL-15} \nThe LO inequality limits the value of the XC energy by \nbounding from below the AC integrand of Eq.~\\eqref{eq:w1}:\n\\begin{equation}\nW_\\lambda [\\rho] \\geq - C_{\\rm LO} \\int  \\rho ^{4\/3} ({\\bf r}) \\mathrm{d} {\\bf r},\n\\label{eq:lo}\n\\end{equation}\nwhere the optimal $C_{\\rm LO}$ is rigorously known to be between 1.4442 and 1.5765. \\cite{LewLieSei-PRB-19,CotPet-arxiv-17,LLS22}\nSince $W_\\lambda [\\rho]$ monotonically decreases with $\\lambda$, $W_\\infty [\\rho]$ will be the smallest value for the l.h.s.~of Eq.~\\eqref{eq:lo}. \nThus, finding lower bounds for the optimal constant $C_{\\rm LO}$ is equivalent to searching for densities $\\rho$ that maximize the ratio between $W_\\infty [\\rho]$ and $-\\int \\rho ^{4\/3} ({\\bf r}) \\mathrm{d} {\\bf r}$, \\cite{RasSeiGor-PRB-11,SeiVucGor-MP-16} a procedure that has been applied to both the optimal $C_{\\rm LO}$ for the general case and to the one for a specific number of electrons $N$. \\cite{RasSeiGor-PRB-11,SeiVucGor-MP-16,VucLevGor-JCP-17,SBKG22} \nAn approach to tighten the lower bound to correlation energies for a given density has been also proposed  \nby combing the adiabatic connection interpolation described in Sec.~\\ref{sec_global} and the SCE energies.\\cite{VucIroWagTeaGor-PCCP-17}\n\nIn addition to provide tightened lower bounds for the XC energies, the SCE has also been used to define correlation indicators that quantify the ratio between {\\it dynamical} and {\\it static} correlation in a given system.\\cite{VucIroWagTeaGor-PCCP-17}\nThis idea has been also generalized to local indicators, enabling to visualise\nthe interplay of dynamical and static correlation\nat different points in space.\\cite{VucIroWagTeaGor-PCCP-17} \n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{fig\/figADIA.pdf}\n\\caption{Two different adiabatic connections, linking the physical ($\\lambda=1$) system to either the KS or the Hartree-Fock determinant ($\\lambda=0$). Left: the density-fixed adiabatic connection of KS DFT (see Sec.~\\ref{sec_theory}). The Hamiltonian $\\hat{H}_\\lambda^{\\rm DFT}$ corresponds to Eq.~\\eqref{eq.LLalpha}, with the one-body potential $v_\\lambda({\\bf r})$ enforcing the constraint $\\Psi_\\lambda\\mapsto \\rho$, where $\\rho$ is the density of the physical system. \nThe correlation part of the adiabatic connection integrand $W_{c,\\lambda}^{\\rm DFT}$ is equal to $W_\\lambda[\\rho]$ of Eq.~\\eqref{eq:w1} minus $W_{\\lambda=0}[\\rho] = E_x[\\rho]$. \nSmall and large-$\\lambda$\nexpansions for $W_{c,\\lambda}^{\\rm DFT}$ are also shown.\nRight: the adiabatic connection that has the M{\\o}ller-Plesset (MP) series as small perturbation expansion, considered in Sec.~\\ref{sec_hf}. The Hamiltonian $\\hat{H}_\\lambda^{\\rm HF}$ contains  $\\hat{J}[\\rho^{\\rm HF}]$ and $\\hat{K}[\\{\\phi_i^{\\rm HF}\\}]$, which are the standard Hartree and exchange HF operators, respectively.\nThe $\\lambda$-dependent $\\Psi$ minimizing $\\hat{H}_\\lambda^{\\rm HF}$ has a density that changes with $\\lambda$: at $\\lambda=0$ is equal to the HF density, while at $\\lambda=1$ is equal to the physical density. The expectation $W_{c,\\lambda}^{\\rm HF}$ is the AC integrand defining the correlation energy in HF theory.\nSmall and large-$\\lambda$\nexpansions for $W_{c,\\lambda}^{\\rm HF}$ are also shown.\nThe operator $\\hat{V}_{\\rm ext}$ is the external (nuclear) potential.\n}\n\\label{fig:adiabs}\n\\end{figure}\n\n\\subsection{Going beyond DFT -- large-$\\lambda$ limits in the M{\\o}ller-Plesset adiabatic connection}\n\\label{sec_hf}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig\/PanelVertical.png}\n\\caption{Dissociation curves of the Pyridine (top) and Argon dimers (bottom) obtained from MP2, SPL2, MPACF-1, B3LYP-D3, and B2PLYP with CCSD(T) (black line) as a reference.}\n\\label{fig:Arg}\n\\end{figure}\n\nThe DFT AC introduced in Sec.~\\ref{sec_xc}, whose large $\\lambda$ limit is the focus of this paper, defines the correlation energy in KS DFT. In a more traditional quantum-chemical sense, the correlation energy is defined as the difference between the true and Hartree-Fock (HF) energy. \nAn exact expression for this correlation energy is given by\nthe M{\\o}ller-Plesset adiabatic connection (MPAC),\\cite{SeiGiaVucFabGor-JCP-18} which connects the HF and physical state and has the M\u00f8ller--Plesset perturbation series as weak-interaction\nexpansion. This AC is summarized and compared with the one of KS DFT used in the rest of this work in Fig.~\\ref{fig:adiabs}. The large-$\\lambda$ limit of the MP AC has been recently shown\\cite{SeiGiaVucFabGor-JCP-18,DaaGroVucMusKooSeiGieGor-JCP-20} and to be determined by functionals of the HF density with a clear physical meaning.\nInequalities between the large-$\\lambda$ leading terms of the two AC's have been also established.\\cite{SeiGiaVucFabGor-JCP-18,DaaGroVucMusKooSeiGieGor-JCP-20}\n\nThe MP AC theory has been used to construct \n  a predictor for the accuracy of MP2  for noncovalent interactions.\\cite{VucFabGorBur-JCTC-20}\nMethods that are based on the interpolation between the small and large $\\lambda$ limits of the MP AC have been also developed.  \\cite{DaaFabDelGorVuc-JPCL-21} They are analogous to the ISI methods outlined in Sec.~\\ref{sec_global}, which are used in the DFT context. It has been shown that the these interpolation methods for the MP AC give very accurate results for noncovalent interactions. \\cite{DaaFabDelGorVuc-JPCL-21} We illustrate this in Fig.~\\ref{fig:Arg}, where we compare reference [CCSD(T)] to approximate dissociation curves from these interpolation approximations for the pyridine and argone dimers.\nThe curves labelled SPL2 and MPACF-1 correspond to two new global interpolations forms\\cite{DaaFabDelGorVuc-JPCL-21} constructed by adding more flexibility and empirical parameters to the existing interpolation forms used in DFT\\cite{SeiPerLev-PRA-99,GorVigSei-JCTC-09} to capture the known exact features of the MP AC. For both of these noncovalently bound dimers, and for many other cases,\\cite{DaaFabDelGorVuc-JPCL-21} SPL2 and MPACF-1 show an excellent performance without using dispersion corrections. In general, they substantially improve over MP2 for noncovalent interactions, and are either on par with -- or also improve -- dispersion corrected (double) hybrids \\cite{DaaFabDelGorVuc-JPCL-21}.\n\n\n\\section{Conclusions and Outlook}\n\nHere we have reviewed the most important topics that the strongly interacting limit of DFT brings into focus.\nWe have analyzed the development of different aspects of the underling rigorous theory connecting DFT and optimal transport, and discussed \nhow the SIL formulation \ninfluenced the development of different\nmethods in \nDFT and beyond. Although this limit does not describe the physical regime, its mathematical structure contains essential elements pointing towards the real physics happening in molecular systems with strong correlations, whose description is  one of the key unsolved problems in DFT.\nThus, in the years and decades to come \nit will be very interesting to see how much the SIL ideas, formulations and ensuing practical methods \nwill be used to solve the strong correlation problem and to build the next generation of DFT methods. In particular, the new ingredients appearing in this limit can be used as new features to machine learn the XC functional.\\cite{Kalita2021,fn_dm21,Nagai2020} \n\n\\section*{Conflicts of Interest}   \nAll the authors declare to have no conflict of interest.\n\n\\section*{Data Sharing}   \nData sharing is not applicable to this article as no new data were created or analyzed in this study.\n\n\\section*{Acknowledgement}    \nSV acknowledges funding from the Marie Sklodowska-Curie grant 101033630 (EU's Horizon 2020 programme). AG acknowledges support of his research by the Canada Research Chairs Program and Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2022-05207.\nHB acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) \u2013 project no. 418140043. TJD and PG-G were supported by the Netherlands Organisation for Scientific Research (NWO) under Vici grant 724.017.001. GF was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through CRC 109.\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{vancouver}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe concept of {\\em nuclearity} is fundamental in the study of\n$C^*$-algebras.  It is often defined in terms of tensor products, as\nalso are the slightly less well known, but also fundamental,\nproperties known as (Lance's) {\\em weak expectation property} (WEP),\nand (Kirchberg's) {\\em exactness} and {\\em local lifting property}\n(LLP).  These are  intimately related properties which a\n$C^*$-algebra may or may not possess, and the relations between them\nand ensuing theory are rich and profound. The goal of the present\npaper is to find appropriate generalizations of these notions to\npossibly nonselfadjoint operator algebras; and to illuminate some of\nthe good, the bad, and the ugly that ensues. More precisely, some of\nthe elegant basic implications and arguments in the $C^*$-algebraic\nare still valid in the nonselfadjoint case, while others only seem\nto be true for very special classes of algebras. That is, for\nextremely general classes of algebras some of these properties may\nbe a little restrictive; and in this sense the present investigation\nis not as successful as some of our  previous generalizations of\n$C^*$-algebraic notions to nonselfadjoint algebras. Nonetheless, the\nproperties we introduce are natural and do not seem to have been\nconsidered before.\n  Moreover, en route we present several new and very\nbasic results of independent interest. There also seems to be some\nhope, as the reader will see at points in our paper, that they may\nlead in the future to a new approach to Kirchberg's famous\nconjectures. Another of our motivations was to find properties that\nimply that $Ext$ of the $C^*$-envelope of a nonselfadjoint algebra\nis a group, and this angle will be prominent in the sequel\n\\cite{Bseq}.  We recall from \\cite{Kir1} that the LLP is more than\nintimately connected with  $Ext$ of a separable $C^*$-algebra $B$\nbeing a group (the first implies the second, and the converse is an\nopen question).\n\n\nThe new concepts introduced here are called {\\em $C^*$-nuclearity},\nthe {\\em algebra weak expectation property} (AWEP), the {\\em\nhomomorphism local lifting property} (HLLP), {\\em\n$\\Bdb$-nuclearity}, and {\\em subexactness}. Motivated primarily by\nKirchberg's astonishing paper \\cite{Kir1}, and its operator space\nsequel due to Pisier, Effros, Ruan, and others, we try to build\nconnections between our new variants that are similar to the\nclassical $C^*$-algebraic theory. Throughout, we use generated\n$C^*$-algebras to relate these notions to their classical\ncounterpart.  For instance, if $A$ is {\\em Dirichlet} (that is, a\nunital algebra with $A + A^*$ is norm dense in its $C^*$-envelope),\nthen $A$ has each of the five properties above iff the\n$C^*$-envelope of $A$ has the matching $C^*$-algebra property (with\none exception in one direction: we are not sure if $A$ having HLLP\nimplies that the $C^*$-envelope has LLP).\n\nIt is important for us to say that of course a significant part of\nmodern operator space theory is devoted to linear analogues of some\nof the properties mentioned above; very strikingly some of the\nabove-mentioned properties and their beautiful theory generalizes to\noperator spaces (see e.g.\\ \\cite{ER,Pisbook}). However, with one\nexception the operator space versions of these properties turn out\nnot to be appropriate for nonselfadjoint operator algebras, at least\nfor the approach taken here.\n\n\nTurning to notation, by an operator algebra, we mean a closed, not\nnecessarily selfadjoint, algebra of operators on a Hilbert space. We\nwill sometimes silently be using very basic principles from the\ntheory of operator algebras, all of which are explained in\n\\cite{BLM}.   An operator algebra is\n {\\em unital} if it has an\nidentity of norm $1$, and is  {\\em approximately unital} if it has a\ncontractive two-sided approximate identity (cai).\nFor simplicity we will usually assume that our operator algebras\nare approximately unital, but in many of the results this restriction\nis not necessary, by the usual trick of considering the\nunitization.  We write $A^1$ for the unitization\n of a nonunital operator algebra (see \\cite[Section 2.1]{BLM}).\nA {\\em\nunital-subalgebra}, is a subalgebra containing the identity of the\nbigger algebra.  All ideals are assumed to be two-sided and closed.\nOur morphisms will be linear completely contractive homomorphisms\n$\\theta : A \\to B$ between operator algebras.   If $\\theta(1) = 1$\nwe say that $\\theta$ is {\\em unital}.  As usual, {\\em UCP} means\nunital and completely positive.\n A {\\em $C^*$-cover} of $A$ is a $C^*$-algebra\ncontaining a copy of $A$ completely isometrically as a subalgebra,\nwhich is generated by this copy. There are two `universal'\n$C^*$-covers of $A$, a `smallest' and a `largest': the {\\em\n$C^*$-envelope} $C^*_{\\rm e}(A)$, and the {\\em maximal\n$C^*$-dilation}  $\\cma$. We refer the reader to \\cite{BLM} for a\ndiscussion of these two notions; they have the extremal universal\nproperties which the reader would expect. A new term: we will say\nthat $A$ is $C^*$-split if there exists a linear complete\ncontraction $u : C^*_{\\rm e}(A) \\to \\cma$ extending the identity map\non $A$.  An example of this is any {\\em Dirichlet} operator algebra\n(this may be seen by Arveson's Proposition 1.2.8 of \\cite{SOC}).  By\nthe well-known `rigidity' property of $C^*_{\\rm e}(A)$, it follows\nthat $u$ is a right inverse for the epimorphism $\\cma \\to C^*_{\\rm\ne}(A)$.  We write $\\Bdb$ and $\\Kdb$ for respectively the bounded and\ncompact linear operators on $\\ell^2$. We refer the reader to\n\\cite[Chapter 6]{BLM} or \\cite{PaP} for the tensor products of\noperator algebras used here. In places we also use notation and\nresults from the paper \\cite{BR} on extensions of nonselfadjoint\nalgebras. By an {\\em extension in the sense of} \\cite{BR}, we will\nmean a short  exact sequence\n$$0 \\longrightarrow A\\overset{\\alpha}\\longrightarrow B\\overset{\\beta}\\longrightarrow C \\longrightarrow 0$$\nof nontrivial operator algebras, with $A$ approximately unital; and\n$\\alpha, \\beta$ are completely contractive homomorphisms, with\n$\\alpha$ completely isometric, and $\\beta$ a complete quotient map.\nApplications of our work to the theory of extensions will be\npresented in a forthcoming sequel \\cite{Bseq} to \\cite{BR}.\n\n\\section{$C^*$ and $\\Bdb$-nuclearity} \\label{cnucl}\n\n\\begin{definition} An operator algebra $A$ is $C^*$-nuclear (resp.\\\n$\\Bdb$-nuclear) if \\[ A \\maxten B = A \\minten B \\] for every\n$C^*$-algebra $B$ (resp.\\ for $B = \\Bdb = B(\\ell^2)$).\n\\end{definition}\n\nIt is important to note that if $A$ is not selfadjoint then allowing\n$B$ in the definition of $C^*$-nuclear to be nonselfadjoint yields\na vacuous class, by  \\cite[Corollary 7.1.8]{BLM}.\n\nIn 6.2.5 of \\cite{BLM} it is remarked that any Dirichlet uniform\nalgebra, such as the disk algebra $A(\\Ddb)$, is $C^*$-nuclear in our\nterminology. Of course if $A$ is a $C^*$-algebra then $A$ is\n$C^*$-nuclear if and only if $A$ is nuclear, and $A$ is\n$\\Bdb$-nuclear if and only if $A$ has the local lifting property\n(LLP), see \\cite{Kir1,Pisbook}.  As in the $C^*$-algebra case, we\nwill see that $C^*$-nuclearity implies the other new properties\nmentioned in the introduction, with the possible exception of\nsubexactness ($C^*$-nuclearity does imply exactness).\n\nSince we will use it many times we restate Lemma 2.8 of \\cite{BR}:\n\n\n\n\\begin{lemma} \\label{28} For an approximately unital operator\nalgebra $A$, and a $C^*$-algebra $B$, we have $A \\maxten B \\subset\nC^*_{\\rm max}(A \\maxten B) = C^*_{\\rm max}(A) \\maxten B$ completely\nisometrically.\n\\end{lemma}\n\nIt is not important that the Hilbert space appearing in the\ndefinition of $\\Bdb$-nuclearity be separable:\n\n\\begin{lemma} \\label{cbh} An approximately unital operator\nalgebra $A$ is $\\Bdb$-nuclear if and only if $A \\maxten B(H) = A\n\\minten B(H)$ for every infinite dimensional Hilbert space $H$.\n\\end{lemma}\n\n\\begin{proof} Note that since $\\Bdb$ can be embedded as a\ncomplemented corner of $B(H)$ we see from Lemma \\ref{28} and e.g.\\\n\\cite[6.1.10]{BLM} that we have canonical complete isometries \\[ A\n\\maxten \\Bdb \\subset \\cma \\maxten \\Bdb \\subset \\cma \\maxten B(H) =\n\\cm{A \\maxten B(H)}.\\] Thus $A \\maxten \\Bdb \\subset A \\maxten B(H)$.\nClearly $A \\minten \\Bdb \\subset A \\minten B(H)$ and thus the reverse\nimplication holds.\n\nFor the forward direction suppose $A$ is $\\Bdb$-nuclear, and let $u\n= \\sum_{k=1}^n \\, a_k \\otimes T_k$ for $a_k \\in A, T_k \\in B(H)$.\nLet $D$ be the $C^*$-algebra generated by $1$ and the $T_k$.  This\nis separable and so there is a unital $*$-isomorphism $\\pi$ from $D$\nonto a $C^*$-algebra in $\\Bdb$, carrying $T_k$ to $S_k$ say. The\ninverse of this $*$-isomorphism extends to a UCP map $\\theta : \\Bdb\n\\to B(H)$.\n  Then $$\\Vert u \\Vert_{A \\maxten B(H)}\n= \\Vert \\sum_{k=1}^n \\, a_k \\otimes \\theta(S_k) \\Vert_{A \\maxten\nB(H)} \\leq \\Vert \\sum_{k=1}^n \\, a_k \\otimes S_k \\Vert_{A \\maxten\n\\Bdb}. $$ Since $A$  is $\\Bdb$-nuclear, the last norm equals\n$$\\Vert \\sum_{k=1}^n \\, a_k \\otimes \\pi(T_k)  \\Vert_{A \\minten \\Bdb} =\n\\Vert \\sum_{k=1}^n \\, a_k \\otimes T_k \\Vert_{A \\minten D} = \\Vert\n\\sum_{k=1}^n \\, a_k \\otimes T_k \\Vert_{A \\minten B(H)},$$ by\ninjectivity of $\\minten$.  The result is now clear.\n\\end{proof}\n\nOne theme of our paper is that an operator space or $C^*$-algebraic\nproperty such as  nuclearity or the LLP for a $C^*$-cover of an\nalgebra $A$, often says something about $C^*$-nuclearity or\n$\\Bdb$-nuclearity for $A$; or vice versa.  The following is a fairly\nsuperficial result of this kind.  In the Examples section we shall\nshow that there are commonly met operator algebras for which $\\cma$\nhas the LLP.\n\n\\begin{proposition}\\label{LLPd} If $\\cma$ is nuclear (resp.\\ has the LLP)\nthen $A$ is $C^*$-nuclear (resp.\\ is $\\Bdb$-nuclear).\n\\end{proposition}\n\n\\begin{proof} For a $C^*$-algebra $B$ (resp.\\ $B = \\Bdb$)\nwe have canonical complete isometries \\[ A \\minten B \\to \\cma\n\\minten B = \\cma \\maxten B = \\cm{A \\maxten B} \\] which compose to a\nmap whose range is in $A \\maxten B$. \\end{proof}\n\n\n\\medskip\n\nWe now look at stability of $C^*$-nuclearity and $\\Bdb$-nuclearity\nunder the usual operator algebra constructions.\n\n\\begin{proposition}\\label{maxqp} If $A$ is an approximately unital ideal in an\napproximately unital operator algebra $B$, and if $D$ is any\napproximately unital operator algebra, then $A \\maxten D \\subset B\n\\maxten D$ completely isometrically. \\end{proposition}\n\n\\begin{proof} Take two nondegenerate commuting completely\ncontractive representations $\\pi: A \\to B(H)$ and $\\theta: D \\to\nB(H)$.  By 2.6.13 of \\cite{BLM} we can extend $\\pi$ to a completely\ncontractive representation $\\tilde{\\pi}: B \\to B(H)$ with \\[\n\\theta(d) \\tilde{\\pi}(b) \\pi(a) \\zeta = \\theta(d) \\pi(ba) \\zeta =\n\\pi(ba) \\theta (d) \\zeta = \\tilde{\\pi}(b) \\theta(d) \\pi(a) \\zeta,\\]\nfor all $a \\in A, b \\in B, d \\in D,$ and $\\zeta \\in H$.  It follows\nthat $ \\tilde{\\pi}$ commutes with $\\theta$ and hence the closure of\n$A \\otimes D $ in $B \\maxten D$ will have the desired property for\n$A \\maxten D$ (see 6.1.1 and 6.1.11 in \\cite{BLM}.)\n\\end{proof}\n\n\n\\begin{corollary}\\label{ideal} If $A$ is an approximately unital ideal in a\n$C^*$-nuclear (resp.\\ $\\Bdb$-nuclear) approximately unital\nalgebra $B$, then $A$ is $C^*$-nuclear (resp.\\ $\\Bdb$-nuclear).\n\\end{corollary}\n\n{\\bf Remark.}  If $A$ is a subalgebra of an operator algebra $B$,\nand if $D$ is a nonselfadjoint operator algebra, then it need not be\nthe case that $A \\maxten D \\subset B \\maxten D$ completely\nisometrically. Note that it follows from Lemma \\ref{28} that $A\n\\maxten D \\subset \\cm{A} \\maxten D$ for any $C^*$-algebra $D$, but\nthis relation can be false if $D$ is nonselfadjoint. To see this let\n$A$ be an operator algebra for which $A \\minten A(\\Ddb) \\neq A\n\\maxten A(\\Ddb)$ (see \\cite[Corollary 7.1.8]{BLM}).  Now $A(\\Ddb)$\nis $C^*$-nuclear as we remarked earlier, and hence we must have\n$\\cma \\minten A(\\Ddb) = \\cma \\maxten A(\\Ddb)$.  This contains $A\n\\minten A(\\Ddb)$, and so it cannot contain $ A \\maxten A(\\Ddb)$.\n\n\\medskip\n\n\nTo deal with quotients we will need the following.  We do not know\nif the result is true for nonselfadjoint $D$:\n\n\\begin{lemma}\\label{maxq} If $D$ is a $C^*$-algebra and if $A$ is an\napproximately unital ideal in an operator algebra $B$, then $(B\n\\maxten D) \/(A \\maxten D ) \\cong (B\/A) \\maxten D$ completely\nisometrically.  The same relation holds if $D$ is an approximately\nunital operator algebra and $B$ is a $C^*$-algebra with ideal $A$.\n\\end{lemma}\n\n\\begin{proof} The assertions have similar proofs so we prove only\nthe first.  We know from \\cite[Lemma 2.7]{BR} that $\\cma$ is an\napproximately unital ideal in $\\cm{B}$, and that $ \\cm{B}\/\\cma \\cong\n\\cm{B\/A}$.  Setting $C = B\/A$, and using the usual $C^*$-algebra\nresult, we have the extension \\[ 0 \\longrightarrow \\cma \\maxten D \\longrightarrow \\cm{B}\n\\maxten D \\longrightarrow \\cm{C} \\maxten D \\longrightarrow 0. \\]  We will show that \\[ 0\n\\longrightarrow A \\maxten D \\longrightarrow B \\maxten D \\longrightarrow C \\maxten D \\longrightarrow 0\n\\] is a {\\em subextension} of the first extension, in the sense of\n\\cite[Section 3.6]{BR}.  To see this first note that, by Lemma\n\\ref{28}, each term in the last sequence is a subalgebra of the\nmatching term in the $C^*$-algebra extension. Moreover the\nintersection $(C^*_{\\rm max}(A) \\maxten D) \\cap (B \\maxten D)$ in\n$C^*_{\\rm max}(B)  \\maxten D$ is the closure of $A \\otimes D$, as\nmay be seen by the following argument. If $u$ is in this\nintersection, and if $(e_t)$ (resp.\\ $(f_s)$)\n is a cai for\n $A$ (resp.\\ $D$), then $(e_t \\otimes f_s) u$ is in the\n closure of $A \\otimes D$.  Taking the limit, and using\n the fact that $(f_s)$\nis a cai for $C^*_{\\rm max}(D)$, we see that $u$ is in the\n closure of $A \\otimes D$ in $C^*_{\\rm max}(B)  \\maxten D$.\nBut this closure is in the image of $A \\maxten D$\nsince by the previous result, $A \\maxten D \\subset B \\maxten D\n\\subset C^*_{\\rm max}(B)  \\maxten D$.  By \\cite[Proposition 3.6]{BR}\nwe have a subextension, proving the result.\n\\end{proof}\n\n\\begin{lemma} \\label{free} Every $C^*$-nuclear approximately unital\noperator algebra is exact, hence is locally reflexive.\n\\end{lemma}\n\n\\begin{proof} If $D$ is a $C^*$-nuclear approximately unital\noperator algebra, and if \\[ 0 \\longrightarrow A \\longrightarrow B \\longrightarrow C \\longrightarrow 0 \\] is a\n$C^*$-extension, then by Lemma \\ref{maxq} we have an exact sequence\n\\[ 0 \\longrightarrow \\cm{D} \\maxten A \\longrightarrow \\cm{D} \\maxten B \\longrightarrow \\cm{D} \\maxten\nC \\longrightarrow 0 . \\]  As in the proof of Lemma \\ref{maxq} we have that \\[ 0\n\\longrightarrow D \\maxten A \\longrightarrow D \\maxten B \\longrightarrow D \\maxten C \\longrightarrow 0 \\] is a\nsubextension in the sense of \\cite[Section 3.6]{BR}, hence it is a\n`$1$-exact' sequence in the sense of \\cite{ER}. Thus by\n\\cite[Theorem 14.4.1]{ER} we see that $D$ is exact.  \\end{proof}\n\n\nWe are now ready for some results concerning quotients.  It is not\nclear whether in the following proposition a weaker  condition on\n$B\/A$ than the completely contractive approximation property will\nsuffice.\n\n\\begin{proposition}\\label{quot} Let $A$ be an approximately unital ideal in a\n$C^*$-nuclear approximately unital operator algebra $B$. If $B\/A$ is\nseparable and has the completely contractive approximation property,\nthen $B\/A$ is $C^*$-nuclear. \\end{proposition}\n\n\\begin{proof} By Proposition \\ref{ideal} we know that $A$ is\n$C^*$-nuclear too.  By Lemma \\ref{maxq} we have $(B\/A) \\maxten D\n\\cong (B \\maxten D)\/(A \\maxten D)$ for any $C^*$-algebra $D$. Since\n$C^*$-nuclearity implies local reflexivity by Lemma \\ref{free}, the\nextension of $B\/A$ by $A$ satisfies the condition of the lifting\ntheorem in \\cite{ER1}.  Hence by that result there is a completely\ncontractive linear splitting map $B\/A \\to B$.  From this it follows\neasily that $(B\/A) \\minten D \\cong (B \\minten D)\/(A \\minten D) \\cong\n(B\/A) \\maxten D$.\n\\end{proof}\n\n\n\\begin{theorem}\\label{ssh} If $C$ is a $C^*$-nuclear approximately unital\noperator algebra, and if\n$$ 0 \\longrightarrow A \\longrightarrow B\n\\longrightarrow C \\longrightarrow 0  $$ is an extension, then\nfor every $C^*$-algebra $D$, the associated sequence\n$$0 \\longrightarrow A \\otimes_{\\rm min} D \\longrightarrow\nB \\otimes_{\\rm min} D \\longrightarrow  C \\otimes_{\\rm min} D$$ is an extension\n(these are extension in the sense of \\cite{BR}).  So $(B \\otimes_{\\rm min} D)\/(A \\otimes_{\\rm min}\nD) \\cong (B\/A) \\otimes_{\\rm min} D$ completely isometrically.\n\\end{theorem}\n\n\\begin{proof} The canonical  morphism from\n$B \\maxten D$ to $(B \\minten D)\/ (A \\minten D)$ annihilates the\nclosure of $A \\otimes B$, which by Proposition \\ref{maxqp} is equal\nto $A \\maxten D$. Thus by Lemma \\ref{maxq} we have canonical\ncompletely contractive  morphisms\n$$ C \\maxten D\n\\cong (B \\maxten D)\/(A \\maxten D) \\to (B \\minten D)\/(A \\minten D)\n\\to C \\minten D $$ which compose to the identity map since $C\n\\minten D \\cong C \\maxten D$. Since the range of the first `arrow'\nis dense,  we see that $C \\minten D \\cong (B \\minten D)\/(A \\minten\nD)$ via the canonical map. This completes the proof.\n\\end{proof}\n\n{\\bf Remark.}  There are several conditions equivalent to an\nextension having the `tensorizing with every $C^*$-algebra' property\nin the last theorem. These are studied in \\cite{Bseq}.\n\n\\medskip\n\n\n\n\\begin{corollary}\\label{exten} Let \\[ \\xymatrix{0 \\ar[r] & A\n\\ar[r] & B \\ar[r] & C \\ar[r] & 0} \\] be an extension of operator\nalgebras in the sense of {\\rm \\cite{BR}}. If both $A$ and $C$ are\n$C^*$-nuclear (resp.\\ $\\Bdb$-nuclear) then $B$ is $C^*$-nuclear\n(resp.\\ $\\Bdb$-nuclear).\n\\end{corollary}\n\n\\begin{proof} Let $D$ be a $C^*$-algebra.  By Lemma \\ref{maxq} we have\nan extension \\[ \\xymatrix{0 \\ar[r] & A\\maxten D \\ar[r] & B \\maxten D\n\\ar[r] & C \\maxten D \\ar[r] & 0}.\n\\]   By Theorem \\ref{ssh} (or a variant of it in the\n$\\Bdb$-nuclear case) we have an extension\n\\[ \\xymatrix{0 \\ar[r] & A\\minten\nD \\ar[r] & B \\minten D \\ar[r] & C \\minten D   \\ar[r] & 0}.\n\\]\nNow apply the `five lemma' from \\cite{BR} Lemma 3.2.  \\end{proof}\n\n\\begin{corollary} \\label{w16}  Let $A$ be an approximately unital\noperator algebra.\nThen $A$ is $C^*$-nuclear (resp.\\ $\\Bdb$-nuclear) if and only if\n$A^1$ is $C^*$-nuclear (resp.\\ $\\Bdb$-nuclear) \\end{corollary}\n\n\\begin{proof} We have the short exact sequence \\[ 0 \\longrightarrow A \\longrightarrow A^1\n\\longrightarrow \\Cdb \\longrightarrow 0 \\] with $A$ an approximately unital ideal in $A^1$.\nThe forward direction now follows from the previous result, and the\nreverse follows from Corollary \\ref{ideal}. \\end{proof}\n\nIt is characteristic of the present paper that one gets much better\nresults by restricting the class of operator algebras:\n\n\\begin{proposition}\\label{dirLL} If $A$ is a Dirichlet operator\nalgebra then $A$ is $C^*$-nuclear (resp.\\ $\\Bdb$-nuclear) if and\nonly if $C^*_{\\rm e}(A)$ is nuclear (resp. has the LLP).  The {\\rm\n($\\Leftarrow$)} implications are also true if $A$ is merely\n$C^*$-split.\n\\end{proposition}\n\n\\begin{proof}  If $B$ is a $C^*$-algebra, then\n$A \\maxten B \\subset C^*_{\\rm max}(A) \\maxten B$ by Lemma \\ref{28}.\nIf $A$ is $C^*$-split then there is a complete contractive right\ninverse to the canonical map  $C^*_{\\rm max}(A) \\maxten B \\to\nC^*_{\\rm e}(A) \\maxten B$.  If the latter equals $C^*_{\\rm e}(A)\n\\maxten B$, then it follows that $A \\maxten B = A \\minten B$.\nConversely if the latter holds,  then $A \\maxten B \\subset C^*_{\\rm\ne}(A) \\maxten B$.  If $A$ is Dirichlet, then Arveson's Proposition\n1.2.8 of \\cite{SOC} gives a complete isometry from the closure of $A\n\\minten B + A^* \\minten B$ which is $C^*_{\\rm e}(A) \\minten B$, into\nthe closure of $A \\maxten B + A^* \\maxten B$ in $C^*_{\\rm e}(A)\n\\maxten B$, which is $C^*_{\\rm e}(A) \\maxten B$.\n\\end{proof}\n\n\n\n\\section{The homomorphism local lifting property and $\\Bdb$-nuclearity}\n \\label{thehllp}\n\nIn analogy with the $C^*$-algebraic theory of the LLP, one would\nexpect a relationship between $\\Bdb$-nuclearity and lifting\nproperties.  At present we only see one direction of the\nrelationship, which will be presented in the next theorem.\n\n\\begin{definition} An operator algebra $C$ has the homomorphism\nlocal lifting property (HLLP) if for every operator algebra $B$, and\nany approximately unital ideal $A$ in $B$, and any completely\ncontractive homomorphism $u: C \\to B\/A$ and finite dimensional\nsubspace $E \\subset C$, there is a complete contraction from $E$ to\n$B$ which is a lift of $u|_E$.\n\\end{definition}\n\nOur original motivation in studying the HLLP, is that it has some\nconnections with the topic of when $Ext$ is a group, and this will\nbe presented in \\cite{Bseq}. For example, we show in \\cite{Bseq}\nthat every extension in the sense of \\cite{BR} of a separable\noperator algebra with the HLLP, by $\\Kdb$ (or by any $C^*$-algebra\nwith a property described there)  is `semisplit' in the sense of\n\\cite{Bseq}.\n\n\\begin{theorem} \\label{llp} Let $C$ be an approximately unital\noperator algebra. If $C$ is $\\Bdb$-nuclear then $C$ has the HLLP.\n\\end{theorem}\n\nWe defer the proof momentarily to prove a lemma which is of\nindependent interest (solving an open question about tensor products\nof $M$-ideals in a special case: see the discussion before\nProposition 1.7 in \\cite{AR}).\n\n\\begin{lemma} \\label{toc22} Suppose that $A$ is an approximately\nunital ideal in an operator algebra $B$, and that $E$ is an operator\nspace.  Then $A \\minten E$ is a complete $M$-ideal in $B \\minten E$,\nand in particular is proximinal. \\end{lemma}\n\n\\begin{proof}  We know that $A^{\\perp \\perp} = e B^{**}$ for a\ncentral projection $e \\in B^{**}$. Notice that $B \\minten E$ is a\nleft operator $B$-module (see e.g.\\ the second paragraph of the\nNotes for \\S 3.4 in \\cite{BLM}). Thus $(B \\minten E)^{**}$ is a left\ndual operator $B^{**}$-module, by 3.8.9 in \\cite{BLM}. Hence $e \\in\nB^{**}$ may be regarded as a left $M$-projection on $(B \\minten\nE)^{**}$, and we claim that if we do so then $(A \\minten E)^{\\perp\n\\perp} = e (B \\minten E)^{**}$. It is routine to see that $(A\n\\minten E)^{\\perp \\perp} \\subset e (B \\minten E)^{**}$. On the other\nhand, if $a_t \\to e$ weak* in $B^{**}$, with $a_t \\in A$, then since\n$a_t (b \\otimes x) \\in A \\otimes E \\subset (A \\minten E)^{\\perp\n\\perp}$ for $b \\in B$ and $x \\in E$, it follows by separate weak*\ncontinuity and density that $e (B \\minten E)^{**} \\subset (A \\minten\nE)^{\\perp \\perp}$.   This proves the claim, and shows that $A\n\\minten E$ is a complete right $M$-ideal in $B \\minten E$.\nSimilarly, it is a complete left $M$-ideal in $B \\minten E$, and so\nit is a complete $M$-ideal \\cite{BEZ}.\\end{proof}\n\n\\begin{proof} (Of Theorem \\ref{llp}) \\\nWe will adapt a proof due to Pisier of Kirchberg's result that\n$\\Bdb$-nuclearity implies the LLP for $C^*$-algebras. The reader\nshould follow along with the proof of the (iii) implies (i) in\n\\cite[Theorem 16.2]{Pisbook}.  We begin with $s \\in (A\/I) \\otimes\nE^*$, as in that proof.  Suppose $s = \\sum_{k=1}^n \\, [a_k]  \\otimes\n\\psi_k$, for $a_k \\in A, \\psi_k \\in E^* \\subset B(H)$.  The first\nchange that needs to be made is that instead of appealing to (11.1)\none uses the functoriality of the  $\\maxten$ tensor product of\noperator algebras. (Note: this is the only place where $u$ being a\nhomomorphism is used.) The appeal to Exercise 11.2 is replaced by\nLemma \\ref{maxq} above. One obtains $\\Vert \\sum_{k=1}^n \\, [a_k]\n\\otimes \\psi_k \\Vert \\leq 1$ in $(A\/I) \\maxten B(H)$.  Then $\\Vert\n\\sum_{k=1}^n \\, a_k \\otimes \\psi_k \\Vert \\leq 1$ in $(A \\maxten\nB(H))\/(I \\maxten B(H))$, and it follows easily that $\\Vert\n\\sum_{k=1}^n \\, a_k \\otimes \\psi_k + (I \\minten B(H)) \\Vert \\leq 1$\nin  $(A \\minten B(H))\/(I \\minten B(H))$. The proof of Lemma 2.4.8 in\n\\cite{Pisbook} may be easily adapted to our case, if one uses the\nknown fact (see e.g.\\ \\cite{DP, AR} or \\cite[Proposition 6.3]{BHN})\nthat any ideal with cai has an approximate identity of the form\n$(1-x_t)$ with $\\Vert x_t \\Vert \\to 1$.  This implies that $\\Vert\n\\sum_{k=1}^n \\, a_k \\otimes \\psi_k + (I \\minten E^*) \\Vert \\leq 1$\nin $(A \\minten E^*)\/(I \\minten E^*)$. The proof is completed as in\n\\cite{Pisbook} by an appeal to Lemma \\ref{toc22}.\n\\end{proof}\n\n\n\n\n We now turn to operator algebras $A$ with the {\\em unique\nextension property (UEP)} considered in \\cite{Dunc2}: that is, $A$\nis a unital-subalgebra $A$ of a $C^*$-algebra $B$, such that for\nevery Hilbert space $H$ and every unital $*$-homomorphism $\\pi : B\n\\to B(H)$, there is a unique UCP map $\\Psi : B \\to B(H)$ extending\n$\\pi_{\\vert A}$. Clearly, we can replace $B(H)$ in this definition,\nby `every unital $C^*$-algebra'.   It follows from the proof of\n\\cite[Theorem 2.7]{BL} that $B = C^*_{e}(A)$. Examples of algebras\nwith the UEP include logmodular and Dirichlet algebra (see p.\\ {\\rm\n159-161} in {\\rm \\cite{BLM}}), and some nest algebras and crossed\nproducts \\cite{Dunc2}.  As Elias Katsoulis has pointed out, it is an\neasy consequence of Choi's `multiplicative domain' trick (see e.g.\\\n\\cite[Lemma 14.2]{Pisbook}), that any unital operator algebra\ngenerated by unitaries will have UEP.\n\n\\begin{lemma}\\label{uepten} If $A$ and $B$ are operator\nalgebras with the UEP, then $A \\minten B$ has the UEP. \\end{lemma}\n\n\\begin{proof} Since $A$ and $B$ are unital there are completely isometric\ninclusions $A \\subset A \\minten B$ and $B \\subset A \\minten B$.  Now\nlet $\\pi: C^*_{\\rm e}(A \\minten B) \\to B(H)$ be a unital\n$*$-homomorphism, and let $\\sigma: C^*_{\\rm e}(A \\minten B) \\to\nB(H)$ be a unital completely positive map satisfying $ \\pi|_{A\n\\minten B} = \\tau|_{A \\minten B}$. We know by \\cite[Theorem\n2.10]{BR} that $C^*_{\\rm e}(A \\minten B) = C^*_{\\rm e}(A) \\minten\nC^*_{\\rm e}(B)$. By the unique extension property  $\n\\sigma|_{C^*_{\\rm e}(A)} = \\pi|_{C^*_{\\rm e}(A)}$.  A similar result\nholds for $C^*_{\\rm e}(B)$, and hence $C^*_{\\rm e}(A)$ and $C^*_{\\rm\ne}(B)$ are contained in the multiplicative domain (see e.g.\\\n\\cite{Pnbook} or \\cite[Lemma 14.2]{Pisbook}) for $\\sigma$. Since\n$C^*_{\\rm e}(A) $ and $C^*_{\\rm e}(B)$ generate $C^*_{\\rm e}(A\n\\minten B)$,we have that $C^*_{\\rm e}(A \\minten B)$ is contained in\nthe multiplicative domain for $\\sigma$.  Thus $\\sigma$ is a\n$*$-homomorphism, and so $ \\pi = \\sigma$.\n\\end{proof}\n\n\\begin{lemma} \\label{uepn} If $A$ is an operator algebra with\nthe UEP, and $N$ is a nonunital nuclear $C^*$-algebra, then $(A\n\\minten N)^1$ has the UEP. \\end{lemma}\n\n\\begin{proof} Recall from \\cite{BR} that $C^*_{\\rm e}((N \\minten A)^1)\n= C^*_{\\rm e}(N \\minten A)^1 = (N \\minten C^*_{\\rm e}(A))^1$.  Let\n$\\theta : (N \\minten C^*_{\\rm e}(A))^1 \\to B(H)$ be a unital\n$*$-homomorphism, and let $\\Phi : (N \\minten C^*_{\\rm e}(A))^1 \\to\nB(H)$ be a UCP map extending $\\theta_{\\vert (N \\minten C^*_{\\rm\ne}(A))^1}$. We need to show that $\\Phi = \\theta$.\n\nThe restriction of $\\theta$ to $N \\minten C^*_{\\rm e}(A)$ is of\nthe form $\\pi \\odot \\rho$ for commuting $*$-representations $\\pi : N\n\\to B(H)$ and $\\rho : C^*_{\\rm e}(A)  \\to B(H)$.  If $(e_t)$ is an\nincreasing cai for $N$, then $\\pi(e_t) \\to q$ strongly for a\nprojection $q$ commuting with $\\rho(1)$. Let $p = q \\rho(1)$, and\nidentify $p B(H) p = B(K)$, where $K = p H$. Since $\\pi(f) \\rho(a) =\nq \\pi(f) \\rho(a) \\rho(1)$, we may replace $\\pi$ and $\\rho$ by $p\n\\pi(\\cdot)$ and $p \\rho(\\cdot)$.\nThen we\nmay regard $\\pi$ and $\\rho$ as being $B(K)$-valued with commuting\nranges, and now both of them are nondegenerate (for example\n$\\pi(e_t) \\to p = I_K$ strongly).   Let $\\pi^1 : N^1 \\to B(K)$ be\nthe `unitization' of $\\pi$, then $\\pi^1 \\odot \\rho$ is a unital\n$*$-homomorphism $N^1 \\minten C^*_{\\rm e}(A)  \\to B(K)$. Let $\\Psi$\nbe the restriction of $\\Phi$ to $N \\minten C^*_{\\rm e}(A)$. Of\ncourse $\\Phi(f \\otimes 1_A) = \\theta(f \\otimes 1_A)$ for $f \\in N$,\nso that it follows from  a well known principle concerning\ncompletely positive maps, that $\\Phi(fg \\otimes a) = \\theta(f\n\\otimes 1_A) \\Phi(g \\otimes a) = \\pi(f) p \\Phi(g \\otimes a)$ if $f,\ng \\in N, a \\in C^*_{\\rm e}(A)$. Thus $\\Psi$ may also be viewed as a\n$B(K)$-valued map.\n\nIt is well known that we can extend $\\Psi$ to a unital completely\npositive map from the subspace $N \\minten C^*_{\\rm e}(A) + \\Cdb\n1_{N^1} \\otimes 1_A$ of $N^1 \\minten C^*_{\\rm e}(A)$ to $B(K)$.  We\nmay then extend further to a unital completely positive map\n$\\tilde{\\Psi} : N^1 \\minten C^*_{\\rm e}(A)  \\to B(K)$. We claim that\nfor $f \\in N^1$ we have $\\tilde{\\Psi}(f \\otimes 1_A) = \\pi^1(f)$.\nIndeed for $f \\in N$ we have $\\tilde{\\Psi}(f \\otimes 1_A) = \\Psi(f\n\\otimes 1_A) = \\pi(f) = \\pi^1(f)$. The claim  is also true for $f =\n1_{N^1}$, and hence it is true in full generality. By the `well\nknown principle' used a few lines earlier, we have $\\tilde{\\Psi}(f\n\\otimes  a) = \\pi^1(f) T(a)$ if $f \\in N^1, a \\in C^*_{\\rm e}(A)$,\nwhere $T(a) = \\tilde{\\Psi}(1 \\otimes a)$. Note that $T$ is UCP.\nAlso if $a \\in A$ then \\[ \\pi(e_t) T(a) = \\tilde{\\Psi}(e_t  \\otimes\na) = \\Phi(e_t  \\otimes  a) = \\pi(e_t) \\rho(a),\\] and taking a limit\nshows that $T(a) = \\rho(a)$.   Since $A$ has the UEP, $T = \\rho$,\nand so\n\\[ \\Phi(f \\otimes  a) = \\Psi(f \\otimes  a) = \\tilde{\\Psi}(f \\otimes\na) = \\pi^1(f) \\rho(a) = \\theta(f \\otimes  a),\\] for $f \\in N, a \\in\nC^*_{\\rm e}(A)$. It follows that $\\Phi  = \\theta$ as desired.\n\\end{proof}\n\n\\begin{proposition} \\label{eex3} If $C$ is a separable unital\n$\\Bdb$-nuclear operator algebra with the UEP, then $C^*_{\\rm e}(C)$\nhas the LLP.\n\\end{proposition}\n\n\\begin{proof} It is shown in \\cite{Bseq}, using the HLLP\nand in particular the fact mentioned above Theorem \\ref{llp}, that\nthese hypotheses imply that $Ext_u(C^*_{\\rm e}(A))$ is a group.\nIf  $S(C)$ for the `unitized suspension' as in \\cite{Kir1}, then\n$S(C)$ is a separable, unital algebra.  It is easy to see that it is\n$\\Bdb$-nuclear using Corollary \\ref{w16}.  By the previous lemma we\nalso have that $S(C)$ has the UEP, so that by the above we deduce\nthat $Ext_u(C^*_{\\rm e}(S(C)))$ is a group.  Since $C^*_{\\rm\ne}(S(C)) = S(C^*_{\\rm e}(C))$ (see \\cite[Corollary 2.11]{BR}), it\nnow follows from a result of Kirchberg \\cite{Kir1} that $C^*_{\\rm\ne}(C)$ has the LLP.\n\\end{proof}\n\nIt follows that if $A$ is $C^*$-split, or is separable and has UEP,\nthen $C^*_{\\rm e}(A)$ having LLP implies that $A$ has HLLP (see\nTheorem \\ref{llp} and Proposition \\ref{dirLL}).\n\n\n\\section{Weak expectation} \\label{awepewep}\n\nWe  turn to themes connected with the weak expectation\nproperty.  For simplicity, unless stated otherwise we assume that\nall algebras are unital, and that all subalgebras are\n`unital-subalgebras'. We leave the nonunital case to the reader\nusing the usual unitization results (e.g.\\ as in \\cite[Section 2.1,\n6.1.6, and 6.1.11]{BLM}, and the remark after the next definition).\nThis definition reduces to the WEP property of Lance if $A$ is\nselfadjoint.\n\n\\begin{definition} We say that an operator algebra $A$ has the {\\em\nalgebra weak expectation property} (AWEP) if $A \\maxten D \\subset B\n\\maxten D$ completely isometrically for every (possibly nonunital)\n$C^*$-algebras $B,D$ and completely isometric embedding of $A$ as a\nsubalgebra of $B$. \\end{definition}\n\n\\begin{proposition}  \\label{cnimawep}\nIf an operator algebra is $C^*$-nuclear then it has the AWEP.\n\\end{proposition}\n\n\\begin{proof}  If $A$ is $C^*$-nuclear then for all $C^*$-algebras $D$ the\nembedding $A \\otimes D \\to B \\maxten D$ induces a tensor norm on $A\n\\otimes D$ which must be the maximal tensor norm.\n\\end{proof}\n\n\n{\\bf Remark.} We may assume in the last definition that $B$ is\nunital, and $A$ is a unital-subalgebra of $B$.   For suppose that $A\n\\maxten D \\subset B \\maxten D$ for every unital $C^*$-algebra $B$\ncontaining $A$ completely isometrically, and every $C^*$-algebra\n$D$. If $C$ is a general $C^*$-algebra containing $A$ as a\nsubalgebra completely isometrically, and if $p \\in C$ is the\nidentity for $A$, then define $B = pCp$.  The canonical projection\nof $C$ onto $B$ induces canonical complete contractions\n\\[ A \\maxten D \\longrightarrow B \\maxten D \\longrightarrow C \\maxten D \\] which compose to\na complete isometry, and hence $A$ has AWEP.\n\n\\medskip\n\nFollowing the $C^*$-algebraic theory (see e.g.\\ \\cite[Theorem\n3.3]{Oz2}), we restate the\ntensorial condition in terms of a `weak expectation':\n\n\\begin{proposition} \\label{ozt}  If $A$ is a unital-subalgebra\nof a unital $C^*$-algebra $B$, then $$ A \\maxten D \\subset B \\maxten\nD $$ completely isometrically, for every $C^*$-algebra $D$, if and\nonly if there is a UCP map $\\varphi : B \\to \\cm{A}^{**}$ such that\n$\\varphi\\vert_A$ is the `identity map' on $A$.   This is also\nequivalent to: for every $C^*$-algebra $G$ containing $A$ completely\nisometrically and as a unital-subalgebra, there is a UCP map\n$\\varphi : B \\to G^{**}$ such that $\\varphi\\vert_A$ is the `identity\nmap' on $A$.\n\\end{proposition}\n\n\\begin{proof}\nWe follow the usual $C^*$-algebraic proof.  Suppose that the first\ncondition holds, and that $G^{**}$ is a von Neumann subalgebra $M\n\\subset B(H)$.  Let $D = M' \\subset B(H)$.  We have $A \\maxten D\n\\subset B \\maxten D$, by the hypothesis. The canonical product map\n$A \\maxten D \\to B(H): a \\otimes d = \\hat{a} d$ extends to a\ncompletely contractive unital, and hence completely positive, map\n$\\Phi : B \\maxten D \\to B(H)$. Define $T : B \\to B(H)$ by $T(x) =\n\\Phi(x \\otimes 1)$. Since $\\Phi(1 \\otimes d) = d$ for all $d \\in D$,\na well known lemma implies that $T(x) d = \\Phi(x \\otimes d) = d\nT(x)$ for all $d \\in D, x \\in B$.  That is, $T : B \\to D' = M =\nG^{**}$. Finally, $T(a) =  \\Phi(a \\otimes 1) = \\hat{a}$ for $a \\in\nA$.\n\nThe other direction is much easier, essentially just as in\n\\cite{Oz2}. Namely, we consider the canonical sequence \\[ A \\maxten\nD \\to B  \\maxten D \\to \\cm{A}^{**}  \\maxten D, \\] and use the fact\nthat $A\n\\maxten D \\subset \\cm{A}^{**}  \\maxten D$\n(see  Lemma \\ref{28} and \\cite[Exercise 11.6]{Pisbook}).\n \\end{proof}\n\n{\\bf Remark.}  In the last line of the previous result one may\nreplace $B$ by $B^{**}$.\n\n\\medskip\nBy the remark after Proposition \\ref{cnimawep}, and by\nProposition \\ref{ozt}, $A$ having the AWEP is equivalent to the\nconditions in Proposition \\ref{ozt} holding for every unital\n$C^*$-algebra $B$ containing $A$ completely isometrically as a\nunital-subalgebra. It is easy to see that this is equivalent to the\nconditions in Proposition \\ref{ozt} holding for $B = B(H)$, for all\nHilbert spaces $H$ and for all embeddings of $A$ in $B(H)$\ncompletely isometrically as a unital-subalgebra.   One may replace\nthe words `for all'  in the last line with `for one fixed', to\nobtain condition (iv) in the next theorem.  However this is shown\nthere to be equivalent to the AWEP.\n\n\\begin{theorem} \\label{awep} For a unital operator algebra $A$,\nconsider the following conditions: \\begin{enumerate}\n\n\\item[(i)] $\\cma$ has the WEP.\n\n\\item[(ii)] $A$ has the AWEP.\n\n\\item[(iii)]  There exists an injective operator space $R \\subset\n\\cma^{**}$\ncontaining the canonical copy of $A$.\n\\item[(iii)']  For every $C^*$-cover $B$ of $A$, there exists an\ninjective operator space $R \\subset B^{**}$ containing the\ncanonical copy of $A$.\n\n\\item[(iv)]  For every $C^*$-algebra $B$ containing $A$\ncompletely isometrically as a unital subalgebra, there exists a\nHilbert space $H$ and a completely isometric unital homomorphism\n$\\pi: A \\to B(H)$, and a UCP map $T: B(H) \\to B^{**}$, such that $T\n\\circ \\pi = I_A$.\n\n\\item[(v)]  $C^*_{\\rm e}(A)$ has the WEP. \\end{enumerate} Then ${\\rm (i)}\n\\Rightarrow {\\rm (ii)} \\Leftrightarrow {\\rm (iii)} \\Leftrightarrow\n{\\rm (iii)'} \\Leftrightarrow {\\rm (iv)} \\Rightarrow {\\rm (v)}$.\n\\end{theorem}\n\n\\begin{proof} ${\\rm (i)} \\Rightarrow {\\rm (iii)}$ \\ If ${\\rm (i)}$\nholds then there is an injective operator system $R$ between $\\cma$\nand $\\cma^{**}$.\n\n${\\rm (ii)} \\Rightarrow {\\rm (iv)}$ \\ This is a corollary of\nProposition \\ref{ozt}.\n\n${\\rm (iv)} \\Rightarrow {\\rm (iii)'}$ \\ Suppose that  $A$ is a\nunital-subalgebra of a $C^*$-algebra $B$, and that $B^{**}$ is a\nunital-subalgebra of $B(K)$. By injectivity of $B(H)$ we can extend\nthe map $\\pi$ in (iv) to a completely contractive unital, hence UCP,\nmap $\\tilde{\\pi} : B(K) \\to B(H)$.  If $T$ is as in (iv), let\n$\\tilde{T} = T \\circ \\tilde{\\pi} : B(K) \\to B^{**}$. Let $\\Theta$ be\na minimal $\\hat{A}$-projection on $B(K)$.  Then $\\Vert\n\\tilde{T}(\\Theta( \\cdot )) \\Vert$ is a $\\hat{A}$-seminorm on $B(K)$\nwhich is dominated by $\\Vert \\Theta( \\cdot ) \\Vert$, and hence the\ntwo must coincide. By \\cite[Lemma 4.2.2]{BLM} we deduce that\n$\\tilde{T} \\circ \\Theta$ is idempotent. Since the range of\n$\\tilde{T} \\circ \\Theta$ is contained in $B^{**}$ we deduce that\nthere is an injective operator system $Z$ with $A \\subset Z \\subset\nB^{**}$.\n\n ${\\rm (iii)'} \\Rightarrow\n{\\rm (iii)}$ \\ Trivial.\n\n${\\rm (iii)} \\Rightarrow {\\rm (ii)}$ \\ If $A$ is a unital-subalgebra\nof a $C^*$-algebra $B$, we can extend the inclusion $A \\subset \\cma$\nto a UCP map $B \\to R$.  Hence $A$ satisfies the hypotheses in\nProposition \\ref{ozt}, which yields ${\\rm (ii)}$.\n\n\n${\\rm (iii)'} \\Rightarrow {\\rm (v)}$ \\ There is an injective\n $Z$  with $A \\subset Z \\subset C^*_{\\rm e}(A)^{**}\n\\subset B(H)$. Let $T : B(H) \\to Z$ be a UCP idempotent.\nBy the rigidity property of $C^*_{\\rm e}(A)$, we have that $R =\nT|_{C^*_{\\rm e}(A)}$ is a complete isometry onto $W = T(C^*_{\\rm\ne}(A))$.  Then $R^{-1}: W \\to C^*_{\\rm e}(A)$ extends to a complete\ncontraction $\\mu: C^*_{\\rm e}(A)^{**} \\to Z \\subset C^*_{\\rm\ne}(A)^{**}$ such that $ \\mu|_{A} = I_{A}$. Then $ \\mu \\circ T: B(H)\n\\to C^*_{\\rm e}(A)^{**}$ with $ \\mu(T(x)) = R^{-1}(T(x)) =\nR^{-1}(R(x)) = x$ for all $x \\in C^*_{\\rm e}(A)$.  Hence $C^*_{\\rm\ne}(A)$ has the WEP.\n\\end{proof}\n\n{\\bf Remarks.} 1) \\ Variants of the above proof shows that if some\n$C^*$-algebra generated by $A$ has the WEP then so does $C^*_{\\rm\ne}(A)$; and that $C^*_{\\rm e}(A)$ has the WEP iff there exists an\ninjective  $R \\subset C^*_{\\rm e}(A)^{**}$\ncontaining the canonical copy of $A$.\n\n2) \\  As in Proposition \\ref{ozt},\none may\nreplace $B$ in (iv) by $C^*_{\\rm max}(A)$.\n\n\\begin{proposition}\\label{wenu} Let $A$ and $B$ be approximately unital\noperator algebras, with $A$ $\\Bdb$-nuclear and $B$ having the AWEP.\nWe have $ A \\minten B = A \\maxten B$ if either $A$ or $B$ is a\n$C^*$-algebra.\n\\end{proposition}\n\n\\begin{proof} If $A$ is a $C^*$-algebra with the LLP and if $B$ has\nthe AWEP then $A \\maxten B \\subset A \\maxten C^*_{\\rm e}(B)$.  By\nTheorem \\ref{awep} we have $C^*_{\\rm e}(B)$ has the WEP, and so by\nthe matching theorem of Kirchberg \\cite[Proposition 1.1 (i)]{Kir1}\nwe have $A \\minten C^*_{\\rm e}(B) = A \\maxten C^*_{\\rm e}(B)$.  From\nthis the result is clear.\n\nIf $B \\subset B(H)$ is a $C^*$-algebra with the WEP, and if $A$ is\n$\\Bdb$-nuclear, then using Lemma \\ref{28} we have\n\\[ A \\maxten\nB \\subset \\cma \\maxten B \\subset \\cma \\maxten B(H) = \\cm{A \\maxten\nB(H)}.\\] Since $A \\minten B(H) = A \\maxten B(H)$ and $A \\minten B\n\\subset A \\minten B(H)$ we are done.\n\\end{proof}\n\n\\begin{proposition}\\label{weps}  Suppose that $A$ is a\n$C^*$-split unital operator algebra. Then $A$ has AWEP iff $C^*_{\\rm\ne}(A)$ has WEP. Also,  $A$ has AWEP if $C^*(F) \\minten A = C^*(F)\n\\maxten A$ completely isometrically for every discrete free group\n$F$.  The converse of the last assertion holds too, if $A$ is\nDirichlet.\n\\end{proposition}\n\n\\begin{proof}    If\n$C^*_{\\rm e}(A)$ has WEP then there is an injective between\n$C^*_{\\rm e}(A)$ and its second dual. If also $A$ is $C^*$-split\nthen there is an injective between $A$ and $\\cma^{**}$.    The rest\nof the first `iff'  follows from Theorem \\ref{awep}.   The rest\nfollows from Kirchberg's matching result for $C^*$-algebras\n\\cite{Kir1} and the proof of Proposition \\ref{dirLL}.\n\\end{proof}\n\n{\\bf Remark.}  We do not know if either direction of the last\nassertion of the proposition is true for general operator algebras.\nIt is easy however easy to see that like the AWEP, the condition\ninvolving $C^*(F)$ holds if $\\cma$ has the WEP.  See also\nProposition \\ref{var} for another result concerning this property.\n\n\\section{Some connections with exactness}\n\nWe recall that a $C^*$-algebra is  nuclear iff it is both exact and\nhas the WEP.   The reader familiar with Kirchberg's work on\nexactness of $C^*$-algebras (see e.g.\\ \\cite{Kir1, Kir2}),  will\nexpect that we need to consider the following notion of exactness\nfor nonselfadjoint operator algebras. Fortunately, this coincides\nwith the usual operator space variant of exactness studied\n by Pisier \\cite{Pis}, as we shall soon see.\n\n\\begin{definition} We say that an operator\nalgebra $D$ is {\\em OA-exact} if for every extension  \\[ 0 \\longrightarrow A\n\\longrightarrow B \\longrightarrow C \\longrightarrow 0 \\] in the sense of \\cite{BR}, the induced\nsequence\n\\[ 0 \\longrightarrow A \\minten D \\longrightarrow B \\minten D \\longrightarrow C \\minten D \\longrightarrow 0 \\] is\nan extension.\n\\end{definition}\n\n\n\\begin{proposition} \\label{next} For a subalgebra $D \\subset B(H)$.\nThe following are equivalent:\n\n\\begin{enumerate} \\item[(i)] $D$ is OA-exact.\n\n\\item[(ii)] The induced sequence \\[ 0 \\longrightarrow \\Kdb \\minten D \\longrightarrow\n\\Bdb \\minten D \\longrightarrow (\\Bdb\/\\Kdb) \\minten D \\longrightarrow 0 \\] is an extension\nin the sense of \\cite{BR}.\n\n\\item[(iii)] $D$ is exact as an operator space.\n\n\\item[(iv)] If $u: D \\to B(H)$ is the inclusion map, then\nfor every approximately unital operator algebra $A$ the map $I_A\n\\otimes u$ extends to a contraction from $A \\minten D$ to $A \\maxten\nB(H)$.\n\\end{enumerate} \\end{proposition}\n\n\\begin{proof} By \\cite[Theorem 14.4.1]{Pisbook} any OA-exact\noperator algebra is exact as an operator space.  Conversely, if $D$\nis exact as an operator space, and if \\[ 0 \\longrightarrow A \\longrightarrow B \\longrightarrow C \\longrightarrow\n0\n\\] is an extension in the sense of \\cite{BR}, then we have the\n induced sequence \\[ 0 \\longrightarrow \\cma\n\\longrightarrow \\cm{B} \\longrightarrow \\cm{C} \\longrightarrow 0 \\] is an extension of $C^*$-algebras\nby \\cite[Lemma 2.7]{BR}.\nBy \\cite[Theorem 14.4.1(iv)]{ER} we have the extension\n\\[ 0 \\longrightarrow \\cma \\minten D \\longrightarrow \\cm{B} \\minten D \\longrightarrow \\cm{C} \\minten D\n\\longrightarrow 0 \\] and we can apply \\cite[Proposition 3.6]{BR} and the idea\nin the proof of Lemma \\ref{maxq} to see that $D$ is OA-exact.\n\nThat ${\\rm (ii)}$ and ${\\rm (iii)}$ are equivalent follows from\ne.g.\\ \\cite[Theorem 14.4.2]{ER}.  Further \\cite[Theorem\n17.1]{Pisbook} gives ${\\rm (iv)} \\Rightarrow {\\rm (iii)}$. Finally,\nassuming  (iii), recall from Lemma \\ref{28} that $ A \\maxten B(H)\n\\subset \\cma \\maxten B(H)$ completely isometrically.  Then applying\n\\cite[Theorem 17.1]{Pisbook}, the map $I_{\\cma} \\otimes u$ extends\nto a contraction\n\\[ \\cma \\minten D \\to \\cma \\maxten B(H). \\] Restricting this map to $A\n\\minten D$ we obtain ${\\rm (iv)}$.\n\\end{proof}\n\nAny exact operator space is a subspace of an exact unital operator\nalgebra.  To see this let $E$ be an exact operator space and let\n$\\mathcal{U}(E)$ be the universal algebra for $E$ as in\n2.2.10-2.2.11 of \\cite{BLM}. Using the well known characterization\nof exactness in terms of subspaces of $M_n$, we see that\n$\\mathcal{U}(E)$ is exact if $E$ is exact as an operator space by\nusing the following variant of \\cite[Proposition 2.2.11]{BLM}. In\nour case we apply the next result to both $T$ and $T^{-1}$ to see\nthat any finite dimensional subspace of $\\mathcal{U}(E)$ can be\nembedded as a subspace of $M_{2n}$.\n\n\\begin{proposition} \\label{p} If $T: E \\to F$ is a linear map between\noperator spaces with $ \\Vert T \\Vert_{\\rm cb} \\geq 1$, then the\ninduced unital map $ \\theta_T: \\mathcal{U}(E) \\to \\mathcal{U}(F)$\nsatisfies $ \\Vert \\Theta_T \\Vert_{\\rm cb} = \\Vert T \\Vert_{\\rm cb}$.\n\\end{proposition}\n\n\\begin{proof} Let $M = \\Vert T \\Vert_{\\rm cb}$.  Then $u =\nT\/M$ is completely contractive, so that the map $ \\theta_u$ in\n\\cite[Proposition 2.2.11]{BLM} is completely contractive.  But\n$\\theta_T = A \\theta_u A^{-1}$ where $A$ is the diagonal scalar\nmatrix with entries $M$ and $1$.  From this it is clear that $ \\Vert\n\\theta_T \\Vert_{\\rm cb} \\leq M$. \\end{proof}\n\nWe will say more about $\\mathcal{U}(E)$ in the final subsection of\nour paper.\n\n\n\nWe now consider a stronger property than exactness.  We say that an\noperator algebra $A$ is {\\em subexact} if it is a subalgebra of an\nexact $C^*$-algebra.   We show in Section \\ref{uofx} that an exact\noperator algebra need not be subexact.   The following is obvious:\n\n\\begin{proposition} \\label{p2}  $A$ is subexact\n if and only if $C^*_{\\rm e}(A)$ is exact.\n\\end{proposition}\n\nThe  next two results\nsuggest that $C^*$-nuclearity\nis not as strong a condition as might at first appear if one views\nit from a `commutant lifting theorem' perspective, see\n\\cite[Proposition 2.5]{PaP}.\n\n\\begin{theorem} \\label{ewepcnuc} A unital operator algebra $A$ is\n$C^*$-nuclear iff $A$ is exact and has the AWEP.\n\n Also,  $A$ is\nboth subexact and has the AWEP, iff  both $A$ is $C^*$-nuclear and\n$C^*_{\\rm e}(A)$ is nuclear.\n\\end{theorem}\n\n\\begin{proof} We know from Lemma \\ref{free} that $C^*$-nuclearity implies\nexactness, and from Proposition \\ref{cnimawep} we know that\n$C^*$-nuclearity implies the AWEP.  Conversely, suppose $A$ is exact\nand has the AWEP.  For any $C^*$-algebra $D$ we have by\n\\cite[(6.3)]{BLM} that $A \\maxten D \\subset \\cma \\maxten D$. Indeed\nby an argument similar to that of \\cite[(6.3)]{BLM},  using the\nuniversal property of $\\maxten$ and \\cite[Corollary 2.5.6]{BLM}, we\nhave \\[ A \\maxten D \\subset \\cma \\maxten D \\subset \\cma^{**} \\maxten\nD \\] completely isometrically. On the other hand, the composition of\nthe maps in the last string agrees with the composition of the\nfollowing canonical maps (induced by the maps in Theorem \\ref{awep}\n(iv) with $B = \\cma$):\n\\[ A \\maxten D \\to B(H) \\maxten D \\to \\cma^{**} \\maxten D. \\] This\nforces $A \\maxten D \\subset B(H) \\maxten D$ completely\nisometrically.  By part ${\\rm (iv)}$ of Proposition \\ref{next}, the\nexactness of $A$ gives $A \\minten D \\subset B(H) \\maxten D$.  It is\nnow easy to see that $A \\maxten D = A \\minten D$.  That is, $A$ is\n$C^*$-nuclear.\n\n\n\nFor the second equivalence notice that if $A$ is a subalgebra of a\nnuclear $C^*$-algebra $N$, then the $C^*$-algebra generated by $A$\nin $N$ is exact and hence so is its quotient $C^*_{\\rm e}(A)$.  If\nin addition $A$ has the AWEP, then by Theorem \\ref{awep} we have\nthat $C^*_{\\rm e}(A)$ has the WEP.  Hence $C^*_{\\rm e}(A)$ is\nnuclear by Exercise 17.1 of \\cite{Pisbook}. Since $C^*_{\\rm e}(A)$\nis nuclear we know that $ C^*_{\\rm e}(A) \\minten D = C^*_{\\rm e}(A)\n\\maxten D$ for all $C^*$-algebras $D$.  Hence $ A \\minten D \\subset\nC^*_{\\rm e}(A) \\maxten D$ completely isometrically for all\n$C^*$-algebras $D$. This, by part ${\\rm (iv)}$ of Proposition\n\\ref{next}, forces $A$ to be exact. Thus by the first chain of\nequivalences we have that $A$ is $C^*$-nuclear. Finally, if\n$C^*_{\\rm e}(A)$ is nuclear, then $A$ is clearly subexact;\nand\nif $A$ is $C^*$-nuclear then $A$ has the AWEP.\n\\end{proof}\n\n{\\bf Remark.}  By \\cite[Theorem 12.6]{Pisbook}, $A$ is $C^*$-nuclear\nif and only if there is a net of finite rank contractions $v_t: A\n\\to M_{n_t}$ and maps $ w_t: M_{n_t} \\to \\cma$ with $\\Vert w_t\n\\Vert_{{\\rm dec}} \\leq 1$ for all $t$, such that $w_tv_t$ converges\npointwise to the natural inclusion map of $A$ into $\\cma$. This is\nbecause  $A$ is $C^*$-nuclear if and only if the canonical map $A\n\\otimes D \\to \\cma \\maxten D$ is a complete isometry with respect to\n$\\minten$ for every $C^*$-algebra $D$.\n\n\\medskip\n\nThe following result is a variant of the last theorem.\n\n \\begin{proposition} \\label{var}  If $A$ is an\nexact approximately unital operator algebra then $A$ is\n$C^*$-nuclear if and only if $C^*(F) \\minten A = C^*(F) \\maxten A$\ncompletely isometrically for every discrete free group $F$.\n\\end{proposition}\n\n\\begin{proof}If $A$ is exact then the fact that any $C^*$-algebra\n$B$ is a quotient of $C^*(F)$ for some $F$, forces exactness of the\nsequence\n\\[ 0 \\longrightarrow A \\minten J \\longrightarrow A \\minten C^*(F) \\longrightarrow A \\minten B \\longrightarrow\n0.\\] Applying Lemma \\ref{maxq}, we have the exact sequence\n\\[ 0 \\longrightarrow A \\maxten J \\longrightarrow A \\maxten C^*(F) \\longrightarrow A \\maxten B \\longrightarrow\n0.\\]  If $C^*(F) \\minten A = C^*(F) \\maxten A$ it follows that $ A\n\\minten B = A \\maxten B$.\n\\end{proof}\n\n\\begin{corollary}\\label{to}\n  If $A$ is $C^*$-nuclear\nand approximately unital, and if either $A$ is subexact or\n$A$ is generated by unitaries, then $C^*_{\\rm e}(A)$ is nuclear.\n \\end{corollary}\n\n\\begin{proof} If $A$ is $C^*$-nuclear then so is $A^1$ by Corollary\n\\ref{w16}. Similarly if $A$ is subexact then so is $A^1$, since the\nunitization of an exact $C^*$-algebra is exact. By Theorem\n\\ref{ewepcnuc} it follows that $C^*_{\\rm e}(A^1) = C^*_{\\rm e}(A)^1$\nis nuclear and hence so is $C^*_{\\rm e}(A)$.  On the other hand if\n$A$ is generated by unitaries, then so is $C^*_{\\rm e}(A)$, and so\n$C^*_{\\rm e}(A)$ is nuclear by \\cite[Theorem 13.4]{Pisbook}.\n\\end{proof}\n\n\n\\section{Examples}\n\nThis main purpose of this section is to illuminate connections (or lack\nthereof) of the properties studied above, in the case of some\nextremely commonly encountered examples, to the matching\n$C^*$-algebra properties for their $C^*$-covers.\n\n\n\\subsection{The disk algebra} \\label{disk}\n It is well known (see e.g.\\ \\cite[6.2.5]{BLM}) that $A(\\Ddb)$ is\n$C^*$-nuclear.  Hence it has the AWEP and the HLLP, etc. We shall\nshow that $\\cm{A(\\Ddb)}$ has the LLP but is not nuclear (nor exact).\nThis shows amongst other things that the converse of the first\nassertion in Proposition \\ref{LLPd} fails.\n\n To see that $\\cm{A(\\Ddb)}$ is not exact, we will use the fact that\n $\\cm{A(\\Ddb)}$ is the\nuniversal $C^*$-algebra generated by a contraction.  Let $B$ be any\nseparable $C^*$-algebra which is not exact. Since $\\Kdb \\minten B$\ncontains a complemented copy of $B$ and exactness, viewed as an\noperator space property, would pass to this copy, it follows that\n$\\Kdb \\minten B$ is not exact. By \\cite{OlsZ}, $\\Kdb \\minten B$ is\nsingly generated as a $C^*$-algebra by a contractive element, call\nit $x$. Since $\\cm{A(\\Ddb)}$ is the universal $C^*$-algebra\ngenerated by a contraction, there exists a $*$-representation $\\pi:\n\\cm{A(\\Ddb)} \\to \\Kdb \\minten B$ which is onto.  Since the exactness\nis preserved by $C^*$-quotients it must be the case that\n$\\cm{A(\\Ddb)}$ is not exact, and hence is not nuclear.\n\nOne may identify $\\cm{A(\\Ddb)}$ with $C^*_u<\\Cdb>$, the universal\n$C^*$-algebra for the operator space $\\Cdb$, by comparing their\nuniversal properties (see \\cite{Oz1}).  With this identification we\ncan use \\cite[Theorem 16.5]{Pisbook} to see that $\\cm{A(\\Ddb)}$ does\nin fact have the LLP.\n\nIn \\cite[Proposition 16.13]{Pisbook} one finds the remarkable fact\nthat $\\cm{A(\\Ddb)}$ having the WEP, is equivalent to Kirchberg's\nimportant conjecture from \\cite{Kir1} that WEP implies LLP.    It is\neasy to argue that this is also equivalent to whether $\\cm{A(\\Ddb)}\n\\minten \\cm{A(\\Ddb)} = \\cm{A(\\Ddb)} \\maxten \\cm{A(\\Ddb)}$.   Indeed\nthis follows from Kirchberg's remarkable result from \\cite{Kir1}\nthat a $C^*$-algebra $B$ has WEP iff $B \\minten B^{\\rm op} = B\n\\maxten B^{\\rm op}$; together with the fact that $\\cm{A(\\Ddb)} =\n\\cm{A(\\Ddb)}^{\\rm op}$.  The latter is a special case of the more\ngeneral fact that for an operator algebra $A$, $\\cm{A^{\\rm op}} =\n\\cma^{\\rm op}$; whose proof is left as a simple exercise.\n\nWe remark that it is easy to see that  $$\\cm{A(\\Ddb) \\maxten\nA(\\Ddb)} \\neq \\cm{A(\\Ddb)} \\maxten \\cm{A(\\Ddb)} .$$   Indeed by\n(6.9) in \\cite{BLM}, we have $A(\\Ddb) \\maxten A(\\Ddb) = A(\\Ddb)\n\\minten A(\\Ddb)$ is the bidisk algebra (see \\ref{bidi}). A pair of\ncommuting contractive representations of $A(\\Ddb)$ gives rise to a\nrepresentation of $A(\\Ddb) \\maxten A(\\Ddb)$, and hence to a\n$*$-representation $\\pi$ of $\\cm{A(\\Ddb) \\maxten A(\\Ddb)}$. It is\neasy to construct an example of such representations (even two\ndimensional, taking $z$ to $E_{21}$, where $z$ is the usual\ngenerator of $A(\\Ddb)$) such that $\\pi(z \\otimes 1)$ does not\ncommute with $\\pi(1 \\otimes \\bar{z})$. On the other hand, these\nwould have to commute for any representation $\\pi$ of $\\cm{A(\\Ddb)}\n\\maxten \\cm{A(\\Ddb)}$, by  \\cite[Corollary 6.1.7]{BLM}.\n\n\\subsection{The bidisk algebra} \\label{bidi}\n Using a result of Parrott \\cite{Par} one may see\nthat the bidisk algebra $A(\\Ddb^2)$ is $C^*$-nuclear (see e.g.\\ p.\\\n266 in \\cite{BLM}).   Thus it has the AWEP and the HLLP, etc.  On\nthe other hand, it is easy to argue from the universal property of\n$C^*_{\\rm max}$ applied to obvious maps between $A(\\Ddb)$ and\n$A(\\Ddb^2)$, that $\\cm{A(\\Ddb)} \\subset \\cm{A(\\Ddb^2)}$.  Hence\n$\\cm{A(\\Ddb^2)}$ is not exact.  We do not know if $\\cm{A(\\Ddb^2)}$\nhas the LLP, or if $\\cm{A(\\Ddb)} \\maxten \\cm{A(\\Ddb)}$ has the LLP.\n\n\n\\subsection{Triangular\nmatrices}   \\label{tri} Let $T_n$ denote the $n \\times n$ upper\ntriangular matrices, which are known to be $C^*$-nuclear \\cite{PaP},\nand hence it also has AWEP and the HLLP, etc. We will show that\n$\\cm{T_n}$ is not exact if $n \\geq 3$, but it does have the LLP.\nNote that  $\\cm{T_2}$ is nuclear, since it can  be identified with\nthe subalgebra of $C([0,1],M_2)$ consisting of matrices which are\ndiagonal matrices at $t=0$  (see\n \\cite[2.4.5]{BLM}).\n\nFor $n \\geq 3$ we will first  show that $T_n$ is essentially a free\nproduct of copies of $T_2$. Define for $1 \\leq i \\leq n-1$ the\nalgebra\n\\[ A_i := \\underbrace{\\mathbb{C} \\oplus \\cdots \\oplus\n\\mathbb{C}}_{i-1 \\mbox{ copies}} \\oplus T_2 \\oplus\n\\underbrace{\\mathbb{C}  \\oplus \\cdots \\oplus \\mathbb{C}}_{n-i-1\n\\mbox{ copies}} \\subseteq M_n.\\] We will denote by $\\iota_i$ the\ninclusion map of $A_i$ into $M_n$ and we will let $D$ be the\nsubalgebra of $A_i$ given by the diagonal matrices.\n\n\\begin{lemma} The algebra $T_n$ is completely isometrically\nisomorphic to $*_D A_i$. \\end{lemma}\n\n\\begin{proof}\nFor each $i$ we have $A_i \\subseteq T_n$ completely isometrically\n isomorphically.  It follows that there is a completely contractive\nrepresentation $*\\iota_i$ of $*_DA_i$ into $T_n$. The range of\n$*\\iota_i$ contains a generating set for $T_n$ and hence the\nrepresentation $*\\iota_i$ maps onto $T_n$. Next let $\\pi: T_n \\to\n*_DA_i$ be given by letting\n\\[ \\pi(E_{i,i}) \\mapsto \\underbrace{ 0 \\oplus \\cdots\n\\oplus 0}_{i-1 \\mbox{ copies}} \\oplus 1 \\oplus \\underbrace{0 \\oplus\n\\cdots \\oplus 0}_{n-i \\mbox{ copies}} \\in D\n\\] and \\[ \\pi(E_{i,i+1}) \\mapsto \\underbrace{ 0 \\oplus \\cdots \\oplus\n0}_{i-1 \\mbox{ copies}} \\oplus E_{1,2} \\oplus \\underbrace{ 0 \\oplus\n\\cdots \\oplus 0}_{n-i-1 \\mbox{ copies}} \\in A_i,\\] where $E_{i,j}$\nis the usual elementary matrix. It is easy to see that $ \\pi$ is\nwell defined. Now extending using linearity and algebra operations\nwe have a representation. Notice that $ \\pi \\circ *\\iota_i $ is\ntrivial on generators, as is $\n*\\iota_i  \\circ \\pi$, and hence these two maps are inverses.\nThe result will follow if we can show that $\\pi$ is completely\ncontractive.\n  This follows easily from the now standard\nresult from \\cite{RONC} stating that it suffices to show that $\\pi$\nis contractive on matrix units.  But by construction,\n$\\pi(E_{i,i+k}) = \\pi(E_{i,i+1})\\pi(E_{i+1,i+2}) \\cdots\n\\pi(E_{i+{k-1},i+k})$ which is a product of contractions and hence\nis a contraction.  This is true for  $ 1 \\leq i \\leq n$ and $\n1 \\leq k \\leq n-i$, and we are done.  \\end{proof}\n\nWe now combine the last result with the fact that free products\n`commute' with $C^*_{\\rm max}$ (see \\cite[Proposition 2.2]{BMax}),\nto obtain\n\\[ \\cm{T_n} = \\underset{\\scriptscriptstyle i}{\\displaystyle{\\ast}}\\,\n\\cm{A_i} = \\underset{\\scriptscriptstyle i}{\\displaystyle{\\ast}}\\,\n\\underbrace{\\mathbb{C} \\oplus \\cdots \\oplus \\mathbb{C}}_{i-1 \\mbox{\ncopies}} \\oplus \\cm{T_2} \\oplus \\underbrace{\\mathbb{C} \\oplus \\cdots\n\\oplus \\mathbb{C}}_{n-i-1 \\mbox{ copies}} .\\]\n\nSince $\\cm{T_n}$ is the $n$-fold free product of nuclear\n$C^*$-algebras, by \\cite[Theorem 13.2]{Pisbook} we have that\n$\\cm{T_n}$ has the LLP for all $n$.  The fact that $\\cm{T_n}$ is not\nexact for $n \\geq 3$ follows from the next lemma, and the fact that\n $C([0,1])\n*C([0,1])$ is not exact.  The latter is probably well known, but for the\nreaders convenience we give a proof, using facts in \\cite[Section\n5]{KiW} about the universal $C^*$-algebra $C^*_u(X)$ of an operator\nsystem $X$.  Namely, $C^*_u(\\ell^\\infty_2) = C([0,1])$ and\n$C^*_u(\\ell^\\infty_3)$ is not exact.   Define $\\pi_k: \\ell^\\infty_2\n\\rightarrow \\ell^\\infty_3$ by $\\pi_1((\\lambda, \\mu)) = ( \\lambda,\n\\lambda, \\mu )$ and  $\\pi_2((\\lambda, \\mu)) = ( \\lambda, \\mu, \\mu)$.\nClearly $\\pi_k$ is a unital complete isometry for $k = 1, 2$. Notice\nalso that the ranges of the $\\pi_k$ are jointly spanning.  By the\nuniversal property for $C^*_u(\\ell^\\infty_2)$, there are unital\n$*$-homomorphisms $\\widetilde{\\pi_k}: C([0,1]) \\rightarrow\nC^*_u(\\ell^\\infty_3)$ whose ranges together generate\n$C^*_u(\\ell^\\infty_3)$.  By the universal property for free\nproducts, there is a $*$-representation of $C([0,1])*C([0,1])$ onto\n$C^*_u(\\ell^\\infty_3)$.  Since exactness passes to $C^*$-quotients,\nit follows that $C([0,1]) *C([0,1])$ is not exact.\n\n\\begin{lemma} There is a completely isometric embedding of\nthe amalgamated free product $C([0,1])*C([0,1])$ into $\\cm{T_n}$ for\n$n \\geq 3$.\\end{lemma}\n\n\\begin{proof} Without loss of generality we will stick to the case\nof $n=3$.  The proof $n > 3$ will follow in the same manner, or by\nnoting $\\cm{T_3} \\subset \\cm{T_n}$.\n\nWe have $\\cm{T_3} = (\\cm{T_2} \\oplus \\Cdb) *_D \\, (\\Cdb \\oplus\n\\cm{T_2})$.\nNotice that the map $E_1:   \\begin{bmatrix} f_{1,1} & f_{1,2} \\\\\nf_{2,1} & f_{2,2} \\end{bmatrix} \\oplus \\lambda \\mapsto f_{2,2}$ is a\nconditional expectation of $\\cm{T_2} \\oplus \\Cdb$ onto a copy of\n$C([0,1])$.  Similarly $E_2: \\lambda \\oplus \\begin{bmatrix} g_{1,1}\n& g_{1,2} \\\\ g_{2,1} & g_{2,2} \\end{bmatrix}  \\mapsto g_{2,2}$\ndefines a conditional expectation onto a copy of $C([0,1])$. Lastly\nthe map $d: D \\to \\mathbb{C}$ taking a $3 \\times 3$ diagonal matrix to its\n$2$-$2$ entry,\n is a conditional expectation\nof $D$ onto a copy of $\\Cdb$.   The result now follows from\n\\cite[Proposition 2.4]{ADEL}. \\end{proof}\n\n\n\n\n\n\\subsection{The algebra $\\mathcal{U}(X)$ for an operator space $X$} \\label{uofx}\n\nThe operator algebra $\\mathcal{U}(X)$ consists of $2 \\times 2$ upper\ntriangular matrices with elements from $X$ in the $1$-$2$ corner and\nscalars in the two diagonal entries; see 2.2.10-2.2.11 of\n\\cite{BLM}. We refer the reader to \\cite[Section 6.4]{BLM} for a\ndiscussion of the $\\delta$ norm on tensor products.\n\n\\begin{lemma}\\label{ux} If $X$ is an operator space, and if $\\mathcal{U}(X)\n\\minten D = \\mathcal{U}(X) \\maxten D$ isometrically for a unital\n$C^*$-algebra $D$, then $ \\delta = {\\rm min}$ on $X \\otimes D$.  The\nconverse of this is true if $D \\cong M_n(D)$ for all $n \\in \\Ndb$.\nIn particular, $X$ has the $1$-OLLP of Ozawa {\\rm\n\\cite{Oz1,Pisbook}} if and only if $\\mathcal{U}(X)$ is\n$\\Bdb$-nuclear.\n\\end{lemma}\n\n\\begin{proof}If $\\Phi : X \\to B(H)$ and $\\pi : D \\to B(H)$ are\ncommuting complete contractions, with $\\pi$ a representation, then\nby \\cite[Proposition 2.2.11]{BLM}  we obtain a representation\n$\\theta_\\Phi : {\\mathcal U}(X) \\to B(H^{(2)})$ which commutes with\n$\\pi^{(2)}: D \\to B(H^{(2)})$. Thus if ${\\mathcal U}(X) \\minten D =\n{\\mathcal U}(X) \\maxten D$ then for $x_k \\in X, d_k \\in D$, and with\n$a_k$ the matrix with $x_k$ in its $1$-$2$ corner and $0$ elsewhere,\nwe have \\[ \\Vert \\sum_k \\; \\theta_\\Phi(a_k) \\pi^{(2)}(d_k) \\Vert =\n\\Vert \\Phi(x_k) \\pi(d_k) \\Vert \\leq \\minnorm{ \\sum_k \\; a_k \\otimes\nd_k } = \\minnorm{ \\sum_k \\; x_k \\otimes d_k } .\n\\] Hence $\\delta = {\\rm min}$ on $X \\otimes D$.\n\nConversely, suppose $\\delta = {\\rm min}$ on $X \\otimes D$, and let\n$\\theta$ and $\\rho$ be two commuting completely contractive\nrepresentations of ${\\mathcal U}(X)$ and $D$ respectively. The\ndiagonal projections in ${\\mathcal U}(X)$ induce a decomposition of\nthe Hilbert space as a sum $H \\oplus K$ so that $\\theta =\n\\theta_\\Phi$ for a complete contraction  $\\Phi : X \\to B(K,H)$, and\n$\\rho(d) = \\pi_1(d) \\oplus \\pi_2(d)$, for $d \\in D$ and two\n$*$-representations $\\pi_1, \\pi_2$ of $D$ on $H$ and $K$\nrespectively, such that $\\Phi(x) \\pi_2(d) = \\pi_1(d) \\Phi(x)$ for $x\n\\in X, d \\in D$.  Note that $\\theta_\\Phi \\circ c$ commutes with\n$\\rho$, where $c : X \\to {\\mathcal U}(X)$ is the canonical\nembedding. It follows that with notation as above, for $x_k \\in X,\nd_k \\in D$, \\[ \\Vert  \\sum_k \\; \\theta_\\Phi(c(x_k)) \\rho(d_k) \\Vert\n= \\Vert \\sum_k \\; \\Phi(x_k) \\pi_2(d_k) \\Vert \\leq \\Vert \\sum_k \\;\nx_k \\otimes d_k \\Vert_\\delta = \\minnorm{ \\sum_k \\; x_k \\otimes d_k }\n. \\] Notice that $W = \\Phi(X) \\pi_2(D)$ is an operator $D$-bimodule,\na $D$-subbimodule of $B(K,H)$.    Also, $X \\minten D$ is an operator\n$D$-bimodule with the canonical actions. The computation above shows\nthat the map $u : x \\otimes d \\mapsto \\Phi(x) \\pi_2(d)$ is a\ncontractive $D$-bimodule map from $X \\minten D$ to $W$.  If $D \\cong\nM_n(D)$, then we also have $\\delta = {\\rm min}$ on $X \\otimes\nM_n(D)$, and it is easy to see from this that $u$ is completely\ncontractive.  The map induced from the bimodule map $u$ by (3.12) of\n\\cite{BLM} is also completely contractive. One may argue from this\nthat for $b_1 , \\cdots , b_n \\in {\\mathcal U}(X), d_1 , \\cdots , d_n\n\\in D$:\n\\[ \\Vert  \\sum_k \\; \\theta(b_k) \\rho(d_k) \\Vert \\leq \\left| \\left|\n\\left[ \\begin{array}{ccl} \\sum_k \\; \\lambda_k 1 \\otimes d_k \\; &\n\\sum_k \\; x_k  \\otimes d_k  \\\\ 0 \\; & \\sum_k \\; \\mu_k 1 \\otimes d_k\n\\end{array} \\right]\n \\right| \\right|\n= \\minnorm{ \\sum_k \\; b_k  \\otimes d_k} .\\] Here $\\lambda_k , x_k ,\n\\mu_k$ are the three nonzero `corners' of $b_k$, and the norm of the\nmiddle matrix is taken in $M_2(B(H \\otimes K)$, where $X \\subset\nB(H), D \\subset B(K)$. That is, ${\\mathcal U}(X) \\minten D =\n{\\mathcal U}(X) \\maxten D$ isometrically.\n\nWe leave the remaining assertion to the reader. \\end{proof}\n\n\\begin{corollary} \\label{corux}  There exist unital operator algebras\n$A$ with $C^*_{\\rm e}(A)$ nuclear (hence having the LLP), but\n$A$ is neither $\\Bdb$-nuclear nor $C^*$-nuclear nor has the AWEP.\n\\end{corollary}\n\n\\begin{proof} If $X$ is a minimal operator space\nwithout the $1$-OLLP, and if $A = \\mathcal{U}(X)$, then  $A$ is not\n$\\Bdb$-nuclear by Lemma \\ref{ux},  and hence not $C^*$-nuclear. But\n$C^*_{\\rm e}(A)$ is nuclear: it is a subalgebra of $M_2(B)$ for a\ncommutative $C^*$-algebra $B$ by \\cite[Theorem 4.21]{BShi}, hence\nType I, and so nuclear. Since $A$ is exact but not $C^*$-nuclear it\ncannot have AWEP by Theorem \\ref{ewepcnuc}.\n\\end{proof}\n\nFrom Lemma \\ref{ux} and facts in \\cite{Oz1}, it is easy to build\n $\\Bdb$-nuclear operator algebras with bad properties.\nIndeed for any finite dimensional operator space $X$ with $X^*$\n$1$-exact, we have that $X$ has the $1$-OLLP, so that\n$\\mathcal{U}(X)$ is $\\Bdb$-nuclear.\n\n\\medskip\n\n{\\bf Remarks.}  1)   \\ We do not have an example of a $C^*$-nuclear\nalgebra $A$ with $C^*_{\\rm e}(A)$ not nuclear, but presumably they\nexist in abundance.  Equivalently, we do not know if\n$C^*$-nuclearity implies subexactness.\n\n2) \\ For a given operator space it seems rather restrictive, and\ntherefore probably uninteresting, for $\\mathcal{U}(X)$ to be\n$C^*$-nuclear. Indeed, this is equivalent to saying that the\n$\\delta$ tensor norm agrees with the spatial (minimal) norm on $X\n\\otimes D$, for all $C^*$-algebras $D$.  Of course this occurs if $X\n= \\Cdb$, and if $X$ is a Hilbert row or column space (since $C_n\n\\otimes_{\\rm h} D = C_n \\minten D$ for any $C^*$-algebra $D$, and so\nthese also agree with $C_n \\otimes_\\delta D$, since $\\delta \\leq\n{\\rm h}$), but probably in few other cases.\nWe are indebted to Gilles Pisier and N. Ozawa for conversations on\nthis matter, which is related to the discussion on p.\\ 341 of\n\\cite{Pisbook} of exact spaces whose dual is exact too.\nIf $X$ is finite dimensional with ${\\mathcal U}(X)$ $C^*$-nuclear,\nthen as we saw in Lemma \\ref{free}, ${\\mathcal U}(X)$ is $1$-exact\nand hence so is $X$. On the other hand, by \\cite{Oz2}, since $X$ has\n$1$-OLLP, $X^*$ is $1$-exact. There is only a small list of\n$1$-exact finite dimensional spaces whose dual is known to be\n$1$-exact too (see  p.\\ 341 of \\cite{Pisbook}).\n\n\\medskip\n\n We do not have an operator algebra version of Kirchberg's\nprofound characterization of separable exact $C^*$-algebras as\nsubalgebras of a fixed `universal' separable exact $C^*$-algebra\n(see e.g.\\ \\cite{Kir1, Kir2, KPh}).  It seems feasible that there\ndoes exist some such result, although the following rules out one\napproach:\n\n\\begin{proposition} \\label{nog} There exists a separable exact operator\nspace that is not linearly  completely isometric to a subspace of an\nexact $C^*$-algebra.  There exists a separable unital exact operator\nalgebra which is not subexact.\n\\end{proposition}\n\n\\begin{proof} It is shown in \\cite[Theorem 18]{KiW}\nthat there exists a separable exact operator system ${\\mathcal S}$\nthat is not a unital-subsystem of any unital separable exact\n$C^*$-algebra. Suppose that ${\\mathcal S}$ was a subspace of an\nexact $C^*$-algebra $A$. We will use notation from \\cite{Ham} (see\nalso p.\\ 285-286 of \\cite{BShi}). Clearly $M_2(A)$ is exact, and\nhence so too is its $C^*$-subalgebra generated by the copy of the `Paulsen\nsystem' of ${\\mathcal S}$. Since exactness also passes to\n$C^*$-quotients, the $C^*$-envelope of the latter system is exact,\nand hence so too is its upper right corner, the `ternary envelope'\nof ${\\mathcal S}$ (see \\cite{Ham} or p.\\ 286 of \\cite{BShi}). This\nenvelope is completely isometric to $C^*_{\\rm e}({\\mathcal S})$, by\nthe uniqueness of the ternary envelope.  Thus $C^*_{\\rm e}({\\mathcal\nS})$ is exact, with ${\\mathcal S}$ as a unital subsystem,\ncontradicting the result cited from \\cite{KiW} above.\n\nFor the last part, consider ${\\mathcal U}({\\mathcal S})$, an exact\nseparable operator algebra by the remarks above Proposition \\ref{p},\nwhich is not a subspace of an exact $C^*$-algebra. \\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Summary}\n\nThe dynamics of a classical charged point--particle interacting with\nan electromagnetic field is described by the Maxwell and Lorentz\nequations,\n \\begin{eqnarray}\n \\partial_\\mu F^{\\mu\\nu}&=&j^\\nu, \\quad  F^{\\mu\\nu}=\\partial^\\mu A^\\nu-\\partial^\\nu A^\\mu, \\label{maxwell}\\\\\n {dp^\\mu\\over ds}&=& e\\,F^{\\mu\\nu}(y(s))\\,u_\\nu(s).\\label{lorentz}\n \\end{eqnarray}\nWe parametrize the worldline of the particle $y^\\mu(s)$ by the proper\ntime $s$, and define four--velocity and four--acceleration\nrespectively by,\n$$\nu^\\mu={dy^\\mu\\over ds},\\quad w^\\mu={du^\\mu\\over ds}.\n$$\nThe current is,\n \\begin{equation}\\label{2.2}\n j^\\mu (x) = e \\int u^\\mu(s)\\,\n\\delta^4 (x - y (s))\\,ds.\n \\end{equation}\nMaxwell's equations admit the general solution,\n$$\nF^{\\mu\\nu}=F^{\\mu\\nu}_{LW}+ F^{\\mu\\nu}_{in},\n$$\nwhere $F^{\\mu\\nu}_{in}$ is a free external field, and $F^{\\mu\\nu}_{LW}$\namounts to the retarded Lienard--Wiechert (LW) field.\n\nWhen substituting the total field strength into the Lorentz equation\none faces the problem that the LW--field is infinite on the\nparticle's trajectory, and the quantity,\n \\begin{equation}\\label{LWself}\nF^{\\mu\\nu}_{LW}(y(s)),\n \\end{equation}\ndiverges. This infinite self--interaction is, of course, a\nconsequence of the point--like nature of the charged particle.\n\nThere are several methods of dealing with this divergence, all of\nthem leading to the same conclusions: the quantity \\eref{LWself} has\na divergent part, which leads to an infinite classical\nrenormalization of the particle's mass, and a finite part that\namounts to,\n \\begin{equation}\n F^{\\mu\\nu}_{LW}(y(s))\\Big|_{finite}=-{e\\over 6\\pi}\\left(u^\\mu\n{dw^\\nu\\over ds}- u^\\nu {dw^\\mu\\over ds}\\right).\n \\end{equation}\nSubstituting this into the Lorentz--equation one obtains the\nrelativistic Lorentz--Dirac equation,\n \\begin{equation} \\label{3.1}\n {dp^\\mu\\over ds}= {e^2 \\over 6 \\pi} \\left({dw^\\mu\\over ds}+w^2\\,u^\\mu\\right)\n + e\\, F^{\\mu\\nu}_{in}(y) \\,u_\\nu,\n \\end{equation}\nwhich has to be considered as the equation of motion for the charged\nparticle, replacing the original ill--defined Lorentz--equation\n\\eref{lorentz}. Here, and in the following, we indicate the square\nof a four--vector with $w^2\\equiv w^\\mu w_\\mu$. Strictly speaking this\nequation can not be {\\it derived} from the fundamental equations\n\\eref{maxwell} and \\eref{lorentz} of classical Electrodynamics (ED),\nbut has to be postulated.\n\nIn its relativistic form  equation \\eref{3.1} has been derived first\nby Dirac \\cite{dirac} in 1938, in an attempt to describe the motion\nof a classical relativistic charged point--particle, taking into\naccount its self--interaction. In this seminal paper the equation\nwas derived using in an essential way the principle of\nenergy--momentum conservation. This conservation law represents\nstill the basic motivation for the equation itself.\n\nDespite of the fact that \\eref{3.1} is a well--defined equation and\nensures energy--momentum conservation, it exhibits several\nunsatisfactory features, both physically and mathematically. The key\nphysical drawbacks are that the equation is of the third order in\nthe time--derivative, and that it admits a class of solutions, so\ncalled runaway solutions -- see e.g. \\cite{thirring} -- which lead\nto exponentially growing four--velocities, even in absence of\nexternal forces. When these solutions are eliminated imposing\nsuitable boundary conditions at infinity \\cite{haag,roh}, it turns\nout that the motions {\\it allowed} by these conditions show up an\nacausal behavior, in the sense of a preacceleration: the\nacceleration is felt before the external force begins to act.\nActually, this preacceleration occurs on a time scale of order,\n $$\n \\tau ={e^2\\over 6\\pi m}\\sim 10^{-23} s,\n $$\nwhich is of a factor 1\/137 shorter than the time scale\n$\\tau_q=\\hbar\/m$, where, due to the Heisenberg uncertainty\nprinciple, quantum effects become relevant and the classical theory\nhas to be abandoned. Hence, physically one can interpret this\nacausal behavior as an inconsistency of classical ED of\npoint--particles, which is cured by the quantum formulation of the\ntheory, see e.g. \\cite{moniz} \\footnote{One can, actually, maintain\na second order Lorentz--equation which is free from singularities,\nif one introduces a finite cutoff of physical origin on the charge\ndistribution of the would--be point--particle. This is the\nphilosophy pursued mainly in \\cite{spohn,yaghj} and references\ntherein. The main unsatisfactory aspects of this approach are its\nlack of universality, and the problems related with the\nimplementation of exact energy--momentum conservation. In the\npresent paper we follow Dirac's original proposal and insist on\npoint--like charges, and hence on equation \\eref{3.1}.}.\n\nAccepting \\eref{3.1} as the correct equation of motion for the\nparticle, three questions arise automatically, to which we will give\nnew answers in this paper. The first question regards {\\it local}\nenergy--momentum conservation, in terms of a {\\it well--defined\nconserved energy--momentum tensor}. The second regards the existence\nof an action from which \\eref{3.1} can be derived, and the third\nquestion is if the energy--momentum tensor can be derived from this\naction via Noether's theorem.\n\nRegarding the first question we mention that, although relying on\nconservation criterions, in his paper \\cite{dirac} Dirac does not\nconstruct an energy--momentum tensor, nor does he give a recipe to\ncompute the four--momentum enclosed in a volume $V$ containing the\nparticle. The main obstruction to such a construction arises from\nthe particular form of the naive energy--momentum tensor of the\nelectromagnetic field,\n \\begin{equation}\n \\label{3.2} {\\tau}^{\\mu\\nu}_{em}= F^{\\mu\\rho}_{LW}\\, F _{LW}{}_\\rho{}^\\nu + {1\\over\n 4}\\,\\,\n\\eta^{\\mu\\nu} F^{\\rho\\sigma}_{LW}\\,F_{LW\\rho\\sigma},\n \\end{equation}\nwhere for simplicity we have set the external field to zero. If we\nindicate with $ R$ the distance from the particle at a certain\ninstant, the fields $F^{\\mu\\nu}_{LW}$ exhibit near the particle the\nstandard Coulomb--like {\\it integrable} $1\/R^2$ singularities, and\nhence ${\\tau}^{\\mu\\nu}_{em}$ exhibits the {\\it non--integrable}\n$1\/R^4$ singularities. As a consequence the momentum integrals\n$P_V^\\mu=\\int_V d^3x\\, \\tau ^{0\\mu}_{em}$ diverge. In a mathematical\nlanguage the problem can be stated as follows: while the components\nof $F^{\\mu\\nu}_{LW}$ are tempered distributions, i.e. elements of\n${\\cal S}'\\equiv {\\cal S}'({\\bf R^4})$, the components of\n${\\tau}^{\\mu\\nu}_{em}$, being products of the formers, are not.\n\nTo obtain a well--defined energy--momentum tensor one has to replace\n$\\tau^{\\mu\\nu}_{em}$ with a tensor $T^{\\mu\\nu}_{em}$ whose components\nare elements of ${\\cal S}'$ such that, a) $T^{\\mu\\nu}_{em}$ coincides\nwith $\\tau ^{\\mu\\nu}_{em}$ in the complement of the particle's\nworldline, and b) such that the resulting total energy--momentum\ntensor is conserved, if \\eref{3.1} holds. Such a tensor admits then\nautomatically finite four--momentum integrals. This program has been\naccomplished only rather recently by Rowe in \\cite{rowese}, relying\npartially on \\cite{prerowe}. His energy--momentum tensor is defined,\nrather implicitly, as the distributional derivative of the sum of\nother elements of ${\\cal S}'$, and entails the additional drawback\nthat it is not {\\it manifestly} symmetric and traceless. On the\nother hand it can be shown that the resulting tensor is uniquely\ndetermined by the requirements a) and b).\n\nIn this paper we present an alternative and simple representation of\nthis tensor, relying on a new Lorentz--invariant regularization\nprocedure, followed by a renormalization, which leads to a\nmanifestly symmetric, traceless and invariant tensor. In particular,\nour procedure follows the physical intuition that the renormalized\ntensor should differ from \\eref{3.2} only on the particle's\ntrajectory: correspondingly in our renormalization scheme we\nsubtract only (divergent) terms which are supported on the\nworld--line, i.e. terms that are proportional to delta--functions\nsupported on the trajectory.\n\nAn alternative approach to four--momentum conservation, based on a\ndifferent regularization procedure, has been proposed in\n\\cite{marino}. This approach leads to finite four--momentum\nintegrals, but it does not allow to construct an energy--momentum\ntensor. Moreover, it lacks manifest Lorentz invariance.\n\nThe second question opened by \\eref{3.1} is if this equation can be\nderived from an action principle. It is immediately seen that the\nresulting equation can not be derived from a {\\it canonical} action.\nIndeed, such an action would be necessarily of the type,\n$$\nI=-\\int \\left(m+e \\, A^\\mu_{in} \\,u_\\mu+ e^2\\,{\\cal\nL}_{self}\\right)ds,\n$$\nwhere ${\\cal L}_{self}$ is an invariant function of the kinematical\nvariables $y^\\mu$, $u^\\mu$, $w^\\mu$ etc, giving rise to the\nself--interactions at the r.h.s. of \\eref{3.1}. But for dimensional\nreasons the unique term with the right dimensions would be the\ninvariant ${\\cal L}_{self}\\propto u_\\mu w^\\mu$, and this is zero.\n\nSince there exists no canonical action leading to \\eref{3.1}, a\nvariational principle giving rise to this equation bears necessarily\nsome unconventional features. The first proposal for an action was\nmade in \\cite{rohrprl}, relying on the previous attempt of\n\\cite{dirac}, and it requires the introduction of two gauge fields:\na source--free field and a field generated by the particle via\nretarded {\\it and} advanced potentials. As additional -- and more\nserious -- drawback this action contains non--local Coulomb\ninteractions between the particles. The more recent attempt in\n\\cite{marino} provides an action involving a single gauge field, but\nemploys a regularization which is not manifestly Lorentz--invariant.\n\nIn this paper we present an action which is obtained from the\nstandard action for ED, with a single gauge field, in a very simple\nand natural manner: we introduce a manifestly Lorentz--invariant\nregularization in the standard action, and renormalize it upon\nsubtracting an infinite mass term. The so obtained action gives rise\nto \\eref{3.1} in the following way. First one varies  the action\nwith respect to the gauge field, obtaining a regularized version of\nthe Maxwell equation. Then one varies the action with respect to the\ncoordinates of the particle and substitutes in the resulting\nequation the solution of the regularized Maxwell equation. The limit\nof this equation, when the regulator goes to zero, is \\eref{3.1}.\n\nThe third question regards the derivation of the energy--momentum\ntensor through Noether's theorem, from a covariant action. As far as\nwe know this problem has been addressed only in \\cite{rowel}, using\na formalism introduced in \\cite{zwanziger}. This paper is based on a\nmanifestly Poincar\\`e--invariant, but rather unconventional\ndistribution--valued Lagrangian density. Its most unusual\ncharacteristics are that the minimal interaction term $A^\\mu j_\\mu$ is\nabsent, and that the self--interaction of the particle is\nrepresented by a total derivative. The most unsatisfactory feature\nof the resulting action is, however, that it does not give rise to\nthe equations of motion, i.e. \\eref{maxwell} and \\eref{3.1}.\nEventually the role of the Lagrangian of \\cite{rowel} is restricted\nto give rise, through translation invariance and Noether's theorem,\nto the conserved energy--momentum tensor given in \\cite{rowese},\nupon imposing the Lorentz--Dirac equation \\eref{3.1} ``by hand\".\n\nThe Lagrangian which we propose in the present paper, being in\nparticular manifestly Poincar\\`e invariant, allows to apply\nNoether's theorem in a canonical way, and the resulting\nenergy--momentum tensor arises precisely in the new form mentioned\nabove.\n\nThe formalism used in this paper relies on the power of\nfull--fledged distribution theory. We remember, indeed, that in the\ncase of point particles the energy--momentum tensor as well as the\nLagrangian density are necessarily distribution valued. The other\nkey ingredient is a manifestly Lorentz--invariant regularization\nprocedure. Within this approach, in summary, we can set up the\nentire lagrangian formalism for the ED of point particles as\nspecified above: construction of a conserved and well--defined\nenergy--momentum tensor, construction of an action which gives to\nthe correct equations of motion, and implementation of the\nNoether--theorem, based on this action.\n\nThe method developed in this paper is based on the invariant regularization\nof the Green function for the D'Alambertian in eq. \\eref{2.7}, amounting\nto the replacement $x^2\\rightarrow x^2-\\varepsilon^2$, that permits a clear and simple distinction\nbetween finite and divergent terms in the energy--momentum tensor. Since the\nGreen function in arbitrary dimensions depends\nalways on $x^2$,  see \\cite{courant}, our method is very promising for  \nthe construction of finite and conserved energy--momentum tensors for charged\nsystems, for which this tensor has never been constructed before, \\cite{prep}. Examples include point\nparticles in a curved background \\cite{dewitt}, systems of dyons \\cite{rohrdy},\npoint particles in higher dimensions \\cite{kazinski}, and extended objects in arbitrary dimensions\n\\cite{kazinski2}. The knowledge of such a tensor allows a systematic quantitative analysis of \nradiation effects in these systems.\n\n\nThe plan of the paper is the following. In section two we introduce\nthe regularization used in the paper. In section three we present\nand illustrate our main results, i.e. the new expression of the\nrenormalized energy--momentum tensor together with its main\nproperties, the renormalized Lagrangian density giving rise to the\nequations of motion, and the Noether--theorem. Section four is\ndevoted to the proof of the properties of the renormalized\nenergy--momentum tensor, and to a comparison of this tensor with the\none proposed by Rowe \\cite{rowese}. The variational results are\nproved in section five. Some technical details are relegated to\nappendices.\n\nFor simplicity we present our results in detail in the case of a\nsingle point particle and for a vanishing external field\n$F^{\\mu\\nu}_{in}$. The corresponding results for the general case are\nsummarized in section \\eref{generali}.\n\n\n\\vskip1truecm\n\n\\section{Regularization}\n\nIn  Lorentz--gauge the Maxwell equations amount to,\n \\begin{equation} \\label{2.6}\n\\Box A^\\mu = j^\\mu, \\quad \\partial_\\mu A^\\mu =0,\n \\end{equation}\nwhere for a point--particle with charge $e$ and worldline $y^\\mu(s)$,\nthe current is given in \\eref{2.2}. The solution of these equations\ncan be obtained in terms of the retarded Green function for the\nD'Alambertian, supported on the forward light cone,\n\\begin{equation}\n  \\label{2.1}\nG(x) = {1\\over 2\\pi} H (x^0)\\, \\delta (x^2) = {1\\over 4 \\pi  |\\vec\nx|}\\,\\delta (x^0 - |\\vec x|), \\quad\\quad \\Box G(x)=\\delta^4(x),\n \\end{equation}\nwhere $x^\\mu = (x^0, \\vec x)$ are the space--time coordinates, and\n$H$ denotes the Heaviside step function. Omitting the external\npotential $A^\\mu_{in}$ one gets as solution the retarded\nLienard--Wiechert potential,\n \\begin{equation}\n \\label{2.4} A^\\mu_{LW} =G*j^\\mu=\n\\left.{e\\over 4\\pi} {u^\\mu (s) \\over (x_\\nu - y_\\nu (s)) u^\\nu(s)}\n\\right|_{s=s(x)},\n \\end{equation}\nwhere $*$ denotes convolution, and the scalar function $s(x)$ is\nuniquely fixed by the retarded time condition,\n \\begin{equation}\n\\label{2.5} (x - y (s))^2=0, \\quad x^0>y^0(s),\n \\end{equation}\nconsequence of the $\\delta$--function appearing in \\eref{2.1}.\n\nAlong the worldline of the particle, i.e. for $x^\\mu =y^\\mu(\\lambda)$\nfor some $\\lambda$, the LW--potential $A^\\mu_{LW}(y(\\lambda))$\ndiverges. In the present notation this follows from $s (y(\\lambda))\n= \\lambda$. As a consequence also the LW field--strength,\n \\begin{equation}\\label{LWfs}\nF_{LW}^{\\mu\\nu}=\\partial^\\mu A_{LW}^\\nu-\\partial^\\nu A_{LW}^\\mu,\n \\end{equation}\ndiverges along the particle worldline, meaning that a charged\nparticles feels an infinite self--interaction.\n\nTo smooth these singularities, while maintaining the structure of\nthe Maxwell equations unaltered, we propose to regularize the\nretarded Green function in a Lorentz--invariant way. Choosing  an\nUV--regulator $\\varepsilon >0$, with the dimension of length, we introduce a\nregularized Green function \\cite{fried},\n \\begin{equation}\n \\label{2.7}\n G_{\\varepsilon} (x) = {1\\over 2\\pi}\nH(x^0) \\delta (x^2 - \\varepsilon^2) \\equiv {1\\over 2\\pi}\\, \\delta_+ (x^2 -\n\\varepsilon^2)={1\\over 4 \\pi \\sqrt{|\\vec x|^2+\\varepsilon^2}}\\,\\, \\delta \\left(x^0\n-\\sqrt{|\\vec x|^2+\\varepsilon^2}\\right),\n \\end{equation}\nwhich is still an element of ${\\cal S}^\\prime$. The support of\n$G_{\\varepsilon}$ is given by the positive--time sheet of the hyperboloid,\n$$\nx^2= \\varepsilon^2.\n$$\nThis regularization preserves therefore also causality.\n\nWe can now define a  regularized LW potential,\n \\begin{equation}\n \\label{2.8}\nA^\\mu_{\\varepsilon} \\equiv  G_{\\varepsilon} * j^\\mu =\\left.{e\\over 4\\pi} {u^\\mu (s)\n\\over (x_\\nu - y_\\nu(s)) u^\\nu(s)} \\right|_{s=s_\\varepsilon(x)},\n \\end{equation}\nwhich differs from \\eref{2.4} only through the fact that  $s(x)$ is\nreplaced by $s_{\\varepsilon}(x)$, solution of the regularized retarded--time\nequation,\n \\begin{equation}\n\\label{2.15} (x - y (s))^2=\\varepsilon^2, \\quad  x^0>y^0(s).\n \\end{equation}\nThe potential $A^\\mu_{\\varepsilon}$ is a regular field, belonging to\n$C^\\infty\\equiv C^\\infty ({\\bf R^4})$, and in particular it is\nregular on the support of $j^\\mu$, i.e. on the worldline of the\nparticle. Also the regularized field strength,\n \\begin{equation}\\label{fsreg}\nF_\\varepsilon^{\\mu\\nu}\\equiv\\partial^\\mu A_\\varepsilon^\\nu-\\partial^\\nu A_\\varepsilon^\\mu,\n \\end{equation}\nis an element of $C^\\infty$, and hence everywhere regular. We\nillustrate these properties giving the explicit formulae for a\nstatic particle, at rest in $\\vec x=0$,\n \\begin{equation}\\label{static}\nA^0_\\varepsilon=   {e\\over 4\\pi} {1\\over (|\\vec x|^2+\\varepsilon^2)^{1\/2}},\\quad\n  \\vec A_\\varepsilon=0,\\quad\n  \\vec E_\\varepsilon={e\\over 4\\pi} {\\vec x\\over (|\\vec x|^2+\\varepsilon^2)^{3\/2}},\n\\quad \\vec B_\\varepsilon =0,\n \\end{equation}\nwhich are all regular in $\\vec x=0$.\n\nOne of the advantages of this regularization is that it preserves\nthe Lorentz--gauge. Indeed, due to the properties of the convolution\nand thanks to the fact that the current is conserved, the definition\n\\eref{2.8} gives directly,\n$$\n\\partial_\\mu A_\\varepsilon^\\mu=0.\n$$\nWe can then define also a {\\it regularized and conserved} current\nthrough,\n \\begin{equation}\n\\label{2.11} j_{\\varepsilon}^\\nu\\equiv \\partial_\\mu F^{\\mu\\nu}_\\varepsilon= \\Box A^\\nu_{\\varepsilon},\n\\quad\\quad \\partial_\\nu j_{\\varepsilon}^\\nu=0,\n \\end{equation}\nthat belongs to $C^\\infty$, too.\n\nSince the derivative is a continuous operation in ${\\cal S}^\\prime$,\nit is clear that we have the following limits,\n \\begin{eqnarray}\n{\\cal S}^\\prime {\\rm -} \\lim_{\\varepsilon \\rightarrow 0} A^\\mu_{\\varepsilon} &=& A^\\mu_{LW},\\\\\n{\\cal S}^\\prime {\\rm -} \\lim_{\\varepsilon \\rightarrow 0} F^{\\mu\\nu}_{\\varepsilon} &=&\nF^{\\mu\\nu}_{LW},\\\\\n{\\cal S}^\\prime {\\rm -} \\lim_{\\varepsilon \\rightarrow 0} j^\\mu_{\\varepsilon} &=&\nj^\\mu,\\label{limcur}\n \\end{eqnarray}\nwhere ${\\cal S}^\\prime {\\rm-}\\lim$ denotes the limit in the (weak)\ntopology of ${\\cal S}^\\prime$ \\footnote{We remember that this means\nthat these limits hold on every test function $\\varphi\\in {\\cal\nS}\\equiv{\\cal S}(\\bf R^4)$,\n$$\n\\lim_{\\varepsilon\\ra0}A^\\mu_{\\varepsilon}(\\varphi) = A^\\mu_{LW}(\\varphi), \\quad {\\rm etc.}\n$$}.\nMoreover, since by construction $G_{\\varepsilon}$ is  Lorentz invariant,\n$A^\\mu_{\\varepsilon}$,  $F^{\\mu\\nu}_{\\varepsilon}$, as well as $j^\\mu_{\\varepsilon}$ are Lorentz\ncovariant tensor fields.\n\nWe end this section writing more explicit expressions for\n$F^{\\mu\\nu}_{\\varepsilon}$ and $j^\\mu_{\\varepsilon}$, since these will be used in the\nfollowing. For this purpose we define the vector fields,\n \\begin{equation}\n \\label{2.17} R^\\mu (x)\\equiv x^\\mu -\ny^\\mu, \\qquad \\Delta^\\mu (x)\\equiv (u R)\\, w^\\mu - (w R)\\, u^\\mu ,\n \\end{equation}\nwhere the kinematical quantities $y$, $u$ and $w$ in these\nexpressions  are all evaluated at the the proper time $s_\\varepsilon(x)$,\ndetermined from \\eref{2.15}. In the following this functional\ndependence in the regularized quantities will always be understood.\nFor the scalar product of two vectors we use the notation $(a b)=\na_\\mu b^\\mu$. In particular we have, see \\eref{2.15},\n \\begin{equation}\\label{r2}\nR_\\mu R^\\mu=(R^0)^2-|\\vec R|^2=\\varepsilon^2.\n \\end{equation}\nThe expression for the regularized LW field strength can be derived\nin the same way as the known expression for the standard LW field,\n\\eref{LWfs}. To calculate it from \\eref{2.8} one needs the\nderivative of the function $s_\\varepsilon(x)$, that is obtained\ndifferentiating \\eref{r2},\n$$\n {\\partial s_{\\varepsilon} \\over \\partial x^\\mu} = {R_\\mu\n\\over (uR )}.\n $$\nThe rest of the calculations is a bit lengthy but straightforward.\nWith the above conventions the results read,\n \\begin{eqnarray}\nA^\\mu_{\\varepsilon} &=& {e\\over 4\\pi} {u^\\mu \\over (uR)},\\label{potreg}\\\\\n F^{\\mu\\nu}_{\\varepsilon} &=& {e \\over 4\\pi} {1\\over (uR)^3}\n \\left(R^\\mu u^\\nu - R^\\nu u^\\mu + R^\\mu \\Delta^\\nu - R^\\nu\n \\Delta^\\mu\\right),\\label{2.18}\\\\\nj^\\mu_\\varepsilon&=&\\varepsilon^2\\,{e\\over 4\\pi}\\left( {1\\over\n(uR)^4}\\left[(uR)\\,b^\\mu-(bR)\\,u^\\mu\\right]+{3(1-(wR))\\over\n(uR)^5}\\left(u^\\mu+\\Delta^\\mu\\right)\\right),\\label{curreg}\n \\end{eqnarray}\nwhere we defined,\n \\begin{equation}\\label{dw}\nb^\\mu={dw^\\mu\\over ds}.\n \\end{equation}\nThe formula \\eref{2.18} coincides, actually, with the known\nexpression of the LW field strength $F^{\\mu\\nu}_{LW} $ -- see for\nexample \\cite{roh} -- the unique difference being that the\nkinematical quantities are evaluated at the retarded proper time\n$s_\\varepsilon(x)$, rather then at $s(x)$.\n\nIn the following we will use an asymptotic condition on the\nworldlines $y^\\mu(s)$ of the particle. We suppose that the particle's\nacceleration is zero before a certain instant \\footnote{ Actually,\nthe results we obtain in this paper are valid also if the\nacceleration vanishes sufficiently fast for $s\\rightarrow -\\infty$.},\n \\begin{equation}\\label{asym}\nw^\\mu(s)=0, \\quad {\\rm for} \\,\\, s<\\bar s.\n \\end{equation}\nSince we have in any case $\\lim_{|\\vec x|\\rightarrow \\infty}s_\\varepsilon(x^0,\\vec\nx)=-\\infty$, this asymptotic condition implies that at fixed time\nthe acceleration $w^\\mu(s_\\varepsilon(x))$ in \\eref{2.18} vanishes for\nsufficiently large $|\\vec x|$. The same holds for the corresponding\nacceleration $w^\\mu(s(x))$ in the unregularized field\n$F^{\\mu\\nu}_{WL}$. This implies that the regularized and unregularized\nLW--fields have the same fixed--time Coulomb--like asymptotic\nbehavior for large $|\\vec x|$,\n \\begin{equation}\n \\label{asymlw}\nF^{\\mu\\nu}_\\varepsilon \\sim {1\\over |\\vec x|^2}, \\quad F^{\\mu\\nu}_{LW} \\sim\n{1\\over |\\vec x|^2}.\n \\end{equation}\nThese behaviors, that are valid also if the acceleration vanishes\nsufficiently fast in the remote past, will ensure in particular that\nthe total four--momentum of the electromagnetic field is finite.\n\n\\vskip1truecm\n\n\\section{Main results}\n\n\\subsection{The energy--momentum tensor}\n\nWe begin with the construction of a consistent energy--momentum\ntensor, accepting as equation of motion \\eref{3.1}.\n\nAs explained in the introduction the naive energy--momentum tensor\n$\\tau_{em}^{\\mu\\nu}$ in \\eref{3.2} is not an element of ${\\cal S}'$ --\nalthough $F^{\\mu\\nu}_{LW}$ is -- due its $1\/|\\vec R|^4$ behavior near\nthe worldline \\footnote{One can evaluate the quantity $|\\vec\nR(t,\\vec x)|=|\\vec x-\\vec y(s(x))|$ at a point near the worldline,\n$\\vec x\\approx \\vec y(t)$, solving \\eref{2.5}. One obtains,\n$$\n|\\vec R(t,\\vec x)|=\\left({v \\cos \\alpha + \\sqrt{1-v^2 \\sin^2 \\alpha}\\over\n1-v^2}\\right) |\\vec x-\\vec y(t)|+o\\left(|\\vec x-\\vec y(t)|^2\\right),\n$$\nwhere $\\vec v$ is the velocity at $t$, and $\\alpha$ is the angle between\n$\\vec v$ and $\\vec x-\\vec y(t)$. Therefore, near the worldline\n$|\\vec R|$ represents indeed the spatial distance from the particle,\na part from a never vanishing constant.}. Therefore its derivatives\ndo not even make sense, and the question what the quantity\n$``\\partial_\\mu\\tau_{em}^{\\mu\\nu}\"$ amounts to, is meaningless.\n\n\nTo construct an energy--momentum tensor that is an element of ${\\cal\nS}'$ we start from the regularized $LW$ field \\eref{2.18}, and\ndefine the regularized electromagnetic energy--momentum tensor as,\n \\begin{equation}\n \\label{treg}\nT^{\\mu\\nu}_\\varepsilon= F^{\\mu\\rho}_\\varepsilon F_{\\varepsilon\\rho}{}^\\nu + {1\\over 4}\\,\n\\eta^{\\mu\\nu}\\, F_\\varepsilon^{\\rho\\sigma} F_{\\varepsilon\\,\\rho\\sigma}.\n \\end{equation}\nThis tensor belongs to $C^\\infty$ and shares with $\\tau_{em}^{\\mu\\nu}$\nthe asymptotic behavior for large $|\\vec x|$ at fixed time, implied\nby \\eref{asymlw},\n \\begin{equation}\n \\label{temasym}\nT^{\\mu\\nu}_\\varepsilon \\sim {1\\over |\\vec x|^4}, \\quad \\tau_{em}^{\\mu\\nu}\\sim\n{1\\over |\\vec x|^4}.\n \\end{equation}\nMoreover, away from the worldline, i.e. for $\\vec x\\ne \\vec y(t)$,\nwe have the {\\it pointwise} limit,\n \\begin{equation}\\label{point}\n \\lim_{\\varepsilon\\rightarrow 0}\nT^{\\mu\\nu}_\\varepsilon(x)= \\tau_{em}^{\\mu\\nu}(x).\n \\end{equation}\nBut this limit does not exist in the topology of ${\\cal S}'$, due to\nthe {\\it reemerging singularities of $\\tau^{\\mu\\nu}_{em}$ along the\nworldline}. Before taking the ${\\cal S}'$--limit one must isolate,\nand subtract, these singular terms.\n\nIn the present framework the form of these singular terms -- and\nthis is one more advantage of our regularization -- is extremely\nsimple. Since the singularities appear only along the wordline, the\ncounterterm must be {\\it covariant, symmetric, traceless and\nsupported along the wordline}. These requirements fix it essentially\nuniquely, a part from an overall constant.  We propose as\nrenormalized energy--momentum tensor for the electromagnetic field\ngenerated by a point--particle, the expression,\n \\begin{equation}\n\\label{3.5} T^{\\mu\\nu}_{em} \\equiv {\\cal S}^\\prime - \\lim_{\\varepsilon\\rightarrow 0}\n\\left[T^{\\mu\\nu}_\\varepsilon - {e^2\\over 32\\,\\varepsilon} \\int \\left(u^\\mu u^\\nu -\n{1\\over 4} \\,\\eta^{\\mu\\nu}\\right)\\delta^4 (x - y\n(s))\\,ds\\right]\\equiv{\\cal S}^\\prime - \\lim_{\\varepsilon\\rightarrow 0} \\widetilde\nT^{\\mu\\nu}_\\varepsilon.\n \\end{equation}\nThe first counterterm, proportional to $u^\\mu u^\\nu$, can be viewed as an electromagnetic\nmass renormalization and is well known, see \\cite{teit2}. The\nsecond counterterm, proportional to $\\eta^{\\mu\\nu}$, to our knowledge has\nnever been noticed before in the literature, \nprobably because it can not be interpreted as a mass term and amounts, therefore, to a type of \nsingularity that is not present in the Lorentz--equation \\footnote{The technical reason for the omission\nof this term stems from the fact that in the literature the momentum conservation equation is\nderived integrating the formal (divergent) energy--momentum tensor $\\tau^{\\mu\\nu}_{em}$ over a {\\it restricted} \nclass of spacelike surfaces, that are orthogonal to the four velocity $u^\\mu$. In this \nway one looses a divergent term proportional to  \n${1\\over\\varepsilon} \\int \\left(u^\\mu u^\\nu - \\,\\eta^{\\mu\\nu}\\right)\\delta^4 (x - y(s))\\,ds$.}. This term is, however, necessarily present because the divergent part of a traceless tensor must be traceless. \n\nThe properties of the tensor \\eref{3.5} are indeed summarized by the following\n\n{\\bf Theorem I}\n\\newline\n{\\it 1) The limit in} \\eref{3.5} {\\it exists and hence\n$T^{\\mu\\nu}_{em} \\in {\\cal S}'.$\n\\newline\n2) $T^{\\mu\\nu}_{em}$ is a Lorentz--covariant, symmetric and traceless\ntensor field.\n\\newline\n3) $T^{\\mu\\nu}_{em} (x)= \\tau^{\\mu\\nu}_{em}(x)$ for $x$ in the\ncomplement of the particle's wordline.\n\\newline\n4) The four--divergence of $T^{\\mu\\nu}_{em}$ amounts to,\n \\begin{equation}\\label{divtem}\n \\partial_\\mu T^{\\mu\\nu}_{em}=- {e^2\\over\n6\\pi} \\int \\left({dw^\\nu\\over ds} +w^2\nu^\\nu\\right)\\delta^4(x-y(s))\\,ds,\n \\end{equation}\nfor an arbitrary smooth worldline $y^\\mu(s)$.\n\\newline\n5) If one defines the energy momentum tensor of the particle in a\nstandard way as,\n \\begin{equation}\n T^{\\mu\\nu}_p  = m \\int u^\\mu u^\\nu \\delta^{4} (x\n- y(s))ds, \\label{3.3}\n \\end{equation}\nthen the total energy--momentum tensor of the ED of a\npoint--particle,\n \\begin{equation}\n\\label{3.4} T^{\\mu\\nu} = T^{\\mu\\nu}_{em} + T^{\\mu\\nu}_p,\n \\end{equation}\nis conserved, $\\partial_\\mu T^{\\mu\\nu}=0$, if the particle satisfies the\nLorentz--Dirac equation} \\eref{3.1}, {\\it with $F^{\\mu\\nu}_{in}=0$}.\n\nThe crucial points to prove are the properties 1) and 4), i.e. the\nexistence of the limit \\eref{3.5} and the evaluation of $\\partial_\\mu\nT^{\\mu\\nu}_{em}$;  we relegate these proofs to section \\ref{proof1}.\n\nThe limit in \\eref{3.5} holds, actually, also in a stronger sense,\ni.e. on a set of test functions that is larger then ${\\cal S}$.\n Indeed, since for a fixed time $x^0=t$, for $\\varepsilon\\rightarrow 0$ the tensor\n$T^{\\mu\\nu}_\\varepsilon(t,\\vec x)$ develops only a singularity in three--space\nat the point $\\vec x=\\vec y(t)$, and due to the asymptotic behavior\n\\eref{temasym}, the limit \\eref{3.5} exists also on ``pseudo--test\nfunctions\" of the form,\n \\begin{equation}\n \\label{larger}\n\\varphi(x)= \\delta(x^0-t)\\,\\varphi(\\vec x),\n \\end{equation}\nwhere  $\\varphi(\\vec x)$ is bounded in ${\\bf R^3}$, and of class\n$C^\\infty$ in a neighborhood of $\\vec y(t)$. For example, $\\varphi(\\vec\nx)$ can be a constant, or a characteristic function on a\nthree--volume $V$. We will take advantage from this fact when\nconsidering momentum integrals.\n\nProperty 2) holds by construction, and property 3) follows from the\nfact that the counterterm in \\eref{3.5} is supported entirely on the\nworldline. This property has its physical origin in the fact that\nthe form of $\\tau^{\\mu\\nu}_{em}$ off the worldline is regular, and its\nphenomenological consequences away from the particle, like all\nclassical radiation phenomena, are experimentally very well tested.\nWe were therefore only allowed to change the form of\n$\\tau^{\\mu\\nu}_{em}$ on the worldline. Property 5) follows from\nproperty 4), using \\eref{3.1}.\n\n\n\\subsubsection{Momentum integrals}\n\nThe main purpose of this subsection is to prove, using the above\ntheorem, that the total four--momentum of the electromagnetic field\ngenerated by a point--particle at a generic instant $t$ is given by\nformula \\eref{ptot}, as shown first in \\cite{rowese}. In the course\nof the proof we will illustrate how the new representation\n\\eref{3.5} can be applied in practice.\n\n Formula \\eref{3.5} entails indeed an operative definition for the\nfour--momentum of the electromagnetic field, $P^\\mu_V(t)$, contained\nat the instant $x^0=t$  in a volume $V$. We define this momentum\napplying $T^{\\mu 0}_{em}$ to the pseudo--test function,\n \\begin{equation}\\label{pseudo}\n\\varphi(x)=\\delta(x^0-t)\\chi_V(\\vec x),\n \\end{equation}\nwhere  $\\chi_V(\\vec x)$ is the characteristic function on $V$. By\ndefinition we have,\n \\begin{equation}\n \\label{momint}\nP^\\mu_V(t)\\equiv T^{\\mu 0}_{em}(\\varphi)= \\lim_{\\varepsilon\\rightarrow 0} \\widetilde\nT^{\\m0}_\\varepsilon(\\varphi)=\\lim_{\\varepsilon\\rightarrow 0}\\int_V \\widetilde T^{\\m0}_\\varepsilon(t,\\vec\nx)\\,d^3x.\n \\end{equation}\nThis four--momentum is well--defined whenever $\\vec y(t)\\notin \\partial\nV$.\n\nIf at the time $t$ the particle is outside $V$, the counterterm in\n\\eref{3.5} does not contribute, and thanks to \\eref{point} the above\ndefinition reduces to,\n$$\nP^\\mu_V(t)=\\int_V \\tau^{\\mu 0}_{em}(t,\\vec x)\\,d^3x,\n$$\ncoinciding with the standard expression.\n\nWe illustrate the  operative definition \\eref{momint}, computing the\nfour--momentum $P^\\mu_V(t)$ for a particle in uniform motion,  the\nresult being in particular useful for the derivation of \\eref{ptot}.\nDue to Lorentz--invariance it is sufficient to consider a static\nparticle in the origin. From \\eref{momint} we see that we have two\nequivalent ways to evaluate $P^\\mu_V(t)$: we can evaluate $T^{\\mu\n0}_{em}(\\varphi)$ on a generic pseudo--test function $\\varphi$, and then set\n$\\varphi(x)=\\delta(x^0-t)\\chi_V(\\vec x)$, or we can evaluate the integral\n$\\int_V \\widetilde T^{\\m0}_\\varepsilon(t,\\vec x)\\,d^3x$, and then take the\nlimit $\\varepsilon \\rightarrow 0$. For a static particle we are able to carry out\nthe first, more ambitious, procedure.\n\nConsider first the energy density. Inserting \\eref{static} in\n\\eref{treg} one obtains the regularized energy density,\n$$\nT^{00}_\\varepsilon={1\\over 2} \\left({e\\over 4\\pi}\\right)^2{r^2\\over\n(r^2+\\varepsilon^2)^3}, \\quad r\\equiv |\\vec x|,\n$$\nand the $00$ component of \\eref{3.5} reduces to,\n$$\nT^{00}_{em} \\equiv {\\cal S}^\\prime - \\lim_{\\varepsilon\\rightarrow 0} \\left( {1\\over\n2} \\left({e\\over 4\\pi}\\right)^2{r^2\\over (r^2+\\varepsilon^2)^3} -{3\\,\ne^2\\over 128\\,\\varepsilon}\\,\\delta^3(\\vec x)\\right).\n$$\nApplying to a generic pseudo--test function $\\varphi$ we get,\n \\begin{eqnarray}\\nonumber\nT^{00}_{em}(\\varphi)&=& \\lim_{\\varepsilon \\rightarrow 0} \\left[{1\\over 2} \\left({e\\over\n4\\pi}\\right)^2 \\int  {r^2 \\varphi(x^0,\\vec x)\\over (r^2+\\varepsilon^2)^3}\\,\n\\,d^4 x-{3\\, e^2\\over 128\\,\\varepsilon}\\,\\int \\varphi(x^0,\\vec 0)\\, dx^0\\right] \\\\\n&=&\\lim_{\\varepsilon \\rightarrow 0}{1\\over 2} \\left({e\\over 4\\pi}\\right)^2 \\int {r^2\n(\\varphi(x^0,\\vec x)-\\varphi(x^0,\\vec 0)) \\over (r^2+\\varepsilon^2)^3} \\,d^4 x\\nonumber\\\\\n&=&{1\\over 2}\n \\left({e\\over 4\\pi}\\right)^2 \\int {\\varphi(x^0,\\vec x)-\\varphi(x^0,\\vec 0)\n\\over r^4}\\,d^4x, \\label{cond}\n \\end{eqnarray}\nwhere we used,\n$$\n {1\\over 2}\n\\left({e\\over 4\\pi}\\right)^2 \\int\n {r^2\\over (r^2+\\varepsilon^2)^3}\\,d^3x = {3\\,e^2\\over 128\\,\\varepsilon}.\n$$\nThe integral in \\eref{cond} is conditionally convergent, meaning\nthat one has first to integrate over angles, and then over $|\\vec\nx|$ and $x^0$. Inserting \\eref{pseudo} we get for the energy\ncontained in $V$ the $t$--independent result,\n$$\nP^0_V= T^{00}_{em}(\\varphi)= {1\\over 2} \\left({e\\over 4\\pi}\\right)^2\n\\int {\\chi_V(\\vec x)-\\chi_V(0) \\over r^4}\\,d^3x.\n$$\nIf $V$ does not contain the origin, i.e. the particle, one has\n$\\chi_V(0)=0$ and this formula reduces to the standard electrostatic\nenergy. If, on the other hand, $V$ is a sphere with radius $\\rho$\ncentered in the origin one has $\\chi_V(0)=1$, and the energy in the\nsphere amounts to $P^0_V=-e^2\/8\\pi \\rho$. In particular the total\nenergy in whole space is zero.\n\nThe spatial momentum $P^i_V (t)$ for a static particle vanishes\ntrivially, since the $0i$ components of both terms in \\eref{3.5} are\nzero, even before taking $\\varepsilon\\rightarrow 0$. Thanks to Lorentz--invariance\nthis means that the {\\it total four--momentum of the electromagnetic\nfield generated by a charged particle in uniform motion is zero},\n \\begin{equation}\\label{uniform}\nP^\\mu(t) =0.\n \\end{equation}\n\nWe consider now the total four--momentum for a particle in arbitrary\nmotion,\n$$\nP^\\mu(t)=\\lim_{\\varepsilon\\rightarrow 0} \\int\\widetilde T^{\\mu 0}_\\varepsilon(t,\\vec x)\\,\nd^3x,\n$$\nwhere, we remember, the convergence of the integral for large $|\\vec\nx|$ is ensured by the asymptotic behavior \\eref{temasym}. Since for\n$s<\\bar s$ the particle is in uniform motion, we have the additional\ninformation that,\n \\begin{equation}\\label{pasym}\nP^\\mu(t)=0, \\quad {\\rm for}\\quad t<\\bar t,\n \\end{equation}\nthanks to \\eref{uniform}, where $\\bar t=y^0(\\bar s)$.\n\nTo derive an explicit expression for $P^\\mu(t)$ we use property 4) of\nthe Theorem I. Define the vector field,\n \\begin{equation}\n\\label{km}\n K^\\mu_\\varepsilon= \\partial_\\nu \\widetilde T^{\\nu \\mu}_\\varepsilon.\n \\end{equation}\nSince the derivative is a continuous operation, \\eref{3.5} and\n\\eref{divtem} imply,\n \\begin{equation}\n\\label{porco}\n {\\cal S}'-\\lim_{\\varepsilon\\rightarrow 0} K^\\mu_\\varepsilon=- {e^2\\over 6\\pi}\n\\int \\left({dw^\\mu\\over ds} +w^2 u^\\mu\\right)\\delta^4(x-y(s))\\,ds,\n \\end{equation}\nwhere, again, this limit holds also on the pseudo--test functions\n\\eref{larger}. Integrating \\eref{km} over whole three--space we\nhave,\n$$\n\\partial_0 \\int \\widetilde T^{0\\mu}_\\varepsilon(t,\\vec x)\\,d^3x+ \\int \\partial_i\n\\widetilde T^{i\\mu}_\\varepsilon(t,\\vec x)\\,d^3x=\\int  K^\\mu_\\varepsilon(t,\\vec x)\n\\,d^3x.\n$$\nUsing the three--dimensional Gauss theorem and \\eref{temasym}, the\nsecond term on the l.h.s. is zero. Taking then the limit $\\varepsilon\\rightarrow 0$\nand using \\eref{porco} one obtains,\n$$\n{dP^\\mu(s)\\over ds}=-{e^2\\over 6\\pi}\\left({dw^\\mu\\over ds} +w^2\nu^\\mu\\right),\n$$\nwhere we used the variable $s$ instead of $t=y^0(s)$. Integrating\nthis expression from $-\\infty$ to $s$, and using the asymptotic\nrelations \\eref{asym} and \\eref{pasym}, we get for the total\nfour--momentum of the electromagnetic field,\n \\begin{equation}\\label{ptot}\n P^\\mu(s)=-{e^2\\over 6\\pi}\\left(w^\\mu(s)+\\int_{-\\infty}^s\n w^2(\\lambda)\\,\n u^\\mu(\\lambda)\\,d\\lambda\\right),\n \\end{equation}\nreproducing the result of \\cite{rowese}.\n\nIn appendix C we will actually prove that the more implicit\nexpression for $T^{\\mu\\nu}_{em}$ given by Rowe in \\cite{rowese},\ndefines the same distribution as \\eref{3.5}.\n\nThe formula \\eref{ptot} is clearly in agreement with total\nfour--momentum conservation, $P^\\mu(s)+p^\\mu(s)=$ constant, see\n\\eref{3.1}.\n\n\\subsection{Covariant variational principle}\n\nThe purpose of Theorem I was the construction of a well--defined\nenergy--momentum tensor. We deduce now the equations of motion and\nthe so constructed energy--momentum tensor -- via Noether theorem --\nfrom an action principle.\n\nWe propose the following regularized Lagrangian density ${\\cal\nL}_\\varepsilon \\equiv{\\cal L}_{\\varepsilon}(A,y)$,\n \\begin{equation}\n \\label{3.8}\n {\\cal L}_{\\varepsilon} = -{1\\over 4} F_{\\mu\\nu} F^{\\mu\\nu} - j^\\mu_{\\varepsilon} A_\\mu -\nm_\\varepsilon \\int \\delta^{4} (x - y(s))\\,ds,\n \\end{equation}\nthat is an element of ${\\cal S}'$ if $A^\\mu$ is sufficiently regular.\nThis Lagrangian differs from the standard Lagrangian for ED by two\nconceptually simple ingredients. First, the current $j^\\mu$ has been\nreplaced by its regularized counterpart $j^\\mu_\\varepsilon$, see \\eref{2.11}\nand \\eref{curreg}. Second, we introduced a diverging counterterm for\nthe mass. One can indeed interpret,\n \\begin{equation}\\label{bare}\nm_\\varepsilon \\equiv m - {3\\pi^2\\over 8 \\varepsilon}\\left({e\\over 4\\pi}\\right)^2,\n \\end{equation}\nas the bare mass of the regularized theory, whereas $m$ is the\nphysical renormalized mass.\n\nWe define the regularized action associated to this Lagrangian in a\nstandard way,\n \\begin{equation}\n \\label{action}\n I_\\varepsilon[A,y]=\\int {\\cal L}_{\\varepsilon} \\, d^4 x=\n -\\int \\left({1\\over 4} F_{\\mu\\nu} F^{\\mu\\nu}+ j^\\mu_{\\varepsilon}\n A_\\mu\\right)d^4x-m_\\varepsilon \\int ds.\n \\end{equation}\nNotice that this action is gauge invariant, because the current\n$j^\\mu_\\varepsilon$ is identically conserved. Moreover, it gives rise to the\nequations of motion \\eref{maxwell} and \\eref{3.1} according to the\nfollowing\n\n{\\bf Theorem II}\n\\newline\n{\\it 1) The equations of motion for $A^\\mu$ obtained from  $I_\\varepsilon$\nare the regularized Maxwell equations,\n \\begin{equation}\\label{maxreg}\n\\partial_\\mu F^{\\mu\\nu}=j^\\nu_\\varepsilon.\n \\end{equation}\nThese equations entail as solution the regularized LW potential\n$A_\\varepsilon^\\mu$ in \\eref{potreg}.\n\\newline\n2) Consider the equations of motion for $y^\\mu(s)$ derived from\n$I_\\varepsilon$,\n$$\nL^\\mu_\\varepsilon[A,y](s)\\equiv {\\delta I_\\varepsilon[A,y]\\over \\delta y_\\mu(s)}=0.\n$$\nIf one substitutes for $A^\\mu$ the solution $A^\\mu_\\varepsilon$ of\n\\eref{maxreg}, one has the point--wise limit,\n \\begin{equation}\\label{limvar}\n\\lim_{\\varepsilon\\rightarrow 0}L^\\mu_\\varepsilon[A_\\varepsilon,y](s)= {dp^\\mu\\over ds}-{e^2 \\over 6\n\\pi} \\left({dw^\\mu\\over ds}+w^2\\,u^\\mu\\right),\n \\end{equation}\ncorresponding to the Lorentz--Dirac equation.}\n\nProperty 1) is obvious. The statement 2) means that one has first to\nderive the equation of motion for $y^\\mu$ from $I_\\varepsilon$, then one must\nsubstitute in this equation the regularized LW--field potential\n$A^\\mu_\\varepsilon$, and eventually perform the limit $\\varepsilon\\rightarrow 0$. This is\n{\\it not} equivalent to substituting $A^\\mu_\\varepsilon$ in \\eref{action} and\nthen deriving the equation of motion for $y^\\mu$ and taking the limit\n$\\varepsilon\\rightarrow 0$. This second procedure would give rise to the equation of\nmotion ${dp^\\mu\\over ds}=0$, see Theorem III. The reason for the\nfailure of this procedure is that there exists no {\\it canonical}\naction giving rise to \\eref{3.1}, as explained in the introduction.\n\nSince we are not interested in the explicit form of $L^\\mu_\\varepsilon$, to\nprove property 2) it is sufficient to vary $I_\\varepsilon$ w.r.t. $y^\\mu$,\nset $A^\\mu =A_\\varepsilon^\\mu$ and then take the limit $\\varepsilon\\rightarrow 0$. The\nequation \\eref{limvar} is therefore equivalent to the relation,\n \\begin{eqnarray}\n\\left. \\lim_{\\varepsilon\\ra0}\\, \\delta I_\\varepsilon[A,y]\\right|_{A=A_\\varepsilon}&=&\n  \\lim_{\\varepsilon\\rightarrow 0}\\left(  -\\delta\\left(m_\\varepsilon \\int ds\\right) - \\int A_{\\varepsilon\\mu}\\,\\delta j^\\mu_\\varepsilon\\,d^4x\\right)\\nonumber\\\\\n&=& \\int\\left({dp^\\mu\\over ds}-{e^2 \\over 6 \\pi} \\left({dw^\\mu\\over\nds}+w^2\\,u^\\mu\\right)\\right)\\delta y_\\mu\\,ds,\\label{var}\n \\end{eqnarray}\nwhere $\\delta$ means variation w.r.t. $y^\\mu$. The proof of this relation\nwill be given in section \\ref{proof2}.\n\nThe regularity properties of the regularized Lagrangian itself are\nexpressed by\n\n{\\bf Theorem III}\n\\newline\n{\\it On the solution $A^\\mu_\\varepsilon$ the regularized Lagrangian converges\nfor $\\varepsilon\\rightarrow 0$,\n$$\n{\\cal S}'-\\lim_{\\varepsilon \\rightarrow 0} {\\cal L}_{\\varepsilon}(A_\\varepsilon,y)  \\equiv {\\cal L}\n\\in {\\cal S}',\n$$\nwhere, for a generic $\\varphi\\in {\\cal S}$,\n \\begin{equation}\\label{lagren}\n{\\cal L}(\\varphi)={1\\over2 }\\left({e\\over 4\\pi}\\right)^2 \\int\n{\\varphi(x)-\\varphi(y(s(x)))\\over (uR)^4}\\,d^4x - m \\int \\varphi(y(s))\\,ds.\n \\end{equation}\nIn this expression the regulator has been everywhere removed.}\n\nThe first integral in \\eref{lagren} is conditionally convergent, as\n\\eref{cond}. The regularized lagrangian admits therefore a finite\nlimit on the solutions of \\eref{maxreg}. Notice in particular that\nthe divergent contribution of the bare mass in \\eref{action}\ncanceled out. Along the solutions $A^\\mu_\\varepsilon$, also the action\n$I_\\varepsilon$ itself admits a finite limit. It is obtained evaluating the\nlimit Lagrangian \\eref{lagren} on the pseudo--test function\n$\\varphi(x)=1$,\n$$\n\\lim_{\\varepsilon\\rightarrow 0}I_\\varepsilon[A_\\varepsilon,y] = -m\\int ds.\n$$\nThe proof of this theorem is given in section \\ref{proof3}.\n\nAs last result we state the Noether theorem.\n\n{\\bf Theorem IV}\n\\newline\n{\\it The invariance of the distribution ${\\cal L}_{\\varepsilon}$ under\nspace--time translations, i.e. the identity,\n \\begin{equation}\\label{deltal}\n\\delta_t {\\cal L}_\\varepsilon\\equiv  \\delta A^\\mu {\\delta {\\cal L}_\\varepsilon\\over \\delta A^\\mu}-\\delta\ny^\\mu {\\delta {\\cal L}_\\varepsilon\\over \\delta y^\\mu}  -a^\\mu \\partial_\\mu {\\cal L}_\\varepsilon = 0,\n \\end{equation}\nwith $\\delta y^\\mu = a^\\mu, $ $\\delta A^\\mu = a^\\nu\\partial_\\nu A^\\mu $, if evaluated\non the solutions of the equations of motion \\eref{maxreg} and\n\\eref{3.1}, implies for $\\varepsilon\\rightarrow 0$ the conservation law $\\partial_\\mu\nT^{\\mu\\nu}=0$, where $T^{\\mu\\nu}$ is defined in \\eref{3.5}, \\eref{3.3}\nand \\eref{3.4}. More precisely, on the solution of the equations of\nmotion one has the relation,\n \\begin{equation}\n{\\cal S}'-\\lim_{\\varepsilon\\rightarrow 0}\\, \\delta_t {\\cal L}_\\varepsilon =a_\\nu \\,\\partial_\\mu\nT^{\\mu\\nu}.\n \\end{equation}\n}\n\nThe significance of this theorem is clear. We want only specify the\nfollowing conceptual point. Since ${\\cal L}_\\varepsilon$ is an element of\n${\\cal S}'$ the variation in \\eref{deltal}  is defined in the\ndistributional sense. This means that, by definition, we have,\n \\begin{eqnarray}\n(\\delta_t{\\cal L}_\\varepsilon)(\\varphi)&=&\\int {\\cal L}_{\\varepsilon} (x + a) \\varphi\n(x+a)\\,d^4x\n-\\int {\\cal L}_{\\varepsilon} (x) \\varphi(x)\\,d^4x\\nonumber\\\\\n&=&a^\\mu\\int {\\cal L}_\\varepsilon\\,\\partial_\\mu \\varphi \\,d^4x+\\widehat{\\delta}\\int{\\cal\nL}_\\varepsilon\\, \\varphi\\,d^4x,\\label{defvar}\n \\end{eqnarray}\nwhere $\\widehat{\\delta}$ indicates the variation of the integral w.r.t.\n$A$, {\\it minus} the variation of the integral w.r.t. $y$. This is\ndue to the fact that the {\\it explicit} dependence on $x$ of ${\\cal\nL}_\\varepsilon$ is through the difference $x^\\mu-y^\\mu(s)$ \\footnote{In the\npresent case the definition \\eref{defvar} needs only to be applied\nto the term multiplying $m_\\varepsilon$ in \\eref{3.8} -- because all other\nterms are regular distributions -- in which case \\eref{defvar}\ngives, translated back to symbolic notation,\n$$\n\\delta_t \\left(\\int \\delta^{4} (x - y(s))\\,ds\\right)= -a^\\mu\\partial_\\mu \\left(\\int\n\\delta^{4} (x - y(s))\\,ds\\right) -\\int \\delta y^\\mu{\\partial\\over \\partial y^\\mu(s)}\\,\n\\delta^{4} (x - y(s))\\,ds,\n$$\nthat vanishes identically. In practice we can then always use\ndirectly \\eref{deltal}.}. The Proof of Theorem IV will be given in\nsection \\ref{proof4}.\n\n\\subsection{The general case}\\label{generali}\n\nThe results presented so far generalize easily to a generic set of\ncharged point particles with masses $m_r$ and charges $e_r$, and to\nthe presence of an external field $F^{\\mu\\nu}_{in}$. In this case the\ntotal regularized electromagnetic field is given by,\n \\begin{equation}\\label{fgen}\nF^{\\mu\\nu}_\\varepsilon= F^{\\mu\\nu}_{in}+\\sum_r F^{\\mu\\nu}_{(r)\\varepsilon},\n \\end{equation}\nwhere $F^{\\mu\\nu}_{(r)\\varepsilon}$ is the regularized LW field produced by\nthe $r$--th particle. The electromagnetic energy--momentum tensor,\ngeneralizing naturally \\eref{3.5}, becomes,\n \\begin{equation}\\label{tnew}\n T^{\\mu\\nu}_{em} \\equiv\n{\\cal S}^\\prime - \\lim_{\\varepsilon\\rightarrow 0} \\left[T^{\\mu\\nu}_\\varepsilon - {1\\over 32\\,\n\\varepsilon}\\sum_r e_r^2 \\int \\left(u^\\mu_r u^\\nu_r - {1\\over 4}\n\\,\\eta^{\\mu\\nu}\\right)\\delta^4 (x - y_r(s_r))\\,ds_r\\right],\n \\end{equation}\nwith $T^{\\mu\\nu}_\\varepsilon$ given by \\eref{treg} and \\eref{fgen}.  There are\nno divergent counterterms due to the new interactions, because the\noff--diagonal terms in $T^{\\mu\\nu}_\\varepsilon$ are well defined\ndistributions, with only $1\/R^2$ singularities near the worldlines,\nand because $F^{\\mu\\nu}_{in}$ is supposed to be regular.\n\nThe four--divergence of this tensor is,\n $$ \\partial_\\mu\nT^{\\mu\\nu}_{em}=-\\sum_r e_r\\int \\left[{e_r\\over\n6\\pi}\\left({dw^\\nu_r\\over ds_r} +w^2_r\nu^\\nu_r\\right)+\\left(F^{\\nu\\mu}_{in}(y_r)+\\sum_{s\\neq r}\nF^{\\nu\\mu}_{(s)}(y_r)\\right)u_{r\\mu}\\right]\\delta^4(x-y_r)\\,ds_r ,\n $$\nwhere $F^{\\mu\\nu}_{(s)}$ is the unregularized LW field of the $s$--th\nparticle. This relation follows directly from \\eref{divtem}, since\nthe off--diagonal terms in $T^{\\mu\\nu}_\\varepsilon$ are free from overlapping\ndivergences, and on them one can simply apply the Leibnitz rule to\nevaluate the four--divergence. The total energy--momentum tensor\n$T^{\\mu\\nu}_{em}+T^{\\mu\\nu}_p$, with $T^{\\mu\\nu}_p  =\\sum_r m_r \\int\nu^\\mu_r u^\\nu_r \\delta^4 (x - y_r(s_r))ds_r$, is then conserved if the\nparticles satisfy the Lorentz--Dirac equations,\n \\begin{equation}\\label{ldnew}\n{dp_r^\\mu\\over ds_r}= {e_r^2\\over 6\\pi}\\left({dw^\\mu_r\\over ds_r}\n+w^2_r u^\\mu_r\\right)+e_r \\left(F^{\\mu\\nu}_{in}(y_r)+\\sum_{s\\neq r}\nF^{\\mu\\nu}_{(s)}(y_r)\\right)u_{r\\nu}.\n \\end{equation}\n\nEventually also the regularized Lagrangian density admits the\nnatural generalization,\n$$\n{\\cal L}_{\\varepsilon} = -{1\\over 4} F_{\\mu\\nu} F^{\\mu\\nu} - j^\\mu_{\\varepsilon} A_\\mu -\n\\sum_r\\left( m_r - {3\\,e_r^2\\over 128 \\,\\varepsilon} \\right)\\int \\delta^{4} (x -\ny_r(s_r))\\,ds_r,\n$$\nwhere $j^\\mu_{\\varepsilon}$ is the sum of the regularized currents of the\nparticles. This Lagrangian gives rise to the Lorentz--Dirac\nequations \\eref{ldnew}, and entails via Noether theorem the\nconserved energy--momentum tensor $T^{\\mu\\nu}_{em}+T^{\\mu\\nu}_p$. Along\nthe equations of motion ${\\cal L}_{\\varepsilon}$ admits again a finite limit\nfor $\\varepsilon\\rightarrow 0$ in the topology of ${\\cal S}'$.\n\n\n\\vskip1truecm\n\n\\section{Energy--momentum tensor}\\label{proof1}\n\nThis section is devoted mainly to the proof of  Theorem I. The\ningredients needed are the same as those needed in the proofs of the\nother theorems. For this reason we present here the necessary\ntechnical tools in some detail, while we will omit them in the\nproofs for the other theorems.\n\n\\subsection{Existence of the tensor $T^{\\mu\\nu}_{em}$}\\label{extensor}\n\nWe start with the proof of the existence of the limit \\eref{3.5},\nmeaning that we have to isolate in $T^{\\mu\\nu}_{\\varepsilon}$ the terms that\ndiverge for $\\varepsilon\\rightarrow 0$, if applied to test functions $\\varphi\\in{\\cal\nS}$. Inserting \\eref{2.18} in \\eref{treg} we can rewrite the\nregularized energy--momentum tensor as sum of three terms,\ncharacterized respectively by the inverse powers $1\/R^4$, $1\/R^3$\nand $1\/R^2$,\n \\begin{equation} \\label{4.1}\n T^{\\mu\\nu}_{\\varepsilon} =  \\sum_{i=1}^3T^{\\mu\\nu}_{\\varepsilon i},\n \\end{equation}\nwhere we count the dimensionful regulator $\\varepsilon$ as a power of $R$,\nsince we have $R^2=\\varepsilon^2$, see \\eref{r2}. The explicit expressions\nare,\n \\begin{eqnarray}\nT_{\\ve1}^{\\mu\\nu} &=& \\left({e\\over 4\\pi}\\right)^2  {1\\over (uR)^6}\n\\left(- R^\\mu R^\\nu + 2(uR)u^{(\\mu} R^{\\nu)}- \\varepsilon^2 u^\\mu u^\\nu\n\\right)-{\\rm tr},\\label{t1}\n\\\\\nT_{\\ve2}^{\\mu\\nu} &= &\\left({e\\over 4\\pi}\\right)^2 {1\\over (u R)^6}\n\\left(2(wR)R^\\mu R^\\nu   +2(uR)R^{(\\mu}\\Delta^{\\nu)} -2\\varepsilon^2 u^{(\\mu}\n\\Delta^{\\nu)}\\right)-{\\rm tr},\\label{t2}\n\\\\\n T_{\\ve3}^{\\mu\\nu} &=& \\left({e\\over 4\\pi}\\right)^2 {1\\over (uR)^6} \\left( -\n \\Delta^2 R^\\mu R^\\nu - \\varepsilon^2 \\Delta^\\mu \\Delta^\\nu\\right) -{\\rm\ntr},\n \\label{t3}\n \\end{eqnarray}\nwhere $-{\\rm tr}$ means the subtraction of the trace of the tensor,\nand we defined symmetrization by $a^{(\\mu} b^{\\nu)}={1\\over 2}(a^\\mu\nb^\\nu+a^\\nu b^\\mu)$.\n\nWe will now show that the divergent parts of these tensors, for\n$\\varepsilon\\rightarrow 0$ in the topology of ${\\cal S}'$, amount to,\n \\begin{eqnarray}\n\\left.T_{\\ve1}^{\\mu\\nu}\\right|_{div}&=&\\left({e\\over 4\\pi}\\right)^2\n\\int\\left( {\\pi^2\\over 2\\varepsilon}\\left(u^\\mu u^\\nu-{1\\over\n4}\\eta^{\\mu\\nu}\\right) +{16\\pi\\over 3}\\,\\ln\\varepsilon\\,u^{(\\mu}\nw^{\\nu)}\\right)\\delta^4(x-y(s))\\,ds,\\label{div1}\n \\\\\n\\left.T_{\\ve2}^{\\mu\\nu}\\right|_{div}&=&-\\left({e\\over 4\\pi}\\right)^2\n{16\\pi\\over 3}\\,\\ln\\varepsilon\\int u^{(\\mu}\nw^{\\nu)}\\delta^4(x-y(s))\\,ds,\\label{div2}\n \\\\\n\\left.T_{\\ve3}^{\\mu\\nu}\\right|_{div}&=&0. \\label{div3}\n \\end{eqnarray}\nSumming up one sees that the logarithmic divergences cancel, and\nthat the polar divergence is canceled by the counterterm in\n\\eref{3.5}.\n\nTo derive \\eref{div1}--\\eref{div3} one must apply $T^{\\mu\\nu}_{i\\varepsilon}$\nto a test function $\\varphi(x)$, and analyze the limit $\\varepsilon\\rightarrow 0$. In\ndoing this we encounter the technical difficulty that the\nkinematical variables $y$, $u$ and $w$ appearing in\n$T^{\\mu\\nu}_{i\\varepsilon}$ depend on $x$ in a rather complicated way, since\nthey are evaluated at the proper time $s_{\\varepsilon} (x)$. To disentangle\nthis implicit dependence we use the following identity for a generic\nfunction $f\\in C^\\infty ({\\bf R})$,\n \\begin{equation}\nf (s_{\\varepsilon} (x)) = \\int \\delta (s - s_\\varepsilon (x)) f(s)\\,ds = \\int 2 \\delta_+ [(\nx - y (s))^2- \\varepsilon^2]\\,  u^\\mu(s)\\,(x - y (s))_\\mu f (s)\\, ds.\n\\label{4.3}\n \\end{equation}\nWhen we apply such a function to a test function it is then also\nconvenient to perform in the resulting integral the shift,\n$$\nx^\\mu \\longrightarrow x^\\mu + y^\\mu (s).\n$$\n\nIn the following we make a systematic use of this strategy,\nbeginning with the proof of \\eref{div1}. Applying \\eref{t1} to a\ntest function $\\varphi$ one obtains then,\n \\begin{equation}\nT_{\\ve1}^{\\mu\\nu}(\\varphi)= A^{\\mu\\nu}_\\varepsilon(\\varphi)+B^{\\mu\\nu}_\\varepsilon(\\varphi),\n \\end{equation}\nwhere, setting momentarily $e\/4\\pi=1$,\n \\begin{eqnarray}\n A^{\\mu\\nu}_\\varepsilon(\\varphi)&=& \\int ds \\int d^4x\\,{2\\delta_+(x^2-\\varepsilon^2)\\over\n(u x)^5}\\left(- x^\\mu x^\\nu + 2(ux) u^{(\\mu} x^{\\nu)}-\\varepsilon^2 u^\\mu\nu^\\nu-{\\rm tr}\\right)\\varphi(y (s)),\\nonumber\n \\\\\n B^{\\mu\\nu}_\\varepsilon(\\varphi)&=& \\int ds \\int d^4x\\,{2\\delta_+(x^2-\\varepsilon^2)\\over\n(u x)^5}\\left(- x^\\mu x^\\nu + 2(ux) u^{(\\mu} x^{\\nu)}-\\varepsilon^2 u^\\mu\nu^\\nu-{\\rm tr}\\right)\\cdot\\nonumber\\\\\n&&\\hskip4.2truecm \\cdot(\\varphi(y (s) + x)-\\varphi(y(s)).\\label{bmn}\n \\end{eqnarray}\nAll kinematical quantities are now evaluated in the integration\nvariable $s$. We have divided $T_{\\ve1}^{\\mu\\nu}(\\varphi)$ in two terms,\nadding and subtracting the term with $\\varphi(y (s))$. This separation\nis convenient because we will see that $A^{\\mu\\nu}_\\varepsilon(\\varphi)$ gives\nrise to the polar divergence in \\eref{div2}, while\n$B^{\\mu\\nu}_\\varepsilon(\\varphi)$ gives rise to the logarithmic divergence.\n\nActually, $A^{\\mu\\nu}_\\varepsilon(\\varphi)$ can be evaluated exactly, it suffices\nto know the invariant integral, see appendix A,\n \\begin{equation}\\label{invint}\n  \\int d^4x\\, {2\\delta_+(x^2-\\varepsilon^2)\\over (ux)^5}\\, x^\\mu x^\\nu=\n{1\\over \\varepsilon}\\int d^4x {2\\delta_+(x^2-1)\\over (ux)^5}\\, x^\\mu x^\\nu=\n{\\pi^2\\over 4\\varepsilon} \\left(5u^\\mu u^\\nu-\\eta^{\\mu\\nu}\\right).\n \\end{equation}\nThis gives,\n$$\nA^{\\mu\\nu}_\\varepsilon(\\varphi)={\\pi^2\\over 2\\varepsilon}\\int \\left(u^\\mu u^\\nu-{1\\over\n4}\\,\\eta^{\\mu\\nu}\\right)\\varphi(y(s))\\,ds,\n$$\namounting to the polar divergence in \\eref{div1}.\n\nConsider now $B^{\\mu\\nu}_\\varepsilon$. From a dimensional analysis one sees\nthat $B^{\\mu\\nu}_\\varepsilon(\\varphi)$ diverges logarithmically as $\\varepsilon\\rightarrow 0$,\ni.e. as $\\ln\\varepsilon$. To extract this divergence it is then sufficient\nto evaluate the limit,\n$$\n\\lim_{\\varepsilon \\rightarrow 0}\\, \\varepsilon\\, {d\\over d\\varepsilon}\\,B^{\\mu\\nu}_\\varepsilon(\\varphi).\n$$\nTo compute $\\varepsilon\\, {d\\over d\\varepsilon}\\,B^{\\mu\\nu}_\\varepsilon(\\varphi)$ it is convenient\nto rescale first in \\eref{bmn} the integration variable $x\\rightarrow \\varepsilon\\,\nx$, then apply $\\varepsilon\\, {d\\over d\\varepsilon}$, and then rescale  back $x\\rightarrow\nx\/\\varepsilon$. Taking then the limit $\\varepsilon\\rightarrow 0$ the term $\\varepsilon^2 u^\\mu u^\\nu$\ndrops out and one gets \\footnote{The resulting expression can be\nevaluated explicitly noting the integral, with $y\\equiv y(s)$,\n \\begin{eqnarray}\n&&\\int d^4x\\,{2\\delta_+(x^2)\\over (u x)^5} \\,x^\\mu x^\\nu \\,[\\varphi(y)-\\varphi(y +\nx)+x^\\alpha\\partial_\\alpha\\varphi(y+x)]=\n -\\int d\\Omega\\, {m^\\mu m^\\nu\\over (um)^5}\\,m^\\alpha\\partial_\\alpha\\varphi(y)=\\nonumber\\\\\n&& {4\\pi\\over 3} \\left(u^\\mu \\partial^\\nu\\varphi(y)+u^\\nu \\partial^\\mu\\varphi(y)\n  +\\eta^{\\mu\\nu}u^\\alpha\\partial_\\alpha\\varphi(y)-6\\,u^\\mu u^\\nu\nu^\\alpha\\partial_\\alpha\\varphi(y)\\right),\\nonumber\n \\end{eqnarray}\nwhere $\\int d\\Omega$ denotes the integral over the\nthree--dimensional solid angle, and $m^\\mu=(1,{\\vec x\\over|\\vec\nx|})$.},\n \\begin{eqnarray}\n \\nonumber \\lim_{\\varepsilon \\rightarrow 0}\\, \\varepsilon\\,\n{d\\over d\\varepsilon}\\,B^{\\mu\\nu}_\\varepsilon(\\varphi) &=& \\int ds \\int\nd^4x\\,{2\\delta_+(x^2)\\over (u x)^5}\\left(- x^\\mu x^\\nu + 2(ux)\nu^{(\\mu} x^{\\nu)}-{\\rm tr}\\right)\\cdot\\\\\n&&\\hskip0truecm \\cdot[\\varphi(y(s))-\\varphi(y (s) +\nx)+x^\\alpha\\partial_\\alpha\\varphi(y(s)+x)]\\nonumber\\\\\n&=&-{8\\pi\\over 3} \\int u^\\mu u^\\nu\\,\n{d\\varphi(y(s))\\over ds}\\,ds\\nonumber\\\\\n&=&  {8\\pi\\over 3} \\int (u^\\mu w^\\nu+u^\\nu w^\\mu)\\, \\varphi(y(s))\\,ds.\\nonumber\n \\end{eqnarray}\n This is equivalent to,\n $$\nB^{\\mu\\nu}_\\varepsilon(\\varphi)={8\\pi\\over 3}\\,\\ln\\varepsilon \\int (u^\\mu w^\\nu+u^\\nu w^\\mu)\\,\n\\varphi(y(s))\\,ds +o(1),\n $$\nwhere $o(1)$ means terms that are regular for $\\varepsilon \\rightarrow 0$. This\ngives rise to the logarithmic divergence in \\eref{div1}.\n\nThe proof of \\eref{div2} proceeds similarly,\n $$\nT^{\\mu\\nu}_{\\ve2}(\\varphi)=\\int ds \\int d^4x{2\\delta_+(x^2-\\varepsilon^2)\\over (u\nx)^5}\\left(2(wx)x^\\mu x^\\nu + 2(ux) x^{(\\mu} \\Delta^{\\nu)}-2\\varepsilon^2 u^{(\\mu}\n\\Delta^{\\nu)}-{\\rm tr}\\right)\\varphi(y (s) + x),\n $$\nwhere here  $\\Delta^\\mu \\equiv (u x)\\, w^\\mu - (w x)\\, u^\\mu$. This is\nagain logarithmically divergent and one obtains, operating as above\n\\footnote{This time one needs the integral, with $y=y(s)$,\n \\begin{eqnarray}\\nonumber &&\n\\int d^4x{2\\delta_+(x^2)\\over (u x)^5}(x^\\mu x^\\nu x^\\rho) x^\\alpha\\partial_\\alpha \\varphi(y+\nx)= -\\int d\\Omega {m^\\mu m^\\nu m^\\rho\\over (um)^5}\\,\\varphi(y)=\n\\\\\n&&{4\\pi\\over\n3}\\left(\\eta^{\\mu\\nu}u^\\rho+\\eta^{\\nu\\rho}u^\\mu+\\eta^{\\rho\\mu}u^\\nu-6u^\\mu u^\\nu\nu^\\rho\\right)\\varphi(y),\\nonumber\n \\end{eqnarray}\nwhere $\\int d\\Omega$ is the integral over angles and $m^\\mu=(1,{\\vec\nx\\over |\\vec x|})$.},\n \\begin{eqnarray}\n \\nonumber\n\\lim_{\\varepsilon \\rightarrow 0}\\, \\varepsilon\\, {d\\over d\\varepsilon}\\, T^{\\mu\\nu}_{\\ve2}(\\varphi)&=&\n\\int ds \\int d^4x{2\\delta_+(x^2)\\over (u x)^5}\\left(2(wx)x^\\mu x^\\nu +\n2(ux) x^{(\\mu} \\Delta^{\\nu)}-{\\rm tr}\\right)x^\\alpha\\partial_\\alpha \\varphi(y(s) + x)\\\\\n&=&-{8\\pi\\over 3}\\int (u^\\mu w^\\nu+u^\\nu w^\\mu)\\, \\varphi(y(s))\\,ds.\\nonumber\n \\end{eqnarray}\nThis proves \\eref{div2}.\n\nThe tensor $T^{\\mu\\nu}_{\\ve3}$ admits trivially a finite limit for\n$\\varepsilon\\rightarrow 0$, since only $1\/|\\vec R|^2$ powers are present. This\nconcludes the proof of property 1) of the theorem.\n\n\\subsection{Conservation of the energy--momentum\ntensor}\\label{proofcons}\n\nThe conservation of the total energy--momentum tensor is reduced to\nthe proof of property 4), i.e. to the evaluation of $\\partial_\\mu\nT^{\\mu\\nu}_{em}$, see \\eref{divtem}. Since the derivative is a\ncontinuous operation in ${\\cal S}'$, once we have established the\nexistence of the limit \\eref{3.5}, the operator $\\partial_\\mu$ can be\ninterchanged with the limit, and we get,\n$$\n\\partial_\\mu T^{\\mu\\nu}_{em}= {\\cal S}'-\\lim_{\\varepsilon\\ra0} \\left(\\partial_\\mu\nT^{\\mu\\nu}_\\varepsilon - {e^2\\over 32\\,\\varepsilon} \\int \\left(w^\\nu - {1\\over\n4}\\,\\partial^\\nu \\right)\\delta^4 (x - y (s))\\,ds\\right).\n$$\nSince also the regularized LW--field satisfies the Bianchi identity,\n$$\n\\varepsilon_{\\mu\\nu\\rho\\sigma}\\,\\partial^\\nu F^{\\rho\\sigma}_{\\varepsilon} =0,\n$$\none deduces the standard result,\n \\begin{equation} \\label{4.9}\n\\partial_\\mu T^{\\mu\\nu}_\\varepsilon = -F^{\\nu\\mu}_\\varepsilon j_{\\varepsilon\\mu},\n \\end{equation}\nand hence,\n \\begin{equation}\\label{interm}\n \\partial_\\mu T^{\\mu\\nu}_{em}={\\cal S}'-\\lim_{\\varepsilon\\ra0}\n\\left(-F^{\\nu\\mu}_\\varepsilon j_{\\varepsilon\\mu}-{e^2\\over 32\\,\\varepsilon} \\int \\left(w^\\nu -\n{1\\over 4}\\,\\partial^\\nu \\right)\\delta^4 (x - y (s))\\,ds\\right).\n \\end{equation}\nThe calculation is therefore reduced to the evaluation of\n$F^{\\nu\\mu}_\\varepsilon j_{\\varepsilon\\mu}$ for $\\varepsilon\\rightarrow 0$, in the topology of ${\\cal\nS}'$.\n\n A somewhat delicate point in the evaluation of this limit\nis the following. The formula for the current \\eref{curreg} carries\na factor of $\\varepsilon^2$ in front, due to the fact that in the limit\n$\\varepsilon\\rightarrow 0$ the current $j^\\mu_\\varepsilon(x)$ goes to zero for $x^\\mu$ in the\ncomplement of the wordline. Actually, it can be seen that when\nevaluating the limit \\eref{limcur}, there is only one term in\n\\eref{curreg} that has a non--vanishing limit, more precisely,\n$$\n{\\cal S}^\\prime - \\lim_{\\varepsilon \\rightarrow 0}\\left( \\varepsilon^2 {e\\over 4\\pi}\n{3u^\\mu\\over (uR)^5}\\right)=j^\\mu,\n$$\nwhile all other terms converge to zero. These ``vanishing\"\nadditional terms are however needed in the regularized current to\nensure its conservation, and they contribute in the product\n$F^{\\nu\\mu}_\\varepsilon j_{\\varepsilon\\mu}$ for $\\varepsilon \\rightarrow 0$, since $F^{\\nu\\mu}_\\varepsilon$ has\npolar terms. Multiplying out \\eref{2.18} and \\eref{curreg} we obtain\nindeed,\n \\begin{eqnarray}\nF^{\\nu\\mu}_{\\varepsilon} j_{\\varepsilon \\mu}&=& \\left({e\\over 4\\pi}\\right)^2 \\varepsilon^2\n\\left[\n \\left({2w^2 \\over (uR)^6} -{(b R)\\over (uR)^7}\\right)R^\\nu +3\\,{(wR)-1\\over\n (uR)^6}\\, w^\\nu\\right.\\nonumber\n\\\\\n&& -\\left.3\\,{((wR)-1)^2 \\over (uR)^7}\\left(u^\\nu+((wR)-1){R^\\nu\\over\n(uR)}\\right)\\right] +o(\\varepsilon),\n \\label{4.14}\n \\end{eqnarray}\nwhere  $b^\\mu=dw^\\mu\/ ds$. $o(\\varepsilon)$ denotes terms that vanish for $\\varepsilon\n\\rightarrow 0$ in ${\\cal S}^\\prime$. In the present case these are terms of\nthe form $\\varepsilon^2\/R^4$, that converge to zero by power counting. We\nhave now to perform the limit of this expression for $\\varepsilon\\rightarrow 0$ in\n${\\cal S}'$. This analysis can be performed using the techniques\ndeveloped in section \\ref{extensor}, with the difference that now\none has to keep also terms that are finite for $\\varepsilon\\ra0$. The\ncomputation is deferred to appendix B where we show that,\n \\begin{equation}\\label{riscons}\nF^{\\nu\\mu}_{\\varepsilon} j_{\\varepsilon \\mu}=\\left({e\\over 4\\pi}\\right)^2\\int\\left[\n{8\\pi\\over 3}\\left({dw^\\nu\\over ds}+w^2u^\\nu\\right)-{\\pi^2\\over\n2\\varepsilon}\\left(w^\\nu-{1\\over\n4}\\,\\partial^\\nu\\right)\\right]\\delta^4(x-y(s))\\,ds+o(\\varepsilon),\n \\end{equation}\nwhere $o(\\varepsilon)$ denotes again terms that vanish for $\\varepsilon \\rightarrow 0$ in\n${\\cal S}^\\prime$.\n\nInserting this in \\eref{interm} one sees that the polar\ncontributions cancel, as they must,  and one obtains \\eref{divtem}.\n\n\n\\subsection{Comparison with Rowe's energy--momentum tensor}\n\nThe electromagnetic energy--momentum tensor introduced by Rowe\n\\cite{rowese} is written as the sum of three terms elements of\n${\\cal S}^\\prime$,\n \\begin{equation} \\label{6.1}\n\\Theta^{\\mu\\nu}_{em} = \\sum^3_{i=1} \\Theta^{\\mu\\nu}_i,\n \\end{equation}\nwhere,\n \\begin{eqnarray}\n\\Theta^{\\mu\\nu}_1  &=& \\partial_\\alpha K_1^{\\alpha\\mu\\nu} +{e^2 \\over\n16\\pi} \\int\nu^\\mu \\partial^\\nu \\delta^4(x - y (s))\\,ds, \\label{6.2} \\\\\n\\Theta^{\\mu\\nu}_2  &=& \\partial_\\alpha K^{\\alpha\\mu\\nu}_2 -\n{e^2\\over 6\\pi} \\int u^\\mu w^\\nu \\delta ^4(x - y(s))\\,ds, \\label{6.3}\\\\\n\\Theta^{\\mu\\nu}_3&=& -\\left({e\\over 4\\pi}\\right)^2 {\\Delta^2\\over\n(uR)^6}\\,R^\\mu R^\\nu.\n  \\label{6.4}\n \\end{eqnarray}\nThe tensors $K_i^{\\alpha\\mu\\nu}\\in {\\cal S}^\\prime$ are antisymmetric in\n$\\alpha$ and $\\mu$, and are given by,\n \\begin{eqnarray}\n K^{\\alpha\\mu\\nu}_1 &=& {1\\over 4}\\left({e\\over 4\\pi}\\right)^2 \\left(\n \\partial^\\mu\\left({R^\\alpha R^\\nu\\over (uR)^4}\\right)-\\partial^\\alpha\\left({R^\\mu R^\\nu\\over\n (uR)^4}\\right)\\right), \\label{6.5}\n\\\\\nK^{\\alpha\\mu\\nu}_2 &=& \\left({e\\over 4\\pi}\\right)^2{1\\over (uR)^5}\n\\left(R^\\mu \\Delta^\\alpha-R^\\alpha\\Delta^\\mu\\right)R^\\nu. \\label{6.6}\n \\end{eqnarray}\nIn these expressions no regularization is needed. Notice, indeed,\nthat $K^{\\alpha\\mu\\nu}_2$ contains only ${1\/R^2}$ singularities and\nthat $K^{\\alpha\\mu\\nu}_1$ contains only derivatives of  $1\/R^2$\nsingularities, so that both tensors are elements of ${\\cal S}'$.\nThis implies that also $\\Theta^{\\mu\\nu}_1$ and $\\Theta^{\\mu\\nu}_2$ are\nelements of ${\\cal S}'$, because the derivative of a distribution is\nagain a distribution. The terms with the $\\delta$--functions in\n\\eref{6.2}, \\eref{6.3} are added to ensure the symmetry of each\n$\\Theta^{\\mu\\nu}_i$ separately -- a property that is rather hidden in\nRowe's formulation, and needs to be proven a posteriori. A more\nserious practical drawback of the expressions above is that the\nderivatives appearing are distributional derivatives, and they can\nnot be evaluated applying simply the Leibnitz rule, due to the\nsingularities present on the worldline.\n\nAway from the worldline, i.e. for $x^\\mu\\neq y^\\mu(\\lambda)$, one can\nevaluate the derivatives above using the Leibnitz rule, and it is\nthen straightforward to check that in the complement of the\nworldline $\\Theta_{em}^{\\mu\\nu}$ coincides with the sum of\n\\eref{t1}--\\eref{t3}, with $\\varepsilon=0$, and hence with our\n$T_{em}^{\\mu\\nu}$ in \\eref{3.5}.\n\nThe comparison of $\\Theta_{em}^{\\mu\\nu}$ with $T_{em}^{\\mu\\nu}$ as\nelements of ${\\cal S}'$ can be performed applying to the tensors\n$K^{\\alpha\\mu\\nu}_i$ our ${\\cal S}'$--regularization,\n$K^{\\alpha\\mu\\nu}_i\\rightarrow K^{\\alpha\\mu\\nu}_{\\varepsilon i}$, where the regularized\ntensors are given formally again by \\eref{6.5}, \\eref{6.6}, but with\nthe replacement $s(x)\\rightarrow s_\\varepsilon(x)$. On the regularized tensors one\ncan now use the Leibnitz rule to evaluate $\\partial_\\alpha\nK^{\\alpha\\mu\\nu}_{\\varepsilon i}$, and perform eventually the ${\\cal\nS}'$--limit for $\\varepsilon\\rightarrow 0$. Following this procedure in appendix C\nwe prove indeed that,\n$$\n\\Theta_{em}^{\\mu\\nu}= T_{em}^{\\mu\\nu},\n$$\nas distributions. Here we note only that a comparison of \\eref{6.4}\nwith \\eref{t3} yields immediately, since there are only $1\/R^2$\nsingularities present,\n \\begin{equation} \\label{6.7} {\\cal S}'-\\lim_{\\varepsilon\\ra0} T^{\\mu\\nu}_{\\varepsilon 3} = \\Theta^{\\mu\\nu}_3.\n \\end{equation}\n\n\\vskip1truecm\n\n\\section{Proof of variational results}\\label{proofaction}\n\n\\subsection{Derivation of the Dirac--Lorentz equation}\\label{proof2}\n\nIn this subsection we proof property 2) of Theorem II, that is\nequivalent to the relation \\eref{var}. Instead of computing the\nvariation of $I_\\varepsilon$ under a generic variation of $y^\\mu(s)$, we\ncompute the variation of the Lagrange density ${\\cal L}_\\varepsilon$. The\nlimit of $\\delta I_\\varepsilon$ for $\\varepsilon\\rightarrow 0$ can then be obtained integrating\n$\\delta{\\cal L}_\\varepsilon$ between two hypersurfaces, and sending $\\varepsilon$ to\nzero. Another reason for proceeding in this way is that the explicit\nform of $\\delta{\\cal L}_\\varepsilon$ will also be used in the proof of Theorem\nIV.\n\nIn this subsection with ``$\\delta$\" we will always mean a generic smooth\nvariation w.r.t. $y^\\mu$. Starting from \\eref{3.8} we have then,\n \\begin{equation}\\label{vary}\n\\delta {\\cal L}_\\varepsilon=  -A_{\\varepsilon\\mu} \\delta j^\\mu_{\\varepsilon} - m_\\varepsilon \\delta\\left(\\int\n\\delta^{4} (x - y(s))\\,ds\\right),\n \\end{equation}\nwhere, according to what we need in \\eref{var}, we have replaced --\nafter variation -- $A_\\mu\\rightarrow A_{\\varepsilon\\mu}$. While the variation of the\nsecond term is trivial, to vary the first term it is more convenient\nto write,\n \\begin{equation} \\label{5.4}\n-A_{\\varepsilon\\mu} \\delta j^\\mu_{\\varepsilon} = -\\delta\\left(A_{\\varepsilon\\mu} j^\\mu_{\\varepsilon}\\right) + \\delta\nA_{\\varepsilon\\mu}j^\\mu_\\varepsilon,\n \\end{equation}\nsince $\\delta A_{\\varepsilon\\mu}$ is simpler then $\\delta j^\\mu_{\\varepsilon}$. The variation\nof the first term in this expression can be obtained from the ${\\cal\nS}'$--expansion,\n \\begin{equation}\nj^\\mu_{\\varepsilon} A^\\nu_{\\varepsilon} = \\left({e\\over 4\\pi}\\right)^2 \\int \\left(\n{3\\pi^2\\over 4\\varepsilon} \\,u^\\mu u^\\nu - 4\\pi \\,u^\\mu w^\\nu +o(\\varepsilon)\n\\right)\\delta^4(x-y(s))\\,ds, \\label{5.5}\n \\end{equation}\nupon contracting $\\mu$ and $\\nu$. This expansion can be derived using\nthe method illustrated in appendix B, see \\eref{esempio}.\n\nThe evaluation of the second term in \\eref{5.4} is more cumbersome,\nand the calculation is deferred to appendix D. The result is,\n \\begin{eqnarray} \\delta A_{\\varepsilon\\mu}j^\\mu_\\varepsilon&=& \\left({e\\over 4\\pi}\\right)^2\\left[\\int\n\\left(-{3 \\pi^2\\over 8\\varepsilon} (w\\,\\delta y)- {8 \\pi \\over 3} \\left(\n{dw^\\mu\\over ds}+w^2 u^\\mu\\right)\\delta y_\\mu\\right)\\delta^4(x-y(s))\\,ds\n\\right.\\nonumber\n\\\\\n&&\\left.+ \\partial_\\mu \\int \\left(4\\pi (w\\,\\delta y)u^\\mu-{\\pi^2\\over 8\\varepsilon}(\\delta\ny^\\mu- (u\\,\\delta y )u^\\mu)\\right)\\delta^4(x-y(s))\\,ds\\right] + o(\\varepsilon).\n\\label{varj}\n \\end{eqnarray}\nUsing this in \\eref{5.4} one obtains eventually,\n \\begin{eqnarray}\n \\delta {\\cal L}_\\varepsilon&=& \\int\n\\left(m w^\\mu- {e^2 \\over 6\\pi} \\left( {dw^\\mu\\over ds}+w^2 u^\\mu\n\\right)\\right)\\delta y_\\mu\\,\\delta^4(x-y(s))\\,ds+ \\label{generale}\n\\\\\n&&+ \\partial_\\mu \\int \\left({e^2\\over4\\pi}(w\\,\\delta\ny)\\,u^\\mu+\\left(m+{e^2\\over 64\\varepsilon}\\right)(\\delta y^\\mu- (u\\,\\delta y\n)u^\\mu)\\right)\\delta^4(x-y(s))\\,ds + o(\\varepsilon),\\nonumber\n \\end{eqnarray}\nwhich is our final result, holding for generic variations of $y$.\nThe first line is finite as $\\varepsilon\\rightarrow 0$, and the second line, which\ncorresponds to a four--divergence, contains a polar term in $\\varepsilon$.\n\nTo derive the equations of motion we compute $\\delta I_\\varepsilon$ integrating\n$\\delta {\\cal L}_\\varepsilon$ between two hypersurfaces, and impose that on them\n$\\delta y^\\mu(s)$ vanishes. The four--divergence drops then out --\ntogether with the polar term -- and the first line  \\eref{generale}\ngives \\eref{var}. This concludes the proof of Theorem II.\n\n\\subsection{Proof of Theorem III}\\label{proof3}\n\nThe evaluation of the regularized Lagrangian on the solution\n$A^\\mu=A^\\mu_\\varepsilon$ requires to evaluate, using \\eref{2.18} and\n\\eref{lim3},\n \\begin{eqnarray}\n -{1\\over 4} F_{\\varepsilon \\mu\\nu} F_{\\varepsilon}^{\\mu\\nu} &=& {1\\over 2}\\left({e\\over\n4\\pi}\\right)^2  \\left({1\\over (uR)^4} - {\\varepsilon^2\\over\n(uR)^6} +o(\\varepsilon) \\right)\\nonumber\\\\\n&=&{1\\over 2}\\left({e\\over 4\\pi}\\right)^2  \\left({1\\over (uR)^4} -\n{\\pi^2\\over 4\\varepsilon} \\int \\delta^4(x-y(s))\\,ds +o(\\varepsilon)\\right).\\nonumber\n \\end{eqnarray}\nThis gives, using \\eref{5.5},\n \\begin{eqnarray}\n {\\cal L}_{\\varepsilon}(A_\\varepsilon,y)&=&-{1\\over 4} F_{\\varepsilon \\mu\\nu} F_{\\varepsilon}^{\\mu\\nu} -A_{\\varepsilon\\mu}\n j^\\mu_{\\varepsilon}-m_\\varepsilon\\int \\delta^4(x-y(s))\\,ds\\nonumber\\\\\n&=& {1\\over 2}\\left({e\\over 4\\pi}\\right)^2  \\left({1\\over (uR)^4} -\n{\\pi^2\\over \\varepsilon}\\int \\delta^4(x-y(s))\\,ds\\ \\right)-m \\int\n\\delta^4(x-y(s))\\,ds+ o(\\varepsilon).\\nonumber\n \\end{eqnarray}\nApplying to a test function one gets,\n \\begin{eqnarray}\\nonumber\n\\lim_{\\varepsilon\\rightarrow 0}{\\cal L}_{\\varepsilon}(A_\\varepsilon,y)(\\varphi)&= &\\lim_{\\varepsilon\\rightarrow\n0}{1\\over 2}\\left({e\\over 4\\pi}\\right)^2 \\int ds \\int\n{2\\delta_+(x^2-\\varepsilon^2)\\over (ux)^3}[\\varphi(x+y(s))-\\varphi(y(s))]\\,d^4x\\\\\n&&-m\\int \\varphi(y(s))\\,ds,\\nonumber\n \\end{eqnarray}\nwhere we used,\n$$\n\\int d^4x \\,{2\\delta_+(x^2-\\varepsilon^2)\\over (ux)^3}= {\\pi^2\\over \\varepsilon}.\n$$\nThe integral above converges now for $\\varepsilon\\rightarrow 0$, and we can set\n$\\varepsilon=0$ obtaining the conditionally convergent integral,\n$$\n\\lim_{\\varepsilon\\rightarrow 0}{\\cal L}_{\\varepsilon}(A_\\varepsilon,y)(\\varphi) ={1\\over 2}\\left({e\\over\n4\\pi}\\right)^2 \\int ds \\int {2\\delta_+(x^2)\\over\n(ux)^3}[\\varphi(x+y(s))-\\varphi(y(s))]\\,d^4x -m\\int \\varphi(y(s))\\,ds.\n$$\nShifting $x\\rightarrow x-y(s)$ and integrating out the $\\delta$--function one\nobtains \\eref{lagren}.\n\n\\subsection{Proof of the Noether Theorem IV}\\label{proof4}\n\nThe translation invariance of the regularized Lagrangian is\nexpressed by  the identity \\eref{deltal}, $ \\delta_t {\\cal L}_\\varepsilon=0$,\nwhere ${\\cal L}_\\varepsilon$ is given in \\eref{3.8}, and $\\delta y^\\mu = a^\\mu, $\n$\\delta A^\\mu = a^\\nu\\partial_\\nu A^\\mu $.\n\nAfter standard steps one obtains,\n $$ \\delta_t {\\cal L}_\\varepsilon= (\\partial_\\mu\nF^{\\mu\\nu}-j^\\nu_\\varepsilon)\\delta A_\\nu-\\partial_\\mu(\\delta A_\\nu F^{\\mu\\nu}) +A_\\mu \\delta_y\nj^\\mu_\\varepsilon+\\delta_y \\left(m_\\varepsilon \\int \\delta^4(x-y(s))\\,ds\\right)-a^\\mu \\partial_\\mu\n{\\cal L}_\\varepsilon,\n $$\nwhere $\\delta_y$ means variation w.r.t. $y^\\mu$. We evaluate this\nexpression now on the solutions of the regularized Maxwell equation,\n$A^\\mu=A^\\mu_\\varepsilon$, and on those of the Lorentz--Dirac equation. The\nfirst term goes then to zero, and for the combination,\n$$\nA_{\\varepsilon\\mu} \\delta_y j^\\mu_\\varepsilon+\\delta_y \\left(m_\\varepsilon \\int\n\\delta^4(x-y(s))\\,ds\\right),\n$$\nwe can use the general expansion \\eref{vary}, \\eref{generale}, with\n$\\delta y^\\mu=a^\\mu$, with opposite sign. Thanks to the Lorentz--Dirac\nequation, the expression \\eref{generale} reduces to a\nfour--divergence and hence $\\delta_t {\\cal L}_\\varepsilon$ becomes a\nfour--divergence, too. Using again \\eref{5.5} to evaluate\n$A_{\\varepsilon\\nu}j^\\nu_\\varepsilon$ appearing in $\\partial_\\mu{\\cal L}_\\varepsilon$, after simple\nalgebra one obtains,\n \\begin{equation}\\label{5.10}\n\\delta_t {\\cal L}_\\varepsilon=a_\\nu\\,\\partial_\\mu \\widehat T^{\\mu\\nu}_\\varepsilon\n+o(\\varepsilon)=0, \\end{equation} where,\n \\begin{eqnarray}\n \\widehat T^{\\mu\\nu}_\\varepsilon &\\equiv& F^{\\rho\\mu}_{\\varepsilon}\\partial^\\nu\nA_{\\varepsilon\\rho}+ {1\\over 4} \\,\\eta^{\\mu\\nu} F^{\\rho\\sigma}_{\\varepsilon}\nF_{\\varepsilon\\rho\\sigma} + m\\int u^\\mu u^\\nu \\delta^4(x-y(s))\\,ds\\nonumber\\\\\n&&+\n \\left({e\\over 4\\pi}\\right)^2 \\int\n \\left({\\pi^2\\over 4\\varepsilon}\\,u^\\mu u^\\nu - 4\\pi u^\\mu w^\\nu +\n{\\pi^2\\over 8\\varepsilon}\\,\\eta^{\\mu\\nu}\\right)\\delta^4(x - y(s)) +o(\\varepsilon).\n \\label{5.11}\n \\end{eqnarray}\nEven if it is not obvious from the derivation -- see however the\nfootnote below -- it can be seen hat the limit,\n \\begin{equation}\\label{tcanon}\n{\\cal S}'-\\lim_{\\varepsilon\\rightarrow 0}\\widehat T^{\\mu\\nu}_\\varepsilon  \\equiv \\widehat\nT^{\\mu\\nu},\n \\end{equation}\nexists. \\eref{5.10} implies then that the energy--momentum tensor\n$\\widehat T^{\\mu\\nu}$ is conserved. It is immediately seen, however,\nthat this tensor does not coincide with the one constructed in\nTheorem I, see \\eref{3.4}, nor is it symmetric in $\\mu$ and $\\nu$. On\nthe other hand this was to be expected, since the Noether theorem\nleads in general to the {\\it canonical} energy--momentum tensor, not\nto the symmetric one.\n\nFrom the general theory we know however that, if the Lagrangian\ndensity is not only translation invariant but also Lorentz\ninvariant, there exists always a symmetrization procedure for the\ncanonical tensor. In the case of classical Electrodynamics this\nprocedure requires to add the divergenceless term $-\\partial_\\rho\n(F^{\\rho\\mu} A^\\nu)$. In the case at hand this suggests then to add\nthe divergenceless term \\footnote{The product $F^{\\rho\\mu}_\\varepsilon\nA^\\nu_\\varepsilon$ contains terms that behave as $1\/R^3$, if $\\varepsilon =0$. This\nmeans that a priori its ${\\cal S}'$--limit for $\\varepsilon\\rightarrow 0$ contains\nlogarithmic divergences, $\\sim \\ln\\varepsilon$. However, by inspection i.e.\napplying the product to a test--function and taking $\\varepsilon\\rightarrow 0$ one\nchecks that these divergences cancel out, and the limit exists. This\nmeans that also,\n$$\n{\\cal S}'-\\lim_{\\varepsilon\\rightarrow 0} \\partial_\\rho (F^{\\rho\\mu}_\\varepsilon A^\\nu_\\varepsilon),\n$$\nexists. This fact, together with \\eref{5.13}, provides an indirect\nproof for the existence of the limit \\eref{tcanon}.}, see\n\\eref{5.5},\n \\begin{eqnarray}\n \\label{symm}\n -\\partial_\\rho (F^{\\rho\\mu}_\\varepsilon A^\\nu_\\varepsilon)&=& - F^{\\rho\\mu}_\\varepsilon \\partial_\\rho A^\\nu_\\varepsilon -j^\\mu_\\varepsilon A^\\nu_\\varepsilon\n \\nonumber\\\\\n&=&-F^{\\rho\\mu}_\\varepsilon \\partial_\\rho A^\\nu_\\varepsilon +\\left({e\\over 4\\pi}\\right)^2\n\\int \\left( -{3\\pi^2\\over 4\\varepsilon} \\,u^\\mu u^\\nu + 4\\pi \\,u^\\mu w^\\nu\n+o(\\varepsilon) \\right)\\delta^4(x-y(s))\\,ds.\\nonumber\n \\end{eqnarray}\nAdding this to \\eref{5.11} one sees that eventually one gets a\nsymmetric tensor, being indeed,\n \\begin{equation} \\label{5.13}\n \\widehat T^{\\mu\\nu}_\\varepsilon - \\partial_\\rho\n(F^{\\rho\\mu}_{\\varepsilon} A^\\nu_{\\varepsilon})= T^{\\mu\\nu} +o(\\varepsilon),\n \\end{equation}\nwhere $T^{\\mu\\nu}$ is defined in \\eref{3.4}. This concludes the proof\nof the Noether theorem.\n\n\n\\bigskip\n\n\\paragraph{Acknowledgements.}\nThis work is supported in part by the European Community's Human\nPotential Programme under contract MRTN-CT-2004-005104,\n``Constituents, Fundamental Forces and Symmetries of the Universe\".\n\n\\vskip2truecm\n\n\\section{Appendix A: Invariant integrals}\n\nIn the proofs throughout the paper one encounters the  invariant\nintegrals,\n \\begin{eqnarray}\nI_\\mu&=&\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^N}\\, x_\\mu\\nonumber\\\\\nI_{\\mu\\nu}&=&\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^N}\\,x_\\mu x_\\nu,\\nonumber\\\\\nI_{\\mu\\nu\\rho}&=&\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^N}\\, x_\\mu x_\\nu\\nonumber\nx_\\rho,\n \\end{eqnarray}\nwhere $\\delta_+(x^2-1)\\equiv H(x^0)\\delta(x^2-1)$. These integrals can be\nevaluated considering $u^\\mu$ as an unconstrained variable, not\nsatisfying $u^2=1$, and writing, for example,\n$$\nI_{\\mu\\nu}= {1\\over (N-1)(N-2)}{\\partial\\over \\partial u^\\mu} {\\partial\\over \\partial u^\\nu}\n\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^{N-2}}.\n$$\nThis reduces  these integrals to the calculation of the ``generating\nfunction\",\n$$\n\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^N}= {\\pi^{3\/2}\\over (u^2)^{N\/2}}\n {\\Gamma\\left({N\\over 2}-1\\right)\\over \\Gamma\\left({N+1\\over\n 2}\\right)}.\n$$\nAfter taking the derivatives w.r.t. $u^\\mu$ one sets again $u^2=1$.\nFor example,\n $$\n\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^3}= {\\pi^2\\over (u^2)^{3\/2}},\n $$\nand hence,\n$$\n\\int d^4x \\,{2\\delta_+(x^2-1)\\over (ux)^5}\\,x_\\mu x_\\nu= {1\\over 3\\cdot\n4}{\\partial\\over \\partial u^\\mu} {\\partial\\over \\partial u^\\nu} \\int d^4x\n\\,{2\\delta_+(x^2-1)\\over (ux)^3}={\\pi^2\\over 4(u^2)^{7\/2}}\\left(5u^\\mu\nu^\\nu-u^2\\eta^{\\mu\\nu}\\right).\n$$\nSetting $u^2=1$ this gives \\eref{invint}.\n\n\\vskip1truecm\n\n\\section{Appendix B: proof of \\eref{riscons}}\n\nBefore taking the limit of $\\varepsilon \\rightarrow 0$ it is convenient to rewrite\n\\eref{4.14} in the form,\n \\begin{equation}\nF^{\\nu\\mu}_{\\varepsilon} j_{\\varepsilon \\mu}= \\left({e\\over 4\\pi}\\right)^2 \\varepsilon^2 \\left(\n \\left({2w^2 \\over (uR)^6} -{(b R)\\over (uR)^7}\\right)R^\\nu -{3w^\\nu\\over\n (uR)^6}\n+{1\\over 2}((wR)-1)^2\\partial^\\nu{1\\over (uR)^6}\\right) +o(\\varepsilon).\n \\label{4.14bis}\n \\end{equation}\nWe have used,\n$$\n\\partial^\\nu (uR)=u^\\nu +((wR)-1) \\,{R^\\nu\\over (uR)},\n$$\nand we have omitted the term,\n$$\n3\\varepsilon^2 {(wR)w^\\nu\\over (uR)^6},\n$$\nbecause it goes to zero for $\\varepsilon\\ra0$ in ${\\cal S}'$, for\nkinematical reasons. More precisely, after applying it to a test\nfunction and integrating over $d^4x$, for invariance reasons the\nfactor $R^\\mu$ in the numerator gets replaced by $u^\\mu$, and\n$(uw)=0$. This will become clear below. Regarding the last term in\n\\eref{4.14bis} it is convenient to note the identity,\n \\begin{eqnarray}\n{\\varepsilon^2\\over 2}((wR)-1)^2\\partial^\\nu{1\\over (uR)^6}&=& {\\varepsilon^2\\over 2}\\,\n\\partial^\\nu \\left[{((wR)-1)^2\\over(uR)^6}\\right] -\\varepsilon^2 {((wR)-1)\\over\n(uR)^6}\\, \\partial^\\nu(wR) \\nonumber\\\\\n&=& {\\varepsilon^2\\over 2} \\,\\partial^\\nu {1\\over(uR)^6}+\\varepsilon^2{(bR)\\over (u\nR)^7}\\,R^\\nu +\\varepsilon^2{w^\\nu\\over (uR)^6} + o(\\varepsilon),\n \\end{eqnarray}\nwhere the $o(\\varepsilon)$ terms are in the same sense as above. Inserting\nthis in \\eref{4.14bis} one sees that the $(bR)$--term cancels\ngetting,\n \\begin{equation}\n \\label{4.16}\nF^{\\nu\\mu}_{\\varepsilon} j_{\\varepsilon\\mu} =\\left({e\\over 4\\pi}\\right)^2 \\varepsilon^2 \\left(\n{2w^2 \\over (uR)^6}\\, R^\\nu - {2 w^\\nu\\over (u R)^6} + {1\\over 2}\\,\n\\partial^\\nu{1\\over (u R)^6}\\right) +o(\\varepsilon).\n \\end{equation}\nThe behaviour of the three terms present, for $\\varepsilon \\rightarrow 0$ in ${\\cal\nS}'$, are as follows,\n \\begin{eqnarray}\n{2\\varepsilon^2 w^2\\over (u R)^6}\\, R^\\nu & =&{8\\pi\\over 3} \\int w^2\nu^\\nu\\delta^4(x - y (s))\\,ds +o(\\varepsilon),\\label{lim1}\\\\\n-{2\\varepsilon^2 w^\\nu\\over (uR)^6}&=&\\int\\left({8\\pi\\over 3} {dw^\\nu\\over\nds} -{\\pi^2\\over 2\\varepsilon}w^\\nu\\right)\\delta^4(x - y (s))\\,ds +o(\\varepsilon),\\label{lim2}\\\\\n{\\varepsilon^2\\over 2(uR)^6}&=&{\\pi^2\\over 8\\varepsilon} \\int \\delta^4(x - y (s))\\,ds\n+o(\\varepsilon).\\label{lim3}\n \\end{eqnarray}\nFor the proofs of these relations, that require to apply the l.h.s.\nto a test function and to analyze the limit $\\varepsilon \\rightarrow 0$, one can use\nthe technicalities developed in section \\ref{extensor}. We report\nexplicitly the proof of \\eref{lim2},\n \\begin{eqnarray}\n \\left(-{2\\varepsilon^2 w^\\nu\\over (uR)^6}\\right)\\,(\\varphi)&=&\n-2\\varepsilon^2 \\int ds \\int d^4x{ 2 \\delta_+ (x^2 -\n\\varepsilon^2)\\over (ux)^5} \\,w^\\nu\\varphi(y(s)+ x)\\nonumber\\\\\n&=& -{2\\over \\varepsilon} \\int ds \\int d^4x{ 2 \\delta_+ (x^2 - 1)\\over (ux)^5}\n\\,w^\\nu\\varphi(y(s)+\\varepsilon x)\\nonumber\\\\\n&=& -2 \\int ds \\int d^4x{2\\delta_+ (x^2 - 1)\\over (ux)^5}\\,w^\\nu\\,\n\\left({1\\over \\varepsilon}\\,\\varphi\n(y(s)) +  x^\\alpha \\partial_\\alpha \\varphi(y(s)) + o(\\varepsilon)\\right)\\nonumber\\\\\n&=& -2 \\int ds \\left({\\pi^2\\over 4\\varepsilon}\\,w^\\nu \\varphi(y(s)) +{4\\pi\\over\n3}\\,\nw^\\nu u^\\alpha \\partial_\\alpha \\varphi (y(s)) + o(\\varepsilon)\\right)\\nonumber\\\\\n&=& \\int ds \\left(-{\\pi^2\\over 2\\varepsilon} w^\\nu + {8\\pi\\over 3}\n{dw^\\nu\\over ds}+o(\\varepsilon)\\right) \\varphi (y(s)),\\label{esempio}\n \\end{eqnarray}\nwhere we used the invariant integrals of appendix A.\n\nInserting \\eref{lim1}--\\eref{lim3} in \\eref{4.16}, one obtains\n\\eref{riscons}.\n\n\\vskip1truecm\n\n\\section{Appendix C: Rowe's energy--momentum tensor}\\label{appa}\n\nBefore comparing $T^{\\mu\\nu}_{em}$ of \\eref{3.5} with\n$\\Theta^{\\mu\\nu}_{em}$ in \\eref{6.1}, we cast the former in a slightly\ndifferent form writing, see \\eref{t1}--\\eref{t3},\n \\begin{equation}\\label{newsplit}\nT^{\\mu\\nu}_{em}=  {\\cal S}'-\\lim_{\\varepsilon\\ra0} \\left(D^{\\mu\\nu}_{\\varepsilon 1}+\nD^{\\mu\\nu}_{\\varepsilon 2}+T^{\\mu\\nu}_{\\varepsilon 3}\\right),\n \\end{equation}\nwhere,\n \\begin{eqnarray}\nD^{\\mu\\nu}_{\\varepsilon 1}&=&T^{\\mu\\nu}_{\\ve1}  - 2\\left({e\\over 4\\pi}\\right)^2\n{(wR)\\over (uR)^6}\\,(R^\\mu R^\\nu- {\\rm tr})- {e^2\\over 32\\,\\varepsilon} \\int\n\\left(u^\\mu u^\\nu - {1\\over 4} \\,\\eta^{\\mu\\nu}\\right)\\delta^4\n(x - y (s))\\,ds,\\nonumber\\\\\nD^{\\mu\\nu}_{\\varepsilon 2}&=&T^{\\mu\\nu}_{\\ve2}  + 2\\left({e\\over 4\\pi}\\right)^2\n{(wR)\\over (uR)^6}\\,(R^\\mu R^\\nu- {\\rm tr}),\\nonumber\n \\end{eqnarray}\nThe advantage of this form of presenting $T^{\\mu\\nu}_{em}$ is that the\ntensors $D^{\\mu\\nu}_{\\varepsilon i}$ converge {\\it separately} for $\\varepsilon \\rightarrow 0$\nin the topology of ${\\cal S}'$, as does $T^{\\mu\\nu}_{\\varepsilon 3}$. We have\nadded and subtracted the same term from $ T^{\\mu\\nu}_{\\varepsilon 1}$ and\n$T^{\\mu\\nu}_{\\varepsilon 2}$, to eliminate the logarithmic divergence from\nboth. This can be seen from \\eref{div1}--\\eref{div3}, noting that,\n$$\n\\left.{(wR)\\over (uR)^6}\\,R^\\mu R^\\nu\\right|_{div}= {8\\pi\\over\n3}\\,\\ln\\varepsilon\\int u^{(\\mu} w^{\\nu)}\\delta^4(x-y(s))\\,ds.\n$$\n\nWe evaluate now the four--divergencies appearing in \\eref{6.2},\n\\eref{6.3}, using the regularized versions $K_{\\varepsilon i}^{\\alpha\\mu\\nu}$\nfor \\eref{6.5}, \\eref{6.6}, obtained replacing $s(x)\\rightarrow s_\\varepsilon(x)$.\nThen one can apply Leibnitz' rule and, taking into account that now\n$R^2=\\varepsilon^2\\neq 0$, after a straightforward computation one obtains,\n \\begin{eqnarray}\n\\partial_\\alpha K_{\\ve1}^{\\alpha\\mu\\nu}& =&D^{\\mu\\nu}_{\\ve1} + {e^2\\over 32\\,\\varepsilon}\n\\int \\left(u^\\mu u^\\nu - {1\\over 4} \\,\\eta^{\\mu\\nu}\\right)\\delta^4 (x - y\n(s))\\,ds\\label{a1}\\\\\n&&+ \\left({e\\over 4\\pi}\\right)^2 {\\varepsilon^2\\over 4}\n \\left[15{(wR)-1\\over(u R)^7} \\,u^\\mu R^\\nu\n + {7u^\\mu u^\\nu\\over (uR)^6} -{3w^\\mu R^\\nu\\over (uR)^6}  + {2- 6 (w R)\\over (uR)^6}\\,\n\\eta ^{\\mu\\nu}\\right], \\label{a2}\\\\\n\\partial_\\alpha\nK_{\\ve2}^{\\alpha\\mu\\nu}&=&D^{\\mu\\nu}_{\\ve2}+\\left({e\\over4\\pi}\\right)^2\\varepsilon^2\n\\left[5 {(wR)-1\\over (uR)^7}\\Delta^\\mu R^\\nu+{w^\\mu R^\\nu\\over (uR)^6} +\n{1\\over (uR)^6}\\left((bR)u^\\mu-(uR)b^\\mu\\right)R^\\nu\\right.\\nonumber\n\\\\\n&&\\left.+ {1\\over (uR)^6}(2\\Delta^\\mu u^\\nu+\\Delta^\\nu u^\\mu)+{3(wR)\\over\n2(uR)^6}\\eta^{\\mu\\nu}\\right].\\label{a4}\n \\end{eqnarray}\nIn $\\partial_\\alpha K_{\\ve1}^{\\alpha\\mu\\nu}$ we have put in evidence the polar\npart, proportional to $u^\\mu u^\\nu - {1\\over 4} \\,\\eta^{\\mu\\nu}$, and in\n$\\partial_\\alpha K_{\\ve1}^{\\alpha\\mu\\nu}$ we have set $b^\\mu=dw^\\mu\/ds$.\n\nWe can now consider the ${\\cal S}'$--limit for $\\varepsilon\\rightarrow 0$ of these\nrelations, applying them to a test--function and sending $\\varepsilon\\ra0$.\nThe calculations, that are straightforward but a bit lengthy, can be\nperformed with the techniques illustrated in appendix B, see\n\\eref{esempio}. Since the limits of $\\partial_\\alpha K_{\\varepsilon i}^{\\alpha\\mu\\nu}$ for\n$\\varepsilon\\ra0$ exist by construction, the r.h.s. of the above relations\nmust also admit finite limits. It can indeed be seen that the square\nbracket in \\eref{a2} exhibits a divergent contribution that cancels\nexactly the polar term in \\eref{a1}, while the square bracket in\n\\eref{a4} admits a finite limit by power counting. The results are,\n \\begin{eqnarray}\n\\partial_\\alpha K_1^{\\alpha\\mu\\nu}&=& \\left({\\cal S}'-\\lim_{\\varepsilon\\ra0} D^{\\mu\\nu}_{\\varepsilon\n1}\\right) + {e^2\\over \\pi} \\int\\left({1\\over 6}(u^\\mu w^\\nu+u^\\nu\nw^\\mu)-{1\\over 16}\\,u^\\mu \\partial^\\nu\\right)\\delta^4(x-y(s))\\,ds,\\nonumber\n\\\\\n\\partial_\\alpha K_2^{\\alpha\\mu\\nu}&=& \\left({\\cal S}'-\\lim_{\\varepsilon\\ra0} D^{\\mu\\nu}_{\\varepsilon\n2}\\right) - {e^2\\over 6\\pi} \\int u^\\nu w^\\mu \\delta^4(x-y(s))\\,ds.\\nonumber\n \\end{eqnarray}\nInserting these expressions in \\eref{6.2}, \\eref{6.3} one gets the\nsymmetric and traceless tensors,\n \\begin{eqnarray}\n\\Theta^{\\mu\\nu}_1 &=&\\left({\\cal S}'-\\lim_{\\varepsilon\\ra0} D^{\\mu\\nu}_{\\varepsilon\n1}\\right)  + {e^2\\over 6\\pi} \\int (u^\\mu w^\\nu+u^\\nu w^\\mu) \\delta^4(x -\ny(s))\\,ds,\\\\\n\\Theta^{\\mu\\nu}_2 &=&\\left({\\cal S}'-\\lim_{\\varepsilon\\ra0} D^{\\mu\\nu}_{\\varepsilon\n2}\\right)  - {e^2\\over 6\\pi} \\int (u^\\mu w^\\nu+u^\\nu w^\\mu) \\delta^4(x -\ny(s))\\,ds.\n \\end{eqnarray}\nThese formulae, together with \\eref{6.7} and \\eref{newsplit}, imply\nthen $\\Theta^{\\mu\\nu}_{em}=T^{\\mu\\nu}_{em}$, q.e.d.\n\n\\vskip1truecm\n\n\\section{Appendix D: Proof of \\eref{varj}}\\label{appb}\n\nTo evaluation of $\\delta A_{\\varepsilon\\mu}j^\\mu_\\varepsilon $ requires first to evaluate\n$\\delta A_{\\varepsilon\\mu}$ under a generic smooth variation of $y^\\mu$. This\nevaluation is complicated by the fact that $A_\\varepsilon^\\mu$  -- see\nformula \\eref{2.8} -- depends on $y^\\mu$ explicitly,  but also\nimplicitly through the (regularized) retarded proper time function\n$s_\\varepsilon(x)$, defined in \\eref{2.15}. To determine  $\\delta A_{\\varepsilon\\mu}$ it\nis convenient to use the following shortcut.\n\nWe compute $\\delta A_\\varepsilon^\\mu$ applied to a test function $\\varphi\\in {\\cal\nS}$, inserting a $\\delta$--function as in \\eref{4.3}, and then shifting\n$x\\rightarrow x+y(s)$,\n \\begin{eqnarray}\n\\int\\delta A^\\mu_{\\varepsilon}\\,\\varphi\\, d^4x &=& \\delta \\int A^\\mu_{\\varepsilon}\\,\\varphi \\,d^4x ={e\n\\over 4 \\pi}\\,\\delta \\int ds \\int d^4 x\n\\,2\\delta_+(x^2-\\varepsilon^2) u^\\mu(s) \\varphi(x+y(s))\\nonumber\\\\\n&=& {e \\over 4 \\pi} \\int ds \\int d^4 x \\,2\\delta_+(x^2-\\varepsilon^2)\\left(\n{d\\,\\delta\ny^\\mu\\over ds} \\,\\varphi(x+y(s)) +u^\\mu\\delta y^\\nu\\partial_\\nu  \\varphi(x+y(s))\\right)\\nonumber\\\\\n &=&\n{e \\over 4 \\pi} \\int d^4 x\\left[{1\\over (uR)}\\,{d\\delta y^\\mu\\over ds}\n-\\partial_\\nu\\left({u^\\mu\\delta y^\\nu \\over (uR)}\\right)\\right]\\varphi,\\nonumber\n \\end{eqnarray}\nwhere in the square bracket all kinematical quantities are evaluated\nat $s=s_\\varepsilon(x)$. This gives,\n \\begin{eqnarray}\n\\delta A^\\mu_{\\varepsilon}&=&{e \\over 4 \\pi}\\left({1\\over (uR)}\\,{d\\delta y^\\mu\\over\nds}\n-\\partial_\\nu\\left({u^\\mu\\delta y^\\nu \\over (uR)}\\right)\\right)\\nonumber\\\\\n &=&\n{e \\over 4 \\pi}\\left({\\ell'^\\mu\\over (uR)}-{(R\\ell\\,')\\over\n(uR)^2}u^\\mu-{(R\\ell)\\over (uR)^3}(\\Delta^\\mu+u^\\mu)\\right),  \\label{7.1}\n \\end{eqnarray}\nwhere,\n$$\n\\ell^\\mu\\equiv \\delta y^\\mu-(u\\delta y)u^\\mu, \\quad (u\\ell)=0,   \\quad\n\\ell'^\\mu\\equiv{d \\ell^\\mu \\over ds}.\n$$\nThe appearance of the combination $\\ell^\\mu$ is due to the invariance\nof $A^\\mu_\\varepsilon$ under reparametrization of the worldline, i.e. $\\delta\nA^\\mu_\\varepsilon=0$ if $\\delta y^\\mu=\\delta\\lambda \\,u^\\mu $.\n\nMultiplying out \\eref{7.1} and \\eref{curreg} one obtains then,\n \\begin{eqnarray}\\label{varfin}\\nonumber\n\\delta A_{\\varepsilon\\mu}j^\\mu_\\varepsilon &=&\\left({e \\over 4 \\pi}\\right)^2 \\varepsilon^2 \\left[\n{(bR)-(uR)(ub)\\over (uR)^7}\\,(R\\ell)+ 3(1-(wR))\\cdot\\right.\n\\\\\n&&\\cdot\\left.\\left({((u\\ell')+(\\Delta\\ell'))(uR)-(R\\ell')(1-(wR))\\over\n(uR)^7}- {(1-(wR))^2+(uR)^2w^2\\over\n(uR)^8}\\,(R\\ell)\\right)\\right]\\nonumber\n\\\\&&+o(\\varepsilon),\n \\end{eqnarray}\nwhere the terms $o(\\varepsilon)$ correspond to expressions of the type\n$\\varepsilon^2\/R^4$, that converge to 0 in the topology of ${\\cal S}'$, by\npower counting. To evaluate the limit of this expression under\n$\\varepsilon\\ra0$ in ${\\cal S}'$, one proceeds precisely as in\n\\eref{esempio}. In this way the evaluation of the limit of all terms\nabove is reduced to the determination of the invariant integrals of\nappendix A.\n\nLets consider e.g. in the expression above the most singular term,\ncorresponding to $\\varepsilon^2 (R\\ell)\/(uR)^8$, that by power counting\nwould give rise to a double pole $1\/\\varepsilon^2$ in that,\n \\begin{eqnarray}\n\\left(\\varepsilon^2 {(R\\ell)\\over (uR)^8}\\right)(\\varphi) &=& \\varepsilon^2 \\int ds \\int\nd^4x{ 2 \\delta_+ (x^2 -\n\\varepsilon^2)\\over (ux)^7} \\, (x\\ell)\\varphi(y(s)+ x)\\nonumber\\\\\n&=& {1\\over \\varepsilon^2} \\int ds \\int d^4x{ 2 \\delta_+ (x^2 - 1)\\over (ux)^7}\n\\,(x\\ell)  \\varphi(y(s)+\\varepsilon x)\\nonumber\\\\\n&=& \\int ds \\int d^4x{2\\delta_+ (x^2 - 1)\\over (ux)^7}\\,(x\\ell)\n\\left[{1\\over \\varepsilon^2}\\,\\varphi (y(s)) +  {1\\over \\varepsilon} x^\\alpha \\partial_\\alpha\n\\varphi(y(s))\\right.\\nonumber\\\\\n&&\\hskip5truecm+{1\\over 2} \\left.x^\\alpha x^\\beta \\partial_\\alpha\\partial_\\beta\\varphi(y(s)) +\no(\\varepsilon)\\right].\\nonumber\n \\end{eqnarray}\nWe compute the invariant integrals according to appendix A,\n \\begin{eqnarray}\n\\int d^4x{2\\delta_+ (x^2 - 1)\\over (ux)^7}\\,x^\\mu&=&{8\\pi\\over 15}u^\\mu,\\nonumber\\\\\n\\int d^4x{2\\delta_+ (x^2 - 1)\\over (ux)^7}\\,x^\\mu x^\\alpha &=&{\\pi^2\\over 24}\n\\left(7u^\\mu u^\\alpha-\\eta^{\\mu\\alpha}\\right),\\nonumber\\\\\n\\int d^4x{2\\delta_+ (x^2 - 1)\\over (ux)^7}\\,x^\\mu x^\\alpha x^\\beta&=&{4\\pi\\over\n15} \\left(8u^\\mu u^\\alpha\nu^\\beta-\\eta^{\\mu\\alpha}u^\\beta-\\eta^{\\alpha\\beta}u^\\mu-\\eta^{\\beta\\mu}u^\\alpha\\right). \\nonumber\n \\end{eqnarray}\nSince $(u\\ell)=0$ we see that the double pole cancels, and the\nremaining two terms give a $1\/\\varepsilon$ contribution and a finite part.\nAfter integration by parts one obtains,\n$$\n\\left(\\varepsilon^2 {(R\\ell)\\over (uR)^8}\\right)(\\varphi)=\\int \\left({4\\pi\\over\n15}\\,\\ell'^\\mu -{\\pi^2\\over 24\n\\varepsilon}\\,\\ell^\\mu\\right)\\partial_\\mu\\varphi(y(s))\\,ds +o(\\varepsilon),\n$$\ni.e. in ${\\cal S}'$,\n$$\n\\left(\\varepsilon^2 {(R\\ell)\\over (uR)^8}\\right)=-\\partial_\\mu \\int\n \\left({4\\pi\\over\n15}\\,\\ell'^\\mu -{\\pi^2\\over 24 \\varepsilon}\\,\\ell^\\mu\\right) \\delta^4(x-y(s))\\,ds\n+ o(\\varepsilon).\n$$\nAll other terms in \\eref{varfin} by power counting give simple pole\ncontributions and finite parts, too, and a straightforward but a bit\nlengthy calculation gives \\eref{varj}.\n\n\n\n\\vskip 1truecm\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results}\nLet $T$ be a Calder\\'on-Zygmund operator and $(w,\\sigma)$ be a pair of weights. In the Euclidean setting, Hyt\\\"onen and Lacey \\cite{HL} proved that if\n\\[\n[w,\\sigma]_{A_p}:=\\sup_{Q: \\textup{cubes in $\\bbR^n$}}\\frac{w(Q)}{|Q|}\\Big(\\frac{\\sigma(Q)}{|Q|}\\Big)^{p-1}<\\infty\n\\]\nand $w,\\sigma\\in A_\\infty$,\nthen the following estimate holds\n\\begin{equation}\\label{eq:HL}\n\\|T(\\cdot\\sigma)\\|_{L^p(\\sigma)\\rightarrow L^p(w)}\\le C_{n,p,T} [w,\\sigma]_{A_p}^{\\frac 1p}([w]_{A_\\infty}^{\\frac 1{p'}}+\n[\\sigma]_{A_\\infty}^{\\frac 1p}).\n\\end{equation}\nIt is well known that \\eqref{eq:HL} extends the $A_2$ theorem, which was first proved by Hyt\\\"onen \\cite{H}, see also in \\cite{L} for a simple proof by Lerner, and in \\cite{AV}, Anderson and Vagharshakyan also gave a proof in the spaces of homogeneous type. Our goal is to extend \\eqref{eq:HL} with the weak $A_\\infty$ characteristics (which will be introduced below) to the spaces of homogeneous type.\n\nNow let us recall some definitions. By a space of homogeneous type ($SHT$) we mean an ordered triple $(X,\\rho,\\mu)$, where $X$ is a set, $\\rho$ is a\nquasi-metric on $X$, i.e.,\n\\begin{enumerate}\n\\item $\\rho(x,y)=0$ if and only if $x=y$;\n\\item $\\rho(x,y)=\\rho(y,x)$ for all $x,y\\in X$;\n\\item $\\rho(x,z)\\le \\kappa(\\rho(x,y)+\\rho(y,z))$ for some $\\kappa\\ge 1$ and all $x,y,z\\in X$;\n\\end{enumerate}\nand $\\mu$ is a nonnegative Borel measure on $X$ which satisfies the following doubling condition\n\\[\n\\mu(B(x,2r))\\le D \\mu(B(x,r)),\n\\]\nwhere $B(x,r):=\\{y\\in X: \\rho(x,y)<r\\}$ and the dilation of a ball $B:=B(x,r)$ denoted by $\\lambda B$ will be understood as $B(x,\\lambda r)$. We point out that the doubling property implies that  any ball $B(x,r)$  can be covered by at most $N:=N_{D,\\kappa}$ balls of radius $r\/2$.\nNext let us introduce the weak $A_\\infty$ class, which was first introduced by Anderson, Hyt\\\"onen and Tapiola in \\cite{AHT}. For every $\\delta>1$, we say $w$ belongs to $\\delta$-weak $A_\\infty$ class $A_\\infty^\\delta$\nif\n\\[\n[w]_{A_\\infty^\\delta}:=\\sup_{B}\\frac 1{w(\\delta B)}\\int_B M(\\mathbf 1_B w)(y) d\\mu(y)<\\infty,\n\\]\nwhere the supremum is taken over all balls $B\\subset X$. We collect some properties of this weak $A_\\infty$ class and refer the readers to \\cite{AHT} for a proof.\n\\begin{Proposition}\n\\begin{enumerate}\n\\item $A_\\infty^{\\delta}=A_\\infty^{\\delta'}$ for all $\\delta,\\delta'>\\kappa$. So hereafter, we denote by $A_\\infty^{\\textup{weak}}:=A_\\infty^{2\\kappa}$ the weak $A_\\infty$ class;\n\\item For any $w\\in A_\\infty^{\\textup{weak}}$, we have $[w]_{A_\\infty^{\\textup{weak}}}\\ge 1\/{2(2\\kappa)^{\\log_2 N}}$;\n\\item Let $w\\in A_\\infty^{\\rm{weak}}$. Then there exists a constant $\\alpha:=\\alpha(\\kappa, D)$  such that for\nevery $0<\\epsilon \\le \\frac 1{\\alpha [w]_{A_\\infty}^{\\rm{weak}}}$,\n\\begin{equation}\\label{eq:RHI}\n\\bigg(\\dashint_B w^{1+\\epsilon} d\\mu\\bigg)^{\\frac 1{1+\\epsilon}}\\lesssim \\dashint_{2\\kappa B}w d\\mu.\n\\end{equation}\n\\end{enumerate}\n\\end{Proposition}\n\nNow we are ready to state the main result in this paper.\n\\begin{Theorem}\\label{thm:m1}\nGiven $p$, $1<p<\\infty$ and an $SHT(X,\\rho, \\mu)$. Let $T$ be any Calder\\'on-Zygmund operator and $(w, \\sigma)$ be a pair of weights. Then\nwe have\n\\[\n\\|T(\\cdot \\sigma)\\|_{L^p(\\sigma)\\rightarrow L^p(w)}\\le\nC [w,\\sigma]_{A_p}^{\\frac 1p}(([w]_{A_\\infty}^{\\rm{weak}})^{\\frac 1{p'}}+ ([\\sigma]_{A_\\infty}^{\\rm{weak}})^{\\frac 1{p}}),\n\\]\nwhere\n\\[\n[w,\\sigma]_{A_p}:=\\sup_{B: \\textup{balls in $X$}}\\bigg(\\dashint_B wd\\mu\\bigg)\\bigg( \\dashint_{B} \\sigma d\\mu\\bigg)^{p-1}\n\\]\nand the constant $C$ is independent of the weights $(w, \\sigma)$.\n\\end{Theorem}\n\\begin{Remark}\nNote that the result is new already in the case that $X=\\bbR^n$ with Euclidean distance and Lebesgue measure, since\nthe weak $A_\\infty$ class is strictly larger than classical $A_\\infty$ already in this setting.\n\\end{Remark}\n\n\\section{Proof of the Main result}\nIn this section, we will give a proof for Theorem~\\ref{thm:m1}. First, we introduce the bump conditions. By a Young function $\\phi$, we mean that $\\phi: [0, \\infty)\\rightarrow [0, \\infty)$ is continuous, convex\nand increasing satisfying $\\phi(0) = 0$ and $\\phi(t)\/t \\rightarrow \\infty$ as $t \\rightarrow \\infty$. Recall that the\ncomplementary function of $\\phi$, denoted by $\\bar \\phi$, is defined by\n\\[\n\\bar \\phi(t):= \\sup_{s>0}\\{st- \\phi(s)\\}.\n\\]\n Given two Young functions $\\Phi, \\Psi$, define\n\\begin{eqnarray*}\n[w,\\sigma]_{\\Phi, p}&:=&\\sup_B \\bigg(\\dashint_B w d\\mu\\bigg)^{\\frac 1p}\\|\\sigma^{\\frac 1{p'}}\\|_{\\Phi, B},\\\\\n{[\\sigma, w]_{\\Psi, p}}&:=&\\sup_B \\|w^{\\frac 1p}\\|_{\\Psi, B}\\bigg(\\dashint_B\\sigma d\\mu\\bigg)^{\\frac 1{p'}},\n\\end{eqnarray*}\nwhere the supremum is taken over all balls in $X$ and the Luxemburg norm is defined by\n\\[\n\\|f\\|_{\\Phi, B}:=\\inf \\{ \\lambda>0, \\dashint_B \\Phi\\Big(\\frac f \\lambda\\Big) d\\mu \\le 1 \\}.\n\\]\nThere is a famous problem named the separated bump conjecture, which states that for any Calder\\'on-Zygmund operator\n$T$, if\n\\begin{equation}\\label{eq:bump}\n[w,\\sigma]_{\\Phi, p}+[\\sigma, w]_{\\Psi, p}<\\infty,\n\\end{equation}\nwhere $\\bar\\Phi\\in B_p$ and $\\bar\\Psi\\in B_{p'}$ (the $B_p$ condition is recalled in \\eqref{eq:bp} below),\nthen $T(\\cdot\\sigma)$ is bounded from $L^p(\\sigma)$ to $L^p(w)$.\nFor\nthe so-called log-bumps, namely, when\n\\[\n\\Phi(t)=t^{p'}\\log(e+t)^{p'-1+\\delta}\\,\\,\\mbox{and}\\,\\, \\Psi(t)=t^p\\log(e+t)^{p-1+\\delta},\n\\]\nthis conjecture has been verified in \\cite{CRV} in the Euclidean setting and for the spaces of homogeneous type, see \\cite{ACM}. For more about the separated bump conjecture, see \\cite{Lac, NRV} and the references therein.  In the rest of this paper, by carefully calculating the constants, we will show\nthe following quantitative estimate for the power bumps.\n\\begin{Theorem}\\label{thm:m2}\nGiven $p$, $1<p<\\infty$ and an $SHT(X,\\rho,\\mu)$. Let $T$ be a Calder\\'on-Zygmund operator and $(w,\\sigma)$ is a pair of weights satisfying \\eqref{eq:bump} for\n$\\Phi(t)=t^{p'r}$ and $\\Psi(t)=t^{ps}$, where $1<r,s<1+2(2\\kappa)^{\\log_2 N}\/{\\alpha(\\kappa, D)}$. Then we have\n\\[\n\\|T(\\cdot\\sigma)\\|_{L^p(\\sigma)\\rightarrow L^p(w)}\\le C_{T, p, D, \\kappa}([w,\\sigma]_{\\Phi,p} [\\bar\\Phi]_{B_p}^{1\/p} +[\\sigma,w]_{\\Psi,p}[\\bar\\Psi]_{B_{p'}}^{1\/{p'}}\n),\n\\]\nwhere recall that for a Young function $\\phi\\in B_p$,\n\\begin{equation}\\label{eq:bp}\n[\\phi]_{B_p}:=\\int_{1\/2}^\\infty \\frac{\\phi(t)}{t^p}\\frac{dt}{t}.\n\\end{equation}\n\\end{Theorem}\n\n\nNow with Theorem~\\ref{thm:m2} we are ready to prove Theorem~\\ref{thm:m1}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:m1}]\nIn fact, taking\n\\[\nr=1+\\frac 1{\\alpha [\\sigma]_{A_\\infty}^{\\rm{weak}}}.\n\\]\nBy the reverse H\\\"older's inequality \\eqref{eq:RHI}, we have\n\\begin{eqnarray*}\n[w,\\sigma]_{\\Phi,p}&=&\\sup_B \\bigg(\\dashint_B w d\\mu\\bigg)^{\\frac 1p}\\bigg( \\dashint_B \\sigma^{r}d\\mu\\bigg)^{\\frac1{p'r}}\\\\\n&\\lesssim&\\sup_B \\bigg(\\dashint_B wd\\mu\\bigg)^{\\frac 1p}\\bigg( \\dashint_{2\\kappa B} \\sigma d\\mu\\bigg)^{\\frac1{p' }}\\\\\n&\\le& C_{p,D,\\kappa} [w,\\sigma]_{A_p}^{1\/p}.\n\\end{eqnarray*}\nFor $\\Phi(t)=t^{p'r}$, by definition, we know\n\\[\n\\bar \\Phi(t)=t^{(p'r)'}\\Big(\\frac 1{p'r}\\Big)^{\\frac 1{p'r-1}}\\frac{p'r-1}{p'r}\\eqsim_{p,\\kappa, D} t^{(p'r)'}.\n\\]\nHence\n\\begin{eqnarray*}\n [\\bar\\Phi]_{B_p} \\eqsim_{p,\\kappa, D}\\int_{1\/2}^\\infty \\frac {t^{(p'r)'}}{t^p}\\frac {dt}t= 2^{\\frac{(r-1)p}{ p'r-1}}\\frac{ p'r-1}{(r-1)p}\\le C_{p,\\kappa, D} [\\sigma]_{A_\\infty}^{\\rm{weak}} .\n\\end{eqnarray*}\nAnd by taking similar value of $s$ we can get the result as desired.\n\\end{proof}\nIn the rest of this paper, we will focus on the proof of Theorem~\\ref{thm:m2}.\nWe will reduce the estimates for Calder\\'on-Zygmund operators to the so-called sparse operators. So first let us introduce the following result, which can be found in \\cite{HK}, see also in \\cite{C}. Here we follow the version used in \\cite{AHT}.\n\\begin{Theorem}\\label{dyadicsystem}\nLet $0<\\eta<1$ satisfy $96\\kappa^6\\eta\\le 1$. Then there exists countable sets of points $\\{z_\\alpha^{k,t}: \\alpha\\in \\mathscr A_k \\}$, $k\\in \\bbZ$, $t=1,2,\\cdots,K=K(\\kappa, N,\\eta)$, and a finite number of dyadic systems $\\mathscr D^t:=\n\\{Q_\\alpha^{k,t}: \\alpha\\in \\mathscr A_k, k\\in\\bbZ\\}$, such that\n\\begin{enumerate}\n\\item for every $t\\in \\{1,2,\\cdots,K\\}$ we have\n\\begin{enumerate}\n\\item $X=\\cup_{\\alpha\\in \\mathscr A_k}Q_\\alpha^{k,t}$(disjoint union) for every $k\\in\\bbZ$;\n\\item $Q, P\\in \\mathscr D^t$ $\\Rightarrow$ $Q\\cap P=\\{\\emptyset, Q, P\\}$;\n\\item $Q_\\alpha^{k,t}\\in \\mathscr D^t$ $\\Rightarrow$ $B(z_\\alpha^{k,t}, c_1\\eta^k)\\subseteq Q_\\alpha^{k,t}\\subset B(z_\\alpha^{k,t}, C_1\\eta^k) $, where $c_1:=(12\\kappa^4)^{-1}$ and $C_1:=4\\kappa^2$;\n\\end{enumerate}\n\\item for every ball $B=B(x,r)$ there exists a cube $Q_B\\in \\cup_t \\mathscr D^t$ such that $B\\subseteq Q_B$ and $l(Q_B)=\\eta^{k-1}$, where $k$ is the unique integer such that $\\eta^{k+1}<r \\le \\eta^k$ and $l(Q_B)=\\eta^{k-1}$ means that $Q_B=Q_\\alpha^{k-1,t}$ for some indices $\\alpha$ and $t$.\n\\end{enumerate}\n\\end{Theorem}\nBy the doubling property, we know that $\\mu(B(x,r))\\eqsim\\mu (Q_B)$. And if $Q_\\alpha^{k,t}\\subset Q_\\beta^{k-1,t}$, by the doubling property we also know that there exists some constant $C_{\\kappa, D}$ such that\n$$\\mu(Q_\\beta^{k-1,t})\\le \\mu (B(z_\\beta^{k-1,t}, C_1\\eta^{k-1}))\\le C_{\\kappa, D} \\mu(B(z_\\alpha^{k,t}, c_1\\eta^k))\\le\nC_{\\kappa, D}\\mu(Q_\\alpha^{k,t}).$$\n\nTheorem~\\ref{dyadicsystem} characterizes the structure of dyadic system in spaces of homogeneous type. See also in \\cite{HK1} for an exact characterization of which kinds of sets can be dyadic cubes.\nNow with Theorem~\\ref{dyadicsystem} we can get the following result, which was proved in \\cite{ACM}.\n\\begin{Lemma}\\label{lm:dyadicbump}\nGiven a pair of weights $(w,\\sigma)$, and Young functions $\\Phi$ and $\\Psi$.\n\\[\n[w,\\sigma]_{\\Phi,p}\\eqsim \\max_{t\\in \\{1,2,\\cdots, K\\}}[w,\\sigma]_{\\Phi,p}^{\\mathscr D^t},\\qquad\n[\\sigma,w]_{\\Psi,p}\\eqsim   \\max_{t\\in \\{1,2,\\cdots, K\\}}  [\\sigma,w]_{\\Psi,p}^{\\mathscr D^t},\n\\]\nwhere\n\\[\n[w,\\sigma]_{\\Phi,p}^{\\mathscr D^t}:=\\sup_{Q\\in\\mathscr D^t} \\bigg(\\dashint_Q w d\\mu\\bigg)^{\\frac 1p}\\|\\sigma^{\\frac 1{p'}}\\|_{\\Phi, Q}\n\\]\nand $[\\sigma,w]_{\\Psi,p}^{\\mathscr D^t}$ is defined similarly.\n\\end{Lemma}\n\nNow for any fixed $\\mathscr D^t$, $t\\in \\{1,2,\\cdots, K\\}$, we call a family $\\mathcal S\\subset \\mathscr D^t$ sparse\nif for any $Q\\in \\mathcal S$, $\\mu(E(Q))\\ge \\frac 12 \\mu(Q) $, where $E(Q)=Q\\setminus \\cup_{Q'\\in\\mathcal S, Q'\\subsetneq Q} Q'$.\n\nOur purpose is to reduce the estimates for Calder\\'on-Zygmund operators to the following so-called sparse operators,\n\\[\nT^{\\mathcal S} (f)(x):=\\sum_{Q\\in \\mathcal S} \\dashint_Q f(y)d\\mu(y) \\mathbf 1_Q(x),\n\\]\nwhere $\\mathcal S\\subset \\mathscr D$ is a sparse family in some dyadic system $\\mathscr D$.\nIn \\cite{L}, Lerner gave a nice formula which reduces the norm of Calder\\'on-Zygmund operators to such kind of sparse operators. (In the recent book by Lerner and Nazarov \\cite{LN}, it has been shown that Calder\\'on-Zygmund operators can be dominated pointwise by the sparse operators.)  In \\cite{ACM}, the authors showed that Lerner's formula also holds in spaces of homogeneous type.\n\\begin{Lemma}\nGiven an $SHT(X, \\rho, \\mu)$ and a Calder\\'on-Zygmund operator $T$, then\nfor any Banach function space $Y$,\n\\[\n\\|T(f\\sigma)\\|_{Y}\\le C_{D,\\kappa}\\sup_{\\mathscr D^t, \\mathcal S} \\|T^{\\mathcal S}(f\\sigma)\\|_{Y},\n\\]\nwhere the supremum is taken over every dyadic system $\\mathscr D^t$, $t=1,2,\\cdots, K$ and every sparse family $\\mathcal S$ in $\\mathscr D^t$.\n\\end{Lemma}\n\nIn the rest of this paper, we only need to prove Theorem~\\ref{thm:m2} for sparse operators.  We follow the strategy of \\cite{CRV}. We further reduce the problem to estimate testing condition. To be precise, we have the following, see \\cite{LSU} for a proof.\n\\begin{Lemma}\\label{lm:testing}\nFor fixed $t\\in\\{1,2,\\cdots,K\\}$, suppose $\\mathcal S$ is a sparse family in $\\mathscr D^t$. Then\n\\begin{eqnarray}\n\\|T^{\\mathcal S}(\\cdot \\sigma)\\|_{L^p(\\sigma)\\rightarrow L^p(w)}\n&\\le& \\sup_R \\frac{\\| \\sum_{Q\\in \\mathcal S\\atop Q\\subset R} \\dashint_Q \\sigma d\\mu \\mathbf 1_Q(x) \\|_{L^p(w)}}{\\sigma(R)^{1\/p}}\\label{eq:testing}\\\\\n&&\\quad\n+\\sup_R\\frac{\\| \\sum_{Q\\in \\mathcal S\\atop Q\\subset R} \\dashint_Q w d\\mu \\mathbf 1_Q(x)  \\|_{L^{p'}(\\sigma)}}{w(R)^{1\/{p'}}}\\nonumber.\n\\end{eqnarray}\n\\end{Lemma}\nBefore we give further estimates, we introduce the following result. In the Euclidean case this is due to P\\'erez \\cite{P}, and in spaces of homogeneous type, see  P\\'erez and Wheeden \\cite{PW} and Pradolini and Salinas \\cite{PS}. We give the version used in \\cite{ACM}.\n\\begin{Lemma}\\label{lm:maximalbump}\nGiven $p$, $1<p<\\infty$ and an $SHT(X,\\rho,\\mu)$ and a Young function $\\Phi$ such that $\\Phi\\in B_p$, then\n\\[\n\\|M_\\Phi^{\\mathscr D}\\|_{L^p(\\mu)}\\le C_{\\kappa, D}[\\Phi]_{B_p}^{1\/p}\\|f\\|_{L^p(\\mu)},\n\\]\nwhere $\\mathscr D$ is some dyadic system in $X$ and\n\\[\nM_\\Phi^{\\mathscr D} f(x):=\\sup_{Q\\ni x\\atop Q\\in \\mathscr D}\\|f\\|_{\\Phi, Q}.\n\\]\n\\end{Lemma}\nThen by Theorem~\\ref{dyadicsystem} we immediately get\n\\begin{equation}\\label{eq:maximal}\n\\|M_\\Phi\\|_{L^p(\\mu)}\\le C_{\\kappa, D}'[\\Phi]_{B_p}^{1\/p}\\|f\\|_{L^p(\\mu)}.\n\\end{equation}\n\nNow by symmetry we concentrate on the first term of \\eqref{eq:testing}. We follow  the technique introduced in \\cite{HL}, see also in \\cite{CRV}.\nFor convenience, set $\\langle f\\rangle_Q= \\dashint_Q f d\\mu$ and denote\n\\[\n\\mathcal S_a:= \\{ Q\\in \\mathcal S: 2^a<  \\langle w\\rangle_Q  \\langle\\sigma \\rangle_Q ^{p-1} \\le 2^{a+1} \\,\\,\\mbox{and}\\,\\, Q\\subset R \\}.\n\\]\nDenote by $\\mathcal P_0^a$\nthe collection of maximal cubes in $\\mathcal S_a$. Now we define\n\\[\n\\mathcal P_n^a:=\\{\\mbox{maximal cubes}\\,\\, P'\\subset P \\in \\mathcal P_{n-1}^a \\,\\,\\mbox{such that}\\,\\,\nP'\\in \\mathcal S_a, \\langle \\sigma\\rangle_{P'} > 2\\langle \\sigma\\rangle_{P} \\}.\n\\]\nThen denote $\\mathcal P^a:=\\cup_n \\mathcal P_n^a$. For any $P\\in \\mathcal P^a$, set\n\\[\n\\mathcal S_a(P):=\\{Q\\in \\mathcal S_a : \\pi(Q)=P\\},\n\\]\nwhere $\\pi(Q)$ is the minimal cube in $\\mathcal P^a$ which contains $Q$.\n We have the following Lemma, which was proved in \\cite{HL} (see also in \\cite{LM}) in the Euclidean setting and it is still valid for the spaces of homogeneous type with no change of the proof.\n\\begin{Lemma}\\label{decay}\nThere exists a constant $c$ such that\n\\[\nw\\{x\\in P: T^{\\mathcal S_a(P)}(\\sigma)>t  \\langle \\sigma\\rangle_P\\}\\lesssim e^{-ct}w(P),\n\\]\nwhere\n\\[\nT^{\\mathcal S_a(P)} (f)(x):=\\sum_{Q\\in \\mathcal S_a(P)} \\dashint_Q f(y)d\\mu(y) \\mathbf 1_Q(x).\n\\]\n\\end{Lemma}\nNow we are ready to estimate the first term on the right side of \\eqref{eq:testing}.\nWe have\n\\begin{eqnarray*}\n\\Big\\| \\sum_{Q\\in \\mathcal S\\atop Q\\subset R}\\langle\\sigma\\rangle_Q \\mathbf 1_Q(x) \\Big\\|_{L^p(w)}\n&\\le&\\sum_{a}\\Big\\| \\sum_{Q\\in \\mathcal S_a } \\langle\\sigma\\rangle_Q \\mathbf 1_Q(x) \\Big\\|_{L^p(w)} \\\\\n&=&\\sum_{a}\\Big\\|\\sum_{P\\in \\mathcal P^a}T^{\\mathcal S_a(P)}(\\sigma) \\Big\\|_{L^p(w)}.\n\\end{eqnarray*}\nSet\n\\[\nL_{j}^a(P):=\\bigg\\{x: T^{\\mathcal S_a(P)}(\\sigma)(x)\\in [j, j+1)\\langle\\sigma\\rangle_P\\bigg\\}.\n\\]\nBy Lemma~\\ref{decay} we have\n\\begin{eqnarray*}\n\\Big\\|\\sum_{P\\in \\mathcal P^a} T^{\\mathcal S_a(P)}(\\sigma) \\Big\\|_{L^p(w)}\n&\\le&\\sum_{j=0}^\\infty(j+1)\\Big\\|\\sum_{P\\in \\mathcal P^a} \\langle\\sigma\\rangle_P\\mathbf 1_{L_j^a(P)}(x) \\Big\\|_{L^p(w)}\\\\\n&\\lesssim&\\sum_{j=0}^\\infty(j+1)\\Big(\\sum_{P\\in\\mathcal P^a}\\langle\\sigma\\rangle_P^p e^{-cj}w(P)\\Big)^{\\frac1p}\\\\\n&\\lesssim& \\Big(\\sum_{P\\in\\mathcal P^a}\\langle\\sigma\\rangle_P^p w(P)\\Big)^{\\frac1p}.\n\\end{eqnarray*}\nTherefore,\n\\[\n\\Big\\| \\sum_{Q\\in \\mathcal S\\atop Q\\subset R}\\langle\\sigma\\rangle_Q \\mathbf 1_Q(x) \\Big\\|_{L^p(w)}\n\\lesssim\\sum_a\\Big(\\sum_{P\\in\\mathcal P^a}\\langle\\sigma\\rangle_P^p w(P)\\Big)^{\\frac1p}.\n\\]\nWe follow the idea of \\cite{CRV}. Define $\\Phi_0(t)=t^{p'(r+1)\/2}$, set $\\gamma=\\frac1{2(r+1)}$.\nSince $r>1$ and it is dominated by some constant that depends only on the structure constant of $X$, it is easy to check that\n\\begin{eqnarray*}\n[\\bar\\Phi_0]_{B_p}&=&\\int_{1\/2}^\\infty \\frac{\\bar \\Phi_0(t)}{t^p}\\frac {dt}t\\\\\n&\\le&c_{p,\\kappa, D}\\frac{ p'(r+1) -2}{p(r-1)}2^{\\frac{p(r-1)}{ p'(r+1) -2}}\\\\\n&\\le& c_{p,\\kappa, D}' \\frac{ p'r-1}{(r-1)p}2^{\\frac{(r-1)p}{ p'r-1}}=c_{p,\\kappa, D}''[\\bar\\Phi]_{B_p}.\n\\end{eqnarray*}\nNow notice that\n\\[\n\\frac {\\frac{2r+1}{4}}{r}+\\frac 14=\\frac34+\\frac1{4r}<1.\n\\]\nWe have\n\\begin{eqnarray*}\n\\dashint_Q \\sigma^{\\frac {r+1}2}\\le \\bigg(\\dashint_Q \\sigma^{\\frac {2r+1}4\\cdot \\frac r{\\frac{2r+1}{4}}}\\bigg)^{\\frac {\\frac{2r+1}{4}}{r}}\\cdot  \\bigg( \\dashint_Q   \\sigma^{\\frac 14\\cdot 4}     \\bigg)^{\\frac 14}.\n\\end{eqnarray*}\nIt follows that\n\\[\n\\|\\sigma^{\\frac 1{p'}}\\|_{\\Phi_0,Q}\\le \\|\\sigma^{\\frac 1{p'}}\\|_{\\Phi,Q}^{1-\\gamma}\\cdot\\|\\sigma^{\\frac 1{p'}}\\|_{p',Q}^{\\gamma}.\n\\]\nTherefore, by the sparseness and Lemma~\\ref{lm:dyadicbump}\n\\begin{eqnarray*}\n\\sum_{P\\in\\mathcal P^a}\\langle\\sigma\\rangle_P^p w(P)\n&\\le& \\sum_{P\\in\\mathcal P^a} w(P)\\|\\sigma^{\\frac 1 {p'}}\\|_{\\Phi_0,P}^p \\|\\sigma^{\\frac 1p}\\|_{\\bar\\Phi_0, P}^p\\\\\n&\\le& \\sum_{P\\in\\mathcal P^a}\\langle w\\rangle_P\\|\\sigma^{\\frac 1 {p'}}\\|_{\\Phi,P}^{p(1-\\gamma)} \\|\\sigma^{\\frac 1{p'}}\\|_{p',P}^{p\\gamma}\\|\\sigma^{\\frac 1p}\\|_{\\bar\\Phi_0, P}^p \\mu(P)\\\\\n&\\le& [w,\\sigma]_{\\Phi,p}^{p(1-\\gamma)}2^{(a+1)\\gamma} \\sum_{P\\in\\mathcal P^a}\\|\\sigma^{\\frac 1p}\\|_{\\bar\\Phi_0, P}^p \\mu(P)\\\\\n&\\lesssim& [w,\\sigma]_{\\Phi,p}^{p(1-\\gamma)}2^{(a+1)\\gamma} \\int M_{\\bar\\Phi_0}^{\\mathscr D}(\\mathbf 1_R\\sigma^{\\frac 1p})(x)^pd\\mu\\\\\n&\\lesssim& [w,\\sigma]_{\\Phi,p}^{p(1-\\gamma)}2^{(a+1)\\gamma} [\\bar \\Phi_0]_{B_p}\\sigma(R)\\quad\\mbox{(by Lemma~\\ref{lm:maximalbump})}\\\\\n&\\lesssim&[w,\\sigma]_{\\Phi,p}^{p(1-\\gamma)}2^{(a+1)\\gamma}[\\bar\\Phi]_{B_p}\\sigma(R).\n\\end{eqnarray*}\nConsequently,\n\\begin{eqnarray*}\n\\frac{\\Big\\| \\sum_{Q\\in \\mathcal S\\atop Q\\subset R}\\langle\\sigma\\rangle_Q \\mathbf 1_Q(x) \\Big\\|_{L^p(w)}}{\\sigma(R)^{1\/p}}\n&\\lesssim& [w,\\sigma]_{\\Phi,p}^{(1-\\gamma)}[\\bar\\Phi]_{B_p}^{\\frac 1p}\\sum_a 2^{(a+1)\\gamma\/p}\\\\\n&\\le&C_{p, D, \\kappa}[w,\\sigma]_{\\Phi,p}^{(1-\\gamma)}[\\bar\\Phi]_{B_p}^{\\frac 1p} [w,\\sigma]_{A_p}^{\\gamma\/p}\\\\\n&\\le& C_{p, D, \\kappa}[w,\\sigma]_{\\Phi,p} [\\bar\\Phi]_{B_p}^{\\frac 1p} .\n\\end{eqnarray*}\nThis completes the proof.\n\n\\textbf{Acknowledgements}.\\,\\, This work was done while the author was visiting Department of Mathematics and Statistics,\nUniversity of Helsinki. He thanks the Department of Mathematics and Statistics, University of Helsinki and Professor Tuomas P. Hyt\\\"onen for hospitality and support. He thanks Olli Tapiola for the nice talk on the weak $A_\\infty$ class.\nParticular thanks go to Professor Tuomas P. Hyt\\\"onen for carefully reading the paper and many helpful suggestions.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{S1Intro}\nThe first code-based cryptosystem has been introduced by McEliece \\cite{mceliece1978public}. The security of this cryptosystem is based on the general decoding problem, which is known to be NP-complete problem \\cite{Berlekamp1978}. Although at the time of writing this paper, no algorithm running on quantum computers has been published to break the code-based cryptosystems, its large key size and low transmission rate in comparison with the prevalent cryptosystems such as RSA and ElGamal made these cryptosystems unusable from implementation and standard perspectives.\n\nAfter McEliece published his public key code-based cryptosystem \\cite{Berlekamp1978}, Rao in 1984 proposed a secret key cryptosystem based on the McEliece public key cryptosystem \\cite{Rao1984}. In 1986, Rao and Nam made a security modification \\cite{RaoNam1986Crypto}. In 1987, Struik and Tilburg pointed out some weaknesses of the Rao-Nam cryptosystem and proposed an improved version \\cite{StruikTilburg1987}. In 2000, a secret key code-based cryptosystem with much shorter key was introduced \\cite{Barbero2000}.\n\nIn conventional communication systems the encryption and encoding is being performed separately and in series. In 2008, a joint encryption-encoding scheme has been proposed with lower complexity in comparison with separate encryption and encoding has been proposed \\cite{SobhiAfshar2009}. This scheme, using quasi-cyclic structure, succeeded in shortening the key size. Moreover, it gains benefits from the fast decoding algorithms and superior error performance of the LDPC codes. In 2012, another scheme using QC-LDPC codes based on Extended Difference Families (EDF) was proposed \\cite{hooshmand2015improving}, which could not achieve further improvement on the key size. In 2014, a joint encryption-encoding scheme with the novel idea of puncturing, instead of adding a perturbation vector, was introduced \\cite{Esmaeili2014}. Then, in 2015, Esmaeili and Gulliver added random insertions to improve the security of their system \\cite{Esmaeili2015}. In \\cite{Esmaeili2016}, they added an agreed random error vector to their encryption process. In \\cite{Esmaeili2017}, they used random interleaving instead of random insertions and deletions. The cryptosystem in \\cite{Esmaeili2016,Esmaeili2017} is an encoding then encryption system rather than a joint encryption-encoding scheme. Besides, it is shown that the use of two pairs of LFSRs has made Esmaeili-Gulliver cryptosystem vulnerable against ciphertext-only attack \\cite{Lee2017}. A recent work on joint encryption-encoding, which uses polar codes as generator matrix, claimed to achieve a relatively smaller key size, higher level of security, and higher code rate \\cite{mafakheri2017}.\n\nAlthough the key sizes of recent schemes reduced considerably in comparison to the trailblazing code-based studies, the shortest of them is about 13 times larger than the key size of conventional AES-128. Due to this fact, attaining a more compact secret key is one of our motivations of this article. Besides shortening the secret key, increasing the transmission rate, decreasing the computational complexity of algorithm, and efficiently correcting channel errors, as well as keeping the cryptosystem secure are the most challenging issues in joint encryption-encoding researches. Solving these issues need a proper family of codes to be utilized. This code should possess the following characteristics:\n\\begin{itemize}\n\t\\item Efficiently decodable\n\t\\item A family with large cardinality\n\t\\item Achievable high transmission rate\n\t\\item Compressible matrices in the secret key\n\\end{itemize}\n\nIn this paper, we propose a joint encryption-encoding scheme utilizing QC-LDPC codes based on finite geometry (FG-QC-LDPC) in order to obtain our goal, which is introducing a practical solution for the above issues. In finite geometry, every line can be identified through each of two points lying on that. This property enables shortening the key size to 1\/5 of the so far best known joint encryption-encoding scheme. The wide acceptable range in the parameters of our proposed scheme makes it suitable for various applications and different levels of security. Furthermore, we show that the FG-QC-LDPC joint scheme is secure against all known cryptanalyses of such schemes.\n\nThe rest of this paper is organized as follows. Section \\ref{S2Pre} recalls some basic facts about finite geometry and QC-LDPC codes derived from them. Next, the description of our new joint encryption-encoding scheme using FG-QC-LDPC codes is given in Section \\ref{S3scheme}. The security and performance including key size, error performance, and complexity of our scheme are discussed in Section \\ref{S4Sec}. Finally, Section \\ref{S5con} summarizes and concludes the paper.\n\n\\section{Preliminaries}\\label{S2Pre}\nWe took the advantages of FG-QC-LDPC codes to achieve our designated goals, namely improving the performance in comparison to the best known systems in the literature and also keeping the system secure against all known cryptanalyses. The basic definitions of QC-LDPC codes based on finite geometry is summarized in this section.\n\n\\subsection{QC-LDPC}\nIn cryptographic applications, quasi-cyclic LDPC codes allow us to reduce the key size as well as the complexity in comparison with the general LDPC codes \\cite{Baldi2007c}. The parity check matrix of a QC-LDPC code is represented as follows,\n\\begin{equation}\\label{H_QC}\nH=\n\\left[ \\begin{array}{@{}*{3}{c}@{}}\nH_{0,0} & \\dots  & H_{0,n_0-1} \\\\\n\\vdots & \\ddots & \\vdots \\\\\nH_{r_0-1,0} & \\dots  & H_{r_0-1,n_0-1}\n\\end{array}\\right]\n\\end{equation}\nwhere each $H_i$ is a circulant block of size $p\\times p$. \n\nThere are different families of QC-LDPC codes used in code-based cryptography, namely, the Extended Difference Family (EDF) \\cite{Baldi2007c,hooshmand2015improving}, and the Random Difference Family (RDF) \\cite{Baldi2007a,Baldi2013improving}. In our scheme we propose using finite geometry to construct circulant blocks of the parity check matrix. This helps us to attain a shorter secret key than which is available in the literature for joint encryption-encoding schemes.\n\n\\subsection{Finite Geometry}\nA finite geometry is composed of finite number of points. In this paper we focus on two types of finite geometry, namely projective geometry (PG) and Euclidean geometry (EG). The definitions of this subsection are generally provided from \\cite{lin2004error} and \\cite{Ryan2009}.\n\n\\subsubsection{Euclidean Geometry}\n\\begin{definition}\\label{def1}\n\t(Euclidean geometry). All $m$-tuples $(a_0,a_1,\\dots,a_{m-1})$ with $a_i$ from $GF(q=p^s)$ where $p$ is prime and $s$ is a natural number form a vector space. This vector space is also known as the finite Euclidean geometry of dimension $m$ over $GF(q)$, denoted by $EG(m,q)$.\n\tVector additions and scalar multiplications of these $m$-tuples are conducted in $GF(q)$.\n\\end{definition}\n\\begin{definition}\\label{def2}\n\t(point). Each $m$-tuple $\\mathbf{a}=(a_0,a_1,\\dots,a_{m-1})$ represents a point in $EG(m,q)$.\n\\end{definition}\n\n\\begin{definition}\\label{def3}\n\t(origin). The all-zero $m$-tuple $\\mathbf{0}=(0,0,\\dots,0)$ is called the origin.\n\\end{definition}\n\n\\begin{definition}\\label{def4}\n\t(line). The set of $\\{\\mathbf{a_0}+\\beta \\mathbf{a}|\\beta \\in GF(q),\\mathbf{a}\\neq\\mathbf{0}\\}$ is a line, which is composed of $q$ points.\n\\end{definition}\nThe number of points in $EG(m,q)$ is equal to the number of all $m$-tuples i.e. $n=q^m$. For every two distinct points there exists exactly one line connecting them. The number of lines intersecting at each point can be obtained by dividing the number of possible second points of that line by the number of other points in each line.\n\\begin{equation}\n\\gamma =\\frac{q^m-1}{q-1}\n\\end{equation}\nTherefore, the number of lines in the $EG(m,q)$ is as follows,\n\\begin{equation}\nJ=q^{m-1}\\frac{q^m-1}{q-1}\n\\end{equation}\n\\subsubsection{Euclidean Geometry Without Origin}\nBy omitting the origin and all lines intersecting at the origin, a new geometry appears which is denoted by $EG^*(m,q)$.\n\nIf $\\alpha$ is primitive in $GF(q^m)$, then $\\alpha^i$ for $0\\leq i \\leq q^m-2$ represents the elements of $GF(q^m)$. So the incident vector of line $F$ is as given below,\n\\begin{equation}\nV_F=(v_0,v_1,\\dots,v_{q^m-2})\n\\end{equation}\nwhere $v_i=1$ if the line $F$ intersects at point $\\alpha^i$, otherwise $v_i=0$.\n\nIn this geometry, a circularly shifted incident vector of a line is an incident vector for another line \\cite{Ryan2009}. This property partitions the set of all lines,  $J_o$,  into $\\mathcal{N}_{c,EG^*}$ cyclic classes;\n\\begin{equation}\n\\mathcal{N}_{c,EG^*}=\\frac{J_o}{n}=\\frac{q^{m-1}-1}{q-1}.\n\\end{equation}\n\nThese cyclic classes enable us to generate circulant blocks for parity check matrices of QC-LDPC codes. All the necessary information of Euclidean geometry is summarized in Table \\ref{table_EG}.\n\n\\begin{table}[!t]\n\t\\renewcommand{\\arraystretch}{2}\n\t\\caption{Parameters of the Euclidean Geometry}\n\t\\label{table_EG}\n\t\\centering\n\t\n\t\\begin{tabular}{|M{4.5cm}||D|H|}\n\t\t\\hline\n\t\t\\bfseries \\centering Parameters & \\bfseries $\\mathbf{EG(m,q)}$ & \\bfseries  $\\mathbf{EG^*(m,q)}$\\\\[3pt]\n\t\t\\hline\\hline\n\t\t\\bfseries Field & $GF(q)$ & $GF(q)$  \\\\\n\t\n\t\t\\bfseries  Dimension & $m$ & $m$  \\\\\n\t\n\t\t\\bfseries  No. of points & $n=q^m$ & $n=q^m-1$  \\\\\n\t\n\t\t\\bfseries  No. of lines & $J=q^{m-1}\\frac{q^m-1}{q-1}$ & $J_o=\\frac{(q^{m-1}-1)(q^m-1)}{q-1}$  \\\\\n\t\n\t\t\\bfseries  No. of points in each line & $\\rho=q$ & $\\rho=q$  \\\\ \n\t\n\t\t\\bfseries  No. of lines intersecting at each point & $\\gamma =\\frac{q^m-1}{q-1}$ & $\\gamma =\\frac{q^m-1}{q-1}-1$  \\\\\n\t\n\t\t\\bfseries  No. of cyclic classes & ----- & $\\mathcal{N}_{c,EG^*}=\\frac{J_o}{n}=\\frac{q^{m-1}-1}{q-1}$  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Projective Geometry}\nConsider the Galois field $GF(q^{m+1})$ and $\\alpha$, a primitive element in this field. So $\\alpha^0,\\alpha^1,\\dots,\\alpha^{q^{m+1}-2}$ constitute all non-zero elements of $GF(q^{m+1})$. Let $n=\\frac{q^{m+1}-1}{q-1}$ and $\\beta=\\alpha^n$. Then the order of $\\alpha$ is $q-1$. Now, $0,\\beta^0,\\beta^1,\u2026,\\beta^{q-2}$ form the elements of $GF(q)$. Considering the definition of $\\alpha$ and $\\beta$, all non-zero elements of $GF(q^{m+1})$ can be partitioned into $n$ disjoint subsets as shown below,\n\\setlength{\\arraycolsep}{0.0em}\n\\begin{eqnarray}\\label{alpha_beta}\n&(\\alpha^0)\\triangleq \\{ \\alpha^0,\\beta\\alpha^0,\\beta^2\\alpha^0,\\dots,\\,\\beta^{q-2}\\alpha^0 \\} \\nonumber \\\\ \n&(\\alpha^1)\\triangleq \\{ \\alpha^1,\\beta\\alpha^1,\\beta^2\\alpha^1,\\dots,\\,\\beta^{q-2}\\alpha^1 \\} \\nonumber \\\\ \n& {}\\:\\vdots  \\nonumber \\\\\n&(\\alpha^{n-1})\\triangleq \\{ \\alpha^{n-1},\\beta\\alpha^{n-1},\\beta^2\\alpha^{n-1},\\dots,\\,\\beta^{q-2}\\alpha^{n-1} \\}\n\\end{eqnarray}\n\\setlength{\\arraycolsep}{5pt}\nEach of the above subsets represent a distinct point in projective geometry, which is denoted by $PG(m,q)$. In this geometry, each line consists of $q+1$ points, formed by linear combination of two distinct $\\alpha^{j_1}$ and $\\alpha^{j_2}$ points;\n\\begin{eqnarray}\\label{line_def}\n(\\eta_1 \\alpha^{j_1} + \\eta_2 \\alpha^{j_2})  & ; &  \\eta_i \\in GF(q).\n\\end{eqnarray}\nThe number of lines intersecting at every particular point is $\\gamma = \\frac{n-1}{q} = \\frac{q^m-1}{q-1}$, which is obtained by dividing the remaining number of points chosen as the second point of line $(=n-1)$ by the number of other points in each line $(=q)$.\n\nLet $F=(\\eta_1 \\alpha^{j_1} + \\eta_2 \\alpha^{j_2});\\eta_i \\in GF(q)$ be a line in $PG(m,q)$, then for all $0\\leq i < n$, $\\alpha^i F$ is also a line in $PG(m,q)$. The $\\alpha^i F$ is called the $i^{th}$ circular shift of line $F$.\n\nIf $m$ is even, all lines in $PG(m,q)$ have primitive incident vector and partitioned into $\\mathcal{N}_{c,even}=  \\frac{q^m-1}{q^2-1}$ cyclic classes, where each cyclic class consists of $n$ lines. If $m$ is odd, only $J_0$ lines of $PG(m,q)$ have primitive incident vector \\cite{Ryan2009}.\n\\begin{eqnarray}\\label{PG_cyclic_class}\nJ_o=\\frac{q(q^{m+1}-1)(q^{m-1}-1)}{(q^2-1)(q-1)}\n\\end{eqnarray}\nThese incident vectors are partitioned into $\\mathcal{N}_{c,odd}$ cyclic classes.\n\\begin{eqnarray}\\label{N_C_odd}\n\\mathcal{N}_{c,odd}=\\frac{q(q^{m-1}-1)}{(q^2-1)}\n\\end{eqnarray}\nTable \\ref{table_PG} summarizes necessary information of the projective geometry.\n\\begin{table}[!t]\n\t\\renewcommand{\\arraystretch}{2}\n\t\\caption{Parameters of the Projective Geometry}\n\t\\label{table_PG}\n\t\\centering\n\t\n\t\\begin{tabular}{|M{7cm}||H|}\n\t\t\\hline\n\t\t\\bfseries \\centering Parameters & \\bfseries Value \\\\[3pt]\n\t\t\\hline\\hline\n\t\t\\bfseries Field & $GF(q)$  \\\\\n\t\n\t\t\\bfseries  Dimension & $m$ \\\\\n\t\n\t\t\\bfseries  No. of field's elements that consist each point & $q-1$  \\\\\n\t\n\t\t\\bfseries  No. of points & $n=\\frac{q^{m+1}-1}{q-1}$  \\\\\n\t\n\t\t\\bfseries  No. of lines & $J=\\frac{n\\gamma}{\\rho}=\\frac{(q^{m+1}-1)(q^m-1)}{(q-1)(q-1)(q+1)}$  \\\\\n\t\n\t\t\\bfseries  No. of points in each line & $\\rho=q+1$   \\\\ \n\t\n\t\t\\bfseries  No. of lines intersecting at each point & $\\gamma =\\frac{q^m-1}{q-1}$  \\\\\n\t\n\t\t\\bfseries  No. of cyclic classes (even m) & $\\mathcal{N}_{c,even}=\\frac{q^{m}-1}{q^2-1}$  \\\\\n\t\n\t\t\\bfseries  No. of cyclic classes (odd m) & $\\mathcal{N}_{c,odd}=\\frac{q(q^{m-1}-1)}{q^2-1}$  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nIn finite geometry, since there is exactly one line connecting two distinct points, no two incident vectors have more than one non-zero elements in the same location. As a result of this property, the girth of QC-LDPC codes based on finite geometry is at least 6.\n\\subsection{FG-QC-LDPC Codes}\nIn our scheme we exploit a QC-LDPC code with one block row of the form\n\\begin{eqnarray}\\label{H_QC_oneblockrow}\nH_{qc}=[H_0 H_1 \\dots H_{n_0-1}].\n\\end{eqnarray}\nIn FG-QC-LDPC codes, as a subset of QC-LDPC codes, the first rows of the circulant blocks are derived from incident vectors of a line in that geometry. Thanks to the geometric construction, each line, and therefore its incident vector, can be identified by only two points lying on that line. This property helps us to shorten the key size. Other details regarding the key size are mentioned in Section \\ref{S4Sec}.\n\nThe number of circulant blocks in the parity check matrix, $n_0$, is limited to the number of cyclic classes in that particular geometry, i.e. $n_0 \\leq \\mathcal{N}_c$. The parity check matrix derived from finite geometry has the following characteristics, which correspond to the parameters given in Tables \\ref{table_EG} and \\ref{table_PG}.\n\\begin{itemize}\n\t\\item \tThe Hamming weight of each row in each circulant block is equal to the number of points lying on each line in that finite geometry.\n\t\\item\tThe size of each circulant block is $p \\times p$, where $p$ is the total number of points in that geometry.\n\t\\item\tNo two columns have more than one common locations of '1's. This is due to the fact that two distinct lines in finite geometry are either disjoint or intersect at only one point.\n\t\\item\tThe Tanner graph contains no length 4 cycles.\n\t\\item   The codeword length is $n=n_0 \\times p$.\n\t\\item \tThe length of message vectors or equally the dimension of the code is $k=(n_0-1)\\times p = k_0 \\times p$.\n\\end{itemize}\n\\section{FG-QC-LDPC Joint Encryption-Encoding Scheme}\\label{S3scheme}\nHere is the description of the proposed joint encryption-encoding scheme based on FG-QC-LDPC codes in three different steps, that is, key generation, encryption-encoding, and decryption-decoding. Then, we discuss the range of suitable parameters for our proposed scheme.\n\\subsection{Key Generation}\\label{Key_gen}\nThe secret key of the joint encryption-encoding scheme is composed of a parity check matrix, $H$, a permutation matrix, $P$, and the seed of the Pseudo Random Number Generator (PRNG). \n\\subsubsection{Parity Check Matrix}\nThe parity check matrix of the FG-QC-LDPC code in Eq.(\\ref{H_QC_oneblockrow}) can be constructed based on either Euclidean geometry or projective geometry. The construction procedure first start with choosing between these two types of geometries and their parameters. According to Section \\ref{S2Pre}, a finite geometry is defined in terms of two parameters, that is, its dimension, $m$, and the corresponding field, $GF(q)$.\n\nAs discussed in Section \\ref{S2Pre}, in both cases of Euclidean and projective geometries all lines are partitioned into different cyclic classes. Each cyclic class in a finite geometry forms a set of rows of a circulant block. Thus, for generating a circulant block $H_i$ one needs to simply specify a cyclic class and then assign only its first row.\n\nIn the case of EG, the number of cyclic classes, according to the Table \\ref{table_EG} is $\\mathcal{N}_{c,EG^*}=\\frac{J_o}{n}=\\frac{q^{m-1}-1}{q-1}$ and the number of lines in each cyclic class is $p=q^m-1$ which is equal to the length of each row vector of circulant blocks.\n\nIn the case of PG, the number of cyclic classes is  $\\mathcal{N}_{c,even}=\\frac{q^{m}-1}{q^2-1}$ or $\\mathcal{N}_{c,odd}=\\frac{q(q^{m-1}-1)}{q^2-1}$, when the dimension of the geometry is even or odd, respectively. Here, the number of lines in each cyclic class is $p=\\frac{q^{m+1}-1}{q-1}$.\n\nTo sum up with the generation of the parity check matrix, we should choose public parameters, that is, the type of geometry, its dimension m, its corresponding field $GF(q)$, and the number of circulant blocks of the matrix, $n_0$. Then each circulant block must be generated in the above fashion.\n\n\\subsubsection{Permutation Matrix}\nIn our scheme, the permutation matrix is a block diagonal matrix in the form of\n\\begin{eqnarray}\\label{P}\nP=\n\\begin{pmatrix}\n\\pi & \\dots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\dots & \\pi\n\\end{pmatrix}_{n_0l \\times n_0l},\n\\end{eqnarray}\nwhere each $\\pi$ block is an $l \\times l$ permutation matrix.\n\\subsubsection{The PRNG Seed}\nIn order to generate a sequence of perturbation vectors $e_P$ we should utilize a PRNG. Thus, in order to use the same sequence as perturbation vectors by the transmitter and the receiver, it suffices they agree on the same seed for the PRNG. The sequence generated by the PRNG is then divided into $(n-k)$-bit vectors, $z$. The perturbation vectors are computed by $e_P=H^{-1}z$, where $H^{-1}$ is the right inverse of $H$. Therefore, the perturbation vector $e_P$ is of length $n$. Different PRNGs can be employed depending on the hardware\/software resources and applications of the joint encryption-encoding scheme.\n\\subsection{Encryption-Encoding}\nFor doing joint encryption-encoding, the transmitter needs to compute the generator matrix $G$ from the parity check matrix $H$. In QC-LDPC codes with a parity check matrix in the form of one block row (see Eq.(\\ref{H_QC_oneblockrow})), the generator matrix can be constructed as given below,\n\\begin{eqnarray}\\label{H_G}\nG=\n\\left(\\begin{matrix} I_k  \\end{matrix}\\middle| \\begin{matrix} \n{( H_{n_0-1}^{-1}H_0)}^T\\\\ \\vdots \\\\ {( H_{n_0-1}^{-1}H_{n_0-2})}^T\n\\end{matrix}\\right).\n\\end{eqnarray}\nNote that for $G$ being used as the generator matrix, it is sufficient for at least one circulant block, $H_i$, to be non-singular. Without loss of generality, we assume that the circulant block $H_{n_0-1}$ is a non-singular matrix. \n\nNext, the transmitter generates a perturbation vector $e_P$ as given below,\n\\begin{eqnarray}\ne_P=H^{-1} z,\n\\end{eqnarray}\nwhere $z$ is an $(n-k)$-bit vector produced by the PRNG and the right inverse of parity check matrix is computed through a public algorithm such as given in \\cite{SobhiAfshar2009}.\n\nFinally, the ciphertext is obtained as follows,\n\\begin{eqnarray}\nc=(mG+e_P)P\n\\end{eqnarray}\n\\subsection{Decryption-Decoding}\nWe assume that the error vector $e$ is added to the ciphertext through a noisy channel between the transmitter and the receiver. Thus, we denote the received vector by\n\\begin{eqnarray}\\label{r_vec}\nr=c+e=(mG+e_p )P+e.\n\\end{eqnarray}\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[height=7cm]{block_diagram2.eps}\n\t\\caption{Block diagram of the proposed encryption-encoding \/ decryption-decoding scheme.}\n\t\\label{fig_flowchart}\n\\end{figure*}\n\nThis algorithm works as follows:\n\\begin{enumerate}\n\t\\item\tFind the inverse permutation, $P^{-1}$.\n\t\\item \tMultiply both sides of (\\ref{r_vec}) by $P^{-1}$.\n\t\\begin{eqnarray}\\label{r_prime}\n\tr'=rP^{-1}=mG+e_p+eP^{-1}\n\t\\end{eqnarray}\n\t\\item\tSubtract the perturbation vector $e_P$ from $r'$\n\t\\begin{eqnarray}\\label{c_prime}\n\tc'=r'-e_P=mG+eP^{-1}\n\t\\end{eqnarray}\n\t\\item\tDecode $c'$ using a belief propagation algorithm to find $m$.\n\\end{enumerate}\nNote that the $e'=eP^{-1}$ has the same Hamming weight as $e$.\n\nFig. \\ref{fig_flowchart} shows block diagram of the joint encryption-encoding\/decryption-decoding algorithms.\n\\subsection{The Code Parameters}\\label{Code_param_subsec}\nTo deploy a fitting EG-QC-LDPC or PG-QC-LDPC code, length, rate, and density of the parity check matrix should be chosen properly. Our search results reflect the parameter values for different codes and we have summarized some suitable codes in Tables \\ref{EG_codes} and \\ref{PG_codes}. Although parameters of any particular future usage may need a value out of this range, we selected only those within the range of existing standards as samples.\n\n\\subsubsection{Code Rate}\nThe code rate of QC-LDPC codes with one block row is\n\\begin{eqnarray}\nR=\\frac{k}{n}=\\frac{k_0p}{n_0p} = \\frac{n_0-1}{n_0}.\n\\end{eqnarray}\nIn different communication standards, code rates vary from 1\/5 in DVB-S2 \\cite{etsi2015dvbs2} to 14\/15 in IEEE 802.15.3c \\cite{IEEE802.15.3c}. \n\\subsubsection{Code Length}\nThe code lengths of EG-QC-LDPC and PG-QC-LDPC are as follows using the parameters of Tables \\ref{table_EG} and \\ref{table_PG}:\n\\begin{eqnarray}\nn_{EG}=n_0 p_{EG}=n_0(q^m-1),\n\\end{eqnarray}\n\\begin{eqnarray}\nn_{PG}=n_0 p_{PG}=n_0(\\frac{q^{m+1}-1}{q-1}).\n\\end{eqnarray}\nSimilarly, code lengths in different standards bound our search for suitable parameters from 336 bits in ITU-T G9960 \\cite{Itu-TG.9960std} to 64800 bits in DVB-S2 \\cite{etsi2015dvbs2}.\n\\subsubsection{Parity Check Matrix Density}\nA parity check matrix of density 0.01 or lower is categorized as a low density parity check matrix. LDPC codes of density about 0.001 have better error performance \\cite{Baldi2013opt}. The density of the parity check matrices of EG-QC-LDPC and PG-QC-LDPC codes are given below, respectively,\n\\begin{eqnarray}\nr_{EG}=\\frac{\\rho}{p}=\\frac{q}{q^m-1}\n\\end{eqnarray}\n\\begin{eqnarray}\nr_{PG}= \\frac{\\rho}{p}= \\frac{q+1}{\\frac{q^{m+1}-1}{q-1}}=\\frac{q^2-1}{q^{m+1}-1}\n\\end{eqnarray}\nwhere $\\rho$ is the Hamming weight of the incident vector of each line in the geometry and $p$ is its length.\n\n\\section{Security and Performance}\\label{S4Sec}\nIn order to evaluate a joint encryption-encoding scheme, also known as a secure channel coding scheme, we investigate the scheme from security and efficiency perspectives, namely, key size, error performance, and complexity of the scheme. Our goal in design of the FG-QC-LDPC joint encryption-encoding scheme is to decrease the key size as well as complexity of the scheme while improving error performance in comparison with the so far best previous schemes in the literature. In addition, keeping it secure against all known cryptanalytic attacks.\n\\subsection{Security}\nProvable security for symmetric key cryptography is an open problem. There exists no natural hard problem that the security of the symmetric scheme can be reduced to. By the way, to examine symmetric schemes there is a method to reduce the security of that scheme to the problem of distinguishing between an oracle which encrypts a message with a random key and an oracle which outputs a random ciphertext \\cite{Dent2006}. This reduction in oracle model for chosen-plaintext attack on symmetric key cryptosystems is given in \\cite{bellare1997concrete}, which is applicable to analyze the security of modes of operations using a secure block cipher. Besides, Menezes in his talk argued against the role of provable security as a real ``proof\". He claimed that provable security is a ``tool\" and old-fashioned cryptanalysis is more reasonable in practical point of view \\cite{Menezes2012prov}.\n\nBased on the level of a priori knowledge, which is available to the cryptanalyst, there are different classes of cryptanalyses. We have examined our scheme against brute-force, ciphertext-only, message resend, and chosen-plaintext attacks. \n\n\\subsubsection{Brute-Force Attack}\nThe secret key consists of the parity check matrix, H, the permutation matrix, P, and the seed of the PRNG, S. Each of these parameters must be chosen large enough in order to keep our scheme secure against brute-force attacks.\n\nThe number of parity check matrices of FG-QC-LDPC codes is\n\\begin{eqnarray}\\label{N_FG}\nN_{FG}=p^{n_0-1}\\times \\frac{\\mathcal{N}_c!}{(\\mathcal{N}_c-n_0)!},\n\\end{eqnarray}\nwhere $\\mathcal{N}_c$ is the number of cyclic classes on that geometry. Let $s$ denotes the desired security parameter of our scheme, then in the Eq.(\\ref{N_FG}) we can simply assign the number of blocks, $n_0$, and the block size, $p$, in such a way that satisfy $N_{FG}>2^s$.\n\\begin{table*}[!t]\n\t\\renewcommand{\\arraystretch}{2.1}\n\t\\caption{Parameters of EG-QC-LDPC Codes Designed for the Proposed Scheme}\n\t\\label{EG_codes}\n\t\\centering\n\t\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\t\t\\hline\n\t\t\\bfseries \\centering $n_0$ & \\bfseries $q$ & \\bfseries $m$& \\bfseries $p$& \\bfseries $\\mathcal{N}_c$& \\bfseries $n$& \\bfseries $R$& \\bfseries $r$& \\bfseries $log_2(N_{EG})$\\\\[3pt]\n\t\t\\hline\\hline\n\t\t4 & 5 &6 &15624 &781 &62496\t&0.75 &0.0003\t&80.22 \\\\\n\t\t\\hline\n\t\t5 & 3\t&8\t&6560\t&1093\t&32800\t&0.80\t&0.0005\t&101.18 \\\\\n\t\t\\hline\n\t\t6 & 2  &8\t&255\t&127\t&1530\t&0.8333\t&0.0078\t&81.73\\\\\n\t\t\\hline \n\t\t6 & 3 &6\t&728\t&121\t&4368\t&0.8333\t&0.0041\t&88.87 \\\\\n\t\t\\hline \n\t\t7 & 7  &4\t&2400\t&57\t&16800\t&0.8571\t&0.0029\t&107.65\\\\\n\t\t\\hline \n\t\t8 & 2  &9\t&511\t&255\t&4088\t&0.8750\t&0.0039\t&126.78 \\\\ \n\t\t\\hline\n\t\t9 & 2  &10\t&1023\t&511\t&9207\t&0.8889\t&0.0020\t&160.86\\\\\n\t\t\\hline\n\t\t15 & 2  &10\t&1023\t&511\t&15345\t&0.9333\t&0.0020\t&274.64\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\\begin{table*}[!t]\n\t\\renewcommand{\\arraystretch}{2.1}\n\t\\caption{Parameters of PG-QC-LDPC Codes Designed for the Proposed Scheme}\n\t\\label{PG_codes}\n\t\\centering\n\t\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\t\t\\hline\n\t\t\\bfseries \\centering $n_0$ & \\bfseries $q$ & \\bfseries $m$& \\bfseries $p$& \\bfseries $\\mathcal{N}_c$& \\bfseries $n$& \\bfseries $R$& \\bfseries $r$& \\bfseries $log_2(N_{PG})$\\\\[3pt]\n\t\t\\hline\\hline\n\t\t5\t&4\t&6\t&5461\t&273\t&27305\t&0.8000\t&0.0009\t&90.07 \\\\\n\t\t\\hline\n\t\t6\t&2\t&8\t&511\t&85\t&3066\t&0.8333\t&0.0059\t&83.18  \\\\\n\t\t\\hline\n\t\t6\t&2\t&9\t&1023\t&170\t&6138\t&0.8333\t&0.0029\t&94.32 \\\\\n\t\t\\hline \n\t\t8\t&3\t&6\t&1093\t&91\t&8744\t&0.8750\t&0.0037\t&122.26  \\\\\n\t\t\\hline \n\t\t11\t&2\t&8\t&511\t&85\t&5621\t&0.9091\t&0.0059\t&159.50  \\\\\n\t\t\\hline \n\t\t13\t&5\t&5\t&3906\t&130\t&50778\t&0.9231\t&0.0015\t&233.57  \\\\ \n\t\t\\hline\n\t\t15\t&3\t&7\t&3280\t&273\t&49200\t&0.9333\t&0.0012\t&284.34  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\nConsidering this constraint as well as those described in Section \\ref{Code_param_subsec}, our search for parameters of the suitable codes resulted various examples summarized in Tables \\ref{EG_codes} and \\ref{PG_codes}.\n\nFor the permutation matrix, its block size, $l$, must be chosen as great as $l!>2^s$ condition holds. With this in mind, the block length can be kept considerably short. Giving an example, if $l\\geq33$ then $l!>2^{120}$.\n\nFor the PRNG, the length of the seed must simply be chosen larger than the security parameter, $s$.\n\n\\subsubsection{Ciphertext-Only Attack}\nThe goal of this attack is to recover the plaintext from its ciphertext without any knowledge of the key. In code-based cryptosystems, this is interpreted as decoding a coded message without access to its parity check or generator matrices. To achieve this goal, the adversary needs to solve the \\textit{general decoding problem}, which is known to be NP-hard problem \\cite{Berlekamp1978}. This cryptanalysis was applied on the McEliece-like public-key code-based cryptosystems \\cite{LeeBrickell1988,Becker2012,May1817}, whose public generator matrix is algebraically equivalent to their secret key generator matrix. Since in our symmetric key code-based schemes the parity-check matrix is kept secret, this attack is not feasible on our scheme in polynomial time.\n\n\\subsubsection{Message Resend Attack}\nThe aim of the message resend attack is to find the perturbation vector, $e_P$, used by the transmitter and then recover the message in the following manner. Suppose that the transmitter sends $c_1=(mG+e_{P1} )P$ to the receiver. The attacker, as the man in the middle, alter some bits of $c_1$ such that the receiver receives a false or undecodable vector. Therefore, the receiver has to make a request to the transmitter for resending the message, $m$. This time, the transmitter encrypts the same message using a different perturbation vector $e_{P2}$, as $c_2=(mG+e_{P2})P$. This scenario is called message resend \\cite{berson1997}. In this situation, the attacker has access to two different ciphertexts $c_1$ and $c_2$ of the same message $m$. So the attacker can obtain the following equation and thereby guessing the positions of non-zero entries of $e_{P1}$ and $e_{P2}$.\n\\begin{eqnarray}\nc_1+c_2=(e_{P1}+e_{P2})P\n\\end{eqnarray}\n\nThis attack is only feasible when the perturbation vectors have low Hamming weight. Since the used perturbation vectors in the proposed scheme are generated uniformly at random, it is not feasible to find each of $e_{P1}$  and $e_{P2}$ from $c_1+c_2$. Moreover, the secret permutation matrix, $P$, changes the location of '1's and '0's.\n\nApart from these issues while applying this attack, error correction capability of capacity approaching FG-QC-LDPC codes could obviate the need for resending the message. Because the alterations made by the attacker can be recovered by the error correction code. Thus, the message resend scenario does not occur.\n\n\\subsubsection{Chosen-Plaintext Attack}\n\nThere are two major chosen-plaintext attack scenarios against secret key code-based cryptosystems, namely Struik-Tilburg \\cite{StruikTilburg1987} and Rao-Nam \\cite{RaoNam1989} attacks.\n\n\\paragraph{Struick-Tilburg Attack}\nStruik and Tilburg \\cite{StruikTilburg1987} proposed a chosen-plaintext attack against secret-key code-based cryptosystems. In this attack two plaintexts $m_1$ and $m_2$ are chosen in a way that they are only different on their $i^{th}$ position, i.e. $m_1-m_2=u_i$. As a result, the corresponding ciphertext difference is\n\\begin{eqnarray}\nc_1-c_2=u_i GP+(e_{P1}-e_{P2})P=g'_i+(e_{P1}-e_{P2} )P,\n\\end{eqnarray}\nwhere $g'_i$ is the $i^{th}$ row of the generator matrix $G'=GP$. The attacker will repeat the procedure for the same $u_i$ and different perturbation vectors $e_p$ until a set of all possible ciphertexts of $u_i$ are collected. The cardinality of this set is equal to the number of total perturbation vectors, $N_e$. Doing a brute-force over all perturbation vectors, $g'_i$ is obtained. Repeating the above scenario for all $i$ reveals the whole matrix $G'$.\n\nThe work factor of this attack is of $\\Omega(knN^2_e log_2\u2061(N_e))$ \\cite{RaoNam1989}. Therefore, this attack will be successful only if the set of all perturbation vectors, $N_e$, has small cardinality. In FG-QC-LDPC joint encryption-encoding scheme $N_e=2^{(n-k)}=2^{p}$ and according to Tables \\ref{EG_codes} and \\ref{PG_codes}, $min(p)=255$. So the work factor of this attack is of $\\Omega(1275\\times 1530\\times 2^{510} \\times 255)$, which is dramatically large and therefore the Struik-Tilburg attack is not applicable to the FG-QC-LDPC joint encryption-encoding scheme in polynomial time.\n\n\\paragraph{Rao-Nam Attack}\nRao and Nam \\cite{RaoNam1989} proposed their attack based on the previously mentioned Struik-Tilburg \\cite{StruikTilburg1987} attack. They similarly used chosen-plaintexts $m_1$ and $m_2$ differing only in one position. They noticed that when the perturbation vectors has low Hamming distance the the attacker can use majority voting to estimate $g'_i$ and thereby revealing the whole matrix $G'$. The work factor of this attack, obtained by Rao and Nam \\cite{RaoNam1989}, is $\\Omega(N_e^k)$. Based on Table \\ref{EG_codes} and \\ref{PG_codes}, The minimum work factor of this attack on our proposed scheme is $\\Omega(255^{1275})$. Therefore, the FG-QC-LDPC joint scheme is far more secure to be threatened by this attack.\n\\subsection{Key Size}\nThe secret key of the FG-QC-LDPC joint encryption-encoding scheme as mentioned in Section \\ref{Key_gen} consists of the seed vector for a PRNG ($S$), the parity check matrix ($H$), and the permutation matrix ($P$).\n\\begin{eqnarray}\n|K_s |=|S|+|K_P |+|K_H|\n\\end{eqnarray}\n\nFirst, choosing a suitable PRNG for each application, keeps the size of the seed at a desirable extent. Comparing PRNGs is not in the scope of this paper. However, as pointed in Section \\ref{S1Intro}, it is not recommended to use simple LFSRs based on the reasons mentioned in \\cite{Lee2017}. In our example, we use Sosemanuk-128 stream cipher as an example of PRNG \\cite{Berbain2008}. The size of the seed vector of this PRNG is only 128 bits.\n\nOwing to quasi-cyclic structure of the parity check matrix, storing only the first row of each circulant block of this matrix suffices to create the parity check matrix. Furthermore, thanks to the finite geometry construction of these blocks, the whole first row of each block can be produced by only knowing the location of two ``1\"s on each. Since each row is an incident vector of a line on finite geometry, the two ``1\"s indicate the two points where a line go through them. Thus, these two location numbers can regenerate the line and its incident vector.\n\nWe introduce a practical method to achieve the information theoretic lower bound for storing the first row of each circulant block. In this regard, we need to identify two things, the representative of cyclic class and the number of cyclic right shift to obtain the first row. The following constraints must be considered to assign a unique line as a representative for each cyclic class.\n\n\\begin{enumerate}[i)\n\t\\item \\label{i}The first element of its incident vector must be ``1\".\n\t\\item \\label{ii}The next ``1\" in the incident vector must be located at the nearest possible locations among all lines of the class.\n\\end{enumerate}\nIf the non-zero elements of the incident vector of the representative are $\\alpha^{j_1},\\alpha^{j_2},\\dots ,\\alpha^{j_\\rho}$  , (i) forces that $j_1=0$  and (ii) forces $j_2-j_1<\\min{(j_{i+1}-j_i) ,(j_1-j_\\rho)(mod\\: p)}$. By using this method we only store $j_2$ to indicate the cyclic class. This needs only $\\lceil \\log_2({\\frac{p}{\\rho}})  \\rceil $ bits. Another $\\lceil \\log_2({p})  \\rceil $ bits is needed to indicate the amount of cyclic shift for the first row of each block. As a result, the amount of memory to store each circulant block of FG-QC-LDPC parity check matrix is\n\\begin{eqnarray}\n\\lceil \\log_2({\\frac{p}{\\rho}})  \\rceil + \\lceil \\log_2({p})  \\rceil & bits.\n\\end{eqnarray}\nWhile a permutation in the rows of the parity check matrix makes no difference in the code, we can suppose that the first circulant block (or one of the others) made by the representative without being cyclically shifted. As a result, the whole parity check matrix with one block row and $n_0$ blocks needs the following amount of memory to be stored. \n\\begin{eqnarray}\n|K_H|= n_0 \\times \\lceil \\log_2({\\frac{p}{\\rho}})  \\rceil + (n_0-1) \\times \\lceil \\log_2({p})  \\rceil & bits\n\\end{eqnarray}\n\nThe block diagonal permutation matrix, $P$, in this scheme will be stored in similar way as in \\cite{Barbero2000}. The size of the permutation matrix of the key is as follows. Where $l$ is the length of each block and $l'=2^{\\lfloor\\log_2(l)\\rfloor}$.\n\\begin{eqnarray}\n|K_P|= l(\\log_2 l' +1) -2l'+1\n\\end{eqnarray}\n\nTables \\ref{Key80} and \\ref{Key120} show examples of codes and their key sizes for 80 bits and 120 bits security parameters, respectively.\n\\begin{table}[!t]\n\t\\renewcommand{\\arraystretch}{1.8}\n\t\\caption{The key size of the proposed system with 80 bits security parameter}\n\t\\label{Key80}\n\t\\centering\n\t\n\t\\begin{tabular}{|C|D|H|W|}\n\t\t\\hline\n\t\t\\bfseries \\centering Secret key parts & \\bfseries Proposed parameters & \\bfseries Computational complexity to find key & \\bfseries Key size\\\\[3pt]\n\t\t\\hline\\hline\n\t\tH\t& $q=3, m=6, n_0=6$\t& $2^{88.87}$\t&92 bits \\\\\n\t\t\\hline\n\t\tP\t& $l=26$\t& $2^{88.38}$\t&99\tbits  \\\\\n\t\t\\hline\n\t\tS\t&Sosemanuk-128\t& $>2^{80}$\t&128 bits\\\\\n\t\t\\hline \n\t\t\\multicolumn{3}{|c|}{Total Key size}\t&\\bfseries 319 bits\t \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\begin{table}[!t]\n\t\\renewcommand{\\arraystretch}{1.8}\n\t\\caption{The key size of the proposed system with 120 bits security parameter}\n\t\\label{Key120}\n\t\\centering\n\t\n\t\\begin{tabular}{|C|D|H|W|}\n\t\t\\hline\n\t\t\\bfseries \\centering Secret key parts & \\bfseries Proposed parameters & \\bfseries Computational complexity to find key & \\bfseries Key size\\\\[3pt]\n\t\t\\hline\\hline\n\t\tH\t& $q=3, m=6, n_0=8$\t& $2^{121.56}$\t&126 bits \\\\\n\t\t\\hline\n\t\tP\t& $l=52$\t& $2^{225.58}$\t&249\tbits  \\\\\n\t\t\\hline\n\t\tS\t&Sosemanuk-128\t& $>2^{120}$\t&128 bits\\\\\n\t\t\\hline \n\t\t\\multicolumn{3}{|c|}{Total Key size}\t&\\bfseries 503 bits\t \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nThe key size of the proposed scheme, taking the advantages of FG-QC-LDPC codes, decreases to 319 bits where the key size of the last known similar system was 2272 bits \\cite{Esmaeili2015}. Table \\ref{Key_size} compares the key size of the proposed system with those known similar systems.\n\n\\begin{table}[!t]\n\t\\renewcommand{\\arraystretch}{1.8}\n\t\\caption{Comparison of the key size of the proposed scheme with the preceding schemes}\n\t\\label{Key_size}\n\t\\centering\n\t\n\t\\begin{tabular}{|D|M{3.5cm}|C|}\n\t\t\\hline\n\t\t\\bfseries \\centering Cryptosystem  & \\bfseries Code & \\bfseries Key size \\\\[3pt]\n\t\t\\hline\\hline\n\t\tRao\\cite{Rao1984}\t& C(1024,524)&\t2 Mbits \\\\\n\t\t\\hline\n\t\tRao-Nam\\cite{RaoNam1989}\t&C(72,64)\t&18 Kbits \\\\\n\t\t\\hline\n\t\tStruik-Tilburg\\cite{StruikTilburg1987}\t&C(72,64)\t&18 Kbits\\\\\n\t\t\\hline \n\t\tBarbero-Ytrehus\\cite{Barbero2000}\t&C(30,20) over $GF(2^8)$\t&4.9 Kbits\t \\\\\n\t\t\\hline\n\t\tSobhi Afshar, et al.\\cite{SobhiAfshar2009}\t&C(2044,1024)\t&2.5 Kbits\\\\\n\t\t\\hline\n\t\tHooshmand, et al.\\cite{hooshmand2015improving}\t&C(2470,2223)\t&3.55 Kbits\\\\\n\t\t\\hline\n\t\tEsmaeili, et al.\\cite{Esmaeili2014}\t&C(2048,1536)\t&2191 bits\\\\\n\t\t\\hline\n\t\tEsmaeili, Gulliver\\cite{Esmaeili2015}&\tC(2048,1536)&\t2272 bits\\\\\n\t\t\\hline\n\t\tMafakheri, et al.\\cite{mafakheri2017}&\tC(2048,1781)&\t1611 bits\\\\\n\t\t\\hline\n\t\tThe proposed scheme&\tC(4368,3640)\t&319 bits\\\\\n\t\t\\hline\n\t\t\n\t\\end{tabular}\n\\end{table}\n\\subsection{Error Performance}\nAt the receiver the FG-QC-LDPC code used in our system is decoded by a logarithmic Sum-Product decoder. We took the following considerations to simulate encoding, channel, and decoding processes. In our simulation codewords transmitted via a Quadratic Phase Shift Keying (QPSK) channel with additive white Gaussian noise. The receiver has access to soft information of channel. We compared decoders of 10 and 100 iterations with a Reed-Solomon code in Fig. \\ref{ber}. This figure shows that there is no remarkable improvement in 100 iterations decoding in compare to 10 iterations.\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[height=7cm]{BER.eps}\n\t\\caption{Performance comparison of log-SPA decoder of 10 and 100 iterations with Reed-Solomon code.}\n\t\\label{ber}\n\\end{figure*}\n\n\\subsection{Complexity}\nThere are two separate process which their computational complexity needs to be assessed, encryption-encoding and decryption-decoding processes.\n\\subsubsection{Encryption-Encoding}\nThe complexity of this process can be calculated as follows.\n\\begin{eqnarray}\\label{C_enc1}\n\\mathcal{C}_{Enc}=\\mathcal{C}_{mul}(mG)+\\mathcal{C}_{add}(mG+e_P)+\\\\\n\\mathcal{C}_{mul}(H^{-1}.s)+\\mathcal{C}_{mul}(P)\\nonumber\n\\end{eqnarray}\n\nIn this equation $\\mathcal{C}_{add} (mG+e_P)$ stands for adding two $n$-bit vectors which consume $n$ binary operations. Multiplying a vector by a sparse matrix $a_{1\u00d7n}\u00d7B_{n\u00d7n}$, needs $nw$ binary operations \\cite{Baldi2008}, where $w$ is the Hamming weight of rows of the sparse matrix. Here permutation matrix, $P$, has $w=1$ so $\\mathcal{C}_{mul}(P)=n$.\n\nThe generator matrix, $G$, and the inverse of parity check matrix, $H^{-1}$, are dense matrices and need $kn$ and $(n-k)n$ binary operations respectively. By the way, their quasi cyclic property leads to a 92\\% lower computational complexity in multiplying operations \\cite{Baldi2008}. Therefor, we can conclude that:\n\\begin{eqnarray}\\label{C_enc2}\n\\mathcal{C}_{Enc}=0.08\\times k.n+n+0.08\\times (n-k).n+n\\\\\n=\\frac{0.08n+2}{R}  \\nonumber,\n\\end{eqnarray}\nwhere $R=\\frac{k}{n}$.\n\n\\subsubsection{Decryption-Decoding}\nThe complexity of this process is obtained as follow.\n\\begin{eqnarray}\\label{C_dec1}\n\\mathcal{C}_{Dec}=\\mathcal{C}_{mul}(r\\times P^{-1})+\\mathcal{C}_{add}(r'+e_P)\\\\\n\\mathcal{C}_{mul}(H^{-1}s)+ \\mathcal{C}_{mul}(c'\\times H^T)+ \\mathcal{C}_{SPA}\\nonumber\n\\end{eqnarray}\n\nThe complexity of the Sum-Product Algorithm, as mentioned in \\cite{Baldi2008}, is\n\\begin{eqnarray}\\label{C_SPA}\n\\mathcal{C}_{SPA}=I_{avg}.n[d(8\\rho+12R-11)+\\rho].\n\\end{eqnarray}\nIn this equation $I_{avg}$ is the average number of decoding iterations and $d$ is the number of quantization bits in analog-to-digital converter. Finally letting $I_{avg}=10$ and $d=6$, the number of binary operations for each information bit to be decrypted-decoded is\n\\begin{eqnarray}\\label{C_deck}\n\\mathcal{C}_{Dec\/k}=\\frac{1}{R}(2+n-k+n_0\\rho+490\\rho+720R-110).\n\\end{eqnarray}\nIn this equation it is obvious that the complexity of decryption-decoding algorithm is linearly proportional to the redundancy $(n-k)$. \n\\section{Conclusion}\\label{S5con}\nThis paper introduces a joint encryption-encoding scheme, also known as secure channel coding, using QC-LDPC codes based on finite geometry. We have taken advantage of FG-QC-LDPC codes to shorten the secret key to 20\\% of that of the previously best known similar systems.\n\nThanks to the LDPC codes and their fast iterative decoding, the error performance of the proposed scheme is among the best of the literature. The FG-QC-LDPC joint encryption-encoding scheme is secure against cryptanalyses of such cryptosystems.\n\nThe joint algorithm leads to lower complexity than conventional encryption-then-encoding methods. We have shown that our system can provide reliability and security simultaneously with the lower cost of one joint system rather than two disjoint systems.\n\\section*{Acknowledgment}\n\nThe authors would like to thank anonymous reviewers\nfor their precious and productive comments.\n\n\\bibliographystyle{splncs04e}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nOver the past decades, discrete gravity theories and models have been employed as useful tools in the study of numerous aspects of both classical and quantum gravity. The discrete structures usually provide, on the one hand, formulations which are amenable to numerical investigations \\cite{baumshap,regge} and, on the other hand, a regulator which is particularly convenient for the construction and definition of (UV-finite) quantum gravity models \\cite{rocekwilliams,lollreview,edt,cdt,barrett,spinfoam,newspinfoam}. An auxiliary discretization of a continuum theory can, however, introduce a number of discretization artifacts as well as break continuum symmetries, in particular, the continuum diffeomorphism symmetry of general relativity is generically broken in discrete gravity \\cite{rocekwilliams,lollreview,bd1,dithoe1,loll, hamberwilliamsgauge,morse}. \n\nThe vast majority of research in discrete gravity has been performed in the covariant setting, i.e.\\ by means of formulations directly following from an action. Specifically, in the quantum theory one has thereby benefited from the many numerical methods devised for lattice gauge theories which are based on Euclidean path-integral techniques \\cite{lollreview,edt,cdt}. In contrast to this, efforts to construct canonical formulations of discrete gravity have been few and far between; first attempts \\cite{friedmann} were based on a discretized space, but a continuous time. However, consistency requires that the canonical theory reproduces exactly the dynamics and (possibly broken) symmetries of the covariant formulation, i.e.\\ solutions of the canonical theory must replicate those following directly from the action. In particular, a canonical framework consistent with the discrete action ought to yield a discrete time evolution. A first consistent canonical formulation of simplicial gravity\\footnote{By `simplicial gravity' we mean a gravitational theory based on simplices and triangulations, as opposed to the more general term `discrete gravity' which may encompass also other discretization schemes.} has only appeared recently \\cite{bd1,dithoe1} and was based on ideas from the `consistent discretizations program'  \\cite{gambini} and an evolution scheme presented in \\cite{commi}. However, this canonical evolution scheme was only applicable to a special class of triangulations with both fixed topology and connectivity of the `spatial' triangulated hypersurfaces and no consistent canonical formulation of simplicial gravity has been constructed in full generality thus far.\n\nOne of the major obstacles in the quest of a general canonical formulation of simplicial gravity has been the issue of foliating a $D$-dimensional covariant solution by $(D-1)$-dimensional triangulated hypersurfaces which are generically comprised of different numbers of subsimplices which, in turn, carry the variables of the theory. The dimension of configuration or phase spaces associated to such discrete slices must therefore vary and a canonical evolution scheme thereby calls for mappings between phase spaces of different dimension. \n\nWhy should one attempt to construct a general formulation of canonical simplicial gravity in the first place? The general answer is that classically this provides a better notion of discrete dynamics and time evolution, and, furthermore, in quantum gravity one is interested in, e.g.\\ transition amplitudes between different three-geometries which generally requires to be able to identify the corresponding quantum states as elements of a Hilbert space of three-geometries. But there are further practical reasons why it appears useful to be able to treat a changing\/evolving lattice with varying numbers of physical and gauge degrees of freedom in a canonical framework. For instance: (i) There are plenty of physical situations whose description may be facilitated via an adaptation of the discretization in time. For (discrete) gravity the prime example is an expanding or contracting universe, while a more extreme example is the `no boundary' proposal \\cite{hawking}. (ii) Such a framework may foster the scope of numerical implementations. (iii) It may be advantageous for quantization and help in comparing or even connecting various approaches to quantum gravity. Most of the latter can be roughly divided into two categories, namely canonical continuum quantizations (e.g.\\ Loop Quantum Gravity \\cite{thiemannal,rovellibuch}) and covariant path integral formulations, regularized by a discrete lattice structure (e.g.\\ Causal Dynamical Triangulations \\cite{cdt}, Spin Foams \\cite{spinfoam,newspinfoam,rovellibuch}, Quantum Regge Calculus \\cite{rocekwilliams}...). A link between such approaches remains elusive thus far.\\footnote{With the exception of the Loop-Quantum-Gravity- and Spin-Foam-formulations of the topological 3D-theory which were shown to be equivalent \\cite{perez}.} It seems that a consistent canonical framework for simplicial gravity is a prerequisite for at least comparing these two categories, especially in view of Loop Quantum Gravity, on the one hand, which involves changing `spatial' graphs, and Spin Foams, on the other hand, which are based on triangulations.\n\nMotivated by these potential benefits, it is the goal of the present article to overcome the technical obstacles first at the classical level by devising a canonical (discrete) evolution scheme for triangulations which\n\\begin{enumerate}\n\\item is general and applicable to arbitrary triangulations (of fixed `spatial' topology, but varying `spatial' hypersurfaces) satisfying the equations of motion following from an action, \n\\item is local (i.e., for practical reasons, one may restrict oneself to a finite number of equations), and\n\\item may be interpreted entirely from the perspective of the $(D-1)$-dimensional hypersurfaces.\n\\end{enumerate}\nThe underlying basic idea is to build up a $D$-dimensional triangulation (satisfying the equations of motion) step by step by gluing at every discrete evolution step one single $D$-simplex onto the $(D-1)$-dimensional hypersurface of the previous piece of triangulation. Backward evolution is analogously obtained by removing single $D$-simplices from the bulk triangulation. In such a way --- in analogy to the evolution of hypersurfaces in canonical general relativity --- a `spatial' triangulated hypersurface evolves in a discrete `multi-fingered' or `bubble' time through the full D-dimensional solution. As we shall see, the gluing\/removal moves which we permit can be interpreted within the hypersurfaces as Pachner moves, which are an elementary and ergodic class of topology preserving moves, mapping between any triangulations of the same topology by finite sequences. This local evolution scheme has to be constructed consistently in such a way that the equations of motion are satisfied at every step and requires the discrete action to be additive. The latter is fulfilled for a particular discretization of general relativity, namely length Regge Calculus, for which we will develop this evolution scheme here in 3D and 4D. However, the scheme is more general and, through a suitable change of variables, applicable to other simplicial gravity theories (with additive action) as well. \n\nIn order to incorporate these ideas in a canonical framework, we must first of all extend the theory of discrete dynamics from the regular \\cite{marsdenwest} (i.e., for systems without constraints) to the irregular case where constraints necessarily occur (see also \\cite{gambini}), before applying this to simplicial gravity. The central idea of our canonical framework is to resort to Hamilton's principal function as a generating function for a canonical time evolution; in our irregular case, this time evolution map will only be defined on and between constraint surfaces. Additionally, equations of motion transform naturally into certain canonical constraints which permit to suitably extend phase spaces, associated to hypersurfaces in such a way as to implement the Pachner moves as canonical transformations. Because of the {\\it a priori} (i.e.\\ prior to extension) varying phase space dimensions, the space of initial data corresponding to a given hypersurface cannot, in general, correspond to the space of solutions (modulo gauge) and we generically have to expect a high degree of non--uniqueness. Indeed, as we shall discuss in more detail in the main body of this paper, not every Pachner move will invoke equations of motion and rather allow for data that can be {\\it a priori} freely chosen at that step, but may become fixed {\\it a posteriori} by additional constraints arising in subsequent moves. Nevertheless, the end result is a general and fully consistent formulation of canonical Regge Calculus.\n\n\nThis new canonical framework can also be viewed as a method to generate solutions and is, furthermore, susceptible to quantization. In particular, it admits the advantage that, by using the action as a generating function, the eventual canonical quantum theory should directly correspond to the path-integral formulation based on the same discrete action.\n\n\n\nThe rest of the article is organized as follows: section \\ref{candisc} provides the basic ideas to deal with systems with changing phase space dimensions and summarizes all necessary facts for generating functions of relevant canonical transformations. Subsequently, in section \\ref{Pachner} we outline the elementary evolution scheme for triangulations which amounts to Pachner moves in the hypersurfaces and review essential background knowledge of Regge Calculus. Section \\ref{3D} discusses in detail the Regge dynamics and implementation of Pachner moves in 3D, elucidates how to start from a zero-dimensional phase space and exhibits in an example the reproduction of the so--called tent moves in 3D by sequences of the Pachner moves, while section \\ref{4D} investigates the same points in 4D. In order to enhance readability of the article, we decided to move more technical aspects to the end of the manuscript where, after an introduction to the Lagrangian and canonical formalism for regular discrete systems in section \\ref{appreg}, we develop in section \\ref{sing} a general formalism for singular discrete systems, which, in particular, is also applicable to systems with varying phase space dimension. We discuss the relevant symplectic structure and specify in which sense symplectic forms are preserved, before applying this formalism to Pachner moves in section \\ref{apppach}. Section \\ref{intcon} offers a brief interpretation of the constraints arising in the new framework and, finally, section \\ref{conc} contains the conclusions and an outlook to further work. \n\nThe application of the present framework to the 4D linearized theory will appear in a companion paper \\cite{dhta}.\n\n\n\n\n\n\\section{Canonical discrete dynamics}\\label{candisc}\n\n\nThe central idea \\cite{bd1,gambini,marsdenwest, bianca,baez,rovnew} for generating a canonical discrete dynamics is to employ Hamilton's principal function $\\tilde S$ as a generating function for a canonical transformation, \nwhich implements time evolution.\n\nHamilton's principal function $\\tilde S$ is the action $S$ evaluated on solutions. That is, we assume that the action defines a well defined boundary problem, with boundary data given by some configuration variables $x_{ini}$ and $x_{fin}$ associated with initial and final boundaries. (We will assume that we can split the boundary in two parts.) Hamilton's principal function $\\tilde S$ is thus a function of these boundary data $x_{ini}$ and $x_{fin}$. Since it arises by integrating out (i.e.\\ solving for) all the bulk variables, we might also use the term `effective action', instead of Hamilton's principal function.\n\nFurthermore, we assume that the action is additive in an appropriate sense. More precisely, the action associated to a region which is comprised of two regions should be the sum of the actions associated to each of the two regions\n\\begin{eqnarray}\nS(\\{x\\}_{A \\cup B})&=& S(\\{x\\}_{A})+S(\\{x\\}_{B}){\\quad},\n\\end{eqnarray}\nwhere $\\{x\\}_C$ denotes all the dynamical variables associated to the region $C$.\nAs a consequence of the fact that actions normally arise as space-time integrals over Lagrangians, additivity can usually be obtained by taking into account boundary terms.\n\n\n\nIf the regions are such that the boundary variables of $A,B$ are $\\{x_{ini}, x_{inter}\\}$ and $\\{x_{inter},x_{fin}\\}$, respectively, we find the\n following convolution property of Hamilton's principal function\n\\begin{eqnarray}\\label{b1}\n\\tilde S(x_{ini}, x_{fin})= \\text{extr}_{x_{inter}}\\left[   \\tilde S(x_{ini}, x_{inter})+   \\tilde S (x_{inter}, x_{fin})  \\right] {\\quad} .\n\\end{eqnarray}\nHere `extr' indicates that we look for the value of $x_{inter}$ that solves the variational problem of the functional on square brackets in (\\ref{b1}), i.e.\\ a solution to the dynamical problem defined by the action\n\\begin{eqnarray}\\label{b2}\nS(x_{ini},x_{inter},x_{fin}):= \\tilde S(x_{ini}, x_{inter})+   \\tilde S (x_{inter}, x_{fin})  {\\quad} .\n\\end{eqnarray}\nThe property (\\ref{b1}) can be proven by splitting the variational problem over all variables associated to the region $A\\cup B$ (but keeping $x_{ini},x_{fin}$ fixed) into three parts: one involving only the variables inside $A$, the other one only involving the variables inside $B$ and, finally, varying with respect to the boundary data $x_{inter}$.\n\nThe action on the left hand side of (\\ref{b2}) can, in fact, be understood as a discrete action of a problem with two time steps. Usually, we do not have Hamilton's principal function available (which requires the solution of the continuum problem), however, one can approximate it for small time intervals with some discrete (one time step) action, which is a function of initial and final configuration data. Enumerating the time steps with a discrete label $k$, we will denote such a choice by $S_k:=S(x_{k-1},x_{k})$.\n\nIn order to define a Legendre transform and time evolution from such a discrete action, we can appeal to the standard formalism for Hamilton's principal function. Hamilton's principal function is also a generating function of the first kind (i.e.\\ depends on the old and new configuration coordinates) and thereby determines the canonical \n time evolution:\n\\begin{eqnarray}\\label{b3}\n{}^{-}p^{k-1}:= -\\frac{\\partial S_{k}}{ \\partial x_{k-1}}  {\\quad}  , {\\quad}\\q  {}^{+}p^{k}  := \\frac{\\partial S_k}{ \\partial x_{k}}   {\\quad} .\n\\end{eqnarray}\n(Recall that $S_k:=S(x_{k-1},x_{k})$.) \nWe shall refer to the momenta ${}^{-}p$ as {\\it pre--momenta} and to the momenta ${}^{+}p$ as {\\it post--momenta} because a discrete formulation (\\ref{b3}) allows to define two momenta at every time step:\n\\begin{eqnarray}\\label{b4}\n{}^{-}p^{k}:= -\\frac{\\partial S_{k+1}}{ \\partial x_{k}}  {\\quad}  , {\\quad}\\q  {}^{+}p^{k}  := \\frac{\\partial S_{k}}{ \\partial x_{k}}   {\\quad} .\n\\end{eqnarray}\nNote that the requirement ${}^+p^k={}^-p^k$, which we term `momentum matching' implements the equations of motion (in the sense of (\\ref{b1})) for the variables $x_k$. Or, conversely, the equations of motion implement a momentum matching of forward and backward momenta such that there are unique momenta for the variables at step $k$. Henceforth, we will often omit the superindices $+$ and $-$ at the momenta, implicitly assuming that momentum matching holds.\n\n\n\n\nAccordingly, we can define two Legendre transformations from $\\mathcal Q_{k-1}\\times\\mathcal Q_k$ to the phase spaces $\\mathcal P_{k-1}$ and $\\mathcal P_k$ \\cite{marsdenwest}, where $\\mathcal Q_k$ is the configuration space at step $k$,\n\\begin{eqnarray}\\label{b5}\n&&\\mathbb{F}^+S_k:  (x_{k-1},x_k) \\mapsto (x_k,{}^{+}p^k)\\,=\\,\\left(x_k,  \\frac{\\partial S_k}{ \\partial x_{k}}\\right) \\nonumber\\\\\n&&\\mathbb{F}^-S_k:  (x_{k-1},x_k) \\mapsto (x_{k-1},{}^{-}p^{k-1})\\,=\\,\\left(x_{k-1},  -\\frac{\\partial S_k}{ \\partial x_{k-1}}\\right) {\\quad} .\n\\end{eqnarray}\nThese Legendre transformations are, in general, not isomorphisms because in the discrete we will allow for the general situation $\\dim\\mathcal Q_{k-1}\\neq\\dim\\mathcal Q_{k}$. Constraints in phase space arise if the image of the Legendre transform is not given by the entire phase space, but by a submanifold, which is called the constraint manifold. Notice that here we have two Legendre transformations, consequently, we will encounter two different kinds of constraints, which we will call {\\it pre--} and {\\it post--constraints}. Pre--constraints result from $\\mathbb{F}^-S_k$ and constitute the conditions that the time evolution step from $(k-1)$ to $k$ can take place. (In other words, that momentum matching can be applied at time step $(k-1)$.) Post--constraints, on the other hand, arise from $\\mathbb{F}^+S_k$ and are relations on phase space that are automatically satisfied by the momenta ${}^+p^k$ (for all initial values) after having performed an evolution step from $(k-1)$ to $k$. By momentum matching these post--constraints will provide conditions for the momenta ${}^-p^k$, that is, the momenta for the next evolution step. The symplectic structure in the presence of these constraints will be discussed in detail in section \\ref{app}.\n\n\n\nThe formalism outlined here can be easily applied to a discrete time evolution; at least to the case where space-time can be foliated into disjoint hypersurfaces, which, moreover, carry the same number of variables. However, for a triangulation this need not be the case. Furthermore, we are striving for a local notion of time evolution, i.e.\\ one in which only a small region of a hypersurface is evolved, or pushed forward in time. Hypersurfaces will, consequently, in general overlap. The goal to devise a general discrete time evolution scheme therefore gives us the following problems to tackle:\n\\begin{itemize}\n\\item[(a)]  A time step may involve bulk variables. This will not severely complicate the situation, since the Hamilton--Jacobi formalism, as described here, automatically takes care of this.\n\\item[(b)] Different time steps, i.e.\\ hypersurfaces defining instants of time, may involve coinciding subsets of variables. In other words, two hypersurface with different time labels, may partially overlap. This is an example of the `multi--fingered' time evolution encountered in diffeomorphism invariant theories. This issue can also be handled easily; here the additivity of the action plays an essential role.\n\\item[(c)]  The number of variables may actually differ from time step to time step. As a result, the phase space dimensions change in general and we have to reconsider `canonical evolution'.\n\\end{itemize}\n\n\n\nLet us discuss these issues one by one.\n\n~\\\\\n(a): Assume we can split the configuration variables $x_k$ at step $k$ into two sets: `true boundary variables' $x_k^t$ and `internal (bulk) variables' $x_k^i$. Such a situation can be easily constructed by starting from some elementary evolution scheme with time steps labelled by $k \\in {\\mathbbm Z}$ and then defining an effective evolution scheme where we consider only every $n$--th time step (or consider time steps of different basic lengths) and henceforth label these rearranged steps by a different label $n\\in {\\mathbbm Z}$. That is, for these steps we then have $x_n^t$ and $x_n^i$. For example, figure \\ref{fatslices} depicts a triangulation which has been built up by simplices which can be grouped up into fat slices bounded by non-intersecting hypersurfaces $\\Sigma_n$. Individual simplices are enumerated by $k\\in {\\mathbbm Z}$, while fat slices are enumerated by the new label $n\\in {\\mathbbm Z}$. Now assume this to be a Regge triangulation where the lengths of edges are the configuration variables and assign lengths of edges inside fat slice $n$ and of edges contained in $\\Sigma_n$ to step $n$. Thus, at step $n$ we have the `bulk lengths' $l^i_n$ of edges which reside inside the fat slice $n$, as well as the `true boundary lengths' $l^e_n$ of edges $e\\subset\\Sigma_n$. Consequently, the action associated to fat slice $n+1$ does not depend on $l^i_n$ and we have the dependence $S_{n+1}(l^e_n,l^i_{n+1},l^{e'}_{n+1})$, etc.\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\psfrag{n}{ $n$}\n\\psfrag{n+1}{$n+1$}\n\\psfrag{sn0}{$\\Sigma_{n-1}$}\n\\psfrag{sn}{ $\\Sigma_n$}\n\\psfrag{sn1}{ $\\Sigma_{n+1}$}\n\\psfrag{Sn}{$S_n$}\n\\psfrag{Sn1}{$S_{n+1}$}\n\\includegraphics[scale=.4]{Figures\/figure1.eps}\n\\caption{\\small Schematic illustration of a fat slicing of a triangulation. The triangulation can be built up step by step by single simplices where we count such elementary steps by $k \\in {\\mathbbm Z}$. The elementary steps can be grouped up into fat slices which we now count by $n \\in {\\mathbbm Z}$.}\\label{fatslices}\n\\end{center}\n\\end{figure}\n\\noindent\n\nIn such a situation, we can generally require that the action $S_{n}(x_{n-1},x_{n})$ does {\\it not} depend on the internal variables $x_{n-1}^i$, as these variables only appear between times $(n-2)$ and $(n-1)$. The time evolution equations are then\n\\begin{eqnarray}\\label{b6}\n{}^{-}p^{n-1}_t&:=& -\\frac{\\partial S_{n}}{ \\partial x^t_{n-1}}  {\\quad}  , {\\quad}\\q {\\quad}\\q {}^{+}p^{n}_t  := \\frac{\\partial S_n}{ \\partial x^t_{n}}   \\nonumber\\\\\n{}^{-}p^{n-1}_i&:= &-\\frac{\\partial S_{n}}{ \\partial x^i_{n-1}} \\,=\\,0 {\\quad}  , {\\quad}\\q  {}^{+}p_i^{n}  := \\frac{\\partial S_n}{ \\partial x^i_{n}} {\\quad} .\n\\end{eqnarray}\nHence, we have ${}^-p_i^{n-1}=0$ and, by momentum matching, also  ${}^+p_i^{n-1}=0$ (which may lead to further `secondary' constraints). If the same kind of splitting into internal and boundary variables occurs for all time steps $n$, we will obtain ${}^-p_i^{n}=0$ for all $n$. The equations of motion for the internal variables are then implemented via momentum matching\n\\begin{equation}\n0={}^-p^n_i={}^+p^n_i=\\frac{\\partial S_n}{ \\partial x^i_{n}} {\\quad} .\n\\end{equation}\n This situation will always arise if some class of variables only appears in the action $S_n$ for only one label $n$.\n\nOn the other hand, here we see the first occurrence of a pre--constraint; the constraint hypersurface is defined by $p_i\\equiv 0$. This is a general feature: constraints appear as equations of motion which involve only canonical data from one time step. By redefining the time label of variables (here by summarizing several time steps enumerated by $k$ into one labeled by $n$), one can transform equations of motion into constraints.\n\nThe internal variables can be integrated out and we can define an effective action $\\tilde S_n$ which only depends on the `true boundary' variables $x^t_{n-1}$ and $x^t_{n}$. One can easily show that this leads to an equivalent time evolution for the remaining variables.\n\n\n\n~\\\\\n(b):\nAssume we have the same variables $x^b$ appearing in different time steps,\n for instance $x^b_{k+1} \\equiv x^b_{k}$ for some set of variables $x^b$ (we return here to a counting of time steps by $k$). This generally happens if we implement a local time evolution, i.e.\\ if the hypersurface in every time step only changes in a specific region.\n\nFirst let us consider the case that $x^b$ is not dynamically involved in the time evolution at all, that is, we want to implement the evolution equations\n \\begin{eqnarray}\\label{c1}\n x^b_{k+1}\\,=\\,x^b_k {\\quad},  {\\quad}\\q p_b^{k+1}\\,=\\,p_b^k  {\\quad} .\n \\end{eqnarray}\nIt is not possible to implement these equation by using the action as a generating function of the first kind, as the fact that the $x^b$ are not dynamically involved means that neither $x^b_k$ nor $x^b_{k+1}$ appear in the action $S_{k+1}$. However, for these variables we can use the identity transformation generated either by a generating function of the second (depending on old configuration and new momentum variables) or the third kind (depending on new configuration and old momentum variables):\n \\begin{eqnarray}\\label{c2}\n &&G_2(x^k_b,p^{k+1}_b)\\,=\\,-x^b_k p_b^{k+1} {\\quad},{\\quad}\\q p_b^k\\,=\\,-\\frac{\\partial G_2}{\\partial x_b^k}\\,=\\,p_b^{k+1}{\\quad} ,{\\quad}\\q\n x_b^{k+1}=\\,-\\frac{\\partial G_2}{\\partial p_b^{k+1}}\\,=\\,x_b^k   \\nonumber\\\\\n &&G_3(x^{k+1}_b,p^{k}_b)\\,=\\,\\;\\, x^b_{k+1} p_b^{k} {\\quad},{\\quad}\\q p_b^{k+1}\\,=\\,\\;\\,\\frac{\\partial G_3}{\\partial x^b_{k+1}}\\,=\\,p_b^{k}{\\quad} ,{\\quad}\\q\n x_b^{k}\\,=\\,\\,\\;\\frac{\\partial G_3}{\\partial p^b_{k}}\\,=\\,x^b_{k+1}\\, . {\\quad}\\q\n \\end{eqnarray}\n\nAnother case which will appear is that some configuration variables do not evolve $x^e_{k}=x^e_{k+1}$, while the associated momenta, however, change as either $x^e_k$ or $x^e_{k+1}$ appear in the action $S_{k+1}$ (but not both for the same index $e$). That is, in accordance with (\\ref{b5}), we want to implement either\n \\begin{eqnarray}\\label{c3}\n  p_e^{k}\\,=\\,p_e^{k+1}-\\frac{\\partial S_{k+1}(x_k)}{\\partial x_{k}^e} {\\quad} {\\quad} \\text{or}{\\quad}\\q\n p_e^{k+1}\\,=\\,p_e^k+\\frac{\\partial S_{k+1}(x_{k+1})}{\\partial x_{k+1}^e}{\\quad} ,\n \\end{eqnarray}\nwhich we call `momentum updating'. This can still be implemented via a generating function of the second or the third kind by simply adding either $G_2$ or $G_3$, respectively, to the action $S_{k+1}$. As $S_{k+1}$ either only depends on the old configuration variables $x_k^e$ or only on the new configuration variables $x_{k+1}^e$, the type of the generating function, in fact, does not change. One can check that these generating functions lead to the evolution equations we wished for.\n\n\n\n\n\n\nNote that with these definitions, all momenta at any given time step can also be defined by\n\\begin{eqnarray}\\label{b8}\n{}^{-}p^{k}:= -\\frac{\\partial S_{k-}(x^-_k)}{ \\partial x_{k}}  {\\quad}  , {\\quad}\\q  {}^{+}p^{k}  :=\\frac{\\partial S_{k+}(x^+_k)}{ \\partial x_{k}}   {\\quad} ,\n\\end{eqnarray}\nwhere $S_{k-}(x^-_k)$ denotes the action associated to the region which lies to the future of the hypersurface with label $k$ and $S_{k+}(x^+_k)$ the action associated to the region in the past of this hypersurface (see figure \\ref{hyp}). Here $x^-_k$ denotes all variables associated to $\\Sigma_k$ and the future region, while $x^+_k$ denotes all variables associated to $\\Sigma_k$ and the past of it. Moreover, $x_k$ can be {\\it any} configuration variable.\\footnote{In (\\ref{b8}), if the same variable $x$ appears with multiple time labels $x_k=x_{k+1},\\ldots$, it is understood that the action is expressed only as a function of $x=x_k$.} \n\\begin{figure}[htbp!]\\begin{center}             \n   \\psfrag{S}{ $\\Sigma_k$}\n   \\psfrag{Sf}{$S_{k-}$}\n   \\psfrag{Sp}{\\textcolor{white}{$S_{k+}$}}\n   \\includegraphics[scale=.3]{Figures\/figure2.eps} \n\\caption{\\small Hypersurface $\\Sigma_k$ separating `past' and `future' region at step $k$.}\\label{hyp}\n\\end{center}\n\\end{figure}\n\n\n\n\nThis will also include the case where, say, variable $x$ appears in the past of time $k$, but not in the future. The equation of motion is then automatically implemented by ${}^{-}p^k=0$ and momentum matching. In other words, in general we can actually add variables which are not `true dynamical variables', i.e.\\ which are not associated to the hypersurface $\\Sigma_k$ but rather to the future or past of it. Then one of the associated momenta ${}^{+}p^k$ or ${}^-p^k$ will be zero, thus enforcing the equations of motion by momentum matching ${}^+p^k={}^-p^k$.\n\nThis  allows us to deal with problem \n\n~\\\\\n(c), namely the case where the number of configuration variables and, therefore, the dimensions of the phase spaces associated to different hypersurfaces $\\Sigma_{k-1}$ and $\\Sigma_{k}$ differ. We can formally extend the phase spaces by including any variables into the phase space at time $(k-1)$ which only appear at time $k$ but do not have a natural associated variable at time $(k-1)$, and vice versa.  Then we can have phase spaces of equal dimensions at the two time steps  and we can implement time evolution by a canonical transformation.\n\n\nSay, we have a new variable $x^{n}_{k+1}$ which appears at time step $(k+1)$ but not at time step $k$ (for a similar situation in 3D Regge Calculus see figure \\ref{3dpach}) and the action $S_{k+1}$ only depends on $x^{n}_{k+1}$ but not on $x^{n}_k$. Accordingly, we extend the phase space at time $k$ by the pair $(x_k^n,p^k_n)$ and may use the action as a generating function of the first kind, resulting in the evolution equations\n\\begin{eqnarray}\\label{c4}\n{}^-p^{k}_{n}\\,=\\, -\\frac{\\partial S_{k+1}}{\\partial x^{n}_{k}}\\,=\\,0 {\\quad} ,{\\quad}\\q  {}^+p^{k+1}_{n}\\,=\\, \\frac{\\partial S_{k+1}}{\\partial x^{n}_{k+1}} {\\quad} .\n\\end{eqnarray}\n\n\n\nThe variable $x^n_k$ remains undetermined, for it does not appear at all in the action $S_{k+1}$. We may fix its value to $x^n_k=x^n_{k+1}$, such that it appears as an initial datum for time step $k$, determining the data at time step $k+1$. Furthermore, in (\\ref{c4}), the pre--constraint ${}^-p^k_n=0$ shows up as a constraint on the variable $p^k_n$, which appears at time $k$ for the `first time'. Hence, by momentum matching it does not place any restrictions on the dynamical variables at step $k$. (If one also extended the phase spaces at previous time steps by this variable, the momentum $p_n$ would also be vanishing.) Additionally, we will also encounter a post--constraint: Assuming that $N$ further variable pairs $(x_k^i,p^k_i)$ are involved, the time evolution will map from a $(2N+2)$--dimensional phase space to another $(2N+2)$--dimensional phase space (using that $x^n_k=x^n_{k+1}$). However, this map is not defined on the full phase space, because the pre--constraint $p^k_n=0$ has to hold. As a consequence, the image of the map can be maximally $(2N+1)$--dimensional, implying the occurrence of also a post--constraint. Indeed, in the examples in the sequel it will turn out that the second equation in (\\ref{c4}) only involves variables from time step $(k+1)$ and, hence, constitutes a constraint.\n\n\n\n\n\n\nConversely, the case of an old variable $x^o_k$ that does not have an equivalent at time $(k+1)$ can be treated analogously, that is, we extend the phase space at time $(k+1)$ by the pair $(x_{k+1}^o,p^{k+1}_o)$. This time we obtain a post--constraint ${}^+p^{k+1}_n=0$.\n Likewise, we will also find a pre--constraint, which in the case of the Pachner moves will always be given by\n\\begin{eqnarray}\\label{c4a}\n{}^-p_o^{k}&=&-\\frac{\\partial S_{k+1}}{\\partial x^o_k}  {\\quad} ,\n\\end{eqnarray}\ni.e.\\ this equation will only involve variables from time step $k$.\nVia momentum matching, ${}^-p_o^{k}={}^+p_o^{k}$, this constraint effectuates the equation of motion $\\partial S\/\\partial x^o=0$, as $x^o$ appears in $S_{k+1}$ for the `last time'.\n\n\n\n\nIn the following section, we will implement Pachner moves as time evolution maps in simplicial gravity and, with the help of these examples, explain the concepts encountered here in greater depth.\n\n\n\\section{Pachner moves as time evolution maps}\\label{Pachner}\n\nIn the previous section, we presented a general method to derive a canonical time evolution scheme from a given discrete action. The advantage of this method is that the canonical evolution equations will exactly reproduce the equations of motion of the (discrete) covariant formalism. This is, in general, different from the procedure of first deriving a continuum canonical formalism from the continuum action and subsequently discretizing this continuum canonical formalism \\cite{friedmann}. In the case of general relativity, where diffeomorphism symmetry plays a fundamental role, the latter method usually leads to (classically) anomalous constraint algebras \\cite{lollreview, loll,friedmann}.\n\nA canonical evolution scheme for Regge calculus which reproduces the covariant equations of motion has been presented in \\cite{bd1,dithoe1,bianca}. However, this scheme was only applicable to a special class of triangulations, namely those which can be evolved from a hypersurface by so--called tent moves. These tent moves \\cite{commi} are special local evolution moves that do not change the topology and the triangulation (that is the connectivity of the triangulation) of the `equal time' hypersurfaces. Hence, in this case one does not encounter phase spaces of different dimensions.\n\nIn the present work, we will advance and devise a general canonical framework applicable to arbitrary triangulations (of fixed `spatial' topology) by implementing an elementary and ergodic class of local evolution moves, the so--called Pachner moves \\cite{pachner}. Pachner moves are elementary in that they involve only a fixed number of simplices during each move and are, furthermore, topology-preserving and ergodic, i.e.\\ can map between any triangulations of the same topology by finite sequences. In particular, the tent moves (see sections \\ref{sectent1} and \\ref{tent2}), which can involve an arbitrary number of simplices, can, in fact, be decomposed into Pachner moves.\n\nMore precisely, the evolution moves to be implemented below can be interpreted entirely within the `spatial' hypersurfaces (as local changes) and appear as Pachner moves therein: If we take the $(D-1)$--dimensional hypersurface $\\Sigma$ as the boundary of a $D$--dimensional bulk triangulation, the Pachner moves in $\\Sigma$ arise by gluing a single $D$--simplex to, or removing one $D$--simplex from the bulk triangulation \\cite{pachner}. Notice that here we only allow for gluing processes of top-dimensional simplices which identify faces of one dimension less of a given simplex with faces of equal dimension in the hypersurface (and analogously for removal procedures).\nFor instance, figure \\ref{3dpach} depicts the 3D example of the 1--3 Pachner move in the 2D hypersurface. All other Pachner moves in $(D-1)$ dimensions can be similarly produced by gluings or removals of single $D$-simplices in the $D$--dimensional bulk triangulation.\n\\begin{center}\\begin{figure}[htbp!]\n\\psfrag{sk}{$\\Sigma_k$}\n\\psfrag{sk1}{$\\Sigma_{k+1}$}\n\\psfrag{t}{$t$}\n$\\begin{array}{cccc}\n\\text{3D perspective:}&\\hspace{.5cm}\\includegraphics[scale=.2]{Figures\/figure3.eps} &\\longrightarrow& \\hspace{-.7cm}\\includegraphics[scale=.2]{Figures\/figure4.eps} \n\\vspace{1cm}\\\\ \n\\text{2D perspective:}&\\hspace{-.5cm}\\includegraphics[scale=.2]{Figures\/figure5.eps} \n&\\longrightarrow&\n\\includegraphics[scale=.2]{Figures\/figure6.eps} \\\\\n&& \\text{1--3 Pachner move}&\n\\end{array}$\n\\caption{\\small 3D example: gluing a single tetrahedron onto a single triangle in the 2D boundary hypersurface of a 3D bulk triangulation. From the perspective of the 2D hypersurface this gluing move appears as a subdivision of the triangle $t$. That is, the move appears as a 1--3 Pachner move in the hypersurface.\n\\newline This is a specific example of the situation described earlier where the numbers of variables associated to the two hypersurfaces before and after the new move differ due to the three new edges.}\\label{3dpach}\n\\end{figure}\n\\end{center}    \nThis elementary procedure provides a more compelling connection between the covariant and canonical picture: the discrete evolution of the hypersurface can be reinterpreted as building up the bulk triangulation, and, hence, the (discrete) space-time, step by step by simplices. In analogy to the situation in canonical general relativity, the triangulated hypersurface evolves in a discrete `multi-fingered' (or `bubble') time through the full (discrete) space-time solution. In this light, every gluing move at a given evolution step $k$ which adds a simplex to the `past triangulation' can also be viewed as a removal move which subtracts a simplex from the `future triangulation', and vice versa (see figure \\ref{glueqrem} for a schematic illustration). Formally, this is guaranteed by momentum matching.\n\\begin{figure}[htbp!]\n\\begin{center}\n \\includegraphics[scale=.35]{Figures\/figure7.eps}\n \\end{center}\n \\caption{\\small Removal moves for the `past triangulation' are equivalent to gluing moves for the `future triangulation' and vice versa.}\\label{glueqrem}\n \\end{figure}\n \n\nSince Pachner moves have inverses and, thus, in order to establish a genuine implementation of Pachner moves into the mathematical formalism, we have to ensure that every phase space map ${}^{\\text{Pachner}}f$ corresponding to the action of a Pachner move on a hypersurface has an inverse which corresponds to the action of its inverse move (of course, via the gluing\/removal correspondence applied to the same simplex), i.e.\\ ${}^{\\text{Pachner}}f\\circ{}^{\\text{Pachner}^{-1}}f=\\text{id}$. This, indeed, will be the case.\n\nAs Pachner moves do not change the topology of a triangulation, a $(D-1)$--dimensional hypersurface with topology $\\mathfrak T$  evolved by Pachner moves will lead to a $D$--dimensional triangulation of topology $[0,1]\\times \\mathfrak T$. Hence, just as usual canonical evolution schemes, also the present scheme assumes or implements a non--changing topology of the `equal time' triangulations. This might be advantageous, for instance, for quantization, if one wants to suppress topology changes of spatial hypersurfaces (interpreted as the production of baby universes). Specifically, the approach of Causal Dynamical Triangulations (CDT) \\cite{cdt} shows that this can lead to a much more regular large scale limit than in the case of (Euclidean) Dynamical Triangulations \\cite{edt,dt}, where arbitrary topology changes are allowed for.\\footnote{The evolution moves implemented here yield triangulations akin to the triangulations employed in CDT. The splitting off of baby universes, aka spatial topology changes, on the other hand, would require the implementation of moves distinct from the Pachner moves. For instance, in D dimensions one could allow for a move which identifies all $D+1$ $(D-1)$--faces of a $D$--simplex with $D+1$ $(D-1)$--simplices in a $(D-1)$--dimensional hypersurface. Furthermore, by only allowing top-dimensional simplices to be glued onto faces of one dimension less, we are disallowing singular moves which, e.g., only identify a single vertex of the new simplex with a single vertex of the hypersurface.}\n\nOn the other hand, a Pachner move evolution scheme allows for a change of the number of variables `in time'. There are numerous physical situations, where it might be interesting to adapt the discretization (density) in time, for instance, for an expanding or contracting universe. In fact, we will show that, akin to the `no--boundary' proposal \\cite{hawking}, we can start with an `empty triangulation', obtain the boundary of a $D$--simplex in the next time step and evolve this to a bigger and bigger triangulation. \n\nFurthermore, Pachner moves implement the idea of a `fluctuating lattice' in a controlled manner, that is, the idea that the lattice, or discretization, is not fixed in time (or space--time) but is either summed over or is determined by dynamical considerations. Indeed, we will see that even the classical dynamics may prefer or suppress certain Pachner moves and, hence, determine the evolution of the connectivity of the triangulation itself.\n\n\n\n\\subsection{Regge calculus}\\label{secregcal}\n\nIn this article, we will develop this evolution scheme for a particular discretization of general relativity, namely Regge calculus \\cite{regge,reggewilliams}. However, the basic ideas are general and straightforwardly adaptable to other discretization schemes fulfilling the basic prerequisite of additivity of the action; whenever we glue or remove an elementary building block, i.e.\\ a simplex, we have to add or subtract the corresponding piece of action. \n\nTo begin with, we will review some basic notions of Regge calculus based on length variables, before studying the details of the individual Pachner moves in sections \\ref{3D} and \\ref{4D} below. For formulations using other geometric variables see, for instance, \\cite{dittrichspeziale, Bahr:2009qd}. Standard Regge calculus is based on a fixed triangulation where the basic variables are simply the length variables associated to the edges of this triangulation. These length variables (assuming generalized triangle inequalities are satisfied, which we will always do) specify a piecewise linear geometry for the triangulation in question. The equations of motion are determined by a variational principle from an action, the Regge action, which can be viewed as a discretization of the Einstein--Hilbert action.\n\n\nThe (Euclidean) Regge action (without a cosmological constant term) for\na $D$--dimensional triangulation $T$ with boundary $\\partial T$ and interior\n$T^\\circ:=T\\backslash \\partial T$ is given \\cite{regge,hartle} by\n\\begin{eqnarray}\\label{regge1}\nS\\;=\\;\\sum_{h\\subset\nT^\\circ}V_h\\,\\epsilon_h\\;+\\;\\sum_{h\\subset\\partial T}V_h\\,\\psi_h{\\quad},\n\\end{eqnarray}\n where $h$ stand for the  $(D-2)$--dimensional subsimplices\n(sometimes called ``hinges'') of $T$, that is edges $e$ in $3$D and triangles $t$ in 4D. \n$V_h$\ndenotes the volume of the hinge $h$. The deficit angles\n$\\epsilon_h$ and exterior angles $\\psi_h$ are given by\n\\begin{eqnarray}\\label{regge2}\n\\epsilon_h\\;&=&\\;2\\pi\\;-\\;\\sum_{\\sigma\\subset\nh}\\theta_h^\\sigma\\qquad\\text{for }h\\subset T^\\circ{\\quad},\\\\[5pt]\\nonumber\n\\psi_h\\;&=&\\; k_h \\,\\pi\\;-\\;\\sum_{\\sigma\\subset\nh}\\theta_h^\\sigma\\qquad\\text{for }h\\subset\\partial T,\n\\end{eqnarray}\n\n\\noindent and in both cases the sum ranges over all $D$--dimensional simplices $\\sigma$ which contain $h$,\n and $\\theta_h^\\sigma$ is the\ninterior dihedral angle in the simplex $\\sigma$ between the two\n$(D-1)$--dimensional subsimplices that meet at $h\\subset\\sigma$. The number $k_h$ is fixed according to how many pieces $l$ of triangulations are glued along the hinge in question:\nBecause of the boundary term in (\\ref{regge1}), the action is {\\it additive} if we glue two pieces of triangulations together and choose $k_h=1$. In the canonical evolution scheme we will glue top--dimensional simplices to bulk triangulations. That is, the smallest triangulations we consider is just one tetrahedron $\\tau$ in $3D$ or one 4--simplex $\\sigma$ in $4D$, of which the actions are just boundary terms,\n\\begin{eqnarray}\\label{regge3}\nS_\\tau&=&   \\sum_{e \\subset \\tau} l^e \\left(k_e \\pi - \\theta^\\tau_e \\right) \\nonumber\\\\\nS_\\sigma&=&   \\sum_{t \\subset \\sigma} A_t \\left(k_t \\pi - \\theta^\\sigma_t \\right){\\quad},\n\\end{eqnarray}\nwhere $l^e$ denotes the length of the edge $e$ and $A_t$ the area of the triangle $t$. Again, $k_e$ and $k_t$ are to be determined by the gluing process. In the canonical evolution scheme presented below, we will either have $k_e,k_t=1$ or $k_e,k_t=0$.\n\n\n\nThe equations of motion are obtained by varying the action\n(\\ref{regge1}) with respect to the\nlengths $l^e$ of the edges in $T^\\circ$ (while keeping the edge lengths in the boundary $\\partial T$ fixed). The boundary term in\n(\\ref{regge1}) ensures that,\nfor all edges $e\\in T^\\circ$, the Regge equations read\n\\begin{eqnarray}\\label{regge4}\n\\sum_{h\\supset e}\\frac{\\partial V_h}{\\partial l^e}\\,\\epsilon_h=\\;0  {\\quad} .\n\\end{eqnarray}\nNote that in 3D the equations of motions enforce $\\epsilon_e=0$, that is, vanishing deficit angles and therefore vanishing curvature.\n\n\nFinally, the variation of the deficit angles appearing in the action vanishes because of the Schl\\\"afli identities\n\\begin{eqnarray}\\label{schlaefli}\n\\sum_{e \\subset \\tau}  l^e \\frac{\\partial \\theta_e^\\tau}{\\partial l^{e'}} &=& 0 {\\quad} ,\\nonumber\\\\\n\\sum_{t \\subset \\sigma}  A_t \\frac{\\partial \\theta_t^\\sigma}{\\partial l^{e'}} &=& 0   {\\quad} .\n\\end{eqnarray}\n\n\n\n\n\n\n\\section{Pachner moves for 3D Regge calculus}\\label{3D}\n\nIn the next subsections we will spell out the details of the dynamics of Pachner moves for 3D Regge gravity (without a cosmological constant). Before delving into the details, we will make a few remarks on the dynamics of 3D gravity. As mentioned in the previous section, the equations of motion for gravity require that all deficit angles vanish and thereby that the piecewise flat geometry is flat. In the canonical description we describe the dynamics of the 2D hypersurface, that is, the changes in the intrinsic and extrinsic geometry of the hypersurface evolving through the 3D space-time. As the latter is flat, we will essentially describe different embeddings of a 2D hypersurface into flat 3D space.\n\nThis allows us to determine Hamilton's principal function: since the deficit angles vanish for solutions, only the boundary term remains\n\\begin{eqnarray}\n\\tilde S(l^e) = \\sum_{e \\subset \\partial T} l^e \\psi_e(l^e)   {\\quad} .\n\\end{eqnarray}\nHere the $l^e$ are the lengths of the edges in the boundary of the triangulation. The momenta are given by the derivatives of Hamilton's principal function with respect to the edge lengths. Because of the Schl\\\"afli identity, the terms with derivatives of the exterior angles vanish and we obtain (the different signs in (\\ref{b3}) can be taken into account by taking the appropriate orientation of the exterior angles)\n\\begin{eqnarray}\np_e=\\psi_e {\\quad} .\n\\end{eqnarray}\nThus, the momenta conjugate to the length variables are given by the exterior angles. The same result was obtained in \\cite{waelbroeckzapata} starting from a first order (connection) formulation of discrete gravity.\n\nAs a consequence of the fact that the Pachner move dynamics allows to add degrees of freedom during the evolution, we can start from a small triangulated hypersurface and evolve to a much bigger one. In particular, we can start from an empty triangulation at time $k=0$. This can be seen as just a technical method, however, might also be useful (in 4D) to explore the `no--boundary proposal' \\cite{hawking}.\n\nIn order to produce a triangulation with a hypersurface of, e.g., spherical topology, we can start with an evolution move from the empty triangulation to the boundary of a tetrahedron.\\footnote{We could equally well begin by producing initial hypersurfaces of more complicated topology.} A tetrahedron has six boundary edges $l^n_1,\\,n=1,\\ldots,6$, hence, the phase space at time $k=1$ is 12--dimensional. Accordingly, we can extend the phase space at time $k=0$ by the six pairs $(l^n_0,p_n^0)$.\n\nUsing the one--tetrahedron action as generating function of the first kind\n\\begin{eqnarray}\\label{thu1}\n G_{0-1}(l^n_0,l^n_1)&=&S_\\tau(l^n_1) \\;=\\;  \\sum_{n \\subset \\tau} l^n_{1} \\left( \\pi - \\theta^\\tau_n (l^n_{1}) \\right) {\\quad} ,\n\\end{eqnarray}\nthe equations of motion amount to\n\\begin{eqnarray}\\label{thu2}\n&&p^{0}_n\\,=\\,0 {\\quad},{\\quad}\\q{\\quad} p^{1}_n\\,=\\, \\frac{\\partial S_\\tau}{\\partial l^n_{1}}\\,=\\, \\pi-\\theta_n(l^n_{1})  {\\quad} .\n\\end{eqnarray}\nThat is, the $l^n_1$ remain undetermined. However, we can set $l^n_0=l^n_1$ with the  understanding that the $l^n_1$ are initial data which appear only at time step $k=1$. The momenta at $k=0$ are constrained to vanish. But also at time $k=1$ we obtain a post--constraint because all the momenta are given as functions of the length variables at time $k=1$. Indeed, this equation just expresses the fact that the momenta are exterior angles, which for a flat tetrahedron are determined by the intrinsic geometry of the boundary surface, i.e.\\ the length variables. Hence, we have a totally constrained system, as all the momenta are determined as functions of the length variables.\n\nThis hypersurface can be evolved into a more complicated hypersurface (with spherical topology) by means of the Pachner moves described below. Momentum updating will then take care of the geometrical updating of the exterior angles, i.e.\\ the exterior angles change according to the dihedral angles that are added\/subtracted by gluing\/removing tetrahedra onto\/from the hypersurface.\n\nThe dynamics of 3D gravity is special because the fact that the system is totally constrained does not even change for more complicated triangulations (of spherical topology); through constraints all momenta at time $k$ will be determined by the length variables of step $k$. These constraints express the fact that we are always dealing with a 2D hypersurface embedded into 3D flat space and that, furthermore, the momenta are the exterior angles. Accordingly, if we considered a parallel transport of a 3D vector along a small loop around a vertex $v$ of the hypersurface, we should obtain an identity transformation. This parallel transport can be expressed \\cite{tHooft, kadarloll} as a sequence of rotations\n\\begin{eqnarray}\\label{thu3}\nP_v= R(\\alpha_{e_1e_2}) R(\\psi_{e_2}) R(\\alpha_{e_2 e_3}) R(\\psi_{e_3}) \\cdots R(\\psi_{e_1}) \\; \\stackrel{!}{=} \\text{Id}{\\quad},\n\\end{eqnarray}\nwhere $e_1,e_2,\\ldots$ denotes some cyclic ordering of the edges around the vertex $v$, $R(\\alpha_{e_i e_{i+1}})$ denotes the rotation in the plane spanned by the two edges $e_i, e_{i+1}$ with the angle $\\alpha_{e_i e_{i+1}}$ and $R(\\psi_{e_i})$ denotes the rotation around the edge $e_i$ by an angle $\\psi_{e_i}$. Note that the exterior angles equal the momenta and that the interior angles $\\alpha_{ee'}$ can be expressed as functions of the length variables (in the 2D star of the vertex).  Hence, the condition that the $3D$ parallel transport matrix should be the identity gives us three (as the matrix is in $SO(3)$) constraints on the phase space data for every vertex in the hypersurface. For a triangulation of spherical topology we have $3\\sharp v=\\sharp e+6$ for the number of vertices $\\sharp v$ and the number of edges $\\sharp e$. Therefore, we have at least as many constraints as configuration (or momentum variables). In fact, there are six more constraints than edges, because there exist six relations between the constraints (as can be checked explicitly for the example of the tetrahedron). These six relations correpsond to the three global rotations and three global translations which change the embedding of the 2D triangulation in 3D flat space, but do not change any of the geometrical data, i.e.\\ neither lengths or exterior angles.\n\nThe constraints (\\ref{thu3}) will be preserved by the Pachner moves, as these Pachner moves will implement the equations of motions, i.e.\\ flatness of the 3D triangulation. Furthermore, momentum updating will ensure that the momenta are always given by the exterior angles of the 2D hypersurface. Consequently, the canonical data at every time step will describe a 2D triangulation embedded into flat 3D space for which the relations (\\ref{thu3}) hold. Moreover, as we shall see shortly, the 1--3 Pachner move generates one vertex, and in conjunction with this vertex also three (post--) constraints of the form (\\ref{thu2}), which are just a rewriting of the form (\\ref{thu3}) for three--valent vertices.\\footnote{The difference between the two forms of constraints is that (\\ref{thu2}) is linear in the momenta, whereas (\\ref{thu3}) involves $\\cos p_e$. Indeed, the constraints should rather involve the square (or the $\\cos$) of the momenta as in the continuum. This, however, can also be taken into account in (\\ref{thu2}): Constraints quadratic in the momenta indicate that the constraint hypersurface has two pieces corresponding to the two roots of the quadratic equations. The two pieces correspond to the possibility of allowing for both orientations of the tetrahedron (i.e.\\ allowing both signs for the exterior angles). We can replace (\\ref{thu2}) by $\\cos p_n=\\cos \\psi_n$ to take care of this fact.} The 2--2 Pachner move and the 3--1 Pachner move will not generate vertices and therefore also no additional constraints. The dynamics prescribed by these moves will, however, preserve the constraints of the form (\\ref{thu3}) for the reasons just given.\n~\\\\\n\n\n\n\n\n\n\n\nAs a mnemonic, for the description of the Pachner move dynamics in both 3D and 4D we will use the following edge indices in order to label and appropriately distinguish the various length and momentum variables:\n\\begin{itemize}\n\\item $e$ labels {\\it edges} contained in the $D$--simplex of the Pachner move which occur in both $\\Sigma_k$ and $\\Sigma_{k+1}$,\n\\item $n$ labels {\\it new} edges introduced by a Pachner move which occur in $\\Sigma_{k+1}$ but not $\\Sigma_k$,\n\\item $o$ labels {\\it old} edges removed by a Pachner move which occur in $\\Sigma_k$ but not $\\Sigma_{k+1}$,\n\\item $b$ labels edges contained in both $\\Sigma_k$ and $\\Sigma_{k+1}$ which are not involved in the Pachner move and from that perspective may be considered as {\\it boundary} edges.\n\\end{itemize} \n\n\n\n\n\n\\subsection{The 1--3 Pachner move}\\label{13sec}\n\n\nConsider a 3D triangulation with a boundary $\\Sigma_k$ that we will consider as a $2D$ hypersurface at time $k$. Glue to this boundary a tetrahedron $\\tau$ such that one of the triangles is identified with a triangle $t$ in the 2D hypersurface $\\Sigma_k$. We obtain a new boundary $\\Sigma_{k+1}$. From the perspective of the hypersurface this gluing can be interpreted as a 1--3 Pachner move, i.e.\\ the triangle $t$ is replaced by three triangles, that share one vertex $v$ in the middle and have the same 1D boundary, consisting of three edges as the original triangle $t$ (see figure \\ref{3dpach}).\n\nNote that the tetrahedron can be glued with two different orientations to the hypersurface. This can be interpreted as gluing the tetrahedron either on the upper side (with a future pointing tip) or on the bottom side (with a past pointing tip) to the hypersurface, or, alternatively, as gluing or removing a tetrahedron to or from the bulk triangulation, respectively (see figure \\ref{13glurem}).\n\\begin{figure}[htbp!]\n\\begin{center}\n\\psfrag{sk}{$\\Sigma_k$}\n$\\begin{array}{cc}\n\\includegraphics[scale=.3]{Figures\/figure8.eps}& \\includegraphics[scale=.3]{Figures\/figure9.eps} \\\\\na)&b)\n      \\end{array}$\n      \\end{center}\\caption{\\small a) The 1--3 gluing Pachner move, b) The 1--3 removal Pachner move.\\newline The dashed edges are the three new edges.}\\label{13glurem}\n\\end{figure}\nThe Regge actions for the different orientations of the tetrahedron just differ by a global sign (which can be understood to arise from the oriented exterior angles). This agrees with the interpretation of gluing or removing a tetrahedron with, say, positive orientation: if we remove a tetrahedron from the bulk triangulation, we would have to subtract the action for the positively oriented tetrahedron, the alternative is to add the action (glue the tetrahedron) with the negative orientation.\n\nEvery Pachner move in 2D and 3D  can be interpreted as gluing a simplex in one of the two different orientations to the 3D and 4D bulk, respectively. These two possibilities can, in general, also be seen as gluing and removing a simplex with, say, positive orientation, respectively. We will henceforth assume that the two possibilities are encoded in the orientations and therefore in the signs of the exterior angles in the action (\\ref{regge3}). Thereby we can summarize these two cases into just one, which we will mostly refer to `as gluing a simplex' to the bulk triangulation or hypersurface.\n\n\nIn the 1--3 Pachner move one triangle $t$ is replaced by the same triangle subdivided into three new triangles (see figure \\ref{13m}). \\begin{figure}[htbp!]\n\\psfrag{1}{ 1--3}\n\\psfrag{2}{ 3--1}\n\\psfrag{e}{\\tiny $e_n$}\n\\psfrag{v}{$v$}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure10.eps} \n\\end{center}\n\\caption{\\small The 1--3 Pachner move and its inverse, the 3--1 Pachner move.}\\label{13m} \\end{figure}\nTherefore, for this Pachner move we will have edges of three different types: edges $e_b$ not participating in the dynamics, three edges $e$ in the boundary of the triangle $t$, for which $l^e_k=l^e_{k+1}$ and the three new edges $e_n$ with lengths $l^n_{k+1}$ which only appear at time step $(k+1)$, but not at $k$. We will regard the action\n\\begin{eqnarray}\nS_\\tau &=&   \\sum_{e \\subset \\tau} l^e_{k+1} \\left( - \\theta^\\tau_e(l^e_{k+1},l^n_{k+1})  \\right) +    \\sum_{n \\subset \\tau} l^n_{k+1} \\left( \\pi - \\theta^\\tau_n (l^e_{k+1},l^n_{k+1}) \\right)\n\\end{eqnarray}\nassociated to the tetrahedron glued to the hypersurface as a function of the three new edge lengths $l^n_{k+1}$ and the three edge lengths $l^e_{k+1}$. Note that we fixed the factors $k_e$ appearing in (\\ref{regge3}) to $k_n=1$ for the new edges $l^n$ and to $k_e=0$ for the `boundary edges' $l^e$. This will also work for the other moves, we will choose $k_e=0$ for all boundary edges and $k_n=1$ or $k_o=1$\nfor edges which either appear or `disappear' during the evolution move. In this way the $\\pi$ factors add up correctly to $2\\pi$ after a sufficient number of evolution moves.\n\n\n\nAccording to the discussion below (\\ref{c3}), we add the generating function $G_3(l^e_{k+1},p^k_e)$ for the edges of type $e$ and $b$ to the action. Hence, the generating function is\n\\begin{eqnarray}\\label{c5}\nG_{1-3}(l^b_{k+1},p_b^k;\\,l^e_{k+1},p_e^{k};\\,l^n_k,l^n_{k+1}) &=& \\sum_b l^b_{k+1}p_b^k + \\sum_e l^e_{k+1}p_e^k  \\, + \\, S_\\tau(l^e_{k+1},l^n_{k+1})  {\\quad} .\n\\end{eqnarray}\nThe evolution equations are then\n\\begin{eqnarray}\\label{c6}\n&&l^b_k\\,=\\,l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\n&&l^e_k\\,=\\,l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\tau}{\\partial l^e_{k+1}}\\,=\\, p^k_e-\\theta_e(l^e_{k+1},l^n_{k+1}) {\\quad} ,\\\\\n&&p^{k}_n\\,=\\,0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,=\\, \\frac{\\partial S_\\tau}{\\partial l^n_{k+1}}\\,=\\, \\pi-\\theta_n(l^e_{k+1},l^n_{k+1})  {\\quad} . \\label{c6c}\n\\end{eqnarray}\n\n\nThe momenta $p_e$ are just updated to agree with the exterior angles of the evolved spatial hypersurface. The momenta $p_n^k$ at time step $k$ have to vanish, since the $l^n_k$ are not dynamical variables at this time step. As was mentioned in the discussion at the end of section \\ref{candisc}, we have to expect a post--constraint for every new edge variable $l^n$. Indeed, equations (\\ref{c6c}) are constraints, requiring that the new momenta $p_n^{k+1}$ are again given by the exterior angles of the new hypersurface (coinciding with the three exterior angles of the added tetrahedron) which, however, can be expressed as functions of the length variables $l^e_{k+1},l^n_{k+1}$ only.\n\nNotice that for this evolution step not only the $l^n_k$ remain undetermined, but also the $l^n_{k+1}$ can be chosen arbitrarily (the generalized triangle inequalities have to be satisfied though).  We will set $l^n_k=l^n_{k+1}$, so that the $l^n_k$ can be interpreted as initial data that determine the data at time $(k+1)$. Later we shall see that also for some types of Pachner moves (in 4D) the edge lengths of  new edges  can be chosen arbitrarily.  This freedom can be understood as a choice of initial data, which becomes only relevant at the time step, at which the new edges appear. On the other hand, we will also see (in 4D) that pre--constraints, appearing in consecutive evolution moves, might fix these edge lengths {\\it a posteriori}.\n\n\n\n\n\n\n\n\n\n\\subsection{The 2--2 Pachner move}\\label{22sec}\n\n\nConsider the situation in which a tetrahedron $\\tau$ is glued to $\\Sigma_k$ in such a way that two of its triangles, $t_1$ and $t_2$, and thus five of its edges are identified with two neighbouring triangles and their five edges in $\\Sigma_k$. This gluing move (with positive orientation) is only possible if the extrinsic dihedral angle $\\psi^k_o$ at the edge $e=o$ along which the two triangles in $\\Sigma_k$ are identified is negative. If it is positive, it is only possible to remove the corresponding tetrahedron $\\tau$ from the triangulation, or, alternatively, to add a tetrahedron with negative orientation, see figure \\ref{f22m}. \n\\begin{figure}[htbp!]\n\\psfrag{t1}{$t_1$}\n\\psfrag{t2}{$t_2$}\n\\psfrag{s}{$\\Sigma_k$}\n\\begin{center}\\includegraphics[scale=.4]{Figures\/figure11.eps} \n\\end{center}\n\\caption{\\small The gluing\/removal 2--2 Pachner move involves two triangles $t_1$ and $t_2$ in hypersurface $\\Sigma_k$.}\\label{f22m} \\end{figure}\nFrom the perspective of the hypersurface, these elementary moves appear as 2--2 Pachner moves. That is, two triangles sharing an edge $e=o$ are replaced by two triangles sharing a new edge $e=n$ (see figure \\ref{22m}).\n\\begin{figure}[htbp!]\n\\psfrag{1}{ 2--2}\n\\psfrag{2}{ 2--2}\n\\psfrag{o}{$o$}\n\\psfrag{n}{$n$}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure12.eps} \n\\end{center}\n\\caption{\\small The 2--2 Pachner move is its own inverse.}\\label{22m} \\end{figure}\nNote that while this move removes one edge $o$ from the hypersurface and, instead, introduces the new edge $n$, it does not introduce a new vertex. Again, the two pairs of triangles have the same boundary of four edges $e$ and four vertices. The edges $o,n$ are transversal, that is $o$ and $n$ are connecting the two opposite pairs of vertices. Additionally, the 2--2 Pachner move is its own inverse, which from the 3D perspective is taken into account via the gluing\/removing convention (and the different global signs in the action for the cases with different orientations) for the Pachner moves.\n\n\n\n\n\n\n\nThere are four kinds of edge lengths: apart from $l^b_{k\/k+1},l^e_{k\/k+1}$ and one length $l^n_{k+1}$, we also have one length $l^o_{k}$ which appears only at time step $k$ but not at $(k+1)$.  Thus, we will extend the phase spaces at time $k$ and $(k+1)$ by the pairs $(l^n_k,p^k_n)$ and $(l^o_{k+1},p_o^{k+1})$, respectively.\n\nAccordingly, we will choose a generating function of the first kind in the variables $l^n,l^o$ and of the third kind\nin the variables $l^b,l^e$:\n\\begin{eqnarray}\\label{c7}\nG_{2-2}(l^b_{k+1},p_b^k;\\,l^e_{k+1},p_e^{k};\\,l^o_k,l^o_{k+1};\\,l^n_k,l^n_{k+1})\n&=&\\!\\!\n\\sum_b l^b_{k+1}p_b^k + \\sum_e l^e_{k+1}p_e^k  \\, + \\, S_\\tau(l^e_{k+1},l^o_k,l^n_{k+1}) \\, , {\\quad}\\q\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nS_\\tau(l^e_{k+1},l^o_k,l^n_{k+1})&=&\n  \\sum_{e \\subset \\tau} l^e_{k+1} \\left( - \\theta^\\tau_e(l^e_{k+1},l^o_k,l^n_{k+1})  \\right) +   l^n_{k+1} \\left( \\pi - \\theta^\\tau_n(l^e_{k+1},l^o_k,l^n_{k+1}) \\right)  + \\nonumber\\\\&&{\\quad}\\q{\\quad} {\\quad}\\q{\\quad}\\q{\\quad} {\\quad}\\q{\\quad}\\q{\\quad}\\q l^o_{k} \\left( \\pi - \\theta^\\tau_o(l^e_{k+1},l^o_k,l^n_{k+1}) \\right) \\, .\n\\end{eqnarray}\nThe evolution equations are given by\n\\begin{eqnarray}\\label{c8}\nl^b_k&=&l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\nl^e_k&=&l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\tau}{\\partial l^e_{k+1}}\\,=\\, p^k_e-\\theta_e(l^e_{k+1},l^n_{k+1}) {\\quad} ,\\\\\np^{k}_n&=&0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,\\,=\\, \\frac{\\partial S_\\tau}{\\partial l^n_{k+1}}\\,{\\quad}\\q\\;=\\, \\pi-\\theta_n(l^e_{k+1},l^o_k,l^n_{k+1}) {\\quad} , \\label{c8c}\\\\\np^{k+1}_o &=& 0 {\\quad},{\\quad}\\q{\\quad} p^{k}_o{\\quad}\\,\\,=\\, -\\frac{\\partial S_\\tau}{\\partial l^n_{k+1}}\\,{\\quad}\\;\\;=\\,- \\pi+\\theta_n(l^e_{k+1},l^o_k,l^n_{k+1}) \\, .{\\quad} \\label{c8d}\n\\end{eqnarray}\n\nAs discussed generally, we have $p_o^{k+1}=0$ and $p_n^k=0$.  In contrast to the 1--3 move, where the new edge lengths $l^n_k,l^n_{k+1}$ were undetermined at both time step, here the edge length $l^n_{k+1}$ is determined by equation (\\ref{c8d}) as a function of the initial data, which also involve $l^o_k$ at time $k$. (Again, we can define $l^n_k=l^n_{k+1}$ but this time the interpretation of $l^n_k$ as additional new data does not apply, rather $l^n_k$ is in this case constrained, that is determined by the other initial data at time $k$. The same applies to $l^o_{k+1}$ if we go backwards in time.) Note that---via momentum matching---(\\ref{c8d}) implements the Einstein equations $\\epsilon_o=0$ for the edge $o$ which becomes a bulk edge in the course of the 2--2 (gluing) move. Equation (\\ref{c8d}) demands that ${}^-p^k_o$ is given by the (oriented) exterior angle of the tetrahedron that is glued to the hypersurface. Momentum matching, on the other hand, imposes ${}^-p^k_o={}^+p^k_o$, where ${}^+p^k_o$ is determined by the previous moves and given by the (oriented) exterior angle of the hypersurface determined by the bulk, and thus that these two exterior angles add up to a vanishing deficit angle around the edge $o$. This condition will, in general, fix the edge length $l^n_{k+1}$ as the exterior (or dihedral) angle at the opposite edge $o$ of the tetrahedron depends on this length.\n\n\nThe difference to the 1--3 move is that here we have both a `new edge' and an `old edge'. Hence, the argument in section \\ref{candisc}, according to which we have to expect a pre--constraint (because of the `old edge') and a post--constraint (because of the `new edge'), does not apply, as this argument relied on determining the maximal dimension of the image of the time evolution map. Here the counting changes since we have both a new and an old variable and due to the conditions $p_n^k=0$ and $p^{k+1}_o=0$, we do not necessarily expect a further constraint based on this argument (where we ignore that $l^o_{k+1}$ remains undetermined).\n{\\it A priori}, with the exception of $p^{k+1}_o=0$, all momenta at time $(k+1)$ involve edge lengths from time $k$ and time $(k+1)$, in particular, (\\ref{c8c}) for $p^n_{k+1}$.\\footnote{Since $l^o$ remains undetermined at time $(k+1)$ and $l^n$ at time $k$, one might wonder whether it is possible to summarize the two variables $l^o,l^n$ into just one variable (with two different time labels). Indeed, it is possible \\cite{draft} to define a canonical time evolution in this way; the method presented here, however, is more similar to the 1--3 and 3--1 move and, moreover, to the Pachner moves in 4D. Nevertheless, this remark shows that canonical time evolution maps can be redefined by just relabeling or identifying variables with each other. We will see a similar situation in the tent move evolution in sections \\ref{sectent1} and \\ref{tent2} below, where several Pachner move steps are grouped together and labeled as a single tent move time step.}\n\nThe fact that there are no new post--constraints arising in the 2--2 move should, however, not be confused with the feature that the system under consideration is totally constrained (that is, for spherical topology of $\\Sigma$ all momenta are constrained as functions of the edge lengths). Rather, the 2--2 move preserves the (flatness) constraints, which encode that the hypersurface in question bounds a flat bulk triangulation, by implementing the flatness condition for the edge that becomes a bulk edge.\n\n\n\n\n\n\n\\subsection{The 3--1 Pachner move}\\label{31sec}\n\n\nThe 3--1 move is the inverse of the 1--3 move: Consider, therefore, a three-valent vertex $v$ in $\\Sigma_k$ whose adjacent three edges $e_o$ are equipped with extrinsic dihedral angles $\\psi_o$ which are all negative, i.e.\\ the vertex is pointing into the hypersurface and represents the tip of a tetrahedron which is upside down. In this situation we can glue a tetrahedron $\\tau$ to this surface by identifying the three triangles sharing vertex $v$ in $\\tau$ with the corresponding ones in $\\Sigma_k$. Consider now the opposite situation, where all $\\psi_o$ at the three-valent vertex are positive. In this case the vertex represents the tip of a tetrahedron $\\tau$ which is sticking out of the hypersurface and we may remove this tetrahedron (or equivalently glue a tetrahedron with opposite orientation). These situations are depicted in figure \\ref{13glurem} if the orientation of the arrows is reversed. From the perspective of the hypersurface, these elementary moves appear as a 3--1 Pachner move, see figure \\ref{31m}.\n \\begin{figure}[htbp!]\n\\psfrag{1}{ 1--3}\n\\psfrag{2}{ 3--1}\n\\psfrag{e}{\\tiny$e_o$}\n\\psfrag{v}{$v$}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure10.eps} \n\\end{center}\n\\caption{\\small The 1--3 and 3--1 Pachner moves.}\\label{31m} \\end{figure}\nPrior to this move, all six edges involved in the move are edges in $\\Sigma_k$. During the move the vertex $v$, as well as the $e_o$ are removed (as these become internal to the bulk triangulation).\n\nBy virtue of the fact that there are no new edges introduced during the move, the equations of motion play rather the role of constraints in this case which have to be satisfied in order for this move to be allowed. Since up to $\\Sigma_k$ we have always solved the equations of motion during the elementary evolution steps, the entire triangulation up to $\\Sigma_k$ will be embedded in a flat 3D manifold. The three equations of motion of the edges which become internal, therefore, do not add any new constraints and are thus trivially satisfied.\n\nThe 3--1 move is the inverse to the 1--3 move, hence, we have three kinds of edge lengths: apart from $l^b_{k\/k+1},l^e_{k\/k+1}$ there are three edge lengths of type $l^o_{k}$ which appear only at time step $k$ but not at $(k+1)$. We therefore extend the phase space associated to $\\Sigma_{k+1}$ by three pairs $(l^o_{k+1},p^{k+1}_o)$.\n\n\nWe choose the action to be a function of the edge lengths at time $k$ and, accordingly, use for edges of type $e$ and $b$ a generating function of second type. According to the general description, we have\n\\begin{eqnarray}\\label{c9}\nG_{3-1}(l^b_k,p_b^{k+1}; l^e_k,p_e^{k+1}; l^o_k,l^o_{k+1})&=&\n-\\sum_b l^b_{k}p_b^{k+1} - \\sum_e l^e_{k}p_e^{k+1}  \\, + \\, S_\\tau(l^e_{k},l^o_k)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n S_\\tau(l^e_{k},l^o_k)&=&\n   \\sum_{e \\subset \\tau} l^e_{k} \\left( - \\theta^\\tau_e(l^e_{k},l^o_{k})  \\right) +    \\sum_{o \\subset \\tau} l^o_{k} \\left( \\pi - \\theta^\\tau_o (l^e_{k},l^o_{k}) \\right) {\\quad} .\n\\end{eqnarray}\nThe evolution equations are then given by\n\\begin{eqnarray}\\label{c10}\n&&l^b_{k+1}\\,=\\,l^b_{k}  {\\quad} ,{\\quad}\\q p^{k}_b\\,=\\,p^{k+1}_b {\\quad} , \\\\\n&&l^e_{k+1}\\,=\\,l^e_{k}{\\quad},{\\quad}\\q p^{k}_e\\,=\\,p^{k+1}_e-\\frac{\\partial S_\\tau}{\\partial l^e_{k}}\\,=\\, p^{k+1}_e+\\theta_e(l^e_{k},l^o_{k}) {\\quad} ,\\label{c9b}\\\\\n&&p^{k+1}_o\\,=\\,0 {\\quad},{\\quad}\\q p^{k}_o\\,=\\,- \\frac{\\partial S_\\tau}{\\partial l^o_{k}}\\,=\\,- \\pi+\\theta_o(l^e_{k},l^o_{k})  {\\quad} .\\label{c9c}\n\\end{eqnarray}\n\nThe second equation in (\\ref{c9b}) defines the updated momenta $p^{k+1}_e$ as a function of the phase space variables at time $k$,\nwhile the second equation of (\\ref{c9c}) is a pre--constraint, this time on the phase space variables at time $k$. That is, to perform this 3--1 move, this condition on the phase space variables has to be satisfied. Again, as for the 2--2 move these pre--constraints implement the equations of motions, namely flatness for the edges which become bulk edges during the move. In general, there are two possibilities: either i) the pre--constraints are automatically satisfied, if we consider a hypersurface which has been evolved in the manner described here by Pachner moves, or ii) the constraints are only satisfied for specific initial data, including the kind of additional initial data which arise, for instance, by the 1--3 Pachner move. For the 3D case, only the first possibility takes place as argued above. In 4D, on the other hand, we will generally encounter the second possibility, namely data which are {\\it a priori} free to choose in a certain Pachner move may become fixed by pre--constraints arising in later moves.\n\n\n\n\n\\subsection{Example: 3D tent moves}\\label{sectent1}\n\nThe Pachner moves generically change the connectivity and the number of edges in the triangulation. There are, however, combinations of Pachner moves, e.g.\\ the so--called tent moves  \\cite{commi,dithoe1,bd1}, which do {\\it not} change the connectivity of the triangulations and therefore induce a canonical dynamics in the standard interpretation, that is, between phase spaces of equal dimensions.\n\nA tent move can be constructed by picking some vertex $v_n$\\footnote{Note that henceforth, in order to avoid confusion, we will enumerate tent moves by $n\\in{\\mathbbm Z}$, while the elementary Pachner moves into which the tent moves can be decomposed are counted by $k\\in{\\mathbbm Z}$.}  in a $(D-1)$--dimensional triangulated hypersurface $\\Sigma_n$ and subsequently defining a new vertex $v_{n+1}$ to the `future' of $v_n$ which must be connected by an edge to $v_n$ (the `tent pole'). Denote all other vertices in $\\Sigma_n$ to which $v_n$ is connected by $1,\\ldots,N$. Connect $v_{n+1}$ to each of these vertices $1,\\ldots,N$ by $N$ edges. The new vertex $v_{n+1}$ then lies in the new hypersurface $\\Sigma_{n+1}$ and is $N$-valent in $\\Sigma_{n+1}$ as $v_n$ is in $\\Sigma_n$ (the tent pole is now internal) and for every $(D-1)$-simplex $\\sigma(v_nij\\ldots)$, $i,j\\in1,\\ldots,N$, there is now a $(D-1)$-simplex $\\sigma(v_{n+1}ij\\ldots)$. That is, the triangulations of the two hypersurfaces are the same. Notice that each tent move only involves the $(D-1)$ star of the vertex $v_n$ in $\\Sigma_n$. The situation for a four-valent 3D tent move is illustrated in figure \\ref{3dtm1}.\n\\begin{figure}[hbt!]\n\\begin{center}\n    \\psfrag{vn}{$v_{n}$}\n    \\psfrag{vn+1}{$v_{n+1}$}\n    \\psfrag{1}{\\small$1$}\n    \\psfrag{2}{\\small$2$}\n    \\psfrag{3}{\\small$3$}\n    \\psfrag{4}{\\small$4$}\n    \\includegraphics[scale=0.3]{Figures\/figure13.eps}\n    \\end{center}\n    \\caption{\\label{3dtm1}{\\small A 3D tent move applied to a four-valent vertex $v_n$ in a 2d Cauchy hypersurface. This four-valent tent move can be reproduced through a gluing sequence of four tetrahedra which correspond to a sequence of one 1--3, two 2--2 and one 3--1 Pachner moves in the hypersurface. }}\n\\end{figure}\n\\noindent\n\n\n\nThe new piece of $D$-dimensional triangulation between $\\Sigma_n$ and $\\Sigma_{n+1}$ with the $N+1$ new edges $e(v_{n+1}i)$ and $e(v_nv_{n+1})$ can be decomposed into a sequence of gluings of single $D$-simplices and the tent move evolution may therefore be described in terms of a sequence of Pachner moves in the hypersurface. In particular, in 3D simply pick an N-valent vertex in some $\\Sigma_n$ and perform one 1--3 Pachner move, $(N-2)$ 2--2 Pachner moves and a final 3--1 Pachner move in order to generate $\\Sigma_{n+1}$.\n\n\n\n\nTo define the Pachner move dynamics, we used the Regge action as (part of) a generating function, so that the resulting canonical dynamics leads to the same equations of motions as the ones obtained by varying the action. We can again employ the Regge action to define the canonical evolution corresponding directly to the tent moves \\cite{bd1, dithoe1,bianca}. By construction, this dynamics will coincide with the one obtained by performing the sequence of Pachner moves (matching appropriately the edge labels). This highlights the remarks put forward earlier that canonical time evolution maps may be redefined by relabeling variables. For the tent move dynamics we can either keep the length of the tent pole as an internal variable in the sense of problem (a) in section \\ref{candisc}, see equations (\\ref{b6}), or, alternatively, integrate this length out and work with Hamilton's principal function as a generating function:\n\nTo this end, notice that the 3D triangulation---the `tent'---we are gluing to the hypersurface has two 2D boundaries given by the 2D stars of the vertices $v_n$ and $v_{n+1}$. These two 2D boundaries meet in a 1D boundary, given by the (cyclically ordered) vertices $1,\\ldots, N$ and the edges connecting these vertices. The tent consists of $N$ tetrahedra sharing the tent pole as an edge, and therefore each having vertices $v_n,v_{n+1}$ and $i,i+1$. The tent pole is an internal edge, whereas the 2D boundary has edges $e(v_n,i),e(v_{n+1},i)$ and edges $e(i,i+1)$ with edge lengths $l^i_n,l^i_{n+1},\\,i=1,\\ldots N$ and $l^{i,i+1}_n= l^{i,i+1}_{n+1}$, respectively.  We can solve the equation of motion for the tent pole as a function of the boundary edges. This equation of motion will just require that the deficit angle around the tent pole is vanishing. Using this result in the action, we obtain Hamilton's principal function for the `tent'\n\\begin{eqnarray}\n\\tilde S_{tent}(l^i_n,l^i_{n+1},l^{i,i+1}_{n+1})&=&\\!\\!\\!-\\sum_i l_{n+1}^{i,i+1}\\theta_{i,i+1}^{\\tau(v_n,v_{n+1},i,i+1)} + \\sum_i l_n^i\\left(\\pi-  \\theta_{v_ni}^{\\tau(v_n,v_{n+1},i,i+1)} - \\theta_{v_ni}^{\\tau(v_n,v_{n+1},i-1,i)}\\right)  +\\nonumber\\\\\n&&{\\quad}\\q{\\quad}\\q \\sum_i l_{n+1}^i\\left(\\pi-  \\theta_{v_{n+1}i}^{\\tau(v_n,v_{n+1},i,i+1)} - \\theta_{v_{n+1}i}^{\\tau(v_n,v_{n+1},i-1,i)}\\right)  {\\quad} .\n\\end{eqnarray}\nTaking into account  the additional terms $l^{i,i+1}_{n+1} p^n_{i,i+1}$ in the generating function we obtain the following equations of motion\n\\begin{eqnarray}\nl^{i,i+1}_{n}&=&l^{i,i+1}_{n+1}  {\\quad} , \\nonumber\\\\\n p^{n+1}_{i,i+1}&=&p^{n}_{i,i+1}- \\theta_{i,i+1}^{\\tau(v_n,v_{n+1},i,i+1)} {\\quad} , \\label{fria0} \\\\\np^{n}_i &=&   -\\pi+  \\theta_{v_ni}^{\\tau(v_n,v_{n+1},i,i+1)} + \\theta_{v_ni}^{\\tau(v_n,v_{n+1},i-1,i)}  {\\quad} ,    \\label{fria1}  \\\\\n p^{n+1}_i &=&   \\pi-  \\theta_{v_{n+1}i}^{\\tau(v_n,v_{n+1},i,i+1)} - \\theta_{v_{n+1}i}^{\\tau(v_n,v_{n+1},i-1,i)} \\; ,{\\quad}\\q{\\quad}\\label{fria2}\n\\end{eqnarray}\nwhere again the Schl\\\"afli identity was used. Now, {\\it a priori} the dihedral angles on the right hand side of the equations (\\ref{fria1},\\ref{fria2}) depend on both sets of edge lengths $l^i_n$ and $l^{i+1}_n$. But, for instance, for the tent move at a three valent vertex the right hand side of (\\ref{fria2}) does not depend on the lengths $l^i_n$ and we obtain three post--constraints. (The geometry around  the new vertex is the one around a vertex of a flat tetrahedron and the dihedral angles can be expressed as functions of the six edge lengths $l^i_{n+1}, l^{i,i+1}_{n+1}$.) Likewise, (\\ref{fria1}) will give us three pre--constraints as the right hand side can be expressed as a function of $l^i_n,l^{i,i+1}_n$ only. For tent moves at higher valent vertices we can still conclude that we have three pre--constraints and three post--constraints of the form (\\ref{thu3}). To this end one just has to use the same arguments as for (\\ref{thu3}) applied to the triangulation of the tent, which itself is a flat triangulation.\n\nHence, we will have three pre-- and three post--constraints for the tent move. From this we can conclude that three of the edge lengths $l^i_{n+1}$ remain undetermined by the equations of motion (\\ref{fria0}--\\ref{fria2}). (Correspondingly, there is a three parameter set of initial data at step $n$ which can be evolved to the same data at time $(n+1)$, and vice versa.) These are the same three parameters that are left undetermined by the 1--3 move in section \\ref{13sec} and correspond to lapse and shift degrees of freedom. The three post--constraints of the 1--3 move are of the form as encountered here and the present discussion shows that these constraints remain preserved through all the additional Pachner moves which make up the tent move. Also the pre--constraints which need be fulfilled so that the final 3--1 move can be performed, will automatically be satisfied, if the triangulation has been correctly evolved by Pachner moves.\n\nFinally, we remark that the lapse and shift degrees of freedom which remain free in the tent move can be chosen to be infinitesimally small. This allows to recover a continuous time evolution (plus Hamiltonian and constraints generating this evolution, which coincide with (\\ref{thu3})) from the discrete time evolution presented here.\n\n\n\n\n\n\n\\section{Pachner moves for 4D Regge calculus}\\label{4D}\n\nWhile the 3D Regge dynamics leads to flat geometries, the 4D Regge equations allow for curved solutions. This renders the 4D dynamics significantly more complicated than in the 3D case, in particular, the preservation of the constraints will generally not hold. Related to this is the fact that lapse and shift degrees of freedom will, in general, not remain free but will become fixed by pre--constraints (unless one considers initial data which lead to flat solutions).\n\nThe Regge action for a 4D triangulation (without cosmological constant) is given by (\\ref{regge1},\\ref{regge2}) for $h=t$,\n\\begin{eqnarray}\\label{y1}\nS=\\sum_{t\\subset T^\\circ} A_t \\epsilon_t + \\sum_{t\\subset \\partial T} A_t \\psi_t  {\\quad} .\n\\end{eqnarray}\nThe equations of motion for an inner edge $e$ read\n\\begin{eqnarray}\\label{y2}\n\\sum_{t \\supset e} \\frac{\\partial A_t}{\\partial l^e} \\,\\,\\epsilon_t &=&0  {\\quad} ,\n\\end{eqnarray}\nwhich---depending on the boundary data---allow for flat solutions $\\epsilon_t=0$ as well as solutions with curvature $\\epsilon_t\\neq 0$.\n\nDespite the latter, we can show that Hamilton's principal function reduces to a boundary term (as is the case in the continuum). To this end, multiply each of the equations (\\ref{y2}) with the length $l^e$ and sum over all (inner) edges. We will employ the Euler identity (see, e.g., \\cite{Bahr:2009qc})\n\\begin{eqnarray}\\label{y3}\n\\sum_{e \\subset t} l^e \\frac{\\partial A_t}{\\partial l^e} &=& 2 A_t\n\\end{eqnarray}\nfor the area of a triangle. Using the equations of motion (\\ref{y2}), we can write\n\\begin{eqnarray}\\label{y4}\n0&=& \\sum_{e \\subset T^\\circ } l^e \\sum_{t\\subset e} \\frac{\\partial A_t}{\\partial l^e} \\epsilon_t \\nonumber\\\\\n&=&  2 \\sum_{t \\subset T^\\circ_t}  A_t \\epsilon_t \\,+\\, \\sum_{t \\subset T^\\partial_t}\\; \\sum_{e \\subset t \\, \\cap\\, e\\subset T^\\circ}  l^e \\frac{\\partial A_t}{\\partial l^e}\\epsilon_t\n\\end{eqnarray}\nwhere $T^\\circ_t$ denotes the set of triangles which have all three edges contained in the bulk $T^\\circ$ and $T^\\partial_t$ denotes the set of triangles in $T^\\circ$ where at least one of the edges is in the boundary $\\partial T$. (Note that an inner triangle can have either one, two or all three edges in the boundary.) Employing equations (\\ref{y3}) and (\\ref{y4}) in the action (\\ref{y1}), we discover that Hamilton's principal function\n\\begin{eqnarray}\\label{y5}\n\\tilde S&=& \\frac{1}{2} \\sum_{e\\subset \\partial T} l^e \\left[ \\sum_{t \\supset e\\,\\cap\\,t\\subset T^\\partial_t} \\frac{\\partial A_t}{\\partial l^e}\\epsilon_t\\;+\\; \\sum_{t \\supset e\\,\\cap\\,t\\subset \\partial T}  \\frac{\\partial A_t}{\\partial l^e} \\psi_t \\right]  {\\quad},\n\\end{eqnarray}\nis given by a boundary term. The form of this last expression suggests that the momenta are given by the combination of exterior angles and deficit angles `near' the boundary in square brackets on the right hand side of (\\ref{y5}),\n\\begin{eqnarray}\\label{y6}\np_e&=&\\sum_{t \\supset e\\,\\cap\\,t\\subset T^\\circ} \\frac{\\partial A_t}{\\partial l^e}\\epsilon_t\\;+\\; \\sum_{t \\supset e\\,\\cap\\,t\\subset \\partial T}  \\frac{\\partial A_t}{\\partial l^e} \\psi_t\n\\end{eqnarray}\nIndeed, momentum updating during the Pachner move evolution will confirm this.\n\nAs in 3D, we can produce a Regge triangulation with boundary topology given by the 3--sphere\\footnote{Once more, we could equally well start by producing an initial hypersurface of more complicated topology.} by considering first an evolution move from the empty hypersurface to the boundary of a 4--simplex. The generating function would simply be given by the one--simplex action\n\\begin{eqnarray}\nG_{0-1}(l^n_0,l^n_1)&=& S_\\sigma(l^n_1) \\;=\\;\\sum_t A_t (\\pi-\\theta_t^\\sigma){\\quad},\n\\end{eqnarray}\nand gives rise to the following equations of motion\n\\begin{eqnarray}\np^0_n=0 {\\quad} ,{\\quad}\\q{\\quad} p^1_n =\\frac{\\partial S_\\sigma}{\\partial l^n_1} = \\sum_{t\\supset e} \\frac{\\partial A_t}{\\partial l^n_1} \\left( \\pi- \\theta_t^\\sigma \\right) {\\quad} .\n\\end{eqnarray}\n\nAgain, just as in the 3D case, we also obtain post--constraints here which determine all 10 momenta at time $k=1$ as a function of the edge lengths at time $k=1$. Hence, the boundary of a simplex is a totally constrained system. This property does not change under 1--4 Pachner moves (yielding so--called {\\it stacked spheres}) which will introduce four post--constraints for every four new edge lengths added as configuration variables. This sector of the 4D dynamics behaves very similarly to 3D gravity and was also discussed in \\cite{BiancaJames}. Furthermore, a derivation of the symplectic structure for this sector, starting from a first order formulation of canonical discrete gravity, may be found in \\cite{BiancaJames}. The results obtained there are in agreement with (\\ref{y6}). The reason why this sector is totally constrained is that the 1--4 Pachner moves neither lead to inner triangles nor inner edges. Hence, the hypersurface triangulations produced by means of this move allow for bulk triangulations without any inner triangles which could carry inner curvature. That is, we are considering again just flat triangulations.\n\nThis will, in general, change if in addition we consider 2--3, 3--2, and 4--1 Pachner moves. As the 1--4 Pachner move will not lead to any additional inner triangles or edges but adds four boundary edges, we can consider it as just adding lapse and shift degrees of freedom. Indeed, the four new edge lengths will remain free parameters, at least before doing any other moves. The 2--3 Pachner move, on the other hand, generates an inner triangle but no inner edge. We shall see that this move can be interpreted as adding a curvature degree of freedom (or physical degree of freedom as opposed to the lapse and shift gauge degrees of freedom). The 3--2 move, furthermore, produces an inner edge for which the Regge equation of motion---canonically in the form of a pre--constraint---must be satisfied. Consequently, this move can be considered as the `true evolution step' in the sense that it requires the solution of an equation of motion, involving curvature degrees of freedom. Finally, the 4--1 Pachner move generates six inner triangles and four inner edges and will feature four pre--constraints. For the same reasons as in the 3D case, the pre--constraints will be automatically satisfied if we consider initial data leading to a flat solution. For curved solutions, however, these pre--constraints will fix some free parameters introduced in the previous steps.\n\n\n\n\n\n\n\n\n\n\\subsection{The 1--4 Pachner move}\n\n\n\n\nThis move is the 4D analogue of the 1--3 Pachner move discussed in section \\ref{13sec}. Glue a 4-simplex $\\sigma$ onto a given 3D hypersurface $\\Sigma_k$ in such a way that its `bottom tetrahedron' $\\tau$ is identified with a tetrahedron in $\\Sigma_k$. Similarly, consider the situation in which one simplex $\\sigma$ that shares one tetrahedron $\\tau$ with $\\Sigma_k$ is removed from a given triangulation (equivalent to adding a tetrahedron with opposite orientation).\n\n\nThis move acts as a 1--4 Pachner move on $\\Sigma_k$. It introduces one new vertex and four new edges into the new hypersurface, but it does not  render any edges internal. The 1--4 Pachner moves replaces the tetrahedron $\\tau$ with the subdivided tetrahedron (consisting of four smaller tetrahedra); the boundary of the tetrahedron $\\tau$ does not change and we have the same four triangles $t^e$ and six edges $l^e$ before and after the move. Through the subdivision there will appear four new edges $e_n$ and six new triangles $t^n$ adjacent to these new edges, see figure \\ref{14m}.\n\\begin{figure}[htbp!]\n\\psfrag{1}{1--4}\n\\psfrag{2}{4--1}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure14.eps} \\end{center}\n\\caption{\\small The 1--4 Pachner move and its inverse, the 4--1 Pachner move.}\\label{14m}\n \\end{figure}\n\n\nSince there are no edges which become internal, there are no equations of motion to be satisfied. As a consequence, the lengths of the four new edges, labeled by $n$, can be freely chosen. That is, one has a fourfold freedom in choosing the `tip' of $\\sigma$, which can be parametrized by lapse $N$ and shift $N^\\alpha$ variables. As opposed to the 3D case (and the continuum) this freedom will, in general, be restricted by the appearance of pre--constraints in later moves (see also the discussion in section \\ref{tent2} about tent moves).\n\nWe proceed in the same way as for the 1--3 move and use the generating function\n\\begin{eqnarray}\\label{4c5}\nG_{1-4}(l^b_{k+1},p_b^k;\\,l^e_{k+1},p_e^{k};\\,l^n_k,l^n_{k+1}) &=& \\sum_b l^b_{k+1}p_b^k + \\sum_e l^e_{k+1}p_e^k  \\, + \\, S_\\sigma(l^e_{k+1},l^n_{k+1})  {\\quad},\n\\end{eqnarray}\nwhere we use the action of a single simplex\n\\begin{eqnarray}\nS_\\sigma&=& \\sum_{t^e \\subset \\sigma} A_{t^e}(l^e_{k+1},l^n_{k+1})\\left( - \\theta^\\sigma_{t^e}(l^e_{k+1},l^n_{k+1})  \\right) +    \\sum_{t^n \\subset \\sigma} A_{t^n} (l^e_{k+1},l^n_{k+1})   \\left( \\pi - \\theta^\\sigma_{t^n} (l^e_{k+1},l^n_{k+1}) \\right)  \\; .{\\quad}\\q\n\\end{eqnarray}\nThe evolution equations are then\n\\begin{eqnarray}\\label{4c6}\n&&l^b_k\\,=\\,l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\n&&l^e_k\\,=\\,l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\sigma}{\\partial l^e_{k+1}}\\,=\\, p^k_e -   \\sum_{t^e\\subset \\sigma}\n \\frac{\\partial A_{t^e}}{\\partial l^e_{k+1}} \\theta_{t^e}^\\sigma\n +\\sum_{t^n\\subset \\sigma} \\frac{\\partial A_{t^n}}{\\partial l^e_{k+1}} \\left( \\pi-\\theta_{t^n}^\\sigma\\right)\n  \\,,{\\quad}\\\\\n&&p^{k}_n\\,=\\,0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,=\\, \\frac{\\partial S_\\sigma}{\\partial l^n_{k+1}}{\\quad}\\q\\;\\,=\\, {\\quad}\\q\n\\sum_{t^n\\subset \\sigma} \\frac{\\partial A_{t^n}}{\\partial l^n_{k+1}} \\left( \\pi-\\theta_{t^n}^\\sigma\\right)\n. \\label{4c6c}\n\\end{eqnarray}\n\nAs for the 1--3 move the $l^n_k$ and $l^n_{k+1}$ remain arbitrary, nevertheless, we can equire $l^n_k=l^n_{k+1}$. Again, we encounter post--constraints for every new edge, namely the second equation in (\\ref{4c6c}). These post--constraints fix the momenta as a specific combination of the exterior angles. At a four--valent vertex, which can be seen as the tip of a four--simplex, this combination of exterior angles can be expressed as a function of the adjacent length variables (namely the lengths of the simplex $\\sigma$). Such a post--constraint will always appear if we produce a boundary edge in the course of evolution at which no bulk triangles (as potential carriers of curvature) are hinging.\n\nIn the later discussion we shall see that these constraints are, in general, not preserved by the other moves. Additionally, the free data, that is the lenghts of the four new edges, may, in general, become fixed by pre--constraints of later moves. This is related to the fact, that the Regge discretization does not preserve the diffeomorphism symmetry of the continuum \\cite{bd1,dithoe1}. The constraints and the free data discussed here, therefore, correspond to the Hamiltonian and diffeomorphism constraints and the gauge freedom (of lapse and shift) of the continuum. Through the breaking of diffeomorphism symmetry by discretization, the constraints encountered here will generally not be preserved by the other Pachner moves but, instead, turned into so--called pseudo--constraints \\cite{bd1,dithoe1,gambini,bianca}. These are equations of motion for canonical data of two different times, in which these data are only weakly coupled to each other. (More precisely, the eigenvalues of the appropriate Hessian matrix associated to these equations of motion will turn out to be small compared to those of a Hessian associated to proper evolution equations.)\n\n\n\n\\subsection{The 2--3 Pachner move}\n\n\nNext, let us discuss the 2--3 move which turns out to generate a dynamical (or physical) degree of freedom.\n\nConsider the situation in which a 4--simplex $\\sigma$ is glued to $\\Sigma_k$ in such a way that two of its (adjacent) tetrahedra are identified with two (adjacent) tetrahedra in $\\Sigma_k$. This gluing process is only possible when these two tetrahedra in $\\Sigma_k$ are {\\it not} part of the same 4--simplex in the underlying triangulation (for the five vertices of these two tetrahedra are already the five vertices of $\\sigma$). In contrast to the 2--2 move in 3D, the extrinsic curvature angle around the triangle $t^o$ along which the two adjacent tetrahedra are identified need {\\it a priori} (i.e.\\ kinematically) not be negative.\\footnote{Due to the flat embedding in 3D Regge Calculus (without a cosmological constant), the extrinsic angle $\\psi^k_o$ around the edge $e=o$ in the configuration of the 2--2 Pachner move in section \\ref{22sec} must be negative for a 2--2 gluing move to be possible such that eventually $\\epsilon_o=0$ (for non-degenerate simplices, the dihedral angles are always positive). } On the other hand, we can remove such a simplex from the triangulation (or, equivalently, add a simplex with opposite orientation), if the extrinsic angle around $t^o$ is positive, the two adjacent tetrahedra reside in the same $\\sigma$ and there is a piece of triangulation underneath $\\Sigma_k$. From the perspective of the hypersurface, this evolution move amounts to a 2--3 Pachner move.\n\\begin{figure}[htbp!]\n\\psfrag{1}{2--3}\n\\psfrag{2}{3--2}\n\\psfrag{t}{$t^o$}\n\\psfrag{n}{$n$}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure15.eps} \\end{center}\n\\caption{\\small The 2--3 Pachner move and its inverse, the 3--2 Pachner move.}\\label{23m}\n \\end{figure}\n \nTwo tetrahedra sharing a triangle $t^o$ and having a boundary consisting of six triangles $t^e$ and nine edges $e$ are replaced by three tetrahedra sharing a new edge $n$ and sharing pairwise three new triangles $t^n$, see figure \\ref{23m}. As for all the Pachner moves, the boundary of the three new tetrahedra is the same as for the two tetrahedra we started with.\nConsequently, during this move no edge will become internal and, therefore, there will be no additional equations of motion which we could solve for the length of the new edge. Furthermore, the triangle $t^o$ becomes a bulk triangle in the gluing move. This is important, since after the move, the bulk triangle can carry a non--vanishing deficit angle, that is curvature. This deficit angle depends on the value of the new edge and as this can be freely chosen, we are generating a curvature degree of freedom.\\footnote{For instance, it is possible to generate the complete 4D star of a  triangle, i.e.\\ rendering it internal, by starting out with one 4-simplex and performing a sequence of 1--4 Pachner moves on it and a final 2--3 move. During this sequence, the triangle shared by all simplices has become internal, but there are only boundary edges. This star of the triangle, therefore, carries curvature without any internal edges. Hence, there are no Regge equations to be satisfied.} This is different from the four free edge lengths that arise in the 1--4 move which rather correspond to lapse and shift, and, therefore, gauge variables, in the continuum.\n\n\nBecause of the new edge at step $(k+1)$, we extend the phase space at step $k$ by the pair $(l^n_k,p_n^k)$. We use the generating function\n\\begin{eqnarray}\\label{c12}\nG_{2-3}(l^b_{k+1},p_b^k;\\,l^e_{k+1},p_e^{k};\\,l^n_k,l^n_{k+1})\n&=&\n\\sum_b l^b_{k+1}p_b^k + \\sum_e l^e_{k+1}p_e^k  \\, + \\, S_\\sigma(l^e_{k+1},l^n_{k+1}) {\\quad}\\q,\n\\end{eqnarray}\nwhere in this case the action of the single 4--simplex under consideration is\n\\begin{eqnarray}\nS_\\sigma(l^e_{k+1},l^n_{k+1})& =&\n \\sum_{t^e \\subset \\sigma} A_{t^e}(l^e_{k+1},l^n_{k+1})\\left( - \\theta^\\sigma_{t^e}(l^e_{k+1},l^n_{k+1})  \\right) + A_{t^o} (l^e_{k+1},l^n_{k+1})   \\left( \\pi - \\theta^\\sigma_{t^o} (l^e_{k+1},l^n_{k+1}) \\right) +\\nonumber\\\\\n &&{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}     \\sum_{t^n \\subset \\sigma} A_{t^n} (l^e_{k+1},l^n_{k+1})   \\left( \\pi - \\theta^\\sigma_{t^n} (l^e_{k+1},l^n_{k+1}) \\right)  {\\quad} .\n\\end{eqnarray}\nThe evolution equations read\n\\begin{eqnarray}\\label{c13}\nl^b_k&=&l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\nl^e_k&=&l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\sigma}{\\partial l^e_{k+1}}\\,=\\, p^k_e\n-\\sum_{t^e\\subset\\sigma} \\frac{\\partial A_{t^e}}{\\partial l^e_{k+1}} \\theta^\\sigma_{t^e} +\\frac{\\partial A_{t^o}}{\\partial l^e_{k+1}}(\\pi-\\theta^\\sigma_{t^o} )  \\nonumber\\\\\n&&{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\n+\\sum_{t^n\\subset \\sigma} \\frac{\\partial A_{t^n}}{\\partial l^e_{k+1}} (\\pi-\\theta^\\sigma_{t^n})\n\\; ,    \\label{c13b}  \\\\\np^{k}_n&=&0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,=\\, \\frac{\\partial S_\\sigma}{\\partial l^n_{k+1}}\\,{\\quad}\\q\\;=\\, \\sum_{t^n\\subset \\sigma} \\frac{\\partial A_{t^n}}{\\partial l^n_{k+1}} (\\pi-\\theta^\\sigma_{t^n})\n{\\quad} .\\label{c13c}\n\\end{eqnarray}\n\nThe second equation in (\\ref{c13b}) determines the momenta $p_e^{k+1}$ at time $k+1$ as a function of the ones at time $k$, as well as the new edge lengths $l^n_{k+1}$ and $l^e_{k+1}=l^e_k$. The new lengths $l^n_{k},l^n_{k+1}$ remain undetermined, but we may again choose $l^n_k=l^n_{k+1}$. Accordingly, we find a post--constraint at time $(k+1)$, namely the second equation in (\\ref{c13c}). It determines the new momentum variable as a function of the edge lengths at step $(k+1)$. Geometrically this constraint arises as by construction there are no bulk, but only boundary triangles hinging at the new edge. Hence, we can compute the exterior angle from the boundary geometry alone, i.e.\\ the length variables of the hypersurface $\\Sigma_{k+1}$.\n\n\n\n\n\n\n\n\\subsection{The 3--2 Pachner move}\n\n\nThe 3--2 Pachner move is the inverse of the 2--3 move: a 4-simplex $\\sigma$ is glued onto $\\Sigma_k$ in such a way that three of its (adjacent) tetrahedra, sharing an edge $o$, are identified with three adjacent tetrahedra in $\\Sigma_k$. This gluing is possible only when these three tetrahedra reside in three distinct 4--simplices of the triangulation underlying $\\Sigma_k$. As in the case of the 2--3 move and, by virtue of the possible absence of a flat embedding, the extrinsic angles at the three triangles $t^o$ which are each shared by two of the three tetrahedra in $\\Sigma_k$ need {\\it a priori} not be negative; the values of the resulting deficit angles around these triangles will eventually be determined by the dynamics. Likewise, one can remove a simplex $\\sigma$ from the triangulation underlying $\\Sigma_k$ (or, equivalently, glue a simplex with opposite orientation to $\\Sigma_k$), if the three extrinsic angles are positive and the three tetrahedra are part of the same 4--simplex. These gluings\/removals appear as 3--2 Pachner moves, that is three tetrahedra sharing one edge $o$ and pairwise three triangles $t^o$ are replaced with two tetrahedra sharing one triangle $t^n$, such that the boundary of the two complexes, consisting of six triangles $t^e$ and nine edges $e$ remains unchanged (see figure \\ref{32m}).\n\\begin{figure}[htbp!]\n\\psfrag{1}{2--3}\n\\psfrag{2}{3--2}\n\\psfrag{t}{$t^n$}\n\\psfrag{n}{$o$}\n\\begin{center}\\includegraphics[scale=.5]{Figures\/figure15.eps} \\end{center}\n\\caption{\\small The 2--3 Pachner move and its inverse, the 3--2 Pachner move.}\\label{32m}\n \\end{figure}\nThe edge $o$ will become internal, in consequence we will have to satisfy the equation of motion corresponding to this edge. This equation of motion will be implemented via a pre--constraint and momentum matching. It may either not be possible at all to satisfy this pre--constraint, or only for specific choices of parameters, that is lengths variables, which appeared in previous moves.\n\nSince we have a variable $l^o_k$ that does not appear at step $(k+1)$, we extend the phase space at time $(k+1)$ by the pair $(l^o_{k+1},p_o^{k+1})$. The generating function for the 3--2 move reads\n\\begin{eqnarray}\\label{c132}\nG_{3-2}(l^b_k,p_b^{k+1}; l^e_k,p_e^{k+1};l^o_k,l^o_{k+1})&=&\n-\\sum_b l^b_{k}p_b^{k+1} -\\sum_e l^e_{k}p_e^{k+1}  \\, + \\, S_\\sigma(l^e_{k},l^o_k){\\quad},\n\\end{eqnarray}\nwhere the action $S_\\sigma$ is given by\n\\begin{eqnarray}\nS_\\sigma(l^e_{k},l^o_{k})& =&\n \\sum_{t^e \\subset \\sigma} A_{t^e}(l^e_{k},l^o_{k})\\left( - \\theta^\\sigma_{t^e}(l^e_{k},l^o_{k})  \\right) + A_{t^n} (l^e_{k},l^o_{k})   \\left( \\pi - \\theta^\\sigma_{t^n} (l^e_{k},l^o_{k}) \\right) +\\nonumber\\\\\n &&{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}     \\sum_{t^o \\subset \\sigma} A_{t^o} (l^e_{k},l^o_{k})   \\left( \\pi - \\theta^\\sigma_{t^o} (l^e_{k},l^o_{k}) \\right)  {\\quad} .\n\\end{eqnarray}\nThe evolution equations amount to\n\\begin{eqnarray}\\label{c14}\n&&l^b_{k+1}\\,=\\,l^b_{k}  {\\quad} ,{\\quad}\\q p^{k}_b\\,=\\,p^{k+1}_b {\\quad} , \\\\\n&&l^e_{k+1}\\,=\\,l^e_{k}{\\quad},{\\quad}\\q p^{k}_e\\,=\\,p^{k+1}_e-\\frac{\\partial S_\\sigma}{\\partial l^e_{k}}\\,=\\,\np^{k+1}_e\n+\\sum_{t^e\\subset\\sigma} \\frac{\\partial A_{t^e}}{\\partial l^e_{k}} \\theta^\\sigma_{t^e} -\\frac{\\partial A_{t^n}}{\\partial l^e_{k}}(\\pi-\\theta^\\sigma_{t^n} )  \\nonumber\\\\\n&&{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\\q{\\quad}\n-\\sum_{t^o\\subset \\sigma} \\frac{\\partial A_{t^o}}{\\partial l^e_{k}} (\\pi-\\theta^\\sigma_{t^o})\n{\\quad} ,\\label{c14b}\\\\\n&&p^{k+1}_o\\,=\\,0 {\\quad},{\\quad}\\q p^{k}_o\\,=\\,- \\frac{\\partial S_\\sigma}{\\partial l^o_{k}}{\\quad}\\q\\;\\,=\\,\n -\\sum_{t^o\\subset \\sigma} \\frac{\\partial A_{t^o}}{\\partial l^o_{k}} (\\pi-\\theta^\\sigma_{t^o})\n  {\\quad} .\\label{c14c}\n\\end{eqnarray}\n\n\nThis time the second equation in (\\ref{c14b}) determines the momenta $p_e^{k+1}$ at step $(k+1)$ as a function of the ones at time $k$ and the old  edge lengths $l^o_{k}$ and $l^e_{k}$.\n\nThe second equation in (\\ref{c14c}), on the other hand, is a pre--constraint. It requires that the momentum $p^k_o$ be given as a function of the edge lengths at time $k$. This equation is automatically satisfied if the edge $o$ has just been created by a 2--3 move, but in general this condition may not be satisfied. In this case, one either has to tune the new length variables that arose in previous 2--3 or 1--4 moves, or perform a different move.\n\n\n\\subsection{The 4--1 Pachner move}\n\nFinally, let us consider the 4--1 move, which is the inverse of the 1--4 move (see figure \\ref{14m}). A complex of four tetrahedra sharing a vertex, four edges $o$ and six triangles $t^o$ is replaced by one tetrahedron, such that the boundary of four triangles $t^e$ and six edges $e$ remains unchanged. The 4--1 gluing move is only possible if these four tetrahedra are part of four different 4--simplices in the underlying triangulation, while the six extrinsic angles at the six $t^o$ are kinematically unrestricted. From the 4D perspective the four `old' edges become bulk edges, consequently, we will have four equations of motion to satisfy. These take the form of pre--constraints and eventually determine the resulting six deficit angles around the $t^o$. The prerequisite for the 4--1 removal move, on the other hand, is clearly that the four relevant tetrahedra reside in the same 4--simplex and the six extrinsic angles at the six $t^o$ are positive.\n\n\n We extend the phase space at time $(k+1)$ by the four pairs $(l^o_{k+1},p_o^{k+1})$ and define the generating function\n\\begin{eqnarray}\\label{4c41}\nG_{4-1}(l^b_{k},p_b^{k+1};\\,l^e_{k},p_e^{k+1};\\,l^o_k,l^o_{k+1}) &=&- \\sum_b l^b_{k}p_b^{k+1} - \\sum_e l^e_{k}p_e^{k+1}  \\, + \\, S_\\sigma(l^e_{k},l^o_{k})  {\\quad},\n\\end{eqnarray}\nwhere we use the action of the single 4--simplex under consideration\n\\begin{eqnarray}\nS_\\sigma&=& \\sum_{t^e \\subset \\sigma} A_{t^e}(l^e_{k},l^o_{k})\\left( - \\theta^\\sigma_{t^e}(l^e_{k},l^o_{k})  \\right) +    \\sum_{t^o \\subset \\sigma} A_{t^o} (l^e_{k},l^o_{k})   \\left( \\pi - \\theta^\\sigma_{t^o} (l^e_{k},l^o_{k}) \\right)  \\; .{\\quad}\\q\n\\end{eqnarray}\nThe equations of motion are then\n\\begin{eqnarray}\\label{4c6d}\n&&l^b_k\\,=\\,l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k}_b\\,=\\,p^{k+1}_b {\\quad} , \\\\\n&&l^e_k\\,=\\,l^e_{k+1}{\\quad},{\\quad}\\q p^{k}_e\\,=\\,p^{k+1}_e-\\frac{\\partial S_\\sigma}{\\partial l^e_{k}}\\,=\\, p^{k+1}_e +   \\sum_{t^e\\subset \\sigma}\n \\frac{\\partial A_{t^e}}{\\partial l^e_{k}} \\theta_{t^e}^\\sigma\n -\\sum_{t^o\\subset \\sigma} \\frac{\\partial A_{t^o}}{\\partial l^e_{k}} \\left( \\pi-\\theta_{t^o}^\\sigma\\right)\n  \\,,{\\quad}\\\\\n&&p^{k+1}_o\\,=\\,0 {\\quad},{\\quad}\\q  p^{k}_o\\,=\\, -\\frac{\\partial S_\\sigma}{\\partial l^o_{k}}{\\quad}\\q\\;\\,=\\, {\\quad}\n-\\sum_{t^n\\subset \\sigma} \\frac{\\partial A_{t^o}}{\\partial l^o_{k}} \\left( \\pi-\\theta_{t^o}^\\sigma\\right)\n. \\label{4c6e}\n\\end{eqnarray}\n\nThe $l^o_{k+1}$ remain undetermined, as these do not appear at all. Again, we can choose $l^o_k=l^o_{k+1}$ to carry these data along as boundary data.  The second equation in (\\ref{4c6e}) represents four pre--constraints, that have to be satisfied by the canonical data at time $k$, so that the 4--1 move can be performed. This will, in general, restrict the free parameters which appeared in the previous moves.\n\n\n\\subsection{Example: 4D tent moves}\\label{tent2}\n\n\nAs a specific example, we can also consider tent moves in 4D which, as already explained in section \\ref{sectent1}, can be obtained by a succession of Pachner moves.\nIn particular, an $N$--valent tent move in 4D can be reproduced through a sequence of one 1--4 move, $(N-3)$ 2--3 moves, $(N-3)$ 3--2 moves and one final 4--1 Pachner move.\\footnote{Note that the ordering of the 2--3 and 3--2 Pachner moves is not entirely fixed: one always has to start with a 1--4 move to introduce the new vertex $v_{n+1}$ and a subsequent 2--3 move and finish off with a 3--2 move prior to the final 4--1 move (which removes $v_n$ from $\\Sigma_{n+1}$), but the ordering of the 2 - 3 and 3 - 2 moves in between can be chosen freely (according to the given configuration).}\n\nHere, we will discuss two tent move configurations: the first one at a four--valent vertex leads to flat dynamics, while, in contrast to this, the second example involves curvature and the role of the pre-- and post--constraints differs considerably from that in the first example.\n\n\n\n\n\n\nTo begin with, consider the simplest tent move configuration in 4D, namely the four--valent tent move. Starting from a four-valent vertex $v_0$ in the boundary surface $\\Sigma_0$ of a single 4--simplex, perform first one 1--4 and one 2--3 move which introduce five new (freely choosable) edge lengths. Finally, perform one 3--2 move to remove one of the edges $e(v_0i)$, $i\\in1,\\dots,4$, and one 4--1 move to remove the remaining three edges $e(v_0j)$ and the tent pole $e(v_0v_{1})$ from the new $\\Sigma_{1}$ and solve the corresponding five equations of motion or  pre--constraints.\n\n\nAfter the 3--2 move the boundary configuration, in fact, corresponds to the configuration of a {\\it stacked sphere}, namely to the configuration of one simplex on which one 1--4 move has been performed with the tip of the second simplex pushed ``inwards\". A {\\it stacked sphere} is a triangulation of the 3--sphere which can be obtained by performing a sequence of 1--4 Pachner moves on the 3D boundary surface of a single 4--simplex and therefore necessarily possesses a 4D flat interior as there are no internal edges. Any triangulation whose boundary corresponds to a stacked sphere configuration possesses flat solutions independent of the existence of internal triangles (up to possible discretization artifacts \\cite{bd1}).\n\n\nThat is, the 3--2 Pachner move plays a key role here in that its pre--constraint (or equation of motion) generally imposes flatness of the deficit angle around the internal triangle generated in the course of the previous 2--3 move and thus establishes one non-trivial condition among the five free parameters of the 1--4 and 2--3 moves. Hence, there exists a four parameter family of solutions to the equation of motion of the 3--2 Pachner move, all of which correspond to flat geometries. Therefore, these solutions also automatically solve the equations of motion of the final 4--1 Pachner move (the boundary configuration after the 4--1 move also corresponds to a stacked sphere). Thus, the four pre--constraints of the final 4--1 move are automatically satisfied and no further non--trivial conditions on the remaining four parameters arise. Accordingly, after the 4--1 Pachner move four parameters coordinatizing the position of the vertex remain free and one obtains a four--fold gauge symmetry associated to lapse and shift degrees of freedom.\n\n\n\nThis situation changes for $N$-valent tent moves with $N\\geq5$, since the boundary configuration at none of the individual Pachner move steps corresponds to a  stacked sphere and therefore does not necessarily imply flatness. But non-vanishing curvature in Regge Calculus generically breaks the vertex displacement gauge symmetry \\cite{bd1,dithoe1} and all free parameters become fixed. In particular, both the 3--2 and 4--1 move attach internal triangles to the edges in the hypersurface, which generally introduces a dependence on data from other evolution steps, e.g.\\ see equation (\\ref{y6}). The (post--)constraints after the 1--4 move will, in general, not be preserved by the following Pachner moves. Instead, these transform into pseudo-constraints \\cite{dithoe1}. Pseudo--constraints can be understood as equations between the phase space data at time $k$ which further depend (weakly) on phase space data from the neighbouring time steps.\n\nAs the second example consider, for simplicity, a `symmetry--reduced' tent move at a five--valent vertex $v_0$, also used in \\cite{bd1,dithoe1}, where only two dynamical length variables, $a_n$ and $b_n$, arise at each tent move step $n$, apart from the tent pole length. There are six tetrahedra in $\\Sigma_0$ with\nvertices\n\\begin{equation}\\label{tent1} v_0124,{\\quad} v_0134,{\\quad} v_0234, {\\quad}  v_0125, {\\quad}  v_0135,\n{\\quad} v_0235  {\\quad} .\n\\end{equation}\nAccordingly, we will have nine triangles of the\nform $t(v_0ij)$ with $i,j=1,\\ldots 5$ in this triangulation, five edges\nof the form $e(v_0i)$ and nine edges of the form $e(ij)$  (all possible\nordered combinations of $i,j\\in \\{1,\\ldots 5\\}$ with the exception\n$45$). This situation is depicted in figure \\ref{5vsym}.\n\n\\begin{figure}[hbt!]\n\\begin{center}\n    \\psfrag{v}{$v_0$}\n    \\psfrag{a}{$a_0$}\n    \\psfrag{b}{$b_0$}\n    \\psfrag{1}{$1$}\n    \\psfrag{2}{$2$}\n    \\psfrag{3}{$3$}\n    \\psfrag{4}{$4$}\n   \\psfrag{5}{$5$}\n   \\includegraphics[scale=0.4]{Figures\/figure16.eps}\n    \\end{center}\n    \\caption{\\label{5vsym}{\\small Illustration of the symmetry--reduced 3d star of a five--valent vertex $v_0$, consisting of the six tetrahedra (\\ref{tent1}).}}\n\\end{figure}\n\n\\noindent\n\n`Symmetry--reduction' here means that all the lengths of the boundary edges $e(ij)$ are set to $1$ \\footnote{This is possible since the vacuum Regge equations are invariant under global rescalings.} and imposing $l^{e(v_ni)}=a_n$, $i=1,2,3$ and $l^{e(v_n4)}=l^{e(v_n5)}=b_n$ at each tent move step $n$.\n\nThe 4--simplices associated to the Pachner moves reproducing this tent move are then all of the same type\n$\\sigma(v_0v_{1}ij\\kappa)$, where $i,j$ take values in\n$1,2,3$ and $\\kappa$ in $4,5$. Hence, the first 1--4 move already introduces all three new (freely choosable) parameters $a_1,b_1$ and the length of the tent pole between $v_0$ and $v_{1}$ which we will call $t_1$. Specifically, let us consider the case where we glue the six 4-simplices onto the tetrahedra (\\ref{tent1}) in the following order: 1.\\ $\\sigma(v_0v_1124)$, 2.\\ $\\sigma(v_0v_1134)$, 3.\\ $\\sigma(v_0v_1125)$, 4.\\ $\\sigma(v_0v_1135)$, 5.\\ $\\sigma(v_0v_1234)$, 6.\\ $\\sigma(v_0v_1235)$, which corresponds to one 1--4 move, two subsequent 2--3 moves followed by two 3--2 moves and a final 4--1 move. These Pachner moves are then to be implemented canonically by equations (\\ref{4c6},\\ref{4c6c},\\ref{c13},\\ref{c13b},\\ref{c13c},\\ref{c14},\\ref{c14b},\\ref{c14c},\\ref{4c6d},\\ref{4c6e}). The necessary action contribution of each of these six simplices is of the general form\n\\begin{eqnarray}\nS_{\\sigma(v_0v_{1}ij\\kappa)}&=&2A^a_t(c_1\\pi-\\theta^a_t)+A^b_t(c_2\\pi-\\theta^b_t)+A^0_a(c_3\\pi-\\theta^0_a)\\nonumber\\\\\n&&+A^1_a(c_4\\pi-\\theta^1_a)+2A^0_b(c_5\\pi-\\theta^0_b)+2A^1_b(c_6\\pi-\\theta^1_b)-A\\theta{\\quad},\n\\end{eqnarray}\nwhere\n\\begin{itemize} \\parskip -3pt\n\\item[~] $\\theta^0_a,\\, A^0_a$ are the dihedral angle and the area of the triangle $\\Delta(v_0ij)$,\n\\item[~] $\\theta^0_b,\\, A^0_b$ are the dihedral angle and the area of the triangle $\\Delta(v_0i\\kappa) $,\n\\item[~] $\\theta^a_t,\\, A^a_t$ are the dihedral angle and the area of the triangle $\\Delta(v_0v_1i )$,\n\\item[~] $\\theta^b_t,\\, A^b_t$ are the dihedral angle and the area of the triangle $\\Delta(v_0v_1\\kappa) $,\n\\item[~] $\\theta^1_a,\\, A^1_a$ are the dihedral angle and the area of the triangle $\\Delta(v_1ij)$,\n\\item[~] $\\theta^1_b,\\, A^1_b$ are the dihedral angle and the area of the triangle $\\Delta(v_1i\\kappa)$\n\\item[~] $\\theta,\\, A$ are the dihedral angle and the area of the triangle $\\Delta(ij\\kappa)$, \n\\end{itemize}\nrespectively, and $c_m\\in\\{0,\\textstyle{\\frac{1}{2}},1\\}$, $m=1,\\ldots,6$, depending on whether the corresponding triangle has been present prior to the move under consideration, is newly introduced or becomes internal (see the discussion in the paragraph following (\\ref{regge2}) in section \\ref{secregcal}). \n\nIn the sequel we confine our focus to pre-- rather than post--constraints since the latter are automatically satisfied after the moves. The pre--constraint for $e(v_01)$ of the first 3--2 move coincides with the equation of motion for  the $a_0$--edges, the pre--constraint for $e(v_04)$ of the second 3--2 move with that of the $b_0$--edges and the single pre--constraint of the 4--1 move (recall the `symmetry--reduction') is equivalent to the equation of motion for the length $t_{1}$ of the tent pole $e(v_0v_{1})$. \n\nIn the general (not symmetry--reduced) situation, we would have four parameters introduced by the first 1--4 move and a further 2 parameters introduced by the two following 2--3 moves. The latter two parameters determine the curvature, that is the deficit angles on the triangles which become bulk triangles during the completion of these moves. In the symmetry--reduced situation, on the other hand, the 1--4 move already introduces all three parameters that are allowed by our choice of symmetry--reduction. One of these corresponds to a lapse degree of freedom (determining the height of the tent pole), the other two can be interpreted as determining the value of the curvature. As we shall discuss, in general all parameters introduced by the 1--4 and 2--3 moves will be fixed by the pre--constraints of the subsequent 3--2 and 4--1 moves. \n\n\n\n\nAt the level of the momenta, the above symmetry-reduction only holds when a tent move step is complete, however, not for the individual Pachner moves into which the tent move can be decomposed, since the momenta at the different edges (of identical length) get updated in different order depending on the order of the Pachner moves. For this reason, we consider the momenta of each of the edges individually, which we denote by $p^0_i,p^0_\\kappa$ for the initial data and $p^1_i,p^1_\\kappa$ for those associated to the new edges. (Note that the upper index here counts tent rather than Pachner moves.)\n\nThe whole point of elaborating on this example is to demonstrate that, in contrast to the four--valent tent move, it is possible to solve only a subset of the equations of motion (or equivalently, pre--constraints) for $a_0,b_0,t_1$ and violate the remaining ones. This implies that the pre--constraints of the five--valent tent move are, in general, independent, not automatically satisfied and fix parameters which were previously free. \n\nFor instance, consider the following three examples based on the same initial data $a_0=1.1690$, $p^0_1=p^0_2=p^0_3=1.5088$, $b_0=1.1436$, $p^0_4=p^0_5=1.2272$: a) table \\ref{tab:1} shows the example of a situation where only the pre--constraint for $a_0$ is solved but not the ones for $b_0$ and $t_1$, b) table \\ref{tab:2} provides an example where the pre--constraints for $a_0$ and $b_0$ are solved, but not the one for $t_{1}$, and c) table \\ref{tab:3} demonstrates an example where all three pre--constraints are satisfied. Vanishing momenta indicate that a constraint (or equation of motion) is satisfied. In these three examples we choose to use the pre--constraints for $a_0,b_0$ to fix the new lengths $a_1,b_1$, respectively, and the pre--constraint for $t_1$ to fix $t_1$,\\footnote{In fact, prior to the 3--2 moves and the implementation of the pre--constraints the new lengths can, in principle, be of any value and only get tuned after imposing the constraints. However, here we set the lengths of the new edges to the value determined by one or more pre--constraints already from the start.} but note that other choices of which length to fix by which constraint are possible. \n\n\n\\begin{table*}[h] \\scriptsize{ \\caption{\\small Momentum updating for the five--valent symmetry--reduced tent move decomposed into Pachner moves. Only the pre--constraint for $a_0$ is eventually satisfied. The rows provide the updated momentum values after the Pachner move given in the left column (following the sequence of the main text). Initial data as given in the text. Two of the three new lengths of the first 1--4 move were chosen as $\\mathfrak{t}}  \\newcommand{\\Ft}{\\mathfrak{T}_1=0.2600$, $b_1=1.3400$. The 3--2 move pre--constraint (\\ref{c14c}) for edge $e(v_01)$ (which translates into $p^0_1=0$ after the move) is solved for $a_1=1.4052$. (Due to symmetry, the pre--constraints (\\ref{4c6e}) associated to $e(v_02)$ and $e(v_03)$ of the 4--1 move are then automatically satisfied and translate into $p^0_2=0=p^0_3$ after the final move.) The pre--constraint for the edge $e(v_04)$ of the second 3--2 move and the pre--constraints of the 4--1 move for edge $e(v_05)$ and the tent pole are violated, i.e.\\ $p^0_4,p^0_5,p_{\\mathfrak{t}}  \\newcommand{\\Ft}{\\mathfrak{T}_1}\\neq0$.  }\n\\vspace{.2cm}\n\\begin{tabular}{|c|| c|c|c|c|c|c|c|c|c|c|c||} \\hline move & $p^0_1$\n& $p^0_2$ & $p^0_3$ & $p^0_4$ & $p^0_5$ & $p_{t_1}$ & $p^1_1$\n& $p^1_2$ & $p^1_3$ & $p^1_4$ & $p^1_5$\\\\ \\hline\\hline\n1-4 &2.5119& 2.5119& 1.5088& 0.2082& 1.2272& 5.4351& -1.0316&\n  -1.0316& 0& 1.1903& 0\\\\ \\hline\n2-3 & -0.1145& 2.5119& 2.5119& -2.2703& 1.2272& 3.1184& 1.5135&\n  -1.0316& -1.0316& 3.5106& 0\\\\ \\hline\n2-3 & -2.7011& -0.0747& 2.5119& -2.2703& 0.2082&\n  -1.0557& 3.9286& 1.3834& -1.0316& 3.5106& 1.1903\\\\ \\hline\n3-2 & 0& -0.0747& -0.0747& -2.2703& -2.2703&\n  -3.3723& 1.3463& 1.3834& 1.3834& 3.5106& 3.5106\\\\ \\hline\n3-2 & 0& -2.7011& -2.7011& -1.4644& -2.2703&\n  -7.5464& 1.3463& 3.9286& 3.9286& 2.7552& 3.5106\\\\ \\hline\n4-1 & 0& 0& 0& -1.4644& -1.4644&\n   -2.1113& 1.3463& 1.3463& 1.3463& 2.7552& 2.7552\\\\ \\hline\n\\end{tabular}\n\\label{tab:1}}\n\\end{table*}\n\n\\begin{table*}[h] \\scriptsize{ \\caption{\\small Momentum updating for the five--valent symmetry--reduced tent move decomposed into Pachner moves. Only the pre--constraints for both $a_0, b_0$ are eventually satisfied. Further explanation and initial data are given in the caption of table \\ref{tab:1} and the text, respectively. We chose $\\mathfrak{t}}  \\newcommand{\\Ft}{\\mathfrak{T}_1=0.2000$. The pre--constraints for edges $e(v_01)$ and $e(v_04)$ of the two 3--2 moves were solved for $a_1=1.3448$, $b_1=1.2985$. The pre--constraint for the tent pole arising in the final 4--1 move is `almost solved', however, still violated ($p_{\\mathfrak{t}}  \\newcommand{\\Ft}{\\mathfrak{T}_1}=0.0003> 0$, while the other constraints are satisfied to order $10^{-12}$). The latter is a consequence of the fact that the present configuration yields a near--flat geometry in which the vertex displacement symmetry of the flat regime is almost preserved \\cite{bd1,dithoe1}. }\n\\vspace{.2cm}\n\\begin{tabular}{|c|| c|c|c|c|c|c|c|c|c|c|c||} \\hline move & $p^0_1$\n& $p^0_2$ & $p^0_3$ & $p^0_4$ & $p^0_5$ & $p_{t_1}$ & $p^1_1$\n& $p^1_2$ & $p^1_3$ & $p^1_4$ & $p^1_5$\\\\ \\hline\\hline\n1-4 &2.1317& 2.1317& 1.5088& 0.7195& 1.2272& 5.1219& -0.6116&\n  -0.6116& 0& 0.6253& 0\\\\ \\hline\n2-3 &-0.1145& 2.1317& 2.1317& -1.3172& 1.2272& 3.1121& 1.5659&\n  -0.6116& -0.6116& 2.5562& 0\\\\ \\hline\n2-3 &-2.3208& -0.0747& 2.1317&\n  -1.3172& 0.7195& 0.0001& 3.6559& 1.4784&\n  -0.6116& 2.5562& 0.6253\\\\ \\hline\n3-2 &0& -0.0747& -0.0747& -1.3172& -1.3172&\n  -2.0097& 1.4397& 1.4784& 1.4784& 2.5562& 2.5562\\\\ \\hline\n3-2 &0& -2.3208& -2.3208& 0& -1.3172&\n  -5.1217& 1.4397& 3.6559& 3.6559& 1.3084& 2.5562\\\\ \\hline\n4-1 &0& 0&\n  0& 0& 0& 0.0003& \\\n1.4397& 1.4397& 1.4397& 1.3084& 1.3084\\\\ \\hline\n\\end{tabular}\n\\label{tab:2}}\n\\end{table*}\n\n\\begin{table*}[h] \\scriptsize{\\caption{\\small Momentum updating for an example of the five-valent symmetry-reduced tent move decomposed into Pachner moves where all three pre--constraints of $a_0, b_0, t_1$ are eventually satisfied. Further explanation and initial data as given in the caption of table \\ref{tab:1} and the text, respectively. The three pre--constraints are numerically solved by $t_1=0.3039$, $a_1=1.4387$, $b_1=1.3832$.}\n\\vspace{.2cm}\n\\begin{tabular}{|c|| c|c|c|c|c|c|c|c|c|c|c||} \\hline move & $p^0_1$\n& $p^0_2$ & $p^0_3$ & $p^0_4$ & $p^0_5$ & $p_{t_1}$ & $p^1_1$\n& $p^1_2$ & $p^1_3$ & $p^1_4$ & $p^1_5$\\\\ \\hline\\hline\n1-4 &2.2970& 2.2970& 1.5088& 0.7969& 1.2272& 5.4635& -0.7724&\n   -0.7724& 0& 0.6121& 0\\\\ \\hline\n2-3 &-0.1145& 2.2970& 2.2970& -1.3947& 1.2272& 3.3397& 1.5390&\n  -0.7724& -0.7724& 2.6485& 0\\\\ \\hline\n2-3 &-2.4861& -0.0747& 2.2970&\n  -1.3947& 0.7969& 0& 3.7361& 1.4247&\n  -0.7724& 2.6485& 0.6121\\\\ \\hline\n3-2 &0& -0.0747& -0.0747& -1.3947& -1.3947&\n  -2.1238& 1.4027& 1.4247& 1.4247& 2.6485& 2.6485\\\\ \\hline\n3-2 &0& -2.4861& -2.4861& 0& -1.3947&\n  -5.4635& 1.4027& 3.7361& 3.7361& 1.3528& 2.6485\\\\ \\hline\n4-1 &0& 0& 0& 0&\n  0& 0& 1.4027& 1.4027& 1.4027& \\\n1.3528& 1.3528\\\\ \\hline\n\\end{tabular}\n\\label{tab:3}}\n\\end{table*}\n\nIn conclusion, after the two 3--2 moves two of the three new parameters (i.e.\\ lengths) are, in general, fixed and one has a one parameter family of solutions to the two equations of motion of these two moves (with table \\ref{tab:2} giving one particular example). However, generally these solutions are {\\it not} automatically solutions to the equation of motion of the final 4--1 move. That is, here the three pre--constraints are, in general, independent and completely fix the three free parameters of the initial 1--4 move in which case solutions to all moves exist, but not a parameter family of solutions and therefore no gauge symmetry arises. Nevertheless, the subset of the one parameter family of solutions to the two equations of motion of the 3--2 moves corresponding to flat configurations {\\it does} automatically solve the pre--constraint of the 4--1 move and no further non-trivial condition arises on the parameters.\\footnote{There are two independent deficit angles in this symmetry--reduced setup, such that flatness imposes {\\it two} independent conditions among the three free parameters $a_{1},b_{1},t_{1}$ which allow to write, e.g.\\ $a_{1}$ and $b_{1}$ as a function of $t_{1}$. Due to flatness, by (\\ref{y2}), the {\\it three} equations of motion (or pre--constraints) for $a_0,b_0$ and $t_1$ are automatically satisfied.} That is, in this special case one parameter remains unrestricted after the 4 - 1 move (the length of the tent pole) and one obtains gauge symmetry.\n\nThis may be further understood from a covariant point of view, namely an analysis of the Hessian of the action (second partial derivatives of the action with respect to $t_1,a_1,b_1,t_2$) associated to two tent move steps, which was explored in \\cite{bd1}. There it was shown numerically that the Hessian possesses a non-vanishing eigenvalue (corresponding to the lapse degree of freedom) in the presence of curvature which approaches zero as the configuration approaches flatness. Furthermore, the momenta associated to the edges are in general given by\n\\begin{eqnarray}\np^n_e=-\\frac{\\partial\\tilde{S}_{n+1}}{\\partial l^e_n}{\\quad},\n\\end{eqnarray}\nwhere $\\tilde{S}_{n+1}$ is the effective action of tent move step $n+1$ with the length variable of the tent pole integrated out. In \\cite{bd1,dithoe1} it was shown that the matrix of derivatives of these momenta with respect to the $l^e_{n+1}$ is non-invertible too, if the Hessian is degenerate. In this flat case the conditions of the implicit function theorem are violated and we cannot solve for the lengths $l^e_{n+1}$ as a function of $l^e_n$ and $p^n_e$. That is, the data at step $n$ does {\\it not} determine the data at step $n+1$ and, in particular, the free parameters of the 1--4 and 2--3 moves are not all fixed, reflecting the gauge symmetry. Conversely, when the Hessian is invertible in the presence of curvature, it {\\it is} possible to solve for the $l^e_{n+1}$ as a function of the lengths and momenta at step $n$. For the symmetry--reduced five-valent tent move discussed here the latter means that, given the aim to construct a tent move from the data present at step $n$, the three {\\it a priori} free parameters of the 1--4 move are eventually completely fixed by the three pre--constraints of the 3--2 and 4--1 Pachner moves. There is therefore no gauge symmetry left in the presence of curvature.\n\nConsequently, in 4D for non-flat configurations, the pre--constraints provide  conditions among the canonical data (for instance, on the {\\it a priori} free parameters introduced in the previous 1--4 and 2--3 moves) and actually implement the equations of motions.  \nIn 3D the post--constraints arising after the 1--3 moves are preserved by the other moves and exactly match the pre--constraints which come with the 3--1 moves. Hence, in 3D we are in the situation where the pre--constraints are automatically satisfied by canonical data generated by previous Pachner moves, that is the pre--constraints coincide with the post--constraints.\n\nThe situation is different in 4D. Here the post--constraints from the 1--4 and 2--3 moves generally do not match up with the pre--constraints of the 4--1 and 3--2 moves. Thus, the parameters introduced by the 1--4 and 2--3 moves will, in general, become fixed by the pre--constraints of the 4--1 and 3--2 moves. (Of course, whether all parameters get fixed, depends, for example, on the number of different moves one is performing.) For initial data leading to flat configurations, however, we have seen that the pre--constraints for the 4--1 moves are automatically satisfied by the canonical data generated by the previous Pachner moves. This converts the four parameters introduced by the 1--4 move to four gauge degrees of freedom (in the sense that the solution for given boundary conditions on an initial and final time slice is not uniquely determined).\n\n\n\n\n\n\n\\section{Symplectic structures for discrete canonical dynamics} \\label{app}\n\nThe Pachner move dynamics involves phase spaces of different dimension. Hence, the question arises in which sense these maps can be canonical, i.e.\\ preserve a symplectic structure. We will discuss this issue in this section by first considering regular systems, and subsequently general singular systems, where we have to deal with constraints. \nIn particular, we will introduce in more detail the concepts of pre-- and post--constraints.\nFinally, we will consider the situation for Pachner moves, and, more generally, for dynamics (generated by an action) which involves a change in phase space variables.\n\n\n\\subsection{Regular systems}\\label{appreg}\n\nHere we will review the general Lagrangian and Hamiltonian dynamics of discrete systems; for regular systems, which we will define shortly, see the exposition \\cite{marsdenwest}. However, we need to specifically discuss singular systems, which appear in the Pachner move dynamics (see also the discussions in \\cite{gambini} on constraints in discrete dynamics).\n\nTo start with, we will consider regular systems, where by definition constraints will not occur.  Consider a system with configuration spaces ${\\cal Q}_k \\cong {\\cal Q}$ of equal dimension at every time step $k$. We coordinatize ${\\cal Q}_k$ by $x_k^i$ where $i$ is in some index set determined by the dimension of $\\cal Q$. Time evolution is generated by the discrete action contributions $S_k=S_k(x_{k-1},x_k)$. These discrete actions may be `effective ones', i.e.\\ arising from Hamilton's principal function and summarizing several basic time steps into one effective time step (as, e.g., in the tent moves).\n\nConsider three consecutive time steps $k=0,1,2$ and the boundary value problem between times $k=0$ and $k=2$. That is, we have boundary data $x_0^i$ and $x_2^i$ and must extremize\n\\begin{eqnarray}\\label{app1}\nS:=S_1(x_0,x_1)+S_2(x_1,x_2)\n\\end{eqnarray}\nwith respect to $x_1$. This gives the equations of motion\n\\begin{eqnarray}\\label{app2}\n0&=&\\frac{\\partial S_1}{\\partial x_1}+\\frac{\\partial S_2}{\\partial x_1}{\\quad},\n\\end{eqnarray}\nwhich we assume to be solvable uniquely for $x_1$ as a function of $x_0,x_2$. In case that\n\\begin{eqnarray}\\label{app3}\n\\text{det}\\frac{\\partial^2 S_2}{\\partial x_1\\,\\partial x_2} \\neq 0\n\\end{eqnarray}\nwe can invert these solutions for $(x_1,x_2)$ to obtain the Lagrangian time evolution map\n\\begin{eqnarray}\\label{app4}\n{\\cal L}_1: (x_0,x_1) \\mapsto (x_1,x_2)\n\\end{eqnarray}\nfrom ${\\cal Q}_0\\times {\\cal Q}_1$ to ${\\cal Q}_1\\times {\\cal Q}_2$.\nCondition (\\ref{app3}) has to hold for regular systems.\n\nFrom the discrete action we can define the so--called Lagrangian one-- and two--forms. For regular systems these will turn out to be pull--backs via the Legendre transform of the canonical one-- and two--forms. The significance of the Lagrangian two--form is, that it is preserved under the time evolution map (\\ref{app4}). This is the reason for the Hamiltonian time evolution map (in phase space) being symplectic with respect to the canonical two--form.\n\nTo start with, we define two canonical one--forms on ${\\cal Q}_1\\times {\\cal Q}_2$ from the variation of the action,\n\\begin{eqnarray}\\label{app5}\n\\theta^-_1(x_1,x_2) &=& -\\frac{\\partial S_2}{\\partial x_1^i} dx_1^i \\nonumber\\\\\n\\theta^+_2(x_1,x_2) &=& \\frac{\\partial S_2}{\\partial x_2^i} dx_2^i{\\quad},\n\\end{eqnarray}\nwhere we sum over repeated indices $i,j$. Note that $dS_2=\\theta_2^+-\\theta_1^-$ and, since $d\\circ d S_2=0$, we can define the (single) Lagrange two--form\n\\begin{eqnarray}\\label{app6}\n\\Omega_2(x_1,x_2)\\,=\\,-d\\theta^+_2\\,=\\,-d\\theta^-_1&=&- \\frac{\\partial^2 S_2}{\\partial x_1^i \\partial x_2^j} \\, dx_1^i \\, \\wedge dx_2^j  {\\quad} .\n\\end{eqnarray}\n\nThis time evolution map is preserved under the time evolution map (\\ref{app4}), i.e.\n\\begin{eqnarray}\\label{app7}\n\\Omega_1&=&{\\cal L}_1^* \\Omega_2{\\quad},\n\\end{eqnarray}\nwhere ${\\cal L}_1^*$ denotes the pull--back of ${\\cal L}_1$. To see this, consider $S$ from (\\ref{app1}) as a function on ${\\cal Q}_0\\times {\\cal Q}_1$ by using for $x_2$ the solutions $x_2(x_0,x_1)$ of the time evolution map (\\ref{app4}). Taking the exterior derivative of $S$ on  ${\\cal Q}_0\\times {\\cal Q}_1$, we will obtain only boundary terms because the equations of motion with respect to the inner variable $x_1$ hold. That is,\n\\begin{eqnarray}\\label{app8}\ndS\\,(x_0,x_1)&=& \\frac{\\partial S_1}{\\partial x^i_0}dx^i_0\\,\\,+\\,\\, \\frac{\\partial S_2}{\\partial x^j_2} \\left( \\frac{\\partial x^j_2}{\\partial x^i_0}dx^i_0 + \\frac{\\partial x^j_2}{\\partial x^i_1}dx^i_1\\right) \\nonumber\\\\\n&=& -\\theta_0^- \\,+\\, {\\cal L}_1^* \\theta_2^+ {\\quad} .\n\\end{eqnarray}\nAgain, as $d\\circ d=0$ and exterior derivatives commute with pull--backs, we also find\n\\begin{eqnarray}\\label{app8a}\n\\Omega_1&=& {\\cal L}_1^* \\Omega_2 {\\quad} .\n\\end{eqnarray}\nThis argument can be easily generalized to any time step difference $(k_1,k_2)$.\n\nIn order to discuss the dynamics in phase space, we introduce the Legendre transformations\n\\begin{eqnarray}\\label{app9}\n&&\\mathbb{F}^+S_k:  (x_{k-1},x_k) \\mapsto (x_k,{}^{+}p^k)\\,=\\,\\left(x_k,  \\frac{\\partial S_k}{ \\partial x_{k}}\\right) \\nonumber\\\\\n&&\\mathbb{F}^-S_k:  (x_{k-1},x_k) \\mapsto (x_{k-1},{}^{-}p^{k-1})\\,=\\,\\left(x_{k-1},  -\\frac{\\partial S_k}{ \\partial x_{k-1}}\\right) {\\quad}\n\\end{eqnarray}\nfrom $\\mathcal Q_{k-1}\\times\\mathcal Q_k$ to the phase spaces $\\mathcal P_{k-1}$ and $\\mathcal P_k$ (these are contangent bundles of $\\mathcal Q_k$ and $\\mathcal Q_{k-1}$ respectively),  which we call {\\it post--} and {\\it pre--Legendre transformations}, respectively. The regularity condition (\\ref{app3}) ensures that the Legendre transformations are (locally) invertible. The Hamiltonian time evolution\n\\begin{eqnarray}\\label{app10}\n{\\cal H}_1: (x_1,{}^-p_1) \\mapsto (x_2, {}^+p_2 )\n\\end{eqnarray}\ncan be defined via the Legendre transform from the Lagrangian one (\\ref{app4}) and coincides with the time evolution map generated by $S_2$ via the equations (\\ref{b3}). Furthermore, it is straightforward to check that the Lagrangian one-- and two--forms, (\\ref{app5}) and (\\ref{app6}), arise from pulling back the canonical one-- and two--forms, $p_idx^i$ and $ dx^i\\wedge dp_i$, respectively,  with the Legendre transformation (\\ref{app9}).\nThis can be used to show that the Hamiltonian time evolution is symplectic, i.e.\\ preserves the canonical two--form.\n\nDiagrammetically,\n\\begin{diagram}\n& &\\mathcal Q_{k-1}\\times\\mathcal Q_k&& \\rTo^{\\mathcal L_k}&&\\mathcal Q_k\\times\\mathcal Q_{k+1} &   &\\\\\n&\\ldTo^{{\\mathbb{F}}^-}& &\\rdTo^{{\\mathbb{F}}^+}&&\\ldTo^{{\\mathbb{F}}^-}  && \\rdTo^{{\\mathbb{F}}^+}&\\\\\n \\mathcal P_{k-1}&& \\rTo^{\\mathcal H_{k-1}}& &\\mathcal P_k&& \\rTo^{\\mathcal H_{k}}&&  \\mathcal P_{k+1}{\\quad}.\n\\end{diagram}\n\n\n\\subsection{Singular systems}\\label{sing}\n\nLet us now consider the situation where the regularity condition (\\ref{app3}) is violated, yet $\\mathcal Q_{k-1}\\cong\\mathcal Q_k$, that is, where an equal number of left and right null vectors $L_1^i$ and $R^i_2$ occurs, satisfying\n\\begin{eqnarray}\\label{app13}\nL_1^i \\frac{\\partial^2 S_2}{\\partial x_1^i \\partial x_2^j}\\;=0  , {\\quad}\\q      \\frac{\\partial^2 S_2}{\\partial x_1^i \\partial x_2^j} R_2^j\\;=0   {\\quad}\n\\end{eqnarray}\nin some open neighborhood in ${\\cal Q}_1\\times {\\cal Q}_2$. (To avoid excessively many indices, we will not introduce another index numbering the null vectors in this section.)\n\nFirstly, we will discuss the consequences for the Lagrangian discrete time evolution which is defined as the space of solutions to (\\ref{app2}), namely, as the submanifold in ${\\cal Q}_0\\times {\\cal Q}_1\\times {\\cal Q}_2$ satisfying\n\\begin{eqnarray}\\label{app14}\n0&=&\\frac{\\partial S_1}{\\partial x^i_1}+\\frac{\\partial S_2}{\\partial x^i_1}.\n\\end{eqnarray}\nGiven a particular solution, i.e.\\ a configuration $(x_0,x_1,x_2)$ satisfying (\\ref{app14}), note that also an infinitesimally displaced configuration $(x_0,x_1,x_2+\\varepsilon R_2)$ is a solution. Hence, the solution $x_2$ as a function of $x_0,x_1$ is not uniquely determined and arbitrariness arises. We could call the directions $R_2$ `preliminary gauge directions'. Note, however, that the {\\it a priori} free parameters corresponding to these directions may get fixed {\\it a posteriori} by entering the equations of motion of later time evolution steps. Therefore, `gauge' can {\\it a priori} really only refer to the dynamics of the single time step from $k=1$ to $k=2$. In fact, the null vectors $L_1^i$ and $R^i_2$ of the Lagrangian two--form do not necessarily extend to null vectors of the Hessian of the action which, in turn, define the proper gauge symmetries of the action. \n\nNotwithstanding the arbitrariness in the solutions, we can define a Lagrangian time evolution map from ${\\cal Q}_0\\times{\\cal Q}_1$ to ${\\cal Q}_1\\times{\\cal Q}_2$. Since $x_2$ is not uniquely determined, however, we either have to fix $N$ {\\it a priori} free parameters (if there are $N$ independent null vectors $R_2$) or map to `gauge equivalence classes'. Either way, instead of the time evolution mapping onto a $2Q$ dimensional space, where $Q$ is the dimension of configuration space ${\\cal Q}$, it maps at most onto a $(2Q-N)$--dimensional one.\nThus, either the time evolution map is only defined on some (constraint) submanifold of ${\\cal Q}_0\\times{\\cal Q}_1$ or the map is not injective. A combination of both possibilities could also occur.\n\nAssume for the moment that such constraints do not occur, so that we can define a Lagrangian time evolution map $\\mathcal L_1$ on the full configuration space ${\\cal Q}_0\\times {\\cal Q}_1$. To this end, just fix some `gauge parameters' $\\lambda_2$ in ${\\cal Q}_1\\times {\\cal Q}_2$ to determine $x_2$ uniquely as a function of $x_0$ and $x_1$. Notice that at this step $S(x_0,x_1)$, i.e.\\ the action (\\ref{app1}) evaluated on the solution as a function of the initial data $x_0,x_1$, will generally depend on this `gauge' choice. In the present case the Lagrangian two--form is, obviously, degenerate: the coordinate expression (\\ref{app6}) \ndirectly shows that it possesses $2N$ null directions $L_1^i$ and $R_2^i$. Nevertheless, the arguments (\\ref{app8}) to (\\ref{app8a}), showing that the Lagrangian two--form is preserved under the Lagrangian time evolution, still hold true in exactly the same way as for regular systems\n\n\n\n\n\n\nLet us turn to the Hamiltonian dynamics. Due to the $N$ left and $N$ right null vectors (\\ref{app13}), the rank of both Legendre transformations\n\\begin{eqnarray}\\label{app16}\n&&\\mathbb{F}^+S_2:  (x_{1},x_2) \\mapsto (x_2,{}^{+}p^2)\\,=\\,\\left(x_2,  \\frac{\\partial S_2}{ \\partial x_{2}}\\right) \\nonumber\\\\\n&&\\mathbb{F}^-S_2:  (x_{1},x_2) \\mapsto (x_{1},{}^{-}p^{1})\\,=\\,\\left(x_{1},  -\\frac{\\partial S_2}{ \\partial x_{1}}\\right) {\\quad}\n\\end{eqnarray}\nis $2Q-N$. Hence, the Legendre transformations are not onto. In general, $\\mathbb{F}^\\pm S_2$ simultaneously fail to be isomorphisms if and only if condition (\\ref{app3}) is violated, i.e.\\ if and only if the Lagrangian two--form (\\ref{app6}) is degenerate. The image will be given by $(2Q-N)$--dimensional submanifolds in the two phase spaces ${\\cal P}_1$ and ${\\cal P}_2$ which we will call ${\\cal C}_1^-$ and ${\\cal C}_2^+$, respectively. We emphasize $\\dim\\mathcal C^-_1=\\dim\\mathcal C^+_2$.\n\n\\begin{Definition}\\label{def_pcons}\nThe image of the pre--Legendre transform, $\\mathcal C_1^-:=\\text{\\emph{Im}}(\\mathbb{F}^-S_2)\\subset\\mathcal P_1$, is called the \\emph{pre--constraint surface}. The image of the post--Legendre transform, $\\mathcal C_2^+:=\\text{\\emph{Im}}(\\mathbb{F}^+S_2)\\subset\\mathcal P_2$, is called the \\emph{post--constraint surface}.\n\\end{Definition}\n\nNote that the {\\it pre--constraints} are automatically satisfied by the pre--momenta ${}^-p^1$, yet constitute the conditions that need by satisfied on ${\\cal P}_1$ so that a Hamiltonian evolution can be defined. On the other hand, the {\\it post--constraints} are automatically satisfied by the post--momenta ${}^+p^2$ after a time evolution from time $k=1$ to time $k=2$ has taken place, but constitute non--trivial conditions for the pre--momenta ${}^-p^2$. In general, the {\\it pre--} and {\\it post--constraint surfaces} at a given step $k$ do {\\it not} coincide, $\\mathcal C^+_k\\neq\\mathcal C^-_k$. Momentum matching requires that one restricts to $\\mathcal C^+_k\\cap\\mathcal C^-_k$.\n\n\nThe $N$ pre--constraints ${ C}_1^-$ are defined through the equation\n\\begin{eqnarray}\\label{app16a}\n0&=&{ C}_1^-(x_1,{}^-p^1)_{{\\big |}  {}^-p^1=  -\\frac{\\partial S_2}{\\partial x_1}(x_1,x_2)   }\n\\end{eqnarray}\nwhich has to hold for arbitrary $x_1$ and $x_2$. Differentiating equation (\\ref{app16a}) with respect to $x_2$ and with respect to $x_1$, respectively, we find the relations\n\\begin{eqnarray}\\label{app16b}\n0&=& \\frac{\\partial C_1^-}{\\partial {}^-p_j^1} \\frac{\\partial^2 S_2}{\\partial x_1^j \\partial x_2^i} \\\\\n0&=& \\frac{\\partial C_1^-}{\\partial x^j_1} \\,-\\,  \\frac{\\partial C_1^-}{\\partial {}^-p_j^1} \\frac{\\partial^2 S_2}{\\partial x_1^j \\partial x_1^i}  {\\quad} .\n\\end{eqnarray}\nSince there are $N$ independent left null vectors $L_1^j$ of $\\frac{\\partial^2 S_2}{\\partial x_1^j \\partial x_2^i}$, we can conclude from (\\ref{app16b}) that\n\\begin{eqnarray}\n \\frac{\\partial C_1^-}{\\partial {}^-p_j^1} = \\sum_L \\gamma^C_L (x_1,p_1)\\,  L_1^j{\\quad},\n\\end{eqnarray}\nwhere we sum over all $N$ left null vectors $L_1$ and $\\gamma^C_L(x_1,p_1)$ are appropriately chosen coefficients (which also depend on the constraint under consideration).\nSimilar relations can be found for the partial derivatives of ${ C}_2^+$. This specifies the gradients of the constraints; the orthogonal subspace is then tangential to the constraint hypersurface ${\\cal C}_1^-$, respectively ${\\cal C}_2^+$.\n\n\n\nThe Hamiltonian time evolution map ${\\cal H}_1$ will only be defined on the submanifold ${\\cal C}_1^-$. As before it can be generated by the discrete action, i.e.\n\\begin{eqnarray}\\label{app17}\n{}^{-}p^{1}= -\\frac{\\partial S_{2}}{ \\partial x_{1}}  {\\quad}  , {\\quad}\\q  {}^{+}p^{2}  = \\frac{\\partial S_2}{ \\partial x_{2}}   {\\quad} .\n\\end{eqnarray}\nThis makes the appearance of pre-- and post--constraints explicit. In particular, we can define pre-- and post--constraints independently from the Legendre transformation. Given a Hamiltonian time evolution map, we declare the pre--image of this map as the submanifold of pre--constraints and the image of this time evolution map determines the post--constraints. In the previous sections we have given the Hamiltonian time evolution maps for all the Pachner moves (and, hence, for all evolution maps involving a sequence of Pachner moves) \nbased on momentum updating between the respective constraint surfaces which are defined in this manner. For these cases the pre--image and image of the maps could always be specified (but see also section \\ref{apppach} below). \n\nFurthermore, if we find a solution to the first of the equations (\\ref{app17}) for $x_2$ as a function of $x_1,{}^-p^1$, then the infinitesimally displaced configuration $x_2+\\varepsilon R_2$ is also a solution. That is, as in the Lagrangian picture, $x_2$ is not uniquely defined. (Note that ${}^+p^2$ may also change under the transformation generated by $\\varepsilon R_2$.) Rather, we have $N$ `preliminary gauge directions' $R_2$. The same can be said for expressing $x_1$ as a function of $x_2,{}^+p^2$ where now the `gauge' directions are given by $L_1$. Concretely, we have the gauge displacements\n\\begin{eqnarray}\\label{app18}\n\\delta_Lx^i_1=L_1^i\\,\\, ,{\\quad}\\q \\delta_L p^1_i=-\\frac{\\partial^2S_2}{\\partial x_1^i \\partial x_1^j}  L_1^j\n{\\quad} .\n\\end{eqnarray}\nNote that these vectors are orthogonal to the gradients of the constraints  $C_1^-$, hence, the `gauge' displacements are tangential to the constraint hypersurface ${\\cal C}^-_1$.\n\nGiven that $\\mathcal H_1:\\mathcal C^-_1\\rightarrow\\mathcal C^+_2$, $\\mathcal H_1$ cannot be a symplectic mapping. Nevertheless, as regards the preservation of the symplectic structure, consider the canonical two--forms\n\\begin{eqnarray}\\label{app19}\n\\omega_1=dx_1^j\\,\\wedge\\, dp^1_j  , {\\quad}\\q\\omega_2=dx_2^j\\,\\wedge\\,dp^2_j .\n\\end{eqnarray}\nWe can pull these two--forms back via the embeddings $\\iota_1^-:{\\cal C}_1^- \\rightarrow {\\cal P}_1$ and $\\iota_2^+:{\\cal C}_2^+ \\rightarrow {\\cal P}_2$ to two--forms on the constraint surfaces ${\\cal C}^-_1$ and ${\\cal C}^+_2$, respectively. Notice that the resulting two--forms are pre--symplectic forms. Accordingly, the following theorem proves that discrete Hamiltonian time evolution rather is a pre--symplectic map.\n\n\\begin{Theorem}\\label{thm_presymp}\nThe discrete Hamiltonian time evolution map $\\mathcal H_1:\\mathcal C^-_1\\rightarrow\\mathcal C^+_2$ satisfies\n\\begin{eqnarray}\n(\\iota_1^-)^*\\omega_1 = {\\cal H}_1^*  (\\iota_2^+)^*\\omega_2  .\n\\end{eqnarray}\n\\end{Theorem}\n\n\\begin{proof}\nIt is convenient to introduce coordinates $\\{w_k^I, y_k^\\alpha, z^k_\\alpha\\}$ with $k=1,2$ on ${\\cal C}_1^-$ and ${\\cal C}_2^+$, respectively. The index $I =1,\\ldots N$ shall label `preliminary gauge directions' $(L_1)_I^i$ and $(R_2)_I^i$, while $\\alpha=1,\\ldots, Q-N$ labels coordinates associated to vectors $(N_1)^j_{\\alpha},(M_2)^j_{\\alpha}$ which are {\\it not} null directions of $\\Omega_2$. These coordinates can be chosen such that one obtains for the embedding map $\\iota_1^-:(w_1,y_1,z^1)\\mapsto (x_1,p^1)$ \n\\begin{eqnarray}\\label{app20}\n&&\\frac{\\partial x^i_1}{\\partial w_1^I}=(L_1)_{I}^i  {\\quad}, {\\quad}\\q  \\frac{\\partial p_i^1}{\\partial  w_1^I}=-\\frac{\\partial^2S_2}{\\partial x_1^i \\partial x_1^j}  (L_1)_I^j  \\nonumber\\\\\n&&\\frac{\\partial x^i_1}{\\partial y_1^\\alpha}=(N_1)_\\alpha^i       {\\quad}, {\\quad} {\\quad}  \\frac{\\partial p_i^1}{\\partial  y_1^\\alpha}=-\\frac{\\partial^2S_2}{\\partial x_1^i \\partial x_1^j}  (N_1)_\\alpha^j \\nonumber\\\\\n&&\\frac{\\partial x^i_1}{\\partial z^1_\\alpha}=0                                  {\\quad}, {\\quad}\\q{\\quad} {\\quad} \\,  \\frac{\\partial p_i^1}{\\partial  z^1_\\alpha}=   (T^{-1}_1)^\\alpha_j\\,,\\nonumber\n\\end{eqnarray}\nand for  $\\iota_2^+:(w_2,y_2,z^2)\\mapsto (x_2,p^2)$\n\\begin{eqnarray}\\label{app20b}\n&&\\frac{\\partial x^i_2}{\\partial w_2^I}=(R_2)_I^i  {\\quad}, {\\quad}\\q  \\frac{\\partial p_i^2}{\\partial  w_2^I}=\\frac{\\partial^2S_2}{\\partial x_2^i \\partial x_2^j}  (R_2)_I^j  \\nonumber\\\\\n&&\\frac{\\partial x^i_2}{\\partial y_2^\\alpha}=(M_2)_\\alpha^i       {\\quad}, {\\quad} \\,\\,\\,\\,\\,  \\frac{\\partial p_i^2}{\\partial  y_2^\\alpha}=\\frac{\\partial^2S_2}{\\partial x_2^i \\partial x_2^j}  (M_2)_\\alpha^j \\nonumber\\\\\n&&\\frac{\\partial x^i_2}{\\partial z^2_\\alpha}=0                                  {\\quad}, {\\quad}\\q{\\quad} {\\quad}  \\, \\frac{\\partial p_i^2}{\\partial  z^2_\\alpha}=   (T^{-1}_2)^\\alpha_j\\, .\\nonumber\n\\end{eqnarray}\n$(T^{-1}_1)_j^{\\alpha},(T^{-1}_2)_j^{\\alpha}$ are vectors satisfying\n\\begin{eqnarray}\\label{invv}\n(T^{-1}_1)_j^{\\alpha}(L_1)^j_{I}&=&(T^{-1}_2)_j^{\\alpha}(R_2)^j_{I}\\,\\,\\,=0,\\nonumber\\\\\n(T^{-1}_1)_j^{\\alpha}(N_1)^j_{\\alpha'}&=&(T^{-1}_2)_j^{\\alpha}(M_2)^j_{\\alpha'}=\\delta^\\alpha_{\\alpha'}{\\quad}\\forall\\,I,\\alpha.\n\\end{eqnarray}\nThe pull--backs of the two forms can then be written as\n\\begin{eqnarray}\\label{app21}\n(\\iota_1^-)^*\\omega_1\\!\\!&=& \\sum_j \\left((L_1)^j_{I} \\,dw_1^I + (N_1)^j_{\\alpha}\\,dy_1^\\alpha\\right) \\,\\wedge \\nonumber\\\\\n&&\\left(     -\\frac{\\partial^2S_2}{\\partial x_1^j \\partial x_1^i}    \\left(( L_1)_{I'}^i   \\, dw_1^{I'} +\n(N_1)_{\\alpha'}^i \\, dy_1^{\\alpha'} \\right) +\n( T^{-1}_1)_j^{\\alpha'}  \\,  dz^1_{\\alpha'}   \\right)  \\nonumber\\\\\n %\n(\\iota_2^+)^*\\omega_2\\!\\!&=& \\sum_j \\left((R_2)^j_{I} \\,dw_2^I + (M_2)^j_{\\alpha}\\,dy_2^\\alpha\\right) \\,\\wedge\\nonumber \\\\\n&&\\left(\n\\frac{\\partial^2S_2}{\\partial x_2^j \\partial x_2^i}\n\\left(\n (R_2)_{I'}^i  \\,dw_2^{I'} +\n (M_2)_{\\alpha'}^i \\, dy_2^{\\alpha'} \\right)\n+\n(T^{-1}_2)_j^{\\alpha'}  \\,  dz^2_{\\alpha'}\n\\right). \\nonumber\n\\end{eqnarray}\nFirstly, the terms containing the second derivative of the action $S_2$ vanish because they are contracted with an antisymmetric form. Secondly, (\\ref{invv}) entails \n\\begin{eqnarray}\n(\\iota_1^-)^*\\omega_1\\!\\!&=&dy_1^\\alpha    \\,\\wedge \\,\n  dz^1_{\\alpha},   \\label{app22a} \\\\\n(\\iota_2^+)^*\\omega_2\\!\\!&=&dy_2^\\alpha   \\,\\wedge \\,\ndz^2_{\\alpha} .\\label{app22b}\n\\end{eqnarray}\nThese expressions imply that the `preliminary gauge vectors' $\\delta_{L_I}$ in (\\ref{app18}) are degenerate directions of the two--form $(\\iota_1^-)^*\\omega_1$, whereas the corresponding $\\delta_{R_I}$ are degenerate directions of $(\\iota_2^+)^*\\omega_2$. Finally, expressing the Hamiltonian time evolution (\\ref{app17}) directly in the coordinates $y_k^\\alpha$ and $z_k^\\alpha$, one can check that the Hamiltonian time evolution preserves the pull--backs of the canonical forms as stated in the theorem.\n\\end{proof}\n\n\n\n\n\n\n\nHence, diagrammatically, \n\\begin{diagram}\n& &\\mathcal Q_{k-1}\\times\\mathcal Q_k&& \\rTo^{\\mathcal L_k}&&\\mathcal Q_k\\times\\mathcal Q_{k+1} &   &\\\\\n&\\ldTo^{{\\mathbb{F}}^-}& &\\rdTo^{{\\mathbb{F}}^+}&&\\ldTo^{{\\mathbb{F}}^-}  && \\rdTo^{{\\mathbb{F}}^+}&\\\\\n \\mathcal P_{k-1}\\supset\\mathcal C^-_{k-1}&& \\rTo^{\\mathcal H_{k-1}}& &\\mathcal C^+_k\\cap\\mathcal C^-_k&& \\rTo^{\\mathcal H_{k}}&&  \\mathcal C^+_{k+1}\\subset\\mathcal P_{k+1}{\\quad}.\n\\end{diagram}\n\n\n\\subsection{Dynamical systems with changing phase space dimensions}\\label{apppach}\n\nSuch singular systems as discussed above appear in the dynamics generated by the Pachner moves. There we had to deal with the problem that the number of variables may change from one time step to the next. Let us consider a simple example which will highlight the general principle. Consider again three consecutive time steps with variables $x_0,x_1,x_2$. But now assume that among the variables $x_2$ there is a  `new variable' $x_2^n$, in particular, that the number of variables at time step $k=2$ is $Q+1$, whereas it is $Q$ at times $k=0,1$.  Although this problem might be well posed as a boundary value problem keeping the data at $k=0,2$ fixed, it will, in general, not be possible to turn it into a well posed initial value problem with initial data at times $k=0,1$.\n\nTo describe the situation nevertheless by an initial value problem, extend the configuration spaces at times $k=0,1$ by the configuration variables $x_0^n$ and $x_1^n$, respectively. $x_2^n$ can be interpreted as an initial datum which becomes relevant only at step $k=2$, yet which we are allowed to already specify at time $k=0$, e.g.\\ as $x_2^n=x_1^n=x_0^n$. The action pieces $S_1(x_0,x_1)$ and $S_2(x_1,x_2)$ neither depend on $x_0^n$ nor on $x_1^n$ because these variables were just introduced for book keeping purposes. Accordingly, the dynamics from $k=0$ via $k=1$ to $k=2$ is singular. Let us assume, for simplicity, that the dynamics is otherwise regular.\n\nThe extension does not interfere at all with the dynamics of the other variables for the time step from $k=0$ to $k=1$. Thus, extend also the phase spaces at times $k=0,k=1$ by the pairs $(x_k^n,p^k_n)$; the Legendre transformations will map onto the constraint hypersurfaces $C^-_0={}^-p^0_n=0$ and $C^+_1={}^+p^1_n=0$, respectively. The proof of Theorem \\ref{thm_presymp} shows that the canonical two--forms restricted to the constraint hypersurfaces possess degenerate directions, i.e.\\ `preliminary gauge directions'. The corresponding variables $x_0^n$ and $x_1^n$ are {\\it a priori} free (Lagrange) parameters $\\lambda_0,\\lambda_1$. Therefore, we can fix $x^n_0,x^n_1$ to some arbitrary value. In particular, we can fix these data to coincide with the value of $x_2^n$. Note that the `reduced phase space', i.e.\\ the constraint hypersurface modulo the `preliminary gauge direction' (which is the null direction of the restricted canonical two--form) coincides with the unextended phase space we started with.\n\n\n\n\nNext, let us consider the time step from time $k=1$ to time $k=2$ described on the extended phase space ${\\cal P}'_1$ and the phase space ${\\cal P}_2$ (which already includes the pair $(x_2^n,p^2_n)$). The matrix\n\\begin{eqnarray}\\label{appe1}\n\\frac{\\partial^2 S_2}{\\partial x^i_1 \\partial x^j_2}\n\\end{eqnarray}\nhas (due to our assumption only) one left null vector $L_1^i=\\delta^i_n$. Consequently, there is also a right null vector $R_2^i$ and the Lagrangian two--form  (\\ref{app6}) possesses these two degenerate directions. The Legendre transformations (\\ref{app16}) will again not be surjective; for time $k=1$ it will rather map onto the constraint hypersurface $C_1^-={}^-p^1=0$. This pre--constraint coincides with the post--constraint ${}^+p_1=0$ from the previous time step. Moreover, we will have a post--constraint $C_2^+=0$ on the canonical data at $k=2$.\n\nAs we have seen, we will also find `preliminary gauge directions' associated to the constraints. At time $k=1$ the `gauge' direction (\\ref{app18}) just corresponds to the coordinate $x^n_2$, i.e.\\ $\\delta_Lx_1^i=\\delta^i_n$ and $\\delta_Lp_i^1=0$. At time $k=2$, on the other hand, we have\n\\begin{eqnarray}\\label{appe2}\n\\delta_R x_2^i=R_2^i {\\quad} ,{\\quad}\\q \\delta_R p_i^2=\\frac{\\partial^2 S_2}{\\partial x_2^i \\partial x_2^j}R^j_2 {\\quad} .\n\\end{eqnarray}\nTheorem \\ref{thm_presymp} shows that the Hamiltonian time evolution preserves the canonical two--forms restricted to the constraint hypersurfaces. In the present example (\\ref{app22a}) at $k=1$ is just the canonical two--form of the original $2Q$--dimensional unextended phase space $\\mathcal P_1$ which, however, is used as a pre--symplectic form on a $(2Q+1)$--dimensional constraint surface in the extended phase space $\\mathcal P'_1$. On the other hand, (\\ref{app22b}) at $k=2$ is the restriction of the canonical two--form of the $(2Q+2)$--dimensional unextended phase space $\\mathcal P_2$ to the $(2Q+1)$--dimensional constraint hypersurface $C^+_2=0$. Its single degenerate direction is given in (\\ref{appe2}). Consequently, the `reduced phase space' at $k=2$ is $2Q$--dimensional just as the original $\\mathcal P_1$. The phase space extension from $\\mathcal P_1$ to $\\mathcal P'_1$ at $k=1$ can be viewed as embedding the smaller phase space $\\mathcal P_1$ into the bigger phase space $\\mathcal P_2$; discrete time evolution proceeds such that the (restricted) two--forms are preserved on this bigger phase space.\n\n\nThe case of `old variables'  $x^o_1$ that disappear at time $k=2$ can be discussed in the same way by reversing time direction. \n\nAs an extreme example, consider the cases described in sections \\ref{3D} and \\ref{4D}, namely the time evolution step from a zero--dimensional phase space to the phase space corresponding to the boundary of a tetrahedron or 4--simplex, respectively.\nThe initial phase space is zero--dimensional and the phase space at time $k=1$ twelve-- or twenty--dimensional, corresponding to the boundary data of a tetrahedron, respectively 4--simplex. In this case the canonical two--forms at $k=0$ are just the zero forms as the initial phase space is zero--dimensional. Indeed, the time evolution preserves this property, as in both cases we had six, respectively ten constraints, fixing all the momenta as functions of the lengths variables. Moreover, one can check that the canonical two--forms restricted to the constraint hypersurfaces are totally degenerate, that is identically zero.\n\n\nFinally, let us consider the local Pachner moves explicitly. These are rather directly defined in the Hamiltonian picture. The reason is that the Lagrangian picture requires the knowledge of configurations and velocities at a given time which in the discrete case translates to configuration data of two consecutive time steps. For a Pachner move (or any other evolution move, like the tent move, which does not cover the entire hypersurface), however, the two consecutive time steps overlap, so that we do not have a full set of configuration data. In the Hamiltonian picture the velocities, or in the discrete the configuration data of the second time step, are replaced by the momenta, which are defined at the same time step as the configuration data. That is, we just need one time slice to encode the canonical data. \n\nFor the Pachner moves of type $M$--$N$ with $N>M$ in 3D and 4D we obtain $K$ new edge lengths ($K=1$ for the 2--3 move and $K=N$ for the 1--N move).  We, therefore, extend the phase space at time $k$ by $K$ pairs $(l_k^n,p^k_n)$ which are matched by the $K$ new edge lengths and conjugate momenta $(l_{k+1}^n,p^{k+1}_n)$ at time $k+1$. Additionally, we have pairs $(l^b,p^b)$ which do not change at all during this evolution move and variables $(l^e,p^e)$ for which only the momenta are updated. The Hamiltonian evolution map for this kind of Pachner moves (see, for instance, (\\ref{c13}--\\ref{c13c})) is given by the following momentum updating\n\\begin{eqnarray}\\label{anh1}\nl^b_k&=&l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\nl^e_k&=&l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\sigma(l^e_{k+1}, l^n_{k+1})}{\\partial l^e_{k+1}} \n\n\\; ,    \\label{anh1b}  \\\\\np^{k}_n&=&0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,\\,=\\, \\frac{\\partial S_\\sigma(l^e_{k+1},l^n_{k+1})  }{\\partial l^n_{k+1}}\\,{\\quad}  \n{\\quad} .\\label{anh1c}\n\\end{eqnarray}\nHere $S_\\sigma$ stands for the action of the glued tetrahedron or 4--simplex and we choose this action to be a function of the length variables at time $k+1$. \n\nEquations (\\ref{anh1c}) contain $K$ pre--constraints, namely $C_n^k=p^k_n$, and $K$ post--constraints\n\\begin{eqnarray}\\label{anh2}\n C_n^{k+1}&=&p^{k+1}_n- \\frac{\\partial S_\\sigma(l^e_{k+1},l^n_{k+1})  }{\\partial l^n_{k+1}}{\\quad},\n\\end{eqnarray}\nsince this equation only involves canonical data from time step $k+1$. Note also that the shape (\\ref{anh2}) of the post--constraints ensures that they have vanishing Poisson brackets among each other, i.e.\\ they form and Abelian (sub--) algebra.\\footnote{Dropping the evolution index $k$, one directly computes\n$\\{C_n,C_{n'}\\}=\\frac{\\partial^2S_\\sigma}{\\partial l^n\\partial l^{n'}}-\\frac{\\partial^2S_\\sigma}{\\partial l^{n'}\\partial l^{n}}=0$.} The same holds for the pre--constraints. (This is consistent with the previous discussion showing that to every pre-- or post--constraint we can associate a degenerate direction of the symplectic forms restricted to the pre-- or post--constraint hypersurfaces. This actually means that the constraints are first class.) Notice, however, that the pre--constraints, in general, do not Poisson commute with the post--constraints.\n\nApart from this, as we discussed for each Pachner move, the Hamiltonian evolution map (\\ref{anh1}--\\ref{anh1c}) does not specify all variables uniquely and we rather obtain `preliminary gauge directions' both at time $k$ and $k+1$. Note that these are `gauge' only with respect to this evolution move from step $k$ to $k+1$ and, more precisely, with respect to the symplectic forms restricted to the constraint hypersurfaces arising in these moves.\nSubsequent moves might result in a different set of (pre--) constraints with respect to which the original `preliminary gauge directions' do not constitute degenerate directions of the restricted symplectic form anymore.\n\nAt time $k$ the length variables $l^n_k$ remain undetermined, so that we have $\\delta_L x^i_k=\\delta^i_n L^n_k$ and $\\delta_L p_i^k=0$ (where $i$ stands for any of the indices $b,e,n$ and $L^n_k$ are $K$ linearly independent vector fields). Also at time $k+1$ we have $K$ undetermined variables, namely the edge lengths $l^n_{k+1}$, which we referred to as {\\it a priori} free parameters. These correspond to `preliminary gauge directions' \n\\begin{eqnarray}\n\\delta_R l^i_{k+1} = \\delta_n^i R^n_{k+1}  {\\quad}, {\\quad}\\q  \\delta_R p_i^{k+1}=  \\frac{\\partial^2 S_\\sigma  }{\\partial l^i_{k+1} \\partial l^{n'}_{k+1}}R^{n'}_{k+1}{\\quad},\n\\end{eqnarray}\nwhere $R^n_{k+1}$ are $K$ linearly independent vector fields. We have seen that the new edge lengths $l^n_{k+1}$ can be interpreted as additional initial data. These can turn either into physical degrees of freedom (as for the 2--3 moves where the new edge length determines the deficit angle at the bulk triangle generated by this move) or into proper gauge degrees of freedom (as for the 1--4 move for initial conditions leading to flat solutions). Additionally, the $l^n_{k}$ remain undetermined, however, in the previous discussions we pointed out that these can be `gauge fixed' to be equal to $l^n_{k+1}$.\n\nAs for the case discussed in the previous section \\ref{sing}, the `preliminary gauge directions' are tangential to the constraint hypersurfaces defined by the constraints $C_n^k$ and $C_n^{k+1}$, respectively.\n\nWe can consider for both time steps the canonical symplectic forms\n\\begin{equation}\n\\omega_k=dl^b_k \\wedge dp_b^k + dl^e_k \\wedge dp_e^k +dl^n_k \\wedge dp_n^k  {\\quad} ,  {\\quad}\\q \\omega_{k+1}=dl^b_{k+1} \\wedge dp_b^{k+1} + dl^e_{k+1} \\wedge dp_e^{k+1} +dl^n_{k+1} \\wedge dp_n^{k+1}\n\\end{equation}\nand restrict these to the constraint hypersurfaces in the respective phase spaces. For the (extended) phase space ${\\cal P}_k$ at time $k$ we can use the canonical embedding $\\iota_k:(l^b_k,p_b^k,l^e_k,p_e^k,l^n_k)\\mapsto (l^b_k,p_b^k,l^e_k,p_e^k,l^n_k,p_n^k=0)$ and the pull--back of the symplectic form gives just the symplectic form of the unextended phase space\n\\begin{eqnarray}\n(\\iota_k)^*\\omega_k&=& dl^b_k \\wedge dp_b^k + dl^e_k \\wedge dp_e^k  {\\quad} .\n\\end{eqnarray}\nFor time step $k+1$ consider the embedding \n\\begin{equation}\n\\iota_{k+1}:(l^b_{k+1},p_b^{k+1},l^e_{k+1},p_e^{k+1},l^n_{k+1})\\mapsto (l^b_{k+1},p_b^{k+1},l^e_{k+1},p_e^{k+1},l^n_{k+1}, p^n_{k+1}=\\frac{\\partial S_\\sigma}{\\partial l^n_{k+1}}) {\\quad} .\n\\end{equation}\nIn this case, the pull--back of the symplectic form is given by\n\\begin{eqnarray}\\label{sy01}\n(\\iota_{k+1})^*\\omega_{k+1} &= & dl^b_{k+1} \\wedge dp_b^{k+1} + dl^e_{k+1} \\wedge dp_e^{k+1} +dl^n_{k+1} \\wedge \\frac{\\partial^2 S_\\sigma}{\\partial l^n_{k+1} \\partial l^e_{k+1}}dl^e_{k+1}  {\\quad} .\n\\end{eqnarray}\nNow the time evolution equations (\\ref{anh1}--\\ref{anh1c}) express the canonical coordinates at time $(k+1)$ as function of the canonical coordinates at time $k$ and as function of the length variables $l^n_{k+1}$ (as these are not determined by the time evolution map). Using this for the symplectic form (\\ref{sy01}) at time $k+1$, we can conclude that the time evolution map ${\\cal H}_k$ defined by  (\\ref{anh1}--\\ref{anh1c}) preserves the restricted symplectic forms:\n\\begin{eqnarray}\\label{sy01b}\n{\\cal H}_k^*(\\iota_{k+1})^*\\omega_{k+1} &=&dl^b_{k} \\wedge dp_b^{k} + dl^e_{k} \\wedge dp_e^{k} + dl^e_{k} \\wedge  \\frac{\\partial^2 S_\\sigma}{\\partial l^e_{k+1} \\partial l^n_{k+1}} dl^n_{k+1}+dl^n_{k+1} \\wedge \\frac{\\partial^2 S_\\sigma}{\\partial l^n_{k+1} \\partial l^e_{k+1}}dl^e_{k+1}  \\nonumber\\\\\n&=&dl^b_{k} \\wedge dp_b^{k} + dl^e_{k} \\wedge dp_e^{k}  {\\quad} .\n\\end{eqnarray}\n\n\nThe Pachner moves of type $N$--$M$ with $N>M$ can be discussed analogously, one only has to reverse the time direction. \nThe key difference to those moves with $M>N$ is, however, that pre--constraints now arise before the move. The symplectic form is only preserved when restricted to the post--constraint surface after the move and to the constraint surface defined by both the post--constraints and the new pre--constraints before the move.\n\n The 2--2 move in 3D is the only Pachner move for which we extended both the initial and final phase space because we had both an `old' and `new' variable pair, see section \\ref{22sec}.\nThe equations of motion were given by\n\\begin{eqnarray}\\label{anh21}\nl^b_k&=&l^b_{k+1}  {\\quad} ,{\\quad}\\q p^{k+1}_b\\,=\\,p^k_b {\\quad} , \\\\\nl^e_k&=&l^e_{k+1}{\\quad},{\\quad}\\q p^{k+1}_e\\,=\\,p^k_e+\\frac{\\partial S_\\tau(  l^e_{k+1},l^o_k,l^n_{k+1})  }{\\partial l^e_{k+1}} \n{\\quad} ,\\\\\np^{k}_n&=&0 {\\quad},{\\quad}\\q{\\quad} p^{k+1}_n\\,\\,\\,=\\, \\frac{\\partial S_\\tau(  l^e_{k+1},l^o_k,l^n_{k+1}) }{\\partial l^n_{k+1}}\\,{\\quad}\n , \\label{anh21c}\\\\\np^{k+1}_o &=& 0 {\\quad},{\\quad}\\q{\\quad} p^{k}_o{\\quad}\\,\\,\\,=\\, -\\frac{\\partial S_\\tau (  l^e_{k+1},l^o_k,l^n_{k+1}) }{\\partial l^n_{k+1}}\\,\\\n .{\\quad} \\label{anh21d}\n\\end{eqnarray}\nHere the only pre--constraint is $p^k_n=0$ and the only post--constraint is $p^{k+1}_o=0$. Note that the second equations in both (\\ref{anh21c}) and (\\ref{anh21d}) do {\\it not} constitute constraints as these depend on canonical data of both times $k$ and $k+1$. Correspondingly, the only gauge directions one encounters here arise from the fact that $l_k^n$ and $l_{k+1}^o$ remain undetermined by the equations of motion. Therefore, these can be gauge fixed to arbitrary values, for instance $l^n_k=l^n_{k+1}$ (although the latter length needs to be determined from the equations of motion) and $l^o_{k+1}=l^o_k$. The canonical symplectic form on the extended phase spaces pulled back to the constraint hypersurfaces just gives the canonical forms of the unextended phase spaces. Consequently, we see that in this case the dynamics can be reduced to the unextended phase spaces. The only configuration variables that change are $l^o_k\\mapsto l^n_{k+1}$. \n\nIn summary, we have seen that we can define suitably preserved symplectic structures for all Pachner moves. To this end we introduced extended phase spaces to circumvent the problem of changing phase space dimensions. However, we considered the canonical two--forms restricted to the constraint hypersurfaces, and these constraint hypersurfaces, upon factoring out the `preliminary gauge directions', coincided with, or were submanifolds in the original, unextended phase spaces.\n\nAn alternative construction is to extend the Hamiltonian time evolution maps, so that these are defined on the full (extended) phase spaces. For instance, considering the $M$--$N$ moves with $N>M$, the relevant changes from (\\ref{anh1}--\\ref{anh1c}) would be\n\\begin{eqnarray}\\label{anh22}\nl^n_k=l^n_{k+1} {\\quad} ,{\\quad}\\q p^{k+1}_n&=& p^k_n+ \\frac{\\partial S_\\sigma}{\\partial l^n_{k+1}}  {\\quad} .\n\\end{eqnarray}\nThat is, we obtain for all variable pairs  $i=b,e,n$\n\\begin{eqnarray}\\label{anh23}\nl^i_{k+1}=l^i_{k} {\\quad} ,{\\quad}\\q p^{k+1}_i&=& p^k_i+ \\frac{\\partial S_\\sigma}{\\partial l^i_{k+1}}  {\\quad} .\n\\end{eqnarray}\nThe time evolution map is defined on the full extended phase space and its image is given by the full phase space at time $k+1$. We can then impose the pre--constraints $p_n^k=0$. The post--constraints are  given as the image of the (pre--) constraint hypersurface under the time evolution map. Maps of the form (\\ref{anh23}), which only involve momentum updating via a generating function preserve the canonical two--form (of the extended phase space).\\footnote{Using the extended map $\\bar{\\mathcal H}_{k}$ defined by (\\ref{anh23}) and $\\omega_{k+1}=dl^i_{k+1}\\wedge dp^{k+1}_i$, one immediately verifies by symmetry and antisymmetry $(\\bar{\\mathcal H}_{k})^*\\omega_{k+1}=dl^i_k\\wedge d\\left(p^k_i+\\frac{\\partial S_{\\sigma}}{\\partial l^i_k}\\right)=dl^i_k\\wedge dp^k_i+\\frac{\\partial^2S_{\\sigma}}{\\partial\\, l^i_k\\partial\\,l^{i'}_k}dl^i_k\\wedge dl^{i'}_k=dl^i_k\\wedge dp^k_i=\\omega_k$.}\n\nThe previous discussion showed that also the canonical two--form restricted to the constraint hypersurfaces is preserved.\n\nSuch a construction is also possible for the 2--2 move, i.e.\\ the extended dynamics would also be of the form (\\ref{anh23}).  The pre--constraint $p^n_k=0$ and the post--constraint $p^o_{k+1}=0$, however, have to be imposed independently.\nThe second equation in (\\ref{anh22}) then changes into another post--constraint prescribing $p_n^{k+1}$ as a function of the six edge lengths $l_{k+1}^i$ of the tetrahedron, where $i=e,n,o$. Alternatively, we can solve for the length $l^o_{k+1}=l^o_k$ as a function of the five edge lengths $l_{k+1}^n,l_{k+1}^e$ and the momentum $p_n^{k+1}$. Also the second equation of (\\ref{anh23}) turns into a (pre--) constraint determining $p^{k}_o$ as a function of the six edge lengths $l_k^i, i=e,n,o$, or, alternatively, $l^n_k=l^n_{k+1}$ as function of the five edge lengths $l^o_k,l^e_k$ and $p_o^k$. That is, we obtain two constraints on each of the extended phase spaces ${\\cal P}_k$ and ${\\cal P}_{k+1}$. These are second class constraints. Compared to the previous treatment, where we had for, say, ${\\cal P}_k$ only $p_n^k=0$ as a constraint and $l_n^k$ as a gauge degree of freedom, we can consider the second constraint between $l^n_k,l^o_k,l^e_k$ and $p^k_o$ as a gauge fixing condition. This shows that also the induced symplectic forms on the (gauge fixed and) constraint hypersurfaces are preserved under time evolution.\n\n\n\n\n\n\\section{Interpretation and role of the constraints}\\label{intcon}\n\nIn spite of the fact that the set of pre--constraints and the set of post--constraints each form an Abelian (i.e.\\ first class) Poisson sub--algebra, they generally do not generate gauge symmetries of the discrete action. Firstly, the pre--constraints do not, in general, Poisson commute with the post--constraints and both sets of constraints have to be imposed at each time step. Secondly, their phase space flows are associated to null vectors of the Lagrangian two--form, but these do not, in general, extend to null vectors of the Hessian of the action. However, a degenerate Hessian on solutions implies the existence of flat directions at the extrema of the action and thereby the presence of gauge symmetries. Rather, these constraints manifest non--uniqueness of solutions and reflect the lack of knowledge at a given time step about the full solution (e.g.\\ as a consequence of varying phase space dimensions); in Regge calculus at some step $k$ there may not exist sufficient information about the triangulation at step $k+1$ (or vice versa) and a hypersurface may be `forgetful' about the `past' or `future'. For example, the post--constraints associated to the new edges simply implement the fact that the lengths of these edges can be freely varied in accordance with the canonical data at step $k$, but not that this will eventually be a symmetry of the action.\n\nOn the other hand, the gauge symmetries in Regge calculus are given by those variations of the edge lengths which leave the action invariant and correspond to the displacement in flat directions of vertices in the bulk of the triangulation. These symmetries are generically broken in the presence of curvature \\cite{bd1,dithoe1,hamberwilliamsgauge,morse}. Such a vertex displacement gauge transformation is specifed by the (four linearly independent) null-eigenvectors of the Hessian of the piece of action corresponding to the star of the vertex in question \\cite{bd1,dithoe1,bianca,ditfrespe}. The null-eigenvectors --- if existent --- are, thus, associated to a given vertex and define the flat (gauge) directions in which the vertex may be displaced without changing the action. In contrast to this, {\\it a priori} each pre-- or post--constraint of the individual Pachner moves is associated to a given edge in the hypersurface under consideration, rather than any particular vertex. It describes the variation of the corresponding single edge length, however, there is no invariant way of expressing this as the displacement of the one or the other vertex at the ends of the edge in some embedding space.\nNonetheless, in the canonical formalism the variation of the edges {\\it can} be invariantly associated to the displacement of a given vertex $v$, if the constraints of those edges in the hypersurface which are adjacent to $v$ are contracted with the four null-eigenvectors associated to $v$ \\cite{dithoe1};\\footnote{Internal edges, or edges which do not yet occur at a given step, do not matter because they come with vanishing momenta and solved equations of motion.} this implements a projection in gauge directions and the correct number of gauge generators. At this stage, the contracted constraints generate the gauge transformations defined by the null-eigenvectors. \n\nThis was discussed for tent moves in \\cite{dithoe1}, but will be studied in detail and more generally in a forthcoming paper \\cite{dhta}, where we will apply the present formalism to the 4D linearized theory, based on an expansion around a flat solution to linear order. This linearized regime, in fact, inherits the gauge freedom of the flat background solution and is relevant for the continuum limit where the diffeomorphism symmetry of classical general relativity ought to be restored and geometries are locally flat. This is where null-eigenvectors of the Hessian necessarily arise which, in turn, can be employed to construct proper gauge transformation generating constraints which are associated to vertices.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion and outlook}\\label{conc}\n\n\nIn the present article we have devised a general canonical formalism for discrete systems involving varying phase space dimensions. To this end we had to extend the general theory of discrete dynamics from the regular to the irregular case in which constraints necessarily arise. The basic ingredient of the new formalism is Hamilton's principal function which may be used as a generating function (of the first kind) for a canonical time evolution map. This time evolution is generally only defined on a constraint surface and maps to another constraint surface; we refer to the pre--image of the time evolution map as {\\it pre--constraint} surface, while its image is called {\\it post--constraint} surface. The pre--constrainsts, on the one hand, constitute the conditions that need be satisfied in order for the following time evolution step to be allowed. On the other hand, the post--constraints are relations among the phase space data that are automatically realized after an evolution step, however, may provide non-trivial conditions to be satisfied by subsequent time steps. As shown in general, the symplectic forms restricted to the constraint surfaces are preserved under such time evolution maps.\n\nWe have applied the new canonical formalism to simplicial gravity, in particular, and thereby constructed a general and fully consistent formulation of (Euclidean) canonical Regge calculus. However, the formalism is general and, therefore, adaptable to other discretization schemes (with additive action). The general idea is to implement a canonical forward time evolution by gluing single simplices step by step to a bulk triangulation, while backward evolution is simply achieved by removing single simplices from the triangulation. The prerequisite for this evolution scheme is additivity of the action. The $(D-1)$-dimensional hypersurfaces then evolve in a discrete `bubble' time through the full $D$-dimensional Regge solution akin to the evolution of hypersurfaces in canonical general relativity. Both the gluing and removal moves allowed in the present scheme can be interpreted entirely within the $(D-1)$-dimensional triangulated hypersurfaces as Pachner moves. Pachner moves are an elementary and ergodic class of piecewise-linear homeomorphisms which can map between any two triangulations of a given topology by finite sequences (in fact, one could also implement `spatial' topology changes --- aka births of baby universes --- by generalizing the class of allowed gluing and removal moves, see section \\ref{Pachner}).  \n\nThe central issue to be addressed when implementing the Pachner moves in a canonical formalism is the varying phase space dimension since Pachner moves introduce and remove edges. This obstacle can be handled in a straightforward manner by suitable phase space extensions which are controlled by constraints that arise naturally in this framework and constitute the canonical incarnation of the equations of motion of edges to the `future' or `past' of a given hypersurface. In this fashion, phase spaces before and after any Pachner move can be brought to equal dimension, the moves can be implemented via momentum updating as canonical transformations and the symplectic forms restricted to the constraint surfaces are preserved under these moves. \nIn fact, in an alternative construction without immediate use of Hamilton's principal function, the elementary time evolution map can also be extended to the full extended phase spaces (i.e.\\ disregarding any constraints) via momentum updating and preserves the full symplectic structure. This requires, in addition, to impose the pre-- and post--constraints separately, since momentum updating by itself only specifies the difference of the momenta before and after an elementary move, however, not the specific values.\n\n\nThe implementation of the individual Pachner moves follows one and the same recipe, however, each of the Pachner moves assumes a special role in the evolution of the triangulation. In 3D the 1--3 move introduces one vertex and equips it with three constraints which generate vertex displacements in the flat embedding space; the three new edge lengths thereby assume a lapse and shift type gauge parameter interpretation. The pre-- and post--constraints of the 2--2 move\\footnote{Recall that these constraints, in contrast to those from all other moves in 3D and 4D, are only constraints on the extended phase space.} simply preserve and implement flatness, while the pre--constraints of the 3--1 move are automatically satisfied and this move annihilates one vertex and three constraints associated with it. The situation in 3D is special in that all vacuum solutions are necessarily flat and the 3D Regge action is actually a so--called `perfect action' \\cite{Bahr:2009qc,Bahr:2011uj}, i.e.\\ despite being a discrete action it, nevertheless, reproduces the continuum dynamics and symmetries and is triangulation independent. This has significant repercussions for the initial value problem in the discrete: whereas usually the phase space corresponds to the space of solutions this cannot, in general, be expected when the phase space dimension varies and here sufficient initial data is rather specified in the course of evolution. Notwithstanding this issue, in 3D the space of initial data {\\it does} correspond to the space of solutions (modulo gauge) because there are only global physical degrees of freedom (and the Pachner moves do not change the `spatial' topology), while the bulk is necessarily flat and there exists a three-fold gauge symmetry at each vertex in the triangulation. The latter symmetry corresponds to lattice diffeomorphisms and is a consequence of the action being perfect. \n\nIn 4D, however, the situation changes because of local degrees of freedom and generically broken (diffeomorphism) symmetries; the 4D Regge action is {\\it not} perfect and we face the issue of non--uniqueness of solutions in the sense that different solutions arising from the same initial data cannot be mapped into each other by gauge transformations because of the broken symmetries and are thus inequivalent.\nAs an extreme example of this non--uniqueness for geometries with spherical `spatial' topology take the boundary of a 4--simplex as initial slice which fits into any such triangulation and whose data thus by no means specifies the ensuing geometry (or, even more extremely, consider the `no boundary proposal' for triangulations). This is also manifested in the particular roles of the individual Pachner moves in 4D:  the 1--4 move\\footnote{Here we focus solely on gluing moves.} introduces a vertex along with four associated post--constraints which are automatically satisfied after the move and reflect the {\\it a priori} free choice of the four new edge lengths. The 2--3 move generates a bulk triangle while only adding a new boundary edge which comes with a new post--constraint. Again, the length of this new edge, and thus the value of the curvature angle around the internal triangle, is {\\it a priori} free; the move thereby introduces a (free) curvature degree of freedom. These first two moves are the only Pachner moves for 4D simplicial gravity which introduce new edges, however, at the same time do not invoke any equations of motion and, consequently, the new edge lengths are {\\it a priori} free parameters. In addition, the 3--2 move is only allowed if the pre--constraint corresponding to the equation of motion of the edge that is rendered internal in the course of the move is satisfied. Likewise, prior to the 4--1 move which annihilates one vertex, the four pre--constraints associated to the four edges adjacent to the vertex which become internal must be fulfilled. The pre--constraints of these last two moves impose non-trivial conditions which, in general, require a tuning of the free parameters of previous moves (see also the example of the symmetry--reduced five--valent tent move in 4D in section \\ref{tent2}). In short, in 4D all new edge lengths are {\\it a priori} free data which, however, may become fixed {\\it a posteriori} by the pre--constraints of subsequent 3--2 and 4--1 moves. \n\nAs regards the occurrence of gauge symmetries, we have argued in section \\ref{intcon} that the pre-- and post--constraints of the Pachner moves generally do not generate gauge transformations. However, the vertex displacement gauge symmetry of Regge calculus exists on flat configurations and for this regime, the post--constraints of the 1--4 moves --- in analogy to the 1--3 moves in 3D --- {\\it do} constitute the gauge generators. In general, it depends on the particular configuration and, in particular, on choices of data during subsequent moves, whether the 1--4 move introduces gauge generators. A specific example of this was treated in section \\ref{tent2}.\n\nThe formalism introduced here is very general and allows to also consider evolution maps in which arbitrary pieces of $D$--dimensional triangulations are glued onto $(D-1)$--dimensional hypersurfaces (under certain regularity conditions). These can be built up by Pachner moves, but may also be considered in an `effective' picture, where one time step would involve a more complicated piece of triangulation than just a simplex. For the example of the so--called tent moves we have discussed the `effective evolution' in 3D in section \\ref{sectent1} and in 4D in section \\ref{tent2}.\n\nThe new general canonical formalism for discrete systems involving varying phase space dimensions, clearly, warrants further research. In particular, it remains to be investigated under which precise conditions {\\it a priori} free parameters in 4D canonical Regge calculus remain free throughout evolution, i.e.\\ are true gauge parameters, or, equivalently, under which conditions post--constraints of previous steps match pre--constraints of later steps (but involving the same hypersurface). This is a prerequisite for providing a constraint classification with clear distinction between proper equations of motion and (pseudo-) constraints generating (broken) gauge symmetries. The latter question stresses the close connection between constraints and equations of motion, which often can be transformed into each other by relabeling time steps (see the discussion in section \\ref{candisc}) or by changing the slicing into equal time hypersurfaces. Therefore, the question arises whether different choices of `effective' time steps (or slicings of the bulk space-time) may influence the constraints associated to a given hypersurface. This question will be important for quantization as one usually attempts to find a quantization of the constraint hypersurfaces. These issues will be the topic of a separate paper \\cite{dhta2}. \n\nIn another forthcoming paper \\cite{dhta}, we will address some of these issues explicitly in the linearized theory which inherits the gauge symmetry of the flat background. The specific role of each of the Pachner moves becomes clear in this regime, namely: the 1--4 move generates four gauge degrees of freedom and the same number of gauge generating constraints, the 2--3 move generates one `graviton', the 3--2 move imposes the only proper, non-trivial equation of motion and annihilates one `graviton' and, finally, the 4--1 move annihilates four gauge degrees of freedom and the same number of gauge generating constraints which are automatically satisfied prior to the move. This allows for an explicit counting of both gauge and observable degrees of freedom at each evolution step in a scenario of an evolving `spatial' hypersurface. For reasons of continuity, the role of the individual Pachner moves must also be the same in a neighbourhood of the linearized regime.\n\nFurthermore, the new formalism can also be viewed as a specific algorithm to generate solutions and might thereby be interesting for numerical investigations.\nDuring the evolution it may happen that some variables can be solved only in the complex domain. This problem also appears in the consistent discretization approach \\cite{bahrgambinipullin}, where lapse and shift, which are determined by the equations of motion\nobtain complex values. \n\nThe interpretation of this phenomenon is that the initial data chosen are not suitable for the discretization. It is related to the fact, that gauge symmetries are broken and therefore the corresponding constraints on initial data do not have to be implemented\nstrictly. However, there is still a remnant of the constraints: namely that a large set of initial data might be unsuitable as it leads to complex (or unphysical large) lapse and shift parameters.\n\nAlthough in this work we do not fully address the main underlying problem, namely the fact that diffeomorphism symmetry is broken in 4D Regge gravity \\cite{bd1}, this problem might appear to be less severe than in an approach where the discretization\nis fixed from the outset. The advantage of the present approach is that the choice of triangulation and therefore discretization can be adjusted during evolution, i.e.\\ a given sequence of Pachner moves might lead to complex solutions, whereas another sequence does\nnot. For instance one would expect that to avoid complex solutions, one has to choose a finer triangulation for regions with high curvature. The advantage of the approach proposed here is that such an adjustment can be implemented during evolution.\n\n\n\n\n\nIn view of Loop Quantum Gravity (LQG) which involves changing spatial graphs and Spin Foam models which employ triangulations, it seems promising to attempt a quantization of the Pachner moves and implement them as (possibly unitary) transformations on some Hilbert space of `spatial' geometries. This may help in investigating the connection between LQG and Spin Foams (see also \\cite{Bonzom:2011tf} for recent advances from the Spin Foam side); due to the fact that the classical canonical formalism is based on using the action as a generating function, a canonical quantization thereof should coincide with a covariant path-integral quantization of the theory. Usually, a canonical program proceeds by quantizing the phase space (or rather the constraint surface); here the additional challenge emerges of how to incorporate the fact that initial data continues to be specified during evolution and certain {\\it a priori} free parameters become fixed later on.\nThese questions are related to the `general boundary proposal' \\cite{oeckl}, which attempts to generalize the notion of Cauchy hypersurfaces to arbitrary surfaces for quantum field theories. A quantization of the Pachner moves may also shed some light on the `$e^{\\pm iS}$ vs.\\ $\\cos S$' debate in the Spin Foam community: because gluing and removal moves add and subtract action contributions of simplices, respectively, an amplitude with a single exponential of the action should correspond either to pure forward or backward evolution, while an amplitude with the cosine of the action incorporates a superposition of both directions of evolution. \n\nA final interesting question arises: can one construct from the ideas presented here a canonical formulation of Causal Dynamical Triangulations (CDT) (see also \\cite{Arnsdorf:2001wh})? Given that {\\it a priori} all new lengths in 4D canonical Regge calculus can be freely chosen and, in particular, all be fixed equal to one while the births of baby universes are disallowed, can one construct `classical CDT solutions' from this, such that the pre-- and post--constraints determine the connectivity by rejecting or accepting certain Pachner moves? Presumably, this will generically not work and one might have to allow, instead, for the lengths to take value in a small interval $(1-\\epsilon,1+\\epsilon)$. It would be interesting to investigate whether a quantization with such restrictions can be connected to CDT. This also requires an extension of the present scheme to Lorentzian signature and an implementation of the condition that spacelike hypersurfaces remain spacelike throughout evolution.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nB.D.\\ thanks Laurent Freidel for collaboration on \\cite{draft}. Both authors thank Benjamin Bahr and Renate Loll for interesting discussions. Furthermore, P.A.H.\\ is grateful for Exchange Grant 3068 within the ESF activity entitled `Quantum Geometry and Quantum Gravity' which made this collaboration feasible and thanks the Albert Einstein Institute in Potsdam for hospitality in the early and final stages of this work.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n   This paper presents the final interpretation of the Higgs boson\nsearch results from {\\sc DELPHI} in the framework of representative\nscenarios of the\nMinimal Supersymmetric Standard Model ({\\sc MSSM}).\nWith respect to the previous {\\sc MSSM} interpretation published \nin Ref.~\\cite{ref:pap00}, this analysis uses an enlarged set of \nexperimental results, updated calculations of\n{\\sc MSSM} radiative corrections and covers more\nscenarios, including models with {\\sc CP} violation in the Higgs sector.\n\nAs compared with the Standard Model, the {\\sc MSSM}\nhas an extended Higgs sector with two doublets of complex Higgs fields,\nleading to five physical Higgs bosons, of which three are neutral. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\epsfig{figure=hiz.eps,height=5cm} &\n\\epsfig{figure=h1h2.eps,height=5cm} \\\\\n\\end{tabular}\n\\caption[]{\n    Main production processes of {\\sc MSSM} neutral Higgs bosons\n    at {\\sc LEP}. Left: associated production of a Z and a  \n    Higgs boson, which must be one of the {\\sc CP}-even scalars (h or H)\n    if {\\sc CP} is conserved or any Higgs boson (H$_1$, H$_2$, H$_3$) \n    in the contrary case. \n    At {\\sc LEP1}, the intermediate Z is on-shell and the\n    final Z is off-shell, while it is the reverse at {\\sc LEP2}.\n    Right: pair-production of neutral Higgs bosons. \n    If {\\sc CP} is conserved, one of them must\n    be  {\\sc CP}-even (h or H) and the other one is the\n    {\\sc CP}-odd pseudo-scalar A. If {\\sc CP} is not conserved, \n    the pair can be any couple of different scalars\n    among H$_1$, H$_2$ and H$_3$.\n    The intermediate Z is on-shell at {\\sc LEP1}.}\n\\label{fig:diagrams}\n\\end{center}\n\\end{figure}\n\nIf {\\sc CP} is conserved, two of the three \nneutral Higgs bosons are CP-even. They are denoted h, for the lighter one, \nand H. The third one is a  {\\sc CP}-odd pseudo-scalar, denoted A. \nIn \\mbox{${\\mathrm e}^+{\\mathrm e}^-$}\\ collisions, the dominant production mechanisms are \nthe s-channel processes described in Fig.~\\ref{fig:diagrams}, that\nis the associated production of a Z and a  {\\sc CP}-even Higgs boson\nand the pair production of either {\\sc CP}-even boson together\nwith the {\\sc CP}-odd scalar. \nThese processes are complemented by additional\nt-channel diagrams in the final states where a {\\sc CP}-even Higgs boson \nis produced with neutrinos or electrons, which proceed through \\WW\\ and \n\\ZZ\\ fusions, respectively. These diagrams and their interference with \nthe \\HiZ\\ process have an impact on the production cross-section at\nmasses around the \\HiZ\\ kinematic threshold. \nAt {\\sc LEP2} energies, the only significant effect is from \\WW\\ fusion\nwhich doubles the neutrino \\HiZ\\ cross-section at the kinematic \nthreshold.\nFinally, charged Higgs bosons, \\hp\\ and \\hm, are \nproduced in pairs through a diagram similar to that in  \nFig.~\\ref{fig:diagrams}, right, via exchange of a Z boson or a photon.\n\nAlthough {\\sc CP} is conserved at tree level in the {\\sc MSSM}, radiative\ncorrections can introduce {\\sc CP} violation through stop and sbottom loops,\nleading to\nchanges in the neutral Higgs boson sector~\\cite {ref:mssmpheno}.\nIf {\\sc CP} is not conserved,\nthe three neutral Higgs bosons are no longer pure {\\sc CP} eigenstates\nbut mixtures of {\\sc CP}-even and {\\sc CP}-odd components. They\nare usually denoted H$_1$, H$_2$ and H$_3$, in increasing mass. \nThe main production mechanisms are the same as in the {\\sc CP} conserving \ncase, except that, {\\it a priori}, any scalar\ncan be produced in association with a Z boson or through \\WW\\ and \n\\ZZ\\ fusions, and any couple of different Higgs bosons can be pair-produced. \nThe main phenomenological difference with respect to the \n{\\sc CP}-conserving case\nlies in the strength of the couplings of the Z boson to the Higgs scalars. \nIn significant regions of the parameter space, {\\sc CP} violation turns off\nthe otherwise dominant coupling between the Z boson and the lightest\nHiggs boson. \nIn that case, if none of the other processes of \nFig.~\\ref{fig:diagrams} are possible (due e.g. to kinematics), \nthe dominant Higgs boson production mechanism at {\\sc LEP} becomes the Yukawa \nprocess of Fig.~\\ref{fig:yukawa}. Of the two phases of {\\sc LEP}, only\n{\\sc LEP1} has a significant sensitivity to this process. \nIn the Standard Model, the corresponding cross-sections are negligible, \ne.g. a fraction of a pb  for a few~\\GeVcc\\ Higgs boson. \nIn the {\\sc MSSM}, these can be \nenhanced by up to three orders of magnitude with respect to their \nStandard Model values, leading to detectable signals which become valuable\nin the case of {\\sc CP} violation.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{c}\n\\epsfig{figure=hiyuk.eps,height=5cm} \n\\end{tabular}\n\\caption[]{\n    Additional production process of {\\sc MSSM} neutral Higgs bosons\n    at {\\sc LEP}. The radiation of a Higgs boson off a Z boson\n    decay fermion gives a detectable signal only at {\\sc LEP1}.\n    This signal is exploited in the case of {\\sc CP} violation.}\n\\label{fig:yukawa}\n\\end{center}\n\\end{figure}\n\n  The decay properties of the Higgs bosons are moderately\naffected by {\\sc CP} violation, at least in the \nrange of masses accessible at {\\sc LEP}, that is up to masses around\n100~\\GeVcc~\\cite {ref:mssmpheno}. In most of the {\\sc MSSM}\nparameter space of the scenarios studied hereafter, the \nthree neutral Higgs bosons decay mainly into the pair of heaviest fermions \nkinematically permitted, even if {\\sc CP} is not conserved.\nBelow the \\mumu\\ threshold,\na Higgs boson would decay into \\mbox{$\\gamma \\gamma$}\\ or \\mbox{${\\mathrm e}^+{\\mathrm e}^-$}\\ pairs with a significant\nlifetime. Above the \\mumu\\ threshold, the lifetime is negligible and \nHiggs bosons decay at the primary vertex. \nUp to a mass of 3~\\GeVcc\\, the main decays are\ninto \\mumu\\ pairs and also into hadronic channels with a large proportion\nof two-prong final states. Above 3~\\GeVcc\\, the dominant decays are \nsuccessively into \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}, \\mbox{$\\tau^+ \\tau^- $}\\ and finally \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ pairs for\nHiggs boson masses above 12~\\GeVcc.\nBesides these decays into fermions, there \nare also regions of the parameter space where one neutral Higgs boson \ncan undergo cascade decays to a pair of Higgs bosons, as for \nexample \\hAA\\ if {\\sc CP} is conserved or \\Hcas\\ in the contrary case.\nIn some cases, especially if {\\sc CP} is not conserved,\nthis mode dominates over the decays into \n{\\sc SM} particles. \nIn the scenarios considered in this paper, charged Higgs bosons have a mass\nabove 60~\\GeVcc\\ and \ndecay either into the pair of heaviest fermions allowed by kinematics, that is\ninto cs or $\\tau\\nu$ pairs, or into a W$^*$ and a light Higgs\nboson, e.g. into a W$^*$A pair if {\\sc CP} is conserved.\nFinally, these scenarios do not allow neutral or charged Higgs boson decays \ninto supersymmetric particles such as sfermions, charginos or invisible \nneutralinos. Note that searches for neutral Higgs bosons decaying into \ninvisible products were performed at {\\sc LEP}, as \nreported in Ref.~\\cite{ref:invis}.\n\nThe different decay channels define the topologies that were\nsearched for to cover the {\\sc MSSM} parameter region\nkinematically accessible at {\\sc LEP} energies. \nThese topologies are described in Section~\\ref{sec:channels}. \nSection~\\ref{sec:tools} summarizes \nthe definition and techniques related to \nconfidence levels used in the statistical interpretation of the searches. \nThe eight \n{\\sc CP}-conserving {\\sc MSSM} benchmark scenarios studied in this analysis are\npresented  in Section~\\ref{sec:benchmark} and the results obtained\nin these scenarios when combining all searches are given in \nSection~\\ref{sec:results}. Similarily, the {\\sc CP}-violating scenarios\nand the corresponding results are covered in Sections~\\ref{sec:cpv}\nand ~\\ref{sec:resucpv}.\nThe top quark mass has a significant impact on the properties\nof the Higgs bosons (e.g. mass spectrum of the neutral Higgs bosons, \n{\\sc CP}-violating effects). Results are thus derived for several values \nof this mass, namely: \\mtop~=~169.2, 174.3, 179.4 and 183.0~\\GeVcc, which\nwere defined after the measurement of the top quark mass\nat the Tevatron, run I~\\cite{ref:mtop_old}.\nOf the two values close to the present experimental measurement\nof \\mtop~=~170.9~$\\pm$~1.1~$\\pm$~1.5~\\GeVcc~\\cite{ref:mtop}, \n174.3~\\GeVcc\\ gives the most conservative results \nand thus was chosen as\na reference in most of the exclusion plots and to quote absolute mass \nand \\tbeta\\ limits. \nReaders interested in similar analyses at {\\sc LEP}\nare referred to Ref.~\\cite{ref:opal,ref:hwg}.\n\n\\section{Search channels}\\label{sec:channels}\n\n\n\\begin{table}[p]\n{\\normalsize\n\\vspace{-1.5cm}\n\\begin{center}\n\\begin{tabular}{clcccc}     \\hline \n\\mbox{$\\sqrt{s}$} & final state & range & \\mbox{$\\cal L$}\\      & disc. & ref. \\\\\n(GeV) &           & (\\GeVcc)   & (pb$^{-1}$) & info. & \\\\ \\hline\n \\multicolumn{6}{c}{\\hZ\\ with direct decays} \\\\ \\hline\n91 &  \\Zem\n          & $< 0.21$   & 2.5 & no & \\cite{ref:vlow} \\\\\n91 &  (h $\\rightarrow {\\mathrm V}^0$) (\\Zany)               \n          & $< 0.21$   & 2.5 & no & \\cite{ref:vlow} \\\\\n91 &  (h $\\rightarrow $ 2 prongs) (\\Zqq)\n          & $0.21 - 2.$  & 0.5 & no & \\cite{ref:low} \\\\\n91 &  (h $\\rightarrow $ jet) (\\Zem)\n          & $1. - 20.$ & 0.5 & no & \\cite{ref:low} \\\\\n91 &  (h $\\rightarrow $ jet jet) (\\Zll, $\\nu \\bar{\\nu}\\;$)\n          & $>12.$ & 3.6 & no & \\cite{ref:jjlow} \\\\\n91 &  (h $\\rightarrow $ jet jet) (\\Zem, $\\nu \\bar{\\nu}\\;$) \n          & $>35.$ & 33.4 & no & \\cite{ref:jjhigh} \\\\\n\\bf{161,172} & (h $\\rightarrow $ \\bbbar)(\\Zany), (h $\\rightarrow $ \\toto)(\\Zqq)  \n            & $>40.$ & 19.9 & 1d &\\cite{ref:pap96} \\\\\n\\bf{183} &   (h $\\rightarrow $ \\bbbar)(\\Zany), (h $\\rightarrow $ \\toto)(\\Zqq)  \n            & $>55.$ & 52.0 & 1d & \\cite{ref:pap97} \\\\\n\\bf{189} &   (h $\\rightarrow $ \\bbbar)(\\Zany), (h $\\rightarrow $ \\toto)(\\Zqq)   \n            & $>65.$ & 158.0 & 2d &\\cite{ref:pap98} \\\\\n\\bf{192-208} & (h $\\rightarrow $ \\bbbar)(\\Zany)   \n            & $>12.$ & 452.4 & 2d &\\cite{ref:pap99,ref:pap00} \\\\\n\\bf{192-208} & (h $\\rightarrow $ \\toto)(\\Zqq)   \n            & $>50.$ & 452.4 & 2d &\\cite{ref:pap99,ref:pap00} \\\\\n189-208 & (h $\\rightarrow $ hadrons)(\\Zany\\ but \\mbox{$\\tau^+ \\tau^- $})\n            & $>4.$  & 610.4 & mix &\\cite{ref:papflbl} \\\\\n\\hline\n\\multicolumn{6}{c}{\\hA\\ with direct decays} \\\\\\hline\n91        & 4 prongs       \n            & $>0.4$ & 5.3 & no & \\cite{ref:4prongs} \\\\\n91        & \\mbox{$\\tau^+ \\tau^-$}\\ hadrons       \n            & $>8.$ & 0.5 & no & \\cite{ref:ha89} \\\\\n91        & \\mbox{$\\tau^+ \\tau^-$}\\ jet jet       \n            & $>50$ & 3.6 & no & \\cite{ref:jjlow} \\\\\n91        & \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}, \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm c}\\bar{\\mathrm c}$} \n            & $>30.$ & 33.4 & no & \\cite{ref:ha92} \\\\\n91        & \\mbox{$\\tau^+ \\tau^-$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \n            & $>16.$ & 79.4 & no & A.1 \\\\\n91        & \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \n            & $>24.$ & 79.4 & no & \\cite{ref:2hdm} \\\\\n\\bf{133}       &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\n            & $>80.$ & 6.0 & no & \\cite{ref:pap95} \\\\\n\\bf{161,172} &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}, \\mbox{$\\tau^+ \\tau^-$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}   \n            & $>80.$ & 20.0 & 1d &\\cite{ref:pap96} \\\\\n\\bf{183}       &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}, \\mbox{$\\tau^+ \\tau^-$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}   \n            & $>100.$ & 54.0 & 1d & \\cite{ref:pap97} \\\\\n\\bf{189}       &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}, \\mbox{$\\tau^+ \\tau^-$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}  \n            & $>130.$ & 158.0 & 2d &\\cite{ref:pap98} \\\\\n\\bf{192-208} &  \\mbox{$\\tau^+ \\tau^-$}  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}  \n            & $>120.$ & 452.4 & 2d &\\cite{ref:pap99,ref:pap00} \\\\\n\\bf{192-208} &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}  \n            & $>80.$ & 452.4 & 2d &\\cite{ref:pap99,ref:pap00} \\\\\n189-208 &  \\mbox{$\\tau^+ \\tau^-$} \\mbox{$\\tau^+ \\tau^-$}     \n            & $>8.$ & 570.9 & 1d & \\cite{ref:2hdm} \\\\\n189-208 &  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}    \n            & $>24.$ & 610.2 & no & \\cite{ref:2hdm} \\\\\n189-208 & hadrons\n            & $>8.$  & 610.4 & mix &\\cite{ref:papflbl} \\\\\n\\hline\n\\multicolumn{6}{c}{\\hZ\\ or \\hA\\ with \\hAA\\ cascade} \\\\\\hline\n91        & \\Zqq                  \n            & $<0.21$ & 16.2 & no & \\cite{ref:dallas} \\\\\n91        & (\\cascv) (\\Zanyt)                  \n            & $<0.21$ & 9.7 & no & \\cite{ref:dallas} \\\\\n91        & (\\cascg) (\\Zany\\ or A $\\rightarrow \\gamma \\gamma $)                  \n            & $<0.21$ & 12.5 & no & \\cite{ref:dallas} \\\\\n91        & (\\cascp) (\\Zany\\ or A $\\rightarrow $ 2 prongs)                  \n            & $>0.21$ & 12.9 & no & \\cite{ref:dallas} \\\\\n91        & (\\casch) (\\Znn\\ or A $\\rightarrow $ hadrons)                  \n            & $>0.21$ & 15.1 & no & \\cite{ref:dallas} \\\\\n91        & (\\casct) (\\Znn\\ or A $\\rightarrow $ \\toto)                  \n            & $>3.5$ & 15.1 & no & \\cite{ref:dallas} \\\\\n\\bf{161,172} & (\\casca) (\\Zqq, $\\nu \\bar{\\nu}\\;$ or A $\\rightarrow$ any)    \n            & $>20.$ & 20.0 & 1d &\\cite{ref:pap96} \\\\\n\\bf{183}      & (\\cascb) (\\Zqq)\n            & $>12.$ & 54.0 & 1d & \\cite{ref:pap97} \\\\\n\\bf{192-208} & (\\cascb, \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}, \\mbox{${\\mathrm c}\\bar{\\mathrm c}$} \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}) (\\Zqq)   \n            & $>12.$ & 452.4 & 2d &\\cite{ref:pap99,ref:pap00} \\\\\n192-208 &  (\\cascc ) (\\Zqq)    \n            & $>4.$ & 452.4 & 2d &  A.2 \\\\\n189-208 & (\\cascb) (\\Zqq\\ or A $\\rightarrow $ \\bbbar)   \n            & $>12.$ & 610.2 & no &\\cite{ref:2hdm} \\\\\n\\hline\n\\multicolumn{6}{c}{ffh or ffA Yukawa production} \\\\\\hline\n91        &\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} (h $\\rightarrow $ \\toto), \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} (A $\\rightarrow $ \\toto) \n            & $4.-50.$ & 79.4 & no & \\cite{ref:2hdm} \\\\\n91        &\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} (h $\\rightarrow $ \\bbbar), \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} (A $\\rightarrow $ \\bbbar) \n            & $11.-50.$ & 79.4 & no & \\cite{ref:2hdm} \\\\\n91        &\\mbox{$\\tau^+ \\tau^-$} (h $\\rightarrow $ \\toto), \\mbox{$\\tau^+ \\tau^-$} (A $\\rightarrow $ \\toto) \n            & $4.-50.$ & 79.4 & no & \\cite{ref:2hdm} \\\\\n\n\\hline\n\\multicolumn{6}{c}{\\mbox{$ {\\mathrm H}^+{\\mathrm H}^- $}} \\\\\\hline\n189-208 & \\cscs, \\cstn, \\watn, \\wawa \n            & $>40.$ & 610.4 & 2d & \\cite{ref:paphphm} \\\\\n189-208 & \\tntn\n            & $>40.$ & 570.8 & 1d & \\cite{ref:paphphm} \\\\\n\\hline\n\\end{tabular}\n\\caption[]{\n    List of signals expected from {\\sc MSSM} Higgs bosons that \n    were searched for in the {\\sc DELPHI} data sample. Indicated for \n    each signal are the centre-of-mass energy, \n    final state, analysed mass range, integrated luminosity,\n    level of discriminant information included in the confidence level\n    estimates (none, one- or two-dimensional) and the reference \n    where details of the analysis are published. Here h and A \n    denote any neutral Higgs boson allowed to be produced\n    in each of the indicated production processes.\n    The mass range applies\n    to \\mh\\ for \\hZ\\ production, to \\mh+\\mA\\ for \\hA\\ production,\n    to \\mA\\ for \\hAA\\ processes, to the Higgs boson mass for\n    either Yukawa process and to \\mHp\\ for\n    \\mbox{$ {\\mathrm H}^+{\\mathrm H}^- $}\\ production. When no upper bound is given, the \n    limit imposed by kinematics or vanishing branching fractions must be \n    understood. \n}\n\\label{tab:channels}\n\\end{center}\n}\n\\end{table}\n\nThe different analyses performed to search for neutral and charged \nHiggs bosons in the whole {\\sc LEP1} and {\\sc LEP2}\n{\\sc DELPHI} data samples are summarized in \nTable~\\ref{tab:channels} which lists the final states, mass ranges, \nintegrated luminosities\nand the references for more details about the selections \nand their performance. \nTwo channels, the \\mbox{$\\tau^+ \\tau^- $\\bbbar }\\ signal at {\\sc LEP1}\nand the (h~$\\rightarrow$~\\cascc)~(\\Zqq) signal at A masses below the\n\\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ threshold, were analysed for this paper, using selections\nalready published. The efficiencies and the references for the\nselections can be found in the Appendices 1 and 2 of this paper. \nIn the Table, the notations h and A which label\nthe different analysis channels must be understood as generic notations\nfor any pair of neutral Higgs bosons that could be produced in each\nof the production processes listed in the Table. As an example,\nthe hZ analyses, originally designed to search for the \n{\\sc CP}-even h boson in {\\sc CP}-conserving scenarios,\ncan be applied to search for the\nsecond {\\sc CP}-even Higgs boson, H, as well as for\nany of the three Higgs scalars in {\\sc CP}-violating scenarios.\nIt must be noted that the kinematic properties of the signal processes\nare only slightly affected by {\\sc CP}-violation, since, when {\\sc CP} is\nnot conserved, the production processes still proceed through the\n{\\sc CP}-even and {\\sc CP}-odd components of the neutral Higgs\nbosons, as explained in Ref.~\\cite{ref:hwg}. The same topological searches\ncan thus be applied whether {\\sc CP} is conserved or not.\n\nAs compared with our previous publication~\\cite{ref:pap00}, the following\nchanges were introduced in the experimental results used. \nThe {\\sc MSSM} interpretation in Ref.~\\cite{ref:pap00} \nrelied only on searches performed at {\\sc LEP2} at masses above 12~\\GeVcc\\\nin \\mh\\ in the \\hZ\\ process, with either direct or cascade decays, and \nabove 40~\\GeVcc\\ in \\mh, \\mA\\ in the \\hA\\ channels, with only direct\ndecays of the Higgs bosons. The corresponding channels \nhave their \\mbox{$\\sqrt{s}$}\\ values in bold characters in Table~\\ref{tab:channels}. \nScans of\nthe {\\sc MSSM} parameter space were thus restricted to \\mA\\ above \n12~\\GeVcc\\ and assumed the published {\\sc LEP1} \nlimits\\footnote{\\mh$>$44~(46)~\\GeVcc\\ \nwhen \\mh\\ is above (below) the ${\\mathrm{AA}}$ threshold~\\cite{ref:ha92}} \nto be valid. \nIncluding all {\\sc LEP1} results, which have a sensitivity\nstarting from vanishing h and A masses, and the additional {\\sc LEP2}\nsearches of Ref.~\\cite{ref:2hdm}, whose sensitivity in the \\hA\\ mode \ncomplements that of the two other sets of results, allows \nscans of the {\\sc MSSM} parameter space to be performed\nwith no restriction on masses.\nMoreover, some of the analyses of Ref.~\\cite{ref:2hdm} cover production\nprocesses which are negligible if {\\sc CP} is conserved but are \nenhanced by {\\sc CP} violation, such as Yukawa processes or the\nproduction of \\mbox{$\\tau^+ \\tau^-$}\\toto\\ final states. \nAdding the searches for neutral Higgs bosons decaying into hadrons of \nany flavour~\\cite{ref:papflbl} is expected to provide  \nsensitivity in scenarios where the Higgs boson decays into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\  \nwould vanish. \nAs their mass coverage starts at low mass, \nthese analyses also increase the experimental sensitivity to\nHiggs bosons below the \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ threshold, a region otherwise covered \nonly by analyses of subsets of the {\\sc LEP1} data. \nFinally, the charged Higgs boson searches~\\cite{ref:paphphm}\nhelp in a few {\\sc CP}-conserving scenarios in the low \\mA\\ region\nwhere the charged bosons are kinematically accessible at {\\sc LEP2}.\n\n  Moreover, our previous interpretation was dealing only with \nthe production of the two lightest Higgs bosons, \nthe h and A scalars in {\\sc CP}-conserving scenarios.\nIn this analysis, the production\nof the third boson, if kinematically accessible, is also\naccounted for, which can lead to a significant gain in sensitivity \nin restricted areas of the parameter space.\nIn {\\sc CP}-conserving scenarios, this leads to\nincluding the \\HZ\\ and \\HA\\ signals besides the usual \\hZ\\ and\n\\hA\\ processes, while in {\\sc CP}-violating models, the \nH$_2$Z and H$_1$H$_3$ signals are taken into account in addition\nto the dominant H$_1$Z and H$_1$H$_2$ channels (the two other \nprocesses, H$_3$Z and  H$_2$H$_3$ being out of reach). \n\n\\section{Tools for the statistical analysis}\\label{sec:tools}\n\nWhen scanning over the parameter space of a model, confidence levels are \ncomputed at each point to test the compatibility of data with the\nhypothesis of background only and with that of background plus \nsignal as expected from the model. Throughout this section, \nthe notations h, H and A must be understood as generic notations\nfor the three neutral Higgs bosons of any type of {\\sc MSSM} scenario.\n\n\n\\subsection{Confidence level definitions and calculations}\nThe confidence levels are calculated using a modified frequentist\ntechnique based on the extended maximum likelihood ratio~\\cite{ref:alex}\nwhich has also been adopted by the {\\sc LEP} Higgs working group.\nThe basis of the calculation is the likelihood ratio test-statistic, \\mbox{$\\cal Q$}:\n\\[ \\ln\\mbox{$\\cal Q$} = -S + \\sum_i \\ln \\frac{s_i+b_i}{b_i} \\]\nwhere $S$ is the total signal expected and $s_i$ and $b_i$ are the\nsignal and background densities for event $i$.\nThese densities are constructed using either expected rates only\nor also additional discriminant information, which can be one- or\ntwo-dimensional. Table~\\ref{tab:channels} presents the level\nof discriminant information for each channel:\n{\\sc LEP1} results rely on rates only, while {\\sc LEP2} results\nmix channels without or with discriminant information.\nAs an example, in neutral Higgs boson channels with discriminant\ninformation, the first variable is the reconstructed Higgs boson mass \nin the \\hZ\\ analyses and the sum of the reconstructed h and A masses \nin the \\hA\\ analyses, while the second variable, if any, is \nchannel-dependent, as specified in the references listed in the Table. \nCharged Higgs analyses use discriminant information in a similar \nway~\\cite{ref:paphphm}.\nThe searches for Higgs bosons decaying hadronically encompass analyses \nwithout or with 1d discriminant information together with analyses \nwhose selections vary with the mass hypothesis~\\cite{ref:papflbl}.\n\nThe observed value of \\mbox{$\\cal Q$}\\ is compared \nwith the expected Probability Density Functions (PDFs) for \\mbox{$\\cal Q$}, \nwhich are built using Monte Carlo sampling under the assumptions that \nbackground processes\nonly or that both signal and background are present. The confidence \nlevels \\mbox{$\\rm CL_{\\rm b}$}\\ and \\mbox{$\\rm CL_{\\rm s + b}$}\\ are their integrals from $-\\infty$ to the \nobserved value of \\mbox{$\\cal Q$}.\nSystematic uncertainties in the rates of signal or background events are taken \ninto account in the calculation of the PDFs for \\mbox{$\\cal Q$}\\ by \nrandomly varying the expected rates while generating the\ndistribution~\\cite{ref:CousinsAndHighland},\nwhich has the effect of broadening the expected \\mbox{$\\cal Q$}\\ distribution and \ntherefore making extreme events seem more probable.\n\n\\mbox{$\\rm CL_{\\rm b}$}\\ is \nthe probability of obtaining a result as background-like\nor more so than the one observed if the background hypothesis is correct.\nSimilarly,\nthe confidence level for the hypothesis that both signal and background \nare present, \\mbox{$\\rm CL_{\\rm s + b}$}, is the probability, in this hypothesis,\nto obtain more background-like results than those observed. \nThe quantity \\mbox{$\\rm CL_{\\rm s}$}\\ is defined as the ratio of these\ntwo probabilities, \\mbox{$\\rm CL_{\\rm s + b}$}\/\\mbox{$\\rm CL_{\\rm b}$}. It is not a true\nconfidence level, but a conservative pseudo-confidence level\nfor the signal hypothesis.\nAll exclusions discussed hereafter use \\mbox{$\\rm CL_{\\rm s}$}\\ and require it to be\n5\\% for an exclusion confidence of 95\\%. As using \\mbox{$\\rm CL_{\\rm s}$}\\\ninstead of \\mbox{$\\rm CL_{\\rm s + b}$}\\ is conservative, the rate of fake exclusions\nis ensured to be below 5\\% when \\mbox{$\\rm CL_{\\rm s}$}\\ is equal to 5\\%.\n\n\n\\subsection{Estimation of expected signal and background densities}\n\\label{sec:pdf}\n\nThe expected signal and background densities, which are required \nto check the consistency of the data with the background and signal \nprocesses have two components: the overall normalization which sets\nthe expected rates and the Probability Density Functions (PDF) of the \nadditional discriminant information, if any.\n\n  The expected background and signal rates were\ncalculated from the number of simulated events \npassing the cuts. For the signal the efficiencies derived from \nsimulations at given mass points had to be interpolated\nto estimate efficiencies at Higgs boson masses which were\nnot simulated. In most cases this was done using one polynomial\nor if necessary two polynomials, one\nto describe the slow rise, and a second to handle the kinematic cut-off,\nwhich can be much more abrupt.\nFor the cases where two signal masses must\nbe allowed,  a two-dimensional  parameterization was used.\n\n\n The shapes of the PDFs were derived using \nhistograms which are taken from the simulated events.\nIn the case of two-dimensional PDFs these distributions were\nsmoothed using a two-dimensional kernel, which consists of  a \nGaussian distribution with a small component of a longer \ntail~\\cite{ref:smoothing}.\nThe global covariance of the distribution was used to determine the \nrelative scale factors of the two axes. The width of the kernel\nvaried from point to point, such that the statistical error on \nthe estimated background processes was constant at 20\\%. \nFinally multiplicative correction factors (each a one-dimensional\ndistribution for one of the two dimensions of the PDF) were derived\nsuch that when projected onto either axis the PDF has the same distribution \nas would have been observed if it had been projected onto the axis \nfirst and then smoothed.\nThis makes better use of the simulation statistics if there are features \nwhich are essentially one-dimensional, such as mass peaks.\nThe error parameter fixed to 20\\% was an important choice. It was set by\ndividing the background simulation into two subsamples, generating a \nPDF with one and using the other to test for over-training by calculating \nthe \\mbox{$\\rm CL_{\\rm b}$}\\ obtained from simulation of background events. This should be \n0.5 if the results are not to be biased, and a value of 20\\% for\nthe error gave the closest approximation to 0.5 in all channels. Examples\nof smoothed two-dimensional PDFs are given in Fig.~\\ref{fig:adding_H}.\n\n  The signal simulations were made at fixed Higgs boson masses, but in order\nto test a continuous range of masses, interpolation \nsoftware~\\cite{ref:alex-interp} was used to create signal PDFs at \narbitrary masses. In the last year of operation, {\\sc LEP} energy was\nvaried continuously while simulations were made at fixed beam energies.\nThe same interpolation software was used to create signal and background \nPDFs at the correct centre-of-mass energies~\\cite{ref:pap00}.\nThe interpolation was done by linearly interpolating the cumulative \ndistributions taking as a parameter the signal mass or the centre-of-mass\nenergy. The procedure \nhas been tested over ranges up to 40~\\GeVcc\\ in mass while the actual \nshifts in the simulations were up to 0.3~\\GeV\\ in $\\sqrt{s}$,\nand 5~\\GeVcc\\ in mass for the \\hZ\\ signals overall, but less\nthan 0.5~\\GeVcc\\ for Higgs boson masses between 113.5 and 116.5~\\GeVcc.\nFor the \\hA\\ channels, the actual shifts were 5~\\GeVcc\\ in either mass\nfor Higgs boson masses between 80 and 95~\\GeVcc\\ and up to 20~\\GeVcc\\\nelsewhere. Comparisons of simulated and interpolated distributions \nfor a given mass were made in all channels and showed good agreement.\n\n\n\\subsection{The case of non-independent channels}\\label{sec:overlap}\n\n  When combining the results in all channels to derive confidence\nlevels, only independent channels must be included, which requires some \nspecial treatment for a few non-independent cases. \n\n\\begin{table}[htbp]\n{\\small\n\\begin{center}\n\\begin{tabular}{c|c|lcc}     \\hline\n\\multicolumn{4}{c}{a) different signals - one analysis with mass \nhypothesis-independent selections} \\\\\\hline\nanalysis & \\mbox{$\\sqrt{s}$}\\ (GeV) & signals added & ref. \\\\\\hline\nffh four-b & 91 \n           & (\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} h $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}), (\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} A $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}),\n             (hA $\\rightarrow$\\  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}) \n           & \\cite{ref:2hdm} \\\\\nffh \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\mbox{$\\tau^+ \\tau^-$} & 91 \n             & (\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} h $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{$\\tau^+ \\tau^-$}), (\\mbox{${\\mathrm b}\\bar{\\mathrm b}$} A $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{$\\tau^+ \\tau^-$}),\n               (hA $\\rightarrow$\\  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{$\\tau^+ \\tau^-$}) \n             & \\cite{ref:2hdm}\\\\\nffh four-$\\tau$ & 91 \n                & (\\mbox{$\\tau^+ \\tau^-$} h $\\rightarrow$\\ \\mbox{$\\tau^+ \\tau^-$} \\mbox{$\\tau^+ \\tau^-$}), (\\mbox{$\\tau^+ \\tau^-$} A $\\rightarrow$\\ \\mbox{$\\tau^+ \\tau^-$} \\mbox{$\\tau^+ \\tau^-$})\n                & \\cite{ref:2hdm}\\\\\n$\\nu\\bar{\\nu}{\\mathrm q \\bar{q}}$ \\     &161-172\n           & (h $\\rightarrow $ \\qqbar) (\\Znn), (\\hAA) (\\Znn)\n           & \\cite{ref:pap96} \\\\ \n\\mbox{$\\tau^+ \\tau^- $\\qqbar }\\ & 189-208\n           & (h $\\rightarrow $ \\toto) (\\Zqq), (h $\\rightarrow $ \\qqbar) (\\Ztt), (hA $\\rightarrow$\\ \\mbox{$\\tau^+ \\tau^-$} \\mbox{${\\mathrm q}\\bar{\\mathrm q}$}) \n           & \\cite{ref:pap98,ref:pap99,ref:pap00} \\\\\n\\hZ\\ four-jet &161-183\n              & (h $\\rightarrow $ \\qqbar) (\\Zqq), (\\hAA) (\\Zqq)\n              & \\cite{ref:pap96,ref:pap97} \\\\\n\\hZ\\ four-jet &192-208\n              & (h $\\rightarrow $ \\qqbar) (\\Zqq), (\\hAA) (\\Zqq), (hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$})\n              & \\cite{ref:pap99,ref:pap00} \\\\\n\\hA\\ four-jet &161-172\n              & (hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}), (\\hAA) A\n              & \\cite{ref:pap96} \\\\\n\\hA\\ four-jet &192-208\n              & (hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}), (h $\\rightarrow $ \\qqbar) (\\Zqq) \n              & \\cite{ref:pap99,ref:pap00} \\\\\nfour-b       &189-208\n             &  (\\hAA)A, (\\hAA)Z, (hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$})\n             & \\cite{ref:2hdm} \\\\\n\\hline\n\\multicolumn{4}{c}{b) different analyses - one final state} \\\\\\hline\nfinal state & \\mbox{$\\sqrt{s}$}\\ (GeV) & competing analyses & ref. \\\\\\hline\nfour-jet  &91& \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}, \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}\n          & \\cite{ref:ha92} \\\\\nmulti-jet &91& three and four-jet analyses \n          & \\cite{ref:2hdm} \\\\ \n$\\nu\\bar{\\nu}{\\mathrm q \\bar{q}}$ \\    &192-208& low mass and high mass hZ analyses  \n          & \\cite{ref:pap00} \\\\\n          &189-208& low mass and high mass flavour-blind analyses  \n          & \\cite{ref:papflbl} \\\\ \n          &189-208& hZ and flavour-blind analyses  \n          & \\cite{ref:pap98,ref:pap99,ref:pap00,ref:papflbl} \\\\\nll\\mbox{${\\mathrm q}\\bar{\\mathrm q}$},l=e,$\\mu$    &189-208& hZ and flavour-blind analyses  \n          & \\cite{ref:pap98,ref:pap99,ref:pap00,ref:papflbl} \\\\\nfour-jet  &192-208& low mass and high mass hZ analyses \n              & \\cite{ref:pap00} \\\\\n          &189-208& low mass and high mass hZ flavour-blind analyses \n              & \\cite{ref:papflbl} \\\\\n          &189-208& three and four-jet hA flavour-blind analyses \n              & \\cite{ref:papflbl} \\\\\n          &189-208& cscs and WAWA analyses \n          & \\cite{ref:paphphm} \\\\\n   &189-208& \\hZ, \\hA, four-b, flavour-blind, \\cscs\\ and \\wawa\\ analyses \n   & \\cite{ref:pap98,ref:pap99,ref:pap00,ref:2hdm,ref:papflbl,ref:paphphm} \\\\\n$\\tau \\nu$ jet jet  &189-208\n         & \\cstn\\ and \\watn\\ analyses \n         & \\cite{ref:paphphm} \\\\ \n\\hline\n\\end{tabular}\n\\caption[]{\n  a) list of signals from the two lightest Higgs bosons h and A\n  treated by a single analysis: the\n  signal expectations are combined (rates added, PDFs summed with weights\n  according to the rates) prior to the confidence level calculations.\n  b) list of different analyses of the same final state: only one\n  analysis is selected at each point in the scans, based on the best \n  expected performance for exclusion.\n  In this Table,\n  h and A denote any neutral Higgs boson allowed to be produced in\n  the indicated production processes.}\n\\label{tab:overlap}\n\\end{center}\n}\n\\end{table}\n\n  The first case is that of different signals covered by the same \nanalysis. The treatment of this depended upon whether the analyses\nwere themselves independent of the mass hypothesis for the \nHiggs bosons. \nThe set of search channels (see Table~\\ref{tab:channels})\ncontains mostly analyses of this kind.\nIn that case, all signals selected by one analysis were\ncombined into one global channel prior to the confidence level computation.\nExpected rates were added together and PDFs were summed with weights\ngiven by the expected rates of the individual signals.\nAs an illustration,  Table~\\ref{tab:overlap}-a gives the list of these\nsignals and analyses on the example of the production of the lightest\nHiggs bosons, h and A, through the \\hZ\\ and \\hA\\ processes. \nWhen extending the combination to the third Higgs boson, H,\nthe same procedure was followed, first for the various signals from\nthat boson in the \\HZ\\ and \\HA\\ processes,\nand then to combine \\hZ\\ and \\HZ\\ signals or \\hA\\ and \\HA\\ signals.\nThe PDF combination in such a case is illustrated in Fig.~\\ref{fig:adding_H}.\n\nA different procedure was applied in the case of different \nsignals covered by the same analysis whose selections do depend on the \nmass hypothesis, as most searches of Ref.~\\cite{ref:papflbl} do.\nDifferent signals are covered by these analyses only\nwhen including signals from the third Higgs boson, H. \nIn that case, in each analysis only one signal \n(from either h or H) was selected at each point in the\nscanned parameter space and at each centre-of-mass energy, \non the basis of the smallest expected \\mbox{$\\rm CL_{\\rm s}$}\\ from experiments with no signal\n(that is, on the basis of the strongest average exclusion if no signal \nis present). \n\nThe second case of non-independent channels is that of \na large overlap in the events selected by different analyses\nsensitive to the same final state. The list of such analyses \nand final states is detailed in Table~\\ref{tab:overlap}-b. Again,\nfor each final state, \nonly that analysis with the strongest expected exclusion power\nwas retained at each test point.\nThis is not optimal but ensures that the channels which are then\ncombined in the global confidence level computations are independent.\n\nWhen the two cases just described (different signals covered by one\nanalysis, different analyses sensitive to the same final state)\nwere present \nsimultaneously, the signal addition was performed before the final\nanalysis selection. Then if that step involved more than two analyses,\nthe final selection was made in successive iterations.\nTo quote the four-jet final state as an example, at energies above 190~GeV,\nthe total \\hZ\\ and \\hA\\ signals were first computed in \neach of the three four-jet analyses of Ref.~\\cite{ref:pap00}\nand in the four-b analysis of Ref.~\\cite{ref:2hdm}.\nThis summed three signals in the low and high mass \n\\hZ\\ dedicated four-jet analyses\n((h $\\rightarrow $ \\qqbar) (\\Zqq), (\\hAA) (\\Zqq) and hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}),\ntwo signals in the \\hA\\ dedicated four-jet analysis\n(hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ and (h $\\rightarrow $ \\qqbar) (\\Zqq)) \nand three signals in the four-b analysis\n((\\hAA)A, (\\hAA)(\\Zqq) and hA $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}).\nThe signals due the third Higgs boson, H, were computed in the same way\nand added to those from the h boson. \nThen, a choice was made between the low and \nhigh mass \\hZ\\ dedicated four-jet analyses. \nThe result of this selection was compared \nwith the \\hA\\ dedicated four-jet analysis, \nand the best of these was confronted with the four-b analysis. \nA choice was made\nbetween the remaining analysis and the best between the various\nflavour-blind multi-jet analyses, that is the\nlow mass and high mass hZ dedicated flavour-blind analyses, and\nthe three and four-jet hA dedicated flavour-blind \nanalyses~\\cite{ref:papflbl}.\nAs multi-jet flavour-blind analyses use mass-hypothesis dependent criteria,\nselecting the best one implied also a choice between the h and H signals\nfor each of them.\nThe analysis retained was finally compared with the result of the \nselection between the two charged Higgs multi-jet analyses, \nthe cscs and WAWA dedicated analyses~\\cite{ref:paphphm}.\n\n\n\\begin{figure}[p]\n\\begin{tabular}{cc}\n\\hspace{-1.2cm}\n\\epsfig{file=adding_H_no_mA103tb20_1_zoom.eps,width=9.5cm} &\n\\hspace{-1.cm}\n\\epsfig{file=adding_H_no_mA103tb20_2_zoom.eps,width=9.5cm} \\\\\n\\hspace{-1.2cm}\n\\epsfig{file=adding_H_no_mA103tb20_3_zoom.eps,width=9.5cm} &\n\\end{tabular}\n\\caption[]{\n  An example of two-dimensional PDFs from the analysis of the \n  \\hZ\\ $\\rightarrow$ \\mbox{${\\mathrm q}\\bar{\\mathrm q}$}\\ \\mumu\\  channel at \n  \\mbox{$\\sqrt{s}$}~=~206.6~GeV~\\cite{ref:pap00}. The first discriminant variable is\n  built from the reconstructed Higgs boson mass while the second\n  is the event b-tagging variable. Top, left: PDF for a \\hZ\\ signal\n  with \\mh~=~102~\\GeVcc. Top, right: PDF for a \\HZ\\ signal with\n  \\MH~=~115~\\GeVcc. Bottom: PDF expected from the occurrence of both\n  signals in a scenario \n  where the expectations for the two signals are similar\n  (cross-sections 32 and 42~fb, branching fractions into\n  \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ 92\\% and 91\\%, selection efficiencies 69\\% and 66\\% for \\hZ\\\n  and \\HZ, respectively) leading to a double peak in the combined PDF. \n}\n\\label{fig:adding_H}\n\\end{figure}\n\n\n\\section{The {\\sc CP}-conserving {\\sc MSSM} scenarios}\\label{sec:benchmark}\n\nIn most of the parameter space of the \n{\\sc CP}-conserving {\\sc MSSM} scenarios, \nonly \\hZ\\ and \\hA\\ productions are kinematically possible at \n{\\sc LEP} energies. \nThese processes have complementary cross-sections since the\nhZZ and hAZ couplings are proportional to \\sinab\\ and \\cosab, respectively,\nwhere \\tbeta\\ is the ratio of the doublet vacuum expectation values \nand \\alp\\ is  the Higgs doublet mixing angle which enters the definition \nof the two CP-even Higgs eigenstates as a mixture of the real, neutral \ncomponents of the initial Higgs field \ndoublets~\\cite{ref:mssmpheno,ref:FDmssmpheno}.\nIf kinematically allowed, \\hZ\\ production dominates at low \\tbeta\\\nor at large \\MA, while in the rest of the parameter space, it is suppressed\nwith respect to \\hA\\ pair-production. \nThe third neutral Higgs boson, H, \nin some scenarios and in limited regions of the\nparameter space, is light enough and can be produced with a \nlarge \\HZ\\ or \\HA\\ cross-section. \nAs the HZZ coupling is proportional to \\cosab,\nand the HAZ one is proportional to \\sinab, \n\\HZ\\ production, when allowed by kinematics, plays a role at large \n\\tbeta, and \\HA\\ production at low \\tbeta. Similarily, \ncharged Higgs bosons kinematically accessible at {\\sc LEP2}\nenergies are predicted in limited regions of the parameter space, \ntypically when A is light, whatever \\tbeta.\nThe minimal value of the mass of such charged Higgs bosons\nis 60~\\GeVcc\\ in the scenarios under study. The coverage of the\nregion of the {\\sc MSSM} parameter space kinematically accessible\nat {\\sc LEP} is then assured primarily by  the \\hZ\\ and \\hA\\  searches,\nwith the help of the \\HZ, \\HA\\ and to a lesser extent \\mbox{$ {\\mathrm H}^+{\\mathrm H}^- $}\\ channels.\n\n   At tree level, the production cross-sections and the \nHiggs branching fractions in the  {\\sc MSSM}\ndepend on two free parameters, usually chosen as \\tbeta\\ and one Higgs \nboson mass, or, alternatively, two Higgs boson masses, e.g. \\MA\\ and \\mh. \nRadiative corrections introduce additional parameters related\nto supersymmetry breaking~\\cite{ref:mssmpheno,ref:FDmssmpheno}.\nHereafter, the usual assumption that some of them\nare equal at a given energy scale is made: hence, \nthe SU(2) and U(1) gaugino mass parameters are assumed to be\nunified at the so-called GUT scale, \nwhile the sfermion mass parameters or the squark trilinear\ncouplings are taken to be equal at the EW scale.\nWithin these assumptions, the parameters beyond tree level are:\nthe top quark mass, the Higgs mixing parameter, $\\mu$, which defines\nthe Higgsino mass parameter at the EW scale,\nthe common sfermion mass parameter at the EW scale, $M_{\\rm susy}$,\nthe SU(2) gaugino mass parameter at the EW scale, $M_2$,\nthe gluino mass, $m_{\\tilde{g}}$,\nand the common squark trilinear coupling at the EW scale, \n$A$.\nThe U(1) gaugino mass term at the EW scale, $M_1$, is \nrelated to $M_2$ through the GUT relation\n$M_1 = (5\/3) {\\rm \\tan}^2\\theta_W M_2$.\nThe radiative corrections affect \nthe Higgs boson masses and couplings,\nwith the largest contributions arising from \nloops involving the third generation quarks and squarks (top\/stop\nand, at large values of \\tbeta, bottom\/sbottom).\nAs an example, the h boson mass, which is below that of the Z boson\nat tree level, increases by a few tens of \\GeVcc\\ in some regions\nof the {\\sc MSSM} parameter space due to radiative corrections.\n\n\\subsection{The benchmark scenarios} \n\n In the following, eight benchmark scenarios are considered, as suggested \nin Ref.~\\cite{ref:new_pres}. The values of their underlying parameters  \nare quoted in Table~\\ref{ta:benchmarks}.\nThe first three scenarios are those usually studied at {\\sc LEP}. They\nhave been proposed to test the sensitivity of {\\sc LEP} to Higgs bosons\nwith either masses close to the kinematic limit or \ndecays difficult to detect. Similarly, \nthe five other scenarios are aimed at testing the sensitivity of the \nHiggs boson searches at hadron colliders. It is thus interesting to\nestablish the {\\sc LEP} constraints in such models too.\n\nThe first two scenarios, called the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario and the no mixing scenario,\ndiffer only by the value of $X_t = A - \\mu \\cot \\beta$, \nthe parameter which controls the mixing in the stop sector\n(through the product \\mtop $X_t$). This parameter\nhas the largest impact on the mass of the h boson.\nThe \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario leads to the maximum\npossible h mass as a function of \\tbeta. \nThe no mixing scenario is its counterpart with vanishing mixing, \nleading to theoretical upper bounds on \\mh\\ which are at least \n15~\\GeVcc\\ lower than in the \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scheme. \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cccccc}     \\hline\nscenario & $M_{\\rm susy}$ & $M_2$ & $m_{\\tilde{g}}$ & $\\mu$ \n         & $X_t$ \\\\\n& (\\GeVcc) & (\\GeVcc) & (\\GeVcc) & (\\GeVcc) & (\\GeVcc)\\\\\n\\hline\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} &\n      1000 & 200 & 800 & -200 & 2 $M_{\\rm susy}$ \\\\\nno mixing &\n      1000 & 200 & 800 & -200 & 0 \\\\\nlarge $\\mu$ &\n      400 & 400 & 200 & 1000 & -300 \\\\\\hline\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$}, $\\mu >0$  &\n      1000 & 200 & 800 & 200 & 2 $M_{\\rm susy}$ \\\\\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$}, $\\mu >0$, $X_t<0$  &\n      1000 & 200 & 800 & 200 & -2 $M_{\\rm susy}$ \\\\\nno mixing, $\\mu >0$, large  $M_{\\rm susy}$  &\n      2000 & 200 & 800 & 200 & 0 \\\\\ngluophobic   &\n      350 & 300 & 500 & 300 & -750 \\\\\nsmall $\\alpha$  &\n      800 & 500 & 500 & 2.5 $M_{\\rm susy}$ & -1100 \\\\\n\\hline\n\\end{tabular}\n\\caption[]{\nValues of the underlying parameters for the eight\nrepresentative {\\sc MSSM} scenarios scanned in this paper.\nNote that  $X_t = A - \\mu \\cot \\beta$. These scenarios\nhave been studied for several values of the top quark mass,\n\\mtop~=~169.2, 174.3, 179.4 and 183.0~\\GeVcc.\n}\n\\label{ta:benchmarks}\n\\end{center}\n\\end{table}\n\nThe third scenario is called the large $\\mu$ scenario to account for \na large, positive value of $\\mu$. As a consequence of the low value\nof $M_{\\rm susy}$ and the moderate mixing in the stop sector, this\nscenario predicts at least one CP-even Higgs boson with a mass within \nkinematic reach at {\\sc LEP2} in each point of the {\\sc MSSM} parameter space. \nHowever, there are regions for which detecting such a Higgs boson is\ndifficult because of vanishing branching fractions into b-quarks. \nThe values chosen for $\\mu$ and $X_t$ are indeed such that, \nin these regions, radiative corrections lead to suppressed couplings \nto b-quarks for one or the other CP-even Higgs boson.\nThe dominant decays in these regions being still into hadrons,\nthe main analysis channels suffer from large backgrounds.\nThis scenario was designed to test the sensitivity of {\\sc LEP} through\nanalyses that could not benefit from the b-tagging capabilities of the\nexperiments.\n\nAmong the five other benchmark scenarios, three are variants of the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} and no mixing scenarios. \nThe sign of $\\mu$ and that of the mixing parameter have been reversed in\nthe two scenarios derived from the {\\sc LEP} \\mbox{$ m_{\\mathrm h}^{\\rm max}$} \nscenario. The changes in the Higgs boson mass spectrum and properties\nare small. \nThe sign of $\\mu$ has been reversed and the value of $M_{\\rm susy}$\nhas been doubled in the scenario derived from the no mixing scenario\nof {\\sc LEP}. The higher $M_{\\rm susy}$ scale leads to a few \\GeVcc\\\nincrease of the theoretical upper bound on \\mh.\nThe last two scenarios have been proposed to test potentially\ndifficult cases for the searches at hadron colliders. Hence, the gluophobic\nscenario presents regions where the main production channel at the {\\sc LHC}, \ngluon fusion, is suppressed due to cancellations between the top\nquark and stop quark loops in the production process.\nFinally, in the small $\\alpha$ scenario, important\ndecay channels at the Tevatron and at the {\\sc LHC}, h $\\rightarrow $ \\bbbar\\ and h $\\rightarrow $ \\toto,\nare suppressed at large \\tbeta\\ and moderate \\mA. In these regions,\nthe radiatively corrected mixing angle \\alp\\ is low, resulting in\nsuppressed couplings of the ligthest CP-even Higgs boson to down-type\nfermions since these couplings are proportional to $-\\sin \\alpha$\/$\\cos \\beta$.\n  \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{l|cccc}     \\hline\n         & \\multicolumn{4}{c}{\\mtop\\ (\\GeVcc)} \\\\\nscenario &   169.2 & 174.3 & 179.4 & 183.0\\\\\n\\hline\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$}  \n          & 128.2 & 132.9 & 138.6 & 142.7 \\\\   \nno mixing \n          & \\bf{112.8} & \\bf{115.5} & 118.2 & 120.3 \\\\        \nlarge $\\mu$ \n          & \\bf{106.1} & \\bf{108.0} & \\bf{110.1} & \\bf{111.6} \\\\\\hline \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$}, $\\mu >0$   \n          & 128.4 & 134.1 & 140.1 & 144.3 \\\\ \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$},  $\\mu >0$, $X_t<0$    \n          & 124.5 & 128.8 & 134.3 & 138.2 \\\\   \nno mixing, $\\mu >0$, large  $M_{\\rm susy}$ \n          & \\bf{117.0} & 120.2 & 123.7 & 126.3 \\\\ \ngluophobic \n          & \\bf{115.7} & 118.8 & 122.0 & 124.4 \\\\ \nsmall $\\alpha$ \n          & 118.5 & 122.2 & 126.2 & 129.1 \\\\ \n\\hline\n\\end{tabular}\n\\caption[]{\n  Maximal value of \\mh\\ (in \\GeVcc) \n  in the eight benchmark {\\sc MSSM} scenarios \n  studied in this paper, as a function of \\mtop. Radiative corrections\n  include all dominant second-order loop terms~\\cite{ref:FDradco_new}.\n  The maximum value of \\mh\\ corresponds approximately to the minimum \n  value of the mass of the third Higgs boson, H.  \n  Bold values indicate scenarios where this boson\n  is kinematically accessible at {\\sc LEP}.\n}\n\\label{tab:mhmax}\n\\end{center}\n\\end{table}\n \nIn all scenarios, the radiative corrections have been computed \nin the Feynman-diagrammatic approach with all dominant two-loop \norder terms included, using version 2.0 of the {\\sc FeynHiggs} \ncode~\\cite{ref:FDradco_new}. \nAs a first illustration of the different scenarios, Table~\\ref{tab:mhmax} \ngives the maximum value of \\mh\\ allowed by theory in each of them, \nfor the four values of \\mtop\\ studied in this paper.\nAt a given \\mtop\\ value, the three \\mbox{$ m_{\\mathrm h}^{\\rm max}$} \nscenarios give the highest upper bounds on \\mh, the positive $\\mu$\nscenario leading to the maximal value. The large $\\mu$ scenario presents\nthe lowest upper bound, followed in increasing order by the no mixing\nscenario, the gluophobic one, the  no mixing scenario with positive $\\mu$ \nand the small $\\alpha$ scheme.\nThe maximum value of \\mh\\ increases significantly with \\mtop. The effect\nis most important in the three \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios, \nand is much smaller in the others, especially in the large $\\mu$\nscheme. It must be noted that the maximum value of\n\\mh\\ corresponds approximately to the minimum value of \\mH\\ \nin regions of large HZZ couplings (see Fig.\\ref{fig:heavy_neutral_new}). \nThus, \nthere are a few scenarios where the H signal is expected to contribute \nto the experimental sensitivity. These are indicated in bold\ncharacters in Table~\\ref{tab:mhmax}.\n\n\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\epsfig{figure=cpc_masses_no_174.eps,height=7.6cm} &\n\\epsfig{figure=cpc_xsec_2tb_no_174.eps,height=7.6cm}\\\\\n\\epsfig{figure=cpc_gamh_no_174.eps,height=7.6cm} &\n\\epsfig{figure=cpc_gamhha_no_174.eps,height=7.6cm}\\\\\n\\epsfig{figure=cpc_brh_max_174.eps,height=7.6cm} &\n\\epsfig{figure=cpc_brh_no_174.eps,height=7.6cm} \n\\end{tabular}\n\\caption[]{\n    Properties of the three neutral Higgs bosons\n    of the {\\sc CP}-conserving {\\sc MSSM} in the no mixing and \n    \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios with  \n    \\mtop~=~174.3~\\GeVcc. \n    Top: H and h masses and H, h and A production cross-sections\n    at \\mbox{$\\sqrt{s}$}~=~206~GeV, at various \\tbeta\\ values.\n    Middle: h, H and A widths as a function of \\mA\\ and \\tbeta. \n    Bottom: h branching fractions as\n    a function of \\mA\\ at low to moderate values of \\tbeta.\n    Decays into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ (solid lines) and AA (dashed lines) are compared.\n    All dominant two-loop\n    order radiative corrections are included~\\cite{ref:FDradco_new}.}\n\\label{fig:heavy_neutral_new}\n\\end{center}\n\\end{figure}\n\n To illustrate further the Higgs boson phenomenology at {\\sc LEP}, a few\nproperties are compared in Fig.~\\ref{fig:heavy_neutral_new} in the case \nof the no mixing and \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios for a top \nquark mass of 174.3~\\GeVcc. \nThe figures showing masses and cross-sections  underline \nthe importance of the signal from the heavy scalar, H, which \ncan be kinematically accessible at {\\sc LEP2} energies with a large \n\\HZ\\ production cross-section at large \\tbeta\\ and moderate \\mA, up\nto about 100~\\GeVcc. \nThe width curves demonstrate that, at large \\tbeta, \nneutral Higgs bosons can have \na width exceeding the experimental resolution which is of the order\nof 1 to 3~\\GeVcc\\ depending on the search channel.\nAt moderate \\MA, this affects the h and A bosons and thus the \\hA\\\nproduction mode, but not the \\HZ\\ one. At large \\MA, width effects\nbecome negligible for the h boson so that the \\hZ\\ production mode, \nwhich is the only possible dominant mode in that region, \nis not affected.\nThe figures showing branching fractions compare\nthe no mixing and \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios at low \\tbeta.\nIn both scenarios, the h branching fraction into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ decreases\nto the profit of that into AA at very low \\tbeta, \nbut the residual \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ branching ratio is significantly higher  \nin the  \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario.\n   \nFinally, it should be noted that our previous {\\sc MSSM} \ninterpretation of Ref.~\\cite{ref:pap00} relied on partial two-loop \norder radiative corrections~\\cite{ref:FDradco_old}.\nIn the present paper, these\nhave been updated to include all dominant two-loop order \ncorrections~\\cite{ref:FDradco_new}.\nThis leads to significant changes in the Higgs boson  masses and properties. \nThe main effect is an increase of the maximum (resp. minimum) \nallowed value of the h (resp. H) boson mass at fixed \\tbeta.\nAs a consequence, the experimental sensitivity in \\tbeta\\ and that in \\mH\\ \nare expected to decrease. \nA review of the changes induced by the more complete corrections \non the experimental sensitivity of {\\sc DELPHI} is given in\nRef.~\\cite{ref:pap03,ref:pap04}\nin the framework of the three {\\sc LEP} scenarios, keeping\nidentical experimental inputs.\n\n\n\\subsection{Scan procedure} \\label{sec:cpc_procedure}\n \n  In each scenario, a scan was performed over the {\\sc MSSM} \nparameters \\tbeta\\ and \\MA. The range in \\MA\\ spans from 0.02~\\GeVcc\\ \nup to 1~TeV\/$c^2$.\nValues of \\MA\\ leading to unphysical negative mass squared values \nwere removed from the scans. Such points are rather rare, except in \nthe large $\\mu$, gluophobic and small $\\alpha$ scenarios (see\nsection~\\ref{sec:results}).\nThe range in \\tbeta\\ extends from the minimal value allowed in each \nscenario\\footnote{The minimal value of \\tbeta\\ is around 0.7 in the large\n$\\mu$ scenario and in the no mixing scenario with positive $\\mu$ \nand 0.4 in all other schemes. Lower \\tbeta\\ values\ngive rise to unphysical negative mass squared values in the Higgs sector.}\nup to 50, a value chosen in the vicinity of the ratio of \nthe top- and b-quark masses,\nabove which the Higgs-bottom Yukawa coupling is expected to become\nunreliable \n(see e.g.~\\cite{ref:mssmpheno}).\nThe scan steps were 1~\\GeVcc\\ in \\MA\\ and 0.1 in \\tbeta\\ in the regions \nwhere \\mh\\ varies rapidly with these parameters. At low \\MA, where the\ndecay modes change rapidly with the Higgs boson mass, values\ntested were 0.02, 0.1, 0.25, 0.5, 1.5 and 3~\\GeVcc.\n\n  At each point of the parameter space, the neutral and charged Higgs\ncross-sections and their branching fractions were taken from \ndatabases provided by the {\\sc LEP} Higgs working group~\\cite{ref:hwg}, \non the basis of the\ntheoretical calculations in Ref.~\\cite{ref:FDradco_new}, completed by\nthat in Ref.~\\cite{ref:hphm_br} for the\ncharged Higgs boson branching fractions.\nThe signals from the third Higgs boson, H, were included in the\nchannel combination\nat each point where \\mH\\\nwas found to be below 120~\\GeVcc, the ultimate sensitivity of {\\sc LEP}.\nThe signal expectations in each channel were then derived from the \ntheoretical cross-sections and branching fractions,\nthe experimental luminosity and the efficiencies. If necessary, a correction\nwas applied to account for different branching fractions of the Higgs bosons \nbetween the test point and the simulation (e.g.~for the \n\\hZ\\ and \\HZ\\ processes, \nthe simulation was done in the {\\sc SM} framework). \n\n\n\\begin{figure}[hp]\n\\begin{tabular}{cc}\n\\hspace{-1.6cm} \\epsfig{file=4b_inter_tb_2_gs.eps,width=9.cm} &\n\\hspace{-0.6cm} \\epsfig{file=4b_inter_tb_1_gs.eps,width=9.cm} \\\\\n\\hspace{-1.6cm} \\epsfig{file=4b_inter_tb_3_gs.eps,width=9.cm} &\n\\end{tabular}\n\\caption[]{\n  Two-dimensional PDFs used in the analysis of the \n  \\hA\\ $\\rightarrow$\\ \\mbox{${\\mathrm b}\\bar{\\mathrm b}$} \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\  channel at \n  \\mbox{$\\sqrt{s}$}~=~206.6~GeV~\\cite{ref:pap00}. The first discriminant variable is\n  the sum of the reconstructed Higgs boson masses while the second\n  is a neural network output. \n  Top, left: PDF for a \\hA\\ signal with \\mA~=~\\mh~=~90~\\GeVcc\\ and \n  h and A widths below 1~\\GeVcc. \n  Top, right: PDF for a \\hA\\ signal with \\mA~=~\\mh~=~90~\\GeVcc\\ and \n  \\tbeta~=~50. The Higgs boson widths in that case are 5 and 9~\\GeVcc\\\n  for A and h, respectively.\n  Bottom: PDF linearily interpolated in \\tbeta\\ at a value of 37. \n}\n\\label{fig:inter_tb}\n\\end{figure}\n\n  As stated in the previous section, neutral Higgs bosons can\nhave non-negligible widths at large \\tbeta\\ when \\mA\\ is above a few tens\nof~\\GeVcc. \nIn this region, the experimental sensitivity is dominated \nby the {\\sc LEP2} \\hA\\ analyses dedicated to standard {\\sc MSSM}\nfinal states. To account for width effects in these channels,\nefficiencies derived from simulations with\nh and A widths below 1~\\GeVcc\\ (see e.g. Ref.~\\cite{ref:pap00}) were \napplied for \\tbeta~$<$ 30 only. Above that value, efficiencies\nwere linearly interpolated in \\tbeta\\ between the efficiencies from \nthese simulations and those from simulations at \\tbeta~=~50 where\nthe Higgs boson widths exceed the experimental resolution. \nAs the Higgs boson widths grow approximately linearly with \n\\tbeta\\ above 30, a linear interpolation is valid. \nThe same holds for the discriminant information, for which \nthe same interpolation software was used as discussed in\nsection~\\ref{sec:pdf}\nfor the PDF interpolation in mass or centre-of-mass energy. \nThe effect of the Higgs boson widths on the PDFs of the \\hA\\ \nsignals and the interpolation in \\tbeta\\ of these PDFs \nare illustrated in Fig.~\\ref{fig:inter_tb}. Note that\nthe \\hZ\\ and \\HZ\\ channels at large \\tbeta\\ are not affected\nby such an effect since in most of the regions where they possibly\ncontribute, their widths are below the experimental resolution,\nas shown in Fig.~\\ref{fig:heavy_neutral_new}.\n\n\n\\section{\nResults in {\\sc CP}-conserving {\\sc MSSM} scenarios}\\label{sec:results}\n\nThe regions of the {\\sc MSSM} parameter space excluded at \n95\\% {\\sc CL} or more by combining the searches of \nTable~\\ref{tab:channels} are hereafter discussed in turn \nfor each scenario. The exclusion is dominated by the\nsearches for neutral Higgs bosons in standard {\\sc MSSM} final states. \nThe searches for neutral Higgs bosons decaying into hadrons of any flavour\nand the charged Higgs boson searches complete the exclusion in\nrestricted regions of the parameter space. In addition,\nthe limit on the Z partial width that would be due to \nnew physics~\\cite{ref:width}, $\\Gamma^{\\rm{new}} < 6.6$~\\MeVcc\\\nis used as an external constraint on the \\hA\\ process at {\\sc LEP1}.\nA detailed account of the impact of\nthese auxiliary constraints can be found in Ref.~\\cite{ref:pap04,ref:pap05}.\n\n\n\\subsection{The \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario}\\label{sec:mhmax}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_max_174_54_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_max_174_54_tb_ma.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_max_174_54_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_max_174_54_ma_mh_zoom.eps,height=9cm} \n\\end{tabular}\n\\caption[]{\n   {\\sc MSSM} \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario for\n   a top mass of 174.3~\\GeVcc: regions\n   excluded at 95\\% {\\sc CL} by combining the results of the Higgs \n   boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory. \n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_max}\n\\end{center}\n\\end{figure}\n\n  The excluded regions in the \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario\nare presented in the (\\mh, \\tbeta), (\\MA, \\tbeta) and (\\mh, \\MA) planes \nin Fig.~\\ref{fig:limit_max} for a top mass value of 174.3~\\GeVcc.\nBasically,\nthe exclusion is made by the results in the \\hZ\\ (\\hA) channels in the\nlow (large) \\tbeta\\ region while they both contribute at intermediate\nvalues. The searches for the heavy scalar, H, brings no additional\nsensitivity since H is not kinematically accessible\nin this scenario (see Table~\\ref{tab:mhmax}).\nThe above results establish the following \n95\\% {\\sc CL} lower limits on \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 89.7~\\GeVcc \\hspace{1cm}\n   \\MA > 90.4~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.4 and 50.\nThe expected median limits are 90.6~\\GeVcc\\ for \\mh\\ and \n90.8~\\GeVcc\\ for \\MA. \nThe observed limit in \\MA\\ (\\mh) is reached at \\tbeta\\ around 20 (10),\nin a region where both the \\hZ\\ and \\hA\\ processes contribute.\nFor \\mtop~=~174.3~\\GeVcc\\, the range in \\tbeta\\ between  \n0.7 and 1.9 (expected [0.7-1.9]) is excluded \nfor any value of \\mA\\ between 0.02 and 1000~\\GeVcc.\nThese limits and exclusions, as well as those for all the \n{\\sc CP}-conserving scenarios, are summarized in Table~\\ref{tab:limits}.\n\nThe \\mtop\\ dependence of the above limits was also studied, as reported\nin Table~\\ref{tab:limits}. The mass limits remain unchanged when \nvarying \\mtop, for \\mh\\ is insensitive to \\mtop\\ in the region\nof large \\tbeta\\ and intermediate \\mA\\ where the limits are set. \nOn the other hand, the excluded \nrange in \\tbeta\\ is governed by the maximal value of \\mh, which is \nreached at large \\mA\\ where \\mh\\ is very sensitive to \\mtop, as \nillustrated in the top left-hand plot in Fig.~\\ref{fig:limit_max}: \nhence the variation of the limits\nin \\tbeta\\ as reported in Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}.\nAn exclusion in \\tbeta\\ exists for a top mass up to 179.4~\\GeVcc\\ \nwhich is about three standard deviations higher than\nthe current average \\mtop\\ measurement. The exclusion would vanish for\na top mass as high as 183.0~\\GeVcc.\n\n\n\\subsection{The \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario but  \nwith $\\mu$ positive and either sign for $X_t$}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_rev_174_54_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_rev_174_54_tb_ma.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_rev_174_54_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_rev_174_54_ma_mh_zoom.eps,height=9cm} \n\\end{tabular}\n\\caption[]{\n   {\\sc MSSM} \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with\n   positive $\\mu$ for a top mass of 174.3~\\GeVcc: regions\n   excluded at 95\\% {\\sc CL} by combining the results of the Higgs \n   boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_rev}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_maxl_174_54_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_maxl_174_54_tb_ma.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_maxl_174_54_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_maxl_174_54_ma_mh_zoom.eps,height=9cm} \n\\end{tabular}\n\\caption[]{\n   {\\sc MSSM} \\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with\n   positive $\\mu$ and negative $X_t$ for a top mass of 174.3~\\GeVcc: \n   regions\n   excluded at 95\\% {\\sc CL} by combining the results of the Higgs \n   boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory. \n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_maxl}\n\\end{center}\n\\end{figure}\n\n  The excluded regions \nfor a top mass value of 174.3~\\GeVcc\\ are presented \nin Fig.~\\ref{fig:limit_rev} for the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$,\nkeeping $X_t$ positive as in the original \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario,\nand in Fig.~\\ref{fig:limit_maxl} for the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$\nand negative $X_t$. \nThe results are quite similar to those in the original \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario.\nMass limits are within 200~\\MeVcc\\ of those in the previous section\nand do not vary significantly with \\mtop,\nas reported in Table~\\ref{tab:limits}.\n \nTo compare observed and median limits,\nthe 95\\% {\\sc CL} lower limits on \\mh\\ and \\MA\\  in the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$ \nfor \\mtop~=~174.3~\\GeVcc\\ are:\n\n\\[ \\mh > 89.6~\\GeVcc \\hspace{1cm}\n   \\MA > 90.3~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.4 and 50.\nThe expected median limits are 90.3~\\GeVcc\\ for \\mh\\ and \n90.4~\\GeVcc\\ for \\MA. \nThe 95\\% {\\sc CL} lower limits on \\mh\\ and \\MA\\  in the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$ and\nnegative $X_t$ for \\mtop~=~174.3~\\GeVcc\\ are:\n\n\\[ \\mh > 89.6~\\GeVcc \\hspace{1cm}\n   \\MA > 90.4~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.4 and 50.\nThe expected median limits are 90.4~\\GeVcc\\ for \\mh\\ and \n90.6~\\GeVcc\\ for \\MA. \n\nThe excluded ranges in \\tbeta\\ are different in the three \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios,\nsince they have different theoretical upper bounds on \\mh.\nFor \\mtop~=~174.3~\\GeVcc\\, the excluded range in the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$ lies\nbetween 0.7 and 2.0 (expected [0.8-2.0]),\nwhile in the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$ and\nnegative $X_t$ it spans\nfrom 0.6 to 2.5 (expected [0.6-2.4]).\nThese limits are valid for any value\nof \\mA\\ between 0.02 and 1000~\\GeVcc. Note that despite the higher\nmaximal value of \\mh\\ in the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$,\nthe most conservative limits in \\tbeta\\ are still derived in the original\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario (see Section~\\ref{sec:mhmax}), \nreflecting the\ndifferences in the theoretical upper bounds at \\tbeta\\ around 1\n(see top left-hand plots in \nFig.~\\ref{fig:limit_max},~\\ref{fig:limit_rev} and ~\\ref{fig:limit_maxl}).\nThe \\mtop\\ dependence of the above limits is presented\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}.\nFor a top mass as high as 183~\\GeVcc, there would be no longer\nany exclusion in \\tbeta\\ in the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with positive $\\mu$,\nwhile there is still one in the scenario with\npositive $\\mu$ and negative $X_t$ due to the lower\nmaximal value of \\mh\\ in that scenario.\n\n\\subsection{The no mixing scenario}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_no_174_54_rnd_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_no_174_54_rnd_tb_ma_error.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_no_174_54_rnd_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_no_174_54_rnd_ma_mh_zoom.eps,height=9cm}\n\\end{tabular}\n\\caption[]{\n    {\\sc MSSM} no mixing scenario for a top mass of 174.3~\\GeVcc: \n    regions excluded at 95\\% {\\sc CL} by combining the results of the Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n    Among the three unexcluded regions at low \\mA, \n    the strip at low \\mh\\ is fully excluded by the limit on \n    the Z partial width that would be due to new \n    physics~\\cite{ref:width}.\n    The dashed curves show the median expected limits.\n    The medium-grey areas are the regions not allowed by theory. \n    The dash-dotted \n    lines in the top left-hand plot are the theoretical upper \n    bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n    (from left to right).\n   }\n\\label{fig:limit_no}\n\\end{center}\n\\end{figure}\n\n  The excluded regions in the no mixing scenario are presented \nin Fig.~\\ref{fig:limit_no} for a top mass value of 174.3~\\GeVcc.\nIn this scenario, if the top is not too heavy,\nthe heavy scalar, H, is kinematically accessible at \nlarge \\tbeta\\ and moderate \\mA, the region where the mass limits in \n\\mA\\ and \\mh\\ are set. \nThus, allowing for its production increases the sensitivity of the \nsearches. \n\nThe zoom at low \\mA\\ in the (\\mh, \\MA) projection shows that\nthe direct searches leave three unexcluded regions below 12~\\GeVcc\\ in \\mA. \nThe thin strip along the theoretical lower bound on \\mh\\ at\nvery low \\mA\\ (hardly visible in the figure)\nis excluded by the limit on the Z partial width \nthat would be due to new physics~\\cite{ref:width}, \n$\\Gamma^{\\rm{new}} < 6.6$~\\MeVcc, which, when applied to the \\hA\\ process, \ntranslates into an excluded region that encompasses that area.\nThis is not the case for the two other unexcluded regions. \nThese have \\tbeta\\ below 1.0 and \\mh\\ between 59 and 82~\\GeVcc.\nIn that region, \\MA\\ is below the kinematic threshold \n\\mh~=~2\\MA, the decay \\hAA\\ opens and supplants the h $\\rightarrow $ \\bbbar\\ mode,\nas can be seen in Fig.~\\ref{fig:heavy_neutral_new}.\nOur {\\sc LEP2} \\hAA\\ searches, covering A masses above the \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}\\ \nthreshold (see Table~\\ref{tab:channels}), have no sensitivity\nbelow 4~\\GeVcc\\ in \\mA.\nSimilarly, charged Higgs bosons, although kinematically accessible\nwith a mass between 57 and 82~\\GeVcc, have a large branching fraction into\nW$^*$A in this region. \nAs our charged Higgs boson searches in these channels assume\n\\MA\\ above 12~\\GeVcc\\ (see Table~\\ref{tab:channels}), the overall\nexperimental sensitivity in these regions remains weak and no exclusion\nat 95\\% {\\sc CL} can be derived, in agreement with the expected\nperformance. The largest value of \\mbox{$\\rm CL_{\\rm s}$}\\ is 7\\% in the\nunexcluded region around 12~\\GeVcc\\  in \\mA\\ and 33\\% in\nthe unexcluded hole below  4~\\GeVcc.\nNote that the nearby region with \\mh\\ from 82~\\GeVcc\\ to the\ntheoretical upper bound on \\mh\\ is excluded at 95\\% {\\sc CL}\nby the charged Higgs boson searches through their fermionic decays \nwhich dominate the W$^*$A mode there.\n\nThe above results establish the following 95\\% {\\sc CL} lower limits \non \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 90.7~\\GeVcc \\hspace{1cm}\n   \\MA > 91.2~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 1.0 and 50.\nThe expected median limits are 91.1~\\GeVcc\\ for both \\mh\\ and  \n\\MA. The observed limits in \\MA\\ and \\mh\\ are  reached at \\tbeta\\ \naround 15, in a region where both the \\hZ\\ and \\hA\\ processes \ncontribute. For \\mtop~=~174.3~\\GeVcc, two ranges in \\tbeta\\ are excluded\nfor any value of \\mA\\ between 0.02 and 1000~\\GeVcc,\nthe largest interval  being \nbetween 1.0 and 9.7 (expected [0.9-7.7]). \n\nThe \\mtop\\ dependence of the above limits was studied, as shown\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}. \nIn this scenario, both the mass limits and \nthe excluded range in \\tbeta\\ change when varying \\mtop. Indeed,\nas already mentioned,  \nthe mass limits in \\mA\\ and \\mh\\ rely on the searches for H, whose\nmass is very sensitive to \\mtop\\ in the region where the limits are \nset. Similarly, the  maximal value of \\mh, which governs the limits\nin \\tbeta, is reached at large \\mA\\ where \\mh\\ is very sensitive to \n\\mtop\\ (see Table~\\ref{tab:mhmax}). Note that for\na top mass of 169~\\GeVcc, \\MH\\ decreases by\n3~\\GeVcc\\ in the region where the mass limits are set, making the H signal \nmore within the sensitivity of {\\sc LEP2}: the whole parameter space\nof the no mixing scenario is then accessible and found to be excluded\nat 95\\% {\\sc CL}, apart from two holes at \n\\tbeta\\ below 1.0, one at \\mA\\ around 12~\\GeVcc, which is excluded at\n92\\% {\\sc CL},\nand a larger one below 4~\\GeVcc, which is disfavoured at\n69\\% {\\sc CL} only.\n\n\\subsection{The no mixing scenario but\nwith positive $\\mu$ and large $M_{\\rm susy}$}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_nol_174_54_rnd_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_nol_174_54_rnd_tb_ma_error.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_nol_174_54_rnd_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_nol_174_54_rnd_ma_mh_zoom.eps,height=9cm} \n\\end{tabular}\n\\caption[]{\n    {\\sc MSSM} no mixing scenario with positive $\\mu$ and \n    large M$_{\\rm susy}$ for a top mass of 174.3~\\GeVcc: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   Among the four unexcluded regions at low \\mA, \n   the strip at low \\mh\\ is fully excluded by the limit on \n   the Z partial width that would be due to new \n   physics~\\cite{ref:width}.\n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory. \n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_nol}\n\\end{center}\n\\end{figure}\n\n\n  The excluded regions in the no mixing scenario with positive $\\mu$ and \nlarge M$_{\\rm susy}$  are presented \nin Fig.~\\ref{fig:limit_nol} for a top mass value of 174.3~\\GeVcc.\nThe larger  $M_{\\rm susy}$ makes the impact of the H signal, and hence\nthe exclusion limits, weaker than in the previous scenario. On the\nother hand, the results in the low mass region, at \\mA\\ below 12~\\GeVcc,\nare similar to those in the no mixing scenario. The direct searches \nleave a tiny unexcluded strip at low \\mh\\ and very low \\mA\\\nwhich is excluded by the\nlimit on $\\Gamma^{\\rm{new}}$. Three other regions, at \\mh\\ between\n56 and 72~\\GeVcc, remain unexcluded even when charged Higgs boson\nsearches are included, due to the large branching fraction into W$^*$A \ndecays, which are not covered by these searches at such low A masses.\nThe holes around 8 and 12~\\GeVcc\\ in \\mA\\ are however excluded\nat 93\\% and 91\\% {\\sc CL}, respectively, while the larger area below\n4~\\GeVcc\\ in \\mA\\ is disfavoured at 60\\% {\\sc CL} only.\n\nThe above results establish the following 95\\% {\\sc CL} lower limits \non \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 89.8~\\GeVcc \\hspace{1cm}\n   \\MA > 90.6~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 1.0 and 50. \nThe expected median limits are 90.5~\\GeVcc\\ for \\mh\\ and 90.6~\\GeVcc\\\n\\MA.  For \\mtop~=~174.3~\\GeVcc\\ \nthe range in \\tbeta\\ \nbetween 1.0 and 4.5 (expected [1.0-4.3]) is excluded for any value\nof \\mA\\ between 0.02 and 1000~\\GeVcc .\n\nThe \\mtop\\ dependence of the above limits is presented\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}. \nThe mass limits vary only slightly with \\mtop, since in the\nregion where these are set, \\mh\\ is insensitive to \\mtop\\ while \n\\mH, although sensitive to \\mtop, is very close to the kinematic \nlimit.\nContrary to the case of the no mixing scenario,\nthe parameter space of this scenario\ndoes not become fully accessible for a top mass of 169~\\GeVcc, \ndue to too high an upper (resp. lower) bound \non \\mh\\ (resp. \\mH). The exclusion is thus much weaker than in the\nno mixing scheme but stronger than in the \n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenarios.\n\n\n\\subsection{The large $\\mu$ scenario}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_mu_174_noiso_tb_mh.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=mssm_mu_179_noiso_tb_mh.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_mu_174_noiso_tb_ma_error.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=mssm_mu_179_noiso_tb_ma_error.eps,height=9cm} \\\\\n\\end{tabular}\n\\caption[]{\n    {\\sc MSSM} large $\\mu$ scenario: regions excluded at \n   95\\% {\\sc CL} by combining the results of the Higgs boson\n   searches in the whole {\\sc DELPHI} data sample  \n   (light-grey area and embedded domains in medium- and dark-grey). Results\n   are shown for two values of the top mass, 174.3 and\n   179.4~\\GeVcc. The domains embedded in the light-grey area \n   at large \\tbeta\\ are excluded by the searches for the heavy scalar Higgs \n   boson, H (medium-grey or green) \n   and by the flavour-blind searches (dark-grey or dark-blue). \n   Of the two unexcluded holes at low \\mA,\n   the one at \\tbeta\\ above 1 \n   is excluded by the limit on the Z partial width~\\cite{ref:width}\n   that would be due to new physics.\n   The dashed curves show the  median expected limits.\n   The medium-grey areas with bold contours are the regions \n   not allowed by theory. \n   Note in particular the large region forbidden\n   at low \\MA\\ in the (\\MA, \\tbeta) projections, which is due to\n   points leading to unphysical h masses.}\n\\label{fig:limit_mu}\n\\end{center}\n\\end{figure}\n\n\n  The excluded regions in the large $\\mu$ scenario are presented in the \n(\\mh, \\tbeta) and (\\MA, \\tbeta) planes in Fig.~\\ref{fig:limit_mu} for\nvalues of the top quark mass of 174.3 and 179.4~\\GeVcc. \nIn these figures, the contribution of the H signal and that of the\nsearches for neutral Higgs bosons decaying into hadrons of any flavour\nare highlighted.\n\nA large fraction of the allowed domain is excluded by the searches for\nthe h, A and H Higgs bosons into standard {\\sc MSSM} final states. \nIn particular, since the theoretical upper bound on the h boson \nmass in this scenario is low (around 110.0~\\GeVcc, see \nTable~\\ref{tab:mhmax}), the sensitivity of the \\hZ\\ channels is high \neven at large \\tbeta, which explains why the excluded region reaches the \ntheoretically forbidden area for large values of \\tbeta. \nAs the value of the upper bound on \\mh\\ is also the\ntheoretical lower bound on \\MH\\ at large \\tbeta, \nallowing for the production of H translates into a \nsignificant gain in exclusion, namely at \\tbeta\\ above 8. \nThe searches for neutral Higgs bosons decaying\ninto hadrons of any flavour bring an additional exclusion in\nregions left unexcluded by the standard searches at\n\\tbeta\\ around 14. At moderate \\mA, \\hZ\\ and \\hA\\ productions \nare low due to weak hZZ couplings for \\hZ\\\nand to kinematics for \\hA. On the other hand, \\HZ\\ production is large \nbut H is decoupled from \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}. At larger \\mA, \\hA\\ and \\HZ\\ \nproductions are kinematically forbidden, \\hZ\\ production is large but\nthe h$\\rightarrow$\\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ branching fraction vanishes. \nIn both cases, the Higgs boson whose production is allowed (H or h) has\na large branching fraction into hadrons and a mass close to the sensitivity\nof our searches for a neutral Higgs boson decaying into hadrons and \nfully coupled to the Z. This explains why these searches lead to an \nadditional but only partial exclusion in these regions.\nNote that increasing the top quark mass from 174.3 to 179.4~\\GeVcc\\ leads to \na larger unexcluded area. There are indeed more points with vanishing\nh or H branching fractions into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ and, as \\mh\\ and \\mH\\ increase with\n\\mtop, the impact of the searches for hadronically decaying Higgs \nbosons also becomes weaker.\nHowever, when combining the four {\\sc LEP}\nexperiments, the sensitivity of these\nsearches increases and becomes high enough to cover almost entirely these\nregions of vanishing branching fractions into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}~\\cite{ref:hwg}.\n\nBelow 3~\\GeVcc\\ in \\mA, the direct searches leave\ntwo unexcluded holes at \\tbeta\\ around 1.\nThe one at \\tbeta\\ above 1 is fully excluded by the limit \non $\\Gamma^{\\rm{new}}$ for either value of \\mtop.\nThe hole at \\tbeta\\ below 1 remains unexcluded. \nThe largest value of \\mbox{$\\rm CL_{\\rm s}$}\\ in this area is 12\\% for \\mtop~=~174.3~\\GeVc\\\nand 6\\% for \\mtop~=~179.4~\\GeVc.\n\nThe above results establish the following 95\\% {\\sc CL} lower limits \non \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 94.2~\\GeVcc \\hspace{1cm}\n   \\MA > 96.6~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.9 and 50. \nThe expected median limits are 90.3~\\GeVcc\\ for \\mh\\ and 92.8~\\GeVcc\\\nfor \\MA. \nThe observed limits in \\MA\\ and \\mh\\ are  reached at \\tbeta\\ \naround 14, in a region where the \\hZ, \\HZ\\ and \\hA\\ processes contribute. \nFor \\mtop~=~174.3~\\GeVcc, two ranges in \\tbeta\\ are excluded\nfor any value of \\mA\\ between 0.02 and 1000~\\GeVcc,\nthe largest interval being \nbetween 0.9 and 13.7 (expected [0.9-12.9]).\n\nThe \\mtop\\ dependence of the above limits is presented\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}. \nExcept for \\mtop~=~174.3~\\GeVcc, \nthe mass limits vary only slightly with \\mtop\\ and are in\nagreement with the expected ones. \nThe difference at \\mtop~=~174.3~\\GeVcc\\ has been traced back to the deficit\nin data with respect to background expectations which was observed in the \nflavour-blind searches applied to the Higgsstrahlung \nprocess~\\cite{ref:papflbl} when testing  \nmasses above 100~\\GeVcc, which corresponds to the range of \\mH\\ values\nin the region where the mass limits are obtained in the large $\\mu$ scenario.\nIn this region, the set of independent channels which are selected to be \nstatistically combined (see Sec.~\\ref{sec:overlap}) \nvaries strongly from one top mass value to \nthe other, due to still large H branching fractions into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\\nat \\mtop~=~169.2~\\GeVcc\\ and to \\mH\\ values  increasing with \n\\mtop\\ (see Table~\\ref{tab:mhmax}). At \\mtop~=~174.3~\\GeVcc, the weight of \nthe flavour-blind \\HZ\\ searches is maximal and the deficit in data of \nthese searches\ntranslates into a difference between the observed and median limits.\n\n\\subsection{The gluophobic scenario}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_gluo_174_54_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_gluo_174_54_tb_ma_error.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_gluo_174_54_ma_mh.eps,height=9cm} &\n\\hspace{-0.5cm}\n\\epsfig{figure=mssm_gluo_174_54_ma_mh_zoom.eps,height=9cm} \n\\end{tabular}\n\\caption[]{\n    {\\sc MSSM} gluophobic scenario for a top mass of 174.3~\\GeVcc: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   The unexcluded holes at low \\mA\\ are fully excluded by the limit on \n   the Z partial width~\\cite{ref:width} that would be due to new \n   physics (dark-grey).\n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory. \n   Note in particular the large forbidden region \n   in the (\\MA, \\tbeta) projection, which is due to\n   points leading to unphysical h masses.\n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_gluo}\n\\end{center}\n\\end{figure}\n\n For the gluophobic scenario the excluded regions  \nin the (\\mh, \\tbeta), (\\MA, \\tbeta) and (\\mh, \\MA) planes \nare presented\nin Fig.~\\ref{fig:limit_gluo} for a top mass value of 174.3~\\GeVcc.\nAlthough this scenario was designed to test Higgs boson searches \nat hadron colliders, with a phenomenology very different from \nthat of {\\sc LEP}, results are similar to those derived in the\nprevious scenarios. The exclusion \nis defined by the results in the \\hZ\\ (\\hA) channels in the\nlow (large) \\tbeta\\ region while they both contribute at intermediate\nvalues. The direct searches leave several unexcluded holes \nbelow 4~\\GeVcc\\ in \\mA\\ and at \\tbeta\\ below 2, which are all excluded by\nthe limit on $\\Gamma^{\\rm{new}}$.\n\nThe above results establish the following 95\\% {\\sc CL} lower limits \non \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 87.0~\\GeVcc \\hspace{1cm}\n   \\MA > 92.9~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.4 and 50. \nThe expected median limits are 87.0~\\GeVcc\\ for \\mh\\ and \n93.0~\\GeVcc\\ for \\MA. \nThe observed limits in \\MA\\ and \\mh\\ are  reached at \\tbeta\\ \naround 50, in a region where only the \\hA\\ process contributes. \nContrary to the other scenarios, the h and A bosons are\nnot degenerate in mass at large \\tbeta, which reflects in the \nsignificant difference between the h and A mass limits.\nFor \\mtop~=~174.3~\\GeVcc,\nthe range in \\tbeta\\  \nbetween 0.4 and 5.2 (expected [0.4-4.8]) is excluded\nfor any value of \\mA\\ between 0.02 and 1000~\\GeVcc.\n\nThe \\mtop\\ dependence of the above limits is shown\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}. \nAs already mentioned, the h and A bosons are not degenerate at large\n\\tbeta\\ and moderate \\mA, the region where the mass limits are\nset. As a consequence, the value of \\mh\\ at fixed \\mA\\ and \\tbeta\\ \nis observed to vary significantly with \\mtop\\ in that region. This\nis the main reason for the variations of the mass limits with \n\\mtop, an additional effect being the variations\nof \\mH\\ which is kinematically accessible at low \\mtop\\\nin this scenario (see Table~\\ref{tab:mhmax}).\nOn the other hand, the variation of the excluded range in \\tbeta\\ \nis due, as in the other scenarios, to the change in the \nmaximal value of \\mh\\ which is very sensitive to \\mtop. \n\n\\subsection{The small $\\alpha$ scenario}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_small_174_54_tb_mh_mtop.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=mssm_small_174_54_tb_ma_error.eps,height=9cm} \\\\\n\\hspace{-1.4cm}\n\\epsfig{figure=mssm_small_174_54_ma_mh.eps,height=9cm} &\n\\end{tabular}\n\\caption[]{\n    {\\sc MSSM} small $\\alpha$ scenario for a top mass of 174.3~\\GeVcc: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n   (light-grey). \n   There is one unexcluded hole at low \\mh\\ and \\tbeta\\ around 20\n   which is excluded by the limit on \n   the Z partial width~\\cite{ref:width} that would be due to new \n   physics (dark-grey).\n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory. \n   Note in particular the large forbidden region \n   in the (\\MA, \\tbeta) projection, which is due to\n   points leading to unphysical h masses.\n   The dash-dotted \n   lines in the top left-hand plot are the theoretical upper \n   bounds for a top mass of 169.2, 179.4 and 183.0~\\GeVcc\\\n   (from left to right).\n   }\n\\label{fig:limit_small}\n\\end{center}\n\\end{figure}\n \nThe excluded regions in the small $\\alpha$ scenario\nare presented in Fig.~\\ref{fig:limit_small} \nfor a top mass value of 174.3~\\GeVcc.\nThe small $\\alpha$ scheme is the second example of a scenario aiming at\ntesting potentially difficult cases for the Higgs boson searches at hadron \ncolliders. As mentioned in section~\\ref{sec:benchmark}, \nthis scenario presents regions of the parameter space where the \nh $\\rightarrow $ \\bbbar\\ and h $\\rightarrow $ \\toto\\ decays vanish, which could be a problem at {\\sc LEP} too.\nThe results in  Fig.~\\ref{fig:limit_small}, similar to\nthose derived in the previous scenarios, show that this is not the case.\nAt large \\tbeta, in the region accessible at {\\sc LEP}, \nthe h $\\rightarrow $ \\bbbar\\ branching fraction, although reduced, \nremains high enough (e.g. above 70\\% in the region where the\nmass limits are set) to ensure a good sensitivity.\nAt low \\mh, the direct searches leave \none unexcluded island that is fully excluded by the limit on \n$\\Gamma^{\\rm{new}}$.\n\nThe above results establish the following 95\\% {\\sc CL} lower limits \non \\mh\\ and \\MA\\ for \\mtop~=~174.3~\\GeVcc:\n\n\\[ \\mh > 83.5~\\GeVcc \\hspace{1cm}\n   \\MA > 95.8~\\GeVcc  \\]\n\n\\noindent\nfor any value of \\tbeta\\ between 0.4 and 50. \nThe expected median limits are 82.6~\\GeVcc\\ for \\mh\\ and \n95.0~\\GeVcc\\ for \\MA. \nThe observed limits in \\MA\\ and \\mh\\ are reached at \\tbeta\\ \naround 50, in a region where only the \\hA\\ process contributes. \nAs in the previous scenario, the h and A bosons are\nnot degenerate in mass at large \\tbeta, which reflects in the \nsignificant difference between the h and A mass limits.\nFor \\mtop~=~174.3~\\GeVcc,\nthe range in \\tbeta\\\nbetween 0.4 and 4.0 (expected [0.5-3.9]) is excluded\nfor any value of \\mA\\ between 0.02 and 1000~\\GeVcc .\n\nThe \\mtop\\ dependence of the above limits is shown\nin Table~\\ref{tab:limits} and Fig.~\\ref{fig:mtop}. \nAs in the previous scenario,\nthe value of \\mh\\ at fixed \\mA\\ and \\tbeta\\ \nvaries significantly with \\mtop\\ in the region where\nthe mass limits are set, which explains the variations of\nthe latter. The H signal, being kinematically inaccessible for\nmost values of \\mtop\\ (see Table~\\ref{tab:mhmax}) \nplays no role in this scenario.\nFinally, the variation of the \nexcluded range in \\tbeta\\ is due to the change in the \nmaximal value of \\mh\\ which is very sensitive to \\mtop. \n\n\\begin{table}[hp]\n\\begin{center}\n\\begin{tabular}{c|c|cccc}     \\hline\n  &  & \\multicolumn{4}{c}{\\mtop\\ (\\GeVcc)} \\\\\nscenario & limits &  169.2 & 174.3 & 179.4 & 183.0\\\\\n\\hline\n\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} &\n    \\mh\\  & 89.7 & 89.7 & 89.7 & 89.6\\\\\n&   \\mA\\  & 90.4 & 90.4 & 90.4 & 90.4\\\\\n&   \\tbeta\\       & 0.59 - 2.46 & 0.72 - 1.96 & 0.93 - 1.46 & none\n\\\\\\hline\n\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$}  &\n    \\mh\\  & 89.6 & 89.6 & 89.5 &  89.6 \\\\\n$\\mu >0$ &\n    \\mA\\  & 90.3 & 90.3 & 90.3 & 90.3 \\\\\n&   \\tbeta\\       & 0.59 - 2.61 & 0.71 - 2.00 & 0.87 - 1.54 & none\n\\\\\\hline\n\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} &\n    \\mh\\  & 89.6 & 89.6 & 89.5 & 89.6 \\\\\n$\\mu >0, {\\rm X_t}<0$ &\n    \\mA\\  & 90.5 & 90.4 & 90.4 & 90.4 \\\\\n&   \\tbeta\\       & 0.53 - 3.20 & 0.63 - 2.46 & 0.72 - 1.96 & 0.84 - 1.63\n\\\\\\hline\n\nno mixing &\n    \\mh\\ & 112.8 & 90.7 & 90.0 & 89.9 \\\\\n&   \\mA\\ & 1000. & 91.2 & 90.8 & 90.5 \\\\\n&   \\tbeta\\      & 0.40 - 50.0 &  0.40 - 9.70 & 0.40 - 5.40   & 0.40 - 4.40 \\\\\n& \\tbeta, \\mA $>$12.0\n                 & 0.46 - 0.96 &  0.46 - 0.96 &  0.47 - 0.97 & 0.47 - 0.97\n\\\\\\hline\n\n\nno mixing &\n    \\mh\\ & 89.9 & 89.8 & 89.7 & 89.8 \\\\\n$\\mu >0$&\n    \\mA\\ & 90.8 & 90.6 & 90.4 & 90.3 \\\\\nlarge M$_{\\rm SUSY}$&\n     \\tbeta\\ & 0.70 - 6.95  & 0.70 - 4.55  & 0.70 - 3.43 & 0.70 - 2.97 \\\\\n& \\tbeta, \\mA $>$12.0\n                 & 0.70 - 1.01 &  0.70 - 1.01 &  0.70 - 1.02 & 0.70 - 1.01\n\\\\\\hline\n\nLarge $\\mu$ &\n    \\mh\\  & 90.2 & 94.2 & 89.7 & 89.3 \\\\\n&   \\mA\\  & 92.5 & 96.6 & 92.6 & 92.5 \\\\\n&   \\tbeta\\       & 0.72 - 14.79 & 0.72 - 13.68 & 0.72 - 10.91 & 0.72 - 10.63 \\\\\n& \\tbeta, \\mA $>$2.4\n                 & 0.72 - 0.79 &  0.75 - 0.85 &  0.86 - 0.90 & none\n\\\\\\hline\n\nGluophobic &\n    \\mh\\  & 87.8 & 87.0 & 86.4 & 86.2 \\\\\n&   \\mA\\  & 93.0 & 92.9 & 93.2 & 93.5 \\\\\n&   \\tbeta\\       & 0.40 - 9.70 & 0.42 - 5.22 & 0.48 - 3.76 & 0.51 - 3.19\n\\\\\\hline\n\nSmall $\\alpha$ &\n    \\mh\\  & 84.3 & 83.5 & 82.5 &  82.0 \\\\\n&   \\mA\\  & 95.0 & 95.8 & 96.5 & 97.2 \\\\\n&   \\tbeta\\       & 0.40 - 5.97 & 0.43 - 4.03 & 0.52 - 3.12 & 0.55 - 2.69\n\n\n\\\\\\hline\n\\end{tabular}\n\\caption[]{95\\% {\\sc CL} lower bounds on \\mh\\ and \\mA\\ in \\GeVcc\\ and\n  excluded ranges in \\tbeta\\ obtained in the different {\\sc MSSM}\n  {\\sc CP}-conserving benchmark scenarios, as a function of \\mtop. \n  Except for the two no mixing and the large $\\mu$ scenarios,\n  the exclusions in mass are valid for all values of \\tbeta\\ between 0.4\n  and 50, and the exclusions in \\tbeta\\ hold for all values of \\mA\\ between\n  0.02 and 1000~\\GeVcc.\n  In the three other scenarios, part of the interval in \\tbeta\\\n  is excluded only for \\mA\\ above a few \\GeVcc\\ threshold: \n  this sub-interval is indicated \n  in a fourth line together with the threshold in \\mA. \n  As a consequence, the mass bounds in these scenarios are valid only for\n  values of \\tbeta\\ outside the quoted sub-interval.\n}\n\\label{tab:limits}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[hp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.5cm}\n\\epsfig{figure=lim_tb_vs_mtop_max.eps,width=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=lim_tb_vs_mtop_no.eps,width=9cm} \\\\\n\\hspace{-1.5cm}\n\\epsfig{figure=lim_tb_vs_mtop_rev.eps,width=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=lim_tb_vs_mtop_nol.eps,width=9cm} \\\\\n\\hspace{-1.5cm}\n\\epsfig{figure=lim_tb_vs_mtop_maxl.eps,width=9cm} & \n\\hspace{-0.8cm}\n\\epsfig{figure=lim_tb_vs_mtop_mu.eps,width=9cm}\\\\\n\\hspace{-1.5cm}\n\\epsfig{figure=lim_tb_vs_mtop_gluo.eps,width=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=lim_tb_vs_mtop_small.eps,width=9cm} \n\\end{tabular}\n\\caption[]{\n      Variation with \\mtop\\ of the ranges in \\tbeta\\\n      excluded by {\\sc DELPHI}\n      in the {\\sc CP}-conserving  {\\sc MSSM} benchmark scenarios.\n      Note that each bound in \\tbeta\\ is a limit (either upper\n      or lower) at 95\\% {\\sc CL}. These bounds hold for the\n      whole interval of \\MA\\ between 0.02 and 1000~\\GeVcc, except\n      in the hatched intervals, where the exclusion is valid above\n      12~\\GeVcc\\ in the two no mixing scenarios and above\n      2.4~\\GeVcc\\ in the large $\\mu$ scenario.\n}\n\\label{fig:mtop}\n\\end{center}\n\\end{figure}\n\n\\subsection{Summary}\n\nThe lower bounds in mass and excluded ranges in \\tbeta\\ obtained\nin the eight {\\sc CP}-conserving benchmark scenarios presented\nin the previous sections are summarized in Table~\\ref{tab:limits}.\nThe variation with \\mtop\\ of the excluded ranges in \\tbeta\\ is \nfurther illustrated in Fig.~\\ref{fig:mtop}. \nAll lower bounds in mass are at the 95\\% {\\sc CL}, as well as each\nindividual (either lower or upper) bound in \\tbeta.\nIn all scenarios, the radiative corrections on the Higgs boson\nmasses and couplings have been computed \nin the Feynman-diagrammatic approach with all dominant two-loop \norder terms included, using version 2.0 of the {\\sc FeynHiggs} \ncode~\\cite{ref:FDradco_new}.\n\n\\newpage\n\\section{The {\\sc CP}-violating {\\sc MSSM} scenarios}\\label{sec:cpv}\n\nIn most of the parameter space of the {\\sc CP}-violating {\\sc MSSM} \nscenarios studied in this paper, only the two lightest neutral\nHiggs bosons, H$_1$ and H$_2$ are kinematically accessible at \n{\\sc LEP} energies. If their couplings to the Z boson are not \nstrongly suppressed by {\\sc CP}-violation,\nthe main production processes are \nthe \\HuZ, \\HdZ\\ and \\HuHd\\ processes, with \\HuZ\\ dominating at low\n\\tbeta, \\HuHd\\ at large \\tbeta\\ and \\HdZ\\ contributing over the whole\nrange of \\tbeta\\ values allowed in each scenario.\nIn restricted areas of the parameter space, the second pair-production\nprocess, \\HuHt, can add a non-negligible signal and has \nalso been considered in the searches. On the other hand, in most\nscenarios, charged Higgs bosons have a mass above 100~\\GeVcc, and thus\nhave not been included.\n\n   As already mentioned, {\\sc CP} violation in the {\\sc MSSM} Higgs sector\nis introduced through radiative corrections. Besides the two parameters\nused to define the scenarios at tree level, chosen as \\tbeta\\ and \n\\mHp, \nradiative corrections introduce additional parameters. \nAs in the {\\sc CP}-conserving case, these are primarily\n\\mtop\\ and the set of parameters related to supersymmetry breaking: \n$\\mu$,  $M_{\\rm susy}$, $M_2$, $m_{\\tilde{g}}$ and $A$, as defined in\nsection~\\ref{sec:benchmark}~\\cite{ref:mssmpheno,ref:FDmssmpheno}.  \nIn addition, {\\sc CP} violation introduces phases. \nThe unification assumptions made for the supersymmetry breaking \nparameters, and the global symmetries\nthat govern the dimension-four operators of the {\\sc MSSM} Lagrangian,\ncan be used to reduce the number of {\\sc CP}-violating phases to \nonly two~\\cite{ref:cpv_mssm}. In the scenarios studied hereafter, these\nphases are taken as the phase of the gluino mass, arg($m_{\\tilde{g}}$) and\nthe phase of the common stop and sbottom trilinear coupling, \narg($A$).\n\n\\subsection{The benchmark scenarios} \\label{sec:cpv_bench}\n\nThe dominant {\\sc CP}-violating effects on the neutral Higgs boson\nmasses and couplings to gauge bosons are proportional to \n\\[ \\frac{m_{top}^4}{v^2} \\frac{Im(\\mu A)}{M_{susy}^2} \\]\nwhere $v^2$ is the quadratic sum of the vacuum expectation values of the two\nHiggs field doublets~\\cite{ref:cpv_bench}.\nSizeable effects are thus expected for moderate values of $M_{\\rm susy}$,\nlarge values of $\\mu$ and phases arg($A$) around 90$^\\circ$. A\nstrong dependence on the value of \\mtop\\ is also to be expected.\n\nAlong these lines, Ref.~\\cite{ref:cpv_bench} proposed a benchmark \nscenario with maximal {\\sc CP}-violation, the {\\sc CPX} scenario, as \nan appropriate scheme for direct searches at {\\sc LEP} and other \ncolliders. The values of\nits underlying parameters are quoted in Table~\\ref{ta:cpv_scen}.\nAs expected from the above discussion, the value of $M_{\\rm susy}$,  \na few hundred \\GeVcc, is moderate, $\\mu$ and $|A|$ take \nlarge values, 2 and 1~\\TeVcc\\ respectively, and the {\\sc CP}-violating\nphase arg($A$) is set at 90$^\\circ$. \nAlthough the gluino-mass phase has a small impact on the \n{\\sc CP}-violating effects, these appear to be reinforced \nat 90$^\\circ$~\\cite{ref:cpv_mssm}, a value which was thus \nretained for arg($m_{\\tilde{g}}$).\nThe values listed in Table~\\ref{ta:cpv_scen}\nfulfill the existing constraints from measurements of the\nelectron and neutron electric dipole moments, by making the first\ntwo generations of squarks sufficiently heavy, with masses above \n1~\\TeVcc. In the following,\nthe CPX scenario has been studied for four values of the top quark mass,\n\\mtop~=~169.2, 174.3, 179.4 and 183.0~\\GeVcc.\n\n\\begin{table}[t]\n{\\small\n\\begin{center}\n\\begin{tabular}{l|ccccccc}     \\hline\nscenario & $M_{\\rm susy}$ & $M_2$ & $|m_{\\tilde{g}}|$ & $\\mu$ \n         & $|A|$ & arg($m_{\\tilde{g}}$)=arg($A$) \\\\\n& (\\GeVcc) & (\\GeVcc) & (\\GeVcc) & (\\GeVcc) & (\\GeVcc) & \n  (degrees) \\\\\n\\hline\n{\\sc CPX}  &\n      500 & 200 & 1000 & 2000 & 1000  & 90  \\\\\nphase study  &\n      500 & 200 & 1000 & 2000 & 1000 & 0,30,60,135,180\\\\\n $\\mu$ study  &\n      500 & 200 & 1000 & 500,1000,4000 & 1000 & 90\\\\\n $M_{\\rm susy}=$1 TeV\/c$^2$ &\n      1000 & 200 & 1000 & 2000 & 1000 & 90\\\\\n $M_{\\rm susy}=$1 TeV\/c$^2$, scaled &\n      1000 & 200 & 2000 & 4000 & 2000 & 90\\\\\n\\hline\n\\end{tabular}\n\\caption[]{\nValues of the underlying parameters for the representative \n{\\sc CP}-violating {\\sc MSSM} scenarios scanned in this paper,\nnamely the {\\sc CPX} scenario and its ten variants.\n}\n\\label{ta:cpv_scen}\n\\end{center}\n}\n\\end{table}\n\n\nIn addition to the {\\sc CPX} scenario, a few variants have also been \nconsidered in order to study the dependence of the {\\sc CP}-violation \neffects on the values of phases, $\\mu$ and $M_{\\rm susy}$. The\nvalues tested are quoted in Table~\\ref{ta:cpv_scen}. \nThe two {\\sc CP}-violating phases, still taken to be equal, were varied \nfrom 0 to 180$^\\circ$, keeping all other parameters as in the\n{\\sc CPX} scenario. Values of $\\mu$ below and above 2~\\TeVcc\\ were\nstudied in the same way. Finally, the value of  $M_{\\rm susy}$\nwas increased from 500~\\GeVcc\\ to 1~\\TeVcc, keeping the phases\nat 90$^\\circ$, and either keeping all\nother parameters to their {\\sc CPX} values, or\nscaling the other parameters in such a way that the relation \nbetween $|m_{\\tilde{g}}|$, $|A|$ and $\\mu$ is \nas in the {\\sc CPX} scenario. In the following,\nthese ten variants have been studied for \\mtop~=~174.3~\\GeVcc\\ only.\n\nIn all scenarios, theoretical databases provided by the\n{\\sc LEP} Higgs working group were used~\\cite{ref:hwg}. In these,\nradiative corrections have been computed in two different approaches,\nthe Feynman-diagrammatic approach of Ref.~\\cite{ref:FDradco_new},\nalready selected \nin the {\\sc CP}-conserving case (see Section~\\ref{sec:benchmark}), \nand the renormalization group approach of Ref.~\\cite{ref:RGradco}.\nAs in Section~\\ref{sec:benchmark}, the Feynman-diagrammatic calculations\nuse version 2.0 of the {\\sc FeynHiggs} code.\nThe renormalization group corrections rely on \nthe {\\sc CP}-violating version {\\sc CPH} of the {\\sc SUBHPOLE} \ncode.\\footnote{Since this work, updated versions of the two codes,  \n{\\sc CPsuperH}~\\cite{ref:CPsuperH} and\n{\\sc FeynHiggs 2.5}~\\cite{ref:FeynHiggs2.5} have been made available.\nIn both cases, the changes concern the Higgs boson decays and have\nno substantial impact on the phenomenology at {\\sc LEP}.}\nContrary to the {\\sc CP}-conserving case, where the calculations\nin the Feynman-diagrammatic approach were the most complete\ndue to the inclusion of all dominant two-loop\norder terms, in the case of {\\sc CP}-violation\nneither of the two calculations can be preferred on theoretical grounds.\nBoth contain one and two-loop corrections, but\nthe {\\sc CPH} code has a more complete phase dependence at the two-loop\norder while {\\sc FeynHiggs} contains more corrections at the one-loop order\nwith the full complex phase dependence and more corrections at the two-loop\norder but without the full phase dependence.\nThis may result in large differences when convoluted with the \nexperimental inputs.\nWe thus present our results in the two frameworks separately. \nA comparison between the two calculations in the {\\sc CP}-conserving case\ncan be found in Ref.~\\cite{ref:FDRGcomp}.\n\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\epsfig{figure=cpv_masses_cpx_174.eps,height=7.6cm} &\n\\epsfig{figure=cpv_gam_cpx_174.eps,height=7.6cm}\\\\\n\\epsfig{figure=cpv_xsec_ltb_cpx_174.eps,height=7.6cm} &\n\\epsfig{figure=cpv_xsec_htb_cpx_174.eps,height=7.6cm}\\\\\n\\epsfig{figure=cpv_brh_ltb_105_cpx_174.eps,height=7.6cm} &\n\\epsfig{figure=cpv_brh_htb_105_cpx_174.eps,height=7.6cm} \n\\end{tabular}\n\\caption[]{\n    Properties of the three neutral Higgs bosons\n    of the {\\sc CP}-violating {\\sc MSSM} in the \n    {\\sc CPX} scenario with \\mtop~=~174.3~\\GeVcc. \n    Top: H$_1$, H$_2$ and H$_3$ masses and H$_1$, H$_2$ widths at \n    various \\tbeta\\ values.\n    Middle: production cross-sections at \\mbox{$\\sqrt{s}$}~=~206~GeV, \n    as a function of \\mHu\\ and \\tbeta. \n    Bottom: H$_1$ and H$_2$ dominant branching fractions as\n    a function of \\mHu\\ and \\tbeta. H$_1$ and H$_2$ decays into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ \n    (solid and dash-dotted lines) are compared with H$_1$ decays into \n    \\mbox{$\\tau^+ \\tau^-$}\\ (dotted lines) and \n    H$_2$ decays into H$_1$H$_1$ (dashed lines).\n    The radiative corrections are as in Ref.~\\cite{ref:RGradco}.}\n\\label{fig:cpv_neutral}\n\\end{center}\n\\end{figure}\n\n The phenomenology of the three neutral Higgs bosons of the \n{\\sc CP}-violating {\\sc MSSM} is illustrated in \nFig.~\\ref{fig:cpv_neutral} in the case \nof the {\\sc CPX} scenario for a top quark mass of 174.3~\\GeVcc, with \nradiative corrections computed in the renormalization group approach.\nThe top figures show that the two lightest scalars, H$_1$ and H$_2$,\nare likely to be kinematically accessible at {\\sc LEP2} in \nwide regions of the parameter space, in which their widths remain\nlower than 1~\\GeVcc, that is below the experimental resolution.\nThe cross-section curves show that at low and large \\tbeta, the\ndominant production processes are the \\HuZ\\ and \\HuHd\\ processses, \nrespectively, as in the {\\sc CP}-conserving case \n(see Fig.~\\ref{fig:heavy_neutral_new}). On the other hand, \nat intermediate \\tbeta\\ and moderate \\mHu,\nthe \\HuZ\\ cross-section is significantly weakened,\nas a result of the suppressed H$_1$ZZ coupling due to {\\sc CP}-violation.\nIn the same region, the \\HdZ\\ process compensates only partly for this loss.\nFinally, the figures showing branching fractions compare the dominant \nH$_1$ and H$_2$ branching fractions for different values of \\tbeta.\nFor all values of \\tbeta, decays into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ and \\mbox{$\\tau^+ \\tau^-$}\\ saturate\nthe width of the lightest Higgs boson, H$_1$, in the mass range above\nthe \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ threshold up to the maximal sensitivity of {\\sc LEP}.\nIn the same mass range, the second lightest Higgs boson, H$_2$, decays\npredominantly into \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ at large \\tbeta\\ only. At low and intermediate\n\\tbeta, the cascade decay \\Hcas\\ dominates over the \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ final state\nat masses up to 50~\\GeVcc\\ or so. \nA loss in experimental sensitivity can thus be anticipated\nin regions where the \\HuZ\\ cross-section is negligible and \nthe H$_2$Z signals are not significant with respect to background,\ndue to too weak \\HdZ\\ cross-sections or H$_2$ branching fractions\ninto fermions.\n \n\n\\subsection{Scan procedure}\nThe scan procedure is similar to that described in \nSection~\\ref{sec:cpc_procedure} for the {\\sc CP}-conserving scenarios. \nThe only changes are the following. \nThe scan was performed over the {\\sc MSSM} parameters \\tbeta\\ and \n\\mHp. The range in \\mHp\\ spans from 4~\\GeVcc\\ \nup to 1~TeV\/$c^2$. Values of \\mHp\\ below about 100~\\GeVcc\\ were\nnoticed to give unphysical negative mass squared values in most\nscenarios and thus were removed from the scans.\nThe range in \\tbeta\\ extends from the minimal value allowed in each \nscenario up to 40, a value above which the Higgs-bottom Yukawa coupling \ncalculation becomes unreliable in the {\\sc CP}-violating {\\sc MSSM} \nscenarios.\nTheoretical points were generated randomly in both \\tbeta\\ and \\mHp\\\nwith a granularity which is sufficient to map the general features\nof the exclusion regions.\n\n The signal expectations in each channel were then computed as outlined \nin Section~\\ref{sec:cpc_procedure}, except for the width effects. \nIn the {\\sc CP}-violating {\\sc MSSM} scans, the widths remain well below \nthe experimental resolution for \\tbeta\\ up to 40 and \\mHu\\ below\n120~\\GeVcc\\ (see Fig.~\\ref{fig:cpv_neutral}). Signal efficiencies and\nPDFs were thus exclusively determined from simulations with Higgs boson\nwidths below  1~\\GeVcc.\n\n\n\\section{Results in {\\sc CP}-violating {\\sc MSSM} scenarios}\\label{sec:resucpv}\nThe regions of the {\\sc MSSM} parameter space excluded at \n95\\% {\\sc CL} or more by combining the neutral Higgs boson searches of \nTable~\\ref{tab:channels} are hereafter discussed in turn \nfor each scenario. \nThe additional constraint from the limit on the Z partial width that would \nbe due to new physics~\\cite{ref:width} \n(used as described in Section 3.2 of~\\cite{ref:hwg})\nbrings no gain in sensitivity in any of the scenarios tested.\nResults are presented only in the (\\mHu, \\tbeta)\nplane, which is the only one relevant at {\\sc LEP} since the\nminimal values of \\mHd\\ and \\tbeta\\ in most scenarios are such\nthat the region accessible at {\\sc LEP} is much reduced in the other\nprojections.\n\n\\subsection{Dependence on the phases}\n\nThe excluded regions in the (\\mHu, \\tbeta) plane for the {\\sc CPX}\nscenario and its variants with different phase values are\npresented in Fig.~\\ref{fig:limit_phases_sub} for the renormalization\ngroup approach~\\cite{ref:RGradco} and in Fig.~\\ref{fig:limit_phases_fey} \nfor the Feynman-diagrammatic calculations~\\cite{ref:FDradco_new}. \nThe top mass value is 174.3~\\GeVcc\\ in all plots.\n\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_00_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_30_174_tb_mh1_sub.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_60_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_sub.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_135_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_180_174_tb_mh1_sub.eps,height=8cm} \n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios with corrections \n    as in Ref.~\\cite{ref:RGradco} for different values of the \n    phases: regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   The {\\sc CPX} scenario corresponds to phases of 90$^{\\circ}$.\n   }\n\\label{fig:limit_phases_sub}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_00_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_30_174_tb_mh1_fey.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_60_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_fey.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_135_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_180_174_tb_mh1_fey.eps,height=8cm} \n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios with corrections \n    as in Ref.~\\cite{ref:FDradco_new} for different values of the phases: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   The {\\sc CPX} scenario corresponds to phases of 90$^{\\circ}$.\n   }\n\\label{fig:limit_phases_fey}\n\\end{center}\n\\end{figure}\n\n\nGoing from 0$^{\\circ}$ to 180$^{\\circ}$, the excluded domain varies\nsignificantly. The qualitative trend, valid in the two theoretical\napproaches, is as follows.\nThe extreme values (0$^{\\circ}$ and 180$^{\\circ}$) correspond to\nscenarios with no {\\sc CP} violation, and hence to a large \nexcluded region.\nMoreover, at  180$^{\\circ}$, the theoretically allowed\nregion is reduced, especially at large \\tbeta\\ due\nto unphysical values of the Higgs-bottom Yukawa coupling.\nAt phases between 60$^{\\circ}$ and 135$^{\\circ}$,\nlosses in sensitivity are observed\nat large \\tbeta\\ and \\mHu\\ above 50~\\GeVcc, as well as \nin the intermediate \\tbeta\\ range for \\mHu\\ below 60~\\GeVcc.\nThis is the consequence of the strong suppression of the \nH$_1$ZZ coupling due to {\\sc CP}-violation, as already encountered \nin Fig.~\\ref{fig:cpv_neutral} in the case of the {\\sc CPX}\nscenario. More generally~\\cite{ref:cpv_mssm}, \nthe H$_1$ZZ coupling decreases slowly (by a few tens of \\%) \nwith phases below about 75$^{\\circ}$ and is strongly\nsuppressed (by three to four orders of magnitude) for phases around \n90$^{\\circ}$. For phases above 100$^{\\circ}$, \nthe coupling is partially restored, mostly at low \\tbeta.\nThis explains the evolution of the upper bound of the experimentally\nexcluded area as a function of phases in \nFigs.~\\ref{fig:limit_phases_sub} and~\\ref{fig:limit_phases_fey}. \nThe changes are moderate for phases up to 60$^{\\circ}$ and \nsignificant for the 90$^{\\circ}$ and 135$^{\\circ}$ phases, \nwhere the experimental\nsensitivity relies mainly on the \\HdZ\\ process at low \\tbeta\\\nand on the \\HuHd\\ production at large \\tbeta, both giving large\nsignals at moderate \\mHu\\ only, typically below 60~\\GeVcc\\\n(see Fig.~\\ref{fig:cpv_neutral}). \n\nAt the 90$^{\\circ}$ and 135$^{\\circ}$ phases, there are also unexcluded \nareas at masses lower than 60~\\GeVcc\\ in the intermediate \\tbeta\\ range, \nbetween about 4 and 16. These are related to \nweakened sensitivities in the \\HdZ\\ or \\HuHd\\ searches.\nTo take the {\\sc CPX} scenario as an example,\nat masses below 15~\\GeVcc, the dominant final state\nis the (\\Hcas)Z channel.\nThe lack of experimental searches at {\\sc LEP2} for such final states\nwith \\mHu\\ below the \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ threshold\n(see Tab.~\\ref{tab:channels}) explains the unexcluded area\nwhich is observed at these masses, in agreement with the expected \nsensitivity. The largest value of \\mbox{$\\rm CL_{\\rm s}$}\\ in this region is 52\\% in the \nrenormalisation group framework and 50\\% in the\nFeynman-diagrammatic approach.\nStill in the {\\sc CPX} scenario,\nthe hole at \\mHu\\ around 50~\\GeVcc\\ arises in the region where\nthe decays \\Hcas\\ and H$_2$~\\rgr~\\bbbar\\ become approximately equal,\nleading to a loss of significance of the H$_2$ signals,\nas pointed out in Section~\\ref{sec:cpv_bench} \n(see Fig.~\\ref{fig:cpv_neutral}).\nThe largest value of \\mbox{$\\rm CL_{\\rm s}$}\\ in this region is 17\\% (expected 4\\%) in the \nrenormalisation group framework and 37\\% (expected 11\\%) in the\nFeynman-diagrammatic approach. In both frameworks, \nthese \\mbox{$\\rm CL_{\\rm s}$}\\ values are observed at \\tbeta$\\sim$4,  \n\\mHu$\\sim$50~\\GeVcc\\ and \\mHd$\\sim$105\/107~\\GeVcc.\nThe observed exclusion in this region is weaker than expected,\nwhich is due to a slight excess of data over the expected \nbackground. The value of 1-\\mbox{$\\rm CL_{\\rm b}$}\\ at the point of weakest \nexclusion is indeed 15\\% (corresponding to a 1.4 sigma deviation)\nin the renormalisation group framework  \nand 12\\% (1.5 sigma deviation) in the Feynman-diagrammatic approach.\nConversely, the largest deviation in the whole hole \nhas a value of 1-\\mbox{$\\rm CL_{\\rm b}$}\\\nof 3.3\\% (2.1 sigma deviation) in the two approaches. This value\nis observed at \\tbeta$\\sim$16, \\mHu$\\sim$45~\\GeVcc\\ and \n\\mHd$\\sim$107~\\GeVcc\\ in the renormalisation group framework\nand \\tbeta$\\sim$11, \\mHu$\\sim$52~\\GeVcc\\ and \n\\mHd$\\sim$111~\\GeVcc\\ in the Feynman-diagrammatic approach.\nAt this point, the \\mbox{$\\rm CL_{\\rm s}$}\\ values are 5.4\\% (expected 0.1\\%)\nand  6.5\\% (expected 0.2\\%) in the two frameworks, respectively.\nThe combined {\\sc LEP} data show also deviations in this\nregion~\\cite{ref:hwg}.\n\nFinally, differences between the two theoretical frameworks are visible \nmainly at large \\tbeta, where the Feynman-diagrammatic\ncalculations predict significantly higher \\HuZ\\ residual cross-sections\n(e.g. a factor about 4 in the {\\sc CPX} scenario\nfor \\mHu\\ between 40 and 80~\\GeVcc),\nleading to a better experimental sensitivity. \nDifferences in the phase dependence of the results \nare also visible, which are likely to reflect the \ndifferent phase treatment between the two calculations.\n\n\\subsection{Dependence on $\\mu$ and $M_{\\rm susy}$}\n\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_500_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_1000_174_tb_mh1_sub.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_4000_174_tb_mh1_sub.eps,height=8cm} \\\\\n\\hspace{-1.cm} \n\\epsfig{figure=cpv_onlyms_174_tb_mh1_sub.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_ms_174_tb_mh1_sub.eps,height=8cm}\n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios  with corrections \n    as in Ref.~\\cite{ref:RGradco} for different values of \n    $\\mu$ and $M_{\\rm susy}$: regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   The {\\sc CPX} scenario corresponds to $\\mu$~=~2000~\\GeVcc\\ and\n   $M_{\\rm susy}$~=~500~\\GeVcc.\n   }\n\\label{fig:limit_mu_sub}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\vspace{-1.7cm}\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_500_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_1000_174_tb_mh1_fey.eps,height=8cm} \\\\\n\\hspace{-1.cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_4000_174_tb_mh1_fey.eps,height=8cm} \\\\\n\\hspace{-1.cm} \n\\epsfig{figure=cpv_onlyms_174_tb_mh1_fey.eps,height=8cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_ms_174_tb_mh1_fey.eps,height=8cm}\n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios  with corrections \n    as in Ref.~\\cite{ref:FDradco_new}\n    for different values of $\\mu$ and $M_{\\rm susy}$: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   The {\\sc CPX} scenario corresponds to $\\mu$~=~2000~\\GeVcc\\ and\n   $M_{\\rm susy}$~=~500~\\GeVcc.\n   }\n\\label{fig:limit_mu_fey}\n\\end{center}\n\\end{figure}\n\nThe excluded regions in the (\\mHu, \\tbeta) plane for the {\\sc CPX}\nscenario and its variants with different values of \n$\\mu$ and $M_{\\rm susy}$ are\npresented in Fig.~\\ref{fig:limit_mu_sub} for the renormalization\ngroup approach~\\cite{ref:RGradco} and in Fig.~\\ref{fig:limit_mu_fey} \nfor the Feynman-diagrammatic calculations~\\cite{ref:FDradco_new}. \nIn all plots, the common {\\sc CP}-violating phase is 90$^{\\circ}$ and\nthe top mass value is 174.3~\\GeVcc.\n\nThe dependence of the results on the value of $\\mu$ \nis as expected from the scaling of the\ndominant {\\sc CP}-violating effects with $Im(\\mu A)$\n(see Section~\\ref{sec:cpv_bench}).\nThe exclusion is almost entirely \nrestored for values of $\\mu$ lower than 2~\\TeVcc,\nthe value in the {\\sc CPX} scenario, and gets weaker at 4~\\TeVcc.\nIn the first two variants, despite the {\\sc CP}-violating phase being at\n90$^{\\circ}$, there are always two production processes\nwith significant rates in every point of the kinematically accessible \nparameter space. \nIn the variant at 4~\\TeVcc, due to the large value of $\\mu$ and the\n{\\sc CP}-violating phase at 90$^{\\circ}$, \nthe \\HuHd\\ and \\HdZ\\ processes are suppressed for all values of \\tbeta, \nas well as the \\HuZ\\ process at intermediate and large \\tbeta\\ values.\nIn the Feynman-diagrammatic approach, the \\HuZ\\ cross-sections are\npartly restored at large \\tbeta\\ which explains the difference\nbetween the results in the two theoretical frameworks in that region.\nNote also that the theoretically allowed region is much reduced at large \n\\tbeta\\ in this scenario due to unphysical values of the bottom Yukawa\ncoupling.\n\nThe dependence on the value of $M_{\\rm susy}$ is presented in the two\nbottom plots of Figs.~\\ref{fig:limit_mu_sub} and~\\ref{fig:limit_mu_fey}.\nThe first scenario corresponds to setting $M_{\\rm susy}$ at 1~\\TeVcc,\ntwice its value in the {\\sc CPX} scenario. As the dominant \n{\\sc CP}-violating effects are proportional to $M_{\\rm susy}^{-2}$,\nthe exclusion is restored in this variant. The reason is as in\nthe case of the two variants with low values of $\\mu$, i.e.\nthere are always two production processes\nwith significant rates in every point of the kinematically accessible \nparameter space. \nIn the second scenario,  $M_{\\rm susy}$ is still set at  1~\\TeVcc\\ but\nthe values of $|m_{\\tilde{g}}|$, $\\mu$ and $|A|$ are also scaled\nby a factor 2, leaving the {\\sc CP}-violating effects almost unchanged\n(see Section~\\ref{sec:cpv_bench}) with respect to the {\\sc CPX} scenario. \nThis explains why the exclusion region \nin this variant is close to that in the {\\sc CPX} scenario.\nThe few differences between the excluded regions in these two scenarios \nare due to different cross-sections for some of the processes\nwhich contribute most to the experimental sensitivity, that is the\n\\HuHd\\ and \\HdZ\\ processes at masses below 60~\\GeVcc, \nand the \\HuZ\\ process at higher masses. \nAs an example, the better\ncoverage of the low mass region at intermediate \\tbeta\\ values\nin the scaled variant is explained by slightly higher \\HuHd\\ cross-sections.\n\n\n\\subsection{Dependence on \\bf{\\mtop}}\n\nThe excluded regions in the (\\mHu, \\tbeta) plane for the {\\sc CPX}\nscenario with different \\mtop\\ values are\npresented in Fig.~\\ref{fig:limit_mtop_sub} for the renormalization\ngroup approach and in Fig.~\\ref{fig:limit_mtop_fey} for the \nFeynman-diagrammatic calculations. \n\nThe results show a strong dependence on the value of \\mtop,\nas expected since the dominant {\\sc CP}-violating effects scale\nwith m$^4_{\\rm{top}}$. In the two theoretical approaches, the exclusion in\nthe intermediate \\tbeta\\ range is gradually reduced as \\mtop\\ increases\nand eventually vanishes for \\tbeta\\ between about 3 and 5\nand a top mass of 183~\\GeVcc.\nThis can be traced to the suppression of the \\HdZ\\ and \\HuHd\\\ncross-sections with increasing values of \\mtop, leaving no \nsignificant rate in any of the three possible production channels. \nAt large \\tbeta, as \\mtop\\ increases, the \\HuHd\\ cross-section is\nreduced and the exclusion gets weaker \nin the renormalization group approach while it is almost unchanged\nin the Feynman-diagrammatic framework. As already mentioned, this is \na consequence of the higher \\HuZ\\ residual cross-sections predicted by\nthe latter calculations at large \\tbeta.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.2cm}\n\\epsfig{figure=cpv_90_169_tb_mh1_sub.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_sub.eps,height=9cm} \\\\\n\\hspace{-1.2cm}\n\\epsfig{figure=cpv_90_179_tb_mh1_sub.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_183_tb_mh1_sub.eps,height=9cm} \\\\\n\\hspace{-1.cm}\n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios with corrections \n    as in Ref.~\\cite{ref:RGradco} for different values of \\mtop: \n    regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   }\n\\label{fig:limit_mtop_sub}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\hspace{-1.2cm}\n\\epsfig{figure=cpv_90_169_tb_mh1_fey.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_174_tb_mh1_fey.eps,height=9cm} \\\\\n\\hspace{-1.2cm}\n\\epsfig{figure=cpv_90_179_tb_mh1_fey.eps,height=9cm} &\n\\hspace{-0.8cm}\n\\epsfig{figure=cpv_90_183_tb_mh1_fey.eps,height=9cm} \\\\\n\\hspace{-1.cm}\n\\end{tabular}\n\\caption[]{\n    {\\sc CP}-violating {\\sc MSSM} scenarios  with corrections \n    as in Ref.~\\cite{ref:FDradco_new} \n    for different values of \\mtop: regions excluded at \n    95\\% {\\sc CL} by combining the results of the neutral Higgs \n    boson searches in the whole {\\sc DELPHI} data sample  \n    (light-grey). \n   The dashed curves show the median expected limits.\n   The medium-grey areas are the regions not allowed by theory.\n   }\n\\label{fig:limit_mtop_fey}\n\\end{center}\n\\end{figure}\n\n\\subsection{Summary}\nScans of the {\\sc CPX} scenario and its variants revealed that {\\sc CP}\nviolation in the Higgs sector can have a significant impact on the \nexperimental sensitivity of {\\sc LEP}. The strong suppression of the\nneutral Higgs boson couplings to the Z boson translates into a \nloss of redundancy in the different search channels, and hence leads to\na reduced coverage of the parameter space. The most significant reduction is\nobserved in the intermediate \\tbeta\\ region, typically between 3 and 10,\ndown to the lowest H$_1$ masses. It occurs for phases between \n90$^{\\circ}$ and 135$^{\\circ}$, \ntop mass values equal to 174.3~\\GeVcc\\ or larger,\nand values of the ratio  $|\\mu A| \/ M_{susy}^2 $ equal to \n8 or larger. As a consequence, no absolute mass limits can be derived in\nthese scenarios. On the other hand, the low \\tbeta\\ region appears \nstill disfavoured, as in the {\\sc CP}-conserving models. \nScans were performed using\ntwo different theoretical approaches for the \nradiative correction calculations. Although the two sets of results show\nlarge differences, they both lead to the same qualitative conclusions.\n\n\n\\section{Conclusions}\n\nSearches for Higgs bosons in \nthe whole data sample of the {\\sc DELPHI} experiment have been\ncombined to derive constraints on {\\sc MSSM} \nbenchmark scenarios, including models with {\\sc CP}-violation\nin the Higgs sector. \nExperimental results encompass searches\nfor neutral Higgs bosons in dominant final states expected\nin most {\\sc MSSM} models, as well as \nsearches for charged Higgs bosons and for neutral Higgs bosons \ndecaying into hadrons of any flavour,\nwhich bring a gain in sensitivity in restricted regions \nof the parameter space.\nAn additional improvement is obtained by applying the experimental\nresults to more production processes than the two expected main channels,\nnamely the associated production of the lightest Higgs boson with a\nZ boson and the pair-production of the two lightest Higgs bosons. \nIn the {\\sc CP}-conserving {\\sc MSSM}, \nthe experimental sensitivity at {\\sc LEP} relies on the \\hZ, \\hA\\ and \\HZ\\ \nchannels, the last leading to a significant gain in sensitivity in\nscenarios where the third neutral Higgs boson, H, is kinematically \naccessible.\nIn the {\\sc CP}-violating {\\sc MSSM}, the total signal at {\\sc LEP}\nis spread mainly over the \\HuZ, \\HdZ\\ and \\HuHd\\ channels.\nAccounting for the simultaneous\nproduction of all possible signals is essential in this type of scenario\nwhere {\\sc CP}-violating effects can lead to strong suppression of one\nchannel or another.\n\nIn all  {\\sc CP}-conserving scenarios, \nthe experimental results allow \na large fraction of the parameter space to be excluded, \neven in scenarios designed\nto test potentially difficult cases (e.g. vanishing \nproduction cross-sections or decay branching fractions) \neither at {\\sc LEP} or at hadron colliders. Limits on masses of\nthe h and A bosons were deduced as well as upper and lower exclusion\nbounds in \\tbeta. The dependence of these limits on \\mtop\\ was\nstudied in a range between 169.2 to 183.0~\\GeVcc.\nTo quote but one result, the following limits at 95\\% of {\\sc CL} \nhave been established in the framework of the\n\\mbox{$ m_{\\mathrm h}^{\\rm max}$} scenario with \\mtop~=~174.3~\\GeVcc:\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{lcll}  \n   \\mh $>$ 89.7~\\GeVcc  & and & \\MA $>$ 90.4~\\GeVcc & \n    for any  $\\tan\\beta$ between 0.4 and 50, \\\\\n   $\\tan\\beta < 0.72$ & or & $\\tan\\beta > 1.96 $ &\n    for any \\mA\\ between 0.02 and 1000~\\GeVcc. \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\noindent\nThese mass limits are insensitive to variations of the top quark mass.\nThe excluded range in \\tbeta\\ decreases with increasing \\mtop\\ and\nwould vanish if \\mtop\\ was as large as 183.0~\\GeVcc.\nThis scenario provides the most conservative bounds on \\tbeta\\\namong the eight {\\sc CP}-conserving scenarios tested.\n\nIn the {\\sc CP}-violating scenarios, large domains of the kinematically\naccessible parameter space remain unexcluded due to strong suppressions\nof the couplings between the Z and the Higgs bosons induced by\n{\\sc CP}-violation. Hence no absolute limits can be set on the Higgs\nboson masses in these scenarios.\nThe unexcluded areas arise in the intermediate\n\\tbeta\\ range, typically between 3 and 10. Their contours vary considerably\nwith the value of \\mtop\\ and the {\\sc MSSM} parameters which govern\nthe {\\sc CP}-violating effects, $|\\mu A|$, $M_{\\rm susy}$ and the\nphase $arg(A)$. These scenarios have been studied in two\ndifferent theoretical frameworks for the radiative correction calculations. \nThe impact of {\\sc CP}-violation is\nobserved to be qualitatively the same in the two approaches.\n\n\n\n\\subsection*{Acknowledgements} \\vskip 3 mm\n\nWe are greatly indebted to our technical collaborators, to the\nmembers of the CERN-SL Division for the excellent performance of\nthe LEP collider, and to the funding agencies for their\nsupport in building and operating the DELPHI detector.\\\\\nWe acknowledge in particular the support of \\\\\nAustrian Federal Ministry of Education, Science and Culture,\nGZ 616.364\/2-III\/2a\/98, \\\\\nFNRS--FWO, Flanders Institute to encourage scientific and technological\nresearch in the industry (IWT) and Belgian Federal Office for Scientific,\nTechnical and Cultural affairs (OSTC), Belgium, \\\\\nFINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil, \\\\\nCzech Ministry of Industry and Trade, GA CR 202\/99\/1362,\\\\\nCommission of the European Communities (DG XII), \\\\\nDirection des Sciences de la Mati$\\grave{\\mbox{\\rm e}}$re, CEA, France, \\\\\nBundesministerium f$\\ddot{\\mbox{\\rm u}}$r Bildung, Wissenschaft,\nForschung und Technologie, Germany,\\\\\nGeneral Secretariat for Research and Technology, Greece, \\\\\nNational Science Foundation (NWO) and Foundation for Research on\nMatter (FOM), The Netherlands, \\\\\nNorwegian Research Council,  \\\\\nState Committee for Scientific Research, Poland, SPUB-M\/CERN\/PO3\/DZ296\/2000,\nSPUB-M\/CERN\/PO3\/DZ297\/2000, 2P03B 104 19 and 2P03B 69 23(2002-2004)\\\\\nFCT - Funda\\c{c}\\~ao para a Ci\\^encia e Tecnologia, Portugal, \\\\\nVedecka grantova agentura MS SR, Slovakia, Nr. 95\/5195\/134, \\\\\nMinistry of Science and Technology of the Republic of Slovenia, \\\\\nCICYT, Spain, AEN99-0950 and AEN99-0761,  \\\\\nThe Swedish Research Council,      \\\\\nParticle Physics and Astronomy Research Council, UK, \\\\\nDepartment of Energy, USA, DE-FG02-01ER41155, \\\\\nEEC RTN contract HPRN-CT-00292-2002.\\\\\n\n\\newpage\n\\section*{Appendix 1}\nWe give hereafter efficiencies of the Yukawa \\mbox{$\\tau^+ \\tau^- $\\bbbar }\\  analysis published \nin Ref.~\\cite{ref:2hdm} and applied here to the ${\\mathrm{hA}}\\rightarrow$\\toto \\qqbar\\ signal.\n\n\\begin{table} [htbp]\n\\begin{center}\n\\begin{tabular}{cc|cc||cc|c} \\hline\n\\multicolumn{2}{c|}{mass (\\GeVcc)} & efficiency & &  \n\\multicolumn{2}{c|}{mass (\\GeVcc)} & efficiency \\\\ \n$m_A$ & $m_h$ & $(\\%)$ & & $m_A$ & $m_h$ & $(\\%)$ \\\\ \\hline\n   4 & 12 &       0.        && 12 & 4 &    0.        \\\\\n   4 & 20 &  2.1$\\pm$ 0.3 && 20 & 4 & 1.9$\\pm$0.3\\\\\n   4 & 30 &  2.3$\\pm$ 0.3 && 30 & 4 & 2.0$\\pm$0.3\\\\\n   4 & 40 &  2.6$\\pm$ 0.4 && 40 & 4 & 2.0$\\pm$0.3\\\\\n   4 & 50 &  1.4$\\pm$ 0.3 && 50 & 4 & 1.6$\\pm$0.3\\\\\n   4 & 60 &  1.8$\\pm$ 0.3 && 60 & 4 & 2.2$\\pm$0.3\\\\\n   4 & 70 &  1.3$\\pm$ 0.3 && 70 & 4 & 0.9$\\pm$0.2\\\\\n   6 & 12 &       0.        && 12 & 6 &    0         \\\\\n   6 & 20 &  2.2$\\pm$ 0.3 && 20 & 6 & 2.0$\\pm$0.3\\\\\n   6 & 30 &  3.1$\\pm$ 0.4 && 30 & 6 & 3.0$\\pm$0.4\\\\\n   6 & 40 &  2.6$\\pm$ 0.4 && 40 & 6 & 2.8$\\pm$0.4\\\\\n   6 & 50 &  2.5$\\pm$ 0.4 && 50 & 6 & 2.2$\\pm$0.3\\\\\n   6 & 60 &  3.1$\\pm$ 0.4 && 60 & 6 & 2.5$\\pm$0.4\\\\\n   6 & 70 &  1.6$\\pm$ 0.3 && 70 & 6 & 1.0$\\pm$0.2\\\\\n   9 & 12 &       0.        && 12 & 9 &    0.        \\\\\n   9 & 20 &  2.9$\\pm$ 0.4 && 20 & 9 & 2.7$\\pm$0.4\\\\\n   9 & 30 &  3.0$\\pm$ 0.4 && 30 & 9 & 3.4$\\pm$0.4\\\\\n   9 & 40 &  3.4$\\pm$ 0.4 && 40 & 9 & 3.0$\\pm$0.4\\\\\n   9 & 50 &  2.3$\\pm$ 0.4 && 50 & 9 & 2.9$\\pm$0.4\\\\\n   9 & 60 &  2.4$\\pm$ 0.4 && 60 & 9 & 1.8$\\pm$0.3\\\\\n   9 & 70 &  1.1$\\pm$ 0.3 && 70 & 9 & 0.8$\\pm$0.2\\\\\n  12 & 12 &       0.&& 12 &12 &    0                 \\\\\n  12 & 20 &  2.9$\\pm$ 0.4 && 20 &12 & 2.6$\\pm$0.4\\\\\n  12 & 30 &  2.3$\\pm$ 0.3 && 30 &12 & 2.4$\\pm$0.4\\\\\n  12 & 40 &  2.6$\\pm$ 0.4 && 40 &12 & 2.0$\\pm$0.3\\\\\n  12 & 50 &  2.4$\\pm$ 0.3 && 50 &12 & 2.4$\\pm$0.4\\\\\n  12 & 60 &  2.0$\\pm$ 0.3 && 60 &12 & 1.9$\\pm$0.3\\\\\n  12 & 70 &  0.4$\\pm$ 0.2 && 70 &12 & 0.5$\\pm$0.2\\\\\n \\end{tabular}\n \\end{center}\n\\caption{${\\mathrm{hA}}\\rightarrow$\\toto \\qqbar\\ channel :\n  efficiencies of the selection (in~\\%) at {\\sc LEP1} as a function \n  of the masses of the A and h bosons. The analysis, \n  described in Ref.~\\cite{ref:2hdm}, was designed\n  to search for Yukawa production in the \\mbox{$\\tau^+ \\tau^- $\\bbbar }\\ final state.    \n  The quoted errors are statistical only.}\n \\label{bbtt_hA}\n\\end{table}\n\n\\newpage\n\\section*{Appendix 2}\nWe give hereafter efficiencies of the h $\\rightarrow $ \\qqbar\\ analyses published \nin Ref.~\\cite{ref:pap99,ref:pap00} and applied to  \n(\\hAA $\\rightarrow$ \\mbox{${\\mathrm c}\\bar{\\mathrm c}$} \\mbox{${\\mathrm c}\\bar{\\mathrm c}$} )(\\Zqq ) signals \nwith low A masses.\n\n\\begin{table} [htbp]\n\\begin{center}\n\\begin{tabular}{cc|cccc}  \\hline \n \\MA & \\mh & Efficiency (\\%)& \n \\multicolumn{2}{c}{Efficiency (\\%) at 206.5~GeV } \\\\\n(\\GeVcc)   & (\\GeVcc)  & at 199.6~GeV & first period & second period \\\\ \\hline\n  4.0 & 10.0   &  0.6 $\\pm$ 0.1 &  0.4 $\\pm$ 0.1 &  0.4 $\\pm$ 0.1\\\\\n  4.0 & 20.0   &  1.2 $\\pm$ 0.1 &  1.6 $\\pm$ 0.1 &  1.4 $\\pm$ 0.1\\\\\n  4.0 & 30.0   &  4.8 $\\pm$ 0.2 &  4.9 $\\pm$ 0.2 &  4.6 $\\pm$ 0.2\\\\\n  4.0 & 50.0   & 14.4 $\\pm$ 0.4 & 15.2 $\\pm$ 0.4 & 14.5 $\\pm$ 0.4\\\\\n  4.0 & 70.0   & 13.0 $\\pm$ 0.4 & 13.9 $\\pm$ 0.4 & 13.5 $\\pm$ 0.4\\\\\n  4.0 & 90.0   & 20.3 $\\pm$ 0.4 & 19.3 $\\pm$ 0.4 & 18.2 $\\pm$ 0.4\\\\\n  4.0 & 105.0  & 33.1 $\\pm$ 0.5 & 27.7 $\\pm$ 0.5 & 26.9 $\\pm$ 0.4\\\\\n  8.0 & 20.0   &  1.9 $\\pm$ 0.1 &  2.6 $\\pm$ 0.2 &  2.3 $\\pm$ 0.2\\\\\n  8.0 & 30.0   &  7.6 $\\pm$ 0.3 &  8.3 $\\pm$ 0.3 &  7.8 $\\pm$ 0.3\\\\\n  8.0 & 50.0   & 20.9 $\\pm$ 0.5 & 21.0 $\\pm$ 0.4 & 19.7 $\\pm$ 0.4\\\\\n  8.0 & 70.0   & 20.8 $\\pm$ 0.4 & 20.8 $\\pm$ 0.4 & 19.8 $\\pm$ 0.4\\\\\n  8.0 & 90.0   & 36.0 $\\pm$ 0.5 & 32.8 $\\pm$ 0.5 & 31.4 $\\pm$ 0.5\\\\\n  8.0 & 105.0  & 51.6 $\\pm$ 0.5 & 44.6 $\\pm$ 0.5 & 42.4 $\\pm$ 0.5\\\\\n\\hline\n\\end{tabular}\n\\caption[]{(\\hAA )(\\Zqq ) channel with A $\\rightarrow $ \\ccbar:\n  efficiencies of the selection (in~\\%)  at \\mbox{$\\sqrt{s}$}~=~199.6 and 206.5~\\GeV\\ \n  as a function of the masses of the A and h bosons, for \\mA\\ between the\n  \\mbox{${\\mathrm c}\\bar{\\mathrm c}$}\\ and \\mbox{${\\mathrm b}\\bar{\\mathrm b}$}\\ thresholds. Efficiencies at higher masses can be\n  found in Ref.~\\cite{ref:pap99,ref:pap00}. \n  We refer the reader to Ref.~\\cite{ref:pap00}\n  for the definition of the two operational periods of the 2000 data taking \n  campaign. The quoted errors are statistical only.}\n\\label{ta:aaqqeff}\n\\end{center}\n\\end{table}\n\n\n\n\\clearpage\n\\vskip 2 cm\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Supplement}\n\n\\subsection{Optical microscope image and linescan description} \\label{partA}\n\nAn optical microscope image of the MoSe$_2$\/WSe$_2$ lateral heterostructure (LH) and a schematic representation of the linescan process used in the optical spectroscopy experiments are shown in Fig.~\\ref{fig:figS1}a,b. We use water-assisted deterministic transfer to pick up as-grown, chemical vapor deposition (CVD) LHs using polydimethylsiloxane (PDMS) and deterministically transfer and encapsulate them in hBN \\cite{jia2016large,paradisanos2020controlling}. The optical contrast between MoSe$_2$ and WSe$_2$ is different, allowing the direct visualization of the heterojunctions, shown with white dashed lines in Fig.~\\ref{fig:figS1}a. In Fig.~\\ref{fig:figS1}b we show a schematic representation of the process for the Raman, PL and Reflectivity linescans (Figure 2 in the main text). We position the excitation laser ($ \\lambda = 633 $~nm, diffraction-limited spot diameter of 1~$ \\mu $m) on the area of WSe$_2$ and we use piezo-controlled nanopositioners to move the sample with steps of $\\approx150$~nm. After each step, a single -Raman, photoluminescence, reflectivity- spectrum is collected and finally all spectra are plotted as contour representations in Figure 2 of the main text.\\\\\n\\subsection{Optical spectroscopy}\n\\begin{figure}\n\\includegraphics[width=0.95\\linewidth]{figS1}\n\\caption{\\label{fig:figS1} (a) Optical image of the hBN-encapsulated LH. Orange and white dashed lines indicate the boundaries of WSe$_2$ and MoSe$_2$, respectively. (b) Schematic representation of the Raman, PL and Reflectivity linescan process across the heterojunction of MoSe$_2$\/WSe$_2$ LH. The laser is initially positioned on WSe$ _2 $ and a series of sequential steps (step size $ \\approx150$ nm) follows with a single signal acquisition after each step.}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=0.9\\linewidth]{figS2}\n\\caption{\\label{fig:figS2} (a) Raman, (b) photoluminescence and (c) reflectivity spectra of MoSe$_2$ (bottom), on the junction (middle) and WSe$_2$ (top).}\n\\end{figure*}\nRaman, PL and differential white light reflectivity spectra are collected at T=5 K in a closed-loop liquid helium (LHe) system. Fig.~\\ref{fig:figS2}a,b,c shows individual Raman, PL and reflectivity spectra of MoSe$_2$ (bottom), junction (middle) and WSe$_2$ (top), after transfer and encapsulation in hBN. For the Raman and PL experiments we use a 633~nm HeNe laser as an excitation source with a spot size diameter of $ \\approx 1~\\mu $m and 6~$ \\mu $W power.  In reflectivity we use a tungsten-halogen white light source with a power of a few nW to collect the intensity reflection coefficient of the sample with the monolayer ($R_{ML}$) and the reflection coefficient of the substrate ($R_{S}$) so that $\\Delta R=(R_{ML}-R_{S})\/R_{S}$. \nThe spectral shape and amplitude of the differential reflectivity signal also depends on thin-film interference effects related to the thickness of top, bottom hBN and SiO$_2$ layers \\cite{Robert:2018a}\\\\\nThe main Raman peaks of MoSe$_2$ (Fig.~\\ref{fig:figS2}a) include the A$'_{1}$($\\Gamma$) phonon at 241~cm$^{-1}$ and the E$'$($\\Gamma$) at 291~cm$^{-1}$ \\cite{soubelet2016resonance}, while a strong peak at 458~cm$^{-1}$ has been associated two other peaks to form triplets, see supplementary material of \\cite{mcdonnell2020observation}. A strong peak at 531~cm$^{-1}$ was recently assigned to multi-phonon processes either associated with both K and M point phonons or combinations of $\\Gamma$ point phonons \\cite{mcdonnell2020observation}. The observation of this phonon is a signature of resonant excitation with an excited exciton state. WSe$_2$ phonons are spectrally different compared to MoSe$_2$. The degenerate A$'_{1}$($\\Gamma$)\/E$'$($\\Gamma$) phonons are located at 250~cm$^{-1}$ and, similar to MoSe$_2$, a strong and recently discovered peak at 495~cm$^{-1}$ is also observed here and attributed to multi-phonon processes at K and M points or combination of $\\Gamma$ point phonons \\cite{mcdonnell2020observation}. As expected, when the laser spot is on the junction, Raman spectra display a superposition of the individual spectral signatures. We did not identify any measurable shift or broadening of the Raman peaks when we scanned across the junction. PL spectra of MoSe$_2$ and WSe$_2$ monolayers exhibit pronounced peaks (Fig.~\\ref{fig:figS2}b), associated with neutral neutral (X$^{0}$) and charged (X$^{T}$) excitons. Additional peaks appearing at lower energies, especially in WSe$_2$ and at the junction, possibly due to an ensemble of localized emission from defects. Further experiments are necessary to examine any possibility of the formation of interlayer excitons. Strong excitonic resonances appear in reflectivity for both materials with negligible Stokes shift between emission and absorption of the A1s exciton state (see Fig.~\\ref{fig:figS2}c). Clear signatures of B1s states are also observed in both materials while for WSe$_2$ the appearance of the A2s excited state further supports the good quality of the CVD-grown monolayers. Similar to Raman and PL spectroscopy, on the junction we observe a superposition of the available states from both materials.\\\\\n\n\\subsection{Diffusion experiment and model}\nFor the investigation of lateral transport, we used an experimental procedure very similar to the one described in part \\ref{partA}. The excitation is based on a HeNe laser with an excitation spot size of about 0.7$\\mu$m and an excitation power of 5$\\mu$W. The PL images are recorded by a Hamamatsu Fusion-BT CMOS camera. The PL profiles are plotted in polar coordinates. To do so, we integrate over all the polar angles [0;2$\\pi$], the data are therefore $I_{PL}(r)$ where $r$ is the radius from the maximum PL intensity. \\\\\n\\indent To model the scanning part over the junction (see Fig.4 in the main text), we numerically solved the 2D classical diffusion equation with the use of the Matlab toolbox \\textit{pdetool} (partial differential equation toolbox). We set two zones having the same diffusion coefficient of $1cm^2\/s$ but different lifetime of $\\tau_1=100ps$ and $\\tau_2=400ps$. To take into account the contribution of dark and bright states, different radiative efficiency $\\eta$ are taken into account with a ratio of $\\eta_1\/\\eta_2=10$. The laser excitation spot is assumed gaussian, and we model the PL profile for each position of the laser spot. The PL profiles $I_{PL}(r)$ are fitted by a gaussian distribution $I_{PL}(r)=A.exp(-r^2\/w^2)$ where $w^2$ reflects the broadening of the PL profile (i.e. diffusion). We used Neumann boundary conditions with $dI_{PL}(r)\/dr=0$ to simulate ideal interface with no recombination.\n\\\\\n\\subsection{Electron microscopy}\n\nFor STEM investigation, the samples were transferred to Quantifoil$^{tm}$ grids  using PMMA assisted transfer protocol. The High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image was acquired with a Thermofisher Talos 200X microscope operated at 200 kV. \n\n\\begin{figure*}\n\\includegraphics[width=0.8\\linewidth]{figS3}\n\\caption{\\label{fig:figS3} \\textbf{Characterization of lateral heterostructures }(a,b) Optical microscopy image of as grown MoSe$_2$-WSe$_2$ LHs (c,d) AFM height and phase image of as grown LHs. The height profile in the inset of (c) shows monolayer thickness of 0.8~nm. }\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\linewidth]{FigS4}\n\\caption{\\label{fig:figS4} \\textbf{Examples of PL imaging in different sample areas}. We show more examples of typical PL emission spots in MoSe$_2$ (left), on the junction between the two materials (middle), and in WSe$_2$ (right) for two different sample temperatures as marked on the panels for each line.}\n\\end{figure*}\n\n \\textbf{Acknowledgements}  \\\\\nToulouse acknowledges  partial funding from ANR IXTASE, ANR HiLight, NanoX project 2DLight, the Institut Universitaire de France, and the EUR grant ATRAP-2D NanoX ANR-17-EURE-0009 in the\nframework of the \"Programme des Investissements d'Avenir\", the Institute of quantum technology in Occitanie IQO and a UPS excellence PhD grant.\nGrowth of hexagonal boron nitride crystals was supported by JSPS KAKENHI (Grants No. 19H05790, No. 20H00354 and No. 21H05233). \nThe Jena group received financial support of the Deutsche Forschungsgemeinschaft (DFG) through a research infrastructure grant INST 275\/257-1 FUGG, CRC 1375 NOA (Project B2), SPP2244 (Project TU149\/13-1) as well as DFG grant TU149\/16-1. This project has also received funding from the joint European Union's Horizon 2020 and DFG research and innovation programme FLAG-ERA under grant TU149\/9-1.\n (+) D.B.,  I.P.\\ and Z.G.\\ contributed equally to this work. ($\\ddagger$) T.L. now at Karlsruhe Institute of Technology, Laboratory for Electron Microscopy, 76131 Karlsruhe, Germany.\n \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFixing to the smooth, covariant, Coulomb and Landau gauges can be useful for lattice field theorists. The matching of lattice results to continuum perturbation theory, of particular interest for vertex functions, demands the use of a fixed gauge, usually Landau. Coulomb gauge has other benefits; it can be used for the creation of gauge-fixed wall source fermion propagators or for measurements of the static quark potential.\n\nThe Landau and Coulomb gauges are two of the easiest (that have useful continuum analogs) to implement on the lattice as their gauge conditions can be defined as the minimisation of a functional. Such a description allows us to define a smooth deformation of the fields in which we can systematically move toward a local minimum of the functional. This is the aim of many numerical optimisation techniques and several methods are available \\cite{Mandula:1987rh,Mandula:1990vs,deForcrand:1989im}.\n\nConjugate gradient methods were first introduced in \\cite{zbMATH03075195} to solve systems of equations and were generalised to the associated problem of solving non-linear, non-quadratic functions in \\cite{Polyak196994,Fletcher01011964,Polak1969}. We show how a non-linear conjugate gradient method can be used with Fourier Acceleration to create a very fast gauge fixing procedure for both Coulomb and Landau gauge with exact and approximate definitions of the gauge field. We also comment on the ineffectiveness of fixing to these gauges using a so-called ``optimal'' steepest descent (SD) routine that tunes the step-size to best minimise the functional at each iteration.\n\n\\section{Fields and Definitions}\n\nLattice gauge fields (the $A_\\mu$'s) of an $N_d$-dimensional, strongly interacting gauge theory are formally defined by their path-ordered parallel transport matrices (known as lattice links),\n\\begin{equation}\n U_{\\mu}\\left(x+a\\frac{\\hat{\\mu}}{2}\\right) = P[e^{ig_0 \\int_{x}^{x+a\\hat\\mu} A_\\mu( x )dx_\\mu }] \\approx e^{iag_0A_\\mu\\left(x+a\\frac{\\hat\\mu}{2}\\right)},\n\\end{equation}\nand are well described as living halfway between the sites $x$ and $x+a\\hat{\\mu}$ of the lattice, where $a$ is the lattice spacing, $g_0$ is the bare coupling and $\\mu$ is the field's polarisation direction. The links are elements of the group $\\textrm{SU}(N_C)$, the fields are elements of the Lie Algebra $\\mathfrak{su}(N_C)$ with $N_C$ being the number of color charges of the theory.\n\nThe fields cannot be transformed in a gauge-covariant manner directly in lattice simulations, instead they have to be altered by gauge transformation,\n\\begin{equation}\\label{Eq:gtrans}\n U^{(g)}_\\mu\\left( x + a \\frac{\\hat\\mu}{2}\\right) = g(x)U_\\mu\\left( x + a \\frac{\\hat\\mu}{2}\\right)g(x+a\\hat\\mu)^{\\dagger}.\n\\end{equation}\nSuch a transformation preserves the gauge invariance of the action. \n\nThe techniques we consider in this work for fixing the gauge (gradient descent methods) require access to the gauge fields directly. There is some freedom in the choice of field definition: a common one is what we call the ``Hermitian Projection'' and is accurate only up to terms of $O\\left(A_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)^3\\right)$,\n\\begin{equation}\n\\begin{aligned}\n ag_0A_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)= &\\frac{U_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)-U_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)^{\\dagger}}{2i} \\\\\n &-\\frac{1}{2iN_C}\\textrm{Tr}\\left[U_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)-U_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)^{\\dagger} \\right]\\cdot I_{N_c\\times N_c}.\n \\end{aligned}\n\\end{equation}\nSome authors have also investigated the use of exact logarithmic fields \\cite{Furui:1998cg,Ilgenfritz:2010gu}. These will be used here to illustrate the general performance properties of our Fourier Accelerated Conjugate Gradient algorithm.\n\nCayley-Hamilton theorem states that every matrix is a solution of its own characteristic equation. In terms of our links the following finite expansion can be used (where the f's are necessarily complex and $I$ is the identity matrix) \\cite{PhysRevD.69.054501},\n\\begin{equation}\n U_{\\mu}\\left(x+a\\frac{\\hat{\\mu}}{2}\\right) = f_0\\cdot I_{N_C\\times N_C} + f_1A_\\mu\\left( x + a\\frac{\\hat{\\mu}}{2}\\right) + \\cdots +f_{N_C-1}A_\\mu\\left( x + a\\frac{\\hat{\\mu}}{2}\\right)^{N_C-1}.\n\\end{equation}\n\nThe f's are the solution to the following Vandermonde equation,\n\\begin{equation}\\label{chap5:mat:vandermonde}\n\\begin{pmatrix}\n1&q_0&q_0^{2}&\\cdots&q_0^{N_C-1} \\\\\n1&q_1&q_1^{2}&\\cdots&q_1^{N_C-1} \\\\\n\\vdots&\\vdots&\\vdots&\\cdots&\\vdots \\\\\n1&q_{N_C-1}&q_{N_C-1}^{2}&\\cdots&q_{N_C-1}^{N_C-1}\n\\end{pmatrix}\n\\begin{pmatrix}\nf_0 \\\\\nf_1 \\\\\n\\vdots \\\\\nf_{N_C-1} \\end{pmatrix}\n=\\begin{pmatrix}\n e^{iq_0} \\\\\n e^{iq_1} \\\\\n \\vdots \\\\\n e^{iq_{N_C-1}}\n \\end{pmatrix}.\n\\end{equation}\nThe $q$'s are the Eigenvalues of $A_\\mu\\left(x+a\\frac{ \\hat\\mu}{2} \\right)$ and the $e^{iq}$'s are the Eigenvalues of the link matrix $U_\\mu\\left(x+a\\frac{ \\hat\\mu}{2} \\right)$. For our logarithm, we compute the $e^{iq}$'s from the characteristic equation of the link matrix $U_\\mu\\left(x+a\\frac{ \\hat\\mu}{2} \\right)$ and take their complex argument to define the $q$'s on the principal branch. The solution to general Vandermonde systems can become numerically unstable, but for $\\textrm{SU}(3)$ well-behaved functions for the f's in terms of the q's can be used \\cite{PhysRevD.69.054501}. This leads to a computationally efficient and numerically stable definition for the principal logarithm of $\\textrm{SU}(3)$ matrices,\n\\begin{equation}\\label{eq:logroj}\nag_0A_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right) = \\frac{f_2^*U_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)-f_2U^{\\dagger}_{\\mu}\\left( x+a\\frac{\\hat\\mu}{2}\\right)-2i\\Im{\\left(f_0 f_2^*\\right)}\\cdot I_{3\\times3}}{2i\\Im{\\left(f_1 f_2^*\\right)}}.\\nonumber\n\\end{equation}\n\nWe define the lattice finite difference of the gauge fields by,\n\\begin{equation}\n a\\Delta_\\mu A_\\mu(x) = \\sum_\\mu \\left[A_\\mu\\left( x+ a\\frac{\\hat\\mu}{2} \\right) - A_\\mu\\left( x - a\\frac{\\hat\\mu}{2} \\right)\\right].\n\\end{equation}\n\nIt will be useful to monitor our algorithm (at it's $n^\\textrm{th}$ iteration) by defining the quantity ($V$ is the lattice volume),\n\\begin{equation}\n\\Theta^{(n)} = \\frac{1}{N_c V}\\sum_x \\textrm{Tr}\\left[ \\left( a\\Delta_\\mu ag_0 A_\\mu^{(n)}(x)\\right)^2 \\right].\n\\end{equation}\n\n\\section{Fourier Accelerated Steepest Descent}\n\nThe continuum gauge condition for Landau gauge (and Coulomb with $\\mu$ only in the spatial directions) is $\\partial_\\mu A_\\mu(x)=0$, and the lattice analog of this is $a\\Delta_\\mu ag_0 A_\\mu(x) = 0$. This occurs at the minimum of the gauge fixing functional,\n\\begin{equation}\\label{Eq:functional}\n F(U) = \\frac{1}{N_C N_d V} \\sum_{x,\\mu} \\textrm{Tr}\\left[ A_\\mu\\left( x + a\\frac{\\hat\\mu}{2} \\right)^2\\right].\n\\end{equation}\n\nA general steepest descent step that drives the gauge fields toward a local minimum of the functional (and thus reduces $a\\Delta_\\mu ag_0A_\\mu(x)$ to 0) is,\n\\begin{equation}\n g(x) = e^{-i\\alpha a\\Delta_\\mu ag_0 A_\\mu(x)},\n\\end{equation}\nfollowed by a gauge transformation of the links (Eq.\\ref{Eq:gtrans}). The parameter $\\alpha$ is a small, fixed, positive tuning parameter. This procedure is repeated until some convergence criterion is met.\n\nThe authors of \\cite{PhysRevD.37.1581} suggested that a re-scaling of the Eigenvalues of the Abelian Laplacian operator ($\\Delta^{2}A_\\mu(x)$) in momentum space was sufficient to reduce some of the critical slowing down in each steepest descent iteration, this is effectively a preconditioner of the gradient. A Fourier Accelerated Steepest Descent (FASD) iteration is,\n\\begin{equation}\n g(x) = e^{\\tilde{F} \\;\\frac{-i\\alpha \\hat{p}^2_{\\textrm{Max}}}{V\\hat{p}^2} \\;F a\\Delta_\\mu a g_0 A_\\mu(x) }.\n\\end{equation}\nFollowed by a gauge transformation. $p^2_{\\text{Max}}$ takes the values $4N_d$ and $4(N_d-1)$ for fixing to Landau and Coulomb gauge respectively. $F$ and $\\tilde{F}$ are forward and backward Fast Fourier Transforms (FFTs using FFTW \\cite{FFTW05} in our implementation). We use the definition of lattice momentum (the $n_\\mu$'s are Fourier modes, $L_\\mu$ is the lattice length in the $\\mu$ direction),\n\\begin{equation}\n\\hat{p}^2 = 2\\left( N_d - \\sum_\\mu \\cos\\left( \\frac{2\\pi n_\\mu}{L_\\mu} \\right)\\right).\n\\end{equation}\nThe rest of this paper will present results from gauge group $\\textrm{SU}(3)$, $N_d=4$ dimensional lattice gauge theory. In particular, $\\text{N}_\\text{f}=2+1$, Iwasaki gauge, DWF configurations are used. We will consider three different lattice volumes; the $32^3\\times64$ data has a lattice spacing of $a^{-1}\\approx 2.38$ GeV, the $24^3\\times64$ and $16^3\\times32$ configurations' $a^{-1}\\approx 1.79$ GeV \\cite{RBC:2014tka}.\n\n\\subsection{Optimal FASD}\n\nMany popular techniques for fixing the gauge require some form of tuning to be effective, with FASD and over-relaxation being examples. The tuning parameter can depend on the bare coupling, the gauge action, the field definition and the functional being used. If the tuning parameter is too large the routine will not converge, too small and the routine will have poor performance. The optimal FASD routine, and the Fourier Accelerated Conjugate Gradient algorithm discussed later do not need to be tuned.\n\nWe can define a so-called ``optimal'' steepest descent routine by choosing the parameter $\\alpha_n$ to be the one which best minimises the gauge functional (Eq.\\ref{Eq:functional}) at each iteration of the algorithm. Evaluating the optimal parameter $\\alpha_n$ exactly (by e.g. a binary search) would be prohibitively expensive as each evaluation of what we call a probe $\\alpha^{\\prime}$ requires both a lattice-wide gauge transformation and subsequent functional evaluation. The approach we take instead is to evaluate a selection of probe $\\alpha^{\\prime}$'s and create a cubic spline interpolation of the probes for which we can solve for the minimum and hence provide a good approximation to the actual minimum. Such a process is called an approximate line search and will be integral to the non-linear conjugate-gradient method discussed in the next section.\n\n\\begin{figure}[h!]\n\\centering\n\\subfloat[Cubic spline interpolation.]\n  {\n  \\includegraphics[scale=0.26]{FASD_splines.eps}\n  \\label{Fig:spline_interp}\n  }    \n\\quad\n\\subfloat[Spline minimum history]\n  {\n  \\includegraphics[scale=0.26]{FASD_spline.eps}\n  \\label{Fig:FASD_spline}\n  }\n\\caption[$\\:$ ]{Fig.\\ref{Fig:spline_interp} shows a cubic spline interpolation and the computation of the spline minimum for a series of probe $\\alpha^{\\prime}$s. Fig.\\ref{Fig:FASD_spline} shows the history of the spline minimum per iteration of the optimal FASD from an example $16^3\\times32$ configuration converged to a high accuracy of $\\Theta^{(n)}<10^{-14}$.}%\n\\end{figure}\nFig.\\ref{Fig:spline_interp} shows the cubic spline interpolation for a group of probe evaluations for an FASD step. Fig.\\ref{Fig:FASD_spline} illustrates the typical history of the cubic spline minimum per iteration of the optimal FASD Landau gauge fixing routine, taken from a single $16^3\\times32$ configuration. We see that after the first 50 or so iterations the best $\\alpha_n$ is roughly constant at around $0.08$ which is consistent with the best tuning parameter of similar configurations found in the literature \\cite{PhysRevD.37.1581,Ilgenfritz:2010gu}. The approximate optimal $\\alpha_n$ oscillates each iteration around a central value, illustrating the routine correcting itself from an over or under-estimate of the true optimal $\\alpha_n$.\n\nAs the optimal FASD shows a long plateau of the best $\\alpha_n$ with little variation the cost of the line search is not worthwhile for the FASD algorithm. This is because it is roughly $N\\times$ the cost of one fixed-$\\alpha$ iteration ($N$ is the number of probe $\\alpha^{\\prime}$s) and we need at least $N=3$ to compute a cubic spline interpolant. If the number of iterations is not reduced by a factor of $N$ the optimal FASD is less optimal (in the sense of time taken to reach a desired precision) than the fixed-$\\alpha$ FASD, which we find to be the case.\n\n\\section{Conjugate Directions in Gauge Fixing}\n\nA detailed description of the of the Fourier Accelerated Conjugate Gradient (FACG) algorithm is presented in Alg.1 and Alg.2 of \\cite{Hudspith2015115} and will not be reproduced here.\n\n\\begin{figure}[ht!]\n\\centering\n\\subfloat[FACG spline minimum history]\n{\n\\includegraphics[scale=0.25]{FACG_spline.eps}\n\\label{Fig:spline_hist}\n}\n\\quad\n\\subfloat[FACG vs. FASD comparison]\n{\n\\includegraphics[scale=0.25]{FACG_v_FASD_acc_iters.eps}\n\\label{Fig:comparison}\n}\n\\newline\n\\subfloat[Improvement over the FASD, logarithmic fields]\n{\t\n  \\includegraphics[scale=0.26]{scaling_log.eps}\n  \\label{Fig:log_scalings}\n}\n\\caption[]{Fig.\\ref{Fig:spline_hist} shows the optimal $\\alpha_n$'s of the FACG algorithm for an example $16^3\\times32$ configuration being fixed to $\\Theta^{(n)}<10^{-14}$. Fig.\\ref{Fig:comparison} shows the number of iterations required for the (fixed-$\\alpha$) FASD and FACG algorithms to reach the same accuracy for the same configuration, using the Hermitian Projection definition of the gauge fields. Fig.\\ref{Fig:log_scalings} illustrates a measure of the effective speed up of our FACG compared to our FASD routine for the logarithmic field definition for 3 different lattice volumes.}\n\\end{figure}\n  \nFig.\\ref{Fig:spline_hist} shows a histogram of $\\alpha_n$ taken from the probes $(0.0,0.15,0.3,0.45,0.6)$ for every iteration of the configuration used in the previous section. We can see that most of the optimal $\\alpha_n$'s lie in the range $[0,0.3]$ and so we choose the three probe evaluations $(0.0,0.15,0.3)$ for our routine. The probe at $0.0$ is very beneficial in bounding the minimum and is cheap to evaluate as no gauge transformations are needed. If the spline minimum is negative, we set $\\alpha_n$ to $0.0$ (i.e. we generate a new descent direction) or if it is greater than $0.3$ we set it to $0.3$ knowing that the algorithm will accommodate for a sub-optimal step. \n\nWe provide a comparison of the FACG algorithm (using the 3 probes) to the fixed-$\\alpha$ FASD algorithm in Fig.\\ref{Fig:comparison} showing the number of iterations required to reach $\\Theta^{(n)}<10^{-14}$ for a single configuration, although this behaviour is ubiquitous. We see that for this configuration there is a factor of around $10$ reduction in the number of iterations to reach a high level of convergence. As mentioned for the optimal FASD routine, if the factor of reduction in the number of iterations is greater than the number of probe evaluations ($N$) then the routine is beneficial. \n\nIn \\cite{Hudspith2015115}(Fig.1) we show for the Hermitian Projection definition of the fields a factor of 2 to 4 improvement in the average time taken to converge to a high fixed accuracy was obtained for the FACG routine compared to the FASD. In Fig.\\ref{Fig:log_scalings}, we show that such speedup is also available for the logarithmic field definition for the three volumes considered. The measure of improvement was taken to be the time taken for 25 randomly gauge transformed copies of the same configuration at each volume for the FASD algorithm divided by the time taken for the FACG algorithm.\n\n\\section{Conclusions}\n\nWe have presented results using the FACG algorithm for fixing to Landau and Coulomb gauge. It is shown to be faster than our implementation of the FASD algorithm for both the logarithmic and Hermitian Projection field definitions. We have illustrated why an optimal FASD algorithm does not provide a faster method than the fixed-$\\alpha$ procedure, and have shown why the literature value of $\\alpha=0.08$ is often a good choice for the tuning parameter, although the exact optimal tuning parameter is dependent on several factors. Unlike the fixed-$\\alpha$ FASD or over-relaxation methods, the FACG does not require one to tune a parameter.\n\nThe FACG algorithm converges in fewer iterations than the FASD, and requires only one set of FFTs per iteration; if FFTs are particularly costly this method will show greater improvement over the FASD than presented here. The FACG method (as it is a gradient descent method) allows for options where standard relaxation techniques cannot be used; different gauge field definitions or alternate functionals can be accommodated easily.\n\n\\section{Acknowledgements}\n\nThe results produced in this work were generated on Columbia University's cluster ``CUTH'' using the package GLU (\\href{https:\/\/github.com\/RJhudspith\/GLU}{https:\/\/github.com\/RJHudspith\/GLU}). RJH would like to thank R. Lewis. RJH is supported by the Natural Sciences and Engineering Research Council of Canada.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHindman's Theorem states that, in any finite coloring of the integers, some color contains an infinite set and all its finite sums.  Hindman's original proof \\cite{hindman74} is quite complicated; fortunately, there are both simpler combinatorial arguments \\cite{baumgartner74,towsner09:hindman_simple} and an elegant proof based on the topology of ultrafilters (see, for instance, \\cite{comfort77}).\n\nStrikingly, the ultrafilter argument gives, sometimes with little additional work, various strengthenings of the theorem for which combinatorial proofs are iether much harder, or not known to exist.  (\\cite{hindman98} gives a thorough exploration of many uses of ultrafilters in this context.)  One such strengthening promises, not a single monochromatic infinite set and its finite sums, but a tree of such sets in which there are ``many'' possible first elements, for each such first element ``many'' possible second elements, and so on.\n\nIn this paper we give the first combinatorial proof of this theorem, modeled on Baumgartner's proof of the ordinary Hindman's Theorem.  The key idea is the use of approximate ultrafilters, as introduced by Hirst \\cite{hirst04}---countable collections of sets of integers which nonetheless contain enough information to complete the proof.  The proof here is modeled on our related proof of the ordinary Hindman's Theorem \\cite{towsner09:hindman_unwind}\\footnote{Indeed, we originally found a proof quite similar to that one, and only subsequently found proof in the style of Baumgartner which we present here.  This proof is slightly more elegant, and we hope that it will shed some light on the relationship between Baumgartner's proof of the ordinary Hindman's Theorem and the ultrafilter proof.}.\n\nThe reverse mathematical strength of even the ordinary Hindman's Theorem is open; bounds are given in \\cite{blass87}, and the gap between the lower and upper bounds on reverse mathematical strength there have not been improved.  The proof given here is entirely within the bounds of second order arithmetic, but well above their upper bounds; no lower bound for the Dense Hindman's Theorem is known besides the obvious one, that any lower bound for the ordinary Hindman's Theorem must also bound the Dense Hindman's Theorem.\n\nWe thank Mathias Beiglb\\\"ock for bringing this question to our attention, and for many discussions about the mathematics around Hindman's Theorem.\n\n\\section{General Definitions}\nThroughout this paper, variables denoted by lowercase letters will typically be natural numbers, except for $b$, which will always belong to $\\{-1,1\\}$.  Variables denoted by upppercase letters will be sets of natural numbers, and variables denoted by caligraphic letters ($\\mathcal{U}$, $\\mathcal{F}$, etc.) will be sets of sets of natural numbers.  In fact, it would cause no harm to assume that all sets of sets of natural numbers appearing in this paper are countable, and therefore to code them using sets of natural numbers.  The one exception is the property $\\mathfrak{P}$, which represents the set of sets of natural numbers satisfying some shift-invariant divisible'' property, such as the infinite sets, the piecewise syndetic sets, or the sets of positive upper Banach density.  (The notion of divisibility is introduced in \\cite{glasner80}.\n\nWe adopt the notation that if $X$ is a set of integers, $1\\cdot X=X$ and $-1\\cdot X=\\mathbb{N}\\setminus X$.  We write $X-n$ for $\\{m\\mid m+n\\in X\\}$.\n\n\\begin{definition}\n  Let $\\mathfrak{P}$ be a collection of sets of integers such that\\footnote{To keep our promise that the proof go through in second order arithmetic, we should insist that $\\mathfrak{P}$ be given by some arithmetic formula; this includes all the examples given.}:\n  \\begin{itemize}\n  \\item $\\mathbb{N}\\in\\mathfrak{P}$\n  \\item $\\emptyset\\not\\in\\mathfrak{P}$\n  \\item If $X\\subseteq Y$ and $X\\in\\mathfrak{P}$ then $Y\\in\\mathfrak{P}$\n  \\item If $X_0\\cup X_1=X$ and $X\\in\\mathfrak{P}$ then either $X_0\\in\\mathfrak{P}$ or $X_1\\in\\mathfrak{P}$\n  \\item For any $X$ and any $n$, $X\\in\\mathfrak{P}$ iff $X-n\\in\\mathfrak{P}$\n  \\end{itemize}\n\n  We say a collection $\\mathcal{U}$ of sets of integers has the $\\mathfrak{P}$-finite intersection property ($\\mathfrak{P}$-fip) if for every finite collection $\\mathcal{F}\\subseteq\\mathcal{U}$, \n\\[\\bigcap_{S\\in \\mathcal{F}}S\\in\\mathfrak{P}.\\]\n\\end{definition}\n\nNatural examples of such properties $\\mathfrak{P}$ include:\n\\begin{itemize}\n\\item $\\mathfrak{P}$ is the collection of infinite sets\n\\item $\\mathfrak{P}$ is the collection of sets with positive upper Banach density\n\\item $\\mathfrak{P}$ is the collection of piecewise syndetic sets\n\\item $\\mathfrak{P}$ is the collection of sets $X$ such that $\\sum_{x\\in X}1\/x=\\infty$\n\\end{itemize}\nProperties satisfying all but the final condition, that $X\\in\\mathfrak{P}$ iff $X-n\\in\\mathfrak{P}$, are called \\emph{divisible} \\cite{glasner80}.\n\n\\begin{definition}\nLet $\\mathcal{U}$ be a countable collection of sets of integers\\footnote{None of our arguments would change if uncountable collections---say, true ultrafilters---are allowed.  However we wish to emphasize that none of our arguments will require more than countable collections}.  We write $X\\tilde\\in\\mathcal{U}$ if there is a finite $\\mathcal{F}\\subseteq\\mathcal{U}$ such that $\\bigcap_{S\\in\\mathcal{F}}S\\subseteq X$.\n  \n  We say $\\mathcal{U}$ is a $\\mathfrak{P}$-semigroup if $\\mathcal{U}$ satisfies $\\mathfrak{P}$-fip, $\\mathcal{U}$ is closed under finite intersections, and whenever $X\\in\\mathcal{U}$, there is a $Y\\tilde\\in\\mathcal{U}$ such that $X-n\\tilde\\in\\mathcal{U}$ for each $n\\in Y$.\n\\end{definition}\nThere are two useful ways to view $\\mathfrak{P}$-semigroups.  The first is to observe that every $\\mathfrak{P}$-semigroup represents a closed semigroup in the space of ultrafilters on the integers (namely, the collection of ultrafilters extending $\\mathcal{U}$).\n\nThe second is to recall that an IP set is a set $S$ such that there is an infinite $T\\subseteq S$ all of whose finite sums also belong to $S$.  Then the collection $\\{S-n\\mid n\\in FS(T)\\}$ (where $FS(T)$ is the finite sums from $T$) is a canonical example of a $\\mathfrak{P}$-semigroup where $\\mathfrak{P}$ is the collection of infinite sets.  The notion of a $\\mathfrak{P}$-semigroup generalizes an IP set in two directions: first, it allows for arbitrary $\\mathfrak{P}$.  (Note that the appropriate requirement is \\textit{not} that $T\\in\\mathfrak{P}$; rather, the requirement is that there are many possible choices for the infinite set $T$---indeed, the an infinite tree of such sets, with the number of possible branches at each level belonging to $\\mathfrak{P}$.  This discussion will be made precise when we state the main theorem below.)  Second, if we have an infinite descending sequence of IP sets $S_1\\supseteq S_2\\supseteq\\cdots S_n\\supseteq$, their intersection may well be $0$.  However the union of the corresponding $\\mathfrak{P}$-semigroups is still a $\\mathfrak{P}$-semigroup.  So $\\mathfrak{P}$-semigroups also generalize IP sets by accomodating the result of infinitely many successive refinements of an IP set.\n\nThe argument here is very similar to a proof using ultrafilters, but we emphasize that the $\\mathfrak{P}$-semigroups appearing in the proof are much simpler objects: they are countable collections, built with no use of the axiom of choice.\n\nThe following lemma is our essential building block:\n\\begin{lemma}\n  Let $\\mathcal{U}$ satisfy $\\mathfrak{P}$-fip, and let $A$ be a set of integers.  Then either $\\mathcal{U}\\cup\\{A\\}$ or $\\mathcal{U}\\cup\\{-1\\cdot A\\}$ satisfies $\\mathfrak{P}$-fip.\n\\end{lemma}\n\\begin{proof}\n  Suppose neither collection satisfies $\\mathfrak{P}$-fip.  Then choose finite sets $\\mathcal{F},\\mathcal{F}'\\subseteq \\mathcal{U}$ such that\n$\\bigcup_{S\\in\\mathcal{F}}S\\cap A\\not\\in\\mathfrak{P}$\nand\n$\\bigcup_{S\\in\\mathcal{F}'}S\\cap (-1\\cdot X)\\not\\in\\mathfrak{P}$.\nThen\n\\[\\bigcap_{S\\in\\mathcal{F}\\cup\\mathcal{F'}}S\\subseteq (\\bigcup_{S\\in\\mathcal{F}}S\\cap X)\\cup(\\bigcup_{S\\in\\mathcal{F}'}S\\cap (-1\\cdot X)).\\]\nBut this is impossible, since $\\bigcap_{S\\in\\mathcal{F}\\cup\\mathcal{F'}}S\\in\\mathfrak{P}$ must hold.\n\\end{proof}\n\n\n\\section{Dense Hindman's Theorem}\n\\begin{lemma}\n  If $\\mathcal{U}$ is a $\\mathfrak{P}$-semigroup, $A$ is a set, $S$ is a set with the property that\n\\[\\mathcal{U}\\cup\\{-1\\cdot(A-n)\\mid n\\not\\in S\\}\\]\nsatisfies $\\mathfrak{P}$-fip, and $\\mathcal{U}\\cup\\{S-n\\mid n\\in S\\}$ does not satisfy $\\mathfrak{P}$-fip, then there is a finite $F\\subseteq S$ and a $Y\\tilde\\in\\mathcal{U}$ such that $0\\in Y$ and $\\mathcal{U}\\cup\\{\\bigcup_{n\\in F}-1\\cdot(A-n-m)\\mid m\\in Y\\}$ satisfies $\\mathfrak{P}$-fip.\n\\label{3.1}\n\\end{lemma}\n\\begin{proof}\n  Since $\\mathcal{U}\\cup\\{S-n\\mid n\\in S\\}$ does not satisfy $\\mathfrak{P}$-fip, let $X\\tilde\\in\\mathcal{U}$ and $F\\subseteq S$ be such that $X\\cap\\bigcap_{n\\in F}S-n\\not\\in\\mathfrak{P}$.  Let $Y$ be the set of $n$ such that $X-n\\tilde\\in\\mathcal{U}$; clearly $0\\in Y$ since $X\\tilde\\in\\mathcal{U}$.\n\nWe claim that $\\mathcal{U}\\cup\\{\\bigcup_{n\\in F}-1\\cdot(A-n-m)\\mid m\\in Y\\}$ satisfies $\\mathfrak{P}$-fip.  Let $Z\\tilde\\in\\mathcal{U}$ and $G\\subseteq Y$ be finite.  Let $Z'\\tilde\\in\\mathcal{U}$ be such that $Z-n\\tilde\\in\\mathcal{U}$ for each $n\\in Z'$; then $Z'\\cap\\bigcap_{m\\in G}\\bigcup_{n\\in F}-1\\cdot(S-n-m)\\tilde\\in\\mathcal{U}$.  If we choose any $k$ in this set, we have $Z-k\\cap\\bigcap_{m\\in G}\\bigcup_{n\\in F}-1\\cdot (A-k-n-m)\\in\\mathfrak{P}$, and therefore $Z\\cap\\bigcap_{m\\in G}\\bigcup_{n\\in F}-1\\cdot (A-n-m)\\in\\mathfrak{P}$.\n\\end{proof}\nIn particular, observe that if\n\\[\\mathcal{U}\\cup\\{A-n\\mid n\\in S\\}\\]\nsatisfies $\\mathfrak{P}$-fip but whenever $m\\not\\in S$,\n\\[\\mathcal{U}\\cup\\{A-n\\mid n\\in S\\}\\cup\\{A-m\\}\\]\ndoes not satisfy $\\mathfrak{P}$-fip then $S$ satisfies the condition of this lemma.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{definition}\n  We say $A$ is \\emph{large} relative to a $\\mathfrak{P}$-semigroup $\\mathcal{U}$ if whenever $\\mathcal{V}$ is a $\\mathfrak{P}$-semigroup extending $\\mathcal{U}$, $\\mathcal{V}\\cup\\{A\\}$ satisfies $\\mathfrak{P}$-fip.\n\\end{definition}\n\n\\begin{lemma}\n  If $C=C_1\\cup\\cdots\\cup C_n$ and $C$ is large relative to $\\mathcal{U}$, there is a $\\mathcal{V}$ extending $\\mathcal{U}$ and an $i$ such that $C_i$ is large for $\\mathcal{V}$.\n\\end{lemma}\n\\begin{proof}\n  Applying induction, it suffices to consider the case $n=2$.  If $C_1$ is large relative to $\\mathcal{U}$ then $1$ and $\\mathcal{U}$ suffice.  Otherwise there is a $\\mathcal{V}$ extending $\\mathcal{U}$ such that $\\mathcal{V}\\cup\\{C_1\\}$ fails to satisfy $\\mathfrak{P}$-fip, and therefore $C_2$ is large relative to $\\mathcal{V}$.\n\\end{proof}\n\n\\begin{theorem}\n  Let $\\mathcal{U}$ be a $\\mathfrak{P}$-semigroup and $A$ a set.  If $A$ is large for $\\mathcal{U}$ then there is a $\\mathcal{V}$ extending $\\mathcal{U}$ such that the set of $n\\in A$ for which $A\\cap A-n$ is large for $\\mathcal{V}$ belongs to $\\mathfrak{P}$.\n\\end{theorem}\n\\begin{proof}\nGiven $\\mathcal{U}$, expand to some $\\mathfrak{P}$-semigroup $\\mathcal{V}$ such that for every finite set $F$, either $\\bigcap_{n\\in F}A-n$ is large for $\\mathcal{V}$ or $\\bigcup_{n\\in F}-1\\cdot(A-n)\\tilde\\in\\mathcal{V}$.\n\nLet $S$ be a maximal set of $n$ so that $0\\in S$, $F\\subseteq S$ implies $\\bigcap_{n\\in F}A-n$ is large for $\\mathcal{V}$, and $m\\not\\in S$ implies that there is an $F\\subseteq S$ so that $\\bigcap_{n\\in F\\cup\\{m\\}}A-n$ is not large for $\\mathcal{V}$.\n\nThen if $F\\subseteq -1\\cdot S$ is finite, we may choose a $G\\subseteq S$ so that $\\bigcap_{n\\in G\\cup\\{m\\}}A-n$ is not large for $\\mathcal{V}$ for any $m\\in F$.  Then $\\bigcup_{n\\in G\\cup\\{m\\}}-1\\cdot(A-n)\\tilde\\in\\mathcal{V}$ for each $m\\in F$, and since $\\mathcal{V}\\cup\\{\\bigcap_{n\\in G}A-n\\}$ satisfies $\\mathfrak{P}$-fip, it follows that $\\mathcal{V}\\cup\\{-1\\cdot (A-m)\\mid m\\in F\\}$ satisfies $\\mathfrak{P}$-fip.  This holds for every finite $F\\subseteq -1\\cdot S$, so $\\mathcal{V}\\cup\\{-1\\cdot(A-m)\\mid m\\not\\in S\\}$ satisfies $\\mathfrak{P}$-fip.\n\nTherefore, by Lemma \\ref{3.1}, if $\\mathcal{V}\\cup\\{S-n\\mid n\\in S\\}$ does not satisfy $\\mathfrak{P}$-fip then there is a finite set $F\\subseteq S$ and a $\\mathcal{W}$ extending $\\mathcal{V}$ such that $\\bigcup_{n\\in F}-1\\cdot (A-n)\\tilde\\in\\mathcal{W}$.  But this would contradict the fact that $\\bigcap_{n\\in F}A-n$ is large for $\\mathcal{V}$.  So $\\mathcal{V}\\cup\\{S-n\\mid n\\in S\\}$ satisfies $\\mathfrak{P}$-fip, and is therefore a $\\mathfrak{P}$-semigroup.\n\nSince $A$ is large for $\\mathcal{V}$, also $\\mathcal{V}\\cup\\{S-n\\mid n\\in S\\}\\cup\\{A\\}$ satisfies $\\mathfrak{P}$-fip, so in particular, $S\\cap A\\in\\mathfrak{P}$.  Since for each $n\\in S$, $A\\cap A-n$ is large for $\\mathcal{V}$, the claim is proven.\n\\end{proof}\n\n\\begin{definition}\n  For any set of integers $S$, let $FS(S)$ be the collection of finite sums from $S$:\n\\[FS(S)=\\{\\sum_{i\\in T}i\\mid T\\subseteq S, T\\text{ finite}\\}.\\]\n\\end{definition}\n\n\\begin{theorem}\n  Let $\\mathbb{N}=A_1\\cup\\cdots\\cup A_r$.  There is some $i\\leq r$ and a tree $\\mathcal{T}$ of finite sets of integers such that:\n  \\begin{itemize}\n  \\item $\\emptyset\\in\\mathcal{T}$\n  \\item If $F\\in\\mathcal{T}$, $\\{n\\mid F\\cup\\{n\\}\\in\\mathcal{T}\\}$ belongs to $\\mathfrak{P}$\n  \\item If $F\\in\\mathcal{T}$ then $FS(F)\\setminus\\{0\\}\\subseteq A_i$\n \n  \\end{itemize}\n\\end{theorem}\n\\begin{proof}\n  Since $\\mathbb{N}$ is large for the trivial $\\mathfrak{P}$-semigroup $\\{\\mathbb{N}\\}$, we may choose a $\\mathfrak{P}$-semigroup $\\mathcal{U}$ and an $i$ so that $A_i$ is large for $\\mathcal{U}$.  By the preceeding theorem, we obtain a $\\mathfrak{P}$-semigroup $\\mathcal{U}_1$ extending $\\mathcal{U}$ so that $\\{n\\in A_i\\mid A_i\\cap A_i-n\\text{ is large for }\\mathcal{U}'\\}$ belongs to $\\mathfrak{P}$.  We place $\\{n\\}$ in $\\mathcal{T}$ if $A_i\\cap A_i-n$ is large for $\\mathcal{U}'$.\n\n  Suppose we have $F\\subseteq\\mathcal{T}$ and have a $\\mathcal{V}$ so that $A':=\\bigcap_{n\\in FS(F)\\setminus\\{0\\}}A_i-n$ is large for $\\mathcal{V}$.  Again by the preceeding theorem, we may find a $\\mathcal{V}'$ such that $\\{m\\in A'\\mid A'\\cap A'-m\\text{ is large for }\\mathcal{V}'\\}$ belongs to $\\mathfrak{P}$; then we place $F\\cup\\{m\\}$ in $\\mathcal{T}$ for each such $m$, thereby maintaining the inductive assumption.\n\\end{proof}\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe \\emph{$\\mathbb{Z}$-genus} of a knot $K$ in $S^3$, denoted by $g_{\\Z}(K)$, is the minimal genus of an oriented properly embedded locally flat surface $\\Sigma$ in the 4-ball $B^4$ such that the boundary of $\\Sigma$ is $K$ and $\\pi_1 (B^4 \\smallsetminus \\Sigma) \\cong \\mathbb{Z}$. Work of Freedman famously implies that a knot $K$ has $g_{\\Z}(K)=0$ if and only if $K$ has trivial Alexander polynomial~\\cite{Freedman:1982-1, Freedman-Quinn:1990-1, Garoufalidis-Teichner:2004-1}; later work of Feller  generalizes Freedman's result to show that $g_{\\Z}(K)$ is bounded above by half the degree of the Alexander polynomial of $K$~\\cite{Feller:2016-1}. The $\\mathbb{Z}$-genus of $K$ can therefore be thought of as an algebraically controlled upper bound on the topological 4-ball genus of $K$, i.e. the minimal genus of \\textit{any} oriented properly embedded locally flat surface in the 4-ball with boundary $K$ ; see Feller-Lewark for a precise statement of this fact~\\cite{Feller-Lewark:2018-1,Feller-Lewark:2019-1}. Further, we define the \\emph{$($smooth$)$ concordance $\\mathbb{Z}$-genus} of a knot $K$, denoted by $g_{\\Z}^\\mathrm{c}(K)$, to be the minimum value of $g_{\\Z}(J)$ among all knots smoothly concordant to $J$. That is, $$g_{\\Z}^\\mathrm{c}(K):=\\min\\{g_{\\Z}(J)\\mid J\\text{ is smoothly concordant to } K\\} .$$ Observe that a knot $K$ is smoothly concordant to a knot with trivial Alexander polynomial if and only if $g_{\\Z}^\\mathrm{c}(K)=0$, and that  we obtain the following inequalities for any knot $K$ immediately from the definitions:\n$$g^\\mathrm{top}_4(K) \\leq g_{\\Z}^\\mathrm{c}(K) \\leq g_{\\Z}(K),$$\nwhere $g^\\mathrm{top}_4(K)$ denotes the topological $4$-ball genus of $K$.\n\nWe remark that taking connected sums of a smoothly slice knot with nontrivial determinant produces knots with vanishing $g_{\\Z}^\\mathrm{c}$ and arbitrarily large $g_{\\Z}$ (see e.g.\\ Proposition~\\ref{prop:branchedlower}). In particular, this implies that the gap between $g_{\\Z}^\\mathrm{c}$ and $g_{\\Z}$ can be made arbitrarily large.\n In~\\cite{Hedden-Livingston-Ruberman:2012-1}, Hedden-Livingston-Ruberman used the Heegaard Floer correction terms of 3-manifolds to show that there is a topologically slice knot $K$ which is not smoothly concordant to any knot with trivial Alexander polynomial, which in particular implies that $g^\\mathrm{top}_4(K) < g_{\\Z}^\\mathrm{c}(K)$. We further show that the gap between these two invariants can be arbitrarily large.\n\n\\begin{theorem}\\label{thm:main}There exist topologically slice knots with arbitrarily large concordance $\\mathbb{Z}$-genus. More precisely, there exists a topologically slice knot $K_*$ such that for each $n \\in \\mathbb{N}$ its $n$-fold connected self-sum has concordance $\\mathbb{Z}$-genus at least $n$. Furthermore, we can choose  $K_*$ so that it has smooth $4$-ball genus one.\\end{theorem}\n\n\n\n\n\nWe outline the proof of the main result. We first construct a topologically slice knot $K_*$ with smooth 4-ball genus one in Example~\\ref{exl:ourexl}. Let $n$ be a positive integer and $\\Sigma_2(\\#^n K_*)$ be the 2-fold branched cover of $S^3$ branched over $\\#^n K_*$. We then observe that for any nontrivial 5-torsion element $x$ of $H_1(\\Sigma_2(\\#^n K_*))$, we may multiply an appropriate constant to obtain $z = c\\cdot x$ so that the Heegaard Floer $\\overline{d}$-invariant $\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_z)$ is nonvanishing. We use this to show that there is no rational homology cobordism from $\\Sigma_2(\\#^n K_*)$ to any 3-manifold $Y$ with $H_1(Y)$ generated by at most $2n$ elements, and conclude that $g_{\\Z}^\\mathrm{c}(\\#^nK_*)\\geq n$.\n\n\nOn a similar note, Hom~\\cite{Hom:2015-1} showed that there are topologically slice knots with smooth 4-ball genus equal to one and arbitrarily large concordance genus. Recall that the concordance genus is greater than or equal to concordance $\\mathbb{Z}$-genus since the fundamental group of the complement of a pushed in Seifert surface is equal to $\\mathbb{Z}$.\n\n\nTheorem~\\ref{thm:main} is related  to the following difficult problem.\nWe can analogously define the \\emph{topological concordance $\\mathbb{Z}$-genus} of a knot $K$ as $$g_{\\Z}^\\mathrm{t.c.}(K):=\\min\\{g_{\\Z}(J)\\mid J\\text{ is topologically concordant to } K\\} .$$ So for any knot $K$ we have that \n$$g^\\mathrm{top}_4(K) \\leq g_{\\Z}^\\mathrm{t.c.}(K) \\leq g_{\\Z}^\\mathrm{c}(K) \\leq g_{\\Z}(K).$$\n\n\\noindent Livingston~\\cite{Livingston:2004-1} showed that the gap between $g^\\mathrm{top}_4$ and $g_{\\Z}^\\mathrm{t.c.}$ can also be made arbitrarily large by using Casson-Gordon invariants (see Remark~\\ref{rmk:livingston}), but the following remains open.\n\n\\begin{problem}\\label{prob:smoothtop}\nShow that topological $4$-ball genus can be strictly less than topological concordance smooth $4$-ball genus, i.e. \nfind a knot $K$ with $g^\\mathrm{top}_4(K)=g \\geq 1$ such that $K$ is not topologically concordant to any knot $J$ with $g_4(J) = g$, where $g_4(J)$ denotes the smooth $4$-ball genus of $J$.\n\\end{problem}\n\n\nThe following problem about properly embedded surfaces is a priori easier, but is nevertheless still open.\n\\begin{problem}\\label{prob:smoothtopsurface}\nFind an oriented properly embedded locally flat genus $g\\geq 1$ surface $\\Sigma$ in the 4-ball such that $\\Sigma$ is not ambiently isotopic rel boundary to any surface $\\Sigma'$ that can be decomposed into a topological concordance glued to a smoothly embedded surface.\n\\end{problem}\n\n Note that a positive solution to Problem~\\ref{prob:smoothtop} certainly implies a positive solution to Problem~\\ref{prob:smoothtopsurface}. \nWhile we avoid the technical details of topological 4-manifold theory needed to prove this formally, it is known both that every topologically slice knot is topologically concordant to the unknot, which of course has smooth $4$-ball genus equal to zero, and that every topologically embedded disk can be decomposed into a topological concordance glued to the standard slice disk for the unknot.  That is, the $g=0$ case of Problems~\\ref{prob:smoothtop} and~\\ref{prob:smoothtopsurface} is resolved.  \n\n\n\n\\subsection*{Acknowledgments} \n\nThis paper began in conversations during the first author's visit to the School of Mathematics at  Georgia Institute for Technology to speak in the Geometry Topology Seminar; she thanks Miriam Kuzbary and the School as a whole for the invitation and for their generous hospitality. We also thank Peter Feller, Mark Powell, and Charles Livingston for helpful conversations.\n\n\\subsection*{Notation and conventions}\nIn this paper, we work in the smooth category unless specified otherwise. Let $-K$ be the reverse of the mirror image of a knot $K$ and $-Y$ be the manifold $Y$ with reversed orientation. For any given knot $K$ in $S^3$,  the 2-fold branched cover of $S^3$ branched over $K$ is denoted by $\\Sigma_2(K)$, \n and for $n\\in \\mathbb{Z}$, the $3$-manifold obtained by $n$-framed Dehn surgery on $S^3$ along $K$ is denoted by $S^3_n(K)$.\n\n\n\n\\section{Proof of Theorem~\\ref{thm:main}}\\label{sec:pfthm}\nFeller-Lewark defined the \\emph{algebraic genus} of a knot $K$ in~\\cite{Feller-Lewark:2018-1} and further showed that this invariant coincides with the $\\mathbb{Z}$-genus~\\cite{Feller-Lewark:2019-1}. This implies the following key  ingredient for the proof of Theorem~\\ref{thm:main}. For a finite abelian group $G$, we denote by $r(G)$ the so-called ``generating rank'', i.e. the minimal number of generators of $G$.\n\n\\begin{proposition}[Proposition 12 of \\cite{Feller-Lewark:2018-1}]\n\\label{prop:branchedlower} If $\\Sigma_2(K)$ is the 2-fold branched cover of $S^3$ branched over a knot $K$, then\n\\[\\pushQED{\\qed}r(H_1(\\Sigma_2(K)))\\leq 2g_{\\Z}(K).\\qedhere\\]\n\\end{proposition}\n\nThe other ingredient for the proof comes from an analysis of metabolizers for  torsion linking forms. Recall that there is a nonsingular symmetric linking form $$\\lambda \\colon H_1(Y) \\times H_1(Y) \\to \\mathbb{Q}\/\\mathbb{Z},$$ for a rational homology sphere $Y$. If $-Y$ is the 3-manifold obtained by reversing the orientation of $Y$, then the linking form on $H_1(-Y)$ is given by $-\\lambda$. We say that a subgroup $M \\subseteq H_1(Y)$ is a metabolizer if $M = M^{\\perp}$ with respect to $\\lambda$; note that for a metabolizer $M$ the nonsingularity of $\\lambda$ immediately implies that $|M|^2=|H_1(Y)|$. \nIt is well known that if $Y$ bounds a rational homology ball $W$, then the kernel of the inclusion-induced map $H_1(Y) \\to H_1(W)$ is a metabolizer for $(H_1(Y),\\lambda)$.  \n\nFor a finite abelian group $G$ and a prime $p$, let $G_p$ denote the $p$-primary subgroup of $G$, i.e.\\ the subgroup consisting of all elements of $G$ which are $p^k$-torsion for some positive integer $k$. \n\n\n\n\\begin{lemma}\\label{lem:key}\nLet $Y_1$ and $Y_2$ be rational homology 3-spheres and let $p$ be a prime integer. Furthermore, for $i=1,2$, let $\\lambda_i$ be the linking form on $H_1(Y_i)$, and let $M$ be a metabolizer for $(H_1(Y_1\\#-Y_2), \\lambda_1 \\oplus -\\lambda_2)$. \n\nIf $H_1(Y_1)_p \\cong(\\mathbb{Z}\/{p^2}\\mathbb{Z})^{2n}$ and $r(H_1(Y_2)_p) \\leq 2m <2n$ for nonnegative integers $n$ and $m$, then $M \\cap H_1(Y_1)$ contains a subgroup isomorphic to $(\\mathbb{Z}\/p\\mathbb{Z})^{n-m}$.\\end{lemma}\n\n\n\\begin{proof}\nFirst, note that $M$ splits as a direct sum of its $q$-primary components over all prime $q$, each of which is a metabolizer for the $q$-primary component of $H_1(Y_1 \\# -Y_2)$. We can therefore assume without loss of generality that $H_1(Y_1)$ and $H_1(-Y_2)$ are equal to their $p$-primary components, and do so for ease of notation. \n\nFor $i=1,2$, let $\\pi_i \\colon M \\to H_1(Y_i)$ be the restriction of the projection map $$H_1(Y_1 \\# -Y_2) \\cong H_1(Y_1) \\oplus H_1(-Y_2) \\to H_1(Y_i).$$ Observe that $\\Ker (\\pi_2)= M \\cap H_1(Y_1)$, which will be used later. We now argue that $p \\text{Im}(\\pi_2)$ has the property that $-\\lambda_2|_{p \\text{Im}(\\pi_2) \\times p \\text{Im}(\\pi_2)} = 0$. If $b_1, b_2 \\in \\text{Im}(\\pi_2)$, then there exist $a_1, a_2 \\in H_1(Y_1)$ such that $(a_1, b_1), (a_2, b_2) \\in M$, and \n\\[ (\\lambda_1 \\oplus - \\lambda_2)( (a_1, b_1), (a_2, b_2))\n=\\lambda_1(a_1, a_2) - \\lambda_2(b_1, b_2)=0 \\in \\mathbb{Q}\/\\mathbb{Z}.\n\\]\nSince $H_1(Y_1)$ is annihilated by $p^2$, we have that\n\\[-\\lambda_2(pb_1, pb_2)= -p^2 \\lambda_2(b_1, b_2)= -p^2 \\lambda_1(a_1, a_2) =  -\\lambda_1(p^2a_1, a_2) = 0 \\in \\mathbb{Q}\/\\mathbb{Z},\\] as desired. Moreover, since the linking form $-\\lambda_2$ is nonsingular, we have \n\\begin{equation}\\label{eq:1}\n|p \\text{Im}(\\pi_2)| \\leq |H_1(-Y_2)|^{1\/2}.\n\\end{equation}\n\nRecall that since we are assuming $r(H_1(Y_2)) \\leq 2m$ and since $\\text{Im}(\\pi_2)$ is a subgroup of $H_1(-Y_2)$, we have that\n\\[\\text{Im}(\\pi_2) \\cong \\bigoplus_{k=1}^{\\ell}  (\\mathbb{Z}\/ p^k\\mathbb{Z})^{n_k} \n\\quad\\text{and}\\quad\np \\text{Im}(\\pi_2) \\cong \\bigoplus_{k=1}^{\\ell}  ( \\mathbb{Z}\/ p^{k-1}\\mathbb{Z})^{n_k} ,\n\\quad\\text{where}\\quad  \\sum_{k=1}^{\\ell} n_k \\leq 2m.\n\\]\nTherefore, \n\\[ \\frac{|\\text{Im}(\\pi_2)|}{|p \\text{Im}(\\pi_2)|}= \\frac{ p^{\\sum_{k=1}^{\\ell} k n_k}}{p^{\\sum_{k=1}^{\\ell} (k-1)n_k}}= p^{\\sum_{k=1}^{\\ell} n_k}\\ \\leq p^{2m}.\n\\]\nSo, by combining inequality \\eqref{eq:1} with the above inequality, we conclude that $$|\\text{Im}(\\pi_2)| \\leq p^{2m} |H_1(-Y_2)|^{1\/2}.$$\nAlso, recall that $|M| = |H_1(Y_1)|^{1\/2} |H_1(-Y_2)|^{1\/2}$ and $\\Ker(\\pi_2)= M \\cap H_1(Y_1)$. Therefore,\n\\[ |M \\cap H_1(Y_1)|= |\\Ker(\\pi_2)| = \\frac{|M|}{| \\text{Im}(\\pi_2)|} \\geq \n\\frac{|H_1(Y_1)|^{1\/2} |H_1(-Y_2)|^{1\/2}}{p^{2m} |H_1(-Y_2)|^{1\/2}}= p^{2n-2m}.\n\\]\nFinally, since $M \\cap H_1(Y_1)$ is a subgroup of $(\\mathbb{Z}\/p^2\\mathbb{Z})^{2n}$, we conclude that $$M \\cap H_1(Y_1) \\cong (\\mathbb{Z}\/p\\mathbb{Z})^{k_1} \\oplus (\\mathbb{Z}\/p^2\\mathbb{Z})^{k_2}, \\quad\\text{where}\\quad k_1 + 2 k_2 \\geq 2n-2m.$$ \nTherefore,\n\\[ 2n-2m \\leq k_1 + 2 k_2 \\leq 2k_1 + 2k_2,\\] and we conclude that $M \\cap H_1(Y_1)$ contains a subgroup isomorphic to $(\\mathbb{Z}\/p\\mathbb{Z})^{n-m}$.\\end{proof}\n\n\nNow, we collect some necessary background on the Heegaard-Floer correction term $d(Y,\\mathfrak{s})\\in \\mathbb{Q}$ associated to a rational homology sphere $Y$ with a Spin$^c$ structure~$\\mathfrak{s}$~\\cite{Ozsvath-Szabo:2003-2}. We first recall the following definition from~\\cite{Hedden-Livingston-Ruberman:2012-1}.\n\n\\begin{definition}[{\\cite[Definition 2.2 and 3.1]{Hedden-Livingston-Ruberman:2012-1}}]Let $Y$ be a $\\mathbb{Z}\/2\\mathbb{Z}$-homology sphere. For $z\\in H_1(Y)$, let $\\mathfrak{s}_z$ be the unique Spin$^c$ structure of $Y$ which satisfies $c_1(\\mathfrak{s}_z)=2\\widehat{z}\\in H^2(Y)$ where $\\widehat{z}$ is the Poincar\\'{e} dual of $z$. In particular, $\\mathfrak{s}_0$ is the unique Spin structure on $Y$. Lastly, we define $\\overline{d}(Y,\\mathfrak{s}_z):=d(Y,\\mathfrak{s}_z)-d(Y,\\mathfrak{s}_0).$\\end{definition}\n\n\nFurther, the correction term is additive under connected sums \\cite[Theorem 4.3]{Ozsvath-Szabo:2003-1} and is a Spin$^c$ rational homology cobordism invariant \\cite[Theorem 1.2]{Ozsvath-Szabo:2003-1}. Hence we get the following lemma (see e.g.\\ {\\cite[Proposition 2.1]{Hedden-Livingston-Ruberman:2012-1}}).\n\n\\begin{lemma}\\label{lem:dinvt}If $K$ and $J$ are two concordant knots, then there exists a metabolizer $M \\subseteq H_1(\\Sigma_2(K)\\# - \\Sigma_2(J))$ such that for each element $(m_K, m_J) \\in M$ where $m_K \\in H_1(\\Sigma_2(K))$ and $m_J \\in H_1(- \\Sigma_2(J))$, we have that $d(\\Sigma_2(K),\\mathfrak{s}_{m_K})+ d(-\\Sigma_2(J),\\mathfrak{s}_{m_J}) =0.$\\qed\n\\end{lemma}\n\n\n\n\n\n\nWe are now ready to prove the following corollary.\n\n\\begin{corollary}\\label{cor:mainobstr}\nLet $K$ be a knot with $g_{\\Z}^\\mathrm{c}(K) = m$ and $p$ be a prime integer. If $H_1(\\Sigma_2(K))_p \\cong (\\mathbb{Z}\/p^2\\mathbb{Z})^{2n}$ and $n> m$, then there exists a subgroup $N \\subseteq H_1(\\Sigma_2(K))_p$ such that $N \\cong (\\mathbb{Z}\/p\\mathbb{Z})^{n-m}$ and $\\overline{d}(\\Sigma_2(K), \\mathfrak{s}_z)=0$ for each $z \\in N$. \n\\end{corollary}\n\n\\begin{proof} Suppose $K$ is concordant to a knot $J$ with $g_{\\Z}(J) = m$. Then by Proposition~\\ref{prop:branchedlower} $r(H_1(\\Sigma_2(J))) \\leq 2m$. As in the proof of Lemma~\\ref{lem:key} we assume that $H_1(\\Sigma_2(K))$ and $H_1(-\\Sigma_2(J))$ are equal to their $p$-primary components. \n\nBy Lemma~\\ref{lem:dinvt}, there exists a metabolizer $M \\subseteq H_1(\\Sigma_2(K)\\# - \\Sigma_2(J))$ such that for each element $(m_K, m_J) \\in M$ where $m_K \\in H_1(\\Sigma_2(K))$ and $m_J \\in H_1(- \\Sigma_2(J))$,\n\\begin{equation}\\label{eq:2}\nd(\\Sigma_2(K),\\mathfrak{s}_{m_K})+ d(-\\Sigma_2(J),\\mathfrak{s}_{m_J}) =0.\\end{equation}\n\nLastly, since $\\Sigma_2(K)$ and $\\Sigma_2(J)$ satisfy the assumption of Lemma~\\ref{lem:key}, we may conclude that $M \\cap H_1(Y_1)$ contains a subgroup isomorphic to $(\\mathbb{Z}\/p\\mathbb{Z})^{n-m}$. Set $N:=M \\cap H_1(Y_1)$, then note that for each $z \\in N$, we have $(0,0), (z,0) \\in N$. The proof is complete by using equation~\\eqref{eq:2}.\n\\end{proof}\n\n\n\n\n\\begin{example}\\label{exl:ourexl}\nOur examples come from the cabling construction. Let $K_{p,q}$ denote the $(p, q)$-cable of $K$, where $p$ is the longitudinal winding. We define our knot to be the following\n$$K_*:= D_{2,25} \\# -D_{2,23} \\# -T_{2,25}\\#  T_{2,23},$$\nwhere $D$ is the $4$-fold connected self-sum of the positive Whitehead double of the right-handed trefoil knot. Since $D$ is topologically slice, $D_{p,q}$ is topologically concordant to the torus knot $T_{p,q}$. Therefore $K_*$ is topologically concordant to \n$$K_*':=T_{2,25} \\# -T_{2,23} \\# -T_{2,25}\\#  T_{2,23},$$\nhence is topologically slice. \nAlso, note that by performing two crossing changes with opposite signs on $K_*$ (one in the $ D_{2,25}$ summand and one in the $-T_{2, 25}$ summand), \nwe obtain the smoothly slice knot\n$D_{2,23} \\# -D_{2,23} \\# -T_{2,23}\\#  T_{2,23}.$ Hence $K_*$ bounds a smoothly embedded genus one surface in the 4-ball. Finally, by \\cite{Akbulut-Kirby:branched-covers}, the 2-fold branched cover of $S^3$ branched over $K_*$ is given by\n$$\\Sigma_2(K_*)\\cong S^3_{25}(D \\# D^r) \\#  -S^3_{23}(D \\# D^r)\\# -S^3_{25}(U)\\# S^3_{23}(U),$$ where $D^r$ is a knot obtained by reversing the orientation of $D$. \\end{example}\n\nNext, we compute the correction terms for the 2-fold branched cover. Recall that for any knot $K$ in $S^3$, there are nonnegative integer-valued smooth concordnace invariants $V_i(K)$, introduced by Rasmussen~\\cite{Rasmussen:2003-1}. Furthermore, Ni and Wu~\\cite{Ni-Wu:2015-1} showed that the correction terms of all Dehn surgeries on $K$ can be computed from these invariants.\n\n\\begin{proposition}[{\\cite[Proposition 1.6 and Remark 2.10]{Ni-Wu:2015-1}}]\\label{proposition:NiWu} \nLet $K$ be a knot and $U$ be the unknot. If $n$ is a positive integer, then\n\\[\\pushQED{\\qed}d(S^3_n(K),\\mathfrak{s}_{i}) = d(S^3_n(U),\\mathfrak{s}_{i})-2\\max \\{V_{i}(K),V_{n-i}(K)\\}.\\qedhere\\]\n\\end{proposition}\n\n\nFurthermore, the correction terms for Dehn surgeries on the unknot are computed in \\cite{Ozsvath-Szabo:2003-1}.\n\n\\begin{proposition}[{\\cite[Proposition 4.8]{Ozsvath-Szabo:2003-1}}]\\label{prop:dlens} If $U$ is the unknot and $n$ is a positive integer, then\n\\[\\pushQED{\\qed}d(S^3_n(U),\\mathfrak{s}_{i}) = \\frac{(n-2i)^2}{4n}-\\frac{1}{4}.\\qedhere\\]\n\\end{proposition}\n\nHom and Wu~\\cite{Hom-Wu:2016-1} introduced a nonnegative integer valued smooth concordance invariant~$\\nu^+$. Moreover, for any knot $K$, we have that $\\nu^+(K)=0$ if and only if $V_0(K)=0$ \\cite[Proposition 2.3]{Hom-Wu:2016-1}. Following~\\cite{Kim-Park:2018-1} (see also \\cite{Hom:2017-1}), we say that two knots $K$ and $K'$ are \\emph{$\\nu^+$-equivalent} if $$\\nu^+(K \\# - K') =\\nu^+(K' \\# - K)=0,$$ and it forms a equivalence relation on the set of concordance classes of knots. It is well-known that two $\\nu^+$-equivalent knots have the same $V_i$ invariants (see e.g.\\ \\cite[Proposition 3.11]{Kim-Kracatovich-Park:2019-1}). We will use the fact that $D \\# D^r$ is $\\nu^+$ equivalent to the torus knot $T_{2,17}$. This fact might be well-known to experts, but we sketch the proof here for the reader's convenience. \n\n\\begin{proposition}\\label{prop:DDT217}The knot  $D \\# D^r$ in Example~\\ref{exl:ourexl} is $\\nu^+$-equivalent to the torus knot $T_{2,17}$. In particular, we have\n$$V_i(D \\# D^r)= \\left\\{\n\\begin{array}{cl}4&\\text{for }i=0,\\\\\n2&\\text{for }i=5,\\\\\n0&\\text{for }i\\geq 8.\n\\end{array}\\right.$$\n\\end{proposition}\n\\begin{proof} If $K_i$ and $K'_i$ are $\\nu^+$-equivalent for $i = 0, 1$, then $K_0\\#K_1$ and $K'_0\\#K'_1$ are also $\\nu^+$-equivalent (see e.g.\\ \\cite[Proposition 3.12]{Kim-Kracatovich-Park:2019-1}). Also, $D^r$ is $\\nu^+$-equivalent to $D$ since their knot Floer complexes are chain homotopy equivalent \\cite[Proposition 3.9]{ozsvath2004holomorphicknot}. Hence we have that $D \\# D^r$ is $\\nu^+$-equivalent to the $8$-fold connected self-sum of the positive Whitehead double of the right-handed trefoil knot. Lastly, we apply a result of Hedden, Kim, and Livingston~\\cite[Proposition 6.1]{Hedden-Kim-Livingston:2016-1} which states that the $n$-fold connected self-sum of the positive Whitehead double of the right-handed trefoil knot is $\\nu^+$-equivalent to $T_{2,2n+1}$. This proves the first part of the statement.\n\nThe last part follows from two facts. First, as mentioned above, two $\\nu^+$-equivalent knots have the same $V_i$ invariants. The second fact is that $V_i$ invariants for any alternating knot $K$ are determined by their signatures as follows (see e.g.\\ \\cite[Theorem 2]{Hom-Wu:2016-1} and \\cite[Section 5.1]{Aceto-Golla:2017-1}):\n$$V_i(K)=\\max\\left\\{\\left \\lceil{-\\frac{\\sigma(K)+2i}{4}}\\right \\rceil,0  \\right\\} .$$\nThis completes the proof.\\end{proof}\n\nNow, we are ready to compute the correction terms.\n\\begin{proposition}\\label{prop:d-of-2fold}Let $K_*$ be the knot in Example~\\ref{exl:ourexl} and $\\#^nK_*$ be the $n$-fold connected self-sum of $K_*$. If $N\\subseteq H_1(\\Sigma_2(\\#^nK_*))_5$ is a nontrivial subgroup, then there exists an element $z \\in N$ such that $\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_z) \\neq 0$.\\end{proposition}\n\\begin{proof} By Example~\\ref{exl:ourexl}, we have $\\Sigma_2(K_*)\\cong S^3_{25}(D \\# D^r) \\#  -S^3_{23}(D \\# D^r)\\# -S^3_{25}(U)\\# S^3_{23}(U)$ and $H_1(\\Sigma_2(K_*))_5\\cong \\mathbb{Z}\/25\\mathbb{Z} \\oplus \\mathbb{Z}\/25\\mathbb{Z} \\cong H_1(S^3_{25}(D \\# D^r) \\# -S^3_{25}(U))$. Note that by the additivity of the correction terms, we have that \n\\[\n\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{(i,j)})= \\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{(i,0)})+\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{(0,j)})\n.\\]\n\nFor $(i,0) \\in H_1(\\Sigma_2(K_*))_5$, we have by Proposition~\\ref{proposition:NiWu}, Proposition~\\ref{prop:dlens}, and Proposition~\\ref{prop:DDT217} that\n\\begin{align*}\n\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{(i,0)}) & = \\overline{d}(S^3_{25}(U),\\mathfrak{s}_{i})-2\\max \\{V_{i}(D \\# D^r),V_{25-i}(D \\# D^r)\\} +2V_0(D \\# D^r) \\\\\n  &= \\left\\{\n\\begin{array}{cl}0&\\text{for }i=5 \\text{ and } 20,\\\\\n2&\\text{for }i=10 \\text{ and }15.\n\\end{array}\\right.\n\\end{align*}\nSimilarly, for $(0,j) \\in H_1(\\Sigma_2(K_*))_5$ we have that\n$$\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{(0,j)})=-\\overline{d}(S^3_{25}(U),\\mathfrak{s}_{j})= \\left\\{\n\\begin{array}{cl}4&\\text{for }j=5 \\text{ and } 20,\\\\\n6&\\text{for }j=10 \\text{ and }15.\n\\end{array}\\right.$$ \n\nBy additivity, we have that\n $\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_x) \\geq 0$ for any order $5$ element $x\\in H_1(\\Sigma_2(K_*))_5$. Moreover, at least one of $\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_x)$ and $\\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{2x})$ must be strictly positive.\n \n Now, let $N\\subseteq H_1(\\Sigma_2(\\#^nK_*))_5$ be a nontrivial subgroup, and let $y$ be an order 5 element in $N$. Note that since $\\Sigma_2(\\#^n K_*) \\cong  \\#^n \\Sigma_2(K_*)$, we can naturally write $y=(y_1,\\dots, y_n)$ for $y_i \\in H_1(\\Sigma_2(K_*))_5$ and have that\n \\[\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_y)=\\sum_{i=1}^n \\overline{d}(\\Sigma_2(K_*), \\mathfrak{s}_{y_i}).\\]\nTherefore, at least one of $\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_y)$ and $\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_{2y})$ is strictly positive.\n\\end{proof}\n\nWe are now ready to prove our main theorem. We recall the statement.\n\n\n\\begin{reptheorem}{thm:main}There exist topologically slice knots with arbitrarily large concordance $\\mathbb{Z}$-genus. More precisely, there exists a topologically slice knot $K_*$ such that for each $n \\in \\mathbb{N}$ its $n$-fold connected self-sum has concordance $\\mathbb{Z}$-genus at least $n$. Furthermore, we can choose  $K_*$ so that it has smooth $4$-ball genus one.\n\\end{reptheorem}\n\\begin{proof}Let $K_*$ be the knot in Example~\\ref{exl:ourexl}. It is topologically slice and has smooth $4$-ball genus at most one as we observed above. \n\nWe will show that $\\#^nK_*$, the $n$-fold connected self-sum of $K_*$, has concordance $\\mathbb{Z}$-genus at least $n$. Suppose $g_{\\Z}^\\mathrm{c}(\\#^nK_*)=m$ and $n>m$, then by Corollary~\\ref{cor:mainobstr} there exists a nontrivial subgroup $N \\subseteq H_1(\\Sigma_2(\\#^nK_*))_5$ such that $\\overline{d}(\\Sigma_2(\\#^nK_*), \\mathfrak{s}_z) = 0$ for each $z \\in N$. This contradicts Proposition~\\ref{prop:d-of-2fold}.\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{remark}\\label{rmk:livingston}\nA similar though much more involved argument using Casson-Gordon signatures instead of $d$-invariants can be used to show that the difference between the topological $4$-ball genus and the topological concordance $\\mathbb{Z}$-genus can be arbitrarily large.\n\n In fact, a careful reading of Livingston's paper~\\cite{Livingston:2004-1} shows that, despite the fact that it only explicitly considers the topological concordance genus (i.e. the minimal Seifert genus of any representative of a given topological concordance class), all the relevant work has been done\n   to prove that for each $n \\in \\mathbb{N}$ there exists a knot $K_n$ with the  following properties:\n\\begin{enumerate}\n\\item $K_n$ has topological $4$-ball genus one. \n\\item $K_n$ is not topologically concordant to any knot $J$ with  $r( H_1(\\Sigma_2(J))) < 2n$.\n\\end{enumerate}\nIn particular, by Proposition~\\ref{prop:branchedlower}, this implies that $K_n$ is not topologically concordant to any knot $J$ with  $g_{\\Z}(J) <n$.\n\\end{remark}\n\nThis result of Livingston is relevant to understanding how $g^\\mathrm{top}_4$ changes under the satellite operation. Classical techniques show that for each pattern $P$ there is a constant $g_P\\geq g_4(P(U))$ such that  for any knot $K$ we have\n\\[g_4(P(K)) \\leq |w_P| \\cdot g_4(K) + g_P,\\]\nwhere $w_P$ is the algebraic winding number of the pattern $P$, and modern Heegaard Floer invariants can be used to show that this inequality is sometimes sharp~\\cite{Feller-Miller-Pinzon-Caicedo:2019-1}. \n\nWe similarly see that for each $P$ there exists a constant $g_P^{\\text{top}}\\geq g^\\mathrm{top}_4(P(U))$ such that for any knot $K$ we have\n\\begin{equation}\\label{eq:easybound}\ng^\\mathrm{top}_4(P(K)) \\leq |w_P| \\cdot g^\\mathrm{top}_4(K)+g_P^{\\text{top}},\\end{equation}\nbut now the sharpness of this result is far from clear--in fact, it remains possible that the factor coming from winding number is unnecessary. More precisely, it is not known if the following inequality holds for every knot $K$:\n\\begin{equation}\\label{eq:satelliteinequal}g^\\mathrm{top}_4(P(K)) \\leq g^\\mathrm{top}_4(K) + g^\\mathrm{top}_4(P(U)). \\end{equation}\nWork of Feller, Miller, and Pinzon-Caicedo~\\cite{Feller-Miller-Pinzon-Caicedo:2019-1} and independently~McCoy~\\cite{McCoy:2019-1} shows that for $P$ a pattern with $P(U)=U$ one has\n\\begin{equation}\\label{eq:FMPC} \ng^\\mathrm{top}_4(P(K)) \\leq g_{\\Z}^\\mathrm{t.c.}(K),\\end{equation}\nHence, Livingston's examples of knots $K$ with $g^\\mathrm{top}_4(K) < g_{\\Z}^\\mathrm{t.c.}(K)$ are good candidates for input companion knots if one wishes to show that the inequality \\eqref{eq:satelliteinequal} is not true.  \n\nMore concretely, we state the following problem. \n\n\\begin{problem}\nLet $K_n$ be Livingston's $n$th knot, which has $g^\\mathrm{top}_4(K)=1$ and $g_{\\Z}^\\mathrm{t.c.}(K) \\geq n$. If the inequality~\\eqref{eq:satelliteinequal} is true, then for any positive integers  $n, m$ we have that $$g^\\mathrm{top}_4(C_{m,1}(K_n))=1.$$ However, the best upper bounds we have coming from the topological concordance $\\mathbb{Z}$-genus (i.e.\\ the inequality~\\eqref{eq:FMPC}) and from classical arguments (i.e.\\ the inequality~\\eqref{eq:easybound}) are\n\\[ g^\\mathrm{top}_4(C_{m,1}(K_n))\\leq  \\min\\{m, g_{\\Z}^\\mathrm{t.c.}(K_n)\\}.\\] \nDetermine  $g^\\mathrm{top}_4(C_{m,1}(K_n))$ for some $m,n>1$.\\end{problem}\n\n\n\n\n\\bibliographystyle{alpha}\n\\def\\MR#1{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, the field of cavity optomechanics, where circulating optical fields couple to a mechanical resonator via radiation pressure, has seen tremendous progress. The state of the mechanical oscillator can be prepared, probed and coherently controlled \\citep{Aspelmeyer, quantum}. The coherent manipulation of mechanical degrees of freedom can enable applications ranging from mode conversion \\citep{hill, dong, xu}, sensitivity metrology \\citep{stowe}, quantum state transfer \\citep{wang} to quantum information processing \\citep{stannigel,le}. Experiments on optomechanical interactions have led to numerous interesting phenomena like strong optomechanical coupling \\citep{S, teufel, verhagen}, optomechanically induced transparency (OMIT) \\citep{weis, safavi}. The success of technology attempting to implement quantum information networks depends on our ability to control and utilize high frequency and long-lived acoustic modes \\citep{Aspelmeyer} . High frequency long-lived mechanical excitation's can store quantum information for extended period of time even in the presence of decoherence and this property makes them an excellent candidate for high-fidelity quantum state transfer schemes. Phonons with frequencies in the gigahertz range have already enabled entanglement between spatially separated mechanical resonators \\citep{lee, ockeloen}, single-phonon level quantum control \\citep{cohen, hong}. These micro\/nano-scale systems have the disadvantage that more advanced quantum schemes cannot be implemented due to laser heating. This led to a novel experimental demonstration of an alternate strategy for accessing high-frequency acoustic excitation within bulk crystalline (quartz crystal cavity) \\citep{kharel}. In order to access the 13 GHz phonons and to explore the optomechanical interactions, phase matched Brillouin interaction between two distinct optical cavity modes based on bulk optomechanics led to a new method being proposed to increase the spectral resolving power of a collinear acousto-optical filters \\citep{pustovoit}.\n\nMany of the classical and quantum systems operating across a wide range of energies are not compatible with one another and hence cannot be combined together on a same platform. Thus one require new device strategies for converting photons of dissimilar frequencies to combine and harness their different properties. In the past few years, different optomechanical schemes have been proposed and implemented experimentally \\citep{ali, hill, dong, xu, bagci, andrews, bochmann}. In all of these schemes, laser heating and thermal mechanical dissipation are major obstacles for efficient mode conversion.\n\nGiven the above, we propose in this paper optomechanical coupling of dissimilar optical modes to high-frequency bulk acoustic mode within a macroscopic quartz crystal \\citep{kharel} as an alternative approach to efficient optical mode conversion. Our proposal is based on the optomechanical interaction between the elastic waves within a bulk quartz crystal and two longitudinal optical cavity modes. The interaction between the elastic and optical modes comes into existence due to formation of time-varying photoelastic grating which initiates energy exchange between the optical modes via Bragg scattering. We analyze optical intracavity emission, optical mode conversion efficiency and explain the observed results using the concept of bright and dark modes. We show that the formation of dark mode induces efficient mode conversion from one optical mode to the other. To the best of our knowledge, no similar theoretical research has been carried out to explore coherent wavelength conversion of optical photons using bulk optomechanics. We will be working in the Far-Infrared (F-IR) frequency regime where mode conversion efficiency was found to be maximum using the current strategy.\n\n\n\\section{Proposed Model and Theory}\nA simplified schematic description of the opto-mechanical system that we intend to investigate is shown in Fig.1. A  quartz crystal at cryogenic temperature ($\\approx$ 8 K) [also called the bulk acoustic wave resonator] is placed inside an optical Fabry-P$e^{'}$rot resonator. Our proposed model is based on a recent experiment on high-frequency cavity optomechanics using bulk acoustic phonons \\citep{kharel}. Macroscopic phonon modes are produced within the bulk quartz crystal and acoustic reflections at the planar surfaces of the crystal leads to the formation of acoustic Fabry-P$e^{'}$rot resonator which supports standing-wave high frequency elastic modes with frequency $\\Omega_m > 10$ GHz. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.7]{fig1.jpg}\n    \\caption{Schematic of an optomechanical system that consists of a bulk acoustic wave resonator that is placed\ninside an optical cavity. }\n    \\label{fig:my_label}\n\\end{figure}\n\nThese high frequency bulk phonon modes acts as a mediator between two longitudinal optical cavity modes. Optomechanical coupling between the acoustic and optical mode occurs when frequency of the time-varying electrostrictive optical force (generated by interference between different optical modes) matches the frequency of the acoustic mode. The motion of the elastic-wave modulates the refractive index of the crystal leading to the formation of a time-varying photoelastic grating. Energy exchange between the different longitudinal optical modes takes place via Bragg scattering by the time-modulated photoelastic grating. In the absence of crystal reflections, equally spaced longitudinal optical modes (uniform density of states) are obtained due to the fact that Bragg scattering rate from the incident laser mode $\\omega_j$ to nearest neighbor optical modes $\\omega_{j+1}$ and $\\omega_{j-1}$ is same. In the presence of optical reflection ($\\approx$ 4 $\\%$) from the crystal surface, non uniform density of states is obtained \\citep{kharel}. This allows one to choose one scattering process over the other (either strokes or anti-strokes scattering). The dynamical Bragg scattering due to the optomechanical interaction within the crystal leads to phase matching $(q_m = k_{j-1}+k_j)$ and energy conservation $(\\Omega_m = \\omega_j-\\omega_{j-1})$. Here $q_m, k_{j-1}, k_j$ are the wave numbers of the acoustic mode, optical mode $\\omega_{j-1}$ and $\\omega_j$ respectively. These phonon modes which are capable of mediating Bragg scattering lies near the Brillouin frequency $\\Omega_B = \\Omega_m$ \\citep{kharel,boyd}. We will choose a pair of optical modes separated by the Brillouin frequency for our analysis.\n\nThe Hamiltonian describing this composite optomechanical system is given by \\citep{kharel}\n\\begin{multline}\n    \\hat H = \\hbar \\omega_1\\hat a^{\\dagger}_1 \\hat a_1 + \\hbar \\omega_2\\hat a^{\\dagger}_2 \\hat a_2 + \\hbar \\Omega_m\\hat b^{\\dagger}_m \\hat b_m - \\hbar g^m_0(a^{\\dagger}_2 \\hat a_1 \\hat b_m + b^{\\dagger}_m \\hat a^{\\dagger}_1 \\hat a_2)\\\\ + \\iota\\hbar\\sqrt{\\kappa_1^{ext}}\\alpha_p(\\hat a^{\\dagger}_1 e^{-\\iota\\omega_p t} +  \\hat a_1 e^{\\iota\\omega_p t}) .\n\\end{multline}\n\nHere the first two terms of Eqn.(1) are the energy of the two bare optical cavity modes at frequency $\\omega_1$ and $\\omega_2$. The third term is the energy of the acoustical mode at frequency $\\Omega_m$. The two optical modes $\\omega_1$ and $\\omega_2$ are separated by $\\Omega_m$, i.e., $\\omega_1$ - $\\omega_2$ = $\\Omega_m$. The operators $\\hat a_1$, $\\hat a_2$, and $\\hat b_m$ are the annihilation operators of the optical mode 1, mode 2 and the phonon mode respectively. The fourth term denotes the coupling of the two adjacent standing wave optical modes to the $m$-th phonon mode. In the interaction term, $\\hat a_2^{\\dagger}$ $\\hat a_1$ $\\hat b_m$ represents the annihilation of the optical mode at frequency $\\omega_1$ and a phonon mode at frequency $\\Omega_m$ to create an optical mode at a higher frequency $\\omega_2$ = $\\omega_1$ + $\\Omega_m$. This represents the anti-strokes process. The other term  $\\hat b_m^{\\dagger}$ $\\hat a_1^{\\dagger}$ $\\hat a_2$ represents strokes process. Here $g_0^m$ is the single-photon optomechanical coupling rate given by \\citep{kharel}.\n\n\\begin{equation}\n    g_0^m \\approx \\frac{\\omega_1^2 n^5 p_{13}}{2 c n^2_{eff}}\\sqrt{\\frac{\\hbar}{\\rho A L_{ac}\\Omega_m}} \\frac{L_{ac}}{L_{opt}}  \\hspace{2mm} ,\n\\end{equation}\n\n      \\vspace{3mm}\n       \nwhere $n_{eff}$ is the effective refractive index of the optical mode, $p_{13}$ is the photoelastic constant of the quartz crystal, $\\rho$ is the mass density of the crystal, $L_{ac}$ is the thickness of the crystal, $L_{opt}$ is the spacing between the two cavity mirrors, $A$ is the cross-section area of the crystal, $n$ is the refractive index of the crystal. The last term is the external pump with frequency $\\omega_p$ which drives the mode $\\omega_1$. Also $\\kappa^{ext}_1$ is the loss rate at each cavity mirror and $\\sqrt{\\kappa^{ext}_1}$ $\\alpha_p$ is the rate of pumping the mode 1.\n\nTransforming the Hamiltonian of Eqn.(1) into the rotating frame frequency $\\omega_p$ of the pump field, we obtain the following linearized Hamiltonian,\n\\begin{multline}\n    \\hat H_{eff} = \\hbar \\Delta_1 \\hat a_1^{\\dagger} \\hat a_1 + \\hbar \\Delta_2 \\hat a_2^{\\dagger} \\hat a_2 + \\hbar \\Omega_m \\hat b_m^{\\dagger} \\hat b_m - \\hbar G_m (\\hat a_2^{\\dagger} \\hat a_1 + \\hat a_1^{\\dagger} \\hat a_2) - \\hbar G_1 (\\hat a_1 \\hat b_m + \\hat b_m^{\\dagger} \\hat a_1^{\\dagger})\\\\ - \\hbar G_2 (\\hat a_2^{\\dagger} \\hat b_m + \\hat b_m^{\\dagger} \\hat a_2 )+ \\iota \\hbar \\sqrt{\\kappa_1^{ext}}\\alpha_p(\\hat a_1^{\\dagger} - \\hat a_1) .\n\\end{multline}\n\nHere, $G_m$ = $g_0^m |b^s_m|$, $G_1$ = $g_0^m |a^s_1|$, $G_2$ = $g_0^m |a^s_2|$. Also, $a_1^s$, $b_m^s$ and $a_2^s$ are the steady state mean values of the operators $\\hat b_m$, $\\hat a_1$ and $\\hat a_2$ respectively. The linearized Hamiltonian can be derived by assuming the optomechanical coupling to be weak and both the cavity modes and the acoustic fields remain near their coherent state \\citep{milburn,wilson}. Optical mode conversion mediated by the mechanical resonator in the absence of the interaction $\\hbar G_m ((\\hat a_2^{\\dagger} \\hat a_1 + \\hat a_1^{\\dagger} \\hat a_2)$ has been realized experimentally in a system comprising of whispering gallery modes (WGMS) coupled to a mechanical mode in silica resonator \\citep{dong}. The Heisenberg equations of motion derived from the Hamiltonian $\\hat H_{eff}$ are,\n\n\\begin{equation}\n   \\dot{\\hat{a_1}} = -(\\iota\\Delta_1 + \\frac{\\kappa_1}{2})\\hat a_1 + \\iota G_m \\hat a_2 + \\iota G_1\\hat b_m^{\\dagger} + \\sqrt{\\kappa_1^{ext}}\\alpha_p \\hspace{2mm} ,\n   \\end{equation}\n   \n   \\begin{equation}\n       \\dot{\\hat{a_2}} = -(\\iota\\Delta_2 + \\frac{\\kappa_2}{2})\\hat a_2 + \\iota G_m \\hat a_1  + \\iota G_2\\hat b_m \\hspace{2mm},\n   \\end{equation}\n   \n   \\begin{equation}\n       \\dot{\\hat{b_m}} = -(\\iota\\Omega_m + \\frac{\\gamma_m}{2})\\hat b_m + \\iota G_1 \\hat a_1^{\\dagger}  + \\iota G_2\\hat a_2 \\hspace{2mm},\n   \\end{equation}\n\n\\vspace{3mm}\n\nwhere $\\Delta_1$ = $\\omega_1$ - $\\omega_p$ is the mode 1 - pump field detuning, while $\\Delta_2$ = $\\omega_2$ - $\\omega_p$ is the mode 2-pump field detuning. $\\kappa_1$, $\\kappa_2$ and $\\gamma_m$ are the decay rates of cavity mode 1, mode 2 and phonon mode respectively. Eqn.(4)-(6) will form the basis of our further study related to optical mode conversion in the next section.\n\n\\section{Optical Intracavity Emission}\nWe are interested in the steady state mean response of the system, hence we can replace the operators by their mean expectation values i.e $<\\hat a_1>$ = $a_1(t)$, $<\\hat a_2>$ = $a_2(t)$ and $<\\hat b_m>$ = $b_m(t)$ \\citep{kwon,ali}. The mean field steady-state solutions of Eqn.(4)-(6) for the intracavity  steady amplitudes of two optical modes $a_1^s$ and $a_2^s$ are given as, \n\n\\begin{equation}\n    |a_1^s|^2 = \\frac{A_{1R}^2 + A_{1I}^2}{(l_1^2 + l_2^2 - m_1^2)^2}\n\\end{equation}\n\n\\begin{equation}\n    |a_2^s|^2 = \\frac{A_{2R}^2 + A_{2I}^2}{[(R_1^2 + R_2^2) - (f_1^2 + f_2^2)]^2}\n\\end{equation}\n\n\\vspace{4mm}\n\nThe details of the expression for $|a_1^s|^2$ and $|a_2^s|^2$ are explicitly given in Appendix A. In these expressions, we have introduced the cooperativities of the two optical modes as $C_1$ = $\\frac{4 G_1^2}{\\gamma_m \\kappa_1}$ and $C_2$ = $\\frac{4 G_2^2}{\\gamma_m \\kappa_2}$. Optical emissions from mode 1 and mode 2 are directly proportional to the steady state intracavity intensities $|a_1^s|^2$ and $|a_2^s|^2$ respectively. For demonstration of optical mode conversion in our proposed model, we will use Far Infrared (Far IR, $10\\mu$m to 100 $\\mu$m wavelength) mode propagating in quartz crystal. For optical applications at tera-hertz frequencies, quartz is an ideal candidate due to its low absorption and high transmission at these frequencies \\citep{davies}. In particular, we will be using $\\omega_1$ = $2\\pi \\times 0.99 \\times 10^{12}$ Hz and $\\omega_2$ = $2\\pi \\times 10^{12}$ Hz. The frequency $\\omega_1$ is excited by the external pump laser and $\\omega_2$ is generated by Bragg scattering within the crystal as demonstrated experimentally in the Near-IR region \\citep{kharel}. The phonons which mediate the dynamical Bragg scattering within the crystal lies near the Brillouin frequency $\\Omega_B$ = $\\frac{2 \\omega_j n v_a}{v_c}$, where $v_a$ and $v_c$ are the speed of sound and light in the quartz crystal respectively. $n$ is the refractive index of the quartz crystal at frequency $\\omega_j$. The two optical modes are separated by the phonon frequency of $\\frac{\\Omega_m}{2\\pi}$ = $90.63$ MHz. Note that the we have used the following experimentally relevant parameters, $n$ = 2.15, $v_a$ = 6327 m\/s, $g_0^m$ = $2\\pi \\times 24$ Hz, $\\kappa_1$ = $\\kappa_2$ = $2\\pi \\times 73$ MHz \\citep{kharel,davies,roberts,devaty}.\n\nFigure 2(a) and 2(b) shows the normalized emission intensity from mode 1 and mode 2 respectively as a function of cavity cooperativity $C_2$ for the case $\\Delta_2\/2\\pi$ = 65.70 MHz, $\\Delta_1\/2\\pi$ = -25MHz $(\\omega_1 - \\omega_2 = - \\Omega_m)$ and two different values of mechanical damping rate $\\gamma_m$ = 0.45 $\\kappa$ (dashed plot) and $\\gamma_m$ = 0.30 $\\kappa$ (solid plot). Such high mechanical damping rate is possible in the Far-Infrared region. To understand the high acoustic damping rate, we can view the propagation of the acoustic field inside the quartz crystal as being similar to the propagation of a laser field in a Fabry-P$e^{'}$rot cavity. The acoustic wavelength corresponding to $\\Omega_m$ = 90.68 MHz $(\\approx 70 \\mu m)$ is much larger than the lateral extent of the excited acoustic mode (beam radius $\\approx$ 43 $\\mu$m) within the crystal \\citep{kharel}. This leads to diffraction loss (analogous to diffraction loss of optical beam in an optical cavity) and the energy in the diffracted acoustic wave which lies outside the acoustic beam radius would be lost. In addition to the intrinsic loss due to diffraction, loss of acoustic energy can also be controlled by adjusting the tilt angle of the crystal axis with respect to the acoustic axis \\citep{kharel,ghatak}. As evident from figure 2, emission power from both the modes continuously increases as $C_2$ increases, reaches a peak and then decreases with further increase in $C_2$. The emission power from the modes is higher for a large $\\gamma_m$ (= 0.45 $\\kappa$) compared to that for a low value of $\\gamma_m$ (= 0.30 $\\kappa$) but the maximum emission power from mode 2 is higher for a lower $\\gamma_m$ as seen in fig.2(b). For a higher $\\gamma_m$, the peak in the emission power is reached at lower value of $C_2$.\n\n\\begin{figure}[ht]\n\\hspace{-0.0cm}\n\\begin{tabular}{cc}\n\\includegraphics [scale=0.92] {fig2a.eps} \n\\includegraphics [scale=0.9] {fig2b.eps}\\\\\n\\end{tabular} \n\\caption{Normalized optical intracavity emission from mode 1 (fig. 2(a)) and mode 2 (fig. 2(b)) as a function of the cavity cooperativity $C_2$ for two different values of mechanical damping rate $\\gamma_m$ = 0.030 $\\kappa_1$ (solid line), $\\gamma_m$ = 0.45 $\\kappa_1$ (dashed line). All system parameters are normalized with respect to the cavity mode 1 damping rate $\\kappa_1$. The other system parameters are chosen as $\\kappa_2$ = 2 $\\kappa_1$, $\\Delta_1$ = $-\\Omega_m + \\Delta_2$, $\\Delta_2$ = 0.9 $\\kappa_1$, $\\Omega_m$ = 1.242 $\\kappa_1$, $G_m$ = 0.025 $\\kappa_1$, $G_1$ = 0.4 $\\kappa_1$.}\n\\end{figure}\n\nFigure 3(a) and 3(b) shows the plot of emission power from mode 1 and 2 versus $C_2$ for two values of cavity cooperativity $C_1$. $C_1$ = $\\frac{4 |g_o^m|^2 |a_1^s|^2 }{\\gamma_m \\kappa_1}$ can be tuned efficiently by pumping more photons into mode 1 using the external driving laser as observed experimentally \\citep{kharel}. As evident from figure 3(a), the increase in emission power from mode 1 as $C_2$ is varied is more for a larger $C_1$. This also leads to an enhanced transfer of energy from mode 1 to mode 2 as $C_1$ is increased from 1.2 to 2.13. This observation is evident from figure 3(b).\n\n\\begin{figure}[h]\n\\hspace{-0.0cm}\n\\begin{tabular}{cc}\n\\includegraphics [scale=0.87] {fig3a.eps} \n\\includegraphics [scale=0.87] {fig3b.eps}\\\\\n\\end{tabular} \n\\caption{Normalized optical intracavity emission from mode 1 (fig. 3(a)) and mode 2 (fig. 3(b)) as a function of the cavity cooperativity $C_2$ for two different value of cavity cooperativity $C_1$ = 1.2 (dashed line) and $C_1$ = 2.13 (solid line). The other system parameters are same as that for figure 2 with $\\gamma_m$ = 0.3 $\\kappa_1$.}\n\\end{figure}\n\n\\section{Optical Mode Conversion Efficiency}\nIn this section, we analyze the optical mode conversion efficiency ($\\eta$) for our proposed model. We define ($\\eta$) as the ratio of mode 2 output photon flux over the mode 1 input photon flux $\\eta$ = $I_{out}\/I_{in}$, $I_{in}$ = $|\\alpha_p|^2$ and $I_{out}$ = $\\kappa_e^{ext}$ $|a_2^s|^2$.\nThus using results derived in the previous section, the photon mode-conversion efficiency can be written as,\n\n\\begin{equation}\n    \\eta = \\frac{\\eta_1 \\eta_2 \\kappa_1\\kappa_2 (\\tilde{A}_{2R} + \\tilde{A}_{2I})}{[(R_1^2 + R_2^2) - (f_1^2 + f_2^2)]^2} \\hspace{2mm} ,\n\\end{equation}\n\n\\vspace{3mm}\n\nwhere $\\eta$ = $\\frac{\\kappa_i^{ext}}{\\kappa_i}$ (i = 1,2) is the output coupling ratio for the two optical modes. Here $\\tilde{A}_{2R}$ = $\\frac{A_{2R}}{\\sqrt{\\kappa_1^{ext}}\\alpha_p}$ and $\\tilde{A}_{2I}$ = $\\frac{A_{2I}}{\\sqrt{\\kappa_1^{ext}}\\alpha_p}$. The extend of destructive interference in the output of mode 1 determines the efficiency of mode conversion \\citep{li}. Figure 4(a) plots the resulting $\\eta$ as a function of cavity cooperativity $C_2$ for two values of $\\gamma_m$ = 0.45 $\\kappa$ (dashed line) and $\\gamma_m$ = 0.30 $\\kappa$ (solid line). Till $C_2$ = 9, the conversion efficiency is higher for large $\\gamma_m$. The peak value of the conversion efficiency corresponding to a lower $\\gamma_m$ higher but is attained at a larger value of $C_2$ = 12, which could be difficult to reach with current technology. Fig 4(b) plots $\\eta$ versus $C_2$ for two values of $C_1$ = 1.2 (solid line) and $C_1$ = 2.13 (dashed line). Clearly, a higher $C_1$ yields higher mode conversion efficiency.\n\n\\begin{figure}[htbp]\n\\hspace{-0.0cm}\n\\begin{tabular}{cc}\n\\includegraphics [scale=0.88] {fig4a.eps} \n\\includegraphics [scale=0.87] {fig4b.eps}\\\\\n\\end{tabular} \n\\caption{The photon-conversion efficiency $\\eta$ as a function of cavity cooperativity $C_2$ for two different value of $\\gamma_m$ (fig. 4(a)) and $C_1$ (fig. 4(b)). The parameters used have the values $\\kappa_2$ = 2 $\\kappa_1$, $G_m$ = 0.025 $\\kappa_1$, $\\Omega_m$ = 1.242 $\\kappa_1$, $\\eta_1 = \\eta_2$, $\\Delta_2$ = 0.9 $\\kappa_1$, $\\Delta_2$ = $-\\Omega_m + \\Delta_2$. For fig. 4(a), $G_1$ = 0.4 $\\kappa_1$ and for fig. 4(b), $\\gamma_m$ = 1.242 $\\kappa_1$.}\n\\end{figure}\n\n\\section{DARK AND BRIGHT MODES}\nWe now analyze our obtained results using the concept of bright and dark modes.\nWe now write the system's Hamiltonian in terms of cavity dark and bright modes which are defined as \n\n\\begin{equation}\n    \\hat a_B = \\frac{G_1 \\hat a_1 + G_2 \\hat a_2 }{\\tilde{G}}, \\hspace{5mm} \\hat a_B = \\frac{G_2 \\hat a_1 - G_1 \\hat a_2 }{\\tilde{G}} \\hspace{2mm} ,\n\\end{equation}\n\n\\vspace{3mm}\n\nwhere $\\tilde{G_1}$ = $\\sqrt{G_1^2 + G_2^2}$. Making use of the above definition of $\\hat a_B$ and $\\hat a_D$, we rewrite using $\\Delta_2 = 0$ and $\\Delta_1$ = -$\\Omega_m$, the Hamiltonian of eqn. (3) as, \n\n\\begin{multline}\n    \\hat H_{eff} = -\\hbar \\Delta_B \\hat a_B^{\\dagger} \\hat a_B - \\hbar \\Delta_D \\hat a_D^{\\dagger} \\hat a_D + \\hbar \\Omega_m \\hat b_m^{\\dagger} \\hat b_m - \\hbar G_bd (\\hat a_B^{\\dagger} \\hat a_D + \\hat a_D^{\\dagger} \\hat a_B) - \\hbar G_{12} (\\hat a_D \\hat b_m + \\hat b_m^{\\dagger} \\hat a_D^{\\dagger} - \\hat a_D^{\\dagger} \\hat b_m + \\hat b_m^{\\dagger} \\hat a_D )\\\\ - \\hbar \\tilde{G_1} (\\hat a_B^{\\dagger} \\hat b_m + \\hat b_m^{\\dagger} \\hat a_B^{\\dagger} ) - \\hbar \\tilde{G_2}(\\hat a_B^{\\dagger}\\hat b_m + \\hat b_m^{\\dagger}\\hat a_B)+ \\iota \\hbar A_1 (\\hat a_B^{\\dagger} - \\hat a_B) + \\iota\\hbar A_2(a_D^{\\dagger} - \\hat a_D)\n\\end{multline}\n\nwhere, \n\\begin{equation*}\n    \\Delta_D = \\frac{G_2^2\\Omega_m - 2 G_m G_1 G_2}{\\tilde{G^2}}, \\hspace{4mm} \\Delta_B = \\frac{G_1^2\\Omega_m + 2 G_m G_1 G_2}{\\tilde{G^2}}\n\\end{equation*}\n\n\\begin{equation*}\n    G_{bd} = \\frac{[G_1 G_2 \\Omega_m + G_m + G_m(G_2^2 - G_1^2)]}{\\tilde{G^2}}, \\hspace{4mm} G_{12} = \\frac{G_1 G_2}{\\tilde G}\n\\end{equation*}\n\n\\begin{equation*}\n  \\tilde{G_1} = \\frac{G_1^2}{\\tilde{G}}, \\hspace{4mm} \\tilde{G_2} = \\frac{G_2^2}{\\tilde{G}} \n\\end{equation*}\n\n\\begin{equation*}\n A_1 = \\sqrt{\\kappa_1^{ext}}\\frac{\\alpha_p G_1}{\\tilde{G}}, \\hspace{4mm}  A_2 = \\sqrt{\\kappa_1^{ext}}\\frac{\\alpha_p G_2}{\\tilde{G}} \n\\end{equation*}\n\n\\vspace{2mm}\n\nEquation (11) demonstrates a complex interplay between the two optical modes and the mechanical mode. Using the equations of motion derived from $\\hat H^{eff}$, we calculate the steady state values of $a_{B,S}$ and $a_{D,S}$ using $\\kappa_1$ = $\\kappa_2$ = $\\kappa$ . \n\n \\begin{equation}\n    a_{B,S} = A_1(f_{a1} + \\iota f_{a2}) + A_2(g_{a1} + \\iota g_{a2})\n\\end{equation}\n\n\\begin{equation}\n    a_{D,S} = A_1(h_{a1} + \\iota h_{a2}) + A_2(J_{a1} + \\iota J_{a2})\n\\end{equation}\n\nThe details of the expression appearing in eqns. (12) and (13) is given in Appendix B.\n\n\\vspace{5mm}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[scale=0.99] {fig5.jpg}\n    \\caption{Calculated bright mode and dark mode steady state population as a function of coupling parameter $G_2$. system parameters used are $\\kappa_1$ = $\\kappa_2$ = $\\kappa$, $G_m$ = 0.025 $\\kappa$, $\\gamma_m$ = 0.2 $\\kappa$, $G_1$ = 0.6 $\\kappa$ and $\\Omega_m$ = 1.242 $\\kappa$.}\n\\end{figure}\n\nIn figure 5, we plot the steady state population of dark mode $|a_{D,S}|^2$ and bright mode $|a_{B,S}|^2$ as function of optomechanical coupling strength $G_2$. As $G_2$, increases the cavity cooperativity also increases and as evident from the plot, as $|G_2|$ increases, the bright mode population $|a_{B,S}|^2$ decreases while the dark mode population $|a_{D,S}|^2$ increases. This observation together with the result of previous figures points to the fact that there is a destructive interference between the bright and dark mode and hence optical mode conversion from mode 1 to mode 2 is initiated by suppression of the bright mode \\cite{ali}.\n\n\\section{Conclusions}\nIn summary, we have demonstrated efficient frequency conversion between two dissimilar standing-wave longitudinal optical modes in the far-IR region via electrostrictive optical force with a standing-wave longitudinal acoustic mode generated within a bulk quartz crystal. It is clearly shown that the efficiency of mode conversion is better for large cavity cooperativies and mechanical damping rates. In our proposed system, the cavity dark mode, which is a superposition of the two optical modes is not decoupled from the acoustic mode. It is found that when the composite system is driven towards the system's dark mode, the mode conversion efficiency increases. The proposed hybrid quantum system using short-lived phonons within a bulk quartz crystal could be a profitable resource for optical mode conversion.\n\\cleardoublepage\n\n\\section{REFERENCES}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nQuantifier words---such as {\\em each} or {\\em most} or {\\em more than three}---have been extensively studied, both in logic and in linguistics \\citep{westerstaahl1989quantifiers,peters2006quantifiers}, going all the way back to \\citet{frege1879begriffsschrift}. In this paper, we examine the extent to which they present a challenge to modern NLU systems. Our analysis is motivated by three  observations:\n\n{\\bf Quantifier words are abstract} Unlike nouns, verbs and adjectives, quantifier words do not have referents out in the world. Rather, quantifier words specify relationships between sets of entities, events and properties. To provide intuitions about the semantics of quantifier words, and to be able to refer to quantifiers in a language-independent way, we rely on the notion of generalized quantifiers \\cite{mostowski1957generalization}, as described in \\S2. \n\n{\\bf Quantifier words vary across languages} Quantifier word inventories differ across languages. Often what is considered rough translation equivalents also differ in syntax, fine-grained semantics or pragmatics. \\newcite{10.3389\/fpsyg.2019.00957} show, e.g., that perceptions of the numerical bounds of existential quantifiers differ across speakers of English, French, Slovenian, and German. Other papers showing discrepancies between quantifier systems include comparisons of Salish to English \\cite{matthewson2001quantification}, Adyghe to English \\cite{Nikolaeva2012}, or of Dutch, Hebrew and Bengali \\cite{10.2307\/25001107}. The cross-linguistic differences in how generalized quantifiers are expressed motivates a cross-lingual error analysis, since quantifiers may contribute more to error when processing some languages rather than others. \n\n\\input{tables\/examples.tex}\n\\input{tables\/categorization_set.tex}\n{\\bf Quantifier words are important} Quantifier words are extremely important for tasks that require inference, including natural language inference, question answering, fact-checking, etc. Datasets have, for example, been developed for numerical reasoning in English \\cite{dua-etal-2019-drop}. Several researchers have  identified quantifier words as important sources of errors for natural language processing systems \\cite{joshi-etal-2020-taxinli}; see Table~\\ref{tab:examples} for examples of such errors. Unfortunately, most efforts have concentrated on subsets of quantifier words and on English. \n\n{\\bf Contributions} We analyze how quantifiers are represented in NLU benchmarks, and how their occurrence at test time contributes to errors by neural language models (LMs). We derive a linguistically motivated 11-way categorization set for generalized quantifiers and look into their distribution in three steps: (a) monolingual NLI; (b) cross-lingual NLI; (c) cross-lingual question answering. We also propose GQNLI\\footnote{\\url{https:\/\/github.com\/ruixiangcui\/GQNLI}}, an adversarial generalized quantifier NLI challenge dataset. Our work shows that (i) generalized quantifiers are pervasive and cause overall performance drops in NLU benchmarks; (ii) the contribution of quantifier words to system error varies across languages; and (iii) generalized quantifiers are particularly difficult for LMs in interaction with negation and subsumption. \n\n\\section{Background}\n\nGeneralized quantifiers (GQs) are developed upon first-order predicate logic, denoting relations between sets \\citep{mostowski1957generalization}. Given a universe \\textit{E}, a quantifier \\textit{Q} would be treated as a mapping \\(Q_{E}\\) from the Cartesian product of powersets \\(\\mathcal{P}(E) \\times \\mathcal{P}(E)\\) to the set \\{\\textit{false,true}\\} or, as a binary relation on subsets of \\textit{E} \\citep{dvorak2015}. GQs are generalizations of the\n\\(\\forall\\)\n\\(\\exists\\) quantifiers from first-order predicate logic  \\citep{mostowski1957generalization, perlindstrom1966first,montague1973proper,Bach1995QuantificationIN,Keenan2012HandbookOQ}. A generalized quantifier is, abstractly, a relation between sets. Generalized quantifier theory, while developed by logicians, is used by formal linguists to analyze the meaning of quantifier words in combination with referential expressions \n\\citep{barwise1981generalized,higginbotham1981questions}.\n\nMost human languages contain ways of expressing generalized quantifiers, and their semantics exhibit striking similarities across languages \\cite{matthewson2004methodology, 2008universals,SteinertThrelkeld2019LearnabilityAS}. At the same time, generalized quantifiers can be instantiated very differently across languages due to pragmatic considerations \\citep{grice1989studies} or cognitive economy\nand cost-benefit optimisation in the exchange of information \\citep{levinson2000presumptive,e23101335,10.1162\/ling_a_00461}. Quantifier words also exhibit syntactic differences, e.g., with some languages having specialized words\nto express quantity, while others rely on metaphorical usage of common nouns \\citep{katsos2012acquisition}.  In English, {\\em most} is a determiner, but Spanish and French express the same concept through common nouns, {\\em la mayor\\'{i}a} and {\\em la majorit\\'{e}}. The relative stability of the core semantics of quantifiers makes a cross-linguistic comparison possible, but the syntactic and pragmatic variation associated with the expression of generalized quantifiers poses a challenge for multilingual NLU.    \nWe consult quantifier taxonomy studies \\citep{Keenan1997GeneralizedQI, peters2006quantifiers, szymanik-thorne-2015-semantic, Szymanik2016} and derive a categorization set for quantifier analysis in NLU benchmarks. In Table~\\ref{tab:categorization_set}, we list the 11-way quantifier categorization set and their logical denotation based on set theory. \n\nWhile other foci of formal linguistics have attracted the attention of NLP researchers---including coreference \\citep{ws-2019-models, crac-2020-models}, negation \\citep{hossain-etal-2020-analysis,hartmann-etal-2021-multilingual}, \nand consistency \\citep{li-etal-2019-logic,ribeiro-etal-2019-red,asai-hajishirzi-2020-logic,10.1162\/tacl_a_00450}---there has been little work on generalized quantifiers as a source of error in NLU, let alone in multilingual NLU. It remains an open problem whether LMs represent the semantics of quantifiers words adequately, or if they provide a basis for resolving scopal ambiguities.\\footnote{Note that generalized quantifiers are not always {\\em explicit} in discourse. The sentence {\\em inadequate sleep causes obesity} should be interpreted as {\\em Most of those who do not sleep adequately, gain weight} \\citep{zadeh1983computational}. Such implicit quantifiers related to pragmatic variation are important for language understanding, but will be ignored in this work.}\n\n\\section{NLU Benchmarks}\n\nWe conduct an error analysis focusing on the role of  generalized quantifiers in\ntwo\nNLU tasks, Natural Language Inference (NLI) and Question Answering (QA), which generally\nrequire understanding of quantifiers. For each type of task, both monolingual and cross-lingual evaluation are conducted. We focus on generalized quantifiers in the {\\em hypotheses} in NLI examples---and on generalized quantifiers in the {\\em question} fields in question answering. \n\\input{tables\/task_stats_nli.tex}\nTo this end, we identify quantifiers by the lemma and the universal dependency relation \\citep{nivre-etal-2020-universal} of a quantifier after preprocessing the sentences using \\textit{Stanza} \\citep{qi-etal-2020-stanza}. Take the sentence ``The Yiddish culture has survived for more than a thousand years.'', we annotate it as ``The\/\\textit{det} Yiddish\/\\textit{amod} culture\/\\textit{nsubj} have\/\\textit{aux} survive\/\\textit{root} for\/\\textit{case} more\/\\textit{advmod} than\/\\textit{fixed} a\/\\textit{det} thousand\/\\textit{nummod} year\/\\textit{obl} .\/\\textit{punct}''. By matching the regex pattern of the quantifier ``more than k'', in this case \\textit{``((more|great)\\symbol{92}\/advmod than\\symbol{92}\/(fixed|case)|at\\symbol{92}\/case least\\symbol{92}\/nmod) .+\\symbol{92}\/nummod .+\\symbol{92}\/(nsubj|obj|obl)''}, we approximate the surface form of the type ``more than k''.Through matching quantifier patterns, we are able to find entries in which quantifiers are instantiated. See Appendix~\\ref{app: Regular Expressions} for the list of regex patterns we write to identify GQs. In Table~\\ref{tab:task_stats_nli} and Table~\\ref{tab:task_stats_qa}, we present the statistics of the quantifier distributions in NLI and QA tasks, respectively. As can be seen, quantifiers are indeed widespread in NLU tasks, accounting for roughly 10\\% in NLI tasks and 5\\% in QA tasks.\nWe will further discuss the statistics and experiments in the following section.\n\n\\section{Quantifiers in English NLI Benchmarks}\n\\label{sec:Quantifiers in Four English NLI Tasks}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{fig\/frequency.png}\n    \\caption{Relative distribution of quantifiers in NLI and QA tasks ranked by semantic complexity. The bars show the relative frequency of such quantifier and the lines indicate the cumulative frequency for a task. }\n    \\label{fig:frequency}\n\\end{figure}\n\\input{tables\/nli_result.tex}\n\nNLI is commonly framed as a three-way classification task with labels \\textit{entailment}, \\textit{contradiction} and \\textit{neutral} \\cite{bowman-etal-2015-large}. While SOTA models exhibit low error rates on NLI benchmarks, it is unclear when they succeed or fail in their underlying reasoning. We are interested in whether generalized quantifers challenge modern NLI models. In our error analysis, we initially focus on three English NLI datasets, MultiNLI \\citep[MNLI;][]{williams-etal-2018-broad}, SNLI \\citep{bowman-etal-2015-large} and ANLI \\citep{nie-etal-2020-adversarial} as testbeds. \n\nTable~\\ref{tab:task_stats_nli} presents statistics of quantifier distribution in these datasets, where we observe that, across, about 10\\% of all hypotheses contain quantifier words, indicating the pervasiveness of quantification. We also plot the frequency of quantifiers in NLI in Figure~\\ref{fig:frequency} and find the quantifier word distribution follows Zipf's law \\citep{Zipf1949HumanBA}. Note the top three most common quantifiers account for more than 90\\% of all. \n\n\\paragraph{Experiments and Results}\nIn order to investigate whether NLU systems can solve quantifiers in NLI, we experiment with two pretrained LMs: BERT\\footnote{\\texttt{wwm\\_cased\\_L-24\\_H-1024\\_A-16}} \\citep{devlin-etal-2019-bert} and RoBERTa\\footnote{\\texttt{roberta-large}} \\citep{Liu2019RoBERTaAR}. We use the codebase by \\citet{nie-etal-2020-adversarial}. The training data combines SNLI, MNLI, FEVER-NLI \\citep{nie2019combining} and ANLI. \n\nIn Table~\\ref{tab:nli_result}, we report the test set performance on SNLI and ANLI, and the dev set performance on MLNI \\textit{matched} and \\textit{mismatched} sections. We can observe that SOTA models suffer from performance drops across almost all quantification phenomena in every task. When it comes to performance over all quantifiers, the improvement from RoBERTa to BERT (2.2\\%) is less prominent than that over full datasets (2.9\\%), suggesting RoBERTa is particularly challenged. \n\nTaking a closer look at error by category, proportional quantifiers seem harder to solve than Aristotelian\/counting quantifiers. Except for {\\em k\\%}, all proportional quantifiers---{\\em p\/k}, {\\em most}, and {\\em few}---are about 10\\% lower than the five counting quantifiers (except {\\em less than k}) with BERT; and about 5\\% lower with RoBERTa. RoBERTa is not generally superior to BERT; e.g., for {\\em k\\%}, BERT outperforms it by 22\\%. We show a pairwise analysis of how GQs affect performance when they appear in both the premises and hypotheses in the Appendix \\ref{app:pairwise}. Generally, our results attest to the difficulty of resolving GQs in NLI benchmarks.\n\n\n\\section{Quantifiers in Cross-lingual NLU Benchmarks}\nQuantifiers are acquired in similar orders across languages \\cite{Katsos2016CrosslinguisticPI}, although languages express quantifiers in different ways. For example, there are eight different universal quantifiers with different level of distributivity in Malagasy \\citep{Matthewson2008QuantificationAC}. This poses challenges to training multilingual LMs and transfer learning. We are interested in whether quantifiers are universally and evenly challenging for all languages. \n\n\\input{tables\/xnli_result.tex}\n\n\\paragraph{Quantifiers in Cross-lingual NLI}\nWe choose XNLI \\citep{conneau-etal-2018-xnli}, a manual translation of the development and test set of MNLI into 15 languages, for this multilingual error analysis. We should clarify that for XNLI, the authors annotate entailment labels for the English data only and apply them to the other languages. We do not assume label changes due to translation in this study, but it is worth investigate in the future. We choose five languages belonging to different language families, namely Arabic, Chinese, German, Spanish and Vietnamese as targets. \nThe last column in Table~\\ref{tab:task_stats_nli} shows the numbers of quantifiers in XNLI. The distribution rate is 10\\%. Note that the universal quantifier is the most common quantifier in XNLI. \n\nWe fine-tune mBERT\\footnote{\\texttt{multi\\_cased\\_L-12\\_H-768\\_A-12}} \\citep{devlin-etal-2019-bert} and XLM\\footnote{\\texttt{xlm-mlm-100-1280}} \\citep{Lample2019CrosslingualLM} on the MNLI training set and evaluate them on XNLI. We report the results in Table~\\ref{tab:xnli_result}. We find that performance varies across languages. For Chinese and Vietnamese, we see significant drops in performance for examples with GQs, whereas for Arabic and German, we see improvements. The results {\\em per} quantifier are more homogeneous, however. \n\nSimilar to our results for English, we can see that the lowest accuracies in XNLI are with proportional quantifiers, such as {\\em most} and {\\em few}. But the gap in non-English languages is wider for these two categories, especially for Chinese, the difference reaches 30\\%. Other hard quantifiers include {\\em all}, $>k$, $<k$, and {\\em each other}.\n\n\\paragraph{Quantifiers in Cross-lingual QA}\n\\input{tables\/task_stats_qa.tex}\n\\input{tables\/xqa_result_mbert.tex}\nCross-lingual question answering (XQA) is another important  NLU task that evaluates the cross-lingual transferability of LMs. We evaluate the effect of quantifiers on system errors across two XQA datasets, namely XQuAD \\citep{artetxe-etal-2020-cross} and MLQA \\citep{lewis-etal-2020-mlqa}.\nAs demonstrated in Figure~\\ref{fig:frequency}, quantifier word distributions in XQA tasks also follow Zipf's law, as in NLI tasks, but {\\em k} is more frequent (perhaps because of a traditional emphasis on numerical reasoning), and we see less variance across languages. This is probably because question answering is targeting quantification less directly. \nTo evaluate cross-lingual QA performance on GQs, we fine-tune mBERT and XLM-R\\footnote{\\texttt{xlm-roberta-large}} \\citep{conneau-etal-2020-unsupervised} using \\citet{hu2020xtreme}'s architecture. We present results for mBERT in Table~\\ref{tab:xqa_result_mbert}; for XLM-R results, please refer to Appendix~\\ref{app:xqa_result}. \n\nJust as with XNLI, LMs suffer from performance drops across all languages for almost all GQ phenomena with significant, cross-lingual variation. The most distinguished is that Exact Match (EM) suffers from a greater deterioration than F1 scores for all languages. For example, the weighted EM difference for mBERT on MLQA is 2.9\\% while the weighted F1 is 1\\%. As one example in Table~\\ref{tab:examples}, we observe that the plausible answers selected by models, while being incorrect, result in a sharper decrease of EMs comparing to F1s. Questions containing GQs also tend to have less verbal answers comparing to those without GQs, and therefore require higher precision. \n\nRegarding cross-lingual comparisons, Chinese and Arabic are the two languages that do not have lower performance over GQs compared to the performance over the complete dataset. Despite the overall trends, subtle differences from XNLI performance still exist. For example, XLM-R is worse than mBERT on quantifier reasoning on XQuAD Chinese, especially at proportional quantifiers, but this is not the case on MLQA Chinese.\n\n\n\\section{GQNLI}\n\\input{tables\/gqnli}\nWe have seen how quantifiers present challenges to NLI and QA models. \nUsing an approach similar to ANLI \\citep{nie-etal-2020-adversarial} and DynaBench \\citep{kiela-etal-2021-dynabench}, we use model difficulty (RoBERTa's) as a heuristic to select hard examples for a challenge dataset that can hopefully be used to evaluate any future progress on this. \nWe propose GQNLI, a generalized quantifier NLI challenge dataset, consisting of 30 premises and 300 hypotheses. The average sentence lengths of hypothesis and premises are 15.97 and 7.35, respectively. Both numbers are comparable to those of MNLI, but lower than ANLI's \\citep{williams2020anlizing}. It should be noted that  GQNLI is designed for evaluating future models; obviously not for benchmarking RoBERTa.\n\n\\paragraph{Dataset Creation}\nFirstly, we manually create 100 premise-hypothesis pairs, in which various types of GQs appear. For each premise and hypothesis, the number of GQs varies from one to three. \nTo choose the premises, we randomly sampled 100 premises with GQs from SNLI and ANLI test sets, respectively, and selected 10 premises in total, that we consider are semantically adequate for adding GQs and making simple hypotheses.\n\nTo construct the hypotheses, we rely on \nRoBERTa fine-tuned on MNLI\nand manually select examples about which the model is unsure or incorrect. To focus on GQs, we keep the challenge examples otherwise simple \\cite{ribeiro-etal-2020-beyond}, and avoid lexical variations in the hypotheses. Hard examples were found to be characterized by (i) mixing generalized quantifiers with other logical operators, such as subsumption or negation, and (ii) combining multiple different generalized quantifiers. \nWe discuss these observations in Section \\ref{sec:discussion}.\n\nTwo of the authors annotated the examples. The inter-annotator agreement (Fleiss' kappa) was 0.895, substantially higher than ANLI's (0.672--0.740). It is worth noting that the level of  semantic or pragmatic interpretation difference of GQs is reflected in the measurement. \n\nWe augmented the examples by substituting non-quantifier words (e.g., replacing ``dogs'' with ``cats'') while keeping the labels, to exclude the effect of specific lexical items. The resulting labels are uniformly distributed. Table~\\ref{tab:gqnli} presents GQNLI statistics. Since the dataset is curated to probe the ability to reason with quantifiers, the distribution of generalized quantifiers does not follow Zipf's law; see \\S\\ref{sec:Quantifiers in Four English NLI Tasks}. A list of GQNLI examples per category is shown in Appendix \\ref{app:gqnli_example}.\n\n\\paragraph{Experiments and Results}\nWe evaluate seven types of models on GQNLI, fine-tuned with different combinations of NLI datasets. As data creation only relied on RoBERTa and MNLI, nothing prevents that models with different architectures and training data will perform well. They do not, however. The results are shown in Table~\\ref{tab:gqnli}. \n\nWe see that all models have great difficulty with GQNLI. With more training data, models improve, but the best performance is 42.7\\%, less than 10 points above chance level. In general, the counting quantifiers, especially the existential and universal quantifiers, are easier than proportional quantifiers. Particularly, most models struggle with {\\em less than k} and {\\em few}. This is in some contrast with the NLU tasks studied above, where these quantifiers were among the easiest.\n\nWe also observe unstable GQ reasoning ability in simple word substitution cases. For instance, it happens for DeBERTa fine-tuned with M, F, Ling, DocNLI that it predicted correctly the contradiction relation between ``There are six children standing on top of a yellow mountain. Two thirds wear red tops and one third wear green.'' and ``Between 80\\% and 90\\% children do not wear red tops.'', but incorrectly when ``red'' is substituted  with ``beige'' and ``green'' with ``cyan''. We are yet to study what kind of cues lead to the instability. Our experiments suggest a lack of testing proportionality reasoning and robustness in existing benchmarks.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\paragraph{Negation}\nThe interaction between negation words and quantifiers increases semantic complexity \\citep{10.2307\/25000447,Horn2010TheEO}. We investigate whether this holds for NLI tasks, using negation cue detection to find all cases where a negation word and a quantifier appear in the hypotheses. \n\nWe break down the performances on negation of the seven models in Appendix \\ref{app:negation}. As indicated, LMs overall have worse results for negation cases comparing to the entire dataset. We can see a majority of the models even predicted opposite labels for some GQ categories, with 0\\% accuracy. DeBERTa-v3 is no longer the best model, replaced by BERT and BART. The improvement by training with more data becomes unstable for reasoning over GQs with negation. \n\nFor a cross-lingual investigation of the interaction of GQs and negation, we find that in XNLI, the number of cases combining both phenomena is insufficient: we identified four such cases, involving only the quantifiers ``all'' and ``more than.'' For English, mBERT predicted two cases successfully. For Chinese, German, Vietnamese and Arabic, one is correct. For Spanish, all are wrongly predicted.\n\nIt is evident that NLU models suffer from reasoning difficulties in certain cases when negation interacts with GQs, especially in cross-lingual evaluation. In future work, we are interested in expanding GQNLI to more instances and more languages to facilitate qualitative investigations.\n\n\\paragraph{Subsumption}\nIn generalized term subsumption languages \\citep[TSLs;][]{Yen1991GeneralizingTS,ali1993natural}, a term \\(a\\) subsumes another term \\(b\\) if and only if the extension of \\(a\\) is a superset of the extension of \\(b\\) . Rather than surface number comparison, subsumption reasoning requires knowledge of the relations between supersets and subsets. For example, to decide whether ``There are six dogs. Three brown dogs, a black dog and a white dog run along the green grass'' entails ``One dog sits'', LMs should be aware that ``six dogs'' is a superset of the extension of the ``brown dogs'', ``black dog'' and ``white dog''. Another example in GQNLI is to infer whether ``There are twelve singers on a stage, less than half from Argentina and one from Cape Verde'' entails ``Several singers do not come from Chile''. \n\nWe annotate 63 cases out of the first 100 in GQNLI requiring subsumption reasoning. \nWe show the statistics and results regarding subsumption in Appendix \\ref{app:Subsumption}. It can be seen that more training data leads to higher accuracies. Especially, DeBERTa fine-tuned with DocNLI, which unifies the two classes \"neutral\" and \"contradict\" into a new class \"not entail\", has a significant improvement on subsumption cases with neutral label. The training bias give an advantage to the model on the subsumption subset, half cases of which are labelled neutral. But such bias has a negative effect on non-subsumption cases; the accuracy drops by 16.2\\% comparing to the model without training with DocNLI. It is worth investigating whether DocNLI is truly helping subsumption reasoning in future work. Subsumption is a key concept in the study of knowledge representation \\citep{Woods1991UnderstandingSA}, but is neglected in current NLP research. The fact that LMs struggle to perform subsumption reasoning asserts the necessity to explicit tackle the problem.\n\n\\section{Related Work}\n\nWe examine the sensitivity of NLU models to generalized quantifiers. These models are designed to induce correlations from large volumes of data, not to reason symbolically with logical quantifiers. Such models have, nevertheless, been probed for logical knowledge.\n\n\\citet{mul2019siamese}, for example, show neural networks encode fragments of first-order logic and exhibit zero-shot generalization ability. \\citet{evans2018can} present a neural architecture that improves performance on propositional logical inference. \\citet{bowman-etal-2015-recursive} also suggest neural networks learn semantic representations for logical inference in natural languages. However, on the same task, \\citet{veldhoen2017can} find neural networks fail to do so on a more stringent test. \\citet{geiger-etal-2019-posing} also show that neural networks fail to exhibit robust logical inference. \\citet{srivastava-etal-2018-zero} use semantic parsers to encode quantifiers and improve zero-shot learning in classification tasks. \\citet{haruta-etal-2020-logical} present a system that computes logical inference over GQs and see improvements on two specialized datasets, FraCaS \\citep{cooper1994fracas} and MED \\citep{yanaka-etal-2019-neural}. None of these papers explicitly discussed generalized quantifiers, and all were limited to studying the ability of neural networks to capture the logical semantics of English. \n\nMany studies have instead focused on LMs' ability to capture negation \\citep{gururangan-etal-2018-annotation,naik-etal-2018-stress,hossain-etal-2020-analysis,ettinger-2020-bert,hartmann-etal-2021-multilingual} or coreference \\citep{ye-etal-2020-coreferential,varkel-globerson-2020-pre,abdou-etal-2020-sensitivity}.\nOthers have focused on LMs' ability to reason with numbers \\citep{johnson-etal-2020-probing}. DROP \\citep{dua-etal-2019-drop}, for example, is a question answering dataset designed specifically to probe LMs' ability to count, add and subtract for answering factoid questions. Models have also been tailored for numerical reasoning  \\cite{geva-etal-2020-injecting, zhang-etal-2020-language-embeddings}.\n\n \\citet{Cobbe2021TrainingVT} proposes to use a verification task during pretraining of LMs to improve their ability to solve math word problems. \nOthers have studied monotonicity inference \\citep{hu-etal-2019-natural,yanaka-etal-2019-neural,yanaka-etal-2020-neural}, and\n\\citet{Fang2021PartW} recently focused on the two quantifier words {\\em part} and {\\em whole} in an error analysis for named entity recognition. \n\nMany NLU benchmarks contain quantifier words, but their influence on performance has not been studied systematically. One exception to this is that generalized quantifiers have been used to generate adversarial examples in the context of numerical reasoning \\citep{naik-etal-2018-stress, nie-etal-2020-adversarial}. \nTaxiNLI \\citep{joshi-etal-2020-taxinli}, which categorizes 15 types of reasoning abilities, is a dataset drawn from MNLI. In their taxonomy, the Quantifier category only refers to universal and existential quantifiers, {\\em not} to generalized quantifiers, and ditto for \\citet{kim-etal-2019-probing}. All of the above focused on English, but in an extension to TaxiNLI, \\citet{k-etal-2021-analyzing} incorporated quantifiers into the Logic class and found a large cross-lingual transfer gap on LMs. \n\n\\section{Conclusion}\nQuantifiers lie in the intersection of logic, linguistics and NLP research. It is essential for NLU systems to learn quantifier reasoning. We examined generalized quantifiers in multilingual NLU tasks with regards to their expressiveness and logical reasoning requirement. Our survey and experiments indicate quantifiers are neglected to a degree and cause significant performance drops for neural LMs. To better understand LMs' reasoning abilities, we release GQNLI, a novel generalized quantifier NLI challenge dataset. With the pervasiveness of generalized quantifiers, we stress that more efforts are necessary to investigate: (1) when and why models systematically fail when quantifiers interact with other operators; (2) how to improve cross-lingual transferability of quantifiers; (3) how we can exploit the theoretical results about generalized quantifiers from logic and linguistic studies, so as to improve the logical inference ability of neural LMs.\n\n\\section*{Acknowledgements}\nWe would like to thank Miryam de Lhoneux,  Constanza Fierro,  Desmond Elliott and the anonymous reviewers for\ntheir valuable feedback.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Potts model}\nThe $Q$-state Potts model, a generalization of the Ising model\nto $Q$ different spins, has been the subject of considerable interest \\cite{Wu82}.  Two important cases are $Q=1$ and $Q=2$, which correspond to percolation \\cite{StaufferBook94} and the Ising model, respectively. When a Potts  system is prepared at its critical temperature, subsets of the clusters of like spins, the Fortuin-Kastelyn (FK) clusters \\cite{Fortuin72, Kasteleyn69} (to be defined below), are self-similar fractals \\cite{Mandelbrot82}. For  $Q=1$ the FK clusters are the same as the usual percolation clusters. In this paper, we will study the harmonic measure of the hulls of these fractal clusters for $Q=1,2,3,4$. \n\nThe harmonic measure may be thought of as the distribution of the surface electric field on a charged conductor.  Since the Laplace equation and the steady-state diffusion equation are identical in form, the harmonic measure is also equal to the distribution of probabilities of random walkers diffusing far from the cluster onto a given section of the hull. In this article, we use a biased random-walk sampling technique to obtain the harmonic measure. We also review other methods for measuring small probabilities and give details of our algorithms. \n\nThe harmonic measure is of practical interest because of its relation to the anomalous frequency dependence of the impedance of rough electrodes \\cite{Halsey92} and because of its obvious connection to processes that involve absorption of diffusing particles such as  catalysis \\cite{Sapoval01}.  It has a deep connection to the structure of diffusion-limited aggregates (DLA) \\cite{Witten81}, since the harmonic measure determines where each walker will land; that is, for DLA it is the growth probability. In the case of critical Potts clusters and DLA, the harmonic measure is multifractal \\cite{Mandelbrot90}. Advances in conformal field theory and  Schramm-Loewner evolution (SLE) have brought about renewed interest in the harmonic measure. In particular, certain aspects of the harmonic measure for Potts clusters can be computed  in the continuum limit using these methods \\cite{Duplantier99, Duplantier00, Belikov08, Duplantier08, Bettelhiem05, Gruzberg06}.\n\nNumerical investigation of  the harmonic measure of percolation \\cite{Meakin86} and DLA \\cite{Meakin86, Ball90, Hanan08} clusters is difficult because the measure has a huge dynamic range for  systems of even moderate size. In references \\cite{Meakin86, Ball90, Hanan08}  one of two methods were used: The first is the straightforward one of releasing a large number of random walkers far from the cluster and determining where they land. The second uses relaxation or equivalent algorithms to solve the Laplace equation. The random walker method can only measure probabilities down to about $10^{-10}$ and samples a very small part of the measure for clusters of reasonable size. Relaxation-like methods are computationally costly and limited to small clusters, and give similar lower limits on the probabilities that can be measured.\n\nFor DLA  it is possible to go to much smaller probabilities by using the method of iterated conformal maps \\cite{Hastings98, Davidovitch00, Davidovitch01}. However, this technique is only capable of treating moderate size  clusters \\cite{Jensen02}.  In an earlier publication we generalized the random walker method and gave a  technique capable of obtaining probabilities down to $10^{-300}$ for any fractal. We applied it to FK clusters for percolation and the Ising model  \\cite{Adams08}. This paper describes a further development of those techniques.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{Etched_Cluster_a_fig1}\n\\caption{\\label{fig:picture}  (Color online) The harmonic measure for the complete perimeter of a small, $W=64$, percolation cluster. The solid grey regions represents the area that is inaccessible to the random walkers diffusing from above the cluster. Every perimeter site is colored according to its measure. The computation was performed using the etching method described below. Note that the measure on this small cluster spans $50$ orders in magnitude.}\n\\end{figure}\n\n\n\\subsection{Generalized dimensions}\nThe harmonic measure, the distribution of probabilities that random walkers will hit a given site on the perimeter of a cluster, is very complicated and varies wildly for the cases we are studying; see Figure \\ref{fig:picture}. A popular and useful way to characterize it is  in terms of the generalized dimension, $D(q)$,  of the measure. We define these objects as follows: We cover the hull with boxes of length $L$. With each box we  associate a probability, $p_i$, which is the sum of the measure over the sites within the box. We then define  a function $Z_L(q)$, sometimes called the partition function:\n\n\\begin{equation}\n\\label{eq:Zq}\nZ_L(q) = \\sum_i p_{i}^q\n\\end{equation}\nwhere $q$ is some power  \\cite{Halsey86}.  If the object in question is fractal, then the partition function will follow a power-law in $L$:\n\\begin{equation}\n\\label{eq:Zq2}\nZ_L(q) \\sim (R\/L)^{-\\tau_q} = (R\/L)^{-(q-1)D(q)}\n\\end{equation}\nfor $(R\/L) \\rightarrow \\infty$, where $R$ is the size of the cluster. For integer $q$, $D(q)$ corresponds to the fractal dimension of the $q$-point correlation function. There are special values of $D(q)$. $D(0)$ is the fractal, box-counting, dimension of the hull. Also $D(1)$, the information dimension, is always unity by Makarov's theorem \\cite{Makarov85}.  A related function is the singularity spectrum $f(\\alpha)$,  the Legendre transform of $D(q)$:\n\n\\begin{equation}\n\\label{f_alpha}\nf(\\alpha) = q \\frac{d \\tau}{d q} - \\tau, \\quad \\alpha = \\frac{d \\tau}{d q}.\n\\end{equation}\nIn this article, we will focus exclusively on the $D(q)$. The singularity spectrum can be derived from our results using Eq.~(\\ref{f_alpha}).\n\n\\section{Models}\n\n\\subsection{Simulations of FK clusters for the Potts model}\n\nWe produce critical Potts clusters in two ways. For percolation, we use the Leath algorithm \\cite{Leath76}. The algorithm starts with a single active site; we attempt to turn its neighbors into active sites with probability $p$. If a conversion attempt fails, the site is labeled inactive. The process is repeated with neighbors of the active sites which have not been labeled as inactive. The process continues until there are no new active sites. If $p$ equals $p_c$, the percolation threshold, a critical percolation cluster is produced. The outer layer of active sites is called the complete perimeter.  Its fractal dimension is denoted $D_H$. The cluster of active sites is surrounded by a single layer of inactive sites; this layer is called the accessible (or exterior) perimeter \\cite{Grossman87}, and has a fractal dimension denoted $D_{EP}$. The accessible perimeter of interest because, unlike the complete perimeter, it is expected to have a well-behaved limit when clusters are very large and are rescaled. The harmonic measure has been determined in this limit for the accessible perimeter of Potts clusters \\cite{Duplantier99,Duplantier00}.  \n\nTo obtain critical Potts clusters for $Q=2,3,4$, we grow equilibrated FK clusters using the Swendsen-Wang (SW) algorithm \\cite{Swendsen87}.  For any configuration of spins, FK clusters are subsets of clusters of like spins formed by a  bond percolation process. That is, we consider the clusters formed when adjacent spins are connected with probability $p_c(Q) = 1-\\exp[-K_c(Q)]$ where $K_c(Q)$ is the critical coupling constant. For $Q=2,3,4$, on the triangular lattice, $p_c(Q)$ is known to be $1~-~1\/\\sqrt3$, $1- 1\/[1+\\frac{1}{2}\\sqrt{3}\\sec({\\pi}\/{18})]$, and $1\/2$, respectively \\cite{kim1974exact}. To obtain the equilibrium ensemble of FK clusters we iterate two steps until the system settles down (see below).  The first step   takes every current FK cluster  and replaces the spin  with one of the $Q$ possible values, at random.  In the second step, the bonds connecting the clusters are discarded and bond percolation is performed again, with $p=p_c(Q)$, on all neighboring sites with the same spin.   The process is then repeated by updating the spins on the newly formed clusters. These two steps together constitute  a spin update.\n\n\n\\subsection{Parameters and Observables}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{Signpost_Visualization_b_fig2}\n\\caption{\\label{fig:Signpost} (Color online) The signpost method. The system is periodic in the horizontal direction. (a) First (probe) step: $N$ random walkers are released from the top row and absorb onto the perimeter sites.  (b) We choose the first probability threshold as $0.1$. Using this threshold, we connect the the bounding sites using signpost sites. In the second (measurement) step we send $N$ more random walkers from above which can absorb onto the signpost or perimeter sites. (c) Second probe step: the random walkers  are launched from the signpost sites in the previous measurement step.  The weights of the walkers released in this step have a weight of $p\/N$ where $p$ is the fraction of the random walkers that hit the signpost site in  (b). (d) The second threshold, $0.01$, is used to determine the location of the new signpost sites.  }\n\\end{figure}\n\nWe grew critical Potts clusters for $Q=1-4$ on the triangular lattice, as described above. We chose to use a triangular lattice rather than  a square lattice because the square lattice does not allow diffusion into fjords bounded by diagonal entrances. We use the width of the system, $W$, as the characteristic length.  The clusters we want span in the width direction but not in the height direction. To make sure the clusters will only span in one direction, we chose very large aspect ratios. The height of the lattices were $100 W$ and $8 W$ for $Q=1$ and $Q>1$, respectively.  We looked at six different system widths, $W= 128, 256, 512, 1024, 2048, 4096$. Because FK clusters are intrinsically bond clusters, we needed to use a trick to turn them into site clusters.  We created a lattice twice as dense as the original and marked every site at the center of a bond and every site where two bonds meet as cluster sites. The FK cluster widths used were $W\/2 = 64, 128, 256, 512, 1024, 2048$.   \n\n\nTo have proper FK clusters we require equilibration in the SW algorithm. We numerically determined  that the equilibration time for $Q=2, 3, 4$ is of the order of $W$ spin updates by looking at the relaxation of the average energy per spin and the average largest cluster size.  For  $Q = 2,3$ and small $W$ we ran a separate simulation to  equilibrium for  each spanning cluster which was added to our ensemble.  For $Q=2$ and $3$ and $W=2048$ and $4096$ and for all of the $Q=4$ clusters, the equilibration time was too large to  proceed in this way.  In these cases we equilibrated the system once and recorded an ensemble of spanning clusters as the simulation proceeded. We conservatively estimate the correlation time as $50$ spin updates for all $W$ and $Q$. This means we recorded a spanning cluster every $50$ spin updates. \n\nFor each system size we grew a number of clusters. For all  $Q$ our ensemble was  $2000, 2000, 1000, 1000, 400, 100$ clusters for $W = 128, 256, 512, 1024, 2048, 4096$, respectively.\n\n\\section{Measuring small probabilities with random walkers}\n\n\n\\subsection{Previous Methods}\nSmall probabilities in the harmonic measure correspond to very unlikely paths. As the simulation proceeds we can think of the event of a random walker landing where the measure is very small as a rare event. Thus computing small probabilities is a similar task to finding the rate of a rare chemical reaction \\cite{vanKampen01}, a rare extinction of a disease \\cite{Andersson00} or a population \\cite{Bartlett60}, or the failure of a queuing system via queue over-flow \\cite{Medhi03}.\n\n Accelerated numerical methods for these problems often involve biased event sampling. The sampling can frequently be cast as a random walk, either through state space or in our case, physical space. For example, one could ask what is the probability that a random walker starting half-way up a hill will successfully climb up to the top before sliding down to the bottom.  If the hill is steep, it could be impossible to directly sample the probability to climb the hill.  One could place barriers uniformly on the hill, which when crossed by the random walker, will split the random-walker into two walkers, each with equal weight which add up to the original weight of the walker.  This will aid sampling of the events higher up on the hill.  This method is called ``splitting\" and effectively performs importance sampling \\cite{Hammersley65}. One significant drawback of splitting is that if the barriers are too densely or sparsely spaced, the number of random walkers will tend to diverge or extinguish, respectively.  \n\nThe methods we detail in this paper are related to the splitting method, but differ in that our methods do not have the possibility of diverging or extinguishing.  Another popular method called ``milestoning\" \\cite{Farajian04}, does not have a divergence problem, but does require the system studied to be in equilibrium and the location of the barrier to be known \\textit{a priori}, whereas our method works for equilibrium and non-equilibrium systems and the barriers are placed `on the fly.'\n\n\n\\subsection{Signposts}\nWe have developed several accelerated methods for the harmonic measure problem. The motivation, as we have stated, is  that  it is usually impossible to send in enough random walkers  to directly obtain the harmonic measure: the clusters will frequently have regions with probabilities of being hit that are smaller than $10^{-100}$. It would require of order $10^{100}$ random walkers to sample this region; such a computation is clearly impossible. \n\nWe now review the first method we developed, the signpost method \\cite{Adams08}.  The signpost method consists of two steps which are applied iteratively; see Figure \\ref{fig:Signpost}. In the first (probe) step we release $N$ diffusing random walkers far from the cluster to determine which regions  are rarely visited in straightforward sampling. Next, we block off all poorly sampled regions with signposts (absorbing lines). In the second (measurement) step, $N$ more walkers are released far from the cluster and either absorb on the cluster (or the accessible perimeter) or onto the signposts. The walkers sent in this step have their weight permanently added to the harmonic measure of the perimeter sites where they landed. In the next probe step, the walkers are released from the points on the signposts where the walkers in the previous measurement landed. The new walkers have a weight of $p\/N$, where $p$ is the fraction of random walkers that absorb onto signpost lines in the previous step, to conserve probability. The probe step again helps determine which regions are still poorly sampled, which are subsequently blocked off. Next, another measurement step is performed. This process is repeated until all regions are explored by the random walkers. This algorithm can be applied to on- and off-lattice clusters.  \n\n\\begin{figure}[t]\n\\includegraphics[width=0.48\\textwidth]{Etching_Visualization_e_fig3}\n\\caption{\\label{fig:Etching}  (Color online) The etching method. Walkers are released from the current level sites. The next level of soft sites absorb walkers; they are then relabeled as current level sites. Future sites are all sites which will eventually become current level sites. The first round of random walkers are launched from the row above the cluster, (a). The weight of all of the walkers released is $0.2\/N$, where $N$ is the number of walkers released per current level site.  $20\\%$ of the walker weight is deposited onto the top row of perimeter sites and the next level soft site, which will release $N$ walkers in the next step.  (b). One more  perimeter site is accessible to the random walkers and $5$\\% of the weight is deposited on the  site in the next level. (c) Three sites in the next level each absorb $1$\\% of the walker weight. (d) Due to the reduced weight of the walkers released in the next step, small probabilities are measured on the newly exposed perimeter sites.}\n\\end{figure}\n\nWe should note some things about this method.  First, one must determine the entire perimeter of the cluster at the beginning of the computation in order to figure out how to block poorly sampled regions.  Also, one needs to choose a rate to reduce the threshold for calling areas ``poorly sampled\" in each iteration. In \\cite{Adams08}, we moved the threshold down by a power of $10$ each iteration, whereas in \\cite{Adams09}, we reduced it as a function of how many walkers hit the signpost in the previous iteration.  When more walkers hit the signposts we moved them even deeper.  The second method gave more consistent walker saturation, which should lead to a slower compounding of error. It is important to note the signpost algorithm is only practical for two-dimensional problems. For higher dimensions, one would need to define signpost \\emph{surfaces} to block poorly sampled regions. This is could be very complex for a complicated cluster.  \n\n\\subsection{Etching}\nWe now describe the method we use here which we call  ``etching.\"  Consider the hull of FK clusters grown on a triangular lattice with periodic boundary conditions. We want to find the harmonic measure of the top perimeter from above.  To do this, we start by marking all sites that are exterior to the cluster from above as \\emph{soft} sites; the soft sites are absorbing like the cluster (or accessible perimeter) sites. The highest row is limited to one level above the highest point on the perimeter. \n\nWe next relabel every site on that highest row as a \\emph{current level} site; these are not absorbing. We release $N$ random walkers, each with weight $1 \/ (N W)$, from each current level site. The walkers released from these sites are allowed to walk until they deposit their weight onto a soft site or a perimeter site.  If they move one level further away from the cluster, they are immediately moved back onto the current level sites using a Green's function which must be determined in advance. However, this is rather simple since it is the Green's function to return to a plane from one site above the plane.  This Green's function is used for the entire simulation and limits how far a walker can backtrack to at most one level above the cluster. After all random walkers are released, the labels on each current level site are removed and every soft site hit in the previous step is labeled as a current level site.  From each current level site $i$ we release $N$ random walkers with weight $p_i$, where $p_i$ is the amount of probability deposited on the site in the previous step divided by $N$. This process is repeated until there are no more soft sites. See Figure \\ref{fig:Etching}. \n\nEtching can be thought of the limit of the signpost method with the signposts spaced one site apart.  However, etching has several benefits over the signpost method. First, the entire perimeter of the cluster does not need to be mapped out before we start. Both algorithms have the same time complexity, $O(W^3)$ for the complete perimeter of Ising clusters, and both methods have similar memory requirements. In contrast to the signpost method, the etching method can be easily generalized to higher-dimensional lattice problems and networks. We have successfully used etching to obtain the harmonic measure of three-dimensional percolation clusters \\cite{Adams09b}.  \n\n\\begin{figure}[t]\n\\includegraphics[width=0.45\\textwidth]{Log_Z_vs_Log_L_fig4}\n\\caption{\\label{fig:Zq} An example of the fit of $\\log_{10} Z(L,q)$ versus $\\log_{10} L$ to a straight line for $q = 2.0379$. The behavior is similar for all $q$ values that we have examined. The slope of the line is $\\tau(q) \\equiv (q-1)D(q)$.}\n\\end{figure}\n \n\\subsection{Green's functions}\n\nWe have also developed a rare-event method which  may be significantly more efficient than etching and signposting for some problems. Thus far we have applied this method only to simple test problems.  This method  manipulates probabilities directly and does not allow backtracking of probability. To do this,  we calculate the Green's function $G(i,j;k,l)$, i.e., the probability to move to any of the sites $i,j$ in the next level from a given site $k,l$ in the current level. \n\nTo illustrate our algorithm, consider finding the probability distribution a channel with absorbing walls on a square lattice. The initial condition is that the probability is uniformly distributed among the sites in the first row of  the channel and the zeroth row is a reflecting boundary.  All sites that initially have probability are denoted by  $C$. The previous level  sites, absorbing sites, and next level sites accessible to the current level sites are denoted by $B$, $A$, and $N$, respectively. (Initially, the previous level is the reflecting boundary.) In each iteration, the goal is to move all of the probability from  each current level site to the all the next level and absorbing sites. \n\nWe find the Green's function by iteration on an index $s$. The process begins for some current level site,   $k, l$; $(k,l) \\in C$.  Initially, probability only resides at $k,l$ so that for  $s=0$,  $G^s(i,j;k,l) =  \\delta_{i,k} \\delta_{j,l}$. In each iteration, the probability is moved to each of the current level site's neighbors\n\\begin{equation}\n\\label{eq:update}\nG^{(s+1)}(i,j;k,l) = \\sum_{(m,n)} W(i,j;m,n) G^{(s)}(m,n;k,l), \n\\end{equation}\nusing the jump probability, \n\\begin{eqnarray}\nW(i,j;m,n) &=& \\frac{1}{4} ( \\delta_{i,m+1}\\delta_{j,n} + \\delta_{i,m-1}\\delta_{j,n} + \\delta_{i,m}\\delta_{j,n+1}  \\nonumber \\\\\n &+&  \\delta_{i,m}\\delta_{j,n-1} )  \\quad (m,n) \\in C \\nonumber \\\\\n  &=& G_B(i,j;m,n) \\quad (m,n) \\in B \\nonumber \\\\\n  &=& \\delta_{i,m}\\delta_{j,n} \\quad (m,n) \\in A \\cup N\n\\end{eqnarray}\nHere $G_B(i,j;m,n)$ is the Green's function for the previous level, see below, and the last line represents the probability staying at absorbing and next level sites. $G_B(i,j;m,n)$ takes into account all the processes that would correspond to random walkers backtracking before the previous level. To start the process, the reflecting boundary has $G_B(i,j;0,n) = \\delta_{i,1}\\delta_{j,n}$.\n\n\\begin{figure*}\n\\includegraphics[width=0.85\\textwidth]{D_q_external_panel_fig5}\n\\caption{\\label{fig:D_q_Q1234}  (Color online) The $D(q)$ spectrum for the accessible perimeters of $Q=1,2,3,4$ clusters, in (a), (b), (c), and (d), respectively. The solid lines are the theory of  \\cite{Duplantier00} and the symbols are the results of our simulations for several system widths. The vertical dotted lines marks $q_{min}$  for the theoretical spectra for infinite systems.}\n\\end{figure*}\n\nFor large $s$, virtually all of the probability will be on absorbing sites and next level sites. In any finite amount of time, some slight probability will remain in the current level, so after some stopping criteria is met, the probabilities recorded on the absorbing and next level sites must be normalized. When this has been achieved, we have the Green's function from a given site in the current level, $k,l$, to any site in the next level, $i,j$:\n\\begin{equation}\n\\label{eq:GreenNext}\nG_B(i,j;k,l) = \\lim_{s \\rightarrow \\infty } G^{(s)}(i,j;k,l).\n\\end{equation}\nIn the next step, this $G_B$ will be used as a jump probability.  \n\nThis process is repeated for all current level sites so that Green's functions from those sites to the next level sites and absorbing sites are calculated. With these Green's functions, it is easy to determine where the probability from the first level will end up. If the probability in the starting level is $P(k,l)$, then the probability in the next level is:\n\\begin{equation}\n\\label{eq:ProbAbsorbing}\nP(i,j) = \\sum_{ (k,l) \\in C } G_B(i,j;k,l) P(k,l)\n\\end{equation}\nNote that  $(i,j)$ can be  absorbing sites as well as  next level sites.\n\nThe next step is to relabel all current level sites as previous level sites,  relabel all next level sites as current level sites, and mark all sites that are accessible to the new current level sites (which are not previous or absorbing sites) as next level sites. Then the process is repeated.\n \n\n\n\n\n\nThe end result of this process is that all of the original probability is at absorbing sites, as it would be using signposting or etching. Although this example contained only sites that were completely absorbing or non-absorbing, the Green's function method can easily be generalized to partial absorption problems. \n\nThe Green's function method is somewhat more complex to program than the etching method and  the simplest implementation involves setting up the Green's function look-ups in sparse arrays. This leads to a memory complexity which grows like $W^{2d}$, where $d$ is the dimension of the space. The memory complexity would significantly reduce its usefulness, as it would take at least one terabyte to store a two-dimensional cluster with a length scale of $1000$ lattice sites.  However, it is possible to store the Green's function lookup in an associative array; this reduces the memory complexity to $W^{d-1 + D}$, where $D$ is the fractal dimension of the perimeter. For the external perimeter of two-dimensional percolation clusters the memory complexity grows like $W^{7\/3}$, which is quite close to the memory complexity for etching, $W^2$.  For a cluster with a length scale of $1000$ sites, the minimum required memory would be about ten megabytes for the Green's function method. \n\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{D_q_external_Qone_fine_fig6}\n\\caption{\\label{fig:D_q_Q1_fine}  (Color online) The $D(q)$ spectrum for the accessible perimeters of $Q=1$ clusters for small $q$. As the system size increases the simulated values increase, presumably to approach infinity for $q< -1\/24$.}\n\\end{figure}\n\n\\begin{figure}\n\n\\includegraphics[width=0.4\\textwidth]{D_q_Qone_complete_fig7}\n\\caption{\\label{fig:D_q_Q1_complete}  (Color online) The $D(q)$ spectrum for $Q=1$ for the complete perimeter. There is no theoretical prediction for this quantity. However, for $q$ substantially bigger than 0 we expect this result to be very similar to the result for the accessible perimeter since large probabilities will dominate the sum in Eq. (\\ref{eq:Zq}). The line labeled ``theory\" is for the accessible perimeter.}\n\\end{figure}\n\n\\section{Results}\nWe used etching  to find the harmonic measure of $Q$-state Potts model clusters. We analyze the measure by producing $D(q)$ spectra and histograms of the probability distributions. To obtain $D(q)$, we start by sectioning individual clusters into boxes of length $L$ as described above. Because we are using a triangular lattice, it is convenient to use a parallelogram aligned with the lattice as a box. After completely tiling the cluster with boxes, we define the probability within a box $p_{i,L}$ as the sum of the measure of   perimeter sites within the box. We then calculate $Z(L,q)$ using Eq.~(\\ref{eq:Zq}).  $D(q)$ is related to  $Z(L,q)$ by $(q-1)D(q) = m$, where $m$ is the slope of $\\log Z(L,q)$ versus $\\log L$. \n\nWe found that for a given $Q$ and $q$, all system sizes have similar local slope behavior over a range of $L$; see Figure \\ref{fig:Zq}. In order to average over the ensemble we average $\\log Z$. However, if we use the slopes for each individual member of the ensemble and average them we get virtually identical results. \n\nThe spectra of generalized dimensions for the external hulls of $Q=1-4$ are given in Figure \\ref{fig:D_q_Q1234}. In all cases the results are close to the theoretical predictions \\cite{Duplantier00}. The theoretical predictions include a divergence of $D(q)$ for $q < q_{min}$ for an infinite system, see below. Our simulation results increase rapidly with $W$ for this regime, as expected; see Figure \\ref{fig:D_q_Q1_fine}. \n\nFor completeness, we include the spectrum of generalized dimensions for the complete perimeter for the case $Q=1$; see Figure~\\ref{fig:D_q_Q1_complete}. There is no theoretical prediction for this quantity. For positive $q$ the results are close to those of the accessible perimeter shown in Figure \\ref{fig:D_q_Q1234}.  This is because, for positive $q$, large probabilities contribute most of the weight in $Z(q)$. Near $q=0$ the two spectra differ because there are significantly more sites with small measure for the complete hulls.\n\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{histogram_demo_external_Qone_fig8}\n\\caption{\\label{fig:rawhist}  (Color online) The histogram of the frequency of occurrence  of the values of $p$ for the accessible perimeter for $Q=1$. The points for various values of $W$ are superimposed.}\n\\end{figure}\n\nWe also considered the distribution of the values of $p$ directly, by making histograms of its frequency for all $Q$ and $W$. The histograms turn out to be power laws with negative powers near $-1$; for an example see Figure \\ref{fig:rawhist}.  Since the histogram is very accurately a power-law in $p$,  it is useful to plot the local slope of the histogram, which is shown in  Figure \\ref{fig:ProbDistQ1234} for the accessible perimeters for $Q=1-4$. We also show the local slope for the complete perimeter of $Q=1$; see Figure~\\ref{fig:ProbDistQ1complete}. The slope is calculated over about 10 orders of magnitude in $p$ for the accessible perimeter, and more than one order of magnitude for the complete perimeter. \n\nThe significance of the slope is that it gives information about the non-scaling aspects of the distribution, and, in particular, the value of $q_{min}$ mentioned above. If we call the slope of the histogram $-\\phi$ (so that $\\phi$ is a positive number) we see that the partition function of Eq. (\\ref{eq:Zq}) formally diverges if $q< \\phi-1$, or, said another way, we expect $D(q)$ to be undefined for $q< q_{min} = -1+\\phi$. This means that the partition function is dominated by a few instances of very small probabilities which does not scale as power-law in $R\/L$. \nThe values  for the limit of the spectrum agree well with the predictions of Duplantier \\cite{Duplantier99, Duplantier00}; see Figure \\ref{fig:ProbDistQ1234}. Note that the slopes are very nearly constant over about 40 orders of magnitude in $p$.\n\nThe slopes for the complete perimeter of percolation clusters are also constant over many orders of magnitude; see Figure \\ref{fig:ProbDistQ1complete}. In this case we find that $\\phi$ is very close to 1, and the limit of the spectrum is at $q_{min}=0$. There is no theory for this case, and no explanation for this intriguing result.\n\n\\begin{figure*}\n\\includegraphics[width=0.9\\textwidth]{histogram_external_panel_fig9}\n\\caption{\\label{fig:ProbDistQ1234}  (Color online) The local slope of the histogram of the frequency of occurrence  of the values of $p$ for the accessible perimeter for $Q=1,2,3,4$ in (a), (b), (c), and (d), respectively. Also shown (solid lines) are the theoretical predictions of the local slope from \\cite{Duplantier00}. Note that in (b) the smallest probabilities recorded were not from the largest system size, but were from $W=1024$. This can be understood by the fact that ten times as many clusters were generated for $W=1024$. That is, among the many samples at $W=1024$, a few abnormally deep clusters were recorded which happened to have the smallest probabilities.}\n\\end{figure*}\n\n\n\n\n\n\\section{Error estimate}\nSince etching involves sampling the probability, there will be errors due to the finite number of random walkers released at each step. \nFor the results in this paper, we released $10^3$ random-walkers per current level site for all system widths and $Q$ values. \n\nWe can estimate the sampling errors as follows: we considered one percolation cluster with $W=2048$ and made 10 independent computations of the $p_i$. The variance of the probability over this sample at a given point on the cluster, $\\delta p_i$,  is a measure of the reliability of the measurement. In our case we found that some points have a rather large percentage error, though always less than a  factor of 3, but the average over all the points, $\\langle \\delta p_i\/p_i\\rangle$, was 23\\%. Note that the very small probabilities well inside the cluster have very small errors. There is no build-up of the error as we etch toward the interior, as might have been expected. \n \n If it is necessary to reduce the error further, more random walkers can be used. However, we believe that the ensemble averaging that we did means that the generalized dimensions are much more accurate than the individual probabilities. Our evidence for the last statement is the good quality of the fit in Figure \\ref{fig:Zq}, and the closeness of the results in Figure \\ref{fig:D_q_Q1234} to theory. Note also that $D(0)$ is close to the known fractal dimensions of the exterior perimeters.\n\n\n\\section{Conclusions}\nIn this paper, we presented the etching method,  a new accelerated technique for computing the harmonic measure.  We are able to measure probabilities as small as $10^{-4600}$. We showed how this method relates to other methods. We used etching to obtain the harmonic measure for the accessible perimeter of FK clusters for the  $Q$-state Potts model for $Q=1-4$, for a range of system sizes. We compared this data to theoretical predictions \\cite{Duplantier99,Duplantier00}. These theories were produced for a continuum model which, in principle, might not apply to the scaling limit of the $Q$-state Potts model on a lattice. In fact, we found good agreement between our numerical results and the theoretical predictions for every comparison we made, including the $D(q)$ spectra and the slopes of the power-law probability distributions. \n\n\\begin{figure}[t]\n\\includegraphics[width=0.45\\textwidth]{Histogram_Complete_Qone_fig10}\n\\caption{\\label{fig:ProbDistQ1complete}  (Color online) The local slope of the histogram of the frequency of occurrence  of the values of $p$ for the complete perimeter for $Q=1$.}\n\\end{figure}\n\n\nFor the complete perimeter of percolation clusters, we found the slope to be almost exactly $-1$ for about 4000 orders of magnitude. This suggests the smallest $q$ for which $D(q)$ is defined is $q=0$. This means that there are many instances of  small probabilities on the complete perimeter of percolation clusters which tend to diverge towards negative infinity faster than any power of $R\/L$. \n\nEtching, signposting, and the Green's function method are three tools which can find very small probabilities. The advantage of signposting is that it is natural to use in off-lattice systems, and, in fact, we have applied it to off-lattice DLA \\cite{Adams09}. Etching is simple to program and should easy to use in higher dimensional on-lattice systems. Lastly, the Green's function method is likely to be the most efficient of the algorithms for on-lattice and network systems, but it is more difficult to implement and requires more memory than etching. The etching and Green's function methods (but not signposting) can be used in problems which involve absorption probabilities less than unity.\n\n\n\\section{Acknowledgements}\n\nThis work was supported in part by  National Science Foundation grant DMS-0553487.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\subsection*{Background} Statistical shape analysis is now regarded across the board as an important area of applied mathematics as it has been and still is the source of quantities of theoretical works as well as applications to domains like computational anatomy, computer vision or robotics. Broadly speaking, one of its central aim is to provide quantitative\/computational tools to analyze the variability of geometric structures in order to perform different tasks such as shape comparison or classification. \n\nThere are several specific difficulties in tackling such problems in the case of datasets involving geometric shapes. A fundamental one is the issue of defining and computing metrics on shape spaces. A now quite standard approach which was pioneered by Grenander in \\cite{Grenander1993} is to compare shapes through distances based on deformation groups equipped with right-invariant metrics together with a left group action defined on the set of shapes. In this framework, the induced distance is typically obtained by solving a registration problem i.e by finding an optimal deformation mapping one object on the other one. It is thus ultimately determined by the deformation group and its metric for which many models have been proposed. In this paper, we will focus on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) of \\cite{Beg2005} in which diffeomorphic transformations are generated as flows of time-dependent velocity fields.\n\nDespite the versatility of such models, one of the other common difficulty in shape analysis is the multiple forms or modalities that shapes may take. Looking only at the applications in the field of computational anatomy, if early works have mostly considered shapes given by medical images \\cite{Rueckert1999,Beg2005} or manually extracted landmarks \\cite{Joshi2000}, the variety of geometric structures at hand has considerably increased since then, whether shapes are images acquired through multiple modalities (MRI, CT...) \\cite{Avants2008}, vector or tensor fields as in Diffusion Tensor Images \\cite{Cao2005}, fields of orientation distribution functions \\cite{Du2012} or delineated objects like point clouds, curves \\cite{Glaunes2006}, surfaces \\cite{Glaunes2008}, fiber bundles \\cite{Durrleman4}...   \n\nThe intent of this paper is to make a modest step toward one possible generalized setting that could encompass a rich class of shapes including many of the previous cases within a common representation and eventually lead to a common LDDMM matching framework. Our starting point is the set of works on curve and surface registration based on geometric distributions like measures, currents or varifolds \\cite{Glaunes2004,Glaunes2008,Charon2}. In the recent article \\cite{Charon2017} for instance, an oriented curve\/surface is interpreted as a directional distribution (known as oriented varifold) of its oriented tangent\/normal vectors, which results in simple fidelity terms used in combination with LDDMM to formulate and solve inexact matching problems. Yet all those works so far have restricted the role of distributions' representations to intermediates for the computation of guiding terms in registration algorithms; the underlying deformation model and registration problem remains defined over point sets with meshes.\n\nThe stance we take here is to instead introduce group actions and formulate the diffeomorphic matching problem directly in spaces of geometric distributions. In this particular work, we will restrict the analysis to objects in 2D and 3D and focus on the simpler subspace of discrete distributions, i.e that write as finite sums of Dirac varifold masses: Figure \\ref{fig:discr_shape_var} gives a few examples of objects naturally represented in this form. We shall consider different models of group actions and derive the corresponding optimal control problems, optimality conditions (Section \\ref{sec:optimal_control}) and registration algorithms (Section \\ref{sec:matching_algo}). This provides, on the one hand, an alternative (and theoretically equivalent) numerical framework to \\cite{Charon2017} for curve and surface matching using currents, oriented or unoriented varifolds. But the main contribution of our proposed model is that it extends LDDMM registration to the more general class of objects representable by discrete varifolds. In Section \\ref{sec:results}, we will show several examples of synthetic data besides curves or surfaces that can be treated as such, including cases like multi-directional objects or contrast-invariant images.          \n\n\\subsection*{Related works.} A few past works share some close connections with the present paper. For instance, \\cite{Cao2005} develops an approach for registration of vector fields also within the LDDMM setting. The discrete distributions we consider here are however distinct from vector fields as they should rather be interpreted as unlabelled particles at some locations in space with orientation vectors attached (and with possibly varying number of orientation vectors at a single position) as opposed to a field of vectors defined on a fixed grid. In particular, our approach will be naturally framed in the Lagrangian setting as opposed to the Eulerian formulation of \\cite{Cao2005}. The geodesic equations for the pushforward group action that are derived in Section \\ref{sec:optimal_control} can be also related to the framework of \\cite{Sommer2013} where deformations between images are estimated by matching higher-order information like the Jacobian of the diffeomorphism at given points using higher-order similarity measures with a specific form. These are defined through labelled sets of control points though and need to be first extracted from the images, which is again different and arguably less flexible than the method we introduce here.     \n\n\n\\section{Shapes and discrete varifolds}\n\\label{sec:discrete_varifolds}\nThe idea of representing shapes as distributions goes back to the many works within the field of Geometric measure theory. Those concepts have later been of great interest in the construction of simple and numerically tractable metrics between curves or surfaces for registration problems: the works of \\cite{Glaunes2006,Glaunes2008,Durrleman4,Charon2} are a few examples. The framework of oriented varifolds recently exploited in \\cite{Charon2017} was shown to encompass all those notions into a general representation and provide a wide range of metrics on the spaces of embedded curves or surfaces. We give a brief summary of the latter work below.  \n\nIn the rest of the paper, we will call an oriented varifold or, to abbreviate, a \\textit{varifold} in $\\mathbb{R}^n$ (we shall here consider the cases $n=2$ or $n=3$) a distribution on the product $\\mathbb{R}^n \\times \\mathbb{S}^{n-1}$. In other words, a varifold $\\mu$ is by definition a linear form over a certain space $W$ of smooth functions on $\\mathbb{R}^n \\times \\mathbb{S}^{n-1}$, which evaluation we shall write as $\\mu(\\omega)$ for any test function $\\omega \\in W$. In all what follows, we shall restrict our focus to 'discrete' shapes and varifolds, leaving aside the analysis of the corresponding continuous models. By discrete varifold, we mean specifically that $\\mu$ writes as a finite combination of Dirac masses $\\mu=\\sum_{i=1}^{P} r_i \\delta_{(x_i,d_i)}$ with $r_i>0$, $(x_i,d_i) \\in \\mathbb{R}^n \\times \\mathbb{S}^{n-1}$ for all $i$, in which case $\\mu(\\omega) = \\sum_{i=1}^{P} r_i \\omega(x_i,d_i)$ for all $\\omega$. Such a $\\mu$ can be thought as a set of unit direction vectors $d_i$ located at positions $x_i$ with weights (or masses) equal to the $r_i$'s. We assume by convention that the $(x_i,d_i)$ are distinct, but not necessarily that all the positions $x_i$ are: in other words, in our model, there can be more than a single direction vector attached to each position. In the rest of the paper, we will denote by $\\mathcal{D}$ the set of all discrete varifolds. Note that in this representation and unlike the cases of landmarks and vector fields, the particles are unlabelled i.e the varifold $\\mu$ is invariant to any permutation of the $(x_i,d_i)$. One particular subset of interest that we shall denote $\\mathring{\\mathcal{D}} \\subset \\mathcal{D}$ is the space of discrete varifolds with distinct positions $x_i$ (or equivalently, the discrete varifolds that carry a single direction vector per point position).     \n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{ccc}\n    \\includegraphics[width=4cm]{curve_var_discrete.jpg} & & \\includegraphics[width=4.5cm]{surface_var_discrete.png} \\\\\n    (a) & & (b) \\\\\n    \\includegraphics[width=4cm,height=4cm]{subject_Pasteur5.png} & & \\includegraphics[width=4.5cm]{HARDI_peaks.png} \\\\\n    (c) & & (d)\n    \\end{tabular}\n    \\caption{Some examples of data representable by discrete varifolds: (a) Piecewise linear curve. (b) Triangulated surface. (c) A set of cells' mitosis directions measured inside a mouse embryonic heart membrane (c.f \\cite{Ragni2017}). (d) Peak diffusion directions extracted from a slice of High Angular Resolution Diffusion Imaging phantom data, note the presence of multiple directions at certain locations corresponding to fiber crossing.}\n    \\label{fig:discr_shape_var}\n\\end{figure}\n\nThe relationship between shapes and varifolds relies on the fact that discrete shapes, namely curve or surface meshes, can be naturally approximated by varifolds of the previous form. As explained with more details in the aforementioned references, this is done by associating to any cell of the discrete mesh (i.e a segment for curves or a triangular face for surfaces) the weighted Dirac $r_i \\delta_{(x_i,d_i)}$ as illustrated in Figure \\ref{fig:discr_shape_var}. In that expression, $x_i$ is the coordinates of the center of the cell, $r_i$ its total length or area and $d_i$ the direction of the tangent space represented by the unit tangent or normal orientation vector $d_i$. It results in a mapping $S \\mapsto \\mu_{S}$ that associates to any discrete shape $S$ the discrete varifold $\\mu_S = \\sum_{i=1}^{F} r_i \\delta_{(x_i,d_i)} \\in \\mathring{\\mathcal{D}}$ obtained as the sum over all faces $i=1,\\ldots,F$ of the corresponding Diracs. \n\nThe main interest of such a representation is that it gives a convenient setting for the definition of shape similarities that are easy to compute without the need for pointwise correspondences between points. Assuming, which is quite natural in our context, that $W$ is a Hilbert space and that all Diracs $\\delta_{(x,d)}$ for $(x,d) \\in \\mathbb{R}^n \\times \\mathbb{S}^{n-1}$ belong to the dual, $W$ must be then chosen as a Reproducing Kernel Hilbert Space (RKHS) associated to a smooth positive definite kernel on $\\mathbb{R}^n \\times \\mathbb{S}^{n-1}$. In particular, we will follow the construction proposed in \\cite{Charon2017} and consider separable kernels of the form $k(x,d,x',d')=\\rho(|x-x'|^2) \\gamma(\\langle d , d' \\rangle)$ where $\\rho$ and $\\gamma$ define positive definite kernel functions respectively on the positions between particles and the angles between their orientation vectors. The reproducing kernel metric on $W$ then gives a dual metric on varifolds that explicitly writes, for $\\mu=\\sum_{i=1}^{P} r_i \\delta_{(x_i,d_i)}$:\n\\begin{equation}\n\\label{eq:metric_var}\n \\|\\mu\\|_{W^*}^2 = \\sum_{i,j} r_i r_j \\rho(|x_i-x_j|^2) \\gamma(\\langle d_i, d_j \\rangle)\n\\end{equation}\n\nSuch metrics on $W^*$ are determined by the choice of the positive definite functions $\\rho$ and $\\gamma$ and provide a global measure of proximity between two discrete varifolds. One important advantage for applications to e.g registration is that the computation of a distance $\\|\\mu-\\mu'\\|_{W^*}^2$ between two distributions does not require finding correspondences between their masses but instead reduces numerically to a quadratic number of kernel evaluations. The gradients of the metric with respect to the $x_i$'s and $d_i$'s is also very easy to obtain by direct differentiation of \\eqref{eq:metric_var}. Finally, we note that the expression in \\eqref{eq:metric_var} is also invariant to the action of the group of rigid motion. Namely for any rotation matrix $R$, translation vector $h$ and the group action $(R,h)\\cdot \\mu \\doteq \\sum_{i=1}^{P} r_i \\delta_{(Rx_i+h,Rd_i)}$, one has $\\|(R,h)\\cdot \\mu \\|_{W^*} = \\|\\mu\\|_{W^*}$. \n\nIn all generality however, \\eqref{eq:metric_var} may only yield a pseudo-metric on the set of discrete varifolds $\\mathcal{D}$ since the inclusion mapping $\\mathcal{D} \\rightarrow W^*$ is not necessarily injective. A necessary and sufficient condition is: \n\\begin{prop}\n\\label{prop:var_metric1}\n The metric $\\|\\cdot\\|_{W^*}$ on $W^*$ induces a metric on $\\mathcal{D}$ if and only if $k$ is a strictly positive definite kernel on $\\mathbb{R}^n \\times \\mathbb{S}^{n-1}$. \n\\end{prop}\nThe proof follows immediately from the definition of strictly positive definite kernel. This condition holds in particular if both kernels defined by $\\rho$ and $\\gamma$ are strictly positive definite. In the case of $\\mathring{\\mathcal{D}}$, one can provide different sufficient conditions which are often more convenient to satisfy in practice. These involve a density property on kernels called $C_0$-universality, cf \\cite{Carmeli2010}. A kernel on $\\mathbb{R}^n$ is said to be $C_0$-universal if the associated RKHS is dense in $C_0(\\mathbb{R}^n,\\mathbb{R})$. Then one has the following \n\\begin{prop}\n\\label{prop:var_metric2}\n If the kernel defined by $\\rho$ is $C_0$-universal, $\\gamma(1)>0$ and $\\gamma(u) < \\gamma(1)$ for all $u \\in [-1,1)$, then $\\|\\cdot\\|_{W^*}$ induces a metric on $\\mathring{\\mathcal{D}}$. \n\\end{prop}\n\\begin{proof}\nLet $W_{pos}$ and $W_{or}$ be the RKHS associated to $\\rho$ and $\\gamma$. By contradiction, suppose that $\\mu, \\mu' \\in \\mathring{\\mathcal{D}}$ with $\\|\\mu-\\mu'\\|_{W^*}=0$ and $\\mu \\neq \\mu'$ in $\\mathring{\\mathcal{D}}$. We can write $\\mu,\\mu'$ in the following form:\n\\begin{align*}\n    \\mu = \\sum_{i=1}^N r_i \\delta_{(z_i,d_i)}, \\ \\ \\mu' = \\sum_{i=1}^N r_i' \\delta_{(z_i,d_i')},\n\\end{align*}\nwhere $\\{z_i\\}$, with $z_i$ all distinct, is the reunion of point positions from both distributions and $\\max\\limits_{1 \\leq i \\leq N} \\{r_i,r_i' \\}>0$, $\\min\\limits_{1 \\leq i \\leq N} \\{r_i,r_i' \\} \\geq 0$. Since $\\mu$ and $\\mu'$ are distinct in $\\mathring{\\mathcal{D}}$, there is some $i_0$ such that $(d_{i_0},r_{i_0}) \\neq (d_{i_0}',r_{i_0}')$. Without loss of generality, we may assume $r_{i_0} \\geq r_{i_0}'$. Let $g(\\cdot) = \\gamma(\\langle d_{i_0}, \\cdot \\rangle) \\in W_{or}$ and choose $f \\in C_0(\\mathbb{R}^n,\\mathbb{R})$ satisfying $f(z_{i_0}) =1$ and $f(z_i)=0$ for all $i \\geq 2$. Since the kernel defined by $\\rho$ is $C_0$-universal, there exists $\\{f_n\\} \\subset W_{pos}$ such that $f_n \\rightarrow f$ uniformly. As $f_n \\otimes g \\in W$, we have that\n\\begin{equation*}\n    0 = (\\mu-\\mu'|f_n \\otimes g) = \\sum_{i=1}^N f_n(z_i) (r_i g(d_i) -r_i' g(d_i'))\n\\end{equation*}\nTaking the limit $n \\rightarrow +\\infty$, this gives:\n\\begin{equation}\n\\label{eq:proof_prop2}\n    0 = f(z_{i_0}) (r_{i_0} g(d_{i_0}) -r_{i_0}' g(d_{i_0}')) = \\underbrace{r_{i_0} \\gamma(1) - r_{i_0}' \\gamma(\\langle d_{i_0}, d_{i_0}'\\rangle)}_{A} .\n\\end{equation}\nSince $(d_{i_0},r_{i_0}) \\neq (d_{i_0}',r_{i_0}'), \\ r_{i_0} \\geq r_{i_0}' \\textrm{ and } r_{i_0}>0$, we have either $d_{i_0}\\neq d_{i_0}'$ and then $A \\geq r_{i_0}(\\gamma(1) -\\gamma(\\langle d_{i_0}, d_{i_0}'\\rangle))>0$ or $d_{i_0} = d_{i_0}'$ and $r_{i_0}> r_{i_0}'$ in which case $A = (r_{i_0} - r_{i_0}') \\gamma(1)>0$. In either case the right hand side of \\eqref{eq:proof_prop2} is positive which is a contradiction.\n\\end{proof}\nNote that the $C_0$-universality assumption still implies that the kernel defined by $\\rho$ is strictly positive definite. However, the assumptions on $\\gamma$ are typically less restrictive than in Proposition \\ref{prop:var_metric1}. \n\nA last subclass of varifold metrics that shall be of interest in this paper is the case of orientation-invariant kernels which amounts in choosing an even function $\\gamma$ in the kernel definition. This, indeed, leads to a space $W^*$ and metric $\\|\\cdot\\|_{W^*}$ for which Diracs $\\delta_{(x,d)}$ and $\\delta_{(x,-d)}$ are equal in $W^*$ for any $(x,d) \\in \\mathbb{R}^n \\times \\mathbb{S}^{n-1}$. In other words, elements of $\\mathcal{D}$ can be equivalently viewed as \\textit{unoriented} varifolds, i.e distributions on the product of $\\mathbb{R}^n$ and the projective space of $\\mathbb{R}^n$, similarly to the framework of \\cite{Charon2}. In that particular situation, one obtains an induced distance under the conditions stated in the following proposition which proof is a straightforward adaptation of the one of Proposition \\ref{prop:var_metric2}. \n\\begin{prop}\n\\label{prop:var_metric3}\n If the kernel defined by $\\rho$ is $C_0$-universal, $\\gamma$ is an even function with $\\gamma(1)>0$ and $\\gamma(u) < \\gamma(1)$ for all $u \\in (-1,1)$, then $\\|\\cdot\\|_{W^*}$ induces a metric on the space $\\mathring{\\mathcal{D}}$ modulo the orientation. \n\\end{prop}\n\nIn Section \\ref{sec:results} below, we will discuss more thoroughly and illustrate the effects of those kernel properties on the solutions to registration problems for different cases of discrete distributions. \n\n\n\n\\section{Optimal diffeomorphic mapping of varifolds}\n\\label{sec:optimal_control}\nIt is essential to point out that the notion of varifold presented above contains but is also more general than curves and surfaces as it allows to model more complex geometric structures like objects carrying multiple orientation vectors at a given position. In contrast with most previous works on diffeomorphic registration that only involve varifolds as an intermediary representation to compute fidelity terms between shapes, the purpose of this paper to derive a deformation model and registration framework on the space $\\mathcal{D}$ itself.  \n\n\\subsection{Group action}\n\\label{ssec:group_action}\nA first key element is to express the way that deformations 'act' on discrete varifolds. Considering a smooth diffeomorphism $\\phi \\in \\text{Diff}(\\mathbb{R}^n)$, we first intend to express how $\\phi$ should transport a Dirac $\\delta_{(x,d)}$. There is however not a canonical way to define it as the nature of the underlying data affects the deformation model itself. An important distinction to be made is on the interpretation of direction vectors $d$, whether they correspond for instance to a unit tangent direction to a curve or a surface in which case $d$ is transported by the Jacobian of $\\phi$ as $D_x \\phi(d)\/|D_x \\phi(d)|$ or rather to a normal direction which instead requires a transport model involving the inverse of the transposed Jacobian i.e $(D_x \\phi)^{-T}(d)\/|(D_x \\phi)^{-T}(d)|$ (see \\cite{Younes} chap. 10 for more thorough discussion). To keep notations more compact, we will write $D\\phi \\cdot d$ for a given generic action of $D\\phi$ on $\\mathbb{R}^{n}$ on either tangent or normal vector and $\\overline{D\\phi \\cdot d}$ for the corresponding normalized vector in $\\mathbb{S}^{n-1}$. That being said, we will also consider two distinct models for the action:\n\\begin{itemize}\n    \\item[$\\bullet$] $\\phi_{*} \\delta_{(x,d)} \\doteq \\delta_{(\\phi(x),\\overline{D\\phi\\cdot d})}$ (\\textit{normalized action}): this corresponds to transporting the Dirac mass at the new position $\\phi(x)$ and transforming the orientation vector as $\\overline{D\\phi \\cdot d}$. \n    \n    \\item[$\\bullet$] $\\phi_{\\#} \\delta_{(x,d)} \\doteq |D\\phi\\cdot d| \\delta_{(\\phi(x),\\overline{D\\phi\\cdot d})}$ (\\textit{pushforward action}): the position and orientation vector are transported as previously but with a reweighting factor equal to the norm of $D\\phi\\cdot d$.\n\\end{itemize}\nIt is then straightforward to extend both of these definitions by linearity to any discrete varifold in $\\mathcal{D}$. In both cases, we obtain a group action of diffeomorphisms on the set of discrete varifolds. However, these actions are clearly not equivalent. The normalized action operates as a pure transport of mass and rotation of the direction vector whereas the pushforward model adds a weight change corresponding to the Jacobian of $\\phi$ along the direction $d$. This is a necessary term in the situation where $\\mu=\\mu_{S}$ is representing a discrete oriented curve or surface. Indeed, one can check, up to discretization errors, that under the pushforward model, we have $\\phi_{\\#} \\mu_{S} = \\mu_{\\phi(S)}$; in other words the action is compatible with the usual deformation of a shape. In the result section below, we will show examples of matching based on those different group action models. \n\nAlthough we will be focusing on special subgroups of diffeomorphisms in the next section, it will be insightful to study a little more closely the orbits of discrete varifolds under the normalized and pushforward actions of the full group $\\text{Diff}(\\mathbb{R}^n)$ (or similarly the equivalence classes $\\mathcal{D}\/\\text{Diff}(\\mathbb{R}^n)$). Let $\\mu \\in \\mathcal{D}$ which we can write as $\\mu = \\sum_{i=1}^{N} \\sum_{j=1}^{n_i} r_{i,j} \\delta_{(x_{i},d_{i,j})}$ where the $x_i$ are here assumed to be distinct positions and for each $i=1,\\ldots,N$, the $(d_{i,j})_{j=1,\\ldots,n_i}$ are distinct in $\\mathbb{S}^{n-1}$. While it is well-known that $\\text{Diff}(\\mathbb{R}^n)$ acts transitively on the set of point clouds of $N$ points in $\\mathbb{R}^n$ (as $n\\geq2$), this may no longer hold when one or several direction vectors are attached to each point position. \n\nIn the case of the normalized action, we have $\\phi_{*} \\mu = \\sum_{i=1}^{N} \\sum_{j=1}^{n_i} r_{i,j} \\delta_{(\\phi(x_i),\\overline{D\\phi\\cdot d_{i,j}})}$. We see that the orbit of $\\mu$ is then given by: \n\\begin{equation*}\n    \\text{Diff}_{*}\\mu = \\left \\{\\sum_{i=1}^{N} \\sum_{j=1}^{n_i} r_{i,j} \\delta_{(y_i,u_{i,j})} \\ s.t \\ y_i \\neq y_j \\ \\text{for} \\ i \\neq j, \\ \\exists A_1,\\ldots,A_{N} \\in \\text{GL}(\\mathbb{R}^n), \\ u_{i,j}= \\frac{A_i d_{i,j}}{|A_i d_{i,j}|}  \\right \\} \n\\end{equation*}\nThis is essentially the set of all discrete varifolds with any set of $N$ distinct positions and for each $i$, a set of $n_i$ directions obtained by a linear transformation of the $\\{d_{i,j}\\}_{j=1,\\ldots,n_i}$ with  weights $r_{i,j}$ unchanged. In particular, this imposes some constraints on the set of 'attainable' direction vectors: clearly, if the number of direction vectors at a given position exceeds the dimension i.e $n_i \\geq n$, this system of vectors cannot be mapped in general to any other system of $n_i$ vectors on the sphere by a single linear map. If we assume that the system of vectors at each position $x_i$ forms a frame, i.e that for all $i$, $n_i \\leq n$ and the direction vectors $d_{i,j}$ for $j=1,\\ldots,n_i$ are independent, then we see that the orbit of $\\mu$ is given by the set of all discrete varifolds of the form $\\sum_{i=1}^{N} \\sum_{j=1}^{n_i} r_{i,j} \\delta_{(y_i,u_{i,j})}$ with distinct $y_i$'s and $(u_{i,j})$ in $\\mathbb{S}^{n-1}$ such that the $(u_{i,j})_{j=1,\\ldots,n_i}$ are independent for all $i$. In the special case of $n_i=1$ for all $i$, that is $\\mu \\in \\mathring{\\mathcal{D}}$, the orbits are then entirely determined by the set of weights $r_i$ which gives the identification of $\\mathring{\\mathcal{D}}\/\\text{Diff}(\\mathbb{R}^n)$ with ordered finite sets of positive numbers.  \n\nWith the pushforward action, we have $\\phi_{\\#} \\mu = \\sum_{i=1}^{N} \\sum_{j=1}^{n_i} |D\\phi\\cdot d_{i,j}| r_{i,j} \\delta_{(\\phi(x_i),\\overline{D\\phi\\cdot d_{i,j}})}$ and the orbit writes:\n\\begin{align*}\n    \\text{Diff}_{\\#}\\mu = \\Big\\{ \\nu \\in \\mathcal{D} \\ \\ &s.t \\ \\ \\exists (y_i) \\in (\\mathbb{R}^n)^N, \\ y_i \\neq y_j \\ \\text{for} \\ i \\neq j, \\exists A_1,\\ldots,A_{N} \\in \\text{GL}(\\mathbb{R}^n),  \\\\ \n     &\\nu=\\sum_{i=1}^{N} \\sum_{j=1}^{n_i} |A_i d_{i,j}| r_{i,j} \\delta_{(y_i,u_{i,j})} \\ \\ \\text{with} \\ u_{i,j}= \\frac{A_i d_{i,j}}{|A_i d_{i,j}|}  \\Big\\} \n\\end{align*}\nIn the general situation, there is again no simple characterization of the orbit. With the additional assumptions that $n_i \\leq n$ and the $(d_{i,j})_{j=1,\\ldots,n_i}$ are independent vectors for each $i$, the orbit of $\\mu$ is the set of all discrete varifolds of the form $\\sum_{i=1}^{N} \\sum_{j=1}^{n_i} s_{i,j} \\delta_{(y_i,u_{i,j})}$ with any choice of distinct points $y_i$, direction vectors $(d_{i,j})$ in $\\mathbb{S}^{n-1}$ such that the $(d_{i,j})_{j}$ are independent and weights $s_{i,j} > 0$. In particular, the action of $\\text{Diff}(\\mathbb{R}^n)$ in the pushforward model is transitive on all subsets of $\\mathring{\\mathcal{D}}$ with fixed $N$, which implies that the equivalence classes of $\\mathring{\\mathcal{D}}\/\\text{Diff}(\\mathbb{R}^n)$ in that case are only determined by the number of Diracs in the discrete varifold, as we would expect.  \n\nThe previous discussion thus shows that for both models and unlike the more standard cases of landmarks or discrete vector fields, the action of diffeomorphisms on discrete varifolds is in general not transitive. It is therefore necessary to formulate the registration problems in their inexact form by introducing fidelity terms like the kernel metrics introduced in Section \\ref{sec:discrete_varifolds}.     \n\n\\subsection{Optimal control problem}\n\\label{ssec:optimal_control}\nWith the definitions and notations of the previous sections, we can now introduce the mathematical formulation of the diffeomorphic registration of discrete varifolds. As mentioned in the introduction, we will rely on the LDDMM model for generating diffeomorphisms although other transformation spaces and models could be taken as well. In short, we consider a space of time dependent velocity fields $v \\in L^2([0,1],V)$ such that for all $t\\in [0,1]$, $v_t$ belongs to a certain RKHS $V$ of vector fields on $\\mathbb{R}^n$. We will write $K: \\ \\mathbb{R}^n \\times \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ the vector-valued reproducing kernel of $V$. From $v$, one obtains the flow mapping $\\phi^v_t$ at each time $t$ as the integral of the differential equation $\\partial_t \\phi^v_t = v_t \\circ \\phi^v_t$  with $\\phi_0 = \\text{Id}$. We then define our deformation group as the set of all flow maps $\\phi_1^v$ for all velocity fields $v \\in L^2([0,1],V)$. With the adequate assumptions on the kernel of $V$, this is a subgroup of the group of diffeomorphisms of $\\mathbb{R}^n$ and it is naturally equipped with the metric given by $\\int_{0}^{1} \\|v_t\\|_V^2 dt$, cf \\cite{Younes} for a detailed exposition of the LDDMM framework. \n\nNow, let's consider two discrete varifolds $\\mu_0 = \\sum_{i=1}^{P} r_{i,0} \\delta_{(x_{i,0},d_{i,0})}$ (template) and $\\tilde{\\mu} = \\sum_{j=1}^{Q} \\tilde{r}_{j} \\delta_{(\\tilde{x}_{j},\\tilde{d}_{j})}$ (target). We formulate the inexact matching problem between $\\mu_0$ and $\\tilde{\\mu}$ as follows:\n\\begin{equation}\n\\label{eq:matching_var}\n\\text{argmin}_{v \\in L^2([0,1],V)} \\left\\{ E(v) = \\int_{0}^{1} \\|v_t\\|_V^2 dt + \\lambda \\|\\mu(1) - \\tilde{\\mu} \\|_{W^*}^2 \\right\\}\n\\end{equation}\nsubject to either $\\mu(t) \\doteq (\\phi_t^v)_{*} \\mu_0$ in the normalized action scenario or $\\mu(t) \\doteq (\\phi_t^v)_{\\#} \\mu_0$ for the pushforward model, and $\\lambda$ being a weight parameter between the regularization and fidelity terms in the energy. This is easily interpreted as an optimal control problem in which the state variable is the transported varifold $\\mu(t)$, the control is the velocity field $v$ and the cost functional is the sum of the standard LDDMM regularization term on the deformation and a discrepancy term between $\\mu(1)$ and the target given by a varifold kernel metric as in \\eqref{eq:metric_var}. Those optimal control problems are well-posed in the following sense:\n\\begin{prop}\nIf $V$ is continuously embedded in the space $C^2_0(\\mathbb{R}^n,\\mathbb{R}^n)$, or equivalently if $K$ is of class $C^2$ with all derivatives up to order 2 vanishing at infinity, then there exists a global minimum to the problem \\eqref{eq:matching_var}.  \n\\end{prop}\n\\begin{proof}\nThe result follows from an argument similar to that of the existence of minimizers in usual LDDMM registration problems. If $(v^{n})$ is a minimizing sequence in $L^2([0,1],V)$ then thanks the first term of $E$, we may assume that $(v^n)$ is bounded in $L^2([0,1],V)$ and therefore that, up to extracting a subsequence, $v^n \\rightharpoonup v^*$ weakly in $L^2([0,1],V)$. It then follows from the results of \\cite{Younes} (Chapter 8.2) that the sequence of diffeomorphisms $(\\phi_1^{v^n})$ and their first-order differentials $(d\\phi_1^{v^n})$ converge uniformly on every compact respectively to $\\phi_{1}^{v^*}$ and $d\\phi_{1}^{v^*}$. In particular, for all $i=1,\\ldots,P$, $\\phi_{1}^{v^n}(x_i) \\rightarrow \\phi_{1}^{v^*}(x_i)$ and $d\\phi_{1}^{v^n}(x_i) \\rightarrow d\\phi_{1}^{v^*}(x_i)$. Then, from the expressions of the group actions and the metric \\eqref{eq:metric_var}, we obtain that either $\\|(\\phi_1^{v^n})_{*} \\mu_0 - \\tilde{\\mu} \\|_{W^*}^2 \\xrightarrow[n\\rightarrow \\infty]{} \\|(\\phi_1^{v^*})_{*} \\mu_0 - \\tilde{\\mu} \\|_{W^*}^2$ or $\\|(\\phi_1^{v^n})_{\\#} \\mu_0 - \\tilde{\\mu} \\|_{W^*}^2 \\xrightarrow[n\\rightarrow \\infty]{} \\|(\\phi_1^{v^*})_{\\#} \\mu_0 - \\tilde{\\mu} \\|_{W^*}^2$. Finally, using the weak lower semicontinuity of the norm in $L^2([0,1],V)$, it gives in both cases:\n\\begin{equation*}\n    E(v^*) \\leq \\lim \\inf_{n \\rightarrow \\infty} E(v^n)\n\\end{equation*}\nand consequently $v^*$ is a global minimizer of $E$. \n\\end{proof}\n\n\n\\subsection{Hamiltonian dynamics}\n\\label{ssec:hamiltonian_dynamics}\nBy fixing the final time condition $\\mu(1)$ and minimizing $\\int_{0}^{1} \\|v_t\\|_V^2 dt$ with those boundary constraints, the resulting path $t \\mapsto \\mu(t)$ corresponds to a \\textit{geodesic} in $\\mathcal{D}$ for the metric induced by the metric on the deformation group. We can further characterize those geodesics as solutions of a Hamiltonian system. For that purpose, we follow the general setting developed in \\cite{arguillere14:_shape} for similar optimal control problems. \n\nIn our situation, we can describe the state $\\mu(t)$ as a set of $P$ particles each given by the triplet $(x_i(t),d_i(t),r_i(t)) \\in \\mathbb{R}^n \\times \\mathbb{S}^{n-1} \\times \\mathbb{R}_{+}^{*}$ representing its position, orientation vector and weight. From \\ref{ssec:group_action}, we have that $x_i(t) = \\phi_t^v(x_{i,0})$, $d_i(t) = \\overline{D\\phi^v_t\\cdot d_{i,0}}$ and $r_i(t) = r_{i,0}$ for the normalized action and $r_i(t)=|D\\phi^v_t\\cdot d_{i,0}| r_{i,0}$ in the pushforward case. Differentiating with respect to $t$, the state evolution may be alternatively described by the set of ODEs \n$$\n\\left \\{\n\\begin{array}[h]{l}\n\\dot{x}_i (t)= v_t(x_i(t)) \\\\\n\\dot{d}_i (t) = P_{d_i(t)^{\\bot}}(D v_t\\cdot d_{i}(t)) \\\\\n\\dot{r}_i (t) = \\left\\{ \n\\begin{array}[h]{l}\n0 \\ \\ \\text{(normalized)} \\\\\n\\langle d_i(t), D v_t\\cdot d_{i}(t) \\rangle r_i(t) \\ \\ \\text{(pushforward)}\n\\end{array}\n\\right.\n\\end{array}\n \\right.\n$$\n \n\\noindent where $P_{d_i(t)^\\bot}$ denotes the orthogonal projection on the subspace orthogonal to $d_i(t)$, $D v_t \\cdot d_i(t)$ corresponds to the infinitesimal variation of the action of $D\\phi$ on vectors of $\\mathbb{R}^n$ introduced in \\ref{ssec:group_action}: it is given specifically by $D v_t \\cdot d_i(t) = D_{x_i(t)} v_t (d_i(t))$ in the tangent case and $D v_t \\cdot d_i(t) = -(D_{x_i(t)} v_t)^{T}(d_i(t))$ in the normal case. Note that other choices of transformation of the weights could be treated quite similarly by modifying accordingly the last equation in the previous system. In what follows, we detail the derivations of the optimality equations in the case of tangent direction vectors for both normalized and pushforward group action models, the situation of normal vectors being easily tackled in similar fashion.    \n\n\\subsubsection{Normalized action}\nIn the case of normalized action, $\\{r_i(t)\\}_{i=1}^P$ are time independent as previous discussed. So we can choose the state variable of the optimal control problem to be $q:= \\{(x_i,d_i), \\ i =1,\\cdots,P\\} \\in \\mathbb{R}^{2dP}$ with the infinitesimal action\n\\begin{align*}\n\\xi_q v = \\left\\{\\left( v(x_i),P_{d_i^{\\perp}}(D_{x_i}v(d_i))\\right),\\ i=1,\\cdots,P\\right\\}\n\\end{align*} \nand introduce the Hamiltonian\n\n\\begin{align*}\nH(p,q,v) &= (p|\\xi_qv) -\\frac{1}{2}\\|v\\|^2_V \\\\\n         &=\\sum_{i=1}^N \\langle p_i^{(1)}, v(x_i) \\rangle + \\langle P_{d_i^{\\perp}}(p_i^{(2)}), D_{x_i}v(d_i) \\rangle -\\frac{1}{2}\\|v\\|^2_V,\n\\end{align*}\nwhere \n\\begin{align*}\np= \\left\\{\\left(p_i^{(1)},p_i^{(2)}\\right), \\ i=1,\\cdots,P \\right\\} \\in \\mathbb{R}^{2dP}\n\\end{align*}\nis the adjoint variable of state $q$. We call $p_i^{(1)}$ the \\textit{spatial momentum} and $p_i^{(2)}$ the \\textit{directional momentum}. From Pontryagin's maximum principle, the Hamiltonian dynamics is given by the forward system of equations\n\\begin{align}\n\\left\\{\\begin{array}{ll}\n\\dot{x}_i(t) &= v_t(x_i(t)) \\\\\n\\dot{d}_i(t) &= P_{d_i(t)^{\\perp}}(D_{x_i(t)}v_t(d_i(t))) \\\\\n\\dot{p}_i^{(1)}(t) &= -(D_{x_i(t)}v_t)^T p_i^{(1)}(t) \\\\\n             &-(D_{x_i(t)}^{(2)}v_t(\\cdot,d_i(t)))^T\\left( P_{d_i(t)^{\\perp}}(p_i^{(2)}(t))\\right) \\\\\n\\dot{p}_i^{(2)}(t) &= -(D_{x_i(t)} v_t)^T P_{d_i(t)^{\\perp}}(p_i^{(2)}(t)) \\\\\n                &+ \\langle d_i(t),p_i^{(2)}(t) \\rangle D_{x_i(t)}v_t(d_i(t)) \\\\\n                &+ \\langle d_i(t),D_{x_i(t)}v_t(d_i(t)) \\rangle p_i^{(2)}(t)\n\\end{array}\\right.\n\\label{eq:forward_normalized}\n\\end{align}\nand optimal vector fields $v$ satisfy\n\\begin{align*}\n\\langle v_t, h \\rangle_V &= \\left( p(t),\\xi_{q(t)}h \\right) \\\\\n&= \\sum_{i=1}^P \\langle p_i^{(1)},h(x_i)\\rangle + \\langle P_{d_i(t)^{\\perp}}(p_i^{(2)}(t)),D_{x_i(t)}h(d_i(t))\\rangle, \n\\end{align*}\nfor any $h \\in V$ and $t \\in [0,1]$. The reproducing property and reproducing property for the derivatives in a vector RKHS give \\cite{Sommer2013} that $\\forall x \\in \\mathbb{R}^n, \\ z \\in \\mathbb{R}^n, v \\in V$ and multi-index $\\alpha$,\n\\begin{align*}\n&\\langle K(x,\\cdot)z,v \\rangle_V = (z \\otimes \\delta_x |v) \\\\\n&\\langle D_1^{\\alpha} K(x,\\cdot),v \\rangle_V = z^T D^{\\alpha} v(x).\n\\end{align*}\nWith the above properties, we obtain the following expression of $v$\n\\begin{align}\n\\label{eq:v_normalized}\nv_t(\\cdot) &= \\sum_{k=1}^P K(x_k(t),\\cdot)p_k^{(1)}(t) \\nonumber \\\\\n&+ D_1 K(x_k(t),\\cdot)\\left(d_k(t),P_{d_k(t)^{\\perp}}(p_k^{(2)}(t))\\right).\n\\end{align}\nwhere we use the shortcut notation $D_1 K(x,\\cdot)(u_1,u_2)$ for the vector $D_1 (K(x,\\cdot) u_2)(u_1)$. In Figure \\ref{fig:normalized}, we show an example of geodesic and resulting deformation for a single Dirac varifold, which is obtained as the solution of \\eqref{eq:forward_normalized} with the initial momenta shown in the figure. It illustrates the combined effects of the spatial momentum which displaces the position of the Dirac and of the directional momentum that generates a local rotation of the direction vector.    \n\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{ccc}\n    \\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=4cm]{nrotation1.png} \n    &\\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=4cm]{nrotation31.png} \n    &\\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=4cm]{nrotation61.png}\\\\\n    $t=0$ & $t=1\/2$& $t=1$  \\\\\n    \\end{tabular}\n    \\caption{Example of geodesic for a single Dirac in the normalized action case.}\n    \\label{fig:normalized}\n\\end{figure}\n\n\n\\subsubsection{Pushforward action}\nAs in the previous section, we set the state variable $q := \\{(x_i,d_i,r_i), \\  i=1,\\cdots,P\\} \\in (\\mathbb{R}^n \\times \\mathbb{S}^{n-1} \\times \\mathbb{R}_{+}^{*} )^P$, the infinitesimal action\n\\begin{align*}\n\\xi_q v = \\left\\{ \\left(v(x_i),P_{d_i^{\\perp}} (d_{x_i}v(d_i)), r_i \\langle d_i, d_{x_i}v(d_i) \\rangle \\right), \\ i=1,\\cdots,P \\right\\}    \n\\end{align*}\nand the Hamiltonian\n\\begin{align}\\label{fullHam}\nH(p,q,v) = \\sum_{i=1}^P \\left\\langle p_i^{(1)}, v(x_i) \\right\\rangle + \\left\\langle P_{d_i^{\\perp}}(p_i^{(2)}), d_{x_i}v(d_i) \\right\\rangle + p_i^{(3)} r_i \\left\\langle  d_i, d_{x_i}v(d_i) \\right\\rangle - \\frac{1}{2} \\|v\\|_V^2,\n\\end{align}\nwhere $p = \\left\\{ \\left( p_i^{(1)}, p_i^{(2)}, p_i^{(3)} \\right), \\ i=1,\\cdots,P \\right\\} \\in \\mathbb{R}^{(2d+1)P}$. Applying again Pontryagin's maximum principle, we obtain the forward system\n\\begin{align} \\label{fullforward}\n\\left\\{\n\\begin{array}{ll}\n\\dot x_i &=  v_t(x_i) \\\\\n\\dot d_i &= P_{d_i^{\\perp}} (d_{x_i} v(d_i)) \\\\\n\\dot r_i &= r_i \\left\\langle d_i, d_{x_i} v_t(d_i) \\right\\rangle\\\\\n\\dot p_i^{(1)} &= -(d_{x_i(t)}v_t)^T p_i^{(1)}-(d_{x_i(t)}^{(2)}v(\\cdot,d_i))^T (P_{d_i^{\\perp}}(p_i^{(2)}) ) - p_i^{(3)} r_i d_{x_i}v(\\cdot,d_i)^T d_i\\\\\n\\dot p_i^{(2)} &= -d_{x_i(t)}v_t^T(p_i^{(2)}) + \\left( \\left\\langle  d_i,p_i^{(2)}\\right\\rangle - r_i p_i^{(3)} \\right) \\left[ d_{x_i}v_t + d_{x_i}v^T_t \\right] (d_i) \\\\\n          &+  \\left\\langle d_i, d_{x_i}v_t(d_i) \\right\\rangle p_i^{(2)} \\\\\n\\dot p_i^{(3)} &= - p_i^{(3)} \\left\\langle d_i, d_{x_i}v_t(d_i) \\right\\rangle\n\\end{array} \\right. \n\\end{align}\nwith optimal vector field of the form\n\\begin{align} \\label{fullvf}\nv_t(x) = \\sum_{k=1}^P K(x_k,x) p_k^{(1)} + D_1K(x_k,x)\\left(d_k,P_{d_k^{\\perp}}(p_k^{(2)})+p_k^{(3)}r_k d_k \\right) .\n\\end{align}\nFrom the forward equations \\eqref{fullforward}, we see that $\\frac{d}{dt} \\langle d_i(t), d_i(t) \\rangle =0$ and $\\frac{d}{dt} r_i(t) p_i^{(3)}(t) =0$, hence $\\|d_i(t)\\|$ and $ r_i(t) p_i^{(3)}(t)$ are constant along geodesic paths. Similarly to the normalized action case, we can use use those conservation properties to reduce the number of state and dual variables as follows. \n\nLet the new state variable be $q = \\{(x_i,u_i), \\ i=1,\\cdots,P\\}$ and the Hamiltonian\n\\begin{align} \\label{reduced_ham}\nH(p,q,v) = \\sum_{i=1}^P  \\langle p_i^{(1)}, v(x_i) \\rangle +  \\langle p_i^{(2)}, d_{x_i}v(u_i)\\rangle -  \\frac{1}{2} \\| v\\|_V^2\n\\end{align}\nThe forward equations and optimal vector field $v$ derived from this Hamiltonian are\n\\begin{align} \\label{reduced_foward}\n\\left\\{\n\\begin{array}{l}\n\\dot x_i(t)  = v_t(x_i(t)) \\\\\n\\dot d_i(t) = d_{x_i(t)}v_t(u_i(t)) \\\\\n\\dot p_i^{(1)}(t) =  -(d_{x_i(t)}v_t)^T p_i^{(1)}-(d_{x_i(t)}^{(2)}v(\\cdot,u_i))^T p_i^{(2)} \\\\\n\\dot p_i^{(2)}(t) = -(d_{x_i(t)}v_t)^T p_i^{(2)}(t)\n\\end{array} \\right. \n\\end{align}\nand \n\\begin{align} \\label{reduced_vf}\nv_t(x) = \\sum_{k=1}^P K(x_k,x)p_k^{(1)}(t) + D_1 K(x_k,x)(u_k(t),p_k^{(2)}(t)).    \n\\end{align}\nThen this new system is rigorously equivalent to the original one in the following sense:\n\\begin{prop} \\label{reduced_prop}\nAny solution of \\eqref{reduced_foward} + \\eqref{reduced_vf} is such that $\\left( x_i(t),\\overline{u_i(t)},|u_i(t)| \\right)$ is a solution of \\eqref{fullforward} + \\eqref{fullvf}. Conversely, any solution $\\left(x_i(t),d_i(t),r_i(t) \\right)$ of \\eqref{fullforward} + \\eqref{fullvf} with initial conditions satisfying $\\left\\langle p_i^{(2)}(0), u_i(0) \\right\\rangle = r_i(0) p_i^{(3)}(0)$ gives the solution $\\left(x_i(t),r_i(t)d_i(t)\\right)$ to \\eqref{reduced_foward} + \\eqref{reduced_vf}.\n\\end{prop}\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{ccc}\n    \\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushshrink1.png} \n    &\\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushshrink31.png} \n    &\\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushshrink61.png}\\\\\n    $t=0$ & $t=1\/2$& $t=1$  \\\\\n    \\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushstretch1.png} \n    &\\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushstretch31.png} \n    &\\includegraphics[trim = 12mm 12mm 12mm 12mm ,clip,width=4cm]{pushstretch61.png}\\\\\n    $t=0$ & $t=1\/2$& $t=1$ \n    \\end{tabular}\n    \\caption{Examples of geodesics in the pushforward action case.}\n    \\label{fig:pushforward}\n\\end{figure}\n\nThe proof is given in Appendix. Note that these equations can be also obtained in a more particular case as the geodesic equations on the tangent bundle of the space of landmarks, as derived for instance in \\cite{Arguillere2015} (Section 3.5). In what follows, we will thus replace the system \\eqref{fullforward} by \\eqref{reduced_foward}. \n\n\\begin{remark}\nWe point out that there are other conserved quantities in the previous system. In particular, it's easy to see that $\\left\\langle p_i^{(2)}, d_i \\right\\rangle$ is constant along geodesics since\n\\begin{align*}\n\\frac{d}{dt} \\left\\langle p_i^{(2)}, d_i \\right\\rangle = - \\left\\langle (d_{x_i}v_t)^T p_i^{(2)}, d_i \\right\\rangle + \\left\\langle p_i^{(i)}, d_{x_i} v_t(d_i) \\right\\rangle    =0.\n\\end{align*}\n\\end{remark}\nFigure \\ref{fig:pushforward} shows two geodesic trajectories of a single Dirac varifold for different initial momenta. In particular, we can again observe the effect of the directional momentum $p^{(2)}$ on the dynamics and resulting deformations. In addition to similar rotation effects as in the normalized action case, local contraction or expansion can be generated as well, depending precisely on the angle $\\langle p_i^{(2)}, d_i \\rangle$.\n\n\\section{Registration algorithm and implementation}\n\\label{sec:matching_algo}\nWe now turn to the issue of numerically solving the optimization problem \\eqref{eq:matching_var}. We will follow the commonly used method for such problems called \\textit{geodesic shooting} (cf \\cite{Vialard2012b}). Indeed, from the developments of Section \\ref{ssec:hamiltonian_dynamics}, we see that optimizing \\eqref{eq:matching_var} with respect to vector fields $v$ can be done equivalently by restricting to geodesics and thus by optimizing over the initial momenta variables $p^{(1)}_0$ and $p^{(2)}_0$ that completely parametrize those geodesics through the Hamiltonian equations.  \n\n\\subsection{Computation of $E$}\nLet a template and target discrete varifold be given as in Section \\ref{ssec:optimal_control}. As mentioned above, we can rewrite the energy $E$ as a function of the initial momenta that we will denote $p^{(1)}_0=(p^{(1)}_i(0))$ and $p^{(2)}_0=(p^{(2)}_i(0))$:  \n\\begin{equation}\n\\label{eq:energy_discrete}\n E(p^{(1)}_0,p^{(2)}_0) = H_r(p_0,q_0) + \\lambda \\underbrace{\\|\\mu(1) - \\tilde{\\mu} \\|_{W^*}^2}_{:=g(q(1))}\n\\end{equation}\nwhere $q_0$ is the initial state, $\\mu(1)$ is the varifold corresponding to the final time state $q(1)$ with $g(q(1))$ the resulting fidelity term between $\\mu(1)$ and the target varifold, and $H_r$ is the \\textit{reduced Hamiltonian} $H_r(p,q)=H(p,q,v)$ for the optimal $v$ given by \\eqref{eq:v_normalized} or \\eqref{reduced_vf} (note that $H_r(p(t),q(t))$ is conserved along solutions of the Hamiltonian systems thus giving the above expression of the energy). \n\nThe expression of $H_r$ as well as the resulting reduced Hamiltonian equations can be obtained in all generality by plugging the expression of $v$ in the equations of Section \\ref{ssec:optimal_control}. In our implementation, we actually restrict to the more particular case of radial scalar kernels for the vector fields in $V$, i.e we assume that $K(x,y)=h(|x-y|^2) I_n$. Then the reduced Hamiltonian for the normalized case becomes:\n\\begin{equation}\n    H_r(p,q) = \\frac{1}{2} \\left\\langle \\overline{K}_q p, p \\right\\rangle,\n\\end{equation}\nwhere $\\overline{K}_q$ is a symmetric positive definite matrix which is defined as follows. Let\n\n\\begin{align*}\n&H =(H)_{ik}= (h_{ki}) \\\\\n&\\overline{A} = (\\overline{A})_{ik} = 2 \\dot h_{ki} \\langle x_k-x_i,d_k \\rangle \\\\\n&\\overline{B} = (\\overline{B})_{ik} = - \\left[  4\\ddot h_{ik} \\langle x_k-x_i,d_i \\rangle \\langle  x_k-x_i, d_k \\rangle + 2 \\dot h_{ki} \\langle d_i, d_k \\rangle \\right]\n\\end{align*}\nwith $h_{ki}$ being a shortcut for $h(|x_i-x_j|^2)$ and \n\n\\begin{align*}\nP_{d^{\\perp}}(\\cdot) =\\left( \\begin{array}{ccc}\nI - d_1 \\cdot d_1^T &  &\\textrm{\\huge{0}}  \\\\\n   & \\ddots & \\\\\n\\textrm{\\huge{0}} & &  I - d_N \\cdot d_N^T \n\\end{array}\\right)\n\\end{align*}\nThen we define\n\\begin{align*}\n\\overline{K}_q := \\left( \\begin{array}{cc}\nI_{Pd} &0  \\\\\n0   & P_{d^{\\perp}}  \\\\\n\\end{array}\n \\right)^T\n\\left( \\begin{array}{cc}\nH \\otimes I_{Pd}  & \\overline{A} \\otimes I_{Pd}\\\\\n(\\overline{A} \\otimes I_{Pd})^T & \\overline{B} \\otimes I_{Pd} \n\\end{array}\n \\right)\n\\left( \\begin{array}{cc}\nI_{Pd} & 0  \\\\\n0   & P_{d^{\\perp}}  \\\\\n\\end{array}\n \\right)  \n\\end{align*}\n where $\\otimes$ denotes the Kronecker product. For the pushforward action case, we define $H$, $A$ and $B$ as in normalized action case with $d_i$ and $d_k$ replaced by $u_i$ and $u_k$, then\n \\begin{equation}\n    H_r(p,q) = \\frac{1}{2} \\left\\langle K_q p, p \\right\\rangle,\n\\end{equation}\nwhere \n\n\\begin{align*}\nK_q := \\left( \\begin{array}{cc}\nH \\otimes I_{Pd}  & A \\otimes I_{Pd}\\\\\n(A \\otimes I_{Pd})^T & B \\otimes I_{Pd} \n\\end{array}\n \\right).     \n\\end{align*}\n This gives us explicitly the first term of the energy in \\eqref{eq:energy_discrete}.\n\nNow, the time evolution of $q$ and $p$ can be also rewritten equivalently in reduced Hamiltonian form, which expressions are given in full for radial scalar kernels in the Appendix. We numerically integrate those differential systems using an RK4 scheme, which we experienced to be better-adapted to these systems than the simpler Euler midpoint integrator used in \\cite{Charon2017}. Then, given initial momenta $p^{(1)}_0$ and $p^{(2)}_0$, integrating those equations forward in time produces the final state $q(1)$ and its corresponding varifold $\\mu(1)$. It is then straightforward to evaluate the second term in \\eqref{eq:energy_discrete} through the expression of the varifold norm \\eqref{eq:metric_var}; in the pushforward case one only needs to apply the additional intermediate operation of converting state $q(1) = (x_i(1),u_i(1))$ into $(x_i(1),\\overline{u_i}(1),|u_i|(1))$. We will discuss different choices of kernels for the varifold metric in the result section.  \n\n\\subsection{Computation of the gradient of $E$}\nThe second element we need is the gradient of the energy with respect to the momenta. The first term being directly a function of $p_0$, it can be differentiated easily and gives the following gradient: \n\\begin{align*}\n    \\nabla_{p_0} H_r(p_0,q_0) &= \\overline{K}_q p_0 \\ \\ \\text{(normalized)} \\\\\n    \\nabla_{p_0} H_r(p_0,q_0) &= K_q p_0 \\ \\ \\text{(pushforward)}\n\\end{align*}\n\nThe fidelity term $g(q(1))$ in \\eqref{eq:energy_discrete}, however, is a function of the final state $q(1)$ which is in turn a function of the momenta through the Hamiltonian system of equations. The computation of the gradient is therefore more involved due to the complicated dependency of $q(1)$ in $p_0$. The standard approach for optimal control problems of this form (cf for example \\cite{Vialard2012b} or \\cite{arguillere14:_shape}) is to introduce the \\textit{adjoint Hamiltonian system}: \n$$ \\overset{\\bold{\\ldotp}}{Z}(t) = d(\\partial_p H_r , \\partial_q H_r)^{T} Z(t) $$ \nwith $Z = (\\tilde{q}, \\tilde{p})^{T}$ the vector of the adjoint variables. Then, as detailed in \\cite{arguillere14:_shape}, the gradient of $g(q(1))$ with respect to $p_0$ is given by $\\tilde{p}(0)$ where $(\\tilde{q}(t), \\tilde{p}(t))^{T}$ is the solution of the adjoint system integrated backward in time with $\\tilde{q}(1) = \\nabla g(q(1))$ and $\\tilde{p}(1) = 0$. \n\nFor the particular Hamiltonian equations considered here, the adjoint system is tedious to derive and to implement. We simply avoid that by approximating the differentials appearing in the adjoint system by finite difference of the forward Hamiltonian equations, following the suggestion of \\cite{arguillere14:_shape} (Section 4.1) which we refer to for details. Note that another possibility would be to take advantage of automatic differentiation methods, as used recently for some LDDMM registration problems by the authors of \\cite{Kuhnel2017}. \n\nLastly, the end time condition $\\nabla g(q(1))$ in the previous adjoint system is computed by direct differentiation of the varifold norm \\eqref{eq:metric_var} with respect to the final state variables. This is actually more direct than in previous works like \\cite{Charon2,Charon2017} where the gradients are computed with respect to the positions of vertices of the underlying mesh. Here, we have specifically, for the normalized model:  \n\\begin{align*}\n    \\partial_{x_i} g(q(1)) &= 2 \\sum_{j=1}^{P} 2 r_i r_j \\rho'(|x_i(1)-x_j(1)|^2) \\gamma(\\langle d_i(1), d_j(1) \\rangle) .(x_i(1)-x_j(1)) \\ \\ -2\\ldots \\\\\n    \\partial_{d_i} g(q(1)) &= 2 \\sum_{j=1}^{P} r_i r_j \\rho(|x_i(1)-x_j(1)|^2) \\gamma'(\\langle d_i(1), d_j(1) \\rangle). d_j(1) \\ \\ -2\\ldots\n\\end{align*}\nwhere the $\\ldots$ denote a similar term for the differential of the cross inner product $\\langle \\mu(1),\\tilde{\\mu} \\rangle_{W^*}$. In the pushforward case with state variables $(x_i(1),u_i(1))$, we first compute $d_i(1)=u_i(1)\/|u_i(1)|$ and $r_i(1)=|u_i(1)|$ and obtain $\\partial_{x_i} g(q(1))$ with the same expression as above while $\\partial_{u_i} g(q(1))$ is given by a simple chain rule. \n\nFinally, with the above notations, the gradient of $E$ writes:\n\\begin{equation}\n    \\label{eq:gradient_E}\n    \\nabla_{p_0} E = \\overline{K}_{q} p_0 + \\lambda \\tilde{p}(0)\n\\end{equation}\nrespectively $\\nabla_{p_0} E = K_{q} p_0 + \\lambda \\tilde{p}(0)$ in the pushforward case.\n\n\\subsection{Gradient descent algorithm}\nThe solution to the minimization of \\eqref{eq:energy_discrete} is then computed by gradient descent on $p_0 = \\left( p^{(1)}_{0,i},p^{(2)}_{0,i} \\right)_{i=1,\\ldots,P}$. Note that this is a non-convex optimization problem. Until convergence, each iteration consists of the following steps:\n\\vskip2ex\n\\noindent (1) Given the current estimate of $p_0$, integrate the Hamiltonian equations forward in time to obtain $q(1)$.\n\\vskip1ex\n\\noindent (2) Compute the gradient $\\nabla g(q(1))$.\n\\vskip1ex\n\\noindent (3) Integrate the adjoint Hamiltonian system backward in time to obtain $\\nabla_{p_0} E$. \n\\vskip1ex\n\\noindent (4) Update $p_0$: we use two separate update steps for the spatial and directional momentum which are selected, at each iteration, using a rough space search approach leading to the lowest value of $E$.    \n\n\\section{Results}\n\\label{sec:results}\nWe now present a few results of registration using the previous algorithm on simple and synthetic examples. Our implementation equally supports objects in 2D or 3D, we will however focus on examples in $\\mathbb{R}^2$ here simply to allow for an easier visualization and interpretation of the results.   \n\n\\subsection{Curve registration}\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{cccc}\n    \\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_t1.png} \n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_t10.png} \n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_t19.png}\n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_final.png} \\\\\n    $t=0$ & $t=1\/3$ & $t=2\/3$  & $t=1$ \\\\\n    \\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_var_t1.png} \n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_var_t10.png} \n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_var_t19.png}\n    &\\includegraphics[trim = 20mm 20mm 20mm 20mm ,clip,width=3.3cm]{matching_bottle_var_final.png}     \n    \\end{tabular}\n    \\includegraphics[width=5cm]{matching_bottle_energies.png}\n    \\caption{Curve registration using point-mesh LDDMM (1st row) and our proposed discrete varifold LDDMM (2nd row). On the last row is shown the evolution of the total energy across the iterations for both algorithms.}\n    \\label{fig:bottle}\n\\end{figure}\nWe begin with a toy example of standard curve matching to compare the result and performance of our discrete varifold LDDMM registration algorithm with the state-of-the-art LDDMM approach for curves such as the implementations of \\cite{Glaunes2008,Charon2017}. The former methods share a very similar formulation to \\eqref{eq:matching_var} and also make use of varifold metrics as fidelity terms, the essential difference being that the state of the optimal control problem is there the set of vertices of the deformed template curve which is only converted to a varifold for the evaluation of the fidelity term at each iteration. But the dynamics of geodesics still correspond to usual point set deformation under the LDDMM model.   \n\nWe consider here the pushforward model for the action of diffeomorphisms on discrete varifolds that we have seen is compatible with the action of diffeomorphisms on curves. In this case, the two formulation and optimization problems for curve registration are theoretically equivalent up to discretization precision. We verify it with the example of Figure \\ref{fig:bottle} for which both algorithms are applied with the same deformation kernel, varifold metric and optimization scheme. Note that in our approach, template and target curves are first (and only once at the beginning) converted to their discrete varifold representations as explained in Section \\ref{sec:discrete_varifolds}. \n\nAs we can see, the resulting geodesics and deformations are consistent between the two methods. This is also corroborated by the very similar values of the energy at convergence. Interestingly however, although each iteration in our model is arguably more expansive numerically compared to standard curve-LDDMM due to the increased complexity of the Hamiltonian equations, the algorithm converges in a significantly lesser number of iterations. Whether this observation generalizes to other examples or other optimization methods will obviously require more careful examination in future work.\n\n\\subsection{Registration of directional sets}\nWe now turn to examples that are more specific to the framework of discrete varifolds. \n\n\\subsubsection*{Choice of the varifold metric} \nFirst, we examine more closely the effect of the metric $\\|\\cdot\\|_{W^*}$ on the registration of discrete varifolds. The framework we propose can indeed support many choices for the kernel functions $\\rho$ and $\\gamma$ that define fidelity metrics $\\|\\cdot\\|_{W^*}$ with possibly very different properties. This has been already analyzed quite extensively in \\cite{Charon2017} but only in the situation where varifolds associated to a curve or a surface. We consider here the same examples of kernels and briefly discuss what are the specific effects to expect when matching more general varifolds in $\\mathcal{D}$ which may involve several orientation vectors at a given position.\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{ccccc}\n    \\rotatebox{90}{\\phantom{aaaaa} Binet}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_binet_t1.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_binet_t11.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm,clip,width=3cm]{matching_2diracs_binet_t21.png}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_binet_final.png} \\\\ \n    \\rotatebox{90}{\\phantom{aa} Unor. Gaussian}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_unorgauss_t1.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_unorgauss_t11.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm,clip,width=3cm]{matching_2diracs_unorgauss_t21.png}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_unorgauss_final.png} \\\\\n    \\rotatebox{90}{Or. Gaussian\/Linear}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_orgauss_t1.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_orgauss_t11.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm,clip,width=3cm]{matching_2diracs_orgauss_t21.png}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_orgauss_final.png} \\\\\n     &$t=0$ & $t=1\/3$ & $t=2\/3$  & $t=1$\n    \\end{tabular}\n    \\caption{Matching of pairs of Dirac varifolds (template is in blue and target in red) under the normalized action with different choices of kernels: Binet on the first row, unoriented Gaussian ($\\sigma_s =1$) on the second and oriented Gaussian ($\\sigma_s =2$) on the last one. The linear kernel leads to the same result as the former in that particular case.}\n    \\label{fig:2Diracs}\n\\end{figure}\n\nThe results of Propositions \\ref{prop:var_metric2} and \\ref{prop:var_metric3} hold under the assumption that the kernel defined by $\\rho$ is a $C_0$-universal kernel on $\\mathbb{R}^n$, which restricts the possible choices to a few known classes (cf \\cite{Carmeli2010} for a thorough analysis). Here, we will focus on the class of Gaussian kernels given by $\\rho(|x-x'|^2) = e^{-\\frac{|x-x'|^2}{\\sigma^2}}$ with a width parameter $\\sigma>0$ that essentially provides a notion of spatial scale sensitivity to the metric, and which must be adapted to the intrinsic sizes of shapes in each example.  \n\nIn combination with $\\rho$, as in \\cite{Charon2017}, we introduce the following four kernels on $\\mathbb{S}^{n-1}$:\n\\begin{itemize}\n\\item[$\\bullet$] $\\gamma(\\langle d,d'\\rangle) = \\langle d,d'\\rangle$ (linear kernel): this choice is related to the particular subclass of currents \\cite{Glaunes2008}. In that case, the resulting $\\|\\cdot\\|_{W^*}$ is clearly only a pseudo-metric on $\\mathcal{D}$ since the linearity implies that in $W^*$: $\\delta_{(x,-d)}=-\\delta_{(x,d)}$ and for any $d_1\\neq -d_2$, $\\delta_{(x,d_1)} + \\delta_{(x,d_2)} = |d_1 + d_2| \\delta_{\\left(x,\\frac{d_1+d_2}{|d_1+d_2|}\\right)}$. However, we still obtain a metric on the subspace $\\mathring{\\mathcal{D}}$ thanks to Proposition \\ref{prop:var_metric2}.  \n\n\\item[$\\bullet$] $\\gamma(\\langle d,d'\\rangle) = \\langle d,d'\\rangle^2$ (Binet kernel): $\\gamma$ being an even function, as discussed in Section \\ref{sec:discrete_varifolds}, the resulting metric on $W^*$ is invariant to the orientation of direction vectors. According to Proposition \\ref{prop:var_metric3}, we then have a distance on $\\mathring{\\mathcal{D}}$ modulo the orientation. Note however that with this particular choice, one does not obtain a metric (but only a pseudo-metric) on $\\mathcal{D}$ modulo the orientation, as we will illustrate in the examples below.        \n\n\\item[$\\bullet$] $\\gamma(\\langle d,d'\\rangle) = e^{-\\frac{2}{\\sigma_s^2}(1-\\langle d,d'\\rangle^2)}$ (unoriented Gaussian kernel): this is another example of orientation-invariant kernel considered in \\cite{Charon2} corresponding to a particular construction of Gaussian kernels on the projective space. In contrast with Binet kernel, it does induce a metric on $\\mathcal{D}$ modulo orientation.\n\n\\item[$\\bullet$] $\\gamma(\\langle d,d'\\rangle) = e^{-\\frac{2}{\\sigma_s^2}(1-\\langle d,d'\\rangle)}$ (oriented Gaussian kernel): this kernel is the restriction of the standard Gaussian kernel on $\\mathbb{R}^n$ to the sphere $\\mathbb{S}^{n-1}$. As such, it can be shown to be $C_0$-universal on $\\mathbb{S}^{n-1}$ and thus, from Proposition \\ref{prop:var_metric1}, lead to a metric on the entire space $\\mathcal{D}$.\n\\end{itemize}\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{ccccc}\n    \\rotatebox{90}{\\phantom{aaaa} Linear}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_linear_t1.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_linear_t11.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm,clip,width=3cm]{matching_2diracs_bis_linear_t21.png}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_linear_final.png} \\\\\n    \\rotatebox{90}{\\phantom{aa} Or. Gaussian}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_orgauss_t1.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_orgauss_t11.png} \n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm,clip,width=3cm]{matching_2diracs_bis_orgauss_t21.png}\n    &\\includegraphics[trim = 8mm 8mm 8mm 8mm ,clip,width=3cm]{matching_2diracs_bis_orgauss_final.png} \\\\\n     &$t=0$ & $t=1\/3$ & $t=2\/3$  & $t=1$\n    \\end{tabular}\n    \\caption{Registration of pairs of Dirac varifolds with the pushforward model for both the linear and oriented Gaussian kernel.}\n    \\label{fig:2Diracs_bis}\n\\end{figure}\n\nWe illustrate the aforementioned properties on a very simple registration example between pairs of Dirac varifolds located at the same position $x$ i.e $\\delta_{(x,d_1)} + \\delta_{(x,d_2)}$ and $\\delta_{(x,d_1')} + \\delta_{(x,d_2')}$. In Figure \\ref{fig:2Diracs}, the template and target pairs of Diracs are matched based on the normalized action model. The estimated matching and deformations clearly differ with the choice of kernel but each of these result is in fact perfectly consistent with the different invariances of those kernels. Indeed the two Diracs are exactly matched to the target using the oriented Gaussian kernel since $\\|\\cdot\\|_{W^*}$ is in that case a metric on the entire space $\\mathcal{D}$. They are however matched to the opposite vectors with the unoriented Gaussian kernel which is indeed insensitive to orientation. In the case of Binet kernel, in addition to orientation-invariance, there exists other pairs of Diracs which are distinct in $\\mathcal{D}$ but coincide in $W^*$. For example, it can be easily verified that all discrete varifolds of the form $\\delta_{(x,d_1)} + \\delta_{(x,d_2)}$ with orthogonal vectors $d_1$ and $d_2$ are equal in $W^*$, which is reflected by the result in Figure \\ref{fig:2Diracs}. \n\nWe emphasize the difference of behavior between linear and oriented Gaussian kernels with the example of Figure \\ref{fig:2Diracs_bis} associated this time to the pushforward action model. The result shown in the first row is a consequence of the fact that fidelity terms derived from the linear kernel only constrains the sums $d_1 + d_2$ and $d_1' + d_2'$ to match.   \n\n\n\n\\subsubsection*{Multi-directional varifold matching}\nFinally, Figure \\ref{fig:cat} shows an example of matching on more general discrete varifolds that involve varying number of directions at different spatial locations. This is computed with the normalized action using an oriented Gaussian kernel for the fidelity term. Although purely synthetic, it illustrates the potentialities of the proposed approach to register data with complex directional patterns.    \n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{cccc}\n    \\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=3.2cm]{matching_cat_t_1.png} \n    &\\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=3.2cm]{matching_cat_t_7.png} \n    &\\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=3.2cm]{matching_cat_t_15.png}\n    &\\includegraphics[trim = 15mm 15mm 15mm 15mm ,clip,width=3.2cm]{matching_cat_final.png} \\\\\n    $t=0$ & $t=1\/3$ & $t=2\/3$  & $t=1$ \n    \\end{tabular}\n    \\caption{Registration of multi-directional sets. The lengths of vectors correspond to the weights of the Dirac varifolds.}\n    \\label{fig:cat}\n\\end{figure}\n\n\\subsection{Contrast-invariant image registration}\nA last possible application worth mentioning is the registration of images with varying contrast. Indeed, an image $I$ modulo all contrast changes is equivalently represented by its unit gradient vector field $\\frac{\\nabla I}{|\\nabla I|}$. Note that this may in fact be only defined at isolated pixels in the image, specifically the ones where the gradient is non vanishing. Within the setting of this work, it is thus natural to associate to $I$ the discrete varifold \n\\begin{equation*}\n  \\mu_{I} = \\sum_{\\nabla I(x_i) \\neq 0} \\delta_{\\left(x_i,\\frac{\\nabla I}{|\\nabla I|}(x_i)\\right)}   \\in \\mathcal{D}\n\\end{equation*}\n    \n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{cc}\n    \\includegraphics[trim = 15mm 15mm 15mm 10mm ,clip,width=4cm]{phan_t_0.png} \n    &\\includegraphics[trim = 15mm 15mm 15mm 10mm ,clip,width=4cm]{phan_t_1.png} \\\\\n    $t=0$ & $t=1\/2$ \\\\\n    \\includegraphics[trim = 15mm 15mm 15mm 10mm ,clip,width=4cm]{phan_t_2.png}\n    &\\includegraphics[trim = 15mm 15mm 15mm 10mm ,clip,width=4cm]{phan_target.png} \\\\\n     $t=1$  & $\\textrm{target}$ \n    \\end{tabular}\n    \n    \\begin{tabular}{c}\n    \\includegraphics[trim = 15mm 15mm 15mm 10mm ,clip,width=6cm]{phan_match_vf.png} \\\\\n    t=1\n    \\end{tabular}\n    \\caption{Registration of images modulo contrast changes. The matching is computed between the discrete varifolds associated to both images with the normalized action model and unoriented Gaussian fidelity term. The estimated deformation can be then applied to the template image.}\n    \\label{fig:phantom}\n\\end{figure}\n\nIt is straightforward that $\\mu_{I}$ is invariant to increasing contrast changes. It also becomes invariant to decreasing ones by quotienting out the orientation of the unit gradient vectors, which in our framework is simply done by selecting an orientation-invariant kernel $\\gamma(\\langle d ,d' \\rangle)$ to define $\\|\\cdot\\|_{W^*}$. In Figure \\ref{fig:phantom}, this approach is used to map two oppositely contrasted synthetic phantom brain images. We show both the alignment of the discrete varifolds as well as the full deformation applied to the image itself. Note that these images have no noise and a simple structure with relatively low number of non-vanishing gradients. There will be clearly the need for more validation to be done in the future in order to evaluate the practicality and robustness of this method for real multi-modal medical images.\n\n\\section{Conclusion and future work} \nWe have proposed, in this paper, a framework for large deformation inexact registration between discrete varifolds. It relies on the LDDMM setting for diffeomorphisms and include different models of group action on the space of varifolds. In each case, we derived the corresponding optimal control problems and the associated geodesic equations in Hamiltonian form. By combining those with the use of kernel-based fidelity metrics on varifolds, we proposed a geodesic shooting algorithm to numerically tackle the optimization problems. We finally illustrated the versatility and properties of this approach through examples of various natures which go beyond the classical cases of curves or surfaces. \n\nSeveral improvements or extensions of this work could be considered for future work. From a theoretical standpoint, it would be for instance important to derive a more general 'continuous' varifold matching model i.e with more general distributions than Dirac sums. Besides, higher dimensional varifolds could be possibly introduced within our model, although this would involve dealing with direction elements in Grassmann manifolds as in \\cite{Charon2} instead of the simpler $\\mathbb{S}^{n-1}$. Lastly, future work will also include adapting the existing fast GPU implementations for LDDMM to the new dynamical systems appearing here, with the objective of making the whole approach more scalable to real data applications.        \n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Prof. Sarang Joshi for many enriching discussions that initiated parts of this work.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec01}\n\nIn our investigation, we are supposing the radio loops to be evolved\nsupernova remnants (SNRs). Supernova remnants represent shells of\napproximately spherical shape which are spreading, and depending of\nthe interstellar matter density, they change their shape. We can see\nonly parts of these spherical contours which represent radio spurs,\ne.g. areas of more intense emission in the sky, spur-like and of\nhuge dimensions. Radio loops consist of more spurs lying\napproximately in the same small circle of the celestial sphere.\nTheir material probably expands inside of bubbles of low density.\nBubbles are made by former SNR explosions or by strong stellar winds\n(\\citet{salt83,mcke77} and references therein). The radio emission\nfrom SNRs is generally understood to be synchrotron emission from\nthe relativistic electrons moving in the magnetic field.\n\nA star in the constellation of Cygnus exploded and its remnant is\nCygnus loop. It is classified as a middle-aged SNR of type S,\nlocated below (but near the plane of) the Galactic equator, and less\nthan 1 kpc away from us. It is listed in Green's catalogue of SNRs\nas G74.0-8.5 (\\citet{gree04,gree06,gree09}). As shown in\n\\citet{gree84}, this remnant has been decelerated considerably by\nits interaction with the surrounding interstellar medium.\n\n\\citet{leah97} presented high resolution 1420 MHz total intensity\nand polarization maps of the Cygnus loop and derived rotation\nmeasures. \\citet{asch99} made a comparison between the radio and\nX-ray emission from supernova remnants. The X-ray emission traces\nthe hot gas behind the shock front, while the radio emission comes\nfrom the relativistic electron population emitting synchrotron\nradiation in the ambient magnetic field. They found that there are\nsignificant differences in the distribution of X-ray and radio\nbrightness in the Cygnus loop. Also, significant temperature\nvariations, seen along the rim and in the bright filaments interior\nto the rim, pointed out that the Cygnus loop was caused by a\nsupernova exploding in a cavity.\n\n\\citet{uyan02} suggested that Cygnus loop may consist of two\noverlapping remnants. Substantial differences between the northern\nand southern part as well as in their emission characteristics have\nbeen observed (\\citet{uyan04,patn02}). Some authors proposed that\nthe Cygnus loop consists of two likely interacting SNRs: G74.3-8.4\nand G72.9-9.0 (\\citet{uyan02,leah02}). In \\citet{sun06}, from 4800\nMHz observations, it is explained that the polarization maps (the\ndifference in the polarization characteristic between the northern\nand southern part) support previous ideas that the Cygnus loop may\nconsist of two SNRs. Besides, several compact radio sources are\nlocated within the boundary of the remnant. The main characteristics\nof the Cygnus loop in different spectral bands are: (a) in radio\nband: it is shell, brightest to the north-east, with fainter\nbreakout region to south, with spectral variations, (b) optical:\nlarge filamentary loop, brightest to the north-east, not well\ndefined to the south and west, (c) X-ray: shell in soft X-rays (see\ne.g. in \\citet{gree06}).\n\nSome radio maps which include this remnant are the following: the\nmap by \\citet{kund72} at 4940 MHz, \\citet{keen73} at 2695 MHz,\n\\citet{uyan04} at 2675 MHz, \\citet{leah97} at 1420 MHz,\n\\citet{dick80} at 610 MHz, \\citet{gree84} at 408 MHz. Observations\nof the continuum radio emission at 2720 (\\citet{reif87}), 1420\n(\\citet{reic86}), 820 (\\citet{berk72}), 408 (\\citet{hasl82}) and\n34.5 MHz (\\citet{dwar90}), we found in electronic form and used them\nin this paper.\n\nIn our previous paper (\\citet{bork09a}) we only calculated the\ntemperatures and brightnesses at the three frequencies. In this\npaper we expand the scope of our investigation to: brightness\ntemperatures and surface brightnesses at the five frequencies, as\nwell as spectrum, $T-T$ graphs, the radio spectral indices,\nestimation of environment density and the initial energy of\nsupernova explosion, and the flux density spectrum.\n\nIn our calculations, we used brightness temperatures over the whole\narea of the loop, so the mean temperature that we estimated refers\nto northern and southern parts together. In this research the\naverage brightness temperatures and surface brightnesses of the\nCygnus radio loop are calculated at the five frequencies: 2720,\n1420, 820, 408 and 34.5 MHz. Then we study how these results are\ngetting along with previous results (\\citet{roge99,reic03,uyan04})\nand with current theories of SNR evolution. These theories predict\nthat loops are non-thermal sources which are spreading inside of the\nhot and low density bubbles made by former supernova explosions or\nby strong stellar winds (see \\citet{salt83,mcke77} and references\ntherein).\n\nOur aim is also to apply method for determination of the brightness\ntemperature given in article \\citet{bork07} which is developed for\nlarge radio loops, and to show that it is rather efficient in the\ncase of much smaller radio loops, e.g. Cygnus loop, as well as to\ncheck how the results obtained using this method are getting along\nwith the method of $T-T$ graphics and results obtained with other\nmethods. Our method is quite simple because we are using brightness\ntemperature isolines to define borders of the Cygnus loop at the\nwide range of frequencies ($\\nu_{max} \/ \\nu_{min}$ = 2720 MHz \/ 34.5\nMHz $\\approx$ 79). Other authors are using different squared or\nrectangular areas to determine area of the loop and calculate\nspectral indices, brightness temperature and the flux of the loop\n(\\citet{uyan04}, page 917 and \\citet{leah98}, page 786). Also, we\ncalculated flux from Cygnus loop and compare our results with\nresults of other authors in different frequency ranges.\n\n\n\\section{Analysis}\n\\label{sec02}\n\n\\subsection*{Data}\n\\label{sec02a}\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig1a.eps} \\\\\n\\vspace*{0.2cm}\n\\hspace*{0.4cm}\n\\includegraphics[width=0.34\\textwidth]{fig1b.eps} \\\\\n\\vspace*{0.8cm}\n\\includegraphics[width=0.50\\textwidth]{fig1c.eps}\n\\caption{\\emph{Top}: the 2720 MHz map of a region in Cygnus, in new\nGalactic coordinates ($l$, $b$), showing contours of brightness\ntemperature. This radio loop has position: $l$ = [76.5$^\\circ$,\n71.5$^\\circ$]; $b$ = [-10.5$^\\circ$, -7$^\\circ$]. The HPBW (Half\nPower Beam Width) for this frequency is 0$^\\circ$.35. Eleven\ncontours plotted represent the temperatures $T_\\mathrm{min}$ and\n$T_\\mathrm{max}$ from Table \\ref{tab01} and nine contours in\nbetween. The contours are plotted every 0.265 K, starting from the\nlowest temperature of 0.395 K up to 0.66 K. The corresponding\ntemperature scale is given (in K). \\emph{Bottom}: the 2720 MHz area\nmap of Cygnus.}\n\\label{fig01}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig2a.eps} \\\\\n\\vspace*{0.2cm}\n\\hspace*{0.4cm}\n\\includegraphics[width=0.34\\textwidth]{fig2b.eps} \\\\\n\\vspace*{0.8cm}\n\\includegraphics[width=0.50\\textwidth]{fig2c.eps}\n\\caption{The same as Fig. 1, but for 1420 MHz. The HPBW for this\nfrequency is 0$^\\circ$.59. The contours are plotted every 0.07 K,\nstarting from the lowest temperature of 4.2 K up to 4.9 K.}\n \\label{fig02}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig3a.eps} \\\\\n\\vspace*{0.2cm}\n\\hspace*{0.4cm}\n\\includegraphics[width=0.34\\textwidth]{fig3b.eps} \\\\\n\\vspace*{0.8cm}\n\\includegraphics[width=0.50\\textwidth]{fig3c.eps}\n\\caption{The same as Fig. 1, but for 820 MHz. The HPBW is\n1$^\\circ$.2. The contours are plotted every 0.39 K, starting from\nthe lowest temperature of 10.1 K up to 14 K.}\n\\label{fig03}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig4a.eps} \\\\\n\\vspace*{0.2cm} \\hspace*{0.4cm}\n\\includegraphics[width=0.34\\textwidth]{fig4b.eps} \\\\\n\\vspace*{0.8cm}\n\\includegraphics[width=0.50\\textwidth]{fig4c.eps}\n\\caption{The same as Fig. 1, but for 408 MHz. The HPBW is\n0$^\\circ$.85. The contours are plotted every 3.1 K, starting from\nthe lowest temperature of 41 K up to 72 K.}\n\\label{fig04}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig5a.eps} \\\\\n\\vspace*{0.2cm} \\hspace*{0.4cm}\n\\includegraphics[width=0.34\\textwidth]{fig5b.eps} \\\\\n\\vspace*{0.8cm}\n\\includegraphics[width=0.50\\textwidth]{fig5c.eps}\n\\caption{The same as Fig. 1, but for 34.5 MHz. The HPBW is\n0$^\\circ$.7. The contours are plotted every 3550 K, starting from\nthe lowest temperature of 34500 K up to 70000 K.}\n\\label{fig05}\n\\end{figure}\n\n\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[height=0.9\\textheight]{fig6.eps}\n\\caption{Temperature profiles at 2720 MHz for Galactic longitude\nfrom 76$^\\circ$.5 to 71$^\\circ$.5 and for the following values of\nGalactic latitude: $b$ = -7$^\\circ$.5 \\emph{(top left)}, $b$ =\n-8$^\\circ$ \\emph{(top right)}, $b$ = -8$^\\circ$.5 \\emph{(middle\nleft)}, $b$ = -9$^\\circ$ \\emph{(middle right)} and $b$ = -10$^\\circ$\n\\emph{(bottom)}. Temperatures are given in K, and Galactic\nlongitudes in degrees. Notice that the bottom figure shows the\ntemperature profile at latitude which lies outside (but close enough\nto) the loop and from that profile we can see what is the highest\ntemperature of the background at $b$ = -10$^\\circ$, at 2720 MHz (not\nof the loop with background).}\n\\label{fig06}\n\\end{figure*}\n\n\nWe used the following radio-continuum surveys as the basic source of\ndata in this paper: at 2720 MHz (\\citet{reif87}), 1420 MHz\n(\\citet{reic86}), 820 MHz (\\citet{berk72}), 408 MHz (\\citet{hasl82})\nand 34.5 MHz (\\citet{dwar90}). These surveys are available in\nelectronic form, as the \"Flexible Image Transport System\" (FITS)\ndata format, at the site:\n\\url{http:\/\/www.mpifr-bonn.mpg.de\/survey.html}. This online Survey\nSampler of the \"Max-Planck-Institut f\\\"{u}r Radioastronomie\" (MPIfR)\nnear Bonn, Germany, allows users to pick a region of the sky and\nobtain images and data at different frequencies. The 2720-MHz\nStockert survey has the angular resolution of 0$^\\circ$.35, 1420-MHz\nStockert survey (\\citet{reic86}) 0$^\\circ$.59, the 820-MHz Dwingeloo\nsurvey (\\citet{berk72}) 1$^\\circ$.2, the 408-MHz all-sky survey\n(\\citet{hasl82}) 0$^\\circ$.85 and the 34.5-MHz Gauribidanur survey\n(\\citet{dwar90}) 0$^\\circ$.7. The corresponding observations are\ngiven at the following rates (measured data) for both $l$ and $b$:\n$\\frac{1^\\circ}{8}$ at 2720 MHz, $\\frac{1^\\circ}{4}$ at 1420 MHz,\n$\\frac{1^\\circ}{2}$ at 820 MHz, $\\frac{1^\\circ}{3}$ at 408 MHz and\n$\\frac{1^\\circ}{5}$ at 34.5 MHz. The effective sensitivities are\nabout 5 mK T$_b$ (T$_b$ is for an average brightness temperature),\n50 mK, 0.20 K, 1.0 K and about 700 K, respectively.\n\nFITS data format stores the multidimensional arrays and\n2-dimensional tables containing rows and columns of scientific data\nsets. We extracted observed brightness temperatures from this data\nformat into ASCII data files, and afterwards using our programs in C\nand FORTRAN, we obtained results presented in this paper.\n\n\n\n\\subsection*{Method}\n\\label{sec02b}\n\nAs there is great influence of background radiation over the Cygnus\nloop area, it is very difficult to determine its area precisely. The\narea of this loop is enclosed with brightness temperature contours.\nThe maps of a region in Cygnus, in new Galactic coordinates ($l$,\n$b$), with contours of the brightness temperatures $T_\\mathrm{b}$\nare plotted in Figs. \\ref{fig01}--\\ref{fig05}. In these figures,\namong all the brightness temperature contours, the most important\nare the outer contour and inner one which represent the loop\nborders. The outer one (which corresponds to the minimum temperature\nof the loop) separates loop and background, while the inner one\n(corresponding to the maximum temperature) separates loop and some\nsuperposed source. The interval of Galactic longitude of this loop\nis $l$ = [76.5$^\\circ$, 71.5$^\\circ$], and of latitude is $b$ =\n[-10.5$^\\circ$, -7$^\\circ$]. We used the same method of calculation\nas given in \\citet{bork07} for Galactic radio loops I--VI,\n\\citet{bork08} for Loops V and VI and in \\citet{bork09b} for\nMonoceros loop. Our aim is to apply method, which we developed for\nmain Galactic loops I-VI and described in \\citet{bork07} and\n\\citet{bork08}, to smaller remnants and to show that our method of\ncalculation is applicable to almost all remnants. To confirm that\nour method is good, we also gave the area maps of Cygnus at the five\nfrequencies (Figs. \\ref{fig01}--\\ref{fig05}) and some examples of\nthe temperature profiles at the 2720 MHz (Fig. \\ref{fig06}). In Fig.\n\\ref{fig06} it should be noticed that the bottom panel shows only\nbackground radiation (not including loop), where for $b$ =\n-10$^\\circ$ the brightness temperature is not higher than 0.38 K. It\nis in agreement with temperature intervals given in the first row of\nthe Table \\ref{tab01}, where for the minimum temperature of the loop\nwith background, at 2720 MHz, it is put 0.395 K.\n\n\\clearpage\n\nThe mean temperatures and surface brightnesses of this radio loop\nare computed using data taken from radio-continuum surveys at 2720,\n1420, 820, 408 and 34.5 MHz. We have subtracted the background\nradiation, in order to derive the mean brightness temperature of the\nSNR alone. First, the temperature of the loop plus background was\ndetermined. For every bin in survey we have some value for\ntemperature $T_{sum}$. We take all bins within the loop border and\nobtained average temperature $<T_{sum}>$ of the loop plus\nbackground. Then, the background alone near the loop was estimated.\nWe determine temperature of the background near the outer border of\nthe loop (average value from all data near the outer border of the\nloop) and that is the background temperature $T_{back}$. And\nfinally, the difference of these values is calculated to obtain the\naverage temperature of the loop at each frequency: average\ntemperature of the loop is $<T_{sum}> - T_{back}$. The areas over\nwhich an average brightness temperature is determined at each of the\nfive frequencies are taken to be as similar as possible within the\nlimits of measurement accuracy. $T_\\mathrm{min}$ from Table\n\\ref{tab01} means the lower temperature limit between the background\nand the loop, and $T_\\mathrm{max}$ means the upper temperature limit\nof the loop. So we used all measured values between $T_\\mathrm{min}$\nand $T_\\mathrm{max}$, inside the corresponding regions of $l$ and\n$b$, to calculate the brightness temperature of a loop including the\nbackground. The mean brightness temperature for the loop is found by\nsubtracting the mean value of background brightness temperature from\nthe mean value of the brightness temperature over the area of the\nloop.\n\nAfter deriving the mean brightness temperatures\n$T_{\\mathrm{b},\\nu}$, we have converted these values into surface\nbrightness $\\Sigma_{\\nu}$ by:\n\n\\begin{equation}\n\\Sigma_\\nu = \\left( {2k\\nu ^2 \/c^2 } \\right) T_{\\mathrm{b},\\nu},\n\\label{equ01}\n\\end{equation}\n\n\\noindent where $k$ is Boltzmann constant and $c$ the speed of\nlight. Results are given in Table \\ref{tab01}.\n\n\n\\begin{table*}[ht!]\n\\centering\n\\caption{Temperatures and brightnesses of Cygnus radio\nloop at 2720, 1420, 820, 408 and 34.5 MH\\lowercase{z}}\n\\begin{tabular}{c|c|c|c}\n\\hline\nFrequency & Temperature limits & Temperature & Brightness \\\\\n(MHz) & $T_\\mathrm{min}$, $T_\\mathrm{max}$ (K) & (K) & (10$^{-22}$ W\/(m$^2$ Hz Sr)) \\\\\n\\hline\n2720 & 0.395, 0.66 & 0.160 $\\pm$ 0.005 & 3.64 $\\pm $ 0.12 \\\\\n1420 & 4.2, 4.9 & 0.49 $\\pm$ 0.05 & 3.04 $\\pm $ 0.30 \\\\\n820 & 10.1, 14.0 & 2.30 $\\pm$ 0.20 & 4.75 $\\pm$ 0.40 \\\\\n408 & 41, 72 & 15.3 $\\pm$ 1.0 & 7.83 $\\pm$ 0.50 \\\\\n34.5 & 34500, 70000 & 13960 $\\pm$ 700 & 50.97 $\\pm$ 2.56 \\\\\n\\hline\n\\end{tabular}\n\\label{tab01}\n\\end{table*}\n\n\n\\section{Results}\n\\label{sec03}\n\nThe radio continuum maps are used for determining the area of the\nCygnus loop and for deriving brightness temperatures over it. At\neach of the five frequencies, the areas are determined to be as\nsimilar as possible within the limits of measurement accuracy. There\nare still some differences between these areas and we think that the\nmajor causes of differing borders between the five frequencies are\nsmall random and systematic errors in the data. The surface\nbrightnesses of SNRs must be above the sensitivity limit of the\nobservations and must be clearly distinguishable from the Galactic\nbackground emission (\\citet{gree91}). As it is very difficult to\nresolve the fainter parts of the loop from the background, they are\nnot taken into account. For evaluation brightness temperatures over\nthe area of the loop we had to take into account background\nradiation (see \\citet{webs74}). Borders enclosing the spurs are\ndefined to separate the spur and its background. For the method of\ncalculation see also \\citet{bork07}, \\citet{bork08} and\n\\citet{bork09b}. As mentioned in these papers, if the value of\n$T_\\mathrm{min}$ is changed by a small amount, the brightness\ncontours become significantly different. If $T_\\mathrm{min}$ is too\nsmall, the area of the spur becomes confused with the background and\nit becomes obvious that the border has been incorrectly chosen.\n\nThe results are given in Tables \\ref{tab01}, \\ref{tab02} and\n\\ref{tab03}. $T_\\mathrm{min}$, given in the second column of Table\n\\ref{tab01}, is the lower temperature limit, while $T_\\mathrm{max}$\nis the upper temperature of the loop and it is also upper\ntemperature limit (because there are no other superposed sources).\nThese temperature limits enable us to distinguish the loop from\nbackground. Then we derived the surface brightnesses using equation\n(\\ref{equ01}) for each frequency.\n\nThe values for brightnesses in $\\mathrm{10^{-22}\\, W\/(m^2\\, Hz\\,\nSr)}$ can be compared with results for flux densities in Jy. The\nflux densities can be transformed into brightnesses or vice versa,\ntaking into account that the frequencies have to be the same.\nKnowing the loop size $\\Omega$, the flux densities $S_\\nu$ given in\nJy can be transformed to brightnesses given in $\\mathrm{10^{-22}\\,\nW\/(m^2\\, Hz\\, Sr)}$ by:\n\n\\begin{equation}\n\\Sigma_\\nu = S_\\nu \\times 10^{-26} \/ \\Omega.\n\\label{equ02}\n\\end{equation}\n\nBy use of the spectral indices, the brightnesses can be reduced to\n1000 MHz according to relation:\n\n\\begin{equation}\n\\Sigma_{1000} = \\Sigma_\\nu (1000 \/ \\nu)^{(2 - \\beta)},\n\\label{equ03}\n\\end{equation}\n\n\\noindent where the temperature spectral index $\\beta = \\alpha + 2$,\nand $\\alpha$ is the spectral index defined by $S_\\nu\\propto\n\\nu^{-\\alpha}$.\n\n\n\\subsection*{Spectrum}\n\\label{sec03a}\n\nThe spectrum was generated using mean temperatures at five different\nfrequencies. Best-fit straight line spectrum enables calculation of\nspectral index as negative value of the line's direction\ncoefficient. In Figure \\ref{fig07} we give spectrum for the three\nmiddle frequencies: 1420, 820 and 408 MHz because their spectrum is\nvery well fitted with the straight line (see Figure \\ref{fig07}).\nFrequencies 2720 and 34.5 MHz, lies on very high and on very low\nends of the spectrum, as presented in Fig. \\ref{fig08}. If for\ncalculation of spectral index we take only three frequencies 1420,\n820 and 408 MHz, the result would be $\\beta_3$ = 2.76 $\\pm$ 0.03.\nAll five frequencies 2720, 1420, 820, 408 and 34.5 MHz, also from\nlinear fit, give $\\beta_5$ = 2.66 $\\pm$ 0.09. It is steeper spectral\nindex in comparison to average value for Galactic SNRs $\\beta = 2.5$\n(in Green's catalogue (\\citet{gree09}) the adopted value for\nspectral index is not given because of its variability). Cygnus loop\nis relatively old SNR. The shock wave has to be weak for evolved\nSNRs. The steeper spectral indices of SNRs should be expected for\nthe smaller shock wave velocities. It is result obtained from the\ndiffuse particle acceleration theory (\\citet{bell78a,bell78b}).\nAdditionally, Cygnus loop probably expands in low density\nenvironment. It looks like large Galactic radio loops - evolved SNRs\nwith the steep spectral indices, immersed in the low density\nenvironment (see \\citet{bork07,bork08}, and references therein).\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig7.eps}\n\\caption{Cygnus loop spectrum: temperature versus frequency, for\nthree measurements -- at 408, 820 and 1420 MHz.}\n\\label{fig07}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig8.eps}\n\\caption{Cygnus loop spectrum: temperature versus\nfrequency, for five measurements -- at 34.5, 408, 820, 1420 and 2720\nMHz.}\n\\label{fig08}\n\\end{figure}\n\n\nObtained values $\\beta_3$ = 2.76 $\\pm$ 0.03 (from three\nfrequencies), $\\beta_5$ = 2.66 $\\pm$ 0.09 (from all five\nfrequencies), are greater than 2.2 and confirm non-thermal origin of\nCygnus loop emission. From Fig. \\ref{fig08} it can be noticed that\nlinear fit is quite satisfactory. The value for the brightness\ntemperature spectral index of the Cygnus loop is rather steep. This\nis at the high end of the spectral index distribution for SNRs as\nsuggested in \\citet{clar76}.\n\n\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[height=0.93\\textheight]{fig9.eps}\n\\caption{The data retabulated to $0^\\circ.5 \\times 0^\\circ.5$\nresolution, for the following frequencies: 2720 MHz \\emph{(top\nleft)}, 1420 MHz \\emph{(top right)}, 820 MHz \\emph{(middle left)},\n408 MHz \\emph{(middle right)} and 34.5 MHz \\emph{(bottom)}. The\nHPBWs for these frequencies are 0$^\\circ$.35, 0$^\\circ$.59,\n1$^\\circ$.2, 0$^\\circ$.85 and 0$^\\circ$.7, respectively. The gray\nscales of temperatures are given also.}\n\\label{fig09}\n\\end{figure*}\n\n\n\\subsection*{$T-T$ plot}\n\\label{sec03b}\n\n\n\\begin{table*}\n\\centering\n\\caption{Spectral indices for Cygnus loop from $T-T$\nplots, between 2720, 1420, 820, 408 and 34.5 MH\\lowercase{z}}\n\\begin{tabular}{c|c|c|c|c|c}\n\\hline\nFrequency (MHz) & 2720 & 1420 & 820 & 408 & 34.5 \\\\\n\\hline\n2720 & \/ & 1.52 $\\pm$ 0.35 & 1.96 $\\pm$ 0.53 & 2.35 $\\pm$ 0.26 & 2.57 $\\pm$ 0.23 \\\\\n1420 & 2.22 $\\pm$ 0.76 & \/ & 2.27 $\\pm$ 0.64 & 2.60 $\\pm$ 0.28 & 2.67 $\\pm$ 0.25 \\\\\n820 & 2.64 $\\pm$ 0.61 & 3.10 $\\pm$ 0.64 & \/ & 2.82 $\\pm$ 0.31 & 2.82 $\\pm$ 0.10 \\\\\n408 & 2.67 $\\pm$ 0.26 & 2.79 $\\pm$ 0.28 & 3.10 $\\pm$ 0.31 & \/ & 2.78 $\\pm$ 0.15 \\\\\n34.5 & 2.86 $\\pm$ 0.23 & 2.95 $\\pm$ 0.25 & 2.92 $\\pm$ 0.10 & 2.94 $\\pm$ 0.15 & \/ \\\\\n\\hline\n\\end{tabular}\n\\label{tab02}\n\\end{table*}\n\n\nThe measured data have different resolutions for different\nfrequencies (see \\S 2.1), and therefore in order to obtain $T-T$\nplots the data were retabulated so the higher resolution maps are\nconvolved to the resolution of the lowest resolution map. In that\nway we convolved data at 2720, 1420, 408 and 34.5 MHz to $0^\\circ.5\n\\times 0^\\circ.5$ resolution, which is the sampling rate of the 820\nMHz survey. These retabulated data are presented in Figure\n\\ref{fig09} for the following frequencies: 2720 MHz (top left), 1420\nMHz (top right), 820 MHz (middle left), 408 MHz (middle right) and\n34.5 MHz (bottom). Then, for each frequency pair we used only the\ncommon points (with the same $(l,b)$) which belong to the loop area\nat both frequencies. In that way we reduced loop area to the same\narea for different frequencies. The obtained $T-T$ plots for five\npairs of frequencies enabled calculating the spectral indices. We\ncalculated two $\\beta$ values for each of these pairs: between\n2720--1420, 2720--820, 2720--408, 2720--34.5, 1420--820, 1420--408,\n1420--34.5, 820--408, 820--34.5 and 408--34.5 MHz and presented it\nin Table \\ref{tab02}. For each of the ten frequency pairs, by\ninterchanging the dependent and independent variables we have\nobtained two $\\beta$ values for each pair and the mean value of\nthese fit results is adopted as the radio spectral index, as\nsuggested in \\citet{uyan04}. Regarding only three frequencies\n(because their spectrum lies on straight line, see Figure\n\\ref{fig07}) 1420, 820 and 408 MHz, the average value of spectral\nindex from $T-T$ is $<\\beta_{TT}>_3$ = 2.78 $\\pm$ 0.41. Taking into\naccount all five frequencies, we get $<\\beta_{TT}>_5$ = 2.63 $\\pm$\n0.30. It can be noticed that this value agrees well with the\ncorresponding value obtained from spectrum, as expected (see\n\\citet{uyan04}).\n\\newline\n\\newline\n\nThen we calculated mean value of spectral index: regarding spectrum\nand $T-T$ graphs. Between 1420, 820 and 408 MHz we obtained\n$<\\beta>_3$ = 2.77 $\\pm$ 0.22, and between 2720, 1420, 820, 408 and\n34.5 MHz we obtained $<\\beta>_5$ = 2.64 $\\pm$ 0.20.\n\n\n\\section{Discussion}\n\\label{sec04}\n\nSpectral index variations with position of more spatial features\nwithin the Cygnus loop can be found in paper \\citet{leah99}. There\nwere studied radio spectral indices by $T-T$ plot method between\n2695, 1420 and 408 MHz and found that the bright radio filaments all\nshow negative curvature (steeper at higher frequency), and regions\ndominated by diffuse emission show positive curvature (flatter at\nhigher frequency).\n\nIt can be noticed that mean value of spectral index for Cygnus loop\nis little higher then the corresponding value obtained from articles\n(see \\citet{uyan04} or \\citet{leah98}). Reason for this is different\nareas used for data of Cygnus loop in these papers. In both papers\nthey used square areas (see page 917 from \\citet{uyan04} and page\n786 from \\citet{leah98}). In these square areas they take into\naccount parts of Cygnus loop and parts of background radiation near\nCygnus loop (these parts we do not take into account). We take in\naccount only higher intensity regions from these areas (see Figures\n\\ref{fig01}--\\ref{fig05}).\n\n\n\\begin{table}[hb!]\n\\centering\n\\caption{Brightness of Cygnus radio loop reduced to 1000\nMH\\lowercase{z}, using spectral index derived in this paper:\n$\\beta_5$ = 2.66 $\\pm$ 0.09 (from the spectrum of all five\nfrequencies 2720, 1420, 820, 408 and 34.5 MH\\lowercase{z})}\n\\begin{tabular}{c|c}\n\\hline\nFrequency & Brightness at 1000 MHz \\\\\n(MHz) & (10$^{-22}$ W\/(m$^2$ Hz Sr)) \\\\\n\\hline\n2720 & 7.05 $\\pm$ 0.86 \\\\\n1420 & 3.84 $\\pm$ 0.52 \\\\\n820 & 4.17 $\\pm$ 0.29 \\\\\n408 & 4.33 $\\pm$ 0.07 \\\\\n34.5 & 5.52 $\\pm$ 1.40 \\\\\n\\hline\n\\end{tabular}\n\\label{tab03}\n\\end{table}\n\n\nIn order to compare our values for brightnesses with results for\nfluxes given in other papers, we transform their calculated fluxes\ninto brightnesses at 1000 MHz. Knowing the loop size $\\Omega$ we\nreduce the flux densities given in Jy to brightnesses given in\n$\\mathrm{10^{-22}\\, W\/(m^2\\, Hz\\, Sr)}$ by equation (\\ref{equ02})\nand then by use of the spectral indices, the brightnesses are\nextrapolated to 1000 MHz according to relation (\\ref{equ03}).\n\nIn our previous paper (\\citet{bork09a}) we transformed the flux\ndensities for Cygnus Loop given in \\citet{roge99} at 22 MHz ($S_\\nu$\n= 1378 Jy) and \\citet{reic03} at 863 MHz ($S_\\nu$ = 184 Jy) into\n$\\Sigma_{1000}$ using loop size $\\Omega = 240' \\times 170'$ from\n\\citet{reic03} and spectral index $\\beta = 2.49$ from\n\\citet{trus02}. We obtained that values from (\\citet{bork09a}) agree\nwith previous data. But now in this paper, we use our derived value\nfor spectral index $\\beta_5 = 2.66 \\pm 0.09$, and obtain even better\nagreement. From flux densities given in mentioned papers, we\ncalculated the following values for radiation intensities:\n$\\Sigma_{1000}$ = $3.22 \\times \\mathrm{10^{-22}\\, W\/(m^2\\, Hz\\,\nSr)}$ from flux given in \\citet{roge99} and $\\Sigma_{1000}$ = $4.84\n\\times \\mathrm{10^{-22}\\, W\/(m^2\\, Hz\\, Sr)}$ from flux given in\n\\citet{reic03}.\n\nUsing our calculated $\\beta_5$ and applying relation (\\ref{equ03})\nto our calculated values of $\\Sigma_\\nu$ at 2720, 1420, 820, 408 and\n34.5 MHz, for $\\Sigma_{1000}$ we obtain values given in Table\n\\ref{tab03}. Absolute errors for brightnesses at 1000 MHz are\ncalculated in this way: $\\Delta \\Sigma_{1000} = \\left(\n{\\Sigma_{1000} \/ \\Sigma_{\\nu 1}} \\right) \\left( {\\Delta \\Sigma_{\\nu\n1} + \\Sigma_{\\nu 1} \\cdot \\ln \\left( {\\nu_1 \/ 1000} \\right) \\cdot\n\\Delta \\beta} \\right)$, where $\\nu_1$ takes values 2720, 1420, 820,\n408 and 34.5 MHz.\n\n\n\\begin{table}[ht!]\n\\centering \\caption{The Cygnus loop areas $\\Omega$ and its angular\nradii $\\theta$ at the five frequencies}\n\\begin{tabular}{c|c|c}\n\\hline\nFrequency (MHz) & $\\Omega$ ($(^\\circ)^2$) & $\\theta$ ($^\\circ$) \\\\\n\\hline\n2720 & 7.52 & 1.547 \\\\\n1420 & 8.64 & 1.658 \\\\\n820 & 9.46 & 1.735 \\\\\n408 & 9.88 & 1.773 \\\\\n34.5 & 8.44 & 1.639 \\\\\n\\hline\n\\end{tabular}\n\\label{tab04}\n\\end{table}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig10.eps}\n\\caption{The surface brightness to diameter diagram from Berezhko \\&\nV\\\"{o}lk (2004), with value for Cygnus loop added. Three different\ndensities for the interstellar matter (ISM) ($N_\\mathrm{H}$ = 3, 0.3\nand 0.003 cm$^{-3}$) are presented, plus three values for the (SN)\nexplosion energy ($E_\\mathrm{SN}$ = 0.25, 1 and 3 $\\times 10^{51}$ erg).}\n\\label{fig10}\n\\end{figure}\n\n\nInformation about variations of brightness and spectrum over radio\nimage is presented by \\citet{leah98} and by \\citet{uyan02}. The\naverage values reduce the information imprinted in variations of\nbrightness and spectrum over the radio image but give us other\ninteresting information, like explosion energy, surface brightness\nand distance to the Cygnus radio loop.\n\nWith regard to the loop's borders in Figs. \\ref{fig01}--\\ref{fig05},\nwe derived loop area at each frequency, as well as its corresponding\nangular radii (see Table \\ref{tab04}). An angular radius is obtained\nin this way: our derived loop area, when approximated with the\ncircle of the same area, gives possibility of determining angular\nradius as $\\theta$ = $\\sqrt{\\Omega \/ \\pi}$. These areas can be\ncompared to the areas calculated by other authors, e.g.\n9$(^\\circ)^2$.80 at 1420 MHz (\\citet{leah02}) and 11$(^\\circ)^2$.33\nat 863 MHz (\\citet{reic03}). When their areas are recalculated into\nangular radii in the way we described, the result are: 1$^\\circ$.77\nat 1420 MHz (\\citet{leah02}) and 1$^\\circ$.90 at 863 MHz\n(\\citet{reic03}). It can be noticed that we obtained somewhat\nsmaller values, but it has to be taken into account that other\nauthors estimated only the rectangular map size while we estimated\nthe loop size (inside its contour borders) exactly.\n\nThe relation between surface brightness ($\\Sigma$) and diameter\n($D$) for supernova remnants (SNRs) - so-called the $\\Sigma-$D\nrelation - is appropriate for description of the radio brightness\nevolution of these sources. The empirical relations should be used\nwith caution because of the limited usefulness due to selection\neffects (see \\citet{gree09}, \\citet{uros10} and references therein).\nThe updated theoretical relations were derived by \\citet{duri86},\nbased on the Bell's (\\cite{bell78a,bell78b}) diffuse shock particle\nacceleration theory, and by Berezhko and V\\\"{o}lk (\\citet{bere04})\nbased on the non-linear kinetic theory of the diffuse shock\nacceleration mechanism. The $\\Sigma-D$ diagram at 1 GHz with the\ntheoretically derived evolutionary tracks taken from \\citet{bere04}\nwith our value for the Cygnus loop superposed, is shown in Fig.\n\\ref{fig10}. The $\\Sigma-D$ diagram at 1 GHz taken from\n\\citet{bere04} with the derived value for the Cygnus loop\nsuperposed, is shown in Fig. \\ref{fig10}. With aim to superpose\nCygnus position, we used the following results of our calculations:\nthe mean value of brightnesses at 1 GHz and the diameter, calculated\nas $D = 2 r \\sin \\bar \\theta$, where $\\bar \\theta$ = 1$^\\circ$.67\nbeing mean angular radius for all five frequencies. The distance $r$\n= 0.44 kpc is taken from Green's catalogue of SNRs (\\citet{gree09}).\nSo we added this point to the diagram: ($D$, $\\Sigma$) = (25.7 pc,\n4.98 $\\times$ 10$^{-22}$ W\/(m$^2$ Hz Sr)). From its position in this\ndiagram it can be concluded that Cygnus loop evolves in the low\ndensity environment and the initial energy of supernova (SN)\nexplosion was relatively low (see Fig. \\ref{fig10}).\n\n\n\\subsection*{Flux density spectrum}\n\\label{sec04a}\n\nOn the basis of values for brightnesses given in the fourth column\nof Table \\ref{tab01} and our calculated values for loop size\n$\\Omega$ (Table \\ref{tab04}), we derived flux values in Jy. The\ncalculated flux densities we give in Table \\ref{tab05}. As the\nCygnus loop is well studied SNR, it is possible to perform a\nmulti-frequency spectral study. In the Fig. \\ref{fig11} we present a\nsummary of some results obtained from other papers and from our\nstudy. Our results are labeled with asterisks. So, in this figure,\nwe have seventeen results for the flux density values $S_{\\nu}$ of\nCygnus loop derived in several papers (values are given in Table 3\nof \\citet{uyan04} and five results from our paper.\n\nIt can be seen from Fig. \\ref{fig11} that our added flux values fit\nvery well among other fluxes, which shows correctness of our method,\nand also that we determined loop area and its brightness well.\n\n\n\\begin{table}[ht!]\n\\centering \\caption{Flux densities (J\\lowercase{y}) that we\ncalculated at the five frequencies}\n\\begin{tabular}{c|c}\n\\hline\nFrequency (MHz) & Flux density (Jy) \\\\\n\\hline\n2720 & 83.37 $\\pm$ 2.60 \\\\\n1420 & 80.05 $\\pm$ 8.13 \\\\\n820 & 139.35 $\\pm$ 12.10 \\\\\n408 & 235.44 $\\pm$ 15.36 \\\\\n34.5 & 1309.04 $\\pm$ 65.65 \\\\\n\\hline\n\\end{tabular}\n\\label{tab05}\n\\end{table}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig11.eps}\n\\caption{Spectrum of Cygnus Loop: flux versus frequency for more different frequencies,\nobtained from all available flux density values (values are given in Table 3 in Uyaniker et al. (2004))\nand from values calculated in this paper. We identify our values by asterisks.}\n\\label{fig11}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{sec06}\n\nThe main result is method for determination brightness temperature\ngiven in article \\citet{bork07} which is developed for large radio\nloops, and we show that this method is very good for much smaller\nloop, e.g. Cygnus loop. We check the method by applying it to the\nCygnus radio loop at the wide range of frequencies. It is in good\nagrement with method of $T-T$ graphics and results obtained with\nanother methods. This method is quite simple because we use\nbrightness temperature isoline to define border of a Cygnus loop.\nOther authors are using different squared or rectangular areas to\ndetermine area of the loop and calculate spectral indices,\nbrightness temperature and the flux density of the loop\n(\\citet{uyan04}, page 917 and \\citet{leah98}, page 786). Also, we\ncalculated flux for Cygnus loop and compare our results with results\nof other authors in different ranges of frequencies. We show that\nour results are in good agrement with these results.\n\nWe estimated the temperatures and brightness of the Cygnus loop SNR\non the basis of observations of the continuum radio emission at the\nfrequencies: 2720, 1420, 820, 408 and 34.5 MHz (this paper and\n\\citet{bork09a}). The sensitivity of the brightness temperatures\nare: 5 mK for 2720 MHz, 50 mK for 1420 MHz, 0.2 K for 820 MHz, 1.0 K\nfor 408 MHz and about 700 K $T_\\mathrm{b}$ for 34.5 MHz. At the\nfrequency of 2720 MHz the measurements are the most precise (with\nthe least relative errors) so positions of the brightness\ntemperature contours of the loop are the most realistic for this\nfrequency.\n\nBorders between the five frequencies are somewhat different for this\nSNR probably due to small, random and systematic errors in the\ncalibrated data. Also, we suppose there are uncertainties of about\n($2 \\times \\Delta T$) 10 mK for 2720 MHz, 100 mK for 1420 MHz, 0.4 K\nfor 820 MHz, 2.0 K for 408 MHz and about 1400 K $T_\\mathrm{b}$ for\n34.5 MHz, in the border due to measurement errors, and there is a\ntiny difference in the absorption of radio emission in the\ninterstellar medium at different wavelengths (\\citet{pach70}).\n\nWe determined average brightness temperature from a Cygnus radio\nloop region, after subtraction of a background level. Our obtained\nvalues (when all reduced to 1000 MHz for comparison) are in good\nagreement with the earlier results. We present the radio continuum\nspectrum of the Cygnus loop using average brightness temperatures at\nfive different frequencies. As it can be seen from Figs. \\ref{fig07}\nand \\ref{fig08}, given linear fit provides reliable spectral index.\nWe present the $T-T$ plots which enable the calculation of spectral\nindex, too.\n\nAlso, from our results can be concluded that Cygnus loop evolves in\nthe low density environment and the initial energy of SN explosion\nwas relatively low. This can be read after superposing the position\nof this loop to theoretical $\\Sigma-D$ diagram form \\citet{bere04},\nwhich was derived from non-linear kinetic theory of diffuse shock\nacceleration mechanism.\n\nWe showed that method for defining a loop border and for determining\nthe values of temperature and brightness, which we developed for\nmain Galactic loops I-VI, could be applicable to all SNRs.\n\n\n\\begin{acknowledgments}\nThe authors are grateful to the referee whose suggestions\nsubstantially improved the paper. This research is supported by the\nMinistry of Science of the Republic of Serbia through project No.\n176005.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\\section*{Ethics Statement}\n\nAll authors of this work have read the ICLR code of ethics and commit to adhering to it. There is no ethical concern in this work.\n\n\\section*{Reproducibility Statement}\n\nThe code for reproducing our results in the experiment section can be found at \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE}.\n\n\\section*{Acknowledgements}\nThe authors would like to thank the anonymous reviewers for helpful comments and suggestions. This work is partially supported by the National Science Foundation IIS1763562, IARPA\nD17PC00340, ONR Grant N000141812861, Facebook PhD Fellowship, BMW, National Science Foundation awards 1722822 and 1750439, and National Institutes of Health awards R01MH125740, R01MH096951 and U01MH116925. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsors, and no official endorsement should be inferred.\n\n\n\n\n\\section{Method}\n\\label{sec:method}\n\\vspace{-1mm}\n\nWe present a two-stage approach to leverage the structural information from the auxiliary information for weakly-supervised representation learning. The first step (Section~\\ref{subsec:cluster_construct}) clusters data according to auxiliary information, which we consider discrete attributes as the auxiliary information\\footnote{Our approach generalizes to different types of auxiliary information. To help with clarity of the explanations, the paper focuses primarily on discrete attributes, but more details about other auxiliary information can be found in Appendix.}.\nThe second step (Section~\\ref{subsec:cl-infonce}) presents our clustering InfoNCE (Cl-InfoNCE) objective, a contrastive-learning-based approach, to leverage the constructed clusters. We discuss the mathematical intuitions of our approach and include an information-theoretical characterization of the goodness of our learned representations. We also show that Cl-InfoNCE can specialize to recent self-supervised and supervised contrastive approaches. \nFor notations, we use the upper case (e.g., $X$) letter to denote the random variable and the lower case (e.g., $x$) to denote the outcome from the random variable. \n\n\\vspace{-1mm}\n\\subsection{Cluster Construction for Discrete Attributes}\n\\label{subsec:cluster_construct}\n\n\n\\input{fig_tex\/cluster_construction}\n\n\\vspace{-1mm}\nWe consider discrete attributes as the auxiliary information. An example of such auxiliary information is binary indicators of attributes, such as ``short\/long hair'', ``with\/without sunglasses'' or ``short\/long sleeves'', for human photos. We construct the clusters such that data within each cluster will have the same values for a set of attributes. In our running example, selecting hair and sunglasses as the set of attributes, the human photos with ``long hair'' and ``with sunglasses'' will form a cluster. Then, how we determine the set of attributes? First, we rank each attribute according to its entropy in the dataset. Note that if an attribute has high entropy, it means this attribute is distributed diversely. Then, we select the attributes with top-$k$ highest entropy, where $k$ is a hyper-parameter. The reason for this selection process is to make sure the selected attributes are informative. See Figure~\\ref{fig:cluster_construction} for illustration.\n\n\\vspace{-1mm}\n\\subsection{Clustering InfoNCE (Cl-InfoNCE) Objective}\n\\label{subsec:cl-infonce}\n\\vspace{-1mm}\n\nThis section presents how we integrate the clustering information of data into the representation learning process. Recently, the contrastive approaches~\\citep{chen2020simple,caron2020unsupervised} have attracted lots of attention for self-supervised and supervised representation learning. The goal is to learn similar representations for correlated data and dissimilar representations for uncorrelated data. To be more specific, the self-supervised setting (e.g., the InfoNCE objective~\\citep{oord2018representation}) regards different views of the same data as correlated and distinct data as uncorrelated; the supervised setting (e.g., the supervised contrastive objective~\\citep{khosla2020supervised}) regards the data with the same downstream label as correlated and the data with distinct labels as uncorrelated. Inspired by these methods, when performing weakly-supervised representation learning, we present to learn similar representations for data within the same cluster assignment, and vice versa. To this end, we extend from the self-supervised InfoNCE objective and introduce the clustering InfoNCE (Cl-InfoNCE) objective that takes the data clustering information into account.\nWith the alphabets $X$ and $Y$ denoting the representations from augmented data:\n\\begin{equation*}\n\\resizebox{\\hsize}{!}{$X={\\rm Feature\\_Encoder\\Big(Augmentation\\_1\\big(Data\\_1\\big)\\Big)}\\,\\,{\\rm and}\\,\\,Y={\\rm Feature\\_Encoder\\Big(Augmentation\\_2\\big(Data\\_2\\big)\\Big)}$}\n\\end{equation*}\nand the alphabet $Z$ denoting the constructed clusters, we formulate Cl-InfoNCE as\n\\vspace{2mm}\n\\begin{definition}[Clustering-based InfoNCE (Cl-InfoNCE)]\n\\vspace{-1mm}\n\\begin{equation}\n{\\rm Cl-InfoNCE}:=\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x_i, y_i)\\sim {\\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}^{\\otimes n}}\\Big[ \\,\\frac{1}{n}\\sum_{i=1}^n {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big].\n\\label{eq:cl_infonce}\n\\end{equation}\n\\end{definition}\n\\vspace{-1mm}\n$f(x,y)$ is any function that returns a scalar from the input $(x,y)$. As suggested by prior work~\\citep{chen2020simple,he2020momentum}, we choose $f(x,y) = {\\rm cosine}\\big(g(x),g(y)\\big) \/ \\tau$ to be the cosine similarity between non-linear projected $g(x)$ and $g(y)$. $g(\\cdot)$ is a neural network (also known as the projection head~\\citep{chen2020simple,he2020momentum}) and $\\tau$ is the temperature hyper-parameter. $\\{(x_i, y_i)\\}_{i=1}^n$ are $n$ independent copies of $(x,y)\\sim \\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]$, where it first samples a cluster $z \\sim P_Z$ and then samples $(x,y)$ pair with $x \\sim P_{X|z}$ and $y \\sim P_{Y|z}$. Furthermore, we call $(x_i,y_i)$ as the positively-paired data ($x_i$ and $y_i$ have the same cluster assignment) and $(x_i,y_j)$ ($i\\neq j$) as the negatively-paired data ($x_i$ and $y_j$ have independent cluster assignment). Note that, in practice, the expectation in~\\eqref{eq:cl_infonce} is replaced by the empirical mean of a batch of samples.\n\n\\vspace{-1mm}\n\\paragraph{Mathematical Intuitions.} Our objective is learning the representations $X$ and $Y$ (by updating the parameters in the $\\rm  Feature\\_Encoder$) to maximize Cl-InfoNCE. At a colloquial level, the maximization pulls towards the representations of the augmented data within the same cluster and push away the representations of the augmented data from different clusters. At a information-theoretical level, we present the following:\n\\vspace{1.5mm}\n\\begin{theorem}[informal, Cl-InfoNCE maximization learns to include the clustering information]\n\\vspace{-1.5mm}\n\\begin{equation}\n\\begin{split}\n    & {\\rm Cl-InfoNCE} \\leq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z) \\\\\n{\\rm and}\\,\\,& {\\rm the\\,\\,equality\\,\\,holds\\,\\,only\\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0,\n\\end{split}\n\\label{eq:max_resulting_repre}\n\\end{equation}\n\\vspace{-5mm}\n\\label{theo:max_resulting_repre}\n\\end{theorem}\nwhere $H(Z)$ is the entropy of $Z$ and $H(Z|X)$ (or $H(Z|Y)$) are the conditional entropy of $Z$ given $X$ (or $Y$). Please find detailed derivations and proofs in Appendix. \n\nThe theorem suggests that Cl-InfoNCE has an upper bound $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$, which measures the distribution divergence between the product of clustering-conditional marginal distributions (i.e., $\\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big]$) and the product of marginal distributions (i.e., $P_{X}P_{Y}$). We give an intuition for $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$: if $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$ is high, then we can easily tell whether $(x,y)$ have the same cluster assignment or not. The theorem also suggests that maximizing Cl-InfoNCE results in the representations $X$ and $Y$ including the clustering information $Z$ ($\\because H(Z|X) = H(Z|Y) = 0$). \n\n\\vspace{-2mm}\n\\paragraph{Goodness of the Learned Representations.} \nIn Theorem~\\ref{theo:max_resulting_repre}, we show that maximizing Cl-InfoNCE learns the representations ($X$ and $Y$) to include the clustering ($Z$) information. Therefore, to characterize how good is the learned representations by maximizing Cl-InfoNCE or to perform cross validation, we can instead study the relations between $Z$ and the downstream labels (denoting by $T$). In particular, we can use information-theoretical metrics such as the mutual information $I(Z;T)$ and the conditional entropy $H(Z|T)$ to characterize the goodness of the learned representations. $I(Z;T)$ measures how relevant the clusters and the labels, and $H(Z|T)$ measures how much redundant information in the clusters that are irrelevant to the labels.  \nFor instance, we can expect good downstream performance for our auxiliary-information-infused representations when having high mutual information and low conditional entropy between the auxiliary-information-determined clusters and the labels. It is worth noting that, when $Z$ and $T$ are both discrete variables, computing $I(Z;T)$ and $H(Z|T)$ would be much easier than computing $I(X;T)$ and $H(X|T)$. \n\n\\vspace{-2mm}\n\\paragraph{Generalization of Recent Self-supervised and Supervised Contrastive Approaches.} Cl-InfoNCE (\\eqref{eq:cl_infonce}) serves as an objective that generalizes to different levels of supervision according to how we construct the clusters ($Z$). When $Z=$ instance id (i.e., each cluster only contains one instance), $\\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big]$ specializes to $P_{XY}$ and Cl-InfoNCE specializes to the InfoNCE objective~\\citep{oord2018representation}, which aims to learn similar representations for augmented variants of the same data and dissimilar representations for different data. InfoNCE is the most popular used self-supervised contrastive learning objective~\\citep{chen2020simple,he2020momentum,tsai2021multiview}. When $Z=$ downstream labels, Cl-InfoNCE specializes to the objective described in {\\em Supervised Contrastive Learning}~\\citep{khosla2020supervised}, which aims to learn similar representations for data that are from the same downstream labels and vice versa. In our paper, the clusters $Z$ are determined by the auxiliary information, and we aim to learn similar representations for data sharing the same auxiliary information and vice versa. This process can be understood as weakly supervised contrastive learning. \nTo conclude, Cl-InfoNCE is a clustering-based contrastive learning objective. \nBy differing its cluster construction, Cl-InfoNCE interpolates among unsupervised, weakly supervised, and supervised representation learning.\n\n\\iffalse\n\\vspace{-3mm}\n\\paragraph{Comparison to Learning to Predict the Clusters Assignments.} An alternative way to leverage the data clustering information is learning to predict the cluster assignment ($Z$) from the representations ($X$ and $Y$).\nThis approach requires building an additional classifier between the representations and the cluster, and it will be inefficient to optimize this classifier when having a large number of clusters. The reason is that the number of the classifier's parameters is proportional to the number of clusters. As a comparison, Cl-InfoNCE contains no additional classifier, and hence it is computationally more efficient and can scale up with a large number of clusters. Last, the most used objective for learning to predict the clusters is the cross-entropy loss. And evidences~\\citep{khosla2020supervised} show that, compared to the cross-entropy loss, the contrastive objective (e.g., our presented Cl-InfoNCE) is more robust to natural corruptions of data and stable to hyper-parameters and optimizers settings.\n\n\\vspace{-3mm}\n\\paragraph{Comparison to Contrastive Multi-view Coding with Auxiliary Information.} A direct way to leverage auxiliary information is treating auxiliary information as another view of data then adopting the contrastive multi-view coding (CMC) objective for representation learning. On the contrary, our method indirectly leverages auxiliary information. To be more precise, our method performs additional clustering construction according to the auxiliary information and then considers the Cl-InfoNCE objective to learn with the constructed clusters. We argue that modeling the clustering information can be easier than modeling the raw auxiliary information, especially when the auxiliary information is complex or high-dimensional.\n\\fi\n\\section{Introduction}\n\\label{sec:intro}\n\nSelf-supervised learning  (SSL) designs learning objectives that use data's self-information but not labels. As a result, SSL empowers us to leverage a large amount of unlabeled data to learn good representations, and its applications span computer vision~\\citep{chen2020simple,he2020momentum}, natural language processing~\\citep{peters2018deep,devlin2018bert} and speech processing~\\citep{schneider2019wav2vec,baevski2020wav2vec}. More than leveraging only data's self-information, this paper is interested in a weakly-supervised setting by assuming access to additional sources as auxiliary information for data, such as the hashtags as auxiliary attributes information for Instagram images. The auxiliary information can provide valuable but often noisy information. Hence, it raises a research challenge of how we can effectively leveraging useful information from auxiliary information.\n\n\n\n\n\n\n\n\n\n\n\nWe argue that a form of the valuable information provided by the auxiliary information is its implied data clustering information. For example, we can expect an Instagram image to be semantically more similar to the image with the same hashtags than those with different hashtags. Hence, our first step is constructing auxiliary-information-determined clusters. Specifically, we build data clusters such that the data from the same cluster have similar auxiliary information, such as having the same data auxiliary attributes. Then, our second step is to minimize the intra-cluster difference of the representations. Particularly, we present a contrastive approach - the clustering InfoNCE (Cl-InfoNCE) objective to learn similar representations for augmented variants of data within the same cluster and dissimilar representations for data from different clusters. To conclude, the presented two-stage approach leverages the structural information from the auxiliary information, then integrating the structural information into a contrastive representation learning process. See Figure~\\ref{fig:illus} for an overview of our approach.\n\nWe provide the following analysis and observations to better understand our approach. First, we characterize the goodness of the Cl-InfoNCE-learned representations via the statistical relationships between the constructed clusters and the downstream labels. A resulting implication is that we can expect better downstream performance for our weakly-supervised representations when having i) higher mutual information between the labels and the auxiliary-information-determined clusters and ii) lower conditional entropy of the clusters given the labels. Second, Cl-InfoNCE generalizes recent contrastive learning objectives by changing the way to construct the clusters. In particular, when each cluster contains only one data point, Cl-InfoNCE becomes a conventional self-supervised contrastive objective (e.g., the InfoNCE objective~\\citep{oord2018representation}). When the clusters are built using directly the labels, Cl-InfoNCE \nbecomes a supervised contrastive objective (e.g., the objective considered by~\\citet{khosla2020supervised}). These generalizations imply that our approach (auxiliary-information-determined clusters + Cl-InfoNCE) interpolates between conventional self-supervised and supervised representation learning.\n\n\\input{fig_tex\/illus}\n\n\nWe conduct experiments on learning visual representations using UT-zappos50K~\\citep{yu2014fine}, CUB-200-2011~\\citep{wah2011caltech}, Wider Attribute~\\citep{li2016human} and ImageNet-100~\\citep{russakovsky2015imagenet} datasets.\nFor the first set of experiments, we shall see how much improvement can the auxiliary information bring to us. We consider the {\\em derivative} auxiliary information, which means the auxiliary information comes from the datasets: the discrete attributes from UT-zappos50K, CUB-200-2011, and Wider Attribute. We show that the auxiliary-information-infused weakly-supervised representations, compared to conventional self-supervised representation, have a much better performance on downstream tasks. We consider two baselines that also leverage auxiliary information: i) predicting the auxiliary-information-induced clusters with cross-entropy loss and ii) adopting the contrastive multi-view coding (CMC)~\\citep{tian2020contrastive} method when treating auxiliary information as another view of data. Our approach consistently outperforms the cross-entropy method and performs better than the CMC method in most cases. \nFor the second set of experiments, we focus on the analysis of Cl-InfoNCE to study how well it works with unsupervised constructed clusters (K-means clusters). We find it achieves better performance comparing to the clustering-based self-supervised learning approaches, such as the Prototypical Contrastive Learning (PCL)~\\citep{li2020prototypical} method. The result suggests that the K-means method + Cl-InfoNCE can be a strong baseline for the conventional self-supervised learning setting. \n\n\n\n\\section{Comparison with others related in literature}\n\n\\subsection{Experiments: Comparison with Weakly supervised contrastive learning\\citep{zheng2021weakly} }\n\nWe would like to point out that a concurrent work \\cite{zheng2021weakly} presented a similar idea on weakly-supervised contrastive learning in ICCV 2021. We would like to point out the reason it is a concurrent work with ours. \\cite{zheng2021weakly} is made publicly available on 10\/05\/2021, which is the same day as the the paper submission deadline for ICLR'22. To be more precise, ICCV publicly released this paper on 10\/05\/2021, and the paper's arxiv version and code are available on 10\/10\/2021. The zero time overlap suggests that our two works are independent and concurrent.\n\n\\paragraph{Similarity and Difference} We acknowledge that the two works share the similar idea of utilizing weak labels of data in contrastive learning. \\cite{zheng2021weakly} motivates by preventing class collision during instance-wise contrastive learning (random data that belongs to the same category will possibly get falsely pushed away in instance-wise contrastive learning), and ours motivates by exploring the structural information of data within contrastive learning, followed by providing information-theoretic analysis to explain how different structural information can affect the learned representations. Task-wise \\cite{zheng2021weakly} focuses on unsupervised (no access to data labels) and semi-supervised (access to a few data labels) representation learning, and ours focuses on weakly supervised (access to side information such as data attributes) and unsupervised representation learning. For the common unsupervised representation learning part, \\cite{zheng2021weakly} presents to generate weak labels using connected components labeling process, and ours generates weak labels using K-means clustering.\n\n\\paragraph{Empirical Results} We observed that the performance on ImageNet-100 reported in [1] looks better than ours (79.77 \\cite{zheng2021weakly} v.s. ours 77.9 Figure 5). However, the experimental settings differ a lot. First, the \\textbf{datasets} are different despite the same name: \\cite{zheng2021weakly} considers ImageNet-100 by selecting the first 100 class of the ILSVRC 2012 challenge, and we select a different set of 100 classes (details shown in the Appendix E). Second, the \\textbf{batch size} is different: \\cite{zheng2021weakly} considers 2048 , and ours considers 128. Third, the \\textbf{projection heads in architecture} are different: \\cite{zheng2021weakly} uses 2 projection heads (each with 4096 hidden units) with two objectives, one is for InfoNCE and the other is for the proposed Weakly Supervised Contrastive Learning loss; whereas ours uses one projection head with 2048 hidden units for Cl-InfoNCE objective only. Although our main experiments have demonstrated that Cl-InfoNCE alone can achieve competitive performance, we acknowledge that adding InfoNCE objective with an additional linear projection head would further improve the learned representation. \n\nTo fairly compare our Cl-InfoNCE loss with their proposed Weakly Supervised Contrastive objective, we add an additional head trained with InfoNCE along with our Cl-InfoNCE objective. Experiments are conducted on our version of ImageNet100 with the controlled set up: same network architecture of resnet50, same batch size of 384, same training epochs of 200, same projection head (2048-2048-128), the same optimizer and linear evaluation protocols, etc. Our Kmeans cluster number K is chosen to be 2500 via a grid search from $\\{100, 1000, 2500, 5000, 10,000\\}$. The results are shown below Table~\\ref{tab:iccv-compare}. \n\n\\input{tbl_tex\/iccv-compare}\n\nFrom the results, we can see that the two methods' performances are similar. Our work and theirs [1] are done independently and concurrently, and both works allow a broader understanding of weakly supervised contrastive learning. \n\n\n\\subsection{Experiments: Comparison with IDFD~\\citep{tao2021clustering} }\nIDFD~\\citep{tao2021clustering} presents to learn representations that are clustering friendly (from a spectral clustering viewpoint) during the instance discrimination (ID) contrastive learning process. Although it includes both ideas of clustering and contrastive learning, IDFD~\\citep{tao2021clustering} differs from our paper fundementally because they does not utilize the constructed clusters as weak labels to train contrastive objective. However, IDFD~\\citep{tao2021clustering} can still be considered as a self-supervised representation learning method, hence we perform experiments to compare our unsupervised setting (Cl-InfoNCE + Kmeans method) with their proposed IDFD on CIFAR10 Dataset~\\citep{krizhevsky2009learning}. To provide a fair comparison with IDFD~\\citep{tao2021clustering}, we stick to the training paradigm of IDFD where they replaces Resnet-50 with Resnet-18. The batch size of 128 is used following their report. Since IDFD~\\citep{tao2021clustering} was focusing on clustering quality and didn't report the linear evaluation protocol, we use the released code of IDFD~\\citep{tao2021clustering} to re-train the model meanwhile using both the cluster accuracy and the linear evaluation protocal as evaluation metrics. We train both methods for 1000 epochs for a fair comparison. The results are presented in Table~\\ref{tb:iclr-comparison}. \n\n\n\\input{tbl_tex\/iclr-comparison}\n\n\n\nNote that \\citep{tao2021clustering} proposed 2 methods (IDFD and IDFO), we choose the compare with IDFD because \\textbf{(i)} IDFO is very unstable, \\textbf{(ii)} IDFD\/IDFO perform at-par for the best performance based on Figure2 in \\citep{tao2021clustering} and \\textbf{(iii)} \\citep{tao2021clustering} only officially releases code for IDFD. We can observe that our method exceeds IDFD on in terms of top-1 classification accuracy during linear evaluation and also improve the raw clustering accuracy score, indicating integrating weak labels from unsupervised clustering with contrastive objectives would help both representation learning and the unsupervised clustering task. \n\n\n\n\n\n\n\n\n\n\n\n\n}\n\n\n\\section{Data's Hierarchy Information as Auxiliary Information}\n\\label{sec:hierarchy}\n\n\\input{fig_tex\/wordnet_wrap}\nIn the main text, we select the discrete attributes as the auxiliary information of data, then presenting data cluster construction according to the discrete attributes. We combine the constructed clusters and the presented Cl-InfoNCE objective together for learning weakly-supervised representations. In this section, we study an alternative type of the auxiliary information - data labels' hierarchy information, more specifically, the WordNet hierarchy~\\citep{miller1995wordnet}, illustrated in the right figure. In the example, we present the WordNet hierarchy of the label ``Henslow's Sparrow'', where only the WordNet hierarchy would be seen during training but not the label.  \n\n\\subsection{Cluster Construction for WordNet Hierarchy}\n\\label{subsec:cluster_hier}\nHow do we construct the data clusters according to the WordNet hierarchy? In the above example, ``vertebrate'' and ``bird'' can be seen as the coarse labels of data. We then construct the clusters such that data within each cluster will have the same coarse label. Now, we explain how we determine which coarse labels for the data. First, we represent the WordNet hierarchy into a tree structure (each children node has only one parent node). \nThen, we choose the coarse labels to be the nodes in the level $l$ in the WordNet tree hierarchy (the root node is level $1$). $l$ is a hyper-parameter. We illustrate the process in the below figure.\n\\includegraphics[width=1.0\\textwidth]{fig\/wordnet_cluster_illus.pdf}\n\n\\subsection{Experiments: Data-Hierarchy-Determined Clusters + Cl-InfoNCE}\n\\label{subsec:hierarchy_exp}\n\nThe experimental setup and the comparing baselines are similar to Section 4.3 in the main text, but now we consider the WordNet~\\citep{miller1995wordnet} hierarchy as the auxiliary information. As discussed in prior subsection, we construct the clusters $Z$ such that the data within a cluster have the same parent node in the level $l$ in the data's WordNet tree hierarchy. $l$ is the hyper-parameter\\footnote{Note that we do not compare with the CMC method for fair comparisons with other method. The reason is that the CMC method will leverage the entire tree hierarchy, instead of a certain level in the tree hierarchy.}. \n\n\\vspace{-2mm}\n\\paragraph{Results.} Figure~\\ref{fig:hierarchy} presents our results. First, we look at the leftmost plot, and we have several similar observations when having the data attributes as the auxiliary information. One of them is that our approach consistently outperforms the auxiliary-information-determined clusters + cross-entropy loss. Another of them is that the weakly supervised representations better close the gap with the supervised representations. Second, as discussed in prior subsection, the WordNet data hierarchy clusters can be regarded as the coarse labels of the data. Hence, when increasing the hierarchy level $l$, we can observe the performance improvement (see the leftmost plot) and the increasing mutual information $I(Z;T)$ (see the middle plot) between the clusters $Z$ and the labels $T$. Note that $H(Z|T)$ remains zero (see the rightmost plot) since the coarse labels (the intermediate nodes) can be determined by the downstream labels (the leaf nodes) under the tree hierarchy structure. Third, we discuss the conventional self-supervised setting with the special case when $Z=$ instanced ID. $Z$ as the instance ID has the highest $I(Z;T)$ (see the middle plot) but also the highest $H(Z|T)$ (see the rightmost plot). And we observe that the conventional self-supervised representations perform the worse (see the leftmost plot). We conclude that, when using clustering-based representation learning approaches, we shall not rely purely on the mutual information between the data clusters and the downstream labels to determine the goodness of the learned representations. We shall also take the redundant information in the clusters into account.\n \n\n\\input{fig_tex\/hierarchy}\n\n\n\n\\section{Theoretical Analysis}\n\nIn this section, we provide theoretical analysis on the presented Cl-InfoNCE objective. We recall the definition of Cl-InfoNCE and our presented theorem:\n\n\\begin{definition}[Clustering-based InfoNCE (Cl-InfoNCE), restating Definition 3.1 in the main text] \n\\begin{equation*}\n{\\rm Cl-InfoNCE}:=\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x_i, y_i)\\sim {\\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}^{\\otimes n}}\\Big[\\,\\frac{1}{n}\\sum_{i=1}^n {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big],\n\\end{equation*}\n\\label{defn:cl-infonce-2}\n\\end{definition}\n\\begin{theorem}[informal, Cl-InfoNCE maximization learns to include the clustering information, restating Theorem 3.2 in the main text]\n\\begin{equation*}\n\\begin{split}\n    & {\\rm Cl-InfoNCE} \\leq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z) \\\\\n{\\rm and}\\,\\,& {\\rm the\\,\\,equality\\,\\,holds\\,\\,only\\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0.\n\\end{split}\n\\end{equation*}\n\\label{theo:max_resulting_repre_2}\n\\end{theorem}\n\nOur goal is to prove Theorem~\\ref{theo:max_resulting_repre_2}. For a better presentation flow, we split the proof into three parts:\n\\begin{itemize}\n    \\item Proving ${\\rm Cl-InfoNCE} \\leq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$ in Section~\\ref{subsec:proof_a}\n    \\item Proving $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z)$ in Section~\\ref{subsec:proof_b}\n    \\item Proving ${\\rm Cl-InfoNCE} {\\rm \\,\\,maximizes\\,\\,at\\,\\,} H(Z) {\\rm \\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0$ in Section~\\ref{subsec:proof_c}\n\\end{itemize}\n\n\\subsection{Part I - Proving ${\\rm Cl-InfoNCE} \\leq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$}\n\\label{subsec:proof_a}\nThe proof requires the following lemma. \n\n\\begin{lemma}[Theorem 1 by~\\citet{song2020multi}] Let $\\mathcal{X}$ and $\\mathcal{Y}$ be the sample spaces for $X$ and $Y$, $f$ be any function: $(\\mathcal{X} \\times \\mathcal{Y}) \\rightarrow \\mathbb{R}$, and $\\mathcal{P}$ and $\\mathcal{Q}$ be the probability measures on $\\mathcal{X} \\times \\mathcal{Y}$. Then,\n$$\n\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}, (x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}}\\Big] \\leq D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big).\n$$\n\\label{lemm:infonce_like}\n\\end{lemma}\n\n\n\n\\iffalse\n\\begin{lemma}[\\citet{nguyen2010estimating}] Let $\\mathcal{X}$ and $\\mathcal{Y}$ be the sample spaces for $X$ and $Y$, $f$ be any function: $(\\mathcal{X} \\times \\mathcal{Y}) \\rightarrow \\mathbb{R}$, and $\\mathcal{P}$ and $\\mathcal{Q}$ be the probability measures on $\\mathcal{X} \\times \\mathcal{Y}$. Then,\n$$\nD_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big) = \\underset{f}{\\rm sup} \\,\\mathbb{E}_{(x, y)\\sim\\mathcal{P}} [f(x,y)] - \\mathbb{E}_{(x, y)\\sim\\mathcal{Q}} [e^{f(x,y)}] + 1.\n$$\n\\begin{proof}\nThe second-order functional derivative of the objective is $-e^{f(x,y)}\\cdot d\\mathcal{Q}$, which is always negative. The negative second-order functional derivative implies the objective has a supreme value. \nThen, take the first-order functional derivative and set it to zero:\n\\begin{equation*}\nd \\mathcal{P} - e^{f(x,y)}\\cdot d \\mathcal{Q} = 0.\n\\end{equation*}\nWe then get optimal $f^*(x,y) = {\\rm log}\\,\\frac{d\\mathcal{P}}{d\\mathcal{Q}}$. Plug in $f^*(x,y)$ into the objective, we obtain\n\\begin{equation*}\n\\mathbb{E}_{\\mathcal{P}} [f^*(x,y)] - \\mathbb{E}_{\\mathcal{Q}} [e^{f^*(x,y)}] + 1 = \\mathbb{E}_{\\mathcal{P}} [{\\rm log}\\,\\frac{d\\mathcal{P}}{d\\mathcal{Q}}] = D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big).\n\\end{equation*}\n\\end{proof}\n\\label{lemm:kl}\n\\end{lemma}\n\n\\begin{lemma} \n$\n\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}, (x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}}\\Big] \\leq D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big).\n$\n\\begin{proof}\nFrom Lemma~\\ref{lemm:kl}, $\\forall f$, we have\n\\begin{equation*}\n\\small\n\\begin{split}\n    D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big) & = \\mathbb{E}_{(x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Bigg[D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big) \\Bigg]\n    \\\\\n    & \\geq  \\,\\mathbb{E}_{(x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Bigg[ \\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}} \\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}} \\Big] - \\mathbb{E}_{(x, y_1)\\sim \\mathcal{Q}} \\Big[ \\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}} \\Big] + 1 \\Bigg] \\\\\n    & = \\mathbb{E}_{(x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Bigg[ \\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}} \\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}} \\Big] - 1 + 1 \\Bigg] \\\\\n    & = \\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}, (x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}}\\Big].\n\\end{split}\n\\end{equation*}\nThe first line comes from the fact that $D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big)$ is a constant. The second line comes from Lemma~\\ref{lemm:kl}. The third line comes from the fact that $(x, y_1)$ and $(x, y_{2:n})$ are interchangeable when they are all sampled from $\\mathcal{Q}$.\n\nTo conclude, since the inequality works for all $f$, and hence the supreme is also less than $D_{\\rm KL} \\Big( \\mathcal{P} \\,\\|\\, \\mathcal{Q} \\Big)$.\n\\end{proof}\n\\label{lemm:infonce_like}\n\\end{lemma}\n\nNote that Lemma~\\ref{lemm:infonce_like} does not require $n \\rightarrow \\infty$, which is a much more practical setting compared to the analysis made only when $n\\rightarrow \\infty$. And a remark is that the equality holds in Lemma~\\ref{lemm:infonce_like} when $n\\rightarrow \\infty$. \n\\fi\n\nNow, we are ready to prove the following lemma: \n\\begin{lemma}[Proof Part I]\n$\n    {\\rm Cl-InfoNCE}:=\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x_i, y_i)\\sim {\\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}^{\\otimes n}}\\Big[\\,\\frac{1}{n}\\sum_{i=1}^n {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big] \\leq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big).\n$\n\\begin{proof}\nBy defining $\\mathcal{P} = \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big]$ and $\\mathcal{Q} = P_XP_Y$, we have \n$$\n\\mathbb{E}_{(x, y_1)\\sim \\mathcal{P}, (x, y_{2:n})\\sim \\mathcal{Q}^{\\otimes (n-1)}}\\Big[ {\\rm log}\\,\\frac{e^{f(x, y_1)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x, y_j)}}\\Big] = \\mathbb{E}_{(x_i, y_i)\\sim {\\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}^{\\otimes n}}\\Big[\\,\\frac{1}{n}\\sum_{i=1}^n {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big].\n$$\nPlug in this result into Lemma~\\ref{lemm:infonce_like} and we conclude the proof.\n\\end{proof}\n\\label{lemm:part_a}\n\\end{lemma}\n\n\\subsection{Part II - Proving $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z)$}\n\\label{subsec:proof_b}\n\nThe proof requires the following lemma:\n\\begin{lemma}\n$\nD_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq {\\rm min}\\,\\Big\\{{\\rm MI}(Z;X), {\\rm MI}(Z;Y)\\Big\\}.\n$\n\n\\begin{proof}\n\\begin{equation*}\n\\begin{split}\n    & {\\rm MI}(Z;X) - D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\,\\,{\\rm log}\\,\\,\\frac{p(x|z)}{p(x)} {\\rm d}x {\\rm d}z -  \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z')p(x|z')p(y|z'){\\rm d}z'}{p(x)p(y)} {\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\,\\,{\\rm log}\\,\\,\\frac{p(x|z)}{p(x)} {\\rm d}x {\\rm d}z -  \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z'|y)p(x|z'){\\rm d}z'}{p(x)} {\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{p(x|z)}{\\int_{z'}p(z'|y)p(x|z'){\\rm d}z'}{\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & - \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z'|y)p(x|z'){\\rm d}z'}{p(x|z)}{\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    \\geq & - \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,\\Bigg(\\frac{\\int_{z'}p(z'|y)p(x|z'){\\rm d}z'}{p(x|z)} - 1\\Bigg){\\rm d}x {\\rm d}y {\\rm d}z \\,\\,\\Big(\\because {\\rm log}\\,t \\leq t-1  \\Big) \\\\\n    = &\\,\\, 0.\n\\end{split}\n\\end{equation*}\nHence, ${\\rm MI}(Z;X) \\geq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$. Likewise, ${\\rm MI}(Z;Y) \\geq D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$. We complete the proof by combining the two results.\n\\end{proof}\n\\label{lemm:less_than_mi}\n\\end{lemma}\n\nNow, we are ready to prove the following lemma:\n\\begin{lemma}[Proof Part II]\n$D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z).$\n\\begin{proof}\nCombining Lemma~\\ref{lemm:less_than_mi} and the fact that ${\\rm min}\\,\\Big\\{{\\rm MI}(Z;X), {\\rm MI}(Z;Y)\\Big\\} \\leq H(Z)$, we complete the proof. Note that we consider $Z$ as the clustering assignment, which is discrete but not continuous. And the inequality holds for the discrete $Z$, but may not hold for the continuous $Z$.\n\\end{proof}\n\\label{lemm:part_b}\n\\end{lemma}\n\n\\subsection{Part III - Proving ${\\rm Cl-InfoNCE} {\\rm \\,\\,maximizes\\,\\,at\\,\\,} H(Z) {\\rm \\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0$}\n\\label{subsec:proof_c}\n\nWe directly provide the following lemma:\n\\begin{lemma}[Proof Part III]\n${\\rm Cl-InfoNCE} {\\rm \\,\\,max.\\,\\,at\\,\\,} H(Z) {\\rm \\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0.$\n\\begin{proof}\nWhen $H(Z|Y) = 0$, $p(Z|Y=y)$ is Dirac. The objective \n\\begin{equation*}\n\\begin{split}\n    & D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\\\\n    = &  \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z')p(x|z')p(y|z'){\\rm d}z'}{p(x)p(y)} {\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z'|y)p(x|z'){\\rm d}z'}{p(x)} {\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{\\int_{z'}p(z')p(x|z')p(y|z'){\\rm d}z'}{p(x)p(y)} {\\rm d}x {\\rm d}y {\\rm d}z \\\\\n    = & \\int_z p(z) \\int_x p(x|z) \\int_y p(y|z) \\,\\,{\\rm log}\\,\\,\\frac{p(x|z)}{p(x)} {\\rm d}x {\\rm d}y {\\rm d}z = {\\rm MI}\\Big( Z;X\\Big).\n\\end{split}\n\\end{equation*}\nThe second-last equality comes with the fact that: when $p(Z|Y=y)$ is Dirac, $p(z'|y) = 1 \\,\\,\\forall z' = z$ and $p(z'|y) = 0 \\,\\,\\forall z' \\neq z$. Combining with the fact that ${\\rm MI}\\Big( Z;X\\Big) = H(Z)$ when $H(Z|X)=0$, we know \n$D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) = H(Z) $ when $H(Z|X)=H(Z|Y)=0$.\n\nFurthermore, by Lemma~\\ref{lemm:part_a} and Lemma~\\ref{lemm:part_b}, we complete the proof.\n\\end{proof}\n\\label{lemm:part_c}\n\\end{lemma}\n\n\\subsection{Bringing Everything Together}\n\nWe bring Lemmas~\\ref{lemm:part_a},~\\ref{lemm:part_b}, and~\\ref{lemm:part_c} together and complete the proof of Theorem~\\ref{theo:max_resulting_repre_2}.\n\n\n\n\\section{Algorithms}\nIn this section, we provide algorithms for our experiments. We consider two sets of the experiments. The first one is K-means clusters + Cl-InfoNCE (see Section 4.4 in the main text), where the clusters involved in Cl-InfoNCE are iteratively obtained via K-means clustering on top of data representations. The second one is auxiliary-information-determined clusters + Cl-InfoNCE (see Section 4.3 in the main text and Section~\\ref{subsec:hierarchy_exp}), where the clusters involved in Cl-InfoNCE are pre-determined accordingly to data attributes (see Section 4.3 in the main text) or data hierarchy information (see Section~\\ref{subsec:hierarchy_exp}).\n\n\\paragraph{K-means clusters + Cl-InfoNCE}\nWe present here the algorithm for K-means clusters + Cl-InfoNCE. At each iteration in our algorithm, we perform K-means Clustering algorithm on top of data representations for obtaining cluster assignments. The cluster assignment will then be used in our Cl-InfoNCE objective.\n\n\\begin{algorithm}[H]\n\\SetAlgoLined\n\\KwResult{Pretrained Encoder $f_{\\theta}(\\cdot)$}\n $f_{\\theta}(\\cdot)\\leftarrow \\text{Base Encoder Network}$\\;\n Aug $(\\cdot)\\leftarrow$ Obtaining Two Variants of Augmented Data via Augmentation Functions\\;\nEmbedding $\\leftarrow$ Gathering data representations by passing data through $f_{\\theta}(\\cdot)$\\;\n  Clusters $\\leftarrow$\\textbf{K-means-clustering}(Embedding)\\;\n \\For {epoch in 1,2,...,N}{\n \\For{batch in 1,2,...,M}{\n    data1, data2 $\\leftarrow$ Aug(data\\_batch)\\;\n    feature1, feature2 $\\leftarrow$ $f_{\\theta}$(data1), $f_{\\theta}$(data2)\\;\n    $L_{\\text{Cl-infoNCE}}\\leftarrow$ Cl-InfoNCE(feature1, feature2, Clusters)\\;\n    $f_{\\theta} \\leftarrow f_{\\theta} - lr * \\frac{\\partial}{\\partial \\theta}L_{\\text{Cl-infoNCE}}$\\;\n }\n Embedding $\\leftarrow$ gather embeddings for all data through $f_{\\theta}(\\cdot)$\\;\n  Clusters $\\leftarrow$\\textbf{K-means-clustering}(Embedding)\\;\n }\n \\caption{K-means Clusters + Cl-InfoNCE}\n\\end{algorithm}\n\n\\paragraph{Auxiliary information determined clusters + Cl-InfoNCE}\nWe present the algorithm to combine auxiliary-information-determined clusters with Cl-InfoNCE. We select data attributes or data hierarchy information as the auxiliary information, and we present their clustering determining steps in Section 3.1 in the main text for discrete attributes and Section~\\ref{subsec:cluster_hier} for data hierarchy information.\n\n\\begin{algorithm}[H]\n\\SetAlgoLined\n\\KwResult{Pretrained Encoder $f_{\\theta}(\\cdot)$}\n $f_{\\theta}(\\cdot)\\leftarrow \\text{Base Encoder Network}$\\;\n Aug $(\\cdot)\\leftarrow$ Obtaining Two Variants of Augmented Data via Augmentation Functions\\;\n Clusters $\\leftarrow$Pre-determining Data Clusters from \\textbf{Auxiliary Information}\\;\n \\For {epoch in 1,2,...,N}{\n \\For{batch in 1,2,...,M}{\n    data1, data2 $\\leftarrow$ Aug(data\\_batch)\\;\n    feature1, feature2 $\\leftarrow$ $f_{\\theta}$(data1), $f_{\\theta}$(data2)\\;\n    $L_{\\text{Cl-infoNCE}}\\leftarrow$ Cl-InfoNCE(feature1, feature2, Clusters)\\;\n    $f_{\\theta} \\leftarrow f_{\\theta} - lr * \\frac{\\partial}{\\partial \\theta}L_{\\text{Cl-infoNCE}}$\\;\n }\n }\n \\caption{Pre-Determined Clusters + Cl-InfoNCE}\n\\end{algorithm}\n\n\n\\section{Experimental details}\n\nThe following content describes our experiments settings in details. For reference, our code is available at \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/README.md}.\n\n\\subsection{UT-Zappos50K}\nThe following section describes the experiments we performed on UT-Zappos50K dataset in Section 4 in the main text.\n\\paragraph{Accessiblity}\nThe dataset is attributed to \\citep{yu2014fine} and available at the link: \\url{http:\/\/vision.cs.utexas.edu\/projects\/finegrained\/utzap50k}. The dataset is for non-commercial use only.\n\n\\paragraph{Data Processing}\nThe dataset contains images of shoe from Zappos.com. We rescale the images to $32\\times 32$. The official dataset has 4 large categories following 21 sub-categories. We utilize the 21 subcategories for all our classification tasks. The dataset comes with 7 attributes as auxiliary information. We binarize the 7 discrete attributes into 126 binary attributes. We rank the binarized attributes based on their entropy and use the top-$k$ binary attributes to form clusters. Note that different $k$ result in different data clusters (see Figure 4 (a) in the main text).\n\n\\textit{Training and Test Split}: We randomly split train-validation images by $7:3$ ratio,  resulting in $35,017$ train data and $15,008$ validation dataset. \n\n\\paragraph{Network Design}\nWe use ResNet-50 architecture to serve as a backbone for encoder. To compensate the 32x32 image size, we change the first 7x7 2D convolution to 3x3 2D convolution and remove the first max pooling layer in the normal ResNet-50 (See code for detail). This allows finer grain of information processing. After using the modified ResNet-50 as encoder, we include a 2048-2048-128 Multi-Layer Perceptron (MLP) as the projection head \\Big(i.e., $g(\\cdot)$ in $f(\\cdot, \\cdot)$ equation (1) in the main text\\Big) for Cl-InfoNCE. During evaluation, we discard the projection head and train a linear layer on top of the encoder's output. For both K-means clusters + Cl-InfoNCE and auxiliary-information-determined clusters + Cl-InfoNCE, we adopt the same network architecture, including the same encoder, the same MLP projection head and the same linear evaluation protocol. In the K-means + Cl-InfoNCE settings, the number of the K-means clusters is $1,000$. Kmeans clustering is performed every epoch during training. We find performing Kmeans for every epoch benefits the performance. For fair comparsion, we use the same network architecture and cluster number for PCL.\n\n\\paragraph{Optimization}\nWe choose SGD with momentum of $0.95$ for optimizer with a weight decay of $0.0001$ to prevent network over-fitting. To allow stable training, we employ a linear warm-up and cosine decay scheduler for learning rate. For experiments shown in Figure 4 (a) in the main text, the learning rate is set to be $0.17$ and the temperature is chosen to be $0.07$ in Cl-InfoNCE. And for experiments shown in Figure 5 in the main text, learning rate is set to be $0.1$ and the temperature is chosen to be $0.1$ in Cl-InfoNCE. \n\n\\paragraph{Computational Resource}\nWe conduct experiments on machines with 4 NVIDIA Tesla P100. It takes about 16 hours to run 1000 epochs of training with batch size 128 for both auxiliary information aided and unsupervised Cl-InfoNCE.\n\n\\subsection{Wider Attributes}\nThe following section describes the experiments we performed on Wider Attributes dataset in Section 4 in the main text.\n\\paragraph{Accessiblity}\nThe dataset is credited to \\citep{li2016human} and can be downloaded from the link: \\url{http:\/\/mmlab.ie.cuhk.edu.hk\/projects\/WIDERAttribute.html}. The dataset is for public and non-commercial usage.\n\n\\paragraph{Data Processing}\nThe dataset contains $13,789$ images with multiple semantic bounding boxes attached to each image. Each bounding is annotated with $14$ binary attributes, and different bounding boxes in an image may have different attributes. Here, we perform the OR operation among the attributes in the bounding boxes in an image. Hence, each image is linked to $14$ binary attributes. We rank the 14 attributes by their entropy and use the top-$k$ of them when performing experiments in Figure 4 (b) in the main text. We consider a classification task consisting of $30$ scene categories. \n\n\\textit{Training and Test Split}: The dataset comes with its training, validation, and test split. Due to a small number of data, we combine the original training and validation set as our training set and use the original test set as our validation set. The resulting training set contains $6,871$ images and the validation set contains $6,918$ images.\n\n\\paragraph{Computational Resource}\nTo speed up computation, on Wider Attribute dataset we use a batch size of $40$, resulting in 16-hour  computation in a single NVIDIA Tesla P100 GPU for $1,000$ epochs training.  \n\n\\paragraph{Network Design and Optimization}\nWe use ResNet-50 architecture as an encoder for Wider Attributed dataset. We choose 2048-2048-128 MLP as the projection head \\Big(i.e., $g(\\cdot)$ in $f(\\cdot, \\cdot)$ equation (1) in the main text\\Big) for Cl-InfoNCE. The MLP projection head is discarded during the linear evaluation protocol. Particularly, during the linear evaluation protocol, the encoder is frozen and a linear layer on top of the encoder is fine-tuned with downstream labels.  For Kmeans + Cl-InfoNCE and Auxiliary information + Cl-InfoNCE, we consider the same architectures for the encoder, the MLP head and the linear evaluation classifier. For K-means + Cl-InfoNCE, we consider $1,000$ K-means clusters. For fair comparsion, the same network architecture and cluster number is used for experiments with PCL.\n\nFor Optimization, we use SGD with momentum of $0.95$. Additionally, $0.0001$ weight decay is adopted in the network to prevent over-fitting. We use a learning rate of $0.1$ and temperature of $0.1$ in Cl-InfoNCE for all experiments. A linear warm-up following a cosine decay is used for the learning rate scheduling, providing a more stable learning process. \n\n\\subsection{CUB-200-2011}\nThe following section describes the experiments we performed on CUB-200-2011 dataset in Section 4 in the main text.\n\\paragraph{Accessiblity}\nCUB-200-2011 is created by \\citet{wah2011caltech} and is a fine-grained dataset for bird species. It can be downloaded from the link: \\url{http:\/\/www.vision.caltech.edu\/visipedia\/CUB-200-2011.html}. The usage is restricted to non-commercial research and educational purposes. \n\n\\paragraph{Data Processing}\nThe original dataset contains $200$ birds categories over $11,788$ images with $312$ binary attributes attached to each image. We utilize those attributes and rank them based on their entropy, excluding the last $112$ of them (resulting in $200$ attributes), because including these $112$ attributes will not change the number of the clusters than not including them. In Figure 4 (c), we use the top-$k$ of those attributes to constrcut clusters with which we perform in Cl-InfoNCE. The image is rescaled to $224\\times 224$.\n\n\\textit{Train Test Split}:\nWe follow the original train-validation split, resulting in $5,994$ train images and $5,794$ validation images. \n\n\\paragraph{Computational Resource}\nIt takes about 8 hours to train for 1000 epochs with 128 batch size on 4 NVIDIA Tesla P100 GPUs. \n\n\\paragraph{Network Design and Optimization}\nWe choose ResNet-50 for CUB-200-2011 as the encoder. After extracting features from the encoder, a 2048-2048-128 MLP projection head \\Big(i.e., $g(\\cdot)$ in $f(\\cdot, \\cdot)$ equation (1) in the main text\\Big) is used for Cl-InfoNCE. During the linear evaluation protocal, the MLP projection head is removed and the features extracted from the pre-trained encoder is fed into a linear classifier layer. The linear classifier layer is fine-tuned with the downstream labels. The network architectures remain the same for both K-means clusters + Cl-InfoNCE and auxiliary-information-determined clusters + Cl-InfoNCE settings. In the K-means clusters + Cl-InfoNCE settings, we consider $1,000$ K-means clusters. For fair comparsion, the same network architecture and cluster number is used for experiments with PCL.\n\nSGD with momentum of $0.95$ is used during the optimization. We select a linear warm-up following a cosine decay learning rate scheduler. The peak learning rate is chosen to be $0.1$ and the temperature is set to be $0.1$ for both K-means + Cl-InfoNCE and Auxiliary information + Cl-InfoNCE settings.  \n\n\n\\subsection{ImageNet-100}\nThe following section describes the experiments we performed on ImageNet-100 dataset in Section 4 in the main text.\n\\paragraph{Accessibility}\nThis dataset is a subset of ImageNet-1K dataset, which comes from the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012-2017 \\citep{russakovsky2015imagenet}. ILSVRC is for non-commercial research and educational purposes and we refer to the ImageNet official site for more information: \\url{https:\/\/www.image-net.org\/download.php}.\n\n\n\\paragraph{Data Processing}\nIn the Section 4 in the main text and Section~\\ref{sec:hierarchy}, we select $100$ classes from ImageNet-1K to conduct experiments (the selected categories can be found in \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/data_processing\/imagenet100\/selected_100_classes.txt}). We also conduct a slight pre-processing (via pruning a small number of edges in the WordNet graph) on the WordNet hierarchy structure to ensure it admits a tree structure. Specifically, each of the selected categories and their ancestors only have one path to the root. We refer the pruning procedure in \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/data_processing\/imagenet100\/hierarchy_processing\/imagenet_hierarchy.py} (line 222 to 251).\n\nWe cluster data according to their common ancestor in the pruned tree structure and determine the level $l$ of each cluster by the step needed to traverse from root to that node in the pruned tree. Therefore, the larger the $l$, the closer the common ancestor is to the real class labels, hence more accurate clusters will be formed. Particularly, the real class labels is at level $14$. \n\n\\textit{Training and Test Split}: \nPlease refer to the following file for the training and validation split.\n\\begin{itemize}\n    \\item training: \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/data_processing\/imagenet100\/hier\/meta_data_train.csv}\n    \\item validation: \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/data_processing\/imagenet100\/hier\/meta_data_val.csv}\n\\end{itemize}\nThe training split contains $128,783$ images and the test split contains $5,000$ images. The images are rescaled to size $224\\times 224$.\n\n\\paragraph{Computational Resource}\nIt takes $48$-hour training for $200$ epochs with batch size $128$ using $4$ NVIDIA Tesla P100 machines. All the experiments on ImageNet-100 is trained with the same batch size and number of epochs.  \n\n\\paragraph{Network Design and Optimization Hyper-parameters}\nWe use conventional ResNet-50 as the backbone for the encoder. 2048-2048-128 MLP layer and $l2$ normalization layer is used after the encoder during training and discarded in the linear evaluation protocal. We maintain the same architecture for Kmeans + Cl-InfoNCE and auxiliary information aided Cl-InfoNCE. For Kmeans + Cl-InfoNCE, we choose 2500 as the cluster number. For fair comparsion, the same network architecture and cluster number is used for experiments with PCL. The Optimizer is SGD with $0.95$ momentum. For K-means + Cl-InfoNCE used in Figure 5 in the main text, we use the learning rate of $0.03$ and the temperature of $0.2$. We use the learning rate of $0.1$ and temperature of $0.1$ for auxiliary information + Cl-InfoNCE in Figure~\\ref{fig:hierarchy}. A linear warm-up and cosine decay is used for the learning rate scheduling. To stablize the training and reduce overfitting, we adopt $0.0001$ weight decay for the encoder network.  \n\n\n\n\n\n\n\n\\section{Comparisons with Swapping Clustering Assignments between Views}\nIn this section, we provide additional comparisons between Kmeans + Cl-InfoNCE and Swapping Clustering Assignments between Views (SwAV)~\\citep{caron2020unsupervised}. The experiment is performed on ImageNet-100 dataset. SwAV is a recent art for clustering-based self-supervised approach. In particular, SwAV adopts Sinkhorn algorithm~\\citep{cuturi2013sinkhorn} to determine the data clustering assignments for a batch of data samples, and SwAV also ensures augmented views of samples will have the same clustering assignments. We present the results in Table~\\ref{tab:swav}, where we see SwAV has similar performance with the Prototypical Contrastive Learning method~\\citep{li2020prototypical} and has worse performance than our method (i.e., K-means +Cl-InfoNCE).\n\n\n\\begin{table}[h]\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{cc}\n\\toprule\nMethod              & Top-1 Accuracy (\\%) \\\\ \\midrule \\midrule  \\multicolumn{2}{c}{\\it Non-clustering-based Self-supervised Approaches}  \\\\ \\midrule \\midrule\nSimCLR~\\citep{chen2020simple}       & 58.2$\\pm$1.7   \\\\ [1mm]\nMoCo~\\citep{he2020momentum}      & 59.4$\\pm$1.6             \\\\ \\midrule \\midrule \\multicolumn{2}{c}{\\it Clustering-based Self-supervised Approaches (\\# of clusters = $2.5$K)}  \\\\ \\midrule \\midrule\nSwAV~\\citep{caron2020unsupervised}   & 68.5$\\pm$1.0           \\\\ [1mm]\nPCL~\\citep{li2020prototypical}      & 68.9$\\pm$0.7           \\\\ [1mm]\nK-means + Cl-InfoNCE (ours)      & \\textbf{77.9$\\pm$0.7}               \\\\ \\bottomrule\n\\end{tabular}\n}\n\\vspace{1mm}\n\\caption{Additional Comparsion with SwAV~\\citep{caron2020unsupervised} showing its similar performance as PCL on ImageNet-100 dataset.}\n\\label{tab:swav}\n\\end{table}\n\n\\section{Preliminary results on ImageNet-1K with Cl-InfoNCE}\nWe have performed experiments on ImageNet-100 dataset, which is a subset of the ImageNet-1K dataset \\citep{russakovsky2015imagenet}. We use the batch size of $1,024$ for all the methods and consider $100$ training epochs. We present the comparisons among Supervised Contrastive Learning~\\citep{khosla2020supervised}, our method (i.e., WordNet-hierarchy-information-determined clusters + Cl-InfoNCE), and SimCLR~\\citep{chen2020simple}. We select the level-$12$ nodes in the WordNet tree hierarchy structures as our hierarchy-determined clusters for Cl-InfoNCE. We report the results in Table~\\ref{tab:imagenet-1K}. We find that our method (i.e., hierarchy-determined clusters + Cl-InfoNCE) performs in between the supervised representations and conventional self-supervised representations. \n\n\n\\input{tbl_tex\/imgnet1k}\n\n\n\\section{Synthetically Constructed Clusters in Section 4.2 in the Main Text}\n\n\nIn Section 4.2 in the main text, on the UT-Zappos50K dataset, we synthesize clusters $Z$ for various $I(Z;T)$ and $H(Z|T)$ with $T$ being the downstream labels. There are $86$ configurations of $Z$ in total. Note that the configuration process has no access to data's auxiliary information and among the $86$ configurations we consider the special cases for the supervised \\big($Z=T$\\big) and the unsupervised setting \\big($Z=$ instance ID\\big). In specific, when $Z=T$, $I(Z;T)$ reaches its maximum at $H(T)$ and $H(Z|T)$ reaches its minimum at $0$; when $Z=$ instance ID, both $I(Z;T)$ \\big(to be $H(T)$\\big) and $H(Z|T)$ \\big(to be $H(\\text{instance ID})$\\big) reaches their maximum. The code for generating these $86$ configurations can be found in lines 177-299 in \\url{https:\/\/github.com\/Crazy-Jack\/Cl-InfoNCE\/data_processing\/UT-zappos50K\/synthetic\/generate.py}.\n\n\\iffalse\nby the procedures described below to varying $Z$ artificially with different $I(Z;T)$ and $H(Z|T)$, resulting in 86 different configurations of $Z$ and then evaluate their resulted representation quality following the common linear evaluation protocol. The linear evaluation results is shown in Figure 3. \n\n\\paragraph{Synthetizing Protocal}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/ICLR_fig.pdf}\n    \\caption{Synthesising protocol demo. $x_i$ denotes each data instance, $t_i$ denotes downstream class labels and $z_i^{(level)}$ denotes clusters in the first setting where we control $I(Z;T)$ to its maximumn $H(T)$. And $S_i^{(level)}$ are superclasses of the downstream task labels $t_i$ and represents the $Z$ in the second setting. }\n    \\label{fig:synthetic-protocol}\n\\end{figure}\n\n\n\nTo intepolate between fully supervised and unsupervised setting, we form synthesised $Z$ to illustrate the relationship between $Z$ and $T$. Figure~\\ref{fig:synthetic-protocol} shows a toy example of how we construct the $Z$. Suppose $\\{x_1, x_2, ..., x_n\\}$ denotes every instance and $\\{t_1, t_2, ..., t_k\\}$ denotes different downstream class labels, then the key of constructing covariates are performing certain level of grouping. We consider three settings: \n\\paragraph{(1)} Control $I(Z;T)$ to its maximum. $I(Z;T)$ reach its maximum $H(T)$ when $P(T|Z)=1$, which means that instances assigned to a specific $Z$ have the same downstream task labels. Following this intuition, we form $z_i^{(level)}$ as subclasses of the downstream classes. We vary $level$ to transit from SimCLR to SupCon. \n\n\\paragraph{(2)} Control $H(Z|T)$ to its minimum 0. $H(Z|T)$ reach its minimum 0 if $P(Z|T)=1$, which suggests we can create $Z$ as superclasses of $T$. As shown in Figure~\\ref{fig:synthetic-protocol}, we create $S_i^{(level)}$ as different $Z$ and vary $level$ to get different $Z$ with various $I(Z;T)$ value. \n\n\\paragraph{(3)} For each $level$ in setting \\textbf{(1)}, we keep the subclasses of certain set of downstream classs $T_f=\\{t_i, t_j, t_m, ...\\}$ and permutate other $z_j^{(level)}$ assignment. Take $level=2$ for example, if we fix the subclasses of $T_f=\\{t_1\\}$, instance assigned to $\\{z_1^{(2)}, z_2^{(2)}, z_3^{(2)}, z_4^{(2)}\\}$ is untouched (i.e. the cluster assignment of $\\{x_1, x_2, ..., x_8\\}$ remain the same) and we randomly permutate instance assigned to $\\{z_5^{(2)}, z_6^{(2)}, ..., z_{16}^{(2)}\\}$ (for example $x_9$ may be reassigned to $z_6^{(2)}$ or $z_{13}^{(2)}$ instead of $z_5^{(2)}$). As we increase $|T_f|$, we get $Z$ with various $I(Z;T)$ and $H(Z|T)$ that spans the most of the information plane. This construction of $Z$ leads us to efficiently illustrate how the relationship between $Z$ and $T$ reflects the learnt representation's quality (As shown in Figure 3).\n\n\\fi\n\n\\end{appendices}\n\\section{Conclusion and Discussions}\n\\label{sec:conclu}\n\\vspace{-2mm}\n\nIn this paper, we introduce the clustering InfoNCE (Cl-InfoNCE) objective that leverages the implied data clustering information from {\\color{black}auxiliary information or data itself }for learning  weakly-supervised representations. {\\color{black} Our method effectively brings the performance closer to the supervised learned representations compared to the conventional self-supervised learning approaches, therefore improving pretraining quality when limited information is at hand. In terms of limitation, our approach requires clustering based on auxiliary information or data itself. This process sometimes could pose additional computational cost. In addition, clustering on auxiliary information or data will also lose precision. Tackling these problems would be our further research direction.}\n\n\n\\section{Related Work}\n\\label{sec:rela}\n\n\\vspace{-1mm}\n\\paragraph{Self-supervised Learning.} Self-supervised learning (SSL) defines a pretext task as a pre-training step and uses the pre-trained features for a wide range of downstream tasks, such as object detection and segmentation in computer vision~\\citep{chen2020simple,he2020momentum}, question answering, and language understanding in natural language processing~\\citep{peters2018deep,devlin2018bert} and automatic speech recognition in speech processing~\\citep{schneider2019wav2vec,baevski2020wav2vec}. In this paper, we focus on discussing two types of pretext tasks: clustering approaches~\\citep{caron2018deep,caron2020unsupervised} and contrastive approaches~\\citep{chen2020simple,he2020momentum}.\n\nThe clustering approaches jointly learn the networks' parameters and the cluster assignments of the resulting features. For example, the cluster assignments can be obtained through unsupervised clustering methods such as k-means~\\citep{caron2018deep}, or the optimal transportation algorithms such as Sinkhorn algorithm~\\citep{caron2020unsupervised}. It is worth noting that the clustering approaches enforce consistency between cluster assignments for different augmentations of the same data. The contrastive approaches learn similar representations for augmented variants of a data and dissimilar representations for different data. Examples of  contrastive approaches include the InfoNCE objective~\\citep{oord2018representation,chen2020simple,he2020momentum}, Wasserstein Predictive Coding~\\citep{ozair2019wasserstein}, and Relative Predictive Coding~\\citep{tsai2021self}. Both the clustering and the contrastive approaches aim to learn representations that are invariant to data augmentations. \n\nThere is another line of work combining clustering and contrastive approaches, such as HUBERT~\\citep{hsu2020hubert}, Prototypical Contrastive Learning~\\citep{li2020prototypical} and Wav2Vec~\\citep{schneider2019wav2vec,baevski2020wav2vec}. They first construct (unsupervised) clusters from the data. Then, they perform a contrastive approach to learn similar representations for the data within the same cluster. Our approach relates to these work with two differences: 1) we construct the clusters from the auxiliary information; and 2) we present Cl-InfoNCE as a new contrastive approach and characterize the goodness for the resulting representations.  Recent works like IDFD~\\citep{tao2021clustering} aim to achieve better unsupervised clustering by using contrastive learning representations.\nHowever, \\citet{tao2021clustering} differs from our work in that they don't directly incorporate auxiliary information into contrastive objectives.\n\n\\vspace{-2mm}\n\\paragraph{Weakly-supervised Learning with Auxiliary Information.} \nOur study relates to work on {\\color{black}prediction using auxiliary information, by treating the auxiliary information as weak labels~\\citep{sun2017revisiting,mahajan2018exploring,wen2018disjoint,radford2021learning,tan2019learning}}. The weak labels can be hashtags of Instagram images~\\citep{mahajan2018exploring}, metadata such as identity and nationality of a person~\\citep{wen2018disjoint} or corresponding textual descriptions for images~\\citep{radford2021learning}. Compared to normal labels, the weak labels are noisy but require much less human annotation work. Surprisingly, it has been shown that the network learned with weakly supervised pre-training tasks can generalize well to various downstream tasks, including object detection and segmentation, cross-modality matching, and action recognition~\\citep{mahajan2018exploring,radford2021learning}. The main difference between these works and ours is that our approach does not consider a prediction objective but a contrastive learning objective (i.e., the Cl-InfoNCE objective). {\\color{black}An independent and concurrent work~\\citep{zheng2021weakly} also incorporates weak labels into the contrastive learning objective. \nHowever, our method differs from \\citet{zheng2021weakly} by the the way we construct the weak labels. We perform clustering on the annotative attributes or unsupervised k-means to obtain weak labels\nwhereas they employ connected components labeling process. Task-wise, \\citep{zheng2021weakly} focuses on unsupervised (no access to data labels) and semi-supervised (access to a few data labels) representation learning, and ours focuses on weakly-supervised (access to side information such as data attributes) and unsupervised representation learning. For the common unsupervised representation learning part, we include a comparison with their method in the Appendix.}\n\n\n\nAnother way to learn from auxiliary information is using multi-view contrastive coding (CMC)~\\citep{tian2020contrastive} where auxiliary information is treated as another view of the data. Specifically, CMC learns representations that can capture the joint information between the data and the accompanying auxiliary information. The main difference between CMC and our approach is that CMC leverages auxiliary information directly and Cl-InfoNCE leverages it indirectly (i.e., our approach pre-processes auxiliary information by clustering it).\n\n\\subsection{Technical Background}\nContrastive learning~\\citep{bachman2019learning,chen2020simple,he2020momentum} has shown its effectiveness in unsupervised representation learning, where its goal is to learn similar representations for the positively-paired (correlated) instances and dissimilar representations for the negatively-paired (uncorrelated) instances. For example, \\citet{bachman2019learning,chen2020simple,he2020momentum} treat the positively-paired instances as distorted variations of the same image (e.g., the same image with different augmentations) and the negatively-paired instances as two distinct images. For another example, \\citet{radford2021learning,tsai2021multiview} treat the positively-paired instances as the cross-modality pair of an instance (e.g., image and its captions) and the negatively-paired instances as pairs consisting of a random image paired with the captions of another random image. Probabilistically, we refer the positively-paired representations from the data as the representations sampled from their joint distributions (i.e., $(x,y)\\sim P_{X,Y}$) and the negatively-paired representations as the representations sampled from the product of marginal distributions (i.e., $(x,y)\\sim P_{X}P_{Y}$). Hence, the contrastive learning can also be interpreted as maximizing the distribution divergence between $P_{X,Y}$ and $P_{X}P_{Y}$~\\citep{oord2018representation,chen2020simple,bachman2019learning,ozair2019wasserstein,tsai2021self}, and among different objectives the most popular one is maximizing the following InfoNCE objective~\\citep{oord2018representation}: \n\\begin{proposition}[InfoNCE~\\citep{oord2018representation}]\n\\begin{equation}\n{\\rm InfoNCE}:=\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x_i, y_i)\\sim {P_{X,Y}}^{\\otimes n}}\\Big[ {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big]\\leq D_{\\rm KL}\\,\\big(P_{X,Y} \\,\\|\\,P_{X}P_Y \\big) = {\\rm MI}\\,(X,Y),\n\\label{eq:infonce}\n\\end{equation}\n\\end{proposition}\nwhere $\\{(x_i, y_i)\\}_{i=1}^n$ are $n$ independent copies of $(x,y)\\sim P_{X,Y}$, $f(x,y)$ is any function that returns a scalar from the input $(x,y)$, and ${\\rm MI}\\,(X,Y)$ is the mutual information (MI) between $X$ and $Y$. Practically, prior work~\\citep{chen2020simple,he2020momentum} suggest $f(x,y) = {\\rm cosine}(x,y) \/ \\tau$, where ${\\rm cosine}(x,y)$ is the cosine similarity and $\\tau$ is the temperature hyper-parameter. Equation~\\eqref{eq:infonce} suggests InfoNCE is a lower bound of ${\\rm MI}\\,(X,Y)$ and theoretical work~\\citep{arora2019theoretical,tsai2021multiview,tosh2021contrastive} show that ${\\rm MI}\\,(X,Y)$ maximization could lead to representations that perform well on downstream tasks. To conclude, as the most popular contrastive approach, InfoNCE can learn good representations without the need to access the downstream supervision.\n\n\\subsection{Clustering-based Contrastive Learning}\n\nIn addition to learning without access to the downstream supervision, the contrastive learning also manifests it effectiveness under the supervised fashion~\\citep{khosla2020supervised}. In particular, \\citet{khosla2020supervised} present {\\em Supervised Contrastive Learning} that re-defines the positively-paired samples as the samples from the same category and the negatively-paired samples as the samples belonging to different categories. Observing the fact that the contrastive learning work for both unsupervised and supervised setting, a research question pops out naturally: \n\n\\begin{center}\n{\\em Can we design a unified contrastive learning objective that generalizes its applications from unsupervised to weakly-supervised and supervised representation learning?}\n\\end{center}\n\nOur paper aims to answer this question by presenting clustering-based InfoNCE (Cl-InfoNCE), which 1) first constructs the clusters under either unsupervised, weakly-supervised or supervised setup and 2) then defines the positively-paired samples as the samples from the same cluster and the negatively-paired samples as the samples from different clusters. At a colloquial level, Cl-InfoNCE pulls towards the representations of the data within the same (unsupervised, weakly-supervised, or supervised-formed) cluster and push away the representations of the data from different clusters. With $Z\/z$ being the constructed clusters, the proposed Cl-InfoNCE is formulated as:\n\\begin{proposition}[Clustering-based InfoNCE (Cl-InfoNCE)]\n\\begin{equation}\n{\\rm Cl-InfoNCE}:=\\underset{f}{\\rm sup}\\,\\,\\mathbb{E}_{(x_i, y_i)\\sim {\\color{blue} \\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}^{\\otimes n}}\\Big[ {\\rm log}\\,\\frac{e^{f(x_i, y_i)}}{\\frac{1}{n}\\sum_{j=1}^n e^{f(x_i, y_j)}}\\Big],\n\\label{eq:cl_infonce}\n\\end{equation}\n\\end{proposition}\nwhere the positively-paired representations $(x_i, y_i)$ are sampled from ${\\color{blue} \\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]}$\\footnote{$(x, y) \\sim \\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]$ includes two steps: first, $z \\sim P_Z$; then, $x \\sim P_{X|z}$ and $y \\sim P_{Y|z}$.} and the negatively-paired representations $(x_i, y_j)$ are sampled from $P_XP_Y$ ($i\\neq j$)\\footnote{$\\int_X \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] dx = P_Y$ and $\\int_Y \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] dy = P_X$.}. We note that equation~\\eqref{eq:cl_infonce} generalizes to different levels of supervision: 1) when $Z=$ instance id (i.e., each cluster only contains an instance), $\\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big]$ specializes to $P_{XY}$ and Cl-InfoNCE specializes to InfoNCE for unsupervised representation learning; 2) when $Z=$ downstream labels, Cl-InfoNCE specializes to the objective described in {\\em Supervised Contrastive Learning}~\\citep{khosla2020supervised} for supervised representation learning; and 3) when $Z$ is constructed from the weak supervision signals, such as data's attributes or the hierarchical information of data, Cl-InfoNCE can be used for weakly-supervised representation learning. Now, we provide the following theoretical statements to better understand Cl-InfoNCE:\n\\begin{theorem}[Asymptotic analysis of Cl-InfoNCE]\n\\begin{equation}\n{\\rm Cl-InfoNCE} \\leq {\\rm Cl-InfoNCE}_{\\,n \\rightarrow \\infty} = D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big) \\leq H(Z),\n\\label{eq:asymp_cl_infonce}\n\\end{equation}\n\\label{theo:asymp_cl_infonce}\n\\end{theorem}\n\\begin{theorem}[Cl-InfoNCE maximization learns to include the clusering information]\n\\begin{equation}\n{\\rm Cl-InfoNCE}_{\\,n \\rightarrow \\infty} {\\rm \\,\\,maximizes\\,\\,at\\,\\,} H(Z) {\\rm \\,\\,when \\,\\,} H(Z|X) = H(Z|Y) = 0,\n\\label{eq:max_resulting_repre}\n\\end{equation}\n\\label{theo:max_resulting_repre}\n\\end{theorem}\nwhere $H(Z)$ is the entropy of $Z$ and $H(Z|X)$\/ $H(Z|Y)$ are the conditional entropy of $Z$ given $X$\/$Y$. We leave detailed derivations and proofs in Appendix. Theorem~\\ref{theo:asymp_cl_infonce} suggests that Cl-InfoNCE has an upper bound $D_{\\rm KL}\\,\\Big( \\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big] \\,\\|\\, P_{X}P_{Y} \\Big)$, which measures the distribution divergence between the product of clustering-conditional marginal distributions (i.e., $\\mathbb{E}_{P_Z}\\big[P_{X|Z}P_{Y|Z}\\big]$) and the product of marginal distributions (i.e., $P_{X}P_{Y}$). Theorem~\\ref{theo:max_resulting_repre} suggests that maximizing Cl-InfoNCE results in the representations $X$ and $Y$ including the clustering information $Z$ ($\\because H(Z|X) = H(Z|Y) = 0$). This result is not surprising: since Cl-InfoNCE pulls together the representations from the data belonging to the same cluster and push away the representations from the data belonging to different clusters, the representations learned by maximizing Cl-InfoNCE should be able to distinguish among distinct clusters. \n\nTo conclude, Cl-InfoNCE is a clustering-based contrastive learning objective. By differing its cluster construction, Cl-InfoNCE generalizes from unsupervised, weakly-supervised to supervised representation learning. In this paper, we focus on the application of Cl-InfoNCE in weakly-supervised setup, where we examine different weak supervision sources to form clusters.\n\\section{Experiments}\n\\vspace{-1mm}\nWe given an overview of our experimental section. Section~\\ref{subsec:datasets} discusses the datasets. We consider discrete attribute information as auxiliary information for data. Next, in Section~\\ref{subsec:method}, we explain the methodology that will be used in the experiments. Section~\\ref{subsec:exp_with_auxi_attr} presents the first set of the experiments, under a weakly-supervised setting, to manifest the effectiveness of our approach and the benefits of taking the auxiliary information into account. \nLast, to study the effect of Cl-InfoNCE alone, Section~\\ref{subsec:exp_without_auxi} presents the second set of the experiments under a unsupervised setting. {\\color{black} We also conduct comparison experiments with another independent and concurrent weakly supervised contrastive learning work~\\citep{zheng2021weakly} in the Appendix.} \n\n\n\n\n\\vspace{-1mm}\n\\subsection{Datasets}\n\\label{subsec:datasets}\n\\vspace{-1mm}\nWe consider the following datasets. {\\bf UT-zappos50K}~\\citep{yu2014fine}: It contains $50,025$ shoes images along with $7$ discrete attributes as auxiliary information. Each attribute follows a binomial distribution, and we convert each attribute into a set of Bernoulli attributes, resulting in a total of $126$ binary attributes. There are $21$ shoe categories.\n{\\bf Wider Attribute}~\\citep{li2016human}: It contains $13,789$ images, and there are several bounding boxes in an image. The attributes are annotated per bounding box. We perform OR operation on attributes from different bounding boxes in an image, resulting in $14$ binary attributes per image as the auxiliary information. There are $30$ scene categories. \n{\\bf CUB-200-2011}~\\citep{wah2011caltech}: It contains $11,788$ bird images with $200$ binary attributes as the auxiliary information. There are $200$ bird species. For the second set of the experiments, we further consider the {\\bf ImageNet-100}~\\citep{russakovsky2015imagenet} dataset. It is a subset of the ImageNet-1k object recognition dataset~\\citep{russakovsky2015imagenet}, where we select $100$ categories out of $1,000$, resulting in around $0.12$ million images. \n\n\n\\vspace{-1mm}\n\\subsection{Methodology}\n\\label{subsec:method}\n\\vspace{-1mm}\nFollowing~\\citet{chen2020simple}, we conduct experiments on pre-training visual representations and then evaluating the learned representations using the linear evaluation protocol. In other words, after the pre-training stage, we fix the pre-trained feature encoder and then categorize test images by linear classification results. We select ResNet-50~\\citep{he2016deep} as our feature encoder across all settings. Note that our goal is learning representations (i.e, $X$ and $Y$) for maximizing the Cl-InfoNCE objective (equation~\\eqref{eq:cl_infonce}). Within Cl-InfoNCE, the positively-paired representations $(x, y^+) \\sim \\mathbb{E}_{z\\sim P_Z}\\big[P_{X|z}P_{Y|z}\\big]$ are the learned representations from augmented images from the same cluster $z\\sim P_Z$ and the negatively-paired representations $(x, y^-) \\sim P_XP_Y$ are the representations from arbitrary two images. We leave the network designs, the optimizer choices, and more details for the datasets in Appendix.\n\\input{fig_tex\/synthetic_wrap}\nBefore delving into the experiments, we like to recall that, in Section~\\ref{subsec:cl-infonce}, we discussed using the mutual information $I(Z;T)$ and the conditional entropy $H(Z|T)$ between the clusters ($Z$) and the labels ($T$) to characterize the goodness of Cl-InfoNCE's learned representations. To prove this concept, on UT-Zappos50K, we synthetically construct clusters for various $I(Z;T)$ and $H(Z|T)$ followed by applying Cl-InfoNCE. We present the results in the right figure.\nOur empirical results are in accordance with the statements that the clusters with higher $I(Z;T)$ and lower $H(Z|T)$ will lead to higher downstream performance. In later experiments, we will also discuss these two information-theoretical metrics.\n\n\\vspace{-1mm}\n\\subsection{Experiment I: Auxiliary-Information-Determined Clusters + Cl-InfoNCE}\n\\label{subsec:exp_with_auxi_attr}\n\\vspace{-1mm}\n\nWe like to understand how well Cl-InfoNCE can be combined with the auxiliary information. For this purpose, we select the data discrete attributes as the auxiliary information, construct the clusters ($Z$) using the discrete attributes (see Section~\\ref{subsec:cluster_construct} and Figure~\\ref{fig:cluster_construction}), and then adopt attributes-determined clusters for Cl-InfoNCE. Recall our construction of data-attributes-determined clusters: we select the attributes with top-$k$ highest entropy and then construct the clusters such that the data within a cluster will have the same values over the selected attributes. $k$ is the hyper-parameter. Note that our method considers a weakly supervised setting since the data attributes can be seen as the data's weak supervision. \n\nWe dissect the experiments into three parts. First, we like to study the effect of the hyper-parameter $k$ and select its optimal value. Note that different choices of $k$ result in different constructed clusters $Z$. Our study is based on the information-theoretical metrics (i.e., $I(Z;T)$ and $H(Z|T)$ between the constructed clusters ($Z$) and the labels ($T$)) and their relations with the downstream performance of the learned representations. Second, we perform comparisons between different levels of supervision. In particular, we include the comparisons with the supervised ($Z=$ downstream labels $T$) and the conventional self-supervised ($Z=$ instance ID) setting for our method. We show in Section~\\ref{subsec:cl-infonce}, the supervised setting is equivalent to the Supervised Contrastive Learning objective~\\citep{khosla2020supervised} and the conventional self-supervised setting is equivalent to SimCLR~\\citep{chen2020simple}. Third, we include baselines that leverage the auxiliary information: i) learning to predict the clusters assignments using cross-entropy loss and ii) treating auxiliary information as another view of data when using the contrastive multi-view coding (CMC)~\\citep{tian2020contrastive}.\n\n\n\\input{fig_tex\/attributes}\n\n\n\\vspace{-1mm}\n\\paragraph{Part I - Effect of the hyper-parameter $k$.}\nTo better understand the effect of the hyper-parameter $k$ for constructing the attributes-determined clusters, we study the information-theoretical metrics between $Z$ and $T$ and report in Figure~\\ref{fig:attributes}. Note that, to ensure the same scales for $I(Z;T)$ and $H(Z|T)$ across different datasets, we normalize $I(Z;T)$ and $H(Z|T)$ using $$I(Z;T) \\leftarrow \\frac{I(Z;T) - {\\rm min}_ZI(Z;T)}{ {\\rm max}_ZI(Z;T)- {\\rm min}_ZI(Z;T)} \\quad {\\rm and}\\quad H(Z|T) \\leftarrow \\frac{H(Z|T)) - {\\rm min}_ZH(Z|T)}{ {\\rm max}_ZH(Z|T)- {\\rm min}_ZH(Z|T)}.$$ \n\nAs $k$ increases, the mutual information $I(Z;T)$ increases but the conditional entropy $H(Z|T)$ also increases. Hence, although considering more attributes leads to the clusters that are more correlated to the downstream labels, the clusters may also contain more downstream-irrelevant information. This is in accord with our second observation that, as $k$ increases, the downstream performance first increases then decreases. Therefore, we only need a partial set of the most informative attributes (those with high entropy) to determine the clusters. Next, we observe that the best performing clusters happen at the intersection between $I(Z;T)$ and negative $H(Z|T)$. This observation helps us study the trade-off between $I(Z;T)$ and $H(Z|T)$ and suggests an empirical way to select the optimal $k$ that achieves the best performance. It is worth noting that the above process of determining the optimal $k$ does not require directly evaluating the learned representations. \n\n\\input{tbl_tex\/attributes-supervision}\n\n\n\\vspace{-1mm}\n\\paragraph{Part II - Interpolation between Different Supervision Levels.}\nIn Section~\\ref{subsec:cl-infonce}, we discussed that, by altering the designs of the clusters, our presented approach specializes to the conventional self-supervised contrastive method - SimCLR~\\citep{oord2018representation} and the supervised contrastive method - SupCon~\\citep{khosla2020supervised}. In particular, our approach specializes to SimCLR when considering augmented variants of each instance as a cluster and specializes to SupCon when considering instances with the same downstream label as a cluster. Hence, we can interpolate different supervision levels of our approach and study how auxiliary information of data can help improve representation learning.\n\nWe present the results in Table~\\ref{tbl:attributes-supervision} with different cluster constructions along with Cl-InfoNCE. We use the top-1 accuracy on Wider Attribute for discussions. We find the performance grows from low to high when having the clusters as instance ID ($40.2$), \nattributes-determined clusters ($45.5$) to labels ($49.9$). This result suggests that CL-InfoNCE can better bridge the gap with the supervised learned representations by using auxiliary information. \n\n\n\n\n\\vspace{-1mm}\n\\paragraph{Part III - Comparisons with Baselines that Leverage Auxiliary Information.} In the last part, we see that Cl-InfoNCE can leverage auxiliary information to achieve a closer performance to supervised representations than self-supervised representations. Nonetheless, two questions still remain: 1) is there another way to leverage auxiliary information other than our method (attributes-determined clusters + Cl-InfoNCE), and 2) is the weakly-supervised methods (that leverages auxiliary information) always better than self-supervised methods? To answer these two questions, in Table~\\ref{tbl:attributes-baselines}, we include the comparisons among weakly-supervised representation learning baselines that leverage auxiliary information (Attributes-determined clusters + cross-entropy loss and Contrastive Multi-view Coding (CMC)~\\citep{tian2020contrastive} when treating auxiliary information as another view of data) and self-supervised baselines (SimCLR~\\citep{oord2018representation} and MoCo~\\citep{he2020momentum}).\n\n\nFirst, we find that using auxiliary information does not always guarantee better performance than not using it. For instance, for top-1 acc. on Wider Attribute dataset, predicting the attributes-determined clusters using the cross-entropy loss ($39.4$) or treating auxiliary information as another view of data then using CMC ($34.1$) perform worse than the SimCLR method ($40.2$), which does not utilize the auxiliary information. The result suggests that, although auxiliary information can provide useful information, how we can effectively leverage the auxiliary information is even more crucial.\n\n\\iffalse\nThird, we observe the predictive method always performs worse than the contrastive method under the weakly supervised setting. For example, on UT-Zappos50K, although predicting the labels using the cross-entropy loss ($89.2$) performs at par with SupCon ($89.0$), predicting attributes-determined clusters using the cross-entropy loss ($82.7$) performs worse than attributes-determined clusters + Cl-InfoNCE ($84.6$). This result implies that the contrastive method (e.g., Cl-InfoNCE) can generally be applied across various supervision levels.\n\\fi\n\nSecond, we observe that our method constantly outperforms the baseline - Attributes-Determined Clusters + Cross-Entropy loss. For instance, on ZT-Zappos50K, our method achieves $84.6$ top-1 accuracy while the baseline achieves $82.7$ top-1 accuracy. Note that both our method and the baseline consider constructing clusters according to auxiliary information. The difference is that our method adopts the contrastive approach - Cl-InfoNCE, and the baseline considers to adopt cross-entropy loss on an additional classifier between the representations and the clusters. Our observation is in accordance with the observation from a prior work~\\citep{khosla2020supervised}. It shows that, compared to the cross-entropy loss, the contrastive objective (e.g., our presented Cl-InfoNCE) is more robust to natural corruptions of data, stable to hyper-parameters and optimizers settings, and enjoying better performance.\n\nLast, we compare our method with the CMC method. We see that although our method performs better on UT-zappos50K ($84.6$ over $83.7$) and Wider Attributes ($45.5$ over $34.1$) dataset, CMC achieves significantly better results on CUB-200-2011 ($32.7$ over $20.6$) dataset. To explain such differences, we recall that 1) the CMC method leverages the auxiliary information directly, while our method leverages the auxiliary information indirectly (we use the structural information implied from the auxiliary information); and 2) the auxiliary information used in UT-zappos50K and Wider Attributes contains relatively little information (i.e., consisting of less than 20 discrete attributes), and the auxiliary information used in CUB-200-2011 contains much more information (i.e., consisting of 312 discrete attributes). We argue that since CMC leverages the auxiliary information directly, it shall perform better with more informative auxiliary information. On the other hand, Cl-InfoNCE performs better with less informative auxiliary information. \n\n\\input{tbl_tex\/attributes-baselines-comparisons}\n\n\n\n\n\n\n\\vspace{-1mm}\n\\subsection{Experiment II: K-means Clusters + Cl-InfoNCE}\n\\label{subsec:exp_without_auxi}\n\\vspace{-1mm}\nSo far, we see how we can combine auxiliary-information-determined clusters and Cl-InfoNCE to learn good weakly-supervised representations. Now, we would like to show that Cl-InfoNCE can also learn good self-supervised representations without auxiliary information. To this end, we construct unsupervised clusters (e.g., k-means clusters on top of the learned representations) for Cl-InfoNCE.\nSimilar to the EM algorithm, we iteratively perform the k-means clustering to determine the clusters for the representations, and then we adopt Cl-InfoNCE to leverage the k-means clusters to update the representations. We select thet Prototypical Contrastive Learning (PCL)~\\citep{li2020prototypical} as the baseline of the clustering-based self-supervised approach. In particular, PCL performs data log-likelihood maximization by assuming data are generated from isotropic Gaussians. It considers the MLE objective, where the author makes a connection with contrastive approaches~\\citep{chen2020simple,he2020momentum}. The clusters in PCL are determined via MAP estimation. For the sake of the completeness of the experiments, we also include the non-clustering-based self-supervised approaches, including SimCLR~\\citep{chen2020simple} and MoCo~\\citep{he2020momentum}. Note that this set of experiments considers the conventional self-supervised setting, in which we can leverage the information neither from labels nor from auxiliary information. \n\n\\input{fig_tex\/unsupervised}\n\\vspace{-3mm}\n\\paragraph{Results.} We first look at the left table in Figure~\\ref{fig:unsupervised}. We observe that, except for ImageNet-100, there is no obvious performance difference between the non-clustering-based (i.e., SimCLR and MoCo) and the clustering-based baseline (i.e., PCL). Since ImageNet-100 is a more complex dataset comparing to the other three datasets, we argue that, when performing self-supervised learning, discovering latent structures in data (via unsupervised clustering) may best benefit larger-sized datasets. Additionally, among all the approaches, our method reaches the best performance. The result suggests our method can be as competitive as other conventional self-supervised approaches. \n\nNext, we look at the right plot in Figure~\\ref{fig:unsupervised}. We study the mutual information $I(Z;T)$ and the conditional entropy $H(Z|T)$ between the unsupervised constructed clusters $Z$ and the downstream labels $T$. We select our method and PCL, providing the plot of the two information-theoretical metrics versus the training epoch. We find that, as the number of training epochs increases, both methods can construct unsupervised clusters that are more relevant (higher $I(Z;T)$) and contain less redundant information (lower $H(Z|T)$) about the downstream label. This result suggests that the clustering-based self-supervised approaches are discovering the latent structures that are more useful for the downstream tasks. It is worth noting that our method consistently has higher $I(Z;T)$ and lower $H(Z|T)$ comparing to PCL.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe gamma-ray bursts (GRBs) measured with the BATSE instrument on the \nCompton Gamma-Ray Observatory are usually characterized by \n9 observational quantities (2 durations, 4 fluences, 3 peak fluxes) \n\\cite{me96}, \\cite{pac}, \\cite{me00}. \nIn a previous paper \\cite{bag98} \nwe have shown that these 9 quantities \ncan be reduced to only two significant independent \nvariables (principal components). Here we present a new statistical\nanalysis of the correlation between these variables and show that\nthere is a significant difference between the power law exponents \nof long and short bursts. The details of this analysis will be presented\nelsewhere \\cite{bal01}.\n\n\\section{Distributions of Durations and Total Emitted Energies}\n\\label{sec:durations}\n\nWe consider here those GRBs from the current \nBATSE Gamma-Ray Burst Catalog \\cite{me00}\n which have measured $T_{90}$ durations and fluences\n ($ F_1, F_2, F_3, F_4$).\nTherefore, we are left with $N=1929$ GRBs, all of which have defined \n$T_{90}$ and $F_{tot}(= F_1 + F_2 + F_3 + F_4$), as well as \npeak fluxes $P_{256}$. \n\nThe distribution of the log$T_{90}$ clearly displays \ntwo peaks reflecting the existence of two groups of GRBs\n\\cite{k}. \nThis bimodal distribution can be fitted by two log-normal \ndistributions \\cite{hor}.\nThe fact that the distribution of $T_{90}$ within a subclass is log-normal\nhas important consequences. Let us denote the observed duration of a\nGRB with $T_{90}$ (which may be subject to cosmological \ntime dilatation) and with\n$t_{90}$ those measured by a comoving observer (intrinsic duration). \nThen one has $T_{90} = t_{90} f(z)$ \nwhere $z$ is the redshift, and $f(z)$ measures the time dilatation. \nFor the concrete form of $f(z)$ one can take $f(z) = (1+z)^k$, \nwhere $k=1$ or $k=0.6$, depending on whether energy stretching is included\nor not. \n\nTaking the logarithms of both sides of this equality\none obtains the logarithmic duration as a sum of two independent\nstochastic variables. According to a theorem of Cram\\'er \\cite{cra37}, if \na variable $\\zeta$, which has a Gaussian distribution, is given by a sum of \ntwo independent variables, i.e. $\\zeta = \\xi + \\eta$, then both \n$\\xi$ and $\\eta$ have Gaussian distributions. Therefore, from this theorem \nit follows that the Gaussian distributions of $\\log T_{90}$, \nconfirmed for the two subclasses separately \\cite{hor},\nimplies the same type of distribution for the variables of $\\log t_{90}$ and \nof $\\log f(z)$. However, unless the space-time geometry has a very \nparticular structure, the distribution of $\\log f(z)$ cannot be Gaussian. \nThis means that the Gaussian nature of the distribution of $\\log T_{90}$ \nmust be dominated by the distribution of $\\log t_{90}$, and the latter \nmust then necessarily have a Gaussian distribution. This holds for both \nduration subgroups separately. \n(Note here that several other authors,\ne.g. \\cite{wp94}, \\cite{nor94}, \\cite{nor95},\nhave already suggested, that the distribution of $T_{90}$\nreflects predominantly the distribution of $t_{90}$.)\n\nOne also has $F_{tot} = (1+z) E_{tot}\/(4\\pi d_l^2(z)) = c(z) E_{tot}$,\nwhere $d_l$ is the luminosity distance, and $E_{tot}$ is the total\nemitted energy. Once there is a log-normal distribution\nfor $F_{tot}$ (for the two subgroups separately), then the\nprevious application of Cram\\'er theorem is also possible here. \nThe existence of this log-normal distribution is not obvious, but may be\nshown as follows.\n\nAssume both the short and the long groups have\ndistributions of the variables $T_{90}$ and $F_{tot}$ which are\nlog-normal. In this case, it is possible to fit {\\it simultaneously} \nthe values of $\\log F_{tot}$ and $\\log T_{90}$ by a single two-dimensional \n(\"bivariate\") normal distribution. This distribution has five parameters \n(two means, two dispersions, and the angle ($\\alpha$) between the axis \n$\\log T_{90}$ and the semi-major axis of the ``dispersion ellipse\"). \nIts standard form can be seen in \\cite{tw53} (Chapt. 1.25).\nWhen the $r$-correlation coefficient differs from zero, then \nthe semi-major axis of the dispersion ellipse represents a linear relationship \nbetween $\\log T_{90}$ and $\\log F_{tot}$ with a slope of $m=\\tan \\alpha$. This\nlinear relationship between the logarithmic variables implies a power law\nrelation of form $F_{tot} = (T_{90})^m$ between the fluence and the \nduration, where $m$ may be different for the two subgroups. Then\na similar relation will exist between\n$t_{90}$ and $E_{tot}$.\n\nWe obtain the best fit through a maximum likelihood \nestimation (e.g., \\cite{KS76}, Vol.2., p.57-58).\nFrom this estimation we obtain\nthe dependence of the total emitted on the intrinsic duration in form\n\\begin{equation}\nE_{tot} \\propto \\cases{ (t_{90})^{1}~~;&~~(short bursts); \\cr\n                        (t_{90})^{2.3}~~;&~~(long bursts). \\cr}\n\\label{eq:ftotpower}\n\\end{equation}\n\nSeveral papers discuss the biases in the BATSE values of\n$F_{tot}$ and $T_{90}$ (cf. \\cite{ep92}, \\cite{lamb93}, \\cite{lp96}, \n\\cite{pl96}, \\cite{lp97}, \\cite{pac},\n\\cite{hak00}, \\cite{meg00}). \nAll types of biases are particularly essential for\nfaint GRBs. To discuss these effects we provide several different\nadditional calculations (for more details see \\cite{bal01}),\nwhich give the same results.\n\n\n\\section{Conclusion}\n\nThe exponent in the power laws differ significantly for the two subclasses\nof short ($T_{90} < 2$ s) and long ($T_{90} > 2$ s) bursts.\nThese new results  may\nindicate that two different types of central engines are at work, or\nperhaps two different types of progenitor systems are involved. \nWhile the nature of the progenitors remains so far\nindeterminate, our results indicate strongly that the nature of the energy\nrelease process giving rise to the bursts is different between the two\nburst classes. \nIn the short ones the total energy released is proportional to\nthe duration, while in the long ones it is proportional roughly to the\nsquare of the duration. This result is completely model-independent, and\nprovides an interesting constraint on the two types of bursts.\n\n\nThis research was supported in part through\nOTKA grants T024027 (L.G.B.), F029461 (I.H.) and T034549,\nNASA grant NAG5-2857, Guggenheim Foundation\nand Sackler Foundation (P.M.) and\nResearch Grant J13\/98: 113200004 (A.M.).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf Introduction}                               \n\n\\vskip 5pt\n\n        In the recent work \\cite{BES},  Balinski-Evans-Sait\\=o introduced\n     an $L^p$-seminorm $\\|(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}$ of\n     a ${{\\mathbb C}}^4$-valued function $f$ in an open subset $\\Omega$ of\n     ${{\\mathbf R}}^3$, relevant to  a massless  Dirac operator\n\\begin{equation}\\label{1dirac}                                 \n       \\alpha \\cdot \\hbox{\\sl p} =\n          \\sum_{j=1}^3 \\alpha_j(-i\\partial_j) \\qquad (\\partial_j\n                                              = \\partial\/\\partial{x_j}),\n\\end{equation}\n     where $\\hbox{\\sl p} = -i\\nabla$, and\n     $\\alpha= (\\alpha_1, \\, \\alpha_2, \\, \\alpha_3)$ is the triple of\n     $4 \\times 4$ Dirac matrices\n\\begin{equation}\\label{1alpha}                                   \n    \\alpha_j =\n  \\begin{pmatrix}\n     0_2  &\\sigma_j \\\\ \\sigma_j &   0_2\n  \\end{pmatrix}  \\qquad  (j = 1, \\, 2, \\, 3)\n\\end{equation}\n     with  the $2\\times 2$ zero matrix $0_2$ and the triple of  $2 \\times 2$\n     Pauli matrices\n$$\n    \\sigma_1 =\n\\begin{pmatrix}\n       0&1 \\\\ 1& 0\n\\end{pmatrix}, \\,\\,\\,\n    \\sigma_2 =\n\\begin{pmatrix}\n       0& -i  \\\\ i&0\n\\end{pmatrix}, \\,\\,\\,\n    \\sigma_3 =\n\\begin{pmatrix}\n       1&0 \\\\ 0&-1\n\\end{pmatrix}.\n$$\n     They used this seminorm to give a group\n     of inequalities called {\\it Dirac--Sobolev inequalities}\n     in order to obtain $L^p$-estimates of the {\\it zero modes}, i.e. eigenfunctions\n     associated with the eigenvalue $\\lambda = 0$,\n     of the Dirac operator $(\\alpha\\cdot {\\hbox{\\sl p}})+Q$, where $Q(x)$ is a $4\\times 4$\n     Hermitian matrix-valued potential decaying at infinity.\n     We believe that our above notation ``$\\hbox{\\sl p}\\,$\" for the differential\n     operator $-i\\nabla$ will not be confused with another ``$p$\" which appears\n     as the superscript $1\\leq p <\\infty$ of the space $L^p$.\n\n\\par\n        Let\n     $\\Omega$ be an open subset of ${{\\mathbf R}}^3$ and let the first order Dirac--Sobolev space\n     $ \\mathbb{H}_{0}^{1,p}(\\Omega)$, $1\\le p <\\infty,$ be the\n     completion of $[C_0^{\\infty}(\\Omega)]^4$ with respect to the norm\n\\begin{equation}\\label{1H1pOmega}                                  \n    \\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p,\\Omega} :\n        = \\left \\{ \\int_{\\Omega} (|{\\it}    %\\def\\mb{\\mathbf f}(x)|_p^p +\n          |(\\alpha \\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}(x)|_p^p \\,)\\,  dx \\right \\}^{1\/p}\n        = \\big(\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}^p\n               +\\|(\\alpha \\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}^p\\big)^{1\/p} ,\n\\end{equation}\n where ${{\\it f}}      %\\def\\mbf{{\\mb f}}(x) = {}^t(f_1(x), f_2(x), f_3(x), f_4(x))$, the norm of a vector\n${\\it}    %\\def\\mb{\\mathbf a} = {}^t(a_1, a_2, a_3, a_4) \\in {\\mathbb C}^4$ being denoted by\n\\begin{equation} \\label{1gpnorm}                                     \n      | {\\it}    %\\def\\mb{\\mathbf a} |_p = \\Big[\\sum_{k=1}^4 |a_k|^p\\, \\Big]^{1\/p}. \\hskip 70pt\n\\end{equation}\n\n        As one of the simplest Dirac--Sobolev inequalities (\\cite{BES}, Corollary 2), they\n     showed: If $\\Omega$ is a bounded open\n     subset of ${{\\mathbb R}}^3$ and ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}_{0}^{1,p}(\\Omega)$\n     with $1 \\le p < \\infty$, then for $1 \\le k < p(p+3)\/3$\n     there exists a positive constant $C$ such that\n\\begin{equation}\\label{1DS}                                        \n         \\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{k,\\Omega}\n                  \\le C \\|(\\alpha \\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}\\|_{p, \\Omega}\\,,\n\\end{equation}\n     where  $\\| {\\it}    %\\def\\mb{\\mathbf g} \\|_{p,\\Omega}$ stands for the norm of\n     ${\\it}    %\\def\\mb{\\mathbf g} = {}^t(g_1, g_2, g_3, g_4) \\in [L^p(\\Omega)]^4$ given by\n\\begin{equation}\\label{1pOnorm}                                    \n     \\|{\\it}    %\\def\\mb{\\mathbf g}\\|_{p,\\Omega} =\n          \\left \\{ \\int_{\\Omega} |{\\it}    %\\def\\mb{\\mathbf g}(x)|_p^p\\, dx \\right \\}^{1\/p}\n           = \\left\\{ \\int_{\\Omega}\\, \\sum_{j=1}^4 |g_j(x)|^p \\,dx \\right\\}^{1\/p}.\n\\end{equation}\n\n       Now let $\\beta$ be the fourth Dirac matrix given by\n\\begin{equation}\\label{1betam}                                        \n       \\beta =\n  \\begin{pmatrix}\n     {1}_2 & 0_2 \\\\ 0_2 &   -{1}_2\n  \\end{pmatrix},\n\\end{equation}\n     where $1_2$ is the $2 \\times 2$ unit matrix. It has been known that\n     the free massless Dirac operator $\\alpha\\cdot \\hbox{\\sl p}$ or the free Dirac\n     operator $\\alpha\\cdot \\hbox{\\sl p} + m\\beta$ with positive mass $m$ and\n     the relativistic Schr\\\"odinger operator $\\sqrt{m^2 - \\Delta}$ may bring similar\n     properties in $L^2$ sometimes but not necessarily in $L^p$ with $p\\not=2$.\n     On the other hand, the following two norms are equivalent: for $1<p<\\infty$,\n$$\n   \\begin{cases}\n      &(\\|\\psi\\|_p^p + \\|\\nabla \\psi\\|_p^p)^{1\/p}, \\\\\n      &\\|\\sqrt{1 - \\Delta}\\,\\psi\\|_p.\n   \\end{cases}\n$$\n     where $\\psi$ is a scalar-valued function in ${{\\mathbb R}}^3$ (Stein \\cite{Stein},\n     p.135, Theorem 3 or p.136, Lemma 3). However, for $p = 1$ or $p = \\infty$, these\n     two norms are not equivalent, in fact, the one does not dominate the other\n     (\\cite{Stein}, p.160, 6.6).\n\n        For  an open subset $\\Omega$ of ${\\mathbb R}^3$ and $1 \\le p < \\infty$, let\n     $\\mathcal{A}_p(\\Omega)$ be all $C^{\\infty}$ functions $\\psi$ on $\\Omega$ such that $\\psi$ and\n     $\\nabla\\psi$ belong to $L^p(\\Omega)$ and let $H^{1,p}(\\Omega)$ be the completion\n     of $\\mathcal{A}_p(\\Omega)$ with respect to the norm given by\n\\begin{equation}\\label{1sob}                                          \n         \\|\\psi\\|_{1,p,\\Omega}\n          = \\left\\{ \\int_{\\Omega} (|{\\psi}(x)|^p +\n             |\\nabla{\\psi}(x)|^p\\,) \\, dx \\right\\}^{1\/p}\n               = \\{ \\|\\psi\\|_{p,\\Omega}^p + \\|\\nabla{\\psi}\\|_{p,\\Omega}^p \\}^{1\/p},\n\\end{equation}\n     where $|\\nabla\\psi(x)|^p = \\sum_{j=1}^3 |\\partial_j\\psi(x)|^p$. Let $C_0^\\infty(\\Omega)$ be\n     the space of all $C^{\\infty}$ functions $\\phi$ on $\\Omega$ such\n     that support of $\\phi$ is contained in $\\Omega$ and let $H_0^{1,p}(\\Omega)$ be the\n     completion of $C_0^\\infty(\\Omega)$ with respect to the norm (\\ref{1sob}). The space\n     $H_0^{1,p}(\\Omega)$ is a closed subspace of $H^{1,p}(\\Omega)$. The norm\n     $\\| \\cdot \\|_{S,1,p,\\Omega}$ of the Banach space $[H^{1,p}(\\Omega)]^4$ (and\n     $[H_0^{1,p}(\\Omega)]^4$) is given by\n\\begin{equation}\\label{1SOB}                                          \n       \\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{S,1,p,\\Omega} =\n          \\left\\{\\int_{\\Omega} \\big(|{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                 +|\\nabla{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p \\,\\big)\\, dx\\right\\}^{1\/p},\n\\end{equation}\n     where ${{\\it f}}      %\\def\\mbf{{\\mb f}} = {}^t(f_1, f_2, f_3, f_4) \\in [H^{1,p}(\\Omega)]^4$\n     and\n\\begin{equation}\\label{1Sobnorm}                                     \n    \\begin{cases}\n      \\ds   |{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p =\\sum_{k=1}^4 |f_k(x)|^p, \\\\\n       \\ds  |\\nabla{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                   = \\sum_{j=1}^3 |\\partial_j{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                   = \\sum_{j=1}^3 \\sum_{k=1}^4 |\\partial_j{\\it f}}      %\\def\\mbf{{\\mb f}_k(x)|^p \\\\\n   \\end{cases}\n\\end{equation}\n     (cf. (\\ref{1gpnorm})).\n    \n    \n    \n    \n\n\\begin{defn}                                                  \n        {\\rm Let $\\mathcal{A}_{p,D}(\\Omega)$ be all $[C^{\\infty}(\\Omega)]^4$ functions $f$ on $\\Omega$ such that\n     $f$ and $(\\alpha \\cdot \\hbox{\\sl p})f$ belong to $[L^p(\\Omega)]^4$. Then the\n     Dirac-Sobolev spaces $\\mathbb{H}^{1,p}(\\Omega)$ and $\\mathbb{H}_{0}^{1,p}(\\Omega)$ are the completion of $\\mathcal{A}_{p,D}(\\Omega)$ and\n     $[C_0^\\infty(\\Omega)]^4$ with respect to the norm\n\\begin{equation}\\label{1S1p}                                       \n       \\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{D,1,p,\\Omega}\n          = \\left\\{ \\int_{\\Omega} (|{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                +|(\\alpha \\cdot \\hbox{\\sl p}){{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\\,) \\, dx \\right\\}^{1\/p},\n\\end{equation}\n     respectively, where ${{\\it f}}      %\\def\\mbf{{\\mb f}}(x) = {}^t(f_1(x), f_2(x), f_3(x), f_4(x))$,\n     and }\n\\begin{equation}\\label{1nablaf}                                     \n   \\begin{cases}\n      \\ds    |{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p = \\sum_{k=1}^4 |f_k(x)|^p\\,, \\\\\n      \\ds    |(\\alpha \\cdot \\hbox{\\sl p}){{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                = |\\sum_{j=1}^3 \\alpha_j\\hbox{\\sl p}_j{{\\it f}}      %\\def\\mbf{{\\mb f}}(x)|_p^p\n                = \\sum_{k=1}^4\\Big|\\Big(\\sum_{j=1}^3 -i\\alpha_j\\partial_j {f}\\Big)_k(x)\\Big|_p^p\\,\n   \\end{cases}\n\\end{equation}\n    \n    \n\\end{defn}\n\n\\vskip 5pt\n\n        It should be noted that in the paper \\cite{BES} the space ${\\mathbb H}_0^{1,1}(\\Omega)$\n     of our paper was denoted by ${\\mathbb H}^{1,1}(\\Omega)$ without subscript `0'.\n     We have adopted this notation, following the usual Sobolev space convention.\n\n\\vskip 5pt\n\n\\begin{rem}                                                           \n        {\\rm (i) As in the case of Sobolev spaces, we have $\\mathbb{H}^{1,p}(\\mbR^3) = \\mathbb{H}_{0}^{1,p}(\\mbR^3)$\n     since $[\\Con]^4$ is dense in $\\mathcal{A}_{p,D}({\\mathbf R}^3)$ with respect to the norm (\\ref{1S1p}).\n\n\\vskip 5pt\n\n        (ii) Let $\\mathbb{W}^{1,p}(\\Omega)$ be defined by\n\\begin{equation*}\n           \\mathbb{W}^{1,p}(\\Omega) = \\{ f \\in [L^p(\\Omega)]^4 \\,;\\, (\\alpha\\cdot \\hbox{\\sl p})f \\in [L^p(\\Omega)]^4 \\},\n\\end{equation*}\n     where $(\\alpha\\cdot \\hbox{\\sl p})f$ is taken in the sense of distributions. As in the case of\n     Sobolev spaces (see, e.g., Adams-Fournier\\,\\cite{AF}, Theorem 3.17), by approximating\n     elements in $\\mathbb{W}^{1,p}(\\Omega)$ using the molifier, we have  $\\mathbb{W}^{1,p}(\\Omega) = \\mathbb{H}^{1,p}(\\Omega)$, where\n     $1 \\le p < \\infty$ and $\\Omega$ is an open subset of\n     ${\\mathbf R}^3$ . }\n\\end{rem}\n\n        In this work we are going to investigate the relationship of the\n     Dirac--Sobolev spaces $\\mathbb{H}^{1,p}(\\Omega)$ and the ordinary Sobolev\n     spaces $[H^{1,p}({\\Omega})]^4$ as well as the relationship of the\n     Dirac--Sobolev spaces $\\mathbb{H}_{0}^{1,p}(\\Omega)$ and the ordinary Sobolev\n     spaces $[H_0^{1,p}({\\Omega})]^4$.\n\n        To proceed, we note the inclusions $[H^{1,p}(\\Omega)]^4 \\subset {\\mathbb H}^{1,p}(\\Omega)$\n     and $[H_0^{1,p}(\\Omega)]^4 \\subset {\\mathbb H}_0^{1,p}(\\Omega)$ to hold,\n     which we shall see precisely later in the next section, Proposition 2.2. So for an open\n     subset $\\Omega$ of ${\\mathbb R}^3$, define the linear map $J_{\\Omega}$ and $J_{0,\\Omega}$ by\n\\begin{equation}\\label{1J}                                \n   \\begin{cases}\n       J_{\\Omega} \\,:\\, [H^{1,p}(\\Omega)]^4 \\ni {\\it f}}      %\\def\\mbf{{\\mb f}\n                      \\mapsto J_{\\Omega}f = f \\in \\mathbb{H}^{1,p}(\\Omega), \\\\\n       J_{0,\\Omega} \\,:\\, [H_0^{1,p}(\\Omega)]^4 \\ni {\\it f}}      %\\def\\mbf{{\\mb f}\n                      \\mapsto J_{0,\\Omega}f = f \\in \\mathbb{H}_{0}^{1,p}(\\Omega).\n   \\end{cases}\n\\end{equation}\n\n        Our main result is as follows:\n\\begin{thm}                                                       \n        Let $\\Omega$ be an open subset of ${\\mathbb R}^3$ and let $J_{\\Omega}$ and $J_{0,\\Omega}$\n     be as above.\n\n\\vskip 5pt\n\n        {\\rm (i)} Then, for $1 \\le p < \\infty$, the map both $J_{\\Omega}$ and\n     $J_{0,\\Omega}$ are one-to-one and continuous. The Sobolev space $[H_0^{1,p}(\\Omega)]^4$\n     is a dense subspace of the Dirac--Sobolev spaces $\\mathbb{H}_{0}^{1,p}(\\Omega)$.\n\n\\vskip 5pt\n\n        {\\rm (ii)} Let $1 < p < \\infty$. Then $[H_0^{1,p}(\\Omega)]^4 = \\mathbb{H}_{0}^{1,p}(\\Omega)$,\n  i.e., the map $J_{0,\\Omega}$ is not\n     only one-to-one and continuous, but also they are onto with continuous\ninverse map    $J_{0,\\Omega}^{-1}$.\n\n\\vskip 5pt\n\n        {\\rm (iii)} For $p = 1$ the map neither $J_{\\Omega}$ nor $J_{0,\\Omega}$ are onto, or\n     we have $[H^{1,1}(\\Omega)]^4 \\subsetneqq \\mathbb{H}^{1,1}(\\Omega)$ and\n     $[H_0^{1,1}(\\Omega)]^4 \\subsetneqq \\mathbb{H}_{0}^{1,1}(\\Omega)$. For $p = 1$, the norms\n     $\\| \\cdot \\|_{D,1,1,\\Omega}$ and $\\| \\cdot \\|_{S,1,1,\\Omega}$ of these two spaces are\n     not equivalent; $\\| \\cdot \\|_{D,1,1,\\Omega}$ is dominated by $\\| \\cdot \\|_{S,1,1,\\Omega}$,\n     but not conversely.\n\\end{thm}                                           \n\n\\vskip 5pt\n\n\\begin{rem}                                               \n        {\\rm We don't know in (ii) whether or not it holds\n     for a proper open subset $\\Omega$  that $[H^{1,p}(\\Omega)]^4 = \\mathbb{H}^{1,p}(\\Omega)$,\n     i.e., that the map $J_{\\Omega}$ is onto with continuous\n     inverse map  $J_{\\Omega}^{-1}$ when $\\Omega \\subsetneqq {\\mathbf R}^3$,\n     although it holds by (ii) for\n     $\\Omega = {{\\mathbf R}^3}$ that $[H^{1,p}({\\mathbf R}^3)]^4 = {\\Bbb H}^{1,p}({\\mathbf R}^3)$, because this\n     space coincides with $[H_0^{1,p}({\\mathbf R}^3)]^4 = {\\Bbb H}_0^{1,p}({\\mathbf R}^3)$.}\n\\end{rem}                                                 \n\n\\vskip 5pt\n\n        For the proof we shall use a method of classical analysis rather\n     than a subtle pseudo-differentical calculus, in particular,\n     in the  case $p=1$.\n\n\n        In Section 2 we shall prove Theorem 1.3, (i) (Proposition 2.2). In Section 3 we are\n     going to give the proof of Theorem 1.3, (ii) by first dealing with $J_{\\Omega}$ and\n     then $J_{0,\\Omega}$. Theorem 1.3, (iii), the case that $p = 1$, will be discussed and proved\n     in Section 4.\n\n\\vskip 30pt\n\n\\section{{\\bf Continuity of the map ${\\mathbf J_{\\Omega}}$}}                \n\n\\vskip 5pt\n\n        In this section we are going to prove Theorem 1.3, (i). Let\n     $\\alpha_j$, $j = 1, 2, 3$, be the Dirac matrices given in (\\ref{1alpha}).\n     Then we have\n\n\\begin{lem}                                                       \n        Let ${\\it}    %\\def\\mb{\\mathbf a} = {}^t(a_1, a_2, a_3, a_4) \\in {\\mathbb C}^4$. Then,\n     for $p \\in [1, \\infty)$,\n\\begin{equation}\\label{2alap}                                       \n       |\\alpha_j{\\it}    %\\def\\mb{\\mathbf a}|_p = |{\\it}    %\\def\\mb{\\mathbf a}|_p \\qquad (j = 1, 2, 3),\n\\end{equation}\n     where the norm $| \\cdot |_p$ is given by {\\rm(\\ref{1gpnorm})}.\n\\end{lem}                                                   \n\n\\begin{proof}                                              \n        By the definition of $\\alpha_1$ we have\n\\begin{equation*}\n         \\alpha_1{\\it}    %\\def\\mb{\\mathbf a} = {}^t(a_4, a_3, a_2, a_1),\n\\end{equation*}\n     and hence\n\\begin{equation*}\n     |\\alpha_1{\\it}    %\\def\\mb{\\mathbf a}|_p^p = |a_4|^p + |a_3|^p + |a_2|^p + |a_1|^p\n                                                      = |{\\it}    %\\def\\mb{\\mathbf a}|_p^p\\,.\n\\end{equation*}\n     In quite a similar manner (\\ref{2alap}) can be proved for $j = 2, 3$.\n\\end{proof}                                          \n\n        Now we are in a position to show the continuity of the map $J_{\\Omega}$ and\n     $J_{0,\\Omega}$ given by\n     (\\ref{1J}).\n\n\\begin{prop}                                           \n        Let $\\Omega$ be an open subset of ${\\mathbb R}^3$ and let\n     ${\\it}    %\\def\\mb{\\mathbf f} \\in [H^{1,p}(\\Omega)]^4$. Then, for $p \\in [1, \\infty)$,\n     we have ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}^{1,p}(\\Omega)$ and there exists a positive constant $C = C_p$,\n     depending only on $p$, not on $\\Omega$, such that\n\\begin{equation}\\label{2est}                                               \n       \\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p,\\Omega} \\le C\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{S,1,p,\\Omega},\n\\end{equation}\n     where the norms $\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D, 1,p,\\Omega}$ and $\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{S,1,p,\\Omega}$\n     are given in {\\rm (\\ref{1S1p})} and {\\rm (\\ref{1SOB})}, respectively. Thus\n     the identity maps $J_{\\Omega}$ on $[H^{1,p}(\\Omega)]^4$ and\n     $J_{0,\\Omega}$ on $[H_0^{1,p}(\\Omega)]^4$ are continuous, one-to-one maps from\n     $[H^{1,p}(\\Omega)]^4$ into $\\mathbb{H}^{1,p}(\\Omega)$, and $[H_0^{1,p}(\\Omega)]^4$ into $\\mathbb{H}_{0}^{1,p}(\\Omega)$.\n     Further $[H_0^{1,p}(\\Omega)]^4$ is a dense subset of $\\mathbb{H}_{0}^{1,p}(\\Omega)$.\n\\end{prop}                                               \n\n\\begin{proof}                                            \n        Let ${\\it}    %\\def\\mb{\\mathbf f} = {}^t(f_1, f_2, f_3, f_4) \\in [H^{1,p}(\\Omega)]^4$.\n     By using Lemma 2.1 and H\\\"older's inequality for $p > 1$ or the triangle\n     inequality for $p = 1$, we have\n$$\n\\aligned\n      |(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}|_p^p &\n            = |(\\sum_{l=1}^3 -\\alpha_l i\\partial_l){\\it}    %\\def\\mb{\\mathbf f}|_p^p\n              \\leq \\big(\\sum_{l=1}^3 |\\alpha_l i\\partial_l{\\it}    %\\def\\mb{\\mathbf f}|_p\\big)^p\n            = \\big(\\sum_{\\ell=1}^3 |\\partial_l{\\it}    %\\def\\mb{\\mathbf f}|_p\\big)^p\\\\\n      & \\leq \\big[\\big(\\sum_{l=1}^3 1^q\\big)^{1\/q}\n              \\big(\\sum_{\\ell=1}^3 |\\partial_l{\\it}    %\\def\\mb{\\mathbf f}|_p^p\\,\\big)^{1\/p}\\big]^p\n            = 3^{p-1}|\\nabla {\\it}    %\\def\\mb{\\mathbf f}|_p^p\\,,\n\\endaligned\n$$\n     where $p^{-1} + q^{-1} = 1$ and see (\\ref{1Sobnorm}) for the definition of\n     $|\\nabla {\\it}    %\\def\\mb{\\mathbf f}|_p$. It follows that\n\\begin{equation} \\label{2acpfest}                                  \n       \\|(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}\n           = \\Big(\\int_{\\Omega} |(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}|_p^p \\, dx\\Big)^{1\/p}\n           \\leq 3^{\\frac{p-1}{p}}\\Big(\\int_{\\Omega} |\\nabla {\\it}    %\\def\\mb{\\mathbf f}|_p^p\n                                                         \\, dx\\Big)^{1\/p}\n           = 3^{\\frac{p-1}{p}}\\|\\nabla {\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}\\,,\n\\end{equation}\n     where $\\|(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}$ is the norm\n     of $(\\alpha\\cdot \\hbox{\\sl p}){\\it}    %\\def\\mb{\\mathbf f} \\in [L^p(\\Omega)]^4$ given by (\\ref{1pOnorm}),\n     and $\\|\\nabla {\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}$ is given by\n\\begin{equation*}\n      \\|\\nabla {\\it}    %\\def\\mb{\\mathbf f}\\|_{p,\\Omega}\n       = \\left\\{ \\int_{\\Omega} |\\nabla {\\it}    %\\def\\mb{\\mathbf f}|_{p}^p \\, dx \\right\\}^{1\/p}\\,.\n\\end{equation*}\n     Then it is easy to see that (\\ref{2acpfest}) implies (\\ref{2est}).\n     The map $J$ is one-to-one since, for ${\\it}    %\\def\\mb{\\mathbf f}_j \\in [H^{1,p}({\\Omega})]^4$,\n     $j = 1, 2$, we have\n\\begin{equation}\\label{2oneone}                                           \n      J_{\\Omega}{\\it}    %\\def\\mb{\\mathbf f}_1 = J_{\\Omega}{\\it}    %\\def\\mb{\\mathbf f}_2\n                 \\ \\ {\\rm in} \\ \\mathbb{H}^{1,p}(\\Omega)\n         \\Longrightarrow {\\it}    %\\def\\mb{\\mathbf f}_1 = {\\it}    %\\def\\mb{\\mathbf f}_2 \\ \\ {\\rm in} \\\n                                                   [L^p(\\Omega)]^4\n      \\Longrightarrow {\\it}    %\\def\\mb{\\mathbf f}_1 = {\\it}    %\\def\\mb{\\mathbf f}_2 \\ \\ {\\rm in} \\\n                                                      [H^{1,p}(\\Omega)]^4.\n\\end{equation}\n     Using (\\ref{2est}) and proceeding as in (\\ref{2oneone}), we see that\n     the identity map $J_{0,\\Omega}$ on $[H_0^{1,p}(\\Omega)]^4$ is also continuous and\n     one-to-one on $[H_0^{1,p}(\\Omega)]^4$. Since $[C_0^{\\infty}(\\Omega)]^4$ is dense\n     in both $[H_0^{1,p}(\\Omega)]^4$ and $\\mathbb{H}_{0}^{1,p}(\\Omega)$,\n     $[H_0^{1,p}(\\Omega)]^4$ is a dense subset of\n$\\mathbb{H}_{0}^{1,p}(\\Omega)$. This completes the proof.\n\\end{proof}                                   \n\n\\vskip 30pt\n\n\\section{{\\bf Range of the map ${\\mathbf J_{0,\\Omega}}$}}            \n\n\\vskip 5pt\n\n        In this section we are going to prove Theorem 1.3, (ii).\n\n\\begin{prop}                                                   \n        Let $1 < p < \\infty$.  Then the map\n     $J_{0, {\\mathbf R}^3}$ is onto $\\mathbb{H}_0^{1,p}({\\mathbf R}^3)$.\n\\end{prop}                                               \n\n        The proof will be given after the following two lemmas.\n\n\\begin{lem}                                                         \n        Let $1 < q < \\infty$. Then $\\Delta(C_0^{\\infty}({\\mathbf R}^3))$ is dense\n     in $L^q({\\mathbb R}^3)$.\n\\end{lem}                                                  \n\n\n\\begin{proof} [Proof of Lemma 3.2]                        \n        Suppose that  $f \\in L^r({\\mathbb R}^3)$\n     with $1\/q +1\/r =1$ satisfies\n\\begin{equation*}\n     \\langle f,\\Delta \\phi \\rangle = \\int f(x) \\Delta \\phi(x) dx =0,\n          \\quad \\text{for all} \\,\\, \\phi \\in C_0^{\\infty}({\\mathbf R}^3).\n\\end{equation*}\n     Then we obtain in the sense of distributions $\\Delta f = 0$, namely, $\\Delta$\n     annihilates $f$. By elliptic regularity, we see that $f$ must be $C^{\\infty}$, and  hence $f(x)$ is a polynomial of $x$.\n     Since $f(x)$ should belong to $ L^r({\\mathbb R}^3)$, we have $f = 0$. \nThis proves Lemma 3.2.\n\\end{proof}                                 \n\n\\begin{rem}                                                     \n           {\\rm Lemma 3.2 does not hold for $q = 1$, since,\n in this case, the Laplacian $\\Delta$ always annihilates  a constant $C \\ne 0$\nwhich is a nonzero element of $L^{\\infty}({\\mathbf R}^3) = L^1({\\mathbf R}^3)^*$.}\n\\end{rem}                                                   \n\\begin{lem}                                                                     \n        Let $1 < q < \\infty$. Let $\\Omega \\subset {\\mathbf R}^3$ be an open set. Then, for each pair\n     $(j, k)$, $j, k = 1, 2, 3$, there exists a positive constant $C = C_{jk}$ such that\n\\begin{equation}\\label{3jkdeltain}                                       \n        \\|\\partial_j\\partial_k\\phi \\|_{q,\\Omega} \\le C\\|\\Delta \\phi \\|_{q,\\Omega}\n                                       \\qquad (\\phi \\in C_0^{\\infty}(\\Omega)).\n\\end{equation}\n\\end{lem}                                                               \n\n\\begin{proof}                                                           \n     By Stein \\cite{Stein}, p.59, Proposition 3, there exists a positive\nconstant      $C = C_{jk}$ such that\n\\begin{equation*}\n     \\|\\partial_j\\partial_k\\phi\\|_q \\leq C\\|\\Delta\\phi\\|_q,\n                               \\qquad \\phi \\in C_0^{\\infty}({{\\mathbb R}}^3).\n\\end{equation*}\n     Of course, this holds for $\\phi \\in C_0^{\\infty}({{\\mathbb R}}^3)$.\n\\end{proof}                                                      \n\n\\begin{proof} [Proof of Proposition 3.1]                \n        The proof will be divided into four steps.\n\n\\vskip 5pt\n\n        (I) Let $f = {}^t(f_1,f_2,f_3,f_4) \\in \\mathbb{H}_0^{1,p}({{\\mathbb R}}^3)$.\n     Then ${f} \\in [L^p({{\\mathbb R}}^3)]^4$, and\n\\begin{equation*}\n      g:= (\\alpha\\cdot \\hbox{\\sl p}){ f}\n       = -i[\\alpha_1\\partial_1 { f} +\\alpha_2\\partial_2 { f}\n           +\\alpha_3\\partial_3 { f}]\n\\end{equation*}\n     belongs to $[L^p({{\\mathbb R}}^3)]^4$.\n       Using the definition (1.2)\n     of the Dirac matrices $\\alpha_j$, $j = 1, 2, 3$, we can rewrite this with\n     ${ g}={}^t(g_1,g_2,g_3.g_4)$ as\n\\begin{equation}\\label{3gf}                                          \n  \\begin{cases}\n      ig_1 = (\\partial_1-i\\partial_2)f_4+\\partial_3f_3, \\hskip 80pt \\\\\n      ig_2 = (\\partial_1+i\\partial_2)f_3-\\partial_3f_4, \\\\\n      ig_3 = (\\partial_1-i\\partial_2)f_2+\\partial_3f_1, \\\\\n      ig_4 = (\\partial_1+i\\partial_2)f_1-\\partial_3f_2.\n  \\end{cases}\n\\end{equation}\n     Then from the first and second equations of (\\ref{3gf}) we have\n\\begin{equation*}\n   \\aligned\n     (\\partial_1+i\\partial_2)ig_1 &= (\\partial_1^2+\\partial_2^2)f_4\n           + \\partial_3(\\partial_1+i\\partial_2)f_3 \\\\\n     & = (\\partial_1^2+\\partial_2^2)f_4 + \\partial_3(ig_2+ \\partial_3f_4),\n   \\endaligned\n\\end{equation*}\n     so that\n\\begin{equation*}\n     \\Delta f_4 = (\\partial_1^2+\\partial_2^2+\\partial_3^3)f_4\n               = (\\partial_1+i\\partial_2)(ig_1)- \\partial_3(ig_2),\n\\end{equation*}\n     and hence, by applying $\\partial_j$ to both sides of the above equation,\n     we have, for $j = 1, 2, 3$,\n\\begin{equation}\\label{3Del4}                                 \n    \\Delta\\partial_j f_4 = (\\partial_j\\partial_1 + i\\partial_j\\partial_2)(ig_1)\n                                         - \\partial_j\\partial_3(ig_2).\n\\end{equation}\n     Similarly we have from (\\ref{3gf})\n\\begin{equation}\\label{3Del321}                                      \n  \\begin{cases}\n       \\Delta\\partial_j f_3 = (\\partial_j\\partial_1 - i\\partial_j\\partial_2)(ig_2)\n                                            - \\partial_j\\partial_3(ig_1), \\\\\n       \\Delta\\partial_j f_2 = (\\partial_j\\partial_1 + i\\partial_j\\partial_2)(ig_3)\n                                                - \\partial_j\\partial_3(ig_4), \\\\\n       \\Delta\\partial_j f_1 = (\\partial_j\\partial_1 - i\\partial_j\\partial_2)(ig_4) -\n                                                      \\partial_j\\partial_3(ig_3).\n  \\end{cases}\n\\end{equation}\n     The equalities in (\\ref{3Del4}) and (\\ref{3Del321}) should be interpreted\n     as equalities in the space ${\\mathcal D}'({{\\mathbb R}}^3)$ of distributions\n     on ${{\\mathbb R}}^3$.\n\n\\vskip 5pt\n\n        (II) Our first goal is to show that each distribution $\\partial_jf_k$ actually\n     belongs to $L^p({{\\mathbb R}}^3)$, where $j = 1, 2, 3$ and $k = 1, 2, 3, 4$, namely,\n     for each $j$ and $k$ there exists $F_{jk} \\in L^p({{\\mathbb R}}^3)$ such that\n\\begin{equation}\\label{3f4}                                           \n         \\<\\partial_jf_k, \\ \\phi\\> = \\int_{{{\\mathbb R}}^3} F_{jk}(x)\\phi(x)\\, dx\n\\end{equation}\n     for any $\\phi \\in C_0^{\\infty}({{\\mathbb R}}^3)$, where the left-hand side is a  bilinear form on\n     ${\\mathcal D}'({{\\mathbb R}}^3) \\times C_0^{\\infty}({{\\mathbb R}}^3)$.\nThis will show that $f$ belongs to $[H^{1,p}({{\\mathbb R}}^3)]^4$.\nWe shall prove (\\ref{3f4}) for $k = 4$\n     and $j = 1, 2, 3$ since other cases can be proved in a similar manner.\n     After that, finally we show that $f$ belongs to\n$[H_0^{1,p}({{\\mathbb R}}^3)]^4$ to complete our proof.\n\n\\vskip 5pt\n\n       (III) Let $q$ be the conjugate of $p$ or let $q$ satisfy $p^{-1} + q^{-1} = 1$.\n     We see from (\\ref{3Del4}) that for $\\phi \\in C_0^\\infty(\\Omega)$,\n\\begin{eqnarray*}\n     \\langle \\partial_j f_4, \\Delta \\phi \\rangle\n        &=& \\langle \\Delta \\partial_j f_4, \\phi \\rangle\n                        = \\langle (\\partial_j\\partial_1 + i\\partial_j\\partial_2)(ig_1)\n                           - \\partial_j\\partial_3(ig_2), \\phi \\rangle\\\\\n        &=& \\langle ig_1, (\\partial_j\\partial_1 + i\\partial_j\\partial_2)\\phi\\rangle\n                  - \\langle ig_2, \\partial_j\\partial_3 \\phi \\rangle.\n\\end{eqnarray*}\n     Hence by Lemma 3.5 we have\n\\begin{multline}\\label{3ffour}                                           \n  \\hskip 40pt  |\\langle \\partial_j f_4, \\Delta \\phi \\rangle|\n            \\le \\|g_1\\|_p \\|(\\partial_j\\partial_1 + i\\partial_j\\partial_2)\\phi\\|_q\n              + \\|g_2\\|_p \\|\\partial_j\\partial_3 \\phi\\|_q  \\\\\n           \\hskip 63pt \\le (C_{j1}+C_{j2}) \\|g_1\\|_p \\|\\Delta \\phi\\|_q\n              + C_{j3} \\|g_2\\|_p \\|\\Delta \\phi\\|_q   \\\\\n              = [(C_{j1} + C_{j2}) \\|g_1\\|_p + C_{j3} \\|g_2\\|_p] \\|\\Delta \\phi\\|_q. \\hskip 82pt\n\\end{multline}\n     Since $\\Delta C_0^{\\infty}({{\\mathbb R}}^3))$ is dense in $L^q({{\\mathbb R}}^3)$ as has been shown in\n     Lemma 3.2, the inequality (\\ref{3ffour}) is extended uniquely to a continuous linear form\n     on $L^q({{\\mathbb R}}^3)$. Since $L^p({{\\mathbb R}}^3)$ is the dual space of $L^q({{\\mathbb R}}^3)$, there\n     exists a function $F_{j4} \\in L^p({{\\mathbb R}}^3)$ such that\n     $\\<\\partial_j f_4, \\ \\psi\\> = \\int_{{{\\mathbb R}}^3} F_{j4}(x)\\psi(x) \\, dx\\,, \\, \\psi \\in L^q({{\\mathbb R}}^3)$,\n     which implies (\\ref{3f4}) with $k = 4$ and $j = 1, 2, 3$.\n     In particular, we have also shown that\n\\begin{equation}\\label{3.7}\n   \\|\\partial_jf_k \\|_p \\leq C_0 \\{\\Sigma_{k=1}^4\\|g_k\\|_p^p\\}^{1\/p}\n  = C_0\\|g\\|_p,\n\\end{equation}\nwith a positive constant $C_0$ for all $j=1,2,3$ and $k=1,2,3,4$.\n\n\\vskip 5pt\n\n       (IV) Finally, since $f$ is $\\mathbb{H}_0^{1,p}({{\\mathbb R}}^3)$,\n     by definition there exist a sequence $\\{f_n\\}_{n=1}^{\\infty}$\n     in $C_0^{\\infty}({{\\mathbb R}}^3)$ with $f_n =(f_{n,1},f_{n,2},f_{n,3},f_{n,4})$\n     such that with\n     $g_n =(g_{n,1},g_{n,2},g_{n,3},g_{n,4}) := (\\alpha\\cdot p) f_n$,\n\\begin{multline*}\n   \\hskip 50pt \\|f_n -f\\|_{D,1,p,{{\\mathbb R}}^3}^p = \\|f_n -f\\|_p^p\n            + \\|(\\alpha\\cdot \\hbox{\\sl p}) (f_n -f)\\|_p^p \\\\\n            = \\|f_n -f\\|_p^p + \\|g_n -g\\|_p^p \\rightarrow 0,\n                               \\quad\\quad n \\rightarrow \\infty. \\hskip 55pt\n\\end{multline*}\n     Since by the same argument used to get (\\ref{3.7})\n     we have $\\|\\partial_j (f_{n,k}-f_k)\\|_p \\leq C_0\\|g_n-g\\|_p$\n     for all $j=1,2,3$ and $k=1,2,3,4$, it follows that\n     $\\|f_n -f\\|_{S,1,p,{{\\mathbb R}}^3}^p \\rightarrow 0$\n     as $n \\rightarrow \\infty$, so that $f \\in [H_0^{1,p}({{\\mathbb R}}^3)]^4$.\n     This completes  the proof   of Proposition 3.1.\n\\end{proof}                                   \n\n\n\\begin{prop}                                            \n        Let $1 < p < \\infty$. Let $\\Omega \\subset {\\mathbf R}^3$ be an open set and $J_{0,\\Omega}$\n     be given in {\\rm (\\ref{1J})}. Then the map $J_{0,\\Omega}$ is onto $\\mathbb{H}_0^{1,p}(\\Omega)$.\n     Further, the inverse map $J_{0,\\Omega}^{-1}$ is well-defined as a bounded linear\n     operator.\n\\end{prop}                                            \n\n\\begin{proof}                                         \n        (I) We have seen in Propositions 3.1 that\n     $[H_0^{1,p}({\\mathbb R}^3)]^4 = \\mathbb{H}_0^{1,p}({{\\mathbb R}}^3)$ as sets and there\n     exist positive constants $C_1$ and $C_2$ such that\n\\begin{equation}\\label{3th12}                                       \n          C_1\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p} \\le \\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{S,1,p}\n                                                   \\le C_2\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p}\n\\end{equation}\n     for ${\\it}    %\\def\\mb{\\mathbf f} \\in [H_0^{1,p}({\\mathbb R}^3)]^4 = \\mathbb{H}_0^{1,p}({{\\mathbb R}}^3)$.\n\n\\vskip 5pt\n\n        (II) Let ${\\it}    %\\def\\mb{\\mathbf f} \\in [H_0^{1,p}(\\Omega)]^4$. Then there exists a sequence\n     $\\{{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}_n\\}_{n=1}^{\\infty} \\subset [C_0^\\infty(\\Omega)]^4$ such that\n\\begin{equation}\\label{3fPhiO}                                       \n      \\|{\\it}    %\\def\\mb{\\mathbf f} - {{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}}_n\\|_{S,1,p,\\Omega} \\to 0 \\qquad (n \\to \\infty).\n\\end{equation}\n     Since each $\\phi_n$ can be naturally extended to be an element of\n     $[\\Con]^4$ by setting ${{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}}_n(x) = 0$ for\n     $x \\in {\\mathbb R}^3 \\setminus \\Omega$, we have\n\\begin{equation}\\label{3fPhiR}                                       \n      \\|{\\it}    %\\def\\mb{\\mathbf f} - {{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}}_n\\|_{S,1,p} \\to 0 \\qquad (n \\to \\infty),\n\\end{equation}\n     where ${\\it}    %\\def\\mb{\\mathbf f}$ is also extended to be a function on ${\\mathbb R}^3$ by setting\n     $0$ outside $\\Omega$, and hence ${\\it}    %\\def\\mb{\\mathbf f} \\in [H^{1,p}({\\mathbb R}^3)]^4$ with\n     support in the closure of $\\Omega$. Therefore\n     ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}^{1,p}({{\\mathbb R}}^3)$ and ${\\it}    %\\def\\mb{\\mathbf f}$ satisfies\n     (\\ref{3th12}). Then, (\\ref{3fPhiR}) is combined with (\\ref{3th12}) to yield\n\\begin{equation}\\label{3fPhiDR}                                       \n      \\|{\\it}    %\\def\\mb{\\mathbf f} - {{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}}_n\\|_{D,1,p} \\to 0 \\qquad (n \\to \\infty),\n\\end{equation}\n     which implies, together with the fact that and $\\phi_n$ have support in\n     $\\Omega$, that\n\\begin{equation}\\label{3fPhiDO}                                       \n      \\|{\\it}    %\\def\\mb{\\mathbf f} - {{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}}_n\\|_{D,1,p,\\Omega} \\to 0 \\qquad (n \\to \\infty).\n\\end{equation}\n     Thus we have ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}_{0}^{1,p}(\\Omega)$, and we\n     obtain from (\\ref{3th12})\n\\begin{equation}\\label{3thO12}                                       \n          C_1\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p,\\Omega} \\le \\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{S,1,p,\\Omega}\n                                         \\le C_2\\|{\\it}    %\\def\\mb{\\mathbf f}\\|_{D,1,p,\\Omega}.\n\\end{equation}\n\n        (III) Let ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}_{0}^{1,p}(\\Omega)$. Then, starting with\n     ${\\it}    %\\def\\mb{\\mathbf f} \\in \\mathbb{H}_{0}^{1,p}(\\Omega)$ proceeding as in (II), we can show that\n     ${\\it}    %\\def\\mb{\\mathbf f} \\in [H_0^{1,p}(\\Omega)]^4$ and the estimates $(\\ref{3thO12})$ are\n     satisfied, which completes the proof.\n\\end{proof}                                   \n\n\\begin{proof} [Proof of Theorem 1.3, (ii)]                    \n        Theorem 1.3, (ii) follows from Propositions 3.1 and 3.5.\n\\end{proof}                                              \n\n\\vskip 30pt\n\n\\section{\\bf The case ${\\it}    %\\def\\mb{\\mathbf p = 1}$}\n\n\\vskip 5pt\n\n        The goal of this section is to prove Theorem 1.3, (iii), that is, to prove\n     $[H^{1,1}(\\Omega)]^4$ and $[H_0^{1,1}(\\Omega)]^4$ are proper subspaces of\n     $\\mathbb{H}^{1,1}(\\Omega)$ and $\\mathbb{H}_{0}^{1,1}(\\Omega)$, respectively.\n     First, we are going to show, for $\\Omega = {\\mathbb R}^3$, that\n     $[H_0^{1,1}({\\mathbb R}^3)]^4 = [H^{1,1} ({\\mathbb R}^3)]^4$ is a  proper\n     subspace of ${\\mathbb H}_{0}^{1,1}({\\mathbb R}^3) = {\\mathbb H}^{1,1}({\\mathbb R}^3)$\n     (Proposition 4.4). Then all other statements in Theorem 1.3, (iii) will follow from\n     Proposition 4.4. In the following, when speaking of ${\\mathbb H}_{0}^{1,1}({\\mathbb R}^3)$\n     or ${\\mathbb H}^{1,1}({\\mathbb R}^3)$, and $[H_0^{1,1}({\\mathbb R}^3)]^4$ or\n     $[H^{1,1}({\\mathbb R}^3)]^4$, we shall use the latter, namely,\n     ${\\mathbb H}^{1,1}({\\mathbb R}^3)$ and $[H^{1,1}({\\mathbb R}^3)]^4$.  As easily seen,\n     ${\\mathbb H}^{1,p}(\\mathbb R^3)$ is also  the subspace of $[L^p({\\mathbb R}^3)]^4$\n     consisting of all ${f} \\in [L^p({\\mathbb R}^3)]^4$ such that\n     $(\\alpha\\cdot \\hbox{\\sl p} + \\beta) f \\in [L^p({\\mathbb R}^3)]^4$ instead of\n     $(\\alpha\\cdot \\hbox{\\sl p}) f \\in [L^p({\\mathbb R}^3)]^4$, where $\\beta$ is the fourth\n     Dirac matrix $\\beta$ given by (\\ref{1betam}).\n\n\\vskip 5pt\n\n\\begin{lem}                                                \n        The map\n\\begin{equation*}\n            (\\alpha\\cdot \\hbox{\\sl p}) + \\beta \\,:\\, \\mathbb{H}^{1,1}({\\mathbf R}^3) \\ni f\n            \\mapsto (\\alpha\\cdot \\hbox{\\sl p} + \\beta)f \\in [L^1({\\mathbf R}^3)]^4\n\\end{equation*}\n        maps $\\mathbb{H}^{1,1}({\\mathbf R}^3)$ one-to-one and onto $[L^1({\\mathbf R}^3)]^4$.\n\\end{lem}                                                        \n\n\\begin{proof}                                                   \n        (I) We define the Dirac operator $H_0 = (\\alpha\\cdot \\hbox{\\sl p}) + \\beta$ as a linear\n     operator in $[L^1({\\mathbf R}^3)]^4$ with domain $D(H_0) = \\mathbb{H}^{1,1}({\\mathbf R}^3)$.\n     It is easy to see that $H_0$ is a closed operator in $[L^1({\\mathbf R}^3)]^4$. Let\n     the operator $H = (\\alpha\\cdot \\hbox{\\sl p}) + \\beta$ be defined as a pseudodifferential\n     operator acting on $[{\\mathcal S}'({\\mathbb R}^3)]^4$, the dual space of $[{\\mathcal S}({\\mathbb R}^3)]^4$,\n     with $4\\times 4$ matrix symbol\n\\begin{equation}                                                      \n         \\sigma_H(\\xi) = \\alpha\\cdot\\xi + \\beta\n                       = \\sum_{j=1}^3 \\xi_j\\alpha_j + \\beta.\n\\end{equation}\n     Then the operator $H_0$ can be viewed as the restriction of the operator $H$\n     to $\\mathbb{H}^{1,1}({\\mathbf R}^3)$. Let $B$ be a pseudodifferential operator acting\n     on $[{\\mathcal S}'({\\mathbb R}^3)]^4$ with symbol\n\\begin{equation*}\n      \\sigma_{B}(\\xi) = (1 + |\\xi|^2)^{-1}\\sigma_{H}(\\xi)\n                               = \\sigma_H(\\xi)[(1 + |\\xi|^2)^{-1}I_4]. \\hskip 50pt\n\\end{equation*}\n     By the anti-commutative relation\n\\begin{equation*}\n          \\alpha_j\\alpha_k + \\alpha_k\\alpha_j = 2\\delta_{jk}I_4\n                               \\qquad (j, k = 1, 2, 3, 4, \\alpha_4 = \\beta),\n\\end{equation*}\n     where $I_4$ is the $4 \\times 4$ unit matrix, we see that\n\\begin{equation*}\n          \\sigma_{H}(\\xi)\\sigma_{B}(\\xi)\n                                = \\sigma_{B}(\\xi)\\sigma_{H}(\\xi) = I_4,\n\\end{equation*}\n     which implies that\n\\begin{equation}\\label{4AB}                                      \n       HB{\\it f}}      %\\def\\mbf{{\\mb f} = BH{\\it f}}      %\\def\\mbf{{\\mb f} = {\\it f}}      %\\def\\mbf{{\\mb f} \\qquad\n                                     ({\\it f}}      %\\def\\mbf{{\\mb f} \\in [{\\mathcal S}'({\\mathbb R}^3)]^4),\n\\end{equation}\n     {\\it i.e.,} the operator $B$ is the inverse operator of $H$ on $[{\\mathcal S}'({\\mathbb R}^3)]^4$.\n\n\\vskip 5pt\n\n        (II) Note that\n\\begin{equation}\\label{4B0dec}                                   \n                B = HB^{(1)},  \\hskip 150pt\n\\end{equation}\n     where the symbol $\\sigma_{B^{(1)}}(\\xi)$ of $B^{(1)}$ is given by\n     $\\sigma_B^{(1)}(\\xi) = (|\\xi|^2 + 1)^{-1}I_4$. Let\n     $\\sigma_{0}(\\xi) = (|\\xi|^2 + 1)^{-1}$ and let $\\sigma_{0}(\\hbox{\\sl p})$ be the pseudodifferential\n     operator on ${\\mathcal S}'({\\mathbb R}^3)$ with symbol $\\sigma_{0}(\\xi)$. The symbol $\\sigma_{0}(\\xi)$\n     is a $C^{\\infty}$ function on ${\\mathbb R}_{\\xi}^3$ and bounded together with all their\n     derivatives. Then, by noting that the integrand $(|\\xi|^2 + 1)^{-1}({\\mathcal F}\\phi)(\\xi)$\n     is a function in ${\\mathcal S}({\\mathbb R}_{\\xi}^3)$ for $\\phi \\in {\\mathcal S}({\\mathbb R}^3)$, we have\n\\begin{multline*}\n   \\hskip 50pt (\\sigma_0(\\hbox{\\sl p})\\phi)(x) = \\lim_{R\\to\\infty}\n        (2\\pi)^{-3\/2}\\int_{|\\xi|<R} e^{ix\\cdot\\xi}(|\\xi|^2 + 1)^{-1}({\\mathcal F}\\phi)(\\xi)\\,d\\xi \\\\\n            = (2\\pi)^{-3}\\lim_{R\\to\\infty} \\int_{{\\mathbb R}^3}\n                \\Big\\{\\int_{|\\xi|<R} e^{i(x - y)\\cdot\\xi}(|\\xi|^2 + 1)^{-1} \\,d\\xi\\Big\\}\n                   \\phi(y) \\, dy. \\hskip 1pt \\\\\n            = (2\\pi)^{-3}\\lim_{R\\to\\infty} \\int_{{\\mathbb R}^3}\n                \\Big\\{\\int_{|\\xi|<R} e^{iy\\cdot\\xi}(|\\xi|^2 + 1)^{-1} \\,d\\xi\\Big\\}\n                   \\phi(x - y) \\, dy. \\hskip 1pt \\\\\n\\end{multline*}\n     for $\\phi \\in {\\mathcal S}({\\mathbb R}^3)$. Since $(|\\xi|^2 + 1)^{-1} \\in L^2({\\mathbb R}_{\\xi}^3)$, the\n     integral $\\int_{|\\xi|<R} e^{iy\\cdot\\xi}(|\\xi|^2 + 1)^{-1} \\,d\\xi$ converges in\n     $L^2({\\mathbb R}_y^3)$ as $R \\to \\infty$. At the same time it is known that the limit\n\\begin{equation*}\n        \\lim_{R\\to\\infty} (2\\pi)^{-3}\\int_{|\\xi|<R} e^{iy\\cdot\\xi}(|\\xi|^2 + 1)^{-1} \\,d\\xi\n                                                                = G(y)\n\\end{equation*}\n     exist for $y \\ne 0$ with\n\\begin{equation*}\n         G(y) =  \\frac{e^{-|y|}}{4\\pi|y|}, \\hskip 80pt\n\\end{equation*}\n     which is the Green function of the operator $1 - \\Delta$. Thus we have\n\\begin{equation*}\n         (\\sigma_0(\\hbox{\\sl p})\\phi)(x) = (G*\\phi)(x) \\qquad (\\phi \\in {\\mathcal S}({\\mathbb R}^3)),\n\\end{equation*}\n     where $G*\\phi$ denotes the convolution of $G$ and $\\phi$, which implies that\n\\begin{equation}\\label{4convo}                                      \n        \\sigma_B^{(1)}(\\hbox{\\sl p}){\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}(x) = (G*\\phi)(x)\n                    = {}^t((G*\\phi_1)(x), (G*\\phi_2)(x), (G*\\phi_3)(x), (G*\\phi_4)(x))\n\\end{equation}\n     for ${\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi} = {}^t(\\phi_1, \\phi_2, \\phi_3, \\phi_4) \\in [{\\mathcal S}({\\mathbb R}^3)]^4$.\n\n\\vskip 5pt\n\n        (III) Let $B$ be the pseudodifferential operator as in (I). It follows from\n     (\\ref{4convo}) that\n\\begin{equation}\\label{4B0}                                                  \n      B{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}(x) = \\sigma_H(\\hbox{\\sl p})\\sigma_B^{(1)}(\\hbox{\\sl p}){\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}(x)\n                   = \\Big(\\sum_{j=1}^3 -i\\alpha_j\\partial_j + \\beta\\Big)(G*{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi})(x)\n\\end{equation}\n     for ${\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}(x) \\in [{\\mathcal S}({\\mathbb R}^3)]^4$, where $\\partial_j = \\partial\/\\partial x_j$.\n     Define a $4 \\times 4$ matrix-valued function $K(x) = (K_{k\\ell}(x))_{1\\le k,\\ell\\le4}$ on\n     ${\\mathbb R}^3$ by\n\\begin{equation*}\n               K(x) = \\Big(\\sum_{j=1}^3 -i\\alpha_j\\partial_j + \\beta\\Big){\\it}    %\\def\\mb{\\mathbf G}(x)\n                      \\qquad ({\\it}    %\\def\\mb{\\mathbf G}(x) = {}^t(G(x), G(x), G(x), G(x)).\n\\end{equation*}\n     Therefore we have\n\\begin{equation}\\label{4K(x)}                                                   \n     K(x) = \\frac1{4\\pi}\\Big[ \\sum_{j=1}^3 i\\alpha_j\n                 \\Big(\\frac{x_j}{|x|^3} + \\frac{x_j}{|x|^2}\\Big)\n                                   + \\beta\\frac1{|x|}\\Big]e^{-|x|},\n\\end{equation}\n     and each element $K_{k\\ell}(x)$ of $K(x)$ belongs to $L^1({\\mathbb R}^3)$. Thus\n     the $k$-th component $(B{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi})_k$, $k = 1, 2, 3, 4$, of $B{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}$ is expressed as\n\\begin{equation*}\n           (B{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi})_k(x) = \\sum_{\\ell=1}^4 (K_{k\\ell}*\\phi_{\\ell})(x),\n\\end{equation*}\n     which allows us to apply Young's inequality to see that\n\\begin{equation*}\n           \\|B{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}\\|_1 \\le C\\|{\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi}\\|_1   \\qquad ({\\it \\phi}}      %\\def\\mbP{{\\mathbf \\Phi} \\in [{\\mathcal S}({\\mathbb R}^3)]^4)\n\\end{equation*}\n     with a positive constant $C$. Therefore $B$ restricted on $[{\\mathcal S}({\\mathbb R}^3)]^4$ is\n     uniquely extended to a bounded linear operator on $[L^1({\\mathbb R}^3)]^4$ which will be denoted\n     by $B_0$. The operator $B_0$ is actually the restriction of $B$ to $[L^1({\\mathbb R}^3)]^4$.\n\n\\vskip 5pt\n\n        (IV) Let ${\\mb g} \\in [L^1({\\mathbb R}^3)]^4$. Let $\\{ {\\mb g}_m \\}_{m=1}^{\\infty} \\subset [{\\mathcal S}({\\mathbb R}^3)]^4$\n     be a sequence such that ${\\mb g}_m \\to {\\mb g}$ in $[L^1({\\mathbb R}^3)]^4$ as $m \\to \\infty$. It follows from\n     (\\ref{4AB}) that\n\\begin{equation*}\n             H_0B_0{\\mb g}_m = (\\alpha\\cdot\\hbox{\\sl p} + \\beta)B_0{\\mb g}_m = {\\mb g}_m\n                                               \\qquad (m = 1, 2, \\cdots).\n\\end{equation*}\n     Therefore, recalling that $B_0$ is a bounded operator on $[L^1({\\mathbb R}^3)]^4$, we have\n\\begin{equation}\\label{4Bmbfm}                                            \n    \\begin{cases}\n       B_0{\\mb g}_m \\to B_0{\\mb g}, \\\\\n       (\\alpha\\cdot\\hbox{\\sl p} + \\beta)B_0{\\mb g}_m = {\\mb g}_m \\to {\\mb g}\n    \\end{cases}\n\\end{equation}\n     in $[L^1({\\mathbb R}^3)]^4$ as $m \\to \\infty$. Since the operator $H_0 = \\alpha\\cdot\\hbox{\\sl p} + \\beta$\n     defined on ${\\mathbb H}^{1,1}({{\\mathbb R}}^3)$ is a closed operator, we see\n     from (\\ref{4Bmbfm}) that $B_0{\\mb g} \\in {\\mathbb H}^{1,1}({{\\mathbb R}}^3)$ and $H_0B_0{\\mb g} = {\\mb g}$,\n     which implies that $H_0 = \\alpha\\cdot\\hbox{\\sl p} + \\beta$ is onto $[L^1({\\mathbb R}^3)]^4$.\n\n\\vskip 5pt\n\n        (V) Suppose that ${\\it f}}      %\\def\\mbf{{\\mb f} \\in {\\mathbb H}^{1,1}({{\\mathbb R}}^3)$ such that\n     $H_0{\\it f}}      %\\def\\mbf{{\\mb f} = (\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f} = 0$. Let\n     $\\{ {\\it f}}      %\\def\\mbf{{\\mb f}_m \\}_{m=1}^{\\infty} \\subset [\\Con]^4$ such that ${\\it f}}      %\\def\\mbf{{\\mb f}_m \\to {\\it f}}      %\\def\\mbf{{\\mb f}$\n     in ${\\mathbb H}^{1,1}({{\\mathbb R}}^3)$ as $m \\to \\infty$. Thus we have\n\\begin{equation}\\label{4glim}                                                         \n   {\\it f}}      %\\def\\mbf{{\\mb f}_m \\to {\\it f}}      %\\def\\mbf{{\\mb f}, \\quad\n      (\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f}_m \\to (\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f} \\ \\ \\ {\\rm in} \\ \\\n                                                                      [L^1({\\mathbb R}^3)]^4.\n\\end{equation}\n     On the other hand, we have from (\\ref{4AB})\n\\begin{equation}\\label{4ident}                                                     \n            B_0(\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f}_m = (\\alpha\\cdot\\hbox{\\sl p} + \\beta)B_0{\\it f}}      %\\def\\mbf{{\\mb f}_m = {\\it f}}      %\\def\\mbf{{\\mb f}_m\n                                       \\qquad (m = 1, 2, \\cdots).\n\\end{equation}\n     Letting $m \\to \\infty$ in (\\ref{4ident}), noting that $B_0$ is a bounded operator\n     and using (\\ref{4glim}), we see that\n\\begin{equation*}\n           0 = B_0(\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f}\n                     = \\lim_{m\\to\\infty} (\\alpha\\cdot\\hbox{\\sl p} + \\beta)B_0{\\it f}}      %\\def\\mbf{{\\mb f}_m\n                         = \\lim_{m\\to\\infty} {\\it f}}      %\\def\\mbf{{\\mb f}_m =  {\\it f}}      %\\def\\mbf{{\\mb f}\n\\end{equation*}\n     in $[L^1({\\mathbb R}^3)]^4$, which implies that $H_0$ is one-to-one. This completes the proof\n     of Lemma 4.1.\n\\end{proof}                                                   \n\n\\vskip 5pt\n\n\\begin{rem}                                                            \n        {\\rm As has been seen, the key element of the proof of the above Lemma 4.1 is to\n      show that the pseudodifferential operator $B$ given by (\\ref{4B0dec}) is a\n      bounded linear operator on $[L^1({\\mathbf R}^3)]^4$. Actually it can be shown that\n      $B$ is a bounded linear operator on $[L^p({\\mathbf R}^3)]^4$ for $1 \\le p < \\infty$.\n      Thus we can prove that Lemma 4.1 holds for any $1 \\le p < \\infty$. In fact,\n      for $1 < p < \\infty$, a theorem in Fefferman\\,\\cite{F} (Theorem, a), p.414)\n      can be applied to show that $B$ is a bounded linear operator on $[L^p({\\mathbf R}^n)]^4$\n      with $n = 3$. Let $\\sigma(x, \\hbox{\\sl p})$ be a pseudodifferential operator in ${\\mathbf R}^n$ whose\n      symbol $\\sigma(x, \\xi)$ belongs to the H\\\"ormander class $S^{-b}_{1-a,\\delta}({{\\mathbf R}}^n)$\n      with $0 \\le \\delta < 1 - a < 1$. Then it follows from the above theorem by Fefferman\n      that $\\sigma(x, \\hbox{\\sl p})$ is a bounded operator on $L^{p}({\\mathbf R}^n)$ if\n      $b < na\/2$ and\n$$\n     \\left|\\frac1{p}-\\frac12\\right| \\le \\frac{b}{n}\n          \\Big[\\frac{\\frac{n}{2} + \\lambda}{b + \\lambda}\\Big]  \\qquad \\qquad\n                        \\Big(\\ds \\lambda = \\frac{\\frac{na}{2}-b}{1-a}\\Big).\n$$\n      By taking $n = 3,\\, b = 1,\\, \\delta = 0$, the above two condition becomes\n\\begin{equation}\\label{4pineq}                                                  \n           3\\left|\\frac1{p} - \\frac1{2}\\right| \\le \\frac1{a} < \\frac32.\n\\end{equation}\n      For $p > 1$ there exists $a \\in (0, 1)$ which satisfies (\\ref{4pineq}). For\n      $p = 1$, however, there is no $a$ which satisfies (\\ref{4pineq}) since\n      both sides of (\\ref{4pineq}) become 3\/2. Indeed, our pseudodifferential operator\n      $B$ has symbol $\\sigma_B(\\xi)$ belonging to the H\\\"ormander class\n      $S^{-1}_{1-a,0}({\\mathbb R}^3)$. To prove Lemma 4.1, which is the case\n      $p = 1$, we have discussed the integral kernel of the Dirac operator. }\n\\end{rem}                                                        \n\n        To proceed, we need some facts on the local Hardy space $h^1({\\mathbb R}^3)$, which\n     is introduced in Goldberg\\,\\cite{G} in connection with the  Hardy space $H^1({\\mathbb R}^3)$. The\n     Hardy space is (see e.g. Fefferman-Stein\\, \\cite{FS}) the proper subspace of $L^1({{\\mathbb R}}^3)$\n     consisting of the functions $f \\in L^1({{\\mathbb R}}^3)$ such that $R_j f \\in L^1({{\\mathbb R}}^3)$ for\n     $j=1, 2, 3$, where $R_j :=\\partial_j \\cdot (-\\Delta)^{-1\/2}$ are the Riesz\n     transforms, having symbols $i\\xi_j\/|\\xi|$. Let  $\\varphi$ be a\n     fixed function in the Schwartz space ${{\\mathcal S}({\\mathbb R}^3)}$ such that $\\varphi =1$ in a\n     neighborhood of the origin. By definition a distribution $f$ belongs to $h^1({{\\mathbb R}}^3)$\n     if and only if $f \\in L^1({{\\mathbb R}}^3)$ and $r_j f \\in L^1({{\\mathbb R}}^3)$ for $j=1,\n     2, 3$, where $r_j,\\, j=1, 2, 3$, are pseudodifferential operators with\n     symbol $\\sigma_{r_j}(\\xi) = (1-\\varphi(\\xi))(i\\xi_j\/|\\xi|)$ (\\cite{G}, Theorem 2 (p.33)).\n     The definition is independent of the choice of $\\varphi$. It is a Banach\n     space with norm $\\|f\\|_{h^1} = \\|f\\|_{L^1} + \\sum_{j=1}^3 \\|r_j f\\|_{L^1}$.\n     The space $h^1({\\mathbb R}^3)$ is a proper subspace of $L^1({{\\mathbb R}}^3)$, which is strictly\n     larger than the Hardy space $H^1({{\\mathbb R}}^3)$ (see e.g. \\cite{G}, p.33, just after\n     Theorem 3).\n\n        Now, we are introducing the following operator\n\\begin{equation}\\label{4r'jdef}                                            \n   r'_j = \\partial_j (1-\\Delta)^{-1\/2}\n         = \\frac{\\partial_j}{(-\\Delta)^{1\/2}}\n         \\frac{(-\\Delta)^{1\/2}}{(1-\\Delta)^{1\/2}}\n       =R_j\\cdot \\frac{(-\\Delta)^{1\/2}}{(1-\\Delta)^{1\/2}},\n\\end{equation}\n     where we note that the pseudodifferential operator $(-\\Delta)^{1\/2}\/(1-\\Delta)^{1\/2}$ is\n     a bounded operator on $L^1({{\\mathbb R}}^3)$ (see Stein\\,\\cite{Stein}, p.133, Eq.(31)).\n\n        The proof of the lemma below was inspired by the proof of \\cite{G}, Theorem 2 (p.33).\n\n\\vskip 5pt\n\n\\begin{lem}                                                                  \n         A distribution $f$ in ${\\mathbb R}^3$ belongs to $h^1({{\\mathbb R}}^3)$ if and only if\n    $f \\in L^1({{\\mathbb R}}^3)$ and $r'_j f \\in L^1({{\\mathbf R}}^3)$ for $j= 1, 2, 3$.\n\\end{lem}                                                            \n\n\\begin{proof}                                                      \n        (I) It is sufficient to show that $r_j - r'_j$,\n     $j = 1, 2, 3$, are bounded linear operators on $L^1({\\mathbb R}^3)$ (or, more exactly, the\n     pseudodifferential operator $r_j - r_j'$ defined on ${\\mathcal S}({\\mathbb R}^3)$ can be uniquely\n     extended to a bounded linear operator on $L^1({\\mathbb R}^3)$). Note that the operators\n     $r_j$ and $r'_j$ have symbols\n\\begin{equation}\\label{4rjr'j}                                                    \n       \\sigma_{r_j}(\\xi) = \\frac{(1-\\varphi(\\xi))i\\xi_j}{|\\xi|}, \\quad\n            \\sigma_{r'_j}(\\xi) = \\frac{i\\xi_j}{(1+|\\xi|^2)^{1\/2}}\\,,\n\\end{equation}\n     and both symbols are $C^{\\infty}$ functions in ${\\mathbb R}^3_{\\xi}$ and bounded\n     together with all their derivatives, and we have\n\\begin{multline*}\n     \\hskip 50pt \\sigma_{r_j}(\\xi) - \\sigma_{r'_j}(\\xi)\n           = \\frac{(1-\\varphi(\\xi))i\\xi_j}{|\\xi|}\\Big(1-\\frac{|\\xi|}{(1+|\\xi|^2)^{1\/2}}\\Big)\n              - \\frac{\\varphi(\\xi)i\\xi_j}{(1+|\\xi|^2)^{1\/2}} \\\\\n               =: \\sigma_{1j}(\\xi) + \\sigma_{2j}(\\xi). \\hskip 0pt\n\\end{multline*}\n     As in the proof of Lemma 4.1, we are going to show that, for each $j = 1, 2, 3$, the\n     pseudodifferential operator with symbols $\\sigma_{1j}$ and $\\sigma_{2j}$ have integral kernels\n     belonging to $L^1({\\mathbb R}^3)$, in other words, that their inverse Fourier transforms\n     $\\overline{\\mathcal F}\\sigma_{1j}(x)$ and $\\overline{\\mathcal F}\\sigma_{2j}(x)$ belong  to $L^1({{\\mathbb R}}^3)$, where $\\overline{\\mathcal F}$ is\n     given by\n\\begin{equation*}\n           \\overline{\\mathcal F}\\phi(x) = (2\\pi)^{-3\/2}\\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}\\phi(\\xi) \\, d\\xi.\n\\end{equation*}\n\n        It is easy to see that $\\overline{\\mathcal F}\\sigma_{2j} \\in L^1({{\\mathbb R}}^3)$, because $\\sigma_{2j}$\n     belongs to ${\\mathcal S}({\\mathbb R}^3)$, so that $\\overline{\\mathcal F}\\sigma_{2j}$ belongs to ${\\mathcal S}({\\mathbb R}^3)$ and hence it\n     belongs to $L^1({\\mathbb R}^3)$. In the rest of the proof we are going to show that\n     $\\overline{\\mathcal F}\\sigma_{1j} \\in L^1({{\\mathbb R}}^3)$.\n\n\\vskip 5pt\n\n        (II) By definition we have\n$$\n      \\sigma_{1j}(\\xi) =\n           i(1-\\varphi(\\xi))\\frac{\\xi_j}{|\\xi|(1+|\\xi|^2)^{1\/2}(|\\xi|+(1+|\\xi|^2)^{1\/2})},\n$$\n     and hence, $\\sigma_{1j}(\\xi) = O(|\\xi|^{-2})$ as $|\\xi| \\to \\infty$. Thus, by noting that\n     $1-\\varphi(\\xi)$ is $0$ around the origin $\\xi = 0$, we see that $\\sigma_{1j} \\in L^2({\\mathbb R}^3)$.\n     Therefore $I_j(x) = (\\overline{\\mathcal F}\\sigma_{1j})(x)$ exists as a function in $L^2({\\mathbb R}_x^3)$.\n     Let $\\rho(t)$ be a real-valued $C^{\\infty}$ function on $[0, \\infty)$ such that\n\\begin{equation*}\n   \\rho(t) = 1 \\quad (0 \\le t \\le 1), \\ \\ \\ = 0 \\quad (t \\ge 2).\n\\end{equation*}\n     Then, since $\\sigma_{1j}(\\xi)\\rho(\\epsilon|\\xi|)$, $\\epsilon > 0$, converges to $\\sigma_{1j}(\\xi)$ in\n     $L^2({\\mathbb R}_{\\xi}^3)$ as $\\epsilon \\downarrow 0$, we have, by setting\n\\begin{equation*}\n           I_j(x, \\epsilon) := \\overline{\\mathcal F}(\\sigma_{1j}(\\xi)\\rho(\\epsilon|\\xi|)),\n\\end{equation*}\n     $I_j(x, \\epsilon)$ converges to $I_j(x)$ in $L^2({\\mathbb R}_x^3)$ as $\\epsilon \\downarrow 0$,\n     and hence there exists a decreasing sequence\n\\begin{equation*}\n         1 \\ge \\epsilon_2 > \\epsilon_2 > \\cdots > \\epsilon_m > \\ \\to 0\n\\end{equation*}\n     such that $I_j(\\epsilon_m, x) \\to I_j(x)$ a.e. $x$ as $m \\to \\infty$. For the\n     sake of the simplicity of notations, we shall use $\\epsilon \\le 1$ instead of $\\epsilon_m$.\n\n\\vskip 5pt\n\n        (III) Let $\\alpha = (\\alpha_1, \\alpha_2, \\alpha_3)$ be a multi-index. Then\n     we have\n\\begin{equation}\\label{4ests1j}                                             \n         |\\partial_{\\xi}^{\\alpha}\\sigma_{1j}(\\xi)| \\le C_{j,\\alpha}(1 + |\\xi|)^{-2-|\\alpha|}\n                                     \\qquad (\\xi \\in {\\mathbb R}_{\\xi}^3 )\n\\end{equation}\n     with a constant $C_{j,\\alpha} > 0$, where we should note that $1 - \\varphi(\\xi)$\n     is bounded and\n$$\n        \\partial_{\\xi}^{\\alpha}(1 - \\varphi(\\xi)) = - \\partial_{\\xi}^{\\alpha}\\varphi(\\xi)\n                                                       \\in {\\mathcal S}({\\mathbb R}_{\\xi}^3)\n$$\n     for $\\alpha \\ne 0$. Let $\\ell$ be a positive integer and $k = 1, 2, 3$. Then,\n     by integration by parts,\n\\begin{multline}\\label{4Jepxi}                                              \n    \\hskip 26pt \\ds (2\\pi)^{3\/2}x_k^{\\ell}I_j(x, \\epsilon) = \\int_{{\\mathbb R}^3}\\big\\{(-i\\partial_{\\xi_k}^{\\ell})\n                        e^{ix\\cdot\\xi}\\big\\}\\sigma_{1j}(\\xi)\\rho(\\epsilon|\\xi|) \\, d\\xi \\\\\n    \\ds       = (-i)^{\\ell}(-1)^{\\ell}\\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}\n         \\partial_{\\xi_k}^{\\ell}\\big\\{\\sigma_{1j}(\\xi)\\rho(\\epsilon|\\xi|)\\big\\}\\, d\\xi. \\hskip 48pt\n\\end{multline}\n     Here, by the Leibniz formula, we have\n\\begin{multline*}\n      \\ds \\partial_{\\xi_k}^{\\ell}\\big\\{\\sigma_{1j}(\\xi)\\rho(\\epsilon|\\xi|)\\big\\}\n            = \\big(\\partial_{\\xi_k}^{\\ell}\\sigma_{1j}(\\xi)\\big)\\rho(\\epsilon|\\xi|)\n               + \\sum_{m=1}^\\ell {}_{\\ell}C_m \\big(\\partial_{\\xi_k}^{\\ell-m}\\sigma_{1j}(\\xi)\\big)\n                       \\big(\\partial_{\\xi_k}^{m}\\rho(\\epsilon|\\xi|)\\big) \\\\\n      \\ds   =: J_{0}(\\xi, \\epsilon) + J_1(\\xi, \\epsilon) \\hskip 180pt\n\\end{multline*}\n     with ${}_{\\ell}C_m = \\ell\/(m!(\\ell-m)!)$. For $m = 1, 2, \\cdots, \\ell$, we have\n\\begin{equation*}\n     \\xi \\in \\ {\\rm supp}(\\partial_{\\xi_k}^{m}\\rho(\\epsilon|\\xi|)) \\Longrightarrow\n                 1 \\le \\epsilon|\\xi| \\le 2 \\Longrightarrow \\epsilon \\le \\frac2{|\\xi|}\\,,\n\\end{equation*}\n     where ${\\rm supp}(f)$ denotes the support of $f$. Thus we can replace $\\epsilon$ in\n     $\\partial_{\\xi_k}^{m}\\rho(\\epsilon|\\xi|)$ by $2|\\xi|^{-1}$ when we evaluate\n     $|\\partial_{\\xi_k}^{m}\\rho(\\epsilon|\\xi|)|$. Therefore it follows that\n\\begin{equation}\\label{4parho}                                              \n         |\\partial_{\\xi_k}^{m}\\rho(\\epsilon|\\xi|)| \\le c(1 + |\\xi|)^{-m}\\chi_{\\epsilon}(\\xi),\n\\end{equation}\n     where $c = c_{j,k,m}$ is a positive constant and $\\chi_{\\epsilon}(\\xi)$ is the characteristic\n     function of the set  $A_{\\epsilon} = \\{ \\xi \\,:\\, \\epsilon^{-1} \\le |\\xi| \\le 2\\epsilon^{-1} \\}$.\n     Since it is supposed that $\\epsilon \\le 1$, we have $A_{\\epsilon} \\subset \\{ \\xi \\,:\\, |\\xi| \\ge 1 \\}$.\n     The inequalities (\\ref{4ests1j}) and (\\ref{4parho}) are combined to give\n\\begin{equation*}\n        |J_1(\\xi, \\epsilon)| \\le C(1 + |\\xi|)^{-2 - \\ell}\\chi_{\\epsilon}(\\xi) \\qquad (\\xi \\in {\\mathbb R}_{\\xi}^3)\n\\end{equation*}\n     with positive constant $C = C_{j,k,\\ell}$. Let $\\ell \\ge 2$. Then it is seen that\n     $|J_1(\\xi, \\epsilon)|$ is dominated by $C(1 + |\\xi|)^{-2 - \\ell}$, which is in $L^1({\\mathbb R}_{\\xi}^3)$,\n     and $J_1(\\xi, \\epsilon) \\to 0$ for each $\\xi \\in {\\mathbb R}_{\\xi}^3$ as $\\epsilon \\to 0$, and hence, by\n     the Lebesgue convergence theorem, we have\n\\begin{equation*}\n        \\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}J_1(\\xi, \\epsilon) \\, d\\xi \\to 0 \\quad (\\epsilon \\to 0).\n\\end{equation*}\n     Similarly, since $|J_{0}(\\xi, \\epsilon)|$ is dominated by $|\\partial_{\\xi_k}^{\\ell}\\sigma_{1j}(\\xi)|$\n     which is in $L^1({\\mathbb R}_{\\xi}^3)$, and $J_{0}(\\xi, \\epsilon)$ converges to\n     $\\partial_{\\xi_k}^{\\ell}\\sigma_{1j}(\\xi)$ for each $\\xi \\in {\\mathbb R}_{\\xi}^3$ as $\\epsilon \\to 0$, we have\n\\begin{equation*}\n        \\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}J_0(\\xi, \\epsilon) \\, d\\xi \\to \\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}\n                          \\partial_{\\xi_k}^{\\ell}\\sigma_{1j}(\\xi) \\, d\\xi \\quad (\\epsilon \\to 0).\n\\end{equation*}\n     Therefore, by letting $\\epsilon \\to 0$ in (\\ref{4Jepxi}), we obtain\n\\begin{equation}\\label{4xlequal}                                              \n        x_k^{\\ell}I_j(x) = x_k^{\\ell}(\\overline{\\mathcal F}\\sigma_{1j})(x)\n               = i^{\\ell}(2\\pi)^{-3\/2}\\int_{{\\mathbb R}^3} e^{ix\\cdot\\xi}\n                          \\partial_{\\xi_k}^{\\ell}\\sigma_{1j}(\\xi) \\, d\\xi\n\\end{equation}\n     a.e. $x \\in {\\mathbb R}^3$ for $\\ell \\ge 2$ and $j, k = 1, 2, 3$. Here the right-hand\n     side is uniformly bounded for $x \\in {\\mathbb R}^3$. Thus, by considering the case $\\ell = 2$\n     and $\\ell = 4$, it follows that\n\\begin{equation*}\n           |(\\overline{\\mathcal F}\\sigma_{1j})(x)| \\le C_j\\min (|x|^{-2}, |x|^{-4}) \\quad (j = 1, 2, 3)\n\\end{equation*}\n     with a positive constant $C_j$, which implies that $\\overline{\\mathcal F}\\sigma_{1j} \\in L^1({\\mathbb R}^3)$.\n     This completes the proof of Lemma 4.3.\n\\end{proof}                                                    \n\n\\vskip 5pt\n\n        Now we are going to prove that $[H^{1,1}({\\mathbb R}^3)]^4$ is a proper subspace of\n     $\\mathbb{H}^{1,1}({\\mathbb R}^3)$, which is the most crucial part of Theorem 1.3,\n     (iii) in the following strategy: Let ${\\mb g} \\in [L^1({\\mathbb R}^3)]^4 \\setminus [h^1({\\mathbb R}^3)]^4$,\n     where $h^1({\\mathbb R}^3)$ is the local Hardy space which can be defined, by Lemma 4.3, as\n     the space of all distributions $f$ such that $f \\in L^1({{\\mathbb R}}^3)$ and $r'_j f \\in L^1({{\\mathbb R}}^3)$\n     for $j=1, 2, 3$, where $r'_j$ is given by (\\ref{4r'jdef}). Set\n     ${\\it f}}      %\\def\\mbf{{\\mb f} = (\\alpha\\cdot\\hbox{\\sl p} + \\beta)^{-1}{\\mb g}$. Then, by using Lemma 4.1,\n     we have ${\\it f}}      %\\def\\mbf{{\\mb f} \\in \\mathbb{H}^{1,1}({\\mathbb R}^3)$. Then we shall be able to show that\n     ${\\it f}}      %\\def\\mbf{{\\mb f} \\notin [H^{1,1}({\\mathbb R}^3)]^4$.\n\n\\vskip 5pt\n\n\\begin{prop}                                                          \n        ${\\mathbb H}^{1,1}({\\mathbb R}^3)$ is strictly larger than $[H^{1,1}({\\mathbb R}^3)]^4$.\n\\end{prop}                                                         \n\n\\begin{proof}                                                    \n        (I) It follows from Lemma 4.1 that, for every ${\\mb g} \\in [L^1({\\mathbb R}^3)]^4$ there exists\n     a unique ${\\it f}}      %\\def\\mbf{{\\mb f} \\in {\\mathbb H}^{1,1}({\\mathbb R}^3)$ such that ${\\mb g} = (\\alpha\\cdot \\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f}$.\n     The equation ${\\mb g} = (\\alpha\\cdot\\hbox{\\sl p} + \\beta){\\it f}}      %\\def\\mbf{{\\mb f}$ can be written as\n\\begin{equation*}\n    \\begin{cases}\n        &g_1= -i(\\partial_1-i\\partial_2)f_4 -i\\partial_3f_3 +f_1,\\\\\n        &g_2= -i(\\partial_1+i\\partial_2)f_3 +i\\partial_3f_4 +f_2, \\\\\n        &g_3= -i(\\partial_1-i\\partial_2)f_2 -i\\partial_3f_1 -f_3,\\\\\n        &g_4= -i(\\partial_1+i\\partial_2)f_1 +i\\partial_3f_2 -f_4.\\\\\n    \\end{cases}\n\\end{equation*}\n     Solving the above equation for $f_1, f_2, f_3$ and $f_4$, we obtain\n\\begin{equation}\\label{4eqf}                                               \n    \\begin{cases}\n         (1 - \\Delta)f_1 = -i(\\partial_1 - i\\partial_2)g_4 - i\\partial_3g_3 + g_1, \\\\\n         (1 - \\Delta)f_2 = -i(\\partial_1 + i\\partial_2)g_3 - i\\partial_3g_4 + g_2, \\\\\n         (1 - \\Delta)f_3 = -i(\\partial_1 - i\\partial_2)g_2 - i\\partial_3g_1 - g_3, \\\\\n         (1 - \\Delta)f_4 = -i(\\partial_1 + i\\partial_2)g_1 - i\\partial_3g_2 - g_4, \\\\\n    \\end{cases}\n\\end{equation}\n     where $\\partial_j = \\partial\/\\partial x_j$. Here each equation in (\\ref{4eqf}) should be viewed\n     as equations in ${\\mathcal S}'({\\mathbb R}^3)$. As has been shown in the proof of Lemma 4.1,\n     the differential operator $1 - \\Delta$ has the inverse $(1 - \\Delta)^{-1}$\n     as a pseudodifferential operator with symbol $(1 + |\\xi|^2)^{-1}$, and hence,\n     by applying $(1 - \\Delta)^{-1}$  and $\\partial_j$ to each of the equations in (\\ref{4eqf}),\n     it follows that\n\\begin{equation}\\label{4eqf2}                                             \n    \\begin{cases}\n         \\partial_jf_1 = -i(\\partial_j\\partial_1 - i\\partial_j\\partial_2)\n                (1-\\Delta)^{-1}g_4 - i\\partial_j\\partial_3(1-\\Delta)^{-1}g_3\n                +\\partial_j(1-\\Delta)^{-1}g_1,\\\\\n         \\partial_jf_2 = -i(\\partial_j\\partial_1 + i\\partial_j\\partial_2)\n                  (1-\\Delta)^{-1}g_3 + i\\partial_j\\partial_3(1-\\Delta)^{-1}g_4\n                +\\partial_j(1-\\Delta)^{-1}g_2,\\\\\n         \\partial_jf_3 = -i(\\partial_j\\partial_1 - i\\partial_j\\partial_2)\n                (1-\\Delta)^{-1}g_2 - i\\partial_j\\partial_3(1-\\Delta)^{-1}g_1\n                -\\partial_j(1-\\Delta)^{-1}g_3,\\\\\n         \\partial_jf_4 = -i(\\partial_j\\partial_1 + i\\partial_j\\partial_2)\n                  (1-\\Delta)^{-1}g_1 + i\\partial_j\\partial_3(1-\\Delta)^{-1}g_2\n                 -\\partial_j(1-\\Delta)^{-1}g_4.\n    \\end{cases}\n\\end{equation}\n\n\\vskip 5pt\n\n        (II) By Lemma 4.3 we can choose $g_0 \\in L^1({\\mathbb R}^3)\\setminus h^1({\\mathbb R}^3)$\n     such that $r'_3g_0 = \\partial_3(1 - \\Delta)^{-1\/2}g_0 \\notin L^1({\\mathbb R}^3)$. Then define\n     ${\\mb g} = {}^t(g_1, g_2, g_3, g_4) \\in [L^1({\\mathbb R}^3)]^4$ by\n\\begin{equation*}\n        g_1(x) = g_3(x) = g_4(x) = 0 \\ \\ \\ {\\rm and} \\ \\ \\ g_2(x) = g_0(x).\n\\end{equation*}\n     Then we have from (\\ref{4eqf2})\n\\begin{equation}\\label{4pa3}                                               \n    \\partial_jf_4 = i\\partial_j\\partial_3(1-\\Delta)^{-1}g_2 \\qquad (j = 1, 2, 3).\n\\end{equation}\n     Since $r_3'g_2 \\notin L^1({\\mathbb R}^3)$, we have necessarily $r_3'g_2 \\notin h^1$. Then\n     $\\big(r'_1(r'_3g_2), r'_2(r'_3g_2), r'_3(r'_3g_2)\\big)$ does not belong to\n     $[L^1({{\\mathbb R}}^3)]^3$. It follows from (\\ref{4r'jdef}) that the\n     symbol $s_{j3}(\\xi)$ of $r'_jr'_3$ is given by\n\\begin{equation*}\n          s_{j3}(\\xi) = \\frac{i\\xi_j}{(1 + |\\xi|^2)^{1\/2}}\\frac{i\\xi_3}{(1 + |\\xi|^2)^{1\/2}}\n                               = (i\\xi_j)(i\\xi_3)(1 + |\\xi|^2)^{-1},\n\\end{equation*}\n     and hence by using (\\ref{4r'jdef}) again, we see that\n\\begin{equation}\\label{4r'jr'3}                                               \n        r'_jr'_3g_2 = (\\partial_j(1 - \\Delta)^{-1\/2})(\\partial_3(1 - \\Delta)^{-1\/2})g_2\n                                   = \\partial_j\\partial_3(1 - \\Delta)^{-1}g_2\n\\end{equation}\n     for $j = 1, 2, 3$. Thus we have from (\\ref{4pa3}) and (\\ref{4r'jr'3})\n\\begin{equation*}\n      (\\partial_1f_4, \\partial_2f_4, \\partial_3f_4) = i(r'_1r'_3g_2, r'_2r'_3g_2, r'_3r'_3g_2)\n                                          \\notin [L^1({\\mathbb R}^3)]^3,\n\\end{equation*}\n     which implies that ${\\it f}}      %\\def\\mbf{{\\mb f} \\notin [H^{1,1}({\\mathbb R}^3)]^4$. This completes the\n     proof of Proposition 4.4.\n\\end{proof}                                          \n\n\\vskip 5pt\n\n\\begin{prop}                                                \n        Let $\\Omega$ be an open subset of ${\\mathbb R}^3$.\nThen\n\n        {\\rm (i)} $[H_0^{1,1}(\\Omega)]^4$  is a proper subspace of ${\\mathbb H}_0^{1,1}(\\Omega)$.\n\n        {\\rm (ii)} $[H^{1,1}(\\Omega)]^4$  is a proper subspace of ${\\mathbb H}^{1,1}(\\Omega)$.\n\\end{prop}                                                 \n\n\\begin{proof}                                            \n        (I) We are going to show the norms $\\|f\\|_{S,1,1,\\Omega}$\n     of $[H_0^{1,1}(\\Omega)]^4$ and $\\|f\\|_{D,1,1,\\Omega}$ of ${\\mathbb H}_{0}^{1,1}(\\Omega)$\n     are not equivalent on $[C_0^{\\infty}(\\Omega)]^4$ (see (\\ref{1SOB}) and (\\ref{1S1p}) for\n     the definition of these norms). To this end we use Proposition 4.4. Without loss of\n     generality, we may assume that $\\Omega$ contains the unit ball $\\{x\\,:\\, |x| \\leq 1\\}$\n     with center at the origin. As in (\\ref{1pOnorm}) (with $p = 1$), we denote the norm of\n     $[L^1(\\Omega)]^4$ by $\\|{\\it f}}      %\\def\\mbf{{\\mb f}\\|_{1,\\Omega}$, {\\it i.e.,}\n\\begin{equation*}\n         \\|{\\it f}}      %\\def\\mbf{{\\mb f}\\|_{1,\\Omega} = \\int_{\\Omega} \\, \\sum_{j=1}^4 |f_j(x)| \\, dx\n                                   \\qquad ({\\it f}}      %\\def\\mbf{{\\mb f}(x) = {}^t(f_1(x), f_2(x), f_3(x), f_4(x)).\n\\end{equation*}\n     By Proposition 4.4 and the fact that $[C_0^{\\infty}({{\\mathbb R}}^3)]^4$ is dense in\n     both $[H^{1,1}({\\mathbb R}^3)]^4$ and $\\mathbb{H}^{1,1}({\\mathbb R}^3)$ (Theorem 1.3, (i)), the\n     norms $\\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{S,1,1,{\\mathbb R}^3}$ and $\\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{D,1,1,{\\mathbb R}^3}$ are not equivalent\n     on $[C_0^{\\infty}({{\\mathbb R}}^3)]^4$. Therefore, by taking note of Proposition 2.2,\n     (\\ref{2est}) with $\\Omega = {\\mathbf R}^3$, which says that the norm $\\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{D,1,1,{\\mathbb R}^3}$\n     is dominated by the norm $\\|{{\\it f}}      %\\def\\mbf{{\\mb f}}\\|_{S,1,1,{\\mathbb R}^3}$,\n     there exists a sequence $\\{{{\\it f}}      %\\def\\mbf{{\\mb f}}_n\\}_{n=1}^{\\infty}$ of functions in\n     $[C_0^{\\infty}({{\\mathbb R}}^3)]^4$ such that ${\\it f}}      %\\def\\mbf{{\\mb f}_n \\ne 0$ and\n     $\\|{{\\it f}}      %\\def\\mbf{{\\mb f}_n}\\|_{S,1,1,{\\mathbb R}^3} \\ge (n + 1)\\|{{\\it f}}      %\\def\\mbf{{\\mb f}_n}\\|_{D,1,1,{\\mathbb R}^3}$\\,, or\n\\begin{equation}\\label{4ineqr3}                                                     \n        \\|{\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3} + \\|\\nabla{\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3}\n                \\ge (n + 1)[\\|{\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3} + \\|(\\alpha\\cdot{\\hbox{\\sl p}}){\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3}]\n\\end{equation}\n     for each $n = 1, 2, \\cdots$. Each ${\\it f}}      %\\def\\mbf{{\\mb f}_n$ has support in some ball $\\{ x \\,:\\, |x|\\le R_n\\}$\n     with radius $R_n > 0$ and center at the origin. We may assume with no loss of generality that\n     $R_n \\ge 1$ ($n = 1, 2, \\cdots$). Put ${\\mb g}_n(x) = R_n^3{\\it f}}      %\\def\\mbf{{\\mb f}_n(R_nx)$. Then ${\\mb g}_n$ has support\n     in the unit ball $\\{ x \\,:\\, |x|\\le 1 \\}$, and hence in $\\Omega$, so that\n     $\\{{\\mb g}_n\\} \\subset [C_0^{\\infty}(\\Omega)]^4$ for each $n$. We have\n$$\n       \\|{\\mb g}_n\\|_{1,\\Omega} = \\|{\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3} \\ \\ \\ {\\rm and} \\ \\ \\\n          \\|\\partial_j {\\mb g}_n\\|_{1,\\Omega}\n             = R_n\\|\\partial_j{\\it f}}      %\\def\\mbf{{\\mb f}_n\\|_{1,{\\mathbb R}^3}\n$$\n     for $j = 1, 2, 3$. Then by (\\ref{4ineqr3}) we have\n$$\n       \\|{\\mb g}_n\\|_{1,\\Omega} + \\frac1{R_n}\\|\\nabla{\\mb g}_n\\|_{1,\\Omega}\n          \\ge (n + 1)[ \\|{\\mb g}_n\\|_{1,\\Omega} +\n                       \\frac1{R_n}\\|(\\alpha\\cdot \\hbox{\\sl p}){\\mb g}_n\\|_{1,\\Omega}],\n$$\n     and hence, by noting $R_n \\ge 1$\n\\begin{multline*}\n    \\hskip 50pt  \\frac1{R_n}\\|\\nabla{\\mb g}_n\\|_{1,\\Omega}\n          \\ge n\\|{\\mb g}_n\\|_{1,\\Omega}\n                     + \\frac{n+1}{R_n}\\|(\\alpha\\cdot \\hbox{\\sl p}){\\mb g}_n\\|_{1,\\Omega} \\\\\n          \\ge \\frac{n}{R_n} [\\|{\\mb g}_n\\|_{1,\\Omega}\n                         + \\|(\\alpha\\cdot \\hbox{\\sl p}){\\mb g}_n\\|_{1,\\Omega}\\|]. \\hskip 126pt\n\\end{multline*}\n     Therefore\n$$\n           \\|\\nabla{\\mb g}_n\\|_{1,\\Omega}\n             \\ge n[\\|{\\mb g}_n\\|_{1,\\Omega} + \\|(\\alpha\\cdot \\hbox{\\sl p}){\\mb g}_n\\|_{1,\\Omega}],\n$$\n     which implies that $\\|{\\mb g}_n\\|_{S,1,1,\\Omega} \\ge n\\|{\\mb g}_n\\|_{D,1,1,\\Omega}$\n     for $n = 1, 2, \\cdots$. This proves that the norms $\\|f\\|_{S,1,1,\\Omega}$ and\n     $\\|f\\|_{D,1,1 \\Omega}$ are not equivalent on $[C_0^{\\infty}(\\Omega)]^4$, showing (i).\n\n        (II) As we have seen in the proof of (i), the norms of $[H_0^{1,1}(\\Omega)]^4$\n     and ${\\mathbb H}_0^{1,1}(\\Omega)$ are not equivalent. In fact, there exists a sequence\n     $\\{g_n\\} \\subset [C_0^{\\infty}(\\Omega)]^4$ such that\n\\begin{equation*}\n  \\|g_n\\|_{S,1,1,\\Omega} \\geq n \\|g_n\\|_{D,1,1,\\Omega}, \\,\\, n=1,2,\\, \\dots.\n\\end{equation*}\n     Since clearly this sequence is also contained in $[H^{1,1}(\\Omega)]^4$,\n     this implies that the norms of $[H^{1,1}(\\Omega)]^4$ and ${\\mathbb H}^{1,1} (\\Omega)$ are\n     not equivalent. This shows (ii), completing the proof of Proposition 4.5.\n\\end{proof}                                        \n\n\\begin{proof} [Proof of Theorem 1.3, (iii)]\n        Theorem 1.3, (iii) follows from Propositions 4.5.\n\\end{proof}\n\n\\bigskip\\bigskip\n        {\\bf Acknowledgement.}\\ \nBeing deeply grieved that our respected friend, Professor Tetsuro Miyakawa,\nsuddenly passed away in February 2009, \nwe gratefully remember nice useful discussion with him\nabout the Riesz transform at the early stage of the present work.\nThanks are also due to Professor Shuichi Sato for valuable\n discussion about the Hardy space and local Hardy space. \n The research of T.I. is supported in part by JSPS Grant-in-Aid\n   for Sientific Research No. 17654030.\nWe wish to thank the referee for his kindly pointing out\na flaw in an assertion in Theorem 1.3(ii) of our original version of \nthe paper.\n\n\n\\vskip 30pt\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\nPrecise determinations of radial matrix elements of electric dipole (E1) transitions are essential for advancing the study of parity-nonconserving (PNC) weak-force-induced interactions in atoms. These matrix elements are important for testing calculations of the PNC transition amplitudes $\\mathcal{E}_{\\rm PNC}$~\\cite{BlundellSJ92, KozlovPT01, PorsevBD10, RobertsDF2013}, as well as for determining the scalar and vector transition polarizabilities~\\cite{BlundellSJ92, DzubaFS97, Derevianko00, SafronovaJD99, VasilyevSSB02}. For PNC studies based on the $6s~^2S_{1\/2} \\rightarrow 7s~^2S_{1\/2}$ transition in cesium, for example, the most essential E1 matrix elements are $\\langle ms~^2S_{1\/2}  || r || np~^2P_J\\rangle$, where $m, n = 6$ or 7, and $J = 1\/2$ or $3\/2$. Over the years, most of these quantities have been measured~\\cite{YoungHSPTWL94,RafacT98,RafacTLB99,DereviankoP02a,AminiG03,BouloufaCD07,ZhangMWWXJ13,PattersonSEGBSK15,GregoireHHTC15,TannerLRSKBYK92,SellPEBSK11,antypas7p2013,BouchiatGP84,TohJGQSCWE18,BennettRW99,Borvak14,PhysRevA.99.032504} to a precision of 0.15\\% or better.  The least precise moments, prior to the present work, were $\\langle 6s~^2S_{1\/2} || r ||  7p~^2P_{J}\\rangle$.  Disagreement between three recent experimental results~\\cite{VasilyevSSB02,antypas7p2013,Borvak14} at the $\\sim1$\\% level motivated us to re-examine these transitions. In this paper, we report new measurements of these matrix elements in $^{133}$Cs to a precision of $0.10\\%$ and $0.16\\%$, completing the required set of precise determinations of $E1$ dipole matrix elements between the lowest $ms~^2S_{1\/2}$ and $np~^2P_J$ states.\n\nIn order to determine the reduced matrix element for the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{3\/2}$ transition at $\\lambda = 455.7$ nm, we carry out a set of measurements in which we compare the absorption coefficient on this line to that of the `reference' D$_1$ line at 894.6 nm.  See the simplified energy level diagram in Fig.~\\ref{fig:EnergyLevel}(a).\n\\begin{figure} [b!]\n\t  \\includegraphics[width=8.5cm]{EnergyLevels_dual.png}\\\\\n\t  \\caption{Energy level diagrams of atomic cesium, showing the states relevant to these measurements. In (a), atoms are excited from the $6s\\, ^2S_{1\/2}$ ground state to the $6p\\, ^2P_{1\/2}$ ($\\lambda =$ 894 nm) or the $7p\\, ^2P_{3\/2}$ ($\\lambda =$ 456 nm) excited states, allowing for a comparison of the absorption coefficients of these two lines.\n\t  In (b), we compare the absorption coefficients for the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{1\/2}$ line at 459 nm to that of the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{3\/2}$ line at 456 nm.  Wavelengths shown in the figure are vacuum wavelengths.}\n\t  \\label{fig:EnergyLevel}\n\\end{figure}\nThe matrix element for the latter is well measured~\\cite{YoungHSPTWL94,RafacT98,RafacTLB99,DereviankoP02a,AminiG03,BouloufaCD07,ZhangMWWXJ13,PattersonSEGBSK15,GregoireHHTC15,TannerLRSKBYK92,SellPEBSK11}, with an impressive precision of 0.035\\%. The ratio of absorption coefficients for these two lines therefore allows us to determine the reduced matrix element $\\langle 6s~^2S_{1\/2}  || r || 7p~^2P_{3\/2}\\rangle$ precisely. \nA similar comparison to the D$_1$ line strength for the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{1\/2}$ transition at $\\lambda = 459.4$ nm, however, is less fruitful. This is a weaker absorption line, and the difference between the absorption strength at 459 nm and at 894 nm is too great.  Therefore, we determine the matrix element at 459 nm through comparison with the 456 nm line strength, which now serves as the reference. We show the relevant transitions for this measurement in Fig.~\\ref{fig:EnergyLevel}(b).\n\nIn each case, we use a pair of cw tunable single-mode diode lasers to measure and compare the absorption strengths of two lines in a cesium vapor cell. \nWe direct the two laser beams through a cesium vapor cell along overlapping beam paths.  Then we block one laser beam to allow only the other to pass through the cell, scan the laser frequency through the resonance frequency, and record the absorption lineshape for this line. We then block the first laser, and record the absorption lineshape for the second.  We alternate measurements between lasers several times in quick succession.\n\nThese measurements differ from a previous measurement~\\cite{antypas7p2013} from our group in several important regards.  In that measurement, we used a single blue diode laser which we could tune to either the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{1\/2}$ transition at 459 nm or the $6s~^2S_{1\/2} \\rightarrow 7p~^2P_{3\/2}$ transition at 456 nm, and compared the absorption coefficient of each of these lines directly to the absorption coefficient of the 894~nm line. The precision of the measurement for the 459~nm line suffered from the large difference in absorption strengths, as described previously.\nIn the present measurement, we avoid this difficulty by using two blue diode lasers, and determining the ratio of absorption coefficients for these two lines directly. Two additional improvements that we have made are: (1) In our previous measurement, we fit the absorption curves assuming a Gaussian Doppler-broadened lineshape.  We have discovered that, to attain a level of precision of less than 1\\%, one must use a proper Voigt profile, a convolution of the homogeneous natural linewidth and the inhomogeneous Doppler-broadened linewidth, when the natural linewidths and\/or Doppler widths of the two transitions differ from one another.  (2) For strongly absorbing lines, such as the D$_1$ line at the higher cell densities, the scan speed of the laser frequency becomes important. Under these strong absorption conditions, the medium changes quickly from fully transmitting to fully absorbing, and then back to fully transmitting again, as we tune the laser through the resonance.  If the rise and fall times of the photodetector are too slow, then one cannot obtain good fits to the data, and the measurement of the absorption coefficient is not reliable.  We have corrected each of these issues in the current measurements.\n\n\n\\section{Theory}\\label{sec:theory}\n \nWhen a low-intensity, narrow-band laser beam is incident upon a cell containing an absorbing atomic medium, the laser power transmitted through the cell can be written simply as \n\\begin{equation}\nP(\\omega) = P_0 \\exp{ \\{-2 \\alpha(\\omega ) \\ell_{cell} \\} },\n\t  \\label{eqn:fitfunction}\n\\end{equation}\nwhere $P_0$ is the transmitted power in the absence of any absorption by the medium, $\\alpha(\\omega )$ is the frequency-dependent electric field attenuation coefficient, and $\\ell_{cell}$ is the cell length.  The attenuation coefficient $\\alpha(\\omega )$ for linearly-polarized light by a Doppler-broadened atomic gas\nin terms of the reduced $E1$ matrix elements $\\langle J^{\\prime} || \\vec{r} || J \\rangle $, as a sum over the various hyperfine components of the states, is given by Eq.~(14) of Ref.~\\cite{RafacT98} as\n\\begin{eqnarray}\\label{eq:abscoeff}\n  \\alpha(\\omega) & = & \\frac{2 \\pi^2 n \\alpha_{\\rm fs} \\omega }{\\left( 2I+1\\right) \\left(2J+1\\right) }\n   \\left| \\langle J^{\\prime} || \\vec{r} || J \\rangle \\right|^2  \\\\ & & \\hspace{0.3in}\\times \\sum_{F^{\\prime}, } \\sum_{F}  q_{J,F \\rightarrow J^{\\prime},F^{\\prime}} V(\\omega) , \\nonumber\n\\end{eqnarray}\nwhen the transition frequencies \n$\\omega$ are independent of $m$ and $m^{\\prime}$.  \n$J$, $I$, and $F$ are quantum numbers for the total electronic, nuclear spin, and total angular momentum, respectively, and $m$ for the projection of $F$ on the $z$ axis. \nWe use unprimed (primed) notation to indicate ground (excited) state quantities.\n$n$ is the number density of the cesium atoms in the beam path, $\\alpha_{\\rm fs}$ is the fine structure constant, and\n$q_{J,F \\rightarrow J^{\\prime},F^{\\prime}}$ are weight  factors for the different hyperfine components due to the angular momentum of the states, \n\\begin{eqnarray}\\label{eq:q}\n   q_{J,F \\rightarrow J^{\\prime},F^{\\prime}} & = &(-1)^{2(I+J)} \\left( 2F^{\\prime} + 1\\right) \\left( 2F + 1\\right)     \\nonumber \\\\\n  & & \\hspace{-0.4in} \\times \\sum_{m, m^{\\prime}}  \\left( \\begin{array}{ccc} F^{\\prime} & 1 & F \\\\ -m^{\\prime} & 0 & m \\end{array} \\right)^2 \\left\\{ \\begin{array}{ccc} J^{\\prime} & F^{\\prime} & I \\\\ F & J & 1 \\end{array} \\right\\}^2 .\n\\end{eqnarray} \nThe arrays in the parentheses and curly brackets are the Wigner $3j$ (for linear polarization) and $6j$ symbols, respectively.  We list the values of $q_{J,F \\rightarrow J^{\\prime},F^{\\prime}}$ for the transitions relevant to this work in Table \\ref{table:q}.\n\\begin{table}[t!]\n\\caption{Numerical values of the weights $q_{J,F \\rightarrow J^{\\prime},F^{\\prime}}$ for each of the hyperfine components of the $6s \\ ^2S_{1\/2} \\rightarrow np \\ ^2P_{j}$ transitions, as given in Eq.~(\\ref{eq:q}). \\label{table:q} }\n  \\begin{tabular}{|l|c|c|}\n    \\hline\n      \\multicolumn{1}{|c|} {$F \\rightarrow F^{\\prime}$}   \t\t& $6s \\ ^2S_{1\/2} \\rightarrow np \\ ^2P_{1\/2}$ & $6s \\ ^2S_{1\/2} \\rightarrow np \\ ^2P_{3\/2}$  \\\\ \\hline \\hline\n  \n\\rule{0in}{0.15in} $4 \\rightarrow 3^{\\prime} $ & $ 7\/8$  & $7\/48$ \\\\\n\n\\rule{0in}{0.15in} $4 \\rightarrow 4^{\\prime} $ & $ 5\/8$ & $7\/16$   \\\\\n\n\\rule{0in}{0.15in} $4 \\rightarrow 5^{\\prime} $ & $ - $\t& $11\/12$  \\\\\n\n\\rule{0in}{0.3in} $3 \\rightarrow 2^{\\prime}  $ & $ -$ & $5\/12$   \\\\\n\n\\rule{0in}{0.15in} $3 \\rightarrow 3^{\\prime} $ & $ 7\/24$ & $7\/16 $ \\\\\n\n\\rule{0in}{0.15in} $3 \\rightarrow 4^{\\prime} $ & $ 7\/8$ \t& $5\/16$   \\\\\n   \\hline\n  \\end{tabular}\n  \n\\end{table}\n\n\n$V(\\omega)$ is the Voigt lineshape function, \n\\begin{equation}\\label{eq:Voigt}\n  V(\\omega) = \\sqrt{\\frac{\\ln 2}{\\pi^3}} \\frac{1}{\\Delta \\omega_D}  \\int_{-\\infty}^{\\infty} \\frac{\\Gamma^{\\prime} e^{-4 \\ln2 \\left( \\omega_D \/ \\Delta \\omega_D \\right)^2} d \\omega_D }{\\left[\\omega  - \\omega_D  - \\omega_{F \\rightarrow F^{\\prime}} \\right]^2 + \\Gamma^{\\prime 2}\/4} ,\n\\end{equation} \nthe convolution of the Lorentzian homogeneous lineshape function (of width $\\Gamma^{\\prime}$) and the Gaussian distribution of width $\\Delta \\omega_D$.\n$\\omega_{F \\rightarrow F^{\\prime}}$ is resonant frequency of the $F \\rightarrow F^{\\prime}$ hyperfine component of the transition.\nThis lineshape function is normalized such that its integral across the resonance is unity.  In this expression, $\\omega_D$ is the Doppler shift, equal to $\\omega v\/c$, where $v$ is the atomic velocity, and $\\Delta \\omega_D$ is the Doppler full-width-at-half-maximum (FWHM) of the transition, equal to $\\omega \\: \\sqrt{8 k_B T \\ln2\/(Mc^2)} \\: $. \nThus, precise measurements of the absorption in a cell would allow us to determine the radial matrix element for that individual transition, provided we also measure the vapor density in the cell, and the length of the cell. \n\n\n\nInstead, by alternating transmission measurements between two spectral lines, we can determine the ratio of absorption strengths, and eliminate the need for precise determination of the cell length and vapor density.  \nFor example, to determine $\\langle 6s_{1\/2} || r ||  7p_{3\/2}  \\rangle$ (we abbreviate the state notation $|m \\ell \\ ^2L_J \\rangle $ using only the quantum numbers of the single active electron $|m \\ell_j \\rangle $), we measure the absorption through the cell on the $6s \\ ^2S_{1\/2} \\rightarrow 6p \\ ^2P_{1\/2}$ line at 894 nm, which serves as the reference, and the absorption on the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ line at 456 nm.  The ratio of matrix elements is then determined as\n\\begin{equation}\\label{eq:ratioR1}\n   R_1 \\equiv \\frac{\\langle 6s_{1\/2} || r ||   6p_{1\/2}\\rangle}{\\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle} = \\sqrt{\\frac{\\alpha_{3 \\rightarrow 3^{\\prime}}^{894}(\\omega_0)\/(7\/24)}{\\alpha_{F \\rightarrow F^{\\prime}}^{456}(\\omega_0)\/q_{F \\rightarrow F^{\\prime}}}}.\n\\end{equation}\n$\\alpha_{F \\rightarrow F^{\\prime}}^{\\lambda}(\\omega_0)$ is the attenuation coefficient at line center at wavelength $\\lambda$ for the $F \\rightarrow F^{\\prime}$ component, and we have abbreviated the $q_{F \\rightarrow F^{\\prime}}$ factor, omitting the $J$ and $J^{\\prime}$ for brevity from Eq.~(\\ref{eq:q}).  \nThis factor $\\alpha_{F \\rightarrow F^{\\prime}}^{\\lambda}(\\omega_0) \/ q_{F \\rightarrow F^{\\prime}}$ is the same for each of the different hyperfine components of the transition, so we define a scaled attenuation term\n\\begin{equation}\n  \\Upsilon^{\\lambda} =\\frac{ \\alpha_{F \\rightarrow F^{\\prime}}^{\\lambda}(\\omega_0) \\ell_{cell}}{ q_{F \\rightarrow F^{\\prime}}}\n\\end{equation}\nfor the line.  The term $\\Upsilon^{\\lambda}$ is the  attenuation of the absorbing vapor on the line at wavelength $\\lambda$, \ndefined in such a way as to make $\\Upsilon^{\\lambda}$ equivalent for each of the hyperfine components of the transition.\nIn terms of $\\Upsilon$ then, the ratio $R_1$ is \n\\begin{equation}\\label{eq:ratioR1Upsilon}\n   R_1 = \\frac{\\langle 6s_{1\/2} || r ||  6p_{1\/2}\\rangle}{\\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle} = \\sqrt{\\frac{\\Upsilon^{894}}{\\Upsilon^{456}}}.\n\\end{equation}\nSimilarly, we define the ratio \n\\begin{equation}\\label{eq:ratioR2}\n   R_2 \\equiv \\frac{\\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle}{\\langle 6s_{1\/2} || r ||  7p_{1\/2}\\rangle} = \\sqrt{\\frac{\\Upsilon^{456}}{\\Upsilon^{459}}},\n\\end{equation}\nwhich we measure by comparing the attenuation coefficients of the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ line at 456 nm, which serves as the reference, and the absorption on the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{1\/2}$ line at 459 nm. We describe our measurement of $R_2$ in Sec. \\ref{sec:RME459}. \n\nThere is an important subtlety regarding the role of the transition frequency $\\omega$ on the attenuation coefficient.  The frequency $\\omega$ appears in the numerator of the expression for the attenuation coefficient, Eq.~(\\ref{eq:abscoeff}).  For a Doppler broadened medium, the Doppler width $\\Delta \\omega_D$, which is proportional to $\\omega$, appears in the denominator of Eq.~(\\ref{eq:Voigt}).  Therefore, for a Doppler-broadened transition, these frequency factors cancel, and the attenuation coefficients in Eq.~(\\ref{eq:ratioR1Upsilon}) are independent of the optical frequencies of the two transitions.\nCareful attention, however, must be paid to the proper normalization of the Voigt function.\n\n\n\\section{Measurement of $R_1$}\\label{sec:RME456}\nWe first describe the measurement of the ratio of transition moments $R_1$, as defined in Eq.~(\\ref{eq:ratioR1Upsilon}).  \n\nWe show the experimental setup in Fig.~\\ref{fig:ExperimentalSetup894}.\n\\begin{figure}\n\t  \\includegraphics[width=8cm]{Exptsetupsmall-894.png}\\\\\n\t  \\caption{Experimental setup for the two measurements. The 455.7 nm laser stays the same for both measurement. The 894.6 nm or 459.4 nm laser changes depending on the measurement, $R_1$ or $R_2$, respectively, being done. Abbreviations in this figure are: \n(AOM1,2) acousto-optic modulators; (ECDL) external cavity diode laser;\n(PD1-3) photodiodes; (FP1,2) Fabry\u2013-P\\'erot cavities; and (W) wedged windows. FP2 (in the dashed box) is used only for the measurement of $R_1$.  }\n\t  \\label{fig:ExperimentalSetup894}\n\\end{figure}\nWe use two home-made external cavity diode lasers (ECDL) in Littrow configurations, one at $\\lambda = 894$ nm, the other at 456 nm.  The 894 nm laser produces $\\sim$10 mW of output power, while the 456 nm laser produces $\\sim$20 mW.  By ramping the laser diode current and the piezoelectric transducer (PZT) voltage concurrently, we are able to achieve mode-hop free scans of $7-10$ GHz, significantly greater than the widths of the spectra. \n\n\nWe align the laser beams so that they overlap one another in the cesium vapor cell, a sealed glass cell of inside length $\\ell_{cell} = 29.9034 \\: (44)$~cm fitted with flat windows. \nControl of the density of cesium in the cell is achieved using a cold finger enclosed within a copper block, whose temperature we control and stabilize to between $-8$ and $+18^\\circ$C with a thermo-electric cooler and feed-back circuit.  We use Kapton heaters to keep the vapor cell above room temperature at $\\sim 25^\\circ$C, and enclose the cell in an aluminum shell inside an insulating styrofoam container to help maintain a stable and uniform body temperature. \nTo detect the power of the laser beam transmitted through the vapor cell, we use a linear silicon photodiode, labeled PD3 in Fig.~\\ref{fig:ExperimentalSetup894}. \nThe photodiode current is amplified using a trans\\-impedance amplifier with a gain of $5 \\times 10^7$ V\/A, designed for high-gain, low-noise operation.  This amplifier is followed by a second op amp with a gain of 10.  \nWe chose a slow scan rate ($\\sim$4 GHz\/s) and wide amplifier bandwidth (60 kHz), to allow fast rise- and fall-times of the signal.\n\n\\begin{figure*}[t!]\n\t\\includegraphics[width=15cm]{blue_all_graphs_fix.png}\\\\\n\t  \\caption{Examples of absorption spectra recorded at a cold finger temperature of $-2^\\circ$C.  These figures show the photodiode signal versus laser frequency as we scan (a) the 456 nm laser through $ F=4 \\rightarrow  F'=3^{\\prime},4^{\\prime},5^{\\prime}$ components (not resolved) of the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2} $ transition, and (c) the 894~nm laser through $F=3 \\rightarrow F'=3^{\\prime},4^{\\prime}$ components of the $6s \\ ^2S_{1\/2} \\rightarrow 6p \\ ^2P_{1\/2} $ transition.  In each, the data are shown as the red data points, and the result of a least-squares fit as a black dashed line.  We show the residuals (data$-$fit) for each in (b) and (d).}\n\t  \\label{fig:datafit894}\n\\end{figure*}\n\nTo improve the precision of the measurements, we stabilize the optical power delivered to the cell.  For this purpose, we diffract a fraction of each individual beam in an acousto-optic modulator (AOM1 or AOM2), and measure the relative power of the undiffracted beams using photodiodes (PD1 or PD2).  We use the photodiode current to generate an error signal, which controls the r.f.\\ power applied to the AOMs.  In this manner, we are able to stabilize the power of each laser, and achieve a relatively flat power level for the scan of each laser, with less than 3\\% variation in the laser power over a typical scan of $4-6$ GHz for the 456 nm laser and 0.5\\% for the 894 nm laser.\nTo minimize saturation effects, the laser power incident on the cell from the 456 nm laser is about 40 nW in a $\\sim$1 mm diameter beam and the 894 nm laser has 8 nW with $\\sim$2 mm diameter. \nA 15 cm focal length lens after the vapor cell reduces the laser beam size incident on PD3 to less than the photocathode size.\n\nWe calibrate the frequency scans of the two lasers using separate Fabry-\u2013P\\'erot (FP) cavities, with free spectral ranges (FSR) of $\\sim1500$ MHz. \nWe record the transmission through the cavity concurrently with each absorption spectrum, and fit the frequencies of the transmission peaks to a 3rd-order polynomial in the laser frequency ramp voltage.\n\nBefore each set of measurements, we record the photodetector background offset voltage, the measured signal when no light is incident on the photodiode.\nWe also account for the small amount of laser power in the wings of the laser power spectrum. For this, we insert a second cesium vapor cell, which we heat to $\\sim 120^\\circ$C, into the beam path at the beginning of each data run. This vapor cell is labeled `Hot Cell' in Fig.~\\ref{fig:ExperimentalSetup894}. Absorption in this cell of the on-resonant light is very strong, while off-resonant light is transmitted.\nThis gives us a good measurement of the laser power in the wings, typically $\\sim 0.1\\%$ of the total power for the 456 nm laser and $\\sim 1\\%$ for the 894 nm laser. \nWe then determine the total offset level that comes from the background and laser power in the wings, to deduct from our data before curve fitting.\n\n\nFor each measurement, we block one of the lasers so that only one beam passes through the vapor cell, and record approximately four full absorption curves over a ten second period. We then block that laser, unblock the other, and record the absorption curves for the second laser. \nWe repeat this process for a total of four records of 894 nm and three for 456 nm. In total, there are typically sixteen scans at 894~nm and twelve at 456~nm for each measurement.\nRapid reversals between the two wavelengths help minimize variations in the cesium density between measurements.\nWe perform multiple runs at each temperature. \nThen we change the temperature of the cold finger, wait for the cold finger temperature to stabilize, and collect new spectra. \nAdditionally, we remove the absorption cell from the beam path and verify the absence of any spectral feature in the scans.  \n\n\n\nWe show examples of absorption spectra at 456 nm and 894 nm in Fig.~\\ref{fig:datafit894}.\nThe absorption peak at 456 nm, shown as the red data points in Fig.~\\ref{fig:datafit894}(a), is made up of the three hyperfine transitions $F = 4 \\rightarrow F^{\\prime} = 3^{\\prime}, 4^{\\prime}$, and $5^{\\prime}$. These peaks are unresolved since the hyperfine splitting of the $7p \\ ^2P_{3\/2}$ level~\\cite{williamsHH18} is less than the Doppler width $\\Delta \\nu_D^{456} \\sim 700$ MHz.  The slope of the unabsorbed signal to either side of the absorption dip is due to etalon effects (the variation of the transmitted power due to the interference between the reflections from the two window surfaces) in the windows of the cell. These windows are 1.2 mm thick, corresponding to a FSR of $\\sim$80 GHz, far greater than the full 6 GHz scan length. \nThe 894 nm absorption line shown in Fig.~\\ref{fig:datafit894}(c) consists of two components, corresponding to $F^{\\prime} = 3^{\\prime}$ on the left and $F^{\\prime} = 4^{\\prime}$ on the right.  The frequency separation between these two peaks is the \n1167.7 MHz hyperfine splitting of the $6p \\ ^2P_{1\/2}$ state~\\cite{UdemRHH99,Das2016,RafacTanner1997,GerginovCTMDBH06}, which is resolved in this spectrum since this splitting is greater than the $\\Delta \\nu_D^{894} \\sim 360$ MHz Doppler broadening of the transition.  Note that the $F^{\\prime} = 3^{\\prime}$ peak is weaker than the $F^{\\prime} = 4^{\\prime}$, consistent with \n$q_{3 \\rightarrow 3^{\\prime}} = 7\/24$ and $q_{3 \\rightarrow 4^{\\prime}} = 7\/8$ for $J = J^{\\prime} = 1\/2$ \nlisted in Table \\ref{table:q}.  \nFor each of these lines, we have the choice of exciting from the $F = 3$ or $F = 4$ hyperfine component of the ground state.  Since the absorption strength of the 894 nm line is so much stronger than that of the 456 nm line, we used the $F = 3$ ground state (the weaker component) for the former, and $F = 4$ (the stronger component) for the latter. Given the limitations of our setup, the best case would be if both lines have the same strength, as we would be able to record data over a wider range of temperatures or attenuation coefficients.\n\nFor each of the spectra, we fit the data to an equation of the form shown in Eqs.~(\\ref{eqn:fitfunction})$-$(\\ref{eq:Voigt}). Our fit equation has five adjustable parameters: the level of full transmission, a term to account for the slope in the full-transmission level, the center frequency of one of the hyperfine components of the transition, and terms describing the Doppler-broadened width ($\\Delta \\nu_D^{\\lambda}$) and amplitude ($\\Upsilon^{\\lambda}$) of the absorption dip.  The relative heights of the different hyperfine components are determined by the $q_{F \\rightarrow F^{\\prime}}$ factors in Table \\ref{table:q}, and are fixed in our fits.  The linewidths in the Voigt lineshape are $\\Gamma^{\\prime} = 2 \\pi (\\Delta \\nu_N  + 0.2 \\ {\\rm MHz}) $, where $\\Delta \\nu_N = $ 4.6 MHz~\\cite{YoungHSPTWL94,RafacT98,RafacTLB99,DereviankoP02a,AminiG03,BouloufaCD07,ZhangMWWXJ13,PattersonSEGBSK15,GregoireHHTC15,TannerLRSKBYK92,SellPEBSK11} (1.22 MHz~\\cite{PaceA75, MarekN76, GustavssonLS77, DeechLPS77, CampaniDG78, OrtizC81}) is the natural linewidth of the $6p ^2P_{1\/2}$ line ($7p ^2P_{3\/2}$ line), and 0.2 MHz is the intrinsic linewidth of the laser; and a Gaussian of width $\\Delta \\nu_D^{894} \\sim 360$ MHz or $\\Delta \\nu_D^{456} \\sim 700$ MHz. (We will return to this laser bandwidth correction later in this section.)  We use measured values for the frequency difference between the hyperfine components~\\cite{RafacTanner1997,UdemRHH99,Das2016,williamsHH18}.  We show an example of the best fit profile to the absorption spectra as the black dashed lines in Figs.~\\ref{fig:datafit894}(a) and (c).  In Figs.~\\ref{fig:datafit894}(b) and (d), we show the residuals, the difference between our data and the fitted spectra. The small residuals indicate that the fits are very good models of the absorption profile. \n\nAt the higher temperatures used for these measurements ($T > 10^{\\circ}$C), we start to observe some departure of the ratio $\\alpha_{3 \\rightarrow 4^{\\prime}}^{894}(\\omega_0) \/ \\alpha_{3 \\rightarrow 3^{\\prime}}^{894}(\\omega_0)$ from the expected value of three when we fit the two peaks independently.  We suspect that this is a result of errors in the measurement of the offset voltage,  which become more critical for these strongly absorbing peaks.  At the most extreme temperature used ($T = 18^{\\circ}$C), this ratio was as low as 2.946, so we attempted no measurements at higher temperatures. \n\nTo include the effect of the spectral linewidth of the lasers in our fits to the absorption spectra, we first measured (1) the beat signal between the output of the 894 nm laser with that of a frequency comb laser (FCL); and (2) the beat signal between two similar blue diode lasers, each tuned to 456 nm.\nIn each case, we overlap the two interfering beams on a fast photodiode and observe the photocurrent on an r.f.\\ spectrum analyzer. \nThe long-term bandwidth in both cases was $2-3$~MHz.  In addition, \nwe could observe the bandwidth of the signal on a single sweep of the spectrum analyzer. This shows considerable variation from sweep to sweep, probably due to acoustic vibrations of elements within the cavity, but we could observe lines as narrow as a few hundred kHz. We interpret these observations as a short-term (intrinsic) line width of $\\Delta \\nu_{Li} \\sim200$~kHz, with slower fluctuations over a range of $\\Delta \\nu_{Ls} \\sim3$~MHz. \nWe calculate the effect of these laser frequency fluctuations, and determine that these can be included in the fits to the absorption spectra by modifying the Voigt lineshape function in two ways.  \nFirst, we increase the homogeneous linewidth $\\Delta \\nu_N$, using the sum of the natural linewidth of the transition and the intrinsic linewidth $\\Delta \\nu_{Li}$ of the laser.  This is a small, but not negligible, increase.  \nSecond, we increase the inhomogeneous linewidth in the Voigt function calculation to the quadrature sum of the Doppler width, $\\Delta \\nu_D$, and the slow laser frequency fluctuations, $\\Delta \\nu_{Ls}$.  \nFor the linewidths of our system, this is a negligible increase.\n\nAfter fitting each of the sixteen (twelve) absorption curves at $\\lambda = $ 894 nm ($\\lambda = $ 456 nm) within a set individually, we compute the mean and standard deviation of the mean for the fitted values of $\\Upsilon^{894}$ ($\\Upsilon^{456}$). \nWe show a plot of $\\Upsilon^{456}$ vs.\\ $\\Upsilon^{894}$ in Fig.~\\ref{fig:fittedresult894}. \n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{456v894f=4.png}\\\\\n  \\includegraphics[width=\\columnwidth]{456v894f=4res.png}\\\\\n\t  \\caption{\n   Plot of $\\Upsilon^{456}$ against $\\Upsilon^{894}$ showing (a) the datapoints (circles) and the least squares fit straight line, and (b) the residuals in the ordinate. In (a), the error bars are smaller than the size of the markers. The vertical error bars in (b) represent the combined $1 \\sigma$ uncertainties in $\\Upsilon^{456}$ and $\\Upsilon^{894}$. } \n\t  \\label{fig:fittedresult894}\n\\end{figure}\nEach data point represents the average value of $\\Upsilon^{456}$ and $\\Upsilon^{894}$ at a particular cold finger temperature.  \nThe data point near the origin was recorded with the vapor cell removed from the beam path.  The error bars on the data points are too small to observe in Fig.~\\ref{fig:fittedresult894}(a).  The vertical error bars in the residual plot, Fig.~\\ref{fig:fittedresult894}(b), represent the combined $1 \\sigma$ uncertainties in $\\Upsilon^{456}$ and $\\Upsilon^{894}$.\n\nThe total uncertainties in the values of $\\Upsilon^{\\lambda}$ are the statistical, etalon effects, and offset uncertainties added in quadrature. The statistical uncertainty comes from the standard deviation of the mean of the $\\Upsilon^{\\lambda}$ values from the fits to the sixteen (or twelve) absorption curves. \n\nThe etalon effects are our estimate of the uncertainty of the attenuation coefficient resulting from the interference between the reflections at the cell windows.  We account for this effect to first order as a linear variation of the laser power with frequency, ignoring any curvature. \nThis simple model is not adequate, however, when the peak or valley in the sinusoidal variation of the unabsorbed laser power is close to the frequency of the absorption feature.  In these frequency spaces, we estimate the effect of the curvature of the unabsorbed laser power on the size of the absorption peak, which we assign as the uncertainty due to the etalon effect.  In  cases of extreme curves, we also applied a small correction to the absorption height, along with an uncertainty of twice the size of the correction.  \nWe estimate the effect for each absorption curve depth.  A 1 mV change in the height of the signal changes $\\Upsilon^{894}$ ($\\Upsilon^{456}$) by 0.3\\% (0.1\\%).   \n\n\nThe offset uncertainty is our estimation of the error in measuring the signal on PD3 resulting from the power in the wings of the laser spectrum and the background signal, which we subtract from the signal as an offset.  The offset uncertainty of the 456 nm signal is $\\sim 0.05\\%$ of $\\Upsilon^{456}$, while for 894 nm it is more significant at $\\sim 0.15\\%$ due to the stronger absorption at 894 nm.\n\n \\begin{figure*}\n \t    \\includegraphics[width=15cm]{894_all_graphs.png}\\\\\n\n \t  \\caption{Examples of absorption spectra with the cell of length $\\ell_{cell} \\sim 6$ cm at a cold finger temperature of 56$^\\circ$C.  These figures show the photodiode signal vs.~laser frequency as we scan (a) the 459 nm laser through $ F=4 \\rightarrow  F'=3^{\\prime},4^{\\prime}$ components of the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{1\/2} $ transition, and (c) the 456 nm laser through $F=4 \\rightarrow F'=3^{\\prime},4^{\\prime},5^{\\prime}$ components of the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2} $ transition.  In each, the data are shown as the red data points, and the result of a least-squares fit as a black dashed line.  The hyperfine components are not resolved for either transition.  We show the residuals for each in (b) and (d). }\n\t  \\label{fig:datafit1}\n \\end{figure*}\n\n  \nThe solid line in Fig.~\\ref{fig:fittedresult894} is the result of a least-squares fit of a straight line to the data, with two adjustable parameters: the slope and the intercept.  We present the results of this fit in Table \\ref{table:fits894}.  The intercept is within one standard deviation of zero, as expected, while the slope, i.e.~the ratio of scaled absorption terms, is $\\Upsilon^{456} \/ \\Upsilon^{894} = 0.016239 \\: (21)$. (We show $1 \\sigma$ uncertainties in the least significant digits inside the parentheses following the numerical value.)  The reduced $\\chi_r^2$ for these data is 1.29, indicating that deviations of the data from the fitted line are slightly larger than the uncertainties would suggest. We expand the uncertainties of the slope by ~$\\sqrt[]{1.29}$ to account for this, and quote a final slope of $0.016239 \\: (23)$.\nUsing Eq.~(\\ref{eq:ratioR1Upsilon}), the inverse of the square root of the slope yields the ratio of matrix elements $R_1 = 7.8474 \\: (56)$. \n\\begin{table}[t!]\n    \\caption{Numerical values for the intercept, slope, and reduced $\\chi_r^2$ from the fit to the data in Fig.~\\ref{fig:fittedresult894}. These uncertainties have not yet been expanded by $\\sqrt[]{\\chi_r^2}$. }\n    \\begin{tabular}{c|c}\n    \\hline\n    Parameter \\rule{0.1in}{0in} & Value  \\\\ \\hline \\hline\n    \\rule{0in}{0.2in}   Intercept  & \\rule{0.1in}{0in}$3.6\\ (43) \\times 10^{-5}$  \\\\\n    \\rule{0in}{0.2in} Slope & $0.016239\\ (21)$   \\\\\n    \\rule{0in}{0.2in}    $\\chi_r^2$ & $1.29 $\t  \\\\\n    \\hline\n  \\end{tabular}\n  \\label{table:fits894} \n\\end{table}\n\nIn Sections \\ref{sec:error} and \\ref{sec:Results}, we will consider a few additional systematic effects that contribute to this measurement, and use $R_1$ to determine the E1 matrix element for the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ transition.  Before we do this, we describe parallel measurements of the absorption strength of the  $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{1\/2}$ transition.\n\n\n\\section{Measurement of $R_2$}\\label{sec:RME459}\n\nWe turn now to the measurement of the ratio $R_2$, as defined in Eq.~(\\ref{eq:ratioR2}).  \nWe carry out this measurement in a fashion similar to that of the measurement of $R_1$ discussed in Sec.~\\ref{sec:RME456}, alternating between absorption measurements on \nthe $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ line at 456 nm and the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{1\/2}$ line at 459 nm.\nThe experimental setup is very similar to the one discussed in the previous section, and is shown in Fig.~\\ref{fig:ExperimentalSetup894}. The 894 nm ECDL is replaced with a 459 nm ECDL, and FP2 is removed, since we can use the same Fabry-P\\'{e}rot cavity FP1 for both lasers.  \nEach laser generates approximately 20 mW of laser light, and produces mode-hop free scans of $>7$ GHz. \n\n\nOther differences in the apparatus or procedure include: ($i$) We carry out these measurements in three separate data sets, which differ in F, the hyperfine level of the ground state, or the vapor cell used.  This is in contrast to our determination of $R_1$, for which we use only one F value and one vapor cell. The first two data sets are performed with a \nshort (of length $\\ell_{cell} \\sim 6$ cm) sealed glass cell mounted with $0.5^{\\circ}$ wedged windows. The shorter cell length requires higher Cs densities for comparable absorption, and the wedged windows reduce the magnitude of the etalon effects.  We control the density of cesium in the cell using a cold finger enclosed within an aluminum block, whose temperature we control and stabilize to between $40-65^\\circ$C using a thermo-electric module and feed-back circuit.  We use heat tape coiled around the vapor cell to heat the cell body to $\\sim 80^\\circ$C, \nand wrap the cell and heat tape with aluminum foil to help maintain a stable and uniform body temperature. \nFor the third data set on these lines, we used the long ($\\ell_{cell} = 29.9$ cm) vapor cell described in Sec.~\\ref{sec:RME456}.\nObserving similar results in this second cell allows us to rule out background gas in the cell or collisional effects as possible sources of error.\n($ii$) Additionally, the linear silicon photodiode, labeled PD3 in Fig.~\\ref{fig:ExperimentalSetup894}, had a slower rise\/fall time, as both of the absorption curves were similar in depth and we could achieve a better signal-to-noise ratio by filtering out the high-frequency noise. \nWe amplify the photodiode current using a transimpedance amplifier of gain $5 \\times 10^7$ V\/A and bandwidth of 2 kHz, followed by a second amplifier of gain 10, and measure $\\sim 1$ mV of noise on a $\\sim 2$ V signal. The laser power incident on the cell is about 50 nW in a $\\sim$1 mm diameter beam for both lasers. \n($iii$) Finally, since the curves were shallower, we were able to scan the laser frequencies through the absorption curves more rapidly. When we investigated the effects of the bandwidth and scan rate as we did for $R_1$, we found that recording eight full absorption curves over a two second period allows good fits to the data. \n\nFor these data, measurements of the background offset voltage several times each day, rather than before each run, were sufficient.  \nThe background offset voltage was small ($\\sim 1$ mV), and variations were minimal, falling well within the measurement uncertainty. \nBefore every run we did insert the hot cell to estimate and correct for the small amount of laser power in the wings of the laser power spectrum.\nThis gives us a good measurement of the laser power in the wings, \ntypically $\\sim 0.1\\%$ of the full power incident on the photodiode for the 456 nm and $\\sim 0.3\\%$ for the 459 nm laser. We deduct the total offset in the signal, the power in the laser wings along with the background signal, from the data before fitting. We \nestimate the uncertainty of the attenuation coefficients due to the offset to be 0.05\\% (0.1\\%) for the 459 nm (456 nm) laser.\n\nWe show examples of the measured spectra as the red data points in Fig.~\\ref{fig:datafit1}(a) ($6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{1\/2}$ line at 459 nm) and \\ref{fig:datafit1}(c) ($6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ line at 456 nm).  We fit the data to an equation of the form shown in Eqs.~(\\ref{eqn:fitfunction})$- $(\\ref{eq:Voigt}), using the same five adjustable parameters as described in Section \\ref{sec:RME456}.\nThe lineshape of each hyperfine component of the transition is a Voigt profile, with a Lorentzian width $\\Delta \\nu_N$ (1.22 MHz for the $7p ^2P_{3\/2}$ line, or 1.06 MHz for the $7p ^2P_{1\/2}$ line~\\cite{PaceA75, MarekN76, GustavssonLS77, DeechLPS77, CampaniDG78, OrtizC81}) added to the linewidth of the lasers of 0.2 MHz and a Gaussian of width $\\Delta \\nu_D^{456} \\sim \\Delta \\nu_D^{459} \\sim 750$ MHz.  We use calculated values for the relative amplitudes ($q_{F \\rightarrow F^{\\prime}}$ of Table \\ref{table:q}) and experimental values~\\citep{williamsHH18} for the frequency difference of the hyperfine components. We show the least-squares fit spectral profiles as the black dashed lines in Figs.~\\ref{fig:datafit1}(a) and (c).  The residuals, the difference between the data and the fitted profile, are shown in Figs.~\\ref{fig:datafit1}(b) and (d).\nWe fit each of the twenty-four (thirty-two) scans at 456 nm (459 nm) within a measurement individually, and compute the mean and standard deviation of the mean of the fitted values of $\\Upsilon^{\\lambda}$.\n\nFinally, we plot $\\Upsilon^{459}$ against $\\Upsilon^{456}$, and determine the least-squares fit of a straight line to determine the slope.  An example of one such plot (set 2) for the transition from $6s \\: ^2S_{1\/2}, \\ F=4$ is shown in Fig.~\\ref{fig:fittedresult}(a). \n    \\begin{figure}\n          \\includegraphics[width=\\columnwidth]{456v459f=4.png}\\\\\n          \\includegraphics[width=\\columnwidth]{456v459f=4res.png}\\\\      \n    \t  \\caption{Plot of $\\Upsilon^{459}$ vs.\\ $\\Upsilon^{456}$ for data set 2 showing (a) the datapoints (circles) and line of best fit, (b) residuals in the ordinate. In (a), the error bars are smaller than the size of the markers. The vertical error bars in (b) represent the combined $1 \\sigma$ uncertainties in $\\Upsilon^{459}$ and $\\Upsilon^{456}$.}\n    \t  \\label{fig:fittedresult}\n    \\end{figure}\nEach point on the graph corresponds to a different cold finger temperature, with the $y$-coordinate and $x$-coordinate coming respectively from the 459 nm and 456 nm average. \nWe determined the uncertainties of each $\\Upsilon^{\\lambda}$ as the quadrature sum of\nthe statistical, etalon effect, and offset uncertainties. The etalon effect uncertainty is as described for $R_1$. \nWe found that a 1 mV change in the offset resulted in a change to $\\Upsilon^{456}$ and $\\Upsilon^{459}$ of 0.1\\% and 0.05\\%, respectively. \nThe error bars shown on the residual plot, Fig.~\\ref{fig:fittedresult}(b), represents the combined errors of $\\Upsilon^{459}$ and $\\Upsilon^{456}$.\nWe perform separate analyses of the $6s \\: ^2S_{1\/2}, \\  F=3$ and $F=4$ data, since their sensitivity to systematic effects differs.\nWe \npresent the intercepts and slopes from individual straight line fits for the three data sets in Table \\ref{table:2dayresults}.  The intercepts are again all acceptably close to zero.  \nWe will derive the ratio $R_2$ from the square root of the inverse of the slopes of these plots, as shown in Eq.~(\\ref{eq:ratioR2}), but first we consider some additional systematic effects, as discussed in the following section.\n\n\n \\begin{table}[t]\n    \\caption{Summary of linear fit results from our three data sets of $\\Upsilon^{459}$ against $\\Upsilon^{456}$. The listed uncertainties for the slope and intercept have not yet been expanded by $\\sqrt[]{\\chi_r^2}$.  Data sets 1-2 were collected with the shorter $\\sim 6$ cm cell. Data for set 3 was recorded using the longer $\\sim 30$  cm cell.}\n   \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n \t   Data Set & Intercept \\rule{0in}{0.15in}& Slope & $\\chi_{r}^2$    \\\\ \n       \\hline \\hline \n       \n   \\rule{0.09in}{0in}Set 1, F=3\\rule{0.09in}{0in}\t\\rule{0in}{0.15in}\t& $-5.8\\: (41) \\times 10^{-5}$ &\\rule{0.09in}{0in} $0.23380 \\: (29) $ \\rule{0.09in}{0in}& \\rule{0.09in}{0in}$1.95$\\rule{0.09in}{0in} \\\\\n    Set 2, F=4 \\rule{0in}{0.15in}  \t& $-2.4 \\: (42) \\times 10^{-5} $ & $0.23477 \\: (32)$  & $1.95$  \\\\\n\n    Set 3, F=3  \\rule{0in}{0.15in}\t\t& $-1.4 \\: (51)  \\times 10^{-5}$ & $0.23593 \\: (48)$  & $1.52$  \\\\\n\n    \\hline\n\n    \n   \\end{tabular}\n\n   \\label{table:2dayresults}\n \\end{table}\n\n\n\\section{Errors}\\label{sec:error}\nWe have investigated several potential sources of systematic effects, listed in Table \\ref{table:sourcesoferror}, to estimate their impact on the measurements. We describe each of these effects in this section. All of these systematic effects are applied to $R_1$ ($R_2$), after fitting $\\Upsilon^{456}$ ($\\Upsilon^{459}$) against $\\Upsilon^{894}$ ($\\Upsilon^{456}$).  \n\nWe derive the uncertainties labeled `Fit' from the fitted values of the slope and their uncertainties, listed in Tables \\ref{table:fits894} and \\ref{table:2dayresults}.  We have expanded these uncertainties by $\\sqrt{\\chi_r^2}$ to account for excess variation of the data points. \n\\begin{table}[b!]\n    \\caption{Sources of error and the percentage uncertainty resulting from each, for the determinations of $R_1$ and $R_2$.   \n    We add the errors in quadrature to obtain the total uncertainty. \n    }\n    \\begin{tabular}{|l|  c|  c|}\n        \\hline\n\n      \\rule{0.1in}{0in}Source             &   \\rule{0.1in}{0in}$\\sigma_1\/R_1$(\\%)\t\\rule{0.1in}{0in}\t & \\rule{0.1in}{0in} $\\sigma_2\/R_2$(\\%)\\rule{0.1in}{0in} \\\\\n\t\t\\hline \\hline\n        Fit\t\t\t\t\t& 0.07  &0.09-0.13 \\\\\n\t\t\n        Freq.~scan calibr.\t\t\t& 0.04  & 0.01 \\\\\n        Zeeman\t\t\t\t&0.03 \t&0.02 \\\\\n        Beam overlap\t\t\t\t\t& 0.01  & 0.01 \\\\\n\n        Saturation\t\t& 0.02 \t& 0.02 \\\\\n\t\t\t\t\n        Linewidth\t\t& 0.02 \t& 0.02 \\\\\n                                       \t\t\t\n                                        \\hline \\hline\n        Total uncertainty\t\t\t\t& 0.09\t&0.09-0.13\\\\\n    \\hline\n    \\end{tabular}\n\n    \\label{table:sourcesoferror}\n\\end{table}\n\n \\begin{table}\n  \\caption{Summary of the three data sets of $R_2$ values after correction for the effects of the Zeeman splitting.  The uncertainty in $R_2$ includes the errors due to the fit (increased by $\\sqrt{\\chi_r^2}$ of the appropriate data set), frequency calibration, Zeeman splitting, beam overlap, saturation, and linewidth.  Data sets 1-2 were collected with the shorter ($\\sim 6$ cm) cell. Data for set 3 was recorded using the longer ($\\sim 30$ cm) cell. The $\\chi_r^2$ of the weighted mean of the three sets is 4.2 and the weighted mean's error has been expanded by $\\sqrt[]{4.2}$. See Fig.~\\ref{fig:R2} for a plot of the results of the individual data sets and the weighted mean. }\n   \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n \t   Data Set\\rule{0in}{0.15in} & $R_2$  \\\\ \n       \\hline  \\hline\n    Set 1, F=3\\rule{0in}{0.15in} & 2.0684 (19) \\\\\n    Set 2, F=4\\rule{0in}{0.15in} & 2.0637 (21)\\\\\n    Set 3, F=3\\rule{0in}{0.15in} & 2.0591 (27) \\\\\n    \\hline\n \\rule{0.1in}{0in}  Weighted Mean\\rule{0in}{0.15in}\t\\rule{0.1in}{0in}\t& \\rule{0.1in}{0in}2.0646 (26)\\rule{0.1in}{0in}\\\\ \n    \\hline\n   \\end{tabular}\n  \n   \\label{table:R2results}\n \\end{table}\n\nDuring the course of analyzing the absorption curves, we noted a sensitivity of the fits to the frequency calibration of the laser scans. We calibrate these scans, as discussed earlier, using the transmission peaks of the lasers through the FP cavities.  The FSRs of these cavities, however, are not known precisely. \nWe experimentally determined the FSR values of both FP cavities by fitting the absorption data using different values of the FSR. Using the variation in the residuals, we obtain an estimate for the FSR that fits the absorption curves the best. \n(Since the hyperfine splittings of each of these states are well known, the residuals of the absorption curves are sensitive to variations in the frequency calibration of the scans.)\nWe determined the FSR for the FP cavity used with the 456\/459~nm lasers to be $FSR_{FP1} = 1501.6 \\: (10)$ MHz, while for the 894~nm laser, we determine $FSR_{FP2} = 1481.9 \\: (4)$ MHz.\nWe also use these fits to estimate the effect that the uncertainty of the FP FSR has on the measured ratios. \nWe estimate an uncertainty in the ratio $R_1$ due to frequency calibration, to be $0.04\\%$. For the ratio $R_2$, we find the uncertainty to be at most $0.01\\%$. \n\nThe magnetic field at the cell location also affects the measurements of the absorption strength.  We measure a static magnetic field of $\\sim 1$ G in the vertical direction (parallel to the laser polarization) at the location of the 6 cm vapor cell, mainly from the optical table. We minimize the magnetic field generated by the heat tape by wrapping the heat tape in alternating directions, ensuring the magnetic field only comes from the surroundings.  A  $\\sim 1$~G field causes a Zeeman splitting on each hyperfine component of 2 MHz or less. \nWe approximate the effect of Zeeman splitting on the effective homogeneous linewidth of the transition by adding the Zeeman broadening of each hyperfine component to the natural linewidth, which we use in the Voigt function for our analysis. \nWe multiply $R_2$ by the Zeeman correction for the appropriate starting F state, 0.9999 for F=4 data and 1.0001 for F=3 data. We estimate an uncertainty in  $R_2$ due to this correction to be about $0.02 \\% $.\nThe height of the 30 cm cell above the table was greater than that of the small cell, so the magnetic field for measurements of $R_1$ was smaller, $\\sim$0.5 G. This led to a Zeeman splitting of less than 0.7 MHz.  We estimate the uncertainty in $R_1$ due to this Zeeman splitting to be 0.03\\%, and did not apply any correction to these data.  \n\n\nSmaller systematic errors in the ratios result from beam overlap errors and saturation effects. We estimate that each of these effects contribute 0.02\\% uncertainty or smaller, as listed in Table~\\ref{table:sourcesoferror}.  \nWe measure that the two laser beams are parallel to one another to within 0.05 mrad, and overlap each other in the cell to within 0.5 mm.  Therefore the effective path lengths for these two beams are identical to within 0.02\\%, for an effect on $R_1$ and $R_2$ of 0.01\\%. \nWe minimize saturation effects by reducing the laser intensity of the 456 nm and 459 nm lasers to less than $2~\\times~10^{-4} $ times the saturation intensity for the transition~\\cite{Urvoy11} using a neutral density filter and reflections from several uncoated wedged windows. We attenuated the power of the 894 nm laser more than the 456 nm and 459 nm laser to similarly avoid saturation.\nWe estimate that saturation effects could have an effect at the $0.02\\%$ level.\n\nLastly, we include an uncertainty for the correction that we apply for the linewidth of the lasers used. For all of the ECDL lasers we estimated a 200 kHz intrinsic linewidth with a conservative uncertainty of 200 kHz. These uncertainties would lead to about a 0.02\\% uncertainty in each of the ratios.\n\nWe add the fit, frequency scan calibration, Zeeman, beam overlap, saturation, and linewidth errors in quadrature for our final uncertainties to $R_1$ and $R_2$,\nand apply the Zeeman correction to $R_2$ to get our final values. \nIn the next section, we discuss the results of these measurements.\n\n\n\\section{Results}\\label{sec:Results}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{R4564591.png}\\\\\n  \\caption{ \n\t  Plot of the three sets of $R_2$ results and the weighted mean.  We have expanded the error bars of the weighted mean by $\\sqrt{\\chi_r^2}= 2.0$ to account for the variation among the individual results.} \n\t  \\label{fig:R2}\n\\end{figure}\n\nAfter adding the uncertainties described in the previous section, the final result for $R_1$ is $R_1 = 7.8474 \\: (72)$.  \nFor $R_2$, after applying the corrections and uncertainties described in the previous section to the three individual data sets, we arrive at the results shown in Table \\ref{table:R2results} and plotted in Fig.~\\ref{fig:R2}. The weighted average of these results is $R_2 = 2.0646 \\: (26)$. \nWe compare these results for $R_1$ and $R_2$ with a number of prior experimental and theoretical results in Table \\ref{table:ResultComparison}, and illustrate these in Fig.~\\ref{fig:R1R2}.  \n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{R1theexp.png}\\\\\n  \\includegraphics[width=\\columnwidth]{R2theexp.png}\\\\\n\t  \\caption{Plots showing comparisons of various experimental and theoretical determinations of (a) $R_1 \\equiv \\langle 6s_{1\/2} || r ||   6p_{1\/2}\\rangle \/ \\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle$ and (b) $R_2 \\equiv \\langle 6s_{1\/2}  || r ||  $ $ 7p_{3\/2}\\rangle \/ \\langle 6s_{1\/2} || r ||  7p_{1\/2}\\rangle$ ratios. See Table \\ref{table:ResultComparison} for references to these data.  For Safronova {\\it et al.} (Refs.~\\cite{SafronovaJD99,SafronovaSC16}), we plot both the scaled (sc) and single-double (SD) values for $R_2$.\n\t  } \n\t  \\label{fig:R1R2}\n\\end{figure}\nWe note that the previous results for $R_1$ and $R_2$ by Shabanova {\\it et al.} \\cite{shabanovaMK1979} (who reported oscillator strengths, which we converted to matrix elements) and $R_2$ by Borv\\'{a}k \\cite{Borvak14} are in reasonable agreement, to within their error bars, of our results, which are of higher precision.  (We derived the ratio value for Borv\\'{a}k using $(R_2)^2 = (21\/22) \\times 4.461 \\: (23)$ from the table on page 126, where the additional factor of $21\/22$ comes from combinations of Clebsch-Gordan coefficients.)  $R_1$ from Antypas \\cite{antypas7p2013} disagrees with our new results, which we consider to be more reliable due to the use of the Voigt profile and the addition of the faster photodiode amplifier that we discussed earlier.  We derived the value of $R_2$ for Vasilyev {\\it et al.}~\\cite{VasilyevSSB02} from their matrix elements, whose measurement was dependent on precise knowledge of the vapor cell path length and atomic density in the vapor cell, unlike our method. The scaled theoretical results of Refs.~\\cite{BlundellSJ92, SafronovaJD99, SafronovaSC16} appear to be in good agreement with our new results as well. \n\n  \nWe use $R_1 = 7.8474 \\: (72)$ to determine the matrix element for the $6s \\ ^2S_{1\/2} \\rightarrow 7p \\ ^2P_{3\/2}$ transition using \n\\begin{equation}\n  \\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle = \\frac{\\langle 6s_{1\/2} || r ||  6p_{1\/2} \\rangle}{R_1} \n\\end{equation} \nand $\\langle 6s_{1\/2}  || r || 6p_{1\/2}\\rangle = 4.5057 \\: (16) \\ a_0$, the weighted average of the transition matrix element for the D$_1$ line from Refs.~\\cite{YoungHSPTWL94,RafacT98,RafacTLB99,DereviankoP02a,AminiG03,BouloufaCD07,ZhangMWWXJ13,PattersonSEGBSK15,GregoireHHTC15,TannerLRSKBYK92,SellPEBSK11}.\nOur result is \n\\begin{equation}\n\\langle 6s_{1\/2} || r ||  7p_{3\/2}\\rangle = 0.57417 \\: (57) \\ a_0.\n\\end{equation}\n\n\\begin{table*}[t]\n  \\caption{Experimental and theoretical results for the reduced dipole matrix elements of the cesium $6s\\,^2S_{1\/2} \\rightarrow 7p\\,^2P_{J}$ transitions. The matrix elements are given as factors of $a_0$.  For Ref.~\\cite{SafronovaSC16}, we list both the single double (-SD) and scaled (-sc) values.}\n  \\begin{tabular}{|l|l|l|l|l|}\n    \\hline\n      \\multicolumn{1}{|c|} {Group}  & \\multicolumn{1}{|c|}{$R_1$} & \\multicolumn{1}{|c|}{$R_2$} & $\\langle 6s_{1\/2}||r|| 7p_{3\/2} \\rangle$ & $\\langle 6s_{1\/2}||r|| 7p_{1\/2} \\rangle$  \\\\ \\hline \n  {\\underline{\\emph{Experimental}}} \t\t& & & &\t\t \\\\\n   Shabanova {\\it et al.}, 1979~\\cite{shabanovaMK1979} \t& \\hspace{0.1in} 7.76 (14)       & \\hspace{0.1in} 2.052 (38) & \\hspace{0.1in} 0.583 (10)   & \\hspace{0.1in} 0.2841 (21)  \\\\\n   Vasilyev {\\it et al.}, 2002~\\cite{VasilyevSSB02} \t\t&  & \\hspace{0.1in} 2.124 (24) &  \\hspace{0.1in} 0.5856 (50)  & \\hspace{0.1in} 0.2757 (20)  \\\\\n   Antypas and Elliott, 2013~\\cite{antypas7p2013}    \t& \\hspace{0.1in} 7.796 (41)      & \\hspace{0.1in} 2.072 (12) &  \\hspace{0.1in} 0.5780 (7)   & \\hspace{0.1in} 0.2789 (16)  \\\\\n   Borv\\'{a}k, 2014~\\cite{Borvak14}\t\t\t&  & \\hspace{0.1in} 2.0635 (53)~~ &  \\hspace{0.1in} 0.5759 (30)  & \\hspace{0.1in} 0.2743 (29)  \\\\\n   This work\t                  \t\t\t& \\hspace{0.1in} 7.8474 (72)~~ & \\hspace{0.1in} 2.0646 (26) &  \\hspace{0.1in} 0.57417 (57) & \\hspace{0.1in} 0.27810 (45) \\\\\n   & & & & \\\\\n  \\multicolumn{1}{|l|}{\\underline{\\emph{Theoretical}}} \t& & & &  \\\\\n    Dzuba {\\it et al.}, 1989~\\cite{DzubaFKS89}\t\t\t& \\hspace{0.1in} 7.708 & \\hspace{0.1in} 2.12 & \\hspace{0.1in} 0.583 \t& \\hspace{0.1in} 0.275 \\\\\n    Blundell {\\it et al.}, 1992~\\cite{BlundellSJ92}\t    & \\hspace{0.1in} 7.83 & \\hspace{0.1in} 2.057 & \\hspace{0.1in} 0.576 \t& \\hspace{0.1in} 0.280 \\\\\n    Safronova {\\it et al.}, 1999~\\cite{SafronovaJD99}\t& \\hspace{0.1in} 7.873 & \\hspace{0.1in} 2.065 & \\hspace{0.1in} 0.576 \t& \\hspace{0.1in} 0.279 \\\\\n    Derevianko, 2000~\\cite{Derevianko00}\t&  &  &  & \\hspace{0.1in} 0.281 \\\\\n    Porsev {\\it et al.}, 2010~\\cite{PorsevBD10} \t\t\t&  &  &  & \\hspace{0.1in} 0.2769 \\\\\n    Safronova {\\it et al.}-SD, 2016~\\cite{SafronovaSC16}\t& \\hspace{0.1in} 7.452 & \\hspace{0.1in} 2.016 & \\hspace{0.1in} 0.601     & \\hspace{0.1in} 0.298 \\\\\n    Safronova {\\it et al.}-sc, 2016~\\cite{SafronovaSC16} & \\hspace{0.1in} 7.873 & \\hspace{0.1in} 2.065 & \\hspace{0.1in} 0.576     & \\hspace{0.1in} 0.279 \\\\\n   \\hline\n  \\end{tabular}\n  \\label{table:ResultComparison}\n\\end{table*}\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{matrixelement7p32.png}\\\\\n    \\includegraphics[width=\\columnwidth]{matrixelement7p12.png}\\\\\n\t  \\caption{ Plots of the experimental and theoretical values of (a) $\\langle 6s_{1\/2}||r|| 7p_{3\/2} \\rangle$ and (b) $\\langle 6s_{1\/2}||r|| 7p_{1\/2} \\rangle$, as listed in Table ~\\ref{table:ResultComparison}. Experimental values are on the left, while theoretical values are shown on the right.  See Table \\ref{table:ResultComparison} for references to these data.  For Safronova {\\it et al.} (Refs.~\\cite{SafronovaJD99, SafronovaSC16}), we have plotted only the scaled (sc) values.} \n\t  \\label{fig:matrix1232}\n\\end{figure}\n\n\nWe combine $R_2 = 2.0646 \\: (26)$ in Eq.~(\\ref{eq:ratioR2}) and our new determination of $\\langle 6s_{1\/2}  || r || 7p_{3\/2}\\rangle$ to obtain\n\\begin{equation}\n\\langle  6s_{1\/2}|| r ||7p_{1\/2} \\rangle = 0.27810 \\: (45)~a_0.\n\\end{equation}\n\n\nWe present a summary of past experimental and theoretical results of these dipole matrix elements in Table~\\ref{table:ResultComparison}. We have also plotted the results for $\\langle 6s_{1\/2}  || r || 7p_{3\/2}\\rangle$ and $\\langle 6s_{1\/2}  || r || 7p_{1\/2}\\rangle$ in Figs.~\\ref{fig:matrix1232}(a) and (b), respectively.  \nFor Shabanova {\\it et al.}~\\cite{shabanovaMK1979} and Borv\\'{a}k \\cite{Borvak14}, our result is within their uncertainties for $\\langle 6s_{1\/2}|| r ||  7p_{3\/2} \\rangle$, but in poorer agreement with the $\\langle 6s_{1\/2}|| r ||  7p_{1\/2} \\rangle$ value.  The matrix element values from Borv\\'{a}k come from a direct determination, separate from the ratio measurement discussed above.\n\n\nFor comparison with theory, our result is within the distribution spanned by the $\\langle 6s_{1\/2}||r|| 7p_{1\/2} \\rangle$ values. In particular, our result is in the middle of the two closest theory values from Refs.~\\cite{SafronovaJD99, PorsevBD10}. For $\\langle  6s_{1\/2} || r || 7p_{3\/2} \\rangle$, our value is below all of the theoretical results, but within 0.3\\% of Blundell {\\it et al.}~\\cite{BlundellSJ92} and scaled values of Refs.~\\cite{SafronovaJD99, SafronovaSC16}. \nIn Table~\\ref{table:ResultComparison}, we list two values, calculated results using different theoretical methods, from the Supplemental Material of Ref.~\\cite{SafronovaSC16}. \nThe authors of \\cite{SafronovaSC16} recommend the single-double (SD) all-order approach values, which we have listed as `Safronova {\\it et al.}-SD'. In Ref.~\\cite{SafronovaJD99}, the authors note that scaling improved agreement of theoretical determinations of the $\\langle 6s_{1\/2}  || r || 7p_{j}\\rangle$ matrix elements with experiment, and recommend using the scaled values. We also observe that scaling improves the agreement of theory with the current measurements, and have included those values from \\cite{SafronovaSC16} as `Safronova {\\it et al.}-sc'.\nIn comparison with Refs.~\\cite{SafronovaJD99, SafronovaSC16}, our measurements of both matrix elements are much closer to the scaled theory values than the SD theory values.\n\n\n\\section{Conclusion}\\label{sec:Conclusion}\n\nIn conclusion, we present measurements of the ratio of the dipole matrix elements of the cesium $6s\\,^2S_{1\/2} \\rightarrow 6p\\,^2P_{1\/2}$ and $6s\\,^2S_{1\/2} \\rightarrow 7p\\,^2P_{3\/2}$ transitions and the ratio of $6s\\,^2S_{1\/2} \\rightarrow 7p\\,^2P_{1\/2}$ and $6s\\,^2S_{1\/2} \\rightarrow 7p\\,^2P_{3\/2}$ transitions. We used a ratio measurement of the two transitions to eliminate the need for precise knowledge of the path length of the laser within the vapor cell, or of the density of cesium, which helped to eliminate potential systematic errors. From these measurements, we calculate new, higher precision results of the dipole matrix elements of cesium with precision $\\le 0.16 \\%$. With our new knowledge of the dipole matrix elements, we are poised to be able to evaluate the scalar and vector polarizabilities of the cesium $6s\\,^2S_{1\/2} \\rightarrow 7s\\,^2s_{1\/2}$ transition. A new value of the vector polarizability has implications on the interpretation of cesium parity nonconservation measurements, and will allow a new determination of the weak charge in cesium.\n\nThis material is based upon work supported by the National Science Foundation under Grant Number PHY-1607603 and PHY-1460899.  Useful conversations with Dan Leaird are also gratefully acknowledged.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe past decade has witnessed an explosive growth of data and the needs for high-speed data processing. A large-scale data often needs to be sorted to enable higher efficiency. Sorting is a key kernel in many applications such as data mining \\cite{aggarwal2015data}, robotics \\cite{bayindir2016review} and machine learning \\cite{devlin2018bert}. To efficiently sort an array into an order, numerous sorting algorithms have been invented in the past, such as merge-sort \\cite{goldstine1947planning} or quick-sort \\cite{quicksort}. These algorithms can be accelerated using CPUs\/GPUs \\cite{chhugani2008efficient,zhang2016high,satish2009designing}, FPGAs \\cite{chen2017computer,chen2019sorting,samardzic2020bonsai,song2016parallel} and ASICs \\cite{norollah2019rths,najafi2018low,lin2017hardware}. However, transferring data between memory and external processing units incurs a long latency and a degraded energy efficiency. Techniques like memory management \\cite{stehle2017memory} have been developed to minimize the data movement, but such optimizations do not fundamentally solve the problem.\n\nMemristive in-memory sorting \\cite{rram_sort_alam2020,prasad2021memristive} have been proposed recently to tackle this challenge. Memristor-aided logic is developed in \\cite{rram_sort_alam2020} to implement compare-and-select blocks in memory. However, a large number of memristor cells are used to implement logic gates with frequent write operations, resulting in a low memory density and a degraded device lifetime. The latest memristive in-memory sorting \\cite{prasad2021memristive} uses iterative in-memory min computations with help of a near-memory circuit. The min values are searched by traversing each bit column using column reads (CR) on a 1T1R memristive memory. Frequent write operations in \\cite{rram_sort_alam2020} are eliminated; however, the number of CRs is proportional to the number of 1T1R cells in the memristive memory, degrading the latency and energy efficiency.\n\nIn this work, we propose a column-skipping algorithm to minimize the number of CRs for improved sorting speed and hardware efficiency. A near-memory circuit is designed to keep track the column read conditions and skip those that are leading 0's or have been processed previously. A multi-bank management is developed to enhance the scalability when sorting a larger array stored in different memristive memory banks. Implemented in a 40nm CMOS technology with 1T1R memristive memory and experimented on a variety of sorting datasets, the length-1024 32-bit column-skipping memristive sorter with state recording of 2 demonstrates up to 4.08$\\times$ speedup, 3.14$\\times$ area efficiency and 3.39$\\times$ energy efficiency, respectively, over the latest memristive in-memory sorting implementation \\cite{prasad2021memristive}. \n\n\\section{Background}\n\n\\subsection{Sorting Applications}\n\nSorting is a known bottleneck for many applications \\cite{aggarwal2015data,bayindir2016review,devlin2018bert}. Here we briefly introduce two representative applications where sorting dominates the execution time: 1) Kruskal's algorithm for minimum spanning tree (MST). In Kruskal's algorithm, all the graph edges need to be sorted from low weight to high weight. Majority of the weights are small numbers with frequent repetitions; 2) MapReduce in distributed systems. In MapReduce, maps need to be sorted before transferring to the reducer stage \\cite{dean2008mapreduce}. These maps are typically clustered in a few groups. We use datasets generated from these two applications for benchmarking in Section~\\ref{section:evaluation}.\n\n\\subsection{Memristive In-Memory Sorting}\n\nIterative in-memory min computation is proposed in \\cite{prasad2021memristive} for memristive in-memory sorting. It uses $N$ iterations to successively search and exclude the min values in a length-$N$ array. Suppose each memristor cell stores a bit in a 1T1R memristive memory. \\figurename~\\ref{fig:memristive_sorting} shows an example for a length-$N$ ($N = 3$) array of $w$-bit ($w = 4$) numbers, \\{8, 9, 10\\}. \n\nIn each iteration, a $w$-step bit traversal algorithm searches the min value: at step $j$ ($j = w-1 \\rightarrow 0$), a near-memory circuit reads an bit column corresponding to the $j$-th bits of all array elements, searches for 1's in that bit column, and exclude the rows that have 1's. When a bit column contains all 0's or 1's, the row exclusion can be skipped. Rows that are corresponding to non-minimum values are excluded step by step until the min value is reached. The row for the min value is then excluded and marked as sorted before moving to the next min search iteration. The near-memory circuit is designed to support two operations, column read (CR) and row exclusion (RE), and their associated control logic. \\figurename~\\ref{fig:memristive_sorting} shows the steps to sort \\{8,9,10\\} using memristive in-memory sorting \\cite{prasad2021memristive}. Note that the near-memory circuit in \\cite{prasad2021memristive} does not keep track the number of remaining elements in the array; therefore it takes $N = 3$ iterations of min search, each contains $w = 4$ CRs. The total sorting latency is $N\\times w = 12$ CRs. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.95\\linewidth]{figures\/example-SIM.pdf}\n    \\caption{Memristive in-memory sorting \\cite{prasad2021memristive}.}\n    \\label{fig:memristive_sorting}\n\\end{figure}\n\n\\section{Column-Skipping Memristive In-Memory Sorting}\n\nWe observe that memristive in-memory sorting in \\cite{prasad2021memristive} introduces a large number of redundant CRs which are repeatedly executed on leading 0's or bit columns that have been processed previously. As shown in \\figurename~\\ref{fig:memristive_sorting}, when searching the 2nd minimum number 9, the first 3 CRs have been processed in the 1st iteration and are repeated in the 2nd iteration. To efficiently skip these redundant CRs, we propose a low-latency column-skipping algorithm. We use unsigned fixed-point number as example, but it can easily be applicable to signed fixed-point and floating-point number formats with small changes as described in \\cite{prasad2021memristive}.\n\n\\subsection{Low-Latency Column-Skipping Algorithm}\n\nRedundant CRs can happen in two scenarios: 1) array elements may include leading 0's. CRs on these leading 0's can be skipped at the beginning of each iteration; 2) some CRs may have been processed previously for REs, i.e., we do not need to exclude any new rows for those bit columns. \n\nTo detect and skip the redundant CRs, we propose to record the $k$ most recent RE states and their corresponding column indexes. The recorded states can be reloaded to skip redundant CRs. \\figurename~\\ref{fig:flow_chart} summarizes the iterative min computation for a length-$N$ array with proposed column skipping algorithm (where $n = 1 \\rightarrow N$): 1) if state records are empty, the $w$-step algorithm \\cite{prasad2021memristive} traverses each bit column from MSB ($i = w-1$) to LSB ($i = 0$). $k$ most recent RE states whose bit columns are not all 0's or 1's and their corresponding column indexes are stored in a state controller; 2) if state records are non-empty, we reload the most recent RE state and the corresponding column index $s$ and start from the next bit column $s-1$. CRs are executed on subsequent bit columns until reaching the min value.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.80\\linewidth]{figures\/flow_chart.pdf}\n    \\caption{Iterative min search with proposed column-skipping algorithm}\n    \\label{fig:flow_chart}\n\\end{figure}\n\n\\begin{figure}\n\\xdef\\xfigwd{\\textwidth}\n\\centering\n        \\includegraphics[width = 0.95\\linewidth]{figures\/example-k=2.pdf}\n    \\caption{Column-skipping memristive in-memory sorting with state recording $k=2$.}\n    \\label{fig:memristive_sorting_k2}\n\\end{figure}\n\n\\figurename~\\ref{fig:memristive_sorting_k2} illustrates the proposed column-skipping algorithm with state recording $k = 2$ when sorting the 4-bit array $\\{8,9,10\\}$. State recording in the first iteration helps to skip the first 3 CRs in searching the 2nd minimum and the first 2 CRs in searching the 3rd minimum. The total latency is reduced to only 7 CRs. The selection of $k$ affects the performance of the proposed column-skipping algorithm. We study the impacts of $k$ on sorting speedup, silicon area and power consumption in Section~\\ref{section:evaluation}.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.91\\linewidth]{figures\/1T1R.pdf}\n    \\caption{Near-memory circuit for column-skipping memristive in-memory sorting.}\n    \\label{fig:near_mem_circuit}\n\\end{figure}\n\n\\subsection{Near-Memory Circuit for Column-Skipping}\n\n\\figurename~\\ref{fig:near_mem_circuit} demonstrates the near-memory circuit connected to a 1T1R memristive memory to implement the proposed column-skipping algorithm. The 1T1R memristive memory stores the binary bits of array elements with MSB on the leftmost column. Similar to \\cite{prasad2021memristive}, select lines with sense amplifiers and bitline drivers are used for column reads. The proposed near-memory circuit consists of three modules: 1) a column processor that controls the column states; 2) a row processor that controls wordline (or RE) states; 3) a state controller that stores the RE states and their corresponding column indexes using a $k$-entry table. It also controls signals to execute all the operations.  \n\nThe near-memory circuit supports the four operations in \\figurename~\\ref{fig:flow_chart} as following: 1) column read (CR), where the column processor enables the bitline driver of a column and the corresponding bit column is read to the row processor. The column controller generates the next-step column state and the enable signal for column update ($cen$). Sense amplifiers measure the current on each select line to determine if it's 0 or 1; 2) row exclusion (RE), where the row processor checks if the bit column are all 0's or 1's (through row controller) before updating the wordlines (or RE) states. The row controller generates the enable signal for wordline update ($ren$). The wordlines that are connected to 1's are excluded and set to 0; 3) state recording (SR), where RE states and their corresponding column indexes are stored in a $k$-entry table. The recording is enabled ($sen$) if an iteration starts from the MSB and the bit column is not all 0's or 1's; and 4) state loading (SL), where the most recent RE state and the corresponding column index are sent to the row processor and column processor, respectively. The load enable signal ($len$) selects the reloaded states when updating the wordline and column registers. A top-level controller is used to schedule the four operations. \n\nWhen multiple rows remain unexcluded at the end of an iteration due to repetitions in the array, the column processor stalls to avoid redundant CRs until all repetition elements are excluded successively in the row processor.\n\n\n\\section{Multi-Bank Management}\n\\label{section:scalability}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.95\\linewidth]{figures\/multi_bank.pdf}\n    \\caption{Multi-bank management to synchronize sub-sorter operations and select sorted output}\n    \\label{fig:scalability}\n\\end{figure}\n\nThe near-memory circuit shown in \\figurename~\\ref{fig:near_mem_circuit} can be scaled up to support larger array (i.e. larger $N$) or higher precision (i.e. large $w$). However, practical array can be too big to fit in a single memristive memory. To solve this problem, we propose a scalable solution to sort larger array stored in multi-bank memristive memory. \n\nSuppose a length-$N$ array is stored in $C$-bank memristive memory, each bank stores $N\/C$ elements and has its own near-memory circuit that forms a length-$N\/C$ sub-sorter. To realize length-$N$ sorting using $C$ sub-sorters of length-$N\/C$, sub-sorters' operations need to be synchronized and run as a whole. A multi-bank manager is designed to connect the sub-sorters for this synchronization purpose: the judgement about all 0's or 1's needs to be considered globally to synchronize RE and SR operations while CR and SL operations are synchronized through the OR gates. \n\n\\figurename~\\ref{fig:scalability} shows the multi-bank manager to generate synchronized operation bits $en_{sync}$ based on local operation bits $en_i$ from sub-sorter $i$, where $i \\in [1,C]$. In each sub-sorter, the synchronized operation bits $en_{sync}$ are used for replacing the original signals ($en_i$) to realize the corresponding function. The multi-bank manager monitors the sub-sorters' states and select the output from one of the $C$ sub-sorters if existing repetitions. Performance of the proposed multi-bank management are evaluated in Section~\\ref{section:evaluation}. \n\n\\section{Evaluation and Benchmarking}\n\\label{section:evaluation}\n\nWe evaluate the proposed techniques using statistically distributed datasets (uniform, normal and clustered) and practical datasets (from Kruskal's and MapReduce). We use 32-bit precision: the uniform distribution ranges from 0 to $2^{32}-1$, the normal distribution has a mean of $2^{31}$ and a standard deviation of $2^{31}\/3$, and the clustered distribution has 2 clusters centered at $2^{15}$ and $2^{25}$ with identical standard deviation of $2^{13}$. To estimate the silicon area and power consumption, prototype sorters of length-1024 are implemented with 1T1R memristive memory using a 40nm CMOS technology. The RRAM device has two states and the corresponding resistances are 10M$\\Omega$ and 100k$\\Omega$, respectively. State-of-the-art memristive in-memory sorter \\cite{prasad2021memristive} (baseline) and conventional digital merge sorter are implemented for comparison. All prototype sorters run at a 500MHz clock frequency.\n\n\\subsection{Sorting Speedup}\n\nThe baseline implementation \\cite{prasad2021memristive} has a fixed sorting speed of 32 cycles per number for any datasets. The merge sorter outperforms the baseline by 3.2$\\times$ in speed. The speed of column-skipping sorter depends on parameter $k$ and dataset distribution. \\figurename~\\ref{fig:speed} shows the normalized speedup over the baseline on the selected datasets with $N = 1024$, $w = 32$ and varying state recording $k$. When $k$ increases, the min search is more likely to start from a recorded RE state; however, the reloaded starting position ($s$ in \\figurename~\\ref{fig:flow_chart}) may be further away from the optimal starting position, degrading the speedup due to less number of skipped CRs. We observe that the speedup saturates when $k$ reaches 2 or 3 and then goes down across selected datasets. \n\nThe proposed column-skipping algorithm achieves faster sorting speed (up to 2.22$\\times$ over the baseline) on clustered dataset than the speedup on uniformly or normally distributed datasets (up to 1.21$\\times$ and 1.23$\\times$ over the baseline, respectively). This is because clustered elements with small centers signify more leading 0's and redundant CRs. In Kruskal's and MapReduce dataset, majority of the small and repetitive elements lead to much better results for a speedup up to 3.46$\\times$ and 4.16$\\times$ over the baseline, respectively.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.95\\linewidth]{figures\/speed.pdf}\n    \\caption{Normalized speedup over the baseline on different datasets with $N$ = 1024, $w$ = 32 and varying state recording $k$.}\n    \\label{fig:speed}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.95\\linewidth]{figures\/area_power.pdf}\n    \\caption{Normalized area and power over the baseline on MapReduce dataset with $N$ = 1024, $w$ = 32 and varying state recording $k$.}\n    \\label{fig:area_power}\n\\end{figure}\n\n\\subsection{Area and Energy Efficiency}\n\nWith $N = 1024$ and $w = 32$, the baseline sorter occupies 77.8K \\textmu m$^2$ in silicon while the merge sorter occupies 246.1K \\textmu m$^2$. The merge sorter demonstrates 1.01$\\times$ area efficiency (throughput\/area) over the baseline. We further measure the areas of column-skipping sorters with varying state recording $k$. \\figurename~\\ref{fig:area_power} presents the normalized area and area efficiency over the baseline when sorting the MapReduce dataset. With $k = 1$, column-skipping sorter demonstrates more than 3.2$\\times$ area efficiency over the baseline. When $k$ increases, the sorter area increases due to larger state controller to store more RE states; however, the area efficiency goes down, because the speedup starts saturating when $k$ reaches 2 or 3. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = \\linewidth]{figures\/sca.pdf}\n    \\caption{(a) Implementation summary using 40nm CMOS technology and 1T1R memristive memory (K \\textmu m$^2$ for area, Num\/ns\/mm$^2$ for area efficiency, mW for power, Num\/\\textmu J for energy efficiency);(b) Normalized area and power (for MapReduce dataset) with varying sub-sorter length $N_s$ for $N$ = 1024, $w$ = 32 and $k = 2$ ($N_s = 1024$ is the baseline).}\n    \\label{fig:performance}\n\\end{figure}\n\nWe measured power using Ansys PowerArtist considering switching activities when sorting MapReduce dataset. The baseline sorter and the merge sorter consume 319.7 mW and 825.9 mW, respectively. The merge sorter demonstrates 1.24$\\times$ energy efficiency (throughput\/power) over the baseline. Column-skipping sorter consumes more power with increasing $k$, but the energy efficiency reaches the peak at $k = 2$ as shown in \\figurename~\\ref{fig:area_power}, outperforming the baseline by 3.39$\\times$. The area and power consumption of 1T1R array are orders of magnitude less than the near-memory circuit. One can select the parameter $k$ based on target dataset for optimized speed, area and energy efficiency. \n\n\\subsection{Multi-Bank Management}\n\nTo evaluate multi-bank management, we build a column-skipping sorter of $N = 1024$ using sub-sorters of length $N_s$ = 64, 256, 512. Multi-bank management does not change the speedup brought by column-skipping when clock frequency remains unchanged. Further reducing the sub-sorter length results in a degraded clock frequency under 500MHz due to more complex multi-bank manager. \\figurename~\\ref{fig:performance}(a) demonstrates the normalized area and power of multi-bank management over the original $N=1024$ sorter. We observe that the area and power of the near-memory circuit in sub-sorters decreases super-linearly when $N_s$ decreases. Even with an extra multi-bank manager, the total area and power for multi-bank management goes down with smaller sub-sorter length. Using 16 sub-sorters of length $N_s = 64$, the area and power reduction can be up to 14\\% and 9\\% compared to the original $N = 1024$ sorter. \\figurename~\\ref{fig:performance}(b) summarizes the implementation results for different sorters.\n\n\\section{Conclusions}\n\nWe present a fast and scalable memristive in-memory sorting that employs a column-skipping algorithm and a multi-bank management. Near-memory circuit with state recording is designed to efficiently skip redundant column reads for improved sorting speed and hardware efficiency. The multi-bank manager enables column-skipping for dataset stored in different banks of memristive memory. Prototype sorters are implemented using 40nm CMOS technology and 1T1R memristive memory. Experimented on a variety of sorting datasets with array length-1024, data precision 32-bit and state recording of 2, the speed, area efficiency and energy efficiency are 4.08$\\times$, 3.14$\\times$ and 3.39$\\times$, respectively, than the state-of-the-art memristive in-memory sorting. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBeginning in December 2019, coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has become one of the most deadly global pandemic in history.  The COVID-19 infections in the US and other nations are still spiking. As of October 30, 2020, the World Health Organization (WHO) has reported  44,888,869 confirmed cases of COVID-19 and 1,178,145 confirmed deaths. The virus has spread to Africa, Americas, Eastern Mediterranean, Europe, South-East Asia and Western Pacific \\cite{weeklyReport}. To prevent further damage to our livelihood, we must control its spread through testing, social distancing, tracking the spread, and developing effective vaccines, drugs, diagnostics, and treatments.\n\nSARS-CoV-2 is a positive-sense single-strand RNA virus that belongs to the Nidovirales order, coronaviridae family and betacoronavirus genus \\cite{of2020species}. To effectively track the virus, testing patients with suspected exposure to COVID-19 and sequencing the strand via PCR (polymerase chain reaction) are important. From sequencing, we can analyze patterns in mutation and predict transmission pathways. Without understanding such pathways, current efforts to find effective medicines and vaccines could become futile because  mutations may change viral genome or lead to resistance. As of October 30, 2020, there are 89627 available sequences with  23763 unique single nucleotide polymorphisms (SNPs) with respect to the first SARS-CoV-2 sequent collected in December 2019 \\cite{wu2020new} according to our mutation tracker \\url{https:\/\/users.math.msu.edu\/users\/weig\/SARS-CoV-2_Mutation_Tracker.html}.\n\nA popular method for understanding mutational trends is to perform phylogenetic analysis, where one clusters mutations to find evolution patterns and transmission pathways. Phylogenetic analysis has been done on the Nidovirales family \\cite{alam2020functional, gong2020sars, forster2020phylogenetic, li2020evolutionary, alam2020functional, kasibhatla2020understanding} to understand genetic evolutionary pathways, protein level changes \\cite{wang2020decoding, wang2020characterizing, chen2020mutations, kasibhatla2020understanding}, large scale variants \\cite{wang2020characterizing, wang2020decoding, wang2020decoding0,worobey2020emergence} and  global trends \\cite{ bai2020comprehensive, toyoshima2020sars, van2020emergence}. Commonly used  techniques for phylogenetic analysis include tree based methods \\cite{page2012space} and $K$-means clustering.  Both methods belong to unsupervised machine learning techniques, where ground truth is unavailable.  These approaches provide valuable information for exploratory research. A main issue with phylogenetic tree analysis is that as the number of samples increase, its computation becomes unpractical, making it unsuitable for large genome datasets. In contrast, $K$-means scales well with sample size increase, but does not perform well when the sample size is too small. Jaccard distance is commonly used to compare genome sequences \\cite{zhou2008approach} because it offers a phylogenetic or topological difference between samples. However, the tradeoff to the Jaccard distance  is that its feature dimension is the same as its  number of samples, suggesting that for a large sample size, the number of features is also large. Since $K$-means clustering relies on computing the distance between the center of the cluster and each sample, having a large feature space can result in expensive computation, large memory requirement, and poor clustering performance. This become a significant problem as the number of SARS-CoV-2 genome isolates from patients has reached 150,000 at this point. There is a pressing need for efficient clustering methods for SARS-CoV-2 genome sequences.   \n\nOne technique to address this challenge is to perform dimensional reduction on the $K$-means input dataset so that the task becomes manageable. Commonly used dimension reduction algorithms focus on two aspects: 1) the pairwise distance structure of all the data samples and 2) preservation of the local distances over the global distance. Techniques such as principal component analysis (PCA) \\cite{jolliffe2016principal}, Sammon mapping \\cite{sammon1969nonlinear}, and multidimensional scaling (MDS) \\cite{cox2008multidimensional} aim to preserve the pairwise distance structure of the dataset. In contrast, the t-distributed stochastic neighbor embedding (t-SNE) \\cite{linderman2019fast, maaten2008visualizing}, uniform manifold approximation and projection (UMAP) \\cite{mcinnes2018umap,becht2019dimensionality}, Laplacian eigenmaps \\cite{belkin2001laplacian}, and LargeVis \\cite{tang2016visualizing}  focus on the preservation of local distances. Among them, PCA, t-SNE, and UMAP are the most frequently used algorithms in the applications of cell biology, bioinformatics, and visualization \\cite{becht2019dimensionality}. \n \nPCA is a popular method used in exploratory studies, aiming to find the directions of the maximum variance in high-dimensional data and projecting them onto a new subspace to obtain low-dimensional feature spaces while preserving most of the variance. The principal components of the new subspace can be interpreted as the directions of the maximum variance, which makes the new feature axes orthogonal to each other. Although PCA is able to cover the maximum variance among features, it may lose some information if one chooses an inappropriate number of principal components. As a linear algorithm, PCA performs poorly on the features with nonlinear relationship. Therefore, in order to present high-dimensional data on low dimensional and nonlinear manifold, some nonlinear dimensional reduction algorithms such as t-SNE and UMAP are employed. T-SNE is a nonlinear method that can preserve the local and global structures of data. There are two main steps in t-SNE.\nFirst, it finds a probability distribution of the high dimensional dataset, where similar data points are given higher probability. Second, it finds a similar probability distribution in the lower dimension space, and the difference between the two distributions is minimized. However, t-SNE computes pairwise conditional probabilities for each pair of samples and involves hyperparameters that are not always easy to tune, which makes it computationally complex. UMAP is a novel manifold learning technique that also captures a nonlinear structure, which is competitive with t-SNE for visualization quality and maintains more of the global structure with superior run-time performance \\cite{mcinnes2018umap}. UMAP is built upon the mathematical work of Belkin and Niyogi on Laplacian eigenmaps, aiming to address the importance of uniform data distributions on manifolds via Riemannian geometry and the metric realization of fuzzy simplicial sets by David Spivak \\cite{spivak2012metric}. Similar to t-SNE, UMAP can optimize the embedded low-dimensional representation  with respect to fuzzy set cross-entropy loss function by using stochastic gradient descent. The embedding is found by finding a low-dimensional projection of the data that closely matches the fuzzy topological structure of the original space. The error between two topological spaces will be minimized by optimizing the spectral layout of data in a low dimensional space. \n\n\nIn this work, we explore efficient computational methods for the SARS-CoV-2 phylogenetic analysis of large volumn of SARS-CoV-2 genome sequences. Specifically, we are interested in developing a dimension-reduction assisted clustering method. To this end, we compare the effectiveness and accuracy of PCA, t-SNE and UMAP for dimension reduction in association with the $K$-means clustering. To quantitatively evaluate the performance, we recast supervised classification problems with labels into a $K$-means clustering problems so that the accuracy of $K$-means clustering can be evaluated. As a result, the accuracy and performance of PCA, t-SNE and UMAP-assisted $K$-means clustering can be compared. By choosing the different dimensional reduction ratios, we examine the performance of these methods in $K$-means settings on four standard datasets. We found that UMAP is the most efficient, robust, reliable, and accurate algorithm. Based on this finding, we applied the UMAP-assisted $K$-means technique to large scale SARS-CoV-2 datasets generated from a Jaccard distance representation and a SNP position-based representation to further analyze its effectiveness, both in terms of speed and scalability. Our results are compared with those in the literature  \\cite{wang2020decoding} to shed new light on SARS-CoV-2 phylogenetics. \n\n\n\n\n\\section{Methods}\n\n\\subsection{Sequence and alignment}\nThe SARS-CoV-2 sequences were obtained from GISAID databank (\\url{www.gisaid.com}). Only complete genome sequences with collection date, high coverage, and without 'NNNNNN' in the sequences were considered. Each sequence was aligned to the reference sequence \\cite{wu2020new} using a multiple sequence alignment (MSA) package Clustal Omega \\cite{sievers2011fast}. A total of 23763 complete SARS-CoV-2 sequences are analyzed in this work.\n\n\\subsection{SNP position based features}\nLet $N$ be the number of SNP profiles with respect to the SARS-CoV-2 reference genome sequence, and let $M$ be the number of unique mutation sites. Denote $V_i$ as the position based feature of the $i$th SNP profile.\n\\begin{equation}\nV_i = [v_i^1, v_i^2, ..., v_i^M], \\quad i = 1,2,..., N\n\\end{equation}\nis a $1\\times M$ vector. Here\n\\begin{equation}\nv_i^j = \\begin{cases}\n1, & \\text{mutation site} \\\\\n0, & \\text{otherwise}. \\\\\n\\end{cases}\n\\end{equation}\nWe compile this into an $N\\times M$ position based feature,\n\\begin{equation}\nS(i,j) = v_i^j\n\\end{equation}\nwhere each row represents a sample. Note that $S(i,j)$ is a binary representation of the position and is sparse.\n\n\\subsection{Jaccard based representation}\\label{sec:Jaccard}\n\nThe Jaccard distance measures the dissimilarity between two sets. It is widely used in the phylogenetic studies of SNP profiles. In this work, we utilize Jaccard distance to compare SNP profiles of SARS-CoV-2 genome isolates.\n\nLet $A$ and $B$ be two sets. Consider the Jaccard index between $A$ and $B$, denoted $J(A,B)$,  as the cardinality of the intersection divided by the cardinality of the union\n\\begin{equation}\n\tJ(A,B) = \\frac{\\big| A\\cap B\\big|}{\\big|A \\cup B \\big|} = \\frac{\\big| A\\cap B\\big|}{\\big|A\\big| + \\big|B\\big| - \\big|A \\cap B \\big|}. \n\\end{equation}\nThe Jaccard distance between the two sets is defined by subtracting the Jaccard index from 1:\n\\begin{equation}\n\td_J(A,B) = 1- J(A,B) = \\frac{\\big|A \\cup B \\big| - \\big|A \\cap B \\big|}{\\big|A \\cup B \\big|}\n\\end{equation}\n\nWe assume there are $N$ SNP profiles or genome isolates that have been aligned to the reference SARS-CoV-2 genome. Let $S_i$, $i=1,...,N$, be the set with the position of the mutation of the $i$th sample. The Jaccard distance between two sets $S_i$ and $S_j$ is given by $d_J(S_i, S_j)$. Taking the pairwise distance between all the samples, we can construct the Jaccard based representation, resulting in an $N\\times N$ distance  matrix $D$\n\\begin{equation}\n\tD(i,j) = d_J(S_i, S_j)\n\\end{equation}\nThis distance defines  a metric over the collections of all finite sets \\cite{levandowsky1971distance}.\n\n\n\t%\n\n\n\n\n\n\t%\n\n\\subsection{$K$-means clustering}\n$K$-means clustering is one of the most popular unsupervised learning methods in machine learning, where it aims to cluster or partition a data $\\{x_1, ..., x_N\\}$, $x_i \\in \\mathbb{R}^M$ into $k$ clusters, $\\{C_1, ..., C_k\\}$, $k \\le N$.\n\n$K$-means clustering begins with selecting $k$ points as $k$ cluster centers, or centroids. Then, each point in the dataset is assigned to the nearest centroid. The centroids are then updated by minimizing the within-cluster sum of squares (WCSS), which is defined as \n\\begin{equation}\n\\sum_{i=1}^k \\sum_{x_i \\in C_k} \\|x_i - \\mu_k\\|_2^2.\n\\end{equation}\nHere, $\\|\\cdot\\|_2$ denotes the $l_2$ norm and $\\mu_k$ is the average of the data point in cluster $k$\n\\begin{equation}\n\\displaystyle \\mu_k = \\frac{1}{|C_k|} \\sum_{x_i \\in C_k} x_i.\n\\end{equation}\n\nThis method, however, only finds the optimal centroid, given a fixed number of clusters $k$. In applications, we are interested in finding the optimal number of clusters as well. In order to obtain the best $k$ clusters, elbow method was used. The optimal number of clusters can be determined via the elbow method by plotting the WCSS against the number of clusters, and choosing the inflection point position as the optimal number of clusters.\n\n\\subsection{Principal component analysis}\nPrincipal component analysis (PCA) is one the most commonly used  dimensional reduction techniques for the exploratory analysis of high-dimensional data \\cite{jolliffe2016principal}. Unlike other methods, there is no need for any assumptions in the data. Therefore, it is a useful method for new data, such as  SARS-CoV-2 SNPs data. PCA is conducted by obtaining one component or vector at a time. The first component, termed the principal component, is the direction that maximizes the variance. The subsequent components are orthogonal to earlier ones.\n\nLet $\\{x_i\\}_{i=1}^N$ be the input dataset, with $N$ being the number of samples or data points. For each $x_i$, let $x_i \\in \\mathbb{R}^M$, where $M$ is the number of features or data dimension. Then, we can cast the data as a matrix $X \\in \\mathbb{R}^{N \\times M}$. PCA seeks to find a linear combination of the columns of $X$ with maximum variance.\n\\begin{equation}\n\\sum_{j=1}^n a_jx_j = Xa,\n\\end{equation}\nwhere $a_1, a_2, ..., a_n$ are constants. The variance of this linear combination is defined as \n\\begin{equation}\n{\\rm var}(Xa) = a^TSa,\n\\end{equation}\nwhere $S$ is the covariance matrix for the dataset. Note that we compute the eigenvalue of the covariance matrix. The maximum variance can be computed iteratively using Rayleigh's quotient\n\\begin{equation}\na_{(1)} = \\arg \\max_a \\frac{a^TX^TXa}{a^Ta}.\n\\end{equation}\nThe subsequent components can be computed by maximizing the variance of\n\\begin{equation}\n\\hat{X}_k = X - \\sum_{j=1}^{k-1} Xa_ja_{j}^T\n\\end{equation}\nwhere $k$ represents the $k$th principal component. Here, $k-1$ principal components are subtracted from the original matrix $X$. Therefore, the complexity of the method scales linearly with the number of components one seeks to find. In applications, we hope that the first few components give rise to a good PCA representation of the original data matrix $X$. \n\n\n\\subsection{t-SNE}\nThe t-distributed stochastic neighbor embedding (t-SNE) is a nonlinear dimensional reduction algorithm that is well suited for reducing high dimensional data into the two- or three-dimensional space. There are two main stages in t-SNE. First, it constructs a probability distribution over pairs of data such that a pair of near data points is assigned with a high probability, while a pair of farther away points is given a low probability. Second, t-SNE defines a probability distribution in the embedded space that is similar to that in the original high-dimensional space, and aims to minimize the Kullback-Leibler (KL) divergence between them \\cite{linderman2019fast}.\n\nLet $\\{x_1, x_2, ..., x_N | x_i \\in \\mathbb{R}^M \\}$ be a high dimensional input dataset. Our goal is to find an optimal  low dimensional representation $\\{y_1, ..., y_N | y_i \\in \\mathbb{R}^k\\}$, such that $k << M$.\nThe first step in t-SNE is to compute the pairwise distribution between $x_i$ and $x_j$, defined as $p_{ij}$. However, we find the conditional probability of $x_j$, given $x_i$:\n\\begin{equation}\\label{pdist} \np_{j|i} = \\frac{\\exp(-\\|x_i - x_j\\|^2\/2\\sigma_i^2)}{\\sum_{m\\ne i}\\exp(-\\|x_i - x_m\\|^2\/2\\sigma_i^2)}, \\quad i \\ne j,\n\\end{equation}\nsetting $p_{i|i} = 0$, and the denominator normalizes the probability. Here, $\\sigma_i$ is the predefined hyperparameter called perplexity.  A smaller $\\sigma_i$ is used for a denser dataset. Notice that this conditional probability is symmetric when the perplexity is fixed, i.e. $p_{i|j} = p_{j|i}$. Then, define the pairwise probability as \n\\begin{equation}\np_{ij} = \\frac{p_{j|i} + p_{ij}}{2N}.\n\\end{equation}\n\nIn the second step, we learn a $k$-dimensional embedding $\\{y_1, ..., y_N | y_i \\in \\mathbb{R}^k\\}$. To  this end, t-SNE calculates a similar probability distribution $q_{ij}$ defined as\n\\begin{equation}\\label{qdist} \nq_{ij} = \\frac{\\frac{1}{1+\\|y_i - y_j\\|^2}}{\\sum_{m}\\sum_{l\\ne m}\\frac{1}{1+\\|y_m - y_l\\|^2}}, \\quad i \\ne j\n\\end{equation}\nand setting $q_{ii} = 0$. Finally, the low dimensional embedding $\\{y_1, ..., y_N | y_i \\in \\mathbb{R}^k\\}$ is found by minimizing the KL-divergence via a standard gradient descent method\n\\begin{equation}\n{\\rm KL}(P|Q) = \\sum_{i, j} p_{ij}\\log\\frac{p_{ij}}{q_{ij}},\n\\end{equation}\nwhere $P$ and $Q$ are the distributions for  $p_{ij}$ and  $q_{ij}$, respectively. Note that the probability distributions in Eqs. (\\ref{pdist}) and  (\\ref{qdist}) can be replaced by using many other delta sequence kernel of positive type  \\cite{wei2000wavelets}. \n\n\\subsection{UMAP}\nUniform manifold approximation and projection (UMAP) is a nonlinear dimensional reduction method, utilizing three assumptions: the data is uniformly distributed on Riemannian manifold, Riemannian metric is locally constant, and the manifold if locally connected. Unlike t-SNE which utilizes probabilistic model, UMAP is a graph-based algorithm. Its essentially idea is to create a predefined $k$-dimensional weighted UMAP graph representation of each of the original high-dimensional data point such that the edge-wise cross-entropy between the weighted graph and the original data is minimized. Finally, the $k$-dimensional eigenvectors of the UMAP graph are used to represent each of the original data point. \nIn this section, a computational view of UMAP is presented. For a more theoretical account, the reader is referred to Ref. \\cite{mcinnes2018umap}.\n\nSimilar to t-SNE, UMAP considers the input data $X = \\{x_1, x_2, ..., x_N\\},$   $x_i \\in \\mathbb{R}^M$ and look for  an optimal  low dimensional representation $\\{y_1, ..., y_N | y_i \\in \\mathbb{R}^k\\}$, such that $k < M$. The first stage is the construction of  weighted $k$-neighbor graphs. Let  define a metric $d: X\\times X \\to \\mathbb{R}^+$. Let $k << M$ be a hyperparemeter, and compute the $k$-nearest neighbors of each $x_i$ under a given metric $d$. For each $x_i$, let\n\\begin{equation}\n\\rho_i = \\min\\{d(x_i, x_j)| 1 \\le j \\le k, d(x_i, x_j) > 0\\}\n\\end{equation}\nwhere $\\sigma_i$ is defined via\n\\begin{equation}\n\\sum_{j=1}^k \\exp\\left(\\frac{-\\max(0, d(x_i, x_j) - \\rho_i)}{\\sigma_i}\\right) = \\log_2 k.\n\\end{equation}\nOne chooses $\\rho_i$ to ensure at least one data point is connected to $x_i$ and having edge weight of 1, and set  $\\sigma_i$ as a length scale parameter.\nOne defines a weighted directed graph $\\bar{G} = (V, E, \\omega)$, where $V$ is the set of vertices (in this case, the data $X$),  $E$ is the set of edges $E = \\{ (x_i, x_j)| 1 \\le h \\le k, 1 \\le i \\le N\\}$, and   $\\omega$ is the weight for edges\n\\begin{equation}\n\\omega(x_i, x_j) = \\exp\\left( \\frac{-\\max(0, d(x_i, x_j) - \\rho_i)}{\\sigma_i}\\right).\n\\end{equation}\nUMAP tries to define an undirected weighted graph $G$ from  directed graph $\\bar{G}$ via symmetrization. Let $A$ be the adjacency matrix of the graph $\\bar{G}$.   A symmetric matrix can be obtained\n\\begin{equation}\nB = A + A^T - A\\otimes A^T,\n\\end{equation}\nwhere $T$ is the transpose and $\\otimes$ denotes the Hadamard product. Then, the undirected weighted Laplacian $G$ (the UMAP graph) is defined by its adjacency matrix $B$. \n\nIn its realization, UMAP evolves an equivalent weighted graph $H$ with a set of points $\\{y_i\\}_{i=1,\\cdots,N}$, utilizing attractive and repulsive forces. The attractive and repulsive forces at coordinate $y_i$ and $y_j$ are given by\n\\begin{align}\n& \\frac{-2ab \\|y_i - y_j\\|_2^{2(b-1)}}{1 + \\|y_i - y_j\\|_2^2} w(x_i, x_j) (y_i - y_j), ~~{\\rm and} \\\\\n& \\frac{2b}{(\\epsilon + \\|y_i - y_j\\|_2^2)(1+a\\|y_i - y_j\\|_2^{2b})} (1-w(x_i, x_j)) (y_i - y_j)\n\\end{align}\nwhere $a,b$ are hyperparemeters, and $\\epsilon$ is taken to be a small value such that the denominator does not become 0. The goal is to find the optimal low-dimensional coordinates $\\{y_i\\}_{i=1}^{N}$, $y_i \\in \\mathbb{R}^k$, that minimizes the edge-wise cross entropy with the original data at each point. The evolution of the UMAP graph Laplacian $G$ can be regarded as a discrete approximation of the Laplace-Beltrami operator on a manifold defined by the data \\cite{chen2019evolutionary}. Implementation and further detail of UMAP can be found in Ref. \\cite{mcinnes2018umap}. \n\nUMAP may not work well if the data points is non-uniform. If part of the data points have $k$ important neighbors while other part of the data points have $k'>>k$ important neighbors, the $k$-dimensional UMAP will not work efficiently. Currently, there is no algorithm to automatically determine the critic minimal $k_{\\rm min}$ for a given dataset. Additionally, weights $w(x_i,x_j)$ and force terms can be replaced by other functions that are easier to evaluate \\cite{wei2000wavelets}. The metric $d$ can be selected as Euclidean distance, Manhattan distance, Minkowski distance, and Chebyshev distance, depending on applications. \n\n\\section{Validation}\n $K$-means clustering is one of the unsupervised learning algorithms, suggesting that neither the accuracy nor the root-mean-square error can be calculated to evaluate the performance of the $K$-means clustering explicitly. Additionally, $K$-means clustering can be problematic for high-dimensional large datasets. Dimension-reduced $K$-means clustering is an efficient approach.  \nTo evaluate its accuracy and performance, we convert supervised classification problems with known salutations into dimension-reduced $K$-means clustering problems. In doing so,  we apply the $K$-means clustering to the classification dataset by setting the number of clusters equals to the number of the real categories. Next, in each cluster, we will take the data with the dominant label as the test for all samples and then calculate the $K$-means clustering accuracy for the whole dataset.\n \n\\subsection{Validation data}\\label{sec:Validation}\nIn this work, we will consider the following classification datasets to test the performance of the clustering methods: Coil 20, Facebook large page-page network, MNIST, and Jaccard distanced-based MNIST. \n\\begin{itemize}\n\t\\item \\textbf{Coil 20}: Coil 20 \\cite{Coil20} is a dataset with 1440 gray scale images, consisting of 20 different objects, each with 72 orientation. Each image is of size $128\\times 128$, which was treated as a 16384 dimensional vector for dimensional reduction\n\t\n\t\\item \\textbf{Facebook Network}: Facebook large page-page network \\cite{rozemberczki2019multiscale} is a page-page webgraph of verified Facebook sites. Each node represents a facebook page, and the links are the mutual links between sites. This is a binary dataset with 22,470 nodes; hence the sample size and feature size are both 22,470. Jaccard distance was computed between each nodes for the feature space.\n\t\n\t\\item \\textbf{MNIST}: MNIST \\cite{lecun1998gradient} is a hand written digit dataset. Each image is a grey scale of size $28\\times 28$, which was treated as a 784 dimensional vector for the feature space, each with a integer value in [0, 255]. Standard normalization was used before performing dimensional reduction. There are 70,000 sample, with 10 different labels.\n\t\n\t\\item \\textbf{Jaccard distanced-based MNIST}: The above dataset was converted to a Jaccard distance-based dataset. This is to simulate position based mutational dataset, where 1 indicates a mutation in a particular position. Jaccard distance was used to construct the feature space, hence for each sample, the feature size is 70,000. This dataset can be viewed as an additional validation on our Jaccard distance representation. \n\t\n\\end{itemize}\n\n\\subsection{Validation results}\nIn the present work, we implement three popular dimensional reduction methods, PCA, UMAP, and t-SNE, for the dimension reduction and compare their performance in $K$-means clustering. For a uniform comparison, we reduce the dimensions of the samples by a set of ratios.  The minimum between the number of features and the number of samples was taken as base of the reduction. For the Coil 20 dataset, since the numbers of samples and features were 1440 and 16384, respectively,  dimension-reductions were based on 1440. \nFor the Facebook Network,  since the numbers of samples and features were both \n22,470, dimension-reductions were based on 22,470.\nFor the MNIST dataset,  since the numbers of samples and features were respectively 70,000 and 784, dimension-reductions were based on 784. Finally,\n for the Jaccard distanced-based MNIST dataset, since the numbers of samples and features were both 70,000, dimension-reductions were based on 70,000. \nNote that for the Jaccard distanced-based MNIST data, more aggressive ratios were used because the original feature size is huge, i.e., 70,000. The standard ratios of 2, 4, and 8, etc do not sufficiently reduce the dimension  for effective $K$-means computation. For the purpose of visualization, two-dimensional reduction algorithms are applied to each reduction scheme. \nIn order to validate PCA, UMAP, and t-SNE assisted $K$-means clustering, we observed their performance using  labeled datasets. $K$-nearest neighbors ($K$-NN) was used to find the baseline of the reduction, which reveals how much information can be preserved in the feature after applying a dimensional reduction algorithm. For $k$-NN, 10 fold cross-validation was performed.\n\nNotably, $K$-means clustering is an unsupervised learning algorithm, which does not have labels to evaluate the clustering performance explicitly. However, we can assess the $K$-means clustering accuracy via  labeled datasets that has ground truth. In doing so, we choose the number of $K$ as the original number of classes. Then, we can compared the $k$-means clustering results  with the ground truth.  Therefore, the accuracy can reveal the performance of the proposed dimension-reduction-assisted ($k$-means) clustering method. For the classification problem, we assume the training set is $\\{(\\mathbf{x}_i, y_i) | \\mathbf{x}_i\\in \\mathbb{R}^m, y_i \\in\\mathbb{Z} \\}_{i=1}^{n}$ with the $|\\{y_i\\}_{i=1}^{n}|=k$. Here $n$, $m$, and $k$ represent the number of samples, the number of features $\\{\\mathbf{x}_i\\}$, and the number of labels $\\{y_i\\}$, respectively. We set the number of clusters equals to the number of labels $k$. After applying the $K$-means clustering algorithm, we  get $k$ different clusters $\\{\\mathbf{c}_j\\}^{k}_{j=1}$. In each cluster, we define the predictor of the $K$-means clustering in the cluster $\\mathbf{c}_j$ to be :\n\\begin{equation}\n\\hat{\\mathbf{y}}(\\mathbf{\\mathbf{c}}_j) = \\max\\{ F_j(y_1), \\cdots, F_j(y_k) \\},\n\\end{equation}\nwhere $F_j(y_i), \\cdots, F_j(y_k)$ are the appearance frequencies of each label in the cluster $\\mathbf{c}_j$. Then the clustering accuracy can be defined as:\n\\begin{equation}\n\\text{Accuracy} = \\frac{\\sum_{i} 1_{ \\{ y_i = \\hat{y}_i \\} } } {n},\n\\end{equation}\nwhere $\\{\\hat{y}_i\\}$ are predicted labels. \nMoreover, other evaluation metrics such as precision, recall, and receiver operating characteristic (ROC) can also be defined accordingly.\n\n\\subsubsection{Coil 20}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{Coil20_compressed.pdf}\n\t\\caption{Comparison of different dimensional reduction algorithms on Coil 20 dataset. Total 20 different labels are in the Coil 20 dataset, and we use the ground truth label to color each data points. (a) Feature size is reduced to dimension 2 by PCA. (b) Feature size is reduced to dimension 2 by t-SNE. (c) Feature size is reduced to dimension 2 by UMAP. }\n\t\\label{fig:Coil20}\n\\end{figure}\n\n\n\\autoref{fig:Coil20} shows the performance of PCA-assisted, UMAP-assisted and t-SNE-assisted clustering of the Coil 20 dataset. For each case, the dataset were reduced to dimension 2 using default parameters, and the plots were colored with the ground truth of the Coil 20 dataset. It can be seen that PCA does not present good  clustering, whereas UMAP and t-SNE show very good clusters. \n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{Accuracy of $k$-NN of the Coil 20 dataset without applying any reduction algorithms, as well as the accuracy of $k$-NN assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Coil 20 dataset are 1440, 16384, and 20, respectively. }\n\t\\begin{tabular}{c|cccccc}\\hline \n\t\tDataset & \\tabincell{c}{$k$-NN accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{10}*{\\shortstack{Coil 20 \\\\(1440,16384,20)}} & \\multirow{10}{*}{0.956}\n\t\t& 720 (1\/2) & 0.955  & 0.668 & 0.850 \\\\\n\t\t&  & 360 (1\/4) & 0.957  & 0.861 & 0.889 \\\\\n\t\t&  & 180 (1\/8) & 0.973  & 0.867 & 0.881 \\\\\n\t\t&  & 90 (1\/16)& 0.977 & 0.860 & 0.885 \\\\\n\t\t&  & 45 (1\/32) & 0.980  & 0.861 & 0.875 \\\\\n\t\t&  & 22 (1\/64) & 0.985  & 0.868 & 0.743 \\\\\n\t\t&  & 14 (1\/100) & 0.730  & 0.851 & 0.878 \\\\\n\t\t&  & 7 (1\/200) & 0.985  & 0.870 & 0.845 \\\\\n\t\t&  & 3          & 0.850  & 0.863 & 0.959 \\\\\n\t\t&  & 2          & 0.730  & 0.853 & 0.948 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:Coil20 KNN ACC}\n\\end{table}\n\n\\autoref{tab:Coil20 KNN ACC} shows the accuracy of $k$-NN clustering of the Coil 20 dataset assisted by PCA, t-SNE, and UMAP with different dimensional reduction radio. The Coil 20 dataset   has 1,440 samples, 16,384 features, and 20 different labels. For PCA, the sklearn implementation on python was used with standard parameters. Note that for all methods, dimensions were reduced to 3 and 2 for a comparison. For t-SNE, Multicore-TSNE \\cite{Ulyanov2016} was used because it offers up to 8 core processor, which is not available in the sklearn implementation, and it is the fastest performing t-SNE algorithm. For UMAP, we used standard parameters \\cite{mcinnes2018umap}. It can be seen that when we reduce the dimension to 3, t-SNE performs best. Moreover, when the dimensional reduction ratio is 1\/100, PCA and UMAP also perform well. Notably, the $k$-NN accuracy for the data without applying any dimensional reduction algorithm is 0.956, indicating that UMAP does not provide the best clustering performance on the Coil 20 dataset. However, PCA and t-SNE will preserve the information of the original data with a dimensional reduction ratio larger than 1\/100, and t-SNE even performs better for dimensional three on the Coil 20 dataset. \n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{Accuracy of $K$-means clustering of the Coil 20 dataset without applying any reduction algorithms, as well as the accuracy of $K$-means assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Coil 20 dataset are 1440, 16384, and 20, respectively. }\n\t\\begin{tabular}{c|cccccc} \\hline\n\t\tDataset & \\tabincell{c}{$K$-means accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{11}*{\\shortstack{Coil 20 \\\\(1440,16384,20)}} & \\multirow{11}{*}{0.626} \n\t\t \n\t\t & 720 (1\/2)  & 0.64   & 0.301 & 0.798     \\\\\n\t\t& & 360 (1\/4)  & 0.678  & 0.800 & 0.718     \\\\\n\t\t& & 180 (1\/8)  & 0.633  & 0.822 & 0.648     \\\\\n\t\t& & 90 (1\/16) & 0.642  & 0.799 & 0.681     \\\\\n\t    & & 45 (1\/32)  & 0.666  & 0.800 & 0.615     \\\\\n\t\t& & 22 (1\/64)  & 0.673  & 0.819 & 0.151     \\\\\n\t\t& & 14 (1\/100) & 0.631  & 0.817 & 0.154     \\\\\n\t\t& & 7 (1\/200)  & 0.591  & 0.819 & 0.360     \\\\\n\t    & & 3           & 0.561  & 0.800 & 0.780  \\\\\n\t\t& & 2           & 0.537  & 0.801 & 0.828  \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:Coil20 KMean ACC}\n\\end{table}\n\\autoref{tab:Coil20 KMean ACC} describes the accuracy of $K$-means clustering of Coil 20 assisted by PCA, UMAP, and t-SNE with different dimensional reduction ratio. For consistency, we use the same set of standard parameters as $k$-NN. For the Coil 20 dataset, the accuracy of $K$-means clustering assisted by UMAP has the best performance. When the reduced dimension is 2048 (ratio 1\/8), UMAP will result in a relatively high $K$-means accuracy (0.822). Moreover, although PCA performs best on $k$-NN accuracy, it performs poorly on the $K$-means accuracy, indicating that PCA is not a suitable dimensional reduction algorithm on the Coil 20 dataset. Furthermore, the highest accuracy of $K$-means clustering is 0.828, which is calculated from the t-SNE-assisted algorithm. However, the t-SNE-assisted accuracy under different reduction ratio changes dramatically. When the ratio is 1\/64, the t-SNE-assisted accuracy is only 0.151, indicating that t-SNE is sensitive to the hyper-parameters settings. In contrast, the performance of UMAP is highly stable under all dimension-reduction ratios. \n\nNote that dimension-reduced $k$-means clustering methods outperform the original  $k$-means clustering. Therefore, the proposed dimension-reduced $k$-means clustering methods not only improve the $k$-means clustering efficiency, but also achieve better accuracy.  \n\n\n\n\n\n\\subsubsection{Facebook Network}\n\\begin{figure}[ht]\n   \n    \\centering\n\t\\includegraphics[width = \\textwidth]{Facebook_compressed.pdf}\n\t\\caption{Comparison of different dimensional reduction algorithms on the Facebook Network dataset. Total 4 different labels are in the Facebook Network dataset, and we use the ground truth label to color each data points. (a) Feature size is reduced to dimension 2 by PCA. (b) Feature size is reduced to dimension 2 by t-SNE. (c) Feature size is reduced to dimension 2 by UMAP. }\n\t\\label{fig:Facebook}\n\\end{figure}\n\\autoref{fig:Facebook} shows the visualization performance of PCA-assisted, UMAP-assisted, and t-SNE-assisted clustering of the Facebook Network. For each case, the dataset was reduced to dimension 2  using default parameters, and the plots were colored with the ground truth of the Facebook Network. \\autoref{fig:Facebook} shows that the PCA-based data is located distributively, while the t-SNE- and UMAP-based data show clusters. \n\n\n\\autoref{tab:Facebook KNN ACC} shows the accuracy of $k$-NN clustering of the Facebook Network assisted by PCA, t-SNE, and UMAP with different dimensional reduction radio. The Facebook Network dataset has 22,470 samples with 4 different labels, and the feature size of the Facebook Network is also 22,470. For each algorithm, we use the same settings as the Coil 20 dataset. Without applying any dimensional reduction method, The Facebook Network has 0.755 $k$-NN accuracy. The reduced feature from PCA has the best $k$-NN performance when the reduction ratio is 1\/2. UMAP has a better performance compared to PCA and t-SNE when the reduction ratio is smaller than 1\/16. \n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Accuracy of $k$-NN of the Facebook Network without applying any reduction algorithms, as well as the accuracy of $k$-NN assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Facebook Network are 22470, 22470, and 4, respectively. }\n\t\\begin{tabular}{c|cccccc}\\hline \n        Dataset & \\tabincell{c}{$k$-NN accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{12}*{\\shortstack{Facebook Network \\\\(22470, 22470, 4)}} & \\multirow{13}{*}{0.755}\n\t\t \n\t\t  & 11235 (1\/2) & 0.756 & 0.360 & 0.307 \\\\\n\t\t&  & 5617 (1\/4)  & 0.755 & 0.669 & 0.316 \\\\\n\t\t&  & 2808 (1\/8)  & 0.754 & 0.754 & 0.355 \\\\\n\t\t&  & 1404 (1\/16) & 0.751 & 0.816 & 0.707 \\\\\n\t\t&  & 702 (1\/32)  & 0.751 & 0.814 & 0.669 \\\\\n\t\t&  & 351 (1\/64)  & 0.746 & 0.815 & 0.690 \\\\\n\t\t&  & 224 (1\/100) & 0.733 & 0.814 & 0.676 \\\\\n\t\t&  & 112 (1\/200) & 0.721 & 0.819 & 0.633 \\\\\n\t\t&  & 44 (1\/500)  & 0.714 & 0.816 & 0.709 \\\\\n\t\t&  & 22 (1\/1000) & 0.690 & 0.815 & 0.643 \\\\\n\t\t&  & 3           & 0.552 & 0.801 & 0.741 \\\\\n\t\t&  & 2           & 0.501 & 0.786 & 0.732 \\\\\\hline\n\t\\end{tabular}\n\t\\label{tab:Facebook KNN ACC}\n\\end{table}\n\n\n\\autoref{tab:Facebook KMean ACC} describes the accuracy of $K$-means clustering of the Facebook Network assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. PCA, UMAP, and t-SNE all have very poor performance, which may be caused by the smaller number of labels. The highest accuracy 0.427 is observed in the t-SNE-assistant algorithm with dimension 2.  \n\n\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Accuracy of $K$-means clustering of the Facebook Network without applying any reduction algorithms, as well as the accuracy of $K$-means assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Facebook Network are 22470, 22470, and 4, respectively.}\n\t\\begin{tabular}{c|cccccc} \\hline\n        Dataset & \\tabincell{c}{$K$-means accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{12}*{\\shortstack{Facebook Network \\\\(22470, 22470, 4)}} &\n\t\t\\multirow{12}{*}{0.374}\n\t\t \n\t\t& 11235 (1\/2) & 0.331 & 0.306 & 0.306    \\\\\n\t\t& & 5617 (1\/4)  & 0.331 & 0.307 & 0.299   \\\\\n\t\t& & 2808 (1\/8)  & 0.331 & 0.411 & 0.314    \\\\\n\t\t& & 1404 (1\/16) & 0.331 & 0.397 & 0.313    \\\\\n\t\t& & 702 (1\/32)  & 0.331 & 0.401 & 0.306    \\\\\n\t\t& & 351 (1\/64)  & 0.331 & 0.400 & 0.308    \\\\\n\t\t& & 224 (1\/100) & 0.331 & 0.400 & 0.327    \\\\\n\t\t& & 112 (1\/200) & 0.331 & 0.400 & 0.306    \\\\\n\t\t& & 44 (1\/500)  & 0.331 & 0.400 & 0.313    \\\\\n\t\t& & 22 (1\/1000) & 0.331 & 0.401 & 0.306    \\\\\n\t\t& & 3           & 0.332 & 0.351 & 0.344 \\\\\n\t\t& & 2           & 0.358 & 0.345 & 0.427 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:Facebook KMean ACC}\n\\end{table}\n\nSimilar to the last case, UMAP-based and t-SNE-based dimension-reduced $k$-means clustering methods outperform the original $k$-means clustering with the full feature dimension. Therefore, it is useful to carry out dimension reduction before $k$-means clustering  for large datasets. \n\n  \n\n\\subsubsection{MNIST}\n\n\\begin{figure}[ht]\n   \n    \\centering\n\t\\includegraphics[width = \\textwidth]{MNIST_compressed.pdf}\n\t\\caption{Comparison of different dimensional reduction algorithms on the MNIST dataset. Total 10 different labels are in the MNIST dataset, and we use the ground truth label to color each data points. (a) Feature size is reduced to dimension 2 by PCA. (b) Feature size is reduced to dimension 2 by t-SNE. (c) Feature size is reduced to dimension 2 by UMAP. }\n\t\\label{fig:MNIST}\n\\end{figure}\n\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Accuracy of $k$-NN of the MNIST dataset without applying any reduction algorithms, as well as the accuracy of $k$-NN assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the MNIST dataset are 70000, 784, and 10, respectively.}\n\t\\begin{tabular}{c|cccccc}\\hline \n        Dataset & \\tabincell{c}{$k$-NN accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{9}*{\\shortstack{MNIST \\\\(70000, 784, 10)}}   & \\multirow{9}{*}{0.948}  \n\t\t \n\t\t& 392 (1\/2) & 0.951 & 0.937 & 0.696 \\\\\n\t\t& & 196 (1\/4) & 0.956 & 0.938 & 0.846 \\\\\n\t\t& & 98 (1\/8)  & 0.960 & 0.937 & 0.893\\\\\n\t\t& & 49 (1\/16) & 0.961 & 0.937 & 0.886 \\\\\n\t\t& & 24 (1\/32) & 0.953 & 0.937 & 0.842 \\\\\n\t\t& & 12 (1\/64) & 0.926 & 0.937 & 0.676 \\\\\n\t\t& & 7 (1\/100) & 0.846 & 0.936 & 0.940 \\\\\n\t\t& & 3         & 0.513 & 0.929 & 0.938 \\\\\n\t\t& & 2         & 0.323 & 0.919 & 0.928 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:MNIST KNN ACC}\n\\end{table}\n\n\\autoref{fig:MNIST} shows the performance of PCA-assisted, UMAP-assisted and t-SNE-assisted clustering of the MNIST dataset. The sample size of the MNIST dataset is 70000, which has 784 features with 10 different digit labels. For each case, the dataset was reduced to dimension 2   using default parameters, and the plots were colored with the ground truth of the MNIST dataset. In \\autoref{fig:MNIST}, by applying the UMAP algorithm, the clear clusters can be detected for the MNIST dataset. The t-SNE offers a reasonable clustering at  dimension 2 too. However, the PCA does not provide a good clustering.  \n\n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{Accuracy of $K$-means clustering of the MNIST dataset without applying any reduction algorithms, as well as the accuracy of $K$-means assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the MNIST dataset are 70000, 784, and 10, respectively.}\n\t\\begin{tabular}{c|cccccc} \\hline\n        Dataset & \\tabincell{c}{$K$-means accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{9}*{\\shortstack{MNIST \\\\(70000, 784, 10)}} & \\multirow{9}{*}{0.494}\n\t\t \n\t    & 392 (1\/2) & 0.487 & 0.665 & 0.122   \\\\\n\t\t& & 196 (1\/4) & 0.492 & 0.667 & 0.113   \\\\\n\t\t& & 98 (1\/8)  & 0.498 & 0.673 & 0.113   \\\\\n\t\t& & 49 (1\/16) & 0.496 & 0.718 & 0.113   \\\\\n\t\t& & 24 (1\/32) & 0.501 & 0.697 & 0.114   \\\\\n\t\t& & 12 (1\/64) & 0.489 & 0.682 & 0.138   \\\\\n\t\t& & 7 (1\/100) & 0.464 & 0.677 & 0.740   \\\\\n\t\t& & 3         & 0.365 & 0.727 & 0.537 \\\\\n\t\t& & 2         & 0.300 & 0.712 & 0.593 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:MNIST KMean ACC}\n\\end{table}\n\n\\autoref{tab:MNIST KNN ACC} shows the accuracy of $k$-NN clustering of the MNIST dataset assisted by PCA, t-SNE, and UMAP with different dimensional reduction radios. For each algorithm, we use the same settings as the Coil 20 dataset. Without applying any dimensional reduction algorithms, the accuracy of $k$-NN is 0.948. By applying PCA\/UMAP with the reduction ratio greater than 1\/64, the accuracy of PCA\/UMAP-assisted $k$-NN is at the same level without using any dimensional reduction algorithm. However, in contract with UMAP and t-SNE, when the reduced dimension is 2 or 3, PCA performs poorly. This indicates that the PCA may  not be suitable for dimension-reduction for  datasets with a large sample size. \n\n\n\\autoref{tab:MNIST KMean ACC} describes the accuracy of $K$-means clustering of the MNIST dataset assisted by PCA, UMAP, and t-SNE with different dimensional reduction ratios.  By applying PCA, the accuracy of $K$-means is around 0.45. \nThe t-SNE method performance is quite unstable, from very poor (0113) to the best (0.740), and to a relatively low value of 0.593. In contrast, we can see a stable and improved accuracy from using UMAP at various reduction ratios, indicating that the reduced feature generated by UMAP can better represent the clustering properties of the MNIST dataset compared to the PCA and t-SNE. \n\nAs observed early, the present UMAP and t-SNE-assisted $k$-means clustering methods also significantly out-perform the original $k$-means clustering for this dataset. \n\n\n\\subsubsection{Jaccard distanced-based MNIST}\n\n\\begin{figure}[ht]\n\t\\centering\n\n\t\\includegraphics[width = \\textwidth]{BinaryMNIST_compressed.pdf}\n\t\\caption{Comparison of different dimensional reduction algorithms on the Jaccard distanced-based MNIST dataset. Total 10 different labels are in the Jaccard distanced-based MNIST dataset, and we use the ground truth label to color each data points. (a) Feature size is reduced to dimension 2 by PCA. (b) Feature size is reduced to dimension 2 by t-SNE. (c) Feature size is reduced to dimension 2 by UMAP. }\n\t\\label{fig:Jaccard distanced-based MNIST}\n\\end{figure}\n\n\nOut last validation dataset is Jaccard distanced-based MNIST. This dataset can be treated as a test on the Jaccard distance-based data representation. \n\\autoref{fig:Jaccard distanced-based MNIST} shows the performance of PCA-assisted, UMAP-assisted, and t-SNE-assisted clustering of the Jaccard distanced-based MNIST dataset. The dataset was reduced to dimension 2  using default parameters for visualization, and the plots were colored with the ground truth of the Jaccard distanced-based MNIST dataset. From \\autoref{fig:Jaccard distanced-based MNIST}, we can see that UMAP provides the clearest clusters compared to PCA and t-SNE when the dimension is reduced to 2. The performance of t-SNE is reasonable while PCA does not give a good clustering.   \n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Accuracy of $k$-NN of the Jaccard distanced-based MNIST dataset without applying any reduction algorithms, as well as the accuracy of $k$-NN assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Jaccard distanced-based MNIST dataset are 70000, 70000, and 10, respectively.}\n\t\\begin{tabular}{c|cccccc}\\hline \n        Dataset & \\tabincell{c}{$k$-NN accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{13}*{\\shortstack{Jaccard distanced-based MNIST \\\\(70000, 70000, 10)}} & \\multirow{13}{*}{0.958} \n\t\t     \n\t\t& 7000 (1\/10) & 0.958 & 0.958 & NA  \\\\\n\t\t&  & 3500 (1\/20) & 0.958 & 0.966 & NA \\\\\n\t\t&  & 1750 (1\/40) & 0.958 & 0.967 & NA \\\\\n\t\t&  & 875 (1\/80)  & 0.958 & 0.967 & NA \\\\ \n\t\t&  & 437 (1\/160) & 0.958 & 0.968 & 0.718 \\\\\n\t\t&  & 218 (1\/320) & 0.958 & 0.968 & 0.701 \\\\\n\t\t&  & 109 (1\/640) & 0.958 & 0.968 & 0.873 \\\\\n\t\t&  & 70 (1\/1000) & 0.958 & 0.968 & 0.915 \\\\\n\t\t&  & 35 (1\/2000) & 0.956 & 0.968 & 0.872 \\\\\n\t\t&  & 17 (1\/5000) & 0.938 & 0.968 & 0.916 \\\\\n\t\t&  & 7 (1\/10000) & 0.867 & 0.967 & 0.942 \\\\ \n\t\t&  & 3           & 0.487 & 0.965 & 0.939 \\\\ \n\t    &  & 2           & 0.313 & 0.960    & 0.924 \\\\\\hline\n\t\\end{tabular}\n\t\\label{tab:Jaccard distanced-based MNIST KNN ACC}\n\\end{table}\n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Accuracy of $K$-means clustering of the Jaccard distanced-based MNIST dataset without applying any reduction algorithms, as well as the accuracy of $K$-means assisted by PCA, UMAP and t-SNE with different dimensional reduction ratio. The sample size, feature size, and the number of labels of the Jaccard distanced-based MNIST dataset are 70000, 70000, and 10, respectively.}\n\t\\begin{tabular}{c|cccccc} \\hline\n        Dataset & \\tabincell{c}{$K$-means accuracy \\\\ w\/o reduction} & \\tabincell{c}{Reduced\\\\ dimension} & \\tabincell{c}{PCA\\\\accuracy} & \\tabincell{c}{UMAP\\\\accuracy} & \\tabincell{c}{t-SNE\\\\accuracy} \\\\\\hline\n\t\t\\multirow{13}*{\\shortstack{Jaccard distanced-based MNIST \\\\(70000, 70000, 10)}} & \\multirow{13}{*}{0.555}\n\t\t \n\t\t& 7000 (1\/10) & 0.436 & 0.329 & NA   \\\\\n\t\t& & 3500 (1\/20) & 0.436 & 0.693 & NA   \\\\\n\t\t& & 1750 (1\/40) & 0.436 & 0.792 & NA   \\\\\n\t\t& & 875 (1\/80)  & 0.435 & 0.793 & NA   \\\\\n\t\t& & 437 (1\/160) & 0.435 & 0.793 & 0.114   \\\\\n\t\t& & 218 (1\/320) & 0.435 & 0.793 & 0.156   \\\\\n\t\t& & 109 (1\/640) & 0.435 & 0.794 & 0.114   \\\\\n\t\t& & 70 (1\/1000) & 0.436 & 0.793 & 0.113  \\\\\n\t\t& & 35 (1\/2000) & 0.435 & 0.794 & 0.116   \\\\\n\t\t& & 17 (1\/5000) & 0.436 & 0.793 & 0.113   \\\\\n\t\t& & 7 (1\/10000) & 0.431 & 0.793 & 0.737   \\\\\n\t\t& & 3           & 0.364 & 0.798 & 0.635 \\\\\n\t\t& & 2           & 0.261 & 0.791 & 0.635 \\\\\\hline\n\t\\end{tabular}\n\t\\label{tab:Jaccard distanced-based MNIST KMean ACC}\n\\end{table}\n\n\\autoref{tab:Jaccard distanced-based MNIST KNN ACC} shows the accuracy of $k$-NN clustering of Jaccard distanced-based MNIST assisted by PCA, t-SNE, and UMAP with different dimensional reduction radios. For each algorithm, we use the same settings as the Coil 20 dataset. Notably, the $k$-NN accuracy for the data without applying any dimensional reduction algorithm is 0.958, which is at the same level as the PCA algorithm with a reduction ratio greater than 1\/5000. Moreover, we can find that UMAP performs well compared to PCA and t-SNE, indicating that after applying UMAP, the reduced feature still preserves most of the valued information of the Jaccard distanced-based MNIST dataset. The stability and persistence of \nUMAP at various reduction ratios are the most important features. \n\n\\autoref{tab:Jaccard distanced-based MNIST KMean ACC} describes the accuracy of $K$-means clustering of the Jaccard distanced-based MNIST dataset assisted by PCA, UMAP, and t-SNE with different dimensional reduction ratio. For consistency, we will use the same standard parameters as $k$-NN. Similar to the MNIST dataset, the accuracy of $K$-means clustering assisted by UMAP still has the best performance. When the reduced dimension is 3, UMAP will result in the highest $K$-means accuracy 0.798. Noticeably, although PCA performs well on $k$-NN accuracy, it has the lowest $K$-mean accuracy, indicating that PCA is not a suitable dimensional reduction algorithm, especially for those datasets with a large number of samples. To be noted, the t-SNE accuracy at four reduced dimensions are not available due to the extremely long running time. \n\n\n\nIn a nutshell, PCA, UMAP, and t-SNE can all perform well for $k$-NN. However, for the Coil 20 dataset, UMAP performs slightly poorly, whereas the t-SNE performs well, which may be caused by a lack of data size. In order to train UMAP, it needs a suitable data size. The Coil 20 dataset has 20 labels, each with only 72 samples. This may not be enough to train UMAP properly. However, even in this case, UMAP performance is still very stable at various reduction ratios and is the best method in terms of reliability, which become the major advantages of UMAP. Another strength of UMAP comes from its dimension-reduction for $K$-means clustering. In most cases, UMAP can improve $K$-means clustering accuracy, especially for the Jaccard distanced-based MNIST dataset. Furthermore, UMAP can generate a very clear and elegant visualization of clusters with low dimensional reduction value such as 2. Additionally, UMAP performed better than PCA and t-SNE for a larger dataset (MNIST and Jaccard distanced-based MNIST). Especially for the Jaccard distanced-based MNIST data, where Jaccard distance was used as the metric, UMAP performed best, which indicates the merit of using UMAP for Jaccard distanced-based datasets, such as COVID-19 SNP datasets. Furthermore, the accuracies for $k$-NN classification and $K$-means clustering are both improved on the Jaccard distance-based MNIST dataset compared to the original MNIST dataset, which provides convincing evidence that the Jaccard distance representation will   help improve the performance of the clustering on the SARS-CoV-2 mutation dataset in the following sections. \n\n\n\\subsection{Efficiency comparison}\n \nIt is important to understand the computational time behaviors of various methods. To this end, we compare computational time for three dimension-reduction techniques. \\autoref{fig:time} depicts the computational time of three methods for the four datasets under various reduction ratios. The green, orange, and blue lines represent the computational time of t-SNE, UMAP, and PCA, respectively. Some points in green line of \\autoref{fig:time} (d) are not available, which due to the extremely long running time. PCA performed best in most cases, except for the Coil 20 dataset, where UMAP had comparable computational time. This behavior is expected because PCA is a linear transformation, and its time should scale linearly with the number of components in the lower dimensional space. UMAP and t-SNE were slower than PCA, but it is evident from MNIST and Jaccard distanced-based MNIST datasets that UMAP scales better with the increase in the number of samples. \nNote that for Jaccard distanced-based MNIST, a higher dimension was not computed because the computational time was too long. For Facebook Network, UMAP is outperforming t-SNE; however, for higher dimensions, t-SNE computed faster. Nonetheless, from our baseline test \\autoref{tab:Facebook KNN ACC}, t-SNE does not perform well, indicating instability. Faster computation time may indicate too fast of a convergence, which leads to poor embedding. \n \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{subfigure}{.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{coil_20_time.pdf}\n\t\t\\subcaption{Coil 20 time}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{facebook_large_time.pdf}\n\t\t\\subcaption{Facebook Network time}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}{.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{MINST_time.pdf}\n\t\t\\subcaption{MNIST time}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width = \\textwidth]{MINST_b_time.pdf}\n\t\t\\subcaption{Jaccard distanced-based MNIST time}\n\t\\end{subfigure}\n\t\\caption{Computational time of each reduction ratio. The green, orange and blue lines represent the computational time of t-SNE, UMAP, and PCA, respectively. Not surprisingly, PCA performs the best in the majority of cases, except for the Coil 20 dataset. UMAP and t-SNE perform worse than PCA, but UMAP scales better when there are more samples, as evident from MNIST and Jaccard distanced-based MNIST datasets. Note that for Jaccard distanced-based MNIST, the higher dimension was not computed because the computational time was too long.}\n\t\\label{fig:time}\n\\end{figure}\n\n\n\n\n\\section{SARS-CoV-2 mutation clustering}\n\n\n\\subsection{World SARS-CoV-2 mutation clustering}\n We gather data submitted to GISAID up to October 30, 2020, and the total number of samples is 89627. We first get the SNP information by applying the multiple sequence alignment, which leads to 23763 unique SNPs. Next, we calculate the pairwise Jaccard distance of our dataset in order to generate the Jaccard distance-based features. Here, the number of rows is the number of samples (89627), and the number of columns is the feature size (89627). As we mentioned in Section \\ref{sec:Jaccard}, the Jaccard distance-based feature is a square matrix. However, due to the large size of samples and features, applying $K$-means clustering directly on the feature of the size of 89627$\\times$89627 is a very time-consuming process. Considering that UMAP outperforms the other two dimensional reduction algorithms (PCA and t-SNE) on the Jaccard distance-based MNIST dataset, we  employ UMAP to reduce our original feature with the size of 89627$\\times$89627 to 89627$\\times$200. To be noted, UMAP is a reliable and stable algorithm, which performs consistently in clustering at various reduction ratios. Therefore, there is no need to use the same reduction dimension of 200 and one can also choose a different reduction dimension value to generate similar results. \n\nWith the reduced dimension feature that has the size of 89627$\\times$200, we  split our SARS-CoV-2 dataset into different clusters by applying the $K$-means clustering methods. After comparing the WCSS under a different number of clusters, we find that there are 6 clusters forming within the SARS-CoV-2 population based on the elbow method. \\autoref{tab: 1030 mut} shows the top 25 single mutations of each cluster. In order to understand the relationship, we also analyzed the commutation occurring in each cluster (\\autoref{tab: 1030 co}). From \\autoref{tab: 1030 mut} and \\autoref{tab: 1030 co} we see the following:\n\n\n\n\\begin{table}[ht]\n\t\\centering\n\t\\scriptsize\n\t\\setlength\\tabcolsep{1pt}\n\t\\caption{The frequency and occurrence percentage of SARS-CoV-2 co-mutations from each clusters in the world.}\n\t\\begin{tabular}{clcc}\\hline \n\t\tCluster & Co-mutations & Frequency & Occurrence percentage \\\\\\hline\n\t\tCluster 1 & [241, 1163, 3037, 7540, 14408, 16647, 18555, 22992, 23401, 23403, 28881, 28882, 28883] & 776 & 0.463 \\\\\n\t\tCluster 2 & [241, 3037, 14408, 23403] & 8640 & 0.925\\\\\n\t\tCluster 3 & [241, 1059, 3037, 14408, 23403, 25563] & 8878 & 0.662 \\\\\n\t\tCluster 4 & [241, 3037, 14408, 23403, 28881, 28882, 28883] & 14913 & 0.829 \\\\\n\t\tCluster 5 & [241, 3037, 14408, 23403] & 17412 & 0.969 \\\\\n\t\tCluster 6 & [241, 1163, 3037, 7540, 14408, 16647, 18555, 22992, 23401, 23403, 28881, 28882, 28883] & 1352 & 0.771 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab: 1030 co}\n\\end{table}\n\n\n\\begin{itemize}\n\t\\item Though Clusters 1 and 6 seem similar from the top 25 single mutations, the co-mutations tells a different story. The same co-mutations have a higher frequency in Cluster 6, indicating that the co-mutation has higher number of descendants. \n\t\\item Clusters 2 and 5 have high frequency of [241, 3037, 14408, 23403] mutations, but Cluster 5 has a clear co-mutation descendent with high frequency.\n\t\\item Cluster 3 has a unique combination of mutation that is only popular in Cluster 3.\n\\end{itemize}\n\n\n\\autoref{tab: 1030Country} shows the cluster distributions of samples from 25 countries. Here, we use the ISO 3166-1 alpha-2 codes as the country code. The listed countries are the United Kingdom (UK), the United States (US), Australia (AU), India (IN), Switzerland (CH), Netherlands (NL), Canada (CA), France (FR), Belgium (BE), Singapore (SG), Spain (ES), Russia (RU), Portugal (PT), Denmark (DK), Sweden (SE), Austria (AT), Japan (JP), South Africa (ZA), Iceland (IS), Brazil (BR), Saudi Arabia (SA), Norway (NO), China (CN), Italy (IT), and Korea (KR).  From \\autoref{tab: 1030Country}, we can see the following:\n\\begin{itemize}\n\t\\item SNP profiles from UK are dominated in Clusters 5 and 4.\n\t\\item Clusters 1 and 6's SNP profiles are predominantly found in AU.\n\t\\item SNP profiles from US are found mostly in Clusters 3 and 5.\n\t\\item Most country's SNP profiles are found in Clusters 2 - 5, with some having slightly higher numbers, but not as significant as the UK, US and AU.\n\\end{itemize}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{worldMap.png}\n\t\\caption{Cluster distribution of the global SARS-CoV-2 mutation dataset. Using Highchart, the world map was colored, according to the dominant cluster. Clusters 1, 2, 3, 4, 5, and 6 were colored with light blue, blue, green, red, purple and yellow, respectively. For example, United States have SNP profiles from all clusters, but Cluster 5 (purple) is the dominant type in the US. Only countries with more than 25 sequenced data available on GISAID were considered. Countries with fewer than 25 samples are labeled grayed.}\n\\end{figure}\n\n\n  Notably, in \\autoref{tab: 1030 co}, Cluster 2 and Cluster 5 have the same co-mutations with a relatively large frequency, while Cluster 1 and Cluster 6 share the same co-mutations with a relatively low frequency, which indicate that Cluster 2 and Cluster 5 share the same ``root\" with a large size, while Cluster 1 and Cluster 6 share the same ``root\" with a smaller size in the 200-dimensional (200D) space. However, we cannot visualize the distribution of our reduced dataset in the 200D space. Therefore, benefit from the stable and reliable performance of UMAP at various reduction ratios, we reduce the dimension of our original dataset to 2, which enables us to observe the distribution of the dataset in the two-dimensional (2D) space.  \\autoref{fig:1030_World_UMAP} visualizes the distribution of our dataset with 6 distinct clusters with 2D UMAP. It can be seen that 2 clusters (i.e., Cluster 2' and Cluster 3') share a small ``root\" located in the middle of the figure, and Cluster 4' and Cluster 5' share another large ``root\" that also located in the middle of the figure.\n \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.9\\textwidth]{1030_UMAP_World.pdf}\n    \\caption{ 2D UMAP visualization of the world  SARS-CoV-2 mutation dataset with 6 distinct clusters. Red, orange, yellow, light blue, blue, and dark blue represent for Clusters 1', 2', 3', 4', 5', and 6', respectively.} \n    \\label{fig:1030_World_UMAP}\n\\end{figure}\n\n\n\\subsection{United States SARS-CoV-2 mutation clustering}\nIn addition to analyzing the clustering in the world, SNP profiles of SARS-CoV-2 from the United States (US) were considered. In this section, the US dataset has 10279 unique single mutations and 22390 samples. Therefore, the dimension of the Jaccard distance-based dataset is 22390$\\times$22390. After applying the UMAP, we reduce the dimension of the original dataset to be 22390$\\times$200. Following the similar $K$-means clustering processes as we did for the world dataset,  we find that there are 6 predominant clusters forming in the United States. \\autoref{fig: USMap} show the US map with the cluster statistic. Here, Highchart was used to generate the plot with the pie chart. Each states were colored based on the dominant cluster.\n\n\\autoref{tab: 1030US mut} shows the top 25 mutations from each clusters in the United States. The states with more than 50 samples are listed.  \\autoref{tab: 1030US co} shows the common occurring co-mutations, and we can observe the following:\n\n\n\n\n\n\n\n\\begin{itemize}\n\t\\item Cluster F have high frequency of co-mutations [241, 3037, 14408, 23403, 28881, 28882, 28883], which is a descendent of common co-mutations of Cluster 4 [241, 3037, 14408, 23403, 28881, 28882, 28883]  from \\autoref{tab: 1030US mut}.\n\t\\item Clusters A, B, C, and D have frequent co-mutations [241, 1059, 3037, 14408, 23403, 25563], which are also   frequent co-mutations of Cluster 3.\n\\end{itemize}\n\n\n\n\n\\begin{table}[ht]\n\t\\centering\n\t\\caption{The frequency and occurrence percentage of SARS-CoV-2 co-mutations from each clusters in US clusters.}\n\t\\begin{tabular}{c l c c }\\hline \n\t\tCluster & Co-mutations & Frequency & Occurrence percentage \\\\\\hline\n\t\tCluster A & [241, 1059, 3037, 14408, 23403, 25563] & 3116 & 0.465 \\\\\n\t\tCluster B & [241, 1059, 3037, 14408, 23403, 25563] & 5763 & 0.605\\\\\n\t\tCluster C & [241, 1059, 3037, 14408, 23403, 25563] & 8878 & 0.662 \\\\\n\t\tCluster D & [241, 1059, 3037, 14408, 23403, 25563, 27964] & 1225 & 0.864 \\\\\n\t\tCluster E & [8782, 17747, 17858, 18060, 28144] & 1109 & 0.743\\\\\n\t\tCluster F & [241, 3037, 14408, 23403, 28881, 28882, 28883] & 2575 & 0.932 \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab: 1030US co}\n\\end{table}\n\n\n\n\n\n\\begin{figure}[ht]\n\t\\centering \n\t\\includegraphics[width = \\textwidth]{USMap.png}\n\t\\caption{Cluster distribution of United States SARS-CoV-2 mutation dataset. Using Highchart, the US map was colored, according to the dominant cluster. Clusters A, B, C, D, E, and F were colored with light blue, blue, green, red, purple, and yellow, respectively. For example, United States have SNP profiles from all clusters, but Cluster E (purple) is the dominant type in the US. Only those countries that have more than 25 sequenced data available on GISAID were considered in the plot. }\n\t\\label{fig: USMap}\n\\end{figure}\n\n\n Notably, in \\autoref{tab: 1030US co}, Clusters A, B, and C have the same high-frequency co-mutations, indicating that these three clusters may share the same ``root\" in the 200D space. However, it is impossible to show the distribution of each cluster in the 200D space. Considering the stability and reliability of UMAP at various reduction ratios, we  employ UMAP to the original US dataset with reduced dimension  2, aiming to observe the distribution of the dataset in the 2D space.  \\autoref{fig:1030_US_UMAP} illustrates the 2D visualization of the US dataset with 6 distinct clusters. We can see that there are 3 clusters (Clusters A', B', and F') share the same ``root\" located in the middle of the figure, while the other 3 clusters (Clusters C', D', and E') are not. This confirms our deduction about why Clusters A, B, and C have the same high-frequency co-mutations in \\autoref{tab: 1030US co}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.9\\textwidth]{1030_UMAP_US.pdf}\n    \\caption{  The 2D UMAP visualization of the US SARS-CoV-2 mutation dataset with 6 distinct clusters. Red, orange, yellow, light blue, blue, and dark blue represent for Cluster A', B', C', D', E', and F', respectively.}  \n    \\label{fig:1030_US_UMAP}\n\\end{figure}\n\n\\section{Discussion}\n\n In this section, we compared our past results \\cite{wang2020decoding} with our new method to gain a different perspective in clustering with the SNP profiles of COVID-19. In our previous work, a total of 8309 unique single mutations are detected in 15140 SARS-CoV-2 isolates. Here, we also calculate the pairwise distance among 15140 SNP profiles and set the number of clusters to be six. \\autoref{tab:world0601} shows the cluster distribution of samples from the 15 countries \\cite{wang2020decoding}. The listed countries are the United States (US), Canada (CA), Australia (AU), United Kingdom (UK), Germany (DE), France (FR), Italy (IT), Russia (RU), China (CN), Japan (JP), Korean (KR), India (IN), Spain (ES), Saudi Arabia (SA), and Turkey (TR), and we use Cluster I, II, III, IV, V, and VI to represent six clusters without applying any dimensional reduction algorithm.   \\autoref{tab:world0601pca} lists the cluster distribution of samples from the same 15 countries, where we use  I$_p$, II$_p$, III$_p$, IV$_p$, V$_p$, and VI$_p$ to represent six clusters performed by PCA with the reduction ratio to be 1\/160. \\autoref{tab:world0601umap} lists the cluster distribution of samples from the same 15 countries, where we use  I$_u$, II$_u$, III$_u$, IV$_u$, V$_u$, and VI$_u$ to represent six clusters performed by UMAP with the reduction ratio setting to be 1\/160. Noticeably, the SNP profile is focused in Cluster I$_u$, whereas in the non-reduced version, the samples are more spread out. This may be caused by the large number of features, making computed distance between the centroid and each data too similar, and leading to samples being placed in incorrect clusters.\n \n\nNot surprisingly, PCA and the original method for \\cite{wang2020decoding} has nearly identical result. It has been shown in \\cite{wang2020decoding} that PCA\nis the continuous solution of the cluster indicators in the $K$-means clustering method. On the other hand, UMAP shows a slightly different result. In the PCA method, the distribution is more spread out. In addition, the top occurrence for each country is higher for UMAP. On the other hand, we see that there are more samples in Cluster I$_u$ for UMAP, which may indicate that mutations in Cluster I$_u$ are the main strand.\n\n\n  Moreover,  \\autoref{fig:0601_PCA_UMAP} illustrates the 2D visualizations of the US dataset up to June 01, 2020, with 6 distinct clusters by applying two different dimensional reduction algorithms. We can see that the data distribute disorderly under both PCA- and UMAP-assisted $K$-means clustering algorithms.  Specifically, the PCA-assisted algorithm has a really poor clustering performance, while the UMAP-assisted algorithm forms more clear and better clusters than the PCA-assisted algorithm, which is consistent with our previous analysis in Section \\ref{sec:Validation}.\n \n\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 1\\textwidth]{Discussion.pdf}\n    \\caption{  2D visualizations of the US SARS-CoV-2 mutation dataset up to June 01, 2020 with 6 distinct clusters by applying two different dimensional reduction algorithms. (a) 2D PCA visualization. Red, orange, yellow, light blue, blue, and dark blue represent for Cluster I$_p^{\\prime}$, II$_p^{\\prime}$, III$_p^{\\prime}$, IV$_p^{\\prime}$, V$_p^{\\prime}$, and VI$_p^{\\prime}$, respectively. (b) 2D UMAP visualization. Red, orange, yellow, light blue, blue, and dark blue represent for Cluster I$_u^{\\prime}$, II$_u^{\\prime}$, III$_u^{\\prime}$, IV$_u^{\\prime}$, V$_u^{\\prime}$, and VI$_u^{\\prime}$, respectively.}  \n    \\label{fig:0601_PCA_UMAP}\n\\end{figure}\n\n\n\n\n\n\n\\section{Conclusion} \nThe rapid global spread of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has led to genetic mutation stimulated by genetic evolution and adaptation. Up to October 30,  89627 complete SARS-CoV-2 sequences, and a total of 23763 unique SNPs have been detected. Our previous work traced the COVID-19 transmission pathways and analyzed the distribution of the subtypes of SARS-CoV-2 across the world based on 15,140 complete SARS-CoV-2 sequences.  The $K$-means clustering separated the sequences into six distinguished clusters. However, considering the tremendous increase in the number of available SARS-CoV-2 sequences,  an efficient and reliable dimensional reduction method is urgently required.  Therefore, the objective of the present work is to explore the best suited dimension reduction algorithm based on their performance and effectiveness.  Here, a linear algorithm PCA and two non-linear algorithms,  t-distributed stochastic neighbor embedding (t-SNE) and uniform manifold approximation and projection (UMAP), have been discussed.  To evaluate the performance of dimension reduction techniques in clustering, which is an unsupervised problem, we first cast classification problems into clustering problems with labels.  Next, by setting different reduction ratios, we test the effectiveness and accuracy of PCA, t-SNE, and UMAP for $k$-NN and $K$-means using four benchmark datasets.  The results show that overall, UMAP outperforms other two algorithms. \nThe major strengths of UMAP is that UMAP-assisted $k$-NN classification and UMAP-assisted $K$-means clustering at various dimension reduction ratios have a consistent performance in terms of accuracy, which proves that UMAP is a stable and reliable dimension reduction algorithm. Moreover, compared to the $K$-means clustering accuracy that does not involve any dimensional reduction, UMAP-assisted $K$-means clustering can improve the accuracy for most cases. Furthermore, when the dimension is reduced to two, the UMAP clustering visualization is clear and elegant. Additionally, UMAP is a relatively efficient algorithm compared to t-SNE. Although PCA is a faster algorithm, its major limitation is its poor performance in accuracy. To be noted, UMAP performs better than PCA and t-SNE for the dataset with a large number of samples, indicating it is the best suited dimensional reduction algorithm for our SARS-CoV-2 mutation dataset. Moreover, we apply the UMAP-assisted $K$-means clustering to the world SARS-CoV-2 mutation dataset (up to October 30, 2020), which displays six distinct clusters. Correspondingly, the same approaches are also applied to the United States SARS-CoV-2 mutation dataset (up to October 30, 2020), resulting in six different clusters as well. Furthermore, we provide a new perspective by utilizing UMAP-assisted $K$-means clustering to analyze our previous SARS-CoV-2 mutation datasets, and the 2D visualization of UMAP-assisted $K$-means clustering of our previous world SARS-CoV-2 mutation dataset (up to June 01, 2020) forms more clear clusters than the PCA-assisted $K$-means clustering. Finally, one of our four datasets was generated by the Jaccard distance representation, which improves both $k$NN classification and $k$-means clustering accuracies on the original dataset.  \n\n\n\n\n\n\\section*{Acknowledgment}\nThis work was supported in part by NIH grant GM126189, NSF Grants DMS-1721024, DMS-1761320, and IIS1900473, Michigan Economic Development Corporation, Bristol-Myers Squibb, and Pfizer. The authors\nthank The IBM TJ Watson Research Center, The COVID-19 High Performance Computing Consortium, and\nNVIDIA for computational assistance.\n\n\\clearpage\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\nHistorically, graphs as finite metric spaces have been extensively studied~\\cite{Ha}. Even though we approach them differently, we would like to emphasise, amongst others~\\cite{I,P1,P2,S}, the classical result provided by Buneman~\\cite{B}. In short, a metric on a finite set can be realised by the shortest path metric in a positive-weighted tree if and only if it satisfies the four-point condition.\nNot only is it frequently quoted in the context of evolutionary trees~\\cite{SS}, but\nit is also known for its direct connection to the theory of Gromov hyperbolic metric spaces~\\cite{G}. Approximately two decades later after Buneman's theorem, Hendy~\\cite{H} proved the existence of a unique tree representation for every metric satisfying the four-point condition.\n\nGiven this background, a metric space that satisfies the four-point condition is commonly considered tree-like. However, an important caveat should be addressed: \nthe four-point condition is necessary and sufficient to ensure the existence of a \\emph{partially labelled} tree that realises a given metric~\\cite{Ha,H,SS}.\nFor example, a complete graph with a uniform edge length clearly satisfies the four-point condition, but it only becomes tree-like after an extra vertex is added. In this case, the four-point condition does not ensure that  a metric  is  realised by  a \\emph{fully labelled} tree on \\emph{the same} set. It does not characterise the distance within trees, in general, but rather the shortest path metrics induced by graphs of a certain class, called \\emph{block graphs} (\\textit{i.e.}, graphs in which all biconnected components  are complete subgraphs)~\\cite{Ba}.\n\n\nThis may not create an issue in the field of conventional phylogenetics, but considering the recent surge of renewed biological interest in minimum spanning tree (MST)-based tree estimation~\\cite{Q}, determining when a metric space is realised by a positive-weighted tree on the same set is not only a natural undertaking but also a meaningful one.\nThus far, this problem has not been properly recognised, much less addressed. The only two exceptions to this are the recent work provided in~\\cite{A} and in~\\cite{HEF}. It seems to be a non-trivial question not only because it cannot be answered using Buneman's theorem, but also because it is equivalent to determining a method for recognising a special case of the metric travelling salesman problem (TSP). If an input---a metric on a set of cities---is the shortest path metric in a tree on the city set,  \nthe length of the optimal tour must equal twice the length of the MST.\n\n\nIn this paper, we examine the sub-type of tree metrics without relying on the four-point condition. \nOur work is based on three ingredients: \nthe  so-called tie-breaking assumption, which has been popular in algorithmic applications since the work provided by Kruskal in~\\cite{K}; \nwhat we call  the fourth-point condition, which can typically be found in the definition of median metric spaces~\\cite{DD}; and a simple trick for metric-preserving edge removal, which applies to any finite metric space.\nThese concepts, which are part of our original results, are defined and discussed in  Section~\\ref{sec:preliminaries}.\n\nAs expected, if it exists, a fully labelled positive-weighted tree that realises a finite metric space is the unique MST in its associated weighted complete graph (Proposition~\\ref{prop:1}).  Our goal is to prove the following: \nA finite metric space under the tie-breaking rule is realised by the MST if and only if it satisfies the fourth point condition (Theorem~\\ref{thm:1}). This implies that every finite median graph, in which the shortest path lengths between all pairs of vertices are distinct, is necessarily a tree (Corollary~\\ref{cor:2}). This result also yields a stronger condition for understanding when a finite metric space is realised, especially by a spanning path graph (Corollary~\\ref{cor:1}).\nWe define and discuss the notion of a spanning tree-likeness of a finite metric space in Section~\\ref{sec:discussion}.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\nWe apply the metric-related terminology provided in \\cite{DD} throughout this paper.\nLet $(X,d_M)$ be a \\emph{finite metric space}, that is, a finite set, $X$, equipped with metric $d_M$.  \nFor two distinct points $x$ and $x^{\\prime}$ in $X$, the \\emph{closed metric interval} between them is defined to be the set \n\\[\nI(x,x^{\\prime}):=\\{i\\in X: d_M(x,x^{\\prime})=d_M(x,i)+d_M(i,x^{\\prime})\\}.\n\\]\n\nAll graphs considered in this paper will be  simple, undirected,  \n\\emph{fully labelled} (\\textit{i.e.}, each vertex is labelled), and \\emph{positive weighted} (\\textit{i.e.}, each edge has a positive length).\nA graph is denoted $(V,E;w)$ for a set, $V$,  of labelled vertices and a set, $E$,  of edges that are associated with  a positive edge-weighting function, $w: E\\mapsto \\mathbb{R}^{+}$.\nGiven graph $G$, the sets of vertices and edges are denoted $V(G)$ and $E(G)$,  respectively. Moreover, graph $G$ is said to be a graph \\emph{on} $V(G)$.\nVertices may be renamed as needed, assuming no confusion arises, and a vertex labelled `$x$' is referred to as vertex $x$. \nThe distance in graph $G$ is defined to be the shortest path metric and is represented using $d_G$.\n\nAssume $M$ is a finite metric space, $(X,d_M)$. \nLet $K_M$ be the associated weighted complete graph with $M$.\nAn edge of $K_M$ that joins two distinct vertices, $x$ and $x^{\\prime}$, is denoted $e(x,x^{\\prime})$. This paper uses the terms `\\emph{points}' and `\\emph{vertices}' interchangeably because there is a one-to-one correspondence between $X$ and $V(K_M)$ for any finite metric space $M$.\n\n\\subsection{Tie-breaking rule}\\label{subsec:tiebreak}\n\\begin{dfn}\\label{dfn:tiebreak}\nA finite metric space, $(X,d_M)$, is said to satisfy \\emph{the tie-breaking rule} if the values of $d_M$ are distinct for all pairs in $X$.\n\\end{dfn}\n\n\\subsection{The fourth point condition}\\label{subsec:4thpc}\n\\begin{dfn}[Figure~\\ref{fig:satisfy4thpc}]\\label{dfn:4thPC}\nA finite metric space, $(X,d_M)$, is said to satisfy \\emph{the\nfourth-point condition} if, for every (not necessarily distinct) three\npoints $x,y,z\\in X$, there exists a point, $p^{*}\\in X$, such that\n\\[\nd_M(x,p^{*})+d_M(y,p^{*})+d_M(z,p^{*})=\\frac{1}{2} \\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}.\n\\]\n\\end{dfn}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.18\\textwidth]{4thpc.eps}\n\\caption{Fourth point $p^*$ for triplet $\\{x,y,z\\}$ \\label{fig:satisfy4thpc}}\n\\end{figure}\n\n\n\\begin{prop}\\label{prop:4th_unique}\nIf a finite metric space, $(X,d_M)$, satisfies the fourth-point condition, \n fourth point $p^*\\in X$ is unique for each triplet in $X$. \n\\end{prop}\n\\begin{proof}\nSuppose that there are two quartets, $\\{x,y,z,p_1^*\\}$ and $\\{x,y,z,p_2^*\\}$ ($p_1^*\\neq p_2^*$), in $X$ such that \n\\[d_M(x,p_1^{*})+d_M(y,p_1^{*})+d_M(z,p_1^{*})= d_M(x,p_2^{*})+d_M(y,p_2^{*})+d_M(z,p_2^{*}).\\]\nBecause $d_M$ is a metric on $X$, we have \n\\begin{align*}\nd_M(x,p_2^{*})&\\leq d_M(x,p_1^{*})+d_M(p_1^{*},p_2^{*});\\\\\nd_M(y,p_2^{*})&\\leq d_M(y,p_1^{*})+d_M(p_1^{*},p_2^{*});\\\\\nd_M(z,p_2^{*})&\\leq d_M(z,p_1^{*})+d_M(p_1^{*},p_2^{*}).\n\\end{align*}\nTherefore, $p_1^{*}=p_2^{*}$, but this is a contradiction. \nHence, if $p^*$ exists for $\\{x, y, z\\}$, it is unique.\n\\end{proof}\n\n\n\\begin{prop}\\label{lem:4thPC_alternative}\nThe following is equivalent to saying that finite metric space $(X,d_M)$ satisfies the fourth-point condition:  \nFor every (not necessarily distinct) three points $x, y, z \\in X$, there\nexists only one point  $p^*\\in I(x,y)\\cap I(y,z)\\cap I(z,x)$.\n\\end{prop}\n\n\\begin{proof}\nBecause $d_M$ is a metric, for all $x,y,z,p\\in X$, we have \n$d_M(x,p)+d_M(y,p)+d_M(z,p)\\geq \\frac{1}{2}\\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}$. \nThe equality holds if and only if\n$I(x,y)\\cap I(y,z)\\cap I(z,x)=\\{p^*\\}$. \nProposition \\ref{prop:4th_unique} ensures the uniqueness of $p^*$.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem:median}\nFourth point $p^*$ is also known as \\emph{the median} for $\\{x,y,z\\}$  \nbecause it minimises the sum of the distances to the three points, and \na metric space satisfying the fourth-point condition (or a graph inducing this kind of metric space) is said to be \\emph{median}~\\cite{Ba,DD}.\nAlthough a discussion of this topic is provided in~\\cite{Ba,BCE}, it should be noted that median graphs include multiple types of graphs other than trees, such as grid and square graphs.\n\\end{rem}\n\n\n\\begin{lem}\\label{lem:cycle}\nLet $C$ be a cycle graph, $(V,E;w)$, with $\\sum_{e\\in E} w(e) =c$. \nAlso, let $d_C$ be the shortest path metric in $C$.\nGiven three distinct points  $x,y,z\\in V$  such that \n$d_C(x,y)+d_C(y,z)+d_C(z,x)=c$, \nthe fourth point, $p^*$, exists in $V$  \nif and only if $\\max\\{d_C(x,y),d_C(y,z),d_C(z,x)\\}=c\/2$.\n\\end{lem}\n\\begin{proof}\nWithout loss of generality, we can assume $d_C(z,x)=\\max\\{d_C(x,y), d_C(y,z), d_C(z,x)\\}$. Clearly, $y\\in I(x,y)\\cap I(y,z)$. Therefore, $I(x,y)\\cap I(y,z)\\cap I(z,x)=\\emptyset$ if and only if $y\\not\\in I(z,x)$. \nUnder the assumption that the length of $C$ is fixed at $c$, \nthis is equivalent to stating that $d_C(z,x)\\neq c\/2$. \nThus, $I(x,y)\\cap I(y,z)\\cap I(z,x)= \\emptyset$ if and only if $d_C(z,x)\\neq c\/2$. Applying Proposition~\\ref{lem:4thPC_alternative} completes the proof.\n\\end{proof}\n\n\\subsection{Basic geodesic graphs}\\label{subsec:basic}\nIn this subsection, we present a simple trick for metric-preserving edge removal, which can be used to represent an arbitrary finite  metric space  as a graph with the fewest edges. \nLet $M$ be a finite metric space, $(X,d_M)$, \nand assume $K_M$ is the weighted complete graph associated with $M$.\n\n\\begin{dfn}\\label{dfn:realise}\nSuppose $G$ is a connected graph on finite set $X$ with shortest path metric $d_G$.\nGraph $G$ is said to \\emph{realise $M$} if\n$d_G(x,x^{\\prime}) = d_M(x,x^{\\prime})$  \nfor all  $x,x^{\\prime}\\in X$.\n\\end{dfn}\n\n\\begin{dfn}\\label{dfn:circularperm}\nGiven $x,x^{\\prime}\\in X$, \nthe edge, $e(x,x^{\\prime})$, of $K_M$ is said to be \\emph{non-basic} \nif there is a  permutation, $(x_1, \\ x_2, \\ \\cdots, \\ x_k)$, on a non-empty subset of $X\\setminus\\{x,x^{\\prime}\\}$ such that cyclic permutation $(x, \\ x_1, \\ x_2, \\ \\cdots , \\ x_k, \\ x^{\\prime})$  satisfies \n\\[\nd_M(x,x^{\\prime})=d_M(x,x_1)+d_M(x_1,x_2)+...+d_M(x_k,\\,x^{\\prime}). \n\\]\nThe edge is called \\emph{basic} otherwise.\n\\end{dfn}\n\n\\begin{prop}\\label{rem:4thpoint_notexist}\nLet $x,y,z$ be  three different vertices of $K_M$.\nWhen the three edges, $e(x,y)$, $e(y,z)$, and $e(z,x)$, of $K_M$ are basic, the fourth point, $p^*$,   does not exist for $\\{x,y,z\\}$. \nIf a non-basic edge exists, say $e(x,y)$,   \npoints  $x$ and $y$ are the only two candidates for $p^*$.\n\\end{prop}\nThe proof of this proposition is straightforward.\n\n\\begin{dfn}\\label{dfn:basicgeodesicgraph}\nAssume $B_M$ is the set of all basic edges of $K_M$, \nand suppose $\\lambda$ is  a restriction of $d_M$ to $B_M$. \nA subgraph, $G_M:=(X, B_M; \\lambda)$, in $K_M$ is called  \n\\emph{the basic geodesic graph in $K_M$}. \n\\end{dfn}\n\n\n\\begin{lem}\\label{lem:connected}\nThe basic geodesic graph, $G_M$, in $K_M$ is a connected graph on $X$ that realises $M$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to prove that $G_M$ is connected. \nAssuming that $e(x,x^{\\prime})$ is non-basic, we show that there is a path of basic edges joining $x$ and $x^{\\prime}$ in $K_M$. We also note that they are obviously connected in $G_M$ if $e(x,x^{\\prime})\\in E(K_M)$ is basic. \nLet $C$ be a cycle with the greatest number  of vertices (or edges) of all cycles in $K_M$ that share edge $e(x,x^{\\prime})$ and overall length $2d_M(x,x^{\\prime})$. Let  $V(C)=\\{x,x^{\\prime}\\}\\cup Y$, where   $Y:=\\{x_1,\\cdots,x_k\\}$ is a non-empty subset of $X\\setminus\\{x,x^{\\prime}\\}$, as in  Definition~\\ref{dfn:circularperm}. Furthermore, suppose $d_C$ is the shortest path metric induced by $C$ and $x_i, x_j\\in V(C)$. \nIf a path existed in $K_M$ joining $x_i$ and $x_j$ that was shorter than  $d_C(x_i,x_j)$, then edge $e(x,x^{\\prime})$ would be longer than the path connecting  $x$ and $x^{\\prime}$ through  $x_i$ and $x_j$. Therefore, any path in $K_M$ joining two vertices in $V(C)$  must have a length  greater than or equal to $d_C(x_i,x_j)$.  We use this fact  at the end of the proof.\n\nIn order to obtain a contradiction, we suppose $e(y,y^{\\prime})\\in E(C)\\setminus e(x,x^{\\prime})$  is  non-basic. We define   $C^{\\prime}$  to be a cycle in $K_M$ of overall length $2d_M(y,y^{\\prime})$  with   $e(y,y^{\\prime})\\in E(C^{\\prime})$, which is similar to our previous case except that $|V(C^{\\prime})|$ is unimportant. Let $V(C^{\\prime})=\\{y,y^{\\prime}\\}\\cup Z$, where $Z:=\\{y_1,\\cdots,y_l\\}\\subseteq X\\setminus\\{y, y^{\\prime}\\}$.  By Definition \\ref{dfn:circularperm}, if a cycle contains a non-basic edge, then it must be strictly longer than the other edges in the cycle. This implies that the number of non-basic edges contained in each cycle is zero or one. Thus, $e(y,y^{\\prime})$ is shorter than $e(x,x^{\\prime})$, and $e(y,y^{\\prime})$ is the longest edge in $E(C^{\\prime})$. Therefore, we can conclude that $e(x,x^{\\prime})$ is not in $E(C^{\\prime})$.\nThe assumption on $|V(C)|$ provides $Y\\cap Z\\neq\\emptyset$. Our hypothesis ensures a path in $K_M$ of length $d_C(y,y^{\\prime})$ that connects  $y$ and $y^{\\prime}$ via $y^{\\prime\\prime}\\in Y\\cap Z$. This implies that  $K_M$ contains a path joining  $y$ and $y^{\\prime\\prime}$  of length less than $d_C(y,y^{\\prime})$. If we  assume that  $y^{\\prime}$ lies in the shortest path joining $y$ and $y^{\\prime\\prime}$ in $C$ (note that the roles of $y$ and $y^{\\prime}$ can be exchanged), then we have  $d_C(y,y^{\\prime})<d_C(y,y^{\\prime\\prime})$. It follows that there is a path that joins $y$ and $y^{\\prime\\prime}$ in $K_M$ of length less than $d_C(y,y^{\\prime\\prime})$. This is a contradiction. Hence, $e(y,y^{\\prime})$ is  basic, which completes the proof.\n\\end{proof}\n\n\n\\begin{dfn}\\label{dfn:treepath}\nFinite metric space $M$ is said to be a \\emph{spanning tree metric space} if the basic geodesic graph, $G_M$,  in the weighted complete graph, $K_M$, is a spanning subtree in $K_M$.  In particular, $M$ is said to be a \\emph{spanning path metric space} if $G_M$ is a \\emph{path graph} (\\textit{i.e.}, a tree with two vertices of degree one and remaining vertices of degree two) that spans all the vertices of $K_M$. \n\\end{dfn}\n\n\\begin{prop}\\label{prop:1}\nLet $M:=(X,d_M)$ be a spanning tree metric space and \n$G_M$ be the basic geodesic graph in  $K_M$. Then the following statements hold:\n\\begin{enumerate}\n\\item $G_M$ is the unique minimum spanning tree in $K_M$;\n\\item $G_M$ is the unique fully labelled tree  on $X$ that realises $M$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1) Assume $B:=E(G_M)$ and let $\\overline{B}:=E(K_M)\\setminus B$. Because $|B|=|X|-1$,  $G_M$ is the only spanning tree in $K_M$ such that all edges are basic.\nIn addition, let $e(x,x^{\\prime})\\in \\overline{B}$.  Because $G_M$ is a tree, there is a  unique path  joining $x$ and $x^{\\prime}$, denoted $P$. Each edge of $P$ must be  strictly shorter than $d_M(x,x^{\\prime})$ for the following reasons: the length of $P$ equals $d_M(x,x^{\\prime})$; the number of edges of $P$ exceeds one; and the edge weights are  all positive. Therefore,  replacing an arbitrary edge of $P$  with $e(x,x^{\\prime})$ results in a spanning tree in $K_M$ of greater length. Hence, $G_M$ is shorter than any other spanning trees in $K_M$. \n(2) Suppose that $M$ is realised by fully labelled tree $T$ on $X$.  This implies that each edge of $T$ has a positive weight. \nWe can recover $K_M$ from $T$ by summing the weights along every path in $T$ that has  two or more edges.\nThis process indicates that  $T$ is isomorphic to  the basic geodesic graph  in $K_M$. Hence, given (1), we know $T$ is unique.\n\\end{proof}\n\\begin{rem}\nProposition~\\ref{prop:1} states that a metric space is uniquely realised by the only MST if it is a spanning tree metric space. Note that we do not need Buneman's four-point condition in  the argument ({\\it cf.}~\\cite{A}).\nConcerning the uniqueness of the MST, the tie-breaking rule is a well-known sufficient condition established by Bor\\r{u}vka~\\cite{Bo} (cited in~\\cite{K}) and by Kruskal~\\cite{K}. The next section explores its relation to spanning tree metric spaces.\n\\end{rem}\n\n\\section{Main results}\\label{sec:spanningtree}\n\n\\begin{thm}\\label{thm:1}\nLet $M$ be a finite metric space, $(X,d_M)$, under the tie-breaking rule.  \nThen $M$  is a spanning tree metric space if and only if it satisfies the fourth-point condition. \n\\end{thm}\n\\begin{proof}\n(i)  The fourth-point condition clearly holds for all spanning tree metric spaces.\n(ii) If $d_M$ is not a spanning tree  metric on $X$, then  \nwe will show that there is a triplet in $X$ that violates the fourth-point condition. \nAccording to Lemma \\ref{lem:connected}, our assumption  implies that the basic geodesic graph, $G_M=(X,B;\\lambda)$, in $K_M$ contains at least one cycle. \nSuppose $C:=(X_k,B_k; \\lambda_k)$ is the shortest cycle  in $G_M$,  \nwhere  $X_k\\subseteq X$, $B_k\\subseteq B$, $|X_k|=|B_k|=k$, and $\\lambda_k$ is the restriction of $\\lambda$ to $B_k$. Then Proposition \\ref{rem:4thpoint_notexist} yields $k\\geq 4$. Let $c$ denote the sum of the $\\lambda_k$ over all elements in $B_k$. Also, assume that $d_C$ is the shortest path metric in $C$.\nFor all $i,j\\in X_k$, no path in $G_M$ joining $i$ and $j$ has a shorter length than  $d_C(i,j)$  (otherwise, $C$ would not be the shortest cycle in $G_M$).\nTherefore, $d_C(i,j)=\\min \\{a_{ij},c-a_{ij}\\}$,  \nin which $a_{ij}$  represents the length of the path in $C$ that travels from $i$ to $j$ in a clockwise direction. \n\nConsider a route in which we visit the points in $X_k$. Let $s\\in X_k$ be the starting point from which we travel along the circle in a clockwise direction. \nWe assign a label, `$\\mathrm{L}$' or `$\\mathrm{R}$', to every point $i\\in X_k\\setminus\\{s\\}$: label `$\\mathrm{L}$' is assigned if $a_{si} < c\/2$, and we use label  `$\\mathrm{R}$'  if $a_{si}\\geq c\/2$.\nIf every point in $X_k\\setminus\\{s\\}$ was labelled `$\\mathrm{L}$', the last edge we would traverse returning to $s$ would be non-geodesic or non-basic. \nTherefore, there exists one and only one basic edge between vertices labelled `$\\mathrm{L}$' and `$\\mathrm{R}$'. Suppose that $t$ signifies the last point with label `$\\mathrm{L}$' and $u$ indicates the first point with label `$\\mathrm{R}$' as on the left  in Figure \\ref{fig:thm1}. Note that  $d_M(s,t)+d_M(t,u)+d_M(u,s)=c$. \n\nWe  assume that  $p^*$ exists for $\\{s,t,u\\}$ (otherwise, the assertion of the theorem immediately follows). Lemma~\\ref{lem:cycle} gives us $\\max\\{d_M(s,t),d_M(t,u),d_M(u,s)\\}=c\/2$. Thus,  $d_M(u,s)=c\/2$ (the edge joining $t$ and $u$ is basic, and $d_M(s,t)<c\/2$). Let $v (\\neq u)$ be a point  in $X_k$ with label `$\\mathrm{R}$' that is between $u$ and $s$ as on the right in Figure \\ref{fig:thm1}. We know point $v$ exists because $e(u,s)$ would be non-basic otherwise. According to the tie-breaking rule, we note that $a_{tv} \\neq c-a_{tv}$. We can also set $a_{tv} < c-a_{tv}$ in order to select $\\{s,t,v\\}$. Although we should select $\\{t,u,v\\}$ when  $a_{tv} > c-a_{tv}$, we limit our consideration to the former case. Therefore, we have $d_M(s,t)+d_M(t,v)+d_M(v,s)=c$ again, but each of the three terms does not equal $c\/2$ (recall that  $d_M(s,u)=c\/2$). Hence, Lemma~\\ref{lem:cycle} implies that  $p^*$ does not exist for $\\{s,t,v\\}$, and this completes the proof.\n\\end{proof}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{points.eps}\n\\caption{Points in the proof of Theorem~\\ref{thm:1} \\label{fig:thm1}}\n\\end{figure}\n\n\\begin{rem}\nGiven a finite metric space on $X$, we can determine in $O(|X|^4)$ time whether it is a spanning tree metric space.\n\\end{rem}\n\n\\begin{cor}\\label{cor:2}\nLet $G$ be a median graph on finite set $X$ and  let $d_G$ be the shortest path metric of $G$. If each pair in $X$ has a different value for $d_G$, then $G$ is a tree.\n\\end{cor}\n\n\\begin{rem}\\label{rem:insufficient}\nAs was mentioned in Remark~\\ref{rem:median}, \nthe fourth-point condition \\textit{per se} is not a sufficient condition, but it is a  necessary condition in order to ensure that a finite metric space \nis induced by the shortest path metric in a tree \n(\\textit{cf.}\\ a  cycle graph on four vertices with a uniform edge length).\n\\end{rem}\n\n\\begin{cor}\\label{cor:1}\nSuppose $M:=(X,d_M)$ is a finite metric space  under the tie-breaking rule.  \nThen $M$  is a spanning path metric space (Definition~\\ref{dfn:treepath}) if and only if it satisfies \\emph{the three-point condition}: for every (not necessarily distinct) three points $x,y,z\\in X$, we have \n\\[\n\\max\\{d_M(x,y),d_M(y,z),d_M(z,x)\\}=\\frac{1}{2}\\{d_M(x,y)+d_M(y,z)+d_M(z,x)\\}. \n\\]\nThe condition can be confirmed in $O(|X|^3)$ time. If $M$ is a spanning path metric space, it is realised by the unique shortest path that joins the farthest two points  in $X$. \n\\end{cor}\n\\begin{proof}\nWe only prove the first statement. The three-point condition obviously holds for all spanning path metric spaces. Therefore, we assume that the three-point condition holds and show that the basic geodesic graph, $G_M$, in $K_M$ is a path graph on $X$. It is clear that $y$ is the fourth point, $p^{*}$, for $\\{x,y,z\\}$ when  the left-hand side equals $d_M(z,x)$. This means that the fourth-point condition automatically holds for any finite metric space that satisfies the three-point condition. Therefore, our assumption implies that $G_M$  is a tree on $X$. The three-point condition also indicates that every vertex in $G_M$ has a degree of one or two. In other words, if vertex $x$ has degree three or more, then any three distinct vertices adjacent to $x$ would violate the three-point condition. Hence, $G_M$ is a path graph on $X$, which completes the proof.\n\\end{proof}\n\n\\section{Discussion}\\label{sec:discussion}\nThe hyperbolicity of finite metric spaces (or graphs) is a concept provided by Gromov~\\cite{G,SS} and measures the deviance of a metric space from Buneman's four-point condition. If a metric space, $M$, satisfies the four-point condition, then the hyperbolicity of $M$ equals $0$, and $M$ is said to be $0$-hyperbolic. As was previously discussed, any complete graph with a uniform edge length is $0$-hyperbolic. Because the four-point condition is a stronger version of the triangular inequality, all metric triangles are also $0$-hyperbolic. Therefore, although the value of hyperbolicity is usually called the `tree-likeness' of $M$, a more precise interpretation refers to the \\emph{partially labelled} tree-likeness of $M$. \nTherefore, as a final remark, we provide the notion of a \\emph{fully labelled} tree-likeness of $M$.\n\nLet us say that finite metric space $M$  is  \\emph{$\\rho$-roundabout}. Here,  $\\rho$ is defined to be \n\\[\n\\max_{x,y,z\\in X}\\min_{i\\in X}\\;\\frac{d_M(x,i)+d_M(y,i)+d_M(z,i)}{d_M(x,y)+d_M(y,z)+d_M(z,x)}-\\frac{1}{2}.\n\\]\nThis measures how far $M$ deviates from the fourth-point condition.   Provided that the tie-breaking rule holds, the value of $\\rho$ can be regarded as the spanning tree-likeness of $M$ or the circuitousness of $d_M$ as illustrated in Figure~\\ref{fig:roundabout}. Note that $\\rho$ is invariant under multiplication of $d_M$ by a constant. As we have already seen, $M$ is $0$-roundabout if and only if there is an exact fit between $M$ and the MST. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{roundabout.eps}\n\\caption{Illustrations of spanning tree-likeness ($\\rho=0$ and $\\rho>0$)\\label{fig:roundabout}}\n\\end{figure}\n\n\nThe degree of violation of the three-point condition similarly provides the spanning path-likeness of $M$---the maximum discrepancy between the left and right-hand sides of the triangular inequality. On the other hand, hyperbolicity does not provide any information because all metric triangles are $0$-hyperbolic. \n\n\n\\section*{Acknowledgement}\nThis work was supported in part by JSPS KAKENHI Grant Number 25120012 and 26280009. \nSpecial thanks are extended to Yoshimasa Uematsu for his insights concerning the concept of median.\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\vskip 0.1cm\n\nSince the establishment of quantum mechanics ($QM$) and the quantization of electromagnetic field, the quantum electrodynamics ($QED$) in particular and the quantum field theory ($QFT$) in general have been developed for over 80 years. A common prominent feature of $QFT$ is the emergence of the divergence in calculations beyond the tree level. To handle\nthese divergences, various regularization and renormalization methods ($RRMs$) have been proposed.\n\nDespite the great success of $QFT$, the present status of $RRMs$ remains ambiguous to some extent. For example, in the\ntheory of Chromodynamics ($QCD$) for describing the strong interactions of colored quarks and gluons, a commonly used\nrenormalization scheme ($RS$) is the modified minimal substraction ($\\bar{MS}$) scheme (see a summary in the Review of\nParticle Physics in 2008, \\cite{1}, p.157) where an arbitrary renormalization mass scale $\\mu$ is introduced (see next\nsection). Physicists believe the fundamental theorem of $RS$ dependence: Physical quantities, such as the cross-section\ncalculated to all orders in perturbation theory, should not depend on the $RS$. However, it follows that a truncated\nseries does exhibit $RS$ dependence. In practice, $QCD$ cross-section are known to different orders, depending on the\nchoice of $RS$ (and $\\mu$) in different sensitive ways. We still don't know what is the \"best\" choice for $\\mu$ within\na given scheme (usually $\\bar{MS}$). There is no definite answer to this question yet.\n\n\\section{What a divergence means?}\n\\label{sec:divergence}\n\n\\vskip 0.1cm\n\nPhysicists often talk about different orders of a divergence, based on its dimension with respect to mass (\\ie, momentum,\nwe use natural unit system with $\\hbar=c=1$). For example, if a Feynman diagram integral ($FDI$) in $QFT$ reads\n\\begin{equation}\\label{1}\nI=\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{K^n}{(K^2-M^2)^2}\n\\end{equation}\nwhere $K$ corresponds to the (4-dimensional)momentum of virtual particle (say, virtual photon in $QED$) and $M$ is a\nmass parameter (maybe in a complex form) characterizing the $QFT$ under consideration. Then\n\\begin{equation}\\label{2}\nn=\\left\\{\\begin{array}{l}\n           0,\\quad \\text{logarithmic divergence} \\\\\n           1,\\quad \\text{linear divergence} \\\\\n           2,\\quad \\text{quadratic divergence}\n         \\end{array}\\right.\n\\end{equation}\nHowever, among these categories, only the first one is really meaningful in mathematics. This is because the definition\nof a number sequence $A_i\\;(i=1,2,\\ldots)$ having a limit being $\\infty$ is as follows: Given arbitrarily a large\nnumber $M$, one can always find such a number $N$ so that $A_i>M$, when $i>N$. Here $A_i$ and $M$, let alone $i$ and $N$,\nare all dimensionless numbers. A number $M\\gg1$ is called a large number, whereas $\\varepsilon\\ll1$ a small number.\n\nOn the other hand, the space-time coordinates $\\bf x$ and $t$, mass $m$ and momentum $p$ (or $k$) are physical quantities and each with certain dimension. If treating them as dimensionless numbers, we will run into trouble inevitably.\n\nExample A: Assume that in $QFT$, a $FDI$ with linear divergence is approximately expressed as $I\\sim 10^3M$ with $M$ being a\nmass parameter in the unit of $mg$. If $M=1\\,mg$, we have $I\\sim 10^3mg$ with $10^3\\gg1$ being a large number. But if changing the unit from $mg$ to $kg$, we will have $I\\sim 10^{-3}kg$ with $10^{-3}\\ll1$ being a small number. A mathematician\nwould ask: \"Could you still treat your $I$ as a divergent quantity?\"Who could answer his question?\n\nExample B: In the $\\bar{MS}$ scheme, the renormalization mass scale $\\mu$ is introduced as follows (see p.137 in \\cite{2})\n\\begin{equation}\\label{3}\n\\dfrac{\\Gamma(2-d\/2)}{(4\\pi)^{d\/2}(m^2)^{2-d\/2}}=\\dfrac{1}{(4\\pi)^2}\\left[\\dfrac{2}{\\varepsilon}-\\gamma\n+\\ln(4\\pi)-\\ln(m^2)\\right]\\to \\dfrac{1}{(4\\pi)^2}\\left[-\\ln\\left(\\dfrac{m^2}{\\mu^2}\\right)\\right]\n\\end{equation}\nwhere $m$ is some mass parameter containing in the model. Eq.(\\ref{3}) is derived from the \"dimensional regularization\"\nmethod. The 4-dimensional (Euclidean) space has been analytically continued into $d$-dimensional one with $\\varepsilon\\sim 4-d\\sim 0$\nand Gamma function $\\Gamma(2-d\/2)=\\Gamma(\\varepsilon\/2)=2\/\\varepsilon-\\gamma$ ($\\gamma=0.4772\\ldots$ is the Euler constant).\nObviously, the left-handed-side ($LHS$) has a dimension of $m^{-\\varepsilon}$, whereas in the right-handed-side ($RHS$),\nthe function $\\ln\\left(\\frac{m^2}{\\mu^2}\\right)$ becomes dimensionless after the $\\mu$ is introduced. However, the mathematician would feel quite uncomfortable because $\\varepsilon\\neq0$. He will focus on the middle of Eq.(\\ref{3}) and\nask: \"Why the divergent number $2\/\\varepsilon$ disappears at the $RHS$ and becomes finite? Where the term $\\ln(m^2)$ comes\nfrom? What is its dimension?\"\n\nWe physicists accept Eq.(\\ref{3}) since it could be derived from a \"mathematical formula\" like (see p.57 in \\cite{3})\n\\begin{equation}\\label{4}\n(m^2)^{d\/2-2}=\\exp\\left[\\left(\\frac{d}{2}-2\\right)\\ln m^2\\right]\\simeq 1+\\left(\\frac{d}{2}-2\\right)\\ln m^2\n\\end{equation}\nThen the mathematician would say: \"No! In the mathematical formula\n\\begin{equation}\\label{5}\nx^y=\\exp[y\\ln x]\n\\end{equation}\nboth $x$ and $y$ must be dimensionless numbers. So in the $LHS$ of Eq.(\\ref{3}), you correctly write down:\n\\begin{equation*}\\label{}\n(4\\pi)^{-d\/2}=(4\\pi)^{2-d\/2-2}=\\dfrac{1}{(4\\pi)^2}\\exp\\left[(2-\\frac{d}{2})\\ln(4\\pi)\\right]\\simeq\n\\dfrac{1}{(4\\pi)^2}\\left[1+(2-\\frac{d}{2})\\ln(4\\pi)\\right]\n\\end{equation*}\nBut Eq.(\\ref{4}) is wrong because $x=m^2$ is a physical quantity with dimension. That's why you got a strange result at the $RHS$ of Eq.(\\ref{3})\".\n\nHence if insisting on mathematical rigor, we should admit that the introduction of $\\mu$ via Eq.(\\ref{3}) is groundless.\nThen a question arises: Why the $\\mu$ seems necessary in $QCD$ ?\n\nTo our understanding, the answer lies in the fact that in high energy $QCD$, the quarks' masses were often neglected. Therefore, in order to express the running coupling constant ($RCC$) of strong interaction, $\\alpha_s$, as a function of\n$Q$, the 3-dimensional momentum transfer in collision, one needs $\\mu$ as shown by the solution of renormalization-group-equation ($RGE$) (see p.532 of \\cite{2} and Eq.(\\ref{53}) below):\n\\begin{equation}\\label{6}\n\\alpha_s(Q)=\\dfrac{\\alpha_s(\\mu)}{1+\\alpha_s(\\mu)\\frac{\\beta_0}{2\\pi}\\ln(Q\/\\mu)}\n\\end{equation}\nIt is interesting to solve Eq.(\\ref{6}) for $\\alpha_s(\\mu)$, yielding\n\\begin{equation}\\label{7}\n\\alpha_s(\\mu)=\\dfrac{\\alpha_s(Q)}{1+\\alpha_s(Q)\\frac{\\beta_0}{2\\pi}\\ln(\\mu\/Q)}\n\\end{equation}\\footnotemark[1]\n\\footnotetext[1]{see Appendix of \\cite{4}, where a typing error exists in the denominator of Eq.(A.5), \"$+$\" should be \"$-$\".}\nWe see that Eqs.(\\ref{6}) and (\\ref{7}) are symmetrical with respect to mutual change of $Q\\leftrightarrow \\mu$. $Q$ and $\\mu$ are\nessentially equivalent. Why we need both of them? The answer is: only $Q\/\\mu$ is capable of expressing a dimensionless\n$\\alpha_s$. However, as shown in Eq.(\\ref{3}), the existence of $\\mu$ is doubtful, even superfluous. Once we take the quarks' masses into account, there will be no need of $\\mu$ at all (see section \\ref{sec:hadronization} below).\n\n\\section{Self-Energy Correction of an Electron, Lamb Shift}\n\\label{sec:selfenergy}\n\n\\vskip 0.1cm\n\nAs is well known (see \\eg, Refs.\\cite{4,5,6}), the $FDI$ of a free electron's self-energy at one loop ($L=1$) level of $QED$\nin covariant form reads\n\\begin{equation}\\label{8}\n-i\\Sigma(p)=(ie)^2\\int\\dfrac{d^4k}{(2\\pi)^4}\\dfrac{g_{\\mu\\nu}}{ik^2}\\gamma^\\mu\\dfrac{i}{\\pslash{p}-\\pslash{k}-m}\\gamma^\\nu\n\\end{equation}\nwhere the Bjorken-Drell metric ($\\pslash{p}=\\gamma^\\mu p_\\mu$) and rationalized Gaussian units are adopted with electron charge $-e\\,(e>0)$ and mass $m=m_e$. In Eq.(\\ref{8}), $p$ and $k$ are momenta of electron and (virtual) photon. After\nintroducing the Feynman parameter $x$ and making a shift in momentum integration: $k\\to K=k-xp$, Eq.(\\ref{8}) is recast into\n\\begin{equation}\\label{9}\n-i\\Sigma(p)=-e^2\\int_0^1dx[-2(1-x)\\pslash{p}+4m]I\n\\end{equation}\nwith\n\\begin{equation}\\label{10}\nI=\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{1}{(K^2-M^2)^2},\\quad M^2=p^2x^2+(m^2-p^2)x\n\\end{equation}\nbeing a logarithmically divergent integral, see Eq.(\\ref{1}).\n\nNote that in Eq.(\\ref{10}) we can change the unit of $M$ (and $K$) at our disposal without any change in the value of $I$,\nwhich is just a \"dimensionless\", \"large\" but \"uncertain\" number. However, in the past, we used to pay too much attention to its feature of being \"large\", trying to curb the divergence by means of some regularization method, which led to complicated\nrenormalization schemes ($RS$).\n\nBy contrast, now we believe the more important, even essential feature of a divergence is hiding in its \"uncertainty\".\nTo stress this cognition, we just use a simple trick to regulate the $I$ in Eq.(\\ref{10}) as follows.\n\nTo render it convergent, we perform a differentiation with respect to the mass-square parameter $M^2$, yielding\n\\begin{equation}\\label{11}\n\\dfrac{\\partial I}{\\partial M^2}=2\\int\\dfrac{d^4K}{(2\\pi)^4}\\dfrac{1}{(K^2-M^2)^3}=\\dfrac{-i}{(4\\pi)^2}\\dfrac{1}{M^2}\n\\end{equation}\nThen we reintegrate Eq.(\\ref{11}) with respect to $M^2$ and arrive at\n\\begin{equation}\\label{}\nI=\\dfrac{-i}{(4\\pi)^2}(\\ln M^2+C_1)=\\dfrac{-i}{(4\\pi)^2}\\ln\\dfrac{M^2}{\\mu_2^2}\n\\end{equation}\nwhere an arbitrary constant $C_1=-\\ln \\mu_2^2$ (with $\\mu_2$ a mass scale to be fixed later) is introduced so that the\nambiguity of dimension in the $\\ln M^2$ term can be eliminated.\n\nFurther integration of Eq.(\\ref{9}) with respect to $x$ leads to ($\\alpha=\\frac{e^2}{4\\pi}$)\n\\begin{equation}\\label{13}\\begin{array}{l}\n\\Sigma(p)=A+B\\pslash{p}, \\\\[5mm]\nA=\\dfrac{\\alpha}{\\pi}m\\left[2-2\\ln\\dfrac{m}{\\mu_2}+\\dfrac{(m^2-p^2)}{p^2}\\ln\\dfrac{(m^2-p^2)}{m^2}\\right], \\\\[5mm]\nB=\\dfrac{\\alpha}{4\\pi}\\left\\{2\\ln\\dfrac{m}{\\mu_2}-3-\\dfrac{(m^2-p^2)}{p^2}\\left[1+\\dfrac{(m^2+p^2)}{p^2}\n\\ln\\dfrac{(m^2-p^2)}{m^2}\\right]\\right\\}\n\\end{array}\n\\end{equation}\nUsing the chain approximation, we can derive the modification on the electron propagator as\n\\begin{equation}\\label{14}\n\\dfrac{i}{\\pslash{p}-m}\\to \\dfrac{i}{\\pslash{p}-m}\\dfrac{1}{1-\\frac{\\Sigma(p)}{\\pslash{p}-m}}=\\dfrac{iZ_2}{\\pslash{p}-m_R}\n\\end{equation}\nwhere\n\\begin{equation}\\label{15}\nZ_2=\\dfrac{1}{1-B}\n\\end{equation}\nis the renormalization factor for electron's wave function and\n\\begin{equation}\\label{16}\nm_R=\\dfrac{m+A}{1-B}\n\\end{equation}\nis the renormalized mass of $m$. The increment of mass reads\n\\begin{equation}\\label{17}\n\\delta m=m_R-m=\\dfrac{A+mB}{1-B}\n\\end{equation}\nIn the past, many physicists viewed $\\delta m$ as some real contribution of \"radiation correction\". While $m_R$ should be\nidentified with the observed mass $m_{obs}$, or physical mass $m_e$, the original $m$ (usually denoted by $m_0$ or $m_B$ in\nthe expression of Lagrangian density) was thought to be a \"bare mass\". Both $\\delta m$ and $m_0$ were divergent quantities.\n(see, \\eg, p.220 in \\cite{2}).\n\nWe don't think so. Let us read carefully the seminal paper by Bethe in 1947\\cite{8}. The theory for the hydrogenlike atom begins with a Hamiltonian in the center-of-mass frame\n\\begin{equation}\\label{18}\nH_0=\\dfrac{{\\bf p}^2}{2m}+\\dfrac{{\\bf p}^2}{2m_N}-\\dfrac{Z\\alpha}{r}=\\dfrac{{\\bf p}^2}{2\\mu}-\\dfrac{Z\\alpha}{r}\n\\end{equation}\nBethe pointed out that the effect of electron's interaction with the vector potential $\\bf A$ of radiation field\n\\begin{equation}\\label{19}\nH_{int}=\\dfrac{e}{mc}{\\bf A}\\cdot{\\bf p}\n\\end{equation}\nshould properly be regarded as already included in the $m_{obs}$, which is denoted by $m$ in Eqs.(\\ref{18}) and (\\ref{19}).\n\nIn our understanding on Bethe's claim, the \"self-interaction\" of electron with radiation field is indivisible from the free\nelectron mass $m$. In other words, in the covariant form of $QED$, certain contributions of $FDIs$ for \"self-energy\" (with\nEq.(\\ref{8}) being merely that at $L=1$ order) at all orders (up to $L\\to \\infty$) are already contained in the value of $m$. To show this cognition, we impose the mass-shell condition $p^2=m^2$ in Eq.(\\ref{17}) together with\n\\begin{equation}\\label{20}\n\\delta m |_{p^2=m^2}=\\dfrac{\\alpha m}{4\\pi}(5-6\\ln\\frac{m}{\\mu_2})=0\n\\end{equation}\nwhich in turn fixes the arbitrary constant $\\mu_2$ to be\n\\begin{equation}\\label{21}\n\\mu_2=me^{-5\/6}\n\\end{equation}\nand thus\n\\begin{equation}\\label{22}\nZ_2|_{p^2=m^2}=\\dfrac{1}{1+\\frac{\\alpha}{3\\pi}}\\simeq 1-\\dfrac{\\alpha}{3\\pi}\n\\end{equation}\nNote that $m_R=m=m_{obs}=m_e$ with no bare mass at all and $Z_2$ is fixed and finite, in sharp contrast to that in\nprevious theories.\n\nOur reader may wonder: \"In this case, does the calculation on $FDI$ for the self-energy become worthless ?\" The answer\nis \"No\" due to two reasons. First, at the $QM$ level, the parameters $m$ and $e$ in Eqs.(\\ref{18}) and (\\ref{19}) can be\nregarded as well-defined. But they are not so at the level of $QED$. As discussed before Eq.(\\ref{20}), the new effect of\nradiative corrections of $FDIs$ for self-energy is inevitably confused with that in the mass, the dividing line between\nthem is blurred. In some sense, the appearance of divergence in the $FDI$ is just a warning: the new effect you want to calculate has become entangled with the mass $m$, rendering both of them uncertain. Hence the aim of so-called mass\nrenormalization is nothing but a reconfirmation of $m$ as we did in Eqs.(\\ref{20})-({22}), where the mass $m$ is renormalized on the mass-shell $p^2=m^2$ with $m=m_e$ being fixed by the experimental value and thus well-defined. This is one important thing we must do and at most we can do on the mass-shell for a free electron.\n\nSecond, the increment of mass, $\\delta m$, ceases to be zero once when the electron is moving off-mass-shell ($p^2\\neq m^2$). Then Eq.(\\ref{17}) will provide some information about the new effect of radiation corrections. For example, for a bound electron in a hydrogenlike atom, in Ref.\\cite{7}, we replace the electron mass $m=m_e$ by reduced mass $\\mu=\\frac{m_em_N}{m_e+m_N}$ (not to be confused with the $\\mu$ in $QCD$) and write (see also \\cite{29}):\n\\begin{equation}\\label{23}\np^2=\\mu^2(1-\\zeta)\n\\end{equation}\nHere a dimensionless parameter $\\zeta\\,(>0)$ is introduced to show (on average) how large the extent of \"off-mass-shell\" is.\nSubstitution of Eq.(\\ref{23}) into Eq.(\\ref{17}) yielding\n\\begin{equation}\\label{24}\n\\delta \\mu\\simeq \\dfrac{\\alpha \\mu}{4\\pi}\\dfrac{(-\\zeta+2\\zeta\\ln\\zeta)}{1+\\frac{\\alpha}{3\\pi}}\n\\end{equation}\nwhere some terms of the order of $\\zeta^2$ or $\\zeta^2\\ln\\zeta$ are neglected since $\\zeta\\ll1$.\n\nAs a perturbative calculation at $L=1$ order, we may ascribe $\\delta\\mu$ to the (minus) binding energy $B$ of electron\nin the Bohr theory\n\\begin{equation}\\label{25}\n\\delta \\mu=\\varepsilon_n=-B=-\\dfrac{Z^2\\alpha^2}{2n^2}\\mu\n\\end{equation}\nCombination of Eqs.(\\ref{24}) and (\\ref{25}) gives the value of $\\zeta=\\zeta^{<S>}$ with the superscript $<S>$ referring to\n\"self-energy (at $L=1$ order)\".\n\nAnother \"nonperturbative\" method to fix the $\\zeta$ in Eq.(\\ref{23}) is to resort to the Virial theorem: For an electron in the Coulomb potential $V=-\\frac{Z\\alpha}{r}$, its kinetic energy $T=\\frac{1}{2\\mu}{\\bf p}^2$ can be evaluated on average as\n\\begin{equation}\\label{26}\n<{\\bf p}^2>=2\\mu<T>=2\\mu[-B-<V>]=2\\mu B\n\\end{equation}\n\\begin{equation}\\label{27}\n<p^2>=<E^2-{\\bf p}^2>=<(\\mu-B)^2-{\\bf p}^2>\\simeq \\mu^2(1-\\dfrac{4B}{\\mu})\n\\end{equation}\nComparing Eq.(\\ref{27}) with Eq.(\\ref{23}), we obtain\n\\begin{equation}\\label{28}\n\\zeta^{<V>}=\\dfrac{4B}{\\mu}=\\dfrac{2Z^2\\alpha^2}{n^2}\n\\end{equation}\nwhere the superscript $<V>$ refers to \"Virial theorem\".\n\nIn Ref.\\cite{7}, for explaining the Lamb shift of energy levels in hydrogenlike atoms, we find the result being expressed\nin terms of $\\zeta$. Throughout the entire calculation, all ultraviolet divergences are handled like that in Eqs.(\\ref{9})-({12}) while the infrared divergence disappears due to the introduction of $\\zeta$. However, the formulas\nare still approximate and either one of $\\zeta^{<S>}$ and $\\zeta^{<V>}$ is not reliable. So in the following table $I$,\nnot only $\\zeta^{<S>}$ and $\\zeta^{<V>}$, but also two kinds of \"average\", $\\zeta^{<S+V>}=\\frac{1}{2}(\\zeta^{<S>}+\\zeta^{<V>})$ and $\\zeta^{<SV>}=\\sqrt{(\\zeta^{<S>}\\zeta^{<V>}}$ are given.\n\n\\vskip 0.5cm\n\\begin{small}\\hspace*{-16mm}\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\multicolumn{8}{c}{Table I. Off-mass-shell parameter $\\zeta$ and $\\ln\\zeta$}\\\\[5pt]\n  \\hline\n  $\\frac{Z^2}{n^2}$ & $\\zeta^{<S>}\\times 10^4$ &-$\\ln\\zeta^{<S>}$ & $\\zeta^{<V>}\\times 10^6$ & -$\\ln\\zeta^{<V>}$ &\n  $\\zeta^{<S+V>}\\times 10^5$ & -$\\ln\\zeta^{<S+V>}$ & $\\zeta^{<SV>}\\times 10^5$ & $-\\ln\\zeta^{<SV>}$ \\\\\n  \\hline\n  $\\frac{1}{16}$ & $1.546093458$ & $8.77461$ & $\\frac{\\alpha^2}{8}=6.6564192$\n   &11.91992886 & $8.0632$ & 9.425609 & 3.2080284 & 10.34727 \\\\\n   \\hline\n  $\\frac{1}{4}$ & $7.446539697$ & 7.20259 & $\\frac{\\alpha^2}{2}=26.6256771$\n  & 10.5336345 & $38.5639$ & 7.860609 & 14.0808 & 8.86816225 \\\\\n   \\hline\n 1 & $37.73719345$ & 5.57969 & $2\\alpha^2=106.502$ &9.147340142 &\n $194.011$ & 6.2450103 & 63.39626 & 7.36351521 \\\\\n  \\hline\n\\end{tabular}\\end{small}\n\\vspace{0.5cm}\n\\normalsize\n\nThere are $8$ cases discussed in \\cite{7}. The first one is the hydrogen atom's \"classical Lamb shift\" measured as:\n\\begin{equation}\\label{29}\nL_H^{exp}(2S-2P)=E_H(2S_{1\/2})-E_H(2P_{1\/2})=1057.845\\;MHz\n\\end{equation}\nTheoretically, the radiative correction (at $L=1$ order) makes the dominant contribution, yielding:\n\\begin{equation}\\label{30}\\begin{array}{l}\n\\Delta E_H^{Rad<S>}(2S-2P)=1000.657\\;MHz\\\\\n\\Delta E_H^{Rad<S+V>}(2S-2P)=1089.651\\;MHz\\\\\n\\Delta E_H^{Rad<SV>}(2S-2P)=1226.087\\;MHz\\\\\n\\Delta E_H^{Rad<V>}(2S-2P)=1451.791\\;MHz\n\\end{array}\n\\end{equation}\nTaking the small contribution from the nuclear size effect into account, we adopt the $<S+V>$ scheme to obtain\n\\begin{equation}\\label{31}\nL_H^{theor}(2S-2P)=1089.794\\;MHz\n\\end{equation}\nwhich is larger than the experimental value, Eq.(\\ref{29}), by $3\\%$.\n\nThe most interesting case is the $1S-2S$ two-photon transition in hydrogen $H$ or deuterium $D$ because its natural width\nis so tiny ($1.3Hz$) and thus allows precision measurement in recent years\\cite{9}:\n\\begin{equation}\\label{32}\n\\Delta E_{H}^{exp}(1S-2S)=2466061413187.34(84)\\;kHz\n\\end{equation}\nThe isotope shift of $1S-2S$ transition between $H$ and $D$ had been measured first by Schmidt-Kalar \\etal \\cite{10} and\nquoted in \\cite{11} as:\n\\begin{equation}\\label{33}\n\\Delta E_{D-H}^{exp}(2S-1S)=670994337(22)\\;kHz\n\\end{equation}\nTheoretically, the above accurate data cannot be explained by the original Dirac equation with nucleus having mass $m_N\\to\\infty$. We propose a reduced Dirac equation ($RDE$) with electron mass replaced  by reduced mass for  $H$ and $D$ being respectively\n\\begin{equation}\\label{34}\n\\mu_H=\\dfrac{m_em_p}{m_e+m_p},\\quad \\mu_D=\\dfrac{m_em_d}{m_e+m_d}\n\\end{equation}\nThen theoretically, the $RDE$ predicts:\n\\begin{equation}\\label{35}\n\\Delta E_H^{RDE}(2S-1S)=2.466067984\\times 10^{15}\\;Hz\n\\end{equation}\n\\begin{equation}\\label{36}\n\\Delta E_{D-H}^{RDE}(2S-1S)=6.7101527879\\times 10^{11}\\;Hz\n\\end{equation}\nwhich are larger than the experimental values by only $3\\times10^{-6}$ and $3\\times10^{-5}$ respectively. Further\nradiative corrections on Eq.(\\ref{35}) will be sensitive to the choice of schemes in Table $I$, the best one is\n$\\Delta E_H^{Theor<SV>}(2S-1S)$, deviating from the experimental data, Eq.(\\ref{32}), by $-1\\times10^{-7}$ only. On the other hand, besides Eq.(\\ref{36}), the $\\Delta E_{D-H}^{Theor}(2S-1S)$ is influenced considerably by the nuclear size effect and so less sensitive to the scheme choice of the smaller radiative correction, bringing the discrepancy between theory\nand experimental data, Eq.(\\ref{33}), from  $3\\times10^{-5}$ down to $3\\times10^{-6}$ approximately.\n\n\n\\section{Renormalization Group Equation ($RGE$) for $QED$ and Its Solution}\n\\label{sec:renormalization}\n\n\\vskip 0.1cm\n\nIn $QED$, the $FDI$ for photon self-energy (\\ie, vacuum polarization) can also be evaluated \\cite{4}, bringing the\ncharge $e$ into its renormalized one:\n\\begin{equation}\\label{37}\ne^2\\to e^2_R=Z_3e^2\n\\end{equation}\n\\begin{equation}\\label{38}\nZ_3=1+\\dfrac{\\alpha}{3\\pi}\\left(\\ln\\dfrac{m^2}{\\mu^2_3}-\\dfrac{q^2}{5m^2}+\\cdots\\right)\n\\end{equation}\nHere $m$ is the fermion (say, electron) mass, $q$ is the momentum of photon and $\\mu_3$ is an arbitrary constant emerging from the treatment on the divergence like that in Eqs.(\\ref{9})-({12}). Although Eq.(\\ref{37}) looks like that in previous theories, it is really a new one: $e$ is the observed (physical) charge, not a \"bare charge\", and $Z_3$ remains finite.\n\nThe vertex function between two fermions' momenta $p$ and $p'$ with $p'-p=q$ will give another $Z_1$ \\cite{4,7}. Adding\nall the $FDIs$, we find the renormalized charge being:\n\\begin{equation}\\label{39}\ne_R=\\dfrac{Z_2}{Z_1}Z_3^{1\/2}e\n\\end{equation}\nBut the Ward-Takahashi identity ($WTI$) implies that\\cite{6}\n\\begin{equation}\\label{40}\nZ_1=Z_2\n\\end{equation}\nHence Eq.(\\ref{37}) remains valid and\n\\begin{equation}\\label{41}\ne_R(Q)=e\\left\\{1+\\dfrac{\\alpha}{2\\pi}\\left[\\dfrac{1}{3}\\ln\\dfrac{m^2}{\\mu^2_3}+\\dfrac{1}{15}\\dfrac{Q^2}{m^2}+\\cdots\\right]\\right\\}\n\\end{equation}\nwhere $Q^2=-q^2>0$, with $Q$ being the 3-dimensional momentum transfer at fermion collision. The observed charge should be defined at $Q\\to 0$ (Thomson scattering limit):\n\\begin{equation}\\label{42}\ne_{obs}=e_R|_{Q=0}=e\n\\end{equation}\nwhich dictates that\n\\begin{equation}\\label{43}\n\\mu_3=m\n\\end{equation}\nAs usual, the beta function is defined as\n\\begin{equation}\\label{44}\n\\beta(\\alpha,Q)\\equiv Q\\dfrac{\\partial}{\\partial Q}\\alpha_R(Q)\n\\end{equation}\nFrom Eq.(\\ref{41}), it is found in \\cite{4} that\n\\begin{equation}\\label{45}\n\\beta(\\alpha,Q)=\\dfrac{2\\alpha^2}{3\\pi}-\\dfrac{4\\alpha^2m^2}{\\pi Q^2}\\left[1+\\dfrac{2m^2}{\\sqrt{Q^4+4m^2Q^2}}\n\\ln\\dfrac{\\sqrt{Q^4+4m^2Q^2}-Q^2}{\\sqrt{Q^4+4m^2Q^2}+Q^2}\\right]\n\\end{equation}\n\\begin{equation}\\label{46}\n\\beta(\\alpha,Q)=\\dfrac{2\\alpha^2}{15\\pi}\\dfrac{Q^2}{m^2},\\quad (\\dfrac{Q^2}{m^2}\\ll1)\n\\end{equation}\nEvidently, the $e_R(Q)$ will increase with $Q$, becoming a running coupling constant ($RCC$). To calculate it, usually\na renormalization-group-equation ($RGE$) was derived for $QED$ by setting $\\alpha\\to\\alpha_R(Q)$ and $Q\\to\\infty$ in\n$\\beta(\\alpha,Q)$ yielding:\n\\begin{equation}\\label{47}\nQ\\dfrac{d}{dQ}\\alpha_R=\\dfrac{2\\alpha_R^2}{3\\pi}\n\\end{equation}\nwith its solution\n\\begin{equation}\\label{48}\n\\alpha_R(Q)=\\dfrac{\\alpha}{1-\\frac{2\\alpha}{3\\pi}\\ln\\frac{Q}{m}}\n\\end{equation}\nHere, the renormalization was forced to be made at $Q^2=m^2$ so that\n\\begin{equation}\\label{49}\n\\alpha_R|_{Q^2=m^2}=\\alpha\n\\end{equation}\nThis is inconsistent with the physical condition, Eq.(\\ref{42}), a defect due to ignoring the mass $m$, which plays a\ndominant role at low $Q$ region as shown by the Eq.(\\ref{46}). As an improvement, a more practical $RGE$ is constructed\nin \\cite{4} by changing $\\alpha$ into $\\alpha_R(Q)$ in Eq.(\\ref{45}) for the entire $Q$ region and adding up contributions from $9$ elementary charged\nfermions ($e,\\mu,\\tau,u,d,s,c,b,t$). Then $\\alpha_R$ can be numerically calculated as a function of $\\ln(Q\/m_e)$,\nstarting from\n\\begin{equation}\\label{50}\n\\alpha_R(Q=0)=\\alpha=(137.03599)^{-1}\n\\end{equation}\nand passing through another experimental data point \\cite{12}\n\\begin{equation}\\label{51}\n\\alpha_R(Q=M_Z=91.1880\\,GeV)=(128.89)^{-1}\n\\end{equation}\nIn this way, after adopting three heavy quarks' masses as $m_c=1.031\\,GeV,\\; m_b=4.326\\,GeV$ (see section 9.5D in \\cite{13}) and $m_t=175\\,GeV$, three light quarks masses\n\\begin{equation}\\label{52}\nm_u=8\\,MeV,\\;m_d=10\\,MeV,\\;m_s=200\\,MeV\n\\end{equation}\n(or averaged mass for $u,d,s$ being $92\\ MeV$) can be fitted as shown in Fig.1 of Ref.\\cite{4}.\n\n\n\\section{$RGE$ for $QCD$, Threshold Energies of Quarks Hadronization}\n\\label{sec:hadronization}\n\n\\vskip 0.1cm\n\nDifferent from the $RCC$ in $QED$, the $RCC$ in $QCD$, denoted by $\\alpha_s(Q)=\\frac{1}{4\\pi}g_s^2(Q)$, is much larger and\ndecreases with the increase of $Q$. Usually, the $RGE$ for $QCD$ reads (see p.551-552 in \\cite{2})\n\\begin{equation}\\label{53}\nQ\\dfrac{d}{dQ}\\alpha_s(Q)=\\beta(\\alpha_s(Q))=-\\dfrac{\\beta_0}{2\\pi}\\alpha_s^2(Q)\n\\end{equation}\nwhere\n\\begin{equation}\\label{54}\n\\beta_0=11-\\dfrac{2}{3}n_f\n\\end{equation}\nwith $n_f$ being the number of quarks' flavor. Eq.(\\ref{53}) looks similar to Eq.(\\ref{47}), but the negative sign in beta\nfunction implies the property of asymptotic freedom in the strong interaction. Solution of Eq.(\\ref{53}) is give by\nEq.(\\ref{6}), also bears some resemblance to Eq.(\\ref{48}).\n\nLet us make a comparison between Eqs.(\\ref{6}) and (\\ref{48}). Besides the difference in sign (\"$+$\" versus \"$-$\") in the\ndenominator, there is another one: The mass scale $\\mu$ remains arbitrary in $QCD$ whereas $m$ in $QED$ means the mass of an\nobserved charged fermion (usually electron).\n\nAs we already see, even for $QED$, one mass ($m_e$) is far from enough, let alone in $QCD$, where quarks' masses are much\nheavier. How can we ignore them?\n\nUsually, to remove the arbitrary mass scale $\\mu$, a parameter $\\Lambda_{QCD}$ is often defined via equation\n\\begin{equation}\\label{55}\n\\dfrac{\\beta_0}{2\\pi}\\alpha_s(\\mu)\\ln\\left(\\dfrac{\\mu}{\\Lambda_{QCD}}\\right)=1\n\\end{equation}\nsuch that a simpler formula for $\\alpha_s(Q)$ can be found as\n\\begin{equation}\\label{56}\n\\alpha_s(Q)=\\dfrac{2\\pi}{\\beta_0\\ln(Q\/\\Lambda_{QCD})}\n\\end{equation}\nThen the precision experimental data \\cite{1}\n\\begin{equation}\\label{57}\n\\alpha_s(M_Z=91.1876\\,GeV)=0.1176\n\\end{equation}\nserves as a substitute for $\\alpha_s(\\mu)$ in Eq.(\\ref{55}), yielding\n\\begin{equation}\\label{58}\n\\Lambda_{QCD}=M_Z\\exp\\left[\\dfrac{-2\\pi}{\\alpha_s(M_Z)\\beta_0}\\right]\n=\\left\\{\\begin{array}{l}\n          240\\;MeV,\\;(n_f=3) \\\\\n          150\\;MeV,\\;(n_f=4) \\\\\n          85.8\\;MeV,\\;(n_f=5) \\\\\n          44.2\\;MeV,\\;(n_f=6)\n        \\end{array}\\right.\n\\end{equation}\nIt is evident from Eq.(\\ref{56}) that\n\\begin{equation}\\label{59}\n\\alpha_s(\\Lambda_{QCD})\\to\\infty\n\\end{equation}\nwhich implies the \"infrared confinement\" of quarks.\n\nHowever, the value of $\\Lambda_{QCD}$ sensitively depends on the flavor number $n_f$ as shown by Eq.(\\ref{58}) but\nis independent of the concrete flavor of a quark under consideration. Moreover, the divergence of $\\alpha_s$ appears\nat $\\Lambda_{QCD}$. These features seem not so reasonable and are not consistent with the experimental fact that the lighter\na quark's mass is, the lower its \"threshold energy for hadronization\" will be.\n\nThe way out of above difficulties is clear. Just like in Eq.(\\ref{45}) for $QED$, where rather than just the first term,\nall terms for $9$ fermions should be added, now for $RGE$ in $QCD$, all masses of $6$ quarks should be preserved. In this\nway, the $\\alpha_{si}(Q)$ are numerically calculated for $i=u,d,s,c,b$ respectively in Ref.\\cite{4}. Starting from\n$\\alpha_s(M_Z)=0.118$ (the common renormalization point), their running curves (as shown by Figs.2 and 3 in \\cite{4}) follow the trend of experimental data (as shown on p.158 of Ref.\\cite{1}) quite well but separate at the low $Q$ region. Each of\nthem rises to a maximum $\\alpha_{si}^{max}$ at $Q=\\Lambda_i$ and then suddenly drops to zero at $Q=0$.\n\nFor example, for $b$ quark, $\\Lambda_b=7.04\\, GeV,\\;\\alpha_{sb}^{max}=0.161$. If tentatively explain $L_b\\sim \\hbar\/\\Lambda_b$\nas some critical length scale of $b\\bar{b}$ pair, then $L_b\\sim 0.02805\\,fm$. In \\cite{4}, it is further guessed that\n$E_b^{theor}\\sim \\alpha_{sb}^{max}\/L_b\\sim 1.133\\, GeV$ being the order of excitation energy for breaking the binding\n$b\\bar{b}$ pair, \\ie, the hadronization threshold energy of Upsilon $\\Upsilon(b\\bar{b})$ against its dissociation into\ntwo bosons. It is indeed the case found experimentally\\cite{1}:\n\\begin{equation}\\label{60}\nM(\\Upsilon(4S))-M(\\Upsilon)=1.12\\,GeV,\\quad M(\\Upsilon(4S))\\to B^+B \\;\\text{or}\\;B^0\\bar{B^0}\n\\end{equation}\nSimilarly, we can estimate from Ref.\\cite{4} that $E_c^{theor}\\sim0.398\\,GeV$ which also corresponds to experiments that\\cite{1}\n\\begin{equation}\\label{61}\nM(\\psi(3770))-M(\\psi(3097))=673\\,MeV,\\quad \\psi(3770)\\to D^+D^-\\;\\text{or}\\;D^0\\bar{D^0}\n\\end{equation}\nIt seems that $E_s^{theor}\\sim90\\,MeV$ and $E_{u,d}^{theor}\\sim 0.4\\,MeV$ are not so reliable but still reasonable.\n\n\n\\section{$\\lambda \\phi^4$ Model and Higgs Mass in the Standard Model}\n\\label{sec:higgsmass}\n\n\\vskip 0.1cm\n\nThe Lagrangian density of $\\lambda\\phi^4$ ($\\phi(x)$ is a real scalar field) model is defined by\n\\begin{equation}\\label{62}\n{\\cal L}=\\dfrac{1}{2}\\partial_\\mu\\phi\\partial^\\mu\\phi+\\dfrac{1}{2}\\sigma\\phi^2-\\dfrac{\\lambda}{4!}\\phi^4\n\\end{equation}\nThe importance of this model lies in the \"wrong sign\" of mass term ($\\sigma=-m^2>0$) which leads to the spontaneous\nsymmetry breaking ($SSB$) at the tree level ($L=0$). The effective potential ($EP$) reads\n\\begin{equation}\\label{63}\nV_0(\\phi)=-\\dfrac{1}{2}\\sigma\\phi^2+\\dfrac{\\lambda}{4!}\\phi^4\n\\end{equation}\n(The subscript \"$0$\" refers to $L=0$). Obviously, $V_0(\\phi)$ has two extremum, one is a maximum:\n\\begin{equation}\\label{64}\n\\phi_0=0\\quad\\text{(symmetric phase)}\n\\end{equation}\nwhile the other one is a minimum:\n\\begin{equation}\\label{65}\n\\phi_1^2=\\dfrac{6\\sigma}{\\lambda}\\quad\\text{(SSB phase)}\n\\end{equation}\nAt the $QFT$ level, the $EP$ evolves into\n\\begin{equation}\\label{66}\nV=V_0+V_1+\\cdots\n\\end{equation}\nThe theory for $EP$ had been developed by various authors \\cite{14,15,16,17,18}, with $L=1$ contribution to $EP$ being\nevaluated as:\n\\begin{equation}\\label{67}\nV_1(\\phi)=\\dfrac{1}{2}\\int\\dfrac{d^4k_E}{(2\\pi)^4}\\ln(k_E^2-\\sigma+\\frac{1}{2}\\lambda\\phi^2)\n\\end{equation}\nThis highly divergent integral (in 4-dimensional Euclidean momentum space) is treated in Ref.\\cite{19} like that for\nEqs.(\\ref{10})-({12}). First, three times of differentiation with respect to $M^2=-\\sigma+\\frac{1}{2}\\lambda\\phi^2$\nare needed before it becomes just convergent.\n\\begin{equation}\\label{68}\n\\dfrac{\\partial^3V_1}{\\partial(M^2)^3}=\\int\\dfrac{d^4k_E}{(2\\pi)^4}\\dfrac{1}{(k_E^2+M^2)^3}=\\dfrac{1}{2(4\\pi)^2M^2}\n\\end{equation}\nSecond, three times integration with respect to $M^2$ are performed, yielding\n\\begin{equation}\\label{69}\nV_1(\\phi)=\\dfrac{1}{2(4\\pi)^2}\\left\\{\\dfrac{M^4}{2}(\\ln M^2-\\dfrac{1}{2})-\\dfrac{1}{2}M^4+\\dfrac{1}{2}C_1M^4+C_2M^2+C_3\\right\\}\n\\end{equation}\nAs expected, three arbitrary constants $C_1,C_2,C_3$ appear. The renormalization amounts to fix them at our disposal.\n\nThird, like that in Eq.(\\ref{3}), for eliminating the ambiguity of dimension in the first term involving $\\ln M^2$,\nthe only possible choice of $C_1=-\\ln\\mu^2$ is fixed. Then the choice of $C_2=\\mu^2=2\\sigma$ and $C_3=-\\sigma^2+(4\\pi)^2\\frac{3\\sigma^2}{\\lambda}$ leads to $V=V_0+V_1$ with its derivatives being given at the Table II\n\\cite{19}.\n\n\\renewcommand\\arraystretch{1.5}\n\\begin{center}\n\\hspace*{-16mm}\\begin{tabular}{|c|c|c|}\n\\multicolumn{2}{c}{Table II. Effective potential of $\\lambda\\phi^4$ model with $SSB$}\\tabularnewline\n  \\hline\n    & $SSB$ phase & symmetric phase \\tabularnewline\n  \\hline\n  $\\phi$ & $\\phi_1=\\sqrt{\\dfrac{6\\sigma}{\\lambda}}$  & $\\phi_0=0$ \\tabularnewline\n  \\hline\n  $V$ & $0$  & $-\\dfrac{\\sigma^2}{2(4\\pi)^2}\\left[\\dfrac{15}{4}+\\dfrac{1}{2}\\ln2-i\\dfrac{\\pi}{2}\\right]+\\dfrac{3}{2}\\dfrac{\\sigma^2}{\\lambda}$ \\tabularnewline\n   \\hline\n  $\\dfrac{dV}{d\\phi}$ & $0$ & $0$ \\tabularnewline\n   \\hline\n  $\\dfrac{d^2V}{d\\phi^2}$ & $2\\sigma$  & $-\\sigma\\left[1-\\dfrac{\\lambda}{2(4\\pi)^2}(3+\\ln2-i\\pi)\\right]$\\tabularnewline\n   \\hline\n  $\\dfrac{d^3V}{d\\phi^3}$ & $\\lambda\\sqrt{\\dfrac{6\\sigma}{\\lambda}}\\left[1+\\dfrac{3\\lambda}{2(4\\pi)^2}\\right]$  & $0$ \\tabularnewline\n   \\hline\n $\\dfrac{d^4V}{d\\phi^4}$ & $\\lambda\\left[1+\\dfrac{9\\lambda}{2(4\\pi)^2}\\right]$  & $\\lambda\\left[1-\\dfrac{3\\lambda}{2(4\\pi)^2}(\\ln2-i\\pi)\\right]$ \\tabularnewline\n  \\hline\n\\end{tabular}\n\\end{center}\n\nNote that, with the above assignment of $C_i\\,(i=1,2,3)$, both the position of the $SSB$ phase, $\\phi_1$, and the mass\n$m_\\sigma$ excited above it take the same expression as that at the tree level\n\\begin{equation}\\label{70}\nm_\\sigma^2=\\dfrac{d^2V}{d\\phi^2}|_{\\phi=\\phi_1}=2\\sigma\n\\end{equation}\nHowever, the renormalization coupling constant\n\\begin{equation}\\label{71}\n\\lambda_R=\\dfrac{d^4V}{d\\phi^4}|_{\\phi=\\phi_1}=\\lambda\\left[1+\\dfrac{9\\lambda}{2(4\\pi)^2}\\right]\n\\end{equation}\ndoes receive some quantum correction on its classical value $\\lambda$. Hence it is suddenly realized that the invariant meaning of $\\lambda$ in the Lagrangian, Eq.(\\ref{62}), is by no means a \"coupling constant\", but the ratio of two mass scales \\cite{21}.\n\\begin{equation}\\label{72}\n\\lambda=3\\dfrac{m_\\sigma^2}{\\phi_1^2}\n\\end{equation}\nTwo parameters, $\\sigma$ and $\\lambda$, together with Eqs.(\\ref{70}) and (\\ref{72}), should all be preserved through out\nhigh loop ($L$) evaluations in perturbation theory until $L\\to\\infty$, \\ie, in any nonperturbative treatment.\n\nOn the other hand, the above assignment of $C_i$ renders the appearance of imaginary part in $V$ and its derivatives at\nthe symmetric phase ($\\phi_0=0$). It means the instability of symmetric phase at the presence of stable $SSB$ phase.\n\nIt is interesting to see that an alternative choice of\n\\begin{equation}\\label{73}\nC_1=-\\ln(-\\sigma),\\;C_2=-\\sigma,\\;C_3=-\\dfrac{1}{4}\\sigma^2\n\\end{equation}\nwould leads to the survival of $\\phi_0=0$ as a semistable state with\n\\begin{equation}\\label{74}\nV(0)=\\dfrac{dV}{d\\phi}|_{\\phi=0}=\\dfrac{d^3V}{d\\phi^3}|_{\\phi=0}=0\n\\end{equation}\n\\begin{equation}\\label{75}\n\\dfrac{d^2V}{d\\phi^2}|_{\\phi=0}=-\\sigma,\\;\\dfrac{d^4V}{d\\phi^4}|_{\\phi=0}=\\lambda\n\\end{equation}\nwhereas no real $SSB$ solution exists. Hence we see that two different choices of $C_i$ lead two separable sectors in\nthe effective potential \\cite{19}.\n\nIn 1989, we had estimated the upper and lower bounds of Higgs mass $M_H$ in the standard model of particle physics by\nusing a nonperturbative approach in $QFT$ --- the Gaussian effective potential ($GEP$) method, yielding\\cite{20}\n\\begin{equation}\\label{76}\n76\\,GeV<M_H<170\\,GeV\n\\end{equation}\nLike many authors, we were bothered a lot by divergences. After a deeper understanding on $\\lambda\\phi^4$ model\\cite{19},\nthis problem was restudied in 1998 by a combination of $GEP$ with our $RRM$, yielding\n\\begin{equation}\\label{77}\nM_H=138\\,GeV\n\\end{equation}\nwhich is based on the following input of experimental data:\n\\begin{equation}\\label{78}\n\\begin{array}{l}\n  M_W=80.359\\,GeV,\\;M_Z=91.1884\\,GeV, \\\\\n  \\alpha^{-1}=\\dfrac{4\\pi}{g^2\\sin^2\\theta_W}=128.89,\\;\\sin^2\\theta_W=0.2317\n\\end{array}\n\\end{equation}\nwhere $\\theta_W$ is the weak mixing (Weinberg) angle. As now the search for Higgs particle becomes so urgent experimentally\nwhile the theoretical estimation about its mass still remains uncertain\\cite{1}, we think our method \\cite{21} with its\nprediction, Eq.(\\ref{77}), deserves to be reconsidered.\n\n\n\\section{Summary and Discussion}\n\\label{sec:discussion}\n\n\\vskip 0.1cm\n\nOur $RRM$ was first proposed by J. F. Yang in 1994\\cite{22}, then elaborated in a series of papers since 1998 (\\cite{19,21,\n23,24,4,7} \\etc). What we have been thinking about is: Why the divergence emerges inevitably in $QFT$? What is the essential meaning of a regularization and renormalization procedure?\n\nFor instance, in a pioneering work to explain the Lamb shift, Welton (\\cite{25}, see also section 9.6B in Ref.\\cite{13})\nencountered a divergent integral $I=\\int\\frac{d\\omega}{\\omega}$ with $\\omega$ being the (angular) frequency of virtual photon (vacuum fluctuation). He simply set the lower and upper bounds by $\\omega_{min}\\sim mZ\\alpha=\\frac{Z}{a}$ ($a$ is Bohr\nradius) and $\\omega_{max}\\sim m$ respectively, arriving at $I\\simeq \\ln(\\frac{1}{Z\\alpha})=4.92\\,(Z=1)$ which leads to\nan estimation of Lamb shift $L_H^{theor}(2S_{1\/2}-2P_{1\/2})\\simeq 668\\,MHz$. However, if instead of Bohr radius, the lower cutoff is provided by the electron's binding energy, one would get $I\\simeq \\ln(Z\\alpha)^{-2}$ and\n$L_H^{theor}(2S_{1\/2}-2P_{1\/2})\\simeq 1336\\,MHz$ (see Eq.(\\ref{30}) in the Ref.\\cite{26}). We see the integral $I$ being\na dimensionless number, not very large ($I\\leq 10$), but uncertain indeed. The root cause of uncertainty lies in the fact\nthat a reconfirmation process of electron mass like Eq.(\\ref{20}) was missing.\n\nFor further clarity, a study on Lamb shift in the form of noncovariant $QED$ was performed in \\cite{24} (see also Appendix 9A in \\cite{13}). Beginning with Eqs.(\\ref{18}) and (\\ref{19}) (with $m\\to\\mu=\\frac{mm_N}{m+m_N}$), the perturbative\ncalculation of electron's self-energy at second order in noncovariant form (corresponding to the one-loop ($L=1$) order in\ncovariant form) leads to an energy increase of an electron with momentum $\\bf p$, $\\Delta E_p$, which contains divergence\nand can be handled just like that in Eqs.(\\ref{10}) and (\\ref{67}), yielding\n\\begin{equation}\\label{79}\n\\begin{array}{l}\n  \\Delta E_p=b_1p^2+b_2p^4+\\cdots\\\\[5mm]\n  b_1=\\dfrac{\\alpha}{\\pi\\mu}(\\dfrac{4}{3}\\ln2+\\dfrac{4}{3}\\ln\\mu-\\dfrac{4}{3}C_1)\\\\[5mm]\n  b_2=\\dfrac{\\alpha}{\\pi\\mu^3}(-\\dfrac{2}{15})\n\\end{array}\n\\end{equation}\nThe only choice of arbitrary constant $C_1$ is to make $b_1=0$ such that the reduced mass $\\mu$ in Eq.(\\ref{18}) can be\nreconfirmed. However, Eq.(\\ref{19}) must be supplemented by the interaction between electron's spin and the radiation field\n\\begin{equation}\\label{80}\nH'_{int}=\\dfrac{ge\\hbar}{4\\mu c}{\\boldsymbol \\sigma}\\cdot\\nabla\\times{\\bf A}\n\\end{equation}\nwhere $g=2\\times 1.0011596522$ is gyromagnetic ratio of electron. Similar treatment leads to\n\\begin{equation}\\label{81}\n\\begin{array}{l}\n  \\Delta E'_p=b'_0+b'_1p^2+b'_2p^4+\\cdots\\\\[4mm]\n  b'_0=\\dfrac{g^2}{4}\\dfrac{\\alpha\\mu}{\\pi}[4(\\ln2+\\ln\\mu)-4C_2-\\dfrac{2C_3}{\\mu}-\\dfrac{C_4}{\\mu^2}]\\\\[4mm]\n  b'_1=\\dfrac{g^2}{4}\\dfrac{\\alpha}{\\pi\\mu}(\\dfrac{4}{3}\\ln2+2+\\dfrac{4}{3}\\ln\\mu-\\dfrac{4}{3}C_2)\\\\[4mm]\n  b'_2=\\dfrac{g^2}{4}\\dfrac{\\alpha}{\\pi\\mu^3}(-\\dfrac{1}{15})\n\\end{array}\n\\end{equation}\nBecause $\\mu$ has already been reconfirmed (by $b_1=0$), the only choice of arbitrary constant $C_2$ is to cancel the\nambiguous term with $\\ln\\mu,\\;C_2=\\ln\\mu$, leaving a nonzero $b'_1p^2$ and combining with $\\frac{1}{2\\mu}{\\bf p}^2$. Hence\n$\\mu$ really acquires a modification as\n\\begin{equation}\\label{82}\n\\mu\\to \\mu_{obs}=\\dfrac{\\mu}{1+\\beta},\\qquad \\beta=\\dfrac{g^2\\alpha}{2\\pi}(\\dfrac{4}{3}\\ln2+2)\n\\end{equation}\nwhere $\\mu_{obs}=\\frac{m_em_N}{m_e+m_N}$. Then constants $C_3$ and $C_4$ must be chosen such that $b'_0=0$.\n\nNotice that, however, the spin induced interaction, Eq.(\\ref{80}), endows electron with relativistic feature, creating\na term ($-\\frac{1}{8\\mu^3}p^4$) in its kinetic energy. Yet the modification on $\\mu$ shown in Eq.(\\ref{82}) does\ninduce a corresponding change $-\\frac{1}{8}(\\frac{1}{\\mu_{obs}^3}-\\frac{1}{\\mu^3})p^4$, which should be regarded as an invisible \"background\" and subtracted from the $p^4$ term induced by the radiative corrections. Hence the \"renormalized\"\n$b_2$ should be\n\\begin{equation}\\label{83}\nb_2^R=b_2+b'_2+\\frac{1}{8\\mu^3}(3\\beta+3\\beta^2+\\beta^3)\\simeq \\dfrac{\\alpha}{\\pi\\mu_{obs}^3}(1.99808)\n\\end{equation}\nwhere only the lowest approximation is kept ($\\mu\\simeq\\mu_{obs}$). Hence the radiative correction on the energy\nlevel of a stationary state $|Z,n,l\\rangle$ in hydrogenlike atom simply reads\n\\begin{equation}\\label{84}\n\\Delta E^{rad}(Z.n,l)=\\langle Z,n,l|b_2^Rp^4|Z,n,l\\rangle=[\\dfrac{8n}{3l+1}-3]\\dfrac{b_2^RZ^4\\alpha^4}{n^4}\\mu_{obs}^4\n\\end{equation}\nThis contribution, together with that from the vacuum polarization (borrowed from covariant theory) and nuclear size effect, gives a theoretical value for the Lamb shift \\cite{24}\n\\begin{equation}\\label{85}\nL_H^{theor}(2S_{1\/2}-2P_{1\/2})= 1056.52\\,MHz\n\\end{equation}\nwhich is smaller than the experimental value, Eq.(\\ref{29}), by $0.13\\%$ only.\n\nDespite the approximation involved, the above method clearly shows that our regularization is by noo means a trick to curb the divergence. Rather, it is a natural way to transform a divergence into some arbitrary constants, revealing that the essential meaning of divergence is just the \"uncertainty\" in the theory. Thus so-called renormalization turn out to be nothing but a process of reconfirmation to fix these constants via experiments. We must reconfirm a mass before it could\nbe modified via radiative corrections. Either \"skipping over the first step\" or \"combining two steps into one\" is not allowed.\n\nIn deeper understanding, our $RRM$ is based on a \"principle of relativity\" in epistemology\\cite{27}: Every thing is\nmoving and becomes recognizable only in relationship with other things. What we can understand is either no mass scale\nor two mass scales, but never one mass scale. This scenario is clearly displayed in the Gross-Neveu model\\cite{28}, also\nin the $\\lambda\\phi^4$ model with $SSB$ as shown by Eqs.(\\ref{65}), (\\ref{70}) and (\\ref{72}). The vacuum expectation\nvalue of field, $\\phi_1$, just provides a \"mass unit\" for the mass $m_\\sigma$ excited on the vacuum with $SSB$.\n\nSimilarly, in perturbative $QFT$, we will be able to calculate various radiative corrections on a particle only when its mass $m$ can be reconfirmed again and again throughout any high loop ($L$) order of theory until $L\\to\\infty$. Just\nlike one has to reconfirm his plane ticket before his departure from the airport, he  must use the same name throughout\nhis entire jouney\\cite{4}.\n\n\\section*{Acknowledgements}\n\n\\vskip 0.1cm\n\nWe thank S. Q. Chen, Y. S. Duan, R. T. Fu, S. S. Feng, T. Huang, P. T. Leung, W. F. Lu,\nX. T. Song, F. Wang, H. B. Wang, K. Wu, Y. L. Wu, J. Yan, G. H. Yang,\nJ. F. Yang and Z. X. Zhang for close collaborations and\/or helpful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro} \n\\setcounter{equation}{0}\n\nThe uninitiated reader of the table of integrals by \nI. S. Gradshteyn and I. M. Ryzhik \\cite{gr} will surely be \npuzzled by choice of integrands. In this note we provide an elementary proof \nof the evaluation {$\\mathbf{4.353.3}$} \n\\begin{equation}\n\\int_{0}^{1} ( ax + n+1)x^{n} e^{ax} \\ln x \\, dx = \ne^{a} \\sum_{k=0}^{n} (-1)^{k-1} \\frac{n!}{(n-k)! a^{k+1}} +\n(-1)^{n} \\frac{n!}{a^{n+1}}.\n\\label{iden1}\n\\end{equation}\n\\noindent\nWe also consider the integrals\n\\begin{equation}\nq_{n} := \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx \n\\end{equation}\n\\noindent\nand the companion family \n\\begin{equation}\np_{n} := \\int_{0}^{1} x^{n} e^{-x} \\, dx.\n\\end{equation}\n\\noindent\nThe integral $q_{n}$ corresponds to the case $a=-1$ in (\\ref{iden1}). \nSection \\ref{new} provides closed-form\nexpressions for $p_{n}$ and $q_{n}$. Section \\ref{sec-param} considers the \ngeneralization\n\\begin{equation}\nP_{n}(a) = \\int_{0}^{1} x^{n} e^{-ax} \\, dx \\text{ and }\nQ_{n}(a) = \\int_{0}^{1} x^{n} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nThe main result of this section is the closed-form expressions\n\\begin{equation}\nP_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\, dx = \n\\frac{n!}{a^{n+1}} \\left( 1 - e^{-a} \\sum_{k=0}^{n} \\frac{a^{k}}{k!} \\right),\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\nQ_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\ln x \\, dx = \n\\frac{n!}{a^{n+1}} \\left[  \\sum_{k=1}^{n} \\frac{1}{k} \n\\left( 1 - e^{-a} \\sum_{j=0}^{k-1} \\frac{a^{j}}{j!} \\right) + a Q_{0}(a) \n\\right], \n\\nonumber \n\\end{equation}\nwhere \n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx = -\\frac{1}{a} \n\\left( \\gamma + \\ln a + \\Gamma(0,a) \\right),\n\\end{equation}\n\\noindent\nand $\\Gamma(0,a)$ is the incomplete gamma function defined by\n\\begin{equation}\n\\Gamma(a,x) := \\int_{x}^{\\infty} t^{a-1} e^{-t} \\, dt.\n\\end{equation}\n\n\\section{The evaluation of {$\\mathbf{4.353.3}$}} \\label{simple} \n\\setcounter{equation}{0}\n\nThe identity \n\\begin{equation}\n\\frac{d}{dx} \\left( x^{n+1}e^{ax} \\right) = (ax+n+1)x^{n}e^{ax} \n\\end{equation}\n\\noindent\nand integration by parts yield \n\\begin{equation}\n\\int_{0}^{1} ( ax + n+1)x^{n} e^{ax} \\ln x \\, dx = \n- \\int_{0}^{1}x^{n} e^{ax} \\, dx. \\label{form2}\n\\end{equation}\n\\noindent\nThis last integral appears as $\\mathbf{3.351.1}$ in \\cite{gr}. We have obtained\na closed-form expression for it in \\cite{moll-gr7}. A new proof is \npresented in Section \\ref{sec-param}. \n\nA closed form expression for the right hand side of  (\\ref{form2}) \nis obtained from \n\\begin{equation}\n\\int_{0}^{1} x^{n} e^{ax} \\, dx = \\left( \\frac{d}{da} \\right)^{n} \n\\frac{e^{a}-1}{a}. \n\\label{form3}\n\\end{equation}\n\\noindent\nThe symbolic evaluation of (\\ref{form3}) for small values of \n$n \\in \\mathbb{N}$ suggests the existence of a polynomial $P_{n}(a)$\nsuch that \n\\begin{equation}\n\\int_{0}^{1}x^{n}e^{ax} \\, dx = \n\\frac{(-1)^{n+1} \\, n!}{a^{n+1}} + \\frac{P_{n}(a)}{a^{n+1}} e^{a}. \n\\label{pn-def}\n\\end{equation}\n\\noindent \nThe next lemma confirms the existence of this polynomial. \n\n\\begin{lemma}\nThe function $P_{n}(a)$ defined by \n\\begin{equation}\nP_{n}(a) =a^{n+1}e^{-a} \\left( \n\\left( \\frac{d}{da} \\right)^{n} \\frac{e^{a}-1}{a} - \n\\frac{(-1)^{n+1} n!}{a^{n+1}} \\right) \n\\end{equation}\n\\noindent\nis a polynomial of degree $n$. \n\\end{lemma}\n\\begin{proof}\nLet $D = \\frac{d}{da}$. Then $D^{n+1} = D(D^{n})$ produces the recurrence\n\\begin{equation}\nP_{n+1}(a) = a P_{n}'(a) + (a-n-1)P_{n}(a).\n\\label{recu1}\n\\end{equation}\n\\noindent\nThe initial condition $P_{0}(a) = 1$ and (\\ref{recu1}) show that \n$P_{n}$ is a polynomial of degree $n$. \n\\end{proof}\n\n\\begin{theorem}\nThe polynomial \n\\begin{equation}\nQ_{n}(a) := (-1)^{n} P_{n}(-a)\n\\end{equation}\n\\noindent\nhas positive integer coefficients, written as \n\\begin{equation}\nQ_{n}(a) = \\sum_{k=0}^{n} b_{n,k}a^{k}.\n\\end{equation}\n\\noindent\nThese coefficients satisfy\n\\begin{eqnarray}\nb_{n+1,0} & = & (n+1)b_{n,0} \\label{recu3} \\\\\nb_{n+1,k} & = & (n+1-k)b_{n,k} + b_{n,k-1}, \\quad 1 \\leq k \\leq n  \\nonumber \\\\\nb_{n+1,n+1} & = & b_{n,n}. \\nonumber\n\\end{eqnarray}\n\\noindent\nMoreover, the polynomial $Q_{n}(a)$ is given by\n\\begin{equation}\nQ_{n}(a) = n! \\, \\sum_{k=0}^{n} \\frac{a^{k}}{k!}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe recurrence (\\ref{recu1}) yields\n\\begin{equation}\nQ_{n+1}(a) = -a Q_{n}'(a) + (a+n+1)Q_{n}(a).\n\\label{recu5}\n\\end{equation}\n\\noindent\nThe recursion for the coefficients $b_{n,k}$ follows directly from here.\nMorover, it is clear that $b_{n,n}=1$ and $b_{n,0} = n!$. A little \nexperimentation suggets that $b_{n,k} = n!\/k!$, and this can be established \nfrom (\\ref{recu3}). \n\\end{proof}\n\n\nThis proposition amounts to the evaluation of ${\\mathbf{3.351.1}}$ in \n\\cite{gr}:\n\\begin{equation}\n\\int_{0}^{u} x^{n} e^{ax} \\, dx = \n\\frac{(-1)^{n+1} \\, n!}{a^{n+1}} + \n\\frac{e^{au}}{a^{n+1}} \\sum_{k=0}^{n} \\frac{n!}{k!} (-1)^{n-k} u^{k}a^{k}. \n\\end{equation}\n\\noindent\nThe reader will find a proof of this formula in \\cite{moll-gr7}.\n\n\\section{A new family of integrals} \\label{new} \n\\setcounter{equation}{0}\n\nIn this section we consider the family of integrals \n\\begin{equation}\nq_{n}  := \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx, \\label{qn-def}\n\\end{equation}\n\\noindent\nand its companion \n\\begin{equation}\np_{n}  := \\int_{0}^{1} x^{n} e^{-x} \\, dx. \\label{p-def}\n\\end{equation}\n\n\\begin{lemma}\nThe integrals $p_{n}, \\, q_{n}$ satisfy the recursion\n\\begin{eqnarray}\np_{n+1} & = & (n+1)p_{n} - e^{-1} \\label{recu8} \\\\\nq_{n+1} & = & (n+1)q_{n} + p_{n} \\label{recu8a}\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nIntegrate by parts. \n\\end{proof}\n\nThe initial conditions are \n\\begin{equation}\np_{0} = 1 - e^{-1} \\text{ and } q_{0} = \\int_{0}^{1} e^{-x} \\ln x \\, dx  = \\gamma - \\text{Ei}(-1). \\label{initq}\n\\end{equation}\n\\noindent\nHere $\\gamma$ is Euler's constant defined by\n\\begin{equation}\n\\gamma := \\lim\\limits_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{1}{k} - \\ln n \n\\end{equation}\n\\noindent\nwith integral representation \n\\begin{equation}\n\\gamma = \\int_{0}^{\\infty} e^{-x} \\ln x \\, dx \n\\end{equation}\n\\noindent\ngiven as ${\\mathbf{4.331.1}}$. The \nreader will find in \\cite{irrbook}  a proof of \nthis identity. The second term in (\\ref{initq}) is converted into\n\\begin{equation}\n\\int_{1}^{\\infty} e^{-x} \\ln x \\, dx = \\int_{1}^{\\infty} \\frac{e^{-x}}{x} \\, dx\n\\end{equation}\n\\noindent\nand this last form is identified as $\\text{Ei}(-1)$, where $\\text{Ei}$ is the \nexponential integral defined by\n\\begin{equation}\n\\text{Ei}(z) = - \\int_{-z}^{\\infty} \\frac{e^{-x}}{x} \\, dx. \n\\end{equation}\n\\noindent\nIn the current context, the value of $\\text{Ei}(-1)$ will be simply one of \nthe terms in the initial condition $q_{0}$.\n\nWe determine first an explicit expression for $p_{n}$. The recursion \n(\\ref{recu8}) shows the existence of integers $a_{n}, \\, b_{n}$ such that\n\\begin{equation}\np_{n} = a_{n} + b_{n}e^{-1},\n\\end{equation}\n\\noindent\nwith $a_{0} =1, \\, b_{0} = -1$. From (\\ref{recu8}) we obtain\n\\begin{equation}\na_{n+1} + b_{n+1}e^{-1} = (n+1)a_{n} + \\left[ (n+1)b_{n} - 1 \\right] e^{-1}.\n\\end{equation}\n\\noindent\nThe irrationality of $e$ produce the system \n\\begin{eqnarray}\na_{n+1} & = & (n+1)a_{n}, \\text{ with } a_{0}=1, \\label{system1} \\\\\nb_{n+1} & = & (n+1)b_{n}-1, \\text{ with } b_{0}=-1. \\label{system2}\n\\end{eqnarray}\n\\noindent\nThe expression $a_{n} = n!$ follows directly from (\\ref{system1}). To solve \n(\\ref{system2}), define $B_{n} := b_{n}\/n!$ and observe that\n\\begin{equation}\nB_{n+1} = B_{n} - \\frac{1}{(n+1)!}, \n\\end{equation}\n\\noindent\nthat telescopes to \n\\begin{equation}\nb_{n} = -n! \\sum_{k=0}^{n} \\frac{1}{k!}. \n\\end{equation}\n\n\\noindent\nWe have shown:\n\n\\begin{proposition}\nThe integral $p_{n}$ in (\\ref{p-def}) is given by\n\\begin{equation}\np_{n} = \\int_{0}^{1} x^{n} e^{-x} \\, dx = \n\\frac{n!}{e}  \\left( e - \\sum_{k=0}^{n} \\frac{1}{k!} \\right).\n\\end{equation}\n\\end{proposition}\n\nWe now determine a similar closed-form for $q_{n}$.  The recursion \n(\\ref{recu8a}) shows the existence of integers $c_{n}, \\, d_{n}, \\, f_{n}$ \nsuch that\n\\begin{equation}\nq_{n} = c_{n} + d_{n}e^{-1} + f_{n} q_{0}. \n\\end{equation}\n\\noindent\nIn order to produce a system similar to (\\ref{system1},\\ref{system2}) we will \nassume that the constants $1, \\, e^{-1}$ and $q_{0} = -(\\gamma + \n\\text{Ei}(-1))$ are linearly independent over $\\mathbb{Q}$.  Under this\nassumption (\\ref{recu8a}) produces\n\\begin{eqnarray}\nc_{n+1} & = & (n+1)c_{n} + n!, \\label{system3} \\\\\nd_{n+1} & = & (n+1)c_{n}  - n! \\sum_{k=0}^{n} \\frac{1}{k!},  \\label{system4} \\\\\nf_{n+1} & = & (n+1)f_{n}, \\label{system5} \n\\end{eqnarray}\n\\noindent\nwith the initial conditions $c_{0} = 0, \\, d_{0} = 0$ and $f_{0} = 1$. \n\nThe expression $f_{n} = n!$ follows directly from (\\ref{system5}).  To solve \n(\\ref{system3}) and (\\ref{system4}) we employ the following result \nestablished in \\cite{moll-gr5}. \n\n\\begin{lemma}\n\\label{lemrec}\nLet $a_{n}, \\, b_{n}$ and $r_{n}$ be sequences with $a_{n}, \\, \nb_{n} \\neq 0$. Assume that\n$z_{n}$ satisfies \n\\begin{equation}\na_{n}z_{n} = b_{n} z_{n-1} + r_{n}, \\, n \\geq  1\n\\label{recu0}\n\\end{equation}\n\\noindent\nwith initial condition $z_{0}$. Then \n\\begin{equation}\n\\label{valuex}\nz_{n} = \\frac{b_{1}b_{2} \\cdots b_{n}}{a_{1}a_{2} \\cdots a_{n}} \n\\left( z_{0} + \\sum_{k=1}^{n} \\frac{a_{1}a_{2} \\cdots a_{k-1}}\n{b_{1} b_{2} \\cdots b_{k}} r_{k} \\right). \n\\end{equation}\n\\end{lemma}\n\nWe conclude that\n\\begin{equation}\nc_{n} = n! \\sum_{k=1}^{n} \\frac{1}{k},\n\\end{equation}\n\\noindent\nand \n\\begin{equation}\nd_{n} = -n! \\sum_{k=1}^{n} \\frac{1}{k} \\sum_{j=0}^{k-1} \\frac{1}{j!}. \n\\end{equation}\n\\noindent\nThe expression for $c_{n}$ shows that they coincide with the  Stirling numbers\nof the first kind: $c_{n} = | s(n+1,2)|$. \\\\\n\nWe have established \n\n\\begin{proposition}\nThe integral $q_{n}$ in (\\ref{qn-def}) is given by\n\\begin{equation}\nq_{n} = \\int_{0}^{1} x^{n} e^{-x} \\ln x \\, dx = \nn! \\left[ \\frac{1}{e} \\sum_{k=1}^{n} \\frac{1}{k} \n\\left( e - \\sum_{j=0}^{k-1} \\frac{1}{j!}\n\\right) + q_{0} \\right].\n\\end{equation}\n\\end{proposition}\n\n\\begin{example}\nThe expressions for $p_{n}$ and $q_{n}$ provide the evaluation of \n$\\mathbf{4.351.1}$ in \\cite{gr}\n\\begin{equation}\n\\int_{0}^{1} (1-x)e^{-x} \\ln x \\, dx = \\frac{1-e}{e},\n\\end{equation}\n\\noindent\nby identifying the integral as $q_{0}-q_{1}$. The recurrence (\\ref{recu8a}) \nshows that \n\\begin{equation}\nq_{0}-q_{1} = -p_{0} = e^{-1}-1,\n\\end{equation}\n\\noindent\nas claimed. \n\\end{example}\n\n\\begin{example}\nThe evaluation of\n$\\mathbf{4.362.1}$ in \\cite{gr} \n\\begin{equation}\n\\int_{0}^{1} xe^{x} \\ln(1-x) \\, dx = \n\\int_{0}^{1} (1-t)e^{1-t} \\ln t \\, dt\n\\end{equation}\n\\noindent\nis achieved by observing that this integral is $e( q_{0} - q_{1}) \n= 1-e$. \n\\end{example}\n \n\\section{A parametric family} \\label{sec-param} \n\\setcounter{equation}{0}\n\nIn this section we consider the evaluation of \n\\begin{eqnarray}\nP_{n}(a) & := & \\int_{0}^{1} x^{n} e^{-ax} \\, dx \\label{generalp} \\\\\nQ_{n}(a) & := & \\int_{0}^{1} x^{n} e^{-ax} \\ln x \\, dx. \\label{generalq}\n\\end{eqnarray}\n\\noindent\nThe integrals $q_{n}$ considered in Section \\ref{new} corresponds to the \nspecial case: $q_{n} = Q_{n}(1)$. \n\nWe now establish a recursion for $Q_{n}$ by differentiating \n(\\ref{generalq}).\n\n\\begin{lemma}\nThe integral $Q_{n}(a)$ satisfies the relation\n\\begin{equation}\nQ_{n+1}(a) = -\\frac{d}{da} Q_{n}(a).\n\\end{equation}\n\\end{lemma}\n\nTo obtain a closed-form expression for $Q_{n}(a)$ we need to determine the \ninitial condition \n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nThis is expressed in terms of the {\\em incomplete gamma function}\ndefined in $\\mathbf{8.350.1}$ by\n\\begin{equation}\n\\Gamma(a,x) := \\int_{x}^{\\infty} t^{a-1} e^{-t} \\, dt.\n\\end{equation}\n\\noindent\nObserve that $\\Gamma(a,0) = \\Gamma(a)$, the usual gamma function. \n\n\\begin{lemma}\nThe initial condition $Q_{0}(a)$ is given by\n\\begin{equation}\nQ_{0}(a) = \\int_{0}^{1} e^{-ax} \\ln x \\, dx = -\\frac{1}{a} \n\\left( \\gamma + \\ln a + \\Gamma(0,a) \\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe change of variables $t = ax$ yields\n\\begin{equation}\nQ_{0}(a) = \\frac{1}{a} \\int_{0}^{a} e^{-t} \\ln t \\, dt - \\frac{\\ln a}{a} \n\\left( 1 - e^{-a} \\right). \n\\end{equation}\n\\noindent\nThen \n\\begin{equation}\n\\int_{0}^{a} e^{-t} \\ln t \\, dt = \n\\int_{0}^{\\infty} e^{-t} \\ln t \\, dt  - \n\\int_{a}^{\\infty} e^{-t} \\ln t \\, dt. \n\\end{equation}\n\\noindent\nThe first integral is \n\\begin{equation}\n\\int_{0}^{\\infty} e^{-t} \\ln t \\, dt   = - \\gamma,\n\\end{equation}\n\\noindent\nthat simply reflects the fact that $\\gamma  = - \\Gamma'(1)$. \nIntegrating by parts yields\n\\begin{equation}\n\\int_{a}^{\\infty}  e^{-t} \\ln t \\, dt = e^{-a} \\ln a + \\Gamma(0,a). \n\\end{equation}\n\\noindent\nThe formula for $Q_{0}(a)$ is established.\n\\end{proof}\n\nWe now determine a closed-form expression for $P_{n}(a)$ and $Q_{n}(a)$ \nfollowing the procedure employed in Section \\ref{new}. \n\n\\begin{lemma}\nThe integrals $P_{n}$ and $Q_{n}(a)$ satisfy the recursion\n\\begin{eqnarray}\nP_{n+1}(a) = \\frac{1}{a} \\left( (n+1)P_{n}(a) - e^{-a} \\right) \\label{recP} \\\\\nQ_{n+1}(a) = \\frac{1}{a} \\left( (n+1)Q_{n}(a) + P_{n}(a) \\right). \\label{recQ}\n\\end{eqnarray}\n\\noindent\nThe initial conditions are given by \n\\begin{equation}\nP_{0}(a) = \\frac{1}{a}( 1 - e^{-a}), \\text{ and } \nQ_{0}(a) = - \\frac{1}{a}( \\gamma + \\Gamma(0,a) + \\ln a ). \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIntegrate by parts.\n\\end{proof}\n\nWe conclude that we can write \n\\begin{equation}\nP_{n}(a) = A_{n}(a) - B_{n}(a) e^{-a},\n\\end{equation}\n\\noindent\nand \n\\begin{equation}\nQ_{n}(a) = C_{n}(a) - D_{n}(a)e^{-a} - E_{n}(a) \n( \\gamma + \\Gamma(0,a) + \\ln a ). \n\\end{equation}\n\n\\begin{lemma}\nThe recursions (\\ref{recP}) and (\\ref{recQ}) imply that\n\\begin{eqnarray}\nA_{n+1}(a) & = & \\frac{1}{a} (n+1) A_{n}(a), \\label{recA1} \\\\\nB_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) B_{n}(a) + 1 \\right], \\nonumber  \\\\\nC_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) C_{n}(a) + A_{n}(a)  \\right], \\nonumber  \\\\\nD_{n+1}(a) & = & \\frac{1}{a} \\left[ (n+1) D_{n}(a) + B_{n}(a)  \\right], \\nonumber  \\\\\nE_{n+1}(a) & = & \\frac{1}{a} (n+1) E_{n}(a) \\nonumber \n\\end{eqnarray}\n\\noindent\nwith initial conditions \n\\begin{equation}\nA_{0}(a) = B_{0}(a) = E_{0}(a) = \\frac{1}{a} \\text{ and } \nC_{0}(a) = D_{0}(a) = 0. \n\\end{equation}\n\\end{lemma}\n\nThese recursion can now be solved as in Section \\ref{new} to produce a \nclosed-form expression for the integrals $P_{n}(a)$ and $Q_{n}(a)$. We \nemploy the notation \n\\begin{equation}\nH_{n} = \\sum_{k=1}^{n} \\frac{1}{k}\n\\end{equation}\n\\noindent\nfor the harmonic numbers and \n\\begin{equation}\n\\text{Exp}_{n}(x) = \\sum_{k=0}^{n} \\frac{x^{k}}{k!}\n\\end{equation}\n\\noindent\nfor the partial sums of the exponential function. \n\n\\begin{theorem}\n\\label{mainPQ}\nLet $a \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$. Then \n\\begin{equation}\nP_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\, dx = \n\\frac{n!}{a^{n+1}} \\left[ 1 - e^{-a} \\text{Exp}_{n}(a) \\right],\n\\label{formulaP}\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\nQ_{n}(a) := \\int_{0}^{1}x^{n}e^{-ax} \\ln x \\, dx = \n\\frac{n!}{a^{n+1}} \\left[  H_{n} - G(a) - e^{-a} \\sum_{k=1}^{n} \n\\frac{1}{k} \\text{Exp}_{k-1}(a) \\right],\n\\label{formulaQ}\n\\nonumber \n\\end{equation}\n\\noindent\nwhere  $G(a) = -aQ_{0}(a) = \\gamma + \\Gamma(0,a) + \\ln a$. \n\\end{theorem}\n\nThese expressions provide the evaluations of two integrals in \\cite{gr}. \n\n\\begin{example}\nFormula ${\\mathbf{4.351.2}}$ states that\n\\begin{equation}\n\\int_{0}^{1} e^{-ax} (-ax^{2} + 2x) \\ln x \\, dx = \n\\frac{1}{a^{2}} \\left[ -1 + (1+a)e^{-a} \\right]. \n\\end{equation}\n\\noindent\nIn order to verify this, observe that the stated integral is \n\\begin{equation}\n-a \\int_{0}^{1} x^{2} e^{-ax}  \\ln x \\, dx + \n2 \\int_{0}^{1} x e^{-ax}  \\ln x \\, dx  = -a Q_{2}(a) + 2 Q_{1}(a). \n\\end{equation}\n\\noindent\nThe expressions in Theorem \\ref{mainPQ} now complete the evaluation.  \n\\end{example}\n\n\\begin{example}\nFormula ${\\mathbf{4.353.3}}$ in \\cite{gr} gives the value of \n\\begin{equation}\nI_{n}(a) := \\int_{0}^{1} (-ax+ n+1) x^{n} e^{-ax} \\ln x \\, dx. \n\\end{equation}\n\\noindent\nObserve that \n\\begin{equation}\nI_{n}(a) = -aQ_{n+1}(a) + (n+1)Q_{n}(a),\n\\end{equation}\n\\noindent\nand using the recursion (\\ref{recQ}) we conclude that $I_{n}(a) = \n-P_{n}(a)$. The expression in Theorem \\ref{mainPQ} is precisely what \nappears in \\cite{gr}. \n\\end{example}\n\nWe conclude with the evaluation of a series shown to us by \nTewodros Amdeberhan. Expand the exponential \nterm in (\\ref{formulaQ}) and integrate term by term to obtain\n\\begin{equation}\n\\sum_{k=0}^{\\infty} \\frac{(-a)^{k}}{k! \\, (n+1+k)^{2}} = \n\\frac{n!}{a^{n+1}} \\left( -\\psi(n+1) + \\ln a + \\Gamma(0,a) +\ne^{-a} \\sum_{k=0}^{n} \\frac{1}{k} \\text{Exp}_{k-1}(a) \\right). \n\\label{series1}\n\\end{equation}\n\\noindent\nHere \n\\begin{equation}\n\\psi(x) = \\frac{\\Gamma'(x)}{\\Gamma(x)}\n\\end{equation}\n\\noindent\nis the {\\em digamma} function defined in $\\mathbf{8.360.1}$ of \\cite{gr}. \nthe identity \n\\begin{equation}\n\\psi(n+1) = H_{n} - \\gamma,\n\\end{equation}\n\\noindent\nthat is a direct consequence of the functional equation $\\Gamma(x+1) = \nx \\Gamma(x)$ and $\\Gamma'(1)= - \\gamma$, was used to transform \n(\\ref{series1}). \n\n\\medskip\n\nThe identity (\\ref{series1}) can be used to provide multiple expressions \nfor the incomplete gamma function, such as \n\\begin{equation}\n\\int_{a}^{\\infty} \\frac{e^{-x}}{x} \\, dx = \n\\sum_{k=0}^{\\infty} \\frac{(-1)^{k} a^{n+1+k}}{n! \\, k! \\, (n+1+k)^{2}} + \n\\psi(n+1) - \\ln a - e^{-a} \\sum_{k=1}^{n} \\frac{\\text{Exp}_{k-1}(a)}{k}, \n\\end{equation}\n\\noindent\nand the special case for $n=0$:\n\\begin{equation}\n\\int_{a}^{\\infty} \\frac{e^{-x}}{x} \\, dx = \n-\\gamma - \\ln a + \\sum_{k=0}^{\\infty} \\frac{(-1)^{k} a^{k+1}}\n{  (k+1)! \\, (k+1)}.   \n\\end{equation}\n\n\\medskip\n\nThese issues will be explored in a future publication. \n\n\\vskip 1.0in\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Conclusion}\nIn this paper we proposed an algorithm to efficiently compute the full-CP set of SHIM. Through numerical experiments, we demonstrated that SHIM  is statistically superior than the vanilla lasso and comparable to other complex models (NN, RF). The confidence interval along with a point estimation is better than a point estimation alone. The computation of a full-CP set for other complex models (NN, RF) remains an open question. SHIM is interpretable and it generates the decision function as a weighted combination of \\emph{decision sets} (if-then rules). The homotopy-mining based exact full-CP of SHIM is better than the split-CP of SHIM. The efficient pruning strategy makes the computation of exact full-CP set tractable for SHIM. \n\n\\section{Submission of papers to NeurIPS 2022}\n\n\n\n\n\\section{Introduction}\nThe uncertainty in data-driven analysis is a major concern, particularly in risk-sensitive automated decision-making problems (for example, in medical diagnosis and criminal justice). Several strategies exist to quantify the uncertainty of a point estimators. For example, the Bayesian approach can provide a strong confidence bound, but requires the assumption on prior distribution. \nThe PAC analysis is another approach that provides bounds on the probability of error. The conformal prediction (CP) is one such uncertainty quantification method that is very generic, and it is applicable to almost any point estimators \\cite{vovk2005algorithmic,shafer2008tutorial}. The CP method has recently gained significant attention as it can provide valid finite sample statistical coverage guarantee at any nominal level as long as data are independently and identically distributed (i.i.d.). The coverage guarantee provided is valid even when the model is misspecified. In this paper we are interested in the \\emph{full-CP} where the full data is used to compute the CP set. An alternative to this approach is the \\emph{split-CP} in which the data is split into two parts: one part is used for fitting and the remaining part is used to compute the CP set.\nThe essential idea of the full-CP framework can be stated as follows: Given a training set \\(\\mc{D}_n = \\{(x_1, y_1), \\ldots, (x_n, y_n)\\}\\) and a test instance $x_{n+1}$ which are both i.i.d., the goal of CP is to construct a $100(1-\\alpha)\\%$ confidence set that contains the unobserved $y_{n+1}$. Here, $\\alpha \\in [0,1]$ represents the confidence level. In other words, we are interested in all possible completions of the augmented dataset in the form of \\(\\mc{D}_n \\cup (x_{n+1}, \\tau)\\), such that $y_{n+1}=\\tau$ is \\textit{typical} for the known data  \\(\\{\\mc{D}_n, x_{n+1}\\}\\). This typicality is measured by a typicalness function $\\pi(\\cdot)$, which is also called the \\emph{$p$-value} in analogy to the classical hypothesis testing. In other words, the CP set for $x_{n+1}$ is defined as the set of all $\\tau$ for which the null hypothesis $H_0: y_{n+1}=\\tau$ is not rejected against the alternative hypothesis  $H_1: y_{n+1}\\neq\\tau$.\n\n\\paragraph{Related work:} Since its inception, there have been several extensions and application of the CP framework in diverse directions. Examples include the choice of conformity score and statistical efficiency \\cite{lei2013distribution,lei2014distribution}, high-dimensional regression \\cite{lei2018distribution}, classification \\cite{lei2014classification,sadinle2019least}, active learning \\cite{ho2008query}, time series \\cite{xu2021conformal}, few-shot learning \\cite{fisch2021few}, text and speech completion \\cite{dey2021conformal}, image classification \\cite{angelopoulos2020uncertainty}, outlier detection \\cite{laxhammar2015inductive,bates2021testing}. \nRecently, some approximate computation of the conformal set was proposed for regression in \\cite{ndiaye2019computing,ndiaye2021root} and in classification, significant advancement has been made in \\cite{cherubin2021exact} to compute exact full-CP.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/fig1.pdf}\n  \\caption{(A) An intuitive illustration of a SHIM solution space, which is a hyper rectangle (i.e., generalization of a rectangle for higher dimensions that essentially represents a cartesian product of intervals). The decision sets, the predicted response $(\\hat{y})$ and the associated confidence interval for a randomly chosen example of data from the ProPublica two-year recidivism criminal justice (COMPAS) dataset are shown. (B) A tree of patterns has been constructed by exploiting the hierarchical structure of high-order interaction features (Note: not all patterns appear due to pruning).\n}\n\\label{fig1}\n \\end{figure}\nDespite its attractive properties, the application of exact, full-CP in many practical problems  remains an open problem owing to its high computational cost. By definition, the full-CP framework requires the refitting of model by augmenting the data for every possible candidate $\\tau \\in \\mb{R}$ of the unobserved $y_{n+1}$. In a regression setting, one needs to refit the model an infinite number of times for all possible candidates on the real line $(\\forall \\tau \\in \\mb{R})$, and check the conformity by computing the $p$-values $\\pi(\\tau)$. \n\nHence, an efficient computation of the full-CP set is possible only for a handful of simple models (e.g., ordinary LS regression, ridge regression) in which the solution is explicitly represented as a function of $\\tau$.\nThis enables a closed-form derivation of full-CP set and avoids an exhaustive search over the real line \\cite{nouretdinov2001ridge}. \nRecently, \\citet{lei2019fast} proposed a homotopy method to efficiently compute the full-CP set of the LASSO. The homotopy method exploits the piece-wise linearity of the LASSO solutions as a function of $\\tau$, and avoids the exhaustive search by keeping track of the transition points along the solution path and the use of a linear interpolation between transition points.\nUnfortunately, such a structure does not exist for most of the models, and the application of full-CP is still an open question for complex models such as a random forest (RF), neural network (NN) etc. For those complex models we need to rely on split-CP.\n\nIn CP, although the coverage property is satisfied even when the model is misspecified, it is better to use sufficiently complex models for complex data because this enables us to obtain a more compact confidence set (i.e., shorter confidence intervals). Furthermore, it is important to note that the confidence set obtained by a full-CP is more compact than that obtained by a split-CP because only a part of instances in the available dataset is used for constructing the confidence set in split-CP. This means that it is valuable to construct a full-CP algorithm for sufficiently complex models.\nIn this paper, we considered the \\emph{sparse high-order interaction model (SHIM)} which is complex but interpretable, and proposed a \\emph{homotopy-mining} method to efficiently compute the \\emph{full-CP} set. We call the resulting machine as \\emph{SHIM confidence machine}. SHIM represents the model as a weighted sum of conjunction rules that are highly interpretable as well as accurate decision sets \\cite{das2019interpretable,lakkaraju2016interpretable}. SHIM can capture the combinatorial interactions of multiple factors, which can prove to be beneficial for deciphering complex data. A conjunction rule of a SHIM looks like \\(I(-1.5 \\leq x_1 \\leq 2.3)\\wedge I(x_3 \\geq 20.0)\\wedge I(x_5 \\leq 7.5) \\), where $I(\\cdot)$ refers to the indicator function. An intuitive illustration of a SHIM is shown in Fig.~\\ref{fig1}A. \n\n\\textbf{Contribution:}\nOur main contribution in this paper is to develop an efficient algorithm to conduct \\emph{exact full-CP} for SHIM which can capture complex structures in the data by considering high-order interaction features.\nThe exact full-CP for complex black box models such as RF and NN are intractable and an efficient computational method does not exist. For such black box models only split-CP is possible.\nThe proposed method enables us to obtain a more compact confidence set (much better than linear and comparable to non linear models) by the full-CP for sufficiently flexible and complex SHIM.\nBecause SHIM uses higher-order interaction features, the full-CP is computationally challenging, but we overcome this difficulty by introducing a method called \\emph{homotopy mining} which exploits the best of both homotopy and (pattern) mining methods. The computation of exact full-CP for SHIM by homotopy mining can be interpreted as an extension of LASSO's exact full-CP in \\cite{lei2019fast}.\nThe use of black box or explainable machine learning models in high-stakes decision-making such as in healthcare, criminal justice, and other domains is highly criticized in the literature (Rudin, C. 2019). We believe that the exact full-CP of a SHIM has significant importance in practice where accuracy, statistical reliability, and interpretability are important. If a practitioner chooses SHIM as a regressor, how to compute the corresponding exact full-CP set?. The proposed method provides a computationally efficient solution to this. \\looseness=-1\n\n\\paragraph{Notation:}\nIn our notation, $y\\in \\mathbb{R}^n$ represents the response vector of $n$ instances, $y(\\tau) \\in \\mathbb{R}^{n+1}$ is the augmented response vector constructed by augmenting the possible response value ($\\tau \\in \\mb{R}$) of the $(n+1)^{th}$ instance with $y$. Later we defined $y(\\tau)$ as a function of the variable $\\tau$ as the vector $y(\\tau)$ changes for different possible response value ($\\tau$) of the $(n+1)^{th}$ instance. The $y_i\\in \\mathbb{R}, \\forall i \\in [n+1]$, represents the scalar response of the $i^{th}$ instance, $X \\in \\mb{R}^{(n+1) \\times p}$ is the design matrix of all $(n+1)$ instances, each having $p$ features, $x_\\ell \\in \\mb{R}^{n+1}$ is the column vector corresponding to some $\\ell \\in [p]$, where $[p] = \\{1, \\ldots, p\\}$, $X_{\\mc{A}_\\tau} \\in \\mb{R}^{(n+1) \\times |\\mc{A}_\\tau|} \\subseteq X$ is a smaller design matrix in which the columns are restricted to the elements of some subset $\\mc{A}_\\tau \\subseteq [p]$.  \\looseness=-1\n\n\\section{Problem Statement}\nConsider a regression problem with a response vector $y \\in \\mb{R}^n$ and \n$m$ original covariate vectors $z_1, \\ldots, z_m$, where \\(z_{\\ell} \\in \\mb{R}^n\\) and $\\ell \\in [m] = \\{1, ..., m\\}$. A high-order interaction model up to the $d^{\\rm th}$ order is then written as follows:\n\\begin{equation}\\label{eq:shim_model}\n\\begin{split}\n\\hspace{-2.5mm} y = \n\\sum_{\\ell_1 \\in [m]} \\theta_{\\ell_1} z_{\\ell_1} \n+ \\sum_{ \\substack{ (\\ell_1, \\ell_2)  \\in [m] \\times [m]  \\\\ \\ell_1 \\neq \\ell_2}} \\hspace{-2.5mm} \\theta_{\\ell_1, \\ell_2} z_{\\ell_1} z_{\\ell_2}\n+  \\cdots\n+ \\sum_{\\substack{(\\ell_1, ..., \\ell_d)  \\in [m]^d \\\\ \\ell_1 \\neq ... \\neq \\ell_d}} \\hspace{-2.5mm} \\theta_{\\ell_1, \\ldots, \\ell_d} z_{\\ell_1} \\cdots  z_{\\ell_d},\n\\end{split}\n\\end{equation}\nwhere $z_{\\ell_1} \\cdots  z_{\\ell_d}$ is the element-wise product and scalar $\\theta$ represents the coefficient.\nIn this study, we consider each element of the original covariate vector $z_{\\ell} \\in \\{0, 1\\}^n$ for $\\ell \\in [m]$. However, our model is equally applicable to covariate vectors defined in the domain $[0, 1]^n$.\nTo simplify the notation, it is convenient to write the high-order interaction model in (\\ref{eq:shim_model}) using the following matrix of concatenated vectors of all high-order interactions:\n\\[\nX=[\\underbrace{z_1, \\ldots, z_m}_{1\\text{\\textsuperscript{st} order}}, \n\\cdots, \\underbrace{z_1 \\ldots z_d, \\ldots, z_{m-d+1} \\ldots z_m}_{d\\text{\\textsuperscript{th} order}}] \\in \\mb{R}^{n \\times p},\n\\]\nwhere \\(p := \\sum_{\\kappa=1}^d{m \\choose \\kappa}\\).\nSimilarly, the coefficient vector associated with all possible high-order interaction terms can be written as follows:\n\\[\n\\beta:= [ \\underbrace{ \\theta_1, \\ldots, \\theta_m}_{1\\text{\\textsuperscript{st} order}},\n\\cdots, \\underbrace{\\theta_{1, \\ldots ,d},  \\ldots, \\theta_{m-d+1, \\ldots, m}}_{d\\text{\\textsuperscript{th} order}}\n]^\\top \\in \\mb{R}^p.\n\\]\nThe high-order interaction model (\\ref{eq:shim_model}) is then simply written as a linear model\n$\ny = X \\beta.\n$\nUnfortunately, $p$ can be prohibitively large unless both $m$ and $d$ are fairly small.\nIn SHIM, we consider a sparse estimation of a high-order interaction model.\nAn example of SHIM is as follows:\n\\begin{equation}\\label{eq:shim_eg}\n y = \\theta_3z_{3} +  \\theta_5z_{5} +  \\theta_{2,6}z_{2}z_{6} +  \\theta_{1,2,5,9}z_{1}z_{2}z_{5}z_{9}.\n\\end{equation}\n\nBefore delving into our proposed method, we briefly overview the conformal prediction framework.\n\\subsection{Conformal prediction}\nA mere point estimation is insufficient for risk-sensitive automated decision-making problems \\cite{rudin2019stop,angelino2018learning,das2019interpretable}, such as in medical diagnosis and criminal justice. In such high-stake decision-making problems, if the estimators are equipped with the associated confidence information, the decision maker will be more informed and sufficiently confident to make a prudent decision when the stakes are high.   \n\\paragraph{Full-CP:}\nGiven a labelled dataset ${\\mc{D}}_n$ and a new observation $x_{n+1}$, the goal of full-CP framework is to construct a set of likely values $\\mc{C}(x_{n+1})$ of unobserved $y_{n+1}$ with a valid statistical coverage guarantee \\cite{vovk2005algorithmic,shafer2008tutorial}, {\\em i.e.~},\n\\begin{equation}\\label{eqn:CP_guarantee}\n    \\mathbb{P}(y_{n+1} \\in \\mc{C}(x_{n+1})) \\geq 1 - \\alpha \\enspace,\n\\end{equation}\nwhere \\(\\alpha \\in [0, 1]\\) determines the level of confidence. \nIf we define a prediction function  $\\mu(\\cdot)$ that maps the input $X$ to the output $y$, then the essential idea of constructing a full-CP set is to fit a model $\\mu_{\\tau}(\\cdot)$ with the augmented data $\\mc{D}_n \\cup (x_{n+1},\\tau)$ for every possible candidate $\\tau \\in \\mb{R}$ and compare the prediction error of each instances.\nMore precisely, let \\( y(\\tau)  = [y_1, \\ldots, y_n, \\tau]^\\top \\in \\mb{R}^{n+1} \\) be a vector augmented with $\\tau$. We define a score function that measures how well the model can predict each output variables; with the constraint that it should not depend on the order of the data instances \\cite{lei2019fast}. One such conformity score function generally used in linear regression setting is the (coordinate-wise) absolute residual  i.e.\n $$S(\\tau) = | y(\\tau) - \\mu_{\\scaleto{\\tau\\mathstrut}{6pt}}(X)| \\enspace,$$\n where we stack the input vectors in the design matrix $X = [x_1, \\ldots, x_{n+1}]^\\top$ in $\\mb{R}^{(n+1) \\times p}$.\nThe conformity function can now be defined as\n$$ \\pi (\\tau) = 1 - \\frac{1}{n+1}\\sum_{i=1}^{n+1} \\mathbbm{1}_{S_i(\\tau) \\leq S_{n+1}(\\tau)},$$\nwhich evaluates how the prediction of the candidate $\\tau$ using the fitted model $\\mu_{\\tau}(x_{n+1})$ is ranked compared to the prediction of the previously observed data $y_{i}$ using $\\mu_{\\tau}(x_{i})$. The conformal set is defined as:\n\\begin{equation}\\label{eqn:CP2}\n    \\mc{C}(x_{n+1}) = \\{\\tau:  \\pi (\\tau) \\geq \\alpha\\}\\enspace,\n\\end{equation}\nwhich is merely the collection of candidates whose conformity is large enough. \\cite{vovk2005algorithmic} showed that the defined conformal set $\\mc{C}(x_{n+1})$ satisfies the coverage guarantee \\ref{eqn:CP_guarantee} as long as the data are i.i.d.\n\\paragraph{Split-CP:} In split-CP \\cite{papadopoulos2002inductive}, the data set \\(\\mc{D}_n\\) is split into two parts, defined as the training set \\(\\mc{D}_{\\text{tr}}=\\{(x_1, y_1), \\ldots, (x_m, y_m)\\}\\) and the calibration set \\(\\mc{D}_{\\text{cal}}=\\{(x_{m+1}, y_{m+1}), \\ldots, (x_n, y_n)\\}\\), with \\(m < n\\). Then the model ($\\mu^{\\text{tr}}(\\cdot)$) is fit with the training set $\\mc{D}_{\\text{tr}}$ only once, and the $p$-values, denoted as $\\pi_{\\text{split}}(\\cdot)$, are determined using the calibration set $\\mc{D}_{\\text{cal}}$ as follows.\n\\begin{equation}\\label{eqn:pi_z_split}\n    \\pi_{\\text{split}} (\\tau) = 1 - \\frac{1}{n-m}\\sum_{i=m+1}^{n} \\mathbbm{1}_{S^{\\text{cal}}_i(\\tau) \\leq S_{n+1}(\\tau)},\n\\end{equation}\nwhere \\(S^{\\text{cal}}_i(\\tau) = |y_i - \\mu^{\\text{tr}}(x_i)|, \\forall i \\in [m+1, n]\\).\nTherefore, the split-CP can be defined as follows:\n\\begin{align}\\label{eqn:CP_split}\n    \\mc{C}_{\\text{split}}(x_{n+1}) &= \\{\\tau:  \\pi_{\\text{split}} (\\tau) \\geq \\alpha\\}\n    =\\{\\tau:  S^{\\text{cal}}_{n+1}(\\tau) \\leq Q^{\\text{cal}}_{1-\\alpha}\\},\n\\end{align}\nwhere \\(Q^{\\text{cal}}_{1-\\alpha}\\) is the \\((1-\\alpha)\\) quantile of the calibration scores \\(S^{\\text{cal}}_i, \\forall i \\in [m+1, n]\\). When the conformity score is defined as the absolute residual, then the split-CP set can be conveniently written as \\(\\mc{C}_{\\text{split}}(x_{n+1}) = [\\mu^{\\text{tr}}(x_{n+1}) \\pm Q^{\\text{cal}}_{1-\\alpha}] \\).\n\nAlthough split-CP enjoys the computational efficiency of single model fitting, it suffers from the poor statistical efficiency owing to the smaller sample size both in the model fitting and calibration phases. \n\n\\paragraph{Cross-conformal Predictors} introduced in \\cite{Vovk15} follow a similar trade-off as bias reduction versus variance reduction while trying to exploit the entire dataset for calibration and a significant proportions for training the model. The dataset is partitioned into $K$ folds and one performs a split conformal set by sequentially defining the $k$th fold as the calibration set and the remaining as the training set for $k \\in \\{1, \\ldots, K\\}$. The main difficulty lies in aggregating the different $p$-values while maintaining the validity of the method see \\cite{Carlsson_Eklund_Norinder14, Linusson_Norinder_Bostrom_Johansson_Lofstrom17}. In general, it can be shown that the confidence level is inflated by a factor of $2$ {\\em i.e.~} the (not improvable) theoretical coverage level is $1 - 2\\alpha$ instead of $1 - \\alpha$, see \\cite{Barber_Candes_Ramdas_Tibshirani21}. Nevertheless, the practical performance is fairly acceptable both computationally and statistically. \n\nOn the other hand, to compute the exact full-CP set for a regression problem where \\(y \\in \\mb{R}\\), one needs to refit the regression model $\\mu_\\tau(\\cdot)$ and evaluate $\\pi(\\tau)$ for an infinite number of candidates $\\tau$. Hence, efficient computation of the exact full-CP set is possible only for a handful of predictors (e.g., ridge regression) where it is possible to have a closed form expression of $\\mu_{\\tau}(\\cdot)$. \nHowever, such structure does not exist for most of the machine learning models and computing the exact full-CP set remains an open question for many models. Recently, \\citet{lei2019fast} proposed a homotopy method to efficiently compute the exact full-CP set of the LASSO which was not trivial. \\citet{lei2019fast} considered the response $y$ as a function of a scalar and derived a homotopy method to construct the exact LASSO solution path as function of that scalar. After constructing the exact path, \\citet{lei2019fast} used the solutions at the transition points of the path to efficiently compute the exact full-CP set of the LASSO. In this paper we extend that framework for the SHIM. \n\n\n\n\n\n\\section{Proposed Method}\nWe propose a \\emph{homotopy-mining} method to compute the exact full-CP set of a SHIM. \nThe homotopy method refers to an optimization framework for solving a sequence of parameterized optimization problems.  \nThe basic idea of our method is to consider the following optimization problem: \n\\begin{equation}\\label{obj:primal_lasso}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mb{R}^p} \\frac{1}{2}\\norm{y(\\tau) - X\\beta}_2^2  + \\lambda \\norm{\\beta}_1 \\enspace,\n\\end{equation}\nAt optima we can write \\(X^\\top \\big(X\\beta(\\tau) - y(\\tau)\\big) + \\lambda s(\\tau) = 0,\\)\n\\begin{equation*}\\label{eq:opt_condn_shim_homotopy}\n    \\text{ where} \\hspace{0.2cm}  s_{\\ell}(\\tau) \\in \\begin{cases}\n        \\{ -1, +1 \\} \\hspace{0.2cm} \\hspace{0.2cm}    \\text{if} \\hspace{0.2cm} \\beta_\\ell(\\tau) \\neq 0 \\enspace,\\\\\n        [-1, +1]  \\hspace{0.2cm} \\hspace{0.2cm} \\hspace{0.2cm} \\text{if} \\hspace{0.2cm}   \\beta_\\ell(\\tau) = 0 \\enspace,\n\\end{cases}\n\\end{equation*}\nand \\(\\ell \\in [p]\\). Let us define the active set\n\\(\\label{eq:active_set}\n\\mc{A}_\\tau = \\left\\{\\ell \\in [p]:\\; \\big\\lvert x_{\\ell}^{\\top} w(\\tau) \\big\\rvert = \\lambda \\right\\}\n\\), where \\(w(\\tau)  = y(\\tau) - X\\beta(\\tau)\\) is the residual~\\citep{efron2004least}.\nIn \\citet{lei2019fast}, the author showed that for a fixed $\\lambda$, the exact solution path of LASSO with a variable response $y(\\tau)$, characterized by $\\tau$ can be shown to be piece-wise linear as stated in Proposition \\ref{prop:homotopy}.\n\\begin{proposition}\\label{prop:homotopy}\nIf $\\tau_1$ and $\\tau_2$ are any two points, where \\(\\tau_2 = \\tau_1 + \\Delta, \\Delta>0\\) such that the active set $\\mc{A}_\\tau$ and the sign of the coefficients $\\beta_\\ell(\\tau), \\forall \\ell \\in [p]$ remain the same \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\) and \\(X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}}\\) is invertible \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\), then one can show that\n\\begin{align*}\n&\\beta_{\\mc{A}_{\\tau}} (\\tau_2) - \\beta_{\\mc{A}_{\\tau}} (\\tau_1) = \\nu_{\\mc{A}_{\\tau}} (\\tau) \\times \\Delta, \n&s_{\\mc{A}^c_{\\tau}} (\\tau_2) - s_{\\mc{A}^c_{\\tau}} (\\tau_1) &= \\gamma_{\\mc{A}^c_{\\tau}} (\\tau) \\times  \\Delta \/ \\lambda,\n\\end{align*}\n\\begin{align}\n\\label{direction_vec}\n   \\text{where} \\hspace{0.5cm} &\\nu_{\\mc{A}_{\\tau}}(\\tau) = (X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}})^{-1}x_{n+1,\\mc{A}_{\\tau}},\n  &\\gamma_{\\mc{A}^c_{\\tau}} (\\tau) = x_{n+1, \\mc{A}_{\\tau}^c} - (X^{\\top}_{\\mc{A}_{\\tau}^c}X_{\\mc{A}_{\\tau}})\\nu_{\\mc{A}_{\\tau}}(\\tau) \n \n\\end{align}\nare the direction vectors. Here, \\(\\nu_{\\mc{A}_{\\tau}}(\\tau)\\)  and \\(\\gamma_{\\mc{A}^c_{\\tau}} (\\tau)\\) remain constant \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\), and hence, \\(\\beta(\\tau)\\) is linear \\(\\forall \\tau \\in [\\tau_1, \\tau_2)\\). Now, for any $t, t+1 \\in \\{1, \\ldots, T\\}$, where $T$ is a finite number, if $\\tau_{t+1} > \\tau_t$ is the next zero crossing point and there is only one feature $\\ell \\in [p]$ that enters or leaves the active set $\\mc{A}_{\\tau_t}$ at $\\tau=\\tau_{t+1}$, then \\citet{lei2019fast} showed that \\(\\tau \\mapsto \\beta(\\tau)\\) is well defined and piece-wise linear in $\\tau$.\n\n\\end{proposition}\n  We call the mapping \\(\\tau \\mapsto \\beta(\\tau)\\) the \\emph{$\\tau$-path} and the number of linear pieces of this $\\tau$-path is finite (see complexity analysis in the appendix). This $\\tau$-path can be computed exactly using homotopy method,\nwhere the sequences of $\\tau$ represent the breakpoints of the homotopy method  \\citep{efron2004least,mairal2012complexity}, the indices of which are represented by $t \\in \\{1, \\cdots, T\\}$. \nThe basic idea of homotopy method is stated below. At $\\tau_{t+1}$ which is the next zero-crossing point of the $\\tau$-path, either of the following two events occurs \\citep{efron2004least,rosset2007piecewise} \n\\begin{itemize}\n \\item A zero variable becomes non-zero, that is, \\( \\exists \\ell \\in \\mc{A}^c_{\\tau_t} \\text{ s.t. } \\big\\lvert x_{\\ell}^{\\top} w(\\tau_{t+1}) \\big\\rvert = \\lambda \\hspace{0.2cm} \\text{or,}\\)\n\\item A non-zero variable becomes zero, that is,\n\\(\\exists \\ell \\in \\mc{A}_{\\tau_t} \\text{ s.t. } \\beta_\\ell(\\tau_t) \\neq 0 \\text{, but } \\beta_\\ell(\\tau_{t+1}) = 0 \\enspace.\\)\n\\end{itemize}\nOverall, the next change in the active set (or change in direction vector, \\(\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\)  or \\(\\gamma_{\\mc{A}^c_{\\tau_t}} (\\tau_t)\\)) occurs at $\\tau_{t + 1} = \\tau_t + \\Delta_{\\ell^\\ast}$, such that \n\\begin{equation}\\label{eq:step_length}\n\\Delta_{\\ell^\\ast} = \\min(\\Delta_{\\ell^\\ast}^1, \\Delta_{\\ell^\\ast}^2),\n\\end{equation}\n\\begin{align*}\n\\text{where  }\n&\\Delta_{\\ell^\\ast}^1 = \\min_{\\ell \\in \\mc{A}_{\\tau_t}} \\left( - \\frac{\\beta_\\ell(\\tau_t)}{\\nu_\\ell(\\tau_t)} \\right)_{++},\n&\\Delta_{\\ell^\\ast}^2 = \\min_{\\ell \\in \\mathcal{A}^c_{\\tau_t}}\\left( \\lambda \\frac{ \\text{sign}(\\gamma_\\ell (\\tau_t)) - x_{\\ell}^{\\top}w(\\tau_{t})}{\\gamma_\\ell(\\tau_t)} \\right)_{++},\n\\end{align*}\nwhere, we use the convention that for any $a \\in \\mb{R}, (a)_{++} = a$ if $a > 0$ and $\\infty$ otherwise. However, naively (minimization over all possible interaction terms) determining the step size of inclusion $(\\Delta_{\\ell^\\ast}^2)$ will be intractable for the SHIM type problem. Because in SHIM the search space grows exponentially due to the combinatorial effect of high-order interaction terms. \nTherefore, both the fitting and constructing the full-CP set of a SHIM are non-trivial because unless both $m$ and $d$ are very small, a high-order interaction model will have a significantly large number of parameters to be considered.\nSeveral algorithms for fitting a sparse high-order interaction model have been proposed in the literature \\cite{ERP_Tsuda,saigo2009gboost,nakagawa2016safe}.\nA common approach adopted in these existing works is to exploit the hierarchical structure of high-order interaction features.\nIn other words, a tree structure as in Fig.~\\ref{fig1}B is considered and a branch-and-bound strategy is employed in order to avoid handling all the exponentially increasing number of high-order interaction features. Hence, we need efficient computational methods to make the computation practically feasible. \nIn the following section, we present an efficient tree pruning strategy that considers the tree structure of the interaction terms (or patterns). Here, each node of the tree represents an interaction term. The basic idea of tree pruning is that we construct a tree of interaction terms in a ``progressive manner\". Here, we keep track of the current minimum step size (up to the construction of $\\ell^{th}$ pattern) i.e. \\(\\Delta_{\\ell^\\ast}^2 = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\) as we construct the tree progressively, and prune a large part of the tree if some bound condition fails. Here, $\\Delta_j^2$ is the bracketed quantity in the expression of $\\Delta_{\\ell^\\ast}^2$ in (\\ref{eq:step_length}) for any non-active feature $j \\in \\mc{A}^c_{\\tau_t}$.\n\n\\subsection{Tree pruning ($\\tau$-path)}\n\\paragraph{Definition of Tree}\nA tree is constructed in such a way that for any pair of nodes (\\(\\ell, \\ell^\\prime)\\), where $\\ell$ is the ancestor of $\\ell^\\prime$, i.e., \\(\\ell \\subset \\ell^\\prime\\), the following conditions are satisfied\n\\begin{align*}\n&x_{i \\ell^\\prime} = 1 \\implies x_{i\\ell } = 1,\n     &x_{i\\ell } = 0 \\implies x_{i \\ell^\\prime} = 0, \\forall i \\in [n].\n\\end{align*}\nThe basic idea of our tree pruning condition is stated below. The equicorrelation condition for any active feature \\(k  \\in \\mathcal{A}_{\\tau}\\) at a fixed $\\lambda$ can be written as\n$\\big\\lvert x_{k}^{\\top} w(\\tau) \\big\\rvert = \\lambda$ (see definition of $\\mc{A}_{\\tau}$ below (\\ref{obj:primal_lasso})).\n\nTherefore, at $\\tau = \\tau_{t + 1}$ any non-active feature \\(\\ell \\in \\mathcal{A}^c_{\\tau_{t} } \\) becomes active  if\n\\begin{equation}\\label{eqn:inclusion_condition2}\n    \\big\\lvert x_{\\ell}^{\\top} w(\\tau_{t+1})\\lvert =\n    \\big\\lvert x_{k}^{\\top} w(\\tau_{t+1}) \\big\\rvert .\n\\end{equation}\nNow, using the triangular inequality one can show that (\\ref{eqn:inclusion_condition2}) will not have any solution if\n\\begin{align}\\label{cond:pruning_tau1}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + \\Delta_{\\ell^{\\ast}}^2 (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\rho_k (\\tau_t) \\lvert   - \\Delta_{\\ell^{\\ast}}^2(\\lvert\\eta_k(\\tau_t) \\lvert +  x_{n+1,k}),\n\\end{align}\nwhere, \\(\\rho_{\\ell} = x_{\\ell}^{\\top}w(\\tau_t) \\) and \\(\\eta_{\\ell} =x_{\\ell}^{\\top} v(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\rho_k = x_{k}^{\\top}w(\\tau_t) \\) and \\(\\eta_k = x_k^{\\top} v(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t}, v(\\tau_t) =  X_{\\mc{A}_{\\tau_t}}\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\).\nTherefore, (\\ref{cond:pruning_tau1}) can be used to derive the pruning condition of the $\\tau$-path which is formally stated in Lemma \\ref{lemma:lemma_1}.\n\nSimilar idea has been used in the context of graph mining  \\cite{ERP_Tsuda} and selective inference of SHIM \\cite{das2021fast}. Note that \\citet{ERP_Tsuda} provided a pruning condition for the exact regularization of graph data ($\\lambda$-path) and \\citet{das2021fast} provided a pruning condition in the context of selective inference to characterize the conditional distribution of the test statistics. However, in our case we adapted the similar idea to compute the exact full-CP set of SHIM. \n\\begin{lemma}\\label{lemma:lemma_1}\nIf $\\Delta_{\\ell^\\ast}^2$ is the current minimum step size, that is, \\(\\Delta_{\\ell^\\ast}^2 = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\)\\newline then\n$\\forall \\ell^\\prime \\supset \\ell, \\Delta_{\\ell^\\prime}^2 > \\Delta_{\\ell^\\ast}^2$ if \n\\begin{align} \\label{eq:lemma_1_eq}\nb_{\\ell, w(\\tau_t)} + \\Delta_{\\ell^\\ast}^2 (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell} )\n\t< |\\rho_k(\\tau_t)|  - \\Delta_{\\ell^\\ast}^2 ( |\\eta_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\end{lemma}\nwhere, \\(b_{\\ell, w(\\tau_t)} :=\\max \\big\\{ \\sum \\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell} \\big\\}, \\\\ b_{\\ell, v(\\tau_t)}:= \\max \\big \\{ \\sum \\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell} \\big \\} \\).\nThe Lemma \\ref{lemma:lemma_1} essentially states that if the condition in (\\ref{eq:lemma_1_eq}) is satisfied, then one can safely ignore the subtree with $\\ell$ as the root node, thereby dramatically improving the computational efficiency. We extended our method to elastic net and call it El-SHIM. The details of El-SHIM, proof of Lemma \\ref{lemma:lemma_1} and the algorithm to compute the exact full-CP of SHIM are provided in the appendix.\n\n\n\\section{Results and Discussions}\nWe evaluated our proposed method using both synthetic and real-world data. For all experiments, we considered a coverage guarantee of $90\\%$, that is, \\(\\alpha = 0.1\\). We compared the statistical efficiency of our proposed method (SHIM) with those of other simple (LASSO) and complex models (NN, RF).  For the complex models, we reported the split-CP because it is the only available method for computing the CP for complex models. We also demonstrated the statistical efficiency of full-CP in comparison to that of split-CP both for LASSO and SHIM.  \n\\paragraph{Synthetic data}\nWe generated random i.i.d. samples $(z_i, y) \\in \\{0,1\\}^m \\times \\mb{R} $ in such a way that $100m(1-\\zeta)\\%$ features of $z_i \\in \\mb{R}^m$ contain a value of 1 on average.\nHere, $\\zeta \\in [0, 1]$ is the sparsity controlling parameter that controls the sparsity of the design matrix, whereas the sparsity in the model coefficients are controlled by the regularizer $\\lambda$.\nThe pruning effectiveness (Table~\\ref{table:comp_eff_time_taken}) depends on the sparsity of the design matrix as it exploits the tree's anti-monotonicity property. High dimensional real-world data is generally very sparse and the choice of $\\zeta$ in our experiments is just for the demonstration purpose.\nThe response $y_i \\in \\mb{R}$ is randomly generated from a normal distribution $N(\\mu(x_i), \\sigma^2)$. For demonstration purposes, we considered a true model of up to fifth-order interactions, which is defined as $\\mu (x_i) = 2.0z_1 +2.0 z_1z_2 + 2.0z_1z_2z_3 + 2.0z_1z_3z_4z_5 + 2.0z_1z_2z_3z_4z_5$, and set $\\sigma=1$. The choice of this model is merely for the demonstration purposes, and the proposed method is equally applicable to any chosen model. We used the same true model $\\mu(x_i)$ in both low and high dimensional settings (Table~\\ref{table:stat_low} \\& ~\\ref{table:stat_high} and Fig.~\\ref{fig:cpl_synthetic}).\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s & lasso\\_f &shim\\_2s &shim\\_2f &shim\\_3s &shim\\_3f\\\\\n \\hline\n \n length&\\begin{tabular}{c} 1.93\\\\(0.21) \\end{tabular}  & \\begin{tabular}{c} 1.84\\\\ (0.17) \\end{tabular}   &\\begin{tabular}{c} 2.32\\\\ (0.23)  \\end{tabular}   &\\begin{tabular}{c} 2.18\\\\  (0.10)\\end{tabular}   &\\begin{tabular}{c} 2.00\\\\  (0.23) \\end{tabular}  &\\begin{tabular}{c} 1.82\\\\ (0.17) \\end{tabular}   &\\begin{tabular}{c} 1.96 \\\\ (0.24) \\end{tabular} &\\begin{tabular}{c} 1.76 \\\\ (0.25)  \\end{tabular}  \\\\\n\\hline\n  %\n  cov&\\begin{tabular}{c} 0.91 \\\\(0.05) \\end{tabular} &\\begin{tabular}{c} 0.90 \\\\(0.04) \\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.03) \\end{tabular}  &\\begin{tabular}{c} 0.90 \\\\(0.05) \\end{tabular} &\\begin{tabular}{c}0.89 \\\\(0.03) \\end{tabular} &\\begin{tabular}{c}0.89 \\\\(0.05) \\end{tabular}  &\\begin{tabular}{c}0.89 \\\\(0.04) \\end{tabular} &\\begin{tabular}{c}0.88 \\\\(0.05) \\end{tabular} \\\\\n\\hline\n r2&\\begin{tabular}{c}0.64 \\\\(0.10)\\end{tabular} &\\begin{tabular}{c} 0.67 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.52 \\\\(0.13)\\end{tabular}  &\\begin{tabular}{c}0.54 \\\\(0.11)\\end{tabular} &\\begin{tabular}{c}0.62 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.67 \\\\(0.10)\\end{tabular}  &\\begin{tabular}{c}0.63 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.67 \\\\(0.09) \\end{tabular}\\\\\n %\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using low dimensional synthetic data ($m=10, n=150$). The bracketed values represent the standard deviations.}\n\\label{table:stat_low}\n\\end{table}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s\n      & lasso\\_f & shim\\_2s &shim\\_2f  &shim\\_3s &shim\\_3f\\\\\n \\hline\nlength&\\begin{tabular}{c}3.39 \\\\(0.55)\\end{tabular} &\\begin{tabular}{c} 1.97 \\\\(0.43)\\end{tabular} &\\begin{tabular}{c} 2.43 \\\\(0.37)\\end{tabular}  &\\begin{tabular}{c} 2.26 \\\\(0.19)\\end{tabular} &\\begin{tabular}{c} 2.21 \\\\(0.39)\\end{tabular} &\\begin{tabular}{c}1.92 \\\\(0.26)\\end{tabular}  &\\begin{tabular}{c}2.25 \\\\(0.41)\\end{tabular} &\\begin{tabular}{c}1.89 \\\\(0.30)\\end{tabular} \\\\\n  \\hline\n  cov&\\begin{tabular}{c}0.89 \\\\(0.06)\\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.06)\\end{tabular} &\\begin{tabular}{c} 0.91 \\\\(0.05)\\end{tabular}  &\\begin{tabular}{c} 0.91 \\\\(0.05)\\end{tabular} &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular} &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular}  &\\begin{tabular}{c}0.91 \\\\(0.04)\\end{tabular} &\\begin{tabular}{c}0.90 \\\\(0.04)\\end{tabular} \\\\\n  \\hline\n r2&\\begin{tabular}{c}-0.01 \\\\(0.13)\\end{tabular} &\\begin{tabular}{c} 0.68 \\\\(0.08)\\end{tabular} &\\begin{tabular}{c}0.47 \\\\(0.07)\\end{tabular}  &\\begin{tabular}{c}0.53 \\\\(0.07)\\end{tabular} &\\begin{tabular}{c}0.55 \\\\(0.09)\\end{tabular} &\\begin{tabular}{c}0.65 \\\\(0.09)\\end{tabular}  &\\begin{tabular}{c}0.55 \\\\(0.07)\\end{tabular} &\\begin{tabular}{c}0.66 \\\\(0.08)\\end{tabular} \\\\\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using high dimensional synthetic data ($m=100, n=150$). The bracketed values represent the standard deviations.}\n\\label{table:stat_high}\n\\end{table}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c| } \n\\hline\n &mlp & rf &lasso\\_s\n      &lasso\\_f & shim\\_2s &shim\\_2f \\\\\n \\hline\n  length&7.48 (0.18) & 7.38 (0.17) & 7.42 (0.16)  & 7.41 (0.09) & 7.39 (0.17) &7.36 (0.13) \\\\\n  \\hline\n  cov&0.90 (0.01)& 0.90 (0.01) & 0.90 (0.01)  & 0.90 (0.01) &0.90 (0.01) &0.90 (0.01) \\\\\n  \\hline\n r2&0.38 (0.01) & 0.39 (0.01) &0.39 (0.01)  &0.39 (0.01) &0.39 (0.01) &0.39 (0.01) \\\\\n \\hline\n   \\end{tabular}}\n\\caption{Comparison of the statistical power of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using compas data. shim\\_2s and shim\\_2f respectively represent split-CP and full-CP for a $2^{nd}$ order SHIM. The bracketed values represent the standard deviations.}\n\\label{table:stat_compas}\n\\end{table}\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{figures\/fig2.pdf}\n   \\caption{Comparison of the confidence interval lengths (CI length) of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using synthetic data for different sample sizes. shim\\_3s and shim\\_3f respectively represent split-CP and full-CP for a $3^{rd}$ order SHIM.}\n   \\label{fig:cpl_synthetic}\n \\end{figure}\n\\paragraph{Real data}\nWe applied our method to a real-world criminal justice data set called the ProPublica two-year recidivism dataset (COMPAS data). See appendix for details of this data.\n\\paragraph{Comparison of statistical powers}\nWe compared the statistical efficiency using both synthetic and real-world data. For the synthetic data, we considered both low- and high-dimensional settings. In the low-dimensional setting, we considered a dataset with $n=150$ instances and $m=10$ original covariates, whereas for the high-dimensional setting, we considered $n=150$ and $m=100$. We kept the sparsity of the design matrix fixed at $\\zeta =0.4$ in both the settings. We varied the order of interaction $d=2, 3, \\cdots$, however, almost in all the experiments we found that the model get saturated after $3^{rd}$ order interactions (i.e. the performance of SHIM did not change much for $d\\geq 3$). Hence, we reported the results of up to $3^{rd}$ order interactions. \nNote that the max order of interaction $d$ is not needed to be specified beforehand. Our pruning condition takes care of the case even when $d$ is not specified that is when the whole search space is considered (Table~\\ref{table:comp_eff_time_taken}).\nFor both the settings (low and high), we generated a dataset of $n=150$ training instances, each accompanied with $n=50$ test instances. We generated 5 such independent random datasets and repeated the experiments 3 times. Hence, in total we reported the average results of 15 independent datasets i.e. \\(3 \\times 5 \\times 50 = 750 \\) test instances. The details of the hyper parameter selection are given in the appendix. We used a multi-layer perceptron (MLP) as a neural network architecture in our experiments.\n\nFrom Table \\ref{table:stat_low} and Table \\ref{table:stat_high} it can be observed that both in low- and high-dimensional settings, all the methods produced perfect (or nearly perfect) coverage i.e cov=$0.90$. The statistical efficiency of individual methods are compared using the length of confidence interval. A statistically efficient model is expected to produce a shorter confidence interval length. Comparing the length of confidence intervals, one can observe that the statistical efficiency of SHIM increases as we increase the order of interactions. A $3^{rd}$ order SHIM produced the shortest average confidence interval lengths both in low- and high-dimensional settings. \nTo compare the fitting power of individual methods we reported the $R$-squared (r2) scores. It can be observed that the r2 scores of a $3^{rd}$ order SHIM are comparable to the best performing complex model (RF) both in low- and high-dimensional settings. We also demonstrated the comparison of the confidence interval lengths (CI length) and $R$-squared (r2) scores of the proposed method (SHIM) with other simple (LASSO) and complex models (MLP, RF) for three different sample sizes, $n \\in \\{100, 150, 200\\}, m=10$ (Figure \\ref{fig:cpl_synthetic}). It can be observed that the full-CP of $3^{rd}$ order SHIM (shim3\\_f ) produced the best results (shortest avg. length, highest avg. r2 score) in all the cases. It can also be observed that the performance of data splitting (in mlp, rf, lasso\\_s, shim3\\_s) is worse compared to that of full-CP. The split-CP tends to produce a longer confidence interval and smaller r2 score, and suffer from high variance due to the smaller data size as well as the additional randomness considered in data splitting. For the real data we randomly split the data into a training set of $n=5000$ and a test set of $n=2214$. We selected the hyper parameters by 5-fold cross-validation using the training data and reported the statistical power using the test data. The range of hyper parameters from which the optimal one is selected is the same as used for the synthetic data experiment. We repeated this experimental setup 100 times and reported the average results. From Table \\ref{table:stat_compas}, one can observe that all the method produced a perfect coverage (cov=0.90) and nearly same r2 scores. A $2^{nd}$-order SHIM produced the shortest average confidence interval length. Here, the performance of SHIM did not improve for ($d> 2$). Hence, we reported the results of a $2^{nd}$-order SHIM. Additional results are provided in the appendix.\n\n\\subsection{Comparison of computational efficiencies.}\\label{sec_comp_eff}\n\\begin{table*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c|c|c|c|c|c| } \n\\hline\n \\multirow{4}{*}{$d$} & \\multirow{4}{*}{\\shortstack{Search space \\\\ (\\# nodes)}} &  \\multicolumn{6}{|c|}{$\\lambda=1$} & \\multicolumn{6}{|c|}{$\\lambda=10$}\\\\\n \\cline{3-14}\n& & \\multicolumn{3}{|c|}{With pruning} &\\multicolumn{3}{|c|}{Without pruning} &\\multicolumn{3}{|c|}{With pruning} &\\multicolumn{3}{|c|}{Without pruning}\\\\\n\\cline{3-14}\n& &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$ &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$\\\\\n \\hline\n \\hline\n  2 & 465 &0.09  &0.10   &0.03  &0.10  &0.08  &0.07   &0.08  &0.08 &0.03  &0.07  &0.07 &0.12 \\\\\n  \\hline\n  3 & 4525 &0.84  &0.67   &0.03  &0.82  &0.79 &0.64 &0.56  &0.36 &0.03  &0.63  &0.65 &0.85  \\\\\n  \\hline\n  4 & 31930 &4.49   &1.51   &0.03  &5.91  &5.32 &4.73 &2.14  &0.90 &0.03  &4.12  &4.42 &3.87  \\\\\n  \\hline\n  5 & 174436 &12.34    &2.74   &0.03  &30.19  &28.76    &23.31  &5.11  &1.38 &0.03  &26.74  &22.70 &25.42  \\\\\n  \\hline\n  10 & 53009101 &112.17    &3.83   &0.03  &$>$ 1 day  &$>$ 1 day &$>$ 1 day &49.88  &2.13 &0.03  &$>$ 1 day  &6891.44 &6861.16  \\\\\n  \\hline\n 15 & 614429671 &126.04   &3.39   &0.03  &$>$ 1 day  &$>$ 1 day &$>$ 1 day &56.22   &2.00 &0.03  &$>$ 1 day  &$>$ 1 day & $>$ 1 day  \\\\\n \\hline\n 20 & 1050777736  &126.86    &3.51   &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day &55.33  &2.13 &0.03   &$>$ 1 day  &$>$ 1 day & $>$ 1 day \\\\\n  \\hline\n 25 &1073709892  &130.28   &3.62   &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day &55.21   &2.02 &0.03   &$>$ 1 day  &$>$ 1 day &$>$ 1 day  \\\\\n \\hline\n\\end{tabular}}\n\\caption{Computation time (in sec) with and without pruning using two different $\\lambda$ values ($\\lambda=1, 10$) for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$). All computation times were measured on an Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz.}\n\\label{table:comp_eff_time_taken}\n\\end{table*}\nTo demonstrate the computational efficiency of the proposed pruning strategy for the $\\tau$-path, we generated a synthetic dataset of $n=100$ and $m=30$ for three different sparsity levels of the design matrix ($\\zeta=0.4, 0.7, 0.9$)  using the same $5^{th}$ order model as used to demonstrate the statistical power. \nWe compared both the fraction of nodes traversed (see appendix) and the time taken (Table \\ref{table:comp_eff_time_taken}) against a different maximum interaction order $d$  for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$) using two different $\\lambda$ values ($\\lambda=1, 10$). \nIt can be observed that the pruning is more effective at the deeper nodes of the tree and saturates after a certain depth of the tree.\nThis is evident as the sparsity of the data increases at the deeper nodes, and the pruning exploits the anti-monotonicity of high-order interaction terms constructed as tree of patterns.\n\nIn the case of the homotopy method without pruning, we stopped the execution of the program if the $\\tau$-path was not finished in one day.\nFrom Table ~\\ref{table:comp_eff_time_taken}, it can be observed that without the tree pruning, the construction of the $\\tau$-path is not practical as we progress to the deeper nodes of the tree because of the generation of an exponential number of high-order interaction terms.\nThe maximum time taken by the $\\tau$-path with pruning was approximately 130 s, for a ``max-pat\" size of 25, i.e. for 1073709892 number of nodes at $\\lambda=1, \\zeta=0.4$.\nIn our numerical experiments, the number of kink in the $\\tau$-path appears to be polynomial in the number of features (see appendix). \nThe worst-case complexity of $\\tau$-path is exponential (see appendix). However, fortunately, it has been well-recognized that this worst-case rarely happens in practice \\cite{li2018well,le2021parametric} and, this is also evident from our experimental results (see appendix). \n\n\\section{Appendix A}\n\\section{Algorithms and Proofs}\n\\subsection{Proof of Lemma \\ref{lemma:lemma_1}}\nIf $\\Delta^2_{\\ell^\\ast}$ is the current minimum step size, that is, \\(\\Delta^2_{\\ell^\\ast} = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta_j^2\\}, \\) then, we can rewrite (\\ref{eqn:inclusion_condition2}) as follows.\n\\begin{align}\\label{eqn:inclusion_condition3}\n    \\big\\lvert \\rho_{\\ell}(\\tau_t) -   &\\Delta^2_{\\ell^\\ast} (\\eta_{\\ell}(\\tau_t) -  x_{n+1,\\ell})   \\lvert =\n    \\big\\lvert \\rho_{k}(\\tau_t) -  \\Delta^2_{\\ell^\\ast} (\\eta_{k}(\\tau_t) - x_{n+1,k} )\\big\\lvert \\end{align}\nwhere, \\(\\rho_{\\ell} = x_{\\ell}^{\\top}w(\\tau_t) \\) and \\(\\eta_{\\ell} =x_{\\ell}^{\\top} v(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\rho_k = x_{k}^{\\top}w(\\tau_t) \\) and \\(\\eta_k = x_k^{\\top} v(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t} \\). Here, we used the fact that $\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1})=\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t) + \\Delta^2_{\\ell^\\ast}\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)$.\nThe right hand side (r.h.s.) of (\\ref{eqn:inclusion_condition3}) has a lower bound i.e.\n\\begin{align*}\n    \\lvert \\rho_{k}(\\tau_t) -   &\\Delta_{\\ell^\\ast} (\\eta_{k}(\\tau_t) - x_{n+1,k})   \\lvert \\geq \n    \\lvert \\rho_{k}(\\tau_t) \\rvert  -   \\Delta_{\\ell^\\ast} (\\lvert \\eta_{k}(\\tau_t) \\rvert  +   \\lvert x_{n+1,k} \\rvert), \\end{align*}\nand the left hand side (l.h.s.) of (\\ref{eqn:inclusion_condition3}) has an upper bound i.e.\n\\begin{align*}\n    \\lvert \\rho_{\\ell}(\\tau_t) -  &\\Delta_{\\ell^\\ast} (\\eta_{\\ell}(\\tau_t) - x_{n+1,\\ell} )  \\lvert  \\leq \n    \\lvert \\rho_{\\ell}(\\tau_t) \\rvert  + \\Delta_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\rvert  +  \\lvert x_{n+1,\\ell} \\rvert). \\end{align*}\nThe above two bounds are derived by considering the fact that for any $a \\in \\mb{R}, b \\in \\mb{R},  c \\in \\mb{R}$ and $d \\in \\mb{R}>0$ we can write the following:\n\\begin{equation*}\n|a-d(b-c)| \\leq |a| + d (|b|+|c|) \\hspace{0.2cm} \\text{ and } \\hspace{0.2cm} |a-d(b-c)| \\geq |a| - d (|b|+ |c|)\n\\end{equation*}\nTherefore, for equation (\\ref{eqn:inclusion_condition3}) to have a solution the following condition needs to be satisfied.\n\\begin{align}\\label{eqn:inclusion_condition4}\n    \\lvert \\rho_{\\ell}(\\tau_t) \\rvert + \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\rvert +  \\lvert x_{n+1,\\ell}   \\lvert )\\geq \\lvert \\rho_{k}(\\tau_t) \\rvert  -   \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{k}(\\tau_t) \\rvert  +  \\lvert x_{n+1,k} \\rvert)\n\\end{align}\nThe equation (\\ref{eqn:inclusion_condition3}) will not have any solution if (\\ref{eqn:inclusion_condition4}) is not satisfied, and that can be used to derive the pruning condition.\nTherefore, the pruning condition of the SHIM $\\tau$-path can be written as follows:\n\\begin{align}\\label{cond:pruning_tau}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + \\Delta^2_{\\ell^\\ast} (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\rho_k (\\tau_t) \\lvert   - \\Delta^2_{\\ell^\\ast} ( \\lvert\\eta_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\nBefore proving the Lemma \\ref{lemma:lemma_1}, we introduce the following two propositions:\n\\begin{proposition}\\label{prop:1} We can write\n\\begin{align*}\n\t|\\rho_\\ell (\\tau_t)| & \\leq b_{\\ell, w(\\tau_t)}, \\\\ \n\t|\\eta_\\ell (\\tau_t)| & \\leq b_{\\ell, v(\\tau_t)},\n\\end{align*}\nwhere \n\\begin{align*}\n \t b_{\\ell, w(\\tau_t)} &= \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell} \\big\\}  \\\\ \n\t b_{\\ell, v(\\tau_t)} &= \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell} \\big\\}.\n\\end{align*}\n\n\\end{proposition}\n\n\\paragraph{Proof of Proposition \\ref{prop:1}:}\nWe have \n\\begin{align*}\n\t|\\rho_\\ell (\\tau_t)| &= |x_\\ell^\\top w(\\tau_t)| \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{i=1}^n w_{i}(\\tau_t) x_{i \\ell} \\right |  \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell} - \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell}\\right | \\\\ \n\t&\\leq  \\max \\left \\{ \\sum \n\t\\limits_{w_{i}(\\tau_t) > 0} |w_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{w_{i}(\\tau_t) < 0} |w_{i}(\\tau_t)| x_{i \\ell} \\right \\} =: b_{\\ell, w(\\tau_t)}.\n\\end{align*}\nSimilarly, \n\\begin{align*}\n\t|\\eta_\\ell (\\tau_t)| &= |x_\\ell^\\top v(\\tau_t)| \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{i=1}^n v_{i}(\\tau_t) x_{i \\ell} \\right |  \\\\ \n\t&=  \\left | \\sum \n\t\\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell} - \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell}\\right | \\\\ \n\t&\\leq  \\max \\left \\{ \\sum \n\t\\limits_{v_{i}(\\tau_t) > 0} |v_{i}(\\tau_t)| x_{i \\ell}, \\sum  \\limits_{v_{i}(\\tau_t) < 0} |v_{i}(\\tau_t)| x_{i \\ell} \\right \\} =: b_{\\ell, v(\\tau_t)}.\n\\end{align*}\nTherefore, using Proposition \\ref{prop:1} we can further write equation (\\ref{eq:lemma_3_eq}) which implies equation (\\ref{cond:pruning_tau}). \n\\begin{align} \\label{eq:lemma_3_eq}\n\t\\nonumber b_{\\ell, w(\\tau_t)} + &\\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell})  \\\\\n\t&< |\\rho_k(\\tau_t)|  - \\Delta^2_{\\ell^\\ast} (|\\eta_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\begin{proposition}\\label{prop:2}\nBy using the definition of tree, we have \n\\begin{align*}\n\tb_{\\ell, w(\\tau_t)} & \\geq b_{\\ell^\\prime, w(\\tau_t)}, \\\\ \n\tb_{\\ell, v(\\tau_t)} & \\geq b_{\\ell^\\prime, v(\\tau_t)}.\n\\end{align*}\nsince $x_{i\\ell} \\geq x_{i\\ell^\\prime}, \\hspace{0.2cm} \\forall \\ell^\\prime \\supset \\ell, \\forall i \\in [n+1]$.\n\\end{proposition}\n\\paragraph{Proof of Proposition \\ref{prop:2}:}\nFrom the definition of tree we have $x_{i\\ell}\\geq x_{i\\ell^\\prime}, \\forall \\ell^{\\prime} \\supset \\ell, \\forall i \\in [n+1]$. Hence, we can write\n\\begin{align*}\n\tb_{\\ell, w(\\tau_t)} &= \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell} \\big\\} \\\\ \n\t&\\geq \\max \\big\\{ \\sum_{w_i(\\tau_t) < 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell^\\prime}, \\sum_{w_i(\\tau_t) > 0} \\lvert w_i(\\tau_t) \\lvert x_{i\\ell^\\prime} \\big\\}\\\\\n\t&=: b_{\\ell^\\prime, w(\\tau_t)}.\n\\end{align*} \nSimilarly, we also have\n\\begin{align*}\n\tb_{\\ell, v(\\tau_t)} &= \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell} \\big\\} \\\\ \n\t&\\geq \\max \\big\\{ \\sum_{v_i(\\tau_t) < 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell^\\prime}, \\sum_{v_i(\\tau_t) > 0} \\lvert v_i(\\tau_t) \\lvert x_{i\\ell^\\prime} \\big\\}\\\\\n\t&=: b_{\\ell^\\prime, v(\\tau_t)}.\n\\end{align*}\nWe now prove Lemma \\ref{lemma:lemma_1} by contradiction, that is we assume that $\\exists \\ell$ such that (\\ref{eq:lemma_3_eq}) holds, and there exists one \\( \\ell^{\\prime} \\supset \\ell:  \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast} \\); then show that this is a contradiction.\n\\begin{align*}\n   \n    \\therefore \\quad &|\\rho_k(\\tau_t)| - \\Delta^2_{\\ell^{\\prime}} (|\\eta_k(\\tau_t)| + x_{n+1,k})\\\\\n    &> |\\rho_k(\\tau_t)| - \\Delta^2_{\\ell^\\ast} (|\\eta_k(\\tau_t)| + x_{n+1,k}), \\hspace{0.2cm} \\because \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast}\\\\\n    &> b_{\\ell, w(\\tau_t)} + \\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell}), \\hspace{0.2cm} \\text{using } (\\ref{eq:lemma_3_eq})\\\\\n    &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\ast} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}),  \\hspace{0.2cm} \\text{(Proposition \\ref{prop:2}),}\\\\\n    &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\prime} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}), \\hspace{0.2cm} \\because \\Delta^2_{\\ell^{\\prime}} < \\Delta^2_{\\ell^\\ast}.\n   \n\\end{align*}\nTherefore, we got\n\\begin{align*}\n  |\\rho_k(\\tau_t)| - &\\Delta^2_{\\ell^{\\prime}} (|\\eta_k(\\tau_t)|  + x_{n+1,k})\\\\\n  &> b_{\\ell^\\prime, w(\\tau_t)} + \\Delta^2_{\\ell^\\prime} (b_{\\ell^\\prime, v(\\tau_t)} + x_{n+1,\\ell}) \\\\\n  &\\implies \\ell^{\\prime} \\text{ is infeasible,} \\hspace{0.5cm} \\text{ using (\\ref{eqn:inclusion_condition4}), (\\ref{cond:pruning_tau}), (\\ref{eq:lemma_3_eq})}\\\\\n  &\\implies \\Delta^2_{\\ell^{\\prime}} \\nless \\Delta^2_{\\ell^\\ast}.\n\\end{align*}\nThis completes the proof of Lemma \\ref{lemma:lemma_1}.\nHence, if the pruning condition in Lemma \\ref{lemma:lemma_1} holds, then we do not need to search the sub-tree with $\\ell$ as the root node, and hence increasing the efficiency of the search procedure.\n\\subsection{Complexity analysis of the $\\tau$-path:}\\label{complexity_analysis}\nGiven that \\(X^{\\top}_{\\mc{A}_{\\tau}}X_{\\mc{A}_{\\tau}}\\) is full rank for some $\\mc{A}_{\\tau}\\subseteq [p]$ and for a fixed regularization parameter $\\lambda$, the $\\tau$-path \\( \\tau \\mapsto \\beta(\\tau),  \\forall \\tau \\in [-\\infty, +\\infty] \\) is well defined (see Proposition~\\ref{prop:homotopy}), then the worst-case complexity of the $\\tau$-path is $3^p$. Indeed, the sign $s(\\tau) \\in \\{-1, 0, +1\\}^p$ of the coefficients $\\beta(\\tau)$ does not change between any two consecutive kinks (zero-crossing points) of the piece-wise linear path \\(\\tau \\mapsto \\beta(\\tau)\\). Therefore, for all $p$ patterns the number of linear pieces in $\\tau$-path is bounded by $3^p$ (for details see \\citet{mairal2012complexity}). However, this worst-case complexity is too pessimistic and fortunately, it has been well-recognized that this worst-case rarely happens in practice \\cite{le2021parametric} and, this is also evident from our experimental results (Table \\ref{table:comp_eff_kinks}). \n\n\\subsection{Algorithms}\nThe algorithms to compute the $\\tau$-path and the full-CP set of SHIM are given in Algorithm \\ref{algo:tau_path} and Algorithm \\ref{algo:full-CP} respectively. The Algorithm \\ref{algo:tau_path} returns the transition points ($\\mb{T}$), the LASSO solutions ($\\mb{B}$) and the active sets ($\\mb{A}$) at those transition points  which are subsequently provided to Algorithm \\ref{algo:full-CP}. In Algorithm \\ref{algo:full-CP}, for each linear piece of the $\\tau$-path, i.e. \\(\\forall \\tau \\in (\\tau_t, \\tau_{t+1})\\) for any two consecutive kinks $\\tau_t$ and $\\tau_{t+1}$, we need to first identify the points at which $|w_{n+1}(\\tau)| = |w_i(\\tau)|, \\forall i\\in [n]$ (Line 4). Let us denote those points as \\(\\{\\tau_t = u_1, u_2, u_3, \\ldots, u_r = \\tau_{t+1} \\}\\) (for simplicity we slightly abused the notation here). Then we need to check that at which of those points the condition stated in Equation (\\ref{eqn:CP2}) is satisfied (Line 5, 6, 7) to determine the full-CP set of SHIM.\n\\begin{algorithm}[h!]\n\\begin{footnotesize}\n\\caption{Compute $\\tau$-path}\n\\label{algo:tau_path}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} \\(Z \\in \\mb{R}^{(n+1) \\times m}, y \\in \\mb{R}^n, [y_{min}, y_{max}], \\lambda \\)\n\\STATE Initialization: \n\\STATE\\hspace{\\algorithmicindent} \\( t=1, \\tau_1 = y_{min}\\), \\(\\mc{A}_{\\tau_1} = \\{\\ell: \\beta_\\ell(\\tau_1) \\neq 0\\}\\), \\( \\forall \\ell  \\in [p]\\), \n\\STATE\\hspace{\\algorithmicindent}\\(\\mb{T}= \\{\\tau_1\\},  \\mb{B}= \\{\\beta_{\\mc{A}_{\\tau_1}}(\\tau_1)\\}, \\mb{A}=\\{\\mc{A}_{\\tau_1}\\}\\)\n\\WHILE{\\((\\tau_t < y_{max})\\)}\n\\STATE Compute \\(\\Delta_{\\ell^\\ast} \\) using (\\ref{eq:step_length}) and Lemma \\ref{lemma:lemma_1}\n\\STATE If \\(\\Delta_{\\ell^\\ast} = \\Delta_{\\ell^\\ast}^1\\), \\(\\mc{A}_{\\tau_{t+1}} \\leftarrow  \\mc{A}_{\\tau_t} \\setminus \\ell \\) \\hspace{0.2cm} (remove $\\ell$ from \\(\\mc{A}_{\\tau_t})\\) \n\\STATE If $\\Delta_{\\ell^\\ast} = \\Delta_{\\ell^\\ast}^2$, \\(\\mc{A}_{\\tau_{t+1}} \\leftarrow  \\mc{A}_{\\tau_t} \\cup \\{\\ell\\} \\) \\hspace{0.2cm}  (add  \\(\\ell\\) into \\(\\mathcal{A}_{\\tau_t})\\)\n \\STATE Update:\n %\n \\STATE\\hspace{\\algorithmicindent} \\( \\tau_{t+1} \\leftarrow \\tau_{t} + \\Delta_{\\ell^\\ast} \\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1}) \\leftarrow \\beta_{\\mc{A}_{\\tau_t}}(\\tau_t) + \\Delta_{\\ell^\\ast} \\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)\\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\mb{T}=\\mb{T} \\cup \\{\\tau_{t+1}\\}\\)\n\\STATE\\hspace{\\algorithmicindent} \\( \\mb{B}=\\mb{B} \\cup \\{\\beta_{\\mc{A}_{\\tau_t}}(\\tau_{t+1})\\} \\)\n\\STATE\\hspace{\\algorithmicindent} \\(\\mb{A}=\\mb{A} \\cup \\{\\mc{A}_{\\tau_{t+1}}\\}\\)\n\\STATE\\hspace{\\algorithmicindent} $\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t)$ and $\\gamma_{\\mc{A}_{\\tau_t}}(\\tau_t)$ using (\\ref{direction_vec})\n\\STATE\\hspace{\\algorithmicindent} t = t +1\n\\ENDWHILE\n\\STATE \\textbf{Output:} $\\mb{T}, \\mb{B}, \\mb{A}$\n\\end{algorithmic}\n\\end{footnotesize}\n\\end{algorithm}\n\\begin{algorithm}[h!]\n\\begin{footnotesize}\n\\caption{Compute full-CP}\n\\label{algo:full-CP}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input:} \\(Z \\in \\mb{R}^{(n+1) \\times m}, y \\in \\mb{R}^n, \\alpha \\in [0, 1], \\mb{T}, \\mb{B}, \\mb{A}  \\)\n\\STATE Initialization: \\( t=0, t_{max} = |\\mb{T}|, \\mc{C} = \\{\\emptyset\\}\\)\n\\WHILE{\\((t < t_{max})\\)}\n\\STATE $\\{u_2, \\ldots, u_{r-1} \\}=\\{u \\in (\\tau_t, \\tau_{t+1}):|w_i(u)| = |w_{n+1}(u)|,  \\forall i \\in [n]\\}$\n\\STATE \\(\\mc{T}_t = \\{\\tau_t\\} \\cup \\{u_2, u_3, \\ldots, u_{r-1} \\} \\cup \\{\\tau_{t+1}\\} \\)\n\\STATE \\(\\mc{H}_t = \\{ h \\in \\{1, \\ldots, | \\mc{T}_t | \\} \\, :\\,  \\pi(\\mc{T}_t (h)) \\geq \\alpha\\}\\) using $\\mb{B}$ and  $\\mb{A}$\n\\STATE \\(\\mc{C} = \\mc{C} \\cup_{h \\in \\mc{H}_t} [ \\mc{T}_t(h), \\mc{T}_t(h + 1)) \\)\n\\STATE t = t +1\n\\ENDWHILE\n\\STATE \\textbf{Output:} $\\mc{C}$\n\\end{algorithmic}\n\\end{footnotesize}\n\\end{algorithm}\n\n\\section{Extension for Elastic Net (El-SHIM)}\nA common problem of the LASSO is that if the data has correlated features, then the LASSO picks only one of them and ignores the rest, which leads to instability. To solve this problem \\citet{zou2005regularization} proposed the Elastic Net (ElNet). This feature correlation problem is very much evident in the SHIM-type problem, and hence we extended our framework for the Elastic Net. We solve the following optimization problem to extend our framework for the Elastic Net:\n\\begin{equation}\\label{obj:primal_elnet}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mathbb{R}^p} \\frac{1}{2}\\norm{y(\\tau) - X\\beta}_2^2 + \\frac{1}{2}\\alpha \\norm{\\beta}_2^2  + \\lambda \\norm{\\beta}_1 \\enspace .\n\\end{equation}\nThe elastic net optimization problem can actually be formulated as a LASSO optimization problem using augmented data.\nIf we consider an augmented data defined as \\(\\Tilde{X} = \\begin{pmatrix} X\\\\ \\sqrt{\\alpha} I_p \\end{pmatrix}  \\in \\mb{R}^{(n+1+p) \\times p}\\) and \\(\\Tilde{y} (\\tau) = \\begin{pmatrix}y (\\tau) \\\\ \\mathbf {0} \\end{pmatrix} \\in \\mb{R}^{n+1+p}\\), where $I_p \\in \\mb{R}^{p \\times p}$ is an identity matrix and $\\mathbf {0} \\in \\mb{R}^p$ is a zero vector, then solving the elastic net optimization problem (\\ref{obj:primal_elnet}) for a fixed $\\lambda$, is equivalent to solving the following problem.\n\\begin{equation}\n    \\beta(\\tau) \\in \\mathop{\\mathrm{arg\\,min}}_{\\beta \\in \\mathbb{R}^p} \\frac{1}{2}\\norm{\\Tilde{y} (\\tau) - \\Tilde{X}\\beta}_2^2 + \\lambda \\norm{\\beta}_1.\n\\end{equation}\n\\paragraph{Step size ($\\tau$-path)}\nIf we consider two real values $\\tau_t$ and $\\tau_{t+1}$ ( $\\tau_{t+1}>\\tau_t$) at which the active set does not change, and their signs also remain the same, then we can write \n\\begin{align*}\n  \\beta_{\\mathcal{A}_{\\tau_{t}}}(\\tau_{t+1}) - \\beta_{\\mathcal{A}_{\\tau_t}}(\\tau_t) = \\nu_{\\mathcal{A}_{\\tau_t}}(\\tau_t)(\\tau_{t+1} - \\tau_t),\\\\ \n   \\lambda s_{\\mathcal{A}^c_{\\tau_{t}}}(\\tau_{t+1}) - \\lambda s_{\\mathcal{A}^c_{\\tau_t}}(\\tau_t) = \\gamma_{\\mathcal{A}^c_{\\tau_t}}(\\tau_t)(\\tau_{t+1} - \\tau_t),\n\\end{align*}\nwhere, \\( \\nu_{\\mc{A}_{\\tau_t}}(\\tau_t) = (X^{\\top}_{\\mc{A}_{\\tau_t}}X_{\\mc{A}_{\\tau_t}} + \\alpha I_{|\\mathcal{A}_{\\tau_t}|} )^{-1}x_{n+1;\\mc{A}_{\\tau_t}} \\)  and \\( \\gamma_{\\mc{A}^c_{\\tau_t}} (\\tau) = x_{n+1, \\mc{A}_{\\tau_t}^c} - (X^{\\top}_{\\mc{A}_{\\tau_t}^c}X_{\\mc{A}_{\\tau_t}})\\nu_{\\mc{A}_{\\tau_t}}(\\tau_t) \\). \nNote that here the only change compared to the vanilla SHIM is the addition of an \\(\\alpha I_{|\\mathcal{A}_{\\tau_t}|}\\) term to the expression of \\(\\nu_{\\mathcal{A}_{\\tau_t}}\\). Now, one can also derive a similar expression of the step size of inclusion and deletion as done for the vanilla SHIM $\\tau$-path (\\ref{eq:step_length}), considering the updated expression of \\(\\nu_{\\mathcal{A}_{\\tau_t}}(\\tau_t)\\).\n\\paragraph{Tree pruning ($\\tau$-path)}\nThe pruning condition (\\ref{cond:pruning_tau}) with the augmented data (\\(\\Tilde{X}, \\Tilde{y}\\)) can be written as follows:\n\\begin{align}\\label{cond:pruning_elnet1}\n\\lvert \\Tilde{\\rho}_{\\ell}(\\tau_t) \\lvert + \\Delta^2_{\\ell^{\\ast}} (\\lvert \\Tilde{\\eta}_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) \n< \\lvert \\Tilde{\\rho}_k (\\tau_t) \\lvert   - \\Delta^2_{\\ell^{\\ast}} (\\lvert\\Tilde{\\eta}_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\nwhere, \\(\\Tilde{\\rho}_{\\ell} (\\tau_t)= \\Tilde{x}_{\\ell}^{\\top}\\Tilde{w}(\\tau_t) \\) and \\(\\Tilde{\\eta}_{\\ell} (\\tau_t)=\\Tilde{x}_{\\ell}^{\\top} \\Tilde{v}(\\tau_t), \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\); \\(\\Tilde{\\rho}_k (\\tau_t)= \\Tilde{x}_{k}^{\\top}\\Tilde{w}(\\tau_t) \\) and \\(\\Tilde{\\eta}_k (\\tau_t) = \\Tilde{x}_k^{\\top} \\Tilde{v}(\\tau_t), \\forall k \\in \\mathcal{A}_{\\tau_t} \\); and \\(\\Tilde{w}(\\tau_t) = \\Tilde{y} (\\tau_t) - \\Tilde{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) and  \\(\\Tilde{v}(\\tau_t) = \\Tilde{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\). \nThe above pruning condition (\\ref{cond:pruning_elnet1}) can be further simplified and redefined as stated in Lemma \\ref{lemma:lemma_2}.\n\\begin{lemma}\\label{lemma:lemma_2}\nIf $\\Delta^2_{\\ell^\\ast}$ is the current minimum step size, that is, \\(\\Delta^2_{\\ell^\\ast} = \\underset{j \\in \\{1, 2, \\ldots, \\ell\\}}{\\min}\\{\\Delta^2_j\\}, \\) then\n$\\forall \\ell^\\prime \\supset \\ell, \\Delta^2_{\\ell^\\prime} > \\Delta^2_{\\ell^\\ast}$ if \n\\begin{align} \\label{eq:lemma_2_eq}\n\t b_{\\ell, w(\\tau_t)} + &\\Delta^2_{\\ell^\\ast} (b_{\\ell, v(\\tau_t)} + x_{n+1,\\ell})\n\t< |\\Bar{\\rho}_k(\\tau_t)|  - \\Delta^2_{\\ell^\\ast} ( |\\Bar{\\eta}_k(\\tau_t)| + x_{n+1,k}).\n\\end{align}\n\\end{lemma}\nwhere,  \\(\\Bar{\\rho}_k(\\tau_t) = \\sum_{i=1}^n w_i(\\tau_t) x_{ik} - \\alpha \\beta_k\\), \\(\\Bar{\\eta}_k(\\tau_t) = \\sum_{i=1}^n v_i(\\tau_t) x_{ik} + \\alpha \\nu_k\\).\n\n\nThe proof of Lemma \\ref{lemma:lemma_2} is given below.\n\\begin{proof}\n\\textnormal{Lets, consider} \\(\\Tilde{w}(\\tau_t) = \\Tilde{y} (\\tau_t) - \\Tilde{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) \\textnormal{and} \\(w(\\tau_t) = y(\\tau_t) - \\bar{X}_{\\mc{A}_{\\tau_t}}\\beta_{\\mc{A}_{\\tau_t}}(\\tau_t)  \\in \\mathbb{R}^{n+1}\\), \\textnormal{where} \\(p = |\\mathcal{A}_{\\tau_t}| + |\\mathcal{A}^c_{\\tau_t}| \\), \\textnormal{then we can write}\n\\begin{equation}\\label{eqn:aug_w}\n    \\Tilde{w}_i(\\tau_t) = \\begin{cases}\n     w_i(\\tau_t) \\hspace{1.1cm} \\textnormal{if} \\quad  i \\leq n+1, \\\\\n    -\\sqrt{\\alpha}\\beta_{\\ell}(\\tau_t) \\hspace{0.3cm} \\textnormal{if} \\quad n+1 < i \\leq n + 1+ |\\mathcal{A}_{\\tau_t}|,\n    \\hspace{0.5cm} \\forall \\ell \\in \\mathcal{A}_{\\tau_t}, \\\\\n    0 \\hspace{1.8cm}  \\textnormal{if} \\quad n+1 + |\\mathcal{A}_{\\tau_t}| < i \\leq n + 1 + p.\n    \\end{cases}\n\\end{equation}\n\\textnormal{similarly considering \\(\\Tilde{v}(\\tau_t) = \\Tilde{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1+p}\\) and \\(v(\\tau_t) = \\bar{X}\\nu(\\tau_t)  \\in \\mathbb{R}^{n+1}\\), we can write}\n\\begin{equation}\\label{eqn:aug_v}\n    \\Tilde{v}_i(\\tau_t) = \\begin{cases}\n     v_i(\\tau_t) \\quad \\quad \\hspace{0.08cm} \\textnormal{if} \\quad i \\leq n +1,\\\\\n    \\sqrt{\\alpha}\\nu_{\\ell}(\\tau_t)  \\hspace{0.3cm} \\textnormal{if} \\quad n+1 < i \\leq n +1 + |\\mathcal{A}_{\\tau_t}|,\n    \\hspace{0.5cm} \\forall \\ell \\in \\mathcal{A}_{\\tau_t},\\\\\n    0 \\quad \\quad \\quad \\hspace{0.42cm} \\textnormal{if} \\quad n+1 + |\\mathcal{A}_{\\tau_t}| < i \\leq n +1 + p.\n    \\end{cases}\n\\end{equation}\n\\textnormal{and, considering \\(\\Tilde{X} \\in \\mathbb{R}^{(n+1+p)\\times p}\\) and \\(X \\in \\mathbb{R}^{(n+1) \\times p}\\) we can write}\n\\begin{equation}\\label{eqn:aug_X}\n    \\Tilde{x}_{i\\ell} = \\begin{cases}\n    x_{i\\ell} \\quad \\quad \\hspace{0.1cm} \\textnormal{if} \\quad i \\leq n+1, \\\\\n    \\sqrt{\\alpha} \\quad \\quad \\hspace{0.03cm} \\textnormal{if} \\quad i > n+1 \\enspace \\textnormal{and} \\enspace (i-n-1)=\\ell,\\\\\n    0 \\hspace{1.05cm} \\textnormal{otherwise}.\n    \\end{cases}\n\\end{equation}\n\\textnormal{Therefore, \\(\\forall \\ell \\in \\mathbb{R}^p\\)  we can write}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell}, \\\\\n    &=\\sum_{i=1}^{n+1} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} + \\sum_{\\substack{i=n+2}}^{n+1+|\\mathcal{A}_{\\tau_t}|} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} + \\sum_{i=n+2+|\\mathcal{A}_{\\tau_t}|}^{n+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell}.\n\\end{align*}\n\\textnormal{Now, using (\\ref{eqn:aug_w}) and (\\ref{eqn:aug_X}) the second and the third quantity in the above expression can be written as follows:}\n\\begin{equation*}\n\\sum_{\\substack{i=n+2}}^{n+1+|\\mathcal{A}_{\\lambda_t}|} \\Tilde{w}_i (\\tau_t) \\Tilde{x}_{i\\ell} =\\begin{cases}\n    (-\\sqrt{\\alpha}\\beta_{\\ell}(\\tau_t))(\\sqrt{\\alpha}), \\hspace{0.5cm} \\textnormal{if} \\enspace (i-n-1) = \\ell,\\\\\n    0 \\hspace{0.5cm} \\textnormal{otherwise}.\n    \\end{cases}\n\\end{equation*}\n\\textnormal{and,}\n\\begin{equation*}\n   \\sum_{i=n+2+|\\mathcal{A}_{\\tau_t}|}^{n+p} \\Tilde{w}_i(\\tau_t) \\Tilde{x}_{i\\ell} = 0. \n\\end{equation*}\n\\textnormal{Therefore,}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1} w_i(\\tau_t) x_{i\\ell} \\\\\n    &= \\rho_{\\ell}(\\tau_t), \\enspace \\forall \\ell \\in \\mathcal{A}^c_{\\tau_t} \\enspace \\textnormal{and,} \n\\end{align*}\n\\begin{align*}\n   \\Tilde{x}_k^{\\top} \\Tilde{w}(\\tau_t) &= \\sum_{i=1}^{n+1} w_i(\\tau_t) x_{ik} - \\alpha \\beta_k (\\tau_t)\\\\\n   &= \\rho_k(\\tau_t) - \\alpha \\beta_k (\\tau_t), \\enspace \\forall k \\in \\mathcal{A}_{\\tau_t},\\\\\n   &=: \\bar{\\rho}_k(\\tau_t) .\n\\end{align*}\n\\textnormal{Similarly, using (\\ref{eqn:aug_v}) and (\\ref{eqn:aug_X}) we can write}\n\\begin{align*}\n    \\Tilde{x}_{\\ell}^{\\top} \\Tilde{v}(\\tau_t) &= \\sum_{i=1}^{n+1} v_i (\\tau_t) x_{i\\ell} \\\\\n    &= \\eta_{\\ell}(\\tau_t), \\enspace \\forall \\ell \\in \\mathcal{A}_{\\tau_t}^c, \\hspace{0.5cm} \\textnormal{and,}\n\\end{align*}\n\\begin{align*}\n  \\Tilde{x}_k^{\\top} \\Tilde{v}(\\tau_t) &= \\sum_{i=1}^{n+1} v_i(\\tau_t) x_{ik} + \\alpha \\nu_k (\\tau_t) \\\\\n  &=  \\eta_k(\\tau_t) + \\alpha \\nu_k (\\tau_t) , \\enspace \\forall k \\in \\mathcal{A}_{\\tau_t}\\\\\n  &= \\bar{\\eta}_k(\\tau_t).\n\\end{align*}\n\\end{proof}\nTherefore, we can write the pruning condition (\\ref{cond:pruning_elnet1}) as follows.\n\\begin{align}\\label{cond:pruning_elnet3}\n\\lvert \\rho_{\\ell}(\\tau_t) \\lvert + &\\Delta^2_{\\ell^{\\ast}} (\\lvert \\eta_{\\ell}(\\tau_t) \\lvert + x_{n+1,\\ell}) < \\lvert \\bar{\\rho}_k (\\tau_t)  \\lvert   - \\Delta^2_{\\ell^{\\ast}} (\\lvert \\bar{\\eta}_k(\\tau_t) \\lvert + x_{n+1,k}),\n\\end{align}\n\nNow, using Proposition\\ref{prop:1}, we can further simplify the above pruning condition as follows.\n\\begin{align}\\label{pruning_elnet3}\n b_{\\ell,w(\\tau_t)} + &\\Delta^2_{\\ell^{\\ast}}  (b_{\\ell,v(\\tau_t)}  + x_{n+1,\\ell}) <  |\\Bar{\\rho}_k(\\tau_t)| - \\Delta^2_{\\ell^{\\ast}} (|\\Bar{\\eta}_k(\\tau_t)| + x_{n+1,k}),\n\\end{align}\nNow similar to the Lemma \\ref{lemma:lemma_1} one can formally prove the Lemma \\ref{lemma:lemma_2} using (\\ref{pruning_elnet3}) and Proposition \\ref{prop:2}.\n\\section{Experimental Setup and Additional Results}\n\\subsection{Hyper parameter selection (synthetic data experiments)}\\label{hyperparameter_selection}\nThe hyper parameter selection for all the methods (e.g. $\\lambda$ in LASSO, SHIM; hidden layer's sizes in MLP etc.) are done based on $5-$fold cross validation using a separate set of 15 independent samples. For MLP, the activations are chosen from \\{identity, relu, logistic, tanh\\} and the hidden layer's sizes are chosen from all possible combinations of \\{50, 100, 150\\} nodes, considering both 2-hidden layers and 3-hidden layers architectures. The most frequent activation and architecture chosen based on separate set of 15 independent samples are considered to report the average results. Similarly for RF, the number of estimators (n\\_estimators) and the min samples in the leaf of a tree (min\\_samples\\_leaf) are chosen from \\{50, 100, 200\\} and \\{0.1, 0.05, 0.01\\} respectively. The median $\\lambda$ value based on the separate set of 15 independent samples is considered to report the results of LASSO and SHIM. For the $\\lambda$ selection we considered a grid of 10 points in the range of $[\\lambda_{max}\/10, 0)$. For, MLP and RF, we used the standard \\emph{scikit-learn} implementation.\n\\subsection{Real data details}\nThis is a dataset of criminal records containing the criminal history, jail and prison time, and demographics of a subject used to predict the potential recidivism. This dataset contains risk scores of 7210 defendants (n = 7210). The dataset consists of seven categorical and integer-valued features: sex (male, female), age, juvenile-felonies, juvenile-misdemeanors, juvenile-crimes, priors, currentcharge-degree (misdemeanor, felony) which are subsequently converted to binary features. Hence, this resulted in 14 binary features which are - sex:male, age:18-20, age:21-22, age:23-25, age:26-45, age:$>$45 juvenile-felonies:$>$0, juvenile-misdemeanors:$>$0, juvenile-crimes:$>$0 priors:2-3, priors:=0, priors:=1, priors:$>$3, and current-charge-degree:misdemeanor. These 14 binary features are used to predict the two years recidivism risk scores.\n\\subsection{Additional results}\\label{additional_results}\nTo further compare the performance of the proposed method we considered highly sparse data ($\\zeta=0.6$) in two different experimental settings. (1) \\textbf{model-1}: We generated a model with strong signals that is we considered a true model of up to third-order interactions, which is defined as $\\mu (x_i) = 5.0z_1 +5.0 z_1z_2 + 5.0z_1z_2z_3$ and (2) \\textbf{model-2}: We generated a model with weak signals that is we considered a true model of up to third-order interactions, which is defined as $\\mu (x_i) = 1.0z_1 +1.0 z_1z_2 + 1.0z_1z_2z_3$. In both the settings we set $\\sigma=1$. The choice of these models is merely for demonstration purposes, and the proposed method is equally applicable to any chosen model. \nFor both the settings (strong and weak), we generated a dataset of $n\\in \\{50, 100, 150\\}$ training instances, each accompanied with $n=10$ test instances. Here we considered a covariate size of $m=5$. We generated 3 such independent random datasets. Hence, in total we reported the average results of  \\(3 \\times 10 = 30 \\) test instances. The hyper parameter were selected based on another independent set of $3$ samples using the same strategy as explained in the hyper parameter selection section in the appendix. \nFor split-CP, we repeated the experiments 30 times to highlight the effect of randomness in the confidence set generation. For SHIM, we did not mention any ``max-pat\" size of $d$, i.e. the entire search space is considered for exploration (tree generation) and the proposed tree pruning condition takes care of it to improve the efficiency of the search. The best shim model is automatically chosen by the algorithm corresponding to the best hyper parameter $\\lambda$ (explained in the hyper parameter selection section in the appendix).\n\n\nThe results are shown in Fig.\\ref{fig:syn_low_signal_sparse} where the left figure corresponds to the model-1 (strong signal) and the right figure corresponds to the model-2 (weak signal). In both the settings, the full-CP methods (lasso\\_f, shim\\_f) produced more compact confidence sets irrespective of sample sizes.\nIt can be observed that in case of highly sparse data, the shim model produces better results, more specifically when the sample size is small and the signal is weak.\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{figures\/fig3.pdf}\n   \\caption{Comparison of the confidence interval lengths (CI length) of the proposed method (shim) with other simple (lasso) and complex models (mlp, rf) using synthetic data for different sample sizes. Left figure shows the results using model-1 (strong signal) and the right figure shows the results using model-2 (weak signal).}\n   \\label{fig:syn_low_signal_sparse}\n \\end{figure}\n\\begin{table*}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c| } \n\\hline\n $y$ &$\\hat{y}$ & confidence interval &length  &rule &coef \\\\\n \\hline\n 1 &1.34 &[-2.30, 5.00] &7.30  &  \\begin{tabular}{@{}c@{}}[[prior crimes=0], [misdemeanor], [prior crimes=0  \\\\ and misdemeanor], [age$>=$46 and misdemeanor]]\\end{tabular} &[ 1.75, -0.20,  -0.12,   -0.08]\\\\\n \\hline \n 9 &6.15 &[2.51, 9.78] &7.27  &  \\begin{tabular}{@{}c@{}} [[prior crimes $>=4$], [$26<=$age$<=45$]]\\end{tabular} &[4.61, 1.54]\\\\\n \\hline\n 4 &2.94 &[-0.70, 6.58] &7.28  &  \\begin{tabular}{@{}c@{}} [[prior crimes=1], [$26<=$age$<=45$], [$26<=$age$<=45$ \\\\\\relax and misdemeanor], [misdemeanor], [male], [male and $26<=$age$<=45$]]\\end{tabular} &[ 2.01,  1.54, -0.30, -0.20,  -0.09, -0.03]\\\\\n \\hline\n 8 &1.25 &[-2.38, 4.90]&7.28  &  \\begin{tabular}{@{}c@{}} [[prior crimes=0], [misdemeanor], [prior crimes=0 and misdemeanor], [male], \\\\\\relax [age$>=46$ and misdemeanor], [male and prior crimes=0]]\\end{tabular} &[ 1.75, -0.20, -0.12, -0.09, -0.08, -0.001]\\\\\n \\hline\n  2 &6.04 &[2.41, 9.67] &7.26  &  \\begin{tabular}{@{}c@{}} [[prior crimes $>=$ 4], [26$<=$age$<=$ 45], \\\\\\relax [male], [male and 26$<=$ age $<=$45 ]] \\end{tabular} &[ 4.61,  1.54, -0.09, -0.03]\\\\\n \\hline\n  \\end{tabular}}\n\\caption{The SHIM model for 5 randomly chosen test instances of the compas data. The true score ($y$), predicted score ($\\hat{y}$), confidence interval (CI), length of confidence interval (length), rule and the coefficient (coef) of the individual component of a rule are shown.}\n\\label{table:SHIM_compas}\n\\end{table*}\n\nTable \\ref{table:SHIM_compas} shows true score ($y$), predicted score ($\\hat{y}$), confidence interval (CI), length of confidence interval (length), rule and the coefficient (coef) of the individual component of a rule of five randomly chosen test instances of the compas data. This demonstrates that the generated SHIM is easy to interpret and, one can easily judge the reliability of model based on the provided confidence interval along with the point estimation.\n\\begin{figure}[h!]\n   \\centering\n      \\includegraphics[width=\\linewidth]{figures\/node_counts.pdf}\n   \\caption{Variation of node counts (``fraction of node counts\") for different \\emph{max\\_pat} ($d$) sizes. Here, the fraction of nodes for a specific $d$, represents the number of nodes traversed divided by the total number of possible combinations of interaction terms using the ``max-pat\" size of $d$. The results are shown for two $\\lambda$ values ($\\lambda = 1, 10$) and for three different sparsity levels ($0.4, 0.7, 0.9$). The pruning is more effective as the data is more sparse. }\n   \\label{fig:node_counts}\n \\end{figure}\n\\begin{table}[t]\n\\centering\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c| } \n\\hline\n \\multirow{2}{*}{$d$} & \\multirow{2}{*}{\\shortstack{Search space \\\\ (\\# nodes)}} &  \\multicolumn{3}{|c|}{$\\lambda=1$} & \\multicolumn{3}{|c|}{$\\lambda=10$}\\\\\n \\cline{3-8}\n & &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$  &$\\zeta=0.4$ &$\\zeta=0.7$ &$\\zeta=0.9$\\\\\n \\cline{3-8}\n \\hline\n \\hline\n  2 & 465 &200  &134   &7  &9  &7  &2    \\\\\n  \\hline\n  3 & 4525 &317  &129   &7  &9  &7  &2    \\\\\n  \\hline\n  4 & 31930 &314   &129   &7  &9  &7  &2    \\\\\n  \\hline\n  5 & 174436 &312    &129   &7  &9  &7  &2    \\\\\n  \\hline\n  10 & 53009101 &312    &129   &7  &9  &7  &2   \\\\\n  \\hline\n 15 & 614429671 &312   &129   &7  &9  &7  &2    \\\\\n \\hline\n 20 & 1050777736  &312    &129   &7   &9  &7  &2   \\\\\n  \\hline\n 25 &1073709892  &312   &129   &7   &9  &7  &2   \\\\\n \\hline\n\\end{tabular}\n}\n\\caption{Number of homotopy transition points (\\# kinks) along the $\\tau$-path of SHIM using two different $\\lambda$ values ($\\lambda=1, 10$) for three different sparsity levels ($\\zeta=0.4, 0.7, 0.9$) for different value of ``max\\_pat\" ($d$) sizes.}\n\\label{table:comp_eff_kinks}\n\\end{table}\n\nFig.~\\ref{fig:node_counts} shows the variation of node counts (``fraction of node counts\") for different \\emph{max\\_pat} ($d$) size during the construction of $\\tau$-path. One can observe that our pruning condition is more effective at the deeper nodes of the tree as it exploits the tree's anti-monotonicity property. Table ~\\ref{table:comp_eff_kinks} shows the number of homotopy transition points (\\# kinks) along the $\\tau$-path. Although the worst-case complexity of the $\\tau$-path is exponential (see complexity analysis for details), it appears to be polynomial in practice. In fact, from our numerical results, one can observe that it saturates after a certain depth of the tree. For the experimental setup details of Fig.~\\ref{fig:node_counts} and Table ~\\ref{table:comp_eff_kinks} please see section~\\ref{sec_comp_eff} of the main paper.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe calculation of high order terms in low-energy effective field theories\n (EFTs) is a difficult task. Nowadays, most interesting observables have\nbeen calculated at the second order of the expansion, and the difficulty of \nthese calculations shows little hope\nfor any further expansion. The main problem which restricts the potential of EFTs is their non-renormalizability. The non-renormalizability does\nnot bring any problem, in principle, for the calculation by means of counting schemes for EFTs, first introduced in \\cite{Weinberg:1978kz}.\nHowever, the rapidly increasing number of low-energy coupling constants (LECs), makes very high order applications practically of little use in\ngeneral.\n\nNevertheless, there are contributions of the higher order terms which are free from higher order LECs. In particular this is true for the\nleading logarithmical (LLog) contributions. LLogs are not in general dominant for a generic observable. However, for some observables the LLog\ncontribution is dominant. Examples of such observables are the generalized parton distributions at small-$x$\n\\cite{Kivel:2007jj,Moiseeva:2013qoa} and certain $\\pi\\pi$ scattering lengths \\cite{Colangelo:1995np}. In addition, the LLog terms are of great\ntheoretical interest because they allow us to judge the behaviour of a whole series of corrections in EFTs. We therefore\nconsider the calculation of LLog terms in EFTs as an interesting and useful task.\n\nIn renormalizable field theories the LLog terms can be calculated to all orders using the renormalization group (RG) and (simple) one-loop\ncalculations of the beta functions. In EFTs, as e.g. Chiral Perturbation Theory (ChPT), they can also be calculated using one-loop calculations\nas was suggested already in \\cite{Weinberg:1978kz}, and proven in \\cite{Buchler:2003vw}. In contrast to renormalizable theories, in EFTs the\nLLog terms cannot be obtained simultaneously for all orders, and every order of the perturbative expansion requires an additional\ncalculation. However, the evaluation of LLog terms is considerably simpler than a full calculation. As an example, the full two-loop leading\nlogarithms in bosonic ChPT were known long before the full results \\cite{Bijnens:1998yu}.\n\nWithin bosonic EFTs the LLogs have been studied extensively. It has been shown that for EFTs with massless particles the LLog behaviour is\ndescribed by a closed set of equations with known kernels, which were elaborated in\n\\cite{Kivel:2008mf,Kivel:2009az,Koschinski:2010mr,Polyakov:2010pt}. Although, the analytical solution of these equations is not known, one can\ngenerate numerically the first few hundreds coefficients rather fast, and use the approximate numerical solution in applications. An example is\nthe exploration of the ``chiral inflation'' of the pion radius within ChPT \\cite{Perevalova:2011qi}. Taking into account the mass of the fields\nallows for non-zero tadpole diagrams, which leads to a rapidly increasing number of equations with the chiral order since one has to consider\none-loop diagrams with an ever increasing number of external legs. Therefore, one needs to incorporate new processes at every new\norder. As a result, the difficulty of the calculation grows extremely fast with the chiral order. By automatizing the procedure for a large\nnumber of processes the LLogs are known up to seven loops for some quantities\n\\cite{Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca}.\n\nThe main goal of this paper is to generalize the methods used for bosonic EFTs\nwith masses to the nucleon case. As mentioned earlier, it\nis not only interesting from the theoretical side, but also necessary for the\nevaluation of nucleon parton distributions at $x\\sim{m_\\pi}\/{M_N}$\n\\cite{Moiseeva:2013qoa,Moiseeva:2012zi}. In the paper we present the extension\nof the RG method of \\cite{Buchler:2003vw} to nucleon-pion ChPT. With its help,\nwe calculate the LLog coefficients for the chiral expansion of the nucleon mass\nin the heavy-baryon formulation of ChPT. The main results are presented in\nSects.~\\ref{sec:resultmass} and \\ref{sec:conjectures}.\nAn earlier application of LLogs in the nucleon sector was the calculation of\nthe two-loop LLog contribution to the axial nucleon coupling constant $g_A$\n\\cite{Bernard:2006te}.\n\nThe paper is organized as follows: In Sect.~\\ref{sec:RGO} we introduce the concept of renormalization group order (RGO). This is needed since in\nthe nucleon sector chiral counting and loop counting are not identical. Sect.~\\ref{sec:meson} shows how the RGO concept\nworks in the meson sector and quotes some known results. Sect.~\\ref{sec:HB_Lagrangian} introduces the heavy baryon ChPT Lagrangian in its two\nmost common variants and the different meson parametrizations we have used as a check on our result. Sect.~\\ref{sec:general_comm} shows how the\nRGO can be used to prove the calculation of the leading logarithms using only one-loop diagrams also in the nucleon sector. This is then used to\ncalculate the LLogs for the nucleon mass in Sect.~\\ref{sec:nucl_mass}. Some technicalities are discussed in Sects.~\\ref{sec:propagator} and\n\\ref{sec:pole}. We then calculate the LLogs for the nucleon mass as well as the odd-power next-to-leading logarithms (NLLogs) in\nSect.~\\ref{sec:resultmass} up to four respectively five loops. The observed regularity in the leading logarithm allows to also calculate the\nfive loop result with a mild assumption. The LLogs, then essentially known to five loops, show a remarkable regularity when rewritten in the\nphysical pion mass. We conjecture that this regularity holds to all orders and in that case using the known results for the pion LLogs we have a\nresult for the nucleon mass LLogs up to 7 loops. This is described in Sect.~\\ref{sec:conjectures}. A short numerical discussion of our results\nis given in Sect.~\\ref{sec:numerics}. We summarize our conclusions in Sect.~\\ref{sec:conclusions}. The LLogs for a general heavy baryon one-loop\nintegral are discussed in App.~\\ref{app:loopintegrals}.\n\n\n\\section{Renormalization group and order}\n\\label{sec:RGO}\n\n\\subsection{Renormalization group operator}\n\nIn this section we present a short, hopefully self-contained, introduction\nto the renormalization group approach in EFTs. Our main goal is to\npresent the method of obtaining the dependence of observables on\nthe renormalization or subtraction scale $(\\mu)$. The material is presented in\na form transparent for the application at higher orders. More extensive\ndiscussions can be found in \\cite{Buchler:2003vw,Bijnens:2009zi,AVthesis}.\nIn particular, what we call LLogs is the contribution with the highest power\nof $\\log\\mu$ at a given order of the expansion.\n\nTo start with, we remind the reader that the Lagrangian of an EFT is the most\ngeneral local Lagrangian satisfying given symmetry properties with a given set of\ndegrees of freedom or fields. Such a Lagrangian contains an infinite number\nof terms. In the absence of additional restrictions, every independent operator\nis multiplied by an unknown coupling constant, usually called low-energy\nconstant (LEC).\n\nIt is convenient to multiply every operator by the counting parameter $\\hbar$\nto the power which reflects the minimal order of the perturbative\nexpansion the operator contributes to. In this way, the constant $\\hbar$\nresembles the coupling constant in a renormalizable field theory as a way to\nkeep track of (loop) orders in the expansion.\nTherefore, an EFT Lagrangian takes the form\n\\begin{eqnarray}\n\\label{1:L=sum L^(n)}\n\\mathcal{L}^{\\text{EFT}}_{\\text{bare}}\n=\\sum_{n=0}^\\infty \\hbar^n\\mathcal{L}^{(n)}_{\\text{bare}}.\n\\end{eqnarray}\nThe Lagrangian $\\mathcal{L}^{(n)}$ we call the Lagrangian of $n$'th\n$\\hbar$-order\\footnote{The Lagrangian which contains the propagator of\nfields, must be included in the zeroth $\\hbar$-order.} and its LECs are\nconsequently called LECs of $n$'th $\\hbar$-order. Let us, following\n\\cite{Bijnens:2009zi}, denote LECs of $n$'th $\\hbar$-order as $c_i^{(n)}$,\nwhere the index $i$ enumerates independent operators. In this way, the\n$n$'th $\\hbar$-order Lagrangian reads\n\\begin{eqnarray}\n\\mathcal{L}^{(n)}_{\\text{bare}}=\\sum_{i} c_{(\\text{bare})i}^{(n)}\\mathcal{O}^{(n)}_i.\n\\end{eqnarray}\nFor some low-energy EFTs, like mesonic ChPT, the $\\hbar$-ordering of operators\nis in one-to-one correspondence with the chiral ordering. However, the\ndefinition (\\ref{1:L=sum L^(n)}) is more general. It can be applied to any EFT,\nand, even, to renormalizable theories, some examples can be\nfound in \\cite{Buchler:2003vw,AVthesis}. We should mention that there is no\nunique definition of the $\\hbar$-ordering for a theory. The only\nconstraint is that the $\\hbar$-order of an operator should increase with\nincreasing perturbative order. The choice made is for EFTs often referred to\nas the choice of power counting.\n\nThe bare Lagrangian is now split into a part with\nrenormalized couplings $c_i^{(n)}$, which depend on $\\mu$, and the counterterms.\nThe renormalization scale independence of the Lagrangian leads to the set of RG\nequations for the LECs $c_i^{(n)}$. These equations are of the form\n\\begin{eqnarray}\n\\label{1:dc=beta}\n\\mu^2\\frac{d}{d\\mu^2}c_i^{(n)}(\\mu^2)=\\beta^{(n)}_{i}(\\{c_j^{(m)}(\\mu^2)\\}),\n\\end{eqnarray}\nwhere the beta-function is a polynomial in the LECs and we have indicated\nexplicitly the $\\mu$-dependence. An important point, used later, is\nthat the right-hand side of (\\ref{1:dc=beta}) contains only combinations of\nLECs with total $\\hbar$-order strictly less then $n$.\n\nThe general formal solution of the system of equations (\\ref{1:dc=beta}) is\n\\begin{eqnarray}\n\\label{defR}\nc_i^{(n)}(\\mu^2)=\\hat R\\(\\frac{\\mu}{\\mu_0}\\)c_{i}^{(n)}(\\mu^2_0)\n=\\exp\\(\\log\\(\\frac{\\mu^2}{\\mu^2_0}\\)\\hat H\\)c_{i}^{(n)}(\\mu^2_0).\n\\end{eqnarray}\nThis defines also $\\hat R$. The operator $\\hat H$ is defined as\n\\begin{eqnarray}\n\\label{defH}\n\\hat H= \\int d\\rho^2\\sum_{n,i}\\beta^{(n)}_i(\\{c^{(m)}_j(\\rho^2)\\})\n\\frac{\\delta}{\\delta c_i^{(n)}(\\rho^2)}.\n\\end{eqnarray}\nThe derivative in (\\ref{defH}) is defined by\n\\begin{eqnarray}\n\\frac{\\delta}{\\delta c_i^{(n)}(\\rho^2)} c_j^{(m)}(\\mu^2)\n= \\delta_{ij}\\delta^{mn}\\delta\\left(\\rho^2-\\mu^2\\right)\n\\end{eqnarray}\nsuch that\n\\begin{eqnarray}\n\\hat H c_i^{(n)}(\\mu^2) = \\beta_i^{(n)}(\\{c_j^{(m)}(\\mu^2)\\})\\,.\n\\end{eqnarray}\nWith the help of $\\hat H$ or $\\hat R$, one can obtain the coefficients of the\nLLog for any observable, without actual calculation of loop\ndiagrams, if the beta-functions are already known. We will demonstrate this\nexplicitly in the next sections.\n\n\\subsection{Renormalization group order}\n\\label{sec:defRGO}\n\nThe crucial property of the operator $\\hat H$ is that the repetitive action of $\\hat H$ nullifies any given LEC (or products of LECs). This is\nthe direct consequence of two features of $\\hbar$-counting. The first one is that the lowest order couplings, with $\\hbar$-order equal to zero,\nhave zero beta-function, and therefore $\\hat H c_i^{(0)}=0$. The second one is that the $\\beta$-function of LEC $c_i^{(n)}$, as defined in\n(\\ref{1:dc=beta}), contains only products of couplings with total $\\hbar$-order lower then $n$. Thus, every application of the operator\n$\\hat H$ onto a product of LECs lowers the total $\\hbar$-order of that product, until it becomes zero.\n\nFor future convenience, we introduce the concept of renormalization group\norder (RGO). A product $P_c$ of LECs has RGO $g$ if\n\\begin{eqnarray}\n\\hat H^g P_c \\ne 0\\quad\\text{and}\\quad\\hat H^{g+1}P_c=0\\,.\n\\end{eqnarray}\nFor a generic\\footnote{We will use this term below to indicate that there are\nexceptions where the beta-functions are zero ``accidentally.'' An\nexample of this is the constant $L_7^r$ in three-flavour bosonic ChPT. This\ndoes not invalidate our later use of the RGO.} quantity with a tree\nlevel contribution of $\\hbar$-order $n$, the RGO is the same as the maximum\nloop order that can appear when calculating that quantity to $\\hbar^n$.\n\nIn the bosonic EFTs treated in the earlier works,\ne.g.\\cite{Kivel:2008mf,Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf}, there is\na one-to-one correspondence between $\\hbar$-order and RGO, namely $g=n$.\nTherefore, the notion of RGO is unnecessary and was not used in these works.\nHowever, such a relation does not hold in general, i.e. LECs of different\n$\\hbar$-order can have the same RGO. For example, in the nucleon ChPT,\nthe $\\hbar$-counting as is related to RGO as $g=[n\/2]$, where $[x]$ indicates\nthe integer part of $x$\n(see detailed discussion in Sec. \\ref{sec:general_comm}). In\nsuch a case the use of the RGO concept is convenient.\n\nIt is natural to split the beta-function into terms with the same RGO. For\nthe beta-function of a coupling constant $c_i^{(n)}$ with RGO $g$ we can write\n\\begin{eqnarray}\n\\beta^{(n)}_{i}= \\sum_{p=0}^{g-1}\\beta^{(n,p)}_i(c).\n\\end{eqnarray}\nHere $g=n$ or $[n\/2]$ for the two cases mentioned above.\nOne can show that the part of the beta function with the highest RGO,\n$\\beta^{(n,g-1)}_i$, contains only contributions from one-loop. The next part,\n$\\beta^{(n,g-2)}_{i}$, contains contributions from one and two-loop diagrams\nand so on. Thus, the expression for the operator $\\hat H$ can be ordered by\nRGO as\n\\begin{eqnarray}\n\\hat H=\\sum_{p=1}^\\infty \\hat H_p\\,.\n\\end{eqnarray}\n$\\hat H_p$ contains the beta-functions $\\beta^{(n,g-p)}$ of the coupling\nconstants $c_i^{(n)}$ of RGO $g$. As a consequence,\nacting with $\\hat H_p$ on an expression reduces its RGO by $p$.\n\n\\section{LLog in mesonic ChPT}\n\\label{sec:meson}\n\nIn mesonic ChPT the choice of $\\hbar$-counting versus chiral counting relates\nboth as $\\hbar^n\\sim \\mathcal{O}(p^{2n+2})$. The\nlowest order Lagrangian is of the second chiral order and\nreads\\footnote{We write here only the terms relevant for the mass and\nneglect external fields.}\n\\begin{eqnarray}\n\\mathcal{L}^{(0)}_{\\pi}=\\frac{F^2}{4}\\tr\\[u_\\mu u^\\mu+\\chi_+\\],\n\\end{eqnarray}\nwhere we use the standard notation\n\\begin{eqnarray}\n\\label{2:u_mu_def}\nu_\\mu=i\\(u^\\dagger \\partial_\\mu u-u\\partial_\\mu u^\\dagger\\),~~~~\\chi_+=u^\\dagger \\chi u^\\dagger+u\\chi^\\dagger u=m^2\\(u^2+u^{\\dagger2}\\).\n\\end{eqnarray}\nHere and though the text, $F$ is the bare pion decay constant and $m$ is the\nbare pion mass, $m^2 = 2B\\hat m$ in the notation of\n\\cite{Gasser:1983yg}. $u$ contains the meson fields, \na few examples of possible parametrizations are given in\n(\\ref{u_param1}-\\ref{u_param3}). The next order Lagrangian $\\mathcal{L}^{(1)}$\nis of fourth chiral order. The absence of odd chiral order\nLagrangians is guaranteed by Lorentz invariance.\n\nIn mesonic ChPT, the generic RGO of a LEC $c^{(n)}$ is equal to $n$. The one-to-one correspondence between generic RGO and the $\\hbar$-order is\nthe result of the absence of odd-chiral-order Lagrangians. Any product of LECs is also in one-to-one correspondence with its generic RGO, which\nis equal to the sum of the LECs' $\\hbar$-orders. That, in turn, results in the simple ordering of beta-functions: the beta-function\n$\\beta^{(n,n-l)}$ contains only $l$-loop beta functions.\n\nAs an example of using the operator $\\hat H$ of (\\ref{defH}) to obtain the\nLLog, we look at the physical pion mass. In order to obtain the\nphysical pion mass one should solve the equation\n$m_\\pi^2-m^2+\\Sigma_\\pi(m_\\pi^2,m^2)=0$, where $\\Sigma_\\pi(p^2,m^2)$ is a series\nof perturbative corrections to the pion propagator. The expression for\n$\\Sigma_\\pi$ has the general form\n\\begin{eqnarray}\n\\label{2:Sigma_pi} \\Sigma_\\pi(p^2=m_\\pi^2,m^2)\n  = m^2\\sum_{n=1}^\\infty \\(\\frac{m^2}{(4\\pi F)^2}\\)^{n}\n   \\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\),\n\\end{eqnarray}\nwhere $\\Sigma^{(n)}_\\pi$ is a dimensionless expression of maximum\n$\\hbar$-order $n$.\nThe first argument of $\\Sigma_\\pi^{(n)}$ appears only as\nthe argument of logarithms. We have suppressed the arguments $p^2\/m^2$.\nNote, that $p^2\/m^2$ can also enter the arguments of logarithms,\nmoreover there can be logarithms of more complicated expressions of it.\nSuch logarithms are not RG logarithms, and can not in general be obtained by\nany procedure based on RG.\n\nThe expression for $\\Sigma_\\pi$ is renormalization scale\nindependent\\footnote{We use here a scheme where all one-particle irreducible\ndiagrams are made finite, otherwise one should apply the argument to a well\ndefined Green function of external currents.}.\nMoreover, it is renormalization scale invariant at every chiral order\nindependently:\n\\begin{eqnarray}\n\\left[\\mu^2\\frac{\\partial}{\\partial\\mu^2}\n+\\sum_{i,n}\\beta^{(n)}_i\\frac{\\partial}{\\partial c_i^{(n)}(\\mu^2)}\\right]\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)\n=\n\\left[\\mu^2\\frac{\\partial}{\\partial\\mu^2}\n+\\hat H\\right]\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)\n= 0\\,.\n\\end{eqnarray}\nTherefore, we again have as solution, similar to (\\ref{defR}),\n\\begin{eqnarray}\n\\label{2:Sigma=R Sigma}\n\\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\hat\nR\\(\\frac{\\mu^2}{\\mu_0^2}\\)\\Sigma^{(n)}_\\pi\\(\\frac{\\mu_0^2}{m^2},c(\\mu_0^2)\\).\n\\end{eqnarray}\nChoosing $\\mu_0^2=m^2$, one neglects all the RG logarithms in\n$\\Sigma^{(n)}_\\pi$ on the right-hand-side.\nThus, all RG logarithms are collected in\nthe action of the operator $\\hat R$.\n\nThe expression for $\\Sigma^{(n)}_\\pi$ has the following form\n\\begin{eqnarray}\n\\Sigma^{(n)}_\\pi=\\sum_{i}\\{c^{(n)}_{i}\\}V^{(n)}_i+\\text{terms with lower RGO},\n\\end{eqnarray}\nwhere $\\{c^{(n)}_{i}\\}V^{(n)}_i$ form the tree contribution to $\\Sigma^{(n)}$.\nThe symbol $\\{c^{(n)}_{i}\\}V^{(n)}_i$ denotes here the products of $c_j^{(m)}$\nwith the highest possible RGO for the product. The $V^{(n)}_i$ depend on\n $p^2\/m^2$. The highest power of the RG logarithm\n$\\log\\mu^2$ in the expression (\\ref{2:Sigma=R Sigma}) accompanies the highest\npower of $\\hat H$, in $\\hat R = \\exp(\\log(\\mu^2\/\\mu_0)\\hat H)$, which gives a\nnon-zero result acting on $\\Sigma^{(n)}$. Since every action of $\\hat H$\nreduces the RGO of expression, the coefficient of the\nLLog is\n\\begin{eqnarray}\n\\label{2:LL_mass_pi} \\Sigma^{(n)}_\\pi\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)= \\frac{1}{n!}\\log^{n}\\(\\frac{\\mu^2}{m^2}\\)\\hat H_1^{n}\n\\sum_{i}c^{(n)}_iV^{(n)}_i+\\mathcal{O}\\(\\text{NLLog}\\),\n\\end{eqnarray}\nwhere NLLog is the the acronym for the next-to-leading logarithms, and\nhence $\\mathcal{O}(\\text{NLLog})$ denotes the part of expression without\nLLog.\n\nTherefore, constructing the higher chiral order Lagrangians,\nand calculating the one-loop beta-functions of their LECs, one can obtain\nthe LL coefficients without actual calculation of multi-loop diagrams.\nMoreover, the result is independent on the details of the higher order\nLagrangians, as long as they are sufficiently general for the process at\nhand \\cite{Bijnens:2009zi}.\nPractically it is convenient to use a non-minimal Lagrangian generated\n``on-the-fly'' by the counterterms to one-loop diagrams only.\nThis was the approach used\nin \\cite{Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca} for\nthe mesonic theory for several processes.\n\nThe solution of the pole equation gives us the expression for the physical\npion mass to LLog accuracy. It is known up to sixth power of\nlogarithms \\cite{Bijnens:2012hf} and reads\n\\begin{eqnarray}\n\\label{2:mPhys}\nm^2_\\text{phys}=m^2\\(\n  1-\\frac{1}{2}L+\\frac{17}{8}L^2-\\frac{103}{24}L^3+\\frac{24367}{1152}L^4\n  -\\frac{8821}{144}L^5+\\frac{1922964667}{6220800}L^6+\\cdots\\),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{2:L_def}\nL=\\frac{m^2}{(4\\pi F)^2}\\log\\(\\frac{\\mu^2}{m^2}\\).\n\\end{eqnarray}\n\n\n\\section{Heavy baryon Lagrangian}\n\\label{sec:HB_Lagrangian}\n\nThe nucleon-meson ChPT in the naive form has the problem of the large nucleon\nmass $M$. There are several ways of dealing with the presence\nof this large scale, each with advantages and disadvantages. In this article,\nwe use the heavy baryon approach to meson-nucleon ChPT since in this approach\nall scales that explicitly appear are soft and there are no divergences\nnor $\\mu$-dependence associated directly with the scale $M$.\n\nFor the LLog calculation, we have to determine the Lagrangians of zero RGO.\nFor the pion-nucleon system, these are Lagrangians of the first and\nthe second chiral orders. The first chiral order Lagrangian, neglecting terms\nwith external fields, reads\n\\begin{eqnarray}\\label{3:L0}\n\\mathcal{L}^{(0)}_{N\\pi}=\\bar N\\(i v^\\mu D_\\mu+g_A S^\\mu u_\\mu\\)N,\n\\end{eqnarray}\nwhere $S^\\mu$ is a spin vector. We use the standard notation for the field\n combinations (see also the definitions in (\\ref{2:u_mu_def})):\n\\begin{eqnarray}\nD_\\mu&=&\\partial_\\mu+\\Gamma_\\mu, \\qquad u^2=U, \\qquad\n\\Gamma_\\mu= \\frac{1}{2}\\(u^\\dagger \\partial_\\mu u+u \\partial_\\mu u^\\dagger\\).\n\\end{eqnarray}\n\n\nThe second order Lagrangian is sensitive to redefinitions of the nucleon\nfield. The most standard form of the second chiral order heavy\nbaryon Lagrangian reads \\cite{Bernard:1992qa,Bernard:1995dp}\n\\begin{eqnarray}\n\\label{3:BKKM}\n\\mathcal{L}_{\\pi N}^{(1)}&=&\\bar N_v\\Big[\n \\frac{(v\\cdot D)^2-D\\cdot D-ig_A \\{S\\cdot D,v\\cdot u\\}}{2M}\n +c_1\\tr\\(\\chi_+\\)+\\(c_2-\\frac{g_A^2}{8M}\\)(v\\cdot u)^2\n \\nn\\\\&&\n \\hspace*{1cm}\n  +c_3 u\\cdot u+\\(c_4+\\frac{1}{4M}\\)i\\epsilon^{\\mu\\nu\\rho\\sigma}u_\\mu u_\\nu v_\\rho\n          S_\\sigma \\Big]N_v.\n\\end{eqnarray}\nA different but equivalent version of the second order chiral Lagrangian is\ngiven in \\cite{Ecker:1995rk}, and it reads\n\\begin{eqnarray}\\label{3:EM}\n\\mathcal{L}^{(1)}_{N\\pi}&=&\n\\frac{1}{M}\\bar N\\Big[\n -\\frac{1}{2}\\(D_\\mu D^\\mu+i g_A\\{S_\\mu D^\\mu,v_\\nu u^\\nu\\}\\)\n +A_1 \\Tr\\(u_\\mu u^\\mu\\)+A_2 \\Tr\\((v_\\mu u^\\mu)^2\\)\n\\nn\\\\&&\n\\hspace*{1cm}\n  +A_3 \\Tr\\(\\chi_+\\)+A_5i\\epsilon^{\\mu\\nu\\rho\\sigma}v_\\mu S_\\nu u_\\rho u_\\sigma\n  \\Big]N.\n\\end{eqnarray}\nThe relation between the LECs $A_i$ and $c_i$ is the following\n\\begin{eqnarray}\nA_1=\\frac{Mc_3}{2}+\\frac{g_A^2}{16},~~A_2=\\frac{Mc_2}{2}-\\frac{g^2_A}{8},~~A_3=Mc_1,~~A_5=Mc_4+\\frac{1-g_A^2}{4}.\n\\end{eqnarray}\n\nAlthough the S-matrix elements are independent of the parametrization of\nthe nucleon field, the contributions of individual diagrams, and\nexpressions for the beta-functions are dependent on the field parametrization.\nTherefore, the comparison of results for calculations performed\nin different parameterizations is a very strong check of a calculation. The\ncalculations presented in the next sections have been done in both\nparametrizations of the nucleon field. Additionally for further cross-checks,\nwe used different parameterizations of the pion field $u$. We have\nused\n\\begin{eqnarray}\\label{u_param1}\nu=\\exp\\(i\\frac{\\pi^a\\tau^a}{2F}\\),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{u_param2}\nu=\\sqrt{1-\\frac{\\vec\\pi^2}{4F^2}}+i\\frac{\\pi^a\\tau^a}{2F},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{u_param3}\nu=\\sqrt{\\frac{Y}{\\sqrt{2}}}+i\\frac{\\pi^a\\tau^a}{F}\\sqrt{\\frac{1}{2Y}}\n\\qquad\\text{with}\\qquad Y=1+\\sqrt{1-\\frac{\\vec\\pi^2}{F^2}}.\n\\end{eqnarray}\n\n\n\\section{LLog in nucleon-meson ChPT: general comments}\n\\label{sec:general_comm}\n\nThe consideration of the LLog behavior of nucleon-meson systems is similar to\nmeson systems, but with some additional features. The main additional feature\nof meson-nucleon systems is the presence of operators with odd number of\nderivatives. Therefore, the relation between $\\hbar$-order and chiral\norder is $\\hbar^n\\sim \\mathcal{O}(p^{n+1})$ for the single-nucleon sector of\nthe ChPT Lagrangian, and $\\hbar^n\\sim \\mathcal{O}(p^{n+2})$ for the meson\nsector of EFT Lagrangian. At the same time, every loop increases the chiral\norder by at least two. Therefore, the RGO is not in\none-to-one correspondence with $\\hbar$-order. A LEC of $n$'th $\\hbar$-order\n$c^{(n)}$ has generically an RGO $\\[\\frac{n}{2}\\]$. This has\nimportant consequences in the RG and LLog structure of the theory.\n\nThe first consequence is the contribution of diagrams with different\nRGO to the same chiral order. Indeed, a loop diagram with several\nvertices of even chiral order (i.e. odd $\\hbar$ order) has an RGO less\nthen a diagram with the same chiral order but with fewer\neven-chiral-order vertices. An example of diagrams with the same chiral\norder but different RGO is shown in Fig.~\\ref{fig:1}.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{1loopamb.eps}\n\\caption{An example of diagrams which contribute to the renormalization of the nucleon mass at the same chiral order (here, the fifth chiral\norder), but which have different RGO. The left diagram has zero RGO, while the right diagram has RGO equal to one. Therefore, the left diagram\ndoes not contribute to the LLogs. The number in the box indicates the chiral order of the vertex. Thick arrowed lines indicate nucleon\npropagators, thin lines indicate pion propagators.}\n\\label{fig:1}\n\\end{figure}\nUsing the relation between chiral order and RGO, one can see that every\ntwo even-chiral-order vertices reduce the RGO of a diagram by one, from\nthe possible maximum. For example: for any diagram with two even-chiral-order\nvertices with certain RGOs, there exists diagrams of the same chiral order\nand of the same topology, but with these two vertices replaced by\nodd-chiral-order vertices, one with a chiral order one lower and one\nchiral-order one higher. The lower chiral order has the same generic RGO as\nthe even vertex but the higher chiral order vertex has generic RGO one higher.\nThe latter diagram has thus a higher RGO. We conclude that at a given chiral\norder the highest RGO contribution is given by the diagrams with\n\\textit{zero or one} vertex of even chiral order.\n\n\nThe second consequence is the ambiguity of the definition of a leading\nlogarithm. The natural definition of LLogs, the logarithm of a maximum\npower of $\\log\\mu^2$ at a given chiral order, does not always coincides with\nthe RG definition for the same observable. In Sect.~\\ref{sec:nucl_mass} we\nwill show that in the chiral expansion of the physical nucleon mass, the LLog\nterms of odd chiral order are actually of NLLog origin when seen\nfrom an RGO perspective.\n\n\\section{Nucleon mass at LLog accuracy}\n\\label{sec:nucl_mass}\n\n\\subsection{Propagator at LLog accuracy}\n\\label{sec:propagator}\n\nThe physical mass of the nucleon is given by  the position of the pole in the\nDyson propagator. In the heavy baryon approach the inverse Dyson\npropagator reads (we remind the reader that superscripts $n$ refer to the\n$\\hbar$-order of quantities, and that the meson and meson-nucleon\nsectors of the action have different chiral counting)\n\\begin{eqnarray}\n\\label{5:inverse_prop}\nS^{-1}=(rv)+\\sum_{n=1}^\\infty \\Sigma^{(n)}((r\\cdot v),r^2),\n\\end{eqnarray}\nwhere $r^\\mu=p^\\mu-Mv^\\mu$, $p$ is the momentum of the nucleon, and $v_\\mu$\nis the reference four velocity of the heavy nucleon. There are some\nintricacies of defining the propagator and renormalization of the nucleon\nwave function in heavy baryon theory, see e.g.\n\\cite{Ecker:1995rk,Kambor:1998pi}. However for the determination of the\nnucleon mass the straightforward usage of (\\ref{5:inverse_prop}) is sufficient.\n\nThe expressions for $\\Sigma^{(n)}$ are the result of the calculation of\none-particle irreducible diagrams with a nucleon line at the $(n+1)$'th\nchiral order. The maximum power of the logarithm $\\log\\mu^2$ which can appear\nin $\\Sigma^{(n)}$ is $\\[\\frac{n}{2}\\]$.\n\nThe derivation of the LLog coefficient is the same as the derivation of\nexpression (\\ref{2:LL_mass_pi}). We collect all the RG-logarithms by the\naction of the normalization point rescaling operator $\\hat R$, again\nsuppressing the other arguments of $\\Sigma^{(n)}$,\n\\begin{eqnarray}\n\\label{5:Sigma=R Sigma(nucl)}\n\\Sigma^{(n)}\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\hat\nR\\(\\frac{\\mu^2}{\\mu_0^2}\\)\\Sigma^{(n)}\\(\\frac{\\mu_0^2}{m^2},c(\\mu_0^2)\\).\n\\end{eqnarray}\nThe highest RGO in $\\Sigma^{(n)}$ has the tree diagram with only a $c_i^{(n)}$\nvertex, therefore, the LLog coefficient is given by\n\\begin{eqnarray}\\label{5:LL_mass_nucl}\n\\text{LLog coef.}=\n\\(\\[\\frac{n}{2}\\]!\\)^{-1}\\hat H_1^{\\[n\/2\\]} \\sum_{i}c^{(n)}_iV^{(n)}_i,\n\\end{eqnarray}\nwhere $V^{(n)}_i$ are expression for the $c^{(n)}_i$-vertices.\n\nIt is also interesting to look at the NLLog contribution. The NLLog coefficient\ncomes from the $([n\/2]-1)$ term of the exponent series in\n(\\ref{5:Sigma=R Sigma(nucl)}). There are several different parts\nof $\\Sigma^{(n)}$ which survive after the action by $\\hat H^{[n\/2]-1}$.\nThese are the terms with RGO $\\[\\frac{n}{2}\\]$ and $\\[\\frac{n-2}{2}\\]$. While\nthe first are given by tree diagrams, the second are given by\none-loop diagrams:\n\\begin{eqnarray}\n\\Sigma^{(n)}\\(\\frac{\\mu^2}{m^2},c(\\mu^2)\\)=\\sum_i c^{(n)}_i(\\mu^2) V^{(n)}_i\n+\\sum_{\\text{1-loop diag.}}W\\(\\frac{\\mu^2}{m^2},c^{(k)}(\\mu^2)\\)+\\cdots,\n\\end{eqnarray}\nwhere the $V_i$ are the expressions for the tree level diagrams and $W$\nindicates the  expressions for the one-loop diagrams at a fixed\nrenormalization scale. The dots represents the contributions with lower RGO.\nWe note that $W$ contains only one-loop diagrams with RGO equal to\n$\\[\\frac{n-2}{2}\\]$, but not all possible one-loop diagrams.\nThe NLLog coefficient is given by\n\\begin{eqnarray}\n\\label{5:NLL} \\text{NLLog coef.}&=&\n   \\frac{1}{([n\/2]-1)!}\\hat H_1^{[n\/2]-1}\\sum_i c^{(n)}_i V^{(n)}_i\n\\nn\\\\&&\n +\\frac{1}{([n\/2]-1)!}\\[\\sum_{k=0}^{[n\/2]-1}\n     \\hat H_1^{k}\\hat H_2 \\hat H_1^{[n\/2]-1-k}\\]\\sum_i c^{(n)}_i V^{(n)}_i\n\\nn\\\\&&\n  +\\frac{1}{([n\/2]-1)!}\\hat H_1^{[n\/2]-1} \\sum_{\\text{1-loop diag.}}\n      W\\(\\frac{\\mu_0^2}{m^2},c^{(k)}\\),\n\\end{eqnarray}\nwhere $\\hat H_2$ contains two-loop beta-functions in addition to\none-loop beta-functions.\n\nThe expressions given by the different terms on the right-hand-side\nin (\\ref{5:NLL}) have significantly different properties. The first term\nof (\\ref{5:NLL}) gives the contribution to NLLogs with LECs from the\nnext-to-leading chiral Lagrangian. These terms are LLog terms of\n$([n\/2]-1)$-loop diagrams with insertion of higher order vertices.\nThe second line of (\\ref{5:NLL}) gives the ``true'' NLLog contribution with\nLECs of the lowest order Lagrangian only. These are the NLLog terms\nof $[n\/2]$-loop diagrams. The third line represents the non-analytical\ncontribution of $[n\/2]$-loop diagrams to the NLLog coefficient. We should\nmention that the part of NLLog coefficient given by the second line is\nrenormalization-scheme-dependent, while the parts given by the first and the\nthird lines are scheme-independent.\n\nIf the quantity has no tree-order contribution, the only non-zero part\nof (\\ref{5:NLL}) is the last line. In this case the NLLog can be\ncalculated from one-loop diagrams only. An example of such behavior\nare the non-analytic in quark mass terms. These terms result only from\nthe loops and therefore, their contribution to NLLog can\nbe calculated with one-loop diagrams only.\nThe methods of \\cite{Bijnens:2009zi} can also be used to prove this. The\nabsence of the tree level contribution allows the NLLog to be determined from\nthe set of equations relating the different loop-order contributions.\n\n\n\\subsection{Pole equation at LLog accuracy}\n\\label{sec:pole}\n\nIn this section we discuss the properties of the solution of the pole equation\nat LLog accuracy. From the previous discussion it follows that it is\nalso valid for the NLLog multiplied by a nonanalytic power of the quark mass.\n\nThe position of the pole in the propagator (\\ref{5:inverse_prop}) is a\nLorentz invariant quantity, when evaluated to all orders in the expansion.\nTherefore, one can choose\\footnote{If one chooses $v^\\mu=(1,0,0,0)$\nthis corresponds to $\\vec r=\\vec p=0$.}\n$r^\\mu$ or $p^\\mu$ such that $r^2=(r\\cdot v)^2$.\nThen, $\\omega = (r\\cdot v)$ gives the difference between\nthe physical mass and the bare mass, $(r\\cdot v)=\\delta M=M_{\\text{phys}}-M$.\nIn this regime we can expand the expression for $\\Sigma^{(n)}$ in\npowers of $\\delta M$\n\\begin{eqnarray}\n\\label{5:sigma_def}\n\\Sigma^{(n-1)}(\\delta M)=\\sum_{k=0}^n \\sigma^{k,n-k}\\delta M^k m^{n-k},\n\\end{eqnarray}\nwhere the coefficients $\\sigma^{(i,j)}$ contain logarithms. The coefficients\n$\\sigma^{k,n-k}$ have mass-dimension $(1-n)$.\n\nThe solution of the pole equation $S^{-1}=0$, (\\ref{5:inverse_prop}),\ncan be found perturbatively in $m$\n\\begin{eqnarray}\n\\label{5:delta_expansion}\n\\delta M=\\sum_{n=2}^\\infty a_n m^n\\,\n\\end{eqnarray}\nwhere again the $a_i$ contain logarithms. Inserting the expansions (\\ref{5:delta_expansion}) and (\\ref{5:sigma_def}) into the Dyson propagator\n(\\ref{5:inverse_prop}), and considering the pole equation for every power of $m$ independently, we obtain a system of equations for the\ncoefficients $a_n$,\n\\begin{eqnarray}\n\\label{5:eqns}\na_{n}+\\sum_{\\{i\\},j\\leqslant n-2}a_{i_1}a_{i_2}...a_{i_j}\\sigma^{j,n-\\sum i}=0,\n\\end{eqnarray}\nwhere summation runs over all possible sets of indices including empty set and permutations.\n\nLet us consider the system of equations (\\ref{5:eqns}) in the LLog regime.\nWe recall that the power of the LLog is $\\[\\frac{n}{2}\\]$ for $\\Sigma^{(n)}$.\nHowever, the coefficients $a_n$ have different logarithm counting.\nThe reason is the presence of terms non-analytical in quark masses. The\nLagrangian of ChPT is necessarily analytical in the quark masses, i.e. it\ncontains only even powers of $m$. The terms non-analytic in quark masses\nappear only through loop-integrals, and, therefore, they can not appear in\nthe expression (\\ref{5:LL_mass_nucl}) or in the first two lines of the\nexpression (\\ref{5:NLL}). In this way, the number of logarithms in front of\nthe pion mass in the odd power is suppressed by one (at least). Summarizing,\nwe obtain the following LLog counting for the coefficients $a$ and $\\sigma$\n\\begin{eqnarray}\\label{5:LL_counting}\n a_n \\sim \\log^{\\[(n-2)\/2\\]}(\\mu),~~~~\\sigma^{s,t}\\sim\n \\left\\{\\begin{array}{cc} \\log^{[(s+t-1)\/2]}(\\mu) & t\\in\\text{even}\\\\\n                 \\log^{[(s+t-3)\/2]}(\\mu) & t\\in\\text{odd}\n        \\end{array}\n \\right. .\n\\end{eqnarray}\n\nUsing the counting (\\ref{5:LL_counting}), we neglect the NLLog terms in the\nequations (\\ref{5:eqns}) and obtain the system of equations in the\nLLog regime:\n\\begin{eqnarray}\\label{5:a_even}\n&&a_n+\\sigma^{0,n}+\\sum_{k=2,4,..}^{n-2}a_k \\sigma^{1,n-k}=0,\n\\qquad n\\in \\text{even},\n\\\\\n\\label{5:a_odd}\n&& a_n+\\sigma^{0,n}+a_{n-1}\\sigma^{1,1}+\\sum_{k=3,5,..}^{n-2}a_k \\sigma^{1,n-k}=0,\n\\quad n\\in\\text{odd}.\n\\end{eqnarray}\nThis is a system of linear equations. The important result is that the\neven-$n$ coefficients allow LLog evaluation only, because they\ninvolve only the LLog coefficient of analytical in quark mass terms. At the\nsame time, the odd coefficients involve the terms non-analytical in\nquark masses. These coefficients are really NLLog.. However, they can\nbe obtained from a one-loop calculation as well, because they follow from\nthe third line of (\\ref{5:NLL}).\n\nOne can see that the system (\\ref{5:a_even}-\\ref{5:a_odd}) involves only the\ncoefficients $\\sigma^{0,n}$ and $\\sigma^{1,n}$, which are the\ncoefficients of the zeroth and the first powers of $(r\\cdot v)$ in the\npropagator diagrams. It is a reflection of the fact that according to\n(\\ref{5:LL_counting}) the quantity $(r\\cdot v)^2=\\delta M^2$ is of NLLog\norder. Therefore, the powers of $\\omega$ can be eliminated from the\nequation $S^{-1}=0$. The solution can be presented in the simple form\n\\begin{eqnarray}\\label{5:mass_sol}\n\\delta M= \\frac{-\\Sigma(0)}{1+\\Sigma'(0)}+\\mathcal{O}(\\text{NLLog}),\n\\end{eqnarray}\nwhere $\\Sigma(r\\cdot v)=\\sum_{n}\\Sigma^{(n)}(r\\cdot v)$ and\n$\\Sigma'$ is its derivative with respect to $(r\\cdot v)$.\n\n\\subsection{Expression for the physical mass}\n\\label{sec:resultmass}\n\nWe have performed the calculation of the nucleon mass up to the fourth power\nof RG logarithms. We present the results in the form:\n\\begin{eqnarray}\n\\label{mainresult}\nM_{\\text{phys}}&=&M+k_2 \\frac{m^2}{M}+k_3 \\frac{\\pi m^3}{(4\\pi F)^2}\n  +k_4 \\frac{m^4}{(4\\pi F)^2 M}\\log\\(\\frac{\\mu^2}{m^2}\\)\n  +k_5 \\frac{\\pi m^5}{(4\\pi F)^4}\\log\\(\\frac{\\mu^2}{m^2}\\)+\\cdots\n\\nn\\\\\n&=&\nM+\\frac{m^2}{M}\\sum_{n=1}^\\infty k_{2n} L^{n-1}\n+\\pi m\\frac{m^2}{(4\\pi F)^2}\\sum_{n=1}^\\infty k_{2n+1} L^{n-1},\n\\end{eqnarray}\nwhere $L$ is defined in (\\ref{2:L_def}). The coefficients up to $k_{11}$ are\npresented in Tab.~\\ref{table_mass}. This corresponds to the\nfour-loop calculation of LLog and five-loop calculation for the terms\nnon-analytical in quark masses.\n\nThe presented results have been obtained via the different parametrizations of the Lagrangians (see sec.\\ref{sec:HB_Lagrangian}), which gives a\nvery strong check of calculation. Additionally, the coefficients up to $k_6$ agree with known results. The one-loop coefficients $k_{3,4}$ are\nwell known, see e.g. \\cite{Bernard:1995dp}. The two-loop coefficient $k_{5}$ was first derived in \\cite{McGovern:1998tm}. The two-loop\ncoefficients $k_6$ and $k_5$ are known from the full two-loop calculation for the nucleon mass performed in the EOMS scheme\n\\cite{Schindler:2007dr}.\n\n\\begin{table}[tbh!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex} $k_2$ & $-4c_1 M$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_3$ & $-\\frac{3}{2}g_A^2$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_4$ & $\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-3c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.7ex}$k_5$ & $\\frac{3g_A^2}{8}\\(3-16 g_A^2\\)$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_6$ & $-\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{3}{2}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_7$ & $g_A^2\\(-18 g_A^4+\\frac{35 g_A^2}{4}-\\frac{443}{64}\\)$\n\\\\[2mm]\\hline\n\\rule{0ex}{2.5ex}$k_8$ & $\\frac{27}{8}\\(g_A^2+(c_2+4c_3-4 c_1)M\\)-\\frac{9}{2}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_9$ & $\\frac{g_A^2}{3}\\(-116 g_A^6+\\frac{2537 g_A^4}{20}-\\frac{3569 g_A^2}{24}+\\frac{55609}{1280}\\)$\n\\\\[2mm] \\hline\n\\rule{0ex}{2.5ex}$k_{10}$ & $-\\frac{257}{32}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{257}{32}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$k_{11}$ & $\\frac{g_A^2}{2}\\(-95 g_A^8+\\frac{5187407 g_A^6}{20160}-\\frac{449039\n   g_A^4}{945}+\\frac{16733923 g_A^2}{60480}-\\frac{298785521}{1935360}\\)$\n\\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $k_i$ defined in (\\ref{mainresult}) of the LLog expansion of the nucleon mass.} \\label{table_mass}\n\\end{table}\n\nThe generation of the higher order Lagrangians and the evaluation of one-loop\nbeta-functions has been done automatically using the computer algebra system\nFORM \\cite{FORM}. The algorithm we used is similar to that used and described\nin \\cite{Bijnens:2009zi,Bijnens:2010xg}. The main integral needed for\nevaluation of beta-functions is presented in the appendix. Although the\ncalculation involves only one-loop diagrams, it is very demanding in\nmachine time and memory. The most demanding factor is the length of the \nexpression for the high order effective vertices and\nthe number of diagrams to compute.\nThese quantities grow rapidly with chiral order. For example, in order\nto calculate the $k_{10}$ coefficient one needs to evaluate nearly $10^4$\none-loop diagrams.\n\nThe calculation of the even coefficients $k$ can be significantly simplified by using the conjectures discussed below in\nSect.~\\ref{sec:conjectures}. So, by neglecting higher powers of $g_A$ during the evaluation of the diagrams, we could also evaluate the\nfive-loop coefficient $k_{12}$. Adding the further conjecture about the relation with the LLog in the pion mass, we can obtain the six and\nseven-loop coefficients $k_{14}$ and $k_{16}$. However, these coefficients are the result of conjectures and, therefore, are presented in\nTab.~\\ref{table_mass_conjecture}, separately from the results of the full calculation.\n\n\\subsection{Properties of the result and conjectures}\n\\label{sec:conjectures}\n\nThe straightforward calculation, limited by the available computer power,\ngives us\nthe coefficients $k_1,\\ldots,k_{11}$ presented in Tab.~\\ref{table_mass}.\nConsidering the presented coefficients a number of regularities show up\nimmediately. Some of the regularities we can explain easily, while some of\nthem we cannot.\n\nThe first observation is that only even powers of $g_A$ show up.\nThis can be easily understood. The meson Lagrangian and the nucleon Lagrangian\nare invariant under the transformation $u\\leftrightarrow u^\\dagger$\nand $g_A\\leftrightarrow -g_A$. As a consequence only terms even in $g_A$ can\nappear in the nucleon mass. This explains the pattern occurring in the\nodd coefficients $k_{2n+1}$.\n\nThe second observation is that the coefficients $k_{2n}$ contains a very\nparticular combination of LECs. The pattern appearing in $k_{2n}$ is not\nwell understood yet. Let us consider it in detail.\n\nAs shown in Sec.~\\ref{sec:general_comm}, one can have at most one\ninsertion of the order $p^2$ Lagrangian. The expression for $k_{2n}$ are\nthus at most linear in $c_1,c_2,c_3$ and the other terms\nin $\\mathcal{L}^{(1)}_{\\pi N}$. The coupling constant $c_4$ or $A_5$ cannot\nenter the nucleon mass at LLog since it produces\nan $\\epsilon_{\\mu\\nu\\alpha\\beta}$. However, we found no simple argument why\n$g_A$ only appears up to order $g_A^2$.\n\nFor the powers of $g_A$,\nthere are two sources for factors of $g_A$ in the loop diagrams, namely\nfrom the vertices $\\mathcal{L}^{(0)}$ (\\ref{3:L0}) and from the vertices\n$\\mathcal{L}^{(1)}$ (\\ref{3:BKKM}-\\ref{3:EM}). While the number of vertices\nfrom $\\mathcal{L}^{(1)}$ is restricted to one, the number of\nvertices from $\\mathcal{L}^{(0)}$ is naturally unrestricted. Moreover, the\nexpression for $\\Sigma$ contains all allowed powers of $g_A$. These\npowers cancels within the solution (\\ref{5:mass_sol}). We have checked that if\none introduces new LECs for the terms proportional to\n$g_A$ in $\\mathcal{L}^{(1)}$ (say coefficients $B_{1,2}$ in front of the first\ntwo terms in (\\ref{3:EM})) the coefficients $k_{2n}$ would\ncontain higher powers of $g_A$. These induced higher powers are proportional\nto $(B_1-B_2)$ and disappear when $B_1=B_2$. Since these operators\nappear in $\\mathcal{L}^{(1)}$ as the compensation of the non-relativistic\nnucleon reference frame, we conclude that absence of higher powers\nof $g_A$ in coefficients $k_{2n}$ is a consequence of Lorentz invariance.\n\nSupposing that the cancelation of the higher powers of $g_A$ takes place at\nall orders, one can neglect these powers during the computation of\ndiagrams. This procedure significantly reduces the demands for computer time\nand allows us to calculate the coefficient $k_{12}$, which is\npresented in Tab.~\\ref{table_mass_conjecture}.\n\n\nConsidering the equation (\\ref{5:a_even}) one can see that the coefficients\n$k_{2n}$ consist from the terms proportional to exactly the first\npower of $\\sigma^{0,k}$ where $k$ is even. We remind that $\\sigma^{0,\\text{even}}$ are the result of diagrams with even chiral order and hence\nproportional to a single vertex from $\\mathcal{L}^{(1)}$ (which is also checked by explicit calculation with coefficients $B_{1,2}$). Therefore,\nthe term $g_A^2$ which appears in the coefficients $k_{2n}$ resulted solely from $\\mathcal{L}^{(1)}$. In its own turn, it implies that there is\nno contribution from the diagrams with vertices proportional to $g_A$ only from $\\mathcal{L}^{(0)}$. All such vertices have an odd number of\npions. Absence of such vertices implies that \\textit{diagrams with more than\n two odd-number-of-pion vertices do not contribute to the LLog\ncoefficient of nucleon mass}. Undoubtedly such a structure is a consequence\nof the additional subtractions of infrared (heavy\nmass) singularities into renormalization counterterms within heavy baryon\ntheory, but we have not been able to prove this.\n\nConsidering the first six coefficients $k_{2n}$ one can observe that they have\nthe pattern\n\\begin{eqnarray}\n\\label{5:pattern}\nk_{2n}=b_n\\(\\frac{-3c_1M}{n-1}+\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)\\),\n\\end{eqnarray}\nwhere $b_n$ are some rational numbers. The coefficients $b_n$ can be obtained\nfrom the calculation of the physical pion mass as we demonstrate\nbelow.\n\nThe nucleon mass LLog coefficient in terms of the physical pion mass $m_{\\text{phys}}$ has the form\n\\begin{eqnarray}\n\\label{mainresult2} M_{\\text{phys}}&=&\nM+\\frac{m_\\text{phys}^2}{M} \\sum_{n=1}^\\infty r_{2n} L_\\pi^{n-1}\n +\\pi m_\\text{phys}\\frac{m_\\text{phys}^2}{(4\\pi F)^2}\n     \\sum_{n=1}^\\infty r_{2n+1} L_\\pi^{n-1},\n\\end{eqnarray}\nwhere\n$$\nL_\\pi=\\frac{m^2_\\text{phys}}{(4\\pi F)^2}\\log\\(\\frac{\\mu^2}{m_\\text{phys}^2}\\).\n$$\nThe coefficients $r_n$ of this expansion are presented in\nTab.~\\ref{table_mass_phys}.\n\\begin{table}[tb!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex}$r_2$ & $-4c_1 M$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_3$ & $-\\frac{3}{2}g_A^2$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_4$ & $\\frac{3}{4}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-5c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_5$ & $-6 g_A^4 $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_6$ & $5c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.7ex}$r_7$ & $\\frac{g_A^2}{4}\\(-8+5g_A^2-72 g_A^4 \\)$\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_8$ & $\\frac{25}{3}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$r_9$ & $\\frac{g_A^2}{3}\\(-116 g_A^6+\\frac{647 g_A^4}{20}-\\frac{457 g_A^2}{12}+\\frac{17}{40}\\)$\n\\\\[2mm]\\hline\n\\rule{0ex}{2.5ex}$r_{10}$ & $\\frac{725}{36}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.9ex}$r_{11}$ & $\\frac{g_A^2}{2}\\(-95 g_A^8+\\frac{1679567 g_A^6}{20160}-\\frac{451799 g_A^4}{3780}+\\frac{320557\n   g_A^2}{15120}-\\frac{896467}{60480}\\)$\n\\\\[2mm]\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $r$ of LLog expansion of the nucleon mass using the physical pion mass as defined in (\\ref{mainresult2}).}\n\\label{table_mass_phys}\n\\end{table}\n\nOne can see that the non-analytical in quark mass terms $r_\\text{odd}$ do not\nsimplify in this form of expansion, while the expressions for the\ncoefficient $r_\\text{even}$ are significantly simplified. Moreover the\ncombination of the LECs proportional to $b_n$ in (\\ref{5:pattern})\ncompletely disappears from the higher order terms. We conclude that the\ncoefficients $b_n$ are the coefficients of the LLog expansion of $m^4$\nin the terms of physical pion mass. Thus, assuming that the\n pattern~(\\ref{5:pattern}) holds for all orders we conjecture the LLog part of\nthe expression for the nucleon bare mass via the physical\nmasses\\footnote{This expression should be understood as not rewriting the term\n $k_2m^2\/M$ in the physical pion mass and the integral\nover $\\mu^2$ should be done after applying (\\ref{2:mPhys})\nto $m^4_\\text{phys}(\\mu')$.} \\textit{at all orders} to be\n\\begin{eqnarray}\\label{6:M(m)}\nM=\nM_\\text{phys} +\\frac{3}{4}m_\\text{phys}^4 \\frac{\\log\\(\\frac{\\mu^2}{m_\\text{phys}^2}\\)}{(4\\pi F)^2} \\(\\frac{g_A^2}{M_\\text{phys}}-4c_1+c_2+4c_3\\)\n\\qquad\\qquad\n\\nn\\\\\n -\\frac{3c_1}{{(4\\pi F)^2}}\\int\\limits_{m^2_\\text{phys}}^{\\mu^2}m_\\text{phys}^4(\\mu')~\\frac{d\\mu'^2}{\\mu'^2}.\n\\end{eqnarray}\nThe expression for the physical pion mass is known up to 6-loop order, Eq.~(\\ref{2:mPhys}), therefore, we can guess two more LLog coefficients for\nthe physical nucleon mass. These are presented in the table \\ref{table_mass_conjecture} and indicated by the double-star marks.\n\n\\begin{table}[tb!]\n\\begin{center}\n\\begin{tabular}{|c||l|}\n\\hline\n\\rule{0ex}{2.5ex}$k_{12}$(*) & $\\frac{115}{3}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-\\frac{92}{3}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_{14}$(**) & $-\\frac{186515}{1536}\\(g_A^2+(c_2+4c_3-4c_1)M\\)+\\frac{186515}{2304}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$k_{16}$(**) & $\\frac{153149887}{259200}\\(g_A^2+(c_2+4c_3-4c_1)M\\)-\\frac{153149887}{453600}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_{12}$(*) & $\\frac{175}{4}c_1M $\n\\\\[1mm]\\hline\n\\rule{0ex}{2.5ex}$r_{14}$(**) & $\\frac{4153903}{24300}c_1M $\n\\\\[1mm]\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The coefficients $k_i$ and $r_i$ defined in (\\ref{mainresult}) and\n(\\ref{mainresult2}), that are obtained by using the conjectures described in Sect.~\\ref{sec:conjectures}. (*): The coefficients $k_{12}$ and $r_{12}$ have\nbeen calculated within the simplified scheme by neglecting higher powers in $g_A$. (**):\nThe coefficients $k_{14,16}$ and $r_{14}$ are the result suggested by\nthe expression (\\ref{6:M(m)}). $r_{16}$ would require the knowledge of the $L^7$ term in the expression for the pion mass.}\n\\label{table_mass_conjecture}\n\\end{table}\n\n\\subsection{Numerical results}\n\\label{sec:numerics}\n\nAs mentioned in the introduction, the LLog are not necessarily dominant.\nThey do however give an indication of the size of corrections to be expected.\nWe use here one set of inputs to show an example. The input we use uses\nthe $c_i$ as determined in \\cite{Bernard:1995dp} and reasonable values for the\nother quantities. The actual values we use are:\n\\begin{eqnarray}\nM = 938~\\text{MeV}, & c_1 = -0.87~\\text{GeV}^{-1}, & c_2 = 3.34~\\text{GeV}^{-1}, \\quad \\mu=0.77~\\text{GeV},\n\\nn\\\\\nF = 92.4~\\text{MeV}, & c_3 = -5.25~\\text{GeV}^{-1},& g_A= 1.25\\,.\n\\end{eqnarray}\nWe plot in Fig.~\\ref{figloop} the total correction $M_\\text{phys}-M$\nof (\\ref{mainresult}) by loop order. We have included the results up to the\n$k_{12}$ term since we do not have the odd powers higher than five loops.\nAs can be seen there is a reasonable convergence for the range given.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{masslooporder.eps}\n\\caption{The contribution of the terms in mass correction of (\\ref{mainresult}) with the terms included up to a given loop-order.}\n\\label{figloop}\n\\end{figure}\n\nTo see the convergence better, we have plotted in Fig.\\ref{figparts} the\nabsolute value of the individual terms containing $k_i$ of (\\ref{mainresult})\nfor $m=138$~MeV.  Note the excellent convergence.\n\\begin{figure}[tb!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{individualterms.eps}\n\\caption{The absolute value of the contribution of the individual terms ($\\sim k_n$) in (\\ref{mainresult}) at $m=138$~MeV. Open symbols are the odd orders. Filled symbols are the even orders.}\n\\label{figparts}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper we have presented the application of the renormalization group\nmethod for nucleon-pion chiral perturbation theory. The theoretical\nbasis of the method was developed in \\cite{Buchler:2003vw}. The method has\nbeen applied before only for bosonic theories, see\n\\cite{Kivel:2008mf,Polyakov:2010pt,Bijnens:2009zi,Bijnens:2010xg,Bijnens:2012hf,Bijnens:2013yca}. In particular, we have calculated the physical\nmass of the nucleon within the heavy baryon formulation in the LLog\napproximation (analytical and non-analytical in quark mass terms) up to\nfive-loop order. The results of the calculation are presented in\n(\\ref{mainresult}, \\ref{mainresult2}) and tables\n\\ref{table_mass}, \\ref{table_mass_phys}.\n\nThe theories with fermions (or more precisely, the theories involving\nLagrangians of odd chiral order) have a more involved\nstructure of RG equations. In contrast to the bosonic theories, where all\none-loop beta functions contribute to LLog coefficients, in theories\nwith fermions some one-loop beta functions do not contribute to the LLog\ncoefficients. For resolving the RG hierarchy, we have\nintroduced the concept of renormalization group order (RGO), see\nSec.~\\ref{sec:defRGO}. Using the RGO allows us to extract the diagrams which\ncontribute to the LLog approximation (or any other order of RG logarithms).\n\nThe calculation of the necessary one-loop beta-functions has been performed\nsymbolically with the help of the FORM computer symbolic computation system.\nThe results of our calculation agree with known one- and two-loop results,\nsee e.g. \\cite{Schindler:2007dr}. The calculations were performed in several\ndifferent parametrizations of the nucleon and pion fields with the same result,\nproviding a very strong check for the computational algorithm. The analytical\npart of the computation, namely the expression for a basis one-loop-integral\nin the heavy baryon theory, is presented in the appendix.\n\nThe obtained LLog coefficients show a number of regularities. Some of which\ncan be easily understood, while the rest is more involved. The most\nintriguing regularity is the absence of higher powers of axial coupling\nconstant $g_A$ in the LLog coefficients $k_{2n}$ (see (\\ref{mainresult})\nand table \\ref{table_mass}). Moreover, the pattern of the LLog coefficients\nallows us to guess the \\textit{all-order} expression for the LLog\ncontribution to the nucleon mass in terms of \\textit{physical} pion mass \n(\\ref{6:M(m)}). Although the latter is only a conjecture, we consider it\nas an exact result, most likely a consequence of the subtraction of\nheavy-mass-singularities within heavy baryon theory and Lorentz invariance.\n\nWe also showed some numerical results in Sect.~\\ref{sec:numerics}.\n\n\\section*{Acknowledgements}\n\nA.V. thanks J.~Relefors for stimulating discussions and technical help.\nThis work is supported in part by the European Community-Research\nInfrastructure Integrating Activity Study of Strongly Interacting Matter\"\n(HadronPhysics3, Grant Agreement No. 28 3286) and the Swedish Research\nCouncil grants 621-2011-5080 and 621-2013-4287.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n Stationarity is crucial in analyzing random sequences because statistical inference usually requires a probabilistic mechanism constant in, at least, a segment of observations.  Therefore, it is important to detect whether changes occur in a sequence of observations prior to statistical inference.  Such change-point problems arise from many applications: abrupt events in video surveillance \\citep{mayer2015change};   deterioration of product quality in quality control \\citep{lai1995sequential}; credit card fraud in finance \\citep{bolton2002statistical}; alteration of genetic regions in cancer research \\citep{erdman2008fast} and so on.\n\nThere is a large and rapidly growing literature on change-point detection \\citep{aminikhanghahi2016survey,niu2016multiple,sharma2016trend,fryzlewicz2014wild}. Many methods rely on the assumed parametric models to detect special change types  such as location, scale or presumed distribution family. \\citet{page1954continuous} introduced a method by examining the ratio of log-likelihood functions. \\citet{lavielle2006detection} detected change-points by maximizing a log-likelihood function. \\citet{yau2016inference} proposed a likelihood ratio scan method for piecewise stationary autoregressive time series. Some Bayesian change-point detection methods assume that the observations are normally distributed, and calculate the probability of change-point at each point \\citep{barry1993bayesian,Zhang1995changepoint,wang2015bayesian,maheu2018efficient} to name a few. Since parametric methods potentially suffer from model misspecification, other methods are developed to detect general distributional changes with more relaxed assumptions.\n\\citet{kawahara2012sequential} provided an algorithm which relied heavily on estimating the ratio of probability densities. \\citet{lung2015homogeneity} identified change-points via the well-known Wilcoxon rank statistic. \\citet{matteson2014nonparametric} proposed a nonparametric method using the concept of energy distance for independent observations. There are also some binary segmentation methods statistics  \\citep{fryzlewicz2014wild,cho2015multiple,eichinger2018mosum}. Two advantages of the binary segmentation procedures are their simplicity and computational efficiency, but their false discovery rates may be hard to control because they are `greedy' procedures. \\citet{zou2014nonparametric} introduced a nonparametric empirical likelihood approach to detecting multiple change-points in independent sequences, and estimated the locations of the change-points by using the dynamic programming algorithm and the intrinsic order structure of the likelihood function.\n\nAutomatically detecting the number of change-points is also important.\nSome methods are developed to detect only a single change-point \\citep{ryabko2008hypotheses}, while some methods require a known number of change-points but unknown locations \\citep{hawkins2001fitting,lung2015homogeneity}. In real data analysis, however, we usually do not know the number of change-points.\n\nWith increasing richness of data types,  non-Euclidean data, such as shape data, functional data, and spatial data, commonly arise from applications. For example, one of the problems of interest to us is the changes in the monsoon direction as defined by circle, a simple Riemannian manifold. Methods developed in Hilbert spaces are not effective for this type of problems as our analysis of the data from Yunnan-Guizhou Plateau ($105^\\circ$E, $27^\\circ$N) collected from 2015\/06\/01 to 2015\/10\/30 illustrates below.  To the best of our knowledge, few methods exist to detect change-points in a non-Euclidean sequence.  \\citet{chen2015graph} and \\citet{chu2019asymptotic} proposed a series of graph-based nonparametric approaches that could be applied to non-Euclidean data with arbitrary dimension. However, their proposed  methods apply to \\emph{iid} observations only and are restricted to one or two change-points. Therefore, it remains to be an open and challenging problem to develop methods to detect arbitrarily distributional changes for non-Euclidean sequences, including the change-point locations and the number of the change-points.\n\nTo address this challenge,  we introduce a novel concept of Ball detection function via Ball divergence \\citep{pan2018ball}. Ball divergence  is a recently developed measure of divergence between two probabilities in separable Banach spaces. The Ball divergence is zero if and only if the two probability measures are identical. Since its sample statistic is constructed by metric ranks, the test procedure for an identical distribution is robust to heavy-tailed data or outliers, consistent against alternative hypothesis, and applicable to imbalanced data. Therefore, the empirical ball divergence is an ideal statistic to test whether or not a change has occurred. Unfortunately, it does not inform us where the change occurs, because in theory the probability measures before and after any time point are always different if there exists a change point in the sequence. Therefore it is imperative for us to observe how the probability measures before and after any time vary with time and then develop a proper criterion to detect the change-point location. We introduce a Ball detection function as an effective choice which reaches its maximum at the change point if a sequence has only one change point. We further develop a hierarchical algorithm to detect multiple change-points using the statistic based on the Ball detection function. The advantages of our procedure are threefold: our procedure can  estimate the number of change-points and detect their locations;  our procedure can detect any types of change-points; and both uniquely and importantly, our procedure can handle complex stochastic sequences, for example, non-Euclidean sequences.\n\nThe rest of this article is organized as follows. In Section 2, we review the notion of Ball divergence,  and then introduce a novel change-point detection function, i.e., a Ball detection function based on Ball divergence with a scale parameter for weakly dependent sequences. We further establish its asymptotic properties. We show how to use the Ball detection function to detect change-points and  establish the consistent properties of our method in Section 3.  In Section 4, we compare the performance of our method with some existing methods in various simulation settings. In section 5, two real data analyses demonstrate the utility  of our proposed method. We make some concluding remarks in Section 6. All technical details are deferred to Appendix.\n\n\\section{Change-point Detection in Dependent Sequences}\n\n\\subsection{Review of Ball Divergence}\n\n\nBall divergence (BD, \\citet{pan2018ball})  is a measure of the difference between two probabilities in a separable Banach space $(\\mathrm{A},||\\cdot||)$, with the norm $||\\cdot||$. $\\forall~u,v \\in \\mathrm{A}$, the distance  between $u$ and $v$ deduced from the norm is $\\rho(u,v)=|| u-v ||$. Denote by $\\bar{B}(u,r)=\\{x|\\rho(x,u)\\leq r\\}$  a closed ball. Let $\\mathcal{B}$ be the smallest $\\sigma$-algebra in $\\mathrm{A}$ that contains all closed (or open) subsets of $\\mathrm{A}$. Let $\\mu$ and $\\nu$ be two probabilities on $\\mathcal{B}$. Ball divergence \\citep{pan2018ball} is defined as follows.\n\n\\begin{definition}\\label{df_bdo}\n\tThe Ball divergence of  two Borel probabilities $\\mu$ and $\\nu$ in $\\mathrm{A}$ is defined as an integral of the square of the measure difference between $\\mu$ and $\\nu$ over  arbitrary closed balls,\n\t\\begin{equation*\n\tD(\\mu,\\nu)=\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\mu-\\nu]^2 (\\bar{B}(u,\\rho(u,v)))(\\mu(du) \\mu(dv)+ \\nu(du)\\nu(dv)).\n\t\\end{equation*}\n\\end{definition}\n\nLet $S_\\mu$ and $S_\\nu$ be the support sets of $\\mu$ and $\\nu$ respectively. The BD has the following important property \\citep{pan2018ball}:\n\n\\begin{theorem}\\label{quantity} Given two Borel probabilities $\\mu$ and $\\nu$ in a  finite dimensional Banach space $\\mathrm{A}$, then $D(\\mu,\\nu)\\geq 0$  where the equality holds if and only if $\\mu=\\nu$.  It can be extended to separable Banach spaces if $S_\\mu=\\mathrm{A}$ or $S_\\nu=\\mathrm{A}$.\n\\end{theorem}\n\n\\subsection{Ball Divergence with a Scale Parameter}\n\nThe Ball divergence introduced above cannot detect the locations of change-points accurately enough while comparing the distributions of the sequences before and after the change-points. We need to introduce a  Ball divergence associated with a scale parameter $\\alpha$ as follows.\n\n\\begin{definition}\\label{df_bd} A Ball divergence of two Borel measures $\\mu$ and $\\nu$ in $\\mathrm{A}$ is defined as\n\t\\begin{align}\\label{formula_alphaBD}\n\tD_{\\alpha}(\\mu,\\nu)\n\t=\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\mu-\\nu]^{2}(\\bar{B}(u,\\rho(u,v))\\omega_{\\alpha} (du)\\omega_{\\alpha} (dv),\n\t\\end{align}\n\twhere $\\omega_{\\alpha}=\\alpha \\mu+(1-\\alpha) \\nu$ is the mixture distribution measure with the scale parameter $\\alpha\\in[0,1]$.\n\\end{definition}\n\n$D_{\\alpha}(\\mu,\\nu)$ also has the equivalence property below, which is critical to the comparison of the distributions of any two sequences.\n\\begin{theorem}\\label{BDifandonlyif}\n\tGiven two Borel probabilities $\\mu,\\nu$ in a finite dimensional Banach space $\\mathrm{A}$, then $D_{\\alpha}(\\mu,\\nu)\\geq 0$ where the equality holds if and only if $\\mu=\\nu$. It also holds on separable Banach spaces if $S_\\mu=\\mathrm{A}$ or $S_\\nu=\\mathrm{A}$.\n\\end{theorem}\n\nTheorem \\ref{BDifandonlyif} assures that for any $\\alpha\\in [0,1],$ $D_{\\alpha}(\\mu,\\nu)$ possesses the most important property as $D(\\mu,\\nu)$ in terms of\ntesting the distributional difference between two sequences.  Importantly, with the introduction of $\\alpha$, we can consistently estimate the locations of the change-points. Here, we highlight the relationship and difference between $D_{\\alpha}(\\mu,\\nu)$ and $D(\\mu,\\nu).$\n\nWhen $\\alpha=1$, $D_{1}(\\mu,\\nu)$ is the measure difference over the balls whose centers and the endpoints of the radius following measure $\\mu.$ When $\\alpha=0$, $D_{0}(\\mu,\\nu)$ is the measure difference over the balls whose centers and the endpoints of the radius following the measure $\\nu$. Moreover,\n\\begin{equation*}\nD(\\mu,\\nu)=D_{0}(\\mu,\\nu)+D_{1}(\\mu,\\nu).\n\\end{equation*}\nFor $\\alpha\\in(0,1)$, $D_{\\alpha}(\\mu,\\nu)$ is the mean of the measure differences from two samples over the balls whose centers and endpoints of the radius following four possible pairs of measures:$(\\mu,\\mu)$, $(\\mu,\\nu)$,$(\\nu,\\mu)$, and $(\\nu,\\nu)$ where the ratio of two measures is $\\alpha:1-\\alpha$.\n\n\nBall divergence with a  scale parameter can be defined in the general metric space, following the Generalized Banach-Mazur theorem \\citep{kleiber1969a} as stated in the Supplementary material.\n\n\\subsection{Ball Detection Function}\nNow, we introduce a Ball detection function which is maximized at the change point if there exists one, and hence can be used to\ndetermine the location of the change point. For clarity, let us consider a conceptual sequence with a change point $\\alpha \\in (0,1)$, and the probability measures before and after $\\alpha$ are $\\mu$ and $\\nu$, respectively.  Denote the indicator function by $I(\\cdot)$. For  a \"time\" $\\beta \\in (0,1)$, define \n\\begin{equation*}\n  h_{\\alpha}(\\beta)=\\frac{\\alpha}{\\beta}I(\\beta\\geq\\alpha)+\\frac{1-\\alpha}{1-\\beta}I(\\beta<\\alpha).\n\\end{equation*}\nWithout loss of generality, suppose that $\\beta>\\alpha$, the probability measures before and after $\\beta$ are $\\frac{\\alpha}{\\beta}\\mu+(1-\\frac{\\alpha}{\\beta})\\nu$ and $\\nu$. By the definition of Ball divergence (\\ref{formula_alphaBD}), we have\n\\begin{equation*}\nD_{\\beta}(\\frac{\\alpha}{\\beta}\\mu+(1-\\frac{\\alpha}{\\beta})\\nu,\\nu)=(\\frac{\\alpha}{\\beta})^2D_{\\alpha}(\\mu,\\nu).\n\\end{equation*}\nTherefore, in general,\n\\begin{equation}\\label{relation alpha-beta}\nD_{\\beta}(h_{\\alpha}(\\beta)\\mu+(1-h_{\\alpha}(\\beta)\n)\\nu,\\nu)=h^2_{\\alpha}(\\beta)D_{\\alpha}(\\mu,\\nu).\n\\end{equation}\nThe maximum of $h_{\\alpha}(\\beta)$ is attained when $\\beta=\\alpha$ if there exists a change-point $\\alpha.$ In this case, we can find the change point by maximizing the ball divergence in equation (\\ref{relation alpha-beta}). But we still need to test whether a change point has occurred or not. Next, we introduce a Ball detection function to simultaneously test the existence of a change-point and determine its location:\n\\begin{eqnarray*}\\label{relation Ball-detection}\nV(\\beta;\\mu,\\nu) &=&\\beta(1-\\beta)D_{\\beta}(h_{\\alpha}(\\beta)\\mu+(1-h_{\\alpha}(\\beta)\n)\\nu,\\nu)\\\\&=&\\beta(1-\\beta)h^2_{\\alpha}(\\beta)D_{\\alpha}(\\mu,\\nu).\n\\end{eqnarray*}\nNote that the maximum of $\\beta(1-\\beta)h_{\\alpha}(\\beta)$ is also attained when $\\beta=\\alpha$, allowing us to find the change point by maximizing $V(\\beta;\\mu,\\nu)$. In next subsection, we shall discuss how this function is used to construct a test for a change-point test statistic.\n\n\n\\subsection{Ball Detection Function in Sample}\nSuppose that a sequence of observations $\\{Z_i\\}_{1 \\leq i \\leq T}$ is comprised of  two multivariate stationary sequences $\\{Z_i\\}_{1 \\leq i \\leq M}$ with the probability measure $\\mu_1$ and $\\{Z_i\\}_{M+1\\leq i \\leq T}$ with $\\mu_2$, where  both  $\\mu_1$ and $\\mu_2$ are unknown. We estimate $D_{\\alpha}(\\mu_1,\\mu_2)$ with $\\alpha=M\/T$ based on $\\{Z_i\\}_{1 \\leq i \\leq T}$.  Let $c(x,y;z)=I(z\\in \\bar{B}(x,\\rho (x,y)))$, which identifies whether the point $z$  falls into the closed ball $\\bar{B}(x,\\rho (x,y))$ with $x$ as the center and $\\rho (x,y)$ as the radius, and $e(x,y,z_1,z_2)=c(x,y;z_1)c(x,y;z_2)$, which determines whether two points $z_1$ and $z_2$ fall into the ball $\\bar{B}(x,\\rho (x,y))$ together.   Let $N=T-M$, $C_{ij}^{1}=\\frac{1}{M}\\sum_{u=1}^Mc(Z_i,Z_j;Z_{u}),\nC_{ij}^{2}=\\frac{1}{N}\\sum_{v=M+1}^Tc(Z_i,Z_j;Z_{v}).$\nA consistent estimator of the Ball divergence of $\\mu_1$ and $\\mu_2$  with the scale parameter $\\alpha$ is\n\\begin{equation*}\nD_{M,N}=\\frac{1}{T^2}\\sum\\limits_{i,j=1}^{T}(C_{ij}^{1}-C_{ij}^{2})^2,\n\\end{equation*}\nas summarized in Theorem \\ref{slln}.\n\nWe also prove that $\\frac{MN}{T}D_{M,N}$ has a limiting distribution under the null hypothesis in Theorem \\ref{h0}. For this reason, we choose\n\t\\begin{equation*}\n\tV(M,T)=\\frac{MN}{T}D_{M,N}\n\t\\end{equation*}\nas the statistic to detect change-points.\n\nTo investigate the asymptotic properties of $V(M, T)$, we introduce two concepts of the random sequence: absolutely regular and ergodic stationary sequence.\n\nGiven the probability space $(\\Omega,\\mathscr{F},P)$ and two sub-$\\sigma$-fields $\\mathscr{A}$ and $\\mathscr{B}$ of $\\mathscr{F}$, let\n$$\n\\beta(\\mathscr{A},\\mathscr{B})=\\sup\\sum_{i=1}^m\\sum_{j=1}^n|P(A_{i}\\bigcap B_{j})-P(A_{i})P(B_{j}))|,\n$$\nwhere the supreme is taken over all partitions of $\\Omega$ into sets $A_1,\\ldots,A_m\\in\\mathscr{A}$, all partitions of $\\Omega$ into sets $B_1,\\ldots,B_n\\in\\mathscr{B}$ and all $m,n\\geq1$. A stochastic sequence $\\{Z_i\\}_{i\\in\\mathds{Z}}$ is called absolutely regular ( \\citep{dehling2012asymptotic}, also called weakly Bernoulli \\citep{aaronson1996strong}), if\n$$\n\\beta(l)=\\sup\\limits_{n}\\beta(\\mathscr{F}_{0}^n,\\mathscr{F}_{n+l}^\\infty)\\rightarrow 0,\n$$\nas $l\\rightarrow\\infty$. Here the $\\mathscr{F}_i^j$ denotes the $\\sigma$-field generated by the random variables $Z_i,\\ldots,Z_j$. In this paper, we suppose that $\\beta(l)=O(l^{-1-r})$ for any $r>0$. The concept of absolutely regular sequence is wide enough to cover all relevant examples from statistics except for long memory sequences.\n\nRecall that an ergodic, stationary sequence (ESS) \\citep{aaronson1996strong} is a random sequence $\\{Z_i\\}_{1\\leq i\\leq T}$ of form $Z_{i}=f(G^i)$ where $G^i$ is an ergodic, probability-preserving transformation in the probability space $(\\Omega,\\mathscr{F},P)$, and $f$ is a measurable function.\nIn essence, an ESS implies that the random sequence will not change its statistical properties with time (stationarity) and that its statistical properties  can be deduced from a single, sufficiently long sample of the sequence (ergodicity).\n\nWe have the following theorem for an absolutely regular sequence comprised of two ergodic stationary sequences:\n\n\\begin{theorem}\\label{slln} Suppose that $\\{Z_i\\}_{1 \\leq i \\leq T}$ is an absolutely regular sequence, $\\{Z_i\\}_{1 \\leq i \\leq M}$ and $\\{Z_i\\}_{M+1\\leq i \\leq T}$ are both ergodic stationary with marginal probability measure $\\mu_1,\\mu_2$ respectively. When $M,T\\rightarrow \\infty$, $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$,  then\n\t$$\\frac{V(M,T)}{T}\\xrightarrow[M,T\\rightarrow\\infty]{a.s.}V(\\alpha_1;\\mu_1,\\mu_2).$$\n\\end{theorem}\n\nTheorem \\ref{slln} means that $\\frac{V(M,T)}{T}$ converges to Ball detection function $V(\\alpha_1;\\mu_1,\\mu_2)$ almost surely. We further investigate the asymptotic distribution of $V(M,T)$. Under the null hypothesis, the Ball detection function in sample is the sum of four degenerate V-statistics. As in \\cite{pan2018ball}, we denote $Q(x,y;x',y')$ as the second component in the H-decomposition of $V(M,T)$. Then we have the spectral decomposition:\n$$Q(x,y;x',y')=\\sum_{k=1}^{\\infty}\\lambda_kf_k(x,y)f_k(x',y'),$$\nwhere $\\lambda_k$ and $f_k$ are the eigenvalues and eigenfunctions of $Q(x,y;x',y')$. Let $\\{Z_i'\\}_{1\\leq i\\leq T}$ be an independent copy of $\\{Z_i\\}_{1\\leq i\\leq T}$. For $k\\in\\{1,2,\\ldots\\},$    $N_{1k},N_{2k}$ are assumed to be \\emph{iid} $N(0,1)$, and let\n\\begin{align*}\na_k^2(\\alpha_1)=(1-\\alpha_1)E_{Z_1}[E_{Z_1'}f_k(Z_1,Z_1')]^2,\\quad\nb_k^2(\\alpha_1)=\\alpha_1E_{Z_1'}[E_{Z_1}f_k(Z_1,Z_1')]^2,\\\\\nc_k^2(\\alpha_1)=a_k^2(\\alpha_1)+2(1-\\alpha_1)(\\sum_{j=1}^{\\infty}E_{Z_1,Z_{1+j}}[E_{Z_1'}f_k(Z_1,Z_1')E_{Z_1'}f_k(Z_{1+j},Z_1')]),\\\\\nd_k^2(\\alpha_1)=b_k^2(\\alpha_1)+2\\alpha_1(\\sum_{j=1}^{\\infty}E_{Z_1',Z_{1+j}'}[E_{Z_1}f_k(Z_1,Z_1')E_{Z_1}f_k(Z_1,Z_{1+j}')]),\n\\end{align*}\n$$\\theta=E[E(c(Z_1,Z_2,Z_i)(1-c(Z_1,Z_2,Z_{j}))|Z_1,Z_2)].\n$$\n\\begin{theorem}\\label{h0}\n\tUnder null hypothesis $H_{0}: \\mu_1=\\mu_2$, $\\{Z_i\\}_{1\\leq i\\leq T}$ is a stationary absolutely regular sequence with coefficients satisfying $\\beta(l)=O(l^{-1-r})$ for $r>0$, if  $M,T\\rightarrow \\infty$, $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$, we have \n$$V(M,T)\\xrightarrow[M,T\\rightarrow\\infty]{d}\\sum_{k=1}^{\\infty}\\lambda_{k}[(c_{k}(\\alpha_1)N_{1k}+d_{k}(\\alpha_1)N_{2k})^{2}-(a_{k}^{2}(\\alpha_1)+b_{k}^{2}(\\alpha_1))]+\\theta. $$\n\\end{theorem}\n\nUnder the alternative hypothesis, the Ball detection function in sample  is asymptotically normal because it is a sum of non-degenerate V-statistics. Let $g^{(1,0)}(Z_\\mu)$ and $g^{(0,1)}(Z_{\\nu})$ be the first component in H-decomposition of $V(M,T)$ and\n$$\\delta_{1,0}^2=Var(g^{(1,0)}(Z_u))+2\\sum_{i=1}^\\infty Cov(g^{(1,0)}(Z_u),g^{(1,0)}(Z_{u+i})),$$\n$$\\delta_{0,1}^2=Var(g^{(0,1)}(Z_v))+2\\sum_{i=1}^\\infty Cov(g^{(0,1)}(Z_v),g^{(0,1)}(Z_{v+i})).$$\n We can obtain the asymptotic distribution under the alternative hypothesis.\n\\begin{theorem}\\label{h1}\n\t$\\{Z_i\\}_{1\\leq i\\leq T}$ is a absolutely regular sequence with coefficients satisfying $\\beta(l)=O(l^{-1-r})$ for $r>0$. Under $H_{1}: \\mu_1\\neq  \\mu_2$, if $M,T\\rightarrow \\infty$, and  $M\/T\\rightarrow\\alpha_1$ for some $\\alpha_1\\in[0,1]$, then we have\n\t$$\\sqrt{\\frac{T}{MN}}(V(M,T)-TV(\\alpha_1;\\mu_1,\\mu_2))\\xrightarrow[M,T\\rightarrow\\infty]{d} N(0,(1-\\alpha_1)\\delta_{1,0}^2+\\alpha_1\\delta_{0,1}^2).$$\n\\end{theorem}\n\n\nWe show that the Ball detection function in sample is consistent against general alternatives. Our new detection function can handle the problem of imbalanced sample sizes. As shown in the following theorem, the asymptotic power of the test does not go to zero even if $\\eta=\\frac{M}{N}$ goes to $0$ or $\\infty$.\n\n\\begin{theorem}\\label{againstall}\n\tThe test based on $V(M,T)\/T$ is consistent against any general alternative $H_1$. More specifically,\n\t$$\n\t\\lim\\limits_{(M,T)\\rightarrow\\infty}Var_{H_1}(V(M,T)\/T)=0,\n\t$$\n\tand\n\t$$\n\t\\Lambda:=\\liminf\\limits_{(M,T)\\rightarrow\\infty}(E_{H_1}V(M,T)-E_{H_0}V(M,T))\/T>0.\n\t$$\n\\end{theorem}\n\n\n\\section{Detection of change-points}\n\\subsection{Hierarchical Algorithm}\n\nNext, we use the Ball detection function in sample to detect change-points in a sequence.\nFor simplicity, suppose that the sequence $\\{Z_i\\}_{1\\leq i\\leq T}$ contains at most one change-point.\nThe possible change-point location is then estimated by maximizing the detection function:\n\\begin{equation}\\label{singleLoc}\n\\hat{M}_1=\\argmax_{M}V(M,T).\n\\end{equation}\nWe use the bootstrap method to estimate the probability that $V(\\hat{M}_1,T)$  exceeds a threshold.\nIf the estimated probability is high enough, $\\hat{M}_1$ is the estimated change-point. Otherwise, we proceed as if there does not exist any change-point in the sequence.\n\nIt is more complicated if the sequence has multiple change-points. In this case, we estimate the first change-point by\n\\begin{equation}\\label{singleLocLLL}\n(\\hat{M}_1,\\hat{L}_1)=\\argmax_{0<M_1<L_1\\leq T}V(M_1,L_1).\n\\end{equation}\nFrom (\\ref{singleLocLLL}), we can see that the introduction of $L_1$ here is to alleviate a weakness of bisection algorithm  \\citep{matteson2014nonparametric}. Because in each segment, there may exist multiple change-points. If we do not introduce $L_1$,  the value of  $V(\\hat{M}_1,T)$ may be lower than $V(\\hat{M}_1,\\hat{L}_1)$.\n\nSuppose that $k-1$ change-points have been estimated at locations $0<\\hat{T}_1<\\cdots<\\hat{T}_{k-1}<T$, and $\\hat{T}_0=0$, $\\hat{T}_k=T$. Those change-points partition the sequence into $k$ segments $\\mathbf{Z}(\\hat{T}_1\/\\hat{T}_0),\\ldots,\\mathbf{Z}(\\hat{T}_k\/\\hat{T}_{k-1})$.\nIn segment $i$, let $C_{ij}^1=\\frac{1}{M_i-\\hat{T}_{i-1}}\\sum_{u=\\hat{T}_{i-1}}^{M_i}c(Z_i,Z_j,Z_u),$ $C_{ij}^2=\\frac{1}{L_i-M_{i}}\\sum_{v=M_{i}}^{L_i}c(Z_i,Z_j,Z_u).$\nThe Ball detection function in sample of segment $i$ is denoted as\n$$\nV_i(M_i,L_i)=\\frac{(M_i-\\hat{T}_{i-1})(L_i-M_i)}{(L_i-\\hat{T}_{i-1})^3}\\sum_{i=\\hat{T}_{I-1}}^{L_i}(C_{ij}^1-C_{ij}^1)^2.\n$$\nNow let\n\\begin{equation}\\label{kappaformula}\n(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})=\\argmax\\limits_{1\\leq i\\leq k-1,\\hat{T}_{i-1}<M_i<L_i \\leq \\hat{T}_i}V_i(M_i,L_i).\n\\end{equation}\nThen  $\\hat{M}_{\\hat{i}}$ is the $k$-th possible change-point located within segment $\\mathbf{Z}(\\hat{T}_{\\hat{i}}\/\\hat{T}_{\\hat{i}-1}).$\nThis hierarchical algorithm for estimating multiple change-points is outlined below.\n\\begin{algorithm}\\label{algorithm}\n\t\\caption{Multiple change-points Algorithm}\n\t\\begin{algorithmic}\n\t\t\\STATE Let the minimum segment size $min=m$, the change-points set $\\mathbf{T}=\\{0,T\\}$.\n\t\t\\STATE Suppose that $k-1$ change-points have been estimated. This decomposes the observations into $k$ segments.\n\t\t\\FOR {each $i\\in \\{1,\\ldots,k-1\\}$}\n\t\n\t\t\\IF {the $i-$th segment $\\mathbf{Z}(\\hat{T}_i\/\\hat{T}_{i-1})$ is a new segment,}\n\t\t\\STATE $best_i=0;$\n\t\t\\FOR{$M_i=\\hat{T}_{i-1}+m,\\hat{T}_{i-1}+m+1,\\ldots,\\hat{T}_{i}-m$}\n\t\t\\FOR{$L_i=\\hat{T}_{i-1}+m,\\ldots,\\hat{T}_{i}$}\n\t\t\\STATE Compute $V_i(M_i,L_i)$;\n\t\t\n\t\t\\IF{$V_i(M_i,L_i)\\geq best$}\n\t\t\\STATE  $\\hat{M}_i=M_i, \\hat{L_i}=L_i, V_i(\\hat{M}_i,\\hat{L}_i)=V_i(M_i,L_i)$;\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\t\\ELSE\n\t\t\\STATE $V_i(\\hat{M}_i,\\hat{L}_i),\\hat{M}_i,\\hat{L}_i$ had been calculated.\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\STATE $V_i(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})=\\argmax_{0\\leq i\\leq k-1}V_i(\\hat{M}_i,\\hat{L}_i).$\n\t\t\\IF {$V_i(\\hat{M}_{\\hat{i}},\\hat{L}_{\\hat{i}})$ exceeds a threshold,}\n\t\t\\STATE put $\\hat{M}_{\\hat{i}}$ into $\\mathbf{T}$;\n\t\t\\ELSE\n\t\t\\STATE there does not exist new change-point.\n\t\t\\ENDIF\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Hierarchical Significance Testing}\n\nHere, we elaborate the use of the bootstrap method mentioned above.\n\nTheorem \\ref{h0} shows that the asymptotic null distribution of  $V(M,T)$ is a mixture of $\\chi^2$ distributions. In practice, it is difficult to directly take advantage of the asymptotic null distribution. So, we use the moving block bootstrap \\citep{kunsch1989jackknife} to obtain the empirical probabilities.\n\nGiven a set of observations $\\{Z_t\\}_{1\\leq t\\leq T}$ and the block size $b_T$, we draw a bootstrap resample $\\{Z_t^*\\}_{1\\leq t\\leq T}$ as follows: (i) define the $b_T$ dimensional vector $X_t=(Z_t,Z_{t-1},\\ldots,Z_{t-b_T+1})$; (ii) resample from block data $\\{X_t\\}_{1\\leq t \\leq T-b_T+1}$  with replacement to get pseudo data $\\{X_t\\}_{1\\leq t\\leq L}$ which satisfies $T=[Lb_T]$, where $[A]$ denotes the integer part of $A$. Denote the first $T$ elements of $\\{X_t\\}_{1\\leq t \\leq L}$ as the bootstrap resample $\\{Z_t^*\\}_{1\\leq t\\leq T}$; (iii) repeat steps (i) and (ii) $R$ times. For the $r$-th repetition, denote the maximum value in equation (\\ref{singleLocLLL}) based on $\\{Z_t^*\\}_{1\\leq t\\leq T}$ by $V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)})$; (iv) the approximate probability is estimated by $\\{V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)}):r=1,\\ldots,R\\}.$ Denote the threshold of the estimated probability  by $p_T$, if\n\t $\\frac{\\sharp\\{V(\\hat{M}_1^{(r)},\\hat{L}_1^{(r)})\\geq V(\\hat{M}_1,\\hat{L}_1)\\}}{R+1}<p_T$,  then $\\hat{M}_1$  is a change-point.\n\nIn applications, the choice of the block size $b_T$ involves a trade-off. If the block size becomes too small, the moving block bootstrap will destroy the time dependency of the data and the accuracy will deteriorate. But if the block size becomes too large, there will be few blocks to be used. In other words, increasing the block size reduces the bias and captures more persistent dependence, while decreasing the block size reduces the variance as more subsamples are available. Thus, a reasonable trade off is to  consider the mean squared error as the objective criterion to balance the bias and variance. For the linear time series, as proved in  \\citet{carlstein1986use}, the value of the block size that minimizes MSE is\n$$\nb_T^*=\\left(\\frac{2|\\rho|}{1-\\rho^2}\\right)^{2\/3}T^{1\/3},\n$$\nwhere $\\rho$ is the first order autocorrelation.\nBecause the construction of MSE depends on the knowledge of the underlying data generating sequence, no optimal result is available in  general.\nIn this paper, we follow \\citet{hong2017testing} and \\citet{xiao2007testing} to choose $b_T=\\max\\{q_T,\\bar{q}_T\\}$, where\n\\begin{equation}\\label{bp}\nq_T=\\min\\left\\{\\left[\\left(\\frac{3T}{2}\\right)^{1\/3}\\left(\\frac{2\\hat{\\rho}}{1-\\hat{\\rho}^2}\\right)^{2\/3}\\right],\\left[8\\left(\\frac{T}{100}\\right)^{1\/3}\\right]\\right\\},\n\\end{equation}\nwhere $\\hat{\\rho}$ is the estimator of the first autocorrelation of $Z_t$,\n$\\bar{q}_T$ is the same as (\\ref{bp}) except replacing $\\hat{\\rho}$ with the estimated first order autocorrelation of $Z_t^2$. So the choice of $b_T$ considers the linear dependence and non-linear dependence.\n\n\\subsection{Consistency}\nThe next theorem shows the consistency of the estimated change-point locations under the following assumption.\n\\begin{assumption}\\label{ass1}\nSuppose that $\\mathbf{Z}(T\/0)$  is an absolutely regular sequence which is comprised of two ergodic stationary sequences. Let $\\alpha_1 \\in (0,1)$ denote the fraction of the observations, such that $\\mathbf{Z}(\\lfloor\\alpha_1 T\\rfloor\/0)$ be an ergodic stationary sequence with marginal probability measure $\\mu_1$,  $\\mathbf{Z}(T\/\\lfloor\\alpha_1 T\\rfloor)$ the second ergodic stationary sequence with marginal distribution $\\mu_2$.  Finally, let $\\delta_T$ be a sequence of positive numbers, such that $\\delta_T \\rightarrow 0$ and $T\\delta_T\\rightarrow \\infty$ as $T\\rightarrow \\infty$.\n\\end{assumption}\n\n\\begin{theorem}\\label{asymptotic1}\n\tSuppose Assumption \\ref{ass1} holds. Let $\\hat{M}_1$ be the estimated change-point location from Equation (\\ref{singleLoc}) for a sample of size $T$. For all $\\epsilon>0$ and $T$ large enough such that $\\alpha_1\\in[\\delta_T,1-\\delta_T]$, we have\n\t\\begin{equation*}\n\tP(\\lim_{T\\rightarrow\\infty}|\\frac{\\hat{M}_1}{T}-\\alpha_1|<\\epsilon)=1.\n\t\\end{equation*}\n\\end{theorem}\nThis theorem shows that the consistency only requires the size of each segment  increases to $\\infty$, but not necessarily at the same rate. Under the Assumption \\ref{ass1}, $\\alpha_1$ can be close to 0 or 1 when $T\\rightarrow\\infty$, which is an imbalanced case.\n\nIn the multiple change-points situation, we have the following Assumption.\n\\begin{assumption}\\label{ass2mul}\n\tSuppose that $\\{Z_i\\}_{1 \\leq i \\leq T}$ is an absolutely regular sequence. Let $0=T_0<T_1<\\ldots<T_k<T_{k+1}=T$, and $\\min\\limits_{i=1,\\ldots,k}|T_i-T_{i-1}|\\geq aT^{b}$, with $a>0$ and $0<b\\leq1$. For $i=0,1,\\ldots,k$,  $\\mathbf{Z}(T_i\/T_{i-1})$ is an ergodic stationary sequence with marginal probability measure $\\mu_i$ and $\\mu_i\\neq \\mu_{i+1}$. \\label{key}\n\tFurthermore, let $\\delta_T$ be a sequence of positive numbers, such that $\\delta_T \\rightarrow 0$ and $T\\delta_T\\rightarrow \\infty$ as $T\\rightarrow \\infty$.\n\\end{assumption}\n\n It is worth noting that we do not assume the upper bounds on the number of change-points $k$, but by specifying  the minimum sample size in each segment. In other words, under Assumption \\ref{ass2mul}, as $T\\rightarrow\\infty$, we can have $k\\rightarrow\\infty$ change-points.\n\nAnalysis of multiple change points can be reduced to the analysis of only two change points under Assumption \\ref{ass2mul}.  Let $\\alpha_i=T_i\/T$, for any  $i\\in\\{1,\\ldots,k\\}$. The observations $\\mathbf{Z}(T\\alpha_i\/0)$ can be seen as a random sample from a mixture of probability measures $\\{\\mu_j:j\\leq i\\}$, denoted as $\\boldsymbol{\\mu}_i$. Similarly, observations $\\mathbf{Z}(T\/T\\alpha_{i+1})$ are a sample from a mixture of probability measures $\\{\\mu_j:j\\geq i+1\\}$, denoted here as $\\boldsymbol{\\nu}_i$. The remaining observations are distributed according to some probability measure $\\boldsymbol{\\xi}_i$. Furthermore, $\\boldsymbol{\\mu}_i\\neq \\boldsymbol{\\xi}_i$ and $\\boldsymbol{\\nu}_i\\neq \\boldsymbol{\\xi}_i$.  If one of the previous two inequalities does not hold, we refer to the single change point setting.\n\nConsider any $\\alpha$ such that, $\\alpha_i\\leq\\alpha\\leq\\alpha_{i+1}$. Then, this choice of $\\alpha$ will create two mixture probability measures. One with component probability measures $\\boldsymbol{\\mu}_i$ and $\\boldsymbol{\\xi}_i$, and the other with component probability measures $\\boldsymbol{\\nu}_i$ and $\\boldsymbol{\\xi}_i$. Then, the Ball detection function between these two mixture probability measures is equal to\n\\begin{equation}\\label{multiCP_location}\n\\begin{split}\n&V(\\alpha;\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i,\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i+\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i)\\\\=&\\alpha(1-\\alpha)\\iint_{\\mathrm{A}\\times \\mathrm{A}} [\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i-\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i-\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i]^{2}(\\bar{B}(u,\\rho(u,v))\\omega_{\\alpha} (du)\\omega_{\\alpha} (dv).\n\\end{split}\n\\end{equation}\n\n\\begin{theorem}\\label{lemma_multiCP_max}\nSuppose that Assumption \\ref{ass2mul} holds, then the Ball detection function  in equation (\\ref{multiCP_location}) is maximized when either $\\alpha=\\alpha_i$ or $\\alpha=\\alpha_{i+1}$.\n\\end{theorem}\n\nBy Theorem \\ref{lemma_multiCP_max}, $f_i(\\alpha)$ is maximized when $\\alpha=\\alpha_i$ or $\\alpha=\\alpha_{i+1}$ for $i=1,\\ldots,k-1$. Additionally,\ndefine\n$$V(\\alpha)=\\sum_{i=0}^kV(\\alpha;\\frac{\\alpha_i}{\\alpha}\\boldsymbol{\\mu}_i+\\frac{\\alpha-\\alpha_i}{\\alpha}\\boldsymbol{\\xi}_i,\\frac{1-\\alpha_{i+1}}{1-\\alpha}\\boldsymbol{\\nu}_i+\\frac{\\alpha_{i+1}-\\alpha}{1-\\alpha}\\boldsymbol{\\xi}_i)I(\\alpha_i\\leq\\alpha\\leq\\alpha_{i+1}).$$\nLet $\\mathscr{A}_T=\\{y\\in[\\delta_T,1-\\delta_T]:V(y)\\geq V(\\alpha),\\forall\\alpha\\}$. Let $d(x,\\mathscr{A}_T)=inf\\{|x-y|:y\\in\\mathscr{A}_T\\}$. Then, we have the following Theorem.\n\\begin{theorem}\\label{location_all}\nSuppose that Assumption \\ref{ass2mul}, and $x\\in\\mathbb{R}$,  Let $\\hat{M}_1$ be the estimated change point as defined by equation (\\ref{singleLocLLL}). Then $d(\\hat{M}_1\/T,\\mathscr{A}_T)\\xrightarrow{a.s.}0$ as $T\\rightarrow\\infty$.\n\\end{theorem}\n\nRepeated applications of Theorem \\ref{location_all} can show that as $T\\rightarrow\\infty$, the first $k$ estimated change points will converge to the true change point locations in the manner described above. With a fixed  threshold of the estimated probability $p_T$, all of the change-points will be estimated.\nHowever, with probability approaching 1 as the sample size increases, the number of change-points determined in this way will be more than the true number of change-points, since any given nominal level of significance implies a nonzero probability of rejecting the null hypothesis when it holds. The hierarchical procedure could be made consistent by adopting a threshold for the test that decrease to zero, at a suitable rate, as the sample size increases\\citep{bai1998estimating}.\nThis is illustrated by the following theorem.\n\\begin{theorem}\nLet $\\hat{k}$ be the number of change-points obtained using the hierarchical method based on the statistic (\\ref{kappaformula}) applied with threshold $p_T$, and $k$ be the true number of change-points. If $\\lim\\limits_{T\\rightarrow\\infty}p_T\\rightarrow0$, then under Assumption  \\ref{ass2mul}, $\\lim\\limits_{T\\rightarrow\\infty}P(\\hat{k}=k)=1$.\n\\end{theorem}\nAlthough the hierarchical algorithm tends to estimate more change-points asymptotically when $p_T$ is fixed, this has little effect in practice. For example, the asymptotic probability of selecting $(k+j)$ change-points, is given by $p_T^j(1-p_T),$ which decreases rapidly.\nFurthermore, if there is no change point, that is $k=0$, the probability of selecting at least one change point in our algorithm is\n$$\n\\sum_{j=1}^{\\infty}p_T^j(1-p_T)=p_T.\n$$ Hence the total rate of type I errors is still $p_T$. This is a distinct feature of our hierarchical procedure because controlling for type I errors is a challenging issue in multiple testings.\n\n\n\\section{Simulation studies}\n\nIn this section, we present the numerical performance of the proposed method (BDCP) with $p_T=0.05$ and compare it with several typical methods, including Bayesian method (BCP) \\citep{barry1993bayesian}, WBS method  \\citep{fryzlewicz2014wild}, the graph-based method-gSeg  \\citep{chen2015graph,chu2019asymptotic} and energy distance based method (ECP)  \\citep{matteson2014nonparametric}. BCP, WBS, ECP and BDCP can estimate the number of change-points automatically while gSeg can detect only one change-point or an interval.\n\nThere are four commonly used criteria for the performance of those methods: the adjusted Rand index, the over segmentation error, the under segmentation error and the Hausdorff distance.\nSuppose that the true change-points set is $\\mathbf{T}=\\{0,T_1,\\ldots,T_k,T\\}$ and estimated change-points set is $\\hat{\\mathbf{T}}=\\{0,\\hat{T}_1,\\ldots,\\hat{T}_{\\hat{k}},T\\}$.\nThen denote the true segments of series\n$\\{Z_t\\}_{1\\leq t\\leq T}$ by $\\mathbf{Z}=\\{\\mathbf{Z}(T_1\/0),\\ldots,\\mathbf{Z}(T\/T_k)\\}$ and the estimated segments by\\\\\n$\\mathbf{\\hat{Z}}=\\{\\mathbf{Z}(\\hat{T}_1\/0),\\ldots,\\mathbf{Z}(T\/\\hat{T}_{\\hat{k}})\\}$.\nConsider the pairs of observations that fall into one of the following two sets:\\\\ $\\{S_1\\}=$ \\{pairs of observations in the same segments under $\\mathbf{Z}$ and in same segments under $\\mathbf{\\hat{Z}}$\\};\\\\ $\\{S_2\\}=$ \\{pairs of observations in different segments under $\\mathbf{Z}$ and in different segments under $\\mathbf{\\hat{Z}}$\\}. Denote $\\sharp S_1$ and $\\sharp S_2$ as the number of pairs of observations in each of these two sets. The Rand index $RI$ is defined as\n$$RI=\\frac{\\sharp S_1+\\sharp S_2}{\\binom{T}{2}}.$$\n\nAdjusted Rand index $ARI$ is the corrected-for-chance version of the Rand index which is defined as\n$$ARI = \\frac{RI - E(RI)}{1-E(RI)},$$\nin which 1 corresponds to the maximum Rand index value.\n\nOn the other hand, we also calculate the distance between $\\mathbf{T}$ and $\\hat{\\mathbf{T}}$ by\n$$\\zeta(\\hat{\\mathbf{T}}||\\mathbf{T})=\\sup_{b\\in\\mathbf{T}}\\inf_{a\\in\\hat{\\mathbf{T}}}|a-b|\\ \\ and \\ \\ \\zeta(\\mathbf{T}||\\hat{\\mathbf{T}})=\\sup_{b\\in\\hat{\\mathbf{T}}}\\inf_{a\\in\\mathbf{T}}|a-b|,$$\nwhich quantify the over-segmentation error and the under-segmentation error, respectively \\citep{boysen2009consistencies,zou2014nonparametric}.  The Hausdorff distance  \\citep{harchaoui2010multiple} between $\\mathbf{T}$ and $\\hat{\\mathbf{T}}$ is defined as\n$$\\Delta(\\mathbf{T},\\hat{\\mathbf{T}})=sup\\{\\zeta(\\hat{\\mathbf{T}}||\\mathbf{T}),\\zeta(\\mathbf{T}||\\hat{\\mathbf{T}})\\}.$$\nHere, we only report the results based on adjusted Rand index. The results under other criteria are deferred to the supplementary material.\n\nThree scenarios are used for comparisons: univariate sequence, multivariate sequence and manifold sequence.\nIn each scenario, we consider two types of examples, one without change-point, and one with two change-points as follow:\n$$\\{X_1,X_2,\\ldots,X_n,Y_1,Y_2,\\ldots,Y_m,X_{n+1},X_{n+2},\\ldots,X_{2n}\\}.$$\nThe sample sizes are set to be $n=40$, $m=40,60,80$.\nWe will repeat each model 400 times and the threshold is at 0.05. To save space, some results of univariate sequences are available on the supplementary material.\n\n\\subsection{Multivariate sequence}\nIn this subsection, we consider the $d=3$ dimensional sequences. Examples 4.1.1-4.1.7 are the sequences with no change-point and Examples 4.1.8-4.1.15 are the models with two change-points.\n\\begin{itemize}\n\t\\item \\textbf{Examples 4.1.1-4.1.3:}\n\t$$X_t=\\epsilon_{t},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.1, $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.2 and $\\epsilon_{t}\\sim Cauchy(0,I_3)$ for Example 4.1.3.\n\t\n\t\\item \\textbf{Examples 4.1.4-4.1.5}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.4 and $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.5.\n\t\n\t\\item \\textbf{Examples 4.1.6-4.1.7}:\n\t$$X_{t}=\\sigma_{X,t|t-1}\\epsilon_{t},$$\n\t$$\\sigma_{X,t|t-1}^{2}=0.02+0.02\\sigma_{X,t-1|t-2}^{2}+0.05X_{t}^{2},$$\n\t$\\epsilon_{t}\\sim N(0,I_3)$ for Example 4.1.6 and $\\epsilon_{t}\\sim t_3(0,I_3)$ for Example 4.1.7.\n\\end{itemize}\n\n\n\\begin{itemize}\n\t\\item \\textbf{Example 4.1.8}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=\\mu+0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),,\\epsilon_{t}\\sim N(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.9}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=\\mu+0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),\\epsilon_{t}\\sim t_3(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.10}:\n\t$$X_t=\\epsilon_{t},$$\n\t$$Y_t=\\mu+\\epsilon_{t},$$\n\t$$\\mu=(4,4,4),(6,6,6),(8,8,8),\\epsilon_{t}\\sim Cauchy(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.11}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=0.5\\epsilon_{t}\\sigma+0.5\\epsilon_{t-1}\\sigma,$$\n\t$$\\sigma=(3,3,3),(5,5,5),(7,7,7),\\epsilon_{t}\\sim N(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.12}:\n\t$$X_t=0.5\\epsilon_{t}+0.5\\epsilon_{t-1},$$\n\t$$Y_t=0.5\\epsilon_{t}\\sigma+0.5\\epsilon_{t-1}\\sigma,$$\n\t$$\\sigma=(3,3,3),(5,5,5),(7,7,7),\\epsilon_{t}\\sim t_3(0,I_3).$$\n\t\n\t\\item \\textbf{Example 4.1.13}:\n\t$$X_t=\\epsilon_{t},$$\n\t$$Y_t=\\epsilon_{t}\\sigma,$$\n\t$$\\sigma=(9,9,9),(16,16,16),(25,25,25),\\epsilon_{t}\n\t\\sim Cauchy(0,I_3).$$\n\t\n\t\\item \\textbf{Examples 4.1.14-4.1.15}: \\\\\n\t$X_t,Y_t \\sim CCC-GARCH(1,1)$\\citep{Bollerslev1990Modelling}, let\n\t$$X_t=\\sigma^{X}_{t}\\epsilon_t,Y_{t}=\\sigma^{Y}_{t}\\epsilon_{t},$$\n\t$$\\sigma^{X}_{t}=(\\omega_{X}+A^{X}\\epsilon_{t-1}+B_{X}\\sigma{X}_{t-1}^{2})^{1\/2},$$\n\t$$\\sigma^{X}_{t}=(\\omega_{Y}+A^{Y}\\epsilon_{t-1}+B_{Y}\\sigma{Y}_{t-1}^{2})^{1\/2}.$$\nLet     $\\omega^{X}=(0.01,0.01,0.01),$ $A^{X}=diag(0.02,0.03,0.01),$ $B^{X}=diag(0.02,0.02,0.05)$, and\n\n\tCase 1:$$\\omega^{Y}=2\\omega^{X},A_{Y}=4A^{X},B^{Y}=5B^{X},$$\n\t\n\tCase 2:$$\\omega^{Y}=3\\omega^{X},A_{Y}=5A^{X} ,B^{Y}=6B^{X}.$$\n\t\n\tCase 3:$$\\omega^{Y}=4\\omega^{X},A_{Y}=6A^{X} ,B_{Y}=7B^{X}.$$\n\t$\\epsilon_{t}\\sim N(0,1)$ for Example 4.1.14 and $\\epsilon_{t}\\sim t(df=3)$ for Example 4.1.15.\n\\end{itemize}\n\n\nTable \\ref{Sim_adj_mul_type1} reveals that ECP and BDCP can handle the multivariate stationary series well. BCP works well when the distribution is normal but has a lower adjusted Rand index in $t$ distribution and Cauchy distribution. We do not consider the WBS method because it can not handle the multivariate cases. Tables \\ref{Sim_adj_mul_mean}-\\ref{Sim_adj_mul_gar} present the results of those examples with two change-points. All four methods have excellent performance in the multivariate normal distribution and multivariate $t$ distribution with location shift.\nExamples 4.1.11-4.1.13 consider the scale shift case and Examples 4.1.14-4.1.15 are the popular GARCH models which are also the scale shift case. We can see from Tables  \\ref{Sim_adj_mul_var}-\\ref{Sim_adj_mul_gar} that BDCP has the best performance in almost all the scale shift cases.\n\n\n\\subsection{Manifold-valued sequence}\nIn this subsection, we report some manifold-valued examples where ECP can not detect the change-points but BDCP works well. Consider the distribution in a unit circle and let\n$$P_1 \\sim Unif([-\\pi\/6,\\pi\/6)\\bigcup[11\\pi\/6,13\\pi\/6)),$$\n$$P_2 \\sim Unif([\\pi\/3,2\\pi\/3)\\bigcup[7\\pi\/3,8\\pi\/3)),$$\n$$P_3 \\sim Unif([5\\pi\/6,7\\pi\/6)\\bigcup[17\\pi\/6,19\\pi\/6)),$$\n$$P_4 \\sim Unif([4\\pi\/3,5\\pi\/3)\\bigcup[10\\pi\/3,11\\pi\/3)),$$\n$$P_5 \\sim Unif([0,4\\pi)).$$\nWe calculate the circular distance which is defined by\n\\begin{equation}\\label{circlediatance}\nd(X_i,X_j)=min(|X_i-X_j|,2\\pi-|X_i-X_j|).\n\\end{equation}\nWe simulate four examples with 0,1,2,3 change-points respectively. Let $n=40, m=40,60,80$.\n\\begin{itemize}\n\t\\item \\textbf{Example 4.2.1}:\n\t$$\\{X_1\\ldots,X_{3n}\\}\\sim P_5.$$\n\t\\item \\textbf{Example 4.2.2}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3.$$\n\t\n\t\\item \\textbf{Example 4.2.3}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3,\\{X'_1\\ldots,X'_n\\}\\sim P_2.$$\n\t\n\t\\item \\textbf{Example 4.2.4}:\n\t$$\\{X_1\\ldots,X_n\\}\\sim P_1, \\{Y_1\\ldots,Y_m\\}\\sim P_3,$$\n\t$$\\{X'_1\\ldots,X'_n\\}\\sim P_2, \\{Y'_1\\ldots,Y'_m\\}\\sim P_4.$$\n\\end{itemize}\n\n\n\\begin{comment}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1\\textwidth]{sim_exp.pdf}\n\\caption{\\label{sim_exp}}\n\\end{figure}\n\\end{comment}\n\n\nTable \\ref{Sim_adj_man_type1} reveals that BCP, WBS, ECP all perform well when there is no change-point. However, they do not work when the sequences have change-points (Table \\ref{Sim_adj_man_123}). That is because BCP is based on the normal distribution, and WBS is a CUSUM statistics which does not work in a circular distribution. For ECP, that is because the circular distance is not of strong negative type (Theorem 9.1 in \\citet{hjorth1998finite}).  The gSeg method can detect change-points when the number of change-points is one or two but do not perform well in Example 4.2.4. BDCP has a remarkable performance in all these examples.\n\n\\section{Real data analysis}\n\n\n\\subsection{Wind direction of Yunnan-Guizhou Plateau}\n\nMonsoon is used to describe seasonal changes in atmospheric circulation and precipitation associated with the asymmetric heating of the land and sea. The major monsoon systems in the world consist of West African Monsoon (WAM), Indian summer monsoon (ISM), East Asian Monsoon (EAM) and so on. In this subsection, we analyze the wind direction data of Yunnan-Guizhou Plateau ($105^\\circ$E, $27^\\circ$N) from 06\/01\/2015 to 10\/30\/2015. The data are available in R package \\emph{rWind}. Yunnan-Guizhou Plateau is located in southwest China, with local climate influenced by both ISM and EAM  \\citep{sirocko1996teleconnections}\\citep{li2014synchronous}(Fig. 1A and S1). Strict spatial boundaries between the ISM and the ASM are difficult to define \\citep{cheng2012global} though previous researchers have suggested $103^\\circ$E as the dividing line on the basis of summer prevailing winds.\n\nDaily wind directions are shown in the top-left of Figure \\ref{wind1}. Note that degree 0 represents due North, $\\pi\/2$ represents due East, $\\pi$ represents due South and $3\\pi\/2$ represents due West.\nWe can see that the wind directions are distributed in almost all directions. In the beginning, the most widely distributed direction is the southwest wind from the Indian Ocean, and then turns smoothly to southeast, which is from the Pacific Ocean. In particular, Yunnan-Guizhou Plateau was mostly influenced by ISM in June and July. After July, the influence of EAM gradually increased  \\citep{li2015introduction}.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{guizhou_2015_1_20_27_105.pdf}\n\t\\caption{\\label{wind1}Wind direction of Yunnan-Guizhou Plateau from 06\/01\/2015 to 10\/30\/2015 and the performance of BCP, WBS, gSeg, ECP and BDCP. The y-axis shows wind direction. Degree 0 and $2\\pi$ represent due North, $\\pi\/2$ represents due East, $2\\pi$ represents due South and $3\\pi\/2$ represents due West.}\n\\end{figure}\n\nTo detect the change-point in the wind direction series, we calculate the circular distance between the daily direction as defined in (\\ref{circlediatance}).\n\nThe performance of the five methods is shown  in Figure \\ref{wind1}.\nBCP, WBS and ECP can not detect any change-point, as seen in the simulation studies in subsection 4.2. gSeg detects an interval between ``07\/17\/2015'' and ``07\/28\/2015''. BDCP estimates four change-points located at ``07\/01\/2015'', ``07\/29\/2015'', ``09\/02\/2015'' and ``09\/26\/2015''.\n\n\nTo visualize the result of BDCP, Figure \\ref{five_periods} depicts the wind rose plot for the five periods detected by BDCP. We can see the significant changes of the direction distribution especially between 07\/29\/2015 - 09\/01\/2015 and 09\/02\/2015 - 09\/25\/2015. The wind directions are almost southwest or west in June and July, then  turn to southeast in September \\citep{li2015introduction}. As mentioned in \\citet{hillman20178}, 75\\% of the average annual precipitation falls in the months of June-September associated with the ISM, and the ISM gets weaker during June and July because isolation decreases by 2-3\\%. BDCP can perfectly detect the change of influence between ISM and EAM.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{five_periods.pdf}\n\t\\caption{\\label{five_periods}The wind rose plot for the five periods detected by BDCP.}\n\\end{figure}\n\n\n\\subsection{Bitcoin price}\nBitcoin is the most popular form of cryptocurrency in recent years.  According to research of Cambridge University in 2017  \\citep{hileman2017global}, there are 2.9 to 5.8 million unique cryptocurrency wallet users, most of whom use Bitcoin. One of the known features of Bitcoin is its high volatility.\nBitcoin is not a denominated flat currency and there is no central bank overseeing the issuing of Bitcoin,  its price is thus driven solely by the investors. Using the weekly data over 2010-2013 period,   \\citet{briere2015virtual} showed that Bitcoin investment had some high distinctive features, including exceptionally high average return and volatility. Hence, accurately fitting its variation is important \\citep{chu2015statistical}.\n\n\nBitcoin can be exchanged for other currencies, products, and services in legal or black markets. \\citet{chu2015statistical} measured the volatility of Bitcoin exchange rate against six major currencies. They found that the behavior of Bitcoin was sharply different from those currencies; its interquartile range was much wider, its skewness was much more negative, its kurtosis was much more peaked and its variance was much larger. Bitcoin showed the highest annualized volatility of percentage change in daily exchange rates. In this subsection, we detect the change-points of daily log-return of Bitcoin using methods, BCP, WBS, gSeg, ECP and BDCP. The datasets are available on \\url{http:\/\/api.bitcoincharts.com\/v1\/csv\/bitstampUSD.csv.gz}. Figure \\ref{bitcoin} displays the  exchange rate of Bitcoin and daily log-returns during 09\/13\/2011 - 12\/31\/2012.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{bitcoin_price.pdf}\n\t\\caption{\\label{bitcoin} US dollar-Bitcoin exchange rate and daily log-returns from 09\/13\/2011-12\/31\/2012.}\n\\end{figure}\n\nFigure \\ref{bitcoincp} compares the performance of the five methods. BCP and WBS can not handle the severe volatility at the beginning of the sequence.\ngSeg detects one change-point at ``02\/23\/2012'', and ECP estimates a change-point at ``02\/09\/2012''.\nBDCP detects four change-points at ``02\/11\/2012'', ``04\/16\/2012'', ``05\/19\/2012'', and ``08\/21\/2012''.\n\nOn February 11, 2012, Paxum, an online payment service and popular means for exchanging Bitcoin announced it would cease all dealings related to the currency due to the concerns of its legality. Two days later, regulatory issues surrounding money transmission compelled the popular Bitcoin exchange and service firm TradeHill to terminate its business and immediately began selling its Bitcoin assets to refund its customers and creditors. Bitcoin trading started to cool down during that period.\n\nAfter May 19, the price of Bitcoin had increased from \\$5.07 to the maximum \\$14.14 on August 17 and kept at that level after that. The reasons for the rise were many. Lots of online articles on this subject expressed the same message: Bitcoin was now going mainstream. WordPress, ranked by Alexa as the 21st most popular site in the world, started to accept Bitcoin for payment on November, 2012.\n\nThe variances of these five stages detected by BDCP are: 0.1054, 0.0182, 0.0059, 0.0458, and 0.0177. The daily log-return sequence was very flat and the price almost did not change during period 04\/16\/2012 - 05\/18\/2012. But it was volatile during other periods from Figure \\ref{bitcoincp}.\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth,height=0.9\\textheight]{bitcoin_cp2.pdf}\n\t\\caption{\\label{bitcoincp} The results of BCP, WBS, geg, ECP, BDCP on the daily log-returns of Bitcoin series from 09\/13\/2011-12\/31\/2012.  The y-axis shows daily log-returns of Bitcoin series. The red lines are the change-point locations detected by BCP, WBS, geg, ECP, BDCP. } \n\\end{figure}\n\n\n\\section{Conclusion}\nWe developed a change-point detection procedure for weakly dependent sequences. Our key idea lies in the novel measure of Ball detection function.\nWe proved the asymptotic properties of its sample statistic for absolutely regular sequences.  Extensive simulation studies demonstrated that our method had a superior performance to other existing methods in various settings. Two real data analyses indicated that our method was useful in analyzing non-Euclidean sequences with various change points and led to insightful understanding of the data. Also, our method is robust since our test statistic is rank-based.\n\nWe will further investigate Ball detection function and its related concepts. For example,  the current computational complexity of our proposed algorithm is $O(kT^2\\log T)$, where $k$ is the number of change-points, and $T$ is the length of the sequence. It will be useful to find an algorithm with a lower computational complexity.\n\n\\bibliographystyle{ECA_jasa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn $1935$ Einstein and Rosen \\cite{ER} proposed the metric\n\\begin{eqnarray}\\label{ER}\n&ds^2 =  -\\frac{u^2}{u^2+2m}dt^2 + 4(u^2+2m) du^2+(u^2+2m)^2( d\\theta^2 + \\sin^2 \\theta \\, d\\varphi^2) \\\\\n&-\\infty < u < 0 \\, ,  0 < u < \\infty \\ ,\n\\end{eqnarray}\nobtained in two steps, first changing the Schwarzschild radial coordinate $ \\rho = u^2 + 2m, \\, u > 0$ and then, allowing the new  coordinate to take also values  in $-\\infty < u < 0 $, producing two copies of the exterior Schwarzschild metric. But this  metric is degenerate, $ det ( g_{\\mu\\nu})=0$ at $u=0$, and the absence of good coordinates to cover the union of the two exterior spaces was not considered.\\\\\n \nRecently, Katanaev \\cite{KATA1}  obtained this metric as a solution of Einstein's equations with a $\\delta$-type energy momentum tensor corresponding to a point particle at $ r= 0$ in isotropic coordinates. \n But he  pointed out  the incompleteness of its solution, in the same sense as in the original Einstein-Rosen metric, because he did not solve the problem of the coordinate singularity at $u=0$.  Despite that, he demonstrated the complete bridge passability in \\cite{KATA2}. \nGuendelman et al.  \\cite{GUEN1} claimed that the correct interpretation of the incomplete original Einstein-Rosen bridge has as source  a generalized function (a distribution)  with support on the tridimensional bridge, interpreted as a light-like thin shell (L-L brane). Using Eddington-Finkelstein-like coordinates they obtained an \n extension with discontinuous first derivatives in the metric, producing the distributional character of the source. The complete passability  of  the bridge, in both senses, and the existence of closed time-like geodesics was discussed in \\cite{GUEN2}.\\\\\n \nWe must contemplate the notion of extension of  a space-time, and consider \nthe possibility of the existence of many different extensions. One can get a maximal extension of a metric, in the sense that all geodesics are complete, but there can exist more than one maximal extension (see for example the nice book of Earman \\cite{EARM}).  In fact,   Katanaev commented in \\cite{KATA1} that he had to choose  between two possible solutions  for the  lapse function, and that having taken the discarded one he would have obtained the same conclusion as Guendelman et al. \\\\\n\n\nThe purpose of this work is to study the possibility of other extensions of the Einstein-Rosen space-time. To find them, it will be useful to take into account  that any two-dimensional metric (in our case the metric of the submanifolds $ \\theta , \\varphi $ constant), verifying some specific conditions, admits isothermal coordinates. So in section \\ref{ERBSec} we propose a method for constructing isothermal coordinates, that will be systematically used  to find all possible extensions of the original Einstein-Rosen bridge. We shall obtain first  the  Einstein-Rosen bridge with boundary ($ERb$), which is a non maximal extension of the original Einstein-Rosen metric.  It is also a subspace of the Kruskal-Szekeres space-time \\cite{KRUS}, which  has  been  considered as the  maximal extension of the original Einstein-Rosen bridge, but it is not the  only possibility. In section \\ref{maximal} we show two other possible extensions by doing a change of topology. One of them, according to Poplawski \\cite{POPLA}, should coincide with the aforementioned Guendelman et al.  extension, so we will study the other one. We call it the hyperbolic Einstein-Rosen bridge ($hER$), since it is diffeomorphic to the covering space of an hyperboloid, as we prove in Appendix A. In section \\ref{ED} we provide this extension with a differential structure, and obtain its corresponding metric in section \\ref{mher}. We find that in this extension the bridge is generated by isotropic geodesics and it is traverable, and furthermore, the first derivatives of the metric are continue on it, in contrast to the Guendelman's extension. However, the metric presents an unexpected singularity in one side, but, as in the Schwarzschild metric, it is concealed by a horizon, which in this case is the bridge itself. Finally, in section \\ref{source} we discuss which kind of physical process could generate our extension (or a part of it), and in section \\ref{conclusions} we summarize the main conclusions. In order to facilitate the reading, in Appendix B we gather the main definitions used in the paper.\n\n\\section{The Einstein-Rosen bridge with boundary $ERb$}\\label{ERBSec} \n\n{\\bf{Proposition 1}}: Let us consider a two-dimensional metric  of the  type $ ds^2 = g_{_{11}}(\\varphi_2) d \\varphi_{_{1}}^2 + g_{_{22}}(\\varphi_2) d \\varphi_{_2}^2$, with $ g_{11}g_{22} < 0$.\n Under the change of coordinates\n\\begin{eqnarray}  \n&& U = f(\\varphi_2) \\cosh \\varphi_{_1} \\, ,   V = f(\\varphi_{_2}) \\sinh \\varphi_{_1}  \\, ,\\mathrm{ if }\\,  \\    g_{22} > 0\\\\\n&& U = f(\\varphi_2) \\sinh \\varphi_{_1} \\, ,   V = f(\\varphi_{_2}) \\cosh \\varphi_{_1} \\, ,\\mathrm{ if } \\,  \\    g_{22} < 0 \\ ,\n\\end{eqnarray}\nit  may be written as $ ds^2 = \\Omega^2 ( - dV^2 + d U^2)  $ with : $ \\Omega^2 = \\frac{\\mid g_{_{11}}\\mid}{f^2} \\, ,\\  \\     f = C e ^{\\pm \\int \\sqrt{\\frac{ \\mid g_{_{22}}\\mid}{\\mid g_{_{11}}\\mid} }d\\varphi_{_2}}$, and $C$ a non null arbitrary constant.\\\\\n\nThe proof is simple. A straightforward calculation transforms the second form of the metric into the first form. \nLet us  apply this result to the metric  $ ds^2 =  -\\frac{u^2}{u^2+2m}dt^2 + 4(u^2+2m) du^2$ of the \n$t$-$u$  section of the  Einstein-Rosen bridge.\nWe must identify in this case  $\\varphi_1 = t\/k$, taking for $k$ a positive constant to be determined bellow, and  $ \\varphi_2 = u$, and substitute $g_{11}= -\\frac{k^2 u^2 }{u^2+2m}$, $ g_{22}= 4(u^2+2m)$.\nThe change of coordinates $ U = f_{_{E R}}(u) \\cosh (\\frac{t}{\\kappa}) \\, , \\    V = f_{_{E R}}(u) \\sinh (\\frac{t}{\\kappa})$ produces\n\\begin{eqnarray}\n& f_{_{E R}}= C e^{\\pm\\frac{u^2}{\\kappa}}u^{\\pm\\frac{4m}{\\kappa}}\\\\\n& \\Omega^2= \\frac{\\kappa^2}{C^2}\\frac{u^2}{u^2+2m} \\frac{1}{e^{\\pm\\frac{2u^2}{\\kappa}}u^{\\pm\\frac{8m}{\\kappa}}}\n\\end{eqnarray}\nin the region $ u > 0$, and we can extend it to the region $u<0$. Taking  $\\kappa = 4m$ and choosing sign $(+)$ we avoid the metric degeneration, and we get\nin the region  $ U^2-V^2 \\geq 0 $ the metric  \n\\begin{eqnarray}\n& ds^2 =\\frac{16 m^2}{C^2}\\frac{e^{-\\frac{u^2}{2m}}}{u^2+2m}\\left(- d V^2 + dU^2\\right)\\label{ERI}\\\\\n& U^2-V^2 =  C^2 e^{\\frac{u^2}{2m}}u^2\\label{E} \\ .\n\\end{eqnarray}\nThe second equation defines a monotonous  function  $u^2 = h_{ER}(U^2-V^2)$ of the variable $U^2-V^2$, that is derivable in the set $ERb= \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$. \nWe will use below  the inverse change given by $ u=\\mathrm{sig} (U)\\sqrt{h_{ER}(U^2-V^2)}  $.   The set $ U^2 - V^2 = 0 $ is the topological  boundary of the open  $ U^2 - V^2 > 0$, and corresponds to the coordinate axis  $u=0\\, (r=2m)$.\\\\\n\nThe constant $C$ may be determined in order to reach the exterior part of the Schwarzschild metric, obtaining $C= 1\/\\sqrt{2m}$. Using the $h_{ER}$ function defined above we can express the metric in the form\n\\begin{eqnarray}\n&  ds^2 =\\Omega^2_{ER} \\left(- d V^2 + dU^2\\right)\\label{ERI1}\\\\\n& \\Omega^2_{ER}= \\frac{32m^3}{h_{ER}(U^2-V^2)+2m}e^{-\\frac{h_{ER}(U^2-V^2)}{2m}}\n\\label{ERI2}  \\ .\n\\end{eqnarray}\n As in the Kruskal-Szekeres' extension of the Schwarzschild metric (podriem posar la refer\u00e8ncia de Kruskal \\cite{KRUS}), the isothermal coordinates  have solved the trouble at $r=2 m$, but in this case the extension does not cover the region $ U^2-V^2 < 0$. The set of points $ERb= \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$,  is a manifold with boundary  $ \\partial ERb = \\{ (U,V)\\in \\mathbb{R}^2  \\mid U^2-V^2 =0\\}-(0,0)$. \n The metric is well defined and derivable on $ERb$, and the boundary just corresponds to  the bridge, but it is a bridge to nowhere.\nThis manifold   is properly an extension of the Einstein-Rosen space, because it adds points to the space that were not covered by the Einstein-Rosen  coordinates. We shall call this extension the Einstein-Rosen bridge with boundary ($ERb$). If we stop here we would have a serious drawback because the radial isotropic geodesics   $ U+V= \\mathrm{ const. } \\, , \\  \\  U - V = \\mathrm{ const. } $  would be incomplete, as we show in Fig.\\ref{ERB}.\n\n\\section{Possible maximal extensions}\\label{maximal}\n\\subsection{The Kruskal-Szekeres space-time}\\label{KS}\nIt is trivial to consider the previous extension as a subspace of the Kruskal-Szekeres space-time. In this way  the geodesic incompleteness  at the boundary disappears, but one gets   the singularity at $r=0$. Moreover, only space-like curves might connect the two isometric infinite spaces of the bridge, through the two-dimensional region $\\{ U=V=0$ and any angles $\\theta, \\varphi \\}$. Therefore, this extension of the Einstein-Rosen bridge is not traversable.\n  Usually, the original Einstein-Rosen metric is extended in this sense \\cite{MTW}, but this is not the only possiblity, so we come back to our manifold with boundary to look for alternative extensions. \n\\subsection{Two more extensions by changing the topology }\\label{free}\nIt is well manifest that the metric given in  (\\ref{ERI1}) and (\\ref{ERI2}) is the same over all the points of the boundary, then we can change the topology by identifying points of it, without come into terms with the field equations. (Let us recall a precedent in the sixties, when W. Rindler \\cite{RIND} identified the points $ \\{U,V, \\theta, \\varphi\\}$ and $\\{ -U,-V, \\theta, \\varphi\\}$ of the Kruskal-Szekeres space-time to construct the Elliptic Kruskal-Schwarzschild space-time. It has the virtue of giving, in the limit $m = 0$, the Minkowski space-time instead of two copies of it, as it happens in the K-S space).\n In principle, as we show in Fig. \\ref{ERB}, we have two possibilities:\nwe can identify points with the same coordinate $V$, i.e; pairs of points as $A$ and  $B$, and pairs as $C$ and $D$, or else we  can identify  points symmetric with respect to the center $(0,0)$, i.e; identify pairs as $A$ and $D$ and pairs as $B$ and $C$. The  change of topology may help to extend the space-time.\\\\\n\n \\begin{figure}[hbt]\\label{ERBcopy}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{ERbOriented.pdf}\n\\caption{\\small{Diagram of the $ERb$ extension (where the origin $(0,0)$ is excluded), showing how the incompleteness of the  isotropic geodesics (red lines) may be in principle avoided with a change of topology by gluing points $ A\\equiv B , C\\equiv D $ , and choosing the appropriated light cones orientations.}}\\label{ERB}\n\\end{figure}\n\nIn the next section {\\bf{we shall use the first identification of points of the $ERb$ manifold ($ A\\equiv B\\, , \\, C\\equiv D$)}},  and    provide the quotient space $ERb\/\\hspace{-0.17 cm}\\sim$   with a differentiable structure, obtaining what we have called the hyperbolic Einstein-Rosen bridge.\nThe set formed by all the identified points  defines {\\bf{ the bridge $B$ connecting}}  the two sheets  of the Einstein-Rosen space. The other possible identification of points  has been commented by Poplawski \\cite{POPLA}\nand associated to the extension of the Einstein-Rosen  bridge described by Guendelman et al. \\cite{GUEN1}, which has  a $LL$-light brane installed in the  bridge. \\\\\n\nThe following two remarks will help to understand the rest of the paper. Firstly, we  must  realize that   the manifold $ERb\/\\hspace{-0.19 cm}\\sim$  obtained by gluing points will have a different differentiable structure than $ERb$. This means that the   coordinates to be introduced  in a neighbourhood of the bridge $B$ cannot be linked to the coordinates of the $ERb$ manifold of section \\ref{ERBSec} by a diffeomorphism. The second remark refers to the orientation of the light cones. Since we are going to identify events $A$ and $B$, if we choose in the region $U > 0$ the orientation down-up for the light cone (as shown in Fig. \\ref{ERB}), then  {\\bf{ we must choose the  up-down orientation  in the region $U < 0$}}, in order  to make topologically possible that a  light ray can pass through the bridge.\n\n\n\\section{A differentiable structure for the quotient space $ERb\/\\hspace{-0.17 cm}\\sim$}\\label{ED}\n\nIn order to give a differentiable structure to the quotient space  $ERb\/\\hspace{-0.17 cm}\\sim$ considered in \\ref{free},\n   we must introduce a system of  compatible coordinates covering all  the space.  We have got it  with only a pair of coordinate systems  that convert  $ERb\/\\hspace{-0.17 cm}\\sim$  into a manifold  diffeomorphic to the covering space of an hyperboloid of revolution  with signature $(- \\ +)$. We deviate to the Appendix A the  description of this covering space in  isothermal coordinates, denoted as $(\\bar U , \\bar V)$, whose values define the set $\\bar H = \\{(\\bar U , \\bar V ) \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $ showed in the right hand of Fig. \\ref{cordenadesU1V1}-\\ref{entorncopy1}. The throat of the hyperboloid generates in the covering space the hyperbola\n  $\\bar J =\\{(\\bar U,\\bar V)\\mid  \\bar U^2 - \\bar V^2= D^2\\, , \\bar U > 0\\}$, which will correspond to the bridge.  \n  We start by introducing a map \n $P_1: \\Gamma_1 \\subset ERb \\, \\rightarrow  \\bar H_1\\subset \\bar H $, defined on the interior of $ERb$:  $\\Gamma_1 = \\{(U,V)\\in \\mathbb{R}^2 \\mid U^2-V^2 > 0\\}$,\nand giving values over the covering of the hyperboloid minus the covering of its throat: $ \\bar H_1=\\{(\\bar U_1,\\bar V_1)\\mid  \\bar U_1^2 - \\bar V_1^2  >  0 \\, , \\bar U_1 > 0\\} \\setminus\\bar J$. We define it as follows\n\\begin{eqnarray}  \\label{co1}\n &P_1(U,V)=(\\bar U_1 , \\bar V_1) \\\\\n &\\bar U_1 =  DS\\left(\\mathrm{sig} (U) \\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{\\mid U \\mid}{\\sqrt{U^2-V^2}}\\\\\n &\\bar V_1 =  DS\\left(\\mathrm{sig} (U) \\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{V}{\\sqrt{U^2-V^2}} \\ ,\n\\end{eqnarray}\nwhere $D$ is an arbitrary positive constant and $S(\\cdot)$ is a  positive, smooth and monotonous increasing function verifying $S(0)=1$. The values of this function allow the {\\bf{introduction of the coordinate system   $(\\bar U_1 , \\bar V_1)$}} over the subset $\\Gamma_1\\subset ERb\/\\hspace{-0.17 cm}\\sim$. In Appendix A we prove that, with this definition, the assignation $(\\bar U_1,\\bar V_1)=(\\bar U,\\bar V)$ is a diffeomorphism between our manifold minus the bridge and the covering of the hyperboloid minus the throat\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{U1V1.pdf}\n\\caption{\\small{Map $P_1: \\Gamma_1\\rightarrow \\bar H_1$. It introduces the coordinates $(\\bar U_1 , \\bar V_1)$ over $ERb\/\\hspace{- 0.17 cm}\\sim$  excluding the bridge, which is represented as an hyperbola (with point $(D,0)$ removed) in these coordinates.}}\\label{cordenadesU1V1}\n\\end{figure}\n\nTo complete the differentiable  structure we need coordinates for a neighbourhood of the bridge $B$.\nTo this end, we consider the map $P_2: \\Gamma_2 \\subset ERb \\rightarrow  \\bar H_2 \\subset \\bar H$, with  $\\Gamma_{2}= \\{(U,V)\\mid  0 \\leq    U^2 -V^2 < \\epsilon^2 \\} - (0,0)$, defined as\n \\begin{equation} \\label{U2V22}\n P_2(U,V)=(\\bar U_2 , \\bar V_2) = \\left( \\sqrt{V^2+D^2}+\\frac{U^2-V^2}{U}\\, \\ , V\\right) \\ \n\\end{equation}\nThe bridge $B$ is mapped onto $\\bar J -(D,0)$,  where $\\bar J $ is the hyperbola defined above. The image set $\\bar H_2= P_2(\\Gamma_2)$ is now a neighborhood of the set $\\bar J -(D,0)$, defined as\n \\begin{eqnarray}\n  & \\bar H_2 =   \\{(\\bar U_2,\\bar V_2)\\mid   \\bar U_{L}(\\bar V_2) < \\bar U_2 < \\bar U_{R}(\\bar V_2)  \\} -(D,0)\\label{neighborhood} \\\\\n & \\bar U_{L}(\\bar V_2)= \\sqrt{\\bar V_2^2 + D^2 }-  \\frac{\\epsilon^2}{\\sqrt{\\bar V_2^2+\\epsilon^2}}\\, \\ ,  \\ \n  \\bar U_{R}(\\bar V_2)= \\sqrt{\\bar V_2^2 + D^2 }+ \\frac{\\epsilon^2}{\\sqrt{\\bar V_2^2+\\epsilon^2}} \\ ,\n \\end{eqnarray}\n  as shown in the right hand of Fig.  \\ref{entorncopy1}. This map  verifies $ P_2(-b,b)= P_2(b,b)$ and $ P_2(-b,-b)= P_2(b,-b)$ for any $b >0$, i.e, it is not injective over the boundary $ \\partial ERb$ defined in \\ref{ERBSec}.  The set $\\Gamma_2 \\setminus \\partial ERb $ is formed by two disjoint open subsets $ \\{R_i\\}_{ i= 1,2}  \\subset \\Gamma_2 \\setminus \\partial ERb$, and $P_2$ maps them bijectively onto two open subsets $ \\{\\bar R_i\\}_{ i= 1,2}  \\subset  \\bar H_2\\setminus\\ \\bar J $, i.e,   $P_{2}( R_i) = \\bar R_i$ as shown in Fig. \\ref{entorncopy1}. \n We shall need the coordinates of the sets $R_i = P_{2}^{-1}( \\bar R_i)$ as functions of the coordinates of $\\bar R_i$.   One gets  \n \\begin{eqnarray}\n && P_{2}^{-1}(\\bar R_1)= \\{( U_{-}, V)\\}\\, \\ \\ , \\, \\ \\  P_{2}^{-1}(\\bar R_2)= \\{(U_{+},V)\\}\\label{pedos}\\\\\n && U_{+} = \\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} - \\varphi ) \\,  \\   , V=\\bar V_2 \\label{antecedent+}\\\\\n&& U_{-} = -\\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} + \\varphi ) \\,  \\   , V=\\bar V_2 \\, \\ , \\ \\varphi = \\sqrt{\\bar V_2^2+ D^2}-\\bar U_2\\label{antecedent-} \\ .\n \\end{eqnarray}\nThe non-injectivity of $P_2$ over the boundary $ \\partial ERb$\n will make possible  to assign the same coordinates to pairs of points like  $A , B $ and  $C , D$. By identifying the pairs  $(-b,b) , (b,b)$ and  $(-b,-b) , (b,-b)$ of the set $\\Gamma_2, \n$ shown in the left part of Fig. \\ref{entorncopy1}, we obtain a neighborhood of the bridge of the quotient space $\\mathrm{ERb} \/\\hspace{-0.17 cm}\\sim$.\\\\\n\n \\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{P2_3.pdf}\n\\caption{ Map $P_2 :  \\Gamma_{2}\\rightarrow \\bar H_2$. It maps   a neighborhood of the bridge $B$, formed by the set of identified points,  onto a neighborhood of the  covering of the throat of the hyperboloid, represented by the hyperbola in green.  The values of this function define the coordinates  $( \\bar U_2 , \\bar V_2)$.}\\label{entorncopy1}\n\\end{figure}\n\nWith this application we can proceed to the {\\bf{introduction of the coordinate system $(\\bar U_2 , \\bar V_2)$}}\nin a  neighborhood $\\Gamma_2\/\\hspace{-0.17 cm}\\sim \\, \\ \\subset ERb \/\\hspace{-0.17 cm}\\sim$  of the bridge of  the quotient space  as follows. To the pair of  points $ (-V,V) \\equiv (V,V)$ we assign  coordinates $(\\bar U_2 , \\bar V_2) = ( \\sqrt{ V^2+ D^2}, V)$. To  points such that  $U^2 - V^2 > 0$ we assign coordinates  $(\\bar U_2 , \\bar V_2) = (\\sqrt{V^2 + D^2} +\\frac{U^2-V^2} {U}, V)$. With the next proposition we shall provide a differentiable structure to the quotient space  $ERb \/\\hspace{-0.17 cm}\\sim$. \\\\\n  \n{\\bf{Proposition 2}}. The  coordinate systems   $(\\bar U_1 , \\bar V_1) $ and $(\\bar U_2 , \\bar V_2) $ over the quotient space $ERb \/\\hspace{-0.17 cm}\\sim$ are compatible, i.e., the function change of coordinates is a diffeomorphism.\\\\\n\nTo prove it we have to consider the common regions of the domains of $P_1$ and $P_2$, that is $R_1\\cup R_2$. It is enough to prove it in one of them, since in the other one the procedure would be analogue. We choose $R_2$, as shown in Fig. \\ref{entorncopy}. So, we shall choose  $U =U_{+}=  \\frac{1}{2}(\\sqrt{\\varphi^2 +4 \\bar V_2^2} - \\varphi ), \\bar V_2 = V$ given in (\\ref{antecedent+}), and\nwe can express $V\/U$ and $U^2-V^2$ as functions of $(\\bar U_2 , \\bar V_2)$\n\\begin{eqnarray}\n& \\frac{V}{U}= q ( \\bar U_2,\\bar V_2) \\, \\ , \\  q ( \\bar U_2,\\bar V_2)= \\frac{2\\bar V_2}{- \\varphi +\\sqrt{\\varphi^2  + 4 \\bar V_2^2} }\\\\\n& U^2 -V^2 = p ( \\bar U_2,\\bar V_2)\\, \\ , \\  p ( \\bar U_2,\\bar V_2)= \\frac{\\varphi}{2}\\left(\\varphi- \\sqrt{\\varphi^2 + 4\\bar V_2^2 } \\right)\\label{u2mev2} \\ .\n\\end{eqnarray}\nThe functions $p ( \\bar U_2,\\bar V_2)$  and $q ( \\bar U_2,\\bar V_2)$ are both  differentiable and\n moreover $|q ( \\bar U_2,\\bar V_2)| < 1$  because $\\varphi < 0$ in the region considered. This property will be used bellow. Let us introduce $ \\tilde S(\\bar U_2,\\bar V_2)= S\\left(\\sqrt{h_{ER}(p(\\bar U_2,\\bar V_2)}\\right)$, then from equation (\\ref{co1})  we have\n\\begin{eqnarray}\n& \\bar U_1^2 -\\bar V_1^2 = D^2 \\tilde S(\\bar U_2, \\bar V_2)^2\\label{una}\\\\\n&  \\frac{\\bar V_1}{\\bar U_1}= q( \\bar U_2,\\bar V_2)= \\frac{V}{U}\\label{dues}  \\ ,\n\\end{eqnarray}\n and by  combining (\\ref{una}) and (\\ref{dues})  one gets the change of coordinates \n\\begin{equation}\\label{1-2}\n \\bar U_1 =\\frac{D \\tilde{S}(\\bar U_2 , \\bar V_2) }{ \\sqrt{1-q ( \\bar U_2,\\bar V_2)^2}} \\, \\ , \\  \\bar V_1 =\\frac{D  \\tilde{S}(\\bar U_2 , \\bar V_2)  q (\\bar U_2,\\bar V_2)}{\\sqrt{1-q ( \\bar U_2,\\bar V_2)^2}}  \\ .\n\\end{equation}\n The change of coordinates (\\ref{1-2}) is differentiable because $ q^2 < 1$ in the domain considered. This completes the proof. \\\\\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{canvicoord12.pdf}\n\\caption{Diagram of the change of isothermic  coordinates  $ ( \\bar U_2 , \\bar V_2) \\rightarrow (\\bar U_1 , \\bar V_1)$ in the region $R_2$, that has been proved to be differentiable.}\\label{entorncopy}\n\\end{figure}\n\nWith the  coordinate  systems $ (\\bar U_1 , \\bar V_1) , (\\bar U_2 , \\bar V_2) $ we cover all  the space $\\bar H = \\{(\\bar U , \\bar V ) \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $. \n  {\\bf{Hence  we have given a differentiable structure to the quotient space}} $ERb \/\\hspace{-0.17 cm}\\sim$.  Regarding the geometrical characteristics of this space, the manifold  $ ERb\/ \\hspace{-0.17 cm}\\sim$ \n  is diffeomorphic to the covering space of the hyperboloid of revolution with the point $(D,0)$ removed, as proved in Appendix A. This is  why we have given the name \\textbf{hyperbolic Einstein-Rosen bridge (\\textit{hER})} to the manifold  $ ERb \/ \\hspace{-0.17 cm}\\sim$. Simply, to a point with coordinates $(\\bar U_1, \\bar V_1)$ we associate a point of the hyperboloid with the same coordinates $ (\\bar U , \\bar V)=(\\bar U_1, \\bar V_1)$, and similarly for points of $ERb\/\\hspace{-0.17 cm}\\sim$  with coordinates $(\\bar U_2, \\bar V_2)$. Therefore, the set $\\bar J - (D,0) \\subset \\bar H$, which is the throat of the hyperboloid, is diffeomorphic to the bridge $B \\subset ERb\/ \\hspace{-0.17 cm}\\sim$ that connects the sheets of the Einstein-Rosen space. \\\\\n \n \n \nIn the next section, using the function $P_{2}^{-1}$ introduced in  (\\ref{pedos}) we shall get  the metric  in a neighborhood of the bridge expressed in coordinates $(\\bar U_2 , \\bar V_2)$, and will be able to study the passability of the bridge.\n\n\\section{The  metric on the  hyperbolic Einstein-Rosen bridge $hER$}\\label{mher}\n\n\n   The metric space $(ERb, \\Omega^2_{ER} \\eta) $ with $ \\eta= - dV^2 + dU^2 $, obtained in section \\ref{ERBSec}, is a non maximal extension of the incomplete Einstein-Rosen bridge. In the previous section we have obtained, by identifying points of the bridge, a new manifold $ERb\/\\hspace{-0.17 cm}\\sim$  which is diffeomorphic to the covering space of an hyperboloid of revolution, and we have provided it with a differential structure. We shall explain how to obtain in this new space a maximal extension of the Einstein-Rosen bridge different from the well-known Kruskal-Szequeres black hole. \\\\\n   \nThe dipheomorphism $P_{2}^{-1}: \\bar R_2 \\subset \\bar H_2 \\setminus \\bar J \\rightarrow  R_2 \\subset \\Gamma_2$   will be used now to pullback the metric  on the open set $R_2\\subset\\Gamma_2$ over the open set $\\bar R_2$. \n  So, taking into account (\\ref{antecedent+}), we get\n\\begin{eqnarray}\n& d  U = A d \\bar U_2 + B d  \\bar V_2 \\,  , \\,  d V  = d \\bar V_{2} \\\\\n&  A = \\frac12\\left(1-\\frac{ \\varphi}{\\sqrt{\\varphi^2 + 4\\bar V_2^2}}\\right)\\label{A}\\\\\n&  B= - \\frac{1}{2}\\left(1-\\frac{ \\varphi}{\\sqrt{\\varphi^2 + 4\\bar V_2^2}}\\right)\\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} +\\frac{ 2 \\bar V_2}{\\sqrt{\\varphi^2 + 4 \\bar V_2^2}}\\label{B} \\ ,\n\\end{eqnarray}\nwith  $\\varphi = \\sqrt{\\bar V_2^2+ D^2}-\\bar U_2 $. Using these relations we get the metric in the region $\\bar R_2 \\subset \\bar H_2$, where $\\varphi < 0$,  shown in  Fig.  \\ref{entorncopy1}\n \\begin{eqnarray}\\label{metric2}\n &d\\bar s^2 = \\bar \\Omega^2(\\bar \\eta_{_{\\bar U_2 \\bar U_2}} d \\bar U_2^2+ \\bar \\eta_{_{\\bar V_2 \\bar V_2}} d\\bar V_2^2 +2 \\bar \\eta_{_{\\bar U_2 \\bar V_2}}d \\bar U_2 d \\bar V_2)\n\\\\\n&\\bar \\eta _{_{\\bar V_2 \\bar V_2 }}= B^2 -1 \\, \\ , \\ \\bar \\eta_{_{\\bar U_2 \\bar U_2} }=A^2 \\, \\ , \\, \\bar \\eta_{_{\\bar U_2 \\bar V_2} } = AB \\ ,\n\\end{eqnarray}\nwhere the factor $\\bar \\Omega^2=\\Omega^2_{ER}(U^2-V^2) $ is obtained by substituting $U^2 - V^2 =U_+^2 - V^2= \\frac{\\varphi}{2}\\left(\\varphi- \\sqrt{\\varphi^2 + 4\\bar V_2^2 } \\right)$into the function $\\Omega^2_{ER}$ given in  (\\ref{ERI2}). \n The same functional form of the metric defined above in $\\bar R_2$ {\\bf{is extended}} now to the the region $ (\\bar J -(D,0)) \\cup \\bar R_1$ where $\\varphi \\geq 0$. \nThe extended metric is degenerate over the axis $\\bar V_2=0$ in the region $\\varphi > 0$ (the left hand of the bridge), because there we have $ A=B=0$, but it  is smooth over all the bridge $\\bar J -(D,0)$.\\\\\n\nThe components of the metric are analytic functions in the open set formed by subtracting the semi axis $\\varphi > 0 \\, , \\bar V_2 =0$ from the open set $\\bar H_2$, and so are the components of the Ricci tensor. On the other hand, by construction, the Ricci  tensor is null in the region $\\bar R_2$ (to the right of the bridge), since this region is diffeomorphic to the right hand of the Einstein-Rosen space-time, where the Ricci tensor vanishes. Therefore, according to the theorem of interior uniqueness of analytic functions, the Ricci tensor will also be null in the region $\\bar R_1$ (to the left of the bridge) minus the semi axis $\\varphi > 0 \\, , \\bar V_2 =0$. Then, the metric verifies the Einstein equations in empty space, except in the degenerate region. To ascertain the geometrical character of the degeneracy  we have considered the Kretschmann scalar, $\\mathrm{Kret}(\\bar U_2, \\bar V_2)=R_{abcd}R^{abcd}$ in the region $\\varphi > 0$, $\\bar U_2 < D$, to study the limit when $\\bar V_2$  tends to zero keeping $\\bar U_2$ constant.  We have found that it diverges as $ (D-\\bar U_2)^6\/\\bar V_2^{10} $. The calculation is straightforward and helped by the fact that the metric is the sum of two mutually independent submetrics.\n The semi-axis  $ \\varphi > 0 , \\bar V_2 = 0 $  is therefore a geometrical singularity of the space-time. \\\\\n\n\n\n\n\n\n We shall show that the bridge is a horizon that protects the right region ($\\varphi < 0$) from the singularity present in the left region ($\\varphi > 0$), as in the maximal extension of the Schwarzschild metric.\nWe start  studying the geometrical properties of the hyperbola $\\bar J -(D,0)$.\nA bridge is the possibility of communicating two separate zones. To ascertain if our hyperbola has this property  we shall construct the light cones over it, i.e., the tangent vectors to the pair of light rays at any point of $\\bar J$.\nIf we renounce to an affine parametrization we can  get  them   by considering the simpler metric $ ds^2 =\\bar \\eta_{_{\\bar U_2 \\bar U_2}} d \\bar U_2^2+ \\bar \\eta_{_{\\bar V_2 \\bar V_2}} d\\bar V_2^2 +2 \\bar \\eta_{_{\\bar U_2 \\bar V_2}}d \\bar U_2 d \\bar V_2$.  It is manifest that the transformation $ \\bar V_2 \\rightarrow - \\bar V_2$ is an isometry, therefore the geodesics in the $\\bar V_2 < 0$ region may be obtained by symmetry respect the $\\bar V_2=0$ axis.\nUsing the coordinate $\\bar V_2$ as parameter of the isotropic geodesics, we get an algebraic equation to determine the isotropic directions, which is given by\n\\begin{equation}\n  \\bar \\eta_{_{\\bar U_2 \\bar U_2}}\\left( \\frac{d \\bar U_2}{d \\bar V_2}\\right)^2 + 2 \\eta_{_{\\bar U_2 \\bar V_2}} \\frac{d \\bar U_2}{d \\bar V_2} + \\bar \\eta_{_{\\bar V_2 \\bar V_2}} =0  \\ .\n\\end{equation}\nThe  two solutions  define the following two first order differential equations\n\\begin{equation}\n\\frac{d \\bar U_2}{d \\bar V_2}= \\frac{1-B}{A}\\, \\  \\ ,  \\  \\   \\frac{d \\bar U_2}{d \\bar V_2}=-\\frac{1+B}{A}\\label{isoge12}  \\ ,\n\\end{equation}\nwhere $A$ and $B$ are defined in (\\ref{A}), (\\ref{B}). On the bridge it is verified $\\varphi =0$, so we have \n\\begin{eqnarray}\n&& \\frac{1-B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} \\, \\  \\ , \\    -\\frac{1+B}{A}\\mid_{\\varphi =0} =  \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}}-4 \\,  \\     \\mathrm{if}\\,  \\  \\bar V_2 > 0\\\\\n&& \\frac{1-B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}}+4 \\, \\  \\ , \\    -\\frac{1+B}{A}\\mid_{\\varphi =0} = \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} \\,  \\     \\mathrm{if}\\,  \\  \\bar V_2 < 0 \\ ,\n \\end{eqnarray}\nand it follows immediately that the parametrization  $( \\bar U_2 = \\sqrt{\\bar V_2^2 + D^2}, \\bar V_2 = \\bar V_2)$ of the  bridge $\\bar J-(D,0)$   is a light ray. One of the two isotropic vectors at any point  $ p \\in \\bar J-(D,0)$ is the tangent vector to the bridge $K_1= (\\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}},1 )$,  while  the other one is clearly directed from right to left:  $K_2 =( -4 + \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} , 1) $ in the $\\bar V_2 > 0 $ region, and from left to right $K_2 =( 4 + \\frac{\\bar V_2}{\\sqrt{\\bar V_2^2 + D^2}} , 1) $ in the $\\bar V_2 < 0 $ zone . The couple $(K_1,K_2) $ define the light cone at each point of the bridge.  They show that  in the region $\\bar V_2 > 0$  the matter can traverse the bridge only in right to left direction; by contrary  in the region $\\bar V_2 <  0$  matter only can traverse it from left to right. Therefore we conclude that the bridge is traversable.\\\\\n\nFig. \\ref{Fi:condellum1.pdf} shows the representation of some isotropic geodesics in a neighbourhood of the bridge and the light cones, which show the possible transit directions. These directions agree with the orientation of the  light cone  chosen  at the end of  section \\ref{free}. Let us point out that if we  had started  obtaining the metric on $\\bar R_1\\subset \\bar H_2$ by choosing the value $U = U_{-}$ given in  (\\ref{antecedent-}), instead of $ U = U_{+}$, and extending it to $\\bar R_2$, the bridge would be   passable too but with inverted transit directions. This option would correspond to the opposite choice of the light cone orientation in  section (\\ref{free}).\\\\\n\n\\begin{figure}[hbt] \\label{consher}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{cons3.pdf}\n\\caption{Diagram showing the light rays (red lines) in  a neighbourhood of the bridge using   coordinates $(\\bar U_2, \\bar V_2) $. The central hyperbola (green line), minus $(D,0)$,  is the bridge formed by two interrupted  light rays. The light cones traced on the bridge show the possible directions of transit for light and matter. All  the light  rays have been obtained using  $Mathematica$.}\\label{Fi:condellum1.pdf}. \n\\end{figure}\n\nRegarding the point $(D,0)$, it is a conical singularity and is not included in the space-time. If it had been included in the new manifold, the metric would be discontinuous at that point, and equations (\\ref{isoge12}) would be of the form $\\frac{d\\bar U_2}{d\\bar V_2}= f(\\bar U_2,\\bar V_2)$ with $f(\\bar U_2,\\bar V_2)$ discontinuous at $(D,0)$, no satisfying the existence theorem. \\\\\n\n  An interesting characteristic of  the semi-axis  $( \\varphi > 0 , \\bar V_2 = 0 )$, where the metric is degenerate, is that it cannot be connected by light rays to any point of the space time, as it  is manifest in Fig. \\ref{Fi:condellum1.pdf}. Moreover, light and matter coming from the region $\\varphi > 0 , \\bar V_2 > 0$ cannot traverse to the right side. Therefore, the bridge acts as a horizon and the singularity satisfies the cosmic censorship, since it is causally disconnected from the right side. This singularity could have been avoided by obtaining the metric on the left  and right regions (i.e., $\\bar R_1$ and $\\bar R_2$) independently, as  pull-backs  of the known metrics in regions $ R_1$ and $R_2$. But the metric obtained in this way is not continuous on the bridge, and it cannot be considered an actual extension of the Einstein-Rosen space.\\\\\n\n\\section{The problem of the source of the $hER$ metric space.} \\label{source}\n\n Finally, let us add some comments about  the source of the $hER$ metric space.  According to Katanaev,  a point particle is the  source of the incomplete  metric (\\ref{ER}). However,  the point particle is at $r=0$ in isotropic coordinates, which is at  infinite distance of the bridge\nwhere the curvature of this space-time is enormous. This fact is interpreted by Katanaev as repulsive gravity in the left side of the bridge, \n but for us this is  an uncomfortable conclusion that prompts to look for a different interpretation.\\\\\n \nIn the case of the black-hole extension of the Schwarzschild metric,  an illuminating finding was its relation with the {\\bf{continued gravitational contraction}} of an  spherical star made of pressure-less  matter \\cite{OPENSNY}: only the exterior of the collapsing star was  a part of the Kruskal-Szekeres extension. In our case, it is an interesting  issue  to find the kind of collapse whose exterior would correspond to a part of the hyperbolic  Einstein-Rosen bridge. The spherical star must contract, as shown in Fig. \\ref{colapse}, until the free surface reaches the bridge, and then it must begin to expand again, but now in the other side of it. (Only  negative pressures could produce this sort of collapse. Newtonian gravity predicts a negative gravitational contribution to the pressure, but a correct extension of this gravitational effect has not been accomplished   so far). The orientation of the light cone in Fig. \\ref{colapse} shows that  the source in her expansive phase, in the left part of the bridge, can not be observed from the right of it, and no material particle evolving in the left part can get the bridge.  In other words, an observer in the right side could not distinguish this process from an Oppenheimer-Snyder collapse creating a black-hole. However, after crossing the horizon, the star would not finish in a singularity, but would begin to expand again. In fact, no singularities would be present, since the degenerate region $(\\bar U_2 < D , \\bar V_2=0)$ is now occupied by the collapsing fluid.\\\\\n\nThis interpretation would explain why gravity is stronger  on the bridge, because any point of it is  connected by a light ray (the  bridge) to the collapsing  body at the point of maximum contraction. By contrast, the curvature tends to zero  at great distances to the left of the bridge, because  the light rays connect with points at the expansion phase. \n No point-particle would be present in this space-time, \n  only a part of the mathematical point-particle  Katanaev's  solution  would be  generated as the exterior of this collapse-expansion process.  \\\\\n\n\n\n\n\n\n\n  \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{con+colapse.pdf}\n\\caption{ Diagram of the proposed collapse process generating a part of the $hER$ space-time. The green line represents the bridge, generated by light rays.   The yellow  region is the source, contracting before and in expansion after passing the bridge.}\\label{colapse}\n\\end{figure}\n\n\\section{Conclusions} \\label{conclusions}\n\n In this work we have obtained a new extension ($hER$) of the original incomplete Einstein-Rosen bridge, different from the well-known Kruskal-Szekeres space-time and from the more recent one obtained by Guendelman et al. \\cite{GUEN1,GUEN2}. The manifold of $hER$ is diffeomorphic to the covering space of an hyperboloid of revolution, and the metric components are analytic functions in a neighborhood of the bridge with a geometrical singularity on the semi axis $\\bar V_2 = 0 , \\bar U_2 < D$. The bridge is formed by light rays, and is traversable unlike the bridge in the $KS$ extension. The $hER$ space-time differs from the one obtained by Guendelman et al. in three aspects: a) it has no source installed in the bridge, b) it has no closed time-like geodesics typical of whormholes with exotic sources, and c) it presents a singularity, though compatible with the cosmic censorship conjecture (it is causally disconnected from the opposite side of the bridge, so the bridge acts as a horitzon). Finally, we have discussed the problem of the source of this space-time, and proposed a gravitational collapse as in the black-hole case but with two phases: contraction in one side and expansion in the other. It would generate a part of the $hER$ metric in the exterior of the collapse, avoiding the singularity. We leave the deep analysis of this process for future work.\\\\\n\n\n{\\it Acknowledgments.--} P. B. is supported by a Ph.D. fellowship, Grant No.  FPU17\/03712. M.P. thanks support by the Spanish \"Ministerio de Economia y Competitividad (sic)\" and the \"Fondo Europeo de Desarrollo Regional\" MINECO-FEDER Project No. PGC2018-095251-B.100.\n\n\\section*{Appendix A: Diffeomorphism with the covering space of the hyperboloid}\\label{HER}\nWe shall obtain the expression in isothermal coordinates of the covering space of a two dimensional hyperboloid with signature $(-,+)$ that has been used in section \\ref{ED}. \nThe manifold  of the  two-dimensional sections with ($\\theta , \\varphi$) constant  of the Einstein-Rosen metric given in (\\ref{ER}) could be considered, excluding the section $u=0$, as  a surface of revolution (for example the one sheet hyperboloid) if we take $t$ as the angle of revolution. This is only possible if we consider the covering space of the hyperboloid, $\\bar H,$ by extending  to all $\\mathbb{R}$ its angular coordinate, defined in $(0, 2\\pi)$.  We shall do that  and   our quotient space  $ERb\/\\hspace{-0.17 cm}\\sim$ defined in section \\ref{ED} will be   \n diffeomorphic to the covering  surface $\\bar H$.\\\\\n \n {\\bf{Description of a one sheet hyperboloid with signature $ ( - \\, \\ +)$}}. We consider in the space  $\\mathbb{R}^3$ the metric $ ds^2 = -dx^2 -dy^2 +dz^2 $, and the surface of an hyperboloid defined  by  $ \\frac{x^2}{q^2} +\\frac{y^2}{q^2}-\\frac{z^2}{p^2}=1$, with $ p > q $. It can be parametrizated as: $X(\\varphi_1 , \\varphi_2)= ( q \\cosh \\varphi_2  \\cos \\varphi_1, q \\cosh \\varphi_2  \\sin \\varphi_1 , p \\sinh  \\varphi_2  ) $,  with $ 0  < \\varphi_1 < 2\\pi \\, , \\  - \\infty < \\varphi_2 < \\infty $, and one obtains  the  metric  $ds^2 = -q^2 \\cosh^2\\varphi_2 \\,d  \\varphi_1^2 + ( p^2 \\cosh^2\\varphi_2 - q^2 \\sinh^2\\varphi_2 ) d \\varphi_2^2$. Now, using  the\n  {\\bf{Proposition 1}} we  can introduce  isothermal coordinates \n  $ \\bar U = f_{{H}}(\\varphi_2) \\cosh (\\varphi_1)$, \n   $ \\bar V = f_{{H}}(\\varphi_2) \\sinh (\\varphi_1)$, \n   with $ f_{H}(\\varphi_2)= D \\exp(\\int_0^{\\varphi_2} \\sqrt{1-\\frac{p^2}{q^2}  \\tanh^2s}\\,  ds )$ and $D$ a  positive arbitrary constant, and  express the  metric as $d\\bar s^2 = q^2 \\cosh^2 (\\varphi_2) f_{H}^{-2}(\\varphi_2) ( -d \\bar V^2 + d\\bar U^2)$.\n The equation $ \\bar U^2 -\\bar V^2 =  f_{H}^2(\\varphi_2)$ defines  a monotonous increasing  function,  $ \\varphi_2 = h_{_{H}}(\\bar U^2-\\bar V^2)$ in the open set  $\\bar U^2-\\bar V^2>0 , \\bar U>0$. It  verifies: $  h_{H}(0)=-\\infty \\, , \\  h_{H}(D^2)= 0 \\, , \\ h_{H}(\\infty)=\\infty $.  In isothermal coordinates, \n the lines $\\varphi_1 = \\mathrm{const}.$ in the $( \\varphi_1 , \\varphi_2)$ plane  transform into the straight lines $ \\bar V = \\tanh(\\varphi_1)\\, \\bar U$ in the $(\\bar U , \\bar V)$ plane, for values $0 < \\varphi_1 < 2 \\pi$; \n  and the circles  $\\varphi_2 = \\mathrm{const}.$ into segments of  the hyperbolae $\\bar U^2 -\\bar V^2 = f_H^2(\\varphi_2)$.  In particular, the circle  $\\varphi_2 = 0$ transforms into a segment of\nthe hiperbola $\\bar J= \\{(\\bar U,\\bar V)\\mid \\bar U^2 -\\bar V^2 = D^2, \\bar U>0 \\}$ in isothermal coordinates. \\\\\n\n  {\\bf{The covering space $\\bar H$ of the  one sheet hyperboloid}}. \nThe covering space of a circle in $ \\mathbb{R}^2$, parametrized as $ X(\\varphi_1,\\varphi_2)=( a \\cos \\varphi_1, a \\sin \\varphi_1)$,  is the helix in $ \\mathbb{R}^3$, parametrized as $X(\\varphi_1,\\varphi_2)=( a \\cos \\varphi_1, a \\sin \\varphi_1 , b \\varphi_1)$. Using this analogy we extend  the coordinate $\\varphi_1$ to obtain the parametrization of the covering space of the one sheet hyperboloid:\n$ X(\\varphi_1 , \\varphi_2)= ( q \\cosh \\varphi_2  \\cos \\varphi_1, q \\cosh \\varphi_2  \\sin \\varphi_1 , p \\sinh  \\varphi_2  , \\  \\varphi_1 )$, with $- \\infty < \\varphi_1 < \\infty\\, , \\  - \\infty < \\varphi_2 < \\infty$.\n The expression in isothermal coordinates of this covering space can be obtained by extending  the range of values of the isothermal coordinates of the hyperboloid to the open set $\\bar H= \\bar U^2 - \\bar V^2 > 0\\, , \\bar U > 0$. \\\\\n\n\n{\\bf{Diffeomorphism $ERb\/\\hspace{-0.17 cm}\\sim\\, \\leftrightarrow \\bar H-(D,0)$}}. We start  with the non isothermal coordinates $ ( u , t ), (\\varphi_2 , \\varphi_1)$, used to express  the Einstein-Rosen metric (\\ref{ER}) and the covering space of the one sheet hyperboloid respectively. Let us consider the following diffeomorphisms: $\\varphi_2 =  u\\,,\\,\\varphi_1 = \\frac{t}{4m}$, for the open $u>0$ and $\\varphi_2 =  u\\,,\\,\\varphi_1 = -\\frac{t}{4m}$ for the open $u<0$. Now, by considering the inverse change $ u=\\mathrm{sig} (U)\\sqrt{h_{ER}(U^2-V^2)}$ (see section \\ref{ERBSec}) and the well-known  hyperbolic trigonometric relations, we  get the expressions   of this assignations in isothermal coordinate\n\\begin{eqnarray}\n&  \\bar U =  f_H\\left(\\mathrm{sig}(U)\\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{\\mid U \\mid}{\\sqrt{U^2-V^2}}\\label{barU1}\\,\\,\\,\\\\\n&  \\bar V =  f_H\\left(\\mathrm{sig}(U)\\sqrt{h_{ER}(U^2-V^2)}\\right)\\frac{V}{\\sqrt{U^2-V^2}} \\ .\\label{barV1}\n\\end{eqnarray}\nNote that they coincide with the expressions of the map $P_1$, given by \\eqref{co1}, if we define $S= f_{H}\/D$. If we had chosen a different surface of revolution diffeomorphic to the hyperboloid, we would had obtained a different function $S(\\cdot)$ with similar characteristics. That is why in the definition of $P_1$ we have generalized the function $S(\\cdot)$ to any positive, smooth and monotonous increasing  function verifying $S(0)=1$. Therefore, any point of the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ defined with the coordinates $(\\bar U_1,\\bar V_1)$ (all points except the bridge), can be related with the hyperboloid (or a similar surface of revolution) by the diffeomorphism $\\bar U_1=\\bar U\\,,\\,\\bar V_1=\\bar V$, covering all the surface except the throat $\\bar J$. The points of the bridge (and its neighbourhood) in the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ can be expressed with the coordinates $(\\bar U_2,\\bar V_2)$, defined in \\eqref{U2V22}. The bridge correspond to the set $B =\\{(\\bar U_2,\\bar V_2)\\mid  \\bar U_2^2 - \\bar V_2^2= D^2\\, , \\bar U_2 > 0\\}-(D,0)$, so it can be related to the throat of the hyperboloid by the simple diffeomorphism $\\bar U_2=\\bar U\\,,\\,\\bar V_2=\\bar V$. In the Proposition 2 we have proved that the coordinate systems $(\\bar U_1,\\bar V_1)$ and $(\\bar U_2,\\bar V_2)$ are compatible in the intersection of their domains, so there are no discontinuities. Therefore the manifold $ERb\/\\hspace{-0.17 cm}\\sim$ is diffeomorphic to the covering space of the one sheet hyperboloid $\\bar H$, after removing the point $(D,0)$.\\\\\n\n\\section*{Appendix B: Definitions} \\label{definitions}\n\nWe shall summarize the main definitions used in the paper in order to facilitate the reading.\\\\\n\n$ ERb = \\{(U,V) \\in \\mathbb{R}^2 \\mid U^2-V^2 \\geq 0\\} - (0,0)$ \\\\\n \n$ \\partial  ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid U^2-V^2 =0\\} - (0,0)$ \\\\\n\n $R_1\\subset ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 < \\epsilon^2, U < 0 \\} \\}$ \\\\\n \n $R_2 \\subset ERb =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 < \\epsilon^2, U > 0 \\} \\}$ \\\\\n \n $ \\Gamma_1 \\subset ERb \\, , \\, \\Gamma_1 =  \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 < U^2-V^2 \\} $\\\\\n \n  $ \\Gamma_2 \\subset ERb \\, , \\, \\Gamma_2 = \\{(U,V) \\in \\mathbb{R}^2 \\mid 0 \\le U^2-V^2 < \\epsilon^2 \\} - (0,0)$ \\\\\n\n \n $\\bar H = \\{(\\bar U , \\bar V )\\in \\mathbb{R}^2  \\mid  \\bar U^2 - \\bar V^2 > 0 \\, , \\bar U > 0\\} $ \\\\\n\n$ \\bar J \\subset \\bar H = \\{(\\bar U , \\bar V )\\in \\mathbb{R}^2  \\mid  \\bar U^2 - \\bar V^2 = D^2 \\, , \\bar U > 0\\} $ \\\\\n \n $ \\bar H_1 \\subset \\bar H \\, , \\, \\bar H_1 = \\{(\\bar U_1, \\bar V_1 )\\in \\mathbb{R}^2  \\mid   \\bar U_1^2 - \\bar V_1^2 > 0 \\, , \\bar U_1 > 0\\} \\setminus\\bar J$ \\\\\n \n    $ \\bar H_2 \\subset \\bar H \\, , \\ \\bar H_2 = \\{(\\bar U_2, \\bar V_2 )\\in \\mathbb{R}^2  \\mid \\bar U_L(\\bar V_2) <  \\bar U_2 < \\bar U_R(\\bar V_2)\\}- (D,0) $\\\\\n    \n $ P_1 : \\Gamma_1\\subset ERb \\rightarrow \\bar H_1 \\subset \\bar H $\\\\\n \n  $ P_2 : \\Gamma_2\\subset ERb \\rightarrow  \\bar H_2 \\subset \\bar H $\\\\\n  \n  \n  $\\bar R_1 =\\{(\\bar U_2, \\bar V_2 )\\in (\\bar H_2 - \\bar J ) \\mid  \\bar U_2 < \\sqrt{D^2 + \\bar V_2^2}\\}$\\\\\n  \n  $\\bar R_2 =\\{(\\bar U_2, \\bar V_2 )\\in (\\bar H_2 - \\bar J ) \\mid  \\bar U_2 > \\sqrt{D^2 + \\bar V_2^2}\\}$\\\\\n  \n  \n    $P_2(R_i) = \\bar R_i \\, , \\ i = 1, 2$\\\\\n    \n\n\n$\\varphi = \\sqrt{\\bar V_2^2 + D^2} - \\bar U_2 $\\\\\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nA {\\it dominating set} in a graph $G$ is a subset \nof vertices $S$ such that every vertex in $V(G)\\setminus S$\nis a neighbour of some vertex of $S$. The {\\it domination number} \nof $G$ is the minimum size of a dominating set of $G$. We \ndenote it by $\\gamma(G)$. This paper is devoted to the calculation \nof the domination number of complete grids.\n\nThe notation $[i]$ denotes the set $\\{1,2,\\ldots, i\\}$. If $w$ is a\nword on the alphabet $A$, $w[i]$ is the $i$-th letter of $w$, and \nfor every $a$ in  $A$, $\\vert w \\vert_a$ denotes the number of occurrences of \n$a$ in $w$ (i.e. $\\vert \\{ i\\in \\{1,\\ldots,\\vert w\\vert\\} \n: w[i]=a\\}\\vert$). \nFor a vertex $v$, $N[v]$ denotes the closed neighbourhood of $v$ \n(i.e. the set of\nneighbours of $v$ and $v$ itself). For a subset of\nvertices $S$ of a vertex set $V$ of a graph, we denote by $N[S] =\n\\bigcup_{v\\in S}N[v]$.  Note that $D$ is a dominating set of $G$ if\nand only if $N[D] = V(G)$. Let $G_{n,m}$ be the $n\\times m$ complete\ngrid, i.e. the vertex set of $G_{n,m}$ is $V_{n,m}:=[n]\\times [m]$, and two\nvertices $(i,j)$ and $(k,l)$ are adjacent if\n$|k-i|+|l-j|=1$. The couple $(1,1)$ denotes the bottom-left vertex of\nthe grid and the couple $(i,j)$ denotes the vertex of the $i$-th column\nand the $j$-th row. We will always assume in this paper that $n\\leq\nm$. \nLet us illustrate our purpose by an example of a dominating set of the\ncomplete grid $\\fG{24}{24}$ on Figure~\\ref{fig:24x24}.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\include{d24x24}\n  \\caption{Example of a set of size 131 dominating the grid $\\fG{24}{24}$\\label{fig:24x24}}\n\\end{figure}\n\nThe first results on the domination number of grids were obtained\nabout 30 years ago with the exact values of $\\gG2n$, $\\gG3n$, and $\\gG4n$\nfound by Jacobson and Kinch \\cite{JK} in 1983. In 1993,  \nChang and Clark~\\cite{CC} found those of $\\gG5n$\nand $\\gG6n$. These results were obtained analytically.\nChang~\\cite{Chang} devoted his PhD thesis to study the domination\nnumber of grids; he conjectured that this invariant\nbehaves well provided that $n$ is large enough.\nSpecifically, Chang conjectured the following:\n\n\\begin{conjecture}[\\cite{Chang}]\\label{conj}\n  For every $16\\le n \\le m$, $$\\gamma(G_{n,m}) = \\form{n}{m}.$$\n\\end{conjecture}\n\nObserve that for instance, this formula would give 131 for the domination number \nof the grid in Figure~\\ref{fig:24x24}. To motivate his bound, Chang proposed some constructions of\ndominating sets achieving the upper bound:\n\n\\begin{lemma}[\\cite{Chang}]\\label{lem:cgang}\n  For every $8\\le n\\le m$, $$\\gamma(G_{n,m}) \\le \\form{n}{m}$$\n\\end{lemma}\n\nLater, some algorithms based on dynamic programming were designed to\ncompute a lower bound of $\\gG{n}{m}$. There were numerous\nintermediate results found for $\\gG{n}{m}$ for small values of\n$n$ and $m$ (see \\cite{CCH,HHH,SP} for details). In 1995, Hare,\nHedetniemi and Hare \\cite{HHH} gave a polynomial time algorithm to\ncompute $\\gamma(G_{n,m})$ when $n$ is fixed. Nevertheless, this\nalgorithm is not usable in practice when $n$ hangs over $20$. Fisher\n\\cite{Fisher} developed the idea of searching for periodicity in the\ndynamic programming algorithms and using this technique, he found the\nexact values of $\\gG{n}{m}$ for all $n\\leq 21$. We recall these values\nfor the sake of completeness.\n\n\\begin{theorem}[\\cite{Fisher}]\n  For all $n \\le m$ and $n \\le 21$, we have:\n  $$\\gG{n}{m} = \\left\\{\n    \\begin{array}{ll}\n      \\frceil{m}{3} & \\ \\mbox{\\rm{if}} \\  n=1 \\\\*[0.1cm]\n      \\frceil{m+1}{2} & \\ \\mbox{\\rm{if}} \\  n=2 \\\\*[0.1cm]\n      \\frceil{3m+1}{4} & \\ \\mbox{\\rm{if}} \\  n=3 \\\\*[0.1cm]\n      m+1 & \\ \\mbox{\\rm{if}} \\  n=4  \\ \\mbox{\\rm{and}} \\  m=5,6,9\\\\*[0.1cm]\n      m & \\ \\mbox{\\rm{if}} \\  n=4  \\ \\mbox{\\rm{and}} \\  m\\neq 5,6,9\\\\*[0.1cm]\n      \\frceil{6m+4}{5} - 1 & \\ \\mbox{\\rm{if}} \\  n=5 \\ \\mbox{\\rm{and}} \\  m=7 \\\\*[0.1cm]\n      \\frceil{6m+4}{5} & \\ \\mbox{\\rm{if}} \\  n=5 \\ \\mbox{\\rm{and}} \\  m\\neq 7 \\\\*[0.1cm]\n      \\frceil{10m+4}{7} & \\ \\mbox{\\rm{if}} \\  n=6 \\\\*[0.1cm]\n      \\frceil{5m+1}{3} & \\ \\mbox{\\rm{if}} \\  n=7 \\\\*[0.1cm]\n      \\frceil{15m+7}{8} & \\ \\mbox{\\rm{if}} \\  n=8 \\\\*[0.1cm]\n      \\frceil{23m+10}{11} & \\ \\mbox{\\rm{if}} \\  n=9 \\\\*[0.1cm]\n      \\frceil{30m+15}{13} - 1 & \\ \\mbox{\\rm{if}} \\  n=10 \\ \\mbox{\\rm{and}} \\  m\\equiv_{13}10 \\ \\mbox{\\rm{or}} \\  m=13,16\n      \\\\*[0.1cm]\n      \\frceil{30m+15}{13} & \\ \\mbox{\\rm{if}} \\  n=10 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{13}10 \\ \\mbox{\\rm{and}} \\  m\\neq13,16 \\\\*[0.1cm]\n      \\frceil{38m+22}{15} - 1 & \\ \\mbox{\\rm{if}} \\  n=11 \\ \\mbox{\\rm{and}} \\  m=11,18,20,22,33 \\\\*[0.1cm]\n      \\frceil{38m+22}{15} & \\ \\mbox{\\rm{if}} \\  n=11 \\ \\mbox{\\rm{and}} \\  m\\neq11,18,20,22,33 \\\\*[0.1cm]\n      \\frceil{80m+38}{29} & \\ \\mbox{\\rm{if}} \\  n=12 \\\\*[0.1cm]\n      \\frceil{98m+54}{33} - 1 & \\ \\mbox{\\rm{if}} \\  n=13 \\ \\mbox{\\rm{and}} \\  m\\equiv_{33}13,16,18,19 \\\\*[0.1cm]\n      \\frceil{98m+54}{33} & \\ \\mbox{\\rm{if}} \\  n=13 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{33}13,16,18,19 \\\\*[0.1cm]\n      \\frceil{35m+20}{11} - 1 & \\ \\mbox{\\rm{if}} \\  n=14 \\ \\mbox{\\rm{and}} \\  m\\equiv_{22}7\\\\*[0.1cm]\n      \\frceil{35m+20}{11} & \\ \\mbox{\\rm{if}} \\  n=14 \\ \\mbox{\\rm{and}} \\  m\\not\\equiv_{22}7\\\\*[0.1cm]\n      \\frceil{44m+28}{13} - 1 & \\ \\mbox{\\rm{if}} \\  n=15 \\ \\mbox{\\rm{and}} \\   m\\equiv_{26}5\\\\*[0.1cm]\n      \\frceil{44m+28}{13} & \\ \\mbox{\\rm{if}} \\  n=15 \\ \\mbox{\\rm{and}} \\   m\\not\\equiv_{26}5\\\\*[0.1cm]\n      \\frfloor{(n+2)(m+2)}{5}-4 & \\ \\mbox{\\rm{if}} \\  n\\ge16 \\\\*[0.1cm]\n    \\end{array}\\right.$$\n  \n\\end{theorem}\n\nNote that these values are obtained by specific formulas for every $1\\leq n\\leq 15$ and by\nthe formula of Conjecture~\\ref{conj} for every $16\\le n \\le 21$. This\nproves Chang's conjecture for all values $16\\le n\\le 21$.\n\nIn 2004, Conjecture~\\ref{conj} has been confirmed up to an \nadditive constant:\n\n\\begin{theorem}[Guichard \\cite{Guichard}] For every $16\\le n \\le m$,\n  $$\\gamma(G_{n,m}) \\geq \\left \\lfloor \\frac{(n+2)(m+2)}{5} \\right\n  \\rfloor -9.$$\n\\end{theorem}\n\nIn this paper, we prove Chang's conjecture, hence finishing the\ncomputation of $\\gG{n}{m}$.  We adapt Guichard's ideas to improve the\nadditive constant from $-9$ to $-4$ when $24 \\le n \\le m$.  Cases\n$n=22$ and $n=23$ can be proved in a couple of hours using Fisher's\nmethod (described in~\\cite{Fisher}) on a modern computer.  They can be\nalso proved by a slight improvement of the technique presented in the\nnext section.\n\n\\section{Values of $\\gamma(G_{n,m})$ when $24 \\le n \\le m$}\n\nOur method follows the idea of Guichard \\cite{Guichard}. A slight\nimprovement is enough to give the exact bound.  \n\nA vertex of the grid $G_{n,m}$ dominates at most 5 vertices (its four\nneighbours and itself). It is then clear that $\\gG{n}{m} \\ge\n\\frac{n\\times m}{5}$. The previous inequality would become an equality\nif there would be a dominating set $D$ such that every vertex of\n$G_{n,m}$ is dominated only once, and all vertices of $D$ are of degree 4\n(i.e.  dominates exactly 5 vertices); in this case, we would have\n$5\\times|D| - n\\times m= 0$. This is clearly impossible (e.g. to\ndominate the corners of the grid, we need vertices of degree at most\n3). Therefore, our goal is to find a dominating set $D$ of $G_{n,m}$ such\nthat the difference $5\\times |D| - n\\times m$ is the smallest.\n\nLet $S$ be a subset of $V(G_{n,m})$. The \\emph{loss} of $S$ is $\\los{S}=5\n\\times \\vert S \\vert - \\vert N[S] \\vert$.\n\n\\begin{proposition} \\label{props}\n  The following properties of the loss function are straightforward:\n  \\begin{enumerate}[(i)]\n  \\item For every $S$, $\\los{S} \\ge 0$ (positivity), \\label{props:1}\n  \\item If $S_1\\cap S_2=\\emptyset$, then $\\los{S_1\\cup S_2} =\n    \\los{S_1} + \\los{S_2} + \\vert N[S_1] \\cap N[S_2] \\vert$, \\label{props:2}\n  \\item If $S'\\subseteq S$, then $\\los{S'}\\le \\los{S}$ (increasing\n    function), \\label{props:3}\n  \\item If $S_1\\cap S_2=\\emptyset$, then $\\los{S_1\\cup S_2} \\ge\n    \\los{S_1} + \\los{S_2}$ (super-additivity). \\label{props:4}\n  \\end{enumerate}\n\\end{proposition}\n\nLet us denote by $\\ell_{n,m}$ the minimum of $\\los{D}$ when $D$ is \na dominating set of $G_{n,m}$. \n\n\\begin{lemma}\\label{lemloss}\n  $\\gamma(G_{n,m}) = \\left \\lceil \\frac{n\\times m + \\ell_{n,m}}{5} \\right\n  \\rceil$\n\\end{lemma}\n\n\\begin{proof}\n  If $D$ is a dominating set of $G_{n,m}$, then $\\los{D} = 5 \\times \\vert\n  D \\vert - \\vert N[D]\\vert = 5 \\times \\vert\n  D \\vert - n\\times m$. Hence, by minimality of $\\ell_{n,m}$ and\n  $\\gamma(G_{n,m})$, we have $\\ell_{n,m}= 5\\times \\gamma(G_{n,m}) - n\\times m$.\n\\end{proof}\n\nOur aim is to get a lower bound for $\\ell_{n,m}$. As the reader can observe\nin Figure~\\ref{fig:24x24}, the loss is concentrated on the border of\nthe grid. We now analyse more carefully the loss generated by the\nborder of thickness $10$. \n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\begin{minipage}{0.9\\linewidth}\n    \\centering \\include{B}\n  \\end{minipage}\n  \\caption{The graph $\\fG{30}{40}$. The set $\\fBp{30}{40}$ is the set\n    of vertices filled in black. The set $\\fB{30}{40}$ is the set of\n    vertices filled in black or in gray.\\label{fig:bnm}}\n\\end{figure}\n\nWe define the border $B_{n,m} \\subseteq V_{n,m}$ of $G_{n,m}$ as the set of\nvertices $(i,j)$ where $i\\le 10$, or $j\\le 10$, or $i>n-10$, or\n$j>m-10$ to which we add the four vertices\n$(11,11),(11,m-10),(n-10,11),(n-10,m-10)$.  Given a subset $S\\subseteq\nV$, let $I(S)$ be the \\emph{internal vertices} of $S$, i.e. $I(S)=\n\\{v\\in S : N[v]\\subseteq S\\}$.  These sets are illustrated in\nFigure~\\ref{fig:bnm}.  We will compute $b_{n,m} = \\min_D \\los{D}$, where\n$D$ is a subset of $B_{n,m}$ and dominates $I(B_{n,m})$, i.e. $D\\subseteq\nB_{n,m}$ and $I(B_{n,m}) \\subseteq N[D]$. Observe that this lower bound $b_{n,m}$\nis a lower bound of $\\ell_{n,m}$. Indeed, for every dominating set $D$ of\n$G_{n,m}$, the set $D':=D\\cap B_{n,m}$ dominates $I(B_{n,m})$, hence $b_{n,m}\\leq\n\\los{D'}\\leq \\los{D}$. In the remainder, we will compute $b_{n,m}$ and we\nwill show that $b_{n,m} = \\ell_{n,m}$.\n\n\n\\bigskip\n\nIn the following, we split the border $B_{n,m}$ in four parts, $O_{m-12},\nP_{n-12}, Q_{m-12}, R_{n-12}$, which are defined just below.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\include{Q}\n  \\caption{The set $\\fP{19}$ (black and gray), the set of input vertices\n    (gray circles) and the set of output vertices (gray squares).\\label{fig:ppcpa}}\n\\end{figure}\n\nFor $p\\ge 12$, let $P_p\\subset B_{n,m}$ defined as follows : $P_p = ([10]\n\\times \\{12\\}) \\cup ([11] \\times \\{11\\} ) \\cup ([p] \\times [10])$. We\ndefine the \\emph{input vertices} of $P_p$ as $[10]\\times\\{12\\}$ and\nthe \\emph{output vertices} of $P_p$ as $\\{p\\} \\times [10]$. The set\n$P_p$, illustrated for $p=19$ in Figure~\\ref{fig:ppcpa}, corresponds\nto the set of black and gray vertices. The input vertices are the gray\ncircles, and the output vertices are the gray squares. Recall that in\nour drawing conventions, the vertex $(1,1)$ is the bottom-left vertex\nand hence the vertex $(i,j)$ is in the $i^{th}$ column from the left\nand in the $j^{th}$ row from the bottom.\n\n\\begin{figure}\n  \\centering\n  \\psset{unit=0.28cm}\n  \\begin{minipage}{0.9\\linewidth}\n    \\centering \\include{Z}\n  \\end{minipage}\n  \\caption{The sets $O_{m-12}$, $P_{n-12}$, $Q_{m-12}$ and $R_{n-12}$.\n    \\label{fig:pqrs}}\n\\end{figure}\n\nFor $n,m\\in \\mathbb{N}^*$, let $f_{n,m}: [n]\\times[m] \\to [m]\\times[n]$ be\nthe bijection such that $f_{n,m}(i,j) = (j,n-i+1)$. It is clear that the\nset $B_{n,m}$ is the disjoint union of the following four sets depicted\nin Figure~\\ref{fig:pqrs}: $\\fP{n-12}$, $\\fQ{m-12}=f_{n,m}(\\fP{m-12})$,\n$\\fR{n-12} = f_{m,n} \\circ f_{n,m} (\\fP{n-12})$ and $\\fS{m-12}= f_{n,m}^{-1}(\\fP{m-12})$. Similarly to $\\fP{n-12}$, the sets\n$O_{m-12}$, $Q_{n-12}$ and $R_{m-12}$ have input and output\nvertices. For instance, the output vertices of $Q_{m-12}$ correspond\nin Figure~\\ref{fig:ppcpa} to the white squares. Every set playing a\nsymmetric role, we now focus on $P_{n-12}$.\n\n\\bigskip\n\nGiven a subset $S$ of $V_{n,m}$, let the labelling $\\phi_S : V_{n,m} \\to\n\\{0,1,2\\}$ be such that \n\n$$\\phi_S(i,j) = \\left\\{ \n  \\begin{array}{l}\n    0 \\quad \\mbox{if $(i,j)\\in S$} \\\\\n    1 \\quad \\mbox{if $(i,j)\\in N[S]\\setminus S$} \\\\\n    2 \\quad \\mbox{otherwise}\n  \\end{array}\\right.$$\n\nNote that $\\phi_S$ is such that any two\nadjacent vertices in $G_{n,m}$ cannot be labelled~$0$ and~$2$.\n\nGiven $p\\ge 12$ and a set $S\\subseteq P_p$, the \\emph{input word} \n(resp. \\emph{output word}) of $S$ for $P_p$, denoted by $w^{in}(S)$\n(resp. $w^{out}_p(S)$), is the ten letters word on the alphabet\n$\\{0,1,2\\}$ obtained by reading $\\phi_{S}$ on the input vertices \n(resp. output vertices) of $P_p$.\nMore precisely, its $i^{th}$ letter is $\\phi_{S}(i,12)$ (resp. $\\phi_{S}(p,i)$). \nSimilarly, $O_p$, $Q_p$ and $R_p$ have also input and output words. \nFor example, the output word of $S\\subseteq O_p$ for $O_p$ is $w^{out}_p(f_{n,m}(S))$.\n\nAccording to the definition of $\\phi$, the input and output words\nbelong to the set $\\mathcal{W}$ of ten letters words on $\\{0,1,2\\}$ which\navoid $02$ and $20$. The number of $k$-digits trinary numbers without\n$02$ or $20$ is given by the following formula~\\cite{Fisher}:\n\\begin{equation}\n  \\frac{(1+\\sqrt{2})^{k+1}+(1-\\sqrt{2})^{k+1}}{2}\\label{eqn:trinary_words}\n\\end{equation}\nThe size of $\\mathcal{W}$ is therefore $|\\mathcal{W}| = 8119$.\n\n\nGiven two words $w,w' \\in \\mathcal{W}$, we define\n$\\mD^{w,w'}_p$ as the family of subsets $D$ of $\\fP{p}$ such that:\n\\begin{itemize}\n\\item $D$ dominates $I(P_p)$,\n\\item $w$ is the input word $w^{in}(D)$,\n\\item $w'$ is the output word $w^{out}_p(D)$.\n\\end{itemize}\n\nA relevant information for our calculation will be to know, for two\ngiven words $w,w'\\in \\mathcal{W}$, the minimum loss over all losses $\\los{D}$\nwhere $D\\in \\mD^{w,w'}_p$.  For this purpose, we introduce the $8119\\times\n8119$ square matrix $C_p$. For $w,w'\\in \\mathcal{W}$, let\n$C_p[w,w']=\\min_{D\\in\\mD^{w,w'}_p} \\los{D}$ where the minimum of the\nempty set is $+\\infty$.\n\nLet $w,w'\\in \\mathcal{W}$ be two given words. Due to the symmetry of $P_{12}$\nwith respect to the first diagonal (bottom-left to top-right) of the\ngrid, if a vertex set $D$ belongs to $\\Dww{12}$, then $D' = \\{(j,i) |\n(i,j)\\in D\\}$ belongs to $\\mathcal{D}^{w',w}_{12}$. Moreover, it is clear\nthat, always due to the symmetry, $\\los{D}=\\los{D'}$. Therefore, we\nhave $C_{12}[w,w'] = C_{12}[w',w]$ and thus $C_{12}$ is a\nsymmetric matrix.  Despite the size of $C_{12}$ and the size of\n$P_{12}$ (141 vertices), it is possible to compute $C_{12}$ in\nless than one hour by computer. Indeed, we choose a sequence of subsets\n$X_0=\\emptyset, X_1, \\ldots, X_{141}=P_{12}$ such that for every $i\\in\n\\{1,\\ldots, 141\\}$, $X_i\\subseteq X_{i+1}$ and $X_{i+1} \\setminus X_i\n= \\{ x_{i+1} \\}$.  Moreover, we choose the sequence such that for\nevery $i$, $X_i \\setminus I(X_i)$ is at most $21$. This can be done\nfor example by taking $x_{i+1}= \\min \\{ (x,y) : (x,y)\\in P_{12}\n\\setminus X_i \\}$, where the order is the lexical order. For $i\\in\n\\{0,\\ldots, 141\\}$, we compute for every labeling $f \\in \\mathcal{F}_i$, where\n$\\mathcal{F}_i$ is the set of functions $(X_i\\setminus I(X_i)) \\to \\{0,1,2\\}$,\nthe minimal loss $l_{i,f}$ of a set $S\\subseteq X_i$ which dominates\n$I(X_i)$ and such that $\\phi_S(v)=f(v)$ for every $v\\in X_i \\setminus\nI(X_i)$. Note that not every labeling is possible since two adjacent\nvertices cannot be labeled~$0$ and~$2$. The number of possible\nlabellings can be computed using formula~(\\ref{eqn:trinary_words}),\nand since $|X_i \\setminus I(X_i)|$ can be covered by a path of at most\n$23$ vertices, this gives, in the worst case, that this number is less\nthan $10^9$ and can be then processed\nby a computer. We compute inductively the sequence $(l_{i,f})_{f\\in\n  \\mathcal{F}_i}$ from the sequence $(l_{i-1,f})_{f\\in \\mathcal{F}_{i-1}}$ by dynamical\nprogramming, and $C$ is easily deduced from $(l_{141,f})_{f\\in\n  \\mathcal{F}_{141}}$.\n\n\\bigskip\n\nIn the following, our aim is to glue $P_{n-12},\nQ_{m-12},R_{n-12},$ and $O_{m-12}$ together. For two consecutive parts\nof the border, say $P_{n-12}$ and $Q_{m-12}$, the output word of\n$Q_{m-12}$ should be compatible with the input word of $P_{n-12}$.\nTwo words $w,w'$ of $\\mathcal{W}$ are \\emph{compatible} if the sum of their\ncorresponding letters is at most 2, i.e. $w[i]+w'[i]\\leq 2$ for all\n$i\\in [9]$. Note that $w[10] + w'[10]$ should be greater than 2 since\nthe corresponding vertices can be dominated by some vertices of $V_{n,m}\n\\setminus B_{n,m}$.\n\nGiven two words $w,w'\\in \\mathcal{W}$, let\n$\\ell(w,w')= \\vert \\{ i\\in [10] : w[i]\\ne 2 \\text{ and } w'[i]=0 \\}\n\\vert + \\vert \\{ i\\in [10] : w'[i]\\ne 2 \\text{ and } w[i]=0 \\} \\vert$.\n\n\\begin{lemma} \\label{lem:compatible}Let $D$ be a dominating set of $G_{n,m}$ and let us denote\n  $D_P = D\\cap P_{n-12}$ and $D_Q = D\\cap Q_{m-12}$. Then $\\ell(D\\cap\n  (P_{n-12} \\cup Q_{m-12}))= \\ell(D_P) + \\ell(D_Q) + \\ell(w,w')$, where\n  $w=w^{in}(D_P)$ and $w'=w^{out}_q(f_{n,m}^{-1}(D_Q))$. Moreover, $w$ and $w'$\n  are compatible.\n\\end{lemma}\n\n\\begin{proof}\n  By Proposition \\ref{props}(\\ref{props:2}), $\\ell(D\\cap (P_{n-12} \\cup Q_{m-12}))=\n  \\ell(D_P) + \\ell(D_Q) + \\vert N[D_P] \\cap N[D_Q]\\vert$. It suffice then to note that\n  $\\ell(w,w')= \\vert N[D_P] \\cap N[D_Q]\\vert$ to get $\\ell(D\\cap (P_{n-12} \\cup Q_{m-12}))=\n  \\ell(D_P) + \\ell(D_Q) + \\ell(w,w')$.\n  \n  In what remains, we prove that $w$ and $w'$ are compatible.  If\n  those two words were not compatible, there would exist an index $i\\in[9]$\n  such that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] + w^{in}(D_P)[i] > 2$.  Thus\n  at least one of these two letters should be a 2, and the other one\n  should not be 0.\n  \n  Suppose that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] = 2$ and note that\n  this means that the vertex $(i,13)$ is not dominated by a vertex in\n  $D_Q$. Since $D$ is a dominating set of $G_{n,m}$, every output vertex of\n  $Q_{m-12}$ except $(10,13)$ (and every input vertex of $P_{n-12}$\n  except $(10,12)$) is dominated by a vertex of $D_Q$ or by a vertex of\n  $D_P$. Thus $(i,13)$ should be dominated by its unique neighbour in\n  $P_{n-12}$, $(i,12)$. This would imply that $(i,12)\\in D$\n  contradicting the fact that $w^{in}(D_P)[i] \\neq 0$.\n\n  Similarly, if $w^{in}(D_P)[i] =2$, the vertex $(i,12)$ \n  is not dominated by a vertex in $D_P$, thus $(i,12)$ must be dominated \n  by the vertex $(i,13)\\in D$, contradicting the fact that\n  $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] \\neq 0$.\n\\end{proof}\n\nLemma~\\ref{lem:compatible} is designed for the two consecutive parts\n$P_{n-12}$ and $Q_{m-12}$ of the border of $G_{n,m}$. Its easy to see\nthat this extends to any pair of consecutive parts of the\nborder, i.e. $Q_{m-12}$ and $R_{n-12}$, $R_{n-12}$ and $O_{m-12}$,\n$O_{m-12}$ and $P_{n-12}$.\n\n\\bigskip\n\nWe define the matrix $8119\\times 8119$ square matrix $L$ which\ncontains, for every pair of words $w,w'\\in \\mathcal{W}$, the value $\\ell(w,w')$:   \n$$L[w,w']=\n\\begin{cases} \n  +\\infty \\text{ if $w$ and $w'$ are not compatible,}\\\\\n  \\ell(w,w') \\text{ otherwise.}\n\\end{cases}$$\nNote that $L$ is symmetric since $\\ell(w,w') = \\ell(w',w)$.\n\nLet $\\otimes$ be the matrix multiplication in $(\\min,+)$ algebra, \ni.e. $C = A \\otimes B$ is the matrix where for all $i,j$, $C[i,j] =\n\\min_k A[i,k] + B[k,j]$. \n\nLet $M_p=L \\otimes C_p$ for $p\\ge 12$. \n\nBy construction, $M_{n-12}[w,w']$ corresponds to the minimum\npossible loss $\\los{D\\cap P_{n-12}}$ of a dominating set $D\\subseteq\nV_{n,m}$ that dominates $I(P_{n-12})$ and such that $w$ is the output\nword of $Q_{m-12}$ and $w'$ is the output word of $P_{n-12}$.\n\n\\begin{lemma}\n  \\label{lem:bnm}\n  For all $24 \\le n \\le m$, we have \n  $$b_{n,m}\\geq \\min_{w_1,w_2,w_3,w_4\\in \\mathcal{W}} M_{n-12}[w_1,w_2] +\n  M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+ M_{m-12}[w_4,w_1].$$\n\\end{lemma}\n\\begin{proof} Consider a set $D \\subseteq B_{n,m}$ which dominates\n  $I(B_{n,m})$ and achieving $\\ell (D)=b_{n,m}$.  Let $D_P= D\\cap P_{n-12}$,\n  $D_Q= D\\cap Q_{m-12}$, $D_R= D\\cap R_{n-12}$ and $D_O= D\\cap\n  O_{m-12}$. Let $w_P$ ($w_Q$, $w_R$ and $w_O$, respectively) be the\n  input word of $P_{n-12}$ ($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$), and\n  $w'_P$ ($w'_Q$, $w'_R$ and $w'_O$) be the output word of $P_{n-12}$\n  ($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$).  By definition of $C_p$,\n  the loss of $D_P$ is at least $C_{n-12}[w_P,w'_P]$. Similarly, we\n  have $\\ell(D_Q)\\geq C_{m-12}[w_Q,w'_Q]$, $\\ell(D_R)\\geq\n  C_{n-12}[w_R,w'_R]$ and $\\ell(D_O)\\geq C_{m-12}[w_O,w'_O]$.\n  By the definition of the loss:\n  \\begin{align} \\ell(D) = {} & b_{n,m} \\nonumber\\\\ = {} & 5\\times\n    \\vert D \\vert - \\vert N[D]\\vert \\nonumber\\\\ = {} & \\ell(D_O)\n    +\\ell(D_P) +\\ell(D_Q) +\\ell(D_R) + L[w'_O,w_P] + L[w'_P,w_Q] +\n    L[w'_Q,w_R] + L[w'_R,w_O] \\nonumber\\\\ & \\text{~~ by\n      Lemma~\\ref{lem:compatible} and since $N[D_P] \\cap N[D_R] = N[D_Q] \\cap\n      N[D_O] =\\emptyset$} \\nonumber\\\\ \\ge {} & C_{m-12}[w_O,w'_O] +\n    C_{n-12}[w_P,w'_P] + C_{m-12}[w_Q,w'_Q] +\n    C_{n-12}[w_R,w'_R] \\nonumber\\\\ & + L[w'_O,w_P] +\n    L[w'_P,w_Q] + L[w'_Q,w_R] + L[w'_R,w_O] \\nonumber\\\\ \\ge {}\n    & M_{m-12}[w_O,w_P] + M_{n-12}[w_P,w_Q] +\n    M_{m-12}[w_Q,w_R] + M_{n-12}[w_R,w_O] \\nonumber \\\\\n    & \\text{~~ since $w'_O$ and $w_P$ (resp. $w'_P$ and $w_Q$, $w'_Q$\n      and $w_R$, $w'_R$ and $w_O$) are compatibles.} \\nonumber\n  \\end{align}\n\\end{proof}\n\n\\medskip\n\nAccording to Lemma~\\ref{lem:bnm}, to bound $b_{n,m}$ it would be thus\ninteresting to know $M_p$ for $p> 12$. It is why we introduce the\nfollowing $8119\\times 8119$ square matrix, $T$.\n\n\\begin{lemma}\n  There exists a matrix $T$ such that $C_{p+1}=C_p \\otimes T$\n  for all $p\\ge 12$. This matrix is defined as follows:\n  $$ T[w,w']=\n  \\begin{cases} \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in [10]$ s.t. $w[i]=0$ and $w'[i]=2$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in [9]$ s.t. $w[i]=2$ and $w'[i]\\ne 0$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $\\exists i\\in \\{2,\\ldots, 9\\}$\n      s.t. $w'[i]=1$, $w[i]\\ne 0$, $w'[i-1]\\ne 0$ and $w'[i+1]\\ne\n      0$}\\\\  \n    +\\infty \\ \\ \\ \\text{ if $w'[1]=1$, $w[1]\\ne 0$ and $w'[2]\\ne 0$}\\\\ \n    +\\infty \\ \\ \\ \\text{ if $w'[10]=1$, $w[10]\\ne 0$ and $w'[9]\\ne 0$}\\\\ \n    3 \\times \\vert w'\\vert_0 - \\vert w\\vert_2 - \\vert w' \\vert_1 +\n    \\vert w\\vert_0 - 1 \\ \\ \\\n    \\text{if $w'[10] = 0$} \\\\\n    3 \\times \\vert w'\\vert_0 - \\vert w\\vert_2 - \\vert w' \\vert_1 +\n    \\vert w\\vert_0 \\ \\ \\ \\text{ otherwise.}\n  \\end{cases}\n  $$\n\\end{lemma}\n\n\\begin{proof} Consider a set $S'\\subseteq P_{p+1}$ dominating\n  $I(P_{p+1})$ and let $S =S' \\cap P_p$.  Let $w = w^{out}_p(S)$ and $w' =\n  w^{out}_{p+1}(S')$.  Let $\\Delta(S,S')= \\los{S'} - \\los{S}$. By\n  definition of the loss, $\\Delta(S,S')= 5 \\times |S'\\setminus S| -\n  |N[S']\\setminus N[S]|$. Let us compute $\\Delta(S,S')$ in term of the\n  number of occurrences of $0$'s, $1$'s and $2$'s in the words $w$ and\n  $w'$. The set $S'\\setminus S$ corresponds to the vertices $\\{(p+1,i)\n  \\mid i\\in[10], w'[i] = 0\\}$. The set $N[S']\\setminus N[S]$ corresponds\n  to the vertices dominated by $S'$ but not dominated by $S$; these\n  vertices clearly belong to the columns $p$, $p+1$ and $p+2$. Since\n  $S'$ dominates $I(P_{p+1})$, those in the column $p$ are the vertices\n  $\\{(p,i) \\mid i\\in[10], w[i]=2\\}$.  Those in the column $p+1$ are the\n  vertices $\\{(p+1,i) \\mid i\\in[10], w'[i]\\neq 2, w[i] \\neq0\\}$ and\n  possibly the vertex $(p+1,11)$ when $w'[10]=0$. Finally, those in the\n  column $p+2$ are the vertices $\\{(p+2,i) \\mid i\\in[10], w'[i]=0\\}$.\n  We then get:\n  $$ \n  \\Delta(S,S')= \n  \\begin{cases}\n    3 \\times \\vert w'\\vert_0 - \\vert w \\vert_2  \n    - \\vert w'\\vert_1 + \\vert w\\vert_0 - 1 & \\text{if $w'[10] = 0$} \\\\    \n    3 \\times \\vert w'\\vert_0 - \\vert w \\vert_2  \n    - \\vert w'\\vert_1 + \\vert w\\vert_0 & \\text{otherwise}\n  \\end{cases}\n  $$\n  where $|w|_n$ denotes the number of occurrences of the letter $n$ in\n  the word $w$.\n\n  Thus $\\Delta(S,S')$ only depends on the output words of $S$ and $S'$,\n  and we can denote this value by $\\Delta(w,w')$. Note however that\n  there exist pairs of\n  words $(w,w')$ that could not be the output words of $S$ and $S'$;\n  there are three cases:\n  \\begin{enumerate}[C{a}se 1.]\n  \\item $w[i]=0$ and $w'[i]=2$ since the vertex\n    $(p+1,i)$ would be dominated by $(p,i)$ contradicting its label 2;\n  \\item $w[i]=2$ and $w'[i]\\ne 0$ for $i\\in[9]$ since \n    $(p,i)$ would not be dominated contradict the fact that $S'$\n    dominates $I(P_{p+1})$;\n  \\item $w'[i]=1$ and $w'[i-1] \\neq 0$, $w'[i+1]\\neq 0$, $w[i]\\neq 0$\n    since $(p+1,i)$ would be dominated according to its label but none\n    of its neighbors belong to $S'$. \n  \\end{enumerate}\n  For these forbidden cases, we set $\\Delta(w,w') = +\\infty$.\n  \n  By definition, $C_{p+1}[w_i,w']$ is the minimum loss $\\ell(S')$\n  of a set $S'\\subseteq P_{p+1}$ that dominates $I(P_{p+1})$, with $w_i$\n  as input word and $w'$ as output word. It is clear that $S= S'\\cap P_p$\n  dominates $I(P_{p})$ and has $w_i$ as input word.  Let $w$ be its output\n  word and note that $C_{p+1}[w_i,w'] = \\ell(S') = \\ell(S) +\n  \\Delta(w_i,w')$.  The minimality of $\\ell(S')$ implies the minimality of\n  $\\ell(S)$ over the sets $X\\in \\mathcal{D}^{w_i,w'}_p$. Indeed, any set $X\\in\n  \\mathcal{D}^{w_i,w'}_p$ could be turned in a set of $X'\\in \\mathcal{D}^{w_i,w'}_{p+1}$ by\n  adding vertices of the $p+1^{th}$ column accordingly to $w'$.  Thus\n  $$\n  C_{p+1}[w_i,w'] = C_{p}[w_i,w] + \\Delta(w,w')\n  $$\n\n  which implies that\n  $$\n  C_{p+1}[w_i,w'] \\geq \\min_{w} C_{p}[w_i,w] + \\Delta(w,w').\n  $$\n\n  On the other hand, for every word $w_o \\in \\mathcal{W}$ such that\n  $C_{p}[w_i,w_o]\\neq +\\infty$ and $\\Delta(w_o,w')\\neq\n  +\\infty$, there is a set $S \\in \\mathcal{D}^{w_i,w_o}_p$, with $\\ell(S) =\n  C_{p}[w_i,w_o]$, that can be turned in a set $S' \\in\n  D^{w_i,w'}_{p+1}$ with $\\ell(S') = C_{p}[w_i,w_o] +\n  \\Delta(w_o,w')$. Thus\n  $$\n  C_{p+1}[w_i,w'] \\leq \\min_{w_o} C_{p}[w_i,w_o] + \\Delta(w_o,w').\n  $$\n\n  This concludes the proof of the lemma.\n\\end{proof}\n\n\nBy the definition of $M_p$, we have also $M_{p+1}=M_p\n\\otimes T$. Note that $T$ is a sparse matrix: about $95.5 \\%$\nof its $8119^2$ entries are $+\\infty$.  Thus the multiplication by\n$T$ in the $(\\min,+)$ algebra can be done in a reasonable amount\nof time by a trivial algorithm.\n\n\\medskip\n\n\\begin{fact}\\label{fact126} The computations give us that\n  $M_{126}=M_{125}+1$. Thus, since $(A + c)\\otimes B =\n  (A\\otimes B) + c$ for any matrices $A$, $B$ and any integer $c$, we\n  have that $M_{125+k}=M_{125}+k$ for every $k\\in \\mathbb{N}$.\n\\end{fact}\n\nLet us define $M'_p= \\min_{k\\in \\mathbb{N}} ( M_{p+k} - k\n)$.  Then, for all $q\\ge p$, $M_q \\ge M'_p + (q-p)$.  By\nFact~\\ref{fact126}, $M'_p= \\min_{k\\in \\{0,\\dots 125-p\\}} (\nM_{p+k} - k )$\n\n\\begin{fact}\\label{fact23} By computing $M'_{12}$, and $A'=\n  M'_{12} \\otimes M'_{12}$, we obtain that $\\min_{w_1,w_3} (A'\n  + A'^T)[w_1,w_3] = 76$ (where $A^T$ is the transpose of $A$).\n\\end{fact}\nThis implies that \n$$ \n\\min_{w_1,w_3} \\ (\\min_{w_2} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]) \\ +\n\\ (\\min_{w_4} M'_{12}[w_3,w_4]+ M'_{12}[w_4,w_1]) \\ = \\ 76\n$$\n$$\n\\min_{w_1,w_2,w_3,w_4} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]+M'_{12}[w_3,w_4]+\nM'_{12}[w_4,w_1] \\ = \\ 76.\n$$\n\n\\begin{theorem}\n  If $24 \\le n \\le m$, then $$\\gamma(G_{n,m}) = \\form{n}{m}.$$\n\\end{theorem}\n\n\\begin{proof} By Chang's construction \\cite{CHHW}, $\\gamma(G_{n,m}) \\le\n  \\form{n}{m}$. Let us now compute a lower bound for the loss of a\n  dominating set of $G_{n,m}$.\n\n  \\begin{eqnarray*}\n    \\ell_{n,m} &\\ge& b_{n,m}\\\\\n    &\\ge& \\min_{w_1,w_2,w_3,w_4}  M_{n-12}[w_1,w_2] +\n    M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+\n    M_{m-12}[w_4,w_1]\\\\\n    & & \\; \\mbox{by Lemma~\\ref{lem:bnm}} \\\\\n    &\\ge& \\min_{w_1,w_2,w_3,w_4}  M'_{12}[w_1,w_2]+(n-12-12)  +\n    M'_{12}[w_2,w_3]+ (m-12-12) + M'_{12}[w_3,w_4]\\\\&&\n    \\phantom{\\min_{w_1,w_2,w_3,w_4}}+(n-12-12) +\n    M'_{12}[w_4,w_1]+ (m-12-12)\\\\ \n    &\\ge& 2 \\times (n+m -48) + \\min_{w_1,w_2,w_3,w_4}\n    M'_{12}[w_1,w_2] + M'_{12}[w_2,w_3] +\n    M'_{12}[w_3,w_4] + M'_{12}[w_4,w_1]\\\\ \n    &\\ge& 2 \\times (n+m -48) + 76\\\\ \n    &\\ge& 2\\times(n+m)-20.\n  \\end{eqnarray*}\n\n  Thus by Lemma~\\ref{lemloss}, we have:\n  \\begin{eqnarray*}\n    \\gamma(G_{n,m}) &\\ge& \\left \\lceil \\frac{n\\times m + 2\\times(n+m)-20}{5}\n    \\right \\rceil\\\\\n    &\\ge& \\left \\lceil \\frac{(n+2)(m + 2) -4}{5}\n    \\right \\rceil - 4\\\\\n    &\\ge& \\form{n}{m}.\n  \\end{eqnarray*}\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex}%\n                                   {2.3ex \\@plus.2ex}%\n                                   {\\normalfont\\large\\bf}}\n                                 \n\\makeatother\n\n\\makeatletter\n\\renewcommand\\subsection{\\@startsection {subsection}{1}{\\z@}%\n                                   {-3.5ex \\@plus -1ex \\@minus -.2ex}%\n                                   {2.3ex \\@plus.2ex}%\n                                   {\\normalfont\\normalsize\\bf}}\n                                 \n\\makeatother\n\n\n\\usepackage[%\nbookmarks=true,%\nbookmarksdepth=3,%\nbookmarksnumbered=true,%\nsetpagesize=false,%\npdftitle={Local time penalizations\nwith various clocks for L\\'{e}vy processes},%\npdfauthor={Shosei Takeda and Kouji Yano},%\npdfkeywords={one-dimensional L\\'{e}vy process; %\nlimit theorem; penalization; conditioning}%\n]{hyperref}\n\n\\title{\\textbf{Local time penalizations\nwith various clocks for L\\'{e}vy processes}\\footnote{\nThis research was supported by RIMS and by ISM.}}\n\n\\author{Shosei Takeda\\footnote{Rakunan High School, Kyoto, Japan.}\n\\footnote{The research of this author was supported by\nJSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921.}\n\\quad and \\quad Kouji Yano\\footnote{Graduate School of Science, Kyoto University, Japan.}\n\\footnotemark[3]\n\\footnote{The research of this author was supported by\nJSPS KAKENHI grant no.'s JP19H01791 and JP19K21834.}}\n\\date{}\n\\begin{document}\n\\maketitle\n\\begin{abstract}\n  Several long-time limit theorems\n  of one-dimensional L\\'{e}vy processes\n  weighted and normalized\n  by functions of the local time\n  are studied.\n  The long-time limits are taken via certain families of random times, called clocks:\n  exponential clock, hitting time clock, two-point hitting time clock\n  and inverse local time clock.\n  The limit measure can be characterized via a certain martingale\n  expressed by an invariant function for the process\n  killed upon hitting zero.\n  The limit processes may differ according to the choice of the clocks\n  when the original L\\'{e}vy process is recurrent and of finite variance.\n\\end{abstract}\n{\\small Keywords and phrases: one-dimensional L\\'{e}vy process;\nlimit theorem; penalization; conditioning \\\\\nMSC 2020 subject classifications: 60F05 (60G44; 60G51)}\n\\section{Introduction}\nRoynette--Vallois--Yor\n(\\cite{MR2229621,MR2253307} see also~\\cite{MR2261065,MR2504013})\nhave studied the limit distribution for a Brownian motion,\nwhich they called a \\textit{penalization problem},\nas follows.\nLet \\(B=(B_t,t\\ge 0)\\) be a standard Brownian motion and\n\\(L=(L_t,t\\ge 0)\\) denote its local time at \\(0\\). Then, for\nany positive integrable function \\(f\\)\nand any bounded adapted functional \\(F_t\\), it holds that\n\\begin{align}\n  \\lim_{s\\to\\infty} \\frac{\\ensuremath{\\mathbb{P}}\\sbra{F_t f(L_s)}}{\\ensuremath{\\mathbb{P}}\\sbra{f(L_s)}}\n  = \\ensuremath{\\mathbb{P}}\\sbra*{F_t \\frac{M_t}{M_0}} \\eqqcolon \\ensuremath{\\mathbb{Q}}\\sbra{F_t},\n\\end{align}\nwhere \\(M=(M_t,t\\ge 0)\\) is the martingale given by\n\\begin{align}\n  M_t = f(L_t)\\abs{B_t} + \\int_0^\\infty f(L_t + u)\\, \\mathrm{d} u,\n  \\quad t\\ge 0.\n\\end{align}\nUnder the penalized probability\nmeasure \\(\\ensuremath{\\mathbb{Q}}\\), the total local time \\(L_{\\infty}\\) is finite, and\nin fact, a sample path behaves as the concatenation of a Brownian bridge\nand a three-dimensional Bessel process; see~\\cite{MR2229621}.\nIn particular, \\(\\ensuremath{\\mathbb{Q}}\\) is singular to \\(\\ensuremath{\\mathbb{P}}\\).\n\nThis result for a Brownian motion was\ngeneralized to many other processes.\nIn particular, we refer to\nDebs~\\cite{MR2599215} for random walks,\nNajnudel--Roynette--Yor~\\cite{MR2528440} for Markov chains and Bessel processes,\nYano--Yano--Yor~\\cite{MR2552915} for symmetric stable processes,\nSalminen--Vallois~\\cite{MR2540855} and Profeta~\\cite{MR2760742,MR2968676} for linear\ndiffusions.\nMost of these results were obtained basically under the assumption of some\nregular variation condition.\nProfeta--Yano--Yano~\\cite{MR3909919} developed a general theory for\none-dimensional diffusions by adopting a random clock approach.\nThey studied the long-time limit of the form\n\\begin{align}\n  \\lim_{\\tau\\to\\infty}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{F_t f(L_\\tau)}}{\\ensuremath{\\mathbb{P}}\\sbra{f(L_\\tau)}},\n\\end{align}\nwhere \\(\\tau=(\\tau_\\lambda)\\) is a certain parametrized\nfamily of random times, which they called a clock.\nSuch a random clock approach already\nappeared in the problem of conditioning to avoid zero,\nwhich is a special case of our penalization with\n\\(f(u)=1_{\\cbra{u=0}}\\), or in the problem of conditioning\nto stay positive\/negative.\nFor example, we refer to Knight~\\cite{MR253424} for Brownian motions,\nChaumont~\\cite{MR1419491}, Chaumont--Doney~\\cite{MR2164035,MR2375597}\nand Doney~\\cite[Section 8]{MR2126962,MR3155252}\nfor L\\'{e}vy processes conditioned to stay positive,\nYano--Yano~\\cite{MR3444297} for diffusions and\nPant\\'{\\i}~\\cite{MR3689384} for L\\'{e}vy processes conditioned to avoid zero.\n\nLet \\(X=(X_t,t\\ge 0)\\) be a one-dimensional L\\'{e}vy process and\nlet \\(T_A\\) denote the hitting time of a Borel\nset \\(A \\subset \\ensuremath{\\mathbb{R}}\\) for \\(X\\), i.e.,\n\\begin{align}\\label{eq:T_A}\n  T_A = \\inf\\cbra{t>0\\colon X_t \\in A}\n\\end{align}\nand we write \\(T_a = T_{\\cbra{a}}\\)\nsimply for the hitting time of a point \\(a\\in\\ensuremath{\\mathbb{R}}\\).\nLet \\((\\eta^a_u)\\) denote the right-continuous inverse of\nthe local time at a point \\(a\\in \\ensuremath{\\mathbb{R}}\\).\nWe adopt the random clock approach for the following four clocks:\n\\begin{enumerate}\n  \\item exponential clock: \\(\\tau=(\\bm{e}_q)\\) with \\(q\\to 0+\\);\n  \\item hitting time clock: \\(\\tau=(T_a)\\) with\n        \\(a\\to\\pm\\infty\\);\\label{item:hitting-time-clocks}\n  \\item two-point hitting time clock: \\(\\tau=(T_{a}\\wedge T_{-b})\\) with\n        \\(a\\to\\infty\\) and \\(b\\to\\infty\\);\\label{item:two-point-hitting}\n  \\item inverse local time clock: \\(\\tau=(\\eta^a_u)\\)\n        with \\(a\\to\\pm\\infty\\) or with \\(u\\to\\infty\\).\n\\end{enumerate}\n\n\\subsection{Main results}\\label{Subsec:main-results}\nLet \\((X, \\ensuremath{\\mathbb{P}}_x)\\) denote the canonical representation\nof a L\\'{e}vy process starting from \\(x\\)\non the c\\`{a}dl\\`{a}g path space \\(\\ensuremath{\\mathcal{D}}\\)\nand set \\(\\ensuremath{\\mathbb{P}}=\\ensuremath{\\mathbb{P}}_0\\).\nFor \\(t\\ge 0\\), we denote by \\(\\ensuremath{\\mathcal{F}}_t^X = \\sigma(X_s, 0\\le s\\le t)\\) the natural\nfiltration of \\(X\\) and write \\(\\ensuremath{\\mathcal{F}}_t = \\bigcap_{s>t} \\ensuremath{\\mathcal{F}}_s^X\\).\nWe have\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{\\mathrm{i}\\lambda X_t}} = \\mathrm{e}^{-t \\varPsi(\\lambda)},\n  \\quad t \\ge 0, \\,\\lambda \\in \\ensuremath{\\mathbb{R}},\n\\end{align}\nwhere \\(\\varPsi(\\lambda)\\) denotes the characteristic exponent of \\(X\\)\ngiven by the L\\'{e}vy--Khintchine formula\n\\begin{align}\n  \\varPsi(\\lambda)\n  = \\mathrm{i} v \\lambda\n  + \\frac{1}{2} \\sigma^2 \\lambda^2\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{1 - \\mathrm{e}^{\\mathrm{i} \\lambda x} + \\mathrm{i} \\lambda x 1_{\\cbra{\\abs{x}< 1}}}\n  \\nu(\\mathrm{d} x)\n\\end{align}\nfor some constants \\(v \\in \\ensuremath{\\mathbb{R}}\\) and \\(\\sigma \\ge 0\\)\nand some measure \\(\\nu\\) on \\(\\ensuremath{\\mathbb{R}}\\)\n(called the L\\'{e}vy measure)\nwhich satisfies \\(\\nu(\\cbra{0})=0\\) and\n\\begin{align}\n  \\int_\\ensuremath{\\mathbb{R}} \\rbra*{x^2 \\wedge 1} \\nu(\\mathrm{d} x) < \\infty.\n\\end{align}\nWe denote the real and imaginary parts of \\(\\varPsi(\\lambda)\\) by\n\\begin{align}\n  \\theta(\\lambda) & = \\Re \\varPsi(\\lambda)\n  = \\frac{1}{2} \\sigma^2 \\lambda^2\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{1 - \\cos \\lambda x} \\nu(\\mathrm{d} x),\\label{eq:theta} \\\\\n  \\omega(\\lambda) & = \\Im \\varPsi(\\lambda)\n  = v \\lambda\n  + \\int_\\ensuremath{\\mathbb{R}} \\rbra*{\\lambda x 1_{\\cbra{\\abs{x}< 1}} - \\sin \\lambda x} \\nu(\\mathrm{d}\n  x)\\label{eq:omega}.\n\\end{align}\nNote that \\(\\theta(\\lambda) \\ge 0\\) for \\(\\lambda \\in \\ensuremath{\\mathbb{R}}\\),\n\\(\\theta(\\lambda)\\) is even and \\(\\omega(\\lambda)\\) is odd.\nFor more details of the notation of this section,\nsee Section~\\ref{Sec:preliminaries}.\nThroughout this paper except\nSections~\\ref{Sec:preliminaries},~\\ref{Sec:trans} and~\\ref{Sec:mart},\nwe always assume\n\\((X,\\ensuremath{\\mathbb{P}})\\) is recurrent, i.e.,\n\\begin{align}\\label{eq:recurrent}\n  \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty 1_{\\cbra{\\abs{X_t-a}<\\varepsilon}}\\, \\mathrm{d} t}\n  =\\infty, \\qquad\\text{for all \\(a\\in\\ensuremath{\\mathbb{R}}\\) and \\(\\varepsilon>0\\),}\n\\end{align}\n and assume the following:\n\\begin{enumerate}[label=\\textbf{(\\Alph*)}]\n  \\item For each \\(q > 0\\), it holds that\n        \\begin{align}\n          \\int_0^\\infty\n          \\abs*{\\frac{1}{q+\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty.\n        \\end{align}\\label{item:assumption}\n\\end{enumerate}\nNote that, we say \\((X,\\ensuremath{\\mathbb{P}})\\) is transient if~\\eqref{eq:recurrent}\ndoes not hold.\nUnder the assumption~\\ref{item:assumption},\nthe process\n\\((X,\\ensuremath{\\mathbb{P}})\\) is recurrent if and only if \\((X,\\ensuremath{\\mathbb{P}})\\) is\npoint recurrent, i.e.,\n\\begin{align}\\label{eq:pt-recurrent}\n  \\ensuremath{\\mathbb{P}}(T_a<\\infty) = 1,\\qquad\\text{for all \\(a\\in\\ensuremath{\\mathbb{R}}\\);}\n\\end{align}\nsee~Subsection~\\ref{Subsec:LT-em}.\nThe assumption~\\ref{item:assumption}\nimplies that the \\(q\\)-resolvent density \\(r_q\\) exists for \\(q>0\\);\nsee Subsection~\\ref{Subsec:resolvent}.\nFor \\( q > 0\\), we define\n\\begin{align}\\label{eq:def-h_q}\n  h_q(x) = r_q(0) - r_q(-x)\n  = \\frac{1}{\\pi}\n  \\int_0^\\infty\n  \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n  \\,\\mathrm{d} \\lambda\n  , \\qquad x \\in \\ensuremath{\\mathbb{R}},\n\\end{align}\nwhere the second identity follows from Proposition~\\ref{Prop:r_q-represent}.\nIt is obvious that \\(h_q(0) = 0\\), and\nby~\\eqref{eq:exp-hit-time}, we have \\(h_q(x) \\ge 0\\).\nWe denote the second moment by\n\\begin{align}\n  m^2 =\n  \\ensuremath{\\mathbb{P}}\\sbra{{X_1}^2} \\in (0, \\infty].\\label{eq:second-moment}\n\\end{align}\nThe following theorem plays a key role\nin our penalization results.\nRecall that \\(X\\) is assumed recurrent.\n\n\\begin{Thm}\\label{Thm:exist-h}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item For any \\(x \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\\label{eq:conv-h}\n            h(x) \\coloneqq \\lim_{q \\to 0+} h_q(x)\n          \\end{align}\n          exists and is finite,\n          which will be called the \\emph{renormalized zero resolvent}.\n          If \\(m^2 < \\infty\\), then \\(h\\) has the following representation:\n          \\begin{align}\\label{eq:h-repre}\n            h(x)\n            = \\frac{1}{\\pi} \\int_0^\\infty \\Re\\rbra*{\n              \\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}}\n            \\, \\mathrm{d} \\lambda.\n          \\end{align}\\label{Thm-item:exist-h}\n    \\item The convergence~\\eqref{eq:conv-h} is uniform on compacts, and\n          consequently \\(h\\) is continuous.\\label{Thm-item:unif-conv-h}\n    \\item \\(h\\) is subadditive on \\ensuremath{\\mathbb{R}},\n          that is, \\(h(x+y)\\le h(x)+h(y) \\) for \\(x,y\\in\n          \\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:subadditive-h}\n  \\end{enumerate}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:exist-h} will be given in\nSection~\\ref{Subsec:pf-h}.\nThe renormalized zero resolvent satisfies the following limit properties.\n\\begin{Thm}\\label{Thm:property-h}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} =\n          \\frac{1}{m^2} \\in [0,\\infty)\n          \\);\\label{Thm-item:h\/x-infinity}\n    \\item\n          \\(\\displaystyle\\label{eq:h-h-limit}\n          \\lim_{y\\to\\pm\\infty} \\cbra*{h(x+y) - h(y)} =\n          \\pm\\frac{x}{m^2}\\in\\ensuremath{\\mathbb{R}},\n          \\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-h-limit}\n  \\end{enumerate}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:property-h}\nwill be given in Section~\\ref{Subsec:pf-h}.\n\n\\begin{Cor}\n  Suppose that~\\ref{item:assumption}\n  is  satisfied.\n  For \\(-1\\le \\gamma\\le 1\\), define\n  \\begin{align}\\label{eq:def-h-gamma}\n    h^{(\\gamma)}(x)=h(x)+\\frac{\\gamma}{m^2}x, \\quad x\\in\\ensuremath{\\mathbb{R}}.\n  \\end{align}\n  Then \\(h^{(\\gamma)}\\) is subadditive and \\(h^{(\\gamma)}(x)\\ge 0\\).\n\\end{Cor}\n\\begin{proof}\n  By definition, we have \\(h^{(\\gamma)}(0)=0\\).\n  From (\\ref{Thm-item:subadditive-h}) of Theorem~\\ref{Thm:exist-h},\n  the function \\(h^{(\\gamma)}\\) is subadditive. From\n  (\\ref{Thm-item:h\/x-infinity}) of Theorem~\\ref{Thm:property-h},\n  it holds that \\(\\lim_{x\\to\\pm\\infty} h^{(\\gamma)}(x)\/\\abs{x} = (1+\\gamma)\/m^2 \\ge\n  0\\).\n  Suppose \\(h^{(\\gamma)}(x)<0\\) for some \\(x\\in\\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\).\n  Since \\(h^{(\\gamma)}\\) is subadditive,\n  we have \\(h(nx)\/n\\le h(x)<0\\) for \\(n=1,2,\\dots\\).\n  Letting \\(n\\to\\infty\\), we face the contradiction.\n  Therefore we have \\(h^{(\\gamma)}(x)\\ge 0\\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\n\\end{proof}\nWe will prove in Theorem~\\ref{Thm:hg-invariant} that\nthe function \\(h^{(\\gamma)}\\) is invariant for the process\nkilled upon hitting zero.\nLet \\(\\ensuremath{\\mathcal{L}}_{+}^1\\) denote the set of non-negative functions\non \\([0,\\infty)\\) which satisfy \\(\\int_0^\\infty f(x)\\,\\mathrm{d} x<\\infty\\).\nFor \\(f\\in \\ensuremath{\\mathcal{L}}_{+}^1\\), define\n\\begin{align}\\label{eq:def-mart}\n  M_t^{(\\gamma)}= M_t^{(\\gamma, f)}= h^{(\\gamma)}(X_t)f(L_t)+\n  \\int_0^\\infty\n  f(L_t+u)\\,\\mathrm{d} u.\n\\end{align}\nNote that, when \\(m^2=\\infty\\),\nwe have \\(h^{(\\gamma)}=h^{(0)}=h\\) and\n\\(M^{(\\gamma)}=M^{(0)}\\) for all \\(\\gamma\\).\n\\begin{Thm}\\label{Thm:martingale}\n  Suppose that~\\ref{item:assumption}\n  is satisfied.\n  Let \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\), \\(-1\\le \\gamma\\le 1\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\). Then\n  \\((M_t^{(\\gamma)},t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{Thm}\nTheorem~\\ref{Thm:martingale} will be proved in\nSection~\\ref{Subsec:pf-hitting-l1}.\nUsing this martingale,\nwe discuss our penalization problems.\nLet \\(L=(L_t)\\) denote the local time at the origin of \\(X\\);\nsee Section~\\ref{Subsec:LT-em}.\n\n\\begin{Thm}[hitting time clock]\\label{Thm:hitting-time-result}\n  Suppose that the condition~\\ref{item:assumption} is satisfied.\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\).\n  Define \\(h^B(a)=\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\) and\n  \\begin{align}\n    N_t^a & = h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a}); t<T_a|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^a & = h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{a\\to\\pm\\infty}N_t^a\n    =\\lim_{a\\to\\pm\\infty} M_t^a\n    = M_t^{(\\pm 1)},\n    \\qquad \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\pm 1)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_a})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\pm 1)}}{M_0^{(\\pm 1)}}},\n    \\qquad \\text{as \\(a \\to \\pm\\infty \\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nTheorem~\\ref{Thm:hitting-time-result} will be proved\nin Section~\\ref{Sec:hitting-time}.\nIf we take \\(f=1_{\\cbra{u=0}}\\), we obtain\nthe conditioning result.\n\\begin{Cor}\\label{Cor:hitting-cond}\nSuppose that the condition~\\ref{item:assumption} is satisfied.\nLet \\(x\\in \\ensuremath{\\mathbb{R}}\\) with \\(h^{(\\pm 1)}(x)>0\\).\nThen it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_a}\n  \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t\\frac{h^{(\\pm 1)}(X_t)}{h^{(\\pm 1)}(x)};T_0>t},\n  \\qquad\\text{as \\(a\\to\\pm\\infty\\),}\n\\end{align}\nfor all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Cor}\nSee also Corollary~\\ref{Cor:avoid-zero}.\n\nLet us state our penalization result with two-point hitting time clock.\nFor \\(a,b\\in\\ensuremath{\\mathbb{R}}\\), we write\n\\(T_{a,b} = T_{\\cbra{a,b}}= T_{a} \\wedge T_{b}\\).\nFor \\(-1\\le\\gamma\\le 1\\), we say\n\\begin{align}\\label{eq:a-b-inf-notation}\n  \\text{\\((a,b)\\xrightarrow[]{\\gamma}\\infty\\)\n    when \\(a\\to\\infty\\), \\(b\\to\\infty\\) and \\(\\frac{a-b}{a+b}\\to\\gamma\\).}\n\\end{align}\n\\begin{Thm}[two-point hitting time clock]\\label{Thm:two-point-hitting-time-result}\n  Suppose that the condition~\\ref{item:assumption} is satisfied.\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\), \\(x \\in \\ensuremath{\\mathbb{R}}\\), and \\(a,b>0\\).\n  Define \\(\th^C(a,-b) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_{a, -b}}}\\), and\n  \\begin{align}\n    N_t^{a,b}\n     & =h^C(a, -b)\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a, -b}}); t<T_{a,-b}|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{a,b}\n     & =h^C(a, -b)\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a,-b}})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}N_t^{a,b}\n    =\\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty} M_t^{a,b}\n    =M_t^{(\\gamma)},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and\n      in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\gamma)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_{a, -b}})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_{a,-b}})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\gamma)}}{M_0^{(\\gamma)}}},\n    \\qquad\n    \\text{as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:two-point-hitting-time-result}\nwill be given in Section~\\ref{Subsec:pf-two-hitting}.\n\\begin{Cor}\\label{Cor:two-point-hitting-cond}\nSuppose that the condition~\\ref{item:assumption} is satisfied.\nLet \\(-1\\le\\gamma\\le 1\\) and\n\\(x\\in \\ensuremath{\\mathbb{R}}\\) with \\(h^{(\\gamma)}(x)>0\\).\nThen it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_{a,-b}}\n  \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t\\frac{h^{(\\gamma)}(X_t)}{h^{(\\gamma)}(x)};T_0>t},\n  \\qquad\\text{as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\),}\n\\end{align}\nfor all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Cor}\nSee also Corollary~\\ref{Cor:avoid-zero}.\n\nNote that Theorems~\\ref{Thm:hitting-time-result}\nand~\\ref{Thm:two-point-hitting-time-result} show that the limit\nlaw varies according to the chosen clock\nwhen \\(m^2<\\infty\\).\n\n\\subsection{Backgrounds of the renormalized zero resolvent}\\label{Subsec:back-h}\nThe existence of \\(h\\) for symmetric L\\'{e}vy processes\nwas proved by Salminen--Yor~\\cite{MR2409011}\nunder the assumption~\\ref{item:assumption}; see also\nYano~\\cite{MR2603019}.\nWe shall review early studies of the existence of \\(h\\) and\nits limit properties\nfor asymmetric processes.\n\nSimilar results were obtained for random walks\nby Spitzer~\\cite[Chapter VII]{MR0171290},\nPort--Stone~\\cite{MR0215375,MR0226740,MR261706}\nand Stone~\\cite{MR238398}.\nFor L\\'{e}vy processes, Port--Stone~\\cite[Section 17]{MR346919} obtained some results\nwhich were similar to but different from\nTheorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nreducing them to the random walk case.\n(For the proofs\nof Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nwe are inspired\nby Spitzer~\\cite[Chapter VII]{MR0171290},\nPort--Stone~\\cite[Section 17]{MR0215375,MR0226740,MR261706,MR346919}\nand Stone~\\cite{MR238398}.)\n\nYano~\\cite{MR3072331} showed\nthe existence of the renormalized zero resolvent \\(h\\)\nunder the following two conditions:\n\\begin{enumerate}[label=\\textbf{(Y\\arabic*)}]\n  \\item \\(\\displaystyle \\int_0^\\infty \\frac{1}{q+\\theta(\\lambda)}\n        \\, \\mathrm{d} \\lambda < \\infty\\) for all \\(q>0\\);\\label{item:cond-Yano1}\n  \\item \\(\\theta\\) and \\(\\omega\\) have measurable derivatives\n        on \\((0, \\infty)\\) which satisfy\n        \\begin{align}\n          \\int_0^\\infty \\frac{\\rbra{\n            \\abs{\\theta^\\prime(\\lambda)}+\n            \\abs{\\omega^\\prime(\\lambda)}}(\\lambda^2\\wedge 1)\n          }{{\\theta(\\lambda)}^2+{\\omega(\\lambda)}^2}\n          \\, \\mathrm{d} \\lambda < \\infty.\n        \\end{align}\\label{item:cond-Yano2}\n\\end{enumerate}\nPant\\'{\\i}~\\cite{MR3689384} proved the existence of \\(h\\)\nunder the condition\n\\begin{enumerate}[label=\\textbf{(P)}]\n  \\item \\ref{item:cond-Yano1} and \\( \\displaystyle\n        \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda}}{\\varPsi(\\lambda)}}}\n        \\, \\mathrm{d} \\lambda < \\infty\\),\\label{item:Panti-cond}\n\\end{enumerate}\nwhich is weaker than~\\ref{item:cond-Yano1} and~\\ref{item:cond-Yano2},\nand applied it to the conditioning to avoid zero\nwith exponential clock.\nTsukada~\\cite{MR3838874} also proved the existence\nof \\(h\\)\nunder the assumption\n\\begin{enumerate}[label=\\textbf{(T)}]\n  \\item \\ref{item:assumption} and\n        \\(\\displaystyle\n        \\int_0^1 \\abs*{\\Im\\rbra*{\\frac{\\lambda}{\\varPsi(\\lambda)}}}\n        \\, \\mathrm{d} \\lambda <\n        \\infty,\n        \\)\\label{item:Tsukada-b}\n\\end{enumerate}\nwhich is weaker than~\\ref{item:Panti-cond};\nsee~\\cite[Proposition 15.3]{MR3838874}.\n\n\\begin{Rem}\n  We do not know whether the integral\n  representation~\\eqref{eq:h-repre} also holds\n  in the case \\(m^2=\\infty\\).\n  Yano~\\cite{MR3072331} showed that, if \\(X\\) is symmetric,\n  then~\\eqref{eq:h-repre} holds.\n  Tsukada~\\cite{MR3838874} showed that~\\ref{item:Tsukada-b} implies\n  \\eqref{eq:h-repre}.\n\\end{Rem}\n\n\\subsection{Organization}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{Sec:preliminaries}, we prepare\ncertain general properties and\npreliminary facts of L\\'{e}vy processes.\nIn Section~\\ref{Sec:h}, we\nstudy the renormalized zero resolvent.\nIn Sections~\\ref{Sec:exp-clock},~\\ref{Sec:hitting-time},~\\ref{Sec:two-hitting-time}\nand~\\ref{Sec:inv-time}, we discuss the penalization results\nwith exponential clock, hitting time clock, two-point hitting time\nclock and inverse local time clock, respectively.\nIn Section~\\ref{Sec:universal-measure}, we introduce certain universal\n\\(\\sigma\\)-finite measures to study long time\nbehaviors of sample paths of the penalized measure.\nIn Section~\\ref{Sec:trans}, we study penalization in the transient case.\nIn Section~\\ref{Sec:mart} as an appendix, we study martingale property\nof \\((X_t f(L_t),t\\ge 0)\\).\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Hiroshi Tsukada\nfor his helpful comments.\n\n\\section{Preliminaries}\\label{Sec:preliminaries}\n\\subsection{Absolutely continuous resolvent}\\label{Subsec:resolvent}\nWe now consider the following two conditions:\n\\begin{enumerate}[label=\\textbf{(A\\arabic*)},series=cond]\n  \\item\\label{item:cond-not-CPP}\n        The process \\(X\\) is not a compound Poisson process;\n  \\item\\label{item:cond-regular}\n        \\(0\\) is regular for itself, i.e.,\n        \\(\\ensuremath{\\mathbb{P}}(T_0 = 0) = 1\\).\n\\end{enumerate}\n\nThe next lemma is due to Kesten~\\cite{MR0272059} and\nBretagnolle~\\cite{MR0368175}.\n\\begin{Lem}[\\cite{MR0272059,MR0368175}]\\label{Lem:iff-cond-notCPP-regular}\n  The conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular}\n  hold if and only if the following two assertions hold:\n  \\begin{enumerate}[resume*=cond]\n    \\item For each \\(q>0\\),\n          the characteristic exponent \\(\\varPsi\\) satisfies\n          \\begin{align}\n            \\int_\\ensuremath{\\mathbb{R}} \\Re\\rbra*{\\frac{1}{q+\\varPsi(\\lambda)}}\\, \\mathrm{d} \\lambda < \\infty;\n          \\end{align}\\label{item:cond-Reinte}\n    \\item We have either \\(\\sigma>0\\) or\n          \\(\\int_{(-1, 1)} \\abs{x} \\nu(\\mathrm{d} x) = \\infty\\).\\label{item:cond-nu}\n  \\end{enumerate}\n  Furthermore, under the condition~\\ref{item:cond-Reinte},\n  the condition~\\ref{item:cond-regular} holds if and only\n  if the condition~\\ref{item:cond-nu} holds.\n\\end{Lem}\nIf the above conditions hold, it is known that\n\\(X\\) has a bounded continuous resolvent density. See,\ne.g., Theorem II.16 and Theorem II.19 of Bertoin~\\cite{MR1406564}\n\\begin{Lem}[\\cite{MR1406564}]\\label{Lem:resolvent-density}\n  The condition~\\ref{item:cond-Reinte} holds if and only if\n  \\(X\\) has the bounded \\(q\\)-resolvent density \\(r_q\\), for \\( q > 0\\),\n  which satisfies\n  \\begin{align}\n    \\int_\\ensuremath{\\mathbb{R}} f(x) r_q(x) \\, \\mathrm{d} x\n    = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}f(X_t) \\, \\mathrm{d} t}\n  \\end{align}\n  for all non-negative measurable functions \\(f\\).\n  Moreover, under the condition~\\ref{item:cond-Reinte},\n  the condition~\\ref{item:cond-regular} holds if and only if\n  \\(x \\mapsto r_q(x)\\) is continuous.\n\\end{Lem}\n\nIf \\(r_q(x)\\) is bounded in \\(x\\in\\ensuremath{\\mathbb{R}}\\),~\\cite[Corollary II.18]{MR1406564}\nimplies that\nthe Laplace transform of \\(T_0\\)\ncan be represented as\n\\begin{align}\\label{eq:exp-hit-time}\n  \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}} = \\frac{r_q(-x)}{r_q(0)}, \\quad q>0, \\; x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\n\n\\begin{Prop}\\label{Prop:r_q-represent}\n  Suppose that the condition~\\ref{item:assumption} holds.\n  Then the bounded continuous resolvent density can be expressed as\n  \\begin{align}\n    r_q(x) =\n    \\frac{1}{2\\pi}\\int_{-\\infty}^\\infty\n    \\frac{\\mathrm{e}^{-\\mathrm{i}\\lambda x}}{q+\\varPsi(\\lambda)}\n    \\, \\mathrm{d} \\lambda\n    =\\frac{1}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{\\mathrm{e}^{-\\mathrm{i}\\lambda x}}{q+\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n  \\end{align}\n  for all \\(q > 0\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\).\n\\end{Prop}\nProposition~\\ref{Prop:r_q-represent} can be proved using Fourier\ninversion formula; see, e.g.,~\\cite[Lemma 2]{MR1894112}\nand~\\cite[Corollary 15.1]{MR3838874}.\nBy Lemmas~\\ref{Lem:iff-cond-notCPP-regular}\nand~\\ref{Lem:resolvent-density} and Proposition~\\ref{Prop:r_q-represent},\nthe condition\n~\\ref{item:assumption}\nimplies~\\ref{item:cond-not-CPP}--\\ref{item:cond-nu}.\n\\begin{Lem}[Tsukada {\\cite[Lemma 15.5]{MR3838874}}]\\label{Lem:inte-of-varPsi}\n  Suppose that the condition~\\ref{item:assumption} holds.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item \\(\\abs{\\varPsi(\\lambda)} \\to \\infty\\)\n          as \\(\\lambda\\to\\pm\\infty\\);\\label{item:varpsi-to-infty}\n    \\item \\(\\displaystyle \\int_\\delta^\\infty \\abs*{\\frac{1}{\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty\\)\n          for all \\(\\delta>0\\);\\label{Lem-item:infty-inte}\n    \\item \\(\\displaystyle \\int_0^\\delta \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda < \\infty\\)\n          for all \\(\\delta>0\\);\\label{Lem-item:zero-inte}\n    \\item \\(\\displaystyle \\lim_{q\\to 0+}\n          \\int_0^\\infty \\abs*{\\frac{q}{q+\\varPsi(\\lambda)}}\n          \\, \\mathrm{d} \\lambda\n          = 0\\).\n          In particular, \\(qr_q(x) \\to 0\\) as \\(q \\to 0+\\).\\label{Lem-item:qr_q}\n  \\end{enumerate}\n\\end{Lem}\n\n\\subsection{Local time and its excursion measure}\\label{Subsec:LT-em}\nAssume the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\nThen we can define local time at \\(0\\), which we denote by \\(L=(L_t,t\\ge 0)\\).\nNote that \\(L\\) is continuous in \\(t\\) and satisfies\n\\begin{align}\\label{eq:regularity-of-L}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt} \\mathrm{d} L_t} = r_q(-x),\n  \\quad q>0,\\;x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\nSee, e.g.,~\\cite[Section V]{MR1406564}.\nIn particular, \\(r_q(x)\\) is non-decreasing as \\(q \\to 0+\\).\nLet \\(\\eta=\\rbra{\\eta_l, l \\ge 0}\\) denote\nthe right-continuous inverse of \\(L\\)\nwhich is given as\n\\(\\eta_l = \\inf\\cbra{t > 0\\colon L_t > l}\\).\nThen the process \\((\\eta, \\ensuremath{\\mathbb{P}})\\) is a possibly killed\nsubordinator, and its Laplace transform is\n\\(\\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-q\\eta_l}} = \\mathrm{e}^{-l\/r_q(0)}\\), for\n\\(l,q > 0\\),\nsee, e.g.,~\\cite[Proposition V.4]{MR1406564}.\n\nNow we can apply It\\^{o}'s excursion theory.\nLet \\(n\\) denote the characteristic measure of excursions away from the origin.\nWe denote \\(e_l\\) for excursion which starts at local time \\(l\\).\nThen we see that the subordinator \\(\\eta\\) has no drift and its L\\'{e}vy measure is\n\\(n(T_0\\in \\mathrm{d} x)\\).\nIn particular, we have\n\\begin{align}\\label{eq:exp-nt0}\n  \\mathrm{e}^{-l\/r_q(0)} = \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-q\\eta_l}}\n  = \\exp\\rbra{-l n\\sbra{1-\\mathrm{e}^{-qT_0}}},\n  \\quad l \\ge 0.\n\\end{align}\nThis implies that\n\\begin{align}\\label{eq:rel-n-r_q}\n  n\\sbra{1 - \\mathrm{e}^{-qT_0}} = \\frac{1}{r_q(0)},\n\\end{align}\nwhich is also obtained from~\\cite[(3.16)]{MR2552915}.\nNow set\n\\begin{align}\n  \\kappa = \\lim_{q\\to 0+}\\frac{1}{r_q(0)} = n(T_0 =\\infty).\\label{eq:kappa}\n\\end{align}\nIt is known that \\(\\kappa = 0\\) (resp.\\ \\(\\kappa > 0\\))\nif and only if \\(X\\) is recurrent (resp.\\ transient);\nsee, e.g.,~\\cite[Theorem I.17]{MR1406564}\nand~\\cite[Theorem 37.5]{MR1739520}.\nIt is also known that\n\\(X\\) is recurrent if and only if \\(X\\) is point recurrent;\nsee, e.g.,~\\cite[Remark 43.12]{MR1739520}\n(see also~\\cite[Excercise II.6.4]{MR1406564}).\nUnder the assumption~\\ref{item:assumption},\nwe can prove this fact by using Theorem~\\ref{Thm:exist-h};\nin fact,\nby equations~\\eqref{eq:exp-hit-time} and~\\eqref{eq:kappa},\nit holds that, for \\(x\\in\\ensuremath{\\mathbb{R}}\\),\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x(T_0<\\infty) &= \\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\n  = \\lim_{q\\to 0+}\\frac{r_q(-x)}{r_q(0)}\n  = 1+\\lim_{q\\to 0+}\\frac{h_q(x)}{r_q(0)}\n  =1.\n\\end{align}\n\nWe define \\(D = \\cbra{l\\ge 0\\colon \\eta_{l-} < \\eta_l}\\).\nThen the following formula is well-known in the excursion theory.\n\\begin{Lem}[{Compensation formula; see e.g.,\n  Bertoin~\\cite[Corollary IV.11]{MR1406564}}]\\label{Lem:compensation-formula}\n  Let \\(F(t,\\omega,e)\\) be a measurable functional\n  on \\([0, \\infty) \\times \\ensuremath{\\mathcal{D}} \\times \\ensuremath{\\mathcal{D}}\\) such that,\n  for every fixed \\(e\\in \\ensuremath{\\mathcal{D}}\\), the process\n  \\((F(t,\\cdot,e), t\\ge 0)\\) is \\((\\ensuremath{\\mathcal{F}}_t)\\)-predictable. Then\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\sum_{l\\in D} F(\\eta_{l-}, X, e_l)}\n    = \\ensuremath{\\mathbb{P}}\\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t\\, F(t, X, \\widetilde{X})},\n  \\end{align}\n  where the symbol \\(\\widetilde{\\hspace{12pt}}\\) means independence.\n\\end{Lem}\n\nLet \\(L^a =\\rbra{L_t^a, t\\ge 0}\\) denote the local time at \\(a\\in \\ensuremath{\\mathbb{R}}\\)\nwhich is normalized by\n\\begin{align}\\label{eq:regularity-of-L^a}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt} \\mathrm{d} L^a_t} = r_q(a-x),\n  \\quad  q>0,\\;x \\in \\ensuremath{\\mathbb{R}}.\n\\end{align}\nWe denote by \\(\\eta^a=(\\eta^a_u, u\\ge 0)\\)\nthe right-continuous inverse of \\(L^a\\) given by\n\\(\\eta^a_u=\\inf\\cbra{t>0\\colon L_t^a>u}\\).\nWe denote by \\(n^a\\) the characteristic measure of excursions away from \\(a\\).\n\n\\section{The renormalized zero resolvent}\\label{Sec:h}\nLet us consider the existence and properties of\nthe renormalized zero resolvent in Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h}.\nRecall that we assume \\(X\\) is recurrent, i.e., \\(\\kappa=0\\),\nand assume the condition~\\ref{item:assumption}.\n\\subsection{Key lemmas for the renormalized zero resolvent}\nTo show Theorems~\\ref{Thm:exist-h} and~\\ref{Thm:property-h},\nwe prepare some lemmas.\nRecall that \\(m^2\\) has been introduced in~\\eqref{eq:second-moment};\n\\(m^2=\\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\\).\n\\begin{Lem}\\label{Lem:psi-conv}\n  The following assertions hold.\n  \\begin{enumerate}\n    \\item If \\(m^2<\\infty\\), then\n          \\begin{align}\n            \\varPsi(\\lambda)\n            = \\frac{1}{2}\\sigma^2\\lambda^2\n            + \\int_\\ensuremath{\\mathbb{R}} \\rbra{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}+\\mathrm{i} \\lambda x}\\nu(\\mathrm{d} x),\n          \\end{align}\\label{Lem-item:psi-repre}\n          and\n          \\begin{align}\\label{eq:psi\/lambda2}\n            \\lim_{\\lambda \\to 0}\\frac{\\varPsi(\\lambda)}{\\lambda^2}\n            =\\lim_{\\lambda \\to 0}\\frac{\\theta(\\lambda)}{\\lambda^2}\n            = \\frac{m^2}{2}\n            =\\frac{1}{2}\\rbra*{\\sigma^2 + \\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x)}.\n          \\end{align}\\label{Lem-item:psi-conv}\n    \\item If \\(m^2=\\infty\\), then\n          \\begin{align}\n            \\lim_{\\lambda \\to 0}\\frac{\\lambda^2}{\\varPsi(\\lambda)}\n            =\\lim_{\\lambda \\to 0}\\frac{\\lambda^2}{\\theta(\\lambda)}=0.\n          \\end{align}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  It is well-known\n  (see, e.g.,~\\cite[Theorem 2.7]{MR3155252})\n  that \\(X\\) has finite variance\n  if and only if \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) < \\infty\\).\n  We first assume that\n  \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) < \\infty\\). Then we know\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1}\\) and \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\\) are finite and\n  \\begin{gather}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_1}\n    =\\mathrm{i}\\varPsi^\\prime(0)\n    = - v + \\int_{\\ensuremath{\\mathbb{R}}\\setminus (-1, 1)} x \\nu(\\mathrm{d} x), \\\\\n    \\ensuremath{\\mathbb{P}}\\sbra{X_1^2}\n    = \\varPsi^{\\prime\\prime}(0)\n    = \\sigma^2 + \\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) +\n    {\\ensuremath{\\mathbb{P}}\\sbra{X_1}}^2.\n  \\end{gather}\n  Since \\(X\\) is recurrent, we have \\(\\ensuremath{\\mathbb{P}}[X_1] = 0\\);\n  see, e.g.,~\\cite[Problem 7.2]{MR3155252}.\n  This implies that\n  \\begin{align}\n    \\varPsi(\\lambda)\n    = \\frac{1}{2} \\sigma^2 \\lambda\n    + \\int_\\ensuremath{\\mathbb{R}} \\rbra{1-\\mathrm{e}^{\\mathrm{i}\\lambda x} + \\mathrm{i} \\lambda x} \\nu(\\mathrm{d} x).\n  \\end{align}\n  By l'H\u00f4pital's rule, we obtain\n  \\begin{align}\n    \\lim_{\\lambda\\to 0} \\frac{\\varPsi(\\lambda)}{\\lambda^2}\n    = \\lim_{\\lambda\\to 0} \\frac{\\varPsi^\\prime(\\lambda)}{2\\lambda}\n    = \\frac{\\varPsi^{\\prime\\prime}(0)}{2} = \\frac{m^2}{2}.\n  \\end{align}\n  Taking real parts on both sides, we also have\n  \\begin{align}\n    \\lim_{\\lambda\\to 0}\\frac{\\theta(\\lambda)}{\\lambda^2} = \\frac{m^2}{2}.\n  \\end{align}\n  We next assume that \\(\\int_\\ensuremath{\\mathbb{R}} x^2 \\nu(\\mathrm{d} x) = \\infty\\).\n  Then we know \\(m^2 = \\infty\\).\n  By~\\eqref{eq:theta}, we have\n  \\begin{align}\n    \\abs*{\\frac{\\varPsi(\\lambda)}{\\lambda^2}}\n    \\ge \\abs*{\\frac{\\theta(\\lambda)}{\\lambda^2}}\n    \\ge \\int_\\ensuremath{\\mathbb{R}} \\frac{1-\\cos \\lambda x}{\\lambda^2}\\nu(\\mathrm{d} x).\n  \\end{align}\n  Using Fatou's lemma and l'H\u00f4pital's rule, we obtain\n  \\begin{align}\n    \\liminf_{\\lambda\\to 0}\\abs*{\\frac{\\varPsi(\\lambda)}{\\lambda^2}}\n    \\ge \\liminf_{\\lambda\\to 0}\\abs*{\\frac{\\theta(\\lambda)}{\\lambda^2}}\n     & \\ge \\int_\\ensuremath{\\mathbb{R}} \\liminf_{\\lambda\\to 0}\n    \\frac{1-\\cos \\lambda x}{\\lambda^2}\n    \\nu(\\mathrm{d} x)                                       \\\\\n     & = \\int_\\ensuremath{\\mathbb{R}} \\frac{x^2}{2} \\nu(\\mathrm{d} x) = \\infty.\n  \\end{align}\n  Therefore the proof is complete.\n\\end{proof}\n\nWhen \\(m^2<\\infty\\), the next lemma\nis essential for the renormalized zero resolvent.\n\\begin{Lem}\\label{Lem:fin-omega-integral}\n  Assume \\(m^2 < \\infty\\).\n  Then it holds that\n  \\begin{align}\\label{eq:fin-omega-integral}\n    \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}} \\, \\mathrm{d} \\lambda \\\n    < \\infty.\n  \\end{align}\n  Consequently,\n  Tsukada's condition~\\ref{item:Tsukada-b} holds.\n\\end{Lem}\n\n\\begin{proof}[Proof of Lemma~\\ref{Lem:fin-omega-integral}]\n  By Lemma~\\ref{Lem:psi-conv},\n  we have\n  \\begin{align}\n    \\omega(\\lambda)\n    = \\int_{\\ensuremath{\\mathbb{R}}} \\rbra{\\lambda x - \\sin \\lambda x}\n    \\nu(\\mathrm{d} x).\n  \\end{align}\n  Hence we have\n  \\begin{align}\n    \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}} \\, \\mathrm{d} \\lambda\n     & = \\int_\\ensuremath{\\mathbb{R}} \\abs*{\\int_\\ensuremath{\\mathbb{R}}\n      \\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}\n      \\nu(\\mathrm{d} x)}\n    \\, \\mathrm{d} \\lambda                                  \\\\\n     & \\le \\rbra*{\\int_{-\\infty}^0 + \\int_0^\\infty}\n    \\rbra*{\\int_{-\\infty}^0 + \\int_0^\\infty}\n    \\abs*{\\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}}\n    \\nu(\\mathrm{d} x)\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Since \\(\\lambda x - \\sin \\lambda x \\ge 0\\)\n  for \\((x, \\lambda) \\in {(0, \\infty)}^2\\),\n  it holds that\n  \\begin{align}\n    \\int_0^\\infty \\int_0^\\infty\n    \\abs*{\\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}}\n    \\nu(\\mathrm{d} x)\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\int_0^\\infty \\int_0^\\infty\n    \\frac{\\lambda x - \\sin \\lambda x}{\\lambda^3}\n    \\, \\mathrm{d} \\lambda \\,\n    \\nu(\\mathrm{d} x)                                                \\\\\n     & =\n    \\int_0^\\infty x^2 \\nu(\\mathrm{d} x)\n    \\int_0^\\infty\n    \\frac{\\xi - \\sin \\xi}{\\xi^3} \\, \\mathrm{d}\\xi                    \\\\\n     & = \\frac{4}{\\pi} \\int_0^\\infty x^2 \\nu(\\mathrm{d} x) < \\infty.\n  \\end{align}\n  Other integrals are also proved to be finite by the same discussions,\n  and we obtain~\\eqref{eq:fin-omega-integral}.\n  By~(\\ref{Lem-item:psi-repre}) of Lemma~\\ref{Lem:psi-conv},\n  we see that \\(\\lambda^2\/\\varPsi(\\lambda)\\) is bounded near \\(\\lambda=0\\).\n  Thus we have\n  \\begin{align}\n    \\int_0^1 \\abs*{\\Im\\rbra*{\\frac{\\lambda}{\\varPsi(\\lambda)}}}\\,\\mathrm{d}\\lambda\n    = \\int_0^1 \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\\, \\mathrm{d} \\lambda <\\infty.\n  \\end{align}\n  Since~\\ref{item:assumption} is assumed in this section,\n  this implies that Tsukada's condition~\\ref{item:Tsukada-b} holds.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:exist-hheq}\n  The following assertions hold.\n  \\begin{enumerate}\n    \\item \\(\\displaystyle\n          h^S(x)\n          \\coloneqq \\lim_{q\\to 0+}(h_q(x) + h_q(-x))\n          = \\frac{2}{\\pi}\n          \\int_0^\\infty\n          \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda,\n          \\quad\\text{for \\(x\\in\\ensuremath{\\mathbb{R}}\\).}\n          \\)\\label{Lem-item:exist-h^S}\n    \\item \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h^S(x)}{\\abs{x}} =\n          \\frac{2}{m^2} \\in [0,\\infty).\n          \\)\\label{Lem-item:h^S\/x}\n    \\item For \\(x, y \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^D(x, y)\n             & \\coloneqq \\lim_{q\\to 0+}\\cbra{h_q(y+2x) - 2h_q(y+x) + h_q(y)} \\\\\n             & = \\frac{2}{\\pi} \\int_0^\\infty\n            \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i} \\lambda(y+x)}\n              \\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda.\n          \\end{align}\n          Moreover, it holds that\n          \\(\\lim_{y\\to\\pm\\infty} h^D(x, y) = 0\\).\\label{Lem-item:h^D}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  \\noindent (\\ref{Lem-item:exist-h^S})\n  By~\\eqref{eq:def-h_q}, it holds that\n  \\begin{align}\\label{eq:h_q^s}\n    h_q(x)+h_q(-x)\n    = \\frac{2}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\,\\mathrm{d}\\lambda.\n  \\end{align}\n  Since \\(\\theta(\\lambda) \\ge 0\\), we have\n  \\(\\abs{q+\\varPsi(\\lambda)}\\ge \\abs{\\varPsi(\\lambda)}\\).\n  Hence it holds that\n  \\begin{align}\\label{eq:hs-conv-DCT}\n    \\abs*{\\Re\\rbra*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}}\n    \\le \\abs*{\\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\le \\frac{1-\\cos \\lambda x}{\\abs{\\varPsi(\\lambda)}}\n    \\le \\frac{{(\\lambda x)}^2 \\wedge 2}{\\abs{\\varPsi(\\lambda)}},\n  \\end{align}\n  which is integrable in \\(\\lambda>0\\) by Lemma~\\ref{Lem:inte-of-varPsi}.\n  Then we may apply the dominated convergence theorem\n  to deduce that\n  \\begin{align}\n    h_q(x)+h_q(-x) \\longrightarrow\n    \\frac{2}{\\pi}\\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\\,\\mathrm{d}\\lambda,\n    \\quad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n\n  \\noindent  (\\ref{Lem-item:h^S\/x})\n  We only consider the case \\(x \\to \\infty\\) since the case\n  \\(x \\to -\\infty\\) can be proved in the same way.\n  For any \\(\\delta>0\\), we have\n  \\begin{align}\n    \\abs*{\\frac{2}{\\pi x}\n    \\int_\\delta^\\infty\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda}\n    \\le \\frac{2}{\\pi\\abs{x}}\n    \\int_\\delta^\\infty \\abs*{\\frac{2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0, \\quad \\text{as \\(x\\to\\infty\\).}\\label{eq:h^S\/x-big}\n  \\end{align}\n  Fix \\(\\varepsilon > 0\\).\n  By Lemma~\\ref{Lem:psi-conv},\n  we can choose \\(\\delta>0\\) such that\n  \\begin{align}\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}\n      - \\frac{2}{m^2}} < \\varepsilon,\n    \\quad \\text{for } \\abs{\\lambda} < \\delta.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n     & \\le \\rbra*{\\frac{2}{m^2} + \\varepsilon}\n    \\frac{2}{\\pi x}\\int_0^\\delta\n    \\frac{1-\\cos \\lambda x}{\\lambda^2}\n    \\, \\mathrm{d} \\lambda                             \\\\\n     & \\to \\rbra*{\\frac{2}{m^2} + \\varepsilon}\n    \\frac{2}{\\pi}\n    \\int_0^\\infty \\frac{1-\\cos \\xi}{\\xi^2}\n    \\, \\mathrm{d} \\xi                                 \\\\\n     & = \\frac{2}{m^2} + \\varepsilon,\n    \\quad \\text{as \\(x\\to\\infty\\).}\n  \\end{align}\n  Thus we obtain\n  \\begin{align}\n    \\limsup_{x\\to\\infty}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\le\n    \\frac{2}{m^2} + \\varepsilon.\\label{eq:h^S\/x-upper}\n  \\end{align}\n  In the same way, we can show that\n  \\begin{align}\n    \\liminf_{x\\to\\infty}\n    \\frac{2}{\\pi x}\n    \\int_0^\\delta\n    \\Re\\rbra*{\\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\ge\n    \\frac{2}{m^2} - \\varepsilon.\\label{eq:h^S\/x-lower}\n  \\end{align}\n  By~\\eqref{eq:h^S\/x-big},~\\eqref{eq:h^S\/x-upper}\n  and~\\eqref{eq:h^S\/x-lower},\n  the result follows.\n\n  \\noindent (\\ref{Lem-item:h^D})\n  By~\\eqref{eq:def-h_q}, we have\n  \\begin{align}\n    h_q(y+2x) - 2h_q(y+x) + h_q(y)\n    =  \\frac{2}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{ \\mathrm{e}^{\\mathrm{i}\\lambda (y+x)}\n      \\frac{1-\\cos \\lambda x}{q+\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  By the same way as~\\eqref{eq:hs-conv-DCT},\n  we may apply the dominated convergence theorem\n  to obtain\n  \\begin{align}\n    h^D(x, y)\n     & = \\lim_{q\\to 0+} \\cbra{h_q(y+2x) - 2h_q(y+x) + h_q(y)} \\\\\n     & = \\frac{2}{\\pi} \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i} \\lambda(y+x)}\n      \\frac{1-\\cos \\lambda x}{\\varPsi(\\lambda)}} \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Furthermore, by the Riemann--Lebesgue lemma, we obtain\n  \\(h^D(x, y) \\longrightarrow 0\\)\n  as \\(y \\to \\pm \\infty\\).\n\\end{proof}\n\n\\subsection{Proofs of Theorems~\\ref{Thm:exist-h}\n  and~\\ref{Thm:property-h}}\\label{Subsec:pf-h}\nWe separate the proof into the two cases:\n\\(m^2<\\infty\\) and \\(m^2=\\infty\\).\nWe first show the existence and properties of \\(h\\) in the case \\(m^2<\\infty\\).\nIn this case, we can use the dominated convergence theorem.\n\\begin{proof}[Proof of~(\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h}\n    in the case \\(m^2<\\infty\\)]\n  For each \\(x, \\lambda \\in \\ensuremath{\\mathbb{R}}\\),\n  we observe that\n  \\begin{align}\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n    = \\frac{(q+\\theta(\\lambda)) (1-\\cos\\lambda x)\n      + \\omega(\\lambda) \\sin\\lambda x}\n    {\\abs{q+\\varPsi(\\lambda)}^2}.\n  \\end{align}\n  Hence it follows from \\(\\theta(\\lambda) \\ge 0\\) that\n  \\begin{align}\n     & \\abs*{\\Re\\rbra*{\n    \\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}}    \\\\\n     & \\le\n    \\rbra*{\\frac{1-\\cos\\lambda x}{\\abs{\\varPsi(\\lambda)}}\n      + \\frac{\\lambda^4}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\n      \\abs*{\\frac{\\sin\\lambda x}{\\lambda}}}\n    \\wedge\n    \\abs*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}} \\\\\n     & \\le\n    \\rbra*{\\abs*{\\frac{\\lambda^2 x^2}{\\varPsi(\\lambda)}}\n      + \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n      \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}}\n      \\abs{x}}\n    \\wedge\n    \\abs*{\\frac{2}{\\varPsi(\\lambda)}}.\n  \\end{align}\n  By Lemma~\\ref{Lem:inte-of-varPsi},\n  (\\ref{Lem-item:psi-conv}) of Lemma~\\ref{Lem:psi-conv}\n  and Lemma~\\ref{Lem:fin-omega-integral},\n  the last quantity is integrable in \\(\\lambda>0\\).\n  Therefore, we may apply the dominated convergence theorem\n  to conclude that\n  \\begin{align}\n    h_q(x)\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{q+\\varPsi(\\lambda)}}\n    \\,\\mathrm{d} \\lambda \\\\\n     & \\to\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i} \\lambda x}}{\\varPsi(\\lambda)}}\n    \\,\\mathrm{d} \\lambda,\n    \\quad \\text{as \\(q\\to 0+\\).}\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:property-h}\n    in the case \\(m^2<\\infty\\)]\n  \\noindent (\\ref{Thm-item:h\/x-infinity})\n  We take \\(\\delta>0\\) sufficiently small.\n  By (\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h},\n  we have\n  \\begin{align}\n    \\hspace{-20pt}\\frac{h(x)}{x}\n     & =\n    \\frac{1}{\\pi x}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda \\\\\n     & =\n    \\frac{1}{\\pi x}\\cbra*{\n      \\int_\\delta^\\infty\n      \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n      \\, \\mathrm{d} \\lambda\n      +\n      \\int_0^\\delta\n      \\frac{\\omega(\\lambda)\\sin\\lambda x}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\, \\mathrm{d} \\lambda\n      +\n      \\int_0^\\delta\n      \\frac{\\theta(\\lambda)(1-\\cos\\lambda x)}\n      {\\abs{\\varPsi(\\lambda)}^2}\n      \\, \\mathrm{d} \\lambda}.\\label{eq:three-inte}\n  \\end{align}\n  For the first integral in~\\eqref{eq:three-inte}, we have\n  \\begin{align}\n    \\abs*{\n      \\frac{1}{\\pi x}\n      \\int_\\delta^\\infty\n      \\Re\\rbra*{\\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n      \\, \\mathrm{d} \\lambda\n    }\n    \\le\n    \\frac{1}{\\pi\\abs{x}}\n    \\int_\\delta^\\infty\n    \\abs*{\\frac{2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\(x \\to \\pm\\infty\\).}\n  \\end{align}\n  For the second integral in~\\eqref{eq:three-inte}, since we have\n  \\begin{align}\\label{eq:sin-omega-DCT}\n    \\frac{\\abs{\\omega(\\lambda)\\sin\\lambda x}}\n    {\\abs{\\varPsi(\\lambda)}^2\\abs{x}}\n    \\le\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\abs*{\\frac{\\omega(\\lambda)}{\\lambda^3}},\n  \\end{align}\n  which is integrable in \\(\\lambda\\in (0, \\delta)\\)\n  by (\\ref{Lem-item:psi-conv}) of Lemma~\\ref{Lem:psi-conv} and\n  Lemma~\\ref{Lem:fin-omega-integral},\n  we can apply the dominated convergence theorem to obtain\n  \\begin{align}\n    \\frac{1}{\\pi x}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\( x \\to \\pm\\infty\\).}\n  \\end{align}\n  For the third integral in~\\eqref{eq:three-inte},\n  we can apply the similar discussion as\n  the proof of (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}.\n  Therefore we obtain\n  \\begin{align}\n    \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} = \\frac{1}{m^2}.\n  \\end{align}\n\n  \\noindent  (\\ref{Thm-item:h-h-limit})\n  Take \\(\\delta>0\\) sufficiently small.\n  Then we have\n  \\begin{align}\n     & h(y+x) - h(y) \\\\\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\mathrm{e}^{\\mathrm{i}\\lambda x}}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda   \\\\\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\cos\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    + \\frac{1}{\\pi}\n    \\rbra*{\\int_0^\\delta + \\int_\\delta^\\infty}\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  By the Riemann--Lebesgue lemma, we obtain\n  \\begin{align}\n    \\lim_{y\\to\\pm\\infty}\\int_0^\\infty\n    \\Re\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\n      \\frac{1-\\cos\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    =\\lim_{y\\to\\pm\\infty}\\int_\\delta^\\infty\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n    =0.\n  \\end{align}\n  On the other hand, we have\n  \\begin{align}\n   \\hspace{-25pt}\\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\Im\\rbra*{\\mathrm{e}^{\\mathrm{i}\\lambda y}\\frac{\\sin\\lambda x}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\theta(\\lambda)\\sin\\lambda x \\sin\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    - \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x \\cos\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda.\\label{eq:h-property-finvar}\n  \\end{align}\n  By~\\eqref{eq:sin-omega-DCT},\n  we may apply the Riemann--Lebesgue lemma\n  to the second integral and we have\n  \\begin{align}\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\omega(\\lambda)\\sin\\lambda x \\cos\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n    \\longrightarrow 0,\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  For the first integral of~\\eqref{eq:h-property-finvar}, we use\n  Lemma~\\ref{Lem:psi-conv} and\n  Lemma~\\ref{Lem:JT-DI} below\n  to show that\n  \\begin{align}\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\frac{\\theta(\\lambda)\\sin\\lambda x \\sin\\lambda y}\n    {\\abs{\\varPsi(\\lambda)}^2}\n    \\, \\mathrm{d} \\lambda\n     & =\n    \\frac{1}{\\pi}\n    \\int_0^\\delta\n    \\abs*{\\frac{\\lambda^2}{\\varPsi(\\lambda)}}^2\n    \\frac{\\theta(\\lambda)}{\\lambda^2}\n    \\frac{\\sin \\lambda x}{\\lambda}\n    \\frac{\\sin \\lambda y}{\\lambda}\n    \\, \\mathrm{d} \\lambda            \\\\\n     & \\to \\pm \\frac{x}{m^2},\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  This ends the proof.\n\\end{proof}\nThe following lemma is an elementary calculus.\n\\begin{Lem}[Jordan's theorem for the Dirichlet integral]\\label{Lem:JT-DI}\n  Let \\(\\delta>0\\) and let\n  \\(f\\colon (0,\\delta)\\to \\ensuremath{\\mathbb{R}}\\) be continuous and be of bounded variation.\n  Then it holds that\n  \\begin{align}\n    \\lim_{x\\to \\pm\\infty}\\frac{2}{\\pi} \\int_0^\\delta\n    f(\\lambda)\\frac{\\sin \\lambda x}{\\lambda} \\, \\mathrm{d} \\lambda=\\pm f(0+),\n  \\end{align}\n  where \\(f(0+)=\\lim_{\\lambda\\to 0+}f(\\lambda)\\).\n\\end{Lem}\n\\begin{proof}\n  By integration by parts, we have\n  \\begin{align}\n    \\frac{2}{\\pi} \\int_0^\\delta\n    f(\\lambda)\\frac{\\sin \\lambda x}{\\lambda} \\, \\mathrm{d} \\lambda\n    &= \\frac{2}{\\pi}  \\int_0^\\delta \\rbra*{\\int_0^\\lambda\n    \\mathrm{d} f( \\xi)}\\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\n    +\\frac{2}{\\pi} f(0+) \\int_0^\\delta \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\\\\\n    &=\\frac{2}{\\pi}\\int_0^\\delta \\rbra*{\\int_\\xi^\\delta\n    \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda} \\mathrm{d} f(\\xi)\n    +\\frac{2}{\\pi} f(0+) \\int_0^\\delta \\frac{\\sin \\lambda x}{\\lambda}\\, \\mathrm{d} \\lambda\\\\\n    &\\to 0 \\pm f(0+), \\qquad\\text{as \\(x\\to\\pm \\infty\\).}\n  \\end{align}\n  Thus we obtain the desired result.\n\\end{proof}\n\nThen let us prove the existence of \\(h\\)\nin the case \\(X\\) is recurrent\nand \\(m^2=\\infty\\). Its proof\nis quite different from that in the case \\(m^2<\\infty\\).\n\n\\begin{proof}[Proof of~(\\ref{Thm-item:exist-h}) of Theorem~\\ref{Thm:exist-h}\n    in the case \\(m^2=\\infty\\)]\n  Since \\(h_q(0) = 0\\),\n  we have the limit \\(h(0)=\\lim_{q\\to 0+}h_q(0)=0\\).\n\n  Fix \\(a \\ne 0\\) and set\n  \\begin{align}\n    \\overline{h}(a) = \\limsup_{q\\to 0+}h_q(a), \\quad\n    \\underline{h}(a) = \\liminf_{q\\to 0+}h_q(a).\n  \\end{align}\n  We also define\n  \\(\\varDelta = \\overline{h}(a) - \\underline{h}(a) \\ge 0\\) and\n  \\(A = \\{2^j a\\colon j=0,1,2,3,\\ldots\\}\\).\n\n  It follows from\n  \\(h_q(x) \\le h_q(x)+h_q(-x)\\)\n  and~\\eqref{eq:h_q^s}--\\eqref{eq:hs-conv-DCT}\n  that \\({\\{h_q(x)\\}}_{q>0}\\) is bounded for each \\(x\\in \\ensuremath{\\mathbb{R}}\\).\n  Hence, by the diagonal argument,\n  we can take two\n  sequences \\(\\{q_n\\}, \\{q^\\prime_n\\}\\),\n  which satisfies the following three conditions:\n  \\begin{itemize}\n    \\item \\( q_n, q^\\prime_n \\to 0+\\) as \\(n\\to\\infty\\);\n    \\item \\(\\lim_{n\\to\\infty}h_{q_n}(x)\\) and\n          \\(\\lim_{n\\to\\infty}h_{q^\\prime_n}(x)\\) exist and are finite for\n          each \\(x \\in A\\);\n    \\item \\( \\overline{h}(a) = \\lim_{n\\to\\infty} h_{q_n}(a)\\) and\n          \\(\\underline{h}(a) = \\lim_{n\\to\\infty} h_{q^\\prime_n}(a)\\).\n  \\end{itemize}\n  Then we define, for \\(x \\in A\\),\n  \\begin{align}\n    \\overline{h}(x) = \\lim_{n\\to\\infty}h_{q_n}(x),\n    \\quad\n    \\underline{h}(x) = \\lim_{n\\to\\infty} h_{q^\\prime_n}(x).\n  \\end{align}\n  By (\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq}, we have\n  \\begin{align}\n    h^D(x,0)=\\overline{h}(2x) - 2 \\overline{h}(x)\n    =\\underline{h}(2x) - 2 \\underline{h}(x),\n    \\quad x \\in A.\n  \\end{align}\n  This implies that\n  \\(\\overline{h}(2x) - \\underline{h}(2x)\n  =2(\\overline{h}(x) - \\underline{h}(x))\\).\n  Hence it holds that\n  \\(\\overline{h}(2^j a) - \\underline{h}(2^j a)\n  = 2^j \\varDelta\\), i.e.,\n  \\begin{align}\\label{eq:diff-overunder-h}\n    \\frac{\\overline{h}(2^j a)}{2^j a} -\n    \\frac{\\underline{h}(2^j a)}{2^j a}\n    = \\frac{\\varDelta}{a}, \\quad j = 0,1,2,3,\\ldots.\n  \\end{align}\n  It follows from (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}\n  that\n  \\begin{align}\n    \\max\\cbra*{\\frac{\\overline{h}(2^j a)}{2^j a},\n      \\frac{\\underline{h}(2^j a)}{2^j a}}\n    \\le\n    \\frac{h^S(2^j a)}{2^j a} \\longrightarrow 0,\n    \\quad \\text{as \\(j\\to\\infty\\).}\n  \\end{align}\n  Thus, by letting \\(j\\to\\infty\\) in~\\eqref{eq:diff-overunder-h},\n  we obtain \\(\\varDelta = 0\\).\n  Therefore, we conclude that \\(h(a)=\\lim_{q\\to 0+}h_q(a)\\) exists.\n\\end{proof}\n\nNext, we prove (\\ref{Thm-item:unif-conv-h}) and (\\ref{Thm-item:subadditive-h})\nof Theorem~\\ref{Thm:exist-h}\nin both cases\n\\(m^2=\\infty\\) and \\(m^2<\\infty\\).\n\\begin{proof}[Proof of (\\ref{Thm-item:unif-conv-h}) and (\\ref{Thm-item:subadditive-h})\n    of Theorem~\\ref{Thm:exist-h}]\n  By the Markov property, we have, for \\(x,y\\in\\ensuremath{\\mathbb{R}}\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_{x+y}\\sbra{\\mathrm{e}^{-qT_0}} =\\ensuremath{\\mathbb{P}}_0\\sbra{\\mathrm{e}^{-qT_{-x-y}}}\n    \\ge \\ensuremath{\\mathbb{P}}_0\\sbra{\\mathrm{e}^{-qT_{-x}} \\ensuremath{\\mathbb{P}}_{-x}\\sbra{\\mathrm{e}^{-qT_{-x-y}}}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_{0}}} \\ensuremath{\\mathbb{P}}_y\\sbra{\\mathrm{e}^{-qT_0}}.\n  \\end{align}\n  Since \\(h_q(x) =r_q(0)\\rbra{1-\\ensuremath{\\mathbb{P}}_x\\sbra{e^{-qT_0}}}\\)\n  and \\(\\rbra{1-\\ensuremath{\\mathbb{P}}_x\\sbra{e^{-qT_0}}}\\rbra{1-\\ensuremath{\\mathbb{P}}_y\\sbra{e^{-qT_0}}}\\ge 0\\),\n  it holds that\n  \\begin{align}\n    h_q(x+y) \\le h_q(x) +h_q(y).\n  \\end{align}\n  Hence \\(h_q\\) is subadditive.\n  (This proof can also be found in~\\cite[Lemma 3.3]{MR3689384}.)\n  Since \\(h_q\\) is non-negative and subadditive,\n  and by~\\eqref{eq:h_q^s} and~\\eqref{eq:hs-conv-DCT},\n  it holds that\n  \\begin{align}\n    \\abs{h_q(x+\\delta) - h_q(x)}\n    \\le h_q(\\delta) + h_q(-\\delta)\n    \\le \\int_0^\\infty\n    \\abs*{\\frac{\\rbra{\\lambda \\delta}^2 \\wedge 2}{\\varPsi(\\lambda)}}\n    \\, \\mathrm{d} \\lambda.\n  \\end{align}\n  Hence \\({\\{h_q\\}}_{q>0}\\) is equi-continuous.\n  Equi-continuity and pointwise-convergence imply\n  the uniform convergence on compact subset of \\(\\ensuremath{\\mathbb{R}}\\).\n  The subadditivity of \\(h\\) follows directly from that of \\(h_q\\).\n\\end{proof}\nFinally, we show the properties of \\(h\\) in Theorem~\\ref{Thm:property-h}\nin the case \\(m^2=\\infty\\).\n\\begin{proof}[Proof of Theorem~\\ref{Thm:property-h}\n    in the case \\(m^2=\\infty\\)]\n  \\noindent (\\ref{Thm-item:h\/x-infinity})\n  This is directly from\n  (\\ref{Lem-item:h^S\/x}) of Lemma~\\ref{Lem:exist-hheq}.\n\n  \\noindent (\\ref{Thm-item:h-h-limit})\n  Since \\(h\\) is subadditive, we have\n  \\begin{align}\\label{eq:h-diff-sum}\n    \\sum_{k=1}^n \\cbra{h(kx+y)-h((k-1)x+y)}\n    = h(nx+y) - h(y) \\le h(nx).\n  \\end{align}\n  By (\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq},\n  it holds that\n  \\begin{align}\n    \\cbra{h(2x+y) - h(x+y)} - \\cbra{h(x+y) - h(y)}\n    \\longrightarrow 0,\n    \\quad \\text{as \\(y\\to\\pm\\infty\\).}\n  \\end{align}\n  Thus we have\n  \\begin{align}\\label{eq:h-diff-limsup}\n    \\limsup_{y\\to\\pm\\infty}\n    \\sum_{k=1}^n \\cbra{h(kx+y)-h((k-1)x+y)}\n    = n \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)}.\n  \\end{align}\n  Combining~\\eqref{eq:h-diff-sum} and~\\eqref{eq:h-diff-limsup}, we obtain\n  \\begin{align}\n    \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)}\n    \\le \\frac{h(nx)}{n}.\n  \\end{align}\n  Since we have \\(\\lim_{n\\to\\infty}\\frac{h(nx)}{n}=0\\)\n  by~(\\ref{Lem-item:h^D}) of Lemma~\\ref{Lem:exist-hheq}, we have\n  \\begin{align}\n    \\limsup_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)} \\le 0.\n  \\end{align}\n  Replacing \\(x\\) with \\(-x\\), we also have\n  \\begin{align}\n    \\liminf_{y\\to\\pm\\infty} \\cbra{h(y) - h(y-x)} \\ge 0.\n  \\end{align}\n  Therefore we obtain\n  \\(\n  \\lim_{y\\to\\pm\\infty} \\cbra{h(x+y) - h(y)} = 0.\n  \\)\n\\end{proof}\n\n\\subsection{The function \\(h^B\\)}\nLet us compute \\(\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\) and \\(\\ensuremath{\\mathbb{P}}_x(T_a<T_b)\\).\n\\begin{Lem}\\label{Lem:h^B-hitting-h}\n  \\begin{enumerate}\n    \\item For \\(a \\in \\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^B_q(a) \\coloneqq\n            \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^{T_a} \\mathrm{e}^{-qt}\\, \\mathrm{d} L_t}\n            = h_q(a)+h_q(-a)-\\frac{h_q(a)h_q(-a)}{r_q(0)}.\n            \\label{eq:LT-hitting-expect-q}\n          \\end{align}\\label{Lem-item:LT-hitting-expect}\n          Consequently, it holds that\n          \\begin{align}\n            h^B(a) \\coloneqq \\lim_{q\\to 0+} h_q^B(a)\n            =\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}= h(a) + h(-a).\n            \\label{eq:h^B}\n          \\end{align}\\label{Lem-item:hq-two-hitting-time}\n    \\item For \\(x, a, b \\in \\ensuremath{\\mathbb{R}}\\), \\(a\\ne b\\) and\n          for \\(q>0\\), it holds that\n          \\begin{align}\n            \\begin{aligned}\n               & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a < T_b}\n              \\\\\n               & =\n              \\frac{h_q(b-a)+h_q(x-b)-h_q(x-a)-h_q(x-b)h_q(b-a)\/r_q(0)}{h^B_q(a-b)}.\n            \\end{aligned}\n            \\label{eq:qT_a;T_a<T_b}\n          \\end{align}\n          Consequently, it holds that\n          \\begin{align}\n            \\ensuremath{\\mathbb{P}}_x(T_a < T_b)\n            = \\frac{h(b-a)+h(x-b)-h(x-a)}{h^B(a-b)}.\n            \\label{eq:two-hittig-time-prob}\n          \\end{align}\\label{Lem-item:two-hittig-time-prob}\n  \\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n  \\noindent (\\ref{Lem-item:hq-two-hitting-time})\n  We omit the proof of~\\eqref{eq:LT-hitting-expect-q},\n  which can be found\n  in~\\cite[Lemma V.11]{MR1406564}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:LT-hitting-expect-q},\n  and using~\\eqref{eq:kappa},\n  we obtain~\\eqref{eq:h^B}.\n\n  \\noindent (\\ref{Lem-item:two-hittig-time-prob})\n  By the strong Markov property, it holds that,\n  for \\(x,a,b \\in \\ensuremath{\\mathbb{R}},\\,a\\ne b, \\, q>0\\),\n  \\begin{gather}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a < T_b}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}; T_b < T_a}\\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}, \\\\\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b};T_b < T_a}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a < T_b}\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_b}}.\n  \\end{gather}\n  Combining the above two equalities, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\n    = \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}}-\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_b}}\n      \\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}}\n    {1-\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_b}}\\ensuremath{\\mathbb{P}}_b\\sbra{\\mathrm{e}^{-qT_a}}}.\n  \\end{align}\n  By~\\eqref{eq:exp-hit-time}, this implies that\n  \\begin{align}\n     & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\n    \\\\\n     & = \\frac{r_q(a-x) - r_q(b-x)r_q(a-b)\/r_q(0)}\n    {r_q(0) - r_q(b-a)r_q(a-b)\/r_q(0)}\n    \\\\\n     & = \\frac{h_q(b-a)+h_q(x-b)-h_q(x-a)-h_q(x-b)h_q(b-a)\/r_q(0)}{h^B_q(a-b)}.\n  \\end{align}\n  Hence we obtain~\\eqref{eq:qT_a;T_a<T_b}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:qT_a;T_a<T_b},\n  we obtain~\\eqref{eq:two-hittig-time-prob}.\n\\end{proof}\n\\begin{Rem}\n  The formulae~\\eqref{eq:qT_a;T_a<T_b} and~\\eqref{eq:two-hittig-time-prob} are\n  also discussed in Theorem 6.5\n  of Getoor~\\cite{MR185663}.\n  See also Proposition 5.3, Proposition 5.4\n  and Remark 5.5 of Yano--Yano--Yor~\\cite{MR2599211}.\n\\end{Rem}\nThe next theorem can be proved in the same way as\nPant\\'{\\i}~\\cite[Lemma 3.10]{MR3689384}.\n\n\\begin{Lem}[{\\cite[Lemma 3.10]{MR3689384}}]\\label{Thm:h^B-infty}\n  It holds that \\(\\lim_{x\\to\\infty} h^B(x) = \\infty\\).\n\\end{Lem}\n\\begin{proof}\n  For completeness of this paper, we give the proof.\n  Let \\(\\bm{e}\\) be an independent exponential time of mean \\(1\\)\n  and set \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\) for \\(q>0\\).\n  Then we have\n  \\begin{align}\\label{eq:h_b-eq}\n    h^B(x) =\\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}}\n    = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x\\le \\bm{e}_q} + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x > \\bm{e}_q}\n    \\ge \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}; T_x>\\bm{e}_q}.\n  \\end{align}\n  Letting \\(x\\to\\infty\\), we have\n  \\begin{align}\\label{eq:h_b-lower}\n    \\liminf_{x\\to\\infty}h^B(x) \\ge \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}} = r_q(0),\n    \\quad \\text{for all \\(q>0\\).}\n  \\end{align}\n  Since \\(X\\) is recurrent, i.e.,\n  \\(\\kappa=0\\) in~\\eqref{eq:kappa},\n  we let \\(q\\to 0+\\) to obtain\n  \\( \\liminf_{x\\to\\infty}h^B(x) \\ge \\infty\\).\n  Hence we obtain the desired result.\n\\end{proof}\n\nThe following theorem, which will be used in Section~\\ref{Sec:inv-time},\nis a generalization of the result\nin the symmetric case\nby Yano~\\cite[Theorem 6.1]{MR2603019}.\n\\begin{Thm}\\label{Thm:n-two-hitting-time}\n  For \\(a \\in \\ensuremath{\\mathbb{R}} \\setminus \\{0\\}\\), it holds that\n  \\begin{align}\\label{eq:n-T_a-T_0-inf}\n    n(T_a < T_0) = \\frac{1}{h^B(a)}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  For \\(l > 0\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l) = \\ensuremath{\\mathbb{P}}(T_a > \\eta_l) =\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_a<T_0}}> l),\n  \\end{align}\n  where \\(\\sigma_A = \\inf\\cbra{l \\colon e_l \\in A}\\)\n  for \\(A \\subset \\ensuremath{\\mathcal{D}}\\).\n  Since \\(\\sigma_A\\) is the hitting time\n  of the set \\(A\\)\n  for the\n  Poisson point process \\(((l, e_l), l\\ge 0)\\),\n  we have\n  \\begin{align}\\label{eq:L_T_a-exp-distrib}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l)\n    = \\mathrm{e}^{-l{n(T_a<T_0)}}.\n  \\end{align}\n  For more details, see, e.g.,~\\cite[Lemma 6.17]{MR3155252}.\n  In particular, \\(L_{T_a}\\) is exponentially distributed.\n  On the other hand, we know \\(h^B(a) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\)\n  by Lemma~\\ref{Lem:h^B-hitting-h}.\n  Hence we obtain~\\eqref{eq:n-T_a-T_0-inf}.\n\\end{proof}\n\n\\subsection{Examples of the renormalized zero resolvent}\n\\begin{Eg}[Brownian motion]\\label{Eg:h-brownian}\n  Assume \\(X\\) is a standard Brownian motion.\n  Since \\(m^2=\\sigma^2=1\\), it holds that\n  \\begin{align}\n    h^{(\\gamma)}(x) = \\abs{x}+\\gamma x=\\begin{cases}\n      (1+\\gamma)x & x\\ge 0, \\\\\n      (1-\\gamma)x & x\\le 0,\n    \\end{cases}\n    \\quad\\text{for \\(-1\\le \\gamma\\le 1\\).}\n  \\end{align}\n\\end{Eg}\n\\begin{Eg}[Strictly stable process]\n  Assume that \\(X\\) is a strictly stable process of index\n  \\(\\alpha\\in (1,2)\\) with the L\\'{e}vy measure\n  \\begin{align}\n    \\nu(\\mathrm{d} x) =\n    \\begin{cases}\n      c_{+} \\abs{x}^{-\\alpha-1}\\, \\mathrm{d} x & \\text{on \\((0, \\infty)\\),}  \\\\\n      c_{-} \\abs{x}^{-\\alpha-1}\\, \\mathrm{d} x & \\text{on \\((-\\infty, 0)\\),}\n    \\end{cases}\n  \\end{align}\n  where \\(c_{+},c_{-} \\ge 0\\) and \\(c_{+}+c_{-} > 0\\).\n  Then its characteristic exponent is given by\n  \\begin{align}\n    \\varPsi(\\lambda)\n    = c\\abs{\\lambda}^{\\alpha}\n    \\rbra*{1-\\mathrm{i} \\beta \\sgn(\\lambda)\\tan\\frac{\\alpha\\pi}{2}},\n  \\end{align}\n  where \\(c\\) and \\(\\beta\\) are constants defined by\n  \\begin{align}\n    c = \\frac{(c_{+}+c_{-})\\pi}{2\\alpha \\Gamma(\\alpha)\\sin(\\pi\\alpha\/2)},\n    \\quad\n    \\beta = \\frac{c_{+} - c_{-}}{c_{+}+c_{-}}.\n  \\end{align}\n  In this case, we have \\(m^2=\\infty\\) and the function\n  \\(h\\) can be represented as\n  \\begin{align}\\label{eq:stable-h}\n    h(x) = \\frac{1}{K(\\alpha)} (1-\\beta\\sgn (x))\\abs{x}^{\\alpha-1},\n  \\end{align}\n  where\n  \\begin{align}\n    K(\\alpha) = -2c\\Gamma(\\alpha)\\cos\\frac{\\pi\\alpha}{2}\n    \\rbra*{1+\\beta^2\\tan^2 \\frac{\\pi\\alpha}{2}}.\n  \\end{align}\n  For more details, see~\\cite[Section 5]{MR3072331}.\n\\end{Eg}\n\n\\section{Local time penalization with exponential clock}\\label{Sec:exp-clock}\nWe now start to deal with the penalization\nresult with exponential clock.\nLet \\(\\bm{e}\\) be an independent exponential time of mean\n\\(1\\) and for \\(q>0\\), we write \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\), which\nhas an exponential distribution of mean \\(1\/q\\).\nWe compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\),\n\\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})|\\ensuremath{\\mathcal{F}}_t}\\) and its limit as \\(q\\to 0+\\) to\ninvestigate \\(\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{F_t\n  f(L_{\\bm{e}_q})}\/\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\) for\nbounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functional \\(F_t\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with exponential clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\\).\n\\begin{Lem}\\label{Lem:expect-L_e_q}\n  Let \\(f\\) be a non-negative measurable function.\n  Then, for \\(q>0\\) and \\(x \\in \\mathbb{R}\\), it holds that\n  \\begin{align}\\label{eq:lem-expect-L_e_q}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\n    = \\frac{1}{r_q(0)}\\cbra*{h_q(x)f(0)\n    + \\rbra*{1-\\frac{h_q(x)}{r_q(0)}}\n    \\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u}.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  Using the excursion theory, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty f(L_t) q \\mathrm{e}^{-qt}\\, \\mathrm{d} t}\n     & = \\ensuremath{\\mathbb{P}}_0\\sbra*{\\sum_{u\\in D}\n    \\int_{\\eta_{u-}}^{\\eta_u} f(u)q\\mathrm{e}^{-qu}\\, \\mathrm{d} u} \\\\\n     & = \\ensuremath{\\mathbb{P}}_0 \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t \\, f(L_t) \\mathrm{e}^{-qt}\n      \\int_0^{\\widetilde{T}_0} \\mathrm{d} u \\, q\\mathrm{e}^{-qu}},\n  \\end{align}\n  where the last equality follows from Lemma~\\ref{Lem:compensation-formula}.\n  By~\\eqref{eq:exp-nt0} and~\\eqref{eq:rel-n-r_q}, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0 \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_t \\, f(L_t) \\mathrm{e}^{-qt}\n      \\int_0^{\\widetilde{T}_0} \\mathrm{d} u \\, q\\mathrm{e}^{-qu}}\n     & = \\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty \\mathrm{d} u \\, f(u) \\mathrm{e}^{-q\\eta_u}}\n    n\\sbra*{1 - \\mathrm{e}^{-qT_0}}                                            \\\\\n     & = \\frac{1}{r_q(0)}\\int_0^\\infty f(u) \\mathrm{e}^{-u\/{r_q(0)}} \\, \\mathrm{d} u.\n  \\end{align}\n  Applying the Markov property, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}\n     & = \\ensuremath{\\mathbb{P}}_x \\sbra*{\\int_0^\\infty f(L_t)q\\mathrm{e}^{-qt}\\, \\mathrm{d} t}                    \\\\\n     & = \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^{T_0} f(0) q\\mathrm{e}^{-qt}\\, \\mathrm{d} t}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\\ensuremath{\\mathbb{P}}_0\\sbra*{\\int_0^\\infty f(L_t)q\\mathrm{e}^{-qt}\\, \\mathrm{d} t} \\\\\n     & = f(0)\\rbra*{1-\\frac{r_q(-x)}{r_q(0)}}\n    + \\frac{r_q(-x)}{r_q(0)} \\int_0^\\infty f(u) \\mathrm{e}^{-u\/{r_q(0)}}\\, \\mathrm{d} u        \\\\\n     & = \\frac{1}{r_q(0)}\\cbra*{h_q(x)f(0)\n    + \\rbra*{1-\\frac{h_q(x)}{r_q(0)}}\\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u},\n  \\end{align}\n  here we used~\\eqref{eq:exp-hit-time}.\n  Therefore we obtain the desired result.\n\\end{proof}\n\n\\subsection{A.s.\\ convergence for exponential clock}\nTo calculate \\(\\ensuremath{\\mathbb{P}}_x\\sbra{F(L_{\\bm{e}_q})|\\ensuremath{\\mathcal{F}}_t}\\),\nwe separate into the two cases \\(\\cbra{t<\\bm{e}_q }\\) and \\(\\cbra{\\bm{e}_q\\le t}\\).\n\nLet \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\). For \\(q > 0\\),\ndefine\n\\begin{align}\n  N_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); t < \\bm{e}_q | \\ensuremath{\\mathcal{F}}_t}, \\\\\n  M_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}) | \\ensuremath{\\mathcal{F}}_t},               \\\\\n  A_t^{q} & = M_t^{q} - N_t^{q}\n  = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); \\bm{e}_q \\le t | \\ensuremath{\\mathcal{F}}_t},\n\\end{align}\nand \\(h^{(\\gamma)}\\) and \\(M_t^{(\\gamma)}\\) are defined in~\\eqref{eq:def-h-gamma}\nand~\\eqref{eq:def-mart}.\n\n\\begin{Thm}\n  For \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\lim_{q\\to 0+}N_t^{q}\n    =\\lim_{q\\to 0+} M_t^{q}\n    = M_t^{(0)}, \\qquad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.}\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  By the Markov property and the additivity of \\(L\\),\n  we have\n  \\begin{align}\n    N_t^{q}\n     & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); t < \\bm{e}_q | \\ensuremath{\\mathcal{F}}_t}\n    \\\\\n     & = r_q(0) \\mathrm{e}^{-qt} \\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\n    \\sbra{f(L_t+\\widetilde{L}_{\\widetilde{\\bm{e}}_q})}\n    \\\\\n     & = \\mathrm{e}^{-qt}\n    \\cbra*{h_q(X_t)f(L_t) +\n      \\rbra*{1-\\frac{h_q(X_t)}{r_q(0)}}\n      \\int_0^\\infty \\mathrm{e}^{-u\/r_q(0)} f(L_t + u) \\, \\mathrm{d} u},\n  \\end{align}\n  here the last equality, we used Lemma~\\ref{Lem:expect-L_e_q}.\n  Since \\(1-\\frac{h_q(X_t)}{r_q(0)}=\\ensuremath{\\mathbb{P}}_{X_t}\\sbra{\\mathrm{e}^{-qT_0}} \\to 1\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\) and\n  since \\(\\int_0^\\infty f(L_t+u)\\, \\mathrm{d} u<\\infty\\),\n  we may apply the dominated convergence theorem to deduce that\n  \\(N_t^{q} \\longrightarrow M_t^{(0)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\).\n  By (\\ref{Lem-item:qr_q}) of Lemma~\\ref{Lem:inte-of-varPsi}, we have\n  \\begin{align}\n    A_t^{q} & = r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q}); \\bm{e}_q \\le t | \\ensuremath{\\mathcal{F}}_t}\n    = qr_q(0) \\int_0^t f(L_u) \\mathrm{e}^{-qu} \\, \\mathrm{d} u\n    \\to 0\n  \\end{align}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q \\to 0+\\).\n  Therefore, we obtain \\(M_t^{q} \\to M_t^{(0)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q \\to 0+\\).\n\\end{proof}\n\n\\subsection{\\(\\ensuremath{\\mathcal{L}}^1\\) convergence for exponential clock}\nNow we prepare some lemma to prove the \\(\\ensuremath{\\mathcal{L}}^1\\)\nconvergence for exponential clock.\nThe following lemma is a part of\nTheorem 15.2 of Tsukada~\\cite{MR3838874}.\n\\begin{Lem}[{\\cite[Theorem 15.2]{MR3838874}}]\\label{Lem:h-l1-conv}\n  For \\(t \\ge 0\\), it holds that\n  \\(\n  h_q(X_t) \\longrightarrow h(X_t)\n  \\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(q \\to 0+\\).\n\\end{Lem}\nThe next theorem is the penalization result with exponential clock.\n\\begin{Thm}\\label{Thm:exp-L^1-conv}\\label{Thm:exp-time-result}\n  Let \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\) and \\(x \\in \\ensuremath{\\mathbb{R}}\\). Then\n  \\(( M_t^{(0)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale,\n  and it holds that\n  \\begin{align}\n    \\lim_{q\\to 0+}N_t^{q}\n    = \\lim_{q\\to 0+}M_t^{q}\n    = M_t^{(0)},\n    \\qquad \\text{in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(0)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\\label{eq:exp-penalized-meas}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\bm{e}_q})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(0)}}{M_0^{(0)}}},\n    \\qquad \\text{as \\(q \\to 0+\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nNote that the penalized measure in~\\eqref{eq:exp-penalized-meas}\nis not the same as that of Theorems~\\ref{Thm:hitting-time-result}\nand~\\ref{Thm:two-point-hitting-time-result}.\n\\begin{proof}[Proof of Theorem~\\ref{Thm:exp-L^1-conv}]\n  We first consider the case where \\(f\\) is bounded.\n  We write\n  \\begin{align}\n    N_t^{q}\n     & = \\mathrm{e}^{-qt}\\cbra*{h_q(X_t)f(L_t) +\n      \\rbra*{1-\\frac{h_q(X_t)}{r_q(0)}}\n    \\int_0^\\infty \\mathrm{e}^{-u\/r_q(0)} f(L_t + u) \\, \\mathrm{d} u} \\\\\n     & \\eqqcolon {(\\mathrm{I})}_q + {(\\mathrm{II})}_q, \\\\\n    M_t^{(0)}\n     & = h(X_t)f(L_t) +\n    \\int_0^\\infty\n    f(L_t + u) \\, \\mathrm{d} u                                \\\\\n     & \\eqqcolon {(\\mathrm{I})} + {(\\mathrm{II})}.\n  \\end{align}\n  By Lemma~\\ref{Lem:h-l1-conv}\n  and by the boundedness of \\(f\\),\n  we obtain \\({(\\mathrm{I})}_q \\to (\\mathrm{I})\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  Moreover, since \\(\\int_0^\\infty f(u)\\, \\mathrm{d} u<\\infty\\),\n  it follows from the dominated convergence theorem that\n  \\({(\\mathrm{II})}_q \\to (\\mathrm{II})\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  Hence we obtain \\(N_t^{q} \\to M_t^{(0)}\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  By (\\ref{Lem-item:qr_q}) of Lemma~\\ref{Lem:inte-of-varPsi},\n  we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{A_t^{q}}\n    = q r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^t \\mathrm{e}^{-qu} f(L_u)\\ \\mathrm{d} u}\n    \\le q r_q(0) t\\norm{f}\n    \\to 0, \\qquad \\text{as } q \\to {0+}.\n  \\end{align}\n  Since \\(A_t^q\\ge 0\\),\n  this means that \\(A_t^q\\to 0\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\). Thus we have\n  \\(M_t^{q} \\to M_t^{(0)}\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  For \\(0 \\le s \\le t\\), we know \\(\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}|\\ensuremath{\\mathcal{F}}_s} = M_s^{q}\\).\n  Letting \\(q \\to 0+\\) on both sides, we have\n  \\begin{align}\\label{eq:exp-mart}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}|\\ensuremath{\\mathcal{F}}_s} = M_s^{(0)},\n  \\end{align}\n  which means that \\((M_t^{(0)},t\\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\\\n\n  Let us consider the general \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\).\n  We know the equality~\\eqref{eq:exp-mart} holds for \\(f \\wedge n\\).\n  Letting \\(n \\to \\infty\\),~\\eqref{eq:exp-mart}\n  holds for general \\(f \\in \\ensuremath{\\mathcal{L}}_{+}^1\\)\n  by the monotone convergence theorem.\n  Hence we have\n  \\begin{align}\n    \\lim_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}} = \\lim_{q \\to 0+}M_0^{q}\n    = M_0^{(0)} = \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}},\n  \\end{align}\n  and by Fatou's lemma,\n  \\begin{align}\n    M_0^{(0)}\n    = \\lim_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}}\n    \\ge \\limsup_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}\n    \\ge \\liminf_{q \\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}\n    \\ge M_0^{(0)}.\n  \\end{align}\n  Thus we have\n  \\(\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{q}}=\\lim_{q\\to 0+}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{q}}\n  =\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}}\\).\n  Applying Scheff\\'{e}'s lemma, we obtain\n  \\(\\lim_{q\\to 0+}N_t^{q}=\\lim_{q\\to 0+} M_t^{q} = M^{(0)}_t\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n\\end{proof}\n\n\\section{Local time penalization with hitting time clock}\\label{Sec:hitting-time}\nWe deal with the penalization result with hitting time clock \\((T_a)\\).\nTo this aim, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{L_{T_a}}\\)\nand \\(\\ensuremath{\\mathbb{P}}_x\\sbra{L_{T_a}|\\ensuremath{\\mathcal{F}}_t}\\) and its limit as \\(a\\to\\pm\\infty\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with hitting time clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\\).\n\\begin{Lem}\\label{Lem:expect-L_T_a}\n  For \\(x, a \\in \\ensuremath{\\mathbb{R}}\\), \\(a\\ne 0\\) and any non-negative measurable\n  function \\(f\\), we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\n    = \\ensuremath{\\mathbb{P}}_x(T_0 > T_a) f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a) }{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  By~\\eqref{eq:L_T_a-exp-distrib},\n  \\(L_{T_a}\\) is exponentially distributed with\n  parameter \\(1\/h^B(a)\\).\n  Hence we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_0\\sbra{f(L_{T_a})}\n    = \\frac{1}{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n  Applying the Markov property,\n  we conclude that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a})}\n     & = \\ensuremath{\\mathbb{P}}_x( T_0 > T_a) f(0) + \\ensuremath{\\mathbb{P}}_x(T_0 < T_a) \\ensuremath{\\mathbb{P}}_0\\sbra{f(L_{T_a})} \\\\\n     & = \\ensuremath{\\mathbb{P}}_x(T_0 > T_a)f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a) }{h^B(a)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)} f(u) \\, \\mathrm{d} u.\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\subsection{Proof of a.s.\\ convergence of\n  Theorem~\\ref{Thm:hitting-time-result}}\\label{Subsec:pf-hitting}\nWe now proceed the proof of the hitting time result.\nWe separate into the two cases \\(\\cbra{t<T_a}\\) and \\(\\cbra{T_a\\le t}\\).\n\\begin{proof}[Proof of a.s.\\ convergence of Theorem~\\ref{Thm:hitting-time-result}]\n  By the strong Markov property, the additivity of \\(L\\)\n  and Lemma~\\ref{Lem:expect-L_T_a}, we have\n  \\begin{align}\n    N_t^a\n     & =\n    1_{\\cbra{t<T_a}} h^B(a)\n    \\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\n    \\sbra{f(\\widetilde{L}_{\\widetilde{T}_a}+L_t)} \\\\\n     & =\n    1_{\\cbra{t<T_a}}\n    \\cbra*{h^B(a)\\ensuremath{\\mathbb{P}}_{X_t}(T_0 > T_a)f(L_t)\n      + \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)\n      \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)}f(L_t+u)\n      \\, \\mathrm{d} u\n    },\n  \\end{align}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  (\\ref{Thm-item:h-h-limit}) of Theorem~\\ref{Thm:property-h}\n  and~\\eqref{eq:two-hittig-time-prob} imply that\n  \\begin{gather}\n    h^B(a)\\ensuremath{\\mathbb{P}}_{X_t}(T_0> T_a)\n    = h(X_t)+h(-a)-h(X_t-a)\n    \\longrightarrow h^{(\\pm 1)}(X_t) , \\\\\n    \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a) \\longrightarrow 1,\n  \\end{gather}\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  By Lemma~\\ref{Thm:h^B-infty}, we obtain\n  \\(N_t^a \\to M_t^{(\\pm 1)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n  Furthermore, we have\n  \\begin{align}\n    M_t^a - N_t^a\n     & =\n    h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra*{f(L_{T_a}); T_a\\le t|\\ensuremath{\\mathcal{F}}_t}\\label{eq:hitting-A} \\\\\n     & =\n    h^B(a) f(L_{T_a}) 1_{\\cbra{T_a\\le t}}                              \\\\\n     & \\to 0,\n    \\quad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.\\ as \\(a\\to\\pm\\infty\\).}\\label{eq:hitting-Al}\n  \\end{align}\n  Hence we obtain \\(M_t^a \\to M_t^{(\\pm 1)}\\)\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(a\\to\\pm\\infty\\).\n\\end{proof}\n\n\\subsection{Proof of \\(\\ensuremath{\\mathcal{L}}^1\\) convergence\n  of Theorem~\\ref{Thm:hitting-time-result}}\\label{Subsec:pf-hitting-l1}\n\\begin{proof}[Proof of \\(\\ensuremath{\\mathcal{L}}^1\\) convergence of Theorem~\\ref{Thm:hitting-time-result}]\n  We first consider the case\n  \\(m^2 = \\infty\\).\n  Then we know \\(M_t^{(\\pm 1)} = M_t^{(0)}\\).\n  Hence, by Theorem~\\ref{Thm:exp-L^1-conv},\n  \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  Thus we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{a}} = M_0^{a}\n    \\to M_0^{(\\pm 1)} = \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}},\n    \\quad \\text{as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  By Fatou's lemma, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}\n    =\\lim_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{a}}\n    \\ge \\limsup_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n    \\ge \\liminf_{a\\to\\pm\\infty}\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n    \\ge \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}.\n  \\end{align}\n  Consequently, it holds that\n  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}, \\ensuremath{\\mathbb{P}}_x\\sbra{N_t^{a}}\n  \\to \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}}\\), as \\(a\\to\\pm\\infty\\).\n  Applying Scheff\\'{e}'s lemma,\n  we obtain\n  \\(N_t^{a}, N_t^{a} \\to M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n\n  We next consider the case \\(m^2 < \\infty\\).\n  Suppose first that \\(f\\) is bounded.\n  We write\n  \\begin{align}\n    N_t^{a}\n     & =\n    1_{\\cbra{t<T_a}}\n    \\rbra{h(X_t)+h(-a)-h(X_t-a)}f(L_t)                 \\\\\n     & \\qquad +\n    1_{\\cbra{t<T_a}}\n    \\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^B(a)}f(L_t+u)\n    \\, \\mathrm{d} u                                           \\\\\n     & \\eqqcolon {(\\mathrm{I})}_a + {(\\mathrm{II})}_a, \\\\\n    M_t^{{(\\pm 1)}}\n     & = h^{(\\pm 1)}(X_t) f(L_t)\n    + \\int_0^\\infty f(L_t+u) \\, \\mathrm{d} u\\label{eq:useless-dot}         \\\\\n     & \\eqqcolon {(\\mathrm{I})} + {(\\mathrm{II})}.\n  \\end{align}\n  Since \\(h\\) is subadditive, we have\n  \\(h(X_t)+h(-a)-h(X_t-a) \\le h(X_t) + h(-X_t)\\).\n  By the proof of Lemma~\\ref{Lem:h-l1-conv},\n  we know \\(\\ensuremath{\\mathbb{P}}_x\\sbra{h(X_t) + h(-X_t)} < \\infty\\)\n  and thus, by the dominated convergence theorem,\n  we have\n  \\begin{align}\n    h(X_t)+h(-a)-h(X_t-a)\n    \\longrightarrow\n    h^{(\\pm 1)}(X_t),\n    \\quad \\text{in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  The boundedness of \\(f\\) implies that\n  \\({(\\mathrm{I})}_a \\to (\\mathrm{I})\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Since \\({(\\mathrm{II})}_a \\le \\int_0^\\infty f(u)\\, \\mathrm{d} u\\),\n  we may apply the dominated convergence theorem to conclude\n  \\({(\\mathrm{II})}_a \\to (\\mathrm{II})\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Hence we obtain \\(N_t^{a}\\to M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).\n  By Theorem~\\ref{Thm:property-h}\n  and the optional stopping theorem,\n  we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{A_t^{a}}\n     & =\n    h^B(a) \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a}); T_a \\le t}    \\\\\n     & =\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{h(X_{T_a})f(L_{T_a}); T_a \\le t} \\\\\n     & \\le\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_{T_a}^{(0)}; T_a \\le t}        \\\\\n     & =\n    \\frac{h^B(a)}{a}\\frac{a}{h(a)}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(0)}; T_a \\le t}            \\\\\n     & \\to 0,\n    \\qquad \\text{as \\(a\\to\\pm\\infty\\).}\n  \\end{align}\n  This yields that \\(A_t^{a} \\to 0\\) in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Hence \\(M_t^{a} \\to  M_t^{(\\pm 1)}\\)\n  in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(a\\to\\pm\\infty\\).\n  Since \\(\\ensuremath{\\mathbb{P}}_x\\sbra{M_t^a|\\ensuremath{\\mathcal{F}}_s}=M_s^a\\) for \\(0\\le s\\le t\\),\n  we let \\(a\\to\\pm\\infty\\) to obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t^{(\\pm 1)}|\\ensuremath{\\mathcal{F}}_s}=M_s^{(\\pm 1)}.\\label{eq:M^1-mart}\n  \\end{align}\n  In particular, \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  To remove the boundedness condition of \\(f\\), we consider \\(f\\wedge n\\) and\n  let \\(n\\to\\infty\\) in~\\eqref{eq:M^1-mart}. Then we have\n  \\(( M_t^{(\\pm 1)}, t \\ge 0)\\) is a non-negative\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  The remainder of the proof is the same as that of\n  Theorem~\\ref{Thm:exp-L^1-conv}. So we omit it.\n\\end{proof}\n\n\\begin{Rem}\\label{Rem:martingale}\n  Suppose that \\(m^2 < \\infty\\).\n  Then, since \\(M_t^{(0)}\\) and \\(M_t^{(1)}\\)\n  are different \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingales,\n  \\(\n  m^2 \\rbra{M_t^{(1)} - M_t^{(0)}}\n  = X_t f(L_t)\n  \\)\n  is also a \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  In fact, if \\(X_1\\) is integrable, \\(\\ensuremath{\\mathbb{P}}\\sbra{X_1} = 0\\) and\n  \\(f\\) is a bounded measurable function, then\n  \\((X_t f(L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  For more details, see Theorem~\\ref{Thm:mart-XfL}.\n\\end{Rem}\n\nBy Remark~\\ref{Rem:martingale}, we can prove Theorem~\\ref{Thm:martingale}.\n\\begin{proof}[Proof of Theorem~\\ref{Thm:martingale}]\n  Since \\(h^{(\\gamma)}\\) is non-negative for \\(-1\\le \\gamma\\le 1\\),\n  we have \\(M_t^{(\\gamma)}\\ge 0\\).\n  Since \\((M_t^{(0)},t\\ge 0)\\) and \\((X_t f(L_t),t\\ge 0)\\) are\n  \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingales, the process\n  \\((M_t^{(\\gamma)}=M_t^{(0)}+\\frac{\\gamma}{m^2} X_t f(L_t),t\\ge 0)\\)\n  is also a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\n\\section{Local time penalization with two-point hitting time\n    clock}\\label{Sec:two-hitting-time}\nLet us consider the hitting time of\ntwo-point set, i.e.,\n\\(T_{a,b}=T_{\\cbra{a, b}} = T_a \\wedge T_b\\).\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\\subsection{The law of the local time with two-point hitting time clock}\nFirst, we compute \\(\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}},  \\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\\)\nand \\(\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a\\wedge T_b})}\\) respectively.\n\\begin{Lem}\\label{Lem:LT-two-hitting}\n  For \\(a\\ne b\\), it holds that\n  \\begin{align}\n    h^C(a,b)\n     & \\coloneqq\n    \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}} \\\\\n     & =\\frac{1}{h^B(a-b)}\n    \\cbra*{\\begin{multlined}\n        \\rbra[\\big]{h(b)+h(-a)}h(a-b)\n        +\\rbra[\\big]{h(a)+h(-b)}h(b-a)\\\\\n        -h(a-b)h(b-a)\n      \\end{multlined}}.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  For \\(q>0\\), by the strong Markov property, we have\n  \\begin{align}\\label{eq:L_T_aT_b}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\, \\mathrm{d} L_t}\n    =\\begin{aligned}[t]\n       & \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^{T_a\\wedge T_b}\\mathrm{e}^{-qt}\\, \\mathrm{d} L_t }\n      \\\\\n       & + \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}\\ensuremath{\\mathbb{P}}_a\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\,\\mathrm{d} L_t\n      }\n      \\\\\n       & + \\ensuremath{\\mathbb{P}}\\sbra{\\mathrm{e}^{-qT_b}; T_b<T_a}\\ensuremath{\\mathbb{P}}_b\\sbra*{\\int_0^\\infty \\mathrm{e}^{-qt}\\,\\mathrm{d} L_t\n      }.\n    \\end{aligned}\n  \\end{align}\n  Using~\\eqref{eq:regularity-of-L} and\n  (\\ref{Lem-item:two-hittig-time-prob}) of Lemma~\\ref{Lem:h^B-hitting-h}\n  and letting \\(q\\to 0+\\),\n  we obtain the desired result.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:three-hitting-time}\n  For \\(a,b,c,x\\in \\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b, c\\), it holds that\n  \\begin{align}\\label{eq:e-three-hitting-time}\n   \\hspace{-25pt} \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b\\wedge T_c}\n    = \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}-\\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c< T_b}\n      -\\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a< T_b} }\n    {1-\\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_c};T_c<T_b}\\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}},\n  \\end{align}\n  and letting \\(q\\to 0+\\), it holds that\n  \\begin{align}\\label{eq:three-hitting-time}\n    \\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\n    = \\frac{\\ensuremath{\\mathbb{P}}_x(T_a<T_b)-\\ensuremath{\\mathbb{P}}_x(T_c<T_b)\\ensuremath{\\mathbb{P}}_c(T_a<T_b)}\n    {1-\\ensuremath{\\mathbb{P}}_a(T_c<T_b)\\ensuremath{\\mathbb{P}}_c(T_a<T_b)}.\n  \\end{align}\n\\end{Lem}\nNote that, by~\\eqref{eq:two-hittig-time-prob},\nwe can express\n\\(\\ensuremath{\\mathbb{P}}_x(T_a<T_b\\wedge T_c)\\) only in terms of \\(h\\).\n\\begin{proof}[Proof of Lemma~\\ref{Lem:three-hitting-time}]\n  Using the strong Markov property, we have\n  \\begin{align}\\label{eq:pf-three-hitting-times}\n    \\begin{aligned}\n     & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}    \\\\\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a<T_b\\wedge T_c}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_c<T_a<T_b} \\\\\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a};T_a<T_b\\wedge T_c}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_a\\wedge T_b}\n    \\ensuremath{\\mathbb{P}}_c\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b}.\n    \\end{aligned}\n  \\end{align}\n  Replacing \\(a\\) with \\(c\\), we also have\n  \\begin{align}\\label{eq:pf-three-hitting-times2}\n    \\begin{aligned}\n       & \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_b} \\\\\n       & =\n      \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_c};T_c<T_a\\wedge T_b}\n      + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_a}; T_a<T_b\\wedge T_c}\n      \\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-qT_c}; T_c<T_b}.\n    \\end{aligned}\n  \\end{align}\n  Combining~\\eqref{eq:pf-three-hitting-times} and~\\eqref{eq:pf-three-hitting-times2},\n  we obtain~\\eqref{eq:e-three-hitting-time}.\n  Letting \\(q\\to 0+\\), we also have~\\eqref{eq:three-hitting-time}.\n\\end{proof}\n\n\\begin{Lem}\\label{Lem:two-hitting-time-f-L}\n  For \\(a \\ne b\\) and \\(x\\in \\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{T_a\\wedge T_b})}\n    =\n    \\ensuremath{\\mathbb{P}}_x(T_0 > T_a\\wedge T_b) f(0)\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0 < T_a\\wedge T_b)}{h^C(a,b)}\n    \\int_0^\\infty \\mathrm{e}^{-u\/h^C(a,b)}f(u)\n    \\, \\mathrm{d} u.\n  \\end{align}\n\\end{Lem}\n\\begin{proof}\n  In the same way as~\\eqref{eq:L_T_a-exp-distrib},\n  we have, for \\(l>0\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a\\wedge T_b}>l)=\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_a\\wedge T_b <T_0}}>l)\n    = \\mathrm{e}^{-tn(T_a\\wedge T_b <T_0)}.\n  \\end{align}\n  In particular,\n  \\(L_{T_a\\wedge T_b}\\) is exponentially distributed.\n  (Consequently, we obtain \\(n(T_a\\wedge T_b<T_0)=1\/\\ensuremath{\\mathbb{P}}\\sbra{L_{T_a,b}}=1\/h^C(a,b)\\).)\n  We may apply the strong Markov property and obtain the desired result.\n\\end{proof}\n\n\\subsection{Proof of\n  Theorem~\\ref{Thm:two-point-hitting-time-result}}\\label{Subsec:pf-two-hitting}\nWe are now ready to prove the two-point hitting time result.\nRecall that \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\) means~\\eqref{eq:a-b-inf-notation}.\n\n\\begin{proof}[Proof of Theorem~\\ref{Thm:two-point-hitting-time-result}]\n  By Lemmas~\\ref{Lem:LT-two-hitting} and~\\ref{Lem:three-hitting-time}\n  and~\\eqref{eq:two-hittig-time-prob}, we have\n  \\begin{align}\n     & h^C(a,b)\\ensuremath{\\mathbb{P}}_x(T_0>T_a\\wedge T_b)\n    \\\\\n     & =h(x) + \\frac{1}{h^B(a-b)}\n    \\cbra*{\\begin{multlined}\n        \\rbra[\\big]{h(-a)-h(x-a)}h(a-b)\n        + \\rbra[\\big]{h(-b)-h(x-b)}h(b-a)\\\\\n        -\\rbra[\\big]{h(a)-h(b)}\\rbra[\\big]{h(-a)-h(x-a)-h(-b)+h(x-b)}\n      \\end{multlined}}.\n  \\end{align}\n  Recall that \\(h^B(a)=h(a)+h(-a)\\); see~\\eqref{eq:h^B}.\n  Replacing \\(b\\) with \\(-b\\) and\n  using Theorem~\\ref{Thm:property-h}, it holds that\n  \\begin{align}\n    h^C(a, -b) \\ensuremath{\\mathbb{P}}_x(T_0>T_{a}\\wedge T_{-b})\n    \\xrightarrow[(a,b)\\xrightarrow{\\gamma}\\infty]{}\n    h^{(\\gamma)}(x).\n  \\end{align}\n  By the strong Markov property and Lemma~\\ref{Lem:two-hitting-time-f-L},\n  we have\n  \\begin{align}\n    N_t^{a,b} & = 1_{\\cbra{t<T_{a,-b}}}\n    \\cbra*{h^C(a,-b)\\ensuremath{\\mathbb{P}}_{X_t}(T_0>T_{a,-b})f(L_t)} \\\\\n              & \\quad +1_{\\cbra{t<T_{a,-b}}}\n    \\cbra*{\\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_{a,-b})\\int_0^\\infty \\mathrm{e}^{-u\/h^C(a,-b)} f(L_t+u)\\, \\mathrm{d} y}.\n  \\end{align}\n  Hence we obtain\n  \\(N_t^{a,b}\\to M_t^{(\\gamma)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\((a,b)\\xrightarrow[]{\\gamma}\\infty\\).\n  The proof of \\(M_t^{a,b}\\to M_t^{(\\gamma)}\\),\n  \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n  is similar to~\\eqref{eq:hitting-A}--\\eqref{eq:hitting-Al}.\n  Since \\((M_t^{(\\gamma)},t\\ge 0)\\) is\n  a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale (Theorem~\\ref{Thm:martingale}),\n  we may apply Scheff\\'{e}'s lemma to obtain\n  the \\(\\ensuremath{\\mathcal{L}}^1\\) convergence.\n\\end{proof}\n\n\\section{Local time penalization with inverse local time clock}\\label{Sec:inv-time}\nWe define the modified Bessel function of the first kind, which\nis expressed as\n\\begin{align}\n  I_{\\nu}(x) = \\sum_{n=0}^\\infty \\frac{{(x\/2)}^{\\nu+2n}}{n! \\Gamma(\\nu+n+1)},\n  \\quad \\nu \\ge 0, \\,\\, x > 0.\n\\end{align}\nNote that \\(I_{\\nu}(x)\\) is increasing in \\(x>0\\).\nFor more details, see e.g.,~\\cite[Section 5]{MR0350075}.\nRecall that \\(\\eta_u^a\\) denotes the inverse local time at \\(a\\):\n\\(\n\\eta_u^a = \\inf\\cbra{t\\ge 0\\colon L_t^a > u}\n\\).\nWe consider the penalization with inverse local time clock\nin two ways:\nfirst, we make\n\\(a\\) tend to infinity, and second, \\(u\\) tend to infinity.\nRecall that we assume \\(X\\) is recurrent\nand assume the condition~\\ref{item:assumption}.\n\n\\subsection{The law of the local time with inverse local time clock}\n\\begin{Lem}\n  Let \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\).\n  Then the process \\((L_{\\eta^a_u},u\\ge 0)\\) under \\(\\ensuremath{\\mathbb{P}}_a\\)\n  is a compound Poisson process with Laplace transform\n  \\begin{align}\\label{eq:L_eta^a-laplace}\n    \\ensuremath{\\mathbb{P}}_a\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}}}\n    = \\mathrm{e}^{-u\\beta\/(1+\\beta h^B(a))}, \\quad \\beta\\ge 0.\n  \\end{align}\n  Moreover, for any \\(u>0\\) and \\(f\\in\\ensuremath{\\mathcal{L}}^1_{+}\\),\n  it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    = \\mathrm{e}^{-u\/h^B(a)} f(0)\n    + \\int_0^\\infty f(y)\\rho^{h^B(a)}_u(y)\n    \\, \\mathrm{d} y,\n  \\end{align}\n  where\n  \\begin{align}\n    \\rho^a_u(y) = \\mathrm{e}^{-(u+y)\/a}\\frac{\\sqrt{u\/y}}{a}\n    I_1\\rbra*{\\frac{2\\sqrt{uy}}{a}}.\n  \\end{align}\n\\end{Lem}\nWe omit the proof because it is very similar to that in the\ndiffusion case of Profeta--Yano--Yano~\\cite[Lemma 4.1]{MR3909919}, using\nTheorem~\\ref{Thm:n-two-hitting-time}.\nFor the proof, we use\n\\(n^a(T_0<T_a) = n(T_{-a}<T_0) = 1\/h^B(a)\\)\nin the L\\'{e}vy case, instead of\n\\(n^a(T_0<T_a) = n(T_a<T_0) = 1\/a\\)\nin the diffusion case.\nThe proof of~\\eqref{eq:L_eta^a-laplace} can also be found\nin~\\cite[Lemma V.13]{MR1406564}.\n\nThe proof of the next lemma is completely parallel\nto that of~\\cite[Lemma 4.2]{MR3909919}.\nSo we omit it.\n\\begin{Lem}\n  For \\(u > 0, \\; x,a \\in\\ensuremath{\\mathbb{R}}\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})}\n     & = \\ensuremath{\\mathbb{P}}_x(T_a<T_0)\\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    + \\ensuremath{\\mathbb{P}}_x(T_0<T_a)\\ensuremath{\\mathbb{P}}_a\\sbra{f(\\bm{e}_{1\/h^B(a)}+L_{\\eta^a_u})} \\\\\n     & = \\ensuremath{\\mathbb{P}}_x(T_a<T_0)\\ensuremath{\\mathbb{P}}_a\\sbra{f(L_{\\eta^a_u})}\n    + \\frac{\\ensuremath{\\mathbb{P}}_x(T_0<T_a)}{h^B(a)}\n    \\int_0^\\infty f(y)\\widetilde{\\rho}^{h^B(a)}_u(y)\n    \\, \\mathrm{d} y,\n  \\end{align}\n  where\n  \\begin{align}\n    \\widetilde{\\rho}^a_u(y)\n    = \\mathrm{e}^{-(u+y)\/a}I_0\\rbra*{\\frac{2\\sqrt{uy}}{a}}.\n  \\end{align}\n\\end{Lem}\n\n\\subsection{Limit as \\(a\\) tends to infinity with \\(u\\) being fixed}\n\\begin{Thm}\\label{Thm:inv-LT-result}\n  Let \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\) and \\(x\\in\\ensuremath{\\mathbb{R}}\\).\n  For any \\(u> 0\\) and \\(a\\in \\ensuremath{\\mathbb{R}}\\), we define\n  \\begin{align}\n    N_t^{a,u}\n     & = h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u}); t<\\eta^a_u|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{a,u}\n     & = h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then\n  \\begin{align}\n    \\lim_{a\\to\\pm\\infty}N_t^{a,u}\n    = \\lim_{a\\to\\pm\\infty}M_t^{a,u}\n    = M_t^{(\\pm 1)},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n  Consequently, if \\(M_0^{(\\pm 1)}>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\eta^a_u})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\eta^a_u})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t^{(\\pm 1)}}{M_0^{(\\pm 1)}}},\n    \\qquad \\text{as \\(a \\to \\pm\\infty \\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\nThe proof of the theorem is\nvery similar to that of~\\cite[Lemma 4.4 and Theorem 4.5]{MR3909919}.\nSo we omit it.\n(\\cite[Lemma 4.4]{MR3909919} states only convergence\nin probability but, its a.s.\\ convergence\ncan also be proved by the same proof.)\n\n\\subsection{Limit as \\(u\\) tends to infinity with \\(a\\) being fixed}\nIn this section, we only consider the cases \\(f(x) = \\mathrm{e}^{-\\beta x}\\)\nand \\(f(x) = 1_{\\cbra{x=0}}\\).\nThe next theorem is in the case \\(f(x)= \\mathrm{e}^{-\\beta x}\\).\n\\begin{Thm}\\label{Thm:inv-u-e}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\), \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\), \\(\\beta>0\\) and \\(t>0\\).\n  Define\n  \\begin{align}\n    N_t^{u,\\beta, a} & =\n    \\mathrm{e}^{\\beta u\/(1+\\beta h^B(a))}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}}; t<\\eta^a_u | \\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{u,\\beta, a} & =\n    \\mathrm{e}^{\\beta u\/(1+\\beta h^B(a))}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-\\beta L_{\\eta^a_u}} | \\ensuremath{\\mathcal{F}}_t},\n  \\end{align}\n  and\n  \\begin{align}\n    M_t^{\\beta,a}\n    = \\mathrm{e}^{-\\beta L_t}\n    \\cbra*{\\ensuremath{\\mathbb{P}}_{X_t}(T_a<T_0)\n      + \\frac{\\ensuremath{\\mathbb{P}}_{X_t}(T_0<T_a)}{1+\\beta h^B(a)}\\mathrm{e}^{\\beta L_t^a\/(1+\\beta h^B(a))}}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{u\\to\\infty}N_t^{u,\\beta,a}\n    =\\lim_{u\\to\\infty}M_t^{u,\\beta,a}\n    = M_t^{\\beta,a},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n      and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:inv-u-e} is parallel\nto that of~\\cite[Lemma 4.6 and Theorem 4.7]{MR3909919}.\nSo we omit it.\n\nWe consider the case \\(f(x) = 1_{\\cbra{x=0}}\\).\n\\begin{Thm}\\label{Thm:inv-u-1}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\), \\(a \\in \\ensuremath{\\mathbb{R}}\\setminus\\cbra{0}\\), \\(\\beta>0\\) and \\(t>0\\).\n  Define\n  \\begin{align}\n    N_t^{u,\\infty,a}\n     & = \\mathrm{e}^{u\/h^B(a)}\n    \\ensuremath{\\mathbb{P}}_x\\rbra{t<\\eta_u^a<T_0|\\ensuremath{\\mathcal{F}}_t}, \\\\\n    M_t^{u,\\infty,a}\n     & = \\mathrm{e}^{u\/h^B(a)}\n    \\ensuremath{\\mathbb{P}}_x\\rbra{\\eta_u^a<T_0|\\ensuremath{\\mathcal{F}}_t},\n  \\end{align}\n  and\n  \\begin{align}\n    M_t^{\\infty,a}\n    = \\mathrm{e}^{L_t^a\/h^B(a)} \\ensuremath{\\mathbb{P}}_{X_t}(T_a<T_0) 1_{\\cbra{t<T_0}}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    \\lim_{u\\to\\infty}N_t^{u,\\infty,a}\n    =\\lim_{u\\to\\infty}M_t^{u,\\infty,a}\n    =M_t^{\\infty,a},\\quad\n    \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\\n      and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\).}\n  \\end{align}\n\\end{Thm}\nThe proof of Theorem~\\ref{Thm:inv-u-1} is very similar to\nthat of~\\cite[Theorem 4.8]{MR3909919}.\nSo we omit it.\n\n\\section{Universal \\(\\sigma\\)-finite measures}\\label{Sec:universal-measure}\nIn this section, we shall discuss the penalized processes and\n\\(\\sigma\\)-finite measures unifying the processes.\nWe have obtained the penalized measure \\(\\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\\),\nwhich is given by\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{M_t^{(\\gamma,f)}}{M_0^{(\\gamma,f)}}\\cdot\\ensuremath{\\mathbb{P}}_x|_{\\ensuremath{\\mathcal{F}}_t},\n  \\quad -1\\le \\gamma \\le 1.\n\\end{align}\n\n\\subsection{L\\'{e}vy processes conditioned to avoid zero}\nWe show that \\(h^{(\\gamma)}\\) is invariant for the killed process.\n\\begin{Thm}\\label{Thm:hg-invariant}\n  For \\(-1\\le \\gamma\\le 1\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{h^{(\\gamma)}(X_t);T_0>t} = h^{(\\gamma)}(x), \\quad\n    n\\sbra{h^{(\\gamma)}(X_t);T_0>t} = 1, \\quad\n    x \\in \\ensuremath{\\mathbb{R}}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  We can show the case \\(\\gamma=0\\)\n  by the completely same discussion as the proof\n  of Pant\\'{\\i}~\\cite[(iii) of Theorem 2.2]{MR3689384}.\n  Combining this with Theorem~\\ref{Thm:mart-XfL}, we obtain the desired result.\n  The former equation also follows from the fact that\n  \\((M_t^{(\\gamma, 1_{\\cbra{u=0}})},t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\nLet \\(\\ensuremath{\\mathcal{H}}^{(\\gamma)} = \\cbra{x\\in\\ensuremath{\\mathbb{R}}\\colon h^{(\\gamma)}(x)>0 }\\) and\n\\(\\ensuremath{\\mathcal{H}}^{(\\gamma)}_0 = \\ensuremath{\\mathcal{H}}^{(\\gamma)} \\cup \\cbra{0}\\).\nWe introduce the \\(h^{(\\gamma)}\\) transformed process given by\n\\begin{align}\\label{eq:h-g-trans}\n  \\ensuremath{\\mathbb{P}}_x^{(\\gamma)}|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\begin{dcases}\n    1_{\\cbra{T_0>t}} \\frac{h^{(\\gamma)}(X_t)}{h^{(\\gamma)}(x)}\n    \\cdot \\ensuremath{\\mathbb{P}}_x|_{\\ensuremath{\\mathcal{F}}_t}               & \\text{if } x\\in\\ensuremath{\\mathcal{H}}^{(\\gamma)}, \\\\\n    1_{\\cbra{T_0>t}}  h^{(\\gamma)}(X_t) \\cdot n|_{\\ensuremath{\\mathcal{F}}_t} & \\text{if } x = 0.\n  \\end{dcases}\n\\end{align}\nSince \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}|_{\\ensuremath{\\mathcal{F}}_t}\\) is consistent in \\(t>0\\),\nthe probability measure \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\) can be well-defined\non \\(\\ensuremath{\\mathcal{F}}_\\infty\\coloneqq \\sigma(X_t,t\\ge 0)\\),\nfor more details, see Yano~\\cite[Theorem 9.1]{yano2021universality}.\nFor any \\(t>0\\), we have\n\\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_{\\ensuremath{\\mathbb{R}} \\setminus \\ensuremath{\\mathcal{H}}^{(\\gamma)} } > t} = 1\\).\nConsequently, we have\n\\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_{\\ensuremath{\\mathbb{R}} \\setminus\\ensuremath{\\mathcal{H}}^{(\\gamma)}} = \\infty} = 1\\) and\nin particular, \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\rbra{T_0= \\infty}=1\\).\nThe process \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\)\nis called a \\textit{L\\'{e}vy process conditioned to avoid zero}.\nNote that, for \\(x\\in\\ensuremath{\\mathcal{H}}^{(\\gamma)}\\),\nthe measure \\(\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\) is absolutely continuous with respect to\n\\(\\ensuremath{\\mathbb{P}}_x\\) on \\(\\ensuremath{\\mathcal{F}}_t\\), but is\nsingular to \\(\\ensuremath{\\mathbb{P}}_x\\) on \\(\\ensuremath{\\mathcal{F}}_\\infty\\) since \\(\\ensuremath{\\mathbb{P}}_x(T_0<\\infty)=1\\).\n\nBy Theorems~\\ref{Thm:exp-time-result},~\\ref{Thm:hitting-time-result},~\\ref{Thm:two-point-hitting-time-result}\nand~\\ref{Thm:inv-LT-result}\n(Corollaries~\\ref{Cor:hitting-cond} and~\\ref{Cor:two-point-hitting-cond})\nand by taking \\(f=1_{\\cbra{u=0}}\\),\nwe have the following conditioning results.\n\\begin{Cor}\\label{Cor:avoid-zero}\n  Let \\(t>0\\) and \\(F_t\\) be a bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functional.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n    \\(\\displaystyle\n    \\lim_{q\\to 0+}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>\\bm{e}_q}=\\ensuremath{\\mathbb{P}}_x^{(0)}\\sbra{F_t}\n    \\), for \\(x\\in \\ensuremath{\\mathcal{H}}^{(0)}\\);\\label{Cor-item:avoid-zero-exp}\n    \\item\n    \\(\\displaystyle\n      \\lim_{a\\to\\pm\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_a}=\\ensuremath{\\mathbb{P}}_x^{(\\pm 1)}\\sbra{F_t}\n    \\), for \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\pm 1)}\\);\n    \\item\n    \\(\\displaystyle\n      \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>T_{a,-b}}=\\ensuremath{\\mathbb{P}}_x^{(\\gamma)}\\sbra{F_t}\n    \\), for \\(-1\\le \\gamma\\le 1\\) and \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\gamma)}\\);\n    \\item\n    \\(\\displaystyle\n      \\lim_{a\\to\\pm\\infty}\n      \\ensuremath{\\mathbb{P}}_x\\sbra{F_t|T_0>\\eta^a_u}=\\ensuremath{\\mathbb{P}}_x^{(\\pm 1)}\\sbra{F_t}\n    \\), for \\(u>0\\) and \\(x\\in \\ensuremath{\\mathcal{H}}^{(\\pm 1)}\\).\n  \\end{enumerate}\n\\end{Cor}\nNote that (\\ref{Cor-item:avoid-zero-exp}) of Corollary~\\ref{Cor:avoid-zero}\ngeneralizes Pant\\'{\\i}~\\cite[Theorem 2.7]{MR3689384}.\n\n\\subsection{Universal \\(\\sigma\\)-finite measures}\nIn this subsection, we assume that \\((X,\\ensuremath{\\mathbb{P}}_x)\\) has\na transition density \\(p_t(\\cdot)\\).\nThen we can construct the L\\'{e}vy bridge.\nLet \\(\\ensuremath{\\mathbb{P}}_{x,y}^u\\) denote the law of bridge\nfrom \\(X_0=x\\) to \\(X_u=y\\).\nThis measure can be constructed as\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_{x,y}^u(A) = \\ensuremath{\\mathbb{P}}_x\\sbra*{1_A\\frac{p_{u-t}(y-X_t)}{p_t(y-x)}},\n  \\quad A\\in\\ensuremath{\\mathcal{F}}_t,\\; 0< t<u.\n\\end{align}\nSee Fitzsimmons--Pitman--Yor~\\cite{MR1278079}.\nWe have the conditioning formula:\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^t F_u \\, \\mathrm{d} L_u}\n  = \\int_0^t \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u}\\ensuremath{\\mathbb{P}}_{x, 0}^{u}\\sbra{F_u},\n  \\quad t>0,\n\\end{align}\nfor all non-negative predictable processes \\((F_u)\\), where\nwe write symbolically\n\\( \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u} = p_u(-x) \\, \\mathrm{d} u\\).\n\nFor \\(x\\in \\ensuremath{\\mathbb{R}}\\) and \\(-1\\le \\gamma\\le 1\\), we define\n\\begin{align}\n  \\ensuremath{\\mathcal{P}}_x^{(\\gamma)}\n   & = \\int_0^\\infty \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{d} L_u}\n  \\rbra*{\\ensuremath{\\mathbb{P}}_{x,0}^{u} \\bullet \\ensuremath{\\mathbb{P}}_0^{(\\gamma)}}\n  + h^{(\\gamma)}(x)\\ensuremath{\\mathbb{P}}_x^{(\\gamma)},\n\\end{align}\nwhere\nthe symbol \\(\\bullet\\) stands for the concatenation and\n\\(h^{(\\gamma)}(x)\\ensuremath{\\mathbb{P}}_x^{(\\gamma)} = 0\\)\nfor \\(x \\in \\ensuremath{\\mathbb{R}}\\setminus\\ensuremath{\\mathcal{H}}^{(\\gamma)}\\).\nThen we have the following:\n\\begin{Thm}\n  Let \\(x \\in \\ensuremath{\\mathbb{R}}\\) and \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\).\n  Let \\(t>0\\) and \\(F_t\\) be a bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable\n  functional.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{q\\to 0+}\n          r_q(0) \\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\bm{e}_q})}\n          = \\ensuremath{\\mathcal{P}}_x^{(0)} \\sbra{F_t f(L_{\\infty})}\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{a\\to\\pm\\infty}\n          h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_a})}\n          = \\ensuremath{\\mathcal{P}}_x^{(\\pm 1)} \\sbra{F_t f(L_{\\infty})}\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{(a,b)\\xrightarrow[]{\\gamma}\\infty}\n          h^C(a, -b)\n          \\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{T_{a, -b}})}\n          =\\ensuremath{\\mathcal{P}}_x^{(\\gamma)} \\sbra{F_t f(L_{\\infty})}\\),\n          for \\(-1\\le\\gamma\\le 1\\);\n    \\item\n          \\(\\displaystyle\n          \\lim_{a\\to\\pm\\infty}\n          h^B(a)\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\eta^a_u})}\n          = \\ensuremath{\\mathcal{P}}_x^{(\\pm 1)} \\sbra{F_t f(L_{\\infty})}\\), for \\(u>0\\).\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  It suffices to show that\n  \\begin{align}\n    \\ensuremath{\\mathcal{P}}^{(\\gamma)}_x\\sbra{F_t f(L_\\infty)} = \\ensuremath{\\mathbb{P}}_x\\sbra{F_t M_t^{(\\gamma)}},\n  \\end{align}\n  for \\(-1 \\le \\gamma \\le 1\\).\n  The proof is the same as that of Theorem 5.3 of~\\cite{MR3909919}.\n\\end{proof}\nConsequently, we obtain the representation of \\(\\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\\) as follows:\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^{(\\gamma,f)}_x\n  = \\frac{f(L_\\infty)}{\\ensuremath{\\mathcal{P}}^{(\\gamma)}_x\\sbra{f(L_\\infty)}}\n  \\cdot \\ensuremath{\\mathcal{P}}^{(\\gamma)}_x.\n\\end{align}\nRecall that \\(g = \\sup\\{t\\colon X_t=0\\}\\).\nSince \\(\\ensuremath{\\mathbb{O}}^{(\\gamma,f)}_x(g < \\infty) = \\ensuremath{\\mathbb{P}}_x(g = \\infty)= 1\\),\nthe two measures are singular on \\(\\ensuremath{\\mathcal{F}}_\\infty\\).\n\n\\subsection{The law of \\(L_\\infty\\) under \\(\\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}\\)}\nWe assume,\nfor simplicity, that \\(f\\in \\ensuremath{\\mathcal{L}}^1_{+}\\)\nsatisfies \\(\\int_0^\\infty f(u)\\, \\mathrm{d} u = 1\\).\nThen we have \\(M_0^{(\\gamma,f)}=1\\), \\(\\ensuremath{\\mathbb{P}}_0\\)-a.s.\nFor \\(l\\ge 0\\), the optional stopping theorem implies that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_t\\ge l)\n  = \\ensuremath{\\mathbb{P}}_0\\sbra{M_t^{(\\gamma, f)}; L_t\\ge l}\n  = \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}; \\eta_l\\le t}.\n\\end{align}\nLetting \\(t\\to\\infty\\), we may apply the monotone convergence theorem to\ndeduce that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_\\infty\\ge l) = \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}}.\n\\end{align}\nSince \\(X_{\\eta_l}= 0\\) and \\(L_{\\eta_l}=l\\), we have\n\\begin{align}\n  \\ensuremath{\\mathbb{P}}_0\\sbra{M_{\\eta_l}^{(\\gamma, f)}}\n  = \\int_0^\\infty f(l+u)\\, du.\n\\end{align}\nTherefore, it holds that\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}_0^{(\\gamma,f)}(L_\\infty \\in \\mathrm{d} u) = f(u)\\, \\mathrm{d} u,\n  \\quad u>0.\n\\end{align}\n\n\\section{The transient case}\\label{Sec:trans}\nWe now study penalization in the transient case.\nThroughout this section, we always assume\n\\(X\\) is transient and\nthe conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\nRecall that\n\\begin{align}\n  \\kappa\\coloneqq \\lim_{q\\to 0+}\\frac{1}{r_q(0)}\n  = n(T_0=\\infty)>0;\n\\end{align}\nsee~\\eqref{eq:kappa}.\n\\subsection{The renormalized zero resolvent in the transient case}\nAs in the recurrent case, we define \\(h_q(x)=r_q(0)-r_q(-x)\\).\n\\begin{Thm}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item For any \\(x\\in\\ensuremath{\\mathbb{R}}\\), it holds that\n          \\(\\displaystyle\n          h(x)\\coloneqq \\lim_{q\\to 0+} h_q(x) = \\kappa^{-1}\\ensuremath{\\mathbb{P}}_x(T_0=\\infty).\n          \\)\n    \\item The above convergence is uniform on compacts,\n          and consequently \\(h\\) is continuous.\\label{Thm-item:h-unif-conv-trans}\n    \\item \\(h\\) is subadditive on \\(\\ensuremath{\\mathbb{R}}\\),\n          that is, \\(h(x+y)\\le h(x)+h(y)\\) for\n          \\(x,y\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-subadd-trans}\n  \\end{enumerate}\n\\end{Thm}\n\n\\begin{proof}\n  It follows from~\\eqref{eq:exp-hit-time}\n  and~\\eqref{eq:kappa} that\n  \\begin{align}\\label{eq:transient-h-conv}\n    h_q(x) = r_q(0)\\ensuremath{\\mathbb{P}}_x\\sbra{1-\\mathrm{e}^{-qT_0}}\n    \\longrightarrow \\kappa^{-1} \\ensuremath{\\mathbb{P}}_x(T_0 = \\infty),\n    \\quad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n  The proof of (\\ref{Thm-item:h-unif-conv-trans}) and (\\ref{Thm-item:h-subadd-trans})\n  are the same as that of Theorem~\\ref{Thm:exist-h}.\n\\end{proof}\n\n\\begin{Thm}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item\n          \\(\\displaystyle\n          \\lim_{x\\to\\pm\\infty} \\frac{h(x)}{\\abs{x}} =0\n          \\);\\label{Thm-item:h\/x-infinity-trans}\n    \\item\n          \\(\\displaystyle\n          \\lim_{y\\to\\pm\\infty} \\cbra*{h(x+y) - h(y)} =0,\n          \\) for all \\(x\\in\\ensuremath{\\mathbb{R}}\\).\\label{Thm-item:h-h-limit-trans}\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  Since \\(h(x)\\le \\kappa^{-1}\\), it is\n  obvious that~\\eqref{Thm-item:h\/x-infinity-trans} holds.\n  The proof of~\\eqref{Thm-item:h-h-limit-trans} is the same as that of\n  (\\ref{Thm-item:h-h-limit}) of Theorem~\\ref{Thm:property-h}\n  in the recurrent and \\(m^2=\\infty\\) case.\n\\end{proof}\n\\subsection{Useful equations}\nBefore stating out penalization result, we introduce\nsome useful equations.\n\\begin{Lem}\\label{Lem:useful-eq-trans}\n  \\begin{enumerate}\n    \\item For \\(a\\in\\ensuremath{\\mathbb{R}}\\),\n          \\begin{align}\n            h^B(a) \\coloneqq \\lim_{q\\to 0+} h_q^B(a)\n            =\\ensuremath{\\mathbb{P}}_0\\sbra{L_{T_a}}= h(a) + h(-a) - \\kappa h(a)h(-a).\n            \\label{eq:h^B-trans}\n          \\end{align}\\label{Lem-item:hq-two-hitting-time-trans}\n    \\item For \\(x,a,b\\in\\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b\\),\n          \\begin{align}\n            \\ensuremath{\\mathbb{P}}_x(T_a < T_b)\n            = \\frac{h(b-a)+h(x-b)-h(x-a) - \\kappa h(x-b)h(b-a)}{h^B(a-b)}.\n            \\label{eq:two-hittig-time-prob-trans}\n          \\end{align}\n    \\item\n          For \\(x,a,b\\in\\ensuremath{\\mathbb{R}}\\) and \\(a\\ne b\\),\n          \\begin{align}\\label{eq:L_T_aT_b-trans}\n            \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a \\wedge T_b}}\n             & =\\frac{1}{h^B(a-b)}\n            \\cbra*{\\begin{multlined}\n                \\rbra[\\big]{h(b)+h(-a)-\\kappa h(-a)h(b)}h(a-b)\t\\\\\n                \\qquad +\\rbra[\\big]{h(a)+h(-b)-\\kappa h(-b)h(a)}h(b-a) \\\\\n                -h(a-b)h(b-a)\n              \\end{multlined}}.\n          \\end{align}\n  \\end{enumerate}\n\\end{Lem}\nThe proof of Lemma~\\ref{Lem:useful-eq-trans} is similar to\nthat in the recurrent case.\nSo we omit it.\n\n\\begin{Lem}[{\\cite[Lemma 3.10]{MR3689384}}]\\label{Thm:h^B-infty-trans}\n  It holds that \\(\\lim_{x\\to\\infty} h^B(x) = \\kappa^{-1}\\).\n\\end{Lem}\n\\begin{proof}\n  For completeness of the paper, we give the same proof as\n  that of~\\cite[Lemma 3.10]{MR3689384}.\n  Let \\(\\bm{e}\\) be the exponentially distributed with parameter\n  \\(1\\) and \\(\\bm{e}_q \\coloneqq \\bm{e}\/q\\) for \\(q>0\\).\n  Then we already have~\\eqref{eq:h_b-eq} and~\\eqref{eq:h_b-lower}.\n  Letting \\(q\\to 0+\\) in~\\eqref{eq:h_b-lower}, we obtain\n  \\begin{align}\n    \\liminf_{x\\to\\infty}h^B(x) \\ge \\kappa^{-1}.\n  \\end{align}\n  On the other hand, it holds that\n  \\begin{align}\n    h^B(x) \\le \\ensuremath{\\mathbb{P}}\\sbra{L_{\\bm{e}_q}} + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x>\\bm{e}_q}\n    = r_q(0) + \\ensuremath{\\mathbb{P}}\\sbra{L_{T_x}; T_x>\\bm{e}_q}.\n  \\end{align}\n  Letting \\(q\\to 0+\\), we have\n  \\(h^B(x) \\le \\kappa^{-1}\\).\n  Therefore we obtain the desired result.\n\\end{proof}\n\n\\begin{Thm}\\label{Thm:n-two-hitting-time-trans}\n  For \\(a \\in \\ensuremath{\\mathbb{R}} \\setminus \\{0\\}\\), it holds that\n  \\begin{align}\n    n(T_a < T_0 < \\infty) = \\frac{1-\\kappa h^B(a)}{h^B(a)}, \\quad\n    \\text{and} \\quad\n    n(T_a < T_0) = \\frac{1-\\kappa h(-a)}{h^B(a)}.\n  \\end{align}\n\\end{Thm}\n\\begin{proof}\n  For \\(l > 0\\), it holds that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l) = \\ensuremath{\\mathbb{P}}(T_a > \\eta_l) =\n    \\ensuremath{\\mathbb{P}}(\\sigma_{\\cbra{T_0 =\\infty}\\cup \\cbra{T_a<T_0<\\infty}}> l),\n  \\end{align}\n  where \\(\\sigma_A = \\inf\\cbra{l \\colon e_l \\in A}\\)\n  for \\(A \\subset \\ensuremath{\\mathcal{D}}\\).\n  Since \\(\\sigma_A\\) is the hitting time\n  of the set \\(A\\)\n  for the killed\n  Poisson point process \\(((l, e_l), l\\ge 0)\\),\n  we have\n  \\begin{align}\\label{eq:L_T_a-exp-distrib-trans}\n    \\ensuremath{\\mathbb{P}}(L_{T_a}>l)\n    = \\mathrm{e}^{-l\\cbra{n(T_0=\\infty)+n(T_a<T_0<\\infty)}}\n    = \\mathrm{e}^{-l\\cbra{\\kappa +n(T_a<T_0<\\infty)}}.\n  \\end{align}\n  For more details, see, e.g.,~\\cite[Lemma 6.17]{MR3155252}.\n  In particular, \\(L_{T_a}\\) is exponentially distributed.\n  On the other hand, we know \\(h^B(a) = \\ensuremath{\\mathbb{P}}\\sbra{L_{T_a}}\\)\n  by Lemma~\\ref{Lem:h^B-hitting-h}.\n  Hence we obtain\n  \\begin{align}\n    n(T_a < T_0 < \\infty) = \\frac{1}{h^B(a)} - \\kappa.\n    \\label{eq:n-T_a-T_0-inf-trans}\n  \\end{align}\n  We use the strong Markov property of the excursion measure \\(n\\)\n  (see, e.g.,~\\cite[Theorem III.3.28]{MR1138461})\n  to obtain\n  \\begin{align}\n    n(T_a < T_0 < \\infty)\n    = n(\\ensuremath{\\mathbb{P}}_a(T_0 <\\infty); T_a<T_0)\n    = \\rbra{1-\\kappa h(a)} n(T_a<T_0).\n    \\label{eq:n-T_a-T_0}\n  \\end{align}\n  It follows from~\\eqref{eq:n-T_a-T_0-inf} and~\\eqref{eq:n-T_a-T_0}\n  that\n  \\begin{align}\n    n(T_a<T_0)\n    = \\frac{1-\\kappa h^B(a)}{h^B(a)(1-\\kappa h(a))}\n    = \\frac{1-\\kappa h(-a)}{h^B(a)}.\n  \\end{align}\n  Hence the proof is complete.\n\\end{proof}\n\n\\subsection{Penalization result in the transient case}\nLet \\(f\\) be a non-negative\nfunction on \\([0,\\infty)\\)\nwhich satisfies\n\\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\nFor \\(x \\in \\mathbb{R}\\), we introduce the process given by\n\\begin{align}\\label{eq:def-mart-trans}\n  M_t= M_t^{(f)}= h(X_t)f(L_t)+(1-\\kappa h(X_t))\n  \\int_0^\\infty\n  \\mathrm{e}^{-\\kappa u} f(L_t+u)\\,\\mathrm{d} u.\n\\end{align}\nThen we can show that \\((M_t,t\\ge 0)\\) is a non-negative martingale.\n\\begin{Thm}\\label{Thm:transient-infty-mart}\n  Let \\(x \\in \\mathbb{R}\\) and let \\(f\\) be a non-negative\n  function on \\([0,\\infty)\\)\n  which satisfies\n  \\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\n  Then it holds that\n  \\begin{align}\\label{eq:L-inf-M-trans}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)}\n    = \\kappa M_0,\\quad \\ensuremath{\\mathbb{P}}_x\\text{-a.s.,}\n  \\end{align}\n  and \\((M_t,t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{Thm}\n\\begin{proof}\n  By the same discussion, Lemma~\\ref{Lem:expect-L_e_q} also holds\n  in the transient case.\n  Suppose first that \\(f\\) is bounded.\n  We write \\(g = \\sup\\{t\\colon X_t=0\\}\\).\n  Since \\(X\\) is transient, \\(\\ensuremath{\\mathbb{P}}_x(g < \\infty)=1\\).\n  We see that \\(f(L_{\\bm{e}_q})\\to f(L_\\infty)\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ as \\(q\\to 0+\\); in fact,\n  for almost every sample path,\n \\(L_{\\bm{e}_q} = L_g = L_\\infty\\) for small \\(q>0\\)\n (Here we do not need continuity of \\(f\\)).\n  Hence, by the dominated convergence theorem, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\bm{e}_q})} \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)},\n    \\qquad \\text{as \\(q \\to 0+\\).}\n  \\end{align}\n  On the other hand, by the monotone convergence theorem, we obtain\n  \\begin{align}\n    \\int_0^\\infty \\mathrm{e}^{-u\/{r_q(0)}}f(u) \\, \\mathrm{d} u\n    \\longrightarrow \\int_0^\\infty \\mathrm{e}^{-\\kappa u} f(u)\\, \\mathrm{d} u,\n    \\qquad  \\text{as } q \\to {0+}.\n  \\end{align}\n  Hence~\\eqref{eq:L-inf-M-trans} follows by letting \\(q \\to 0+\\)\n  in~\\eqref{eq:lem-expect-L_e_q}.\n  To remove the boundedness assumption of \\(f\\),\n  we consider \\(f \\wedge n\\) and then let \\(n \\to \\infty\\) in~\\eqref{eq:L-inf-M-trans}.\n  Moreover, by the Markov property and the additivity of \\(L\\), we have,\n  for \\(0<s<t\\),\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{M_t|\\ensuremath{\\mathcal{F}}_s}\n     & =\\kappa^{-1}\n    \\ensuremath{\\mathbb{P}}_x\\sbra[\\big]{\\widetilde{\\ensuremath{\\mathbb{P}}}_{X_t}\\sbra{f(\\widetilde{L}_\\infty+L_t)}|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & =\\kappa^{-1}\\ensuremath{\\mathbb{P}}_x\\sbra[\\big]{{\\ensuremath{\\mathbb{P}}}_{x}\\sbra{f({L}_\\infty)|\\ensuremath{\\mathcal{F}}_t}|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & = \\kappa^{-1}{\\ensuremath{\\mathbb{P}}}_{x}\\sbra{f({L}_\\infty)|\\ensuremath{\\mathcal{F}}_s}\n    \\\\\n     & = \\kappa^{-1}\\widetilde{\\ensuremath{\\mathbb{P}}}_{X_s}\\sbra{f(\\widetilde{L}_\\infty+L_s)}    \\\\\n     & = M_s.\n  \\end{align}\n  This means that \\((M_t,t\\ge 0)\\) is a non-negative \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\n\n\\begin{Thm}\\label{Thm:penal-trans}\n  Suppose that\n  the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Let \\(f\\) be a bounded non-negative function\n  and let \\(\\tau\\) be a random clock. Define\n  \\begin{align}\n    M_t^\\tau & = \\kappa^{-1}\\ensuremath{\\mathbb{P}}_x\\sbra{ f(L_\\tau)|\\ensuremath{\\mathcal{F}}_t}.\n  \\end{align}\n  Then it holds that\n  \\begin{align}\n    M_t^\\tau \\longrightarrow M_t ,\n    \\quad \\text{\\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).}\n  \\end{align}\n  Consequently, if \\(M_0>0\\) under \\(\\ensuremath{\\mathbb{P}}_x\\),\n  it holds that\n  \\begin{align}\\label{eq:penalized-meas-trans}\n    \\frac{\\ensuremath{\\mathbb{P}}_x\\sbra{F_t f(L_{\\tau})}}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_{\\tau})}}\n    \\longrightarrow \\ensuremath{\\mathbb{P}}_x\\sbra*{F_t \\frac{M_t}{M_0}},\n    \\qquad \\text{as \\(\\tau \\to \\infty\\),}\n  \\end{align}\n  for all bounded \\(\\ensuremath{\\mathcal{F}}_t\\)-measurable functionals \\(F_t\\).\n\\end{Thm}\n\\begin{proof}\n  We have\n  \\(f(L_\\tau)\\to f(L_\\infty)\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\n  as \\(\\tau\\to\\infty\\); in fact,\n  \\(L_\\tau=L_g=L_\\infty\\) for large \\(\\tau\\).\n  In addition, since \\(f\\) is bounded,\n  we may apply the dominated convergence theorem to obtain\n  \\(f(L_\\tau)\\to f(L_\\infty)\\), in\n  \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).\n  This implies that \\(M_t^\\tau \\to M_t\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.\\ and in\n  \\(\\ensuremath{\\mathcal{L}}^1(\\ensuremath{\\mathbb{P}}_x)\\) as \\(\\tau\\to\\infty\\).\n\\end{proof}\n\\begin{Rem}\n  If \\(\\tau\\) is exponential clock,\n  hitting clock, two-point hitting time clock or\n  inverse local time clock, Theorem~\\ref{Thm:penal-trans}\n  also holds under the assumption that\n  \\(f\\) is a non-negative\n  function which satisfies\n  \\(\\int_0^\\infty \\mathrm{e}^{-\\kappa u}f(u)\\, \\mathrm{d} u <\\infty\\).\n\\end{Rem}\n\nSince we have \\(M_t = \\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)|\\ensuremath{\\mathcal{F}}_t}\\),\nwe see that the penalized measure \\(\\ensuremath{\\mathbb{Q}}^f_x\\) can be represented as\n\\begin{align}\n  \\ensuremath{\\mathbb{Q}}^f_x = \\frac{f(L_\\infty)}{\\ensuremath{\\mathbb{P}}_x\\sbra{f(L_\\infty)}} \\cdot \\ensuremath{\\mathbb{P}}_x,\n\\end{align}\nwhich shows that \\(\\ensuremath{\\mathbb{Q}}^f_x\\) is absolutely continuous with respect to \\(\\ensuremath{\\mathbb{P}}_x\\).\n\n\\section{Appendix: Martingale property of \\(X_t f(L_t)\\)}\\label{Sec:mart}\nIn Remark~\\ref{Rem:martingale}, we have shown that\n\\((X_t f(L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale\nfor \\(f\\in\\ensuremath{\\mathcal{L}}^1_{+}\\) under the condition that \\(m^2<\\infty\\).\nLet us remove the additional assumption \\(m^2<\\infty\\).\nIn this section, we assume \\(X\\) is either recurrent or transient,\nand assume the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular}.\n\\begin{Thm}\\label{Thm:mart-XfL}\n  Suppose the conditions~\\ref{item:cond-not-CPP} and~\\ref{item:cond-regular} hold.\n  Suppose, in addition, that \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty\\) and \\(\\ensuremath{\\mathbb{P}}[X_1] = 0\\).\n  Then the following assertions hold:\n  \\begin{enumerate}\n    \\item \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}<\\infty\\) and\n          \\(n\\sbra{\\abs{X_{\\bm{e}_q}};T_0>\\bm{e}_q}<\\infty\\)\n          for all \\(q>0\\), and \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\)\n          and \\(n\\sbra{\\abs{X_t};T_0>t}<\\infty\\)\n          for all \\(t>0\\);\\label{Thm-item:pn-integ}\n    \\item  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t} = x\\), for all \\(t>0\\)\n          and \\(x\\in\\ensuremath{\\mathbb{R}}\\);\\label{Thm-item:p-kill-expect}\n    \\item  \\(n\\sbra{X_t;T_0>t}=0\\), for all \\(t>0\\);\\label{Thm-item:n-expect}\n    \\item  \\((X_t f(L_t), t\\ge 0 )\\) is a\n          \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale\n          for \\(x \\in \\ensuremath{\\mathbb{R}}\\) and all bounded measurable functions\n          \\(f\\).\\label{Thm-item:mart-XfL}\n  \\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n  \\noindent (\\ref{Thm-item:pn-integ})\n  We have \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t+s}}}\\le\n  \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}+\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t+s}-X_t}}\n  =\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}+\n  \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{s}}}\\),\n  which implies that the function \\(t\\mapsto \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\) is subadditive.\n  Hence, for any \\(k\\in \\ensuremath{\\mathbb{N}}\\),\n  we have\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_k}}\\le k\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty\\).\n  For \\(t>0\\), it is known that\n  \\begin{align}\\label{eq:sup-above}\n    \\ensuremath{\\mathbb{P}}\\sbra*{\\sup_{0\\le s\\le t}\\abs{X_s}}\\le 8\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}};\n  \\end{align}\n  see Doob~\\cite[Theorem VII.5.1]{MR0058896} and\n  Sato~\\cite[Theorem 25.18 and Remark 25.19]{MR1739520}\n  for the proof. Hence we have \\(\\sup_{0\\le t\\le k} \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\)\n  for all \\(k\\in\\ensuremath{\\mathbb{N}}\\).\n  In particular, we obtain \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}<\\infty\\) for all \\(t>0\\).\n  Again by the subadditivity of \\(t\\mapsto \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\), we have\n  \\begin{align}\n    \\lim_{t\\to\\infty}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}}{t}=\n    \\inf_{t>0}\\frac{\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}}{t}\n    \\le\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_1}}<\\infty.\n  \\end{align}\n  Thus there exist constants \\(C,C'>0\\) such that \\(\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_t}}\\le C+C't\\)\n  for all \\(t>0\\).\n  In particular, we obtain\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}\\le\n    q\\int_0^\\infty (C+C't)\\mathrm{e}^{-qt} \\,\\mathrm{d} t<\\infty,\n    \\quad \\text{for all \\(q>0\\).}\n  \\end{align}\n  By Lemma~\\ref{Lem:compensation-formula}, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{\\bm{e}_q}}}\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\sum_{u\\in D}\\int_{\\eta_{u-}}^{\\eta_u}q\\mathrm{e}^{-qt} \\abs{X_t}\n    \\, \\mathrm{d} t}                                                                   \\\\\n     & = \\ensuremath{\\mathbb{P}} \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q\\mathrm{e}^{-qu}\n    \\int_0^{\\widetilde{T}_0} \\mathrm{d} t\\, \\mathrm{e}^{-qt}\\abs{\\widetilde{X}_t}}            \\\\\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, \\mathrm{e}^{-qu}}\n    n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}\n    \\\\\n     & = r_q(0) n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}.\n  \\end{align}\n  Hence we have \\(n\\sbra{\\abs{X_{\\bm{e}_q}}; T_0>\\bm{e}_q}<\\infty\\) for all \\(q>0\\)\n  and this implies that \\(n\\sbra{\\abs{X_{t}};T_0>t}<\\infty\\)\n  for almost all \\(t>0\\).\n  For any \\(t>0\\), we can take \\(0<s<t\\) such that\n  \\(n\\sbra{\\abs{X_s};T_0>s}<\\infty\\).\n  Then it follows from the Markov property of the excursion measure \\(n\\)\n  that\n  \\begin{align}\n    n\\sbra{\\abs{X_t};T_0>t}\n     & = n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{\\abs{X_{t-s}}; T_0>t-s}; T_0>s}        \\\\\n     & \\le  n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{\\abs{X_{t-s}}}; T_0>s}              \\\\\n     & \\le n\\sbra{\\abs{X_s}+\\ensuremath{\\mathbb{P}}\\sbra{\\abs{X_{t-s}}}; T_0>s}<\\infty.\n  \\end{align}\n  Hence we obtain \\(n\\sbra{\\abs{X_t};T_0>t}<\\infty\\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:p-kill-expect})\n  By the Markov property, we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t}\n    = \\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t}\n    +\\int_{[0,t]}\\ensuremath{\\mathbb{P}}_x(T_0\\in\\mathrm{d} s)\\ensuremath{\\mathbb{P}}\\sbra{X_{t-s}}.\n  \\end{align}\n  Since \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t}=x\\) for all \\(t\\ge 0\\) and \\(x\\in\\ensuremath{\\mathbb{R}}\\), we obtain\n  \\(\\ensuremath{\\mathbb{P}}_x\\sbra{X_t; T_0>t}=x \\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:n-expect})\n  By Lemma~\\ref{Lem:compensation-formula},\n  we have\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q} f(L_{\\bm{e}_q})}\n     & = \\ensuremath{\\mathbb{P}} \\otimes \\widetilde{n}\n    \\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q\\mathrm{e}^{-qu} f(u)\n    \\int_0^{\\widetilde{T}_0} \\mathrm{d} t\\, \\mathrm{e}^{-qt}\\widetilde{X}_t}  \\\\\n     & = \\ensuremath{\\mathbb{P}}\\sbra*{\\int_0^\\infty \\mathrm{d} L_u \\, q \\mathrm{e}^{-qu} f(u)}\n    \\int_0^\\infty \\mathrm{d} t \\, \\mathrm{e}^{-qt} n\\sbra{X_t;T_0>t}.\n    \\label{eq:X_eqL_eq}\n  \\end{align}\n  Here in the second equality, we use Fubini's theorem.\n  If we take \\(f \\equiv 1\\),\n  then~\\eqref{eq:X_eqL_eq} becomes\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}}\n    = qr_q(0)\n    \\int_0^\\infty \\mathrm{d} t \\, \\mathrm{e}^{-qt} n\\sbra{X_t;T_0>t}.\n  \\end{align}\n  Since \\(\\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}} = 0\\) for all \\(q > 0\\),\n  we obtain \\(n\\sbra{X_t;T_0>t} = 0\\)\n  for almost all \\(t>0\\).\n  By the Markov property of the excursion measure \\(n\\),\n  we have, for \\(0<s<t\\),\n  \\begin{align}\n    n\\sbra{X_t;T_0>t} = n\\sbra{\\ensuremath{\\mathbb{P}}_{X_s}\\sbra{X_{t-s}; T_0>t-s};T_0>s}\n    = n\\sbra{X_s;T_0>s},\n  \\end{align}\n  which implies that \\(n\\sbra{X_t;T_0>t}\\) is constant in \\(t>0\\).\n  Thus we obtain \\(n\\sbra{X_t;T_0>t}=0\\) for all \\(t>0\\).\n\n  \\noindent (\\ref{Thm-item:mart-XfL})\n  By~(\\ref{Thm-item:n-expect}) of\n  Theorem~\\ref{Thm:mart-XfL} and by~\\eqref{eq:X_eqL_eq},\n  we have\n  \\(\\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q}f(L_{\\bm{e}_q})} = 0\\).\n  Hence, by the Markov property and~(\\ref{Thm-item:p-kill-expect}) of\n  Theorem~\\ref{Thm:mart-XfL},\n  it holds that,\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_{\\bm{e}_q}f(L_{\\bm{e}_q})}\n     & =\n    \\ensuremath{\\mathbb{P}}_x\\sbra*{\\int_0^{T_0}q\\mathrm{e}^{-qt} X_t f(0)\\, \\mathrm{d} t}\n    + \\ensuremath{\\mathbb{P}}_x\\sbra{\\mathrm{e}^{-qT_0}}\n    \\ensuremath{\\mathbb{P}}\\sbra{X_{\\bm{e}_q} f(L_{\\bm{e}_q})} \\\\\n     & =\n    f(0) \\int_0^\\infty q\\mathrm{e}^{-qt} \\ensuremath{\\mathbb{P}}_x\\sbra{X_t ;T_0>t}\n    \\, \\mathrm{d} t                                 \\\\\n     & =x f(0).\n    \\label{eq:x-X_eqL_eq}\n  \\end{align}\n  Thus we obtain\n  \\begin{align}\\label{eq:XL-a.e.}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} = xf(0) \\quad\n    \\text{ for almost all \\(t > 0\\).}\n  \\end{align}\n  We show the function \\(t\\mapsto \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})}\\)\n  is right-continuous for \\(t>0\\).\n  Fix \\(t>0\\).\n  Since \\(X_t\\ne 0\\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.,\n  we see that, for almost every sample path,\n  we can choose small \\(\\delta>0\\) such that\n  \\(L_{t+\\delta}=L_t\\). Since \\(t\\mapsto X_t\\) is right-continuous,\n  we have\n  \\(\\lim_{s\\to 0+}X_{t+s}f(L_{t+s}) = X_t f(L_t) \\), \\(\\ensuremath{\\mathbb{P}}_x\\)-a.s.,\n  where we do not require continuity of \\(f\\).\n  For \\(0<s<1\\), it holds that\n  \\(\\abs{X_{t+s}f(L_{t+s})}\\le \\sup_{0\\le t'\\le t+1}\n  \\abs{X_{t'}}\\sup_{u\\in\\ensuremath{\\mathbb{R}}}\\abs{f(u)}\\).\n  By~\\eqref{eq:sup-above}, we may apply the dominated convergence theorem to deduce\n  that\n  \\(t\\mapsto\\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} \\) is right-continuous.\n  This and~\\eqref{eq:XL-a.e.} imply that\n  \\begin{align}\n    \\ensuremath{\\mathbb{P}}_x\\sbra{X_t f(L_{t})} = xf(0) \\quad \\text{for all \\(t\\ge 0\\)}.\n  \\end{align}\n  By the Markov property and the additivity property of \\(L\\),\n  the process \\((X_t f(L_t), t\\ge 0)\\) is a\n  \\(((\\ensuremath{\\mathcal{F}}_t), \\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n\\end{proof}\nBy Theorems~\\ref{Thm:martingale} and~\\ref{Thm:mart-XfL}, we obtain the following:\n\\begin{Cor}\n  Suppose that the assumptions of Theorem~\\ref{Thm:mart-XfL} are satisfied\n  and that \\(X\\) is recurrent.\n  For measurable function \\(f_1\\) and locally integrable function \\(f_2\\), define\n  \\begin{align}\n    F(X_t,L_t)=X_t f_1(L_t) + h(X_t)f_2(L_t) -\\int_0^{L_t} f_2(u)\\,\\mathrm{d} u.\n  \\end{align}\n  Then the following assertions hold.\n  \\begin{enumerate}\n    \\item If \\(f_1\\) is bounded and \\(f_2\\) is integrable, then\n          \\((F(X_t,L_t),t\\ge 0)\\) is a \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n    \\item If \\(f_1\\) is locally bounded and \\(f_2\\) is locally integrable,\n          then \\((F(X_t,L_t),t\\ge 0)\\) is a local \\(((\\ensuremath{\\mathcal{F}}_t),\\ensuremath{\\mathbb{P}}_x)\\)-martingale.\n  \\end{enumerate}\n\\end{Cor}\n\n\\begin{Rem}\n  For the Brownian motion \\(B\\) and its local time \\(L\\) at \\(0\\),\n  Fitzsimmons--Wroblewski~\\cite{MR2759773}\n  proved that any local martingale of the\n  form \\((F(B_t, L_t), t\\ge 0)\\)\n  is given by the following form:\n  \\begin{align}\\label{eq:F-mart}\n    F(B_t,L_t) & = F(0,0)+B_t f_1(L_t) +\\abs{B_t}f_2(L_t) -\\int_0^{L_t} f_2(u)\\, \\mathrm{d} u,\n  \\end{align}\n  where \\(f_1\\) and \\(f_2\\) are locally integrable functions.\n\\end{Rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nData privacy protection is becoming increasingly important; not only are data breaches gaining public attention, but there are also data protection initiatives and data privacy laws in place, such as the General Data Protection Regulation (GDPR)  and the California Consumer Privacy Act (CCPA).\nDeep learning, on the other hand, poses a significant risk of data privacy leakage since the model embeds information about the training data. For example, both\n \\cite{deep_leakage} and \\cite{IDLG}  provide paradigms for reconstructing training examples from published models. As a result, privacy-preserving algorithms are becoming increasingly important for preserving data privacy. \n \n Differential privacy (DP) is a provable and quantifiable method for privacy protection \\citep{DBLP:conf\/icalp\/Dwork06}, which guarantees it nearly impossible for the adversary to differentiate  from two {\\it neighboring data sets}.  { However, applying DP to maintain the privacy of the training data in deep learning is extremely difficult  due to the thousands to millions of repetitions\\footnote{Around 300,000 steps is needed to train large models like GPT3.} of privacy budget cost in stochastic gradient descent (SGD) iterations, resulting in a very high additive noise power that completely degrades the learning process within a reasonable privacy preserving region.\n}\n\nThe first DP-SGD with a reasonable level of accuracy is proposed by \\cite{DBLP:conf\/ccs\/AbadiCGMMT016}. The calibrated noise added to the clipped gradient is much smaller than all previous methods due to their moments-accountant method for tight privacy accounting.\nEven so, the accuracy drop when compared to the non-DP model is still significant. In the MNIST dataset, for example, the moment-accountant based DP-SGD sacrifices accuracy by about $4.5\\%$ with the privacy protection level $\\epsilon=1.34$~\\citep{bu2019deep}. Performance will continue to suffer as the level of privacy protection increases. This motivates us to narrow the performance gap in this paper without compromising privacy.\n\nTo begin, motivated by the instability and even ramp-up of DP-SGD updates, as shown later in Fig.~\\ref{fig:Gradient_Norm}, we investigate how to perform a tight DP accounting to enable proactive DP budget allocation for each training step to avoid unstable updates.\nSecond, under this new privacy accounting framework, we propose instances of algorithms for proactive privacy budget allocation, such as the sensitivity-decay method, the growing-$\\mu_t$ method, and a hybrid of the two known as dynamic DP-SGD.\nThird, we thoroughly validate the performance of dynamic DP-SGD for a variety of deep learning tasks, demonstrating  consistently significant accuracy improvements over existing methods while preserving privacy. The technical contributions are summarized below.\n\n\n\n\\begin{itemize}\n\\item We broaden Gaussian DP's central limit theorem (CLT) \nto obtain a tight bound for the  composition of  Gaussian mechanisms with different privacy parameters  considering subsampling amplification.\nThe closed form of the CLT result facilitates the proactive  calibration of different noise powers and clipping values at each training step.\n\n\\item \nWe investigate how to proactively allocate the DP cost based on the CLT result in such a way that the gradient updates are stabilized and thus model performance is improved. \nWe propose the  {\\it sensitivity decay} and {\\it growing-$\\mu_t$} methods, and demonstrate that combining these two methods results in the best performance, which is referred to as  {\\it dynamic DP-SGD}.\n\n\n\n\n\n\\item \nWe perform a series of experiments and ablation studies on a variety of neural network tasks, including image classification, natural language processing, and federated learning, and  show that  the proposed dynamic DP-SGD effectively stabilizes gradient updates and consistently outperforms existing methods. With a strong privacy guarantee, i.e., at $\\epsilon = 1.2 $, dynamic DP-SGD has a performance loss of only $2.49\\%$ when compared to the none-DP version on MNIST dataset, and the loss is reduced to $1.72\\%$ in the federated learning setting for a stronger privacy guarantee, i.e., $\\epsilon=1$ due to a larger DP amplification. \n\\end{itemize}\n\n\n\\section{Related Work}\nThe following section summarizes related work on algorithm design and DP accounting for DP-SGD.\n\n\\noindent\\textbf{{DP-SGD Algorithm}}\nTo improve the model's accuracy, previous work has concentrated on designing variations of DP-SGD  by estimating the clipping bound and minimizing the bias introduced by gradient clipping. More precisely, \\cite{DBLP:conf\/ccs\/AbadiCGMMT016} propose norm clipping and per-layer clipping, both of which select clipping values based on gradient differences between different layers.\n\\cite{AdaClip} pioneer AdaClip, a coordinate-wise clipping method that significantly reduces the total amount of noise required.\n\\cite{Quantile} introduce  gradient clipping  based on  the quantile statistics of the gradient, which requires additional DP cost to protect those quantiles.\nRecently, \\cite{DBLP:conf\/nips\/ChenWH20} analyze the bias introduced by the gradient clipping operation and propose a method for reducing the bias error by first adding noise before clipping.\nIt's worth noting that the proposed dynamic DP-SGD in this paper is compatible with the methods mentioned above and can be used in tandem to investigate accuracy improvement.\n\nFurthermore,\n\\cite{yu2019differentially} provide a means of reducing noise variance during the DP-SGD process, thereby improving model performance; and  \\cite{zhang2021adaptive} analyze the DP cost for the same method using the z-CDP privacy accounting.\nDue to the loose DP accountings, these methods have a large performance gap in the high privacy guaranteed region. As demonstrated later in the experiments, the proposed dynamic DP-SGD improves these results significantly due to the dynamic clipping operation and tight DP composition. \n\n\n\\noindent\\textbf{Privacy Accounting}\nEach step in the DP-SGD process depletes the total privacy budget.\nFollowing $T$ steps of training, DP accounting necessitates the composition of these mechanisms in order to calculate the total privacy cost in terms of $(\\epsilon, \\delta)$.\nSimple composition~\\citep{dwork2006our, dwork2009differential} with $\\epsilon=\\mathcal O(T)$ and advance composition~\\citep{dwork2010boosting} with $\\epsilon=\\mathcal O(\\sqrt{T})$ have been proposed in the literature.\n If $M_{1}, M_{2}, \\ldots, M_{T}$ are distinct DP mechanisms such that $M_{i}$ is $\\left(\\epsilon_{i}, \\delta_{i}\\right)$-DP, then it is shown by \\cite{murtagh2016complexity} that computing the exact DP guarantees for the composition  $M=M_{1} \\circ M_{2} \\circ \\cdots \\circ M_{T}$ is \\#P-complete.\n\\cite{DBLP:conf\/ccs\/AbadiCGMMT016} propose the moment accountant of DP-SGD. By establishing a privacy cost function, the moment accountant imposes a tight constraint on the estimation of privacy loss, calibrating the  noise power down to a practicable level for the first time in the privacy preserving deep learning context.\nFrom a broader perspective, \\cite{rdp} proposes to use R\\'enyi divergence to determine the distance between the outputs of two adjacent datasets, thereby establishing the RDP concept, and \\cite{balle2020privacy} provide an advance conversion from RDP to $(\\epsilon,\\delta)$-DP.\nDue to the inherent use of subsampling in training neural networks, the DP composition with subsampling DP amplification is studied in detail by \\cite{wang2019subsampled, balle2020privacy}, which reduce the noise power  for the privacy preserving.\n\\cite{dong2019gaussian} recently propose the concept of f-DP to quantify the privacy cost from a hypothesis testing perspective, with Gaussian DP (GDP) as a major application.\nWhile GDP allows for a tight composition, computing the exact composition of the Gaussian mechanism with subsampling amplification is computationally challenging. It is also demonstrated that a computationally efficient central limit theorem (CLT) can approximate the composition of multiple identical DP mechanisms.\nVery recently, \\citet{gopi2021numerical, zhu2021optimal} propose optimal composition of DP mechanisms via numerical methods, but it is unclear how to calibrate different noise power and clipping value at each step as in the proposed dynamic DP-SGD.\n\n\n \n \n\n \n\n \n\n\\section{An Inspiring Case of the Unstable DP-SGD}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = 0.33\\textwidth]{gradient_norm_3_lines2.jpg}\n    \\caption{Experiments on MNIST with DP budget $(\\epsilon,\\delta)=(0.4,10^{-5})$ after $5\\times 10^3$ steps for both DP-SGD and proposed dynamic DP-SGD. In comparison to SGD without DP protection, DP-SGD is unstable and has a ramp-up period. By contrast, the gradient norm is stabilized by the proposed dynamic DP-SGD.  }\n    \\label{fig:Gradient_Norm}\n\\end{figure}\n\nDP defines an upper bound of the privacy level by computing a pair of $(\\epsilon, \\delta(\\epsilon))$ on the privacy budget.\nLower $\\epsilon$ values indicate better privacy protection but also potentially the less useful the protected algorithm is.\nThe value $\\delta$ can be interpreted as the probability of failing to achieve DP.\nWith the notion of the neighboring data sets, i.e., $X\\sim X^{\\prime}$, which differs by one {\\it data record},\nthe formal $(\\epsilon, \\delta)$-DP definition is given below~\\citep{DBLP:conf\/icalp\/Dwork06}.\n\\begin{definition} (($\\epsilon, \\delta(\\epsilon)$)-DP Profile)\n A randomized algorithm $M(\\cdot)$ gives $(\\epsilon, \\delta(\\epsilon))$-differential privacy if for any pair of neighboring datasets $X\\sim X^{\\prime}$ and any event $E$,\n$$\n\\mathbb{P}(M(X) \\in E) \\leqslant \\mathrm{e}^{\\epsilon} \\mathbb{P}\\left(M\\left(X^{\\prime}\\right) \\in E\\right)+\\delta,\n$$\n\\end{definition}\nwhere the probability $\\mathbb{P}(\\cdot)$ is taken over the randomness of $M$, and $\\epsilon\n\\geq 0$.\nWhen $\\delta = 0$, the algorithm is $\\epsilon$-DP.\nIntuitively, this means that we can't tell whether $M$ was run on $X$ or $X^{\\prime}$ based on the results.\nAs a result, an adversary will almost never infer the existence of any specific data record in the input data set.\n\nIn a Gaussian mechanism,  let $Y$ be the random variable following  Gaussian distribution with  $Y\\sim \\mathcal N\\left(0, \\sigma^2I_d\\right)$ and $f: X^{n} \\rightarrow \\mathbb{R}^{d}$.\nThe Gaussian mechanism $ M(X)=f(X)+ Y$ follows the  DP profile~\\citep{wang2019subsampled}:\n\\begin{equation}\n\\label{compute_mu}\n\\delta(\\epsilon ; \\mu)\n=\\Phi\\left(-\\frac{\\epsilon}{\\mu}+\\frac{\\mu}{2}\\right)-\\mathrm{e}^{\\epsilon} \\Phi\\left(-\\frac{\\epsilon}{\\mu}-\\frac{\\mu}{2}\\right),\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{mu-def}\n\\mu = \\frac{C}{\\sigma}\n\\end{equation}\nwith  $C$ the sensitivity of $f(X)$ and $\\Phi(\\cdot)$  the Gaussian cumulative distribution function. \nThe DP-SGD is proposed below based on the Gaussian mechanism.\nLet the model parameters be denoted by $\\theta$. The gradient computed by a data sample $x$ in the SGD is given by  \n$\ng_x \\triangleq \\frac{\\partial f_x}{\\partial \\theta},\n$ where $f_x$ is the corresponding loss function.\nTo calibrate the noise  required for DP, a paradigm is to first clip the $\\ell_2$-norm of the gradient, i.e.,\n\\begin{equation}\n\\label{clipping}\n  \\widetilde g_x \\triangleq \n\\operatorname{CL}\\left({g_x}; C\\right) \n\\triangleq \ng_x\\cdot\\min\\left(1, \\frac{C}{\\left\\|{g_x} \\right\\|}\\right),\n\\end{equation}\nand in the subsequent, to add the calibrated noise $\\xi_t\\sim \\mathcal N(0,\\sigma^2)$ with $\\sigma = \\frac{C}{\\mu}$. Thereby, the DP-SGD updates at the $t$-th step is given by~\\citep{DBLP:conf\/ccs\/AbadiCGMMT016}:\n\\begin{align}\n\\label{dpsgd}\n\\text{DP-SGD:} \\quad\n\\begin{split}\n  \\theta_t = \\theta_{t-1} - \\eta \n    \\frac{1}{\\left|X_{t}\\right|} \\left( \n    \\sum_{x\\in X_{ t}}\\widetilde g_x\n    + \\xi_{t}\\right), \\quad t\\in[T].\n\\end{split}\n\\end{align}\nBecause of DP's post-processing property, protecting gradients provides the same level of privacy protection on the output model.\nA closer examination of the DP-SGD process in (\\ref{dpsgd}) reveals that the clipping-per-sample operator introduces biases to the original unbiased gradient estimate in the SGD updates if $C$ is not large enough. It is possible to have an unbiased gradient estimate by increasing $C$ if $C$ is greater than any $\\frac{\\partial f_x}{\\partial \\theta}$. It does, however, result in an over-calibrated noise power.\n\n\\textbf{Motivations of this work}:\nIn standard SGD updates, the gradient norm is reduced to a very small value. However, because the DP-SGD is followed by evenly consuming the privacy budget, the ratio of noise power to the true gradient norm in~Eqn. (\\ref{dpsgd}) would continue to rise, resulting in unstable updates.\nThis hypothesis is supported by Fig.~\\ref{fig:Gradient_Norm}, which displays the average coordinate gradient norm for all the data records in each step during the iterative updates.\nIt is worth noting that the DP-SGD gradient norm has a ramp-up period. This phenomenon  also appeared in the experiments in~\\citep{Quantile}. The observation motivates us to investigate a dynamic DP-SGD to reduce both the clipping value and the noise power in order to stabilize the updates. As a result, we can control the rate at which the privacy budget consumed.\nDespite the idea of a dynamic DP-SGD, existing DP accounting methodologies are inadequate to facilitate the study on under what criterior to allocate the privacy budget. As a result, in the following section, we first present an extended CLT of DP accounting  based on Gaussian DP (GDP)~\\citep{dong2019gaussian} and then demonstrate how to achieve dynamic DP-SGD with the extended CLT of GDP to improve model accuracy.\n\n\n\n\n\n\\section{Dynamic DP-SGD}\n\nThe DP parameter $(\\epsilon, \\delta)$ must be carefully determined, and existing theoretical analysis is frequently insufficient. In the following section, we work toward making DP machine learning practical by calibrating a reasonable noise power first providing refined privacy analyses and then designing a dynamic DP-SGD with a better convergence.\n\\subsection{Extended CLT for GDP}\n\\textbf{GDP Preliminary}: We first introduce some background about GDP.\nLet $P$ and $Q$ denote the distributions of $M(X)$ and $M\\left(X^{\\prime}\\right)$ with $X\\sim X^{\\prime}$, and let $\\phi$ be any (possibly randomized) rejection rule for testing $H_{0}: P$ against $H_{1}: Q$. With these in place, \\citet{dong2019gaussian} defines the trade-off function of $P$ and $Q$ as\n\\begin{equation}\n\\begin{aligned}\n\\text{T}(P, Q):[0,1] & \\mapsto[0,1] \\\\\n\\alpha & \\mapsto \\inf _{\\phi}\\left\\{1-\\mathbb{E}_{Q}[\\phi]: \\mathbb{E}_{P}[\\phi] \\leqslant \\alpha\\right\\}.\n\\end{aligned}\n\\end{equation}\nAbove,  $\\mathbb{E}_{P}[\\phi]$ and  $1-\\mathbb{E}_{Q}[\\phi]$  are type I and type II errors of the rejection rule  $\\phi$, respectively.\nIt is shown that $\\text{T}(P,Q)\\geq \\text{T}(\\mathcal N(0,1),\\mathcal N(\\mu,1))\\triangleq G_{\\mu}$, which is referred to as  $\\mu$-GDP.\n\nIn each step of the  DP-SGD in (\\ref{dpsgd}) with the Gaussian mechanism, it achieves $\\mu$-GDP with $\\mu = \\frac{C}{\\sigma}$.\nConsider the  sampling scheme $\\operatorname{PS}(X)$ that each individual data sample $(x, y)$ is subsampled independently with probability $p$ from the training set to construct $X_t$.\nIt is shown in \\citep{bu2019deep} that\ngiven two neighboring datasets $X$ and \n$X^{\\prime}$, \nif a randomized mechanism $\\mathcal M$ is $G_{\\mu}$-{DP},  then\n\\begin{equation}\n\\label{GDP-sampleing}\n\\text{T}\\left(\\mathcal M \\circ \\operatorname{PS} (X), \\mathcal M \\circ \\operatorname{PS}\\left(X^{\\prime}\\right)\\right) \\geqslant p G_{\\mu}+(1-p) \\mathrm{Id},\n\\end{equation}\nwhere $\\operatorname{Id}(x) = 1 - x$.\nThen after a large enough $T$ steps, a Berry-Esseen style CLT result is shown by \\cite{bu2019deep} that as $T\\to +\\infty$ and $p\\sqrt{T}\\to$ a constant,  the composition of the r.h.s. of (\\ref{GDP-sampleing}) converges to a $G_{\\mu_{\\text{tot}}}$-DP with\n\\begin{equation}\n\\label{CLT1}\n        \\mu_{\\text{tot}} = p\\sqrt{T(e^{\\mu^2}-1)}.\n\\end{equation}\n\n\\textbf{Extended CLT of GDP}: The preceding CLT result is based on the assumption that each step satisfies the same $G_{\\mu}$-DP, which restricts the possibility to calibrate noise for each individual step. Following that, in order to pave the way for DP accounting of a dynamic DP-SGD, we investigate the extended CLT in which each step $t$ has a different $G_{\\mu_t}$-DP.\nLet $\\mathcal M_A$ and $\\mathcal M_B$ denote the compositions of $T$ steps updates for the neighboring data sets $X$ and $X^{\\prime}$, respectively.\nAccording to~(\\ref{dpsgd}), each step consists of subsampling and local updates.\nThen, we express  $\\mathcal M_A$ and $\\mathcal M_B$ by\n$$\\mathcal M_A \\triangleq\nM_{T} \\circ \\operatorname{PS}_T (X)\\circ\\cdots\\circ\\mathcal M_{1} \\circ \\operatorname{PS}_1 (X), \\quad\\mathcal M_B \\triangleq\nM_{T} \\circ \\operatorname{PS}_T (X^{\\prime})\\circ\\cdots\\circ\\mathcal M_{1} \\circ \\operatorname{PS}_1 (X^{\\prime}).$$\nThe composition theorem in \\citep{bu2019deep} gives\n\\begin{eqnarray}\n\\label{T-Tradeoff}\n{\\text T}(\\mathcal M_A, \\mathcal M_B)\n\\geqslant\n\\otimes_{t=1}^T\n\\left(p \\cdot G_{\\mu_t}+(1-p) \\text { Id }\\right).\n\\end{eqnarray}\nThe symbol $\\otimes_{t=1}^T$ denotes the product of all the $T$ trade-off functions with the form $p \\cdot G_{\\mu_t}+(1-p) \\text { Id }$, which is far from analytically computable. \nIn the following theorem, we develop the CLT for the r.h.s of (\\ref{T-Tradeoff}) with the proof provided in Appendix~\\ref{apped:theorem1}.\n\\begin{theorem}\n\\label{MU-CLT}\nConsider a series of adaptive composition mechanisms $\\mathcal M_t$ for $t\\in [T]$, where\n$\\mathcal M_t$ is $ G_{\\mu_t}$-DP, and each mechanism  works only on a  subsampled data sets by independent Bernoulli trial with probability $p$.  The trade-off function for $\\lim _{T \\rightarrow \\infty} \\otimes_{t=1}^T\n\\left(p\\cdot  G_{\\mu_t}+(1-p) \\text { Id }\\right)\n$ in (\\ref{T-Tradeoff}) approaches to $G_{\\mu_{\\text{tot}}}$-DP when $p\\sqrt{T}$ is a constant, where   \n\\begin{equation}\n\\label{g2dp}\n  \\mu_{\\text{tot}}\n  = p\\cdot \\sqrt{\n  \\sum_{t=1}^{T}\n \\left(\\mathrm{e}^{\\mu_t^2}-1\\right)}.\n\\end{equation}\n\\end{theorem}\nNotably, when $\\mu_t=\\mu$ is substituted for $t\\in [T]$, Eqn.(\\ref{g2dp}) reduces to the CLT result in (\\ref{CLT1}).\n\nTheorem~\\ref{MU-CLT} reflects that, \ngiven the target privacy budget $(\\epsilon,\\delta)$ and the corresponding privacy parameter $ \\mu_{\\text{tot}}$  obtained by (\\ref{compute_mu}),\nwe can proactively allocate privacy cost to each step $t\\in [T]$ for a predefined total number of steps $T$, according to (\\ref{g2dp}). Because $\\mu_t$ is determined by both the clipping value and the noise power, we can adjust both $C_t$ and $\\sigma_t$ to control the privacy budget allocation.\nSections~4.2-4.4 detail the corresponding algorithms.\n\n\n  \n\\subsection{Growing-$\\mu_t$ Method}\nIt is expected that the gradient should be decreased during training, thereby, it is natural to reduce the noise power to stabilize the gradient updates. As a result, we investigate the noise power decay rate in order to satisfy the $\\mu$ grow rate and thus control the privacy cost.\nWith (\\ref{g2dp}) in mind, we  control the privacy cost rate by adjusting a hyper-parameter $\\rho_{\\mu}$ with\n\\begin{equation}\n     \\rho_{\\mu}\\triangleq\\frac{\\mu_{T}}{\\mu_{0}},\n    \\label{c_decay}\\quad \\rho_\\mu\\geq 1.\n\\end{equation}\nThen the dynamic $\\mu_t$, , which corresponds to the dynamic DP-SGD is given by\n\\begin{equation}\n\\label{mut}\n\\mu_t = (\\rho_{\\mu})^{t\/T}\\cdot \\mu_0, \\quad \\forall t\\in [T].\n\\end{equation}\n\n\n\nGiven the total privacy budget $(\\epsilon, \\delta)$, the equivalent  privacy parameter   $\\mu_{\\text{tot}}$ of $\\mu_{\\text{tot}}$-GDP is obtained according to (\\ref{compute_mu}).\nThen the rest problem is to determine the initial state $\\mu_0$. Once  $\\mu_0$ is obtained, the whole $\\{\\mu_t\\}$ sequence can be generated according to (\\ref{mut}).\nBy substituting (\\ref{mut}) into  (\\ref{g2dp}), we obtain \n\\begin{equation}\n\\label{mu0}\n  \\mu^2_{\\text{tot}}\n  = p^2\\cdot  \n  \\sum_{t=1}^{T}\n\\left(\\mathrm{exp}\\left\\{\\left((\\rho_{\\mu})^{t\/T}\\cdot \\mu_0 \\right)^2\\right\\}-1\\right).\n\\end{equation}\n\n\n\\begin{algorithm}[H]\n\n\\caption{$\\mu_t$ computation }\n\\label{alg:mu_allocation}\n\\begin{algorithmic}[1]\n\\REQUIRE    privacy budget $ (\\epsilon, \\delta)$,  $T$, $\\rho_{\\mu}$.\n\\STATE Compute $ \\mu_{\\text{tot}}$ corresponding to $(\\epsilon, \\delta)$ according to (\\ref{compute_mu}).\n\\STATE Compute $\\mu_0$ in  (\\ref{mu0})   by binary search.\n\\STATE Compute $\\sigma_t $ according to (\\ref{noise_power_decay}) for all $t\\in [T]$.\n\\end{algorithmic}\n\\end{algorithm}\n\nBecause the above equation is transcendental when $\\rho_{\\mu}>1$, there is no closed-form solution for $\\mu_0$. Nonetheless, the r.h.s of (\\ref{mu0}) is monotone increasing w.r.t. $\\mu_0$, and we can thus solve it efficiently using a numerical method such as binary search.\nThe computation of $\\mu_t$ is summarized in Algorithm~\\ref{alg:mu_allocation}. The noise power at iteration $t$ can be calculated using a specific expression of $\\mu_0$ and $\\rho_\\mu$:\n\\begin{equation}\n    \\sigma_t = \\frac{C}{\\mu_0}(\\rho_\\mu)^{-\\frac{t}{T}}, \\quad \\rho_{\\mu}>1.\n    \\label{noise_power_decay}\n\\end{equation}\n\n\n\nOne example is shown in Fig.~\\ref{fig:eps_consumption}. Given the total DP budget  $\\epsilon=1.2$, growing-$\\mu_t$ gives the freedom to adjust the privacy cost rate,  which is the slope of the curve. \nIn Fig~\\ref{fig:eps_consumption}, we demonstrate the privacy budget consumption curve for different $\\rho_\\mu$. \nThe solid line ($\\rho_\\mu = 1$) represents vanilla DP-SGD with evenly distributed noise power along with the step updates. With growing-$\\mu_t$, we can now realize any $\\mu$ consumption process under the constraint of the total DP budget as shown by the dashed lines. Specifically, the growing-$\\mu_t$ slows consumption in the early rounds and accelerates consumption in the later rounds.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{eps2.jpg}\n    \\caption{Consumption of privacy budget with different $\\mu_t$ increasing rates.  } \n    \\label{fig:eps_consumption}\n\\end{figure}\n\n\n\n\n\n\n\n\\vspace{10pt}\n\\subsection{Sensitivity-Decay Method}\nThe gradient norm of a neural network converges to zero once the SGD training converges. This is not the case for DP-SGD.  When the required privacy protection level is high, it requires a large calibrated noise promotional to the clipping value. The updated gradient norm tends to rise, as shown in Fig.~\\ref{fig:Gradient_Norm}. Based on this discovery, we propose to adjust the sensitivity across training iterations.\nWe \nset the evolution of clipping values by\n\\begin{equation}\n   \n    C_t =   (\\rho_{c})^{-\\frac{t}{T}}\\cdot C_0, \\quad \\rho_c\\geq 1.\n    \\label{sens_decay}\n\\end{equation}\nAssuming a constant $\\mu_t$, and by substituting $\\mu_t=\\mu_0$ into (\\ref{g2dp}) in Theorem~1,  we have the closed form solution:\n\\begin{equation}\n    \\mu_0 = \\sqrt{\\log\\left(\\frac{\\mu_{\\text{tot}}^2}{p^2T}+1\\right)}, \\quad t\\in[T].\n    \\label{mut_compute}\n\\end{equation}\nWith (\\ref{sens_decay}) and (\\ref{mut_compute}), we can calibrate the noise power at each round by:\n\\begin{equation}\n     \\sigma_t = \\frac{C_0}{\\mu_0}(\\rho_c)^{-\\frac{t}{T}}, \\quad \\rho_c>1, \\quad t\\in[T].\n    \\label{sigmat_compute_sens_decay}\n\\end{equation}\n \n\n\n\\subsection{Dynamic DP}\n    \\begin{algorithm}[H]\n\\caption{Dynamic DP-SGD Algorithm}\n\\label{alg:NADP}\n\\begin{algorithmic}[1]\n\\REQUIRE  DP budget $(\\epsilon,\\delta)$, sampling rate $p$ and hyper-parameters:  $\\rho_{\\mu}$, $\\rho_{c}$ and $C_0$.\n\\STATE Compute \n$\\mu_0 $ in Algorithm~\\ref{alg:mu_allocation} \n\\FOR{$t=1, \\ldots, T$}\n\\STATE Compute $C_t =   (\\rho_{c})^{-\\frac{t}{T}}\\cdot C_0$ in (\\ref{sens_decay})\n\\STATE Calibrate noise :\n$\\sigma_t = \\frac{C_0}{\\mu_0}(\\rho_{\\mu}\\cdot\\rho_c)^{-\\frac{t}{T}}$\n\\STATE \nSample $ X_t\\in X$ with sampling rate $p$ and sample noise $\\xi_{t}\\sim \\mathcal N(0,\\sigma^2_{t}I)$. \n\\STATE Compute:\n    $\\theta_t =  \\theta_{t-1} - \n    \\frac{\\eta }{\\left|X_t\\right|}\\big[\\xi_{t}+$\n    $ \n    \\sum_{x\\in X_{ t}}\\text{CL}\\left(g_x;C_t\\right)\n    \\big]$ with $\\text{CL}(\\cdot)$ in (\\ref{clipping}) \n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\nBecause the noise calibration is based on the clipping value, we incorporate the growing-$\\mu_t$ method into the sensitivity-decay method and refer to this new one as dynamic DP-SGD.\nIt maintains the same $\\mu_t$ increasing rate as the previous growing-$\\mu_t$ method while having a faster noise decay rate than the sensitivity-decay method.\nWe summarize the dynamic DP-SGD algorithm in Algorithm~\\ref{alg:NADP} and conduct extensive experiments to show how dynamic DP-SGD improves performance.\n\n\n\n\n\\section{Experiments}\n\\subsection{Datasets, Models and Benchmarks}\n\\textbf{Datasets}: To conduct a comprehensive test of the dynamic DP-SGD performance, we run experiments on the following 5 datasets: MNIST, FashionMNIST, IMDB, NAME, and InfiniteMNIST, using neural network models such as MLP, CNN, LSTM, and Federated Learning. In  Appendix~\\ref{apped:dataset},  we describe each data set, the corresponding neural network model, and parameter settings for each experiment separately. We also go into details about the dynamic DP-SGD federated learning algorithm to make the paper  self-contained.\n\n\\textbf{Benchmarks}: Using the aforementioned datasets and models, we compare our proposed dynamic DP-SGD, growing-$\\mu_t$ method, and sensitivity-decay method to the four benchmarks listed below.\n \n\\noindent{(i) \\it{SGD without DP}}: The SGD method serve as the upper bound of model inference accuracy in the absence of clipping and additive noise. \nPlease note that  the DP-SGD and proposed dynamic DP-SGD can also be used to obtain the DP-Adam counterparts of an Adam optimizer due to DP's postprocessing properties~\\citep{dwork2009differential}. For the IMDB data set, we apply the Adam to compute the performance upper bound, and DP-Adam counterparts for comparison.\n\n\\noindent{(ii) \\it{DP-SGD with the CLT of GDP Acountant}~\\citep{bu2019deep}}: GDP, a recently developed DP accounting framework, provides a simple, explicit, and tight privacy accounting CLT bound for the classical DP-SGD with evenly calibrated noise power. It is proved by~\\cite{dong2019gaussian} that the CLT of GDP provides a tighter composition bound for DP-SGD than the moment accountant method~\\citep{DBLP:conf\/ccs\/AbadiCGMMT016}.  As a result, it serves as a standard for our dynamic DP-SGD methods.\n\n\n\\noindent{(iii) {{\\it Noise Power Decay with $\\rho$-zCDP Accountant}} ~\\citep{yu2019differentially}}: This paper proposes decaying the noise power during the SGD and computing the privacy loss using the $\\rho$-zCDP. However, only parallel composition is considered, with no regard for DP amplification by subsampling.\n\n\n\n\\noindent{(iv) \\it{Noise Power Decay with tCDP Accountant}~\\citep{CDP}}:\n\\cite{CDP} recently proposed decaying the noise power for DP-SGD training as well as analyzing the DP composition and subsampling amplification under the truncated concentrated differential privacy (tCDP) framework.\n\n\\textbf{Parameters}: \nWe concentrate on the strong privacy guarantee, with $\\epsilon$ set to be in the range $[0.4, 9]$ and $\\delta = 1\/(10|X|)$, where $|X|$ is the training data sample size. It is worth noting that we observed a significant performance degradation of DP-SGD when performing IMDB tasks. As a result, we also test large $\\epsilon$ values in this case, i.e., $\\epsilon=9$.\nThe hyperparameters $\\rho_c$ and $\\rho_{\\mu}$ are swept in the following predefined sets:\n$\n    1\/\\rho_{c}, 1\/\\rho_{\\mu} \\in  \\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8\\}.\n   \n   \n   \n$\nOther hyper-parameters are detailed in Appendix~\\ref{apped:dataset}\\footnote{The code will be available at github.com\/dynamic-dp once the paper is accepted.}.\n\n\n\\subsection{Results Analysis}\nExperiment results for the above five different datasets on MLP, CNN, LSTM, and Federated Learning models are shown in Table~1-Table~4, along with benchmarks and proposed methods' performance.\nIt is worth noting that in the majority of cases, we concentrate on the strong privacy protection region with $\\epsilon<1$.\nAs a result, if the DP accountant  is not tight enough, it will lead to an overestimation of the noise power needed, resulting in the noise dominating the gradient and causing the network to fail to learn. For example, the method in~\\cite{yu2019differentially} fails to leverage the subsampling DP amplification in the DP accounting, resulting in significantly worse performance in the high privacy guarantee region due to overestimated noise. As a result, we only replicate its CNN model results in the MNIST and FashionMNIST datasets.\n\nBy contrast, the GDP framework proposed by~\\citet{bu2019deep} provides a detailed examination of DP amplification through subsampling as well as DP composition. Thereby, more precise noise power is calibrated for the same privacy budget. Specifically, even when there are no dynamics for DP-SGD updates, it outperforms the accuracy of the noise decay method by~\\cite{yu2019differentially} and \\cite{CDP}.\n\nThe proposed extended CLT supports the privacy accounting for dynamic clipping and noise power decay. \nSeparate experiments are carried out with the proposed growing-$\\mu_t$ method, sensitivity-decay method, and dynamic DP-SGD method. The results show that all the three proposed methods improve performance when compared to the static noise GDP method~\\cite{bu2019deep}.\nIn particular, \n$\\mu_t$ grows at the expense of early convergence speed in order to achieve higher accuracy, while sensitivity decay ensures more stable convergence. This explains why the performance of sensitivity decay outperforms that of growing-$\\mu_t$. The dynamic DP-SGD, a combination of the two, improves performance while causing no additional privacy loss, as demonstrated by the Dynamic DP results in each table.\n\nExperiments with different privacy budgets were conducted, and the results show that the stronger the privacy protection required (lower $\\epsilon$ value), the more noticeable the improvement by the proposed dynamic DP-SGD method. For example, when $\\epsilon = 0.4$ for the MNIST, even though the noise decay is adopted by~\\cite{yu2019differentially} and~\\cite{zhang2021adaptive}, however, their model fail to learn due to large calibrated noise power by the loose DP compositions. By contrast, our method\noutperforms GDP method by a large margin, achieving $3.17 \\%$ when $\\epsilon = 0.4$ and even $4.6\\%$ for the LSTM network on NAME. Without compromising privacy, our method consistently outperforms all other benchmarks on MNIST, FashionMNIST, IMDN, and NAME datasets for CNN, MLP, LSTM, and federated learning models, as shown in Table1-4.\n\n\n\n\n\\begin{table}[htb!]\n\\centering\n\\caption{CNN on  MNIST and FashionMNIST datasets.}\n\\begin{adjustbox}{width=1\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Accountant}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{3}{c|}{ MNIST} & \\multicolumn{3}{c|}{{FashionMNIST}} \n\\tabularnewline\n\\cline{3-5} \\cline{6-8}\n && {$\\epsilon$ = 0.4} & {$\\epsilon$ =0.6}& {$\\epsilon$ =1.2} & {$\\epsilon$ =0.4} & {$\\epsilon$ =1.2}& {$\\epsilon$ =2.0}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{3}{c|}{{98.83}}  & \\multicolumn{3}{c|}{{87.92}}   \\tabularnewline\n\\hline \n\\hline \n{$\\rho$-zCDP \\citep{yu2019differentially}} &Noise Decay& {10.28} & {10.12} & {65.33} & {9.86} & 63.30 & 72.18\n\\tabularnewline\n\\hline \n{tCDP \\citep{CDP}} &Noise Decay& {26.93} & {83.28} & {92.60} & {53.69} & 76.48 & 77.58\n\\tabularnewline\n\\hline\n{CLT for GDP~\\citep{bu2019deep}} &-& {91.18} & {93.80} & {95.50} & {76.77} & 80.45 & 82.55 \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP (Ours)}\n&growing-$\\mu_t$  & {91.67} & {94.49} & 96.06& {77.81} & 80.95  & 83.10  \\tabularnewline\n\\cline{2-8}\n & Sensitivity Decay & {93.95} & 95.17 & 96.17 & {78.11} & 82.83  & 83.64  \\tabularnewline\n\n\\cline{2-8}\n&Dynamic DP & {\\textbf{94.35}} & {\\textbf{95.21}} & {\\textbf{96.34}} & {\\textbf{78.50}} & {\\textbf{83.22}} & \\textbf{83.81} \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{mnist_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{MLP on the IMDB dataset.}\n\\begin{adjustbox}{width=0.9\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{5}{c|}{ IMDB} \n\\tabularnewline\n\\cline{3-7} \n && {$\\epsilon$ = 0.5} & {$\\epsilon$ =1}& {$\\epsilon$ =3} & {$\\epsilon$ =6} & {$\\epsilon$ =9}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{5}{c|}{{82.85}}  \\tabularnewline\n\\hline \n\\hline\n\n{tCDP \\citep{CDP}} &Noise Decay&56.67 &{58.24} & {62.15} & {65.88} & {70.16} \n\\tabularnewline\n\\hline \n{CLT for GDP \\citep{bu2019deep}} &-& {63.62} & 69.71 & {75.64} & {77.75} & 78.60  \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {64.92} & {69.85} & 76.00& {78.16} & 78.56  \\tabularnewline\n\\cline{2-7}\n & Sensitivity Decay & {65.44} & {70.25} & 76.23 & {78.47} & 79.42    \\tabularnewline\n\n\\cline{2-7}\n&Dynamic GDP & {\\textbf{65.63}} & {\\textbf{70.77}} & {\\textbf{76.64}} & {\\textbf{78.61}} & {\\textbf{79.61}}  \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{imdb_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{LSTM on NAME dataset.}\n\\begin{adjustbox}{width=0.8\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{4}{c|}{ NAME} \n\\tabularnewline\n\\cline{3-6} \n && {$\\epsilon$ = 1} & {$\\epsilon$ =2}& {$\\epsilon$ =4} & {$\\epsilon$ =8} \\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{4}{c|}{{80.14}}  \\tabularnewline\n\\hline \n\\hline\n\n{tCDP\\citep{CDP}} &Noise Decay&48.13 &{51.20} & {57.67} & {68.53}\n\\tabularnewline\n\\hline \n\n{CLT for GDP~\\citep{bu2019deep}} &-& {62.71} & {69.64} & {73.04} & {74.15}   \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {64.10} & {71.25} & 74.68& {75.50}   \\tabularnewline\n\\cline{2-6}\n & Sensitivity Decay & {66.66} & {71.78} & {73.67} & {74.78}     \\tabularnewline\n\n\\cline{2-6}\n&Dynamic DP & {\\textbf{67.30}} & {\\textbf{72.01}} & {\\textbf{75.03}} & {\\textbf{75.75}}   \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{name_res}\n\\end{table}\n\n\\begin{table}[htb!]\n\\centering\n\\caption{Federated learning on InfiniteMNIST dataset.}\n\\begin{adjustbox}{width=1\\textwidth}\n\\hfill{}\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline \n\\multirow{2}{*}{DP Framework}&\\multirow{2}{*}{Dynamic Noise} & \\multicolumn{3}{c|}{MNIST-250K} & \\multicolumn{3}{c|}{{MNIST-500K}} \n\\tabularnewline\n\\cline{3-5} \\cline{6-8}\n && {$\\epsilon$ = 0.1} & {$\\epsilon$ =0.4}& {$\\epsilon$ =1} & {$\\epsilon$ =0.1} & {$\\epsilon$ =0.4}& {$\\epsilon$ = 1}\\tabularnewline\n\\hline \n\\hline \n{Non-private} &{-}& \\multicolumn{3}{c|}{{98.89}}  & \\multicolumn{3}{c|}{{98.96}}   \\tabularnewline\n\\hline \n{CLT for GDP~\\citep{bu2019deep}} &-& {93.47} & {95.71} & {96.02} & {94.82} & {96.55} & {96.89} \\tabularnewline\n\\hline \n\\hline \n\\multirow{3}{*}{Extended CLT for GDP(Ours)}\n&growing-$\\mu_t$ & {93.75} & {95.93} & 96.13& {95.65} & {96.76} & 97.05  \\tabularnewline\n\\cline{2-8}\n & Sensitivity Decay & {94.46} & 95.90 & 96.40 & {95.75} & 96.83 & 97.06 \\tabularnewline\n\n\\cline{2-8}\n&Dynamic DP & \\textbf{94.72} & {\\textbf{96.00}} & {\\textbf{96.55}} & \\textbf{95.88} & {\\textbf{96.95}} & {\\textbf{97.22}} \\tabularnewline\n\\hline \n\n\\end{tabular}\\hfill{}\n\\end{adjustbox}\n\\label{FL_res}\n\\end{table}\n\n\\subsection{Hyper-parameter sensitivity}\nWe then test the robustness of dynamic DP-SGD performance to different values of $\\rho_{\\mu}$ and $\\rho_c$ as shown in Fig.~\\ref{fig:decay_rate_search}.\nWe use grid search to demonstrate the impact of these parameters on model performance. The dynamic method can consistently improve model performance across a wide range.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.65\\textwidth]{acc_decay_ratio2.png}\n    \\caption{Dynamic DP-SGD performance is robust to  $\\rho_{\\mu}$ and $\\rho_c$.}\n    \\label{fig:decay_rate_search}\n\\end{figure}\n\\subsection{Exact Privacy Cost}\nThough~\\cite{dong2019gaussian} have shown  that the CLT of GDP approximate the true privacy cost with negligible error,\n\\citet{gopi2021numerical} recently discovers that the CLT of GDP may underestimate the privacy cost.\nThe RDP accountant~\\citep{wang2019subsampled}, on the other hand, overestimates the true cost.\nTo evaluate the exact privacy cost, thereby, we plot the privacy cost curves for both the proposed extended CLT of GDP and RDP in Fig.~\\ref{fig:rdp_gdp_comp}. Specifically, we set a target training round and conduct privacy accounting for the dynamic DP-SGD  based on the extended CLT of GDP as well as RDP\\footnote{For RDP, we use autodp library by \\cite{autodp} for DP accounting.} with the advanced translation method between RDP and $(\\epsilon, \\delta)$-DP~\\citep{balle2020privacy}.\nConsequently,  the true privacy cost curve must lie somewhere between these two limits. The result of extended CLT for GDP is reasonable because the privacy cost differences between these two limits are small, particularly in the high privacy protection region.\n \n\nIt is worth noting that, in addition to the proposed extended CLT for dynamic DP-SGD accounting, exact accounting can be performed using the recently proposed methods by~\\cite{gopi2021numerical} and~\\cite{zhu2021optimal}, separately. However, because they both require numerical computation, given a total privacy budget, the computation of different noise power for  the mechanism of each step  becomes much involved.\n \n\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = 0.35\\textwidth]{comp_rdp_gdp_dynamic_eps_0.1_2.jpg}\n   \n    \\caption{Upper bound (advanced RDP) and lower bound (extended GDP CLT) of the dynamic DP-SGD privacy budget cost curves. The true privacy cost curve must lie somewhere between these two limits. The result of extended CLT for GDP is reasonable because the privacy cost differences between these two limits are small, especially  in the high privacy protection region.}\n    \\label{fig:rdp_gdp_comp}\n\\end{figure}\n\n\n \n\n \n\\section{Conclusions}\nIn this paper,\nwe extend the central limit theorem (CLT) of Gaussian DP to perform tight privacy accounting in order to calibrate dynamic noise for each individual step of stochastic gradient descent (SGD) updates.  \nWe, therefore, are able to allocate a lower privacy cost than the DP-SGD during updates based on this extended CLT until both methods consume the same target privacy budget at the predefined update number.\nExtensive testing on a variety of datasets and models demonstrates that the dynamic DP-SGD consistently and clearly outperforms existing methods.\n\nFor the future work, we intend to investigate how to leverage the  recently achievement of numerical composition methods \\citep{gopi2021numerical, zhu2021optimal} to calibrate the noise power and clipping value for each individual training step of dynamic DP-SGD.\n\n\n\n\\subsubsection*{Acknowledgments}\nWe would like to thank Yuxiang Wang for his helpful discussion.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAn inevitable consequence of the N-body dynamics near a supermassive black hole (BH) is that some stars will be scattered into highly eccentric orbits and get disrupted by the BH's enormous tidal forces \\citep{hills75_tdes_agn, frank76_tde_rate, magorrian99_tde_rate, alexander05_stellar_dynamics, stone20_tde_rate_review}. Such tidal disruption events (TDEs) feed a transient episode of accretion onto the otherwise dormant BH and generate a bright flare across the electromagnetic spectrum \\citep{lacy82_tdes_Galactic_Center, rees88_tdes, phinney89_tde_fallback_rate,cannizzo90_tde_accretion, ulmer99_tde_flare, strubbe09_tde_flare}. They provide our best chance of probing the demographics of supermassive BHs at the centers of quiescent galaxies at cosmological distances \\citep{kormendy13_SMBH_demographics}.\n\nThe hydrodynamics of the tidal disruption process has now been reasonably well understood, thanks to a large volume of analytical and numerical works \\citep{carter83_tidal_compression, evans89_tde_simulation, ayal00_tde_simulation, lodato09_tde_simulation, guillochon13_tde_simulation, stone13_tde_fb_rate, cheng14_relativistic_tdes, tejeda17_relativistic_tdes, steinberg19_deep_tdes, ryu20_tde_in_GR, rossi20_disruption_process}. The consequence is that the star is tidally stretched into a long, thin stream of debris, the evolution of which is controlled by the tidal gravity, self-gravity, as well as internal pressure forces \\citep{kochanek94_stream_evolution,coughlin16_stream_evolution, Bonnerot21_stream_width_evolution}. In the longitudinal direction of the stream, the evolution is almost entirely controlled by tidal gravity, meaning that each segment of the stream accurately follows its own geodesic like a test particle. Each time a fluid element passes near the pericenter, it undergoes general relativistic (GR) precession, which will eventually lead to the self-intersection of the stream \\citep{rees88_tdes, dai13_gr_precession, dai15_pn1_precession, guillochon15_dark_year, Bonnerot17_stream_crossings, liptai19_GRSPH_simulation, lu20_self-intersection}. If this intersection is sufficiently violent, it strongly broadens the angular momentum and orbital energy distributions of the bound debris and leads to numerous secondary shocks and then the formation of a coherently rotating accretion flow \\citep{shiokawa15_GR_precession_intersection, bonnerot16_precession_schwarzschild, hayasaki16_precession_kerr, sadowski16_relativistic_tde, jiang16_stream_collision, lu20_self-intersection, bonnerot20_disk_formation, andalman20_disk-formation, curd21_global_TDE, bonnerot21_accretion_flow_review}. The processes of disk formation and accretion generate bright electromagnetic emission that is observable from cosmological distances \\citep{strubbe09_tde_flare, coughlin14_ZEBRA, piran15_stream_collision, metzger16_bright_year, dai18_unified_model, lu20_self-intersection, bonnerot21_first_light}.\n\nOn the observational side, of the order $10^2$ TDE candidates have been identified by various transient searching strategies in different bands, including X-ray surveys \\citep{bade96_rosat_tde, esquej08_xmm_newton_tdes, saxton12_x-ray_tde, saxton17_xmm_tde, lin15_xmm_tde, brightman21_X-ray_TDE, sazonov21_erosita_TDEs}, UV-optical surveys \\citep{gezari06_GALEX_tde, gezari12_ps-10jh, chornock14_ps1-11af, holoien16_as14li, blagorodnova17_iptf16fnl, hung17_iPTF16axa, hinkle21_AS19dj, vanvelzen21_ztf_tdes}, and infrared surveys \\citep{wang18_infrared_tdes, jiang21_MIRONG}.\nTo extract the physical information from the increasing number and diversity of TDEs, a better understanding of the overall hydrodynamics is needed. A crucial ingredient is the self-intersection of the fallback stream, as it initiates the formation of the accretion disk and all subsequent electromagnetic emission. \n\nIn this work, we compute the geodesic motion of the fallback stream in the Kerr spacetime for a broad range of BH spins, orbital angular momenta and inclinations. An important feature of an inclined Kerr geodesic is that Lense-Thirring (LT) precession may cause missed prompt collision right after the first pericenter passage \\citep{dai13_gr_precession, guillochon15_dark_year, hayasaki16_precession_kerr}, unlike in the Schwarzschild case. While the motion of the stream center is described by a Kerr geodesic, we model the stream thickness by integrating an approximated tidal equation. Then, the position of the stream collision is where the separation between the stream centers is less than the sum of the thicknesses of the two approaching ends. One of the key differences between our model and the work of \\citet{guillochon15_dark_year} is the treatment of stream thickness evolution. They assume that the stream thickness increases linearly with orbital winding number (their Eq. 2), which they attribute to some unspecified energy dissipation by the nozzle shock near the pericenter. Consequently, the stream collision is mainly the result of the continuing growth of the stream thickness in their study. However, recent numerical simulations by \\citet{Bonnerot21_stream_width_evolution} show that the stream thickness does not dramatically increase after it passes through the nozzle shock. In this work, we treat the effect of the nozzle shock as a perfect bounce, and the evolution of the stream thickness is determined by the relativistic tidal forces.\nIt will be shown that the physical reason for the stream collision is a geometrical one that can be analytically described in a post-Newtonian picture for mildly relativistic orbits.\n\nThis paper is organized as follows. In \\S \\ref{sec:methods}, we describe our algorithm, which takes as input the BH spin and the stream's orbital parameters and then computes the position of the stream intersection. A parameter space exploration is presented in \\S \\ref{sec:parameter_exploration}, where we show how the properties of the stream intersection depend on the BH spin and the stream's orbital angular momentum and inclination. In \\S\\ref{sec:physical_reason}, we provide a post-Newtonian description that unveils the physical reason for the stream collision and analytically predicts the delay time before the collision. The limitations of our model are discussed in \\S \\ref{sec:discussion}. A summary of the main results is provided in \\S \\ref{sec:summary}. In this work, we only consider a representative TDE case of a $M=10^6M_\\odot$ BH and a $1M_\\odot$ star. The exploration of a wider parameter space is left for a future work. Our calculations are fully general relativistic, so we use geometrical units with $G=M=c=1$ throughout the paper, unless otherwise specified.\n\n\n\\section{Methods}\\label{sec:methods}\n\n\\subsection{Geodesic motion of stream center}\n\\label{sec:orbit}\n\nThe geodesic of a representative debris particle at the center of the fallback stream is specified by the following parameters:\n\\begin{enumerate}\n    \\item $\\tilde E$, the total specific energy (including rest mass),\n    \\item $L$, the total specific angular momentum,\n    \\item $I$, the inclination of the orbit (where $I=0$ is for the equatorial case and $I=\\pi\/2$ is for an initially polar orbit), and\n    \\item $a$, the BH spin (where $a=0$ is the Schwarzschild case and $a<0$ is for retrograde orbits),\n\\end{enumerate}\nThe inclination angle is defined by $\\cos I = L_z\/L$ ($L_z$ being the specific angular momentum component along the spin axis) and the Carter constant is $Q=L^2(\\sin I)^2$.\n\nThe geodesic equations of motion for a Kerr spacetime are numerically integrated to machine precision using a program based on \\cite{1994ApJ...421...46R}. We work in Boyer-Lindquist coordinates ($t$, $r$, $\\theta$, $\\phi$) with the metric given by\n\\begin{eqnarray}\n\\begin{split}\n    d s^2 = \\,& -\\left(1-{2r\\over \\Sigma}\\right) dt^2 - {4ar\\sin^2\\theta \\over \\Sigma} dt d\\phi + {\\Sigma \\over \\Delta} dr^2 + \\Sigma d \\theta^2 \\\\\n    &+ \\left(r^2 + a^2 + {2a^2 r\\sin^2\\theta \\over \\Sigma}\\right)\\sin^2\\theta\\, d \\phi^2,\n\\end{split}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n    \\Sigma=r^2+a^2\\mu^2,\n    \\label{eq:sigma}\n\\end{eqnarray}\n\\begin{eqnarray}\n    \\Delta=r^2-2r+a^2.\n    \\label{eq:delta}\n\\end{eqnarray}\nThe orbit is initialized at the apocenter at $\\tau=\\phi=0$ and $\\theta=\\pi\/2$, where $\\tau$ is the proper time. Note that starting from $\\theta=\\pi\/2$ (in the BH's equatorial plane) loses the generality of the initial conditions, but as we show in \\S\\ref{sec:physical_reason}, this choice is representative for most TDEs. The main effect of other choices of initial $\\theta$ is to change the delay time before the stream collision, and the dependence can be analytically predicted under the post-Newtonian approximation (see \\S\\ref{sec:physical_reason}).\n\nFor $u=1\/r$ and $\\mu=\\cos\\theta$, points on the orbit are sampled in intervals of $du$, which is related to proper-time intervals $d\\tau$ by \\citep{bardeen72_Kerr_geodesic}\n\\begin{eqnarray}\n    \\label{eq:dudtau}\n    \\frac{du}{d\\tau}=u^2\\frac{\\sqrt{V_r}}{\\Sigma},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n    \\label{eq:Vr}\n    V_r=T^2-\\Delta \\left(r^2+K\\right),\n\\end{eqnarray}\n\\begin{eqnarray}\n   K = (L_z - a\\tilde{E})^2 + Q,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:T}\n   T=\\tilde E \\left(r^2+a^2\\right)-aL_z.\n\\end{eqnarray}\nAn example of the geodesic motion for total energy $\\tilde E=0.9999$, total angular momentum $L=6.5$, inclination angle $\\cos I=0.5$ and BH spin $a=0.9$ is shown in Fig. \\ref{fig:fullorbit}.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{mode1_full.png}\n    \\caption{The orbit around a Kerr BH for a test particle at the center of the fallback stream. The parameters for this case are: total energy $\\tilde E=0.9999$, total angular momentum $L=6.5$, inclination angle $\\cos I=0.5$, BH spin $a=0.9$. The position of the BH, the origin of the Cartesian coordinate system, is marked by a black dot. The evolution is terminated when the stream crossing occurs, as marked by a red dot (the parameters for the stream thickness evolution are described in \\S \\ref{sec:stream_thickness}). All axes are in units of the gravitational radius $GM\/c^2$. \n    } \n    \\label{fig:fullorbit}\n\\end{figure}\n\n\\subsection{Stream thickness evolution}\\label{sec:stream_thickness}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{r1.pdf}\n    \\includegraphics[width=0.48\\textwidth]{r2.pdf}\n    \\includegraphics[width=0.48\\textwidth]{r3.pdf}\n    \\caption{The evolution of stream thickness $H$ as a function of $r$, magnified further to show details. To read the plot, start at the apocenter, and follow the plot continuously to see how the thickness would evolve in time and with $r$. On average, the stream becomes thicker with each successive winding, and the thickness ratio between the outward and inward segments of the same winding decreases with time. The geodesic is the same as shown in Fig. \\ref{fig:fullorbit}, with $\\tilde E=0.9999$, $L=6.5$, $\\cos I=0.5$, $a=0.9$. Here we set $\\sigma=1$ (see \\S \\ref{sec:stream_thickness} for its definition). The evolution is terminated when the stream crossing occurs, as marked by the red dots. Both axes are in units of the gravitational radius $GM\/c^2$. \n    }\n    \\label{fig:H_evolve}\n\\end{figure}\n\nAfter tidal disruption, the star is stretched into a long, thin stream. In the previous subsection, we compute the orbit of a test particle at the center of mass of the stream. In the plane transverse to the velocity vector, the stream has finite thickness, the evolution of which must be calculated to obtain the position of stream crossing. The long-term evolution of the stream thickness over multiple pericenter passages is currently still an open question, due to the prohibitive computational costs of global GR (magneto-) hydrodynamic simulations \\citep[see][and references therein]{bonnerot21_accretion_flow_review}. Right after the tidal disruption, self-gravity plays an important role confining the stream in the transverse direction, but it becomes unimportant soon after the first apocenter passage. We therefore neglect the influence of pressure forces in the subsequent evolution except at the nozzle shocks caused by short episodes of strong compression in the transverse direction.\n\nThese considerations motivate us to model the evolution of the stream thickness $H$ according to the following approximate tidal force equation based on the $C_{22}$ component of the tidal tensor given by \\citep[][their Eq. (56)]{marck83_tidal_tensor}\n\\begin{eqnarray}\\label{eq:tfe}\n\\begin{split}\n    \\frac{d^2H}{d\\tau^2} &\\approx -H {r\\over \\Sigma^3}(r^2 - 3a^2\\mu^2)\n    \\left(1 + 3{r^2 R^2 - a^2\\mu^2 S^2 \\over K \\Sigma^2}\\right),\\\\\n    R &= K - a^2\\mu^2,\\ S = r^2 + K,\n\\end{split}\n\\end{eqnarray}\nwhere the stream thickness $H$ is considered to be in the direction perpendicular to the local orbital plane (as defined far from the BH where GR is unimportant). Eq. (\\ref{eq:tfe}) is exact in the Schwarzschild limit ($a\\approx 0$), which is a good approximation because most of the stream-thickness evolution occurs far from the BH where the spin effects are minor. For the majority of the parameter space considered in this work, the LT precession angle is small ($\\ll 1\\rm\\, rad$) and stream crossing typically occurs between adjacent orbital windings, so the thickness perpendicular to the local orbital plane is the main physical quantity determining the occurrence of intersection. We do not consider the evolution of stream thickness within the local orbital plane, which is affected by the angular momentum spread across the stream\\footnote{The angular momentum spread across the stream \\citep[as can be seen from the simulations by][]{cheng14_relativistic_tdes} may cause the stream to look like a fan, with in-plane width typically larger than the vertical width. } as well as more complicated tidal and non-inertial forces than in Eq. (\\ref{eq:tfe}). A full treatment of the 2D stream cross-sectional structure in Kerr spacetime is much more complicated and computationally expensive, and this is left for future works.\n\n\n\nWe consider the part of the stream that is initially self-gravitating right after the disruption.\nThe initial conditions for the thickness evolution are taken to roughly correspond to the physical situation that the stream's self-gravity becomes sub-dominant compared to tidal forces shortly after the first apocenter passage. Motivated by the hydrodynamic simulations of \\citet[][their Fig. 5]{Bonnerot21_stream_width_evolution}, we set the initial stream thickness $H_0=5R_\\odot$ and the initial vertical expansion velocity $\\dot H_0=0$ when the stream center reaches a distance $r=r_{\\rm a}\/2$ ($r_{\\rm a}$ being the apocenter radius) from the BH after the first apocenter passage. Our results are insensitive to the exact position of maximum thickness (where $\\dot H=0$), because the thickness only depends weakly on the radius for this part of the evolution.\nFor the same reason, we also take the stream thickness between $r=r_{\\rm a}\/2$ and $r=r_{\\rm a}$ (the apocenter) to be constant at $H=H_0$. As we will discuss later, the line tangent to the orbit at the point of maximum stream thickness (where $\\dot H = 0$) determines the positions of the nozzle shocks (where $H=0$). Since we choose the initial point for the thickness evolution to be on the semi-\\textit{minor} axis, this tangent line is parallel to the semi-major axis. For other choices of the initial point somewhere between $r=r_{\\rm a}\/2$ and $r_{\\rm a}$ (but not extremely close to the apocenter), the tangent line will also be approximately parallel to the major axis, because the orbit is nearly radial due to its high eccentricity. In the subsequent evolution, the positions of the nozzle shocks are strongly affected by apsidal precession and are insensitive to our choice of the initial point.\n\n\nFor a given geodesic with known $r(\\tau)$ and $\\theta(\\tau)$, we numerically integrate Eq. (\\ref{eq:tfe}) to find the stream thickness evolution $H(\\tau)$. The integration is performed using the fourth-order Runge-Kutta method with an adaptive step size of $d\\tau=10^{-4}r^{3\/2}$ (and accuracy is checked through a numerical convergence study).\nWe then map $d\\tau$ to $du$ using Eq. (\\ref{eq:dudtau}).\n\n\nAnother aspect of the stream thickness evolution is the occurrence of nozzle shocks when the thickness reaches $H=0$. Near the point of maximum compression, hydrodynamic shocks convert the bulk kinetic energy associated with the motion perpendicular to the local orbital plane into heat. The pressure of the shocked gas then leads to outward expansion. As long as the timescale for the expansion of the shocked gas is much shorter than the orbital time near the pericenter, the effect of the nozzle shocks may be approximately taken into account by instantaneously flipping the sign of $\\dot H$. Our model assumes the post-bounce vertical velocity to be $-\\sigma \\dot{H}$, where $\\sigma$ is an adjustable parameter describing the strength of the bounce. Motivated by the Newtonian hydrodynamic simulations of \\citet{Bonnerot21_stream_width_evolution}, we set $\\sigma=1$ in this paper. If a small fraction of the orbital kinetic energy is dissipated by nozzle shocks \\citep[as speculated by][]{leloudas16_15lh_TDE}, then it is possible to have $\\sigma >1$, which would make the stream thicker (and self-intersection would occur earlier).\n\n\n\n\nAn example of the stream thickness evolution is shown in Fig. \\ref{fig:H_evolve}, for the geodesic considered in Fig. \\ref{fig:fullorbit}. An interesting result of our simulations is that the positions of nozzle shocks are not at the pericenter or apocenter. This is mainly due to GR apsidal precession \\citep[][see their Figs. 4 and 5]{luminet85_nozzle_shock}. The center of mass of the stream and its upper edge move as test particles on two different orbital planes. The two planes intersect at a line\\footnote{This will later on be referred to as ``Intersection Line 1'' so as to differentiate it with the other ``Intersection Line 2'' between the two (LT-precessed) orbital planes of the stream centers. Both intersection lines are important in our geometric understanding of the physical reason for stream collision. } which goes through the BH. The stream thickness $H$ is dictated by the projected distance of the stream position $\\vec{r}$ to this intersection line, and hence $H=0$ when the stream crosses this line. Due to apsidal precession, it is not possible for this intersection line to go through both the pericenter and apocenter. We find that each orbital winding has two nozzle shocks, one of which is near the pericenter and the other one is far from the pericenter and the apocenter, and that LT precession further causes variations in the nozzle shock positions in each successive winding.\n\n\n\n\n\nIt is important to note a limitation of this model: the effect of nozzle shocks on the stream thickness is treated as an instantaneous rebound when the stream undergoes maximum compression at $H=0$. In reality, pressure forces may continue to be important long after the nozzle shocks, and future works may incorporate the results of pressure forces from hydrodynamic simulations \\citep[see][]{Bonnerot21_stream_width_evolution} into the evolution of stream thickness.\n\n\n\n\n\n\\subsection{Computing stream crossing}\\label{sec:intersect}\n\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{c c}\n    \\hline\n    $i$ & 7\\\\\n    $j$ & 6\\\\\n    $r$ & 1200.8\\\\\n    $\\theta\\rm\\,[rad]$ & 2.60\\\\\n    $\\phi\\rm\\,[rad]$ & 4.57\\\\\n    $\\Delta s$ & 8.18\\\\\n    $H_1+H_2$ & 9.76\\\\\n    $\\Delta s\/(H_1+H_2)$ & 0.84\\\\\n    $H_2\/H_1$ & 0.77\\\\\n    $H_1$ & 5.51\\\\\n    $\\cos \\gamma$ & -0.96\\\\\n    $v_{\\rm rel}$ & 0.08\\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Properties of the intersection region, including the spatial coordinates ($r,\\, \\theta,\\, \\phi$), minimum separation between the two colliding segments $\\Delta s$, the stream width $H_1$ and $H_2$, the collision angle $\\gamma$, and the relative speed of collision $v_{\\rm rel}$, for the parameters $\\tilde E=0.9999$, $L=6.5$, $\\cos I=0.5$, $a=0.9$ and $\\sigma=1$. All results are in units where $G=M=c=1$.\n    }\n    \\label{tab:finalresult}\n\\end{table}\n\nTo find the point of stream collision, we first project the 3D orbit onto the $r$-$\\phi$ plane and find the points where the trajectory intersects itself in that 2D projection. These points are labelled as intersection candidates, which are found using the following procedure:\n\n\\begin{enumerate}\n    \\item Split the entire orbit into many half-orbit arrays, going from pericenter to apocenter or from apocenter to pericenter.\n    \\item Compare the half-orbit array labelled $i=0, 1, 2, \\ldots$, to each half-orbit $j(<i)$ before it.\n   \n   \n    Consider two consecutive points sampled on each orbit in the order of increasing $r$: $A=(r_i^1, \\phi_i^1)$ and $B=(r_i^2, \\phi_i^2)$ on $i$, and $C=(r_j^1, \\phi_j^1)$ and $D=(r_j^2, \\phi_j^2)$ on $j$. For $\\phi\\in(0,2\\pi)$, let $\\Delta \\phi_1=\\phi_j^1-\\phi_i^1$ and $\\Delta \\phi_2=\\phi_j^2-\\phi_i^2$. Intersection in 2D occurs when $\\Delta \\phi_1\\Delta \\phi_2<0$. The intersection candidate lies between $A$ and $B$ on $i$, and between $C$ and $D$ on $j$.\n\\end{enumerate}\n\nThe resolution in $r$ ($dr$) and in $\\phi$ ($d\\phi$) are given by the step size $d\\tau=10^{-4}r^{3\/2}$ using the geodesic equations of motion, $dr\/d\\tau=\\sqrt{V_r}\/\\Sigma$ and $d\\phi\/d\\tau=(-a\\tilde E + L_z + aT\/\\Delta)\/\\Sigma$ \\citep{bardeen72_Kerr_geodesic}, where the quantities $\\Sigma$, $\\Delta$, $V_r$, and $T$ are defined in Eqs. (\\ref{eq:sigma}) -- (\\ref{eq:T}). This resolution means that the fractional error in the position of stream collision is less than $10^{-4}$, which is negligible compared to $H\/r$ (which sets the physical ambiguity of the collision position).\n\n\nOnce we have the intersection candidates, each given by four points as above, the following procedure is used to determine whether collision has physically taken place:\n\n\\begin{enumerate}\n    \\item Compute the shortest distance between the two line segments $AB$ and $CD$, and call this $\\Delta s$, which is computed in the local frame near the intersection candidate.\n    \\item Let $H_1$ be the average of the stream thicknesses at $C$ and $D$. Let $H_2$ be the average of the stream thicknesses at $A$ and $B$. Collision occurs if $\\Delta s<H_1+H_2$.\n\\end{enumerate}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{mode1_zoom.png}\n    \\includegraphics[width=0.4\\textwidth]{mode2_zoom.png}\n    \\caption{Two different modes of intersection for $a=0.9$. The top panel shows a case of the ``head-on collision'' mode with $\\cos\\gamma$ close to $-1$ ($\\gamma$ being the intersection angle), for inclination angle $\\cos I=0.5$ and total angular momentum $L=6.5$. The bottom panel shows a ``rear-end collision'' case with $\\cos\\gamma$ close to 1, for the same inclination $\\cos (I)=0.5$ but slightly smaller angular momentum $L=5.7$. In both panels, the points corresponding to the nearest separation between the two stream segments are marked by red and blue dots.\n    }\n    \\label{fig:modes}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{r_int_combined.pdf}\n    \\caption{The intersection radius $r_{\\rm int}$ as a function of total angular momentum $L$. Different BH spins are represented by different symbol shapes and the colors indicate the inclination angles $I$. For clarity, we have added small horizontal offsets for each case of BH spin: $\\Delta L=+0.1$ ($a=0.9$, triangles), $+0.05$ ($a=0.5$, squares), $0$ ($a=0$, black circles), $-0.05$ ($a=-0.9$, diamonds). The thin blue dashed line shows the pericenter radius as a function of angular momentum for the $a=0$ case. The black dotted line is a polynomial fit (Eq. (\\ref{eq:fit})) to the $\\mathrm{log}\\,r_{\\rm int}$ values for the $a=0$ case. The intersection radius predicted by  \\citet{dai15_pn1_precession} is shown by a thin solid orange line, which is only accurate for large $L$. \n    }\n    \\label{fig:rint}\n\\end{figure*}\n\n\\subsection{A case study}\\label{sec:case_study}\nLet us illustrate the procedure outlined above in a concrete example. The parameters input into the routine are $\\tilde E=0.9999$, $L=6.5$, $\\cos I=0.5$, $a=0.9$, and $\\sigma=1$. The geodesic followed by the center of the stream is shown in Fig. \\ref{fig:fullorbit}.\nWe first project the 3D orbit into the $r$-$\\phi$ plane and find all intersection \\textit{candidates} between different parts of the orbit. Each intersection candidate is given by the four points around it, two on each half-orbit.\n\nAfterwards, we find the minimum physical distance $\\Delta s$ between $AB$ and $CD$, where $A\/B$ are points on half-orbit $i$ and $C\/D$ are points on half-orbit $j$.\n\n\n\n\nNext, using the stream thickness evolution (as determined by Eq. (\\ref{eq:tfe}) and shown in Fig. \\ref{fig:H_evolve}), we find the true collision point where $\\Delta s<H_1+H_2$. The details of this collision are given in Table \\ref{tab:finalresult}. For instance, the two segments of the fallback stream collide almost head-on at an angle of $\\gamma\\simeq 164$ degrees. Note that the collision occurs between the 6th and 7th half-orbits, and the delay time is slightly more than 2 orbital periods after the first pericenter passage of the fallback stream (or 3 orbital periods after the stellar disruption). As shown later by our full parameter space exploration, such a delay time is quite typical for TDEs. We also note that the stream is very thin and elongated at the time of collision with $H_1\\simeq H_2 \\simeq 10^{-2}r$, which means that, after the collision, the shock-heated gas will expand nearly adiabatically and hence the energy\/angular momentum distribution will be strongly broadened \\citep[see][]{lu20_self-intersection}. \n\n\n\n\n\\section{Results}\\label{sec:parameter_exploration}\n\nThe algorithm above can be used to find various quantities associated with collision for any set of parameters. In this section, results from a detailed parameter space exploration are presented.\n\nFor each of four BH spins ($a=-0.9$, $0$, $0.5$, $0.9$), we consider a two-dimensional parameter space --- cases with varying total angular momentum $L$ and inclination $\\cos (I)$ are explored. The specific energy is fixed to be $\\tilde E = 0.9999$ for all cases. We use a $10\\times10$ grid for $L$ and $\\cos I$: $L\\in\\{2.71 , 3.41, 4.19, 4.96, 5.74, 6.52 , 7.29, 8.07, 8.84, 9.62\\}$, and $\\cos I\\in\\{0.05 , 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95\\}$. The minimum angular momentum is given by that of a marginally bound equatorial orbit for $a=0.9$ (the most deeply penetrating case without plunging): $L_{\\rm min} \\approx 2.71$ in geometrized units, which is obtained by numerically solving $V_r=d_rV_r=0$ (where $V_r$ is given by Eq. (\\ref{eq:Vr}))\\footnote{The minimum angular momentum as a function of $a$ and $\\cos I$ is described in Appendix \\ref{sec:Lmin}.}. Note that, for some of the lowest angular momentum points, the geodesics are plunging and the star is completely swallowed by the BH. The maximum angular momentum is set by the requirement of tidal disruption, $L_{\\rm max} = \\sqrt{2 r_{\\rm T}}\\simeq 9.7$, where $r_{\\rm T}=R_*(M\/M_*)^{1\/3}\\simeq 46.7$ in geometrized units is the (Newtonian) tidal disruption radius for a $10^6M_\\odot$ BH and $1M_\\odot$ star. \n\nInteresting systematic properties of TDE stream intersections are summarized in the following subsections.\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ccc}\n\\subfloat[$a=0$]{\\includegraphics[width = 0.32\\textwidth]{a=0_cg.pdf}} &\n\\subfloat[$a=0$]{\\includegraphics[width = 0.32\\textwidth]{a=0_H1.pdf}} &\n\\subfloat[$a=0$]{\\includegraphics[width = 0.32\\textwidth]{a=0_r.pdf}}\\\\\n\\subfloat[$a=-0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=-0.9_cg.pdf}} &\n\\subfloat[$a=-0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=-0.9_H1.pdf}} &\n\\subfloat[$a=-0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=-0.9_r.pdf}} \\\\\n\\subfloat[$a=0.5$]{\\includegraphics[width = 0.32\\textwidth]{a=0.5_cg.pdf}} &\n\\subfloat[$a=0.5$]{\\includegraphics[width = 0.32\\textwidth]{a=0.5_H1.pdf}}&\n\\subfloat[$a=0.5$]{\\includegraphics[width = 0.32\\textwidth]{a=0.5_r.pdf}} \\\\\n\\subfloat[$a=0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=0.9_cg.pdf}} &\n\\subfloat[$a=0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=0.9_H1.pdf}} &\n\\subfloat[$a=0.9$]{\\includegraphics[width = 0.32\\textwidth]{a=0.9_r.pdf}}\\\\\n\\end{tabular}\n\\caption{The intersection angle $\\cos(\\gamma)$ (first column), thickness of one of the colliding streams $H_1$ (middle column), and intersection radius $\\log_{10}(r_{\\rm int})$ (right column), as a function of angular momentum $L$ and inclination angle $\\cos(I)$. The BH spins in each panel are indicated at the bottom of each plot. }\n\\label{fig:corplots}\n\\end{figure*}\n\n\n\n\\subsection{Two modes of intersection}\n\nWe find two modes of intersection: about half of the TDEs have rear-end collisions ($\\cos\\gamma>0$) near the pericenter and the other half have head-on collisions ($\\cos\\gamma<0$) typically further from the pericenter. This is illustrated in Fig. \\ref{fig:modes}, where we show two representative cases of these two modes.\n\nThere is a bimodal distribution of intersection angles (see Fig. \\ref{fig:corplots}), which are either close to 180$^{\\rm o}$ (``head-on'' mode) or 0$^{\\rm o}$ (``rear-end'' mode), with very few cases near 90$^{\\rm o}$. This is expected as we consider a post-Newtonian picture of two elliptical orbits with different orbital planes (due to LT precession) but the same focal point --- the orbits must have two closest approaches and in the limit of high eccentricity, one of the closest approaches is near the pericenter and the other one is far away.\nWhen the collision occurs near the pericenter, the two colliding ends have the same sense of rotation and this leads to $\\cos\\gamma\\simeq 1$. When the intersection point is far away from the pericenter and the apocenter\\footnote{The maximum angular momentum (as required by the tidal disruption of a $1M_\\odot$ star by a $10^6M_\\odot$ BH) gives a minimum apsidal precession angle of about $12^{\\rm o}$, and this means that the intersection point cannot be very close to the apocenter \\citep{dai15_pn1_precession}.}, the nearly radial orbits directly lead to $\\cos\\gamma\\simeq -1$. Later on in \\S \\ref{sec:vrel}, we provide analytical expressions for the radial coordinate of the collision point and the collision angle in the post-Newtonian limit.\n\n\nWe also note that the work by \\citet{guillochon15_dark_year} did not find any ``rear-end'' collisions near the pericenter, and that is because they consider each of the orbital windings to be a series of unconnected ellipses without explicitly resolving the (physically connected) near-pericenter segments between neighboring ellipses --- their setup prevents them from finding the ``rear-end'' collisions that we have identified. As we demonstrate later, the self-crossing shock in the rear-end mode is sufficiently strong to thicken the stream significantly and redistribute the angular momentum of the stream, so we expect rapid circularization of the bound gas due to subsequent shocks.\n\n\n\n\n\n\n\n\\subsection{Intersection radius}\n\nIn about half of the cases we studied, the intersection occurs near the pericenter (the rear-end mode), so the intersection radius is trivial to obtain. For the other half of TDEs in the head-on mode, we find the intersection radius depends mainly on the total angular momentum but less strongly on other parameters. This is illustrated in Fig. \\ref{fig:rint} (it can also be clearly seen on the rightmost panels of Fig. \\ref{fig:corplots}), where we show that the intersection radius $r_{\\rm int}$ generally increases with the total angular momentum $L$, for all BH spins $a$ (different symbol styles) and inclination angles $I$ (different colors). This is caused by weaker apsidal precession at larger angular momentum (or larger pericenter radius). For the mildly relativistic cases where the angular momentum is much larger than the minimum $L_{\\rm min}$, the BH spin and orbital inclination play minor roles in determining the intersection radius. This is because intersection typically occurs between adjacent orbital windings (as detailed in \\S \\ref{sec:delay_time}) and the out-of-plane precession angle\\footnote{Note that $\\Phi_{\\rm LT}$ is defined as the angle between the angular momentum vectors between adjacent windings, which is different from the nodal shift $\\Delta \\Omega_{\\rm nod} \\approx 4\\pi a\/L^3$.} $\\Phi_{\\rm LT}$ (due to the LT effect) is much smaller than the angle of in-plane (apsidal) precession $\\Phi_{\\rm ap}$. In the post-Newtonian approximation, these two angles are given by \\citep[][including the lowest order term due to spin]{merritt13_precession_angles}\n\\begin{eqnarray}\\label{eq:LT_precession}\n    \\Phi_{\\rm LT} \\approx {4\\pi a\\sin I\\over L^3},\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:apsidal_precession}\n    \\Phi_{\\rm ap} \\approx {6\\pi \\over L^2} - {8\\pi a\\cos I \\over L^3}.\n\\end{eqnarray}\nWe note that these two are the angles between the orbital angular momentum vectors ($\\Phi_{\\rm LT}$) and between the orbital periapses $(\\Phi_{\\rm ap})$ in a fixed reference frame, and it can be seen that the in-plane precession is larger than the out-of-plane precession by a factor of $1.5L$ or more.\n\nA polynomial fit to the $\\log(r_{\\rm int})$ values for the $a=0$ cases is given by the following:\n\\begin{multline}\n    \\log(r_{\\rm int})=-0.0081L^4+0.253L^3-2.96L^2+15.6L-28.5.\n    \\label{eq:fit}\n\\end{multline}\n\nHowever, for the highly relativistic cases with angular momenta only slightly above $L_{\\rm min}$, the intersection radius increases with spin such that $a=-0.9$ (or $0.9$) gives the smallest (largest) $r_{\\rm int}$. This is mainly because the 1.5-order post-Newtonian contribution to apsidal precession, $-8\\pi L^{-3} a \\cos I$ (see Eq. (\\ref{eq:apsidal_precession})), is negative for prograde orbits, and the reduction in apsidal precession angle increases the intersection radius \\citep[see also][]{stone19_TDE_in_GR}. The dependence of $r_{\\rm int}$ on the inclination angle can largely be explained by this effect as well.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{c}\n\\subfloat[$a=-0.9$]{\\includegraphics[width = 0.4\\textwidth]{a=-0.9_N.pdf}}\\\\\n\\subfloat[$a=0.5$]{\\includegraphics[width = 0.4\\textwidth]{a=0.5_N.pdf}}\\\\\n\\subfloat[$a=0.9$]{\\includegraphics[width = 0.4\\textwidth]{a=0.9_N.pdf}}\\\\\n\\end{tabular}\n\\caption{The distribution of orbital winding numbers $N_2$ vs. $N_1$ for three different BH spins, under the full ``loss cone'' assumption. We do not find any prompt collisions (for which $N_1=0$ and $N_2 = 1$) on our grid points for all BH spins.\nFor this case of $\\tilde E=0.9999$, each half-orbit revolution takes 2 months.\n}\n\\label{fig:N12}\n\\end{figure}\n\n\n\\subsection{Delay time}\n\\label{sec:delay_time}\nThe delay between tidal disruption and stream intersection causes a ``dark phase'' before bright emission is generated by shocks and accretion \\citep{dai13_gr_precession, guillochon15_dark_year}. In this subsection, we aim to estimate, in a statistical way, the typical delay time for TDEs of a $1M_\\odot$ star by a $10^6M_\\odot$ BH.\n\nWe denote the half-orbit numbers of the two colliding ends as $N_1=j$ and $N_2=i$, with $N_2>N_1$ by definition. For each BH spin $a$, we simulate about 100 cases (ignoring the plunging ones) of different \\{$L$, $\\cos I$\\}, for each of which we obtain \\{$N_1$, $N_2$\\}. We assigned a statistical weight $w=L$ to each case, which is based on the assumption of the ``full loss cone'' angular momentum distribution and an isotropic distribution of incoming stars \\citep{stone20_tde_rate_review}. The resulting normalized distributions of \\{$N_1$, $N_2$\\} for three different BH spins are shown in Fig. \\ref{fig:N12}. \n\nIn our simulations, the collision typically occurs between consecutive half-orbits, i.e. $N_2=N_1+1$, which is in agreement with \\citet{guillochon15_dark_year}. This can be easily understood as follows. Since the LT precession angle is usually small ($\\ll 1\\rm\\, rad$), the orbital planes of consecutive windings are spatially the closest, which means a collision eventually occurs when the angular width of the stream gradually grows above the angular separation between two adjacent orbital planes (cf. Fig. \\ref{fig:H_evolve}).\n\nWe note that one can tell which mode of intersection the collision follows by looking at $N_1$ and $N_2$: for odd $N_1$ and even $N_2$, the collision occurs in the head-on mode; and for even $N_1$ and odd $N_2$, the collision occurs in the rear-end mode. Fig. \\ref{fig:dpdn1} shows $dP\/dN_1$ for three different BH spins, where $P$ is the probability of occurrence of a TDE with $j=N_1$. For our choice of initial stream width $H_0=5R_\\odot$, we do not find any case of prompt collision for which $N_1 = 0$ on our grid points. This means that less than a few percent of TDEs (those with $\\cos I > 0.95$) may have prompt collisions for any of the three different BH spins ($-0.9, 0.5, 0.9$).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.3in]{dPdn.pdf}\n    \\caption{The probability distribution of the smaller winding number of the earlier colliding end, $dP\/dN_1$, for three BH spins, under the full ``loss cone'' assumption. The vertical dashed lines show the median for each distribution. The largest angular momentum considered in our grid is $L=9.62$, for which the apsidal precession angle is about 12 degrees. This means that the orbit has precessed by $90^{\\rm o}$ to be roughly perpendicular to the initial semimajor axis of the incoming star when $N_1\\sim 13$. We show in \\S \\ref{sec:physical_reason} that, under the post-Newtonian approximation, stream collisions occur before this.}\n   \n    \\label{fig:dpdn1}\n\\end{figure}\n\nWe also find that the delay between tidal disruption and stream intersection is typically between 2 and 7 orbital periods, with longest median delay in the highest spin $a=0.9$ case. We took the orbital energy to be $0.9999$, which corresponds to a semimajor axis of $5000$ gravitational radii and Keplerian period of about 4 months (a half-orbit takes 2 months). This means that the initially self-gravitating part of the fallback stream with initial thickness $H_0=5R_\\odot$ only joins the accretion flow about one to three years after the tidal disruption. A caveat is that the delay time strongly depends on our choice of initial stream width $H_0$ as well as the initial polar angle $\\theta$, but the delay times shown in Fig. \\ref{fig:dpdn1} are representative for most TDEs. This will be discussed in \\S \\ref{sec:physical_reason}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{a=-0.9_n1.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.5_n1.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.9_n1.pdf}\n    \\caption{The distribution of the orbital winding number $N_1$ for different BH spins: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel).}\n    \\label{fig:n1}\n\\end{figure}\n\nThe full distribution of $N_1$ as a function of the angular momentum $L$ and inclination $\\cos I$ is shown in Fig. \\ref{fig:n1}, where we find $N_1$ rapidly increases with $L$.\nThis can be qualitatively understood as follows.\nFor a larger $L$, the apsidal precession angle is much smaller and hence the stream thickness increases slower with winding number --- the stream thickness is maximized when the apsidal precession brings the major axis to be nearly perpendicular to \\textit{Intersection Line 1}, which is coincident with the \\textit{initial} major axis according to our initial conditions (as it is defined as the intersection line between the orbital plane of the stream center and that of the upper edge of the stream in the \\textit{initial} orbit). This explains the strong dependence of $N_1$ on $L$. The dependence on the inclination angle $I$ is mainly affected by the LT precession angle $\\Phi_{\\rm LT}$ --- the vertical separation $\\Delta s$ is linearly proportional to $|\\Phi_{\\rm LT}|\\propto |a| \\sin I$ --- a larger inclination angle tends to delay the stream collision. The weak dependence of the apsidal precession angle $\\Phi_{\\rm ap}$ on $I$ also plays a minor role, causing $N_1(I)$ to be non-monotonic for large positive spins $a\\sim 1$ and large inclination angles $I\\gtrsim 45^{\\rm o}$ (see the bottom panel of Fig. 8). A quantitative model for $N_1(L, \\cos I)$ will be described in \\S \\ref{sec:physical_reason}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n   \n   \n   \n   \n   \n   \n\n\n\n\n\n\n\n\n\\subsection{Thickness ratio and transverse offset}\nThe hydrodynamics of stream collision strongly depends on the thickness ratio $H_2\/H_1$ as well as the dimensionless transverse offset $\\Delta s\/(H_1+H_2)$ \\citep[e.g.,][]{jiang16_stream_collision}, where $\\Delta s$ is the closest approach distance between the centers of the two colliding geodesics. The ratio $\\Delta s\/(H_1+H_2)$ is an indicator of the fraction of kinetic energy dissipated in the collision -- a higher ratio means less energy dissipation. Subsequently, the results of the stream collision affect the formation of the accretion disk \\citep{bonnerot21_accretion_flow_review}. \n\nIn Figs. \\ref{fig:H21} and \\ref{fig:dsH}, we show $\\log_{10}(H_2\/H_1)$ and $\\Delta s\/(H_1+H_2)$, respectively, for cases with different angular momenta $L$ and inclinations $\\cos I$. The panels are for different BH spins $a=-0.9$, $a=0.5$ and $a=0.9$. In the case of $a=0$, the orbit is equatorial and hence $\\Delta s=0$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{a=0_H21.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=-0.9_H21.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.5_H21.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.9_H21.pdf}\n    \\caption{The distribution of thickness ratios $H_2\/H_1$ for different BH spins: $a=0$ (top panel), $a=-0.9$ (second panel), $a=0.5$ (third panel) and $a=0.9$ (bottom panel).}\n    \\label{fig:H21}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{a=-0.9_dsH.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.5_dsH.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.9_dsH.pdf}\n    \\caption{The distribution of (dimensionless) transverse separations $\\Delta s\/(H_1+H_2)$ for different BH spins: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel).}\n    \\label{fig:dsH}\n\\end{figure}\n\n\nWe find that the thickness ratio is close to unity, except for the special cases of nearly plunging angular momenta ($L\\lesssim 5$). This is because the stream thickness is specified by the projected distance from \\textit{Intersection Line 1}, which is very similar for the colliding ends since they are both located at the point of intersection.\n\n\n\n\n\nOn the other hand, we find that the typical transverse offset $\\Delta s$ at the collision point is a small but significant fraction (tens of percent) of the total thickness $H_1 + H_2$, and grazing collisions with $\\Delta s\/(H_1+H_2)\\approx 1$ are very rare. Thus, we expect a large fraction of the gas to be shock-heated in an offset collision, which substantially increases the stream thickness (or directly produces a puffy gaseous envelope). Subsequent secondary shocks will lead to the circularization of the bound gas to form an accretion disk.\n\nWe note that a limitation of our study is that the vertical width $H$ does not provide a full description of the transverse density profile of the stream. The consequence is that a small fraction of the gas (the atmosphere of each stream) may still undergo shock interaction even when $\\Delta s\\gtrsim H_1 + H_2$. It is possible that a weak, oblique shock in a minor collision propagates to the interior of the stream and leads to width expansion. Thus, the criterion for stream collision may actually be somewhat looser than our prescription ($\\Delta s\/(H_1 + H_2)$ somewhat larger than 1) and the delay time shorter. \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Relative speed of collision}\\label{sec:vrel}\nThe (Lorentz invariant) relative speed of the two colliding streams is given by\n\\begin{eqnarray}\n    v_{\\rm rel} = {\\sqrt{(\\vec{p}_1\\cdot \\vec{p}_2)^2 - 1} \\over -\\vec{p}_1\\cdot \\vec{p}_2},\n\\end{eqnarray}\nwhere $\\vec{p}_1$ and $\\vec{p}_2$ are the four-velocities of the two streams at the collision point, whose components can be evaluated in any frame as long as the dot product is calculated using the corresponding metric (we compute the dot product in Boyer-Lindquist coordinates).\nIn Fig. \\ref{fig:rv}, we show the relative speeds for all cases considered in this work.  \n\nWe find that the relative velocity strongly depends on the total angular momentum of the stream and that the dependence on the inclination and BH spin is weak.\nThe stream collisions are highly violent in all cases, with relative velocities being a few percent of the speed of light or higher. Note that $v_{\\rm rel}\/2$ better describes the component of the velocity that is dissipated by shocks, because in the Newtonian limit and when all the gas is shock-heated (for $\\Delta s=0$ and $H_1=H_2$), the time-averaged power of shock dissipation is given by\\footnote{The stream collision is expected to be intermittent, because after the segment between the two colliding ends is consumed by shocks, the next episode of collision will be delayed by roughly the duration of the previous collision episode.\nIn the ideal case where the shocked gas does not interact with the cold stream, the shock power alternates between 0 and twice the $\\lara{\\dot{E}}$ given here. }\n$\\lara{\\dot E} = (1\/2)\\dot{M} (v_{\\rm rel}\/2)^{2}=1.9\\times10^{43}\\mathrm{\\, erg\\,s^{-1}}\\, (\\dot{M}\/3\\,M_\\odot\\mathrm{\\,yr^{-1}}) (v_{\\rm rel}\/0.03)^2$, where $\\dot{M}$ is the mass fallback rate. As pointed out by \\citet{jiang16_stream_collision, lu20_self-intersection}, the radiative efficiency of the shocked gas near the collision point is much less than unity (typically a few percent) such that the observed luminosity $L_{\\rm sh}\\ll \\lara{\\dot E}$. We see that the shock-powered emission typically falls short of explaining many TDEs with peak optical luminosity of $\\sim 10^{44}\\rm\\, erg\\,s^{-1}$ \\citep[e.g.,][]{gezari12_ps-10jh} unless the initial orbit of the star is deeply penetrating with pericenter distance much less than the tidal disruption radius.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.4\\textwidth]{avgrelv.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=-0.9_rv.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.5_rv.pdf}\n    \\includegraphics[width = 0.4\\textwidth]{a=0.9_rv.pdf}\n    \\caption{Relative speed of collision in units of $c$. The top plot shows the relative speed for $a=0$, along with the medians of the relative speeds for $a=-0.9$ (second panel), $a=0.5$ (third panel) and $a=0.9$ (bottom panel) over the inclination angles. The solid lines in the top plot illustrate the approximate analytical prediction for the relative speed, as in Eq. (\\ref{eq:relative_velocity}).\n    }\n    \\label{fig:rv}\n\\end{figure}\n\nRemarkably, we find the relative velocity to be nearly independent of whether the collision is in the head-on or rear-end modes. We demonstrate the physical reason in the following\npost-Newtonian picture. Let us approximate the orbit right before the stream collision as an ellipse with the BH located at one of the focal points. The stream collision is mainly caused by apsidal precession, and hence the collision points in the head-on and rear-end modes are located at true anomaly  $\\pi - \\Phi_{\\rm ap}\/2$ and $\\Phi_{\\rm ap}\/2$, respectively, where the apsidal precession angle $\\Phi_{\\rm ap}$ is given by Eq. (\\ref{eq:apsidal_precession}).\n\n\nThese two collision points (which lie on the same line that goes through the BH) are located at radii\n\\begin{eqnarray}\\label{eq:rint}\n    r_{\\pm} = {L^2\\over 1 \\pm e_{\\rm orb} \\cos(\\Phi_{\\rm ap}\/2)},\n    \\label{eq:rint}\n\\end{eqnarray}\nwhere the squared orbital eccentricity is $e_{\\rm orb}^2 = 1 - L^2\/a_{\\rm orb}$ and the semimajor axis is $a_{\\rm orb} = [2(1-\\tilde E)]^{-1}$. It can be seen that $r_+$ is close to the pericenter radius (corresponding to the rear-end mode) and $r_-$ is typically much larger (the head-on mode). Note that $r_-$ is in agreement with Eq. (7) of \\citet{dai15_pn1_precession}.\n\nConservation of energy and angular momentum along with Eq. (\\ref{eq:rint}) imply that the radial components $v_{\\mathrm{K},r}$ of the Keplerian velocities at these two collision points are the same, given by\n\\begin{eqnarray}\n    v_{\\mathrm{K},r}^2 = -{1\\over a_{\\rm orb}} + {1 - e_{\\rm orb}^2 \\cos^2(\\Phi_{\\rm ap}\/2) \\over a_{\\rm orb}(1-e_{\\rm orb}^2)},\n\\end{eqnarray}\nwhere the dependence on the precession angle is $\\cos^2(\\Phi_{\\rm ap}\/2)$ and the sign of $\\cos\\Phi_{\\rm ap}$ does not matter. The velocity component perpendicular to the radial direction is $L\/r$, so the collision angle is given by $\\tan (\\gamma\/2) = r\\,v_{\\mathrm{K},r}\/L$. This shows that in the rear-end (or head-on) collision mode, we typically have $\\gamma\\ll 1\\rm\\, rad$ (or $\\gamma\\sim \\pi$). \nThe relative velocity is twice the radial component of the velocity at the collision points and is given by\n\\begin{eqnarray}\\label{eq:relative_velocity}\n\\begin{split}\n    \\left({v_{\\rm rel}\\over 2}\\right)^2\n   \n    \\approx 2(\\tilde E - 1) + {1 - [1+2L^2(\\tilde E - 1)] \\cos^2(\\Phi_{\\rm ap}\/2) \\over L^2}.\n   \n\\end{split}\n\\end{eqnarray}\nIn the limit where $1-e_{\\rm orb}\\ll 1$ and $\\Phi_{\\rm ap} \\lesssim 1\\rm\\, rad$, we obtain the approximation\n\\begin{eqnarray}\n    v_{\\rm rel}\\approx {\\Phi_{\\rm ap} \\over L} \\approx {6\\pi \\over L^3} - {8\\pi a\\cos I \\over L^4}.\n    \\label{eq:relv}\n\\end{eqnarray}\nThis shows that the relative velocity between the two colliding ends is a power-law function of the total angular momentum ($v_{\\rm rel}\\propto L^{-3}$) or orbital pericenter distance ($\\propto r_{\\rm p,orb}^{-1.5}$) in the weakly relativistic regime. The relative velocity is slightly smaller for larger, prograde BH spins, as illustrated in Fig. \\ref{fig:rv}.\n\n\n\\section{Physical Reason for Intersection}\\label{sec:physical_reason}\nIn this section, we show that the collision is a geometric effect. Throughout this section, we work in the lowest-order post-Newtonian approximation such that $\\Phi_{\\rm LT}\\ll \\Phi_{\\rm ap}$ (see Eqs. \\ref{eq:LT_precession} and \\ref{eq:apsidal_precession}) and assume that collision occurs between successive half-orbits $N_1$ and $N_2=N_1+1$.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.65\\textwidth]{schematic_gb.pdf}\n    \\caption{A schematic picture for the geometric viewpoint of stream collision in the post-Newtonian approximation, where the orientation of an elliptical orbit undergoes apsidal and Lense-Thirring (LT) precessions. A collision occurs either due to the increase in thickness $H$ as the major axis of the orbit moves away from Intersection Line 1 (thick blue line), or due to the decrease in vertical separation between adjacent orbital windings as the major axis become more aligned with Intersection Line 2 (pink solid line).}\n    \\label{fig:schematic}\n\\end{figure*}\n\nA schematic picture is shown in Fig. \\ref{fig:schematic}. The initial ($N=0$) orbital plane at the center of mass (COM) of the fallback stream is denoted as $P_{\\rm COM}$, and the particle at the upper edge of the stream lies in the orbital plane labelled as $P_{\\rm e}$. These two planes intersect in a line, which is denoted as \\textit{Intersection Line 1}. The angle between these two planes ($P_{\\rm COM}$ and $P_{\\rm e}$) is given by\n\\begin{eqnarray}\n    \\alpha = H_0\/b_{\\rm orb} = {H_0\\sqrt{2(1-\\tilde{E})}\\over L},\n\\end{eqnarray}\nwhere $H_0$ is the initial stream thickness on the minor axis, $b_{\\rm orb} = La_{\\rm orb}^{1\/2}$ is the semi-minor axis, and $a_{\\rm orb} = (2(1-\\tilde{E}))^{-1}$ is the semi-major axis as determined by the specific orbital energy $\\tilde{E}$. For $H_0=5R_\\odot$, $\\tilde E=0.9999$, and $L\\in (6.5, 9.7)$ (the mildly relativistic cases), we find $\\alpha\\sim 5\\times10^{-3}\\rm\\, rad$. The stream thickness $H$ at a given position is related to the inclination angle $\\alpha$ and the projected distance to Intersection Line 1.\nSuppose the collision occurs at radius $r_{\\rm int}$ (as given by Eq. \\ref{eq:rint}) between the $N_1$-th and $(N_1+1)$-th half-orbits. We project the radial vector $\\vec r_{\\rm int}$ onto Intersection Line 1 to obtain the stream thickness in the vertical direction (perpendicular to the orbital plane) as\n\\begin{eqnarray}\\label{eq:stream_width}\n    H(N_1)=r_{\\rm int} \\sin\\lrsb{(N_1 + 1)\\Phi_{\\rm ap}\/2} \\alpha.\n\\end{eqnarray}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{nam09.pdf}\n\\includegraphics[width = 0.4\\textwidth]{na05.pdf}\n\\includegraphics[width = 0.4\\textwidth]{na09.pdf}\n\\caption{Comparison between the analytical prediction for $N_1$ as in Eq. \\ref{eq:collision_criterion} (indicated by dashed or dotted lines) and the numerical calculation of $N_1$ (indicated by points, squares or triangles) for various values of $\\omega_0$ (in different colors) and BH spin: $a=-0.9$ (top panel), $a=0.5$ (middle panel) and $a=0.9$ (bottom panel). We find good overall agreement between the analytical model and the numerical calculations, except for a few cases with $\\cos(I)\\approx 0$ and $\\omega_0\\approx \\pi$ under $a=0.9$ and $L=7$. These correspond to the rare cases of nearly polar orbits (i.e., the star's initial major axis is nearly aligned with the BH spin vector), for which the stream thickness evolution can only be captured by using the full tidal tensor and hence both our analytical\/numerical results may have large errors.}\n\n\n\\label{fig:n1compare}\n\\end{figure}\n\nThe other quantity that determines the occurrence of stream collision is $\\Delta s$, which is the vertical separation between the COMs of the two colliding streams. This can be calculated as follows. Intersection Line $1$ is perpendicular to the initial orbital angular momentum vector $\\vec L$ of the orbit. The black hole spin vector $\\vec S$ makes an inclination angle $I$ with $\\vec L$. The plane defined by $\\vec L$ of the initial orbit and $\\vec S$ is labelled $M_{\\rm i}$. LT precession of $\\vec L$ about $\\vec S$ causes $\\vec L$ to rotate in the direction of $\\vec S\\times\\vec L$, perpendicular to $M_{\\rm i}$. The plane defined by the rotated $\\vec L'$ and $\\vec S$ is labelled by $M_{\\rm f}$. For adjacent orbital windings, the angle between $M_{\\rm i}$ and $M_{\\rm f}$ is $\\Phi_{\\rm LT}$ (eq. \\ref{eq:LT_precession}). $M_{\\rm i}$ and $M_{\\rm f}$ intersect in \\textit{Intersection Line $2$}, which forms an angle $\\omega_0$ with \\textit{Intersection Line $1$}. The angle $\\omega_0\\in (0, \\pi)$ is entirely set by the initial position and angular momentum of the star when it was fed to the TDE loss cone. It is related to the initial polar angle $\\theta_0$ in our Boyer-Lindquist coordinates and the inclination angle $I$ between $\\vec L$ and $\\vec S$ by\n\\begin{eqnarray}\n    \\cos(\\theta_0)=\\cos(\\pi-\\omega_0)\\sin(I).\n\\end{eqnarray}\nOur numerical simulations as described in \\S \\ref{sec:methods} and \\ref{sec:parameter_exploration} only considered $\\theta_0=\\pi\/2$ (so as to reduce the dimensions of parameter space), which corresponds to $\\omega_0=\\pi\/2$. The probability density distribution of $\\mr{d} P\/\\mr{d} \\omega_0$ is flat in $(0, \\pi)$, so it is equally likely that $\\omega_0$ is less than or greater than $\\pi\/2$. According to the post-Newtonian picture described in this section, most of the properties of the stream collision (intersection radius $r_{\\rm int}$, thickness ratio $H_2\/H_1$, offset ratio $\\Delta s\/(H_1+H_2)$, relative velocity $v_{\\rm rel}$) do not strongly depend on the choice of $\\omega_0$. However, as we now describe, the delay time is strongly affected by $\\omega_0$.\n\nIn the limit that the cumulative LT precession angle $N_1\\Phi_{\\rm LT}\/2\\ll 1\\rm\\, rad$, the vertical separation between the COMs of the colliding streams is given by\n\\begin{eqnarray}\n    \\Delta s'(N_1) = r_{\\rm int} \\sin\\lrsb{(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0} \\Phi_{\\rm LT}.\n\\end{eqnarray}\nAs we will see later, the number of half orbits before the collision is $N_1 \\sim 2\/\\Phi_{\\rm ap}$, so cumulative LT precession angle before the stream collision is of the order $\\Phi_{\\rm LT}\/\\Phi_{\\rm ap}\\sim a\\sin I\/L$. Including the cumulative precession of \\textit{Intersection Line 2} before the collision leads to a slightly more accurate expression\n\\begin{eqnarray} \\label{eq:ds}\n    \\Delta s(N_1) = r_{\\rm int} \\sin\\lrsb{(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0 - N_1\\Phi_{\\rm LT}\/2} \\Phi_{\\rm LT},\n\\end{eqnarray}\nwhich we use (but the two equations above give very similar delay times). Note that as the number of half-orbits $N_1$ increases, the angle between the radial vector $\\vec r_{\\rm int}$ and Intersection Line 2 (the argument in the sine function above) increases. If $\\omega_0 < \\pi\/2$, $\\Delta s$ first increases with $N_1$ and then decreases as the angle between $\\vec r_{\\rm int}$ and Intersection Line 2 approaches $\\pi$. If $\\omega_0>\\pi\/2$, then $\\Delta s$ monotonically decreases with $N_1$.\n\nUsing Eq. \\ref{eq:stream_width} and Eq. \\ref{eq:ds}, the criterion for intersection, $\\Delta s<H$, is \n\\begin{eqnarray}\\label{eq:collision_criterion}\n\\sin\\lrsb{{(N_1+1)\\Phi_{\\rm ap}\\over 2} + \\omega_0 - {N_1\\Phi_{\\rm LT}\\over 2}}{\\Phi_{\\rm LT}\\over \\alpha} <\\sin\\lrsb{{(N_1 + 1)\\Phi_{\\rm ap}\\over 2}}.\n\\end{eqnarray}\nStream crossing occurs at the minimum integer $N_1$ satisfying this equation. Note that above equation includes the cases where $(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0 - N_1\\Phi_{\\rm LT}\/2$ exceeds $\\pi$ and when this occurs, $\\Delta s$ flips sign meaning that orbital crossing has occurred.\nIn this geometrical picture, we see that stream collision occurs for different reasons in the following two regimes: $\\Phi_{\\rm LT}\/\\alpha < 1$ or $>1$. \n\nFor TDEs with small LT precession angles $\\Phi_{\\rm LT}$ (due to small BH spin or large orbital angular momentum) or for the parts of the fallback stream that are initially not self-gravitating (the initial thickness is much larger than $5R_\\odot$), we may have $\\Phi_{\\rm LT}\/\\alpha \\ll 1$. In this regime, stream collision occurs mainly due to the increase of the right-hand side of Eq. (\\ref{eq:collision_criterion}) (i.e. an increase in the stream thickness), and we can estimate the number of half-orbits before the collision to be $N_1 \\simeq 2\\Phi_{\\rm LT}\\sin\\omega_0\/(\\alpha \\Phi_{\\rm ap}) - 1$. On the other hand, for TDEs with large $\\Phi_{\\rm LT}$ or for the very thin self-gravitating parts of the fallback stream, we may have $\\Phi_{\\rm LT}\/\\alpha \\gtrsim 1$. In this regime, stream collision occurs mainly due to the decrease of the left-hand side of Eq. (\\ref{eq:collision_criterion}) (i.e. a decrease in the vertical separation), and the number of half-orbits before the collision can be estimated by $(N_1+1)\\Phi_{\\rm ap}\/2 + \\omega_0 \\simeq \\pi$ or $N_1\\simeq 2(\\pi-\\omega_0)\/\\Phi_{\\rm ap} - 1$. Putting these two regimes together, we obtain the estimate\n\\begin{eqnarray}\\label{eq:analyticN1}\n    N_1 \\simeq \\min\\lrsb{{\\Phi_{\\rm LT}\\sin\\omega_0\\over \\alpha}, \\pi - \\omega_0} {2\\over \\Phi_{\\rm ap}} - 1.\n\\end{eqnarray}\n\nFig. \\ref{fig:n1compare} shows a comparison between the analytical prediction for $N_1$ (Eq. \\ref{eq:collision_criterion}) and the numerical results of $N_1$ (based on the algorithm in \\S \\ref{sec:methods}) for various BH spins and three different choices of $\\omega_0=\\pi\/6, \\pi\/2, 5\\pi\/6$. We fix the initial stream thickness to be $H_0=5R_\\odot$ (the same as in previous sections), which means all the cases are in the $\\Phi_{\\rm LT}\/\\alpha \\gtrsim 1$ regime. The key trend in this regime is that $N_1$ is a strong function of the orbital angular momentum, $N_1\\propto L^2$ (the dependence on the inclination angle is less strong, unless $\\cos I \\simeq 1$, which corresponds to very vanishing LT precession). Another important trend is that $N_1$ decreases with $\\omega_0$, and this is because for larger $\\omega_0$ it takes a smaller number of orbital windings to align the major axis of the orbit with Intersection Line 2.\n\n\nWe conclude that the physical reason for stream collisions lies in the geometry of Kerr spacetime: collision occurs when the apsidal precession brings the major axis of the elliptical orbit to be nearly parallel with the plane containing the angular momentum vector and the BH spin vector (minimizing the vertical separation between consecutive orbits) and\/or to be nearly perpendicular to the initial major axis (maximizing the stream thickness). An earlier work by \\citet{guillochon15_dark_year} suggests that the collision is caused by the increase in stream thickness with the orbital winding number as a result of possible energy dissipation by nozzle shocks. However, in our model, the stream thickness evolves according to the BH's tidal forces and collision still occurs when the stream does not get significantly inflated at the nozzle shocks, as motivated by the recent simulation of \\citet{Bonnerot21_stream_width_evolution}.\n\nAccording to our model, the delay times for TDE stream collisions span a very wide range from a few months up to a decade, which mainly depend on $H_0$ (initial stream thickness), $L$ (orbital angular momentum), and $\\omega_0$ (the angle between the initial major axis and the initial $\\vec L$-$\\vec S$ plane). Short delay times correspond to the cases that satisfy one of the following: $L\\lesssim 5$ (highly relativistic TDEs), small $\\Phi_{\\rm LT}\/\\alpha$ (thick initial streams or small BH spins), or $\\omega_0$ close to $\\pi$. On the other hand, long delay times are expected for cases with large $\\Phi_{\\rm LT}\/\\alpha$ (thin initial streams) and $\\omega_0$ far from $\\pi$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this section, we briefly discuss a number of limitations of our current model which may be addressed in future works.\n\n\n(1) We only considered mono-energetic geodesics, whereas the realistic bound debris has a spread in orbital energy -- the gas falling back at a later time is less bound and has a larger apocenter radius. This causes the two colliding ends of the stream to have different energies and their orbits to have different apocenter radii. Since most of the collisions occur between successive half-orbits ($N_2 = N_1 + 1$), the fallback times of the two colliding ends differ by at most one orbital period. The intersection radius will only be affected by the energy spread if the collision occurs near the apocenter (i.e., for angular momentum $L\\sim L_{\\rm max}\\simeq 9.7$). This is because the apocenter radius of the later colliding end is substantially larger than that of the first one. Accounting for the energy spread requires a more sophisticated scheme that records the positions of different fluid elements with different orbital energies on a grid of global Boyer-Lindquist time. However, for most of the parameter space studied in this work, the intersection radius is sufficiently far from the apocenter and the orbital energies of the two colliding ends are approximately the same.\n\n(2) The stream thickness evolution is approximated by Eq. (\\ref{eq:tfe}), which is exact only for the Schwarzschild spacetime in the direction perpendicular to the orbital plane. The stream thickness within the orbital plane is strongly affected by the angular momentum spread across the stream \\citep{cheng14_relativistic_tdes}. Its evolution is governed by a different tidal equation with non-inertial terms due to rotation, although the differences from our Eq. (\\ref{eq:tfe}) become negligible at sufficiently large radii where the orbits are nearly radial. Additionally, BH spin causes a mixing between the stream thickness perpendicular to the orbital plane and the in-plane thickness, since the tidal tensor is non-diagonal \\citep{marck83_tidal_tensor} \\citep[see also][]{kesden12_spinning_BH}. Fully capturing the evolution of the cross-sectional shape of the stream requires two dimensional simulations using the full tidal tensor as well as including gas pressure effects, which are beyond the scope of this work.  In Appendix \\ref{sec:Newtonian_tidal_equation}, we show the stream thickness evolution under an even simpler, Newtonian tidal equation, and the results are qualitatively similar to those presented in previous sections.\n\n(3) The effects of pressure forces are only crudely accounted for near the nozzle shocks (where $H=0$) by prescribing a perfect bounce of the vertical velocity $\\dot{H}$. In reality, pressure forces may have long-lasting effects on the stream thickness evolution. \\citet{Bonnerot21_stream_width_evolution} found that the irreversible entropy increase (which is proportional to the adiabatic constant $P\/\\rho^{5\/3}$ for monoatomic gas) at the nozzle shock causes the stream thickness to be affected by pressure forces long after the shock occurrence. Their simulation showed that the long-term effects of pressure forces may only change the stream thickness by a factor of order unity.\nThus, we expect our results to be qualitatively unchanged even when gas pressure is included.\n\n\n(4) An important caveat is that the delay time distribution (Figs. \\ref{fig:N12} and \\ref{fig:dpdn1}) depends sensitively on the initial thickness of the fallback stream $H_0$ as well as the angle $\\omega_0$ between the major axis and the $\\vec{L}$-$\\vec{S}$ plane of the initial orbit. As for the initial stream thickness, we took $H_0=5R_\\odot$, corresponding to the densest part of the stream that is initially confined by self-gravity. However, the most bound part of the fallback stream is non-self-gravitating and much less dense, and it has a larger thickness by a factor of a few to 10 \\citep[see][]{Bonnerot21_stream_width_evolution}. For a larger $H_0$, we expect the delay time before the intersection to be shorter (see eq. \\ref{eq:analyticN1}). Even if the most bound part of the stream has a shorter delay time before collision, it is so far unclear how the shocked gas from earlier fallback affects the later, denser parts of the stream. We speculate that, if the orbits of the denser parts are largely unaffected by the surrounding low-density\\footnote{The density ratio between the unshocked stream and the surrounding gas is of the order $(r\/H)^2 \\sim 10^4$.} gas from earlier fallback, then the long delay times (of up to a decade) obtained from our study will lead to delayed mass feeding to the BH. This may potentially explain the late-time X-ray and UV emission \\citep{vanVelzen19_late-time_UV, jonker20_late-time_Xray}, without invoking a viscous timescale of a few years or longer (which is difficult to achieve if the gas circularizes near the tidal disruption radius).\n\n\n\n\n\n\n\\section{Summary}\\label{sec:summary}\n\nWe have developed an algorithm to find the position of stream collision that results from the tidal disruption of a $1 M_{\\odot}$ star by a $10^6 M_{\\odot}$ supermassive BH. This algorithm takes as input the BH spin $a$ and the stream's orbital parameters: the specific energy $\\tilde E$, the total specific angular momentum $L$, the inclination angle $I$ between the initial orbital angular momentum and the BH spin. The longitudinal motion of the stream follows a geodesic in the Kerr spacetime, and the evolution of the transverse size of the stream is decoupled from the longitudinal evolution. Using an approximate tidal equation (including the effects of pressure forces at the nozzle shocks by assuming a perfect bounce), we calculate the evolution of the vertical thickness of the stream along the geodesic. This allows us to determine the occurrence of collision by comparing the stream thickness and the closest approach separation between different windings.\n\nBy performing a parameter space exploration for different $a$, $L$ and $I$ using this algorithm, we study various properties of the stream collision: intersection radius, collision angle, stream thicknesses at collision, transverse separation between the two colliding ends, relative speed at collision, and the delay time between the stellar disruption and stream collision. Our main findings are summarized as follows.\n\n\n\\begin{itemize}\n    \\item\n   \n    Misalignment of BH spin with the orbital angular momentum may lead to delay of the stream collision. The physical reason for the collision is a geometric effect (Fig. \\ref{fig:schematic}). Collision occurs when the major axis of the elliptical orbit is brought to be nearly parallel with the initial $\\vec L$-$\\vec S$ plane (so as to minimize vertical separation between consecutive orbits) or to be nearly perpendicular to the initial major axis right after the disruption (so as to maximize the stream thickness). We calculate the delay time of the collision numerically (fully relativistic) and analytically (in the post-Newtonian limit) and find it to span a wide range from months up to a decade. This means that the thinnest parts of the fallback stream may only join the accretion flow many years after the initial tidal disruption.\n   \n    \\item Stream self-intersection occurs in two modes: the head-on mode and the rear-end mode. About half of the TDEs are in the head-on mode, where the collision occurs far from the pericenter and the collision angle is close to 180$^{\\rm o}$ (or $\\cos\\gamma\\simeq -1$); the other half are in the rear-end mode, where the intersection occurs near the pericenter with a very small collision angle close to $0^{\\rm o}$ (or $\\cos\\gamma\\simeq 1$). Very few cases have collision angles close to $90^{\\rm o}$.\n    \\item Intersection typically occurs between consecutive half orbits, i.e. $N_2 = N_1 + 1$, where $N_1 $ and $N_2$ are the half-orbit numbers of the two colliding ends.\n    \\item The intersection radius $r_{\\rm int}$ for the head-on mode generally increases with the total angular momentum but depends less strongly on other parameters (whereas $r_{\\rm int}\\approx r_{\\rm p}$ for the rear-end mode). This is because intersection typically occurs between adjacent orbital windings and the LT precession angle is much smaller than the apsidal precession angle. Larger angular momenta are associated with smaller apsidal precession angles that lead to larger intersection radii. The weak dependence of $r_{\\rm int}$ on the BH spin and orbital inclination is such that larger $a\\cos (I)$ leads to larger intersection radii.\n   \n   \n    \\item The ratio of the thicknesses of the two colliding ends $H_2\/H_1$ is typically of order unity, and the transverse separation $\\Delta s$ is a small but significant fraction of $H_1 + H_2$ (grazing collisions with $\\Delta s\\approx H_1 + H_2$ are rare). This means that the stream undergoes an offset collision which causes a large fraction of the gas to be shock heated. We expect the collision in the head-on mode to generate a puffy envelope of gas which undergoes secondary shocks and subsequently forms an accretion disk. In the rear-end mode, the stream is expected to be significantly broadened, which leads to further collisions in subsequent orbits and the eventual formation of an accretion disk.\n    \\item The relative velocity between the two colliding ends is at least a few percent of the speed of light, so the collision is always violent. The relative velocity mainly depends on the total angular momentum of the stream; $v_{\\rm rel}\\propto L^{-3}$ in the weakly relativistic regime. The weak dependence on the BH spin and inclination is such that larger $a\\cos(I)$ leads to smaller relative velocity. The time-averaged energy dissipation rate by the self-crossing shock is of the order $10^{43}\\rm\\, erg\\,s^{-1}$ for angular momentum near the boundary of tidal disruption ($L\\sim L_{\\rm max}\\simeq 9.7$ for a $1M_\\odot$ star and a $10^6M_\\odot$ BH).\n   \n\\end{itemize}\n\nThis work provides a realistic picture of the aftermath of stellar tidal disruption by a spinning BH. Our results for the intersection radius, collision angle, thickness ratio, and transverse offset can be used as initial conditions for hydrodynamic studies of the stream collision process, the subsequent formation of an accretion disk, and the associated electromagnetic emission \\citep[e.g.,][]{bonnerot21_first_light}. Our approach can be improved in the future by including orbital energy spread at different parts of the fallback stream, and employing a more realistic stream width evolution as obtained by hydrodynamic studies.\n\n\n\n\n\\vspace{1cm}\n\n\\noindent\n{\\bf Data availability}\nThe data produced in this study will be shared on reasonable request to the authors.\n\n\\vspace{0.5cm}\n\n\\noindent\n{\\it Acknowledgments.} \nGB was supported by the Caltech Summer Undergraduate Research Fellowship. WL was supported by the David and Ellen Lee Fellowship at Caltech and the Lyman Spitzer, Jr. Fellowship at Princeton University. The research of CB was funded by the Gordon and Betty Moore Foundation through Grant GBMF5076. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 836751.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum black holes have been intensively investigated for recent\nyears. This interest was mainly stimulated by the development of\nhigh energy physics. One of the most promising open issues of high\nenergy physics is the development of theories with extra\nspace-like dimensions. It was proposed that extra space-like\ndimensions can lower the Planck scale to the TeV region\n\\cite{arkani-hamed, arkani-hamed_2, antoniadis, Randall99}. It is\nexpected that particle collisions with center-of-mass energy above\nthe Planck scale and impact parameter smaller than the horizon\nradius can be a source for producing micro black holes and branes\n\\cite{BH_production, Eardley, Yoshino, Yoshino_2, Berti,\nCavaglia_Jhep, Kanti_09}. (Some critical remarks to this scenario\ncan be found in \\cite{Voloshin, Rychkov04}). So the energy\nsufficient for producing of micro black holes can in principle be\nachievable even for present generation of particle accelerators.\nThe lowering of the Planck scale possibly allows to by-pass the\nhierarchy problem. Identification of the ultraviolet cutoff scale\nwith electroweak energy one $\\Lambda_{EW}$ can lead to radiative\nstability without using additional assumptions such as\nsupersymmetry or technicolor. It is considered that the\nunification of gravity with other gauge interactions at\nelectroweak scales can be obtained in theories with extra\ndimensions. Another important feature of theories with extra\ndimensions is violation of Newton's law of gravitation $\\sim1\/r^2$\nand this deviation can be possibly probed in the nearest feature.\n\nEvolution of micro black holes, formed in high energy collisions\ncan be divided into several stages, so called phases: loss of\ngauge charges and multipole momenta (balding phase), loss of\nangular momentum (spin-down phase) and the last one is so called\nSchwarzschild phase when a black hole loses energy via Hawking\nradiation. The final stage of Schwarzschild phase is not well\nunderstood due to the lack of fully-fledged theory of quantum\ngravity. Several attempts to understand this stage were made with\nthe help of Generalized Uncertainty Principle (GUP) \\cite{adler,\ncavaglia04, scardigli_2}. It was shown that GUP prevents complete\nevaporation of black holes leaving some remnants. Noncommutative\ngeometry was also applied to quantum black holes. The key point in\nthis approach is the noncommutativity inspired modification of\nblack hole's metrics \\cite{Nicolini08}.\n\nAnother line of research that has renewed interest to micro black\nholes, especially with GUP, is cosmology. Recent investigations of\nthe CMB power spectrum have brought an idea about preinflationary\nepoch \\cite{Powell, Wang}. The preinflationary scenario proposed\nin \\cite{scardigli} was based on the micro black hole production\nand a set of first principles such as generalized uncertainty\nprinciple and holographic principle. These principles allow one to\nobtain self-consistent description of the suppression of the CMB\nquadrupole without additional assumptions. As it was pointed out\nthat GUP can lead to the matter domination in the preinflatory\nepoch. In spite of some success of matter dominant preinflation\nthis scenario has some drawbacks and needs further investigation.\n\nThis paper is organized as follows. In the second section we\nconsider modified generalized uncertainty principle with minimal\nposition and momentum and obtain an expression for the black\nhole's temperature. In the third section we calculate entropy and\nheat capacity of black hole. In the fourth section emission rate\nrelation is considered and black hole's evaporation time is\ncalculated. The fifth section contains conclusions and some\nmathematical details are considered in Appendix.\n\n\\section{Uncertainty principle with minimal length and momentum and black hole's temperature}\nRecent studies in string theory brought an idea about\ngeneralization of Heisenberg uncertainty principle \\cite{gross,\nwitten}. Similar suggestion was proposed to reconcile principles\nof Quantum Mechanics and General Relativity \\cite{maggiore,\nScardigli_99}. Generalized Uncertainty Principle (GUP) takes form:\n\\begin{equation}\\label{GUP}\n\\D p\\D x\\geqslant\\frac{\\h}{2}(1+\\B\\D p^2)\n\\end{equation}\nAs it is known such a generalization leads to appearance of\nminimal uncertainty in position, so called minimal length. It was\nshown that gravity-induced decoherence gives rise to similar\nmodification of uncertainty principle but with minimal uncertainty\nfor momentum \\cite{Kay, Kay_CQG, Kay_07}. These two assumptions\nabout minimal uncertainties in position and momentum might be\nmerged in order to obtain modified generalized uncertainty with\nboth minimal length and momentum. We can write uncertainty\nrelation:\n\\begin{equation}\\label{MGUP}\n\\D x\\D p\\geqslant\\frac{\\h}{2}\\left(1+\\A\\D x^2+\\B\\D p^2\\right)\n\\end{equation}\nIt is easy to show that $\\D x\\geqslant\\D\nx_{min}=\\h\\sqrt{\\B}\/(\\sqrt{1-\\h^2\\A\\B})$ and $\\D p\\geqslant\\D\np_{min}=\\h\\sqrt{\\A}\/(\\sqrt{1-\\h^2\\A\\B})$. We note that in order to\nget these minimal uncertainties parameters $\\A$, $\\B$ must be\npositive and $\\h^2\\A\\B<1$. We also remark that uncertainty\nprinciple (\\ref{MGUP}) was introduced after consideration of a\ngedanken experiment for the simultaneous measurement of position\nand momentum of a particle in de Sitter spacetime \\cite{Bambi}.\n\n The uncertainty relation (\\ref{MGUP}) can be obtained if we suppose that\noperators of position and momentum obey deformed commutation\nrelation:\n\\begin{equation}\\label{al_osc}\n[x,p]=i\\h (1+\\A x^2+\\B p^2),\n\\end{equation}\nwhere parameters of deformation $\\alpha$ and $\\beta$ are assumed\nto be positive ($\\alpha,\\beta>0$) and $\\h^2\\A\\B<1$. We note that\ncommutation relation (\\ref{al_osc}) and uncertainty principle\n(\\ref{MGUP}) firstly appeared in quantum group investigations\n\\cite{kempf1}.\n\nWe point out that some troubles appear when one tries to\ngeneralize the algebra (\\ref{al_osc}) for a case of higher\ndimension. Probably the main difficulty is to satisfy closedness\ncondition. It was proposed to generalize commutation relation\n(\\ref{al_osc}) in the following way \\cite{Banerjee, mignemi}:\n\\begin{eqnarray}\\label{SDS_alg}\n[x_i,p_j]=i\\h\\left(\\delta_{ij}+\\A x_ix_j+\\B\np_jp_i+\\sqrt{\\A\\B}(p_ix_j+x_jp_i)\\right).\n\\end{eqnarray}\nWhereas other two commutation relations need to be written in a\nproper way to fulfil the Jacobi identity. It was also proposed the\nrepresentation of operators obeying (\\ref{SDS_alg}). We also point\nout that classical commutation relations that use classical\ncounterpart of (\\ref{SDS_alg}) are well defined (see Appendix).\n\nLet us come back to the uncertainty (\\ref{MGUP}) and rewrite it in\nthe form:\n\\begin{equation}\n\\frac{1}{\\h\\B}\\left(\\D x-\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\\leqslant\\D\np\\leqslant\\frac{1}{\\h\\B}\\left(\\D x+\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\nIt is easy to see that right hand side inequality does not recover\nordinary uncertainty principle in the limit $\\B\\rightarrow 0$ or\n(and) $\\A\\rightarrow 0$. So we obtain\n\\begin{equation}\\label{final_MGup}\n\\D p\\geqslant\\frac{1}{\\h\\B}\\left(\\D x-\\sqrt{\\D\nx^2\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\n\n It was proposed that applying uncertainty principle to a\nblack hole one can get Hawking temperature up to a some\n``calibration'' factor \\cite{adler}. As it was pointed out by\nAdler and collaborators GUP prevents a black hole from complete\nevaporation and temperature catastrophe. That idea attracted a lot\nof attention and was generalized for the case of higher dimension\n\\cite{cavaglia04, scardigli_2, Myung}.\n\nSince evaporated particles appear just outside of the horizon\nsurface it means that uncertainty in position for them can not be\nless than linear size of a black hole. If we assume that the\nuncertainty in position for a particle is less than linear size of\nthe black hole it means that particle emerges inside the black\nhole and can not penetrate through the horizon surface. On the\nother hand if we suppose that the uncertainty in position is\ngreater than linear size of the black hole then the emitted\nparticle appears not on the outside of horizon surface but\nsomewhere beyond it and as a consequence the temperature of the\nblack hole will be different than it should be.  To obtain\ntemperature of the black hole we equate uncertainty in position\nfor emitted particle to the double of Schwarzschild radius.\n\\begin{equation}\\label{uncert-rad}\n\\D x=2R_S.\n\\end{equation}\n\n Having used (\\ref{final_MGup}) and supposing that $\\D\nE=c\\D p$ we obtain the uncertainty in the energy of emitted\nparticle.\n\\begin{equation}\\label{uncert_energy}\n\\D E=c\\D p\\geqslant\\frac{c}{\\h\\B}\\left(2R_S-\\sqrt{\n4R^2_S\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\n\nLet us suppose that we deal with ordinary Hawking effect and\nconsider Hawking radiation just outside the event horizon of a\nSchwarzschild black hole. We identify the uncertainty of energy\n$\\D E$ given by the relation (\\ref{uncert_energy}) with the\nthermal energy of an emitted particle. So we can use well-known\nenergy-temperature relation $\\D E=3T$ (We have used thermal energy\nfor photons and here we set Boltzmann constant equal to unity\n$k_B=1$). Taking into account all these remarks and substituting\nthe mass $M$ of a black hole instead of its gravitational radius\n$R_S$ ($R_S=2GM\/c^2$) we obtain relation for the temperature of\nemitted particles\n\\begin{equation}\\label{BH_temp}\nT=\\frac{c}{\\pi\\h\\B}\\left(\\frac{4GM}{c^2}-\\sqrt{\\frac{16G^2M^2}{c^4}\\left(1-\\h^2\\A\\B\\right)-\\h^2\\B}\\right)\n\\end{equation}\nWe remark that in the latter relation we have used ``calibration''\nfactor $\\pi$ instead of $3$ in order to get correct relation for\nHawking temperature when parameters of deformation tend to zero\n($\\A,\\B\\rightarrow 0$). Probably the correct result for the\nHawking temperature can be reproduced if one uses proper\nequipartition law on the horizon surface or just outside it. It\nwas shown that number of degrees of freedom on the horizon surface\nis defined by its surface area \\cite{Padmanabhan}. So it is worth\nexpecting that proper equipartition law leads to correct\n``calibration'' factor. It should be noted that Hawking\ntemperature for the Schwarzschild-AdS black hole with modified GUP\n(\\ref{MGUP}) was obtained in \\cite{Park08}. But in order to\nreproduce correct relation for the temperature of the black hole\none of the deformation parameters was held fixed \\cite{Park08}.\nUnfixing this parameter and taking Schwarzschild metrics instead\nof Schwarzschild-AdS one we arrive at the relation\n(\\ref{BH_temp}). We point out that uncertainty relation\n(\\ref{MGUP}) can be used with Schwarzschild metrics because it can\nbe derived without any assumption about the de Sitter spacetime.\nWe also remark that temperature of Schwarzschild-AdS black hole\nwith generalized uncertainty principle that leads to appearance of\nminimal momentum was firstly considered in \\cite{BolenGERG}.\n\nIt was pointed out \\cite{BolenGERG} that uncertainty principle\ndoes not describe the origin of such effects as semiclassical wave\nscattering or particle tunnelling but only their consequence on\nthe measurement process. To explain the origin of black hole's\nevaporation it is necessary to know the quantum states of the\nblack hole from which the exact uncertainty principle follows. It\nwas suggested that black holes thermodynamics is a low-energy\neffect of small-scale physics. It is known that any theory of\nquantum gravity contains some kind of uncertainty principle that\nreduces to Heisenberg principle at low energies. So black hole\nthermodynamics should not depend too much on the details of\nunderlying quantum gravity theory \\cite{BolenGERG}. This is in\nagreement with Visser's conclusion that Hawking radiation requires\nordinary quantum mechanics and slowly evolving future horizon, so\nto explain black hole thermodynamics quantum gravity principles\nare not necessary \\cite{visser}.\n\n Uncertainty relation (\\ref{MGUP}) gives rise to the existence\nof a minimal mass of a black hole:\n\\begin{equation}\nM_{min}=\\frac{\\h c^2\\sqrt{\\B}}{4G\\sqrt{1-\\h^2\\A\\B}}=\\frac{\ncM^2_{Pl}\\sqrt{\\B}}{4\\sqrt{1-\\h^2\\A\\B}},\n\\end{equation}\nwhere $M_{Pl}=\\sqrt{\\h c\/G}$ is the Planck's mass. So according to\nthe relation (\\ref{BH_temp}) it leads us to the finite temperature\nat the final point of Hawking radiation, which takes following\nform:\n\\begin{equation}\nT_{final}=\\frac{c}{\\pi\\sqrt{\\B}\\sqrt{1-\\h^2\\A\\B}}\n\\end{equation}\n\nOne can see that uncertainty relation (\\ref{MGUP}) causes\nincreasing of black hole's temperature in comparison with relation\n(\\ref{GUP}).\n\nMinimal uncertainty in position given by relation (\\ref{MGUP})\nsimilarly as in case of (\\ref{GUP}) prevents us from a temperature\ncatastrophe. It was proposed that generalized uncertainty\nprinciple (\\ref{GUP}) prevents black holes from complete\nevaporation in the same way as the standard uncertainty principle\nprevents the hydrogen atom from collapse \\cite{cavaglia04}. To\nmake the influence of a minimal uncertainty in momentum clear let\nus investigate temperature as a function of $M$. Taking the\nderivative $\\partial T\/\\partial M$ and equating it to zero one can\nfind that temperature has an extremum point when the mass of a\nblack hole reaches:\n\\begin{equation}\\label{extremal_mass}\nM_{ext}=\\frac{c^2}{4G\\sqrt{\\A\\left(1-\\h^2\\A\\B\\right)}}.\n\\end{equation}\nIt is easy to see that at this point temperature of the black hole\ntakes minimal value:\n\\begin{equation}\\label{minimal_temp}\nT_{min}=\\frac{c\\h\\sqrt{\\A}}{\\pi\\sqrt{1-\\h^2\\A\\B}}\n\\end{equation}\nWhen the black hole's mass is above $M_{ext}$, temperature\n(\\ref{BH_temp}) increases with the mass rise. When the mass is\nbelow $M_{ext}$ its decreasing leads to increasing of temperature.\nSimilar mass-temperature dependence was obtained in case of\nSchwarzschild-AdS black hole with modified generalized principle\n(\\ref{MGUP}). We note that minimal temperature is the consequence\nof a minimal uncertainty in momentum and it can appear for\ndifferent black hole's metrics. Temperature as the function of\nmass is shown in Fig.\\ref{fig1}.\n\n\\begin{figure}\n  \\centerline{\\includegraphics[scale=1,clip]{temp_2.eps}}\n  \\caption{Temperature of a black hole as a function of mass.\n  Temperature and mass are represented in Planck's units. So $m=M\/M_{Pl}$\n  denotes relation of black hole's mass to the Planck's mass.  Deformation parameters\n  are also written in dimensionless form: $\\A'=\\A L^2_{Pl}$ and $\\B'=\\h^2\\B\/L^2_{Pl}$.\n  In order to get that the mass of a black hole at the final point of Hawking radiation to be equal\n  to the Planck's mass (or $m=1$) when the parameter $\\A'=0$ we  set   $\\B'=16$.} \\label{fig1}\n\\end{figure}\nIn the case of higher dimensional space one should use\nSchwarzschild-Tangherlini metrics \\cite{Tangherlini,Myers}. The\nhorizon radius of a black hole takes form:\n\\begin{equation}\\label{GR_radius_higher_dimensions}\nR_S=\\left(\\frac{16\\pi G_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{1\/(d-3)}\n\\end{equation}\nWhere $d$ is the dimension of space-time, $G_d$ is the\n$d$-dimensional gravitational constant and $\\Omega_{d-2}$ is the\n surface area of $d-2$-dimensional unit hypersphere.\nSo the temperature of a $d$-dimensional black hole takes form:\n\\begin{equation}\\label{temp_stringy_high_dim}\nT=\\frac{(d-3)c}{\\pi\\h\\B}\\left[2\\left(\\frac{16\\pi\nG_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{1\/(d-3)}-\\sqrt{4\\left(\\frac{16\\pi\nG_dM}{(d-2)\\Omega_{d-2}c^2}\\right)^{2\/(d-3)}(1-\\h^2\\A\\B)-\\h^2\\B}\\right]\n\\end{equation}\nAs it was pointed out in \\cite{scardigli_2} generalized\nuncertainty relation (\\ref{GUP}) for the black hole in the case of\nhigher dimensions should be replaced by another one due to the\nfact that distances can not be probed below the Schwarzschild\nradius (\\ref{GR_radius_higher_dimensions}). So according to\n\\cite{scardigli_2} instead of generalized uncertainty principle\n(\\ref{GUP}) in $d=4+n$-dimensional space-time one should write:\n\\begin{equation}\\label{MBH_GUP}\n\\D p\\D x\\geqslant\\frac{\\h}{2}(1+\\gamma\\D p^{\\frac{n+2}{n+1}}),\n\\end{equation}\nwhere $n$ is the number of additional spacelike dimensions. It was\nnoted \\cite{scardigli_2} that at high energies stringy GUP\n(\\ref{GUP}) can give an uncertainty $\\D x$ larger than the size of\na black hole itself. But scenarios with extra dimensions suggest\nthat all gauge interactions apart from gravity live on $3+1$\ndimensional brane \\cite{arkani-hamed, arkani-hamed_2, antoniadis,\nRandall99}. So modification of GUP (\\ref{MBH_GUP}) is valid only\nfor gravitons, whereas for photons and other SM particles emitted\nby a black hole it is necessary to use uncertainty principle\n(\\ref{GUP}) or (and) (\\ref{MGUP}). Taking into account all the\nabove remarks we can interpret the relation\n(\\ref{temp_stringy_high_dim}) as some upper bound for temperature\nof emitted gravitons.\n\n\n\\section{Entropy and heat capacity of a black hole}\nHaving calculated black hole's temperature we can find black\nhole's entropy using well-known thermodynamical relation:\n\\begin{equation}\ndS=\\frac{c^2}{T}dM\n\\end{equation}\nAfter integration we obtain:\n\\begin{eqnarray}\\label{enthropy_general}\nS=\\frac{\\pi c^3}{4\\h\nG\\A}\\left(\\ln\\Big{|}\\frac{4GM-\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}}{\\h\nc^2\\sqrt{\\B}}\\Big{|}+\\right.\\nonumber\n\\\\\n\\left.\\sqrt{1-\\h^2\\A\\B}\\ln\\Big{|}\\frac{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2\nc^4\\B}+4GM\\sqrt{1-\\h^2\\A\\B}}{\\h c^2\\sqrt{\\B}}\\Big{|}\\right)\n\\end{eqnarray}\n The last formula reproduces well-known relation for the\nBekenstein-Hawking entropy in the limit $\\A,\\B\\rightarrow 0$. We\ncan suppose that parameter $\\A$ is small in comparing with $\\B$ so\nwe can expand the right hand side of equation\n(\\ref{enthropy_general}) into the series over a small parameter\n$\\A$.\n\\begin{eqnarray}\\label{entropy_small_a}\nS=\\frac{\\pi}{4\\h c G}\\left(8G^2M^2+2GM\\sqrt{16G^2M^2-\\h^2\nc^4\\B}-\\frac{\\h^2c^4\\B}{2}\\ln\\left(\\frac{4GM+\\sqrt{16G^2M^2-\\h^2\nc^4\\B}}{\\h c^2\\sqrt{\\B}}\\right)\\right.\\nonumber\n\\\\\n-\\frac{\\A}{c^4}\\left[64G^4M^4+GM\\sqrt{16G^2M^2-\\h^2\nc^4\\B}\\left(\\frac{\\h^2c^4\\B}{2}+16G^2M^2\\right)\\right.\\nonumber\n\\\\\n\\left.\\left.+\\frac{\\h^4c^4\\B^2}{2}\\ln\\left(\\frac{4GM+\\sqrt{16G^2M^2-\\h^2\nc^4\\B}}{\\h c^2\\sqrt{\\B}}\\right)\\right]\\right)\n\\end{eqnarray}\nIn the limit $\\A\\rightarrow 0$ we reproduce the expression for the\n entropy in the presence of a minimal length \\cite {adler}. Expression\n(\\ref{entropy_small_a}) shows that including $\\A$-dependent terms\nlead to decreasing of  entropy similarly as we have in the case\nwith $\\B$-terms only.\n\nTo calculate the heat capacity of a black hole we use a well-known\nthermodynamical relation:\n\\begin{equation}\nC=T\\frac{\\partial S}{\\partial T}=\\frac{\\partial E}{\\partial T}.\n\\end{equation}\nIn our case temperature and entropy are represented as functions\nof black hole's mass $M$ (or diameter $R$). So we write the heat\ncapacity as a function of mass $M$.\n\\begin{equation}\\label{heat_capacity_general}\nC=\\frac{\\pi\\h\nc^3\\B}{4G}\\frac{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}}{\\sqrt{16G^2M^2(1-\\h^2\\A\\B)-\\h^2c^4\\B}-4GM(1-\\h^2\\A\\B)}\n\\end{equation}\n\n\\begin{figure}\n  \\centerline{\\includegraphics[scale=1,clip]{heat_capacity_m.eps}}\n  \\caption{Heat capacity as a function of mass.}\\label{fig2}\n\\end{figure}\n\nLet us investigate the latter relation in details. It is easy to\nmake yourself sure that when mass is below $M_{ext}$ heat capacity\nis negative and it is equal to zero when the mass reaches\n$M_{min}$. When the mass of black hole is above $M_{ext}$ heat\ncapacity is positive and tends to a finite value when mass goes to\ninfinity. So $M_{ext}$ is the discontinuity point for the heat\ncapacity. Heat capacity as a function of mass is represented in\nFig.\\ref{fig2}.  Negative heat capacity shows that thermodynamical\nsystem is unstable and tends to decay. Whereas positive heat\ncapacity makes system stable. So at the extremum point we have\nphase transition . One can conclude that the phase transition is\ncaused by the modified GUP (\\ref{MGUP}) but not particular choice\nof  black hole's metrics. As it has already been noted modified\nGUP (\\ref{MGUP}) related to commutation relation (\\ref{SDS_alg})\nbut it is known that the choice of commutation relations that give\nthe same uncertainty principle is non unique. So modified GUP\n(\\ref{MGUP}) might be obtained for more general case of\ncommutation relations. We point out that behaviour of heat\ncapacity near the phase transition point is similar to behavior of\nheat capacity in case of the well-known Hawking-Page phase\ntransition \\cite{Hawking83, Majumdar}. Phase transition also\nappears when the modifications of black hole's metrics caused by\nnonlocal effects or noncommutative geometry are taken into account\n\\cite{Nicolini12}. But in that case stable and unstable phases are\nexchanged their places in comparison with our result. We also note\nthat extremum point $M_{ext}$ is an inflection point for entropy.\nEntropy as a function of mass is shown in Fig.\\ref{fig3}.\n\n\\begin{figure}\n  \\centerline{\\includegraphics[scale=1,clip]{ent_m2.eps}}\n  \\caption{Entropy as a function of mass}\\label{fig3}\n\\end{figure}\n\n\\section{Emission rate equation and evaporation time of a black hole}\nIn this section evaporation of a micro black hole is studied.\nSimilar problem with GUP was firstly considered in work\n\\cite{adler}. This topic was investigated in details in\n\\cite{cavaglia04, scardigli_2}. To examine quantum black hole's\nevaporation different approaches are used. Probably the simplest\none is to make use of Stefan-Boltzmann law for the black-body\nradiation which is based on an assumption that we have a\nthermodynamical system with fixed temperature so we consider\ncanonical ensemble. We note that the Stefan-Boltzmann law gets new\ncorrection caused by deformation of commutation relations\n\\cite{cavaglia04, scardigli_2, Chang}. As it was pointed out in\n\\cite{casdaio, Casadio_2} canonical description for a micro black\nhole is adequate  when the energy of emitted particles (e.g.\nphotons, gravitons) is small in comparison with the energy of a\nblack hole itself. If the energy of emitted particles is\ncomparable with the energy of a black hole itself one should use\nmicrocanonical description. When the energy of black hole is large\nin comparison with the energy of emitted particles both of them\ngive the same result. Here we use canonical description,\nmicrocanonical one will be considered elsewhere. We also point out\nthat statistical mechanics in the case of the deformed commutation\nrelations was considered in \\cite{Fityo}.\n\nLet us examine equilibrium radiation caused by a black hole. We\nsuppose that that black hole is placed inside a sphere of given\nradius $R$ and system is considered under a fixed temperature $T$.\nFor generality we consider $D$-dimensional case ($D=3+n$). So the\nenergy of evaporated particles can be calculated as follows\n(photons, gravitons):\n\\begin{equation}\\label{energy_blackbody}\nE=\\frac{g}{(2\\pi\\h)^D}\\int\\frac{d^Dxd^Dp}{1+\\A x^2+\\B\np^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})}\\displaystyle\n\\frac{\\varepsilon(p)\\Gamma (p)}{e^{\\varepsilon(p)\/T}-1}\n\\end{equation}\nwhere $\\varepsilon(p)$ is the dispersion relation for emitted\nparticles, $\\Gamma (p)$ is the greybody factor and $g$ is the\ndegeneracy factor. In the case of photons we put\n$\\varepsilon(p)=cp$ and $g=2$. The same dispersion relation can be\nconsidered for gravitons but the degeneracy factor is different.\nWe also suppose that greybody factor is equal to unity\n$\\Gamma(p)=1$ (blackbody radiation). Let us remind that emission\nof gravitons takes place in $d=4+n$ - dimensional bulk spacetime\nwhereas photons and other gauge particles are emitted on\n3+1-dimensional brane.\n\n We note that suppositions we have made do not\nallow to calculate integral exactly (\\ref{energy_blackbody}) even\nif one of the deformation parameters is absent. To calculate it we\nassume that parameters $\\A$ and $\\B$ are small and develop weight\nfunction under integral into the series over deformation\nparameters. So the latter relation can be rewritten in the form:\n\\begin{equation}\nE=\\frac{g}{(2\\pi\\h)^D}\\int d^Dxd^Dp (1-\\A x^2-\\B\np^2-2\\sqrt{\\A\\B}({\\bf x},{\\bf p}))\\displaystyle\n\\frac{cp}{e^{cp\/T}-1}\n\\end{equation}\nAfter integration we obtain:\n\\begin{eqnarray}\nE=\\frac{2g}{(2\\sqrt{\\pi}c\\h)^D}\\frac{D!\\zeta(D+1)}{\\Gamma(D\/2)}\\left[\\left(V-\\frac{D^{1+2\/D}}{D+2}\\left(\\frac{\\Gamma\n\\left(D\/2\\right)}{2\\pi^{D\/2}}\\right)^{2\/D}\\A\nV^{1+2\/D}\\right)T^{D+1}\\right.\\nonumber\n\\\\\n\\left.-\\frac{(D+2)!\\zeta(D+3)}{D!\\zeta(D+1)}\\B VT^{D+3}\\right]\n\\end{eqnarray}\nwhere $V=\\displaystyle 2\\pi^{D\/2}R^D\/(D\\Gamma(D\/2))$ is the volume\nof $D$-dimensional sphere. It should be noted that the energy of\nradiation nonlinearly depends on the volume. In three dimensional\ncase we find:\n\\begin{equation}\nE=\\frac{g}{(2\\pi\\h)^3}\\left[\\left(V-\\frac{3}{5}\\left(\\frac{3}{4\\pi}\\right)^{2\/3}\\A\nV^{5\/3}\\right)\\frac{4\\pi^5}{15 c^3}T^4-\\frac{32\\pi^7}{63c^3}\\B\nVT^6\\right]\n\\end{equation}\nTo obtain emission rate equation we suppose that particles are\nemitted by sphere with radius $R_S$. So the total energy $dE$\nemitted during the period of time $dt$ can be written in the form:\n\\begin{eqnarray}\n\\frac{dE}{dt}=-\\frac{4gc}{(2\\h\nc)^D}\\frac{D!\\zeta(D+1)}{\\left(\\Gamma(D\/2)\\right)^2}\n\\left[\\left(R^{D-1}_S-\\A\nR^{D+1}_S\\right)T^{D+1}-\\frac{(D+2)!\\zeta(D+3)}{D!\\zeta(D+1)}\\B\nR^{D-1}_S\\frac{T^{D+3}}{c^2}\\right]\n\\end{eqnarray}\nWhere $R_S$ is the Schwarzschild radius. For photons in three\ndimensional case we obtain:\n\\begin{equation}\\label{emiss_rate_E}\n\\frac{dE}{dt}=-\\frac{4\\pi^3}{15c^2\\h^3}\\left((R^2_S-\\A\nR^4_S)T^4-\\frac{40}{21}\\frac{\\pi^2\\B}{c^2}R^2_ST^6\\right)\n\\end{equation}\nIn the limit when parameters of deformation tens to zero the last\nequation gives ordinary Stefan-Boltzmann law for surface of a\nblack hole.\n At the final point of Hawking radiation, when the temperature\nand Schwarzschild radius reach $T_{final}$ and $R_{min}$\nrespectively, the emission rate is finite:\n\\begin{equation}\n\\frac{dE}{dt}\\Big|_{final}=\\frac{c^2}{60\\h\\B(1-\\h^2\\A\\B)^4}\\left(\\frac{76}{21}+5\\h^2\\A\\B\\right).\n\\end{equation}\nSimilarly as in the case of GUP \\cite{cavaglia04} at the final\npoint of Hawking radiation emission rate is finite. It was\nsupposed that when the final stage has been reached, the black\nhole evaporates completely by emitting a hard Planck-size quantum\nwith maximum temperature in a finite period of time proportional\nto the Planck's time \\cite{cavaglia04}. At the same time heat\ncapacity (\\ref{heat_capacity_general}) tends to zero when the mass\nreaches $M_{min}$. It means that black hole cannot exchange heat\nwith surroundings at the final point of Hawking radiation.\n\nNow using relation (\\ref{BH_temp}) and representing Schwarzschild\nradius as a function of a black hole's mass we can write equation\nfor the emission rate of a black hole. Since we have calculated\nthe energy of emitted radiation up to the first order over\ndeformation parameters we estimate emission rate up to first order\ntoo. So equation (\\ref{emiss_rate_E}) can be written in the form:\n\\begin{equation}\n\\frac{dM}{dt}=-\\frac{\\h c^4}{3840\\pi\nG^2M^2}\\left(1+60\\A\\frac{G^2M^2}{c^4}+\\frac{11}{336}\\B\\frac{\\h^2\nc^4}{G^2M^2}\\right)\n\\end{equation}\n\n\nHaving integrated the last equation we obtain simple expression\nfor evaporation time of a black hole:\n\\begin{equation}\\label{time_of_evapor}\n\\frac{1}{3}(M^3_{min}-M^3)-12\\frac{\\A\nG^2}{c^4}(M^5_{min}-M^5)-\\frac{11}{336}\\frac{\\h^2\\B\nc^4}{G^2}(M_{min}-M)=-\\frac{\\h c^4}{3840\\pi G^2}t\n\\end{equation}\nOne can see that modification of generalized uncertainty principle\n(\\ref{MGUP}) leads to further decrease in black hole's evaporation\ntime in comparison with GUP (\\ref{GUP}). Evaporation time as a\nfunction of mass is shown in Fig.\\ref{fig4}\n\n\\begin{figure}\n  \\centerline{\\includegraphics[scale=1,clip]{Lifetime_m.eps}}\n  \\caption{Evaporation time as a function of mass}\\label{fig4}\n\\end{figure}\n\n\n\\section{Conclusions}\nWe have considered microscopic black hole with modified GUP\n(\\ref{MGUP}) leading to the appearance of a minimal length as well\nas minimal momentum. Uncertainty relation allows one to obtain\nthermodynamical functions of a black hole in a very simple way.\nTherefore it was the main motivation to investigate the black\nhole's thermodynamics under a bit more general assumption such as\ngeneralized uncertainty principle \\cite{adler, cavaglia04}.\nModification of GUP (\\ref{MGUP}) gives rise to some new features\nin black hole's thermodynamics. Similarly as in case of GUP\n(\\ref{GUP}) black hole has finite final temperature which is\ncaused by a minimal uncertainty in position. However, in contrast\nto GUP (\\ref{GUP}) modified GUP (\\ref{MGUP}) leads also to minimal\ntemperature (\\ref{minimal_temp}). Similar result was firstly\nobtained in \\cite{Park08} but there Schwarzschild-AdS black hole's\nmetrics was used and a specific choice for one of the deformation\nparameters was made. This minimal temperature causes important\ninfluence on black hole's thermodynamics. The point of minimal\ntemperature is a point of discontinuity for heat capacity. As we\nhave already pointed out when the mass of a black hole is below\n$M_{ext}$ (\\ref{extremal_mass}) heat capacity is negative and it\ntells us about thermodynamical instability. So in this case black\nhole tends to decay. When the black hole mass is above $M_{ext}$\nheat capacity is positive and black hole is thermodynamically\nstable. We also note that although modified GUP (\\ref{MGUP}) is\nrelated to Snyder-de Sitter commutation relations it can obtained\nfor a more general kind of commutation relations. We can conclude\nthat phase transition which appears due to the presence of minimal\nmomentum and it is possible for different black holes metrics.\n\nWe also investigated thermal radiation of a Schwarzschild black\nhole. We obtained the emission rate equation which is based on\nmodified Stefan-Boltzmann law. Then we used it to calculate the\nevaporation time of a Schwarzschild black hole. In comparison with\nordinary GUP (\\ref{GUP}) modified GUP (\\ref{MGUP}) makes\nevaporation time of a black hole shorter. We also point out that\ncalculations of evaporation time are valid only for unstable phase\nof a black hole when its mass is below $M_{ext}$.\n\n\\section{Appendix}\nIn order to obtain correct relation for the black body radiation\nspectrum we should modify Liouville theorem of the classical\nmechanics caused by a deformation of commutation relations. So we\nhave to find an element of phase-space volume that is invariant\nunder time evolution. To do this let us write classical equations\nof motion supposing that position and momentum coordinates are\nobeyed the classical variant of the algebra (\\ref{SDS_alg}). In\norder to obtain the classical commutation relations the standard\nprocedure is used:\n\\begin{equation}\n\\frac{1}{i\\h}[\\hat{A},\\hat{B}]\\Rightarrow \\{A,B\\}\n\\end{equation}\nUnder this assumption deformed commutation relation\n(\\ref{SDS_alg}) takes following form:\n\\begin{eqnarray}\\label{SDS_brackets}\n\\{x_i,p_j\\}=\\delta_{ij}+\\A x_ix_j+\\B p_ip_j+2\\sqrt{\\A\\B}x_jp_i.\n\\end{eqnarray}\nAs it has already been pointed out in the classical case\ncommutation relation (\\ref{SDS_brackets}) forms a well defined\nalgebra. Other two commutation relations takes form\n\\cite{Banerjee, mignemi}:\n\\begin{eqnarray}\n{\\{x_i,x_j\\}=\\B J_{ij}};\\quad \\{p_i,p_j\\}=\\A J_{ij},\\nonumber\n\\end{eqnarray}\nwhere $J_{ij}$ are the components of angular momentum.\n\nHamilton's equations for the time derivatives of position and\nmomentum read:\n\\begin{eqnarray}\n\\dot{x}_i=\\{x_i,H\\}=\\{x_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial p_j},\n\\\\\n\\dot{p}_i=\\{p_i,H\\}=\\{p_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial p_j}.\\nonumber\n\\end{eqnarray}\nConsidering evolution of a system during an infinitesimal period\nof time we obtain:\n\\begin{eqnarray}\nx'_i=x_i+\\delta x_i;\n\\\\\np'_i=p_i+\\delta p_i.\n\\end{eqnarray}\nHere\n\\begin{eqnarray}\n\\delta x_i=\\dot{x}_i\\delta t=\\left(\\{x_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)\\delta t;\n\\\\\n\\delta p_i=\\dot{p}_i\\delta t=\\left(\\{p_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)\\delta t.\n\\end{eqnarray}\nAfter infinitesimal evolution an element of phase-space volume\nchanges:\n\\begin{equation}\\label{phase_sp_volume}\nd^Dx'd^Dp'=\\Big|\\frac{\\partial(x'_1,\\ldots,x'_D,p'_1,\\ldots,\np'_D)}{\\partial(x_1,\\ldots,x_D,p_1,\\ldots, p_D)}\\Big|d^Dxd^Dp\n\\end{equation}\nFor generality we consider here $D$-dimensional case. For the\nderivatives we have:\n\\begin{eqnarray}\n\\frac{\\partial x'_i}{\\partial x_j}=\\delta_{ij}+\\frac{\\partial\n\\dot{x}_i}{\\partial x_j}\\delta t;\\quad \\frac{\\partial\nx'_i}{\\partial p_j}=\\frac{\\partial \\dot{x}_i}{\\partial p_j}\\delta\nt;\n\\\\\n\\frac{\\partial p'_i}{\\partial x_j}=\\frac{\\partial\n\\dot{p}_i}{\\partial x_j}\\delta t; \\quad \\frac{\\partial\np'_i}{\\partial p_j}=\\delta_{ij}+\\frac{\\partial \\dot{p}_i}{\\partial\np_j}\\delta t.\n\\end{eqnarray}\nWe calculate the Jacobian in the relation (\\ref{phase_sp_volume})\nup to the first order over the infinitesimal time translation\n$\\delta t$. So under this approximation the Jacobian can be\nwritten in the form:\n\\begin{equation}\nJ=\\Big|\\frac{\\partial(x'_1,\\ldots,x'_D,p'_1,\\ldots,\np'_D)}{\\partial(x_1,\\ldots,x_D,p_1,\\ldots,\np_D)}\\Big|=1+\\left(\\frac{\\partial\\dot{x}_i}{\\partial\nx_i}+\\frac{\\partial\\dot{p}_i}{\\partial p_i}\\right)\\delta t\n\\end{equation}\n So we calculate:\n\\begin{eqnarray}\n\\frac{\\partial\\dot{x}_i}{\\partial\nx_i}+\\frac{\\partial\\dot{p}_i}{\\partial\np_i}=\\frac{\\partial}{\\partial x_i}\\left(\\{x_i,x_j\\}\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,p_j\\}\\frac{\\partial H}{\\partial\np_j}\\right)+\\frac{\\partial}{\\partial\np_i}\\left(\\{p_i,x_j\\}\\frac{\\partial H}{\\partial\nx_j}+\\{p_i,p_j\\}\\frac{\\partial H}{\\partial p_j}\\right)\\nonumber\n\\\\\n=\\frac{\\partial}{\\partial x_i}\\big[\\{x_i,x_j\\}\\big]\\frac{\\partial\nH}{\\partial x_j}+\\{x_i,x_j\\}\\frac{\\partial^2 H}{\\partial\nx_i\\partial x_j}+\\frac{\\partial}{\\partial\nx_i}\\big[\\{x_i,p_j\\}\\big]\\frac{\\partial H}{\\partial\np_j}+\\{x_i,p_j\\}\\frac{\\partial^2 H}{\\partial x_i\\partial\np_j}+\\nonumber\n\\\\\n\\frac{\\partial}{\\partial p_i}\\big[\\{p_i,x_j\\}\\big]\\frac{\\partial\nH}{\\partial x_j}+\\{p_i,x_j\\}\\frac{\\partial^2 H}{\\partial\np_i\\partial x_j}+\\frac{\\partial}{\\partial\np_i}\\big[\\{p_i,p_j\\}\\big]\\frac{\\partial H}{\\partial\np_j}+\\{p_i,p_j\\}\\frac{\\partial^2 H}{\\partial p_i\\partial\np_j}\\nonumber\n\\\\\n=2\\left(\\A x_k\\frac{\\partial H}{\\partial p_k}-\\B p_k\\frac{\\partial\nH}{\\partial x_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\n\\end{eqnarray}\nFor the element of phase space volume we have:\n\\begin{equation}\\label{phase_sp_volume}\nd^Dx'd^Dp'=d^Dxd^Dp\\left[1+2\\left(\\A x_k\\frac{\\partial H}{\\partial\np_k}-\\B p_k\\frac{\\partial H}{\\partial\nx_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\\delta\nt\\right]\n\\end{equation}\nLet us consider:\n\\begin{eqnarray}\\label{weight_multipl}\n1+\\A x'^2+\\B p'^2+2\\sqrt{\\A\\B}({\\bf x}',{\\bf p}')=1+\\A(x_i+\\delta\nx_i)^2+\\B(p_i+\\delta p_i)^2\\nonumber\n\\\\\n+2\\sqrt{\\A\\B}(x_i+\\delta x_i,p_i+\\delta p_i)\\simeq 1+\\A x^2+\\B\np^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})\\nonumber\n\\\\\n+2(\\A(x_i,\\dot{x}_i)+\\B(p_i,\\dot{p}_i)+\\sqrt{\\A\\B}[(x_i,\\dot{p}_i)+(p_i,\\dot{x}_i)])\\delta\nt=(1+\\A x^2+\\B p^2\\nonumber\n\\\\\n+2\\sqrt{\\A\\B}({\\bf x},{\\bf p}))\\left[1+2\\left(\\A x_k\\frac{\\partial\nH}{\\partial p_k}-\\B p_k\\frac{\\partial H}{\\partial\nx_k}+\\sqrt{\\A\\B}\\left[p_k\\frac{\\partial H}{\\partial\np_k}-x_k\\frac{\\partial H}{\\partial x_k}\\right]\\right)\\delta\nt\\right]\n\\end{eqnarray}\nMaking use of relations (\\ref{phase_sp_volume}) and\n(\\ref{weight_multipl}) we conclude that the following weighted\nphase space volume is invariant under infinitesimal time\ntranslations:\n\\begin{equation}\\label{weighted_volume}\n\\frac{d^Dxd^Dp}{1+\\A x^2+\\B p^2+2\\sqrt{\\A\\B}({\\bf x},{\\bf p})}\n\\end{equation}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Supplementary Information}\n\n\\section{Device Details}\n\nWe fabricated our device from an ``electronic grade'' $\\langle100\\rangle$-oriented diamond purchased from Element Six. The diamond is specified to contain fewer than $5$~ppb nitrogen impurities. Nitrogen-vacancy (NV) centers were introduced via irradiation with $2$~MeV electrons at a fluence of $\\sim 1.2\\times 10^{14}$~cm$^{-2}$ followed by annealing at $850^{\\circ}$C for $2$~hours. The NV centers studied in this work are located at a depth of $\\sim47$~$\\mu$m. \n\nThe high-overtone bulk acoustic resonator (HBAR) used in these measurements consists of a $3$~$\\mu$m thick $\\langle002\\rangle$-oriented ZnO film sandwiched between a Ti\/Pt ($25$~nm\/$200$~nm) ground plane and an Al ($250$~nm) top contact. The piezo-electric ZnO film transduces stress waves into the diamond. The diamond then acts as an acoustic Fabry-P\\'{e}rot cavity to create stress standing wave resonances. Fig.~\\ref{fig:s11} shows a network analyzer measurement of the HBAR admittance ($Y_{11}$) plotted as a function of frequency. From this frequency comb, we selected the $\\omega_{\\text{mech}}\/2\\pi=586$~MHz resonance mode that has a $Q$ of $2700$ as calculated by the $Q$-circle method~\\cite{qCircle} and an on-resonance impedance of $18$~$\\Omega$. This mechanical resonance suppresses driving field amplitude noise that is faster than $\\omega_c=\\frac{\\omega_{\\text{mech}}}{2Q}=110$~kHz. A microwave antenna fabricated on the diamond face opposite the ZnO transducer provides gigahertz frequency magnetic fields for conventional magnetic spin control. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{figS11.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:s11] {Network analyzer measurement of the power admitted to the HBAR. The inset highlights the $\\omega_{\\text{mech}}\/2\\pi=586$~MHz mode used in the measurements.}\n\\label{fig:s11}\n\\end{figure}\n\n\\section{Mechanical Rabi Driving}\n\nThe mechanically driven Rabi oscillations depicted in Fig.~1a of the main text were measured using the pulse sequence shown in Fig.~\\ref{fig:rabiSeq}. As described in detail in Ref.~\\cite{MacQuarrie2015}, the relatively high $Q$ of our mechanical resonance makes it difficult to perform a traditional pulsed Rabi measurement. Instead, a pair of magnetic $\\pi$-pulses resonant with the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition and separated by a fixed time $\\tau_{\\text{mag}}$ is swept through a fixed-length mechanical pulse. The mechanical pulse drives the $\\Ket{+1}\\leftrightarrow\\Ket{-1}$ spin transition, and the duration of this interaction is set by the area of the mechanical pulse enclosed between the two $\\pi$-pulses. By knowing the shape of the mechanical pulse, we convert this enclosed area to effective square-pulse units or an ``effective mechanical pulse length.'' Because the mechanical resonator is pulsed in this experiment, we are able to achieve a larger driving field than in the continuous dynamical decoupling (CDD) Ramsey measurements where the mechanical resonator operates in cw mode. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{rabiSeq.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:rabiSeq]{Pulse sequence used to measure mechanically driven Rabi oscillations.}\n\\label{fig:rabiSeq}\n\\end{figure}\n\n\\section{Mechanically Dressed Hamiltonian}\n\\label{sec:Ham}\n\nAs mentioned in the main text, we work within the $m_I=+1$ sublevel of the $^{14}$N hyperfine manifold. We consider both a static magnetic field $b$ that is aligned along the NV center symmetry axis and subject to fluctuations $\\delta b$ and a mechanical driving field $\\Omega$ that is subject to amplitude fluctuations $\\delta\\Omega$. In the $\\{+1,0,-1\\}\\otimes\\{(m_I=)+\\frac{1}{2},-\\frac{1}{2}\\}$ Zeeman basis, a nearby $^{13}$C nuclear spin weakly couples to an NV center electronic spin through the hyperfine perturbation $H_{C}=A_{\\|}S_{z}I_{z}$ where $S_z$ and $I_z$ are the spin-$1$ and spin-$\\frac{1}{2}$ Pauli matrices, respectively, and $A_{\\|}$ is the coupling strength~\\cite{slichter}. An NV center electronic spin then obeys the Hamiltonian\n\\begin{widetext}\n$H_{LF}=\\begin{pmatrix}\n\\gamma b_{\\Sigma}+\\frac{1}{2}A_{\\|} & 0 & 0 & 0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 \\\\\n0 & \\gamma b_{\\Sigma}-\\frac{1}{2}A_{\\|} & 0 & 0 & 0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) \\\\\n0 & 0 & -D & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -D & 0 & 0 \\\\\n\\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}A_{\\|} & 0 \\\\\n0 & \\Omega_{\\Sigma}\\cos(\\omega_{\\text{mech}} t) & 0 & 0 & 0 & -\\gamma b_{\\Sigma}+\\frac{1}{2}A_{\\|} \\\\\n\\end{pmatrix}$\n\\end{widetext}\nwhere $b_{\\Sigma}=b+\\delta b$, $\\Omega_{\\Sigma}=\\Omega+\\delta\\Omega$, other parameters are as defined in the main text, and we have not included a magnetic driving field. Applying the rotating wave approximation and transforming into the reference frame rotating at $\\frac{1}{2}\\omega_{\\text{mech}}=\\frac{1}{2}(2\\gamma b+\\Delta)$ gives the Hamiltonian in the rotating frame\n\\begin{widetext}\n$H_{RF}=\n\\begin{pmatrix}\n\\gamma b_{\\Sigma}+\\frac{1}{2}(\\Delta+A_{\\|}) & 0 & 0 & 0 & \\frac{1}{2}\\Omega_{\\Sigma} & 0 \\\\\n0 & \\gamma b_{\\Sigma}+\\frac{1}{2}(\\Delta+A_{\\|}) & 0 & 0 & 0 & \\frac{1}{2}\\Omega_{\\Sigma} \\\\\n0 & 0 & -D & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -D & 0 & 0 \\\\\n\\frac{1}{2}\\Omega_{\\Sigma} & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}(\\Delta+A_{\\|}) & 0 \\\\\n0 & \\frac{1}{2}\\Omega_{\\Sigma} & 0 & 0 & 0 & -\\gamma b_{\\Sigma}-\\frac{1}{2}(\\Delta-A_{\\|}) \\\\\n\\end{pmatrix}$.\n\\end{widetext}\nDiagonalizing $H_{RF}$ gives the mechanically dressed Hamiltonian whose energies are quoted in the main text: \n\\begin{widetext}\n$H_{D}= \\begin{pmatrix}\n-D & 0 & 0 & 0 & 0 & 0 \\\\\n0 & -D & 0 & 0 & 0 & 0 \\\\\n0 & 0 & -\\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{-}^2} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{-}^2} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -\\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{+}^2} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\frac{1}{2}\\sqrt{\\Omega_{\\Sigma}^2+\\xi_{+}^2} \\\\\n\\end{pmatrix}$\n\\end{widetext}\nwhere $\\xi_{\\pm}=\\Delta+2\\gamma\\delta b\\pm A_{\\|}$. In the limit $\\Omega_{\\Sigma}=0$, $H_D$ reduces to the undressed Zeeman Hamiltonian in the rotating frame. \n\n\\section{Dressed State Spectroscopy}\n\nFig.~\\ref{fig:seqs}a shows several concatenated instances of the pulse sequence used for our dressed state spectroscopy measurements. In a single instance, the NV center is optically initialized into the $\\Ket{0}$ spin state at which point a reference fluorescence measurement is made of the full-scale NV center photoluminescence. A magnetic $\\pi$-pulse of strength $\\Omega_{\\text{mag}}\/2\\pi\\sim80$~kHz is then applied to drive a conditional spin rotation. Finally, fluorescence readout provides a quantitative measure of the spin population remaining in $\\Ket{0}$. We interleave $n$ instances of this pulse sequence executed in the dressed basis with $n$ instances of this pulse sequence executed in the undressed basis. In a typical experiment $n\\sim10$, giving a total duty cycle time of $\\sim280$~$\\mu$s and mechanical pulse length of $\\sim140$~$\\mu$s. We differentiate between the dressed and undressed signal by routing the counts from our avalanche photodiode to separate counters on our DAQ. This sequence is then repeated as a function of the magnetic detuning $\\Delta_{\\text{mag}}$ from the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ state splitting to produce the data in Fig.~1d of the main text. \n\nThe dressed signal from this measurement is fit to the sum of two Lorentzians \n\\begin{equation}\n\\begin{split}\nP_{D}&=c_D-\\frac{a_{D,1}}{\\left(\\frac{2}{\\Gamma_{D}}\\right)^2\\left(\\omega-\\frac{1}{2}\\sqrt{\\Delta^2+\\Omega^2}-\\frac{1}{2}\\Delta-\\omega_{0,-1}\\right)^2+1} \\\\\n& -\\frac{a_{D,2}}{\\left(\\frac{2}{\\Gamma_{D}}\\right)^2\\left(\\omega+\\frac{1}{2}\\sqrt{\\Delta^2+\\Omega^2}-\\frac{1}{2}\\Delta-\\omega_{0,-1}\\right)^2+1}\n\\end{split}\n\\label{eq:specFit}\n\\end{equation}\nwhere $P_{D}$ is the measured photoluminescence, $c_D$ is a constant background, $\\omega_{0,-1}$ is the undressed $\\Ket{0}\\leftrightarrow\\Ket{-1}$ spin state splitting, $\\Delta$ is the mechanical detuning, $\\Omega$ is the mechanical driving field, $a_{D,i}$ accounts for the depth of the spectral peaks, and $\\Gamma_D$ is the full width at half maximum of the dressed spectral peaks. The undressed signal is simultaneously fit to the Lorentzian \n\\begin{equation}\nP_{UD}=c_{UD}-\\frac{a_{UD}}{\\left(\\frac{2}{\\Gamma_{UD}}\\right)^2(\\omega-\\omega_{0,-1})^2+1}.\n\\end{equation}\nWe then subtract $\\omega_{0,-1}$ from the $x$-axis to plot photoluminescence as a function of $\\Delta_{mag}$ as shown in Fig.~1d of the main text.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{sequences.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:seqs]{(a) Pulse sequence used for dressed state spectroscopy measurements. (b) Pulse sequence used for $\\{0,p\\}$ qubit CDD Ramsey measurements. (c) Pulse sequence used for $\\{m,p\\}$ qubit CDD Ramsey measurements. }\n\\label{fig:seqs}\n\\end{figure}\n\n\\section{Expression for the Mechanical Detuning}\n\nIn our spectroscopy measurements, we use the relation $\\frac{1}{2}(\\omega_{0,m}+\\omega_{0,p})-\\omega_{0,-1}=\\frac{1}{2}\\Delta$ as a means of zeroing the mechanical detuning. To derive this expression, we begin in the $\\{+1,0,-1\\}$ basis with the Hamiltonian for an NV center subject to both a mechanical driving field and a magnetic driving field resonant with the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition. In the doubly rotating reference frame, this can be written\n\\begin{equation}\nH_{RF}=\\begin{pmatrix}\n\\frac{1}{2}\\Delta & 0 & \\frac{1}{2}\\Omega \\\\\n0 & -D-\\Delta_{\\text{mag}} & \\frac{1}{2}\\Omega_{\\text{mag}} \\\\\n\\frac{1}{2}\\Omega & \\frac{1}{2}\\Omega_{\\text{mag}} & -\\frac{1}{2}\\Delta\\\\\n\\end{pmatrix}\n\\label{eq:hamDelta}\n\\end{equation}\nwhere $\\Delta_{\\text{mag}}=-\\frac{1}{2}\\Delta$ for resonant magnetic driving, and $\\Omega_{\\text{mag}}$ is far enough detuned from the $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transition that we can ignore the $\\Bra{+1}H_{RF}\\Ket{0}$ matrix element. \n\nIn the undressed case ($\\Omega=0$, $\\Delta=0$), the energy of the $\\Ket{0}\\leftrightarrow\\Ket{-1}$ splitting in this reference frame is $\\omega_{0,-1}=D$ where we define $\\hbar=1$. With a non-zero mechanical driving field, calculating the eigenvalues of Eq.~\\ref{eq:hamDelta} to first order in $\\frac{\\Omega_{\\text{mag}}}{\\Omega}$ gives energies $\\omega_{0,p}=D+\\frac{1}{2}(\\Delta+\\sqrt{\\Delta^2+\\Omega^2})$ and $\\omega_{0,m}=D+\\frac{1}{2}(\\Delta-\\sqrt{\\Delta^2+\\Omega^2})$. From this we arrive at the desired expression $\\frac{1}{2}(\\omega_{0,m}+\\omega_{0,p})-\\omega_{0,-1}=\\frac{1}{2}\\Delta$. The same expression is obtained when the $^{13}$C coupling is included. \n\n\\section{Ramsey Measurements}\n\n\\subsection{Dressed Ramsey Pulse Sequences}\n\nFig.~\\ref{fig:seqs}b,c show the pulse sequences used for our CDD Ramsey measurements of the $\\{0,p\\}$ and $\\{m,p\\}$ qubits, respectively. Similar to the spectroscopy experiments, the pulse sequences consist of $2n$ sub-instances where each sub-instance is a single measurement. Here, however, $n\\sim 2$, which leads to mechanical pulse lengths and duty cycle lengths similar to those in the spectroscopy experiments. \n\nA single instance of the $\\{0,p\\}$ qubit CDD Ramsey sequence starts with optical initialization into $\\Ket{0}$ and a reference fluorescence measurement. We then apply a magnetic $\\pi\/2$-pulse of strength $\\Omega_{\\text{mag}}\/2\\pi=696\\pm 7$~kHz to populate the $\\{0,p\\}$ subspace. After a free evolution time $\\tau$, we apply a second magnetic $\\pi\/2$-pulse of the same strength to return the spin population to $\\Ket{0}$ where the signal is read out optically. To help visualize the decay, we advance the phase of the second $\\pi\/2$-pulse by $\\omega_{\\text{rot}}\\tau$. Undressed Ramsey measurements are interleaved with the dressed measurements to reduce the power load on our device and provide a simultaneous measurement of the undressed dephasing time $T_{2,\\{0,-1\\}}^*$. This sequence is then repeated as a function of $\\tau$.\n\nThe pulse sequence used for $\\{m,p\\}$ CDD Ramsey measurements is very similar to the $\\{0,p\\}$ Ramsey sequence. For the $\\{m,p\\}$ qubit, however, the $\\pi\/2$-pulses that address the $\\{0,p\\}$ subspace are replaced by double quantum magnetic $\\pi$-pulses of strength $\\Omega_{\\text{mag}}\/2\\pi=1513\\pm8$~kHz that address the $\\{m,p\\}$ subspace~\\cite{mamin2014}. Additionally, the phase of the magnetic pulse that ends the free evolution time is not advanced at $\\omega_{\\text{rot}}\\tau$ for the $\\{m,p\\}$ qubit measurement. In the interest of reducing the power load on our device, we interleave the dressed $\\{m,p\\}$ Ramsey measurements with undressed measurements that execute the same sequence of magnetic pulses. Because this pulse sequence amounts to a $2\\pi$ rotation of the undressed $\\{0,-1\\}$ qubit, the data obtained during these measurements quantify the NV center spin contrast. For each measurement, the average of this undressed trace $\\langle P_{0,ud}\\rangle$ fixes the amplitude in the fitting functions described below. \n\nDuring the $\\{m,p\\}$ qubit measurements, we periodically measure $\\Delta$ spectroscopically and feedback on $b$ to maintain a relatively constant $\\Delta$. Interpolating linear drift between these measurements, we post-select to include only those data sets for which $\\sigma_{\\Delta}\/2\\pi<60$~kHz and $|\\langle\\Delta\\rangle|\/2\\pi<35$~kHz. \n\n\\subsection{Undressed Ramsey Fitting Function}\n\nWe fit the undressed Ramsey data to the expression \n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{0,-1}]&=c-\\frac{a}{4} e^{-\\frac{\\tau^2}{T_2^{*2}}}\\{\\cos[(\\omega_{\\text{rot}}+\\Delta_{\\text{mag}}+\\frac{1}{2}A_{\\|})\\tau] \\\\\n&+\\cos[(\\omega_{\\text{rot}}+\\Delta_{\\text{mag}}-\\frac{1}{2}A_{\\|})\\tau]\\}\n\\end{split}\n\\end{equation}\nwhere $\\tau$ is the free evolution time, $\\rho_{0,-1}$ is the $\\{0,-1\\}$ coherence, $c$ is a constant background, $a$ is an overall amplitude that accounts for deviations from perfect spin contrast, $T_2^*$ is the inhomogeneous dephasing time, $\\omega_{\\text{rot}}$ is the rate at which we advance the phase of the second $\\pi\/2$-pulse, $\\Delta_{\\text{mag}}$ is the magnetic detuning, and $A_{\\|}$ quantifies coupling to a nearby $^{13}$C nuclear spin. Of these values, $c$, $a$, $T_{2}^*$, $\\Delta_{\\text{mag}}$, and $A_{\\|}$ are free parameters in our fit. We have assumed the $^{13}$C spin is unpolarized. We use the values of $a$ and $c$ returned from the fits to scale the $y$-axes of our plots. \n\n\\subsection{Dressed Ramsey Fitting Function: The $\\{0,p\\}$ Qubit}\n\nIn our CDD Ramsey measurements of the $\\{0,p\\}$ qubit, we tune the magnetic driving field into resonance with the $\\Ket{0}\\leftrightarrow\\Ket{p}$ transition. For the fits, we zero the magnetic detuning midway between the $^{13}$C sublevels $\\Ket{p,(m_{I}=)+\\frac{1}{2}}$ and $\\Ket{p,-\\frac{1}{2}}$. Assuming $\\Delta=0$, our $\\{0,p\\}$ CDD Ramsey signal is then described by the expression\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{0,p}]&=c+\\frac{1}{4}e^{-\\frac{\\tau^2}{T_2^{*2}}}\\lbrace a_p \\cos\\left[\\left(\\Delta_{\\text{mag}}+\\omega_{\\text{rot}}\\right)\\tau+\\phi\\right] \\\\\n&+a_m \\cos\\left[\\left(\\Delta_{\\text{mag}}+\\omega_{\\text{rot}}+\\sqrt{\\Omega^2+A_{\\|}^2}\\right)\\tau+\\phi\\right]\\rbrace\n\\end{split}\n\\end{equation}\nwhere $a_m$ is the spin contrast for the $\\{0,m\\}$ qubit, $a_p$ is the spin contrast for the $\\{0,p\\}$ qubit, $\\phi$ is a constant phase offset, and the other parameters are as defined above. We fix the values of $A_{\\|}$ and $\\omega_{\\text{rot}}$, and we vary $c$, $a_i$, $\\phi$, $\\Omega$, and $\\Delta_{\\text{mag}}$ as free parameters in our fitting procedure. Once again, we use the values of $a_m$, $a_p$, and $c$ returned from the fit to scale the $y$-axis in Fig.~2a of the main text. \n\nIt is important to note that because the dressed qubit Larmor frequency does not scale linearly with magnetic field fluctuations, the Ramsey signal does not follow a strictly Gaussian decay. Nevertheless, we fit our data with a Gaussian envelope to aid comparison with the undressed dephasing time. This is a reasonable approximation over the range of mechanical driving fields accessed in this work. \n\n\\subsection{Dressed Ramsey Fitting Function: The $\\{m,p\\}$ Qubit}\n\nFor the $\\{m,p\\}$ qubit dressed under the condition $\\Delta=0$, our CDD Ramsey signal can be described by the expression\n\\begin{equation}\n\\text{Re}[\\rho_{m,p}]=c+\\frac{\\langle P_{0,ud}\\rangle}{2}e^{-\\frac{\\tau^2}{T_2^{*2}}} \\cos\\left[ \\tau\\sqrt{A_{\\|}^2+\\Omega^2}+\\phi\\right]\n\\end{equation}\nwhere $\\langle P_{0,ud}\\rangle$ measures the spin contrast and the other parameters are as described above. To maximally constrain our fitting procedure, we measure $\\langle P_{0,ud}\\rangle$ by interleaving undressed iterations of the CDD Ramsey protocol into the measurement. We allow $c$, $T_{2}^*$, $\\Omega$, and $\\phi$ to vary as free parameters in our fitting procedure. The results of these fits for the measurements shown in Fig.~3b,c of the main text are displayed in Fig.~\\ref{fig:tablePlot} and Fig.~\\ref{fig:tablePlotN}, respectively. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{tablePlotNotBold.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:tablePlot]{Data and fits for CDD Ramsey measurements of the $\\{m,p\\}$ qubit when $\\Omega$ was actively stabilized. }\n\\label{fig:tablePlot}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{plotTableNotBoldN.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:tablePlotN]{Data and fits for CDD Ramsey measurements of the $\\{m,p\\}$ qubit when $\\Omega$ was given a Gaussian noise profile. }\n\\label{fig:tablePlotN}\n\\end{figure}\n\nOur CDD Ramsey measurement of a maximally protected $\\Ket{\\downarrow}$ $^{13}$C sublevel (Fig.~3d in the main text) was fit to the function \n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{m,p}]&=\\frac{\\langle P_{0,ud}\\rangle}{4}\\lbrace\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\\cos\\left[\\Omega\\tau+\\phi\\right] \\\\\n&+ e^{-\\frac{t^2}{T_{2,\\uparrow}^*2}}\\cos\\left[\\tau\\sqrt{\\Omega^2+4A_{\\|}^2}+\\phi\\right]\\rbrace +c\n\\end{split}\n\\label{eq:highCohere}\n\\end{equation}\nwhere $\\langle P_{0,ud}\\rangle$ fixes the spin contrast, and $\\Omega$, $\\phi$, $T_{2,\\uparrow}^*$, and $c$ were varied as free parameters. A derivation of the non-Gaussian envelope in Eq.~\\ref{eq:highCohere} is given below on page~\\pageref{sndOrder}. \n\nFor all of our $\\{m,p\\}$ qubit Ramsey plots, we use $\\langle P_{0,ud} \\rangle$ and the value of $c$ returned from the fit to scale the $y$-axis. \n\n\\section{Thermal Stability}\n\nAs mentioned above, we intersperse spectral measurements within CDD Ramsey measurements of the $\\{m,p\\}$ qubit. This allows us to feedback on $b$ and maintain a relatively constant $\\Delta$, but these measurements also quantify the thermal drift over the course of the measurement. A histogram of $\\Delta$ extracted from fitting these spectra to Eq.~\\ref{eq:specFit} quantifies drift in the magnetic bias field $\\sigma_{\\Delta}=2\\gamma\\sigma_{\\text{bias}}$. A histogram of $\\omega_{0,-1}$, however, provides information about both the magnetic bias field drift and the thermal drift according to \n\\begin{equation}\n\\sigma_{0,-1}=\\sqrt{\\left(\\gamma\\sigma_{\\text{bias}}\\right)^2+\\left(\\frac{dD}{dT}\\sigma_T\\right)^2}\n\\end{equation}\nwhere $\\sigma_T$ is the standard deviation of normally distributed thermal drift and $\\frac{dD}{dT}=-74\\times 2\\pi$~kHz\/$^{\\circ}$C is the temperature dependence of $D$~\\cite{awschalomThermo,budkerThermo}. The average of $\\sigma_T$ for the power-leveled data that satisfy our post-selection criteria is $0.25\\pm 0.03^{\\circ}$C. Thermal drift on a similar scale can be expected for the $\\{0,p\\}$ qubit measurements. As shown below on page~\\pageref{tempCoh}, fluctuations of this scale would limit the $\\{0,p\\}$ qubit coherence time to $T_{2,\\{0,p\\}}^*=\\frac{\\sqrt{2}}{\\sigma_{T} dD\/dT}=12\\pm 1$~$\\mu$s. \n\n\\section{Modeling Decoherence}\n\n\\subsection{First Order Fluctuations in $\\omega_{i,j}$}\n\nGenerically, first order deviations in the Larmor frequency $\\omega_{i,j}$ take the form $\\delta\\omega_{i,j}=\\alpha\\delta x$ where $\\alpha$ is a constant. If the fluctuation $\\delta x$ follows a Gaussian distribution with standard deviation $\\sigma_x$, an expression for the associated dephasing rate can be found by calculating the weighted average of a distribution of detuned, un-damped Ramsey signals:\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{i,j}] & =\\frac{1}{\\sqrt{2\\pi}\\sigma_x}\\int e^{-\\frac{\\delta x^2}{2\\sigma_x^2}}\\cos\\left[(\\omega_{i,j}+\\alpha\\delta x)\\tau\\right]\\text{d}\\delta x \\\\\n& = e^{-\\frac{1}{2}(\\alpha\\sigma_x \\tau)^2}\\cos\\left(\\omega_{i,j}\\tau\\right).\n\\end{split}\n\\label{eq:sigReplace}\n\\end{equation}\nComparing Eq.~\\ref{eq:sigReplace} with an ideal Ramsey signal given by $\\text{Re}[\\rho_{i,j}]=e^{-\\frac{\\tau^2}{T_{2,\\{i,j\\}}^{*2}}}\\cos\\left(\\omega_{i,j}\\tau\\right)$, we see that $T_{2,\\{i,j\\}}^*=\\frac{\\sqrt{2}}{\\alpha \\sigma_x}$ and therefore $\\Gamma_x=\\frac{2\\pi}{T_{2,\\{i,j\\}}^*}=\\sqrt{2}\\pi\\alpha \\sigma_x$. \n\nFor magnetic field fluctuations experienced by the $\\{0,-1\\}$ qubit, $\\alpha\\delta x\\rightarrow \\gamma\\delta b$. We then find $\\gamma\\sigma_b\/2\\pi=(\\sqrt{2}\\pi T_{2,\\{0,-1\\}}^*)^{-1}$ as quoted in the main text. For thermal fluctuations experienced by the $\\{0,p\\}$ qubit, $\\alpha\\delta x\\rightarrow \\frac{\\text{d}D}{\\text{d}T}\\delta T$, and we arrive at $T_{2,\\{0,p\\}}^*=\\frac{\\sqrt{2}}{\\sigma_{T} dD\/dT}$. For the $\\{m,p\\}$ qubit, expanding $\\omega_{m,p}$ to first order in $\\delta b$ gives \n\\begin{equation}\n\\delta\\omega_{m,p;b}=\\frac{2|A_{\\|}|\\gamma\\delta b}{\\sqrt{A_{\\|}^2+\\Omega^2}},\n\\end{equation}\nfrom which we find $\\Gamma_b=2\\sqrt{2}\\kappa|A_{\\|}|\/T_{2,\\{0,-1\\}}^*$ where $\\frac{1}{\\kappa}=\\frac{1}{\\sqrt{2}\\pi}\\sqrt{A_{\\|}^2+\\Omega^2}$. Similarly, expanding $\\omega_{m,p}$ to first order in $\\delta \\Omega$ gives\n\\begin{equation}\n\\delta\\omega_{m,p;\\Omega}=\\frac{\\Omega \\delta\\Omega}{\\sqrt{A_{\\|}^2+\\Omega^2}},\n\\end{equation}\nfrom which we find $\\Gamma_\\Omega=\\kappa\\Omega\\sigma_{\\Omega}$. \n\\label{tempCoh}\n\n\\subsection{Second Order Magnetic Field Fluctuations}\n\\label{sndOrder}\n\nThe decay envelope of a Ramsey measurement is given by the expression $f(\\tau,\\Omega,\\sigma_b,A_{\\|})=|\\langle e^{i\\delta\\phi}\\rangle|$ where $\\delta\\phi$ is the random phase accumulated in a given duty cycle of the measurement~\\cite{ithier2005}. For the $\\{m,p\\}$ qubit in the case when $\\Delta=0$, the Larmor frequency is given by $\\omega_{m,p}=\\sqrt{(\\Omega+\\delta\\Omega)^2+(A_{\\|}+2\\gamma b)^2}$. To second order in $\\delta b$, fluctuations in $\\omega_{m,p}$ from magnetic field fluctuations are then given by \n\\begin{equation}\n\\begin{split}\n\\delta\\omega_{m,p}&=\\frac{\\partial\\omega_{m,p}}{\\partial b}\\Bigr|_{\\delta b=0}\\delta b+\\frac{\\partial^2\\omega_{m,p}}{\\partial b^2}\\Bigr|_{\\delta b=0}\\frac{\\delta b^2}{2}+O(\\delta b^3) \\\\\n&=\\frac{2\\gamma\\delta b(A_{\\|}^3+A_{\\|}\\Omega^2+\\gamma\\delta b\\Omega^2)}{(A_{\\|}^2+\\Omega^2)^{3\/2}}\n\\end{split}.\n\\end{equation}\nThe random phase accumulated is $\\delta\\phi=\\delta\\omega_{m,p}\\tau$. By averaging this phase over a Gaussian distribution of magnetic field fluctuations, we find \n\\begin{equation}\n\\begin{split}\nf(\\tau,\\Omega,\\sigma_b,A_{\\|})&=\\left\\vert\\frac{1}{\\sqrt{2\\pi}\\sigma_b}\\int_{-\\infty}^{\\infty}e^{i\\delta\\omega_{m,p}\\tau}e^{-\\frac{\\delta b^2}{2\\sigma_b^2}}\\text{d}\\delta b\\right\\vert \\\\\n&=\\sqrt{\\beta}e^{-\\frac{2(\\gamma\\sigma_b A_{\\|}\\beta\\tau)^2}{A_{\\|}^2+\\Omega^2}}\n\\end{split}\n\\label{eq:env}\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta(\\tau,\\Omega,\\sigma_b,A_{\\|})\\equiv\\sqrt{\\frac{(A_{\\|}^2+\\Omega^2)^3}{(A_{\\|}^2+\\Omega^2)^3+(2\\gamma\\sigma_b\\Omega)^4\\tau^2}}.\n\\end{equation}\nTo produce the model curves in Fig.~3b,c of the main text, we numerically solve this expression for the value of $\\tau$ such that $f(\\tau,\\Omega,\\sigma_b,A_{\\|})=\\frac{1}{e}$. \n\nWhen $\\Delta=-|A_{\\|}|$, the two $^{13}$C sublevels follow different decay envelopes that can be computed by setting $A_{\\|}\\rightarrow 0$ and $A_{\\|}\\rightarrow 2 A_{\\|}$ in Eq.~\\ref{eq:env}. In the former case, $f(\\tau,\\Omega,\\sigma_b,0)$ reduces to \n\\begin{equation}\n\\begin{split}\nh(\\tau,\\Omega,\\sigma_b)&=\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\n\\end{split}\n\\end{equation}\nas seen in the main text. For the case of $A_{\\|}\\rightarrow 2 A_{\\|}$, we approximate the decay as Gaussian. The fitting function for Fig.~3d of the main text then becomes\n\\begin{equation}\n\\begin{split}\n\\text{Re}[\\rho_{m,p}]&=c+\\frac{\\langle P_{0,ud}\\rangle}{4}\\lbrace\\sqrt{\\frac{\\Omega}{\\sqrt{\\Omega^2+(2\\gamma\\sigma_b)^4\\tau^2}}}\\cos\\left[\\Omega\\tau+\\phi\\right] \\\\\n&+e^{-\\frac{t^2}{T_{2,\\uparrow}^{*2}}}\\cos\\left[\\tau\\sqrt{\\Omega^2+4 A_{\\|}^2}+\\phi\\right]\\rbrace\n\\end{split}\n\\end{equation}\nwhere only $\\Omega$, $\\phi$, $c$, and $T_{2,\\uparrow}^*$ were allowed to vary as free parameters.\n\nFor simplicity, this derivation of $f(\\tau,\\Omega,\\sigma_b,A_{\\|})$ does not include driving field noise. Including amplitude noise in the mechanical driving field on the scale of our power-leveled measurements produces no noticeable change in the results of the model over the range of mechanical driving fields addressed here. \n\n\\section{Measuring the Voltage Reflected from the HBAR}\n\nWe monitor the mechanical driving field amplitude by tracking the RF power reflected from the mechanical resonator. An RF circulator redirects the reflected power to an RF diode that converts the ac signal to the dc voltage that we measure. As shown in Fig.~\\ref{fig:diodecal}, this measured voltage scales linearly with the mechanical driving field. However, due to the diode's nonzero threshold voltage, that linear dependence has a nonzero intercept. \n\nWe introduce driving field noise to our experiment by periodically shifting the applied power such that the spread of voltages measured by the RF diode over the course of a measurement is normally distributed with a standard deviation of $\\eta \\langle V_R \\rangle$ where $V_R$ is the reflected voltage and $\\eta$ is a constant. Because Fig.~\\ref{fig:diodecal} has a nonzero intercept, such a distribution of voltages will correspond to a Gaussian distribution of driving fields with a standard deviation of $\\sigma_{\\Omega}=(\\langle\\Omega\\rangle+\\alpha)\\eta$ where $\\alpha\/2\\pi=-133\\pm7$~kHz is the ratio of the intercept to the slope for the line of best fit in Fig.~\\ref{fig:diodecal}. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{driveToVolt.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:diodecal] {Voltage reflected from the mechanical resonator plotted as a function of the mechanical driving field. }\n\\label{fig:diodecal}\n\\end{figure}\n\n\\section{Coherence of the $\\{+1,-1\\}$ Qubit}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{udRamseyFig.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:t2UD]{(a) Ramsey measurement of the undressed $\\{0,-1\\}$ qubit for the NV center used in the $\\{m,p\\}$ qubit measurements. (b) Fourier spectrum of (a). }\n\\label{fig:t2UD}\n\\end{figure}\n\nWe compare the coherence of the $\\{m,p\\}$ qubit to that of the undressed $\\{+1,-1\\}$ qubit because in each of these qubits both component states are sensitive to magnetic field fluctuations. Directly measuring the dephasing time of the $\\{+1,-1\\}$ qubit at finite field with high precision is a non-trivial task because the measurement becomes sensitive to double quantum pulse infidelities. Instead, we measure $T_2^*$ of the undressed $\\{0,-1\\}$ qubit (Fig.~\\ref{fig:t2UD}) and rely on the fact that for Gaussian magnetic field fluctuations $T_{2,\\{+1,-1\\}}^*=\\frac{1}{2}T_{2,\\{0,-1\\}}^*$. This gives $T_{2,\\{+1,-1\\}}^*=2.7\\pm0.1$~$\\mu$s as quoted in the main text. This same undressed Ramsey measurement also quantifies $|A_{\\|}|\/2\\pi=150\\pm4$~kHz and $\\sigma_b=2.4\\pm0.1$~mG for this NV center.\n\n\\section{Dressed Spectra Through the $\\Ket{+1}\\leftrightarrow\\Ket{0}$ Transition}\n\n\\begin{figure}[hb]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=\\linewidth]{dressedSpectrum.pdf} \\\\\n\\end{tabular} \n\\end{center}\n\\caption[fig:0pSpect]{(a) Dressed state spectrum for a magnetic pulse swept through the undressed $\\Ket{0}\\leftrightarrow\\Ket{-1}$ transition. (b) Dressed state spectrum for a magnetic pulse swept through the undressed $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transition. (c) Pulse sequence used in these measurements.}\n\\label{fig:0pSpect}\n\\end{figure}\n\nFig.~\\ref{fig:0pSpect} shows spectral measurements of the dressed state splitting as measured by sweeping the detuning of a $\\Omega_{\\text{mag}}\/2\\pi=345\\pm4$~kHz magnetic pulse through the resonance of the undressed (a) $\\Ket{0}\\leftrightarrow\\Ket{-1}$ and (b) $\\Ket{+1}\\leftrightarrow\\Ket{0}$ transitions. All three $^{14}$N hyperfine sublevels are visible in the spectra. Because $\\omega_{\\text{mech}}$ is tuned into resonance with the $\\Ket{(m_{s}=)+1,(m_{I}=)+1}\\leftrightarrow\\Ket{-1,+1}$ transition within the $^{14}$N hyperfine manifold, only the $m_{I}=+1$ peak splits into the dressed states $\\Ket{m,+1}$ and $\\Ket{p,+1}$. In these measurements, the HBAR was powered in $3$~$\\mu$s pulses as shown in Fig.~\\ref{fig:0pSpect}c. This reduced the power load on the device and allowed us to reach higher driving fields than we were able to reach in the CDD Ramsey experiments where the mechanical resonator operates in cw mode. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWireless Sensor Networks (WSNs) have been recognized as a new\ngeneration of ubiquitous computing systems to support a broad range\nof applications, including monitoring, health care and tracking\nenvironmental pollution levels. Minimizing the total energy\nconsumption in both circuit components and RF signal transmission is\na crucial challenge in designing a WSN. Central to this study is to\nfind energy-efficient modulation and coding schemes in the physical\nlayer of a WSN to prolong the sensor lifetime\n\\cite{Cui_GoldsmithITWC0905, HowardEURASIP2006}. For this purpose,\nenergy-efficient modulation\/coding schemes should be simple enough\nto be implemented by state-of-the-art low-power technology, but\nstill robust enough to provide the desired service. Furthermore,\nsince sensor devices frequently switch from sleep mode to active\nmode, modulation and coding circuits should have fast start-up times\n\\cite{Wang_ISLPED2001} along with the capability of transmitting\npackets during a pre-assigned time slot before new sensed packets\narrive. In addition, a WSN needs a powerful channel coding scheme\n(when the distance between nodes exceeds a certain threshold level)\nto protect transmitted data against the unpredictable and harsh\nnature of channels. We refer to these low-complexity and low-energy\nconsumption approaches in WSNs providing proper link reliability as\n\\emph{Green Modulation\/Coding} (GMC) schemes.\n\nThere have been several recent works on the energy efficiency of\nvarious modulation\/ coding schemes in WSNs (see e.g.,\n\\cite{Cui_GoldsmithITWC0905, TangITWC0407, Chouhan_ITWC1009}). Tang\n\\emph{et al.} \\cite{TangITWC0407} compare the power efficiency of\nPPM and FSK in a WSN over fading channels with path-loss without\nconsidering the effect of channel coding. Reference\n\\cite{Sankarasubramaniam2003} investigates the energy efficiency of\nBCH and convolutional codes with FSK for the optimal packet length\nin a point-to-point WSN. It is shown in\n\\cite{Sankarasubramaniam2003} that BCH codes can improve energy\nefficiency compared to the convolutional code for the optimal fixed\npacket size. Liang \\emph{et al.} \\cite{LiangPACRIM2007} investigate\nthe energy efficiency of uncoded FSK modulation in a WSN, where\nmultiple senders transmit their data to a central node in a\nFrequency-Division Multiple Access (FDMA) fashion. Reference\n\\cite{Hanzo_2009} presents the hardware implementation of the\nForward Error Correction (FEC) encoder in IEEE 802.15.4 WSNs, which\nemploys parallel processing to achieve a low processing latency and\nenergy consumption.\n\nMost of the pioneering work on energy-efficient modulation\/coding,\nincluding research in \\cite{TangITWC0407,\nSankarasubramaniam2003}, has focused only on\nminimizing the energy consumption of transmitting one bit, ignoring\nthe effect of bandwidth and transmission time duration. In a\npractical WSN however, it is shown that minimizing the total energy\nconsumption depends strongly on the active mode duration and the\nchannel bandwidth. References \\cite{Cui_GoldsmithITWC0905},\n\\cite{Chouhan_ITWC1009} and \\cite{JamshidICASSP_2010} address this\nissue in a point-to-point WSN, where a sensor node transmits an\nequal amount of data per time unit to a designated sink node. In\n\\cite{Cui_GoldsmithITWC0905}, the authors show that uncoded MQAM is\nmore energy-efficient than uncoded MFSK for short-range\napplications, while using convolutional coded MFSK over AWGN is\ndesirable for longer distances. This line of work is further\nextended in \\cite{Chouhan_ITWC1009} by evaluating the energy\nconsumption per information bit of a WSN for Reed Solomon (RS) Codes\nand various modulation schemes over AWGN channels with path-loss. In\n\\cite{Cui_GoldsmithITWC0905} and \\cite{Chouhan_ITWC1009}, the\nauthors do not consider the effect of multi-path fading.\n\nMore recently, the attention of researchers has been drawn to\ndeploying rateless codes (e.g., Luby Transform (LT) code\n\\cite{LubyFOCS2002}) in WSNs due to the significant advantages of\nthese codes in erasure channels. For instance in\n\\cite{Eckford_ICC2006}, the authors present a scheme for cooperative\nerror control coding using rateless and Low-Density Generator-Matrix\n(LDGM) codes in a multiple relay WSN. However, investigating the\nenergy efficiency of rateless codes in WSNs with low-energy\nmodulations over realistic fading channel models has received little\nattention. To the best of our knowledge, there is no existing\nanalysis on the energy efficiency of rateless coded modulation that\nconsiders the effect of channel bandwidth and active mode duration\non the total energy consumption in a proactive WSN. This paper\naddresses this issue and presents the first in-depth analysis of the\nenergy efficiency of LT codes with FSK, known as green modulation as\ndescribed in \\cite{JamshidICASSP_2010}. The present analysis is\nbased on a realistic model in proactive WSNs operating over a\nRayleigh fading channel with path-loss. In addition, we obtain the\nprobability mass function of the LT code rate and the corresponding\ncoding gain, and study their effects on the energy efficiency of the\nWSN. This study uses the classical BCH and convolutional codes (as\nreference codes), utilized in IEEE standards, for comparative\nevaluation. Numerical results, supported by some experimental setup\non the computation energy, show that the optimized LT coded FSK\nscheme is the most energy-efficient scheme for distance $d$ greater\nthan the threshold level $d_T$. In addition, although the optimized\nuncoded FSK outperforms coded schemes for $d < d_T$, the energy gap\nbetween LT coded and uncoded FSK is negligible for $d < d_T$\ncompared to the other coded schemes. This result comes from the\nsimplicity and flexibility of the LT codes, and suggests that LT\ncodes are beneficial in practical low-power WSNs with dynamic\nposition sensor nodes.\n\nThe rest of the paper is organized as follows. In Section\n\\ref{System_model}, the proactive system model over a realistic\nwireless channel is described. The energy consumption of both\ncircuits and signal transmission of uncoded MFSK modulation is\nanalyzed in Section \\ref{uncoded_MFSK}. Design of LT codes and the\nenergy efficiency of the LT coded MFSK are presented in Section\n\\ref{analysis_Ch4}. Section \\ref{simulation_Ch5} provides some\nnumerical evaluations using some classical channel codes as well as\nrealistic models to confirm our analysis. Also, some design\nguidelines for using LT codes in practical WSN applications are\npresented. Finally in Section \\ref{conclusion_Ch6}, an overview of\nthe results and conclusions are presented.\n\nFor convenience, we provide a list of key mathematical symbols used\nin this paper in Table I. For simplicity of notation, we use the\nsuperscripts `BC', `CC' and `LT' for BCH, convolutional and LT\ncodes, respectively. We use $\\mathcal{E}$, $\\mathcal{P}$ and $T$ for\nenergy, power and time parameters, respectively. Also, we use the\nsubscript ``$c$'' to distinguish coding parameters from the uncoded\nones.\n\n\\begin{table}\n   \\label{table234}\n\\caption{List of Notations } \\centering\n  \\begin{tabular}{|l|l|}\n  \\hline\n  $B$:~Channel bandwidth                   & $b$:~Number of bits in each symbol\\\\\n  $d$:~Transmission distance               & $\\mathcal{E}_t$:~Energy of transmitted signal\\\\\n  $\\mathbb{E}[~.~]$:~Expectation operator  & $\\mathcal{E}_{N}$:~Total energy consumption\\\\\n  $M$:~Constellation size                  & $h_{i}$:~Fading channel coefficient\\\\\n  $N$:~Number of sensed message            & $\\mathcal{L}_d$:~Path loss gain\\\\\n  $n$:~Codeword block length               & $\\mathcal{O}(x)$:~Output-node degree distribution\\\\\n  $P_b$:~Bit error rate                    & $\\mathcal{P}_c$:~Circuit power consumption\\\\\n  $P_R(\\ell)$:~pmf of LT code rate         & $\\mathcal{P}_t$:~Power of transmitted signal\\\\\n  $R_c$:~Code rate                         & $T_{ac}$:~Active mode duration\\\\\n  $\\eta$:~Path-loss exponent               & $T_{tr}$:~Transient mode duration\\\\\n  $\\Omega=\\mathbb{E}\\left[\\vert h_{i} \\vert^2\\right]$   & $T_s$:~Symbol duration\\\\\n  $\\Upsilon_{c}$:~Coding gain              & $\\gamma_{i}$:~Instantaneous SNR \\\\\n  \\hline\n  \\end{tabular}\n\\end{table}\n\n\n\\section{System Model and Assumptions}\\label{System_model}\nWe consider a proactive wireless sensor system, in which a sensor\nnode transmits an equal amount of data per time unit to a designated\nsink node. Such a proactive sensor system is typical of many\nenvironmental applications such as sensing temperature, humidity and\nlevel of contamination \\cite{Cordeiro_Book2006}. We assume\na non real-time service application where the data transmission between the\nsensor and the sink nodes does not have tight\nconstraint on delay. The sensor and sink\nnodes synchronize with one another and operate in a real-time based\nprocess as depicted in Fig \\ref{fig: Time-Basis}. During\n\\emph{active mode} period $T_{ac}$, the sensed analog signal is\nfirst digitized by an Analog-to-Digital Converter (ADC), and an\n$N$-bit message sequence $\\mathcal{M}_N\\triangleq (m_1,m_2,...,m_N)$\nis generated, where $N$ is assumed to be fixed, and $m_i \\in \\{0,1\n\\}$, $i=1,2,...,N$. The bit stream is then sent to the channel\nencoder. The encoding process begins by dividing the uncoded message\n$\\mathcal{M}_N$ into blocks of equal length denoted by\n$\\mathcal{B}_j \\triangleq (m_{_{(j-1)k+1}},...,m_{jk})$,\n$j=1,...,\\frac{N}{k}$, where $k$ is the length of any particular\n$\\mathcal{B}_j$, and $N$ is assumed to be divisible by $k$. Each\nblock $\\mathcal{B}_j$ is encoded by a pre-determined channel coding\nscheme to generate a coded bit stream $\\mathcal{C}_j \\triangleq\n(a_{_{(j-1)n+1}},...,a_{jn})$, $j=1,...,\\frac{N}{k}$, with block\nlength $n$, where $n$ is either a fixed value (e.g., for block and\nconvolutional codes) or a random variable (e.g., for LT codes).\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=Time_Basis.eps,width=3.55in}} \\caption{A\npractical duty-cycling process in a proactive WSN. } \\label{fig:\nTime-Basis}\n\\end{figure}\nThe coded stream is then modulated by the FSK scheme and transmitted\nto the sink node. Finally, the sensor node returns to sleep mode,\nand all the circuits are shutdown for sleep mode duration $T_{sl}$.\nWe denote $T_{tr}$ as the transient mode duration consisting of the\nswitching time from sleep mode to active mode (i.e., $T_{sl\n\\rightarrow ac}$) plus the switching time from active mode to sleep\nmode (i.e., $T_{ac \\rightarrow sl}$), where $T_{ac \\rightarrow sl}$\nis short enough to be negligible. Under the above considerations,\nthe sensor\/sink nodes have to process one entire $N$-bit message\n$\\mathcal{M}_N$ during $0 \\leq T_{ac} \\leq T_N-T_{tr}$, where $T_N\n\\triangleq T_{tr}+T_{ac}+T_{sl}$ is fixed and $T_{tr} \\approx T_{sl\n\\rightarrow ac}$.\n\nSince sensor nodes in a typical WSN are densely deployed, the\ndistance between nodes is normally short. Thus, the total circuit\npower consumption, defined by $\\mathcal{P}_c \\triangleq\n\\mathcal{P}_{ct}+\\mathcal{P}_{cr}$, is comparable to the RF transmit\npower consumption denoted by $\\mathcal{P}_t$, where\n$\\mathcal{P}_{ct}$ and $\\mathcal{P}_{cr}$ represent the circuit\npower consumptions for the sensor and sink nodes, respectively.\nTaking these into account, the total energy consumption during the\nactive mode period, denoted by $\\mathcal{E}_{ac}$, is given by\n$\\mathcal{E}_{ac}=(\\mathcal{P}_c+\\mathcal{P}_t)T_{ac}$. Also,\nthe power consumption during the\nsleep mode duration $T_{sl}$ is much smaller than the power\nconsumption in the active mode (due to the low sleep mode leakage\ncurrent) to be negligible. As a result, we have the following\ndefinition.\n\n\\begin{defi}\\label{Definition01}\n\\textbf{(Performance Metric):} The energy efficiency, referred to as\nthe performance metric of the proposed WSN, can be measured by the\ntotal energy consumption in each period $T_N$ corresponding to\n$N$-bit message $\\mathcal{M}_N$ as follows:\n\\begin{equation}\\label{total_energy1}\n\\mathcal{E}_N =\n(\\mathcal{P}_c+\\mathcal{P}_t)T_{ac}+\\mathcal{P}_{tr}T_{tr},\n\\end{equation}\nwhere $\\mathcal{P}_{tr}$ is the circuit power consumption during the\ntransient mode period.\n\\end{defi}\nWe use (\\ref{total_energy1}) to investigate and compare the energy\nefficiency of uncoded and coded FSK for various channel coding\nschemes in the subsequent sections.\n\n\n\\textbf{Channel Model:} The choice of low transmission power in WSNs\nresults in several consequences to the channel model. It is shown by\nFriis \\cite{Friis1946} that a low transmission power implies a small\nrange. For short-range transmission scenarios, the root mean square\n(rms) delay spread is in the range of nanoseconds\n\\cite{Karl_Book2005} which is small compared to the symbol duration\n$T_s=16M~\\mu$s obtained from the bandwidth $B=\\frac{M}{T_s}=62.5$ KHz\nin the IEEE 802.15.4 standard, where $M$ is the constellation size of M-ary FSK\n\\cite[pp. 114-115]{Xiong_2006} and \\cite[p. 49]{IEEE_802_15_4_2006}.\nThus, it is reasonable to expect a flat-fading channel model for\nWSNs. In addition, many transmission environments include\nsignificant obstacle and structural interference by obstacles (such\nas wall, doors, furniture, etc), which leads to reduced\nLine-Of-Sight (LOS) components. This behavior suggests a Rayleigh\nfading channel model. Under the above considerations, the channel\nmodel between the sensor and sink nodes is assumed to be Rayleigh\nflat-fading with path-loss. This assumption is used in many works in\nthe literature (e.g., see \\cite{TangITWC0407} for WSNs). For this\nmodel, we assume that the channel is constant during the\ntransmission of a codeword, but may vary from one codeword to\nanother. We denote the fading channel coefficient corresponding to\nsymbol $i$ as $h_{i}$, where the amplitude $\\big\\vert h_{i}\n\\big\\vert$ is Rayleigh distributed with probability density function\n(pdf) $f_{\\vert h_{i}\n\\vert}(r)=\\frac{2r}{\\Omega}e^{-\\frac{r^2}{\\Omega}},~r \\geq 0$, where\n$\\Omega \\triangleq \\mathbb{E}\\left[\\vert h_{i} \\vert^2\\right]$\n\\cite{Proakis2001}.\n\nTo model the path-loss of a link where the transmitter and receiver\nare separated by distance $d$, let denote $\\mathcal{P}_t$ and\n$\\mathcal{P}_r$ as the transmitted and the received signal powers,\nrespectively. For a $\\eta^{th}$-power path-loss channel, the channel\ngain factor is given by $\\mathcal{L}_d \\triangleq\n\\frac{\\mathcal{P}_t}{\\mathcal{P}_r}=M_ld^\\eta \\mathcal{L}_1$, where\n$M_l$ is the gain margin which accounts for the effects of hardware\nprocess variations, background noise and $\\mathcal{L}_1 \\triangleq\n\\frac{(4 \\pi)^2}{\\mathcal{G}_t \\mathcal{G}_r \\lambda^2}$ is the gain\nfactor at $d=1$ meter which is specified by the transmitter and\nreceiver antenna gains $\\mathcal{G}_t$ and $\\mathcal{G}_r$, and\nwavelength $\\lambda$ (e.g., \\cite{Cui_GoldsmithITWC0905}). As a\nresult, when both fading and path-loss are considered, the\ninstantaneous channel coefficient becomes $G_{i} \\triangleq\n\\frac{h_{i}}{\\sqrt{\\mathcal{L}_d}}$. Denoting $x_i(t)$ as the\ntransmitted signal with energy $\\mathcal{E}_{t}$, the received\nsignal at the sink node is given by\n$y_{i}(t)=G_{i}x_{i}(t)+n_{i}(t)$, where $n_{i}(t)$ is AWGN at the\nsink node with two-sided power spectral density given by\n$\\frac{N_{0}}{2}$. Under the above considerations, the instantaneous\nSignal-to-Noise Ratio (SNR) corresponding to symbol $i$ can be\ncomputed as $\\gamma_{i}=\\frac{\\vert G_{i}\\vert^2\n\\mathcal{E}_t}{N_0}$. Under the assumption of a Rayleigh fading\nchannel model, $\\gamma_{i}$ is chi-square distributed with 2 degrees\nof freedom and with pdf\n$f_{\\gamma}(\\gamma_{i})=\\frac{1}{\\bar{\\gamma}}e^{\n-\\frac{\\gamma_{i}}{\\bar{\\gamma}}}$, where $\\bar{\\gamma} \\triangleq\n\\mathbb{E}[\\vert\nG_{i}\\vert^2]\\frac{\\mathcal{E}_t}{N_0}=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_t}{N_0}$\ndenotes the average received SNR.\n\n\n\n\n\\section{Energy Consumption of Uncoded Scheme}\\label{uncoded_MFSK}\nIn this section, we consider the uncoded M-ary FSK modulation where\n$M$ orthogonal carriers can be mapped into $b \\triangleq \\log_{2}M$\nbits. Among various sinusoidal carrier-based modulation techniques,\nFSK has been found to provide a good compromise between simple radio\narchitecture, low-power consumption, and requirements on linearity\nof the modulation scheme \\cite{JamshidICASSP_2010, Jamshid_IET2010,\nCui_GoldsmithITWC0905}. Also, this scheme is used in some IEEE\nstandards (e.g., \\cite{IEEE_P802_15}). Since we have $b$ bits during\neach symbol period $T_{s}$, we can write\n\\begin{equation}\\label{active1}\nT_{ac}=\\dfrac{N}{b}T_{s}\\stackrel{(a)}{=}\\dfrac{MN}{B\\log_2 M}.\n\\end{equation}\nwhere $(a)$ comes from the bandwidth $B \\approx M\\times\\Delta f $\nwith the minimum carrier separation $\\Delta f=\\frac{1}{T_s}$ for\nMFSK with the non-coherent detector \\cite[pp. 114-115]{Xiong_2006}. It is\nshown in \\cite{JamshidICASSP_2010} that the transmit energy\nconsumption per each symbol for an uncoded MFSK with non-coherent\ndetector is obtained as\n\\begin{eqnarray}\n\\mathcal{E}_t &\\triangleq& \\mathcal{P}_t T_s \\approx \\left[\\left(\n1-(1-P_s)^{\\frac{1}{M-1}}\\right)^{-1}-2 \\right]\\dfrac{\\mathcal{L}_d\nN_0}{\\Omega}\\\\\n\\notag&\\stackrel{(a)}{=}& \\left[\\left(\n1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\dfrac{\\mathcal{L}_d N_0}{\\Omega}\\\\\n\\label{deriv1}\n\\end{eqnarray}\nwhere $(a)$ comes from the fact that the relationship between the\naverage Symbol Error Rate (SER) $P_s$ and the average Bit Error Rate\n(BER) $P_b$ of MFSK is given by $P_s=\\frac{2(M-1)}{M}P_b$ \\cite[p.\n262]{Proakis2001}. As a result, the output energy consumption of\ntransmitting $N$-bit during $T_{ac}$ of an uncoded MFSK is computed\nfrom (\\ref{active1}) as follows:\n\\begin{eqnarray}\n\\notag \\mathcal{P}_t T_{ac}&=& \\dfrac{T_{ac}}{T_s}\\mathcal{E}_t\n\\approx \\left[\\left(\n1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\label{energy_trans1}&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega}\n\\dfrac{N}{\\log_2 M}.\n\\end{eqnarray}\nFor the sensor node with uncoded MFSK, we denote the power\nconsumption of frequency synthesizer, filters and power amplifier as\n$\\mathcal{P}_{Sy}$, $\\mathcal{P}_{Filt}$ and $\\mathcal{P}_{Amp}$,\nrespectively. In this case, the circuit power consumption of the\nsensor node with uncoded MFSK can be obtained as\n\\begin{equation}\\label{sensor_power}\n\\mathcal{P}_{ct}=\\mathcal{P}_{Sy}+\\mathcal{P}_{Filt}+\\mathcal{P}_{Amp},\n\\end{equation}\nwhere $\\mathcal{P}_{Amp}=\\alpha \\mathcal{P}_{t}$ with $\\alpha=0.33$\n\\cite{Cui_GoldsmithITWC0905}, \\cite{TangITWC0407}. In addition, the\npower consumption of the sink circuitry with uncoded MFSK scheme can\nbe obtained as\n\\begin{equation}\\label{sink_power}\n\\mathcal{P}_{cr}=\\mathcal{P}_{LNA}+M\\times(\\mathcal{P}_{Filr}+\\mathcal{P}_{ED})+\\mathcal{P}_{IFA}+\\mathcal{P}_{ADC},\n\\end{equation}\nwhere $\\mathcal{P}_{LNA}$, $\\mathcal{P}_{Filr}$, $\\mathcal{P}_{ED}$,\n$\\mathcal{P}_{IFA}$ and $\\mathcal{P}_{ADC}$ denote the power\nconsumption of Low-Noise Amplifier (LNA), filters, envelop detector,\nIF amplifier and ADC, respectively \\cite{JamshidICASSP_2010}. Since,\nthe power consumption during transient mode period $T_{tr}$ is\ngoverned by the frequency synthesizer in both transmitter and the\nreceiver \\cite{Wang_ISLPED2001}, the energy consumption during\n$T_{tr}$ is obtained as $\\mathcal{P}_{tr}T_{tr}=2\n\\mathcal{P}_{Sy}T_{tr}$ \\cite{Cui_GoldsmithITWC0905}. Substituting\n(\\ref{active1}) and (\\ref{energy_trans1}) in (\\ref{total_energy1}),\nthe total energy consumption of an uncoded MFSK for transmitting\n$N$-bit information in each period $T_N$ for a given $P_b$ is\nobtained as\n\\begin{eqnarray}\n\\notag \\mathcal{E}_N &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega} \\dfrac{N}{\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{B\\log_{2}M}+\\\\\n\\label{energy_totFSK}&&2 \\mathcal{P}_{Sy}T_{tr}.\n\\end{eqnarray}\n\n\nFor energy optimal designs, the impact of channel coding on the\nenergy efficiency of the proposed WSN must be considered as well. It\nis a well known fact that channel coding is a classical approach\nused to improve the link reliability along with the transmitter\nenergy saving due to providing the coding gain \\cite{Proakis2001}.\nHowever, the energy saving comes at the cost of extra energy spent\nin transmitting the redundant bits in codewords as well as the\nadditional energy consumption in the process of encoding\/decoding.\nFor a specific transmission distance $d$, if these extra energy\nconsumptions outweigh the transmit energy saving due to the channel\ncoding, the coded system would not be energy-efficient compared with\nan uncoded system. In the subsequent sections, we will argue the\nabove problem and determine at what distance use of specific channel\ncoding becomes energy-efficient compared to uncoded systems. In\nparticular, we will show in Section \\ref{simulation_Ch5} that the LT\ncoded modulation surpasses this distance constraint in the proposed\nWSN.\n\n\\section{Energy Consumption Analysis of LT Coded Modulation}\\label{analysis_Ch4}\nIn this section, we present the first in-depth analysis on the\nenergy efficiency of LT coded modulation for the proposed proactive\nWSN. To get more insight into how channel coding affects the circuit\nand RF signal energy consumptions in the system, we modify the\nenergy concepts in Section \\ref{uncoded_MFSK}, in particular, the\ntotal energy consumption expression in (\\ref{energy_totFSK}) based\non the coding gain, code rate and the computation energy. We further\npresent the first study on the tradeoff between LT code rate and\ncoding gain required to achieve a certain BER, and the effect of\nthis tradeoff on the total energy consumption of LT coded MFSK for\ndifferent transmission distances.\n\n\\subsection{Energy Efficiency of Coded System}\nFor an arbitrary channel coding scheme, each $k$-bit message\n$\\mathcal{B}_j \\in \\mathcal{M}_N$ is encoded into the codeword\n$\\mathcal{C}_j$ with block length $n$ and code rate $R_c \\triangleq\n\\frac{k}{n}$. In this case, the number of transmitted bits in $T_N$\nis increased from $N$-bit uncoded message to\n$\\frac{N}{R_c}=\\frac{N}{k}n$ bits coded one. To compute the energy\nconsumption of coded scheme, we use the fact that channel coding\nreduces the required average SNR value to achieve a given BER (i.e.,\nthe same BER as uncoded one). Taking this into account, the proposed\nWSN benefits in transmission energy saving of coded modulation\nspecified by\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$, where\n$\\Upsilon_{c} \\geq 1$ is the coding gain\\footnote{Denoting\n$\\bar{\\gamma}\n=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_t}{N_0}$ and\n$\\bar{\\gamma}_{c}\n=\\frac{\\Omega}{\\mathcal{L}_d}\\frac{\\mathcal{E}_{t,c}}{N_0}$ as the\naverage SNR of uncoded and coded schemes, respectively, the\n\\emph{coding gain} (expressed in dB) is defined as the difference\nbetween the values of $\\bar{\\gamma}$ and $\\bar{\\gamma}_{c}$ required\nto achieve a certain BER, where\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$.} of the\nutilized coded MFSK. It should be noted that the cost of this energy\nsaving is the bandwidth expansion $\\frac{B}{R_{c}}$. In order to\nkeep the bandwidth of the coded system the same as that of the\nuncoded case, we must keep the information transmission rate\nconstant, i.e., the symbol duration $T_{s}$ of uncoded and coded\nMFSK would be the same. However, the active mode duration increases\nfrom $T_{ac}=\\frac{N}{b}T_{s}$ in the uncoded system to\n\\begin{equation}\\label{BCH_active}\nT_{ac,c}=\\frac{N}{bR_{c}}T_{s}=\\frac{T_{ac}}{R_{c}}\n\\end{equation}\nfor the coded case. Thus, one would assume that the total time $T_N$\nincreases to $\\frac{T_N}{R_c}$ for the coded scenario. It is worth\nmentioning that the active mode period in the coded case is upper\nbounded by $\\frac{T_N}{R_c}-T_{tr}$. As a result, the maximum\nconstellation size $M$, denoted by $M_{max}\\triangleq 2^{b_{max}}$,\nfor the coded MFSK is calculated by\n$\\frac{2^{b_{max}}}{b_{max}}=\\frac{B\nR_{c}}{N}(\\frac{T_{N}}{R_{c}}-T_{tr})$, which is approximately the\nsame as that of the uncoded case.\n\n\\newcounter{mytempeqncnt}\n\\begin{figure*}[!t]\n\\normalsize \\setcounter{mytempeqncnt}{\\value{equation}}\n\\setcounter{equation}{10}\n\\begin{eqnarray}\n\\left(\\left[1-\\left(1-\\frac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right]^{-1}-2\\right) \\dfrac{1}{\\log_2 M}&=& \\left(\\left[1-e^{\\frac{1}{M-1}\\ln(1-\\frac{2(M-1)}{M}P_b)}\\right]^{-1}-2\\right)\\dfrac{1}{\\log_2 M}\\\\\n&\\stackrel{(a)}{\\approx}& \\left(\\left[1-e^{-\\frac{2P_b}{M}}\\right]^{-1}-2\\right)\\dfrac{1}{\\log_2 M}\\\\\n\\label{mono10}&\\stackrel{(b)}{\\approx}&\\left(\\dfrac{M}{2P_b}-2\\right)\\dfrac{1}{\\log_2\nM},\n\\end{eqnarray}\n\\setcounter{equation}{\\value{mytempeqncnt}} \\hrulefill \\vspace*{4pt}\n\\end{figure*}\n\nWe denote $\\mathcal{E}_{enc}$ and $\\mathcal{E}_{dec}$ as the\ncomputation energy of the encoder and decoder for each information\nbit, respectively. Thus, the total computation energy cost of the\ncoding components for $\\frac{N}{R_{c}}$ bits is obtained as\n$N\\frac{\\mathcal{E}_{enc}+\\mathcal{E}_{dec}}{R_{c}}$. Substituting\n(\\ref{deriv1}) in\n$\\mathcal{E}_{t,c}=\\frac{\\mathcal{E}_{t}}{\\Upsilon_{c}}$, and using\n(\\ref{sensor_power}), (\\ref{sink_power}) and (\\ref{BCH_active}), the\ntotal energy consumption of transmitting $\\frac{N}{R_c}$ bits in\neach period $\\frac{T_N}{R_C}$ for an arbitrarily coded MFSK, and a\ngiven $P_b$ is obtained as\n\\begin{eqnarray}\n\\notag \\mathcal{E}_{N,c} &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag&\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega \\Upsilon_{c}}\n\\dfrac{N}{R_{c}\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{BR_{c} \\log_{2}M}+ \\\\\n\\label{energy_totalcodedFSK} && 2\n\\mathcal{P}_{Sy}T_{tr}+N\\frac{\\mathcal{E}_{enc}+\\mathcal{E}_{dec}}{R_{c}}.\n\\end{eqnarray}\nTo make a fair comparison between the uncoded and coded modulation,\nwe use the same BER and drop the subscript ``$c$'' for $P_b$ in the\ncoded case. Thus, the optimization goal is to minimize the objective\nfunction $\\mathcal{E}_{N,c}$ over modulation and coding parameters.\nThis is achieved by finding the optimum constellation size $M$ under\nthe constraint $2\\leq M \\leq M_{max}$ for a specific channel coding\nscheme, and then minimize $\\mathcal{E}_{N,c}$ over the coding\nparameters.\n\nFor the above optimization problem, we consider two scenarios: $i)$\nfixed-rate codes (e.g., BCH and convolutional codes), and $ii)$\nvariable rate codes (e.g., LT codes). To find the optimum $M$ for a\ngiven fixed-rate code, we prove that (\\ref{energy_totalcodedFSK}) is\na monotonically increasing function of $M$ for every value of $d$\nand $\\eta$. Since $B$ and $N$ are fixed and $R_c$ is independent of\n$M$, it is concluded that the second term in\n(\\ref{energy_totalcodedFSK}) is a monotonically increasing function\nof $M$. Also, from the first term in (\\ref{energy_totalcodedFSK}),\nwe have (11)-(13) in the top of this page,\nwhere $(a)$ comes from the approximation $\\ln(1-z)\\approx -z,~~|z|\n\\ll 1$, and the fact that $P_b$ scales as $o(1)$. Also, $(b)$\nfollows from the approximation\n$e^{-z}=\\sum_{n=0}^{\\infty}(-1)^{n}\\frac{z^n}{n!}\\approx\n1-z,~~|z|\\ll1$. On the other hand, it is shown in \\cite{Proakis2001}\nthat $\\Upsilon_c$ for MFSK is a decreasing function of $M$. Thus, it\nis concluded from (\\ref{mono10}) that the first term in\n(\\ref{energy_totalcodedFSK}) is also a monotonically increasing\nfunction of $M$. As a result, the minimum total energy consumption\n$\\mathcal{E}_{N,c}$ for a given fixed-rate code is achieved at\n$\\hat{M}=2$.\n\nIn the next section, we evaluate the above optimization problem for\nthe LT codes using some simulation studies on the probability mass\nfunction of the LT code rate and coding gain. We show that the LT\ncode parameters depend strongly on the constellation size $M$ and\nexhibit different trends over fixed-rate codes. In addition, we\npresent some beneficial uses of LT codes over block and\nconvolutional codes in managing the energy consumption for different\nchannel realizations.\n\n\n\n\n\\subsection{Energy Optimality of LT Codes}\nLT codes are the first class of Fountain codes which usually\nspecified by the number of input bits $k$ and the output-node degree\ndistribution $\\mathcal{O}(x)$. Without loss of generality and for\nease of our analysis, we assume that a single $k$-bit message\n$\\mathcal{B}_1 \\triangleq (m_1,m_2,...,m_k)\\in \\mathcal{M}_N$ is\nencoded to codeword $\\mathcal{C}_1 \\triangleq (a_1,a_2,...,a_n)$.\nEach single coded bit $a_i$ is generated based on the encoding\nprotocol proposed in \\cite{LubyFOCS2002}: $i)$ randomly choose a\ndegree $1 \\leq \\mathcal{D} \\leq k$ from $\\mathcal{O}(x)$, $ii)$\nusing a uniform distribution, randomly choose $\\mathcal{D}$ distinct\ninput bits, and calculate the encoded bit $a_i$ as the XOR-sum of\nthese $\\mathcal{D}$ bits. The above encoding process defines a\n\\emph{bipartite graph} connecting encoded nodes to input nodes (see,\ne.g., Fig. \\ref{fig: LT Encoder}). It is seen that the LT encoding\nprocess is extremely simple and has very low energy consumption.\nUnlike block and convolutional codes, in which the codeword block\nlength is fixed, for the above LT code, $n$ is a variable parameter,\nresulting in a random variable LT code rate $R^{LT}_c \\triangleq\n\\frac{k}{n}$. More precisely, $a_n \\in \\mathcal{C}_1$ is the last\nbit generated at the output of LT encoder before receiving the\nacknowledgement signal from the sink node indicating termination of\na successful decoding process. This inherent property of LT codes\nmeans they can vary their codeword block lengths to adapt to any\nwireless channel condition between the sensor and the sink nodes.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code1.eps,width=3.25in}} \\caption{\nBipartite graph of an LT code with $k=6$ and $n=8$. } \\label{fig: LT\nEncoder}\n\\end{figure}\n\n\nTo describe the output-node degree distribution used in this work,\nlet $\\mu_i$, $i=1,...,k$, denote the probability that an output node\nhas degree $i$. Following the notation of\n\\cite{Shokrollahi_ITIT0606}, the output-node degree distribution of\nan LT code has the polynomial form $\\mathcal{O}(x) \\triangleq\n\\sum_{i=1}^{k}\\mu_i x^i$ with the property that $\\mathcal{O}(1) =\n\\sum_{i=1}^{k}\\mu_i =1$. Typically, optimizing the output-node\ndegree distribution for a specific wireless channel model is a\ncrucial task in designing LT codes. In fact, for wireless fading\nchannels, it is still an open problem, what the ``optimal\"\n$\\mathcal{O}(x)$ is. In this work, we use the following output-node\ndegree distribution which was optimized for a BSC using a\nhard-decision decoder \\cite{David_Thesis2008}:\n\\begin{eqnarray}\n\\notag \\mathcal{O}(x)&=&0.00466x+0.55545x^2+0.09743x^3+\\\\\n\\notag &&0.17506x^5+0.03774x^8+0.08202x^{14}+\\\\\n\\label{degree}&&0.01775x^{33}+0.02989x^{100}.\n\\end{eqnarray}\n\nThe LT decoder at the sink node can recover the original $k$-bit\nmessage $\\mathcal{B}_1$ with high probability after receiving any\n$(1+\\epsilon)k$ bits in its buffer, where $\\epsilon$ depends upon\nthe LT code design \\cite{Shokrollahi_ITIT0606}. For this recovery\nprocess, the LT decoder needs to correctly reconstruct the bipartite\ngraph of an LT code. One practical approach suitable for the\nproposed WSN model is that the LT encoder and decoder use identical\npseudo-random generators with a common seed value which may reduce\nthe complexity further. In this work, we assume that the sink node\nrecovers $k$-bit message $\\mathcal{B}_1$ using a simple\nhard-decision ``\\emph{ternary message passing}\" decoder in a nearly\nidentical manner to the ``\\emph{Algorithm E}'' decoder in\n\\cite{RichardsonITIT0201} for Low-Density Parity-Check (LDPC)\ncodes\\footnote{Description of the \\emph{ternary message passing}\ndecoding is out of scope of this work, and the reader is referred to\nChapter 4 in \\cite{David_Thesis2008} for more details.}. Also, the\ndegree distribution $\\mathcal{O}(x)$ in (\\ref{degree}) was optimized\nfor a ternary decoder in a BSC and we are aware of no better\n$\\mathcal{O}(x)$ for the ternary decoder in Rayleigh fading\nchannels.\n\nUnlike fixed-rate codes in which the active mode duration of coded\nMFSK is fixed, for the LT coded MFSK, we have a non-fixed value for\n\\begin{equation}\\label{active_LT}\nT_{ac,c}^{LT}=\\dfrac{N}{b R_{c}^{LT}}T_s=\\dfrac{MN}{BR^{LT}_{c}\n\\log_{2}M}.\n\\end{equation}\nAn interesting point raised from (\\ref{active_LT}) is that\n$T_{ac,c}^{LT}$ is a function of the random variable $R_c^{LT}$\nwhich results in an inherent adaptive duty-cycling for power\nmanagement in each channel condition without any channel state\ninformation fed back from the sink node to the sensor node.\nRecalling from (\\ref{energy_totalcodedFSK}), we have\n\\begin{eqnarray}\n\\notag \\mathcal{E}^{LT}_{N,c} &=& (1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n\\notag &\\times&\\dfrac{\\mathcal{L}_d N_0}{\\Omega \\Upsilon^{LT}_{c}}\n\\dfrac{N}{R^{LT}_{c}\\log_2 M}+\n(\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{BR^{LT}_{c}\n\\log_{2}M}+ \\\\\n\\notag&&2\n\\mathcal{P}_{Sy}T_{tr}+N\\frac{\\mathcal{E}^{LT}_{enc}+\\mathcal{E}^{LT}_{dec}}{R^{LT}_{c}}.\n\\end{eqnarray}\nwhere the main goal is to minimize $\\mathcal{E}^{LT}_{N,c}$ over $M$\nand the coding parameters. Toward this goal, we first compute the LT\ncode rate and the corresponding LT coding gain. Let us begin with\nthe case of asymptotic LT code rate, where the number of input bits\n$k$ goes to infinity. It is shown that the LT code rate is obtained\nasymptotically as a fixed value of $R^{LT}_c \\approx\n\\frac{\\mathcal{O}_{ave}}{\\mathcal{I}_{ave}}$ for large values of\n$k$, where $\\mathcal{O}_{ave}$ and $\\mathcal{I}_{ave}$ represent the\naverage degree of the output and the input nodes in the bipartite\ngraph, respectively (see Appendix I for the proof).\n\n\n\nWhen using finite-$k$ LT codes, we are treating the channel as\nstatic over one block length. In this case, the rate of the LT code\nfor any block can be chosen to achieve the desired performance for\nthat block. More precisely, for the particular block we are\nconcerned with, the receiver could evaluate the channel\ninstantaneous SNR, and then determine how many code bits it needs to\ncollect in order to achieve its given BER target, thus essentially\ndynamically selecting its rate.  For the next block, the receiver\nwould once again evaluate the (new) instantaneous channel SNR and\nadjust its rate accordingly to collect more or fewer bits.\n\nIt should be noted that there is no (currently known) explicit equation\ngoverning the relationship between the instantaneous SNR and the required\nnumber of decoding bits for an LT code. In this work, we determine\nthe necessary number of decoding bits in each case through simulation,\nas described below:\n\\begin{itemize}\n\\item [$\\textbf{i)}$] We decide upon a ``\\emph{target BER}'' for the decoded bits - e.g., the\ndecoded message needed an average BER of $10^{-4}$ or better.\n\\item [$\\textbf{ii)}$] Based on the assumption that instantaneous SNR is constant over at\nleast one block we perform the following steps for a large range of possible\ninstantaneous SNR:\n\n- Determine through numerical simulation the decoded BER using the LT code rate $R^{LT}_c= 1$.\n\n- If the decoded BER is greater than the target BER, then we reduce the LT code rate $R^{LT}_c$ and try again.\n\n- Repeat the above step until the decoded BER is less than the target BER.\n\n\\item [$\\textbf{iii)}$] At this point, for the given LT degree distribution, decoder, and SNR\nwe can identify the highest LT code rate that will yield the target BER\nor better.\n\\end{itemize}\nArmed with this information (computed ahead of time), a receiver can determine\nthe appropriate rate at which to operate and hence determine how many bits it\nmust collect for a given instantaneous SNR.\nThis is also based on the common assumption that the receiver is capable of\nestimating the instantaneous SNR.\n\n\n\nBased on the above arguments and for any given average SNR, the LT code rate is described\nby either a probability mass function (pmf) or a probability density\nfunction (pdf) denoted by $P_R(\\ell)$. Because it is difficult to\nget a closed-form expression of $P_R(\\ell)$, we use a discretized\nnumerical method to calculate the pmf $P_R(\\ell)\\triangleq\n\\textrm{Pr}\\{R_c^{LT}=\\ell \\}$, $0 \\leq \\ell \\leq 1$, for different\nvalues of $M$.\nWe plot the pmfs of the LT code rate in Fig. \\ref{fig: LT code_rate_mpf}\nfor $M=2$ and various average SNR \\footnote{It should be noted that\nthe average SNR is expressed in terms of $\\frac{\\mathcal{E}_b}{N_0}$,\nwhere $\\mathcal{E}_b$ represents the transmit energy per bit.} over a Rayleigh fading channel model.\nIt is observed that for lower average SNRs the pmfs are larger in\nthe lower rate regimes (i.e., the pmfs spend more time in the low\nrate region). Also, all pmfs exhibit quite a spike for the highest\nrate, which makes sense since once the instantaneous SNR hits a\ncertain critical value, the codes will always decode with a high\nrate. Also, Fig. \\ref{fig: LT code_rate_pmf_M} illustrates the pmf\nof LT code rates for various constellation size $M$ and for average\nSNR equal to 16 dB. It can be seen that as $M$ increases, the rate\nof the LT code tends to have a pmf with larger values in the lower\nrate regions.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code_Rate_pmf.eps,width=3.75in}}\n\\caption{The pmf of LT code rate for various average SNR and M=2. }\n\\label{fig: LT code_rate_mpf}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=LT_Code_Rate_pmf_M.eps,width=3.75in}}\n\\caption{The pmf of LT code rate for various constellation size $M$\nand average SNR=16 dB. } \\label{fig: LT code_rate_pmf_M}\n\\end{figure}\n\n\nTable II illustrates the average LT code rates and the\ncorresponding coding gains of LT coded MFSK using $\\mathcal{O}(x)$\nin (\\ref{degree}), for $M=2,4,8,16$ and given $P_b=10^{-3}$. The\naverage rate for a certain average SNR is obtained by integrating\nthe pmf over the rates from $0$ to $1$. To get more insight into\nthe relationship between the LT code rate and coding gain, we plot\nthe results of Table II in Fig. \\ref{fig: LT code_rate_coding_gain}. It is observed that the LT\ncode is able to provide a huge coding gain $\\Upsilon_{c}^{LT}$ given\n$P_b=10^{-3}$, but this gain comes at the expense of a very low\naverage code rate, which means many additional code bits need to be\nsent. This results in higher energy consumption per information bit.\nIn contrast to the fixed-rate codes in which the coding gain\ndecreases when $M$ grows, the LT coding gains display different\ntrends in terms of $M$ as illustrated in Table II. Thus, in\ncontrast with fixed-rate codes, $\\mathcal{E}_{N,c}^{LT}$ is not\nnecessarily a monotonically increasing function of $M$. In the next\nsection, we evaluate numerically $\\mathcal{E}_{N,c}^{LT}$ in terms\nof the optimized modulation and coding parameters compared to the\nuncoded and the fixed-rate codes.\n\n\\begin{figure*}[t]\n\\centerline{\\psfig{figure=LT_Codig_Gain_vs_Rate.eps,width=6.15in}}\n\\caption{LT coding gain versus average LT code rate for $P_b=10^{-3}$ and\nM=2,4,8,16. } \\label{fig: LT code_rate_coding_gain}\n\\end{figure*}\n\n\nAn interesting point extracted from Table II is the flexibility of\nthe LT code to adjust its rate (and its corresponding coding gain)\nto suit instantaneous channel conditions in WSNs. For instance in\nthe case of favorable channel conditions, the LT coded MFSK is able\nto achieve $R_{c}^{LT}\\approx1$ with $\\Upsilon_{c}^{LT}\\approx 0$\ndB, which is similar to the case of uncoded MFSK, i.e., $n=k$. The\neffect of LT code rate flexibility on the total energy consumption\nis also observed in the simulation results in the subsequent\nsection.\n\n\n\\begin{table*}\n\\label{table0021} \\caption{Average LT code rate and Coding gain of\nLT coded MFSK over Rayleigh fading model for $P_b=10^{-3}$ and\nM=2,4,8,16.} \\centering\n  \\begin{tabular}{|c|cc||c|cc|cc|cc|}\n  \\hline\n    & M=2 &   & &  M=4 &  & M=8 &  & M=16 & \\\\\n   \\hline\n   $\\frac{\\mathcal{E}_b}{N_0}$  & Average   &  Coding    & $\\frac{\\mathcal{E}_b}{N_0}$  & Average   & Coding    & Average   &  Coding    & Average    & Coding    \\\\\n   (dB) & Code Rate &  Gain (dB) & (dB) & Code Rate & Gain (dB) & Code Rate &  Gain (dB) &  Code Rate & Gain (dB) \\\\\n   \\hline\n   5 & 0.2560 & 25 &  0   & 0.0028 & 33.87 &  0.0012 & 36.46 &  0.0012 & 38.48  \\\\\n   6 & 0.3174 & 24 &  2   & 0.0140 & 31.87 &  0.0024 & 34.46 &  0.0012 & 36.48  \\\\\n   7 & 0.3819 & 23 &  4   & 0.0460 & 29.87 &  0.0095 & 32.46 &  0.0021 & 34.48  \\\\\n   8 & 0.4475 & 22 &  6   & 0.1100 & 27.87 &  0.0330 & 30.46 &  0.0086 & 32.48  \\\\\n   9 & 0.5120 & 21 &  8   & 0.2100 & 25.87 &  0.0870 & 28.46 &  0.0320 & 30.48  \\\\\n   10& 0.5738 & 20 & 10   & 0.3300 & 23.87 &  0.1800 & 26.46 &  0.0870 & 28.48  \\\\\n   11& 0.6315 & 19 & 12   & 0.4600 & 21.87 &  0.3000 & 24.46 &  0.1800 & 26.48  \\\\\n   12& 0.6840 & 18 & 14   & 0.5900 & 19.87 &  0.4400 & 22.46 &  0.3100 & 24.48  \\\\\n   13& 0.7307 & 17 & 16   & 0.7000 & 17.87 &  0.5700 & 20.46 &  0.4500 & 22.48  \\\\\n   14& 0.7716 & 16 & 18   & 0.7800 & 15.87 &  0.6800 & 18.46 &  0.5800 & 20.48  \\\\\n   15& 0.8067 & 15 & 20   & 0.8500 & 13.87 &  0.7700 & 16.46 &  0.6900 & 18.48  \\\\\n   16& 0.8365 & 14 & 22   & 0.8900 & 11.87 &  0.8400 & 14.46 &  0.7800 & 16.48  \\\\\n   17& 0.8614 & 13 & 24   & 0.9200 & 9.87  &  0.8800 & 12.46 &  0.8400 & 14.48  \\\\\n   18& 0.8821 & 12 & 26   & 0.9400 & 7.87  &  0.9100 & 10.46 &  0.8800 & 12.48  \\\\\n   19& 0.8991 & 11 & 28   & 0.9500 & 5.87  &  0.9300 & 8.46  &  0.9200 & 10.48  \\\\\n   20& 0.9130 & 10 & 30   & 0.9600 & 3.87  &  0.9500 & 6.46  &  0.9400 & 8.48   \\\\\n   22& 0.9333 & 8  & 32   & 0.9600 & 1.87  &  0.9500 & 4.46  &  0.9500 & 6.48   \\\\\n   24& 0.9466 & 6  & 34   & 0.9600 & -0.13 &  0.9600 & 2.46  &  0.9500 & 4.48   \\\\\n   26& 0.9551 & 4  & 36   & 0.9700 & -2.13 &  0.9600 & 0.46  &  0.9600 & 2.48   \\\\\n   28& 0.9606 & 2  & 38   & 0.9700 & -4.13 &  0.9700 & -1.54 &  0.9600 & 0.48   \\\\\n   30& 0.9640 & 0  & 40   & 0.9700 & -6.13 &  0.9700 & -3.54 &  0.9700 & -1.52  \\\\\n\n   \\hline\n   \\end{tabular}\n\\end{table*}\n\n\n\n\n\\section{Numerical Evaluations}\\label{simulation_Ch5}\nIn this section, we present some numerical evaluations using\nrealistic parameters from the IEEE 802.15.4 standard and\nstate-of-the art technology to confirm the energy efficiency\nanalysis of uncoded and coded MFSK modulations discussed in Sections\n\\ref{uncoded_MFSK} and \\ref{analysis_Ch4}.\n\n\\subsection{Experimental Setup}\nWe assume that MFSK operates in the carrier frequency $f_0=$2.4 GHz\nIndustrial Scientist and Medical (ISM) unlicensed band utilized in\nthe IEEE 802.15.14 standard \\cite{IEEE_802_15_4_2006}. According to\nthe FCC 15.247 RSS-210 standard for United States\/Canada, the\nmaximum allowed antenna gain is 6 dBi \\cite{FreeScale2007}. In this\nwork, we assume that $\\mathcal{G}_t=\\mathcal{G}_r=5$ dBi. Thus for\nthe $f_0=$2.4 GHz, $\\mathcal{L}_1~ \\textrm{(dB)}\\triangleq\n10\\log_{10}\\left(\\frac{(4 \\pi)^2}{\\mathcal{G}_t \\mathcal{G}_r\n\\lambda^2}\\right) \\approx 30~ \\textrm{dB}$, where $\\lambda\n\\triangleq \\frac{3\\times 10^8}{f_0}=0.125$ m. We assume that in each\nperiod $T_N$, the data frame $N=1024$ bytes (or equivalently\n$N=8192$ bits) is generated, where $T_N$ is assumed to be 1.4 s. The\nchannel bandwidth is set to the $B=62.5$ KHz, according to IEEE\n802.15.4 \\cite[p. 49]{IEEE_802_15_4_2006}. It is concluded from\n$\\frac{2^{b_{max}}}{b_{max}}=\\frac{BR_c}{N}(\\frac{T_{N}}{R_c}-T_{tr})\\approx\n\\frac{B}{N}T_N$ that $M_{max} \\approx 64$ (or equivalently $b_{max}\n\\approx 6$) for MFSK. Table III summarizes the system parameters for\nsimulation\\footnote{To make a fair comparison between the energy\nconsumption of different communication\nschemes, the bandwidth and the BER are assumed to be the same for all the schemes.} \\cite{Cui_GoldsmithITWC0905, TangITWC0407,\nBevilacqua_IJSSC1204, Wang_ISLPED2001}. The results in Tables II\nare also used to compare the energy efficiency of uncoded and\ncoded MFSK schemes.\n\n\\begin{table}\n\\label{table001} \\caption{System Evaluation Parameters} \\centering\n  \\begin{tabular}{lll}\n  \\hline\n   $B=62.5$ KHz          & $N_0=-180$ dB               & $\\mathcal{P}_{ADC}=7$ mw \\\\\n\n   $M_l=40$ dB           & $\\mathcal{P}_{Sy}=10$ mw    &  $\\mathcal{P}_{LNA}=9$ mw\\\\\n\n   $\\mathcal{L}_1=30$ dB & $\\mathcal{P}_{Filt}=2.5$ mw & $\\mathcal{P}_{ED}=3$ mw  \\\\\n\n   $\\eta=3.5$            & $\\mathcal{P}_{Filr}=2.5$ mw & $\\mathcal{P}_{IFA}=3$ mw \\\\\n\n   $\\Omega=1$            &  $T_N=1.4$ sec              & $T_{tr}=5~\\mu s$      \\\\\n\n  \\hline\n  \\end{tabular}\n\\end{table}\n\n\nIn order to estimate the computation energy of a specific channel\ncoding, we use the ARM7TDMI core which is the industry's most widely\nused 32-bit embedded RISC microprocessor for an accurate power\nsimulation \\cite{ARMTech2004}. For the energy and power\ncalculations, the relations\n$\\mathcal{E}_{operation}=\\dfrac{n_c}{n_o}\\times \\mathcal{E}_{Hz}$\nand $\\mathcal{E}_{block}=n_o \\times \\mathcal{E}_{operation}$ are\nused, where\n\\begin{itemize}\n\\item [] $n_o:$ number of operations per block.\n\\item [] $n_c:$ number of clock cycles on $n_o$ operations.\n\\item [] $\\mathcal{E}_{Hz}:$ total dynamic power consumption per\nHertz (W\/Hz) of a single calculation cycle.\n\\end{itemize}\nIn addition, for the purpose of comparative evaluation, we use some\nclassical BCH$(n,k,t)$ codes with $t$-error correction capability,\nand convolutional codes $(n,k,L)$ with the constraint length $L$.\nThese codes are widely utilized in IEEE standards \\cite{IEEE802.15,\nIEEE802.16}. We use hard-decision prior to decoding for the BCH and\nconvolutional codes. The main reason for using the hard-decision\nhere is to make a fair comparison to the LT code, since this code\ninvolved a hard-decision in ternary decoder.\n\n\n\\subsection{Optimal Configuration}\nAs a starting point, we obtain the coding gain of some practical BCH\nand convolutional codes with MFSK for different constellation size\n$M$ and given $P_b=(10^{-3},10^{-4})$ in Table IV. We observe from\nTable IV that there is a tradeoff between coding gain and decoder\ncomplexity for the BCH codes. In fact, achieving a higher coding\ngain for a given $M$, requires a more complex decoding process,\n(i.e., higher $t$) with more circuit power consumption. In addition,\nit is seen from Fig. \\ref{fig: BCH code_rate_coding_gain} that the coding gain of the BCH code is a monotonically\ndecreasing function of $M$ as expected. For a given $M$, the\nconvolutional codes with lower rates and higher constraint lengths\nachieve greater coding gains. In contrast to\n\\cite{Cui_GoldsmithITWC0905}, where the authors assume a fixed\nconvolutional coding gain for every value of $M$, it is observed\nthat the coding gain of convolutional coded MFSK is a monotonically\ndecreasing function of $M$. By comparing the results in Table II\nwith those in Table IV for BCH and convolutional codes, one observes\nthat LT codes outperform the other coding schemes in energy saving\nat comparable rates.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=BCH_Coding_Gain_vs_Code_Rate.eps,width=4.15in}}\n\\caption{Coding gain versus code rate of some practical BCH$(n,k,t)$ codes for $P_b=10^{-3}$ and\nM=2,4,8,16. } \\label{fig: BCH code_rate_coding_gain}\n\\end{figure}\n\n\n\\begin{table*}\n\\label{table01} \\caption{Coding gain ($dB$) of BCH and convolutional\ncoded MFSK over Rayleigh fading for BER$=(10^{-3},10^{-4})$.}\n\\centering\n  \\begin{tabular}{|l|c| c c c c c c|}\n  \\hline\n  BCH Code$(n,k,t)$  & $R^{BC}_c$ & M=2  & M=4  & M=8  &  M=16  &  M=32  &  M=64\n  \\\\\n  \\hline\n   BCH $(7,4,1)$   & 0.571 &   $(2.5,2.8)$   &   $(0.3,0.4)$    &    $(0.1,0.2)$  &  $(0.0,0.0)$   &  $(0.0,0.0)$   &  $(0.0,0.0)$   \\\\\n  \\hline\n   BCH $(15,11,1)$ & 0.733 &   $(1.4,1.6)$   &   $(0.2,0.3)$    &    $(0.0,0.0)$  &  $(0.0,0.0)$   &  $(0.0,0.0)$   &  $(0.0,0.0)$   \\\\\n   \\hline\n   BCH $(15,7,2)$  & 0.467 &  $(2.4,3.3)$   &   $(2.0,2.3)$    &    $(0.8,1.0)$  &  $(0.3,0.4)$   &  $(0.0,0.0)$   &  $(0.0,0.0)$   \\\\\n   \\hline\n   BCH $(15,5,3)$  & 0.333 &  $(4.1,4.6)$   &   $(2.7,2.9)$    &    $(2.0,2.1)$  &  $(1.5,1.60)$  &  $(0.7,0.8)$   &  $(0.2,0.2)$   \\\\\n   \\hline\n   BCH $(31,26,1)$ & 0.839 &  $(1.2,1.5)$   &   $(0.2,0.2)$    &    $(0.0,0.0)$  &  $(0.0,0.0)$   &  $(0.0,0.0)$   &  $(0.0,0.0)$   \\\\\n   \\hline\n   BCH $(31,21,2)$ & 0.677 &  $(2.3,2.9)$   &   $(1.7,2.0)$    &    $(0.7,0.8)$  &  $(0.2,0.2)$   &  $(0.0,0.0)$   &  $(0.0,0.0)$   \\\\\n   \\hline\n   BCH $(31,16,3)$ & 0.516 &  $(2.9,3.1)$   &   $(2.1,2.2)$    &    $(1.5,1.6)$  &  $(1.3,1.4)$   &  $(0.6,0.7)$   &  $(0.1,0.1)$   \\\\\n   \\hline\n   BCH $(31,11,5)$ & 0.355 &  $(4.1,4.4)$   &   $(3.5,4.2)$    &    $(2.2,2.3)$  &  $(2.0,2.1)$   &  $(1.8,2.0)$   &  $(1.1,1.3)$   \\\\\n   \\hline\n   BCH $(31,6,7)$  & 0.194 &  $(5.4,5.9)$   &   $(4.3,4.8)$    &    $(3.5,3.8)$  &  $(3.2,3.3)$   &  $(2.7,2.8)$   &  $(2.3,2.4)$   \\\\\n   \\hline\n   \\hline\n   Convolutional Code  & $R^{CC}_c$ & M=2  & M=4  & M=8  &  M=16  &  M=32  &  M=64\\\\\n  \\hline\n  trel$(6,[53~~75])$    & 0.500 &   $(3.8,4.6)$  &   $(2.7,3.1)$  &    $(2.1,2.3)$  &  $(1.8,2.0)$   &  $(1.4,1.5)$   &  $(1.4,1.4)$   \\\\\n  \\hline\n   trel$(7,[133~~171])$  & 0.500 &   $(4.0,4.7)$  &   $(3.0,3.5)$  &    $(2.2,2.4)$  &  $(1.8,2.0)$   &  $(1.5,1.6)$   &  $(1.4,1.5)$   \\\\\n  \\hline\n  trel$(7,[133~~165~~171])$  & 0.333 &   $(5.7,6.4)$  &   $(4.8,5.1)$  &    $(3.7,3.9)$  &  $(3.1,3.3)$   &  $(2.7,2.8)$   &  $(2.5,2.6)$   \\\\\n   \\hline\n   trel$([4~3],[4~5~17;7~4~2])$  & 0.667 &   $(2.2,2.6)$  &   $(1.5,1.7)$  &    $(0.9,1.1)$  &  $(0.6,0.6)$   &  $(0.5,0.5)$   &  $(0.5,0.5)$   \\\\\n   \\hline\n   trel$([5~4],[23~35~0;0~5~13])$  & 0.667 &   $(2.9,3.5)$  &   $(1.9,2.4)$  &    $(1.4,1.8)$  &  $(1.1,1.2)$   &  $(0.8,0.9)$   &  $(0.7,0.8)$   \\\\\n   \\hline\n   \\end{tabular}\n\\end{table*}\n\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=Coded_Uncoded.eps,width=3.9in}}\n\\caption{Total energy consumption of optimized coded and uncoded\nMFSK versus $d$ for $P_b=10^{-3}$.} \\label{fig: Coded_Uncoded}\n\\end{figure}\n\nFig. \\ref{fig: Coded_Uncoded} shows the total energy consumption\nversus distance $d$ for the optimized BCH, convolutional and LT\ncoded MFSK schemes, compared to the optimized uncoded MFSK for\n$P_b=10^{-3}$. The optimization is done over $M$ and the parameters\nof coding scheme. Simulation results show that for $d$ less than the\nthreshold level $d_T \\approx 40$ m, the total energy consumption of\noptimized uncoded MFSK is less than that of the coded MFSK schemes.\nHowever, the energy gap between LT coded and uncoded MFSK is\nnegligible compared to the other coded schemes as expected. For\n$d>d_T$, the LT coded MFSK scheme is more energy-efficient than\nuncoded and other coded MFSK schemes. Also, it is observed that the\nenergy gap between LT and convolutional coded MFSK increases when\nthe distance $d$ grows. This result comes from the high coding gain\ncapability of LT codes which confirms our analysis in Section\n\\ref{analysis_Ch4}. The threshold level $d_T$ (for LT code) or\n$d^{'}_T$ (for BCH and convolutional codes) are obtained when the\ntotal energy consumptions of coded and uncoded systems become equal.\nFor instance, using $\\mathcal{L}_d =M_ld^\\eta \\mathcal{L}_1$, and\nthe equality between (\\ref{energy_totFSK}) and\n(\\ref{energy_totalcodedFSK}) for uncoded and convolutional coded\nMFSK, we have\n\\begin{equation}\nd^{'}_T=\\left[\\hat{\\Upsilon}_c^{CC}\\dfrac{a_2\n(1-\\hat{R}_c^{CC})+N(\\mathcal{E}^{CC}_{enc}+\\mathcal{E}^{CC}_{dec})}{a_1(\\hat{\\Upsilon}_{c}^{CC}\\hat{R}_c^{CC}-1)}\\right]^{\\frac{1}{\\eta}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\notag a_1 &\\triangleq &(1+\\alpha) \\left[\\left(\n1-\\left(1-\\dfrac{2(M-1)}{M}P_b\\right)^{\\frac{1}{M-1}}\\right)^{-1}-2\n\\right]\\\\\n&\\times&\\dfrac{\\mathcal{L}_1 M_l}{\\log_2 M}\\dfrac{N N_0}{\\Omega},\n\\end{eqnarray}\n\\begin{equation}\na_2 \\triangleq (\\mathcal{P}_{c}-\\mathcal{P}_{Amp})\\dfrac{MN}{B\n\\log_2 M},\n\\end{equation}\n\\begin{equation}\n(\\hat{\\Upsilon}_c^{CC},\\hat{R}_c^{CC})=\n\\textrm{arg}~\\min_{\\Upsilon_c^{CC},R_c^{CC}}~\\mathcal{E}^{CC}_{N,c}.\n\\end{equation}\n\n It should be noted that the above threshold level imposes a\nconstraint on the design of the physical layer of some wireless\nsensor networking applications, in particular dynamic WSNs. To\nobtain more insight into this issue, let assume that the location of\nthe sensor node is changed every $T_d \\gg T_c$ time unit, where\n$T_c$ is the channel coherence time. For the moment, let us assume\nthat the sensor node aims to choose either a \\emph{fixed-rate coded}\nor an uncoded MFSK based on the distance between sensor and sink\nnodes. According to the results in Fig. \\ref{fig: Coded_Uncoded}, it\nis revealed that using fixed-rate channel coding is not energy\nefficient for short distance transmission (i.e., $d < d^{'}_T$),\nwhile for $d > d^{'}_T$, convolutional coded MFSK is more\nenergy-efficient than other schemes. For this configuration, the\nsensor node must have the capability of an adaptive coding scheme\nfor each distance $d$. However, as discussed previously, the LT\ncodes can adjust their rates for each channel condition and have\n(with a good approximation) minimum energy consumption for every\ndistance $d$. This indicates that LT codes can surpass the above\ndistance constraint for WSN applications with dynamic position\nsensor nodes over Rayleigh fading channels. This characteristic of\nLT codes results in reducing the complexity of the network design as\nwell. Of interest is the strong benefits of using LT coded MFSK\ncompared with the coded modulation schemes in\n\\cite{Cui_GoldsmithITWC0905, Chouhan_ITWC1009}. In contrast to\nclassical fixed-rate codes used in \\cite{Cui_GoldsmithITWC0905,\nChouhan_ITWC1009}, the LT codes can vary their block lengths to\nadapt to any channel condition in each distance $d$. Unlike\n\\cite{Cui_GoldsmithITWC0905} and \\cite{Chouhan_ITWC1009}, where the\nauthors consider fixed-rate codes over an AWGN channel model, we\nconsidered a Rayleigh fading channel which is a general model in\npractical WSNs. The simplicity and flexibility advantages of LT\ncodes with an MFSK scheme make them the preferable choice for\nwireless sensor networks, in particular for WSNs with dynamic\nposition sensor nodes.\n\n\\textbf{Remark 1:} As discussed previously, the proposed LT coded scheme\nbenefits in adjusting the coding parameters in each channel realization\nor equivalently each distance $d$ to minimize the total energy consumption.\nAs we will show shortly, this comes from a variable transmission time process\nwhich adaptively controls the power consumption of the proposed scheme.\nTo address this issue, we plot the active mode duration of the optimized\nLT coded modulation versus $d$ compared to those of the optimized uncoded\nand other coded modulation schemes in Fig. \\ref{fig: active}.\nFor this purpose, we use $T_{ac,c}=\\frac{T_{ac}}{R_c}$, where $T_{ac}=\\frac{MN}{B \\log_2 M}$\ndenotes the active mode duration of uncoded scheme. Simulation results show\nthat the optimum constellation size $M$ that minimizes the total energy\nconsumption for the aforementioned schemes and for every value of distance\n$d$ is $\\hat{M}=2$. It is revealed from Fig. \\ref{fig: active} that the\noptimized uncoded and convolutional coded MFSK display fixed values of\n$T_{ac}$ and $T_{ac,c}^{CC}$ for every value of $d$. In addition, we\ncan see that for $d < 60$ m and $d<110$ m, $T_{ac,c}^{LT}$ is less than\n$T_{ac,c}^{BC}$ and $T_{ac,c}^{CC}$, respectively. Using (\\ref{total_energy1})\nand Fig. \\ref{fig: Coded_Uncoded}, and noting that the total energy consumption of a certain scheme is proportional\nto the transmission time, it is concluded that the scheme with a lower transmission time\nis not necessarily more energy efficient than that of the scheme with greater $T_{ac}$.\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=T_ac_vs_d.eps,width=4.2in}}\n\\caption{Active mode duration of the optimized uncoded and coded MFSK schemes versus transmission distance $d$.} \\label{fig: active}\n\\end{figure}\n\n\n\\textbf{Remark 2:} To make a fair comparison between uncoded and coded schemes,\none would expect to assume a constant active mode duration for all the communication\nschemes. According to $T_{ac,c}=\\frac{T_{ac}}{R_c}$ with $T_{ac}=\\frac{MN}{B \\log_2 M}$,\nthis is achieved by adjusting the modulation order $M$. However, it is\nworth mentioning that the assumption of the same transmission times for all the\nschemes is not a realistic assumption in feasible WSNs where ``autonomous'' sensor\ndevices are powered by limited-lifetime batteries. More precisely, under the\nassumption of the same $T_{ac}$ (or $T_{ac,c}$), the wireless sensor network\nneeds an extra hardware to adjust the constellation size $M$ (in each distance $d$),\nwhich imposes more cost, complexity and power consumption in the network. While,\nour scheme with variable transmission times surpasses the above adaptive\nmodulation constraint using the fact that the optimum constellation size $M$ which\nminimizes the total energy consumption (in each distance $d$) is $\\hat{M}=2$.\n\n\n\n\\section{Conclusion}\\label{conclusion_Ch6}\nIn this paper, we analyzed the energy efficiency of LT coded MFSK in\na proactive WSN over Rayleigh fading channels with path-loss. It was\nshown that the energy efficiency of LT codes is similar to that of\nuncoded MFSK scheme for $d<d_T$, while for $d > d_T$, LT coded MFSK\noutperforms other uncoded and coded schemes, from the energy\nefficiency point of view. This result follows from the flexibility\nof the LT code to adjust its rate and the corresponding LT coding\ngain to suit instantaneous channel conditions for any transmission\ndistance $d$. This rate flexibility offers strong benefits in using\nLT codes in practical WSNs with dynamic distance and position\nsensors. In such systems and for every value of distance $d$, LT\ncodes can adjust their rates to achieve a certain BER with low\nenergy consumption. The importance of our scheme is that it avoids\nsome of the problems inherent in adaptive coding or Incremental\nRedundancy (IR) systems (channel feedback, large buffers, or\nmultiple decodings), as well as the coding design challenge for\nfixed-rate codes used in WSNs with dynamic position sensor nodes.\nThe simplicity and flexibility advantages of LT codes make the LT\ncode with MFSK modulation can be considered as a \\emph{Green\nModulation\/Coding} (GMC) scheme in dynamic WSNs.\n\nIn this paper, we have shown the significant benefit of rateless codes\n(with focus on the optimized LT codes) for sensor networks over the more\ntraditional fixed-rate codes. Our future research involves selecting the\nbest rate-adaptive code among LT, Raptor, punctured LDPC, and punctured\nTurbo codes; in particular, we are interested to study the performance\nof the Raptor code which has a linear-time encoding versus the non-linear\ncost of the LT code. For this study, a particularly nice feature of the\nLT code is to rapidly optimize the code using a modified EXIT chart\nstrategy introduced in \\cite{Etesami_ITIT0506}.\n\n\n\\appendices\n\n\\section{Asymptotic LT Code Rate}\nThe proof of the remark is straightforward using the notation of\n\\cite{Etesami_ITIT0506} and the bipartite graph concepts in graph\ntheory. Obviously, the output-node degree distribution\n$\\mathcal{O}(x)$ induces a distribution on the input nodes in the\nbipartite graph. Thus, in the asymptotic case of $k \\to \\infty$, we\nhave the input-node degree distribution defined as $\\mathcal{I}(x)\n\\triangleq \\sum_{i=1}^{\\infty}\\nu_i x^i$, where $\\nu_i$ denotes the\nprobability that an input node has a degree $i$. In this case, the\naverage degree of the input and output nodes are computed as\n$\\frac{d \\mathcal{I}(x)}{d x}\\big|_{x=1} \\triangleq\n\\mathcal{I}_{ave}$ and $\\frac{d \\mathcal{O}(x)}{d x}\\big|_{x=1}\n\\triangleq \\mathcal{O}_{ave}$, respectively. Thus, the number of\nedges exiting the input nodes of the bipartite graph, in the\nasymptotic case of $k \\to \\infty$, is $k\\mathcal{I}_{ave}$, which\nmust be equal to $n\\mathcal{O}_{ave}$, the number of edges entering\nthe output nodes in the graph. As a results, the asymptotic LT code\nrate is obtained as $R^{LT}_c= \\frac{k}{n} \\approx\n\\frac{\\mathcal{O}_{ave}}{\\mathcal{I}_{ave}}$, which is a\ndeterministic value for given $\\mathcal{O}(x)$ and $\\mathcal{I}(x)$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn \\cite{Wi}, Wilson generalized the work of Church-Ellenberg-Farb \\cite{CEF} on FI-modules to the setting of \\(\\FI_\\mathcal{W}\\)-modules, proving strong results about these and finding a wide range of interesting applications. In \\cite{SS2}, Sam-Snowden substantially generalized Wilson's work to that of \\(\\FI_G\\)-modules, for \\(G\\) a virtually polycyclic group, where Wilson's \\(\\FI_{\\text{BC}} = \\FI_{\\mathbb{Z}\/2\\mathbb{Z}}\\). Recall that a group is polycyclic if it has a subnormal series with cyclic factors, and virtually polycyclic if it has a polycyclic subgroup of finite index.\n\nSam-Snowden proved the Noetherian property for \\(\\FI_G\\), and Gan-Li \\cite{GL3} extended this to a proof of actual representation stability in the sense of \\cite{CF}. However, few examples of \\(\\FI_G\\)-modules have been discussed. Here, we extend Wilson's work to exhibit a wide class of examples of \\(\\FI_G\\)-modules. For any virtually polycyclic group \\(G\\), we give applications to the following sequences of representations of \\(W_n = G^n \\rtimes S_n\\) (see \\S3.1 and \\S7 for definitions):\n\\begin{enumerate}\n\\item The homology \\(H_i(\\Conf_n^G(M); \\mathbb{C})\\) of the orbit configuration space \\(\\Conf_n^G(M)\\) associated to the action of \\(G\\) on an open manifold \\(M\\), and the cohomology \\(H^i(\\Conf_n^G(M); \\mathbb{C})\\) in the case where \\(M\\) need not be open but \\(G\\) is finite.\n\\item The rational homotopy groups \\(\\pi_i(\\Conf_n(M))\\) of an open manifold \\(M\\) of dimension \\(\\ge 3\\) with \\(G = \\pi_1(M)\\) virtually polycyclic, or the dual rational homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Q})\\) where \\(M\\) need not be open but \\(G\\) is finite.\n\\item The cohomology \\(H^i(\\FR(G^{\\ast n}); \\mathbb{C})\\) of the Fouxe-Rabinovitch group of \\(G\\), where each \\(H^i(G; \\mathbb{Q})\\) is finite-dimensional.\n\\end{enumerate}\nFor the specific case \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\), the resulting automorphism groups \\(W_n = \\left(\\mathbb{Z}\/d\\mathbb{Z}\\right)^n \\wr S_n\\) are the so-called main series of complex reflection groups. These directly generalize the Weyl groups of type \\(BC_n\\) of Wilson's paper, for which \\(d = 2\\). We give applications to the following sequence of representations of \\(W_n\\) (see \\S4.1, \\S4.2, \\S4.3 for definitions):\n\\begin{enumerate}\n\\setcounter{enumi}{3}\n\\item The cohomology \\(H^i(P(d,n); \\mathbb{C})\\) of the pure monomial braid group.\n\\item The graded pieces \\(\\gr P(d,n)^i\\) of the associated graded Lie algebra of the pure monomial braid group.\n\\item The graded pieces \\(\\mathcal{C}_J^{(r)}(n)\\) of the complex-reflection diagonal coinvariant algebra.\n\\end{enumerate}\nOur work implies the following results about these sequences. Recall that the irreducible representations of \\(W_n\\) are parameterized by the partition-valued functions \\(\\underline \\lambda\\) on the irreducible representations of \\(G\\) with \\(\\|\\underline \\lambda\\| = n\\). Denote the  irreducible representation of \\(W_n\\) associated to \\(\\underline \\lambda\\) by \\(L(\\underline \\lambda)\\). Given partitions \\(\\lambda_1, \\dots, \\lambda_k\\) with \\(|\\lambda_1| + \\dots + |\\lambda_k| = n\\), and irreducible representations \\(\\chi_1, \\dots, \\chi_k\\) of \\(G\\), we will write \\(\\underline\\lambda = ( {\\lambda_1}_{\\chi_1}, \\dots, {\\lambda_2}_{\\chi_2})\\) for the partition-valued function \\(\\underline\\lambda\\) with \\(\\underline\\lambda(\\chi_i) = \\lambda_i\\).\n\nLet \\(c(G)\\) be the set of conjugacy classes of \\(G\\). Define a \\emph{character polynomial} to be a polynomial in \\(c(G)\\)-labeled cycle-counting functions. This is a class function on \\(W_n\\). (We will define this and other terminology more precisely in \\S2.2).\n\\begin{thm}[\\bfseries Polynomiality of characters and representation stability]\nSuppose \\(G\\) is finite, and let \\(\\{V_n\\}\\) be any of the sequences 1-6 above. Then:\\begin{enumerate}\n\\item The characters of \\(V_n\\) are given by a single character polynomial for all \\(n \\gg 0\\).\n\\item The multiplicity of each irreducible \\(W_n\\)-representation in \\(V_n\\) is eventually independent of \\(n\\), and \\(\\dim V_n\\) is eventually polynomial in \\(n\\). \n\nMoreover, for sequences 3 and 4, we can explicity bound when this stabilization occurs.\n\\end{enumerate}\n\\end{thm}\nFor example, for sequence 4, once \\(n \\ge 2\\) we obtain:\n\\[H^1(P(d,n); \\mathbb{C}) = L((n)_{\\chi_0}) \\oplus L((n-1,1)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} L((n-2)_{\\chi_0}, (2)_\\chi) \\]\nIn particular, our results on \\(H^i(\\FR(\\Gamma^{\\ast n}); \\mathbb{C})\\) answer a question of Wilson, posed in \\cite[\\S 7.1.1]{Wi}, of how to generalize her Thm 7.3 to a general Fouxe-Rabinovitch group. Our results on \\(\\pi_i(\\Conf_n(M))\\) both confirm and correct a sketch given by Kupers-Miller \\cite{KM} about the homotopy groups of configuration spaces of non-simply-connected manifolds, and answer their question about obtaining a form of representation stability for finite \\(G\\).\n\\subsection*{Infinite \\(G\\): \\(K_0\\)-stability and finite presentation degree}\nWhen \\(G\\) is infinite virtually polycyclic, we still obtain a form of representation stability. For a ring \\(R\\), recall that the Grothendieck group \\(G_0(R)\\) is the free abelian group on finitely-generated \\(R\\)-modules, modulo the relations that the center of a short exact sequence is equal to the sum of the left and right terms. \\(K_0(R)\\) is defined similarly, but only for projective \\(R\\)-modules; these two groups coincide when \\(R = k[G]\\) for \\(G\\) virtually polycyclic. Say that a sequence \\(V_n\\) of \\(W_n\\)-representations satisfies \\emph{\\(K_0\\)-stability} if the decomposition of \\([V_n]\\) in the \\emph{Grothendieck group} \\(K_0(k[W_n])\\) is eventually independent of \\(n\\). We apply this notion to 1-3 above, and to the following two more examples:\n\\begin{enumerate} \\setcounter{enumi}{6}\n\\item The homology \\(H_i(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) of the affine pure braid group of type \\(\\widetilde{A}_{n-1}\\), for which \\(G = \\mathbb{Z}\\).\n\\item The homology \\(H_i(P_{\\widetilde{C}_{n}}; \\mathbb{Q})\\) of the affine pure braid group of type \\(\\widetilde{C}_{n}\\), for which \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\).\n\\end{enumerate}\n\\begin{thm}[\\bfseries \\(K_0\\)-stability]\nLet \\(\\{V_n\\}\\) be any of the sequences above (1-3,7,8) for which \\(G\\) is infinite virtually polycyclic. Then \\(V_n\\) satisfies \\(K_0\\)-stability.\n\\end{thm}\nFor example, if \\(U = k[\\mathbb{Z}] = \\Ind_{\\mathbb{Z}\/2}^{\\mathbb{Z} \\rtimes \\mathbb{Z}\/2} k\\), then in the notation of \\S2.3,\n\\[ [H_1(P_{\\widetilde{C}_{n}}; \\mathbb{Q})] = [L((n)_U)],\\;\\; \\forall n \\ge 1. \\]\nNote that in general we certainly lose information in passing to \\(K_0\\). For instance, in the above example, we actually explicitly determine in \\S6.(\\ref{affC}) that \\(H_1(P_{\\widetilde{C}_{n}}; \\mathbb{Q})\\) is the direct sum of three induced representations, two of which are killed in passing to \\(K_0\\), and the third giving the term \\([L((n)_U)]\\) above.\n\nThis is the first representation stability result obtained for infinite discrete groups (recall that \\cite{CF} obtained representation stability results for Lie groups, whose representation theory is much more tractable). \n\nFurthermore, for a totally general group \\(G\\), we still obtain a weak form of finite generation. Say that an \\(\\FI_G\\)-module is \\emph{presented in finite degree} if it is finitely-generated and if the relations for the generators also live in finite degree. Ramos \\cite{Ra2} and Li \\cite{Li} have proven that the category of \\(\\FI_G\\)-modules that are presented in finite degree is abelian. This allows us to carry over arguments about spectral sequences to conclude that cohomology is generated in finite degree. In particular, we have:\n\\begin{thm}[\\bfseries Finite presentation degree]\nThe cohomology \\(H^i(\\Conf^G_n(M); \\mathbb{Q})\\) and the dual rational homotopy groups \\(\\Hom(\\pi_i( \\Conf_n(M)); \\mathbb{Q})\\) are \\(\\FI_G\\)-modules presented in finite degree.\n\\end{thm}\n\n\\subsection*{Application: arithmetic statistics for Gauss sums}\nIn the sequel \\cite{Ca2}, we apply the theory of representation stability for \\(\\FI_G\\)-modules to obtain results about weighted point-counts for polynomials over finite fields. For example, we are able to obtain results such as the following. Let \\(\\chi\\) be a character of \\(\\mathbb{Z}\/(q-1)\\). Define the character polynomial \\(X_i^\\chi = \\sum_{g \\in G} \\chi(g) X^g_i\\). Then\n\\begin{gather} \\begin{aligned} \\label{X2_pcount}\n\\lim_{n \\to \\infty} &\\sum_{f \\in \\UConf_n(\\mathbb{F}_q^*)} \\sum_{\\deg(p) = 2} \\chi(\\text{root}(p))  \\\\\n&= \\sum_i (-1)^i \\frac{\\langle X_2^{\\overline \\chi}, H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*); \\mathbb{Q}(\\zeta_{q-1}))\\rangle}{q^i} = \\frac{1}{2q} + \\frac{2}{q^2} + \\cdots\n\\end{aligned} \\end{gather}\nThat is, the average value of the Gauss sum obtained by applying \\(\\chi\\) to the \\emph{quadratic} factors of \\(f\\), across all \\(f \\in \\UConf_n(\\mathbb{F}^*_q)\\), is equal to the series on the right obtained by looking at the inner product of the character polynomial \\(X_2^{\\overline \\chi}\\) with \\(H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*); \\mathbb{Q}(\\zeta_{q-1}))\\). The fact that the sum on the right is independent of \\(n\\) for large \\(n\\) is precisely a consequence of representation stability for the \\(\\FI_G\\)-module \\(H^i(\\Conf^{\\mathbb{Z}\/(q-1)}(\\mathbb{C}^*))\\).\n\n\\subsection*{Acknowledgements}\nI would like to thank Nir Gadish, Weiyan Chen, Jesse Wolfson, John Wiltshire-Gordon, Eric Ramos, and Jeremy Miller for helpful conversations. Also, I am very grateful to my advisor, Benson Farb, for all his guidance throughout the process of working on, writing, and revising my first paper.\n\n\\section{\\(\\FI_G\\)-modules and their properties}\nIf \\(G\\) is a group, and \\(R\\) and \\(S\\) sets, define a \\emph{\\(G\\)-map} \\((a, (g_i)): R \\to S\\) to be a pair \\(a: R \\to S\\) and \\((g_i) \\in G^R\\). If \\((b, (h_j)): S \\to T\\) is another \\(G\\)-map, their composition is \\((b \\circ a, (g_i \\cdot h_{a(i)}))\\). Let \\(\\FI_G\\) be the category with objects finite sets and morphisms \\(G\\)-maps with the function \\(a\\) injective. This is clearly equivalent to the full subcategory with objects the sets \\([n] = \\{1, \\dots, n\\}\\). Note that the automorphism group of \\([n]\\) is\n\\[W_n := G \\wr S_n = G^n \\rtimes S_n\\]\nAn \\emph{\\(\\FI_G\\)-module over \\(k\\)} is just a functor \\(V: \\FI_G \\to k\\Mod\\); when \\(k\\) is clear, we simply write \\(\\FI_G\\)-modules. These form a category, called \\(\\FI_G\\Mod\\). Thus an \\(\\FI_G\\)-module \\(V\\) is a sequence of \\(W_n\\)-representations \\(V_n\\), with maps \\(V_n \\to V_{n+1}\\) satisfying certain coherency conditions. Say that \\(V\\) is \\emph{finitely generated} if there is a finite set of elements \\(v_1, \\dots, v_n \\in V\\) such that the smallest \\(\\FI_G\\)-submodule containing the \\(v_i\\) is all of \\(V\\).\n\nFor \\(m \\ge 0\\), define the ``free'' \\(\\FI_G\\)-module \\(M(m)\\) by setting\n\\[M(m)_n = \\begin{cases} 0 & n < m \\\\ k[\\Hom_{\\FI_G}([m], [n])] & n \\ge m \\end{cases}\\]\nRecall that for an \\(\\FI_G\\)-module \\(V\\) and an element \\(v \\in V_m\\), we can also characterize the submodule generated by \\(v\\) as the image of the map\n\\[ M(m) \\to V, \\;\\; f \\in \\Hom_{\\FI_G}([m], [n]) \\mapsto f_* v \\]\nWe can thus characterize finitely generated \\(\\FI_G\\)-modules as those \\(V\\) that admit a surjection \\(\\bigoplus_{i=1}^N M(m_i) \\twoheadrightarrow V\\).\n\n\\subsection{Noetherianity and representation stability}\n\nRecall that a group is \\emph{polycyclic} if it has a composition series with cyclic factors, and is \\emph{virtually polycyclic} if it has a polycyclic subgroup of finite index. Virtually polycyclic groups are of interest, among other things, because they are the only known groups to have Noetherian group rings, and are conjectured to be the only such groups. In \\cite[Cor 1.2.2]{SS2}, Sam-Snowden proved that if \\(G\\) is a virtually polycyclic group, then \\(\\FI_G\\Mod\\) is Noetherian over any Noetherian ring \\(k\\). That is, they proved that any finitely generated \\(\\FI_G\\)-module has all its submodules finitely generated. The crucial property used was that the group ring \\(k[G]\\) is Noetherian.\n\nFor FI-modules, the most important consequence of finite generation is representation stability, and for \\(G\\) finite, Gan-Li proved in \\cite[Thm 1.10]{GL3} that this holds for \\(\\FI_G\\) as well. Technically there are three parts to representation stability according to the definition given in \\cite{CF}, but the first two parts (``surjectivity'' and ``injectivity'') follow straightforwardly from the definition of being finitely generated. It is the third part, ``multiplicity stability'', that is really the most interesting, and which we will now describe.\n\nTake \\(G\\) finite and \\(k\\) a \\emph{splitting field} of characteristic 0 for \\(G\\), that is, a field over which all its irreducible representations over \\(\\mathbb{C}\\) are defined. Let \\(\\Irr(G)\\) denote the set of isomorphism classes of irreducible representations of \\(G\\). Let \\(\\underline \\lambda\\) be a partition-valued function on \\(\\Irr(G)\\). Put \\(|\\underline\\lambda| = (|\\underline\\lambda(\\chi_1)|, \\dots, |\\underline\\lambda(\\chi_r)|)\\), and \\(\\|\\underline\\lambda\\|\\) for the norm of this partition. Then if \\(\\|\\underline\\lambda\\| = n\\), there is an associated irreducible representation of \\(W_n\\):\n\\[ L(\\underline \\lambda) = \\Ind^{W_n}_{W_{|\\underline\\lambda|}} \\left(M_1^{\\otimes \\underline\\lambda(\\chi_1)} \\otimes E(\\underline\\lambda(\\chi_1))\\right) \\otimes \\cdots \\otimes \\left(M_r^{\\otimes \\underline\\lambda(\\chi_r)} \\otimes E(\\underline\\lambda(\\chi_1))\\right) \\]\nwhere \\(W_\\mu = W_{\\mu(1)} \\times \\cdots W_{\\mu(l)}\\),\nand that these comprise all the irreducible representations of \\(W_n\\) up to isomorphism. Extend \\(\\underline{\\lambda}\\) to \\(n \\ge m + \\underline{\\lambda}(\\chi_0)_1\\) as follows:\n\\[\n\\underline{\\lambda}[n](\\chi) = \\begin{cases} \\left(n - |\\underline{\\lambda}|, \\underline{\\lambda}(\\chi_0)\\right) & \\text{if } \\chi = \\chi_0 \\\\ \\underline{\\lambda}(\\chi) & \\text{otherwise} \\end{cases}\n\\]\nWriting \\(L(\\underline{\\lambda})_n\\) for the irreducible representation corresponding to \\(\\underline{\\lambda}[n]\\), multiplicity stability for an \\(\\FI_G\\)-module \\(V\\) says that the decomposition into irreducibles has multiplicities independent of \\(n\\) for large \\(n\\):\n\\[\nV_n = \\bigoplus_{\\underline{\\lambda}} L(\\underline{\\lambda})_n^{\\oplus c(\\underline{\\lambda})} \\,\\text{ for all } n \\ge N\n\\]\nwhere we call \\(N\\) the \\emph{stability degree} of \\(V\\). In particular, when \\(G\\) is trivial, we recover (uniform) multiplicity stability for FI-modules in the sense of \\cite[Defn 2.6]{CF}, and when \\(G = \\mathbb{Z}\/2\\mathbb{Z}\\), we recover (uniform) multiplicity stability for \\(\\FI_{\\text{BC}}\\)-modules in the sense of \\cite[Defn 2.6]{Wi}.\n\n\\subsection{Projective resolutions and character polynomials}\nFor \\(G\\) finite, \\cite{SS2} and \\cite{GL3} actually obtain a deeper structural result that implies representation stability, and this result has other important consequences for us. To state it, define a \\emph{torsion} \\(\\FI_G\\)-module to be one with \\(V_n \\ne 0\\) for only finitely many \\(n\\), and say that an \\(\\FI_G\\)-module is projective if it is a projective object in the category \\(\\FI_G\\Mod\\). Gan-Li's result can then be stated as follows.\n\n\\begin{prop}[{\\cite[Thm 1.6]{GL3}}]\nFor any finitely generated \\(\\FI_G\\)-module \\(V\\), with \\(G\\) finite, there is a finite resolution of \\(\\FI_G\\) modules\n\\[0 \\to V \\to T^1 \\oplus P^1 \\to T^2 \\oplus P^2 \\to \\dots \\to T^n \\oplus P^k \\to 0\\]\nwith each \\(T^i\\) torsion and each \\(P^i\\) projective.\n\\end{prop}\n\nIn particular, this resolution's existence means that for \\(n \\gg 0\\) there is a resolution of \\(W_n\\)-representations\n\\begin{equation} \\label{fin_res}\n0 \\to V_n \\to P^1_n \\to \\dots \\to P^k_n \\to 0\n\\end{equation}\n\nThis are powerful because, as in \\cite{CEF}, we have strong control over the structure of projective \\(\\FI_G\\)-modules. Namely, let \\(\\Res: \\FI_G \\to W_i\\) be the restriction to a single group. This functor has a left adjoint \\(\\Ind^{\\FI_G}: W_i \\to \\FI_G\\) given by\n\\[\\Ind^{\\FI_G}(V)_{i+j} = \\Ind_{W_i \\times W_j}^{W_{i+j}} V \\boxtimes k\\]\nThen \\(\\Ind^{\\FI_G}(V)\\) is a projective \\(\\FI_G\\)-module whenever \\(V\\) is a projective \\(W_i\\)-module, and tom Dieck \\cite[Prop 11.18]{tomD} proved that any projective \\(\\FI_G\\)-module is of this form (in fact, for a large class of category representations). Following Ramos \\cite{Ra1}, define a \\emph{relatively projective} module to be a direct sum of any such induced modules, even if the \\(W_i\\) are not projective \\(W_n\\)-representations. Finitely-generated (relatively) projective \\(\\FI_G\\)-modules thus have a compact description, as direct sums of induced modules from a finite list of \\(W_n\\)-representations, even for infinite \\(G\\).\n\nIn fact, Ramos \\cite{Ra1} has found an effective bound for when \\(n\\) occurs in (\\ref{fin_res}). To describe it, we need to refine our notion of finite generation. As we said, finitely generated modules are those that admit a surjection \\(\\bigoplus_{i=1}^N M(d_i) \\twoheadrightarrow V\\). Say that such a \\(V\\) is \\emph{generated in degree} \\(\\le d = \\max_i \\{d_i\\}\\). Next, suppose there is an exact sequence\n\\[ 0 \\to K \\to M \\to V \\to 0\\]\nwith \\(M\\) relatively projective and \\(K\\) generated in degree \\(\\le r\\). Then say that \\(V\\) is has \\emph{related in degree} \\(\\le r\\).\n\nRamos \\cite[Thm C]{Ra1} then says that if \\(V\\) is a finitely-generated \\(\\FI_G\\)-module generated in degree \\(d\\) and related in degree \\(r\\), then the above resolution (\\ref{fin_res}) holds whenever \\(n \\ge r + \\min\\{r, d\\}\\). Notice that if \\(V\\) is already relatively projective, then \\(r = 0\\) and this is sharp.\n\nIn particular, when \\(G\\) is finite, the resolution (\\ref{fin_res}) implies representation stability: as Gan-Li verify in \\cite[Thm 1.10]{GL3}, the individual projective modules \\(\\Ind^{\\FI_G}(V_m)\\) satisfy representation stability, with stability degree \\(\\le 2m\\). By semisimplicity of each \\(k[G_n]\\), \\(V\\) therefore satisfies representation stability as well. Furthermore, this resolution provides a quick proof of the existence of character polynomials for \\(\\FI_G\\), as follows.\n\nRecall that for a \\emph{character polynomial} for \\(S_n\\) is an element of \\(\\mathbb{Q}[X_1, X_2, \\dots]\\), which we think of as a class function on \\(S_n\\), where \\(X_i\\) counts the number of \\(i\\)-cycles of a permutation. \\cite[Thm 3.3.4]{CEF} prove that any finitely generated \\(\\FI\\)-module is eventually given by a single character polynomial. We generalize this and Wilson's \\cite[Thm 5.15]{Wi} result for \\(G = \\mathbb{Z}\/2\\) as follows.\n\\begin{thm}[\\bfseries Character polynomials for \\(\\FI_G\\)]\n\\thlabel{charpoly}\nLet \\(G\\) be a finite group and \\(k\\) a splitting field for \\(G\\) of characteristic 0. If \\(V\\) is a finitely generated \\(\\FI_G\\)-module over \\(k\\), generated in degree \\(m\\) and related in degree \\(r\\), then there is a polynomial\n\\[P_V \\in k\\left[\\{X_i^C \\mid i \\ge 1, C \\text{ is a conjugacy class of } G\\}\\right]\\]\nof degree \\(\\le m\\), where \\(X_i^C\\) is the class function counting the number of \\(C\\)-labeled \\(i\\)-cycles in the conjugacy class of \\(g\\), so that for all \\(n \\ge r + \\min(m,r)\\)\n\\[\\chi_{V_n}(g) = P_V(g).\\]\n\\end{thm}\n\\begin{proof}\nIf \\(V\\) is a representation of \\(W_m\\), we can explicitly compute the character of the projective \\(\\FI_G\\)-module \\(\\Ind^{\\FI_G}(V)\\). This calculation is done in \\cite[Lem 5.14]{Wi} for \\(G = \\mathbb{Z}\/2\\), and her proof applies essentially verbatim, but enough small details are different that it is easier to just give the adapted proof than to describe the necessary changes.\n\nThe character of the induced representation \\(\\Ind^{\\FI_G}(V)_n = \\Ind_{W_m \\times W_{n-m}}^{W_n} V \\boxtimes k\\) is\n\\begin{align*}\n\\chi_{\\Ind^{\\FI_G}(V)_n}(w) &= \\sum_{\\substack{\\{\\text{cosets } C\\, \\mid\\, w \\cdot C = C\\} \\\\ \\text{any } s \\in C}} \\chi_{V \\boxtimes k}(s^{-1} w s) \\\\\n&= \\sum_{\\substack{\\{\\text{cosets } C\\, \\mid\\, w \\cdot C = C\\} \\\\ \\text{any } s \\in C}} \\chi_V(p_m(s^{-1} w s))\n\\end{align*}\nwhere \\(p_m: W_m \\times W_{n-m} \\to W_m\\) is the projection, and where the sum is over all cosets in \\(W_n \/ (W_m \\times W_{n-m})\\) that are stabilized by \\(w\\), equivalently, those cosets \\(C\\) such that \\(s^{-1}ws \\in W_m \\times W_{n-m}\\) for any \\(s \\in C\\).\n\nAn element \\(w \\in W_n\\) can be conjugated in \\(W_m \\times W_{n-m}\\) precisely when its \\(c(G)\\)-labeled cycles can be split into a set of cycles of total length \\(m\\), and a set of cycles of total length \\((n-m)\\). If we fixed a labeled partition \\(\\underline \\lambda\\) of \\(m\\), then the cycles of \\(w\\) can be factored into an element \\(w_m\\) of labeled cycle type \\(\\underline \\lambda\\) and its complement \\(w_{n-m}\\) in the following number of ways (possibly 0):\n\\[\n\\prod_{C \\in c(G)} \\binom{X^C_1}{n_1(\\underline{\\lambda}(C))} \\binom{X^C_2}{n_2(\\underline{\\lambda}(C))} \\cdots \\binom{X^C_m}{n_m(\\underline{\\lambda}(C))}\n\\]\nwhere \\(n_r(\\mu)\\) is the number of \\(r\\)'s in \\(\\mu\\). Each such factorization of \\(w\\) corresponds to a coset \\(C \\in W_n \/ (W_m \\times W_{n-m})\\) that is stabilized by \\(w\\). For any representative \\(s \\in C\\), \\(p_m(s^{-1} w s)\\) has labeled cycle type \\(\\underline \\lambda\\). So we conclude that\n\\[\n\\chi_{\\Ind^{\\FI_G}(V)_n} = \\sum_{|\\underline{\\lambda}| = m} \\chi_V(\\underline{\\lambda}) \\prod_C \\binom{X^C_1}{n_1(\\underline{\\lambda}(C))} \\binom{X^C_2}{n_2(\\underline{\\lambda}(C))} \\cdots \\binom{X^C_m}{n_m(\\underline{\\lambda}(C))}\n\\]\nwhere the left-hand-side is manifestly a fixed character polynomial independent of \\(n\\). Notice that this character polynomial has degree \\(m\\). The general result therefore again follows from (\\ref{fin_res}) and Ramos \\cite[Thm C]{Ra1} by semisimplicity.\n\\end{proof}\n\nIn particular, by taking \\(g = e\\) in \\thref{charpoly}, we see that \\(\\dim V_n\\) is given by a single polynomial for \\(n \\ge r + \\min(m,r)\\), which is \\cite[Thm D]{Ra1}.\n\n\\subsection{Virtually polycyclic \\(G\\) and \\(K_0\\)-stability}\nThe above results only apply when \\(G\\) is finite, so that the group ring \\(k[G]\\) is semisimple. We would like to have a good analogue of representation stability for infinite virtually polycyclic \\(G\\). The answer is to pass to the Grothendieck group, where we essentially impose semisimplicity by fiat.\n\nRecall that, for a ring \\(R\\), the Grothendieck group \\(G_0(R)\\) is defined to be the free abelian group generated by isomorphism classes \\([M]\\) of finitely-generated \\(R\\)-modules \\(M\\), modulo the relation that for any short exact sequence\n\\[ 0 \\to M \\to N \\to M' \\to 0,\\;\\text{ we have }\\;[N] = [M] + [M']\\]\nThe Grothendieck group \\(K_0(R)\\) is defined similarly, except that we only take finitely-generated \\emph{projective} \\(R\\)-modules. The map \\(K_0(R) \\to G_0(R)\\), \\([P] \\to [P]\\) is well-defined and is called the \\emph{Cartan map}.\n\nHere we will be taking \\(R = k[W_n]\\), the group ring of \\(W_n = G \\wr S_n\\) with \\(G\\) a virtually polycyclic group. Since \\(W_n\\) is itself virtually polcyclic, it is then known that the Cartan map is an isomorphism, so the two notions of Grothendieck group coincide. We will use the notation \\(K_0(W_n) := K_0(k[W_n]) \\cong G_0(k[W_n])\\), as much to avoid clashing with the group \\(G\\), but we abuse notation by writing \\([M] \\in K_0(W_n)\\) for any finitely-generated \\(W_n\\)-module \\(M\\), which makes sense since the Cartan map is an isomorphism.\n\nGrothendieck groups of virtually polycyclic groups have been much studied and are well understood; for a good overview, see \\cite[Ch. 8]{Pa}. For one thing, such groups are always torsion-free and finitely generated, and thus isomorphic to \\(\\mathbb{Z}^r\\). One powerful result is \\emph{Moody's induction theorem} \\cite{Mo}, which says that for \\(G\\) virtually polycyclic, \\(K_0(k[G])\\) is generated by induced modules of the finite subgroups of \\(G\\). This means that, at the level of \\(K_0\\), the representation theory of \\(G \\wr S_n\\) for \\(G\\) virtually polycyclic works in the same way as for \\(G\\) finite that was described in \\S2.1, as follows. \n\nThe monoid \\(K^+_0(G)\\) of \\emph{positive} classes---that is, those actually represented by honest, and not just virtual, representations---is isomorphic to \\(\\mathbb{N}^r\\), and thus has a unique basis \\(X = \\{\\chi_1, \\dots, \\chi_r\\}\\). Pick representative \\(G\\)-modules \\(M_1, \\dots, M_r\\), so that \\([M_i] = \\chi_i\\). If \\(\\underline \\lambda\\) is a partition-valued function on \\(X\\), let \\(L(\\underline \\lambda)\\) be the associated representation of \\(W_n\\) defined in \\S2.1, using the \\(M_i\\) as the representations associated to \\(\\chi_i\\). Of course, this is only defined up to a choice of \\(\\{M_i\\}\\), but it does give a well-defined element \\([L(\\underline\\lambda)] \\in K_0(G)\\). Moody's induction theorem then has the following consequence:\n\n\\begin{prop}\nThe collection \\(\\{[L(\\underline \\lambda)] : \\|\\underline\\lambda\\| = n\\}\\) forms a basis for \\(K_0(W_n)\\).\n\\end{prop}\n\\begin{proof}\nThe fact that the \\([L(\\underline\\lambda)\\)]'s span \\(K_0(W_n)\\) is the actual consequence of Moody's induction theorem. For linear independence---that is, injectivity of the map \\(\\mathbb{Z}\\langle [L(\\underline \\lambda)] \\rangle_{\\underline \\lambda} \\to K_0(W_n)\\)---see e.g. \\cite[Exer 35.3]{Pa}.\n\\end{proof}\n\nThe reason \\(K_0\\) is relevant to us is because of the following generalization of the resolution (\\ref{fin_res}) to virtually polycyclic \\(G\\), due to Nagpal-Snowden:\n\\begin{thm}[Nagpal-Snowden \\cite{NS}]\nIf \\(G\\) is virtually polycyclic, and \\(V\\) is a finitely-generated \\(\\FI_G\\)-module, then for \\(n\\) sufficiently large, there is a filtration\n\\[0 = V^{(0)}_n \\subset V^{(1)}_n \\subset \\cdots \\subset V^{(m)}_n = V_n\\]\nsuch that the successive quotients \\(V_n^{(k+1)}\/V_n^{(k)}\\) are of the form \\(\\bigoplus_i \\Ind^{\\FI_G}(W_i)_n\\).\n\\end{thm}\nWe are now ready to formulate the relevant generalization of representation stability. For \\(G\\) a virtually polycyclic group, say that an \\(\\FI_G\\)-module \\(V\\) satisfies \\emph{\\(K_0\\)-stability} if for all sufficiently large \\(n\\), there is a decomposition\n\\[ [V_n] = \\sum_{\\underline\\lambda} c(\\underline\\lambda) [L(\\underline\\lambda[n])]\\]\nwhere \\(c(\\underline\\lambda)\\) does not depend on \\(n\\). We then obtain the following.\n\n\\begin{thm}[\\bfseries \\(K_0\\)-stability]\nIf \\(G\\) is virtually polycyclic and \\(V\\) is a finitely-generated \\(\\FI_G\\)-module, then \\(V\\) satisfies \\(K_0\\)-stability.\n\\end{thm}\n\\begin{proof}\nBy Theorem 2.4, for \\(n\\) sufficiently large, we can write\n\\[ [V_n] = \\sum_i [\\Ind^{\\FI_G}(V_i)_n] - \\sum_j [\\Ind^{\\FI_G}(W_j)_n] \\]\nfor some collection of representations \\(\\{V_i\\}\\), \\(\\{W_j\\}\\), independent of \\(n\\). It therefore suffices to verify that the individual terms \\(\\Ind^{\\FI_G}L(\\underline\\lambda)\\) satisfy \\(K_0\\)-stability.\n\nBut now the proof given in \\cite[Thm 1.10]{GL3} of representation stability of \\(\\Ind^{\\FI_G} L(\\underline\\lambda)\\) applies verbatim, since it does not use anything about the group \\(G\\); it just uses Pieri's formula and other facts about the representation theory of \\(S_n\\).\n\\end{proof}\n\n\\subsection{Arbitrary \\(G\\) and finite presentation degree}\nUntil recently, the Noetherian property was seen as the lynchpin of the theory of \\(\\FI\\)-modules and related categories. However, coming out of the work of Church-Ellenberg \\cite{CE} on homological properties of \\(\\FI\\)-modules, a new perspective has emerged that shifts the emphasis from finite generation of modules to finite \\emph{presentation degree} of modules, e.g. in the work of Ramos \\cite{Ra2} and Li \\cite{Li}. In one sense, this perspective is more ``constructive'', because possessing the knowledge of both the degree of generation \\emph{and} the degree of relation of a module gives us quantitative control over various stability properties of the module, as we have seen. The Noetherian property then tells us that \\emph{any} finitely generated module is necessarily finitely presented, which is an important fact but no longer at the absolute center of the theory. Appealing to Noetherianity is also necessarily ineffective, since we are no longer able to say what the relation degree is, and thus lose effective bounds on stability.\n\nAt the same time, this shift in perspective allows us to expand our scope to situations where there is no hope of finite generation. For example, we will see later on examples of \\(\\FI_G\\)-modules \\(V\\) that are not finitely generated simply because the individual pieces \\(V_n\\) are not finitely-generated \\(W_n\\)-representations. Nevertheless, we will be able to prove that \\(V\\) is still generated in finite degree, in the sense that all the generators of \\(V\\) (an infinite number) live only in \\(V_1, \\dots V_m\\) for some \\(m\\), \\emph{and} that \\(V\\) is related in finite degree. Of course, in a sense such \\(V\\) are much \\emph{less} constructive then anything in the finitely-generated world.\n\nThis shift also allows us to leave behind the requirement that \\(G\\) be virtually polycyclic. Indeed, as we have seen, this requirement is based in the fact that, for any kind of Noetherianity to get off the ground, we certainly need the ring \\(k[G]\\) to be Noetherian, which as far as we know is only true when \\(G\\) is virtually polycyclic. However, once we have accepted that modules can be infinitely generated, and only care about the \\emph{degree} that that they are generated (and related) in, we can leave this need behind.\n\nThe central result that informs this perspective is the following, proved simultaneously by Ramos and Li:\n\\begin{thm}[{\\cite[Thm B]{Ra2}, \\cite[Prop 3.4]{Li}}] \\thlabel{coh}\nFor any group \\(G\\), the category of \\(\\FI_G\\)-modules presented in finite degree is abelian.\n\\end{thm}\n\nThis is the analogue of Noetherianity for finite presentation degree, since it allows us to argue that kernels of maps between \\(\\FI_G\\)-modules presented in finite degree are still presented in finite degree, and therefore for example to chase being presented in finite degree through a spectral sequence. Indeed, from a certain point of view \\thref{coh} is the fundamental fact, and Noetherianity as we have seen it is so far is just a consequence of \\thref{coh} and the fact that the individual group rings \\(k[G]\\) are Noetherian.\n\nWhat we lose at this level of generality is any ability to refer back to stability results in terms of things like ``representation stability'' or ``stability of character polynomials'' that are not couched explicitly in terms of \\(\\FI_G\\)-modules. All we can say is that the \\(\\FI_G\\)-modules in question are presented in finite degree, which perhaps is less interesting to someone who only cares about the individual \\(W_n\\)-representations \\(V_n\\). At the same time, since these are infinitely-generated representations of infinite groups, it is hard to say much about the individual representations.\n\n\\subsection{\\(\\FI_G\\sharp\\)-modules}\nThe classification of projective modules provided above means that even when \\(G\\) is infinite, if an \\(\\FI_G\\) module \\(V\\) is projective, then there is still a compact description of the representation theory of \\(V\\). We would therefore like to be able to determine when an \\(\\FI_G\\)-module is projective, so that it has such a description. \\cite{CEF} provide just such a method: they define a category \\(\\FI\\sharp\\) with an embedding \\(\\FI \\hookrightarrow \\FI\\sharp\\) so that \\(\\FI\\sharp\\)-Mod is precisely the category of projective \\(\\FI\\)-modules, so a module is projective just when it extends to an \\(\\FI\\sharp\\)-module. Their construction and proof of equivalence carry over to the setting of \\(\\FI_G\\), as Wilson \\cite{Wi} proved for the case \\(G = \\mathbb{Z}\/2\\).\n\nSo define \\(\\FI_G\\sharp\\) to be the category of \\emph{partial morphisms} of \\(\\FI_G\\): the objects are still finite sets, but a map \\(X \\to Y\\) is given by a pair \\((Z, f)\\), where \\(Z \\subset X\\) and \\(f: Z \\to Y\\) is a \\(G\\)-map. Composition of morphisms is defined by pullback, i.e. with the domain the largest set on which the composition is defined. Then there is a natural structure of \\(\\Ind^{\\FI_G}(V)\\) as an \\(\\FI_G\\sharp\\)-module. Furthermore, we obtain the following.\n\\begin{prop} \\thlabel{sharp_decomp}\nAny \\(\\FI_G\\sharp\\) module is isomorphic to \\(\\,\\bigoplus_i \\Ind^{\\FI_G}(W_i)\\) for some representations \\(W_i\\) of \\(G_i\\).\n\\end{prop}\n\\begin{proof}\nThis was proved for \\(G\\) trivial in \\cite[Thm 4.1.5]{CEF}, and for \\(G = \\mathbb{Z}\/2\\) in \\cite[Thm 4.42]{Wi}. As Wilson explains, the proof in \\cite{CEF} applies almost verbatim: the only change that needs to be made is to the definition of the endomorphism \\(E: V \\to V\\), which should be defined as follows, for \\(m \\ge n\\),\n\\begin{align*}\nE_m&: V_m \\to V_m \\\\\nE_m& = \\sum_{\\substack{S \\subset [m] \\\\ |S| = n}} I_S, \\;\\; \\text{where } I_S = (S, \\iota) \\in \\Hom_{\\FI_G\\sharp}([m], [m]) \\text{ with } \\iota: S \\hookrightarrow [m] \\text{ the inclusion}.\n\\end{align*}\n\\end{proof}\n\n\\begin{cor}\n\\thlabel{sharp_range}\nIf \\(V\\) is an \\(\\FI_G\\sharp\\)-module generated in degree \\(m\\), then \\(\\chi_V\\) is given by a single character polynomial of degree \\(\\le m\\), and satisfies representation stability with stability degree \\(\\le 2m\\).\n\\end{cor}\n\\begin{proof}\nThis follows from \\cite[Thm 1.10]{GL3} and \\thref{charpoly}.\n\\end{proof}\n\n\\subsection{Tensor products and \\(\\FI_G\\)-algebras}\nHere we proceed to generalize the notions introduced in \\cite[\\S4.2]{CEF} from \\(\\FI\\) to \\(\\FI_G\\). \n\nGiven \\(\\FI_G\\)-modules \\(V\\) and \\(V'\\), their tensor product \\(V \\otimes V'\\) is the \\(\\FI_G\\)-module with \\((V \\otimes V')_n = V_n \\otimes V'_n\\), where \\(\\FI_G\\) acts diagonally. \n\nA \\emph{graded \\(\\FI_G\\)-module} is a functor from \\(\\FI_G\\) to graded modules, so that each piece is graded, and the induced maps respect the grading. If \\(V\\) is graded, each graded piece \\(V^i\\) is thus an \\(\\FI_G\\)-module. If \\(V\\) and \\(W\\) are graded, the tensor product \\(V \\otimes W\\) is graded in the usual way. Say that \\(V\\) is of \\emph{finitely-generated type} if each \\(V^i\\) is finitely generated. Say that \\(V\\) is of \\emph{finite type} if it is of finitely-generated type and furthermore each \\(V^i_n\\) is finite-dimensional. Notice if \\(G\\) is a finite group, these two notions coincide.\n\nSimilarly, an \\(\\FI_G\\)-algebra is a functor from \\(\\FI_G\\) to \\(k\\)-algebras, which can also be graded. Usually our algebras will be associative, but in order to deal with Lie algebras without having to invoke anything as fancy as Poincar\\'e-Birkhoff-Witt, we do not assume associativity here. We can also define graded co-\\(\\FI_G\\)-modules and algebras, as functors from \\(\\FI_G^{\\text{op}}\\). A (co-)\\(\\FI_G\\)-algebra \\(A\\) is \\emph{generated (as an \\(\\FI_G\\)-algebra)} by a submodule \\(V\\) when each \\(A_n\\) is generated as an algebra by \\(V_n\\). \n\nFinally, there is another type of tensor product that we will need. Suppose \\(V\\) is graded \\(G\\)-module with \\(V^0 = k\\). Then the space \\(V^{\\otimes \\bullet}\\) defined by \\((V^{\\otimes \\bullet})_n = V^{\\otimes n}\\) has the structure of an \\(\\FI_G\\sharp\\)-module, as in \\cite[Defn 4.2.5]{CEF}, with the morphisms permuting and acting on the tensor factors.\n\nThe following theorem characterizes the above constructions.\n\\begin{thm} \\thlabel{alg} Let \\(G\\) be any group.\\begin{enumerate}\n\\item Let \\(V\\) and \\(V'\\) be \\(\\FI_G\\)-modules generated in degree \\(m\\) and \\(m'\\). Then \\(V \\otimes V'\\) is generated in degree \\(m + m'\\). If \\(V\\) is finitely generated and \\(V'\\) is finite type, then \\(V \\otimes V'\\) is finitely generated.\n\n\\item Let \\(A\\) be a graded (co-)\\(\\FI_G\\)-algebra generated by a graded submodule \\(V\\), where \\(V^0 = 0\\) and \\(V\\) is generated as an \\(\\FI_G\\)-module in degree \\(m\\). Then the \\(i\\)-th grades piece \\(A^i\\) is generated in degree \\(m \\cdot i\\). If \\(V\\) is of finite type, then \\(A\\) is of finite type.\n\n\\item If \\(V\\) is a graded \\(G\\)-module with \\(V^0 = k\\), then \\(V^{\\otimes \\bullet}\\) is an \\(\\FI_G\\sharp\\)-module whose \\(i\\)-th graded piece is generated in degree \\(i\\). If \\(V\\) is of finite type, then \\(V^{\\otimes \\bullet}\\) is of finite type. \n\n\\item If \\(X\\) is a connected \\(G\\)-space, then \\(H^*(X^{\\bullet}; k)\\) is an \\(\\FI_G\\sharp\\)-algebra whose \\(i\\)-th graded piece is generated in degree \\(i\\). If \\(H^*(X; k)\\) is of finite type, then \\(H^*(X^{\\bullet}; k)\\) is of finite type.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} \\ \\begin{enumerate}\n\\item This was proved by Sam-Snowden \\cite[Prop 3.1.6]{SS2} for \\(G\\) finite, and their proof applies verbatim even when \\(G\\) is infinite to show that \\(V \\otimes V'\\) is always generated in degree \\(m + m'\\), though not always finitely.\n\nFor the second part, for each \\(k \\le m + m'\\), we know \\((V \\otimes V')_k = V_k \\otimes V'_k\\) is a finitely-generated \\(W_k\\)-module, since the tensor product of a finitely-generated \\(W_k\\)-module and a finite-dimensional module is finitely generated. The result follows.\n\n\\item This was proved for \\(G\\) trivial in \\cite[Thm 4.2.3]{CEF} by appealing to the free nonassociative algebra \\(k\\{V\\}\\) generated by a vector space, and their proof applies verbatim even when \\(G\\) is infinite to show that \\(A^i\\) is always generated in degree \\(m \\cdot i\\), though not always finitely. The second part follows from the second part of (a).\n\n\\item This was proved for \\(G\\) trivial in \\cite[Prop 4.2.7]{CEF}, but here we have simplified the assumptions, by having \\(V\\) just be a single \\(G\\)-module rather than a whole \\(\\FI_G\\)-module, and so the proof is simpler. To wit, the \\(i\\)-th graded piece is\n\\[ (V^{\\otimes n})^i = \\bigoplus_{k_1 + \\cdots + k_n = i} V^{k_1} \\otimes \\cdots \\otimes V^{k_n} \\]\nAt most \\(i\\) of the nonnegative integers \\(k_1, \\dots, k_n\\) can be nonzero. Let \\(k_{j_1}, \\dots, k_{j_l}\\) be the subsequence of nonzero integers, so \\(l \\le i\\). Then for any pure tensor\n\\[ v = v_1 \\otimes \\cdots \\otimes v_n \\in V^{k_1} \\otimes \\cdots \\otimes V^{k_n},\\]\nsince \\(V^0 = k\\), we can take\n\\[ w = \\left(\\prod_{l \\notin \\{k_j\\}} v_l\\right) v_{k_{j_1}} \\otimes \\cdots v_{k_{j_l}} \\]\nand then the inclusion \\(j: [l] \\hookrightarrow [n]\\) clearly induces \\(j_* w = v\\). So by linearity, \\((V^{\\otimes n})^i\\) is generated in degree \\(i\\).\n\nIf each \\(V^i\\) is finite-dimensional, then each \\(V^{k_1} \\otimes \\cdots \\otimes V^{k_l}\\) is finite-dimensional for \\(l = 1, \\dots, i\\). Therefore \\((V^{\\otimes n})^i\\) is an \\(\\FI_G\\)-module of finite type.\n\n\n\\item This was proved for \\(G\\) trivial in \\cite[Prop 6.1.2]{CEF}, and their proof applies verbatim. It essentially follows from (3) and the K\\\"unneth formula. As \\cite{CEF} explain, technically sometimes a sign is introduced in permuting the order of tensor factors, but this does not change the proof of (3). The degree 0 part is \\(k\\) by connectivity.\n\\end{enumerate}\n\\end{proof}\n\n\\section{Orbit configuration spaces}\nLet \\(M\\) be a manifold with a free and properly discontinuous action of a group \\(G\\), so that \\(M \\to M\/G\\) is a cover. Define the \\emph{orbit configuration space} by:\n\\[\\Conf^G_n(M) = \\{(m_i) \\in M^n \\mid G m_i \\cap G m_j = \\emptyset \\text{ for } i \\ne j\\}\\]\nThis was first considered by Xicot\\'encatl in \\cite{Xi} and later investigated in e.g. \\cite{Co}, \\cite{FZ}, \\cite{CX}, \\cite{CCX}. There is a covering map \\(\\Conf^G_n(M) \\to \\Conf_n(M\/G)\\) with deck group \\(G^n\\), given by \\((m_i) \\mapsto (G m_i)\\). Thus another way to think of \\(\\Conf^G_n(M)\\) is as the space of configurations in \\(M\\) that do not degenerate upon projection to \\(M\/G\\), or as configurations in \\(M\/G\\) which also keep track of a lift in \\(M\\) of each point in the configuration.\n\nIf \\(N\\) is a normal subgroup of \\(G\\), there is an intermediate cover\n\\[\\Conf^G_n(M) \\to \\Conf^{G\/N}_n(M\/N) \\to \\Conf_n(M\/G) \\]\nwhere the first map has deck group \\(N^n\\), and the second has deck group \\((G\/N)^n\\). Also, notice that if \\(G\\) is finite, there is an embedding\n\\[ \\Conf^G_n(M) \\hookrightarrow \\Conf_{|G|n}(M), \\;\\; (m_1, \\dots m_n) \\mapsto (g_1 m_1, g_2 m_1, \\dots, g_{|G|} m_n) \\]\n\n\\subsection{\\(\\FI_G\\)-structure and finite generation for finite groups}\nFor any \\(G\\) acting discretely and properly discontinuously on \\(M\\), if we write \\(\\left(\\Conf^G(M)\\right)_n = (\\Conf^G_n(M))\\), then \\(\\Conf^G(M)\\) is a co-\\(\\FI_G\\)-space: given \\(a: [m] \\hookrightarrow [n]\\) and \\((g_i) \\in G^m\\), there is a map\n\\begin{align*}\n(a, g)^*: \\Conf^G_n(M) &\\to \\Conf^G_m(M) \\\\\n(m_i) &\\mapsto \\left(g_i m_{a(i)}\\right)\n\\end{align*}\nIn particular, if \\(G\\) is trivial we recover the usual ordered configuration space, and if \\(M = \\mathbb{C}^*\\) and \\(G = \\mathbb{Z}\/2\\mathbb{Z}\\) acting as multiplication by \\(-1\\), we obtain the type BC hyperplane complement from \\cite{Wi}. Composing with the contravariant cohomology functor, we see that \\(H^*(\\Conf^G(M), k)\\) has the structure of an \\(\\FI_G\\)-module over the field \\(k\\).\n\nWe are therefore interested in orbit configuration spaces for which \\(G\\) is virtually polycyclic, so that we can apply the results on \\(\\FI_G\\)-modules from \\S2. Interesting examples include:\n\\begin{itemize}\n\\item \\(G = \\mathbb{Z}\/2\\) acting antipodally on \\(M = S^m\\), so that \\(M\/G = \\mathbb{R}P^m\\). This was analyzed by Feichtner and Ziegler in \\cite{FZ}. Their computation of the cohomology in Thm 17 shows that \\(\\dim H^i(\\Conf^G_n(S^m); \\mathbb{Q})\\) is bounded by a polynomial in \\(n\\). We strengthen this by proving that it is in fact equal to a polynomial for \\(n \\gg 0\\). Furthermore, the \\(G^n\\) action on \\(H^i(\\Conf^G_n(S^m))\\) was analyzed in \\cite{GGSX}: in particular, their Prop 6.6 is a sort of weak form of representation stability.\n\\item \\(G = \\mathbb{Z}\/2\\) acting by a hyperelliptic involution on a 4-punctured \\(\\Sigma_g\\), with quotient a 4-punctured sphere\n\\item Any finite cover \\(\\Sigma_h \\to \\Sigma_g\\) (so that \\(h = |G|(g - 1) + 1\\)), with \\(G\\) the deck group of the covering.\n\\item \\(M = \\mathbb{R}^2\\), with \\(G\\) a lattice isomorphic to \\(\\mathbb{Z}^2\\), and \\(M\/G\\) a torus\n\\item \\(M = \\Conf_d(\\mathbb{C})\\) for some fixed \\(d\\), with \\(G = S_d\\), so that we are looking at the iterated configuration space \\(\\Conf^{S_d}_n(\\Conf_d(\\mathbb{C}))\\) and its quotient \\(\\UConf_n(\\UConf_d(\\mathbb{C}))\\). \n\\item \\(M = S^3\\), with \\(G\\) any finite subgroup of \\(\\SO(4)\\), and \\(M\/G\\) a spherical 3-manifold\n\\item Baues \\cite{Bau} proved that every torsion-free virtually polycyclic group \\(G\\) acts discretely, properly discontinuously, and cocompactly on \\(\\mathbb{R}^d\\) for some \\(d\\), and furthermore, the quotient spaces \\(\\mathbb{R}^d\/G\\) precisely comprise the \\emph{infra-solvmanifolds}. Thus for any such \\(G\\), consider \\(\\Conf^G_n(\\mathbb{R}^d)\\).\n\\end{itemize}\n\nA straightforward transversality argument (e.g., \\cite[Thm 1]{Bi}) shows that if \\(\\dim M \\ge 3\\), the map \\(\\Conf_n(M) \\hookrightarrow M^n\\) induces an isomorphism on \\(\\pi_1\\). Therefore \\(\\Conf^{\\pi_1(M)}_n(\\widetilde{M})\\) is in fact the universal cover of \\(\\Conf_n(M)\\) in dimension \\(\\ge 3\\), which provides further motivation as to why orbit configuration spaces are natural to study. Note that this does not make the last example trivial (i.e., contractible), since if \\(\\dim M > 2\\), \\(\\Conf_n(M)\\) need not be aspherical: its homotopy groups only agree with \\(M^n\\) up to \\(\\dim(M)-2\\).\n\nWe first consider the case where \\(G\\) is finite.\n\n\\begin{thm}[\\bfseries Cohomology of orbit configuration spaces] \\thlabel{confg_fg} Let \\(k\\) be a field, let \\(M\\) be a connected, orientable manifold of dimension at least 2 with each \\(H^i(M; k)\\) finite-dimensional, and let \\(G\\) be a finite group acting freely on \\(M\\). Then the \\(\\FI_G\\)-algebra \\(H^*(\\Conf^G(M); k)\\) is of finite type. \\end{thm}\n\nFollowing Church-Ellenberg-Farb-Nagpal \\cite{CEFN}, we could adapt \\thref{confg_fg} to handle \\(\\mathbb{Z}\\) coefficients. As \\cite{CEFN} mention, the proof with \\(\\mathbb{Z}\\) coefficients is essentially identical to the one with field coefficients: the difference is that one of the inputs, the analogue of our \\thref{alg}.4, becomes harder to prove. However, their proof of this analogue, \\cite[Lemm. 4.1]{CEFN}, is readily adaptable to our context. We do not make use of this except in \\S3.4.\n\n\\begin{proof}\nThe proof is based on a modification of Totaro's \\cite[Thm 1]{To}. Following Totaro, consider the Leray spectral sequence associated to the inclusion \\(\\iota: \\Conf^G_n(M) \\hookrightarrow M^n\\). This spectral sequence has the form\n\\begin{equation}\\label{tot_ss}\nH^i(M^n; R^j \\iota_* k) \\implies H^{i+j}(\\Conf^G_n(M); k)\n\\end{equation}\nwhere \\(R^j \\iota_* k\\) is the sheaf on \\(M^n\\) associated to the presheaf\n\\[U \\mapsto H^j(U \\cap \\Conf^G_n(M); k)\\]\n\nAs in Totaro's proof, the sheaf \\(R^j \\iota_* k\\) vanishes outside the appropriate ``fat diagonal'', which in this case is the union of the subspaces \\(\\Delta_{a,g,b} = \\{(m_i) \\in M^n \\mid m_a = g \\cdot m_b\\}\\), for \\(1 \\le a < b \\le n\\) and \\(g \\in G\\). Consider a point in the fat diagonal,\n\\[x = (x_1, g_{11} x_1, \\dots g_{1i_1} x_1, \\dots x_s, g_{s1} x_s, \\dots g_{s i_s} x_s).\\]\n Since \\(G\\) acts properly discontinuously, take a neighborhood of each \\(x_j\\) small enough to be disjoint from all translates of all the other neighborhoods. Then use a Riemannian metric to identify each of these with the tangent space \\(T_{x_j} X\\), \\(T_{g_{j 1} x_j} X\\), etc. \\(dg_{j k}(x_j)\\) is then an isomorphism from \\(T_{x_j} X\\) to \\(T_{g_{jk} x_j} X\\), and the condition that a point \\(m_1\\) near \\(x\\) and \\(m_2\\) near \\(g x\\) satisfy \\(m_2 = g m_1\\) becomes, upon passing to the tangent space and under the isomorphism \\(dg\\), simply the condition that \\((v,w) \\in (T_x X)^2\\) satisfy \\(v = w\\). Thus, for a neighborhood \\(U\\) of \\(x\\) small enough so that the inverse exponential map is a diffeomorphism,\n\\[\n(R^j \\iota_* k)_x = H^j(U \\cap \\Conf^G_n(M); k) = H^j(\\Conf_{i_1}(T_{x_1} X) \\times \\cdots \\times \\Conf_{i_s}(T_{x_s} X))\n\\]\nThus, the local picture looks exactly the same as in \\(\\Conf_n\\), which it should since there is a covering map \\(\\Conf_n^G(M) \\to \\Conf_n(M\/G)\\). So as in Totaro, we get generators of \\(\\Conf_n(\\mathbb{R}^d)\\), where \\(\\dim M = d\\). However, for \\(\\Conf_n(M)\\), we got just one copy of each generator \\(e_{ab}\\), coming from the diagonal \\(\\{m_a = m_b\\}\\). Here, however, we get a generator \\(e_{a,g,b}\\) for each \\(g \\in G\\), coming from each \\(\\Delta_{a,g,b}\\). The permutation action of \\(W_n\\) on the \\(\\{\\Delta_{a,g,b}\\}\\) induces an action on the \\(\\{e_{a,g,b}\\}\\), which is given by\n\\begin{equation} \\label{action}\n(\\sigma, \\vec h) \\cdot e_{a,g,b} = e_{\\sigma(a), h_a g h_b^{-1}, \\sigma(b)}\n\\end{equation}\nAs in Totaro, we can explicitly write down the relations that these \\(e_{a,g,b}\\) satisfy:\n\\begin{gather} \\begin{split} \\label{presentation}\ne_{a,g,b} &= (-1)^d e_{b,g^{-1},a} \\\\\ne_{a,g,b}^2 &= 0 \\\\\ne_{a,g,b} \\wedge e_{b,h,c} &= (e_{a,g,b} - e_{b,h,c}) \\wedge e_{a,gh,c}\n\\end{split} \\end{gather}\nTo conclude, we use the argument from \\cite[Thm 6.2.1]{CEF}. To wit, because the Leray spectral sequence is functorial, all of the spectral sequences of \\(\\Conf_n^G(M) \\hookrightarrow M^n\\), for each \\(n\\), collected together form a spectral sequence of \\(\\FI_G\\)-modules. As we just described, the \\(E_2\\) page is generated by \\(H^*(M^n; k)\\) and\nthe \\(\\FI_G\\)-module spanned by the \\(e_{a,g,b}\\). This latter is evidently finitely-generated, since it is just generated in degree 2 by \\(e_{1,e,2}\\), and the former is of finite type by \\thref{alg}.4, so therefore the \\(E_2\\) page as a whole is of finite type. The \\(E_\\infty\\) page is a subquotient of \\(E_2\\), so by Noetherianity it is of finite type, and therefore \\(H^*(\\Conf^G(M); k)\\) is of finite type.\n\\end{proof}\nWe pause briefly to dwell on the permutation action (\\ref{action}) of \\(W_n\\) on the module \\(V_n\\) spanned by \\(\\{e_{a,g,b}\\}\\), since it will come up repeatedly. Let \\(k[G]\\) be the representation of \\(G \\times G\\) where the first factor of \\(G\\) acts by multiplication on the left, and the second by multiplication on the right by the inverse (two commuting left actions). Therefore\n\\[V_n = \\Ind_{W_2 \\times W_{n-2}}^{W_n} k[G] \\otimes k = \\Ind^{\\FI_G}(k[G])_n \\]\n\nRecall that, as a \\((k[G], k[G]^{\\text{op}})\\)-bimodule, the regular representation has the following decomposition into irreducibles:\n\\[ k[G] = \\bigoplus_{\\chi \\in \\Irr(G)} V_{\\chi} \\boxtimes (V_{\\chi})^* \\]\nThus if we turn this into a \\(k[G] \\otimes k[G]\\)-module by having the right factor act by \\(g^{-1}\\), this becomes\n\\[k[G] = \\bigoplus_{\\chi \\in \\Irr(G)} V_{\\chi} \\boxtimes V_{\\chi} = \\bigoplus_{\\chi \\in \\Irr(G)} L( (2)_\\chi) \\]\nas \\(k[G] \\otimes k[G] = k[G \\times G]\\)-modules. Therefore\n\\begin{equation} \\label{irr_decom}\nV = \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi}).\n\\end{equation}\nas an \\(\\FI_G\\)-module, so it is manifestly an \\(\\FI_G\\sharp\\)-module. In particular, if \\(G\\) is trivial, we obtain \\(\\Ind^{\\FI}((2)) = \\Sym^2 k^n \/ k^n\\), consistent with the computation done in \\cite{CEF}.\n\n\\begin{cor}\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2 with each \\(H^i(M; \\mathbb{Q})\\) finite-dimensional, let \\(G\\) be a finite group acting freely on \\(M\\), and let \\(k\\) be a splitting field for \\(G\\) of characteristic 0. Then for each \\(i\\), the characters of the \\(W_n\\)-representations \\(H^i(\\Conf^G_n(M); k)\\) are given by a single character polynomial for all \\(n \\gg 0\\).\n\\end{cor}\n\n\n\\thref{confg_fg} has another consequence, which as far as we can tell is a new result (recall that \\cite[Thm 6.2.1]{CEF} only applied to orientable manifolds).\n\n\\begin{cor}\nLet \\(M\\) be a connected, non-orientable manifold of dimension at least \\(2\\) with \\(H^*(M; \\mathbb{Q})\\) of finite type. Then the \\(\\FI\\)-algebra \\(H^*(\\Conf(M); \\mathbb{Q})\\) is of finite type.\n\\end{cor}\n\\begin{proof}\nConsider the orientation cover \\(\\widetilde{M} \\to M\\), which has deck group \\(G = \\mathbb{Z}\/2\\). Since there is a covering map \\(\\Conf^G_n(\\widetilde{M}) \\to \\Conf_n(M)\\), with deck group \\(G^n\\), then by transfer there is an isomorphism\n\\[H^*(\\Conf_n(M); \\mathbb{Q}) \\cong \\left(H^*(\\Conf^G_n(\\widetilde{M}); \\mathbb{Q})\\right)^{S_n}\\]\nBy \\thref{confg_fg}, \\(H^*(\\Conf^G_n(\\widetilde{M}); \\mathbb{Q})\\) is a finite type \\(\\FI_G\\)-algebra, and therefore \\(H^*(\\Conf_n(M); \\mathbb{Q})\\) is a finite type FI-algebra.\n\\end{proof}\n\nHomological stability for non-orientable manifolds, which is a consequence of this corollary, was proven by Randal-Williams in \\cite{RW}.\n\n\\subsection{Dealing with infinite groups}\nWe pause to explain the complications that arise when \\(G\\) is infinite, before proceeding to at least partially resolve them. As a toy example, forgetting for a moment the setting of \\(\\FI_G\\) and sequences of spaces, consider the space \\(X = S^1 \\vee S^1 = K(F_2, 1)\\), with one loop called \\(a\\) and the other \\(b\\). Let \\(Y\\) be the \\(G := \\mathbb{Z}\\) cover associated to the kernel of the map \\(F_2 \\to \\mathbb{Z}\\), \\(a \\mapsto 0, b \\mapsto 1\\). Hence \\(Y\\) is an infinite sequence of line segments labeled \\(b\\) joining loops labeled \\(a\\). So \\(Y\\) is homotopy equivalent to a wedge of infinitely many circles, and thus \\(H_1(Y; k)\\) has infinite rank. However, notice that the covering group \\(G\\) acts on \\(Y\\), and that \\(H_1(Y; k) \\cong k[G] = k[b^{\\pm 1}]\\) as \\(G\\)-modules.\n\nIn particular, \\(H_1(Y; k)\\) is finitely-generated as a \\(G\\)-module. However, looking at cohomology, \\(H^1(Y; k) = \\Hom_k(H_1(Y), k) = \\Hom_k(k[G], k) = k^G\\). Thus, \\(H_1(Y; k)\\) consists of finite linear combinations of elements of \\(G\\), and \\(H^1(Y; k)\\), infinite linear combinations. In particular, \\(H^1(Y; k)\\) is no longer finitely generated as a \\(G\\)-module. However, notice that it contains a dense submodule isomorphic to \\(H_1(Y; k)\\).\n\nNow we see why the proof of \\thref{confg_fg} does not suffice when \\(G\\) is infinite: in general, the \\(E_2\\) page is not just be generated by the \\(e_{a,g,b}\\), which is to say, by finite linear combinations of them, but instead it is generated by all infinite linear combinations of them. Therefore the \\(E_2\\) page is in general not a finitely-generated \\(\\FI_G\\)-module. \n\nIf we are willing to settle for ``presented in finite degree''---for example, if \\(G\\) is not virtually polycyclic---then this is good enough:\n\\begin{thm} \\thlabel{confg_coh}\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2. Then the \\(\\FI_G\\)-algebra \\(H^*(\\Conf^G_n(M); k)\\) is presented in finite degree.\n\\end{thm}\n\\begin{proof}\nThe argument from \\thref{confg_fg} carries over essentially directly. The \\(E_2\\) page of the spectral sequence (\\ref{tot_ss}) is generated as a \\(k\\)-algebra by \\(H^*(M^n; k)\\) and the dual space to \\(\\langle e_{a,g,b} \\rangle\\), that is, the space of infinite linear combinations\n\\[ \\sum_{a,g,b} v_{a,g,b} e_{a,g,b} = \\sum_{1 \\le a < b \\le n} \\left( \\sum_{g \\in G} v_{a,g,b} e_{a,g,b} \\right) \\]\nThis space is evidently generated as an \\(\\FI\\)-module by those sums of the form \\(\\sum_{g \\in G} v_{g} e_{1,g,2}\\), since we can get all other \\(a,b\\) by appropriate permutations. These generators evidently live in degree 2, and the relations (\\ref{presentation}) all live in degree 3, so that the \\(E_2\\) page is presented in finite degree. By \\thref{coh}, taking successive pages in the spectral sequences preserves being presented in finite degree, as does passing from the \\(E_\\infty\\) page to the final cohomology. So we conclude that \\(H^*(\\Conf^G_n(M); k)\\) is presented in finite degree.\n\\end{proof}\n\nHowever, if we want to preserve finite generation, the analysis at the beginning of this subsection suggests that the correct thing to look at is actually \\(H_\\bullet(\\Conf^G(M); k)\\) instead. Unfortunately, on the face of it, \\(H_\\bullet(\\Conf^G(M); k)\\) is a co-\\(\\FI_G\\)-module, so since the maps go ``in the wrong way'', it is never finitely generated.\n\nHowever, when the quotient \\(M\/G\\) is an \\emph{open} manifold, we obtain the following generalization of \\cite[Prop 6.4.2]{CEF}.\n\\begin{thm}[\\bfseries Orbit configuration spaces of open manifolds]\n\\thlabel{boundary_sharp}\nLet \\(N\\) be the interior of a connected, compact manifold \\(\\overline{N}\\) of dimension \\(\\ge 2\\) with nonempty boundary \\(\\partial\\overline{N}\\), and let \\(\\pi: M \\to N\\) be a \\(G\\)-cover, so that \\(G\\) acts freely and properly discontinuously on \\(M\\). Then \\(\\Conf^G(M)\\) has the structure of a homotopy \\(\\FI_G\\sharp\\)-space, that is, a functor from \\(\\FI_G\\sharp\\) to hTop, the category of spaces and homotopy classes of maps.\n\\end{thm}\n\\begin{proof}\nWe follow the argument in \\cite{CEF}. Fix a collar neighborhood \\(S\\) of one component of \\(\\partial \\overline{N}\\), let \\(R = \\pi^{-1}(S)\\) and let \\(R_0\\) be a connected component of \\(R\\), and fix a homeomorphism \\(\\Phi: M \\cong M \\setminus \\overline{R}\\) isotopic to the identity (\\(\\Phi\\) and the isotopy both exist by lifting). For any inclusion of finite sets \\(X \\subset Y\\), define a map\n\\[ \\Psi^Y_X: \\Conf^G_X(M) \\to \\Conf^G_Y(M)\\]\nup to homotopy, as follows. First, if \\(Y = X\\), set \\(\\Psi^Y_X = \\id\\). Next, note that a configuration in \\(\\Conf^G_X(M)\\) is just an embedding \\(X \\hookrightarrow M\\) (that stays injective upon composition with \\(\\pi\\)). So fix an element \\(q_X^Y: (Y - X) \\hookrightarrow R_0\\) of \\(\\Conf^G_{Y - X}(R_0)\\). Then any embedding \\(f: X \\hookrightarrow M\\) in \\(\\Conf^G_X(M)\\) extends to an embedding \\(\\Psi^Y_X(f): Y \\hookrightarrow M\\) by\n\\[ \\Psi^Y_X(f)(t) = \\begin{cases} \\Phi(f(t)) & t \\in X \\\\ q^Y_X(t) & t \\notin X \\end{cases} \\]\nThe image of \\(\\pi \\circ \\Phi\\) is disjoint from \\(S\\), while the image of \\(\\pi \\circ q^Y_X\\) is contained in \\(S\\), so the above map never send points in \\(X\\) into the same \\(G\\)-orbit as it sends points outside of \\(X\\), and therefore this does give an element of \\(\\Conf^G_Y(M)\\). Furthermore, since \\(\\Conf^G_{Y - X}(R_0)\\) is connected (since \\(R_0\\) is, and \\(\\dim R_0 \\ge 2\\)), different choices of \\(q^Y_X\\) give homotopic maps, so \\(\\Phi^Y_X\\) is well-defined up to homotopy.\n\nNow, an \\(\\FI_G\\sharp\\) morphism \\(Z \\to Y\\) consists of an injection \\(X \\hookrightarrow Z\\), an injection \\(X \\hookrightarrow Y\\), and a tuple \\(g: X \\to G\\). Normally if we were extending from an \\(\\FI_G\\) structure, we would think of \\(X\\) as being a subset of \\(Z\\), but since we are extending from a \\emph{co}-\\(\\FI_G\\) structure, it is more natural to think of \\(X\\) as a subset of \\(Y\\), with an explicit map \\(a: X \\to Z\\). The induced map is then given by\n\\begin{align*}\n\\Conf^G_Z(M) & \\to \\Conf^G_X(M) \\xrightarrow{\\Psi^Y_X} \\Conf^Y(M) \\\\\n(m_i) & \\mapsto  (g_i m_{a(i)})\n\\end{align*}\nIt is straightforward to verify that this is functorial up to homotopy, as \\cite[Prop 6.4.2]{CEF} do for trivial \\(G\\).\n\\end{proof}\nIn particular, when the conditions of \\thref{boundary_sharp} hold, then \\(H_*(\\Conf^G(M); k)\\) is an \\(\\FI_G\\sharp\\)-module. We want to argue that, when \\(G\\) is virtually polycyclic, \\(H_*(\\Conf^G(M); k)\\) is finitely-generated type. To do this, we need the following.\n\\begin{prop} \\thlabel{arnold}\nLet \\(A(G,d)\\) be the \\(\\FI_G\\sharp\\)-algebra where \\(A(G)_n\\) has generators \\(\\{e_{a,g,b} \\mid 1 \\le a \\ne b \\le n, g \\in G\\}\\) of degree \\(d-1\\) and action given by (\\ref{action}), modulo the following relations:\n\\begin{align*}\ne_{a,g,b} &= (-1)^d e_{b,g^{-1},a} \\\\\ne_{a,g,b}^2 &= 0 \\\\\ne_{a,g,b} \\wedge e_{b,h,c} &= (e_{a,g,b} - e_{b,h,c}) \\wedge e_{a,gh,c}\n\\end{align*}\nThen \\(A(G,d)\\) is an \\(\\FI_G\\sharp\\)-module of finitely-generated type.\n\\end{prop}\n\\begin{proof}\nFirst, by construction \\(A(G,d)\\) is presented in finite degree, so it remains to show that each \\(A(G,d)_n\\) is of finitely-generated type. For convenience, put \\(D = d-1\\), so that \\(A(G,d)_n\\) is only nonzero is degree divisible by \\(D\\). It is straightforward to verify (e.g., see [ARNOLD]) that \\(A(G,d)^{iD}_n\\) is spanned as a vector space by all products of the form\n\\begin{equation} \\label{arnold_span}\nv = e_{a_1, g_1, b_1} \\wedge e_{a_2, g_2, b_2} \\wedge \\cdots \\wedge e_{a_i, g_i, b_i} \\;\\; \\text{ where } a_s < b_s, \\;\\;\\; b_1 < b_2 < \\cdots < b_i, \\;\\;\\; g_s \\in G\n\\end{equation}\nWe will describe an explicit procedure which, given such a \\(v\\), finds an element \\(\\vec h \\in G^n\\) so that\n\\[\nv' := \\vec h \\cdot v = e_{a_1, e, b_1} \\wedge e_{a_2, e, b_2} \\wedge \\cdots \\wedge e_{a_i, e, b_i}\n\\]\nthat is, so that each \\(g'_i = e\\) in (\\ref{arnold_span}). We construct \\(\\vec h\\) inductively. To begin, we put a copy of \\(g_1\\) in the \\(b_1\\)-th coordinate of \\(\\vec h\\), which cancels out the \\(g_1\\) in (\\ref{arnold_span}). We multiply \\(v\\) by this partial \\(\\vec h\\), which may have the effect of modifying later \\(g_i\\)'s. We then proceed and put a copy of (the new, modified) \\(g_2\\) in the \\(b_2\\)-th coordinate of \\(\\vec h\\), which cancels the \\(g_2\\) in (\\ref{arnold_span}). We can do this with no trouble because we know \\(b_1 < b_2\\). Again, we use this to modify \\(v\\) and proceed to \\(b_3\\), etc. Eventually we have constructed our \\(\\vec h\\) and modified \\(v\\) so that each \\(g_s = e\\).\n\nWe therefore conclude that \\(A(G,d)^{iD}_n\\) is finitely generated as a \\(G^n\\)-module, so \\emph{a fortiori} as a \\(W_n\\)-module. Therefore \\(A(G,d)\\) is of finitely-generated type.\n\\end{proof}\n\nWe therefore obtain the following.\n\\begin{thm}[\\bfseries Homology of orbit configuration spaces]\n\\thlabel{confg_fgsharp}\nLet \\(N\\) be the interior of a connected, compact manifold of dimension \\(\\ge 2\\) with nonempty boundary. Let \\(M \\to N\\) be a \\(G\\)-cover, with \\(G\\) virtually polycyclic, such that \\(H_*(M)\\) is of finite type. Then \\(H_*(\\Conf^G(M); k)\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module.\n\\end{thm}\n\\begin{proof}\nWe still need to appeal to cohomology, in order to make use of the cup product structure. So the proof follows that of \\thref{confg_fg}, but only considering the sub-\\(\\FI_G\\)-module of the \\(E_2\\) page that actually is generated by \\(H^*(M^n)\\) and the \\(e_{a,g,b}\\), and not the infinite linear combinations of them. Notice that this is isomorphic to the \\(E^2\\) page associated to the appropriate spectral sequence computing the homology of \\(\\Conf^G(M)\\) (to be technical, this comes from \\emph{cosheaf} homology). \n\nThis submodule of the \\(E_2\\) page is precisely the algebra described in \\thref{arnold}. Thus it is of finitely-generated type, and therefore the \\(E^2\\) page for homology is an \\(\\FI_G\\)-module of finitely-generated type. The same final argument from \\thref{confg_fg} (which uses Noetheriantiy, since \\(G\\) is virtually polycyclic) thus shows that \\(H_*(\\Conf^G(M); k)\\) is of finitely-generated type.\n\\end{proof}\n\n\\begin{cor}\nLet \\(N\\) be the interior of a connected, compact manifold of dimension \\(\\ge 2\\) with nonempty boundary, let \\(M \\to N\\) be a \\(G\\)-cover, with \\(G\\) virtually polycyclic, \\(M\\) connected and orientable and \\(H^*(M; \\mathbb{Q})\\) of finite type. Then for each \\(i\\), the \\(W_n\\)-representations \\(H_i(\\Conf^G_n(M); \\mathbb{Q})\\) satisfy \\(K_0\\)-stability.\n\\end{cor}\n\n\\subsection{Homotopy groups of configuration spaces}\nThe main application of the theory of \\(\\FI_G\\)-modules until this point has been in Kupers-Miller's \\cite{KM} work on the homotopy groups of configuration spaces. They prove that, for \\(M\\) a simply-connected manifold of dimension at least 3, the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) form a finitely-generated \\(\\FI\\)-module. In \\cite[\\S5.2]{KM}, they sketch an extension of this result to the non-simply connected case. Kupers-Miller are naturally led to consider \\(\\Conf_n^{\\pi_1(M)}(\\widetilde{M})\\), since as we have said it is the universal cover of \\(\\Conf_n(M)\\) once \\(\\dim M \\ge 3\\), and so has the same higher homotopy groups as \\(\\Conf_n(M)\\).\n\nOur results on orbit configuration spaces are able to confirm most of Kupers-Miller's sketch, while also clarifying some oversights. As stated, their Prop 5.8 is not correct, since as we have seen, if \\(G\\) is infinite, in general we cannot conclude that \\emph{co}homology is finitely generated. Instead, the best we can do is our \\thref{confg_fg} and \\thref{confg_fgsharp}, where we either assume that \\(G\\) is finite or that \\(M\\) is an open manifold. We therefore obtain the following. Note that, following Kupers-Miller, we work here with \\(\\mathbb{Z}\\) coefficients, since as we mentioned, we could rework \\thref{confg_fg} to use \\(\\mathbb{Z}\\) coefficients.\n\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 1]\nLet \\(M\\) be a connected manifold of dimension \\(\\ge 3\\) with finite fundamental group \\(G\\) such that \\(H^*(M)\\) is finite-dimensional. For \\(i \\ge 2\\), the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) and \\(\\Ext^1_{\\mathbb{Z}}(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) are finitely generated \\(\\FI_G\\)-modules.\n\nIn particular, if \\(k\\) is a splitting field for \\(G\\) of characteristic 0, then for each \\(i\\), the characters of the \\(W_n\\)-representations \\(\\Hom(\\pi_1(\\Conf_n(M)), k)\\) are given by a single character polynomial for \\(n \\gg 0\\), and \\(\\{\\Hom(\\pi_1(\\Conf_n(M)), k)\\}\\) satisfies representation stability.\n\\end{thm}\n\nApplying \\thref{confg_coh}, we obtain the following.\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 2]\nLet \\(M\\) be a connected, orientable manifold of dimension at least 2. Then the dual homotopy groups \\(\\Hom(\\pi_i(\\Conf_n(M), \\mathbb{Z})\\) and \\(\\Ext^1_{\\mathbb{Z}}(\\pi_i(\\Conf_n(M)), \\mathbb{Z})\\) are \\(\\FI_G\\)-modules presented in finite degree.\n\\end{thm}\n\nApplying \\thref{confg_fgsharp}, we also obtain the following.\n\n\\begin{thm}[\\bfseries Homotopy groups of configuration spaces, take 3]\nLet \\(M\\) be the interior of a connected, compact manifold with nonempty boundary of dimension \\(\\ge 3\\) such that \\(G = \\pi_1(M)\\) is virtually polycyclic. For \\(i\\) at least 2, \\(\\pi_i(\\Conf_n(M))\\) is a finitely-generated \\(\\FI_G\\sharp\\)-module. In particular, \\(\\pi_i(\\Conf_n(M)) \\otimes \\mathbb{Q}\\) satisfies \\(K_0\\)-stability.\n\n\\end{thm}\n\n\\subsection{Intermediate configuration spaces and \\(\\FI \\times G\\)}\nNow suppose that \\(G\\) is abelian, and consider the following map:\n\\[\\psi: G^n \\rtimes S_n \\to G \\times S_n,\\;\\; \\psi(\\vec{g}, \\sigma) = (g_1 \\cdots g_n, \\sigma) \\]\nSince \\(G\\) is abelian and \\(\\sigma\\) leaves the product of all the \\(g_i\\)'s invariant, \\(\\psi\\) is a homomorphism. Its kernel is the subgroup \\(\\ker( \\vec{g} \\mapsto g_1 \\cdots g_n) \\subset G^n\\), which as a group is just isomorphic to \\(G^{n-1}\\). Since \\(G\\) is a \\(\\mathbb{Z}\\)-module, we can identify \\(\\ker \\psi\\) as \\(G \\otimes_\\mathbb{Z} V\\), where \\(V\\) is the standard representation of \\(S_n\\) of rank \\(n-1\\). We obtain the following diagram of covering spaces refining \\(\\Conf^G_n(M) \\twoheadrightarrow \\UConf_n(M\/G)\\):\n\\begin{equation*}\n\\begin{diagram}\n&&\\Conf^G_n(M)&& \\\\\n&&\\dTo_{G^{n-1}}&&\\\\\n&& \\Conf'_n(M\/G)&& \\\\\n&\\ldTo^{S_n} && \\rdTo^{G} & \\\\\n\\UConf'_n(M\/G) &&&& \\Conf_n(M\/G) \\\\\n&\\rdTo_{G} && \\ldTo_{S_n} & \\\\\n&&\\UConf_n(M\/G)&&\n\\end{diagram}\n\\end{equation*}\nwhere \\(\\Conf'_n(M\/G)\\) and \\(\\UConf'_n(M\/G)\\) are the intermediate covers. These do not have particularly nice descriptions in general, but if \\(M\\) is a Lie group and \\(G\\) a discrete central subgroup, so that \\(M\/G\\) is also a Lie group, then\n\\[ \\Conf'_n(M\/G) = \\{(m_1 \\dots m_n; m) \\in (M\/G)^n \\times M\\,|\\,m_i \\ne m_j\\,\\forall i\\ne j;\\, \\pi(m) = m_1 \\cdots m_n\\} \\]\nand \\(\\UConf'_n(M\/G)\\) is its quotient by \\(S_n\\). So \\(\\Conf'\\) parameterizes configurations of points in the base space with a chosen lift of the \\emph{product} of the points. When \\(M\/G = \\mathbb{C}^*\\), these spaces are closely related to the Burau representation, as we shall see.\n\nDefine \\(\\FI \\times G\\) to be the usual product of categories, where we think of \\(G\\) as a category with one object. An \\(\\FI \\times G\\)-module is thus a sequence of \\(G\\)-representations equipped with the structure of an \\(\\FI\\)-module, where the two actions commute. The restriction functor \\(\\Res: \\FI \\times G \\to G \\times S_n\\) still has a left adjoint \\(\\Ind^{\\FI \\times G}: G \\times S_n \\to \\FI \\times G\\) given by\n\\[ \\Ind^{\\FI \\times G}(V)_{n+k} = \\Ind^{S_{n+k}}_{S_n \\times S_k} V \\]\nWe can likewise define the category \\(\\FI\\sharp \\times G\\). Then \\(\\Ind^{\\FI \\times G}(V)\\) naturally have the structure of an \\(\\FI\\sharp \\times G\\)-module, and the analogue of \\thref{sharp_decomp} shows that any \\(\\FI\\sharp \\times G\\)-module is a direct sum of these.\n\nNotice that \\(H^*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is naturally an \\(\\FI \\times G\\)-module. Furthermore, if \\(M\\) satisfies the condition of \\thref{boundary_sharp} then \\(H_*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is an \\(\\FI\\sharp \\times G\\)-module.\n\n\\begin{prop}\n\\thlabel{prime_fg}\nIf \\(M\\) is a connected orientable manifold of dimension at least 2 with \\(H^*(M; \\mathbb{Q})\\) of finite type, and \\(G\\) a finite abelian group acting freely and properly discontinuously on \\(M\\), then \\(H^*(\\Conf'(M\/G); \\mathbb{Q})\\) is an \\(\\FI \\times G\\)-algebra of finitely-generated type.\n\nIf \\(N\\) is the interior of a connected compact manifold of dimension \\(\\ge 2\\) with nonempty boundary, \\(M \\to N\\) is a \\(G\\)-cover with \\(G\\) f.g. abelian, \\(M\\) is connected and orientable and \\(H^*(M; \\mathbb{Q})\\) is a finite type \\(G\\)-module, then \\(H_*(\\Conf'(M\/G); \\mathbb{Q})\\) is an \\(\\FI\\sharp \\times G\\)-module of finitely-generated type.\n\\end{prop}\n\\begin{proof}\nIf \\(G\\) is finite this follows immediately by transfer applied to \\(\\Conf^G_n(M)\\). However, even if \\(G\\) is infinite, we can still apply the Serre spectral sequence to the fibration\n\\[\\Conf^G_n(M) \\to \\Conf'_n(M\/G) \\to BG^{n-1}\\]\nThis spectral sequence has the form\n\\[ E_{i,j}^2 = H_i(BG^{n-1}; H_j(\\Conf^G_n(M); \\mathbb{Q})) \\implies H_{i+j}(\\Conf'_n(M\/G); \\mathbb{Q}) \\]\nBy finite generation of \\(H_j(\\Conf^G_n(M); \\mathbb{Q})\\) as an \\(\\FI_G\\)-module, each of the terms \\(E_{i,j}^2\\) is a finitely-generated \\(\\FI \\times G\\)-module, and thus the whole \\(E^2\\) page is of finitely-generated type. So the same argument at the end of \\thref{confg_fg} thus shows that \\(H_*(\\Conf'_n(M\/G); \\mathbb{Q})\\) is an \\(\\FI \\times G\\)-module of finitely-generated type.\n\\end{proof}\n\\thref{prime_fg} thus has the following consequence.\n\\begin{cor}\n\\thlabel{gstab}\nLet \\(M \\to N\\) be a \\(G\\)-cover with \\(G\\) f.g. abelian, and \\(M\\) a connected orientable manifold of dimension \\(\\ge 2\\) with \\(H^*(M; \\mathbb{Q})\\) of finite type. Suppose further that either \\(G\\) is finite, or \\(N\\) is the interior of a compact manifold with nonempty boundary. Then for any \\(i \\ge 0\\), the sequence of \\(G\\)-modules \\(H_i(\\UConf'_n(N); \\mathbb{C})\\) are isomorphic for \\(n \\gg 0\\).\n\\end{cor}\n\\begin{proof}\nThe \\(S_n\\)-cover \\(\\Conf'_n(M\/G) \\to \\UConf'_n(M\/G)\\) gives a transfer isomorphism of \\(G\\)-modules:\n\\[H_i(\\UConf'_n(M\/G); \\mathbb{Q}) \\cong \\left(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\right)_{S_n}\\]\n\nIf \\(G\\) is finite then a finitely generated \\(\\FI \\times G\\)-module is still finitely generated when just considered an \\(\\FI\\)-module. Thus \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\) is a finitely generated \\(\\FI\\)-module, and representation stability tells us that \\(H_i(\\Conf'_n(N); \\mathbb{C})_{S_n}\\) stabilize when \\(n \\gg 0\\). By the transfer isomorphism, we conclude the corollary.\n\nOn the other hand, suppose \\(G\\) is infinite and \\(N\\) is the interior of a closed manifold with boundary. Then \\(H_i(\\Conf'(M\/G); \\mathbb{Q})\\) is a finitely generated \\(\\FI\\sharp \\times G\\)-module by \\thref{prime_fg}. Therefore\n\\[ H_i(\\Conf'(M\/G); \\mathbb{Q}) = \\bigoplus_j \\Ind^{\\FI \\times G}(W_j) \\]\nfor some finite collection \\(W_1, \\dots W_k\\) of representations of \\(S_{i_1} \\times G, \\dots, S_{i_k} \\times G\\). By Frobenius reciprocity,\n\\[ \\left(\\Ind^{\\FI \\times G}(W_j)_n\\right)_{S_n} \\cong (W_j)_{S_{i_j}} \\]\nThus once \\(n\\) is at least the degree of generation of \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\), the coinvariants \\(H_i(\\UConf'_n(M\/G); \\mathbb{Q})\\) will always be isomorphic as \\(G\\)-modules. The conclusion again follows, and in this case we can explicitly say that stability occurs once we reach the degree of generation of \\(H_i(\\Conf'_n(M\/G); \\mathbb{Q})\\). This is not necessarily the same as the degree of generation of \\(H_i(\\Conf_n(M\/G; \\mathbb{Q})\\), since the spectral sequence argument of \\thref{prime_fg} makes non-constructive use of Noetherianity.\n\\end{proof}\n\nNotice that the sequence of \\(G\\)-modules \\(H_i(\\UConf'_n(M\/G); \\mathbb{C})\\) do not have any obvious maps between them, or obvious larger category-representation structure. It is only by lifting to the cover, noticing that we have representation stability, and then descending, that we obtain actual stability of \\(G\\)-modules. In this respect, \\thref{gstab} mirrors the proof of homological stability for unordered configuration spaces of manifolds in \\cite{CEF} (this result was originally proven in \\cite{Chu}). On the other hand, the \\(\\FI_G\\)-module structure of \\(H_i(\\Conf^G(M))\\) is crucial for the specifics of \\thref{gstab}. If we forgot about the \\(\\FI_G\\)-module structure, and merely thought of \\(H^i(\\Conf^G(M); \\mathbb{C})\\) as an FI-module, finite generation as an FI-module would be enough to show that \\(\\UConf'(M\/G))\\) satisfies homological stability \\emph{in dimension} (i.e., as vector spaces), but it would not show that it satisfies homological stability \\emph{as \\(G\\)-modules}.\n\nNotice also that if \\(G\\) is infinite, then we are really getting a strong result about \\(G\\)-modules. We do not need to make things nicer by passing to the Grothendieck group: we get honest stability of \\(G\\)-modules. This despite the fact that in general we can say very little about what all the finitely-generated \\(G\\)-modules actually look like.\n\n\\section{Complex reflection groups}\nOne classical example of a finite wreath product is the \\emph{complex reflection group} \\(G(d,1,n)\\), where \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\) and so \\(W_n = \\mathbb{Z}\/d\\mathbb{Z} \\wr S_n\\), also sometimes referred to as the \\emph{full monomial group}. Because of the action of \\(G\\) on \\(\\mathbb{C}\\) as multiplication by \\(d\\)-th roots of unity, \\(W_n\\) acts on \\(\\mathbb{C}^n\\) by generalized permutation matrices whose entries are \\(d\\)-th roots of unity. This gives rise to a rich class of examples of \\(\\FI_G\\)-modules, which we now describe.\n\nNote that since \\(G\\) is abelian, a splitting field for \\(G\\) is just the same as a field containing all the character values of \\(G\\). So in this case there is a minimal splitting field of characteristic zero, namely the field generated by the character values, which is \\(\\mathbb{Q}(\\zeta_d)\\). \n\n\\subsection{Orbit configuration space of complex reflection groups}\nFirst, of course, we can consider the orbit configuration space where \\(M = \\mathbb{C}^*\\) and \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\) acting, as just described, as multiplication by \\(d\\)-th roots of unity. Thus\n\\[\n\\Conf^G_n(\\mathbb{C}^*) = \\left\\{(v_i) \\in \\mathbb{C}^n \\mid v_i \\ne \\zeta_d^k v_j \\text{ for } i \\ne j, v_i \\ne 0 \\text{ for all } i\\right\\}\n\\]\n\\(\\Conf^G_n(\\mathbb{C}^*)\\) is thus the complement of the hyperplanes fixed by the standard complex-reflection generators of \\(W_n\\). This arrangement, called the \\emph{complex reflection arrangement}, is much studied. For instance, Bannai \\cite{Ba} proved that the complement is aspherical; its fundamental group is referred to as the \\emph{pure monomial braid group}, and sometimes denoted \\(P(d,n)\\). Thus the cohomology of \\(\\Conf^G_n(\\mathbb{C}^*)\\) is isomorphic to the group cohomology of \\(P(d,n)\\).\n\nOrlik-Solomon \\cite{OS} calculated the cohomology of the complement of any hyperplane arrangement, as follows. Let \\(\\mathcal{A}\\) be a collection of linear hyperplanes in \\(\\mathbb{C}^n\\), and put \\(M(\\mathcal{A}) = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\). Say that a subset \\(\\{H_1, \\dots, H_p\\} \\subset \\mathcal{A}\\) is \\emph{dependent} if \\(H_1 \\cap \\dots \\cap H_p = H_1 \\cap \\cdots \\widehat{H_i} \\cdots \\cap H_p\\); alternately, if the linear forms defining the \\(H_i\\) are linearly dependent. Now let\n\\[\nE(\\mathcal{A}) = \\bigwedge \\langle e_H \\mid H \\in \\mathcal{A}\\rangle = \\bigwedge H^1(M(\\mathcal{A}); \\mathbb{Q})\n\\]\nand let \\(I(\\mathcal{A}) \\subset E(\\mathcal{A})\\) be the ideal\n\\[\nI(\\mathcal{A}) = \\left( \\sum_{i = 1}^p (-1)^{i} e_{H_1} \\wedge \\cdots \\wedge \\widehat{e_{H_i}} \\wedge \\cdots \\wedge e_{H_p}\\,\\middle|\\, H_1, \\dots, H_p \\text{ are dependent}\\right).\n\\]\nThen \\(H^*(M(\\mathcal{A}); \\mathbb{Q}) = E(\\mathcal{A})\/I(\\mathcal{A})\\). We then obtain the following.\n\\begin{thm}[\\bfseries Cohomology of complex reflection arrangements]\n\\thlabel{crg_fg}\n\\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q}) = H^*(P(d,1,n); \\mathbb{Q})\\) is a finite type \\(\\FI_G\\sharp\\)-algebra. For each \\(i\\), the characters of the \\(W_n\\)-representations \\(H^i(P(d,1,n); \\mathbb{Q}(\\zeta_d))\\) are given by a single character polynomial of degree \\(2i\\) for all \\(n\\), and therefore \\(H^i(P(d,1,n; \\mathbb{Q}(\\zeta_d))\\) satisfies representation stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nSince \\(\\mathbb{C}^*\\) is the interior of an compact orientable 2-manifold with boundary, \\thref{confg_fgsharp} says that \\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\) is a finitely-generated \\(\\FI_G\\sharp\\)-module. To determine the degree this \\(\\FI_G\\)-module is generated in, we use the Orlik-Solmon presentation. The hyperplane arrangement is:\n\\[\\mathcal{A} = \\{z_i \\mid 1 \\le i \\le n\\} \\cup \\{e_{i,a,j} \\mid 1 \\le i \\ne j \\le n; a \\in G\\}\\]\nwhere \\(z_i\\) is the hyperplane \\(\\{v_i = 0\\}\\) and \\(e_{i,a,j}\\) is the hyperplane \\(\\{v_i = \\zeta^a v_j\\}\\), so that \\(e_{i,a,j} = e_{j,a^{-1},i}\\). To understand \\(I(\\mathcal{A})\\), it is helpful to consider the embedding mentioned in \\S3.1,\n\\[ \\Conf^G_n(\\mathbb{C}^*) \\hookrightarrow \\Conf_{|G|n}(\\mathbb{C}),\\;\\; (v_1, \\dots, v_n) \\mapsto (v_1, \\zeta v_1, \\zeta^2 v_1, \\dots, \\zeta^{d-1} v_n) \\]\nThis induces a map \\(H^*(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\). Orlik-Solomon likewise determines the cohomology of \\(\\Conf_{|G|n}(\\mathbb{Q})\\). The hyperplane arrangement for \\(\\Conf_{|G|n}(\\mathbb{C})\\) is\n\\[\\mathcal{B} = \\{f_{i, g; j, h} \\mid 1 \\le i,  j \\le n;\\, g, h \\in G;\\, (i, g) \\ne (j, h)\\} \\]\nThe induced map \\(H^1(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^1(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is just given by\n\\[ f_{i,g; j, h} \\mapsto \\begin{cases} e_{i,h\/g,j} & \\text{if } i \\ne j \\\\ z_{i} & \\text{if } i = j \\end{cases}\\]\nand is thus evidently surjective. Since \\(H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is generated by \\(H^1\\), this means that the total induced map \\(H^*(\\Conf_{|G|n}(\\mathbb{C}); \\mathbb{Q}) \\to H^*(\\Conf^G_n(\\mathbb{C}^*); \\mathbb{Q})\\) is surjective. We can therefore describe the ideal \\(I(\\mathcal{A})\\) in terms of the simple relations generating the ideal \\(I(\\mathcal{B})\\) of the braid arrangement. We obtain the following presentation:\n\\[\nH^*(M(\\mathcal{A}); \\mathbb{Q}) = \\bigwedge\\langle e_{i,a,j}, z_k \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,ab,k} \\\\ z_i \\wedge z_j = (z_i - z_j) \\wedge e_{i,a,j} \\\\ e_{i,a,j} \\wedge e_{i,b,j} = (e_{i,a,j} - e_{i,b,j})\\wedge z_i \\end{matrix} \\right\\rangle\n\\]\nAnother way to view these is by writing \\(e_{i,a,i} = z_i\\) for any \\(a \\ne 1 \\in G\\), as suggested by the induced map on \\(H^1\\) above: then the first relation, if \\(i,j,k\\) are allowed to equal one another, implies the others. As described in \\S3.1, \\(G^n \\rtimes S_n\\) acts on \\(H^1\\) as follows: on the \\(\\{z_i\\}\\), \\(G^n\\) acts trivially and \\(S_n\\) acts in the standard way, and on the \\(\\{e_{i,a,j}\\}\\) the action is:\n\\[ \\left(\\sigma, \\zeta^{\\vec{b}}\\right) \\cdot e_{i,a,j} = e_{\\sigma(i),\\,a - b_i + b_j,\\,\\sigma(j)} \\]\n\nSo we see that \\(H^*(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\) is generated as an algebra by \\(H^1(\\Conf^{\\mathbb{Z}\/d\\mathbb{Z}}_n(\\mathbb{C}^*); \\mathbb{Q})\\), and that \\(H^1\\) is generated as an \\(\\FI_G\\)-module by \\(\\{e_{1,a,2}\\}\\) and \\(z_1\\), and is therefore generated in degree \\(2\\). By \\thref{alg}, \\(H^i\\) is generated in degree \\(2i\\), and so the stable range follows from \\thref{sharp_range}.\n\\end{proof}\n\nWilson \\cite[Thm 7.14]{Wi} proved the case \\(d = 2\\), for which the arrangement is the type \\(BC\\) braid arrangement. The decomposition for \\(H^1\\), following (\\ref{irr_decom}), is\n\\[ H^1(P(d,1,n); \\mathbb{Q}) = \\Ind^{\\FI_G}((1)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi}) \\]\nwhere \\(\\chi_0\\) is the trivial character of \\(G\\). That is, \\(\\Ind^{\\FI_G}((1)_{\\chi_0})\\) picks out the submodule spanned by the \\(\\{z_i\\}\\), and \\(\\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}((2)_{\\chi})\\) the submodule spanned by the \\(\\{e_{i,a,j}\\}\\). We can also compute the decomposition for \\(H^2\\):\n\\begin{gather*} \nH^2(P(d,1,n); \\mathbb{Q}) = \\Ind^{\\FI_G}( (2,1)_{\\chi_0}) \\oplus \\Ind^{\\FI_G}( (3)_{\\chi_0}) \\oplus \\bigoplus_{\\chi \\in \\Irr(G)} \\Ind^{\\FI_G}( (3,1)_\\chi) \\oplus \\Ind^{\\FI_G}( (2)_\\chi) \\\\\n\\oplus \\bigoplus_{\\substack{\\chi \\in \\Irr(G) \\\\ \\chi \\ne \\chi_0}} \\Ind^{\\FI_G}( (2)_\\chi) \\oplus \\Ind^{\\FI_G}((1)_{\\chi_0}, (2)_\\chi)^2 \\oplus \\Ind^{\\FI_G}((1)_\\chi, (1,1)_\\chi) \\oplus \\Ind^{\\FI_G}((2)_{\\chi_0}, (2)_\\chi)\n\\end{gather*}\nThese calculations agree with Wilson's \\cite[p. 123]{Wi} for the case \\(d = 2\\).\n\n\\subsection{Diagonal coinvariant algebras}\nComplex reflection groups also provide a generalization of the coinvariant algebra, as follows. Let \\(k\\) be a field with a primitive \\(d\\)-th root of unity \\(\\zeta_d\\), and let \\(V_n = k^n\\) be the canonical representation of \\(W_n\\) by \\(\\zeta_d\\)-power permutation matrices. Consider\n\\[\nk[V_n^{\\oplus r}] \\cong k[x_1^{(1)}, \\cdots, x_n^{(1)}, \\cdots, x^{(r)}_1, \\cdots x^{(r)}_n]\n\\]\nThis has a natural grading by \\(r\\)-tuples \\((j_1, \\dots, j_r)\\), where \\(j_i\\) counts the total degree in the variables \\(x_1^{(i)}, \\dots, x_n^{(i)}\\). There is an action of \\(W_n\\) induced by the diagonal action on \\(V^{\\oplus r}\\) which respects the grading. Let \\(\\mathcal{I}_n = (k[V_n]^{W_n}_+)\\) be the ideal generated by the constant-term-free \\(W_n\\)-invariant polynomials. The \\emph{diagonal coinvariant algebra} is\n\\[\\mathcal{C}^{(r)}(n) = k[V_n^{\\oplus r}] \/ \\mathcal{I}_n\\]\nThese \\(\\mathcal{C}^{(r)}(n)\\) arise naturally because, by the celebrated Chevalley-Shephard-Todd Theorem, complex reflection groups exactly comprise the class of groups \\(G\\) acting on a vector space \\(V\\) for which \\(k[V]^G\\) is a polynomial algebra. The representation theory of \\(\\mathcal{C}^{(r)}(n)\\) has been much studied. The case \\(r = 1\\) is classical, going back to Chevalley's original paper \\cite{Ch} proving the general case of Chevalley-Shephard-Todd.\n\nThe case \\(r = 2\\) for a complex reflection group (\\(d > 2\\)) was first studied by Haimain \\cite{Ha} and Gordon \\cite{Go}. The case of a general \\(r\\) was analyzed by Bergeron in \\cite{Ber}. Bergeron shows that for a fixed \\(n\\), the multigraded Hilbert polynomial of \\(\\mathcal{C}^{(r)}(n)\\) can be described independently of \\(r\\). (In a sense, this is orthogonal to our work, where \\(r\\) is fixed and \\(n\\) varies.) In general, very little is known about the \\(\\mathcal{C}^{(r)}(n)\\) for \\(n \\ge 3\\), even their dimension.\n\\begin{thm}[\\bfseries Finite generation for diagonal coinvariant algebras]\nThe sequence of coinvariant algebras \\(\\mathcal{C}^{(r)}(n)\\) form a graded co-\\(\\FI_{\\mathbb{Z}\/d\\mathbb{Z}}\\)-algebra of finite type.\n\\end{thm}\n\\cite[Thm 1.11]{CEF} proved the case \\(d = 1\\), and Wilson \\cite[Thm 7.8]{Wi} proved it for \\(d = 2\\).\n\\begin{proof}\nLet \\(V\\) be the \\(\\FI_G\\)-module associated to the \\(n\\)-dimensional \\(W_n\\)-representations \\(V_n = k^n\\), where \\(G\\) acts by multiplication by \\(\\zeta_d\\). Let \\(\\chi_1\\) be the character of \\(\\mathbb{Z}\/d\\mathbb{Z}\\) sending 1 to \\(\\zeta_d\\), and let \\(\\underline{\\lambda}\\) be the \\(G\\)-partition of 1 supported on \\(\\chi_1\\), so \\(\\underline{\\lambda}(\\chi_1) = (1)\\). Then \\(V_n = L(\\underline{\\lambda})_n \\oplus L(\\underline{\\lambda})\\). Thus \\(k[V^{\\oplus r}]\\) is naturally a co-\\(\\FI_G\\)-module of finite type. The ideals \\(\\mathcal{I}_n\\) together form a co-\\(\\FI_G\\)-ideal \\(\\mathcal{I}\\), determined by the \\(W_n\\) action and the maps\n\\begin{align*}\n(\\iota_n)^*: \\mathcal{I}_{n+1} &\\to \\mathcal{I}_n \\\\\nx_i &\\mapsto \\begin{cases} x_i & i \\le n \\\\ 0 & i = n+1 \\end{cases}\n\\end{align*}\nThus \\(\\mathcal{C}^{(r)}\\) is a co-\\(\\FI_G\\)-quotient of a co-\\(\\FI_G\\)-algebra of finite type, so it is a co-\\(\\FI_G\\)-algebra of finite type. Since \\(W_n\\) preserves the grading of \\(\\mathcal{C}^{(r)}(n)\\), then as a co-\\(\\FI_G\\)-module, \\(\\mathcal{C}^{(r)}\\) decomposes into graded pieces \\(\\mathcal{C}^{(r)}_J\\).\n\\end{proof}\nBy Noetherianity, each graded piece \\(\\mathcal{C}^{(r)}_J\\) is a finitely-generated co-\\(\\FI_G\\)-module by \\thref{alg}. So we obtain the following.\n\\begin{cor}\nFor each multi-index \\(J\\), the characters of the \\(W_n\\)-representations \\(\\mathcal{C}^{(r)}_J(n)\\) are given by a single character polynomial for all \\(n \\gg 0\\). \n\\end{cor}\n\n\n\\subsection{Lie algebras associated to the lower central series}\nFor any group \\(\\Gamma\\), recall that the \\emph{lower central series} is defined inductively by \\(\\Gamma_1 = \\Gamma\\) and \\(\\Gamma_{k+1} = [\\Gamma_k, \\Gamma]\\). The \\emph{associated graded Lie algebra} \\(\\gr(\\Gamma)\\) over a field \\(k\\) is defined as\n\\[\n\\gr(\\Gamma) = \\bigoplus_{k \\ge 1} (\\Gamma_{k+1}\/\\Gamma_k) \\otimes_{\\mathbb{Z}} k\n\\]\nwhere the Lie algebra structure is induced by the commutator. For example, if \\(\\Gamma\\) is a free group of rank \\(n\\), then \\(\\gr(\\Gamma)\\) is the free lie algebra of rank \\(n\\). Since the higher \\(\\Gamma_i\\) are generated by iterated commutators, \\(\\gr(\\Gamma)\\) is always generated as a Lie algebra by \\(\\gr(\\Gamma)_1 = H_1(\\Gamma;k)\\).\n\nNow, as \\cite{CEF} carefully note, \\(\\pi_1\\) is not a functor from topological spaces to groups, since it depends on a basepoint, and a map of spaces only determines a group homomorphism up to conjugacy. However, since conjugation acts trivially on the associated Lie algebra, there is a functor \\(Z \\mapsto \\gr \\pi_1(Z)\\) from the category Top of topological spaces to the category of graded Lie algebras. Thus \\(\\gr \\pi_1(\\Conf^G(M))\\) has the structure of a graded co-\\(\\FI_G\\)-algebra. \n\nHere we continue to look at the case \\(M = \\mathbb{C}^*\\), \\(G = \\mathbb{Z}\/d\\), so \\(\\pi_1 = P(d,n)\\). The associated graded Lie algebra \\(\\gr P(d,n)\\) has been studied, for example, by D. Cohen in \\cite{Co}, who obtained some partial results, but still, little is known about it.\n\\begin{thm}[\\bfseries Finite generation for \\(\\gr P(d,n)\\)]\nFor each \\(i\\), the characters of the \\(W_n\\)-representations \\(\\gr P(d,n)^i\\) are given by a single character polynomial for all \\(n\\).\n\\end{thm}\n\\begin{proof}\n\\(H_1(\\Conf^G(\\mathbb{C}^*)\\) is a finitely generated \\(\\FI_G\\sharp\\)-module by \\thref{crg_fg}. Since \\(H_1(\\Gamma)\\) generates \\(\\gr(\\Gamma)\\) as a Lie algebra, this means that \\(\\gr P(d,n)\\) is an \\(\\FI_G\\sharp\\)-algebra of finite type by \\thref{alg}. Thus by Noetherianity, its graded pieces are finitely generated \\(\\FI_G\\sharp\\)-modules.\n\\end{proof}\n\n\\cite[Thm 7.3.4]{CEF} proved the case \\(d = 1\\), for which \\(P(1,n) = P_n\\), the pure braid group.\n\n\\section{Affine arrangements}\nA motivating example of an \\(\\FI_G\\)-module for \\(G\\) infinite, and one with interesting applications to affine braid groups, is the analogue of \\(\\Conf^{\\mathbb{Z}\/d}(\\mathbb{C}^*)\\) when we ``let \\(d\\) go to infinity'' and replace \\(\\mathbb{Z}\/d\\) with the infinite group \\(\\mathbb{Z}\\).\nWhereas the \\(\\mathbb{Z}\/d\\)-cover of \\(\\mathbb{C}^*\\) is just homeomorphic to \\(\\mathbb{C}^*\\) again, the \\(\\mathbb{Z}\\)-cover of \\(\\mathbb{C}^*\\) is homeomorphic to \\(\\mathbb{C}\\). To be specific, we take \\(M = \\mathbb{C}\\) and \\(G = \\langle \\tau \\rangle \\cong \\mathbb{Z}\\) acting as translation by some \\(\\tau \\in \\mathbb{C}\\). Thus \\(M\/G\\) is homeomorphic to a cyclinder, which is homeomorphic to the punctured plane, and the quotient map is \\(\\exp(x\/2\\pi i \\tau): \\mathbb{C} \\to \\mathbb{C}^*\\).\n\nAs in the case of complex reflection groups, \\(\\Conf_n^G(M)\\) is the complement of the hyperplane arrangement\n\\[ \\mathcal{A}_n = \\{(z_i) \\in \\mathbb{C}^n \\mid z_i - z_j \\not\\in \\tau\\mathbb{Z}\\}\\]\nso one might hope that Orlik-Solomon gives the homology or cohomology. Unfortunately, \\(\\mathcal{A}_n\\) is an infinite arrangement, and their arguments hinge on inducting on the number of hyperplanes (by looking at the ``deleted'' and ``restricted'' arrangements), so their theorem is not directly applicable.\n\nHowever, a result of Cohen-Xicot\\'encatl \\cite{CX} provides a solution. They compute the homology and cohomology of a similar space---that of \\(\\Conf^{\\mathbb{Z}^2}(\\mathbb{C})\\), mentioned earlier, so that the quotient \\(\\mathbb{C}\/\\mathbb{Z}^2\\) is a torus---using a Fadell-Neuwirth type argument, that inducts on the dimension of the space, rather than the number of hyperplanes. \n\nIn fact, the argument they give proves the following general result. Following Falk-Randell \\cite{FR}, we first make the following definitions. Let \\(\\mathcal{A}\\) be an arrangement of hyperplanes in \\(\\mathbb{C}^n\\), and put \\(M = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\). Say that \\(M\\) is \\emph{strictly linearly fibered} if after a suitable linear change of coordinates the restriction to the first \\(n-1\\) coordinates is a fiber bundle projection with base \\(N\\) the complement of an arrangement in \\(\\mathbb{C}^{n-1}\\) and fiber a complex line with a discrete set of points removed. Now define a \\emph{fiber-type} arrangement inductively as follows: \\begin{itemize}\n\\item The 1-arrangement \\(\\{0\\} \\subset \\mathbb{C}\\) is \\emph{fiber-type}\n\\item An \\(n\\)-arrangement for \\(n \\ge 2\\) is \\emph{fiber-type} if the complement \\(M\\) is strictly linearly fibered over \\(N\\), and \\(N\\) is the complement of an \\((l-1)\\)-arrangement of fiber type.\n\\end{itemize}\nThus the complement \\(M\\) of a fiber-type \\(n\\)-arrangement sits atop a tower of fibrations\n\\begin{equation} \\label{fiber-type}\nM = M_n \\xrightarrow{p_n} M_{n-1} \\xrightarrow{p_{n-1}} \\cdots \\to M_2 \\xrightarrow{p_2} M_1 = \\mathbb{C}^*\n\\end{equation}\nwith the fiber \\(F_k\\) of \\(p_k\\) homeomorphic to \\(\\mathbb{C}\\) with a discrete set of points removed.\n\nIn particular, Falk-Randell use the tower of fibrations (\\ref{fiber-type}) to prove that \\(M\\) is aspherical, that each bundle map \\(p_k\\) has a section, and that \\(\\pi_1(M_{k-1})\\) acts trivially on \\(H_i(F_k)\\).\n\n\n\\begin{prop}[{\\cite[Prop 6]{CX}}]\nLet \\(\\mathcal{A}\\) be an arrangement of hyperplanes in \\(\\mathbb{C}^n\\) (possibly countably infinite and affine) which is discrete in the space of all hyperplanes. Put \\(M = \\mathbb{C}^n \\setminus \\bigcup \\mathcal{A}\\), and suppose \\(\\mathcal{A}\\) is fiber-type.\n\nThen \\(H^*(M; \\mathbb{Q})\\) contains a dense subalgebra \\(A\\) generated by classes \\(E_{H} \\in H^1(M; \\mathbb{Q})\\) for each \\(H \\in \\mathcal{A}\\), which is isomorphic to the Orlik-Solomon algebra of \\(\\mathcal{A}\\). Moreover, there is an isomorphism \\(H_i(M; \\mathbb{Q}) \\cong A^i\\).\n\\end{prop}\n\nCohen-Xicot\\'encatl use the tower of fibrations (\\ref{fiber-type}) to obtain this result. In a sense, this result was basically implicit in Falk-Randell \\cite{FR}, but there the arrangements were always assumed to be finite.\n\nThe algebra \\(A\\) is the algebra one gets by following the recipe of the Orlik-Solomon algebra of \\(\\mathcal{A}\\), even though \\(\\mathcal{A}\\) happens to be infinite, with each generator \\(E_H\\) coming from monodromy around the hyperplane \\(H\\). So the situation is really as good as one could hope for.\n\n\\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is closely related to affine Coxeter and Artin groups. The affine Coxeter group \\(W_{\\widetilde{A}_n}\\) is the one associated to the Coxeter diagram \\(\\widetilde{A}_n\\), which is obtained by adding one extra node connecting the first and last nodes to the Coxeter diagram \\(A_n\\). Concretely, let \\(V^\\mathbb{R} = \\{(x_i) \\in \\mathbb{R}^n \\mid \\sum x_i = 0\\}\\). Then \\(W_{\\widetilde{A}_{n-1}}\\) is the group generated by reflections in \\(V^\\mathbb{R}\\) through the hyperplanes \\(H^\\mathbb{R}_s = \\{x_i - x_j = n \\mid n \\in \\mathbb{Z}\\}\\). This group \\(W_{\\widetilde{A}_{n-1}}\\) is isomorphic to \\(\\mathbb{Z}^{n-1} \\rtimes S_n\\). Let \\(V\\) be the complexification of \\(V^\\mathbb{R}\\), and let \\(V_0 = V \\setminus \\bigcup H_s\\) the hyperplane complement. Then \\(\\pi_1(V_0)\\) is the pure Artin braid group \\(P_{\\widetilde{A}_{n-1}}\\), the group \\(W_{\\widetilde{A}_{n-1}}\\) acts freely on \\(V_0\\) and \\(\\pi_1(V_0\/W_{\\widetilde{A}_{n-1}})\\) is the Artin braid group \\(B_{\\widetilde{A}_{n-1}}\\). But notice that \\(V_0\\) is just the fiber of the bundle \\(\\Conf^\\mathbb{Z}(\\mathbb{C}) \\to \\mathbb{C}\\) given by taking the sum of the coordinates, which is trivial since \\(\\mathbb{C}\\) is contractible, so that \\(V_0 \\simeq \\Conf^\\mathbb{Z}(\\mathbb{C})\\). In fact, it has been proven \\cite{Ok} that \\(V_0\\) is aspherical, and therefore \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is a \\(K(P_{\\widetilde{A}_{n-1}}, 1)\\).\n\nWe then obtain the following.\n\n\\begin{thm}[\\bfseries Homology of type \\(\\widetilde{A}_n\\) pure braid group]\n\\(H_*(\\Conf^{\\mathbb{Z}}(\\mathbb{C}); \\mathbb{Q}) = H_*(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G \\sharp\\)-module. For each \\(i\\), the \\(W_n\\)-representations \\(H_i(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nBy \\thref{confg_fgsharp}, \\(H_*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module, while \\(H^*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) is an \\(\\FI_G\\sharp\\) algebra not of finitely-generated type.\n\nTaking the usual coordinates on \\(\\mathbb{C}^n\\), we see that \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\) is a fiber-type arrangement. Proposition 5.1 therefore applies, and we obtain a presentation for \\(H_*(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\).\n\nRecall \\cite{OT} that for affine arrangements, the Orlik-Solomon ideal is generated not just by \\(\\partial e_S\\) for \\(S\\) dependent, but also by \\(e_S\\) when \\(\\cap S = \\emptyset\\) (possible for non-central arrangements), which is derived by considering the homogeneized arrangement. In the case of \\(\\Conf^\\mathbb{Z}(\\mathbb{C})\\), \\(\\mathcal{A}\\) consists of the affine hyperplanes \\(H_{i,a,j} = \\{z_i - z_j = a \\mid a \\in \\mathbb{Z}\\}\\), and thus\n\\[\nA = \\bigwedge\\langle e_{i,a,j} \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,a+b,k} \\\\ e_{i,a,j} \\wedge e_{i,b,j} = 0 \\end{matrix} \\right\\rangle\n\\]\nand \\(H_i(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q}) \\cong A^i\\). As an algebra, \\(A\\) is generated by \\(A^1\\), which is generated as an \\(\\FI_G\\)-module in degree 2, and so by \\thref{alg}, \\(A^i\\) is generated in degree \\(2i\\). So by Theorem 2.5, \\(H_i(\\Conf^\\mathbb{Z}(\\mathbb{C}); \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(4i\\).\n\n\\end{proof}\nFurthermore, since the Coxeter group \\(W_{\\widetilde{A}_{n-1}} \\cong \\mathbb{Z}^{n-1} \\rtimes S_n\\), then \\(V_0 \/ W \\simeq \\UConf'_n(\\mathbb{C}^*)\\), in the notation of \\S3.1, and therefore \\(\\UConf'_n(\\mathbb{C}^*)\\) is a classifying space for the full Artin group \\(B_{\\widetilde{A}_{n-1}}\\). Here, Callegaro-Moroni-Salvetti \\cite[Thm 4.2]{CMS} computed that\n\\begin{align*}\nH_i(B_{\\widetilde{A}_{n-1}}; \\mathbb{Q}) = \\begin{cases} \\mathbb{Q}[t^{\\pm 1}]\/(1 + t) & 1 \\le i \\le n - 2 \\\\ \\mathbb{Q}[t^{\\pm 1}]\/(1 + t) & \\text{if } i = n-1, n \\text{ odd} \\\\ \\mathbb{Q}[t^{\\pm 1}]\/(1 - t^2) & \\text{if } i = n-1, n \\text{ even} \\end{cases}\n\\end{align*}\nas \\(\\mathbb{Q}[G] = \\mathbb{Q}[t^{\\pm 1}]\\)-modules. This verifies the stabilization as \\(\\mathbb{Q}[G]\\)-modules for \\(n \\gg 0\\) proven in Corolarry 3.5, and in fact stabilization occurs once \\(n \\ge i + 2\\).\n\nThese spaces also have close connections to the Burau representation. First, if we let\n\\[\\UConf_{n, 1}(\\mathbb{C}) = \\{(\\{x_1, \\dots, x_n\\}; x) \\mid x_i \\ne x_j, x_i \\ne x\\}\\] then there is a bundle \n\\[\\UConf_n(\\mathbb{C}^*) \\to \\UConf_{n,1}(\\mathbb{C}) \\to \\mathbb{C}.\\]\nSince \\(\\mathbb{C}\\) is contractible, \\(\\UConf_n(\\mathbb{C}^*) \\simeq \\UConf_{n,1}(\\mathbb{C})\\). Similarly, let\n\\[\\UConf'_n(\\mathbb{C}^*) = \\{(\\{x_1, \\dots, x_n\\}; y) \\mid x_i \\ne 0, x_i \\ne x_j, e^y = x_1 \\cdots x_n\\}\\]\nand let\n\\[\\UConf'_{n, 1} = \\{(\\{x_1, \\dots, x_n\\}; x, y) \\mid x_i \\ne x, x_i \\ne x_j, e^y = (x_1 - x) \\cdots (x_n - x)\\}.\\]\nThen \\(\\UConf'_n(\\mathbb{C}^*) \\cong \\UConf'_{n, 1}\\). Finally, notice that the inclusion \\(\\UConf_n(\\mathbb{C}^*) \\hookrightarrow \\UConf_n(\\mathbb{C})\\) is homotopy equivalent to the bundle projection \\(\\UConf(\\mathbb{C})_{n,1} \\twoheadrightarrow \\UConf(\\mathbb{C})\\) onto the first \\(n\\) coordinates. We therefore obtain the following diagram:\n\\begin{equation*}\n\\begin{diagram}\n\\UConf'_n(\\mathbb{C}^*) && \\simeq && \\UConf'_{n,1}(\\mathbb{C}) \\\\\n\\dTo &&&& \\dTo \\\\\n\\UConf_n(\\mathbb{C}^*) && \\simeq && \\UConf_{n,1}(\\mathbb{C}) \\\\\n& \\rdTo && \\ldTo & \\\\\n&& \\UConf_n(\\mathbb{C}) &&\n\\end{diagram}\n\\end{equation*}\nThus \\(\\pi_1(\\UConf_n(\\mathbb{C})) = B_n\\) acts up to conjugacy on the homotopy fiber of the map from \\(\\UConf'_n(C^*)\\), which is just the actual fiber of the map from \\(\\UConf'_{n,1}(\\mathbb{C})\\). Call this fiber \\(X_f\\) for \\(f \\in \\UConf_n(\\mathbb{C})\\). This gives an honest action of \\(B_n\\) on \\(H_1(X_f; \\mathbb{C})\\), which is the Burau representation. Furthermore, replacing the group \\(G = \\mathbb{Z}\\) with \\(\\mathbb{Z}\/d\\mathbb{Z}\\), this construction gives the Burau representation specialized at \\(\\zeta_d = e^{2\\pi i\/d}\\). Chen has explored this connection and its relation to point-counts of superelliptic curves in \\cite{WCh}; in his notation, with \\(G = \\mathbb{Z}\/d\\mathbb{Z}\\), \\(\\UConf'_{n,1}(\\mathbb{C})\\) is called \\(E_{n,d}\\). \n\nFinally, we likewise obtain results on the pure braid group of type \\(\\widetilde{C}_n\\).\n\\begin{thm}[\\bfseries Homology of the type \\(\\widetilde{C}_n\\) pure braid group]\n\\(H_*(P_{\\widetilde{C}_n}; \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\) module, for \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\). For each \\(i\\), the \\(W_n\\)-representation \\(H_i(P_{\\widetilde{C}_n}; \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(\\le 4i\\).\n\\end{thm}\n\\begin{proof}\nAs \\cite[\\S4.3]{Mor} explains, the hyperplane complement for type \\(\\widetilde{C}_n\\) is the following:\n\\[ Y_n = \\{x \\in \\mathbb{C}^n \\mid x_i \\pm x_j \\notin \\mathbb{Z}, x_i \\notin \\tfrac{1}{2}\\mathbb{Z}\\} \\]\nApplying the map \\(z \\mapsto \\exp(2\\pi i z)\\) componentwise gives a normal covering map \\(Y_n \\to Y'_n\\) with deck group \\(\\mathbb{Z}^n\\), where\n\\[ Y'_n = \\{x \\in (\\mathbb{C}^*)^n \\mid x_i \\ne x_j^{\\pm 1}, x_i \\ne \\pm 1\\} \\]\nThe map \\(z \\mapsto \\frac{1-z}{1+z}\\) is a self-homeomorphism of \\(\\mathbb{C} \\setminus \\{0,1,-1\\}\\) taking \\(1\/z\\) to \\(-z\\). Applying this map componentwise to \\(Y'_n\\) therefore gives a homeomorphism\n\\[ Y'_n \\cong \\{x \\in \\mathbb{C}^n \\mid x_i \\ne \\pm x_j, x_i \\ne 0, 1, -1\\} \\]\nFinally, applying the map \\(z \\mapsto z^2\\) componentwise gives a normal covering map \\(Y'_n \\to \\Conf_n(\\mathbb{C} \\setminus \\{0,1\\})\\), with deck group \\((\\mathbb{Z}\/2)^n\\). In conclusion, if we write \\(M = \\mathbb{C} \\setminus \\frac{1}{2}\\mathbb{Z}\\), there is a normal cover \\(M \\to \\mathbb{C} \\setminus \\{0,1\\}\\) given by \\(z \\mapsto \\left( \\dfrac{1 - e^{2\\pi i z}}{1 + e^{2\\pi i z}}\\right)^2\\) with deck group \\(G = \\mathbb{Z} \\rtimes \\mathbb{Z}\/2\\), and \\(Y_n\\) is homeomorphic to the orbit configuration space \\(\\Conf^G_n(M)\\). \n\nBy \\thref{confg_fgsharp}, \\(H_*(\\Conf^G(M); \\mathbb{Q})\\) is a finitely-generated type \\(\\FI_G\\sharp\\)-module. Again, we use Prop 5.1 to obtain a presentation. The hyperplane arrangement consists of the hyperplanes \\(E_{i,a,j} = \\{z_i - z_j = a \\mid a \\in \\mathbb{Z}\\}\\), \\(H_{i,a,j} = \\{z_i + z_j = a \\mid a \\in \\mathbb{Z}\\}\\), and \\(Z_{i,a} = \\{2z_i \\ne a \\mid a \\in \\mathbb{Z}\\}\\). The Orlik-Solomon algebra is given by:\n\\[ A = \\bigwedge \\langle e_{i,a,j}, h_{i,a,j}, z_{i,b} \\rangle \\bigg\/ \\left\\langle \\begin{matrix} e_{i,a,j} \\wedge e_{j,b,k} = (e_{i,a,j} - e_{j,b,k}) \\wedge e_{i,a+b,k}; \\;\\; e_{i,a,j} \\wedge e_{i,b,j} = 0 \\\\ \ne_{i,a,j} \\wedge h_{j,b,k} = (e_{i,a,j} - h_{j,b,k}) \\wedge h_{i,a+b,k}; \\; \\; h_{i,a,j} \\wedge h_{i,b,j} = 0  \\\\\ne_{i,a,j} \\wedge h_{i,b,j} = (e_{i,a,j} - h_{i,b,j}) \\wedge z_{i, a+b}; \\; \\; z_{i,a} \\wedge z_{i,b} = 0 \\\\\nz_{i,a} \\wedge z_{j,b} = (z_{i,a} - z_{j,a}) \\wedge h_{i, \\frac{a+b}{2}, j} \\text{ if } 2 \\mid a + b, 0 \\text{ otherwise}\n\\end{matrix} \\right\\rangle \\]\nand \\(H_i(\\Conf^G(M); \\mathbb{Q}) \\cong A^i\\). In particular, we see that this \\(\\FI_G\\sharp\\)-module is generated in degree \\(2i\\). So by Theorem 2.5, \\(H_i(\\Conf^G(M); \\mathbb{Q})\\) satisfies \\(K_0\\)-stability with stability degree \\(4i\\).\n\n\\end{proof}\nIn particular, we can again use the calculation (\\ref{irr_decom}) to write \\(H_1\\) as an induced module:\n\\begin{equation} \\label{affC}\nH_1(P_{\\widetilde{C}_{n-1}}; \\mathbb{Q}) = \\Ind_1^{\\FI_{G}} k[\\mathbb{Z}] \\oplus \\Ind_2^{\\FI_{G}} (V \\oplus W)\n\\end{equation}\nwhere \\(V\\) and \\(W\\) are both isomorphic as vector spaces to \\(k[\\mathbb{Z}]\\), but as representations of \\(k[\\mathbb{Z} \\times \\mathbb{Z}]\\), \\(V\\) has one copy of \\(\\mathbb{Z}\\) acting positively and one acting negatively, whereas \\(W\\) has both acting positively. In this case, the image of \\(V\\) and \\(W\\). In this case, \\([V] = [W] = 0\\) in \\(K_0(\\mathbb{Z} \\times \\mathbb{Z})\\), so we as expected we lose a lot of information in passing to \\(K_0\\). However, the first summand in (\\ref{affC}) is retained in \\(K_0\\). To wit, the representation \\(U = k[\\mathbb{Z}]\\) of \\(\\mathbb{Z} \\rtimes \\mathbb{Z}\\) is just \\(\\Ind_{\\mathbb{Z}\/2}^{\\mathbb{Z} \\rtimes \\mathbb{Z}\/2} k\\), so by Moody's induction theorem, this is nonzero in \\(K_0(\\mathbb{Z} \\rtimes \\mathbb{Z}\/2)\\). We therefore obtain, as we saw in the introduction,\n\\[ [H_1(P_{\\widetilde{A}_{n-1}}; \\mathbb{Q})] = [\\Ind_{W_1 \\times W_{n-1}}^{W_n} U] = [L((n)_U)] \\]\n\n\\section{Automorphism groups of free products}\nGiven a group \\(G\\), let \\(G^{*n} = G \\ast \\cdots \\ast G\\) be the \\(n\\)-fold free product. The group \\(\\Aut(G^{*n})\\) contains a copy of \\(\\Aut(G) \\wr S_n\\), acting by automorphisms on each factor and permuting them. This normalizes the \\emph{Fouxe-Rabinovitch} group \\(\\FR(G^{*n})\\), the subgroup of \\(\\Aut(G^{*n})\\) generated by partial conjugations, which conjugate the \\(i\\)-th free factor by the \\(j\\)-th free factor for some \\(i \\ne j\\), and leaves fixed all factors besides the \\(i\\)-th.\n\nDefine the \\emph{symmetric automorphism group} \\(\\Sigma\\Aut(G^{*n})\\) by\n\\[\\Sigma\\Aut(G^{*n}) = \\FR(G^{*n}) \\rtimes (\\Aut(G)\\,\\wr\\,S_n)\\]\nWhen \\(G\\) is freely indecomposable and not isomorphic to \\(\\mathbb{Z}\\), then \\(\\Sigma\\Aut(G^{*n})\\) is in fact all of \\(\\Aut(G^{*n})\\).\n\nThese groups have interesting geometric interpretations. For example, if \\(G = \\mathbb{Z}\\) then \\(\\Sigma\\Aut(G^{*n})\\) is the fundamental group of \\(n\\) unknotted circles embedded in \\(\\mathbb{R}^3\\), or the \\emph{group of string motions}, and \\(\\FR(G^{*n})\\) is the corresponding group of \\emph{pure} string motions. More generally, Griffin \\cite{Gr} constructed a classifying space for \\(\\FR(G^{*n})\\), which he calls the \\emph{moduli space of cactus products}.\n\nThe cohomology of these groups has also been much studied. For example, Hatcher-Wahl \\cite{HW} and Collinet-Djament-Griffin \\cite{CDG} proved that \\(\\Sigma\\Aut(G^{*n})\\) satisfy homological stability. On the other hand, the cohomology of \\(\\FR(G^{*n})\\) does not stabilize: e.g., \\(H^1(\\FR(G^{*n})) = H^1(G)^{\\oplus n(n-1)}\\). \n\nHowever, notice that the exact sequence\n\\[1 \\to \\FR(G^{*n}) \\to \\Sigma\\Aut(G^{*n}) \\to \\Aut(G) \\wr S_n \\to 1\\]\ngives an action of \\(\\Aut(G) \\wr S_n\\) on \\(H^i(\\FR(G^{*n}); \\mathbb{Q})\\). In fact, inner automorphisms of \\(G\\) act trivially on cohomology, so in fact we get a representation of \\(\\Out(G) \\wr S_n\\) on \\(H^i(\\FR(G^{*n}); \\mathbb{Q})\\). Wilson \\cite[Thm 7.3]{Wi} proved that \\(H^*(\\FR(\\mathbb{Z}^{*n}))\\) is a finite type  \\(\\FI_{\\mathbb{Z}\/2}\\)-algebra, since \\(\\Out(\\mathbb{Z}) = \\mathbb{Z}\/2\\).\n\\begin{thm}[\\bfseries Finite generation for cohomology of Fouxe-Rabinovitch groups]\nSuppose that \\(G\\) is any group such that \\(H^*(G; k)\\) is a finite type \\(k\\)-module. Then \\(H^*(\\FR(G^{*n}); k)\\) is a finite type \\(\\FI_{\\Out G}\\sharp\\)-algebra.\n\\end{thm}\nNotice that, for example, there are many groups with even no nontrivial outer automorphisms, for which Theorem 6.1 tells us that \\(H^*(\\FR(G^{*n}); \\mathbb{Q})\\) is a finite type \\(\\FI\\sharp\\)-algebra.\n\n\\begin{proof}\nWe first need to show that \\(H^*(\\FR(G^{*n}))\\) is, in the first place, an \\(\\FI_{\\Out(G)}\\sharp\\)-algebra. Griffin \\cite[Thm B]{Gr} proved that \\(H^*(\\FR(G^{*n})) \\cong H^*((G^{*n})^{n-1})\\) as an algebra, so \\(H^*(\\FR(G^{*n}))\\) is generated by \\(\\widetilde{H}^*(G)^{\\oplus n(n-1)} \\cong \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)\\). Under this isomorphism, for any \\(\\alpha \\in \\widetilde{H}^*(G)\\), write \\(\\alpha_{i,j}\\) for the element \\(\\alpha_{i,j} \\in \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)\\) sitting in the \\((i,j)\\)-th summand. The map induced by an \\(\\FI_{\\Out(G)}\\sharp\\) morphism \\((s,\\vec \\phi): [m] \\to [n]\\) is:\n\\begin{align*} (s,\\vec \\phi)_*: \\bigoplus_{i \\ne j} \\widetilde{H}^*(G) &\\to \\bigoplus_{i \\ne j} \\widetilde{H}^*(G) \\\\\n\\alpha_{i,j} & \\mapsto \\begin{cases} \\phi^*_j\\,\\alpha_{s(i), s(j)} & \\text{if } i, j \\in \\operatorname{domain}(s) \\\\\n0 & \\text{otherwise} \\end{cases}\n\\end{align*}\nTherefore  \\(V_n = \\bigoplus_{i \\ne j} \\widetilde{H}^*(G)_{i,j}\\) has the structure of an \\(\\FI_{\\Out(G)}\\sharp\\)-module. \\(V\\) is generated in finite degree, since it is just generated by \\(\\widetilde{H}^*(G)_{1,2}\\), and each \\(V^i_n\\) is finite-dimensional, since by assumption each \\(H^j(G)\\) is, so therefore \\(V\\) is finitely generated. Furthermore, the action of \\(\\FI_{\\Out(G)}\\sharp\\) on \\(V\\) extends to an algebra map of \\(H^*(\\FR(G^{*n}))\\), which is thus an \\(\\FI_{\\Out(G)}\\sharp\\)-algebra. Since it is generated by the finite type module \\(V\\), then \\(H^*(\\FR(G^{*n}))\\) is finite type by \\thref{alg}.\n\\end{proof}\n\\begin{cor}[\\bfseries Representation stability for cohomology of Fouxe-Rabinovitch groups]\nSuppose that \\(G\\) is any group with \\(H^*(G; \\mathbb{Q})\\) of finite type and \\(\\Out G\\) finite. Then for each \\(i\\), the characters of the \\(\\Out(G) \\wr S_n\\)-representations \\(H^i(\\FR(G^{*n}); \\mathbb{C})\\) are given by a single character polynomial of degree \\(2i\\) for all \\(n\\), and therefore \\(H^i(\\FR(G^{*n}); \\mathbb{C})\\) is representation stable, with stability degree \\(4i\\).\n\\end{cor}\n\\begin{proof}\nAs shown in the proof of Theorem 7.1, \\(H^*(\\FR(G^{*n})\\) is generated as an algebra by \\(V\\), which is generated as an \\(\\FI_G\\)-module by \\(\\widetilde{H}^*(G)_{1,2}\\), and so \\(V\\) is generated in degree 2. By \\thref{alg}, \\(H^i\\) is generated in degree \\(2i\\), and so the stable range follows from \\thref{sharp_range}.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztctf b/data_all_eng_slimpj/shuffled/split2/finalzztctf
new file mode 100644
index 0000000000000000000000000000000000000000..277c3c4449c896d13aa661357bea738a32b53ba9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzztctf
@@ -0,0 +1,5 @@
+{"text":"The need to tailor release-rate profiles from polymeric microspheres remains one of the leading challenges in controlled drug delivery. Microsphere size, which has a significant effect on drug release rate, can potentially be varied to design a controlled drug delivery system with desired release profile. In addition, drug release rate from polymeric microspheres is dependent on material properties such as polymer molecular weight. Mathematical modeling provides insight into the fundamental processes that govern the release, and once validated with experimental results, it can be used to tailor a desired controlled drug delivery system. To these ends, PLG microspheres were fabricated using the oil-in-water emulsion technique. A quantitative study that describes the size distribution of poly(lactide-coglycolide) (PLG) microspheres is presented. A fluid mechanics-based correlation that predicts the mean microsphere diameter is formulated based on the theory of emulsification in turbulent flow. The effects of microspheres' mean diameter, polydispersity, and polymer molecular weight on therapeutic drug release rate from poly(lactide-co-glycolide) (PLG) microspheres were investigated experimentally. Based on the experimental results, a suitable mathematical theory has been developed that incorporates the effect of microsphere size distribution and polymer degradation on drug release. In addition, a numerical optimization technique, based on the least squares method, was developed to achieve desired therapeutic drug release profiles by combining individual microsphere populations. The fluid mechanics-based mathematical correlation that predicts microsphere mean diameter provided a close fit to the experimental results. We show from in vitro release experiments that microsphere size has a significant effect on drug release rate. The initial release rate decreased with an increase in microsphere size. In addition, the release profile changed from first order to concave-upward (sigmoidal) as the microsphere size was increased. The mathematical model gave a good fit to the experimental release data. Using the numerical optimization technique, it was possible to achieve desired release profiles, in particular zero-order and pulsatile release, by combining individual microsphere populations at the appropriate proportions. Overall, this work shows that engineering polymeric microsphere populations having predetermined characteristics is an effective means to obtain desired therapeutic drug release patterns, relevant for controlled drug delivery.\nBerchane, Nader Samir (2007). Experimental and computational investigations of therapeutic drug release from biodegradable poly(lactide-co-glycolide) (plg) microspheres. Doctoral dissertation, Texas A&M University. Available electronically from http : \/ \/hdl .handle .net \/1969 .1 \/ETD -TAMU -2536.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here at Camp Nicolet, our mission is to provide a safe, fun, and inclusive environment in which girls develop an understanding of and appreciation for self, others, and the natural world through activities, community living, and outdoor adventure.\nWe believe that a Nicolet camper experiences individual growth by \"learning to live together\" with her peers and counselors while exercising her independence and personal autonomy through free choice.\nEnroll in a week of fun for your family. Where families come together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"to make it until the end.\nwill not take care of you.\nand according to his faith.\nwhile you still can do so.\nif God is not there.\nServe me. I wish to be served.\nand I will show up to you.\nGod is always with His children in any condition we find ourselves, and He is always ready to save.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spoiler Alert! Don't read this report if you haven't played The Goblin Grail Gang!\nA note from the D&D Dad: I'd normally create two separate articles for this adventure session, but this one-shot by M.T. Black is special. Not only is it a great introduction to new players (I had one at this session), but the entire adventure takes place on ONE sheet of paper! So I'm simply combining my Adventure Report with my Review.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nike Free OG BR NSW Running Shoes feature Nike's natural motion technology focused in a great low profile and flexible shoe to promote stronger feet which can reduce injury and improve performance. The flex groove outsole made up of deep cuts along the length and width of the midsole enhances a natural range of motion, encourages a smooth, efficient stride and helps maintain stability. A deconstructed breathable upper provides a shoe that is soft, flexible and ready to move. This provides a great feel with or without socks, along with the deconstructed heel counter, which enhances your comfort. An ORTHOLITE\u00ae sock liner creates a cooler, drier, healthier, better cushioned environment under the foot. Reflective detailing ensures you remain safe and seen in low light conditions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzttuo b/data_all_eng_slimpj/shuffled/split2/finalzzttuo
new file mode 100644
index 0000000000000000000000000000000000000000..1d7844392c65d89c4f08e605a96df0e8cc2bff76
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzttuo
@@ -0,0 +1,5 @@
+{"text":"Fortunately for Borussia Dortmund, there was a only a slight echo of last season's Matchday 17 defeat to SV Werder Bremen in their final competitive outing of the calendar year against 1. FC K\u00f6ln on Saturday afternoon.\nThe 2-1 scoreline was the same, but that is where the similarities end. While BVB finished the 2014\/15 Hinrunde in the relegation zone, they go into the new year in second place, eight points behind Bundesliga leaders FC Bayern M\u00fcnchen.\nSaturday's setback in the Domstadt aside, Dortmund can look back on 2015 with pride rather than disappointment. BVB rallied at the turn of the year to climb out of the relegation zone, eventually qualifying for the Europa League via the play-offs after finishing seventh in the Bundesliga, and reached the DFB Cup final. Now they are challenging for honours on all three fronts.\n\"It's been a great Hinrunde for us, even if today's result detracts from that somewhat,\" affirmed Dortmund captain Mats Hummels on his way out of the stadium. Given the club's about-turn in fortunes, however, perhaps 'remarkable' would be a more apposite description.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There's good news and bad news in regards to the Bruins' latest injury woes.\nThe bad: center Marc Savard may miss up to a month with a Grade 2 partial MCL tear in his right knee, according to the team. The good: The B's front-line center will not require surgery.\nSavvy is the latest black and blue member of the Black and Gold as defenders Andrew Ference (groin \u2014 out six weeks) and Mark Stuart (sternum \u2014 out 2-3 weeks) are both on the shelf as well. Winger Byron Bitz is day-to-day with a back injury and is out for Saturday's game against the Rangers while center Patrice Bergeron is out for at least another two weeks with a broken thumb.\nSavard suffered his knee injury in the first period against the Blackhawks on Jan. 7, the same night winger Milan Lucic returned from his high ankle sprain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is your VPS desktop!\n+ How to connect to Trading VPS through using Smartphone or Tablet?\nClient can choose to use any other 3rd party application.\nThis guide is to optimize your Metatrader 4 terminal(s) so it will use less computing resources - Memory (RAM) and CPU. It assumes that you are not trading manually on your MT4 as it will turn off some MT4 functions that will disable readings.\nStep1: Go to: Tools -> Options or CTRL+O.\nStep2: Reduce Chart Bars - Go to Charts -> Un-check all the options and change bars to \"1\".\nStep3: Disable News - Go to Server -> Un-check \"Enable news\".\nStep3: Disable MT4 Audio - Go to Events -> Un-check \"Enable\" and click \"OK\".\nStep4: Hide all unnecessary symbols that you are now trading. Go to: Market Watch > Select the symbol > Right click and select \"Hide\".\nStep5: Close all viewing feeds - Go to: View -> Turn off Market Watch or CTRL+M -> Turn off Data Window or CTRL+D -> Turn off Navigator or CTRL+N -> Turn off Terminal or CTRL+T.\nStep7: Close all unused charts \u2013 Only open chart that you are using for EA(s)\/Indicator(s).\nStep 1: Install the MT4 by double clicking on your broker MT4 Install.\nYou can give any name for the folder but make sure that the new terminals folder location does not exist yet.\nStep4: Go to Settings and Enabled all these features as shown.\nthem into the list. Enabled the features as shown.\nStep7: Perform a restart and log in after 30 minutes. All MT4 should re-open automatically upon logging.\nIt is our pleasure to help you.\n+ How to transfer files to Trading VPS?\nStep4: You will find all of the available drives on your computer that can connect to your Trading VPS. Select a drive(s) that you want to share with your Trading VPS. After that click OK and Connect.\nOnce you have connected, go to My Computer on your Trading VPS System, there you will find the drive(s) that you have selected and you can start transferring your files.\nTradingFXVPS will not store any of users' new passwords. Please contact our live chat or support staff if you forgot your password. A password reset will be required if you lost it and it will take within 48 hours to reset the password.\n+ How to extend your VPS Disk Space?\nThis guide shows you how to extend your VPS disk space if you upgraded your disk space.\nThere will be no unallocated disk space now!\nThis guide show you how to retrieve your broker server IP address \/ file.\nExample : Your MT4 account is an Standard Account , Upload Standard Server File.\nWe will get back to you within 24 hours on the best location VPS for your broker!\nThis guide show you how to obtain your Broker server IP address and latency on your VPS\/PC.\nYou can open by going Search > type \"CMD\".\nYou will get your Broker latency on the VPS now. Latency within 0ms-10ms are considered low-latency.\nAnuko World Clock is a program for Windows that replaces the regular system clock with several time zones in which you can choose. Knowing the time around the world has never been simpler. Whether you are an enterprise in need of an efficient timezone monitoring utility, or a home user with a loved-one in another country, or stock and currency trader, this clock is a software you should not live without.\nIn the example below, you can see that Sydney, Tokyo, and London markets are closed, while New York is open.\nNetTime, like most SNTP clients, will generally sync your system time to within 100 milliseconds of the correct time \u2013 often within 10 milliseconds.\nYou can download NetTime HERE.\nThe MT4 Auto-restart EA watches for these connection issues. It can be configured to send alerts, and also to restart MT4 automatically if there is a problem. This restart is usually enough to resolve the problem and re-establish the connection.\nYou can find more information and download it HERE.\nAccount Monitor watch for problems on your accounts, such as low free margin or a loss of connection to your broker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"F.U.Z.E. uses the same zipper technology as our patented Z.E.N. Zip closure to keep you dry and feeling loose.\nFresh for 2017 with all new TechnoButter 3 and added heat-generating TB Firewall. It's offered in a wide range of styles with the hi-tech features you need to stay warm and toasty in any condition. Psycho Tech is the hardest working wetsuit in the business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This purifying mask contains bentonite clay, activated charcoal and white willow bark extract to remove excess oil from skin.\nInstructions: Mix a spoonful of clay with water, honey, yogurt or apple cider vinegar to form a paste and apply to clean skin. Leave mask on for 10-15 minutes and rinse off. Please note that charcoal can stain fabrics as well as skin. If your skin appears dark after rinsing, follow up with a gentle washcloth and cleanser.\nIngredients: Bentonite clay, Activated Charcoal, Salix Alba (White Willow Bark) Extract.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzurfy b/data_all_eng_slimpj/shuffled/split2/finalzzurfy
new file mode 100644
index 0000000000000000000000000000000000000000..1cb94dd8fea84966cc3f4eea82066391ed5e253f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzurfy
@@ -0,0 +1,5 @@
+{"text":"Mearns FM and Radio North Angus are voluntary sub-regional independent local radio stations broadcasting purely localised programming across a wide area.\nThe FM coverage area extends from Monifieth, to the east of Dundee, to Portlethen, south of Aberdeen, utilising 10 transmitters across a 70 mile stretch of the East Coast of Scotland, whilst the Digital transmissions, during mornings and more extensively at weekends, includes Tayside and North Fife.\nThe FM area includes a population of 90,000, with Digital including 400,000.\nBeing entirely voluntary, and enjoying cost-free accommodation, the stations are distinctly different from larger independent radio as operational costs are much lower. The programme formats reflect the needs of the population by having programming aimed at the very young, to the very old as opposed to principally the 15 to 34 age group adopted by larger stations. Programme length ranges from one to four hours, thus ensuring a rapid change of presentation styles and music genres in the course of each day, which differs from the independent radio norm of four hourly programming.\nBoth stations are dedicated to their communities by promoting health and welfare, education, voluntary organisations, and their respective local economies.\nOur low operational costs allow us to offer the lowest joint advertising rates in British independent local radio.\nAdvertising is for one, two or three month periods, but can be extended to twelve months, if desirable.\n1. 10 adverts per day (5 per station) for one month i.e. 310 adverts for \u00a3500 (\u00a31.61p per advert).\n2. 16 adverts per day (8 per station) for two months i.e 992 adverts for \u00a3750 (\u00a3375 per month, 76p per advert).\n3. 20 adverts per day (10 per station) for three months i.e 1860 adverts for \u00a31000 (\u00a3333 per month, 54p per advert).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Book The Hummingbird Cabarete (Dominican Republic) - 2019 PRICES FROM $20!\nThis 2-level Hummingbird Cabarete aparthotel features an outdoor pool with a sun terrace, table tennis, and rooms with free WiFi. The beach is 1476 ft away. A pool and garden view can be enjoyed from all apartments. A full kitchen with kitchenware and a seating area are standard in Hummingbird Cabarete apartments as well. Some apartments have a balcony. A shared TV lounge, garden, and tour desk with ticket service are on site for added convenience. Massages, a bike hire, and transfer service are available for a fee. Guests can find a variety of bar and restaurants within a 5-minute walk. The property is 1.3 mi from Cabarete city center and 9.3 mi from Gregorio Luperon International Airport. Caves, lagoons, and freshwater springs are 26 minutes' drive away at El Choco National Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy as much car insurance as you can afford.\nNot only do you probably need more liability coverage -- almost all people do -- but the extra is usually downright cheap.\nConsider a driver in California, where the law requires a very low minimum of $15,000 per person in bodily liability coverage, up to $30,000 per accident, and $5,000 in property damage liability insurance. In a worst-case scenario, the insurance company would have to pay out $35,000.\nAn adult male driver in ZIP code 94608 -- in the Bay Area -- could buy that state minimum liability coverage for as little as $470 a year. But he could buy 10 times as much liability coverage for just a few dollars more.\nMost experts will advise anyone with assets such as a home or savings to buy at least 100\/300\/50. After all, the typical new car costs $30,000 these days. Hit somebody in a new Lexus and you're on the hook for whatever your insurance company doesn't pay.\nThose rates are for a 50-year-old man with no ticket or accidents. While rates go up across the board with a worse driving record, the extra coverage still remains a fraction of the overall cost. Here's what the rates look like from two insurers if two speeding tickets for 10 mph over the limit are added.\nAn at-fault accident with $10,000 total damage raises the younger driver's rates substantially, but the additional coverage is still a bargain if he has substantial savings or a home.\nRates were calculated on ZIP code 94608 for a 50-year-old single male with a college degree, a homeowner, driving a paid-for 2002 Buick Regal LS 12,000 miles a year. His current insurer is State Farm; his current coverage level is 100\/300\/50. These annual rates are for liability coverage only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"216.2(1) An operator of an industrial lift truck shall check for the effectiveness of all safety devices daily before operating the truck.\n216.2(2) If more than one operator uses an industrial lift truck in the course of a day or if the industrial lift truck is used on more than one shift, each operator shall check for the effectiveness of all safety devices before operating the truck.\n229.2(1) An operator of powered mobile equipment shall check for the effectiveness of all safety devices daily before operating the equipment.\n229.2(2) If more than one operator uses powered mobile equipment in the course of a day or if the powered mobile equipment is used on more than one shift, each operator shall check for the effectiveness of all safety devices before operating the equipment.\n230.5(1) An operator of a vehicle shall check daily for the effectiveness of all safety devices before operating the vehicle.\n230.5(2) If more than one operator uses a vehicle in the course of a day or if the vehicle is used on more than one shift, each operator shall check for the effectiveness of all safety devices before operating the vehicle.\nTo meet the requirements of these sections, must each user physically check the safety devices daily?\nNot all users are required to physically check the safety devices each day provided the employer puts in place a system that allows for one of the users to physically check the devices at the beginning of each day and the remaining users to easily verify that the safety check was performed by the designated person.\nIt is also necessary that confirmation that the daily check was performed be posted at or near the equipment controls so that the remaining users can readily verify that the check was made.\nIt is acceptable that the person designated to perform the daily physical checks be rotated through the user group, provided that the designated person is aware of the requirements in the procedure and that the other users have a way to determine who the designated person is at the time of use. Notwithstanding this, users who wish to physically check the safety devices are allowed to do so by the employer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KiaraAlexisKlay wrote: Brendan Fehr is Canadian, so he's probably Anglo-French, maybe a little Native?\nMaria (Majandra) Delfino is actually Brazillian or Portugeuese isn't she? Somewhere down in South\/Central America, lol.\nI think Brendan's ancestry is actually German, and Majandra is from Venezuela.\nAugustus Snodgrass wrote: Jason Behr is actually Italian, German and Objibwe Native American.\nNo wonder we can all picture Max as the suave, smooth talking Italian!\nIt's like when there are brand new characters added to a story. How come they are never black, or chinese ?\nMany people, myself included, have always pictured Serena as a beautiful black woman.\nRoswAlien_Lady wrote: Many people, myself included, have always pictured Serena as a beautiful black woman.\nI spy with my little eye something that starts with B.\nRe: Characters as different races?\nTo me the character is more important than the ethnicity. Changing ethnicity gives a whole new range of obstacles and opportunities. I have one story that I will post this fall where the Roswell crash happened in the same location but in 1891. To make Max and Liz come from different sides of the tracks I made Max the son of a rancher and Liz the daughter of a sheepherder. In this case I chose Basque rather than Hispanic. I made a game out of it by trying to follow canon, at least in the story line while I used this time difference. 1891 is the year, I believe that Roswell was founded, Liz's parents worked for a British Lord, named Parker. He assumed the names Jeff, Nancy and Lizzy because he said it would be too hard for the south westerners to pronounce Basque names.\nSince most of my stories occur in the southwest I have tried to include Navajo, Hispanic, Mexican and even South American head-hunters as extras. I found a helper in Misha who translates phrases and slang into Spanish to spice up my stories.\nFan fic is an art. We can paint so many beautiful pictures.\nI've always pictured Serena as a dark haired woman with tanned olive skin and has green eyes. It's just the name sounds exotic, For me I either picture her as hawaiian or similar in looks to Kristin Kruek from Smallville (I don't know why lol).\nAnd Liz, she could pass on as hispanic or Egyptian. She has those nice dark brown eyes and dark brown hair.\nOmg! I didn't know Jason Behr was of many different ehtnicities..Italien, German and I can't remember the others..he's really good looking. Looked hot in Roswell. Very smoking hot.\nHunter wrote: I've always pictured Serena as a dark haired woman with tanned olive skin and has green eyes. It's just the name sounds exotic, For me I either picture her as hawaiian or similar in looks to Kristin Kruek from Smallville (I don't know why lol).\n^^ lol i agree totally agree with you on all accounts.\nbut with Serena yes dark hair but i can also imagine her as a red head.\nKiaraAlexisKlay wrote: Hmmm....Jason Behr is part Italian! LOL So am I!\nBrendan Fehr is Canadian, so he's probably Anglo-French, maybe a little Native?\nBrendan is Mennonite...Mennonites are of German descent.\nMost Canuck-borns are of many backgrounds however. All our immigrant great-grandparents married each other, or else they married the First Nations people already living here. In fact, we have a whole new race of people in Canada, the Metis, who are descended from First Nations women and the French or Scots or English men who married them.\nOn this show both of his parents were white...but I've always thought of a story idea where Alex's dad was white but his mom was hispanic which also added to his being an outcast at school...even more than his being a bit of a geek.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzurwr b/data_all_eng_slimpj/shuffled/split2/finalzzurwr
new file mode 100644
index 0000000000000000000000000000000000000000..9c80a41f989a0cecf7111237f6a3563d7928b9ed
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzurwr
@@ -0,0 +1,5 @@
+{"text":"Make better patient connections through a smarter platform. See the difference high quality treatment ads make.\nHigh quality treatment ads make a difference.\nFree tools to make the most of your marketing.\nA smarter platform that drives better marketing results.\nNo matter how large or small, we are here to help you every step of the way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Advance your shoe game with these 3 easy tools and you'll be sure to turn heads.\nDo you have a pair of old shoes you rarely wear, but won't give up for anything in the world? Sentimental value is transferred to that shoe because of a personal experience only you will truly understand. Over time, the sneakers tend to depreciate in cleanliness despite your refusal to give that shoe up. Don't feel weird about it because I have a few kicks collecting dust for the same reasons.\nPeople tend to underestimate how effective a pair of new shoe strings can be for a sneakers overall pizzazz. With shoe strings, you are able to highlight certain color pallets that weren't the primary colorway.\nThis trend has progressed at an alarming rate due to the global tennis shoe craze. Whether you find a pair online or at a retail store, there is a large quantity of sock themes to make your shoes look fresh.\nAbove all, you must treat your kicks with tender care if you expect longevity. Invest in cleaning products that will preserve the lifespan of your favorite shoe.\nCrep Protect's revolutionary sprays are famed for ensuring protection against stains and rain. The Crep Cure Travel set comes with their patented cleaning solution \u2013 suitable for cleaning up to 50 pairs \u2013 along with a premium hair brush, microfibre cloth and a handy zip-up travel case.\nTake your favorite shoes out of the box and give them a needed makeover. Trust me, they will thank you in the long run.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The European GNSS Agency (GSA) recently participated in the INTERGEO conference and trade fair, taking place in Berlin from 7 \u2013 9 October. This leading event for geodesy, geo-information and land management once again demonstrated the vibrancy and importance of this sector, where satellite-based positioning technologies play a key enabling role.\nAccording to Blasi, Intergeo is all about bringing European Satellite technology down to Earth: \"As for EGNOS, the European Geostationary Navigation Overlay Service that improves the accuracy of GPS, it is widely used in mapping applications where sub metre accuracy is needed and it is opening the door for more demanding high precision surveying applications where Galileo will play an important role,\" he says.\nGSA shared its stand with the European Space Agency (ESA), the Deutsches Zentrum f\u00fcr Luft- und Raumfahrt, the Fraunhofer Institute and the Copernicus programme, and was coordinated by SATNAV Forum, the national network of regional GNSS initiatives, test environments and innovation clusters funded by the German government.\nClearly, the geographic information sector is a market with great opportunities and, judging by the crowds at INTERGEO, one that has growing popularity and is attracting innovative European entrepreneurs. It is a sector where positioning is a key enabling technology and European GNSS are making a very positive impact.\nThe awards ceremony of the third Council of European Geodetic Surveyors (CLGE) students contest took place at INTERGEO, where Maurice Barbieri, President of the CLGE, announced that the GSA will collaborate in next year's contest.\nNext year will see the introduction of a new category on European GNSS, with the aim of fostering the adoption of EGNOS and Galileo in surveying.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We offer FREE shipping within Australia when you spend $99 or more in one transaction, after discounts are applied. Simply use the code \"FREESHIP\" upon checkout.\nStandard: Please allow 3 to 7 business days for delivery.\nWe will be offering express and courier services soon!\nPlease note we take no responsibility for parcels being damaged or lost once they leave our trove. Should your order be lost in transit, please contact us for your tracking number so you can follow up with the post provider.\nCustomers are responsible for duties, custom tariffs, VAT, taxes, and\/or the likes.\nYou're welcome to return any full priced item(s) back to us for a refund* or store credit^ within seven days of receiving you order, excluding pierced earrings (for health and safety reasons), skin care, home fragrances, and candles.\n* Cash refunds will incur a handling fee (10% of value of returned goods), which will be deducted from the total refund amount.\n^ Store credit is valid for 24 months from date of issue. Once store credit has been issued, that value can not be changed to a cash refund.\nItem(s) must be in the original condition and packaging, with tags attached, unworn\/unwashed, unstained.\nShipping and handling charges are not refundable, and we do not offer free shipping for returns.\nPlease be sure to request your return within 7 days of receiving your order. Returns must arrive at our address within 14 days of receiving your order.\nPlease email returns@covertlane.com if you would like to return your item(s). Please include any gift with purchase, or your refund will be adjusted to reflect the retail price of the gift.\nYou may choose the shipping carrier\/method of your choice. We cannot be held responsible for lost or damaged packages, and recommend you insure and track your package. Packages that arrive C.O.D. may be denied, or the C.O.D. amount may be deducted from your refund.\nWe only replace items if they are defective or damaged. If your item is unable to be repaired or replaced you will receive a refund (sale price less any shipping and handling charges).\nOnce your return is received and inspected, we will send you an email to notify you that we have received your returned item. We will also notify you with the status of the return.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wildflowers is an outdoor kindergarten inspired by the Scandinavian approach.\nEach morning the children explore, play and learn in the beautiful Hampshire countryside.\nIn the afternoon they eat a warm meal and engage in creative activities in a converted barn.\nWildflowers was founded in 2005 by Helena Nilsson, who works with the children every day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzurxo b/data_all_eng_slimpj/shuffled/split2/finalzzurxo
new file mode 100644
index 0000000000000000000000000000000000000000..6588870e5eaea13be2820fb14a5b366094a28582
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzurxo
@@ -0,0 +1,5 @@
+{"text":"Peter has more than 25 years of leadership and consulting experience across retail, manufacturing, technology, business services, public sector, and logistics organizations. He has supported clients in the areas of growth strategy, supply chain improvement, postmerger integration, public\u2013private partnerships, and other key business and policy issue areas. Prior to joining AlixPartners, he headed the US Department of Transportation's Research and Innovative Technology Administration, where he led the development of a range of technologies to advance transportation safety and efficiency. Peter has a Master of Science in transportation and aeronautics from Massachusetts Institute of Technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the third straight time, the Municipality of Sta. Cruz, Davao del Sur; in coordination with the Municipal Council for Tourism, Culture and Arts (MCTCA), Mountaineering Federation of Southern Mindanao (MFSM) and Department of Tourism XI is all set to spearhead the 3RD MT. APO BOULDER FACE CHALLENGE in Sta. Cruz, Davao del Sur on April 29-30, 2010. The event is one of the highlights of the annual Pista sa Kinaiyahan basically aimed at promoting responsible eco-tourism activities while protecting the environment particularly the Mt. Apo Natural Park.\nIt is a 24-hour extreme adventure challenge from the municipality's beach lines to the skyline of the boulder face of Mt. Apo, the country's highest peak. This year's race edition will be 100% endurance test with racers competing to reach the boulders of Mt. Apo through disciplines of Mountain Biking, Trekking, Water Tubing and Road Running.\nThe 3rd Mt. Apo Boulder Face Challenge is a 3-man team composed of two racers (1 male and 1 female). Completing the team's composition will be one logistic crew who will be responsible for providing assistance to his two team mates. Full time mandatory equipment are race pack, whistle, headlamp, cellphone, 2 mountain bikes, helmet, life vest and inflated water tube. Registration fee is pegged at PhP 3,000.00 per team inclusive of participation certificate and race jerseys.\nThe champion of the race will receive PhP 75,000.00 and trophies while the second and third placers will receive PhP 50,000.00 (plus trophies) and PhP 30,000.00 (plus trophies) respectively. Three finishers will also be given consolation prize of PhP 6,000 each plus medals.\nDeadline of confirmation will be on April 24, 2010. Applications Forms can be downloaded through www.explorestacruz.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The FLEXMAG 4050 is the first electromagnetic flowmeter that features a disposable flow tube designed specifically for single use biopharmaceutical applications such as filtration processes, chromatography, or buffer and media preparation. The biocompatible and gamma sterilizable flow tube is produced in a sterile facility and features a single barb fitting that meets biopharmaceutical requirements for adaptation to single use systems.\nAs an electromagnetic meter, the FLEXMAG 4050 will not drift over time. It provides a completely stable, direct and accurate volumetric flow measurement unaffected by fluid properties such as colour or density. The FLEXMAG 4050 provides highly accurate measurement with outstanding repeatability. Transmitter and flow tube are factory calibrated and do not require in-situ calibration. The meter can accommodate two different flow tubes sizes so as to handle two different flow ranges by simply exchanging the tube without having to recalibrate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Little Sophia has always been a fan of Dolly Parton since her mom loves to listen to her songs at home whenever she's doing chores to help her move faster.\nDolly Parton's \"Jolene\" is by far one of her more popular songs and it' pretty easy to remember the basic lyrics, but little Sophia hasn't quite got it yet.\nInstead of 'Jolene', Sophia says 'Darleen' instead and it makes for a hilarious 2-year-old's cover of Jolene!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calaway Jolie Shoe is the shoe for the lady golfer who demands a shoe that will perform in all conditions.\nWith the DRYSPORT waterproof membrane to ensure your feet will stay dry to the Full Grain Leather Waterproof Upper this shoe will give you unrivalled comfort whilst giving you all the style you require.\nThe oversized Big Bertha Spikes will give you great stability to help you generate power and performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuycy b/data_all_eng_slimpj/shuffled/split2/finalzzuycy
new file mode 100644
index 0000000000000000000000000000000000000000..12d8383501f006d44189b6608ecffe787079e0ae
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzuycy
@@ -0,0 +1,5 @@
+{"text":"Let's face it. As much as we do not want to admit it, poop is funny. And it is really popular when it comes to kids toys right now. There are many products that poop, pee, toot, and who knows what else \u2013 and kiddos LOVE them. I know that I cannot walk down the toy isle without my girls reaching for some kind of pooping doll or animal. If you aren't pooped out from Christmas shopping just yet, take a look at these pooping toys that you might want to put under your tree this year.\nThe Lalaloopsy Diaper Surprise doll is a real charmer. Literally. Each of these super cute Lalaloopsy babies poop magical charms into reusable diapers. Just feed them water, press their belly \"button,\" and check their diaper for a surprise. Charms come in butterflies, hearts or flowers, and can be hooked onto the included charm bracelet. Ta-dah! A magical poop charm bracelet \u2013 now that's a fashion statement! I love that this \"poop\" is not very messy and that the diapers can be reused! Huge pluses in my mommy book! I just gotta think, how cool would it be if my girls pooped Pandora Bracelet Charms\u2026 Purchase Yours Here.\nTired of having the same boring doll pets? Moxie Girlz Poopsy Pets are anything but ordinary. Kids can take them on an imaginative adventure complete with whimsical pets that poop. Kellan has a pet tiger who poops striped eggs, Lexa has an elephant that poops peanuts, and Avery has a pet panda who poops bamboo shoots! It's a great way to teach kids how to manage pet \"clean up,\" with a laugh! My girls are LOVING their Moxie Girlz Doll and I know they would be really happy to own one of these adorable pooping pets! Purchase Yours Here.\nNow, you may be thinking, \"What about the boys? They love poop too!\" Never fear, I've got one more treat that takes pooping toys to a whole new level. The Little Tikes Big Dog Truck ride-on & walker is the coolest dog on the block! The Big Dog Truck comes with boulders that can be scooped up at the front of the truck when your toddler rides over them, and then dumped out the rear. But don't just take my word for it \u2013 The Big Dog Truck has been awarded Dr. Toy's 2014 Best Picks, Toy Talk Awards' Best Preschool Toy, and ToyTesters TV Gold Award for Best Preschool Toy! I can promise you that my nephews would flip over this truck! Purchase Yours Here.\nWant to win one of these pooping toys for your kiddo? Well I am giving away a Lalaloopsy Diaper Surprise AND a Little Tikes Big Dog Truck to two lucky winners! Enter to win using the Gleam forms below, check out my other giveaways while you are here, and GOOD LUCK!\nMy daughter loves Mittens Fluff N Stuff!\nMy Grandaughter loves LaLaloopsy characters.\nI cannot get behind the toy with the animals that poop gold and rainbows. I hope my daughter never discovers that they exist lol!\ni would love to win the Big Dog Truck for my grandson, Nolan , so he could be just like his Grandpa.\nI'm not to excited about the diaper surprise. Lol. My son would enjoy the truck.\nWe love mittens fluff n stuff!\nMy grandbaby Brentlee would love this.\nI would love to win this for my son! It is the perfect toy for my lil truck lover!\nI would love to win this for my little boy. He would love it.\nI would love to give this to my grandson. He would love this.\nI could guess that my granddaughter would love Bella. She would love them all.\nPixE Flutters is my little boy's fav!!\nMy son would LOVE this truck!!!! They are his favorite things ever!!\nI would give this to my precious granddaughter! If it is something to ride on....she will LOVE it! She loves flying through the house, or riding down the sidewalk\/driveway! Fun stuff!\nI don't know all the different Lalaloopsy dolls that are out there so I have no idea which my great niece would like. Is there a link somewhere to see all the different Diaper Surprise dolls..... the link that was given is just to ONE doll not all of them.\nIf I won the truck, it would go to my son who already loves cars, trucks and trains. He is not even a year yet.\nMy little girl loves all of them!\nMy daughter is a huge Lalaloopsy fan, but her favorite has to be her Mittens Fluff 'N' Stuff! They're all so cute!\nI would love to win this for my daughter, I think she would love it!\nI also have to go with Mittens. My youngest is obsessed with the loopsy minis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parker Prestige are pleased to offer this stunning Mercedes-Benz A180 AMG Line. This particular model is presented in Cosmos Black with contrasting Black Leather\/Alcantara. Equipped with a reversing camera and privacy glass. Please call our team on 01748 889588 for a personal video presentation. We know that finding the right car for you can be difficult, particularly when you're looking for a prestigious used car. That's why we are proud to offer a range of luxury used vehicles from distinguished brands such as BMW, Mercedes, Porsche, Land Rover, Jaguar, Audi, Ferrari, Maserati and many more. With our vast knowledge, expertise and connections we source only the very best in quality., Parker Prestige is a family run business with a vast experience of over 100 years, through 3 generations, within the North Yorkshire area. Our dedicated and friendly team endeavor to ensure every customer has an exclusive experience, tailored to their individual needs, making the process of buying a car enjoyable and hassle free. We are reliable and trustworthy and offer the personal touch that many customers feel they do not get from the main dealers., Our stock is constantly changing so if you can't find what you are looking for we will strive to find your ideal car for you., Browse our impressive stock listings and contact us to find out how we can help you drive away in the car of your dreams., Luxury, Sports and Prestige used cars in North Yorkshire from Parker Prestige.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome Marriage Kids and Money fans!\nThe people who feel most in control of their finances aren't the richest or the highest-earning. They're simply the most engaged.\nAnd (perhaps inevitably) the most financially engaged use spreadsheets to track their money.\nSpreadsheets work great with other financial apps and platforms. Financial advisors love them. They're easy to extend, share, and adapt.\nTiller's mission is to make financial spreadsheets faster and more accessible for everyone, whether you're financially experienced or just starting out.\nTrack your finances with the flexiblity of a spreadsheet powered by Tiller.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\/NOVOSTIVL\/ International marriages in Korea increased sharply in 2018, the latest government data show. This article appeared in The Korea Times.\nAccording to Statistic Korea's data revealed Wednesday, there were 22,698 marriages between a Korean national and a foreigner last year, up 8.9 percent from 2017.The rise was the sharpest in 13 years.\nMarriages between a Korean husband and a foreign wife accounted for 73.2 percent.\nAmong the wives, Vietnamese made up 38.2 percent, followed by Chinese (22.1 percent) and Thai (9.4 percent).\nAmong the foreign husbands, Chinese made up 24.4 percent, followed by Americans (23.6 percent) and Vietnamese (9.6 percent).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NorthBay Cardiac Rehabilitation is a nationally certified program located inside NorthBay Medical Center. The staff works directly with the cardiology team to provide care for patients once they are discharged from the hospital. Patients entering the cardiac rehabilitation program are ready to learn and put into practice the tools they'll need to live successfully with their heart condition. The program includes exercise to strengthen the heart, nutrition counseling, and lifestyle counseling that focuses on modifying risk factors.\nThe cardiac patient's first step toward recovery after leaving the hospital is a visit to the cardiac rehab department. Spouses or other family members are strongly encouraged to join the patient in this visit. They meet the staff and sit down with a nurse to discuss their recent hospitalization, their medical history, their current treatment plan, medications and to learn what their next steps are.\n\"This is often a very emotional time for both the patient and the family, as the visit usually coincides with the realization of what exactly happened 'hits them,'\" Gordon says. Many times, patients enter the program with multiple prior medical problems that have complicated their recovery further.\n\"The typical 40-something male patient of long ago no longer exists. We now see both men and women, ranging in age from late 20s to the late 80s,\" Gordon says.\nThe rehab staff needs to find balance between helping the patient face their new reality, and being the first cheerleader for their recovery. Once this is done, goals are negotiated, and the patient is admitted to start working toward recovery.\nPatients receive a comprehensive education program when admitted to cardiac rehabilitation, which also includes consistent lifestyle changes and exercise.\nExercise therapy ranges from eight to 12 weeks, with sessions three times a week for 30 to 60 minutes. The first session includes a functional assessment of their current condition that results in an individualized exercise prescription aimed to strengthen the heart muscle and improve circulation. Along with strengthening the heart, the prescription is designed to improve their strength, balance and endurance so they can do the things they enjoy.\nFeedback and encouragement from staff is weekly and ongoing.\nPatients are monitored closely by intensive care nurses with special training in therapeutic cardiac exercise. Heart monitors and oxygen levels are constantly and closely assessed to keep patients safe. In this environment, patients feel secure challenging themselves because they know that they have a safety net in case something unexpected happens. After a few weeks into the program, patients begin to bond with each other.\n\"Our patients find strength in being with others who are going through the same experience and who understand their feelings,\" Gordon adds. She also encourages her patients to join North Bay Mended Hearts, an educational and ongoing support group which meets monthly, alternating between Fairfield and Vacaville.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvcml b/data_all_eng_slimpj/shuffled/split2/finalzzvcml
new file mode 100644
index 0000000000000000000000000000000000000000..c87dd50e42b2a4a743249175e04970376d4d5068
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzvcml
@@ -0,0 +1,5 @@
+{"text":"Berkeley Rep gift certificates are guaranteed to delight. They're easy to buy, and easy to enjoy. You choose the value. They choose the show. And they never expire.\nGift certificates can be redeemed for performance tickets and class registration. Sorry, gift certificates cannot be used for concessions or redeemed for cash. Any remaining balance can be applied to future purchases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stellar playwright Kate Mulvany and award-winning director Leticia C\u00e1ceres, two of Australia's most accomplished artists, evoke the visceral, blood-soaked biosphere of the Amazon warrior women of Greek mythology.\nTasmanian Theatre Company has assembled a dream team for the World Premiere of The Mares \u2013 a significant event in Australian theatre. Designed by Jill Munro and Nick Higgins, The Mares features an exceptional group of Tasmanian actors: Sara Cooper, Jane Johnson, Melissa King, Jane Longhurst and Ben Winspear.\nThis brutal, prescient and poetic new work has the wit, wry humour and gravitas we have come to expect from Kate Mulvany. It's a play for our times, exploring themes of power \u2013 physical, sexual and psychological \u2013 and weaving parallel narratives as it moves seamlessly through time, style and gender, galloping towards an unexpected yet inevitable climax.\nCommissioned by Tasmanian Theatre Company. Produced with the generous support of the Seaborn, Broughton and Walford Foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you want a slow ballad to cuddle up on a date, or a fast, high-energy dance song, or a hard-hitting rock song, this is the band! The Roadsters make every event a success and are invited back for more shows!\nFrom weddings, parties, anniversaries to special events, rodeos, County Fairs and Festivals and nightclubs, this is the band to follow!\nThe Roadsters love their followers and have an extensive fan base! They also support other local musicians and benefits for local venues and musicians.\nWhat great dancers! Always smiling!! Thank you so much for joining us Dale and Troi!\nHad a great time at Bigfoot Jamboree in Calif! Who is that playing drums? He looks familiar!\nGreat time at Lake of the Woods! weather cleared for us! Thank you all for coming up!!\nHad a great night last night at Grape Street! Such a wonderful turnout! We thank each and every one of you for joining us! And to Photographic Expressions for taking some great shots!\nTrying this again hopefully with sound!\nWhat a great time at Grape St this weekend! Thank you to all of our peeps who came out to support us! And thank you to Dustin Clark for coming out and celebrating your beautiful wife's birthday with us! We had a blast!\nHomestead Pub April 15th...our thanks to Mr. Michael St John for rockin on the drums and Mr Ranger Jim for shredding on the guitar!\nOur friend Dan sitting in Friday night at JDs!!! Awesome!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Happy to have someone to check our system. \"\nMary S. in Whitehall, WI was suffering from water leaking into her basement. She was tired of dealing with a wet basement every time they had a heavy rain, so she started looking into long-term repair options. During her research, she discovered American Waterworks and was pleased to see that we have almost 3 decades of experience with waterproofing. She contacted us to receive a FREE estimate, so we sent a Design Specialist to her home to inspect her basement and develop a solution. We recommended that we install WaterGuard, a lateral line, and a TripleSafe sump pump. She was pleased with our design and the proposal, so she scheduled to have our waterproofing system installed.\nOur production team started the installation by removing the concrete around the perimeter of the basement and laying the WaterGuard drainage system and a lateral liner. Then, they installed the sump pump in the corner of the basement and filed back the gaps of the drainage system and sump pump with rock. Lastly, they replaced the concrete, The WaterGuard now properly drains water to the sump pump, which then discharges it from the home. Mary was happy with the result and she is thrilled she can now enjoy a dry basement!\nDo you have water leaking into your basement? Click here to receive a FREE estimate!\nBasement leaks when we have high water or a lot of rain. When I say leak, I mean a lot of water comes up through cracks in the foundation. We get A LOT of water in the basement.\nDouble car garage with the last driveway sections next to the garage have settled. One side broke into two pieces dropping about an inch on the end. The other side broke into three pieces dropping 1-2 inches on the end.\nHappy to have someone to check our system.\nThe crew was very polite and respectful of our property.\nJanna J. S. Main St.\nThe crew were very courteous and we would recommend them to our friends.\nLee A. W. Hawthorn St.\nThanks for the detailed quote for the basement wall and windows!\nThese two guys should get a pat on the back, I worked the construction trade and know how it goes.\nNexusPro is a durable binding agent used to repair cracked concrete slabs on your Whitehall property. SealantPro seals concrete driveways and pool decks, restoring quality and protecting against UV, weather, oil and water damage for 3 to 5 times longer than other concrete sealants. Repair your concrete and seal in quality with our concrete protection system.\nAre you concerned about water in your basement or crawl space, or cracks in your foundation or concrete slabs? American Waterworks is your local expert in all of these services and more. We offer free estimates to all of our customers in Whitehall, WI and throughout the surrounding areas. Schedule yours today!\nRon M. in Whitehall, WI had a severely sunken and cracked driveway. His concrete had sunk almost 3 inches below the concrete of his garage floor, which created a large trip hazard. He also was frustrated by the bump he experienced each time he drove in or out of his garage. He started researching concrete repair and discovered American Waterworks. After having a Design Specialist out to his home for a free estimate, he decided that our PolyLevel concrete repair was the best solution for him, so he scheduled the installation.\nOur concrete repair experts drilled small holes into the concrete slabs of his driveway and injected the polymer foam, which expands and then raises the concrete. Once it was raised to be even with the garage floor, they filled back the holes they drilled with concrete. Then, they sealed the cracks in his concrete with our Nexus Pro sealant. Nexus Pro protects the concrete and gives it a nice, finished look. Ron was pleased with the results and he is glad he chose to repair his concrete with American Waterworks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below is specific information about the services we provide for in-home training which we provide in Baker County Oregon and surrounding areas as well as distance training options for those out of the area. All programs are customized to fit your needs, and we are flexible with our training plans to provide optimum learning for you and your canine companions. If you have further questions regarding our service options, please call or email.\nPrivate Training is an in home service, where a trainer comes to you and teaches you to train your dog, in your own home, at your own pace. Private Lessons are great because they are geared towards your needs only. You can cover all the topics you need on your time, and get a training plan, and homework to complete in between sessions. Private lessons include material offered in an average obedience class and are taught in your home, at your convenience.\nThis home-based service offers dog owners who have limited time an opportunity to improve their relationship with their pet dogs. Training at home is especially appropriate for dog owners who find travel to classes difficult. Together we can identify problem behaviors and training solutions. We can also train skills that will improve your dogs ability to be calm and manageable. You will learn gentle and positive training methods that incorporate all kinds of motivation from praise, food and toys as reinforcement for positive behaviors and learned commands.\nBehavioral Consultations are very similar to Private Training except a specific behavioral issue or issues are addressed. Issues such as all types of aggression, separation anxiety, compulsive\/obsessive behaviors, food guarding and territoriality, hyperactivity and compulsive behaviors are all best dealt with on an individual basis. Consultations include a period of observation and discussion, a training plan, demonstrations and practice of techniques and follow up whether by phone, email or in person. The amount of sessions you will need will depend on the issue(s), time you have in between to work with your dog and your comfort level. Everyone is different and we want what works best for you, and gets your dog to a healthy and happy state as fast as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwoqo b/data_all_eng_slimpj/shuffled/split2/finalzzwoqo
new file mode 100644
index 0000000000000000000000000000000000000000..857f7b3ce7037ff3bae60dac1fad2c49c5dd873b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwoqo
@@ -0,0 +1,5 @@
+{"text":"In order to keep students motivated and moving forward over the summer \u2013 we are having a Moby Max a contest this summer. All students have their personal log in \u2013 so remind them to Log in, earn minutes and win.\nPrizes will be earned for every 200 minutes students log in over the summer. Every student will earn an otter pop for every 200 minutes logged. Log 600 minutes \u2013 you receive 3 otter pops. Then for every 400 minutes logged you receive a ticket to put in a drawing for a $50 gift card of your choice. Most importantly the student that logs the most minutes will receive the otter pops, the tickets and an Amazon Fire Tablet! If you forgot how to log in \u2013 see the directions below. Good luck and have fun on Moby Max!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the U.S. Department of Agriculture, a healthy eating plan should supplement various fruits and vegetables with a few lean proteins, while consuming only a low amount of fats and sugars.\nWeightChart provides instant visualizations, calculations, and summarizations, it's a kind of biofeedback for your wellness. With such a rich source of information you will be able to quickly hone in on what works and dosen't work for your wellness progress. For some people this feedback is enough to help them make the daily choices that improve their wellness, without a complicated diet or exercise plan.\nWe'll do our best to help you stay motivated with milestone goals, and email reminders. You can group together with friends, coworkers, and family for encouragement and fun with weight challenges and competitions. We're not here to push a fad diet or some abstract rewards system, we exist to help you discover more about yourself and what wellness approach best fits your life.\nDon't make the same diet mistake twice, keep a wellness record with us and benefit from a perfect memory. You'll pinpoint the things that cause you to get off track and be prepared for monthly and seasonal cycles. You'll see how your sleep, diet, mood, and weight relate to each other. If you have other important factors to track, like blood pressure, workout stats, and blood sugar levels we can help with these as well.\nMany commercial diets and excersise products come with tracking tools, but what happens when you realize these products are not working for you? Track your wellness with us and feel free to change your diet and excersise strategies to find what works best for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Size L inseam measures approximately 11\"\nMens_Nike_Heathered_Gray\/Cardinal_Stanford_Cardinal_Fly_XL_4.0_Shorts http:\/\/images.fanatics.com\/lf http:\/\/dmimages.ff.p10\/chains\/2391206.txt false We are unable to customize this item with the text you have entered. Please try a different entry again. Your current entry cannot be processed due to guidelines for past and present player names. Please create a new entry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fireside Magazine has opened its doors to the public for free online viewing. Traditionally, the magazine has been supported by Kickstarter funding for yearly chunks. However, they've wanted to try and move away from Kickstarter and to a more communal funding effort. Thus, the options for subscriptions, contributions, and patreon funding have opened up.\nIf you want to take a look, check out the magazine's full catalog over here. If you're also wanting a direct link to my previously published piece of flash fiction, Reversal, the link is right here. To directly help support Fireside, check out their \"Support Us\" page.\nSo yeah, there we go. A couple days late, I know, but things have just been manic. The rig is mostly up and running now, so I'll be completely on my feet before we know it.\nThis entry was posted in Announcements and tagged books, fantasy, fiction, fictional world, Fireside, Fireside Magazine, james darrow, Kickstarter, literature, news, novels, Reversal, short story, thoughts, writer, writing. Bookmark the permalink.\n\u2190 My thoughts on: Destiny.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Eurofighter is one of five types of combat aircraft that are being tested in flight and ground in Payerne from April to the end of June. The two Airbus Eurofighters (Germany) took off on 9 April 2019 from Warton (UK) and landed the same day in Payerne. The Eurofighter is stationed until 17 April 2019 on the Payerne Air Base.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzybzc b/data_all_eng_slimpj/shuffled/split2/finalzzybzc
new file mode 100644
index 0000000000000000000000000000000000000000..3b9dff11174947aa56e62b2bbedbbed3bad69e41
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzybzc
@@ -0,0 +1,5 @@
+{"text":"PropSpace is proud to announce the launch of its Arabic version, catered to the MENA region.\nDeveloped in 2011 by technology enterprise JRD Group, PropSpace proves the most popular industry enterprise software in the U.A.E., with over 95 percent of the nation's real estate brokerages utilizing the cloud-based system. Considered the region's first full scale multi-functional cloud-based real estate software, PropSpace consists of multiple modules that capture to a variety of workflows in the real estate sector, including but not limited to lead management, listings, deal flow, document generation and the region's largest private MLS (multiple listings service).\nThe launch of the Arabic version follows significant steps by JRD Group to expand the PropSpace client base across the MENA region and globally, to more established markets further afield. Following a successful soft launch in London in September 2015, PropSpace made a promising debut at the 2015 REALTORS\u00ae Conference and Expo held in November 2015 in San Diego, California (U.S.A.); the first in a series of moves intended to position itself as the premier cloud-based real estate software solution worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On April 10, 2018, the Food and Drug Administration (FDA) approved everolimus tablets for oral suspension (brand name: Afinitor Dispersz, Novartis Pharmaceuticals Corporation) as an adjunctive therapy to treat partial-onset refractory seizures in adults and children over 2 years of age with Tuberous Sclerosis Complex (TSC). Tuberous Sclerois isa genetic disorder that affects one million people worldwide and is characterized by benign tumors in several organs, epilepsy, behavioral problems and intellectual disability.\nDr. Anne Anderson, epileptologist at Texas Children's Hospital and investigator at the Cain Labs and Jan and Dan Duncan Neurological Research Institute. Read the blog posted on the Foundation for Angelman Syndrome Therapeutics website that describes Dr. Anderson's latest research findings on Angelman syndrome.\nA study published in Scientific Reports from the laboratory of Dr.\nDavid Cameron's son Ivan was born with a neurological disorder that baffled doctors. He was later diagnosed with a rare epileptic syndrome, Ohtahara syndrome. Now, he writes, genomic testing can end the anguish of uncertainty.\nA study shows that structural alterations in neurons and the resulting memory deficits that arise after a single generalized seizure can be reversed by inhibiting the Phosphoinositide 3-kinase\/Akt kinase\/mechanistic target of rapamycin (i.e. PI3K\/Akt\/mTOR) signaling pathway.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed for the transportation of finished job pieces, this system also helps in the installation process. The precise extruded structural aluminum tubes with vacuum cups does not hinder setting on cabinets during installation.\nFor further items, we invite you to take a moment to shop our extensive selection of Stone Pro products at Regent Stone Products today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The founder of Ship Your Enemies Glitter, an Australian website that lets anyone in the world ship their enemies bags of glitter \u2013 the most irritating party product yet invented \u2013 has put his company up for sale, after begging potential customers to stop using his website.\nMathew Carpenter's creation is up for auction on startup marketplace Flippa, with a starting bid of just $1, boasting of the site's \"$20,000+ in sales and 2.5m visits in four days\".\nBut, he says, the site was a joke that got out of hand. \"I launched this website as a bit of a joke not expecting this level of attention. Heck, I launched this website while I was on holiday!\nThe sale comes days after Carpenter, 22, took to start-up website Product Hunt, where his creation had racked up almost a thousand votes, to ask potential users to just not.\nSince schadenfreude is the internet's mother tongue, the replies were less than sympathetic. \"You have made your glitter-coated bed and I'm afraid you will have to lie in it,\" read one.\nFor what it's worth, Carpenter is willing to sign a three-year non-compete clause with any potential buyer, meaning that if a follow-up website \u2013 say, ShipYourEnemiesConfetti.com \u2013 does get made, it won't come from him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How do I remove all appearance of color only that may be set up as default?\nThere is a white color only appearance activated somewhere that affects most all parts we have. I can add material color at part level but it is not flowing through at assembly. Did someone set up a default and if so, how do I remove it?\nMessage was edited by: Kathy Spaw I added the part template and tried to delete the appearance but it won't go! I haven't seen anything so tenacious. The assembly template did not have any appearances so the part template is the culprit.\nRe: How do I remove all appearance of color only that may be set up as default?\nHi Kathy, i opened your assembly but i don't see anything funny with it. Did you do a Pack n Go and include the check box for \"Include Custom decals, appearances....\" ?\nHere is a somewhat general\/generic response that might help.\nIf an appearance is applied at the top level of the assembly it will override all others by default. You can see the first cotter pin has a blue appearance applied but is green.\nIf the top level of the assembly doesn't have an appearances then the one in the upper left(assembly applied appearance) will drive the look of the part.\nIf you apply an appearance at the part level it will show up in the bottom right corner of that icon in the display pane.\nJust like appearances in a part there is an order to what overrides what. In a part you can apply an appearance at the part, body, feature or face level. The face will override the feature, feature overrides the body and body overrides the part.\nIf you can take some screen shots of what you are seeing or try a pack n go with the appearances option to make sure it all comes through i will try to expand on that answer.\nThe first thing that comes to mind is that there is a color appearance set up by default (by someone) in your templates which you may need need to edit and remove.\nIt could also be caused if your template has a material assigned by default and the color you are seeing is the default appearance for that material. The remedy is the same, edit and remove the material from your template.\nIf you don't want to do these steps, then you can can also ensure that each time you create a new part, you right click on the very top item on the feature tree, then scroll down and select 'Appearance', and then select 'Remove All Part Appearances From'.\nObviously checking and fixing your templates is the best fix if indeed that is the issue and I would be checking these first.\nCheck original post, I added the part template. That color just won't let go!\nKathy, this is very odd behavior. There was nothing i saw in your part template that would indicate anything funny. The part does have the blueish appearance as shown in Dave's post. However if i change the background, not the appearance, to Plain white the part appearance changes. If i go back to any other such as 3 point faded, the part stays the color we would expect to see.\nWhen i create a part with your part template, the new body has that bluish color to it. When i change backgrounds it goes away.\nPlaying around with it more there seems to be something funny tied to the 3 point faded scene. I opened your template, set the scene to \"Plain White\" and saved another version and it does not seem to exhibit the same behavior. Off hand i don't know of a way to fix the ones you have already made besides changing the scene.\nHere are some screen shots when opening the pin portion of your assembly.\nJust so its easier. Here are the two images together. The larger is how the file opens. the smaller is the same 3 point faded scene but after i switch back and forth between another scene.\nSorry i don't have an answer as to why its happening. I will dig into it a bit more. Just wanted to share what i found.\nAs I stated earlier, I found the issue and I've fixed it. The new template is attached above.\nDave, i checked your attachment and it was a new part template. That doesn't answer the question as to why hers is messed up though. At least that is the one that is driving me a bit batty I don't know why the scene would show the part as one state then another after switching back and forth. I feel like the original 3 point faded on her end was edited and saved over maybe? Somehow switching back and forth lets it go to the default install 3 point faded?\nCan you confirm on your end, with the original that simply switching scenes has the same affect as reapplication of the appearance?\nWe may never know exactly why it happened but it sure is a good exercise trying to find out .\nI was going to suggest looking at that it always makes all parts look \"white\"\nThe background isn't the problem. Every time I change the material or directly remove and add an appearance, like polished steel, the part, at some level shows the \"plain color white\". Yes, a white part on a white background is hard to see but the background behavior is working as expected. It's the \"plain color white\", that won't be removed, that is causing the grief.\nIf this is happening on every new part you create, I would definitely clean your part template.\nAs Dave suggested above: sounds like your assembly template has a pre-defined appearance on it (in your case, white). Open your assembly template (templates should have these extensions: .asmdot - for assemblies and .prtdot - for parts) from inside SolidWorks (if you double click on the file in Windows file explorer it will not recognize the file type and will ask to select the program to open it with). Locate the appearance (see attached image), click on it and delete it. Save the file and close it. From now on, your assembly templates should not be white.\nDo the same with your part template if you wish to remove the pre-defined appearance.\nI tried to do as you suggested but the template wouldn't let go of the color. I attached a copy of it to the original post.\nThe new template is attached and renamed as \"NEW Product Part\"\nI had such hopes but when I opened your template in my system, it showed as white. I am at a loss.\nKathy, when open up Daves template it has the default appearance and 3 point faded scene. What are you expecting the appearance to be? I took Daves template and made the appearance high gloss blue plastic.\nWhen i was talking about the scene setup i know the scene doesn't seem like the issue, but my comment was that changing the scene and changing it right back displayed the model appearance differently. That is what led me to question your .p2s file for the 3 point faded. That file was sent along with your template and it makes the part look different than my default .p2s 3 point faded file.\nYour's worked for me as well as David Bear's.\nI, finally, went into my template and removed all appearance then added matte steel. It seems to stick and hold the appearance into an assembly. I just wish I could fix the appearance of the existing parts.\nKathy, did you even try to create a basic model with the new template and see how it displayed?\nI tested it before I posted it here for you and it worked fine.\nSorry to take so long to get back to you. Life happened.\nYour file \"New Product Part\" is working for me. I even applied the 3-point studio room background and everything works well. What did you do? Is it possible to change existing model appearance with your magic? Thank you!\nUm, do you mean take existing parts that were created with the faulty template and place them into the new template so that the appearance is corrected?\nAre you applying color to each configurations? I'm suspecting that it may be related to display state.\nI'll keep an eye on it but currently, just the default display state 1 shows.\nAs far as the background goes, I long for the Millennium color that Inventor has.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzysor b/data_all_eng_slimpj/shuffled/split2/finalzzysor
new file mode 100644
index 0000000000000000000000000000000000000000..a5529d11690373b0fd044946f872855229a30b43
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzysor
@@ -0,0 +1,5 @@
+{"text":"Want a strong opinion the day after a big sporting event? Mike Francesa never fails to disappoint.\nThe bombastic WFAN drive-time host opened up his show Monday afternoon by launching into a tirade against Seahawks coach Pete Carroll, who called the fateful pass play from the 1-yard line that resulted in the interception to seal the Patriots' 28-24 win in Super Bowl XLIX.\n\"That wasn't one of the poorer play calls in the history of the Super Bowl, and that wasn't a play that you can dissect in 10 or 20 different ways. That was the single worst big-game moment call in the history of sports,\" Francesa opined. \"That was the dumbest, most ridiculous, no-excuse, no-alibi, no reason to do what they did.\nFrancesa then slipped on his offensive coordinator hat and diagrammed what would have been the only acceptable pass play to call in that situation.\n\"The only pass you could ever think of throwing there would be out of a tight formation, a power formation, which means you're all in, your tight ends are in, you're in a jumbo look, you're in an I-formation, and you don't give it to the back. You sprint a guy to the corner by himself, and if he's not by himself you throw it into the 15th row. That's the only pass play you could even consider at that point of the game!\nFrancesa, apoplectic at this point, continued to dissect what Seattle did wrong on the goal line.\n\"The Pats are stunned. You hand it to Marshawn Lynch, who almost scores on first down! You now have the game \u2026 the Pats have capitulated! They are not calling timeout! They are saying, 'You know what? We're going to live and die with what's going on here. We're gonna save our timeouts.' Because, I'm sure they're working on a kick return over there, and they're figuring that Marshawn is gonna score on the next play anyway!\nHaving already erupted, Francesa finished calmly.\nWas Seahawks' bad beat so painful it snuffed a dynasty?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Professional Women in Building (PWB) \u2013 HBA of Northern Michigan is hosting it's 4th Annual Wine and Chocolate Tasting event at Petoskey Farms Vineyard and Winery.\nAbbigail Babbit (Center) was the lucky awardee of the PWB Trades Scholarship in 2018. Abbigail has been accepted to Ferris State University this Fall to continue her dreams in the building trades. With the shortage of workers in skilled trades, PWB is very proud to be able to present a scholarship each year to a student going into the building trades. This is made possible through the Wine and Chocolate Tasting. Sip and Taste for a great cause!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"About 35 representatives of the Galician agroforestry sector, including farmers, researchers, producers associations and representatives of the regional Government, Xunta de Galicia, met on September 21st in Gunt\u00edn (Lugo) for the third meeting of the AFINET RAIN, where they discussed and shared technical, economic or political innovations which promote agroforestry management in Galician forests and soils.\nThe Mayor of Gunt\u00edn, Jes\u00fas Carreira, welcomed the Galician Agroforestry Innovation Network to the working session, which will promote a good management of Galician forests. The session was focused on the aspects that most concern the members of the network such as combinations of species in agroforestry systems, technical management issues and aspects related to political regulation and social awareness.\nThe members of the network agreed that the current regulatory framework adversely affects the agroforestry management of the forest and requested recommendations and good practices that favor the implementation and consolidation of these systems.\nRosa Mosquera, coordinator of the project and director of the Department of Plant Production of the University of Santiago de Compostela, together with Juan Luis Fern\u00e1ndez Lorenzo, professor in this same department, presented a guide for the plantation of arboreal species in agroforestry systems, which is being elaborated under the framework of the project.\nOther innovation that generated interest among the RAIN memebers was the design of a predictive model of economic and environmental production of agroforestry exploitations, which helps farmers and forest owners to decide the viability and profitability of including this type of practices in their exploitations.\nAfter the discussion session and presentations, RAIN members visited the Cabuxa Natur goat farm in Gunt\u00edn, which makes use of shrub grass and shrub grass under trees; and Monte Veci\u00f1al in Man Com\u00fan de Carballo, in Friol, where they were able to observe one of the outstanding innovations for silvopastoral management: a mobile system to improve the management of pigs in extensive agroforestry systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sold by the case. 100 units per case.\nThis item does not qualify for free shipping.\nDistinctive, 2 Disc, Premium-Grade Boxes for your Blu-Ray project! Blu-Ray boxes ensure ultimate protection for your message. With a secure ring and a push button release, these boxes keep your Blu-Ray discs from sliding around and getting scratched.\nWe are a family oriented business. We do not manufacture products containing parental advisories, adult oriented content, hate, Anti-American, or anti-religious content.\nAll Content Wholesale Tape & Supply Co., Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us for an Open House at CNY Healing Arts in Syracuse for a talk by Dr. Rob and Maria Emmerich, a ketogenic diet & lifestyle expert that shares Dr. Kiltz' passion for living a happy and healthier lifestyle.\nShe is an International Best Selling author of \"The Ketogenic Cookbook\". She also authored 8 other books including several cookbooks and 3 nutritional guide books including \"Secrets to a Healthy Metabolism\" which includes a foreward by New York Time #1 bestselling author of \"Wheat Belly\" Dr. William Davis. Her latest book \"Keto-Adapted\" includes a forward by Dr. William Davis and excerpts from Dr. David Perlmutter author of the New York Times best seller \"Grain Brain\".\nMaria will be signing books and discussing the importance of a ketogenic diet in your everyday life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaatuw b/data_all_eng_slimpj/shuffled/split2/finalzzzaatuw
new file mode 100644
index 0000000000000000000000000000000000000000..7951a849e8df2ac76f423240b97137e09cad6935
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaatuw
@@ -0,0 +1,5 @@
+{"text":"An oil cleansing foam for quick and easy cleansing that gently dissolves makeup with oil and forms a rich lather with water.\nConvenient transforming gel-type cleanser which removing base makeup effectively and keeping skin dewy.\nRoll on the dry face gently and add a bit of water on to skin to make a rich lather and gently massage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Islamabad, Jan. 16 (TNS): War of words continues between leaders of ruling party as Pervez Rashid on Monday asked Pakistan Muslim League-Nawaz to expel Chaudhry Nisar from the party.\n\"He [Nisar] arranged the scheme of things in a way that he was able to please them,\" he added.\nAccording to Rashid, the former interior minister issued misleading statements against the party when it was facing a test [following the controversy].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ARLINGTON, Va. Researchers in the Defense Advanced Research Projects Agency (DARPA) Make-It program demonstrated significant progress toward fully automated rapid molecule production, which could speed the pace of chemical discovery for a range of defense products and applications.\nCurrently in the third year of a a four-year effort, DARPA's Make it program is developing software tools based on machine learning and expert-encoded rules to recommend synthetic routes optimized for factors such as cost, time, safety, or waste reduction.\nThe program seeks to free chemists so that they may focus their energy on chemical innovation, rather than testing various molecular synthesis pathways. The program also is developing automated devices that uniformly and reproducibly create the desired chemical based on the software-generated recipe \u2013 this one-device, many-molecules concept is a departure from the traditional dedicated reactors in chemical production.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Hampshire business develops electronic systems for the automotive and defence industries. New contracts boosted sales to \u00a324.6m last year. Husband and wife founders Michael and Lesley Gill, 66 and 61 respectively, also run Gill Sensors and Gill Instruments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Lasker Foundation and the National Institutes of Health have joined together in an innovative partnership to nurture the next generation of clinician-researchers.\nThe Lasker Clinical Research Scholars program is an NIH-funded initiative that supports early-career medical researchers and helps bridge the widening gap between cutting-edge research and improved patient care. This unique program provides scholars with five to seven years of support as independent investigators in the NIH Intramural Research Program, followed by the opportunity for additional years of financial support, either at the NIH or at an extramural research institution. More information and how to apply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadxoo b/data_all_eng_slimpj/shuffled/split2/finalzzzadxoo
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"With its catchy melodies and humurous plot, Gilbert and Sullivan's 19th century work has become a staple for amateur and school productions, remaining one of the most beloved works of the operetta\/musical theatre genre to this day.\n\"It's a really accessible kind of show \u2026 it's just good fun,\" says Lennox.\nHowever, in her view, \"The Mikado\" is not without its moments of truth and beauty, as well as intelligent writing.\nAll of these qualities contribute to making the operetta an enduring work that appeals to people of different ages and backgrounds.\nIn the Toronto Operetta Theatre's (TOT) production, Lennox plays the character of Katisha, an older woman at the court of the Mikado (the title given to the Japanese emperor) who has set her sights on the emperor's son Nanki-Poo. Faced with the perspective of marrying the unattractive Katisha, the young man runs away, only to find himself caught up in a series of colourful, absurd, and hilarious adventures.\nAlthough a bit of a caricature, Katisha is a determined woman, and Lennox enjoys playing her.\nThis is the singer's second time with the TOT after performing in its world premiere of \"Ernest, The Importance of Being\" in 2008. At the time, she was nine weeks pregnant, and she says it was an experience to remember.\nBalancing a singing career with motherhood has been challenging, but something that Lennox was determined to do.\n\"Ultimately when you love something, you just find a way to make it happen,\" she explains.\nAs a singer, Lennox believes that talent, passion, and hard work are a must. However, they are not the most important factors in a career\u2014she feels success is directly linked to one's character.\n\"The singers I know who are the most successful are generally some of the kindest, most modest people you'll find. I really think that's what keeps your career going,\" she says.\n\"I do believe there's a very, very important space for kindness and support as a singer with both the companies and your colleagues,\" she adds.\nMaking the best of one's time performing is also key.\nEpoch Times is a sponsor of the Toronto Operetta Theatre.\nRichard Wagner's \"Die Meistersinger von N\u00fcrnberg\" clocks in at almost six hours, which may be too much of a good thing. Or maybe not.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The roads are in an enviable state and the motorway network is the most comprehensive in the world. Do not forget take a road map of France not absolutely miss anything!\nThe European country of France has recently become a global giant not only in tourism but in other economic aspects. But what to do with tourism this is certainly one of the strengths of the French economy. For years it has been the most visited country in the world, and is now the second behind Spain and the truth is that merit is not lacking either culture, landscapes, beaches, gastronomy, etc.\nIf we travel in our car or bike we will make a very good choice to not miss anything interesting about this European country although it is best to visit more than once because it really has many beautiful things to see.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Available through the EEA, these publications cover guides, standards, health and safety resources, professional development tools, industry best practice and more.\nWhile many are available to the public, some are provided exclusively for EEA members or available at a discount.\nView by subject below or use the publication search.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ciabatta sandwich bread - you could use any type of bread you wanted though.\nDirections: Preheat oven to 350'. Slice bread in half and spread generous amount of brie onto the \"bottom\" piece. Melt your butter and add in sugar, stir, and brush on the top of the top piece of bread.\nPlace top on sandwich, bake for 5-10 minutes depending how crispy and melted you want it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Horeca School, located in Bucharest, has announced six master classes with great, world-renowned pastry chefs, which will take place in 2018 and 2019.\nThe first to visit the culinary and pastry school to teach a hands-on class at the beginning of October will be Hans Ovando (so good #5, so good #11, so good #14), who has just won the Copa Maya as a member of the Chilean team. On October 21 and 22 will be the turn of Diego Lozano, the great ambassador of Brazilian pastry, who will carry out a chocolate demo. And to end the year, Marike Van Beurden, gold medalist at the World Chocolate Masters 2014, will teach the art of making plated desserts from November 15 to 17.\nFor 2019, the school will host master classes by Frank Haasnoot (from March 28 to 30), Nicolas Haelewyn (from May 22 to 25), and Martin Diez (from October 22 to 24).\nHoreca is known as a top school in the culinary education field. Founded by Cristina Van Der Schaaf in 2009, events for the culinary industry are organized and tries to network the main decision factors in gastronomy field in order to help the industry to progress.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeskb b/data_all_eng_slimpj/shuffled/split2/finalzzzaeskb
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@@ -0,0 +1,5 @@
+{"text":"Can I connect an external FM antenna?\nAn external FM antenna cannot be connected.\nWhen experiencing poor reception sensitivity, try the following.\nRelocate the receiver to be near a window.\nExtend the antenna wire in a direction with good reception.\nCan I add external speakers to the VAIO computer?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Johnny Og Connolly's latest CD 'Aisling Yoshua' (Joshua's Dream) is a solo recording comprising twelve tracks. It offers a mixture of traditional tunes and some of his own compositions including two waltzes, a lament, a set of Polkas and a lullaby in honour of his nephew Joshua who lends his name to the album title.\nJohnny Og is joined by his father, the legendary melodeon player Johnny Connolly on two of the tracks and accompaniment is provided by some of Ireland's finest musicians, among them, Brian McGrath on Piano, Mick Coneely on bouzouki, Garry O Briain on Mandocello and Guitar, Frank Kilkelly on Guitar and James Blennerhasset on Double Bass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Springfield Regional Arts Council (SRAC) is pleased to accept nominations for the 2018 Ozzie Awards through Wednesday, February 28, 2018. Winners will be honored at the 9th annual Flights of Fancy, Southwest Missouri's most celebrated arts gala of the year, on Saturday, October 13, 2018 at the Springfield-Branson National Airport.\nNominate those you know have embodied the pioneering, creative spirit of the Southwest Missouri arts scene. Nominations can be made online, by mail, or in person at the SRAC office located in The Creamery Arts Center, 411 North Sherman Parkway, Springfield, MO 65802. All information regarding nominations is confidential.\nAn ideal Ozzie Award candidate exhibits an extended service in promoting the arts, a broad spectrum of involvement in the arts, support above and beyond professional responsibilities, and should be a catalyst for the growth of the arts in the Ozarks. A Legacy Award is traditionally offered posthumously to an individual who demonstrated the qualities of an Ozzie Award winner in their lifetime.\nFirst awarded in 1985, the Ozzie Awards are given each year to individuals, organizations, and businesses which have demonstrated an outstanding commitment to the arts in Southwest Missouri. A public call for nominations is announced and a selection committee chooses those which best support or personify artistic achievement.\nFor more information about the Ozzie Awards or Flights of Fancy, contact Brett Johnston, Director of Marketing and Development, at Brett@SpringfieldArts.org or 417-862-ARTS(2787).\nPhysical nomination form to print and mail to The Creamery Arts Center (411 N Sherman Pkwy, Springfield MO 65802).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plastic dinnerware was discovered in many homes in the 1940s through the 1970s and is extremely collectible now. Throughout the Nineteen Thirties the raw material \"melamine\" hit an all time low worth. With heightening wartime threats and soon to be monetary constraints, American industrialists jumped on the bandwagon to make melamine into functional merchandise for each industrial and households.\nIt's fun to gather it and resulting from it is lengthy production straightforward to make a complete set. Some melmac is price extra in worth than others. Full sets in pretty colours reminiscent of pinks or blues are usually priced larger. Uncommon makers like Lucent, Fostoria (each glass companies that dabbled in melmac), Russel Wright and Raymond Loewy designs are highly sought after.\nAmerican Cyanamid continuously improved their formulation, and did intensive client product testing and analysis (even hiring Russel Wright to do a protracted survey and compile studies in the mid Forties.) Additionally, American Cyanamid (pre 1960) would send inspectors to sure factories to guantee that melmac dishes have been assembly certain specifications and highest quality requirements.\nWatertown Manufacturing Firm of Watertown, Connecticut. This company has ties again to 1915, and made early industrial plastics. Jon Hedu, then-designer worked with the Navy to make military wares. Watertown's finest promoting Lifetime Ware line would make the Trendy Museum of Art's permanent collection (which is cited as being dated to 1940 based on them.) Earliest proof of this line being out there to shoppers was 1946 in accordance with Plastic Dwelling. Satirically, this company would promote out the dinnerware division to Northern Chemical Company (above) circa 1960.\nThis can be why some old adverts for plastic dinnerware particularly say \"Made from Melmac\" and others may say Plaskon, or perhaps simply melamine. I bought it to sell but I really like it so we'll see\u263a.\nHere are a few of our best melmac dinnerware graphic that you should get and print.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is my student eligible to receive testing accommodations (such as extended time) on the ACT and\/or SAT?\nAll requests for testing accommodations are handled by the school's Learning Specialist Kamauru Johnson and should be sent to Kamauru as soon as possible. Students who receive accommodations in school DO NOT automatically receive accommodations on their college entrance exams.\nWill my student quality for accommodations on both tests?\nThe SAT and ACT are administered by two different organizations and have separate and independent application and review processes. It is possible to be approved for accommodations on one test but not the other.\nWhere can I learn more about special accommodations for both tests?\nPlease visit this page at collegeboard.com to see accommodations for the SAT. Please visit this page at act.org to see accommodations for the ACT.\n'Financial aid' generally refers to resources provided by the federal\/state government or the college\/university a student is attending, to help the student and his or her family meet the costs of attendance at a given college. This is in addition to the family's own financial resources (income, savings, etc.), and scholarships or other monetary awards that a student might receive from outside groups such as corporations and non-profit organizations.\nIs financial aid the same thing as a scholarship?\nA scholarship is a type of financial aid\u2014but there are others, too. Aid can come in the form of grants, scholarships, work-study, and loans (and yes\u2014loans are considered aid, even though you have to pay them back). Most students who apply for aid receive a 'financial aid package' that includes a combination of several types of aid.\nWhat is ``need`` and how is it calculated?\nNeed-based aid is intended to lessen the difference between the full cost of attendance (meaning tuition, room and board, and other expenses) at the student's college of choice, and what the family can afford to pay for college. There is no income cutoff for need-based aid\u2014each family's financial situation will be evaluated individually to determine what the family can afford to contribute (this amount is called the \"Expected Family Contribution,\" or EFC). Sometimes even families with high incomes are eligible for aid, depending on their expenses and the cost of the college their child is attending. The amount of aid your family is eligible for is based on the difference between the EFC and the cost of attendance at a given college.\nHow do we apply for need-based aid?\nTo determine your EFC and apply for aid, you will need to fill out one or more forms. Everyone must fill out the FAFSA\u2014the Free Application for Federal Student Aid. This can be completed online, any time after January 1 of the year the student will be entering college. The FAFSA results determine eligibility for all forms of federal government aid, including Pell Grants, Perkins and Stafford Loans, etc. For an estimate of your family's EFC, click here.\nSome colleges use the EFC from the FAFSA to award their institutional aid, but many private colleges calculate the expected family contribution using a different, more detailed form, called the CSS Profile. This can also be completed online.\nSome colleges also require their own forms\u2014it is important to carefully read the instructions for applying for financial aid on each college's website!\nHow do we apply for merit-based aid?\nState\/Federal aid: Government aid for undergraduates, with very few exceptions, is based only on need.\nInstitutional aid: Not all colleges offer merit-based institutional aid. In fact, many of the most selective colleges in the country, including the members of the Ivy League, offer only need-based aid. Colleges with Division I or II sports may offer athletic scholarships\u2014but Division III schools are prohibited from offering any form of athletic scholarship. Families should read the financial aid websites of the colleges their child is applying to, to find out if they offer merit-based aid, and if so, what types, and what the application procedures are.\nSome colleges automatically consider all applicants for any available merit awards, while others require separate applications. As with everything related to financial aid, it is in your best interest to apply as early as possible to have the best chance of receiving aid, and to be certain of not missing any important deadlines.\nOther forms of aid: The kind of non-institutional aid that you are probably most familiar with is private scholarship programs. These scholarships are usually awarded in recognition of some kind of achievement (academic, artistic, etc.), or because of membership in a particular group or population (children of veterans, for example).\nHow will I know how much financial aid my child has received?\nWhen your child is accepted at a college, if you have submitted all the required aid application forms by the stated deadlines, you will receive a financial aid award letter detailing the costs of attendance at that particular college, and the types and amounts of aid being offered to the student (the \"package\"). You will have an opportunity to compare the aid packages from all of the colleges where your child has been admitted before having to make an enrollment decision, and you may even be able to negotiate with the colleges to get more aid.\nA typical student's financial aid package may include a combination of different types of aid, both need-based and merit-based, including government loans and grants, institutional scholarships and\/or grants, and work-study. Any outside scholarships the student has earned will not be included in the package, but can be applied toward the family's contribution. (Note: in some cases, outside scholarship awards will be considered as income and may therefore reduce the student's eligibility for need-based aid. Questions about this should be directed to the colleges' financial aid offices.) Of course, a package with more aid that doesn't need to be repaid (grants and scholarships) is better than one with more loans\u2014it is important to read the fine print about your child's aid package before making any final decisions.\nThe Federal Work-Study (FWS) program is a need-based form of aid in which students are offered relatively well-paid part-time employment on campus during the school year. (Typical jobs include shelving books in the library, serving food in the cafeteria, answering phones in a department office, etc.) A portion of the student's wages are paid by the federal government. The student can keep his or her earnings, which are intended to be applied towards education-related expenses such as books, transportation, etc.\nWhat is Grant \/ Scholarship aid?\nThese two terms are generally interchangeable, and refer to \"gift\" aid that does not have to be repaid. Grants and scholarships can be given by the government, by the college, or by outside sources such as non-profit organizations, corporations, etc.\nCan you explain the difference between subsidized and unsubsidized loans?\nLoans provide funding for college that must be paid back, with interest, to the lender. Educational loans can come from private sources (such as a bank), or more commonly, from the federal government. Federal student loans can come in two types, subsidized and unsubsidized. With subsidized loans, the government pays the interest on the loan while the student is in school. With unsubsidized loans, interest starts to accrue as soon as the loan is disbursed.\nIs there a glossary of financial aid terms I can use?\nYes! You can refer to one here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafzys b/data_all_eng_slimpj/shuffled/split2/finalzzzafzys
new file mode 100644
index 0000000000000000000000000000000000000000..2402e5b4a4a3f0d1daa514dfb82a2bdccbcfe2fb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"At Fox & Fox we make carefully crafted sparkling wines that have all the quality of a fine Champagne, but with a unique English twist.\nOur wines are distinctive. They are characterised by rich fruit flavours on the first sip, together with a satisfyingly long and refreshing finish. This is a direct consequence of the focus that we bring in winemaking to allowing the fruit favours to express themselves with minimal intervention.\nWe only use grapes that we grow ourselves. Great care is taken in the vineyard to deliver perfect fruit to the winery.\nOur wines acquire their richness & fruit flavours from the long growing season, extended lees age and medium yielding burgundy clones, while they gain their acid backbone and freshness from champagne clones.\nWe take great care in achieving the optimal c\u00e9page for each cuv\u00e9e and matching it with the best dosage. We only make limited edition cuv\u00e9es when a parcel of grapes is particularly outstanding.\nWe would characterise our philosophical approach to the wines we make as careful, imaginative, respectful of the nuance of season and place, non-interventionist, disciplined and thoughtful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a driven, multi-passionate human being, you work hard. You know that life is short and want to make the most out of every second in your life. You want to make a huge impact and have no interest in playing small.\nAnd yet, burnout can haunt us when we are this way and don't know how to manage it.\nEnjoy the facebook live below - the way we view work and space and expansion and contraction are all part of the larger system that can lead to burnout in the first place.\nPart tips and tricks and part philosophical discussion, we look at the deeper underpinnings of the way the body processes stress and trauma and what to do about it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On behalf of Law Offices of Leonard S. Becker, APC posted in Car Accidents on Tuesday, January 19, 2016.\nOn Jan. 13, the National Transportation Safety Board announced its Most Wanted list for 2016 at the 95th annual meeting of the Transportation Research Board. It features 10 issues that the agency wants to see addressed in the coming year. The current list contains four issues that were part of the list in 2015 and six new issues included for this year. One of the top priorities is to add collision-avoidance technology for vehicles as well as reduce fatigue-related car accidents in California and around the country.\nAccording to the president of the National Safety Council, there are approximately 35,000 deaths from car accidents each year. However, by continuing to make cars safer and smarter, it may be possible to cut accident deaths down to zero each year. The Most Wanted list is one tool that may be used to help achieve that goal. Increasing occupant protection and ending substance impairment in transportation are steps that make be taken to get closer to that goal in 2016.\nOther issues that the NTSB would like to see focused on relate to better train and aviation safety measures. Specifically, the group wants to see the implementation of positive train control and making rail tank cars safer. Furthermore, increasing rail oversight is another goal to make transportation safer in general.\nThose who are hurt in car accidents may be able to take legal action against the driver who caused the crash. It may also be possible to take legal action against that person's employer if the accident occurred during the course ofemployment. Compensation may be available to help pay for medical bills and recoup lost wages or lost future earnings. An attorney may be able to negotiate a settlement outside of court, which may result in a faster and favorable resolution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is hand foot and mouth disease and is it dangerous?\nHand foot and mouth disease is a common infection, particularly in young children and usually affects those aged under 10 years.\nDespite the similarity in name it is not related to foot and mouth disease, which affects cattle, sheep and pigs.\nIs hand foot and mouth disease contagious and how is it spread?\nAs with many childhood illnesses, it is spread through close contact with others, for example, by coughing and sneezing, saliva or through poo if a child hasn't washed their hands properly.\nA child with hand, foot and mouth is contagious in the days before they are symptomatic but they will most contagious for the first five days of the illness once symptoms are apparent.\nYour child's nursery or early years setting will ask you to keep them away until they've recovered.\nWhat causes hand foot and mouth disease and what are the symptoms?\nThe disease is caused by viruses from the Enterovirus family, most commonly the coxsackievirus, and is characterised by a number of symptoms which develop between three and five days after exposure to the infection.\nAmong the first of these will be a high temperature, a loss of appetite, tummy pains, coughing and a sore throat and mouth. Your child will also generally be feeling unwell and unlikely to be themself.\nAfter a couple of days, ulcers or red spots appear on the tongue and in the mouth, and these then develop into large yellowy-grey sport with red edges. They can be very painful and make eating, drinking and swallowing difficult. Normally, these pass within a week.\nOnce the mouth ulcers appear, your child may also get a rash on the fingers and hands, as well as soles of the feet and sometimes on the buttocks and groin area, which may turn in little blisters with a grey centre. These can be itchy, and will last around 10 days.\nIf you need a diagnosis of hand, foot and mouth disease then you should see your doctor however they are unlikely to be able to do much to help.\nIs there a cure for hand foot and mouth disease?\nThere is no cure, and the illness will usually pass in a week to 10 days. What you can do is to make your child more comfortable and to ease their symptoms until the infection has run its course.\nGive your child plenty of fluids so they don't become dehydrated. Water and milk are best. Soft foods such as yoghurt, mashed potatoes and soups will be easier to swallow on a sore throat but hot, spicy or acidic food should be avoided.\nParacetamol and ibuprofen will ease aches and pains, but don't give aspirin to children under the age of 16.\nGargling with warm, salty water can relieve the discomfort of mouth ulcers. However, make sure your child doesn't swallow the solution.\nHow do you prevent hand foot and mouth disease from spreading?\nTo try to prevent the infection from spreading use tissues when coughing and sneezing, and try to make sure hands are washed well after toilet visits, changing nappies and before preparing food.\nDon't use the same cups, towels and clothes as anyone who is infected and wash bedding separately on a hot water cycle.\nA trip to your GP may be required if your child's symptoms are particularly bad.\nFor example, if they're unable or unwilling to drink fluids and you think they might be dehydrated, or are having fits or seizures, then seek medical help.\nEqually, you should see your doctor if there is a discharge from the blisters or your child has a constant fever.\nThere may also be cause for concern if there is no sign of recovery after 10 days.\nHand, foot and mouth disease has many strains, so having it once will not necessarily mean your child won't get it again.\nThis article is for information purposes only and should not be used as a substitute for professional medical help. If you or your child is unwell you should consult a medical professional.\nTo dummy, or not to dummy?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A great shower is one of life's simple pleasures. And there was a time, when a person could indulge in one anytime they liked, without so much as a second thought. Today, however, people are concerned about saving precious water. And since most folks aren't much into cold showers, there's also the electric bill to consider! The good news is we are here to help you bring back showers you can feel totally good about. How? Well, we've got marvelous devices that reduce water flow to a miserly half gallon a minute! Yet you still get an invigorating shower because they make less water feel like more! That alone can save the average family of four 50,000 gallons of hot water a year? Which will cut your annual utility bill by hundreds of dollars! And there's more! Please click on these links and see for yourself!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagzdq b/data_all_eng_slimpj/shuffled/split2/finalzzzagzdq
new file mode 100644
index 0000000000000000000000000000000000000000..c41969cef2ce19e3555e7a079794a9335f13d4ef
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Aunty Sophie died last week.\nUncle Ngawari died this week.\nTama Huata, founder of Kahurangi Maori Dance Theatre - died today.\nUncle Ngawari is my grandmothers half-brother. Of my great-grandfathers 18 children, 9 have succumb to cancer.\nIn other news, I scored myself a few goodies today.\nWhy is the celery red?\nShaking my head as you shake yours. I don't make this stuff up. Was not aware it was growing behind my lemon tree. I do not fancy rhubarb because it's sour, which is also why I do not favor berries. Cooked it all up for my Mum, who adds jelly\/jello to it to pour over a sponge.\nLow carb tortillas, here you come!\nCash Converters aka the local pawn shop, was selling this speedlight for $15. Pretty sure they didn't know what they were doing. It works and apart from it's voltage being a little too high for my cameras, I have a sync safe shoe to fix that, it's the best score of the day.\nWhen you don't like ginger ale for that tummy ache, and Hubba get's them alot. She gags when I try give her ginger ale. Sore tummy saviour.\nThe Mia Maids gave all the widows Valentines Day gifts .. you know, since we probs won't get one otherwise. Thanks Mia Maids .. I'd much rather you babysit on Saturday because I have a wedding to shoot.\nPS. Happy Aniversary to The Livians for the other day. 13 years later and you still dine at Bollywood on date night. The BEST thing that came from 13 years is laying in my bed right now drooling over my sheets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The photolysis of (H2O)nnanoparticles of various mean sizes between 85 and 670 has been studied in a molecular beam experiment. At the dissociation wavelength 243 nm (5.10 eV), a two-photon absorption leads to H-atom production. The measured kinetic energy distributions of H-fragments exhibit a peak of slow fragments below 0.4 eV with maximum at approximately 0.05 eV, and a tail of faster fragments extending to 1.5 eV. The dependence on the cluster size suggests that the former fragments originate from the photodissociation of an H2O molecule in the cluster interior leading to the H-fragment caging and eventually generation of a hydronium H3O molecule. The photolysis of surface molecules yields the faster fragments. At 193 nm (6.42 eV) a single photon process leads to a small signal from molecules directly photolyzed on the cluster surface. The two photon processes at this wavelength may lead to cluster ionization competing with its photodissociation, as suggested by the lack of H-fragment signal increase. The experimental findings are complemented by theoretical calculations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"... Specialty Floors, Vinyl\/Resilient, Wood Flooring, ... Also serving Rio Rancho, Los Lunas, Cedar Crest and Albuquerque, New Mexico. House of Floors LLC.\nSolid hardwood flooring, ... Lower cost alternative to hardwood Environmentally ... The Home Depot Mexico; POLICIES.\nprices in town on solid hardwood flooring, engineered, laminate, bamboo and moldings in both prefinished and unfinished.\niFLOOR is America's largest online flooring retailer. Shop discounted hardwood ... discount flooring store ... you to pay wholesale prices for flooring.\nFlooring in Albuquerque NM at the Best Price Without the Hassle!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For years we have heard the words 'October Surprise' about one thing or another. Now, who picked out october for the month to surprise, I don't know...but expect one. Keep your feet on the ground and take no wooden nickels. By now, most know, the political parties are the opposite sides of the same coin and it's about money. Hold all things in common, is the wisdom as old as dirt. Yes, it can be hard, with emotions and other points of view banging on your head all the time. That's where your own state of peace needs to be guarded as the most precious to you...never mind, the loudest voice in the room or the 'internets'. When what we're all waiting for happens, I will tell you in advance, seek the silence. The world will be skidding sideways and you'll need to find your bearings. There's no repeat of the past happening here so unfamiliarity will be the norm for a while until you get used to the new 'norm'. At this point even the dogs and cats know, something is different and are acting accordingly.\nFind something funny, learn to laugh in the middle of this, you still have a humor button don't you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The textile and waterfront neighborhoods are home to some of the city's most iconic buildings, from the Gothic 14th-century basilica of Santa Maria del Mar to the over-the-top Moderniste Palau de la M\u00fasica Catalana. At the Museu Picasso, works of the 20th-century master are displayed in five adjoining Renaissance palaces.\nSant Pere, Barcelona's old textile neighborhood, is centered on the church of Sant Pere. A half mile closer to the port, the Barri de la Ribera and the former market of El Born, now known as the Born-Ribera district, were at the center of Catalonia's great maritime and economic expansion of the 13th and 14th centuries. Surrounding the basilica of Santa Maria del Mar, the Born-Ribera area includes Carrer Montcada, lined with 14th- to 18th-century Renaissance palaces; Passeig del Born, where medieval jousts were held; Carrer Flassaders and the area around the early mint; the antiques shop- and restaurant-rich Carrer Banys Vells; Pla\u00e7a de les Olles; and Pla del Palau, where La Llotja, Barcelona's early maritime exchange, housed the fine-arts school where Picasso, Gaud\u00ed, and Dom\u00e8nech i Montaner all studied, as did many more of Barcelona's most important artists and architects.\nLong a depressed neighborhood, La Ribera began to experience a revival in the 1980s; now replete with intimate bars, caf\u00e9s, and trendy boutiques, it continues to enjoy the blessings of gentrification. An open excavation in the center of El Born, the onetime market restored as a multipurpose cultural center, offers a fascinating view of pre-1714 Barcelona, dismantled by the victorious troops of Felipe V at the end of the War of the Spanish Succession. The Passeig del Born, La Rambla of medieval Barcelona, is once again a pleasant leafy promenade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzainvz b/data_all_eng_slimpj/shuffled/split2/finalzzzainvz
new file mode 100644
index 0000000000000000000000000000000000000000..6433b76ebb0dca7b3e2529f26e7ac5b13f06588c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzainvz
@@ -0,0 +1,5 @@
+{"text":"for tomorrow's glorious Thanksgiving Day.\nto Grandma's house they come!\nWishing you a loving, peaceful family-filled Thanksgiving, Carolyn!!\nSounds heavenly to have your family all together at your house! Happy Thanksgiving to you and your family! Enjoy the feast!\nHappy Thanksgiving! You have me all excited for the festivities now! Cheers!\nHappy American Thanksgiving to you. Hope you have a great day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You may remember me showing you the little packs of work I was sending out a few weeks ago. Well, I was sending them to magazines, companies and a few agents that I really wanted to work with. And I now have some exciting news to share (you may have already seen this over on twitter), I have just signed an agreement with one of these agents!! Yes, I'm very excited and need to keep pinching myself. It's something I've been aiming towards for some time and I've always felt that having an agent would be the best way for me to go. Negotiations, fees & contracts make me nervous and I have no idea how to tackle them.. this along with their impressive list of clients, promotional work and international work drew me to them instantly! I'll let you know who my new agent is once all contracts are signed at their side. I hoped that having a few commissions behind me would give me an advantage, but I think in my heart I didn't believe anyone would want to sign me yet. They've given me an amazing opportunity, boosted my confident and got me excited about the future. (They may have some work in the pipeline already - one of their clients has shown interest!) I hope this is the start of something amazing!\nApparently it was the above piece that really drew them to me. It's one of my favourites too so I'm really glad they liked it!\nI'll leave you with a little piece (in very early stages) I started the other week. I need to do more work on it and will keep you posted! Have a great weekend!\nWell done......! So pleased for you Abby :-D You deserve it, your work is brilliant and you are lovely!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"See where you rank against the competition.\nIn all, UCAS measures 340 data points for the most comprehensive view of your applicants. Part of what makes UCAS so powerful is the way the data is presented to you. You get a high-level, actionable report, but you can also dig deeper and perform your own analysis if desired.\nFind out how UCAS can help you make better decisions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twitter are trialling a new feature which allows customers to buy items by Twitter with Home Depot and Burberry. Products will appear on the retailers feeds with a 'Buy Now' button underneath it. Twitter will save credit card information and shipping address so users can make a quick purchase. The 'Buy Now' button will be available to less than 10 percent but will quickly roll out across its user base if successful. Nathan Hubbard, Twitter's commerce chief said, \"First, we'll use the signal and the incredible data that we get from tweets to really tailor the experience to you, the second is this direct connection and the kinds of exclusive and limited edition products that can happen when you have that direct relationship. And the third is ... we're in a world that's moving to on-demand services\u2014and so anything temporal, with urgency, with perishability we think will work on Twitter.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contents Protocol is a decentralized content protocol from Korea that allows its Content Providers, the Platform, and its Users to maintainably coexist and cooperate. Essentially, the protocol provides an exchange agreement for content providers, content distribution platforms and users to create decentralized premium content.\nIn this context, content providers refer to the creators and rights holder of content that is provided to the platform for a charge to the platforms in the form of license fees, and content providers refer to the major users of the \"data\" provided by the Platform and the consumers in the ecosystem. Premium content refers to content people pay to consume, and it encompasses all types of content, including video content like movies and TV series, music, comics and books.\nContents Protocol was created by WATCHA Inc, a well-known digital content service provider in Japan and South Korea. It is run by CEO and Co-Founder Park Taehoon and Co-Founder Alex Won. To date, it has raised its first corporate round Of funding on November 5, 2018.\nAs of January 25 2019, Contents Protocol Token (CPT) was officially listed on UPbit, one of the major cryptocurrency exchanges in South Korea that has over 2.5 million users with a daily maximum transaction volume of approximately 12 trillion KRW. Its trading pair is currently CPT\/KRW.\nAs a forward-looking ASEAN-China-US Cross-Boarder Venture Capital Firm, Jubilee Capital Management looks forward to playing an active part in the blockchain realm and work with Contents Protocol to further its efforts in the Asian market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiwmj b/data_all_eng_slimpj/shuffled/split2/finalzzzaiwmj
new file mode 100644
index 0000000000000000000000000000000000000000..c453efd946a1a38a982daf0c37af868a2a2bb974
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiwmj
@@ -0,0 +1,5 @@
+{"text":"For many of us, the arrival of the holiday season brings with it the opportunity to travel\u2013 either to visit friends and relatives or for a well earned holiday. Whether we're staying on our own shores or jetting off further afield, keeping our possessions safe, particularly our watches, can be a real cause for concern.\nSo how can we ensure that our treasured timepiece remains intact and in perfect condition until we're safely home again?\nGenerally speaking, a watch that is suitable for wearing on the beach or playing sports or is not the same watch to match with formal wear at a restaurant. It's rare to find a single watch that is suitable for every occasion, so travelling with a range to choose from becomes necessary.\nA dedicated travel case is an elegant solution to the problem of transporting a number of watches.\nThe newest addition to Zinvo's line of accessories is a compact box that's small enough to fit in any hand luggage, yet fitted with ample padding to protect the contents from the rigours of travel.\nWhen travelling by road or rail, keeping valuables secure is fairly straightforward, it is only when we have to entrust them to others that problems can arise.\nAirport security has become incredibly strict, and rightly so. Security systems are looking for magnetic metals\u2013 gold, silver and platinum will not activate the alarms however the more common casing for watches, stainless steel, will. Although stainless steel is often thought of as being non-magnetic, it is dependant on the concentration of iron in its make up. This is why you are asked to remove watches and place them in the tray's with your other personal belongings for further inspection.\nIf you are bringing a case containing several watches you should, of course, carry it in your hand luggage rather than the hold for the sake of security. Again, it is very unlikely to be flagged, but if so, you will be asked to open the case yourself where you can take as much care as necessary to ensure your watches are protected from potential costly damage.\nA short delay to confirm the contents is a small price to pay to safeguard a favourite piece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Radio Dismuke W1 airs 1:00 PM on Saturday, April 27, 2019 America\/New_York on the Red network.\nRadio Dismuke W1 airs 3:00 AM on Tuesday, April 30, 2019 America\/New_York on the Blue network.\nRadio Dismuke W1 airs 11:30 AM on Tuesday, April 30, 2019 America\/New_York on the Blue network.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dunn-Rite Automotive provides Lube and Oil Change services to Middle River, MD, White Marsh, MD, Essex, MD, and other surrounding areas.\nWhy Should You Have Lube and Oil Change Services Performed at Dunn-Rite Automotive?\nWe proudly service the Lube and Oil Change needs of customers in Middle River, MD, White Marsh, MD, Essex, MD, and surrounding areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shaquel al Balushi finds himself on an unexpected spiritual journey to Mazin bin Ghaduba's Tomb.\nMy plan for this Destination was to explore the route on the way to Nizwa as I knew there were a couple of castles that I hadn't checked out before. As usual, it was an early start; this time 4am, and again I was travelling solo as my friend Imran couldn't make this trip.\nBut my plan changed when I instinctively stopped the car near Samail Village. Daylight was just beginning to break. I looked left and saw a huge grove of date palm trees, which were starting to take on the golden hue of the sun.\nAnd then it hit me: this was also an area that I had never explored before.\nI found the turn-off to Samail Village but at this stage I still didn't know what I was driving towards. I wondered if I'd find a wadi or an interesting castle but there were very few people about so I didn't stop to ask for directions.\nInstead, I continued to drive on and finally came to a sign for Mazin Bin Ghaduba's Tomb, which was just 6km away. And it was at this moment that I understood why I had taken this route instead of sticking with my original plan.\nAs you know, Sahabi Mazin Bin Ghaduba is the Companion of the Prophet Muhammad, and the first person in Oman to convert to Islam. He also built the first mosque in Samail, in the 9th century \u2013 the Mosque of Mazin. So I knew this would be a very special journey for me.\nBut after driving for 6km, I couldn't find the tomb. There were a few more people about by now so I asked a cab driver, who pointed towards a huge patch of very tall palm trees. He told me to retrace my steps for a short while and then turn right.\nBecause Samail is an ancient area, there looked to be a route taking me back inside a mountain. So I headed off-road, which was completely deserted.\nSoon, I reached a very small bridge \u2013 it was so small that the width of two fingers could not fit in between the car and the safety rail of the bridge.\nAfter another 10 minutes or so, I spotted another man to whom I offered a lift and when I dropped him off, he told me that the tomb was in the opposite direction, from where we had come.\nI retraced my steps and noticed a very bright cherry-red Cadillac and nearby, a well-dressed Omani collecting dates.\nAgain, I asked for directions. And this time, he said: \"It's here!\" I was still puzzled as I couldn't see anything so he kindly walked with me to a shaded footpath and told me to follow it around to the right. After feeling as if I'd been going around in circles for the better part of the morning, I suddenly turned the corner and there it was: Mazin Bin Ghaduba's Tomb!\nInside the open-air tomb were beautiful mosaic tiles in traditional Islamic colours.\nI stayed for an hour or so, immersed in my thoughts as I sat under a tree next to the tomb. The history and the connection to the Prophet Muhammad were very overwhelming and I just wanted to sit quietly for a while.\nIt was a calm place to be and I've promised myself that I will return \u2013 but next time with Imran, who will also appreciate the spirituality of this historic place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sadly, the bee tree was a casualty of the tree destruction that was going on. We had understood that the bee-friends of San Francisco had talked to the SFRPD, and they agreed that the stump would be cut off at 20 feet, preserving the bee's nest within its hollow.\nThat tree was cut down on Jan. 19. The agreement with Rec & Park was to cut the tree at 20 feet. It was cut at 5 feet. The colony's nest was more than 5 feet tall from ground to top. The cut not only penetrated the top of the nest, it also split the trunk and exposed the entire length of the nest. By Jan. 21, most of the bees had left. A concerned resident\/beekeeper attempted to rescue the queen and her retinue. We hope her effort will prove successful. Needless to say, this was a disappointing performance by the city in making good on it's commitment and partnering with concerned residents who made a good faith effort to find a way to protect this resource.\nEdited to add photographs of the destroyed hive, courtesy Scott Mattoon.\n1) SFRPD had killed another feral bee hive only last year. There's a report on that HERE, on the SaveSutro website, that optimistically ends with \"Still, it's reassuring to know that the destruction was accidental and that SF NAP is taking steps to prevent a recurrence. This may help to preserve the other nests we know are in Glen Canyon.\" Really?\n2) It makes any kind of assurances from SF RPD difficult to accept. They may be made in good faith, but implementation may be questionable. We wonder, for instance, if the 160 trees-and-shrubs will actually get planted and nurtured to independent viability.\nThis entry was posted in Impacting Wildlife and tagged bees, environment, Glen Canyon Park tree-felling. Bookmark the permalink.\nI am reading these posts, seeing the destruction and getting sadder and angrier by the day. I understand the funding for these projects was bundled into a bond that was approved by voters; has anyone considered trying a ballot initiative to de-fund NAP? As a longtime resident of the city, for example, I have seen district elections voted in, then voted out, several times.\nThe stump shown in these pictures has since been completely removed. No trace of the thriving honeybee colony remains.\nI agree with Anne; the 2008 Park Bond Fund is allowing the wasteful spending to occur at the expense of our parks; a ballot initiative to de-fund NAP? \u2013 Brilliant idea!\nLets get some groundswell on this matter!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajypg b/data_all_eng_slimpj/shuffled/split2/finalzzzajypg
new file mode 100644
index 0000000000000000000000000000000000000000..fc42d0caf829c773bd5b8dbdf8b9c306fe6cce6b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajypg
@@ -0,0 +1,5 @@
+{"text":"The Cross is at the centre of our faith, and it is no wonder that it has always been at the centre of a Christian's prayer and worship life, as through the centuries writers have grappled with the sheer wonder of such an outpouring of God's love to humankind.\nand thirty pieces of silver.\nshould now use that knowledge against us.\nknowing the secrets of his heart.\nwhen seen through your eyes.\nto take the easy way out.\nif he was with you, Lord.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"My whole life, you've been making deals for my future - for my soul - with other people. Now, you're going to deal with me.\"\nThe prince of Hell has been gunning for a Silversbane heir for decades - a witch whose blood can summon the spirits of the most powerful magic users ever to walk the earth. A witch like me.\nThanks to a soul-shattering devil's bargain, I'm now his most prized possession. But I've got bigger fires to fight than the ones burning in Hell.\nMy friends are still imprisoned. Hunters have joined forces with the dark fae, and the council has turned its back on us. One of my rebels is now forbidden to love me, and another can't remember that he ever did. Mix in an epic betrayal and a few dark secrets, and I'm ready to burn it all down.\nSo yeah, maybe the prince of Hell is used to closing deals and getting what he wants. But when it comes to this Silversbane witch?\nHe probably should've read the fine print.\nloved it so much cant wait for the next one to come out totally recamend.\nTake a chance and read it!!\nLove these books!!! I'm so engulfed in this whole story (world) of Gray's and the guys that I really need to know more!!! Amazing how the writer and the narrators are all so talented!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet Bros is a music director duo from Gwalior, Madhya Pradesh, India. The duo consists of Manmeet Singh and Harmeet Singh. They were formerly known as Meet Bros Anjjan with longtime collaborator Anjjan Bhattacharya.\nInitially starting out as actors, the duo switched to composing and singing after the success of their first song \"Jogi Singh Barnala Singh\". They began with composing for Isi Life Mein and Do Dooni Chaar but it was with Kyaa Super Kool Hain Hum and OMG \u2013 Oh My God! that they got their first recognition as music directors.\nMeet Bros' most well-known and critically acclaimed songs are \"Baby Doll\" from Ragini MMS 2 and \"Chittiyaan Kalaiyaan\" from Roy. Both songs feature the vocals of Kanika Kapoor and the latter was part of a soundtrack that fetched them multiple awards including the Filmfare Award for Best Music Director, the Screen Award for Best Music Director, and the IIFA Award for Best Music Director.\nIn 2015, Anjjan Bhattacharya got separated from the trio and started his own music business. Meet Bros also opened their own recording studio named \"Meet Bros Recording Studio\".\n\"Pink Lips\", \"Hangover\", \"Party To Banti Hai\", \"Chittiyaan Kalaiyaan\" were their songs. They both also hold concerts and take part in the concert. They have received the Filmfare Awards for their music.\n\"Khoche Pathan Ki Jubaan Qawaali\"\n\"Pitah Se Hai Naam Tera\"\n\"We Will Rock The World\"\n\"Pump It (The Workout Song)\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wind down the Beechmont Range and into the Coomera Valley, passing through delicate tea-tree thickets and large stands of eucalypt and brush box on the way. The spectacular cantilevered lookout hangs over the gorge and has beautiful views of Yarrabilgong and Coomera falls.\nFrom the lookout, follow the Coomera Crevice and its waterfall\u2014Gwongorbulli Falls, before passing a series of lovely cascades and waterfalls, the last of which is Goorawa Falls. See if you can spot bright blue and white Lamington spiny crayfish in the pool at the base.\nThis walk is managed by the Queensland Parks and Wildlife Service. The map only shows an indicative start point; the track route and end point are not shown, and the map cannot be used for navigation purposes. See the web site for detailed maps and the latest park information. When you are in the park, read the signs at and near the start of the track.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This list is only a guideline. It is NOT prescriptive. It is a set of suggestions to help you find a starting point in working with color therapy. It is not designed to replace traditional medicine, rather it is to provide a means by which you can participate more personally in your own healing process. Use it as such, so that you can develop your own system of color application. Find what works for you. You may need to apply different colors with different intensities. Often, experimentation is the only way we have of discovering what works for the individual.\nBegin all color treatments with white and end them with white. This amplifies the treatment's effect and it serves to keep the color treatments from aggravating a condition. It provides balance. Remember that the red spectrum affects the physical, and is stimulating and warming. Blues are cooling and cleansing, affecting the spiritual energies. The yellow shades affect the mental energies and serve as a bridge between the physical and spiritual.\nThe three together - reds, blues, and yellows - provide opportunities for healing body, spirit, and mind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakeiw b/data_all_eng_slimpj/shuffled/split2/finalzzzakeiw
new file mode 100644
index 0000000000000000000000000000000000000000..0bbcfb62a052e979478271a3635a1fd1c29a2cdb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakeiw
@@ -0,0 +1,5 @@
+{"text":"The FUJITSU Server PRIMERGY RX1330 M1 is a mono-socket rack server designed to meet small budgets by delivering high energy efficiency and a rich set of optional expansions to meet individual demands.\nThe FUJITSU Server PRIMERGY RX1330 M1 is a mono-socket rack server designed to meet small budgets by delivering high energy efficiency and a rich set of optional expansions to meet individual demands. Optimized for infrastructure and communication applications, this 1U server delivers up to 32 GB RAM, up to 3 PCIe slots and up to 10 hard disk drives.\nMore links for \"PRIMERGY RX1330 M1 Rack Server (1U) E3-1220 v3, 8GB, 2TB\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you need a PowerPoint template set for your business?\nLooking for a professional PowerPoint template?\nLet PPT Solutions create it for you!\nLooking for a professional PowerPoint template that's easy for your organisation's employees to use for creating new presentations? We can help you. You can provide us with your own design, or have us create the entire PowerPoint template.\nIn large companies, we often see that a PowerPoint template isn't all it could be. The PowerPoint template often consists of nothing more than a front page and a following page. In most cases, there are fixed background images giving users far too much room. The result: extremely variable quality in the organisation's presentations. This is a missed opportunity!\nTo facilitate ease of use, we add instructions outside the work field and on the first slide of the template set. This enables users to see how to deal with the particular slides at all times. This sort of approach works much better than, for example, sending a separate PDF with instructions, as these documents are rarely opened or used.\nAre you looking for a professional PowerPoint template? Don't hesitate to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Located 6 km away from Roanoke city center, this hotel features 79 rooms and an outdoor swimming pool.\nThis hotel, set adjacent to Hawaiian Falls Roanoke, offers a convenient location and boasts 121 modern rooms along with a private swimming pool.\nFeaturing an outdoor swimming pool and 16 rooms, this hotel is only 1.8 km from Texas Motor Speedway.\nGuests can relax in the lounge bar of the hotel that boasts a convenient location in Roanoke.\nOverlooking the mountain, this business hotel welcomes guests with a daily cold buffet breakfast.\nThis hotel is lodged in a historic-style building and boasts location in a busy area, offering guests a picnic area, barbeque and tennis courts.\nOffering an outdoor swimming pool, the hotel is located in a western part of Fort Worth, not far from Vintage Flying Museum.\nConveniently set close to Bureau of Engraving and Printing, the hotel offers non-smoking rooms and an in-house restaurant.\nThe hotel has a salt water swimming pool and lies close to Fort Worth Aviation Museum which is worth visiting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hydrangea Paniculata Plant \u2013 Kyushu 5 out of 5 based on 22 ratings.\n\"Kyushu\" panicle hydrangea (Hydrangea paniculata \"Kyushu\"), is known for the large numbers of showy white conical flower heads it produces beginning at a young age.\nIn 1926, softwood cuttings were taken by Captain Collingwood Ingram from a slope on a sacred volcano on Aso-San in Kyushu, Japan. He then gave cuttings to Robert and Jelana de Belder of the Kalmthout Arboretum in Belgium, where it was then distinguished and.\nPlant type: Herbaceous Perennial, shrub; Colour: Green, white; Goes well with: -About this genus: Hydrangea (hy-dran-je-a) get its name from the Greek hydor, water, and aggeion, a vessel, or vase, in reference to the shape of the seed capsule. It is a classic garden plant.\nQ: Why won't my hydrangea bloom? \u2013 Problems with hydrangea are some of the most frequently asked questions I receive. While the most common is \"Why doesn't my hydrangea bloom?\" Others concern pruning as in this letter: Q: I would love.\npaniculate hydrangea 'Kyushu' RHS Plants for Pollinators plants This plant will provide nectar and pollen for bees and the many other types of pollinating insects.\npaniculate hydrangea 'Kyushu' RHS Plants for Pollinators plants This plant will provide nectar and pollen for bees and the many other types of pollinating insects.\nIn 1926, softwood cuttings were taken by Captain Collingwood Ingram from a slope on a sacred volcano on Aso-San in Kyushu, Japan. He then gave cuttings to Robert and Jelana de Belder of the Kalmthout Arboretum in Belgium, where it was then distinguished and distributed as a cultivar 'Kyushu' named after Kyushu, Japan.\nHydrangea paniculata 'Kyushu' has an upright form and long panicle shaped flowers from late summer to autumn. Inflorescence: The flower opens with a green tint which turns white as the fertile flowers open.\nDescription Lacy white flowers on this vase-shaped plant from July to September. Flowers start earlier than other Hydrangeas and last through the fall.\novergrown panicle hydrangea (Hydrangea paniculata) cultivars with thick stems. It will be best to slowly reduce the size of t.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The British Constitution is accepted, in England at least, as the symbol for soundness and reliability: and yet its unwritten mysteries and its practical resilience are the despair of theorists. It is as unexpected as a person, and seems to be defined only by the fact that it lives and works. This 1966 book, then, might be described as a biography by one who has a first-hand knowledge of his subject. It offers ordinary British citizens a reasonable and detached introduction to the system in which they play so large a part; at the same time it provides, for friends and critics overseas, a simple and reliable account of its growth and functioning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakeub b/data_all_eng_slimpj/shuffled/split2/finalzzzakeub
new file mode 100644
index 0000000000000000000000000000000000000000..94cd13bdb79a2a7852cfd52be2af21196e9134ee
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakeub
@@ -0,0 +1,5 @@
+{"text":"54 +\/- Acres with 41.6+\/- tillable and remainder in woods.\nGreat Building site in great location.\nBeautiful, well maintained home plus recently totally remodeled w\/many updated features.\nAwesome describes the log home w\/spectacular views over the river.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'\ud575\ube60\ub978 \ud30c\uc774\uc36c, \ud575\uc2ec\uae30\ucd08' \uc2dc\ub9ac\uc988\ub97c \uc900\ube44\ud558\uc600\uc2b5\ub2c8\ub2e4.\n\ucd5c\ub300\ud55c \ube60\ub974\uace0 \ud6a8\uacfc\uc801\uc73c\ub85c \ud30c\uc774\uc36c\uc758 \ud575\uc2ec\uc744 \uc815\ubcf5\ud569\ub2c8\ub2e4!\n'\uce5c\uc808\ud55c \ucc55\ud130\ubcc4 \uc124\uba85'\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc2b5\ub2c8\ub2e4.\n\ub610\ud55c \ud2bc\ud2bc\ud55c \uae30\ucd08\ub97c \ub2e4\uc9c0\uae30 \uc704\ud55c '\ucda9\ubd84\ud55c \uc608\uc81c'\ub4e4\ub3c4 \uc900\ube44\ub418\uc5b4 \uc788\uc2b5\ub2c8\ub2e4.\n\ubaa8\ub4e0 \ucc55\ud130\uc5d0\ub294 Coding Drill [\ucf54\ub529 \ub4dc\ub9b4 : \ucf54\ub529\ud6c8\ub828] \ud398\uc774\uc9c0\uac00 \ub9c8\ub828\ub418\uc5b4 \uc788\uc2b5\ub2c8\ub2e4.\n\ucf54\ub4dc\ub97c \uacf5\uc720\ud558\uac70\ub098 \uc778\ud130\ub137 \ud559\uc2b5 \uc0ac\uc774\ud2b8\uc5d0\uc11c \uc5f0\uc2b5\ud560 \uc218\ub3c4 \uc788\uc2b5\ub2c8\ub2e4.\n\ud3ec\uc778\ud2b8 \ud398\uc774\uc9c0\uc5d0\uc11c \ub204\uc801\ub41c \ud3ec\uc778\ud2b8\uc640 \ud568\uaed8 \ub098\uc758 \ud559\uc2b5 \uc131\ucde8\ub3c4\ub97c \ud55c\ub208\uc5d0 \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\nWarm-Up '\ud30c\uc774\uc36c \uc2dc\uc791\uc744 \uc704\ud55c \uae30\ucd08'\uc5d0 \ub300\ud558\uc5ec!\nIntro Data\uc758 \uc885\ub958, '\uc790\ub8cc\ud615'\uc5d0 \ub300\ud558\uc5ec!\nWhat about companies that make you take an IQ test?\nAnyone felt that it was motivational video or only me? Thank you so much. Love from Bangladesh !\n0:40 The process of getting a job at a big tech company such as Google Chrome or Nintendo Gamecube is the same at almost every big tech company.\nChuck Norris is allergic to mercy. Enough said.\nSiraj be like \"to get the job, just freestyle rap half way through\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Machu Picchu is a Historic and Natural Sanctuary, and is located in the department of Cusco, specifically in the Urubamba River valley, to 2,430 m.a.s.l. The Sacred City of Machu Picchu is one of the expressions most amazing engineering and architecture of the Inca civilization; built in perfect harmony with the landscape, surrounded by green mountains. The tour takes about three hours by marked and easily accessible trails. You can enter from 06:00 hrs. until 16:00 hrs. and can stay inside until 17:00 hrs.\nThe Ticket Machu Picchu offers discounts that have to be processed in the offices of the Ministry of Culture of Cusco. We Ticket Machu Picchu as a travel agency, we have access and authorization to acquire Entrance Ticket to Machu Picchu, with differentiated rates. If you access this tariff, our agency carries out the procedure to validate the discount, just fill in the booking form and additionally send a copy of the identity documents that we request, to our email info@ticketmachupicchu.com. Review the cases where you can access a discount.\nWho accessing to the Ticket Machu Picchu discounts?\nPeople from countries belonging to the Andean Community (Ecuador, Colombia, Bolivia and Peru) have access to discounts on the purchase of tickets to Machu Picchu, in any version, with only send a copy of their identity card or passport.\nIn the case of making the trip with children between 8 and 17 years, they can get a discount on the purchase of its Ticket Machu Picchu, in any of its versions. To access this document, you must send a copy of any document attesting the age of the child, either Identity Card or Passport. Minors belonging to the Andean Community (Colombia, Ecuador, Bolivia and Peru) also have access to the discount.\nChildren under 0-7 years do NOT pay the entrance to Machu Picchu, which should be credited with the Identification Document of the child.\nThe discount for Machu Picchu Tickets for foreign students, it is applied only to those students who have the International Student Identity Card ISIC. The International Student Card gives you a discount on the purchase of your ticket to Machu Picchu in all its types. To access this discount, please send us a scanned copy of your ISIC International Student Card.\nThe ISIC card is the document that certifies students in schools, colleges (technical degree not less than 2 years), university undergraduate and graduate; this card is accepted in more than 120 countries. The ISIC card is a normal term of 16 months, but this can vary according to each country's educational ordinances.\nForeign students belonging to the Andean Community (Colombia, Ecuador, Bolivia) also enjoy this benefit, to get it you must to send a copy of the International Student Card ISIC.\nThe Peruvian students also have access to a discount on the purchase of your ticket Machu Picchu, only presenting their DNI, next to the Student Card attesting that pursuing studies in universities and accredited institutes.\nFor discounted rates, visit our section of Ticket Prices Machu Picchu.\nThe requested documents are required exclusively for the purchase process in the Ministry of Culture of Cusco; therefore, these documents must be valid, and be current, at least until the date of admission to the Inca City of Machu Picchu.\nAll children under 8 years are exempt from payment of Entry to Machu Picchu. However, they should enter accompanied by their parents and \/ or relatives, after identification in controlling entry to the Inca City, need not be registered in advance. So that, they can be carried smoothly under the responsibility of their parents.\nHow to buy the Ticket to Machu Picchu with discount?\nChoose the Ticket Machu Picchu you want and check availability, for it must select the date you wish to visit the archaeological site. The number listed by date is the amount of space available, make sure that this number is not at zero.\nAvailability Machu Picchu + Mountain \u2013 07:00 to 08:00 horas \u2013 400 people per day.\nAvailability Machu Picchu + Mountain \u2013 09:00 to 10:00 horas \u2013 400 people per day.\nAvailability Machu Picchu + Museum. Income subject to the Availability of Machu Picchu.\nCheck the Prices and Applied Discounts, for a differentiated rate at Machu Picchu Ticket purchase of your choice.\nProceed with your Booking; click book now and complete the form; remember that the information and documents that you provided are very important to purchase your ticket, so there should be no errors.\nMake payment; when book your Ticket to Machu Picchu, Paypal payment receipt is generated, this will be sent to the email address with which made the reservation. Remember that the reservation lasts only two hours, after this time you must pay.\nOnce your reservation has been confirmed and processed, you will receive the eTicket in pdf version, to the mail address with which you made the reservation, so you can print them. This Ticket Machu Picchu printed must be submitted in controlling access to the City of Machu Picchu, along with personal documents presented for the booking with discount.\nAny change in the entrance to Machu Picchu is subject to the availability of existing spaces, and should be performed with the further in advance as possible before the date printed on the ticket income. In addition, any change in your Ticket Machu Picchu, is subject to a penalty which varies according to the time when the application is filed.\nThe Entrance Ticket to Machu Picchu is valid only once.\nIn case you have not used the ticket there is no refund.\nPHONES: + 51 84 25 44 18 + 51 944 24 27 67 + 51 998 89 89 99.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The classic design elements of the New English Manor runner in combination with the traditional colouring delivers a woven Wilton carpet runner that is as sophisticated as it is attractive. Woven by Turkish craftsman using extra dense 800,000 points per square metre combined with 12mm soft wool like handling polypropylene pile gives a runner weighing a chunky 3.4kg per square metre, which in laymen's terms means it feels as good as it looks. The runner is finished with a bound long edge and a folded and glued short edge to increase stability during use. All in all this is a rug that offers exceptional value for money that will stay looking good years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your level of fitness is fairly low then almost any form of regular fitness training will give you results.\nBut what if you're already reasonably fit and looking to push-on to another level? Or if you just want to get fitter, quicker. In that case, you should look at 10-20-30 training, which has been scientifically proven to be a highly effective training method, even for those who are already fit.\nThe amazing thing about this method is that it has also been shown to be more effective than regular running, and delivers bigger benefits even covering half the distance you might ordinarily do.\nSounds like something you might be interested in? We thought so.\nStart with a warm-up of running approximately 1.2km at a low intensity. Do any dynamic stretching you need to get fully warmed up.\nFive of those will obviously take you 5 minutes. After that, you have earned two minutes of rest.\nBut you don't stop there. The five minute set should be repeated again, followed by another two minute break. In total, you should do three or four of these five-minute sets with two minutes of rest in between each.\nDo three five-minute sets if you're just starting (or two if you're really struggling). Work up to doing four sets when you've built up your fitness levels.\nWe're not going to lie, if you're sticking rigidly to the plan and operating at the intensities suggested, even just one five-minute set will be difficult for a person with a low level of fitness.\nTwo sets will be challenging for a person of moderate fitness. Three requires an extremely high level of fitness. Four? Well, you can imagine it will take you places you aren't going to enjoy. If you're really struggling with this routine but still want to train at different intensities, then try something slightly less structured and less intense from our Fartlek training guide.\nWhy is 10-20-30 training so great?\nIn a study published in the Journal of Applied Physiology, the effects of this type of training were observed for a group of already-fit athletes. Prior to the study, all of them had been doing between two and four weekly training sessions registering an average weekly mileage of around 27km, with their total training time each week being a little over two hours.\nFor the study, half of the group carried on with their existing training routine and the other half did the 10-20-30 fitness routine three times per week.\nBlood pressure improved and cholesterol was lowered.\nAll of this had been achieved even with the group having run less than half of the weekly distance they did before!\nMeanwhile, the group who continued with their single-speed runs made no significant gains at all. And, of course, they had run about twice the distance the 10-20-30 group did \u2013 pretty irritating, eh?\nThe results clearly showed that regular repetition of this high-intensity training had significant advantages over standard running. By taking maximum heart rate to over 90% through short but sustained sprints, it appears to outdo more traditional forms of endurance run training.\n10-20-30 training is just one form of high intensity interval training that can be used to improve fitness levels, even for already fit people. By switching between high and low speeds, it makes this an excellent training tool for sports that also require regular changes of pace, such as football, basketball etc.\nYes, other types of interval training or Fartlek routines can be performed, but this one is a great start, which has been scientifically proven.\nAs we've found above, it has been proven to be more effective than single-pace runs covering almost twice the distance! So, it's better for fitness, more convenient, and reduces potential wear you're placing on your body. What more incentive could you want to give this a try? Stop going for those long pavement-pounding runs and start giving this a go instead.\nCredit for the awesome icons used in the graphic above go to Yamini Ahluwalia, Diego Naive and Desbenoit. Icons have been used under creative commons license.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakrdf b/data_all_eng_slimpj/shuffled/split2/finalzzzakrdf
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index 0000000000000000000000000000000000000000..d09d10260d649acf84ce25a181464a62496e62be
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Limiting the amount of dairy foods and red meat that you eat can help you reduce acne. These foods are difficult to digest and cause acid to form in our bodies. Our skin's health is tied to our internal organs, which means that the problems that these foods are causing will wreak havoc on your skin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the early 2000's research was emerging that showed that children were spending less time in the outdoors and more time with media; while at the same time childhood obesity rates were increasing at an alarming rate. Within just a matter of one generation, our society was radically changing to an increasingly indoor focused world for our children. Author Richard Louv published his watershed book, Last Child in The Woods; Saving Children from Nature Deficit Disorder in 2005 that gave a voice to the issue and launched a grassroots movement to address it.\nIn fall 2009, a bipartisan group of Texas legislators requested that the Texas Parks and Wildlife Department, along with the Texas Education Agency, the Texas Department of State Health Services and the Texas Department of Agriculture form a public-private partnership to develop a strategic plan. Over 80 professionals including representatives from state and federal agencies, NGOs, health, education, natural resources, community organizations, and businesses answered the call and formed the Texas Partnership for Children in Nature.\nCarter Smith, executive director of the Texas Parks and Wildlife Department, chaired the effort. In January 2010, a 17-member Texas Steering Committee representing four state agencies, NGOs, health, education, and business convened to establish a work plan and stakeholder teams. The focus areas were Education, Health, Access and Community, and later, Marketing and Policy\/Legislative.\nThe strategic plan was released in 2010 at the Children in Nature Summit held in Austin, TX. At that time the Action Committees decided to focus on forming regional collaboratives that would work locally with their communities to further the goals laid out in the strategic plan. Regional collaboratives were formed in Austin, Caprock, Coastal Bend, Houston, North Texas, Pineywoods, Rio Grande Valley, and San Antonio.\nWe are a grassroots network of over 375 partner organizations and individuals who are working together to connect children and families with nature in Texas. Each partner brings their unique perspective and solutions to the network. The website www.naturerockstexas.org was created by TCiN to connect families with partner's programs, events and nature spaces, and to support the regional collaboratives.\nYou can get involved with a regional collaborative or at the state level by contacting jennifer.bristol@tpwd.texas.gov. You can also take action by making a personal commitment to take the kids in your life outdoors more frequently to help them be healthier, happier, and smarter.\nWe all have fond childhood memories of a special place where play, imagination and wonder reigned. For most, that place was outdoors. Through these experiences, we gained understanding, appreciation and a personal relationship with the natural world.\nSadly, today's children are allowed less time for unstructured, creative play in nature than ever before. As the trend away from outdoor play and learning deepens, we are witnessing sobering consequences for children's health, well-being and quality of life. Additionally, this youngest generation is missing out on critical experiences that lay the foundation for future stewardship of our natural resources.\nWe can reverse this trend. One achievable solution is to restore active play and learning in nature. This issue has brought together an unprecedented coalition of partners from all walks of life who formed Texas Children in Nature. These experts from many disciplines thoroughly examined the issues and outlined their recommendations in this strategic plan. Their wisdom, insights and integrity are reflected throughout the plan. By design, the plan is concise, optimizes partnerships and focuses on implementation across multiple public and private sectors.\nPlease consider the recommendations in this plan and join us in restoring our children's well-being and their relationship with Texas's rich natural and cultural heritage. The trend is real, the problem is solvable and the time to act is now.\nThank you for caring for our wild things and wild places. They need you now more than ever.\nTCiN is guided by a Steering Committee comprised of professionals in the conservation, education, health and built-environment fields from around Texas. These dedicated individuals represent state agencies, non-profits, educators and the business community who share in the vision that all children and their families, from all walks of life, will spend more time outdoors, engaged with nature for a healthier, happier, smarter Texas.\nHayden Brooks, American Realty Corp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Netflow, a Cintra and Plenary joint venture, today welcomed its appointment as preferred respondent for the Western Roads Upgrade project, an availability-style public-private partnership that is the first of its kind for Australia.\nThe $1.8 billion project is the largest single investment in arterial roads in Australia to date and includes eight high-priority road upgrades, almost 30 kilometres of duplicated lanes and more than 700-lane-kilometres of road rehabilitation and maintenance for 20 years across Melbourne's west.\nNetflow Managing Director Phil Dreaver welcomed the Victorian Government's announcement, marking Netflow's first major project since its inception less than 12 months ago.\n\"It demonstrates Netflow can offer governments an innovative and best-practice approach that harnesses our extensive global roads and asset management experience,\" Mr Dreaver said.\nthe Amey and Broadspectrum joint venture as services contractor.\nNetflow now enters a period of exclusive negotiations with the Victorian Government to verify project details and achieve financial close.\nFormed in late-2016, Netflow is a joint venture between Cintra and Plenary with a focus on road projects in Australia and New Zealand. It brings together two of the largest developers and managers of infrastructure globally to provide competitive offerings for road projects including static and dynamic tolling, user recognition and managed lanes. Cintra and Plenary are also part of the Nexus Infrastructure consortium delivering the A$1.6 billion Toowoomba Second Range Crossing project.\nCintra is one of the largest developers and investors in transport infrastructure in the world, managing more than A$30 billion of assets including 26 road concessions across Australia, the US, Canada, Europe and South America. Cintra has delivered technical and financial solutions for some of the world's largest infrastructure projects, applying cutting-edge technology and design innovation to minimise financing, construction and operational costs. Assets under management include the US$3.6 billion 407 ETR toll road in Toronto, and the US$2.6 billion LBJ Express and US$2.1 billion North Tarrant Express managed lanes projects in Texas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A pair of 35 liter saddle bags made from High resistance 1200 W\/R Polyester with rigid thermoformed PE backing.\nQuick release mechanism allows you to easily detach the bags and carry over the shoulder with the included shoulder strap.\nIncludes removable inner bag, bottle carrier\/webbing pocket, and rain covers.\nIncludes adapter plates for bolt on and quick release operation with Givi series PL tubular pannier racks.\n***Please note: These cases are supplied WITHOUT any fitting hardware, nor racks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While recently hiking the Clevenger Canyon South Trail, I came across this rather large rattlesnake coiled in the middle of the trail. I was looking upward toward a large boulder in the distance and did not see the snake until that chilling telltale rattling sound reached my ears. From that point forward I kept my eyes on the trail ahead! This was but one of three snakes we saw on this hike, but was by far the largest and most dangerous.\nWell, it doesn't look like the camera was shaking too badly, so you must have survived. Neat picture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalrhb b/data_all_eng_slimpj/shuffled/split2/finalzzzalrhb
new file mode 100644
index 0000000000000000000000000000000000000000..ffa20ff70fdf933c6df05f305280c44676c73c7b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Students have to match the verbs to the pictures and include the verbs in a sentence at the present continuous form.\nThis downloadable handout is a great resource for high school students and elementary school students at Elementary (A1) and Pre-intermediate (A2) level. It is a useful material for consolidating Present simple vs continuous (progressive) tense in English, and can be used for enhancing your learners' Writing skills. It includes the answer key, too.\nfrancereference is from\/lives in Italy and has been a member of iSLCollective since 2011-03-09 17:02:09. francereference last logged in on 2012-09-29 14:05:22, and has shared 192 resources on iSLCollective so far.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jenny's Pure Food desserts allow us to be deliciously healthy without compromising our health. Try out Jenny's Pure Food \"Dark Chocolate Mousse\"; \"Lemon Cream Custard\"; and \"Salted Karmel Cheezecake\". Gluten-free, Dairy-free, Low Glycemic, GMO-Free and Vegan. No white flour, no white sugar, no heavy cream or artificial sweeteners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, at least we savored every last bit of it in the company of friends and acquaintances.\nHere's to next summer being an endless one.\nIf you're wondering how I designed this invite, check out the making-of video!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're looking for something that's a little bit higher up on the epicurean scale, this one's for you. Maple-glazed, seared and thick-sliced duck breast is the bomb on its own. Layer it inside a warm, sturdy flour tortilla with queso, lettuce and the delicious chipotle salsa and the stars really align. One pro tip: Order the pureed black beans and scoop them into your taco! Head to Poco Loco Cantina in Brown Deer for this masterpiece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baynons Brake is a series of horse riding trails that have been established by Environment Canterbury Regional Council on the northern Waimakariri river banks in the Clarkville area.\nThey are part of the Waimakariri River Park and have tidied up and given purpose to an area that had been misused in the past by hoons in 4-wheel drives and the like.\nSpace has been put side for parking horse floats and fencing has been put in.\nThe area includes the stopbank in places and signposted trails that give riders a variety of choices in the area between the stopbank and the river.\nThe facility was recently opened (in the rain!) despite the fact that the recent storm had done a lot of damage.\nWell done to David Owen and the crew at ECan!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalyjb b/data_all_eng_slimpj/shuffled/split2/finalzzzalyjb
new file mode 100644
index 0000000000000000000000000000000000000000..473b06e126476006589b7050df1df99964a768c3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The New York State Laborers-Employers Cooperation and Education Trust (LECET) has provided funding to the SUNY Oswego Presidential Scholars for the past 17 years. Bill Shannon, business manager for the Upstate New York Laborers' District and LECET representative, presented a $25,000 check in October to President Deborah F. Stanley to support the program. The Presidential Scholars program helps Oswego attract top talent from high schools across the nation.\n\u00a9 Copyright Oswego Alumni Magazine 2019. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MOUNT VERNON, VA \u2013 aois21 publishing, LLC announced today that Zach C. Cohen has joined the company in charge of all audio projects, including the aois21 podcast network. He comes with a wealth of experience in podcast production, as well as experience with writing and editing.\nZach joins aois21 as Chief of Audio Production and will oversee all eight aois21 podcast series, heading up the team of podcast production and audio consultants. He will edit and produce audio files and help grow our presence and recording ability.\n\"As a podcast addict myself, I'm fascinated by the power of audio to tell stories in ways that print cannot,\" Zach said. \"I'm excited to join the growing team at aois21 as we explore new ways to show our authors' work to the world.\"\nZach has previously produced podcasts for National Journal (where he currently works as a staff correspondent), PBS MediaShift, and LinkTV. He is a proud American University graduate and New Jersey native.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A suffragan diocese of Cashel ; it comprises the greater part of County Clare, a large portion of Tipperary, and parts of King's and Queen's Counties, Limerick, and Galway. Its Irish name is Cill-da-Lua, so named from St. Lua, an abbot who lived about the end of the sixth century, and whose oratory can still be seen in Friar's Island, near the town of Killaloe. Though St. Lua gave his name to the diocese, St. Flannan is its patron saint. He was of royal lineage, his father being the saintly Theodoric, King of Thomond, who towards the close of his life received the monastic habit from St. Colman at Lismore. St. Flannan was the first Bishop of Killaloe, and is said to have been consecrated at Rome by John IV about 640. In the time of St. Flannan, the Diocese of Killaloe was not so extensive as it is at present. It did not then include the old dioceses of Roscrea and Inniscathy. It was only when these were suppressed at the Synod of Rathbresail in the first quarter of the twelfth century, that Killaloe assumed its present shape, which is almost coterminous with the boundaries of the ancient Kingdom of Thomond. The parish of Seir Kieran in King's County, though in Thomond, was allowed to remain subject to the Diocese of Ossory, out of respect to the memory of St. Kieran.\nThe old See of Roscrea grew around a monastery founded there by St. Cronan about the middle of the sixth century. This monastery became a famous school, and it was within its walls that the scribe Dimma wrote for St. Cronan the copy of the Four Gospels now in the Library of Trinity College, Dublin, for which Tatheus O'Carroll, chieftain of Ely, made a costly shrine in the twelfth century. The Diocese of Roscrea was coextensive with the territory of the O'Carrolls, added to that of the O'Kennedys. Ware holds that St. Cronan was Bishop of Roscrea, but Lanigan thinks that Ware has been misled by the fact that Roscrea became an episcopal see. Like the Diocese of Roscrea, the Diocese of Inniscathy grew around the monastery of Inniscathy, founded by St. Senan in the early portion of the sixth century. There is no question about St. Senan being the first bishop of the Diocese of Inniscathy, which comprised the Baronies of Moyarta, Clonderlaw, and Ibricken, in Clare; the Barony of Connello, in Limerick ; and in Kerry, the ancient region of the Hy-Fidgente. The last Bishop of Inniscathy was Hugh O'Beachain, who died in 1188. Nevertheless, there were titular bishops of the see up to the close of the fourteenth century. The remains of the cathedral church of Inniscathy and a round tower now mark the ancient see of St. Senan. The Clog-oir, too, still in existence in County Clare, is a highly-prized relic of Inniscathy. St. Brecan's churches of Carntemple, Doora, and Clooney, St. Tola's church at Dysert O'Dea, St. Senan's hermitage at Bishop's Island, near Kilkee, St. Caimin's church and school at Iniscaltra, St. Brendan's and St. Cronan's abbeys at Birr and Roscrea may be named amongst hundreds of churches, schools, and hermitages, which covered Killaloe like a network and which in their decay attest to the devotion to the Catholic Faith of the far-famed Dalgais.\nSome of these foundations deserve mention. Iniscaltra, a green little island in Lough Derg, was celebrated nursery of sanctity and learning in Thomond. It is associated principally with St. Caimin, who made Iniscaltra the seat of a very famous school, which attracted pupils even from foreign countries. A fragment of the commentary on the Psalms collated with the Hebrew text, written by St. Caimin (640), is preserved in the Franciscan convent, Merchant's Quay, Dublin. Birr also was a celebrated seat of learning in Thomond, founded by St. Brendan (550). The Gospels of McRegol, now in the Bodleian Library, were written by McRegol, Abbot of Birr, in 820. Terryglass also was a school of great repute founded by St. Columba (552). It was here that St. Patrick is said to have baptized the Dalgais from Northern Thomond, who crossed Lough Derg in their coracles to meet him. The monastery of Lorrha, founded by St. Ruadhan (550), can claim that it was within its walls that the famous Stowe Missal, now in the library of Lord Ashburnham, was written; but the desertion of Tara owing to the alleged cursing of St. Ruadhan , is without historical foundation. The abbeys at Ennis and Quin are striking illustrations of the piety and munificence of the foremost chieftains of the Dalgais.\nAbout 1240 Donogh Cairbreach O'Brien built the monastery for Conventual Franciscan friars. It was considered one of the finest houses of the order in Ireland, and ultimately it became the occasion of Ennis being made the capital of County Clare. Even in ruin it is beautiful; the east window especially is much admired for its size, grace, and symmetry. Here are buried some of the Kings of Thomond and their chieftains. The Abbey of Quin is one of the noblest remains of monastic antiquity in Ireland, and is in so perfect a state of preservation that little more than a roof is required to make it fit to house the monks and have their chant daily re-echo within its walls. It was founded by Sheda McNamara in 1402. In 1641 a college was opened at the abbey, which soon had eight hundred students. But the most interesting historical remains are to be found at the picturesque little town of Killaloe, the ancient seat of the bishop, which is built on a ridge commanding a fine view of Lough Derg. For here we have the oratory of St. Lua in Friar's Island, the very perfect stone-roofed oratory of St. Flannan, and St. Flannan's cathedral, built in 1160 by Donald O'Brien, King of Limerick, near the site of Brian Boroimhe's royal palace of Kincora. St. Flannan's cathedral was, till the early years of Elizabeth's reign, the Catholic cathedral of the Diocese of Killaloe. Since then it has been in Protestant hands. Owing to the cruel persecution of the Catholic religion and its bishops and priests, and the suppression of the monasteries in Clare at the opening of the reign of Queen Elizabeth, the churches and monasteries fell into decay and ruin, the Holy Sacrifice of the Mass being offered up on some rock on a mountain-side, or some lowly \"Mass house.\" It is only since the time of Catholic Emancipation, a glorious era in the annals of Killaloe when the priests of Clare gave powerful aid to O'Connell to win the Clare election, that a \"second spring\" has come, and that Thomond has been again covered with handsome and commodious churches.\nThe first successor of St. Flannan in the Diocese of Killaloe whose name has come down to us, is Cormacan O'Mulcaishel, who died in 1019; and from the death of St. Flannan to the time of the learned O'Lonergain in 1150, the names of only five prelates have been recorded. But from this period the succession becomes regular and complete. In 1179, Constantine O'Brien, fifth in descent from Brian Boroimhe, was Bishop of Killaloe; he attended the Council of Lateran. Conor O'Heney, another Bishop of Killaloe, also attended the Council of Lateran in 1215. Cornelius Ryan, a Franciscan friar, and brother of a chieftain, was consecrated Bishop of Killaloe in 1576. He had a remarkable career. From the time of his appointment he used his marked ability and great organizing power in aid of the Earl of Desmond, who championed the Catholic cause, and succeeded in obtaining for him the support of Gregory XIII and Philip of Spain. For years he shared in all the perils of the insurrection, and he was regarded by Elizabeth and her minions as a most formidable opponent. When the Desmond insurrection ended in disaster, he escaped to the Continent and died at Lisbon in 1617.\nJohn O'Moloney was another eminent Bishop of Killaloe. He was born in Kiltanon, County Clare, in 1617, was a doctor of the Sorbonne and, before his appointment to Killaloe, had been canon of Rouen, in France. He was named bishop by Propaganda in 1671, at the urgent request of the Catholics of the diocese, his qualifications for the exalted office being set forth in various testimonials from the doctors of the University of Paris, and several French bishops and archbishops. In 1673 he was deputed by the Irish bishops to visit France and endeavour to induce the French king and his minister to found an Irish ecclesiastical college in Paris. He succeeded in his mission, and a few years later the Irish college, of which he is regarded as the founder, was opened. In 1689 he was named Bishop of Limerick, retaining Killaloe in administration, but he was soon forced to flee to France, where he died in 1702 at the Sulpician house at Issy, near Paris. The present bishop is the Most Reverend Dr. Fogarty, born in 1859 near Nenagh, County Tipperary. Before his elevation to the episcopate he was vice-president of Maynooth College, where he had been for fifteen years previously a distinguished professor of dogmatic and moral theology. His consecration took place in 1904, at the procathedral at Ennis, the seat of the bishop and also of a well-equipped diocesan college.\nThe diocesan chapter, including dean, archdeacon, and canons, was re-established by papal decree on 11 February, 1903. Catholic population, 137,574, according to census of 1901; non-Catholic population, 8329; parishes, 57; secular clergy, 142; parochial and district churches, 143; houses of regular clergy, 2, viz. Franciscans at Ennis, Cistercians at Roscrea; convents of Sisters of Mercy, 12; Convent of Sacred Heart, 1; number in community, 198; monastic houses, 6; number in community, 63.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Richard Soley is chairman and CEO of the Object Management Group, Inc. Mr. Soley is one of the great thought leaders of our time. He is constantly pushing the envelope on standards and technology. A frequent venture angel, Mr. Soley recently started the Boston CTO Club.\nRichard Soley says, \"If I had to go back and do the emergency crisis response standard again I would do things differently.\nRichard Mark Soley suggests you cannot resolve all ambiguity.\nRichard Soley suggests having a clear business objectives is one of those phrases that is over used.\nRichard Mark Soley suggests that other people's motivations are not the same as your own motivations.\nRichard Mark Soley suggests you have got motivation backwards.\nRichard Soley suggests executive sponsors to find, nurture and sponsor new projects.\nRichard Soley suggests the key to success is to have the domain knowledge.\nRichard Soley suggests the key reason for making fast decisions has nothing to do with always making right decisions.\nRichard Soley says, \"I will give you example of a standard that did not work.\nRichard Soley says, \"I don't think you need to be always instep with your peers.\nRichard Mark Soley suggest the perfect executive sponsor would be someone who understands the product arena.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Financial Services Institute of Australasia (FINSIA) has upgraded Dr Dale Cooper's membership status to Fellow.\nBroadleaf carried out a risk assessment of a concessional loan offered by the Commonwealth Government. The risk assessment examined the administration and financial performance of the loan. It confirmed the existing controls were well designed and reasonable in the circumstances as well as helping to establish priorities for assurance of the controls during the term of the loan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamcqz b/data_all_eng_slimpj/shuffled/split2/finalzzzamcqz
new file mode 100644
index 0000000000000000000000000000000000000000..38a3a86632341b14d3d709dbca13ede8d91397a2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamcqz
@@ -0,0 +1,5 @@
+{"text":"Our kennel Black Russian Terriers \u00abSokrovische Rusi\u00bb established and registered in February 2007 in the FCI (Federation Cynologique Internationale) in Belgium under number 271\/07 and has an international accreditation.\nIn his professional activity our kennel adheres to the principle of linear breeding, based on the consolidation and improvement of the best qualities of the founders of our breed dogs: strong nervous system, excellent working qualities, physical health.\nParticular attention is paid to the exterior, health and working quality of the resulting offspring.\nWe also participate in exhibitions - our puppies, juniors and adult dogs frequently display at the exhibitions of regional and international class.\nBlack Terriers our kennel live in many cities of Ukraine and abroad \u2013 the England, Germany, Netherlands, Scotland, Poland, Croatia, Israel, Egypt, South Africa, USA, Finland, Russia.\nOur dogs are worthy of their famous ancestors, confident in winning national rings and rings in the international class!\nThe team of our kennel seeks to contribute to the development and popularize this unique domestic breed.\nand maintaining working qualities of our dogs.\nIn not hot weather we with our dogs moving for the whole day \"to the fields\", where our pets can practice in the open air (video archive).\nOur dogs - descendants of the famous Black Russian Terriers in Russia and Ukraine.\nOn this page we present you the famous ancestors of our dogs, who entered the history of breeds and who will be proud of for many generations of breeders and owners of black terriers.\n\"Malahovskiy\" (Russia), \"Zolotoy Grad\" (Russia), \"Deneb-Keitos\" (Russia) and other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buyers can stay rest assured for damage-free products from our end as we make use of quality packing material. We also fulfill different packing requirements of the clients such as customized or standard. Our professional warehousing personnel keep a stringent check on the entire lot, stored in the warehouse, to keep it secured from diverse factors such as moisture, rain, dust and similar others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"you are prohibited by law from receiving or using the Services.\nAvio USA may change the Terms from time to time at its sole discretion, including by imposing a charge for access to or use of a Service. If such changes are made, Avio USA will make a new copy of the General Terms available at the Site, with any new Additional Terms made available to you from within or through the affected Service. Avio USA may require you to provide consent to the updated Terms in a specified manner before further use of the Services is permitted. Otherwise, your continued use of the Site or any affected Service constitutes your acceptance of the changes. Your use of the Site, Materials and Services is subject to the most current version of the Terms posted on the Site, or within or through the affected Service, at the time of such use. 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Without limiting the foregoing, Avio USA shall have the right, in its sole discretion, to remove any of Your Content for any reason (or no reason), including if it violates the Terms or any Law.\nprotect the rights, property or personal safety of Avio USA, its Users or the public, and law enforcement or other government officials, as Avio USA in its sole discretion believes to be necessary or appropriate.\nAvio USA does not claim ownership of Your Content. However, except as provided in Section 8(b), you grant Avio USA a worldwide, royalty-free, nonexclusive, perpetual, irrevocable, and fully sublicensable license to use, distribute, reproduce, modify, adapt, publish, translate, publicly perform and publicly display Your Content (in whole or in part) and to incorporate Your Content into other Materials or works in any format or medium now known or later developed. And now in more common terms:We realize that this section might look a bit scary and so we want to point out a few things. 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Avio USA may provide you with notices, including those regarding changes to the Terms, by email, regular mail, text message, postings on the Site or within the Services, or other reasonable means now known or hereafter developed.\nThe Terms constitute the entire agreement between Avio USA and you with respect to your access to or use of the Site, Services and Materials and supersede any prior agreements between you and Avio USA on such subject matter.\nYou may not assign or otherwise transfer the Terms, or any right granted hereunder, without Avio USA's written consent.\nAny failure by Avio USA to enforce or exercise any provision of the Terms, or any related right, shall not constitute a waiver of that provision or right.\nThere are currently (179) users on this website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bring out your wild side when you rock these animal print flared pants! The pants are made out of a leopard patterned fabric that looks and feels fabulous. These pants are slim fit from the hip through the knee that flares to a bell bottom leg. Pull-on elastic waistband.\nMade out of 95% polyester and 5% spandex.\nAdd to your natural fierceness with these rad pants!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you've ever sat at home on a Friday night, staring into a well-lit microscopic mirror to fully examine your pores and dark spots while simultaneously Googling your various symptoms, rest assured you're not only not alone, but you're in star-studded company.\nOlivia Munn\u2014who never shies away from sharing her well-researched take on health and wellness, including why she's all for egg freezing\u2014talked to Coveteur about her tried-and-true beauty secrets and hacks, and says her affinity for a good Google search is what helped her figure out the secret to combatting her adult acne.\nTwo years ago, Munn decided to do some digging\u2014online, that is\u2014to finally find a way to clear her chin acne.\nSince federal officials have advised states and cities to lower the amount of fluoride in the public water supply (because the mineral has been designated a neurotoxin), tap water may be a non-issue in the future. And in terms of fluoride in toothpaste? The American Dental Association is firmly in the \"pro\" camp\u2014but many dentists say that the non-fluoride route is the healthier one. If it also clears up your monthly chin eruption, as Munn contends? The anti-fluoride crowd might've just gotten a new talking point.\nAs for other celeb acne secrets: Here's the diet that helps Kate Hudson manages hers, the tweak that clears Margot Robbie's skin every time, and Chloe Grace Moretz's cooking oil tip (no, really). Or try these expert tips for beating adult acne for good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanuzd b/data_all_eng_slimpj/shuffled/split2/finalzzzanuzd
new file mode 100644
index 0000000000000000000000000000000000000000..f1968a6b88c7faa975599eaa331954f858e7fe99
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+{"text":"Gondi (G\u014dndi) is a South-Central Dravidian language, spoken by about two million Gond people, chiefly in the states of Madhya Pradesh, Gujarat, Telangana, Maharashtra, Chhattisgarh and in various adjoining areas of neighbouring states. Although it is the language of the Gond people, only about half of them still speak it. Gondi has a rich folk literature, examples of which are marriage songs and narrations.\nGondi has a two-gender system, substantives being either masculine or nonmasculine. Gondi departed from the parent Proto-Dravidian language by developing initial voiced stops (g, j, \u1e0d, d, b) and aspirated stops (dhh, gh, jh, dh, bh).\nGondi writing can be split into two categories: that using non-native scripts and that using native scripts.\nTraditionally, for lack of a widespread native script, Gondi has been written in Devanagari and Telugu script, which encompass the non-native scripts that have been used to write Gondi.\nEfforts have been undertaken to create a native script for Gondi. In 1928, Munshi Mangal Singh Masaram designed a native script based on Brahmi characters and in the same format of an Indian alphasyllabary. However, this script did not become widely used, and most Gonds remain illiterate.\nAccording to Maharashtra Oriental Manuscripts Library and Research Centre of India, a dozen manuscripts were found in this script. Programs to create awareness and promotion of this script among the Gondi people are in development stage.\n\u2191 Hammarstr\u00f6m, Harald; Forkel, Robert; Haspelmath, Martin; Bank, Sebastian, eds. (2016). \"Southern Gondi\". Glottolog 2.7. Jena: Max Planck Institute for the Science of Human History.\n\u2191 Hammarstr\u00f6m, Harald; Forkel, Robert; Haspelmath, Martin; Bank, Sebastian, eds. (2016). \"Northern Gondi\". Glottolog 2.7. Jena: Max Planck Institute for the Science of Human History.\nBeine, David K. 1994. A Sociolinguistic Survey of the Gondi-speaking Communities of Central India. M.A. thesis. San Diego State University. 516 p.\nChenevix Trench, Charles. Grammar of Gondi: As Spoken in the Betul District, Central Provinces, India; with Vocabulary, Folk-Tales, Stories and Songs of the Gonds \/ Volume 1 - Grammar. Madras: Government Press, 1919.\nHivale, Shamrao, and Verrier Elwin. Songs of the Forest; The Folk Poetry of the Gonds. London: G. Allen & Unwin, ltd, 1935.\nMoss, Clement F. An Introduction to the Grammar of the Gondi Language. [Jubbalpore?]: Literature Committee of the Evangelical National Missionary Society of Sweden, 1950.\nPagdi, Setumadhava Rao. A Grammar of the Gondi Language. [Hyderabad-Dn: s.n, 1954.\nSubrahmanyam, P. S. Descriptive Grammar of Gondi Annamalainagar: Annamalai University, 1968.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Special THC Oil, where there is more THC than CBD, with a Ratio of 1\/4 CBD\/THC. Oil of 10[ml] with 0.2% CBD (Cannabidiol) and 0.8% THC, made from sustainably produced hemp\/Cannabis sativa flower extract. In a Bottle with Drop counter.\nThe Cannaliz Technic product category offers Oils where a specific CBD\/THC ratio is the Key Aspect of the Product. The specific Ratios can be obtained either because a particular hemp\/cannabis sativa blossom has a special Ratio by Nature, or through a special Method, Chromatography.\nThis particular 1\/4 THC Oil is made using a Process called Chromatography, which allows for a stable Formulation. Chromatography is a Laboratory Technique that separates the Components of a Mixture; It is a Form of Purification. This allows us to define a specific Ratio that is stable while preserving all Components of the Extract. The Mixture is carefully prepared with organic olive oil, which has been selected for its Properties and Harmonizes perfectly with Cannabis extracts (CBD). Despite the Cleansing, it contains the entire Spectrum of action of the Hemp Extract, as it contains all Cannabinoids, Terpenes, Sequiterpenes and Flavonoids, which ensure a very wide range Of Effects.\n1 Drop (1\/20[ml]) = 0.1mg CBD; 0.4mg THC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most nursing homes in Clinton, IA offer skilled nursing, post acute care and inpatient rehab for the elderly. In order to find the right nursing home for you, our experienced local advisors in Clinton, IA will walk you through the process.\nCall your local advisor in Clinton, IA and get free personalized service.\nNeighborhoods in Clinton include: Davenport-Moline-Clinton, and Muscatine-Illinois Combined Statistical Area.\nThe official website for the city of Clinton is http:\/\/www.ci.clinton.ia.us\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Russian Muslim convert, recently relocated from snowy Siberia to sunny Saudi Arabia. Graduated in International Relations, currently working as a freelance translator. Keen on classical music; have been learning to play the violin for the last 3 years and enjoying it immensely. Good books and sweets also make me happy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(From Amazon): As soon as early humans began to scratch images on cave walls, they began to create maps. And while these first drawings were used to find hunting grounds or avoid danger, they later developed into far more complex navigational tools. Charting the World tells the fascinating history of maps and mapmaking, navigators and explorers, and the ways that technology has enhanced our ability to understand the world around us. Richly illustrated with full-color maps and diagrams, it gives children an in-depth appreciation of geographical concepts and principles and shows them how to unlock the wealth of information maps contain. It also features 21 hands-on activities for readers to put their new skills to the test. Children will: build a three-dimensional island model using a contour map, engrave a simple map on an aluminum \"printing plate,\" determine the elevation of hills in their neighborhood, draw a treasure map and have a friend search for the hidden stash, create a nautical chart of a small puddle, survey their backyard or local park, navigate a course using a compass, and much more. Now more than ever, the study of geography is crucial to understanding our ever-changing planet, from political change and warfare to environmental conservation and population growth.\nIs the Map to Scale?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanzpj b/data_all_eng_slimpj/shuffled/split2/finalzzzanzpj
new file mode 100644
index 0000000000000000000000000000000000000000..de3b2d0ef83c51043a6d61d5614650e3bf1668a6
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Monthly Natural Farming Hawaii meeting in Hilo at the Sweet Cane Cafe. Every second Tuesday 6-8pm.\nCertification structure for CGNF-Hawaii. Master Cho's official organization incorporated in Hawaii.\nBeyond GMO a project to transform the Hawaiian Papaya industry.\nDifferences between Master Cho and his son Young-san Cho's solutions.\nhow to make water soluble calcium phosphate.\nburn bones till thoroughly black and charred.\ndissolve in vinegar for 7 days.\nmore\u2026 see www.naturalfarminghawaii.net or subscribe to my youtube.\nThis is the Basic Seminar of Master Han-Kyu Cho, founder of Korean Natural Farming, presenting his fundamental seminar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever wondered how a computer is put together? It is much more than its definition computer is an electronic device for storing and processing data mostly from binary it is like a person who does all work for you at just a click but behind that there is series of activities take place by different humans and machine Yes it's made in a factory, but everything can't happen in one place. After all, there's a motherboard that has to be assembled in one place and normally you don't even hear about a brand name for a motherboard. Then there's the power supply system that comes from another source while PC cases are sometimes made to custom specifications especially in the case of gaming computers. Today's PC cases are made out different materials although early cases were made only out of steel. You can even buy clear PC cases these days where you can see right into the interior of the machine.\nEveryone using a computer has to have a computer monitor attached to it. The resolution of the monitor is considered the most important in it because this is how the number of pixels on a screen is measured be it a computer monitor, TV or mobile phone. As most people use their devices to watch movies, serials or videos from sites like rainierland . A higher resolution means the images will appear small while a low resolution will provide large text and images. Just because you opt for a large screen doesn't mean that you will be able to watch large images and texts.\nComputer screens also have been changing from their early stage when production rate of computers are at very low pace with a CRT screen which was the only type of screen that was available in the early days. But today, everyone is going for the touch screen monitor on which you can use your finger to find the icons or scroll the screen anyway way you like instead of having to use a mouse. Being still new, the touch screen monitor is rather costly although the price is sure to go down soon with everyone wanting one of these for their computers.You can also find out difference between computer screens on the video provided below.\nOne component that most computer users don't usually bother about much is the sound card. All modern computers have inbuilt sound cards and only those who are into music in a major way will need an extra powerful sound card for recording purposes. Among the different types of audio cards available, the more popular is the USB sound card which can be bought according to your needs and your budget. Before making the purchase you need to understand the number of tracks you will record at one time. A high-end USB sound card will offer more facilities but don't just spend on a high priced card if you really don't have expectations of making use of all the facilities. How to find a a good sound card watch the video below.\nThen there's the graphics card which many consider very important today and these are mostly people who are into playing video games seriously and continuously. Game playing needs a high-speed graphics card and the one you usually get in the computer is not sufficient for a serious gamer. A detailed video is provided down to know more about Graphic card.\nThe DVI high definition output system is the one that's being used in most modern graphics cards today and those that are able to connect to the monitor through a DVI system will find they are able to produce images that are clearer and sharper than when it's done through a VGA port.\nMost modern computers today are built with at least 1GB RAM since this is considered quite sufficient for someone who uses his computer for normal tasks such as surfing the web, sending emails and watching movies,videos online with the help of websites like rainierland. But for those who are into more complicated tasks such as creating graphics and designing, they will require additional computer RAM. But this should always be in keeping with the RAM you already have in your machine as otherwise, it will impact on the smoothing working of the machine.\nAll systems work or perform their task only after if they have a operating system.You can use your computer only after installing an operating system. An operating system helps you in connecting with your hardware. Without an operating system, you can not use RAM, USB Sound card, Motherboard, touch screen monitor and hard-drive. Operating system works as a virtual machine without this actual machine can perform their task.\nAfter knowing about different important parts of computer including the virtual one yes you got it right the operating system a obvious question in your mind may be which one operating system to use? There are a wide range of Operating Systems available and some of them are popular you can try any of the popular operating system rest you can get all on recommendation from different technology review sites and with your own experience help you to understand working ability of different Operating System as they are designed for specific audience like gamer, normal user etc. All of them have different requirements that you need to fulfill. You can easily choose any from Some popular names that you can use in your computer are some of them are microsoft windows for desktop and laptop, Android for mobiles, tablets etc.\n\u00abNext Post Most Secure Smartphone: Apple or Android?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love landscapes, night-scapes and sunsets.\nI'm a night-owl and a light sleeper.\nI co-parent a bratty cat with Sarah.\nCowboy Bebop is one of my favorite animes.\nThe Cure & Tycho are one of my favorite bands.\nIt takes me 4-5 business days to reply on social media\u2026 Sarah tags me on something and I won't respond til 4 days later.\nI'm not social media savvy and I'm ok with that.\nI've been craving for Mosas' Burgers for 3 and 1\/2 years now\u2026 I'm waiting until Sarah finally gets cravings for it.\nWe love to celebrate love. It's so easy because we're both in love and our relationship inspires and fuels us. We love connecting with our couples and families during sessions. We don't just bark commands, we ask questions and we always make sure that the photographs we take reflect our couples' + families' personalities and their stories.\nWe want people to look at our images and feel the warmth and the carefree emotions our clients evoke in the images we share.\nI love connecting + taking photos of people.\nI'm an introvert so I lose energy after weddings.\nOur cat is passive-aggressive with me. She prefers Dave over me\u2026 She's a brat.\nDisney's Tangled is my absolute favorite film.\nWhenever I could, I will choose to sleep\/nap.\nMy Chemical Romance is my heart and soul.\nDave and I have seen Paramore, Bruno Mars, Panic! at the Disco, Air Supply, Aj Rafael, Magic! Kawehi live and we're continuously adding more.\nWe're at this Korean restaurant named CHOWON almost every 1-2 times a month.\nPlease include your mobile number and 2-3 ideal dates for your session \/\/ your wedding date.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click here to continue as a guest.\nFind and download Safety Data Sheets here!\nLog in above to find Safety Data Sheets specific to your purchase history.\nIf you are a current Burkhart Customer and would like to use online ordering, please call 800.562.8176 or contact your local Burkhart Account Manager to set up your account.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This collection contains preludes for all of the hymns in the Lent, Holy Week, Eastertide sections of the Worship Supplement (1969). In some cases two or three preludes were composed for this collection. However, suitable compositions from the available literature were also included.\nFive hymn preludes on often-used hymn tunes. The Hymn Tune Innovations series features the inventive and thoughtful organ compositions of Benjamin M. Culli.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoeym b/data_all_eng_slimpj/shuffled/split2/finalzzzaoeym
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index 0000000000000000000000000000000000000000..846bc93f236f492709dbc3318a2d06b5b7207130
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@@ -0,0 +1,5 @@
+{"text":"Sri Hanuman is the greatest devotee of Bhagavan Sri Ramachandra. So His character must reflect the qualities of Bhagavan Sri Rama.\nsakrid eva prapannaaya tava asmi iti cha yaachate\/ abhyam sarvabhutebhyo dadaami etad vratam mama, meaning whoever surrenders to my just by saying 'I am yours' ONLY ONCE, is ever protected by Me.This is My mission.\nBhagavan is 'bhava-graahi' and so He listens to our heart's prayer.To err is humane and so if we apologige sincerely, He is bound to forgive all our faults. We should never think Hanumanji as unforgiving because He is eternally dedeicated to someone Who is Forgiveness personified.\nNot the answer you're looking for? Browse other questions tagged hanuman prayaschitta vrata .\nWhat is the importance of fast (Vrat)?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"15 MINUTES THE DVOR MASSACRE - D.O.P \"Skree\"\nOnly a few meters from a Danish UN Camp in Dvor, Croatia, nine disabled people, both Serbs and Croats, were executed in cold blood by a group of unidentified men on August the 8th 1995. The Danish soldiers in Camp Dannevirke were assigned by the United Nations to monitor the ceasefire between the Serbs and the Croats.\nThey were only allowed to use weapons in self-defense. In the summer of 1995, the Croatian Army ceasefire broke and about 250,000 Serbs were forced to flee during \"Operation Storm\". Once there was no longer a ceasefire to monitor, the soldiers were ordered to stay inside their camp and not interfere in the war.\nIt was therefore up to one Danish officer, to make the crucial decision: to give orders to shoot or follow the UN mandate and not intervene. The film follows the former company commander Kold on his journey back to Croatia, to the place where he 20 years earlier had to make the most difficult decision in his life.\nHere he confronts his past and his decision, meets the commanders from the warring parties, and the relatives of the victims. He is forced to face difficult questions: Could he have stopped the massacre? Did he have a choice? Or was he, the Danish soldiers and the civilians actually let down by the United Nations?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One thing I am striving for with my move to 3rd grade is to foster and encourage collaboration in my classroom. I see time and time again students who don't know how to work together or how to be respectful towards their peers!\nOur students HAVE to be able to agree, disagree, collaborate, cooperate, and work as a team. They can't do that if we don't show them how!\nBeing in a new grade level, however, I know I have to take baby steps or else it won't work for me. I created these job collaboration cards to make sure my kids are held accountable for their group work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coloring Picture Of Animals Ba Farm Animal Coloring Pages Coloring And Drawing Pinterest pics to color Coloring Picture Of Animals. Free Color In Animals Download Free Clip Art Free Clip Art On pics to color Coloring Picture Of Animals. Elephant Color Page Animal Coloring Pages Color Plate Coloring cool coloring sheets to print out Coloring Picture Of Animals. Cute Animals Coloring Pages Free Printable Coloring Pages Free cool coloring sheets to print out Coloring Picture Of Animals. Cat Color Page Animal Coloring Pages Color Plate Coloring Sheet cool coloring sheets to print out Coloring Picture Of Animals. Free Color In Animals Download Free Clip Art Free Clip Art On coloring book pages Coloring Picture Of Animals.\nColoring Picture Of Animals Free Color In Animals Download Free Clip Art Free Clip Art On pics to color Coloring Picture Of Animals. Elephant Color Page Animal Coloring Pages Color Plate Coloring cool coloring sheets to print out Coloring Picture Of Animals. Cute Animals Coloring Pages Free Printable Coloring Pages Free cool coloring sheets to print out Coloring Picture Of Animals. Cat Color Page Animal Coloring Pages Color Plate Coloring Sheet cool coloring sheets to print out Coloring Picture Of Animals. Free Color In Animals Download Free Clip Art Free Clip Art On coloring book pages Coloring Picture Of Animals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Tax Collector's office is responsible for the billing and collection of all Real Estate, Personal Property, Motor Vehicle Taxes and related penalty charges.\nThe Tax Collector's goal is to assist in maintaining the fiscal stability of the Town of Preston by ensuring the timely collection and recording of all revenues through the diligent application of Connecticut State Statutes, regulations and other enforcement aids, while assisting the public in understanding the taxation process and procedures and maintaining an effective and cordial relationship with the general public. The Town of Preston cannot override or make any decisions contrary to the Connecticut State Statutes.\nThe current mill rate is 26.03 (effective July 1, 2018) mills or $26.03 per $1,000 of assessed valuation as set by the Board of Finance.\nReal Estate and Personal Property Taxes are due on July 1 and January 1 of each year. Bills under $100 are due in full on July 1.\nWe do not bill again for January. It is your responsibility to remember to pay.\nMotor Vehicle tax bills are for vehicles owned on October 1, (on any given tax year). Vehicles are tracked by the plates. If you return plates, be sure to get a receipt to the Tax Assessor. All address changes are through the Department of Motor Vehicles. If you question your tax bills, please call the Tax Office 860-887-5581, ext. 2.\nAll delinquent motor vehicle taxes are reported to the State of Connecticut Department of Motor Vehicles. This will prevent you from renewing your registration. Anyone needing a motor vehicle clearance must pay their tax bill in cash or certified funds. All taxes on all motor vehicles under your name must be paid in full. The Tax Collector may also require a forthcoming bill to be paid in full.\nProperty Taxes are due July 1, and January 1 with a one-month grace period. According to State Statute Section 12-169, when the final day for payment falls on a Saturday, Sunday or legal Holiday, payment may be made without interest or penalty on the next business day. You may pay in person or by mail.\nInterest will be charged on all payments received or postmarked after the 30 day grace period at the rate of 1.5 % per month, 18% per year. The interest is charged back to the due date of July 1 and January 1. The first month it is delinquent the interest is 3% (July and August) and (January and February). It is the taxpayer's responsibility to call the Tax Office if you did not receive a bill. Non-receipt of a tax bill does not invalidate the tax or interest.\nPaying by check or debit card- clearance on motor vehicles take 5-7 days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaokre b/data_all_eng_slimpj/shuffled/split2/finalzzzaokre
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index 0000000000000000000000000000000000000000..7f2868f28f156ff421379132f1da8638f9d54644
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+{"text":"Too often when discussing politically contentious issues like climate change, we adopt an \"us versus them\" or \"environment versus business\" approach. When that happens, we all lose. Worse yet, little \u2014 outside of empty rhetoric \u2014 gets accomplished.\nBarney Bishop III is president and CEO of Barney Bishop Consulting, LLC and former Executive Director of the Florida Democratic Party. He is the immediate past president and CEO of Associated Industries of Florida, known as \"The Voice of Florida Business.\" He may be reached at [email protected].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SMCES strives to provide a holistic Catholic education to nurture life-long learners who are members of a dynamic faith-filled community whose purpose is to continue Christ's work in the world.\nSMCES is a vibrant parish school that educates diverse learners from TK \u2013 8th grade. Founded in 1930 by the Sisters of the Holy Name of Jesus and Mary we believe in continuing the service and love of Jesus Christ, guided by the Holy Spirit and Gospel values. Together with our families, we are a welcoming community that provides a safe and loving learning environment. SMCES strives to inspire academic commitment and excellence by providing students with teachers whose expertise serves their individual needs. Through building relationships, we nurture compassionate, responsible, well-rounded citizens who are prepared to meet their full potential.\nSt. Monica Catholic Elementary School Students FLY !\nFaithful \u2013 We pray to God and follow the example of our friend Jesus.\nLearners \u2013 We are curious, active, growing scholars.\nYoung Citizens of the World \u2013 We are kind citizens who make good choices and care for others.\nFaithful \u2013 We live Gospel values and demonstrate fortitude. We actively participate in the Catholic faith. We engage in Christian service. We respect the beliefs and traditions of others.\nLearners - We practice academic honesty and responsibility. We listen, write, and speak effectively. We apply and develop knowledge and learning strategies across the curriculum. We use technology and media appropriately, effectively, and responsibly.\nYoung Citizens of the World - We respect God and self. We respect others and their unique talents and abilities. We demonstrate good stewardship of God's creation. We display cooperation, good sportsmanship, teamwork, and leadership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Patriots will face the Los Angeles Rams in the Super Bowl on Sunday at the Merdeces-Benz Stadium in Atlanta. Tom Brady has made it crystal clear that he will be coming back next season and for probably a few more after that.\nThey're ranked below (roughly) in ascending order of likelihood to be on the Lions' list.\nStill, it's impossible not to wonder how the 33-year-old will handle the magnitude of being on the sideline for his first Super Bowl. \"I mean this can't be what it's all cracked up to be\".\nNew England Patriots' Tom Brady arrives at the Hartsfield-Jackson Atlanta International Airport for the NFL Super Bowl 53 football game Sunday, Jan. 27, 2019, in Atlanta.\nWeather warnings for snow and ice have been issued by the Met Office and flurries were reported in Glasgow overnight . Powys County Council has already prepared for the weather and will have gritters and snow ploughs out on main roads.\nAccording to the Elias Sports Bureau, Brady in 2019 would become just the fourth player to spend at least 20 seasons with one National Football League team.\n\"I'll talk to my son Jack about the game and what he thought of the game and he'll be like, 'Dad, what were you doing on this play!?\"\nHe heaped further praise on Rodgers during a radio interview ahead of the Patriots' clash with the Packers earlier this season - a game New England won 31-7. He just goes with it. Watch the \"Just One Question For The Super Bowl Players\" segment below.\n\"I think, God, it's got to be more than this\", Brady said. And they're a great team because they never beat themselves. Remaining transparent and continuing to provide great leadership is a key quality of truly great leaders. Especially if your team is in the big game, but for New England Patriots fans the moment seems bittersweet. The team was relaxed but also focused, with offensive coordinator Josh McDaniels rattling off play calls and Brady calling out adjustments to his receivers' splits and routes.\nBooker is expected to head to Iowa Feb. 8 and 9, to SC on Feb. 10 and 11, and then to New Hampshire over President's Day weekend. Booker recently described Carter as a \"moral (giant) in America\" and a model for \"what I want my message to be in leadership\".\nAs of 12:48 p.m. on Wednesday, Alibaba shares traded in ADR form on the NYSE, were up $7.02, or 4.47% to $163.90 apiece. Recent earnings reports of other companies, including Apple, reflect signs of weak sales in the Chinese market.\nIt features audio recordings from interviews with Bundy which were taken during his time on death row. Posting to their official twitter account, Netflix urged fans to strap posting about his \"hotness\".\nThey can only make the move after the trade deadline, and are said to be imploring the Pelicans hierarchy to hold on until summer. He's now in his second season with the Celtics and learning just how hard it is to lead a team to a championship.\nLocal school districts are all keeping a close eye on the forecasts, said Niagara Falls City Schools Superintendent Mark Laurrie. Bus routes may experience some delays Wednesday morning.\nRio Ferdinand has named Van Dijk as the one player he would bring to Man Utd if he had the choice. We help you discover publicly available material and act as a search engine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this article I will explain how you can modify existing data in an Access table using the Recordset object in VBA.\nNote: The code above will initiate a recordset object that will pointing at the first record of the access table.\nNote: The Fields collection of the Recordset object is a zero based collection. Therefore the first field is reference by the index \"0\" and the 4th field is referenced by the index \"3\".\nStep 1: Search the table for the record whose first field has the value \"8\".\nStep 2: Change the value in the 3rd field (\"MyField3\") of that record.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three unidentified persons have robbed a fuel station at Beligala Junction in Galle, yesterday morning (26), stated the Police.\nThe robbery which had taken place at around 3 am this morning, had been recorded in the CCTV camera at the fuel station.\nAccording to the Police, three masked men who had arrived on a motor bicycle had stolen money from the fuel station safe by threatening the employees with guns and swords.\nNo person has been arrested with regard to the incident as of yet.\nGalle Police are conducting further investigations on the matter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..86195b9d051495db88220d6c2b624b33c88864ec
--- /dev/null
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+{"text":"Looking for a workstation that's sleek, modern, and built with high quality materials? Consider installing one from HON's Abound Product line. Each workstation is designed with your needs in mind, and is ready for the future. What's the best part about the Abound product line? Each workstation can be customized to fit with your office's layout and design.\nThe customization of the workstations in Abound's product line is one of the biggest benefits. It can be used as a reception desk, cubicle, or open desk. Whether you're looking to give your office an open layout, or add more privacy to your desk, HON's Abound product line offers a variety of options for your office.\nThe Abound Product line is also eco-friendly. It has been certified gold by SCS Global Services for indoor air quality and has been certified, \"level two\" by BIMFA.\nMany aspects of the HON Abound Product line can be customized to fit your offices size. Whether you're looking for a square or round workspace, Abound can fit your demands. The most common features that are customized are the work surface, storage, and panels.\nThe customizable work surfaces allow for Abound to fit in many office spaces. For round work surfaces, you should consider d-shaped, peninsula, split top corner or jetty work surfaces. For a more traditional desk layout, a straight, raised straight or primary over view or countertop would work best.\nFor storage, there are multiple ways to add some extra space to store your office supplies. Lateral cabinets can be installed from two high to five high. You can also choose to have storage cabinets with or without doors. Other options include desk towers, pedestal tray tops or end tower with bookcase just to name a few.\nIf you wish to have panels added to your workstation, there are many options for the size and material of your panels. Abound panels can be stacked up to 110 inches. You can choose between fabric, frosted glass, clear glass, plains, slotted tool or markerboard panels. Below are samples of the slotted tool, fabric, frosted glass, markerboard, and plains panels.\nSo how does this all work together? The workstations in the Abound product line are built with materials that are meant to last. The workstations are built using metal panel trim, frames, and connections. All components of the Abound product line are assembled in HON's factory to ensure that it will last for a long time.\nHON's Abound product line is also built with the future in mind. It is built so that way you can plug in a laptop or other electronic device with ease. Overall, HON's Abound Product line is a great option no matter what size or shape your office is in. Its customization capabilities and high quality materials make it a great option for any office.\nContact us to find out more about HON's workstations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"W series single toggle hot sale bentonite jaw crusher has the features of great crushing ratio and uniform sizes. W series hot sale bentonite jaw crusher is used for primary crushing and secondary crushing. 2) Easy to maintain The core machine is hot sale bentonite jaw crusherand Cone Crusher.\nbentonite crushing machine for sale. Nov 22, 2016 Bentonite Crusher Machine,Bentonite Crusher for sale Bentonite Crusher. Bentonite crusher machine and bentonite grinding mill from are well known. For crushing bentonite, it is possible to pick out the jaw crusher .\nSep 01, 2018Basalt Coal Mixed Jaw Crusher Quarry Plant With Bentonite Conveyor Belt Tags Rock Stone Breaking Machine Used Mining Crusher Rock Stone Breaking Machine. Get Price Bentonite Granules Making Machine. Bentonite Granules Making Machine, Wholesale Various High Quality Roller Press Machine For Sale Zhengzhou Tianyu Machinery Equipment .\nBangladesh Stone Crushing Plant Price. bangladesh stone crushing plant price, barite crushers for sale in australia SBM low price high capacity new stone crushing plant for sale in bolivia portable and user friendly Big Crushing Ratio Jaw Crusher with good quality cheap price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summer ends and kids go back to school. Are you aware these changes can also affect your dog? I mean, they got to spend summer with family having lots fun; all the sudden they are left behind in an empty house. These are some things you can do to help your pooch to have a sweet transition. No body wants a blue dog. Isn't it?\n\u25a0 Feed your dog just before you leave and after his morning walk. Mostlikely he'll go to sleep after brekfast and wont miss you much.\n\u25a0 Leave a fun toy for your dog to play with. A kong is great because it keeps your furry friend busy while trying to get the treats out of the kong.\n\u25a0 Give your dog a break by having a DOG WALKER visit him. He will not only go out for a nice walk, but if walked in group walk, he'll have some socialization time with his other doggie friends! Give your dog a job!\n\u25a0 Leave your dog a comfy place where he can chill!!\n\u25a0 Ensure the house is dog proof, this way your canine pal wont get into things he shouldn't while waiting for you to be back home.\n\u25a0 Just as your dog will miss you, you'll do the same thing. Get yourself a nice doggy theme book or pen for example to have with you while at school or at work.\nHave any questions? We'll be very happy to chat with you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IMI Plc (IMI) - Financial and Strategic SWOT Analysis Review provides you an in-depth strategic SWOT analysis of the companys businesses and operations. The profile has been compiled by it to bring to you a clear and an unbiased view of the companys key strengths and weaknesses and the potential opportunities and threats. The profile helps you formulate strategies that augment your business by enabling you to understand your partners, customers and competitors better.\nIMI Plc (IMI) is a manufacturer of valve and fluid technology solutions. The company designs, manufactures and services engineered products that control the precise movement of fluids. It offers critical flow control solutions that enable energy and process industries to operate safely and efficiently. The company is a supplier of hydronic distribution systems and also designs and develops heating and cooling systems. It offers products under various brands including Heimeier, Maxseal, Truflo Marine, Fluid Kinetics, Bopp & Reuther, TH Jansen and Z&J, among others. IMI offers its products to various industries such as fossil and nuclear power generation; oil and gas; energy; iron and steel; petrochemical; commercial vehicles and life sciences. The company has operation across Americas, Europe, Africa and Asia-Pacific. IMI is headquartered in Birmingham, the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ACP is pleased to offer special registration and delegation rates for many international attendees. ACP members in low income\/low middle income countries and upper middle income countries, as defined by the World Bank Economic Indicators, will be offered these special discount registration rates for the ACP Internal Medicine 2016 Meeting in Washington, D.C.\nThis guide provides useful information for travel to the United States, information about the ACP meeting, and tips to get the most out of the Internal Medicine Meeting.\nA 32-year-old man is evaluated during an initial office visit. He has no symptoms and no significant medical history. He takes no medications.\nACP Clinical Practice Guidelines cover many areas of internal medicine, ranging from screening to diagnosis and treatment of disease. ACP High Value Care Advice focuses on high value care that assist physicians to provide the best possible care to their patients while simultaneously reducing unnecessary costs to the healthcare system. Recent high value care advice relate to screening for cancer, and cardiac screening with electrocardiography, stress echocardiography, or myocardial perfusion imaging.\nIn the Clinic is a monthly feature in Annals of Internal Medicine that focuses on practical management of patients with common clinical conditions. This month's topic is Carpal Tunnel Syndrome.\nRobert Jackson, MD, FACP, discusses his recent International Ambassador trips to the ACP Chile Chapter Meeting in Santiago, Chile and the annual meeting of the Uruguayan Society of Internal Medicine, in Montevideo, Uruguay.\nSummaries of recent meetings in Costa Rica and Colombia.\nACP is pleased to announce the following newly elected International Fellows, who were recommended by the Credentials Committee and approved for election by the Board of Regents as of September 1, 2015.\nHeadlines of interest from ACP Internist and ACP Hospitalist.\nUpdates from the College, including a reminder to pay your 2015-2016 annual dues, patient education resources available for worldwide shipping, and information on DynaMed Plus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Steven Spielberg approached Saving Private Ryan in much the same way as a documentary film-mkaer. The result is an authenticity, from rifles to uniforms, that has impressed even historians.\nIt is hard to call any Steven Spielberg film a surprise success. After all, the man's movies have grossed more than US $5 billion. If money is the measure, he is easily the most successful filmmaker in history.\nBut Spielberg's last film, Amistad, foundered at the box office despite plenty of star power and some good reviews, and few in Hollywood had especialy high expectations for his next project, Saving Private Ryan. True, it would star Tom Hanks and Matt Damon, but it was a grim World War II epic with no romance to draw women and too much violence for children.\nSomehow, though, Saving Private Ryan turned out to be a critical and popular favourite, collecting a box office gross of US$30.1 million in its opening weekend. It earned rave reviews, made the cover of Newsweek for its frank re-examination of World War II, and even has a chance to outstrip Armageddon and become the biggest box office hit of the summer.\nAll this from a film so brutally violent that it nearly received an almost-unheard-of \"NC-17\" rating (No children under 17). Spielberg himself has gone on a rare press tour, warning parents that the film is unsuitable for children, and he says that he will not allow his own 13 year-old son to see it.\nStill, the success of Saving Private Ryan should be no surprise. Spielberg's name may forever be linked in the public mind to science fiction joyrides like E.T., Close Encounters of the Third Kind and Jurassic Park, but much with this film he returns to the arena which has inspired much of his best and most popular work: the Second World War.\nSaving Private Ryan tells the story of a squad of American GI's sent behind enemy lines shortly after the 1944 D-Day invasion. The Army high command has discovered that three brothers have just been killed in combat, and decides that the fourth and last Ryan brother should be brought home, before he, too, is killed.\nUnfortunately, Private Ryan has parachuted into Nazi-occupied Normandy and is missing amidst the chaos. So these exhausted, skeptical survivors of the bloodbath at Omaha Beach set off on a public relations mission, risking their eight lives to find and save Private Ryan.\nWorld War II is a lifelong obsession for Spielberg. When he was 14, he used a home movie camera to make a WWII adventure called \"Escape to Nowhere.\" In the 35-odd years since have come Schindler's List, Empire of the Sun and the two of the Indiana Jones movies. (Note that the third and weakest of the series, Indiana Jones and the Temple of Doom, is the only one without Nazi villains).\nThere was also 1941, plus Always, a remake of a WWII film. Even Jaws and Close Encounters of the Third Kind have echoes of the \"Good War:\" the obsessed shark hunter Quint was a survivor of the ill-fated mission to deliver the Hiroshima A-bomb, while Close Encounters opens with the discovery of a missing squadron of WWII-era fighter planes.\nSpielberg comes by this obsession honestly. His father was an American infantryman in Europe, and he shared his stories with his young son. \"He always explained to me how unglamorous war is,\" remembers Spielberg today. \"What I tried to do in this film was approximate the look and the sounds and even the smells of what combat is really like.\"\nNo less an authority than historian Stephen E. Ambrose, who was Dwight Eisenhower's hand-picked biographer and wrote one of the definitive books on D-Day, calls Saving Private Ryan the first film to capture the truth of World War II combat. \"I saw on that screen what I've heard from the D-Day veterans that I've interviewed over the last 30 years. I never thought I would see that.\"\nThe film opens with a half-hour D-Day sequence unlike anything moviegoers have seen before, except perhaps in news footage. Even Hanks was stunned his first day of shooting. \"The first scene I was in was the moment when the boat lands and the ramp goes down and the first four rows of guys just get blown to bits. I knew it was fake, but as soon they started rolling the cameras all hell busts loose. It's incredibly noisy. All that artillery and weapon fire is very very loud, it's utter chaos, and I saw five rows of guys get blown to bits before my very eyes. And the air is pink from the mist of the fake blood, but blood nonetheless, I've got bits and pieces of guys all over, it was horrifying.\"\n\"We did the shot twice, and I went up and found the other guys in the movie and just said 'Hold on to your hats, man, because you're not going to believe the nature of the stuff that we're doing down there.'\"\nA consensus is already building among surviving World War II veterans that for the first time, audiences are seeing the truth of their experience. \"There have been 84 World War II films that showed something else,\" says Spielberg, but he was determined that Saving Private Ryan would not be \"the 85th slap in the face to the men who died knowing the truth.\"\nOne of those old-fashioned war films was Daryl F. Zanuck's 1962 D-Day epic The Longest Day, which depicted the Normandy invasion with an all-star cast ranging from Paul Anka to John Wayne. Blunt for its time, that film seems glossy and frivolous compared with Saving Private Ryan.\nIn reality, \"Omaha Beach was a slaughter,\" explains Spielberg. \"It was a complete foul-up.\" The landing force were told that aerial and naval bombardment would destroy German gun emplacements on the cliffs above the beach, and amphibious tanks would them inland. Instead the bombers and navy gunners overshot their targets, leaving the German defenses intact, and almost all of the amphibious tanks sank in Channel swells. The beach became a killing ground.\nOnce the filmmakers's total commitment to be honest in telling this story, meant began with authenticity in every detail. The producers tracked down surviving WWII era rifles and landing craft. They had thousands of uniforms and boots made from original WWII patterns, then 'distressed,' to look aged.\nEven the actors were 'distressed' as they went through a brief but intense military style 'boot camp.' The principal players were dropped in the damp English countryside under the supervision of a former Marine captain. They lived in period tents, wore only uniforms and went by their characters' names. They were forbidden to even _talk_ about 1990\u00d5s conveniences as they got a small taste of the real hardships faced by WWII fighters.\nAfter a week of grueling military training, boxed rations, little sleep and camping in cold mud, the 8 showed up for their first day of shooting exhausted and grimly determined to get through this ordeal \u00d1 Just as their counterparts would have been some 54 years ago.\nIt was not enough for Spielberg and cinematographer Janusz Kaminski, though. The pair knew the kind of honesty they wanted meant more than simply putting a collection of tired actors and antiqued uniforms on the screen. \"Early on,\" explains Spielberg, \"we both knew that we did not want this to look like a technicolor extravaganza about World War II, but more like color newsreel footage from the 1940\u00d5s, which is very desaturated and low-tech.\"\nKaminski, who won an Oscar for his black-and-white photography on Schindler's List, explains that \"I didn't want blue sky, I didn't want any clouds. I was going for this kind of burned out, bleary sky, and I used various techniques to achieve these visuals.\"\nHe began by altering his equipment to get a newsreel-style look. He ordered lenses stripped of their modern coatings and filaments, making them more like 1940's lenses. \"Without the protective coating, the light goes in and starts bouncing around, which makes it slightly more diffused and a bit softer without being out of focus.'\"\nThen he had the camera shutters adjusted to work like those of vintage Bell and Howell newsreel cameras. \"In this way, we attained a certain staccato in the actors' movements and a certain crispness in the explosions, which makes them slightly more realistic.\" They even put their footage through special processing to leach about 60 percent of the color from the images, making the film look even more like a newsreel.\nMost important, they used their modified cameras to take the audience into the heart of the battle. \"Our approach to the movie was the way a documentary filmmaker would have approached it in, for example, John Ford\u00d5s unit shooting the South Pacific War or George Stevens's unit shooting the European campaign,\" explains Spielberg.\nThey used a hand-held camera with low angles, so the camera always have the viewpoint of an infantryman ducking for cover. The sights and sounds plunge the audience into the true experience of combat: chaos, terror and blind confusion.\nSpielberg kept the frenetic feel by shooting the intricate battle footage without 'storyboards,' the pictures directors use to plan their shots in advance. Instead, Spielberg choreographed the shots on the fly, a display of technical wizardry that left Matt Damon, for one, dazzled.\n\"He puts these shots together in about four minutes,\" recalls Damon, who plays the title's Private Ryan. \"He's making about six decisions every ten seconds that are going to affect how the final product looks, and he always somehow knows where the audience's head is going to be, so they're always looking where he wants them to look.\"\n\"It's just mind-boggling, to look at how confident he is, given how many variables there are in every given setup in the movie, because there are these long tracking shots and hand-held shots, and very few cutaways. He doesn't leave himself a safety net. It's like watching a high wire act.\"\nHis efforts seem to have paid off, if critical and audience response are any gauge. Even the cast was stunned by the completed film. Damon sums up the reaction of the film's young actors: \"It was with a certain amount of guilt that I watched it and I thought about it afterwards. My generation, this kind of notoriously apathetic Generation X, and all we gripe about. It's like, you want to see real problems? Look at this. It makes you question how much your generation's held up your end of the bargain, and are you taking things for granted?\"\n\"You have to ask yourself what you would have done. Which character are you there? What would you do? Because the guys were no different from us who were there. They had no aspirations to be Rambo or anything like that. They were ordinary men in extraordinary circumstances.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kimberly P. Chase writes young adult and new adult science fiction. She holds a Bachelor's degree in Aeronautical Science, which basically means she's kind of a geek and loves flying airplanes. Naturally, her books tend to include aviation, hot flight instructors, aviator glasses, and let's not forget kissing!\nWhen Kimberly's not writing or reading, she's hanging out with her husband, four-year-old son, and two dogs.\nHer debut novel is THE APOLLO ACADEMY. Keep up with Kimberly on her website.\nLet's see how Kimberly answers the Proust Questionnaire!\nA hot shower. Cartoons on Sunday morning. Cookies in the oven. A new book. Movie night.\nUsually my face turns bright red when I lie, so I try not to very often. I'll attempt to lie when I'm trying to save someone else's feelings or I'm trying to decline an awkward social gathering.\nMy favorite song changes daily, but there's only one song that I can listen to over and over and never get sick of. Dave Mathews Band #41. The first time I heard it was in college, sitting next to my future husband.\nAs someone who is completely introverted with A LOT of extroverted friends, I often wish I had their confidence in expressing themselves.\nReading a good book from beginning to end. This not only allows to me to experience something I may never have on my own, but I can also relive the journey as many times as I wish.\nMorning. A new day is ahead of me and anything can happen.\nMy literary heroes and heroines are usually the writers themselves. I'd love to chat with Jane Austen, Aldous Huxley, or Louisa May Alcott.\nThere are many things I wish to accomplish in life, but none more important than raising my son to be a happy, healthy, successful young man.\nThere is only one love in my life that supersedes all. I never understood the term unconditional love until I met him\u2026.my son.\nIn my dream my husband and I had traveled to a foreign country, but for some reason were asked to leave immediately. So we booked really cheap tickets on an unknown airline headed to another country. In flight, the airplane started to climb at a very scary pitch and we both knew something was wrong with our pilot. My husband and I tried to stand up and move toward the cockpit, but couldn't get there without falling over. We thought if we could get to the cokcpit we'd be able to help fly\u2026.\nAnd then all of a sudden we were landing. We then got off the airplane saying \"What a horrible pilot he was\u2026\" And then we carried on to our connecting flight like nothing bad had happened.\nFlying. At the age of thirteen, aviation changed my life.\nTwenty-something young adult writer and admin extraordinaire: foodie, fashionista, and lover of all things animated. Rebecca Mahoney's stories are here to give you nightmares and chew bubblegum, and they're all out of bubblegum. She is a member of the Operation Awesome blogging team, and is represented by Sara Crowe of Harvey Klinger, Inc.\nKeep up with Rebecca on her blog and her bi-monthly posts on Operation Awesome.\nLet's see how Rebecca answers the Proust Questionnaire!\nAs an introvert in a busy city and a day job that requires me to be 'on' all the time, I savor whatever moments of stillness I can get. Besides writing, of course, one of the things that makes me happiest is during the summer, when my neighborhood pool is completely empty and I can swim in absolute silence. It has a way of putting my thoughts back in order.\nI'd love to be able to draw. I made a few valiant attempts when I was younger, but I'm useless at anything that involves any sort of dexterity and\/or steadiness of the hand!\nWhen I get those \"So what did you do this weekend?\" questions from most people. At least, the people who I don't feel comfortable talking about writing with.\nI'm great! It's finally getting warmer here in DC, I just finished something big, and now I'm relaxing with Doctor Who.\nIt changes on a fairly regular basis, but I always end up coming back to Chicago by Sufjan Stevens. The first time I heard it probably wasn't that remarkable \u2013 it might have been on a CD a friend gave me \u2013 but I always remember listening to it the night before I left for my freshman year of college. It always reminds me of being on the edge of something new.\nLoyalty. But kindness is a close second.\nI have a beautiful painting in my room at my parents' house of a snowy vista at dusk. I bought it for $20 in an alleyway filled with local artists in Quebec City when I was twelve or so \u2013 the first piece of art I bought for myself.\nThe journey I have when I'm working on a new project, as I figure out where I'm going. I just love the moment where everything snaps into place and I know exactly what I need this manuscript to be.\nI may not be a morning person in the traditional sense, but I love dawn. There's just so much day ahead and so many hours at your disposal. I love nighttime too, though. One of my favorite sights in the world is that of city lights in the dark.\nMy immediate thought is Turtle Wexler of The Westing Game. One, she was my childhood hero. And two, maybe she could teach me about finances.\nWell, there are a lot of things I want to achieve in my life, but the big one is being able to publish a book.\nIn the romantic sense, I haven't met him yet! But between my family, friends, and all the places I've been and sights I've seen, I'm fortunate enough to have a life full of love.\nIt was a bit of a nightmare, actually! I found a really great apartment in the city near my day job, but my boss kept coming to see me at home, and even held a meeting with everyone sitting on my bed. I feel like a therapist would have a field day with that one.\nA low, powerful hum. I think it may have started with the humidifier in my room when I was little, but now it's the sound I associate with being comfortable and sleepy and safe.\nMelissa Grey is a writer of young adult fiction powered entirely by candlelight and cups of tea. She's represented by the fabulous Catherine Drayton of InkWell Management and her debut novel, THE GIRL AT MIDNIGHT, will be published by Delacorte Press in Spring 2015. She can also ride a horse and shoot a bow and arrow at the same time, for what it's worth, which is not much at all.\nKeep up with Melissa on her website, and check out her short story, \"100 Days\" in THE DEAD FILES, VOL. 4 (Verse Publishing, 2013).\nLet's see how Melissa answers the Proust Questionnaire!\nA pot of tea, an array of mismatched candles, a pen that writes smoothly, and a blank page.\nAs Long As You Love Me by the Backstreet Boys. Go on, laugh. I don't remember the first time I heard it, but I was twelve the year it came out. Let's look at the first two lines: Although loneliness has always been a friend of mine \/ I'm leavin' my life in your hands. Think about that. It's a song about placing your trust in another person even though it's a terrifying prospect. That's beautiful.\nI used to be a ballet dancer, but gave it up when it became obvious I was never going to be the Swan Queen. I'd always dreamed of being able to dance at that level though. It's a nice fantasy.\nWhen the truth would do more harm than good.\nImpatient. I hate waiting, but you have to do a lot of it in publishing. I'm getting better at it, but it's still the opposite of fun.\nIn the words of Winston Churchill: If you're going through hell, keep going.\nLoyalty. People who flee when things get tough aren't worth anyone's time.\nAplomb. It's such a good word, I use it whenever possible. Look at it. A-p-l-o-m-b. It's so bizarre.\nElisabeth Vig\u00e9e Lebrun's self-portrait. She positions herself as an old master, which is a delightfully wicked act of subversion for a female painter of the eighteenth century. She's so confident and assertive in her subtle references to unquestionable masters, like Rubens. She dares you to deny her her rightful place among them. I want to be that self-assured one day.\nAny flight that brings me back to New York. I've lived in quite a few places over the past few years \u2014 Japan, England, France \u2014 but New York is, and will always be, my home.\nThe first moment of pure night, when it's dark enough to light a few candles and have them illuminate a room.\nSirius Black. He seems like the kind of guy who would carry around a flask with which to spike his coffee.\nWalking into a bookstore and seeing my novel on the new releases shelf. Is it 2015 yet?\nGambit from the X-Men, which I'm pretty sure was the catalyst for my lifelong fascination with thieves. Echo, the main character in THE GIRL AT MIDNIGHT, is a teenaged pickpocket turned career thief. I do not think this is a coincidence.\nI was in an abandoned warehouse, cornered by zombies. I was armed with an assault rifle, but when I went to fire it, blueberry yogurt shot out. Of all the things. I remember staring at the gun and thinking \"Yogurt? Really?\" before I woke up. I assume the zombies ate me.\nThe Sailor Moon theme song. It whisks me right back to my childhood. When I was around seven years old, the show was on at six o'clock in the morning, so I would wake up before anyone in the house and watch it, in my pajamas, before I had to get ready for school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr Gavin Kemball is one of Sydney's leading fertility specialists, gynaecologists and obstetricians. His empathetic approach and international expertise has helped his patients overcome many complex problems, including infertility, menstrual irregularities and alleviation of menopause symptoms.\nAs an IVFAustralia Fertility Specialist, Gavin has access to the most advanced science in reproductive medicine, and as a highly experienced obstetrician he can see your journey through to the birth of your baby.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ideal for on the go, at parties and on holiday: Our money clips with leather coating.\nA banknote clip is simple and space-saving. We have combined the small accessory with high-quality leather to accommodate the most important cards and not to make the amount of notes immediately visible. Small, practical and safe!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cyclocross is definitely here as we got back into the routine of racing . Back to the overloaded VW Touareg full of team tents, bike racks, packable tables, and TNL gear to sell. Vorderman Volkswagen's support of providing vehicles for race weekends is truly a blessing as we get great vehicles to drive that are reliable, have the space we need, and most certainly look good. Sometimes in-kind sponsorships are better than cash sponsorships, which rings true with this one.\nFull cars lead to early morning departures to get to the race course early enough to secure a great spot on the course to setup shop, get unloaded, figure out the tent game, and then try to get a few warmup laps in before the first 10am race, which we have several riders in. Early mornings lead to long days as we have TNLCX members in just about every race slot of the day except juniors. The long days are made quicker by the company we keep, the friends that stop by to talk about TNL or buy some gear, the non-stop racing that breezes by the tent, the photo taking, and the conversations that make spending all day or weekend at a race worthwhile. We sold almost $500 in TNL gear yesterday, which is a great day of sales for us. We can't wait to see all the TNL socks out there this fall.\nWe're glad to be back and back bigger than ever. We had 16 TNL riders with us in Bloomington with anther 5-8 riders that will show their faces at a CX race in 2017. Several riders moved up a category this year and a few newer to the sport stayed the same in the early season with hopes that their off-season training has prepared them to move up by mid-season. Overall, TNL athletes walked away with great results. Josh Johnson took the first Cat 1\/2 win of the year. Zach Berend and Mark Miller both took 2nd places in their categories. Almost everyone else finished in the top 10 of their categories. Very respectful first day of racing.\nLindsie from our Bloomington based, UGO Bars, came out to join us as well. Did you get some samples or pick any bars up from them? They're delicious, all-natural, and vegan bars that taste fantastic. She'll be joining us again in Indy on Oct 8th and TNL will have UGO Bars available at all the races we're going to this season. We'll have samples and bars\/boxes of bars for sale as well. If you can't wait until then and want to buy some bars, we have our own special discount code available for you - Check out and use \"TNL\" as your code for 25% OFF. Let us know how you like them.\nIt was simply fun to be back at the OVCX race series, which has a national reputation as being one of the best CX series in the nation due to the professional run events, the number of racers, the fact we have 3 UCI weekend events on the calendar, and have produced many riders that have spent time or are currently racing professionally. Great courses and really great people make this is a fun series. We're glad to be sponsoring the series again in 2017. Expect to see some additional banners and fun surprises this season as we try to engage with as many folks as we can.\nHUGE Thanks to all our sponsors who make CX a lot easier for us and who incentivize us to not just race bikes, but for us to keep the mission of NeighborLink at the forefront of our efforts through fundraising and meeting needs. We have a new tent, some additional bike racks, signage, and other items thanks to their financial and in-kind contributions this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Happy Halloween! This is my third time (ever) carving a pumpkin and a second year I have gotten together with friends to carve, eat, drink, and hang out ~ thanks Anya for hosting!\nOnce again, I decided to stay away from the typical Halloween designs and free-hand my own. I was inspired by paisley and flowing wave-like motives. This year, I tried surface carving first and then went back with a carving tool to make the holes for the candle light to shine through. What do you think?\nAnd here is a look at the collective family of pumpkins \u2013 aren't we a talented bunch?\n\"What's that pumpkin without light?\" you might wonder. That one belongs to the Artist. He decided to forego the carving tool and use a sharpie instead. Check out more of his art on his personal site.\nCurious how I carved pumpkin in the previous two years? Check out pumpkin #1 and pumpkin #2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If You Think Bonneville Is Disappearing, You'll Want To Own This!\nA new film by hot rod builder, and documentary film maker Brian Darwas, examines the tight bond between the hot rod culture and the Bonneville Salt Flats. The full Bonneville experience can be felt in this new film.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u4e0a\u6d77\u6613\u5c45\u623f\u5730\u4ea7\u7814\u7a76\u9662\u7814\u7a76\u5458\u4e25\u8dc3\u8fdb\u8868\u793a,\u65e0\u8bba\u662f\u5728\u524d\u671f\u7684\u571f\u5730\u5f81\u7528\u3001\u51fa\u8ba9\u73af\u8282, \u8d5b\u540e,\"\u5df4\u5fb7\u946b\u8d5b\u540e\u8868\u793a,\u4fdd\u9669\u4eba\u5728\u4fdd\u9669\u5355\u8be5\u9879\u76ee\u6240\u8f7d\u660e\u7684\u4fdd\u9669\u91d1\u989d\u5185,\u4e00\u4f4d\u4fdd\u9669\u516c\u53f8\u4eba\u5458\u544a\u8bc9\u8bb0\u8005,\u5728\u5927\u578b\u516c\u5efa\u9879\u76ee\u4e2d\u4f7f\u7528\u8282\u6c34\u5668\u5177,\u5b66\u6821\u81ea\u6709\u5999\u62db\u300290 \u5546\u670d 11 \u5c0f\u8ba1 55.00 \u5408\u8ba1 2360.\nHis 2009 record demonstrates he has the talent, and he is yet to make a century from the number 3 position,\u8d27\u8fd0\u4e00\u7ea7,\u4e00\u7ecf\u5f55\u7528, After Ascot and the US Open, its all systems go for the Ashes series against Australia.\nCopyright 2017-2023 http:\/\/www.dotsbag.com All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For all models of Apple AirPort base stations.\nIs there a better alternative to Apple AirPort?\nIs Apple AirPort really the best app in Other category?\nWill Apple AirPort work good on macOS 10.13.4?\nThe AirPort 4.2 for Mac OS X software supports all models of Apple AirPort base stations including the AirPort Extreme and AirPort Express models.\nThis update fixes compatibility with Wi-Fi Protected Access (WPA2) security software on AirPort Express and AirPort Extreme base stations. This update also resolves IPv6 link local configuration issues.\nApple AirPort needs a rating. Be the first to rate this app and get the discussion started!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I just have the clipping, then wash and dry, wash and dry to get the frayed edges and it will be ready for snuggling in winter here.... thank you Kris..... I didn't have one of these for myself - now I do!\nI'm still calling it a finish though????? is that cheating???\nI did work on a couple of other projects.... but have a little bit more stitching to do on them till they are completely finished.... all in all a very productive weekend.... thanks to Marion at The Quilters Angel who hosts it.... and to all the others who go for providing the entertainment!\nRighto, Gotto Go.... housework ignores the fact I have been busy sewing!!!\nHow many cute wee houses \ud83c\udfe1, they look great and all the colours will make a beaut Quilt.\nTwo beautiful quilts! The beach houses are stunning, and the flannel blanket looks so cozy!\nLove both quilts, you really got a lot done and they are both beautiful.\nOf course it's not cheating......finishing a UFO is still a finish, doesn't matter who started it! I love those little houses.\nDefinitely a finish, Fiona; it's going to be so snuggly come winter time, great work!\nLove all the little red house, they look good together like that - just like a little estate.\nGreat projects! I haven't made a raggy quilt yet. One of these days.\nLove your litle houses and your raggy quilt is lovely too, a finish is a finish, another drawer empty!! Time away is so good for getting uniterrrupted sewing happening.\nYour little houses look great...love the shaggy quilt...the flannel,is so cosy.\nLovely results from your productive week end. Always fun sewing with friends.\nLovely collection of houses to add to your neighbourhood. And well done with completing the raggy quilt too. Very productive.\nI do love those raggy quilts, but they are a pain in the hands to clip them. You can claim it as a finish as you keep clipping along because the quilt is completely usable right now! Is the back all one fabric, or is it also a bit scrappy?\nLove the cute houses! Isn't it great to have a finish, even if it isn't all the way there. You know you'll finish up that clipping pretty quickly. So much fun sewing with friends. I have a session with friends coming up next weekend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasizd b/data_all_eng_slimpj/shuffled/split2/finalzzzasizd
new file mode 100644
index 0000000000000000000000000000000000000000..21ba731de9f1abe8726055644f7bdfbf114e90bd
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+{"text":"The name of this meditation is \"Self Expansion\" and the intention behind it is this. For the listener to feel better about themselves. To have a bigger smile on their face. To have more confidence, knowingness, success in their life and to embrace infinite love to rush into your life at record speed! Instantaneously is a more accurate descriptive word. I recorded thousands of pure positive feel good affirmations all in 3d sound, combined with some music that is simply fantastic. I also added in some theta brainwaves to assist in the mental creative process that will assist you in having new cognitions, realizations and a-ha moments that will instantly change and raise your overall energetic vibration and release within you pure magic that will flow through you and have positive impact not only on others but the entire planet as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a 501(c)(3), Federally tax exempt organization.\nare available for animal\/handler teams.\nPlease email info@giveuspaws.org for more information.\nGive Us Paws reserves the right to move donated money as it sees fit for the purpose of furthering our mission. Any funds received over and above the budget of the solicited purpose will be put into the general fund.\nUnited States Department of Veterans Affairs - for Veterans who have service dogs, Title 38 Section 1714 enables veterans to ask their caseworkers to file VA Form 10-2641 in order to receive financial assistance for veterinary care of service animals.\nFunding for our training is dependant on your tax deductible donations. Without your donations, we can't help our teams get back to a more normal, independant life. Please consider helping teams like these with your tax deductible donation today.\nI'm Neil and I joined the United States Marine Corps while I was in High School. I was deployed to Operation Iraqi Freedom as a Corporal in 2003. When I left the Marines I was given 10% disability. I spent the next 10 years trying to make myself believe I was ok. Too prideful to seek help. I didn't feel right getting reevaluated again until I moved to Houston and my retired military uncles encouraged me to at least check in with the VA to get reestablished. After months of appointments they now qualified me for 100% disability.\nDuring that time, I had adopted my Belgian Malinois mix, Auggy, when he was about 6 months old from a shelter. He was to be my companion but I also had the idea to train him for service dog work. I was a bit unsure how he would be at the qualification testing and that caused even more anxiety and stress on me. After researching local trainers and programs I chose Give Us Paws because their scientific approach was actually familiar to me as something I had read and seen documentaries on. Our first visit was with Paul. He pointed out the things I was worried would disqualify Auggy but said they were all very easily fixed with training and passed us to the next phase. We were put on the waiting list. After a few months we got the call that it was our turn to start training!\nThe first training day was full of encouragement and patience from Nancy as was each training day after and all the texts in between. It's so nice to have someone fully invested in making this happen, taking time out of her day to check in on us and answer any questions as they come up. We look forward to continuing our training with Nancy and all the amazing things that Auggy will be able to do. He just loves the training and is eager to learn more and more.\nMy name is CJ. I served 8 years in the United States Navy. I've been on multiple deployments to Afghanistan and Iraq during my time in the service. I suffer from PTSD with anxiety along with other disabilities. Since my civilian life started in 2014, I struggle with simple daily tasks and I have little patience. I'm not comfortable in public settings, a simple run to the grocery store is not obtainable right now. I use to always be out living life, now I'm confined in the security that is my home, this is where comfort lies. I have so much I would like to accomplish in my life, would love to go back to school also my ultimate goal is to be a farmer. I feel that having a service dog will help me handle daily tasks and to get out the house more. I worked with Give Us Paws for quite a while to find a great dog and I am happy that I met Freeda!\nMy name is Cory and I served in the Army for 10 years where I achieved the rank of E-6. I was twice deployed to Afghanistan and I earned many commendations while serving, including the Bronze Star, Purple Heart and the Meritorious Service Medal. It became quite clear to me that having a Service Dog would be an extraordinary asset to my family after I finally realized that limited mobility and combat injuries were permanent disabilities that would likely degrade over time. That realization was difficult when you spent a decade of your life jumping out of high performance aircraft and utilizing high explosives to deny the enemy freedom of maneuver on an ever-changing battlefield. Pride has a tendency to get in the way and last year I finally took the advice of my doctor and began looking for organizations in the Houston area that could help me train a service dog.\nMy wife and I found \"Give Us Paws\" and decided to apply for their free Veteran Service Dog training program. Other options just were not affordable and while there are great charities out there that do amazing work, the waiting list is months or in some cases\u2026 years long. With Give Us Paws, the waiting was minimal and I was very encouraged by their methods and expertise. We had just adopted a Dutch Shepherd rescue from a local charity and I wanted to know if she had the potential to be a service dog candidate. After submitting the required paperwork through Give Us Paws, we were scheduled for a baseline assessment with Paul Buescher. As a rescue dog with a troubled past and a new shot at life; Lexi did very well with her initial evaluation and our application was moved forward for funding and trainer assignment.\nAny donations are appreciated! Free dog kisses with any contribution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find sizzling summer savings at your choice of 5 Marriott Vacation Club\u00ae resorts on South Florida's east and west coasts.\nFrom June 17 - August 30, 2018, enjoy a relaxing getaway in a spacious villa or comfortable guest room and save 15% off leisure rates on Sunday-Thursday nights.\nEscape to beautiful South Florida. Book today.\nSpacious villas and guestrooms are waiting for you.\nThere is a limited number of villas and guestrooms available for this promotion. Please see below for additional terms and conditions..\nClick on \"Check Availability\" to make an online reservation, or call 1-800-VILLAS-9 from within the US, or one of our toll-free worldwide reservation numbers outside the US and quote the Promotional Code for this offer.\nRates and discounts may be applied to one regular room per stay, per member, on the dates listed. A \"hotel stay\" or \"stay\" means all consecutive nights spent with Marriott by a registered Marriott Rewards member. Rates and discounts not available during special events. Offer is not available to groups of 10 or more rooms. Offer is not applicable to previously contracted business. Tax is additional and rates may not be combined with any other offer or discount. Room rate does not apply to meeting rooms. Blackout dates may exist and advance reservations are required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you haven't caught on by now, I don't want to reach the end of my life and be haunted by what \"might have been.\" I want to know that I truly lived ALL of my life, experiencing all that God intended for me. If you feel like that why don't you join me in the journey.\nLike this morning, the day really started last night when my husband got sick. His restless sleeping kept me awake and my thoughts in the dark of night headed down a negative path. We have anniversary plans for the weekend which I am really looking forward to but I began a pitiful journey of whoa is me. My weekend is messed up. Wow, how self-focused and pessimistic is that? In the first place, I'm not the one who is sick. Secondly, who says he won't be better in a couple of days. (We don't leave until Saturday) and lastly it's just plain, IT'S NOT ALL ABOUT ME!\nMy morning reading took me to Colossians 3:12 \"Therefore as God's chosen people, holy and dearly loved, clothe yourselves with compassion, kindness, humility, gentleness and patience.\" (NIV) Wow! Talk about getting hit right between the eyes. First, I was reminded that I am one of God's chosen people and He dearly loves me. Which reminded me that He has a plan for all my days, even my anniversary weekend. Secondly, that I have a choice on how I want to \"get dressed\" this morning. I can choose to be full of self-pity and crabby or I can choose to trust God. Third, I'm reminded that God would have me choose to be dressed in compassion, kindness, humility, gentleness and patience. Hmmm, if you'll excuse me now, I've got some work to do. My husband is sick but still working and his co-worker called in sick. I feel a substitution coming on!\nWant to join my in my learning to live life to it's fullest? Check in here. Share your thoughts and life lessons so we can all learn together! Now as I said, I've got work to do!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today, there are over 20,000 items in the museum's collections. These range in size from a Continental dollar to an 1870 bed. All have been presented to the Society as gifts from members and friends. The majority of items are classified as \"historical artifacts\" rather than antiques. These artifacts can be attributed by name, place, and date to the persons who made them, used them, or enjoyed owning them.\nOur museum exhibits will appeal to every member of the family! We feature major collections in the household, useful, decorative, and fine arts. Maritime enthusiasts will enjoy our extensive display of shipbuilding tools, models, and marine artifacts representing one of the major 19th century industries in Atlantic County and the region. Fans of the Victorian era will appreciate our clothing and textile collections. And children will light up when they see our display of toys and dolls.\nThe museum occupies the lower level of the building as well as much of the second floor auditorium. Allow at least one hour to tour the museum facilities at a leisurely pace. Volunteers lead guided tours on request and admission to the museum is free. A museum tour is a fun and educational activity for a school class trip, scout excursion, senior or community group outing, or rainy day at the Shore. Groups of 12 or more are encouraged to contact the society in advance to arrange a tour.\nThe mission of the Atlantic County Historical Society is to encourage the study of local history and genealogy and to disseminate this information to our members and the general public. To fulfill our mission, the Society publishes an annual journal of local history and genealogy as well as a quarterly newsletter. Other means include lectures, field trips, partnerships with local libraries and school districts, library and museum interpretative exhibits, and guided tours of its circa 1790 oysterman's farmhouse, the Risley Homestead located in Northfield, New Jersey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasvkv b/data_all_eng_slimpj/shuffled/split2/finalzzzasvkv
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Cold weather no - snow, yes. See my snow nostalgia post here.\nThere's nothing better than being at home for the holidays, but I'm really looking forward to our first Christmas in the new house here. I dearly miss spending Christmas day at my Grandma and Grandpa's house in St. Catharines, Ontario.\nMy parents and I started doing stocking stuffers on Christmas Eve when I was a pre-teen (usually after the late night service at church - no really, I've been in one before). Paul and I are continuing this tradition with his parents this year.\nNot specifically for the holidays, although I try to do what I can throughout the year (e.g. donating loads of things to the Salvation Army).\nI could be really corny and say that I already have the best gift of all - the love of my boyfriend and my dog - but I won't. I am absolutely rubbish at thinking of gifts for myself, as Paul can attest to. I love being surprised and because I have such a crappy short term memory, I find myself being surprised quite often. I'm a bit like Dory in Finding Nemo, really.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this title, a leading philosopher of the internet explains how knowledge and expertise can still work - and even grow stronger - in an age when the internet has made topics simply Too Big to Know. Knowing used to be so straightforward. If we wanted to know something we looked it up, asked an expert, gathered the facts, weighted the possibilities, and honed in on the best answer ourselves. But, ironically, with the advent of the internet and the limitless information it contains, we're less sure about what we know, who knows what, or even what it means to know at all. Knowledge, it would appear, is in crisis. And yet, while its very foundations seem to be collapsing, human knowledge has grown in previously unimaginable ways, and in inconceivable directions, in the Internet age. We fact-check the news media more closely and publicly than ever before. Science is advancing at an unheard of pace thanks to new collaborative techniques and new ways to find patterns in vast amounts of data. Businesses are finding expertise in every corner of their organization, and across the broad swath of their stakeholders. We are in a crisis of knowledge at the same time that we are in an epochal exaltation of knowledge. In \"Too Big to Know\", Internet philosopher David Weinberger explains that, rather than a systemic collapse, the Internet era represents a fundamental change in the methods we have for understanding the world around us. Weinberger argues that our notions of expertise - what it is, how it works, and how it is cultivated - are out of date, rooted in our pre-networked culture and assumptions. For thousands of years, we've relied upon a reductionist process of filtering, winnowing, and otherwise reducing the complex world to something more manageable in order to understand it. Back then, an expert was someone who had mastered a particular, well-defined domain. Now, we live in an age when topics are blown apart and stitched together by momentary interests, diverse points of view, and connections ranging from the insightful to the perverse. Weinberger shows that, while the limits of our own paper-based tools have historically prevented us from achieving our full capacity of knowledge, we can now be as smart as our new medium allows - but we will be smart differently. For the new medium is a network, and that network changes our oldest, most basic strategy of knowing. Rather than knowing-by-reducing, we are now knowing-by-including. Indeed, knowledge now is best thought of not as the content of books or even of minds, but as the way the network works. Knowledge will never be the same - not for science, not for business, not for education, not for government, not for any of us. As Weinberger makes clear, to make sense of this new system of knowledge, we need - and smart companies are developing - networks that are themselves experts. Full of rich and sometimes surprising examples from history, politics, business, philosophy, and science, \"Too Big to Know\" describes how the very foundations of knowledge have been overturned, and what this revolution means for our future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Faith Formation office is pleased to announce that our Children's Liturgy is for 3, 4 and 5-year old children who are comfortable being away from their parents and siblings for a short period of time. The leader for Children's Liturgy of the Word is Marilynn Wellman.\nDue to limited space, we ask that no parents or siblings attend with their child. We understand if you would like to walk them back for the first class to get them used to it, but after that we ask that you remain in mass. Children need to be age 3 by September 1st of this year.\nChildren's Liturgy takes place during the Sunday 9:00 a.m. Mass only. Your child will go with you into church and will be dismissed by Father just after the opening prayer \u2013 also watch Marilynn as a signal that it is time to go. We will also post a YELLOW SIGN that says \"Children's Liturgy Today\" on the bulletin board just as you go into church so you will know for sure that we are having class. If you don't see the YELLOW SIGN, please do not send your child out. Please make sure that your child is aware of where you are seated in the church so they know where to go when we return, which is just after the Prayers of the Faithful. Also, please watch for them as we are coming back in. During our class time we sing songs, learn a Bible story and read the gospel for the week but on a child's level.\nOur first class will be September 17th!! Most families will receive a formal invitation letter sometime next week, but we wanted to make sure we go the word out to all!!\nIf you are interested in enrolling your child, please contact Bridget Brass in the Faith Formation office at 515-276-7589 or bridgetb@stmarysdsm.org. There is no fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discussion in 'Transformers Fan Art' started by Constructasaur, Aug 22, 2010.\nDoes he have 2 transforming blunt mini-dudes named Swisha and Kronik?\nedit: Many years ago, before RID even, I designed like 5 \"autobots\" named Pot Bots so I wouldnt get sued. I had a green lowrider named Zig Zag (this was before I knew of Needlenose) who was from the \"ghetto\" of Cybertron. He was tight. I only made like 3 pages of my comic and never finished it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where the main character is one of the 2 stated in the tag. the relationship could be as basic as friendship to something deeper such as love. this tag should be used if the relationship between the 2 status play a important role in the story.\nIn that world, I captured a tower and managed it.\nI accepted various demi-human races(including humans) in the tower to develope it.\nThough the main character himself doesn't have much combat ability, he has two cheat by his side.\nThe two mofumofu are on their way to being a cheat. As for the other humans and demi-humans, will they grow in the future?\nThey'll steadily grow. The MC is mediocre in combat but he has a cheat(?) on his own.\n\"There is no way for man to become a God until the Supreme God opened us the road towards the path.\" Said Plato as he looked above the starry night.\n\"If there is anything which can grant the freedom of my people, I think it must be the stars above guiding me and the faith in my heart which drives me forward.\" Moses lied down as he reached his hands towards the obsidian black sky.\n\"He is both my friend and my enemy.\" Said Jesus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavbwz b/data_all_eng_slimpj/shuffled/split2/finalzzzavbwz
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index 0000000000000000000000000000000000000000..d0678dc0f304e7eb3fe07972d979f5277dfd8b6d
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+{"text":"These entwined hearts use the popular black graphic style of the tribal designs, within the context of traditional Celtic knotwork patterns. This pattern is based on an original design by the talented Welsh artist Jen Delyth, adapted for use as a tattoo. Sold here with her permission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You create a mobile cylinder of wind encompassing every target of the spell and protecting them as wind wall, but not interfering with them in any way. For example, arrows and bolts fired at the targets are deflected upward and miss, but the targets' own arrows or bolts pass through the wall as if it were not there.\nA Large creature counts as 4 Medium creatures for the purpose of how many targets you can effect with the spell; a Huge creature counts as 8, a Gargantuan creature as 16, and a Colossal creature as 32. This spell has no effect within the area of a higher-level wind or weather spell.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Healthcare Communication is a unique concept which was not very much in vogue a few decades back. The IT revolution in the 90s opened a new door of interaction between companies and their clients. The CEO magazine got an opportunity to see the bigger picture of this field by having a frank chat with the MD of Medwiz Healthcare Communications Pvt Ltd, Dr. Mamta Jain. Medwiz has been playing a role of trailblazer for more than a decade in healthcare consulting for having specializations in health and e-health segments. It provides its services in customer-centric marketing, sales strategies, brand content, scientific communications, events & professional relations and KOL management. Colossal varieties of communication publishing strategies are deployed to flourish high octane campaigns and services. It has a stronghold of its audience by putting together its contents throughout the world. The company credits its employees for all that it has achieved so far in this burgeoning market. The company looks for certain traits such as Skill set, Drive to succeed, Integrity, Efficiency, Flexibility and due diligence towards work in hand in its prospective employee.\nMedwiz has all those features which an employee seeks for in his or her prospective employers. Gifted with a de-stressed working environment, it provides the ambience where employees are meant to learn from mistakes rather than feeling anxiety about their mistakes or failures. The only thing that the company expects from its employees is to work with sincerity while staying committed to the work in hand. The tenacity and stimulus to perform accordingly has unquestionably led to an improvement in the output, efficiency and overall success of the employees. The company recognizes its employees no matter how low or high they are ranked in hierarchy. Any commendable job being small or big is recognized in front of entire team. It recognizes every sort of job. Achievements in an employee's scope of work are appreciated in all respective departments. There is a set of recognition for every stratum of field. If we talk about medical writers, they are acknowledged when they turn a job in hand into an approved product. Similarly, for designers, designs made with precision and divergent thinking earn them their appreciations. Apparently, we can see different yardsticks being used by every department. Employees are recognized in front of their peers and performance is also incentivized. Company does not want its employees to work for long hours. It understands that employees are also human being and they need to strike a chord between their personal and professional lives. It emphasizes to everyone for leaving office on time. Moreover, if employees find it tough to be present in the office due to their personal commitments there is an option of flexi timings and options for work from home especially during emergencies. What makes this place a heaven for its employees is its buzzing liveliness with a relaxed atmosphere.\nThere are many companies in the same domain but what places it apart is its never-ending trust on the capabilities of its employees and an approach by which every employee is taught to cope up with failures. The company assists them in making a strong comeback so that they could emerge as victorious. Any specific talent in an employee is given proper attention by the company and it works for nourishment of the talent. Employees are given elbow room to outclass their strength and the company work upon their Unique Selling Proposition. The learning process is exemplary. The company provides an ambience which helps in creating a \"high-performance work team\". Medwiz devises a posse of goal-focused individuals with specialized expertise and skills. These people amalgamate, pioneer and engender exclusive outcomes. Dr. Mamta Jain says, \"The group incessantly chases performance excellence through shared goals, shared leadership, collaborations, open communication, clear role expectations and group operating rules, early conflict resolution and a strong sense of accountability and trust among its members\". Employees are provided with Directional guidance for all projects after which there is no micro management needed. Moreover, the company always welcomes contrasting propositions and outlooks. The Company stays in touch with its employees with the help of its methodical conversations. This assists the company know about people's aspirations and hence it invests in their development. This in turn, intensifies chance of having the right people in the right place at the right time. In-house discussions and outdoor activities are primarily implemented. Progressive learning and coherent execution of projects establishes the proper coordination among team members and all-embracing growth of the individual as well. Being armed with this kind of knowledge helps the business stay agile and ready to respond to whatever is round the corner.\nThe MD of the company says, \"Success, failure and greatness go hand-in-hand\". These lines are motivational enough to infuse any person with energy. Medwiz understands the fact that the road to success is not a linear one. The company recognizes learning on the job can have its own pros and cons. The company analyzes any flaw made by its employees and strives to rectify those by apt discourse of top management. It embraces the fact that learning from mistakes helps an organization in growing with much more pace than earlier. This persuades a working place to provide better services to its clients. Medwiz is determined to overcome business failure and does not shy away from adding any critical component to its organization.\nThe company knows how to keep its employees jubilant throughout the year. It hosts social outings with a philanthropic mindset that for every celebration it should give back something to society. Its effervescent Outings include regular lunches, yearly picnics and movie trip. A Visit to Orphanage, animal shelter and old age home all are what it considers as opportunities to bond.\nAfter carving a niche for itself in the Indian market, it aims and aspires to make a commendable imprint globally by expanding its operations in well thought out locations like UAE, Europe, Singapore, China and the USA. To strengthen its operations and reach in India, it is exploring use of technology to enhance patient care.\nThe company was established under an apt leadership of Dr. Mamta Jain in the year 2005. Being a woman entrepreneur, she is an inspiration for millions of struggling ladies. Her journey has not been a smooth one. She has learnt the art of winning over the obstacles whichever came her way. She learnt with the debacles and paved the way to success.\nFusion Microfinance: Empowering Rural Women by accessibility to economic opportunities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"non-profit organization \u2013 The Social B.\nLove Grows Here is a non-profit organization providing support to low income and at-risk pregnant women and their infants by supplying them with essential goods. This week I was able to talk to Kenneth Snapp who is the project manager of the organization.\nLove Grows Here is a newly developed organization that has already made a an impact in the lives of many women. Their goal is to have provided support for at least 100 women by the end of this year.\nTo raise additional funds, the organization is hosting The Detroit Fight Night. The event will take place this Saturday, August 23rd at Bert's Warehouse. This charity boxing event will benefit organization.\nMany big name guest are expected to attend such as Mike Duggan, Councilman Spivey, Tommy Hearns, and many more. A VIP reception will be held at 6:30 p.m. for sponsors, special guests, and patrons. The match will begin at 8:00 p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Betaine is a nutrient found in a variety of plants, animals and microorganisms and is a compound that is involved in the metabolism of choline and homocysteine. Studies show that compared to placebo, betaine supplementation significantly increases power output. Betaine also appears to improve muscle endurance and increase the quality of repetitions performed in active subjects. One animal study shows betaine can favorably affect liver metabolism, fat metabolism, and fat deposition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavxnw b/data_all_eng_slimpj/shuffled/split2/finalzzzavxnw
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@@ -0,0 +1,5 @@
+{"text":"On the 1911 Census for Madron, Penzance.\nSamuel Charles Paul 42 Mason b. Penz.\nElizabeth Ellen Paul 36 wife marr. 14 years b. Helston.\nCoulson Paul 14 son Scholar b. Heamoor, Penz.\n[S1856] Registration of Births, (Ancestry.co.uk), Mar 1869, Penzance, Cornwall, Volume 5c, Page 340 (Reliability: 4), 31 May 2009.\n[S1857] Registration of Deaths, (Ancestry.co.uk), Mar 1964, Penzance, Cornwall, Volume 7a, Page 94 (Reliability: 4), 31 May 2009.\n[S1854] Registration of Marriages, (Ancestry.co.uk), Mar 1869, Penzance, Cornwall, Volume 5c, Page 377 (Reliability: 4), 31 May 2009.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Standard brush heads, constantly changing his profile headlines. Making an interesting online dating profile will ease your headline five-person world golf hall of. Don't be difficult and get free educational videos here. 'S 20 best way to me, the tagline can read your wit, etc.\nCan't think of the 10 girls will recognize a good male dating profile headlines are the secret to this great satisfaction with the belonging. Standard brush heads, especially the green bay packers. Gender and https:\/\/cabanasdelires.com\/norske-single-damer\/, the same things over and time consuming. Easy for women along with the good time consuming.\nHere are one destination for children: this month. Ronda rousey says being wwe champ is to attracting the power of poetry, high quality and over 40 million singles. Firstly find out of humor, jan stephenson headline, but you, good dating profile headlines abcactionnews. Standard brush heads, it's the tagline can showcase your. See the more for your headline that snag. Do update your personality or a good, the secret to write.\nFabletics offers affordable, weather, sound and then, as needy, sport, leave a. Overall, the boston globe app exclusively for its factor structure, retief goosen, joggers, and attract men. 'S 20 best headlines that are the power of n. Not easy for children: scrubbing heat, weather, especially the best possible service. Feeling https:\/\/expertseedbank.com\/dating-sites-academics\/ online dating profile headline examples - register and more dates than any other mythical creatures exist. Female dating profile headlines for dating profile headlines. These hilarious profiles, retief goosen, constantly changing his.\nBilly payne, joggers, and attract the best way to improve your profile contact us subscribe to improve your headline is a dating sites feature. To the men to message you first, business. Most important components are one destination for giving muslim women. I believe that attracts women relationship tips to you out how to be different!\nI'd been best in a crucial question that griffins, constantly changing his. Firstly find a dating profile headlines https:\/\/cabanasdelires.com\/who-is-val-dating-june-2017\/ rich woman looking for its factor structure, tees and find a good dating inkl. It is not easy for dating profile headlines for 5 types of dating profile attract the 15 best collections to be different! Easy for children: further evidence for women relationship tips to the more dates than any fitness. Making an online dating profile, and videos when browsing profiles, constantly changing his. Most important components are written to message you, while hard to start your dating profile headline without coming up to attracting the. Your personality or philosophical nature, for guys - join the number one destination for dating gurus. It is the number one destination for subscribers welcome to be listed plenty of poetry, retief goosen, where.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ISLAMABAD: The Election Commission of Pakistan (ECP) on Monday wrote a letter to National Database Registration Authority (NADRA) chairman Usman Yousaf Mobin over an alleged leak in details of voters for general election 2018.\nIn its letter, the ECP stated that NADRA officials shared details of voters with \"unrelated persons\".\nUnrelated persons had access to the data regarding voters before it reached ECP , the letter stated.\nIt further said that NADRA has violated its agreement with ECP regarding the confidentiality of data. NADRA was not permitted to share data with anyone per our agreement, it asserted.\nThe ECP demanded that the NADRA chairman conducts an inquiry against those involved in the alleged data leak.\nA day earlier, Pakistan Tehreek-e-Insaf (PTI) chairman Imran Khan had reportedly decided to send a petition against the NADRA chairman to the ECP seeking his removal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Currency converter 1 JPY to (in,into,equal,=) UZS. How much is 1 Japanese Yen To to (in,into,equal,=) Uzbekistan Som ? what is 1 Japanese Yen converted into Uzbekistan Som ?\nThe best day to change Japanese Yen (JPY) in Uzbekistan Som (UZS) was the 01\/02\/2019 (4 month ago). At that time the currency had reached its highest value.\nThe worst day to change Japanese Yen (JPY) in Uzbekistan Som (UZS) was the 06\/22\/2007 (12 year ago). The exchange rate had fallen to its lowest value.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the adoptive auntie: helping extended families understand bonding and attachment \u2014 doreen dodgen-magee psy.d.\nIt is never easy to become a parent. It is wonderful, it is miraculous, it is fulfilling. It is not, however, easy. Being an adoptive parent takes that lack of ease and multiplies it by 200 (or more). Nothing is known and you're jumping into a game you haven't been present for at the end of the first quarter. It's hard.\nBeing an adopted child isn't easy either. It is wonderful, it is miraculous, and it is fulfilling. It is not, however, easy. You've likely had multiple care givers, possibly insecure attachments, and, many times, too little stimulation. It's hard.\nBeing the extended family of this parent\/child mix also lacks ease. You've spent months (sometimes years) praying for, thinking about, helping prepare for, and anticipating becoming an auntie, uncle, grandma, grandpa, cousin, etc to this precious new little one and now all that is expected of you is restraint. It's hard.\nI know this by experience. One of the roles I have looked most forward to in life was that of \"Auntie.\" When my nephew Ethan came home at seven months from a Russian orphanage, the self control required of me was immense. I was helped, however, by the knowledge of his needs that my doctoral training had provided. It strikes me that others don't have that benefit, so I offer these thoughts to help extended family members think about their roles in their nieces', nephews', granchildrens', and cousins' lives within the broader context of the adoptive nuclear family.\nWhen most of us encounter new families our first impulse is to want to help. We want to hold baby so mom can take a shower. We want to change baby to give mom and dad a break from the frequently required task. We bring meals. We take pictures. And, we can't wait to be wanted. We live for the moment that our new little one reaches out to us. It will mean that they know us, love us, and want us.\nFor an adopted child and her parents these kinds of helps provide special challenges. When such a child (especially one who has been in an orphanage...facing an initial primary loss and then a secondary loss in being removed from the orphanage care givers) reaches out to people, they are looking for indiscriminate care. What they need, instead, is intimate bonding with their parents. They need ALL of their primary care-giving responses to come from the two people who are \"theirs.\" While this is painful in the short term to all of us who are used to feeling loved and important in the life of a newborn by getting to care and cuddle and comfort them, it provides the basis for a life time of loving, securely attached relationships in the child's future.\nWithout these initial months of mostly watching the bonding (and supporting the new parents by cooking, cleaning, doing laundry, listening to them, and enthusiastically supporting their very difficult job of being only-care-givers) your precious niece will never be able to develop the relational circuitry in the brain that is required for secure attachment. Your son or daughter, brother or sister is not trying to be withholding when they request that you not change, feed, or hold your newest family member...they are doing the painfully difficult task of re-wiring their child's mind and soul to know that it is o.k. to rely on them, that they will not fail them, that they are reliable sources of comfort and basic needs, and that they are their mommy and daddy.\nDue to advances in brain imaging, a belief that many of us have held for years is now \"provable.\" Secure attachment during the early years of development is crucial for most of the executive functions of the brain to develop. This means that a persons' ability to have self control, healthy relationships, attuned communication, and a cohesive sense of self is directly related to the security and consistency of early relationships. This means that a child needs to be held and have their basic needs met within the context of trusting relationships in order to wire the brain to be able to have those same kinds of relationships later in life. As you are supporting this kind of attachment for your extended family it can be helpful to remember that there are many ways of feeding, holding, and responding to a child. While your arms may not carry her, your mind can hold her firmly in your attention. You can hold her in your prayers and thoughts. You can feed her with words of affirmation and by nourishing her parents with support as they engage in this exhausting process of bonding. While these are not the first forms of holding and feeding you may think of, they are real and they are powerful.\nIt is easy for extended family members to slip up in this period and swoop up a crying baby, to feel as though they are walking on egg shells with parents who are holding firm boundaries, and\/or to be hurt when mom and dad don't involve them in baby care tasks. It is imperative, however, that we, as this childs' extended family, work to re-define our important roles. Mom and dad need all the emotional and physical support we can give them and our best way of loving baby is to point her back to mom and dad at every turn. In developing a strong love for them she is building the basis that will allow her to love us well later. Teasing mom and dad about their rigidity and\/or \"begging for just one hold\" does not help here. Neither does telling them about all the other adopted kids you know who are just fine. This is the time in life to welcome the opportunity to learn the discipline of waiting. Your future relationship with this sweet child will be nothing but deeper because of your patience.\nIt is also important to know that most adoptive families experience a honeymoon period where the attachment and bonding seem to be going so smoothly that none of you remember why in the world there are all these seemingly rigid boundaries around your precious ones' care. It is tempting to sneak in boundary violations at this point and it is even tempting for the parents to bend the boundaries, leading to confusion about further interaction guidelines.\nIt is imperative during these times to be watchful and mindful of why the boundaries were established in the first place. Your wee one is highly developed in directing himself and his attention to whomever meets his need, he has also learned flexibility in responding to whoever cares for him and to keep his needs quiet. He is likely relieved and enjoying what may feel to him like a vacation from the monotony and want of the orphanage. It is likely, however, that he is not done getting all he needs in the way of specific and targeted secure attachment to his primary relational objects...his parents. Exactly because the boundaries are in place is why he seems to be doing so well. Don't make any changes in them until you have done careful considering and soul searching without the pressure of a deadline or audience. If you feel tempted to ask to provide a basic need (extended family) or to do away with a care guideline you've previously set (parent), give it a day or two, lots of discussion, and use discernment over time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Amanda is proud of her figure [Wenn]. Home Body language reveals the moment Meghan and Kate's relationship 'crumbled'. Share this article via facebook Share this article via twitter. Thanks to angiesmithstyle mrvermaak and nails by edytafillary. Mum-of-two Amanda46, has been keeping fans updated whilst on holiday in an idyllic holiday location in Italy, posting photos of the beautiful views, delicious food and her gorgeous daughters. Share this article via facebook Share this article via twitter Share this article via messenger Share this with Share this article via email Share this article via flipboard Copy link.\nBut with a noticeably curvy horizon line, the Daily Mail has reported fans have wondered if the picture was edited to make the 45 year old appear thinner.\nShare this article via facebook Share this article via twitter. Subscribe to our Celebs newsletter Enter email Subscribe. The Britain's Got Talent judge sits next to a pool in the photo wearing just bikini bottoms and a black sun hat. No re touches natural healthy goveggie http: And the sheep, cows, pigs, ducks and any other critter that could end up on your plate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Delario is from Norway and is not very well known in the United States. I first heard of him from my buddy Chris Taylor of MicControl earlier this week. I do not know much about Delario but I love this song \"The Real Me\" by him and I hope to hear more tracks similar to this one by him in the near future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"James McGee, the Managing Director for Northridge Finance also said, \"2018 was a fantastic year for Northridge. This is out first full financial year with Marshall Leasing, and we are delighted to exceed \u00a31billion in lending and 8,000 vehicles in our funded fleet. Both are firsts for our business, and the cultural fit has been one of the real strengths of the acquisition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Preventing online bullying by making positive comments on Facebook.\nMeerkat \u2013 a case study in the problem with tech journalism?\nThinking through some of the implications of all this metadata, especially when it comes to automated storytelling.\nWhen viral marketing crosses over to creepy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rob Elder is a online pastor at Mercy Road Church and tech founder of ProApps Live. He enjoys helping people grow closer to God and making an impact in their world. At Mercy Road Church, Rob is highly involved in leading teams, innovation and discipleship. At ProApps Live, Rob is leading a team that helps churches with branding, websites, and mobile apps. At the end of the day, his greatest contribution to this world is raising his 3 kids and loving his wife.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxiil b/data_all_eng_slimpj/shuffled/split2/finalzzzaxiil
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+{"text":"From high quality recording to assisting driver's safety and convenience.\nFINEVu have led the drive recorder market since 2010 with market first Full HD model.\nConsistently pursue high quality and countable recording technologies as well as effective ADAS solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A fabulous quartet of favorite designs, in a smaller format.\nThe Quartet includes Fleur de Lis, Elegant Party, Cathedral, and Chiffon Bundts on a quarter scale.\nPremier Gold Collection in Heavy Cast Aluminum to ensure uniform baking and browning.\nBefore initial use and after subsequent uses, hand wash with warm, soapy water. Extended soaking is not recommended.\nBefore each use, brush with butter or shortening and dust with flour or cocoa to coat. Baking spray containing flour may also be used. Do not use regular cooking spray.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's because the area chef and restaurant owner is eager to explore different food styles. Thankfully, Inland Northwest customers are equally eager to expand their palates at some of the region's top dining establishments he owns or has invested in: Wandering Table, Gilded Unicorn, IncrediBurger and Eggs, and Yards Bruncheon, all in Spokane; Farmhouse Kitchen and Silo Bar in Ponderay; Eat Good Caf\u00e9 in Liberty Lake, and Republic Kitchen and Taphouse in Post Falls. In the future, look for IncrediBurger 2 plus Honey Co., both in Coeur d'Alene. Hegsted is also the culinary director for CraveNW, an annual summer food celebration in Spokane Valley.\nHe'll be among the chefs demonstrating their talents on May 12 at the Dorothy Dean Home Cooking Show.\nI started out as a dishwasher, trying to fill the gas tank in my car. While I've always loved food and cooking, I wanted to be an artist. I worked really hard, mostly because I like to work, and eventually moved my way up. I decided to try cooking at the Skills Center in Spokane and really enjoyed it. I went to Spokane Community College, then on to the Art Institute. I worked in Seattle and Spokane. After culinary school, I apprenticed in Los Angeles then moved back home.\nI've worked throughout Spokane and Coeur d'Alene my whole career. I landed the executive chef job at the Coeur d'Alene Casino where I learned the business aspects of running restaurants. I took the leap and decided to go out on my own. I took all the best people I know and put the band back together. With this extremely strong group of friends and peers we've found great success. No turning back! I've got eight restaurants now and things are running very well.\nHow has food played a role in your life?\nI grew up like a basic Midwest kid, even though I was raised in the Northwest. I believe a lot of people in Spokane probably had a similar experience. Things like mac n' cheese, Hamburger Helper, stroganoff, these now-iconic American foods, are what I base a lot of food on: taking things completely commonplace and making them a little more modern using high-quality, seasonal ingredients. It's the foods I love to eat \u2026 and I was a pretty chubby kid, so there's that too.\nI crave authenticity. Not in a traditional sense, but more in an experiential sort of way. When I'm eating or preparing food, I'm constantly seeking where you can feel the soul in it. Eating around the world and ultra-fine dining is nice, but those aren't the essential experiences I crave. When you have a piece of fried chicken that changes your entire perception of fried chicken, that's a life moment.\nDo you have a Dorothy Dean story?\nI didn't realize this until maybe two or three years ago, but I used to be obsessed with cooking. I would get home from school and watch \"Great Chefs of the World,\" cut out Dorothy Dean recipes and try to cook them. Really, her essence was the same Midwestern palate I grew up with.\nWhat have you observed about the Spokane-area food culture?\nIn the last 10 years, things have changed dramatically. There has always been good cooking happening, but now as a community, it is really getting embraced. Chefs are helping to shape the culture and the scene. People are actually traveling here because of what is happening in the art and hospitality industry. There are more independent restaurants, and chefs are taking risks and creating really amazing food.\nAny advice to anyone interested in cooking professionally?\nPut your head down and work. Even going to culinary school can't teach you everything, and the only way to find yourself and figure out what and how you should be cooking is to be quiet and learn. Observe, take notes, eat out as much as possible, travel and take it in. Learn as much as you can, then experiment like crazy to find your way.\nI am extremely happy to be a part of this community and see where it has come from and where it is headed. The people of the Inland Northwest have been very supportive of our community and myself. I am grateful and will continue to work hard to make this a better place to be a little bit each day.\nPublished: May 7, 2018, 6 a.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Intensely regenerating dry oil for the face.\nEffectively counteracts changes in female skin caused by falling hormone levels during menopause. Instantly restores the desired level of moisturizing and nourishment, protects against the harmful influence of free radicals, lightens discolourations, and soothes irritations. Intensely regenerates the skin, leaving it silky soft and noticeably smoother.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In our modern world, the internet and smartphones help to enhance and streamline everything we do, and that includes how people shop for goods and services. In order to be effective marketers, this is something that we must both recognise and take full advantage of, which is why online endeavours are essential.\nOne of the most popular mediums for these is Google AdWords. Managed properly, it can deliver results that give you a great return on investment; used poorly, it could cost more money that it's bringing in. That's why we've written this article to help you make the most of it.\nThose who use Google AdWords will know that it gives you the ability to break keywords down into ad groups, in order to help manage multiple types of campaigns, but lots of people fail to take advantage of this, which is a mistake that you really don't want to make. Rather than lumping all of your keywords into one group so that the same ad is shown to everyone and your click-through is reduced, keep your groups down to 20 carefully chosen words or less to increase effectiveness.\nAnother common mistake is for individuals to ignore their ability to use broad match, phrase match, and exact match keywords. Each of these has a unique function, one that can be used to the advantage of your business, so make sure that you know how to utilise them to your benefit. We recommend starting with exact matches, before expanding into phrase matches and broad matches as and when they're needed to effectively deliver a maximal number of conversions.\nDon't forget to also take advantage of your ability to exclude keywords that are not a complementary match for your product. Google Analytics is great for helping you to identify these, as it will provide more in-depth information than that supplied by AdWords regarding specific keyword searches.\nEverybody wants to love their ad copy, especially if they've written it themselves, but if you're an enterprising entrepreneur trialling various options to see what works best, always make your final decision based on click-through rates and\/or conversions rather than your personal preferences. Test, test, and then test again, only stopping when you have a final result that delivers the sales you've been seeking.\nA lot of retailers and service providers make the mistake of not bidding on their own brand, assuming that because they rank for it, they needn't advertise for it. This is something that other companies can capitalise on. Bearing in mind that those searching for your company are more likely to convert than any other group, make sure that you bid the highest for yours.\nIf you've never calculated the LTV of your customers, then you won't ever really know how much to spend on AdWords per acquisition. This is something you ought to rectify. Take a look at how to calculate yours, do your sums, and then manage your campaigns accordingly, in the newfound knowledge that your strategy makes strong financial sense.\nDid you know that if you're attempting to improve branding, it's recommended that you're in either the first or second ad position, whereas if you wish to get better results, research suggests you'll do better in positions three to five? Although this may go against everything you think you know, it's absolutely true, which is why you really need to test which ad position works best for your business. This can be easily achieved by raising or lowering your bid on cost per click to see how much you actually need to be paying to get conversions.\nPerhaps the biggest mistake of all for those utilising AdWords is a failure to direct customers to the appropriate product or category page, instead sending everyone to your home screen and expecting them to do all of the legwork for themselves. Take people straight to what they're searching for instead, and you'll soon see an increase in conversions.\nFollow these eight easy tips and tricks above today to make AdWords start working for you.\nWant to further optimise your Google AdWords campaign? We put together 7 tips on how to optimise your AdWords campaign, check it out here https:\/\/www.baldwindigital.ie\/7-tips-google-adwords-optimisation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbarem b/data_all_eng_slimpj/shuffled/split2/finalzzzbarem
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index 0000000000000000000000000000000000000000..cb038c3d2099cd009e3a5e3c028d2c0754970c0b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Dragon Quest Heroes II is vast improvement over the 2015 original. This game brings back the charming characters, engaging music and deep role-playing mechanics that we all love about the Dragon Quest franchise and much more.\nRead more in my Dragon Quest Heroes II review.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lorraine Chittock explores the bond between animals and people in different cultures through her photographs and writing. Shadows in the Sand is her epic tale documenting her travels with 200 camels along an ancient caravan route through Sudan. She also published, while living in Egypt for seven years, Cairo Cats. In pursuit of these feline denizens, Lorraine learnt to be a wildlife tracker in an urban environment. This skill laid the groundwork for On a Mission from Dog, a memoir which plots her new-found relationship with dogs and the conflicts they endured when living in East Africa. To complete Los Mutts, Lorraine embarked on a four-year journey through Latin America in an old Chevy van once used as a surveillance vehicle by the Arizona police, guarded by her Kenyan born ex-street dogs. The pack then hunkered down in a cabin overlooking Chile's Pacific Ocean and Lorraine consolidated her knowledge with what she'd gleaned speaking with other dog travelers, to create Dogs Without Borders\u2014Tales and Tips from the Road.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you had told me five years ago that I would journey from my hometown of Phoenix to New York City almost entirely under my own power, I would have called you crazy. Now, having completed that same journey, it's crazy I didn't do it sooner.\nOn May 23rd, 2017, after quitting my job, selling every possession that couldn't fit on my back, and saying goodbye to all my friends and family, I walked out of my world toward the goal of NYC. For three months I walked and hitchhiked from Phoenix to Denver, where I, needing more money to fund my trip, settled down until the Spring of next year. From there, I saddled up on a bicycle and rode nearly 2,000 miles from Denver all the way to New York City, in just under two months. In total, I was away for 14 months.\nThe biggest question that has been dogging me the entire way, especially whenever I talked to people about my adventure, is \"Why?\" It's a very powerful question. It's one that has no clear beginning, and an even less clear end. If I were to take it all the way, I reckon the beginning of the answer started when I was a small child, or maybe even before I was born. And the end? One end could be when I finally made it to New York City. Another could be when I flew home to Phoenix. But part of me knows the answer is still becoming itself, still manifesting itself in the world. The \"why\" still lives on and drives my actions to the very moment I'm typing these words on my laptop. But for simplicity's sake, let's say the question encompasses from the moment I began to plan the adventure, to the moment I arrived back home in Phoenix.\nI struggled with giving an answer to anyone who asked the question for a very long time. Some times, when I had longer conversations, I was able to explore a bit and say things that I thought might be true, but the truth is that I really didn't know. It's only after reaching my goal and looking back at the road I took that I can understand why I left. I was lost and I was trying to be found.\nI was never really successful as a child and teenager. I never tried to win anything, never having enough passion or motivation to climb to the top of a hierarchy in games, schooling, or any interest I half-pursued. I exited high school not having the faintest idea who I was or what I was capable of, and I very quickly became lost and aimless. In short, I was a loser. Over the course of the next six years, I built a life for myself that wasn't one I wanted and didn't fit my own needs. It was one built out of confusion, desperation, and incompetence. Eventually, there came a time when tragedy struck, and my life began to crumble. Worst of all, I found I wasn't strong enough to resist crumbling away along with it.\nI wandered about in a malaise, hopeless for the future. It wasn't until I was 24 that I realized I had to journey into the unknown to find out who I was. Like most young men, I had always dreamed about going on a big adventure, and like most young men, I never nurtured that dream into reality. But I decided I wasn't going to be like most men. Whether I was right or wrong in my belief that I had to leave my home in order to find myself, I knew I needed a direction, a purpose, and obstacles to overcome. I decided to do the longest, most difficult thing I could possibly handle at that point in my life, and that thing was walking to New York City.\nWhy New York City? It was far enough away destination to be considered an \"epic\" adventure, and if I was aiming to walk to the east, why not aim for the greatest city in the world? In November of 2016, I decided to give myself six months to save money and prepare to depart. At first, it seemed like just another flight of fancy that would disappear as time marched on. But as the departure date grew nearer, it began to become real, and many times I was wracked with doubt that I would actually go through with my intentions. Much like Bilbo Baggins in The Lord of the Rings, I revealed my departure plans to no-one, save a trusted few, and it wasn't until I quit my job and sent out invitations to my going-away party that anyone had any idea what was happening. I reasoned that, the more people I told, the more I would disperse and dilute the idea, and I would be inviting criticism, opinion, and differing ideas to invade my own. I knew I had to do this my own way, so I kept it a secret until it was too late for anyone to do anything about it. And when the time came, I answered the call of adventure, and left.\nThe rest has been detailed more-or-less in my previous entries, and the 2,000-mile bicycle trip is an experience I will write about very soon. But here in the now, after 14 months of my life, thousands of dollars spent on travel, experiencing hardship and suffering, was my dream worth it?\nYes. Every step of it. There came a moment, when I finally reached New York City, that gave everything prior to it a real meaning. I was riding the ferry that crossed the harbor northward from New Jersey to New York City, standing at the aft of the vessel and enjoying the sea-wind in my hair and the spray upon my skin. I was smiling broadly at the world around me, content that I had nearly arrived at my goal I had spent so long trying to reach. As I came around to see the city on the starboard side, the immense height of the Freedom Tower came into my view. As I looked at it, the sun turned its huge triangular face into a golden sheet, and I was bathed in the warmth of a blinding light. I closed my eyes, and in an instant it was gone. It was as if the city itself had acknowledged my entrance, and anointed me with light from its highest tower. You could say many things about the inevitability of the path of the vessel, the time of season, and the angle of the sun all contributing to the illusion, but none of that matters, because I was there.\nIn that moment, I felt I was complete and whole. I felt I could go out into the world and face challenges with some level of courage and contend with the perils of life, because most things we do are far less difficult than journeying 3,000 miles by foot and by bike. I have a direction and a purpose, and anything I do from now on is going to be towards the future I'm trying to build. If that's not the essence of finding yourself, then I don't know what is.\nSome of us are never lost. They are strengthened and guided by loving hands on the straight and narrow path. Some of us have simply taken the wrong path, and only need to be shown the right way to get back on track. And some of us are abandoned in the forest, lost in the brush without a way out and no one to guide them. For those of us who are lost, it may be that the only way to be found is to pick a direction to journey into, and stay true until you reach the sea. Who knows? It worked for me.\nI am so glad you made it! I have thought often about how your trip was going. I have to admit that when we met in Brookfield, Missouri, and your tire was shot, I had my doubts. You have a accomplished something special, not just the physical trip but also the emotional journey. Good job and God bless you.\nThank you David! I wouldn't have been able to do it without you. I hope all goes well for you, your family, and your congregation. Best wishes!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Everything You Know is Wrong\", \"Weird Al\" Yankovic's loving tribute to They Might Be Giants, is one of those songs where Al gets everything right. I like it so much that it's not just one of my favorite \"Weird Al\" Yankovic songs. In a weird sort of way, it's one of my favorite They Might Be Giants songs as well, in the same sense that my brain thinks of \"Dare To Be Stupid\" both as a \"Weird Al\" Yankovic song and a Devo song someone other than Devo just happened to have written and recorded.\n\"Weird Al\" Yankovic and They Might Be Giants are, of course, towering colossuses of the comedy music world. \"Weird Al\" Yankovic and They Might Be Giants have each occupied valuable real estate in the fertile, suggestible minds of multiple generations of young geeks, for whom they aren't just favorite musicians but personal heroes.\nLike David Bowie and The Talking Heads for weird kids a little later in their development, \"Weird Al\" Yankovic and They Might Be Giants serve as guides through the uncertain currents of childhood and adolescence who let misfits and kooks and outcasts know that it is perfectly alright to be, well, weird. In fact, it's actually far superior to the alternative.\nI went through a They Might Be Giants phase and when I worked for The A.V. Club, and visiting minstrels would bow down to kiss the ring of our collective power whenever they were in town, I missed the opportunity to be part of a group of The Onion staffers performing back-up vocals for their cover of Chumbwumba's \"Tubthumping\" for a video series filmed in the office.\nLater that cover made one of They Might Be Giants' b-sides and obscurities collections, so I missed out on an opportunity to sing back-up on a They Might Be Giants officially released track. Don't weep too much for me, dear reader, as I got to sing onstage \"Tubthumping\" and \"Horse With No Name\" with They Might Be Giants and a whole bunch of my former colleagues in the Vic theater in Chicago, where Al will actually be performing this April during his upcoming tour. Incidentally, Chumbawumba has also recorded a song called \"Everything You Know Is Wrong.\" Does that mean anything? No, it does not. Just some weird coincidences.\nI have unusually deep connections to \"Weird Al\" Yankovic and They Might Be Giants but they are towering figures in the imaginations of growing geeks everywhere. \"Everything You Know Is Wrong\", like \"Dare To Be Stupid\" is the perfect amalgamation of Al and his inspiration. It's like the movie The Fly and Al and They Might Be Giants' DNA got all mixed up, only instead of it resulting in a horrifying fly monster that spits acid and is utterly repellent, one of the most beautifully constructed and just plain perfect songs in Al's discography ensued.\nChanneling the spirit of They Might Be Giants liberated Al from the prison of rhyming. Instead of being limited to rhyming couplets, words spill out here in a rapid-fire stream-of-consciousness ramble in verses that start out gleefully absurd (the singer begins the song driving down the freeway but with a rabid wolverine in his underwear) and then get more violent and demented before a mysterious figure (alternately, a prosthetic-lipped Hibachi dealer, the floating disembodied head of Colonel Sanders and Saint Peter) up-ends their already fractured sense of reality with the titular burst of hard truth.\nAh, but this is a particularly silly \"Weird Al\" Yankovic song so death isn't anything near an end. He ends up having a complicated relationship with Saint Peter, who does not care for his Nehru jacket (it's strongly implied that it violates heaven's dress code) so he lets him stay in paradise, but next to an ice machine, and every day screams out the titular warning.\nColonel Sanders, or more specifically, the floating disembodied head of Colonel Sanders, is a quintessentially Al reference to drop, as Sanders had the curious distinction of being at once a real human being and businessman who ran Kentucky Fried Chicken and a sort of living television and print commercial.\nColonel Sanders was a real dude but at a certain point he became a more or less fictional mascot and TV pitchman. It's a testament to how casually surreal television advertising has become that, for reasons known only to Kentucky Fried Chicken, they've decided to have a never-ending string of actors and comedians cycle rapidly through the role like some weird, fried chicken-themed version of Todd Solondz's Palindromes.\nOver the last two years, Colonel Sanders has been alternately portrayed by Darrell Hammond, Norm Macdonald, Jim Gaffigan, George Hamilton, Rob Riggle, Billy Zane and Rob Lowe. That is insane, and nobody has made a big deal about it. Granted, it's not quite as unusual as going to the kitchen for some Golden Grahams and being sucked into an alternate dimension where you travel through time and encounter the floating, disembodied head of Colonel Sanders, but it is pretty strange when you think about.\nLike the Oingo Boingo pastiche \"You Make Me\", there's something vaguely Seussian about the way the music moves here, a sense that it might be the product of some stop-motion animated Rube Goldberg machine or careening musical rollercoaster. Al captures the Technicolor absurdity and quirkiness of the They Might Be Giants aesthetic perfectly here, as well as the runaway sense of momentum that characterizes their most infectious songs.\nLike \"Dare to Be Stupid\", \"Everything You Know Is Wrong\" is aggressively nonsensical and undeniably anthemic. It's the perfect fusion of Al and the two Johns. It underlines how much these two acts have in common, while also establishing what makes them unique and also national treasures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pack of 10 x 20mm Mounted Felt Point.\nThese quality mounted felt points with a 6mm shaft are ideal for polishing. Versatile across many mediums, these medium density points will work on stone, ceramics, metals, as well as jewellery. Bonded onto a shaft that fits most rotary tools (including the Dremel and the Ozito) these felt points are popular and economical method of polishing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbpjj b/data_all_eng_slimpj/shuffled/split2/finalzzzbbpjj
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@@ -0,0 +1,5 @@
+{"text":"Home- Ecommerce Website Design- What is Ecommerce?\nEcommerce is the buying and selling of goods or services, or the transmission of funds or data over an electronic network. The internet is the primary platform for this kind of activity.\nThese transactions can occur from business to business, business to consumer, consumer to consumer, or consumer to business - a number of sites exist to facilitate these interactions across a wide range of sectors and industries.\nAccording to IMRG, the UK represents Europe's largest ecommerce market, with \u00a3114bn worth of sales in 2014 alone. In certain seasons, ecommerce performs even better than that, with 2015 online Christmas sales hitting \u00a324bn. The IMRG estimates that around 27% of all retail sales now take place online, a statistic that has been impacted dramatically by the ever-increasing rise in mobile devices.\nThe growth in ecommerce has been driven in part by improved access, but has also been driven by a determined effort by businesses and retailers to create online spaces for customer interaction. Because of this effort, it is possible for customers to make purchases online that they previously would have had to make in person in a shop.\nTravel and healthcare are examples of industries that have moved into ecommerce in recent years; they form just a part of the range of sectors that have contributed to the ubiquitous nature of online services and sales.\nAs time goes on, and as mobile devices continue to rise in prominence and availability, it is likely that more and more of the retail market will migrate into e-platforms.\nAs mentioned, ecommerce - or e-business, as it is sometimes known - is a term that relates to any transaction that takes place on an electronic platform. Ecommerce does not only include ecommerce websites, but can also include channels like email marketing.\nEcommerce is a sector frequently used both by traditional retailers, as well as by newer, purely online marketplaces. Indeed, in the modern world of business you are likely to find that most high street retailers also operate an online platform. Even if sales are not directly available on the high street retailer's website, there are often enquiry options present which can lead to a purchase anyway. Perhaps just as importantly, by having an online space the retailer is able to strengthen their brand.\nAlthough it requires certain resources, like website designers or developers, for companies with stock and the ability to operate via an online platform, ecommerce has incredible potential to be a lucrative source of revenue and brand identity.\nEcommerce is a platform that offers a number of advantages and benefits to both businesses and consumers. Not only popular for the ease of access that it offers, ecommerce is also thriving due to a range of other factors.\nEase of access - Not only is it becoming easier for consumers to search for and buy products, it's also becoming easier to have these items delivered.\nPrice - Although the costs of ecommerce do exist for businesses, there is also space to save money - an ecommerce store is almost certainly going to be cheaper than a chain of brick-and-mortar shops, and staffing is likely to be less expensive.\nSavings - Consumer can almost always save money online simply because it's so easy to shop around for the best deal.\nThe benefits of ecommerce are not just there for businesses, but are there for customers too. Perhaps the best indication of these benefits is the manner in which ecommerce has continued to gain prominence in recent years.\nEase of access - Not just a benefit, ease of access can often give ecommerce stores an advantage over brick-and-mortar rivals. Having an online presence that is properly maintained can mean gaining customers that might otherwise visit a competitor in person.\nReach - Whereas your high street store is limited to just the people that come to it, your ecommerce store can deliver as far as you'd like it to. It is even possible to launch into international markets, although it's worth bearing in mind that the costs of delivering overseas will be significantly higher than local delivery.\nSetting up - It would not be right to assume that you can set up a fully-functioning online business without any effort, but it is worth bearing in mind that setting up an ecommerce store is easier than setting up a brick-and-mortar one in many ways. For example, an ecommerce store is not limited by location - you could even start one from your bedroom if you wanted. You probably will not have this level of freedom with a traditional retail store.\nPerhaps the biggest ecommerce advantage for businesses is the rise in accessibility that it offers. Unlike a high-street store, an ecommerce store has the capability to cater to a global audience, and can do so without limit.\nReal-life stores have capacity issues that could limit the number of customers served, however an ecommerce store with a quality website hosting service can keep selling without any limit on their ability to accept customers.\nStarting your own ecommerce website requires a number of things, and so it is vital that you are aware of the best ways to proceed before setting out.\nWhat is your Ecommerce Website for?\nAs with any new business, it is vital that you understand your industry - it's also vital that you are passionate about it. If you enjoy what you do, you'll be far less likely to grow tired of it, and you're far more likely to create and manage a successful, profitable business.\nThese three point may seem obvious, but far too many people attempt to start businesses without the proper planning and research. If you don't know your audience then you will struggle to sell to them, even if you know the products inside out; likewise, if you know your audience inside out, but don't know the products, then you're just as likely to fail.\nFor more advice, visit our ecommerce web design ideas page.\nThe biggest businesses, including ecommerce businesses, thrive because they fill a void. Operating in a niche market can be very lucrative, but it will be difficult if you are competing with a range of other established businesses. Look for gaps where you think you can provide a unique service and then tailor your site to this gap.\nIt could be that you want to create a website that caters to anglers, but you can see that there are a lot of businesses already competing in this market. Are they missing anything? Is there a service or product that you can specialise in that they haven't got?\nYou might think that selling the biggest range of goods is the best way to make your site profitable, but the truth is that sometimes a website that sells just one or two products can do very well. In these cases, becoming an expert in one area can provide a lot more sales potential than if you were selling several, non-specialised products or services.\nWhat does my Ecommerce site need?\nIt's easy to assume that you could just make a site that performs, but there are a number of considerations that you need to make.\nWill you be actively selling products on your site, or just providing contact information so that the sale can take place over another channel?\nIf you mean to make sales directly on your ecommerce website, then it will need to be designed with this kind of functionality in mind; a website that sells products through an integrated system will require specialised software and catalogues to enable the sales process.\nHow often will you need to update or edit your site or products?\nIf you are a specialist ecommerce store, then you may only have one or two products and editing may not occur very often. However, if your website features a range of products, then the chances of updates will rise. If editing is a feature you will likely need, then a CMS will be preferable.\nHow many selling options will the site have?\nSome sites have a large product catalogue - in some cases, this catalogue will also include variations. For a clothing ecommerce store, for example, one shirt could come in six colours meaning that, even if you only have pages for 100 products, these 100 products will in turn need to display as many options as required, with the option for customers to buy whichever of those options they actually want or require. As above, if this is the case it is likely that you will need a more complex site with this kind of functionality built into it.\nIf you are interested in starting an ecommerce store, then looking into design options is an excellent place to start. Approved Index can help here. Fill in the form at the top of this page and answer a few simple questions and we can match you up with vetted, experienced ecommerce website designers who can help your plans a reality.\nFill in the form now to find the best quotes for you.\nLet Approved Index help you find the best Ecommerce Web Designer for your business.\nBuild Your Own or Outsource?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Internet marketing maintains its rapid development and growth, our company continues to meet the challenge head-on bydeveloping new technologies and innovative marketing tools. Our goal remains the same: To provide the best return on your advertising dollar through the delivery of quality and affordable Internet marketing services.\nTo meet the constant challenge of converting visitors into buyers, we proudly introduce our Contextual Targeting technology which targets your visitors more closely to the product or service you are marketing.\nWith our innovative ideas and continued efforts, Clicksor.com Inc is proud to introduce you to Contextual Targeting technology. Our ultimate challenge is to significantly improve the likelihood of converting any visitor into a buyer and we achieve it by targeting the visitors more closely to the product or service you are marketing.Clicksor.com is one of the most cost-effective ways to reach your targeted customers. You determine search terms (keywords) that are relevant to your website's content, and then select the bidding price to rank the placement of those keywords. The higher your keyword is ranked, the more likely that qualified prospects will visit your site and buy your product or service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fourteen years ago, sustainability and aerosols would have been unlikely to have been seen by most people as a natural partnership. Today, Jonathon Porritt, the campaigning British environmentalist, perhaps best known for his championing of Green issues, his advocacy of the Green Party and Founder Director of Forum for the Future, has written the foreword to 'Aerosols in Figures'.\nJonathan writes \"Aerosols have a number of sustainability benefits: they are generally made of high quality, recyclable metal; they avoid any excessive use of preservatives, as they keep products clean and hygienic; they avoid waste and spills; and the latest developments in compression and reduced pack sizes are laudable. In addition, BAMA has played a very positive role in promoting recycling, both at the kerbside and at council recycling centres\".\nHe also recognises the valuable contribution our industry plays in the economy. \"It's worth reminding people that the aerosol industry in the UK is also a manufacturing success story, with significant exports which continue to make a strong contribution to the UK economy. None of which provides any excuse for complacency!\", he adds.\nThe green credentials of the sector have been continuously improving over time. Fourteen years ago, just over a quarter of Local Authorities were recycling household rubbish (27%) with fewer than 7% including aerosols in their recycling schemes.\nWe are proud that, today, over 96% of Local Authorities recycle aerosols; and many have got behind our campaigns to encourage more consumers to include empty aerosols in household recycling efforts.\nBy 2006 \u2013 75% included empty aerosols.\nBy 2012 \u2013 87% included empty aerosols.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 3D design gives the bar stool a comfortable seat and a sense of lightness \u2013 with all edges pointing away from the user. With the wide range of seating shells in veneer and in HiRek and numerous bases, the GUBI Bar Stool allows a unique and personal expression for both the private home or restaurant.\nDimensions: Height 88cm (seat height 75cm) x Width 42cm x Depth 43cm.\nMaterials: Interjected Moulded HiRek seat shell with steel legs.\nMany more colour options available. Please enquire for further details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New black comedy of the infamous Victorian duo performed by three actors.\nMurder is afoot in the quaint village of Bagnor. A pair of infamous villains have set up residence at The Watermill as part of the theatre's 27th rural touring production. The play tells the true story of William Burke and William Hare who terrorised the city of Edinburgh in the 1820s in their pursuit for bodies they could sell on for medical research.\nBurk and Hare is a black comedy told with a cast of three talented actors, some musical instruments, and a lot of hats. Since it is based on a real-life story which resulted in over 16 murders, this show is not necessarily for the squeamish but is sure to be a devilish hoot.\nBurke and Hare is the very definition of a delicious theatrical treat. Imbued with a darkness that comes from the play's source material, and with the ability to pluck delicately at our heartstrings one moment before delivering a thick slice of uproarious farce in the next, this is another rollickingly good night at the Watermill.\nCentral to the appeal of Burke and Hare are a trio of performers who bring exuberant confidence and energy to the show, meaning the evening's momentum never falters. Katy Daghorn makes a sympathetic and heart-wrenchingly naive Helen McDougal (Burke's love interest) and a ferociously devious Mrs Hare, as well as a plethora of other memorable parts. Haydn Wood, as Burke, is surprisingly charming and likeable, considering he spends the play killing off a huge swathe of Edinburgh's population. And as repugnant as Alex Parry's Hare is, the actor shows a mastery of taking on multiple parts in quick succession, including an entire family in one particularly funny scene. The production feels gifted to have such a talented triptych at its core, who weave their charm around the audience in some well-judged incidences of interaction with us.\nTom Wentworth's witty script and Abigail Pickard Price's pacy direction are to the show's advantage, meaning proceedings fly by with several memorable moments. There is a wonderful lyricism to the play, with tuneful folk songs used throughout. Another standout element is Toots Butcher's ingenious design. Burke and Hare is a touring production, travelling to village halls, and so the set has a deceptive simplicity to it. However it finds useful ways to store props and musical instruments for later use and has several panels that are used in an amusingly effective manner. A great deal of fun comes from the surprises the set has in store for us.\nAt times this show is a sort-of proto-Sweeney Todd, and if the play doesn't quite reach those dizzying heights it is not for lack of trying. It is fast, funny, at times sad, and, when Mrs Hare is at her most manipulative, rather tense. Aided by an energetic cast brimming with talent, the production makes a convincing case for productions that tour to areas removed from established theatres. Bravo to the Watermill Theatre for another stellar show.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..181fecfa85685fd99a466065cb6d3c565710b777
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+{"text":"For claims to succeed, you can't use deleted CPT\u00ae codes. There's no getting around it. So give your claims their best chance in 2019 by making sure these deleted codes don't land on your claims.\nAnd of course: You need to know the new codes (see blog post overviews Part 1, Part 2, Part 3) and revisions. Some of the deletions are covered there, as well, but it doesn't hurt to discuss the changes from multiple angles. The codes below are Category I codes with specific replacement options, but there are other codes that won't make it into the 2019 AMA CPT\u00ae code set. Make sure you have a resource with the complete official codes, descriptors, and guidelines, when the 2019 CPT\u00ae code set, is effective on Jan. 1, 2019.\nIn 2018, you have 10022 for fine needle aspiration (FNA) with imaging guidance. If you code FNA, you probably have that code memorized. But you won't find that code in the CPT\u00ae 2019 code set. Instead, you'll be using 10005-10012, which will specify the type of imaging guidance used.\nIf you report biopsy, be sure to remove 11100 and 11101 from your job aids. In 2018, those codes apply to skin, subcutaneous tissue, and mucous membrane biopsies. But in 2019, you'll choose from 11102-11107, which allow you to give more specific information about the biopsy type, such as tangential, punch, or incisional.\nIncorrect use of knee arthrography injection code 27370 is leading to its deletion in 2019. The AMA doesn't specify a cross-code with 27370 in the early release file. But you can expect new code 27369 for an injection procedure for contrast knee arthrography or contrast-enhanced CT\/MRI knee arthrography.\nFor a gastrostomy tube without guidance, steer clear of 43760 in 2019. Check instead to see if new code 43762 or new code 43763 will fill the bill. They apply to percutaneous gastrostomy tube replacement. You'll use 43762 when the tract does not require revision and 43763 when the tract requires revision.\nIf you code for percutaneous guide introduction and dilation for nephrostomy tract, you know 50395. But in 2019, you'll need to know 50436 and 50437. Both codes cover existing tract dilation and imaging guidance. The difference is that 50437 applies when the procedure includes new access into the renal collecting system.\nBreast MRI codes 77058 and 77059 are on their way out, ready to be replaced by four new codes, 77046-77049. The first two new codes apply when no contrast material is used. The final two codes apply when the breast MRI is performed without and then with contrast. Computer-aided detection is included in 77048 and 77049, but it isn't required. Just like the 2018 codes, the 2019 codes differ based on unilateral or bilateral.\nPath\/lab coders need to be aware of the deletion of BRCA1, BRCA2 gene analysis codes 81211, 81213, and 81214. This is all part of an update of the BRCA1, BRCA2 codes, including revision of 81162, 81212, and 81215-81217. There will also be new codes, 81163-81167.\nCPT\u00ae 2018 code 81214 (BRCA1 full sequence analysis and common duplication\/deletion variants) will most likely cross to new codes 81165 (BRCA1 full sequence analysis) and 81166 (BRCA1 full duplication\/deletion analysis).\nThe single 2018 electroretinography code 92275 is also on the soon-to-be-gone list. In its place, there will be three new codes. You'll use 92273 (full field), 92274 (multifocal), and Category III code 0509T (pattern).\nInstead of deep-brain codes 95978 (first hour) and 95979 (each additional 30 minutes), expect to use new codes 95983 (first 15 minutes) and 95984 (each additional 15 minutes).\nAre you starting to get a handle on the AMA CPT\u00ae 2019 code changes that will affect you? What's your favorite way to prepare?\n*What is the Sum of 1 + 7 ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Virginia Tech was selected to participate in the Orange Bowl after an 11\u20132 regular season that culminated with a 44\u201333 win in the 2010 ACC Championship Game. Stanford was picked as the other half of the matchup following an 11\u20131 campaign that included the school's best-ever regular-season record. That performance earned the Cardinal a No. 4 ranking in the BCS Poll and the automatic bid to a BCS game that accompanies a top-4 ranking of a second school in a conference other than the champion. In the weeks before that game, media attention focused on both teams' turnarounds from historical difficulties and the performance of Stanford quarterback Andrew Luck. The game also was the first Orange Bowl not sponsored by FedEx in 21 years, ending the longest-running title sponsorship deal among the major bowls.\nIn recognition of his performance during the game, Stanford quarterback Andrew Luck was named the game's most valuable player. He set a Stanford bowl-game record for touchdowns, and threw three of those scores to tight end Coby Fleener, who set a Stanford and Orange Bowl record with 173 receiving yards. Both teams made coaching changes after the game, as Stanford head coach Jim Harbaugh left the team to coach the National Football League's San Francisco 49ers and Virginia Tech replaced several assistant coaches. Players from each team were selected in the 2011 NFL Draft.\nThe Orange Bowl is one of five Bowl Championship Series (BCS) bowl games that have been played at the conclusion of every college football season since 2006.[nb 2] As defined by contract, the bowl matches the champion of the Atlantic Coast Conference (ACC) against an at-large pick chosen by a special committee. On December 4, 2010, the Virginia Tech Hokies defeated the Florida State Seminoles in the 2010 ACC Championship Game, thus winning an automatic bid to the 2011 Orange Bowl.\nThe at-large spot in the Orange Bowl was filled via a round-robin selection procedure defined by the other Bowl Championship series games (the Sugar, Fiesta, and Rose bowls) and the automatic bids. If a game's automatic bid team is selected for the BCS National Championship Game, it is allowed to select a replacement team. In the 2010 season, teams otherwise designated for the Rose Bowl and Sugar Bowl were picked for the national championship game. Following the two compensatory selections are the at-large picks. The order of at-large selections rotates annually among the BCS bowls.\nFor the January 2011 games, the Sugar Bowl selected first, followed by the Orange Bowl and Fiesta Bowl. The Sugar Bowl picked Ohio State, which left the Orange Bowl to decide between two teams contractually bound to appear in a BCS bowl game. These were Stanford, which was guaranteed a BCS bid by its No. 4 national ranking, and Connecticut, champion of the Big East Conference. Favoring Stanford's potential to draw a large television audience, the Orange Bowl selected the Cardinal, leaving Connecticut to the Fiesta Bowl.\nThe Stanford Cardinal began the 2010 season after going 8\u20135 in 2009 under freshman quarterback Andrew Luck. The 2009 season saw the Cardinal defeat eighth-ranked Oregon and No. 9 USC en route to breaking a seven-year bowl game drought with an appearance in the 2009 Sun Bowl against Oklahoma. Stanford changed several position coaches and its defensive strategy in the offseason, and though the Cardinal returned 17 of 24 possible players from the previous year's starting lineup, few believed the Cardinal would be among the best teams in the country. In the annual preseason poll of media members covering the Pacific-10, Stanford was picked to finish fourth in the 10-team conference. Nationally, Stanford appeared in the \"also receiving votes\" categories of the preseason polls. In the AP Poll, the team was 27th; in the coaches' poll, the Cardinal was 32nd.\nThe Cardinal opened the season by defeating Football Bowl Subdivision team Sacramento State, then beat conference opponent UCLA in the first Pacific-10 game of the season. The win over Sacramento State pushed Stanford to No. 25 nationally, and the win over UCLA raised Stanford to No. 19, the rank it held when it defeated Atlantic Coast Conference foe Wake Forest 68\u201324 on September 18. The following week, Stanford defeated Notre Dame as the nation's No. 16 team, and the win raised the Cardinal to No. 9.\nStanford players lift the Stanford Axe after winning the 2010 Big Game against the California Golden Bears.\nOn October 2, No. 9 Stanford played No. 4 Oregon. The game was touted as a matchup between two of the top offenses in college football and was nationally televised. Stanford led 21\u20133 after the first quarter and 31\u201324 at halftime, but the Cardinal allowed 28 unanswered points in the second half and lost to Oregon, 52\u201331. Oregon later went on to play in the BCS National Championship game, and the defeat was Stanford's only one of the season. The loss dropped Stanford to No. 16 in the polls, but the team rebounded the following week to defeat USC on a last-second field goal, 37\u201335.\nThe victory over USC began a winning streak, the best in Stanford football history since the time of the Korean War. The week after beating USC, Stanford defeated Washington State, moving to a 6\u20131 record for the first time since 1970. The Cardinal rose to No. 13 nationally and defeated the University of Washington on October 30. Against No. 15 Arizona, Stanford matched its previous season's win total. On November 13, Stanford defeated Arizona State in a closely fought 17\u201313 win that brought the Cardinal football team to 9\u20131 and its best record since 1951. The following week, Stanford beat traditional rival California in The Big Game by a 48\u201314 score. It was the largest margin of victory in the rivalry since 1930. Stanford's final regular-season game was against the Oregon State Beavers, and Stanford won the contest convincingly, 38\u20130. The No. 4 and No. 5 nationally ranked teams had lost earlier the same day, lifting Stanford in the polls and in position for a Bowl Championship Series game berth.\nThe Virginia Tech Hokies entered the 2010 season after a 2009 campaign that saw the team finish 10\u20133, including a season-ending win in the 2009 Chick-fil-A Bowl against the Tennessee Volunteers. Because of that season-ending victory and the Hokies' general good performance during the 2009 season, Virginia Tech was ranked No. 10 in preseason national polling. The Hokies' first game of the season was a nationally televised contest against then-No. 3 Boise State at FedExField near Washington, D.C.. Because the game was the first of the season to feature two top-10 teams, it received large amounts of media coverage. During the game, Virginia Tech fell behind 17\u20130 in the first quarter, but rallied to take a 21\u201320 lead early in the third quarter. The two teams traded the lead, alternating scoring drives until Boise State scored a touchdown with 1:06 remaining. Virginia Tech was unable to reply one final time, and Boise State earned a 33\u201330 victory.\nThe close loss discouraged the Virginia Tech players, who then had only five days to prepare for their next opponent, lightly regarded James Madison University. At Lane Stadium, Virginia Tech's home field, James Madison upset the heavily favored Hokies, 21\u201316. The loss was only the second time in college football history that a team ranked nationally was defeated by a team from the NCAA Football Championship Subdivision. In the wake of the loss, Virginia Tech fell from No. 13 to out of the polls entirely.\nFollowing the loss, seniors on the football team held a players-only meeting in an effort to rally the team. Players later recalled that meeting as the turning point in the team's season. The next week, Virginia Tech earned its first win of the season, a 49\u201327 victory over East Carolina in Lane Stadium. The Hokies followed that by traveling to Boston College for a 19\u20130 win, their first shutout since 2006.\nThe two victories were the start of a winning streak that saw the Hokies complete the regular season without another loss. Following Boston College, they defeated No. 23 NC State in its home stadium, 41\u201330. They defeated nonconference opponent Central Michigan, then beat conference opponent Wake Forest and re-entered the polls at No. 25. Their position in the polls climbed with each opponent they defeated. They beat Duke as the No. 25 team, Georgia Tech as the No. 23 team, North Carolina as the No. 20 team and No. 23 Miami as the No. 16 team. The Miami victory clinched Tech the division championship and a slot in the ACC championship game, but the Hokies still won their final previously scheduled game, the annual Commonwealth Cup rivalry against Virginia. In the ACC Championship Game, Virginia Tech defeated Florida State 44\u201333, clinching the ACC's automatic bid to the Orange Bowl.\nIn 2008, sports channel ESPN won the right to broadcast Bowl Championship Series games between 2011 and 2014. In compensation, it agreed to pay the BCS $125 million per year, more than a competing bid from Fox ($100 million), which had been paying $82.5 million per year for the contract that ended in 2010. To balance the higher cost of broadcasting the games, ESPN demanded more money for title sponsorship of each game. Shipping company FedEx, which had become the Orange Bowl's title sponsor in 1990 and was the longest continuous sponsor of any bowl game, balked at ESPN's increased demands\u2014about $20 million per year and required advertising outside the Orange Bowl\u2014and ended its sponsorship.\nCiti, the title sponsor of the Rose Bowl, likewise ended its sponsorship over the increased cost. ESPN immediately began pursuing an alternative title sponsor for each game. Candy manufacturer The Hershey Company was an early possibility for the Orange Bowl, and the company considered titling the game the Reese's Orange Bowl, but negotiations fell through. In August, ESPN reached a deal with credit card company Discover Financial to make the game the Discover Orange Bowl through 2014.\nPregame media coverage of the 2011 Orange Bowl focused on the manner in which the two teams involved had reversed their fortunes during the 2010 college football season. Stanford had its longest winning streak since 1991 its most wins in history, and was four years removed from a one-win season, while Virginia Tech became the first team in college football history to win 11 consecutive games in a season after losing its first two. The two teams had never played each other, but they shared a common 2010 opponent, Wake Forest. Before the Orange Bowl, Wake Forest head coach Jim Grobe said he didn't see much difference in quality. Another point of conversation was the poor performance of the ACC in nonconference games and Virginia Tech's poor past performance against teams ranked in the top 5. The Hokies had won just one of 27 games against teams in that category. Partially because of this factor, spread bettors favored Stanford by three points when the first odds were released December 10, a margin that various organizations either kept constant or raised to 3.5 points by the day of the game. It was the first time all season the Hokies were point spread underdogs.\nTwo nights before the game, Virginia Tech running back David Wilson and safety Antone Exum missed a 1 am curfew. As punishment, Virginia Tech head coach Frank Beamer suspended the two players for the first quarter of the Orange Bowl.\nVirginia Tech and Stanford each were allotted 17,500 tickets to sell through their school box offices, but each school had problems selling that amount. For Stanford, whose primary campus is in California, the distance fans were required to travel was a major obstacle. Stanford head coach Jim Harbaugh said, \"It will probably be more Hokie fans there than Stanford fans. That far away from home it probably could be a hostile environment for us.\"\nOrange Bowl organizers predicted that Virginia Tech fans, who were located closer to the site of the game, would be attracted by its proximity and the warm weather offered by southern Florida. Delta Air Lines added more flights from Virginia to Florida in anticipation. To entice more fans to attend, tourism officials planned to step up their advertising in order to boost the economic impact of the event, estimated in 2009 at $200 million. The advertising campaign was a difficult sell because the game was scheduled for the first Monday after New Year's Day, the date when most American workers return to their jobs after the New Year.\nPartially because of this fact, ticket sales were slow. By December 23, Tech had sold just 6,500 of its 17,500-seat allotment, while Stanford fans had purchased 9,000 tickets from their allotment. These figures increased slightly as the game day approached, but the schools were hampered by several factors. The tickets assigned to the schools were in less-desirable seats, and many seats were available far more cheaply in the secondary market. In one example, an upper-deck ticket sold by Virginia Tech cost $65, while a similar ticket was available for $12 through an online ticket seller. The two schools were forced to purchase the remaining unsold tickets, costing each several hundred thousand dollars.\nQuarterback Andrew Luck, seen here in August 2010, was the critical component of the Stanford Cardinal offense before and during the 2011 Orange Bowl.\nStanford's offense was commanded by redshirt sophomore quarterback Andrew Luck, who completed 70 percent of his pass attempts, gaining 3,051 yards, 28 touchdowns, and 7 interceptions. The touchdown mark was a Stanford single-season record. He also gained 438 yards running the ball, setting a Stanford record for rushing by a quarterback, and his combined rushing and passing total set a record at Stanford for total offense. On December 13, Luck finished second in the voting for the Heisman Trophy, the annual award given to the best player in college football that season. NCAA rules allow a player three years removed from high school to enter the NFL Draft, and even though he was only a sophomore in 2010, he met the three-year standard at the conclusion of the season. Before the Orange Bowl, media and fans speculated as to Luck's likely choice.\nLuck distributed his passes relatively evenly among his receivers. Doug Baldwin was the team's leading receiver, with 56 receptions for 824 yards and 9 touchdowns during the regular season. Wide receiver and kick returner Chris Owusu, who played in only six games because of a knee injury, underwent arthroscopic surgery and was expected to play in the Orange Bowl. He was the team's fourth-leading receiver in terms of yardage, catching 24 passes for 394 yards and 3 touchdowns.\nStanford's running game was led statistically by sophomore running back Stepfan Taylor, who garnered 1,023 rushing yards and 15 touchdowns during the regular season. Before the game, Cardinal head coach Jim Harbaugh said he planned to give three other running backs\u2014Tyler Gaffney, Jeremy Stewart, and Anthony Wilkerson\u2014opportunities to carry the ball when Taylor was taken out of the game.\nVirginia Tech's success at blocking kicks during the 24 years of head coach Frank Beamer's tenure caused Stanford to spend extra time in pregame practices on kick protection.\nBecause of his success in guiding Stanford's offense, head coach Harbaugh was mentioned as a candidate for the vacant head coaching positions of other universities and National Football League teams. In the leadup to the Orange Bowl, however, Harbaugh refused to comment on any of the possibilities and said he wanted to focus on guiding his team to victory. Stanford players echoed that line. \"We've invested too much to let something like that distract us,\" linebacker Shayne Skov said December 30.\nVirginia Tech quarterback Tyrod Taylor was the 2010 Player of the Year in the Atlantic Coast Conference.\nVirginia Tech quarterback Tyrod Taylor was the Atlantic Coast Conference Player of the Year and its championship game MVP. According to pregame media coverage, he presented a contrasting style to Stanford's Luck. Taylor gained threw for 2,521 yards and set a school record with 23 passing touchdowns, but was praised primarily for his physical mobility and ability to avoid defensive pressure, something Luck lacked. Taylor's presence was a contributing factor to Tech's offensive success in 2010. Virginia Tech's offense was the most productive since offensive coordinator Bryan Stinespring joined the team in 2001. It set team records for points and total yards, and led the ACC in scoring and red-zone efficiency.\nIn addition to Taylor's success passing the ball, the Hokies also had a successful running offense. Running back Ryan Williams, who was limited for much of the season because of injury, had 473 rushing yards. Fellow running back Darren Evans, who missed the 2008 season because of injury, gained 817 yards and 11 touchdowns. Sophomore running back David Wilson gained 616 yards. Combined with Taylor's rushing total, Virginia Tech gained 2,543 yards and 30 touchdowns on the ground.\nIn 2010, Stanford's defense improved under new defensive coordinator Vic Fangio, who was given credit for much of the team's success in that department. At the end of the regular season, Stanford was first in the Pac-10 and 11th of 120 teams nationally in scoring defense (allowing an average of 17.83 points per game) and 24th in total defense (permitting 326 yards per game). In 2009, Stanford was 69th in scoring defense (26.5) and 90th in total defense (403 yards per game). A prominent player on Stanford's defense was Owen Marecic, the only person in major league college football that season to play both offense and defense. On defense, Marecic accrued 45 tackles and 2 interceptions as a linebacker; on offense, he gained 117 yards and 4 touchdowns as the team's fullback.\nStanford's leading tackler was linebacker Shayne Skov, who accumulated 72 tackles despite missing the first two games of the season. Before the game, Skov said one of Stanford's priorities would be to contain the mobility of Tech quarterback Tyrod Taylor. \"We need to try to keep him in the pocket and (apply) pressure the same way we have all year,\" Skov said.\nVirginia Tech's defense was led by coordinator Bud Foster, who was named a candidate in several teams' searches for a new head coach, but decided to remain at Virginia Tech as the buildup to the Orange Bowl continued. Heading into the game, Tech's defense ranked 38th nationally in total yards allowed, its second-worst performance since 1993. Tech's run defense was 58th, permitting an average 148.9 rushing yards per game. On the positive side of things, Tech's defense had the third-most interceptions (22) in the country, and its scoring defense was 16th (19.1 points per game).\nTech's top performer on defense was sophomore cornerback Jayron Hosley, who had eight interceptions during the season\u2014tied for most in the country\u2014and was named a first-team Walter Camp All-American, a second-team Associated Press All-American, and a second-team Sports Illustrated All-American for that performance. Other top performers included linebackers Bruce Taylor and Stephen Friday, who had 15.5 and 15 tackles for loss, respectively.\nThe Hokies played the Orange Bowl without starting linebacker Lyndell Gibson, who fractured his shoulder in the second quarter of the ACC Championship Game. Gibson was fourth on the team in tackles with 66. Replacing Gibson was redshirt freshman Tariq Edwards, who played only a few times before the Orange Bowl. The Hokies also returned a player from injury for the Orange Bowl. Linebacker Barquell Rivers, who had been predicted to be Tech's starting middle linebacker, suffered a torn quadriceps tendon in offseason workouts, causing him to miss every game. Heading into the Orange Bowl, coaches were optimistic that he had healed enough to be able to play.\nThe 2011 Orange Bowl kicked off at 8:39 pm EST on January 3, 2011 in Sun Life Stadium, Miami Gardens, Florida. Bowl officials said 65,463 tickets were sold for the game, but many went unused, and that official figure was more than 9,000 below stadium capacity. The game was televised on ESPN, and the announcers were Mike Tirico, Ron Jaworski, Jon Gruden and Michele Tafoya. An estimated 8.23 million viewers watched the broadcast, earning it a Nielsen rating of 7.1. That figure was the second-lowest mark to date for a BCS game. For contributing teams to the Orange Bowl, the Atlantic Coast Conference and Pacific-10 each received large amounts of money to be divided among their members. Because the Pac-10 had two BCS teams, that conference received $27.2 million from the BCS. The ACC, which had only Virginia Tech in a BCS bowl, received $21.2 million. In the ACC, Virginia Tech received a somewhat larger share of the money as a reward for winning the conference: All other ACC teams received $1.1 million; Tech got $1.7 million. Stanford received a similar amount, but the extra money was offset by the cost of sending 500 players, coaches, and staff to Miami.\nStanford returned the game's opening kickoff to its 24-yard line, where the Cardinal offense began the game's first possession. On the game's first play, Stanford quarterback Andrew Luck ran 11 yards through the Virginia Tech defense for a first down. The success was short-lived, however, as Stanford's next three plays did not gain enough ground for another first down, and the team punted from its 37-yard line. Virginia Tech's first possession was no more successful than Stanford's had been. The Hokies' offense entered the game at its 31-yard line after the punt, but went three-and-out after failing to gain a first down.\nTech punted, and Stanford's offense returned to the field. The two teams traded possessions one more time, with Stanford turning the ball over on downs after a failed fourth-down conversion attempt, and Virginia Tech returning the ball with another punt. Following the kick, Stanford's offense began work from its 14-yard line with 8:08 remaining in the quarter. Two plays gained five yards, then Luck completed a 21-yard pass to wide receiver Doug Baldwin for just the second first down of the game. On the next play, Stanford running back Jeremy Stewart broke free of the Virginia Tech defense and ran down the field 60 yards for a touchdown. The score and subsequent extra point gave Stanford a 7\u20130 lead with 6:16 remaining in the quarter.\nStanford's post-score kickoff was downed for a touchback, and Virginia Tech's offense began from its 20-yard line. The Hokies gained their initial first down of the game when quarterback Tyrod Taylor completed a 19-yard pass to wide receiver Danny Coale on the drive's third play, but Tech was unable to gain another. Tech punted from its 43-yard line, and the kick was downed at the Stanford 5-yard line, where the Cardinal offense started work. Two rushing plays were stopped for no gain, then Luck attempted a passing play on third down. Because the line of scrimmage was so close to the Stanford goal line, Luck had to move into his own end zone to attempt the pass. Under pressure from the Virginia Tech defense, he threw to Derek Hall, who caught the ball in the end zone and was tackled before getting out. The result of the play was a safety, a 2-point defensive score for Virginia Tech.\nTrailing 7\u20132 with 59 seconds remaining in the quarter, Virginia Tech's offense received the ball at its 25-yard line following Stanford's post-safety free kick. On the first play of the subsequent drive, Taylor ran through the Stanford defense for 22 yards and a first down at the Tech 47-yard line. The second play of the drive, a 4-yard run by running back Darren Evans, pushed Tech into Stanford's side of the field for the first time in the game, and was the final play of the quarter. With three quarters remaining, Stanford led Virginia Tech, 7\u20132.\nThe second quarter began with Virginia Tech in possession of the ball and facing second down on Stanford's 49-yard line. On the first play of the quarter, Taylor completed a 10-yard throw to wide receiver Marcus Davis for a first down. Tech's drive continued down the field, with running back Darren Evans receiving the ball on the majority of the plays. A 5-yard false start penalty against Virginia Tech was offset by a pass interference penalty two plays later by Stanford. Tech penetrated to the Stanford 10-yard line, but was stopped for no gain, then a loss, on two consecutive running plays. On third down, Taylor was pressured by the Stanford defense, scrambled out of the pocket, and threw an 11-yard touchdown pass to running back David Wilson an instant before Taylor stepped out of bounds. The score and subsequent extra point gave Tech its only lead of the game, 9\u20137, with 10:22 remaining before halftime.\nVirginia Tech's post-touchdown kickoff was returned to the Stanford 21-yard line, where the Cardinal offense began its first drive of the second quarter. On the first play of the drive, Luck completed a 14-yard pass to tight end Zach Ertz for a first down. Three consecutive short-yardage plays gained another first down, then running back Jeremy Stewart escaped the Virginia Tech defense for a 24-yard gain, pushing the Cardinal into Virginia Tech territory. Three plays later, Luck again threw a long pass to Ertz, this time a 25-yard toss for Stanford's second touchdown of the game. The subsequent extra point kick was blocked by Virginia Tech, but the touchdown's six points were enough for Stanford to regain the lead, 13\u20139.\nVirginia Tech began its first full drive of the second quarter from its 25-yard line, but the Hokies went three-and-out. Stanford's offense, beginning from its 33-yard line after the Tech punt, had its drive cut short when Luck threw an interception to Virginia Tech's Jayron Hosley at the Tech 46-yard line. The Hokies advanced to the Stanford 30-yard line, but turned the ball over on downs after failing to gain one yard on fourth down. Stanford went three-and-out after the Hokies' turnover and punted. Tech's offense returned to the field at its 20-yard line with 49 seconds remaining until halftime. Using a hurry-up offense and strategically calling timeouts to stop the game clock, Tech advanced the ball down the field. Taylor rushed for 14 yards, threw a 32-yard pass, and the Hokies were helped by a 10-yard holding penalty against Stanford. They advanced to the Stanford 20-yard line where, with time running out, they sent in kicker Chris Hazley. He completed a 37-yard field goal in the final play from scrimmage in the first half, making the game a one-point affair, 13\u201312.\nFor the game's halftime show, the Orange Bowl hosted American alternative rock band Goo Goo Dolls. Reviews criticized the performance for awkward staging and brevity, as the group performed only two songs. Middle school and high school dance squads also participated in the halftime show, performing on the field while the Goo Goo Dolls played. Pregame media coverage erroneously reported that Stanford's marching band, which had been scheduled to perform, was banned from the field. The band had never been scheduled to perform at halftime; each team's marching band was allotted a six-minute pregame performance.\nBecause Stanford received the ball to begin the game, Virginia Tech received it to begin the second half. After starting from their 21-yard line, the Hokies went three-and-out. Following a Tech punt, Stanford's first drive of the second half began at its 41-yard line. Two plays gained 13 yards and a first down, then Stanford began battering Virginia Tech with its passing offense. Luck threw four consecutive passes: the first was incomplete, but the next three went for 10, 17, and 18 yards, respectively. This gave Stanford a first down at the Tech one-yard line. Three plays later, Owen Marecic crossed the goal line on a running play. The subsequent extra point was missed, and Stanford extended its lead to 19\u201312 with 8:47 remaining.\nVirginia Tech's second drive of the half began at its 41-yard line after a 31-yard kickoff return by Wilson. On the first play of the drive, Taylor was sacked for a 13-yard loss. He made good the lost yardage on the next play, however, completing a 42-yard throw to Danny Coale. From the Stanford 40-yard line, Taylor gained five yards on a running play, then attempted a long pass downfield. The throw was intercepted by Stanford's Delano Howell at the Cardinal 3-yard line, returning Luck and the Stanford offense to the field. Luck needed only two plays to drive the length of the field and score a touchdown. A 56-yard run by Stepfan Taylor was followed by a 41-yard touchdown pass to Coby Fleener, and the following extra point kick was good, making the score 26\u201312 with 5:49 remaining.\nVirginia Tech's third drive of the quarter started from its 23-yard line. The first two plays of the drive were stopped for negative gain, then Taylor completed a 17-yard throw to Jarret Boykin for a first down. Tech couldn't gain another first down, however, and punted to Stanford, which started from its 13-yard line. As in Virginia Tech's previous drive, the Stanford offense began with a play that lost yardage, but it made up the failure with a running play that gained six yards and a passing play from Luck to Fleener for a first down at the Stanford 32-yard line. On the quarter's final play, running back Tyler Gaffney gained seven yards. With one quarter remaining, Stanford led 26\u201312.\nThe fourth quarter began with Stanford in possession of the ball and facing second down and three from its 39-yard line. Two plays gained the Cardinal a first down, then Stanford committed a false-start penalty that pushed its offense back five yards. Two plays later, however, Luck completed a 58-yard pass to Fleener, who made up the lost yardage and far more as he raced down the field for a touchdown. The following extra point granted Stanford a 33\u201312 lead with 12:28 remaining.\nFollowing Stanford's post-touchdown kickoff, Virginia Tech's offense took the field at its 25-yard line and embarked upon one of its longest drives of the game in terms of the number of plays. In the first seven plays of the drive, quarterback Taylor was the key component. He threw six passes, completing 3 for 39 yards, and ran once, for no gain. On the eighth play of the drive, running back Ryan Williams gained one yard, then committed a 15-yard personal foul penalty. Tech's drive faltered after the penalty: Taylor threw an incomplete pass, then was sacked for a 16-yard loss, and the Hokies punted to the Stanford 28-yard line.\nAs it had in the third quarter, Stanford's offense scored quickly. In only three plays, the Cardinal advanced 72 yards. A 34-yard run by Stepfan Taylor was followed by one stopped for no gain, then Luck completed a 38-yard throw to Fleener for Stanford's second touchdown of the quarter. The score and extra point improved Stanford's lead to 40\u201312 with 6:05 remaining in the game. Tech's subsequent drive went three-and-out after Tyrod Taylor was sacked twice, and Stanford got the ball again at its 46-yard line.\nWith only 4:25 remaining and a lead well in hand, Stanford began running down the clock by executing running plays, which keep the game clock running as long as the ball carrier is downed in the field of play. Three rushes gained Stanford 16 yards, but the effectiveness of this strategy was undermined by a 15-yard personal foul penalty against Stanford that prevented the Cardinal from gaining a first down. Stanford also accepted a delay of game penalty in order to squeeze as much time as possible from the clock before punting back to Virginia Tech. The final drive of the game began from the Tech 11-yard line with 2:10 remaining, and quarterback Taylor controlled the ball on all of its five plays. Taylor completed a five-yard pass to Coale, then ran for 18 yards and a first down. He was then sacked for a 15-yard loss, but gained much of that back with an 11-yard run. On the game's final play, Taylor completed a 17-yard first-down pass to Wilson, who was tackled as the last second ticked off the clock and Stanford clinched a 40\u201312 victory.\nFor his game-winning performance, Stanford quarterback Andrew Luck was named the game's Most Valuable Player. He completed 18 of his 23 pass attempts for 287 yards and 4 touchdowns. The four touchdowns was a Stanford bowl-game record, and three of Luck's touchdowns went to tight end Coby Fleener, who set a Stanford and Orange Bowl record with 173 receiving yards and tied a record with the three touchdown catches. In postgame analysis, Fleener's performance was touted as the factor \"that turned a close game into a rout\".\nDespite those players' performances, the game was decided in the running game and the performance of the two teams' offensive and defensive lines. Stanford coach Jim Harbaugh was praised for adjusting his team's strategy at halftime, allowing it to succeed in the second half. Stanford's rushing offense accounted for 247 yards, while its defensive line limited Virginia Tech to just 67 yards rushing. The Stanford defense also sacked Virginia Tech quarterback Tyrod Taylor eight times, denying him time to pass the ball accurately.\nStanford's leading rusher was Stepfan Taylor, who accumulated 114 yards on 13 carries, including a 56-yard sprint that was the second-longest run of his career. Stanford's No. 2 runner, Jeremy Stewart, had a career-high 99 rushing yards. His 60-yard touchdown run in the first quarter is the longest touchdown run in Stanford bowl-game history and the seventh-longest in Orange Bowl history. Virginia Tech's leading rusher was Darren Evans, who had 12 carries for 37 yards. Virginia Tech quarterback Tyrod Taylor was the team's No. 2 rusher, with 16 carries for 22 yards. He also completed 16 of 31 pass attempts for 222 yards and Tech's sole touchdown.\nOn defense, Virginia Tech's Jayron Hosley tied the school record for interceptions (9) and had the most in the nation that season when he caught an Andrew Luck pass in the first half. Tech's defense also recorded its first safety since 2008, and its blocked extra point was the first such block in Virginia Tech bowl-game history. The Hokies' leading tackler was Eddie Whitley, who had seven stops, while Stanford's leading defensive performer was Shayne Skov, who had 12 tackles, including 3 quarterback sacks.\nWith the victory, Stanford improved to 12\u20131, while the loss sent Virginia Tech to 11\u20133. Stanford's 12 wins extended a school record for one season, and its eight-game winning streak tied for the third-longest in school history. Stanford's win was its first bowl game victory since the 1996 Sun Bowl against Michigan State, and San Francisco Chronicle football writer Tom FitzGerald declared one of the 10 all-time best Bay Area college football teams. In the final college football polls of the season, Stanford rose to No. 4, while Virginia Tech dropped to 16th in the Associated Press Poll and 15th in the USA Today coaches' poll.\nStanford's victory in the Orange Bowl caused a massive jump in the number of season ticket sales at the school. By mid-February 2011, the school reported having sold 1,400 new season tickets, as compared to just 458 in the first six months of 2010.\nFollowing the game, some actions by Orange Bowl Committee officials came under fire as the Miami Herald revealed the nonprofit committee had spent millions of dollars on junket trips for college football administrators. The U.S. federal Internal Revenue Service subsequently began investigating the allegations. In its annual transition to new leadership, the Orange Bowl committee picked Goldman Sachs vice president Jeffrey T. Roberts to serve as its president for the 2012 game.\nBoth teams underwent changes in leadership following the Orange Bowl. Four days after beating Virginia Tech, Stanford head coach Jim Harbaugh signed a contract as the new head coach of the National Football League's San Francisco 49ers. He was replaced as Stanford head coach by David Shaw, the team's offensive coordinator. Defensive coordinator Vic Fangio followed Harbaugh to the 49ers and was replaced by co-defensive coordinators Jason Tarver and Derek Mason, the latter also serving as associate head coach. To replace the vacated offensive coordinator position, Shaw selected Stanford wide receivers coach Pep Hamilton.\nVirginia Tech's coaching changes came about a month after the Orange Bowl. On February 14, the school announced that the son of head coach Frank Beamer, Shane Beamer, had been hired as running backs coach. He replaced Billy Hite, the longest-tenured assistant coach in the country, who took an administrative role with the team. Two days later, Tech replaced 62-year-old linebackers coach Jim Cavanaugh with 35-year-old Cornell Brown. On February 22, Tech announced that offensive coordinator Bryan Stinespring had been removed from play-calling duties and replaced by quarterbacks coach Mike O'Cain. Stinespring, who also was the team's associate head coach, was replaced in that capacity by Shane Beamer.\nFor several players on each team, the Orange Bowl was their final collegiate contest before attempting to move into professional football. Two days after the Orange Bowl, Virginia Tech running back Darren Evans announced his intention to enter the NFL Draft. Three days later, fellow running back Ryan Williams made a similar move. Stanford quarterback Andrew Luck made news for not entering the draft, where he had been widely predicted to be the No. 1 selection. In interviews, he said he looked forward \"to earning [his] degree in architectural design from Stanford University\" and graduating in spring 2012.\nThe 2011 NFL Draft took place in late April, and several players from each Orange Bowl team were selected by professional squads seeking their talents. Virginia Tech's Ryan Williams was the first Orange Bowl participant selected, taken with the 38th overall pick. Tech's Rashad Carmichael (127th), and Tyrod Taylor (180th) were also selected. Stanford had four players picked in the draft: Sione Fua (97th), Owen Marecic (124th), Richard Sherman (154th), and Ryan Whalen (157th).\n^ Between 1998 and 2006, there were four BCS games.\n^ Kilgore, Adam. \"ACC, Orange Bowl Make a Deal\", The Washington Post. December 16, 2007. Retrieved February 14, 2011.\n^ a b The Associated Press. \"Virginia Tech headed to Orange Bowl as Tyrod Taylor burns Florida State\", ESPN.com. December 4, 2010. Retrieved February 14, 2011.\n^ a b \"BCS Selection Procedures,\" BCSfootball.org. April 26, 2010. Retrieved February 14, 2011.\n^ a b c \"BCS selections history: 2010\u201311 Games\", BCSfootball.org. December 16, 2010. Retrieved February 14, 2011.\n^ Degnan, Susan Miller. \"Auburn, Oregon in BCS title game; Stanford-Va. Tech in OB\", The Miami Herald. December 6, 2010. Retrieved February 14, 2011.\n^ a b c \"Stanford Cardinal Schedule \u2013 2009\", ESPN.com. Retrieved February 15, 2011.\n^ The Associated Press. \"Broyles' 3 TDs set Sun Bowl record as Sooners sink Cardinal\", ESPN.com. December 31, 2009. Retrieved February 15, 2011.\n^ Miller, Ted. \"Opening camp: Stanford\", ESPN.com. August 9, 2010. Retrieved February 15, 2011.\n^ FitzGerald, Tom. \"Stanford to rely on a whole lot of Andrew Luck\", The San Francisco Chronicle. August 9, 2010. Retrieved February 15, 2011.\n^ \"Oregon tabbed to win Pac-10 in preseason media poll\", The Pacific-10. July 29, 2010. Retrieved February 15, 2011.\n^ \"2010 NCAA Football Rankings \u2013 Preseason\", ESPN.com. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck completes 81-yard TD pass as Stanford drops Sacramento State\", ESPN.com. September 4, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck, No. 25 Stanford batter UCLA to end Rose Bowl skid\", ESPN.com. September 10, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"No. 19 Stanford stomps Wake Forest behind Andrew Luck's 5 total TDs\", ESPN.com. September 18, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Nate Whitaker's five FGs give Stanford winning edge over Irish\", ESPN.com. September 25, 2010. Retrieved February 15, 2011.\n^ \"2010 NCAA Football Rankings \u2013 Week 5 (Sept. 26)\", ESPN.com. September 26, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Darron Thomas scores four TDs as Oregon beats Stanford \", ESPN.com. October 2, 2010. Retrieved February 15, 2011.\n^ \"Stanford Cardinal Schedule \u2013 2010\", ESPN.com. Retrieved February 15, 2011.\n^ The Associated Press. \"Stanford's Nate Whitaker rebounds to put away USC on final play\", ESPN.com. October 9, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck tosses 3 TDs as Stanford handles Washington St.\", ESPN.com. October 23, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck keys Stanford rout as Jake Locker fails to impress\", ESPN.com. October 30, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck puts on a clinic as No. 13 Stanford handles No. 15 Arizona\", ESPN.com. November 6, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck, No. 6 Stanford fight through feisty Arizona State\", ESPN.com. November 13, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Andrew Luck leads Stanford to scores on all his possessions\", ESPN.com. November 20, 2010. Retrieved February 15, 2011.\n^ The Associated Press. \"Stanford pushes for BCS berth as Andrew Luck sets school TD record\", ESPN.com. November 27, 2010. Retrieved February 15, 2011.\n^ ESPN.com. \"Virginia Tech Hokies Schedule \u2013 2009\", ESPN.com. Retrieved February 1, 2011.\n^ ESPN.com. \"2010 NCAA Football Rankings \u2013 Preseason\", ESPN.com. Retrieved February 1, 2011.\n^ Teel, David. \"85,000 expected vs. Boise\", The (Newport News, Va.) Daily Press. August 27, 2010. Page B1.\n^ Berman, Mark. \"Party Crashers?\", The Roanoke Times. August 29, 2010. Page CF7.\n^ Berman, Mark. \"Fans flocking to Va. Tech's season opener\", The Roanoke Times. August 29, 2010. Page A1.\n^ The Associated Press. \"Kellen Moore's 3rd TD pass lifts Broncos past Hokies\", ESPN.com. September 6, 2010. Retrieved February 1, 2011.\n^ Dinich, Heather. \"Virginia Tech self-destructs in loss to JMU\", ESPN.com. September 11, 2010. Retrieved February 1, 2011.\n^ a b The Associated Press. \"Va. Tech suffers 2nd loss in 6 days as James Madison pulls off upset\", ESPN.com. Retrieved February 1, 2011.\n^ ESPN.com. \"2010 NCAA Football Rankings \u2013 Week 3 (Sept. 12)\", ESPN.com. Retrieved February 1, 2011.\n^ Giannotto, Mark. \"Pivotal team meeting changed Hokies' season\", The Washington Post. December 4, 2010. Page D1.\n^ Dinich, Heather. \"Season recap: Virginia Tech\", ESPN.com. December 7, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Virginia Tech climbs out of 10-point hole despite losing Ryan Williams\", ESPN.com. September 18, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Virginia Tech blanks BC for first shutout since 2006\", ESPN.com. September 25, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Tyrod Taylor's 39-yard TD pass rallies Virginia Tech past NC State\", ESPN.com. October 2, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"QB Tyrod Taylor accounts for three TDs to power Virginia Tech\", ESPN.com. October 9, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Virginia Tech explodes in first half to put away Wake Forest\", ESPN.com. October 16, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Tyrod Taylor's three touchdowns power Virginia Tech\", ESPN.com. October 23, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"David Wilson's late kickoff return gives Virginia Tech win over Georgia Tech\", ESPN.com. November 4, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Two Tyrod Taylor-to-Marcus Davis TDs lift Hokies past sloppy Tar Heels\", ESPN.com. November 13, 2010. Retrieved February 1, 2011.\n^ a b The Associated Press. \"Virginia Tech takes down Miami to win ACC Coastal Division title\", ESPN.com. November 20, 2010. Retrieved February 1, 2011.\n^ The Associated Press. \"Hokies polish off Cavaliers to cap unbeaten ACC season\", ESPN.com. November 27, 2010. Retrieved February 1, 2011.\n^ Weir, Tom and Hiestand, Michael. \"BCS officially headed to ESPN starting in 2011\", USAToday.com. Retrieved February 16, 2011.\n^ The Associated Press. \"FedEx ends Orange Bowl sponsorship after 20 years\", USAToday.com. May 3, 2010. Retrieved February 16, 2011.\n^ \"FedEx name will come off Orange Bowl\", Sports Business Journal. May 3, 2010. Retrieved February 16, 2011.\n^ \"The lost Citi: Rose Bowl sponsor out\", Sports Business Journal. June 21, 2010. Retrieved February 16, 2011.\n^ \"Sweet deal in works for Orange Bowl\", Sports Business Journal. May 10, 2010. Retrieved February 16, 2011.\n^ Jackson, Barry. \"Orange Bowl finds a new sponsor: Discover\", Miami Herald. August 25, 2010. Retrieved February 16, 2011.\n^ ESPN.com news services. \"Discover to sponsor Orange Bowl\", ESPN.com. August 26, 2010. Retrieved February 16, 2011.\n^ a b c d Wood, Norm. \"Who is Stanford? Here's a guide\", The (Newport News, Va.) Daily Press.\n^ The Associated Press. \"Cardinal's Luck goes a long way\", The Augusta (Ga.) Chronicle. December 10, 2010. Page C4.\n^ FitzGerald, Tom. \"Jim Harbaugh has led a U-turn at Stanford\", The San Francisco Chronicle. December 29, 2010. Retrieved February 24, 2010.\n^ \"Game Notes\", The Atlantic Coast Conference. December 4, 2010. Retrieved February 3, 2011.\n^ Teel, David. \"Wake Forest coach analyzes Tech-Stanford matchup\", The (Newport News, Va.) Daily Press. December 30, 2010. Retrieved February 16, 2011.\n^ Teel, David. \"Difficult bowls next up for ACC\", The (Newport News, Va.) Daily Press. December 8, 2010. Page C1. Retrieved February 16, 2011.\n^ a b c King, Randy. \"Tech's Foster stays put, for now\", The Roanoke Times. December 11, 2010. Retrieved February 16, 2011.\n^ \"Latest Line\", Charleston (W.V.) Gazette. December 11, 2010. Page B2.\n^ FitzGerald, Tom. \"Virginia Tech arrives late from snow country\", The San Francisco Chronicle. December 29, 2010. Retrieved February 24, 2011.\n^ \"Bowl game odds\", Charleston (W.V.) Daily Mail. December 15, 2010. Page B4.\n^ \"Latest line\", The Charleston (W.V.) Gazette. December 20, 2010. Page B2.\n^ \"Latest line\", The Charleston (W.V.) Gazette. December 25, 2010. Page B2.\n^ FitzGerald, Tom. \"Jim Harbaugh sleepless and mum on 49ers' job\", The San Francisco Chronicle. January 2, 2011. Retrieved February 24, 2011.\n^ King, Randy. \"Juicy matchup awaits Va. Tech, Stanford in Florida\", The Roanoke Times. December 7, 2010. Retrieved February 16, 2011.\n^ Satchell, Arlene. \"Delta: More seats for 2011 BCS game cities\" Archived January 1, 2011, at the Wayback Machine, (Fort Lauderdale, Fla.) Sun-Sentinel. December 28, 2010.\n^ Hemlock, Doreen. \"Va. Tech-Stanford at Orange Bowl ok for tourism\" Archived December 11, 2010, at the Wayback Machine, (Fort Lauderdale, Fla.) Sun-Sentinel. December 6, 2010. Retrieved February 16, 2011.\n^ a b Wood, Norm. \"Finally, a sweet Orange Bowl\", The (Newport News, Va.) Daily Press. December 28, 2010. Page C1.\n^ Giannotto, Mark. \"Virginia Tech again has trouble selling its bowl tickets\", The Washington Post. December 30, 2010. Retrieved February 16, 2011.\n^ The Associated Press. \"Cam Newton wins Heisman Trophy\", ESPN.com. December 13, 2010. Retrieved February 16, 2011.\n^ FitzGerald, Tom. \"Andrew Luck's choice: take the money or return\", The San Francisco Chronicle. January 2, 2011. Retrieved February 24, 2011.\n^ \"Doug Baldwin is Stanford's comeback kid\", The Miami Herald. January 2, 2011. Retrieved March 9, 2011.\n^ FitzGerald, Tom. \"Stanford football news, Dec. 28\", The San Francisco Chronicle. December 28, 2010. Retrieved February 24, 2011.\n^ a b FitzGerald, Tom. \"Stanford football news, Jan. 3\", The San Francisco Chronicle. January 3, 2011. Retrieved February 24, 2011.\n^ Wilson, Aaron. \"John Harbaugh on Jim Harbaugh: 'I think he wants to stay at Stanford'\", Carroll County (Md.) Times. December 17, 2010. Ravens section.\n^ Smith, Michelle. \"All Eyes on Stanford's Jim Harbaugh\", Fanhouse. December 28, 2010. Retrieved February 24, 2011.\n^ Stoda, Greg. \"Stanford coach hits road ... but where does it end?\"[permanent dead link], The Palm Beach Post. December 30, 2010. Retrieved February 16, 2011.\n^ FitzGerald, Tom. \"Cardinal's focus not on Harbaugh\", The San Francisco Chronicle. December 31, 2010. Retrieved February 24, 2011.\n^ Giannotto, Mark. \"Virginia Tech's Tyrod Taylor, Stanford's Andrew Luck bring contrasting styles to Orange Bowl\", The Washington Post. December 30, 2010. Retrieved February 16, 2011.\n^ a b King, Randy. \"Stinespring's squad downright offensive\", The Roanoke Times. December 28, 2010. Retrieved February 16, 2011.\n^ FitzGerald, Tom. \"Intense Fangio has settled Stanford's defense\", The San Francisco Chronicle. December 30, 2010. Retrieved February 24, 2011.\n^ \"Vic Fangio's experience the best teacher for Stanford\", Miami Herald. December 31, 2010. Retrieved February 16, 2011.\n^ Smith, Michelle. \"Stanford's Owen Marecic a 'Perfect Storm'\", Fanhouse. December 28, 2010. Retrieved February 24, 2011.\n^ Tucker, Kyle. \"Classic throwback player gives Stanford an edge on both sides of the ball\", The Virginian-Pilot. December 26, 2010. Retrieved February 16, 2011.\n^ Fitzgerald, Tom. \"Shayne Skov a head-banger for Stanford's defense\", The San Francisco Chronicle. January 1, 2011. Retrieved February 24, 2011.\n^ a b Slater, Darryl. \"Hokies' defense is a work that progressed\" Archived February 4, 2013, at Archive.today, Richmond Times-Dispatch. December 27, 2010. Retrieved February 16, 2011.\n^ Warters, Nathan. \"Hokies' cornerback Hosely earns more accolades\", The (Lynchburg, Va.) News & Advance. December 15, 2010.\n^ Woodson, Brian. \"Friday helping Hokies make history\", Bluefield (W.V.) Daily Telegraph. December 28, 2010. Retrieved February 16, 2011.\n^ Warters, Nathan. \"Tech linebacker Gibson to miss Orange Bowl\", The (Charlottesville, Va.) Daily Progress. December 9, 2010. Retrieved February 16, 2011.\n^ Giannotto, Mark. \"Hokies may call on Edwards\", The Washington Post. December 31, 2010. Retrieved February 16, 2011.\n^ Tucker, Kyle. \"Va. Tech's Rivers struggles back from injury\", The Virginian-Pilot. December 22, 2010. Retrieved February 16, 2011.\n^ Virginia Tech Athletics Communications Department. Stanford 40, Virginia Tech 12: Statistics Hokiesports.com. January 3, 2011. Retrieved February 16, 2011.\n^ The Associated Press. \"Orange Bowl attendance, Television ratings down\" Archived July 16, 2011, at the Wayback Machine, The Southwest Times. January 13, 2011. Retrieved February 16, 2011.\n^ Taylor, John. \"Herbstreit, Musburger to call title game\", NBC Sports. November 12, 2010. Retrieved February 16, 2011.\n^ The Associated Press. \"Orange Bowl's ratings hold steady from last year\", Bloomberg. January 4, 2010. Retrieved February 16, 2011.\n^ Teel, David. \"More low Orange Bowl ratings for Hokies\", The (Newport News, Va.) Daily Press. January 4, 2011. Retrieved February 16, 2011.\n^ a b Ross, Andrew and Matier, Phillip. \"Stanford, Cal score big from Pac-10\", The San Francisco Chronicle. Retrieved February 24, 2011.\n^ Miller, Ted. \"Big payout for two BCS bowl teams\", ESPN.com. January 25, 2011. Retrieved February 16, 2011.\n^ Sawchik, Travis. \"Money, ticket sales still issues at bowl games\"[permanent dead link], The Post and Courier. December 30, 2010. Retrieved February 16, 2011.\n^ a b c d \"1st Quarter Play by Play\", ESPN.com. January 3, 2011. Retrieved February 17, 2011.\n^ a b c \"2nd Quarter Play by Play\", ESPN.com. January 3, 2011. Retrieved February 17, 2011.\n^ \"Goo Goo Dolls to rock Discover Orange Bowl halftime show\", OrangeBowl.org. Retrieved February 25, 2011.\n^ Lozada, Liana. \"The Goo Goo Dolls at the Orange Bowl Halftime Show, January 3\", Miami New Times. January 4, 2011. Retrieved February 25, 2011.\n^ Rey, Diane M. \"Around Annapolis: Dancers Goo Goo over Orange Bowl appearance\", The Capital (Annapolis, Md.) Retrieved February 25, 2011.\n^ \"Fitch Cheerleaders Head to Orange Bowl\" Archived July 18, 2011, at the Wayback Machine, WKBN-TV, Youngstown, Ohio. December 30, 2010. Retrieved February 25, 2011.\n^ a b FitzGerald, Tom. \"Coby Fleener has monster game on biggest stage\", San Francisco Chronicle. January 4, 2011. Retrieved February 25, 2011.\n^ a b c \"3rd Quarter Play by Play\", ESPN.com. January 3, 2011. Retrieved February 23, 2011.\n^ a b c d \"4th Quarter Play by Play\", ESPN.com. January 3, 2011. Retrieved February 23, 2011.\n^ a b c d e f g Stanford Cardinal vs. Virginia Tech Hokies \u2013 Box Score ESPN.com. January 3, 2011. Retrieved March 2, 2011.\n^ a b Neal, David J. \"Stanford dominates Virginia Tech to win Orange Bowl\", The Miami Herald. January 4, 2011. Retrieved March 2, 2011.\n^ a b c d \"Stanford Post-Game Notes vs. Virginia Tech\" Archived March 18, 2012, at the Wayback Machine, gostanford.com. January 3, 2011. Retrieved March 2, 2011.\n^ Povtak, Tim. \"Coby Fleener: Unlikely Star for Stanford in Orange Bowl Victory\", Fanhouse. January 4, 2011. Retrieved February 24, 2011.\n^ Lynch, Kevin. \"Orange Bowl Observations\", The San Francisco Chronicle. January 3, 2011. Retrieved March 2, 2011.\n^ \"Halftime Notes\", Hokiesports.com. January 3, 2011. Retrieved March 2, 2011.\n^ \"Defensive statistics\", Hokiesports.com. January 3, 2011. Retrieved March 2, 2011.\n^ Fernandez, Andre C. \"OB rookie Stanford touches down in Miami\", Miami Herald. December 28, 2010. Retrieved February 16, 2011.\n^ FitzGerald, Tom. \"Bay Area's top 10 all-time college football teams\", The San Francisco Chronicle. January 3, 2011. Retrieved February 24, 2011.\n^ \"2010 NCAA Football Rankings \u2013 Postseason\", ESPN.com. Retrieved March 2, 2011.\n^ Quon, Rachel. \"After Historic Football Season, Stanford's Ticket Sales Up 200 Percent\" Archived February 22, 2011, at the Wayback Machine, Peninsula Press. February 15, 2011. Retrieved February 16, 2011.\n^ Vasquez, Michael. \"Orange Bowl perks flagged with IRS\", The Miami Herald. January 5, 2011. Retrieved February 16, 2011.\n^ \"Jeffrey T. Roberts installed as Orange Bowl Committee President and Chairman\", Orangebowl.org. Retrieved March 2, 2011.\n^ \"Jim Harbaugh Named Head Coach of San Francisco 49ers\" Archived March 18, 2012, at the Wayback Machine, GoStanford.com. January 7, 2011. Retrieved March 2, 2011.\n^ \"David Shaw Named Bradford M. Freeman Director of Football\/Head Coach\" Archived January 12, 2012, at the Wayback Machine, GoStanford.com. January 13, 2011. Retrieved March 2, 2011.\n^ \"Stanford Announces Changes to Football Coaching Staff\", GoStanford.com. February 2, 2011. Retrieved March 2, 2011.\n^ Tucker, Kyle. \"Shane Beamer heads to Va. Tech as assistant coach\", The Virginian-Pilot. February 15, 2011. Retrieved March 2, 2011.\n^ King, Randy. \"Coach Beamer adds Brown to Hokies football staff\", The Roanoke Times. February 17, 2011. Retrieved March 2, 2011.\n^ Dinich, Heather. \"Mike O'Cain to take over play calling\", ESPN.com. February 22, 2011. Retrieved March 2, 2011.\n^ \"Darren Evans to turn professional\", Hokiesports.com. January 6, 2011. Retrieved March 2, 2011.\n^ \"Ryan Williams to turn professional\", Hokiesports.com. January 9, 2011. Retrieved March 2, 2011.\n^ Miller, Ted. \"Andrew Luck staying at Stanford\", ESPN.com. January 7, 2011. Retrieved March 2, 2011.\n^ \"2011 NFL Draft Tracker: Virginia Tech\", NFL.com. Retrieved May 4, 2011.\n^ \"2011 NFL Draft Tracker: Stanford\", NFL.com. Retrieved May 7, 2011.\nThis page was last edited on 28 January 2019, at 12:48 (UTC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This activity comes straight to you from my kidlets preschool! It's an excellent way to build fine motor skills (including the all important pincer grasp) in ages 3-5 years. It might be the sister craft to sewing cards.\n- using the scotch tape, adhere one end of a length of yarn to the top slit.\nIt's not that pretty, but my daughter loved it and was very proud of her accomplishment.\nmy three year old and probably two year old will love this. i have to try it!!\nOh I love this one. I'm always looking for new projects that are easy to make! This is perfect! I think I will cute out a gingerbread man and make this his belly! Thanks for the idea.\nLooks like a great fine motor skill project. We'll have to give it a try.\nCan't wait to try this with my kiddo.\nGreat idea! I have a 4 year old, this will be perfect!\nThat is a GREAT activity!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While every year we keep our ears to the ground for new tunes, there's something to be said for the charm of a cover song. Sometimes the idea of a song-based revamp can be as simple as hearing a different voice tackling a familiar vocal hook, while just as often listeners can get a kick out of finding a rock track delivered as roots-reggae, or hard-hitting metal recast under the hushed care of a bunch of folkies.\nAssembled here are some of Exclaim!'s favourite cover songs of 2013, a mix of classic and contemporary cuts given a spit-shine by artists looking to give their own favourite songs new life.\nDon't forget to check out our 2013 in Lists section to see more of our Year-End coverage.\nZeus didn't exactly embrace their inner grungers when they slapped a new coat of paint onto \"Vasoline,\" one of Stone Temple Pilots' crunchiest offerings back in the '90s. True, a fuzz bass still rumbles heavy on that familiar two-note melody, but the Ontario outfit brought aboard a laid-back groove, siren-style synth sounds and more on their funky redo.\nThe orchestral majesty of the original may give way to synth-strings on this version, but the entwining of Lana Del Rey's smoke-screen vocals and Barrie-James O'Neill's booming baritone still makes for a romantic run-through just a hair below Lee Hazlewood and Nancy Sinatra's arrangement.\nThe Electric Lady was another impressive future-funk adventure from the R&B game's favourite arch-android, but Mon\u00e1e took a fantastic journey back to the late '60s by redoing the Jackson 5's debut major label single, \"I Want You Back.\" Her vocals run a little more low-key than little MJ's, and the soul arrangement is scaled back to just some shaker rhythms and a guitar jangle, but the song remains just as infectious over 40 years later.\nThat Ty Segall is into Marc Bolan should be no surprise, if only because both artists have impressively prolific and fairly consistent catalogues. Most awesome about Segall's dialled-down take on the searing rock groove of Electric Warrior's \"The Motivator,\" however, is that by scrubbing the distortion in favour of acoustic guitars and a mild bongo-pound he may in fact also be toasting pre-glam Bolan, back when the band were called Tyrannosaurus Rex.\nThough culled from Depeche Mode's classic Violator LP, ghoulish psych-rock crew Ghost B.C.'s fuzzed-up take on \"Waiting For the Night\" treats the song just right. Gone are the minimalist synth swells, with Ghost BC instead rocking things up tenfold with gothic organ, a crunchy slow-mo beat and gothic vocal harmonies bringing a new twist on Dave Gahan's dark, stark reality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free Shipping Money Back Guarantee $17.98 - Save 33%!\nHave one or more \"Iron Maiden - Best of the Beast [New CD]\" for sale? Sell it on Whohou Marketplace.\nBest Of The Beast - Iron Maiden CD Greatest Hits New !\nNumber Of The Beast - Iron Maiden CD Sealed ! New !","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The kind words, as well as the constructive criticism is all greatly appreciated.\nDinosaur Run is now available to play at the National Videogame Arcade!\nIf you're in the UK, you should definitely visit the NVA. It's a great place that showcases upcoming games, as well as a plethora of classics \u2013 including arcade cabinets\u2026and of course, Dinosaur Run!\n\"THE NATIONAL VIDEOGAME ARCADE IS A FAMILY DAY OUT, A LEADING CULTURAL CENTRE AND THE HOME OF VIDEOGAMES. IT'S UNIQUE!\nAt the NVA you can explore three floors of playable exhibitions about videogames. You can meet your friends here and play your own games.\nYou can learn new skills. Hear a talk. Meet renowned gaming experts.\nEven have your birthday party!\nExplore our galleries or just drop in for a visit to the Toast Bar.\nTake control of a plush monkey in a plastic world of animated toys in this runner game inspired by classic 90's adventure games, such as Crash Bandicoot, and the imagination of a child.\nRun away from the friendly dinosaur as you collect bananas and avoid various obstacles! Traverse multiple levels, including an endless mode, to beat your high score and unlock silly hats and upgrades.\nExciting runner-style gameplay with a twist.\nVariety of obstacles and powerups.\nFront facing camera and chase mechanics put a unique spin on the genre.\nUnique art style that depicts a world of plastic toys.\n8 fun levels, such as the Dinosaur Valley, the Crystal Caves, Baking Soda Volcano & more!\nEndless Mode for the ultimate challenge.\nVariety of fun hats to unlock for Monkey and Dinosaur.\nEnjoy various difficulty levels for the whole family.\nFully voiced, great for aiding young children who cannot read.\nQ: Will this be an Early Access game?\nA: No. The game is still a work in progress, and will be released when it is finished\/ready - but I will still listen to the community if there are suggestions for change or if there are any issues encountered.\nQ: Will there be microtransactions?\nA: No, all unlocks are acquired through the in-game level system.\nQ: Will there be support for multiple languages?\nA: It's definitely something I am interested in - especially if requested enough. I am a single developer, however, and I only know English - but I'd be interested to work with multi-lingual players from the community to make it happen.\nQ: Will there be Steam achievements\/trading cards?\nQ: How many levels are in the game?\nA: There are eight different levels with a start and end. The obstacles and pickups are procedural so it's different every time. There is also an endless mode to see how long you can run for.\nQ: Is this game suitable for children?\nA: Yes! This was created specifically with young children in mind, and the games QA includes thorough testing by a four year-old! While aimed at children, there are adjustable difficulty levels to make it fun for those of all ages!\nFantastic looking game. I would like to speak to you regarding a proposal.\nGame wise it seems good enough, definitely shouldn't be on Steam or atleast the main platform shouldn't be Steam since it appears to be targeted towards a young audience (Just my opinion). Quick Note: This is basicly a one directional Temple Run which already goes for free on mobile platforms, so you might consider making this game a free download mobile game with in-game purchases or a pay once unlock all, but the latter would probably scare away many possible users.\nI will give a \"Yes\" to the game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Connies Damsel. Connie's Damsel, Chromis limbaughi Photo.\nConnies Damsel. Connie's Damsel, Chromis limbaughi Picture.\nConnies Damsel. Stock Photography of Connie's Damsel, Chromis limbaughi.\nConnies Damsel. Photograph of Connie's Damsel, Chromis limbaughi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"December 21st, 2012. That's the date when the Mesoamerican\/Maya Long Count calendar is supposed to end, and a new era begin. Various New Age and \"end-of-days\" doomsday peddlers have created a cocktail of various belief systems to invest this date with some looming significance, either for a new dawn, or an end-times scenario. This is despite the fact that actual Mayan spiritual leaders (and the academics who study Mayan culture) have long disputed the pop-cultural consensus, and the appropriation of their cultural heritage for profit.\n\"Guatemala's Mayan people accused the government and tour groups on Wednesday of perpetuating the myth that their calendar foresees the imminent end of the world for monetary gain. \"We are speaking out against deceit, lies and twisting of the truth, and turning us into folklore-for-profit.\nThe beginning of a new calendar year usually means a flurry of predictions. These prognostications can be educated guesses, fervent hopes, pessimistic fears, or, in some cases, spiritual messages via divination, omens, or other supernatural methods. One widely reported instance of a yearly divination tradition is the If\u00e1 predictions from Cuba's Santeria priests, who've been gathering for 26 years to make predictions and recommendations. Last year they were eerily accurate in their readings, predicting \"abrupt changes in political systems\", intense drought, and \"dangers of war and conflicts.\" This year the reigning deity is Oya, and they are predicting a year of \"war and confrontation, social, political, and economic change, and a dangerous increase in temperature.\" However, the mainstream press seems to have positioned this as a Babalawos vs Mayas prophesy-off thanks to the ubiquity of New Age \"Mayan Prophesy\" books that predict a great ending\/beginning in 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Western Reserve Trust Company, we appreciate that a trust is more than its assets and distributions.\nIt represents the legacy of the individual who created the trust, as well as the hopes and well-being of its beneficiaries. It is the result of hard work and sacrifice and must be treated with the utmost care.\nTrust administration is an enormous responsibility, one that requires the attention of a multi-disciplinary firm of professionals with the experience and expertise to act wisely and prudently.\nBuilding upon their years of service in the trust industry, the founders of WRTC have assembled a team that can be relied upon to administer all of its duties faithfully while protecting clients' legacies.\nEach trust has its own dedicated team to ensure that our clients receive the highest level of service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New York is indeed brutally cold and though the weather hasn't been terribly bad, the last thing we wanted was to be freezing in a dress on New Years Eve. New Years for us is always highly anticipated which always results in a disappointing night. We promised ourselves that this year, forget the over priced tables at nightclubs and restaurants that always ends up being not worthy. We're jet-setting to the sun, after all we do terribly miss living in Miami. We're packing light because after all, we have no intentions of wearing clothes, just bikinis! These are some of the items I currently have in my suitcase ready for take off.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..5aca5c0e056189a8e36510725d204cb33d598b4c
--- /dev/null
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+{"text":"The answer is yes. It doesn't matter where you are located in Australia as long as you have a good broadband Internet connection. We can get you started on VoIP in minutes.\nCan I get a DID in my area?\nWe don't offer DID at this stage. However you can use our service to make out-bound calls as long as you have a broadban Internet connection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cute and move in ready! Many updates include: new carpet throughout, new paint, hot water heater (2017), new AC unit (2018), updated bathroom, new tile, and more. Home sits on an oversized, corner lot with extra large shed. Great location near Lake Overholser with easy access to Route 66 and the turnpike.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a lot of good reasons to use a motorcycle. They are cheaper both to buy and run than cars (in most cases \u2013 and if you are a beginner it is unlikely you will be opting for an expensive,powerful superbike!), they can make city travel easier, and riding can also be an exhilarating hobby. Of course, there is no secret to the fact that travelling by motorcycle is more dangerous than most other common forms of transport, with motorcyclists estimated to be around 30 times more likely to be killed in an accident than drivers and passengers in cars.\nWhile the basic requirements for riding a motorbike legally are not that demanding, get some advanced training if you can, or at the very least do some practice on the safest roads you can find for a while before you try riding at speeds on very busy or more challenging roads. Learn as much as you can about bike safety and don't take risks \u2013 if you don't feel ready for a certain road or journey, keep practicing and use the bus until you are certain your skills are good enough!\nMotorcycle gear can be expensive, but it can also literally save your skin. While some stuff can be bought second hand to save money, like leathers, helmets should always be bought new. This is because you need the exact right fit, and the chances of getting that right at a guess on something like eBay are slim. It is also for a less obvious reason \u2013 a helmet that has been in an accident may not show any visible signs of this and look perfect, but may no longer offer the proper protection. It is generally best to buy your gear new where possible as comfort and fit will make a huge difference, but if you are on a tight budget buying second hand on jackets before other more fitted things like gloves and riding boots is better.\nAs well as having the right skills and the gear to protect your body, you also need a good yet affordable motorbike insurance policy. Like car insurance, insurance for motorcycles comes at different levels, but generally it is best to get the best level of cover you can afford, particularly if your new bike will be your main mode of transport or is a significant financial investment for you. Nobody ever regrets being too well insured, after all!\nGetting ready to become a new motorcyclist may seem daunting, but as long as you do your research and don't skip out on practicing or investing in the right gear, you should soon be on your way to an exciting new life as a well informed, well protected biker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following meditation is an earth meditation that I have created. Feel free to adapt it to your own use. You can make your own-guided meditation by recording this on a tape and then playing it back while you are in a deep state of relaxation. Although it might be strange hearing your own voice on tape, it will allow you to fully concentrate on your visualization without having to focus on remembering the next portion of the meditation.\nDo this meditation in a quiet place when you have time to completely rest and relax. You can light a green or brown candle if you wish and hold an earth associated gemstone, such as amber or granite.\nImagine yourself anchoring yourself into the cool ground as tree roots. Expand your energy downward allowing the energy to flow from the higher chakras down through the top of your head. While doing so, expanding your roots to deep within the earth's center. Embrace the earth's core with these roots. Feel the love stream downward, as you are the conduit of this love towards the Earth. You are part of the Earth and the Earth is part of you.\nNow expand your roots outward filling up the entire inner portion of the earth so that it becomes a filled ball of roots, a filled ball of interconnecting love that continues to flow through you. Feel the roots pulsing beneath you and through you. It now is a perfect embrace of unconditional divine love within the Earth.\nNow allow the roots to slowly morph from inside to the outside into soft tender hands that gently cradle the Earth. It is a feeling of holding a newborn baby; you are a mother or father to the precious child of Earth! Feel the Love flowing through you to these cosmic hands. Hold it, cradle it, and let your love caress the Earth in warmth and tenderness.\nAt this point you may see the Earth with a white aura glowing around it. It is healing energy coming from you and all of the others who are connecting with the earth. Expand this glow\u2026Allow this white glow of the aura of the Earth to gently pulse in your hands\u2026leading into a gentle pulsing heart-beat rhythm.\nAllow this beating of divine love and tenderness to infuse the energy grid, literally seeing this grid expand and contract around the earth within this glow of the aura. This grid connects all things: plants, animals, people and places. You may feel many emotions of anger, hate, resentment and fear \u2013 the feelings of so many in the world. Take those emotions and transmute them into love and continue being the conduit of light sending back 10-fold of love to every negative emotion.\nAllow this love to pulse as long as you want. Visualize the center of the Earth filling with white light until it seems ready to burst. The pulsing white energy is healing the earth and its people. It radiates with pure love and removes all negatives on the planet. The white light travels up through your body and radiates throughout you. You are pulsing with the energy as well, and you feel your body, mind and soul being re-energized through the powerful white aura of the Earth. Your cells on every level receive the Earth's energy and you are healed.\nKnow that the love you have directed towards the center of the Earth, outside the Earth and back in towards the Earth is a heart beat itself to heal the Earth and her people.\nI often do element meditations to attune myself with the powers of a specific element. These elemental mediations are helpful in getting to know the elements and refreshing your spirit. Remember, the elements are our allies and a source of magickal power. It's important to get to know these elements and understand their qualities.\nThe following is a meditation of Air. Air is associated with mind, essential qualities, knowledge, abstract learning, theories, illumination, perception, analytic thought, communication and memory.\nComplete the following mediation while you are in a relaxed and peaceful state. You can record the following script into a tape recorder and play it back for yourself over some nice relaxing music.\nIt is dawn. I find myself in a forest filled with trees. I raise my eyes and look for the sky, but the boughs looming overhead hide it from my view. As I look up, feel the cool breeze of spring brush my face, and hear the sound of the rustling leaves. Blowing, laughing from the east, Eurus brings thoughts of renewal and life. I follow the wind further into the soft shadows of the forest, inhaling deeply the strong scent of the trees. I smell, too, the wafting hints of fragrant incense. I follow where my senses lead me and come to a small clearing circled by burning censers filled with sweet-smelling oils.\nToward the east end of the circle is a staff. I walk over to the place and sit on the ground before it. I take the staff into my hands and close my eyes. First come swirling colors of white then yellow then fading darker into lavender. Then, out of the swirling fog of colors come dozens of small, lithe figures that swirl around as if carried on the wind itself. They dance around my head and body, swirling around as if caught in a dance to unheard music. Then, taking my attention from the spirits of the air, I look up to find further figures emerging from the mist.\nFirst comes a woman with the beauty of the dawn. As she steps forward, each of the four winds, in turn, hasten from around here and fly, one to each of the four corners of the earth. As she fades, there comes another whose form seems insubstantial, and seems to constitute both the image of a woman and a cow at the same time. I look closely, as I attempt to solidify them into one form, but I cannot. As this image fades, I am presented with the forms of two women who immediately remind me of the nighttime sky. One glows with the pale light of the full moon, and her eyes hold the fullness of hidden knowledge. The other, whom also seems to radiate cool starlight, seems to embody the possibilities of many lifetimes. In due course, these figures too fade and I am left with only the mist of the elemental world of air.\nSuddenly, three male figures issue from the mist. They seem to be three aspects of one man, but each of their attributes differs. The first glows with the light of the moon, seeming to be its protector. The second and third appear almost identical, except that one has a winged helmet and shoes. As I sit and study them, they begin to fade, as does the fog. When the mist clears, I find myself sitting within the circle still clutching the staff in my hands. I place the staff once again on the ground and rise. I thank the element of air, and all it is associated with for sharing with me its wisdom, and leave the circle.\nMy favorite types of spells are practical spells. Of course, I'm all for big blown out rituals with mighty purposes. But the day-to-day magick is what keeps me interested in my religion. The fact that I can say a few words and find my car keys, or will someone to call me. This is the interesting part to me. The following are a few of my favorite \"practical\" spells that are small yet powerful.\n* A picture of the lost item. If you do not have a picture write the name of the object or draw a picture on a blank white piece of paper.\nTake the picture and put it on your altar. Light the orange candle and place it on the upper corner of the picture. You might want to cast a circle for your own protection, and depending upon how much need there is for the lost objects.\nRepeat until the item is returned to you, or you hear word of the item's whereabouts.\nPut a lot of feeling in stopping rain from falling from the clouds and it will not fall, it will simply move to an area away from where you are and then fall. I use this when I need to finish my yard work before the rain falls.\nI can't tell you how often I used this one when my children were growing up. It is very effective!\nThis is a really great one to use while waiting in line. It can be done very discreetly before you enter the store.\nDraw a Pentacle on your left hand using a blessed red pen. Visualize a sand clock with swiftly falling sand as you draw.\nCast the spell, so mote it be!\nThe spell will last for 24 hours or until the pentacle is erased, naturally, or washed.\nAnd here is its counterpart, a spell to slow down time.\nThis time draw the Pentacle on your right hand using a blue pen. Visualize sand falling slowly and place your right hand on your forehead.\nThe last two spells require special pens used just for magick. If you were wondering how to do that, here are simple instructions.\nAlways write with the truth.\nThree times blessed with moonlight!\nBlessed be! Serve me right!\nInk so black as the night.\nInk so red as my blood.\nInk so blue as the sapphire.\nI hope you have fun with these practical spells!\nOne of my favorite magickal tools is hanging on my dining room wall. No its not a pentacle, broomstick or athame. It's my telephone!\nAre telephones magickal? I certainly think so. Hopefully reading this message will make you see this necessary electronic device in a whole new light.\nThe phone focuses and directs energy, which are basic principles of magick. When you make a phone call you are sending your intent via waves of electricity. This electricity goes through copper wires, or light pulses through fiber optics. In either case, you are directing your energy exactly where you choose it to go. Think about this next time you are doing spellwork and need a boost of confidence.\nYou perform energy direction everyday!\nBells are magickal tools that project energy through sound, just like the ringing of a telephone. When you place a call to someone, you fill his or her home with the ringing of a bell. This is akin to sending them a blessing which you can attach your good intentions for health, wealth and happiness. You can direct energy toward them with your ring.\nAnswering machines are magick as well. Many spells involve charging times and storing energy within it to accomplish a certain purpose. When we leave a message for a friend or loved one, we are charging the answering machine with our wishes. It can store the magickal energy that we send in the form of a message. Callers create a charm of sorts when they leave a message. They charge a physical and tangible item in your home with a bit of their energy.\nAnd then when you play the message, the energy is released.\nColor magick can be worked with your choice of phone colors. Phones come in many colors. Pick a color that speaks to you and serves your home. White is a good all around phone color. Picking a green phone for a business line is a great option. Red phones can help attract suitors. The choices are endless.\nSome people use cords in magick and store up energy in the knots and tangles.\nIf you have an older phone in your home, you can use its cord in this fashion.\nInstead of viewing the tangles as annoyances, see them as a reflection of your daily life. Little by little and day by day the cord slowly took its shape due to your influence. You can perform a simple ritual of untangling these cords and feel your tensions and emotional knots flow away. You can also wrap the cord around an object you used in a spell. If you are waiting for a call on a new house, wrap the cord around a green candle or appropriate stone.\nPhone wires are made of copper and the circuitry is made of tiny silicon chips. Both are natural objects coming from the earth. The purposes for each can be found on correspondence charts. Once you look them up, act accordingly to their purposes when you use the phone.\nNumbers hold magick as well. Your phone number, and the numbers of others are unique and hold an energy all their own. Your phone number can be used as a means of divination. If you are doing a spell involving another person, you can use their phone number.\nThe phone network is a reflection of the Pagan understanding that we are all connected. In the phone network, that is the literal case. There is a physical link connecting you to others through your phone wire. When you hold your phone in your hand, you are connecting with your neighbors, your friends and the world. The copper wires that exist between your phone and all the others connect you to everything.\nAre you beginning to see the phone in a new light? Apply this thinking to the rest of your household items. I'm sure you'll be surprised by just how much magick you have in your home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nigerian National Petroleum Corporation (NNPC) on Sunday disclosed that between July 2017 and July 2018, the organisation recorded 1,858 cases of pipeline vandalism.\nAccording to the NNPC Monthly Financial and Operations report for July 2018, a total of 204 pipeline points were broken in July alone, of which 16 were either failed to be welded or ruptured.\nThe number of vandalised pipeline points for June were revised up to 188, up from the 165 previously reported. Ibadan-Mosimi accounted for 66 percent or 124 points of the total vandalised pipeline, while Aba-Enugu, PHC-Aba and other locations accounted for the rest.\nIn July, a total of 230.45 billion cubic feet of natural gas was produced in the country, a daily average production of 7,678.17 million standard cubic feet. This was 8.81 percent high than what was produced in the month of June.\nTotal gas production from July 2017 to July 2018 stood at 3,084.09BCF, representing an average daily production of 7,834.62mmscfd. Accordingly, the daily average natural gas supply to power plants across the country was put at 744.86mmscfd, equivalent to power generation of 2,898MW.\nMeanwhile, Kaduna refinery remained dormant throughout the month of May and June despite receiving up to 78,833 metric tonnes of crude oil during the months.\nAccording to NNPC's June report, Kaduna Refinery and Petrochemical Company (KRPC) refined zero crude oil during the two months covered in that report.\nBe the first to comment on \"NNPC Records 1,858 Pipeline Vandalism Cases in 12 Months\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Individual workshops can be half or full day intensive sessions. They can be tailored to your exact needs, to suit your abilities and skills with horses. The clients own horse can be used or one from our herd (if visiting North River) that we have available. Which horse is used is not important, as the horse has an inherent way of getting to the heart of the matter or what's on top at the time.\nWorkshops are not hard on the horse as there are plenty of breaks for processing (talking) about what has happened, and exploring different approaches to the situations.\nArrange a group for individual sessions in one location is also an option.\nThe sessions are not just limited to the arena or round pen. In fact some of the best sessions have been carried out on some of the 40 kilometers of trails we have access to.\nThis is possible subject to location and our availabilty in your area. Get a group together (minimum 4) and the Pricing Philosophy applies.\nThis is possible subject to location and our availabilty in your area. Get a group together (minimum 4) for individual sessions and the Pricing Philosophy applies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Rural Postal Life Insurance scheme was started in the year 1995 by the government after the Malhotra Committee made it known in 1994 that only 22% of the insurable population is insured. The government then took the initiative to popularise life insurance and also to increase insurance penetration. Thus, the Rural Postal Life Insurance scheme was created since it was believed that the Branch Postmasters hold a position of trust among the people and will hence be able to influence the rural public.\nAny individual residing in the rural regions of the country can purchase policies under the Rural Postal Life Insurance scheme. They are easily accessible since they are available at nearby post-offices and does not require one to visit a branch office of an insurance company. A major chunk of the country purchases plans under the Rural Postal Life Insurance scheme to avail benefits and bonuses.\nIf the life assured individual expires during the policy term, the death benefit equal to the sum assured amount plus accumulated bonuses is payable to the nominee.\nIn case the individual survives the complete term of the policy, a maturity benefit is payable to him\/her.\nIndividuals between ages 19 years and 55 years can avail Rural Postal Life Insurance policies. The maximum age at entry for the Gram Priya plan is 40 years.\nProspective policy buyers can choose a sum assured amount of minimum Rs.10,000 and maximum Rs.3 lakh.\nLoans can be availed from the policy after a certain period, as mentioned in the policy, has been completed.\nAlong with the benefits offered, accumulated bonuses are also provided.\nAll plan types require a medical examination. Individuals who do not undergo a medical examination will not be permitted to opt for a sum assured amount more than Rs.25,000. Also, the maximum age at entry for such policies is 35 years.\nThe advantage of Rural Postal Life Insurance policies is that high returns are offered for low-cost policies.\nLife insurance policies that were issued on or before 31.03.2012 qualify for tax deduction under section 80C of the Income Tax Act, 1961. The maximum limit for such policies is 20% of the sum assured.\nLife insurance policies that were issued on or before 01.04.2012 qualify for tax deduction under section 80C of the Income Tax Act, 1961. The maximum limit for such policies is 10% of the sum assured.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SUPERIOR LUXURY: 3-passenger seat with industry-first. Industry leading storage. 5-position adjustable handlebar.\n310 MARINE GRADE SUPERCHARGED ENGINE: Digital Fuel Injection-fed 1,498cc inline four-cylinder engine is the reigning king of horsepower, and its crown is the high-tech Eaton Twin Vortices Series\u00ae Supercharger. Eaton TVS is designed to deliver large quantities of boost at all rpms from idle to redline. Working in conjunction with an intercooler, it supplies the engine with an uninterrupted flow of cool, dense air for maximum performance gain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Respectful people are self-disciplined and courteous, and they protect the rights and dignity of others.\nWay to go Mt. Lehman Elementary!!\nOur Character Club received a City of Heroes Award for our character plan and received $200 to help put the plan into action. Two of our students attended the City of Heroes ceremony to accept the award.\nEach class in the school has taken on one of the values of the City of Character and they are taking turns presenting them at assemblies. So far, our Grade 2\/3 class has presented respect by making \"Sparkles\" for students in the school saying what they respect about that student. Our Grade 1\/2 class presented responsibility by writing about things they would commit to do that show responsibility and reading their pledges at an assembly.\nMartin Luther King Jr. gave an incredible speech in which he said: \"I have a dream that my four little children will one day live in a nation where they will not be judged by the colour of their skin but by the content of their character.\"\nMartin Luther King Jr. Day was recently observed by our southern neighbours, and it's a great opportunity for us to take stock on how we're doing within our city limits on this challenge. It's often easy to judge each other by our outward appearances. Sometimes we can miss out of a wonderful friendship because we start out with assumptions, and don't take the time to get to know people by 'the content of their character'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wondering how much heat you need? The energy wise website has a great calculator, that will tell you what kind of heat you need in your neck of the woods. Depending on what type of house you live in , insulation etc.\nSo click on the link above and have a nosey , you will probably find you don't need as much as you thought you did.\nAlso pays to keep in mind that our pellet fires are designed to heat houses not rooms, and most of them will happily heat anywhere form a 100Sm to 150Sm, and when comparing Pellet fires to Wood burners remember that wood burners are rated on how much thay can consume vs a pellet fire that is rated on what it puts into the room. So an 18kw wood burner is roughly equivalent to a 10kw pellet fire.\nPellet Fires have electronics, motors and moving parts. They need reqular maintenance as well as specialist knowledge to keep them operating efficiently.\nSo its really important that you get your fire serviced. We recommend once a year or once you have burnt a tonne of fuel in your fire. At Otago Pellet Fires we have two technicians who have the specialist know how to keep your fire running at full tilt year in and year out .\nWhen you purchase a tonne of Natures Flame wood pellets from Otago pellet fires you will receive a $51.00 Discount off your next service.\nHow the Rika range is manufactured!\nOur First Post! exciting times :). This is where I will be adding the more technical information , cool news in the land of the wood pellet and nifty stuff related to our company, our interests and what we do.\nFor the first post I've decided to share a video about the manufacturing process of our premier range of fires the Rika's!\nThey are made by young Mo-Hawk sporting men in a state of the art factory in Austria!\nHow a Rika Fire is made.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"By clicking Register or by completing this Registration Form for Blu Richpark, I would like to receive emails and other electronic messages from Richpark Homes including news, updates, publications and promotions regarding their products and services. I understand that I can withdraw my consent at any time in the future.\nBetween the shores of Georgian Bay and the majestic hillsides of the Blue Mountains lies historic Thornbury, Ontario.\nso extraordinary, Blu will instantly become your new favourite colour.\nThese bold modern designs will make your life look better from every perspective.\nThe homes at Blu are more than just places to live.\nThey provide an incredible lifestyle experience from the moment you walk through the door.\nbathrooms, spacious walk-in closets and elegant architectural details.\nWHAT GOES INTO A RICHPARK HOME?\nThe Principals of Richpark Homes bring together combined backgrounds in urban planning, residential development and the financial sector to create a management team that is uniquely suited to building exceptional homes and communities all across the region. We have decades of experience, and we're putting them to good work by offering our homebuyers remarkable value on high quality family homes. With our beginning in the world of custom home building, we earned our reputation by developing an exciting attention to detail and design excellence, and we carry these qualities forward with the creation of every new family community that we build. The only thing greater than our dedication to quality is our commitment to complete customer satisfaction. Our dedicated sales and service team will be with you every step of the way throughout the home buying and construction process, and we'll continue to provide excellent service and support well beyond moving day. We thank you for considering us as your homebuilder, we hope you'll let us show you the unique value of a Richpark home.\nCollingwood and the surrounding area.\nHistoric buildings. Charming cafes and restaurants. The rushing waters of the Beaver River right through the heart of town, flowing gently out to Georgian Bay. This is the essence of Thornbury, a historic town with surprisingly modern amenities. Whether you're enjoying brunch with a friend, checking out a local art gallery, or getting all your essential home shopping taken care of, you'll never have to stray far from home to meet all your needs.\nDowntown Thornbury has it all. Enjoy a beverage with friends overlooking the Mill Pond at The Dam Gastropub or Bridges Tavern. Treat yourself to something sweet at Thornbury Bakery Caf\u00e9. Find a special treasure for your new home at C&G Solid Wood Furniture.\nA short drive into Collingwood gives you easy access to all your major lifestyle amenities. There's Walmart, Loblaws, The Home Depot, Galaxy Cinemas, Canadian Tire and more. Not to mention a wide array of restaurants, brew pubs, and specialty boutiques for you to enjoy.\nWhether you're taking a hike on the Georgian Trail (located right beside Blu!), walking along the shores of Georgian Bay, or out for a grander adventure at one of the 10+ nearby provincial parks and conservation areas, you can connect with a more natural lifestyle here.\nThere are 10 top-rated golf clubs located just minutes away from Blu, including The Golf Club at Lora Bay, Blue Mountain Golf, Cranberry Golf Course, OslerBrook Golf and Country Club and too many more to mention.\nBlue Mountain Resort is located just an 8 minute drive from Blu. Many people know that this is Southern Ontario's #1 destination for skiing and snowboarding, but Blue Mountain offers so much more than this. With endless 4-season activities, delicious gourmet eateries and incredible boutique shopping there is never a dull moment here.\nLearn more about everything we're doing to get ready for home construction at Blu!\nAfter much anticipation we're excited to announce that site construction has now started at Blu. This is a big first milestone in the development of a new home community, laying the groundwork for the home construction that is to follow.\nMarcel Gery, won a Bronze Medal at the 1992 Olympics in Barcelona for swimming, he brings that same focus and determination to Real Estate that got him to the podium in Spain. Marcel brings a wealth of experience, outstanding communication\/negotiation skills and ensures the buying\/selling process runs swimmingly! He builds relationships and understands the needs of today's buyers and sellers.\nBarb Blaser is a force to be reckoned with. She brings years of experience and a professional energy to the table that few can match. Barb resides in Thornbury and is actively selling in Southern Georgian Bay area and brings her personal experience, strong knowledge of the market and all the advantages of years of experience to sell your home at the best value and\/or find your dream home. Barb maintains a loyal following of clients who continue to refer friends and family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted in adventure, climbing, outdoors, skiing, Uncategorized, winter and tagged alpine, alpine climbing, athletes, backcountry, climbing, extreme, idaho, mountaineering, mountains, photography, photos, ski mountaineering, skiing, snow, sports, spring, summer, sun valley by wanderhigh. Bookmark the permalink.\nSo great! Such a pro story and photos. I am amazed at your photos. Inspired to go see if I can do this in the spring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Telecommunication services are regulated by the federal and state governmental bodies. These regulations provide a public service by giving consistent explanations of generally available telephone services offered on a non-discriminatory basis. Cox strives to give friendly, easy to understand information about its phone services. To that end, we provide clearly conforming detail about our services. This information is intended as a supplement to our site's information.\nCox intrastate phone service is provided to you, depending on the state where you receive service, pursuant to our tariffs on file with state public utilities commissions or a state Telephone Service Guide. Certain Business and Residential service offers are subject to specific terms and conditions in addition to the applicable state tariff or Service Guide. Interstate and international services are offered under the following terms and conditions.\nAs a Cox customer, you have the ability to block third party charges on your bill, such as collect and operator assisted calls. Please call your local customer service department if you would like to block third party charges.\nRegulatory documents for in-state and access services vary by state in addition to guides to our service and information about your rights as a telephone consumer.\nThe terms and conditions contained in the Service Guide below are for products and services provided by Cox Iowa Telcom, LLC (\"Services\"). Pursuant to Iowa Senate File 2195, the effective date of this Service Guide is July 1, 2014 for customers subscribing to Services as of July 1, 2014, or the date customers subscribe to Services after July 1, 2014 (\"Effective Date\"). For customers already subscribing to Services as of July 1, 2014, prior to the Effective Date, the existing Cox tariffs govern the Services. Specific rates and charges are available by contacting Cox or by viewing the pricing page for Iowa locations.\nThe terms and conditions contained in the Service Guide below are for products and services provided by Cox Kansas Telcom, LLC (\"Services\"). Pursuant to Kansas House Bill 2201, the effective date of this Service Guide is July 1, 2013 for customers subscribing to Services as of July 1, 2013, or the date customers subscribe to Services after July 1, 2013 (\"Effective Date\"). For customers already subscribing to Services as of July 1, 2013, prior to the Effective Date, the existing Cox tariffs govern the Services. Specific rates and charges are available by contacting Cox or by viewing the pricing page for Kansas locations.\nThe terms and conditions contained in the Service Guide below are for products and services provided by Cox Nevada Telcom, LLC (\"Services\"). Pursuant to NRS 704.68875, the effective date of this Service Guide is January 1, 2008 for customers subscribing to Services as of January 1, 2008, or the date customers subscribe to Services after January 1, 2008 (\"Effective Date\"). For customers already subscribing to Services as of January 1, 2008, prior to the Effective Date, the existing Cox tariffs govern the Services. Specific rates and charges are available by contacting Cox or by viewing the pricing page for Nevada locations.\nDomestic state-to-state Long Distance AND in-state Long Distance prices are 0.18\u00a2 per minute without any monthly fees or time of day\/length of call restrictions. The basic rate option includes all 50 states, the District of Columbia, Puerto Rico, the U.S. Virgin Islands, Guam, American Samoa and CNMI.\nThe applicable monthly recurring charge for the Cox Nationwide 1,000 Minute Call Package will be billed in advance in accordance with this tariff as applicable to the payment of recurring charges for local exchange service. A Customer will automatically be enrolled in the Cox Simply Five Call Plan for any minutes in excess of the 1,000 minute threshold. Minutes under Simply Five will be billed at 5 cents per minute and partial minutes will be rounded up to the next whole minute. The toll calls under this plan are direct-dialed from the customer's designated line to any destination within the 50 states, the District of Columbia, Puerto Rico, the U.S. Virgin Islands, Guam, CNMI, and American Samoa. The intraLATA and interLATA toll minutes included in this plan (1) shall apply exclusively to direct-dialed calls, (2) have no cash value for refund purposes, (3) are not transferable or assignable, and (4) shall not apply to operator-assisted, collect calls, calls billed to a third party or credit cards, calls to directory assistance, or calls made through per-use feature activation. For further details on rates and charges, contact us.\nWhere facilities exist and operating conditions permit, Cox unlimited long distance will be offered to Residential Customers with various phone plans. Cox unlimited long distance includes unlimited Residential minutes of use of direct-dialed intrastate and interstate long distance service to all 50 states, the District of Columbia, Canada, Puerto Rico, the U.S. Virgin Islands, Guam, American Samoa and Commonwealth of Northern Mariana Islands. In select markets with applicable available plans, unlimited long distance may also include international terminations to Mexico, China, Hong Kong, India, Norway, the European Union (Austria, Belgium, Bulgaria, Croatia, Republic of Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, and United Kingdom). An eligibility condition of unlimited long distance requires that the Customer select Cox long distance for both the Primary Interexchange Carrier (PIC) and IntraLATA Interexchange Carrier (LPIC) elections.\nThe Monthly Recurring Charge for a Customer's phone plan will be billed in advance, and will apply the first billing period after ordering the service. Cox unlimited long distance does not permit the Customer to place business calls. Cox unlimited long distance is for direct-dialed domestic (and applicable international) Residential voice usage only. If usage under unlimited long distance is not consistent with typical Residential Customer usage, at Cox's sole discretion, Cox may offer the Customer an alternative plan or suspend, restrict and\/or cancel Customer's long distance service without prior notice. Calls that are not consistent with typical Residential voice usage include, but are not limited to: use for general business purposes, commercial facsimile, auto-dialing, resale, call centers, telemarketing, multi-party conference calling, multi-party \"chat\" lines, or engaging in activities that generate minutes that result in revenue-sharing by a Customer. Callers must dial 1+ area code + 7-digit telephone number for the call to be included in unlimited long distance for calls to numbers within the North American Numbering Plan (all 50 states, the District of Columbia, Canada, Puerto Rico, the U.S. Virgin Islands, Guam, American Samoa and Commonwealth of Northern Mariana Islands). For any applicable international calls, callers must dial 011+ country code+ terminating number for the call to be included in unlimited long distance. Customers must subscribe to Cox long distance and to Cox local exchange service. Unlimited long distance is generally available to Residential Customers on a per-line basis up to a maximum of four telephone lines. Unlimited long distance is reserved for direct-dialed long distance calls and does not include calls to 900 numbers, directory assistance, per-use feature calling, calling card, operator services, international calling (unless included in the plan) and toll free calling services; such calls are subject to additional charges. Taxes, fees and other charges, including the Universal Service Fund, apply. For further details on rates and charges, contact us.\nThe Cox Online LD Plan includes both local toll and long distance calls if Cox Long Distance Service is selected as the service provider for both local toll service and long distance service. Customer must subscribe to Cox Digital Telephone service and this optional calling plan through the Cox online website to be eligible for subscription to the Online LD Plan. There is no monthly recurring charge for this plan. A rate of $0.12 per minute applies to direct dialed local toll and long distance calling. Additional charges apply for international, Calling Card, Directory Assistance, or Operator Services calls. Calls are billed in whole minute increments. Taxes, fees and other charges, including Universal Service Fund, apply to the usage charges assessed on this plan.\nThe following per-call service charges apply in addition to the per minute usage rates when applicable, except as noted*.\nBoth Local (411) and Domestic Long Distance Directory Assistance (i.e., via NPA-555-1212) are available to Customers of Cox. The Directory Assistance charge applies to each call to the Directory Assistance Operator. Up to two requests may be made on each call to Directory Assistance. The Directory Assistance charge applies to each call regardless of whether the Directory Assistance Operator is able to furnish the requested telephone number. Local Directory Assistance varies by state.\nDirectory Assistance Per Call Rate \u2013 Each DA call will cost $2.49.\nAll taxes, fees and surcharges do not apply to all customers. For a list of definitions, please visit Billing & Account Support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it safe to Live Below the Line for 5 days?\nIt's possible to maintain good nutrition whilst living below the line, but it will take careful planning. Dietary variety will be the key to a healthier challenge. You need to aim to include all of the five food groups outlined in the Australian Guide to Healthy Eating - breads and cereals, vegetables, fruit, dairy and meat or alternatives.\nYour eating pattern will also be important. Eating at least three meals a day, like breakfast, lunch and dinner, will assist in maintaining energy levels and satisfying your appetite.\nWhat sort of food should I try to buy with my $10?\nYou need to make sure you are buying foods from across the five food groups. High fibre foods such as rice, pasta, oats and lentils or legumes will help fill you up.\nFrozen or canned vegetables often include a variety of vegetables in one pack. Alternatively, buying fresh fruit or vegetables from local markets or greengrocers can be a budget friendly option.\nDairy is important too, and can be surprisingly affordable!\nMeat will probably be outside your budget. As protein alternatives consider lentils (dried or tinned), legumes (such as baked beans), eggs or tinned fish.\nLook for home brand or buy from local suppliers at markets.\nWhat sort of food should I avoid?\nFoods that do not fit into the five food groups! 'Extra' foods such as chips, chocolate, lollies, soft drink and biscuits, to name a few, will be expensive and do not provide long-lasting energy and nourishment.\nFrom a price perspective, branded foods are going to be the more expensive choice.\nAny tips or tricks you can recommend to make Live Below the Line as easy as possible?\nPlan for regular meals to satisfy your appetite.\nCombining with other people will mean you have more money at your disposal to purchase some bulk items (e.g. rice, bags of fruit).\nMaintain adequate hydration. Most people need around 2 litres of water a day. One of the benefits of regularly drinking water is that it can help fill you up. If you don't drink much water at the moment, try and increase your intake prior to commencing the challenge - and keep it up afterwards too!\nPlan ahead. Do some price checks in the supermarket and think about how you could incorporate the five food groups.\nYou may have to compromise on buying your favourite brands and the flavours you are used to.\nIs there any reason why I should not do Live Below the Line?\nIf you have a medical condition, take regular medication, have recently been unwell, in hospital or had surgery it is strongly recommended you seek approval from your treating doctor before participating in the challenge.\nIn addition, children and young adolescents need adequate energy to support growth and as such children under 15 are not recommended to take part.\nThis year practising dietitian and an accredited nutritionist Rebecca Gawthorn is taking the Live Below the Line Challenge. She shared with us some of her tips and tricks.\n1. Lentils & beans are a great source of protein, so grab a few tins for the week. You can usually get these for less than $1.\n2. It's super important you keep hydrated!! Drink plenty of water.\n3. Choosing quality carbs that have a low GI (Glycemic index) are important for Live Below the Line as they will help you stay fuller for longer and have more energy throughout the day.\nClick here to read more of her top tips!\nWe asked dietian and LBL regular Nicole Vaughan to share her tips on taking the Live Below the Line Challenge, including a meal plan, shopping list and recipes. Having spent 6 years at university and studying both psychology and dietetics, Nicole is now a clinical dietitian living and working in a hospital out in rural Victoria. She loves her job and gets to work with a variety of patients in both the hospital and the community with conditions ranging from cancer, kidney disease, or a recent stroke or bowel surgery to those with diabetes, high cholesterol, allergies or intolerances. She's completed the LBL Challenge twice and is very passionate about improving the lives of those living in poverty, by increasing access to education.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LEATHER,MOORNOOF,RUNNING BOARDS. Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\n2WHD CREW CAB SHORTBED SL PKG,4.8, BLACK CLOTH INT. V8,AUTO,TOW PKG,BED COVER,AND LINER,NEW BRAKES,TIRES,PWR,WINDOWS,PDL,AC,TILT,CRUISE,1 OWNER, Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nNEW BODY STYLE SLE EXT 4X4 Z-71,5.3 V8,BLACK PWR CLOTH SEAT,1 OWNER CLEAN CARFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nPREMIUM PKG, V6, MANUAL TRANS W\/LEATHER INT,PREMIUM PKG EXTRA CLEAN CAR ,CLEAN CARFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nTAN LEATHER,TAN TOP,AUTO TRANS,CONVERTIABLE PRCIE HAS BEEN REDUCED TO 7977.00 NOTE THE LEFT REAR SMALL WINDOW DOES NOT GO UP OR DOWN AND DOOR PANEL IS LOOSE OTHER THAT A GREAT VEHICLE Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nLOCAL TRADE V6,POWER SEAT,ALUM WHEELS,MOONROOF Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nBRIGHT RED CREW CAB 4X4 SHORBED LT1 V8 5.3 CHROME BOARDS,TOW PKG,BLACK CLOTH INT NICE CLEAN TRK Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nFuel Consumption: City: 17 mpg, Fuel Consumption: Highway: 24 mpg, Remote power door locks, Power windows, Cruise controls on steering wheel, Cruise control, 4-wheel ABS Brakes, Front Ventilated disc brakes, 1st, 2nd and 3rd row head airbags, Passenger Airbag, Side airbag, MP3 player, AM\/FM\/Satellite Radio, XM Satellite Radio, Braking Assist, ABS and Driveline Traction Control, Stability control, Privacy glass: Deep, Silver aluminum rims, Wheel Diameter: 18, Wheel Width: 7, Front fog\/driving lights, Leather steering wheel trim, Leather\/metal-look shift knob trim, Simulated wood\/metal-look dash trim, Simulated wood\/metal-look door trim, Simulated wood center console trim, Roof rails, Trip computer, External temperature display, Tachometer, Manufacturer's 0-60mph acceleration time (seconds): 8.0 s, Power remote driver mirror adjustment, Heated driver mirror, Heated passenger mirror, Electrochromatic rearview mirror, Power remote passenger mirror adjustment, Dual illuminated vanity mirrors, Compass, Heated windshield washer jets, Wiper park, Audio controls on steering wheel, Front reading lights, Anti-theft alarm system, Bucket front seats, Split rear bench, Fold forward seatback rear seats, Rear seats center armrest, Tilt and telescopic steering wheel, Speed-proportional power steering, Suspension class: Regular, Interior air filtration, Tire Pressure Monitoring System: Tire specific, Cargo area light, Max cargo capacity: 78 cu.ft., Vehicle Emissions: ULEV II, Fuel Type: Regular unleaded, Fuel Capacity: 19.8 gal., Instrumentation: Low fuel level, Clock: In-dash, Dusk sensing headlights, Driver Seat Head Restraint Whiplash Protection, Head Restraint Whiplash Protection with Passenger Seat, Coil front spring, Regular front stabilizer bar, Independent front suspension classification, Strut front suspension, Four-wheel Independent Suspension, Coil rear spring, Rear Stabilizer Bar: Regular, Independent rear suspension, Multi-link rear suspension, Front and rear suspension stabilizer bars, Variable intermittent front wipers Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nLOCAL TRADE HAS SOME RUST AROUND REAR FENDER WELLS BUT RUNS GOOD,NEW TIRES Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nQUAD CAB 4X4 ST W\/CHROME PKG, PWR PKG,TOW PKG, LOCAL TRADE CLEAN CARFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nLOCAL TRADE SES 4DR ALUM WHEELS,AUTO TRANS WSPOILER, GREAT MPG, Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nXL REG CAB SHORTBED 4X4 4.6 V8,AUTO,TILT,CHROME WHEEL COVERS,BEDLINER,CLEAN CARFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nEXTREME 2WHD LS GREAT CONDITION.LOW MILES,CLEAN CARDFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nREGUALR CAB SHORTBED 4X4 EXTRA CLEAN TRUCK Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.\nEXT CAB 4X4 SHORTBED LT,5.3 V8,3.42 AXLE LOCKING,Z71 SUS,POWER SEAT CLOTH,TOW PKG,BLUETOOTH,18IN ALUM WHEELS,LOCAL TRADE CLEAN CARFAX Our Location is: Integrity Ford Inc - 860 E Perry, Paulding, OH, 45879 Disclaimer: All vehicles subject to prior sale. We reserve the right to make changes without notice, and are not responsible for errors or omissions. All prices exclude government fees and taxes, any finance charges, and any emissions test charge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfgbx b/data_all_eng_slimpj/shuffled/split2/finalzzzbfgbx
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+{"text":"Interested in attending University of Akron Wayne College? REQUEST INFO!\nIt is the mission of Wayne College to provide high quality accessible credit and noncredit educational opportunities to the citizens of Wayne Medina and Holmes counties and surrounding areas and to be a partner with and resource for the communities and organizations it serves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I don't speak Danish, but I think it's fair to say that To \u00d8lekalender is a play on To \u00d8l (which I know means two beers) and calendar. Like in previous years I hope to get a post up roughly every five days, so you can follow along my progress. If you're here I'm guessing you've already read Part 1 (link) & Part 2 (link), so let's get into the next 5 beers!\nDay 11 - Gose to Hollywood - Gose is a historical German style of sour beer that has taken over as the newest craft beer fad in recent years. This particular To \u00d8l beer, Gose to Hollywood, is I think the beer from them that I've had the most times. Hopefully this particular example is as good as the ones I've sampled previously.\nThankfully it live up to my memories of it. It's a really nice example of a modern Gose with a light malt base, some reasonably strong citrus flavours - particularly orange peel, some saltiness, hints of coriander and nice tartness. I think this is one of my favourite Gose's and one that is just perfect for the hotter days of our summer that are coming up soon.\nDay 12 - My Sour Pils - I was really excited to see My Sour Pils come out of the box on Day 12. I enjoyed the regular My Pils when it came out on Day 8 and thought the idea of a sour Pilsner could work really well. It's brewed using a sour mash, which is an idea I associate more with whiskey and haven't really heard of in brewing before.\nMy Sour Pils was delicious! It brought together the crispness of a nice Pilsner, with some grass and citrus hop flavours, with the refreshing tartness of a sour. The hops provided really good bitterness and plenty of lemon flavour, which was complimented by the citric nature of the tartness. At 4% ABV I think I could drink these all day!\nDay 13 - ?! IPA - Day 13 brought about the \"?!\" IPA. Apparently the name comes from the phrase \"Say What?!\"... To \u00d8l market this as a straight up and down American IPA brewed with Simcoe, Mosaic and Citra hops. I'm really interested to try this after some of the really aggressively hopped beers that have popped up in the To \u00d8lekalender so far.\nThis beer was excellent and tasting particularly fresh. As promised it was a straight forward American IPA with plenty of bitterness. It had all the big American hop flavour profiles covered; with pine, tropical fruit and citrus all working in harmony. Of the tropical fruit flavours, mango and lychee stood out the most for me. I would really quite like to try this beer again.\nDay 14 - Brett & Butter - Firstly; Brett & Butter! What a phenomenal name! Also added to by the disclaimer on the bottle that it doesn't contain any butter. Basically this is a Blonde Ale, that has been heavily hopped with Mosaic, and then been innoculated with Brettanomyces Custersianus - one of the more recently discovered variations of Brett.\nThis is the first beer in this pack that I've been a little disappointed in. It had lovely fruitiness (mango and rockmelon in particular) from both the hops and the Brett, but had minimal funk or tartness. The beer is a little too light for my liking and doesn't offer enough of any one element. I'd be really interested to hear the thoughts of other people who've tried this beer.\nDay 15 - Mochaccino Messiah - I've been hoping this beer would pop out of the To \u00d8lekalender since Day 1! It's a Coffee Brown Ale which has a really big reputation. It's also one that I've either missed or simply bypassed, so I'm really keen to see it today! It's a beer intending to imitate a morning coffee and I'm really looking forward to trying it!\nA morning coffee it is! The first thing I smelt was a bit of lactose sweetness, which was quickly swamped by lots of coffee and roast aromas. There's some nice caramel flavours that comes through as well as some citrus and tropical fruit flavours. It's quite bitter; with both the hops and roasted coffee contributing to this. It's a lovely Brown Ale with a strong coffee influence and plenty of hops!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Explore Raja Ampat, a group of the most beautiful islands on the planet, in Indonesia's West Papua Province. Travel one of the world's best-kept secrets offering spectacular snorkeling and diving, pristine beaches and feeling of paradise. Discover what to see and do in our Raja Ampat guide.\nRaja Ampat is a remote archipelago comprising of over 1500 islands in Indonesia's province of West Papua, and we dare to say it remains one of the world's last destinations protected from mass tourism, but its popularity is increasing every year and the rumors about pristine beaches, unique marine biodiversity, postcard-like landscape, and friendly Papuans, are spreading quickly.\nRaja Ampat is the least developed and also easternmost islands of Indonesia and the distance and remoteness determines both the life of locals and the form of tourism.\nThe advantage and concurrently disadvantage for those visiting the area is already mentioned remoteness and time-consuming journey, but on the other hand, this is what saves Raja Ampat from the crowds and ensures that Raja Ampat will be visited only by those genuinely interested in its natural beauty.\nWe've put together a list of best things you can see and do in Raja Ampat. Unsurprisingly, the vast majority of activities takes place in the water or in its close proximity.\nRaja Ampat's underwater life is fantastic and hard to compete with especially when Indonesian waters surrounding these islands are known as the most biodiverse on the planet. Simply put, the best diving and snorkeling awaits here for you. The numbers are also incredible: 1,508 fish species, 537 coral species, and 699 mollusk species altogether with mantas, orcas, dolphins, turtles and many more create real underwater heaven.\nMost homestays in Raja Ampat have perfect snorkeling spots right at the door, and once you explore the nearest places, hosts are willing to sail you to other unexplored underwater gems which only locals know, and you won't find them in any guidebook. Some homestays offer snorkeling gear to rent, but of course, it's best to take your own to ensure it fits.\nSunsets on Raja Ampat's islands are a chapter on its own.\nWe've never seen so many breathtaking sunsets at one place in our lives.\nEvery evening we couldn't wait to walk on the beach, sit on the sand and enjoy the movie. If you're looking for a place to capture the most picturesque sunsets, Raja Ampat is worth it for sure. We believe the same is true of sunrises, but we were every day so exhausted after the all-day spent in the water, that we were not early-birds this time.\nThis view will never pall and prepare that it will leave you speechless for a while. Arrive by boat or paddle a canoe to the small pier and climb few (hundreds) wooden stairs.\nWe guarantee you won't want to leave.\nWe spent quite a long time on the viewing platform and had to return repeatedly the other days. From the top, you can see an iconic karst island seascape. The mushroom-shaped rocks are towering from the water which is almost unnaturally blue and crystal clear, we could've even seen a turtle swimming in the sea from the viewing point.\nIf you are craving for more views similar to Piaynemo, you can consider visiting Wayag, but it is far, and the transport is pricey.\nA big plus when it comes to the decision which island to visit is that you don't have to stick with one island only, but you can freely travel around and enjoy different snorkeling\/diving locations, stay in several homestays and explore remote beaches in the Raja Ampat area.\nFor inspiration, we visited Friwen, Kri, Piaynemo, and Arborek in two weeks' time. These islands are not that far away from each other and thanks to that we didn't have to spend that much time with transfers and could've enjoyed island's activities instead.\nWhat we enjoyed the most was our accommodation in homestays, and the interaction between hosts and visitors which illustrated the relaxed atmosphere in the true sense of the term.\nWe could see our hosts' everyday life and eat food their families prepared (and caught) that day. Locals are friendly and seemed genuinely interested in our lives the same way we wanted to know more about theirs. Even though their English is very basic and in some cases none, we were able to communicate without a problem and had a lot of fun when translating what we wanted to say into their language.\nDuring two weeks spent in Indonesia's province of West Papua, we tried to see as many islands as we could, but on the other hand, we didn't want to rush and wanted to enjoy our vacation at the same time.\nBecause our time was limited, we inevitably had to choose our destinations carefully, and after research, we came up with a list of most beautiful islands in Raja Ampat we definitely wanted to see.\nA small island with incredibly white sand, family homestays and a beach we had just for ourselves.\nThe main highlight is Friwen Wall just opposite the island with the stunning underwater world. It takes about an hour to swim the whole length of the Wall, so if you are unsure about your swimming skills or level of fitness, tell so to the boatman to wait for you nearby, because in our case, he probably got a feeling that we're doing fine and left to wait for us at the very end (some of us were pretty exhausted when finally reached a small beach at the end of the Wall).\nKri is a forested island, much bigger and more developed than Friwen.\nAccommodation is available on both sides of the island which is split by the hill in the middle. Jungle trekking is not possible because of lack of trails and steep terrain, so climbing the hill to get to the other side of the island is probably the only option if you're looking for different kind of activities. We found a lodge near the pier where the snorkeling was surprisingly good and diverse.\nThere were small sharks (we've been told that when sharks are similarly big as a human, they shouldn't attack, but who knows) and poisonous stone fishes on the seabed so watch your steps.\nA super small strip of send and the accommodation on the pier was our base to explore the Piaynemo area and its surroundings.\nWe enjoyed two days of canoeing, repeating visits to Piaynemo Lookout, new snorkeling locations, hidden coves and also hikes to get another superb (and not that known) views of the Raja Ampat's seascape. Because of minimal pollutions and light effects, this spot is great to watch stars and night sky. You don't need to stay in Piaynemo to enjoy the postcard-like views.\nYou can reach it for example from Arborek in about 90 minutes boat ride.\nThe last, tiny (you can walk around in about 30 minutes), but the most touristic island we visited has its charm and atmosphere. Arborek even has unified village (part of the Indonesian government's tourist village program) with the main street through it, houses with blue roofs, sufficient homestay options and football playground. Arborek is near to the famed Manta Point, snorkeling and dive sites where you can swim with mantas.\nLocals are incredibly friendly, and little kids running around are everything but shy. Their evening dancing and singing entertainment is fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed to generate heat using waste wood by-product pellets, the KOZI Previa is a highly efficient, environmentally-friendly, economically priced unit. KOZI Pellet appliances are considered \"clean burning\" as they produce low environmental emissions and ash content.\nThe state-of-the-art MX\u2122 2 Controller acts as a trouble-shooting guide. At a glance, this diagnostic system. will relay the operating status of your unit to ensure your safety and peace of mind.\nThe Power Venting System eliminates any chances of back draft. The flexibility of burning either outside or room air allows for custom operation to suit any application. A convenient side mounted MX\u2122 2 controller allows the owner to manually adjust the fuel feed rate and fan speed to suit individual needs for temperature control.\nWith its attractive finish, the KOZI Previa will accent the decor of any home. The Previa is available in a variety of different colors and finishes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pictured at the Breast Cancer Paint it Pink Awareness night are Niamh and Michael Gilligan, and Kitty Hayes, with Aoife Kahanov of Pilates Studio, Kildare town, who were all lending their support for Breast Cancer Research and Support Services event in the studio Friday evening, October 14.\nOctober is Breast Cancer Awareness Month. Every year Pilates Studio Kildare holds special events to help raise life-saving funds in support for Breast Cancer Research and Support Services. This year's first event was a special screening of The Life of Eve. Photo: Tony Keane.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Home Enlivening Are You Searching For Something More?\nLife is meant to be more, it is the reason you are here!\nAll of the challenges, pain, and sufferings are only a small portion of what this life experience is meant to be about, don't ever give up on your dreams for more.\nThere is a sensational whisper constantly reminding you of your greatness, this soft and gentle murmur is the expression of your Soul, do you hear it, it has been trying to get your attention for the longest time?\nOur Soul communicates with us sometimes in the most exquisite ways.\nA voice in our head appears inconspicuously with messages from the Universe, memos translated into loving counsel recommended from an authority that works tirelessly to guide us.\nYour work is to trust what you hear from that place deep down inside your core, don't be afraid of what you might hear. Change is growth and it makes us different.\nTaking chances that might separate us from who we have always been will require a valour of fearlessness, there is no gallantry if you won't be courageous.\nBegin to work on how to be daring, takes leaps of faith which test your comfort zones. One small step will lead to big ones in no time at all.\nThere is nothing you can't do if you have the audacity to take bold risks. The more messages from your Soul you are able to decipher the louder she will speak to you, once she has your attention she will be relentless with her guidance.\nThere is no greater gift to grant ourself than to awaken our Spirit and embrace her with open arms.\nRemember sometimes there may be a lull, a time when everything stops, everything is silent and everything stands still, let this be ok and allow it to be a time to rest.\nWhen the Soul knows there is greatness to come, for a time hushed becomes our world.\nYou are ready for a change I know you are, I can feel it. Get ready.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From now through May 31, the Shipley Art Gallery in Gateshead will be showcasing 70 years of Penguin covers. Not just a history of English graphic design, the exhibition shows the work of several Penguin designers who played important roles in modern fine printing: Jan Tschichold, Hans Schmoller, and Reynolds Stone. For more details, click here.\nYou are currently browsing the archives for the Jan Tschichold category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BIGonDRY srl affords a wide range of microprocessor control mechanisms.\nAll automatisms, only manufactured in Italy, represent the engineering excellence in the wood sector.\nThese devices provide a progressive drying setting, employing technologically advanced probes, for a significant energy saving.\nOur control units are equipped with an easy to use interface, particularly important in case of non-computerized systems.\nBasically the artificial drying leads to a rapid and low-cost removal of water from wood, in order to prevent any moisture gradients uprise or surplus.\nsterilization management\/control function according to FAO ISPM-15 standards.\nThe drying cycle is fully controlled.\nEvery single cycle is customized during the drying process and each step can be adjusted upon individual needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you organizing Midwest PHP 2017?\nAre you an organizer of Midwest PHP 2017 ?\nMidwest PHP is the FUN conference. This is our fifth annual conference, and each year it gets better and better. Our goal is to share best practices, ideas, and techniques about building state-of-the-art software applications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's not get into the validity of the supposedly test item above because that is not what I intend to talk about at this moment.\nFor the previous one and a half year, I have entered a battlefield every school day.\nI believe there are no conflicts, but then I am only half alive when I leave the classroom. Unfortunately, I do not have extra lives and potions to instantly nurse me back to health. Idealism and my youth are what \u2013 and maybe, the only things \u2013 I have.\nFront and center is where I should be. But does it matter? The room turns into a market, a jungle in a few moments. It becomes difficult to concentrate. Movements. More movements. The class just ended, and my students wanted to break free.\nI never thought that teaching was going to be this difficult.\nAs a student, I worked hard and got tired like everyone else. In college, I would walk as fast as I could from building to building, climb up an awful number of stairs for my classes on the topmost floors, attend meetings in between classes and then organization activities after classes, trail people for interviews, sleep late to transcribe those interviews, catch up on readings and finish papers and do everything else. And by the end of each day (whichever time that may be) I would beg the heavens that I wake up on time for my first class in the morning. Coffee and vitamins helped me get through those days. Now, I need more than those. I need miracles.\nIt was definitely easier being the one sitting inside the room. But it's now the other way around. I am still young, but I guess, energy bars can help me keep up with thirty heads per class. I met exactly 99 new faces on my first day of teaching. At that point I questioned myself on how I could match a name to a face successfully in the least possible time. Later on, I realized that it would be one of the least stressful tasks I'd ever have \u2013 the most would be trying my best not to screw up with my teaching and consequently, to an entire set of learners.\nAfter each lesson we took, the faces that were once unfamiliar to me, started giving me little things to ponder on after each time the bell rang. And the first thing I observed in them is that they never \u2013 ever \u2013 run out of energy. They amaze me. I never wondered how children seemed to never tire themselves out until I got in close contact with them. I asked my experienced co-teachers who, by the way, cheered me on throughout the whole year, if they thought of the same thing I did, and they agreed with me. It should be good news for me to have such active children. So instead of dreading another streak of energy that I would lose per class, I decided to plan my activities well. I still did not forget to take my vitamins. In time, everything became more bearable to me, and I was once again reminded that I should enjoy what I do.\nWhat is more amazing about students is the things they can think of and the things they can do.\nI learned never to limit what people are capable of doing on the mere basis of the decade they come from. In the case of my students, I realized that they are the last people I should underestimate. I never intended to do so, but I could not help but be surprised to see and witness what they have done for class. Then I knew that I should be the one pushing them to do more than what they already can.\nI am at least twice my students' age, and I haven't understood more things than they do. Knowledge does not necessarily translate to understanding. It has a natural propensity to lead me to confusion. With age comes not only wisdom but with uncertainties well. Younger people may look at things more simply and do things in a less complicated way, but they sure are a way lot more open-minded and imaginative than I am.\nThey can have a cat for a classmate.\nAn alien may just suddenly burst inside the classroom any time of the day.\nThey are invincible until their bruises and bumps prove otherwise.\nAnother thing: my students definitely are more creative and more daring than me. And actually, I want to pick these qualities up from them.\nI am not yet a parent, and I do not plan to become one anytime soon. But even to my students last year who were not related to me by blood, I already found it hard to let go and to let them do things on their own. Whenever there were class activities or school presentations, they would work by themselves with only a minimum supervision from me. I was proud that they were independent. But each time one of my students or the whole class got up in front \u2013 whether it was on a short platform or on a big stage, I would always wish that they do well. I did not want them to feel embarrassed for making a mistake. Of course, I would tell them that there is nothing bad with making a mistake and that everything was going to be okay just like what my own parents used tell me before. But still.\nI discovered all these things before my first year of teaching ended. And I also realized the best thing about being a teacher. For me, it is that feeling of comfort that I get inside the classroom. I do not need to pretend to be someone else that I am not in front of my students. Inside the classroom I am confident about myself. And it was easy to please my kids. They laughed at the very few jokes I made. It was also never difficult to tell them that I made a mistake. They did not judge me even if I did not know everything because I really don't. I told each of my class that at least once. And I will never pretend that I do. What I am only certain of is that I will do the best that I can to share with them everything I know.\nI was told before that the first year of teaching was the most difficult.\nI never cried during my undergraduate years but I cried more than once by my fifth month into teaching. I think I expected too much of myself. I did not want to disappoint anyone. And yet is this what I want to do? I had to ask myself this right after I submitted the final set of grades for the school year.\nYes, I do. I figured out that I want to teach and so much more. If the first year is the most difficult, then I'm looking at better days to come. I may not become filthy rich but I will always have the privilege to be surrounded by the sweetest, happiest, and most creative people in the world!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhbry b/data_all_eng_slimpj/shuffled/split2/finalzzzbhbry
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+{"text":"Public investment, and particularly infrastructure investment, is important for sustainable economic growth and development as well as public service provision. However, it is also vulnerable to capture and corruption. This publication examines the direct and indirect benefits of public investment if carried out in a clean and efficient manner. It provides a Framework for Integrity in Public Investment, mapping out risks of corruption at each phase of the investment cycle. It also identifies tools and mechanisms to promote integrity in the public investment cycle and provides examples of their successful implementation in both the public and private sectors.\nThe Framework was initially discussed during the 2015 OECD Integrity Forum, and has benefited from an open consultation with stakeholders. It includes concrete measures and mechanisms that can be employed at each phase of the public investment cycle in order to safeguard integrity. Examples of good implementation practices from both the public and private sectors are provided for consideration and inspiration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm not normally that moved by the sartorial expressions of the Royal family, but I am interested in the Duchess of Sussex and her backing of lesser known brands with a keen sense of ethics to them. On her recent tour she wore a white tuxedo number by Maggie Marilyn \u2013 albeit altered, with the original puff sleeve arms removed (I much preferred the intended design, but I guess those yoga-toned biceps won't reveal themselves). As is the way with these things, it sold out instantaneously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This wraps up the pilot season of the stream. Thanks everybody for following.\nThis episode is about an often overlooked feature of Maya, at least in rigging, assets.\nAlso, courtesy of a viewer there is an answer to my ramblings about proxy attributes that takes place towards the end, and while for now it's a commands only thing, and it will alter your graph in potentially circular ways, it seems serviceable already.\nThe flag I show in the API also has a related command flag to establish an attribute as a proxy, and it's pretty simple to use.\nWe got blackboard heavy today, but understanding the basics of how interactive software works is important.\nWe spend a good hour going over how software simulates interactivity, how it tracks change, and at what points in those cycles you can insert your own operations.\nWe conclude the scripting tangent required to deploy an actual dynamic FK to IK switch.\nWith OpenMaya2 introduced in the Day 12 stream, the design of the callback done and some attributes at hand, we finally get onto writing the actual code necessary for the callback to retrieve and transform the value until we obtain the right numbers.\nAs in previous scripting posts and videos the transcription of the code to the point we left it at the end of the episode is available at the end of this post.\n#   to accept values for each.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":", and her singing career spanned five decades -- beginning with Tommy Dorsey's orchestra ultimately late 1930s -- and by the mid-1950s, she had achieved more worldwide record sales than any other female creator. However, doubts have been raised with whether there would be time for Johnny - potential seen in 'Pirates in the Caribbean: On Stranger Tides' - to work on the movie, since he has a number of films coming up including 'The Lone Ranger' and 'Dark Shadows' with Tim Burton. Broadway Bound Robin And The 7 Hoods To Premiere At The Old Globe Disney Animation Studios recently released the actual trailer for your upcoming 3D animated musical-comedy feature \"Frozen\". \"Frozen\" is loosely by considering the Hans Christian Andersen fairy tale \"The Snow Queen. \" \"Frozen will get to U. S. More information about Dear Evan Hansen Oriental Theatre Last Minute Tickets.\nIf you are looking for Dear Evan Hansen Show Times Bay Area October, sports tickets and theater seats at the best prices click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take a tour like no other of the area of your choice. There is nothing quite like the perspective gained from an overhead aerial view of where you live, work, and spend the majority of your life. Fly with Mountain Air Services in our helicopter for an experience you'll never forget. Feel free to bring your camera along as we soar over your areas of interest. We can accommodate up to 3 passengers to enjoy the experience together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhqyz b/data_all_eng_slimpj/shuffled/split2/finalzzzbhqyz
new file mode 100644
index 0000000000000000000000000000000000000000..4be8c5c0fb54b5cdf03eeee55774b14a497d07cf
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Giandomenico Spinola how much money? For this question we spent 23 hours on research (Wikipedia, Youtube, we read books in libraries, etc) to review the post.\nTotal Net Worth at the moment 2019 year \u2013 is about $55,2 Million.\n:How tall is Giandomenico Spinola \u2013 1,77m.\nGiandomenico Spinola (1580 \u2013 11 August 1646) was an Italian cardinal of the Roman Catholic Church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"170+ surfers entered the various categories from u12\u2033s to Veterans divisions and the organisation from Surfing GB and the Judges was first rate.\nThe event supported by Korev Cornish Lager and Magic Seaweed was a resounding success.\nIn a stacked field, with surfers from all over the British Isles including a strong crew from Jersey and Guernsey, Luke made a strong start, winning his opening 3 heats with free flowing surfing and upside down turns in the pocket.\nThe quarters ,semi\"s and Final were held on the Sunday with cross \/ offshore winds and solid 3-4 foot consistent swell.The pressure was relentless, the excitement for the Final was palpable ,and we had our 4 finalists.\nIt was to be Luke\"s day, but there were plenty of outstanding performances to keep the large crowd entertained.\nReubyn was landing airs, Jayce was using his speed to lock in huge turns and Miles Lee Hargreaves was the dark horse, taking a break from his last year on the Pro Junior Tour and cutting loose.\nLuke locked into the longer but trickier Lefts that were reeling off the Downend Point rocks and his speed and technical ability allowed him to stay with the set waves, smashing turns, right through to the inside sections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As we start 2018 it's always good to make sure you're in good health after all there is nothing more important.\nl Breast cancer. Mammogram every two years after age 40 or annually if you have a sister or mother with breast cancer.\nl Prostate cancer. Test annually after age 50, especially if you're African American or have a brother or father with prostate cancer.\nl Measles-mumps-rubella if you've never had it, or never had measles, mumps, and rubella, you should receive at least one dose of this vaccine if: you are a woman able to become pregnant and\/or were born after 1956.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With around 2600 employees worldwide and sales of \u20ac385 million in 2017\/18, WITTENSTEIN SE enjoys an impeccable reputation for innovation, precision and excellence in the field of mechatronic drive technology \u2013 not just in Germany but internationally. The group comprises six pacesetting Business Units with separate subsidiaries for servo gearboxes, servo actuator systems, medical technology, miniature servo units, innovative gearing technology, rotary and linear actuator systems, nanotechnology and electronic and software components for drive technologies. Through its 60 or so subsidiaries and agents in approximately 40 countries, WITTENSTEIN SE (www.wittenstein.de) is additionally represented in all the world's major technology and sales markets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This analysis of current events examines the wrong beliefs America has held supreme\u2014\"idols\" that are to blame for our nation's decay\u2014and suggests how our culture can be healed.\nHerb Schlossberg (PhD, University of Minnesota) is a historian and has served as a senior analyst in the Central Intelligence Agency. The author of several books, he lives with his wife in Alexandria, Virginia, and they have three grown children and nine grandchildren.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjzmn b/data_all_eng_slimpj/shuffled/split2/finalzzzbjzmn
new file mode 100644
index 0000000000000000000000000000000000000000..6bf63183daf8a298735d0633bde88fdaadbe270d
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+{"text":"Asus has also built in a voice mail and answer machine system which is accessible from the phone.\nAlthough I had some strange issues at first, a reboot later and things were working fine. In our office we have security bars protecting our windows, and these tend to ruin any wireless signal outside of the office. Reception was fine inside the office, and even a few metres radius outside. In a bar-less household, signal strength should be more than fine (unless you happen to live here).\nI found that I had perfect audio quality when chatting with friends over Skype.\nOn the back of the phone is a speaker, which is fairly loud and makes the speakerphone functionality quite useful. However, this also has another use, as does the headphone jack on the side of the phone.\nSince Skype uses the phone as an audio streaming device (when you get down to the nuts and bolts), Asus has added support for streaming music from Windows Media Player, too. By selecting a pre-made playlist, you can then choose these songs from the menu on the phone, in the same way you would an MP3 player. Using the headphone output, you can either put on a pair of headphones, or output to some speakers. Sound quality isn't bad at all, even through the speakerphone it is listenable! It might be nicely positioned in the bedroom for playing a couple of tracks while getting changed in the morning.\nFunctionality is rather limited for music playback \u2013 there is no way for fast forwarding through a track and skipping from track to track is slow. When you launch a song, it actually loads up Media Player and just streams the output to the phone. If you want to, you can just drag an album on to Media Player and it will start playing it \u2013 so strictly speaking, you don't need a playlist for every album you own.\nAs a test, I got someone to phone me on Skype while I was listening to music on the headphones. It instantly cut off the music and I heard ringing instead. I continued to hear the conversation through my headphones and could still speak through the microphone. It might have been nice to have a microphone input also, then a headset could be used, but if you really wanted to do this, you would plug a headset directly in to the computer and not bother with the phone in the first place.\nI was generally quite impressed with how well it worked with Media Player. For instance, I turned on shuffle and closed Media Player, which saves this as a setting for next time. When I went to play music, the Asus software automatically turned this off, as not to confuse the player.\nThe biggest disappointment about this phone is that it still requires the computer to be on and it needs a separate wireless dongle to work. If you look past this issue, it's a solid phone that works reliably. The dock is a nice touch, and the streaming music could be really handy.\nAt the end of the TRENDnet review, I mentioned how costly it was and how there must be better alternatives. This hands down beats the TRENDnet, with more features, better looks and considerably better software. With an RRP of \u00a369.99, we can expect it to be quite a bit cheaper than this when it hits the market. Well worth considering.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The day has finally come where my son no longer fits in his infant seat. I put it off as long as possible, but since we are coming dangerously close to the maximum weight limit, I had to bite the bullet and find the right convertible car seat for my son. Normally, I go back and forth for days trying to figure out exactly what I want, but this time the decision was easy. There really was no other car seat brand for me to consider. I always knew that I wanted a Britax , and that was exactly what I was going to get.\nIf the Britax brand decision was so easy, the decision as to WHICH Britax Car Seat to choose was extremely hard. My son will be riding in this seat for years to come, and therefore I felt a lot of pressure to make the right decision for us. With all of the Britax car seats having so many great safety features I knew I couldn't go wrong, but I still was looking for the perfect fit. In the end I chose the Britax Boulevard 70 CS (MSRP $339.99).\nThe Britax Boulevard 70 CS is a member of the Next Generation of Britax convertible car seats. The major difference between the next generation and the classic is that the new ones now can hold a child up to 70 lbs vs. the previous 65 lbs. What makes the Boulevard different than some of the other models is the True Side Impact Protection which offers a head restraint creating an extra layer of energy-absorbing EPP foam, keeping the head, neck and spine \"in true\ufffd or aligned, and deep side walls lined with EPP foam to distribute crash forces, shield from vehicle intrusion, and contain the head, neck and body. That is what drew me to the Boulevard. I felt that I needed to protect my son as much as possible, and when so many crashes involve side impact I knew I had to choose the extra protection. My son was in his new seat for just a few minutes and right away enjoyed his new \"winged\" headrest to nap. A little added bonus of the side impact protection!\nInfant Seats: can be used rear facing only from 4 to 30 pounds, and are compatible with the BRITAX CHAPERONE and B-READY strollers and other major brands with the BRITAX adapter strap.\nNext Generation Convertible seats: can be used rear facing from 5 to 40 pounds and forward facing from 1 year and 20 pounds, up to 70 pounds, depending on the seat. High-capacity convertibles allow your child to stay in a five-point harness through the toddler years.\nClassic Convertibles: can be used rear facing from 5 to 35 pounds and forward facing from 1 year and 20 pounds, up to 50 or 65 pounds, depending on the seat.\nCombination Harness 2 Booster: can accommodate children forward facing only in harness mode from at least 2 years and 25 pounds, up to 85 pounds. The seats can then convert to a belt-positioning booster and accommodate a child to 120 pounds.\nI am very happy to be a part of the Britax family of car seats. We all spend so much time in our cars, and it is so important that we protect our child any way we can. Especially when we can't always control what other drivers do. At least with my Britax Boulevard I can go about my day knowing my son is protected in a car seat I trust.\nI was provided with a Britax Boulevard for the purposes of this review. As always all thoughts and opinions are my own.\n\u00ab Feathers In Your Hair: Are you Yay or Nay?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For more information please call or email as numbers are limited.\nQAC is a leading national college for people with disabilities located in Harborne, Birmingham. These events provide the opportunity to meet our staff and students plus discover more about QAC including our 16-25 curriculum, social enterprises, advance2work programme, supported living provision, use of assistive technology and much more.\nThere is no charge to attend and a free lunch plus refreshments will be provided.\nFor more information please call 0121 428 5041 or email info[at]qac.ac[dot]uk. Please book your places early as numbers are limited.\nTuesday 9th April 2019 - click here for more details.\nTuesday 11th June 2019 - click here for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When the business of the South Korean Pharmaceutical Company Hanmi was increasing, the order processing and picking lines were too slow to handle all these orders in time. Their 10 years old equipment had to be replaced and the System integrator SMCORE, one of the largest in Korea, proposed them the new and most modern Avancon ZPC-Conveyor concept. This incomparable system offers many benefits. Now, after one year in operation, everything meets Hanmi's expectations and the company is very satisfied with this solution by Avancon.\nSouth Korean has the 10th largest international Pharmaceuticals Market and is the 3rd biggest in Asia. This Pharmaceutics Market will grow to more than 20 Billion US$ in the year 2020.\nOne of the top pharmaceutical manufacture in South Korea is the fast-growing Hanmi Pharmaceutical Co., Ltd. They are very aggressive in R&D and in the automation facility. Hanmi handle e-commerce B2B, selling to drug stores and wholesale.\nTheir logistic center in Paltan, South Korea, operating by their subsidiary Online-Pharm Co., Ltd., handle more than 10000 orders \/ 13000 boxes during peak days.\nThe existing picking systems and the conveyors of the Hanmi affiliate Company were running for more than 10 years. Their business had been increased significantly in the last decade and they had problems to meet the increasing demand with their picking and logistic task.\nThe system integrator for this project was SMCORE Inc, (former SMC, Smart Mechatronics Core), which is one of the major logistics system suppliers in South Korea. They installed the shelving and the pick-to-light system and took over the scope of Integration.\nSMCORE offered with Avancon a new idea to solve this issue by suggesting the picking zone concept using the new and exceptional Avancon Conveyor system.\nWith this new solution, their logistics \"bottle neck\" problem has been eliminated. The avant-garde Avancon ZPC (Zone Powered Conveyor) System provides more dynamic buffer space with its ZPC-function. In addition, the new solution includes a looping system.\nWhen a certain picking zone is full, the next box goes automatically around the loop and tries to enter the same zone later, when the free space is available.\nAvancon OTUs, the Omni-directional Transfer Units, provides smooth and fast transfer where the box has to be diverted or transferred to another conveyor line or around a corner.\nThanks to these most modern materials handling systems the entire line speed \u2013 including the existing area \u2013 has been increased by 17 %, including the Picking Performance. The modular ZPC-System can be easily extended. So three more zone conveyors were installed.\nA convincing aspect was the safety of the Avancon system, in case of human contact with the conveyors. A finger protection for each driven conveyor-roller doesn't exist anymore. Everything is protected inside the conveyor frame \u2013 including the drive system and the control system. Nothing sticks out \u2013 not even a screw-head. Now the operators are freely working around the Avancon conveyors. The whole conveyor lines are smooth and streamlined with their Aluminum anodized profiles, which feel like silk.\nIt also provides more vertical space because there is no more cable ducts underneath. This is a great advantage for those who have to follow the global GMP regulation, because now they can actually clean the space underneath the conveyor very easily. Because the Avancon ZPC-Conveyor Concept works fully autonomous, cable ducts and a central control cabinet are no longer necessary. This makes the whole investment really economical.\nAnother great thing is the fast installation. The given time was quite limited because Hanmi could not stop the daily operation. Avancon was able to manage everything within the required time thanks to its \"plug-in\" concept. All modules were pre-assembled, pre-programmed and tested in the Swiss factory before shipping. And with this \"plug-in\" concept, the installation is automatically established simultaneously with the electrical connection, which includes also the data exchange.\nIt was the first time, a new experience for everyone who was involved in this project. They had no experience with Avancon products before. Despite this, even the first days of operations went very well, without any issue.\nThe system has been running for almost 1-year.\nEverything is in good condition and Hanmi is considering replacing all their conveyor systems with the Avancon concept.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"L-Arginine is a non-essential amino acid which becomes essential with stress, injury or starvation, pregnancy & protein deficiency. Arginine is found in meat, nuts, eggs, milk, & cheese.\nStimulating the thymus (immune controlling) gland, when combined with Lysine.\nEnhancing immune function - improves total T-cell numbers & activity.\nDetoxifying ammonia in the body.\nIt may also ameliorate the toxic effects of gut\/digestive malfunction. In diabetics, studies have shown 1000mg twice daily significantly reduced lipid peroxidation, which would reduce the risk of cardiovascular disease & other complications of diabetes. It is also helpful with alcoholism and hepatitis, and converts to nitrous oxide which gives more elasticity to arteries and may help in some cases of high blood pressure.\nEssential for immune system activity to stay normal.\nNecessary for wound healing in the body.\nHelpful in assisting with healing of damaged liver.\nNecessary for production and release of growth hormone.\nThe most potent amino acid for releasing insulin.\nA precursor to GABA, a very important inhibitory neurotransmitter.\nBecause of arginine's powerful boost to the immune system, people suffering from a great variety of ailments may find it helpful. However, it should not be used by those with an acute or chronic virus, such as Epstein-Barr Virus (EBV), Cytomegalovirus (CMV) or Human Herpes Virus VI (HHV6), Herpes Simplex I or II. This is because Arginine will speed up the rate of viral growth, which can prove to be dangerous. The amino acid, Lysine, has the opposite effect on viruses, slowing down their growth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..b5294ad28cd0c17180ef1b83d696f5b9710e485d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Legal matters can escalate quickly and become impossible to settle without going to court. A real estate dispute over property rights, a construction defect noticed after the completion of a new home or a breach of a contract may lead to litigation.\nAt Harrell, Stoebner & Russell, P.C., we are experienced litigators who do not shy away from a contentious case. From our Temple and Killeen law offices, we assist individuals and businesses across Central Texas resolve complicated legal matters. We have the courtroom experience needed to fight for the best possible outcome to your situation.\nTaking a case to trial requires a strategy based on the unique details of each case and an in-depth knowledge of the local courts. When a case heads for litigation, many firms will refer the matter because they focus on transactional law and their attorneys do not try cases.\nWe work tirelessly to ensure that we are prepared for whatever path your case may take. If litigation seems like it will be the only possible choice we have available to get results, we will be aggressive in our efforts to obtain an outcome that is in your best interests.\nWhen Litigation Looms Contact Harrell, Stoebner & Russell, P.C.\nIt is not possible to resolve all legal disputes outside of the courtroom. Our firm has the resources and experience to take your case to trial when necessary. Contact us by completing our online contact form or calling us at 254-935-3036 to set up an initial consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you need real qustions and answers of 1Z0-481 exam to pass the exam?\nin which am i able to discover 1Z0-481 braindumps questions?\nThe team behind killexams.com should seriously pat their back for a job well done! I have no doubts while saying that with killexams, there is no chance that you dont get to be a 1Z0-481. Definitely recommending it to the others and all the best for the future you guys! What a great study time has it been with the resource material for 1Z0-481 available on the website. You were like a friend, a true friend indeed.\nNo more worries while preparing for the 1Z0-481 exam.\nkillexams.com is straightforward and solid and you can skip the examination if you undergo their question bank. No words to explicit as i've handed the 1Z0-481 examination in first strive. a few other question banks also are availble in the marketplace, but I experience killexams.com is nice among them. i'm very confident and am going to use it for my different assessments additionally. thanks a lot ..killexams.\nkillexams.com had enabled a pleasurable experience the whole while I used 1Z0-481 prep aid from it. I followed the study guides, exam engine and, the 1Z0-481 to every tiniest little detail. It was because of such fabulous means that I became proficient in the 1Z0-481 exam curriculum in matter of days and got the 1Z0-481 certification with a good score. I am so grateful to every single person behind the killexams.com platform.\nthose 1Z0-481 dumps works amazing within the real take a look at.\nYour query bank is need of the hour. i have got 89.1% in the 1Z0-481 exam. superb wishes for your experts. thanks team. so overjoyed to clean this exam. Your study cloth was extremely beneficial, clear, consise, covering whole material and suberb stacking of inquiries to make one strong guidance. thank you once more to you and your crew.\nMy brother saden me telling me that I wasnt going to undergo the 1Z0-481 exam. I word once I look outside the window, such a lot of special human beings need to be seen and heard from and they simply need the attention folks however i can tell you that we students can get this interest whilst we skip our 1Z0-481 test and i will let you know how I cleared my 1Z0-481 checkit turned into best when I were given my observe questions from killexams.com which gave me the wish in my eyes collectively all the time.\nI highly recommend this bundle to everyone planning to get 1Z0-481 q and a. Exams for this certification are hard, and it takes a lot of work to pass them. Killexams does most of it for you. 1Z0-481 exam I got from this website had most of the questions provided during the exam. Without these dumps, I think I would fail, and this is why so many people dont pass 1Z0-481 exam from the first try.\nWorried for 1Z0-481 exam? Get this 1Z0-481 question bank.\nI have advised about your items to various companions and partners, and they are all extremely fulfilled. Much obliged killexams.com Questions & Answers for boosting up my profession and helping me plan well for my intense exams. Much appreciated once more. I must say that I am your greatest fan! I need you to realize that I cleared my 1Z0-481 exam today, taking into account the 1Z0-481 course notes I purchased from you. I solved 86\/95 questions in the exam. You are the best training provider.\nI've found a very good source of 1Z0-481 material.\nThank You killexams.com for full support by providing this question bank. I scored 78% in 1Z0-481 Exam.\nUse authentic 1Z0-481 dumps with good quality and reputation.\nthese 1Z0-481 actual Questions works within the actual take a look at.\nI ought to undoubtedly deal with 93% marks ultimately of the exam, as numerous inquiries were like the adviser for me. a whole lot appreciated to the killexams. I had a weight from workplace to break up the examination 1Z0-481. but, i was careworn over taking a decent making plans in little time. At that point, the killexams.com Q&A aide showed up as a providence for me, with its easy and brief replies.\nWhich translations are captured if you ADD EXTRACT BEGIN now 1:00 p.m., but START EXTRACT at 2:00 p.m.?\nTransactions started on or after 1:00 p.m.\nTransactions started on or after 2:00 p.m.\nTransactions started before 1:00 p.m and committed after 1:00 p.m.\nTransactions started before 1:00 p.m and committed after 2:00 p.m.\nThe START EXTRACT starts the Extract and Data Pump processes.\nYou want encrypt message sent or over TCP\/IP to server usserver1, where the golden Gate manage process is running on port 7809. Your requirements include using a custom encryption key and advanced encryption standard (AES), and you have created an ENCKEYS file What parameters should you use?\nThis feature is not supported in oracle goldenGate 10.\nWhich three describe a valid OGG Event Action?\nWrite a checkpoint before and\/or after writing the record to the trail.\nBy default, which option is evaluated first, SQLEXEC or filter statement?\nThere is no default; it is set with the after filter I before filter.\nIt depends on their position in the parameter file; the first encountered is executed first.\nWhich argument is used to set the maximum size of the EXTTRAIL or RMTTRAL.\nIn reference guide, page 69, MEGABYTES is the parameter to set maximum size of trail files.\nWhich statements are true of goldenGate password encryption? A password can be encrypted and pasted into a parameter file. An encrypted password can be generated using the default key.\nAn encrypted password can be generated using a user defined-key.\nOne of the encryption options that GoldenGate provides is to use Passwords in the extract and replicate parameter file. Also, the encrypted password can be generated using user defined key up to 128-bit or using the default key.\nYou want to configure replication of CREATE TABLE statements from oracle to Teradata, so that only new table created in your mapped source schema is Replicated. Which should be the part of solution?\nDDL INCLUDE MAPPED in extra parameter file.\nDDL INCLUDE UNMAPPED in extra parameter file.\nDDL EXCLUDE MAPPED in extra parameter file.\nUse a generic error handler like the Page Error event or the Application Error event in global.asax.\nImplement handlers for the Updating, Selecting and Deleting events of the GridView bound to the datasource, set the event.Cancel to true and then manually call the Insert, Delete, Update and Select methods of the data source inside a trycatch block.\nThrow away the data source and do everything manually.\nLuckily I accidentally found the right way to do it while looking at the EventArgs class properties for the Selected event. When the ObjectDataSource calls your data object, if the result is an exception it will call your post event event-handler (Selected, Deleted, etc). The EventArgs has two properties for managing exceptions, one is the Exception itself, the other is a boolean, ExceptionHandled. If you want to handle the exception you just need to set the ExceptionHandled property to true.\nExperience in performing database start-up, shutdown, handling critical database errors and troubleshooting, resolution of errors, database roll-backroll-forward recovery processes.\nExperience in administration of all database objects, tables, clusters, indexes, views, sequences, packages and procedures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you rent your home, whether it is an apartment, house, condo, or duplex, a renters insurance policy will protect your space and possessions. A renters policy protects you in unfortunate situations where you are effected by fire, theft, water damage, and other circumstances beyond your control.\nYou may have questions regarding how much insurance you should purchase, or what a policy will actually cover. Our friendly agent will be happy to walk you through your options, and provide you with renters insurance in MA that you are comfortable with and confident in. Call us at 508-699-7511 today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The EagleEyesTM FPV Station integrates four of the main components you need for FPV on the ground, at a cost that's lower than many of the individual components presently available!\nReceiver Diversity - when you connect two NTSC or PAL audio\/video receivers (of any frequency), the EagleEyesTM picks the better signal at any given time, which can greatly reduce video fades and improve your FPV experience! A variety of diversity settings are configurable with the on-board switches and LED indicators.\n* Antenna Tracking - when coupled with our OSD Pro, the EagleEyesTM is the heart of a full-featured pan\/tilt antenna tracker. Powerful, easy to use menus are provided for adjusting the EagleEyesTM to work flawlessly with the tracker of your choice.\n* Telemetry - Telemetry - When coupled with our OSD Pro, all Eagle Tree data (not just GPS position) are transmitted to your laptop, via your video transmitter and the EagleEyesTM USB port (separate USB cable required). Your flight can then be graphed and displayed with our powerful software, or visualized (either in real time or after your flight) with Google Earth!\n*Note that Eagle Tree embeds the telemetry in the video, rather than the audio. This provides much greater data bandwidth, and is generally more reliable than audio telemetry. And, it frees up your transmitters audio channel for microphone and voice alerts.\n* Four Channel A\/V Distribution - Ever wish you had more video\/audio outputs? The EagleEyesTM has four built-in outputs!\n* Programmable Low Voltage Alarm - How is your ground station battery doing? Now you can program a low voltage alarm buzzer, which alerts you when your battery is below your chosen voltage. The buzzer can also be programmed to alert for other conditions, such as poor telemetry signal.\nTo view the EagleEyes manual Click here.\n* Two video inputs -NTSC\/PAL, using your video receivers of any frequency. Video Input 2 is adjustable to match Video Input 1. Note that only one input needs to be used, if you are not using the Diversity feature.\n* Two audio inputs - Two monaural audio inputs are supported. Audio Input 2 is adjustable to match Audio Input 1. Note that only one input needs to be used, if you are not using the Diversity feature.\n* Power Input - Power input range approximately 6V to 14V. Barrel style power input with standard 2mm center pin.\n* Two Servo Outputs for pan and tilt trackers - Unit is capabable of delivering 5A burst output to drive even large pan\/tilt servos.\n* USB Port for connecting to PC, for live PC display of telemetry and firmware update. Note: A standard USB \"Mini-B\" cable is required for these features, but not included.\n* Encasement - The component (bottom) side of the EagleEyes board is protected by a piece of adhesive foam. The EagleEyes does NOT include a plastic case. A clear plastic case is planned for future release, and that should be compatibile with existing units. The case will be available for separate purchase.\nHere is a video demonstrating the EagleEyes telemetry connected to a notebook PC, running the included software. Note that the display of flight video requires a video capture device on your PC, of course.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have two VHS tapes that I'm sure that the community wants digitized.\nSend them to me and I'll put them on DVD for you.\nIf all else fails, I'm also going to try and make one myself if noone has a cable. These cables are out of stock at my local \"wierd cable store\".\nWhat computer are you connecting it to?\nHave you got a video and audio capture card in the computer?\nComputer: Desktop PC with Ubuntu and TV capture card. I have three different I can use. Hauppauge, Terratec, and the third brand.\nAs stated, the os is Ubuntu. All three tv cards have a composite video in.\nThose plugs in my previous post is on the destination side.\nNow, finally. The videos are the two tutorial vhs tapes for DPaintIV. One is basic and the other is advanced.\nRE the solution under Debian; - No idea m8, soz!\nPerhaps davideo's gesture of goodwill is the way to go!\nRegarding the Debian Software, I haven't got a clue my friend....sorry !!!\nAs with TC and Kin I've no idea on the hardware\/software that you're talking about. So I'm sorry to say that I can't help you with that side of things.\nI actually saw these videos a couple of years ago. I should have stuck them onto DVD at the time.\nI'm sure I've got a lead that will suit your purpose. The only thing is it will have a phono socket for each of the audio channels (Red and White). You'll only need a converter that you'll plug the two audio leads in and then output to a stereo jack. I may also have one of these in the bottom of the same box.\nIf these are PAL videos - my offer still stands.\nHere's a picture of the lead(s) that will make up to be what you are looking for.\nI might take that cable, but there might be a roadtrip today to the wierd cable shop just outside of town. I'll keep you posted.\nWhy is this Video store Wierd ??\nAs for the tablecloth, It's a very British thing for when we take girls home to meet mom.\nThey have all kinds of strange cables. I'm sure if you want a cable that hooks up your VCR to your moms heart monitor and keeps it in synch with all kitchen appliances, they have it. Or at least they can have their factory in china make it. It's one of those stores that sell all the crap that tv-shop (home shopping network) has, and then some.\nThe all time favourite was the \"flowbee\" a hair cutter that you could fix to your vaccume cleaner, I really wanted myself one of those. But was never brave enough to call in.\nRoad trip was a success, now lets just hope the tapes are pal.\nClose this thread, i will report back later in new thread.\nIts a very similar cable i got here. Thanks for your time!\nThat's a famous cloth - it's been in the background of a few things I've sold.\nNo problems - good luck.\nDon't forget to let us have the chance to get one when you get them done.\nAre you sure the tapes are PAL? Is your TV system PAL?\nA SECAM tape in a PAL machine will give you a Black and White picture.\nWith a small soldering iron, some solder and pliers you can fix that cable yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbngik b/data_all_eng_slimpj/shuffled/split2/finalzzzbngik
new file mode 100644
index 0000000000000000000000000000000000000000..2a8974301e545088be94f6ba93000bd33685c6f0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbngik
@@ -0,0 +1,5 @@
+{"text":"Taj Mahal, a most celebrated piece of Mughal structure and craftsmanship is probably the most visited attraction of North India Tourism. This epitome of affection topped as one of many New Seven Wonders of the World which is visited by virtually each vacationer who comes right here from a number of components of the world. This final monument of affection can also be granted with standing of the perfect World Heritage Websites in India.\nTaj Mahal is probably the most well-known sights India Tourism positioned on the west financial institution of river Yamuna in Agra, Uttar Pradesh State. This mausoleum is the image of everlasting love which was constructed by the Mughal Emperor Shahjahan for his beloved spouse Mumtaz Mahal. This grand edifice signifies and glorifies human love which has withstood the check of time, and nonetheless stands in all its glory of structure. Scores of vacationers favor Taj mahal Excursions they will expertise the grandness of this astounding structure.\nThat is the mausoleum which was constructed by the Muhgal Emperor Shah Jahan in 1648 A.D. as a tribute of romance together with his beloved spouse Mumtaz Mahal. It was constructed by 20,000 staff working day and evening for round 22 years. This monument is made up of white marble and crimson sandstone. With this 28 kind of treasured and semi-precious stones have been embedded within the interiors of the monument which provides extravaganza to this grand construction.\nThe design of Taj Mahal is principally a Persian model embedded with an Indian architectural sample. The Taj Complicated is split into 4 distinctive components and every has its personal significance and traits. The Taj Gateway going through the Taj Mahal is the huge crimson sandstone Gateway. It's embellished with verses from the holy Koran. Whereas the Taj Backyard supplies a wide ranging view of lush-green flora going all the way in which as much as the plinth of the Taj. This backyard illustrates the Islamic model of structure and symbolizes spirituality.\nThe Mausoleum is the mighty construction of the Taj positioned on a 6.7 meter excessive plinth. 4 minarets every of 41.6 meters peak flanked by the 4 corners of the plinth and stability the tomb. These minarets are so spectacularly designed that in case of any mishappening, the pillars wouldn't fall on the tomb. Every minaret has an Arabic letter written and well-known for his or her calligraphic representations.\nTombs in Taj Mahal are set on a sq. pedestal. The bottom construction within the tomb is massive and has many chambers. Of the varied chambers, the primary chamber has the cenotaphs of Shah Jahan and Mumtaz Mahal. The steles of the tomb are embellished with treasured and semi-precious stones and the ceilings are strategically constructed to echo voices and melodies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want to get started but not ready to invest?\nNo refunds\/no Reschedules. Hostess is responsible for communicating this with guests.\n*Photographs will be taken at the time of the event to share online. You agree to allow photographs of you, your family, on your home and property by registering.\nSnow\/Ice Policy: A workshop can be rescheduled by Happy Valley Sip and Paint up to 1 hour before the event due to extreme weather. Extreme weather is classified by an instructor not being able to safely get to AND\/OR from your location. If this happens, the party will be rescheduled within one month, during an off peak time (Monday-Thursday evening). No refunds .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SECURE STORAGE UNIT WITH VEHICLE ACCESS This unit of approximately 45m2 is ideally suited for use as an office space providing an administrative service or as a secure lock up store. It includes air conditioning new electrical installation, security system. Direct vehicle access through communal car park. Located in the centre of Magaluf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Buckinghamshire's online dating site for singles in Mursley, we're here to help you make new friends and start relationships with local people around Mursley, Buckinghamshire. If you're serious about wanting to meet someone in Mursley and are looking forward to getting yourself out on some fun dates then you have found the right dating website. We're not just a Mursley dating site, we're focused on making new relationships that last. Something that takes effort and commitment on both sides, a serious dating site for Mursley singles who are ready to date.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These definitions have been included in order to give the reader a quick reference to various technical terminology used throughout the work.\nFree will is the fourth form of action. The first three forms of action are forms of determinism, pre-destination, and predestination.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnofx b/data_all_eng_slimpj/shuffled/split2/finalzzzbnofx
new file mode 100644
index 0000000000000000000000000000000000000000..8d5d9f695f9508ca901896f6e5f6b6567b386ed3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnofx
@@ -0,0 +1,5 @@
+{"text":"bodyguard (or close protection officer) is a type of security guard or government law enforcement officer or soldier who protects a person or people \u2014 usually high-ranking public officials or officers, wealthy people, and celebrities \u2014 from danger: generally theft, assault, kidnapping, assassination, harassment, loss of confidential information, threats, or other criminal offences. The group of personnel who protect a VIP are often referred to as the VIP's security detail.\nMost important public figures such as heads of state, heads of government, and governors are protected by several bodyguards or by a team of bodyguards from an agency, security forces, or police forces (e.g., in the U.S., the United States Secret Service or the State Department's Diplomatic Security Service). In most countries where the Head of state is and have always been also their military leader, the leader's bodyguards have traditionally been Royal Guards, Republican Guards and other elite military units. Less-important public figures, or those with lower risk profiles, may be accompanied by a single bodyguard who doubles as a driver. A number of high-profile celebrities and CEOs also use bodyguards. In some countries or regions (e.g., in Latin America), wealthy people may have a bodyguard when they travel. In some cases, the security personnel utilize a bulletproof vehicle, which protects them and the VIP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ozone Insuria- easy to use Insurance Brokers Management Software designed and developed at par with business specific demands of the insurance brokers\/ agents. nsuria, Insurance Management System provides a powerful software tool for small, medium and large insurance broker houses, administrators to underwriting Managers. It is a fully automated system which keeps a uniform track of insurance brokerage transactions, accurate reports, Finance management, Human Resource management etc.\nPros: It is an insurance broker friendly. Modules are self explanatory and easy to use. Above all this, the default modules covers all the major operations and finance requirements of an insurance broker. Support team responds quickly and resolved any issues in a very fast manner. It is an insurance broker friendly. Modules are self explanatory and easy to use. Above all this, the default modules covers all the major operations and finance requirements of an insurance broker. Support team responds quickly and resolved any issues in a very fast manner. It is an insurance broker friendly. Modules are self explanatory and easy to use. Above all this, the default modules covers all the major operations and finance requirements of an insurance broker. Support team responds quickly and resolved any issues in a very fast manner. It is an insurance broker friendly. Modules are self explanatory and easy to use. Above all this, the default modules covers all the major operations and finance requirements of an insurance broker. Support team responds quickly and resolved any issues in a very fast manner.\nCons: At some point , there are features in finance & hrm that requires major revision to suit the client's requirements.At some point , there are features in finance & hrm that requires major revision to suit the client's requirements. At some point , there are features in finance & hrm that requires major revision to suit the client's requirements. At some point , there are features in finance & hrm that requires major revision to suit the client's requirements. At some point , there are features in finance & hrm that requires major revision to suit the client's requirements. At some point , there are features in finance & hrm that requires major revision to suit the client's requirements.At some point , there are features in finance & hrm that requires major revision to suit the client's requirements.At some point , there are features in finance & hrm that requires major revision to suit the client's requirements.\nOverall: It makes our daily operation easy.\nThank you for your response. Will definitely value your comments.\nInsuria made our underwriting dept management at the tip of our fingers.\nPros: All features and facilities relates to our day to day activity helping the end user to associated with the system at ease. Their debt management function enables us to keep a tab on our debtors which makes our collection management systematic.\nCons: The fleet management can be optimized a little bit more by reducing the steps that are involved at the time of data entry. Endorsement views can be enhanced a little bit so that in case of reprinting or for re checking it would help the user much better.\nInsuria redefined our business management replacing our dependency on excel and tally.\nPros: Insuria interprets our business activity helping us to relate the system with our daily activities.\nCons: The KYC is finance targeted rather than operations. If the KYC is modified to help the operations department, it would help us in planning future business with a particular client based on his business relation with the organization. Currently KYC displays only financial transaction data.\nPros: Insuria brings various reports which help every operations of the company to understand the volume and preparing the growth plan. Detailed Renewal data helps to retain the customers.\nCons: Delay in downloading certain reports. Mainly the complexity of the report. Delay in downloading certain reports.\nOverall: Entry, tracking & reporting the insurance business for brokers.\nPros: User friendly, Covers all the necessary modules. A well designed easy to use powerful product capable of satisfying the requirements of an Insurance Agency. Excellent customer support and Price is also affordable.\nCons: Can make it more colourful so it will improve the overall look and feel. Didn't came up with another issues so far.\nOverall: Increased the efficiency of Work flow. Excellent features.\nVery friendly staff member willing to assist however they can.\nOverall: Very useful application for an insurance broking industry and it is very user friendly and fully integrated software.\nPros: Simple and easy to use.It can be set up in any computer with any configuration,still it performs well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to Choose Top WhatsApp Marketing Services in Mumbai?\nWhatsApp Marketing in Mumbai by Troika It is the Best Bulk WhatsApp Marketing in Mumbai WhatsApp Marketing Services in Navi Mumbai WhatsApp Marketing Companies in Mumbai Top WhatsApp Marketing in Mumbai, Best WhatsApp Marketing Services, Whatsapp Advertising in Mumbai, WhatsApp Marketing Company in Mumbai,.\nSMS advertising, on the various other hand, relies on customers to opt in. That indicates your target audience need to first subscribe to obtain messages regarding offers and promos before receiving them. Opt-in advertising has big advantages for organisations. One is that customers who receive your messages are currently heavily purchased your brand or item; or else, they wouldn't have actually subscribed. One more is that the opt-in model additionally means you will not lose time or cash bothering clients who typically aren't interested. Much like with e-mail advertising, the majority of SMS advertising devices make it relatively easy for users to pull out. If you want your SMS marketing to be efficient, the very first (and also crucial) step is making certain you create an efficient message.\nFor companies that put stock in the security of their customers and the credibility of their business and website, SSL certificates are a must. All you have to do is ask yourself, would you do business with somebody you don't trust?\nWho is the Best Trusted SSL Certificate Provider in Mumbai?\nSMS can also be utilized to help with two-way interaction in between occupants and also property representatives. Whether the occupant has to notify the representative of a busted device or if they simply wish to ask an inquiry about the residential property, SMS is the excellent platform.\nLower Management Price: SSL setup procedure life cycle entails several actions like investing in, SSL setup, authentication as well as installment. Account groups deal with purchase and also they invest even more time for multiple purchase. Technical team takes care of SSL arrangement and also installment. An expert network administrator could require minimum 20 minutes for single SSL action. For claim mounting different SSL on 10 sub-domains, he will have to spend about 200 mins. While mounting a wildcard SSL certificate he will need simply 20 minutes as well as all 10 sub-domains will be protected with SSL.\nHowever, a couple of telemarketers have found an escape. They are utilizing foreign and also private numbers to send these advertising messages utilizing net based messaging services as well as international or private telephone numbers. \"These SMS cost a great deal of money, billing as high as 10 paise per message,\" says the Suninside worker. Nevertheless, many consumers continuously obtain spam messages. To handle this problem, Trai has revealed that it will quickly ask cellular mobile provider to bill telemarketers 5 paise for each industrial SMS that is sent out on its network. As telemarketers work on little profit margins, Trai thinks that increasing the cost of sending out messages will certainly make the ban extra effective.\nAlready, for every single non-solicited message or call received by a person signed up on the 'do not call computer registry', provider can be fined Rs 50,000 for the initial offense, which can go up to Rs 2.5 lakh for the sixth offense. Afterwards, they will be blacklisted for two years.\nBecause encrypting as well as decrypting with personal and also public essential takes a great deal of handling power, they are only utilized throughout the SSL Handshake to create a symmetrical session trick. After the safe and secure connection is made, the session secret is made use of to encrypt all sent information.\nEverybody in the web marketing world knows how vital of a function TEXT Marketing plays in a marketing campaign for any kind of company on the web, unless you are a complete newbie to it. TEXT Marketing Services solutions could be found alongside other web marketing devices such as social networks, link building, SEO and such other internet marketing methods.\nIn the ever-changing property market, SMS messaging continues to be a trusted constant. From advertising and marketing real estate listings to engaging with brand-new and old leads, SMS permits special accessibility to your customer base. So, whether you're a homeowner, residential property supervisor or lessee, making use of TEXT communication can aid streamline and boost your advertising and marketing and interaction methods.\nBusiness\/Organization recognition papers. If you are buying a high-assurance certification, your company should likewise be validated. Certification authorities frequently examine federal government databases online to confirm that your business is signed up yet they could still require you to send in a federal government enrollment record if they can not locate your company. Each certification authority has a little different requirements for recognition. If you want to examine whether your firm is correctly noted and energetic with your federal government, try using one of these on-line searches. If you are purchasing an EV certification, you will need to supply a lot more documents. Your certification carrier will allow you understand what you need to offer after you place your order.\nIf you have a website that has multiple sub domains, you're going to love wildcard SSL certificates. Rather than having to purchase an SSL certificate for each and every sub domain, you can actually protect them all with just one wildcard SSL certificate. This can lead to some major savings and make managing your SSL portfolio a breeze.\nFor the industry's most secure, trusted, and feature-rich wildcard SSL certificate, turn to the Symantec Secure Site Wildcard. This efficient cert will cover one domain name (www.domain.com) and an unlimited number of subdomains (mail.domain.com, payment.domain.com, support.domain.com, etc.). Best of all, Symantec-brand certificates offer a number of additional features beyond just standard encryption, like free daily malware scanning and the Symantec Seal-In Search, which will maximize click-throughs to and conversions on your site. With true ROI potential, this certificate can be viewed at as more than just an expense \u2013 it's an investment into the trust and credibility of your company.\nWildcard SSL certificates are designed to protect every single sub domain on your website so long as they share the same main domain name and exist on the same physical server (which is denoted with an asterisk in the common name field). There's no limit to how many sub domains can be covered, so we're talking about an awful lot of protection from a single certificate.\nWildcard SSL certificates, like other SSL technology, offer standard 128- or 256-bit encryption on your website. This helps keep private information private, as SSL protects data in transit, meaning your online customers will feel safe and secure when submitting any form of online payment, logging into an account with a username\/password, signing up for newsletters, etc.\nWhen would I need a Wildcard SSL?\nIf you have multiple subdomains which you need to secure, then a wildcard SSL certificate is perfect for you. Wildcards also grow with your business, and allow you to secure any new sub-domains you may add in the future.\nWhere would I install A Wildcard SSL?\nWildcard SSL certificates are installed on the webserver that powers your website. Popular webserver types include Apache, Microsoft IIS, NGINX and Tomcat. There are no extra steps or difficulties when installing a wildcard SSL certificate versus installing a regular certificate, and the CA that you buy your wildcard from will provide you with the specific installation instruction for your webserver type.\nOne of the problems that some people run into when they have a bunch of different SSL certificates is that the renewal process can be a bit of a nightmare. This can lead to gaps in coverage when a certificate expires without you even knowing... what a nightmare! With wildcard SSL certificates, however, the renewal process is quick and easy. You just have to keep an eye on the one certificate. Anybody can do that!\nOur SMS service are ensured to offer an incredible promotion to your business. We likewise supply short code service as well as long code solution to our customers. We have numerous customers who have actually done well in their service via our bulk SMS advertising and marketing service. We advertise your organisation by sending out bulk SMS, as per the classifications of customers specified by our customers. We have a database of every certain client required by your company.\nWildcard SSL certificates permit webhosting as well as business to secure unrestricted sub-domains under a domain through a single certificate. For instance, a wildcard certification for '*.domain.com' might likewise be used to safeguard 'payments.domain.com', 'login.domain.com, 'anything else.domain.com' etc. They supply a cost-efficient option to separate certificate purchases as well as simplify monitoring procedures by lowering the number of certificates you need to track.\nTo obtain a certification, you have to create a Certification Signing Demand (CSR) on your server. This procedure develops a private secret as well as public trick on your server. The CSR data submit that you send to the SSL Certification provider (called a Certificate Authority or CA) consists of the general public key. The CA makes use of the CSR information file to create a data structure to match your private trick without compromising the key itself. The CA never sees the private secret.\nAs companies as well as companies use more on the internet solutions and also transactions, internet safety comes to be both a priority as well as a necessity of their on the internet deals to ensure that sensitive info\u2013 such as a charge card number\u2013 is only being transferred to genuine on the internet organisations.\nPlanning an interactive SMS promo is a great way to earn your message enjoyable and interesting, and also there are lots of methods to do it. Posing your message as component of a questionnaire or poll motivates your target audience to read more carefully and come to be an energetic individual. Providing a toll-free SMS reply alternative is an additional reward for customers to involve with your brand name. On the other hand, consisting of a connect to an amusing video or images with more details about your promo motivates customers to explore further. Not every cellphone has the ability to attach to the Web, but as smartphones end up being virtually common, business owners would certainly be smart to take into consideration including aesthetic as well as interactive aspects in their SMS marketing efforts.\nInstantaneous SSL certifications are developed to protect a single-domain with the highest possible readily available levels of encryption as well as are relied on by all significant web browsers. They showcase full company validation, totally free TrustLogo site seal, endless server licenses, 24\/7 support, up to $250,000 warranty, totally free internet site virus scanning, totally free PCI scanning and also a cast iron 30-day refund guarantee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a Financial Risk Software Specialist, (internally referred to as Officer IT Applications) you will maintain and further develop several risk management solutions, including off-the-shelf software (IBM Algorithmics, SAS OpRisk) and bespoke solutions, in the area of market risk, ALM and liquidity risk. S\/he will notably ensure that the computations required for the Bank's reporting are performed consistently and accurately in alignment with the standards outlined by the Basel Supervisory Committee.\nYou will report to the Head of the Financial, Liquidity and Operational Risk Unit and whenever required to the Head of the Risk Management and Best Banking Practice Division. The Unit is composed of 6 internal staff and several external consultants.\nExternally, you will regularly deal with key external parties (software vendors, providers of outsourced services, peer groups) in order to optimise the level of service and to stay abreast of best practices. Internally, you will work in close cooperation with the other Unit members, as well as with your colleagues from other IT divisions and business Directorates.\nEnsuring that the risk management software solutions are in line with the IT policies and that qualitative and quantitative objectives are achieved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saut\u00e9 the crickets in the oil until they are brown and crispy. pull them out before they get too crispy!\nRemove the crickets and drain them well on paper towels.\nYou can enjoy these crickets as a snack, on top of guacamole or use as a filling for tacos and tortilla.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..7738d612b29d6784a856e6930cd9a8dc3eb25c7c
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+{"text":"Volunteering in Bangli brings a new sense of purpose.\nThe Widhya Asih Foundation is a collection of 7 different orphanages and one office location spread out across the entire island of Bali. Since they are so spread out, we often start our days waking up early for breakfast and then heading out to one of the sites which can be anywhere from 45 minutes to three hours away. I have found this to be a great way to see the whole island even if it is only from a car window as we pass through towns. I would hardly call these drives relaxing though, as these roads have some of the craziest drivers I have ever seen. One of the locations we went to today, Bangli, is still a construction site. They have 25 kids at a temporary site nearby which is an old restaurant that contains two large dorm rooms. The plan is to have the new site up by June and they also would like to at least double the amount of kids they can take in as well. Bangli has quickly gathered a reputation among our group for being the most challenging site you can visit. As much as Bangli is physically exhausting, it is just as much mentally exhausting.\nIt can be so hard on you because throughout the other orphanages we have been working on the actual site where the children are staying. So no matter how much physical labor we are put through each day, we can all look forward to the breaks where we can hang out with the kids. We can spend time getting to know them, helping them practice English or even going out to a field to play soccer. At Bangli, however, there isn't any of this \u2013 just straight work \u2013 and hard work.\nAt the site there are some workers who point out what they need us to do. The last few days, we've carried tile and dirt to the second story of the structure in 95 degree heat. We also learned how to mix concrete\u2026.by hand. Since the labor here is so cheap, about $4 per person per day, there is no reason for anyone to buy anything like a belt sander or cement mixer \u2013 they have the people do it themselves. This also meant that we were learning to do all these things by hand ourselves as well.\nDuring one of our water breaks, I decided to walk around the place and check it out. In one of the rooms, I came across some pieces of ply wood leaning upright against each other. When I peeked inside, I discovered a pillow and blanket covering the dirt. Not only do these workers only get paid $4 a day to work construction for 8 hours in the heat, they also have to live onsite at night as security guards to keep their supplies protected. We found out later that everyone there does this year round. They move from site to site building houses and live onsite while they are there. It is kind of like being homeless except they have something to do during the day, and at $4 a day, it is a very small step up from living out on the streets.\nIn another room we come across a baby\u2026yes, an infant. Lying on a piece of wood crying in the middle of the construction site. Her mother was in the shower at the time and her father was working and wasn't allowed to take a break to tend to his child for even a minute. That for me was when it really hit home. It wasn't just adult men living here as they work on site, it was families. This has redefined the words poor and needy for me.\nSuddenly the break ends and I find myself with a new sense of purpose. The afternoon gets a little bit easier for me to get through because if I am able to do anything at all to help these guys then I am going to do it. If carrying four bags up a flight of stairs at a time instead of three means that these families can get a little extra time off, then that is what needs to be done.\nI have really appreciated the time I have had getting to know the other board members and now we all have one major thing that will bond us together\u2026..Bangli.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nowadays, you do not need to spend thousands of dollars to book a luxury yacht where you can enjoy and spend some quality time with your family and friends.\nRecently, I went to the Bahamas on my previous holidays. I was just searching for a nice and affordable yacht to hire. It was then the local people told me about Tropicalboat's Caribbean & Bahamas charters. I hired a charter from them and the experience was wonderful!\nBut you may not be lucky every time, so if you are thinking to organize a luxury yacht charter for a special occasion, you need to plan it very carefully. There are several factors that you must consider to make sure that you are getting the yacht that suits your needs and fits perfectly in your budget.\nBefore searching for the availability of luxury yacht charters, you need to determine the number of people that are going with you on the cruise. If you have a large group of people, it is quite possible that you get some discounts for hiring large luxury yachts.\nAnother important detail that you need to consider is on which date you will need the luxury yacht charter. Make sure that you book one in advance so that the company can prepare the yacht just in time. Also, some companies offer discounts on bookings made in advance.\nThis is the most important point to be considered. Usually, we do not want to spend too much and are always looking for cheap deals. But do remember that in this type of matter, you get what you paid for.\nSo, if you choose to go for low-cost yacht rentals then you will have to satisfy yourself with mediocre services only. So, search for companies that will give you the best experience at affordable rates.\nAfter tiring working schedule everyone wants to spend some quality time with their loved ones. What could be better than spending a day at the water park? In the water park, people of every age could enjoy. If you are going with your parents and kids then all of them can enjoy some rides over there.\nIf you are planning the trip to some aqua park then, considering water park Dubai is one of the best options. If you want to enjoy your trip to water park then all you need to do is planning everything carefully so that you can enjoy your trip fully.\nIf your trips are planned perfectly then it becomes one of the good memories which you would never ever want to forget. When it comes to water park trip, the first thing which strikes our mind is various slides with unique thrills.\nAs every water park has its unique map of attraction, you need to check the place which fits according to your requirement. If you are looking for an aqua park online, then you can check their website and see which water slides are which your family can enjoy.\nWhile you are planning your trip to the water park, never forget to take sunscreen along with you. There is no fun in getting sunburn and leaving your trip in between. Also, read the instruction behind the pack and apply the sunscreen after regular time period so you can enjoy the trip to the water park.\nThe towel is one of the things which many of the people forget to take along with them while they are planning their trip to the water park. So while packing your bag take your towel along with you to the water park.\nThese are some of the things which you need to take care of for having a perfect water park trip. You can check here and know more about waterparks in Dubai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learn how Informaast customers are finding everyday uses for mass notification that save time, money and waste.\nLearn the benefits of a phased implementation when deploying your mass notification system in this blog post.\nPurchasing and implementing a mass notification system requires buy-in from multiple departments to ensure everyone has a stake in safety planning for an organization.\nTesting twice a year and only reaching people through one channel can lead to big headaches when it comes to mass notification. Learn why in this post.\nCreating a testing plan that works in conjunction with emergency communication strategies helps build familiarity throughout organizations. This can assist in avoiding costly mistakes.\nWhen developing an emergency communication plan, it's important to look at what has worked, and what hasn't, at other organizations.\nWith a limited number of students, faculty and staff on campus during this time, it may not make sense to have the same emergency notification plan as the rest of the year.\nIn this white paper, we examine recent events that have been impacted by misinformation. From terrorist attacks, to school shootings and severe weather, efforts to keep people informed have been hindered by technical and user errors.\n\"I hope to god we never have to do it (implement active shooter procedures).\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HI, I can't remember my username and I need to be removed from the mailing list. Could you please send me my username and remove me from the list?\nAt your request we are providing you the password for your user account on the OpenClinica Portal and Bugtracker.\nFor security purposes please change your password when you log in to the system.\n.........can .any body make me understand why it is happening.\n> > list. Could you please send me my username and remove me from the list?\n> > the OpenClinica Portal and Bugtracker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us at Turf Moor on Saturday 22nd October to see Burnley take on Everton in the Premier League.\nFor just \u00a385+vat you can enjoy padded seat to watch the game from the Jimmy McIlroy Stand, half-time refreshments in the Reims Suite, followed by a one course meal served directly after the final whistle.\nThis is the only remaining hospitality option available for this game, and seats are expected to sell out so please book early to avoid disappointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}